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((x+y+1)/(x+y))((x+z+2)/(x+z))((x+w+3)/(x+w))>8?
Fractions/Ratios
tagged by: fskilnik@GMATH
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+1 Upvote Post
fskilnik@GMATH wrote:
$$A = {{\left( {x + y + 1} \right)} \over {x + y}} \cdot {{\left( {x + z + 2} \right)} \over {x + z}} \cdot {{\left( {x + w + 3} \right)} \over {x + w}} = \left( {1 + {1 \over {x + y}}} \right)\left( {1 + {2 \over {x + z}}} \right)\left( {1 + {3 \over {x + w}}} \right)\,\,\,\mathop > \limits^? \,\,\,8$$
$$\left( 1 \right)\,\,\,x,y,z,w\,\, \ge 1\,\,\,\,{\rm{ints}}\,\,\,\, \Rightarrow \,\,\,\,\,2 \le x + y\,\,,\,\,x + z\,\,,\,\,x + w\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,\,{1 \over {x + y}} \le {1 \over 2} \hfill \cr
\,\,{1 \over {x + z}} \le {1 \over 2}\,\,\,\,\, \Rightarrow \,\,{2 \over {x + z}}\,\, \le \,\,2 \cdot {1 \over 2}\,\,\,\, \hfill \cr
\,\,{1 \over {x + w}} \le {1 \over 2}\,\,\,\,\, \Rightarrow \,\,{3 \over {x + w}}\,\, \le \,\,3 \cdot {1 \over 2} \hfill \cr} \right.\,\,\,\,\,\,\left( * \right)$$
$$A\,\,\,\mathop < \limits^{\left( * \right)} \,\,\,\left( {1 + {1 \over 2}} \right)\left( {1 + 2 \cdot {1 \over 2}} \right)\left( {1 + 3 \cdot {1 \over 2}} \right) = {3 \over 2} \cdot 2 \cdot {5 \over 2} = {{15} \over 2}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,\,x < y < z < w\,\,\,\,::\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x;y;z;w} \right) = \left( {1;2;3;4} \right)\,\,\,\,\mathop \Rightarrow \limits^{{\rm{by}}\,\,\left( 1 \right)} \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {x;y;z;w} \right) = \left( {{1 \over 4};{1 \over 3};{1 \over 2};1} \right)\,\,\,\, \Rightarrow \,\,\,A = \left( {1 + {{12} \over 7}} \right)\left( {1 + {8 \over 3}} \right)\left( {1 + {{12} \over 5}} \right) > 2 \cdot 3 \cdot 3\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$
The correct answer is (A).
We follow the notations and rationale taught in the GMATH method.
Fabio.
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\begin{document}
\begin{center}
\large
{\textbf{Approximate Consistency and Prediction Algorithms in Quantum
Mechanics} \par}
J. N. \McElwaine \par
\textbf{Summary} \end{center}
This dissertation investigates questions arising in the consistent histories formulation of the quantum mechanics of closed systems. Various criteria for approximate consistency are analysed. The connection between the Dowker-Halliwell criterion and sphere packing problems is shown and used to prove several new bounds on the violation of probability sum rules. The quantum Zeno effect is also analysed within the consistent histories formalism and used to demonstrate some of the difficulties involved in discussing approximate consistency. The complications associated with null histories and infinite sets are briefly discussed.
The possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set for pure initial density matrices is investigated. The problems that arise are explained, and different possible algorithms discussed. Their properties are analysed with the aid of simple models. A set of computer programs is described which apply the algorithms to more complicated examples.
Another algorithm is proposed that selects the consistent set (formed using Schmidt projections) with the highest Shannon information. This is applied to a simple model and shown to produce physically sensible histories. The theory is capable of unconditional probabilistic prediction for closed quantum systems, and is strong enough to be falsifiable. Ideas on applying the theory to more complicated examples are discussed.
\end{document} | arXiv |
A simulation study on matched case-control designs in the perspective of causal diagrams
Hongkai Li1,
Zhongshang Yuan1,
Ping Su1,
Tingting Wang1,
Yuanyuan Yu1,
Xiaoru Sun1 &
Fuzhong Xue1
In observational studies, matched case-control designs are routinely conducted to improve study precision. How to select covariates for match or adjustment, however, is still a great challenge for estimating causal effect between the exposure E and outcome D.
From the perspective of causal diagrams, 9 scenarios of causal relationships among exposure (E), outcome (D) and their related covariates (C) were investigated. Further various simulation strategies were performed to explore whether match or adjustment should be adopted. The "do calculus" and "back-door criterion" were used to calculate the true causal effect (β) of E on D on the log-odds ratio scale. 1:1 matching method was used to create matched case-control data, and the conditional or unconditional logistic regression was utilized to get the estimators (\( \overset{\frown }{\beta } \)) of causal effect. The bias (\( \overset{\frown }{\beta}\hbox{-} \beta \)) and standard error (\( SE\left(\overset{\frown }{\beta}\right) \)) were used to evaluate their performances.
When C is exactly a confounder for E and D, matching on it did not illustrate distinct improvement in the precision; the benefit of match was to greatly reduce the bias for β though failed to completely remove the bias; further adjustment for C in matched case-control designs is still essential. When C is associated with E or D, but not a confounder, including an independent cause of D, a cause of E but has no direct causal effect on D, a collider of E and D, an effect of exposure E, a mediator of causal path from E to D, arbitrary match or adjustment of this kind of plausible confounders C will create unexpected bias. When C is not a confounder but an effect of D, match or adjustment is unnecessary. Specifically, when C is an instrumental variable, match or adjustment could not reduce the bias due to existence of unobserved confounders U.
Arbitrary match or adjustment of the plausible confounder C is very dangerous before figuring out the possible causal relationships among E, D and their related covariates.
In observational studies, confounding factors (C) that are pre-exposure variables associated with the exposure E and the outcome D will distort the estimation of the target causal effect [1–4]. Generally, the magnitude of confounding bias mainly depends on the strength of the effects from confounder C to exposure E and from confounder C to outcome D. If one of these two effects is precisely null, confounding bias does not exist at all. Furthermore, the directions of effect from C to E and from C to D determine the direction of the bias. Usually, confounding factors could mainly lead to three kinds of biases in an attempt to find the causal effect from E to D, including over-estimation, under-estimation, or even missing the direction of the effect [5].
In analytic epidemiology, various strategies could be adopted to remove confounding bias, such as Restriction, Adjustment, Stratification [6, 7], while strategy of matching on confounders C (e.g. matched case-control designs) mainly focuses on improving estimation precision of the effect of E on D, rather than removing confounding bias [8, 9]. For matched case-control designs, matching refers to the selection of controls group that is identical, or nearly so, to the cases group with respect to the distribution of one or more potentially confounding factors. Generally, two matching strategies, including individual matching and frequency matching, could be selected to force the distribution of the matching factors to be identical across groups of individuals [10]. In particular, individual matching involves selection of one or more controls group with matching factor values equal to cases group. From the perspective of causal diagrams, several qualitative studies had suggested that matching on confounders not only fails to remove confounding bias but also adds colliding bias [11–15]. Therefore, it is still necessary to adjust for the matching variables.
However, for obtaining unbiased and precise estimation, it is crucial to choose matching variables correctly and further determine whether they should be adjusted for. For matching variables, matching on common child nodes of exposure and outcome, or mediators of the exposure and outcome will generally lead to irremediable bias [13, 14]. For further adjustment, conditional logistic regression models are customarily used to adjust for matching variables, which just provide conditional rather than causal estimation of odds ratio [16]. Sometimes, unconditional logistic regression models can also be adopted to adjust for matching variables, but they will lead to lower precision for the parameters estimation when the number of matched variables is larger under given limited sample size [17].
In this paper, we performed various quantitative simulations under the following 9 scenarios to illustrate the benefits of correct match and further proper adjustment, and to highlight the consequences of improper match and further inappropriate adjustment. a) C is a confounder for the exposure E and the outcome D (Fig. 1a); b) C is a common cause of E and D with an absence of cause effect between them (Fig. 1b); c) C is an independent cause of D (Fig. 1c); d) C is a cause of E, but has no direct causal effect on D (Fig. 1d); e) C is a common effect (i.e. collider) of E and D (Fig. 1e); f) C is an effect of outcome D (Fig. 1f); g) C is an effect of exposure E (Fig. 1g); h) C is a mediator of causal path from E to D (Fig. 1h); i) C is an instrumental variable for E and D (Fig. 1i). All above scenarios almost involve common roles of C in analytic epidemiology.
Nine simulation scenarios. E, C, D indicate exposure, matching factor, outcome, respectively. Let variable S indicate whether a person is selected for case-control study or not, the square around S indicates the analysis is conditional on individuals having been selected into the matched case-control study. Dashed line C--D show the colliding bias (i.e., selective bias) due to matching on C. S is a collider on C→S←D. Colliding bias will arise if conditioning on colliding node (i.e., S). a) C is a confounder for the exposure E and the outcome D; b) C is a common cause of E and D with an absence of cause effect between them; c) C is an independent cause of D; d) C is a cause of E, but has no direct causal effect on D; e) C is a common effect (i.e. collider) of E and D; f) C is an effect of outcome D; g) C is an effect of exposure E; h) C is a mediator of causal path from E to D; i) C is an instrumental variable for E and D
A brief introduction to causal diagrams and calculation of causal effect
In the past few decades, causal diagrams, one kind of directed acyclic graphs (DAGs), have been widely used to visually summarize hypothetical causal relations among variables of interest. Modern causal diagrams were more recently developed to merger probability theory with path diagrams [2, 18–20]. The resulting theory provides a powerful yet intuitive device for deducing the statistical associations implied by causal relations. Furthermore, given a set of observed statistical associations, a researcher armed with causal diagrams theory can systematically characterize all causal analysis. In causal diagrams, the d-Separation criterion is an essential graphic rule for linking causal relations to statistical associations [20, 21]. They help epidemiologists to draw logically sound conclusions about certain types of statistical relations and facilitate many tasks, such as understanding confounding bias and selection bias [15], choosing covariates for adjustment or match [10], analyzing direct and indirect effects [22], using instrumental variable to estimate causal effect when unobserved confounders exist [23]. In this paper, we used causal diagrams to illustrate the relationships among variables in above 9 scenarios.
Furthermore, do-calculus together with back-door criterion proposed by Pearl [20, 24, 25] were used to calculate the causal effect of exposure (X) on outcome (Y). Given a causal diagram G, together with non-experimental data on a subset V of observed variables in G, we estimate the causal effect of X on Y by calculating P(y|do(X = x)) from a sample estimation of P(V = v). Namely, we aim to estimate what the intervention do(X = x) would have on a set of response variable Y, where X and Y are two subsets of V. For identifying P(y|do(X = x)), the "back-door criterion" [20] was further used to test if a set Z ⊆ V of variables is sufficient, where Z satisfied the following conditions. (i) it blocked every path from X to Y that has an arrow into X ("blocks the back door"); and (ii) no node in Z is a descendant of X. If a set of variable Z satisfies the back-door criterion relative to (X, Y), then the causal effect of X on Y is identifiable and is calculated by the following formula,
$$ P\left(y\Big| do\left(X=x\right)\right)={\displaystyle \sum_ZP\left(y\Big|x,z\right)P(z)} $$
In this paper, this formula was used to calculate the true causal effect β of exposure E on outcome D from source population. It was regarded as a gold standard to assess the bias of estimation in all 9 simulation scenarios.
Simulation scenarios
Figure 1 showed the causal diagrams of 9 simulation scenarios for estimating causal effect of E on D, which illustrated 9 different roles of C respectively. Based on Fig. 1(a) to (i), Monte Carlo simulations were used to generate simulation data. We made the following assumptions for the simulation: 1) all variables are binary following a Bernoulli distribution; 2) the correlations between variables are positive unless otherwise specified; and 3) the association between covariates (E and C) and the outcome D is log-linearly additive effect. Logistic regression models were used to simulate child nodes from their corresponding parent nodes. Take scenario 1 [seeing Fig. 1(a)] as an example, let P(C = 1) = π , then P(E = 1|C) = exp(α 0 + α 1 C)/[1 + exp(α 0 + α 1 C)] for the child node E from its parent node C; similarly, P(D = 1|C, E) = exp(β 0 + β 1 C + β 2 E)/[1 + exp(β 0 + β 1 C + β 2 E)]; where the parameters α 0, β 0 denoted the baseline prevalence of E and D respectively, and each effect parameter (α 1, β 1, β 2) refers to the log-odds ratio conditional on other covariates. The simulated source population with 100,000 subjects was generated from above procedure. 1000 cases were randomly sampled from this simulated source population with D = 1, while 1000 controls were randomly sampled from D = 0; so far none-matched case-control data with 1000 cases and 1000 controls was created. For matched case-control data, we still used the above same 1000 cases as the cases group, for individual with C = 1 in cases group, we matched its control by randomly sampling a subject with C = 1 and D = 0 from the source population; similarly, for individual with C = 0, we matched its control with C = 0 and D = 0 from the source population.
Besides, unconditional and conditional regression models were applied to above two datasets to assess their performances. For non-matched case-control data, both unconditional logistic regression model without adjusting for C, \( \log it\left(p\left(D=1\Big|E\right)\right)={\beta}_0+{\beta}_{{}_1}^{\prime}\mathrm{E} \), (model 1), and with adjusting for C, log it(p(D = 1|E, C)) = β 0 + β ″1 E + β 2C, (model 2), were performed for comparing their bias (\( {\overset{\frown }{\beta}}_1\hbox{-} \beta \), where \( {\overset{\frown }{\beta}}_1 \) was the estimation by the logistic regression models, while β was the true causal effect from source population) and precision by the standard error of \( {\overset{\frown }{\beta}}_1 \) (\( \mathrm{S}\mathrm{E}\left({\overset{\frown }{\beta}}_1\right) \)). For matched case-control data, the following three models were used to compare their bias (\( {\overset{\frown }{\beta}}_1\hbox{-} \beta \)) and precision (\( \mathrm{S}\mathrm{E}\left({\overset{\frown }{\beta}}_1\right) \)): model 3) unconditional logistic regression without adjusting for C; model 4) unconditional logistic regression with adjusting for C; and model 5) conditional logistic regression.
Various simulation scenarios were performed by varying across a target effect parameter [e.g. C → E in Fig. 1(a)] and keeping all others constant to explore the trends of bias (\( {\overset{\frown }{\beta}}_1\hbox{-} \beta \)) and standard error (\( \mathrm{S}\mathrm{E}\left({\overset{\frown }{\beta}}_1\right) \)). 1000 simulations were repeated in each scenario. All simulation studies were conducted using software R from CRAN (http://cran.r-project.org/).
Scenario 1 (C is a confounder for E and D, Fig. 1a)
Theoretically, in this scenario, the confounder C is d-connected with outcome D via two natural paths: C → D and C → E → D, which contribute to the crude association between C and D. Nevertheless, under matched case-control designs, C is unconditionally independent of D due to the identical distribution of C in cases and controls group (i.e. the sum of C → D, C → E → D and C--D is null). Furthermore, the path C--D is of equal magnitude, but opposite direction to the C → E → D and C → D. Therefore, the joint distribution of E, C and D is unfaithful to the DAG of Fig. 1a due to matching on C. As C is a confounder, both paths C → E and C → D will lead to the bias for E on D before matching, while after matching, a new colliding bias path C--D is created and the two bias paths (C → E, C → D) still exist. In this situation, the total bias is contributed by the path of C → E, C → D and C--D [13–15].
Figure 2 showed the simulation results under scenario 1. It indicated that given other parameters fixed and varying across the effect of C → E (Fig. 2a), the bias (\( {\overset{\frown }{\beta}}_1\hbox{-} \beta \)) elevated linearly with effects of C → E increasing in the model without adjusting for C under non-matched case-control designs (model 1), while elevated in the opposite direction with effects of C → E increasing in the model without adjusting for C under matched case-control designs (model 3); after adjusting for C, the bias was approximate to zero in all models of adjustment for C under non-matched case-control designs (model 2) and matched case-control designs (model 4 or model 5). For their precision (Fig. 2c), the \( SE\left({\overset{\frown }{\beta}}_1\right) \) of all above five models increased with effects of C → E increasing, and model 5 obtained largest standard error, followed by model 4, model 2, model 3, model 1. Similarly, given other parameters fixed and varying across C → D (Fig. 2b), the bias (\( {\overset{\frown }{\beta}}_1\hbox{-} \beta \)) still elevated linearly with effects of C → D increasing in model 1, while lowered with effects of C → D increasing in model 3. After adjusting for C, the bias was still nearly approximate to zero in model 2, model 4 or model 5. For their precision (Fig. 2d), the \( SE\left({\overset{\frown }{\beta}}_1\right) \) of all above five models kept stable with effects of C → D increasing, and model 5 attained largest standard error, followed by model 4, model 2, model 1, model 3. These results suggested that confounding bias and colliding bias generally changed in opposite directions and adjustment was indispensable after matching on C.
Bias (upper panels) and standard error (i.e. SE, lower panels) of log transformed odds ratio estimations for different effect sizes of CE and CD. Each line indicated one model. The left panel displayed the bias and standard error on the estimated values of exposure E for different odds ratio (from 1 to 10) of CE respectively. The right panel showed the bias and standard error of estimated values on exposure E for different odds ratio (from 1 to 10) of CD respectively. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
Scenario 2 (C is a common cause of exposure E and outcome D without causal effect between them, Fig. 1b)
It is similar to scenario 1 (Fig. 1a) except that instead of having causal effect between E and D. In this situation, the path C → D leads to the association of C and D in a non-matched case-control designs. But two effect paths of C and D offset each other after matching, that is the effect of C--D is of equal magnitude, but opposite direction to C → D [14].
Simulation showed that (Fig. 3): keeping other parameters constant, and varying across C → E (Fig. 3a), the bias (\( {\overset{\frown }{\beta}}_1\hbox{-} \beta \)) elevated linearly with effects of C → E increasing in the model without adjusting for C under non-matched case-control designs (model 1), while approximate unbiased estimations were got in model 2, model 3, model 4 and model 5. All five models revealed an increased effect with effects of C → E increasing. Among them, model 2 got largest standard error, followed by model 4, model 5, model 3 and model 1. Similarly, as E ← C → D is a confounding path (Fig. 3b), the bias (\( {\overset{\frown }{\beta}}_1\hbox{-} \beta \)) elevated linearly with effects of C → D increasing in model 1, while the bias was almost null after adjustment (model 2, model 3, model 5) or match (model 4). The \( SE\left({\overset{\frown }{\beta}}_1\right) \) revealed a linearly increasing trend for the five models, while the model 2 illustrated largest standard error, followed by model 4, model 5, model 1, model 3. These results indicated that both matching and adjustment could block the bias path E ← C → D, but adjustment for C would lead to lower precision. Therefore, the best choice is the model without adjusting for C in matched case-control designs (model 3) in scenario 2.
Bias (upper panels) and standard error (i.e. SE, lower panels) of log transformed odds ratio estimations for different effect sizes of CE and CD. Each line indicated one model. The left panel displayed the bias and standard error of different odds ratio (from 1 to 10) of CE. The right panel showed the bias and standard error of different odds ratio (from 1 to 10) of CD. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
Scenario 3 (C is a cause of outcome D, Fig. 1c)
As C is not a confounder, C and E are independent causes of D, respectively, the marginal effect from E to D is an unbiased estimator. In this situation, matching on or adjustment for C will inevitably lead to bias for E on D due to conditional on C by matched case-control designs or logistic regression model [14].
As expected, only model without adjusting for C in non-matched case-control designs (model 1) got unbiased and precise estimation (Fig. 4), and both match and adjustment would increase bias and lower precision with effects of C → D increasing in model 2 to model 5.
Bias (left panels) and standard error (i.e. SE, right panels) of log transformed odds ratio estimations for different effect sizes of CD. Each line indicated one model. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
Scenario 4 (C is a cause of exposure E, Fig. 1d)
The C has a direct effect on E and an indirect effect on D through E. So C is not a confounder for E and D. In this situation, if matching on C, a new association is generated between C and D (denoted with C--D). Thus E ← C--D becomes an open bias path for E on D [14]. Simulation results (Fig. 5) supported above deductions, and revealed that only model without adjusting for C in matched case-control designs (model 3) led to bias (Fig. 5a). In matched case-control designs, although the bias could be remedied by adjusting for C, the precision (Fig. 5b) would become lower (model 4 and model 5).
Bias (left panels) and standard error (i.e. SE, right panels) of log transformed odds ratio estimations for different effect sizes of CE. Each line represented one model. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
Scenario 5 (C is a common effect of exposure E and outcome D, Fig. 1e)
In this scenario, as C is not a confounder but a collider, match on or adjustment for C (model 2 to model 5) will generate colliding bias [14, 15]. The simulation results under varying across the effects of E → C and C ← D (Fig. 6) verified that only model without adjusting for C in non-matched case-control designs (model 1) got unbiased estimation.
Bias (left panels) and standard error (i.e. SE, right panels) of log transformed odds ratio estimations for different effect sizes of EC and DC. Each line indicated one model. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
Scenario 6 (C is an effect of outcome D, Fig. 1f)
In this scenario, the C is not a confounder but an effect (child node) of outcome D, so match on or adjustment for C is not necessary. Simulation results showed that both matching on C and adjusting for C did not lead to bias of β (Fig. 7a), but adjustment for C (model 2, model 4 and model 5) led to lower precision (Fig. 7b).
Bias (left panels) and standard error (i.e. SE, right panels) of log transformed odds ratio estimations for different effect sizes of DC. Each line indicated one model. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
Scenario 7 (C is an effect of exposure E, Fig. 1g)
For this scenario, although C is associated with E (E → C) and D (C ← E → D), it is not a confounder. In practice, it is difficult to distinguish it from confounder by statistical association study. Theoretically, matching on this kind of spurious confounders will open bias path E → C--D and thus lead to biased estimation of β. On the other hand, adjustment for C will not lead to biased estimation of β but will lower its precision. Simulation results are concordant with above deductions, which revealed the biased estimation of β (Fig. 8a) by matching on C (model 3), and showed lower precision (Fig. 8b) by adjusting for C (model 2, model 4 and model 5).
Bias (left panels) and standard error (i.e. SE, right panels) of log transformed odds ratio estimations for different effect sizes of EC. Each line indicated one model. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
Scenario 8 (C is a mediator of causal path from E to D, Fig. 1h)
In this scenario, although C is associated with E (E → C) and D (C → D), it is not a confounder but a mediator. Matching on C will block the path E → C, while adjusting for C will block the path C → D. Therefore, either match or adjustment will inevitably block the causal path E → C → D, and thus leads to the biased estimation of β [14]. Both Fig. 9a and b illustrated that only model without adjusting for C in non-matched case-control designs (model 1) got unbiased estimation of β in the situation of varying across effects of E → C and C → D. In these two situations, lower precision of \( {\overset{\frown }{\beta}}_1 \) (Fig. 9c and d) were observed by adjusting for C (model 2, model 4 and model 5).
Bias (upper panels) and standard error (i.e. SE, lower panels) of log transformed odds ratio estimations for different effect sizes of EC and CD. Each line indicated one model. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
Scenario 9 (C is an instrumental variable for E and D, Fig. 1i)
In Fig. 10, we can easily find that C is not a confounder but an instrumental variable (IV), though C is associated with E (C → E) and D (C → E → D). This instrumental variable C can be used to control for the unobserved confounder U for estimating the causal effect of E on D [26]. However, instead of controlling for the confounding effect of U through either matching on or adjusting for C, the biased estimation for effect of E → D could not be reduced. The simulation results (Fig. 10) indicated that all the five models led to similar bias.
Bias (left panels) and standard error (i.e. SE, right panels) of log transformed odds ratio estimations for different effect sizes of CE. Each line indicated one model. Note: model 1, unconditional logistic regression model without adjusting for C for non-matched case-control designs; model 2, unconditional logistic regression model with adjusting for C for non-matched case-control designs; model 3, unconditional logistic regression model without adjusting for C for matched case-control designs; model 4, unconditional logistic regression model with adjusting for C for matched case-control designs; model 5, conditional logistic regression model with adjusting for C for matched case-control designs
From the perspective of causal diagrams, several studies had claimed that matching on confounders C in matched case-control designs can improve estimation precision for the effect of exposure (E) on outcome (D), though it fails to remove confounding effect of C [8, 9]. Therefore, further adjustment for C using conditional or unconditional logistic regression model after matching is widely used to eliminate the confounding bias of C in analytic epidemiology [13, 14]. When C is exactly a confounder for E and D (scenario 1, Fig. 1a), however, our simulation results did not illustrate distinct improvement of precision for estimating effect of E on D by matching on C (model 3) comparing with by non-matching designs (model 1). Nevertheless, the benefit of matching on C was to greatly reduce the bias for estimating the effect of E on D (model 3) though failed to completely remove the bias (Fig. 2a and b). Further adjusting for C using logistic regression model (model 4 or model 5) after matching almost removed the bias (Fig. 2a and b). Our simulation results suggested that further adjusting for C in matched case-control designs is still essential, while adjustment (Fig. 2c and d) by unconditional logistic regression model (model 4) tend to be more precise than by conditional logistic regression (model 5). Similarly, in scenario 2 (Fig. 1b), C also is a confounder though the causal effect from E to D does not exist. In this situation, both matching on or adjusting for C could obtain unbiased estimation of E on D (Fig. 3), but matched case-control designs without adjusting for C (model 3) was the optimal strategy.
In practice, it is usually difficult to identify confounders just from statistical association [7, 27]. 1) In scenario 3 (Fig. 1c), both C and E are independent causes of D, matching on or adjustment for C will inevitably lead to bias for E on D due to conditional on C (Fig. 4) [14]. 2) In scenario 4 (Fig. 1d), C is associated with E (C → E) and D (C → E → D), but not a confounder. In this situation, matching on C, a new association was generated between C and D (denoted with C--D). Thus E ← C--D became an open bias path for E on D, and generated its biased estimation (Fig. 5). Fortunately, further adjustment for C after match could remedy this bias (model 4 and model 5 in Fig. 5) [14]. 3) In scenario 5 (Fig. 1e), C is not a confounder but a collider, match on or adjustment for C (model 2 to model 5) will inevitably generate colliding bias; only non-matched case-control designs without adjusting for C (model 1) got unbiased estimation (Fig. 6a and b) [14, 15]. 4) In scenario 8 (Fig. 1h), C is associated with E (E → C) and D (C → D), it is not a confounder but a mediator. Matching on C will block the path E → C, while adjusting for C will block the path C → D [14]. Therefore, either match or adjustment will inevitably block the causal path E → C → D, and thus lead to the biased estimation of β (Fig. 9). In this situation, only model without adjusting for C in non-matched case-control designs (model 1) got unbiased estimation of β. However, adjustment for C (model 2, model 4 and model 5) would reduce the precision of \( {\overset{\frown }{\beta}}_1 \) (Fig. 9c and d). It was, therefore, dangerous and improper to arbitrarily match on or adjust for the plausible confounder C [28].
Above simulation scenarios (scenario 1, 2, 3, 4, 5, 8) have been explored by shahar and Mansournia et al., but beyond that we proposed three new causal diagrams (scenario 6, 7, 9) with respect to match or adjustment strategies. Our simulation results showed that, for case-control study designs, when C is not a confounder but an effect (child node) of outcome D (scenario 6, Fig. 1f), match on or adjustment for C is not necessary (Fig. 7) in that it did not lead to biased estimation of β (Fig. 7a). In scenario 7 (Fig. 1g), C is associated with E (E → C) and D (C ← E → D), but not a confounder. Matching on this kind of spurious confounders would open bias path E → C--D and thus led to biased estimation of β (Fig. 8). Although adjusting for C did not lead to biased estimation of β, it would reduce precision (Fig. 8). Specifically, when C is an instrumental variable for E and D, although it is associated with E (C → E) and D (C → E → D), matching on or adjusting for it, the biased for effect of E → D could not be reduced (Fig. 10).
In conclusion, for using match or adjustment strategy in case-control studies, investigators should firstly attempt to figure out the possible causal relationships among exposure (E), outcome (D) and their related covariates (C) empirically based on the etiologic and pathological mechanism and then determine whether match or adjustment should be adopted. Otherwise, arbitrary matching on or adjusting for the plausible confounder C is dangerous.
DAGs, directed acyclic graphs; IV, instrumental variable; SE, standard error
Weinberg CR. Toward a clearer definition of confounding. Am J Epidemiol. 1993;137(1):1–8.
Greenland S, Pearl J, Robins JM. Causal diagrams for epidemiologic research. Epidemiology. 1999;10(1):37–48.
Greenland S, Robins JM. Identifiability, exchangeability, and epidemiological confounding. Int J Epidemiol. 1986;15(3):413–9.
VanderWeele TJ, Shpitser I. On the definition of a confounder. Ann Stat. 2013;41(1):196–220.
Hernán MA, Hernández-Díaz S, Werler MM, et al. Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology. Am J Epidemiol. 2002;155(2):176–84.
Pourhoseingholi MA, Baghestani AR, Vahedi M. How to control confounding effects by statistical analysis. Gastroenterol Hepatol Bed Bench. 2012;5(2):79.
Williamson EJ, Aitken Z, Lawrie J, et al. Introduction to causal diagrams for confounder selection. Respirology. 2014;19(3):303–11.
Pearce N. Analysis of matched case-control studies. BMJ. 2016;352:i969.
Kupper LL, Karon JM, Kleinbaum DG, et al. Matching in epidemiologic studies: validity and efficiency considerations. Biometrics. 1981;37(2):271–91.
Stuart EA. Matching methods for causal inference: a review and a look forward. Stat Sci. 2010;25(1):1–21.
Rose S, Laan MJ. Why match? Investigating matched case-control study designs with causal effect estimation. Int J Biostat. 2009;5(1):Article 1.
Brookmeyer RON, Liang KY, Linet M. Matched case-control designs and overmatched analyses. Am J Epidemiol. 1986;124(4):693–701.
Shahar E, Shahar DJ. Causal diagrams and the logic of matched case-control studies. Clin Epidemiol. 2012;4:137–44.
Mansournia MA, Hernan MA, Greenland S. Matched designs and causal diagrams. Int J Epidemiol. 2013;42(3):860–9.
Shahar E, Shahar DJ. Causal diagrams and the logic of matched case– control studies. Clin Epidemiology. 2012;4:137–144.
Breslow NE, Day NE. Conditional logistic regression for matched sets. Statistical Methods in Cancer Research. 1980;1:248–79.
Rahman M, et al. Conditional versus unconditional logistic regression in the medical literature. J Clin Epidemiol. 2003;56(1):101–2.
Joffe M, Gambhir M, Chadeau-Hyam M, Vineis P. Causal diagrams in systems epidemiology. Emerg Themes Epidemiol. 2012;9(1):1.
Pearl J. Causal diagrams for empirical research. Biometrika. 1995;82(4):669–88.
Pearl. Causality: Models, Reasoning, and Inference. 2nd ed. Cambridge University Press; 2009.
Geiger D, Verma TS, Pearl J. d-separation: From theorems to algorithms. arXiv preprint arXiv:1304.1505. 2013
Pearl J. Direct and indirect effects. In: Proceedings of the Seventeenth Conference on Uncertainty and Artificial Intelligence. San Francisco: Morgan Kaufmann; 2001. p. 411–420.
Angrist JD, Imbens GW, Rubin DB. Identification of causal effects using instrumental variables. J Am Stat Assoc. 1996;91(434):444–55.
Pearl J. Causal inference in statistics: an overview. Statistics Surveys. 2009;3:96–146.
Geiger D, Pearl J. On the logic of causal models. arXiv preprint arXiv:1304.2355. 2013
Myers JA, Rassen JA, Gagne JJ, et al. Effects of adjusting for instrumental variables on bias and precision of effect estimates. Am J Epidemiol. 2011;174(11):1213–22.
Jepsen P, Johnsen SP, Gillman MW, et al. Interpretation of observational studies. Heart. 2004;90(8):956–60.
CAS Article PubMed Central Google Scholar
Robinson LD, Jewell NP. Some surprising results about covariate adjustment in logistic regression models. International Statistical Review/Revue Internationale de Statistique. 1991;59(2)227–40.
We would like to thank anonymous reviewers and academic editors for providing us with constructive comments and suggestions and also wish to acknowledge our colleagues for their invaluable work.
This work was supported by grants from National Natural Science Foundation of China (grant number 81573259).
HKL helped conduct the literature review and prepare the Methods and the Discussion sections of the text. ZSY, PS, XRS, TTW and YYY designed the study's simulation strategy. FZX designed the study and directed its implementation. All authors read and approved the final manuscript.
FZX is an professor at Shandong University, China. He is an expert in GWAS analysis and Spatial data analysis. ZSY is a lecturer at the same university and mainly study the GWAS analysis. HKL, PS, XRS, TTW, YYY are graduate students in same university.
Department of Biostatistics, School of Public Health, Shandong University, Jinan City, Shandong Province, People's Republic of China
Hongkai Li, Zhongshang Yuan, Ping Su, Tingting Wang, Yuanyuan Yu, Xiaoru Sun & Fuzhong Xue
Hongkai Li
Zhongshang Yuan
Ping Su
Tingting Wang
Yuanyuan Yu
Xiaoru Sun
Fuzhong Xue
Correspondence to Fuzhong Xue.
Li, H., Yuan, Z., Su, P. et al. A simulation study on matched case-control designs in the perspective of causal diagrams. BMC Med Res Methodol 16, 102 (2016). https://doi.org/10.1186/s12874-016-0206-3
Matched case-control designs
Causal diagrams | CommonCrawl |
Snub order-6 square tiling
In geometry, the snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}.
Snub order-6 square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.3.4.3.4
Schläfli symbols(4,4,3)
s{4,6}
Wythoff symbol| 4 4 3
Coxeter diagram
Symmetry group[(4,4,3)]+, (443)
[6,4+], (4*3)
DualOrder-4-4-3 snub dual tiling
PropertiesVertex-transitive
Images
Symmetry
The symmetry is doubled as a snub order-6 square tiling, with only one color of square. It has Schläfli symbol of s{4,6}.
Related polyhedra and tiling
The vertex figure 3.3.3.4.3.4 does not uniquely generate a uniform hyperbolic tiling. Another with quadrilateral fundamental domain (3 2 2 2) and 2*32 symmetry is generated by :
Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443)
[(4,4,3+)]
(3*22)
[(4,1+,4,3)]
(*3232)
h{6,4}
t0(4,4,3)
h2{6,4}
t0,1(4,4,3)
{4,6}1/2
t1(4,4,3)
h2{6,4}
t1,2(4,4,3)
h{6,4}
t2(4,4,3)
r{6,4}1/2
t0,2(4,4,3)
t{4,6}1/2
t0,1,2(4,4,3)
s{4,6}1/2
s(4,4,3)
hr{4,6}1/2
hr(4,3,4)
h{4,6}1/2
h(4,3,4)
q{4,6}
h1(4,3,4)
Uniform duals
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
See also
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
Footnotes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
\begin{document}
\title{An instability index theory for quadratic pencils and applications} \author{
Jared Bronski
\thanks{E-mail: \href{mailto:[email protected]}{[email protected]}} \\
Department of Mathematics \\
University of Illinois Urbana-Champaign \\
Urbana, IL 61801 \and
Mathew A. Johnson
\thanks{E-mail: \href{mailto:[email protected]}{[email protected]}} \\
Department of Mathematics \\
University of Kansas \\
Lawrence, KS 66045 \and
Todd Kapitula
\thanks{E-mail: \href{mailto:[email protected]}{[email protected]}} \\
Department of Mathematics and Statistics \\
Calvin College \\
Grand Rapids, MI 49546 }
\begin{titlingpage} \usethanksrule \setcounter{page}{0} \maketitle \begin{abstract}
Primarily motivated by the stability analysis of nonlinear waves in second-order in time Hamiltonian systems, in this paper we develop an instability index theory for quadratic operator pencils acting on a Hilbert space. In an extension of the known theory for linear pencils, explicit connections are made between the number of eigenvalues of a given quadratic operator pencil with positive real parts to spectral information about the individual operators comprising the coefficients of the spectral parameter in the pencil. As an application, we apply the general theory developed here to yield spectral and nonlinear stability/instability results for abstract second-order in time wave equations. More specifically, we consider the problem of the existence and stability of spatially periodic waves for the ``good" Boussinesq equation. In the analysis our instability index theory provides an explicit, and somewhat surprising, connection between the stability of a given periodic traveling wave solution of the ``good" Boussinesq equation and the stability of the same periodic profile, but with different wavespeed, in the nonlinear dynamics of a related generalized Korteweg-de Vries equation. \end{abstract}
\cancelthanksrule \renewcommand{0.0pt}{0.0pt}
\pdfbookmark[1]{\contentsname}{toc} \tableofcontents \end{titlingpage}
\section{Introduction}\label{sec:intro}
When analyzing equations arising in mathematical physics and engineering, the question of stability of special families of solutions is of prominent importance as it generally determines those solutions which are most likely to be observed in physical applications. In particular, solutions which are unstable do not naturally, i.e. in absence of a controller, arise in applications except possibly as transient phenomena. Furthermore, stability analysis is often the first step in the study of finer phenomena such as transient behavior, bifurcation, and the ability to control a wave to restrict it to a stable configuration. In this paper, we are primarily motivated by recent studies into the stability of traveling wave solutions of second-order in time Hamiltonian equations of the form \begin{equation}\label{e:ham} \partial_t^2u+\mathcal{L}_xu+\mathcal{N}(u)=0,\quad (t,x)\in\mathbb{R}^2 \end{equation} where $\mathcal{L}_x$ is a self-adjoint linear operator acting on the $x$-variable only, and $\mathcal{N}(u)$ denotes nonlinear terms (e.g., see \citep{angulo,arruda:nsp09,hakkaev:lsa12,stanis}). A fundamental characteristic of such PDE is that they take into account weak effects of both nonlinearity and dispersion, and they arise naturally, for instance, as models for propagation of waves in nonlinear strings and in the study of bi-directional water wave propagation in the small amplitude, long wavelength regime.
In regards to the latter water wave application, an equation of particular interest in this paper (see \autoref{s:gbou} below) is the generalized ``good" Boussinesq (gB) equation \begin{equation}\label{e:gbou} \partial_t^2u-\partial_x^2\left(\partial_x^2u-u+f(u)\right)=0, \end{equation} which is a variant of one of one of the equations formulated by Boussinesq in the 1870's in precisely this physical context. While the nonlinear stability and instability of solitary waves in \eref{e:gbou} is by now well understood (see \citep{bona:geo88,liu}), the stability (whether linear or nonlinear) of the periodic traveling waves have received considerably less attention, and results only exist for very special classes of solutions; namely, those expressible in terms of Jacobi-elliptic functions. One of the main applications of the theory developed in this paper is an explicit connection between periodic traveling waves of \eref{e:gbou} and the stability of the same traveling wave profile (but with a \emph{different} wavespeed) in the nonlinear dynamics governed by the generalized Korteweg-de Vries (gKdV) equation \begin{equation}\label{e:gkdv} \partial_tu+\partial_x\left(\partial_x^2u+f(u)\right)=0. \end{equation} We consider this observation as a major contribution to the theory of traveling periodic waves in \eref{e:gbou}. Indeed, since the stability of periodic traveling waves in gKdV equations has been under intense investigation over the last few years (see, for instance, \citep{angulobona,arruda:nsp09,brj,bronski:ait11,deconinck:ots10,deconinck:tos10,Jkdv}), this connection between the dynamics of gB and gKdV near periodic traveling waves allows one to immediately translate known results for the stability of gKdV periodic waves to results about periodic waves in gB. Applications of this connection will be illustrated in \autoref{s:case1} and \autoref{s:case2} below.
Suppose that $u(x-ct)$ is a traveling wave solution to \eref{e:ham}. When considering small perturbations to this wave of the form $e^{\lambda t}v(x-ct),\,\lambda\in\mathbb{C}$, we are naturally led to quadratic spectral problems of the form \[ \lambda^2 v -2c\lambda\partial_xv+\left(c^2\partial_x^2+\mathcal{L}_x+\mathcal{N}'(u)\right)v=0. \]
More generally, when considering the spectral stability of the given nonlinear dispersive wave $u$ it becomes important to understand the spectrum of quadratic operator pencils of the form
\begin{equation}\label{e:i1} \mathcal{P}_2(\lambda)\mathrel{\mathop:}=\mathcal{A}+\lambda\mathcal{B}+\lambda^2\mathcal{C}, \end{equation} where the operators $\mathcal{A}$ and $\mathcal{C}$ are self-adjoint on some Hilbert space $X$, endowed with an inner-product $\langle\cdot,\cdot\rangle$, and $\mathcal{B}$ is skew-symmetric
on X.
The class of perturbations considered in our applications naturally leads us to assume that
the domain of each of the operators $\mathcal{A},\mathcal{B},\mathcal{C}$ is dense in $X$ and that they each enjoy a particular compactness property
which guarantees that the spectrum of $\mathcal{P}_2(\lambda)$, i.e., the collection of values $\lambda$ for which
$\mathcal{P}_2(\lambda)$ fails to be boundedly invertible, is composed of point spectrum only, that each eigenvalue has finite algebraic multiplicity, and the only accumulation point of the eigenvalues is infinity (see \autoref{l:noessential} below for a precise statement). In this paper $\sigma(\mathcal{P}_2)$ will denote the collection of all eigenvalues for the pencil.
Due to its clear connection and importance in analyzing the spectral stability of traveling wave solutions of equations of the form \eref{e:ham}, our main theoretical results concern extending many previously known results regarding the number of eigenvalues of $\mathcal{P}_2$ with positive real part. For the readers convenience, we now briefly recall the relevant known results. The spectrum was studied in \citep{pivovarchik:oso07} under the assumption that $\mathcal{A}$ has compact resolvent, and $\mathcal{B},\mathcal{C}$ are bounded and positive semi-definite. The operators $\mathcal{B},\mathcal{C}$ are not assumed to have any symmetry properties, however. While it is not shown in that paper, by \autoref{l:noessential} it is then known that the pencil only possesses point eigenvalues, each of which has finite algebraic multiplicity. It is shown in that paper that if $\mathcal{A}$ is positive semi-definite, then all of the spectrum is located in the closed left-half of the complex plane. If $\mathcal{A}$ is not definite, it is shown that if $\mathcal{C}=\mathcal{I}$ and $\mathcal{B}$ is positive definite, then the total number of eigenvalues in the closed right-half of the complex-plane is equal to the number of negative eigenvalues of $\mathcal{A}$. As is seen in, e.g., \autoref{s:3}, it is not necessary that $\mathcal{C}=\mathcal{I}$ in order for that result to hold; indeed, all that is needed is that $\mathcal{C}$ be positive-definite. In \citep{lyong:tsp93} similar results are shown regarding the number of eigenvalues with positive real part under the assumption that $\mathcal{C}=\mathcal{I}$ and $\mathcal{B}$ is positive semi-definite. The spectrum of operators of the form $\mathcal{P}_2$ was studied in \citep{gurski:sdo04} under the assumptions that both $\mathcal{A},\,\mathcal{C}$ are positive definite and self-adjoint, while $\mathcal{B}$ is a negative definite self-adjoint operator (also see \citep{kollar:hmf11} for a generalization). Under assumptions different than those given in \autoref{l:noessential} (in particular, both $\mathcal{A}$ and $\mathcal{C}$ are assumed to be compact) it is shown therein that in $\sigma(\mathcal{P}_2)$ there is an infinite number of positive eigenvalues which have zero as a limit point. Finally, the matrix-valued version of the pencil was studied in \citep{kollar:hmf11,chugunova:oqe09} under the assumption that $\mathcal{C}=\mathcal{I}$ and $\mathcal{A},\,\mathcal{B}$ self-adjoint. Therein a parity index is given relating the number of eigenvalues with positive real part to the number of negative eigenvalues of $\mathcal{A}$. This result strongly depends upon the fact that the operators are matrix-valued, and consequently the Hilbert space under question is finite-dimensional.
One of the main goals of this paper is thus to extend the previous theory regarding the number of eigenvalues (counting multiplicity) with positive real part; see Section \ref{s:3} below. Throughout this analysis, two underlying assumptions will be: \begin{enumerate} \item $\mathrm{n}(\mathcal{A}),\mathrm{n}(\mathcal{C})<+\infty$, where $\mathrm{n}(\mathcal{S})$ refers to the number of negative eigenvalues (counting multiplicity) of the self-adjoint operator $\mathcal{S}$ \item $\mathcal{C}$ is invertible, i.e., $\mathrm{z}(\mathcal{C})=0$, where $\mathrm{z}(\mathcal{S})=\mathop\mathrm{dim}\nolimits[\mathop\mathrm{ker}\nolimits(\mathcal{S})]$ for the self-adjoint operator $\mathcal{S}$. \end{enumerate} Under these assumptions we have the intuition that the sum $\mathrm{n}(\mathcal{A})+\mathrm{n}(\mathcal{C})$ acts as an upper bound on the total number of eigenvalues of $\mathcal{P}_2$ with positive real part. In order to see this, consider the following. If the linear term in the pencil is dropped, then the quadratic pencil is equivalent to the linear pencil \[ \left[\left(\begin{array}{cc}\mathcal{A}&0\\0&\mathcal{C}^{-1}\end{array}\right)-\lambda \left(\begin{array}{rr}0&\mathcal{I}\\-\mathcal{I}&0\end{array}\right)\right] \left(\begin{array}{c}u\\v\end{array}\right)= \left(\begin{array}{c}0\\0\end{array}\right). \] Note that the first term in the pencil is self-adjoint, whereas the second term is skew-symmetric. Linear pencils of this form have been well-studied (e.g., see \citep{haragus:ots08,kapitula:cev04,kapitula:ace05,pelinovsky:ilf05} and the references therein), and for this problem it has firmly been established that the intuition is indeed correct. The technical difficulty is then the inclusion of the linear term, and it is overcome in the establishment of \autoref{thm:index} below.
In particular, in the case where $\mathcal{P}_2(\lambda)$ arises in the stability analysis of a given periodic traveling wave (as described previously), a sufficient condition for spectral stability is that $\mathrm{n}(\mathcal{A})+\mathrm{n}(\mathcal{C})=0$.
In order to achieve an equality relating the number of eigenvalues of $\mathcal{P}_2$ with positive real part to spectral properties of the operators $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ themselves, two additional factors must be taken into account: \begin{enumerate} \item the effect of $\mathcal{A}$ having a nontrivial kernel \item purely imaginary eigenvalues having a negative Krein index (see \autoref{s:3}). \end{enumerate} The first factor arises because in applications the presence of symmetries yields the existence of a nontrivial kernel, while the consideration of the second factor is necessary in order to remove the intuitive inequality and make it an equality.
The paper is organized as follows. In \autoref{s:2} it is shown that under general assumptions, which are natural for the applications we have in mind, there is no essential spectrum for the quadratic operator pencil $\mathcal{P}_2$; in other words, there will be only point eigenvalues, and each eigenvalue will have finite algebraic multiplicity. The proof of this result easily generalizes to polynomial operators of arbitrary order; see \autoref{r:higherorderpencil} below. In \autoref{s:3} the main theoretical result of the paper relating the number of eigenvalues of $\mathcal{P}_2$ with positive real part to the spectral properties of $\mathcal{A}$, $\mathcal{B},$ and $\mathcal{C}$, are stated and proved. Finally, in \autoref{s:4} we apply the theoretical results developed in \autoref{s:2} and \autoref{s:3} to the study of the spectral and orbital stability of nonlinear waves to second-order in time Hamiltonian systems. The first application develops a general theoretical result concerning the stability of steady states in abstract nonlinear wave equations; see \autoref{s:wave}. The second application examines the spectral and nonlinear (orbital) stability of periodic traveling waves in the ``good" Boussinesq equation \eref{e:gbou}; see \autoref{s:gbou}. As described above, of particular interest in our study of the Boussinesq equation is that we establish in \autoref{lem:51} and \autoref{thm:orbstable2} a rigorous connection between a given stationary periodic wave $u$ of the equation \[
\partial_t^2u-2c\partial_{tx}^2u+\partial_x^2\left(\partial_{x}^2u-(1-c^2)u+f(u)\right)=0,\quad |c|<1, \] corresponding to the traveling wave $u(x-ct)$ in the gB, and the stability of the same stationary wave profile in the gKdV, \[ \partial_tu+\partial_x\left(\partial_x^2-\sqrt{1-c^2}\,u+f(u)\right)=0. \]
To illustrate the power of this connection, in \autoref{s:case1} we translate recent results by Bronski et al. \citep{bronski:ait11} concerning the stability of periodic KdV waves with power-law nonlinearity to results for the corresponding Boussinesq equation, allowing one to see the complete stability picture of periodic waves in gB in the cases considered. In \autoref{s:case2}, using known asymptotic analyses worked out in the context of stability theory for gKdV waves (see \citep{brj}), we analyze the stability of periodic waves with power-law nonlinearity which are near (in an appropriate sense) either the solitary wave or equilibrium solutions of gB. The analysis near the solitary wave makes a beautiful connection with the known nonlinear stability/instability results for the nearby solitary waves. Similarly, the analysis near the equilibrium solution provides new insights into the transitions to instability in the periodic context; in particular, we find that for a given power-law nonlinearity stable periodic traveling wave solutions always exist, even when no stable solitary wave exists.
\begin{acknowledgment} TK gratefully acknowledges the support of the Jack and Lois Kuipers Applied Mathematics Endowment, a Calvin Research Fellowship, and the National Science Foundation under grants DMS-0806636 and DMS-1108783. MJ was partially supported by the University of Kansas General Research Fund allocation 2302278. JB gratefully acknowledges support from the National Science Foundation under grant DMS-DMS-0807584. \end{acknowledgment}
\section{Preliminary result: spectra is point only}\label{s:2}
\noindent\textbf{Notation:} In this paper, and particularly in this and the subsequent section, the notion of \textsl{matrix representation} of a self-adjoint operator $\mathcal{S}$ constrained to a subspace $E$, i.e., $S|_{E}$, will often be used. Let $\{e_1,\dots,e_n\}$ be a basis for $E$, and let $P_E:X\mapsto E$ be the orthogonal projection. If the basis is orthonormal, then one can write \[ P_E=\sum_{j=1}^n\langle\cdot,e_j\rangle e_j. \]
Setting $S|_E=P_E\mathcal{S} P_E:E\mapsto E$, the quadratic form \[
\langle u,S|_Eu\rangle=\langle P_Eu,P_E\mathcal{S} P_Eu\rangle=\bm{\mathit{c}}\cdot\bm{\mathit{S}}\bm{\mathit{c}}, \] where $P_Eu=\sum c_je_j,\,\bm{\mathit{c}}=(c_1,\dots,c_n)^\mathrm{T}$, and the symmetric matrix
$\bm{\mathit{S}}\in\mathbb{R}^{n\times n}$ is given by $\bm{\mathit{S}}_{ij}=\langle e_i,\mathcal{S} e_j\rangle$. The matrix $\bm{\mathit{S}}$ is precisely the matrix representation for the self-adjoint operator $\mathcal{S}|_E$. In the rest of this paper the symbol $\mathcal{S}|_E$ will be used to represent the matrix representation $\bm{\mathit{S}}$ of $\mathcal{S}$ constrained to operate on the subspace $E$.
The goal of this section is to demonstrate that for the quadratic pencil of \eref{e:i1}, i.e., \[ \mathcal{P}_2(\lambda)=\mathcal{A}+\lambda\mathcal{B}+\lambda^2\mathcal{C}, \] there is point spectra only, that each eigenvalue has finite multiplicity, and infinity is the only possible limit point of the eigenvalues. The result requires the use of \citep[Theorem~12.9]{markus:itt88}, in which it is shown that the spectrum has the desired properties for the polynomial operator \[ \mathcal{P}_n(\lambda)=\mathcal{I}+\sum_{j=1}^n\lambda^n\mathcal{A}_j \] if each operator $\mathcal{A}_j,\,j=1,\dots,n$, is compact. With this result in mind, first assume that $\mathcal{A}$ is invertible, and rewrite the eigenvalue problem as \[ \mathcal{A}(\mathcal{I}+\lambda\mathcal{A}^{-1}\mathcal{B}+\lambda^2\mathcal{A}^{-1}\mathcal{C})u=0. \] Of course, since $\mathcal{A}$ is nonsingular this problem is equivalent to \[ (\mathcal{I}+\lambda\mathcal{A}^{-1}\mathcal{B}+\lambda^2\mathcal{A}^{-1}\mathcal{C})u=0. \] If the operators $\mathcal{A}^{-1}\mathcal{B}$ and $\mathcal{A}^{-1}\mathcal{C}$ are both compact, then the result immediately follows from \citep[Theorem~12.9]{markus:itt88}.
On the other hand, now suppose that $\mathop\mathrm{ker}\nolimits(\mathcal{A})$ is nontrivial, but that $\mathcal{A}$ has compact resolvent. The proof of the desired result will now be accomplished via the construction and evaluation of a generalized \textsl{Krein matrix}, which was recently introduced in a general form in \citep{kapitula:tks10}. Let $P_\mathcal{A}:X\mapsto\mathop\mathrm{ker}\nolimits(\mathcal{A})$ be the orthogonal projection. Writing $u=a+a^\perp$, where $a\in\mathop\mathrm{ker}\nolimits(\mathcal{A})$ and $a^\perp\in \mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp$, the eigenvalue problem becomes \begin{equation}\label{e:pp2} \mathcal{P}_2(\lambda)a+\mathcal{P}_2(\lambda)a^\perp=0. \end{equation} Defining the complementary projection $P_{\mathcal{A}}^\perp\mathrel{\mathop:}=\mathcal{I}-P_\mathcal{A}$, applying this projection to \eref{e:pp2}, and solving for $a^\perp=P_{\mathcal{A}}^\perp a^\perp$ yields \begin{equation}\label{e:pp3} a^\perp=-(P_{\mathcal{A}}^\perp\mathcal{P}_2(\lambda)P_{\mathcal{A}}^\perp)^{-1}P_{\mathcal{A}}^\perp\mathcal{P}_2(\lambda)a. \end{equation} In the formulation of \eref{e:pp3} it is implicitly being assumed that $\mathcal{P}_2(\lambda)a^\perp\neq0$. If $\mathcal{P}_2(\lambda)a^\perp=0$, then $\lambda$ is an eigenvalue whose eigenfunction is in $\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp$: this follows immediately from \eref{e:pp2} upon setting $a=0$. Since $\mathcal{P}_2(\lambda)a=0$, the potential pole singularity for such a $\lambda$ is removable. If the inner-product with $a$ is now taken in \eref{e:pp2}, then it is seen that \begin{equation}\label{e:pp4} \langle a,\mathcal{P}_2(\lambda)a\rangle+\langle a^\perp,\mathcal{P}_2(\lambda)^\mathrm{a} a\rangle=0, \end{equation} where \[ \mathcal{P}_2(\lambda)^\mathrm{a}=\mathcal{A}-\overline{\lambda}\mathcal{B}+\overline{\lambda}^2\mathcal{C} \] is the adjoint operator for the original pencil. Note that the fact that $\mathcal{A},\mathcal{C}$ are self-adjoint and $\mathcal{B}$ is skew-symmetric was used in this formulation of the adjoint pencil. Substituting the expression for $s^\perp$ given in \eref{e:pp3} into \eref{e:pp4} yields the linear system \begin{equation}\label{e:pp5} \bm{\mathit{K}}_2(\lambda)\bm{\mathit{x}}=\bm{\mathit{0}},\quad
\bm{\mathit{K}}_2(\lambda)\mathrel{\mathop:}=\mathcal{P}_2(\lambda)|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}-
(P_{\mathcal{A}}^\perp\mathcal{P}_2(\lambda)P_{\mathcal{A}}^\perp)^{-1}|_{P_{\mathcal{A}}^\perp\mathcal{P}_2(\lambda)[\mathop\mathrm{ker}\nolimits(\mathcal{A})]}. \end{equation} Here \[ \bm{\mathit{x}}\in\mathbb{C}^{\mathrm{z}(\mathcal{A})},\quad\bm{\mathit{K}}_2(\lambda)\in\mathbb{C}^{\mathrm{z}(\mathcal{A})\times\mathrm{z}(\mathcal{A})}, \] and \[ \mathcal{P}_2(\lambda)[\mathop\mathrm{ker}\nolimits(\mathcal{A})]\mathrel{\mathop:}=\{\mathcal{P}_2(\lambda)a:a\in\mathop\mathrm{ker}\nolimits(\mathcal{A})\}. \] The matrix $\bm{\mathit{K}}_2(\lambda)$ is known as the Krein matrix.
Eigenvalues for the pencil are found either via $\bm{\mathit{x}}=\bm{\mathit{0}}$, which means that if $\lambda$ is an eigenvalue, then the associated eigenfunction satisfies $u\in\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp$, or $\mathop\mathrm{det}\nolimits[\bm{\mathit{K}}_2(\lambda)]=0$. The quest for eigenvalues is now a search for the zeros of the determinant of the Krein matrix. The poles of the Krein matrix are the eigenvalues of the operator \[ P_{\mathcal{A}}^\perp\mathcal{P}_2(\lambda)P_{\mathcal{A}}^\perp= P_{\mathcal{A}}^\perp\mathcal{A} P_{\mathcal{A}}^\perp+\lambda P_{\mathcal{A}}^\perp\mathcal{B} P_{\mathcal{A}}^\perp+ \lambda^2P_{\mathcal{A}}^\perp\mathcal{C} P_{\mathcal{A}}^\perp:\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp\mapsto\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp. \] Since $P_{\mathcal{A}}^\perp\mathcal{A} P_{\mathcal{A}}^\perp:\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp\mapsto\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp$ is invertible, from the earlier argument in this section we know that the operator $P_{\mathcal{A}}^\perp\mathcal{P}_2(\lambda)P_{\mathcal{A}}^\perp$ has point spectra only, each eigenvalue has finite multiplicity, and infinity is the only possible limit point of the eigenvalues. Thus, one can say that $\mathop\mathrm{det}\nolimits[\bm{\mathit{K}}_2(\lambda)]:\mathbb{C}\mapsto\mathbb{C}$ is meromorphic, each singularity is a pole of finite order, and the only possible accumulation point of the poles is infinity. Since $\mathop\mathrm{det}\nolimits[\bm{\mathit{K}}_2(\lambda)]$ is meromorphic, it is then known that each of its zeros is of finite order, and that the only possible accumulation point of the zeros is infinity. Finally, it was demonstrated in \citep{kapitula:tks10} that for linear pencils the order of the zero of $\mathop\mathrm{det}\nolimits[\bm{\mathit{K}}_2(\lambda)]$ is equal to the algebraic multiplicity of the eigenvalue. The proof of that result easily carries over to quadratic pencils, and will be left for the interested reader.
\begin{lemma}\label{l:noessential} Let $P_\mathcal{A}:X\mapsto\mathop\mathrm{ker}\nolimits(\mathcal{A})$ be the orthogonal projection, and let $P_\mathcal{A}^\perp=\mathcal{I}-P_\mathcal{A}$ be the complementary projection. Suppose that the operators \[ (P_\mathcal{A}^\perp\mathcal{A} P_\mathcal{A}^\perp)^{-1}P_\mathcal{A}^\perp\mathcal{B} P_\mathcal{A}^\perp,\, (P_\mathcal{A}^\perp\mathcal{A} P_\mathcal{A}^\perp)^{-1}P_\mathcal{A}^\perp\mathcal{C} P_\mathcal{A}^\perp:\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp\mapsto\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp \] are both compact. Then the spectrum of the quadratic pencil $\mathcal{A}+\lambda\mathcal{B}+\lambda^2\mathcal{C}$ is point spectra only. Furthermore, each eigenvalue has finite multiplicity, and infinity is the only possible limit point of the eigenvalues. \end{lemma}
\begin{remark}\label{r:higherorderpencil} Note that the proof of \autoref{l:noessential} did not require that any of the operators have a symmetry property. Thus, this lemma can be thought of as a general result about quadratic pencils. Indeed, although it will not be proven here, the above argument can be extended to show that for $n^\mathrm{th}$-order polynomial pencils of the form \[ \mathcal{P}_n(\lambda)=\sum_{j=0}^n\lambda^j\mathcal{A}_j, \] if each of the operators $(P_{\mathcal{A}_0}^\perp\mathcal{A}_0P_{\mathcal{A}_0}^\perp)^{-1}P_{\mathcal{A}_0}^\perp\mathcal{A}_jP_{\mathcal{A}_0}^\perp$ is compact for $j=1,\dots,n$, then the spectrum for this pencil will have exactly the same properties as that for the quadratic pencil. \end{remark}
\section{Main result: the instability index theorem}\label{s:3}
The goal here is to derive an instability index theorem for the quadratic pencil.
In applications it may be the case that $\mathcal{C}^{-1}$ is bounded and not compact, and/or $\mathcal{B}$ is not bounded. In order to overcome these technical difficulties (which are associated with the proof only), take a positive definite self-adjoint operator $\mathcal{S}$ which has a compact inverse, and consider the quadratic pencil \[ \widehat{\mathcal{P}}_2(\lambda)\mathrel{\mathop:}=\widehat{\mathcal{A}}+\lambda\widehat{\mathcal{B}}+ \lambda^2\widehat{\mathcal{C}}. \] The operators in this new pencil are related to those of the original pencil by \[ \widehat{\mathcal{A}}\mathrel{\mathop:}=\mathcal{S}^{-1}\mathcal{A}\mathcal{S}^{-1},\quad \widehat{\mathcal{B}}\mathrel{\mathop:}=\mathcal{S}^{-1}\mathcal{B}\mathcal{S}^{-1},\quad \widehat{\mathcal{C}}\mathrel{\mathop:}=\mathcal{S}^{-1}\mathcal{C}\mathcal{S}^{-1}, \] so that \[ \mathcal{P}_2(\lambda)=\mathcal{S}\widehat{\mathcal{P}}_2(\lambda)\mathcal{S}. \] The working assumption is that $\mathcal{B},\,\mathcal{C}$ are $\mathcal{S}$-compact, while $\mathcal{A}$ has $\mathcal{S}$-compact resolvent, where an operator $\mathcal{T}$ is said to be $\mathcal{S}$-compact if the operator $\mathcal{S}^{-1}\mathcal{T}\mathcal{S}^{-1}$ is compact, and is said to have $\mathcal{S}$-compact resolvent if the inverse $\mathcal{S}\mathcal{T}^{-1}\mathcal{S}$ (defined to act on the range space) is compact.
Define the space $X_\mathcal{S}$ by \[ X_\mathcal{S}\mathrel{\mathop:}=\{u\in X:\langle\mathcal{S} u,\mathcal{S} u\rangle<\infty\}. \] It will be assumed that $X_\mathcal{S}\subset X$ is dense. Since $\mathcal{S}^{-1}$ is compact, and therefore bounded, it is clear that $\sigma(\widehat{\mathcal{P}}_2)\subset\sigma(\mathcal{P}_2)$ on $X$. It is also clear that $\sigma(\mathcal{P}_2)$ when considered on the space $X_\mathcal{S}$ is a subset of $\sigma(\widehat{\mathcal{P}}_2)$ when considered on the space $X$. In other words, it is true that \[ \sigma(\mathcal{P}_2)\,\,\mathrm{on}\,\,X_\mathcal{S}\subset \sigma(\widehat{\mathcal{P}}_2)\,\,\mathrm{on}\,\,X\subset \sigma(\mathcal{P}_2)\,\,\mathrm{on}\,\,X. \] Since $X_\mathcal{S}\subset X$ is dense, by \citep[Proposition~3.2]{shkalikov:opa96} it will be the case that \[ \sigma(\mathcal{P}_2)\,\,\mathrm{on}\,\,X_\mathcal{S}= \sigma(\widehat{\mathcal{P}}_2)\,\,\mathrm{on}\,\,X. \] Finally, since $\mathcal{S}^{-1}$ is bounded, it is true that $\mathrm{n}(\mathcal{A})=\mathrm{n}(\widehat{\mathcal{A}})$ and $\mathrm{n}(\mathcal{C})=\mathrm{n}(\widehat{\mathcal{C}})$. In conclusion, from this point forward the ``hat"'s associated with the operators can be dropped, and it will be assumed that the operators satisfy $\mathcal{A}^{-1},\,\mathcal{C},\,\mathcal{B}$ are compact.
The instability index theorem for the quadratic pencil will be proven by constructing an equivalent linear pencil, and then deriving an index theorem for that linear pencil. Upon setting $w=(u,\lambda\mathcal{C} u)^\mathrm{T}$, the quadratic pencil \eref{e:i1} is linearized to become \begin{equation}\label{e:h1} (\mathcal{L}-\lambda\mathcal{J}^{-1})w=0, \end{equation} where \[ \mathcal{L}=\left(\begin{array}{cc}\mathcal{A}&0\\0&\mathcal{C}^{-1}\end{array}\right),\quad \mathcal{J}=\left(\begin{array}{rr}0&\mathcal{I}\\-\mathcal{I}&-\mathcal{B}\end{array}\right). \] Here is where the assumption that $\mathcal{C}$ be invertible comes into play. Since $\mathcal{B}$ is skew-symmetric, \eref{e:h1}, and consequently the pencil \eref{e:i1}, is formally equivalent to the Hamiltonian eigenvalue problem \begin{equation}\label{e:h2} \mathcal{J}\mathcal{L} w=\lambda w, \end{equation} where $\mathcal{J}$ is skew-symmetric and $\mathcal{L}$ is self-adjoint. Note that in this formulation \begin{enumerate} \item $\mathcal{L}$ has compact resolvent \item $\mathcal{J}$ has bounded inverse \item the spectrum satisfies the symmetry $\{\lambda,-\overline{\lambda}\}\subset\sigma(\mathcal{J}\mathcal{L})$, and if all of the operators have zero imaginary part, the symmetry becomes $\{\pm\lambda,\pm\overline{\lambda}\}\subset\sigma(\mathcal{J}\mathcal{L})$. \end{enumerate}
Since $\mathcal{C}$ is invertible, it is clear that for nonzero $\lambda$ the two eigenvalue problems have identical eigenvalues; furthermore, the geometric multiplicities match. As it will be seen, it is also the case that the algebraic multiplicities of these eigenvalues also coincide. We will show that this is true at the origin: the proof will clearly generalize to the case of a nonzero eigenvalue. We must consider the structure of $\mathop\mathrm{gker}\nolimits(\mathcal{J}\mathcal{L})$ and the manner in which it relates to $\mathop\mathrm{gker}\nolimits(\mathcal{P}_2(0))$.
First consider $\mathop\mathrm{gker}\nolimits(\mathcal{P}_2(0))$. It is clear that $\mathop\mathrm{ker}\nolimits(\mathcal{P}_2(0))=\mathop\mathrm{ker}\nolimits(\mathcal{A})$. Following \citet{markus:itt88}, generalized eigenfunctions are found by solving \[ \mathcal{P}_2(0)a_1+\mathcal{P}_2'(0)a_0=0,\quad a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A}). \] This is equivalent to solving \begin{equation}\label{e:h3} \mathcal{A} a_1=-\mathcal{B} a_0, \end{equation} which by the Fredholm alternative has a nontrivial solution if and only if
$\mathcal{B} a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp$. Since $\mathcal{B}$ is skew-symmetric, it is not unreasonable to assume that $\mathcal{B}|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}=\bm{\mathit{0}}$. In this case there is a solution to \eref{e:h3} for any $a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A})$, and a full set of associated eigenfunctions is given by $\mathcal{A}^{-1}\mathcal{B}\mathop\mathrm{ker}\nolimits(\mathcal{A})$, where the obvious notation \[ \mathcal{A}^{-1}\mathcal{B}\mathop\mathrm{ker}\nolimits(\mathcal{A})\mathrel{\mathop:}=\{\mathcal{A}^{-1}\mathcal{B} a:a\in\mathop\mathrm{ker}\nolimits(\mathcal{A})\} \] is being used. The next set of eigenfunctions is found by solving \[ \mathcal{P}_2(0)a_2+\mathcal{P}_2'(0)a_1+\frac12\mathcal{P}_2''(0)a_0=0,\quad a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A}),\,\,a_1=-\mathcal{A}^{-1}\mathcal{B} a_0\in\mathcal{A}^{-1}\mathcal{B}\mathop\mathrm{ker}\nolimits(\mathcal{A}). \] This equation is equivalent to \begin{equation}\label{e:h4} \mathcal{A} a_2=-\mathcal{B} a_1-\mathcal{C} a_0= -(\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})a_0. \end{equation} Again using the Fredholm alternative, it is seen that there is a nontrivial solution to \eref{e:h4} if and only if $(\mathcal{B}\mathcal{A}^{-1}\mathcal{B}-\mathcal{C})a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp$. Upon using the fact that for any $a\in\mathop\mathrm{ker}\nolimits(\mathcal{A})$, \[ \langle\mathcal{B}(\mathcal{A}^{-1}\mathcal{B} a_0),a\rangle=-\langle\mathcal{A}^{-1}\mathcal{B} a_0,\mathcal{B} a\rangle, \] it is seen that if \[ \bm{\mathit{D}}\mathrel{\mathop:}=
(\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}
\] is nonsingular, then $(\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})a_0\notin\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp$ for any $a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A})$. It will henceforth be assumed that $\bm{\mathit{D}}$ is nonsingular.
Now consider $\mathop\mathrm{gker}\nolimits(\mathcal{J}\mathcal{L})$. It is clear that $\mathop\mathrm{ker}\nolimits(\mathcal{L})=(\mathop\mathrm{ker}\nolimits(\mathcal{A}),0)^\mathrm{T}$.
Since \begin{equation}\label{e:h4a} \mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})= \left\{\left(\begin{array}{c}-\mathcal{B} a_0\\a_0\end{array}\right):a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A})\right\}, \end{equation} upon writing $w=(u,v)^\mathrm{T}$ the generalized eigenfunctions are found by solving \[ \mathcal{A} u=-\mathcal{B} a_0,\quad\mathcal{C}^{-1}v=a_0. \] The first equation is precisely \eref{e:h3}, which was seen to have a solution for any $a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A})$. Consequently, there is a generalized eigenspace at $\lambda=0$ which is given by $\{(-\mathcal{A}^{-1}\mathcal{B} a_0,\mathcal{C} a_0)^\mathrm{T}:a_0\in\mathop\mathrm{ker}\nolimits(\mathcal{A})\}$. Since \[ \mathcal{J}^{-1}\left(\begin{array}{c}-\mathcal{A}^{-1}\mathcal{B} a_0\\\mathcal{C} a_0\end{array}\right)= -\left(\begin{array}{c}(\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})a_0\\\mathcal{A}^{-1}\mathcal{B} a_0\end{array}\right), \] the next set of generalized eigenfunctions is found by solving \[ \mathcal{A} u=-(\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})a_0,\quad \mathcal{C}^{-1}v=-\mathcal{A}^{-1}\mathcal{B} a_0. \] The first equation is precisely \eref{e:h4}, which was seen to have no solution under the assumption that $\bm{\mathit{D}}$ is nonsingular.
It is now seen that regarding the algebraic multiplicity of the eigenvalue the Hamiltonian linearization \eref{e:h2} is equivalent to the pencil at $\lambda=0$. Following the same argument for nonzero $\lambda$ it is not difficult to check that this equivalence continues to hold; namely, the location of the eigenvalues, and their multiplicities, are the same for the two systems.
We are now ready to derive the instability index for the linear Hamiltonian system \eref{e:h2}. We must first construct the appropriate closed subspace on which both $\mathcal{J},\,\mathcal{L}$ are nonsingular. Recalling that $P_\mathcal{A}^\perp:X\mapsto\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp$ is the orthogonal projection, let $\Pi_\mathcal{L}^\perp:X\times X\mapsto\mathop\mathrm{ker}\nolimits(\mathcal{A})^\perp\times X=\mathop\mathrm{ker}\nolimits(\mathcal{L})^\perp$ be given by \[ \Pi_\mathcal{L}^\perp\mathrel{\mathop:}=\left(\begin{array}{cc}P_\mathcal{A}^\perp&0\\0&\mathcal{I}\end{array}\right); \] in other words, for $w=(u,v)\in X\times X$ it is true that $\Pi_\mathcal{L}^\perp w=(P_\mathcal{A}^\perp u,v)$.
Define another orthogonal projection by \begin{equation}\label{e:h5a} \Pi_{\mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})}^\perp:X\times X\mapsto[\mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})]^\perp. \end{equation}
Because $\mathop\mathrm{ker}\nolimits(\mathcal{L})\perp\mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})$, the projections $\Pi_\mathcal{L}^\perp,\,\Pi_{\mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})}^\perp$ commute. Upon setting $\Pi\mathrel{\mathop:}=\Pi_\mathcal{L}^\perp\Pi_{\mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})}^\perp$ (note that $\Pi$ being the composition of self-adjoint commuting operators implies that it too is self-adjoint), nonzero eigenvalues for the linearization \eref{e:h2} are found by solving \begin{equation}\label{e:h6} \Pi\mathcal{J}\Pi\cdot\Pi\mathcal{L}\Pi\cdot\Pi w=\lambda\Pi w,\quad \Pi w\in[\mathop\mathrm{ker}\nolimits(\mathcal{L})\oplus\mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})]^\perp \end{equation} (e.g., see \citep[Section~2]{deconinck:ots10}). This is the eigenvalue problem to be studied in the rest of this section.
The goal is to now count the total number of eigenvalues in the open right-half of the complex plane (counting multiplicity), along with a those eigenvalues on the imaginary axis which have negative Krein index. For each $\lambda\in\mathrm{i}\mathbb{R}$ let $E_\lambda$ denote the generalized eigenspace. The negative Krein index of the eigenvalue for the linearized system \eref{e:h2} is given by \[
k_\mathrm{i}^-(\lambda)\mathrel{\mathop:}=\mathrm{n}(\mathcal{L}|_{E_\lambda}), \] and if $k_\mathrm{i}^-(\lambda)=0$, then the eigenvalue is (often) said to have positive Krein signature. Let us now relate this definition to a definition using the quadratic pencil. By the definition of $w$ leading to the linearization \eref{e:h1} one has that \[
\mathcal{L}|_{E_{\lambda}}=(\mathcal{A}+|\lambda|^2\mathcal{C})|_{\Pi_1E_{\lambda}}=
(\mathcal{A}-\lambda^2\mathcal{C})|_{\Pi_1E_{\lambda}}, \] where $\Pi_1:X\times X\mapsto X$ is the projection onto the first component, i.e., $\Pi_1(u,v)^\mathrm{T}=u$, and the second equality follows from the fact that $\lambda\in\mathrm{i}\mathbb{R}$. Now, $\lambda$ being an eigenvalue with associated eigenfunction $u$ means that \[ \mathcal{P}_2(\lambda)u=0\quad\Rightarrow\quad \mathcal{A} u=-\lambda\mathcal{B} u-\lambda^2\mathcal{C} u, \] which in turn implies \[
(\mathcal{A}-\lambda^2\mathcal{C})|_{\Pi_1E_{\lambda}}=
-\lambda\mathcal{P}_2'(\lambda)|_{\Pi_1E_{\lambda}}. \] In conclusion, the negative Krein index for the quadratic pencil is defined to be \begin{equation}\label{e:kindex}
k_\mathrm{i}^-(\lambda)\mathrel{\mathop:}=\mathrm{n}(-\lambda\mathcal{P}_2'(\lambda)|_{E_{\lambda}}), \end{equation} where now $E_\lambda=\mathop\mathrm{gker}\nolimits(\mathcal{P}_2(\lambda))$, i.e., the generalized eigenspace of the quadratic pencil $\mathcal{P}_2(\lambda)$ associated with the eigenvalue $\lambda\in\mathrm{i}\mathbb{R}$.
We are now ready to derive the index formula. For the eigenvalue problem \eref{e:h6} let $k_\mathrm{r}$ represent the number of positive real-valued eigenvalues (counting multiplicity), and let $k_\mathrm{c}$ be the number of complex-valued eigenvalues (counting multiplicity) with positive real part. Furthermore, let the total negative Krein index be given by \[ k_\mathrm{i}^-\mathrel{\mathop:}=\sum_{\sigma(\mathcal{P}_2(\lambda))\cap\mathrm{i}\mathbb{R}}k_\mathrm{i}^-(\lambda). \] Regarding the eigenvalue problem \eref{e:h6} it is known that $(\Pi\mathcal{L}\Pi)^{-1}$ is compact, and that the operator $\Pi\mathcal{J}\Pi$ is bounded with bounded inverse. Indeed, since one can write \[ \Pi\mathcal{J}\Pi=\Pi\left(\begin{array}{rr}0&\mathcal{I}\\-\mathcal{I}&0\end{array}\right)\Pi+ \Pi\left(\begin{array}{rr}0&0\\0&-\mathcal{B}\end{array}\right)\Pi, \] one actually has that both $\Pi\mathcal{J}\Pi$ and $(\Pi\mathcal{J}\Pi)^{-1}$ can be written as the sum of a bounded operator and a compact operator. Thus, upon using the fact that compact operators are uniformly approximated by matrices, when computing an index which takes into the account the (finite) number of negative directions of an operator it is sufficient to consider the case of matrices only. For the eigenvalue problem \eref{e:h6} when the operators are matrices it is known from \citep{haragus:ots08} that \begin{equation}\label{e:h9} k_\mathrm{r}+k_\mathrm{c}+k_\mathrm{i}^-=\mathrm{n}(\Pi\mathcal{L}\Pi). \end{equation} Since all of the quantities are integer-valued, by taking the limit one deduces that the result holds for the full operators.
Before stating the final result, the quantity $\mathrm{n}(\Pi\mathcal{L}\Pi)$ must be computed in terms of the original operator $\mathcal{L}$. It is known (e.g., see \citep[Index Theorem]{kapitula:sif12}) that \begin{equation}\label{e:h7}
\mathrm{n}(\Pi\mathcal{L}\Pi)=\mathrm{n}(\mathcal{L})-\mathrm{n}(\mathcal{L}^{-1}|_{\mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})}). \end{equation} It is clearly the case that \[ \mathrm{n}(\mathcal{L})=\mathrm{n}(\mathcal{A})+\mathrm{n}(\mathcal{C}). \] Furthermore, it is straightforward to verify that \[
\mathcal{L}^{-1}|_{\mathcal{J}^{-1}\mathop\mathrm{ker}\nolimits(\mathcal{L})}=
(\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}. \] The instability index \eref{e:h9} can then be rewritten as \begin{equation}\label{e:h8}
\mathrm{n}(\Pi\mathcal{L}\Pi)=\mathrm{n}(\mathcal{A})+\mathrm{n}(\mathcal{C})-\mathrm{n}((\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}). \end{equation} Combining \eref{e:h9} with \eref{e:h8} yields the following theorem:
\begin{theorem}\label{thm:index} Suppose that the operators satisfy the assumption of \autoref{l:noessential}. Further suppose that there is a self-adjoint and positive operator $\mathcal{S}$ such that \begin{enumerate} \item $X_\mathcal{S}\mathrel{\mathop:}=\{u\in X:\langle\mathcal{S} u,\mathcal{S} u\rangle<\infty\}\subset X$ is dense \item the operators $\mathcal{B},\mathcal{C}$ are $\mathcal{S}$-compact, and the operator $\mathcal{A}$ has an $\mathcal{S}$-compact resolvent. \end{enumerate} Finally, assume that the operator $\mathcal{C}$ is invertible. If \begin{enumerate}
\item[(i)] $\mathcal{B}|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}=\bm{\mathit{0}}_{\mathrm{z}(\mathcal{A})}$, and
\item[(ii)] $(\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}$ is invertible, \end{enumerate} then the total number of eigenvalues in the closed right-half of the complex plane satisfies the instability index \[ k_\mathrm{r}+k_\mathrm{c}+k_\mathrm{i}^-=\mathrm{n}(\mathcal{A})+\mathrm{n}(\mathcal{C})-
\mathrm{n}((\mathcal{C}-\mathcal{B}\mathcal{A}^{-1}\mathcal{B})|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}). \] \end{theorem}
\begin{remark} If $\mathcal{A}$ is nonsingular, then \autoref{thm:index} is precisely the result of \citep[Corollary~3.9]{shkalikov:opa96}. On the other hand, in the event that $\mathcal{A}$ has a nontrivial kernel then it is an improvement over \citep[Theorem~4.2]{shkalikov:opa96}, where it was shown that \[ k_\mathrm{r}+k_\mathrm{c}+k_\mathrm{i}^-\le\mathrm{n}(\mathcal{A})+\mathrm{n}(\mathcal{C}). \] The proof presented here is quite different than that given in \citep{shkalikov:opa96}; in particular, in that paper the analysis takes place on Pontryagin spaces, and the linearization that is studied there is not the Hamiltonian linearization of \eref{e:h2}. \end{remark}
\begin{remark} If the imaginary part of all of the operators is zero, then due to the Hamiltonian eigenvalue symmetry $\{\pm\lambda,\pm\overline{\lambda}\}\subset\sigma(\mathcal{P}_2)$ it is necessarily the case that $k_\mathrm{c}$ and $k_\mathrm{i}^-$ are even-value. Thus, under this assumption $\mathrm{n}(\Pi\mathcal{L}\Pi)$ being odd automatically implies that $k_\mathrm{r}\ge1$. On the other hand, if one or more of the operators has a nontrivial imaginary part, then the Hamiltonian eigenvalue symmetry reduces to $\{\lambda,-\overline{\lambda}\}\subset\sigma(\mathcal{P}_2)$, and no such conclusion can be drawn. \end{remark}
\begin{remark} One consequence of the index is that all but a finite number of eigenvalues are purely imaginary; furthermore, the purely imaginary eigenvalues have positive Krein signature if the modulus is sufficiently large. \end{remark}
\begin{remark} The proof of the index formula \eref{e:h9} in \citep{haragus:ots08} first required the SCS Basis Lemma, in which it was shown that the generalized eigenvectors associated with the linearization \eref{e:h1} formed a basis. Technical assumptions on the operators were needed in order for the SCS Basis Lemma to hold true. Unfortunately, at least in the application discussed later in this paper these technical assumptions do not hold; hence, the alternate proof via the limiting argument. \end{remark}
\section{Applications to second-order in time Hamiltonian systems}\label{s:4}
As discussed in the introduction, quadratic operator pencils arise naturally in when one studies the stability of solutions to second-order in time Hamiltonian systems. In this section, we present two applications of the general theory developed in the previous sections in precisely this context. We begin by considering the stability of periodic waves in an (abstract) nonlinear wave equation posed in a Hilbert space, and conclude with a stability analysis for periodic waves in the so-called ``good" Boussinesq equation.
\subsection{Example: stability in (abstract) nonlinear wave equations}\label{s:wave} One important example of quadratic pencils arises in the study of second-order (in time) Hamiltonian systems (for a specific case of the following discussion, see, e.g., \citep[Section~7]{grillakis:sto90}). Consider a wave equation of the form \begin{equation}\label{e:sh1} \partial_t^2u+\mathcal{H}'(u)=0,\quad \mathcal{H}^{(k)}(u)\mathrel{\mathop:}=\frac{\delta^k\mathcal{H}}{\delta u^k}(u), \end{equation} where $u\in X$, which is a Hilbert space with inner-product $\langle\cdot,\cdot\rangle$. The Hamiltonian $\mathcal{H}:X\mapsto\mathbb{R}$ is assumed to be smooth.
It will be assumed that the Hamiltonian system has symmetries. Let $G$ be a finite-dimensional abelian Lie group with Lie algebra $\mathfrak{g}$. Denote by $\mathop\mathrm{exp}\nolimits(\omega)=\mathrm{e}^\omega$ for $\omega\in\mathfrak{g}$ the exponential map from $\mathfrak{g}$ into $G$, and assume that $\mathcal{T}:G\mapsto L(V)$, where $X\subset V\subset X^*$ (the dual space of $X$), is a unitary representation of $G$ on $V$. It is then the case that $\mathcal{T}'(e)$ maps $\mathfrak{g}$ into the space of closed skew-symmetric operators on $V$ with domain $X$. The notation $\mathcal{T}_\omega\mathrel{\mathop:}=\mathcal{T}'(e)\omega$ for $\omega\in\mathfrak{g}$ will be used to denote the linear skew-symmetric operator which is the generator of the semigroup $\mathcal{T}(\mathrm{e}^{\omega t})$. Using this notation the symmetry assumption becomes that the Hamiltonian satisfies $\mathcal{T}(\omega)\mathcal{H}(u)=\mathcal{H}(\mathcal{T}(\omega)u)$ for all $\omega\in\mathfrak{g}$.
Writing $\bm{\mathit{u}}=(u,v)^\mathrm{T}$, where $v=\partial_tu\in X_1$ (in applications, it is often the case that $X\subset X_1$ is dense), the system \eref{e:sh1} can be written on $X\times X_1$ as the first-order Hamiltonian system \begin{equation}\label{e:sh2} \partial_t\bm{\mathit{u}}=\mathcal{J}\widehat{\mathcal{H}}'(\bm{\mathit{u}}), \end{equation} where \[ \mathcal{J}=\left(\begin{array}{rr}0&\mathcal{I}\\-\mathcal{I}&0\end{array}\right),\quad \widehat{\mathcal{H}}(\bm{\mathit{u}})=\mathcal{H}(u)+\frac12\langle v,v\rangle. \] The system \eref{e:sh2} is invariant under the action $\widehat{\mathcal{T}}(\boldsymbol{\omega})$, where \[ \widehat{\mathcal{T}}(\boldsymbol{\omega})\bm{\mathit{u}}= \left(\begin{array}{rr}\mathcal{T}(\boldsymbol{\omega})&0\\0&\mathcal{T}(\boldsymbol{\omega})\end{array}\right)\bm{\mathit{u}}. \] An $n$-parameter family of conserved quantities for the Hamiltonian system \eref{e:sh2} is induced from the self-adjoint operator $\mathcal{J}^{-1}\widehat{\mathcal{T}}_\omega$, and is given by \[ \mathcal{Q}(\bm{\mathit{u}})\mathrel{\mathop:}=\frac12\langle\mathcal{J}^{-1}\widehat{\mathcal{T}}_\omega\bm{\mathit{u}},\bm{\mathit{u}}\rangle= -\mathop\mathrm{Re}\nolimits\left(\langle\mathcal{T}_\omega u,v\rangle\right). \]
Upon defining the Lagrangian \[ \Lambda(\bm{\mathit{u}})\mathrel{\mathop:}=\widehat{\mathcal{H}}(\bm{\mathit{u}})+\mathcal{Q}(\bm{\mathit{u}}), \] waves to \eref{e:sh2} will be realized as steady-state solutions for the system \begin{equation}\label{e:sh3} \partial_t\bm{\mathit{u}}=\mathcal{J}\Lambda'(\bm{\mathit{u}}), \end{equation} i.e., they are critical points for the Lagrangian. Since \[ \Lambda'(\bm{\mathit{u}})=\widehat{\mathcal{H}}'(\bm{\mathit{u}})+ \mathcal{J}^{-1}\widehat{\mathcal{T}}_\omega(\bm{\mathit{u}})= \left(\begin{array}{c}\mathcal{H}'(u)+\mathcal{T}_\omega v\\v-\mathcal{T}_\omega u\end{array}\right), \] critical points are solutions to \begin{equation}\label{e:sh4} \mathcal{H}'(u)+\mathcal{T}_\omega^2u=0,\quad \mathcal{T}_\omega^2u\mathrel{\mathop:}=\mathcal{T}_\omega(\mathcal{T}_\omega u). \end{equation} It should be noted here that \eref{e:sh3} is equivalent to the second-order problem \begin{equation}\label{e:sh4a} \partial_t^2u+2\mathcal{T}_\omega\partial_tu+\mathcal{H}'(u)+\mathcal{T}_\omega^2u=0. \end{equation}
Suppose that $u=U$ is a solution to \eref{e:sh4} (the $\omega$-dependence of the solution is being suppressed here), so that $\bm{\mathit{U}}=(U,\mathcal{T}_\omega U)^\mathrm{T}$ is a critical point of the Lagrangian. Indeed, further suppose that there is a nonempty open set $\Omega\subset\mathfrak{g}$ such that the solution is smooth in $\omega$ for all $\omega\in\Omega$, and further assume that the isotropy subgroups $\{g\in G:\mathcal{T}(g)U=U\}$ are discrete for all $\omega$. Now consider the spectral and orbital stability of the wave. The linearized problem associated with \eref{e:sh3} is given by \begin{equation}\label{e:sh5} \partial_t\bm{\mathit{u}}=\mathcal{J}\mathcal{L}\bm{\mathit{u}}, \end{equation} where the self-adjoint operator $\mathcal{L}$ is \[ \mathcal{L}\mathrel{\mathop:}=\Lambda''(\bm{\mathit{U}})= \left(\begin{array}{cc}\mathcal{H}''(U)&\mathcal{T}_\omega\\-\mathcal{T}_\omega&\mathcal{I}\end{array}\right). \] The eigenvalue problem for \eref{e:sh5} is given by \[ \mathcal{J}\mathcal{L}\bm{\mathit{u}}=\lambda\bm{\mathit{u}}. \] This eigenvalue problem is the system \[ -\mathcal{H}''(U)u-\mathcal{T}_\omega v=\lambda v,\quad -\mathcal{T}_\omega u+v=\lambda u, \] which after substitution is equivalent to the quadratic pencil \begin{equation}\label{e:sh6} (\mathcal{H}''(U)+\mathcal{T}_\omega^2+2\lambda\mathcal{T}_\omega+\lambda^2\mathcal{I})u=0. \end{equation} In the notation of \eref{e:i1} one has \[ \mathcal{A}=\mathcal{H}''(\phi)+\mathcal{T}_\omega^2,\quad \mathcal{B}=2\mathcal{T}_\omega,\quad\mathcal{C}=\mathcal{I}. \] Note that the operators $\mathcal{A},\,\mathcal{C}$ are self-adjoint, while the operator $\mathcal{B}$ is skew-symmetric. It is interesting to note that the negative index of $\mathcal{L}$ is discussed in \citep[Lemma~1]{kostenko:otd02}, where it is stated that \[ \mathrm{n}(\mathcal{L})=\mathrm{n}(\mathcal{H}''(\phi)+\mathcal{T}_\omega^2). \] The number of negative directions of $\mathcal{L}$ is precisely the number of negative directions associated with the linearization of \eref{e:sh4} about $u=U$.
With respect to the spectrum of the pencil \eref{e:sh6} the result of \autoref{thm:index} says the following. The assumptions associated with the symmetries present in the problem imply that \[ \mathop\mathrm{ker}\nolimits(\mathcal{H}''(U)+\mathcal{T}_\omega^2)=\mathop\mathrm{span}\nolimits\{\mathcal{T}_\omega U\}, \] so that $\mathrm{z}(\mathcal{H}''(U)+\mathcal{T}_\omega^2)=n$. Furthermore, these assumptions imply that \[ \mathcal{T}_\omega:\mathop\mathrm{ker}\nolimits(\mathcal{H}''(U)+\mathcal{T}_\omega^2)\mapsto\mathop\mathrm{ker}\nolimits(\mathcal{H}''(U)+\mathcal{T}_\omega^2)^\perp, \] so that the generalized kernel for the pencil has (at least) dimension $2n$. Since $\mathcal{C}=\mathcal{I}$, under the assumption that the matrix \[
(\mathcal{I}-4\mathcal{T}_\omega(\mathcal{H}''(U)+\mathcal{T}_\omega^2)^{-1}\mathcal{T}_\omega)|_{\mathop\mathrm{span}\nolimits\{\mathcal{T}_\omega U\}} \] is invertible, it will then be the case that the instability index count satisfies \begin{equation}\label{e:sh7} k_\mathrm{r}+k_\mathrm{c}+k_\mathrm{i}^-=\mathrm{n}(\mathcal{H}''(\phi)+\mathcal{T}_\omega^2)-
\mathrm{n}\left((\mathcal{I}-4\mathcal{T}_\omega(\mathcal{H}''(U)+\mathcal{T}_\omega^2)^{-1}\mathcal{T}_\omega)|_{\mathop\mathrm{span}\nolimits\{\mathcal{T}_\omega U\}}\right) \end{equation}
Now that the spectral problem is understood, consider the orbital stability of the wave. This result follows almost immediately for \citep[Theorem~4.1]{grillakis:sto90}. An alternate interpretation of that result is as follows. In the language of that paper the wave is said to be orbitally stable if the reduced Hamiltonian, which is the Hamiltonian restricted to the closed subspace orthogonal to the generalized kernel of $\mathcal{J}\mathcal{L}$, is positive definite. As was discussed in, e.g., \citep{deconinck:ots10,kapitula:ots07,deconinck:tos10}, this condition is equivalent to saying that for the linearized problem \eref{e:sh5} the spectrum is purely imaginary and satisfies $k_\mathrm{i}^-=0$.
Under this spectral assumption, and the compactness assumptions associated with the operators, the wave is then a local minimizer for the Lagrangian, and hence is orbitally stable.
\begin{theorem}\label{thm:orbitalstable} Suppose that for the quadratic pencil \[ \mathcal{P}_2(\lambda)\mathrel{\mathop:}=(\mathcal{H}''(U)+\mathcal{T}_\omega^2)+\lambda(2\mathcal{T}_\omega)+\lambda^2\mathcal{I}, \] which is the spectral problem for the linearization of the second-order Hamiltonian system \eref{e:sh4a} about the steady-state $u=U$, the operators satisfy the assumptions associated with \autoref{thm:index}. Assume that solutions to \eref{e:sh4a} exist globally in time. If the eigenvalues satisfy the instability index count $k_\mathrm{r}=k_\mathrm{i}^-=k_\mathrm{c}=0$ (see \eref{e:sh7}), then the wave is orbitally stable. In other words, for each $\epsilon>0$ there is a $\delta>0$ such that if \[
\|u(0)-U\|_X+\|\partial_t u(0)-\mathcal{T}_\omega U\|_{X_1}<\delta, \] then \[
\sup_{t>0}\inf_{g\in G}\left(\|u(t)-\mathcal{T}(g)U\|_X+\|\partial_tu(t)-\mathcal{T}(g)\mathcal{T}_\omega U\|_{X_1}\right)<\epsilon. \] \end{theorem}
\subsection{Example: periodic waves to the ``good" Boussinesq equation}\label{s:gbou}
The generalized ``good" Boussinesq equation (gB) is of the form \begin{equation}\label{e:a51} \partial_t^2u+\partial_x^2(\partial_x^2u-u+f(u))=0, \end{equation} where $f:\mathbb{R}\mapsto\mathbb{R}$ is smooth. In traveling coordinates, i.e., $\xi=x-ct$ with $c\in(-1,1)$, the gB can be rewritten as \begin{equation}\label{e:a52} \partial_t^2u-2c\partial_{t\xi}^2u+\partial_\xi^2(\partial_\xi^2u-(1-c^2)u+f(u))=0. \end{equation} The interest will be on solutions to \eref{e:a52} which are $2L$-periodic in $\xi$, i.e., $u(\xi+2L,t)=u(\xi,t)$.
In order to study the existence, spectral, and orbital stability problems, it is convenient to recast the gB \eref{e:a51} in a Hamiltonian formulation similar to that of \eref{e:sh2}. Herein this task will be accomplished via a trick presented in \citep{bona:geo88}. The evolution is considered to take place on the space $L^2_\mathrm{per}[-L,+L]$, i.e., the space of square-integrable functions which are $2L$-periodic in $\xi$. The inner-product is the standard one, i.e., \[ \langle f,g\rangle=\int_{-L}^{+L}f(x)\overline{g(x)}\,\mathrm{d} x. \] It is straightforward to check that the original gB \eref{e:a51} is equivalent to the system \begin{equation}\label{e:54} \partial_t\bm{\mathit{u}}=\mathcal{J}\widehat{\mathcal{H}}'(\bm{\mathit{u}}),\quad\bm{\mathit{u}}=(u,v)^\mathrm{T}, \end{equation}
where $\partial_xv=\partial_tu$,
\[ \mathcal{J}=\left(\begin{array}{cc}0&\partial_x\\\partial_x&0\end{array}\right),\quad \widehat{\mathcal{H}}(\bm{\mathit{u}})=\int_{-L}^{+L}\left[\frac12(\partial_xu)^2+\frac12u^2-F(u) +\frac12v^2\right]\,\mathrm{d} x. \] Here $F'(u)=f(u)$. Note that the above formulation of $\widehat{\mathcal{H}}$ is consistent with the formulation of the previous section, i.e., \[ \widehat{\mathcal{H}}(\bm{\mathit{u}})=\mathcal{H}(u)+\frac12\langle v,v\rangle,\quad \mathcal{H}(u)=\int_{-L}^{+L}\left[\frac12(\partial_xu)^2+\frac12u^2-F(u)\right]\,\mathrm{d} x, \] while the skew-symmetric operator $\mathcal{J}$ no longer has the property of having a bounded inverse. The system is invariant under spatial translation, i.e., $\widehat{\mathcal{T}}(\omega)\bm{\mathit{u}}(x,t)=\bm{\mathit{u}}(x+\omega,t)$. Consequently, upon using the fact that on $\mathop\mathrm{ker}\nolimits(\partial_x)^\perp$ it is true that \[ \mathcal{J}^{-1}\widehat{\mathcal{T}}_\omega=\left(\begin{array}{cc}0&1\\1&0\end{array}\right), \] the conserved quantity associated with the spatial translation is given by \[ \mathcal{Q}(\bm{\mathit{u}})=\langle u,v\rangle\qquad\left(=\frac12\partial_t\langle u,u\rangle\right), \] and the Lagrangian for the system is \[ \Lambda(\bm{\mathit{u}})=\widehat{\mathcal{H}}(\bm{\mathit{u}})+c\mathcal{Q}(\bm{\mathit{U}})\quad\Rightarrow\quad \Lambda'(\bm{\mathit{u}})=\left(\begin{array}{c}\mathcal{H}'(u)+cv\\cu+v\end{array}\right). \] In conclusion, the system to be studied is \begin{equation}\label{e:55} \partial_t\bm{\mathit{u}}=\mathcal{J}\Lambda'(\bm{\mathit{u}}), \end{equation} which is equivalent to $\partial_t\bm{\mathit{u}}=\widehat{\mathcal{H}}'(\bm{\mathit{u}})$ in traveling coordinates $\xi=x-ct$.
First consider the existence problem. Since $\mathop\mathrm{ker}\nolimits(\partial_x)=\mathop\mathrm{span}\nolimits\{1\}$, for real-valued parameters $a,b$ the problem is \begin{equation}\label{e:56} -\partial_x^2u+u-f(u)+cv=-a,\quad v+cu=b \end{equation} which is equivalent to \begin{equation}\label{e:57a} -\partial_x^2u+(1-c^2)u-f(u)=-(a+cb). \end{equation} This is a well-studied problem. In order to use the desired geometric formulation of \citet{bronski:ait11}, it will first be necessary to rescale the wave-speed via \begin{equation}\label{e:defch} \hat{c}\coloneqq1-c^2\quad\Rightarrow\quad c=c_\pm\mathrel{\mathop:}=\pm\sqrt{1-\hat{c}}, \end{equation} so that \eref{e:57a} can be rewritten as \begin{equation}\label{e:57} -\partial_x^2u+\hat{c}u-f(u)=-(a+cb). \end{equation} Note that in \eref{e:57} the wave-speed $c$ is either of $c_\pm$. Without loss of generality assume that $b=0$. A periodic steady-state, say $U$, will be a solution to the ODE \begin{equation}\label{e:53} \partial_\xi^2U-\hat{c}U+f(U)=a; \end{equation} hence, the solution will naturally depend on the parameters $a$ and $\hat{c}$. If one sets \[ E=\frac12(\partial_\xi U)^2+V(U,a,\hat{c}),\quad V(U,a,\hat{c})\mathrel{\mathop:}=-aU-\frac12\hat{c}U^2+F(U), \] then under the assumption that $E,\,a,$ and $\hat{c}$ are chosen so that \begin{enumerate} \item $E=V(U,a,\hat{c})$ has (at least) two real roots $U_\pm$ with $U_-<U_+$ \item $V(U,a,\hat{c})<E$ for $U_-<U<U_+$ \end{enumerate} (see \autoref{f:ExistenceCriteria}) there will be a periodic solution with period $2L$, where \[ L=\frac1{\sqrt{2}}\int_{U_-}^{U_+}\frac{\mathrm{d} U}{\sqrt{E-V(U,a,\hat{c})}}. \] As it will be seen, in particular examples the spectral stability of the $2L$-periodic solution will naturally depend upon the parameters $a,\hat{c},E$. The dependence of the solution on these parameters will be implicit in all that follows.
\begin{figure}
\caption{(color online) The criteria that the potential $V(U,a,\hat{c})$ must satisfy relative to the energy $E$ in order for there to exist spatially periodic solutions. Here the energy was chosen so that there exist two distinct periodic solutions. Increasing the energy past a threshold $E^*$, for which there are two homoclinic orbits, yields that the two solutions merge into one periodic solution.}
\label{f:ExistenceCriteria}
\end{figure}
Now consider the stability problem. Let the $2L$-periodic wave found in \eref{e:57} be denoted by $u=U$. Recalling that we set $b=0$, from \eref{e:56} the $v$-component of the wave is given by $v=-cU$, where from \eref{e:defch} one has $c=c_\pm$ can be of either sign for a fixed value of $\hat{c}$. The steady-state solution for \eref{e:55} is then given by \[ \bm{\mathit{u}}=\bm{\mathit{U}}=\left(\begin{array}{c}U\\-cU\end{array}\right). \] Under the mapping \begin{equation}\label{e:58} \bm{\mathit{u}}=\bm{\mathit{U}}+\bm{\mathit{v}} \end{equation} the system \eref{e:55} becomes \[ \partial_t\bm{\mathit{v}}=\mathcal{J}\Lambda'(\bm{\mathit{U}}+\bm{\mathit{v}}). \] The evolution problem must now be considered on a space for which $\mathcal{J}$ has bounded inverse, i.e., on the space of mean-zero functions. Let $\Pi_0:L^2_\mathrm{per}[-L,+L]\mapsto H_0$ be the self-adjoint projection operator \[ \Pi_0u=u-\frac1{2L}\langle u,1\rangle; \] in other words, $\Pi_0$ is the orthogonal projection onto $\mathop\mathrm{ker}\nolimits(\partial_\xi)^\perp$. Here \[ H_0\mathrel{\mathop:}=\{u\in L^2_{\mathrm{per}}[-L,+L]:\langle u,1\rangle=0\}= \mathop\mathrm{ker}\nolimits(\partial_\xi)^\perp. \] When writing $\Pi_0\bm{\mathit{v}}$ it will be implicitly assumed that $\Pi_0$ is being applied to each component of $\bm{\mathit{v}}$. In \citep[Section~2]{deconinck:ots10} it is shown that the proper evolution equation to consider is \begin{equation}\label{e:59} \partial_t\bm{\mathit{v}}=\mathcal{J}\Pi_0\Lambda'(\bm{\mathit{U}}+\bm{\mathit{v}}),\quad\bm{\mathit{v}}(0)=\bm{\mathit{v}}_0, \end{equation} where $\Pi_0\bm{\mathit{v}}_0=\bm{\mathit{v}}_0$ implies that $\Pi_0\bm{\mathit{v}}(t)=\bm{\mathit{v}}(t)$ for all $t>0$. In other words, \eref{e:59} describes the evolution of mean-zero perturbations of the underlying wave. Since the evolution occurs on $H_0\times H_0$, and $\partial_x:H_0\mapsto H_0$ has bounded inverse, in this formulation the operator $\mathcal{J}$ now has bounded inverse.
First consider the spectral stability problem. The linearized eigenvalue problem \[ \lambda\bm{\mathit{v}}=\mathcal{J}\Pi_0\Lambda''(\bm{\mathit{U}})\bm{\mathit{v}},\quad\Pi_0\bm{\mathit{v}}=\bm{\mathit{v}} \] can be rewritten as \[ \partial_x[\Pi_0\mathcal{H}''(U)\Pi_0u+cv]=\lambda v,\quad\partial_x[cu+v]=\lambda u, \] where $u,v\in H_0$. Differentiating the first equation yields \[ \partial_x^2\Pi_0\mathcal{H}''(U)\Pi_0 u+c\partial_x(\partial_xv)=\lambda\partial_xv, \] and substituting the second equation into the first and simplifying gives the quadratic pencil problem \[ \left[\lambda^2-2c\lambda\partial_x+\partial_x^2(\Pi_0(-\mathcal{H}''(U)+c^2)\Pi_0)\right]u=0. \] Since $u\in H_0$, one can write $v=\partial_x^{-1}u\in H_0$, so that the pencil becomes \[ \partial_x\left[\lambda^2-2c\lambda\partial_x-\partial_x(\Pi_0(\mathcal{H}''(U)-c^2)\Pi_0)\partial_x\right]v=0. \] Since $\partial_x$ has bounded inverse, upon setting $\mathcal{L}_2$ to be the well-understood self-adjoint Hill operator \[ \mathcal{L}_2=-\partial_x^2+\hat{c}-f'(U(x)), \] the pencil problem to be studied is \[ \left[\lambda^2-2c\lambda\partial_x-\partial_x(\Pi_0\mathcal{L}_2\Pi_0)\partial_x\right]v=0,\quad v\in H_0. \] Note that in the notation of the previous section, \begin{equation}\label{e:59a} \mathcal{C}=\mathcal{I},\quad\mathcal{B}=-2c\partial_x,\quad\mathcal{A}=-\partial_x(\Pi_0\mathcal{L}_2\Pi_0)\partial_x. \end{equation}
Before proceeding with the spectral analysis, the assumptions on the operators given in \autoref{thm:index} must be verified. First consider $\mathop\mathrm{ker}\nolimits(\mathcal{A})$. Since $\mathcal{L}_2(\partial_\xi U)=0$, it is true that \[ \mathcal{L}_2\Pi_0\cdot\partial_\xi(U-\overline{U})=0,\quad \overline{U}=\frac1{2L}\int_{-L}^{+L}U(x)\,\mathrm{d} x. \] In other words, $U-\overline{U}\in\mathop\mathrm{ker}\nolimits(\mathcal{A})$. In order for $\mathop\mathrm{ker}\nolimits(\mathcal{A})$ to have another linearly independent element, it must be the case that $\mathcal{L}_2^{-1}(1)\in H_0$. It will be henceforth assumed that no other element in the kernel exists, i.e., \begin{equation}\label{e:59b} \langle\mathcal{L}_2^{-1}(1),1\rangle\neq0, \end{equation} so that \[ \mathop\mathrm{ker}\nolimits(\mathcal{A})=\mathop\mathrm{span}\nolimits\{U-\overline{U}\}. \] Letting \[ P_\mathcal{A}:H_0\mapsto\mathop\mathrm{span}\nolimits\{U-\overline{U}\},\quad P_\mathcal{A}^\perp:H_0\mapsto\mathop\mathrm{span}\nolimits\{U-\overline{U}\}^\perp\subset H_0 \] be orthogonal projections, it must be checked that \[ (P_\mathcal{A}^\perp\mathcal{A} P_\mathcal{A}^\perp)^{-1}P_\mathcal{A}^\perp\mathcal{B} P_\mathcal{A}^\perp,\, (P_\mathcal{A}^\perp\mathcal{A} P_\mathcal{A}^\perp)^{-1}P_\mathcal{A}^\perp\mathcal{C} P_\mathcal{A}^\perp \] are compact operators. This immediately follows from the fact that $(P_\mathcal{A}^\perp\mathcal{A} P_\mathcal{A}^\perp)^{-1}$ is compact, and both $\mathcal{B}$ and $\mathcal{C}$ are differentiable operators of lesser order than $\mathcal{A}$. Together, the above considerations verify the hypothesis of \autoref{l:noessential}.
Next, we must verify that the operators $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are $\mathcal{S}$-compact for some compact operator $\mathcal{S}$. For each $\alpha>0$, define the operator \[ \mathcal{S}_\alpha\mathrel{\mathop:}=\Pi_0(\partial_x^2+1)^\alpha\Pi_0 \] acting on $L^2_{\rm per}([-L,+L])$. It is clear that $\mathcal{S}_\alpha^{-1}$ is a compact self-adjoint operator on $H_0$ for each $\alpha>0$; furthermore, it is true that the space \[ X_{\mathcal{S}_\alpha}=\{u\in H_0:\langle\Pi_0(\partial_x^2+1)^{2\alpha}\Pi_0u,u\rangle<\infty\} \] is dense for any $\alpha>0$. Now, clearly the operator $\mathcal{S}_\alpha^{-1}\mathcal{C}\mathcal{S}_\alpha^{-1}=\mathcal{S}_\alpha^{-2}$ is compact for any $\alpha>0$. Regarding the operator $\mathcal{B}$, it is easy to see that $\mathcal{S}_\alpha^{-1}\partial_x\mathcal{S}_\alpha^{-1}$ will be compact as long as $1/4<\alpha$. Finally, the operator $\mathcal{S}_\alpha^{-1}\mathcal{A}\mathcal{S}_\alpha^{-1}$ will have a compact resolvent as long as $0<\alpha<1$. In conclusion, as long as $1/4<\alpha<1$, the operators will be $\mathcal{S}_\alpha$-compact, thus verifying hypothesis (a) and (b) of \autoref{thm:index}.
From the skew-symmetry of the operator and the fact that $U$ is $2L$-periodic it is clear that \[
\mathcal{B}|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}=-2c\langle U-\overline{U},\partial_x(U-\overline{U})\rangle=0. \]
Provided that $(\mathcal{I}-4c^2\partial_x\mathcal{A}^{-1}\partial_x)|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}$ is invertible then, a direct application of \autoref{thm:index}, using the explicit form of the operators given in \eref{e:59a} and noting that $\mathcal{C}=\mathcal{I}$ is clearly a positive definite operator, implies that the index count satisfies
\[ k_\mathrm{r}+k_\mathrm{c}+k_\mathrm{i}^-=\mathrm{n}(\mathcal{A})-
\mathrm{n}((\mathcal{I}-4c^2\partial_x\mathcal{A}^{-1}\partial_x)|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}). \] In order to compute $\mathrm{n}(\mathcal{A})$, first note that \[ \langle u,\mathcal{A} u\rangle=\langle u,-\partial_x(\Pi_0\mathcal{L}_2\Pi_0)\partial_x u\rangle= \langle\partial_x u,\Pi_0\mathcal{L}_2\Pi_0(\partial_x u)\rangle. \] Thus, upon using the fact that $\partial_x:H_0\mapsto H_0$ has a bounded inverse it is clear that \[ \mathrm{n}(\mathcal{A})=\mathrm{n}(\Pi_0\mathcal{L}_2\Pi_0). \] Regarding the quantity on the right, it was shown in \citep[equation~(2.25)]{deconinck:ots10} that if the inequality of \eref{e:59b} holds, then \[ \mathrm{n}(\Pi_0\mathcal{L}_2\Pi_0)=\mathrm{n}(\mathcal{L}_2)-\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle). \] Consequently, it can now be said that \begin{equation}\label{e:510} \mathrm{n}(\mathcal{A})=\mathrm{n}(\mathcal{L}_2)-\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle), \end{equation} so that the index count satisfies \[ k_\mathrm{r}+k_\mathrm{c}+k_\mathrm{i}^-=\mathrm{n}(\mathcal{L}_2)-\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle)-
\mathrm{n}((\mathcal{I}-4c^2\partial_x\mathcal{A}^{-1}\partial_x)|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}). \]
Recalling that $c^2=1-\hat{c}$, the index count is complete once the scalar \[
(\mathcal{I}-4(1-\hat{c})\partial_x\mathcal{A}^{-1}\partial_x)|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})} \] is computed. From \eref{e:59a} one has that \[ \begin{split} \mathcal{I}-4(1-\hat{c})\partial_x\mathcal{A}^{-1}\partial_x&=\mathcal{I}+ 4(1-\hat{c})\partial_x\cdot \partial_x^{-1}(\Pi_0\mathcal{L}_2\Pi_0)^{-1}\partial_x^{-1} \cdot\partial_x\\ &=\mathcal{I}+4(1-\hat{c})(\Pi_0\mathcal{L}_2\Pi_0)^{-1}: \end{split} \] the second line follows from the fact that $\partial_x$ is invertible on $H_0$. Using the characterization of $\mathop\mathrm{ker}\nolimits(\mathcal{A})$, it is then seen that \[
(\mathcal{I}-4(1-\hat{c})\partial_x\mathcal{A}^{-1}\partial_x)|_{\mathop\mathrm{ker}\nolimits(\mathcal{A})}= \langle U-\overline{U},U-\overline{U}\rangle+4(1-\hat{c}) \langle(\Pi_0\mathcal{L}_2\Pi_0)^{-1}(U-\overline{U}),U-\overline{U}\rangle. \] Finally, in the study of the orbital stability of periodic waves for the generalized Korteweg-de Vries equation (gKdV) it was shown in \citep[Section~3]{deconinck:ots10} that \[ \langle(\Pi_0\mathcal{L}_2\Pi_0)^{-1}(U-\overline{U}),U-\overline{U}\rangle=
D_{\mathrm{gKdV}},\quad D_{\mathrm{gKdV}}\mathrel{\mathop:}=\frac{\left| \begin{array}{cc} \langle\mathcal{L}_2^{-1}(U),U\rangle&\langle\mathcal{L}_2^{-1}(U),1\rangle\\ \langle\mathcal{L}_2^{-1}(U),1\rangle&\langle\mathcal{L}_2^{-1}(1),1\rangle \end{array}
\right|} {\langle\mathcal{L}_2^{-1}(1),1\rangle}. \]
\begin{lemma}\label{lem:51} Consider the quadratic pencil \eref{e:59}. If $\langle\mathcal{L}_2^{-1}(1),1\rangle\neq0$, then the stability index is given by \[ k_\mathrm{r}+k_\mathrm{i}^-+k_\mathrm{c}=\mathrm{n}(\mathcal{L}_2)-\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle)- \mathrm{n}(\langle U-\overline{U},U-\overline{U}\rangle+4(1-\hat{c})D_{\mathrm{gKdV}}), \] where \[
D_{\mathrm{gKdV}}\mathrel{\mathop:}=\frac{\left| \begin{array}{cc} \langle\mathcal{L}_2^{-1}(U),U\rangle&\langle\mathcal{L}_2^{-1}(U),1\rangle\\ \langle\mathcal{L}_2^{-1}(U),1\rangle&\langle\mathcal{L}_2^{-1}(1),1\rangle \end{array}
\right|} {\langle\mathcal{L}_2^{-1}(1),1\rangle}, \] and the parameter $\hat{c}$ is related to the original wave-speed $c$ via $c^2=1-\hat{c}$. \end{lemma}
\begin{remark} It was shown in \citep[Theorem~2.6]{deconinck:ots10} that the stability index for the gKdV is given by \[ k_\mathrm{r}+k_\mathrm{i}^-+k_\mathrm{c}=\mathrm{n}(\mathcal{L}_2)-\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle)- \mathrm{n}(D_{\mathrm{gKdV}}); \] hence, when studying the spectrum for periodic waves to gB and gKdV there is an intimate connection in the indices for the two problems. If $D_{\mathrm{gKdV}}<0$, then for \[ 1>\hat{c}>1+ \frac{\langle U-\overline{U},U-\overline{U}\rangle}{4D_{\mathrm{gKdV}}}
\] the stability index for the gB is exactly the same as for the gKdV. Otherwise, there is precisely one more eigenvalue which is counted by the index. On the other hand, if $D_{\mathrm{gKdV}}>0$, then the index for the quadratic pencil is exactly that for the gKdV equation. \end{remark}
\begin{remark} There is a geometric interpretation associated with the quantity $D_{\mathrm{gKdV}}$. The interested reader should consult \citep{bronski:ait11} for more details. \end{remark}
\begin{remark}\label{hakkeavrem} In \citet{hakkaev:lsa12} the spectral stability problem was considered under the additional assumption that $\mathrm{n}(\mathcal{L}_2)=1$. Furthermore, while it is not explicitly stated, they further assume that $\langle\mathcal{L}_2^{-1}(1),1\rangle>0$, so that (in this paper's notation) the index becomes \[ k_\mathrm{r}+k_\mathrm{i}^-+k_\mathrm{c}=1- \mathrm{n}(\langle U-\overline{U},U-\overline{U}\rangle+4(1-\hat{c})D_{\mathrm{gKdV}}). \] The instability criterion in that paper follows from the fact that $k_\mathrm{i}^-,k_\mathrm{c}$ must be even integers, so that $k_\mathrm{i}^-=k_\mathrm{c}=0$, with \[ k_\mathrm{r}=\begin{cases} 1,\quad&\hat{c}>\hat{c}^*\\ 0,\quad&\hat{c}<\hat{c}^*, \end{cases} \] where \[ \hat{c}^*=1+ \frac{\langle U-\overline{U},U-\overline{U}\rangle}{4D_{\mathrm{gKdV}}}.
\] It is not clear if in that paper an explicit connection is shown between the gB and the gKdV. \end{remark}
Now consider the orbital stability problem. The local global well-posedness problem has been studied in, e.g., \citep{farah:otp10,fang:eau96,arruda:nsp09}, and it will henceforth be assumed that the problem can be solved (at least) locally. Depending on the growth rate of the nonlinearity, this implies that the initial data for \eref{e:59} satisfies, e.g., $v_1(0)\in H^1_{\mathrm{per}}[-L,+L]$, and $v_2(0)\in H^{-1}_{\mathrm{per}}[-L,+L]$, where the norm for the latter space is given by \[
\|u\|_{H^{-1}}^2=\sum_{z\in\mathbb{Z}}\frac{|\hat{u}(z)|^2}{1+|z|^2}. \]
The form of the Lagrangian in \eref{e:55} makes clear that in order to control the nonlinear terms the proper space in which to work is $H^1_{\mathrm{per}}[-L,+L]\times L^2_{\mathrm{per}}[-L,+L]$. It will be further assumed that the hypothesis leading to \autoref{lem:51} hold, and that the spectral problem has zero instability index, i.e., $k_\mathrm{r}=k_\mathrm{c}=k_\mathrm{i}^-=0$. As was seen in \citep[Section~2.4]{deconinck:ots10}, this is sufficient in order to conclude that the wave is orbitally stable with respect to the evolution defined by \eref{e:59}.
\begin{proposition}\label{thm:orbstable} Suppose that the IVP for \eref{e:59} is locally well-posed. Further suppose that in addition to what is required for a unique local solution to exist, the mean-free perturbative initial data for the system \eref{e:59} satisfies $\bm{\mathit{v}}(0)\in H^1_{\mathrm{per}}[-L,+L]\times L^2_{\mathrm{per}}[-L,+L]$. If the spectral problem satisfies $k_\mathrm{r}=k_\mathrm{c}=k_\mathrm{i}^-=0$, then the underlying wave is orbitally stable. In other words, for each $\epsilon>0$ there is a $\delta>0$ such that for \eref{e:59}, \[
\|\bm{\mathit{u}}(0)-\bm{\mathit{U}}\|_{H^1_{\mathrm{per}}\times L^2_{\mathrm{per}}}<\delta\quad\Rightarrow\quad
\sup_{t>0}\inf_{\omega\in\mathbb{R}}\|\bm{\mathit{u}}(t)-\widehat{\mathcal{T}}(\omega)\bm{\mathit{U}}\|_{H^1_{\mathrm{per}}\times L^2_{\mathrm{per}}}<\epsilon. \] \end{proposition}
Note that for the original system \eref{e:a51} the requirement on the initial data is \[ u(0)=U+v_1(0),\quad\partial_tu(0)=-c\partial_xU+\partial_xv_2(0), \] where each component $v_j(0)$ has zero mean. A natural question is then: what happens if the initial perturbation is not mean-free? In this case, we now argue for orbital stability with respect to a nearby periodic traveling wave of \eref{e:59}. For related arguments in the contexts of other nonlinear dispersive equations, see the work of \citet{HG07} on the nonlinear Schrodinger equation as well as the works of \citet{Jkdv,Jbbm} on generalized KdV and BBM models, respectively.
To begin, we make a few comments regarding the conserved quantities of \eref{e:54}. As discussed above, even though we set $b=0$ in the analysis, the periodic traveling wave solutions of \eref{e:59}, which are solutions to the ODE \eref{e:57a}, form a five-parameter family of solutions of the form \[ u_{\xi}(x,t)=u(x-ct+\xi;a,E,c,b) \] where $\xi\in\mathbb{R}$ and $(a,E,c,b)$ belong to some open set $\Omega\in\mathbb{R}^4$. Furthermore, the evolution equation \eref{e:54} admits the following two conserved quantities: the momentum (charge) \[ \mathcal{P}(\bm{\mathit{w}})\mathrel{\mathop:}=\int_{-L}^Lw_1w_2\,\mathrm{d} x,\quad \bm{\mathit{w}}\mathrel{\mathop:}=\left(w_1,w_2\right)\in H^1_{\rm per}[-L,+L]\times L^2_{\mathrm{per}}[-L,+L], \] arising from the translation invariance of \eref{e:59}, and the casimirs \[ \mathcal{M}_1(\bm{\mathit{w}})\mathrel{\mathop:}=\int_{-L}^Lw_1\,\mathrm{d} x,\quad \mathcal{M}_2(\bm{\mathit{w}}\mathrel{\mathop:}=\int_{-L}^Lw_2\,\mathrm{d} x,\quad \bm{\mathit{w}}\mathrel{\mathop:}=\left(w_1,w_2\right)\in H^1_{\rm per}[-L,+L]\times L^2_{\mathrm{per}}[-L,+L], \] arising from the fact that $\mathop\mathrm{ker}\nolimits(\mathcal{J})$ is non-trivial. Notice that $\mathcal{P}$ and $\mathcal{M}$ are smooth functionals on $H^1_{\rm per}[-L,+L]\times L^2_{\mathrm{per}}[-L,+L]$ and that, when restricted to the manifold of traveling wave solutions of \eref{e:59} the functionals $\mathcal{M}_1,\,\mathcal{M}_2$, and $\mathcal{P}$ reduce to \[ \quad M_1(a,E,c,b)\mathrel{\mathop:}=\int_0^Tu(x;a,E,c,b)\,\mathrm{d} x,\quad M_2(a,E,c,b):=cM_1(a,E,c,b)-bT, \] and \[ \widetilde P(a,E,c,b)\mathrel{\mathop:}=-P(a,E,c,b)+bM_1(a,E,c,b), \] where here $T=T(a,E,c,b)\,(=2L)$ denotes the period of the wave and $P(a,E,c,b)\mathrel{\mathop:}= c\int_0^Tu(x;a,E,c,b)^2\,\mathrm{d} x$.
Now, consider the case where the means of $v_j(0)$ are small, but non-zero. Using the geometric formalism of Bronski et.al (see \citep{bronski:ait11,brj}) we have the following key lemma.
\begin{lemma}\label{l:geom} With the notation as above, we the equality \[
\left(\mathcal{I}-4(1-\hat{c})\partial_x\mathcal{A}^{-1}\partial_x\right)|_{{\rm ker}(\mathcal{A})} =T{\rm det}\left(\begin{array}{cc}
T_a & M_{1,a}\\
T_E & M_{1,E}
\end{array}\right) {\rm det}\left(\begin{array}{cccc}
T_E & M_{1,E} & \widetilde P_E & M_{2,E}\\
T_a & M_{1,a} & \widetilde P_a & M_{2,a}\\
T_c & M_{1,c} & \widetilde P_c & M_{2,c}\\
T_b & M_{1,b} & \widetilde P_b & M_{2,b}
\end{array}
\right). \] \end{lemma}
\begin{proof} To compute the left hand side, define the function \[ \phi_2(x):=\mathop\mathrm{det}\nolimits\left(\begin{array}{ccc}
u_a & T_a & M_{1,a}\\
u_E & T_E & M_{1,E}\\
u_c & T_c & M_{1,c}
\end{array}\right) \] and satisfies \[ \Pi_0\mathcal{L}_2\Pi_0\phi_2 = \Pi_0\mathcal{L}_2\phi_2 = 2c{\rm det}\left(\begin{array}{cc}
T_a & M_{1,a}\\
T_E & M_{1,E}
\end{array}\right)\left(U-\bar U\right). \] The stated equality now follows by directly calculating \[
\left(\mathcal{I}-4(1-\hat{c})\partial_x\mathcal{A}^{-1}\partial_x\right)|_{{\rm ker}(\mathcal{A})} =\left<\Pi_0 U,\left(1+4c^2\left(\Pi_0\mathcal{L}_2\Pi_0\right)^{-1}\right)\Pi_0 U\right> \] and comparing to the right hand side of the above equality. \end{proof}
From \autoref{l:geom} and the assumption that \[
\left(\mathcal{I}-4(1-\hat{c})\partial_x\mathcal{A}^{-1}\partial_x\right)|_{{\rm ker}(\mathcal{A})} \] is nonsingular at the underlying wave $U$, corresponding say to $(a,E,c,b)=(a_0,E_0,c_0,b_0)$, implies that the map \[ \mathbb{R}^4\ni(a,E,c,b)\mapsto\left(T(a,E,c,b),M_1(a,E,c,b),\widetilde P(a,E,c,b),M_2(a,E,c,b)\right)\in\mathbb{R}^4 \] is a local diffeomorphism from a neighborhood of $(a_0,E_0,c_0,b_0)$ onto a neighborhood of the point \[ (T,M_1,\widetilde P,M_2)(a_0,E_0,c_0,b_0). \] It follows that we can find a curve $[0,1]\ni s\mapsto (a(s),E(s),c(s),b(s))\in\mathbb{R}^4$ with
$(a(0),E(0),c(0),b(0))=(0,0,0,0)$
such that for each $s\in[0,1]$ the function \[ \tilde{u}(x;s)=u\left(x;a_0+a(s),E_0+E(s),c_0+c(s),b_0+b(s)\right) \] is a $T=T(a_0,E_0,c_0,b_0)$-periodic traveling wave solution of \eref{e:a51} and that, moreover the endpoint condition
\[ \begin{split} M_j\left(a_0+a(1),E_0+E(1),c_0+c(1),b_0+b(1)\right)&=\mathcal{M}_j\left(\bm{\mathit{u}}(0)+\bm{\mathit{v}}(0)\right),\quad j=1,2\\ P\left(a_0+a(1),E_0+E(1),c_0+c(1),b_0+b(1)\right)&=\mathcal{P}\left(\bm{\mathit{u}}(0)+\bm{\mathit{v}}(0)\right) \end{split} \] and growth constraint \[
\sup_{s\in(0,1)}\left|\left(a(s),E(s),c(s),b(s)\right)\right|_{\mathbb{R}^4}\lesssim \left\|\bm{\mathit{v}}(0)\right\|_{H^1_{\rm per}(-L,L)\times L^2_{\rm per}(-L,L)} \]
are satisfied.
Assuming $\left\|\bm{\mathit{v}}(0)\right\|_{H^1_{\rm per}\times L^2_{\mathrm{per}}}$ is sufficiently small, it follows that the wave $\tilde{u}(\cdot,1)$ is nonlinearly orbitally stable in the sense described in \autoref{thm:orbstable}, which, by the triangle inequality, implies orbital stability of $U$ to initial perturbations $\bm{\mathit{v}}(0)$ with nonzero, but sufficiently small, mean. Since $\mathcal{M}_j$ and $\mathcal{P}$ are continuous in the $H^1_{\rm per}\times L^2_{\mathrm{per}}$ topology, it follows that we have orbital stability in the standard sense without the restriction to mean-free initial data in \eref{e:59}. This observation yields the following extension of \autoref{thm:orbstable}
\begin{theorem}\label{thm:orbstable2} Suppose that the IVP for \eref{e:59} is locally well-posed. Further, suppose that in addition to what is required for a unique local solution to exist, the perturbative initial data for the system \eref{e:59} satisfies $\bm{\mathit{v}}(0)\in H^1_{\rm per}[-L,+L]\times L^2_{\mathrm{per}}[-L,+L]$. If the spectral problem satisfies $k_r=k_c=k_i^-=0$, then the underlying wave is orbitally stable, i.e., for each $\epsilon>0$ there exists a $\delta>0$ such that for \eref{e:59} we have \[
\|\bm{\mathit{u}}(0)-\bm{\mathit{U}}\|_{H^1_{\mathrm{per}}\times L^2_{\mathrm{per}}}<\delta\quad\Rightarrow\quad
\sup_{t>0}\inf_{\omega\in\mathbb{R}}\|\bm{\mathit{u}}(t)-\widehat{\mathcal{T}}(\omega)\bm{\mathit{U}}\|_{H^1_{\mathrm{per}}\times L^2_{\mathrm{per}}}<\epsilon. \] \end{theorem}
In the next sections, we utilize the above explicit connection of the stability problems for gB and gKdV type equations to make several comments regarding the stability of periodic waves in the gB equation.
\subsubsection{Case study: power law nonlinearity}\label{s:case1}
The stability indices for the gKdV equation were computed in \citet{bronski:ait11} for the case that $f(u)=u^{p+1}$ for some $p\ge1$ and
$|\hat{c}|<1$. Note via \eref{e:defch} that this implies $|c_\pm|<1$. While we will not do it here, the gB can be rescaled so that the influence of $\hat{c}$ is removed from the steady-state problem. This independence is reflected in the existence diagrams. As in the analysis leading to the statement of \autoref{lem:51}, it will be assumed here that $b=0$.
First suppose that $p=1$, which corresponds to the classical gB equation. In \citep[Section~5.1]{bronski:ait11} it is shown that all periodic waves in this model satisfy \[ \mathrm{n}(\mathcal{L}_2)=1,\quad\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle=0,\quad \mathrm{n}(D_{\mathrm{gKdV}})=1. \] Thus, via \autoref{lem:51} it is true that for a given $(a,E)$ there is a critical positive wave-speed $\hat{c}^*_{a,E}<1$ such that $k_\mathrm{i}^-=k_\mathrm{c}=0$ with \[ k_\mathrm{r}=\begin{cases} 1,\quad&\hat{c}>\hat{c}_{a,E}\\ 0,\quad&\hat{c}<\hat{c}_{a,E}. \end{cases} \]
Returning to the original wavespeed $c$, it follows that for any periodic traveling wave solution of the classical gB equation there exists a range of wavespeeds $(1-\hat{c}_{a,E}^*)^{1/2}<|c|<1$ for which the wave is nonlinearly (orbitally) stable, while it is unstable spectrally unstable to perturbations with the same period if $|c|<(1-\hat{c}_{a,E}^*)^{1/2}$. This is consistent with the result of \citep[Theorem~2]{hakkaev:lsa12}, where $\hat{c}_{a,E}$ is explicitly given when $a=0$. In that paper the case of nonzero $a$ was not considered and only spectral instability for
$|c|\leq(1-\hat{c}_{a,E}^*)^{1/2}$ was verified. Here, our calculations compliment this result by verifying that waves traveling with speed greater than $(1-\hat{c}_{a,E}^*)^{1/2}$ are by \autoref{thm:orbstable} indeed nonlinearly stable.
\begin{figure}
\caption{(color online) The configuration space in the $aE$-plane when $p=2$,
$b=0$, and $|\hat{c}|<1$ fixed
(see \citep[Figure~3]{bronski:ait11}). The swallowtail figure divides the plane into regions containing $0,1,$ and $2$ periodic solutions. In region (a) there are two solutions, while all of the other marked regions have one solution. In the unmarked region there are no periodic solutions. The quantities used in the stability calculation are given in an accompanying table.}
\label{f:J_MKdV}
\end{figure}
For examples which are not covered in \citet{hakkaev:lsa12}, e.g., when it is possible for $\mathrm{n}(\mathcal{L}_2)\ge2$, first consider the problem when $p=2$. The table below, which corresponds to \autoref{f:J_MKdV}, can be derived from \citep[Section~5.2]{bronski:ait11}:
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline\rule[-3mm]{0mm}{8mm} Region & $\mathrm{n}(\mathcal{L}_2)$ & $\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle)$ & $\mathrm{n}(D_{\mathrm{gKdV}})$ \\\hline\hline (a) & 1 & 0 & 1\\\hline (b) & 2 & 0 & 1 \\\hline (c) & 2 & 1 & 0\\\hline (d) & 2 & 1 & 1\\\hline (e) & 1 & 0 & 1\\\hline \end{tabular} \end{center}
\noindent From the theoretical result in \autoref{lem:51} it will be the case that in that in regions (a), (d), and (e) there will exist a $0<\hat{c}_{a,E}<1$ such that $k_\mathrm{r}=1$ for $\hat{c}>\hat{c}_{a,E}$ and $k_\mathrm{r}=0$ otherwise; furthermore, it is always true that $k_\mathrm{i}^-=k_\mathrm{c}=0$ in these regions. In region (c) it will be the case that $k_\mathrm{r}=1$ for all $c$ with the other two indices being zero. Finally, in region (b) there will exist a $0<\hat{c}_{a,E}<1$ such that $k_\mathrm{r}=1$ for $-1<\hat{c}<\hat{c}_{a,E}$ with the other two indices being zero, while for $\hat{c}>\hat{c}_{a,E}$ all that can be said is that $k_\mathrm{r}+k_\mathrm{i}^-+k_\mathrm{c}=2$. Notice, however, that by parity we see for speeds $\hat{c}>\hat{c}_{a,E}^*$ in region (b) we have $k_\mathrm{r}=0$ and $k_{\mathrm{i}}^-+k_\mathrm{c}=2$, which allows the possibility that some waves may still be spectrally stable in this region with $k_{\mathrm{i}}^-=2$ or that some waves may be spectrally unstable to perturbations with the same period with $k_\mathrm{c}=2$: such a situation is precluded in the well-studied solitary wave theory.
\begin{remark} In \citep[Theorem~1]{hakkaev:lsa12} it is shown that for one of the two solutions in region (a) of \autoref{f:J_MKdV} with $a=0$ the index satisfies $k_\mathrm{i}^-=k_\mathrm{c}=0$ with $k_\mathrm{r}=1$ for $\hat{c}>\hat{c}_{a,E}$, and $k_\mathrm{r}=0$ for $\hat{c}<\hat{c}_{a,E}$. Furthermore, for this solution the constant $\hat{c}_{a,E}$ is explicitly given when $a=0$. Although they do not consider the case in their paper, the same result holds in region (e). The parameter region which is outside their theory comprises the union of (b), (c), and (d). \end{remark}
\begin{figure}
\caption{(color online) The configuration space in the $aE$-plane when $p=4$ and $|\hat{c}|<1$ (see \citep[Figure~4]{bronski:ait11}). The swallowtail figure divides the plane into regions containing $0,1,$ and $2$ periodic solutions. In region (a) there are two solutions, while all of the other marked regions have one solution. In the unmarked region there are no periodic solutions. The quantities used in the stability calculation are given in an accompanying table.}
\label{f:KdV4_stability_diagram}
\end{figure}
The below table for $p=4$ which corresponds to \autoref{f:KdV4_stability_diagram} can be derived from \citep[Section~5.3]{bronski:ait11}:
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline\rule[-3mm]{0mm}{8mm} Region & $\mathrm{n}(\mathcal{L}_2)$ & $\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle)$ & $\mathrm{n}(D_{\mathrm{gKdV}})$ \\\hline\hline (a) & 1 & 0 & 1\\\hline (a') & 1 & 0 & 1 \\\hline (b) & 1 & 0 & 0\\\hline (c) & 2 & 1 & 0\\\hline (d) & 2 & 0 & 1\\\hline \end{tabular} \end{center}
\noindent From the theoretical result in \autoref{lem:51} it will be the case that in that in regions (a) and (a') there will exist a $0<c^*_{a,E}<1$ such that $k_\mathrm{r}=1$ for $\hat{c}>\hat{c}_{a,E}$ and $k_\mathrm{r}=0$ otherwise; furthermore, it is always true that $k_\mathrm{i}^-=k_\mathrm{c}=0$. In regions (b) and (c) it will be the case that $k_\mathrm{r}=1$ for all $\hat{c}$ with the other two indices being zero. Finally, in region (d) there will exist a $0<\hat{c}_{a,E}<1$ such that $k_\mathrm{r}=1$ for $\hat{c}<\hat{c}_{a,E}$ with the other two indices being zero, while for $\hat{c}>\hat{c}_{a,E}$ all that can be said is that $k_\mathrm{r}+k_\mathrm{i}^-+k_\mathrm{c}=2$.
\subsubsection{Case study, continued: solitary wave and equilibrium solution limits}\label{s:case2}
In this final section, we make some comments regarding the stability of periodic traveling wave solutions of \eref{e:a51} which are either near the solitary wave or near an equilibrium (constant) solution. Throughout this section, we continue to consider \eref{e:a51} with power law nonlinearity $f(u)=u^{p+1}$ for some $p\geq 1$. In this case, we have from \eref{e:57a} that the profile $U$ satisfies the ODE \[ \partial_x^2u=(1-c^2)u-u^{p+1}, \] where, for simplicity, we are restricting our discussion to those waves with $a=b=0$\footnote{As we will see below, this is a natural restriction when considering the limiting case to a solitary wave asymptotic to zero as $x\to\pm\infty$.}. This equation is clearly Hamiltonian, and has critical points $(u,\partial_x u)=(0,0)$, corresponding to a saddle point, and $(u,\partial_x u)=((1-c^2)^{1/p},0)$\footnote{Notice when $p$ is an even integer, the point $(u,\partial_x u)=(-(1-c^2)^{1/p},0)$ is also a critical point. In this discussion, we ignore this additional critical point, noting that any conclusions for periodic waves emerging from the $((1-c^2)^{1/p},0)$ critical point hold also for those emerging from the $(-(1-c^2)^{1/p},0)$ critical point. For general $p\geq 1$, the governing ODE does not admit such negative solutions.}, corresponding to a nonlinear center. Further, for a fixed wavespeed $c\in(-1,1)$, in the two-dimensional $(u,\partial_x u)$ phase plane the nonlinear center $((1-c^2)^{1/p},0)$ is surrounded by a one parameter family of periodic orbits, which are in turn bounded by an orbit which is homoclinic to the saddle point $(0,0)$. These periodic orbits can be parameterized by the ODE energy $E$ determined from the defining relation \[ \frac{1}{2}(\partial_x u)^2 = E+\frac{1-c^2}{2}u^2-\frac{1}{p+2}u^{p+2}. \] The period $T(E,c)$ of these waves inside the homoclinic orbit satisfies \[ \lim_{E\to 0^-}T(E,c)=+\infty,\quad\lim_{E\to E^*(c)^+} T(E,c)=\frac{2\pi}{\sqrt{(1-c^2)p}} \] where $E^*(c)$ is the ODE energy level associated with the equilibrium point $((1-c^2)^{1/p},0)$. Below, we consider the stability of the periodic traveling wave solutions of \eref{e:a51} in both the distinguished limits $E\to 0^-$, corresponding to the solitary wave limit, and $E\to E^*(c)^+$, associated with small amplitude periodic wave trains.
We begin by considering the stability of the periodic traveling waves of \eref{e:a51} in the solitary wave limit. When considering solitary wave solutions which decay to zero as $x\to\pm\infty$ we obtain a two-parameter family of solutions parameterized by wavespeed $c$ and spatial translation, which corresponds to a one-parameter family of homoclinic orbits in the two-dimensional phase space of \eref{e:53} (notice the boundary conditions in this case require $a=E=b=0$). Keeping throughout $a=b=0$ and fixing the wavespeed $c\in(-1,1)$, by the above considerations there exists, up to translations, a one parameter family of ``large period" periodic traveling wave solutions of \eref{e:a51} parameterized by the ODE energy $E$. Furthermore, for fixed $c$ these periodic waves approach locally uniformly on $\mathbb{R}$ an appropriate translate of the limiting solitary wave as $E\to E^-$.
The stability of the limiting solitary waves was investigated by \citet{bona:geo88}, where they applied the abstract theory of \citet{grillakis:sto87} to obtain nonlinear stability of the solitary wave when $1<p<4$ and $p/4<c^2<1$. This stability theory was later complimented by \citet{liu}, obtaining nonlinear instability if either $c^2\leq p/4$ or $p\geq 4$. Next, we show that this phenomenon is observed again for the ``nearby" periodic traveling wave solutions of \eref{e:a51}.
For such periodic waves, it is easy to see that \[ \mathrm{n}(\mathcal{L}_2)-\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle)=1,\quad0<-E\ll 1, \] and, furthermore, \[
\lim_{\tilde E\to 0^-}D_{\mathrm{gKdV}}\Big{|}_{(a,E,c,b)=(0,\tilde E,c,0)}=-\Lambda(c,p)(4-p) \] for some positive constant $\Lambda(c,p)>0$ (for details, see the asymptotic analysis in \citep[Section~3.2]{brj}). It follows that for all\footnote{By the previous example, we see that such a critical wavespeed also exists for
$p=4$.} $1\leq p<4$ there exists a critical wavespeed $c=c^*(p,E)$ such that the periodic traveling wave $u(\cdot;a=0,E,c,b=0)$ with $0<-E\ll 1$ is orbitally stable for $c^*(p,E)<|c|<1$ and spectrally unstable with $k_\mathrm{r}=1$
for $|c|<c^*(p,E)$, while for $p>4$ all such long-period waves waves are spectrally unstable with $k_\mathrm{r}=1$. This is consistent with the solitary wave orbital stability/instability results of \citep{bona:geo88} and \citep{liu}.
Finally, continuing to restrict to $a=b=0$, we consider the stability of small amplitude periodic traveling waves of \eref{e:a51} associated to those periodic orbits of the profile ODE near the nonlinear center $((1-c^2)^{1/p},0)$. Fixing the wavespeed $c\in(-1,1)$, it follows by basic asymptotic analysis near the equilibrium solution $u=(1-c^2)^{1/p}$ that \[ \mathrm{n}(\mathcal{L}_2)-\mathrm{n}(\langle\mathcal{L}_2^{-1}(1),1\rangle)=1 \] and, furthermore, that at the nonlinear center we have \[
D_{\mathrm{gKdV}}(a=0,E^*(c),c,b=0)=-\Omega(c,p)V''\left((1-c^2)^{1/p},0,1-c^2\right)^{-9/2}=-\Omega(c,p)\left((1-c^2)p\right)^{-9/2} \] for some positive constant $\Omega(c_0,p)>0$ (see \citep[Section~5]{Jkdv} for details).
It follows that in a neighborhood of the equilibrium solution $u\equiv(1-c^2)^{1/p}$ there exists a critical wavespeed $c(p,E)$ such that all nearby small-amplitude periodic traveling wave solutions of \eref{e:a51} of the form $u(\cdot;a=0,E,c,b=0)$ with $E_0<E\ll 1$ are orbitally stable for
$c^*(p,E)<|c|<1$, and are spectrally unstable to perturbations with
$k_\mathrm{r}=1$ for $0<|c|<c^*(p,E)$. In particular, the equilibrium solution itself is orbitally stable for all $|c|\in(-1,1)$, implying that $c^*(p,E)\to 0^+$ as $E\to (E^*_0)^-$, i.e. as one approaches the equilibrium solution. Note this is consistent with the numerical calculations of $c^*(p,E)$ presented in \citep{hakkaev:lsa12} in the cases $f(u)=u^2$ and $f(u)=u^3$.
\begin{remark} It is interesting to note that when $p>4$ the waves with $a=b=0$ undergo a transition to instability as one moves from a neighborhood of the equilibrium solution $u\equiv(1-c^2)^{1/p}$ to a neighborhood of the limiting solitary wave.
\end{remark}
\phantomsection \addcontentsline{toc}{section}{\refname}
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A $q$ -ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE
CHENG-YANG GU, VICTOR J. W. GUO
Journal: Bulletin of the Australian Mathematical Society , First View
Published online by Cambridge University Press: 18 July 2019, pp. 1-5
We give a $q$ -analogue of the following congruence: for any odd prime $p$ ,
$$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p),\end{eqnarray}$$
which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author ['A $q$ -analogue of the (L.2) supercongruence of Van Hamme', J. Math. Anal. Appl. 466 (2018), 749–761].
Media and Protest Logics in the Digital Era: The Umbrella Movement in Hong Kong By Francis L. F. Lee and Joseph M. Chan New York and Oxford: Oxford University Press, 2018 ix + 263 pp. £19.99 ISBN 978-0-19-085678-6
Edmund W. Cheng
Journal: The China Quarterly / Volume 238 / June 2019
Genetic and morphology analysis among the pentaploid F1 hybrid fishes (Schizothorax wangchiachii ♀ × Percocypris pingi ♂) and their parents
H. R. Gu, Y. F. Wan, Y. Yang, Q. Ao, W. L. Cheng, S. H. Deng, D. Y. Pu, X. F. He, L. Jin, Z. J. Wang
Journal: animal , First View
Triploid and pentaploid breeding is of great importance in agricultural production, but it is not always easy to obtain double ploidy parents. However, in fishes, chromosome ploidy is diversiform, which may provide natural parental resources for triploid and pentaploid breeding. Both tetraploid and hexaploid exist in Schizothorax fishes, which were thought to belong to different subfamilies with tetraploid Percocypris fishes in morphology, but they are sister genera in molecule. Fortunately, the pentaploid hybrid fishes have been successfully obtained by hybridization of Schizothorax wangchiachii (♀, 2n = 6X = 148) × Percocypris pingi (♂, 2n = 4X = 98). To understand the genetic and morphological difference among the hybrid fishes and their parents, four methods were used in this study: morphology, karyotype, red blood cell (RBC) DNA content determination and inter-simple sequence repeat (ISSR). In morphology, the hybrid fishes were steady, and between their parents with no obvious preference. The chromosome numbers of P. pingi have been reported as 2n = 4X = 98. In this study, the karyotype of S. wangchiachii was 2n = 6X = 148 = 36m + 34sm + 12st + 66t, while that the hybrid fishes was 2n = 5X = 123 = 39m + 28sm + 5st + 51t. Similarly, the RBC DNA content of the hybrid fishes was intermediate among their parents. In ISSR, the within-group genetic diversity of hybrid fishes was higher than that of their parents. Moreover, the genetic distance of hybrid fishes between P. pingi and S.wangchiachii was closely related to that of their parental ploidy, suggesting that parental genetic material stably coexisted in the hybrid fishes. This is the first report to show a stable pentaploid F1 hybrids produced by hybridization of a hexaploid and a tetraploid in aquaculture.
MP07: Diagnosis of elevated intracranial pressure in critically ill adults – a systematic review and meta-analysis
S. Fernando, A. Tran, W. Cheng, B. Rochwerg, M. Taljaard, K. Kyeremanteng, S. English, M. Sekhon, D. Griesdale, D. Dowlatshahi, M. Czosnyka, V. McCredie, E. Wijdicks, S. Almenawer, K. Inaba, V. Rajajee, J. Perry
Journal: Canadian Journal of Emergency Medicine / Volume 21 / Issue S1 / May 2019
Published online by Cambridge University Press: 02 May 2019, p. S44
Introduction: Elevated intracranial pressure (ICP) is a devastating complication of brain injury, such as traumatic brain injury, subarachnoid hemorrhage, intracerebral hemorrhage, ischemic stroke, and other conditions. Delay to diagnosis and treatment are associated with increased morbidity and mortality. For Emergency Department (ED) physicians, invasive ICP measurement is typically not available. We sought to summarize and compare the accuracy of physical examination, imaging, and ultrasonography of the optic nerve sheath diameter (ONSD) for diagnosis of elevated ICP. Methods: We searched Medline, EMBASE and 4 other databases from inception through August 2018. We included only English studies (randomized controlled trials, cohort and case-control studies). Gold standard was ICP≥20 mmHg on invasive ICP monitoring. Two reviewers independently screened studies and extracted data. We assessed risk of bias using Quality Assessment of Diagnostic Accuracy Studies 2 criteria. Hierarchical Summary Receiver Operating Characteristic model generated summary diagnostic accuracy estimates. Results: We included 37 studies (n = 4,768, kappa = 0.96). Of exam signs, pooled sensitivity and specificity for increased ICP were: mydriasis (28.2% [95% CI: 16.0-44.8], 85.9.0% [95% CI: 74.9-92.5]), motor posturing (54.3% [95% CI: 36.6-71.0], 63.6% [95% CI: 46.5-77.8]) and Glasgow Coma Scale (GCS) ≤8 (75.8% [95% CI: 62.4-85.5], 39.9% [95% CI: 26.9-54.5]). Computed tomography findings: compression of basal cisterns had 85.9% [95% CI: 58.0-96.4] sensitivity and 61.0% [95% CI: 29.1-85.6] specificity; any midline shift had 80.9% [95% CI: 64.3-90.9] sensitivity and 42.7% [95% CI: 24.0-63.7] specificity; midline shift≥1cm had 20.7% [95% CI: 13.0-31.3] sensitivity and 89.2% {95% CI: 77.5-95.2] specificity. Finally, pooled area under the ROC curve describing accuracy for ONSD sonography for ICP was 0.94 (95% CI: 0.91-0.96). Conclusion: The absence of any one physical exam feature (e.g. mydriasis, posturing, or decreased GCS) is not sufficient to rule-out elevated ICP. Significant midline shift is highly suggestive of elevated ICP, but absence of shift does not rule it out. ONSD sonography may be useful in diagnosing elevated ICP. High suspicion of elevated ICP may necessitate treatment and transfer to a centre capable of invasive ICP monitoring.
A brief measure of predeath grief in dementia caregivers: the Caregiver Grief Questionnaire
Sheung-Tak Cheng, Duan Yang Ma, Linda C. W. Lam
Journal: International Psychogeriatrics , First View
Published online by Cambridge University Press: 29 April 2019, pp. 1-9
The study of predeath grief is hampered by measures that are often lengthy and not clearly differentiated from other caregiving outcomes, most notably burden. We aimed to validate a new 11-item Caregiver Grief Questionnaire (CGQ) assessing two dimensions of predeath grief, namely relational deprivation and emotional pain.
Cross-sectional survey.
Community and psychogeriatric clinics.
173 Alzheimer (AD) caregivers who cared for relatives with different degrees of severity (63 mild, 60 moderate, and 50 severe).
Besides the CGQ, measures of caregiver burden and depressive symptoms, and care-recipients' neuropsychiatric symptoms and functional impairment were assessed.
Confirmatory factor analysis supported the hypothesized 2-factor over the 1-factor model, and both subscales were only moderately correlated with burden. Two-week test-retest reliabilities were excellent. Caregivers for mild AD reported less grief than those caring for more severe relatives. Z tests revealed significantly different correlational patterns for the two dimensions, with emotional pain more related to global burden and depressive symptoms, and relational deprivation more related to care-recipients' functional impairment. Both dimensions were mildly correlated with neuropsychiatric symptoms (especially disruptive behaviors and psychotic symptoms) of the care-recipient.
Results supported the reliability and validity of the two-dimensional measure of predeath grief. As a brief measure, it can be readily added to research instruments to facilitate study of this important phenomenon along with other caregiving outcomes.
Resolving wave and laminar boundary layer scales for gap resonance problems
H. Wang, H. A. Wolgamot, S. Draper, W. Zhao, P. H. Taylor, L. Cheng
Journal: Journal of Fluid Mechanics / Volume 866 / 10 May 2019
Print publication: 10 May 2019
Free surface oscillations in a narrow gap between elongated parallel bodies are studied numerically. As this represents both a highly resonant system and an arrangement of relevance to offshore operations, the nature of the damping is of primary interest, and has a critical role in determining the response. Previous experimental work has suggested that the damping could be attributed to laminar boundary layers; here our numerical wave tank successfully resolves both wave and boundary layer scales to provide strong numerical evidence in support of this conclusion. The simulations follow the experiments in using wave groups so that the computation is tractable, and both linear and second harmonic excitation of the gap are demonstrated.
Using multiple correspondence analysis to identify behaviour patterns associated with overweight and obesity in Vanuatu adults
Andrew van Horn, Charles A Weitz, Kathryn M Olszowy, Kelsey N Dancause, Cheng Sun, Alysa Pomer, Harold Silverman, Gwang Lee, Leonard Tarivonda, Chim W Chan, Akira Kaneko, J Koji Lum, Ralph M Garruto
The present study evaluates the use of multiple correspondence analysis (MCA), a type of exploratory factor analysis designed to reduce the dimensionality of large categorical data sets, in identifying behaviours associated with measures of overweight/obesity in Vanuatu, a rapidly modernizing Pacific Island country.
Starting with seventy-three true/false questions regarding a variety of behaviours, MCA identified twelve most significantly associated with modernization status and transformed the aggregate binary responses of participants to these twelve questions into a linear scale. Using this scale, individuals were separated into three modernization groups (tertiles) among which measures of body fat were compared and OR for overweight/obesity were computed.
Vanuatu.
Ni-Vanuatu adults (n 810) aged 20–85 years.
Among individuals in the tertile characterized by positive responses to most of or all the twelve modernization questions, weight and measures of body fat and the likelihood that measures of body fat were above the US 75th percentile were significantly greater compared with individuals in the tertiles characterized by mostly or partly negative responses.
The study indicates that MCA can be used to identify individuals or groups at risk for overweight/obesity, based on answers to simply-put questions. MCA therefore may be useful in areas where obtaining detailed information about modernization status is constrained by time, money or manpower.
Gender and time delays in diagnosis of pulmonary tuberculosis: a cross-sectional study from China
H. G. Chen, T. W. Wang, Q. X. Cheng
Journal: Epidemiology & Infection / Volume 147 / 2019
Published online by Cambridge University Press: 22 February 2019, e94
Gender inequality has severe consequences on public health in terms of delay in diagnosis of pulmonary tuberculosis (PTB). In order to explore gender-related differences in diagnosis delay, a cross-sectional study of 10 686 patients diagnosed with PTB in Yulin from 1 January 2009 to 31 December 2014 was conducted. Diagnosis delay was categorised into 'short delay' and 'long delay' by four commonly used cut-off points of 14, 30, 60 and 90 days. Logistic regression analysis was used to analyse gender differences in diagnostic delay. Stratified analyses by smear results, age, urban/rural were performed to examine whether the effect persisted across the strata. The median delay was 31 days (interquartile range 13–65). Diagnostic delay in females at cut-off points of 14, 30, 60 and 90 days had odds ratios (OR) of 0.99 (95% CI 0.91–1.09), 1.09 (95% CI 1.01–1.18), 1.15 (95% CI 1.05–1.26) and 1.18 (95% CI 1.06–1.31), respectively, compared with males. Stratified analysis showed that females were associated with increased risk of longer delay among those aged 30–60 years, smear positive and living in the rural areas (P < 0.05). The female-to-male OR increased along with increased delay time. Further inquiry into the underlying reasons for gender differences should be urgently addressed to improve the current situation.
The Pain Catastrophizing Scale—short form: psychometric properties and threshold for identifying high-risk individuals
Sheung-Tak Cheng, Phoon Ping Chen, Yu Fat Chow, Joanne W. Y. Chung, Alexander C. B. Law, Jenny S. W. Lee, Edward M. F. Leung, Cindy W. C. Tam
Published online by Cambridge University Press: 20 February 2019, pp. 1-10
The Pain Catastrophizing Scale (PCS) measures three aspects of catastrophic cognitions about pain—rumination, magnification, and helplessness. To facilitate assessment and clinical application, we aimed to (a) develop a short version on the basis of its factorial structure and the items' correlations with key pain-related outcomes, and (b) identify the threshold on the short form indicative of risk for depression.
Social centers for older people.
664 Chinese older adults with chronic pain.
Besides the PCS, pain intensity, pain disability, and depressive symptoms were assessed.
For the full scale, confirmatory factor analysis showed that the hypothesized 3-factor model fit the data moderately well. On the basis of the factor loadings, two items were selected from each of the three dimensions. An additional item significantly associated with pain disability and depressive symptoms, over and above these six items, was identified through regression analyses. A short-PCS composed of seven items was formed, which correlated at r=0.97 with the full scale. Subsequently, receiver operating characteristic (ROC) curves were plotted against clinically significant depressive symptoms, defined as a score of ≥12 on a 10-item version of the Center for Epidemiologic Studies-Depression Scale. This analysis showed a score of ≥7 to be the optimal cutoff for the short-PCS, with sensitivity = 81.6% and specificity = 78.3% when predicting clinically significant depressive symptoms.
The short-PCS may be used in lieu of the full scale and as a brief screen to identify individuals with serious catastrophizing.
Reusable blood collection tube holders are implicated in nosocomial hepatitis C virus transmission
Vincent C. C. Cheng, Shuk-Ching Wong, Sally C. Y. Wong, Siddharth Sridhar, Cyril C. Y. Yip, Jonathan H. K. Chen, James Fung, Kelvin H. Y. Chiu, Pak-Leung Ho, Sirong Chen, Ben W. C. Cheng, Chi-Lai Ho, Chung-Mau Lo, Kwok-Yung Yuen
Loneliness interacts with family relationship in relation to cognitive function in Chinese older adults
Ada W. T. Fung, Allen T. C. Lee, S.-T. Cheng, Linda C. W. Lam
Journal: International Psychogeriatrics / Volume 31 / Issue 4 / April 2019
Loneliness and social networks have been extensively studied in relation to cognitive impairments, but how they interact with each other in relation to cognition is still unclear. This study aimed at exploring the interaction of loneliness and various types of social networks in relation to cognition in older adults.
a cross-sectional study.
face-to-face interview.
497 older adults with normal global cognition were interviewed.
Loneliness was assessed with Chinese 6-item De Jong Gierverg's Loneliness Scale. Confiding network was defined as people who could share inner feelings with, whereas non-confiding network was computed by subtracting the confiding network from the total network size. Cognitive performance was expressed as a global composite z-score of Cantonese version of mini mental state examination (CMMSE), Categorical verbal fluency test (CVFT) and delayed recall. Linear regression was used to test the main effects of loneliness and the size of various networks, and their interaction on cognitive performance with the adjustment of sociodemographic, physical and psychological confounders.
Significant interaction was found between loneliness and non-confiding network on cognitive performance (B = .002, β = .092, t = 2.099, p = .036). Further analysis showed a significant interaction between loneliness and the number of family members in non-confiding network on cognition (B = .021, β = .119, t = 2.775, p = .006).
Results suggested that a non-confiding relationship with family members might put lonely older adults at risk of cognitive impairment. Our study might have implications on designing psychosocial intervention for those who are vulnerable to loneliness as an early prevention of neurocognitive impairments.
Large-eddy simulation of flow over a rotating cylinder: the lift crisis at $Re_{D}=6\times 10^{4}$
W. Cheng, D. I. Pullin, R. Samtaney
Journal: Journal of Fluid Mechanics / Volume 855 / 25 November 2018
Print publication: 25 November 2018
We present wall-resolved large-eddy simulation (LES) of flow with free-stream velocity $\boldsymbol{U}_{\infty }$ over a cylinder of diameter $D$ rotating at constant angular velocity $\unicode[STIX]{x1D6FA}$ , with the focus on the lift crisis, which takes place at relatively high Reynolds number $Re_{D}=U_{\infty }D/\unicode[STIX]{x1D708}$ , where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. Two sets of LES are performed within the ( $Re_{D}$ , $\unicode[STIX]{x1D6FC}$ )-plane with $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FA}D/(2U_{\infty })$ the dimensionless cylinder rotation speed. One set, at $Re_{D}=5000$ , is used as a reference flow and does not exhibit a lift crisis. Our main LES varies $\unicode[STIX]{x1D6FC}$ in $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 2.0$ at fixed $Re_{D}=6\times 10^{4}$ . For $\unicode[STIX]{x1D6FC}$ in the range $\unicode[STIX]{x1D6FC}=0.48{-}0.6$ we find a lift crisis. This range is in agreement with experiment although the LES shows a deeper local minimum in the lift coefficient than the measured value. Diagnostics that include instantaneous surface portraits of the surface skin-friction vector field $\boldsymbol{C}_{\boldsymbol{f}}$ , spanwise-averaged flow-streamline plots, and a statistical analysis of local, near-surface flow reversal show that, on the leeward-bottom cylinder surface, the flow experiences large-scale reorganization as $\unicode[STIX]{x1D6FC}$ increases through the lift crisis. At $\unicode[STIX]{x1D6FC}=0.48$ the primary-flow features comprise a shear layer separating from that side of the cylinder that moves with the free stream and a pattern of oscillatory but largely attached flow zones surrounded by scattered patches of local flow separation/reattachment on the lee and underside of the cylinder surface. Large-scale, unsteady vortex shedding is observed. At $\unicode[STIX]{x1D6FC}=0.6$ the flow has transitioned to a more ordered state where the small-scale separation/reattachment cells concentrate into a relatively narrow zone with largely attached flow elsewhere. This induces a low-pressure region which produces a sudden decrease in lift and hence the lift crisis. Through this process, the boundary layer does not show classical turbulence behaviour. As $\unicode[STIX]{x1D6FC}$ is further increased at constant $Re_{D}$ , the localized separation zone dissipates with corresponding attached flow on most of the cylinder surface. The lift coefficient then resumes its increasing trend. A logarithmic region is found within the boundary layer at $\unicode[STIX]{x1D6FC}=1.0$ .
Aluminum oxide free-standing thin films to enable nitrogen edge soft x-ray scattering
Dan Ye, Sintu Rongpipi, Joshua H. Litofsky, Youngmin Lee, Tyler E. Culp, Sang Ha Yoo, Thomas N. Jackson, Cheng Wang, Esther W. Gomez, Enrique D. Gomez
Journal: MRS Communications / Volume 9 / Issue 1 / March 2019
Resonant soft x-ray scattering (RSoXS) leverages chemical specificity to characterize thin films but is limited near the nitrogen edge. The challenge is that commercially available x-ray transparent substrates are composed of Si3N4 and thereby absorb incident x-rays and generate incoherent fluorescence. To overcome this challenge, we designed and fabricated Al2O3 free-standing films for use as RSoXS windows. Al2O3 films offer higher x-ray transmittance and minimal fluorescence near the nitrogen edge. As an example, Al2O3 windows allow for nitrogen RSoXS of conjugated block copolymer thin films that reveal domain spacings, which are not apparent with commercially available Si3N4 substrates.
Generation of uniform transverse beam distributions for high-energy electron radiography
Q.T. Zhao, S.C. Cao, R. Cheng, Y.C. Du, X.K. Shen, Y.R. Wang, J.H. Xiao, Y. Zong, Y.L. Zhu, Y.W. Zhou, Y.T. Zhao, Z.M. Zhang, W. Gai
Journal: Laser and Particle Beams / Volume 36 / Issue 3 / September 2018
High-energy electron radiography (HEER) has been proposed for time-resolved imaging of materials, high-energy density matter, and for inertial confinement fusion. The areal-density resolution, determined by the image intensity information is critical for these types of diagnostics. Preliminary experimental studies for different materials with the same thickness and the same areal-density target have been imaged and analyzed. Although there are some discrepancies between experimental and theory analysis, the results show that the density distribution can indeed be attained from HEER. The reason for the discrepancies has been investigated and indicates the importance of the uniformity in the transverse distribution beam illuminating the target. Furthermore, the method for generating a uniform transverse distribution beam using octupole magnets was studied and verified by simulations. The simulations also confirm that the octupole field does not affect the angle-position correlation in the center part beam, a critical requirement for the imaging lens. A more practical method for HEER using collimators and octupoles for generating more uniform beams is also described. Detailed experimental results and simulation studies are presented in this paper.
Impacts of long-term fertilization on the soil microbial communities in double-cropped paddy fields
H. M. Tang, Y. L. Xu, X. P. Xiao, C. Li, W. Y. Li, K. K. Cheng, X. C. Pan, G. Sun
Journal: The Journal of Agricultural Science / Volume 156 / Issue 7 / September 2018
The response of soil microbial communities to soil quality changes is a sensitive indicator of soil ecosystem health. The current work investigated soil microbial communities under different fertilization treatments in a 31-year experiment using the phospholipid fatty acid (PLFA) profile method. The experiment consisted of five fertilization treatments: without fertilizer input (CK), chemical fertilizer alone (MF), rice (Oryza sativa L.) straw residue and chemical fertilizer (RF), low manure rate and chemical fertilizer (LOM), and high manure rate and chemical fertilizer (HOM). Soil samples were collected from the plough layer and results indicated that the content of PLFAs were increased in all fertilization treatments compared with the control. The iC15:0 fatty acids increased significantly in MF treatment but decreased in RF, LOM and HOM, while aC15:0 fatty acids increased in these three treatments. Principal component (PC) analysis was conducted to determine factors defining soil microbial community structure using the 21 PLFAs detected in all treatments: the first and second PCs explained 89.8% of the total variance. All unsaturated and cyclopropyl PLFAs except C12:0 and C15:0 were highly weighted on the first PC. The first and second PC also explained 87.1% of the total variance among all fertilization treatments. There was no difference in the first and second PC between RF and HOM treatments. The results indicated that long-term combined application of straw residue or organic manure with chemical fertilizer practices improved soil microbial community structure more than the mineral fertilizer treatment in double-cropped paddy fields in Southern China.
Early intervention with faecal microbiota transplantation: an effective means to improve growth performance and the intestinal development of suckling piglets
C. S. Cheng, H. K. Wei, P. Wang, H. C. Yu, X. M. Zhang, S. W. Jiang, J. Peng
Journal: animal / Volume 13 / Issue 3 / March 2019
Recent studies indicate that early postnatal period is a critical window for gut microbiota manipulation to optimise the immunity and body growth. This study investigated the effects of maternal faecal microbiota orally administered to neonatal piglets after birth on growth performance, selected microbial populations, intestinal permeability and the development of intestinal mucosal immune system. In total, 12 litters of crossbred newborn piglets were selected in this study. Litter size was standardised to 10 piglets. On day 1, 10 piglets in each litter were randomly allotted to the faecal microbiota transplantation (FMT) and control groups. Piglets in the FMT group were orally administrated with 2ml faecal suspension of their nursing sow per day from the age of 1 to 3 days; piglets in the control group were treated with the same dose of a placebo (0.1M potassium phosphate buffer containing 10% glycerol (vol/vol)) inoculant. The experiment lasted 21 days. On days 7, 14 and 21, plasma and faecal samples were collected for the analysis of growth-related hormones and cytokines in plasma and lipocalin-2, secretory immunoglobulin A (sIgA), selected microbiota and short-chain fatty acids (SCFAs) in faeces. Faecal microbiota transplantation increased the average daily gain of piglets during week 3 and the whole experiment period. Compared with the control group, the FMT group had increased concentrations of plasma growth hormone and IGF-1 on days 14 and 21. Faecal microbiota transplantation also reduced the incidence of diarrhoea during weeks 1 and 3 and plasma concentrations of zonulin, endotoxin and diamine oxidase activities in piglets on days 7 and 14. The populations of Lactobacillus spp. and Faecalibacterium prausnitzii and the concentrations of faecal and plasma acetate, butyrate and total SCFAs in FMT group were higher than those in the control group on day 21. Moreover, the FMT piglets have higher concentrations of plasma transforming growth factor-β and immunoglobulin G, and faecal sIgA than the control piglets on day 21. These findings indicate that early intervention with maternal faecal microbiota improves growth performance, decreases intestinal permeability, stimulates sIgA secretion, and modulates gut microbiota composition and metabolism in suckling piglets.
Effects of probiotic supplementation on performance traits, bone mineralization, cecal microbial composition, cytokines and corticosterone in laying hens
F. F. Yan, G. R. Murugesan, H. W. Cheng
Journal: animal / Volume 13 / Issue 1 / January 2019
Published online by Cambridge University Press: 22 May 2018, pp. 33-41
Recent researches have showed that probiotics promote bone health in humans and rodents. The objective of this study was to determine if probiotics have the similar effects in laying hens. Ninety-six 60-week-old White Leghorn hens were assigned to four-hen cages based on their BW. The cages were randomly assigned to 1 of 4 treatments: a layer diet mixed with a commercial probiotic product (containing Enterococcus faecium, Pediococcus acidilactici, Bifidobacterium animalis and Lactobacillus reuteri) at 0, 0.5, 1.0 or 2.0 g/kg feed (Control, 0.5×, 1.0× and 2.0×) for 7 weeks. Cecal Bifidobacterium spp. counts were higher in all probiotic groups (P<0.001) compared with the control group. The percentage of unmarketable eggs (cracked and shell-less eggs) was decreased in both 0.5× and 2.0× groups compared with the control group (P=0.02), mainly due to the reduction of shell-less eggs (P=0.05). The increases in tibial and femoral mineral density and femoral mineral content (P=0.04, 0.03 and 0.02, respectively), with a concomitant trend of increases in humerus mineral density and tibial mineral content (P=0.07 and 0.08, respectively), occurred in the 2.0× group. However, the bone remodeling indicators of circulating osteocalcin and c-terminal telopeptide of type I collagen were similar among all groups (P>0.05). In addition, the plasma concentrations of cytokines (interleukin-1β, interleukin-6, interleukin-10, interferon-γ and tumor necrosis factor-α) and corticosterone as well as the levels of heterophil to lymphocyte ratio were similar between the 2.0× group and the control group (P>0.05). In line with these findings, no differences of cecal tonsil mRNA expressions of interleukin-1β, interleukin-6 and lipopolysaccharide-induced tumor necrosis factor-α factor were detected between these two groups (P>0.05). These results suggest that immune cytokines and corticosterone may not involve in the probiotic-induced improvement of eggshell quality and bone mineralization in laying hens. In conclusion, the dietary probiotic supplementation altered cecal microbiota composition, resulting in reduced shell-less egg production and improved bone mineralization in laying hens; and the dietary dose of the probiotic up to 2.0× did not cause negative stress reactions in laying hens.
Imported cases and minimum temperature drive dengue transmission in Guangzhou, China: evidence from ARIMAX model
Q. L. Jing, Q. Cheng, J. M. Marshall, W. B. Hu, Z. C. Yang, J. H. Lu
Journal: Epidemiology & Infection / Volume 146 / Issue 10 / July 2018
Dengue is the fastest spreading mosquito-transmitted disease in the world. In China, Guangzhou City is believed to be the most important epicenter of dengue outbreaks although the transmission patterns are still poorly understood. We developed an autoregressive integrated moving average model incorporating external regressors to examine the association between the monthly number of locally acquired dengue infections and imported cases, mosquito densities, temperature and precipitation in Guangzhou. In multivariate analysis, imported cases and minimum temperature (both at lag 0) were both associated with the number of locally acquired infections (P < 0.05). This multivariate model performed best, featuring the lowest fitting root mean squared error (RMSE) (0.7520), AIC (393.7854) and test RMSE (0.6445), as well as the best effect in model validation for testing outbreak with a sensitivity of 1.0000, a specificity of 0.7368 and a consistency rate of 0.7917. Our findings suggest that imported cases and minimum temperature are two key determinants of dengue local transmission in Guangzhou. The modelling method can be used to predict dengue transmission in non-endemic countries and to inform dengue prevention and control strategies.
LO50: Necrotizing soft tissue infection: diagnostic accuracy of physical examination, imaging and LRINEC score a systematic review and meta-analysis
S. M. Fernando, A. Tran, W. Cheng, M. Taljaard, B. Rochwerg, K. Kyeremanteng, A. J.E. Seely, K. Inaba, J. J. Perry
Introduction: Necrotizing soft tissue infection (NSTI), a potentially life-threatening diagnosis, is often not immediately recognized by clinicians. Delays in diagnosis are associated with increased morbidity and mortality. We sought to summarize and compare the accuracy of physical exam, imaging, and Laboratory Risk Indicator of Necrotizing Fasciitis (LRINEC) Score used to confirm suspected NSTI in adult patients with skin and soft tissue infections. Methods: We searched Medline, Embase and 4 other databases from inception through November 2017. We included only English studies (randomized controlled trials, cohort and case-control studies) that reported the diagnostic accuracy of testing or LRINEC Score. Outcome was NSTI confirmed by surgery or histopathology. Two reviewers independently screened studies and extracted data. We assessed risk of bias using the Quality Assessment of Diagnostic Accuracy Studies 2 criteria. Diagnostic accuracy summary estimates were obtained from the Hierarchical Summary Receiver Operating Characteristic model. Results: We included 21 studies (n=6,044) in the meta-analysis. Of physical exam signs, pooled sensitivity and specificity for fever (49.4% [95% CI: 41.4-57.5], 78.0% [95% CI: 52.2-92.0]), hemorrhagic bullae (30.8% [95% CI: 16.2-50.6], 94.2% [95% CI: 82.9-98.2]) and hypotension (20.8% [95% CI: 7.7-45.2], 97.9% [95% CI: 89.1-99.6]) were generated. Computed tomography (CT) had 88.5% [95% CI: 55.5-97.9] sensitivity and 93.3% [95% CI: 80.8-97.9] specificity, while plain radiography had 48.9% [95% CI: 24.9-73.4] sensitivity and 94.0% [95% CI: 63.8-99.3] specificity. Finally, LRINEC 6 (traditional threshold) had 67.5% [95% CI: 48.3-82.3] sensitivity and 86.7% [95% CI: 77.6-92.5] specificity, while a LRINEC 8 had 94.9% [95% CI: 89.4-97.6] specificity but 40.8% [95% CI: 28.6-54.2] sensitivity. Conclusion: The absence of any one physical exam feature (e.g. fever or hypotension) is not sufficient to rule-out NSTI. CT is superior to plain radiography. The LRINEC Score had poor sensitivity, suggesting that a low score is not sufficient to rule-out NSTI. For patients with suspected NSTI, further evaluation is warranted. While no single test is sensitive, patients with high-risk features should receive early surgical consultation for definitive diagnosis and management.
Seed treatment with glycine betaine enhances tolerance of cotton to chilling stress
C. Cheng, L. M. Pei, T. T. Yin, K. W. Zhang
Journal: The Journal of Agricultural Science / Volume 156 / Issue 3 / April 2018
Chilling injury is an important natural stress that can threaten cotton production, especially at the sowing and seedling stages in early spring. It is therefore important for cotton production to improve chilling tolerance at these stages. The current work examines the potential for glycine betaine (GB) treatment of seeds to increase the chilling tolerance of cotton at the seedling stage. Germination under cold stress was increased significantly by GB treatment. Under low temperature, the leaves of seedlings from treated seeds exhibited a higher net photosynthetic rate (PN), higher antioxidant enzyme activity including superoxide dismutase, ascorbate peroxidase and catalase, lower hydrogen peroxide (H2O2) content and less damage to the cell membrane. Enzyme activity was correlated negatively with H2O2 content and degree of damage to the cell membrane but correlated positively with GB content. The experimental results suggested that although GB was only used to treat cotton seed, the beneficial effect caused by the preliminary treatment of GB could play a significant role during germination that persisted to at least the four-leaf seedling stage. Therefore, it is crucial that this method is employed in agricultural production to improve chilling resistance in the seedling stage by soaking the seeds in GB. | CommonCrawl |
Malondialdehyde and superoxide dismutase levels in patients with epilepsy: a case–control study
Nahed Shehta1,
Amr Elsayed Kamel1,
Eman Sobhy ORCID: orcid.org/0000-0001-8844-87711 &
Mohamed Hamdy Ismail1
Oxidative stress has a significant influence in the initiation and progression of epileptic seizures. It was reported that inhibiting oxidative stress could protect against epilepsy. The aim of the current research is to estimate some biomarkers that reflect the oxidative stress in epileptics, its relation to seizure control as well as to study the impact of antiepileptic drugs (AEDs) on these biomarkers. This case–control study included 62 epileptic patients beside 62 age and gender-matched healthy controls. The epileptic patients subjected to detailed history taking with special regards to disease duration, seizure frequency, and the current AEDs. Laboratory evaluation of serum malondialdehyde (a lipid peroxidation byproduct) and superoxide dismutase (an endogenous antioxidant) were done.
Malondialdehyde (MDA) was significantly higher, and superoxide dismutase (SOD) was lower in epileptic patients than in the controls (p < 0.001). Seizure frequency was directly correlated with MDA (r = 0.584, p < 0.001) while inversely correlated with SOD (r = − 0.432, p = 0.008). High MDA and low SOD were recorded in epileptic patients receiving polytherapy as compared to monotherapy (p < 0.001).
Epileptic patients had higher oxidative stress biomarkers than healthy individuals. Frequent seizures, long disease duration, and AEDs were associated with higher MDA and lower SOD that reflects an imbalance in the oxidant–antioxidant status among these patients.
Epilepsy is a neurological illness affecting over 70 million people worldwide. It is characterized by a continuous tendency for spontaneous (unprovoked) seizures that carries multiple behavioral, cognitive, and psychosocial consequences [1, 2]. The previous experimental and clinical studies had demonstrated the influence of oxidative stress on epilepsy pathogenesis. It was reported that oxidative stress can impact seizure initiation and recurrence [3,4,5].
Oxidative stress refers to an imbalance between generation and degradation of reactive oxygen and nitrogen species [6]. The reactive oxygen species that are generated during cellular metabolism can be neutralized by either endogenous antioxidant enzymes, for example, superoxide dismutase (SOD) [7, 8] or nonenzymatic pathway through molecules with scavenging properties, for example, vitamin E, melatonin, and glutathione [9].
Superoxide dismutase, an intracellular antioxidant enzyme, belongs to metalloenzymes. It stimulates the conversion of superoxide radical to hydrogen peroxide [4]. The SOD level is considered a biomarker that reflects the antioxidant status in various studies [10,11,12]
Lipid peroxidation process denotes to the damage of polyunsaturated fatty acids that caused by oxidative stress and leads to irreversible damage of cell membrane [13,14,15] and changing membrane permeability leading to hyperexcitability [16].
Malondialdehyde is considered as an important byproduct of lipid peroxidation which is formed by oxidation of polyunsaturated lipids [17]. Malondialdehyde was vastly employed as a marker of oxidative stress among many studies [12, 18, 19].
This study aimed to assess SOD and MDA levels among epileptic patients and compare them with controls. In addition, studying the impact of seizure frequency, disease duration, and AEDs on these biomarkers.
Study design and patients
A case–control study that included a total of one hundred and twenty-four subjects; 62 epileptic patients beside 62 age and gender matched healthy individuals as controls. The epileptic group consisted of patients aged 18 to 45 years old. These patients were selected from the outpatient's epilepsy clinic, Neurology Department, Zagazig University from April to December 2020. Epilepsy was diagnosed according to the International League Against Epilepsy (ILAE) 2017 classification system [20, 21].
Inclusion criteria patients aged 18 to 45 years old, on regular antiepileptic drugs either monotherapy or polytherapy. Exclusion criteria: smoking, pregnancy, breastfeeding, psychiatric comorbidity, acute, or chronic medical illnesses, malignancies, and metabolic disorders. Patients who take medications other than AEDs were also excluded from the study.
The research protocol was approved by the Ethics Committee of our institution (ZU-IRB # 6007/9-3-2020). Written consent was taken out from all included subjects.
Clinical and laboratory assessment
All patients were subjected to thorough examination and history taking including disease duration, seizure frequency, seizure control (either controlled or uncontrolled according to the response to AED treatment), and the current antiepileptic medications. Patients who were seizure free during the last year prior to the study were considered controlled while the uncontrolled patients were considered when adequate trials of two tolerated, properly selected antiepileptic drugs (whether used separately or in combination) with proper doses failed to control patient's seizure [22]. Regarding the AEDs treatment, patients who were on single AED were categorized as monotherapy and those on more than one drug were considered polytherapy recipients.
Electroencephalography (EEG) using EB Neuro machine (Italy), according to 10- 20 system of electrode placement and magnetic resonance imaging (MRI) of brain by 1.5 Tesla MR imager (Achieva, Philips Medical System) were done at the time of recruitment to all patients.
Blood sample collection
Venous blood samples were obtained from participants by venipuncture from the antecubital vein using a disposal plastic syringe and collected without using an anticoagulant. After clotting, the blood was centrifuged at 4000 rpm for 15 min. Serum was separated from the blood and stored at – 20 °C until chemical analysis.
Measurement of serum MDA and SOD levels were done at Medical Biochemistry and Molecular Biology Department by calorimetric method according to Ohkawa et al. [23] for MDA (nmol/ml), and to Nishikimi et al. [24] for SOD (U/ml).
Principle of MDA The methododology is based on the reaction of MDA with thiobarbituric acid in acidic medium at temperature of 95 °C for 30 min to form thiobarbituric acid reactive product. The absorbance of the resulting product can be measured spectrophotometrically at 534 nm [23].
Estimation of malondialdhyde
$${\text{Serum Malondialdehyde }} = \left( {{\text{A Sample}} \div {\text{ A Standard}}} \right) \, \times \, 10 {\text{nmol}}/{\text{ml}}.$$
Principle of SOD This test depends on the capability of SOD to inhibit the phenazine methosulphate-mediated reduction of nitro blue tetrazolium dye. The change in the absorbance over 5 min was measured at 560 nm for control (Δcontrol) and for sample (Δsample) at 25 °C. 1.5 U/assay of the purified enzyme produced 80% inhibition [24]
Calculation of SOD
$${\text{Percent inhibition}} = \left[ {\left( {\Delta {\text{ control}}{-} \, \Delta {\text{ sample}}} \right) \div \Delta {\text{ control}}} \right] \times 100$$
$${\text{SOD Activity }}\left( {\text{U/ml}} \right) = \% {\text{ inhibition}} \times 3.75$$
Data analysis was done using IBM SPSS software package version 20.0. (Armonk, NY: IBM Corp). Qualitative data were expressed as number and percentage, Quantitative data were expressed as mean and standard deviation. Chi-square, Student's t test, and Pearson correlation coefficient were used when appropriate. For multiple group comparisons, we used analysis of variance (ANOVA) followed by Tukey's Post Hoc test. p value was set at ≤ 0.05 for significant results.
Sixty-two epileptic patients (36 males and 26 females) and 62 controls (35 males and 27 females) were recruited. The mean age (± SD) was 27.98 ± 6.44 for epileptic patients and 26.63 ± 6.06 years for controls. In the epileptic group, 21 patients with focal and 41 patients with generalized seizures. The mean age of disease onset was 17.85 ± 4.64 years, the mean disease duration was 13.11 ± 8.19 years. 41.9% of patients were well-controlled by AEDs while 58.1% were uncontrolled. Regarding the AEDs, there were 27 (43.5%) patients on monotherapy and 35 patients (56.5%) on polytherapy (Table 1).
Table 1 Demographic, clinical and laboratory data of the studied subjects
Malondialdehyde was significantly higher (p < 0.001) among the epileptic group (5.83 ± 1.46 nmol/ml) than controls (4.80 ± 0.62 nmol/ml). Superoxide dismutase (SOD) was significantly lower (7.59 ± 0.85 U/ml) among epileptics than controls (11.98 ± 1.39 U/ml) (p < 0.001) (Table 1).
There was a direct correlation between seizure frequency and MDA (p < 0.001), and an inverse correlation between seizure frequency and SOD (p = 0.008) (Fig. 1). Regarding the relation of disease duration with laboratory parameters, low SOD and high MDA concentrations were observed among patients with disease duration > 5 years than those ≤ 5 years. No significant difference in levels of MDA or SOD was observed between patients with focal and generalized seizures (Table 2).
Correlation between Seizure frequency and malondialdehyde (MDA) and superoxide dismutase (SOD) levels in the epileptic group
Table 2 Differences in laboratory findings as regard disease duration and seizure type
Regarding AEDs, lower SOD and higher MDA were demonstrated in patients receiving polytherapy than those on monotherapy and healthy controls (Table 3). Nonsignificant difference was detected between individual AEDs as regard MDA or SOD levels (Table 4).
Table 3 SOD and MDA comparison between healthy controls, monotherapy and poly-therapy groups
Table 4 Comparison between individual AEDs as monotherapy regarding laboratory findings
Oxidative stress attracts great interest in the pathogenesis of epilepsy [25]. It could aid in recognizing individuals at risk of developing epilepsy. Hence, it might facilitate the clinical trials concerned with antiepileptogenesis [26].
In the current study, the oxidant status was studied through assessment of MDA as a byproduct of lipid peroxidation, while the antioxidant status was assessed by measuring SOD as an endogenous antioxidant. There was a significant increase in mean values of MDA in epileptic group than controls. This is in consistence with many previous human [11, 12, 18, 19, 27,28,29,30] and experimental studies [25, 31]. A significant (p < 0.001) reduction in mean values of SOD was demonstrated among epileptic group than controls as observed in the previous studies [10,11,12, 29, 32] in which the SOD level among epileptic patients was significantly less than controls.
In contrary, other studies had demonstrated no significant alteration in SOD values between the epileptics and controls [30, 33,34,35]. Moreover, Ercegovac et al. [36] observed a significant elevation of SOD among patients with first seizure. They explained this rise in SOD level as an adaptive mechanism to face the increased free radical generation during seizure.
From the above data, we observed that epileptic patients had an imbalance in the oxidant–antioxidant status. It could be explained as recurrent epileptic seizures can cause oxidative stress and free radicals formation leading to macromolecular damage, neuroinflammation and neurotoxicity [37]. The seizure, as a brain insult, produces free radicals that disturb the mitochondrial function and energy metabolism and lead to enhancement of lipid peroxidation, gliosis and abnormal rearrangements of neural circuits that promote the formation of hyperexcitable networks [38, 39].
On studying the relation between biomarkers of oxidative stress and seizure profile, we found a direct correlation between seizure frequency and MDA. However, an indirect correlation was observed between SOD and seizure frequency. Similarly, Maes et al. [40] found that highly frequent seizures were associated with high levels of oxidative stress markers as MDA.
On stratifying our epileptic patients based on disease duration (> 5 years and ≤ 5 years duration), there was significantly high values of MDA and lower values of SOD in patients with disease duration > 5 years. In contrast, Turkdogan et al. [41] found no relation between antioxidant enzymes or MDA levels and disease duration.
Regarding seizure type of our patients, no significant difference in MDA or SOD values could be observed between patients with focal and generalized seizures. Similarly, Yis et al. [42] found no difference in oxidant and antioxidant biomarkers between patients with generalized and focal epilepsy. This denotes that epilepsy type does not disturb oxidant/antioxidant status in different ways.
In this study, we compared patients on single drug (monotherapy) with those on multiple drugs (polytherapy) to study the effect of antiepileptic drugs, if any, on MDA and SOD. we observed higher MDA and lower SOD levels in polytherapy than monotherapy patients and healthy controls. This finding suggests that currently used antiepileptic drugs did not improve the antioxidant status in those patients and an additional oxidative stress could be induced by AEDs.
Similarly, Iwuozo et al. [43] demonstrated that patients on AED polytherapy had significantly higher MDA and lower SOD levels than AED naïve patients as well as patients on monotherapy. Also, Ethemoglu et al. [37] found higher oxidant and lower antioxidant biomarkers in polytherapy in comparison to monotherapy groups. They believed that the patients on monotherapy consisted of patients with controlled seizures, while patients receiving polytherapy had higher seizure frequency and being uncontrolled. While Menon et al. [18, 44] and Guler et al. [11] recorded no significant alterations in the biomarkers of oxidant–antioxidant status between patients receiving monotherapy and polytherapy.
The oxidative stress induced by AEDs was explained as many conventional AEDs are metabolized to active metabolites able to combine with vital molecules such as lipids and proteins and resulting in impairment of cellular function and structure rather than having a neuroprotective effect [45].
Accumulating evidence suggests that new generations of AEDs are superior on the conventional AEDs in terms of neuroprotection and antioxidant effects by scavenging-free radicals [3, 46]. To address this point, 27 epileptic patients on monotherapy were tested in the current study for MDA and SOD. We found no significant changes between individual AED groups as regard MDA and SOD concentrations. This could be attributed to small number of patients in each individual AED group. Therefore, the superiority of an individual AED on antioxidant status could not be inferred from the results of this study and future studies with large sample size recruiting patients receiving different AEDs including the newer and old AEDs might reveal this issue.
In this study, we observed that epileptic patients had an imbalance in the oxidan–antioxidant status as we found higher MDA and lower SOD levels in patients than healthy individuals. Types of epilepsy did not affect oxidative status and antioxidant enzyme activities. Poor seizure control impaired the oxidant–antioxidant regulatory system. AEDs did not improve the antioxidant status in epileptic patients and an additional oxidative stress could be induced by AEDs. Future research should focus on novel drug treatments that can modify the development and progression of epilepsy through having antioxidant effect.
The data results generated or analyzed during this study are included in this published article.
AEDs:
Antiepileptic drugs
SOD:
MDA:
Malondialdehyde
ILAE:
International league against epilepsy
EEG:
CBZ:
VPA:
Valproate
LMG:
LEV:
Levitracetam
OXC:
Oxcarbazepine
Fisher RS, Acevedo C, Arzimanoglou A, Bogacz A, Cross JH, Elger CE, et al. ILAE official report: a practical clinical definition of epilepsy. Epilepsia. 2014;55(4):475–82.
Thijs RD, Surges R, O'Brien TJ, Sander JW. Epilepsy in adults. The Lancet. 2019;393(10172):689–701.
Shin EJ, Jeong JH, Chung YH, Kim WK, Ko KH, Bach JH, et al. Role of oxidative stress in epileptic seizures. Neurochem Int. 2011;59(2):122–37.
Aguiar CC, Almeida AB, Araújo PV, Abreu RN, Chaves EM, Vale OC, et al. Oxidative stress and epilepsy: literature review. Oxid Med Cell Longev. 2012;2012: 795259. https://doi.org/10.1155/2012/795259.
Vezzani A, Aronica E, Mazarati A, Pittman QJ. Epilepsy and brain inflammation. Exp Neurol. 2013;1(244):11–21.
Fujii H, Nakai K, Fukagawa M. Role of oxidative stress and indoxyl sulfate in progression of cardiovascular disease in chronic kidney disease. Ther Apher Dial. 2011;15(2):125–8.
Small DM, Coombes JS, Bennett N, Johnson DW, Gobe GC. Oxidative stress, anti-oxidant therapies and chronic kidney disease. Nephrology. 2012;17(4):311–21.
Yasui K, Baba A. Therapeutic potential of superoxide dismutase (SOD) for resolution of inflammation. Inflamm Res. 2006;55(9):359–63.
Mirończuk-Chodakowska I, Witkowska AM, Zujko ME. Endogenous non-enzymatic antioxidants in the human body. Adv Med Sci. 2018;63(1):68–78.
Al-Muhammadi MO, Al-Tameemi KM, Kadhim HM. Role of superoxide dismutase enzyme in patients with epilepsy. Med J Babylon. 2015;12(4):1015–9.
Guler SK, Aytac B, Durak ZE, Cokal BG, Gunes N, Durak I, et al. Antioxidative–oxidative balance in epilepsy patients on antiepileptic therapy: a prospective case–control study. Neurol Sci. 2016;37(5):763–7.
Prasad DK, Satyanarayana U, Shaheen U, Prabha TS, Munshi A. Oxidative stress in the development of genetic generalised epilepsy: an observational study in southern Indian population. J Clin Diagn Res JCDR. 2017;11(9):BC05.
Yoshida Y, Umeno A, Shichiri M. Lipid peroxidation biomarkers for evaluating oxidative stress and assessing antioxidant capacity in vivo. J Clin Biochem Nutr. 2013;52(1):9–16.
Ayala A, Muñoz MF, Argüelles S. Lipid peroxidation: production, metabolism, and signaling mechanisms of malondialdehyde and 4-hydroxy-2-nonenal. Oxid Med Cell Longev. 2014;2014: 360438. https://doi.org/10.1155/2014/360438.
Nigar S, Pottoo FH, Tabassum N, Verma SK, Javed MN. Molecular insights into the role of inflammation and oxidative stress in epilepsy. J Adv Med Pharm Sci. 2016;10(1):1–9. https://doi.org/10.9734/JAMPS/2016/24441.
Wong-Ekkabut J, Xu Z, Triampo W, Tang IM, Tieleman DP, Monticelli L. Effect of lipid peroxidation on the properties of lipid bilayers: a molecular dynamics study. Biophys J. 2007;93(12):4225–36.
Sapkota M, Wyatt TA. Alcohol, aldehydes, adducts and airways. Biomolecules. 2015;5(4):2987–3008.
Menon B, Ramalingam K, Kumar RV. Oxidative stress in patients with epilepsy is independent of antiepileptic drugs. Seizure. 2012;21(10):780–4.
Das A, Sarwar MS, Hossain MS, Karmakar P, Islam MS, Hussain ME, et al. Elevated serum lipid peroxidation and reduced vitamin C and trace element concentrations are correlated with Epilepsy. Clin EEG Neurosci. 2019;50(1):63–72.
Fisher RS, Cross JH, French JA, Higurashi N, Hirsch E, Jansen FE, et al. Operational classification of seizure types by the international league against epilepsy: position paper of the ILAE commission for classification and terminology. Epilepsia. 2017;58(4):522–30.
Scheffer IE, Berkovic S, Capovilla G, Connolly MB, French J, Guilhoto L, et al. ILAE classification of the epilepsies: position paper of the ILAE Commission for Classification and Terminology. Epilepsia. 2017;58(4):512–21.
Kwan P, Arzimanoglou A, Berg AT, Brodie MJ, Allen Hauser W, Mathern G, et al. Definition of drug resistant epilepsy: consensus proposal by the ad hoc Task Force of the ILAE Commission on Therapeutic Strategies [published correction appears in Epilepsia. 2010 Sep; 51(9):1922]. Epilepsia. 2010;51(6):1069–77. https://doi.org/10.1111/j.1528-1167.2009.02397.x.
Ohkawa H, Ohishi N, Yagi K. Assay for lipid peroxides in animal tissues by thiobarbituric acid reaction. Anal Biochem. 1979;95(2):351–8.
Nishikimi M, Rao NA, Yagi K. The occurrence of superoxide anion in the reaction of reduced phenazine methosulfate and molecular oxygen. Biochem Biophys Res Commun. 1972;46(2):849–54.
Pauletti A, Terrone G, Shekh-Ahmad T, Salamone A, Ravizza T, Rizzi M, et al. Targeting oxidative stress improves disease outcomes in a rat model of acquired epilepsy. Brain. 2019;142(7): e39. https://doi.org/10.1093/brain/awz130.
Simonato M, Agoston DV, Brooks-Kayal A, Dulla C, Fureman B, Henshall DC, et al. Identification of clinically relevant biomarkers of epileptogenesis—a strategic roadmap. Nat Rev Neurol. 2021;17(4):231–42.
Hamed SA, Abdellah MM, El-Melegy N. Blood levels of trace elements, electrolytes, and oxidative stress/antioxidant systems in epileptic patients. J Pharmacol Sci. 2004;96(4):465–73.
Hamed SA, Hamed EA, Hamdy R, Nabeshima T. Vascular risk factors and oxidative stress as independent predictors of asymptomatic atherosclerosis in adult patients with epilepsy. Epilepsy Res. 2007;74(2–3):183–92.
Nemade ST, Melinkeri RR. Oxidative and antioxidative status in epilepsy. Pravara Med Rev. 2010;2(4):8–10.
Saad K, Hammad E, Hassan AF, Badry R. Trace element, oxidant, and antioxidant enzyme values in blood of children with refractory epilepsy. Int J Neurosci. 2014;124(3):181–6.
Dal-Pizzol F, Klamt F, Vianna MM, Schröder N, Quevedo J, Benfato MS, et al. Lipid peroxidation in hippocampus early and late after status epilepticus induced by pilocarpine or kainic acid in Wistar rats. Neurosci Lett. 2000;291(3):179–82.
Ben-Menachem E, Kyllerman M, Marklund S. Superoxide dismutase and glutathione peroxidase function in progressive myoclonus epilepsies. Epilepsy Res. 2000;40(1):33–9.
Sudha K, Rao AV, Rao A. Oxidative stress and antioxidants in epilepsy. Clin Chim Acta. 2001;303(1–2):19–24.
Verrotti A, Basciani F, Trotta D, Pomilio MP, Morgese G, Chiarelli F. Serum copper, zinc, selenium, glutathione peroxidase and superoxide dismutase levels in epileptic children before and after 1 year of sodium valproate and carbamazepine therapy. Epilepsy Res. 2002;48(1–2):71–5.
Peker E, Oktar S, Arı M, Kozan R, Doğan M, Çağan E, et al. Nitric oxide, lipid peroxidation, and antioxidant enzyme levels in epileptic children using valproic acid. Brain Res. 2009;22(1297):194–7.
Ercegovac M, Jovic N, Simic T, Beslac-Bumbasirevic L, Sokic D, Djukic T, et al. Byproducts of protein, lipid and DNA oxidative damage and antioxidant enzyme activities in seizure. Seizure. 2010;19(4):205–10.
Ethemoglu O, Ay H, Koyuncu I, Gönel A. Comparison of cytokines and prooxidants/antioxidants markers among adults with refractory versus well-controlled epilepsy: a cross-sectional study. Seizure. 2018;1(60):105–9.
Cardenas-Rodriguez N, Huerta-Gertrudis B, Rivera-Espinosa L, Montesinos-Correa H, Bandala C, Carmona-Aparicio L, et al. Role of oxidative stress in refractory epilepsy: evidence in patients and experimental models. Int J Mol Sci. 2013;14(1):1455–76.
Puttachary S, Sharma S, Stark S, Thippeswamy T. Seizure-induced oxidative stress in temporal lobe epilepsy. Biomed Res Int. 2015;20(2015): 745613. https://doi.org/10.1155/2015/745613.
Maes M, Supasitthumrong T, Limotai C, Michelin AP, Matsumoto AK, de Oliveira SL, et al. Increased oxidative stress toxicity and lowered antioxidant defenses in temporal lobe epilepsy and mesial temporal sclerosis: associations with psychiatric comorbidities. Mol Neurobiol. 2020;57:3334–48.
Turkdogan D, Toplan S, Karakoc Y. Lipid peroxidation and antioxidative enzyme activities in childhood epilepsy. J Child Neurol. 2002;17(9):673–6.
Yiş U, Seçkin E, Kurul SH, Kuralay F, Dirik E. Effects of epilepsy and valproic acid on oxidant status in children with idiopathic epilepsy. Epilepsy Res. 2009;84(2–3):232–7.
Iwuozo EU, Obiako OR, Ejiofor JI, Kehinde JA, Abubakar SA. Effect of epilepsy and antiepileptic drugs therapy on erythrocyte malondialdehyde and some antioxidants in persons with epilepsy. West Afr J Med. 2019;36(3):211–6.
Menon B, Ramalingam K, Kumar RV. Low plasma antioxidant status in patients with epilepsy and the role of antiepileptic drugs on oxidative stress. Ann Indian Acad Neurol. 2014;17(4):398.
Grewal GK, Kukal S, Kanojia N, Saso L, Kukreti S, Kukreti R. Effect of oxidative stress on ABC transporters: contribution to epilepsy pharmacoresistance. Molecules. 2017;22(3):365.
Martinc B, Grabnar I, Vovk T. The role of reactive species in epileptogenesis and influence of antiepileptic drug therapy on oxidative stress. Curr Neuropharmacol. 2012;10(4):328–43.
We would like to acknowledge the excellent technical assistance of staff members of Central Research Lab, Medical Biochemistry and Molecular Biology Department, Faculty of Medicine, Zagazig University.
This study received no funding.
Nahed Shehta, Amr Elsayed Kamel, Eman Sobhy & Mohamed Hamdy Ismail
Nahed Shehta
Amr Elsayed Kamel
Eman Sobhy
Mohamed Hamdy Ismail
All authors were involved in crafting the study topic and design. All authors read and approved the final manuscript. NS analyzed and interpreted the data, wrote, and prepared the final manuscript. AEK supervised clinical/laboratory work, interpreted results, and participated in manuscript drafting. ES recruited the patients, carried out clinical/laboratory investigation, collected data and submitted manuscript. MHE supervised clinical/laboratory work, and participated in manuscript drafting. All authors read and approved the final manuscript.
Correspondence to Eman Sobhy.
The study protocol was approved by the ethics committee of the faculty of Medicine, Zagazig University. The reference number is (ZU-IRB # 6007/9-3-2020). The purpose of the study was explained, and an informed written consent was taken before taking any data or doing any investigations. The participants were informed that their participation was voluntary and that they could withdraw from the study at any time without consequences.
Shehta, N., Kamel, A.E., Sobhy, E. et al. Malondialdehyde and superoxide dismutase levels in patients with epilepsy: a case–control study. Egypt J Neurol Psychiatry Neurosurg 58, 51 (2022). https://doi.org/10.1186/s41983-022-00479-5 | CommonCrawl |
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.
In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables.
Statement
Lebesgue's dominated convergence theorem.[1] Let $(f_{n})$ be a sequence of complex-valued measurable functions on a measure space $(S,\Sigma ,\mu )$. Suppose that the sequence converges pointwise to a function $f$ and is dominated by some integrable function $g$ in the sense that
$|f_{n}(x)|\leq g(x)$
for all numbers n in the index set of the sequence and all points $x\in S$. Then f is integrable (in the Lebesgue sense) and
$\lim _{n\to \infty }\int _{S}|f_{n}-f|\,d\mu =0$
which also implies
$\lim _{n\to \infty }\int _{S}f_{n}\,d\mu =\int _{S}f\,d\mu $
Remark 1. The statement "g is integrable" means that measurable function $g$ is Lebesgue integrable; i.e.
$\int _{S}|g|\,d\mu <\infty .$
Remark 2. The convergence of the sequence and domination by $g$ can be relaxed to hold only μ-almost everywhere provided the measure space (S, Σ, μ) is complete or $f$ is chosen as a measurable function which agrees μ-almost everywhere with the μ-almost everywhere existing pointwise limit. (These precautions are necessary, because otherwise there might exist a non-measurable subset of a μ-null set N ∈ Σ, hence $f$ might not be measurable.)
Remark 3. If $\mu (S)<\infty $, the condition that there is a dominating integrable function $g$ can be relaxed to uniform integrability of the sequence (fn), see Vitali convergence theorem.
Remark 4. While $f$ is Lebesgue integrable, it is not in general Riemann integrable. For example, take fn to be defined in $[0,1]$ so that it is one at rational numbers and zero everywhere else (on the irrationals). The series (fn) converges pointwise to 0, so f is identically zero, but $|f_{n}-f|=f_{n}$ is not Riemann integrable, since its image in every finite interval is $\{0,1\}$ and thus the upper and lower Darboux integrals are 1 and 0, respectively.
Proof
Without loss of generality, one can assume that f is real, because one can split f into its real and imaginary parts (remember that a sequence of complex numbers converges if and only if both its real and imaginary counterparts converge) and apply the triangle inequality at the end.
Lebesgue's dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below, however, is a direct proof that uses Fatou’s lemma as the essential tool.
Since f is the pointwise limit of the sequence (fn) of measurable functions that are dominated by g, it is also measurable and dominated by g, hence it is integrable. Furthermore, (these will be needed later),
$|f-f_{n}|\leq |f|+|f_{n}|\leq 2g$
for all n and
$\limsup _{n\to \infty }|f-f_{n}|=0.$
The second of these is trivially true (by the very definition of f). Using linearity and monotonicity of the Lebesgue integral,
$\left|\int _{S}{f\,d\mu }-\int _{S}{f_{n}\,d\mu }\right|=\left|\int _{S}{(f-f_{n})\,d\mu }\right|\leq \int _{S}{|f-f_{n}|\,d\mu }.$
By the reverse Fatou lemma (it is here that we use the fact that |f−fn| is bounded above by an integrable function)
$\limsup _{n\to \infty }\int _{S}|f-f_{n}|\,d\mu \leq \int _{S}\limsup _{n\to \infty }|f-f_{n}|\,d\mu =0,$
which implies that the limit exists and vanishes i.e.
$\lim _{n\to \infty }\int _{S}|f-f_{n}|\,d\mu =0.$
Finally, since
$\lim _{n\to \infty }\left|\int _{S}fd\mu -\int _{S}f_{n}d\mu \right|\leq \lim _{n\to \infty }\int _{S}|f-f_{n}|\,d\mu =0.$
we have that
$\lim _{n\to \infty }\int _{S}f_{n}\,d\mu =\int _{S}f\,d\mu .$
The theorem now follows.
If the assumptions hold only μ-almost everywhere, then there exists a μ-null set N ∈ Σ such that the functions fn 1S \ N satisfy the assumptions everywhere on S. Then the function f(x) defined as the pointwise limit of fn(x) for x ∈ S \ N and by f(x) = 0 for x ∈ N, is measurable and is the pointwise limit of this modified function sequence. The values of these integrals are not influenced by these changes to the integrands on this μ-null set N, so the theorem continues to hold.
DCT holds even if fn converges to f in measure (finite measure) and the dominating function is non-negative almost everywhere.
Discussion of the assumptions
The assumption that the sequence is dominated by some integrable g cannot be dispensed with. This may be seen as follows: define fn(x) = n for x in the interval (0, 1/n] and fn(x) = 0 otherwise. Any g which dominates the sequence must also dominate the pointwise supremum h = supn fn. Observe that
$\int _{0}^{1}h(x)\,dx\geq \int _{\frac {1}{m}}^{1}{h(x)\,dx}=\sum _{n=1}^{m-1}\int _{\left({\frac {1}{n+1}},{\frac {1}{n}}\right]}{h(x)\,dx}\geq \sum _{n=1}^{m-1}\int _{\left({\frac {1}{n+1}},{\frac {1}{n}}\right]}{n\,dx}=\sum _{n=1}^{m-1}{\frac {1}{n+1}}\to \infty \qquad {\text{as }}m\to \infty $
by the divergence of the harmonic series. Hence, the monotonicity of the Lebesgue integral tells us that there exists no integrable function which dominates the sequence on [0,1]. A direct calculation shows that integration and pointwise limit do not commute for this sequence:
$\int _{0}^{1}\lim _{n\to \infty }f_{n}(x)\,dx=0\neq 1=\lim _{n\to \infty }\int _{0}^{1}f_{n}(x)\,dx,$
because the pointwise limit of the sequence is the zero function. Note that the sequence (fn) is not even uniformly integrable, hence also the Vitali convergence theorem is not applicable.
Bounded convergence theorem
One corollary to the dominated convergence theorem is the bounded convergence theorem, which states that if (fn) is a sequence of uniformly bounded complex-valued measurable functions which converges pointwise on a bounded measure space (S, Σ, μ) (i.e. one in which μ(S) is finite) to a function f, then the limit f is an integrable function and
$\lim _{n\to \infty }\int _{S}{f_{n}\,d\mu }=\int _{S}{f\,d\mu }.$
Remark: The pointwise convergence and uniform boundedness of the sequence can be relaxed to hold only μ-almost everywhere, provided the measure space (S, Σ, μ) is complete or f is chosen as a measurable function which agrees μ-almost everywhere with the μ-almost everywhere existing pointwise limit.
Proof
Since the sequence is uniformly bounded, there is a real number M such that |fn(x)| ≤ M for all x ∈ S and for all n. Define g(x) = M for all x ∈ S. Then the sequence is dominated by g. Furthermore, g is integrable since it is a constant function on a set of finite measure. Therefore, the result follows from the dominated convergence theorem.
If the assumptions hold only μ-almost everywhere, then there exists a μ-null set N ∈ Σ such that the functions fn1S\N satisfy the assumptions everywhere on S.
Dominated convergence in Lp-spaces (corollary)
Let $(\Omega ,{\mathcal {A}},\mu )$ be a measure space, $1\leq p<\infty $ a real number and $(f_{n})$ a sequence of ${\mathcal {A}}$-measurable functions $f_{n}:\Omega \to \mathbb {C} \cup \{\infty \}$.
Assume the sequence $(f_{n})$ converges $\mu $-almost everywhere to an ${\mathcal {A}}$-measurable function $f$, and is dominated by a $g\in L^{p}$ (cf. Lp space), i.e., for every natural number $n$ we have: $|f_{n}|\leq g$, μ-almost everywhere.
Then all $f_{n}$ as well as $f$ are in $L^{p}$ and the sequence $(f_{n})$ converges to $f$ in the sense of $L^{p}$, i.e.:
$\lim _{n\to \infty }\|f_{n}-f\|_{p}=\lim _{n\to \infty }\left(\int _{\Omega }|f_{n}-f|^{p}\,d\mu \right)^{\frac {1}{p}}=0.$
Idea of the proof: Apply the original theorem to the function sequence $h_{n}=|f_{n}-f|^{p}$ with the dominating function $(2g)^{p}$.
Extensions
The dominated convergence theorem applies also to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable as above. The assumption of convergence almost everywhere can be weakened to require only convergence in measure.
The dominated convergence theorem applies also to conditional expectations. [2]
See also
• Convergence of random variables, Convergence in mean
• Monotone convergence theorem (does not require domination by an integrable function but assumes monotonicity of the sequence instead)
• Scheffé's lemma
• Uniform integrability
• Vitali convergence theorem (a generalization of Lebesgue's dominated convergence theorem)
Notes
1. For the real case, see Evans, Lawrence C; Gariepy, Ronald F (2015). Measure Theory and Fine Properties of Functions. CRC Press. pp. Theorem 1.19.
2. Zitkovic 2013, Proposition 10.5.
References
• Bartle, R.G. (1995). The Elements of Integration and Lebesgue Measure. Wiley Interscience. ISBN 9780471042228.
• Royden, H.L. (1988). Real Analysis. Prentice Hall. ISBN 9780024041517.
• Weir, Alan J. (1973). "The Convergence Theorems". Lebesgue Integration and Measure. Cambridge: Cambridge University Press. pp. 93–118. ISBN 0-521-08728-7.
• Williams, D. (1991). Probability with martingales. Cambridge University Press. ISBN 0-521-40605-6.
• Zitkovic, Gordan (Fall 2013). "Lecture10: Conditional Expectation" (PDF). Retrieved December 25, 2020.
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
| Wikipedia |
What is the theory of Matrices?
I've just passed high school and studying matrices. I've learned about determinant, transpose and adjoint etc. I've learned the method of finding these things but what's the purpose of finding these things? What actually Matrices do which makes it solving equations easier.
Why determinant is equal to $(ab)-(cd)$ not $(cd)-(ab)$ for the matrix, $\left[\begin{matrix}a&c\\d&b\end{matrix}\right] $ ?
Why inverse of $\rm A$ is equal to $\dfrac{\operatorname{adj}A} {\det A}$ not $\dfrac{\det A} {\operatorname{adj}A}$ ?
I've read many answers like, What is the usefulness of matrices? , they say that matrices do this and that ,but they does not explain how or why ?
Also what is the relation between vectors and matrices?
matrices determinant
A---B
M.AhmadM.Ahmad
$\begingroup$ What textbook are you using? $\endgroup$ – Shaun Aug 27 '17 at 15:33
$\begingroup$ Check out axlers linear algebra done right $\endgroup$ – Lorenzo Aug 27 '17 at 15:34
$\begingroup$ Recommended youtube playlist: youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab $\endgroup$ – tkausl Aug 27 '17 at 16:40
$\begingroup$ @AreaMan Determinants :(. $\endgroup$ – A---B Aug 27 '17 at 19:12
$\begingroup$ Good for you for asking this question! Sadly you won't learn these until after you start college --in some cases after you take linear algebra. But to get you started, the determinant is the scaling factor of the volume of the unit cube after its coordinates are transformed by the matrix. $\endgroup$ – Mehrdad Aug 27 '17 at 23:13
Matrix theory can be viewed as the calculational side of linear algebra. Linear algebra is the theory of vectors, vector spaces, linear transformations between vector spaces, and so on, but if one wants to calculate particular instances, one uses matrix algebra. In part it is a body of notational conventions for how one represents the abstractions described by linear algebra, and in part a collection of recipes for manipulating these notations.
The boundary between MA and LA is not crisp, and a case can be made that there are topics in MA that are not really discussed in LA, so my description above is perhaps simplistic. In particular, the Perron-Frobenius theorem seems inherently bound to matrices and don't seem to have clean abstract linear algebra formulation. Similarly the subject of "total positivity".
kimchi loverkimchi lover
Matrices are compact representations of linear systems of equations.
These types of problems are called "linear" because they are closely related to straight lines (and flat planes in higher dimensions).
Note: Matrices can be used in a large variety of ways, but in this answer, I will focus on their historical relationship to simple algebra problems. It should be noted, however, that some properties of matrices are easier to understand from other perspectives (i.e. from their applications in geometry, etc.).
All of your questions will be answered by the end, but it will take a little time to motivate and justify those answers. So please bear with me.
Remember in grade school when you first learned to solve problems like the following?
$$ 7x+2y=5 \\ 3x-4y=7 $$
Well, how exactly did you solve a problem like this?
By Graphing
One way was to solve both equations for $y$, plot them as two straight lines on a Cartesian plane, find their intersection point, and list the ordered pair corresponding to that point as the answer. Here is a Wolfram Alpha page doing exactly that. The intersection point is $(1,-1)$; therefore, the solution is $x=1$ and $y=-1$.
From this perspective, the "linear" nature of the problem is fairly obvious.
By Substitution
Another standard way to solve this problem is by "substitution" - which involves solving one equation for $y$ in terms of $x$ and then plugging it into the other equation to find $x$ (and then $y$). Like this:
$$ 7x+2y=5\\2y=5-7x\\y=\frac{5}{2}-\frac{7}{2}x\\ \ \ \ \\ \ \ \ \\ 3x-4y=7\\3x-4(\frac{5}{2}-\frac{7}{2}x)=7\\3x-10+14x=7\\17x=17\\x=1\\ \ \ \ \\ y=\frac{5}{2}-\frac{7}{2}(1)\\y=-1 $$
This method has the advantage of being less involved than the graphing method, but it is also more abstract. Calculating the answer is more straightforward, but the connection to geometry is much less apparent. This will be a recurring theme from here on: We will continue to trade obviousness and simplicity for elegant calculations.
By Row Operations
The last method that is normally taught is to perform operations on an entire equation (like multiplying by a number) and then to add it to the other equation. When done thoughtfully, this method dramatically speeds up the problem-solving process. Here's how it could work in this case:
$$ 7x+2y=5 \\ 3x-4y=7\\ \ \ \ \\ 2(7x+2y=5) \rightarrow 14x+4y=10\\ \ \ \\(14x+4y=10)\\+ (3x-4y=7)\\ \rule{4 cm}{0.4pt} \\17x=17\\ \ \ \\ x=1, \text{etc....} $$
Matrices: Gauss-Jordan Elimination
If you have studied a little bit of matrix algebra, then that last method should look familiar. The "Row Operations" method is exactly the same idea as Gauss-Jordan Elimination on an augmented matrix.
Gauss-Jordan Elimination is significantly more abstract than the previous methods because the variables $x$ and $y$ no longer appear in the problem itself. However, all of the coefficents are still there, and that is what matters. The objective in this case is to get the matrix into Reduced-Row Echelon Form. Here is a quick demonstration:
$$\text{Start:}\ \left(\begin{array}{cc|c}7&2&5\\ 3 & -4 & 7 \end{array}\right)\\ \text{Top Row x2:} \ \left(\begin{array}{cc|c} 14 & 4 & 10 \\ 3 & -4 & 7 \end{array}\right)\\ \text{Add Top to Bottom:} \ \left(\begin{array}{cc|c} 14 & 4 & 10 \\ 17 & 0 & 17 \end{array}\right)\\ \text{Bottom Row $\div$17:} \ \left(\begin{array}{cc|c}14 & 4 & 10 \\ 1 & 0 & 1 \end{array}\right)\\ \text{Bottom Row x14:} \ \left(\begin{array}{cc|c}14 & 4 & 10 \\ 14 & 0 & 14 \end{array}\right)\\ \text{Subtract Bottom from Top:} \ \left(\begin{array}{cc|c} 0 & 4 & -4 \\ 14 & 0 & 14 \end{array}\right)\\ \text{Top Row $\div$4:} \ \left(\begin{array}{cc|c} 0 & 1 & -1 \\ 14 & 0 & 14 \end{array}\right)\\ \text{Bottom Row $\div$14:} \ \left(\begin{array}{cc|c} 0 & 1 & -1 \\ 1 & 0 & 1 \end{array}\right)\\ \text{Switch Rows:} \ \left(\begin{array}{cc|c} 1 & 0 & 1 \\ 0 & 1 & -1 \end{array}\right) $$
Matrices: By Inversion
Now that we can see some connection between matrices and linear systems of equations, we might naturally ask how to represent this problem as a matrix equation and whether that matrix equation can be easily solved.
First, let's set up the matrix equation:
$$ \textbf{A}\vec{x}=\vec{b}\\ \ \ \\ \text{Let, } \textbf{A}=\left(\begin{array}{cc} 7 & 2 \\ 3 & -4 \end{array}\right), \ \vec{x}=\left(\begin{array}{c} x \\ y \end{array}\right), \ \text{and} \ \vec{b}=\left(\begin{array}{c} 5 \\ 7 \end{array}\right)\\ \ \ \\ \therefore \ \left(\begin{array}{cc} 7 & 2 \\ 3 & -4 \end{array}\right) \left(\begin{array}{c} x \\ y \end{array}\right) = \left(\begin{array}{c} 5 \\ 7 \end{array}\right) $$
From here, it is obvious that performing matrix multiplication between the matrix and vector on the left-hand side of the equation results in the original system of equations from the very beginning:
$$\left(\begin{array}{cc} 7 & 2 \\ 3 & -4 \end{array}\right) \left(\begin{array}{c} x \\ y \end{array}\right) = \left(\begin{array}{c} 5 \\ 7 \end{array}\right)\\ \left(\begin{array}{c} 7x+2y \\ 3x-4y \end{array}\right) = \left(\begin{array}{c} 5 \\ 7 \end{array}\right) $$
Therefore, one can see that a vector is just an ordered pair turned on its side. The top entry is the $x$-coordinate while the bottom entry is the $y$-coordinate. Thus, our goal in this problem is to determine the components of the unknown vector $\vec{x}=\left(\begin{array}{c} x \\ y \end{array}\right)$. To do that, we must isolate $\vec{x}$ (just like we would if it were a normal variable rather than a variable vector).
Isolating $\vec{x}$ means finding the inverse of $\textbf{A}$. The inverse of a matrix is unique, so I will write down the general form of the inverse for a 2x2 matrix and that will suffice to cover our example.
$$\textbf{A}=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\\ \textbf{A}^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc} d & -b \\ -c & a \end{array}\right)\\ $$
I recommend checking for yourself to see that
$$\textbf{A}\textbf{A}^{-1}=\textbf{A}^{-1}\textbf{A}=\textbf{I}\\ \ \ \\ \frac{1}{ad-bc}\left(\begin{array}{cc} d & -b \\ -c & a \end{array}\right)\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right) $$
For convenience (and because it has many other applications), we define $ad-bc$ to be the "determinant of $\textbf{A}$." In this situation, it is relevant because it is the factor by which the inverse matrix must be divided so as to return the identity matrix when multiplied by $\textbf{A}$.
Now, finally, to solve our problem we multiple the original equation by the inverse matrix and we are done.
$$\textbf{A}\vec{x}=\vec{b}\\ \ \ \\ \textbf{A}^{-1}\textbf{A}\vec{x}=\textbf{A}^{-1}\vec{b}\\ \ \ \\ \textbf{I} \vec{x}=\textbf{A}^{-1}\vec{b}\\ \ \ \\ \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right) \left(\begin{array}{c} x \\ y \end{array}\right)=-\frac{1}{34}\left(\begin{array}{cc} -4 & -2 \\ -3 & 7 \end{array}\right)\left(\begin{array}{c} 5 \\ 7 \end{array}\right)\\ \ \ \\ \left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} 1 \\ -1 \end{array}\right) $$
As you can see, the algebra is now much more complicated than the basic methods that you learned in grade school. However, this trade-off comes with the advantage of being a much more elegant-looking solution.
There are two main reasons for using matrix algebra to solve linear systems of equations. First, the theory of matrices is very broad. It generalizes specific problems into much larger classes of problem types. By generalizing in this way, we can identify similarities between apparently unrelated problems and, therefore, relate their solutions to one another. Second, computers are really good with matrices. If a problem can be solved with a matrix, then a computer can create an approximate answer very, very quickly. And, guess what, many of the world's most pressing problems can be modeled with matrices.
GeoffreyGeoffrey
Matrix theory is linear algebra with the method of the coordinate systems.
As to why the determinant is calculated that way try to compute the area of a square of unitary length side once it is transformed by a matrix (considering two adjacent sides as vectors). Determinant is an operation that can be applied to any linear operator $L: A\rightarrow A$, $A$ is a linear space over a field, and it gives an element of such a field and has such a geometrical interpretation that I asked you to search for.
"Taking the inverse" is again an operation that can be applied to any linear operator that has a non-null determinant.
Adjoint: Given a square matrix the sum of the product of the elements of a row (column) and the corresponding cofactors equals the determinant, while the sum of the product of the elements of a row (column) and the corresponding cofactors of the elements of another row (column) is null. That means that:$$A \operatorname{adj}(A)=\det A$$ where with $\det A$ I mean a diagonal matrix whose elements are all equal to the determinant.
Transpose is an operation that can be applied to any linear application $L: A\rightarrow B$, where $A$ and $B$ are linear space over the same field $\Bbb{K}$: it gives another linear application that describes how scalar linear functional on $B$, $f:B\rightarrow \Bbb{K}$, are mapped into scalar linear functional on $A$, $g:A\rightarrow \Bbb{K}$, as a composition of $f$ with $L$: $g=f\circ L$. In this way instead of performing such computation $f(L(x))$ one can simply do $g(x)$, for any $x\in A$
Matrices in addition to linear application can also be used to describe bilinear and quadratic forms in given coordinate systems.
Vectors are the element of a linear space. Coordinate vectors are their representation once a linear coordinate system has been chosen for such a linear space. With a linear coordinate system a linear space $A$ over a field $\Bbb{K}$ becomes an homomorphic image of a (coordinate) linear space $\Bbb{K}^n$, $n\in\Bbb{N}$ where $n\ge dim_\Bbb{K}A$. If the linear coordinate system is bijective the equality holds. So coordinate vectors are elements of the coordinate linear space. What you have being using till now are coordinate vectors even though you have been calling them simply vectors. By extension usually the numerical representation of a vector is called coordinate vector also when the coordinate system chosen is not linear, but in this case they cannot be used with the algorithm of the matrix theory. Coordinate vectors are also used to represent numerically linear functional once a linear coordinate system is chosen: those are usually represented as row vectors, while the former are represented as column vectors.
tryingtrying
A multiplicative inverse to $A$ is a matrix that when you multiply it with $A$ becomes the identity matrix. That is the definition of inverse.
You can define matrix multiplication or division by scalar, but det./adj would be a scalar divided by matrix and it is in general not defined.
The usefulness of matrices is everywhere. There is today definitely no branch of science or engineering where you will not find use for matrices. But it is difficult to explain why before you learn more about them!
mathreadlermathreadler
I'll take a very... non-Bourbakian approach on this.
Matrices are transformations! You do have an intuition what vectors are? The very first intuition is: if you want to transform your vectors from one coordinate system ("base") to another, you'd need something encoding this transformation. A matrix.
Of course, there is much more to it. We can encode large linear equation systems as matrix-vector product. So, solving these equations is reshuffling the matrix, basically. Think: $Ax=b$, with matrix A and vectors $x$ and $b$, can be solved for $x$ when we manage to invert the matrix $A$. But that's a very crude approach, look up LU decomposition.
Pretty things happen when you study the matrix-vector product, you reach bilinear forms $v^tAv$, scalar products and other funny stuff.
Another issue that might interest you are all the transformations you do (with matrices!) in computer graphics! They basically transform the 3D world representation to the viewport of your screen with a $\mathbb{R}^4$ matrix-vector product. Why four-dimensional? Read up, cannot post a link for a stupid reason.
Oh, and you can generalize matrices to tensors, basically $n$-dimensional matrices, if matrices are 2D.
To conclude: linear algebra is pretty as it is, but it is first of all a tool of a working mathematician. You use it to solve other problems. For example, if you can reduce your problem to a (even very large) linear equation system, you are done.
Oleg LobachevOleg Lobachev
$\begingroup$ Matrices don't have to transform between coordinate systems. $\endgroup$ – Mateen Ulhaq Aug 28 '17 at 1:23
$\begingroup$ Of course not. But as a first intuition I regard this as a good approach. "A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, every linear transformation can be represented by a matrix, and every matrix corresponds to a unique linear transformation. The matrix, and its close relative the determinant, are extremely important concepts in linear algebra, and were first formulated by Sylvester (1851) and Cayley." – Mathworld $\endgroup$ – Oleg Lobachev Aug 28 '17 at 17:08
Matrix computations with matrices (multiplication, and inversion) are very much used in the optimization of of multivariate functions under additional constraints.They speed up optimization tasks enormously especially with high dimensional problems. Optimization is a the tool for using machine learning algorithms to teach computers (machine) to solve problems without given them the instructions to do so, and without human intervention. Machine learning is one important branch of Artificial Intelligence (AI) that's helping automate many difficult but useful tasks such as hand writing and face recognition, fraud detection, medical diagnostics and self driving cars.
HassHass
Be patient. There are many wonderful ideas that come out of matrix theory. The elements of matrices don't have to be real numbers; they can be elements of a finite field, or they can be imaginary or complex numbers. You can do more than just add and multiply matrices; you can get their sines, arctangents, logarithms, square roots, and more. There is a whole branch of computing where special methods are studied to accurately compute the inverse. In probability theory and reliability engineering, matrices represent transition probabilities of a piece of equipment from good to various failed states. Anther thing you can do with matrices is linear programming: finding the best solution for a system subject to constraints. And there is much, much more.
So be patient. You will come to appreciate and love matrices. For now, I advise you to force yourself to learn everything your math class has to offer about matrices; if you do this you will never be sorry.
richard1941richard1941
I've just passed high school and studying matrices. I've learned about determinant, transpose and adjoint etc. I've learned the method of finding these things but what's the purpose of finding these things?
First it is a useful way of aggregating several objects. A way of organizing. A matrix $$ A = (a_{ij}) \quad i \in \{1, \dotsc, m \}, j \in \{ 1, \dotsc, n \} \quad m, n \in \mathbb{N} $$ will allow you to treat $m \times n$ objects under the handle $A$. Or $$ x = (x_i) \quad i \in \{1, \dotsc, m \} \quad m \in \mathbb{N} $$ a (column) vector, which can be interpreted as $m \times 1$ matrix, allows you treat $m$ objects under the handle $x$.
Up to here we could have also written this using functions: $$ a_{ij} = a(i, j) : \{1, \dotsc, m \} \times \{1, \dotsc, n \} \to X \\ x_i = x(i) : \{1, \dotsc, m \} \to X $$ This already has applications, e.g. in a programming language we might find this implemented as data structures (a useful way to handle data), as arrays or e.g. as hash maps, if we index with elements from other sets than whole numbers.
These aggregates become even more useful if one can operate on them and it turns out that they are useful to describe nature in physics, even in the versions with more than two indices and indices ranging over infinite sets.
(To be continued later)
What actually Matrices do which makes it solving equations easier.
Why inverse of $\rm A$ is equal to $\dfrac{\operatorname{adj}A} {\det > A}$ not $\dfrac{\det A} {\operatorname{adj}A}$ ?
mvwmvw
Not the answer you're looking for? Browse other questions tagged matrices determinant or ask your own question.
What is the usefulness of matrices?
Invertible matrices over a commutative ring and their determinants
Determinant of Matrix of Matrices
The determinent of a vector? Where those it come from and then is is useful and true?
What causes commutativity of matrices?
Determinant of adjoint
The determinant of the transposing endomorphism
Prove that Gl(n, F) is a group
The meaning and logic of the determinant of a matrix, and matrices' cofactors
Adjoint matrix times the matrix of coefficients
Questions on the proof of Cayley Hamilton Theorem | CommonCrawl |
# Using arrow functions for concise and elegant code
Arrow functions are a feature introduced in ES6 that provide a more concise and elegant way of writing functions. They are still categorized as functions in JavaScript.
The syntax of an arrow function is as follows:
```javascript
const functionName = (parameter1, parameter2, ...) => {
// function body
return value;
};
```
Here's an example of an arrow function that adds two numbers:
```javascript
const add = (num1, num2) => {
return num1 + num2;
};
```
You can also omit the curly braces and the return keyword if the function body is a single expression:
```javascript
const add = (num1, num2) => num1 + num2;
```
Arrow functions are especially useful when passing them as parameters to other functions or methods. They make the code more concise and readable. For example, you can use an arrow function with the `map` method to add 1 to each element of an array:
```javascript
const numbers = [1, 2, 3];
const incrementedNumbers = numbers.map((num) => num + 1);
console.log(incrementedNumbers); // [2, 3, 4]
```
Remember that arrow functions have lexical scoping for `this`, which means they don't have their own `this` value. Instead, they inherit the `this` value from the surrounding context. This makes arrow functions particularly useful when working with object methods.
- The following arrow function calculates the area of a rectangle given its width and height:
```javascript
const calculateArea = (width, height) => width * height;
```
- The following arrow function checks if a number is even:
```javascript
const isEven = (number) => number % 2 === 0;
```
## Exercise
Write an arrow function called `multiplyByTwo` that takes a number as a parameter and returns the result of multiplying it by 2.
### Solution
```javascript
const multiplyByTwo = (number) => number * 2;
```
# Creating and using classes in ES6
Classes are an important concept in object-oriented programming. They provide a way to define objects with properties and methods. In ES6, classes were introduced as a new syntax for creating objects.
To create a class in ES6, you use the `class` keyword followed by the name of the class. The class can have a constructor method, which is called when a new object is created from the class. Inside the constructor, you can initialize the object's properties.
Here's an example of a class called `Person`:
```javascript
class Person {
constructor(name, age) {
this.name = name;
this.age = age;
}
sayHello() {
console.log(`Hello, my name is ${this.name} and I am ${this.age} years old.`);
}
}
```
In the example above, the `Person` class has a constructor method that takes two parameters: `name` and `age`. Inside the constructor, the `name` and `age` properties of the object are initialized using the `this` keyword. The class also has a `sayHello` method, which logs a greeting message to the console.
To create a new object from a class, you use the `new` keyword followed by the name of the class and any arguments required by the constructor. Here's an example:
```javascript
const person1 = new Person('Alice', 25);
person1.sayHello(); // Output: Hello, my name is Alice and I am 25 years old.
```
In this example, a new `Person` object is created with the name `'Alice'` and age `25`. The `sayHello` method is then called on the `person1` object.
Classes in ES6 can also have static methods, which are methods that are called on the class itself, rather than on instances of the class. Static methods are defined using the `static` keyword. Here's an example:
```javascript
class MathUtils {
static square(number) {
return number * number;
}
}
console.log(MathUtils.square(5)); // Output: 25
```
In this example, the `MathUtils` class has a static method called `square` that calculates the square of a number. The `square` method is called on the class itself, rather than on an instance of the class.
- The following class represents a car:
```javascript
class Car {
constructor(make, model, year) {
this.make = make;
this.model = model;
this.year = year;
}
startEngine() {
console.log(`Starting the engine of ${this.make} ${this.model}`);
}
}
```
- The following class represents a student:
```javascript
class Student {
constructor(name, grade) {
this.name = name;
this.grade = grade;
}
getGrade() {
return this.grade;
}
}
```
## Exercise
Create a class called `Rectangle` with the following properties and methods:
- Properties:
- `width` (a number representing the width of the rectangle)
- `height` (a number representing the height of the rectangle)
- Methods:
- `calculateArea` (a method that calculates and returns the area of the rectangle)
- `calculatePerimeter` (a method that calculates and returns the perimeter of the rectangle)
### Solution
```javascript
class Rectangle {
constructor(width, height) {
this.width = width;
this.height = height;
}
calculateArea() {
return this.width * this.height;
}
calculatePerimeter() {
return 2 * (this.width + this.height);
}
}
```
# Destructuring objects and arrays for efficient data manipulation
Destructuring is a powerful feature in ES6 that allows you to extract values from objects and arrays and assign them to variables in a more concise way. It can make your code more readable and efficient, especially when working with complex data structures.
### Destructuring Objects
To destructure an object, you use curly braces `{}` and specify the names of the properties you want to extract. Here's an example:
```javascript
const person = {
name: 'Alice',
age: 25,
city: 'New York'
};
const { name, age } = person;
console.log(name); // Output: Alice
console.log(age); // Output: 25
```
In this example, the `person` object has three properties: `name`, `age`, and `city`. We use destructuring to extract the `name` and `age` properties and assign them to variables of the same name.
You can also assign the extracted properties to variables with different names by using the syntax `{ property: variable }`. Here's an example:
```javascript
const { name: personName, age: personAge } = person;
console.log(personName); // Output: Alice
console.log(personAge); // Output: 25
```
In this example, the `name` property is assigned to the variable `personName`, and the `age` property is assigned to the variable `personAge`.
### Destructuring Arrays
To destructure an array, you use square brackets `[]` and specify the positions of the elements you want to extract. Here's an example:
```javascript
const numbers = [1, 2, 3, 4, 5];
const [first, second, third] = numbers;
console.log(first); // Output: 1
console.log(second); // Output: 2
console.log(third); // Output: 3
```
In this example, the `numbers` array has five elements. We use destructuring to extract the first three elements and assign them to variables.
You can also use the rest operator (`...`) to assign the remaining elements of an array to a variable. Here's an example:
```javascript
const [first, second, ...rest] = numbers;
console.log(first); // Output: 1
console.log(second); // Output: 2
console.log(rest); // Output: [3, 4, 5]
```
In this example, the `first` and `second` elements are extracted and assigned to variables, and the remaining elements are assigned to the `rest` variable as an array.
- The following object represents a book:
```javascript
const book = {
title: 'The Great Gatsby',
author: 'F. Scott Fitzgerald',
year: 1925
};
```
- The following array represents a list of fruits:
```javascript
const fruits = ['apple', 'banana', 'orange', 'kiwi'];
```
## Exercise
Consider the following object:
```javascript
const person = {
name: 'Alice',
age: 25,
city: 'New York'
};
```
Use destructuring to extract the `name` and `city` properties and assign them to variables.
### Solution
```javascript
const { name, city } = person;
```
# Using the spread operator for merging and copying data
The spread operator is a powerful feature in ES6 that allows you to expand an iterable object into multiple elements. It is denoted by three dots `...`. The spread operator can be used in various contexts, such as merging arrays, copying arrays, and passing arguments to functions.
### Merging Arrays
To merge two or more arrays into a single array, you can use the spread operator. Here's an example:
```javascript
const arr1 = [1, 2, 3];
const arr2 = [4, 5, 6];
const mergedArray = [...arr1, ...arr2];
console.log(mergedArray); // Output: [1, 2, 3, 4, 5, 6]
```
In this example, the spread operator is used to expand `arr1` and `arr2` into individual elements, which are then combined into a new array `mergedArray`.
### Copying Arrays
The spread operator can also be used to create a shallow copy of an array. Here's an example:
```javascript
const originalArray = [1, 2, 3];
const copiedArray = [...originalArray];
console.log(copiedArray); // Output: [1, 2, 3]
```
In this example, the spread operator is used to expand `originalArray` into individual elements, which are then used to create a new array `copiedArray`. Modifying `copiedArray` will not affect `originalArray`, as they are two separate arrays.
### Passing Arguments to Functions
The spread operator can be used to pass multiple arguments to a function. Here's an example:
```javascript
function sum(a, b, c) {
return a + b + c;
}
const numbers = [1, 2, 3];
const result = sum(...numbers);
console.log(result); // Output: 6
```
In this example, the spread operator is used to expand the `numbers` array into individual elements, which are then passed as arguments to the `sum` function.
- The following arrays represent different categories of books:
```javascript
const fictionBooks = ['The Great Gatsby', 'To Kill a Mockingbird', '1984'];
const nonFictionBooks = ['Sapiens', 'Educated', 'Becoming'];
```
- The following array represents a list of fruits:
```javascript
const fruits = ['apple', 'banana', 'orange', 'kiwi'];
```
## Exercise
Consider the following arrays:
```javascript
const arr1 = [1, 2, 3];
const arr2 = [4, 5, 6];
```
Use the spread operator to merge `arr1` and `arr2` into a single array.
### Solution
```javascript
const mergedArray = [...arr1, ...arr2];
```
# Enhancing string manipulation with template literals
Template literals are a powerful feature in ES6 that allow you to create strings with embedded expressions. They are denoted by backticks (` `) instead of single or double quotes. Template literals offer a more concise and flexible way to manipulate strings compared to traditional string concatenation.
### Basic Syntax
To create a template literal, you enclose the string content within backticks. You can then embed expressions within the string using `${}`. Here's an example:
```javascript
const name = 'Alice';
const age = 25;
const message = `My name is ${name} and I am ${age} years old.`;
console.log(message); // Output: My name is Alice and I am 25 years old.
```
In this example, the expressions `${name}` and `${age}` are evaluated and their values are inserted into the string. The resulting string is stored in the variable `message`.
### Multiline Strings
Template literals also make it easy to create multiline strings. In traditional JavaScript, you would need to use line breaks and string concatenation to achieve this. With template literals, you can simply include line breaks within the string. Here's an example:
```javascript
const poem = `
Roses are red,
Violets are blue,
Sugar is sweet,
And so are you.
`;
console.log(poem);
```
In this example, the poem is created as a multiline string using backticks. The line breaks are preserved when the string is printed.
### Expression Evaluation
Expressions within template literals are evaluated as JavaScript code. This means you can include any valid JavaScript expression within `${}`. Here's an example:
```javascript
const x = 5;
const y = 10;
const result = `The sum of ${x} and ${y} is ${x + y}.`;
console.log(result); // Output: The sum of 5 and 10 is 15.
```
In this example, the expression `${x + y}` is evaluated and its value is inserted into the string.
- Consider the following variables:
```javascript
const firstName = 'John';
const lastName = 'Doe';
const age = 30;
```
- You can use template literals to create a string that combines these variables:
```javascript
const message = `My name is ${firstName} ${lastName} and I am ${age} years old.`;
console.log(message); // Output: My name is John Doe and I am 30 years old.
```
## Exercise
Consider the following variables:
```javascript
const product = 'apple';
const quantity = 5;
const price = 0.5;
```
Use template literals to create a string that represents the total cost of buying `quantity` number of `product` at a price of `price` per unit.
### Solution
```javascript
const totalCost = `The total cost of buying ${quantity} ${product}s is $${quantity * price}.`;
```
# ES6 modules and their benefits
ES6 introduced a new module system that allows you to organize your code into separate files and easily share functionality between them. This module system provides several benefits over the traditional approach of using global variables and script tags.
### Creating Modules
In ES6, a module is simply a file that exports functionality using the `export` keyword. You can export variables, functions, classes, or even entire modules. Here's an example:
```javascript
// math.js
export const add = (a, b) => a + b;
export const subtract = (a, b) => a - b;
// main.js
import { add, subtract } from './math.js';
console.log(add(5, 3)); // Output: 8
console.log(subtract(5, 3)); // Output: 2
```
In this example, the `math.js` module exports two functions: `add` and `subtract`. These functions can then be imported and used in the `main.js` module.
### Benefits of Modules
ES6 modules offer several benefits:
- **Encapsulation**: Modules allow you to encapsulate functionality within separate files, making your code more organized and easier to maintain.
- **Reusability**: Modules can be easily reused in different parts of your application or even in different projects.
- **Dependency Management**: Modules make it easy to manage dependencies between different parts of your code. You can specify which modules are required by importing them, and the module system takes care of resolving and loading the dependencies.
- **Better Performance**: The module system only loads the modules that are actually needed, reducing the amount of code that needs to be downloaded and executed.
- **Improved Debugging**: Modules provide a clear and explicit way to define dependencies, making it easier to identify and fix issues in your code.
- Consider the following modules:
```javascript
// utils.js
export const capitalize = (str) => str.toUpperCase();
export const reverse = (str) => str.split('').reverse().join('');
// main.js
import { capitalize, reverse } from './utils.js';
console.log(capitalize('hello')); // Output: HELLO
console.log(reverse('hello')); // Output: olleh
```
- In this example, the `utils.js` module exports two functions: `capitalize` and `reverse`. These functions can then be imported and used in the `main.js` module.
## Exercise
Create a new module called `math.js` that exports a function called `multiply` which takes two parameters and returns their product. Then, import and use the `multiply` function in the `main.js` module to calculate the product of `5` and `3`.
### Solution
```javascript
// math.js
export const multiply = (a, b) => a * b;
// main.js
import { multiply } from './math.js';
console.log(multiply(5, 3)); // Output: 15
```
# Working with Promises for asynchronous programming
Promises are a powerful feature in ES6 that allow you to work with asynchronous code in a more structured and readable way. They provide a way to handle the result or error of an asynchronous operation, such as making an HTTP request or reading a file.
### Creating Promises
To create a promise, you use the `Promise` constructor and pass a function as an argument. This function takes two parameters: `resolve` and `reject`. Inside the function, you perform the asynchronous operation and call `resolve` with the result or `reject` with an error. Here's an example:
```javascript
const fetchData = new Promise((resolve, reject) => {
// Simulate an asynchronous operation
setTimeout(() => {
const data = 'Hello, world!';
resolve(data);
}, 2000);
});
```
In this example, the `fetchData` promise is created with a function that simulates an asynchronous operation using `setTimeout`. After 2 seconds, the promise is resolved with the string `'Hello, world!'`.
### Handling Promises
Once you have a promise, you can use the `then` method to handle the result of the asynchronous operation. The `then` method takes a callback function as an argument, which will be called with the result of the promise. Here's an example:
```javascript
fetchData.then((data) => {
console.log(data); // Output: Hello, world!
});
```
In this example, the callback function passed to `then` is called with the result of the `fetchData` promise, which is the string `'Hello, world!'`.
### Chaining Promises
Promises can also be chained together using the `then` method. This allows you to perform multiple asynchronous operations in sequence. Each `then` callback can return a new promise, which will be resolved before the next `then` callback is called. Here's an example:
```javascript
fetchData
.then((data) => {
console.log(data); // Output: Hello, world!
return data.toUpperCase();
})
.then((data) => {
console.log(data); // Output: HELLO, WORLD!
});
```
In this example, the first `then` callback logs the result of the `fetchData` promise and returns a new promise that converts the data to uppercase. The second `then` callback logs the uppercase data.
- Consider the following promise:
```javascript
const fetchData = new Promise((resolve, reject) => {
// Simulate an asynchronous operation
setTimeout(() => {
const data = 'Hello, world!';
resolve(data);
}, 2000);
});
fetchData.then((data) => {
console.log(data); // Output: Hello, world!
});
```
- In this example, the `fetchData` promise is resolved after 2 seconds and the result is logged to the console.
## Exercise
Create a new promise called `fetchData` that resolves with the number `42` after 1 second. Then, use the `then` method to log the result of the promise to the console.
### Solution
```javascript
const fetchData = new Promise((resolve, reject) => {
// Simulate an asynchronous operation
setTimeout(() => {
const data = 42;
resolve(data);
}, 1000);
});
fetchData.then((data) => {
console.log(data); // Output: 42
});
```
# Iterators and generators for efficient data processing
Iterators and generators are powerful features in ES6 that allow you to work with collections of data in a more efficient and flexible way. They provide a way to iterate over the elements of a collection and perform operations on them.
### Iterators
An iterator is an object that provides a `next` method, which returns the next element in the collection. The `next` method returns an object with two properties: `value`, which is the current element, and `done`, which is a boolean indicating whether the iteration is complete. Here's an example:
```javascript
const numbers = [1, 2, 3];
const iterator = numbers[Symbol.iterator]();
console.log(iterator.next()); // Output: { value: 1, done: false }
console.log(iterator.next()); // Output: { value: 2, done: false }
console.log(iterator.next()); // Output: { value: 3, done: false }
console.log(iterator.next()); // Output: { value: undefined, done: true }
```
In this example, the `numbers` array is converted into an iterator using the `Symbol.iterator` method. The iterator is then used to iterate over the elements of the array using the `next` method.
### Generators
A generator is a special type of function that can be paused and resumed. It is defined using the `function*` syntax. Inside a generator, you can use the `yield` keyword to pause the execution and return a value. Here's an example:
```javascript
function* generateNumbers() {
yield 1;
yield 2;
yield 3;
}
const iterator = generateNumbers();
console.log(iterator.next()); // Output: { value: 1, done: false }
console.log(iterator.next()); // Output: { value: 2, done: false }
console.log(iterator.next()); // Output: { value: 3, done: false }
console.log(iterator.next()); // Output: { value: undefined, done: true }
```
In this example, the `generateNumbers` generator function is defined with three `yield` statements. Each `yield` statement pauses the execution and returns a value. The generator is then used to iterate over the values using the `next` method.
- Consider the following iterator:
```javascript
const numbers = [1, 2, 3];
const iterator = numbers[Symbol.iterator]();
console.log(iterator.next()); // Output: { value: 1, done: false }
console.log(iterator.next()); // Output: { value: 2, done: false }
console.log(iterator.next()); // Output: { value: 3, done: false }
console.log(iterator.next()); // Output: { value: undefined, done: true }
```
- In this example, the `numbers` array is converted into an iterator using the `Symbol.iterator` method. The iterator is then used to iterate over the elements of the array using the `next` method.
## Exercise
Create an iterator called `range` that generates the numbers from `start` to `end` (inclusive). Then, use the iterator to log each number to the console.
### Solution
```javascript
function* range(start, end) {
for (let i = start; i <= end; i++) {
yield i;
}
}
const iterator = range(1, 5);
for (const number of iterator) {
console.log(number);
}
```
# Using ES6 in the browser with transpilers and polyfills
ES6 introduced many new features and syntax improvements, but not all browsers support these features. To use ES6 in the browser, you can use transpilers and polyfills.
### Transpilers
A transpiler is a tool that converts code written in one version of JavaScript into another version. For example, you can use a transpiler to convert ES6 code into ES5 code, which is supported by older browsers. One popular transpiler for ES6 is Babel.
To use Babel, you need to install it and configure it to transpile your code. Once configured, you can write your code using ES6 syntax and Babel will automatically convert it to ES5 syntax. Here's an example:
```javascript
// ES6 code
const add = (a, b) => a + b;
// Transpiled ES5 code
var add = function add(a, b) {
return a + b;
};
```
In this example, the ES6 code is converted into equivalent ES5 code by Babel.
### Polyfills
A polyfill is a piece of code that provides the functionality of a newer JavaScript feature in older browsers that do not support it. For example, you can use a polyfill to provide support for ES6 features in older versions of Internet Explorer.
To use a polyfill, you need to include it in your HTML file before your JavaScript code. The polyfill will add the necessary functionality to the browser if it is not already supported. Here's an example:
```html
<!-- Include the ES6 polyfill -->
<script src="https://cdn.polyfill.io/v2/polyfill.min.js"></script>
<!-- Your ES6 code -->
<script src="main.js"></script>
```
In this example, the ES6 polyfill is included before the `main.js` script. The polyfill will add support for ES6 features to the browser if needed.
- Consider the following ES6 code:
```javascript
const add = (a, b) => a + b;
```
- To use this code in older browsers, you can transpile it to ES5 using Babel:
```javascript
var add = function add(a, b) {
return a + b;
};
```
- This transpiled code can be used in browsers that do not support ES6 syntax.
## Exercise
Assume that the browser you are using does not support the `includes` method for arrays, which was introduced in ES6. Use a polyfill to add support for this method.
### Solution
```html
<!-- Include the ES6 polyfill -->
<script src="https://cdn.polyfill.io/v2/polyfill.min.js"></script>
<!-- Your ES6 code -->
<script>
const numbers = [1, 2, 3];
console.log(numbers.includes(2)); // Output: true
</script>
```
# Real-world examples of ES6 applications
ES6 introduced many new features and syntax improvements that make JavaScript more powerful and expressive. These features can be used in a wide range of real-world applications. Here are a few examples:
### Web Development
ES6 features such as arrow functions, template literals, and destructuring can make web development more efficient and readable. They allow you to write cleaner and more concise code, reducing the amount of boilerplate and improving the overall developer experience.
### Node.js
ES6 features are also widely used in server-side JavaScript development with Node.js. They provide a more modern and expressive way to write server-side code, making it easier to build scalable and maintainable applications.
### Mobile Development
ES6 features can be used in mobile development with frameworks such as React Native. They allow you to write cross-platform code that runs on both iOS and Android devices, reducing the amount of code duplication and improving development efficiency.
### Data Analysis
ES6 features such as iterators and generators can be used in data analysis and processing. They provide a more efficient and flexible way to work with large datasets, allowing you to write code that is easier to understand and maintain.
### Game Development
ES6 features can be used in game development with frameworks such as Phaser. They provide a more modern and expressive way to write game logic, making it easier to build complex and interactive games.
### Internet of Things (IoT)
ES6 features can be used in IoT development with platforms such as Johnny-Five. They provide a more modern and expressive way to write code for IoT devices, making it easier to build connected and intelligent systems.
- Web developers can use ES6 features such as arrow functions, template literals, and destructuring to write cleaner and more efficient code:
```javascript
// Arrow functions
const add = (a, b) => a + b;
// Template literals
const name = 'Alice';
console.log(`Hello, ${name}!`);
// Destructuring
const { x, y } = point;
console.log(`x: ${x}, y: ${y}`);
```
- These features can improve the developer experience and make code easier to read and maintain.
## Exercise
Think about a real-world application that you are familiar with. How could ES6 features be used to improve the development process or the quality of the code?
### Solution
ES6 features could be used in a real-world application such as a social media platform to improve the development process and the quality of the code. For example, arrow functions and template literals could be used to write cleaner and more concise code for handling user interactions and generating dynamic content. Destructuring could be used to extract and manipulate data from complex data structures such as user profiles or posts. Promises could be used to handle asynchronous operations such as fetching data from a server or uploading images. Overall, ES6 features can improve the efficiency, readability, and maintainability of the codebase, leading to a better user experience and faster development cycles.
# Best practices and tips for using ES6 effectively
ES6 introduces many powerful features and syntax improvements that can greatly enhance your JavaScript code. However, to use ES6 effectively, it's important to follow best practices and keep a few tips in mind. Here are some recommendations for using ES6 effectively:
### Use ES6 features where they make sense
ES6 features are not meant to be used in every situation. Some features may be more appropriate for certain scenarios than others. It's important to understand the purpose and benefits of each feature and use them where they make sense. This will ensure that your code remains readable, maintainable, and efficient.
### Stay up-to-date with the latest ES6 features
JavaScript is an evolving language, and new features are constantly being added. It's important to stay up-to-date with the latest ES6 features and improvements. This will allow you to take advantage of new capabilities and improve the quality of your code. Follow reputable sources such as the ECMAScript specification and popular JavaScript blogs to stay informed about the latest developments.
### Use a transpiler for browser compatibility
Not all browsers fully support ES6 features. To ensure that your code works across different browsers, it's recommended to use a transpiler such as Babel. A transpiler will convert your ES6 code into equivalent ES5 code, which is compatible with older browsers. This allows you to use the latest ES6 features without worrying about browser compatibility issues.
### Write modular and reusable code
ES6 introduces the concept of modules, which allow you to organize your code into separate files and import/export functionality as needed. This promotes modularity and reusability, making your code easier to understand and maintain. Take advantage of modules to create clean and modular codebases.
### Use arrow functions wisely
Arrow functions are a powerful feature of ES6, but they should be used judiciously. While arrow functions can make your code more concise, they also have some limitations. For example, arrow functions do not have their own `this` value, so they cannot be used as methods or constructors. It's important to understand the differences between arrow functions and regular functions and use them appropriately.
### Test your code thoroughly
As with any code, it's important to thoroughly test your ES6 code to ensure that it works as expected. Use testing frameworks such as Mocha or Jest to write unit tests for your code. This will help you catch any bugs or issues early on and ensure that your code is reliable and robust.
By following these best practices and tips, you can effectively use ES6 features and improve the quality of your JavaScript code. Remember to always strive for readability, maintainability, and efficiency in your code. | Textbooks |
Virologic failure in HIV-positive adolescents with perfect adherence in Uganda: a cross-sectional study
Julian Natukunda1,2,
Peter Kirabira1,
Ken Ing Cherng Ong2,
Akira Shibanuma2 &
Masamine Jimba2
Tropical Medicine and Health volume 47, Article number: 8 (2019) Cite this article
Adolescents living with human immunodeficiency virus (HIV) die owing to acquired immune deficiency syndrome (AIDS)-related causes more than adults. Although viral suppression protects people living with HIV from AIDS-related illnesses, little is known about viral outcomes of adolescents in sub-Saharan Africa where the biggest burden of deaths is experienced. This study aimed to identify the factors associated with viral load suppression among HIV-positive adolescents (10–19 years) receiving antiretroviral therapy (ART) in Uganda.
We conducted a cross-sectional study among school-going, HIV-positive adolescents on ART from August to September 2016. We recruited 238 adolescents who underwent ART at a public health facility and had at least one viral load result recorded in their medical records since 2015. We collected the data of patients' demographics and treatment- and clinic-related factors using existing medical records and questionnaire-guided face-to-face interviews. For outcome variables, we defined viral suppression as < 1000 copies/mL. We used multivariate logistic regression to determine factors associated with viral suppression.
We analyzed the data of 200 adolescents meeting the inclusion criteria. Viral suppression was high among adolescents with good adherence > 95% (adjusted odds ratio [AOR] 2.73, 95% confidence interval [95% CI, 1.09 to 6.82). However, 71% of all adolescents who did not achieve viral suppression were also sufficiently adherent (adherence > 95%). Regardless of adherence status, other risk factors for viral suppression at the multivariate level included having a history of treatment failure (AOR 0.26, 95% CI, 0.09 to 0.77), religion (being Anglican [AOR 0.19, 95% CI, 0.06 to 0.62] or Muslim [AOR 0.17, 95% CI, 0.05 to 0.55]), and having been prayed for (AOR 0.38, 95% CI, 0.15 to 0.96).
More than 70% of adolescents who experienced virologic failure were sufficiently adherent (adherence > 95). Adolescents who had unsuppressed viral loads in their initial viral load were more likely to experience virologic failure upon a repeat viral load regardless of their adherence level or change of regimen. The study also shows that strong religious beliefs exist among adolescents. Healthcare provider training in psychological counseling, regular and strict monitoring of adolescent outcomes should be prioritized to facilitate early identification and management of drug resistance through timely switching of treatment regimens to more robust combinations.
In many countries, the "treat all" approach is implemented to identify HIV-positive people earlier and get them on treatment. Its goal is to suppress viral replication by maintaining high levels of adherence to antiretroviral therapy (ART). These efforts are part of the global response toward achieving "the third 90" in the 90-90-90 targets, an initiative to end the AIDS epidemic as a public health threat by 2030.
Briefly, the Joint United Nations Program on HIV and AIDS' 90-90-90 campaign aims to have 90% of people living with HIV to know their status, have 90% of people living with HIV who know their status start or maintain their treatment, and have 90% of people on treatment to be virally suppressed by 2020. Achieving viral suppression protects people living with HIV from acquired immune deficiency syndrome (AIDS)-related illnesses and lowers the risk of transmission to others. In 2016, however, only 44% of the people living with HIV who were on treatment had viral suppression globally [1].
To improve viral suppression, adolescents should be more cared for. This is because, globally, up to 150 adolescents die every day due to AIDS-related illnesses. In 2016, 91% of adolescent deaths worldwide were reported in sub-Saharan Africa, and the rate of AIDS-related deaths among this age group have not reduced [2, 3]. In addition, children and adolescents from different parts of the world had worse viral suppression outcomes than adults [4,5,6].
So far, barriers to and factors that promote viral suppression have been identified. Commonly cited barriers among adolescents include pill burden, medication taste, secrecy/stigma, drug toxicity and resistance, clinic-related factors, being sick, missed appointments, loss of a mother, strong religious beliefs, and poor knowledge about HIV, among others [7,8,9,10,11]. On the contrary, promoting factors have also been identified: use of community interventions [12], setting up adolescent-specific care spaces [13], and maintaining adherence > 95% [4]. However, most of the recommendations from these studies require financial support, which is scarcely available in low-income countries, and interventions may collapse once support is withdrawn. Therefore, more cost-effective interventions are needed.
Uganda ranked among the top 20 high burden countries contributing 5% of AIDS-related deaths among adolescents in 2014 [14]. The deaths are attributed to the late diagnosis and poor access to treatment with most perinatally infected children starting treatment later in life. Like other countries, Uganda committed to achieve the 90-90-90 targets by 2020; however, the country is still short on achieving the ambitious target with the third 90 scoring lowest along the cascade. In 2017, about 73% of the people living with HIV knew their HIV status; 67% were enrolled on ART; while almost 60% had achieved viral suppression [15]. To identify people living with HIV who are likely to fail on treatment (including adolescents) and monitor their quality of life, the Uganda Ministry of Health recommended viral load testing for all people who are receiving ART for at least 6 months or more. Children under 15 years continued to have much lower rates of viral suppression compared to adults [4, 15, 16].
Although adherence to ART is important for viral suppression, some children and adolescents are sufficiently adherent but their viral loads remain high, while others have suppressed viral loads despite poor adherence [17,18,19]. In Uganda, information on viral suppression among adolescents aged 10–19 years is limited. This is because the age disaggregation groups adolescents 10–15 years with children aged 0–14 years while those 16–19 years are considered among adults. It is therefore difficult to ascertain outcomes of adolescents on ART with the current data [15]. This study aimed to identify the factors associated with viral suppression among adolescents aged 10–19 years and characteristics of sufficiently adherent adolescents who fail to achieve viral suppression.
We conducted a cross-sectional study of HIV-positive adolescents receiving ART at Jinja Regional Referral Hospital (RRH), Uganda, from August to September 2016. To assess the individual and socio-demographic factors associated with viral suppression, we conducted face-to-face interviews using semi-structured questionnaires. Additionally, using a data extraction tool developed in Excel, we collected secondary (retrospective) data on ART-related factors such as duration of ART, CD4 count at initiation, adherence status (past 12 months), treatment failure, World Health Organization clinical staging, and nutritional status. We developed the data extraction tool based on screening key indicators monitored nationally using the individual medical card, viral load register, and electronic database.
We conducted this study at Jinja Regional Referral Hospital—a public referral center for complicated cases and specialized laboratory tests in the East Central region, Uganda. The hospital runs a separate wing for children, pediatrics, and adolescents. Adolescents living with HIV aged 18 years are transferred to the adult HIV clinic located 3 m away from the main hospital. Health workers regularly offer pre-regimen and adherence counseling, discuss appointment schedules, probe for other illnesses, and explain viral load outcomes to adolescents and caretakers at every clinical review appointment.
In June 2016, over 600 adolescents at Jinja RRH were given ART. Of them, 397 adolescents were actively attending their scheduled clinical review appointments; only 238 adolescents had access to a viral load test. Mass viral load monitoring using viral load testing begun in 2015 at this hospital. Using the viral load national testing algorithm, children and adolescents below 20 years are required to have a viral load done every 6 months [20]. The study period coincided with school holidays where adolescents could be easily seen, and appointments had been scheduled to coincide with the adolescent clinic day.
Study participants
HIV-positive adolescents (10–19 years) who received ART for at least 6 or more months at Jinja RRH and had viral load results in their medical records were included in this study. The viral load results considered in this study were those documented between 1 January 2015 to 31 July 2016. We obtained assent from the caretakers of adolescents aged 10–14 years and consent from adolescents aged 15–19 years. We selected 238 adolescents to participate in the study from the total eligible adolescents enrolled on ART using Open Medical Records System (OpenMRS), an electronic database.
Sample size calculation
Given the total population of 600 HIV-positive adolescents, 397 adolescents (10–19 years) were receiving ART at Jinja RRH's Nalufenya children's and adult HIV clinic. Of the 397 adolescents, 238 had accessed a viral load test in the past 2 years.
Using Sloven's formula
$$ n=\frac{N}{\left(1+{\mathrm{Ne}}^2\right)} $$
The total number of adolescents required for our study was approximately 209. We therefore decided to study all the adolescents, whose files contained viral load results, representing 60% of the study target population.
The primary outcome was viral suppression defined as viral load < 1000 copies/mL [21]. We collected individual viral load results and suppression status from the viral load result forms kept in the medical files. We used the most recent viral load results for adolescents (results documented between January 2015 and July 2016). For other treatment-related factors, we used data from individual ART cards and electronic medical records (OpenMRS) software. For example, we extracted data on adherence status from the ART cards, which tracked the adherence of each adolescent for the past 12 months before their latest viral load test. For adolescents who were on ART for less than 12 months, we calculated the adherence rate based on the total number of months spent on treatment (at least 6 months). Medication adherence in this study was referred to as the degree to which adolescents took their prescribed medicines (includes dosage, time of the day) using pill counts as a primary measure. Adolescents' adherence was described as good, fair, or poor based on the following; G (good) > 95%, F (fair) if between 85% and 94%, and P (poor) < 85% [20]. According to the ART card summary guide (at the bottom of the generic ART card), the level of adherence was estimated using the number of missed doses per month based on a twice daily regimen. For example, if an adolescent missed 3 or less doses in a month, adherence was documented as good; if 4–8 doses were missed, it would be recorded as fair adherence, and 9 or more missed doses implied poor adherence. Other variables from ART cards included age, sex, regimen type [20], CD4 count, appointment keeping, history of treatment failure, nutritional status, clinical stage, opportunistic infection, treatment start date, and counseling after treatment [4, 20].
Sociodemographic variables
We collected information about age and sex using face-to-face questionnaires to cross examine information collected from the medical cards. In addition, we asked questions about level of education, religion, being prayed for, believing in God or Allah healing their illness, having both parents alive, perceived family support [9, 22], level of satisfaction of health services offered, convenience of scheduled appointments, and staying with a family member who is HIV positive [23, 24].
We recruited eight field assistants (particularly those working within the HIV clinic) to collect data and trained them on how to fill the data tools, resolve issues related to obtaining consent, and perform procedures according to ethical standards. The data assistants collected information from the patient cards into a data extraction tool. They then conducted face-to-face interviews and filled out questionnaires as they interacted with adolescents and their care givers during scheduled clinical review appointments. Whereas the field assistants were conversant with the local dialect, they had no formal connections with the respective adolescents, which minimized any bias in responses to the subjects. In addition, while extensive training was offered to the field assistants to fully acquaint them with the data collection procedures, they were ignorant about the study outcome. The counselors further contacted caregivers of adolescents whose appointments had passed or were scheduled on a future date to come with their children to the clinic using phone calls. We validated the data abstraction tool and pretested semi-structured questionnaires before the study.
We coded and entered quantitative data from study tools into a predesigned Excel spreadsheet and performed statistical analyses using STATA version 13.1 (College Station, TX, USA). We also performed descriptive analyses and summarized data on continuous variables such as age, viral load, and mean and their corresponding standard deviations. Then, we summarized the categorical variables including sex, viral suppression, and religion, among others, as proportions. For binary outcomes, we obtained odds ratio for each independent variable, adjusting for suppression. We performed multiple logistic regression analyses and chi-square tests to determine any possible association at 95% confidence level. Since CD4 count at initiation had missing baseline values, we performed multiple imputations (10 imputations) using variables associated with the outcome variable. While we did not have a command for variance inflation factor in stata in the case of multiple imputation, we checked variance inflation factor without multiple imputation and did not find multicollinearity. We included significant (P ≤ 0.05) variables and other variables that were not considered significant but were reported to influence viral outcomes in the final model [4, 9, 13]. Finally, we included all variables with P value < 0.05 from our multivariable analysis as independent predictors of viral suppression in this population.
We obtained a waiver from the International Health Sciences University's Degree Research and Ethics Committee. The reason for the waiver was that the study had no potential to cause harm whether physical or psychological to the participants. The decision was based on the National Guidelines for Research involving humans as research participants developed by the Uganda National Council for Science and Technology. We also obtained permission from the hospital administration to conduct the study. We sought an informed consent for adolescents aged 15 years and above, or informed assent from parents/caretakers of adolescents aged below 15 years. To ensure confidentiality, we assigned study participants with unique identifiers.
Table 1 shows the basic characteristics of adolescents. Overall, 200 adolescents participated in the study. The mean age was 13.2 (standard deviation: 2.4) years, and 52% of them were male adolescents. Of the 200 adolescents, 131 (65.5%) achieved viral suppression. Over 75% of adolescents attended primary level education and received support from their family members such as accommodation, feeding, and education. The HIV status of nearly 80% of the adolescents was known among family members. About 65% of adolescents lived with at least one family member of the same HIV status in one household. Adolescents belonged to either one of the following religious denominations: Catholic, Anglican, Muslim, and others. In addition, 75% of the adolescents attended clinic services in the same space as adults and believed in God healing their illness.
Table 1 Socio-demographic and client-related characteristics of adolescents (n = 200)
Table 2 presents the univariate (regimen-and treatment-related) characteristics of adolescents. Over 50% had been on treatment for 5 or more years while over 80% had good adherence above 95%, had never experienced treatment failure and belonged to clinical stage 1. Overall, about 90% adolescents had a normal nutritional status.
Table 2 Univariate treatment characteristics of adolescents
Table 3 shows the regimen and health-related characteristics of 200 adolescents with or without viral suppression. Both in the suppression and non-suppression groups, more than 50% of adolescents spent at least 5 or more years on ART and were started on the same regimen (AZT + 3TC + NVP). Over 70% of adolescents had good adherence (> 95%) and no history of treatment failure. About 85% of adolescents had WHO clinical stage I, while more than 90% had normal nutritional status. However, CD4 counts of the adolescents at the start of ART varied largely between the two groups. For the suppression group, the proportion of adolescents who achieved viral suppression increased with increasing CD4 counts at ART initiation [16% (CD4 < 200 cells), 23% (CD4 200–499 cells), and 34% (CD4 > 500 cells], while more adolescents in the non-suppression group tended to have lower CD4 counts at ART initiation [30% (CD4 < 200 cells), 22 (CD4; 200–499 cells), and 15% (CD4 > 500 cells)].
Table 3 Treatment- and health-related characteristics of adolescents and viral suppression
Table 4 shows the factors associated with viral suppression among adolescents when all variables were regressed. At multivariate level, an adherence rate of > 95% (AOR 2.73, 95% CI 1.09 to 6.82) was associated with viral suppression. Adolescents were less likely to achieve viral suppression if they belonged to the following categories: those who had a history of treatment failure (AOR 0.26, 95% CI 0.09 to 0.77) and had normal nutritional status (AOR 0.11, 95% CI 0.02 to 0.67). In addition, being Anglican (AOR 0.19, 95% CI 0.06 to 0.62) or Muslim (AOR 0.17, 95% CI 0.05 to 0.55) and having ever been prayed for (AOR = 0.38, P < 0.05) were risk factors of viral suppression.
Table 4 Factors associated with viral suppression among adolescents on ART
Additional file 1: Table S1 shows detailed individual and treatment characteristics of 49 adolescents who were adherent (> 95%) but failed to achieve viral suppression. Of the 49 adolescents, 7 (14.3%) had experienced treatment interruptions (missed medication) at least once in 12 months. The remaining 42 adolescents had medication adherence of 100% across all the 12 months. In this group, male adolescents (63%) constituted the biggest percentage. Likewise, all the 4 (8%) adolescents who were malnourished were men and belonged to clinical stage III. A total of 34 (69%) adolescents had their CD4 counts at ART initiation documented; about 18 adolescents had CD4 < 200 cells/mL, 10 had 200–499 cells/mL, and only 6 had > 500 cells/mL. About 75% of 49 adolescents had been on treatment for 5 or more years.
Out of 200 adolescents, 69 (34.5%) did not achieve viral suppression. Of 69, 71% had a good adherence (above 95%). Altogether, adolescents in the study had a low viral suppression rate than recommended in the 90-90-90 targets despite having good adherence. It is possible that the number of HIV-positive people with good adherence who have poor viral outcomes is increasing, especially in developing countries where treatment options are not regularly updated. This is one of the few studies which reports sufficiently adherent adolescents with no history of treatment interruptions failing to achieve viral suppression. Some of the characteristics of these adolescents include being on a regimen containing nevirapine at initiation and having a history of treatment failure.
In this study, the viral failure rate among adolescents on ART was 34.5%. This rate is higher than that reported for low- and middle-income countries (29%), and in South Africa (19%) [5, 10]. However, a study conducted in Zimbabwe and another recently concluded study in Uganda reported slightly higher rate of viral failure (43%) among children and adolescents on ART [25, 26]. The study results which show increasing rates of HIV drug resistance (HIVDR) are in line with recent findings by WHO showing an increase in levels of HIVDR even in the context of well-managed HIV treatment programs [27].
This study found that an adherence rate of > 95% was positively associated with viral suppression, and 65.5% of adolescents achieved viral suppression. However, even when adherence was more than 95%, 71% of 69 adolescents failed to achieve viral suppression. This failure might be explained as follows: first, from the Additional file 1: Table S1, regimen-specific differences were more pronounced in adolescent males, especially those enrolled on a treatment combination consisting of nevirapine (AZT + 3TC + NVP). From the Additional file 1: Table S1, 57% of the adolescents who experienced virologic failure were on a nevirapine combination. However, apart from the skin rash, no specific adverse events were reported among adolescents on this regimen. Nevirapine use, in the past, was known to be associated with increased risk of ART failure [28, 29]. In sub-Saharan Africa, however, treatment combination options remain largely unchanged for HIV-positive people, and transition to more robust regimens is costly.
Second, the time spent on ART should be considered. Longer exposure to ART (5 years and above) was a common characteristic among adolescents who failed to achieve viral suppression. This finding is consistent with that reported in studies conducted in Northwest Ethiopia [30] and Swaziland [31]. Because long-term ART is common in perinatally infected children transitioning into young adults, viral failure may be explained by the possible accumulation of drug resistance mutations over time (adherence to a failing regimen).
Third, only 14.3% of the 49 adolescents with good adherence who failed to achieve viral suppression had experienced treatment interruptions at least once in 12 months. Missed antiretroviral doses or interruptions in therapy may result in HIV drug resistance hence virologic failure [32]. Some studies have reported low adherence for people with strong spiritual beliefs who tend to delay or miss medication during fasting, after being baptized with holy water [10], or after receiving healing prayers [11]. Although treatment interruptions were indicated by a "fair" or "poor" adherence status in the medical files, there was no specific explanation about the nature of the interruptions and how long they lasted within the month. More so, information about interruptions was obtained via individual self-reports which are subject to bias. Future research should prioritize inventing a daily measure or tool for capturing treatment interruptions.
In this study, 15% of 131 HIV-positive adolescents with poor adherence (< 95%) achieved viral suppression. This finding is consistent with those reported in other studies [17,18,19]. The Uganda Ministry of Health Guidelines recommend intensive adherence counseling for 6 weeks for anyone with high viral loads (> 1000 copies) and a repeat viral load test conducted after the third intensive adherence counseling session. In our study, 168 (84%) adolescents had a repeat viral load result in their files overall. Of these, 61 (78%) of the adolescents had unsuppressed viral load after a repeat viral load. In this case, intensive adherence counseling would not be a priority for such adolescents; instead, efforts should be focused on understanding facilitators of viral suppression among this group. Furthermore, many countries have a limited number of health workers trained in adolescent health services [33]. Training of all health workers in adolescent health service delivery may address existing challenges and provide better understanding of the promoting factors and barriers to suppression among this age group.
Adolescents were less likely to achieve viral suppression if they had previously failed on treatment [4], were Muslim or Anglican, and had ever been prayed for. Contrary to our expectations, WHO clinical stage, duration of treatment, CD4 count [31], family support, and sharing clinic space were not associated with viral suppression. These findings were also contrary to those of other studies conducted in Zimbabwe [13], Tanzania [22], and Ethiopia [28], which indicated that family support and establishment of adolescent-specific clinic spaces [13] improved viral outcomes.
Adolescents have been reported to have better viral outcomes when they had a person living with HIV in the same household or had been switched to a second line treatment [25, 34]. In this study, although the majority (66.5%) had members in their family living with a positive HIV status, there was no significant relationship with viral suppression. In addition, few adolescents (14%) were switched to the second line treatment despite their viral outcome status. Majority (61%) who had not achieved suppression had not been switched, and no markers were in place to prioritize for a change of regimen.
In this study, age and education (school attendance) were not significantly associated with viral load suppression. However, adolescents aged 12 and 16 years were more likely to have an unsuppressed viral load in both the initial and subsequent viral load despite consistently high levels of adherence and family support. This is contrary to the findings from other studies conducted in Uganda where adolescents who attended school were found with lower levels of adherence to treatment and older adolescents had high tendencies of missed scheduled appointments [16, 35].
This study had several limitations. Although our study was cross sectional in nature, we collected viral load results from secondary data. It is possible that the viral suppression status of some adolescents may have changed at the time of the study. We also sampled only those adolescents who had a viral load test result documented in their medical records between January 2015 and July 2016. Other limitations include limited knowledge and data about drug-resistance testing, missing data, the use of sub-optimal treatment regimens, limited sample size, and non-ascertainment of prior treatment interruptions (such as history of adherence since start of ART). For nutrition, we used the most current nutritional status of adolescents documented in the individual ART cards and could not rule out that nutritional status might have been different at time when the viral load test was taken. We restricted the sample to adolescents from the East Central region of Uganda; hence, our results could be different from those conducted in other settings. The need for caretaker's consent was a major obstacle especially when young adolescents reported to the clinic alone.
More than 70% of all adolescents who did not achieve viral suppression were sufficiently adherent (with adherence > 95%). Adolescents who had unsuppressed viral loads in their initial viral load were more likely to experience virologic failure upon a repeat viral load regardless of their adherence level or change of regimen. The study also shows that strong religious beliefs exist among adolescents. Healthcare provider training in psychological counseling, regular and strict monitoring of adolescent outcomes should be prioritized to facilitate early identification and management of drug resistance through timely switching of treatment regimens to more robust combinations.
3TC:
ABC:
AIDS:
AOR:
Adjusted odds ratio
Antiretroviral therapy
EFV:
HIV:
HIVDR:
HIV drug resistance
NVP:
Nevirapine
OpenMRS:
Open Medical Records System
RRH:
Regional Referral Hospital
Joint United Nations Programme on HIV/AIDS (UNAIDS). Ending AIDS; Progress towards 90-90-90 targets. Geneva: UNAIDS; 2017.
World Health Organization. A global research for adolescents living with HIV. Geneva: WHO; 2017.
UNICEF. Analysis of UNAIDS 2016 estimates.
Bulage L, Ssewanyana I, Nankabirwa V, Nsubuga F, Kihembo C, Pande G, et al. Factors associated with virological non-suppression among HIV-positive patients on antiretroviral therapy in Uganda August 2014–July 2015. BMC Infect Dis. 2017;17:326. https://doi.org/10.1186/s12879-017-2428-3.
Boerma RS, Boender TS, Bussink AP, Calis JC, Bertagnolio S, Rinke de Wit TR, et al. Sub-optimal viral suppression rates among HIV-infected children in low-and middle- income countries: a meta-analysis. Clin Infect Dis. 2016; doi.org/10.1093/cid/ciw645.
World Health Organization. Prevent HIV, test and treat all. Geneva: World Health Organization; 2016.
Galea JT, Wong M, Muñoz M, Valle E, Leon SR, Perez DD, et al. Barriers and facilitators to antiretroviral therapy adherence among Peruvian adolescents living with HIV: a qualitative study. PLoS One. 2018;13:e0192791. https://doi.org/10.1371/journal.pone.0192791.
Shubber Z, Mills EJ, Nachega JB, Vreeman R, Freitas M, Bock P, et al. Patient-reported barriers to adherence to antiretroviral therapy: a systematic review and meta-analysis. PLoS Med. 2016;13:e1002183. https://doi.org/10.1371/journal.pmed.1002183.
Okawa S, Kabaghe SM, Mwiya M, Kikuchi K, Jimba M, Kankasa C, et al. Psychological well-being and adherence to antiretroviral therapy among adolescents living with HIV in Zambia. AIDS Care. 2018;30:634–42. https://doi.org/10.1080/09540121.2018.1425364.
McKinney O, Modeste NN, Lee JW, Gleason PC, Maynard-Tucker G. Determinants of antiretroviral therapy adherence among women in Southern Malawi: Healthcare providers' perspectives. AIDS Res Treat. 2014; doi.org/10.1155/2014/489370.
Bereznicki LR, Peterson GM, Bimirew MA, Kassie DM. Barriers and facilitators of adherence to antiretroviral drug therapy and retention in care among adult HIV-positive patients: a qualitative study from Ethiopia. PLoS One. 2014;9:e97353. https://doi.org/10.1371/journal.pone.0097353.
Zanoni BC, Sibaya T, Cairns C, Lammert S, Haberer JE. Higher retention and viral suppression with adolescent-focused HIV clinic in South Africa. PLoS One. 2017;12:e0190260. https://doi.org/10.1371/journal.pone.0190260.
Ferrand RA, Simms V, Dauya E, Bandason T, Mchugh G, Mujuru H, et al. The effect of community-based support for caregivers on the risk of virological failure in children and adolescents with HIV in Harare, Zimbabwe (ZENITH): an open-label, randomized controlled trial. Lancet Child Adolesc Health. 2017;1:175–83. https://doi.org/10.1016/S2352-4642(17)30051-2.
Joint United Nations Programme on HIV/AIDS (UNAIDS). The Gap report: beginning of the end of the AIDS epidemic. Geneva: UNAIDS; 2014.
Ministry of health. Uganda HIV/AIDS country progress report July 2016-June 20. Reaching men, girls and young women to reduce new HIV infections. 2017.
Kamya M, Mayanja-kizza H, Kambugu A, Bakeera-kitaka S, Semitala F, Mwebaze-songa P, et al. Predictors of long-term viral failure among Ugandan children and adults treated with antiretroviral therapy. J Acquir Immune Defic Syndr. 2007;46:187–93. https://doi.org/10.1097/QAI.0b013e31814278c0.
Duarte HA, Harris DR, Tassiopolous K, Leister E, Fabiana S, Ferreira FF, et al. Relationship between viral load and behavioral measures of adherence to antiretroviral therapy in children living with human immunodeficiency virus in Latin America. Braz J Infect Dis. 2015;19:263–71. https://doi.org/10.1016/j.bjid.2015.01.004.
Glass TR, Rotger M, Telenti A, Decosterd L, Csajka C, Bucher HC, et al. Determinants of sustained viral suppression in HIV-infected patients with self-reported poor adherence to antiretroviral therapy. PLoS One. 2012;7:e29186. https://doi.org/10.1371/journal.pone.0029186.
Williams PL, Storm D, Montepiedra G, Nichols S, Kammerer B, Sirois PA, et al. Predictors of adherence to antiretroviral medications in children and adolescents with HIV infection. J Pediatr. 2006;118:e1745–57. https://doi.org/10.1542/peds.2006-0493.
Ministry of Health, Uganda. Consolidated guidelines for prevention and treatment of HIV in Uganda. 2016.
World Health Organization. Technical and operational considerations for implementing HIV viral load testing; Interim update. Geneva: WHO; 2014. https://www.who.int/hiv/pub/arv/viral-load-testing-technical-update/en.
Nlend N, Esther A, Annie Nga M, Suzie Tetang N, Joseph F. Predictors of virologic failure on first-line antiretroviral therapy among children in a referral pediatric center in Cameroon. Pediatr Infect Dis J. 2017;362. https://doi.org/10.1097/INF.0000000000001672.
Buma D, Bakari M, Fawzi W, Mugusi F. The influence of HIV-status disclosure on adherence, immunological and virological outcomes among HIV-infected patients started on antiretroviral therapy in Dar-es-Salaam, Tanzania. J HIV AIDS. 2015;1. doi. org/https://doi.org/10.16966/2380-5536.111.
Agwu AL, Fairlie L. Antiretroviral treatment, management challenges and outcomes in perinatally HIV-infected adolescents. Int AIDS Soc. 2013;16:18579. doi.org/10.7448/IAS.16.1.18579.
Sithole Z, Mbizvo E, Chonzi P, Mungati M, Juru TP, Shambira G, et al. Virological failure among adolescents on ART, Harare City, 2017- a case-control study. BMC Infect Dis. 2018;18:469. https://doi.org/10.1186/s12879-018-3372-6.
Nasuuna E, Kigozi J, Babirye L, Muganzi A, Sewankambo NK, Nakanjako D. Low HIV viral suppression rates following the intensive adherence counseling (IAC) program for children and adolescents with viral failure in public health facilities in Uganda. BMC Public Health. 2018:1048.ty 0 doi.org/10.1186/s12889-018-5964-x.
World Health Organization. HIV drug resistance report 2017. Geneva: WHO; 2017. http://apps.who.int/iris/bitstream/handle/10665/128121/9789241507578_eng.pdf;jsessionid=1FB8247E8E38F2A921196BE73B49DAEDsequence=1.
Geary RS, Gómez-Olivé FX, Kahn K, Tollman S, Norris SA. Barriers to and facilitators of the provision of a youth-friendly health services programme in rural South Africa. BMC Health Serv Res. 2014;14:259. https://doi.org/10.1186/1472-6963-14-259.
Pillay P, Ford N, Shubber Z, Ferrand RA. Outcomes for efavirenz versus nevirapine-containing regimens for treatment of HIV-1 infection: a systematic review and meta-analysis. PLoS One. 2013;8:e68995. 238943930.
Bayu B, Tariku A, Bulti AB, Habitu YA, Derso T, Teshome DF. Determinants of virological failure among patients on highly active antiretroviral therapy in University of Gondar Referral Hospital, Northwest Ethiopia: a case–control study. J HIV AIDS. 2017;9:153–9. https://doi.org/10.2147/HIV.S139516.
Jobanputra K, Parker LA, Azih C, Okello V, Maphalala G, Kershberger B, et al. Factors associated with virological failure and suppression after enhanced adherence counselling, in children, adolescents and adults on antiretroviral therapy for HIV in Swaziland. PLoS One 2015;10:e0116144. https://doi.org/10.1371/journal.pone.0116144.
Nachega JB, Marconi VC, van Zyl GU, Gardner EM, Preiser W, Hong SY, et al. HIV treatment adherence, drug resistance, virologic failure: evolving concepts. Infect Disord Drug Targets. 2011;11(2):167–74.
Lowenthal ED, Ellenberg JH, Machine E, Sagdeo A, Boititswe S, Steenhoff AP, et al. Association between efavirenz-based compared with nevirapine-based antiretroviral regimens and virological failure in HIV-infected children. JAMA. 2013;309:1803–9. https://doi.org/10.1001/jama.2013.3710.
Kyaw NT, Harries AD, Kumar AM, Oo MM, Kyaw KW, Win T, et al. High rate of virological failure and low rate of switching to second-line treatment among adolescents and adults living with HIV on first-line ART in Myanmar, 2005-2015. PLoS One. 2017;12(2):e0171780. https://doi.org/10.1371/journal.pone.0171780.
MacCarthy S, Saya U, Samba C, Birungi J, Okoboi S, Linnemayr S. "How am I going to live?": exploring barriers to ART adherence among adolescents and young adults living with HIV in Uganda. BMC public health. 2018;18(1):1158. https://doi.org/10.1186/s12889-018-6048-7.
We would like to acknowledge hospital staff and study participants in Uganda. We also thank Krishna Poudel and Kalpana Poudel of the University of Massachusetts Amherst, for their critical comments on this article.
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Public Health and Management, Institute of Health, International Health Sciences University, Kampala, Uganda
Julian Natukunda
& Peter Kirabira
Department of Community and Global Health, Graduate School of Medicine, The University of Tokyo, Tokyo, Japan
, Ken Ing Cherng Ong
, Akira Shibanuma
& Masamine Jimba
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NJ made substantial contributions to conception and design, acquisition of data, analysis and interpretation of data. KP made substantial contribution to the design of the data collection tools and interpretation of results. He was also involved in writing some sections of the manuscript. MJ has been involved in revising the manuscript critically for important intellectual content and has given final approval of the version to be published. AS and KICO have made substantial contribution to data analysis, interpretation, and writing of results. All authors read and approved the final manuscript.
Correspondence to Julian Natukunda.
We obtained a waiver from the International Health Sciences University's Degree Research and Ethics Committee. The reason for the waiver was that the study had no potential to cause harm whether physical or psychological to the participants. The decision was based on the National Guidelines for Research involving humans as research participants developed by the Uganda National Council for Science and Technology. We also obtained permission from the hospital administration to conduct the study.
All participants (adolescents or caretakers) involved in the study gave their consent to use the information collected for publication purposes.
Table S1. Key characteristics of adolescents whose adherence level was good (> 95%) but failed to achieve viral suppression (n = 49). (DOCX 13 kb)
Natukunda, J., Kirabira, P., Ong, K.I.C. et al. Virologic failure in HIV-positive adolescents with perfect adherence in Uganda: a cross-sectional study. Trop Med Health 47, 8 (2019) doi:10.1186/s41182-019-0135-z
Viral suppression
Virologic failure | CommonCrawl |
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\renewcommand{\(}{\left(} \renewcommand{\)}{\right)} \renewcommand{\[}{\left[} \renewcommand{\right]}{\right]}
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\begin{document} \title[Concentration on minimal submanifolds for a Yamabe type problem]{Concentration on minimal submanifolds for a Yamabe type problem}
\author{Shengbing Deng} \address[Shengbing Deng]{School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China,
and Departamento de Matem\'atica, Pontificia Universidad Catolica de Chile, Santiago, Chile} \email{[email protected]} \author{Monica Musso} \address[Monica Musso]{Departamento de Matem\'aticas, Pontificia Universidad Cat\'olica de Chile, Santiago, Chile} \email{[email protected]} \author{Angela Pistoia} \address[Angela Pistoia] {Dipartimento SBAI, Universt\`{a} di Roma ``La Sapienza", via Antonio Scarpa 16, 00161 Roma, Italy} \email{[email protected]}
\begin{abstract} We construct solutions to a Yamabe type problem on a Riemannian manifold $M$ without boundary and of dimension greater than $2$, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a non degenerate minimal submanifold of $M$, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of P.D.E.'s on the submanifold with a singular term of attractive or repulsive type is established.
\end{abstract}
\subjclass[2000]{35B10, 35B33, 35J08, 58J05}
\date{\today}
\keywords{supercritical Yamabe type problem, concentration along minimal submanifolds, P.D.E.'s with attractive or repulsive type singularity}\maketitle
\footnotetext{The first author was supported by Fondecyt grant 1130360 and Fondo Basal CMM. The second and the third authors have been partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\'a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). }
\section{Introduction and statement of main results}\label{intro}
Let $(M,g)$ be a compact Riemannian manifold of dimension $m \geq 3$ without boundary. This paper deals with the semilinear elliptic problem
\begin{equation}\label{p} -\Delta _gu+h u=u^{q-1},\ u>0,\ \hbox{in}\ (M,g),\end{equation} where the potential $h\in C^2(M)$ is such that $-\Delta_g+h$ is coercive and the exponent $q>2$.\\
Existence of non-trivial solutions to problem \eqref{p} is strictly related to the position of $q$ with respect to the critical Sobolev exponent $2^*_m:={2m\over m-2}$.
Indeed, in the subcritical case, i.e. $q<2^*_m$, the Sobolev embedding $H^1_g(M)\hookrightarrow L^q_g(M)$ is compact for any $q\in(2,2^*_m)$
and so
\begin{equation}\label{inf}\inf\limits_{u\in H^1_g(M)\atop u\not=0}{\int\limits_M\(|\nabla_g u|^2+h u^2\)d\sigma_g\over \(\int\limits_M
| u|^qd\sigma_g\)^{2/q}}\end{equation} is achieved
and problem \eqref{p} has a non-trivial solution.\\
The critical case, i.e. $q=2^*_m$, has important links with the well known Yamabe problem \cite{y}, namely
find a metric $\widetilde g$ in the conformal class $[g]=\{\phi g:\, \phi \in C^\infty(M),\, \phi>0 \}$
with constant scalar curvature $\kappa.$ This is equivalent to set $\widetilde g= u^{4/\(m-2\)} g$ and to find a solution $u$ to the Yamabe problem \begin{equation}\label{yam} -\Delta _gu+{m-2\over 4(m-1)}S_g u= \kappa u^{m+2\over m-2},\ u>0,\ \hbox{in}\ (M,g),\end{equation} where $S_g$ is the scalar curvature of $(M,g). $
The Yamabe problem on the round sphere $(\mathbb S^{m},g_0)$, equipped with the standard metric $g_0$, plays a crucial role in solving problem \eqref{yam}.
Since the scalar curvature of the round sphere is $m(m-1)$ equation \eqref{yam} reduces to
$$-\Delta _{g_0}u+ {m(m-2)\over 4} u=u^{m+2\over m-2},\ u>0,\ \hbox{in}\ (\mathbb S^m,g_0), $$
which is equivalent (via the stereographic projection) to the problem in the Euclidean space
\begin{equation}\label{rm}-\Delta w=w^{m+2\over m-2},\ w>0,\ \hbox{in}\ \mathbb R^m.\end{equation} Problem \eqref{rm} has infinitely many solutions (see \cite{cgs}),
\begin{equation}\label{bbs} w_{\delta,y}(x):= \alpha_m\({\delta \over \delta^2+|x-y|^2}\)^{m-2\over2},\ x,y\in\mathbb R^m,\ \delta>0,\end{equation} where $\alpha_m:=\(m(m-2)\)^{m-2\over4}.$ In the general case, Aubin in \cite{a} proved that if
$$ \mu_g(M,h):=\inf\limits_{u\in H^1_g(M)\atop u\not=0}\displaystyle{\int\limits_M\(|\nabla_g u|^2+h u^2\)d\sigma_g\over \(\int\limits_M | u|^{2m\over m-2}d\sigma_g\)^{m-2\over m}}$$ is such that \begin{equation}\label{aubin}\mu_g(M,h)<\mu_{g_0}\(\mathbb S^m,{m(m-2)\over 4}\)\end{equation} then $\mu_g(M,h)$ is achieved and so the corresponding critical problem \eqref{p} with $q={2^*_m}$ has a non-trivial solution. The validity of \eqref{aubin} turns out to be strictly related to the position of the potential $h$ with respect to the geometric potential \begin{equation}\label{geopot}\omega(\xi):={m-2\over 4(m-1)}S_g(\xi),\ \xi\in M.\end{equation} Indeed, if \begin{equation}\label{sotto}h(\xi)<\omega(\xi)\ \hbox{for any}\ \xi\in M,\end{equation} it is not difficult to check that condition \eqref{aubin} holds. In the case of the Yamabe problem, i.e. $h\equiv \omega$, condition \eqref{aubin} is also true, but the proof is a delicate issue. It was proved by Trudinger \cite{t} when $\mu_g(M,\omega) \leqslant 0$, by Aubin \cite{a} when $\mu_g(M,\omega) > 0$ and $(M,g)$ is not locally conformally flat and $m\ge6$ and by Schoen \cite{schoen} when $\mu_g (M,\omega)> 0$ and either $(M,g)$ is locally conformally flat or $3\leqslant m\leqslant 5$.\\
We can summarize known results just saying that problem \eqref{p} has a non-trivial solution if either $q<2^*_m$ and no extra assumptions on $h$, or $q=2^*_m$ and $h$ has to satisfy $h\leqslant \omega$ on $M.$ Therefore, it is natural to ask what happens when $$ h(\xi)>\omega(\xi)\ \hbox{for some}\ \xi\in M\ \hbox{or}\ \hbox{$p$ is supercritical, i.e.}\ p>2^*_m.$$\\
A first partial answer was given by Micheletti, Pistoia and V\'etois \cite{mpv} in a perturbative setting. They consider the almost critical problem \begin{equation}\label{ac} -\Delta _gu+h u=u^{q_\epsilon-1},\ u>0,\ \hbox{in}\ (M,g),\ \hbox{with}\ q_\epsilon:=2^*_m\pm\epsilon\end{equation} where $\epsilon$ is a positive small parameter. If $q_\epsilon =2^*_m-\epsilon$ problem \eqref{ac} is said to be slightly subcritical,
while if $q_\epsilon =2^*_m+\epsilon$ it is said to be slightly supercritical. They proved the following result.
\begin{theorem}\label{mpv-teo} [Theorem 1.1, Theorem 1.2, \cite{mpv}] Let $m\ge6$ and $\xi_0\in M$ be a non degenerate critical point of $h-\omega$ (see \eqref{geopot}). \begin{itemize} \item[(i)] If $h(\xi_0)>\omega(\xi_0)$ then there exists $\epsilon_0>0$ such that for any $\epsilon\in(0,\epsilon_0)$ the slightly subcritical problem \eqref{ac} with $q_\epsilon={2^*_m}-\epsilon$ has a solution $u_\epsilon$ such that
$$|\nabla u_\epsilon|^2\rightharpoonup c_m\delta_{\xi_0}\ \hbox{as}\ \epsilon\to0.$$ \item[(ii)] If $h(\xi_0)<\omega(\xi_0)$ then there exists $\epsilon_0>0$ such that for any $\epsilon\in(0,\epsilon_0)$ the slightly supercritical problem \eqref{ac} with $q_\epsilon={2^*_m}+\epsilon$ has a solution $u_\epsilon$ such that
$$|\nabla u_\epsilon|^2\rightharpoonup c_m\delta_{\xi_0}\ \hbox{as}\ \epsilon\to0.$$ \end{itemize} Here $\delta_{\xi_0}$ stands for the Dirac measure supported on $\xi_0$ and $c_m$ is an explicit positive constant depending only on $m.$\end{theorem} The profile of $u_\epsilon$ close to the concentration point $\xi_0$ is given by (see \eqref{bbs})
$$u_\epsilon(x)\approx \alpha_m \({\delta_\epsilon\over \delta_\epsilon^2+|x-\xi_0|^2}\)^{m-2\over2}$$ where the concentration parameter $\delta_\epsilon\sim d \sqrt\epsilon$, as $\epsilon \to 0$, and the positive number $d$ solves \begin{equation}\label{com1} a_m\underbrace{\[h(\xi_0)-\omega(\xi_0)\right]}_{>0}d -{b_m\over d }=0 \quad \hbox{in the slightly sub-critical case}\end{equation}
or \begin{equation}\label{com2}a_m\underbrace{\[h(\xi_0)-\omega S_g(\xi_0)\right]}_{<0}d +{b_m\over d }=0\quad \hbox{in the slightly super-critical case.}\end{equation}
Here $a_m$ and $b_m$ are positive constants which only depend on $m.$\\
This result suggests to explore what happens when the exponent $q$ is close to higher critical exponents. More precisely, for any integer $0\leqslant k\leqslant m-3$ we consider the $(k+1)$-st critical exponent $2^*_{m,k}={2(m-k)\over m-k-2} =2^*_{m-k,0}$, namely the critical exponent for the Sobolev embedding ${\mathrm H}^ 1_h(N)\hookrightarrow {\mathrm L}^{q}_h(N)$
where $(N,h)$ is a $(m-k)-$dimensional Riemannian manifold. In particular, $ 2^*_{m,0}={2 m\over m- 2}$ is the usual Sobolev critical exponent. We know by Theorem \ref{mpv-teo} that problem \eqref{p} when the exponent $q$ approaches the first critical Sobolev exponent $2^*_{m,0}$ has solutions which blow-up at a single point. A set consisting of a single point is a $0-$dimensional submanifold of $M$. We ask: {\it if $q$ approaches the $(k+1)$-st critical exponent $2^*_{m,k}$, do positive solutions blowing-up at $k-$dimensional submanifolds of $M$ exist?} \\ Recently, a partial answer has been given by D\'avila, Pistoia and Vaira in \cite{dpv} when $k=1$ and by Ghimenti, Micheletti and Pistoia in \cite{gmp} in a symmetric setting. Here we deal with the general case. Let us consider the almost $k$-critical problem \begin{equation}\label{pb} -\Delta _gu+h u=u^{q_\epsilon-1},\ u>0,\ \hbox{in}\ (M,g),\ \hbox{with}\ q_\epsilon:=2^*_{m,k}\pm\epsilon\end{equation} where $\epsilon$ is a positive small parameter. If $q=2^*_{m,k}-\epsilon$ problem \eqref{pb} is said to be slightly $k$-th subcritical,
while if $q=2^*_{m,k}+\epsilon$ it is said to be slightly $k$-th supercritical. \\
To state our result we need to introduce some geometric background. Let $K\subset M$ be a $k-$dimensional submanifold. Set $N:=m-k.$ Let us introduce Fermi coordinates in $\mathcal{M}$ near the submanifold $K.$\\ Let $((E_a)_{a=1,\cdots, k},(E_i)_{i=1,\cdots, N })$ be a local oriented and orthonormal frame field along $K$. At points $\xi$ of $K$, $T_\xi \mathcal{M}$ splits as $T_\xi K \oplus N_\xi K$, where $T _\xi K$ is the tangent bundle to $K$ with orthonormal basis $(E_a)_a$ and $N_\xi K$ is the normal bundle, which is spanned by the orthonormal basis $(E_j)_j$. We assume that the normal vectors $(E_i)_i$, $i = 1, \dots, N $, are parallel transported along $K$, namely \begin{equation}\label{eq:parall}
g\left( \nabla_{E_a}E_j\,,E_i\right)=0 \ \hbox{ at }\ \xi,
\ \hbox{for any}\ i,j = 1, \dots, N ,\ a = 1, \dots, k. \end{equation} Here $\nabla$ is the connection associated with the metric $g$. We denote by $\Gamma_a^b(\cdot)$ the 1-forms defined on the normal bundle of $K$ by \begin{equation}\label{eq:Gab}
\Gamma_{ai}^b := \Gamma_a^b(E_i)= g(\nabla_{E_a}E_b,E_i). \end{equation} The {\it minimal condition on $K$ } translates precisely into \begin{equation} \label{minimality} \sum\limits_{a=1}^k\Gamma_{ai}^a = 0\ \hbox{for any}\ i=1, \ldots N. \end{equation} In a neighborhood of $\xi$ in $K$, we consider normal geodesic coordinates \begin{equation}\label{defoff} f(y) : = \exp^K_\xi (\sum\limits_{a=1}^ky_a\, E_a ), \qquad y := (y_{1}, \ldots, y_{k}), \end{equation} where $\exp^K$ is the exponential map on $K$. In a neighborhood of $\xi$ in $\mathcal{M}$, we introduce \begin{equation}\label{eq:fc} \mathfrak F ( y, x) =\exp _{f(y)}( \sum\limits_{i=1}^{N}\,x_i\,E_i); \qquad \quad (y,x) = \left(( y_a)_a,(x_i)_i\right), \end{equation} where $\exp_{f(y) }$ is the exponential map at $f(y) $ in $\mathcal{M}$. It holds true that $f(y)=\mathfrak F(y,0)\in K.$ Let $\tilde g_{ab}$ be the coefficients of the induced metric on $K$ and let
$R_{\a\b\g\delta }$ be the components of the curvature tensor computed at the point $\xi$ of $K.$ {\it Non-degeneracy of $K$ } translates into the fact that the linear system \begin{equation}\label{jacobi} -\Delta_K \Phi_\ell+\sum\limits_{m=1}^N\sum\limits_{a,b=1}^k\(\tilde g^{ab}R_{mab\ell}-\Gamma^b_{am}\Gamma^a_{b\ell}\)\Phi_m=0,\ \ell=1,\dots,N, \end{equation} has only the trivial solution $\Phi=\(\Phi_1,\dots,\Phi_N\)\equiv 0.$\\
\textcolor{blue}{} The Levi-Civita connection $\nabla$ for $g$ induces a connection $\nabla^N$ on the normal bundle $N_\xi K.$ We denote by $$\mathcal R^N:=\sum\limits_{a=1}^k\(R(E_a,\cdot) E_a\)^N, $$ the curvature operator for this connection. The second fundamental form $$B:T_\xi K\times T_\xi K\to N_\xi K,\ B(X,Y)=\(\nabla _X Y\)^N$$ defines a symmetric operator $\mathcal B^N:=B^t\cdot B,$ in terms of the coefficients $\Gamma^b_a:=B(E_b,E_a),$ $$g(\mathcal B^NX,Y)=\sum\limits_{a,b=1}^k\Gamma^b_a(X)\Gamma^a_b(Y),\ X,Y\in T_\xi M.$$ We also use the Ricci tensor $$Ric(X,Y)=\sum\limits_{j=1}^N g\(R(X,E_j)E_j,Y\)+\sum\limits_{a=1}^k g\(R(X,E_a)E_a,Y\),\ X,Y\in T_\xi M.$$ We introduce the quadratic form $$\mathcal Q(X,Y):=\frac 13 Ric(X,Y)-\frac23 g\(\mathcal R^N X,Y\)+g(\mathcal B^NX,Y),\ X,Y\in N_\xi K.$$ Recall that $N=m-k$. We set \begin{equation}\label{hgk} \Omega(\xi):={3(m-k-2)\over4(m-k-1)}\sum\limits_{i=1}^{m-k}\mathcal Q(E_i,E_i),\ \xi\in K. \end{equation} The expression of $\Omega$ in Fermi coordinates is given by $$ \hat \Omega (y) = {3(N-2)\over 4(N-1)}\[\sum\limits_{i,j=1}^{N} \frac13 R_{jiji}(y)+ \sum\limits_{i=1}^{N}\sum\limits_{a,b=1}^k
\(\tilde g^{ab}R_{iaib}(y)+ \Gamma_{ai}^b(y) \Gamma_{bi}^a(y) \)\right] . $$ This function appears in our construction in \eqref{defgy}. Observe that if $k=0$ the function $\Omega$ is nothing but the geometric potential ${m-2\over 4(m-1)} S_g(\xi)$ introduced in \eqref{geopot}. Surprisingly enough, the function defined in \eqref{hgk} is not new in the literature, and it appears in a completely different context in \cite{mmpac} (Section 4). In fact, Mahmoudi, Mazzeo and Pacard in \cite{mmpac} deal with the existence of a family of Constant Mean Curvature submanifolds condensating to a fixed submanifold of a given Riemannian manifold. The limit submanifold has to be a closed non degenerate minimal submanifold. It would be interesting to further investigate the relation between our construction and the results in \cite{mmpac}. \\
Existence of solutions to the almost $k$-th critical problem \eqref{pb} is strictly related to the existence of solutions to the elliptic PDE's with a singularity of {\it attractive} type\begin{equation}\label{choiceofmusub} - \Delta_K d +a_N\underbrace{\left[h(\xi)-\Omega(\xi)\right]}_{>0\ \hbox{in}\ K}d -\frac{b_N}{d}=0,\ d>0\ \hbox{in} \ K \ \hbox{in the slightly $k$-th sub-critical case} \end{equation} or with a singularity of {\it repulsive} type \begin{equation}\label{choiceofmusup} - \Delta_Kd +a_N\underbrace{\left[h(\xi)-\Omega(\xi)\right]}_{<0\ \hbox{in}\ K}d +\frac{b_N}{d}=0,\ \mu>0\ \hbox{in} \ K \ \hbox{in the slightly $k$-th super-critical case.} \end{equation}
Here \begin{equation}\label{abn} a_N=\frac{4(N-1)}{(N-2)(N+2)}\ \hbox{and}\ b_N = {(N-2)^2(N-4) \over 2 (N+2) } .\end{equation}
This relation is new and unexpected. It seems to be the natural extension of conditions \eqref{com1} and \eqref{com2} to higher critical problems.
More precisely, our main result reads as follows.
\begin{theorem}\label{main} Let $K\subset M$ be a closed non-degenerate minimal $k-$dimensional submanifold. Assume $m-k\geqslant 7.$ \begin{itemize} \item[(i)] Assume \eqref{choiceofmusub} has a non-degenerate solution. Then there exists a sequence $\epsilon=\epsilon_ n\to0$ such that the slightly $k$-th sub-critical problem \eqref{pb} with $q_\epsilon={2^*_{m,k}}-\epsilon$ has a solution $u_\epsilon$ such that
$$|\nabla u_\epsilon|^2\rightharpoonup c_{m,k}\delta_{K}\ \hbox{as}\ \epsilon\to0.$$ \item [(ii)] Assume \eqref{choiceofmusup} has a non-degenerate solution. Then there exists a sequence $\epsilon=\epsilon_ n\to0$ such that the slightly $k$-th super-critical problem \eqref{pb} with $q_\epsilon={2^*_{m,k}}+\epsilon$ has a solution $u_\epsilon$ such that
$$|\nabla u_\epsilon|^2\rightharpoonup c_{m,k}\delta_{K}\ \hbox{as}\ \epsilon\to0.$$ \end{itemize} Here $\delta_{K}$ stands for the Dirac measure supported on $K$ and $c_{m,k}$ is an explicit positive constant depending only on $m$ and $k.$\end{theorem} The profile of $u_\epsilon$ close to the submanifold $K$ is given in Fermi coordinate by (see \eqref{bbs})
$$u_\epsilon(y,x)\approx \alpha_{m-k} \({\delta_\epsilon\over \delta_\epsilon^2+|x|^2}\)^{m-k-2\over2}$$ where the concentration parameter $\delta_\epsilon=\delta_\epsilon(y)$ satisfies $\delta_\epsilon\sim d \sqrt\epsilon$ and the positive function $d=d(y),$ defined on $K,$ solves either the attractive singular PDE \eqref{choiceofmusub} in the slightly sub-critical case or the repulsive singular PDE \eqref{choiceofmusup} in the slightly super-critical case. \\
It is important to point out that if $\min\limits_{\xi\in K}\[h(\xi)-\Omega(\xi)\right]>0$ then problem \eqref{choiceofmusub} has a non-degenerate solution as proved in Theorem \ref{th1}. On the other hand, existence of solutions to problem \eqref{choiceofmusup} is a difficult issue, unless we deal with constant function $h(\xi)-\Omega(\xi)$ (see Remark \ref{rm4}). Indeed, as far as we know, there is only one result in the literature, which was proved by del Pino, Man\'asevich and Montero in \cite{demamo} in the case $k=1$, when $\max\limits_{\xi\in K}\[h(\xi)-\Omega(\xi)\right]<0$ (see Theorem \ref{geode}). We would like to stress the fact that existence of solutions to problem \eqref{choiceofmusup} is an interesting open question by itself, which as a by product allows to find solutions to the supercritical problem \eqref{pb}.\\
Another remark is that the result we find suggests that the natural extension to higher critical exponent of the classical Yamabe equation is \begin{equation}\label{hyp} -\Delta_g u+\Omega(\xi) u=u^{m-k+2\over m-k-2},\ u>0\ \hbox{in}\ (M,g). \end{equation} where $\Omega$ is the function defined in \eqref{hgk}. If $k=0$ Problem \eqref{hyp} reduces to the classical Yamabe equation since ${m-k+2\over m-k-2} = { m+2\over m-2}$ and $\Omega (\xi ) = {m-2\over 4(m-1)} S_g(\xi)$, as we already mentioned. A natural open question is thus: {\it does problem \eqref{hyp} have a solution?}\\
Finally, we point out some interesting problems, whose solutions could help in understanding equation \eqref{hyp}.
\begin{itemize} \item[(i)] Theorem \ref{main} holds true when $m\geqslant k+7.$ The question is:
{\it does problem \eqref{pb} have any blowing-up solutions when $3\leqslant m\leqslant k+6$?} The case $k=0$ was completely studied by Druet \cite{do1,do2}. \item[(ii)] Theorem \ref{main} holds true when $h(\xi)\not= \Omega(\xi)$ for any $\xi\in K.$ The question is:
{\it does problem \eqref{pb} have any blowing-up solutions if $h(\xi)= \Omega(\xi)$ at some $\xi\in K$?} The case $k=0$ was extensively studied by Esposito, Pistoia and Vet\'ois \cite{epv} and by Esposito and Pistoia in \cite{ep}. \item[(iii)] Theorem \ref{main} holds true when $q\to 2^*_{m,k}$. The question is:
{\it does problem \eqref{pb} have any solutions if $q= 2^*_{m,k}$?} The case $k=0$ is nothing but the well known Yamabe problem. \end{itemize}
\noindent In the last few years several investigations have been carried out around the possibility of constructing singular limit solutions to non linear elliptic PDEs or problems in geometric analysis, depending on some parameters, whose mass or energy concentrate on sets of high dimension, like curves, surfaces, or higher dimensional sets. We refer the readers to \cite{mmpac,dmp,mmah,mm1,mm2,m,mussoyang,dkwei,demamu,dww,dengmamu} for instance, and the references therein. First contributions on concentration at higher dimensional set for problems involving higher critical Sobolev exponents are contained in the papers \cite{dmp,mussoyang,demamu}.
The general strategy used to prove all the above results is the so-called infinite dimensional version of the Liapunov-Schmidit reduction method. A main ingredient is to construct an approximate solution with arbitrary degree of accuracy in powers of $\epsilon$, in a neighborhood of the submanifold manifold $K$. This approximation is, at main order, a solution of some limit problem, which is independent of some of the variables. After this is done, one builds the desired solution by linearizing the equation around the approximation. The associated linear operator turns out to be invertible with inverse controlled in a suitable norm by certain large negative power of $\epsilon$, provided that $\epsilon$ remains away from certain critical values where resonance occurs. The interplay of the size of the error and that of the inverse of the linearization then makes it possible a fixed point scheme.
The rest of the paper is organized as follows.
We first discuss solvability and non-degeneracy of solutions to problems \eqref{choiceofmusub} and \eqref{choiceofmusup}. This is done in Section \ref{one}. In Section \ref{sec4} we introduce some scaled variables around the submanifold $K$ and we describe the Laplace Beltrami operator in these new variables. Section \ref{aprsol} is devoted to the construction of the approximate solution to our problem using the local coordinates around the sub-manifold $K$ introduced before. To perform this construction we need to invert a linear operator and to estimate the inverse. The proof of this result is postponed to Section \ref{luigi}. In Section \ref{s:linear} we define globally the approximation and we write the solution to our problem as the sum
of the global approximation plus a remaining term. Thus we express our original problem as a non linear problem in the remaining term and we prove our Theorem.
To solve such problem, we need to understand the invertibility properties of another linear operator. To do so we start expanding a quadratic functional
associated to the linear problem.
This is done in Section \ref{linearas}.
\section{Some remarks on PDE's with a singular term}\label{one}
First let us consider the attractive case, i.e. problem \eqref{choiceofmusub}. We can deal with a more general situation.
\begin{theorem}\label{th1}
Let $(M,g)$ be a smooth Riemannian compact manifold without boundary.
Assume $\alpha,\beta\in C^0(M) $ and $\min\limits_M\alpha,\min\limits_M\beta>0.$
Then there exists a non-degenerate solution to
\begin{equation}\label{p1} \left\{\begin{aligned} &-\Delta_g u+\alpha u-{\beta\over u}=0 \quad \hbox{in}\ M\\ & u>0 \quad \hbox{in}\ M.\\ \end{aligned}\right. \end{equation} \end{theorem} \begin{proof}
Let us prove that \eqref{p1} does have a solution. Set $L(u):=-\Delta_g u+\alpha u.$ Let us rewrite problem \eqref{p1} in the following way \begin{equation}\label{p11}L(u)=f(x,u)\ ,\ u>0\ \hbox{in}\ M,\end{equation}
where $L(u)=-\Delta_g u+\alpha u$ and $f(x,u):={\beta\over u}.$ The linear operator $L$ is coercive.
\\ First of all, we prove that problem \eqref{p1} has a lower solution $\underline u$ and an upper solution $\overline u,$ i.e. $$L(\underline u)\leqslant f(x,\underline u)\ \hbox{in}\ M\quad\hbox{and}\ L(\overline u)\geqslant f(x,\overline u)\ \hbox{in}\ M$$ such that $$0<\underline u(x)\leqslant \overline u(x)\ \hbox{for any}\ x\in M.$$
It is enough to consider $\underline u$ and $\overline u$ as positive constant functions and to observe that
$L(c)-f(x,c)<0$ if $c$ is small enough and $L(C)-f(x,C)>0$ if $C$ is large enough.
\\
As a second step, we consider the modified problem
\begin{equation}\label{p12}L(u)=\tilde f(x,u)\ ,\ u>0\ \hbox{in}\ M,\end{equation} where $$\tilde f(x,u):=\left\{\begin{aligned} &{\beta(x)\over \underline u(x)}\ &\hbox{if}\ u(x)<\underline u(x)\\ &{\beta(x)\over u(x)}\ &\hbox{if}\ \underline u(x)\leqslant u(x)\leqslant\overline u(x)\\ &{\beta(x)\over \overline u(x)}\ &\hbox{if}\ u(x)>\overline u(x)\\ \end{aligned}\right.
$$ We point out that any solutions of the modified problem \eqref{p12} is a solution to the problem \eqref{p11}. Indeed, assume $u$ solves \eqref{p12}. We want to show that $\underline u(x)\leqslant u(x)\leqslant\overline u(x)$ for any $x\in M.$ Suppose, by contradiction that $\max\limits_M\(\underline u-u\)>0.$ Then there exists a point $x_0\in M$ such that $\(\underline u-u\)(x_0):=\max\limits_M\(\underline u-u\)>0$ and an open set $\Gamma\subset M$ such that $x_0\in \Gamma $ and $\(\underline u-u\)(x)\geqslant 0$ for any $x\in\Gamma.$ Moreover, the function $ \underline u-u $ solves $$L\(\underline u-u\)\leqslant 0\ \hbox{in}\ \Gamma.$$ Since it achieves a maximum at the point $x_0$ which is in $\Gamma,$ by remark \ref{rm2} we immediately get a contradiction. \\ As a final step, we prove that problem \eqref{p12} has a solution. We remark that $u$ solves problem \eqref{p12} if $u$ is a fixed point of the operator $K(u):= T\(\tilde f(x,u)\),$ $u\in C^0(M)$ where $T$ is defined in Remark \ref{rm3}. By Remark \ref{rm3} and by the definition of $\tilde f(x,u)$ we deduce that $K:C^0(M)\to C^0(M)$ is a compact operator and moreover that there exists $R>0$ such that
$\|K(u)\|_{C^0(M)}<R$ for any $u\in C^0(M).$ Hence for any $t\in[0,1]$, using the homotopy invariance of the Leray-Schauder degree, we get $$\textrm{deg}\(I-K,B(0,R)\)=\textrm{deg}\(I-tK,B(0,R)\)=\textrm{deg}\(I,B(0,R)\)=1$$ and so problem \eqref{p12} has a solution. In order to prove that it is non degenerate, we point out that the linearized equation $$-\Delta_g v+\alpha v+{\beta\over u^2}v=0 \ , \ \hbox{in}\ M.$$ has only the trivial solution, since $\alpha$ and $\beta$ are strictly positive functions on $M.$\\
That concludes the proof. \end{proof}
\begin{remark}\label{rm1}
$L$ satisfies the maximum principle, namely if $u\in H^1(M)$ is such that $Lu\leqslant 0$ in $M$ then $u\leqslant 0$ in $M.$ \end{remark}
\begin{remark}\label{rm2} Assume that for some open set $\Gamma\subset M$ the function
$u\in H^1(M)$ solves $Lu\leqslant 0$ in $\Gamma.$ Then if $u$ achieves its maximum at a point $x\in \Gamma $ then $u(x)\leqslant 0.$ \end{remark}
\begin{remark}\label{rm3} For any $h\in C^0(M)$ there exists a unique $w\in C^2(M)$ such that $Lu=h$ in $M.$ The linear map $T:C^0(M)\to C^0(M)$ defined by $Th=w$ is continuous and compact.\end{remark} \begin{proof} It is enough to remark that the linear map $T:C^0(M)\to C^2(M)$ is continuos, because by standard elliptic regularity theory there exists a constant $c$ which only depends on $M$ and $g$ such that
$$\|Th\|_{C^2(M)}\leqslant c\|h\|_{C^0(M)}\ \hbox{for any}\ h\in C^0(M).$$ Moreover, the embedding $C^2(M) \hookrightarrow C^0(M)$ is compact because of the Ascoli-Arzel\'a Theorem. \end{proof}
As far as it concerns the repulsive case, i.e. equation \eqref{choiceofmusup}, we quote the results obtained by del Pino, Man\'asevich and Montero in \cite{demamo} in the case $k=1$ and by D\'avila, Pistoia and Vaira in \cite{dpv}.
Let us consider the more general problem
\begin{equation}\label{p1bis} \left\{\begin{aligned} &-\Delta_g u+\alpha u+{\beta\over u}=0 \quad \hbox{in}\ M\\ & u>0 \quad \hbox{in}\ M,\\ \end{aligned}\right. \end{equation} where $\alpha,\beta\in C^0(M) ,$ with $\min\limits_M \beta>0.$
\begin{theorem}\label{geode} Let $M$ be a one-dimensional manifold whose length is $\ell.$ Assume \begin{equation}\label{mamamo}-\({(\kappa+1)\pi\over2\ell}\)^2<\max\limits_{\xi\in M}\alpha(\xi)<-\({\kappa\pi\over2\ell}\)^2<0, \end{equation} for some integer $\kappa\ge1$. Then problem \eqref{p1bis} has a solution (see \cite{demamo}).\\ Moreover, it is non-degenerate for most functions $\alpha$'s (see \cite{dpv}). \end{theorem} In the general case, we can only make a few remarks.
\begin{remark}\label{rm4}
\begin{itemize}
\item[(i)] If \eqref{p1bis} has a solution, then $\min\limits_{\xi\in M}\alpha(\xi)<0.$ \item[(ii)] Let $\alpha=a$ and $\beta=b$ be constants. If $a<0$,
then problem \eqref{p1bis} has a constant solution, which is non-degenerate if in addition $-{\lambda_{\kappa+1}}<2a <-{\lambda_\kappa}<0$ holds for some $\kappa.$ Here $\(\lambda_\kappa\)_{\kappa\ge1}$ denotes the sequence of eigenvalues of $-\Delta_Mu=\lambda_\kappa u$ on $M.$ \end{itemize} \end{remark} \begin{proof} To prove (i) it is enough to integrate equation \eqref{p1bis} on $M$, so we get $$ \int\limits_M \alpha (\xi)u(\xi) d\xi +\int\limits_M{\beta(\xi)\over u(\xi)}d\xi=0$$ which implies that $\alpha$ has to be negative somewhere in $M.$ The proof of (ii) follows by straightforward computations.
\end{proof}
It would be really interesting to find conditions on $\alpha$ and $\beta$ which ensure the existence of a solution to problem \eqref{p1bis} in a more general setting.
\section{Laplace-Beltami operator in scaled variables}\label{sec4}
In this section we describe the Laplace-Beltrami operator in some scaled variables, by means of the Fermi coordinates introduced in \eqref{eq:fc}.\\
Let $\mu_\epsilon$ be a positive smooth function $\mu_\epsilon = \mu_\epsilon (y) $ defined on $K$ which we assume to be uniformly bounded, as $\epsilon \to 0$, along $K$. Let also $\Phi_\epsilon$ be a smooth normal section (in $M$) $\Phi_\epsilon\,:K\longrightarrow\,NK$ defined by $
\Phi_\epsilon (y)= \Phi_\epsilon^j(y)E_j $, and we assume that $ \Phi_\epsilon^j(y)$, $ j=1,\cdots,N$, are functions uniformly bounded, as $\epsilon \to 0$, in $K$. Having introduced the above function, we define the following change of variables \begin{equation} \label{ba}
u(\mathfrak F (y, x))=(1+\alpha_\epsilon)(\sqrt{\epsilon}\,\mu_\epsilon (y))^{-\frac{N-2}{2}} \, v\left(\frac{y}{\sqrt{\epsilon} }, \frac{ x-\epsilon \Phi_\epsilon (y)}{\sqrt{\epsilon}\,\mu_\epsilon (y)} \right), \end{equation} where $\mathfrak F (y, x)$ is the change of variables defined in (\ref{eq:fc}). and \begin{equation} \label{b0}
v=v(z,\xi), \quad z= {y \over \sqrt{\epsilon}}, \quad \xi=\frac{ x-\epsilon \Phi_\epsilon}{\sqrt{\epsilon}\,\mu_\epsilon}. \end{equation} In (\ref{ba}) $\alpha_\epsilon$ is a number defined so that $(1+\alpha_\epsilon)^{p\pm\epsilon-1}\epsilon^{\mp\frac{N-2}{4}\epsilon }=1$, that is, \begin{equation}\label{alphaepsilon} \alpha_\epsilon=\epsilon^{\pm\frac{(N-2)^2}{16\pm4(N-2)\epsilon}\epsilon}-1. \end{equation} To emphasize the dependence of the above change of variables on $\mu_\epsilon$ and $\Phi_\epsilon$, we will use the notation \begin{equation} \label{defTT} u={\mathcal T}_{\mu_\epsilon , \Phi_\epsilon} (v) \quad \Longleftrightarrow \quad u\ \ {\mbox {and}} \quad v \quad {\mbox {satisfy (\ref{ba})}}. \end{equation}
Recall that the original variables $(y,x)\in \mathbb R^{k+N}$ are {\it local} coordinates along $K$. Thus we let the variables $(z, \xi )$ vary in the set $\mathcal{D}$ defined by \begin{equation} \label{defD} \mathcal{D} = \left\{ (z, \xi ) \, : \, \sqrt{\epsilon} z \in K,
\quad | \xi | <{\eta \over \sqrt{\epsilon}} \right\} \end{equation} for some small and fixed positive number $\eta$ that will be fixed in the sequel. We will also use the notation $ \mathcal{D} = K_\epsilon \times \hat \mathcal{D}$, where $K_\epsilon = {K\over \sqrt{\epsilon}}$ and \begin{equation} \label{hatD}
\hat \mathcal{D} = \left\{ \xi \, : \, | \xi | < {\eta \over \sqrt{\epsilon}} \right\}. \end{equation}
We note that $\partial\hat{ \mathcal{D}} = \left\{ \xi \in\hat\mathcal{D} \, : \, | \xi | = {\eta \over \sqrt{\epsilon} } \right\}$.
We are interested in computing the Laplace Beltrami operator in the new variables $(z, \xi )$ in terms of the parameter $\epsilon$, of the function $\mu_\epsilon (y)$ and of the normal section $\Phi_\epsilon$.
We have the validity of the following
\begin{lemma} \label{scaledlaplacian} Given the change of variables defined in \eqref{ba}, the following expansion for the Laplace Beltrami operator holds true \begin{equation} \label{lap1}
(1+\alpha_\epsilon)^{-1} \epsilon^{{N+2 \over 4}}\mu_\epsilon^{{N+2 \over 2}} \Delta_g u= {\mathcal A}_{\mu_\epsilon , \Phi_\epsilon } (v) := {\mu_\epsilon^2} \Delta_{K_\epsilon} v + \Delta_{\xi} v + \sum_{\ell=0}^2 {\mathcal A}_\ell v + B(v). \end{equation} Above, the expression ${\mathcal A}_k$ denotes specific differential operators, respectively defined as follows \begin{equation} \label{D0} \begin{array}{rlllll} {\mathcal A}_0 v &= & - \epsilon\,\mu_\epsilon \,\Delta_K
\mu_\epsilon \,\left( \gamma v + D_\xi v \, [\xi] \right) \\[3mm]
& + & \epsilon \,| \nabla_K\mu_\epsilon|^2 \left[ D_{\xi\xi} v \, [ \xi]^2 + 2 (1+\gamma ) D_\xi v [\xi] + \gamma (1+ \gamma ) v \right] \\[3mm] & - & \epsilon^{3\over 2} \mu_\epsilon D_{ \xi}\, v \,[\Delta_K \Phi_\epsilon ] + \epsilon^{3\over 2} \nabla_K \mu_\epsilon \,\cdot\,\left\{ 2D_{\xi\,\xi} v[\xi] + N D_{\xi} v \right\}\, [\nabla_K \Phi_\epsilon ] \\[3mm] &+ & \epsilon^2 D_{\xi\,\xi} v \,[\nabla_K \Phi_\epsilon]^2\\[3mm] & - & 2\, \epsilon^2 \mu_\epsilon \, g^{ab}\,\left[ D_\xi (\partial_{\bar a} v ) [\partial_b \mu_\epsilon \xi] +\epsilon^{1\over 2} D_{\xi} (\partial_{\bar a} v )[ \partial_b \Phi_\epsilon ] +
\gamma \partial_a\mu_\epsilon\, \partial_{\bar b} v \right], \end{array} \end{equation} where we have set $\gamma=\frac{N-2}{2}$,
\begin{equation} \label{D1}
{\mathcal A}_1 \, v = \,-{\epsilon\over 3}\,\, \sum\limits_{i,j} \bigg[\sum\limits_{m,l} R_{mijl} (\mu_\epsilon \xi_m +\sqrt{\epsilon} \Phi_\epsilon^m ) (\mu_\epsilon \xi_l + \sqrt{\epsilon} \Phi_\epsilon^l ) \bigg] \partial^2_{ij} v, \end{equation} and
\begin{equation} \label{D4} {\mathcal A}_2 v =\epsilon \mu_\epsilon \sum\limits_j \bigg[ \sum_s \frac23 R_{mssj}+\sum\limits_{m,a,b}
\big( {\tilde g}^{ab}\, R_{mabj}- \Gamma_{am}^b \Gamma_{bj}^a \big)\bigg](\mu_\epsilon\xi_m+\sqrt{\epsilon} \Phi_\epsilon^m)
\partial_j v. \end{equation} Finally, the operator $B(v)$ can be described as follows: $B(v) = \epsilon^2 \hat B (v)$ \begin{eqnarray*}
\hat {\mathcal{B}}(v)&=&O \left( |\mu_\epsilon \xi + \sqrt{\epsilon} \Phi |^3 \right) \partial^2_{ij} v\\ &+&
O(| \mu_\epsilon + \sqrt{\epsilon} \Phi_\epsilon + \partial_{z_a} ( \mu_\epsilon + \sqrt{\epsilon} \Phi_\epsilon ) |^2 ) O ( v + \xi_i \partial_{\xi_i } v ). \end{eqnarray*}
We recall that the symbols $\partial_{a}$, $\partial_{\over a}$ and $\partial_i$ denote the derivatives with respect to $\partial_{y_a}$, $\partial_{z_a}$ and $\partial_{\xi_i}$ respectively.
\end{lemma}
\begin{proof} Recall that the Laplace-Beltrami operator is defined by $$ \Delta_{ g}=\frac{1}{\sqrt{\det g}}\,\partial_A(\,\sqrt{\det g}\,( g)^{AB}\,\partial_B\,)\,, $$ where indices $A$ and $B$ run between $1$ and $n=N+k$. In other words \begin{equation} \label{zero0}\Delta_{ g} =({g})^{AB}\,\partial^2_{AB}+\partial_A\,({ g})^{AB}\,\partial_B+\partial_A(\,\log{\sqrt{\det g}}\,)\,({g})^{AB}\,\partial_B \end{equation} If now $u$ and $v$ are defined as in (\ref{ba}), we have $$
(1+\alpha_\epsilon)^{-1} \epsilon^{N \over 4} \mu_\epsilon^{N \over 2} \partial_{x_j} u = \partial_{\xi_j } v, \quad (1+\alpha_\epsilon)^{-1} \epsilon^{N+2 \over 4} \mu_\epsilon^{N+2 \over 2} \partial^2_{x_j , x_i} u = \partial^2_{\xi_j , \xi_i } v $$ and \begin{eqnarray*}
(1+\alpha_\epsilon)^{-1} \epsilon^{N+2 \over 4} \mu_\epsilon^{N+2 \over 2} \partial^2_{y_a , y_b} u &=& \mu_\epsilon^2 \partial^2_{z_a z_b} v + \mu_\epsilon^{N-2 \over 2} \partial^2_{z_a z_b} (\mu_\epsilon^{-{N-2 \over 2}} ) v \\ &+& 2 \mu_\epsilon^{N-2 \over 2} \partial^2_{z_a } (\mu_\epsilon^{-{N-2 \over 2}} ) [ \partial_{z_b} v + \partial_{z_b} (\mu_\epsilon^{-1} ) \nabla v \cdot \xi - \sqrt{\epsilon} \nabla v \cdot \partial_{z_b} \Phi_\epsilon ] \\ &+& \partial_{z_a} [ \partial_{z_b} (\mu_\epsilon^{-1} ) \nabla v \cdot \xi - \sqrt{\epsilon} \nabla v \cdot \partial_{z_b} \Phi_\epsilon ] \end{eqnarray*} On the other hand, by our choice of coordinates (\ref{defoff}), on $K$ the metric $g$ splits in the following way \begin{equation}\label{eq:splitovg}
g(q) = g_{ab}(q)\,d y_a\otimes d y_b+ g_{ij}(q)\,dx_i\otimes dx_j, \qquad \quad q \in K. \end{equation} If we denote by $r$ the distance function from $K$, at any point $\mathfrak F (y,x)$ (see (\ref{eq:fc}),
we have $$ \begin{array}{rllll} g_{ij}(y, x)&=\delta_{ij}+\frac{1}{3}\,R_{istj}\,x_s\,x_t\, +\,{\mathcal O}(r^3);\\[3mm]
g_{aj}(y, x)&={\mathcal O}(r^2);\\[3mm]
g_{ab}(y, x)&={\tilde g}_{ab}-\, [ {\tilde g}_{ac}\,\Gamma_{bi}^c+{\tilde g}_{bc}\,\Gamma_{ai}^c ] \,x_i+\left[R_{sabl}+ {\tilde g}_{cd} \Gamma_{a s}^c\, \Gamma_{d l}^b \right]x_s x_l+{\mathcal O}(r^3). \end{array} $$ Here $a=1,...,k$, $i,j=1,...,N$. See \cite{demamu}. Let now $g^\epsilon$ be the scaled metric on $\mathcal{M}_\epsilon=\epsilon^{-1/2}\mathcal{M},$ whose coefficients are defined by $$g_{\alpha , \beta}^\epsilon(z,x)=g_{\alpha,\beta}(\sqrt\epsilon z,\sqrt \epsilon x).$$ For the metric $g^\epsilon$ in the above coordinates $(z,x )$ we have the expansions \begin{eqnarray*} &&g^\epsilon_{ij}=\delta_{ij} + \frac{ \epsilon }{3} \,R_{istj}\,x_s\,x_t
\,+\,{\mathcal O}(\epsilon^{3\over 2} (|x|^3), \quad 1\leqslant i,j\leqslant N;\\[3mm]
&&g_{aj}^\epsilon={\mathcal O}(\epsilon |x|^2)
\quad 1\leqslant a\leqslant k , \, 1\leqslant j\leqslant N;\\[3mm]
&&g_{ab}^\epsilon={\tilde g}^\epsilon_{ab}-\sqrt{\epsilon} \bigg\{{\tilde g}^\epsilon_{ac}\,\Gamma_{bi}^c+{\tilde g}^\epsilon_{bc}\,\Gamma_{ai}^c\bigg\}\,x_i + \epsilon\,\bigg[R_{sabl}+{\tilde g}^\epsilon_{cd}\Gamma_{as}^c\,
\Gamma_{dl}^b \bigg]x_s x_l + {\mathcal O}(\epsilon^{3 \over 2} |x|^3), \\ &&\quad 1\leqslant a,b\leqslant k. \end{eqnarray*} Thus we first conclude that \begin{eqnarray}\label{one1}
(1+\alpha_\epsilon)^{-1} \epsilon^{N+2 \over 4} \mu_\epsilon^{N+2 \over 2} ({g})^{AB}\,\partial^2_{AB} &=& \mu_\epsilon^2 g^{ab} \partial^2_{z_a z_b} v + \partial^2_{jj} v \nonumber \\ &-& {\epsilon \over 3} R_{mijl} (\mu_\epsilon \xi_l + \sqrt{\epsilon} \Phi_\epsilon^l ) (\mu_\epsilon \xi_m + \sqrt{\epsilon} \Phi_\epsilon^m ) \, \partial_{\xi_i \xi_j} v \nonumber \\ &+& {\mathcal B}_0 (v) + {\mathcal B}_1 (v) \end{eqnarray} where \begin{eqnarray*} {\mathcal B}_0 (v) &=& - \epsilon\,\mu_\epsilon \, g^{ab} \partial_{z_a z_b}^2
\mu_\epsilon \,\left( {N-2 \over 2} v + D_\xi v \, [\xi] \right) \\[3mm]
& + & \epsilon \, g^{ab} \partial_{z_a} \mu_\epsilon \partial_{z_b} \mu_\epsilon \left[ D_{\xi\xi} v \, [ \xi]^2 + N D_\xi v [\xi] + {(N-2) N \over 4} v \right] \\[3mm] & - & \epsilon^{3\over 2} \mu_\epsilon g^{ab} \partial^2_{z_a z_b} \Phi_\epsilon^ j \partial_{\xi_j} v+ \epsilon^{3\over 2} g^{ab} \partial_{z_a} \mu_\epsilon \partial_{z_b } \Phi_\epsilon^l \left\{ 2 \partial^2_{\xi_j \xi_l} v \xi_j + N \partial_{\xi_l} v \right\}\\[3mm] &+ & \epsilon^2 g^{ab} \partial_{z_a} \Phi_\epsilon^j \partial_{z_b} \Phi_\epsilon^l \partial^2_{\xi_j \xi_l} v \\[3mm] & - & 2\, \epsilon^2 \mu_\epsilon \, g^{ab}\,\left[ D_\xi (\partial_{\bar a} v ) [\partial_b \mu_\epsilon \xi] +\epsilon^{1\over 2} D_{\xi} (\partial_{\bar a} v )[ \partial_b \Phi_\epsilon ] +
\gamma \partial_a\mu_\epsilon\, \partial_{\bar b} v \right], \end{eqnarray*} and $$
{\mathcal B}_1 (v) =O(\epsilon^2 |\mu_\epsilon \xi + \sqrt{\epsilon} \Phi_\epsilon |^3 ) \partial^2_{\xi_i \xi_j} v. $$
\noindent Moreover \begin{eqnarray} \label{two2}
(1+\alpha_\epsilon)^{-1} \epsilon^{N+2 \over 4} \mu_\epsilon^{N+2 \over 2} \partial_A\,({ g})^{AB}\,\partial_B u &=& \mu_\epsilon^2 \partial_a (g^{ab} ) \partial_{z_b} v \nonumber \\ &+& {\epsilon \over 3} \, \mu_\epsilon R_{liij} (\mu_\epsilon \xi_l +\sqrt{\epsilon} \Phi_\epsilon^l ) \, \partial_{\xi_j} v \nonumber \\ &+& {\mathcal B}_2 (v) \end{eqnarray} where \begin{eqnarray*} {\mathcal B}_2 (v) &=& - \epsilon\,\mu_\epsilon \, \partial_{z_a} (g^{ab}) \partial_{z_b}
\mu_\epsilon \,\left( {N-2 \over 2} v + D_\xi v \, [\xi] \right) \\[3mm] & - & \epsilon^{3\over 2} \mu_\epsilon \partial_{z_a }( g^{ab}) \partial_{ z_b} \Phi_\epsilon^ j \partial_{\xi_j} v \\[3mm]
&+&\epsilon^2 O(| \mu_\epsilon + \sqrt{\epsilon} \Phi_\epsilon + \partial_{z_a} ( \mu_\epsilon + \sqrt{\epsilon} \Phi_\epsilon )|^2 ) O ( v + \xi_i \partial_{\xi_i } v ). \end{eqnarray*} Finally, we recall that we have the validity of the following expansions for the square root of the determinant of $g^\epsilon$ and the log of determinant of $g^\epsilon$ \begin{eqnarray}\label{expdeterminante}
\sqrt{\det g^\epsilon} &=& \sqrt{\det g^\epsilon} \times \\ & & \bigg \{1+ \frac{\epsilon}6 R_{miil} x_m x_l + \frac{\epsilon}2
\bigg( {\tilde g}^{ab}\,R_{mabl}-\Gamma_{am}^c \Gamma_{cl}^a \bigg) x_m x_l
+ \epsilon^{3 \over 2} \mathcal{O}(|x|^3) \biggl\} \nonumber \end{eqnarray} and \begin{eqnarray*}
\log\big(\det g^\epsilon\big) &=& \log\big(\det g^\epsilon\big)+ \frac{\epsilon}3 R_{miil}\,x_m x_l \\ & + &
\epsilon\bigg( {\tilde g}^{ab}\, R_{mabl}-\Gamma_{am}^{c}
\Gamma_{cl}^{a} \bigg)x_m x_l + \mathcal{O}(\epsilon^{3\over 2}|x|^3). \end{eqnarray*} See for instance \cite{demamu}.
So we get \begin{eqnarray} \label{three3}
(1+\alpha_\epsilon)^{-1} \epsilon^{N+2 \over 4} &&\mu_\epsilon^{N+2 \over 2} \partial_A(\,\log{\sqrt{\det g}}\,)\,({g})^{AB}\,\partial_B u = \partial_a(\,\log{\sqrt{\det g}}\,)\,({g})^{ab}\,\partial_b v \nonumber \\ &+& \epsilon \left( {R_{mssj} \over 3} + ({\tilde g}^{ab} R_{mabj} - \Gamma_{am}^c \Gamma_{cj}^a ) \right) (\mu_\epsilon \xi_m + \sqrt{\epsilon} \Phi_\epsilon^m ) \partial_{\xi_j } v \nonumber \\ &+& {\mathcal B}_3 (v). \end{eqnarray} Here ${\mathcal B}_3 (v)$ is a function that can be described as follows \begin{eqnarray*}
{\mathcal B}_3 (v) &=& \epsilon^2 O(| \mu_\epsilon + \sqrt{\epsilon} \Phi_\epsilon + \partial_{z_a} ( \mu_\epsilon + \sqrt{\epsilon} \Phi_\epsilon ) |^2 ) O ( v + \xi_i \partial_{\xi_i } v ). \end{eqnarray*}
\noindent Colleting (\ref{one1}), (\ref{two2}) and (\ref{three3}) in (\ref{zero0}), we get the proof of the Lemma.
\end{proof}
\setcounter{equation}{0} \section{Construction of an approximate solution}\label{aprsol}
Using the local coordinates along the submanifold $K$ introduced in Section \ref{sec4}, after performing the change of variables in \eqref{ba}, the original equation in $u$ reduces locally close to $K_\epsilon $ to the following equation in $v$ \begin{equation} \label{adesso}
-{\mathcal A}_{\mu_\epsilon , \Phi_\epsilon } v + \epsilon\, \mu_\epsilon^2 \, h v- \mu_\epsilon^{\mp\epsilon\frac{N-2}{2}}v^{p\pm\epsilon} =0 , \end{equation} where ${\mathcal A}_{\mu_\epsilon , \Phi_\epsilon }$ is defined in \eqref{lap1} and $p={N+2 \over N-2}$. Let us denote by $\Xi_\epsilon$ the operator given by \begin{equation} \label{Sep} \Xi_\epsilon (v ) := -{\mathcal A}_{\mu_\epsilon , \Phi_\epsilon } v + \epsilon\mu_\epsilon^2 \, hv- \mu_\epsilon^{\mp\epsilon\frac{N-2}{2}}v^{p\pm\epsilon}. \end{equation}
\noindent This section is devoted to build an approximate solution to Problem \eqref{adesso} locally around $K_\epsilon$, in the set $\mathcal{D}=K_\epsilon\times\hat\mathcal{D}$, (see Section \ref{sec4}).
Let $r$ be an integer. For a function $w$ defined in $\mathcal{D}=K_\epsilon \times \hat\mathcal{D}$, we define \begin{equation}\label{eqinftynu}
\|w\|_{\epsilon,r}:=\sup_{(z,\xi)\in K_\epsilon \times \hat\mathcal{D}}\left( \,(1+|\xi|^2)^{r \over 2}|w(z,\xi)| \right). \end{equation} Let $\sigma \in (0,1)$. We define \begin{equation}
\label{normsigma} \| w \|_{\epsilon, r, \sigma} := \| w \|_{\epsilon , r} + \sup_{(z,\xi)\in K_\epsilon
\times \hat\mathcal{D}}\left( \,(1+|\xi|^2)^{r +\sigma\over 2} [w]_{\sigma, B(\xi , 1)} \right) \end{equation} where we have denoted \begin{equation}\label{eqnorms} [w]_{\sigma, B(\xi , 1)} := \sup_{ \xi_1, \xi_2\in B(\xi , 1)}
\frac{|w(z,\xi_2)-w(z,\xi_1)|}{|\xi_1-\xi_2|^\sigma} \end{equation}
The main result of this section is as follows.
\begin{lemma} \label{Construction} There exist $\epsilon_0 >0$, $\eta >0$ in the definition of ${\mathcal D}$ in (\ref{defD}), and a constant $C>0$, such that, for any integer $I $ and for all $\epsilon \in (0, \epsilon_0)$ there exist a smooth function $\mu_{I+1,\epsilon} :K \to \mathbb R$, a smooth normal section $\Phi_{I+1,\epsilon} : K \to NK$, of the form $\Phi_{I+1, \epsilon} (y) = \Phi_{I+1, \epsilon}^j (y) E_j$ \begin{equation} \label{bf2}
\| \mu_{I+1,\epsilon}\|_{ \infty } +\| \partial_a
\mu_{I+1,\epsilon}\|_{\infty } +\|\partial^2_a \mu_{I+1,\epsilon}\|_{ \infty } \leq C \end{equation} \begin{equation} \label{bf3}
\| \Phi_{I+1 ,\epsilon} \|_{ \infty } +\| \partial_a
\Phi_{I+1 , \epsilon} \|_{ \infty } +\|\partial^2_a
\Phi_{I+1 , \epsilon} \|_{ \infty } \leq C, \end{equation} and a positive function $v_{I+1, \epsilon} :K_\epsilon \times \hat\mathcal{D} \to \mathbb R$ such that $$ -{\mathcal A}_{\mu_{I+1,\epsilon} , \Phi_{I+1 , \epsilon}} (v_{I+1 , \epsilon} ) +\epsilon \mu_{I+1,\epsilon}^2 h v_{I+1 , \epsilon} - \mu_{I+1,\epsilon}^{\mp \frac{N-2}{2}\epsilon} v_{I+1, \epsilon}^{p\pm \epsilon} = {\mathcal E}_{I+1 , \epsilon} \quad {\mbox {in}} \quad \mathcal{D} $$ with \begin{equation} \label{boh1}
\| v_{I+1 , \epsilon} - v_{I , \epsilon} \|_{\epsilon , N-4 , \sigma } \leq C \epsilon^{I+\frac{1}{2}} \end{equation} and \begin{equation} \label{bf4}
\| {\mathcal E}_{I+1 , \epsilon} \|_{\epsilon , N-2 , \sigma} \leq C \epsilon^{I+\frac{1}{2}}. \end{equation} We refer to (\ref{lap1}) for the definition of ${\mathcal A}_{\mu_\epsilon , \Phi_\epsilon} $, to (\ref{defD}) for $ K_\epsilon \times \hat\mathcal{D}$. \end{lemma}
The proof of Lemma \ref{Construction} is based on an explicit construction of the functions $\mu_{\epsilon , I+1}$, $\Phi_{\epsilon , I+1}$ and $v_{\epsilon , I+1}$, via an iterative scheme, in the spirit developed in \cite{demamu}. Fix an integer $I>1$, we will define the functions $\mu_{\epsilon , I}$ and $\Phi_{\epsilon, I}$ respectively of the form \begin{equation} \label{muep} \mu_{I,\epsilon}:= \mu_0 + \epsilon \mu_1 + \epsilon^2 \mu_2+ \ldots + \epsilon^{I-1} \mu_{I-1}, \end{equation} and \begin{equation} \label{phiep} \Phi_{\epsilon}: = \Phi_{1,\epsilon} + \Phi_{2,\epsilon}+\ldots + \Phi_{I-1,\epsilon}. \end{equation} to be solutions of certain linear elliptic PDEs on the sub manifold $K$. The solvability of these equations is related to the result contained in Section \ref{one}. At each step $I$, we also define \begin{equation} \label{roma1} v_{I} (z, \xi ) : = w_0 (\xi ) + w_{1,\epsilon} (z, \xi )+ w_{2,\epsilon} (z, \xi ) + w_{3,\epsilon} (z, \xi )+ \ldots + w_{I,\epsilon} (z, \xi ), \end{equation} where each term $w_{j,\epsilon}$ will also be solution of a linear problem, this time defined on $\mathcal{D}$. The function $w_0$ has been already defined as solution to \begin{equation} \label{w0} \Delta u + u^{N+2 \over N-2} =0\quad \mbox{in}\ \mathbb R^N, \end{equation} given explicitely by \begin{equation}
\label{defw0} w_0 (\xi ) = \alpha_N (1+ |\xi|^2 )^{-{N-2 \over 2}} . \end{equation} We consider the domain $\mathcal{D}$ defined as (\ref{defD}) and for function $\phi$ defined on $\mathcal{D}$, an operator of the form $$ L(\phi):=- \Delta_\xi \phi - p w_0^{p-1} \phi +\epsilon\, a(\epsilon z) \phi, $$ where $a$ is a given smooth function $a:K \to \mathbb R$ with $a(y) \geq \lambda >0$ for all $y \in K$.
Let us introduce the functions \begin{equation}
\label{lezetas} Z_j (\xi ) = {\partial w_0 \over \partial \xi_j} , \quad j=1, \ldots , N\quad {\mbox {and}} \quad Z_0 (\xi ) = \xi \cdot \nabla w_0 (\xi ) + \frac {N-2}2 w_0 (\xi ) \end{equation} that are known to be the only bounded solutions to the linearized equation around $w_0$ of problem \eqref{w0} $$ -\Delta \phi - p w_0^{p-1} \phi =0 \quad {\mbox {in}} \quad \mathbb R^{N}. $$ See \cite{bianchiengel}.
Given a function $g:K \times \hat\mathcal{D}\to \mathbb R$ that depends smoothly on the variable $y \in K$, we want to find a linear theory for the following linear problem \begin{equation}\label{eq:eqwd}
\left\{
\begin{array}{ll}
L(\phi)=h, &\ \hbox{ in } \mathcal{D}\\ \phi = 0 & \hbox{ on } \partial \hat \mathcal{D} \\
\int_{\hat\mathcal{D}} \phi (\epsilon z, \xi ) Z_j (\xi ) \, d\xi = 0 &\ \forall z \in K_\epsilon, \quad j=0, \ldots N.
\end{array}
\right. \end{equation}
We have the validity of the following result.
\begin{proposition}\label{linear} Let $r$ be an integer such that $4<r < N$.
Let $a : K \to \mathbb R$ be a smooth function, such that $a (y) \geq \lambda
>0$ for all $y \in K$. Then there exist $\epsilon_0 >0$, $\eta>0$, that depends only on $\sup_{y\in K} |a(y)|$, in the definition of $\mathcal{D}$ in (\ref{defD}), and $C>0$ such that, for any $\epsilon \in (0, \epsilon_0 )$ and for any function $h : K \times \hat\mathcal{D}
\to \mathbb R $ that depends smoothly on the variable $y \in K$, such that $ \| h \|_{\epsilon , r} $ is bounded, uniformly in $\epsilon$, and $$ \int_{\hat\mathcal{D}}h(\epsilon z,\xi)Z_j(\xi)d\xi=0\quad \mbox{for\ all}\ z\in K_\epsilon,\ \ j=0,1,\ldots,N, $$ then there exists a solution $\phi$ of problem (\ref{eq:eqwd}) such that \begin{equation}
\label{est0a} \| D^2_\xi \phi \|_{\epsilon , r , \sigma } + \| D_\xi \phi
\|_{\epsilon , r -1 , \sigma } +\|\phi \|_{\epsilon, r- 2 , \sigma}\leqslant C
\|h\|_{\epsilon,r ,\sigma} \end{equation} Furthermore, the function $\phi$ depends smoothly on the variable $\sqrt{\epsilon} z$, and the following estimates hold true: for any integer $l$ there exists a positive constant $C_l$ such that \begin{equation}
\label{est1a} \| D^l_z \phi \|_{\epsilon , r- 2 , \sigma } \leqslant C_l \left(
\sum_{k\leq l} \|D^k_z h\|_{\epsilon,r , \sigma}\right). \end{equation} \end{proposition}
We postpone the proof of Proposition \ref{linear} to Section \ref{luigi}. We devote the rest of the section to the Proof of Proposition \ref{Construction}.
\begin{proof}[Proof of Proposition \ref{Construction}]
Define \begin{equation} \label{roma2} \hat {\mathcal A} =\Delta_{\mathbb R^N} v + \sum_{\ell=0}^2+ {\mathcal A}_\ell v + B(v) \end{equation} referring to Lemma \ref{scaledlaplacian}, and \begin{equation} \label{esse1} \Xi_\epsilon (u) = - \hat {\mathcal A} u+ \epsilon \mu_\epsilon^2 \, h \, u - \mu_\epsilon^{\mp\epsilon\frac{N-2}{2}}\,u^{p\pm\epsilon} \end{equation}
\noindent \noindent We start with $I=1$ and the construction of $w_{1,\epsilon}$ and $\mu_0$ .\ \ A direct computation gives \begin{eqnarray*} \Xi_\epsilon(v_1) & = & - \hat {\mathcal A} ( w_0+w_{1,\epsilon}) + \epsilon \mu_0^2 \, h \, w_0+ \epsilon \mu_0^2 \, h \,w_{1,\epsilon} - \mu_0^{\mp\frac{N-2}{2}\epsilon}\,( w_0+w_{1,\epsilon}) ^{p+\epsilon}\\ & = & - \hat {\mathcal A} ( w_0+w_{1,\epsilon}) + \epsilon \mu_0^2 \, h \, w_0+ \epsilon \mu_0^2 \, h \,w_{1,\epsilon} \\ &&- \mu_0^{\mp\frac{N-2}{2}\epsilon}\, w_0^{p\pm\epsilon}- (p\pm\epsilon) \mu_0^{\mp\frac{N-2}{2}\epsilon}\,w_0^{p-1\pm\epsilon}w_{1,\epsilon}\\ &&\underbrace{- \mu_0^{\mp\frac{N-2}{2}\epsilon}\,\left[ (w_0+w_{1,\epsilon}) ^{p+\epsilon}-w_0^{p\pm\epsilon}-(p\pm\epsilon) w_0^{p-1\pm\epsilon}w_{1,\epsilon}\right]}\limits_{Q_\epsilon (w_{1 })}\\ & = & - \Delta w_{1,\epsilon}-pw_0^{p-1}w_{1,\epsilon}-\sum_{\ell=0}^2 {\mathcal A}_\ell w_{1,\epsilon} - B(w_{1,\epsilon})- \sum_{\ell=0}^2 {\mathcal A}_\ell w_0 - B(w_0) \\ &&- \underbrace{\left\{ \mu_0^{\mp\frac{N-2}{2}\epsilon}\, w_0^{p\pm\epsilon}-w_0^p\right\} }\limits_{I_1}+ \epsilon \mu_0^2 \, h \, w_0\\ &&- \underbrace{\left\{(p\pm\epsilon) \mu_0^{\mp\frac{N-2}{2}\epsilon}\,w_0^{p-1\pm\epsilon}w_{1,\epsilon}-pw_0^{p-1}w_{1,\epsilon}\right\}}\limits_{I_2}+ \epsilon \mu_0^2 \, h \,w_{1,\epsilon}+Q_\epsilon (w_{1 })\\ & = &- \Delta w_{1,\epsilon}-pw_0^{p-1}w_{1,\epsilon}+ \epsilon \mu_0^2 \, h \,w_{1,\epsilon} + \epsilon \mu_0^2 \, h \, w_0-\underbrace{ \mu_{0}^{\pm\frac{(N-2)^2}{8}\epsilon} \sum_{\ell=0}^2 {\mathcal A}_\ell w_0}\limits_{I_0}\\ &&- \underbrace{ \left\{ \mu_0^{\mp\frac{N-2}{2}\epsilon}\, w_0^{p\pm\epsilon}-w_0^p\right\}}\limits_{I_1}- \underbrace{\left\{(p\pm\epsilon) \mu_0^{\mp\frac{N-2}{2}\epsilon}\,w_0^{p-1\pm\epsilon}w_{1,\epsilon}-pw_0^{p-1}w_{1,\epsilon}\right\}}\limits_{I_2}\\ && - B( w_0) -\sum_{\ell=0}^2 {\mathcal A}_\ell w_{1,\epsilon} - B(w_{1,\epsilon})+Q_\epsilon (w_{1 }). \end{eqnarray*} We next analyze each one of the above terms. Using the expression of the operators $ {\mathcal A}_\ell $, $\ell = 0 , \ldots , 2$, given by Lemma \ref{scaledlaplacian}, we get \begin{eqnarray*} I_0 &=& \epsilon\, \left\{-\mu_0 \,\Delta_K
(\mu_0) \, Z_0+\,| \nabla_K\mu_0|^2 \mathcal{T}_1(w_0) -\mu_0^2(\mathcal{T}_2(w_0)-\mathcal{T}_3(w_0))\right\} \\ &+& O(\epsilon^2) b(\xi ) \end{eqnarray*}
where $b(\xi )$ is a smooth function such that $\| (1+ |\xi|^{N-2} ) b(\xi ) \|_\infty \leq C$, for some constant $C$ independent of $\epsilon$. Furthermore, we recall that $ Z_0=\gamma w_0 + D_\xi w_0 \, [\xi]. $ Also we denoted \begin{eqnarray}\label{mathcala} \mathcal{T}_1(w_0)=D_{\xi\xi} w_0 \, [ \xi]^2 + 2 (1+\gamma ) D_\xi w_0 [\xi] + \gamma (1+ \gamma ) w_0, \end{eqnarray} \begin{eqnarray}\label{mathcalb} \mathcal{T}_2(w_0)= {1 \over 3}\,\, \sum\limits_{i,j} \bigg[\sum\limits_{m,l} R_{mijl}
\xi_m \xi_l \bigg] \partial^2_{ij} w_0,\end{eqnarray} \begin{eqnarray}\label{mathcalc} \mathcal{T}_3(w_0)= \sum\limits_j \bigg[ \sum_s \frac23 R_{mssj}+\sum\limits_{m,a,b}
\big( \,\tilde g^{ab} R_{mabj}- \Gamma_{am}^b \Gamma_{bj}^a \big)\bigg] \xi_m
\partial_j w_0. \end{eqnarray} On the other hand, a direct computation shows that \begin{eqnarray*}
I_1=w_0^p\left [\mu_0^{\mp\frac{N-2}{2}\epsilon}\, w_0^{ \pm\epsilon}-1\right]=\pm \epsilon\, w_0^p\ln w_0 \left( 1 +O(\epsilon^2) \right). \end{eqnarray*} Thus we can write \begin{eqnarray*} \Xi_\epsilon(v_1) & = &
- \Delta_{\mathbb R^{N}} w_{1}
- p w_0^{p-1} w_{1}+\epsilon \mu_0^2 \, h \, w_{1} +\epsilon\, H_1 (z, \xi) + \epsilon L_\epsilon (w_1 )
+ Q_\epsilon (w_{1 }), \end{eqnarray*} where \begin{eqnarray*} H_1 (z,\xi ) &=& \mu_0 \,\Delta_K
(\mu_0) \, Z_0-\,| \nabla_K\mu_0|^2 \mathcal{T}_1(w_0) +\mu_0^2(\mathcal{T}_2(w_0)-\mathcal{T}_3(w_0))\\ & & \\ & &+ \mu_0^2 h w_0\mp w_0^p\ln (w_0)+ \mathcal{E}_{1,\epsilon}, \end{eqnarray*} with $\mathcal{E}_{1, \epsilon}$ is a sum of functions of the form $$ \epsilon \mu_0 \left( \epsilon \mu_0 + \epsilon \partial_a \mu_0 + \epsilon \partial^2_a \mu_0 \right) a (z) b (\xi ) $$ and $a(\epsilon z)$ is a smooth function uniformly bounded, together with its derivatives, as $ \epsilon \to 0$, while the function $b$ is such that $$
\sup_{\xi } (1+|\xi|^{N-2} ) |b (\xi ) |< \infty. $$ The term $ \epsilon L_\epsilon (w_1 )$ is linear in $w_1$, infact it is explicitely given by \begin{equation} \label{mi1} \epsilon L_\epsilon (w_1) =I_2 -\sum_{\ell=0}^2 {\mathcal A}_\ell w_{1,\epsilon} - B(w_{1,\epsilon}) \end{equation} The term $Q_\epsilon (w_{1,\epsilon} )$ is quadratic in $w_{1,\epsilon}$, in fact it is explicitly given by \begin{equation}\label{mi2} \mu_0^{\mp \frac{N-2}{2}\epsilon}\left[(w_0 + w_{1,\epsilon} )^{p\pm \epsilon} - w_0^{p\pm \epsilon} - p w_0^{p-1\pm \epsilon} w_{1,\epsilon}\right]. \end{equation} We ask the function $w_{1,\epsilon}$ to satisfy the following equation \begin{equation}\label{eq:eqw1}
- \Delta_{\mathbb R^{N}} w_{1,\epsilon}
- p w_0^{p-1} w_{1,\epsilon}+\epsilon \mu_0^2 \, h \, w_{1,\epsilon} =-\epsilon\, H_1 (z, \xi), \ \hbox{ in } \mathcal{D} , \quad \phi = 0 \quad {\mbox {on}} \quad \partial \hat \mathcal{D}. \end{equation} Using Proposition \ref{linear}, we see that equation \eqref{eq:eqw1} is solvable if the right-hand side satisfies the orthogonality conditions in (\ref{eq:eqwd}). These conditions, for $j = 1, \dots, N$ are clearly satisfied since both $\xi_j\partial_{j} w_0$ and $\partial^2_{ij} w_0$ are even functions in $ \xi$, while the $Z_i$'s are odd functions in $\xi$ for every $i$. It remains to compute the $L^2$ product of the right-hand side against $Z_0$. Imposing this $L^2$ product equal to zero will define the function $\mu_0$.
We define $\mu_0$ to satisfy, at main order, \begin{equation}\label{chmu0} \int_{\hat \mathcal{D}} H_1 (z, \xi ) Z_0 (\xi ) d\xi= 0 \quad \forall z \in K_\epsilon. \end{equation} Let us be more precise. We have \begin{eqnarray*} &&\int_{\hat \mathcal{D}} H_1 (z, \xi ) Z_0 (\xi ) d\xi= \mu_0(y) \,\Delta_{K}
(\mu_0) \,\int_{\mathbb R^N} Z^2_0(\xi)d\xi\\
&& -\,| \nabla_K\mu_0 |^2\int_{\mathbb R^N}Z_0(\xi) \mathcal{T}_1 (w_0)d\xi +\mu_0^2(y)\int_{\mathbb R^N}(\mathcal{T}_2(w_0)-\mathcal{T}_3(w_0))Z_0(\xi)d\xi\\ & &+ \mu_0^2(y)h(y)\int_{\mathbb R^N} w_0(\xi)Z_0(\xi)d\xi \mp \int_{\mathbb R^N}w_0^p\ln (w_0)Z_0d\xi+O\left(\left(\frac{\epsilon}{\eta^2}\right)^{\frac{N-4}{2}}\right). \end{eqnarray*} Define \begin{eqnarray}\label{c1} c_{1,N}:=\int_{\mathbb R^N} Z^2_0(\xi)d\xi &=&\alpha_N^2\frac{(N-2)^2(N+2)}{2N(N-4)}\omega_N\, I_{N}^{N/2}>0. \end{eqnarray} A direct computation gives that \begin{eqnarray*} \int_{\mathbb R^N}Z_0(\xi) \mathcal{T}_1(w_0)d\xi =0. \end{eqnarray*} Moreover \begin{eqnarray*} \int_{\mathbb R^N}Z_0(\xi) \mathcal{T}_2(w_0)d\xi &=&{1 \over 3}\,\, \sum\limits_{i,j} \sum\limits_{m,l} R_{mijl} \int_{\mathbb{R}^N} \xi_m \xi_l \partial^2_{ij} w_0 Z_0d\xi\nonumber\\&=&{1 \over 3}\,\, \sum\limits_{i,j} R_{jiij} \int_{\mathbb{R}^N} \xi_j \partial_{j} w_0 Z_0d\xi, \end{eqnarray*} because $R_{mijl}$ is antisymmetric (i.e. $R_{mijl}=-R_{imjl}$) and \begin{eqnarray*}\int_{\mathbb{R}^N} \xi_m \xi_l \partial^2_{ij} w_0 Z_0d\xi& =& \alpha_N(N-2) \int_{\mathbb{R}^N} \xi_m
\xi_i \left(-{\delta_{ij}\over (1+|\xi|^2)^{N\over2}}+ {N \xi_i\xi_j\over(1+|\xi|^2)^{N+2\over2}}\right) Z_0d\xi \end{eqnarray*}
and $\int_{\mathbb{R}^N} {\xi_m\xi_l\xi_i\xi_j\over(1+|\xi|^2)^{N+2\over2}} Z_0d\xi$
is symmetric. On the other hand, \begin{eqnarray}\label{c2from} \int_{\mathbb R^N}Z_0(\xi) \mathcal{T}_3(w_0)d\xi &=& \sum\limits_j \bigg[ \sum_s \frac23 R_{mssj}+\sum\limits_{m,a,b}
\big( \, {\tilde g}^{ab} R_{mabj}- \Gamma_{am}^b \Gamma_{bj}^a \big)\bigg] \int_{\mathbb{R}^N} \xi_m
\partial_j w_0 Z_0d\xi\nonumber
\\&=& \sum\limits_j \bigg[ \sum_s \frac23 R_{jssj}+\sum\limits_{j,a,b}
\big( \, {\tilde g}^{ab} R_{jabj}- \Gamma_{aj}^b \Gamma_{bj}^a \big)\bigg] c_{2,N} \end{eqnarray} where \begin{eqnarray}\label{c2} c_{2,N}:= \int_{\mathbb{R}^N} \xi_j
\partial_j w_0 Z_0d\xi= \alpha_N^2\frac{3(N-2)^2}{2N(N-4)}\omega_N\, I_{N}^{N/2}>0. \end{eqnarray} From the above computations we deduce \begin{eqnarray*}\int_{\mathbb R^N}Z_0(\xi) \left(\mathcal{T}_2(w_0)- \mathcal{T}_3(w_0)\right)d\xi = c_{2,N} \left[ \sum_{i,j} \frac13 R_{ijij}+\sum\limits_{j}\sum\limits_{a,b}
\big( \,{\tilde g}^{ab} R_{jajb}+ \Gamma_{aj}^b \Gamma_{bj}^a \big)\right],\end{eqnarray*}
We also have \begin{eqnarray*} \int_{\hat \mathcal{D}} \ln (w_0)w_0^p\, Z_0d\xi& = &\int_{\mathbb{R}^N} \ln (w_0)w_0^p\, Z_0d\xi-\int_{\mathbb{R}^N\backslash\hat \mathcal{D}} \ln (w_0)w_0^p\, Z_0d\xi\\ & = &\int_{\mathbb{R}^N} \ln (w_0)w_0^p\, Z_0d\xi+O\left(\left(\frac{\epsilon}{\eta}\right)^{\frac{N-2}{2}}\right)\\ &: = &c_{4,N} +O\left(\left(\frac{\epsilon}{\eta}\right)^{\frac{N-2}{2}}\right), \end{eqnarray*} where \begin{eqnarray}\label{c4} c_{4,N}:= \frac{N}{(p+1)^2}\int_{\mathbb{R}^N} w_0^{p+1}(\xi)d\xi=\alpha_N^{p+1}\frac{ (N-2)^3}{4N^2}\omega_N\, I_{N}^{N/2}>0. \end{eqnarray} Finally, set \begin{eqnarray}\label{c3} c_{3,N}:= \int_{\mathbb R^N} w_0(\xi)Z_0(\xi)d\xi= -\alpha_N^2\frac{2(N-1)(N-2)}{N(N-4)}\omega_N\, I_{N}^{N/2}<0. \end{eqnarray} We define $\mu_0$ to satisfy \begin{eqnarray}\label{choiceofmu0}
- \Delta_K\mu_0+ {\frac{c_{3,N}}{c_{1,N}}} \, {\mathcal H} (y)\mu_0\pm\frac{c_{4,N}}{c_{1,N}}\frac{1}{\mu_0}=0,\quad {\mbox {in}} \quad \ K. \end{eqnarray} where ${\mathcal {H}}$ is the function defined in Fermi coordinates by \begin{equation}\label{defgy} {\mathcal H} (y) = h(y) -\hat \Omega (y) \end{equation} where $$ \hat \Omega (y) = - {3(N-2)\over 4(N-1)}\[\sum\limits_{i,j=1}^{N} \frac13 R_{jiji}(y)+ \sum\limits_{i=1}^{N}\sum\limits_{a,b=1}^k
\(\tilde g^{ab}R_{iaib}(y)+ \Gamma_{ai}^b(y) \Gamma_{bi}^a(y) \)\right] . $$ The existence of $\mu_0$ is guaranteed by our assumption. With this choice for $\mu_0$, the integral of the right hand side in (\ref{eq:eqw1}) against $Z_{0}$ vanishes on $K$ and this implies the existence of $w_{1,\epsilon}$, thanks to Proposition \ref{linear}. Moreover, it is straightforward to check that $$
\| H_1 (z, \xi ) \|_{\epsilon , N-2 , \sigma } \leq C $$ for some $\sigma \in (0,1)$. Proposition \ref{linear} thus gives that \begin{equation}\label{ew1}
\| D^2_\xi w_{1,\epsilon} \|_{\epsilon , N-2 , \sigma } + \| D_\xi w_{1,\epsilon} \|_{\epsilon , N-3 , \sigma } +\|w_{1,\epsilon} \|_{\epsilon, N-4 , \sigma}\leqslant C \epsilon \end{equation} and that there exists a positive constant $\beta$ (depending only on $ K$ and $N$) such that for any integer $\ell$ there holds \begin{equation}\label{eq:estw1}
\|\nabla^{(\ell)}_{z} w_{1,\epsilon}(z,\cdot)\|_{\epsilon,N-4,\sigma} \leq \beta C_l
\epsilon \qquad \quad \sqrt{\epsilon}z \in K \end{equation} where $C_l$ depends only on $l$, $p$ and $K$.
With this choice of $\mu_{0}$ and $w_{1, \epsilon}$ we get that \begin{equation} \label{mi3}
\| - {\mathcal A}_{\mu_\epsilon , \Phi_\epsilon } v_{1,\epsilon} + \epsilon\mu_0^2 hv_{2,\epsilon}
- \mu_0^{\mp \epsilon\frac{N-2}{2}}\,v_{1,\epsilon}^{p\pm \epsilon} \|_{\epsilon, N-2 , \sigma} \leq C\epsilon. \end{equation} To prove this, we first observe that $$
\| \mu_0^2 \Delta_{K_\epsilon} w_{1,\epsilon} \|_{\epsilon, N-2 , \sigma} \leq C\epsilon, $$
as consequence of (\ref{eq:estw1}). Then we claim that $\| \epsilon L_\epsilon (w_{1 , \epsilon} ) + Q_\epsilon (w_{1,\epsilon} ) \|_{\epsilon , N-2 } \leq C \epsilon$, see (\ref{mi1}) and (\ref{mi2}). Indeed, first observe that \begin{eqnarray*} I_2= (p\pm\epsilon)\mu_0^{\mp\frac{N-2}{2}\epsilon}\,w_0^{p-1\pm\epsilon}w_{1,\epsilon}-pw_0^{p-1}w_{1,\epsilon} = O(\epsilon)w_0^{p-1}w_{1,\epsilon}. \end{eqnarray*}
and then easily we get that $|I_2 | \leq C {\epsilon \over (1+ |\xi | )^{N-2}}$. Analogous consideration gives that $\| Q_\epsilon (w_{1 , \epsilon} ) \|_{\epsilon , N-2} \leq C \epsilon$. Furthermore, using the fact that
$|\xi | \leq \eta \epsilon^{-{1\over 2}}$, we have that $$
\left| \sum_{\ell =0}^2 {\mathcal A}_\ell (w_{1,\epsilon} )+ B(w_{1,\epsilon} ) \right| \leq C {\epsilon \over (1+ |\xi | )^{N-2}}. $$ Estimate (\ref{mi3}) follows from the regularity of the function $w_{1, \epsilon}$.
\noindent Let $I=2$. Then $\mu_{2,\epsilon}=\mu_0+\epsilon \mu_1$, $\Phi_{\epsilon}=\Phi_{1,\epsilon}$, and $v_2= w_0+w_{1,\epsilon}+w_{2,\epsilon}$, where $\mu_0$ and $w_{1,\epsilon}$ have already been constructed in the previous step. Computing $\Xi_\epsilon(v_{2})$ , \begin{eqnarray}\label{sw2} \Xi_\epsilon(v_2)& = & - \Delta_{\mathbb R^{N}} w_{2,\epsilon}
- p w_0^{p-1} w_{2,\epsilon} + \epsilon \mu_{2,\epsilon}^2 \, h \, w_{2,\epsilon} + H_2 (z, \xi) \nonumber \\ &+&\epsilon^2 L_\epsilon (w_{2,\epsilon} )
+ Q_\epsilon (w_{2,\epsilon}), \end{eqnarray} where \begin{eqnarray*} H_2 (z,\xi ) &=& \epsilon^2 \, \left\{\mu_0\, \Delta_K
(\mu_1) \,Z_0 +\mu_1\Delta_K(\mu_0)Z_0- \, \nabla_K(\mu_0)\nabla_K(\mu_1)\mathcal{T}_1(w_0)\right. \\ & &\qquad\qquad \left.+2 \mu_0\,\mu_1(\mathcal{T}_2(w_0)-\mathcal{T}_3(w_0))+ 2\mu_0\, \mu_1h w_0\right.\\ &&\qquad\qquad \left.\pm \frac{(N-2)^2}{16} w_0^p\left(\ln (w_0)+1\right)\right\}\\ &&+\epsilon^{\frac{3}{2}} \, \mu_0 \left\{- \Delta_K\Phi_{1,\epsilon}D_\xi w_0 + {1\over 3}\,\, \sum\limits_{i,j} \sum\limits_{m,l} R_{mijl} ( \xi_m \Phi_{1,\epsilon}^l +\xi_I\Phi_{1,\epsilon}^m) \partial^2_{ij} w_0,\right. \\ & & \qquad\qquad\left.+ \sum\limits_j \bigg[ \sum_s \frac23 R_{mssj}+\sum\limits_{m,a,b}
\big( \,{\tilde g}^{ab} R_{mabj}- \Gamma_{am}^b \Gamma_{bj}^a \big)\bigg] \Phi_{1,\epsilon}^m
\partial_j w_0 \right\}\\ &&+ \epsilon^{\frac{3}{2}} \mathcal{E}_{2,\epsilon}(y,\xi,w_0,w_{1,\epsilon},\mu_0), \end{eqnarray*} and $\mathcal{E}_{2, \epsilon}$ is a sum of functions of the form $$
\left( \mu_0 + \partial_a \mu_0 + \partial^2_a \mu_0 \right) a (\sqrt{\epsilon }z) b (\xi ) $$ and $a(\sqrt{\epsilon }z)$ is a smooth function uniformly bounded, together with its derivatives, as $ \epsilon \to 0$, while the function $b$ is such that $$
\sup_{\xi } (1+|\xi|^{N-2} ) |b (\xi ) |< \infty. $$ Furthermore, we have \begin{eqnarray*} \epsilon^2 L_\epsilon (w_2)& = & - \mu_0^{\mp\frac{N-2}{2}\epsilon}\,\left[ (w_0+w_{1,\epsilon} + w_{2,\epsilon} ) ^{p+\epsilon} \right. \\ & & \left.-(w_0 + w_{1,\epsilon})^{p\pm\epsilon}-(p\pm\epsilon) (w_0 + w_{1,\epsilon})^{p-1\pm\epsilon}w_{2,\epsilon}\right]\\ & - &\sum_{\ell=0}^2 {\mathcal A}_\ell w_{2,\epsilon} - B(w_{2,\epsilon}) \end{eqnarray*} and the term $Q_\epsilon (w_{2,\epsilon} )$ in (\ref{sw2}) is a sum of quadratic terms in $w_{2,\epsilon}$ like $$ (\mu_0+\epsilon\mu_1)^{\mp \frac{N-2}{2}\epsilon}\left[(w_0 + w_{1,\epsilon} + w_{2,\epsilon} )^{p\pm \epsilon} - (w_0 + w_{1,\epsilon} )^{p\pm \epsilon}\right. $$ $$ \left. -(p\pm \epsilon) (w_0 + w_{1,\epsilon} )^{p-1\pm \epsilon} w_{2,\epsilon}\right]. $$
We will choose $w_{2,\epsilon}$ to satisfy the following equation \begin{equation}\label{eq:eqw2}
- \Delta_{\mathbb R^{N}} w_{2,\epsilon}
- p w_0^{p-1} w_{2,\epsilon}+\epsilon \mu_{2,\epsilon}^2 \, h \, w_{2,\epsilon} =- H_2 (z, \xi), \hbox{ in } \mathcal{D} , \quad w_{2, \epsilon} = 0 \quad {\mbox {on}} \quad \partial \hat \mathcal{D}. \end{equation} Thanks to Proposition \ref{linear}, we see that equation \eqref{eq:eqw2} is solvable if the right-hand side is $L^2$-orthogonal to the functions $Z_j$, for $j=0, \ldots , N$. These orthogonality conditions will define the parameters $\mu_{1}$ and the normal section $\Phi_{1,\epsilon}$.
\noindent Projection onto $Z_0$ and choice of $\mu_{1}$:\ \ the function $\mu_1$ is asked to satisfy, at main order, \begin{eqnarray*}
\int_{\hat D } H_{2,\epsilon}Z_0d\xi&=& 0. \end{eqnarray*} Computations similar to the ones already performed to define $\mu_0$ give that $\mu_1$ satisfies \begin{equation}\label{mu11}
-\mu_0(y)\, \Delta_K
\mu_1(y) -\mu_1\Delta_K \mu_0(y) +2\mu_0(y)\,\mu_1(y)g(y) \pm \frac{(N-2)^2}{16}\frac{c_{4,N}}{c_{1,N}}=0 \end{equation}
in $K$, where $g(y)$ is given by (\ref{defgy}), and $c_{i,N}(i=1,2,3,4)$ are defined in (\ref{c1}), (\ref{c2}), (\ref{c3}) and (\ref{c4}). According to our choose of $\mu_0$ satisfies (\ref{choiceofmu0}), then (\ref{mu11}) is equivalent to the following equation \begin{eqnarray}\label{mu11a}
- \Delta_K\mu_1(y)+g(y)\mu_1(y)\mp\frac{c_{4,N}}{c_{1,N}}\frac{1}{\mu_0^2(y)}\mu_1(y)=\mp\frac{(N-2)^2}{16}\frac{c_{4,N}}{c_{1,N}}\frac{1}{\mu_0(y)},\quad \end{eqnarray}
in $K$. The existence of $\mu_1$ is guaranteed by the nondegeneracy of $\mu_0$.
\noindent Projection onto $Z_s$ and choice of $\Phi_{1,\epsilon}$. \ \ Multiplying $H_{2,\epsilon}$ with $Z_s = \partial_s w_0$, integrating over $\hat\mathcal{D}$ and using the fact $w_0$ is even in the variable $\xi$, one obtains \begin{eqnarray}\label{projZl}
\int_{\hat\mathcal{D}}H_{2,\epsilon}\,\partial_s w_0d\xi &=& - \epsilon^{\frac{3}{2}} \, \mu_0 \, \sum_j \, \Delta_K \Phi_{1,\epsilon}^j \,\int_{\hat\mathcal{D}} \partial_jw_0 \partial_s w_0d\xi +\epsilon^{\frac{3}{2}} \, \int_{\hat\mathcal{D}}\mathcal{E}_{2, \epsilon}\partial_sw_0d\xi\nonumber\\[2mm] && +\epsilon^{\frac{3}{2}} \, \, { \mu_0 \over 3}\,\, \sum\limits_{i,j} \sum\limits_{m,l} R_{mijl}\int_{\hat\mathcal{D}} ( \xi_m \Phi_{1,\epsilon}^l +\xi_l \Phi_{1,\epsilon}^m) \partial^2_{ij} w_0\partial_s w_0d\xi \nonumber \\[2mm] &&+ \epsilon^{\frac{3}{2}} \, {2 \mu_0 \over 3} \,\sum\limits_{j , m} \sum_l
R_{mllj} \Phi_{1,\epsilon}^m
\int_{\hat\mathcal{D}}\partial_jw_0\,\partial_sw_0 \\[2mm] &&+ \epsilon^{\frac{3}{2}} \, \,\sum\limits_{j , m}
\sum\limits_{a,b}
\big( \, {\tilde g}^{ab} R_{mabj}- \Gamma_{am}^b \Gamma_{bj}^a \big) \Phi_{1,\epsilon}^m
\int_{\hat\mathcal{D}}\partial_jw_0\,\partial_sw_0.\nonumber \end{eqnarray} First of all, observe that by oddness in $\xi$ we have that $$ \int_{\hat\mathcal{D}} \partial_jw_0 \partial_sw_0=
\delta _{js}\, C_0+O(\epsilon^{\frac{N-2}{2}}), \quad C_0:= \int_{\mathbb R^N} |\partial_lw_0|^2d\xi. $$ Thus $$ - \epsilon^{\frac{3}{2}} \, \mu_0 \, \sum_j \, \Delta_K \Phi_{1,\epsilon}^j \,\int_{\hat\mathcal{D}} \partial_jw_0 \partial_s w_0d\xi = - \epsilon^{\frac{3}{2}} \, \mu_0 \, C_0 \, \Delta_K \Phi_{1,\epsilon}^s + O(\epsilon^{\frac{N-2}{2}}) $$ and $$
\epsilon^{\frac{3}{2}} \, {2 \mu_0 \over 3} \,\sum\limits_{j , m} \sum_l
R_{mllj} \Phi_{1,\epsilon}^m
\int_{\hat\mathcal{D}}\partial_jw_0\,\partial_sw_0 = \epsilon^{\frac{3}{2}} \, {2 \mu_0 \over 3} \, C_0 \, \sum\limits_{ m} \sum_l
R_{mlls} \Phi_{1,\epsilon}^m. $$
On the other hand the integral $\int_{\hat\mathcal{D}} \xi_m \,\partial^2_{ij}w_0 \partial_sw_0$ is non-zero only if, either $i = j$ and $m = s$, or $i = s$ and $j = m$, or $i = m$ and $j = s$. In the latter case we have $R_{mijs}=0$ (by the antisymmetry of the curvature tensor in the first two indices). Therefore, the first term of the second line in (\ref{projZl}) becomes simply \begin{eqnarray*} && 2 \epsilon^{3\over 2} {\mu_0 \over 3} \sum_m \sum_{l} R_{mlls} (- \int \xi_l \partial^2_{ls} w_0 \partial_s w_0 + \int \xi_s \partial^2_{ll} w_0 \partial_s w_0 ) {\Phi }^m_{1,\epsilon} +O(\epsilon^{\frac{N-2}{2}})\\ && =- 2 \epsilon^{3\over 2} {\mu_0 \over 3} \sum_m \sum_{l} R_{mlls} (- \int \xi_l \partial^2_{ls} w_0 \partial_s w_0 + \int \xi_s \partial^2_{ls} w_0 \partial_l w_0 ) {\Phi }^m_{1,\epsilon} +O(\epsilon^{\frac{N-2}{2}}) \\ && =- 2 \epsilon^{3\over 2} {\mu_0 \over 3} \, C_0 \, \sum_m \sum_{l} R_{mlls} {\Phi }^m_{1,\epsilon} +O(\epsilon^{\frac{N-2}{2}}) \end{eqnarray*} where the last identity is consequence of the following fact $$ \int \xi_l \partial^2_{sl} w_0 \partial_s w_0 = -{C_0 \over 2} , $$ which can be proved with a straightforward computation.
Hence formula (\ref{projZl}) becomes simply \begin{eqnarray*} \int_{\hat\mathcal{D}}H_{2,\epsilon}\,\partial_s w_0 d\xi&=& \epsilon^{3\over 2} \, \mu_0 \, C_0\,\left[ - \Delta_K\,\Phi^s_{1,\epsilon}+\sum_m \sum_{a,b} \bigg( \tilde g^{ab}\,R_{mabs}- \Gamma_a^b(E_m) \Gamma_b^a(E_s) \bigg) \,\Phi^m_{1,\epsilon} \right]\\ && +O(\epsilon^{N-{1\over2}} ) \, \Phi^m_{1, \epsilon} +\epsilon^{3\over 2} \int_{\hat\mathcal{D}}\mathcal{E}_{2,\epsilon}\,\partial_s w_0. \end{eqnarray*} We thus obtain that $\int H_{2,\epsilon}(z,\xi, w_0, \dots, w_{1,\epsilon}) Z_l =0$ at main order
if $\Phi_{1,\epsilon}$ satisfies an equation of the form \begin{equation}\label{phi1defa}
\Delta_K \,\Phi^s_{1,\epsilon}- \sum_m \sum_{a,b} \bigg( \tilde g^{ab}\,R_{mabs}- \Gamma_a^b(E_m) \Gamma_b^a(E_s) \bigg) \,\Phi^m_{1,\epsilon} = G_{2,\epsilon}(\sqrt{\epsilon} z), \end{equation} for some expression $G_{2,\epsilon}$ smooth on its argument. Observe that the operator acting on $\Phi_{1,\epsilon}$ in the left hand side is nothing but the Jacobi operator, which is invertible by the non-degeneracy condition on $K$. This implies the solvability of the above equation in $\Phi_{1,\epsilon}$. Furthermore, equation (\ref{phi1defa}) defines $\Phi_{1, \epsilon}$ as a smooth function on $K$, of order $\epsilon$, more precisely we have \begin{equation} \label{ePhi1e}
\| \Phi_{1,\epsilon}\|_{\infty } +\| \partial_a
\Phi_{1,\epsilon}\|_{\infty} +\|\partial^2_a \Phi_{1,\epsilon}\|_{\infty } \leq C . \end{equation}
By our choice of $\mu_{1}$ and $\Phi_{1,\epsilon}$ we have solvability of equation (\ref{eq:eqw2}) in $w_{2,\epsilon}$. Moreover, it is straightforward to check that \begin{eqnarray*}
|H_{2,\epsilon} ( z, \xi ) | &\leq& C \max\left\{\epsilon^{2 } ,\epsilon^{\frac{3}{2}} \right\}{1\over (1+|\xi|)^{N-2}} . \end{eqnarray*} Furthermore, for a given $\sigma \in (0,1)$ we have $$
\| H_{2,\epsilon} \|_{\epsilon, N-2, \sigma} \leq C \epsilon^{\frac{3}{2}} . $$ Proposition \ref{linear} thus gives then that \begin{equation} \label{ew2}
\| D^2_\xi w_{2,\epsilon} \|_{\epsilon , N-2 , \sigma } + \| D_\xi w_{2,\epsilon} \|_{\epsilon , N-3 , \sigma } +\|w_{2,\epsilon} \|_{\epsilon, N-4, \sigma}\leqslant C \epsilon^{\frac{3}{2}} \end{equation} and that there exists a positive constant $\beta$ (depending only on $\Omega, K$ and $n$) such that for any integer $\ell$ there holds $$
\|\nabla^{(\ell)}_{z} w_{2,\epsilon}(z,\cdot)\|_{\epsilon,N-4,\sigma} \leq \beta C_\ell \,
\epsilon^{\frac{3}{2}}, $$ where $C_\ell$ depends only on $\ell$, $p$, $K$.
\noindent Arguing as in the previous step for $I=1$, we see that with this choice of $\mu_{1, \epsilon}$, $\Phi_{1,\epsilon}$ and $w_{2, \epsilon}$ we get that $$
\| - {\mathcal A}_{\mu_\epsilon , \Phi_\epsilon } v_{2,\epsilon} + \epsilon\mu_\epsilon^2 hv_{2,\epsilon}
- \mu_\epsilon^{\mp \epsilon\frac{N-2}{2}}\,v_{2,\epsilon}^{p\pm \epsilon} \|_{\epsilon, N-2 , \sigma} \leq C\epsilon^{\frac{3}{2}}. $$
\noindent Expansion at an arbitrary order. \ \ We take now an arbitrary integer $I$. Let \begin{equation}\label{eqmu}
\mu_{I+1,\epsilon}:= \mu_0+\epsilon \mu_{1}+\epsilon^2\mu_2\cdots +\epsilon^{I-1} \mu_{I-1} +\epsilon^{I} \mu_{I} , \end{equation} \begin{equation}\label{eqPhi} \Phi_{\epsilon}=\Phi_{1,\epsilon}+\cdots +\Phi_{I-1,\epsilon} + \Phi_{I, \epsilon} \end{equation} and \begin{equation} \label{eqWWW} v_{I+1, \epsilon} = w_0 (\xi ) + w_{1, \epsilon} (z, \xi) + \ldots + w_{I, \epsilon} (z, \xi ) + w_{I+1, \epsilon} (z, \xi ) \end{equation} where $\mu_0, \mu_{1} , \cdots , \mu_{I-1,\epsilon}$, $\Phi_{1,\epsilon}, \cdots , \Phi_{I,\epsilon}$ and $w_{1, \epsilon} $, .. , $w_{I, \epsilon}$ have already been constructed following an iterative scheme, as described in the previous steps of the construction.
In particular one has, for any $i=1, \ldots , I-1$ \begin{equation}
\label{emuie} \| \mu_{i}\|_{\infty} +\| \partial_a
\mu_{i}\|_{\infty} +\|\partial^2_a \mu_{i}\|_{\infty } \leq C \end{equation} \begin{equation}
\label{ePhiie} \| \Phi_{i,\epsilon}\|_{\infty } +\| \partial_a
\Phi_{i,\epsilon}\|_{\infty} +\|\partial^2_a \Phi_{i,\epsilon}\|_{\infty } \leq C \epsilon^{i-1} \end{equation} We have \begin{eqnarray} \label{ewi}
\| D^2_\xi w_{i,\epsilon} \|_{\epsilon , N-2 , \sigma } + \|
D_\xi w_{i,\epsilon} \|_{\epsilon , N-3 , \sigma } +\|w_{i,\epsilon} \|_{\epsilon, N-4 , \sigma} \leq C \epsilon^{i-\frac{1}{2}}. \end{eqnarray} and, for any integer $\ell$, $\sqrt{\epsilon}z\in K$, \begin{equation}\label{eq:estwi}
\|\nabla^{(\ell)}_{z} w_{i,\epsilon}(z,\cdot)\|_{\epsilon,N-2,\sigma} \leq \beta C_l \epsilon^{i-\frac{1}{2}}. \end{equation}
The new triplet $(\mu_{I} , \Phi_{I, \epsilon} , w_{I+1 , \epsilon} )$ will be found reasoning as in the construction of $(\mu_{1} , \Phi_{1} , w_{2 , \epsilon} )$. Computing $\Xi_\epsilon (v_{I+1 , \epsilon })$ we get \begin{eqnarray}\label{rivoli5a} &&-\hat {\mathcal A}_{\mu_\epsilon , \Phi_\epsilon } v_{I+1,\epsilon} + \epsilon\mu_{I+1,\epsilon}^2 \, hv_{I+1,\epsilon}- \mu_{I+1,\epsilon}^{\mp\epsilon\frac{N-2}{2}}v_{I+1,\epsilon}^{p\pm\epsilon}\\[3mm] &=& - \Delta_{\mathbb R^{N}} w_{I+1,\epsilon}
- p w_0^{p-1} w_{I+1,\epsilon}+\epsilon \mu_{I+1,\epsilon}^2 \, h \, w_{I+1,\epsilon} + H_{I+1} (z, \xi)
+ Q_\epsilon (w_{I+1,\epsilon}) \nonumber \end{eqnarray} where the function $H_{I+1,\epsilon}$ is given by \begin{eqnarray}\label{wIepsilonl} H_{I+1,\epsilon}&=& \epsilon^{I+1} \left\{\mu_0\, \Delta_K
(\mu_I) \,Z_0 +\mu_I\Delta_K(\mu_0)Z_0- \, \nabla_K(\mu_0)\nabla_K(\mu_I)\mathcal{T}_1(w_0)\right. \nonumber\\ & &\qquad\qquad \left.+2 \mu_0\,\mu_I(\mathcal{T}_2(w_0)-\mathcal{T}_3(w_0))+ 2\mu_0\, \mu_Ih w_0\right.\nonumber\\ &&\qquad\qquad \left.\pm \frac{(N-2)^2}{16} w_0^p\left(\ln (w_0)+1\right)\right\}\nonumber\\ &&+\epsilon^{I+\frac{1}{2}} \, \mu_0 \left\{ -\Delta_K\Phi_{I,\epsilon}D_\xi w_0 + {1\over 3}\,\, \sum\limits_{i,j} \sum\limits_{m,l} R_{mijl} ( \xi_m \Phi_{I,\epsilon}^l +\xi_I\Phi_{I,\epsilon}^m) \partial^2_{ij} w_0,\right. \nonumber\\ & & \qquad\qquad\left.+ \sum\limits_j \bigg[ \sum_s \frac23 R_{mssj}+\sum\limits_{m,a,b}
\big( \, R_{mabj}- \Gamma_{am}^b \Gamma_{bj}^a \big)\bigg] \Phi_{I,\epsilon}^m
\partial_j w_0 \right\}\nonumber\\ &&+ \epsilon^{I+\frac{1}{2}}\mathcal{E}_{I+1,\epsilon}(y,\xi,w_0,w_{1,\epsilon},\cdots, w_{I,\epsilon},\mu_0,\cdots,\mu_{I-1}). \end{eqnarray} In (\ref{wIepsilonl}), $\mathcal{E}_{I+1, \epsilon}$ is a sum of functions of the form $$ \biggl[ \left( \mu_0 + \partial_a \mu_0 + \partial^2_a \mu_0 + \ldots + \mu_{I-1} + \partial_a \mu_{I-1} + \partial^2_a \mu_{I-1} \right) $$ $$
+ \left( \Phi_1 + \partial_a \Phi_1 + \partial^2_a \Phi_1 + \ldots + \Phi_{I-2} + \partial_a \Phi_{I-2} + \partial^2_a \Phi_{I-2} \right) \biggl] a(\sqrt{\epsilon }z) b(\xi ) $$ where $a(\sqrt{\epsilon} z)$ is a smooth function uniformly bounded, together with its derivatives, as $ \epsilon \to 0$, while the function $b$ is such that $$
\sup_{\xi } (1+|\xi|^{N-2} ) |b (\xi ) |< \infty. $$ Finally the term $Q_\epsilon (w_{I,\epsilon} )$ in (\ref{rivoli5a}) is a sum of quadratic terms in $w_{I+1,\epsilon}$ like $$ (\mu_0+\epsilon\mu_1+\cdots+\epsilon^{I}\mu_I)^{\mp \frac{N-2}{2}\epsilon}\times $$ $$\left[(w_0 + w_{1,\epsilon} + w_{2,\epsilon}+\cdots+w_{I+1,\epsilon} )^{p\pm \epsilon} \right. $$ $$ \left.- (w_0 + w_{1,\epsilon} +\cdots+w_{I,\epsilon})^{p\pm \epsilon}\right. $$ $$ \left. -(p\pm \epsilon) (w_0 + w_{1,\epsilon} +\cdots+w_{I,\epsilon})^{p-1\pm \epsilon} w_{I+1,\epsilon}\right] $$ and linear terms in $w_{I,\epsilon}$ multiplied by a term of order $\epsilon$, like $$ p \left( ( w_0 + w_{1,\epsilon} + \ldots + w_{I-1, \epsilon} )^{p-1\pm \epsilon} - w_0^{p-1\pm \epsilon} \right) w_{I,\epsilon}. $$
Consider the following problem \begin{eqnarray}\label{eq:eqwI}
- \Delta_{\mathbb R^{N}} w_{I+1,\epsilon}
- p w_0^{p-1} w_{I+1,\epsilon}+\epsilon \mu_{I+1,\epsilon}^2 \, h \, w_{I+1,\epsilon} & & =-H_{I+1,\epsilon} ( z , \xi ) \quad\hbox{ in } \mathcal{D} \nonumber \\
\quad w_{I+1, \epsilon} = 0 & & \quad {\mbox {on}} \quad \partial \hat \mathcal{D}. \end{eqnarray} Again by Proposition \ref{linear}, the above problem is solvable in $w_{I+1 , \epsilon}$ if $H_{I+1,\epsilon}$ is $L^2$-orthogonal to $Z_j$, $j=0, 1,\cdots,N$. These orthogonality conditions will define the parameters $\mu_{I}$ and the normal section $\Phi_{I,\epsilon}$.
\noindent Projection onto $Z_0$ and choice of $\mu_{I}$. \ \ We define $\mu_{I}$ to make, at main order, \begin{eqnarray*}
\int_{\hat\mathcal{D}} H_{I+1,\epsilon}Z_0d\xi=0. \end{eqnarray*}
The above relation defines $\mu_{I}$ as a smooth function of $\sqrt{\epsilon} z $ in $K$. From estimates (\ref{ewi}) we get that \begin{equation} \label{emu1e}
\| \mu_{I}\|_{ \infty } +\| \partial_a
\mu_{I}\|_{ \infty } +\|\partial^2_a \mu_{I}\|_{ \infty } \leq C \end{equation}
\noindent Projection onto $Z_s$ and choice of $\Phi_{I,\epsilon}$. \ \ Multiplying $H_{I+1,\epsilon}$ with $\partial_s w_0$, integrating over $\hat\mathcal{D}$ and arguing as in the construction of $\Phi_{I, \epsilon}$, we get \begin{eqnarray*}
&&\left(C_0\epsilon^{I+\frac{1}{2}} \, \mu_0 \right)^{-1}\int_{\hat\mathcal{D}}H_{I+1,\epsilon}\,\partial_s w_0 d\xi= - \Delta_K \,\Phi^l_{I,\epsilon}\\
& +&
\sum_m \sum_{a,b} \,\bigg(\tilde g^{ab}\, R_{maal}- \Gamma_a^b(E_m) \Gamma_b^a(E_l) + O(\epsilon^{\frac{N-2}{2}}) \bigg) \,\Phi^m_{I,\epsilon}+ \int_{\hat\mathcal{D}}\mathfrak{G}_{I+1,\epsilon}\,\partial_l w_0. \end{eqnarray*} We then conclude that $H_{I+1,\epsilon}(z,\xi, w_0, \dots, w_{I,\epsilon})$, the right-hand side of (\ref{eq:eqwI}), is $L^2$-orthogonal to $Z_l$ ($l=1,\cdots,N$) if and only if $\Phi_{I,\epsilon}$ satisfies an equation of the form \begin{eqnarray}\label{phi1def} && \Delta_K \,\Phi^l_{I,\epsilon}- \sum_m \sum_{a,b} \bigg( \tilde g^{ab}\,R_{maal}- \Gamma_a^b(E_m) \Gamma_b^a(E_l) + O(\epsilon^{\frac{N-2}{2}}) \bigg) \,\Phi^m_{I,\epsilon} \nonumber\\ &=& \epsilon^{I+\frac{1}{2}} G_{I+1,\epsilon}(\sqrt{\epsilon} z), \end{eqnarray} where $G_{I+1,\epsilon}$ is a smooth function on $K$, uniformly bounded as $\epsilon \to 0$. Using again the non-degeneracy condition on $K$ we have solvability of the above equation in $\Phi_{I,\epsilon}$. Furthermore, taking into account (\ref{rivoli5a}), we get \begin{eqnarray} \label{ePhiIe}
\| \Phi_{I,\epsilon}\|_{ \infty } +\| \partial_a
\Phi_{I,\epsilon}\|_{ \infty } +\|\partial^2_a \Phi_{I,\epsilon}\|_{ \infty
} \leq C \epsilon^{I+\frac{1}{2}}. \end{eqnarray}
By our choice of $\mu_{I+1}$ and $\Phi_{I+1,\epsilon}$ we have solvability of equation (\ref{eq:eqwI}) in $w_{I+1,\epsilon}$. Moreover, it is straightforward to check that $$
|H_{I+1,\epsilon} (\epsilon z, \xi ) | \leq C \epsilon^{I+\frac{1}{2}}{1 \over
(1+|\xi|)^{N-2}} . $$ Furthermore, for a given $\sigma \in (0,1)$ we have $$
\| H_{I+1,\epsilon} \|_{\epsilon, N-2, \sigma} \leq C \epsilon^{I+\frac{1}{2}}. $$ Proposition \ref{linear} gives then that \begin{eqnarray} \label{ewI}
\| D^2_\xi w_{I+1,\epsilon} \|_{\epsilon , N-2 , \sigma } + \|
D_\xi w_{I+1,\epsilon} \|_{\epsilon , N-3 , \sigma } +\|w_{I+1,\epsilon} \|_{\epsilon, N-4 , \sigma}\leq C \epsilon^{I+\frac{1}{2}}, \end{eqnarray} and that there exists a positive constant $\beta$ (depending only on $K$ and $N$) such that for any integer $\ell$ there holds, for $\sqrt{\epsilon}z\in K$,
$$ \|\nabla^{(\ell)} w_{I+1,\epsilon}(z,\cdot)\|_{\epsilon,N-2,\sigma} \leq \beta C_l
\epsilon^{I+\frac{1}{2}}. $$
\noindent With this choice of $\mu_{I}$, $\Phi_{I,\epsilon}$ and $w_{I+1 , \epsilon}$ we obtain that \begin{eqnarray*}
\| -{\mathcal A}_{\mu_\epsilon ,
\Phi_\epsilon } v_{I+1,\epsilon} + \epsilon\mu_{I+1,\epsilon}^2 \, hv_{I+1,\epsilon}- \mu_\epsilon^{\mp\epsilon\frac{N-2}{2}}v_{I+1,\epsilon}^{p\pm\epsilon} \|_{\epsilon , N-2 , \sigma}\leq C \epsilon^{I+\frac{1}{2}}. \end{eqnarray*}
\noindent This concludes our construction.
\end{proof}
\section{A global approximation}\label{s:linear}
Let us recall that if $u$ is a solution to problem (\ref{p}), and we define $$ u(x) =(1+\alpha_\epsilon) \epsilon^{-{N-2 \over 4}} \tilde{u}(\epsilon^{-{1 \over 2}} x ). $$ Then $\tilde{u}$ satisfies the following equation \begin{equation} \label{changea} -\Delta_{g^\epsilon} \tilde{u} +\epsilon h \tilde{u}= \tilde{u}^{{N+2\over N-2} \pm\epsilon} \quad \mbox{ in } \mathcal{M}_\epsilon ; \quad
\tilde{u}>0 \mbox{ in }\mathcal{M}_\epsilon, \end{equation}
where $\Delta_{g^\epsilon}$ denotes the Laplace-Beltrami operator on $\mathcal{M}_\epsilon$ is given by
\begin{equation*} \label{laplacee} \Delta_{g^\epsilon}=\frac{1}{\sqrt{\mbox{det}\ g^\epsilon}}\partial_A(\sqrt{\mbox{det}\ g^\epsilon}(g^\epsilon)^{AB}\partial_B)
\end{equation*}
here indices $A$ and $B$ run between $1$ and $n=N+k$, and $g^\epsilon$ is the scaled metric on $\mathcal{M}_\epsilon$ whose coefficient are defined by \begin{equation*} \label{ge}
g^\epsilon_{\alpha,\beta}(z,x)=g_{\alpha,\beta}(\sqrt{\epsilon} z,\sqrt{\epsilon} x)
\end{equation*}
where $g_{\alpha,\beta}$ are the coefficients of the metric $g$ on $\mathcal{M}$.
Let $\mu_\epsilon (y) $, $\Phi_\epsilon (y)$ and $v_{I+1 , \epsilon}$ be the functions whose existence and properties have been established in Lemma \ref{Construction}. We define locally around $K_\epsilon := {K\over \sqrt{\epsilon}}$ the function \begin{equation} \label{Vdef} \tilde{U}_\epsilon (z, x):= \, \mu_{\epsilon}^{-{N-2 \over 2}} (\sqrt{\epsilon} z) \, v_{I+1 , \epsilon}
\left( z, \, \frac{x-\sqrt{\epsilon} \Phi_\epsilon (\sqrt{\epsilon} z)}{ \mu_\epsilon (\sqrt{\epsilon} z)} \right) \chi_\epsilon (|(x-\sqrt{\epsilon} \Phi_\epsilon (\sqrt{\epsilon} z) )|) \end{equation} where $z \in K_\epsilon$. The function $\chi_\epsilon$ is a smooth cut-off function with \begin{equation}\label{magaly}
\chi_\epsilon (r) = \left\{
\begin{array}{ll}
1, & \hbox{for} \quad r \in [0,2 \epsilon^{-\gamma} ] \\[3mm]
0, & \hbox{for} \quad r \in [3 \epsilon^{-\gamma} , 4\epsilon^{-\gamma} ],
\end{array} \right. \end{equation} and $$
|\chi_\epsilon^{(l)} (r) | \leq C_l \epsilon^{l \gamma}, \quad \forall l\geq 1, $$ for some $\gamma \in ({1\over 2} , 1)$ to be fixed later.
We will use the notation \begin{equation} \label{defTTa} \tilde{u}=\widetilde{{\mathcal T}}_{\mu_\epsilon , \Phi_\epsilon} (\tilde{v}) \end{equation} if and only if $\tilde{u}$ and $\tilde{v}$ satisfy \begin{equation*}
\tilde{u}=\mu_{\epsilon}^{-{N-2 \over 2}} (\sqrt{\epsilon} z) \, \tilde{v} \left( z, \, \frac{x-\sqrt{\epsilon} \Phi_\epsilon (\sqrt{\epsilon} z)}{ \mu_\epsilon (\sqrt{\epsilon} z)} \right) . \end{equation*}
The function $\tilde{U}_\epsilon$ is well defined in a small neighborhood of $K_\epsilon$. We will look for a solution to (\ref{changea}) of the form $$ \tilde{u}_\epsilon = \tilde{U}_\epsilon +\phi. $$ Thus $\phi$ satisfies the following problem \begin{equation}\label{nonlinearproblem}
-\Delta_{g^\epsilon} \phi + \epsilon h \phi- (p\pm\epsilon) \tilde{U}_\epsilon^{p\pm\epsilon-1} \phi =S_\epsilon (\tilde{U}_\epsilon) + N_\epsilon (\phi) \quad \text{ in } \mathcal{M}_\epsilon, \end{equation} where \begin{equation} \label{eomegaeps} S_\epsilon (\tilde{U}_\epsilon )= \Delta_{g^\epsilon} \tilde{U}_\epsilon - \epsilon h\tilde{U}_\epsilon + \tilde{U}_\epsilon^p \end{equation} and \begin{equation} \label{Nomegaeps} N_\epsilon (\phi) = (\tilde{U}_\epsilon + \phi )^{p\pm\epsilon} - \tilde{U}_\epsilon^p - (p\pm\epsilon) \tilde{U}_\epsilon^{p\pm\epsilon-1} \phi. \end{equation} Define $$ L_\epsilon (\phi) = -\Delta_{g^\epsilon} \phi + \epsilon h \phi- (p\pm\epsilon) \tilde{U}_\epsilon^{p\pm\epsilon-1} \phi. $$ We shall solve the Non-Linear Problem (\ref{nonlinearproblem}) by using a fixed point argument based on the contraction Mapping Principle. To do so, we first establish some invertibility properties of the linear problem $$ L_\epsilon (\phi) = f \quad {\mbox {in}} \quad \mathcal{M}_\epsilon , $$ with $f\in L^2 (\mathcal{M}_\epsilon )$. We do this in two steps. First we study the above problem in a strip close to the scaled manifold $K_\epsilon= {K\over \sqrt{\epsilon}}$. Let $\gamma \in ({1\over 2} , 1)$ be the number fixed before in (\ref{magaly}) and consider \begin{equation} \label{omegaepsilongamma} \mathcal{M}_{\epsilon, \gamma} := \{ x \in \mathcal{M}_\epsilon \, : \, {\mbox {dist}}_{g^\epsilon} (x,K_\epsilon ) <2 \epsilon^{-\gamma} \}. \end{equation}
We are first interested in solving the following problem: given $f \in L^2 (\mathcal{M}_{\epsilon , \gamma})$ \begin{equation} \label{lineare}
-\Delta_{g^\epsilon} \phi + \epsilon h \phi- (p\pm \epsilon) \tilde{U}_\epsilon^{p\pm \epsilon-1} \phi =f\quad \text{ in } \mathcal{M}_{\epsilon , \gamma}. \end{equation}
We have the validity of the following result. \begin{proposition}\label{teouffa} There exist a constant $C>0$ and a sequence $\epsilon_l = \epsilon \to 0$ such that, for any $f \in L^2 (\mathcal{M}_{\epsilon , \gamma} )$ there exists a solution $\phi \in H^1_{g^\epsilon}(\mathcal{M}_{\epsilon , \gamma} ) $ to Problem (\ref{lineare}) such that \begin{equation} \label{uffa1}
\| \phi \|_{H^1_{\epsilon}} \leq C \epsilon^{- \max \{2, \frac{k}{2}\}} \| f
\|_{L^2 (\mathcal{M}_{\epsilon , \gamma} )}. \end{equation} \end{proposition}
The proof will be given in Section \ref{linearas}.
Using this, we can get the existence of solution to the linear problem in the whole domain $\mathcal{M}_\epsilon$.
\begin{proposition}\label{glinear} There exist a sequence $\epsilon_l \to 0$ and a positive constant $C >0$, such that, for any $f \in L^2 (\mathcal{M}_{\epsilon_l} )$, there exists a solution $\phi \in H^1 (\mathcal{M}_{\epsilon_l} )$ to the equation $$ L_{\epsilon_l }\phi = f \quad \hbox{in } \mathcal{M}_{\epsilon_l }. $$ Furthermore, \begin{equation} \label{gigio}
\| \phi \|_{H^1_{g^{\epsilon}} (\mathcal{M}_{\epsilon_l} )} \leq C\, \epsilon_l^{-\max \{
2, \frac{k}{2} \}} \| f \|_{L^2 (\mathcal{M}_{\epsilon_l})}. \end{equation} \end{proposition}
\begin{proof} By contradiction, assume that for all $\epsilon \to 0$ there exists a solution $(\phi_\epsilon , \lambda_\epsilon )$, $\phi_\epsilon \not= 0$, to \begin{equation} \label{uno} L_\epsilon (\phi_\epsilon )= \lambda_\epsilon \phi_\epsilon \quad {\mbox {in}} \quad \mathcal{M}_\epsilon , \end{equation} with \begin{equation}\label{due}
|\lambda_\epsilon | \epsilon^{-\max \{ 2, \frac{k}{2}\} } \to 0 , \quad {\mbox {as}} \quad \epsilon \to 0. \end{equation} Let $\eta_\epsilon $ be a smooth cut off function (like the one defined in (\ref{magaly})) so that $$ \eta_\epsilon = 1 \quad\hbox{ if dist$_{g^\epsilon}(z, K_\epsilon ) <{\epsilon^{-\gamma} \over 2}$\qquad and} \quad \eta_\epsilon =0 \quad \hbox{if dist$_{g^\epsilon}(z, K_\epsilon ) > \epsilon^{-\gamma} $.} $$ In particular one has that
$|\nabla_{K_\epsilon} \eta_\epsilon| \leq c
\epsilon^{\gamma}$ and $|\Delta_{K_\epsilon} \eta_\epsilon |\leq c \epsilon^{2\gamma}$ in the whole domain.
Define $\tilde \phi_\epsilon = \phi_\epsilon \eta_\epsilon$. Then $\tilde \phi_\epsilon$ solves \begin{equation}\label{tre}
L_\epsilon (\tilde \phi_\epsilon ) = \lambda_\epsilon \tilde \phi_\epsilon -\nabla_{K_\epsilon} \eta_\epsilon \nabla_{K_\epsilon} a \phi_\epsilon - \Delta_{K_\epsilon} \eta_\epsilon \phi_\epsilon \text{ in } \mathcal{M}_{\epsilon,\gamma}, \end{equation} where $\mathcal{M}_{\epsilon , \gamma}$ is the set defined in (\ref{omegaepsilongamma}). We now apply Proposition (\ref{teouffa}), that guarantees the existence of a sequence $\epsilon_l \to 0$ and a constant $c$ such that \begin{equation}
\label{quattro} \| \tilde \phi_{\epsilon_l} \|_{H^1_{\epsilon_l}} \leq c
\epsilon_l^{-\max \{2,\frac{k}{2}\}} \left[ \lambda_{\epsilon_l} \| \tilde \phi_{\epsilon_l}
\|_{L^2} + \| \nabla_{K_\epsilon} \eta_{\epsilon_l} \nabla_{K_\epsilon} \phi_{\epsilon_l} \|_{L^2} + \|
\Delta_{K_\epsilon} \eta_{\epsilon_l} \phi_{\epsilon_l} \|_{L^2} \right]. \end{equation} Observe now that, in the region where $\nabla_{K_\epsilon} \eta_{\epsilon_l} \not= 0$
and $\Delta_{K_\epsilon} \eta_{\epsilon_l} \not= 0$, the function $\tilde{U}_{\epsilon_l}$ can be uniformly bounded $|\tilde{U}_\epsilon (y) |\leq c \epsilon $, with a positive constant $c$, fact that follows directly from (\ref{Vdef}) and (\ref{boh1}). Furthermore, since we are assuming (\ref{due}), we see that in the region we are considering, namely where $\nabla_{K_\epsilon} \eta_{\epsilon_l} \not= 0$ and $\Delta_{K_\epsilon} \eta_{\epsilon_l} \not= 0$, the function $\phi_{\epsilon_l}$ satisfies the equation $$- \Delta_{K_\epsilon} \phi_{\epsilon_l} + \epsilon_l^2 a_{\epsilon_l} (y)
\phi_{\epsilon_l} = 0$$ for a certain smooth function $a_{\epsilon_l}$, which is uniformly positive and bounded as $\epsilon_l \to 0$. Elliptic estimates give that, in this region, $|\phi_{\epsilon_l} |\leq c e^{-\epsilon_l^{\gamma'}}$, and $|\nabla_{K_\epsilon} \phi_{\epsilon_l} | \leq c e^{-\epsilon_l^{\gamma'}}$ for some $\gamma'>0$ and $c>0$. Inserting this information in (\ref{quattro}), it is easy to see that $$
\| \tilde \phi_{\epsilon_l} \|_{H^1_{\epsilon_l}} \leq c \epsilon_l^{-\max \{2,\frac{k}{2}\}}
\lambda_{\epsilon_l} \| \tilde \phi_{\epsilon_l} \|_{H^1_{\epsilon_l}} (1+ o(1)) $$ where $o(1) \to 0$ as $\epsilon_l \to 0$. Taking into account (\ref{due}) the above inequality gives a contradiction with the fact that, for all $\epsilon$, the function $\phi_\epsilon$ is not identically zero. This concludes the proof. \end{proof}
\begin{proof}[Proof of the main Theorem] By Proposition \ref{glinear}, $\phi \in H^1_{g^\epsilon} (\mathcal{M}_\epsilon )$ is a solution to (\ref{nonlinearproblem}) if and only if $$ \phi = L_\epsilon^{-1} \left( S_\epsilon (\tilde{U}_\epsilon ) + N_\epsilon (\phi ) \right). $$ Notice that \begin{equation}
\label{nonno1} \| N_\epsilon (\phi ) \|_{L^2 (\mathcal{M}_\epsilon )} \leq C \begin{cases}
\| \phi \|_{H^1 (\mathcal{M}_\epsilon )}^p & \text{ for } p\leq 2, \\
\| \phi \|_{H^1 (\mathcal{M}_\epsilon )}^2 & \text{ for } p>2
\end{cases} \quad\qquad \| \phi \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )} \leq 1 \end{equation} and \begin{eqnarray} \label{nonno2}
&&\| N_\epsilon (\phi_1 ) - N_\epsilon (\phi_2 ) \|_{L^2 (\mathcal{M}_\epsilon )} \nonumber\\ &&\leq C \,\begin{cases}
\left( \| \phi_1 \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )}^{p-1} + \| \phi_2 \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )}^{p-1} \right)
\| \phi_1 -\phi_2 \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )} & \text{ for } p\leq 2, \\[3mm]
\left( \| \phi_1 \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )} + \| \phi_2 \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )} \right) \| \phi_1 -\phi_2 \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )} & \text{ for } p>2
\end{cases}, \end{eqnarray}
for any $\phi_1$, $\phi_2$ in $H^1_{g^\epsilon} (\mathcal{M}_\epsilon )$ with $\| \phi_1
\|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )}$, $ \| \phi_2 \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )} \leq 1.$
Defining $T_\epsilon :H^1_{g^\epsilon} (\mathcal{M}_\epsilon ) \to H^1_{g^\epsilon} (\mathcal{M}_\epsilon )$ as $$ T_\epsilon (\phi ) = L_\epsilon^{-1} \left( S_\epsilon (\tilde{U}_\epsilon ) + N_\epsilon (\phi ) \right) $$ we will show that $T_\epsilon$ is a contraction in some small ball in $H^1_{g^\epsilon} (\mathcal{M}_\epsilon ) $. A direct consequence of (\ref{bf4}), we have $$
\|S_\epsilon (\tilde{U}_\epsilon )\|_{L^2(\mathcal{M}_\epsilon)}\leq C\epsilon^{I+\frac{1}{2}}. $$ Using this inequality and by (\ref{nonno1}), (\ref{nonno2}) and (\ref{gigio}), we obtain $$
\| T_\epsilon (\phi ) \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )} \leq C \epsilon^{- \max \{ 2 , \frac{k}{2} \}} \begin{cases}
\left( \epsilon^{I+\frac{1}{2} }+ \| \phi \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )}^p \right) & \text{ for } p\leq 2, \\
\left( \epsilon^{I+\frac{1}{2}} + \| \phi \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )}^2 \right) & \text{ for } p>2.
\end{cases} $$ Now we choose integers $d$ and $I$ so that $$ d> \begin{cases}
{\max \{ 2 ,\frac{ k}{2} \} \over p-1}& \text{ for } p\leq 2, \\
\max \{ 2 ,\frac{ k}{2} \} & \text{ for } p>2
\end{cases} \quad I > d-\frac{1}{2} + \max \{ 2 , \frac{k}{2} \}.
$$
Thus one easily gets that $T_\epsilon$ has a unique fixed point in set
$${\mathcal B} = \{ \phi \in H^1_{g^\epsilon} (\mathcal{M}_\epsilon )\, : \, \| \phi \|_{H^1_{g^\epsilon} (\mathcal{M}_\epsilon )} \leq \epsilon^d \},
$$
as a direct application of the contraction mapping Theorem. This concludes the proof. \end{proof}
\section{A linear problem: proof of Proposition \ref{teouffa}}\label{linearas}
The quadratic functional associated to problem (\ref{lineare}) given by \begin{equation}\label{functional1} E(\phi ) = {1\over 2} \int_{\mathcal{M}_{\epsilon,\gamma}}
(|\nabla_{g^\epsilon} \phi |^2 + \epsilon h \phi^2 - (p\pm \epsilon) \tilde{U}_\epsilon^{p\pm \epsilon-1} \phi^2 ) \end{equation} for functions $\phi \in H^1_{g^\epsilon}(\mathcal{M}_{\epsilon,\gamma}) $.
Let $(y,x)\in \mathbb{R}^{k+N}$ be the local coordinates along $K_\epsilon$ introduced in (\ref{eq:fc}). With abuse of notation we will denote \begin{equation} \label{bb}
\phi (\mathfrak F ( y, x))= \phi(z, x ),\quad \mbox{with}\ \ y=\sqrt{\epsilon}z. \end{equation} Since the original variable $(z, x)\in \mathbb{R}^{k+N}$ are only local coordinates along $K_\epsilon$ we let the variable $(z, x)$ vary in the set $\mathcal{C}_\epsilon$ defined by \begin{equation}\label{ddomain}
\mathcal{C}_\epsilon = \{ (z,x) \ / \ \sqrt{\epsilon} z\in \ K,\quad | x| < \epsilon^{-\gamma} \}. \end{equation} We write $\mathcal{C}_\epsilon=\frac{1}{\sqrt{\epsilon}} K\times\hat {\mathcal{C}_\epsilon}$ where \begin{equation}\label{dddomain}
\hat {\mathcal{C}_\epsilon} = \{ x \ / \ |x| < \epsilon^{-\gamma} \}. \end{equation} Observe that $\hat {\mathcal{C}_\epsilon}$ approaches, as $\epsilon \to 0$, the whole space $\mathbb{R}^N$.
In these new local coordinates, the energy density associated to the energy $E$ in (\ref{functional1}) is given by \begin{equation} \label{e}
\frac12 \left[|\nabla_{g^\epsilon} \phi|^2+\epsilon h \phi^2 - (p\pm\epsilon) \tilde{U}_\epsilon^{p\pm \epsilon-1} \phi^2 \right] \sqrt{\det(g^\epsilon)}, \end{equation} where $\nabla_{g^\epsilon}$ denotes the gradient in the new variables and where $g^\epsilon$ is the metric in the coordinates $(z, x)$. Using the expansions contained in the proof of Lemma \ref{scaledlaplacian}, we have that, if $(z,x)$ vary in $\mathcal{C}_\epsilon$, then, the energy functional (\ref{functional1}) in the new variables (\ref{bb}) is given by \begin{eqnarray}\label{energydensity} E ( \phi) & = &\int_{K_\epsilon \times \hat{\mathcal{C}_\epsilon}} \left(\frac12 (
|\nabla_x \phi|^2 +\epsilon h \phi^2 - (p\pm \epsilon) \tilde{U}_\epsilon^{p\pm \epsilon-1} \phi^2 ) \right) \sqrt{\det(g^\epsilon)} \, dz \, dx\nonumber\\ && -\frac{\epsilon}{6}\int_{K_\epsilon \times \hat{ \mathcal{C}_\epsilon}} R_{islj} x_lx_s \,\partial_{i}\phi\partial_{j}\phi\,\sqrt{\det(g^\epsilon)} \, dz \, dx\\
&&
+\frac12 \int_{K_\epsilon \times \hat{\mathcal{C}_\epsilon}} |\nabla_{K_\epsilon}\phi|^2\,\sqrt{\det(g^\epsilon)} \, dz \, dx+
\int_{K_\epsilon \times \hat{\mathcal{C}_\epsilon}} B(\phi,\phi)\,\sqrt{\det(g^\epsilon)} \, dz \, dx, \nonumber \end{eqnarray} where we denoted by $B(\phi,\phi)$ a quadratic term in $\phi$ that can be expressed in the following form \begin{eqnarray}\label{defBB}
B(\phi,\phi)= O \left( \epsilon^{\frac{3}{2}} | x |^3 \right)\partial_{i} \phi \partial_{j} \phi +{\epsilon }\,|\nabla_{K_\epsilon}\phi|^2\,
O(\sqrt{\epsilon} |x|) +\partial_j \phi \partial_{\bar a}\phi \left(\mathcal{O}(\sqrt{\epsilon} |x
|)\right) \end{eqnarray} and we used the Einstein convention over repeated indices. Furthermore we use the notation $\partial_a = \partial_{y_a} $ and $\partial_{\bar a} = \partial_{z_a}$. A detailed proof of expansion (\ref{energydensity}) can be found in \cite{demamu}.
\noindent Given a function $\phi \in H^1_{g^{\epsilon}}(\mathcal{M}_{\epsilon,\gamma})$, we decompose it as \begin{equation}\label{decomp} \phi= \left[{\delta \over \, \mu_\epsilon} \widetilde{{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} ( Z_0) + \sum_{j=1}^{N} {d^j \over \, \mu_\epsilon} \widetilde{{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} ( Z_j)
+ {e \over \, \mu_\epsilon} \widetilde{{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z) \right] \bar \chi_\epsilon + \phi^\bot \end{equation} where the expression $\widetilde{{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} ( v)$ is defined in (\ref{defTTa}), the functions $Z_0$ and $Z_j$ are already defined in (\ref{lezetas}) and where $Z$ is the eigenfunction, with $\int_{\mathbb{R}^N} Z^2 =1$, corresponding to the unique positive eigenvalue $\lambda_0$ in $L^2 (\mathbb{R}^N)$ of the problem \begin{equation}\label{lambda0} \Delta_{\mathbb{R}^N} \phi + p w_0^{p-1} \phi = \lambda_0 \phi \quad {\mbox {in}} \quad \mathbb{R}^N. \end{equation}
It is worth mentioning that $Z (\xi )$ is even and it has exponential decay of order $O(e^{-\sqrt{\lambda_0} |\xi|} )$ at infinity. The function $\bar \chi_\epsilon$ is a smooth cut off function defined by \begin{equation}
\label{chibar} \bar \chi_\epsilon (x) = \hat \chi_\epsilon \left( \left|\left({ x-
\sqrt{\epsilon}\Phi_\epsilon \over \, \mu_\epsilon}\right) \right| \right), \end{equation} with $\hat \chi(r) = 1$ for $r \in (0,{3\over 2} \epsilon^{-\gamma} )$, and $\chi(r)=0$ for $r>2\epsilon^{-\gamma}$. Finally, in (\ref{decomp}) we have that $\delta = \delta (\sqrt{\epsilon} z)$, $d^j = d^j (\sqrt{\epsilon} z)$ and $e= e(\sqrt{\epsilon} z)$ are function defined in $K$ such that $\forall z\in K_\epsilon$ \begin{equation} \label{orth1} \int_{\hat{\mathcal{C}}_\epsilon} \phi^\bot {\widetilde{{\mathcal T}}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0) \bar \chi_\epsilon d x = \int_{\hat{\mathcal{C}}_\epsilon} \phi^\bot {\widetilde{{\mathcal T}}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_j) \bar \chi_\epsilon = \int_{\hat{\mathcal{C}}_\epsilon} \phi^\bot {\widetilde{{\mathcal T}}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z) \bar \chi_\epsilon=0. \end{equation} We will denote by $(H_\epsilon^1)^\bot$ the subspace of the functions in $H_\epsilon^1$ that satisfy the orthogonality conditions (\ref{orth1}).
A direct computation shows that $$ \delta (\sqrt{\epsilon} z) = {\int \phi {\widetilde{{\mathcal T}}}_{\, \mu_\epsilon , \Phi_\epsilon} (Z_0) \over \, \mu_\epsilon \int Z_0^2} (1+ O(\epsilon )) + O(\epsilon ) (\sum_j d^j (\sqrt{\epsilon} z) + e (\sqrt{\epsilon} z)), \quad $$ $$ d^j (\sqrt{\epsilon} z) = {\int \phi {\widetilde{{\mathcal T}}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_j) \over \, \mu_\epsilon \int Z_j^2} (1+ O(\epsilon )) + O(\epsilon ) (\delta (\sqrt{\epsilon} z) + \sum_{i\not= j} d^i (\sqrt{\epsilon} z) + e (\sqrt{\epsilon} z)), $$ and $$ e(\sqrt{\epsilon} z) = {\int \phi {\widetilde{{\mathcal T}}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z) \over \, \mu_\epsilon \int Z^2} (1+ O(\epsilon)) + O(\epsilon) (\delta (\sqrt{\epsilon} z) +\sum_j d^j (\sqrt{\epsilon} z) ). $$ Observe that, since $\phi \in H_{g^\epsilon}^1$, one easily get that the functions $\delta $, $d^j$ and $e$ belong to the Hilbert space \begin{equation}\label{H1K} {\mathcal H}^1 (K) = \{ \zeta \in {\mathcal L}^2 (K) \, : \, \partial_a \zeta \in {\mathcal L}^2 (K),\quad a=1,\cdots,k \}. \end{equation}
Thanks to the above decomposition (\ref{decomp}), we have the validity of the following expansion for $E(\phi)$.
Observe that in the region we are considering the function $\tilde{U}_\epsilon$ is nothing but $\tilde{U}_\epsilon= {\widetilde{{\mathcal T}}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (v_{I+1 , \epsilon})$, where $v_{I+1, \epsilon}$ is the function whose existence and properties are proven in Lemma \ref{Construction}. For the argument in this part of our proof it is enough to take $I=3$, and for simplicity of notation we will denote by $\hat w$ the function $v_{I+1 , \epsilon}$ with $I=3$. Referring to (\ref{bf4}) we have \begin{equation}\label{hatw} \hat w (z, \xi) = w_0 (\xi) + \sum_{i=1}^4 w_{i,\epsilon} (z, \xi) \end{equation} where $w_0$ is defined by (\ref{roma1}) and \begin{equation}\label{hatw1}
\| D^2_\xi w_{i+1,\epsilon} \|_{\epsilon , N-2 , \sigma } + \|
D_\xi w_{i+1,\epsilon} \|_{\epsilon , N-3 , \sigma } +\|w_{i+1,\epsilon} \|_{\epsilon, N-4 , \sigma}\leqslant C \epsilon^{i+\frac{1}{2}} \end{equation} and, for any integer $\ell$ $$
\|\nabla^{(\ell)}_{y} w_{i+1,\epsilon}(y,\cdot)\|_{\epsilon,N-2,\sigma} \leq \beta C_l \epsilon^{i+\frac{1}{2}} \qquad \quad y = \sqrt{\epsilon} z \in K $$ for any $i=0, 1, 2, 3$.
\begin{theorem} \label{teo4.1} Let $\gamma= 1-\sigma$, for some $\sigma >0$ and small. Assume we write $\phi \in H^1_\epsilon$ as in (\ref{decomp}) and let $d= (d^1 , \ldots , d^{N})$. Then, there exists $\epsilon_0>0$ such that, for all $0<\epsilon <\epsilon_0$, the following expansion holds true \begin{equation}\label{EEE} E(\phi ) = E(\phi^\bot ) + \epsilon^{-\frac{k}{2}} \left[ P_\epsilon (\delta ) +Q_\epsilon (d ) + R_\epsilon (e) \right] +{\mathcal G} (\phi^\bot , \delta , d , e). \end{equation} In (\ref{EEE}) \begin{equation}\label{q0e} P_\epsilon (\delta ) = P (\delta ) + P_1 (\delta) \end{equation} with \begin{eqnarray*}
P (\delta ) &=&{A_\epsilon \over 2} \int_K \epsilon
|\nabla_K ( \delta (1 + o(\epsilon) \beta_1^\epsilon (y) ) )|^2\nonumber\\ && +\epsilon \int_K \delta^2\left(2c_{3,N}h -c_{2,N} \sum\limits_j \bigg[ \sum_s \frac23 R_{mssj}+\sum\limits_{m,a,b}
\big( \,{\tilde g}^{ab} R_{mabj}- \Gamma_{am}^b \Gamma_{bj}^a \big)\bigg]\right)\nonumber\\ && \mp\epsilon c_{4,N}\int_K \frac{\delta^2}{\mu_0} \end{eqnarray*} where $A_\epsilon$ a real number such that $\lim\limits_{\epsilon \to 0 } A_\epsilon = c_{1,N}:= \int_{\mathbb{R}^N} Z_0^2 $, and $c_{2,N},c_{3,N},c_{4,N}$ are given in (\ref{c2})-(\ref{c3}), $\beta_1^\epsilon$ is an explicit smooth function defined on $K$ which is uniformly bounded as $\epsilon \to 0$; furthermore, $P_1 (\delta ) $ is a small compact perturbation in ${\mathcal H}_g^1 (K) $ whose shape is a sum of quadratic functional in $\delta$ of the form $$
\epsilon^{2} \int_K b(y) |\delta |^2 $$ where $b(y)$ denotes a generic explicit function, smooth and uniformly bounded, as $\epsilon \to 0$, in $K$. In (\ref{EEE}), \begin{equation} \label{qje} Q_\epsilon (d ) = Q(d ) + Q_1 (d) \end{equation} with \begin{equation} \label{Qj}
Q (d)= {\epsilon \over 2} C_\epsilon \left( \int_K |\nabla_K ( d (1+ o(\epsilon^2
) \beta_2^\epsilon (y) ) )|^2 + \int_K (\tilde g^{ab}R_{mabl} - \Gamma_a^c (E_m) \Gamma_c^a (E_l) ) d^m d^l \right) \end{equation} where $C_\epsilon$ is a real number such that $\lim_{\epsilon \to 0} C_\epsilon = C:= \int_{\mathbb{R}^N_+} Z_1^2$, $\beta_2^\epsilon$ is an explicit smooth function defined on $K$ which is uniformly bounded as $\epsilon \to 0$ and the terms $R_{maal} $ and $\Gamma_a^c (E_m)$ are smooth functions on $K.$ Furthermore, $Q_1 (d)$ is a small compact perturbation in ${\mathcal H}^1 (K) $ whose shape is a sum of quadratic functional in $d$ of the form $$ \epsilon^{3} \int_K b(y) d^i d^j $$ where again $b(y)$ is a generic explicit function, smooth and uniformly bounded, as $\epsilon \to 0$, in $K$. In (\ref{EEE}), \begin{equation} \label{qe0} R_\epsilon (e) = R(e) + R_1 (e) \end{equation} \begin{equation} \label{Q} R(e) = \epsilon^{-\frac{k}{2}} \left[ {D_\epsilon \over 2} \left( \epsilon^2 \int_K
|\nabla_K ( e (1+ e^{-{\lambda_0 \over 2} \epsilon^{-\gamma}} \beta_3^\epsilon
(y) ) )|^2 -\lambda_0 \int_K e^2 \right)\right] \end{equation} with $D_\epsilon$ a real number so that $\lim_{\epsilon \to 0 } D_\epsilon = D:= \int_{\mathbb{R}^N} Z^2 $, $\beta_3^\epsilon$ an explicit smooth function in $K$, which is uniformly bounded as $\epsilon \to 0$, and $\lambda_0$ the positive number defined in (\ref{lambda0}). Furthermore, $R_1$ is a small compact perturbation in ${\mathcal H}^1 (K) $ whose shape is a sum of quadratic functional in $e$ of the form $$ \epsilon^{3} \int_K b(y) e^2 $$ where again $b(y)$ is a generic explicit function, smooth and uniformly bounded, as $\epsilon \to 0$, in $K$. Finally in (\ref{EEE}) $$ {\mathcal G} : (H^1_{g^\epsilon} )^\bot \times ({\mathcal H}^1 (K) )^{N+1} \to \mathbb{R} $$ is a continuous and differentiable functional with respect to the natural topologies, homogeneous of degree $2$ $$ {\mathcal G} (t \phi^\bot , t \delta , t d , t e ) = t^2 {\mathcal M} ( \phi^\bot , \delta , d , e ) \quad \forall t. $$ The derivative of ${\mathcal G}$ with respect to each one of its variable is given by a small multiple of a linear operator in $(\phi^\bot , \delta , d , e)$ and it satisfies $$
\| D_{(\phi^\bot , \delta , d)} {\mathcal G}(\phi_1^\bot , \delta_1 , d_1 , e_1) - D_{(\phi^\bot , \delta , d)} {\mathcal G}(\phi_2^\bot ,
\delta_2 , d_2 , e_2) \| \leq C \epsilon^{\gamma (N-3)} \times $$ \begin{equation}
\label{lips} \left[ \| \phi_1^\bot - \phi_2^\bot \| + \epsilon^{-\frac{k}{2}} \|
\delta_1 - \delta_2 \|_{{\mathcal H}^1 (K)} + \epsilon^{-\frac{k}{2}} \| d_1 - d_2
\|_{({\mathcal H}^1 (K))^{N-1}} + \epsilon^{-\frac{k}{2}} \| e_1 - e_2 \|_{{\mathcal H}^1 (K)}\right]. \end{equation}
Furthermore, there exists a constant $C>0$ such that \begin{equation} \label{estMM}
\left| {\mathcal G } ( \phi^\bot , \delta , d , e )
\right| \leq C \epsilon^{2} \left[ \| \phi^\bot \|^2 + \epsilon^{-\frac{k}{2}} \left( \|
\delta \|_{{\mathcal H}^1 (K)}^2 + \| d \|_{{\mathcal H}^1 (K)}^2 +
\| e\|_{{\mathcal H}^1 (K)}^2 \right) \right]. \end{equation} \end{theorem}
\begin{proof}
\noindent STEP 1. We claim that there exists $\epsilon_0>0 $ such that for all $0<\epsilon<\epsilon_0$, we have \begin{equation} \label{leQ0} E\left({\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon\right)= \epsilon^{-\frac{k}{2}} P_\epsilon (\delta ), \end{equation} \begin{equation} \label{leQj} E\left({ d^j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_ j) \bar \chi_\epsilon\right)= \epsilon^{-\frac{k}{2}} Q_\epsilon (d^j), \end{equation} \begin{equation} \label{leQ} E\left({e\over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } (Z) \bar \chi_\epsilon\right)= \epsilon^{-\frac{k}{2}} R_\epsilon (e). \end{equation} Define \begin{eqnarray} \label{defFunctionalF}
F (u ) :&=&\int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon}
\left(\frac12 |\nabla_x u|^2 +\frac12 \epsilon hu^2 - \frac{1}{p+1\pm\epsilon} u^{p+1\pm\epsilon} \right) \sqrt{\det(g^\epsilon)} \, dz \, dx \nonumber\\ &&-\frac{\epsilon}{6}\int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} R_{islj}\ x_l\ x_s \,\partial_{i} u \partial_{j} u\sqrt{\det(g^\epsilon)} \, dz \, dx\\
&&+\frac12 \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} |\nabla_{K_\epsilon}u|^2\sqrt{\det(g^\epsilon)} \, dz \, dx
+ \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} B(u,u)\sqrt{\det(g^\epsilon)} \, dz \,
dx.\nonumber \end{eqnarray}
\noindent To prove (\ref{leQ0})), we write for small $t \not= 0$ \begin{eqnarray}\label{tere1} &&\left[ DF({\widetilde{\mathcal T}}_{\, \mu_\epsilon + t \delta , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon) - DF({\widetilde{\mathcal T}}_{\, \mu_\epsilon , \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon )\right] ({\delta \over \mu_\epsilon+t\delta} {\widetilde{\mathcal T}}_{\, \mu_\epsilon+t\delta , \, \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon)\nonumber\\ &=& -2t E ({\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon ) (1+ O(t)). \end{eqnarray} On the other hand, for any $\psi$ \begin{equation} \label{tere2} \left[ DF({\widetilde{\mathcal T}}_{\, \mu_\epsilon + t \delta , \Phi_\epsilon } (\hat w )\bar \chi_\epsilon) - DF({\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon )\right] ( \psi) = \mathfrak{a} (t) - \mathfrak{a} (0) + \mathfrak{b}(t) + \mathfrak{c}(t) \end{equation} where \begin{eqnarray*} \mathfrak{a}(t) &= &\int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon}\left[ \left( \nabla_x {\widetilde{\mathcal T}}_{\, \mu_\epsilon + t \delta , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon \right) \nabla_x \psi + \epsilon h {\widetilde{\mathcal T}}_{\, \mu_\epsilon + t \delta , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon \psi \right.\\ &&\quad \qquad\left.- \left({\widetilde{\mathcal T}}_{\, \mu_\epsilon + t \delta , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon \right)^{p\pm \epsilon} \psi\right]\sqrt{\det(g^\epsilon)} \, dz \, dx \\ &&-\frac{\epsilon}{6} \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} R_{islj} \ x_lx_s \partial_i \left( {\widetilde{\mathcal T}}_{\, \mu_\epsilon + t \delta , \, \Phi_\epsilon } (\hat w )\bar \chi_\epsilon \right) \partial_j \psi\sqrt{\det(g^\epsilon)} \, dz \, dx, \end{eqnarray*} \begin{eqnarray*} \mathfrak{b}(t)&=&\int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} \partial_{\bar a} ( {\widetilde{\mathcal T}}_{\, \mu_\epsilon + t \delta , \, \Phi_\epsilon } (\hat w )\bar \chi_\epsilon ) \partial_{\bar a} \psi \sqrt{\det(g^\epsilon)} \, dz \, dx\\ &&- \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} \partial_{\bar a} ( {\widetilde{\mathcal T}}_{ \mu_\epsilon , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon ) \partial_{\bar a} \psi\sqrt{\det(g^\epsilon)} \, dz \, dx \end{eqnarray*} and \begin{eqnarray*} \mathfrak{c}(t)&=& \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} B({\widetilde{\mathcal T}}_{ \mu_\epsilon + t \delta , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon , \psi) \sqrt{\det(g^\epsilon)} \, dz \, dx\\ &&- \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} B({\widetilde{\mathcal T}}_{ \mu_\epsilon , \bar \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon , \psi)\sqrt{\det(g^\epsilon)} \, dz \, dx. \end{eqnarray*} We now compute $\mathfrak{a} (t) - \mathfrak{a} (0) $ with $\psi = {\delta \over \mu_\epsilon+t\delta} {\widetilde{\mathcal T}}_{\, \mu_\epsilon + t \delta , \, \Phi_\epsilon } (Z_0 ) \bar \chi_\epsilon $. Performing the change of variables $x = (\mu_\epsilon + t \delta ) \xi + \, \sqrt{\epsilon} \Phi_\epsilon$ and using the expansion of ${\delta \over \, \mu_\epsilon+t\delta}=\frac{\delta}{\mu_e}-\frac{\delta^2}{\mu_e^2}t+O(t^2)$, we see that \begin{eqnarray*} &&t^{-1} \left[\mathfrak{a}(t)-\mathfrak{a}(0) \right] \\ &=&- \epsilon\left\{ \int_{K_\epsilon} \delta^2 \left(c_{3,N}h -c_{2,N} \sum\limits_j \bigg[ \sum_s \frac23 R_{mssj}+\sum\limits_{m,a,b}
\big( \, {\tilde g}^{ab} R_{mabj}- \Gamma_{am}^b \Gamma_{bj}^a \big)\bigg]\right) \mp c_{4,N}\int_{K_\epsilon} \frac{\delta^2}{\mu_\epsilon^2} \right\} \\ &&\times\big(1+ O(t)\big) \bigg(1 +O(\epsilon^{\gamma (N-4)})\bigg). \end{eqnarray*}
On the other hand, by the definition of the function $\mathfrak{b}(t)$ above,
a Taylor expansion gives $$
\mathfrak{b}(t) = -t\left( \int_{K_\epsilon \times \hat {\mathcal{C}}_\epsilon} |\nabla_{K_\epsilon} ({\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,
\Phi_\epsilon} ( Z_0 ) \bar \chi_\epsilon )|^2 dzdx \right) \times (1+ O(t)) . $$ Observe now that \begin{eqnarray*} \partial_{\bar a} \left( {\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon \right) &=& (\partial_{\bar a} \delta ) {1 \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon + \delta \partial_{\bar a} ( {1 \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon ) \\ &=& \sqrt{\epsilon} ( \partial_a \delta ) {1 \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 )\bar \chi_\epsilon + \sqrt{\epsilon} \delta ( \partial_a \, \mu_\epsilon ) \partial_{\, \mu_\epsilon} ({1 \over \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon )\\
&&+ \sqrt{\epsilon} \delta ( \partial_a \, \Phi_\epsilon ) \partial_{ \Phi_\epsilon} ({1 \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \bar \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon ). \end{eqnarray*} Since $ \int \left( {1 \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon \right)^2 \, dx= \int \left( {1 \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 ) \bar \chi_\epsilon \right)^2 \, dx= c_{1,N} (1+ o(\epsilon) )$, we conclude that \begin{equation} \label{III1} \mathfrak{b}(t) = - t \epsilon^{-\frac{k}{2}} \left[ A_\epsilon \epsilon \int_K
|\nabla_K ( \delta (1+ o(\epsilon ) \beta_1^\epsilon (y) )) |^2 \right] \end{equation} where $A_\epsilon \in \mathbb{R}$, $\lim_{\epsilon \to 0} A_\epsilon = c_{1, N}= \int_{\mathbb{R}^N} Z_0^2 $ and $\beta_1^\epsilon$ is an explicit smooth function in $K$, which is uniformly bounded as $\epsilon \to 0$. Finally we observe that the last term $\mathfrak{c}(t)$ defined above is of lower order, and can be absorbed in the terms already described.
\noindent Proof of (\ref{leQj}). Let $d $ be the vector field along $K$ defined by $$d (\sqrt{\epsilon} z) = (d^1 (\sqrt{\epsilon} z ) , \ldots , d^{N} (\sqrt{\epsilon} z) ).$$ For any $t$ small and $t\not= 0$, we have (see (\ref{defFunctionalF}))
\begin{eqnarray*}
&&\left[ DF({\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon + t d} (\hat w) \bar \chi_\epsilon) - DF ({\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon } (\hat w) \bar \chi_\epsilon )\right] [\varphi] \\ &=& t D^2 F ({\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon } (\hat w) \bar \chi_\epsilon ) \left[\sum_l { d^l \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_l ) \bar \chi_\epsilon \right] [\varphi] \,\big(1+ O(t)\big) \big(1+ O(\epsilon)\big) \end{eqnarray*} for any function $\varphi \in H^1_{g^\epsilon}$. Choosing $\varphi = { d^j \over \, \sqrt{\epsilon}\mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_j) \bar \chi_\epsilon$ we write \begin{eqnarray} &&\label{leQj2} \left[ DF ({\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon + t d} (\hat w) \bar \chi_\epsilon) - DF ({\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon } (\hat w) \bar \chi_\epsilon )\right] [{ d^j \over \sqrt{\epsilon} \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_j) \bar \chi_\epsilon]\nonumber\\ &=& \mathfrak{a}_2(t) - \mathfrak{a}_2(0) + \mathfrak{b}_2(t) + \mathfrak{c}_2(t) \end{eqnarray} where we have set, for $\psi = { d^j \over \,\sqrt{\epsilon} \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_j) \bar \chi_\epsilon$, \begin{eqnarray*} \mathfrak{a}_2(t)& =& \int \left( \nabla_X {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon +t d } (\hat w ) \bar \chi_\epsilon \right) \nabla_X \psi + \epsilon {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon +t d } (\hat w ) \bar \chi_\epsilon \psi - \left({\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon +t d} (\hat w \bar \chi_\epsilon )\right)^{p\pm\epsilon} \psi\\ & &-\frac{\epsilon}{6} \int R_{islj} \ x_lx_s \partial_i \left( {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon +t d } (\hat w ) \bar \chi_\epsilon \right) \partial_j \psi, \end{eqnarray*} $$ \mathfrak{ b}_2(t) =\int \partial_{\bar a} ( {\widetilde{\mathcal T}}_{ \mu_\epsilon , \, \Phi_\epsilon +t d} (\hat w ) \bar \chi_\epsilon) \partial_{\bar a} ( {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon +t d } (\hat w ) \bar \chi_\epsilon) - \int \partial_{\bar a} ( {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon) \partial_{\bar a} ( {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon) $$ and $$ \mathfrak{c}_2 (t)= \int B({\widetilde{\mathcal T}}_{\bar \mu_\epsilon , \bar \Phi_\epsilon +t d } (\hat w ) \bar \chi_\epsilon , \psi) - \int B({\widetilde{\mathcal T}}_{\bar \mu_\epsilon , \, \Phi_\epsilon } (\hat w ) \bar \chi_\epsilon , \psi). $$ Define $$ {\mathcal R}_{ml}=\bigg( (\tilde g)^{ab}\,R_{mabl}-\Gamma_a^c(E_m) \Gamma_c^a(E_l) \bigg). $$ Performing the change of variables $x= \, \mu_\epsilon \xi + \, \sqrt{\epsilon} (\Phi_\epsilon+td)$, we get \begin{eqnarray*} &&t^{-1} [\mathfrak{a}_2(t) - \mathfrak{a}_2(0) ]\\ &=& \epsilon \left\{ \int { d^j \over \, \mu_\epsilon} \bigg[ \nabla \hat w \nabla Z_j+ \epsilon \, \mu_\epsilon^2 h \hat w - \mu_{\epsilon}^{\mp\frac{N-2}{2}\epsilon} \hat w^{p\pm\epsilon} Z_j\bigg] \right. \\ &&\times ({R_{mijl} \over 6} + {{\mathcal R}_{lm} \over 2} ) [( \mu_\epsilon\xi_m +\, \sqrt{\epsilon}\Phi_{\epsilon, m} ) d^l + (\, \mu_\epsilon \xi_l +\, \sqrt{\epsilon}\Phi_{\epsilon, l}) d^m ] \\ &&\left. - \int { d^j \over \, \mu_\epsilon} {R_{ilsr} \over 6} [(\mu_\epsilon \xi_s +\, \sqrt{\epsilon}\Phi_{\epsilon, s}) d^l + (\mu_\epsilon \xi_l +\, \sqrt{\epsilon}\Phi_{\epsilon, l}) d^s ] \partial _i \hat w \partial_r Z_j \right\}\\ &&\times (1+ O(\epsilon )) (1+ O(t)). \end{eqnarray*} Integration by parts in the $\xi$ variables, using the fact that $\hat {\mathcal C}_\epsilon \to \mathbb{R}^N$ as $\epsilon \to 0$, $R_{irll}=0$, we deduce that \begin{eqnarray}\label{leQj3} t^{-1} [\mathfrak{a}_2(t) - \mathfrak{a}_2(0) ] &= &\epsilon \left\{ - C \int_{K_\epsilon} ({R_{miij} \over 6} + {{\mathcal R}_{mj} \over 2} ) d^j d^m +C \int_{K_\epsilon} {R_{jrrm} \over 3} d^m d^j \right\} \times\nonumber\\ &\times& \big(1+ O(\epsilon )\big) \,\big(1+ O(t)\big) \\&=& \epsilon^{-\frac{k}{2}} \epsilon \left[ - C \int {{\mathcal R}_{mj} \over 2} d^j d^m + O(\epsilon) Q(d) \right] \big(1+ O(t)\big)\nonumber \end{eqnarray} where here we have set $$ C= \int_{\mathbb R^N} Z_1^2 \qquad \hbox{ and } \quad Q(d):=\int_K \pi(y) d^i d^j $$ for some smooth and uniformly bounded (as $\epsilon \to 0$) function $\pi(y)$. To estimate the term $\mathfrak{b}_2$ above we argue as in (\ref{III1}), we get that \begin{equation} \label{leQj4} t^{-1} \mathfrak{b}_2(t) = -\epsilon^{-\frac{k}{2}} \left[ \epsilon C_\epsilon
\int_K |\nabla_K (d^j (1+ \beta_2^\epsilon (y) o(\epsilon) ))|^2 \right] (1+ O(t)). \end{equation} Finally we observe that the last term $\mathfrak{c}_2(t)$ is of lower order, and can be absorbed in the terms described in (\ref{leQj3}) and (\ref{leQj4}).
\noindent
Proof of (\ref{leQ}). To get the expansion in (\ref{leQ}), we compute \begin{equation}\label{tere3} E({e\over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z)) = I + II + III \end{equation} where \begin{eqnarray*} I&=& \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} {e^2 \over \mu_\epsilon^2}
\left(\frac12 ( |\nabla_x {\widetilde{\mathcal T}}_{ \mu_\epsilon , \Phi_\epsilon} (Z) |^2 +\epsilon h {\widetilde{\mathcal T}}_{\mu_\epsilon , \Phi_\epsilon} (Z)^2 - (p\pm\epsilon) \tilde{U}_\epsilon^{p\pm\epsilon-1} {\widetilde{\mathcal T}}_{ \mu_\epsilon , \Phi_\epsilon} (Z)^2 ) \right) \times \\ & & \sqrt{\det g^\epsilon} dz dx \\ &&-\frac{\epsilon}{6}\int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} {e^2 \over \, \mu_\epsilon^2} R_{islj}x_sx_l \,\partial_{i} {\widetilde{\mathcal T}}_{ \mu_\epsilon , \, \Phi_\epsilon} (Z) \partial_{j}{\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z) \,\sqrt{\det g^\epsilon} \, dz \, dx, \\[3mm]
II&=& \frac12 \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} |\nabla_{K} \left(
{e\over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z) \right)|^2
\,\sqrt{\det g^\epsilon} \, dz \, dx\\[3mm] \mbox{and}\\ III&=& \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} B({e\over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z), {e\over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z))\,\sqrt{\det(g^\epsilon)} \, dz \, dx. \end{eqnarray*} Using the change of variables $x = \, \mu_\epsilon \xi + \sqrt{\epsilon} \Phi_\epsilon$ in $I$, we have $$
I= \int {1\over 2} {e^2 \over \, \mu_\epsilon^2} \bigg[ |\nabla Z|^2 - p \hat w^{p-1} Z^2 +\epsilon \, \mu_\epsilon^2 h Z^2 \bigg] \bigg(1+ \epsilon O(e^{-|\xi|})\bigg). $$ Then, recalling the definition of $\lambda_0$ in (\ref{lambda0}), we get \begin{equation} \label{tere4} I=\epsilon^{-\frac{k}{2}} \left[ -{\lambda_0 \over 2} D \int_K e^2 + \epsilon Q(e) \right] \end{equation} where we have set $$ D= \int_{\mathbb{R}^N} Z^2(\xi)\,d\xi \quad \hbox{and }\qquad Q(e):= \int_K \tau(y) e^2\,dy, $$ for some smooth and uniformly bounded, as $\epsilon \to 0$, function $\tau$. On the other hand, using a direct computation and arguing as in (\ref{III1}), we get \begin{equation} \label{ttere4}
II= {D_\epsilon \over 2} \int_{K_\epsilon} |\nabla_{K_\epsilon} e + e^{-\lambda_0
\epsilon^{-\gamma}} \beta_3^\epsilon (\epsilon z) e |^2 =
\epsilon^{-\frac{k}{2}}
\bigg[ {D_\epsilon \over 2} \epsilon \int_{K} |\nabla_{K} (e (1+ e^{-\lambda'\epsilon^{-\gamma}} \beta_3^\epsilon (y) )) |^2 \bigg] \end{equation} where $\beta_3^\epsilon$ is an explicit smooth function on $K$, which is uniformly bounded as $\epsilon \to 0$, while $\lambda'$ is a positive real number. Finally we observe that the last term $III$ is of lower order, and can be absorbed in the terms described in (\ref{tere4}) and (\ref{ttere4}). This concludes the proof of (\ref{leQ}).
\noindent STEP 2. We write \begin{eqnarray*} {\mathcal G} (\phi^\bot , \delta , d, e) &=& E(\phi ) - E(\phi^\bot ) -
E({\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon)
- \sum_{j=1}^{N} E({d_j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_j ) \bar
\chi_\epsilon)\\ &&- E({e \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z ) \bar \chi_\epsilon). \end{eqnarray*} Thus it is clear that the term ${\mathcal G}$ recollects all the mixed terms in the expansion of $E(\phi)$. Indeed, if we define \begin{eqnarray*} m(f_1,f_2)& =& \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} \left(\nabla_x f_1 \nabla_x f_2 - (p\pm\epsilon) U_\epsilon^{p\pm\epsilon-1} f_1f_2 \right) \sqrt{\det(g^\epsilon)} \, dz \, dx \\ & & +\epsilon \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon}hf_1f_2dzdx \\ &&-\frac{\epsilon}{6}\int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} \,R_{islj}x_sx_l\partial_{i}f_1\partial_{j}f_2\,\sqrt{\det(g^\epsilon)} \, dz \, dx\\ &&+ \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon} \partial_{\bar a}f_1\,\partial_{\bar a}f_2\,\sqrt{\det(g^\epsilon)} \, dz \, dx+ \int_{K_\epsilon \times \hat{\mathcal{C}}_\epsilon}
B(f_1,f_2)\,\sqrt{\det(g^\epsilon)} \, dz \, dx\\
&:=& m_1(f_1,f_2)+m_2(f_1,f_2)+m_3(f_1,f_2)+m_4(f_1,f_2)+m_5(f_1,f_2), \end{eqnarray*} for $f_1$ and $f_2$ in $H^1_\epsilon$, then \begin{eqnarray}\label{auxi} {\mathcal G}(\phi^\bot , \delta , d , e) &=& m (\phi^\bot , {\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon ) +\sum_j m (\phi^\bot , {d_j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_j ) \bar \chi_\epsilon )\nonumber \\ &+& m (\phi^\bot , {e \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon ) + \sum_j m ({\delta \over \bar \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon , {d^j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\bar \Phi_\epsilon } ( Z_j ) \bar \chi_\epsilon )\nonumber \\ &+&\sum_{i\not= j} m ({d^j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\bar \mu_\epsilon ,\, \Phi_\epsilon } ( Z_j ) \bar \chi_\epsilon , {d_i \over \bar \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_i ) \bar \chi_\epsilon ) \\ &+& m ({\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\bar \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon , {e \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z ) \bar \chi_\epsilon )\nonumber\\ & +& \sum_j m ({d^j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_j ) \bar \chi_\epsilon , {e \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z ) \bar \chi_\epsilon ).\nonumber \end{eqnarray} One can see clearly that ${\mathcal G}$ is homogeneous of degree $2$ and that its first derivatives with respect to its variables is a linear operator in $(\phi^\bot , \delta, d , e)$. We will then show the validity of estimate (\ref{estMM}). In a very similar way one shows the validity of (\ref{lips}). To prove (\ref{estMM}), we should treat each one of the above terms. Since the computations are very similar, we will limit ourselves to treat the term $$ m:= m ({\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon , {d^j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_j ) \bar \chi_\epsilon ). $$ This term can be written as \begin{equation} m= \sum_{i=1}^5 m_i(f_1,f_2) \end{equation} with $f_1= {\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\bar \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon$ and $f_2= {d^j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_j ) \bar \chi_\epsilon$. Using the fact that the function $Z_0$ solves $$\Delta Z_0+ p w_0^{p-1} Z_0 = 0 \quad \hbox{in}\quad \mathbb R^N, $$ with $\int_{\mathbb{R}^N} \partial_{\xi_N} Z_0 Z_j = 0$ and integrating by parts in the $x$ variable (recalling the expansion of $\sqrt{{\mbox {det}} g^\epsilon}$), one gets \begin{eqnarray*} m_1 &=& \left\{ \int {\delta d^j \over \, \mu_\epsilon^2} [-\Delta {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon } (Z_0) - (p\pm\epsilon) U_\epsilon^{p\pm\epsilon-1} {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0 ) ] \bar \chi_\epsilon^2 {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_j ) \sqrt{{\mbox {det}} g^\epsilon} \right. \\ &+& \left. \int {\delta d^j \over \, \mu_\epsilon^2} \partial_{\xi_N} \left( {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_0) \bar \chi_\epsilon \right) {\widetilde{\mathcal T}}_{\, \mu_\epsilon , \, \Phi_\epsilon} (Z_j ) \,\frac{1}{\, \mu_\epsilon}\,(\epsilon \,tr(H)+O(\epsilon^2))\,\bar \chi_\epsilon \right\} (1+ o(1) ) \end{eqnarray*} where $o(1) \to 0$ as $\epsilon \to 0$. Thus, a H\"older inequality yields $$
| m_1 | \leq C \epsilon^{-\frac{k}{2}} \epsilon^{\gamma (N-2)} \| \delta \|_{{\mathcal L}^2 (K)} \| d^j \|_{{\mathcal L}^2 (K)}. $$ On the other hand, using the orthogonality condition $\int_{\mathbb R^N} Z_0 Z_j =0$, we get $$
|m_2 | \leq C \epsilon \epsilon^{-\frac{k}{2}} (\int_{|\xi|>\epsilon^{-\gamma}} Z_0 Z_j ) \|
\delta \|_{{\mathcal L}^2 (K)} \| d^j \|_{{\mathcal L}^2 (K)} \leq C
\epsilon^{-k} \epsilon^{1+\gamma (N-3) } \| \delta \|_{{\mathcal L}^2 (K)} \|
d^j \|_{{\mathcal L}^2 (K)}. $$ Now, since $\int_{\mathbb R^N} \xi_N \partial_i Z_0 \partial_l Z_j = 0$, for any $i, j, l=1, \ldots , N-1$, one gets \begin{eqnarray*}
|m_3| &\leq &C \epsilon \epsilon^{-\frac{k}{2}} \left( \int_{|\xi|>\epsilon^{-\gamma}} \xi_N
\partial_i Z_0 \partial_l Z_j \right) \| \delta \|_{{\mathcal L}^2
(K)} \| d^j \|_{{\mathcal L}^2 (K)} \\
&\leq& C \epsilon^{-\frac{k}{2}} \epsilon^{1+\gamma (N-2)} \| \delta \|_{{\mathcal L}^2
(K)} \| d^j \|_{{\mathcal L}^2 (K)}. \end{eqnarray*} A direct computation on the term $m_4$ gives \begin{eqnarray*}
|m_4 | &\leq& C \epsilon^{-\frac{k}{2}} \left\{ \epsilon^2 (\int_{|\xi|>\epsilon^{-\gamma}} Z_0 Z_j ) \| \partial_a \delta \|_{{\mathcal L}^2 (K)} \| \partial_a d^j
\|_{{\mathcal L}^2 (K)} \right. \\
&+& \epsilon (\int_{|\xi|>\epsilon^{-\gamma}} Z_0 Z_j ) ( \| \delta
\|_{{\mathcal L}^2 (K)} \| \partial_a d^j \|_{{\mathcal L}^2 (K)} +
\| \partial_a \delta \|_{{\mathcal L}^2 (K)} \| d^j \|_{{\mathcal L}^2 (K)} ) \\
&+& \left. (\int_{|\xi|>\epsilon^{-\gamma}} Z_0 Z_j ) \| \delta
\|_{{\mathcal L}^2 (K)} \| d^j \|_{{\mathcal L}^2 (K)} \right\} \\
&\leq& C \epsilon^{-\frac{k}{2}} \epsilon^{\gamma (N-3)} [ \| \delta \|_{{\mathcal H}^1
(K)}^2 + \| d^j \|_{{\mathcal H}^1 (K)}^2 ]. \end{eqnarray*}
Since $|m_5| \leq C \sum_{j=1}^4 |m_j|$ we conclude that $$
|m| \leq C \epsilon^{-\frac{k}{2}} \epsilon^{\gamma (N-3)} [ \| \delta \|_{{\mathcal H}^1
(K)}^2 + \| d^j \|_{{\mathcal H}^1 (K)}^2 ]. $$ Each one of the terms appearing in (\ref{auxi}) can be estimated to finally get the validity of (\ref{estMM}). This conclude the proof of Theorem (\ref{teo4.1}). \end{proof}
Now, we are going to prove Proposition \ref{teouffa}.
\begin{proof} We define the energy functional associated to Problem (\ref{lineare}) $$ {\mathcal E}: (H^1_{g^{\epsilon}})^\bot \times ( {\mathcal H}^1 (K) )^{N+2} \to \mathbb R $$ by \begin{equation}\label{functional2} {\mathcal E} (\phi^\bot , \delta , d , e) = E(\phi ) -{\mathcal L}_f (\phi ) \end{equation} where $E$ is the functional in (\ref{functional1}) and ${\mathcal L}_f (\phi )$ is the linear operator given by $$ {\mathcal L}_f (\phi ) = \int_{\mathcal{M}_{\epsilon , \gamma}} f \phi. $$ Observe that $$ {\mathcal L}_f (\phi ) = {\mathcal L}_f^1 (\phi^\bot ) + \epsilon^{-\frac{k}{2}} \left[ {\mathcal L}_f^2 (\delta ) + {\mathcal L}_f^3 (d ) +{\mathcal L}_f^4 (e) \right] $$ where ${\mathcal L}_f^1 :H^1_{g^\epsilon} \to \mathbb{R}$, $ {\mathcal L}_f^2 , {\mathcal L}_f^4 \, : \, {\mathcal H}^1 (K) \to \mathbb{R}$ and ${\mathcal L}_f^3 \, : \, ( {\mathcal H}^1 (K) )^{N} \to \mathbb{R}$ with $$ {\mathcal L}_f^1 (\phi^\bot ) = \int_{\mathcal{M}_{\epsilon , \gamma}} f \phi^\bot , \quad \epsilon^{-\frac{k}{2}} {\mathcal L}_f^2 (\delta ) = \int_{\mathcal{M}_{\epsilon , \gamma}} f {\delta \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z_0 ) \bar \chi_\epsilon $$ $$ \epsilon^{-\frac{k}{2}} {\mathcal L}_f^3 (d ) = \sum_{j=1}^{N}\int_{\mathcal{M}_{\epsilon , \gamma}} f {d^j \over \, \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\bar \Phi_\epsilon } ( Z_j ) \bar \chi_\epsilon \quad {\mbox {and}} \quad \epsilon^{-\frac{k}{2}} {\mathcal L}_f^4 (e ) = \int_{\mathcal{M}_{\epsilon , \gamma}} f {e \over \bar \mu_\epsilon} {\widetilde{\mathcal T}}_{\, \mu_\epsilon ,\, \Phi_\epsilon } ( Z ) \bar \chi_\epsilon . $$ Finding a solution $\phi \in H^1_{g^\epsilon}$ to Problem (\ref{lineare}) reduces to finding a critical point $(\phi^\bot , \delta , d , e) $ for ${\mathcal E}$. This will be done in several steps.
\noindent \textrm{\textbf{Step 1}}. We claim that there exist $\sigma >0$ and $\epsilon_0$ such that for all $\epsilon \in (0,\epsilon_0)$ and all $\phi^\bot \in (H^1_{g^\epsilon})^\bot$ then \begin{equation}\label{punto1}
E(\phi^\bot ) \geq \sigma \| \phi^\bot \|^2_{L^2}. \end{equation}
In fact, using the local change of variables (\ref{bb}), together with the expansion of energy $E$ in (\ref{energydensity}), we see that, for sufficiently small $\epsilon>0$ $$ E(\phi^\bot ) \geq {1\over 4} E_0 (\phi^\bot ), $$ with $$ E_0 (\phi^\bot ) = \int_{K_\epsilon \times \hat{\mathcal C}_\epsilon}
\left[ |\nabla_x \phi^\bot |^2 - (p\pm\epsilon) U_\epsilon^{p\pm\epsilon-1} (\phi^\bot)^2 \right]\sqrt{\det(g^\epsilon)}dzdx $$ for any $\phi^\bot = \phi^\bot (\sqrt{\epsilon} z , x),$ with $ z \in K_\epsilon = {1 \over \sqrt{\epsilon}}K$. The set $\hat {\mathcal C}_\epsilon$ is defined in (\ref{dddomain}) and the function $U_\epsilon$ is given by (\ref{Vdef}). We recall that $\hat {\mathcal C}_\epsilon \to \mathbb R^N$ as $\epsilon \to 0$.
We will establish (\ref{punto1}) showing that \begin{equation}\label{pas1}
E_0 (\phi^\bot ) \geq \sigma \| \phi^\bot \|^2_{L^2} \quad \forall \phi^\bot. \end{equation} To do so, we first observe that if we scale in the $z$-variable, defining $\varphi^\bot (y,x) = \phi^\bot ({y\over \sqrt{\epsilon} }, x)$, the relation (\ref{pas1}) becomes \begin{equation}\label{pas2}
E_0 (\varphi^\bot ) \geq \sigma \| \varphi^\bot
\|^2_{L^2}. \end{equation} Thus we are led to show the validity of (\ref{pas2}). We argue by contradiction. Assume that for any $n$, there exist $\epsilon_n \to 0 $ and $\varphi_n^\bot \in (H_{g^{\epsilon_n}}^1)^\bot $ such that \begin{equation}\label{shen}
E_0 (\varphi_n^\bot ) \leq \frac1n \| \varphi_n^\bot
\|^2_{L^2}. \end{equation}
Without loss of generality we can assume that the sequence $(\|
\varphi_n^\bot\|)_n$ is bounded, as $n \to \infty$. Hence, up to subsequences, we have that $$ \varphi_n^\bot \rightharpoonup \varphi^\bot \quad \hbox{in} \quad H^1 (K \times \mathbb R^N )\qquad \hbox{and } \quad \varphi_n^\bot \to \varphi^\bot \quad \hbox{in} \quad L^2 (K \times \mathbb R^N ). $$ Furthermore, using the estimate in (\ref{boh1}) we get that $$
\sup_{y \in K , x \in \mathbb R^N} \left| (1+ |x|)^{N-4} \left[ U_\epsilon ({y\over \sqrt{\epsilon}} , x ) - \mu_0^{-{N-2 \over 2}} (y) w_0 ({ x -\sqrt{\epsilon} \Phi_1 (y) \over\mu_0 (y) } ) \right]
\right| \to 0, $$ as $\epsilon \to 0$, where $\mu_0$ and $\Phi_1 $ are the smooth explicit function defined in (\ref{choiceofmu0}) and (\ref{phi1def}).
Passing to the limit as $n \to \infty$ in (\ref{shen}) and applying dominated convergence Theorem, we get \begin{equation}\label{pas3}
\int_{K \times \mathbb R^N} \left[ |\nabla_x \varphi^\bot
|^2 - p \left( ( \mu_0)^{-{N-2 \over 2}} (y) w_0 ({ x - \sqrt{\epsilon}\Phi_1 (y) \over \mu_0 (y) } \right)^{p-1} (\varphi^\bot)^2 \right] dy dx \leq
0. \end{equation} Furthermore, passing to the limit in the orthogonality condition we get, for any $y \in K$ \begin{equation} \label{shen1} \int_{\mathbb R^N} \varphi^\bot (y, x) Z_0 ({x - \sqrt{\epsilon}\Phi_1 (y) \over \mu_0 (y) } ) dx = 0 , \end{equation} \begin{equation} \label{shen2} \int_{\mathbb R^N} \varphi^\bot (y, x) Z_j ({x - \sqrt{\epsilon}\Phi_1 (y) \over \mu_0 (y) } ) dx = 0 , \quad j=1, \ldots N \end{equation} and \begin{equation} \label{shen3} \int_{\mathbb R^N} \varphi^\bot (y, x) Z ({x - \sqrt{\epsilon}\Phi_1 (y) \over \mu_0 (y) } ) dx = 0 . \end{equation} We thus get a contradiction with (\ref{pas3}), since for any function $\varphi^\bot$ satisfying the orthogonality conditions (\ref{shen1})--(\ref{shen3}) for any $y \in K$ one has $$
\int_{K \times \mathbb R^N } \left[ |\nabla_x \varphi^\bot |^2 - p \left( \mu_0^{-{N-2 \over 2}} (y) w_0 ({x - \sqrt{\epsilon}\Phi_1 (y) \over \mu_0 (y) } ) \right)^{p -1} (\varphi^\bot)^2 \right] dy dx > 0. $$
\noindent \textrm{\textbf{Step 2}}. For all $\epsilon>0$ small, the functional $P_\epsilon (\delta )$ defined in (\ref{q0e}) is continuous and differentiable in
${\mathcal H}^1 (K)$. Furthermore,
$P_\epsilon$ is a small perturbation in $({\mathcal H}^1 (K) )^{N }$ of \begin{eqnarray*}
P (\delta ) &=&{A \over 2} \epsilon \left[ \int_K
|\nabla_K \delta |^2 + a_N \int_K {\mathcal H} \delta^2 \pm \epsilon b_{N}\int_K \frac{\delta^2}{\mu_0} \right] \end{eqnarray*} where $A= \int_{\mathbb{R}^N} Z_0^2 $ and ${\mathcal H} (y)$ defined in (\ref{defgy}). Since we are assuming that $\mu_0$ is a nondegenerate solution to the following problem, \begin{eqnarray*}
- \Delta_K\mu +a_N {\mathcal H} (y)\mu \pm \frac{b_N}{\mu}=0,\quad \ \ in \ K, \end{eqnarray*} the operator $P$ is invertible. Thus, for each $f \in L^2 (\Omega_{\epsilon , \gamma} )$, $$ \delta \in {\mathcal H}^1 (K) \longrightarrow \mathbb R , \quad \delta \longmapsto P_\epsilon (\delta ) - {\mathcal L}_f^3 (\delta ) $$ has a unique critical point $\delta $, which satisfies $$
\epsilon^{-{k\over 2}} \| \delta \|_{{\mathcal H}^1 (K)} \leq
\widetilde{\sigma} \epsilon^{-2} \| f \|_{L^2 (\Omega_{\epsilon , \gamma} )} $$ for some proper $\widetilde{\sigma} >0$.
\noindent \textrm{\textbf{Step 3}}. For all $\epsilon >0$ small, the functional $Q_\epsilon$ defined in (\ref{qje}) is a small perturbation in $({\mathcal H}^1 (K) )^{N }$ of the quadratic form $\epsilon Q_0 (d)$, defined by $$
\epsilon Q_0 (d)= {\epsilon \over 2} C \left[ \int_K |\nabla_K d |^2 + \int_K (\tilde g^{ab}\,R_{mabl} - \Gamma_a^c (E_m) \Gamma_c^a (E_l) ) d^m d^l \right] $$ with $C:= \int_{\mathbb R^N} Z_1^2$ and the terms $R_{maal} $ and $\Gamma_a^c (E_m)$ are smooth functions on $K.$ Recall that the non-degeneracy assumption on the minimal submanifold $K$ is equivalent to the invertibility of the operator $Q_0 (d)$.
A consequence, for each $f \in L^2 (\Omega_{\epsilon , \gamma} )$, $$ d \in ({\mathcal H}^1 (K))^{N} \longrightarrow \mathbb R , \quad d \longmapsto Q_\epsilon (d) - {\mathcal L}_f^3 (d) $$ has a unique critical point $d$, which satisfies $$
\epsilon^{-{k\over 2}} \| d \|_{({\mathcal H}^1 (K))^{N}} \leq
\widetilde{\sigma} \epsilon^{-2} \| f \|_{L^2 (\Omega_{\epsilon , \gamma} )} $$ for some proper $\widetilde{\sigma} >0$.
\noindent \textrm{\textbf{Step 4}}. Let $f \in L^2 (\Omega_{\epsilon , \gamma})$ and assume that $e$ is a given (fixed) function in ${\mathcal H}^1 (K)$. We claim that for all $\epsilon >0$ small enough, the functional ${\mathcal Q} : (H_\epsilon^1)^\bot \times ({\mathcal H}^1 (K) )^N \to \mathbb R$ $$ (\phi^\bot , \delta , d) \to {\mathcal E} (\phi^\bot , \delta , d , e) $$ has a critical point $(\phi^\bot , \delta , d) $. Furthermore there exists a positive constant $C$, independent of $\epsilon$, such that \begin{equation} \label{punto4}
\| \phi^\bot \| + \epsilon^{-{k\over 2}} \bigg[ \| \delta \|_{{\mathcal H}^1 (K)} + \| d \|_{({\mathcal H}^1 (K))^{N}} \bigg] \leq C
\epsilon^{-2} \bigg[ \|f \|_{L^2 (\Omega_{\epsilon , \gamma} )} + \epsilon^{-{k\over 2}} \epsilon^{2} \| e \|_{{\mathcal H^1 (K) } }\bigg]. \end{equation}
To prove the above assertion, we first consider the functional $$ {\mathcal Q}_0 (\phi^\bot , \delta , d) = {\mathcal Q} (\phi^\bot , \delta , d , e) - {\mathcal G} (\phi^\bot , \delta , d , e) $$ where ${\mathcal G} $ is the functional that recollects all mixed terms, as defined in (\ref{EEE}). A direct consequence of Step 1, Step 2 and Step 3 is that ${\mathcal Q}_0$ has a critical point $(\phi^\bot = \phi^\bot (f) , \delta = \delta (f) , d = d(f))$, namely the system $$ D_{\phi^\bot } E (\phi^\bot ) = D_{\phi^\bot} {\mathcal L}_f^1 (\phi^\bot ), \quad \epsilon^{-{k\over 2}} D_\delta P_\epsilon (\delta ) = D_\delta {\mathcal L}_f^2 (\delta ), \quad \epsilon^{-{k\over 2}} D_d Q_\epsilon (d) = D_d {\mathcal L}_f^3 (d) $$ is uniquely solvable in $(H_\epsilon^1 )^\bot \times ({\mathcal H}^1 (K) )^N $ and furthermore $$
\| \phi^\bot \|_{H^1_\epsilon} +\epsilon^{-{k\over 2}} \| \delta \|_{{\mathcal H}^1 (K)} + \epsilon^{-{k\over 2}} \| d \|_{({\mathcal H}^1 (K) )^{N-1}}
\leq C \epsilon^{-2} \| f \|_{L^2 (\mathcal{M}_{\epsilon , \gamma })} $$ for some constant $C>0$, independent of $\epsilon$.
If we now consider the complete functional ${\mathcal Q}$, a critical point of ${\mathcal Qx}$ shall satisfy the system \begin{equation}\label{syst2}
\begin{cases}
D_{\phi^\bot } E(\phi^\bot ) = D_{\phi^\bot} {\mathcal L}_f^1 (\phi^\bot ) + D_{\phi^\bot } {\mathcal G} (\phi^\bot , \delta , d , e) \\
D_\delta P_\epsilon (\delta ) = D_\delta {\mathcal L}_f^2 (\delta ) + D_{\delta } {\mathcal G} (\phi^\bot , \delta , d , e)
\\ D_d Q_\epsilon (d ) = D_d {\mathcal L}_f^3 (d ) + D_{d } {\mathcal G} (\phi^\bot , \delta , d ,
e).
\end{cases} \end{equation} On the other hand, as we have already observed in Theorem \ref{teo4.1}, we have $$
\| D_{(\phi^\bot , \delta , d)} {\mathcal G}(\phi_1^\bot , \delta_1 , d_1 , e_1) - D_{(\phi^\bot , \delta , d)} {\mathcal G}(\phi_2^\bot ,
\delta_2 , d_2 , e_2) \| \leq C \epsilon^{2} \times $$ $$
\left[ \| \phi_1^\bot - \phi_2^\bot \| + \epsilon^{-{k\over 2}} \|
\delta_1 - \delta_2 \|_{{\mathcal H}^1 (K)} + \epsilon^{-{k\over 2}} \|
d_1 - d_2 \|_{({\mathcal H}^1 (K))^{N}} + \epsilon^{-{k\over 2}} \| e_1
- e_2 \|_{{\mathcal H}^1 (K)}\right]. $$ Thus the contraction mapping Theorem guarantees the existence of a unique solution $(\bar \phi^\bot , \bar \delta , \bar d)$ to (\ref{syst2}) in the set $$
\| \phi^\bot \|_{H_\epsilon^1} + \epsilon^{-{k\over 2}} \| \delta \|_{{\mathcal H}^1 (K)} + \epsilon^{-{k\over 2}} \| d \|_{({\mathcal H}^1 (K))^{N}}
\leq C \left[ \epsilon^{-2} \| f \|_{L^2 (\Omega_{\epsilon , \gamma} )} + \epsilon^{2}
\epsilon^{-{k\over 2}} \| e \|_{{\mathcal H}^1 (K)} \right]. $$ Furthermore, the solution $\bar \phi^\bot = \bar \phi^\bot (f,e) $, $\bar \delta =\bar \delta (f,e )$ and $\bar d= \bar d(f,e)$ depend on $e$ in a smooth and non-local way.
\noindent {\rm \textbf{Step 5}}. Given $f\in L^2 (\Omega_{\epsilon , \gamma} )$, we replace the critical point $(\bar \phi^\bot = \bar \phi^\bot (f,e), \bar \delta =\bar \delta (f,e ), \bar d= \bar d(f,e) )$ of ${\mathcal Q}$ obtained in the previous step into the functional ${\mathcal E}(\phi^\bot , \delta , d , e)$ thus getting a new functional depending only on $e \in {\mathcal H}^1 (K)$, that we denote by ${\mathcal F}_\epsilon(e)$, given by \begin{eqnarray*} {\mathcal F}_\epsilon(e) &=& \epsilon^{-{k\over 2}} [ R_\epsilon (e) - {\mathcal L}_f^4 (e) ] + E(\bar \phi^\bot (e) ) -\epsilon^{-{k\over 2}} {\mathcal L}_f^1 (\bar \phi^\bot (e)) +\epsilon^{-{k\over 2}} [ P_\epsilon (\bar \delta (e)) - {\mathcal L}_f^2 (\bar \delta (e) )] \\ &+& \epsilon^{-{k\over 2}} [ Q_\epsilon (\bar d (e)) - {\mathcal L}_f^3 (\bar d (e)) ] + {\mathcal G} (\bar \phi^\bot (e) , \bar \delta (e) , \bar d (e) , e). \end{eqnarray*} The rest of the proof is devoted to show that there exists a sequence $\epsilon = \epsilon_l \to 0$ such that \begin{equation} \label{effe1} D_e {\mathcal F}_\epsilon(e) = 0 \end{equation} is solvable. Using the fact that $(\bar \phi^\bot , \bar \delta , \bar d )$ is a critical point for ${\mathcal Q}$ (see Step 4 for the definition), we have that \begin{equation} \label{effe11} D_e {\mathcal F}_\epsilon (e) = \epsilon^{-{k\over 2}} D_e [ R_\epsilon (e) - {\mathcal L}_f^4 (e) ] + D_e {\mathcal G} (\bar \phi^\bot (e) , \bar \delta (e) , \bar d (e) , e). \end{equation} Define \begin{equation} \label{defLe} {\mathcal L}_\epsilon := \epsilon^{-{k\over 2}} D_e R_\epsilon (e) + D_e {\mathcal G} (\bar \phi^\bot (e) , \bar \delta (e) , \bar d (e) , e), \end{equation} regarded as self adjoint in ${\mathcal L}^2 (K)$. The work to solve the equation $D_e {\mathcal F}_\epsilon (e)=0$ consists in showing the existence of a sequence $\epsilon_l \to 0$ such that $0$ lies suitably far away from the spectrum of ${\mathcal L}_{\epsilon_l}$.
We recall now that the map $$ (\phi^\bot , \delta , d , e ) \to D_e {\mathcal G} (\phi^\bot , \delta , d , e) $$ is a linear operator in the variables $\phi^\bot , \delta , d$, while it is constant in $e$. This is contained in the result of Theorem \ref{teo4.1}. If we furthermore take into account that the terms $\bar \phi^\bot $, $\bar \delta $ and $\bar d $ depend smoothly and in a non-local way through $e$, we conclude that, for any $e \in {\mathcal H}^1 (K)$, \begin{equation} \label{effe2} D_e {\mathcal G} (\bar \phi^\bot (e) , \bar \delta (e) , \bar d (e) , e )[e] = \epsilon^{\gamma (N-3)} \epsilon^{-{k\over 2}} \int_K \left( \epsilon \eta_1 (e) \partial_a e + \eta_2 (e) e \right)^2 \end{equation} where $\eta_1$ and $\eta_2$ are non local operators in $e$, that are bounded, as $\epsilon \to 0$, on bounded sets of ${\mathcal L}^2 (K)$. Thanks to the result contained in Theorem \ref{teo4.1} and the above observation, we conclude that the quadratic from $$ \Upsilon_\epsilon (e) := \epsilon^{-{k\over 2}} D_e R_\epsilon (e) [e] + D_e {\mathcal G} (\bar \phi^\bot (e) , \bar \delta (e) , \bar d (e) , e) [e] $$ can be described as follows \begin{equation} \label{Uptilde} \tilde \Upsilon_\epsilon (e) = \epsilon^{k\over 2} \Upsilon_\epsilon (e ) = \Upsilon^0_\epsilon (e)
- {\bar \lambda_0 } \int_K e^2 + \epsilon \Upsilon_\epsilon^1 (e) \end{equation} where \begin{equation} \label{Up0} \Upsilon_\epsilon^0 (e) = \epsilon^2 \int_K (1+ \epsilon^{\gamma (N-3)}
\eta_1 (e) ) \left| \partial_a \left( e (1+ e^{-\epsilon^{-\lambda'}}
\beta_3^\epsilon (y) ) \right) \right|^2. \end{equation} In the above expression $\bar \lambda_0 $ is the positive number defined by $$ \bar \lambda_0 = (\int_{\mathbb R^N} Z_1^2 )\, \lambda_0, $$ $\Upsilon_e^1 (e)$ is a compact quadratic form in ${\mathcal H}^1 (K)$, $\beta_3^\epsilon$ is a smooth and bounded (as $\epsilon \to 0$) function on $K$, given by (\ref{Q}). Finally, $\eta_1 $ is a non local operator in $e$, which is uniformly bounded, as $\epsilon \to 0$
on bounded sets of ${\mathcal L}^2 (K)$.
Thus, for any $\epsilon >0$, the eigenvalues of $$ {\mathcal L}_\epsilon e = \lambda e , \quad e \in {\mathcal H}^1 (K) $$ are given by a sequence $\lambda_j (\epsilon)$, characterized by the Courant-Fisher formulas \begin{equation}\label{courantFisher} \lambda_j (\epsilon )= \sup_{dim (M)= j-1} \inf_{e \in M^\bot \setminus \{0 \}} {\tilde \Upsilon_\epsilon (e ) \over \int_K e^2 } = \inf_{dim (M)= j } \sup_{e \in M\setminus \{0 \}} {\tilde \Upsilon_\epsilon (e) \over \int_K e^2 }. \end{equation} The proof of Theorem \ref{teouffa} and of the inequality (\ref{uffa1}) will follow then from Step 4 and formula (\ref{punto4}), together with the validity of the following
\begin{lemma} \label{gafa} There exist a sequence $\epsilon_l \to 0 $ and a constant $c>0$ such that, for all $j$, we have \begin{equation}\label{autovalore}
|\lambda_j (\epsilon_l ) | \geq c \epsilon_l^k. \end{equation} \end{lemma}
For the proof of Lemma we refer to \cite{demamu}.
\end{proof}
\section{Proof of Proposition \ref{linear}}\label{luigi}
The proof of this Proposition will be divided into several steps.
{\bf Step 1}. \ \ Let us assume that $\phi$ solves (\ref{eq:eqwd}). We claim that there exists $C>0$ such that \begin{equation}
\label{mar2} \|\phi \|_{\epsilon, r- 2}\leqslant C \|h\|_{\epsilon,r}. \end{equation} By contradiction, assume that there exist sequences $\epsilon_n
\to 0$, $h_n$ with $\| h_n \|_{\epsilon_n , r} \to 0$ and solutions
$\phi_n$ to (\ref{eq:eqwd}) with $\| \phi_n \|_{\epsilon_n , r -2} =1$.
Let $z_n \in K_{\epsilon_n}$ and $\xi_n$ be such that $$
|\phi_n (\epsilon_n z_n , \xi_n )| = \sup |\phi_n (y, \xi )|. $$ We may assume that, up to subsequences, $(\epsilon_n z_n ) \to \bar y $ in $K$. Furthermore, we have by assumption that
$|\xi_n | \leq \eta \epsilon_n^{-{1\over 2}}$.
Let us now assume that there exists a positive constant $R$ such that $|\xi_n |\leq R$. In this case, up to subsequences, one gets that $\xi_n \to \xi_0$. Consider the functions $$\tilde \phi_n ( z, \xi ) = \phi_n ( z , \xi + \xi_n ), \quad {\mbox {for}} \quad (z, \xi) \in K_{\epsilon_n} \times
\{ \xi \in \mathbb R^N \, : \, |\xi | \leq \eta' \epsilon_n^{-{1\over 2}} \} $$ for some $\eta' >0$. This is a sequence of uniformly bounded functions, that converges uniformly over compact sets of $K \times \mathbb R^N$ to a function $\tilde \phi$ solution to $$
- \Delta \tilde \phi - p w_0^{p-1} \tilde \phi =0 \quad \hbox{ in } \mathbb R^{N}
$$ Since the orthogonality conditions pass to the limit, we get that furthermore $$
\int_{\mathbb R^{N} } \tilde \phi (y, \xi ) Z_j (\xi ) \, d\xi = 0 \quad {\mbox {for all}} \quad y \in K, \quad j=0, \ldots N.
$$ These facts imply that $\tilde \phi \equiv 0$, that is a contradiction.
Assume now that $\lim\limits_{n \to \infty} |\xi_n | = \infty$. Consider the scaled function $$
\tilde \phi_n (z, \xi ) = \phi_n (z, |\xi_n | \xi + \xi_n ) $$ defined on the set $$
\tilde \mathcal{D}=\left\{(z,\xi) :\ z \in K_{\epsilon_n}, \, |\xi|<\frac{\eta}{\sqrt{\epsilon_n}|\xi_n|}-\frac{\xi_n}{|\xi_n|},\ \right\}. $$ Thus $\tilde \phi_n$ satisfies the equation $$
\Delta \tilde \phi_n + p \,C_N {|\xi_n |^2 \over (1+| \, |\xi_n | \xi
+ \xi_n |^2 )^2 } \tilde \phi_n -|\xi_n |^2 \epsilon_n a \tilde \phi_n =
|\xi_n |^2 h (z, |\xi_n | \xi +\xi_n )\ \ \mbox{in}\ \tilde \mathcal{D}. $$
Consider first the case in which $\lim\limits_{n \to \infty} \epsilon_n |\xi_n
|^2 = 0$. Under our assumptions, we have that $\tilde \phi_n$ is uniformly bounded and it converges locally over compact sets to $\tilde \phi$ solution to $$
\Delta \tilde \phi = 0, \quad |\tilde
\phi | \leq C |\xi |^{2-r}\quad {\mbox {in}} \quad \mathbb R^{N}\setminus \{ 0 \}.\quad $$ Since $4<r<N$, we conclude that $\tilde \phi \equiv 0 $, which is a contradiction.
Consider now the other possible case, namely that $$\lim_{n \to
\infty} \epsilon_n |\xi_n |^2 = \beta >0.$$ Then, $$
\tilde \mathcal{D}\to \mathcal{S}:=\{\xi \, : \, |\xi| \in [0,L)\} \quad \mbox{as}\ n\to\infty $$ where $L$ is some positive constant. Furthermore, up to subsequences, we get that $\tilde \phi_n$ converges uniformly over compact sets to $\tilde \phi$ solution to $$
\Delta \tilde \phi - \beta a \tilde \phi = 0, \quad |\tilde \phi | \leq C |\xi |^{2-r} \quad {\mbox {in}} \ \mathcal{S}, \quad \tilde \phi = 0 \quad {\mbox {on}} \quad \partial \mathcal{S}. $$ Multiplying equation by $\tilde\phi$, and integrating it over $ \mathcal{S}$ only in $\xi$, we get $$
\int_{\mathcal{S}}(|\nabla \tilde\phi|^2+\beta a \tilde\phi^2)d\xi=0. $$ Thus we conclude that $\tilde \phi \equiv 0$, which is a contradiction. The proof of (\ref{mar2}) is completed.
{\bf Step 2}. \ \ We shall now show that there exists $C>0$ such that, if $\phi$ is a solution to (\ref{eq:eqwd}), then \begin{equation}
\label{mar3} \| D^2_\xi \phi \|_{\epsilon , r } + \| D_\xi \phi \|_{\epsilon , r -1 } +\|\phi \|_{\epsilon, r- 2 }\leqslant C \|h\|_{\epsilon,r }. \end{equation}
For $z \in K_\epsilon$, we have that $\phi$ solves $ -\Delta \phi = \tilde h $
in $ |\xi |< \eta \epsilon^{-{1\over 2}} $ where $|\tilde h |\leq {C
\over (1+|\xi|^{r } )}$, for some constant $C>0$. Elliptic estimates give that $| \phi | \leq
{C \over (1+|\xi|^{r -2 } )}$.
Let us now fix a point $e \in \mathbb R^N$ and a positive number $R>0$. Perform the change of variables $ \tilde \phi (z,t) = \phi (z, Rt +3Re)$, so that $$ \Delta \tilde \phi = {1\over R^{r-2}} \tilde h \quad {\mbox {in}}
\quad |t|\leq 1 $$
where $|\tilde h | \leq {c \over |t+3e|^{r}}$. Elliptic estimates give then that $\| R^{r-2} D^2 \tilde \phi \|_{L^\infty(0,1)} \leq C
\| \tilde h \|_{L^\infty (B(0,2))}$, inequality that translates into $$
\| R^{r} D^2 \phi \|_{L^\infty (B(3Re,R))} \leq C \| (1+|\xi |)^{r}
h \|_{L^\infty (|\xi |\leq \eta \epsilon^{-{1\over 2}} )}. $$ This inequality finally gives $$
\| (1+|\xi |)^{r} D^2 \phi \|_{L^\infty (|\xi |\leq \eta
\epsilon^{-{1 \over 2}} )} \leq C \| (1+|\xi |)^{r} h \|_{L^\infty (|\xi
|\leq \eta \epsilon^{-{1\over 2}} )}. $$ Arguing in a similar way, one gets the internal weighted estimate for the first derivative of $\phi$ $$
\| (1+|\xi |)^{r-1} D \phi \|_{L^\infty (|\xi |\leq \eta
\epsilon^{-{1\over 2}} )} \leq C \| (1+|\xi |)^{r} h \|_{L^\infty (|\xi
|\leq \eta \epsilon^{-{1\over 2}} )}. $$
By using the representation formula for solution $\phi$ to the above equation, we see that $ | \phi | \leq C \epsilon^{{r - 2 \over 2}}$ in $|\xi |<\eta
\epsilon^{-{1\over 2}}$. Furthermore, elliptic estimates give that in this region $ | D\phi | \leq C \epsilon^{{r - 1 \over 2}}$ and $ | D^2 \phi | \leq C \epsilon^{{r \over 2}}$. This concludes the proof of (\ref{mar3}).
{\bf Step 3}. \ \ We shall now show that there exists $C>0$ such that, if $\phi$ is a solution to (\ref{eq:eqwd}), then \begin{equation}
\label{mar4} \| D^2_\xi \phi \|_{\epsilon , r , \sigma } + \| D_\xi \phi
\|_{\epsilon , r -1 , \sigma } +\|\phi \|_{\epsilon, r- 2 , \sigma}\leqslant C
\|h\|_{\epsilon,r ,\sigma} . \end{equation}
From elliptic regularity, we have that if $ \|h\|_{\epsilon,r ,\sigma} \leq C$ then $\|\phi \|_{\epsilon, r- 2 , \sigma} \leq C$. Thus, we write that
$\phi$ solves $ -\Delta \phi = \tilde h $ in $ |\xi |< \eta
\epsilon^{-{1\over 2}} $ where $\|\tilde h\|_{\epsilon,r ,\sigma} \leq C$.
Arguing as in the previous step, we fix a point $e \in \mathbb R^N$ and a positive number $R>0$. Perform the change of variables $ \tilde \phi (z,t) = \phi (z, Rt +3Re)$, so that $$ \Delta \tilde \phi = {1\over R^{r-2}} \tilde h \quad {\mbox {in}}
\quad |t|\leq 1 $$
where $|\tilde h | \leq {c \over |t+3e|^{r}}$. Elliptic estimates give then that $\| R^{r-2} D^2 \tilde \phi \|_{C^{0,\sigma}
(B(0,1))} \leq C \| \tilde h \|_{L^\infty (B(0,2))}$. This implies that $$
R^{r-2} \| D_\xi^2 \tilde \phi \|_{L^\infty (B_1)} + R^{r-2} [D^2 \tilde \phi ]_{\sigma, B(0,1)} \leq C. $$ In particular, we have for any $z \in K_\epsilon$, that $$
R^{r-2} \sup_{y_1 , y_2 \in B(0,1)} {|D^2 \tilde \phi (z, y_1) - D^2
\tilde \phi (z, y_2) | \over |y_1 - y_2 |^\sigma} \leq C. $$ This inequality gets translated in term of $\phi$ as $$
R^{r+\sigma} \sup_{\xi_1 , \xi_2 \in B(\xi,1)} {|D^2 \phi (z,
\xi_1) - D^2 \phi (z, \xi_2) | \over |\xi_1 - \xi_2 |^\sigma} \leq C. $$ In a very similar way, one gets the estimate on $D\phi$. This concludes the proof of (\ref{mar4}).
{\bf Step 4}. \ \ Differentiating equation (\ref{eq:eqwd}) with respect to the $z$ variable $l$ times and using elliptic regularity estimates, one proves that \begin{equation}
\label{est1} \| D^l_y \phi \|_{\epsilon , r- 2 , \sigma } \leqslant C_l \left(
\sum_{k\leq l} \|D^k_y h\|_{\epsilon,r , \sigma}\right) \end{equation} for any given integer $l$.
{\bf Step 5}. \ \ Now we shall prove the existence of the solution $\phi$ to problem (\ref{eq:eqwd}). We consider the Hilbert space $\mathcal{H}$ defined as the subspace of functions $\psi$ which are in $H^1(\mathcal{D})$ such that $\psi = 0 $ on $\partial \hat \mathcal{D} $, and $$ \int_{\hat\mathcal{D}} \psi (\epsilon z, \xi ) Z_j (\xi ) \, d\xi = 0 \ {\mbox {for all}} \quad z \in K_\epsilon, \quad j=0, \ldots N. $$ Define a bilinear form in $\mathcal{H}$ by $$ B(\phi,\psi):=\int_{\hat\mathcal{D}}\psi L\phi. $$ Then problem (\ref{eq:eqwd}) gets weakly formulated as that of finding $\phi\in \mathcal{H}$ such that $$ B(\phi,\psi)=\int_{\hat\mathcal{D}}h\psi\quad \forall\ \psi\in \mathcal{H}. $$ By the Riesz representation theorem, this is equivalent to solve \begin{eqnarray*} \phi= T(\phi)+\tilde{h} \end{eqnarray*} with $\tilde{h}\in \mathcal{H}$ depending linearly on $h$, and $T: \mathcal{H} \rightarrow \mathcal{H}$ being a compact operator. Fredholm's alternative guarantees that there is a unique solution to problem (\ref{eq:eqwd}) for any $h$ provided that \begin{eqnarray}\label{linear7} \phi= T(\phi) \end{eqnarray} has only the zero solution in $\mathcal{H}$. Equation (\ref{linear7}) is equivalent to problem (\ref{eq:eqwd}) with $h=0$. If $h=0$, the estimate in (\ref{est0a}) implies that $\phi=0$.
This concludes the proof of Proposition \ref{linear}.
\end{document} | arXiv |
\begin{document}
\title[Fibonacci-like unimodal inverse limits and the core Ingram conjecture] {Fibonacci-like unimodal inverse limit spaces and the core Ingram conjecture} \author{H.~Bruin \quad and \quad S.~\v{S}timac} \thanks{HB was supported by EPSRC grant EP/F037112/1. S\v{S} was supported in part by the MZOS Grant 037-0372791-2802 of the Republic of Croatia.}
\subjclass[2000]{54H20, 37B45, 37E05}
\keywords{tent map, inverse limit space, Fibonacci unimodal map, structure of inverse limit spaces}
\begin{abstract} We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allow us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to
the proof of the Ingram Conjecture for cores of Fibonacci-like unimodal inverse limits. \end{abstract}
\maketitle
\baselineskip=18pt
\section{Introduction}\label{sec:intro}
A unimodal map is called Fibonacci-like if it satisfies certain combinatorial conditions implying an extreme recurrence behavior of the critical point. The Fibonacci unimodal map itself was first described by Hofbauer and Keller \cite{hofkel0} as a candidate to have a so-called wild attractor. (The combinatorial property defining the Fibonacci unimodal map is that its so-called {\em cutting times} are exactly the Fibonacci numbers $1, 2, 3 ,5, 8, \dots$) In \cite{BKNS} it was indeed shown that Fibonacci unimodal maps with sufficiently large critical order possess a wild attractor, whereas Lyubich \cite{Lyu} showed that such is not the case if the critical order is $2$ (or $\leq 2+\varepsilon$ as was shown in \cite{KN}). This answered a question in Milnor's well-known paper on the structure of metric attracts \cite{Mil}. In \cite{BTams} the strict Fibonacci combinatorics were relaxed to Fibonacci-like. Intricate number-theoretic properties of Fibonacci-like critical omega-limit sets were revealed in \cite{LM} and \cite{BKS}, and \cite[Theorem 2]{BNonlin} shows that Fibonacci-like combinatorics are incompatible with the Collet-Eckmann condition of exponential derivative growth along the critical orbit. This underlines that Fibonacci-like maps are an extremely interesting class of maps in between the regular and the stochastic unimodal maps in the classification of \cite{ALM}.
One of the reasons for studying the inverse limit spaces of Fibonacci-like unimodal maps is that they present a toy model of invertible strange attractors (such as H\'enon attractors) for which as of today very little is known beyond the Benedicks-Carleson parameters \cite{BC} resulting in strange attractors with positive unstable Lyapunov exponent. It is for example unknown if invertible wild attractors exist in the smooth planar context, or to what extent H\'enon-like attractors satisfy Collet-Eckmann-like growth conditions. The precise recurrence and folding structure of H\'enon-like attractors may be of crucial importance to answer such questions, and we therefore focus on these aspects of the structure of Fibonacci-like inverse limit spaces.
A second reason for this paper is to provide a better understanding and the solution of the Ingram Conjecture for cores of Fibonacci-like inverse limit spaces. The original conjecture was posed by Tom Ingram in 1991 for tent maps $T_s : [0, 1] \to [0, 1]$ with slope $\pm s$, $s \in [1, 2]$, defined as $T_s(x) = \min\{sx, s(1-x)\}$: \begin{quote} If $1 \leq s < s' \leq 2$, then the corresponding inverse limit spaces $\underleftarrow\lim([0,s/2],T_s)$ and $\underleftarrow\lim([0,s'/2],T_{s'})$ are non-homeomorphic. \end{quote} The first results towards solving this conjecture have been obtained for tent maps with a finite critical orbit \cite{Kail2,Stim,Betal}. Raines and \v{S}timac \cite{RS} extended these results to tent maps with an infinite, but non-recurrent critical orbit. Recently Ingram's Conjecture was solved for all slopes $s \in [1,2]$ (in the affirmative) by Barge, Bruin and \v Stimac in \cite{BBS}, but we still know very little of the structure of inverse limit spaces (and their subcontinua) for the case that $\mbox{\rm orb}(c)$ is infinite and recurrent, see \cite{BBD, BB, Bsubcontinua}. Also, the arc-component ${\mathfrak C}$ of $\underleftarrow\lim([0,s/2],T_s)$ containing the endpoint $\bar 0 := (\dots,0,0,0)$ is important in the proof of the Ingram Conjecture in \cite{BBS}, leaving open the ``core'' version of the Ingram Conjecture. It is this version that we solve here for Fibonacci-like tent maps:
\begin{maintheorem}\label{mainthm} If $1 \leq s < s' \leq 2$ are the parameters of Fibonacci-like tent-maps, then the corresponding cores of inverse limit spaces $\underleftarrow\lim([c_2, c_1],T_{s})$ and $\underleftarrow\lim([c_2, c_1],T_{s'})$ are non-homeomorphic. \end{maintheorem}
The core version of the Ingram Conjecture was proved already in the postcritically finite case, since neither Kailhofer \cite{Kail2}, nor \v{S}timac \cite{Stim} use the arc-component ${\mathfrak C}$, but work on some other arc-components of the core (although not on the arc-component of the fixed point $(\dots , r, r, r)$ which we use in this paper).
The key observation in our proof is Proposition~\ref{prop:symmetric} which implies that every homeomorphism $h$ maps symmetric arcs to symmetric arcs, not just to quasi-symmetric arcs. (The difficulty that quasi-symmetric arcs pose was first observed and overcome in \cite{RS} in the setting of tent maps with non-recurrent critical point.) To prove Proposition~\ref{prop:symmetric}, the special structure of the Fibonacci-like maps, and especially the special chains it allows, is used. But assuming the result of Proposition~\ref{prop:symmetric}, the proof of the main theorem works for general tent maps.
The paper is organized as follows. In Section~\ref{sec:def} we review the basic definitions of inverse limit spaces and tent maps and their symbolic dynamics. In Section~\ref{sec:homeomorphisms} we introduce salient points, show that any homeomorphism on the core of the Fibonacci-like inverse limit space maps salient points ``close'' to salient points, and using this we prove our main theorem in Section~\ref{sec:maintheorems}. Appendix~\ref{sec:chains} is devoted to the construction of the chains ${\mathcal C}$ having special properties that allow us to prove desired properties of folding structure in Appendix~\ref{sec:furtherlemmas}. In Appendix~\ref{sec:link}, we show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs.
\section{Preliminaries}\label{sec:def}
\subsection{Combinatorics of tent maps} The tent map $T_s : [0,1] \to [0,1]$ with slope $\pm s$ is defined as $T_s(x) = \min\{sx, s(1-x)\}$. The critical or turning point is $c = 1/2$ and we write $c_k = T_s^k(c)$, so in particular $c_1 = s/2$ and $c_2 = s(1-s/2)$. We will restrict $T_s$ to the interval $I = [0, s/2]$; this is larger than the {\em core} $[c_2, c_1] = [s-s^2/2, s/2]$, but it contains both fixed points $0$ and $r = \frac{s}{s+1}$.
Recall now some background on the combinatorics of unimodal maps, see {\em e.g.} \cite{Bknea}. The {\em cutting times} $\{S_k\}_{k \geq 0}$ are those iterates $n$ (written in increasing order) for which the central branch of $T_s^n$ covers $c$. More precisely, let $Z_n \subset [0,c]$ be the maximal interval with boundary point $c$ on which $T_s^n$ is monotone, and let $\mathfrak D_n = T_s^n(Z_n)$. Then $n$ is a {\em cutting time} if $\mathfrak D_n \ni c$. Let ${\mathbb N} = \{ 1,2,3,\dots\}$ be the set of natural numbers and ${\mathbb N}_0 = {\mathbb N} \cup \{ 0 \}$. There is a function $Q : {\mathbb N} \to {\mathbb N}_0$ called the {\em kneading map} such that \begin{equation}\label{eq:Q} S_k - S_{k-1} = S_{Q(k)} \end{equation} for all $k$. The kneading map $Q(k) = \max \{ k-2, 0 \}$ (with cutting times $\{ S_k \}_{k \geq 0} = \{ 1,2,3,5,8,\dots\}$) belongs to the {\em Fibonacci map}. We call $T_s$ {\em Fibonacci-like} if its kneading map is eventually non-decreasing, and satisfies Condition \eqref{eq:cond4} as well: \begin{equation}\label{eq:cond4} Q(k+1) > Q(Q(k)+1) \qquad \text{ for all $k$ sufficiently large.} \end{equation} \begin{remark}\label{rem:cond4} Condition \eqref{eq:cond4} follows if the $Q$ is eventually non-decreasing and $Q(k) \leq k-2$ for $k$ sufficiently large. (In fact, since tent maps are not renormalizable of arbitrarily high period, $Q(k) \leq k-2$ for $k$ sufficiently large follows from $Q$ being eventually non-decreasing, see \cite[Proposition 1]{Bknea}.)
Geometrically, it means that $|c-c_{S_k}| < |c-c_{S_{Q(k)}}|$, see Lemma~\ref{lem:order} and also \cite{Bknea}. \end{remark} \begin{lemma}\label{lem:order} If the kneading map of $T_s$ satisfies \eqref{eq:cond4}, then \begin{equation}\label{eq:order}
|c_{S_k} - c| < |c_{S_{Q(k)}} - c| \quad \text{ and } \quad |c_{S_k} - c| < \frac12 |c_{S_{Q^2(k)}} - c|. \end{equation} for all $k$ sufficiently large. \end{lemma}
\begin{proof} For each cutting time $S_k$, let $\zeta_k \in Z_{S_k}$ be the point such that $T_s^{S_k}(\zeta_k) = c$. Then $\zeta_k$ together with its symmetric image $\hat \zeta_k := 1-\zeta_k$ are closest precritical points in the sense that $T_s^j((\zeta_k,c)) \not\ni c$ for $0 \leqslant j \leqslant S_k$. Consider the points $\zeta_{k-1}$, $\zeta_k$ and $c$, and their images under $T_s^{S_k}$, see Figure~\ref{fig:order}. \begin{figure}
\caption{The points $\zeta_{k-1}$, $\zeta_k$ and $c$, and their images under $T_s^{S_k}$.}
\label{fig:order}
\end{figure} Note that $Z_{S_k} = [\zeta_{k-1},c]$ and $T_s^{S_k}([\zeta_{k-1},c]) = \mathfrak D_{S_k} = [c_{S_{Q(k)}}, c_{S_k}]$. Since $S_{k+1} = S_k + S_{Q(k+1)}$ is the first cutting time after $S_k$, the precritical point of lowest order on $[c,c_{S_k}]$ is $\zeta_{Q(k+1)}$ or its symmetric image $\hat \zeta_{Q(k+1)}$. Applying this to $c_{S_k}$ and $c_{Q(k)}$, and using \eqref{eq:cond4}, we find $$ c_{S_k} \subset (\zeta_{Q(k+1)-1}, \hat \zeta_{Q(k+1)-1}) \subset (\zeta_{Q(Q(k)+1)}, \hat \zeta_{Q(Q(k)+1)}) \subset (c_{S_{Q(k)}}, \hat c_{S_{Q(k)}}). $$
Therefore $|c_{S_k} - c| < |c_{S_{Q(k)}} - c|$. Since $T_s^{S_k}|_{[\zeta_{k-1},c]}$ is affine, also the preimages $\zeta_{k-1}$ and $\zeta_k$ of
$c_{S_{Q(k)}}$ and $c$ satisfy $|\zeta_k - c| < |\zeta_{k-1} - \zeta_k|$. Applying \eqref{eq:cond4} twice we obtain \begin{equation}\label{eq:cond3} Q(k+1) > Q(Q^2(k)+1)+1, \end{equation} for all $k$ sufficiently large. Therefore there are at least two closest precritical points ($\hat \zeta_{Q(Q^2(k)+1)}$ and $\hat \zeta_{Q(Q^2(k)+1)+1}$ in Figure~\ref{fig:order}) between $c_{S_k}$ and $c_{S_{Q^2(k)}}$. Therefore \begin{equation}\label{eq:cond4twice}
|c_{S_k} - c| < |\hat \zeta_{Q(Q^2(k)+1)+1} - c|< \frac12 |\hat \zeta_{Q(Q^2(k)+1)} - c| < \frac12 |c_{S_{Q^2(k)}} - c|, \end{equation} proving the lemma. \end{proof} Not all maps $Q: {\mathbb N} \to {\mathbb N}_0$ nor all sequences of cutting times (as defined in \eqref{eq:Q}) correspond to a unimodal map. As was shown by Hofbauer \cite{Hof1}, a kneading map $Q$ belongs to a unimodal map (with infinitely many cutting times) if and only if \begin{equation}\label{eq:Hofbauer} \{ Q(k+j) \}_{j \geq 1} \geq_{lex} \{ Q(Q^2(k)+j) \}_{j \geq 1} \end{equation} for all $k \geq 1$, where $\geq_{lex}$ indicates lexicographical order. Clearly, Condition \eqref{eq:cond4} is compatible with (and for large $k$ implies) Condition~\eqref{eq:Hofbauer}.
\begin{remark}\label{rem:Hofbauer} The condition $\{ Q(k+j) \}_{j \geq 1} \geq_{lex} \{ Q(l+j) \}_{j \geq 1}$
is equivalent to $|c-c_{S_k}| < |c-c_{S_l}|$. Therefore, because $c_{S_{k-1}} \in (\zeta_{Q(k)-1}, \zeta_{Q(k)})$, we find by taking the $T_s^{S_{Q(k)}}$-images, that $c_{S_k} \in [c_{S_{Q^2(k)}}, c]$ and \eqref{eq:Hofbauer} follows. The other direction, namely that \eqref{eq:Hofbauer} is sufficient for admissibility is much more involved, see \cite{Hof1,Bknea}. \end{remark}
Let $\beta(n) = n - \sup\{ S_k < n\}$ for $n \geq 2$ and find recursively the images of the central branch of $T_s^n$ (the levels in the Hofbauer tower, see {\em e.g.} \cite{Bknea, BBbook}) as \[ \mathfrak D_1 = [0,c_1] \text{ and } \mathfrak D_n = [c_n , c_{\beta(n)}]. \] It is not hard to see that $\mathfrak D_n \subset \mathfrak D_{\beta(n)}$ for each $n$, see \cite{Bknea}, and that if $J \subset [0,s/2]$ is a maximal interval on which $T_s^n$ is monotone, then $T_s^n(J) = \mathfrak D_m$ for some $m \leqslant n$.
The condition that $Q(k) \to \infty$ has consequence on the structure of the critical orbit:
\begin{lemma}\label{lem:Qinfty}
If $Q(k) \to \infty$, then $|\mathfrak D_n| \to 0$ as $n \to \infty$, $c$ is recurrent and $\omega(c)$ is a minimal Cantor set. \end{lemma}
\begin{proof} See \cite[Lemma 2.1.]{Btree}. \end{proof}
\subsection{Definitions for inverse limit spaces} The inverse limit space $K_s = \underleftarrow\lim([0,s/2],T_s)$ is the collection of all backward orbits \[ \{ x = (\dots, x_{-2}, x_{-1}, x_0) : T_s(x_{i-1}) = x_i \in [0,s/2] \text{ for all } i \leq 0\}, \] equipped with metric
$d(x,y) = \sum_{n \leqslant 0} 2^n |x_n - y_n|$ and {\em induced} $($or {\em shift$)$ homeo\-morphism} \[ \sigma(\dots, x_{-2}, x_{-1}, x_0) = (\dots, x_{-2}, x_{-1}, x_0, T_s(x_0)). \] Let $\pi_k : \underleftarrow\lim([0,s/2],T_s) \to [0,s/2]$, $\pi_k(x) = x_{-k}$ be the $k$-th projection map. The \emph{arc-component} of $x \in X$ is defined as the union of all arcs of $X$ containing $x$. Since $0 \in [0,s/2]$, the endpoint $\bar 0 = (\dots, 0,0,0)$ is contained in $\underleftarrow\lim([0,s/2],T_s)$, and the arc-component of $\underleftarrow\lim([0,s/2],T_s)$ of $\bar 0$ will be denoted as ${\mathfrak C}$; it is a ray converging to, but disjoint from the core $\underleftarrow\lim([c_2, c_1],T_s)$ of the inverse limit space. Since $r \in [c_2, c_1]$, the point $\rho = (\dots, r, r, r)$ is contained in $\underleftarrow\lim([c_2, c_1],T_s)$. The arc-component of $\rho$ will be denoted as ${\mathfrak R}$; it is a continuous image of ${\mathbb R}$ and is dense in $\underleftarrow\lim([c_2, c_1],T_s)$ in both directions.
We fix $s \in (\sqrt{2}, 2]$; for these parameters $T_s$ is not renormalizable and $\underleftarrow\lim([c_2, c_1],T_s)$ is indecomposable.
A point $x = (\dots, x_{-2}, x_{-1}, x_0) \in K_s$ is called a {\em $p$-point} if $x_{-p-l} = c$ for some $l \in {\mathbb N}_0$. The number $L_p(x) := l$ is called the {\em $p$-level} of $x$. In particular, $x_0 = T_s^{p + l}(c)$. By convention, the endpoint $\bar 0 = (\dots, 0,0,0)$ of ${\mathfrak C}$ and the point $\rho = (\dots, r, r, r)$ of ${\mathfrak R}$ are also $p$-points and $L_p(\bar 0) = L_p(\rho) := \infty$, for every $p$.
The {\em folding pattern of the arc-component} ${\mathfrak C}$, denoted by $FP({\mathfrak C})$, is the sequence $$ L_p(z^0), L_p(z^1), L_p(z^2), \dots , L_p(z^n), \dots , $$ where $E_p^{{\mathfrak C}} = \{ z^0, z^1, z^2, \dots , z^n, \dots \}$ is the ordered set of all $p$-points of ${\mathfrak C}$ with $z^0 = \bar 0$, and $p$ is any nonnegative integer. Let $q \in {\mathbb N}$, $q > p$, and $E_q^{{\mathfrak C}} = \{ y^0, y^1, y^2, \dots , y^n, \dots \}$. Since $\sigma^{q-p}$ is an order-preserving homeomorphism of ${\mathfrak C}$, it is easy to see that $\sigma^{q-p}(z^i) = y^i$ and $L_p(z^i) = L_q(y^i)$ for every $i \in {\mathbb N}$. Therefore the folding pattern of ${\mathfrak C}$ does not depend on $p$.
The {\em folding pattern of the arc-component} ${\mathfrak R}$, denoted by $FP({\mathfrak R})$, is the sequence \begin{equation}\label{eq:Rr} \dots , L_p(z^{-n}), \dots , L_p(z^{-1}), L_p(z^0), L_p(z^1), \dots , L_p(z^n), \dots , \end{equation} where $E_p^{{\mathfrak R}} = \{ \dots , z^{-n}, \dots , z^{-1}, z^0, z^1, \dots , z^n, \dots \}$ is the ordered set (indexed by ${\mathbb Z}$) of all $p$-points of ${\mathfrak R}$ with $z^0 = \rho$, and $p$ is any nonnegative integer. Since $r > 1/2$, we have $\pi_i(\rho) > 1/2$ for every $i \in {\mathbb N}_0$. It is easy to see that for every $i \in {\mathbb N}_0$, there exists an arc $A = A(i) \subset {\mathfrak R}$ containing $\rho$ such that $\pi_i(A) = [c, c_1]$. Therefore two neighboring $p$-points of $\rho$ have $p$-levels 0 and 1. From now on we assume, without loss of generality, that the ordering on ${\mathfrak R}$, {\em i.e.,} the parametrization of ${\mathfrak R}$, is such that $L_p(z^{-1}) = 0$ and $L_p(z^1) = 1$. Let $q \in {\mathbb N}$, $q > p$, and $E_q^{{\mathfrak R}} = \{ \dots , y^{-n}, \dots , y^{-1}, y^0, y^1, \dots , y^n, \dots \}$ with $y^0 = \rho$. Since $\sigma^{q-p}$ is an order-preserving (respectively, order-reversing) homeomorphism of ${\mathfrak R}$ if $q-p$ is even (respectively, odd), $\sigma^{q-p}(z^i) = y^i$ and $L_p(z^i) = L_q(y^i)$
for every $i \in {\mathbb Z}$. Therefore the folding pattern of ${\mathfrak R}$ does not depend on $p$.
Note that every arc of ${\mathfrak C}$ and of ${\mathfrak R}$ has only finitely many $p$-points, but an arc $A$ of the core of $K_s$ can have infinitely many $p$-points.
We will mostly be interested in the arc-component ${\mathfrak R}$, but also in some other arc-components 'topologically similar' to ${\mathfrak R}$. Therefore, unless stated otherwise, let ${\mathfrak A} \subset \underleftarrow\lim([c_2, c_1],T_s)$ denote an arc-component which does not contain any end-point, such that every arc $A \subset {\mathfrak A}$ contains finitely many $p$-points, and let ${\mathfrak A}$ be dense in the core of $K_s$ in both directions. Let $E_p^{{\mathfrak A}} = (a^i)_{i \in {\mathbb Z}}$ denote the set of all $p$-points of ${\mathfrak A}$, where $a^0 = (\dots, a^0_{-2}, a^0_{-1}, a^0_0) \in {\mathfrak A}$ is the only $p$-point of ${\mathfrak A}$ with $a^0_{-j} \ne c$ for every $j \in {\mathbb N}_0$, and let by convention $L_p(a^0) = \infty$ for every $p$. Also, we abbreviate $E_p := E_p^{{\mathfrak A}}$. The {\em $p$-folding pattern of the arc-component} ${\mathfrak A}$, denoted by $FP_p({\mathfrak A})$, is the sequence $$\dots , L_p(a^{-n}), \dots , L_p(a^{-1}), L_p(a^0), L_p(a^1), \dots , L_p(a^n), \dots .$$ Given an arc $A \subset {\mathfrak A}$ with successive $p$-points $x^0, \dots , x^n$, the {\em $p$-folding pattern} of $A$ is the sequence $$ FP_p(A) := L_p(x^0), \dots , L_p(x^n). $$
An arc $A$ in $\underleftarrow\lim([0,s/2],T_s)$ is said to {\em $p$-turn at $c_n$} if there is a $p$-point $a \in A$ such that $a_{-(p+n)} = c$, so $L_p(a) = n$. This implies that $\pi_p:A \to [0,s/2]$ achieves $c_n$ as a local extremum at $a$. If $x$ and $y$ are two adjacent $p$-points on the same arc-component, then $\pi_p([x,y]) = \mathfrak D_n$ for some $n$, so $\pi_p(x) = c_n$ and $\pi_p(y) = c_{\beta(n)}$ or vice versa. Let us call $x$ and $y$ (or $\pi_p(x)$ and $\pi_p(y)$) {\em $\beta$-neighbors} in this case. Notice, however, that there may be many post-critical points between $\pi_p(x)$ and $\pi_p(y)$. Obviously, every $p$-point of ${\mathfrak C}$ and ${\mathfrak R}$ has exactly two $\beta$-neighbors, except the endpoint $\bar 0$ of ${\mathfrak C}$ whose $\beta$-neighbor (w.r.t.\ $p$) is by convention the first proper $p$-point in ${\mathfrak C}$, necessarily with $p$-level $1$.
\subsection{Chainability and (quasi-)symmetry}
A space is {\em chainable} if there are finite open covers ${\mathcal C} = \{ \ell_i\}_{i = 1}^N$, called {\em chains}, of arbitrarily small {\em mesh} $(\mathop\mathrm{mesh} {\mathcal C} = \max_i \mbox{\rm diam}\, \ell_i)$ with the property that the {\em links} $\ell_i$ satisfy $\ell_i \cap \ell_j \neq \emptyset$ if and only if
$|i-j| \leqslant 1$. The combinatorial properties of Fibonacci-like maps allow us to construct chains ${\mathcal C}_p$ such that whenever an arc $A$ $p$-turns in $\ell \in {\mathcal C}_p$, {\em i.e.,} enters and exits $\ell$ through the same neighboring link, then the projections $\pi_p(x) = \pi_p(y)$ of the first and last $p$-point $x$ and $y$ of $A \cap \ell$ depend only on $\ell$ and not on $A$, see Proposition~\ref{prop:chains}. We will work with the chains ${\mathcal C}_p$ which are the $\pi_p^{-1}$ images of chains of the interval $[0,s/2]$.
\begin{defi}\label{def:sym} An arc $A \subset {\mathfrak A}$ such that $\partial A = \{ u, v \}$ and $A \cap E_p = \{ x^0, \dots , x^{n} \}$ is called \emph{$p$-symmetric}\, if $\pi_p(u) = \pi_p(v)$ and $L_p(x^i) = L_p(x^{n-i})$, for every $0 \leqslant i \leqslant n$. \end{defi} \noindent It is easy to see that if $A$ is $p$-symmetric, then $n$ is even and $L_p(x^{n/2}) = \max \{ L_p(x^i) : x^i \in A \cap E_p \}$. The point $x^{n/2}$ is called the {\em midpoint} of $A$.
It frequently happens that $\pi_p(u) \ne \pi_p(v)$, but $u$ and $v$ belong to the same link $\ell \in {\mathcal C}_p$. Let us call the arc-components $A_u$, $A_v$ of ${\mathfrak A} \cap \ell$ that contain $u$ and $v$ respectively the {\em link-tips} of $A$, see Figure~\ref{fig:link-tips}. Sometimes we can make $A$ $p$-symmetric by removing the link-tips. Let us denote this as $A \setminus \ell\mbox{-tips}$. Adding the closure of the link-tips can sometimes also produce a $p$-symmetric arc. \begin{figure}
\caption{The arc $A$ is neither $p$-symmetric, nor quasi-$p$-symmetric, but both arcs $A \setminus \ell\mbox{-tips}$ and $A \cup \mathop\mathrm{Cl} (\ell\mbox{-tips})$ are $p$-symmetric.}
\label{fig:link-tips}
\end{figure}
\begin{remark}\label{rem_basic}
{\bf (a)} Let $A$ be an arc and $m \in A$ be a $p$-point of maximal $p$-level, say $L_p(m) = L$. Then $\pi_p$ is one-to-one on both components of $\sigma^{1-L}(A \setminus \{ m \})$, so $m$ is the only $p$-point of $p$-level $L$. It follows that between every two $p$-points of the same $p$-level, there is a $p$-point $m$ of higher $p$-level.
{\bf (b)} If $A \owns m$ is the maximal open arc such that $m$ has the highest $p$-level on $A$, then we can write $\mathop\mathrm{Cl} A = [x,y]$ or $[y,x]$ with $L_p(x) > L_p(y)> L_p(m) =: L$, and $\pi_p$ is one-to-one on $\sigma^{-L}(\mathop\mathrm{Cl} A)$. Here $L_p(x) = \infty$ is possible, but if $L_p(x) < \infty$, then $A' := \pi_p \circ \sigma^{-L}(A)$ is a neighborhood of $c$ with boundary points $c_{S_k} = \pi_p \circ \sigma^{-L}(x)$ and $c_{S_l} = \pi_p \circ \sigma^{-L}(y)$ for some $k,l \in {\mathbb N}$ such that $l = Q(k)$. By Lemma~\ref{lem:order} this means that the arc $[x,m]$ is shorter than $[m,y]$. \end{remark}
\begin{defi}\label{quasi-p-symmetric} Let $A$ be an arc of ${\mathfrak A}$. We say that the arc $A$ is \emph{quasi-$p$-symmetric with respect to} ${\mathcal C}_p$ if \begin{itemize} \item[(i)] $A$ is not $p$-symmetric; \item[(ii)] $\partial A$ belongs to a single link $\ell$; \item[(iii)] $A \setminus \ell\mbox{-tips}$ is $p$-symmetric; \item[(iv)] $A \cup \ell\mbox{-tips}$ is not $p$-symmetric. (So $A$ cannot be extended to a symmetric arc within its boundary link $\ell$.) \end{itemize} \end{defi}
\begin{defi}\label{def:linksym} Let $\ell_0, \ell_1, \dots, \ell_k$ be the links in ${\mathcal C}_p$ that are successively visited by an arc $A \subset {\mathfrak A}$, and let $A_i \subset \mathop\mathrm{Cl}({\ell_i})$ be the corresponding maximal subarcs of $A$. (Hence $\ell_i \neq \ell_{i+1}$, $\ell_i \cap \ell_{i+1} \neq \emptyset$ but $\ell_i = \ell_{i+2}$ is possible if $A$ turns in $\ell_{i+1}$.) We call $A$ {\em $p$-link-symmetric} if $\ell_i = \ell_{k-i}$ for $i = 0, \dots, k$. In this case, we say that $A_i$ is $p$-link-symmetric to $A_{k-i}$. \end{defi} \begin{remark} Every $p$-symmetric and quasi-$p$-symmetric arc is $p$-link-symmetric by definition, but there are $p$-link-symmetric arcs which are not $p$-symmetric or quasi-$p$-symmetric. This occurs if $A$ turns both at $A_i$ and $A_{k-i}$, but the midpoint of $A_i$ has a higher $p$-level than the midpoint of $A_{k-i}$ and $i \notin \{ 0, k \}$. Note that for a $p$-link-symmetric arc $A$, if $U$ and $V$ are $p$-link-symmetric arc-components which do not contain any boundary point of $A$, then $U$ contains at least one $p$-point if and only if $V$ contains at least one $p$-point. \end{remark}
Appendix~\ref{sec:furtherlemmas} is devoted to give a precise description of quasi-symmetric arcs and their concatenated components. In Appendix~\ref{sec:link} we use this structure to show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs.
\section{Salient Points and Homeomorphisms}\label{sec:homeomorphisms}
Note that in this section all proofs except the proof of Proposition \ref{prop:symmetric} work in general, only the proof of Proposition~\ref{prop:symmetric} uses the special structure of the Fibonacci-like inverse limit spaces revealed in this paper.
\begin{defi}\label{df:salient}[see \cite[Definition 2.7]{BBS}] Let $(s_i)_{i \in {\mathbb N}}$ be the sequence of all $p$-points of the arc-component ${\mathfrak C}$ such that $0 \leq L_p(x) < L_p(s_i)$ for every $p$-point $x \in (\bar 0 , s_i)$. We call $p$-points satisfying this property \emph{salient}. \end{defi}
For every slope $s > 1$ and $p \in {\mathbb N}_0$, the folding pattern of ${\mathfrak C}$ starts as $\infty \ 0 \ 1 \ 0 \ 2 \ 0 \ 1 \ \dots$, and since by definition $L_p(s_1) > 0$, we have $L_p(s_1) = 1$. Also, since $s_i = \sigma^{i-1}(s_1)$, $L_p(s_i) = i$, for every $i \in {\mathbb N}$. Note that the salient $p$-points depend on $p$: if $p \geq q$, then the salient $p$-point $s_i$ equals the salient $q$-point $s_{i+p-q}$.
\begin{defi}\label{df:Rsalient} Recall that ${\mathfrak R}$ is the arc-component containing the point $\rho = (\dots,r,r,r)$ where $r = \frac{s}{s+1}$ is fixed by $T_s$. Let $(t^i)_{i \in {\mathbb Z}} \subset E^{{\mathfrak R}}_p$ be the bi-infinite sequence of all $p$-points of the arc-component ${\mathfrak R}$ such that for every $i \in {\mathbb N}$ $$ \left\{ \begin{array}{ll} t^0 = \rho, & \\ L_p(t^i) > L_p(x) & \text{ for every $p$-point } x \in (\rho, t^i), \\ L_p(t^{-i}) > L_p(x) & \text{ for every $p$-point } x \in (t^{-i}, \rho). \end{array} \right. $$ \end{defi}
Note that $p$-points $(t^i)_{i \in {\mathbb Z}} \subset {\mathfrak R}$ are defined similarly as salient $p$-points $(s_i)_{i \in {\mathbb N}}$; we call them \emph{${\mathfrak R}$-salient} $p$-points, or simply
\emph{salient} $p$-points when it is clear which arc-component they belong to. There is an important difference between the sets $(s_i)_{i \in {\mathbb N}} \subset {\mathfrak C}$ and $(t^i)_{i \in {\mathbb Z}} \subset {\mathfrak R}$, namely $L_p(s_i) = i$ for every $i \in {\mathbb N}$, whereas $L_p(t^i) \ne |i|$ for all $i \in {\mathbb Z} \setminus \{ 1 \}$.
\begin{lemma} \label{lem:PR1} For $(t^i)_{i \in {\mathbb Z}} \subset {\mathfrak R}$ we have $$ L_p(t^i) = \begin{cases} 2i-1 & \text { if } i > 0, \\ -2i & \text { if } i < 0. \end{cases} $$ \end{lemma}
\begin{proof} Since $r$ is the positive fixed point of $T_s$, the $p$-points closest to $\rho = (\dots, r, r, r)$ have $p$-levels 0 and 1. Also $\sigma(\rho) = \rho$ implies $\sigma({\mathfrak R}) = {\mathfrak R}$. The parametrization of ${\mathfrak R}$, chosen in Section~\ref{sec:def} below \eqref{eq:Rr}, is such that for $\rho \in [x^{-1}, x^1]$ we have $L_p(x^{-1}) = 0$ and $L_p(x^{1}) = 1$, thus $x^1 = t^1$. Since $\sigma(\rho) = \rho \in \sigma([x^{-1}, x^1]) \subset {\mathfrak R}$ and
$\sigma|_{{\mathfrak R}}$ is order reversing, we have $\sigma(x^{-1}) = x^1$, $\sigma(x^1) \prec x^1$, {\em i.e.,} $\sigma([x^{-1}, x^1]) = [x^{-2}, x^1]$ with $L_p(x^{-2}) = 2$. Note that $x^{-2} = t^{-1}$. For the same reason, $\sigma([x^{-2}, x^1]) = [x^{-2}, x^j]$, where $x^j$ is the first $p$-point to the right of $x^1$ such that $L_p(x^j) = 3$, {\em i.e.,} $x^j = t^2$. The claim of the lemma follows by induction. \end{proof}
Analogously, we define ${\mathfrak A}$-salient $p$-points of an arc-component ${\mathfrak A}$ of the core of $K_s$. \begin{defi}\label{df:Asalient} Let $(u^i)_{i \in {\mathbb Z}} \subset E^{{\mathfrak A}}_p = (a^i)_{i \in {\mathbb Z}}$ be the bi-infinite sequence of all $p$-points of the arc-component ${\mathfrak A}$ such that for every $i \in {\mathbb N}$ $$ \left\{ \begin{array}{ll} u^0 = a^0, & \\ L_p(u^i) > L_p(x) & \text{ for every $p$-point } x \in (u^0, u^i), \\ L_p(u^{-i}) > L_p(x) & \text{ for every $p$-point } x \in (u^{-i}, u^0). \end{array} \right. $$ This fixes an orientation on ${\mathfrak A}$; the choice of orientation is immaterial, as long as we make one. \end{defi}
\begin{lemma} \label{lem:PR2} If there exist $J, J', K \in {\mathbb N}_0$ such that for every $j \in {\mathbb N}$, $L_p(u^{J+j}) = 2(K+j)-1$ and $L_p(u^{-(J'+j)}) = 2(K+j)$, then ${\mathfrak A} = {\mathfrak R}$. \end{lemma}
In other words, the asymptotic shape of this folding pattern is unique to ${\mathfrak R}$.
\begin{proof} Let $J, J', K \in {\mathbb N}_0$ be as in the statement of the lemma. Then for every $j \in {\mathbb N}$ we have: \begin{itemize} \item[(i)] $L_p(u^{-(J'+j)}) - L_p(u^{J+j}) = L_p(u^{J+j+1}) - L_p(u^{-(J'+j)}) = 1$, \item[(ii)] $L_p(x) < L_p(u^{J+j})$ for every $p$-point $x \in (u^{-(J'+j)}, u^{J+j})$, and \item[(iii)] $L_p(x) < L_p(u^{-(J'+j)})$ for every $p$-point $x \in (u^{-(J'+j)}, u^{J+j+1})$. \end{itemize} Therefore, $\pi_{p+2(K+j)-1}: [u^{-(J'+j)}, u^{J+j}] \to [c, c_1]$ and $\pi_{p+2(K+j)}: [u^{-(J'+j)}, u^{J+j+1}] \to [c, c_1]$ are bijections, implying that for every $j \in {\mathbb N}$, $FP_p([u^{-(J'+j)}, u^{J+j}])$ and $FP_p([u^{-(J'+j)}, u^{J+j+1}])$ are uniquely determined by $T^{2(K+j)-1}_s$ and $T^{2(K+j)}_s$ respectively. Thus we have the following: $$ \left\{ \begin{array}{l} FP_p([u^{-(J'+j)}, u^{J+j}]) = FP_p([t^{-(K+j)}, t^{K+j}]), \\[2mm] FP_p([u^{-(J'+j)}, u^{J+j+1}]) = FP_p([t^{-(K+j)}, t^{K+j+1}]), \end{array} \right. $$ whence $FP_p([u^{-(J'+j)}, u^{J+j+1}]) = FP_p(\sigma([u^{-(J'+j)}, u^{J+j}]))$ for every $j \in {\mathbb N}$. It follows that $FP_p(\sigma({\mathfrak A})) = FP_p({\mathfrak A}) = FP_p({\mathfrak R})$ implying ${\mathfrak A} = {\mathfrak R}$. \end{proof}
Note that in general $J, J', K$ in the above lemma are not related since $u^0 = a^0$ can be any point, but there exists a point $a \in {\mathfrak A}$ such that for $u^0 = a$, we have $J = J' = K$.
Let $h : \underleftarrow\lim([c_2, c_1],T_{s}) \to \underleftarrow\lim([c_2, c_1],T_{s})$ be a homeomorphism on the core of a (Fibonacci-like) inverse limit space. Let $q, p, g \in {\mathbb N}_0$ be such that ${\mathcal C}_q$, ${\mathcal C}_p$ and ${\mathcal C}_g$ are chains as in Proposition~\ref{prop:chains}, and such that $$ h({\mathcal C}_{q}) \preceq {\mathcal C}_{p} \preceq h({\mathcal C}_{g}). $$ It is straightforward that any $q$-link-symmetric arc $A \subset \underleftarrow\lim([c_2, c_1],T_{s})$ maps to a $p$-link-symmetric arc $h(A) \subset \underleftarrow\lim([c_2, c_1],T_{s})$.
In Appendix~\ref{sec:chains}, we construct special chains by which we are able to describe the structure of
link-symmetric arcs (see Definition~\ref{def:linksym}) precisely. The Fibonacci-like structure, and the extra structure of these chains, allow us to conclude the stronger statement that $q$-symmetric arcs map to $p$-symmetric arcs. This is a rather technical undertaking, but let us paraphrase Remark~\ref{rem:quasi-sym} so as to make this section understandable (although for the fine points we will still refer forward to the appendix).
Link-symmetric arcs tend to be composed of smaller {\em (basic) quasi-symmetric arcs} $A_k$ (see Definition~\ref{basic-quasi-p-symmetric}) that are ordered linearly such that $A_k$ and $A_{k+1}$ overlap, and the midpoint of $A_{k+1}$ is the endpoint of $A_k$. An entire concatenation of such arcs is called {\em decreasing quasi-symmetric} (respectively {\em increasing quasi-symmetric}, see Definition~\ref{def:decreasing}) if the levels of the successive midpoints (also called {\em nodes}) - all contained in, alternately, one of two given links - are decreasing (respectively increasing). The concatenation is called {\em maximal decreasing quasi-symmetric} (respectively {\em maximal increasing quasi-symmetric}, see Definition~\ref{def:maximaldecreasing}) if it cannot be extended to a concatenation with more components. The last endpoint (respectively the first endpoint), namely, of the arc with midpoint of the lowest level, is then no longer a $p$-point.
For a point $x$, we denote a link of ${\mathcal C}_p$ which contains $x$ by $\ell_p^x$, and the arc-component of $\ell_p^x$ which contains $x$ by $A_x$.
\begin{defi}\label{def:extended_arc} Let $x \in E_q^{{\mathfrak A}} \subset {\mathfrak A}$ be a $q$-point, and let $A_{h(x)} \subset \ell_p^{h(x)}$ be the arc-component of $\ell_p^{h(x)}$ which contains $h(A_x)$ (and therefore $h(x)$). Let $a, b \in {\mathbb N}$, $a \leqslant b$, be such that $h(\cup_{i = a}^b \ell_q^i) \subseteq \ell_p^{h(x)}$, $h(\ell_q^{a-1}) \nsubseteq \ell_p^{h(x)}$ and $h(\ell_q^{b+1}) \nsubseteq \ell_p^{h(x)}$. Let $\hat A_x$ be an arc-component of $\cup_{i = a}^b \ell_q^i$ such that $h(\hat A_x) \subseteq A_{h(x)} \subset \ell_p^{h(x)}$. We call $\hat A_x$ the \emph{extended arc-component of the $q$-point} $x$. If a $p$-point $u$ is the midpoint of $A_{h(x)}$, then we write $u \vdash h(x)$. \end{defi}
The extended arc-component $\hat A_x$ is obtained by extending $A_x$ so much on both sides that $h(\hat A_x)$ fits almost exactly in the $p$-link containing $h(A_x)$. Note that the arc-component $A_x$ of a $q$-point $x$ depends on the chain ${\mathcal C}_q$, while the extended arc-component $\hat A_x$ of the $q$-point $x$ also depends on the chain ${\mathcal C}_p$. But we still can define its midpoint as the $q$-point $z \in \hat A_x$ such that $L_q(z) \geqslant L_q(y)$ for every $q$-point $y \in \hat A_x = \hat A_z$. If a $q$-point $x$ is the midpoint of its extended arc-component $\hat A_x$ we call it a \emph{$q_p$-point}.
\begin{proposition}\label{prop:symmetric} Let $x, y \in E_q^{{\mathfrak A}} \subset {\mathfrak A}$ be $q_p$-points and let $u \vdash h(x)$ and $v \vdash h(y)$. Then $L_q(x) = L_q(y)$ implies $L_p(u) = L_p(v)$. \end{proposition}
Since the endpoints of a symmetric arc have the same level, and
$q$-link symmetric arcs are mapped to $p$-link-symmetric arcs by a homeomorphism $h$, Proposition~\ref{prop:symmetric} implies that $h$ maps symmetric arcs to symmetric arcs.
\begin{proof} Without loss of generality we suppose that between $x$ and $y$, there are no $q$-points with $q$-level $L_q(x)$. Then the arc $A = [x, y]$ is $q$-symmetric. The midpoint $m$ of $A$ is a $q_p$-point. Let $w \vdash h(m)$.
Let us assume by contradiction that $L_p(u) \ne L_p(v)$. Then $D = [u, v]$ is not $p$-symmetric with midpoint $w$. Since $A$ is $q$-symmetric, $D$ is $p$-link symmetric. By Proposition~\ref{thm:linksym} and Remark~\ref{rem:quasi-sym}, $D$ is contained either in an extended maximal decreasing/increasing (basic) quasi-$p$-symmetric arc, or in a $p$-symmetric arc which is concatenation of two arcs, one of which is a maximal increasing (basic) quasi-$p$-symmetric arc, and the other one is a maximal decreasing (basic) quasi-$p$-symmetric arc.
{\bf (1)} Let us assume that $D$ is contained in an extended maximal increasing (basic) quasi-$p$-symmetric arc $G$. Let $B'$ and $B$ be the link-tips of $G$, so $G = [B', B]$. Then, by Remark \ref{rem:quasi-sym}, $B'$ does not contain any $p$-point and hence $B' \ne A_u$.
(a) Suppose first that the $p$-point $z \in G$, such that $L_p(z) \geqslant L_p(d)$ for all $p$-points $d \in G$, does not belong to the open arc $(u, v)$. Then $B \ne A_v$.
\begin{figure}
\caption{The relations between points and arcs in ${\mathcal C}_q$ (left), ${\mathcal C}_p$ (right), and ${\mathcal C}_g$ (bottom). }
\label{fig:lem:l}
\end{figure}
Let $b'$ be any point of $B'$ and let $b$ be the midpoint of $B$. Then $b'$ and $b$ are nodes of $G$ (see Remark~\ref{rem:quasi-sym} for the definition of a node). Since $L_p(u) \ne L_p(v)$, $u$ and $v$ are also nodes of $G$, as well as $w$ and $z$.
Let $a \vdash \sigma^{q-g} \circ h^{-1}(b)$ (note that $b$ is a $p_g$-point, {\em i.e.,} $b$ is the midpoint of the extended arc-component $\hat A_b$ such that $\sigma^{q-g} \circ h^{-1}(\hat A_b) \subseteq A_{\sigma^{q-g} \circ h^{-1}(b)} = A_a \subset \ell_g^a \in {\mathcal C}_g$). If the arc-component $A_{\sigma^{q-g} \circ h^{-1}(b')}$ contains a $g$-point, let $a'$ be its midpoint; otherwise let $a'$ be any point of $A_{\sigma^{q-g} \circ h^{-1}(b')}$. Let us consider the arc $H = [a', a]$, see Figure~\ref{fig:lem:l}. Let $x' \vdash \sigma^{q-g} \circ h^{-1}(u)$, $y' \vdash \sigma^{q-g} \circ h^{-1}(v)$, $z' \vdash \sigma^{q-g} \circ h^{-1}(z)$ and $m' \vdash \sigma^{q-g} \circ h^{-1}(w)$. Since ${\mathcal C}_p \prec h({\mathcal C}_g)$, the arc $H$ is $g$-link-symmetric and $g$-points $a', x', m', y', a$ are some of its nodes. Note that $x' = \sigma^{q-g}(x)$ and $y' = \sigma^{q-g}(y)$, thus the arc $[x', y']$ is $g$-symmetric. Since there is at least one node in $H$ on either side of $[x', y']$, Remark~\ref{rem:quasi-sym} says that $H$ is contained in the maximal $g$-symmetric arc $K$ with midpoint $m'$. Therefore the arc $M = \sigma^{-q+g}(K) \supset A$ is $q$-symmetric with midpoint $m$.
Let $j, k \in {\mathbb N}$, $j \leqslant k$, be such that $h(\cup_{i = j}^k \ell_q^i) \subseteq \ell_p^{b'}$, $h(\ell_q^{j-1}) \nsubseteq \ell_p^{b'}$ and $h(\ell_q^{k+1}) \nsubseteq \ell_p^{b'}$. Let $N'$ be an arc-component of $\cup_{i = j}^k \ell_q^i$ such that $h(N') = B' \subset \ell_p^{b'}$. Obviously, $N' \subset M$. Since $M$ is $q$-link symmetric, there exists an arc-component $N$ of $\cup_{i = j}^k \ell_q^i$ such that the arc $[N', N] \subset M$ is $q$-symmetric with midpoint $m$. Then $h(N) \subset h(M)$ is an arc-component of $\ell_p^{b'}$. Since $[N', N]$ is $q$-symmetric, the arc-component $h(N')$ contains a $p$-point if and only if the arc-component $h(N)$ contains a $p$-point. Since $h(N') = B'$, the arc-components $h(N')$ and $h(N)$ do not contain any $p$-point, see Figure~\ref{fig:lem:l}.
On the other hand, the arc $[h(N'), h(N)]$ is $p$-link-symmetric with midpoint $w$. Recall that $w$ is also the midpoint of the arc $D \subset [h(N'), h(N)]$, $D$ is not $p$-symmetric by assumption, and $D \subset G$, where $G$ is an extended maximal increasing (basic) quasi-$p$-symmetric arc. The arc-component $h(N)$ can be contained in the arc $[A_v, B]$, as in Figure~\ref{fig:lem:l}. In this case $h(N)$ does contain at least one $p$-point, a contradiction.
The other possibility is that $h(N)$ is not contained in $[A_v, B]$, {\em i.e.,} $h(N)$ is on the right hand side of $B$. Since $[h(N'), h(N)]$ is $p$-link symmetric and $h(N') = B'$ contains a node $b'$ of $G$, we have that $h(N)$ also contains a node of $G$, say $n$. Hence, on the right hand side of $z$ (which is the $p$-point with the highest $p$-level in $G$), there are at least two nodes, $b$ and $n$. Therefore, by Remark~\ref{rem:quasi-sym}, $G$ is contained in a $p$-symmetric arc with midpoint $z$ and this arc conatins $h(N)$, implying that $h(N)$ does contain at least one $p$-point, a contradiction.
(b) Let us assume now that $B = A_v$. Then $z \in (u, v)$. Let $a', x', m', z', y'$ and $H$ be defined as in case (a). Since $b', u, w, z, v$ are nodes of $G$, we have that $a', x', m', z', y'$ are also nodes of $H$. Moreover, since $[x', y']$ is $g$-symmetric with midpoint $m'$, there is $z'' \in [x', m']$ such that $[z'', z']$ is $g$-symmetric with midpoint $m'$, and $z''$ is a node of $H$. Thus, the arc between nodes $z''$ and $z'$ is $g$-symmetric, and on either side of $[z'', z']$ there is at least one additional node. By Remark~\ref{rem:quasi-sym}, $H$ is contained in the maximal $g$-symmetric arc $K$ with midpoint $m'$, and the arc $M = \sigma^{-q+g}(K) \supset A$ is $q$-symmetric with midpoint $m$. Now the proof follows in the same way as in case (a).
If $D$ is contained in an extended maximal decreasing (basic) quasi-$p$-symmetric arc $G$, the proof is analogous.
{\bf (2)} Let us assume that $D$ is contained in a $p$-symmetric arc $G$ which is concatenation of two arcs, one of which is a maximal increasing (basic) quasi-$p$-symmetric arc, and the other one is a maximal decreasing (basic) quasi-$p$-symmetric arc. Let $B'$ and $B$ be the link-tips of $G$, thus $G = [B', B]$. Then, by Remark \ref{rem:quasi-sym}, $B'$ and $B$ do not contain any $p$-point and hence $B' \ne A_u$ and $B \ne A_v$. If for the midpoint $z$ of $G$ we have $z \not\in (u, v)$, we are in case (1). If $z \in (u, v)$ (note $z \ne m$ since the arc $D$ is not $p$-symmetric), then the proof is analogous to the proof of case (1a) (since $B \ne A_v$). \end{proof}
\begin{defi}\label{defi:bridges} Let $\kappa \in {\mathbb N}$, $\kappa > 2$, be the smallest integer with $c_{\kappa} < c$. It is easy to see that $\kappa$ is odd. Set $$ \Lambda_\kappa :=
{\mathbb N} \setminus \{ 1,3, 5, \dots, \kappa-4\}. $$ \end{defi}
\begin{lemma}\label{lem:bridges} Let $x,y$ be $q$-points of ${\mathfrak A}$. Then there exist $q_p$-points $x'$, $z'$ and $y'$ such that the arc $A = [x',z']$ is $q$-symmetric with midpoint $y'$, $L_q(x') = L_q(z') = L_q(x)$ and $L_q(y') = L_q(y)$ if and only if $L_q(y)-L_q(x) \in \Lambda_\kappa$. \end{lemma}
This is proven in Lemma 46 of \cite{Kail2} and in Lemmas 3.13 and 3.14 of \cite{Stim2}. Although \cite{Kail2} deals with the periodic case and \cite{Stim2} with the finite orbit case, the proofs of the mentioned lemmas work in the general case, as stated above.
\begin{proposition}\label{prop:nonsymmetric} Let $x, y \in E_q^{{\mathfrak A}} \subset {\mathfrak A}$ be $q_p$-points and let $u \vdash h(x)$ and $v \vdash h(y)$. Then $L_q(x) < L_q(y)$ implies $L_p(u) < L_p(v)$. \end{proposition}
\begin{proof} {\bf (1)} Let us first assume that $L_q(y) - L_q(x) \in \Lambda_\kappa$. Then, by Lemma~\ref{lem:bridges}, there exist $q_p$-points $x'$, $z'$ and $y'$ such that the arc $A = [x', z']$ is $q$-symmetric with midpoint $y'$, $L_q(x') = L_q(z') = L_q(x)$, $L_q(y') = L_q(y)$ and between $x'$ and $z'$ there are no $q_p$-points with $q$-level $L_q(x')$. Let $u \vdash h(x)$, $v \vdash h(y)$, $u' \vdash h(x')$, $v' \vdash h(y')$, $w' \vdash h(z')$. By Proposition~\ref{prop:symmetric} we have $L_p(u) = L_p(u') = L_p(w')$, $L_p(v) = L_p(v')$ and between the points $u'$ and $w'$ there are no $p$-points with the $p$-level $L_p(u')$. Therefore, the arc $[u', w']$ is $p$-symmetric with midpoint $v'$, implying $L_p(v) = L_p(v') > L_p(u') = L_p(u)$, which proves the proposition in this case. Note that also we have $L_p(v) - L_p(u) \in \Lambda_\kappa$.
\begin{figure}
\caption{The points $x$ and $y$, their companion arc $A = [x', z']$ and their images under $h$. Dots indicate some shape of the arc $[x,y]$ and $[u,v]$; the shape of $[x,y]$ can be very different from the shape of $[x',y']$ and similar for the shapes of $[u,v]$ and $[u',v']$.}
\label{fig:AhA}
\end{figure}
{\bf (2)} Let us now assume that $\Lambda_\kappa \ne {\mathbb N}$, $L_q(y) - L_q(x) \in \{ 1,3, \dots, \kappa-4\}$, and that for $u \vdash h(x)$ and $v \vdash h(y)$ we have, by contradiction, $L_p(u) > L_p(v)$.
Without loss of generality we suppose that $x$ has the smallest $q$-level among all $q_p$-points which satisfy the above assumption and that, for this choice of $x$, the $q_p$-point $y$ (which also satisfies the above assumption) is such that $L_q(y) - L_q(x) > 0$ is the smallest difference of $q$-levels.
{\bf Claim 1:} $L_q(y) - L_q(x) = 1$.
Let us assume, by contradiction, that $L_q(y) - L_q(x) > 1$, and let $z$ be a $q_p$-point such that $L_q(y) - L_q(z) = 2$. Note first that $L_q(z) \ne L_q(x)$ since $L_q(y) - L_q(x) \ne 2$ by assumption. Therefore, $L_q(z) > L_q(x)$.
Let $w \vdash h(z)$ and recall $u \vdash h(x)$ and $v \vdash h(y)$. By the choice of $q_p$-points $x$ and $y$ and since $L_q(z) - L_q(x) < L_q(y) - L_q(x)$, we have $L_p(w) > L_p(u)$ and $L_p(u) > L_p(v)$, implying $L_p(w) > L_p(v)$.
On the other hand, $L_q(y) - L_q(z) \in \Lambda_\kappa$ and by {\bf (1)} we have $L_p(v) > L_p(w)$, a contradiction. This proves Claim 1.
{\bf Claim 2:} $L_p(u) - L_p(v) = 1$.
Let us assume, by contradiction, that $L_p(u) - L_p(v) > 1$. For a $q_p$-point $z$ let $w$ denote the $p$-point with $w \vdash h(z)$. We will show that the above assumption implies that there is no $q_p$-point $z$ such that $L_p(w) = L_p(v) + 1$. This contradicts assumption that both arc-components ${\mathfrak A}$ and $h({\mathfrak A})$ are dense in $\underleftarrow\lim([c_2, c_1],T_{s})$ in both directions.
By the choice of $q_p$-points $x$ and $y$, for every $q_p$-point $z$ such that $L_q(z) < L_q(x) < L_q(y) = L_q(x) + 1$ we have $L_p(w) < L_p(v)$ and hence $L_p(w) \ne L_p(v) + 1$.
Let $L_q(z) = L_q(x) + 2$. Since $L_q(z) - L_q(x) \in \Lambda_\kappa$, by {\bf (1)} we have $L_p(w) > L_p(u) > L_p(v) + 1$.
Let $L_q(z) = L_q(x) + 3$. Then $L_q(z) - L_q(y) \in \Lambda_\kappa$ (recall $L_q(y) = L_q(x) + 1$ by Claim 1) and again by {\bf (1)} we have $L_p(w) > L_p(v)$ and $L_p(w) - L_p(v) \in \Lambda_\kappa$. Note that $L_p(w) - L_p(v) \ne 1$ since $1 \not\in \Lambda_\kappa$ (recall $\Lambda_\kappa \ne {\mathbb N}$ by assumption). Hence $L_p(w) > L_p(v) + 1$.
It follows now by induction that for every $i \in {\mathbb N}$, $L_q(z) = L_q(x) + 3 + i$ implies $L_p(w) > L_p(v) + 1$. To see this, for a $q_p$-point $z'$ let $w'$ denote the $p$-point with $w' \vdash h(z')$. Take $j \in {\mathbb N}$ such that $L_q(z) = L_q(x) + 3 + i$ implies $L_p(w) > L_p(v) + 1$ for every $i < j$. Let $L_q(z') = L_q(x) + 1 + j$ and $L_q(z) = L_q(x) + 3 + j$. Then $L_q(z) - L_q(z') \in \Lambda_\kappa$ and by {\bf (1)} we have $L_p(w) > L_p(w')$. Since $L_p(w') > L_p(v) + 1$, we have $L_p(w) > L_p(v) + 1$. This proves Claim 2.
{\bf Claim 3:} For a $q_p$-point $z$ let $w$ denote the $p$-point with $w \vdash h(z)$. For every $i \in {\mathbb N}$, $L_q(z) = L_q(x) + 2i$ implies $L_p(w) = L_p(u) + 2i$, and $L_q(z) = L_q(y) + 2i$ implies $L_p(w) = L_p(v) + 2i$.
Let $L_q(z) = L_q(x) + 2 = L_q(y) + 1$. Note first that $L_p(w) \ne L_p(u) + 1$, since by {\bf (1)} $L_q(z) - L_q(x) \in \Lambda_\kappa$ implies $L_p(w) - L_p(u) \in \Lambda_\kappa$. Note also that $L_q(z) - L_q(y) \not\in \Lambda_\kappa$. Therefore, $L_p(w) = L_p(v) + L = L_p(u) - 1 + L$, where $1 < L < \kappa - 2$ is odd.
For $q_p$-points $z'$ and $z''$, let $w'$ and $w''$ denote the $p$-points with $w' \vdash h(z')$ and $w'' \vdash h(z'')$ respectively.
Let us assume that $L_q(z') = L_q(y) + 2$ and $L_p(w') \ne L_p(v) + 2 = L_p(u) + 1$. Then $L_p(w') > L_p(v) + 2$ and for every $q_p$-point $z''$ with $L_q(z'') > L_q(z')$ we have $L_p(w'') > L_p(v) + 2$. This implies that there is no $q_p$-point $z''$ such that $L_p(w'') = L_p(v) + 2 = L_p(u) + 1$, a contradiction. Therefore, $L_p(w') = L_p(v) + 2$, and by Claims 1 and 2, $L_p(w) = L_p(v) + 3 = L_p(u) + 2$. The proof of Claim 3 follows by induction in the same way.
Finally, to complete the proof of the proposition, let us consider $q_p$-point $z$ such that $L_q(z) - L_q(x) = \kappa - 2 \in \Lambda_\kappa$. Then, by Claim 3 (see Figure \ref{fig:level1}), $L_p(w) - L_p(u) = \kappa - 4 \not\in \Lambda_\kappa$, a contradiction.
\begin{figure}
\caption{The configuration of levels that cannot exist. }
\label{fig:level1}
\end{figure}
Therefore, $L_q(x) < L_q(y)$ implies $L_p(u) < L_p(v)$, which proves the proposition. \end{proof}
\section{Proof of the main theorems}\label{sec:maintheorems}
Consider the arc-component ${\mathfrak A} := h({\mathfrak R}) \subset \underleftarrow\lim([c_2, c_1],T_{s})$, and let $E_p^{{\mathfrak A}} = (y^i)_{i \in {\mathbb Z}} \subset {\mathfrak A}$ be the set of all $p$-points of ${\mathfrak A}$ such that $y^0 = h(\rho)$. Let $(u^i)_{i \in {\mathbb Z}} \subset E_p^{{\mathfrak A}}$ be the set of all salient $p$-points of ${\mathfrak A}$, {\em i.e.,} the set of all ${\mathfrak A}$-salient $p$-points, with $u^0 = h(\rho)$. Recall that ${\mathfrak R}$ is dense in $\underleftarrow\lim([c_2, c_1],T_{s})$ in both directions. Since $h$ is a homeomorphism, ${\mathfrak A}$ and in fact $h^i({\mathfrak R})$, $i \in {\mathbb Z}$, are also dense in the core $\underleftarrow\lim([c_2, c_1],T_{s})$ in both directions.
We want to prove that ${\mathfrak A} = {\mathfrak R}$. For a $p$-point $y$ we write $y \approx x$ if $y \in A_x$.
\begin{lemma}\label{lem:some} There exist $M, M' \in {\mathbb Z}$ such that $h(t^i) \approx u^{i+M}$ and $h(t^{-j}) \approx u^{-j-M'}$, for every $i,j \in {\mathbb N}$ with $i+M > 0$, $j+M' > 0$, if $h$ is order preserving, or $h(t^i) \approx u^{-i-M}$ and $h(t^{-j}) \approx u^{j+M'+1}$ if $h$ is order reversing. \end{lemma}
\begin{proof} If $h : {\mathfrak R} \to {\mathfrak A}$ is order reversing, then $h \circ \sigma : {\mathfrak R} \to {\mathfrak A}$ is order preserving, and also if the proposition works for $h \circ \sigma$, it works for $h$. Therefore we can assume without loss of generality that $h$ is order preserving.
Let $j \in {\mathbb N}$, and let $B_j$ be the maximal $q$-symmetric arc with midpoint $t^j$. Since $s > \sqrt 2$, $\rho \in B_j$. Therefore, for every $q_p$-point $x \in (\rho, t^j)$ there exists a $q_p$-point $y \in (t^j, t^{j+1})$, such that the arc $[x, y]$ is $q$-symmetric with midpoint $t^j$ and $L_q(x) = L_q(y)$. Let $u$ and $v$ be $p$-points such that $u \vdash h(x)$ and $v \vdash h(y)$. By Proposition~\ref{prop:symmetric}, we have $L_p(u) = L_p(v)$. Note that for the midpoint $w$ of the arc $[u, v]$ we also have $w \vdash h(t^j)$. This implies, by Remark~\ref{rem_basic} (a), that $L_p(w) > L_p(z)$ for every $z \in (u^0, w)$. Therefore, $w$ is a salient $p$-point, {\em i.e.,} $w \in (u^i)_{i \in {\mathbb N}}$.
Let $k, l \in {\mathbb N}$, $k < l$, be such that $u^k \vdash h(t^j)$ and $u^l \vdash h(t^{j+1})$. We want to prove that $l = k+1$. Let us assume by contradiction that $l > k+1$. Since $L_p(u^{k+1}) > L_p(u^k)$, there exists a $q_p$-point $x \in (t^j, t^{j+1})$ such that $u^{k+1} \vdash h(x)$. But $x \in (t^j, t^{j+1})$ implies $L_q(x) < L_q(t^j)$, contradicting Proposition~\ref{prop:nonsymmetric}.
In this way we have proved that $h(t^i) \approx u^{i+M}$ for some $M\in {\mathbb Z}$ and every $i \in {\mathbb N}$ with $M+i > 0$. In an analogous way we can prove that $h(t^{-i}) \approx u^{-i-M'}$ for some $M' \in {\mathbb Z}$ and for every $i \in {\mathbb N}$ with $M'+i > 0$. \end{proof}
\begin{theorem}\label{thm:ray} Every self-homeomorphism $h$ of $\underleftarrow\lim([c_2, c_1],T_{s})$ preserves ${\mathfrak R}$: $h({\mathfrak R}) = {\mathfrak R}$. \end{theorem}
\begin{proof} Let $h : {\mathfrak R} \to {\mathfrak A}$, as before. We want to prove that ${\mathfrak A} = {\mathfrak R}$. Note that $h \circ \sigma^i : {\mathfrak R} \to {\mathfrak A}$ and $\sigma^i \circ h : {\mathfrak R} \to \sigma^i({\mathfrak A})$ are homeomorphisms for every $i \in {\mathbb Z}$, and $\sigma^i({\mathfrak A}) = {\mathfrak R}$ if and only if ${\mathfrak A} = {\mathfrak R}$. By using $h^{-1}$ instead of $h$ if necessary, we can assume that $M \geqslant 0$ (with $M$ as in Lemma~\ref{lem:some}). Also, instead of studying $h$, we can study $\sigma^{1-a} \circ h : {\mathfrak R} \to \sigma^{1-a}({\mathfrak A})$, where $a = L_p(u^{1+M})$ (recall that $h(t^1) \approx u^{1+M}$). Therefore, without loss of generality we can assume that $h(t^1) \approx u^1$ and $L_p(u^1) = 1$. Recall that $L_q(t^1) = 1$, $L_q(t^{-1}) = 2$ and for every $i \in {\mathbb N}$, $L_q(t^{-i}) - L_q(t^i) = L_q(t^{i+1}) - L_q(t^{-i}) = 1$. If $L_p(u^{-i}) - L_p(u^{i}) = L_p(u^{i+1}) - L_p(u^{-i}) = 1$, then ${\mathfrak A} = {\mathfrak R}$ by Lemma~\ref{lem:PR2}.
Recall that $h(t^{-1}) \approx u^{-1-M'}$, where $M'$ is as in Lemma~\ref{lem:some}. Since $$L_q(t^1) < L_q(t^{-1}) < L_q(t^2) < L_q(t^{-2}) < \cdots,$$ by Proposition~\ref{prop:nonsymmetric} we have $$1 = L_p(u^1) < L_p(u^{-1-M'}) < L_p(u^2) < L_p(u^{-2-M'}) < \cdots < L_p(u^n) < L_p(u^{-n-M'}) < \cdots.$$ Let $L_p(u^n) = 1 + a_1 + b_1 + \cdots + a_{n-1} + b_{n-1}$ and $L_p(u^{-n-M'}) = 1 + a_1 + b_1 + \cdots + a_{n-1} + b_{n-1} + a_n$, for every $n \in {\mathbb N}$ and some $a_1, \dots , a_n, b_1, \dots , b_{n-1} \in {\mathbb N}$. We want to prove that $a_i = b_i = 1$ for every $i \in {\mathbb N}$.
Assume by contradiction that $k \in {\mathbb N}$ is the smallest integer with $a_i = b_i = 1$ for all $i < k$ and $a_k > 1$. Then, by Proposition~\ref{prop:nonsymmetric}, there is no salient $p$-point $u \in (u^i)_{i \in {\mathbb Z}}$ with $L_p(u) = L_p(u^k) + 1$. Thus, Proposition~\ref{prop:symmetric} implies that ${\mathfrak A}$ does not contain any $p$-point with $p$-level $L_p(u^k) + 1$, contradicting that ${\mathfrak A}$ is dense in $\underleftarrow\lim([c_2, c_1],T_{s})$ in both directions. If $k \in {\mathbb N}$ is the smallest integer with $a_i = b_i = 1$ for all $i < k$, $a_k = 1$ and $b_k > 1$, the proof follows in an analogous way. \end{proof}
\begin{remark}\label{rem:M} If $h$ is order-preserving, then by proof of Theorem \ref{thm:ray} we have $M' = M$, where $M$ and $M'$ are as in Lemma~\ref{lem:some}. Also, by Lemma~\ref{lem:PR1}, Lemma~\ref{lem:some} and Theorem~\ref{thm:ray} we have $L_p(u^{i+M}) = 2(i+M)-1 = (2i-1) + 2M = L_q(t^i) + 2M$ for $i > 0$ and $L_p(u^{i-M}) = 2(-i+M) = -2i + 2M = L_q(t^i) + 2M$ for $i < 0$. Moreover, by Proposition \ref{prop:symmetric}, for every $q_p$-point $x$, and for the $p$-point $u$ with $u \vdash h(x)$, we have $L_p(u) = L_q(x) + 2M$. \end{remark}
We finish with the
\begin{proof}[Proof of Theorem~\ref{mainthm}] Let $1 \leq s \leqslant \sqrt 2 < s' \leq 2$. Then $\underleftarrow\lim([c_2, c_1],T_{s})$ is decomposable, $\underleftarrow\lim([c_2, c_1],T_{s'})$ is indecomposable, and the proof follows.
Since Lemmas 2.1 and 2.2 of \cite{BBS} show how to reduce the case $1 \leq s < s' \leq \sqrt 2$ to the case $\sqrt 2 < s < s' \leq 2$, it suffices to prove the latter case.
Let $\sqrt 2 < s < s' \leq 2$. Suppose that there exists a homeomorphism $h : \underleftarrow\lim([c_2, c_1],T_{s'}) \to \underleftarrow\lim([c_2, c_1],T_{s})$. Let $r':=\frac{s'}{s'+1}$ be the positive fixed point of $T_{s'}$ and $\rho':= (\dots, r', r', r') \in C_{s'} = \underleftarrow\lim([c_2, c_1],T_{s'})$. Let ${\mathfrak R}'$ denote the arc-component containing $\rho'$. Let $r$, $\rho$ and ${\mathfrak R}$ be the analogous objects of $C_{s} = \underleftarrow\lim([c_2, c_1],T_{s})$, as before. Take $q, p \in {\mathbb N}_0$ such that $h({\mathcal C}_q) \prec {\mathcal C}_p$. Let
$(t^i)_{i \in {\mathbb Z}}$ be the sequence of salient $q$-points of ${\mathfrak R}'$ with $t^0 = \rho'$.
Let $(u^i)_{i \in {\mathbb Z}}$ be the sequence of salient $p$-points of ${\mathfrak R}$.
Let $f = h^{-1} \circ \sigma \circ h$, and assume by contradiction that $h({\mathfrak R}') = {\mathfrak A} \ne {\mathfrak R}$. Since ${\mathfrak R}$ is the only arc-component in $\underleftarrow\lim([c_2, c_1],T_{s})$ that is fixed by $\sigma$, we have $\sigma({\mathfrak A}) \ne {\mathfrak A}$ implying $f({\mathfrak R}') \ne {\mathfrak R}'$. But this contradicts Theorem~\ref{thm:ray}. Therefore $h({\mathfrak R}') = {\mathfrak R}$.
We want to prove that $FP({\mathfrak R}') = FP({\mathfrak R})$. Without loss of generality we suppose that $h$ is order-preserving and that $M > 0$ (with $M$ as in Remark \ref{rem:M}).
{\bf Claim 1:} Let $l \in {\mathbb N}$ and let $x$ be a $q$-point with $L_q(x) = l$. Then $u := h(x) \in \ell_p^{u^{l+2M}}$ and the arc component $A_u \subset \ell_p^{u^{l+2M}}$ containing $u$, also contains a $p$-point $y$ such that $L_p(y) = l + 2M$.
Note that Claim 1 is the same as Proposition 4.2 (1) of \cite{BBS}. The proof is analogous:
By Remark \ref{rem:M}, Claim 1 is true for all salient $q$-points and for all $q_p$-points. Note that there exists $j \in {\mathbb N}$ such that every $q$-point $x \in [t^{-j}, t^j]$ is also a $q_p$-point. Therefore Claim 1 is true for all $q$-points $x \in [t^{-j}, t^j]$, i.e., for every $q$-point $x \in [t^{-j}, t^j]$ the arc-component $A_{h(x)}$ containing $h(x)$, also contains a $p$-point $y$ such that $L_p(y) = L_q(x) + 2M$. Also $h([t^{-j}, t^j]) = [a_{-j},a_j]$, $u^{-j-2M} \in A_{a_{-j}}$ and $u^{j+2M} \in A_{a_j}$. Let $q$-point $x_1 \in [t^{-j}, t^j]$ be such that the open arc $(x_1,t^{j+1})$ is $q$-symmetric with midpoint $t^j$. Such $x_1$ exists since $L_q(t^{j+1}) - L_q(t^j) = 2$ and $L_q(t^{-j}) - L_q(t^j) = 1$. Then $h((x_1, t^{j+1}))$ is $p$-link-symmetric with midpoint $u^{j+2M}$. Since there exists a unique $p$-point $b_1$ such that the open arc $(b_1, u^{j+1+2M})$ is $p$-symmetric with midpoint $u^{j+2M}$, for every $q$-point $x' \in (t^j, t^{j+1})$ the arc-component $A_{h(x')}$ containing $h(x')$, also contains a $p$-point $y'$ such that $L_p(y') = L_p(y) = L_q(x) + 2M = L_q(x') + 2M$, see Figure \ref{fig:levels}.
\begin{figure}
\caption{The configuration of symmetric arcs. }
\label{fig:levels}
\end{figure}
Let us consider now the arc $h([t^{-j-1}, t^{j+1}]) = [a_{-j-1},a_{j+1}]$, $u^{-j-1-2M} \in A_{a_{-j-1}}$ and $u^{j+1+2M} \in A_{a_{j+1}}$. Let the $q$-point $x_{-1} \in [t^{-j}, t^{j+1}]$ be such that the open arc $(t^{-j-1},x_{-1})$ is $q$-symmetric with midpoint $t^{-j}$. Such $x_{-1}$ exists since $L_q(t^{-j-1}) - L_q(t^{-j}) = 2$ and $L_q(t^{j+1}) - L_q(t^{-j}) = 1$. Therefore $h((t^{-j-1},x_{-1}))$ is $p$-link-symmetric with midpoint $u^{-j-2M}$. Since there exists a unique $p$-point $b_{-1}$ such that the open arc $(u^{-j-1-2M}, b_{-1})$ is $p$-symmetric with midpoint $u^{-j-2M}$, for every $q$-point $x'' \in (t^{-j-1}, t^{-j})$ the arc-component $A_{h(x'')}$ containing $h(x'')$, also contains a $p$-point $y''$ such that $L_p(y'') = L_q(x'') + 2M$, as before. The proof of Claim 1 follows by induction.
{\bf Claim 2:} For $l \in {\mathbb N}_0$ and $i \in {\mathbb N}$, the number of $q$-points in $[t^{-i}, t^i]$ with $q$-level $l$ is the same as the number of $p$-points in $[u^{-i-2M}, u^{i+2M}]$ with $p$-level $l+2M$.
Claim 2 is the same as Proposition 4.2 (2) of \cite{BBS}. The proof is very similar and we omit it.
Claims 1 and 2 show that $$ FP_q([t^{-i}, t^i]) = FP_{p+2M}([u^{-i-2M},u^{i+2M}]) = FP_p([u^{-i},u^i]), $$ for every positive integer $i$, and therefore $FP({\mathfrak R}') = FP({\mathfrak R})$.
This proves the Ingram Conjecture for cores of the Fibonacci-like inverse limit spaces. \end{proof}
\appendix \section{The Construction of Chains}\label{sec:chains}
We turn now to the technical part, {\em i.e.,} the construction of special chains that will eventually allow us to show that symmetric arcs map to symmetric arcs (see Proposition~\ref{prop:symmetric}).
As mentioned before, we will work with the chains which are the $\pi_p^{-1}$ images of chains of the interval $[0,s/2]$. More precisely, we will define a finite collection of points $G = \{ g_0, g_1, \dots, g_N \} \subset [0, s/2]$ such that
$|g_m - g_{m+1}| \leq s^{-p} \varepsilon/2$ for all $0 \leq m < N$
and $|0-g_0|$ and $|s/2-g_N|$ positive but very small. >From this one can make a chain ${\mathcal C} = \{ \ell_n \}_{n = 0}^{2N}$ by setting \begin{equation}\label{eq:links} \left\{ \begin{array}{lll} \ell_{2m+1} = \pi_p^{-1}( (g_m,g_{m+1}) ) & \qquad & 0 \leq m < N,\\ \ell_{2m} = \pi_p^{-1}( (g_m-\delta, g_m+\delta) \cap [0,s/2] )& & 0 \leq m \leq N, \end{array} \right. \end{equation}
where $\min\{ |0-g_0|, |s/2-g_N|\} < \delta \ll \min_m \{ |g_m - g_{m+1}| \}$. Any chain of this type has links of diameter $< \varepsilon$.
\begin{remark}\label{rem:p_points_in_links} We could have included all the points $\cup_{j \leq p} T_s^{-j}(c)$
in $G$ to ensure that $T_s^p|_{(g_m, g_{m+1})}$ is monotone for each $m$, but that is not necessary. Naturally, there are chains of $\underleftarrow\lim([0,s/2],T_s)$ that are not of this form.
For a component $A$ of ${\mathfrak C} \cap \ell$, we have the following two possibilities: \\ \indent (i) ${\mathfrak C}$ goes straight through $\ell$ at $A$, {\em i.e.,} $A$ contains no $p$-point and $\pi_p(\partial A) = \partial \pi_p(\ell)$; in this case $A$ enters and exits $\ell$ from different sides. \\ \indent (ii) ${\mathfrak C}$ turns in $\ell$: $A$ contains (an odd number of) $p$-points $x^0, \dots, x^{2n}$ of which the middle one $x^n$ has the highest $p$-level, and $\pi_p(\partial A)$ is a single point in $\partial \pi_p(\ell)$, in this case $A$ enters and exits $\ell$ from the same side. \end{remark}
Before giving the details of the $p$-chains we will use, we need a lemma.
\begin{lemma}\label{lem:linkDn} If the kneading map $Q$ of $T_s$ is eventually non-decreasing and satisfies Condition \eqref{eq:cond3}, then for all $n \in {\mathbb N}$ there are arbitrarily small numbers $\eta_n > 0$ with the following property: If $n' > n$ is such that $n \in \mbox{\rm orb}_\beta(n')$, then either
$|c_{n'} - c_n| > \eta_n$ or
$|c_{n''} - c_n| < \eta_n$ for all $n \leq n'' \leq n'$ with $n'' \in \mbox{\rm orb}_\beta(n')$. \end{lemma}
To clarify what this lemma says, Figure~\ref{fig:toavoid} shows the configuration of levels $\mathfrak D_k$ that should be avoided, because then $\eta_n$ cannot be found.
\begin{figure}
\caption{Linking of levels $\mathfrak D_{m_i}$ with $\beta(m_1) = \beta(m_2) = \beta(m_3) = \dots = n$. The semi-circles indicates that two intervals have an endpoint in common.}
\label{fig:toavoid}
\end{figure}
\begin{proof} We will show that the pattern in Figure~\ref{fig:toavoid} (namely with $c_{m_1} < c_{m_2} < c_{m_3} < \dots$ and $c_{m_{i-1}} < c_{k_i}$ for each $i$) does not continue indefinitely. To do this, we redraw the first few levels from Figure~\ref{fig:toavoid}, and discuss four positions in $\mathfrak D_{m_1}$ where the precritical point $T_s^{-r}(c) \in \mathfrak D_{m_1}$ of lowest order $r$ could be, indicated by points $a_1, \dots, a_4$, see Figure~\ref{fig:toavoid2}. \begin{figure}
\caption{Linking of levels $\mathfrak D_{m_i}$, $i = 1,2,3$ and three possible positions of the precritical point $a_j = T^{-r}_s(c) \in \mathfrak D_{m_1}$ of lowest order $r$.}
\label{fig:toavoid2}
\end{figure}
{\bf Case $a_1 \in (c_{m_1}, c_{m_2})$:} Take the $r+1$-th iterate of the picture, which moves $\mathfrak D_{m_1}$ and $\mathfrak D_{k_1}$ to levels with lower endpoint $c_1$. then we can repeat the argument, until we arrive in one of the cases below.
{\bf Case $a_2 \in (c_{m_2}, c_{k_1})$:} Take the $r$-th iterate of the picture, which moves $\mathfrak D_{m_1}$, $\mathfrak D_{k_1}$, $\mathfrak D_{m_2}$ and $\mathfrak D_{k_2}$ all to cutting levels and $c_{r+k_2} \in (c, c_{r+k_3})$. But $m_2 > m_1$, whence $k_2 > k_1$, and this contradicts that
$|c_{S_{k_2}} - c| < |c_{S_{k_1}} - c|$.
(If $a_2 \in (c_{m_3}, c_{k_2})$, then the same argument would give that $r+k_2 < r+k_3$ are both cutting times, but $|c-c_{r+k_2}| < |c-c_{r+k_3}|$.)
{\bf Case $a_3 \in (c_{k_1}, c_{m_3})$:} Take the $r$-th iterate of the picture, which moves $\mathfrak D_{m_1}$, $\mathfrak D_{m_2}$ and $\mathfrak D_{k_2}$ to cutting levels, and $\mathfrak D_{m_3}$ to a non-cutting level $\mathfrak D_u$ with $u := m_3+r$ such that \[
S_j := n+r = \beta(u) = \beta(m_2+r) = \beta^2(k_2+r). \] The integer $u$ such that $c_u$ is closest to $c$ is for $u = S_i + S_j$ where $j$ is minimal such that $Q(i+1) > i$, and in this case, the itineraries of $T_s(c)$ and $T_s(c_u)$ agree for at most $S_{Q^2(i+1)+1}-1$ iterates (if $Q(i+1) = j+1$) or at most $S_{Q(j+1)}-1$ iterates (if $Q(i+1) > j+1$). Call $S_h := k_2+r$, then $j = Q^2(h)$ and the itineraries of $T_s(c_{S_h})$ and $c$ agree up to $S_{Q(h+1)}-1$ iterates. By assumption \eqref{eq:cond3}, we have \[ Q(j+1) \leq Q^2(i+1)+1 = Q(j+1)+1 = Q(Q^2(h)+1)+1 < Q(h+1), \] but this means that $\mathfrak D_u$ and $\mathfrak D_{S_h}$ cannot overlap, a contradiction.
{\bf Case $a_4 \in (c_{k_2}, c_n)$:} Then take the $r+1$-st iterate of the picture, which has the same structure, with $c_n$ replaced by $T_s^{r+1}(a_1) = c_1$. Repeating this argument, we will eventually arrive at Case $a_2$ or $a_3$ above.
Therefore we can find $\eta_n$ such that $c_n - \eta_n$ separates $c_n$ from all levels $\mathfrak D_{k_i}$, $\beta^2(k_i) = n$ that intersect $\mathfrak D_{m_1}$. Indeed, in Case $a_2$, we place $c_n-\eta_n$ just to the right of $c_{k_1}$ and in Case $a_3$ (and hence $c_{k_1} \in \mathfrak D_{k_2}$), we place $c_n-\eta_n$ just to the right of $c_{k_2}$. \end{proof} \begin{proposition}\label{prop:chains} Under the assumption of Lemma~\ref{lem:linkDn}, given $\varepsilon > 0$, there exists $p \in {\mathbb N}$ and a chain ${\mathcal C} = {\mathcal C}_p$ of $\underleftarrow\lim([0,s/2],T_s)$ with the following properties: \begin{enumerate} \item The links of ${\mathcal C}$ have diameter $< \varepsilon$. \item For each $n \in {\mathbb N}$, there is exactly one link $\ell \in {{\mathcal C}}$ such that every $x \in \underleftarrow\lim([0,s/2],T_s)$ that $p$-turns at $c_n$ belongs to $\ell$. \item If $y \in \ell$ is a $p$-point not having the lowest
$p$-level of $p$-points in $\ell$, then both $\beta$-neighbors of $y$ belong to $\ell$. \item If $y \not\in \ell$ is a $\beta$-neighbor of $x$ above, then the other $\beta$-neighbor of $y$ either lies outside $\ell$, or has $p$-level $n$ as well. \end{enumerate} \end{proposition} \begin{proof} We will construct the chain ${\mathcal C}$ as outlined in the beginning of this section, see \eqref{eq:links}. So let us specify the collection $G$ by starting with at least $\lceil 2s^p/\varepsilon \rceil$ approximately equidistant points $g_m \in [0,s/2]$ so that no $g_m$ lies on the critical orbit, and then refining this collection inductively to satisfy parts 2.-4.\ of the proposition.
Start the induction with $n = 1$, {\em i.e.,} the point $c_1$. Note that $c_1 \notin G$, so there will be only one link $\ell \in {\mathcal C}$ with $c_1 \in \pi_p(\ell)$. Let $\eta_1 \in (0, s^{-p} \varepsilon/2)$ be as in Lemma~\ref{lem:linkDn}. Then, since each $k$ contains $1$ in its $\beta$-orbit, each $\mathfrak D_k$ intersecting $(c_1-\eta_1, c_1]$ is either contained in $(c_1-\eta_1, c_1]$ or has $c_1$ as lower endpoint ({\em i.e.,} $\beta(k) = 1$). In the latter case, also $\mathfrak D_l \cap (c_1-\eta_1, c_1] = \emptyset$ for each $l$ with $\beta(l) = k$. Hence by inserting $c_1-\eta_1$ into $G$, we can refine the chain ${\mathcal C}$ so that properties 3.\ and 4.\ holds for the link $\ell$ with $\pi_p(\ell) \owns c_1$.
Suppose we have refined the chain to accommodate links $\ell$ such that $\pi_p(\ell) \owns c_i$ for each $i < n$. Then $c_n$ does not belong to the set $G$ created so far, so there will be only one link $\ell \in {{\mathcal C}}$ with $\pi_p(\ell) \owns c_n$. Again, find $\eta_n\in (0, s^{-p}\varepsilon /2)$ as in Lemma~\ref{lem:linkDn} and extend $G$ with $c_n + \eta_n$ if $c_n$ is a local minimum of $T_s^n$ or with $c_n - \eta_n$ if $c_n$ is a local minimum of $T_s^n$.
We skip the induction step if $\mathfrak D_n$ already belongs to complementary interval to
$G$ extended with all point $c_i \pm \eta_i$ created so far. Since $|\mathfrak D_n| \to 0$, the induction will eventually cease altogether, and then the required set $G$ is found. \end{proof}
\section{Symmetric and Quasi-Symmetric Arcs} \label{sec:furtherlemmas}
From now on all chains ${\mathcal C}_p$ are as in Proposition~\ref{prop:chains}. Also, we assume that the slope $s$ is such that $T_s$ is Fibonacci-like and we abbreviate $T := T_s$.
\noindent Suppose $A = [u, v] \subset {\mathfrak A}$ is a quasi-$p$-symmetric arc with $u, v \in \ell$, and let $A_u$ and $A_v$ be arc-components of $\ell$ that contain $u$ and $v$ respectively. We will sometimes say, for simplicity, that the arc $[A_u, A_v]$ between $A_u$ and $A_v$, including $A_u$ and $A_v$, is quasi-$p$-symmetric. \begin{defi}\label{basic-quasi-p-symmetric} A quasi-$p$-symmetric arc $A = [u, v]$ with midpoint $m$ is called {\em basic} if there is no $p$-point $w \in (u, v)$ such that either $[u, w] \subset [u,m]$ or $[w, v] \subset [m,v]$ is a quasi-$p$-symmetric arc. \end{defi} \begin{ex} Let us consider the Fibonacci map and the corresponding inverse limit space. Then the arc-component ${\mathfrak C}$ (as well as an arc-component ${\mathfrak A}$) contains the arc $A = [x^0, x^{33}]$ such that the folding pattern of $A$ is as follows (see Figure~\ref{fig:example}): \begin{equation}\label{eq:fold}
27 \ 6 \ \overbrace{\underbrace{{\bf 1}_2 \ {\bf 14}_3 \ {\bf 1} \ 6 \ {\bf 1}_6}_{\textrm{basic}} \ 0 \ 3 \ 0 \ 1 \ 0 \ 2 \ 0 \ 1 \ 4 \ 1 \ {\bf 9} \ 1 \ 4 \ 1 \ 0 \ 2 \ 0 \ 1 \ 0 \ 3 \ 0 \, \underbrace{1 \ 6 \ {\bf 1}_{30}}_{\textrm{sym}}}^{\textrm{quasi-$p$-symmetric}} \ 0 \ 3 \ 0 \end{equation} (for easier orientation we write sometimes for example $1_2$ which means that the $p$-level 1 belongs to the $p$-point $x^2$). We can choose a chain ${\mathcal C}_p$ such that $p$-points with $p$-levels 1 and 14 belong to the same link. \begin{figure}
\caption{The arc $A$ with folding pattern as in \eqref{eq:fold}, with $p$-points of $p$-level $1$ and $14$ in a single link $\ell$.}
\label{fig:example}
\end{figure} The arc $[x^2,x^6]$ with the folding pattern $1\ 14\ 1\ 6\ 1$ is a basic quasi-$p$-symmetric arc; the arc $[x^2,x^{30}]$ with the folding pattern as in \eqref{eq:fold} under the wide brace is also a quasi-$p$-symmetric but not basic, because it contains $[x^2,x^6]$. Notice also that the arc $[x^3, x^{30}]$ is a quasi-$p$-symmetric arc for which Proposition~\ref{lem:until_lm} and Proposition~\ref{lem:symarc} do not work (see the folding patterns to the left of $[x^3, x^{30}]$ and to the right of $[x^3, x^{30}]$). \end{ex} \begin{lemma}\label{prop2}\label{lem:maxL} Let ${\mathcal C}_p$ be a chain and $[x,y]$ a quasi-$p$-symmetric arc with respect to this chain (not contained in a single link) with midpoint $m$ and such that $L_p(x) \geq L_p(m)$. Let $A_x$ be the link-tip of $[x,y]$ which contains $x$. Then $L_p(m) > L_p(z)$ for all $p$-points $z \in [x,y] \setminus (\{ m \} \cup A_x)$. \end{lemma} \begin{proof} Let $A = [a,b] \owns m$ be the smallest arc with $p$-points $a,b$ of higher $p$-level than $L_p(m)$, say $m \in [a,b]$ and $L_p(m) \leq L_p(a) \leq L_p(b)$. By part (a) of Remark~\ref{rem_basic} we obtain $L := L_p(m) < L_p(a) < L_p(b)$. Since $L_p(x) \geq L_p(m)$, $[x,m]$ contains one endpoint of $A$. We can assume that $[x,m] \setminus A$ is contained in a single link, because otherwise $[x,y] \setminus \ell$-tips is not $p$-symmetric. If $[y,m]$ does not contain the other endpoint of $A$, then the statement is proved.
Let us now assume by contradiction that $A \subset [x,y]$. Again, we can assume that $[y,m] \setminus A$ is contained in a single link, because otherwise $[x,y] \setminus \ell$-tips is not $p$-symmetric. By part (a) of Remark~\ref{rem_basic} once more we have $\pi_{p+L}([a,b]) = [c_{S_l}, c_{S_k}] \owns c = \pi_{p+L}(m)$
for some $k$ and $l = Q(k)$, and $|\pi_{p+L}(a) - c| > |\pi_{p+L}(b) - c|$, see the top line of Figure~\ref{fig:xy}. It follows that $[a,b]$ contains a symmetric open arc $(b',b)$ where $b' \in (a,b)$ is the unique point such that $T(\pi_{p+L}(b')) = T(\pi_{p+L}(b))$. Since $[x,y] \setminus \ell$-tips is $p$-symmetric, $L_p(b) > L_p(m)$ implies $b, b' \in \ell$-tips. Moreover, the arc $[a,b']$ is contained in the same link $\ell$ as $b$.
If $k$ and $l$ are relatively small, then $\pi^{-1}_{p}(c_{S_l})$ and $\pi^{-1}_{p}(c_{S_k})$ belong to different links of ${\mathcal C}_p$, so we can assume that they are so large that we can apply Condition \eqref{eq:cond4}. \begin{figure}
\caption{The image of $\pi_{p+L}([x,y]) \owns c = \pi_{p+L}(m)$ under appropriate iterates of $T$. }
\label{fig:xy}
\end{figure} Let $r = Q(k+1)$ and $r' = Q(l+1)$ be the lowest indices such that the closest precritical points $\hat \zeta_{r'} \in [c_{S_l}, c]$ and $\zeta_r \in [c, c_{S_k}]$. By \eqref{eq:cond4}, $r'= Q(l+1) = Q(Q(k)+1) < Q(k+1) = r$. Consider the image of $[c_{S_l}, c_{S_k}]$ first under $T^{S_{r'}}$ and then under $T^{S_r}$ (second and third level in Figure~\ref{fig:xy}).
By the choice of $r$, we obtain $\pi_{p+L-S_r}([m,b]) = [ c_{S_{k+1}}, c_{S_{Q(k+1)}}]$, and $\pi_{p+L-S_r}([a,b']) \owns c_{S_t}$ for $t = Q(Q(k+1))$. As in \eqref{eq:cond4twice},
$|c_{S_t} - c| > |c_{S_{Q(k+1)}} - c| > |c_{S_{k+1}} - c|$, and taking one more iterate, we see that $[c_{1+S_{k+1}}, c_1] \subset [c_{1+S_{Q(k+1)}}, c_1] \subset [1+c_{S_t}, c_1]$ (last level in Figure~\ref{fig:xy}).
Let $n \in [m,b]$ be such that $\pi_{p+L}(n) = \zeta_r$, see the first level in Figure~\ref{fig:xy}. Since $[a,b']$ belongs to a single link $\ell \in {\mathcal C}_p$, $m \in \ell$ as well. Suppose that $[a,m]$ is not contained in $\ell$. Then there is a maximal symmetric arc $[d', d]$ with midpoint $n$ such that the points $d, d' \notin \ell$. Then the arcs $[d', a]$ and $[d, m]$ both enter the same link $\ell$ but they have different `first' turning levels in $\ell$, contradicting the properties of ${\mathcal C}_p$ from Proposition~\ref{prop:chains}.
This shows that $[a,m] \subset \ell$. In the beginning of the proof we argued that the components of $[x,y] \setminus A$ belong to the same link, so that means that the entire arc $[x,y]$ is contained in a single link, contradicting the assumptions of the proposition. This concludes the proof. \end{proof} \begin{remark} In fact, this proof shows that the $p$-point $b \in \partial A$ of the highest $p$-level belongs to $[m,x]$. Indeed, if $a \in [m,x]$, then because $[m,b]$ has shorter arc-length than $[m,a]$, either $a$ and $b$, and therefore $x$ and $y$ do not belong to the same link $\ell$ (whence $[x,y]$ is not quasi-$p$-symmetric), or the arc $[a,b]$ itself is quasi-$p$-symmetric and contradicts Lemma~\ref{prop2}. \end{remark}
\begin{corollary}\label{cor:qusi_symmetry_type} Let $A = [x,y] \subset {\mathfrak A}$ be a quasi-$p$-symmetric arc with midpoint $m$. Let $A_x$, $A_y$ be the link-tips of $A$ containing $x$ and $y$ respectively. If $x$ is the midpoint of $A_x$, and $y$ is the midpoint of $A_y$, then either $L_p(x) > L_p(m) > L_p(y)$, or $L_p(x) < L_p(m) < L_p(y)$. \end{corollary} \begin{remark} Note that in general there are quasi-$p$-symmetric arcs $[x,y]$ with midpoint $m$ such that $L_p(x) > L_p(y) > L_p(m)$. For example, if a tent map $T_s$ has a preperiodic critical point, then for every quasi-$p$-symmetric arcs $[x,y]$ with midpoint $m$ either $L_p(x) > L_p(y) > L_p(m)$, or $L_p(y) > L_p(x) > L_p(m)$. \end{remark}
\begin{corollary}\label{cor:spiral_linktips} Let $[x,y] \subset {\mathfrak A}$ be a quasi-$p$-symmetric arc with midpoint $m$, not contained in a single link, such that $L_p(x) > L_p(m) > L_p(y)$. If $[m,x]$ is longer than $[y,m]$ measured in arc-length, then there exists a $p$-point $y' \in A_x$ such that $[y,y']$ is $p$-symmetric. \end{corollary} \begin{proof} As in the previous proof, $b \in [x,m]$ and $y \in [m,b']$ and take $y' \in [m,b]$ such that $\pi_{p+L}(y') = \pi_{p+L}(y)$. \end{proof} \begin{remark} If $A_x \ni x$ and $A_y \ni y$ are maximal arc-components of ${\mathfrak A} \cap \ell$ (with still $L_p(x) > L_p(m) > L_p(y)$), and $m_y$ is the midpoint of $A_y$, then there is $y' \in A_x$ such that $[y', m_y]$ is $p$-symmetric.
In other words, when ${\mathfrak A}$ enters and turns in a link $\ell$, then it folds in a symmetric pattern, say with levels $L_1, L_2,\dots, L_{m-1}, L_m, L_{m-1}, \dots, L_2, L_1$. The nature of the chain ${\mathcal C}_p$ is such that $L_1$ depends only on $\ell$. The Corollary~\ref{cor:spiral_linktips} does not say that the rest of the pattern is the same also, but only that if $A \subset {\mathfrak A}$ is such that $A \setminus \ell\mbox{-tips}$ is $p$-symmetric, then the folding pattern at the one link-tip is a subpattern (stopping at a lower center level) of the folding pattern at the other link-tip. \end{remark} \begin{proposition}[Extending a quasi-$p$-symmetric arc at its higher level endpoint] \label{lem:symarc} Let $A = [x,y] \subset {\mathfrak A}$ be a basic quasi-$p$-symmetric arc, not contained in a single link, such that the $p$-points $x, y \in \ell$ are the midpoints of the link-tips of $A$ and $L_p(x) > L_p(y)$. Let $m$ be the midpoint of $A$. Then there exists a $p$-point $m'$ such that the arc $[m, m']$ is (quasi-)$p$-symmetric with $x$ as its midpoint. \end{proposition} \begin{figure}
\caption{The configuration in Proposition~\ref{lem:symarc} where the existence of $p$-point $m'$ is proved. $v$ is the first $p$-point 'beyond' $x$ such that $L_p(v) > L_p(x)$ and $u$ is such that $[u,y]$ is $p$-symmetric with midpoint $m$.}
\label{fig:symarc1}
\end{figure} \begin{remark} The conditions are all crucial in this lemma: \begin{itemize} \item[$(a)$] It is important that $y$ is a $p$-point. Otherwise, if ${\mathfrak A}$ goes straight through $\ell$ at $y$, then it is possible that $x$ is the single $p$-point in $A_x$ (where $A_x$ is the arc-component of ${\mathfrak A} \cap \ell$ containing $x$) and $[v,x]$ is shorter than $[x,m]$, and the lemma would fail. \item[$(b)$] Without the assumption that $[x,y]$ is basic the lemma can fail. If Figure~\ref{fig:example} the quasi-$p$-symmetric arc $[x,y] = [x^3,x^{30}]$ is not basic, and indeed there is no $p$-point $m' \in [x,v] = [x^3,x^{0}]$ with $L_p(m') = L_p(m) = L_p(x^{17}) = 9$. \end{itemize} \end{remark} \begin{proof} Since $[u,y]$ is $p$-symmetric, $L_p(u) = L_p(y) < L_p(m)$ and $x \neq u$. Let $w$ be the first $p$-point `beyond' $y$ such that $L_p(w) > L_p(x)$. Take $L = L_p(x)$; Figure~\ref{fig:config_symarc} shows the configuration of the relevant points on $[w,v]$ and their images under $\pi_p \circ \sigma^{-L}$ denoted by $\tilde{}\ $-accents. Clearly $\tilde x = c$. \begin{figure}
\caption{ The configuration of points on $[w,v]$ and their images under $\pi_p \circ \sigma^{-L}$ and an additional $T^r$.}
\label{fig:config_symarc}
\end{figure}
{\bf Case I:} $|\tilde w - c| < |\tilde v - c|$. Then by Remark~\ref{rem_basic} (b), $\tilde w = c_{S_l}$ and $\tilde v = c_{S_k}$ with $k = Q(l)$. The points $\tilde y, \tilde m, \tilde u$ have symmetric copies $\tilde y', \tilde m', \tilde u'$ ({\em i.e.,} $T(\tilde y) = T(\tilde y')$, etc.) in reverse order on $[c,\tilde v]$, and the pre-image under $\sigma^L \circ \pi_p^{-1}$ of the copy of $\tilde m'$ yields the required point $m'$.
{\bf Case II:} $|\tilde w - c| > |\tilde v - c|$, so in this case, $l = Q(k)$. We can in fact assume that
$|\tilde m - c| > |\tilde v - c|$ because otherwise we can find $m'$ precisely as in Case I. Now take the $p$-point $a' \in (x,v)$ of maximal $p$-level, and let $a \in [m,x]$ be such that their $\pi_p \circ \sigma^{-L}$-images $\tilde a$ and $\tilde a'$ are each other symmetric copies. Let $r$ be such that $T^{r}(\tilde a) = c$; the bottom part of Figure~\ref{fig:config_symarc} shows the image of
$[\tilde m,\tilde v]$ under $T^r$. The point $T^r(\tilde x)$ and $T^r(v)$ are each others $\beta$-neighbors, and since $L_p(v) > L_p(x)$, and by \eqref{eq:cond4}, $|T^r(\tilde x) - c| > |T^r(v)-c|$. Therefore $[T^{r+j}(\tilde x) , T^{r+j}(\tilde a')] \supset [T^{r+j}(\tilde v) , T^{r+j}(\tilde a')]$ for all $j \geq 1$.
If $a, a' \in \ell$, then since $[x,a] \subset \ell$, this would imply that $[a', v] \subset \ell$ as well, contrary to the fact that $x$ is the midpoint of $A_x$.
If on the other hand $a, a' \notin \ell$, then there is a point $u'' \in [m,a]$ such that $T^r(\tilde u'')$ and $T^r(\tilde u)$ are each other symmetric copies. It follows that $[u'', x]$ is a quasi-$p$-symmetric arc properly contained in $[x,y]$, contradicting that $[x,y]$ is basic. \end{proof} \begin{proposition}[Extending a quasi-$p$-symmetric arc at its lower level endpoint]\label{lem:until_lm} Let $A = [x, y] \subset {\mathfrak A}$ be a basic quasi-$p$-symmetric arc, not contained in a single link, such that $x$ and $y$ are the midpoints of the link-tips of $A$ and $L_p(x) > L_p(y)$. Let $m$ be the midpoint of $A$. Then there exists a point $a$ such that $[m, a]$ is a quasi-$p$-symmetric arc with $y$ as the midpoint. \end{proposition} \begin{remark} The assumption that $[x,y]$ is basic is essential. Without it, we would have a counter-example in $[x,y] = [x^3,x^{30}]$ in Figure~\ref{fig:example}. The quasi-$p$-symmetric arc $[x^3,x^{30}]$ is indeed not basic, because $[x^3,x^6]$ is a shorter quasi-$p$-symmetric arc in the figure. There is a point $n = x^{32}$ beyond $y$ with $L_p(n) = L_p(x^{32}) = 3 > 1 = L_p(y)$, making it impossible that $y$ is the midpoint of a quasi-$p$-symmetric arc stretching unto $m$. \end{remark} \begin{proof} \begin{figure}
\caption{The arcs $A$ and $B$ and the relevant points for Proposition~\ref{lem:until_lm}, which is meant to show that the point $n$ does not exist in $B$.}
\label{fig:Lemma9}
\end{figure} A quasi-$p$-symmetric arc is not contained in a single link, so $[x,m] \not\subset \ell$. Let $H = [x,n] \supset A$ be the smallest arc containing a point $n$ `beyond' $y$ with $L_p(n) > L_p(y)$.
Corollary~\ref{cor:spiral_linktips} implies that the arc $[x,m]$ contains a $p$-point $y'$ with $L_p(y') = L_p(y)$. Let $b$ and $b'$ be the $p$-points having the highest $p$-level on the arcs $[y,m)$ and $[y',m)$ respectively. By symmetry, $L_p(b) = L_p(b')$, and possibly $b = y$, $b' = y'$. Let $z \in [x,y']$ be the point closest to $y'$ such that $L_p(z) > L_p(b)$; possibly $z = x$. Since $b' \in [y',m)$, we have \[ L_p(y) = L_p(y') \leqslant L_p(b) = L_p(b') < L_p(m). \] Take $L := L_p(b)$ and let $\tilde H = \pi_p \circ \sigma^{-L}(H)$. Since $y$ is the midpoint of its link-tip, $[y, n] \not\subset \ell$. Therefore $\pi_p^{-1}(c) \cap \sigma^{-L}(H) \supset \{ \sigma^{-L}(b), \sigma^{-L}(b')\}$, and $\tilde z = \pi_p \circ \sigma^{-L}(z)$ and $\tilde n = \pi_p \circ \sigma^{-L}(n)$ have $\tilde m = \pi_p \circ \sigma^{-L}(m)$ as common $\beta$-neighbor, see Figure~\ref{fig:Lemma9b}. \begin{figure}
\caption{The arc $\tilde H$ drawn as multiple curve, its preimage under $T_s^{S_j}$ and the relevant points on them.}
\label{fig:Lemma9b}
\end{figure} Since $L_p(z) > L_p(b)$ there is $k$ such that $\tilde z = c_{S_k}$. Also take $l$ such that $\tilde n = c_{S_l}$ and $j$ such that $\tilde m = c_{S_j}$. Let $\tilde y = \pi_p \circ \sigma^{-L}(y)$ and $\tilde y' = \pi_p \circ \sigma^{-L}(y')$.
We claim that there is a point $a \in [n,m]$ such that \[ \tilde a := \pi_p \circ \sigma^{-L}(a) \in [c_{S_l} , \tilde y] \quad\text{ and }\quad T_s(\tilde a) = T_s(\tilde m). \] Since $c_{S_j}$ is $\beta$-neighbor to both $c_{S_l}$ and $c_{S_k}$, we have three cases: \begin{enumerate} \item
$j = Q(k)$ and $l = Q(j)$, so $l = Q^2(k)$. In this case, Equation~\eqref{eq:cond4} and Remark~\ref{rem:cond4} imply that $|c - c_{S_l}| > |c - c_{S_{Q(k)}}|$, so $[c_{S_l}, c]$ contains the required point $\tilde a$ with $T_s(\tilde a) = T_s(\tilde m)$. By the same token,
$|c_{S_k} - c| < |c_{S_j} - c| = \frac12 |\tilde a - \tilde m|$. Since
$|\tilde y - c| = |\tilde y'- c| < |c_{S_k} - c|$, we indeed obtain that $\tilde a \in [ c_{S_l}, \tilde y]$. \item $j = Q(l)$ and $k = Q(j)$, so $k = Q^2(l)$. Then Remark~\ref{rem:Hofbauer} implies that
$|c - c_{S_k}| > |c - c_{S_l}|$. But this would mean that the arc $[n,m]$ is shorter than $[z, m]$ and in particular that $[y,n] \subset \ell$, contradicting that $y$ is the midpoint of its link-tip. \item $j = Q(k) = Q(l)$. In this case, we pull $\tilde H$ back for another $S_j$ iterates, or more precisely, we look at the arc $\pi_p \circ \sigma^{-S_j-L}(H)$. The endpoints of this arc are $c_{S_{k-1}}$ and $c_{S_{l-1}}$ which are therefore $\beta$-neighbors. If $l - 1 = Q(k-1)$, then we find \[ Q(k) = Q(l) = Q(Q(k-1)+1) \] which contradicts Condition \eqref{eq:cond4} with $k$ replaced by $k-1$. If $k-1 = Q(l-1)$, then we find \[ Q(l) = Q(k) = Q(Q(l-1)+1) \] which contradicts Condition \eqref{eq:cond4} with $k$ replaced by $l-1$. \end{enumerate} This proves the claim.
Suppose now that $\tilde y \neq c$ ({\em i.e.,} $y \neq b$). Then $b, b' \notin \ell$ because $y$ has the largest
$p$-level in its link-tip. Since $|c_{S_k} - c| < |c - \tilde m|$, there is a point $u \in [z,m]$ such that $\tilde u = \pi_p \circ \sigma^{-L}(u) \in [c,\tilde m]$ and $T_s(\tilde u) = T_s(\tilde y)$. This means that $[x,u]$ is a quasi-$p$-symmetric arc properly contained in $[x,m]$, contradicting the assumption that $[x,y]$ is a basic quasi-symmetric arc.
Therefore $y = b$, so there are no $p$-points between $y$ and $m$ of level higher than $L_p(y)$. Instead, the arc $[a,m]$ has midpoint $y$, and is the required quasi-$p$-symmetric arc, proving the lemma. \end{proof} \begin{remark}\label{rem:level-zero} Let $A = [x,y]$ be a basic quasi-$p$-symmetric arc such that $x$ and $y$ are the midpoints of the link-tips of $A$ and $L_p(x) > L_p(y)$. Let $\ell^m$ be the link which contains the midpoint $m$ of $A$, and let $A_m$ be the arc-component of $\ell^m$ containing $m$. Then, by the lemma above, $A \setminus (\ell\mbox{-tips } \cup A_m)$ does not contain any $p$-point $z$ with $L_p(z) \geqslant L_p(y)$. \end{remark}
\section{Link-Symmetric Arcs}\label{sec:link}
\begin{defi}\label{def:decreasing} We say that an arc $[x, y]$ is \emph{decreasing $($basic$)$ quasi-$p$-symmetric} if it is the concatenation of (basic) quasi-$p$-symmetric arcs where the $p$-levels of the midpoints decrease, {\em i.e.,} if there are $p$-points $x = x^0, x^1, x^2, \dots , x^{n-1}$ and $x^n = y$ can be a $p$-point or not, such that the following hold: \begin{itemize} \item[(i)] $[x^{i-1}, x^{i+1}]$ is a (basic) quasi-$p$-symmetric arc with midpoint $x^{i}$, for $i = 1, \dots , n-1$. (By definition of a (basic) quasi-$p$-symmetric arc, the points $x^{2i}$ all belong to a single link, and the points $x^{2i-1}$ belong to a single link as well.) \item[(ii)] $L_p(x^i) > L_p(x^{i+1})$, for $i = 1, \dots , n-1$ (and if $y$ is a $p$-point then also $L_p(x^{n-1}) > L_p(y)$). \end{itemize} Similarly, we say that the arc $[x, y]$ is \emph{increasing $($basic$)$ quasi-$p$-symmetric} if it is the concatenation of (basic) quasi-$p$-symmetric arcs where the $p$-levels of the midpoints increase. \end{defi} \begin{figure}
\caption{Illustration of a basic decreasing quasi-$p$-symmetric arc. The point $y$ is not a $p$-point here; instead, the arc $A$ goes straight through $\hat \ell$ at $y$.}
\label{fig:decreas-quasi-sym}
\end{figure} \begin{ex} Consider the Fibonacci inverse limit space, and let our chain ${\mathcal C}_p$ be such that $p$-points with $p$-levels $1$ and $14$ belong to the same link $\ell$, but $p$-points with $p$-level $9$ are not contained in $\ell$. Since $p$-points with $p$-level 14 belong to the same link $\ell$ as $p$-points with $p$-level $1$, also $p$-points with $p$-levels $22$, $35$, $56$ and $77$ belong to $\ell$. Let $p$-points with $p$-level $43$ belong to the same link as $p$-points with $p$-level $9$. \begin{itemize} \item[(1)] {\bf Example of a basic decreasing quasi-$p$-symmetric arc.} Let $A = [y^0, y^{12}]$ be an arc with the following folding pattern (where the subscripts count important $p$-points): $$ \underbrace{{\bf 1 \ 22 \ 77}_2 \ {\bf 22 \ 1} \ 9 \ 43_6 \ 9 \ {\bf 1}}_{\textrm{basic}} \overbrace{{\bf 22}_9 \ {\bf 1} \ 9_{11} \ {\bf 1}_{12}}^{\textrm{basic}} $$ Let $x^i$ be as in the above definition. Then $x^1 = y^2$, $x^2 = y^6$, $x^3 = y^9$, $x^4 = y^{11}$, and $x^5 = y^{12}$. So $[y^2, y^9]$ is basic quasi-$p$-symmetric with midpoint $y^6$, $[y^6, y^{11}]$ is basic quasi-$p$-symmetric with midpoint $y^9$, and $[y^9, y^{12}]$ is basic quasi-$p$-symmetric with the midpoint $y^{11}$. Also $L_p(y^2) = 77 > L_p(y^6) = 43 > L_p(y^9) = 22 > L_p(y^{11}) = 9 > L_p(y^{12}) = 1$. \item[(2)] {\bf Example of a non-basic decreasing quasi-$p$-symmetric arc.} Let $[y^0, y^{72}]$ be an arc with the following folding pattern: $$
\overbrace{\underbrace{{\bf 1 \ 22 \ 1 \ 56}_3 \, {\bf 1 \ 22 \ 1} \ 9 \ {\bf 1}}_{\textrm{basic}} \ 4 \ 1 \ 0 \ 2 \ 0 \ 1 \ 0 \ 3 \ 0 \ {\bf 1} \ 6 \ {\bf 1 \ 14 \ 1 \ 35}_{23} {\bf 1 \ 14 \ 1} \ 6 \ {\bf 1} \ 0 \ 3 \ 0 \ 1 \ 0 \ 2 \ 0 \ 1 \ 4 \ \underbrace{{\bf 1} \ 9 \ {\bf 1}}}^{\textrm{quasi-$p$-symmetric}} $$ $$ \underbrace{\overbrace{{\bf 22}_{41} {\bf 1} \ 9 \ {\bf 1}}^{\textrm{basic}}
\ 4 \ 1 \ 0 \ 2 \ 0 \ 1 \ 0 \ 3 \ 0 \ {\bf 1} \ 6 \ {\bf 1 \ 14}_{57} {\bf 1}
\ 6 \ {\bf 1} \ 0 \ 3 \ 0 \ 1 \ 0 \ 2 \ 0 \ 1 \ 4 \ \overbrace{{\bf
1} \ 9 \ {\bf 1}_{72}}^{\textrm{sym}}}_{\textrm{quasi-$p$-symmetric}} $$ Let $x^i$ be again as in the above definition. Then $x^1 = y^3$, $x^2 = y^{23}$, $x^3 = y^{41}$, $x^4 = y^{57}$, and $x^5 = y^{72}$. So, arcs $[y^3, y^{41}]$, $[y^{23}, y^{57}]$ and $[y^{41}, y^{72}]$ are quasi-$p$-symmetric, and $L_p(y^3) = 56 > L_p(y^{23}) = 35 > L_p(y^{41}) = 22 > L_p(y^{57}) = 14 > L_p(y^{72}) = 1$. \item[(3)] {\bf Example of an arc that is the concatenation of two quasi-$p$-symmetric arcs (one of them is basic), but is not decreasing quasi-$p$-symmetric.} Let $[y^0, y^{40}]$ be an arc with the following folding pattern: $$
\underbrace{{\bf 1 \ 22 \ 77}_2 \, {\bf 22 \ 1} \ 9 \ 43_6 \, 9 \ {\bf 1}}_{\textrm{basic}}\underbrace{{\bf 22}_9 \, {\bf 1} \ 9_{11} {\bf 1}_{12} 4 \ 1 \ 0 \ 2 \ 0 \ 1 \ 0 \ 3 \ 0 \ {\bf 1} \ 6 \ {\bf 1 \ 14}_{25} {\bf 1} \ 6 \ {\bf 1} \ 0 \ 3 \ 0 \ 1 \ 0 \ 2 \ 0 \ 1 \ 4 \ {\bf 1} \ 9 \ {\bf 1}_{40}}_{\textrm{quasi-$p$-symmetric}} $$ Then $[y^2, y^9]$ is basic quasi-$p$-symmetric with midpoint $y^6$, $[y^6, y^{11}]$ is basic quasi-$p$-symmetric with midpoint $y^9$, and $[y^9, y^{12}]$ is basic quasi-$p$-symmetric with the midpoint $y^{11}$. However, $[y^9, y^{40}]$ is quasi-$p$-symmetric with midpoint $y^{25}$ and $[y^6, y^{25}]$ is neither basic quasi-$p$-symmetric, nor quasi-$p$-symmetric. So $A = [y^0, y^{40}]$ is not a decreasingly quasi-$p$-symmetric arc. Note that $[y^0, y^{12}]$ is a decreasing quasi-$p$-symmetric arc. \end{itemize} \end{ex}
\begin{proposition}\label{lem:not-link-sym} Let $A$ be a non-basic quasi-$p$-symmetric arc. Then there are $k, n, m, d \in {\mathbb N}$, $d < k$, such that $$ A \cap E_p = \{ x^0, \dots , x^k, \dots , x^{k+n}, \dots , x^{k+n+m} \}, $$ $[x^0, x^k]$ is a basic quasi-$p$-symmetric arc with midpoint $x^{k-d}$ and $[x^k, x^{k+n}]$ is $p$-symmetric. Moreover, \begin{itemize} \item[(i)] If $[x^{k+n}, x^{k+n+m}]$ is $p$-symmetric, then $[x^{-k+m/2}, x^{k+n+3m/2}]$ is not $p$-link-symmetric. \item[(ii)] If $[x^{k+n}, x^{k+n+m}]$ is a basic quasi-$p$-symmetric arc, then $A$ is contained in a decreasing quasi-$p$-symmetric arc consisting of at least two quasi-$p$-symmetric arcs. More precisely, $[x^{-k-n/2}, x^{k+n/2}]$ and $[x^{k+n/2}, x^{k+2m+3n/2}]$ are the quasi-$p$-symmetric arcs contained in the decreasing quasi-$p$-symmetric arc $[x^{-k-n/2}, x^{k+2m+3n/2}]$ containing $A$. \end{itemize} \end{proposition} \begin{proof} Since $A$ is a non-basic quasi-$p$-symmetric arc, there is a basic quasi-$p$-symmetric arc which we can label $[x^0,x^k]$. The arc $[x^{k},x^{k+n}]$ in the middle is $p$-symmetric by definition of quasi-$p$-symmetry, and it has the same midpoint $x^{k+n/2}$ as $A$. The arc $[x^{k+n}, x^{k+n+m}]$ could be either $p$-symmetric or basic quasi-$p$-symmetric.
$(i)$ Assume that $[x^{k+n}, x^{k+n+m}]$ is $p$-symmetric. Without loss of generality we can suppose that $x^0$ and $x^{k+n+m}$ are the midpoints of the link-tips of $A$, and also that $x^k$ and $x^{k+n}$ are the midpoints of their arc-components. Since the point $x^{k+n+m/2}$ is the midpoint of the $p$-symmetric arc $[x^{k+n},x^{k+n+m}]$, and the symmetry of the arc $[x^k, x^{k+n}]$ can be extended to the midpoints of its neighboring (quasi-)symmetric arcs, we obtain that $d = m/2$ and the point $x^{k-m/2}$ is the midpoint of the basic quasi-$p$-symmetric arc $[x^0,x^k]$. Proposition~\ref{lem:symarc} implies that we can extend $[x^0, x^{k-m/2}]$ beyond $x^0$ to obtain the arc $[x^{-k+m/2}, x^{k-m/2}]$ which is either $p$-symmetric, or quasi-$p$-symmetric, and hence $p$-link-symmetric.
First, let us assume that $L_p(x^{k+n+m}) = 1$. Let us consider the arc $[x^{k+n+m/2}, x^{k+n+3m/2}]$. Its midpoint $x^{k+n+m}$ has $p$-level $1$. If $L_p(x^{k+n+m-1}) = L_p(x^{k+n+m+1})$, then $L_p(x^{k+n+m-1}) = 0$. Furthermore $x^{k+n+m-1} \ne x^{k+n+m/2}$ since a midpoint cannot have $p$-level zero. It follows that $x^{k+n+m-2}$ and $x^{k+n+m+2}$ have different $p$-levels, and are not in the same link, since by Lemma~\ref{prop2} there is no quasi-$p$-symmetric arc whose both boundary points are $p$-points and whose midpoint has $p$-level $1$.
If $L_p(x^{k+n+m-1}) \ne L_p(x^{k+n+m+1})$ then again $x^{k+n+m-1}$ and $x^{k+n+m+1}$ are not in the same link (by Lemma~\ref{prop2} there is no quasi-$p$-symmetric arc whose both boundary points are $p$-points and whose midpoint has $p$-level $1$). In either case, $[x^{k+n+m/2}, x^{k+n+3m/2}]$ is not $p$-link-symmetric and hence $[x^{-m/2}, x^{k+n+3m/2}]$ is not $p$-link-symmetric. This proves statement (i) in the case that $L_p(x^{k+n+m})=1$.
Now for the general case, let $L := L_p(x^{k+n+m})$. The basic idea is to shift $[x^0,x^{k+n+m}]$ back by $L-1$ iterates, and use the above argument. Note that the arcs $[x^k, x^{k+n}]$ and $[x^{k+n}, x^{k+n+m}]$ are $p$-symmetric and hence $L_p(x^{k+n/2}) > L_p(x^{k+n}) = L_p(x^{k+n+m}) = L$. Then $\sigma^{-L+1}(A)$ is also a quasi-$p$-symmetric arc which is not basic, the arc $\sigma^{-L+1}([x^0, x^k])$ is a basic quasi-$p$-symmetric arc and $L_p(\sigma^{-L+1}(x^{k+n+m})) = 1$. Let $$ \sigma^{-L+1}(A) \cap E_p = \{ u^0, \dots , u^{\hat k}, \dots , u^{{\hat k}+{\hat n}}, \dots , u^{{\hat k}+{\hat n}+{\hat m}} \}, $$ where $u^{\hat i} = \sigma^{-L+1}(x^i)$. (Note that $\hat k \leqslant k$, $\hat n \leqslant n$ and $\hat m \leqslant m$, since not every $\sigma^{-L+1}(x^i)$ needs to be a $p$-point.) Then $G = [u^{-{\hat k}+{\hat m}/2}, u^{{\hat k}+{\hat n}+3{\hat m}/2}]$ is an arc with `boundary arcs' $[u^{-{\hat k}+\hat m/2}, u^{{\hat k}-\hat m/2}]$ and $[u^{k+n+\hat m /2}, u^{k+n+3\hat m/2}]$ and the midpoint of the latter has $p$-level $1$. The above argument shows that this arc cannot be $p$-link-symmetric, and therefore the whole arc $G$ is not $p$-link-symmetric with midpoint $u = \sigma^{-L+1}(x^{k+n/2})$.
We want to prove that $\sigma^j(G)$ is also not $p$-link-symmetric with the midpoint $\sigma^j(u)$ for $j = L - 1$.
Let us assume by contradiction that $\sigma^j(G)$ is $p$-link-symmetric. Since $[x^{-k+m/2}, x^{k-m/2}]$ is $p$-symmetric, also $\sigma^j([u^{{\hat k}+{\hat n}+{\hat m}/2}, u^{{\hat k}+{\hat n}+3{\hat m}/2}])$ is $p$-link-symmetric. But $[u^{{\hat k}+{\hat n}+{\hat m}/2}, u^{{\hat k}+{\hat n}+3{\hat m}/2}]$ has its midpoint at $p$-level $1$, and hence is not $p$-link-symmetric. Therefore, there exists $l < j$ such that $\sigma^l([u^{{\hat k}+{\hat n}+{\hat m}/2}, u^{{\hat k}+{\hat n}+3{\hat m}/2}])$ is not $p$-link-symmetric and $\sigma^{l+1}([u^{{\hat k}+{\hat n}+{\hat m}/2}, u^{{\hat k}+{\hat n}+3{\hat m}/2}])$ is $p$-link-symmetric. By Proposition~\ref{prop:chains}, and since $L_p(\sigma^l(u^{{\hat k}+{\hat n}+{\hat m}})) = l+1 \ne 0$, there exist $v \in \sigma^l([u^{{\hat k}+{\hat n}+{\hat m}/2}, u^{{\hat k}+{\hat n}+{\hat m}}])$ and $w \in \sigma^l([u^{{\hat k}+{\hat n}+{\hat m}}, u^{{\hat k}+{\hat n}+3{\hat m}/2}])$ such that $L_p(v) = L_p(w) = 0$, see Figure~\ref{fig:sym2}.
\begin{figure}\label{fig:sym2}
\end{figure}
Since $\sigma^{l+1}(u^{{\hat k}+{\hat n}+{\hat m}/2})$ and $\sigma^{l+1}(u^{{\hat k}+{\hat n}+3{\hat m}/2})$ belong to the same link and $L_p(\sigma^{l+1}(u^{{\hat k}+{\hat n}+{\hat m}/2})) \ne L_p(\sigma^{l+1}(u^{{\hat k}+{\hat n}+3{\hat m}/2}))$, Proposition~\ref{prop:chains} implies that $\sigma^{l+1}(u^{{\hat k}+{\hat n}+{\hat m}/2})$ and $\sigma^{l+1}(u^{{\hat k}+{\hat n}+3{\hat m}/2})$ belong to the same link as $\sigma(v)$ and $\sigma(w)$. But then $\sigma^{l}(u^{{\hat k}+{\hat n}+{\hat m}/2})$ and $\sigma^{l}(u^{{\hat k}+{\hat n}+3{\hat m}/2})$ belong to the same link as $v$ and $w$, contradicting the choice of $l$.
$(ii)$ The rough idea of this proof is as follows: Whenever $[x^{k+n}, x^{k+n+m}]$ is not $p$-symmetric, there exists $N \in {\mathbb N}$ such that $\sigma^{-N}(A)$ is a basic quasi-$p$-symmetric arc and we can apply Propositions~\ref{lem:symarc} and~\ref{lem:until_lm} to obtain the arc $B \supset \sigma^{-N}(A) $ which is decreasing basic quasi-$p$-symmetric. Then $\sigma^{N}(B) \supset A$ is the required decreasing quasi-$p$-symmetric arc.
Let us assume now that $[x^{k+n}, x^{k+n+m}]$ is basic quasi-$p$-symmetric. Let us denote by $\ell$ the link which contains $x^0$. Then $x^k, x^{k+n}, x^{k+n+m} \in \ell$. We can assume without loss of generality that $x^k$ and $x^{k+n}$ are the $p$-points in the link-tips of $[x^k, x^{k+n}]$ furthest away from the midpoint $x^{k+n/2}$ and, similarly, $x^0$ and $x^{k+n+m}$ are the $p$-points in the link-tips of $[x^0, x^{k+n+m}]$ furthest away from the midpoint $x^{k+n/2}$. Then from the properties of the chain in Proposition~\ref{prop:chains} we conclude that $L_p(x^0) = L_p(x^k) = L_p(x^{k+n}) = L_p(x^{k+n+m})$. Let us denote by $x^a$ and $x^b$ the midpoints of arc-components which contains $x^0$ and $x^{k+n+m}$ respectively. Then $x^a, x^b \in \ell$ and $x^b \ne x^{k+n+m}$. Without loss of generality we can assume that $L_p(x^a) > L_p(x^b)$.
Since $x^{k-d}$ is the midpoint of $[x^0, x^k]$ and $A$ is quasi-$p$-symmetric, $x^{k+n+d}$ is the midpoint of $[x^{k+n}, x^{k+n+m}]$.
By Proposition~\ref{lem:symarc}, $L_p(x^{-d}) = L_p(x^{k-d})$ and $L_p(x^{k+n+d}) = L_p(x^{k+n+m+d})$, see Figure~\ref{fig:not-link-sym}.
Let us denote by $\ell^d$ the link which contains $x^{-d}$, and by $A_d$ the arc-component of $\ell^d$ which contains $x^{-d}$. \\[2mm] {\bf Claim} $x^{-d}$ is the midpoint of its arc-component $A_d$. \\[2mm] Consider the arc $\sigma^{-L+1}(A)$, where $L := L_p(x^b)$. Since $L_p(x^a) > L_p(x^{k+n/2}) > L_p(x^b) = L$, the preimage $\sigma^{-L+1}(A)$ contains the points $\sigma^{-L+1}(x^b)$ with $L_p(\sigma^{-L+1}(x^b)) = 1$, $\sigma^{-L+1}(x^a)$ and $\sigma^{-L+1}(x^{k+n/2})$ is the midpoint of $\sigma^{-L+1}(A)$.
By Corollary~\ref{cor:spiral_linktips} the arc-component containing $x^a$ also contains $p$-points $x'$ and $x''$ with the property that $[x', x'']$ is $p$-symmetric with midpoint $x^a$ and $L_p(x') = L_p(x'') = L_p(x^b)$, Assume also that $x'$ and $x''$ are furthest away from $x^a$ with these properties. Therefore, $\sigma^{-L+1}(A)\cap E_p \supseteq \{ u^0, u^{\hat a}, u^{2\hat a}, u^{2\hat a+\hat n}, u^{2\hat a+ 2\hat n} \}$, where $u^{\hat a} = \sigma^{-L+1}(x^a)$, $u^{2 \hat a+\hat n} = \sigma^{-L+1}(x^{k+n/2})$, $u^{2 \hat a+ 2\hat n} = \sigma^{-L+1}(x^b)$, $u^0 = \sigma^{-L+1}(x')$, $u^{2 \hat a} = \sigma^{-L+1}(x'')$ and $L_p(u^0) = L_p(u^{2\hat a}) = 1$.
Let us suppose that $\sigma^{-L+1}(A)$ is not contained in a single link. Since $\sigma^{-L+1}(x^a)$ and $\sigma^{-L+1}(x^b)$ are contained in the same link, $\sigma^{-L+1}(A)$ is a basic quasi-$p$-symmetric arc. Let $\ell^n$ be the link containing $u^{2\hat a + \hat n}$, and let $A_{2a + n}$ be the arc component of $\ell^n$ containing $u^{2\hat a + \hat n}$. Since $L_p(u^{2\hat a+2\hat n}) = 1$, by Remark \ref{rem:level-zero}, $(u^{2\hat a + \hat n}, u^{2\hat a + 2\hat n}) \setminus A_{2a + n}$ can contain at most one $p$-point and its $p$-level is 0. Therefore $(u^{2\hat a}, u^{2\hat a + \hat n}) \setminus A_{2a + n}$ can also contain at most one $p$-point and its $p$-level is 0. By Proposition~\ref{lem:symarc}, $[u^{-\hat n}, u^{2\hat a+\hat n}]$ is either a $p$-symmetric arc, or a basic quasi-$p$-symmetric arc, see Figure~\ref{fig:not-link-sym}. Let us denote by $A_n$ the arc-component of $\ell^n$ containing $u^{-\hat n}$. Then $(u^{-\hat n}, u^0) \setminus A_n$ also does not contain any $p$-point with non-zero $p$-level.
\begin{figure}
\caption{ The configuration of points on $[x^{-d}, x^{k+n+m+2d}]$ and their images under $\sigma^{-L+1}$ as in $(ii)$.}
\label{fig:not-link-sym}
\end{figure}
Assume by contradiction that $x^{-d}$ is not the midpoint of its arc-component $A_d$. Let us denote the midpoint of $A_d$ by $x$, and let $u := \sigma^{-L+1}(x)$. Since $L_p(x) > L_p(x^{a})$, also $L_p(u) > L_p(u^{\hat a})$. Let $\ell^a$ be the link which contains $u^{\hat a}$, and let $A_a$ be the arc-component of $\ell^a$ containing $u^{\hat a}$. Then $u \in A_n$ and $[u^{-\hat n}, u^{2\hat a+\hat n}]$ is basic quasi-$p$-symmetric. But, since $u^{2\hat a+\hat n} \in \ell^n$ and $\sigma^{L-1}(u^{2\hat a+\hat n}) = x^{k+n/2}$, $x^{k+n/2} \in \ell^d$. Since the arc $[x, x^{k-d}]$ is quasi-$p$-symmetric, $[x^{k-d}, x^{k+n/2}]$ is also quasi-$p$-symmetric and $L_p(x^{a}) > L_p(x^{k-d})$ implies $L_p(x^{k-d}) > L_p(x^{k+n/2})$, a contradiction.
Let us assume now that $\sigma^{-L+1}(A)$ is contained in a single link. Since $L_p(u) > L_p(u^{\hat a})$ and $L_p(u^0) = 1$, we have $\pi_p([u, u^0]) \subset \pi_p([u^{\hat a}, u^0])$. Then $\sigma^{L-1}([u^{\hat a}, u^0]) \subset \ell$ implies $\sigma^{L-1}([u, u^{\hat a}]) \subset \ell$ and hence $[x^{-d}, x^{k-d}] \subset \ell$, a contradiction.
These two contradictions prove the claim.
In the same way we can prove that $x^{k+n+m+d}$ is the midpoint of its arc-component, and by Proposition~\ref{lem:until_lm} the arc $[u^{2\hat a+\hat n}, u^{2\hat a+ 3\hat n}]$ is either $p$-symmetric, or quasi-$p$-symmetric.
So we have proved that the arcs $[u^{-\hat n}, u^{2\hat a+\hat n}]$ and $[u^{2\hat a+\hat n}, u^{2\hat a+ 3\hat n}]$ are both either $p$-symmetric, or quasi-$p$-symmetric. Since $[x^{a}, x^{b}] = \sigma^{L-1}([u^{\hat a}, u^{2\hat a+ 2\hat n}])$ is quasi-$p$-symmetric, the arcs $\sigma^{L-1}([u^{-\hat n}, u^{2\hat a+\hat n}])$ and $\sigma^{L-1}([u^{2\hat a+\hat n}, u^{2\hat a+ 3\hat n}])$ are both either $p$-symmetric, or quasi-$p$-symmetric. This implies that $[x^{-2d-n/2}, x^{k+n/2}]$ and $[x^{k+n/2}, x^{k+n+m+2d+n/2}]$ are contained in the decreasing quasi-$p$-symmetric arc $[x^{-2d-n/2}, x^{k+n+m+2d+n/2}]$ containing $A$. \end{proof} \begin{ex} (Example for $(ii)$ of Proposition~\ref{lem:not-link-sym}.) Let us consider the Fibonacci map and the corresponding inverse limit space. The arc-component ${\mathfrak C}$ contains an arc $A = [x^0, x^{77}]$ with the following folding pattern: $$ 1 \ 9_1 \underbrace{{\bf 1}_2 \, {\bf 22 \ 1 \ 56 \ 1 \ 22 \ 1} \ 9 \ {\bf 1}}_{\textrm{basic}} \ 4 \ 1 \ 0 \ 2 \ 0 \ 1 \ 0 \ 3 \ 0 \ 1 \ 6 $$ $$ \overbrace{\underbrace{{\bf 1}_{22} {\bf 14 \ 1 \ 35 \ 1 \ 14 \ 1} \ 6 \ {\bf 1}}_{\textrm{basic}} \ 0 \ 3 \ 0 \ 1 \ 0 \ 2 \ 0 \ 1 \ 4 \ {\bf 1} \ 9 \ {\bf 1 \ 22 \ 1} \ 9 \ {\bf 1} \ 4 \ 1 \ 0 \ 2 \ 0 \ 1 \ 0 \ 3 \ 0 \ \underbrace{{\bf 1} \ 6 \ {\bf 1 \ 14 \ 1}_{60}}_{\textrm{basic}}}^{\textrm{quasi-$p$-symmetric}} $$
$$ 6 \ 1 \ 0 \ 3 \ 0 \ 1 \ 0 \ 2 \ 0 \ 1 \ 4 \ \underbrace{1 \ 9 \ {\bf 1}_{74}}_{\textrm{sym}} 4_{75} 1 \ 0 $$ We can choose a chain ${\mathcal C}_p$ such that $p$-points with $p$-levels 1, 14, 22, 35 and 56 belong to the same link. Then the arc $[x^{22}, x^{60}]$ is quasi-$p$-symmetric, and it is not basic. The arc $\sigma^{-13}([x^{22}, x^{60}])$ is basic quasi-$p$-symmetric with the folding pattern $1 \ 22 \ 1 \ 9 \ 1$. So we can apply Propositions~\ref{lem:symarc} and~\ref{lem:until_lm} as in the above proof. The arc $[x^2, x^{74}]$ is decreasing quasi-$p$-symmetric. Note that the arc $[x^1, x^{75}]$ is not $p$-link-symmetric. \end{ex} \begin{defi}\label{def:maximaldecreasing} An arc $A = [x, y]$ is called \emph{maximal decreasing (basic) quasi-$p$-symmetric} if it is decreasing (basic) quasi-$p$-symmetric and there is no decreasing (basic) quasi-$p$-symmetric arc $B \supset A$ that consists of more (basic) quasi-$p$-symmetric arcs than $A$.
Similarly we define a \emph{maximal increasing (basic) quasi-$p$-symmetric} arc. \end{defi}
\begin{remark}\label{rem:quasi-sym} (a) Propositions~\ref{lem:symarc} and~\ref{lem:until_lm} imply that $A = [x, y]$ is a maximal decreasing basic quasi-$p$-symmetric arc if and only if $A$ is a decreasing basic quasi-$p$-symmetric and for $x = x^0, x^1, \dots , x^{n-1}, x^n = y$ which satisfy (i) of Definition~\ref{def:decreasing}, there exists a point $x^{-1}$ such that $[x^{-1}, x^1]$ is $p$-symmetric with midpoint $x^0$ and $x^n$ is not a $p$-point. The arc $[x^{-1}, x^n]$ we call the \emph{extended} maximal decreasing basic quasi-$p$-symmetric arc. The points $x^{-1}$, $x = x^0, x^1, \dots , x^{n-1}, x^n = y$ we call the \emph{nodes} of $[x^{-1}, x^n]$.
The analogous statement holds if $A$ is a maximal increasing basic quasi-$p$-symmetric arc: If $A = [x^0, x^{n+1}]$ is an extended maximal increasing basic quasi-$p$ symmetric arc, then $x^0$ is not a $p$-point, $L_p(x^n) > L_p(z)$ for every $p$-point $z \in A$, $z \ne x^n$, and $L_p(x^{n-1}) = L_p(x^{n+1})$.
(b) Let $A = [x^0, x^{n+1}]$ be an extended maximal increasing basic quasi-$p$ symmetric arc. If there exists an additional $p$-point $x^{n+2}$ such that the arc $[x^n, x^{n+2}]$ is quasi-$p$ symmetric with midpoint $x^{n+1}$, Propositions~\ref{lem:symarc} and~\ref{lem:until_lm} imply that $A$ is contained in an $p$-symmetric arc $B = [x^0, x^{2n}]$ where the arc $[x^{n-1}, x^{2n}]$ is an extended maximal decreasing basic quasi-$p$-symmetric arc.
The analogous statement holds if $A$ is a maximal decreasing basic quasi-$p$-symmetric arc. \end{remark}
\begin{lemma}\label{lem:max-quasi-sym} Every (basic) quasi-$p$-symmetric arc $A$ can be extended to a maximal decreasing/increasing (basic) quasi-$p$-symmetric arc $B \supset A$. \end{lemma}
\begin{proof} We take the largest decreasing (basic) quasi-$p$-symmetric arc $B$ containing $A$. The only thing to prove is that there really is a largest $B$. If this were not the case, then there would be an infinite sequence $(x_i)_{i \geqslant 0}$ with $x_0 \in \partial A$, $L_p(x_i) < L_p(x_{i+1})$ and $[x_i, x_{i+2}]$ is a (basic) quasi-$p$-symmetric arc for all $i \geqslant 0$. By the definition of (basic) quasi-$p$-symmetric arc, there are two links $\ell$ and $\hat \ell$ containing $x_i$ for all even $i$ and odd $i$ respectively. (Note that $\ell = \hat \ell$ is possible.) By Lemma~\ref{lem:maxL} for the basic case, the $p$-points in $\bigcup_{i \geqslant 0} [x_0, x_i] \setminus (\ell \cup \hat \ell)$ can only have finitely many different $p$-levels. By the construction in the proof of Proposition~\ref{lem:not-link-sym} $(ii)$, the same conclusion is true for the non-basic case as well. But $\bigcup_{i \geqslant 0} [x_0, x_i]$ is a ray, and contains $p$-points of all (sufficiently high) $p$-levels. Since the closure of $\pi_p(\{ x : L_p(x) \geqslant N\})$ contains $\omega(c)$ for all $N$, this set is not contained in the $\pi_p$-images of the two links $\ell$ and $\hat \ell$ only. So we have a contradiction. \end{proof}
\begin{proposition}\label{thm:linksym} Let $A$ be a $p$-link-symmetric arc with midpoint $m$ and $\partial A = \{ x, y \} \subset E_p$. Then $A$ is $p$-symmetric, or is contained in an extended maximal decreasing/increasing (basic) quasi-$p$-symmetric arc, or is contained in a $p$-symmetric arc which is the concatenation of two arcs, one of which is a maximal increasing (basic) quasi-$p$-symmetric arc, and the other one is a maximal decreasing (basic) quasi-$p$-symmetric arc. \end{proposition}
\begin{proof} Let $A \cap E_p = \{{x}^{-k'}, \dots , {x}^{-1}, {x}^{0}, {x}^{1}, \dots {x}^{k}\}$ and ${x}^{0} = m$. Without loss of generality we assume that ${x}^{-k'}$ and ${x}^{k}$ are the midpoints of the link-tips of $A$. If $L_{p}(x^{-i}) = L_{p}(x^{i})$, for $i = 1, \dots , \min \{ k', k \}$, then the arc $A$ is either $p$-symmetric, or (basic) quasi-$p$-symmetric. Hence in this case the theorem is true.
Let us assume that there exists $j < \min \{ k', k \}$ such that $L_{p}(x^{-i}) = L_{p}(x^{i})$, for $i = 1, \dots , j-1$, and $L_{p}(x^{-j}) \ne L_{p}(x^{j})$. The arc $[x^{-j}, x^{j}]$ is (basic) quasi-$p$-symmetric and by Lemma~\ref{lem:max-quasi-sym} and Remark~\ref{rem:quasi-sym}, there exists an extended maximal decreasing/increasing (basic) quasi-$p$-symmetric arc which contains $[x^{-j}, x^{j}]$. Hence in this case the theorem is also true. \end{proof}
\noindent Faculty of Mathematics, University of Vienna,\\ Nordbergstra{\ss}e 15/Oskar Morgensternplatz 1, A-1090 Vienna, Austria\\ \texttt{[email protected]}\\ \texttt{http://www.mat.univie.ac.at/}$\sim$\texttt{bruin}
\noindent Department of Mathematics, University of Zagreb.\\ Bijeni\v cka 30, 10 000 Zagreb, Croatia\\ \texttt{[email protected]}\\ \texttt{http://www.math.hr/}$\sim$\texttt{sonja}
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March 2014, 13(2): 645-655. doi: 10.3934/cpaa.2014.13.645
Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials
Daxiong Piao 1, and Xiang Sun 1,
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China, China
Received January 2013 Revised July 2013 Published October 2013
In this paper, we consider the boundedness of solutions for a class of impact oscillators with time dependent polynomial potentials, \begin{eqnarray} \ddot{x}+x^{2n+1}+\sum_{i=0}^{2n}p_{i}(t)x^{i}=0, \quad for\ x(t)> 0,\\ x(t)\geq 0,\\ \dot{x}(t_{0}^{+})=-\dot{x}(t_{0}^{-}), \quad if\ x(t_{0})=0, \end{eqnarray} where $n\in N^+$, $p_i(t+1)=p_i(t)$ and $p_i(t)\in C^5(R/Z).$
Keywords: Canonical transformation, Time-dependent polynomial potentials, Moser's small twist theorem., Boundedness of solutions, Impactor oscillators.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Daxiong Piao, Xiang Sun. Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 645-655. doi: 10.3934/cpaa.2014.13.645
R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the Twist Theorem,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 79. Google Scholar
S. Laederich and M. Levi, Invariant curves and time-dependent potentials,, Ergo.Th. and Dynam. Syst., 11 (1991), 365. doi: 10.1017/S0143385700006192. Google Scholar
X. Yuan, Invariant tori of Duffing-type equations,, Adv. in Math. (China), 24 (1995), 375. Google Scholar
X. Yuan, Invariant tori of Duffing-type equations,, J. Differential Equations, 142 (1998), 231. Google Scholar
M. Kunze, "Non-Smooth Dynamical Systems,", in: Lecture Notes in Math., (2000). Google Scholar
H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator,, Physica D, 82 (1995), 117. doi: 10.1016/0167-2789(94)00222-C. Google Scholar
P. Boyland, Dual billiards, twist maps and impact oscillators,, Nonlinearity, 9 (1996), 1411. doi: 10.1088/0951-7715/9/6/002. Google Scholar
M. Corbera and J. Llibre, Periodic orbits of a collinear restricted three body problem,, Celestial Mech. Dynam. Astronom., 86 (2003), 163. doi: 10.1023/A:1024183003251. Google Scholar
D. Qian and P. J. Torres, Periodic motions of linear impact oscilltors via successor map,, SIAM J. Math. Anal., 36 (2005), 1707. doi: 10.1137/S003614100343771X. Google Scholar
D. Qian and X. Sun, Inariant tori for asymptotically linear impact oscillators,, Sci. China: Ser. A Math., 49 (2006), 669. doi: 10.1007/s11425-006-0669-5. Google Scholar
V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts,, Comm. Math. Phys., 211 (2000), 289. doi: 10.1007/s002200050813. Google Scholar
Z. Wang and Y. Wang, Existence of quasiperiodic solutions and Littlewood's boundedness problem of super-linear impact oscillators,, Applied Mathematics and Computation, 217 (2011), 6417. doi: 10.1016/j.amc.2011.01.037. Google Scholar
Z. Wang, Q. Liu and D. Qian, Existence of quasiperiodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators,, Nonlinear Analysis, 74 (2011), 5606. doi: 10.1016/j.na.2011.05.046. Google Scholar
D. Qian, Large amplitude periodic bouncing in impact oscillators with damping,, Proc. Amer. Math. Soc., 133 (2005), 1797. doi: 10.1090/S0002-9939-04-07759-7. Google Scholar
D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity,, Proc. Roy. Soc. Edinburgh, 134 (2004), 201. doi: 10.1017/S0308210500003164. Google Scholar
Z. Wang, C. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators,, J. Nanjing Univ. Math. Biquart., 27 (2010), 17. doi: 10.3969/j.issn.0469-5097.2010.01.003. Google Scholar
R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem,, Proc. London. Math. Soc., 79 (1999), 381. doi: 10.1112/S0024611599012034. Google Scholar
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,, Commun. Math. Phys., 143 (1991), 43. doi: 10.1007/BF02100285. Google Scholar
B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142. doi: 10.1006/jdeq.1998.3553. Google Scholar
L. Jiao, D. Piao and Y. Wang, Boundedness for the general semilinear Duffing equation via the twist theorem,, J. Differential Equations, 252 (2012), 91. doi: 10.1016/j.jde.2011.09.019. Google Scholar
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. wiss, Kl. (1962), 1. Google Scholar
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes Math., 1007 (1983), 677. doi: 10.1007/BFb0061441. Google Scholar
Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017
Huiping Jin. Boundedness in a class of duffing equations with oscillating potentials via the twist theorem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 179-192. doi: 10.3934/cpaa.2011.10.179
Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911
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Alessandro Fortunati, Stephen Wiggins. Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1109-1118. doi: 10.3934/dcdss.2016044
Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations & Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015
Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240
Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053
Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1
Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969
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Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
Rehana Naz, Imran Naeem. Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020122
Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control & Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143
Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041
Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068
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Daxiong Piao Xiang Sun | CommonCrawl |
Springer for Research & Development
Time-domain boundary integral equation modeling of heat transmission problems
Tianyu Qiu1,
Alexander Rieder2,
Francisco-Javier Sayas1 &
Shougui Zhang3
Numerische Mathematik volume 143, pages223–259(2019)Cite this article
This paper investigates the numerical modeling of a time-dependent heat transmission problem by the convolution quadrature boundary element method. It introduces the latest theoretical development into the error analysis of the numerical scheme. Semigroup theory is applied to obtain stability in the spatially semidiscrete scheme. Functional calculus is employed to yield convergence in the fully discrete scheme. We compare the results to a more traditional Laplace domain approach.
Since the inception of the boundary integral equation method, the thermal engineering community has been exploiting its potential in solving transient heat conduction problems [47]. The method is also a popular choice among environmental scientists in the study of pollutant transport problem [26]. Recent applications include photothermal spectroscopy [38] and diffusion in variable media [1, 2]. The method enjoys sustained interest due to its remarkably simple way to handle problems on infinite domains.
Various schemes have emerged to discretize time domain boundary integral equations associated to parabolic problems. The theoretical basis for much of this work originated with [4, 15] and focused on the development of Galerkin discretization of the basic integral equations for problems on the exterior of a bounded domain. While we will focus on a particular class of methods (Galerkin in space, Convolution Quadrature in time), let us mention that there is a large literature on other families of schemes applied to exterior problems for the heat equation [19, 32, 33, 43, 44].
Our context is that of a Convolution Quadrature Boundary Element Method applied to transmission problems for the heat equation. Methods combining CQ and Galerkin BEM for heat diffusion problems appeared first in [29] (a combination of the ideas of CQ first set in [27], with the then recent results on the single layer operator for the heat equation) and then reinterpreted as a Rothe-type method in [14]. From the point of view of the algorithm itself, we here combine a Costabel-Stephan formulation for transmission problems [16, 36], with Galerkin semidiscretization in space and multistep or multistage CQ in time [7, 27, 28]. We set our main goals in the proof of stability and convergence of the method avoiding Laplace domain estimates (more on this later), using instead techniques of evolutionary equations associated to infinitesimal generators of strongly continuous analytic semigroups in certain Hilbert spaces. We obtain three kinds of results: (a) long term stability of the system after semidiscretization in space; (b) optimal order of convergence in time (with explicitly expressed behavior of the bounds with respect to the time variable) for BDF-based CQ schemes; (c) reduced (but higher than stage) order of convergence for RK-based CQ schemes.
Let us next try to clarify what kind of mathematical techniques are used for our analysis and where the novelty of this work lies. To do that, we first need to discuss the mathematical background for the field. The modern analysis for time domain boundary integral equations (TDBIE) traces its roots back to the seminal paper [6], where the bounds for boundary integral operators and layer potentials of the Helmholtz equation are derived by carrying out inversion using a Plancherel formula in anisotropic Sobolev spaces. In the regime of parabolic equations, the paper [29] converts the Laplace domain results to the time domain by bounding the inverse Laplace transform with pseudodifferential calculus, a theoretical tool not available for non-smooth domain problems. As already mentioned, [4, 15] contain the seeds of a space-and-time coercivity analysis for the thermal boundary integral operators using anisotropic Sobolev spaces, while [29] adopts a separate strategy for the space and time variables, focusing on Sobolev regularity in space and Hölder regularity in time. Working on a different parabolic problem (Stokes flow around a moving obstacle), [5] gave an alternative proof of the bounds applicable to non-smooth boundaries, improving past results by revealing how all constants depend on time. Note that the usual approach was the study of mapping properties for the TDBIE and its inverse operator and for the associated potentials. Some kind of Laplace-domain coercivity was used to justify space or space-and-time Galerkin discretization. Instead, the work of [24] understood Galerkin semidiscretization in space as part of the problem set-up and its effects were analyzed as a continuous problem, instead of as a discretization of an existing continuous problem. The time-domain translation of this approach (using any variant of Laplace inversion or Payley–Wiener estimates) typically yields estimates that are suboptimal, due to the passage through the Laplace domain. (This effect can be seen for the TDBIE associated to the wave equation in [40]).
What we do in this paper is related to the purely time-domain analysis of TDBIE initiated in [17] for wave propagation problems. This theory has seen different extensions and refinements: for instance, [37] extends the results to the Maxwell equations and [22] is the realization that a first order in time formulation makes the analysis much simpler. The goal of exploring purely time-domain techniques is multiple. First of all, in comparison with the Laplace domain approach, and when compared on the same type of bounds (the Laplace domain also provides estimates in weighted Sobolev norms that are not available with other techniques), time-domain estimates provide: (a) lower needs for the regularity of the input data to obtain the same estimates (i.e., refined mapping properties); (b) better bounds for the estimates as time grows. The time-domain analysis also emphasizes that a dynamical process is occurring through the entire discretization problem, a process that is hidden when we focus on transfer function estimates directly attached to the boundary integral operators.
The previous two paragraphs dealt with integral formulations, mapping properties, and semidiscretization in time. We now turn our attention to Convolution Quadrature. The multistep version of CQ, considered as a method to approximate causal convolutions and convolution equations, originated in [27]. A multistage version of the method was derived only a couple of years later in [28]. CQ techniques are now widely used in the realm of TDBIE, especially for wave propagation phenomena [8, 25, 34], but they are also useful in the context of TDBIE for parabolic problems [5, 29]. The Laplace domain analysis of CQ has a black-box nature that makes it very attractive: it deals with general families of operators as long as their Laplace transforms (transfer functions) satisfy certain properties. However, as already observed in the seminal work of Lubich, the CQ process applied to TDBIE can be rewritten as the application of a background ODE solver to the associated PDE in the exterior domain. In fact, rewriting the CQ process as a time-stepping procedure expressed through \(\zeta \)-transforms puts into evidence the fact that we are approximating a non-standard evolutionary PDE with non-homogeneous boundary conditions using an ODE solver. Along those lines, this paper offers two non-trivial contributions. First of all, we use functional calculus techniques and classical analysis of BDF methods [45] to show a direct-in-time analysis of BDF–CQ methods applied to the semidiscrete system of TDBIE. Second, we borrow heavily on difficult results by Alonso and Palencia [3] to offer an analysis of RK–CQ methods applied to the same problem. While we can prove estimates that improve the basic stage order (which is what a naive approach would give), our numerical experiments will show that we are still slightly suboptimal and some additional work is needed. We also show that these improved rates are what a Laplace domain analysis [9, 10, 28] would yield, although with less insight into the long-time behavior.
The paper is organized as follows. Section 2 introduces the time domain boundary integral equation formulation (TDBIE) for the heat equation transmission problem. Section 3 proves the stability and convergence of the Galerkin-semidiscretization-in-space scheme. Sections 4 and 5 prove the convergence of BDFCQ and RKCQ in respectively. Finally, Sect. 6 provides several numerical experiments. An appendix presents some needed background material to ease readability.
Notation For Banach spaces X and Y, \({\mathscr {B}}(X,Y)\) will be used to denote the space of bounded linear operators from X to Y. We use standard function space notations: \({\mathscr {C}}^k(I;X)\) for the space of k times continuously differentiable functions of a real variable in the interval I with values in the Banach space X, \(L^2({\mathscr {O}})\) for the space of square integrable functions on a domain \(\mathscr {O}\), and the Sobolev spaces
$$\begin{aligned} H^1({\mathscr {O}})&:= \{ f\in L^2({\mathscr {O}}) \,:\, \nabla f\in L^2({\mathscr {O}})^d\}, \\ H_\varDelta ^1({\mathscr {O}})&:= \{ f\in H^1({\mathscr {O}})\,:\, \varDelta f \in L^2({\mathscr {O}})\}. \end{aligned}$$
If \(\varGamma \) is the boundary of a Lipschitz domain, \(H^{1/2}_\varGamma \) will be the trace space, \(H^{-1/2}_\varGamma \) its dual, and \(\langle \cdot , \cdot \rangle _\varGamma \) will denote the duality product of \(H_\varGamma ^{-1/2}\times H_\varGamma ^{1/2}\). We will use the following convention
$$\begin{aligned} \Vert \cdot \Vert _{{\mathscr {O}}}, \quad \Vert \cdot \Vert _{1,{\mathscr {O}}}, \quad \Vert \cdot \Vert _{1/2,\varGamma }, \quad \Vert \cdot \Vert _{-1/2,\varGamma }, \end{aligned}$$
for the norms of the spaces \(L^2({\mathscr {O}})\), \(H^1({\mathscr {O}})\), \(H^{1/2}_\varGamma \) and \(H^{-1/2}_\varGamma \) respectively. We will not have a special notation for the natural norm of \(H^1_\varDelta (\mathscr {O})\). We will use the same notation (1) for the norms on Cartesian products of several copies of the same spaces. Finally, we will denote \({\mathbb {R}}_+:=[0,\infty )\).
Model problem and TDBIE formulation
We are concerned with a transmission problem for the heat equation in free space in presence of a single homogeneous inclusion. Both the inclusion and the free space medium possess homogeneous and isotropic thermal transmission properties, characterized by two positive constants: \(\kappa \) as the thermal conductivity and \(\rho \) as the density scaled by heat capacity. Let \(\varOmega _-\subset {{\mathbb {R}}^d}(d=2,3)\) be a bounded Lipschitz domain with boundary \(\varGamma \). The normal vector field \(\nu :\varGamma \rightarrow {{\mathbb {R}}^d}\) is defined almost everywhere on the boundary, pointing from the interior \(\varOmega _-\) to the exterior domain \(\varOmega _+:={{\mathbb {R}}^d}{\setminus } \overline{\varOmega _-}\). We can thus define two trace operators \(\gamma ^\pm :H^1({{\mathbb {R}}^d{\setminus }\varGamma })\rightarrow H^{1/2}_\varGamma \), two normal derivative operators \(\partial _\nu ^\pm :H^1_\varDelta ({{\mathbb {R}}^d{\setminus }\varGamma })\rightarrow H^{-1/2}_\varGamma \) and the jumps
$$\begin{aligned}{}[\![ \gamma u ]\!]:=\gamma ^-u-\gamma ^+u, \qquad [\![ \partial _\nu u ]\!]:=\partial _\nu ^-u-\partial _\nu ^+u. \end{aligned}$$
Given \(\beta _0:[0,\infty )\rightarrow H^{1/2}_\varGamma \), and \(\beta _1:[0,\infty )\rightarrow H^{-1/2}_\varGamma \), we look for \(u:[0,\infty ) \rightarrow H_\varDelta ^1({\mathbb {R}}^d\backslash \varGamma )\) satisfying
$$\begin{aligned} \rho {\dot{u}} (t)&= \kappa \varDelta u (t) \qquad \text { in } \varOmega _-, \qquad \forall t\ge 0, \end{aligned}$$
$$\begin{aligned} {\dot{u}}(t)&= \varDelta u(t) \qquad \,\,\,\text { in } \varOmega _+, \qquad \forall t\ge 0, \end{aligned}$$
$$\begin{aligned} \gamma ^- u(t) - \gamma ^+ u(t)&= \beta _0(t) \qquad \quad \text { on } \varGamma , \qquad \,\, \forall t \ge 0, \end{aligned}$$
$$\begin{aligned} \kappa \partial _\nu ^- u (t) - \partial _\nu ^+ u(t)&=\beta _1(t) \qquad \quad \text { on } \varGamma , \qquad \,\,\forall t \ge 0, \end{aligned}$$
(2d)
$$\begin{aligned} u(0)&= 0, \end{aligned}$$
(2e)
where upper dots denote differentiation in time. A non-zero initial condition can be handled by letting it diffuse in free space and changing the transmission conditions. Note that TDBIE cannot deal with non-vanishing initial conditions, since they involve delayed potentials. Physically speaking, the current model could be one where the medium is kept at fixed temperature (we measure the varation of temperature) and heat sources are activated at time zero.
We next give a crash course (based on [40]) on the few ingredients that are needed to have a rigorous setting for the weak definition of the heat boundary integral operators applied in the sense of distributions. Let \(\mathrm F:{\mathbb {C}}_+:=\{s\in \mathbb C\,:\,\mathrm {Re}\,s>0\} \rightarrow X\) be an analytic function such that
$$\begin{aligned} \Vert \mathrm F(s)\Vert _{X}\le C_{\mathrm F}(\mathrm {Re} \,s) |s|^\mu \quad \forall s\in {\mathbb {C}}_+, \end{aligned}$$
where \(C_{\mathrm F}:(0,\infty )\rightarrow (0,\infty )\) is non-increasing and is allowed to blow-up as a rational function at the origin, i.e., there exists a constant \(C>0\) and \(\ell \ge 0\) such that \(C_{\mathrm F}(\sigma )\le C\sigma ^{-\ell }\) when \(\sigma \rightarrow 0\). It is then possible to prove [40, Chapter 3] that there exists an X-valued causal tempered distribution f whose Laplace transform is \(\mathrm F\), i.e., \({\mathscr {L}}\{ f\}=\mathrm F\). The precise set of distributions whose Laplace transforms satisfy the above conditions is described in [40, Chapter 3] and denoted \(\mathrm {TD}(X)\). These Laplace transforms (symbols, or transfer functions) include the ones that appear in parabolic problems (see [5]) where now \(\mathrm F\) is well defined and analytic in \({\mathbb {C}}_\star :={\mathbb {C}}{\setminus } (-\infty ,0]\) and satisfies
$$\begin{aligned} \Vert \mathrm F(s)\Vert _{X}\le D_{\mathrm F}(\mathrm {Re} \,s^{1/2}) |s|^\mu \quad \forall s\in {\mathbb {C}}_\star , \end{aligned}$$
where \(D_{\mathrm F}:(0,\infty )\rightarrow (0,\infty )\) has the same properties as \(C_{\mathrm F}\) above. Since
$$\begin{aligned} \min \{ 1,\mathrm {Re}\, s^{1/2}\} \ge \min \{1,\mathrm {Re}\,s\} \qquad \forall s\in {\mathbb {C}}_+, \end{aligned}$$
a symbol satisfying (4) satisfies (3) with \(C_{\mathrm F}(\sigma ):=D_{\mathrm F}(\min \{1,\sigma \})\). Therefore, if \(\mathrm F\) satisfies (4), it is the Laplace transform of a causal distribution.
Let \(s\in {\mathbb {C}}_\star \), \(\phi \in H^{1/2}_\varGamma \), and \(\lambda \in H^{-1/2}_\varGamma \). The transmission problem
$$\begin{aligned} U\in H^1({{\mathbb {R}}^d{\setminus }\varGamma }), \qquad&\varDelta U-s U=0 \qquad \hbox { in}\ {{\mathbb {R}}^d{\setminus }\varGamma },\\ [\![ \gamma U ]\!]=\phi , \qquad&[\![ \partial _\nu U ]\!]=\lambda ,&\end{aligned}$$
$$\begin{aligned}&U\in H^1({{\mathbb {R}}^d{\setminus }\varGamma }), \quad [\![ \gamma U ]\!] = \phi , \end{aligned}$$
$$\begin{aligned}&(\nabla U,\nabla V)_{{{\mathbb {R}}^d{\setminus }\varGamma }} + s(U,V)_{{{\mathbb {R}}^d}} = \langle \lambda ,\gamma V\rangle _\varGamma \quad \forall V\in H^1({{\mathbb {R}}^d}), \end{aligned}$$
and therefore admits a unique solution. To see that, note that the associated bilinear form is coercive as
$$\begin{aligned} \mathrm {Re}\left( {{\overline{s}}}^{1/2} \left( \Vert \nabla U\Vert _{{{\mathbb {R}}^d{\setminus }\varGamma }}^2+ s \Vert U\Vert _{{{\mathbb {R}}^d{\setminus }\varGamma }}^2\right) \right) =(\mathrm {Re}\, s^{1/2}) \left( \Vert \nabla U\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }^2+ |s| \Vert U\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }^2\right) . \end{aligned}$$
This solution of (5) can be expressed using two \(s-\)dependent bounded operators acting on the data:
$$\begin{aligned} U={\mathrm S}(s)\lambda -{\mathrm D}(s)\phi . \end{aligned}$$
This gives a simultaneous variational definition of the single and double layer heat potentials in the Laplace domain. By definition,
$$\begin{aligned}{}[\![ \gamma ]\!]{\mathrm S}(s)=0, \qquad [\![ \partial _\nu ]\!]{\mathrm S}(s) = {\mathrm I}, \qquad [\![ \gamma ]\!]{\mathrm D}(s)=-{\mathrm I}, \qquad [\![ \partial _\nu ]\!]{\mathrm D}(s)=0. \end{aligned}$$
We can now define the four associated boundary integral operators by taking averages of the traces and normal derivatives of the single and double layer potentials:
$$\begin{aligned} \begin{array}{ll@{\qquad }l} &{} \quad {\mathrm V}(s):=\gamma ^\pm {\mathrm S}(s), &{}\quad \quad {\mathrm K}(s):=\tfrac{1}{2}(\gamma ^-+\gamma ^+){\mathrm D}(s),\\ &{} {\mathrm K}^T(s):=\tfrac{1}{2}(\partial _\nu ^-+\partial _\nu ^+){\mathrm S}(s), &{}\quad \quad {\mathrm W}(s):=-\partial _\nu ^\pm {\mathrm D}(s). \end{array} \end{aligned}$$
Once again by definition, the following limit relations hold:
$$\begin{aligned} \partial _\nu ^\pm {\mathrm S}(s)=\mp \tfrac{1}{2}{\mathrm I}+{\mathrm K}^T(s), \qquad \gamma ^\pm {\mathrm D}(s)=\pm \tfrac{1}{2}{\mathrm I}+{\mathrm K}(s). \end{aligned}$$
Theorem 1
For \(s\in {\mathbb {C}}_+\), denote \(\sigma :=\mathrm {Re}\,s^{1/2}>0\) and \({{\underline{\sigma }}}:=\min \{1,\sigma \}\). There exists a constant C only depending on the boundary \(\varGamma \) for all \(s\in \mathbb C_\star \) such that
$$\begin{aligned} \Vert {\mathrm S}(s)\Vert _{H^{-1/2}_\varGamma \rightarrow H^1({{\mathbb {R}}^d})}&\le C \frac{|s|^{1/2}}{\sigma {{\underline{\sigma }}}^2}, \quad \Vert {\mathrm D}(s)\Vert _{H^{1/2}_\varGamma \rightarrow H^1({{\mathbb {R}}^d{\setminus }\varGamma })} \le C\frac{|s|^{3/4}}{\sigma {{\underline{\sigma }}}^{3/2}},\\ \Vert {\mathrm V}(s)\Vert _{H^{-1/2}_\varGamma \rightarrow H^{1/2}_\varGamma }&\le C \frac{|s|^{1/2}}{\sigma {{\underline{\sigma }}}^2}, \quad \,\, \Vert {\mathrm K}^T(s)\Vert _{H^{-1/2}_\varGamma \rightarrow H^{-1/2}_\varGamma } \le C\frac{|s|^{3/4}}{\sigma {{\underline{\sigma }}}^{3/2}},\\ \Vert {\mathrm K}(s)\Vert _{H^{1/2}_\varGamma \rightarrow H^{1/2}_\varGamma }&\le C\frac{|s|^{3/4}}{\sigma {{\underline{\sigma }}}^{3/2}}, \qquad \Vert {\mathrm W}(s)\Vert _{H^{1/2}_\varGamma \rightarrow H^{-1/2}_\varGamma } \le C\frac{|s|}{\sigma {{\underline{\sigma }}}}. \end{aligned}$$
Since we can write the heat equation in Laplace domain as \(\varDelta u-(s^{1/2})^2 u=0\) for \(s^{1/2}\in {\mathbb {C}}_+\), i.e., \(s\in {\mathbb {C}}_\star \), s in the estimates in [24, Table 1] can be replaced by \(s^{1/2}\). \(\square \)
Applying the results of [40, Chapter 3], we can define the operator-valued distributions in the time domain through the inverse Laplace transform, using an inverse diffusivity parameter \(m>0\)
$$\begin{aligned} {\mathscr {S}}_m&:={\mathscr {L}}^{-1}\{{\mathrm S}(\cdot /m) \} \in \mathrm {TD}(\mathscr {B}(H^{-1/2}_\varGamma ,H^1({{\mathbb {R}}^d}))),\\ {\mathscr {D}}_m&:={\mathscr {L}}^{-1}\{{\mathrm D}(\cdot /m) \} \in \mathrm {TD}(\mathscr {B}(H^{1/2}_\varGamma ,H^1({{\mathbb {R}}^d{\setminus }\varGamma }))),\\ {\mathscr {V}}_m&:={\mathscr {L}}^{-1}\{{\mathrm V}(\cdot /m) \} \in \mathrm {TD}(\mathscr {B}(H^{-1/2}_\varGamma ,H^{1/2}_\varGamma )), \\ {\mathscr {K}}_m&:={\mathscr {L}}^{-1}\{{\mathrm K}(\cdot /m) \} \in \mathrm {TD}(\mathscr {B}(H^{1/2}_\varGamma ,H^{1/2}_\varGamma )),\\ {\mathscr {K}}^T_m&:={\mathscr {L}}^{-1}\{{\mathrm K}^T(\cdot /m) \} \in \mathrm {TD}({\mathscr {B}}(H^{-1/2}_\varGamma ,H^{-1/2}_\varGamma )), \\ {\mathscr {W}}_m&:={\mathscr {L}}^{-1}\{{\mathrm W}(\cdot /m) \} \in \mathrm {TD}(\mathscr {B}(H^{1/2}_\varGamma ,H^{-1/2}_\varGamma )). \end{aligned}$$
When \(m=1\), the subscript will be omitted. As is well known, convolutions in time correspond to multiplications in the Laplace domain. For instance, if the Laplace transform of \(\lambda \in \mathrm {TD}(H^{-1/2}_\varGamma )\) is \(\varLambda ={\mathscr {L}}\{ \lambda \}\), then \({\mathscr {L}}({\mathscr {S}}_m*\lambda )={\mathrm S}(s/m)\varLambda (s)\) and \(\mathscr {S}_m* \lambda \in \mathrm {TD}(H^1({{\mathbb {R}}^d}))\). The convolution operator \(\lambda \mapsto {\mathscr {S}}_m * \lambda \) is the heat single layer potential. More details about the distributional convolution can be found in [40, Section 3.2], with a more general theory given in [46]. The next theorem is the Green's representation theorem for the heat equation, which is a consequence of the analogous result in the Laplace domain.
Given \(\phi \in \mathrm {TD}(H^{1/2}_\varGamma )\) and \(\lambda \in \mathrm {TD}(H^{-1/2}_\varGamma )\), \(u={\mathscr {S}}_m*\lambda -{\mathscr {D}}_m*\phi \) is the unique solution to the problem
$$\begin{aligned}&u\in \mathrm {TD}(H^1_\varDelta ({\mathbb {R}}^d{\setminus }\varGamma )) \quad {\dot{u}}=m\varDelta u, \end{aligned}$$
$$\begin{aligned}&[\![ \gamma u ]\!]=\phi , \quad [\![ \partial _\nu u ]\!]=\lambda . \end{aligned}$$
Even though we will not need them for our numerical scheme, we now write down the explicit expression for the heat layer potentials. The time domain fundamental solution of the heat equation is
$$\begin{aligned} {\mathscr {G}}_m(\mathbf {x},t):= (4\pi mt)^{-d/2} \exp \left( -{|\mathbf {x}|^2\over 4mt}\right) \chi _{(0,\infty )} (t). \end{aligned}$$
The single layer potential is then given by
$$\begin{aligned} ({\mathscr {S}}_m* \lambda ) (\mathbf {x},t):= \int _0^t\int _\varGamma \mathscr {G}_m(\mathbf {x}-\mathbf {y},t-\tau )\lambda (\mathbf {y},\tau )\,\mathrm {d} \varGamma _{\mathbf {y}}\,\mathrm {d} \tau \end{aligned}$$
while the integral form of the double layer potential is
$$\begin{aligned} ({\mathscr {D}}_m* \phi ) (\mathbf {x},t) := \int _0^t\int _\varGamma {\partial \over \partial \nu _{\mathbf {y}}} {\mathscr {G}}_m(\mathbf {x}-\mathbf {y},t-\tau )\phi (\mathbf {y},\tau )\,\mathrm {d} \varGamma _{\mathbf {y}}\,\mathrm {d} \tau . \end{aligned}$$
These integral operators are well defined for smooth enough densities \(\lambda \) and \(\phi \) and they coincide with the distributional defintions. Precise mapping properties in anisotropic space–time Sobolev spaces are given in the fundamental work of Martin Costabel [15].
The transmission problem in the sense of distributions is a weak version of (2). The data are now \(\beta _0\in \mathrm {TD}(H^{1/2}_\varGamma )\) and \(\beta _1\in \mathrm {TD}(H^{-1/2}_\varGamma )\) and we look for \(u\in \mathrm {TD}(H^1_\varDelta ({\mathbb {R}}^d{\setminus }\varGamma )\) satisfying
$$\begin{aligned} \rho {\dot{u}}&= \kappa \varDelta u, \qquad (\text {in } L^2(\varOmega _-)), \end{aligned}$$
$$\begin{aligned} {\dot{u}}&= \varDelta u, \quad \quad \,\,\, (\text {in } L^2(\varOmega _+)), \end{aligned}$$
$$\begin{aligned} \gamma ^- u - \gamma ^+ u&=\beta _0, \qquad \quad (\text {in } H^{1/2}_\varGamma ), \end{aligned}$$
$$\begin{aligned} \kappa \partial _\nu ^- u - \partial _\nu ^+ u&=\beta _1 ,\qquad \quad (\text {in } H^{-1/2}_\varGamma ). \end{aligned}$$
The upper dot is now the distributional differentiation with respect to the time variable. The bracket in the right-hand sides of the equations in (7) clarifies where the equations hold. For instance, when we say \(\rho \dot{u}=\kappa \varDelta u\) in \(L^2(\varOmega _-)\), we mean that both sides of the equation are equal as \(L^2(\varOmega _-)\)-valued distributions. A rigorous understanding of such an equation requires the elementary but careful use of steady-state operators, like distributional differentiation in the space variables or the restriction of a function to a subdomain.
An integral system We finally derive a system of time domain boundary integral equations (TDBIE) that is equivalent to the transmission problem (7). This follows exactly the same pattern as the work of Costabel and Stephan for steady-state (or time-harmonic) problems [16], recently extended to transmission problems for the wave equation [36]. Since the ideas are exactly the same as in those references, we will just sketch the process. We first choose the interior trace and normal derivative of u from (7) as unknowns
$$\begin{aligned} \phi := \gamma ^- u,\qquad \lambda :=\partial _\nu ^- u. \end{aligned}$$
We then define two scalar fields
$$\begin{aligned} u_-&:= {\mathscr {S}}_m * \lambda - {\mathscr {D}}_m * \phi , \quad \text{ with } m:=\rho ^{-1}\kappa , \end{aligned}$$
$$\begin{aligned} u_+&:=- {\mathscr {S}} * (\kappa \lambda - \beta _1)+ {\mathscr {D}} * (\phi -\beta _0), \end{aligned}$$
each of them defined on both sides of the boundary, and related to the solution of (7) by
$$\begin{aligned} u_- = u\chi _{\varOmega _-}, \qquad u_+ = u\chi _{\varOmega _+}, \end{aligned}$$
where the symbol \(\chi _{{\mathscr {O}}}\) is used to the denote the characteristic function of the set \({\mathscr {O}}\). In theory, this doubles the number of unknowns of the problem, even if we know that \(u_-\) and \(u_+\) vanish identically in \(\varOmega _+\) and \(\varOmega _-\) respectively. Later on, it will be clear that the doubling of unknowns is a natural byproduct of semidiscretization in space. The solution of (7) can be reconstructed as \(u=u_-+u_+\), where \(u_-,u_+\in \mathrm {TD}(H^1_\varDelta ({\mathbb {R}}^d{\setminus }\varGamma ))\) satisfy
$$\begin{aligned} \rho {\dot{u}}_-&= \kappa \varDelta u_-, \qquad (\text {in } L^2({{\mathbb {R}}^d{\setminus }\varGamma })), \end{aligned}$$
(10a)
$$\begin{aligned} {\dot{u}}_+&= \varDelta u_+, \qquad \quad (\text {in } L^2({{\mathbb {R}}^d{\setminus }\varGamma })), \end{aligned}$$
(10b)
$$\begin{aligned} \gamma ^+ u_- - \gamma ^- u_+&=0, \qquad \qquad \quad (\text {in } H^{1/2}_\varGamma ), \end{aligned}$$
(10c)
$$\begin{aligned} \kappa \partial _\nu ^+ u_- - \partial _\nu ^- u_+&=0, \qquad \quad \qquad (\text {in } H^{-1/2}_\varGamma ), \end{aligned}$$
(10d)
$$\begin{aligned}{}[\![ \gamma u_- ]\!] + [\![ \gamma u_+ ]\!]&=\beta _0, \qquad \, \qquad (\text {in } H^{1/2}_\varGamma ), \end{aligned}$$
(10e)
$$\begin{aligned} \kappa [\![ \partial _\nu u_- ]\!] + [\![ \partial _\nu u_+ ]\!]&=\beta _1, \qquad \,\qquad (\text {in } H^{-1/2}_\varGamma ). \end{aligned}$$
(10f)
If we now substitute the representation formula (9) in (10c)–(10d), it follows that \(\lambda \) and \(\phi \) satisfy
$$\begin{aligned} \left[ \begin{array}{cc} {\mathscr {V}}_{m}+\kappa {\mathscr {V}} &{}\quad - {\mathscr {K}}_{m} - {\mathscr {K}} \\ {\mathscr {K}}^T_{m}+ {\mathscr {K}}^T &{}\quad {\mathscr {W}}_{m} +\frac{1}{\kappa } {\mathscr {W}} \end{array}\right] * \left[ \begin{array}{c} \lambda \\ \phi \end{array}\right] = \frac{1}{2}\left[ \begin{array}{c} \beta _0 \\ \frac{1}{\kappa } \beta _1 \end{array}\right] + \left[ \begin{array}{cc} {\mathscr {V}} &{}\quad - {\mathscr {K}} \\ \frac{1}{\kappa } {\mathscr {K}}^T &{}\quad \frac{1}{\kappa } {\mathscr {W}} \end{array}\right] * \left[ \begin{array}{c} \beta _1 \\ \beta _0 \end{array}\right] .\nonumber \\ \end{aligned}$$
We summarize the relations between the boundary integral equations and the partial differential equation in the following proposition. Its proof follows from elementary arguments using the jump relations of potentials and the definitions of the associate boundary integral operators.
Assume that \((\lambda ,\phi )\in \mathrm {TD}(H^{-1/2}_\varGamma \times H^{1/2}_\varGamma )\) is a solution of (11) and define \((u_-,u_+)\in \mathrm {TD}(H^1_\varDelta ({\mathbb {R}}^d{\setminus }\varGamma )^2)\) by (9). The pair \((u_-,u_+)\) then is a solution to (10). Additionally \(u:=u_-\chi _{\varOmega _-} + u_+\chi _{\varOmega _+}\in \mathrm {TD}(H^1_\varDelta ({\mathbb {R}}^d{\setminus }\varGamma ))\) is the solution to (7). Reciprocally, if \(u\in \mathrm {TD}(H^1_\varDelta ({\mathbb {R}}^d{\setminus }\varGamma ))\) solves (7), then the pair \((\lambda ,\phi )\) defined by (8) is a solution of (11).
Galerkin semidiscretization in space
In this section we address the semidiscretization in space of the system of TDBIE (11), using a completely general Galerkin scheme, and the postprocessing of the boundary unknowns using the potential expressions (9).
The semidiscrete problem
We start by choosing a pair of finite dimensional subspaces \(X_h\subset H^{-1/2}_\varGamma , Y_h\subset H^{1/2}_\varGamma \). (Note that following [24] we will only need \(X_h\) and \(Y_h\) to be closed.) Their respective polar sets are
$$\begin{aligned} X_h^\circ&:= \{ \phi \in H^{1/2}_\varGamma : \langle \mu ^h, \phi \rangle _\varGamma =0\quad \forall \mu ^h \in X_h \}, \\ Y_h^\circ&:= \{ \lambda \in H^{-1/2}_\varGamma : \langle \lambda , \varphi ^h\rangle _\varGamma =0\quad \forall \varphi ^h \in Y_h \}. \end{aligned}$$
The semidiscrete method looks first for \(\lambda ^h \in \mathrm {TD}(X_h), \phi ^h \in \mathrm {TD}(Y_h)\) satisfying a weakly-tested version of equations (11)
$$\begin{aligned} \left[ \begin{array}{cc} {\mathscr {V}}_{m}+\kappa {\mathscr {V}} &{}\quad - {\mathscr {K}}_{m} - {\mathscr {K}} \\ {\mathscr {K}}_{m}^T+ {\mathscr {K}}^T &{}\quad {\mathscr {W}}_{m} +\frac{1}{\kappa } {\mathscr {W}} \end{array}\right] * \left[ \begin{array}{c} \lambda ^h \\ \phi ^h \end{array}\right] - \frac{1}{2}\left[ \begin{array}{c} \beta _0 \\ \frac{1}{\kappa } \beta _1 \end{array}\right] - \left[ \begin{array}{c@{\qquad }c} {\mathscr {V}} &{}\quad - {\mathscr {K}} \\ \frac{1}{\kappa } {\mathscr {K}}^T &{}\quad \frac{1}{\kappa } {\mathscr {W}} \end{array}\right] * \left[ \begin{array}{c} \beta _1 \\ \beta _0 \end{array}\right] \in X_h^\circ \times Y_h^\circ . \end{aligned}$$
The expression (12) is a compacted form of the Galerkin equations: when we write that the residual of the equations is in \(X_h^\circ \times Y_h^\circ \), we are equivalently requiring the residual to vanish when tested by elements of \(X_h\times Y_h\). Once the boundary unknowns have been computed, the potential representation
$$\begin{aligned} u_-^h = {\mathscr {S}}_m * \lambda ^h - {\mathscr {D}}_m * \phi ^h, \quad u_+^h =- {\mathscr {S}} * (\kappa \lambda ^h - \beta _1)+ {\mathscr {D}} * (\phi ^h -\beta _0), \end{aligned}$$
yields two fields defined on both sides of \(\varGamma \) and satisfying the corresponding heat equations.
If we subtract (12) by (11), we obtain the system satisfied by the error of unknown densities on the boundary
$$\begin{aligned} \left[ \begin{array}{c@{\qquad }c} {\mathscr {V}}_{m}+\kappa {\mathscr {V}} &{}\quad - {\mathscr {K}}_{m} - {\mathscr {K}} \\ {\mathscr {K}}_{m}^T+ {\mathscr {K}}^T &{}\quad {\mathscr {W}}_{m} +\frac{1}{\kappa } {\mathscr {W}} \end{array}\right] * \left[ \begin{array}{c} \lambda ^h-\lambda \\ \phi ^h-\phi \end{array}\right] \in X_h^\circ \times Y_h^\circ . \end{aligned}$$
The error corresponding to the posprocessed fields is easily derived by subtracting (9) from (13),
$$\begin{aligned} e_-^h&:= u_-^h - u_- = {\mathscr {S}}_m * (\lambda ^h-\lambda ) - {\mathscr {D}}_m * (\phi ^h-\phi ), \end{aligned}$$
$$\begin{aligned} e_+^h&:= u_+^h - u_+ = - {\mathscr {S}} * \kappa ( \lambda ^h - \lambda )+ {\mathscr {D}} * (\phi ^h -\phi ). \end{aligned}$$
An exotic transmission problem A transmission problem will encompass the solution of the semidiscrete system (12)–(13) and the associated error system (14)–(15). The problem looks for \(w_-,w_+\in \mathrm {TD}(H^1_\varDelta ({\mathbb {R}}^d{\setminus }\varGamma ))\) such that
$$\begin{aligned}&\rho {\dot{w}}_- = \kappa \varDelta w_-, \quad {\dot{w}}_+ = \varDelta w_+, \end{aligned}$$
$$\begin{aligned}&[\![ \gamma w_- ]\!] + [\![ \gamma w_+ ]\!] =\beta _0, \quad \kappa [\![ \partial _\nu w_- ]\!] +[\![ \partial _\nu w_+ ]\!] =\beta _1, \end{aligned}$$
$$\begin{aligned}&\gamma ^+ w_- - \gamma ^- w_+ \in X_h^\circ , \quad \kappa \partial _\nu ^+ w_- - \partial _\nu ^- w_+ \in Y_h^\circ , \end{aligned}$$
$$\begin{aligned}&[\![ \gamma w_- ]\!] +\phi \in Y_h, \quad [\![ \partial _\nu w_- ]\!] +\lambda \in X_h. \end{aligned}$$
Equation (16a) takes place in \(L^2({{\mathbb {R}}^d{\setminus }\varGamma })\), while all six transmission conditions are equalities of \(H_\varGamma ^{\pm 1/2}\)-valued distributions. It can be shown by Theorem 2 that (12)–(13) is equivalent to the above system with \(\lambda =0,\phi =0\). On the other hand, if we set \(\beta _0=0,\beta _1=0\), then the spatial semidiscrete error \((e_-^h,e_+^h)\) of (15) is the solution to (16). The main results for this section require some additional functional language and will be given in Sect. 3.3, after we have embedded a stronger version of the distributional system (16) in a framework of evolutionary problems on a Hilbert space.
Functional framework
The handling of the double transmission problem (16) (with two fields defined on both sides of the interface) can be carried out with theory of differential equations associated to the infinitesimal generator of an analytic semigroup. Consider first the following spaces
$$\begin{aligned} \varvec{H}:=L^2({{\mathbb {R}}^d{\setminus }\varGamma })^2, \quad \varvec{V}:=H^1({{\mathbb {R}}^d{\setminus }\varGamma })^2, \quad \varvec{D}:=H^1_\varDelta ({{\mathbb {R}}^d{\setminus }\varGamma })^2. \end{aligned}$$
To separate components of the elements of these spaces we will write \(\varvec{w}=(w_-,w_+)\). Given a constant \(c\ne 0\) (we will need \(c\in \{\rho ,\rho ^{-1},\kappa \}\)) we will write \(\varvec{c}\varvec{w}:=(c w_-,w_+)\) for the associated multiplication operator acting on the first component of the vector. We will also consider the following bilinear forms
$$\begin{aligned} (\varvec{w},\varvec{w}')_{\varvec{H}}&:= (\varvec{\rho }\varvec{w},\varvec{w}')_{{\mathbb {R}}^d{\setminus }\varGamma }\,=(\rho w_-,w_-')_{{\mathbb {R}}^d{\setminus }\varGamma }+(w_+,w_+')_{{\mathbb {R}}^d{\setminus }\varGamma },\\ [\varvec{w},\varvec{w}']&:=(\varvec{\kappa }\nabla \varvec{w},\nabla \varvec{w}')_{{\mathbb {R}}^d{\setminus }\varGamma }\,=(\kappa \nabla w_-,\nabla w_-')_{{\mathbb {R}}^d{\setminus }\varGamma }+(\nabla w_+,\nabla w_+')_{{\mathbb {R}}^d{\setminus }\varGamma }, \end{aligned}$$
and four copies of the boundary spaces, equipped with their product duality pairing
$$\begin{aligned} \varvec{H}^{\pm 1/2}_\varGamma :=(H^{\pm 1/2}_\varGamma )^4, \qquad \langle \!\langle \varvec{n},\varvec{d}\rangle \!\rangle _\varGamma :=\sum _{i=1}^4 \langle n_i,d_i\rangle _\varGamma . \end{aligned}$$
The two-sided trace and normal derivative operators
$$\begin{aligned} \varvec{V} \ni \varvec{v} \longmapsto \varvec{\gamma }\varvec{v}&:=(\gamma ^-v_-,\gamma ^+v_-,\gamma ^-v_+,\gamma ^+v_+),\\ \varvec{D} \ni \varvec{v} \longmapsto \varvec{\partial }_\nu \varvec{v}&:=(\partial _\nu ^-v_-,\partial _\nu ^+v_-, \partial _\nu ^-v_+,\partial _\nu ^+v_+), \end{aligned}$$
are remixed to transmission operators
$$\begin{aligned} \varvec{\gamma }_D:=\varTheta _D\varvec{\gamma }:\varvec{V} \rightarrow \varvec{H}^{1/2}_\varGamma , \qquad \varvec{\gamma }_N:=\varTheta _N\varvec{\partial }_\nu :\varvec{D} \rightarrow \varvec{H}^{-1/2}_\varGamma , \end{aligned}$$
where the matrices
$$\begin{aligned} \varTheta _D:=\left[ \begin{array}{cccc} 1 &{}\quad -\,1 &{}\quad 1 &{}\quad -\,1 \\ 0 &{}\quad 1 &{}\quad -\,1 &{}\quad 0 \\ 1 &{}\quad -\,1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{array}\right] , \qquad \varTheta _N:=\left[ \begin{array}{cccc} 1 &{}\quad -\,1 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad -\,1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad -\,1 &{}\quad 0 \\ 1 &{}\quad -\,1 &{}\quad 1 &{}\quad -\,1 \end{array}\right] \end{aligned}$$
$$\begin{aligned} \varTheta _N^\top \varTheta _D=S:=\mathrm {diag}(1,-1,1,-1). \end{aligned}$$
Note that the first three components of \(\varvec{\gamma }_D\) and the last three components of \(\varvec{\gamma }_N\) appear in the transmission conditions of (16) and
$$\begin{aligned} \nonumber (\nabla \varvec{w},\nabla \varvec{v})_{{\mathbb {R}}^d{\setminus }\varGamma }+(\varDelta \varvec{w},\varvec{v})_{{\mathbb {R}}^d{\setminus }\varGamma }&=\langle \!\langle \varvec{\partial }_\nu \varvec{w},S\varvec{\gamma }\varvec{v}\rangle \!\rangle _\varGamma =\langle \!\langle \varTheta _N \varvec{\partial }_\nu \varvec{w},\varTheta _D\varvec{\gamma }\varvec{v}\rangle \!\rangle _\varGamma \nonumber \\&=\langle \!\langle \varvec{\gamma }_N\varvec{w},\varvec{\gamma }_D \varvec{v}\rangle \!\rangle _\varGamma \qquad \forall \varvec{w}\in \varvec{D}, \quad \varvec{v}\in \varvec{V}. \end{aligned}$$
This integration by parts formula can be understood as a different way of rephrasing [36, formula (4.7)]. We finally consider the operator
$$\begin{aligned} \varvec{D} \ni \varvec{w} \longmapsto A_\star \varvec{w}:=\varvec{\rho }^{-1}\varvec{\kappa }\varDelta \varvec{w}= (\rho ^{-1} \kappa \varDelta w_-,\varDelta w_+) \in \varvec{H}, \end{aligned}$$
and the spaces
$$\begin{aligned} \varvec{M}^{1/2}&:=\{ 0\} \times X_h^\circ \times Y_h \times H^{1/2}_\varGamma \qquad \subset \varvec{H}^{1/2}_\varGamma ,\end{aligned}$$
$$\begin{aligned} \varvec{M}^{-1/2}&:=H^{-1/2}_\varGamma \times X_h \times Y_h^\circ \times \{0\} \quad \subset \varvec{H}^{-1/2}_\varGamma , \end{aligned}$$
which are respective polar spaces. In the coming paragraphs we will study the following problem: we are given data
$$\begin{aligned} \varvec{\chi }_D:[0,\infty )\rightarrow \varvec{H}^{1/2}_\varGamma , \qquad \varvec{\chi }_N:[0,\infty )\rightarrow \varvec{H}^{-1/2}_\varGamma , \end{aligned}$$
and we look for \(\varvec{w}:[0,\infty )\rightarrow \varvec{D}\) satisfying
$$\begin{aligned} \dot{\varvec{w}}(t)&=A_\star \varvec{w}(t) \quad \,\,\, \forall t\ge 0, \end{aligned}$$
$$\begin{aligned} \varvec{\gamma }_D \varvec{w}(t)-\varvec{\chi }_D(t)&\in \varvec{M}^{1/2} \qquad \quad \forall t\ge 0, \end{aligned}$$
$$\begin{aligned} \varvec{\gamma }_N\varvec{\kappa }\varvec{w}(t)-\varvec{\chi }_N(t)&\in \varvec{M}^{-1/2} \qquad \, \forall t\ge 0,\end{aligned}$$
$$\begin{aligned} \varvec{w}(0)&=0. \end{aligned}$$
This is a strong form (restricted to the interval \([0,\infty )\) and with strong derivatives, instead of distributional ones) of (16) when we choose \(\varvec{\chi }_D=(\beta _0,0,-\phi ,0)\) and \(\varvec{\chi }_N=(0,-\kappa \lambda ,0,\beta _1)\). The last ingredient for our framework consists of two spaces
$$\begin{aligned} \varvec{V}_h&:=\{\varvec{v}\in \varvec{V}\,:\, \varvec{\gamma }_D \varvec{v}\in \varvec{M}^{1/2}\}, \end{aligned}$$
$$\begin{aligned} \varvec{D}_h&:=\{\varvec{w}\in \varvec{D}\,:\,\varvec{\gamma }_D \varvec{w}\in \varvec{M}^{1/2}, \varvec{\gamma }_N\varvec{\kappa }\varvec{w}\in \varvec{M}^{-1/2}\}, \end{aligned}$$
and the unbounded operator \(A:D(A)\rightarrow \varvec{H}\) given by \(A\varvec{w}:=A_\star \varvec{w}\), when \(\varvec{w}\in D(A):=\varvec{D}_h\). In some future arguments we will find it advantageous to collect the transmission conditions (20b)–(20c) in a single expression, using
$$\begin{aligned} {\mathscr {B}}\varvec{w}:=(\varvec{\gamma }_D\varvec{w},\varvec{\gamma }_N\varvec{\kappa }\varvec{w}), \qquad \varvec{\chi }:=(\varvec{\chi }_D,\varvec{\chi }_N), \qquad \varvec{M}:=\varvec{M}^{1/2}\times \varvec{M}^{-1/2}, \end{aligned}$$
so that (20b)–(20c) can be shortened to \({\mathscr {B}}\varvec{w}(t)-\varvec{\chi }(t)\in \varvec{M}\).
With the above notation:
For every \((\varvec{g},\varvec{\xi }_D,\varvec{\xi }_N)\in \varvec{H}\times \varvec{H}^{1/2}_\varGamma \times \varvec{H}^{-1/2}_\varGamma \), the steady state problem
$$\begin{aligned} \varvec{w}=A_\star \varvec{w}+\varvec{g}, \qquad \varvec{\gamma }_D \varvec{w}-\varvec{\xi }_D \in \varvec{M}^{1/2}, \qquad \varvec{\gamma }_N\varvec{\kappa }\varvec{w}-\varvec{\xi }_N \in \varvec{M}^{-1/2} \end{aligned}$$
admits a unique solution and
$$\begin{aligned} \Vert \varvec{w}\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}+\Vert \varDelta \varvec{w}\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C (\Vert \varvec{g}\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert \varvec{\xi }_D\Vert _{1/2,\varGamma }+\Vert \varvec{\xi }_N\Vert _{-1/2,\varGamma }). \end{aligned}$$
The constant C depends only on \(\varGamma \), \(\rho \), and \(\kappa \).
The operator A is maximal dissipative and self-adjoint.
The operator A is the generator of a contractive analytic semigroup in \(\varvec{H}\).
To prove (a), consider the coercive variational problem:
$$\begin{aligned}&\varvec{w}\in \varvec{V}, \qquad \varvec{\gamma }_D\varvec{w}-\varvec{\xi }_D \in \varvec{M}^{1/2}, \end{aligned}$$
$$\begin{aligned}&(\varvec{w},\varvec{v})_{\varvec{H}}+[\varvec{w},\varvec{v}] =(\varvec{g},\varvec{w})_{{\mathbb {R}}^d{\setminus }\varGamma }+\langle \!\langle \varvec{\xi }_N, \varvec{\gamma }_D\varvec{v}\rangle \!\rangle _\varGamma \quad \forall \varvec{v}\in \varvec{V}_h. \end{aligned}$$
The coercivity constant of the bilinear form in (23b) depends only on the constants \(\kappa \) and \(\rho \) if we use the standard \(H^1({{\mathbb {R}}^d{\setminus }\varGamma })^2\) norm in \(\varvec{V}_h\subset \varvec{V}\). If we test (23) with smooth functions that are compactly supported in \({{\mathbb {R}}^d{\setminus }\varGamma }\), we can prove that \(\varvec{\rho }\varvec{w}=\varvec{\kappa }\varDelta \varvec{w}\). Therefore, by (17), it follows that
$$\begin{aligned} \langle \!\langle \varvec{\gamma }_N\varvec{\kappa }\varvec{w}-\varvec{\xi }_N,\varvec{\gamma }_D\varvec{v}\rangle \!\rangle _\varGamma =0 \qquad \forall \varvec{v}\in \varvec{V}_h. \end{aligned}$$
Since \(\varvec{\gamma }_D:\varvec{V}_h \rightarrow \varvec{M}^{1/2}\) is surjective, this latter condition is equivalent to \(\varvec{\gamma }_N\varvec{\kappa }\varvec{w}-\varvec{\xi }_N\in \varvec{M}^{-1/2}=(\varvec{M}^{1/2})^\circ \).
Note next that
$$\begin{aligned} \langle \!\langle \varvec{\gamma }_N\varvec{\kappa }\varvec{w},\varvec{\gamma }_D \varvec{v}\rangle \!\rangle _\varGamma =0 \qquad \forall \varvec{w} \in \varvec{D}_h, \quad \varvec{v}\in \varvec{V}_h, \end{aligned}$$
and therefore, by (17),
$$\begin{aligned} (A\varvec{w},\varvec{v})_{\varvec{H}}=-[\varvec{w},\varvec{v}] \quad \forall \varvec{w} \in \varvec{D}_h, \quad \varvec{v}\in \varvec{V}_h. \end{aligned}$$
This proves symmetry and dissipativity of A. Taking \(\varvec{\xi }_D=0\) and \(\varvec{\xi }_N=0\) in (a), we easily show that \(I-A:D(A)\rightarrow \varvec{H}\) is surjective and, therefore, A is maximal dissipative and self-adjoint (see [42, Proposition 3.11]).
Finally A is the infinitesimal generator of a contractive semigroup if and only if it is maximal dissipative (see [35, Chapter 1, Theorem 4.3] or [23, Theorem 4.4.3, Theorem 4.5.1]) and the dissipativity and self-adjointness of A show that the semigroup is analytic (see [18, Corollary 4.8]). \(\square \)
$$\begin{aligned} | \varvec{w}|_V:=[\varvec{w},\varvec{w}]^{1/2} \approx \Vert \nabla \varvec{w}\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }, \end{aligned}$$
the identity (24) and a simple computation show that
$$\begin{aligned} |\varvec{w}|_V \le \Vert A\varvec{w}\Vert _H + \Vert \varvec{w}\Vert _H \approx \Vert \varDelta \varvec{w}\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert \varvec{w}\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\qquad \forall \varvec{w}\in D(A), \end{aligned}$$
where in both formulas above the symbol \(\approx \) is used to denote equivalence of norms (seminorms) with constants independent of h. In the sequel we will also use Hölder spaces \(\mathscr {C}^\theta ({\mathbb {R}}_+;X)\) for \(\theta \in (0,1)\), where X is a Hilbert space, and the seminorms
$$\begin{aligned} | f|_{t,\theta ,X} := \sup _{0\le \tau _1 < \tau _2\le t} {\Vert f (\tau _1)- f(\tau _2)\Vert _X \over |\tau _1-\tau _2|^\theta },\qquad t>0. \end{aligned}$$
Let \(\varvec{\chi }:=(\varvec{\chi }_D,\varvec{\chi }_N):{\mathbb {R}}_+\rightarrow \varvec{H}_\varGamma :=\varvec{H}^{1/2}_\varGamma \times \varvec{H}^{-1/2}_\varGamma \) and assume that
$$\begin{aligned} \dot{\varvec{\chi }}\in {\mathscr {C}}^\theta ({\mathbb {R}}_+;\varvec{H}_\varGamma ), \qquad \varvec{\chi }(0)=\dot{\varvec{\chi }}(0)=0. \end{aligned}$$
The problem (20) admits a unique solution satisfying
$$\begin{aligned} \dot{\varvec{w}}= A_\star \varvec{w}\in {\mathscr {C}}^\theta ({\mathbb {R}}_+;\varvec{H}), \qquad \varvec{w}\in {\mathscr {C}}^\theta ({\mathbb {R}}_+;\varvec{V}). \end{aligned}$$
Moreover, there exist constants independent of h such that for all t
$$\begin{aligned} \Vert \varvec{w}(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }&\le C_1 t (\Vert \varvec{\chi }\Vert _{L^\infty (0,t;\varvec{H}_\varGamma )} +\Vert \dot{\varvec{\chi }}\Vert _{L^\infty (0,t;\varvec{H}_\varGamma )}), \end{aligned}$$
$$\begin{aligned} \Vert \varDelta \varvec{w}(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }&\le C_2 \left( \max \{1,t\} \Vert \dot{\varvec{\chi }}\Vert _{L^\infty (0,t;\varvec{H}_\varGamma )} +\theta ^{-1} t^\theta |\dot{\varvec{\chi }}|_{t,\theta ,\varvec{H}_\varGamma }\right) , \end{aligned}$$
$$\begin{aligned} \Vert \nabla \varvec{w}(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }&\le C_3 \left( t\Vert \varvec{\chi }\Vert _{L^\infty (0,t;\varvec{H}_\varGamma )} +\max \{1,t\} \Vert \dot{\varvec{\chi }}\Vert _{L^\infty (0,t;\varvec{H}_\varGamma )} {+}\theta ^{-1} t^\theta |\dot{\varvec{\chi }}|_{t,\theta ,\varvec{H}_\varGamma }\right) . \end{aligned}$$
The proof is based on the decomposition of the solution of (20) into a sum \(\varvec{w}=\varvec{w}_\chi +\varvec{w}_0\), where \(\varvec{w}_\chi \) takes care of the data (using Proposition 2), while \(\varvec{w}_0\) will be handled using a Cauchy problem (Theorem 6).
Let \(\varvec{w}_\chi (t)\) be the solution of (22) with \(\varvec{g}=0\), \(\varvec{\xi }_D=\varvec{\chi }_D(t)\) and \(\varvec{\xi }_N=\varvec{\chi }_N(t)\), and note that \(\dot{\varvec{w}}_\chi =\varvec{w}_{{{\dot{\chi }}}}\in {\mathscr {C}}^\theta (\mathbb R_+;\varvec{D})\), since we have applied a time-independent operator to the transmission data. Let now \(\varvec{f}:=\varvec{w}_\chi -\dot{\varvec{w}}_\chi \in {\mathscr {C}}^\theta ({\mathbb {R}}_+;\varvec{H}),\) which satisfies \(\varvec{f}(0)=0.\) Note that for all \(t\ge 0\)
$$\begin{aligned}&\Vert \varvec{w}_\chi (t)\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}+\Vert \varDelta \varvec{w}_\chi (t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C \Vert \varvec{\chi }(t)\Vert _{\varvec{H}_\varGamma }, \end{aligned}$$
$$\begin{aligned}&\Vert \varvec{f}(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C (\Vert \varvec{\chi }(t)\Vert _{\varvec{H}_\varGamma } +\Vert \dot{\varvec{\chi }}(t)\Vert _{\varvec{H}_\varGamma }), \end{aligned}$$
$$\begin{aligned}&| \varvec{f}|_{\theta ,t,\varvec{H}} \le C( |\varvec{\chi }|_{\theta ,t,\varvec{H}_\varGamma }+ |\dot{\varvec{\chi }}|_{\theta ,t,\varvec{H}_\varGamma }), \end{aligned}$$
with a constant C depending exclusively on the parameters and geometry. Let finally \(\varvec{w}_0:{\mathbb {R}}_+\rightarrow D(A)\) be the solution of
$$\begin{aligned} \dot{\varvec{w}}_0(t)=A\varvec{w}(t)+\varvec{f}(t) \qquad t\ge 0, \qquad \varvec{w}(0)=0. \end{aligned}$$
By Theorem 6, \(\varvec{w}_0\) and therefore \(\varvec{w}\) have the required regularity. Using the bound for \(\varvec{w}_0\) in Theorem 6 and (27a)–(27b), we can easily prove (26a). Using the bound for \(A\varvec{w}_0\) in Theorem 6 and (27a)–(27c), we can prove that
$$\begin{aligned} \Vert \varDelta \varvec{w}(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C \left( \Vert \varvec{\chi }(t)\Vert _{\varvec{H}_\varGamma } + \Vert \dot{\varvec{\chi }}(t)\Vert _{\varvec{H}_\varGamma } +\theta ^{-1} t^\theta |\varvec{\chi }|_{t,\theta ,\varvec{H}_\varGamma } +\theta ^{-1} t^\theta |\dot{\varvec{\chi }}|_{t,\theta ,\varvec{H}_\varGamma }\right) . \end{aligned}$$
Proving (26b) from the above estimate is the result of a simple computation. Finally, in view of (25), the estimate
$$\begin{aligned} \Vert \nabla \varvec{w}(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }&\le C(\Vert \nabla \varvec{w}_\chi (t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+ \Vert \varvec{w}_0(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert A\varvec{w}_0(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }) \end{aligned}$$
$$\begin{aligned}&\le C' (\Vert \varvec{w}_\chi (t)\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }} +\Vert \varvec{w}(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert \varDelta \varvec{w}(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }) \end{aligned}$$
follows and therefore (26c) is a simple consequence of (27a), (26a), and (26b). \(\square \)
Main results on the semidiscrete problem
The first step towards the analysis of the two problems that are hidden in (16) is the reconciliation of the solution of the classical differential equation (20) with a distributional form, where we look for \(\varvec{w}\in \mathrm {TD}(\varvec{D})\) such that
$$\begin{aligned}&\dot{\varvec{w}} =A_\star \varvec{w} \qquad \qquad \qquad \!\!\! (\text{ in } \varvec{H}),\\&{\mathscr {B}}\varvec{w}-\varvec{\eta }\in \varvec{M} \qquad \,\quad (\text{ in } \varvec{H}_\varGamma ), \end{aligned}$$
for given data \(\varvec{\eta }\in \mathrm {TD}(\varvec{H}_\varGamma )\).
Problem (28) has a unique solution.
Let \(\mathrm H=(\mathrm H_D,\mathrm H_N):={\mathscr {L}}\{\varvec{\eta }\}\). The s-dependent transmission problem
$$\begin{aligned}&s\varvec{W}(s) =A_\star \varvec{W}(s), \end{aligned}$$
$$\begin{aligned}&{\mathscr {B}} \varvec{W}(s)-\mathrm H(s) \in \varvec{M}, \end{aligned}$$
is equivalent to the variational problem
$$\begin{aligned}&\varvec{W}(s)\in \varvec{V}, \end{aligned}$$
$$\begin{aligned}&\varvec{\gamma }_D \varvec{W}(s)-\mathrm H_D(s)\in \varvec{M}^{1/2}, \end{aligned}$$
$$\begin{aligned}&[\varvec{W}(s),\varvec{w}]+s(\varvec{V}(s),\varvec{w})_{\varvec{H}}= \langle \!\langle \mathrm H_N(s),\varvec{\gamma }_D\varvec{w}\rangle \!\rangle _\varGamma \qquad \forall \varvec{w}\in \varvec{V}_h, \end{aligned}$$
for all \(s\in {\mathbb {C}}_+\), which can be easily proved using the techniques of the proof of Proposition 2. We will prove that (30) is uniquely solvable and that its solution can be bounded as
$$\begin{aligned} \Vert \varvec{W}(s)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert \nabla \varvec{W}(s)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C(\mathrm {Re}\,s) |s|^\nu \, \Vert \mathrm H(s)\Vert _{\varvec{H}_\varGamma } \qquad \forall s\in {\mathbb {C}}_+, \end{aligned}$$
for some \(\nu \ge 0\) and non-increasing \(C:(0,\infty )\rightarrow (0,\infty )\) that is allowed to grow rationally at the origin. These statements imply that \(\varvec{W}={\mathscr {L}}\{\varvec{w}\}\) where \(\varvec{w}\in \mathrm {TD}(\varvec{D})\) (note that the needed bounds for the Laplacian of \(\varvec{W}(s)\) follow from equation (29)) and \(\varvec{w}\) satisfies (28), which is the inverse Laplace transform of (29).
In order to deal with (30) and (31), we proceed as follows. For fixed \(s\in {\mathbb {C}}_+\), we consider the coercive transmission problem
$$\begin{aligned}&-|s|\varvec{W}_D(s)+A_\star \varvec{W}_D(s) =0, \end{aligned}$$
$$\begin{aligned}&\varvec{\gamma }\varvec{W}_D(s) =\varTheta _D^{-1} \mathrm H_D(s), \end{aligned}$$
and note that the four separate boundary conditions in (32b) are equivalent to the transmission conditions \(\varvec{\gamma }_D \varvec{W}_D(s)=\mathrm H_D(s)\). Using the Bamberger-HaDuong lifting lemma (the original appears in [6] and an 'extension' to non-smooth boundaries can be found as Lemma 2.7.1 in [40]), it follows that
$$\begin{aligned} |\!|\!| \varvec{W}_D(s) |\!|\!|_{|s|}^2&:= \Vert \varvec{\kappa }^{1/2}\nabla \varvec{W}_D(s)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }^2+ |s|\, \Vert \varvec{\rho }^{1/2} \varvec{W}_D(s)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }^2 \nonumber \\&\le C \,\left( \frac{|s|}{\min \{1,\mathrm {Re}\,s\}}\right) ^{1/2} \Vert H_D(s)\Vert ^2_{1/2,\varGamma }. \end{aligned}$$
We then consider the coercive variational problem
$$\begin{aligned}&\varvec{W}_0(s)\in \varvec{V}_h, \end{aligned}$$
$$\begin{aligned}&[\varvec{W}_0(s),\varvec{w}]+s(\varvec{W}_0(s),\varvec{w})_{\varvec{H}} = \langle \!\langle \mathrm H_N(s),\varvec{\gamma }_D\varvec{w}\rangle \!\rangle _\varGamma \end{aligned}$$
$$\begin{aligned}&-[\varvec{W}_D(s),\varvec{w}] -s(\varvec{W}_D(s),\varvec{w})_{\varvec{H}} \quad \forall \varvec{w}\in \varvec{V}_h. \end{aligned}$$
Testing (34b) with \(\varvec{w}=\overline{\frac{s}{|s|}\varvec{W}_0(s)}\), we have the inequalities
$$\begin{aligned} \min \bigg (1,\frac{\mathrm {Re}(s)}{|s|}\bigg )|\!|\!| \varvec{W}_0(s) |\!|\!|_{|s|}^2&\le \frac{\mathrm {Re}(s)}{|s|} [\varvec{W}_0(s),\overline{\varvec{W}_0(s)}] + |s| (\varvec{W}_0(s),\overline{\varvec{W}_0(s)})_{\varvec{H}}\\&\le \left( \Vert \mathrm H_N(s)\Vert _{-1/2,\varGamma }\Vert \varvec{\gamma }_D\varvec{W}_0(s)\Vert _{1/2,\varGamma }\right. \\&\left. \qquad +\, |\!|\!| \varvec{W}_D(s) |\!|\!|_{|s|}|\!|\!| \varvec{W}_0(s) |\!|\!|_{|s|}\right) . \end{aligned}$$
What is left for the proof is very simple indeed. First of all, it is clear that \(\varvec{W}(s):=\varvec{W}_D(s)+\varvec{W}_0(s)\) is the solution to (30). Second, it is simple to see that
$$\begin{aligned} \Vert \varvec{\gamma }_D\varvec{W}_D(s)\Vert _{1/2,\varGamma } \le \frac{C}{\min \{1,\mathrm {Re}\,s^{1/2}\}}|\!|\!| \varvec{W}_D(s) |\!|\!|_{|s|} \le \frac{C}{\min \{1,\mathrm {Re}\,s\}}|\!|\!| \varvec{W}_D(s) |\!|\!|_{|s|}, \end{aligned}$$
which yields a bound for \(|\!|\!| \varvec{W}_0(s) |\!|\!|_{|s|}\) in terms of \(\Vert \mathrm H_N(s)\Vert _{-1/2,\varGamma }\) and \(|\!|\!| \varvec{W}_D(s) |\!|\!|_{|s|}\). Finally (33) can be used to prove (31). \(\square \)
The following process mimics the one in [22] and in [40, Chapter 7]. It involves two aspects: (a) an extension by zero of the data to negative values of the time variable; (b) a hypothesis on polynomial growth of the data. The reason to deal with (a) lies in the fact that the distributional equations (28) are for causal distributions of the real variable, not for distributions defined in the positive real axis. This is due to the fact that the heat potentials and operators have memory terms that involve the entire history of the process including values at time \(t=0\). The extension by zero to negative time will be done through the operator
$$\begin{aligned} E f(t)={\left\{ \begin{array}{ll} f(t), &{} t\ge 0, \\ 0, &{} t< 0.\end{array}\right. } \end{aligned}$$
The reason why (b) is important is the fact that the equation (28) can be shown to have a unique solution in the space \(\mathrm {TD}(\varvec{D})\), which imposes some restrictions on the growth of the solution (and hence the data) at infinity.
Let \(\varvec{\chi }:{\mathbb {R}}_+\rightarrow \varvec{H}_\varGamma \) be continuous and \(\dot{\varvec{\chi }}\in {\mathscr {C}}^\theta ({\mathbb {R}}_+;\varvec{H}_\varGamma )\) be polynomially bounded in the following sense: there exist \(C>0\) and \(m\ge 0\) such that
$$\begin{aligned} \Vert \dot{\varvec{\chi }}(t)\Vert _{\varvec{H}_\varGamma }+|\dot{\varvec{\chi }}|_{t,\theta ,\varvec{H}_\varGamma } \le C t^m \qquad \forall t\ge 1. \end{aligned}$$
Assume also that \(\varvec{\chi }(0)=\dot{\varvec{\chi }}(0)=0\). If \(\varvec{w}\) is the solution to (20), then \(\varvec{v}=E\varvec{w}\) is the solution to (28) with \(\varvec{\eta }=E\varvec{\chi }\).
The hypotheses imply that \(E\varvec{\chi }\in \mathrm {TD}(\varvec{H}_\varGamma )\). The bounds of Proposition 3 imply that \(\varvec{w}\) is polynomially bounded as a \(\varvec{D}\)-valued function and therefore \(\varvec{v}:=E\varvec{w}\in \mathrm {TD}(\varvec{D})\). Finally, since \(\varvec{w}(0)=0\), it follows that \(E\dot{\varvec{w}}=\dot{\varvec{v}}\), which finishes the proof, since E commutes with any operator that does not affect the time variable. \(\square \)
We are almost ready to state and prove the two main results concerning the semidiscrete system: semidiscrete stability and an error estimate. To shorten up some of the expressions to come, we introduce the bounded jump operator
$$\begin{aligned} \varvec{D} \ni \varvec{u}=(u_-,u_+) \longmapsto \varvec{J}\varvec{u}:=([\![ \gamma u_- ]\!], [\![ \partial _\nu u_- ]\!])\in H^{1/2}_\varGamma \times H^{-1/2}_\varGamma =:H_\varGamma . \end{aligned}$$
We also consider the function spaces tagged in the parameter \(\theta \in (0,1)\)
$$\begin{aligned} {\mathscr {B}}^\theta&:=\{ \varvec{\eta }\in {\mathscr {C}}(\mathbb R;H_\varGamma )\,:\, \varvec{\eta }\equiv 0 \text{ in } (-\infty ,0), \,\, \varvec{\eta }|_{{\mathbb {R}}_+}\in {\mathscr {C}}^\theta ({\mathbb {R}}_+;H_\varGamma ),\\&\quad \exists C,m\ge 0 \text{ s.t. } \Vert \varvec{\eta }(t)\Vert _{H_\varGamma }+|\varvec{\eta }|_{t,\theta ,H_\varGamma }\le C t^m \quad \forall t\ge 1\},\\ {\mathscr {B}}^{1+\theta }&:=\{ \varvec{\eta }:{\mathbb {R}} \rightarrow H_\varGamma \,:\, \varvec{\eta }\equiv 0 \text{ in } (-\infty ,0), \,\, \dot{\varvec{\eta }}\in {\mathscr {B}}^\theta \},\\ {\mathscr {U}}^\theta&:=\{\varvec{u}\in {\mathscr {C}}^1({\mathbb {R}};\varvec{H})\,:\, \varvec{u}\equiv 0 \text{ in } (-\infty ,0), \,\, \varvec{u}|_{\mathbb R_+}\in {\mathscr {C}}^\infty ({\mathbb {R}}_+;\varvec{D})\}. \end{aligned}$$
If \(\varvec{\beta }\in {\mathscr {B}}^{1+\theta }\), \(\varvec{\psi }^h=(\phi ^h,\lambda ^h)\) is the solution to the semidiscrete system of TDBIE (12) and \(\varvec{u}^h=(u^h_-,u^h_+)\) is given by the potential representation (13), then \(\varvec{u}^h\in \mathscr {U}^\theta \), \(\varvec{\psi }^h\in {\mathscr {B}}^\theta \), and we can bound
$$\begin{aligned} \Vert \varvec{u}^h(t)\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}+\Vert \varvec{\psi }^h(t)\Vert _{H_\varGamma } \le C \max \{1,t\} \mathrm {Acc}(\varvec{\beta },t,\theta ), \end{aligned}$$
$$\begin{aligned} \mathrm {Acc}(\varvec{\beta },t,\theta ) :=\max _{0\le \tau \le t}\Vert \varvec{\beta }(\tau )\Vert _{H_\varGamma } +\max _{0\le \tau \le t}\Vert \dot{\varvec{\beta }}(\tau )\Vert _{H_\varGamma } + \frac{t^\theta }{\theta }\sup _{0\le \tau _1<\tau _2\le t} \frac{\Vert \varvec{\beta }(\tau _1)-\varvec{\beta }(\tau _2)\Vert _{H_\varGamma }}{|\tau _1-\tau _2|^\theta }, \end{aligned}$$
is a collection of cummulative seminorms in \({\mathscr {B}}^{1+\theta }\).
If \(\varvec{\beta }=(\beta _0,\beta _1)\), we define \(\varvec{\chi }_D:=(\beta _0,0,0,0)\) and \(\varvec{\chi }_N=(0,0,0,\beta _1)\). Let \(\varvec{u}^h\) be the solution to (28) with data \(\varvec{\eta }=(\varvec{\chi }_D,\varvec{\chi }_N)\) and \(\varvec{\psi }^h:=\varvec{J}\varvec{u}^h\). We can identify \(\varvec{u}^h\) and \(\varvec{\psi }^h\) with the solution of (12) and (13). We also note that
$$\begin{aligned} \Vert \varvec{u}^h(t)\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}+\Vert \varvec{\psi }^h(t)\Vert _{H_\varGamma } \le C( \Vert \varvec{u}^h(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert \nabla \varvec{u}^h(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert \varDelta \varvec{u}^h(t)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }). \end{aligned}$$
The bounds in the statement of the theorem follow from the fact that Proposition 5 identifies \(\varvec{u}^h|_{{\mathbb {R}}_+}\) with the solution of (20), with \((\varvec{\chi }_D,\varvec{\chi }_N)|_{{\mathbb {R}}_+}\) as data, and we can thus use the estimates of Proposition 3. \(\square \)
For data \((\beta _0,\beta _1)\in \mathrm {TD}(H_\varGamma )\), we let
\(\varvec{\psi }=(\phi ,\lambda )\) be the solution of the TDBIE (11),
\(\varvec{u}=(u_-,u_+)\) be given by the potential representation (9),
\(\varvec{\psi }^h=(\phi ^h,\lambda ^h)\) be the solution of the semidiscrete TDBIE (12),
\(\varvec{u}^h=(u^h_-,u^h_+)\) be given by the potential representation (13).
If \(\varvec{\psi }\in {\mathscr {B}}^{1+\theta }\), then
$$\begin{aligned} \Vert \varvec{u}^h(t)-\varvec{u}(t)\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}+\Vert \varvec{\psi }^h(t)-\varvec{\psi }(t)\Vert _{H_\varGamma } \le C \max \{1,t\} \mathrm {Acc}(\varvec{\psi }-\varvec{\varPi }_h\varvec{\psi },t,\theta ), \end{aligned}$$
where \(\varvec{\varPi }_h:H_\varGamma \rightarrow Y_h\times X_h\) is the best approximation operator onto \(Y_h\times X_h\).
First of all, note that the errors \(\varvec{\psi }^h-\varvec{\psi }\) can be defined as the solution of the semidiscrete TDBIE (14) and the associated potential errors \(\varvec{e}^h:=\varvec{u}^h-\varvec{u}\) are given by a potential representation (15) using \(\varvec{\psi }^h-\varvec{\psi }\) as input densities. If we define \(\varvec{\chi }_D=(0,0,-\phi ,0)\) and \(\varvec{\chi }_N=(0,-\kappa \lambda ,0,0)\), we can see that \(\varvec{e}^h\) is the solution to (28) with data \(\varvec{\eta }=(\varvec{\chi }_D,\varvec{\chi }_N)\) and that \(\varvec{\psi }^h-\varvec{\psi }=\varvec{J}\varvec{e}^h\). Using the same arguments as in the proof of Theorem 3, we can prove that
$$\begin{aligned} \Vert \varvec{e}^h(t)\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}+\Vert \varvec{J}\varvec{e}^h(t)\Vert _{H_\varGamma }\le C \max \{1,t\} \mathrm {Acc}(\varvec{\psi },t,\theta ). \end{aligned}$$
Note now that we can decompose \(\varvec{\psi }^h-\varvec{\psi }=(\varvec{\psi }^h-\varvec{\varPi }_h\varvec{\psi })-(\varvec{\psi }-\varvec{\varPi }_h\varvec{\psi })\) and consider \(\varvec{\psi }-\varvec{\varPi }_h\varvec{\psi }\) as the exact solution in the argument and \(\varvec{\psi }^h-\varvec{\varPi }_h\varvec{\psi }\) as its Galerkin approximation. In other words, if we input \(\varvec{\varPi }_h\varvec{\psi }\) as exact solution of the TDBIE, then \(\varvec{\varPi }_h\varvec{\psi }\) is also the solution of the semidiscrete TDBIE and the associated error is zero. Therefore, we can rewrite (37) as
$$\begin{aligned} \Vert \varvec{e}^h(t)\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}+\Vert \varvec{J}\varvec{e}^h(t)\Vert _{H_\varGamma }\le C \max \{1,t\} \mathrm {Acc}(\varvec{\psi }-\varvec{\varPi }_h\varvec{\psi },t,\theta ), \end{aligned}$$
which proves (36). \(\square \)
Multistep CQ time discretization
In this section we introduce and analyze Convolution Quadrature schemes, based on BDF time-integrators, applied to the semidiscrete TDBIE (12) and to the potential postprocessing (13). All convolution operators—in the left and right hand sides of (12) and in the retarded potentials in (13)—will be treated with BDF–CQ.
Consider a constant \(k>0\) and a sequence of discrete time steps \(t_n := nk\) for \(n\ge 0\) and let
$$\begin{aligned} \delta (\zeta ):=\sum _{j=1}^q {1\over j} (1-\zeta )^j =: \sum _{j=0}^q \alpha _j \zeta ^j \end{aligned}$$
be the characteristic polynomial of the BDF(q) method. We will allow \(q\le 6\), so that the methods are \(A(\alpha )\)-stable. (A-stability only holds for \(q=1\) and 2.) If we write the \(\zeta \)-transform of a sequence of samples of a causal X-valued function v,
$$\begin{aligned} V(\zeta ) := \sum _{n=0}^\infty v(t_n) \zeta ^n, \end{aligned}$$
$$\begin{aligned} \frac{1}{k} \delta (\zeta ) V(\zeta ) =\frac{1}{k} \sum _{n=0}^\infty \left( \sum _{j=0}^{\min \{q,n\}} \alpha _j v(t_{n-j})\right) \zeta ^n \approx \dot{V}(\zeta ) = \sum _{n=0}^\infty \dot{v}(t_n) \zeta ^n. \end{aligned}$$
Note that the approximation of the derivative is only good when v is a smooth causal function. In terms of the \(\zeta \)-transform of data
$$\begin{aligned} B_0(\zeta ) := \sum _{n=0}^\infty \beta _0(t_n) \zeta ^n,\qquad B_1(\zeta ) := \sum _{n=0}^\infty \beta _1(t_n) \zeta ^n, \end{aligned}$$
and of the fully discrete unknowns
$$\begin{aligned}&\varLambda ^{h,k}(\zeta ) := \sum _{n=0}^\infty \lambda _n^{h,k} \zeta ^n \approx \sum _{n=0}^\infty \lambda ^h(t_n)\zeta ^n,&\quad \varPhi ^{h,k}(\zeta ) := \sum _{n=0}^\infty \phi _n^{h,k} \zeta ^n \approx \phi ^h(t_n) \zeta ^n,\\&U_-^{h,k}(\zeta ) := \sum _{n=0}^\infty u_{-,n}^{h,k} \zeta ^n \approx \sum _{n=0}^\infty u^h_-(t_n)\zeta ^n,&\quad U_+^{h,k}(\zeta ) := \sum _{n=0}^\infty u_{+,n}^{h,k} \zeta ^n \approx \sum _{n=0}^\infty u^h_+(t_n)\zeta ^n, \end{aligned}$$
the fully discrete CQ–BEM equations look for \((\varLambda ^{h,k}(\zeta ),\varPhi ^{h,k}(\zeta ))\in X_h\times Y_h\) such that
$$\begin{aligned} \begin{aligned} \begin{bmatrix} {\mathrm V}(\frac{1}{k m} \delta (\zeta ))+\kappa {\mathrm V}(\frac{1}{k} \delta (\zeta ))&\quad - {\mathrm K}(\frac{1}{k m} \delta (\zeta ))- {\mathrm K}(\frac{1}{k} \delta (\zeta )) \\ {\mathrm K}^T(\frac{1}{k m} \delta (\zeta ))+ {\mathrm K}^T(\frac{1}{k} \delta (\zeta ))&\quad {\mathrm W}(\frac{1}{k m} \delta (\zeta ))+\frac{1}{\kappa } {\mathrm W}(\frac{1}{k} \delta (\zeta )) \end{bmatrix} \begin{bmatrix} \varLambda ^{h,k}(\zeta ) \\ \varPhi ^{h,k}(\zeta ) \end{bmatrix}&\\ {-}\frac{1}{2}\begin{bmatrix} B_0(\zeta ) \\ \frac{1}{\kappa } B_1(\zeta ) \end{bmatrix} - \begin{bmatrix} {\mathrm V}(\frac{1}{k} \delta (\zeta ))&\quad -{\mathrm K}(\frac{1}{k} \delta (\zeta )) \\ \frac{1}{\kappa } {\mathrm K}^T(\frac{1}{k} \delta (\zeta ))&\quad \frac{1}{\kappa } {\mathrm W}(\frac{1}{k} \delta (\zeta )) \end{bmatrix} \begin{bmatrix} B_1(\zeta ) \\ B_0(\zeta ) \end{bmatrix}&\in X_h^\circ {\times } Y_h^\circ , \end{aligned} \end{aligned}$$
and then postprocess their output to build
$$\begin{aligned} U_-^{h,k}(\zeta )&= {\mathrm S}(\tfrac{1}{k m} \delta (\zeta )) \varLambda ^{h,k}(\zeta ) - {\mathrm D}(\tfrac{1}{k m} \delta (\zeta ))\varPhi ^{h,k}(\zeta ), \end{aligned}$$
$$\begin{aligned} U_+^{h,k}(\zeta )&=-{\mathrm S}(\tfrac{1}{k} \delta (\zeta ))(\kappa \varLambda ^{h,k}(\zeta )-B_1(\zeta ))+ {\mathrm D}(\tfrac{1}{k} \delta (\zeta )) (\varPhi ^{h,k}(\zeta )-B_0(\zeta )). \end{aligned}$$
This short-hand exposition of the BDF–CQ method can be easily derived by taking the Laplace transform of (12) and (13) and substituting the Laplace transformed variable s by the discrete symbol \(k^{-1}\delta (\zeta )\). For readers who are not acquainted with CQ techniques, we explain in "Appendix A.2" the meaning of formulas (40) and (41).
We can think of (40)–(41) as a 'frequency-domain' system of BIE followed by potential postprocessing associated to a transmission problem with diffusion parameters \(\rho \kappa ^{-1}\,k^{-1}\delta (\zeta )\) and \(k^{-1}\delta (\zeta )\). Therefore, if we write \(\varvec{u}_n^{h,k}:=(u_{-,n}^{h,k},u_{+,n}^{h,k})\), \(\varvec{\chi }(t):=((\beta _0(t),0,0,0),(0,0,0,\beta _1(t)))\), and recall (39), it follows that
$$\begin{aligned}&\partial _k \varvec{u}_n^{h,k} =A_\star \varvec{u}_n^{h,k}, \end{aligned}$$
$$\begin{aligned}&{\mathscr {B}}\varvec{u}_n^{h,k} -\varvec{\chi }(t_n) \in \varvec{M}, \end{aligned}$$
$$\begin{aligned} \partial _k \varvec{u}_n^{h,k}= \frac{1}{k}\sum _{j=0}^{\min \{q,n\}}\alpha _j \varvec{u}_{n-j}^{h,k} \end{aligned}$$
is the backward derivative associated to the BDF scheme. On the other hand, the semidiscrete solution \(\varvec{u}^h\) satisfies very similar equations at the discrete times
$$\begin{aligned}&\dot{\varvec{u}}^h(t_n) =A_\star \varvec{u}^h(t_n) , \end{aligned}$$
$$\begin{aligned}&{\mathscr {B}}\varvec{u}^h(t_n) -\varvec{\chi }(t_n) \in \varvec{M}. \end{aligned}$$
Therefore, the error \(\varvec{e}_n:=\varvec{u}_n^{h,k}-\varvec{u}^h(t_n)\) satisfies the equations
$$\begin{aligned}&\varvec{e}_n \in \varvec{D}_h=D(A), \end{aligned}$$
$$\begin{aligned}&\partial _k \varvec{e}_n =A\varvec{e}_n+\varvec{\theta }_n, \end{aligned}$$
where \( \varvec{\theta }_n:=\dot{\varvec{u}}^h(t_n)-\partial _k \varvec{u}^h(t_n) \) is the error associated to the finite difference approximation of the time derivative. Note that
$$\begin{aligned} \varvec{J}\varvec{e}_n=\varvec{\psi }_n^{h,k}-\varvec{\psi }^h(t_n) =(\phi _n^{h,k}-\phi ^h(t_n),\lambda _n^{h,k}-\lambda ^h(t_n)). \end{aligned}$$
In what follows, smoothness of a function of the time variable is to be understood as smoothness as a function defined in the entire real line, and vanishing in \((-\infty ,0)\). This imposes zero values for derivatives of the function at time \(t=0\). Non-zero initial conditions can be handled by modifying the CQ scheme, but then the all-time-steps at once strategy that we use in the implementation is no longer applicable.
If \(\varvec{u}^h\) is smooth enough, then for all \(n\ge 0\)
$$\begin{aligned}&\Vert \varvec{u}_n^{h,k}-\varvec{u}^h(t_n)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C t_n k^q \max _{t\le t_n}\left\| \frac{d^{q+1}}{dt^{q+1}}\varvec{u}^{h}(t)\right\| _{{\mathbb {R}}^d{\setminus }\varGamma }, \end{aligned}$$
$$\begin{aligned}&\Vert \nabla \varvec{u}_n^{h,k}-\nabla \varvec{u}^h(t_n)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert \varvec{\psi }_n^{h,k}-\varvec{\psi }^h(t_n)\Vert _{H_\varGamma } \le C t_n k^q \max _{t\le t_n}\left\| \frac{d^{q+2}}{dt^{q+2}}\varvec{u}^{h}(t)\right\| _{{\mathbb {R}}^d{\setminus }\varGamma }. \end{aligned}$$
Following [45, Lemma 10.3], we can show that the solution of the recurrence (44) is
$$\begin{aligned} \varvec{e}_n=k \sum _{j=0}^{n-1} P_j(-k\,A) (\alpha _0I-kA)^{-j-1} \varvec{\theta }_{n-j} \end{aligned}$$
(here \(\alpha _0=\delta (0)\) is the leading coefficient of the BDF derivative), where \(\{ P_j\}\) is a sequence of polynomials with \(\mathrm {deg}\,P_j\le j\) and
$$\begin{aligned} \sup _{z>0}\left| \frac{P_j(z)}{(\alpha _0+z)^{j+1}}\right| \le C \qquad \forall j \ge 0. \end{aligned}$$
The proof of (47) is purely algebraic, based only on the fact that \( \alpha _0I-kA \) can be inverted. The rational functions in (48) are bounded at infinity and have all their poles at \(-\alpha _0<0\), which allows us to apply Theorem 7 and show that
$$\begin{aligned} \Vert \varvec{e}_n\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C k \sum _{j=1}^{n} \Vert \varvec{\theta }_{j}\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C t_n \max _{1\le j\le n} \Vert \varvec{\theta }_j\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }. \end{aligned}$$
However, if \(\varvec{u}^h\) is smooth enough (as a function from \(\mathbb R\) to \(\varvec{H}=L^2({{\mathbb {R}}^d{\setminus }\varGamma })^2)\), Taylor expansion yields
$$\begin{aligned} \Vert \varvec{\theta }_j\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C k^q \max _{t_{j-q}\le t\le t_j}\left\| \frac{d^{q+1}}{dt^{q+1}}\varvec{u}^{h}(t)\right\| _{{\mathbb {R}}^d{\setminus }\varGamma }, \end{aligned}$$
which proves (46a).
Note now that
$$\begin{aligned} \partial _k \varvec{\theta }_n=\partial _k \dot{\varvec{u}}^h(t_n)-\ddot{\varvec{u}}^h(t_n) -(\partial _k^2\varvec{u}^h(t_n)-\ddot{\varvec{u}}^h(t_n)) \end{aligned}$$
and that \(\varvec{f}_n:=\partial _k\varvec{e}_n\in \varvec{D}_h\) satisfies the recurrence \( \partial _k \varvec{f}_n =A\varvec{f}_n+\partial _k\varvec{\theta }_n. \) Using a simple argument on Taylor expansions and Theorem 7, it follows that
$$\begin{aligned} \Vert \varvec{f}_n\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C t_n \max _{1\le j\le n} \Vert \partial _k\varvec{\theta }_j\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C t_n k^{q} \max _{t\le t_n}\left\| \frac{d^{q+2}}{dt^{q+2}}\varvec{u}^{h}(t)\right\| _{{\mathbb {R}}^d{\setminus }\varGamma }. \end{aligned}$$
This can be used to give a bound for \(A_\star \varvec{e}_n=\partial _k\varvec{e}_n-\varvec{\theta }_n\) and (25) then proves that
$$\begin{aligned} \Vert \varvec{e}_n\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}+\Vert \varDelta \varvec{e}_n\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C t_n k^{q} \max _{t\le t_n}\left\| \tfrac{\mathrm d^{q+1}}{\mathrm d t^{q+1}}\varvec{u}^{h}(t)\right\| _{{\mathbb {R}}^d{\setminus }\varGamma }. \end{aligned}$$
Using this bound, (45), and the boundedness of \(\varvec{J}\), (46b) follows. \(\square \)
Multistage CQ time discretization
In this section we introduce and analyze some Runge–Kutta based Convolution Quadrature (RKCQ) schemes for the full discretization of the semidiscrete system of TDBIE (12) and the potential postprocessing (13). Some background material and references on RKCQ and the needed Dunford calculus can be found in "Appendices A.3 and A.4".
The algorithm and some observations
We consider an implicit s-stage RK method with Butcher tableau
$$\begin{aligned} \begin{array}{c|c} \mathbf {c} &{} {\mathscr {Q}}\\ \hline &{} \\ &{} \mathbf {b}^T \end{array} \qquad \begin{array}{c|c@{\qquad }c@{\qquad }c@{\qquad }c} c_1 &{} a_{11} &{} a_{12} &{} \cdots &{} a_{1s} \\ c_2 &{} a_{21} &{} a_{22} &{} \cdots &{} a_{2s} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ c_s &{} a_{s1} &{} a_{s2} &{} \cdots &{} a_{ss} \\ \hline &{} b_1 &{} b_2 &{} \cdots &{} b_s \end{array} \end{aligned}$$
and stability function
$$\begin{aligned} r(z) := 1+z\mathbf {b}^T({\mathscr {I}}-z{\mathscr {Q}})^{-1}\mathbf {1}, \end{aligned}$$
where \(\mathbf {1}={(1,\ldots ,1)}^{T} \in {\mathbb {R}}^s\) and \({\mathscr {I}}\) is the \(s\times s\) identity matrix. We will assume the following hypotheses on the RK method:
The method has (classical) order p and stage order \(q\le p-1\). We exclude methods where \(q=p\) for simplicity. (For instance, the one-stage backward Euler formula, which was covered as the BDF(1) method in the previous section, is not included in this exposition.)
The method is A-stable, i.e., the matrix \(\mathscr {I}-z{\mathscr {Q}}\) is invertible for \(\mathrm {Re} z\le 0\) and the stability function satisfies \(|r(z)|\le 1\) for those values of z.
The method is stiffly accurate, i.e., \(\mathbf {b}^T {\mathscr {Q}}^{-1} = (0,0,\ldots ,0,1)=:{\mathbf {e}}_s^T.\) This implies that \(\lim _{|z|\rightarrow \infty } |r(z)|=0\). Assuming the usual simplifying hypothesis for RK schemes \(\mathscr {Q}{\mathbf {1}}={\mathbf {c}}\), stiff accuracy implies that \(c_s=1\), that is, the last stage of the method is the step. Stiff accuracy also implies that (cf. [10, Lemma 2])
$$\begin{aligned} r(z)&= {\mathbf {b}}^T{\mathscr {Q}}^{-1}{\mathbf {1}} + z{\mathbf {b}}^T (\mathscr {I}-z{\mathscr {Q}})^{-1}{\mathbf {1}} \\&= {\mathbf {b}}^T {\mathscr {Q}}^{-1} ({\mathscr {I}}-z{\mathscr {Q}}+ z{\mathscr {Q}}) ({\mathscr {I}}-z{\mathscr {Q}})^{-1}{\mathbf {1}} \\&= \mathbf {b}^T{\mathscr {Q}}^{-1} ({\mathscr {I}} - z{\mathscr {Q}})^{-1} \mathbf {1}. \end{aligned}$$
The matrix \({\mathscr {Q}}\) is invertible. This hypothesis and A-stability imply that the spectrum of \({\mathscr {Q}}\) is contained in \({\mathbb {C}}_+\).
Examples of the Runge–Kutta methods satisfying all the hypotheses above are provided by the family of s-stage Radau IIA methods with order \(p=2s-1\) and stage order \(q=s\).
Given a function of the time variable, we will write
$$\begin{aligned} w(t_n+\mathbf {c}k) := \begin{bmatrix} w(t_n+c_1k) \\ w(t_n+c_2k) \\ \vdots \\ w(t_n+c_s k)\end{bmatrix} \end{aligned}$$
to denote the s-vectors with the samples at the stages in the time interval \([t_n,t_{n+1}]\). We sample the boundary data and collect the vectors of time samples in formal \(\zeta \)-series
$$\begin{aligned} B_0(\zeta ) := \sum _{n=0}^\infty \beta _0(t_n+\mathbf {c}k) \zeta ^n,\qquad B_1(\zeta ) := \sum _{n=0}^\infty \beta _1(t_n+\mathbf {c}k) \zeta ^n. \end{aligned}$$
The unknowns for the fully discrete method can be collected in
$$\begin{aligned}&\varLambda ^{h,k}(\zeta ) := \sum _{n=0}^\infty \lambda ^{h,k}_n \zeta ^n \approx \sum _{n=0}^\infty \lambda ^h(t_n{+}\mathbf {c}k)\zeta ^n,&\varPhi ^{h,k} (\zeta ):= \sum _{n=0}^\infty \phi ^{h,k}_n \zeta ^n \approx \sum _{n=0}^\infty \phi ^h(t_n{+}\mathbf {c}k) \zeta ^n,\\&U_-^{h,k} (\zeta ):= \sum _{n=0}^\infty U_{-,n}^{h,k} \zeta ^n \approx \sum _{n=0}^\infty u^h_-(t_n{+}\mathbf {c}k)\zeta ^n,&U_+^{h,k} (\zeta ):= \sum _{n=0}^\infty U_{+,n}^{h,k} \zeta ^n \approx \sum _{n=0}^\infty u^h_+(t_n{+}\mathbf {c}k)\zeta ^n, \end{aligned}$$
The pairs \((\lambda ^{h,k}_n,\phi ^{h,k}_n)\in X_h^s\times Y_h^s\) are computed using (40) where the symbol \(\delta \) is now the matrix-valued RK differentiation operator
$$\begin{aligned} \delta (\zeta ) = \left( {\mathscr {Q}}+\frac{\zeta }{1-\zeta }\mathbf 1{\mathbf {b}}^T\right) ^{-1}={\mathscr {Q}}^{-1} - \zeta {\mathscr {Q}}^{-1} \mathbf {1}\mathbf {b}^T {\mathscr {Q}}^{-1}, \end{aligned}$$
(these two matrices can be easily seen to be equal using the stiff accuracy hypothesis) and the testing condition has to be modified, imposing that the residual is in \((X_h^s\times Y_h^s)^\circ \equiv (X_h^\circ )^s\times (Y_h^\circ )^s\). The discrete potentials at the different stages can be computed using (41), with the new definition of \(\delta (\zeta )\). In all the expressions for the RK–CQ fully discrete equations, analytic functions are evaluated at \(k^{-1}\delta (\zeta )\) via Dunford calculus (see "Appendix A.3"). This is meaningful since the spectrum of \(\delta (\zeta )\) lies in \({\mathbb {C}}_+\) for \(\zeta \) small enough, which is due to the fact that \(\delta (\zeta )\) is a small (rank-one) perturbation of \({\mathscr {Q}}^{-1}\) and the spectrum of \({\mathscr {Q}}\) is in \({\mathbb {C}}_+\) (see hypotheses (b) and (d) above).
Before we embark ourselves in the error analysis of the fully discrete method, which involves using quite non-trivial results from [3], we are going to make some important remarks that will be pertinent to the analysis.
Using classical results on interpolation spaces on (bounded and unbounded) Lipschitz domains and the identification of Sobolev spaces with or without Dirichlet condition for low order indices (see [30, Theorem 3.33, Theorem B.9, Theorem 3.40]) it follows that for all \(\mu <1/2\)
$$\begin{aligned} H^1({\mathbb {R}}^d{\setminus }\varGamma )\equiv H^1(\varOmega _-)\times H^1(\varOmega _+)&\subset H^\mu (\varOmega _-)\times H^\mu (\varOmega _+) =H^\mu _0(\varOmega _-)\times H^\mu _0(\varOmega _+) \\&= [L^2(\varOmega _-),H^2_0(\varOmega _-)]_{\mu /2} \times [L^2(\varOmega _+),H^2_0(\varOmega _+)]_{\mu /2}\\&\equiv [L^2({\mathbb {R}}^d),H^2_0(\mathbb R^d{\setminus }\varGamma )]_{\mu /2}, \end{aligned}$$
and therefore for all \(\nu < 1/4\):
$$\begin{aligned} \varvec{V}=H^1({\mathbb {R}}^d{\setminus }\varGamma )^2 \subset [\varvec{H},H^2_0(\mathbb R^d{\setminus }\varGamma )^2]_\nu \subset [\varvec{H},D(A)]_\nu =D((I-A)^\nu ) . \end{aligned}$$
Because of the hypotheses on the RK method, the rational function \(R(z):=r(-z)\) satisfies the conditions of Theorem 7 (it is bounded at infinity and has all its poles in the negative real part complex half-plane). We also know that the operator \(-A\) is self-adjoint and non-negative (Proposition 2(c)). Therefore,
$$\begin{aligned} \Vert r(kA)\Vert _{\varvec{H}\rightarrow \varvec{H}}=\Vert R(-kA)\Vert _{\varvec{H}\rightarrow \varvec{H}} \le \sup _{z>0}|R(z)|=\sup _{z<0}|r(z)|= 1, \end{aligned}$$
where we have used A-stability of the RK scheme. Consequently
$$\begin{aligned} \sup _n \Vert r(kA)^n\Vert _{\varvec{H} \rightarrow \varvec{H}}\le 1 \qquad \forall n,\quad \forall k>0. \end{aligned}$$
The entries of the matrix-valued rational function \(z\mapsto ({\mathscr {I}}+z{\mathscr {Q}})^{-1}\) are rational functions with poles in \(\{ z\,:\,\mathrm {Re}\,z<0\}\) and converging to zero as \(|z|\rightarrow \infty \). Using Theorem 7 in a similar way to how we proved (51) above, it follows that
$$\begin{aligned} \Vert ({\mathscr {I}}\otimes I-k{\mathscr {Q}} \otimes A)^{-1}\Vert _{\varvec{H}^s \rightarrow \varvec{H}^s}\le C \qquad \forall k>0. \end{aligned}$$
Here we have used traditional tensor product notation for Kronecker products of \(s\times s\) matrices by operators.
Error estimates
The following proposition basically says that using RKCQ in the semidiscrete system of integral equations and then in the potential representation is equivalent to applying RK to the transmission problem associated to the heat equation satisfied by the potential fields.
If \(\varvec{U}^{h,k}_n:=(U^{h,k}_{-,n},U^{h,k}_{+,n})\in \varvec{D}^s\) is the vector of the internal stages of the RKCQ method in \([t_n,t_{n+1}]\), then
$$\begin{aligned} \varvec{U}^{h,k}_n-k({\mathscr {Q}}\otimes A_\star ) \varvec{U}^{h,k}_n&={\mathbf {1}}{\mathbf {e}}_s^T \varvec{U}^{h,k}_{n-1}, \end{aligned}$$
$$\begin{aligned} {\mathscr {B}} \otimes \varvec{U}^{h,k}_n -\varvec{\chi }(t_n+{\mathbf {c}}\,k)&\in \varvec{M}^s. \end{aligned}$$
If we collect data and potential fields in
$$\begin{aligned} U^{h,k}(\zeta )&:=(U^{h,k}_-(\zeta ),U^{h,k}_+(\zeta ))\in \varvec{D}^s\\ \varXi (\zeta )&:=((B_0(\zeta ),0,0,0),(0,0,0,B_1(\zeta ))\in \varvec{H}_\varGamma ^s, \end{aligned}$$
it then follows that
$$\begin{aligned}&k^{-1}\delta (\zeta ) U^{h,k}(\zeta ) =({\mathscr {I}} \otimes A_\star ) U^{h,k}(\zeta ), \end{aligned}$$
$$\begin{aligned}&{\mathscr {B}}\otimes U^{h,k}(\zeta )-\varXi (\zeta ) \in \varvec{M}^s. \end{aligned}$$
(See the Appendix of [31] for a detailed argument showing why this holds in the context of wave equations. Those ideas can be used almost verbatim for the heat equation.) Equation (55a) is equivalent to
$$\begin{aligned} U^{h,k}(\zeta )-\zeta {\mathbf {1}}{\mathbf {e}}_s^T U^{h,k}(\zeta ) =k({\mathscr {Q}}\otimes A_\star ) U^{h,k}(\zeta ). \end{aligned}$$
Looking at the time instances of the discrete-in-time equations encoded in the \(\zeta \)-transformed equations (56) and (55b), the result follows. \(\square \)
(\(L^2\) error estimate for the steps) Let \(\varvec{u}^{h,k}_n=\varvec{e}_s^T\varvec{U}^{h,k}_n\) be the n-th step approximation provided by the RKCQ method and assume that \(u^h\) is smooth enough as a causal function. We have the bounds: if \(p=q+1\), then
$$\begin{aligned} \Vert \varvec{u}^{h,k}_n-\varvec{u}^h(t_n)\Vert _{{\mathbb {R}}^d}\le C k^{q+1} c(\varvec{u}^h,t_n), \end{aligned}$$
whereas if \(q\le p+2\) and \(\varepsilon \in (0,1/4]\),
$$\begin{aligned} \Vert \varvec{u}^{h,k}_n-\varvec{u}^h(t_n)\Vert _{{\mathbb {R}}^d}\le C_\varepsilon k^{q+1+\frac{1}{4}-\varepsilon } c(\varvec{u}^h,t_n). \end{aligned}$$
$$\begin{aligned} c(\varvec{u}^h,t_n):=(1+ t_n)\sum _{\ell =q+1}^{p+1}\max _{ t\le t_n} \left\| \tfrac{\mathrm d^\ell }{\mathrm d t^\ell }\varvec{u}^h(t)\right\| _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}. \end{aligned}$$
If \({\mathscr {P}}: \varvec{H}_\varGamma \rightarrow \varvec{M}^\bot \) is the orthogonal projection onto \(\varvec{M}^\bot \) (the orthogonality is with respect to the \(\varvec{H}_\varGamma \) inner product), then the semidiscrete equations (see (20)) are equivalent to
$$\begin{aligned} \dot{\varvec{u}}^h(t)&=A_\star \varvec{u}^h(t), \end{aligned}$$
$$\begin{aligned} {\mathscr {P}}{\mathscr {B}} \varvec{u}^h(t)&= {\mathscr {P}} \varvec{\chi }(t), \end{aligned}$$
$$\begin{aligned} \varvec{u}^h(0)&=0. \end{aligned}$$
(Note that the second to last condition is equivalent to \(\mathscr {B}\varvec{u}(t)-\varvec{\chi }(t)\in \varvec{M}\).) If we apply the RK method to Eq. (58), and we recall that the method is stiffly accurate, we obtain that the computation of the internal stages is given by
$$\begin{aligned}&\varvec{U}^{h,k}_n ={\mathbf {1}}{\mathbf {e}}_s^T \varvec{U}^{h,k}_{n-1} +k({\mathscr {Q}}\otimes A_\star ) \varvec{U}^{h,k}_n , \end{aligned}$$
$$\begin{aligned}&{\mathscr {P}}{\mathscr {B}} \otimes \varvec{U}^{h,k}_n ={\mathscr {P}}\otimes \varvec{\chi }(t_n+{\mathbf {c}}\,k). \end{aligned}$$
These equations are clearly equivalent to Eq. (54), which have been shown to be equivalent to the equations satisfied by the fields obtained in the RKCQ method. In summary, we are dealing here with the direct application of the RK method to equations (58).
This result is now a consequence of one of the main theorems of [3]. Unfortunately the reader will be now teleported from the middle of this proof to the core of a highly technical article. We will just give a translation guide to help with the application of the results of that paper. In our context, and taking advantage of our limitation to methods with \(p\ge q+1\), the key theorem of [3] is Theorem 2. This result is stated for problems with homogeneous boundary conditions, but Section 4 of [3] explains how to handle the non-homogeneous boundary conditions and why the result still holds. The following translation table
$$\begin{aligned} \begin{array}{r||c|c|c|c|c|c|c|c|c|c|c|} {[}3] &{} {{\widetilde{A}}} &{} \partial &{} D({{\widetilde{A}}}) &{} D(A) &{} S_\alpha &{} Z_\alpha &{} \rho _k(T) &{} m &{} \nu &{} \nu ^* &{} \theta \\ \hline &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ \text{ Here } &{} A_\star &{} {\mathscr {P}}{\mathscr {B}} &{} \varvec{D} &{} \varvec{D}_h &{} (-\infty ,0] &{} \emptyset &{} 1 &{} 0 &{} \min \{p-q-1,\frac{1}{4}-\varepsilon \} &{} 1 &{} 0 \end{array} \end{aligned}$$
can be used to navigate [3, Theorem 2] and relate it to our particular problem. Note that when \(p\ge q+2\), it is important to have \(\varvec{D}\subset \varvec{V} \subset [\varvec{H},D(A)]_{1/4-\varepsilon }\) with bounded embeddings as established in (50). This is a hypothesis needed in the application of the theorems (it determines the admissible choices of \(\nu \)) and allows us to eliminate the estimates in terms of interpolated norms and write them in the natural Sobolev norm of \(\varvec{V}\). \(\square \)
We note that the error for the case \(p=q+1\) can be written with the \(\varvec{H}=L^2({{\mathbb {R}}^d{\setminus }\varGamma })^2\) norm in the right-hand-side. We will keep the \(\varvec{V}= H^1({{\mathbb {R}}^d{\setminus }\varGamma })^2\) overestimate in this case for simplicity.
(\(L^2\) error for the internal stages) With \(c(\varvec{u}^h,t_n)\) defined as in Proposition 7, we have the bounds
$$\begin{aligned} \Vert \varvec{U}^{h,k}_n-\varvec{u}^h(t_n+{\mathbf {c}}\,k)\Vert _{{{\mathbb {R}}^d}} \le C k^{q+1} c(\varvec{u}^h,t_n). \end{aligned}$$
Let \(\varvec{E}_n:=\varvec{U}^{h,k}_n-\varvec{u}^h(t_n+{\mathbf {c}}\,k)\in \varvec{D}_h^s=D(A)^s\) and note that
$$\begin{aligned} \varvec{E}_n-k({\mathscr {Q}}\otimes A) \varvec{E}_n ={\mathbf {1}}{\mathbf {e}}_s^T \varvec{E}_{n-1}+k {\mathscr {Q}} \varvec{\varTheta }_n, \end{aligned}$$
$$\begin{aligned} \varvec{\varTheta }_n:=&\, \dot{\varvec{u}}^h(t_n+{\mathbf {c}}\,k) - k^{-1} \mathscr {Q}^{-1} (\varvec{u}^h(t_n+{\mathbf {c}}\,k)- {\mathbf {1}}{\mathbf {e}}_s^T\varvec{u}^h(t_{n-1}+{\mathbf {c}}\,k)) \\ =&\, \dot{\varvec{u}}^h(t_n+{\mathbf {c}}\,k) - k^{-1} {\mathscr {Q}}^{-1} (\varvec{u}^h(t_n+{\mathbf {c}}\,k)- {\mathbf {1}} u^h(t_n)). \end{aligned}$$
Using Taylor expansions and the fact that a method with stage order q satisfies \(\ell {\mathscr {Q}}{\mathbf {c}}^{\ell -1}={\mathbf {c}}^\ell \) for \(1\le \ell \le q\) (powers of a vector are taken componentwise), we can easily prove that
$$\begin{aligned} \Vert \varvec{\varTheta }_n\Vert _{{\mathbb {R}}^d}\le C k^q \max _{t_n\le t\le t_{n+1}} \left\| \tfrac{\mathrm d^{q+1}}{\mathrm dt^{q+1}}\varvec{u}^h(t)\right\| _{{\mathbb {R}}^d}. \end{aligned}$$
Therefore, by (53), we can bound
$$\begin{aligned} \Vert \varvec{E}_n\Vert _{\varvec{H}_s} \le&C ( \Vert \varvec{u}^{h,k}_{n-1}-\varvec{u}^h(t_{n-1})\Vert _{{\mathbb {R}}^d}+k \Vert \varvec{\varTheta }_n\Vert _{{\mathbb {R}}^d}) \end{aligned}$$
and the result follows by applying (61) and Proposition 7. \(\square \)
(\(H^1\) error and estimates for boundary unknowns) With the definition of \(c(\varvec{u}^h,t_n)\) given in Proposition 7, we have the estimates
$$\begin{aligned} \Vert \phi ^{h,k}_n-\phi ^h(t_n+{\mathbf {c}} k)\Vert _{1/2,\varGamma } +\Vert \varvec{U}^{h,k}_n-\varvec{u}^h(t_n+{\mathbf {c}}\, k)\Vert _{1,{{\mathbb {R}}^d{\setminus }\varGamma }}&\le C k^{q+\frac{1}{2}} c(\varvec{u}^h,t_n),\\ \Vert \lambda ^{h,k}_n-\lambda ^h(t_n+{\mathbf {c}}\,k)\Vert _{-1/2,\varGamma }&\le C k^q c(\varvec{u}^h,t_n). \end{aligned}$$
From (60), we have
$$\begin{aligned} A\otimes \varvec{E}_n={\mathscr {Q}}^{-1}({\mathscr {Q}}\otimes A)\varvec{E}_n= k^{-1}(\varvec{E}_n-{\mathbf {1}}{\mathbf {e}}_s^T\varvec{E}_{n-1})-\varvec{\varTheta }_n, \end{aligned}$$
and therefore
$$\begin{aligned} \Vert A_\star \otimes \varvec{U}^{h,k}_n-A_\star \otimes \varvec{u}^h(t_n+\mathbf c\,k)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }=\Vert A\otimes \varvec{E}_n\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C k^q c(\varvec{u}^h,t_n), \end{aligned}$$
by Proposition 8 and (61). Therefore, by (24)
$$\begin{aligned} |\varvec{U}^{h,k}_n-\varvec{u}^h(t_n+{\mathbf {c}}\,k)|_{\varvec{V}^s} =|\varvec{E}_n|_{\varvec{V}^s} \le \Vert \varvec{E}_n\Vert _{\varvec{H}^s}^{1/2}\Vert A\otimes \varvec{E}_n\Vert _{\varvec{H}^s}^{1/2} \le C k^{q+\frac{1}{2}} c(\varvec{u}^h,t_n). \end{aligned}$$
The result is therefore clear, since the errors in the boundary quantities can be derived from jump relations applied to \(\varvec{E}_n\) and the \(H^1({{\mathbb {R}}^d{\setminus }\varGamma })^2\) norm can be bounded by the sum of the \(\varvec{V}\) seminorm and the \(\varvec{H}\) norm. \(\square \)
Laplace domain analysis
In this section we provide an alternative analysis for the Runge–Kutta Convolution Quadrature approximation. It is of a different flavor than the time-domain approach followed thus far, but provides slightly improved estimates for the convergence rates.
Lemma 1
For \(s \in {\mathbb {C}}_+\), let \(\varvec{W}(s)\in V_h\) solve (30), and set \(\big (\phi (s),\lambda (s)\big ):= \varvec{J} \varvec{W}(s)\).
The following estimates hold:
$$\begin{aligned}&|s|^{1/2} \Vert \varvec{W}(s)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }+\Vert \nabla \varvec{W}(s)\Vert _{{\mathbb {R}}^d{\setminus }\varGamma }\le C \frac{|s|^{5/4}}{\mathrm {Re}(s)} \, \Vert \mathrm H(s)\Vert _{\varvec{H}_\varGamma }, \end{aligned}$$
$$\begin{aligned}&\Vert {\lambda (s)}\Vert _{H^{-1/2}_\varGamma } \le C \frac{|s|^{3/2}}{\mathrm {Re}(s)} \, \Vert \mathrm H(s)\Vert _{\varvec{H}_\varGamma }, \end{aligned}$$
$$\begin{aligned}&\Vert {\phi (s)}\Vert _{H^{1/2}_\varGamma } \le C \frac{|s|^{5/4}}{\mathrm {Re}(s)} \, \Vert \mathrm H(s)\Vert _{\varvec{H}_\varGamma } \end{aligned}$$
$$\begin{aligned}&\Vert {\phi (s)}\Vert _{\varGamma } \le C \frac{|s|^{1}}{\mathrm {Re}(s)} \, \Vert \mathrm H(s)\Vert _{\varvec{H}_\varGamma } . \end{aligned}$$
(62) follows by inspection of the proof of Proposition 4. The trace estimates then follow from the standard trace theorem in the case of the \(H^{1/2}\)-norm of \(\phi \), and the trace theorem for weighted norms in [40, Proposition 2.5.1] (note that we are using \(c:=\sqrt{|s|}\)) for \(\lambda \). To get the \(L^2\)-estimate of \(\phi \) we use a multiplicative trace estimate (see [13, est. (1.6.2)]) to get
$$\begin{aligned} \Vert {\phi \Vert }_{\varGamma }&\lesssim \Vert {\varvec{W}(s)}\Vert _{{{\mathbb {R}}^d{\setminus }\varGamma }}^{1/2}\Vert {\nabla \varvec{W}(s)}\Vert _{{{\mathbb {R}}^d{\setminus }\varGamma }}^{1/2} \le |{s}|^{-1/4} \big (|{s}|^{1/2}\Vert {\varvec{W}(s)}\Vert _{{{\mathbb {R}}^d{\setminus }\varGamma }} + \Vert {\nabla \varvec{W}(s)}\Vert _{{{\mathbb {R}}^d{\setminus }\varGamma }}\big ). \end{aligned}$$
\(\square \)
Corollary 1
We expect the following convergence rates for the Runge–Kutta CQ-approximations:
$$\begin{aligned} \Vert {\lambda ^h(t_n)-\lambda ^{h,k}_n}\Vert _{H^{-1/2}_\varGamma }&\le C k^{\min \{q+1/2,p\}}, \\ \Vert {\phi ^h}(t_n) - \phi ^{h,k}_n\Vert _{H^{1/2}_\varGamma }&\le C k^{\min \{q+3/4,p\}}, \quad \Vert {\phi ^h}(t_n) - \phi ^{h,k}_n\Vert _{\varGamma } \le C k^{\min \{q+1,p\}}. \end{aligned}$$
The constants depend on the data, the time t and the geometry, but not on k or h.
As was already established in the proof of Proposition 4, (30) corresponds to the Laplace transformation of (28). Using the theory developed in [10], estimates in the Laplace domain of the form
$$\begin{aligned} |K(s)|\le C \frac{|s|^{\mu }}{\mathrm {Re}(s)^\nu } \end{aligned}$$
yield convergence of the CQ approximation of K of order \({\mathscr {O}}(k^{\min (q+1+\nu -\mu ,p)})\). The Corollary follows from Lemma 1. \(\square \)
Numerical experiments
In order to confirm our theoretical findings, we conduct some numerical experiments in \({\mathbb {R}}^2\). For this we combine a frequency-domain Galerkin-BEM code with the fast CQ-algorithm from [11]. Note that a faster implementation can be achieved using the Fast and Oblivious CQ method [41]. For the discretization in space we use piecewise polynomial spaces; for \(X_h\) we use discontinuous polynomials of degree p and for \(Y_h\) we use globally continuous piecewise polynomials of degree \(p+1\). We denote these pairings as \({\mathscr {P}}_p-{\mathscr {P}}_{p+1}\).
Testing on a manufactured solution We start our experiments with the case where the exact solution can be computed analytically. This allows us to test the predicted convergence rates from Sects. 4 and 5.
Since we are mainly interested in the performance of the CQ-schemes, i.e., the discretization in time we try to use a spatial discretization of higher order than the time discretization, while keeping the ratio k / h of timestep size and mesh-width constant. This means, whenever we cut the timestep size in half, we perform a uniform refinement of the spatial grid. See Table 1 for the degrees used. We note that for sufficiently smooth solutions, we expect to observe convergence rates in space of order \({\mathscr {O}}(k^{p+1})\) for the quantity \(\Vert \lambda - \lambda ^h\Vert _{L^2(\varGamma )} + \Vert \phi -\phi ^h\Vert _{H^{1/2}(\varGamma )}\) when using the space \({\mathscr {P}}_p - {\mathscr {P}}_{p+1}\) (see e.g. [39]).
The domain \(\varOmega _-\) is the quadrialteral with vertices (0, 0), (1, 0), (0.8, 0.8), (0.2, 1). The thermal transmission constants are chosen to be \(m:=\rho ^{-1}\kappa =0.8, \kappa :=1.2\). We can then prescribe a solution by picking a source point outside the polygon \(\mathbf {x}^{\mathrm {sc}}=(1.5,1.6)\) and defining:
$$\begin{aligned} u_-(\mathbf {x},t)={1\over 4\pi m t}\exp \left( -{|\mathbf {x}-\mathbf {x}^{\mathrm {sc}}|^2\over 4m t} \right) \qquad u_+(\mathbf {x},t)=0. \end{aligned}$$
For our computations, we solve the system up to the fixed end-time \(T=4\).
When using a Runge–Kutta method, we expect a reduction of order phenomenon, which depends on the norm under consideration, see Proposition 9. Since they are easier to compute, we would like to consider the \(L^2\)-errors for the Dirichlet and Neumann traces. For the Dirichlet-trace \(\phi \), the \(L^2(\varGamma )\)-norm is weaker than the \(H^{1/2}_\varGamma \)-norm in Proposition 9. We therefore expect a slightly higher rate of convergence. Namely, if we use the multiplicative trace estimate (see e.g. [13, est. (1.6.2)]) we get:
$$\begin{aligned} \Vert {\phi (t_n) - \phi ^{h,k}(t_n)}\Vert _{\varGamma }&\lesssim \Vert \varvec{u}(t_n) - \varvec{u}^{h,k}(t_n)\Vert _{0,{\mathbb {R}}^d}^{1/2} \Vert \varvec{u}(t_n) - \varvec{u}^{h,k}(t_n)\Vert _{1,{\mathbb {R}}^d {\setminus } \varGamma }^{1/2}. \end{aligned}$$
Considering the error quantity
$$\begin{aligned} E^{\phi }:=\max _{\max _{j \varDelta t \le T}}{\Vert {\phi (t_n) - \phi ^{h,k}(t_n)}\Vert _{\varGamma }} \end{aligned}$$
we therefore expect a rate of \(p_{e,\phi }:=\min \left\{ q+3/4,\frac{3}{4}p+\frac{q}{4}\right\} \) (up to arbitrary \(\varepsilon > 0\)) for the convergence in time.
For the normal derivative, the \(L^2\) norm is stronger than what is covered by our theory. Therefore, we also include an estimation for the true \(H^{-1/2}_\varGamma \)-norm. We thus consider the following two quantities:
$$\begin{aligned} E^{\lambda ,0}&:=\max _{j \varDelta t \le T}\Vert \lambda (t_j) - \lambda ^{h,k}(t_j)\Vert _{\varGamma }, \\ E^{\lambda ,-\frac{1}{2}}&:=\max _{j \varDelta t \le T}\Vert \varPi _{L^2}\lambda (t_j) - \lambda ^{h,k}(t_j)\Vert _{V(1)}, \end{aligned}$$
where \(\varPi _{L^2}\) denotes the \(L^2\)-projection onto the boundary element space \(X_h\), and \(\Vert {\cdot }\Vert ^2_{V(1)}:={\langle {V(1)\cdot ,\cdot \rangle }}\) is the norm induced by the Galerkin discretization of the operator V(1). Since the exact solution is smooth and the space-discretization error is taken to be of higher order than the time discretization, this should give a good estimate for the \(H^{-1/2}\)-error. We predict rates of \(p_{e,\lambda }:=q\).
Table 1 Expected convergence rates and space discretization used for the different Convolution Quadrature methods
For multistep methods there is no reduction of order phenomenon, and we expect to see the full convergence order. We collect the expected rates in Table 1.
When compared to the Laplace domain analysis in Sect. 5.3, our estimates in Proposition 9 do not appear to be sharp.
Using more refined techniques, one can show the convergence rate of \(\Vert {\mathrm { A} ( \varvec{u}^{h}(t_n) - \varvec{u}^{h,k}(t_n))\Vert }_{0,{\mathbb {R}}^d {\setminus } \varGamma }\) to be \({\mathscr {O}}(k^{q+1/4})\). Carefully taking traces, one can then recover the same convergence rates as obtained by the Laplace-domain method. Due to the technicalities involved, we postpone proving such estimates for general semigroup approximations to a separate upcoming article.
The Laplace theory gives the following predicted convergence rates:
$$\begin{aligned} {\widetilde{p}}_{e,\phi }:={\min \left\{ q+1,p\right\} } \qquad {\widetilde{p}}_{e,\lambda }:={q+1/2}, \end{aligned}$$
for the \(L^2\)-error of \(\phi \) and the \(H^{-1/2}_\varGamma \)-error of \(\lambda \) respectively. We include these rates in Table 1.
Comparison of different CQ methods
In Fig. 1 we observe that the Runge–Kutta based methods slightly outperform the expected rates \(p_{e,\phi }\) and \(p_{e,\lambda }\). Instead we get very good correspondence to the Laplace domain rates \({\widetilde{p}}_{e,\phi }\) and \({\widetilde{p}}_{e,\lambda }\). For multistep methods we see the full convergence rate, as was predicted in Sect. 4.
A simulation We finally illustrate the use of our method for a simulation without known analytic solution. We choose the domain \(\varOmega _-\) as a horseshoe shaped polygon, with a high conductivity parameter \(\kappa :=100\). The density \(\rho \) was set to 1. We then placed point-sources of the form
$$\begin{aligned} u^{\text {src}}(\mathbf {x},t) :={\left\{ \begin{array}{ll} \displaystyle \sum _{j=0}^{m}{{1\over 4\pi t}\exp \left( -{|\mathbf {x}-\mathbf {x}^{\mathrm {sc}}_j|^2\over 4t} \right) } &{}\text {for }\quad \mathbf {x} \in \varOmega _+, \\ 0 &{} \text { in }\quad \varOmega _-, \end{array}\right. } \end{aligned}$$
on points \(\mathbf {x}^{\mathrm {sc}}_j\) uniformly distributed on a circle. Making the ansatz for the solution \(u^{\text {tot}}=u^{\text {src}} + u\), we can compute u using our numerical method and recover \(u^{\text {tot}}\) by postprocessing. See Fig. 2a for the geometric setting and initial condition. (Note: in order to avoid the singularity at \(t=0\), we shifted the functions by a small time \(t_{\text {lag}}:=0.001\)).
We then solved the evolution problem up to the final time \(T=1\). We used the BDF(4) method with a step-size of \(k:=1/2048\). We used \({\mathscr {P}}_3 - {\mathscr {P}}_4\) elements with \(h \approx 1/64\). The results can be seen in Fig. 2.
Time evolution of the heat distribution
We have presented a collection of fully discrete methods for transmission problems associated to the heat equation in free space. The problem is reformulated as a system of time domain boundary integral equations associated to the heat kernel, thus reducing the computational work to the interface between the materials. The system is discretized with Galerkin BEM in the space variable and Convolution Quadrature (of multistep or multistage type) in time. Part of our work has consisted in dealing with the error analysis for the fully discrete method directly in the time domain, thus avoiding estimates based on Laplace transforms. All the results have been presented for the case of a single inclusion, but the extension to multiple inclusions is straightforward. The case where the inclusion has piecewise constant material properties is more complicated and will be the aim of future work.
Abreu, A.I., Canelas, A., Mansur, W.J.: A CQM-based BEM for transient heat conduction problems in homogeneous material and FGMs. Appl. Math. Model. 37(3), 776–792 (2013)
Al-Jawary, M.A., Ravnik, J., Wrobel, L.C., Škerget, L.: Boundary element formulations for the numerical solution of two-dimensional diffusion problems with variable coefficients. Comput. Math. Appl. 64(8), 2695–2711 (2012)
Alonso-Mallo, I., Palencia, C.: Optimal orders of convergence for Runge–Kutta methods and linear, initial boundary value problems. Appl. Numer. Math. 44(1–2), 1–19 (2003)
Arnold, D.N., Noon, P.J.: Coercivity of the single layer heat potential. J. Comput. Math. 7(2), 100–104 (1989). [China–US seminar on boundary integral and boundary element methods in physics and engineering (Xi'an, 1987–88)]
Bacuta, C., Hassell, M.E., Hsiao, G.C., Sayas, F.-J.: Boundary integral solvers for an evolutionary exterior Stokes problem. SIAM J. Numer. Anal. 53(3), 1370–1392 (2015)
Bamberger, A., Duong, T.H.: Formulation variationnelle pour le calcul de la diffraction d'une onde acoustique par une surface rigide. Math. Methods Appl. Sci. 8(4), 598–608 (1986)
Banjai, L.: Multistep and multistage convolution quadrature for the wave equation: algorithms and experiments. SIAM J. Sci. Comput. 32(5), 2964–2994 (2010)
Banjai, L., Laliena, A.R., Sayas, F.-J.: Fully discrete Kirchhoff formulas with CQ–BEM. IMA J. Numer. Anal. 35(2), 859–884 (2015)
Banjai, L., Lubich, C.: An error analysis of Runge–Kutta convolution quadrature. BIT Numer. Math. 51(3), 483–496 (2011)
Banjai, L., Lubich, C., Melenk, J.M.: Runge–Kutta convolution quadrature for operators arising in wave propagation. Numer. Math. 119(1), 1–20 (2011)
Banjai, L., Sauter, S.: Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. 47(1), 227–249 (2008/09)
Banjai, L., Schanz, M.: Wave propagation problems treated with convolution quadrature and BEM. In: Langer, U., et al. (eds.) Fast Boundary Element Methods in Engineering and Industrial Applications. Lecture Notes Applied Computational Mechanics, vol. 63, pp. 145–184. Heidelberg, Springer (2012)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)
Chapko, R., Kress, R.: Rothe's method for the heat equation and boundary integral equations. J. Integral Equ. Appl. 9(1), 47–69 (1997)
Costabel, M.: Boundary integral operators for the heat equation. Integral Equ. Oper. Theory 13(4), 498–552 (1990)
Costabel, M., Stephan, E.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106(2), 367–413 (1985)
Domínguez, V., Sayas, F.-J.: Some properties of layer potentials and boundary integral operators for the wave equation. J. Integral Equ. Appl. 25(2), 253–294 (2013)
Engel, K.-J., Nagel, R.: A Short Course on Operator Semigroups. Universitext. Springer, New York (2006)
Greengard, L., Lin, P.: Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9(1), 83–97 (2000)
Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory Advances and Applications, vol. 169. Birkhäuser Verlag, Basel (2006)
Hassell, M., Sayas, F.-J.: Convolution quadrature for wave simulations. In: Higueras, I., et al. (eds.) Numerical Simulation in Physics and Engineering. SEMA SIMAI Springer Series, vol. 9, pp. 71–159. Cham, Springer (2016)
Hassell, M.E., Qiu, T., Sánchez-Vizuet, T., Sayas, F.-J.: A new and improved analysis of the time domain boundary integral operators for the acoustic wave equation. J. Integral Equ. Appl. 29(1), 107–136 (2017)
Kesavan, S.: Topics in Functional Analysis and Applications. Wiley, New York (1989)
Laliena, A.R., Sayas, F.-J.: Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves. Numer. Math. 112(4), 637–678 (2009)
Li, J., Monk, P., Weile, D.: Time domain integral equation methods in computational electromagnetism. In: de Castro, A.B., Valli, A. (eds.) Computational Electromagnetism. Lecture Notes in Mathematics, vol. 2148, pp. 111–189. Berlin, Springer (2015)
Liggett, J., Liu, P.: The Boundary Integral Equation Method for Porous Media Flow. Allen & Unwin, London (1983)
Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52(2), 129–145 (1988)
Lubich, C., Ostermann, A.: Runge–Kutta methods for parabolic equations and convolution quadrature. Math. Comput. 60(201), 105–131 (1993)
Lubich, C., Schneider, R.: Time discretization of parabolic boundary integral equations. Numer. Math. 63(4), 455–481 (1992)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Melenk, J.M., Rieder, A.: Runge–Kutta convolution quadrature and FEM–BEM coupling for the time-dependent linear Schrödinger equation. J. Integral Equ. Appl. 29(1), 189–250 (2017)
Messner, M., Schanz, M., Tausch, J.: A fast Galerkin method for parabolic space–time boundary integral equations. J. Comput. Phys. 258, 15–30 (2014)
Messner, M., Schanz, M., Tausch, J.: An efficient Galerkin boundary element method for the transient heat equation. SIAM J. Sci. Comput. 37(3), A1554–A1576 (2015)
Monegato, G., Scuderi, L., Stanić, M.P.: Lubich convolution quadratures and their application to problems described by space–time BIEs. Numer. Algorithms 56(3), 405–436 (2011)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)
Qiu, T., Sayas, F.-J.: The Costabel–Stephan system of boundary integral equations in the time domain. Math. Comput. 85(301), 2341–2364 (2016)
Qiu, T., Sayas, F.-J.: New mapping properties of the time domain electric field integral equation. ESAIM Math. Model. Numer. Anal. 51(1), 1–15 (2017)
Rapún, M.-L., Sayas, F.-J.: Boundary element simulation of thermal waves. Arch. Comput. Methods Eng. 14(1), 3–46 (2007)
Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics, vol. 39. Springer, Berlin (2011). (Translated and expanded from the 2004 German original )
Sayas, F.-J.: Retarded Potentials and Time Domain Integral Equations: A Roadmap. Springer Series in Computational Mathematics, vol. 50, 1st edn. Springer, Berlin (2016)
Schädle, A., López-Fernández, M., Lubich, C.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28(2), 421–438 (2006)
Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol. 265. Springer, Dordrecht (2012)
Tausch, J.: A fast method for solving the heat equation by layer potentials. J. Comput. Phys. 224(2), 956–969 (2007)
Tausch, J.: Nyström discretization of parabolic boundary integral equations. Appl. Numer. Math. 59(11), 2843–2856 (2009)
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25, 2nd edn. Springer, Berlin (2006)
Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, London (1967)
Wrobel, L., Brebbia, C.: The boundary element method for steady state and transient heat conduction. Numer. Methods Therm. Problems 1, 58–73 (1979)
Open access funding provided by Austrian Science Fund (FWF).
Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA
Tianyu Qiu
& Francisco-Javier Sayas
Institut für Analysis und Scientific Computing, TU Wien, 1040, Vienna, Austria
Alexander Rieder
College of Mathematics Science, Chongqing Normal University, Chongqing, People's Republic of China
Shougui Zhang
Search for Tianyu Qiu in:
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Correspondence to Alexander Rieder.
Alexander Rieder: Funded by the Austrian Science Fund (FWF) (Grant W1245 and F65).
Francisco-Javier Sayas: Partially funded by NSF (Grant DMS 1216356).
Shougui Zhang: Supported by China Scholarship Council.
A background material
A.1 A result on abstract evolution equations
(Cauchy problems for analytic semigroups) Let X be a Hilbert space, \(B:D(B)\rightarrow X\) be self-adjoint and maximal dissipative, and let \(f\in {\mathscr {C}}^\theta ({\mathbb {R}}_+; X)\) with \(\theta \in (0,1)\) and \(f(0)=0\). The nonhomogeneous initial value problem
$$\begin{aligned} {\dot{w}}_0(t) = Bw_0(t) + f(t)\quad t\ge 0,\quad w_0(0) = 0, \end{aligned}$$
has a unique solution \(w_0\in \mathscr {C}^{1+\theta }({\mathbb {R}}_+;X)\cap {\mathscr {C}}^\theta ({\mathbb {R}}_+;D(B))\) and
$$\begin{aligned} \Vert w_0(t)\Vert _X \le \int _0^t \Vert f(\tau )\Vert _X \,\mathrm {d} \tau , \qquad \Vert B w_0(t)\Vert _X \le t^\theta \theta ^{-1}| f|_{t,\theta , X} + 2\Vert f(t)\Vert _X. \end{aligned}$$
We will use results from [35, Section 4.3] concerning non-homogeneous problems associated to analytic semigroups. By [35, Chapter 4, Theorem 3.5(iii)] the initial value problem (68) has a unique solution with the given regularity and this solution is given by the variation of constants formula
$$\begin{aligned} w_0(t)=\int _0^t T(t-\tau ) f(\tau )\mathrm d\tau , \end{aligned}$$
where \(\{ T(t)\}\) is the associated contractive semigroup. Since B is selfadjoint, it follows that \(\Vert tBT(t)\Vert \le 1\) for all \(t>0\) (see [23, Theorem 4.5.2] for instance). To bound the norm of \(Bw_0(t)\) we proceeed as in the proof of [35, Chapter 4, Theorem 3.2] and decompose
$$\begin{aligned} Bw_0(t)&=B\int _0^t T(t-\tau )(f(\tau )- f(t))\,\mathrm {d}\tau + B\int _0^t T(\tau ) f(t)\,\mathrm {d}\tau \\&=\int _0^t B T(t-\tau )(f(\tau )- f(t))\,\mathrm {d}\tau +T(t) f(t)-f(t). \end{aligned}$$
$$\begin{aligned} \int _0^t \Vert BT(t-\tau ) (f(\tau )-f(t))\Vert _X\mathrm d\tau \le |f|_{t,\theta ,X}\int _0^t\frac{1}{(t-\tau )^{1-\theta }}\mathrm d\tau , \end{aligned}$$
the result follows easily. \(\square \)
A.2 BDF–CQ
Let \(\delta \) be the backward differentiation symbol introduced in (38). When \(\mathrm F:{\mathbb {C}}_\star \rightarrow \mathscr {B}(X_1,X_2)\) is an operator-valued analytic function,
$$\begin{aligned} \mathrm F(\tfrac{1}{k} \delta (\zeta )) = \sum _{n=0}^\infty \omega _n^{\mathrm F} (k) \zeta ^n \end{aligned}$$
is given by the Taylor expansion of the left hand side about \(\zeta =0\) since \(\delta (\zeta )\) takes values in the domain of \(\mathrm F\) for small \(|\zeta |\). (This is due to the \(A(\theta )\)-stability of the BDF formulas of order less than or equal to six.) We thus obtain a sequence of operator-valued coefficients
$$\begin{aligned} \omega _n^{\mathrm F}(k)\in {\mathscr {B}}(X_1,X_2). \end{aligned}$$
Then given \(\mathrm G(\zeta ):=\sum _{n=0}^\infty g_n \zeta ^n : {\mathbb {C}}_\star \rightarrow X_1\), when we multiply
$$\begin{aligned} \mathrm F (\tfrac{1}{k} \delta (\zeta )) \mathrm G(\zeta ) = \sum _{n=0}^\infty \sum _{m=0}^n \left( \omega _{n-m}^{\mathrm F}(k) g_m \right) \zeta ^n, \end{aligned}$$
we are just computing the discrete causal convolution of the sequence of operators \(\{ \omega _n^{\mathrm F}(k) \}\) to the discrete sequence \(\{g_n\}\). Computational strategies for efficient implementation of multistep CQ (in the context we find it in (40)–(41)) can be found in the literature. The lecture notes [21] contain a simple introduction to the topic.
A.3 Rudiments of functional calculus
We here introduce some minimun requirements on (Dunford–Riesz) functional calculus needed for our work. First of all, here is a result concerning bounds for rational functions of non-negative self-adjoint operators. The theorem is a consequence of more general results related to functions of operators, which can be found in general introductions to functional calculus (see, for instance, [20, Corollary 7.1.6, Theorem 2.2.3]).
Let \(B:D(B)\subset X\rightarrow X\) be a self-adjoint non-negative operator in a Hilbert space X. Let \(P,Q\in {\mathscr {P}}({\mathbb {C}})\) be two polynomials such that the rational function \(R:=P/Q\) is bounded at infinity (\(\mathrm {deg}\,P\le \mathrm {deg}\,Q\)) and has all its poles in \(\{s\in {\mathbb {C}}\,:\, \mathrm {Re}\,s<0\}\). Then \(R(B):X\rightarrow X\) is bounded and
$$\begin{aligned} \Vert R(B)\Vert _{X\rightarrow X}\le \sup _{z>0} |R(z)|. \end{aligned}$$
We will also need the evaluation of analytic functions on matrices, using Dunford calculus. Let \(\mathrm F:{\mathscr {O}}\subset \mathbb C\rightarrow X\) be an analytic function defined on a simply connected open set of \({\mathbb {C}}\). Let \({\mathscr {M}}\) be a matrix whose spectrum is contained in \({\mathscr {O}}\). We then define
$$\begin{aligned} \mathrm F({\mathscr {M}}):=\frac{1}{2\pi \imath } \oint _C (z\mathscr {I}_s-{\mathscr {M}})^{-1}\otimes \mathrm F(z)\mathrm dz, \end{aligned}$$
where C is any simple positively oriented open contour in \({\mathscr {O}}\) surrounding the spectrum of \({\mathscr {M}}\).
A.4 RK–CQ
Let \(\mathrm F:{\mathbb {C}}_+\rightarrow {\mathscr {B}}(X_1,X_2)\) be an operator-valued analytic function and consider the expansion (70), where now \(\delta (\zeta )\in {\mathbb {R}}^{s\times s}\) is defined by (49). Since for \(\zeta \) small \(\delta (\zeta )\) is a small perturbation of \({\mathscr {Q}}^{-1}\) and \({\mathscr {Q}}\) has its spectrum contained in \({\mathbb {C}}_+\), then \(\mathrm F(k^{-1}\delta (\zeta ))\) can be defined using functional calculus as in "Appendix A.3". The expansion is then a simple Taylor expansion about the origin for an analytic function with values in \({\mathscr {B}}(X_1,X_2)^{s\times s}\). The coefficients of this expansion can be given by the Cauchy integrals
$$\begin{aligned} \omega _n^{\mathrm F} (k)=\frac{1}{2\pi \imath }\oint _C (z\mathscr {I}_s-k^{-1} {\mathscr {Q}}^{-1})^{-(n+1)}\otimes \mathrm F(z)\mathrm dz \in {\mathscr {B}}(X_1,X_2)^{s\times s}, \end{aligned}$$
where C is any simple positively oriented closed contour in \({\mathbb {C}}_+\) surrounding the spectrum of \({\mathscr {Q}}^{-1}\). Discrete causal convolutions in the form (71) can be made now for sequences \(\{ g_n\}\) in \(X_1^s\). A simple practical introduction to Dunford calculus related to RK–CQ methods can be found in [21, Section 6.3]. Practical computational strategies involve diagonalizing \(\delta (\zeta )\), cf. [7, 12, 21].
Qiu, T., Rieder, A., Sayas, F. et al. Time-domain boundary integral equation modeling of heat transmission problems. Numer. Math. 143, 223–259 (2019) doi:10.1007/s00211-019-01040-y
Issue Date: 01 September 2019
DOI: https://doi.org/10.1007/s00211-019-01040-y
Mathematics Subject Classification
65N38 | CommonCrawl |
\begin{document}
\title[Nevanlinna families] {Invariance theorems for Nevanlinna families} \author{Vladimir Derkach} \author{Seppo Hassi} \author{Mark Malamud}
\address{Department of Mathematics, National Pedagogical University, Kiev, Pirogova 9, 01601, Ukraine} \email{[email protected]} \address{Department of Mathematics and Statistics \\ University of Vaasa \\ P.O. Box 700, 65101 Vaasa \\ Finland} \email{[email protected]}
\email{[email protected] }
\subjclass[2000]{Primary 30E20, 47A07, 47A10, 47A56; Secondary 30C40, 30C80, 47B10, 47B44}
\keywords{Holomorhic operator-valued function, Herglotz-Nevanlinna function, harmonic function, Harnack's inequality, maximum principle, nonnegative kernel, sesquilinear form, spectrum, resolvent, Schatten-von Neumann classes.}
\begin{abstract} A complex function $f(z)$ is called a Herglotz-Nevanlinna function if it is holomorphic in the upper half-plane ${\mathbb C}_+$ and maps ${\mathbb C}_+$ into itself. By a maximum principle a Herglotz-Nevanlinna function which takes a real value $a$ in a single point $z_0\in {\mathbb C}_+$ should be identically equal to $a$. In the present note we prove similar invariance results both for the point and the continuous spectra of an operator-valued Herglotz-Nevanlinna function with values in the set of bounded or unbounded linear operators (or relations) in a Hilbert space. The proof of this invariance result for continuous spectrum is based on Harnack's inequality. This inequality is systematically used to characterize operator-valued Herglotz-Nevanlinna functions with form-domain invariance property for their imaginary parts or Herglotz-Nevanlinna functions with values in the Schatten-von Neumann classes. \end{abstract}
\maketitle
\section{Introduction} The class of Herglotz-Nevanlinna functions plays an important role in function theory, probability theory, mathematical physics, etc. In particular, the m-function of a Sturm-Liouville operator on a half-line belongs to this class; \cite{Titch62}, \cite{Co}. Similarly, the M-function of an elliptic operators is an operator-valued Herglotz-Nevanlinna function; see \cite{AmrPear04}. Also the Kre\u{\i}n's formula for (generalized) resolvents involves an another source for various important applications of this class. In particular, Kre\u{\i}n's formula allows to parametrize sets of solutions of various classical interpolation and moment problems with a parameter ranging over the class of Herglotz-Nevanlinna families; cf. \cite{Kr46}, \cite{KL71}. For basic properties of Herglotz-Nevanlinna functions see e.g. the surviews in~\cite{KacK}, \cite{Br}, \cite{Don74}.
The class $R[\cH]$ of Herglotz-Nevanlinna functions with values in the set $\cB(\cH)$ of bounded linear operators in a separable Hilbert space $\cH$ is defined as follows. \begin{definition} \label{def:Rfunc0} An operator-valued function $F({z})$ holomorphic on $ {\dC \setminus \dR}$, with values in $\cB(\cH)$ is said to belong to the class $R[\cH]$, if: \begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})} \item for every ${z} \in \dC_+ (\dC_-)$
the operator $F({z})$ is dissipative
(resp. accumulative); \item $F({z})^*=F(\bar {z})$, ${z} \in {\dC \setminus \dR}$; \end{enumerate} \end{definition} In what follows an operator $T\in\cB(H)$ is called dissipative
(resp. accumulative), if its imaginary part \[ {\rm Im\,} (T)=\frac{1}{2i}(T-T^*) \] is a nonnegative (resp. nonpositive) operator in $\cH$; cf. \cite{Kato}, \cite{Ph0}.
Each operator-valued function $F\in R[\cH]$ admits the following integral representation \begin{equation} \label{INTrep}
F({z})
= B_0 + B_1{z}+\int_{\dR}\left(\frac{1}{t-{z}}-\frac{t}{t^2+1}\right)\,
d\Sigma(t), \end{equation}
where $B_0= B_0^*\in\cB(\cH)$, $0\le B_1=B_1^*\in\cB(\cH)$, and $\Sigma(\cdot)$ is a $\cB(\cH)$-valued operator measure, such that
\begin{equation} \label{INT_cond} K_\Sigma := \int_{\dR}\,\frac{d\Sigma(t)}{t^2+1}\in\cB(\cH).
\end{equation}
Here integral in~\eqref{INT_cond} is uniformly convergent in $\cB(\cH)$; cf. \cite{Br}, \cite{KacK}, \cite{GesTs00}.
The next result summarizes some invariance results on the spectra properties of operator-valued functions $F\in R[\cH]$ (cf.~\cite[Proposition~1.2]{DM97}).
\begin{theorem}\label{prop:Inv} Let $F\in R[\cH]$, $z_0\in\dC_+$ and $a=\bar a$. Then the following equivalences hold: \begin{enumerate}
\item [(1)] $0\in\sigma_p({\rm Im\,}(F(z_0)))\Longleftrightarrow 0\in\sigma_p({\rm Im\,}(F(z)))$ for all $z\in\dC_+$;
\item [(2)] $0\in\sigma_c({\rm Im\,}(F(z_0)))\Longleftrightarrow 0\in\sigma_c({\rm Im\,}(F(z)))$ for all $z\in\dC_+$;
\item [(3)] $0\in\rho({\rm Im\,}(F(z_0)))\Longleftrightarrow 0\in\rho({\rm Im\,}(F(z)))$ for all $z\in\dC_+$;
\item [(4)] $a\in\sigma_p(F(z_0))\Longleftrightarrow a\in\sigma_p(F(z))$ for all $z\in\dC_+$;
\item [(5)] $a\in\sigma_c(F(z_0))\Longleftrightarrow a\in\sigma_c(F(z))$ for all $z\in\dC_+$;
\item [(6)] $a\in\rho(F(z_0))\Longleftrightarrow a\in\rho(F(z))$ for all $z\in\dC_+$. \end{enumerate} \end{theorem}
The following two subclasses of the class $R[\cH]$ appear in the theory of $Q$-functions of symmetric operators, \cite{KL73}, and in the boundary triplet approach to the extension theory of symmetric operators, \cite{DM91,DM95}. \begin{equation}\label{eq:R_su}
\begin{split}
R^s[\cH]&=\left\{F(\cdot) \in R[\cH]:\,{\xker\,} {\rm Im\,} F({z})=\{0\}\mbox{ for all
}{z}\in{\dC \setminus \dR}\,\right\};\\
R^u[\cH]&=\left\{F(\cdot) \in R^s[\cH]:\,0 \in \rho({\rm Im\,} F({z}))\mbox{ for all
}{z}\in{\dC \setminus \dR}\,\right\}. \end{split} \end{equation}
It follows from Theorem~\ref{prop:Inv} that each of the subclasses $R^s[\cH]$ and $R^u[\cH]$ can be single out by a single condition: \begin{equation}\label{eq:R_su2}
\begin{split}
R^s[\cH]&=\left\{F \in R[\cH]:\,{\xker\,} {\rm Im\,} F(i)=\{0\}\,\right\};\\
R^u[\cH]&=\left\{F \in R^s[\cH]:\,0 \in \rho({\rm Im\,} F(i))\,\right\}. \end{split} \end{equation} The classes $R^u[\cH]$, $R^s[\cH]$, and $R[\cH]$ are ordered by inclusion \begin{equation} R^u[\cH] \subset R^s[\cH] \subset R[\cH]. \end{equation}
It follows from~\eqref{eq:R_su2} and Theorem~\ref{prop:Inv} that the operator function $F(z)$ with the integral representation~\eqref{INTrep} belongs to the class $R^s[\cH]$ (or $R^u[\cH]$), if $0\not\in\sigma_p(\Sigma_0)$ ($0\in\rho(\Sigma_0)$, respectively).
Recall that every Nevanlinna family can be realized (uniquely up to unitary equivalence) as a Weyl function (or Weyl family) of a (minimal unitary) boundary relation; see \cite{DHMS06}, \cite[Theorem~3.9]{DHMS12}. Various other subclasses of the class $\wt R(\cH)$ appearing above and below can be characterized in boundary triplet and boundary relation context. Each property of $R$-function in Theorem~1.2 when treated as a Weyl function of a (generalized) boundary triplet has its geometrical counterpart. For instance, the class of $R^u[\cH]$-functions is known to characterize the class of Weyl functions corresponding to ordinary boundary triplets of $A^*$, where $A$ is a not necessarily densely defined symmetric operator in $\mathfrak H$ (see \cite{KL73}, \cite{DM91,DM95}).
The class $ R^s[\cH]$ gives a precise characterization of
Weyl functions of symmetric operators, corresponding to the so-called $B$-generalized boundary triplets (see \cite{DM95}, \cite{DHMS06}).
The paper is organized as follows. For later purposes a proof of items (1) -- (4) in Theorem~\ref{prop:Inv} will be presented in Section~2, while items (5) and (6) will be treated in Section~3, where all of these invariance results are extended to the class $R(\cH)$ of Herglotz-Nevanlinna functions with values in the set $\cC(\cH)$ of closed linear operators and to the class $\wt R(\cH)$ of Nevanlinna families. The proof of the first half of these invariance results is based on the maximum principle for Herglotz-Nevanlinna functions or alternatively for contractive holomorphic operator functions. The rest is proven then with the help of the Harnack's inequality for harmonic functions. It is emphasized that no realization results via operator or functional models, or boundary triplets methods, for functions from these classes of operator functions are involved in the given arguments.
Harnack's inequality is systematically used in Section~4 to characterize invariance properties of operator-valued harmonic functions as well as for Herglotz-Nevanlinna functions whose imaginary parts have a so-called form-domain invariance property and for Herglotz-Nevanlinna functions with values in the Schatten-von Neumann classes. For instance, by applying such analytic arguments it is shown (see Proposition \ref{prop6.2}) that under certain additional assumptions a Herglotz-Nevanlinna function $F(\cdot)$ whose imaginary part is bounded at one point admits a representation
\begin{equation}\label{6.9B} F(z) = G(z) + T, \quad z\in \C_+,
\end{equation}
where $G(\cdot)\in R[\cH]$ and $T=T^*\in \cC(\cH)$ if and only if $F_I(z_0)\in \cB(\cH)$ for some $z_0\in\mathbb C_+$. Functions of the form \eqref{6.9B} with $G(\cdot)\in R^s[\cH]$ characterize Weyl functions of generalized boundary triplets with a selfadjoint operator $A_0= A^*\lceil {\xker\,}\Gamma_0$; see \cite[Section~4]{DHMS06} , \cite[Theorem 7.39]{DHMS12}. Similar functions appear also in a connection with the so-called quasi-boundary triplets introduced and investigated in \cite{BeLa07}. In fact, an arbitrary function of the form, \[
F(z)=T+F_0(z), \quad z\in {\dC \setminus \dR}, \] where $T$ is a symmetric densely defined operator (not necessarily self-adjoint or even closed) on $\cH$ and $F_0(\cdot)$ belongs to the class $R[\cH]$. Such $F(\cdot)$ appears as a Weyl function of a so-called almost $B$-generalized boundary triplet (possibly multi-valued); a concept that is originated in a forthcoming paper \cite{DHM15} by the authors. Such functions play a central role in the study of form-domain invariant Nevanlinna families (see Definition \ref{Nevforminv} below). Characteristic properties of such functions are investigated in \cite{DHM15} within boundary triplet (and boundary relation) setting: in particular, it is shown therein that a Herglotz-Nevanlinna function having a form-domain invariant imaginary part can be realized as the Weyl function of a unitary boundary triplet (boundary relation) $\{\cH,\Gamma_0,\Gamma_1\}$ with an essentially selfadjoint kernel $A_0={\xker\,} \Gamma_0$. Such functions appear in applications, e.g. in the study of local point interactions, \cite{KosMMM,MN2012}, in PDE setting as M-functions, Dirichlet-to-Neumann maps, and their analogs; see e.g. \cite[Section 7.7]{Post12} for a treatment of the Zaremba problem.
Finally, in Section~5 we present several examples which reflect various invariance properties of associated Herglotz-Nevanlinna functions and stability properties of quadratic forms generated by the imaginary parts of such functions.
\section{Preliminaries}
\subsection{Linear relations in Hilbert spaces} Let $\cH$ be a separable Hilbert space. The set of bounded (closed) linear operators in $\cH$ is denoted by $\cB(\cH)$, ($\cC(\cH)$, respectively).
Recall that a linear relation $T$ in $\cH$ is a linear subspace of $\cH \times \cH$. Systematically a linear operator $T$ will be identified with its graph. The set of closed linear relations in $\cH$ is denoted by $\wt \cC(\cH)$. It is convenient to interpret the linear relation $T$ as a multi-valued linear mapping from $\cH$ into $\cH$. For a linear relation $T\in\wt \cC(\cH)$ the symbols ${\rm dom\,} T$, ${\xker\,} T$, ${\rm ran\,} T$, and ${\rm mul\,} T$ stand for the domain, kernel, range, and the multi-valued part, respectively. The inverse $T^{-1}$ is a linear relation in $\cH$ defined by $\{\,\{f',f\}:\,\{f,f'\}\in T\,\}$. The adjoint $T^*$ is the closed linear relation from $\cH$ defined by (see~\cite{AI}, \cite{Ben}, \cite{Co}) \[
T^*=\{\,\{h,k\} \in \cH \oplus \cH :\,
(k,f)_{\cH}=(h,g)_{\cH}, \, \{f,g\}\in T \,\}. \] The sum $T_1+T_2$ and the componentwise sum $T_1 \wh + T_2$ of two linear relations $T_1$ and $T_2$ are defined by \[ \begin{split}
&T_1+T_2=\{\,\{f, g+h\} :\, \{f,g\} \in T_1, \{f,h\} \in T_2\,\}, \\
&T_1 \hplus T_2=\{\, \{f+h,g+k\} :\, \{f,g\} \in T_1, \{h,k\}\in T_2\,\}. \end{split} \] If the componentwise sum is orthogonal it will be denoted by $T_1 \oplus T_2$. Moreover, $\rho(T)$ ($\hat\rho(T)$) stands for the set of regular (regular type) points of $T$. The closure of a linear relation $T$ will be denoted by ${\rm clos\,} T$.
Recall that a linear relation $T$ in $\cH$ is called \textit{symmetric} (\textit{dissipative} or \textit{accumulative}) if $\mbox{Im }(h',h)=0$ ($\ge 0$) or $\le 0$, respectively) for all $\{h,h'\}\in T$. These properties remain invariant under closures. By polarization it follows that a linear relation $T$ in $\cH$ is symmetric if and only if $T \subset T^*$. A linear relation $T$ in $\cH$ is called \textit{selfadjoint} if $T=T^*$, and it is called \textit{essentially selfadjoint} if ${\rm clos\,} T=T^*$. A dissipative (accumulative) linear relation $T$ in $\cH$ is called maximal dissipative (maximal accumulative) if it has no proper dissipative (accumulative) extensions.
Assume that $T$ is closed. If $T$ is dissipative or accumulative, then ${\rm mul\,} T \subset {\rm mul\,} T^*$. In this case the orthogonal decomposition $\cH=({\rm mul\,} T)^\perp \oplus {\rm mul\,} T$ induces an orthogonal decomposition of $T$ as \[
T=T_s \oplus T_\infty, \quad T_\infty=\{0\} \times {\rm mul\,} T, \quad
T_s=\{\, \{f,g\} \in T:\, g \perp {\rm mul\,} T\,\}, \] where $T_\infty$ is a selfadjoint relation in ${\rm mul\,} T$ and $T_s$ is an operator in $\cH \ominus {\rm mul\,} T$ with ${\rm \overline{dom}\,} T_s={\rm \overline{dom}\,} T=({\rm mul\,} T^*)^\perp$, which is dissipative or accumulative. Moreover, if the relation $T$ is maximal dissipative or accumulative, then ${\rm mul\,} T={\rm mul\,} T^*$. In this case the orthogonal decomposition $({\rm dom\,} T)^\perp={\rm mul\,} T^*$ shows that $T_s$ is a densely defined dissipative or accumulative operator in $({\rm mul\,} T)^\perp$, which is maximal (as an operator); see e.g. \cite[Sec.~3,~Cor.~4.16 ]{HdSSz2009}. In particular, if $T$ is a selfadjoint relation, then there is such a decomposition where $T_s$ is a selfadjoint operator (densely defined in $({\rm mul\,} T)^\perp$).
\subsection{Nevanlinna families}
\begin{definition}
A family of linear relations $\cF({z})$, ${z} \in {\dC \setminus \dR}$, in a Hilbert space $\cH$ is called a \textit{Nevanlinna family} if: \begin{enumerate}
\item[(NF1)] for every ${z} \in \dC_+ (\dC_-)$
the relation $\cF({z})$ is maximal dissipative
(resp. accumulative); \item[(NF2)] $\cF({z})^*=\cF(\bar {z})$, ${z} \in {\dC \setminus \dR}$; \item[(NF3)] for some (and hence for all) ${w} \in \dC_+ (\dC_-)$ the
operator family $(\cF({z})+{w})^{-1} (\in [\cH])$ is
holomorphic in ${z} \in \dC_+ (\dC_-)$. \end{enumerate} \end{definition}
The condition $(\cF({z})+{w})^{-1} (\in [\cH])$ implies that $\cF({z})$ is maximal dissipative
or accumulative relation $\cF({z})$, ${z} \in {\dC \setminus \dR}$, and thus, in particular, closed. The \textit{class of all Nevanlinna families} in a Hilbert space is denoted by $\wt R(\cH)$. If the multi-valued part ${\rm mul\,} \cF({z})$ of $\cF(\cdot) \in \wt R(\cH)$ is nontrivial, then it is independent of ${z} \in {\dC \setminus \dR}$, so that \begin{equation} \label{ml}
\cF({z})=\cF_s({z}) \oplus \cF_\infty, \quad \cF_\infty=\{0\}
\times {\rm mul\,} \cF({z}), \quad {z}\in\dC\setminus\dR, \end{equation} where $\cF_s({z})$ is a Nevanlinna family of single-valued linear relations in $\cH \ominus {\rm mul\,} F({z})$, \cite{KL71}.
Clearly, if $\cF(\cdot) \in \wt R(\cH)$, then $\cF_\infty \subset \cF({z}) \cap \cF({z})^*$ for all ${z}\in\dC\setminus\dR$.
\subsection{Class $R(\cH)$} Class $R(\cH)$ is defined below as a class of all single-valued Nevanlinna families. \begin{definition} \label{def:Rfunc} An operator-valued function $F({z})$, ${z} \in {\dC \setminus \dR}$, with values in $\cC(H)$ is said to belong to the class $R(\cH)$, if: \begin{enumerate}
\item[(NF1)] for every ${z} \in \dC_+ (\dC_-)$
the operator $F({z})$ is maximal dissipative (resp. accumulative); \item[(NF2)] $F({z})^*=F(\bar {z})$, ${z} \in {\dC \setminus \dR}$; \item[(NF3)] for some (and hence for all) ${w} \in \dC_+ (\dC_-)$ the
operator function $(F({z})+{w})^{-1} (\in [\cH])$ is
holomorphic in ${z} \in \dC_+ (\dC_-)$. \end{enumerate} \end{definition}
The following subclass \[ R[\cH]=\left\{F(\cdot) \in R(\cH):\,{\rm dom\,} F({z})=\cH\mbox{ for all }{z}\in{\dC \setminus \dR}\,\right\} \] of the class $R(\cH)$ consist of operator-valued functions $F({z})$ with values in $\cB(\cH)$.
Analogous to~\eqref{eq:R_su}, the following subclasses of the class $R(\cH)$ can be determined \[ \begin{split}
R^s(\cH)&=\left\{F(\cdot) \in R(\cH):{\xker\,}({\rm Im\,} F({z})) =\{0\}\mbox{ for all
}{z}\in{\dC \setminus \dR}\,\right\};\\
R^u(\cH)&=\left\{F(\cdot) \in R(\cH):\,\cF({z}) \hplus \cF({z})^*=\cH^2\mbox{ for all
}{z}\in{\dC \setminus \dR}\,\right\},\\ \end{split} \] where $\cF(z)$ is the graph of the linear operator $F(z)$. As will be shown in Proposition~\ref{Runif+} (see also~\cite[Proposition~2.18]{DHMS06}) the classes $R^u(\cH)$ and $R^u[\cH]$ coincide.
The Nevanlinna functions in $R^s(\cH)$ and $R^u[\cH]$ will be called \textit{strict} and \textit{uniformly strict}, respectively.
These definitions give rise to the following chain of inclusions: \begin{equation} \label{scheme} R^u(\cH) \subset R^s(\cH) \subset R(\cH) \subset \wt R(\cH). \end{equation} In the infinite-dimensional situation each of the inclusions in~\eqref{scheme} is strict.
It follows from the integral representation~\eqref{INTrep} for every $F\in R[\cH]$ the kernel \begin{equation}\label{eq:Kern_F}
{\mathbf N}_F(z,w):=\left\{\begin{array}{cc}
{\displaystyle\frac{F(z)-F( w)^*}{z-\bar w}}, & \mbox{if }\, z\ne\bar w; \\
F'(z) & \mbox{if }\, z=\bar w.
\end{array}\right. \end{equation} is nonnegative in $\dC_+\cup\dC_-$. This observation leads to a couple of the invariance results in the class $R[\cH]$, which were stated in Theorem~\ref{prop:Inv} and can be considered to be well known. For the convenience of the reader, a short proof for items (1)-(4) is given; the rest will follow from a more general Theorem~\ref{invar} given below.
\begin{proof}[Proof of Theorem~\ref{prop:Inv} (1)-(4)]
(1) Assume that $0\in\sigma_p({\rm Im\,}(F(z_0)))$ and ${\rm Im\,}(F(z_0))h_0=0$ for some $h_0\in\cH$, $h_0\ne 0$. The matrix \begin{equation}\label{eq:Im_F0}
\left(
\begin{array}{cc}
\left({\mathbf N}_F(z_0,z_0)h_0,h_0\right) & \left({\mathbf N}_F(z_0,z)h_0,h_0\right) \\
\left({\mathbf N}_F(z_0,z_0)h_0,h\right) & \left({\mathbf N}_F(z,z)h_0,h_0\right) \\
\end{array}
\right)
\end{equation} is nonnegative for all $z\in\dC_+\cup\dC_-$, $z\ne \bar z_0$. Since the left-upper corner of this matrix vanishes, then also $\left({\mathbf N}_F(z,z_0)h_0,h_0\right)=0$ and hence \begin{equation}\label{eq:Im_F2} (F(z)h_0,h_0)=(F(z_0)^*h_0,h_0)=(h_0,F(z_0)h_0) \quad\mbox{for all}\quad z\in\dC_+\cup\dC_-\,\,(z\ne \bar z_0).
\end{equation} Hence, $({\rm Im\,}(F(z))h_0,h_0)=0$ and since ${\rm Im\,}(F(z))\geq 0$ (or $\le 0)$ this gives ${\rm Im\,}(F(z))h_0=0$.
(2) Assume that $0\in\sigma({\rm Im\,}(F(z_0)))$ and ${\rm Im\,}(F(z_0))h_n\to 0$ as $n\to\infty$ for some sequence $h_n\in\cH$, such that
$\|h_n\|=1$. Then for all $z\in\dC_+\cup\dC_-$, $z\ne \bar z_0$, \[
\left|({\mathbf N}_F(z,z_0)h_n,h_n)\right|^2\le \left({\mathbf N}_F(z_0,z_0)h_n,h_n\right)
\left({\mathbf N}_F(z,z)h_n,h_n\right)\to 0 \quad (n\to\infty). \] This implies that $({\rm Im\,}(F(z))h_n,h_n)\to 0$. Hence,
$\|({\rm Im\,}(F(z))^{1/2}h_n\|^2\to 0$ (if e.g. ${\rm Im\,} z>0$) and therefore also ${\rm Im\,}(F(z))h_n\to 0$ as $n\to\infty$.
(3) This statement is implied by (1) and (2).
(4) Assume that $F(z_0)h_0=ah_0$ for some $h_0\in\cH$, $h_0\ne 0$. Then the left--upper corner of the matrix in~\eqref{eq:Im_F0} equals to 0 and therefore~\eqref{eq:Im_F2} holds. Hence one obtains $F(z)h_0=ah_0$ for all $z\in{\dC \setminus \dR}$.
The proof of (5) and (6) is postponed until Theorem~\ref{invar} (v), (vi). \end{proof}
\section{Nevanlinna pairs}
In abstract eigenvalue depending boundary value problems Nevanlinna family is often represented via its counterpart -- a Nevanlinna pair, see e.g. \cite{DM95}, \cite{CDR}, \cite{DHMS1}. In this section connections between Nevanlinna families and Nevanlinna pairs are investigated in the general Hilbert space setting.
Every closed linear relation $T$ in a separable Hilbert space $\cH$ can be represented as \begin{equation} \label{tau00}
T=\{\,\{\Phi h,\Psi h\}:\, h\in\cL \,\}, \end{equation} where $\cL$ is a parameter Hilbert space and the operators $\Phi$, $\Psi$ belong to $[\cL,\cH]$. To show this it is enough to take $T$ as $\cL$ and the projections $\pi_1$, $\pi_2$ onto the first and the second components of $T\subset\cH\times\cH$ as $\Phi$ and $\Psi$. Clearly, each pair $\{\Phi,\Psi\}$ of operators in $[\cL,\cH]$ gives rise to a linear relation $T$ in $\cH$ via~\eqref{tau00}. In the infinite-dimensional case (${\rm dim\,}\cH=\infty$) the parameter Hilbert space $\cL$ can be taken to be equal to $\cH$. Note that when $\rho(T)$ is not empty and ${z}_0 \in \rho(T)$ then \[
T=\{\,\{(T-{z}_0)^{-1}h, (I+{z}_0(T-{z}_0)^{-1})h\} :\,
h \in \cH\,\}, \] so that $\cL=\cH$ and there is a natural choice for the pair $\{\Phi, \Psi\}$ in $\cB(\cH)$.
\iffalse In what follows it is convenient to interpret the Hilbert space $\cH^2=\cH\oplus\cH$ as a Kre\u{\i}n space $(\cH^2,J_\cH)$ whose inner product is determined by the fundamental symmetry \begin{equation}\label{jh}
J_\cH=\left(
\begin{array}{cc}
0 &-iI_\cH \\
iI_\cH & 0 \\
\end{array}
\right). \end{equation} The following connections between linear relations in the Hilbert space $\cH$ and subspaces in the Kre\u{\i}n space $(\cH^2,J_\cH)$ will be useful.
\begin{proposition}[\cite{AI}, \cite{Ph0}] \label{rem1.6} Let $T$ be a linear relation in the Hilbert space $\cH$. Then \begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})} \item $T$ is symmetric (selfadjoint)
if and only if $T$ is a neutral (hypermaximal neutral)
subspace of $(\cH^2,J_\cH)$; \item $T$ is dissipative (maximal dissipative)
if and only if $T$ is a nonnegative
(maximal nonnegative) subspace of $(\cH^2,J_\cH)$; \item $T$ is a accumulative (maximal accumulative)
if and only if $T$ is a nonpositive
(maximal nonpositive) subspace of $(\cH^2,J_\cH)$. \end{enumerate} \end{proposition} \fi
For linear relations given by the equation~\eqref{tau00} its properties can be characterized in terms of the pair $\{\Phi, \Psi\}$.
\begin{proposition}[\cite{DHMS1}]\label{CRIT} Let $T$ be a linear relation $T$ in $\cH$, defined by~\eqref{tau00}. Then:
\begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})} \item the adjoint $T^*$ is a linear relation given by \begin{equation} \label{tau01}
T^*=\{\,\{h,h'\}\in\cH^2:\, \Psi^*h-\Phi^*h'=0 \,\}. \end{equation} \item $T$ is a dissipative (accumulative) relation if and only if \begin{equation} \label{Acor1}
-i(\Phi^*\Psi- \Psi^*\Phi)\ge 0, \quad ( \le 0); \end{equation} \item $T$ is symmetric if and only if \begin{equation} \label{cor2}
\Phi^*\Psi- \Psi^*\Phi=0; \end{equation} \end{enumerate} If, additionally, ${\xker\,} \Phi\cap{\xker\,}\Psi=\{0\}$, then
\begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})} \item[(iv)] ${z}\in\rho(T)$ if and only if the operator $\Psi-{z}\Phi$ has a bounded inverse; \item[(v)] $T$ is maximal dissipative (accumulative) if and only if~\eqref{Acor1} holds and the operator $\Psi+i \Phi$ ($\Psi-i \Phi$) has a bounded inverse; \item[(vi)] $T$ is selfadjoint if and only if~\eqref{cor2} holds and the operators $\Psi\pm i \Phi$ have bounded inverses. \end{enumerate} \end{proposition}
\begin{proof} (i) For $\{g,g'\}\in T^*$ and arbitrary $h\in \cH$ one has the equality \[ 0=(g',\Phi h)-(g,\Psi h)=(\Phi^*g'-\Psi^*g,h), \] which implies $\Psi^*g-\Phi^*g'=0$.
(ii), (iii) If $T$ is symmetric then for $\{\Phi h,\Psi h\}\in T$ one obtains \[ 0=-i[(\Psi h,\Phi h)-(\Phi h,\Psi h)] =-i( (\Phi^*\Psi- \Psi^*\Phi)h,h), \quad h\in\cH, \] and conversely. Similarly, one obtains the conditions~\eqref{Acor1} for dissipative and accumulative linear relations.
(iv) It follows from\eqref{tau00} that \begin{equation} \label{Tinv} T-{z}=\{\,\{\Phi h,(\Psi-{z}\Phi) h\}:\, h\in\cL \,\}, \end{equation} Assume that ${z}\in\rho(T)$ and $(\Psi-{z}\Phi) h=0$. Then $\Psi h=\Phi h=0$ and, hence, $h=0$ (by the assumption ${\xker\,} \Phi\cap{\xker\,}\Psi=\{0\}$). Since ${\rm ran\,} (\Psi-{z}\Phi)={\rm ran\,}(T-{z})=\cH$, it follows $0\in\rho(\Psi-{z}\Phi)$. Similarly, if $0\in\rho(\Psi-{z}\Phi)$ one obtains from~\eqref{Tinv} that ${z}\in\rho(T)$ and $(T-{z})^{-1}=\Phi(\Psi-{z}\Phi)^{-1}$.
(v), (vi) are immediate from (ii), (iii) and (iv). \end{proof}
Now let a family of linear relations $\cF(\cdot)$ be represented in the form \begin{equation} \label{tau1} \cF({z})=\{\Phi({z}),\Psi({z})\}:=
\{\,\{\Phi({z})h,\Psi({z})h\}:\, h\in\cH\,\}, \end{equation} where $\Phi(\cdot)$, $\Psi(\cdot)$ is a pair of holomorphic operator functions on $\dC_+\cup\dC_-$. In the case when $\cF(\cdot)$ is a Nevanlinna family the corresponding pair $\{\Phi(\cdot),\Psi(\cdot)\}$ in the representation~\eqref{tau1} is called the Nevanlinna pair.
\begin{definition} \label{npair} A pair $\{\Phi,\Psi\}$ of $\cB({\cH})$-valued functions $\Phi(\cdot)$, $\Psi(\cdot)$ holomorphic on ${\dC \setminus \dR}$ is said to be a Nevanlinna pair if: \begin{enumerate}
\item[(NP1)] ${\rm Im\,} \Phi({z})^*\Psi({z})/{\rm Im\,} {z} \geq 0$, ${z}\in \dC_+\cup\dC_-$; \item[(NP2)] $\Psi(\bar {z})^*\Phi({z})-\Phi(\bar {z})^*\Psi({z})=0$, ${z} \in {\dC \setminus \dR}$; \item[(NP3)] $0\in\rho(\Psi({z})\pm i\Phi({z}))$, ${z} \in\dC_\pm$. \end{enumerate} \end{definition}
Two Nevanlinna pairs $\{\Phi(\cdot),\Psi(\cdot)\}$ and $\{\Phi_1(\cdot),\Psi_1(\cdot)\}$ are said to be \textit{equivalent}, if they generate the same graph $\cF(z)$ in \eqref{tau1} for every $z\in{\dC \setminus \dR}$. In fact, the formula~\eqref{tau1} establishes a one-to-one correspondence $\{\Phi,\Psi\}\mapsto \cF$ between the equivalence classes of Nevanlinna pairs and Nevanlinna families $\cF(\cdot)\in \wt R(\cH)$; cf. \cite[Proposition~2.4]{DHMS1}.
\begin{proposition}[\cite{DHMS1}]\label{propnp} Let $\{\Phi,\Psi\}$ be a Nevanlinna pair of $\cB(\cH)$-valued functions on $\dC_+\cup\dC_-$, and let $\cF(\cdot)$ be defined by \eqref{tau1}. Then $\cF(\cdot)$ is a Nevanlinna family.
Conversely, if $\cF(\cdot)\in \wt R(\cH)$ then there exists a Nevanlinna pair $\{\Phi,\Psi\}$ of $\cB(\cH)$-valued functions on $\dC_+\cup\dC_-$ such that \eqref{tau1} holds. \end{proposition}
\begin{proof} Let $\{\Phi,\Psi\}$ be a Nevanlinna pair. Then it follows from (NP1), (NP3) and Proposition~\ref{CRIT} that the linear relation $\cF({z})$ is maximal dissipative (maximal accumulative) for all ${z}\in\dC_+$ (${z}\in\dC_-$). The assumption (NP2) concerning $\{\Phi,\Psi\}$ means that $\cF(\bar{z})\subset \cF({z})^*$. According to \eqref{tau01} \begin{equation} \label{Mstar} \cF({z})^* =\{\,\{h,h'\}\in\cH^2:\, \Psi({z})^*h-\Phi({z})^*h'=0 \,\}, \end{equation} and, hence \[
\cF({z})^*\pm i=\left\{ \left\{h,g \right\}:
(\Psi({z})^*\pm i\Phi({z})^*)h=\Phi({z})^*g \right\}. \] Using (NP3) one obtains ${\xker\,}(\cF({z})^*\pm i)=\{0\}$ and ${\rm ran\,}(\cF({z})^*\pm i)=\cH$ for ${z}\in\dC_\mp$. Similarly ${\xker\,}(\cF(\bar{z})\pm i)=\{0\}$ and ${\rm ran\,}(\cF(\bar{z})\pm i)=\cH$, ${z}\in\dC_\mp$. Hence, $(\cF({z})^*\pm i)^{-1}$ and $(\cF({\bar z}) \pm i)^{-1}$, ${z}\in\dC_\mp$, both are everywhere defined operators and, thus, the inclusion $\cF(\bar{z})\subset \cF({z})^*$ must hold as an equality $\cF(\bar{z})= \cF({z})^*$, ${z}\in{\dC \setminus \dR}$. This proves (NF2) and (NF3) with $w=\pm i$.
Conversely, assume that $\cF(\cdot)\in \wt R(\cH)$. Define $\Phi(\cdot)$ and $\Psi(\cdot)$ by \[
\Phi({z})=(\cF({z})\pm i)^{-1}, \quad
\Psi({z})=I\mp i(\cF({z})\pm i)^{-1}, \quad {z}\in \dC_\pm. \] Then $\cF(\cdot)$ has the representation \eqref{tau1}. The property (NF3) implies that $\Phi(\cdot)$, $\Psi(\cdot)$ are holomorphic on $\dC_+\cup\dC_-$ with the values in $\cB(\cH)$. Clearly, $\Psi({z})\pm i\Phi({z})=I$ and hence (NP3) holds. Moreover, the symmetry condition (NP2) is obvious and the positivity condition (NP1) follows from (NF1) in view of \[
\frac{((\Phi({z})^*\Psi({z})
-\Psi({z})^*\Phi({z}))h,h)}{{\rm Im\,} {z}}
=\frac{{\rm Im\,} (\Psi({z})h,\Phi({z})h)}{{\rm Im\,} {z}}
\geq 0. \] This completes the proof. \end{proof}
Let $\{\Phi,\Psi\}$ be a Nevanlinna pair and let $\cF(\cdot)$ be a family of linear relations associated with the Nevanlinna pair $\{\Phi,\Psi\}$. The Cayley transform $\cC({z})$ of $\cF({z})$ is given by \begin{equation} \label{Cayley}
\cC({z})
=(\Psi({z})-i\Phi({z}))(\Psi({z})+i\Phi({z}))^{-1}\quad({z}\in\dC_+). \end{equation} The operator-valued function $\cC({z})$ belongs to the Schur class ${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}({\mathcal H})$, i.e. $\cC({z})$ is holomorphic on $\dC_+$ and takes values in the set of contractive operators on ${\mathcal H}$ for all ${z}\in\dC_+$. For every operator-valued function $\cC\in{\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}({\mathcal H})$ the kernel \begin{equation}\label{eq:Ker_K}
{\sf K}({z},\,{w})=\frac{I-\cC({w})^*\cC({z})}{-i({z}-\bar{{w}})},\quad z,w\in\dC_+. \end{equation} is nonnegative on $\dC_+$ in a sense that for every choice of $z_j\in\dC_+$ and $h_j\in\cH$ $(j=1,\dots,n)$ the quadratic form \[ \sum_{j=1}^n\left({\sf K}({z_i},\,{z_j})h_i,h_j\right)\xi_i\bar\xi_j \] is nonnegative. \begin{proposition} \label{Nkernel} Let $\{\Phi,\Psi\}$ be a Nevanlinna pair. Then the following kernel is nonnegative on $\dC_+$: \[
{\sf N}_{\Phi,\Psi} ({z},{w})
=\frac{\Phi({w})^* \Psi({z})-
\Psi({w})^* \Phi({z})}{{z}-\bar {w}}, \quad
{z},{w} \in \dC_+. \] \end{proposition} \begin{proof} Let $\cF(\cdot)$ be a family of linear relations associated with the Nevanlinna pair $\{\Phi,\Psi\}$ and let $\cC({z})$ be the Cayley transform of $\cF({z})$ given by~\eqref{Cayley}. It follows from the equality \begin{equation}\label{eq:KerK_N}
{\sf K}({z},\,{w}) =2(\Psi({w})+i\Phi({w}))^{-*}{\sf N}_{\Phi,\Psi}({z},{w})
(\Psi({z})+i\Phi({z}))^{-1} \end{equation} that the kernel ${\sf N}_{\Phi,\Psi}({z},{w})$ is nonnegative on $\dC_+$; cf. \cite{SNF}.
\end{proof}
Nonnegativity of the kernel $ {\sf N}_{\Phi,\Psi}$ implies the following properties for Nevanlinna families and reflect maximum principle in the class $\wt R(\cH)$.
\begin{proposition}\label{NFmaxmod} Let $\{\Phi(\cdot), \Psi(\cdot)\}$ be a Nevanlinna pair and let $\cF(\cdot)\in\wt R(\cH)$ be the corresponding Nevanlinna family. Let $z_0\in{\dC \setminus \dR}$ be fixed and let $S$ be a symmetric relation in $\cH$. Then \begin{equation}\label{Sinvar}
S\subset \cF(z_0) \quad \Rightarrow \quad S\subset \cF(z) \quad
\text{for every } z\in{\dC \setminus \dR}. \end{equation} In particular, \begin{equation}\label{eq:F_capF*0}
\cF({z})\cap \cF(\bar{z})\equiv \cF({z_0})\cap \cF(\bar{z_0}) \quad ({z}\in {\dC \setminus \dR}) \end{equation} is a maximal symmetric subspace $S$ satisfying the inclusion $S\subset \cF(z)$ for some (equivalently for every) $z\in{\dC \setminus \dR}$. Moreover, \begin{equation}\label{eq:F_cap_F*}
\cF({z})\cap \cF(\bar{z})=\{\,\{\Phi(z)u,\Psi(z)u\}:\,
u\in {\xker\,}({\sf N}_{\Phi,\Psi}({z},{z}))\,\} \quad (z\in\dC_+). \end{equation} \end{proposition}
\begin{proof} Assume that $S\subset\cF(z_0)$ and let $\{\Phi(z_0)h_0,\Psi(z_0)h_0\}\in S$ and $\{\Phi(z)h,\Psi(z)h\}\in \cF(z)$ with $z\neq \bar{z_0}$. By Proposition \ref{Nkernel} the matrix \[ \begin{pmatrix} ({\sf N}_{\Phi,\Psi}({z}_0,{z}_0)h_0,h_0) & ({\sf N}_{\Phi,\Psi}({z}_0,{z})h_0,h) \\ ({\sf N}_{\Phi,\Psi}({z},{z}_0)h,h_0) & ({\sf N}_{\Phi,\Psi}({z},{z})h,h) \end{pmatrix} \quad(h\in \cH,\,\, h_0\in\cH_0) \] is nonnegative, and since $S$ is symmetric the left-upper corner equals to 0. This implies that $({\sf N}_{\Phi,\Psi}({z}_0,{z})h_0,h)=0$ for all $h\in \cH$ or, equivalently, \[ (\Psi(z_0)h_0,\Phi(z)h)_\cH=(\Phi(z_0)h_0,\Psi(z)h)_\cH \] for all $h\in \cH$. Therefore, $\{\Phi(z_0)h_0,\Psi(z_0)h_0\}\in \cF(z)^*=\cF(\bar{z})$ for all $z\neq \bar{z_0}$. This proves \eqref{Sinvar}.
Since $\cF({z})\cap \cF(\bar{z})=\cF({z})\cap\cF(z)^*$, this subspace is symmetric and hence its maximality as a symmetric subset of $\cF(z)$ follows from \eqref{Sinvar}. Moreover, the invariance property \eqref{eq:F_capF*0} is also immediate from \eqref{Sinvar}.
Finally, the equality \eqref{eq:F_cap_F*} follows from the general formula $F\cap F^*=F{\upharpoonright\,}{\xker\,}({\rm Im\,} F)$ for a linear relation $F$ with ${\rm mul\,} F={\rm mul\,} F^*$; see \cite[Section 5.1]{HdSSz2009}. In fact, here in view of \eqref{tau01} $\{\Phi(z)u,\Psi(z)u\}\in \cF(z)^*$ precisely, when \[
\Psi(z)^*\Phi(z)u -\Phi(z)^*\Psi(z)u=0, \] or, equivalently, ${\sf N}_{\Phi,\Psi}({z},{z})u=0$. \end{proof}
\begin{corollary}\label{NFCor} Let $\{\Phi(\cdot), \Psi(\cdot)\}$ and $\cF(\cdot)\in\wt R(\cH)$ be as in Proposition \ref{NFmaxmod} and let $z_0\in{\dC \setminus \dR}$ and $a\in\dR$.
Then the following statements hold for all $z\in{\dC \setminus \dR}$: \begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})}
\item ${\xker\,} (\cF(z)-a)={\xker\,} (\cF(z_0)-a)$;
\item ${\xker\,} (\cF(z)-\cF(\bar{z}))=\{f\in \cH:\, \{f,g\}\in \cF({z_0})\cap \cF(\bar{z_0})\}$;
\item ${\xker\,} (\cF(z)^{-1}-\cF(\bar{z})^{-1})=\{g\in \cH:\, \{f,g\}\in \cF({z_0})\cap
\cF(\bar{z_0})\}$;
\item ${\rm mul\,} \cF(z)={\rm mul\,} \cF(z_0)$. \end{enumerate} \end{corollary} \begin{proof} The statements (i) and (iv) follow from Proposition \ref{NFmaxmod}, since ${\xker\,} (\cF(z)-a)$ with $a\in\dR$ and ${\rm mul\,} \cF(z)$ are symmetric subspaces of $\cF(z)$.
On the other hand, the formulas (ii) and (iii) clearly hold when $z=z_0$. The independence from $z\in{\dC \setminus \dR}$ follows from \eqref{eq:F_capF*0}. \end{proof}
The statements of the following lemma are based on the maximum principle for the class ${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}({\mathcal H})$ and, apparently, are well-known. \begin{lemma}\label{lem:MaxP}
Let $\cC(\cdot)\in{\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}({\mathcal H})$, ${z}_0\in\dC_+$, $\alpha\in\dC$ and $|\alpha|=1$. Then the following statements hold: \begin{enumerate}
\item[(1)] If $0\in\sigma_p({{\sf K}({z}_0,\,{z}_0)})$, then $0\in\sigma_p({{\sf K}({z},\,{z})})$ for all ${z}\in\dC_+$ and in this case \begin{equation*} {\xker\,} {\sf K}({z}_0,\,{z}_0)={\xker\,} {\sf K}({z},\,{z}); \end{equation*}
\item[(2)] If $0\in\rho({{\sf K}({z}_0,\,{z}_0)})$, then $0\in\rho({{\sf K}({z},\,{z})})$ for all ${z}\in\dC_+$.
\item[(3)] If $0\in\sigma_c({{\sf K}({z}_0,\,{z}_0)})$, then $0\in\sigma_c({{\sf K}({z},\,{z})})$ for all ${z}\in\dC_+$. \item[(4)] If $\alpha\in\sigma_p({\cC({z}_0)})$, then $\alpha\in\sigma_p({\cC({z})})$ for all ${z}\in\dC_+$ and in this case \begin{equation}\label{eq:KerId0} {\xker\,} (\cC({z}_0)-\alpha)={\xker\,} (\cC({z})-\alpha); \end{equation} \item[(5)] If $\alpha\in\rho({\cC({z}_0)})$, then $\alpha\in\rho({\cC({z})})$ for all ${z}\in\dC_+$; \item[(6)] If $\alpha\in\sigma_c({\cC({z}_0)})$, then $\alpha\in\sigma_c({\cC({z})})$ for all ${z}\in\dC_+$. \end{enumerate} \end{lemma} \begin{proof} (1) Let $0\in\sigma_p({{\sf K}({z}_0,\,{z}_0)})$ and let ${\sf K}({z}_0,\,{z}_0)h_0=0$ for some ${z}_0\in\dC_+$ and $h_0\ne 0$. Then the following matrix \[ \begin{pmatrix} ({\sf K}({z}_0,{z}_0)h_0,h_0) & ({\sf K}({z}_0,{z})h_0,h) \\ ({\sf K}({z},{z}_0)h,h_0) & ({\sf K}({z},{z})h,h) \end{pmatrix}, \] is nonnegative for all ${z}\in\dC_+$, $h\in{\mathcal H}$ and hence $({\sf K}({z}_0,{z})h_0,h)=0$ for all
$h\in{\mathcal H}$. This implies that the contraction $T=\cC({z})^*\cC({z}_0)$ has an eigenvector $h_0$ corresponding to the eigenvalue 1: $Th_0=h_0$. Therefore, $h_0$ is also an eigenvector for the operator $T^*=\cC({z}_0)^*C({z})$ ($T^*h_0=h_0$) and, hence, \[
\|h_0\|=\|\cC({z}_0)^*\cC({z})h_0\|\le\|\cC({z})h_0\|. \] This implies the inequality \[ (0\le)({\sf K}({z},{z})h_0,h_0)=(h_0,h_0)-(\cC({z})h_0,\cC({z})h_0)\le 0, \] which means that ${\sf K}({z},{z})h_0=0$. This proves the statement (1).
(2) Let $0\in\rho({{\sf K}({z}_0,\,{z}_0)})$ for some ${z}_0\in\dC_+$. Then the operator $\cC({z}_0)$ is a strict contraction and the space ${\mathcal H}^2$ admits the decomposition \begin{equation}\label{eq:decom}
{\mathcal H}^2={\rm ran\,}\left( \begin{array}{c}
I \\
\cC({z}_0) \\ \end{array} \right) \dotplus {\rm ran\,}\left( \begin{array}{c}
\cC({z}_0)^* \\
I \\ \end{array} \right). \end{equation}
Assume that $0\in\sigma_c({{\sf K}({z},\,{z})})$. Then there exists a sequence $h_n\in{\mathcal H}$, such that $\|h_n\|=1$ and ${\sf K}({z},\,{z})h_n\to 0$ as $n\to\infty$. Using the decomposition~\eqref{eq:decom}, one obtains \begin{equation} \label{eq:decom1} \left( \begin{array}{c}
h_n \\
\cC({z})h_n \\ \end{array} \right)= \left( \begin{array}{c}
h_n' \\
\cC({z}_0)h_n' \\ \end{array} \right) \dotplus {\rm ran\,}\left( \begin{array}{c}
\cC({z}_0)^*h_n'' \\
h_n'' \\ \end{array} \right), \end{equation} where $h_n',h_n''\in{\mathcal H}$. Since the matrix \[ \begin{pmatrix} ({\sf K}({z}_0,{z}_0)h_n',h_n') & ({\sf K}({z}_0,{z})h_n',h_n) \\ ({\sf K}({z},{z}_0)h_n,h_n') & ({\sf K}({z},{z})h_n,h_n) \end{pmatrix}, \] is nonnegative, then \[ ({\sf K}({z}_0,{z})h_n',h_n)\to 0 \] as $n\to\infty$. Using~\eqref{eq:decom1} one obtains \[ ((I-\cC({z}_0)^*\cC({z}_0))h_n',h_n')\to 0. \] By the assumption $0\in\rho({{\sf K}({z}_0,\,{z}_0)})$ this implies $h_n'\to 0$. Next, the equality \[ ((I-\cC({z})^*\cC({z}))h_n,h_n)=((I-\cC({z}_0)^*\cC({z}_0))h_n',h_n') -((I-\cC({z}_0)\cC({z}_0)^*)h_n'',h_n'') \] yields \[ ((I-\cC({z}_0)\cC({z}_0)^*)h_n'',h_n'')\to 0, \] which, in view of the condition $0\in\rho(I-\cC({z}_0)^*\cC({z}_0))$
implies that $h_n''\to 0$. Therefore, $h_n\to 0$ as $n\to\infty$ and this contradicts the equalities $\|h_n\|=1$.
(3) If $0\in\sigma_c({{\sf K}({z}_0,\,{z}_0)})$, then by (1) and (2) $0\not\in\sigma_p({{\sf K}({z},\,{z})})\cup\rho({{\sf K}({z},\,{z})})$ and hence
$0\in\sigma_c({{\sf K}({z},\,{z})})$.
(4) Let $\alpha\in\sigma_p({\cC({z}_0)}$ $(|\alpha|=1)$ and $\cC({z}_0)h_0=\alpha h_0$ for some vector $h_0\ne 0$. Then $h_0$ is an eigenvector for the contraction $T=\cC({z})^*$, corresponding to the eigenvalue $\alpha^{-1}$ and for the contraction $T^*=\cC({z})$, corresponding to the eigenvalue $\alpha^{-*}=\alpha$. This proves the equality~\eqref{eq:KerId0}.
(5) Let $\alpha\in\rho(\cC({z}_0))$ $(|\alpha|=1)$. Then for every ${z}\in\dC_+$ and $u\in\cH$ the harmonic function \[ h_u({z}):={\rm Re\,}\{ e^{-i\arg(\alpha)}\left((\alpha-\cC({z}))u,u\right)\}\ge 0 \]
is nonnegative and satisfies the inequality $h_u({z}_0)\ge q\|u\|^2>0$ for some $q\in(0,1)$. By Harnack's inequality (cf. Section 4 below) for every ${z}\in\dC_+$ there are positive constants $c_1({z})$ and $c_2({z})$, such that \[ c_1({z})h_u({z}_0)\le h_u({z})\le c_2({z})h_u({z}_0). \] It is emphasized that constants $c_1(z)$ and $c_2(z)$ do not depend on $u\in \cH$. Therefore, \[
h_u({z})\ge qc_1({z})\|u\|^2>0\quad\mbox{for all}\quad u\in\cH \] and hence $\alpha\in\rho(\cC({z}))$.
(6) Let $\alpha\in\sigma_c({\cC({z}_0)})$
$(|\alpha|=1)$. Then by (4) and (5) $\alpha\not\in\sigma_p({{\cC}({z})})\cup\rho({{\cC}({z})})$. Moreover, $\alpha\not\in\sigma_r({{\cC}({z})})$, since otherwise we would have $\bar\alpha\in\sigma_p({{\cC}({z})^*})$ and hence $\bar\alpha\in\sigma_p({{\cC}({z}_0)^*})$ which contradicts the assumption $\alpha\in\sigma_c({\cC({z}_0)})$. This completes the proof of (6). \end{proof}
In order to adapt the above statements to the class $\wt R(\cH)$ we will need the following lemma connecting the spectral properties of a Nevanlinna family $\cF(\cdot)\in\wt R(\cH)$ with the spectral properties of its Cayley transform $\cC(z)$. \begin{lemma} \label{lem:F_C} Let $\{\Phi(\cdot), \Psi(\cdot)\}$ be a Nevanlinna pair, let $\cF(\cdot)\in\wt R(\cH)$ be the corresponding Nevanlinna family and let the operator function $\cC(z)$ and the kernel ${\sf K}({z},\,{w})$ be defined by~\eqref{Cayley} and~\eqref{eq:Ker_K}. Let $a\in \dR$, $\alpha=(a-i)(a+i)^{-1}$ and ${z}\in \dC_+$. Then the following equivalences hold: \iffalse\begin{eqnarray}\label{}
0\in \sigma_p({\sf N}_{\Phi,\Psi} ({z},{z}))&\Longleftrightarrow &0\in \sigma_p({\sf K}({z},{z}));\\
0\in \sigma_c({\sf N}_{\Phi,\Psi} ({z},{z}))&\Longleftrightarrow &0\in \sigma_c({\sf K}({z},{z}));\\
0\in\rho({\sf N}_{\Phi,\Psi} ({z},{z}))&\Longleftrightarrow &0\in\rho({\sf K}({z},{z}));\\
a\in \sigma_p(\cF({z}))&\Longleftrightarrow &\alpha\in \sigma_p(\cC({z}));\\
a\in \sigma_c(\cF({z}))&\Longleftrightarrow &\alpha\in \sigma_c(\cC({z}));\\
a\in \rho(\cF({z}))&\Longleftrightarrow &\alpha\in \rho(\cC({z}));\\
\cF({z})\in \cB(\cH)&\Longleftrightarrow & 1\in \rho(\cC(z)). \end{eqnarray} \fi \begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})} \item $0\in \sigma_p({\sf N}_{\Phi,\Psi} ({z},{z}))\Longleftrightarrow 0\in \sigma_p({\sf K}({z},{z}))$; \item $0\in \sigma_c({\sf N}_{\Phi,\Psi} ({z},{z}))\Longleftrightarrow 0\in \sigma_c({\sf K}({z},{z}))$; \item $0\in \rho({\sf N}_{\Phi,\Psi} ({z},{z}))\Longleftrightarrow 0\in \rho({\sf K} ({z},{z}))$; \item $a\in \sigma_p(\cF({z}))\Longleftrightarrow \alpha\in \sigma_p(\cC({z}))$; \item $a\in \sigma_c(\cF({z}))\Longleftrightarrow \alpha\in \sigma_c(\cC({z}))$; \item $a\in \rho(\cF({z}))\Longleftrightarrow \alpha\in \rho(\cC({z}))$; \item $\cF({z})\in \cB(\cH)\Longleftrightarrow 1\in \rho(\cC(z))$. \end{enumerate} \end{lemma} \begin{proof} The equivalences (i)-(iii) are implied by the identity~\eqref{eq:KerK_N}.
Notice, that $a\in \sigma_p(\cF({z}))$ if and only if \begin{equation}\label{eq:PsiPhi}
(\Psi(z)-a\Phi(z))u=0\quad\mbox{for some }\quad u\in\cH\setminus\{0\}. \end{equation} If \eqref{eq:PsiPhi} holds then by (NP3) $h:=(\Psi(z)+i\Phi(z))u=(a+i)\Phi(z)u\ne 0$ and in view of~\eqref{Cayley} \[
\cC(z)h=(a-i)\Phi(z)u=\frac{a-i}{a+i}h=\alpha h. \] Therefore, $\alpha\in \sigma_p(\cC({z}))$. Conversely, if $\alpha\in \sigma_p(\cC({z}))$, and $\cC(z)h=\alpha h$ for some $h\in\cH\setminus\{0\}$, then~\eqref{eq:PsiPhi} holds for $u=(\Phi(z)+i\Phi(z))^{-1}h(\ne 0)$ and hence $a\in \sigma_p(\cF({z}))$. This proves (iv).
The equivalences (v)-(vi) follows from the equality~\eqref{Cayley} and the equivalences \[ a\in \sigma_c(\cF({z}))\Longleftrightarrow {\xker\,}(\Psi(z)-a\Phi(z))=\{0\},\,\mbox{ and }\,{\rm ran\,}(\Psi(z)-a\Phi(z))\,\mbox{ is dense in }\, \cH; \] \[ a\in \rho(\cF({z}))\Longleftrightarrow 0\in\rho(\Psi(z)-a\Phi(z)). \]
Similarly, (vii) follows from the equality~\eqref{Cayley} and the equivalence \[ \cF({z})\in \cB(\cH)\Longleftrightarrow 0\in\rho(\Phi(z)). \qedhere \] \end{proof}
\begin{theorem} \label{invar} Let $\{\Phi(\cdot), \Psi(\cdot)\}$ be a Nevanlinna pair and let $\cF(\cdot)\in\wt R(\cH)$ be the corresponding Nevanlinna family. Let ${z}_0\in \dC_+$ and let $a \in \dR$. Then the following statements hold: \begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})}
\item if $0\in \sigma_p({\sf N}_{\Phi,\Psi} ({z}_0,{z}_0))$,
then $0\in \sigma_p({\sf N}_{\Phi,\Psi}({z},{z}))$ for all ${z}\in {\dC \setminus \dR}$;
\item if $0\in \sigma_c({\sf N}_{\Phi,\Psi} ({z}_0,{z}_0))$,
then $0\in \sigma_c({\sf N}_{\Phi,\Psi}({z},{z}))$ for all ${z}\in {\dC \setminus \dR}$;
\item if $0\in \rho({\sf N}_{\Phi,\Psi} ({z}_0,{z}_0))$, then
$0\in \rho({\sf N}_{\Phi,\Psi} ({z},{z}))$ for all ${z}\in {\dC \setminus \dR}$;
\item if $a\in \sigma_p(\cF({z}_0))$, then
$a\in \sigma_p(\cF({z}))$ for all ${z}\in {\dC \setminus \dR}$ and in this case \[ {\xker\,} (\cF({z})-a)={\xker\,} (\cF({z_0})-a); \]
\item if $a\in \sigma_c(\cF({z}_0))$, then
$a\in \sigma_c(\cF({z}))$ for all ${z}\in {\dC \setminus \dR}$;
\item if $a\in \rho(\cF({z}_0))$, then
$a\in \rho(\cF({z}))$ for all ${z}\in {\dC \setminus \dR}$;
\item if $\cF({z}_0)\in \cB(\cH)$, then
$\cF({z})\in \cB(\cH)$ for all ${z}\in {\dC \setminus \dR}$; \item ${\rm mul\,}(\cF({z}))$ does not depend on ${z}\in {\dC \setminus \dR}$. \end{enumerate} \end{theorem}
\begin{proof} Statements (i), (iv) and (viii) have been derived already in Corollary~\ref{NFCor}.
Statements (ii) and (iii) follow from Lemma~\ref{lem:MaxP} (2)-(3) and Lemma~\ref{lem:F_C} (ii)--(iii).
Statements (iv) -- (vii), follow from Lemma~\ref{lem:MaxP} (4)-(6) and Lemma~\ref{lem:F_C} (iv) -- (vii).
\end{proof}
\begin{remark}\label{RemInvariant} The invariance results in Theorem~\ref{invar} can be obtained from the realization of a Nevanlinna family as a Weyl family of a boundary relation. Such an approach for proving these facts was used in \cite{DHMS06}; see, in particular, \cite[Lemma 4.1, Prop. 4.18]{DHMS06}. Also other models giving realizations for Nevanlinna families can be used in establishing such invariance results; we mention, in particular, the functional models which can be found from \cite{BDHdS2011,BHdS2009}. \end{remark}
\begin{proposition} \label{rwi1} Let $\{\Phi, \Psi\}$ be a Nevanlinna pair and let $\cF(\cdot)\in\wt R(\cH)$ be the corresponding Nevanlinna family. Let ${z}\in \dC_+$. Then: \begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})} \item
$\cF(\cdot)\in R^s(\cH)$ if and only if $0\not\in\sigma_p({\sf N}_{\Phi,\Psi}({z},{z}))$;
\item
$\cF(\cdot)\in R^u(\cH)$ if and only if $0\in\rho({\sf
N}_{\Phi,\Psi}({z},{z}))$. \end{enumerate} \end{proposition} \begin{proof} (i) Let $h\in{\xker\,}{\sf N}_{\Phi,\Psi}({z},{z})$, that is $(\Phi({z})^* \Psi({z})-\Psi({z})^* \Phi({z}))h=0$. Then it follows from~\eqref{Mstar} that \[ \{\Phi({z})h, \Psi({z})h\}\in \cF({z})\cap \cF({z})^* \] and, therefore, $h=0$ if and only if $\cF({z})\cap \cF({z})^*=\{0\}$.
(ii) Let $f$, $f'\in\cH$ and let $h$, $g$ satisfy the equations \begin{equation} \label{DHMS06} \Phi({z})h+\Phi(\bar{z})g=f, \quad \Psi({z})h+\Psi(\bar{z})g=f', \end{equation} Then it follows from (NP2) that \begin{equation} \label{TRANS2} {\sf N}_{\Phi,\Psi}({z},{z})h=\Psi({z})^*f-\Phi({z})^*f', \quad {\sf N}_{\Phi,\Psi}(\bar{z},\bar{z})g =\Psi(\bar{z})^*f-\Phi(\bar{z})^*f'. \end{equation} Assume that $0\in\rho({\sf N}_{\Phi,\Psi}({z},{z}))$. Then it follows from~\eqref{TRANS2} and Theorem~\ref{invar} that the system~\eqref{DHMS06} has a unique solution for all $f$, $f'\in\cH$ and, therefore, $\cF({z})\hplus \cF({z})^*=\cH^2$.
Conversely, let $\cF(\cdot)\in R^u(\cH)$ and thus the system~\eqref{DHMS06} has a unique solution for all $f$, $f'\in\cH$. Then it follows from the first equation in~\eqref{TRANS2} and the hypothesis (NP3) that ${\rm ran\,} {\sf N}_{\Phi,\Psi}({z},{z})=\cH$. This implies that $0\in\rho({\sf N}_{\Phi,\Psi}({z},{z}))$. \end{proof}
\begin{proposition} \label{Runif+} Let $ \cF(\cdot)\in R^u(\cH)$. Then $ \cF({z})\in\cB(\cH)$ and $\cF({z})^{-1}\in\cB(\cH)$ for every ${z}\in{\dC \setminus \dR}$. In particular, the following equality holds $R^u(\cH)=R^u[\cH]$. \end{proposition}
\begin{proof} Let $\{\Phi,\Psi\}$ be a Nevanlinna pair associated to $ M$. It is enough to prove that $\Phi({z})$ and $\Psi({z})$ are invertible. Now assume, for instance, that $\Phi({z})h_n\to 0$
for some sequence $h_n\in\cH$, $\|h_n\|=1$. This together with $0\in\rho({\sf N}_{\Phi,\Psi}({z},{z}))$ shows that for some $\alpha>0$ one has \[
\alpha \le({\sf N}_{\Phi,\Psi}({z},{z})h_n,h_n)_\cH\to 0, \] a contradiction. Since ${\rm ran\,} \Phi({z})={\rm dom\,} \cF({z})$ (${\rm ran\,} \Psi({z})={\rm ran\,} \cF({z})$) is dense in $\cH$, one concludes that $\Phi({z})$ must be invertible. A similar argument shows that $\Psi({z})$ is invertible. \end{proof}
We finish this section with some further properties of Nevanlinna pairs. The next statement can be found e.g. from \cite[Proposition~2.4]{DHMS1}.
\begin{lemma} Two Nevanlinna pairs $\{\Phi,\Psi\}$ and $\{\Phi_1,\Psi_1\}$ are equivalent if and only if $\Phi_1 ({z})=\Phi({z})\chi({z})$ and $\Psi_1({z})=\Psi({z})\chi({z})$ for some operator function $\chi(\cdot)\in\cB(\cH)$ which is holomorphic and invertible on $\dC_+\cup\dC_-$. \end{lemma} \begin{proof} By definition the Nevanlinna pairs $\{\Phi,\Psi\}$ and $\{\Phi_1,\Psi_1\}$ are equivalent if and only if the ranges of the block operators \[ T({z})=\begin{pmatrix}\Phi({z})\\ \Psi({z})\end{pmatrix}, \quad \wt T_1({z})=\begin{pmatrix}\Phi_1({z})\\ \Psi_1({z})\end{pmatrix} \] coincide with the graph of the corresponding Nevanlinna family $\cF(z)$, $z\in{\dC \setminus \dR}$. It is well known (from Douglas' lemma) that the equality ${\rm ran\,} T({z})={\rm ran\,} T_1({z})$ implies the existence of a bounded operator $\chi(z)\in\cB(\cH)$ such that $T({z})=T_1({z})\chi(z)$. Thus, $\Phi({z})=\Phi_1({z})\chi(z)$ and $\Psi({z})=\Psi_1({z})\chi(z)$, and hence \[
\Phi({z})\pm i\Psi({z})=(\Phi_1({z})\pm i\Psi_1({z}))\chi(z), \quad
z\in{\dC \setminus \dR}. \] In view of (NP3) this implies that $\chi(z)$ is bounded with bounded inverse and holomorphic in $z\in\dC_\pm$. \end{proof}
As follows from Proposition~\ref{NPpr} the conditions (NP3) and (NF3) can be replaced, for instance, by \[
0\in \rho(\Psi({z})+{w}\Phi({z}))\quad\mbox{and}\quad
0\in \rho(\cF({z})+{w} I), \] respectively, for some (equivalently for every) ${w} \in \dC_\pm$ and for all ${z}$ in the same halfplane as ${w}$. Moreover, the following more general statement holds.
\begin{proposition} \label{NPpr} Let $\{\Phi(\cdot),\Psi(\cdot)\}$ be a Nevanlinna pair, let $W$ be a unitary operator in the Kre\u{\i}n space $(\cH^2,J_\cH)$, and let \[ \begin{pmatrix}\wt\Phi({z})\\ \wt\Psi({z})\end{pmatrix} =W\begin{pmatrix}\Phi({z})\\ \Psi({z})\end{pmatrix}. \] Then $\{\wt\Phi(\cdot),\wt\Psi(\cdot)\}$ is also a Nevanlinna pair. In particular, if $X=X^*\in\cB(\cH)$, $Y$ is an invertible operator in $\cB(\cH)$ and $M(\cdot)\in R^u[\cH]$, each of the following pairs is also a Nevanlinna pair: \begin{equation} \label{NPs} \{\Phi(\cdot),\Psi(\cdot)+X\Phi(\cdot)\},\quad \{Y^{-1}\Phi(\cdot),Y^{*}\Psi(\cdot)\},\quad \{-\Psi(\cdot),\Phi(\cdot)\},\quad \{\Phi(\cdot),\Psi(\cdot)+M(\cdot)\Phi(\cdot)\}. \end{equation} \end{proposition}
\begin{proof} Consider $\cF({z})$ and $\wt\cF({z})$ as the ranges of the block operators \[ T({z})=\begin{pmatrix}\Phi({z})\\ \Psi({z})\end{pmatrix}, \quad \wt T({z})=\begin{pmatrix}\wt\Phi({z})\\ \wt\Psi({z})\end{pmatrix}. \] Then the kernel ${\sf N}_{\Phi, \Psi} ({z},{w})$ can be represented as follows: \begin{equation} \label{Nkern2}
{\sf N}_{\Phi, \Psi} ({z},{w})
=\frac{T({w})^*J_\cH T({z})}{-i({z}-\bar{{w}})}. \end{equation} The properties (NP1), (NP2) for $\{\wt\Phi(\cdot),\wt\Psi(\cdot)\}$ are implied by the equalities \begin{equation} \label{NN2}
{\sf N}_{\wt\Phi, \wt\Psi} ({z},{w})
=\frac{\wt T({w})^*J_\cH\wt T({z})}{-i({z}-\bar{{w}})}
=\frac{ T({w})^*J_\cH T({z})}{-i({z}-\bar{{w}})}
={\sf N}_{\Phi, \Psi} ({z},{w}). \end{equation} The graph $\cF({z})$ can be treated as a a maximal nonnegative subspace of the Kre\u{\i}n space $(\cH^2,J_\cH)$ for ${z}\in\dC_+$; see \cite[Section~2]{DHMS06}. Since $\wt \cF({z})$ is the range of $\wt T({z})$ it has the same property and, therefore, $\wt \cF(\cdot)\in\wt R_\cH$. By Proposition~\ref{propnp} $\{\wt\Phi,\wt\Psi\}$ is a Nevanlinna pair.
Applying this statement to the pair $\{\Phi,\Psi\}$ and the matrices \[ W=\begin{pmatrix} I & 0\\X & I\end{pmatrix},\quad W=\begin{pmatrix} Y^{-1} & 0\\0 & Y^*\end{pmatrix},\quad W=\begin{pmatrix} 0 & -I\\I & 0\end{pmatrix} \] one shows that the first three pair in~\eqref{NPs} are Nevanlinna pairs. The properties (NP1), (NP2) for the pair $\{\wt\Phi(\cdot),\wt\Psi(\cdot)\} =\{\Phi(\cdot),\Psi(\cdot)+M(\cdot)\Phi(\cdot)\}$ are implied by the identity \[
{\sf N}_{\wt\Phi, \wt\Psi} ({z},{w})
= {\sf N}_{\wt\Phi, \wt\Psi} ({z},{w}) +\Phi({w})\frac{M({z})-M({w})^*}{{z}-\bar{w}}\Phi({z}). \] To show that the operator $\wt\Psi({z})+i \wt\Phi({z})$ is invertible for some ${z}\in\dC_+$, set $X=\mbox{Re }M({z})$, $Y=\mbox{Im }M({z})$ and apply the previous statement to the pairs: \[ \{\Phi_1({z}),\Psi_1({z})\}= \{(Y+I)^{1/2}\Phi({z}),(Y+I)^{-1/2}\Psi({z})\}, \] \[ \{\Phi_2({z}),\Psi_2({z})\}= \{\Phi_1({z}),\Psi_1({z})+(Y+I)^{-1/2}X(Y+I)^{-1/2}\Phi_1({z})\}. \] Since these pairs are maximal dissipative it follows that the operator \[ \wt\Psi({z})+i \wt\Phi({z})=(Y+I)^{1/2}(\Psi_2({z})+i \Phi_2({z})) \] is also invertible. \end{proof}
\begin{remark}\label{later0} The connection between Nevanlinna pairs $\{\Phi(\cdot),\Psi(\cdot)\}$ and Nevalinna families $\cF(\cdot)\in \wt R(\cH)$ in Proposition~\ref{propnp} implies some invariance properties for the pair $\{\Phi(\cdot),\Psi(\cdot)\}$ via Theorem~\ref{invar}. We indicate here a couple of the underlying connections. \begin{enumerate} \def\rm (\roman{enumi}){\rm (\roman{enumi})} \item If $\{\Phi(\cdot),\Psi(\cdot)\}$ corresponds to $\cF(\cdot)$ then the transformed pair $\{\Psi(\cdot), -\Phi(\cdot)\}$ corresponds to the inverse $-\cF(z)^{-1}$ and, moreover, \[
{\mathbf N}_{\Psi,-\Phi}(z,w)={\mathbf N}_{\Phi,\Psi}(z,w),\quad z,w\in{\dC \setminus \dR}. \] \item The kernels of $\Phi(z)$ and $\Psi(z)$ do not depend on $z\in{\dC \setminus \dR}$; namely, \[
{\rm mul\,} \cF(z)={\rm mul\,} (\cF(z)\pm iI)={\rm mul\,} \{\Phi(z), \Psi(z)\pm i
\Phi(z)\}={\xker\,} \Phi(z), \] and, similarly, ${\xker\,} \Phi(z)={\xker\,} \cF(z)$.
\item It is not difficult to see that the condition $0\in \rho({\mathbf N}_{\Phi,\Psi}(z,z))$ implies that ${\xker\,} \Phi(z)={\xker\,} \Psi(z)= 0$ and that ${\rm ran\,} \Phi(z)$ and ${\rm ran\,} \Psi(z)$ are closed. Moreover, \[ ({\rm ran\,} \Phi(z))^\perp=({\rm dom\,} \cF(z))^\perp={\rm mul\,} \cF(z)^*={\rm mul\,} \cF(z)={\xker\,} \Phi(z)=\{0\}. \] Consequently, ${\rm ran\,} \Phi(z_0)=\cH$ and hence $0\in \rho(\Phi(z))$. Then in view of (i) we have also $0\in \rho(\Psi(z))$. The Nevanlinna kernel for $\cF(\cdot)$ as defined in \eqref{eq:Kern_F} is given by \[
{\mathbf N}_\cF(z,w)
=(\Phi(w))^{-*}
({\mathbf N}_{\Phi,\Psi}(z,w))(\Phi(z))^{-1}, \quad z\ne\bar w\,\,
z,w\in{\dC \setminus \dR}, \] and there is a similar formula for $-\cF(\cdot)^{-1}$.
Therefore, \[
0\in \rho({\mathbf N}_{\Phi,\Psi}(z,z)) \quad \Leftrightarrow\quad
\cF(\cdot)\in R^u[\cH] \quad \Leftrightarrow\quad -\cF(\cdot)^{-1}\in R^u[\cH]. \] \end{enumerate} \end{remark}
\section{Invariance theorems for harmonic operator-valued functions and quadratic forms}
Let us recall from \cite[Section 7.1]{Kato} the definition of a boundedly holomorphic function $T(\cdot)$ with values in the set $\cC(\cH)$ of closed (not necessarily bounded) operators acting in $\cH$.
\begin{definition}\label{def_Kato} Let $T(\kappa)$ be a family of operators with values in $\cC(\cH)$ and defined in a neighborhood of $\kappa_0\in\mathbb C$ and let $\zeta\in\rho\bigl(T(\kappa_0)\bigr)$. The family $T(\cdot)$ is called holomorphic at $\kappa_0\in\C$ if $\zeta\in\rho\bigl(T(\kappa)\bigr)$ and the resolvent
$R(\zeta,\kappa)=\bigl(T(\kappa)-\zeta\bigr)^{-1}$ is boundedly holomorphic in $\kappa$ for $|\kappa-\kappa_0|$ small enough.
\end{definition}
It is shown in \cite[Theorem 7.1.3]{Kato} that in this case the resolvent $R(\zeta,\kappa)$ of the family $T(\kappa)$ is holomorphic in both variables $(\zeta,\kappa)$ in an appropriate domain in $\C^2$.
The following definition of holomorphic $R$-function is crucial in the sequel.
\begin{definition}\label{def_strong_holomorphy} A function $F\in R(\cH)$ will be called a \emph{strongly holomorphic function} if the following two conditions are satisfied:
\begin{enumerate}
\item [(i)] the set
\begin{equation}\label{6.2A}
\mathcal D(F) := \cap_{z\in\mathbb C_+}{\rm dom\,} F(z)\qquad \text{is dense in}\qquad \cH. \end{equation}
\item [(ii)] vector function $F(z)u$ is holomorphic in a domain $\Omega\subset\mathbb C_+\cup \R$ for each $u\in\mathcal D(F)$. \end{enumerate} \end{definition}
\begin{remark} Note that in general, the domain of holomorphy $\Omega$ in $(ii)$ might be broader than the corresponding domain in Definition \ref{def_Kato}. Namely, in general conditions (i) and (ii) do not imply the local boundedness of the resolvent $(F(\cdot)+i)^{-1}$ at real points. For instance, consider the function
\begin{equation*} F(z)=Az, \quad 0 \le A=A^*\in \cC(\cH)\setminus {\mathcal B}(\cH) ,\quad {\rm dom\,} F(z)={\rm dom\,} A \not = \cH, \quad z\in\mathbb C\setminus\{0\}, \end{equation*}
and ${\rm dom\,} (F(0))=\cH$. It is easily seen that ${\rm Im\,} F(z)= Ay\ge 0$ for $z=x+iy\in\mathbb C_+$ and conditions (i), (ii) are satisfied with $\Omega = \mathbb C_+\cup \R$ and $\mathcal D(F)= {\rm dom\,} A$.
However, $F(\cdot)$ is not holomorphic at zero in the sense of Definition \ref{def_Kato}. Indeed, $F(z)^{-1} = z^{-1}A^{-1}$ is not boundedly holomorphic at zero even if $A^{-1}\in\cB(\cH)$.
This example shows that the domain of holomorphy $\Omega$ in Definition \ref{def_strong_holomorphy}$(ii)$ might be broader than the corresponding domain in Definition \ref{def_Kato}.
\end{remark}
An unbounded Nevanlinna function is not in general strongly holomorphic. In fact, in the next example an extreme situation of a Nevanlinna function is constructed, such that the domains of $F(\lambda)$ and $F(\mu)$ have a zero intersection: \[
{\rm dom\,} F(\lambda)\cap {\rm dom\,} F(\mu)=\{0\} \text{ for all }
\lambda,\mu \in {\dC \setminus \dR}. \]
\begin{example}\label{Ex4A} Let $A$ and $B\ge 0$ be two bounded selfadjoint operators on a Hilbert space $\cH$ with ${\xker\,} A={\xker\,} B=\{0\}$ and such that \[
{\rm ran\,} A\cap {\rm ran\,} B=\{0\}. \] Then $M(z)=A-\frac{1}{z}\, B$, $z\neq 0$, is a Nevanlinna function from the class $R[\cH]$. The transform $F(z):=-M(z)^{-1}$ is an operator-valued Nevanlinna function in the class $R(\cH)$: \[
F(z):=-\left(A-\frac{1}{z}\, B\right)^{-1}, \quad z\in{\dC \setminus \dR}.
\] To consider the domain of $F(z)$ at two points $\lambda,\mu\in{\dC \setminus \dR}$ and assume that there is a nonzero vector $k\in {\rm dom\,} F(\lambda)\cap{\rm dom\,} F(\mu)$. This means that $k=M(\lambda)f=M(\mu)g$ for some $f,g\in\cH$, i.e., \[
(A -\frac{1}{\lambda}\,B)f=(A -\frac{1}{\mu}\,B)g \] or, equivalently, \[
A(f -g)=B \left(\frac{1}{\lambda}f-\frac{1}{\mu}g\right). \] Since ${\rm ran\,} A\cap{\rm ran\,} B=\{0\}$ and ${\xker\,} A={\xker\,} B=\{0\}$, we get $f=g$ and $\frac{1}{\lambda}f=\frac{1}{\mu}g$ which leads to $\lambda=\mu$. Therefore, \[
{\rm dom\,} F(\lambda)\cap {\rm dom\,} F(\mu)=\{0\}, \quad \lambda\neq \mu,\,\,
\lambda,\mu \neq 0. \]
Recall that there exist bounded nonnegative selfadjoint operators on a Hilbert space $\cH$ with ${\xker\,} A={\xker\,} B=\{0\}$ and such that \[
{\rm ran\,} A\cap {\rm ran\,} B=\{0\}, \quad {\rm ran\,} A^{1/2}={\rm ran\,} B^{1/2}; \] see e.g. \cite[Example, p.278]{FW} for an example of such operators. Then it follows from ${\rm ran\,} A^{1/2}={\rm ran\,} B^{1/2}$ that there exists a bounded and boundedly invertible positive operator $C$ such that \[
A=B^{1/2} C B^{1/2}. \] Hence, this choice of $A$ and $B$ implies that $M(z)=B^{1/2} (C- \frac{1}{z})B^{1/2}$ and \[
F(z)=-B^{-1/2} \left(C- \frac{1}{z}\right)^{-1}B^{-1/2}=B^{-1/2}\wt F(z)B^{-1/2}, \] where $\wt F(z)=-(C- \frac{1}{z})^{-1}$ satisfies, $\wt F(\cdot),-\wt F^{-1}(\cdot)\in R^u[\cH]$; cf. Remark \ref{later0}~(iii). Consequently, for every $z\in {\dC \setminus \dR}$ the form \[
{\mathfrak t}_{F(z)}[u,v]:=\frac{1}{z-\bar z}\,[(F(z)u,v)-(u,F(z)v)]
=({\mathbf N}_{\wt F}(z,z)B^{-1/2}u,B^{-1/2}v), \quad u,v\in {\rm dom\,} F(z), \] is closable and its closure has the same formula which is defined on a constant domain ${\rm ran\,} B^{1/2}$. Nevanlinna functions with this property are studied systematically in a forthcoming paper \cite{DHM15} by the authors and they are called form-domain invariant Nevanlinna functions. \end{example}
The general definition of the class of form-domain invariant Nevanlinna families $\cF(\cdot)\in \wt R(\cH)$ reads as follows; cf. \cite{DHM15} for a treatment of such functions as Weyl functions of boundary triplets and boundary pairs (or boundary relations).
\begin{definition}\label{Nevforminv} A Nevanlinna family $\cF(\cdot)\in \wt R(\cH)$ is said to be form-domain invariant if its operator part $F_s(\cdot)\in R(\cH_s)$ is form-domain invariant, which means that the quadratic form ${\mathfrak t}_{F_s(\lambda)}$ in $\cH_s$ generated by the imaginary part of $F_s(\lambda)$ via \[
{\mathfrak t}_{F_s(\lambda)}[u,v]=\frac{1}{\lambda-\bar\lambda}\,[(F_s(\lambda)u,v)-(u,F_s(\lambda)v)],
\] is closable for all $\lambda\in{\dC \setminus \dR}$ and the closure of the form ${\mathfrak t}_{F_s(\lambda)}$ has a constant domain. \end{definition}
In what follows the set of nonnegative harmonic functions in $\mathbb C_+$ is denoted by $Har_+(\mathbb C_+)$. In this section we will systematically make use of the classical Harnack's inequality: Given a pair of points $z_1, z_2\in\mathbb C_+$, there exists positive constants $c_j = c_j(z_1,z_2),\ j\in\{1,2\},$ such that
\begin{equation}\label{6.0} c_1 h(z_1)\le h(z_2)\le c_2 h(z_1), \qquad h(\cdot) \in Har_+(\mathbb C_+).
\end{equation}
It is emphasized that the constants $c_1$ and $c_2$ do not depend on $h(\cdot)\in Har_+(\mathbb C_+)$.
With any strongly holomorphic $R$-function $F(\cdot)\in R(\cH)$ one associates a family of the quadratic forms $\mathfrak t(z)[\cdot]$ given by
\begin{equation}\label{6.1} \mathfrak t(z)[u] := {\rm Im\,} (F(z)[u]) := {\rm Im\,}\bigl(F(z)u, u\bigr)\ge 0, \qquad u\in{\rm dom\,}\bigl(F(z)\bigr), \quad z\in\mathbb C_+.
\end{equation}
Equipping ${\rm dom\,} F(z)$ with the inner product
\[ (u, v)_{+,z} = (u, v)_{\cH} + \mathfrak t(z)[u, v],\qquad u, v\in{\rm dom\,} F(z), \]
we obtain a pre-Hilbert space $\cH'_+(z)$. The corresponding energy space is denoted by $\cH_+(z)$, i.e., it is the completion of
$\cH'_+(z)$ with respect to the norm $\|\cdot\|_{+,z}$.
Recall that the form $\mathfrak t(z)$ is called closable if the norms $\|\cdot\|_{\cH}$ and $\|\cdot\|_{+,z}$ are compatible. The latter means that the completion of $\cH'_+(z)$ holds within $\cH,$ i.e. $\cH_+(z)\subset\cH.$ The form $\mathfrak t(z)$ is called closed if it is closable and $\cH_+(z) = \cH'_+(z)$, i.e. $\cH'_+(z)$ is a Hilbert space.
In the following proposition we investigate certain stability properties of the family \eqref{6.1}.
\begin{proposition}\label{prop_form_stability} Let $F(\cdot)\in R(\cH)$ be strongly holomorphic. Assume in addition that for some $z_0\in\mathbb C_+$ the form $\mathfrak t(z_0)$ is closable and $\mathcal D(F)$ is a core for its closure $\overline {\mathfrak t}(z_0)$. Then:
\item[\;\;\rm (i)]
The form $\mathfrak t(z)$ is closable for any $z\in\mathbb C_+;$
\item[\;\;\rm (ii)] $\mathcal D(F)$ is a core for the closure $\overline {\mathfrak t}(z)$ of the form ${\mathfrak t}(z)$ for each $z\in\mathbb C_+.$ Moreover, the corresponding energy spaces $\cH_+(z)$, $z\in\mathbb C_+,$ coincide algebraically and topologically,
\begin{equation}\label{4.6} \cH_+(z) = \cH_+(z_0), \qquad z\in\mathbb C_+.
\end{equation}
In particular,
\begin{equation}\label{4.6A} \mathcal D[F] := \cap_{z\in \C_+} \cH_+(z) = \cH_+(z_0).
\end{equation}
\item[\;\;\rm (iii)] For any pair $u,v\in \mathcal D[F]$ the function $\overline {\mathfrak t}(\cdot)[u,v]$ is a harmonic (hence real analytic) function in $\C_+$.
\end{proposition}
\begin{proof} (i) and (ii). Lets us show that the form $\mathfrak t_z$ is closable for any $z\in\mathbb C_+$. Assume that $h_n\in\mathcal D$ and
\begin{equation}\label{6.3}
\lim_{n\to\infty}\|u_n\| = 0 \quad \text{and} \quad \mathfrak t(z)[u_n - u_m]\to 0\quad \text{as} \quad n,m\to\infty.
\end{equation}
Then by the Harnack's inequality there exist positive constants $c_1 = c_1(z_0,z),$ $c_2= c_2(z_0,z),$ depending only on $z_0,z,$ and such that
\begin{equation}\label{6.4} 0\le c_1\mathfrak t(z_0)[u_n - u_m] \le \mathfrak t(z)[u_n - u_m] \le c_2 \mathfrak t(z_0)[u_n - u_m], \quad u_n\in\mathcal D(F).
\end{equation}
Combining \eqref{6.3} with the first inequality in \eqref{6.4} we get $ \mathfrak t(z_0)[u_n - u_m]\to 0$ as $n,m\to\infty$. Since in addition $\lim_{n\to\infty}\|u_n\| = 0$ and the form $\mathfrak t(z_0)$ is closable, one has $\mathfrak t(z_0)[u_n]\to 0$ as $n\to\infty$. In turn, the second inequality in \eqref{6.4} with $u_m=0$ yields
\begin{equation}\label{6.5} \lim_{n\to\infty}\mathfrak t(z)[u_n] = 0,\qquad z\in\mathbb C_+.
\end{equation}
Thus, \eqref{6.3} implies \eqref{6.5}. This means the closability of $\mathfrak t_z$, hence the identical embedding $\cH'_+(z)\hookrightarrow\cH$ is extended to a (continuous) embedding of the energy space $\cH_+(z)$ into $\cH$. Moreover, it follows from \eqref{6.4} that the norms in $\cH'_+(z)$ and $\cH'_+(z_0)$ are equivalent. Hence completing the spaces $\cH'_+(z)$ and $\cH'_+(z_0)$ we conclude that $\cH_+(z)$ and $\cH_+(z_0)$ coincide algebraically and topologically.
(iii) Since ${\mathcal D}} \def\cE{{\mathcal E}} \def\cF{{\mathcal F}(F)$ is a core for $\overline {\mathfrak t}(z_0)$, it follows form the definition of the closure that for any $u\in \mathcal D[F] = \cH_+(z_0)$ there exists $u_n\in {\mathcal D}} \def\cE{{\mathcal E}} \def\cF{{\mathcal F}(F)$ such that
$$ \overline {\mathfrak t}(z_0)[u] = \lim_{n\to\infty} \mathfrak t(z_0)[u_n], \quad u_n\in\mathcal D(F). $$
It follows from \eqref{6.4} (with $u_m=0$) that $\overline {\mathfrak t}(z)[u] = \lim_{n\to\infty} \mathfrak t(z)[u_n]$ uniformly on compact subsets of $\C_+.$ So, by the first Harnack theorem, $\overline {\mathfrak t}(\cdot)[u]$ is a nonnegative harmonic function in $\C_+.$ Using the polarization identity one proves that $\overline {\mathfrak t}(\cdot)[u,v]$ is also harmonic for any pair $u, v\in \mathcal D[F].$
\end{proof}
Proposition \ref{prop_form_stability} makes it possible to introduce the imaginary part of the function $F(\cdot)$.
\begin{definition}\label{def_imaginary_part} Let $F(\cdot)\in R(\cH)$ satisfy the conditions of Proposition \ref{prop_form_stability}. Denote by $F_I(z)$, $z\in \mathbb C_+,$ the nonnegative self-adjoint operator associated with the closed form $\overline {\mathfrak t}(z)$ in accordance with the first representation theorem (see \cite[Theorem 6.2.1]{Kato}).
\end{definition}
Note that the operator $F_I(z) = F_I(z)^*$ is a self-adjoint extension of the operator
$$ F'_I(z) := (F(z) -F(z)^*)/2i, \quad {\rm dom\,}(F'_I(z)) = {\rm dom\,}(F(z))\cap {\rm dom\,}(F(z)^*), $$
which is only nonnegative symmetric not necessarily essentially self-adjoint.
\begin{remark} (i) Note that in accordance with the second representation theorem (see \cite[Theorem 6.2.23]{Kato})) and Definition \ref{def_imaginary_part} equalities \eqref{4.6}-\eqref{4.6A} can be rewritten as
\begin{equation}\label{4.11} \cH_+(z) = {\rm dom\,}(F_I(z)^{1/2}) = {\rm dom\,}(F_I(z_0)^{1/2}) = \cH_+(z_0) = {\mathcal D}} \def\cE{{\mathcal E}} \def\cF{{\mathcal F}[F]\quad \text{for each}\quad z\in \C_+.
\end{equation}
Here the spaces $\cH_+(z)$ and ${\rm dom\,}(F_I(z)^{1/2})$ (equipped with the graph norm) coincide algebraically and topologically.
(ii) Proposition \ref{prop_form_stability} shows that the family $F(\cdot)$ is a holomorphic family in $\C_+$ of the type $(B)$ in the sense of T. Kato \cite[Section 7.4.2]{Kato}
\end{remark}
\begin{proposition}\label{prop6.2} Let $F(\cdot)\in R(\cH)$ and let the conditions of Proposition \ref{prop_form_stability} be satisfied.
Assume also that $F_I(z_0)\in \cB(\cH)$ for $z_0\in\mathbb C_+$. Then:
\item[\;\;\rm (i)] $F_I(\cdot)$ takes values in $\cB(\cH);$
\item[\;\;\rm (ii)] The function $F(\cdot)$ admits a representation
\begin{equation}\label{6.9A} F(z) = G(z) + T \quad z\in \C_+,
\end{equation}
where $G(\cdot)\in R[\cH]$ and $T=T^*\in \cC(\cH)$.
\end{proposition}
\begin{proof} (i) Since $F\in R(\cH)$ is a strongly holomorphic function, the family
$$ \mathcal F=\{{\rm Im\,}\bigl(F(\cdot)u,u\bigr): u\in\mathcal D(F)\} $$
is well defined and constitutes the family of nonnegative harmonic functions, $\mathcal F\subset Har_+(\mathbb C_+)$. Fix $z\in\mathbb C_+$. Then, by Proposition \ref{prop_form_stability}(ii) the form $ t(z)[\cdot]$ is closable and by the Harnack's inequality \eqref{6.0},
\begin{equation*}
0\le \mathfrak t(z)[u] := {\rm Im\,}\bigl(F(z)u, u\bigr) \le c_2{\rm Im\,}\bigl(F(z_0)u, u\bigr)
\le c_2\|F_I(z_0)\|\cdot\|u\|^2, \quad u\in \mathcal D(F).
\end{equation*}
It follows that the form $\mathfrak t(z)$ is bounded on $\mathcal D(F)$. Since $\mathcal D(F)$ is a core for $\mathfrak t(\cdot)$, the form $\mathfrak t(z)$ admits a bounded continuation on $\cH$ and by the Riesz representation theorem,
\begin{equation}\label{6.11A} \mathfrak t(z)[u, v] = \bigl(T(z)u, v\bigr)_{\cH}, \qquad 0\le T(z) = T^*(z) \in \cB(\cH), \quad u,v\in \cB(\cH).
\end{equation}
Using the polarization identity we obtain from \eqref{6.1} that
\[ \mathfrak t(z)[u, v]= (2i)^{-1}\left(\bigl(F(z)u, v\bigr) - \bigl(u, F(z)v\bigr)\right), \qquad u, v \in {\rm dom\,} F(z). \]
Combining this identity with \eqref{6.11A} we derive
\[ \bigl((F(z)-i T(z))u, v\bigr) = \bigl(u, (F(z) - iT(z))v\bigr),\qquad u,v \in {\rm dom\,} F(z). \]
Since ${\rm dom\,} F(z)$ is dense in $\cH$, it follows that $v\in {\rm dom\,}\bigl(F(z)^* + iT(z)\bigr)$, i.e. \[ {\rm dom\,} F(z)\subset {\rm dom\,}\bigl(F(z)^* + iT(z)\bigr). \] On the other hand, since $T(z) = T^*(z)$ is bounded, then \[ {\rm dom\,}\bigl(F(z)^* + iT(z)\bigr)= {\rm dom\,} F(z)^*\quad\mbox{ and }\quad {\rm dom\,} F(z)\subset {\rm dom\,} F(z)^*. \] By symmetry,
\[ {\rm dom\,} F(z)^* = {\rm dom\,} F(\overline z) \subset {\rm dom\,} F^*(\overline z) = {\rm dom\,} F(z) \]
Thus, ${\rm dom\,} F(z)^* = {\rm dom\,} F(z)$ and the imaginary part $F_I(\cdot) := (2i)^{-1}\bigl(F(\cdot)-F^*(\cdot)\bigr)$ \\ of $F(\cdot)$ is well defined and
\begin{equation}\label{6.12} F_I(z)u = T(z)u, \quad u\in{\rm dom\,} F(z) (\supset \mathcal D(F)).
\end{equation}
Hence $F_I(z)$ is bounded and its closure coincides with $T(z)$.
(ii) Being a nonnegative harmonic $\cB(\cH)$-valued function in $\mathbb C_+$, $T(\cdot)$ admits a representation
\begin{equation}\label{Int_rep-n} T(z)= B_0 + B_1 y + \int_{\mathbb R}\frac{y}{(x-t)^2+y^2}d\Sigma(t),
\end{equation}
where $B_j = B_j^*\in\cB(\cH)$, $j\in \{0,1\}$, $B_1\ge 0$, and $\Sigma(\cdot)$ is the $\cB(\cH)$-valued operator measure satisfying
\begin{equation}\label{Measuer_Condition} K_\Sigma := \int_{\mathbb R}(1+t^2)^{-1}d\Sigma(t)\in\cB(\cH).
\end{equation}\label{6.13}
Define $R[\cH]$-function $G(\cdot)$ by setting
\begin{equation}\label{6.14} G(z) = B_0 + B_1 z + \int_{\mathbb R}\left(\frac{1}{t-z}-\frac{1}{1+t^2}\right)\,d\Sigma(t).
\end{equation}
Further we let
\begin{equation}\label{6.18} G_1(z) := F(z) - G(z) \end{equation}
and note that $G_1(\cdot)$ is holomorphic in $\mathbb C_+$. Moreover, it follows from \eqref{6.12}, \eqref{6.13} and \eqref{6.14} that
\[ {\rm Im\,} \bigl(G_1(z)u, u\bigr) = 0, \quad z\in\mathbb C_+, \quad u\in\mathcal D(F). \]
Hence the operator $G_1(z)$ is symmetric for any $z\in\mathbb C_+$. Let us show that $G_1(z)$ is self-adjoint. Since $F(z)$ is $m$-dissipative for $z\in\mathbb C_+$ and $G(\cdot)$ takes values in $\cB(\cH),$ one has $\rho\bigl(G_1(z)\bigr)\cap\mathbb C_- \not = \emptyset$.
Further, since $F(z)^*$ is $m$-accumulative for $z\in\mathbb C_+$ and $G(z)^*\in\cB(\cH)$, we get
\[ G_1(z)^*=F(z)^*-G(z)^* \qquad \text{and}\qquad \rho\bigl(G_1(z)^*\bigr)\cap\mathbb C_+\not = \emptyset. \]
Thus, $G_1(z) = G_1(z)^*$ for any $z\in\mathbb C_+$. Being holomorphic in $\mathbb C_+$, the operator-valued function $G_1(\cdot)$ is constant, $G_1(z)=T=T^*,\ z\in\mathbb C_+$. Combining this with \eqref{6.18} leads to \eqref{6.9A}.
\end{proof}
Next we investigate the invariance property of real continuous spectrum.
Recall that $\lambda_0\in \sigma_c(T)$ if $\lambda_0\not \in \sigma_p(T)$ and there exists a non-compact (quasi-eigen) sequence $f_n\in {\rm dom\,}(T)\subset \cH$ such that
$$
\lim_{n\to\infty} \|(T- \lambda_0)f_n\| =0. $$
\begin{proposition}\label{prop_Harmonic_cont_and_point_spec} Let $F\in R(\cH)$ and satisfy the conditions of Proposition \ref{prop_form_stability}. Let also $F_I(\cdot)$ be its imaginary part in the sense of Definition \ref{def_imaginary_part}. Then the following holds:
\item[\;\;\rm (i)] If $a = \overline{a}\in\sigma_c\bigl(F_I(z_0)\bigr)$ for some $z_0\in\mathbb C_+$, then
\[ a\in\sigma_c\bigl(F_I(z)\bigr)\qquad \text{for}\qquad z\in\mathbb C_+; \]
\item[\;\;\rm (ii)] If $a = \overline{a}\in\sigma_p\bigl(F_I(z_0)\bigr)$ for some $z_0\in\mathbb C_+$, then
\[ a\in\sigma_p\bigl(F_I(z)\bigr)\quad\text{and}\quad ker\bigl(F_I(z)-a)= {\xker\,}\bigl(F_I(z_0)-a)\quad \text{for}\quad z\in\mathbb C_+. \]
\end{proposition}
\begin{proof} (i) Without loss of generality we can assume that $a=0.$ Since $0\in\sigma_c\bigl(F_I(z_0)\bigr)$, there exists an non-compact (quasi-eigen) sequence $\{v_k\}_{k\in \mathbb N}\in {\rm dom\,}(F_I(z_0))$ such that
\begin{equation}\label{4.25}
\|v_k\|=1\quad \mbox{ and }\quad \lim_{k\to\infty}\|F_I(z_0)v_k\|=0.
\end{equation}
By Proposition \eqref{prop_form_stability}(ii), $\{v_k\}_{k\in \N}\in {\rm dom\,}(F_I(z_0))\subset {\mathcal D}} \def\cE{{\mathcal E}} \def\cF{{\mathcal F}[F]$. Using Definition \ref{def_imaginary_part} and relation \eqref{4.11} one rewrites the right-hand side of inequality \eqref{6.4} as
\begin{equation}\label{6.4New}
0 \le \|F_I(z)^{1/2}v_k\|^2 = \mathfrak t(z)[v_k] \le c_2
\mathfrak t(z_0)[v_k]= \|F_I(z_0)^{1/2}v_k\|^2, \quad v_k\in\mathcal D[F]= \cH_+(z).
\end{equation}
Combining \eqref{4.25} with \eqref{6.4New} and noting that the sequence $\{v_k\}$ is not compact, one gets that $0\in\sigma_c\bigl(F_I(z)^{1/2}\bigr)$. Hence $0\in\sigma_c\bigl(F_I(z)\bigr)$.
(ii) Let $a=0 \in\sigma_p\bigl(F_I(z_0)\bigr)$ and $u\in {\xker\,}\bigl(F_I(z_0)\bigr)$. Hence $u\in {\xker\,}\bigl(F_I(z_0)^{1/2} \bigr)$. By Proposition~\eqref{prop_form_stability}~(ii), $u\in {\rm dom\,}(F_I(z))\subset {\mathcal D}} \def\cE{{\mathcal E}} \def\cF{{\mathcal F}[F] = {\rm dom\,}(F_I(z_0)^{1/2})$ for each $z\in \C_+$. It follows from \eqref{6.4New} with $u$ in place of $v_k$ that $F_I(z)^{1/2}u = 0$. Hence $F_I(z)u = 0$ for each $z\in \C_+$.
\end{proof}
Next we slightly improved Proposition \ref{prop_Harmonic_cont_and_point_spec}(i).
\begin{proposition}\label{prop_Harmonic_cont_spec} Let $F\in R(\cH)$ and satisfy the conditions of Proposition \ref{prop_form_stability}. Let also $F_I(\cdot)$ be its imaginary part in the sense of Definition \ref{def_imaginary_part}. If $a = \overline{a}\in\sigma_c\bigl(F_I(z_0)\bigr)$ for some $z_0\in\mathbb C_+$, then
\begin{equation*} a\in\sigma_c\bigl(F_I(z)\bigr)\qquad \text{for}\qquad z\in\mathbb C_+.
\end{equation*}
Moreover, a quasi-eigen sequence can be chosen to be common for all $F_I(z),\ z\in \mathbb C_+.$
\end{proposition}
\begin{proof} (i) First we reduce the proof to the case of $R$-function with bounded imaginary part.
In fact, this sequence $\{v_k\}_{k\in \mathbb N}\in {\rm dom\,}(F_I(z_0))$ in \eqref{4.25} can be chosen to be orthonormal. Passing if necessary to a subsequence $\{v_{n_k}\}_{k\in\mathbb N},$ we can assume that \[
\sum^{\infty}_{k=1}\|F_I(z_0)v_k\|^2 := \alpha_{F}(z_0)<\infty. \]
Since $\mathcal D(F)$ is a core for the form $\mathfrak t(z_0)$, it is dense in ${\rm dom\,}(F_I(z_0))(\subset \cH)$ equipped with the graph's norm. So, there exists a sequence $\{u_k\}^{\infty}_1\subset\mathcal D(F)$ such that
\begin{equation}\label{4.26}
\sum^{\infty}_{k=1}\|u_k - v_k\|^2<\infty \quad \text{and}\quad
\sum^{\infty}_{k=1}\|F_I(z_0)(u_k - v_k)\|^2 =: \beta_F(z_0)<\infty.
\end{equation}
Let $\cH_0$ be a subspace spanned by the sequence $\{u_k\}_1^\infty$, $\cH_0:= {\rm span\,}\{u_k\}^{\infty}_1$. It is known (see \cite[Theorem 6.2.3]{GK65}) that the system $\{u_k\}^{\infty}_1$ forms (after possible replacement of a finite number of vectors by another system of linearly independent vectors) a Riesz basis in $\cH_0$. Assume for convenience that such replacement is not needed, i.e. the system $\{u_k\}_1^\infty$ itself constitutes the Riesz basis in $\cH_0$. Denote by $P_0$ the orthoprojection in $\cH$ onto $\cH_0$ and put
\begin{equation*} F_{0}(\cdot) := P_0 F(\cdot)\lceil\cH_0 \quad \text{and}\quad F_{I,0}(z) := P_0 F_{0,I}(z)\lceil\cH_0, \quad z\in\mathbb C_+.
\end{equation*}
First we show that $F_{I,0}(\cdot)\in R[\cH_0]$. Indeed, since the system $\{u_k\}_1^\infty$ forms the Riesz basis in $\cH_0$, any $u\in \cH_0$ admits a decomposition $u=\sum_k c_k u_k$ with $c:= \{c_k\}_1^\infty \in l^2(\mathbb N)$. Clearly,
\begin{eqnarray*}
\|F_{I,0}(z_0)\sum^n_{k=1}c_k u_k \|^2 &\le & \left(\sum^n_{k=1}|c_k|\cdot\|F_{I,0}(z_0)u_k \|\right)^2 \nonumber \\
&\le &(\sum^n_1|c_k|^2)\bigl(\sum^n_1\|P_0F_I(z_0)u_k\|^2\bigr)\nonumber \\
&\le & 2(\alpha_{F}(z_0) + \beta_{F}(z_0)))\cdot\|c\|_{l^2}^2, \quad n\in \mathbb N.
\end{eqnarray*}
Hence $F_{I,0}(z_0) \in [\cH_0]$. Moreover, it is easily seen that $F_{0}(\cdot)$ satisfies the conditions of Proposition~\ref{prop_form_stability} together with $F(\cdot)$. Thus, by Proposition \ref{prop6.2}, $F_{I,0}(\cdot)$ takes values in $\cB(\cH_0)$.
(ii) It follows from \eqref{4.25} and \eqref{4.26} that the sequence $\{u_k\}_{k\in \N}$ is a quasi-eigen sequence for the operator $F_{I,0}(z_0)$ corresponding to the point $a=0,$ i.e. it is bounded, non-compact and
\begin{equation}\label{4.27}
\lim_{k\to\infty}\|F_{I,0}(z_0)u_k\|=0.
\end{equation}
Define a family of scalar nonnegative harmonic functions $h_k(\cdot) := \bigl(F_{I,0}(\cdot)u_k,u_k\bigr)$ in $\mathbb C_+$. Since the sequence $\{u_k\}_{k\in \N}$ is bounded, relation \eqref{4.27} yields
\[ \lim_{k\to\infty}h_k(z_0) = \lim_{k\to\infty} \bigl(F_{I,0}(z_0)u_k, u_k\bigr) = 0. \]
By the Harnack inequality \eqref{6.0}, this relation implies similar relation for any $z\in\mathbb C_+$ (cf.~\eqref{6.4}),
\[
\lim_{k\to\infty}h_k(z) = \lim_{k\to\infty} \bigl(F_{I,0}(z)u_k, u_k\bigr) = \lim_{k\to\infty} \|F_{I,0}^{1/2}(z)u_k\|^2 = 0, \qquad z\in\mathbb C_+. \]
Since $F_{I,0}(\cdot)$ takes values in $\cB(\cH_0)$, the latter implies
$\lim_{k\to\infty} \|F_{I,0}(z)u_k\| =0$ which proves the result.
\end{proof}
\begin{corollary}\label{cor6.4} Let $F(\cdot)\in R(\cH)$ and $F(z_0)\in \cB(\cH)$ for some $z_0\in\mathbb C_+$. Then $F(\cdot)\in R[\cH]$.
\end{corollary}
\begin{proof} By Proposition \ref{prop6.2}, $F(\cdot)= F_1(\cdot) + T$ where $F_1(\cdot)\in R[\cH]$ and $T=T^*$. Since \[ F(z_0)=F_1(z_0) + T\in\cB(\cH)\quad\mbox{ and }\quad F_1(z_0)\in\cB(\cH), \] the operator $T$ is bounded and $F\in R[\cH]$.
\end{proof}
Next we present another proof of statement (iv) in Theorem~\ref{invar}.
\begin{proposition}\label{prop6.5} Let $F(\cdot)\in R(\cH)$, $a =\overline{a}\in\sigma_p\bigl(F(z_0)\bigr)$ for $z_0\in\mathbb C_+$. Then
\[ a\in \sigma_p \bigl(F(z)\bigr)\qquad \text{and}\qquad {\xker\,}\bigl(F(z)-a)= {\xker\,}\bigl(F(z_0)-a) \qquad z\in\mathbb C_+. \]
\end{proposition}
\begin{proof} Since $F(\cdot) - a\in R(\cH)$ for any $a\in \mathbb R$, we can assume without loss of generality that $a=0$. Let us put $G(\cdot) : = -\bigl(F(\cdot)+ i\bigr)^{-1}$. Since $F(z)$ is $m$-dissipative for $z\in \mathbb C_+$, $G(\cdot) \in R[\cH]$. Moreover, due to the classical estimate \[
\|G(z)\| = \|\bigl(F(z) + i\bigr)^{-1}\| \le 1, \qquad z\in \C_+, \] $G(\cdot)$ is a contractive holomorphic operator-valued function in $\mathbb C_+.$
Hence its imaginary part $G_I(\cdot)$ is also contractive, $0\le G_I(z) \le 1,\ z\in\mathbb C_+$. Further, let us assume that $u_0 \in {\xker\,}\bigl(F(z_0)\bigr)$ and for definiteness $\|u_0\|=1$. Then $h(\cdot) := \bigl(G_I(\cdot)u_0,u_0\bigr)$ is a scalar nonnegative contractive harmonic function in $\mathbb C_+$. Moreover, since $\bigl(F(z_0)+i\bigr)u_0 = iu_0$ we get
\begin{equation*}
G_I(z_0)u_0=u_0\quad \text{and}\quad h(z_0) = \bigl(G_I(z_0)u_0,u_0\bigr) = \|u_0\|^2 = 1.
\end{equation*}
According to the Maximum Principle applied to the contractive harmonic function $h(\cdot)$, one gets $h(z) = h(z_0) = 1$, $z\in\mathbb C_+$. Rewriting this identity in the form
\begin{equation*} \bigl((I - G_I(z))u_0, u_0\bigr) = 0,\quad z\in\mathbb C_+,
\end{equation*}
and noting that $I-G_I(z)\ge 0$, we derive $G_I(z)u_0=u_0,\ z\in\mathbb C_+$. Since $G(\cdot)$ is contractive, the previous identity yields $G(z)u_0 = iu_0$, i.e. $F(z)u_0=0$ for $z\in\mathbb C_+$. \end{proof}
\begin{corollary} Assume the conditions of Proposition \ref{prop6.5} and let $\cH_0 :={\xker\,}\bigl(F(i)-a\bigr)$. Then $\cH = \cH_0\oplus\cH_1$ and $F(\cdot)$ admits the following orthogonal decomposition
\begin{equation*} F(z)=aI_{\cH_0}\oplus F_a(z),
\end{equation*}
where $F_a(\cdot)\in R(\cH_1)$ and ${\xker\,}\bigl(F_a(z)-a\bigr)=\{0\},\ z\in\mathbb C_+$.
\end{corollary}
\begin{proposition}\label{prop_continuous_spec} Let $F\in R(\cH)$ satisfy the conditions of Proposition \ref{prop_form_stability}, and let $a = \overline{a}\in\sigma_c\bigl(F(z_0)\bigr)$ for some $z_0\in\mathbb C_+$. Then
\begin{equation*} a\in\sigma_c\bigl(F(z)\bigr)\qquad \text{for}\qquad z\in\mathbb C_+.
\end{equation*}
Moreover, a quasi-eigen sequence can be chosen to be common for all $F(z),\ z\in \mathbb C_+.$
\end{proposition}
\begin{proof} Repeating the procedure applied in the proof of Proposition \ref{prop_Harmonic_cont_spec} one reduces the proof to the case of $F=F_0$ with values in $\cB(\cH).$
Therefore $F_0(\cdot)$ admits the integral representation \eqref{6.14}
with $B_0\ge 0,\ B_1=B^*_1\in[\cH_0]$ and $\Sigma(\cdot)$ being the $\cB(\cH_0)$-valued operator measure satisfying condition \eqref{Measuer_Condition}.
Clearly, $K_\Sigma \ge 0$ and $K_\Sigma\in \cB(\cH_0)$.
Repeating the reasoning of Proposition \ref{prop_Harmonic_cont_spec} one shows that there exists a quasi-eigen sequence for $F_{0}(z_0)$ and such that $\{u_k\}\subset {\mathcal D}} \def\cE{{\mathcal E}} \def\cF{{\mathcal F}(F)$, i.e.
\begin{equation}\label{4.27NevFun}
\lim_{k\to\infty}\|F_{0}(z_0)u_k\|=0.
\end{equation}
Setting \[ H(z):= {\rm Im\,} F_0(z) := (2i)^{-1}\bigl(F_0(z) - F_0(z)^*\bigr) \] one defines a family of scalar nonnegative harmonic functions $h_n(\cdot) := \bigl(H(\cdot)u_n,u_n\bigr)$ in $\mathbb C_+$. It follows from \eqref{4.27NevFun} that
$\lim_{n\to\infty}h_n(z_0) = \lim_{n\to\infty} \bigl(H(z_0)u_n, u_n\bigr) = 0.$
By Proposition \ref{prop_Harmonic_cont_spec} similar conclusion holds for any $z\in\mathbb C_+$, i.e.
\begin{equation}\label{6.30} \lim_{n\to\infty}h_n(z) = \lim_{n\to\infty} \bigl(H(z)u_n, u_n\bigr) = 0, \qquad z\in\mathbb C_+.
\end{equation}
On the other hand, it follows from \eqref{Int_rep-n} with account of \eqref{Measuer_Condition} that
\begin{equation}\label{6.32} H(i) = -i\bigl(F_0(i) - B_0\bigr) = B_1 + K_{\Sigma}.
\end{equation}
Combining \eqref{6.30} with \eqref{6.32} and noting that $B_1\ge 0$ and $K_{\Sigma}>0$, one gets
\begin{equation}\label{6.33}
\lim_{n\to\infty}\|B_1u_n\| = \lim_{n\to\infty}\|K_{\Sigma}u_n\| = 0.
\end{equation}
Further, for any fixed $z\in\mathbb C_+$ we set
\begin{equation}\label{6.34}
c_2(z):=\max_{t\in\mathbb R}\left|\frac{1+z t}{t-z}\right|.
\end{equation}
Combining \eqref{Int_rep-n} with \eqref{Measuer_Condition}, applying the Cauchy-Bunyakovskii inequality for integrals, and taking the notation \eqref{6.34} into account we derive (cf. \cite[Section 7]{MalMal03})
\begin{eqnarray*}
|\bigl((F_0(z)- B_1z - B_0)u, v\bigr)|^2
&\le & \left|\int_{\mathbb R}\frac{1+z t}{(t-z)(1+t^2)}\,d \bigl(\Sigma(t)u, v\bigr)\right|^2 \nonumber \\ &\le & c_2(z)^2 \int_{\mathbb R} \frac{1}{1+t^2}\,d \bigl(\Sigma(t)u, u\bigr) \int_{\mathbb R} \frac{1}{1+t^2}\,d \bigl(\Sigma(t)v, v\bigr) \nonumber \\ &\le & c_2(z)^2 (K_{\Sigma}u, u)(K_{\Sigma}v, v) \nonumber \\
&= & c_2(z)^2 \|K_{\Sigma}^{1/2}u\|^2\cdot \|K_{\Sigma}^{1/2}v\|^2.
\end{eqnarray*}
This "weak" estimate is equivalent to the following "strong" one
\begin{equation}\label{6.36}
\|\bigl(F_0(z)- B_1z - B_0\bigr)u\| \le c_2(z)\|K_{\Sigma}^{1/2}\|\cdot \|K_{\Sigma}^{1/2}u\|,\quad z\in\mathbb C_+.
\end{equation}
Inserting in this inequality $u=u_n$ and taking into account \eqref{6.33} yields
\begin{equation}\label{6.41}
\lim_{n\to\infty} \|\bigl(F_0(z) - B_0\bigr)u_n\| = 0, \qquad z\in\mathbb C_+.
\end{equation}
Setting here $z=z_0$ and using the assumption $ \lim_{n\to\infty}\|F_0(z_0)u_n\| = 0,$ we get $\lim_{n\to\infty}B_0u_n =0$. Finally, combining this relation with \eqref{6.41} implies
\begin{equation*} \lim_{n\to\infty}F_0(z)u_n = 0, \quad z\in\mathbb C_+.
\end{equation*}
Since the sequence $\{u_n\}_{n\in \mathbb N}$ is non-compact, the latter means that $0\in\sigma_c\bigl(F_0(z)\bigr)$. Hence $0\in\sigma_c\bigl(F(z)\bigr)$ and the result is proved.
\end{proof}
\begin{proposition} Let $F(\cdot)\in R[\cH]$ and $F(z_0)\in \mathfrak S_p(\cH)$ for some $z_0\in\mathbb C_+$ and $p\in (0, \infty]$. Then $F(\cdot)$ takes values in $ \mathfrak S_p(\cH)$,
\begin{equation*} F(\cdot): \mathbb C_+ \to \mathfrak S_p(\cH).
\end{equation*}
\end{proposition}
\begin{proof} Since $F(\cdot)\in R[\cH]$, it admits integral representation \eqref{6.14}. According to \eqref{6.36} the following estimate holds
\[
|\bigl((F(z) - B_1z - B_0)u, v\bigr)| \le c_2(z)\|K_{\Sigma}^{1/2}u\|\cdot \|K_{\Sigma}^{1/2}v\|,\quad u,v \in \cH, \quad z\in\mathbb C_+, \]
where $K_{\Sigma}\ge 0$ is a nonnegative bounded operator in $\cH$ given by \eqref{Measuer_Condition}. This estimate is equivalent to the following representation
\begin{equation}\label{6.44} F(z) - B_1z - B_0 = K^{1/2}_{\Sigma}T(z)K^{1/2}_{\Sigma}, \quad z\in\mathbb C_+,
\end{equation}
where $T(z)$ is an operator-valued function with values in
$\cB(\cH)$ and $\|T(z)\|\le c_2(z)$.
On the other hand, setting as above $H(z):= {\rm Im\,} F(z) := F_I(z)$, applying the Harnack's inequality \eqref{6.0}, and taking \eqref{6.32} into account, we obtain
\begin{eqnarray*} C_1\bigl(H(z_0)u, u\bigr) &\le & \bigl(H(i)u, u\bigr) = \bigl(B_1u, u\bigr) + \bigl(K_{\Sigma}u, u\bigr) \nonumber \\ &\le & C_2\bigl(H(z_0)u, u\bigr), \qquad u\in\cH. \end{eqnarray*}
It follows that there exists an operator $T_0 \in\cB(\cH)$ with bounded inverse $T^{-1}_0\in\cB(\cH)$ and such that
\begin{equation*} B_1 + K_{\Sigma} = T_0 H(z_0)T_0^* = T_0 F_I(z_0)T_0^*\in\mathfrak S_p(\cH).
\end{equation*}
Since both operators $B_1$ and $K_{\Sigma}$ are nonnegative, one gets
\begin{equation*} s_j(B_1) = {\lambda}_j(B_1) \le {\lambda}_j(B_1+K_{\Sigma}) = s_j(B_1 + K_{\Sigma}), \quad j\in\mathbb N.
\end{equation*}
Hence $B_1\in\mathfrak S_p(\cH)$ and $K_{\Sigma}\in \mathfrak S_p(\cH)$. Combining these inclusion with \eqref{6.44} we get \[ F(z)- B_0\in \mathfrak S_p(\cH). \] Setting here $z=z_0$, yields $B_0\in\mathfrak S_p(\cH)$. Thus, $F(z) \in \mathfrak S_p(\cH)$ for any $z\in\mathbb C_+$.
\end{proof}
\begin{corollary} Let $F(\cdot)\in R[\cH]$ and $F_I(z_0)\in \mathfrak S_p(\cH)$ for some $z_0\in\mathbb C_+$ and $p\in (0, \infty]$. Then $F_I(\cdot)$ takes values in $ \mathfrak S_p(\cH)$,
\begin{equation*} F(\cdot): \mathbb C_+ \to \mathfrak S_p(\cH).
\end{equation*}
\end{corollary}
\begin{remark}\label{rem_general_ideal} The result is valid for any two-sided ideal $\mathfrak S(\cH)$ instead of $\mathfrak S_p(\cH)$. In particular, for any $F(\cdot)\in R[\cH]$ the following implication holds
\begin{equation*} s_j\bigl(F(z_0)\bigr)= O(j^{-1/p}) \Longrightarrow s_j\bigl(F(z)\bigr)= O(j^{-1/p}), \quad z\in \C_+.
\end{equation*}
\end{remark}
\section{Examples}
\begin{example} Let $\varphi(\cdot)$ be a scalar $R$-function and $\cH =L^2(0, \infty).$ Consider an operator-valued function $F_\varphi(\cdot)$ given by
\[ F_\varphi(z) u = -\frac{d^2u}{dx^2},\qquad {\rm dom\,} (F_\varphi(z)) = \{u\in W^2_2(\mathbb R_+) :\ u'(0) = \varphi(z)u(0)\}, \quad z \in \C_+. \]
Clearly, $F(\cdot)\in R(\cH)$ and
$$ \mathcal D(F) := \cap_{z\in\mathbb C_+}{\rm dom\,} F(z) = W^2_{2,0}(\mathbb R_+) := \{u\in W^2_{2}(\mathbb R_+): u(0)= u'(0)=0\} $$
is dense in $\cH$. The corresponding family of closed quadratic forms reads as follows
\begin{equation}\label{quadr_form}
{F}_{\varphi(z)} [u] = \int_{\R_+}|u'(x)|^2dx + \varphi(z)|u(0)|^2, \quad u\in {\rm dom\,} ({F}_{\varphi(z)}) = W^1_2(\mathbb R_+), \quad z \in \mathbb C_+.
\end{equation}
However, the imaginary parts of these forms constitute a family of non-closable (singular) forms
\begin{equation*} {\mathfrak t}_{\varphi(z)} [u] := {\rm Im\,} {F}_{\varphi(z)} [u] =
{\rm Im\,}{\varphi(z)}\cdot|u(0)|^2, \quad u\in W^1_2(\mathbb R_+), \quad z \in \mathbb C_+.
\end{equation*}
In accordance with Proposition \ref{prop_form_stability} they are non-closable for all $z\in \C_+$ simultaneously.
On the other hand, taking the real part of the form \eqref{quadr_form} one gets
\begin{equation}\label{real_part_form} {\mathfrak r}_{\varphi(z)} [u] := {\rm Re\,} {F}_{\varphi(z)} [u] =
\int_{\R_+}|u'(x)|^2dx + {\rm Re\,}{\varphi(z)}\cdot|u(0)|^2, \quad u\in W^1_2(\mathbb R_+), \quad z \in \mathbb C_l.
\end{equation}
If $\varphi(\cdot)\in S_+$, then this form is nonnegative for each $z\in \C_l,$ hence $F_\varphi(\cdot)\in S_+(\cH)$. This example demonstrates Proposition \ref{prop_form_stability} applied to the real part of $F_\varphi(\cdot)$ in place of its imaginary part: the form ${\mathfrak r}_{\varphi(z)}$ is closed for each $z\in \C_l,$
$$ \cH_{+,r}(z) = \cH_{+,r}(z_0) = W^1_2(\mathbb R_+), \ z\in\mathbb C_l, \quad \text{and}\quad \mathcal D_r[F] = W^1_2(\mathbb R_+). $$ Moreover, the operator associated with the form \eqref{real_part_form} is given by \[ F_{\varphi(z),R}(z) u = -\frac{d^2u}{dx^2},\quad {\rm dom\,} (F_{\varphi(z),R}(z)) = \{u\in W^2_2(\mathbb R_+) : u'(0) = ({\rm Re\,}\varphi(z))u(0)\}, \ z \in \C_l. \]
In accordance with Definition \ref{def_imaginary_part} (in fact, with its real counterpart) $F_{\varphi(z),R}(\cdot)$ is the real part of the function $F_{\varphi(z)}(\cdot)\in S_+(\cH)$.
It is easily seen that $a\in\sigma_c(F_\varphi(z))$ for each $z\in \C\setminus \R_+$ and $a\ge0.$ This fact correlates with Propositions \ref{prop_continuous_spec}
for $z\in \C_+$ and $z\in \C_l$, respectively.
\end{example}
\begin{example} Let $\varphi(\cdot)$ and $\cH$ be as above. Define an operator-valued function $G_\varphi(\cdot)$ by
\[ G_\varphi(z) f = -i\frac{d^2u}{dx^2},\qquad {\rm dom\,} (G_\varphi(z)) = \{u\in W^2_2(\mathbb R_+) :\ u'(0) = \varphi(z)u(0)\}, \quad z \in \C_+. \]
Clearly, $\rho(G_\varphi(z))\not = \emptyset$ for each $z \in \C_+.$ Furthermore, the corresponding family of quadratic forms have the description
\begin{equation}\label{6.7}
{G}_{\varphi(z)} [u] = i\int_{\R_+}|u'(x)|^2dx + \varphi(z)|u(0)|^2, \quad u\in {\rm dom\,} {G}_{\varphi(z)} = {\rm dom\,} (G_\varphi(z)).
\end{equation}
It follows that the form ${G}_{\varphi(z)} [\cdot]$ is dissipative for each $z \in \C_+$, hence $G(\cdot)\in R(\cH)$ and $\mathcal D(G) = W^2_{2,0}(\mathbb R_+)$ is dense in $\cH$. Taking imaginary part in \eqref{6.7} we get
$$ {\mathfrak t}_{\varphi(z)}[u] := {\rm Im\,} {G}_{\varphi(z)}[u] =
\int_{\R_+}|u'(x)|^2dx + {\rm Im\,} \varphi(z)|u(0)|^2, \quad {\rm dom\,} {\mathfrak t}_{\varphi(z)} = {\rm dom\,} (G_\varphi(z)). $$
This form is closable and its closure is given by
$$ {\overline {\mathfrak t}}_{\varphi(z)}[u] := {\rm Im\,} {G}_{\varphi(z)}[u]
= \int_{\R_+}|u'(x)|^2dx + {\rm Im\,} \varphi(z)|u(0)|^2, \quad {\rm dom\,} {\overline {\mathfrak t}}_{\varphi(z)} = W^1_2(\mathbb R_+), \ z \in \mathbb C_+. $$
The latter is in accordance with Proposition \ref{prop_form_stability}:
$$ \cH_+(z) = W^1_2(\mathbb R_+), \quad z\in\mathbb C_+, \quad \text{and}\quad \mathcal D[G_{\varphi}] = W^1_{2}(\mathbb R_+). $$
The operator associated with the form ${\overline {\mathfrak t}}_{\varphi(z)}$ (the imaginary part of the operator $G_{\varphi}(z)$ in the sense of Definition \ref{def_imaginary_part}) is given by
\[ G_{\varphi(z),I} u = - \frac{d^2u}{dx^2},\quad {\rm dom\,} (G_{\varphi(z),I}) = \{u\in W^2_2(\mathbb R_+): u'(0) = ({\rm Im\,} \varphi(z)) u(0)\}, \ z \in \C_+. \]
According to Proposition \ref{prop_Harmonic_cont_and_point_spec}, $a\in\sigma_c(G_{\varphi(z),I})$ for each $z\in \C_+$ and $a\ge0.$ Moreover, by Proposition \ref{prop_continuous_spec}, $0\in\sigma_c(G_{\varphi(z)})$ for each $z\in \C_+.$
\end{example}
\begin{example}\label{ex_finite_int-val} Let $\varphi(\cdot)$ be a scalar $R$-function and $\cH =L^2(0, 1).$ Consider an operator-valued function $G_\varphi(\cdot)$ given by
\[ G_\varphi(z) u = -i\frac{d^2u}{dx^2},\quad {\rm dom\,} (G_\varphi(z)) = \{u\in W^2_2(0,1) :\ u'(0) = \varphi(z)u(0),\ u(1)=0\}, \quad z \in \C_+. \]
It is easily seen that $\rho(G_\varphi(z))\not = \emptyset$ for each $z \in \C_+$ and the operator $G_\varphi(z)$ has discrete spectrum. Moreover, the corresponding quadratic form is
\begin{equation}\label{6.7B}
{G}_{\varphi(z)} [u] = \int_0^1 |u'(x)|^2\,dx + \varphi(z)|u(0)|^2, \quad u\in {\rm dom\,} {G}_{\varphi(z)} = {\rm dom\,} (G_\varphi(z)).
\end{equation}
Clearly, the form is dissipative, hence $G(\cdot)\in R(\cH)$ and
$$ \mathcal D(G) := \cap_{z\in\mathbb C_+}{\rm dom\,} G(z) = \{u\in W^2_{2}(0,1): u(0)= u'(0)=u(1) =0\} $$
is dense in $\cH$. Taking imaginary part in \eqref{6.7B} one obtains a nonnegative closable form ${\mathfrak t}_{\varphi(z)}[\cdot]$ defined on ${\rm dom\,} (G_\varphi(z))$. Its closure is given by
$$ {\overline {\mathfrak t}}_{\varphi(z)}[u] := {\rm Im\,} {G}_{\varphi(z)}[u]
= \int_0^1 |u'(x)|^2dx + {\rm Im\,} \varphi(z)|u(0)|^2, \quad {\rm dom\,} ({\overline {\mathfrak t}}_{\varphi(z)}) = {\widetilde W}^2_{2,0}(0,1), \ z \in \mathbb C_+,
$$
where ${\widetilde W}^2_{2,0}(0,1) := \{u\in W^1_2(0,1) :\ u(1)=0\}$.
The latter is in accordance with Proposition \ref{prop_form_stability}:
$$ \cH_+(z) = {\widetilde W}^2_{2,0}(0,1), \quad z\in\mathbb C_+, \quad \text{and}\quad \mathcal D[G_{\varphi}] = {\widetilde W}^2_{2,0}(0,1). $$
The operator associated with the form ${\overline {\mathfrak t}}_{\varphi(z)}$ (the imaginary part of $G_{\varphi}(z)$) is given by
\[ G_{\varphi,I}(z) u = - \frac{d^2u}{dx^2},\ {\rm dom\,} (G_{\varphi,I})(z) = \{u\in W^2_2(\mathbb R_+): u'(0) - ({\rm Im\,} \varphi(z)) u(0)= u(1)=0\}. \]
Since the spectrum of $G_{\varphi}(z)$ is discrete, $\sigma_c(G_{\varphi}(z)) = \sigma_c(G_{\varphi,I}(z))=\emptyset$ for each $z\in \C_+$. This fact is in accordance with Propositions \ref{prop_continuous_spec} and \ref{prop_Harmonic_cont_and_point_spec}.
Moreover, the estimate $s_j((G_{\varphi(z)})^{-1}) = O(j^{-2})$, $j\in \N$, holds for each $z\in \C_+.$ This fact correlates with Remark \ref{rem_general_ideal}.
\end{example}
\end{document} | arXiv |
{{#invoke:Hatnote|hatnote}} {{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }}
Books of the
MatthewTemplate:·MarkTemplate:·LukeTemplate:·John
1 CorinthiansTemplate:·2 Corinthians
GalatiansTemplate:·Ephesians
PhilippiansTemplate:·Colossians
1 ThessaloniansTemplate:·2 Thessalonians
1 TimothyTemplate:·2 Timothy
TitusTemplate:·Philemon
HebrewsTemplate:·James
1 PeterTemplate:·2 Peter
1 JohnTemplate:·2 JohnTemplate:·3 John
New Testament manuscripts
Template:Navbar
Template:Chapters in the Gospel of John Template:Content of John Template:John
The Gospel According to John (also referred to as the Gospel of John, the Fourth Gospel, or simply John; Greek: Τὸ κατὰ Ἰωάννην εὐαγγέλιον{{#invoke:Category handler|main}}, to kata Ioannen euangelion) is one of the four canonical gospels in the Christian Bible. In the New Testament it traditionally appears fourth, after the synoptic gospels of Matthew, Mark and Luke. John begins with the witness and affirmation of John the Baptist and concludes with the death, burial, resurrection, and post-resurrection appearances of Jesus.
Template:Bibleref2 states that the book derives from the testimony of the "disciple whom Jesus loved" and early church tradition identified him as John the Apostle, one of Jesus' Twelve Apostles. The gospel is closely related in style and content to the three surviving Epistles of John such that commentators treat the four books,[1] along with the Book of Revelation, as a single body of Johannine literature. According to most modern scholars, however, the apostle John was not the author of any of these books.[2]
Raymond E. Brown has proposed the development of a tradition from which the gospel arose.[3] The discourses seem to be concerned with issues of the church-and-synagogue debate at the time when the Gospel was written.[4] It is notable that, in the gospel, the community appears to define itself primarily in contrast to Judaism, rather than as part of a wider Christian community.[5] Though Christianity started as a movement within Judaism, Christians and Jews gradually became bitterly opposed.[6]
1.1 Authorship
2.1 Order of material
2.2 Signs Gospel
2.3 Discourses
2.4 Inspiration
2.5 Trimorphic Protennoia
2.6 Date
2.7 Textual history and manuscripts
2.7.1 Egerton gospel
2.8 Position in the New Testament
3 Narrative summary (structure and content)
3.1 Hymn to the Word
3.2 Seven signs
3.3 Last teachings and death
4 Characteristics
4.1 Christology
4.1.1 Jesus' divine role
4.1.2 Logos
4.2 John the Baptist
4.3 Jews
4.4 Gnostic elements
4.5 Historical reliability
4.7 Chronology of Jesus' ministry
4.7.1 Two-year ministry
4.7.2 Cleansing of the Temple
4.7.3 Earlier baptizing ministry in Judea
4.7.4 Repeated visits to Jerusalem
4.7.5 Date of the crucifixion
5 Compared with the synoptics
5.1 Comparison chart of the major gospels
7 Representations
{{#invoke:main|main}} The Gospel of John was written in Greek by an anonymous author.[7][8][9][10][11][12][13][14][15] According to Paul N. Anderson, the gospel "contains more direct claims to eyewitness origins than any of the other Gospel traditions".[16] F. F. Bruce argues that 19:35 contains an "emphatic and explicit claim to eyewitness authority".[17] Bart D. Ehrman, however, does not think the gospel claims to have been written by direct witnesses to the reported events.[9][18][19]
The gospel identifies its author as "the disciple whom Jesus loved." Although the text does not name this disciple, by the beginning of the 2nd century, a tradition had begun to form which identified him with John the Apostle, one of the Twelve (Jesus' innermost circle). Although some notable New Testament scholars affirm traditional Johannine scholarship,[20][21] the majority do not believe that John or one of the Apostles wrote it,[22][23][24][25][26][27] and trace it instead to a "Johannine community" which traced its traditions to John; the gospel itself shows signs of having been composed in three "layers", reaching its final form about 90–100 AD.[28][29] According to Victorinus[30]Template:Fv and Irenaeus,[31]Template:Fv the Bishops of Asia Minor requested John, in his old age, to write a gospel in response to Cerinthus, the Ebionites and other Jewish Christian groups which they deemed heretical.[32] This understanding remained in place until the end of the 18th century.[33]
The earliest manuscripts to contain the beginning of the gospel (Papyrus 66 and Papyrus 75), dating from around the year 200, entitled "The Gospel according to John".
According to some, the Gospel of John developed over a period of time in various stages,[34] summarized by Raymond E. Brown as follows:[35]
An initial version based on personal experience of Jesus;
A structured literary creation by the evangelist which draws upon additional sources;
The final harmony that presently exists in the New Testament canon, around 85–90 AD.[36]
Within this view of a complex and multi-layered history, it is meaningless to speak of a single "author" of John, but the title perhaps belongs best to the evangelist who came at the end of this process.[37] The final composition's comparatively late date, and its insistence upon Jesus as a divine being walking the earth in human form renders it highly problematical to scholars who attempt to evaluate Jesus' life in terms of literal historical truth.[38][39]
Order of material
Stained glass depiction of St. John at St. Matthew's German Evangelical Lutheran Church in Charleston, South Carolina.
Among others, Rudolf Bultmann suggested[40] that the text of the gospel is partially out of order; for instance, chapter 6 should follow chapter 4:[41]
4:53 So the father knew that it was at the same hour, in the which Jesus said unto him, Thy son liveth: and himself believed, and his whole house.
4:54 This is again the second miracle that Jesus did, when he was come out of Judaea into Galilee.
6:1 After these things Jesus went over the sea of Galilee, which is the sea of Tiberias.
6:2 And a great multitude followed him, because they saw his miracles which he did on them that were diseased.
Chapter 5 deals with a visit to Jerusalem, while chapter 7 opens with Jesus again in Galilee because "he would not walk in Judaea, because the Jews sought to kill him," a consequence of the incident in Jerusalem described in chapter 5. There are more proposed rearrangements.
Signs Gospel
Template:Further2 One possible construction of the "internal evidence" states that the Beloved Disciple wrote an account of the life of Jesus,Template:Bibleref2c-nb but that this disciple died unexpectedly, necessitating that a revised gospel be written.Template:Bibleref2c-nb It may be that John "is the source" of the Johannine tradition but "not the final writer of the tradition."[42] Therefore, scholars are no longer looking for the identity of a single writer but for numerous authors whose authorship has been absorbed into the gospel's development over a period of time and in several stages.[34][35][43]
The hypothesis of the Gospel being composed in layers over a period of time had its start with Rudolf Bultmann in 1941. Bultmann suggested[40] that the author(s) of John depended in part on an author who wrote an earlier account. This hypothetical "Signs Gospel" listing Christ's miracles was independent of, and not used by, the synoptic gospels. It was believed to have been circulating before the year 70 AD. Bultmann's conclusion was so controversial that heresy proceedings were instituted against him and his writings. (See: Depiction of Jesus and more detailed discussions linked below.)
Nevertheless, scholars such as Raymond Edward Brown continue to consider this hypothesis a plausible possibility. They believe the original author of the Signs Gospel to be the Beloved Disciple. They argue that the disciple who formed this community was both an historical person and a companion of Jesus Christ. Brown goes one step further by suggesting that the Beloved Disciple had been a follower of John the Baptist before joining Jesus.[35]
The author may have used a source consisting of lengthy discourses,[44] but this issue has not been clarified.[45]
The author has Jesus foretell that new knowledge will come to his followers after his death.[46] This reference indicates that the author may have included new information, not previously revealed, that is derived from spiritual inspiration rather than from historical records or recollection.[46]
Trimorphic Protennoia
{{#invoke:main|main}} In terminology close to that found in later Gnostic works, one tract, generally known as the Trimorphic Protennoia, must either be dependent on John or the other way round.[47]
{{#invoke:main|main}} The gospel was apparently written near the end of the 1st century.[48][49] Bart Ehrman argues that there are differences in the composition of the Greek within the Gospel, such as breaks and inconsistencies in sequence, repetitions in the discourse, as well as passages that he believes clearly do not belong to their context, and believes that these suggest redaction.[50]
The so-called "Monarchian Prologue" to the Fourth Gospel supports AD 96 or one of the years immediately following as to the time of its writing.[51] Scholars set a range of c. 90–100.[52] The gospel was already in existence early in the 2nd century.[53] It is thought that the Gospel of John was composed in stages (probably two or three).[54] Since the middle of the 2nd century writings of Justin Martyr use language very similar to that found in the Gospel of John, the Gospel is considered to have been in existence at least at that time.[55] The Rylands Library Papyrus P52, which records a fragment of this gospel, is usually dated to the first half of the 2nd century.[56]
Conservative scholars consider internal evidences, such as the lack of the mention of the destruction of the Temple and a number of passages that they consider characteristic of an eyewitness,[57] sufficient evidence that the gospel was composed before 100 and perhaps as early as 50–70.[58] In the 1970s, scholars Leon Morris and John A.T. Robinson independently suggested such earlier dates for the gospel's composition.[59][60][61] Evidence supporting this position comes from the New Testament scholar Daniel Wallace.[62] The strongest argument for this position appears to be that the word ἐστιν ("is" in John 5:2, "Now there is at Jerusalem by the sheep market a pool, which is called in the Hebrew tongue Bethesda...") cannot be a historical present.
The noncanonical Dead Sea Scrolls suggest an early Jewish origin, having parallels and similarities to the Essene Scroll and Community Rule.[63] Many phrases are duplicated in the Gospel of John and the Dead Sea Scrolls.[64] These are sufficiently numerous to challenge the theory that the Gospel of John was the last to be written among the four Gospels or that it shows marked non-Jewish influence.[65]
Textual history and manuscripts
The Rylands Papyrus is perhaps the earliest New Testament fragment; dated from its handwriting to about 125.
Probably the earliest surviving New Testament manuscript, Rylands Library Papyrus P52, is a Greek papyrus fragment discovered in Egypt in 1920 (now at the John Rylands Library, Manchester). Although P52 has no more than 114 legible letters, it must come from a substantial codex book; as it is written on both sides in a generously scaled script, with Template:Bibleref2 on one side and Template:Bibleref2-nb on the other. The surviving text agrees with that of the corresponding passages in the Gospel of John, but it cannot necessarily be assumed that the original manuscript contained the full Gospel of John in its canonical form. Metzger and Aland list the probable date for this manuscript as c. 125[66][67] but the difficulty of estimating the date of a literary text based solely on paleographic evidence must allow potentially for a range that extends from before 100 to well into the second half of the 2nd century. P52 is small, and although a plausible reconstruction can be attempted for most of the fourteen lines represented, the proportion of the text of the Gospel of John for which it provides a direct witness is so small that it is rarely cited in textual debate.[68][69] Other notable early manuscripts of John include Papyrus 66 and Papyrus 75, in consequence of which a substantially complete text of the Gospel of John exists from the beginning of the 3rd century at the latest. Hence the textual evidence for the Gospel of John is commonly accepted as both earlier and more reliable than that for any other of the canonical Gospels.
Much current research on the textual history of the Gospel of John is being done by the International Greek New Testament Project.
Egerton gospel
The mysterious Egerton Gospel appears to represent a parallel but independent tradition to the Gospel of John. According to scholar Ronald Cameron, it was originally composed some time between the middle of the 1st century and early in the 2nd century, and it was probably written shortly before the Gospel of John.[70] Scholar Robert W. Funk and the Jesus Seminar place the Egerton fragment in the 2nd century, perhaps as early as 125, which would make it as old as the oldest fragments of John.[71]
Position in the New Testament
In the standard order of the canonical gospels, John is fourth, after the three interrelated synoptic gospels Matthew, Mark and Luke. In the earliest surviving gospel collection, Papyrus 45 of the 3rd century, it is placed second in the order Matthew, John, Luke and Mark, an order which is also found in other very early New Testament manuscripts. In syrcur it is placed third in the order Matthew, Mark, John and Luke.[72]
Narrative summary (structure and content)
Jesus giving the Farewell Discourse (John 14–17) to his eleven remaining disciples, from the Maesta by Duccio, 1308–1311.
Template:Chapters in the Gospel of John After the prologue,Template:Bibleref2c the narrative of the gospel begins with verse 6, and consists of two parts. The first partTemplate:Bibleref2c-nb relates Jesus' public ministry from John the Baptist recognizing him as the Lamb of God to the raising of Lazarus and Jesus' final public teaching. In this first part, John emphasizes seven of Jesus' miracles, always calling them "signs." The second partTemplate:Bibleref2c-nb presents Jesus in dialogue with his immediate followersTemplate:Bibleref2c-nb and gives an account of his Passion and Crucifixion and of his appearances to the disciples after his Resurrection.Template:Bibleref2c-nb In the "appendix",Template:Bibleref2c-nb Jesus restores Peter after his denial, hints at how Peter would die, and declines to answer Peter's question about the fate of the Beloved Disciple.
Raymond E. Brown, a scholar of the social environment where the Gospel and Letters of John emerged, labeled the first and second parts the "Book of Signs" and the "Book of Glory", respectively.[73]
Hymn to the Word
This prologue is intended to identify Jesus as the eternal Word (Logos) of God.[74] Thus John asserts Jesus' innate superiority over all divine messengers, whether angels or prophets.[75] Here John adapts the doctrine of the Logos, God's creative principle, from Philo, a 1st-century Hellenized Jew.[75]
Philo had adopted the term Logos from Greek philosophy, using it in place of the Hebrew concept of Wisdom (sophia) as the intermediary (angel) between the transcendent Creator and the material world.[75] Some scholars argue that the prologue was taken over from an existing hymn and added at a later stage in the gospel's composition.[74]
Seven signs
Template:Rellink This section recounts Jesus' public ministry.[74] It consists of seven miracles or "signs," interspersed with long dialogues and discourses, including several "I am" sayings. The miracles culminate with his most potent, the raising of Lazarus from the dead.[75] In John, it is this last miracle, and not the temple incident, that prompts the authorities to have Jesus executed.
Last teachings and death
{{#invoke:see also|seealso}} This section opens with an account of the Last Supper that differs significantly from that found in the synoptics.[75] Here, Jesus washes the disciples' feet instead of ushering in a new covenant of his body and blood.[75] This account of foot washing might refer to a local tradition by which foot washing served as a Christian initiation ritual rather than baptism.[76] John then devotes almost five chapters to farewell discourses. Jesus declares his unity with the Father, promises to send the Paraclete, describes himself as the "true vine," explains that he must leave (die) before the Holy Spirit comes, and prays that his followers be one.
The Jesus Seminar has argued that verses John 14:30–31 represent a conclusion, and that the next three chapters have been inserted into the text later. This argument considers the farewell discourse not to be authentic, and postulates that it was constructed after the death of Jesus.[77] However, scholars such as Herman Ridderbos see John 14:30–31 as a "provisional ending" just to that part of the discourse and not an ending to the entire discourse.[78] In 2004 Scott Kellum published a detailed analysis of the literary unity of the entire Farewell Discourse and stated that it shows that it was written by a single author, and that its structure and placement within the Gospel of John is consistent with the rest of that gospel.[79][80]
John then records Jesus' arrest, trial, execution, and resurrection appearances, including "doubting Thomas." Significantly, John does not have Jesus claim to be the Son of God or the Messiah before the Sanhedrin or Pilate, and he omits the midday darkness and the earthquake that is said in Matthew to have accompanied Jesus' death. The gospel also omits Jesus' ascension. John's revelation of divinity is Jesus' triumph over death, the eighth and greatest sign.[75]
Template:Bibleref2, in which the Beloved Disciple is said to be the author, is commonly assumed to be an appendix, probably added to allay concerns after the death of the Beloved Disciple.[75] Chapter 21 states that there had been a rumor that the End would come before the Beloved Disciple died.
Though the three Synoptic Gospels share a considerable amount of text, over 90% of John's Gospel is unique to it.[81] The synoptics describe much more of Jesus' life, miracles, parables, and exorcisms. However, the material unique to John is notable, especially in its effect on later Christianity.
As a gospel, John is a story about the life of Jesus. The Gospel can be divided into four parts:[82]
The Book of signs
The Book of exaltation (Passion narrative)
The Epilogue.[83]
The PrologueTemplate:Bibleref2c is a hymn identifying Jesus as the divine Logos. The Book of SignsTemplate:Bibleref2c-nb recounts Jesus' public ministry, and includes the signs worked by Jesus and some of his teachings. The Passion narrativeTemplate:Bibleref2c-nb recounts the Last Supper (focusing on Jesus' farewell discourse), Jesus' arrest and crucifixion, his burial, and resurrection. The EpilogueTemplate:Bibleref2c records a resurrection appearance of Jesus to the disciples in Galilee.
Following on from the "higher criticism" of the 19th century, scholars such as Adolf von Harnack[84] and Raymond E. Brown[35] have questioned the gospel of John as a reliable source of information about the historical Jesus.[85][86]
According to one scholar, John portrays Jesus Christ as "a brief manifestation of the eternal Word, whose immortal spirit remains ever-present with the believing Christian."[87] The book presents Jesus as the divine Son of God, and yet subordinate to God the Father.[88] This gospel gives more focus to the relationship of the Son to the Father than the other gospels. It also focuses on the relation of the Redeemer to believers, the announcement of the Holy Spirit as the Comforter (Greek Paraclete), and the prominence of love as an element in Christian character.
Jesus' divine role
In the synoptics, Jesus speaks often about the Kingdom of God; his own divine role is obscured (see Messianic secret). In John, Jesus talks openly about his divine role. He says, for example, that he is the way, the truth, and the life.Template:Bibleref2c-nb He echoes Yahweh's own statements with several "I am" declarations that also identify him with symbols of major significance.[89] He says, "I am":
"the bread of life"Template:Bibleref2c-nb
"the light of the world"Template:Bibleref2c-nb
"the gate of the sheep"Template:Bibleref2c-nb
"the good shepherd"Template:Bibleref2c-nb
"the resurrection and the life"Template:Bibleref2c-nb
"the way, the truth, and the life"Template:Bibleref2c-nb and
"the true vine"Template:Bibleref2c-nb
Critical scholars think that these claims represent the Christian community's faith in Jesus' divine authority but doubt that the historical Jesus actually made these sweeping claims.[75] John Shelby Spong has argued that the "I Am" statements are in reference to YHWH, and interprets Template:Bibleref2 ("He that believeth on me, believeth not on me, but on him that sent me") as meaning that Jesus expressly denied being God.[90]
John also promises eternal life for those who believe in Jesus.Template:Bibleref2c
{{#invoke:main|main}} In the Prologue, John identifies Jesus as the Logos (Word). A term from Greek philosophy, it meant the principle of cosmic reason. In this sense, it was similar to the Hebrew concept of Wisdom, God's companion and intimate helper in creation. The Jewish philosopher Philo merged these two themes when he described the Logos as God's creator of and mediator with the material world. The evangelist adapted Philo's description of the Logos, applying it to Jesus, the incarnation of the Logos.[28]
The opening verse of John is translated as "the Word was with God and the Word was God" in all "orthodox" English Bibles.[91] There are alternative views. The New World Translation of the Holy Scriptures of Jehovah's Witnesses has, "the Word was with God, and the Word was a god." The Scholar's Version of the gospel, developed by the Jesus Seminar, loosely translates the phrase as "The Logos was what God was," offered as a better representation of the original meaning of the evangelist.[92]
{{#invoke:main|main}} John's account of the Baptist is different from that of the synoptic gospels. John is not called "the Baptist."[74] John's ministry overlaps with that of Jesus, his baptism of Jesus is not explicitly mentioned, but his witness to Jesus is unambiguous.[74] The evangelist almost certainly knew the story of John's baptism of Jesus and he makes a vital theological use of it.[93] He subordinates John to Jesus, perhaps in response to members of the Baptist's sect who denied Jesus' superiority.[75]
In John, Jesus and his disciples go to Judea early in Jesus' ministry when John has not yet been imprisoned and executed by Herod. He leads a ministry of baptism larger than John's own. The Jesus Seminar rated this account as black, containing no historically accurate information.[92] Historically, John likely had a larger presence in the public mind than Jesus.[94]
{{#invoke:main|main}} In his Jerusalem speeches, John's Jesus makes unfavorable references to the Jews (the Ioudaioi, a term with a range of meanings). It has been argued that these references may constitute a rebuttal on the part of the author against Jewish criticism of the early Church.[28] Yet the Gospel of John collectively describes the enemies of Jesus as "the Jews". In none of the other gospels do "the Jews" demand, en masse, the death of Jesus; instead, the plot to put him to death is always presented as coming from a small group of priests and rulers, the Sadducees. John's gospel is thus the primary source of the image of "the Jews" acting collectively as the enemy of Jesus.[95]
Some scholars have attempted to counter charges that the Gospel of John is anti-Semitic by arguing that its author most likely considered himself Jewish, did not deny that Jesus and his disciples were all Jewish, and was probably speaking to a largely Jewish community.[96] While passages in John have been used to support anti-semitism, these passages reflect a dispute within Judaism, and it is highly questionable whether the evangelist himself was anti-semitic.[97]
Gnostic elements
{{#invoke:see also|seealso}} Though not commonly understood as Gnostic, many scholars, perhaps most notably Rudolf Bultmann, have forcefully argued that the Gospel of John has elements in common with Gnosticism.[75] Christian Gnosticism did not fully develop until the mid-2nd century, and so 2nd-century Proto-Orthodox Christians concentrated much effort in examining and refuting it.[98] To say John's Gospel contained elements of Gnosticism is to assume that Gnosticism had developed to a level that required the author to respond to it.[99] Bultmann, for example, argued that the opening theme of the Gospel of John, the pre-existing Logos, was actually a Gnostic theme. Other scholars, e.g. Raymond E. Brown have argued that the pre-existing Logos theme arises from the more ancient Jewish writings in the eighth chapter of the Book of Proverbs, and was fully developed as a theme in Hellenistic Judaism by Philo Judaeus.
Comparisons to Gnosticism are based not in what the author says, but in the language he uses to say it, notably, use of the concepts of Logos and Light.[100] Other scholars, e.g. Raymond E. Brown, have argued that the ancient Jewish Qumran community also used the concept of Light versus Darkness. The arguments of Bultmann and his school were seriously compromised by the mid-20th century discoveries of the Nag Hammadi library of genuine Gnostic writings (which are dissimilar to the Gospel of John) as well as the Qumran library of Jewish writings (which are often similar to the Gospel of John).
Gnostics read John but interpreted it differently from the way non-Gnostics did.[101] Gnosticism taught that salvation came from gnosis, secret knowledge, and Gnostics did not see Jesus as a savior but a revealer of knowledge.[102] Barnabas Lindars asserts that the gospel teaches that salvation can only be achieved through revealed wisdom, specifically belief in (literally belief into) Jesus.[103]
Raymond Brown contends that "The Johannine picture of a savior who came from an alien world above, who said that neither he nor those who accepted him were of this world,Template:Bibleref2c-nb and who promised to return to take them to a heavenly dwellingTemplate:Bibleref2c-nb could be fitted into the gnostic world picture (even if God's love for the world in Template:Bibleref2-nb could not)."[104] It has been suggested that similarities between John's Gospel and Gnosticism may spring from common roots in Jewish Apocalyptic literature.[105]
Historical reliability
The differences between the Synoptics and John were acknowledged in the early Church.[106] Around AD 200, Clement of Alexandria noted that John's gospel was a "spiritual gospel", distinct from the Synoptics.[107] However, there is some degree of debate regarding Clement's exact meaning of "spiritual gospel"; care must be taken not to ascribe to his phrase modern prejudices or expectations. Critical scholarship in the 19th century distinguished between the "biographical" approach of the synoptics and the "theological" approach of John, and began to disregard John as a historical source. Current scholarship, however, emphasizes that all four gospels are both biographical and theological.[108]
According to the majority viewpoint for most of the 20th century, Jesus' teaching in John is largely irreconcilable with that found in the synoptics, and perhaps most scholars consider the Synoptic Gospels to be more accurate representations of the teaching of the historical Jesus.
There are notable exceptions to this perception, e.g. the story of the calling of the first disciples. In the Synoptic Gospels the account of Jesus' calling of his first disciples from among Galilean fishermen belongs to the paranormal or unexplained. In the first chapter of the Gospel of John, by contrast, John the Baptist personally pointed those first apostles to Jesus. Furthermore, the Synoptic Gospels contradict each other about the number of times that Jesus visited Jerusalem, and so should not be given instant precedence in comparison with the Gospel of John which maintains that Jesus visited Jerusalem multiple times.[109]
The teachings of Jesus in John are distinct from those found in the synoptic gospels.[39] Thus, since the 19th century many Jesus scholars have argued that only one of the two traditions could be authentic.[110] J. D. G. Dunn comments on historical Jesus scholarship, "Few scholars would regard John as a source for information regarding Jesus' life and ministry in any degree comparable to the synoptics."[111][112] E. P. Sanders concludes that the Gospel of John contains an "advanced theological development, in which meditations on the person and work of Christ are presented in the first person, as if Jesus said them."[113] Sanders points out that the author would regard the gospel as theologically true as revealed spiritually even if its content is not historically accurate[113] and argues that even historically plausible elements in John can hardly be taken as historical evidence, as they may well represent the author's intuition rather than historical recollection.[113] The scholars of the Jesus Seminar identify the historical inferiority of John as foundational to their work.[114] Geza Vermes discounts all the teaching in John when reconstructing his view of "the authentic gospel of Jesus."[115]
While a large number of 20th-century biblical critics argue that the teaching found in John does not go back to the historical Jesus, they usually agree that gospel is not entirely without historical value.[116] Several of its independent elements are historically plausible,[117] such as Jesus being executed before Passover, as John reports.[117][118] Former followers of John the Baptist probably joined Jesus' movement.[117] It has become generally accepted that certain sayings in John are as old or older than their synoptic counterparts, that John's representation of things around Jerusalem is often superior to that of the synoptics, and that its presentation of Jesus in the garden and the prior meeting held by the Jewish authorities are possibly more historically accurate than their synoptic parallels.[119]
Throughout the 20th century a minority of prominent scholars, such as John A.T. Robinson, have argued that John is as historically reliable as the synoptics. Robinson wrote that, where the gospel narrative accounts can be checked for consistency with surviving material evidence, the account in John is commonly the more plausible.[120] Robinson further wrote that it is generally easier to reconcile the various synoptic accounts within John's narrative framework than to explain John's narrative within the framework of any of the synoptics,[121] and that when in the gospel Jesus and his disciples are described as travelling around identifiable locations the journeys can always be plausibly reconstructed on a map,[122] which is not the case for any synoptic gospel. Scholars such as D. A. Carson, Douglas J. Moo, and Craig Blomberg, often agree with Robinson.[123][124] Henry Wansbrough writes: "Gone are the days when it was scholarly orthodoxy to maintain that John was the least reliable of the gospels historically."[125]
Some scholars today believe that parts of John represent an independent historical tradition from the synoptics, while other parts represent later traditions.[126] The Gospel was probably shaped in part by increasing tensions between synagogue and church, or between those who believed Jesus was the Messiah and those who did not.[127]
Chronology of Jesus' ministry
{{#invoke:main|main}} A distinctive feature of the Gospel of John is that it provides a very different chronology of Jesus' ministry from that in the synoptics. E.P. Sanders suggests that John's chronology, even when ostensibly more plausible, should nevertheless be treated with suspicion on the grounds that the Synoptic accounts are otherwise superior as historic sources. C.H. Dodd proposes that historians may mix and match between John and the synoptics on the basis of whichever appears strongest on a particular episode. Robinson says that John's chronology is consistently more likely to represent the original sequence of events.
Robinson offers three arguments for preferring the chronology of John's Gospel to that of the synoptics. First, he argues that John's account of Jesus' ministry is always consistent, in that seasonal references always follow in the correct sequence, geographical distances are always consistent with indications of journey times, and references to external events always cohere with the internal chronology of Jesus' ministry. He claims that the same cannot be claimed for any of the three Synoptic accounts. For example, the harvest-tide story of Template:Bibleref2 is shortly followed by reference to green springtime pasture at Template:Bibleref2-nb. Again, the historically consistent reference to the period of the temple construction in Template:Bibleref2, may be contrasted with the impossibility of reconciling Luke's account of the census of Template:Bibleref2 with historic records of Quirinius's governorship of Syria. Second, Robinson appeals to the critical principle, widely applied in textual study, that the account is most likely to be original that best explains the other variants. He argues that it would be relatively easy to have created the Synoptic chronology by selecting and editing from John's chronology; whereas expanding the Synoptic chronology to produce that found in John, would have required a wholescale rewriting of the sources. Third, Robinson claims that elements consistent with John's alternative chronology can be found in each of the Synoptic accounts, whereas the contrary is never the case. For example, Mark's explicit claim that the Last Supper was a Passover meal is contradicted by his statement that Joseph of Arimathea bought a shroud for Jesus on Good Friday; which would not have been possible if it were a festival day.
Two-year ministry
In John's Gospel, the public ministry of Jesus extends over rather more than two years. At the start of his ministry, Jesus is in Jerusalem for Passover,Template:Bibleref2c then he is in Galilee for the following Passover,Template:Bibleref2c-nb before going up to Jerusalem again for his death at a third Passover.Template:Bibleref2c-nb The synoptics, by contrast, only explicitly mention the final Passover, and their accounts are commonly understood as describing a public ministry of less than a year. Recent studies in ancient narrative historiography argue that it is possible for John's Gospel to record multiple Passovers—as historical testimony not theological literary-devices—and yet not represent three years, as it was not uncommon for ancient historians to organize their histories without an absolute timeline.[128] If true, this would mean John's chronology is much closer to Synoptic chronology than often assumed.
In favour of the Synoptic chronology, E.P. Sanders observes that a short ministry accords with the careers of other known prophetic figures of the time—who appear in the desert, raise large scale public interest, but soon come to a bloody end at the hand of the Roman military. In favour of the two-year ministry, John Robinson points out that both Matthew and Luke imply that Jesus was preaching in Galilee for at least one Passover during his ministry. The Temple taxTemplate:Bibleref2c is only collected at Passover; moreover, the massacred Galileans of Template:Bibleref2 would appear to have been in Jerusalem for Passover.
Cleansing of the Temple
{{#invoke:main|main}} In John, Jesus drives the money changers from the Temple at the start of his ministry, whereas in the Synoptic account this occurs at the end, immediately after Palm Sunday. In favor of the later dating of the synoptics, Geza Vermes says that this event provides a clear context and pretext for Jesus' arrest, trial and execution. It makes more sense to suppose that events proceeded quickly. But Robinson says that all three Synoptic accounts explain the reluctance of the Temple authorities to arrest Jesus on the spot, as being due to their fear of popular support for John the Baptist. Some believe this would make more sense while the Baptist was still alive.
Earlier baptizing ministry in Judea
In Template:Bibleref2 of the Gospel of John, Jesus, following his encounter with John the Baptist, undertakes an extended and successful baptizing ministry in Judea and on the banks of the River Jordan; initially as an associate of the Baptist, latterly more as a rival. In the Synoptic accounts, Jesus retreats into the wilderness following his baptism, and is presented as gathering disciples from scratch in his home country of Galilee; following which he embarks on a ministry of teaching and healing, in which baptism plays no part. In favour of the Synoptic account is the clear characterisation of Jesus and his disciples in all the Gospels as predominantly Galilean. Against this, Robinson points out that all the synoptics are agreed that, when Jesus arrives in Jerusalem in the week before his death, he already has a number of followers and disciples in the city, notably Joseph of Arimathea, and the unnamed landlord of the upper room, who knows Jesus as 'the Teacher'.
Repeated visits to Jerusalem
In John, Jesus not only starts his ministry in Jerusalem, he returns there for other festivals, notably at Template:Bibleref2 and at Template:Bibleref2-nb. As noted above, E.P Sanders regards the short, sharp prophetic career as having greater verisimilitude. Against this John Robinson notes the numerous instances in the Synoptic account of Jesus' final days in Jerusalem, when it is implied that he has been there before. In (Template:Bibleref2 and Template:Bibleref2), Jesus appears to recall several previous preaching ministries in Jerusalem, when his message had been generally spurned.
Date of the crucifixion
{{#invoke:main|main}} In the Jewish calendar, each day runs from sunset to sunset, and hence the Last Supper (on the Thursday evening), and Jesus' crucifixion (on Friday afternoon), both fell on the same day. In John, this day was the 14th of Nisan in the Jewish calendar; that is the day on the afternoon of which the Passover victims were sacrificed in the Temple, which was also known as the Day of Preparation. The Passover meal itself would then have been eaten on the Friday evening (i.e. the next day in Jewish terms), which would also have been a Sabbath. In the Synoptic accounts, the Last Supper is a Passover meal, and so Jesus' trial and crucifixion must have taken place during the night time and following afternoon of the festival itself, the 15th of Nisan. In favour of the Synoptic chronology is that in the earliest Christian traditions relating to the Last Supper in the first letter of Paul to the Corinthians, there is a clear link between the Last Supper and the Passover lamb. However, Paul also calls Christ "our passover", "sacrificed for us" (Template:Bibleref2), and if as according to John Jesus died on the afternoon of the 14th this was when the passover lambs were slaughtered.[129] Colin Humphreys and W. Graeme Waddington favor the date of Friday April 3, 33 from a combination of astronomical and historical reasons, which would have been the 14th rather than the 15th of Nisan.[129] Also in favor of John's chronology is the near universal modern scholarly agreement that the Synoptic accounts of a formal trial before the Sanhedrin on a festival day are historically impossible. By contrast, an informal investigation by the High Priest and his cronies (without witnesses being called), as told by John, is both historically possible in an emergency on the day before a festival, and accords with the external evidence from Rabbinic sources that Jesus was put to death on the Day of Preparation for the Passover. Astronomical reconstruction of the Jewish Lunar calendar tends to favor John's chronology, in that the only year during the governorship of Pontius Pilate when the 15th Nisan is calculated as falling on a Wednesday/Thursday was AD 27, which appears too early as the year of the crucifixion, whereas the 14th of Nisan fell on a Thursday/Friday in both AD 30 and 33.[130]
Compared with the synoptics
John 8:32 is inscribed at the entrance to Southwest Texas Junior College in Uvalde, Texas.
The Book of John is significantly different from the Synoptic Gospels:
Jesus is identified with the divine Word ("Logos") and the Word is called theos ("god" in Greek).[131]
The Gospel of John gives no account of the Nativity of Jesus, unlike Matthew and Luke, and his mother, while frequently mentioned, is never identified by name. John does assert that Jesus was known as the "son of Joseph" in Template:Bibleref2-nb.
In chapter 7:41–42, and again in 7:52, John records some of the crowd of Pharisees dismissing the possibility of Jesus' being the Messiah, on the grounds that the Messiah must be a descendent of David and born in Bethlehem, stating that Jesus instead came out of Galilee (as is stated in the Gospel of Mark); John made no effort to refute or correct this, and this has been advanced as implying that John rejected the synoptic tradition of Jesus' birth in Bethlehem.
The Pharisees, portrayed as more uniformly legalistic and opposed to Jesus in the synoptic gospels, are instead portrayed as sharply divided; they debate frequently in the Gospel of John's accounts. Some, such as Nicodemus, even go so far as to be at least partially sympathetic to Jesus. This is believed to be a more accurate historical depiction of the Pharisees, who made debate one of the tenets of their system of belief.[132]
John makes no mention of Jesus' baptism,[114] but quotes John the Baptist's description of the descent of the Holy Spirit as a dove, as happens at Jesus' baptism in the other gospels.
John the Baptist publicly proclaims Jesus to be the Lamb of God. The Baptist recognizes Jesus secretly in Matthew, and not at all in Mark or Luke. The Gospel of John also has John the Baptist deny that he is Elijah, whereas Mark and Matthew identify him with Elijah.
The Cleansing of the Temple appears near the beginning of Jesus' ministry. In the synoptics this occurs soon before Jesus is crucified.
John contains four visits by Jesus to Jerusalem, three associated with the Passover feast. This chronology suggests Jesus' public ministry lasted two or three years. The synoptic gospels describe only one trip to Jerusalem in time for the Passover observance.
Jesus washes the disciples' feet instead of the synoptics' ritual with bread and wine (the Eucharist)
No other women are mentioned going to the tomb with Mary Magdalene.
John does not contain any parables.[133] Rather it contains metaphoric stories or allegories, such as The Shepherd and The Vine, in which each individual element corresponds to a specific group or thing.
Major synoptic speeches of Jesus are absent, including the Sermon on the Mount and the Olivet discourse.[134]
While the synoptics look forward to a future Kingdom of God (using the term parousia, meaning "coming"), John presents a more "realized eschatology".[135]Template:Clarify
The Kingdom of God is mentioned only twice in John.[136] In contrast, the other gospels repeatedly use the Kingdom of God and the Kingdom of Heaven as important concepts.
The exorcisms of demons are never mentioned as in the synoptics.[114][136]
John never lists all of the Twelve Disciples and names at least one disciple (Nathanael) whose name is not found in the synoptics; Nathanael appears to parallel the apostle Bartholomew found in the synoptics, as both are paired with Philip in the respective gospels. While James and John are prominent disciples in the synoptics, John mentions them only in the epilogue, where they are referred to not by name but as the "sons of Zebedee."
Thomas the Apostle is given a personality beyond a mere name, as "Doubting Thomas".
Comparison chart of the major gospels
The material in the comparison chart is from Gospel Parallels by B. H. Throckmorton, The Five Gospels by R. W. Funk, The Gospel According to the Hebrews, by E. B. Nicholson and The Hebrew Gospel and the Development of the Synoptic Tradition by J. R. Edwards.
Matthew, Mark, Luke
Gospel of the Hebrews
New Covenant The central theme of the Gospels – Love God with all your heart and your neighbor as yourself[137] The central theme – Love is the New Commandment given by JesusTemplate:Bibleref2c Secret knowledge, love your friends[138] The central theme – Love one another[139]
Forgiveness Very important – particularly in MatthewTemplate:Bibleref2c and LukeTemplate:Bibleref2c AssumedTemplate:Bibleref2c Not mentioned Very important – Forgiveness is a central theme and this gospel goes into the greatest detail[140]
The Lord's Prayer In Matthew and Luke but not Mark Not mentioned Not mentioned Important – "mahar" or "tomorrow"[141][142]
Love and the poor Very important – The rich young man[143] Assumed[144] Important[145] Very important – The rich young man[146]
Jesus starts his ministry Jesus meets John the Baptist and is baptized[147] Jesus meets John the Baptist[148] N/A – Speaks of John the Baptist[149] Jesus meets John the Baptist and is baptized. This gospel goes into the greatest detail[150]
Disciples-inner circle Peter, Andrew, James and John[151] Peter, Andrew, the Beloved Disciple[152] Thomas, James the Just[153] Peter, Andrew, James and John[150]
Disciples-others
Philip, Bartholomew, Matthew, Thomas, James, Simon the Zealot, Judas Thaddaeus, and Judas[152]
Philip, Nathanael, Thomas, Judas not Iscariot and Judas Iscariot[152]
Peter,[154] Matthew, Mariam
Matthew, James the Just (Brother of Jesus), Simon the Zealot, Thaddaeus, Judas[155]
Possible Authors Unknown;[156] Mark the Evangelist and Luke the Evangelist The Beloved Disciple[157] Thomas[158] Matthew the Evangelist[159]
Virgin birth account In Matthew and Luke, but not Mark[160] Not mentioned Not mentioned Not mentioned
Jesus' baptism Described Not mentioned N/A Described in great detail[150]
Preaching style Brief one-liners; parables Essay format, Midrash Sayings, parables Brief one-liners; parables
Storytelling Parables[161] Figurative language and Metaphor[162] Proto-Gnostic, hidden, parables[163] Parables[164]
Jesus' theology 1st century liberal Judaism[165] Critical of Jewish Authorities[166] Proto-Gnostic 1st century Judaism[167]
Miracles Many miracles Seven Signs N/A Fewer but more credible miracles[168]
Duration of ministry 1 year[169] 2 years (three Passovers mentioned) N/A 1 year[169]
Location of ministry Mainly Galilee Mainly Judea, near Jerusalem N/A Mainly Galilee
Last Supper Body and Blood=Bread and wine Interrupts meal for foot washing N/A Hebrew Passover is celebrated but details are N/A Epiphanius[170]
Burial shroud A single piece of cloth Multiple pieces of cloth, as was the Jewish practice at the timeTemplate:Bibleref2c N/A Given to the High Priest[171]
Resurrection Mary and the Women are the first to learn Jesus has arisenTemplate:Bibleref2c Template:Bibleref2c Template:Bibleref2c John adds detailed account of Mary Magdalene's experience of the ResurrectionTemplate:Bibleref2c Not applicable, as Gospel of Thomas is a collection of the "sayings" of Jesus, not the events of his life In the Gospel of the Hebrews is the unique account of Jesus appearing to his brother, James the Just[172]
John was written somewhere near the end of the 1st century, probably in Ephesus, in Roman Asia. The tradition of John the Apostle was strong in Asia, and Polycarp of Smyrna reportedly knew him. Like the previous gospels, it circulated separately until Irenaeus proclaimed all four gospels to be scripture.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}
Although the Church Fathers Polycarp and Ignatius of Antioch did not mention this gospel,[173] it is thought that its ideas are reflected in their writings.[174] The gospel appears to have been familiar to Papias of Hierapolis,[175][176] and was used by other early Christians, including the author of the Muratorian Canon,[177] the Christians of Vienna and Lugdunum,[178] Theophilus of Antioch,[179] Tatian[180] and Justin Martyr.[181]
Papyrus 52 ( P 52 {\displaystyle {\mathfrak {P}}^{52}} ), Papyrus 90 ( P 90 {\displaystyle {\mathfrak {P}}^{90}} ), Papyrus 75 ( P 75 {\displaystyle {\mathfrak {P}}^{75}} ), and Papyrus 66 ( P 66 {\displaystyle {\mathfrak {P}}^{66}} ) are early papyri containing parts of the Gospel of John.
In the 2nd century, the two main, conflicting expressions of Christology were John's Logos theology, according to which Jesus was the incarnation of God's eternal Word, and adoptionism, according to which Jesus was "adopted" as God's Son. Christians who rejected Logos Christology were called "Alogi," and Logos Christology won out over adoptionism.
The Gospel of John was the favorite gospel of Valentinus, a 2nd-century Gnostic leader.[134] His student Heracleon wrote a commentary on the gospel, the first gospel commentary in Christian history.[134]
In the Diatesseron, the content of John was merged with the content of the synoptics to form a single gospel that included nearly all the material in the four canonical gospels.
When Irenaeus proposed that all Christians accept Mark, Matthew, Luke, and John as orthodox, and only those four gospels, he regarded John as the primary gospel, due to its high Christology.[134]
Jerome translated John into its official Latin form, replacing various older translations.
Although harmonious with the Synoptic Gospels and probably primitive (the Didascalia Apostolorum definitely refers to it and it was probably known to Papias), the Pericope Adulterae is not part of the original text of the Gospel of John.[182]
Bede translating the Gospel of John on his deathbed, by James Doyle Penrose, 1902
The Gospel of John has influenced Impressionist painters, Renaissance artists and classical art, literature and other depictions of Jesus, with influences in Greek, Jewish and European history.
It has been depicted in live narrations and dramatized in productions, skits, plays, and passion plays, productions, as well as on film. The most recent film portrayal being that of 2003's The Gospel of John, directed by Philip Saville and narrated by Christopher Plummer, and starring Henry Ian Cusick as Jesus.
Free Grace theology
Gospel harmony
Last Gospel
List of Bible verses not included in modern translations
List of Gospels
Textual variants in the Gospel of John
↑ Lindars 1990 p. 63.
↑ "Although ancient traditions attributed to the Apostle John the Fourth Gospel, the Book of Revelation, and the three Epistles of John, modern scholars believe that he wrote none of them." Harris, Stephen L., Understanding the Bible (Palo Alto: Mayfield, 1985) p. 355
↑ Ehrman, Bart D.. Jesus, Interrupted, HarperCollins, 2009. ISBN 0-06-117393-2
↑ Bruce Chilton and Jacob Neusner, Judaism in the New Testament: Practices and Beliefs (New York: Routledge, 1995), 5. "by their own word what they (the writers of the New Testament) set forth in the New Testament must qualify as a Judaism. ... [T]o distinguish between the religious world of the New Testament and an alien Judaism denies the authors of the New Testament books their most fiercely held claim and renders incomprehensible much of what they said."
↑ E P Sanders, The Historical Figure of Jesus, (Penguin, 1995) page 63 - 64.
↑ Bart D. Ehrman (2000:43) The New Testament: a historical introduction to early Christian writings. Oxford University Press.
↑ 9.0 9.1 Bart D. Ehrman (2005:235) Lost Christianities: the battles for scripture and the faiths we never knew Oxford University Press, New York.
↑ Geoffrey W. Bromiley (1995:287) International Standard Bible Encyclopedia: K-P MATTHEW, GOSPEL ACCORDING TO. Wm. B. Eerdmans Publishing. Quote: "Matthew, like the other three Gospels is an anonymous document."
↑ Donald Senior, Paul J. Achtemeier, Robert J. Karris (2002:328) Invitation to the Gospels Paulist Press.
↑ Keith Fullerton Nickle (2001:43) The Synoptic Gospels: an introduction Westminster John Knox Press.
↑ Ben Witherington (2004:44) The Gospel code: novel claims about Jesus, Mary Magdalene, and Da Vinci InterVarsity Press.
↑ F.F. Bruce (1994:1) The Gospel of John Wm. B. Eerdmans Publishing.
↑ Patrick J. Flannagan (1997:16) The Gospel of Mark Made Easy Paulist Press
↑ Paul N. Anderson, The Riddles of the Fourth Gospel, p. 48.
↑ F. F. Bruce, The Gospel of John, p. 3.
↑ Bart D. Ehrman (2004:110) Truth and Fiction in The Da Vinci Code: A Historian Reveals What We Really Know about Jesus, Mary Magdalene, and Constantine. Oxford University Press.
↑ Bart D. Ehrman (2006:143) The lost Gospel of Judas Iscariot: a new look at betrayer and betrayed. Oxford University Press.
↑ Anderson 2007, p. 19."These facts pose a major problem for the traditional view of John's authorship, and they are one of the key reasons critical scholars reject it."
↑ Lindars, 1990, p. 20."It is thus important to see the reasons why the traditional identification is regarded by most scholars as untenable."
↑ The New Interpreter's Dictionary of the Bible: Volume 3 Abingdon Press, 2008. p. 362 "Presently, few commentators would argue that a disciple of Jesus actually wrote the Fourth Gospel,..."
↑ Marilyn Mellowes The Gospel of John From Jesus to Christ: A Portrait of Jesus' World. PBS 2010-11-3. "Tradition has credited John, the son of Zebedee and an apostle of Jesus, with the authorship of the fourth gospel. Most scholars dispute this notion;..."
↑ D. A. Carson, Douglas J. Moo. An introduction to the New Testament. Zondervan; 2 New edition. 2005. Pg 233 "The fact remains that despite support for Johannine authorship by a few front rank scholars in this century and by many popular writers, a large majority of contemporary scholars reject this view."
↑ "To most modern scholars direct apostolic authorship has therefore seemed unlikely." "John, Gospel of." Cross, F. L., ed. The Oxford dictionary of the Christian church. New York: Oxford University Press. 2005
↑ 28.0 28.1 28.2 Harris 1985 pp. 302–10. "John."
↑ Harris 1985 pp. 367–432. "Glossary."
↑ Victorinus, CA 11.I
↑ Irenaeus AH 3.11
↑ Hill 2004 pp. 391, 444.
↑ 34.0 34.1 Anderson 2007 p. 77.
↑ 35.0 35.1 35.2 35.3 Brown 1997 pp. 363–4. Cite error: Invalid <ref> tag; name "Brown" defined multiple times with different content
↑ Lindars 1990 p. 20. "It is the evangelist who comes at the end of the process who is the real author of the Fourth Gospel."
↑ Harris, Stephen L., Understanding the Bible. Palo Alto: Mayfield. 1985. p. 268.
↑ 39.0 39.1 so that "it is primarily in the Synoptics that we must seek information about Jesus." Sanders, E. P. The historical figure of Jesus. Penguin, 1993. p. 57.
↑ 40.0 40.1 Das Evangelium des Johannes, 1941 (translated as The Gospel of John: A Commentary, 1971)
↑ Wikisource: John in KJV
↑ Anderson 2007 p. 78.
↑ The Muratorian fragment (c. 180) states that while John was the primary author, several people were involved, that mutual revision was part of the original intent of the authors, and that the editors included the apostle Andrew. Geza Vermes, The authentic gospel of Jesus, London, Penguin Books. 2004. A note on sources, p. x–xvii.
↑ Funk 1993 p. 542–8. "Glossary."
↑ Theissen 1998. Ch. 2. "Christian sources about Jesus."
↑ 46.0 46.1 Sanders, E. P. The historical figure of Jesus. Penguin, 1993. Chapter 6, Problems with primary sources. p 57-77.
↑ 'The time of origin is to be put around the turn of the century.' Theissen, Gerd and Annette Merz. The historical Jesus: a comprehensive guide. Fortress Press. 1998. translated from German (1996 edition). p. 36.
↑ '[T]he Gospel circulated abroad during the first half of the 2nd century but was probably composed about 90—100 CE.' Harris, Stephen L., Understanding the Bible. Palo Alto: Mayfield. 1985. p. 303.
↑ Ehrman, Bart. A Brief Introduction to the New Testament. Oxford University Press, USA. 2004. ISBN 0-19-516123-8. p. 164–5.
↑ Fonck, Leopold. "Gospel of St. John." The Catholic Encyclopedia. Vol. 8. New York: Robert Appleton Company, 1910. 7 Aug 2009.
↑ Bruce 1981 p. 7.
↑ Livingstone, E. A. The Concise Oxford Dictionary of the Christian Church. Oxford University Press, USA, 2006. ISBN 978-0-19-861442-5. p. 313
↑ Mark Allan Powell. Jesus as a figure in history. Westminster John Knox Press, 1998. ISBN 0-664-25703-8/978-0664257033. p. 43.
↑ Justin Martyr NTCanon.org. Retrieved 25 April 2007.
↑ Nongbri, Brent, 2005. "The Use and Abuse of P52: Papyrological Pitfalls in the Dating of the Fourth Gospel." Harvard Theological Review 98:23–52.
↑ McMenamin, Mark A. S., "The historical Jesus," Homiletic and Pastoral Review CIX, 2008:6.
↑ Stegall, Thomas L. "Reconsidering the Date of John's Gospel," Chafer Theological Seminary Journal 14.2 (2009): 70–103.
↑ Morris 1995 p. 59.
↑ Robinson 1977 pp. 284, 307.
↑ "[Robinson's] later books, which argue that all the Gospels, incl. Jn., are very early, have not carried widespread conviction." "Robinson, John Arthur Thomas." Cross, F. L., ed. The Oxford dictionary of the Christian church. New York: Oxford University Press. 2005.
↑ Rule of the Community. "And by His knowledge, everything has been brought into being. And everything that is, He established by His purpose; and apart from Him nothing is done."
↑ Leon Morris, The Gospel according to John, The New International Commentary on the New Testament (Grand Rapids, MI: Wm. B. Eerdmans Publishing Co., 1995), 28.
↑ D. A. Carson, The Gospel according to John, The Pillar New Testament Commentary (Leicester, England; Grand Rapids, MI: Inter-Varsity Press; W.B. Eerdmans, 1991), 59.
↑ Bruce M. Metzger. The text of the New Testament: its transmission, corruption, and restoration. Oxford University Press, 1992. ISBN 0-19-507297-9. p.56
↑ Kurt Aland, Barbara Aland. The Text of the New Testament: an Introduction to the Critical Editions and to the Theory and Practice of Modern Textual Criticism. Wm. B. Eerdmans, 1995. ISBN 0-8028-4098-1/978-0802840981. p.99
↑ Tuckett p. 544. Skypoint.com
↑ Ronald Cameron, editor. The Other Gospels: Non-Canonical Gospel Texts, 1982
↑ Funk 1993 p. 543.
↑ Thomas Spencer Baynes, The Encyclopædia Britannica: A Dictionary of Arts, Sciences, and General Literature, 9th Ed., Vol. 5. A. & C. Black, 1833 pp.13
↑ 74.0 74.1 74.2 74.3 74.4 Cross 2005. "John, Gospel of."
↑ 75.00 75.01 75.02 75.03 75.04 75.05 75.06 75.07 75.08 75.09 75.10 Harris 1985.
↑ Johnson, Maxwell E. "The Apostolic Tradition" in The Oxford History of Christian Worship. Oxford University Press, USA. 2005. page 32-75. ISBN 0-19-513886-4
↑ Funk, Robert W., Roy W. Hoover, and the Jesus Seminar. The five gospels. HarperSanFrancisco. 1993. "Introduction," p 1-30.
↑ The Gospel according to John by Herman Ridderbos 1997 ISBN 978-0-8028-0453-2 pages 510–512
↑ John, Jesus, and History, Volume 2 by Paul N. Anderson, Felix Just, Tom Thatcher 2007 ISBN 1589832930 page 273
↑ The Unity of the Farewell Discourse by L. Scott Kellum 2004 ISBN 0567080765 pages 1–6
↑ Marshall, Celia Brewer and Celia B. Sinclair. A Guide Through the New Testament. Westminster John Knox Press, 1994. ISBN 0-664-25484-5
↑ The Cradle, the Cross, and the Crown: An Introduction to the New Testament by Andreas J. Köstenberger, L. Scott Kellum 2009 ISBN 978-0-8054-4365-3 page 305
↑ C. Marvin Pate, et al. "The Story of Israel: a biblical theology" (InterVarsity Press: Downers Grove, 2004), 153.
↑ Adolf von Harnack What is Christianity? Lectures Delivered in the University of Berlin during the Winter-Term 1899–1900 "In particular, the fourth Gospel, which does not emanate or profess to emanate from the apostle John, cannot be taken as an historical authority in the ordinary meaning of the word. The author of it acted with sovereign freedom, transposed events and put them in a strange light, drew up the discourses himself, and illustrated 22 great thoughts by imaginary situations. Although his work is not altogether devoid of a real, if scarcely recognizable, traditional element, it can hardly make any claim to be considered an authority for Jesus' history; only little of what he says can be accepted, and that little with caution. On the other hand, it is an authority of the first rank for answering the question, What vivid views of Jesus' person, what kind of light and warmth, did the Gospel disengage?"
↑ Gospel of Saint John, in Catholic Encyclopedia 1910
↑ Harris 1985 p. 268. John's biography is "highly problematical to scholars."
↑ Harris p. 304.
↑ Harris pp. 302–10.
↑ John Shelby Spong. Jesus for the Non-Religious
↑ New International Version (and Today's New International Version), New American Standard Bible, Amplified Bible, New Living Translation, King James Version, Young's Literal Translation, Darby Translation, and Wycliffe New Testament, to name a few.
↑ 92.0 92.1 Funk 1998 pp. 365–440. "John."
↑ Barrett, C. K. The Gospel According to St. John: An Introduction with Commentary and Notes on the Greek Text. Westminster John Knox Press, 1978. p. 16
↑ Funk 1998 p. 268. "John the Baptist."
↑ Donald Senior, The passion of Jesus in the Gospel of John, Liturgical Press, 1991 (pp 155–156)
↑ "The Fourth Evangelist is still operating within a context of intra-Jewish factional dispute, although the boundaries and definitions themselves are part of that dispute. It is clear beyond doubt that once the Fourth Gospel is removed from that context, and the constraints of that context, it was all too easily read as an anti-Jewish polemic and became a tool of anti-semitism. But it is highly questionable whether the Fourth Evangelist himself can fairly be indicted for either anti-Judaism or anti-semitism." J.G.Dunn. The Question of Anti-Semitism in the New Testament Writings of the Period. Jews and Christians: the parting of the ways, A.D. 70 to 135. Wm. B. Eerdmans Publishing, 1999. p. 209.
↑ Roger E. Olson, The Story of Christian Theology, p. 36; InterVarsity Press, Downers Grove, IL, 1999
↑ Brown 1997 p. 375.
↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
↑ Harris, Stephen L., Understanding the Bible. Palo Alto: Mayfield. 1985. "John" p. 302-310
↑ For example, see the discussion in Saeed Hamid-Khani, Revelation and Concealment of Christ, WUNT 120 (Tubingen: Mohr Siebeck, 2000), 1–4.
↑ "All four Gospels should be regarded primarily as biographies of Jesus, but all four have a definite theological aim." Lindars 1990 p. 26.
↑ 'John, however, is so different that it cannot be reconciled with the synoptics except in very general ways (e.g., Jesus lived in Palestine, taught, healed, was crucified and raised). The greatest differences, though, appear in the methods and content of Jesus' teaching. Scholars have largely chosen the Synoptic Gospels' version of Jesus' teaching.' "Jesus Christ." Encyclopædia Britannica. 2010. Encyclopædia Britannica Online. 15 Nov 2010 [1].
↑ 'It is impossible to think that Jesus spent his short ministry teaching in two such completely different ways, conveying such different contents, and that there were simply two traditions, each going back to Jesus, one transmitting 50 per cent of what he said and another on the other 50 per cent, with almost no overlaps. Consequently, for the last 150 years or so scholars have had to choose. They have almost unanimously, and I think entirely correctly, concluded that the teaching of the historical Jesus is to be sought in the synoptic gospels and that John represents an advanced theological development, in which meditations on the person and work of Christ are presented in the first person, as if Jesus said them.' Sanders, E. P. The historical figure of Jesus. Penguin, 1993. p. 70–71.
↑ Sanders, E. P. The historical figure of Jesus. Penguin, 1993.
↑ James D. G. Dunn, Jesus Remembered, Eerdmans (2003), page 165
↑ 113.0 113.1 113.2 Sanders, E. P. The historical figure of Jesus. Penguin, 1993. p. 71.
↑ 114.0 114.1 114.2 Funk 1993 pp. 1–30. "Introduction."
↑ Vermes, Geza. The authentic gospel of Jesus. London, Penguin Books. 2004.
↑ Theissen, Gerd and Annette Merz. The historical Jesus: a comprehensive guide. Fortress Press. 1998. translated from German (1996 edition). Chapter 2. Christian sources about Jesus.
↑ 117.0 117.1 117.2 Theissen 1998 pp. 36–7.
↑ Marianne Meye Thompson, The Historical Jesus and the Johannine Christ in Culpepper, R. Alan, and Black, C. Clifton, eds. Exploring the Gospel of John. Westminster John Knox Press, 1996. p. 28
↑ Henry Wansbrough, The Four Gospels in Synopsis, The Oxford Bible Commentary, pp. 1012–1013, Oxford University Press 2001 ISBN 0-19-875500-7
↑ Robinson 1977 p. 201.
↑ Robinson 1977 p. 53.
↑ "Introduction to the New Testament", chapter on John, by D. Carson and D. Moo, Zondervan Books (2005)
↑ Craig L. Blomberg, Historical Reliability of the Gospels (1986, Inter-Varsity Press)
↑ Henry Wansbrough says: "Gone are the days when it was scholarly orthodoxy to maintain that John was the least reliable of the gospels historically." The Four Gospels in Synopsis, The Oxford Bible Commentary, pp. 1012–1013, Oxford University Press 2001 ISBN 0-19-875500-7
↑ Brown 1997 pp. 362–4.
↑ For example, Douglas Estes cites the work of Thucydides, Herodotus, Tacitus and many others from the ancient world who wrote historiography that was often linear but not necessarily absolute (absolute time being perhaps first promoted by Joseph Scaliger, the early critical historian); see Estes, Douglas. The Temporal Mechanics of the Fourth Gospel: A Theory of Hermeneutical Relativity in the Gospel of John, BIS 92 (Leiden: Brill, 2008).
↑ 129.0 129.1 *Template:Cite doi {{#invoke:Citation/CS1|citation |CitationClass=journal }}
↑ Ehrman, Bart D.. Misquoting Jesus: The Story Behind Who Changed the Bible and Why. HarperCollins, 2005. ISBN 978-0-06-073817-4
↑ Neusner, Jacob. Invitation to the Talmud: a Teaching Book (1998): 8
↑ 134.0 134.1 134.2 134.3 Pagels, Elaine. Beyond belief: the secret gospel of Thomas. New York: Random House. 2003. ISBN 0-375-50156-8
↑ "Biblical Literature." Encyclopædia Britannica Online. The Fourth Gospel
↑ 136.0 136.1 {{#invoke:citation/CS1|citation |CitationClass=book }}
↑ In the Synoptic Gospels this is the "Greatest Commandment" that sums up all of the "Law and the Prophets"
↑ Log 25
↑ The Lord says to his disciples: "And never be you joyful, except when you behold one another with love." Jerome, Commentary on Ephesians
↑ In the Gospel of the Hebrews, written in the Chaldee and Syriac language but in Hebrew script, and used by the Nazarenes to this day (I mean the Gospel of the Apostles, or, as it is generally maintained, the Gospel of Matthew, a copy of which is in the library at Caesarea), we find, "Behold the mother of the Lord and his brothers said to him, 'John the Baptist baptizes for the forgiveness of sins. Let us go and be baptized by him.' But Jesus said to them, 'in what way have I sinned that I should go and be baptized by him? Unless perhaps, what I have just said is a sin of ignorance.'" And in the same volume, "'If your brother sins against you in word, and makes amends, forgive him seven times a day.' Simon, His disciple, said to Him, 'Seven times in a day!' The Lord answered and said to him, 'I say to you, Seventy times seven.' " Jerome, Against Pelagius 3.2
↑ In the so-called Gospel of the Hebrews, for "bread essential to existence," I found "mahar", which means "of tomorrow"; so the sense is: our bread for tomorrow, that is, of the future, give us this day. Jerome, Commentary on Matthew 1
↑ In Matthew's Hebrew Gospel it states, 'Give us this day our bread for tomorrow." Jerome, On Psalm 135
↑ Template:Bibleref2, Template:Bibleref2 and Template:Bibleref2
↑ Template:Bibleref2
↑ Jesus said "Blessed are the poor, for to you belongs the Kingdom of Heaven" Log 54
↑ The second rich youth said to him, "Rabbi, what good thing can I do and live?" Jesus replied, "Fulfill the law and the prophets." "I have," was the response. Jesus said, "Go, sell all that you have and distribute to the poor; and come, follow me." The youth became uncomfortable, for it did not please him. And the Lord said, "How can you say, I have fulfilled the Law and the Prophets, when it is written in the Law: You shall love your neighbor as yourself and many of your brothers, sons of Abraham, are covered with filth, dying of hunger, and your house is full of many good things, none of which goes out to them?" And he turned and said to Simon, his disciple, who was sitting by Him, "Simon, son of Jonah, it is easier for a camel to go through the eye of a needle than for the rich to enter the Kingdom of Heaven. "Origen, Commentary on Matthew 15:14
↑ Template:Bibleref2, Template:Bibleref2, Template:Bibleref2-nb
↑ Gospel of Thomas, Logion 46
↑ 150.0 150.1 150.2 Epiphanius, Panarion 30:13
↑ Matt 10:1, Mk 6:8, Lk 9:3
↑ 152.0 152.1 152.2 Template:Bibleref2, Template:Bibleref2-nb, Template:Bibleref2-nb, Template:Bibleref2-nb, Template:Bibleref2-nb
↑ Log 1–114
↑ Epiphanius, Panarion 30:13, Jerome, On Illustrious Men, 2
↑ Although several Fathers say Matthew wrote the Gospel of the Hebrews they are silent about Greek Matthew found in the Bible. Modern scholars are in agreement that Matthew did not write Greek Matthews which is 300 lines longer than the Hebrew Gospel (See James Edwards the Hebrew Gospel)
↑ Suggested by Irenaeus first
↑ Preface to the Gospel of Thomas
↑ They too accept Matthew's gospel, and like the followers of Cerinthus and Merinthus, they use it alone. They call it the Gospel of the Hebrews, for in truth Matthew alone in the New Testament expounded and declared the Gospel in Hebrew using Hebrew script. Epiphanius, Panarion 30:3
↑ Matt 1:18
↑ Log 109
↑ Similar to beliefs taught by Hillel the Elder. (e.g., "golden rule") Hillel Hillel the Elder
↑ Template:Bibleref2; Template:Bibleref2-nb
↑ Similar to beliefs taught by Hillel the Elder. (e.g. "golden rule") Hillel Hillel the Elder
↑ Jerome, Commentary on Matthew 2
↑ 169.0 169.1 Events leading up to Passover
↑ Epiphanius, Panarion 30:22
↑ Jerome, On Illustrious Men, 2
↑ Elaine Pagels (Beyond belief: the Secret Gospel of Thomas [London: Pan Books, 2005], p. 149) suggests that Polycarp 'may not have known John's Gospel' or that, 'at any rate, he chose not to mention it, as far as we know'. Regarding Ignatius, Paul R. Trebilco (The Early Christians In Ephesus From Paul To Ignatius [Mohr Siebeck: Tubingen, 2004], p. 678) says, 'Such silence does not necessarily mean that he does not know of a "John" who wrote the Fourth Gospel'. Brian H. Edwards (Why 27?: How can we be sure that we have the right books in the New Testament? [Darlington: Evangelical Press, 2007], p. 115) comments, 'as so often has to be said, the silence on a particular [canonical] book cannot necessarily be taken as anything more than that the writer had no need to quote from it'.
↑ 'John, Gospel of St.' in F. L. Cross and E. A. Livingstone (eds), The Oxford Dictionary of the Christian Church (Oxford: OUP, 1997)
↑ Paul R. Trebilco, The Early Christians In Ephesus From Paul To Ignatius (Mohr Siebeck: Tubingen, 2004), p. 247
↑ Brian H. Edwards, Why 27?: How can we be sure that we have the right books in the New Testament? (Darlington: Evangelical Press, 2007), p. 95
↑ T. Herbert Bindley, The Epistle of the Gallican Churches (London: SPCK, 1900), p. 15
↑ To Autolycus, Book III, ch. 22
↑ Cf. his Diatessaron
↑ I Apol. 61.4
↑ 'These verses... are certainly not part of the original text of St. John's Gospel.' Cross, F. L., ed. The Oxford dictionary of the Christian church. New York: Oxford University Press. 2005
{{#invoke:citation/CS1|citation
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|CitationClass=book }} Called by F.F. Bruce "the most important work to appear in this field in a generation".[1]
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Online translations of the Gospel of John:
The Gospel of St. John the Apostle, Douay Rheims Bible Version with commentaries by Bishop Challoner
Bible Gateway 35 languages/50 versions at GospelCom.net
Unbound Bible 100+ languages/versions at Biola University
Online Bible at gospelhall.org
Text of the Gospel with textual variants
The Egerton Gospel: text. Compare it with Gospel of John
The Gospel according to John - Audiobook - King James Version
Gospel According to John, Encyclopædia Britannica Online.
A textual commentary on the Gospel of John Detailed textcritical discussion of the 300 most important variants of the Greek text (PDF, 376 pages)
Papyrus fragment of John at the John Rylands Library; illustrated.
John Rylands papyrus: text, translation, illustration and a bibliography of the discussion
John, Gospel of St. in the 1911 Encyclopædia Britannica – collected comments
Conflicts Between the Gospel of John & the Remaining Three (Synoptic) Gospels on ReligiousTolerance.com.
David Robert Palmer, Translation from the Greek
John Henry Bernard, Alan Hugh McNeile, A critical and exegetical commentary on the Gospel according to St. John, Continuum International Publishing Group, 2000.
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↑ Bruce 1981 p. 59.
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Notations and definitions
State-dependent fractional model
Parameter identification and sensitivity analysis
Model validation
Fractional-order modelling of state-dependent non-associated behaviour of soil without using state variable and plastic potential
Yifei Sun1Email authorView ORCID ID profile and
Changjie Zheng2
Advances in Difference Equations20192019:83
It was found that the constitutive behaviour of granular soil was dependent on its density and pressure (i.e. material state). To capture such state dependence, a variety of state variables were empirically proposed and introduced into the existing plastic potential functions, which inevitably resulted in the complexity and meaninglessness of some model parameters. The purpose of this study is to theoretically investigate the state-dependent non-associated behaviour of granular soils without using predefined plastic potential and state variable. A novel state-dependent non-associated model for granular soils is mathematically developed by incorporating the stress-fractional operator into the bounding surface plasticity. Unlike previous studies using empirical state variables, the soil state and non-associativity in this study are considered via analytical solution, where a state-dependent plastic flow rule and the corresponding hardening modulus without using additional plastic potentials are obtained. Possible mathematical connection with a well-known empirical state variable is also discussed. The non-associativity between plastic flow and loading directions as well as material hardening is found to be controlled by the fractional-derivative order. To validate the proposed approach, a series of drained and undrained triaxial test results of different granular soils are simulated and compared, where a good agreement between the model predictions and the corresponding test results is observed.
Fractional plasticity
Constitutive relations
State dependence
Granular soils
It has been widely acknowledged that the strength and deformation behaviour of granular soil, such as sand and rockfill, is significantly dependent on its density and pressure (material state) [1]. Before the proper consideration of state dependence during constitutive modelling, different model parameters were often required for modelling the stress-strain behaviour of granular soils with different initial densities or subjected to different confining pressures [2–4].
For the purpose of better understanding and unified constitutive modelling of granular soils, a variety of different empirical state variables have been suggested, such as the state ratio of the difference between the threshold and current void ratios to the difference between the threshold and critical void ratios [5], the ratio of the current to critical void ratios [6], the disturbance in disturbed state concept [7], the stress ratio of the current to critical mean effective stresses [8] and the most widely used state variable (ψ) defined by the difference between the current and critical state void ratios [9]. It can be found that developing a reasonable state-dependent non-associated plastic flow rule (or stress-dilatancy relationship) has been of the utmost importance in recent years. One of the popular approaches was to modify the existing stress-dilatancy equations, for example, the Cam-clay (CC) stress-dilatancy equation [10] and Rowe's stress-dilatancy equation [11], by empirically incorporating ψ [12–16]. Undeniably, this approach can always substantially improve the model performance; however, the empirical correlation between the state variable and the existing constitutive parameters would inevitably result in more model parameters. It was found that the fractional mechanics was an efficient way to capture the relaxation [17], diffusion [18–20], and stress-strain [14, 21] behaviour of materials. To reduce the number of model parameters without the loss of modelling capability, Sun and Shen [21] proposed a non-associated plastic flow rule for granular soil by simply conducting fractional-order derivatives of the yielding surface, where the obtained vector (plastic flow direction) was no longer normal to the yielding surface, even without using an additional plastic potential. This non-normality increased as the fractional order (α) decreased [14, 22, 23]. To consider the state dependence, the state-dependent fractional plasticity model was then proposed [14] by empirically incorporating ψ, which however made the parameters of the obtained stress-dilatancy equation lack physical meaning.
This study attempts to theoretically investigate the state-dependent non-associated stress-strain behaviour of granular soils. A state-dependent non-associated constitutive model without using any empirical state variables and plastic potentials is developed by using strict mathematics. Instead of modelling the dependence of soil state by empirically incorporating state variables, analytical derivations of the state-dependent plastic flow rule and the associated hardening rule are presented. As the fractional derivative is defined in integral form, the soil state is captured through the integrating range from the lower limit (current stress state) to the upper limit (critical stress state). This paper is divided into four main parts: Sect. 2 defines the basic constitutive relations and the relevant fractional derivative used in this study; Sect. 3 develops a novel state-dependent constitutive model without using plastic potential, where the state-dependent fractional plastic flow rule is analytically derived; Sect. 4 presents the identification and sensitivity analysis of model parameters; Sect. 5 provides the model validation against a series of laboratory test results of different granular soils; Sect. 6 concludes the study. For the sake of simplicity, all the derivations and discussions in this study are limited to homogenous and isotropic materials.
2 Notations and definitions
2.1 Constitutive relations
Following the conventional assumption in soil mechanics, compressive stress and strain are considered as positive while the extensive ones are negative. All the stress mentioned in this study is effective stress obtained by using Terzhgi's effective stress principle. In the elastoplastic model, there are four main parts, i.e. the elastic stiffness tensor (E), plastic loading tensor (m), plastic flow tensor (n) and hardening modulus (H). Accordingly, the total strain (\(\varepsilon _{ij}\)) can be decomposed into the following elastic (\(\varepsilon _{ij}^{e}\)) and plastic (\(\varepsilon _{ij}^{p}\)) parts:
$$ \Delta \varepsilon _{ij} = \Delta \varepsilon _{ij}^{e} + \Delta \varepsilon _{ij}^{p}, $$
where \(i, j = 1, 2, 3\). Δ indicates increment, while the superscripts e and p imply the elastic and plastic components, respectively. Based on Hooke's law, the incremental elastic strain tensor (\(\Delta \varepsilon _{ij}^{e}\)) can be correlated to the incremental effective stress tensor (\(\Delta \sigma '_{ij}\)) by
$$ \Delta \sigma '_{ij} = E_{ijkl}\Delta \varepsilon _{kl}^{e}, $$
where \(E _{\mathit{ijkl}}\) denotes the fourth-order elastic stiffness tensor, which can be defined as follows [24]:
$$ E_{ijkl} = ( K - 2G / 3 )\delta _{ij}\delta _{kl} + G ( \delta _{ik}\delta _{jl} + \delta _{il}\delta _{jk} ), $$
in which \(\delta _{ij}\) is the Kronecker delta. K and G are the bulk and shear moduli, respectively, which can be expressed by using [25]
$$\begin{aligned}& K = \frac{1 + e}{\kappa } p', \end{aligned}$$
$$\begin{aligned}& G = \frac{3(1 - 2\nu )}{2 + 2\nu } K, \end{aligned}$$
where κ is the gradient of the swelling line in the \(e - \ln p'\) plane; e is the current void ratio of the sample; ν is Poisson's ratio. \(p' = \sigma '_{ij}\delta _{ij} / 3\) is the mean effective principal stress. In addition, the generalised shear stress \(q = \sqrt{3 / 2s_{ij}s_{ij}}\), in which \(s_{ij} = \sigma '_{ij} - \sigma '_{kk}\delta _{ij} / 3\), is the deviatoric stress tensor. The corresponding volumetric strain (\(\varepsilon _{v}\)) can be defined as \(\varepsilon _{v} = \varepsilon _{ij}\delta _{ij}\), while the generalised shear strain \(\varepsilon _{s} = \sqrt{2 / 3e_{ij}e_{ij}}\), where \(e_{ij} = \varepsilon _{ij} - \varepsilon _{v}\delta _{ij}\), is the deviatoric strain tensor.
In plasticity approach, the incremental plastic strain tensor (\(\Delta \varepsilon _{ij}^{p}\)) can be related to incremental effective stress tensor such that [26]
$$ \Delta \varepsilon _{ij}^{p} = \frac{1}{H}n_{ij}m_{kl} \Delta \sigma '_{kl}, $$
where the plastic loading tensor (\(m _{ij}\)) is normal to the yielding surface and thus can be determined by conducting first-order derivative of the yielding surface; the plastic flow tensor (\(n _{ij}\)) is non-normal to the yielding surface for non-associative geomaterials and can be determined by conducting fractional-order derivative of the yielding surface, as shown in Sumelka and Nowak [23] and Sun et al. [27]; the hardening modulus (H) for granular soils involves both the size hardening and a relative position of the yielding surface with respect to the bounding surface, and it will be defined later. Combining Eqs. (2)–(6), the following elastoplastic constitutive relation can be given:
$$ \Delta \sigma '_{ij} = \biggl[ E_{ijkl} - \frac{E_{ijct}n_{ct}E_{klrs}m _{rs}}{H + m_{ct}E_{ctab}n_{ab}} \biggr]\Delta \varepsilon _{kl}. $$
2.2 Fractional derivative and yielding surface
There are many different definitions of fractional derivatives [17–20, 28, 29], where each of them is in integral form and somehow complex. Therefore, only some of the functions, for example, the power-law function, have certain analytical solution, whereas the rest need numerical approximation [30, 31]. Following the fractional plasticity [14, 22, 32], the following well-known Caputo's left-sided (Eq. (8)) and right-sided (Eq. (9)) fractional derivatives [33, 34] are used in this study:
$$\begin{aligned}& {}_{\sigma '_{c}}D_{\sigma '}^{\alpha } f\bigl(\sigma ' \bigr) = \frac{1}{\varGamma (n - \alpha )} \int _{\sigma '_{c}}^{\sigma '} \frac{f^{(n)}(\chi )\,d \chi }{(\sigma ' - \chi )^{\alpha + 1 - n}},\quad \sigma ' > \sigma '_{c}, \end{aligned}$$
$$\begin{aligned}& {}_{\sigma '}D_{\sigma '_{c}}^{\alpha } f\bigl(\sigma ' \bigr) = \frac{( - 1)^{n}}{ \varGamma (n - \alpha )} \int _{\sigma '}^{\sigma '_{c}} \frac{f^{(n)}( \chi )\,d\chi }{(\chi - \sigma ')^{\alpha + 1 - n}},\quad \sigma '_{c} > \sigma ', \end{aligned}$$
where D (\(= \partial ^{\alpha } /\partial \sigma ^{\prime \,\alpha } \)) denotes partial derivation to obtain the fractional stress gradient on the yielding function f. It is noted that Caputo's fractional derivative has singular kernel, which may not be applicable in a more specific case, e.g. capturing material heterogeneities and structures with different scales. A possible solution of this limit is to propose a new fractional derivative without singular kernel, for example, the Caputo–Fabrizio derivative [35] and Yang–Srivastava–Machado derivative [28]. It is worthwhile pointing out that the Yang–Srivastava–Machado fractional derivative [28] is a well-known extension of the Riemann–Liouville fractional derivative with singular kernel, which has great potential in solving local thermal and mechanical problems. However, in this study, only mechanical phenomena related to plasticity are considered; therefore, the original Caputo's fractional derivative [33, 34] is sufficient and effective [35]. Following Sun et al. [27, 36], the modified Cam-clay (MCC) yielding function is used
$$ f = \bigl(2p' - p'_{0}\bigr)^{2} + \biggl( \frac{2q}{M} \biggr)^{2} - p_{0} ^{\prime 2} = 0, $$
where the critical state stress ratio M can be expressed as [37]
$$ M = M_{c} \biggl[ \frac{2c^{4}}{1 + c^{4} - (1 - c^{4})\sin (3\theta )} \biggr]^{1 / 4} $$
in which \(M _{c}\) denotes the critical state stress ratio for triaxial compression; c denotes the ratio between the critical state stress ratio for triaxial extension (\(M _{e}\)) and \(M _{c}\). The Lode angle (θ) is expressed as \(\theta = \arccos \{ 9s_{ij}s_{jk}s _{ki} / [ 2(3s_{lr}s_{lr} / 2)^{3 / 2} ] \} / 3\). \(\theta \in [ - \pi / 6,\pi / 6]\), where \(\theta = \pi / 6\) corresponds to triaxial compression, whereas \(\theta = - \pi / 6\) corresponds to triaxial extension. \(p'_{0}\) controls the size (hardening) of the yielding surface. Furthermore, in Eq. (9), the gamma function is defined as follows: \(\varGamma (x) = \int _{0}^{\infty } e^{ - \tau } \tau ^{x - 1}\,d \tau \). α is the fractional order and should be at least less than two due to the thermodynamic restriction for a positive plastic dissipation [27]. As discussed in Sun and Shen [21], α represents the extent of non-associativity between plastic flow and loading directions as well as the degree of state dependence of material deformation. \(\sigma '\) is the current effective stress, while \(\sigma '_{c}\) is the corresponding critical-state stress. The use of \(\sigma '_{c}\) for integral limit is attributed to the basic assumption from the critical state soil mechanics [10]: upon shearing, soil would deform continuously and finally approach the critical state represented by the critical state lines (CSLs) in the \(e - \ln p'\) and \(p' - q\) planes, respectively, as follows:
$$\begin{aligned}& p'_{c} = p_{r}\exp \biggl( \frac{e_{\varGamma } - e}{\lambda } \biggr), \end{aligned}$$
$$\begin{aligned}& q_{c} = q + M\bigl(p' - p'_{c} \bigr), \end{aligned}$$
where \(p_{r}\) is the unit pressure; λ is the gradient of the CSL in the \(e - \ln p'\) plane. \(e_{\varGamma } \) denotes the intercept of the CSL at \(p'= 1\mbox{ kPa}\). \(p'\) and \(p'_{c}\) are the mean effective principal stresses at the current and the corresponding critical states, respectively, while q and \(q_{c}\) are the deviator stresses at the current and corresponding critical states, respectively. It should be noted that in this study we connect the current stress point \((p', q)\) with the corresponding CSL by using the critical state stress ratio M as shown in Fig. 1. Before soil reaches the critical state, there are three possible locations of the current stress point in relation to the CSL in the \(p' - q\) plane, i.e. on the "dry" side (above), on the "wet" side (below) and on the CSL. If the current state is on the CSL, then \(p' = p'_{c}\) and \(q = q_{c}\), where only one critical-state stress point is indicated, and the material is currently undergoing critical state flow. However, in the rest two cases, it is observed that \(p'_{c}\) and \(q_{c}\) should represent two independent critical-state stress points (A and B) on the CSL. As shown in Fig. 1, stress point A implies a soil with dilation trend, while stress point B represents a soil with contraction trend. \(p'_{c}\) is obtained by extending a horizontal parallel line from the current stress points (A or B) with regard to the \(p'\)-axis and intersecting the CSL, while \(q _{c}\) is obtained by intersecting a vertical parallel line extended from the current stress points (A or B) with regard to the q-axis. The relative location of the current stress point to the critical state line defines which equation, (8) or (9), will be used. More specifically, if \(p' > p'_{c}\) or \(q > q_{c}\), Eq. (8) should be used and vice versa. However, as demonstrated in [36], even if different positions (or definitions) of the current stress point (or fractional derivatives) were used, a unique state-dependent plastic flow rule was obtained without using any plastic potential and empirical state parameters in the end.
Schematic show of the possible location of current stress point
3 State-dependent fractional model
3.1 State-dependent plastic flow
In this study, the state-dependent fractional plastic flow tensor (\(n _{ij}\)) is defined as
$$ n_{ij} = \frac{1}{\sqrt{1 + d_{g}^{2}}} \biggl[ \frac{d_{g}}{3} \delta _{ij} + \frac{3s_{ij}}{2q} \biggr], $$
where \(d _{g}\) is the stress-dilatancy ratio, which has several different definitions by different researchers, for example, the CC expression [26, 38], elliptic expression [12, 39], Rowe's expression [6, 40], etc. However, no matter which kind of dilatancy equation was used in the classical plasticity model, an empirical state index was necessarily incorporated for the unified modelling of state-dependent behaviour of geomaterials subjected to a wide range of densities and pressures [3]. In this study, a three-dimensional state-dependent dilatancy ratio without using empirical state index and plastic potential can be derived by following Sun et al. [27, 36], where the unified stress-dilatancy ratio (\(d _{g}\)) can be obtained by using Eqs. (8) and (9) to perform fractional-order derivatives on the MCC surface such that
$$ \begin{aligned}[b] d_{g} &= - \frac{{}_{p'}D_{p'_{c}}^{\alpha } f(p')}{{}_{q_{c}}D_{q} ^{\alpha } f(q)} = - \frac{{}_{p'_{c}}D_{p'}^{\alpha } f(p')}{{}_{q}D _{q_{c}}^{\alpha } f(q)} \\ &= M^{1 + \alpha } \frac{(p' - p'_{c}) + (2 - \alpha )(p'_{c} - p'_{0}/2)}{(q - q_{c}) + (2 - \alpha )q_{c}}, \end{aligned} $$
where the material flow is found to be influenced by several factors, including the Lode angle (θ) via M, current stress \((p', q)\) and critical-state stress \((p'_{c}, q _{c})\); most importantly, it is also determined by the stress distances from the current state to the corresponding critical state (\(p' - p'_{c}\) and \(q - q _{c}\)). It is easy to find that \(d_{g} = 0\) at critical state where \(p' = p'_{c}\) and \(q = q_{c}\), indicating no plastic volumetric strain. The fractional gradient on MCC, by using Eqs. (8) and (9), intrinsically considers the state information of soil during shearing. Therefore, unlike classical isotropic and anisotropic plasticity models [41–45] where the influence of material state on plastic flow requires the empirical incorporation of a state parameter (e.g. ψ), a state-dependent plastic flow is mathematically developed in this study. By using Eq. (15), the effect of α on the stress-dilatancy behaviour of granular soil during shearing can be obtained. As shown in Sun et al. [36],the stress-dilatancy curve exhibited clockwise rotation coupled with a downward shifting as α increased. It reduces to the classical MCC stress-dilatancy model [10] with \(\alpha = 1\).
It should be also noted that there are only two chances for the stress-dilatancy ratio \(d_{g} = 0\). One is at the critical state and the other is at the phase transformation state. At critical state where \(p' = p'_{c}\) and \(q = q_{c}\), the stress-dilatancy ratio is automatically equal to zero, indicating no plastic volumetric strain. At the phase transformation state, \(d _{g} = 0\) can be ensured by a proper value of α, which can be defined by rearranging Eq. (16) as follows:
$$ \alpha = \frac{2M^{2}p'_{ct} - 2\eta _{t}^{2}p'_{t}}{2M^{2}p'_{ct} - M ^{2}p'_{t} - \eta _{t}^{2}p'_{t}}, $$
in which \(p'_{ct} = p_{r}\exp ((e_{\varGamma } - e_{t})/\lambda )\); \(e_{t}\), \(p'_{t}\) and \(\eta _{t}\) are the void ratio, effective mean principal stress and stress ratio, respectively, at the phase transformation state. Due to the use of the vertical (\(q - q _{c}\)) and horizontal (\(p' - p'_{c}\)) distances through α, the phase transformation line is not fixed but changed with the initial material state. For further clarification, an analogy of the function of α with that of the state parameter (m) in Li and Dafalias [1] can be made, where m was used to capture the state-dependent plastic flow and ensured the variation of the phase transformation line with material state. Therefore, a state-dependent plastic flow is analytically developed.
3.2 Possible mathematical connection with \(I_{p}\)
Another interesting finding on the mathematical connection with the empirical state variable \(I _{p}\) [8] can be made by further substituting Eq. (13) into Eq. (15). Then, the above stress-dilatancy equation can be rearranged as follows:
$$ \begin{aligned}[b] d_{g} &= M^{1 + \alpha } \frac{(I_{p} - 1) + (2 - \alpha ) [ 1 - I _{p}/2 - I_{p}\eta ^{2} / (2M^{2}) ]}{(1 - I_{p})M + (2 - \alpha ) [ q/p'_{c} + M(I_{p} - 1) ]} \\ &= M^{\alpha } \frac{ [ \alpha /2 - (1 - \alpha /2)\eta ^{2} / M ^{2} ]I_{p} + (1 - \alpha )}{(1 - I_{p}) + (2 - \alpha ) [ q/Mp'_{c} + (I_{p} - 1) ]}, \end{aligned} $$
where \(I_{p} = p'/p_{c}\) is the state pressure index defined by Wang et al. [8]. A mathematical connection between the derived stress-dilatancy ratio and the empirical state pressure index \(I _{p}\) is observed. Unlike other studies [46] that empirically incorporated \(I _{p}\), the current study reflects the dependence of state pressure from the natural mathematical derivation. However, it should be noted that the relation between \(d _{g}\) and \(I _{p}\) in particular ties this model to other models [1, 13–15, 41, 47] that used \(I _{p}\) or ψ [9] with the added advantage of deriving such relation by conducting fractional derivative of a specific (e.g. MCC) yield surface. Does this connection also generally exist when using other yielding surfaces? Further studies need to be carried out. Nevertheless, it should be emphasized that due to the integral definition of the fractional derivative, the fractional approach proposed in this study intrinsically considers the state information from current state to critical state, as indicated in the initial definition of the fractional derivatives shown in Eqs. (8) and (9). One can also theoretically derive a state-dependent stress-dilatancy equation by conducting fractional derivative of other yielding surfaces.
3.3 Bounding surface and loading direction
For the sake of simplicity, the bounding surface (f̄) is assumed to have the same shape as the yielding surface, i.e.
$$ \bar{f} = \bigl(2\bar{p}' - \bar{p}'_{0} \bigr)^{2} + \biggl( \frac{2\bar{q}}{M} \biggr)^{2} - \bar{p}_{0}^{\prime 2} = 0, $$
where \(\bar{p}'_{0}\) is the intercept of the bounding surface with the abscissa, controlling the size of the bounding surface. The image stress point \((\bar{p}', \bar{q})\) on the bounding surface can be expressed by employing a scalar ρ as
$$\begin{aligned}& \bar{p}' = \rho \bar{p}'_{0} \end{aligned}$$
$$\begin{aligned}& \bar{q} = \rho \eta \bar{p}'_{0}, \end{aligned}$$
in which the stress ratio η can be defined by using the radial mapping rule [39] as follows:
$$ \eta = \frac{q}{p'} = \frac{\bar{q}}{\bar{p}'}. $$
In bounding surface plasticity [48], the loading tensor (\(m _{ij}\)) is normal to the bounding surface and therefore can be obtained by conducting first-order derivative of the bounding surface function as follows:
$$ m_{ij} = \frac{1}{\sqrt{1 + d_{f}^{2}}} \biggl[ \frac{d_{f}}{3} \delta _{ij} + \frac{3s_{ij}}{2q} \biggr], $$
where the loading ratio \(d _{f}\) is formulated as
$$ d_{f} = \frac{M^{2} - \eta ^{2}}{2\eta }. $$
Further substituting Eqs. (19)–(21) into Eq. (18), the scalar (ρ) which determines the image stress point can be obtained as follows:
$$ \rho = \frac{1}{1 + (\eta /M)^{2}}. $$
In addition, the position of the initial bounding surface (\(\bar{p}'_{0i}\)) can be further obtained by intersecting the normal compression and swelling lines in the \(e - p'\) plane as follows:
$$ \bar{p}'_{0i} = 2p_{r}\exp \biggl[ \frac{e_{\varGamma } - e_{0} - \kappa \ln p'_{ic}}{\lambda - \kappa } \biggr], $$
where \(e _{0}\) is the initial void ratio prior to shearing. \(p'_{ic}\) is the initial confining pressure. The evolution of the bounding surface (\(\bar{p}'_{0}\)) can be further obtained as follows:
$$ \bar{p}'_{0} = \bar{p}'_{0i}\exp \biggl( \frac{1 + e_{0}}{\lambda - \kappa } \varepsilon _{v}^{p} \biggr). $$
Note that detailed derivations of Eqs. (25) and (26) can be found in Sun and Shen [21] and Sun et al. [49], thus not repeated here for simplicity.
3.4 Hardening modulus
According to Dafalias [48], the hardening modulus H is determined by both the size (hardening) of the bounding surface and the distance between the loading and bounding surfaces. H was also observed to depend on the material state where a state variable was usually empirically incorporated, for example, in Wang et al. [8]. It is usually decomposed into two components [12, 39, 47]:
$$ H = H_{b} + H_{\delta }, $$
where \(H_{b}\) is determined by applying consistency condition on the bounding surface that experiences isotropic hardening:
$$ H_{b} = - \frac{\partial \bar{f}}{\partial \bar{p'}_{0}}\frac{\partial \bar{p'}_{0}}{\partial \varepsilon _{v}^{p}}\frac{n_{v}}{ \Vert \frac{ \partial \bar{f}}{\partial \bar{\boldsymbol{\sigma }}} \Vert } = \frac{1 + e_{0}}{\lambda - \kappa } \frac{\bar{p'}M_{c}^{2}d_{g} / \sqrt{d _{g}^{2} + 1}}{\sqrt{(2\rho - 1)^{2}M_{c}^{4} + 4\rho ^{2}\eta ^{2}}}. $$
It is easy to find that \(H _{b}\) is state-dependent due to the dependence of \(d _{g}\) on material state. \(H_{\delta } \) is related to the ratio between the distance (δ) from the current stress point to the image stress point and the distance (\(\delta _{\max } \)) from the stress origin to the image stress point [21, 39]:
$$ H_{\delta } = h_{0}p'\frac{1 + e_{0}}{\lambda - \kappa } \frac{\delta }{\delta _{\max } - \delta }, $$
where \(h_{0} = h_{1}e - h_{2}\); \(h _{1}\) and \(h _{2}\) are material constants. As expected, the hardening modulus \(H = + \infty \) at the load onset where \(\delta _{\max } - \delta \to 0\), \(\eta = 0\) and \(\dot{\eta } = 0\), indicating a state of no plastic strain. \(H = 0\) at the critical state where \(p' = p'_{c}\), \(q = q_{c}\), \(\bar{p}'_{0} = p'_{0}\).
4 Parameter identification and sensitivity analysis
There are totally nine parameters (\(M _{c}\), λ, \(e_{\varGamma } \), c, α, \(h _{1}\), \(h _{2}\), κ, ν) in the proposed state-dependent model, which can be all determined from traditional triaxial tests. Detailed elaborations on parameter identification and sensitivity analysis are given below.
Similar to the classical plasticity models, there are four critical state parameters, i.e. \(M _{c}\), λ, \(e_{\varGamma } \) and c. The critical state stress ratio (\(M _{c}\)) is determined by measuring the gradient of the critical state line in the \(p' - q\) plane. With the increase of \(M _{c}\), the predicted shear strength of granular soil increases while the volumetric dilation decreases, as shown in Fig. 2. Note that the other model parameters used for simulations in Fig. 2 are: \(\lambda = 0.057\), \(e_{\varGamma } = 0.64\), \(c = 1\), \(\alpha = 1.12\), \(h _{1} = h _{2} = 2.5\), \(\kappa = 0.0013\) and \(\nu = 0.25\).
Effect of \(M _{c}\) on the stress-strain behaviour of granular soil
λ and \(e_{\varGamma } \) are the critical state parameters in the \(e - \ln p'\) plane. λ can be obtained by measuring the gradient of the critical state line in the \(e - \ln p'\) plane, while \(e_{\varGamma } \) can be determined by the intercept of the critical state line at \(p' = 1\). c can be determined by further conducting a triaxial extension test to obtain the critical state stress ratio (\(M _{e}\)) for triaxial extension and then \(c = M _{e} /M _{c}\). As shown in Fig. 3, with the increase of λ, the predicted peak stress and volumetric dilation decrease. However, as shown in Fig. 4, with the increase of \(e_{\varGamma } \), the predicted peak stress and volumetric dilation increase. Note that the other model parameters used for simulations in Figs. 3 and 4 are: \(M _{c} = 1.7\), \(c = 1\), \(\alpha = 1.12\), \(h _{1} = h _{2} = 2.5\), \(\kappa = 0.0013\) and \(\nu = 0.25\).
Effect of λ on the stress-strain behaviour of granular soil
Effect of \(e_{\varGamma } \) on the stress-strain behaviour of granular soil
There is one new parameter, the fractional order α, that is distinct from the ones in the traditional plasticity model. It controls the state-dependent plastic flow of granular soil. Thus, it can be obtained by using the stress-dilatancy ratio at the phase transformation state, i.e. Equation (16). As shown in Fig. 5, with the increase of α, the predicted peak stress decreases while the volumetric dilation increases. Samples modelled by a higher α reach critical state more quickly; a transition from the strain softening behaviour to strain hardening behaviour is also observed when α increases from 1.0 to 1.4. Note that the other model parameters used for simulations in Fig. 5 are: \(M _{c} = 1.7\), \(c = 1\), \(\lambda = 0.057\), \(e_{\varGamma } = 0.64\), \(h _{1} = h _{2} = 2.5\), \(\kappa = 0.0013\) and \(\nu = 0.25\).
Effect of α on the stress-strain behaviour of granular soil
The hardening parameter \(h _{0}\) is correlated with the void ratio through \(h _{1}\) and \(h _{2}\). As illustrated in Li and Dafalias [1] as well as Sun and Shen [21], they can be obtained by fitting the \(\varepsilon _{1} - q\) relationship of samples with different void ratios. The effect of \(h _{0}\) on the stress-strain behaviour of granular soil can be found in Fig. 6, where an increasing shear strength coupled with an increasing volumetric dilation is found. Note that the other model parameters used for simulations in Fig. 6 are: \(M _{c} = 1.7\), \(c = 1\), \(\lambda = 0.057\), \(e_{\varGamma } = 0.64\), \(\alpha = 1.12\), \(\kappa = 0.0013\) and \(\nu = 0.25\).
Effects of \(h _{1}\) and \(h _{2}\) on the stress-strain behaviour of granular soil
The elastic constant κ can be determined by measuring the gradient of the swelling line in the \(e - \ln p'\) plane. As observed in Fig. 7, with the increase of κ, the peak deviator stress increases, while the initial deviator stress increases more rapidly with a lower κ. In addition, a distinct difference is observed in the relationship between \(\varepsilon _{v}\) and \(\varepsilon _{1}\). At the initial loading stage where \(\varepsilon _{1}\) is small, a large volumetric contraction is observed for soils with high κ. However, the effect of κ on \(\varepsilon _{v}\) gradually becomes insignificant when \(\varepsilon _{1}\) increases, where the plastic parts of \(\varepsilon _{1}\) and \(\varepsilon _{v}\) become dominant.
Effect of κ on the stress-strain behaviour of granular soil
Poisson's ratio ν defines the lateral deformation ability of the material and can be obtained by using the following equation during the initial loading stage [12]:
$$ \nu \approx - \frac{\varepsilon _{3}}{\varepsilon _{1}}. $$
The effect of ν was found to be limited and thus not presented here for simplicity. However, according to the theory of elasticity, a higher lateral deformation could be observed with increasing ν. Detailed values of each model parameter used for predicting the stress-strain behaviour of different granular soils are listed in Table 1. It is noted that for simulating triaxial tests with \(\theta = \pi /6\), c will not be used.
Model parameters
κ (10−3)
\(M_{c}\)
\(e_{\varGamma } \)
\(h_{1}\)
Ottawa sand [50]
Sacramento River sand [51]
Tacheng rockfill [12]
\(0.53+0.59e_{0}\)
5 Model validation
This section demonstrates the model's ability to capture the state-dependent constitutive behaviour of different granular soils, including rockfill and sand with different initial states. A series of triaxial tests results of Tacheng rockfill [12], Sacramento River sand [51] and Ottawa sand [50] are simulated in Figs. 8–14. Drained stress-strain behaviour of Tacheng rockfill [12] with four different initial void ratios is simulated in Figs. 8–11, while Figs. 12–14 present the model predictions of the drained and undrained triaxial behaviour of Sacramento River sand [51] and Ottawa sand [50]. It is noted that the all the test results are represented by discrete data points, while continuous lines are used for model predictions.
Model predictions of the drained behaviour of Tacheng rockfill with \(e _{0} = 0.189\) (data from Xiao et al. [12])
Model predictions of the drained behaviour of Sacramento River sand [51]
Xiao et al. [12] reported a series of drained triaxial test results of Tacheng rockfill with different initial void ratios. The material mainly consisted of sub-angular to rounded particles with a median diameter (\(d _{50}\)) of 23 mm and a coefficient of uniformity (\(C _{u}\)) of 5.4. Samples were prepared by layered compaction to have a diameter around 300 mm and a height around 600 mm. Initial void ratios and the corresponding confining pressures can be found in Figs. 8–11 and thus not repeated here. It is observed from Figs. 8–11 that even without using the state variable and plastic potential function, the proposed model can well simulate with state-dependent constitutive behaviour of Tacheng rockfill subjected to different initial states (confining pressures and void ratios). The strain hardening and softening behaviour as well as the corresponding volumetric contraction and dilation behaviour of Tacheng rockfill can be all reasonably captured.
Lee and Seed [51] conducted a comprehensive study on the drained and undrained triaxial behaviour of Sacramento River sand under different initial conditions. The Sacramento River sand mainly consisted of sub-round quartz aggregates and occasional shell fragments with a \(d _{50}\) of 0.22 mm and a \(C _{u}\) equal to 1.45. The minimum and maximum void ratios were tested to be 0.61 and 1.03, respectively. Samples were prepared to have an initial diameter around 42.67 mm and a height around 103.63 mm. The initial void ratio of 0.87 is used for simulating the drained test results, whereas the initial void ratios of 0.86, 0.86, 0.86 and 0.85 are used for simulating undrained tests carried out under the confining pressures of 98, 294, 490 and 1069 kPa, respectively. Figure 12 shows the model simulations of the drained constitutive behaviour of Sacramento River sand, where a good agreement between the model simulations and the corresponding test results can be observed. Figure 13 shows the model simulations of the undrained behaviour of Sacramento River sand subjected to different initial conditions. As the axial strain increases, the simulated deviator stresses increase until reaching critical (steady) state flow, which is in good agreement with the corresponding test results, which further validates the newly developed state-dependent stress-dilatancy relationship.
Model predictions of the undrained behaviour of Sacramento River sand [51]
Figure 14 presents the simulation results of undrained constitutive behaviour of uniform Ottawa sand [50] that mainly consisted of round/sub-round quartz aggregates. Test results of Ottawa sand with the initial void ratios of 0.793, 0.793 and 0.805 under the respective corresponding confining pressures of 348, 475 and 550 kPa are simulated. It is found that the model predictions match well with the test results. The simulated deviator stress increases initially until reaching a peak value and then decreases significantly at the critical state, indicating a state of static liquefaction of the material. Samples with the same initial void ratios approach the same deviator stress with further shearing, which can be all reasonably characterised by the proposed fractional plasticity model.
Model predictions of the undrained behaviour of Ottawa sand [50]
6 Conclusions
It was found that the stress-strain behaviour of granular soil was state-dependent. To capture such state dependence, many sophisticated constitutive models have been proposed by incorporating empirical state variables into additional plastic potential functions. To simplify the modelling approach, a novel state-dependent fractional plasticity model without using predefined state variable and plastic potential was proposed by conducting fractional-order derivatives of the yielding function. Detailed identifications and sensitivity analysis of model parameters were then carried out. To further validate the developed model, a series of drained and undrained test results of different granular soils were simulated and discussed. The main findings are summarized as follows:
Without using predefined state variables and plastic potential functions, a novel state-dependent stress-dilatancy equation was analytically derived by using the fractional-order plasticity theory. Dependence of the non-associated flow on material state was modelled through rigorous mathematical definition of the fractional stress gradient.
Possible mathematical connections between the proposed state-dependent dilatancy equation and the state pressure index by Wang et al. [8] were also discussed, where the dependence of state pressure on the stress-dilatancy phenomenon of granular soil was analytically proved.
The extent of non-associativity and hardening modulus were influenced by the material state via the vertical and horizontal distances from the current stress state to the corresponding critical stress state in the \(p' - q\) plane.
With the increase of the fractional order, the predicted peak stress decreased while the volumetric dilation increased. Samples modelled by a higher fractional order reached critical state more quickly; the transition from the strain softening behaviour to strain hardening behaviour also increased with the increase of the fractional order.
Model parameters can be all determined from traditional triaxial test results. It was found that the proposed state-dependent fractional plasticity model can well capture the stress-strain behaviour of different granular soils subjected to a variety of loading conditions.
The authors would like to thank Prof. Yannis F. Dafalias for his invaluable suggestions and Prof. Wen Chen for his lifelong inspiration.
All the data in this study were collected from published literatures, which were appropriately cited.
The financial support provided by the National Natural Science Foundation of China (Grant Nos. 41630638, 51679068), the National Key Basic Research Program of China ("973" Program) (Grant No. 2015CB057901) and the China Postdoctoral Science Foundation (Grant No. 2017M621607) is appreciated.
The first author formulated the main ideas and equations of the paper, the second author helped to prepare the figures. All the authors read and approved the final manuscript.
Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, College of Civil and Transportation Engineering, Hohai University, Nanjing, China
School of Civil Engineering, Chongqing University, Chongqing, China
Li, X., Dafalias, Y.: Dilatancy for cohesionless soils. Geotechnique 50(4), 449–460 (2000). https://doi.org/10.1680/geot.2000.50.4.449 View ArticleGoogle Scholar
Dafalias, Y.F., Taiebat, M.: SANISAND-Z: zero elastic range sand plasticity model. Geotechnique 66(12), 999–1013 (2016). https://doi.org/10.1680/jgeot.15.P.271 View ArticleGoogle Scholar
Javanmardi, Y., Imam, S.M.R., Pastor, M., Manzanal, D.: A reference state curve to define the state of soils over a wide range of pressures and densities. Geotechnique 68(2), 95–106 (2018). https://doi.org/10.1680/jgeot.16.P.136 View ArticleGoogle Scholar
Yang, J., Li, X.: State-dependent strength of sands from the perspective of unified modeling. J. Geotech. Geoenviron. Eng. 130(2), 186–198 (2004). https://doi.org/10.1061/(ASCE)1090-0241(2004)130:2(186) View ArticleGoogle Scholar
Ishihara, K.: Liquefaction and flow failure during earthquakes. Geotechnique 43(3), 351–451 (1993). https://doi.org/10.1680/geot.1993.43.3.351 View ArticleGoogle Scholar
Wan, R., Guo, P.: A simple constitutive model for granular soils: modified stress-dilatancy approach. Comput. Geotech. 22(2), 109–133 (1998). https://doi.org/10.1016/S0266-352X(98)00004-4 View ArticleGoogle Scholar
Desai, C.S.: Mechanics of Materials and Interfaces: The Disturbed State Concept. CRC Press, Boca Raton (2000) View ArticleGoogle Scholar
Wang, Z., Dafalias, Y., Li, X., Makdisi, F.: State pressure index for modeling sand behavior. J. Geotech. Geoenviron. Eng. 128(6), 511–519 (2002). https://doi.org/10.1061/(ASCE)1090-0241(2002)128:6(511) View ArticleGoogle Scholar
Been, K., Jefferies, M.G.: A state parameter for sands. Geotechnique 35(2), 99–112 (1985). https://doi.org/10.1016/0148-9062(85)90263-3 View ArticleGoogle Scholar
Schofield, A., Wroth, P.: Critical State Soil Mechanics. McGraw-Hill, London (1968) Google Scholar
Rowe, P.W.: The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 269(1339), 500–527 (1962). https://doi.org/10.1098/rspa.1962.0193 View ArticleGoogle Scholar
Xiao, Y., Liu, H., Chen, Y., Jiang, J.: Bounding surface model for rockfill materials dependent on density and pressure under triaxial stress conditions. J. Eng. Mech. 140(4), 04014002 (2014). https://doi.org/10.1061/(ASCE)EM.1943-7889.0000702 View ArticleGoogle Scholar
Mortara, G.: A constitutive framework for the elastoplastic modelling of geomaterials. Int. J. Solids Struct. 63, 139–152 (2015). https://doi.org/10.1016/j.ijsolstr.2015.02.047 View ArticleGoogle Scholar
Sun, Y., Xiao, Y.: Fractional order plasticity model for granular soils subjected to monotonic triaxial compression. Int. J. Solids Struct. 118–119, 224–234 (2017). https://doi.org/10.1016/j.ijsolstr.2017.03.005 View ArticleGoogle Scholar
Najma, A., Latifi, M.: Predicting flow liquefaction, a constitutive model approach. Acta Geotech. 12(4), 793–808 (2017). https://doi.org/10.1007/s11440-016-0517-x View ArticleGoogle Scholar
Tasiopoulou, P., Gerolymos, N.: Constitutive modelling of sand: a progressive calibration procedure accounting for intrinsic and stress-induced anisotropy. Geotechnique 66(9), 754–770 (2016). https://doi.org/10.1680/jgeot.15.P.284 View ArticleGoogle Scholar
Yang, X.: New rheological problems involving general fractional derivatives with nonsingular power-law kernels. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 19(1), 45–52 (2018) MathSciNetGoogle Scholar
Saad, K.M., Baleanu, D., Atangana, A.: New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger's equations. Comput. Appl. Math. 37(4), 5203–5216 (2018) MathSciNetView ArticleGoogle Scholar
Saad, K.M., Atangana, A., Baleanu, D.: New fractional derivatives with non-singular kernel applied to the Burgers equation. Chaos 28(6), 063109 (2018). https://doi.org/10.1063/1.5026284 MathSciNetView ArticleMATHGoogle Scholar
Saad, K., Al-Sharif, E.H.: Analytical study for time and time-space fractional Burgers' equation. Adv. Differ. Equ. 2017(1), 300 (2017). https://doi.org/10.1186/s13662-017-1358-0 MathSciNetView ArticleMATHGoogle Scholar
Sun, Y., Shen, Y.: Constitutive model of granular soils using fractional order plastic flow rule. Int. J. Geomech. 17(8), 04017025 (2017). https://doi.org/10.1061/(ASCE)GM.1943-5622.0000904 View ArticleGoogle Scholar
Sumelka, W., Nowak, M.: Non-normality and induced plastic anisotropy under fractional plastic flow rule: a numerical study. Int. J. Numer. Anal. Methods Geomech. 40(5), 651–675 (2016). https://doi.org/10.1002/nag.2421 View ArticleGoogle Scholar
Sumelka, W., Nowak, M.: On a general numerical scheme for the fractional plastic flow rule. Mech. Mater. 116, 120–129 (2018). https://doi.org/10.1016/j.mechmat.2017.02.005 View ArticleGoogle Scholar
Heidarzadeh, H., Oliaei, M.: Development of a generalized model using a new plastic modulus based on bounding surface plasticity. Acta Geotech. 13(4), 925–941 (2018). https://doi.org/10.1007/s11440-017-0599-0 View ArticleGoogle Scholar
Wheeler, S.J., Näätänen, A., Karstunen, M., Lojander, M.: An anisotropic elastoplastic model for soft clays. Can. Geotech. J. 40(2), 403–418 (2003). https://doi.org/10.1139/t02-119 View ArticleGoogle Scholar
Pastor, M., Zienkiewicz, O.C., Chan, A.H.C.: Generalized plasticity and the modelling of soil behaviour. Int. J. Numer. Anal. Methods Geomech. 14(3), 151–190 (1990). https://doi.org/10.1002/nag.1610140302 View ArticleMATHGoogle Scholar
Sun, Y., Gao, Y., Zhu, Q.: Fractional order plasticity modelling of state-dependent behaviour of granular soils without using plastic potential. Int. J. Plast. 102, 53–69 (2018). https://doi.org/10.1016/j.ijplas.2017.12.001 View ArticleGoogle Scholar
Yang, X., Srivastava, H.M., Machado, J.A.T.: A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm. Sci. 20(2), 753–756 (2016) View ArticleGoogle Scholar
Yang, X., Gao, F., Machado, J.A.T., Baleanu, D.: A new fractional derivative involving the normalized sinc function without singular kernel. Eur. Phys. J. Spec. Top. 226, 3567–3575 (2017) View ArticleGoogle Scholar
Sun, Y., Gao, Y., Chen, C.: Critical-state fractional model and its numerical scheme for isotropic granular soil considering state-dependence. Int. J. Geomech. 13(9), 04018202 (2018). https://doi.org/10.1061/(ASCE)GM.1943-5622.0001353 View ArticleGoogle Scholar
Yang, X.-J., Gao, F., Ju, Y., Zhou, H.-W.: Fundamental solutions of the general fractional-order diffusion equations. Math. Methods Appl. Sci. 41(18), 9312–9320 (2018). https://doi.org/10.1002/mma.5341 View ArticleGoogle Scholar
Sumelka, W.: Fractional viscoplasticity. Mech. Res. Commun. 56, 31–36 (2014). https://doi.org/10.1016/j.mechrescom.2013.11.005 View ArticleMATHGoogle Scholar
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967). https://doi.org/10.1111/j.1365-246X.1967.tb02303.x View ArticleGoogle Scholar
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1998) MATHGoogle Scholar
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015). https://doi.org/10.12785/pfda/010201 View ArticleGoogle Scholar
Sun, Y., Gao, Y., Shen, Y.: Mathematical aspect of the state-dependent stress-dilatancy of granular soil under triaxial loading. Geotechnique 69(2), 158–165 (2018). https://doi.org/10.1680/jgeot.17.t.029 View ArticleGoogle Scholar
Sheng, D., Sloan, S.W., Yu, H.S.: Aspects of finite element implementation of critical state models. Comput. Mech. 26(2), 185–196 (2000). https://doi.org/10.1007/s004660000166 View ArticleMATHGoogle Scholar
Kan, M.E., Taiebat, H.A.: Application of advanced bounding surface plasticity model in static and seismic analyses of Zipingpu Dam. Can. Geotech. J. 53(3), 455–471 (2015). https://doi.org/10.1139/cgj-2015-0120 View ArticleGoogle Scholar
Bardet, J.P.: Bounding surface plasticity model for sands. J. Eng. Mech. 112(11), 1198–1217 (1986). https://doi.org/10.1061/(ASCE)0733-9399(1986)112:11(1198) View ArticleGoogle Scholar
Yu, H., Khong, C., Wang, J.: A unified plasticity model for cyclic behaviour of clay and sand. Mech. Res. Commun. 34(2), 97–114 (2007). https://doi.org/10.1016/j.mechrescom.2006.06.010 View ArticleMATHGoogle Scholar
Yin, Z.Y., Chang, C.S., Hicher, P.Y., Karstunen, M.: Micromechanical analysis of kinematic hardening in natural clay. Int. J. Plast. 25(8), 1413–1435 (2009). https://doi.org/10.1016/j.ijplas.2008.11.009 View ArticleMATHGoogle Scholar
Yin, Z.-Y., Chang, C.S., Hicher, P.-Y.: Micromechanical modelling for effect of inherent anisotropy on cyclic behaviour of sand. Int. J. Solids Struct. 47(14–15), 1933–1951 (2010). https://doi.org/10.1016/j.ijsolstr.2010.03.028 View ArticleMATHGoogle Scholar
Dafalias, Y.F.: Must critical state theory be revisited to include fabric effects? Acta Geotech. 11(3), 479–491 (2016). https://doi.org/10.1007/s11440-016-0441-0 View ArticleGoogle Scholar
Leoni, M., Karstunen, M., Vermeer, P.A.: Anisotropic creep model for soft soils. Geotechnique 58(3), 215–226 (2008). https://doi.org/10.1680/geot.2008.58.3.215 View ArticleGoogle Scholar
Seidalinov, G., Taiebat, M.: Bounding surface SANICLAY plasticity model for cyclic clay behavior. Int. J. Numer. Anal. Methods Geomech. 38(7), 702–724 (2014). https://doi.org/10.1002/nag.2229 View ArticleGoogle Scholar
Xiao, Y., Liu, H., Chen, Y., Jiang, J.: Bounding surface plasticity model incorporating the state pressure index for rockfill materials. J. Eng. Mech. 140(11), 04014087 (2014). https://doi.org/10.1061/(ASCE)EM.1943-7889.0000802 View ArticleGoogle Scholar
Chen, Q., Indraratna, B., Carter, J., Rujikiatkamjorn, C.: A theoretical and experimental study on the behaviour of lignosulfonate-treated sandy silt. Comput. Geotech. 61, 316–327 (2014). https://doi.org/10.1016/j.compgeo.2014.06.010 View ArticleGoogle Scholar
Dafalias, Y.F.: Bounding surface plasticity. I: mathematical foundation and hypoplasticity. J. Eng. Mech. 112(9), 966–987 (1986). https://doi.org/10.1061/(ASCE)0733-9399(1986)112:9(966) View ArticleGoogle Scholar
Sun, Y., Indraratna, B., Carter, J.P., Marchant, T., Nimbalkar, S.: Application of fractional calculus in modelling ballast deformation under cyclic loading. Comput. Geotech. 82, 16–30 (2017). https://doi.org/10.1016/j.compgeo.2016.09.010 View ArticleGoogle Scholar
Sasitharan, S., Robertson, P.K., Sego, D.C., Morgenstern, N.R.: State-boundary surface for very loose sand and its practical implications. Can. Geotech. J. 31(3), 321–334 (1994). https://doi.org/10.1139/t94-040 View ArticleGoogle Scholar
Lee, K.L., Seed, H.B.: Drained strength characteristics of sands. J. Soil Mech. Found. Div. 93(6), 117–141 (1967) Google Scholar | CommonCrawl |
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Why does a force not do any work if it's perpendicular to the motion?
I have a book that says the Moon's orbit is [in this context assumed to be] circular. The Earth does no work on the moon. The gravitational force is perpendicular to the motion. Why is there no work done if support force is perpendicular to the motion?
newtonian-mechanics forces energy rotational-dynamics work
Don Branson
avito009avito009
$\begingroup$ The perpendicular (to veloicity) component provide change in direction only, not the magnitude in velocity that in turns the kinetic energy. See another answer that has some sorts of linkage. $\endgroup$ – Ng Chung Tak Feb 6 '17 at 15:21
$\begingroup$ The Earth does do work on the moon because in fact the force is not perpendicular. The Earth is rotating faster than the moon is going around the earth, and so the tidal bulge on the side closest to the moon is slightly in front of the moon. Therefore the force is not perpendicular, and the moon is slightly accelerated by the Earth, and therefore the Earth is slightly decelerated by the moon. Thus our day is getting slightly longer all the time, and the moon is getting faster. Exercise: when will this process stop? Exercise: if the moon is getting faster, is the lunar month getting shorter? $\endgroup$ – Eric Lippert Feb 6 '17 at 18:06
$\begingroup$ The moon is being accelerated by the tidal forces but it's not getting faster because the speed it gains is quickly traded for height. Welcome to the strange world of orbital mechanics. $\endgroup$ – Peter Green Feb 7 '17 at 3:30
$\begingroup$ Exactly; and so the lunar month is getting longer, and the day is getting longer. So when does the process end? Original poster, see if you can work it out. Also: what relevant quantities are conserved by this exchange between the Earth and the moon? $\endgroup$ – Eric Lippert Feb 7 '17 at 15:40
$\begingroup$ Because the resultant of the triangle of forces in the direction of the motion is zero. $\endgroup$ – user207421 Feb 7 '17 at 17:10
As explained by SchrodingersCat, mathematically work is proportional to the scalar product of force and line element. Therefore any forces acting perpendicular to the path do not contribute to the work.
Now you might want to ask why work is defined like this. I would like to justify this definition taking your example of the moon.
In physics work is intimately related to energy: basically if you want to change the energy of an object you need to do work on it. Now in the case of the moon there are two relevant energies, (1) kinetic energy of the moon related to the magnitude (but not direction) of the moon's velocity, i.e. its speed; and (2) gravitational energy related to the position of the moon in the earth's gravitational field; this one depends on the distance moon-earth.
For (1), since perpendicular forces do not change the magnitude of velocity (only their direction), the perpendicular force should not enter into the equation of work (since it does not contribute to the energy change).
For (2) if you displace the moon always perpendicular to the direction of the gravitational force, you stay at the same distance, i.e. at the same gravitational potential energy. Therefore such perpendicular displacements do not change the energy and should not enter in the expression for work.
$\begingroup$ Work is the change in kinetic energy, so gravitational potential energy (2) shouldn't be included here. Indeed if the Earth's gravity is the only force acting on an object, then (1)+(2) will be constant for that object, so this reasoning would lead to the conclusion that gravity never does work on the object. $\endgroup$ – stewbasic Feb 7 '17 at 1:51
$\begingroup$ @stewbasic: In my understanding one can also speak of work when e.g. lifting an object in gravity, i.e. when increasing its potential energy. I cannot follow your second argument. In the example Earth's gravity is the only force acting on the object/moon. What do you mean by (1)+(2) will be constant? $\endgroup$ – user1583209 Feb 7 '17 at 2:13
$\begingroup$ Suppose I drop a ball from rest (in vacuum). During its descent it gains kinetic energy (1) and loses the same amount of potential energy (2), so the total energy (1)+(2) is unchanged. Gravity has done positive work on the ball, and the amount is the change in kinetic energy (1). $\endgroup$ – stewbasic Feb 7 '17 at 5:31
$\begingroup$ @stewbasic If I lift a box from the floor ontop a table I do not change the kinetic energy but I increase its potential energy. And I also do a work on it. Actually, I increase the kinetic energy from zero to some arbitrary value and then I decrease it back to zero while I increase the potential energy all the time. $\endgroup$ – Crowley Feb 7 '17 at 13:04
$\begingroup$ @Crowley in that scenario the positive work done by the force you apply cancels with the negative work done by the force of gravity. You could arrive at the same answer by ignoring the work done by gravity, instead including change in gravitational potential energy (which effectively captures the work done by gravity). But this alternate method won't work for the OP's question which is specifically asking about work done by gravity. $\endgroup$ – stewbasic Feb 7 '17 at 22:24
Actually, work or, "work" as we mean in physics, has been mathematically defined as $$W=\int_{x_1}^{x_2}\vec F\cdot d\vec{s}$$ So if $\vec{F}$ and $d\vec{s}$ are orthogonal vectors, that is, the angle between the vectors is $90^\circ$, then according to the above formula, no work is done, physically; in this case, by the earth on the moon.
SchrodingersCatSchrodingersCat
$\begingroup$ The larger question is "why is work defined using that scaler product?" Is there physical behavior reasoning behind that definition? Yes, there is. $\endgroup$ – Bill N Feb 7 '17 at 18:35
$\begingroup$ @BillN It is the same sort of questions like "Why is blue colour called blue?" He who defined the work found that the scalar product of force and displacement is useful and named it. Story ends. $\endgroup$ – Crowley Feb 8 '17 at 9:53
As @SchrodingersCat explained, there will be no work on the system if the force is orthogonal to the displacement. However, I would like to elaborate the answer a little further.
What is the physical meaning of representing work done on a body by the dot product of the force and displacement of the object?
An object can move even without a force (Newton's first law says about this), the motion however will be an unaccelerated one. So a body could displace even if there is no force. If you have studied classical mechanics, you may have heard that it is the linear momentum which is the generator of translation, not force.
To say a work is to be done by a force on the object, it should have some effect[1] ..... on the object, right? Any dynamical property (like in this case, the effect of a force) is represented by a change in position coordinate of the object (the order of which varies according to the dynamical quantity) w.r.t time, because position is something very fundamental in dynamics.
If the force has some effect on the object (which is acceleration, of course), then this force contributes some displacement along the direction of applied force (even if there is motion already in some other direction). In such a case, the effect of force on the body can be measured by taking the component of the resultant (or net, if you insist) displacement along the direction of the applied force. So, work done by a force is defined as the product of the applied force with the displacement component caused by this force. This can be achieved by taking the dot product of the the two vectors-Force and the net displacement.
So, what does it mean no work is done if force and displacement are orthogonal?
In Euclidean geometry, orthogonal vectors implies mutually perpendicular vectors. However, the actual sense is that the two vectors are independent. This means that one has no common component with the other, which according to the above discussions states that one vector has no effect on the other. So, the displacement occurred is not due to the force given. Geometrically, that is possible only if force and displacement are perpendicular so that their dot product vanishes. This is why there is no work done on the system if the applied force and the resultant displacement are perpendicular.
But, that perpendicular force could affect the direction of motion of the body (since a force on a body should accelerate it somehow). So, no work is needed to change the direction of a body, even though it happens only by a force. In such a case, there is no displacement due to the force applied, only a change in direction- the effect of which is defined by the torque on the body (the rotational analogue of force).
[1]: "effect" in the present context is used to imply anything that can contribute to work. We cannot say that the force has no effect on the object. It could accelerate the body, even if no displacement has happened due to that force, which is by changing the direction of motion.
edited Feb 9 '17 at 1:48
UKHUKH
The basic physics principle regarding the energy of a system is that the energy changes when work is done on the system, or the system does work on a different system. The work can either increase (positive work) or decrease (negative work) the total energy of the system.
Forces are the agents of work. Only external forces can change the total energy of a system. Internal forces cause exchanges of energy between pieces of the system.
Let's consider a system, the Moon (only). The gravitational pull of the Earth can do work on the system. That would change the total energy of the Moon, which in this case would simply be a change in kinetic energy. What does a change of kinetic energy mean? It means that the speed of the object has changed.
If the Moon is moving in a circular orbit, then the instantaneous velocity of the Moon is always perpendicular to the instantaneous acceleration, so according to the (correct) definition of work in other answers, $$W=\int \vec{F}\cdot\mathrm{d}\vec{s},$$ the work is zero.
You have asked
Why is there no work done if support force is perpendicular to the motion?
That is, what is the physics behavior that says the perpendicular force doesn't change the energy?
Because the change in energy (by work being done) is a change in the kinetic energy, the force must change the speed of the objects in the system. Accelerations which are perpendicular to the instantneous velocity only change the direction, not the speed. In order to change the speed there must be a component of acceleration which is not perpendicular.
An aside on potential energy
Potential energy is a system energy. If your system is the Moon only, there is no gravitational potential energy. If your system is the Earth and Moon, then one can consider the gravitational energy due to the interaction of the two. But when accounting for work, it's an either/or situation: Either you measure energy as the sum of kinetic and potential, or you consider kinetic only, modified by the work done by the gravitational interaction. You cannot count both simultaneously, because a potential energy change is defined as the negative of the work done by a conservative force (in this case, gravitational), when relative positions of the interacting objects change.
Bill NBill N
As others have mentioned, work (in 3D) on a system in defined as the dot product between the force on the system and its displacement. If they are perpendicular, no work is done.
In terms of intuition, what it means is that the force "redirects" the system (the moon, in your case) but in such as way that some of the particles are sped up and some are slowed down in the redirecting such that the net result is that the total energy of the moon is the same as it was earlier. The moon is forced to move in a circular, but in such a way that its energy remains the same at every point (at least, in the extremely simple model that you are assuming).
Ben SBen S
I see that most of the posters have answered this question in terms of the usual definition of work. That's fine as far as it goes, but to many people the definition of work seems somewhat arbitrary in the first place.
An alternative would be to treat the work-energy theorem $$W_\text{net} = \Delta T \;,$$ (with $T$ the kinetic energy) as an expected behavior (because it is integral in building the conservation law from Newtonian principles and the conservation principle is so useful) and use that to deduce the form that work must take and thus show the reason for the scalar product.
What follows is just an outline.
The straight-line version gives us $W_\parallel = F_\text{net} \,\Delta x$.
Uniform circular motion shows us that speed (and therefore kinetic energy) are not changed by forces perpendicular to the direction of motion. That is $W_\perp = 0$.
We may then break any net force into its parallel and perpendicular components and note that the work comes wholly from the parallel one so that writing the work in terms of the scalar product becomes a natural step.
This also leads us to the main physical content of that definition: forces applied perpendicularly to the direction of motion don't chnage the speed of the object and are in that way different from forces applied along (or against) the direction of motion.
At a higher level of sophistication one would employ Noether's theorem as a postulate and work from there.
dmckee --- ex-moderator kittendmckee --- ex-moderator kitten
$\begingroup$ I noticed no one brought up Stokes' theorem either, since any field that can be reduced to $\oint_{C} \ \mathbf{F} \cdot d\mathbf{l} = 0$ is by definition conservative (i.e., path independent). $\endgroup$ – honeste_vivere Feb 8 '17 at 0:37
$\begingroup$ Very interesting your answer, I asked a question based on this answer $\endgroup$ – user143115 Feb 9 '17 at 6:41
$\begingroup$ @sofky's question: physics.stackexchange.com/questions/310620/… $\endgroup$ – Helen Apr 5 '17 at 6:44
I think you are right @Shrodingers cat. It would be better to clarify the answer this way.
Work done (w) = F . d
= F d Cos θ
At 90 degrees, θ = 90 and Cos 90 = 0,
W = F x d x Cos 90
= F x d x 0 = 0 Joule.
So, when the force is applied perpendicularly to the surface, the work done will be zero.
$\begingroup$ Only that you don't apply it perpendicular to any surface but perpendicular to the line element, perpendicular to the path the moon takes. $\endgroup$ – user1583209 Feb 6 '17 at 15:18
As others have already explained. Well its because there is no displacement in the direction of the gravitational force. It is assumed that the orbit stays the same during our observation. The Earth is pulling the moon towards the centre but the moon is moving in a circular orbit, with no displacement towards the centre.
A simple analogy is a block released from a particular height. Now as it falls down gravitational force does work on it, by pulling it down through a distance and raising its kinetic energy. Now suppose as it moves down you apply a constant horizontal force to the right, now the block is moving slanted path, because of the resultant force due to the two forces. Now if we consider the horizontal motion, gravitational force is not the cause of that, it is simply your hand. So your hand does work for the horizontal displacement, not the vertical one which is done by gravity.
So, in the case of the moon, Earth's gravitational force is not the cause of why it moves in a circle, it only provides the centripetal force necessary and centripetal force causes no whatsoever displacement to the moon.
AllenAllen
In case you are interested in a more mathematical approach, this can actually be proven entirely using geometry. To proof this you need to to know a few things about vectors.
If $\vec v=(v_x,v_y,v_z)$ is the velocity vector of the moon then $v_x$, $v_y$ and $v_z$ represent the components of the speed in the x, y and z direction.
You can calculate the speed of the moon by calculation the length of the velocity vector $$v=|v|=\sqrt{v_x^2+v_y^2+v_z^2}$$ If you square the length you get $v^2=v_x^2+v_y^2+v_z^2$, which I will use later. Lastly the dot product between two vectors is defined to be $$\vec a\cdot\vec b=a_xb_x+a_yb_y+a_zb_z=|a||b|\cos \alpha$$ $\alpha$ is the angle between the two vectors. This means that the dot product is zero if the two vectors are perpendicular
If no work is done on the moon the kinetic energy must be constant. So the derivative of the kinetic energy must be zero. So we take the derivative by applying our $v^2$ substitution and taking the constants out of the derivative. $$\frac{dKE}{dt}=\frac{d}{dt}(\tfrac{1}{2}mv^2)=\tfrac{1}{2}m\cdot\frac{d}{dt}(v_x^2+v_y^2+v_z^2)$$ Then apply the chain rule to each of the components. $$\frac{d}{dt}(v_x^2+v_y^2+v_z^2)=2v_x\frac{dv_x}{dt}+2v_y\frac{dv_y}{dt}+2v_z\frac{dv_z}{dt}=2v_xa_x+2v_ya_y+2v_za_z$$ In which we recognize the dot product: $$\frac{d}{dt}v^2=2\vec v\cdot\vec a$$ So the derivative of the kinetic energy becomes $$\frac{dKE}{dt}=m\vec v\cdot\vec a$$
If the orbit is circular, the velocity is always perpendicular to the acceleration (and the force). So the kinetec energy doesn't change and no work is done on the moon.
AccidentalTaylorExpansionAccidentalTaylorExpansion
Not the answer you're looking for? Browse other questions tagged newtonian-mechanics forces energy rotational-dynamics work or ask your own question.
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Intuitive explanation to why force perpendicular to velocity results in circular path
Does constant perpendicular force always cause uniform circular motion? | CommonCrawl |
Next: Printing and Saving Plots, Previous: Manipulation of Plot Windows, Up: High-Level Plotting [Contents][Index]
15.2.8 Use of the interpreter Property
All text objects—such as titles, labels, legends, and text—include the property "interpreter" that determines the manner in which special control sequences in the text are rendered.
The interpreter property can take three values: "none", "tex", "latex". If the interpreter is set to "none" then no special rendering occurs—the displayed text is a verbatim copy of the specified text. Currently, the "latex" interpreter is not implemented for on-screen display and is equivalent to "none". Note that Octave does not parse or validate the text strings when in "latex" mode—it is the responsibility of the programmer to generate valid strings which may include wrapping sections that should appear in Math mode with '$' characters.
The "tex" option implements a subset of TeX functionality when rendering text. This allows the insertion of special glyphs such as Greek characters or mathematical symbols. Special characters are inserted by using a backslash (\) character followed by a code, as shown in Table 15.1.
Besides special glyphs, the formatting of the text can be changed within the string by using the codes
\bf Bold font
\it Italic font
\sl Oblique Font
\rm Normal font
These codes may be used in conjunction with the { and } characters to limit the change to a part of the string. For example,
xlabel ('{\bf H} = a {\bf V}')
where the character 'a' will not appear in bold font. Note that to avoid having Octave interpret the backslash character in the strings, the strings themselves should be in single quotes.
It is also possible to change the fontname and size within the text
\fontname{fontname} Specify the font to use
\fontsize{size} Specify the size of the font to use
The color of the text may also be changed inline using either a string (e.g., "red") or numerically with a Red-Green-Blue (RGB) specification (e.g., [1 0 0], also red).
\color{color} Specify the color as a string
\color[rgb]{R G B} Specify the color numerically
Finally, superscripting and subscripting can be controlled with the '^' and '_' characters. If the '^' or '_' is followed by a { character, then all of the block surrounded by the { } pair is superscripted or subscripted. Without the { } pair, only the character immediately following the '^' or '_' is changed.
Greek Lowercase Letters
\alpha \beta \gamma
\delta \epsilon \zeta
\eta \theta \vartheta
\iota \kappa \lambda
\mu \nu \xi
\o \pi \varpi
\rho \sigma \varsigma
\tau \upsilon \phi
\chi \psi \omega
Greek Uppercase Letters
\Gamma \Delta \Theta
\Lambda \Xi \Pi
\Sigma \Upsilon \Phi
\Psi \Omega
Misc Symbols Type Ord
\aleph \wp \Re
\Im \partial \infty
\prime \nabla \surd
\angle \forall \exists
\neg \clubsuit \diamondsuit
\heartsuit \spadesuit
"Large" Operators
\int
\pm \cdot \times
\ast \circ \bullet
\div \cap \cup
\vee \wedge \oplus
\otimes \oslash
\leq \subset \subseteq
\in \geq \supset
\supseteq \ni \mid
\equiv \sim \approx
\cong \propto \perp
\leftarrow \Leftarrow \rightarrow
\Rightarrow \leftrightarrow \uparrow
\downarrow
\lfloor \langle \lceil
\rfloor \rangle \rceil
\neq
\ldots \0 \copyright
\deg
Table 15.1: Available special characters in TeX mode
15.2.8.1 Degree Symbol
Conformance to both TeX and MATLAB with respect to the \circ symbol is impossible. While TeX translates this symbol to Unicode 2218 (U+2218), MATLAB maps this to Unicode 00B0 (U+00B0) instead. Octave has chosen to follow the TeX specification, but has added the additional symbol \deg which maps to the degree symbol (U+00B0). | CommonCrawl |
When Rachel divides her favorite number by 7, she gets a remainder of 5. What will the remainder be if she multiplies her favorite number by 5 and then divides by 7?
Let $n$ be Rachel's favorite number. Then $n \equiv 5 \pmod{7}$, so $5n \equiv 5 \cdot 5 \equiv 25 \equiv \boxed{4} \pmod{7}$. | Math Dataset |
\begin{document}
\title{Latent Magic: An Investigation into Adversarial Examples Crafted in the Semantic Latent Space}
\begin{abstract}
Adversarial attacks against Deep Neural Networks(DNN) have been a crutial topic ever since \cite{goodfellow} purposed the vulnerability of DNNs. However, most prior works craft adversarial examples in the pixel space, following the $l_p$ norm constraint. In this paper, we give intuitional explain about why crafting adversarial examples in the latent space is equally efficient and important. We purpose a framework for crafting adversarial examples in semantic latent space based on an pre-trained Variational Auto Encoder from state-of-art Stable Diffusion Model\cite{SDM}. We also show that adversarial examples crafted in the latent space can also achieve a high level of fool rate. However, examples crafted from latent space are often hard to evaluated, as they doesn't follow a certain $l_p$ norm constraint, which is a big challenge for existing researches. To efficiently and accurately evaluate the adversarial examples crafted in the latent space, we purpose \textbf{a novel evaluation matric} based on SSIM\cite{SSIM} loss and fool rate.Additionally, we explain why FID\cite{FID} is not suitable for measuring such adversarial examples. To the best of our knowledge, it's the first evaluation metrics that is specifically designed to evaluate the quality of a adversarial attack. We also investigate the transferability of adversarial examples crafted in the latent space and show that they have superiority over adversarial examples crafted in the pixel space. \end{abstract}
\section{Introduction}
Deep Neural Networks(DNN) have achieve high(often state-of-art) performance in various tasks. However, research about the robustness of DNNs \cite{goodfellow} have shown that they are not reliable for out-of-distribution inputs. In image classification tasks, we can reduce the classification success rate to a amazingly low level(sometimes even lower than random) by adding carefully crated noise to the input images. The noise added is confined to a certain degree, often quasi-perceptible by human. The existence of adversarial examples is a big threat to the reliability of DNNs. Therefore, it's crutial to expose as many blind spots of DNNs as possible.
Many methods have been purposed. The most common method is to add small perturbations to the original images based on direct gradient ascent on the loss function(aka iterative/single step methods). Many iterative/single-step optimization methods have been purposed and achieve high success rate. For examples, BIM \cite{kurakin2016adversarial}, I-FGSM \cite{kurakin2017adversarial}, MI-FGSM \cite{dong2018boosting}, PGD \cite{PGD} have all reached high performance. There are also generator-based methods that train a noise generator against a target model. Results (\cite{gen1},\cite{gen2},\cite{gen3} )have shown that generator-based methods have strong cross-model transferability. However, most prior works manipulate images in the pixel space, which often create spatially regular perturbation to the images. Though the perturbation is confined to a certain degree and is claimed imperceptible, human can still distinguish the pattern of noise in the adversarial examples. A preturbed image is given below as {}, with a common constraint $l_\infty<=16$.
Rather than noising images directly in the pixel space, some works have explore adding noise beyond traditional methods. AdvGAN++\cite{ADVGAN++},ATGAN\cite{ATGAN} explore the possibility to add noise in the down-sampled space.\cite{NATUREADV} explicitly search adversarial examples in the latent space created by GAN. \cite{DataPoisonGen} purpose to find adversarial examples in the latent space created by Variational Auto Encoder(VAE). However, we argue that these methods are purposed before a large, pre-trained encoder-decoder model exists. Thus the latent space they generated are relatively coarse and the adversarial examples they crafted are not as effective as those crafted in the pixel space. A more recent work \cite{SDM} train a VAE model to project images into a latent space as part of the Stable Diffusion Model. Results in \cite{SDM} shows the latent space created by the model is highly semantical and precise. We take advantage of them by utilizing their VAE as our pre-trained encoder-decoder structure. Therefore, we're able to make use of the semantic latent space and create purturbed images. In section \ref{results} , we show that our method can achieve a comparable attack rate as those crafted in the pixel space, while our noise being much more imperceptible to human. A comparison is given below as figure \ref{fig:compare}.
\begin{figure}
\caption{Comparison between adversarial examples crafted in pixel space and latent space. Under latent attack, the perturbation is more covert, and the noise is highly semantic. PGD attack is under a noise budget of $l_\infty<16$}
\label{fig:compare}
\end{figure}
However, as shown in \cite{Concurrent}, preturbed images decoded from latent space are extremely hard to follow the common $l_p$ norm constraint. As illustrated in figure \ref{0.82}, the perturbation is almost imperceptible while the $l_\infty$ distance is larger than the normal constraint. Thus a new evaluation metric is needed for the adversarial examples crated from latent space. Existing methods includes FID score \cite{FID}, Structural Similarity Index Measure (SSIM) and Peak Signal-to-Noise Ratio(PGNR)\cite{genlatenttrash}. In section\ref{analyze_metric}, we discuss the weakness of those pre-existing evaluation metrics, and purpose our novel metric based on SSIM loss and the fool rate. To the best of our knowledge, our work is \textbf{the first} to systemetically explore the quality of adversarial attack on latent space.
\begin{figure}
\caption{A adversarial example crafted from latent space. The perturbation is almost imperceptible, comparing to a normal PGD attack with $l_\infty=0.82$ }
\label{0.82}
\end{figure}
As we are disturbing images in a semantic level, we expect the perturbation to have a high level of transferability. Thus in section\ref{transfer} we investigate the cross-model transferability of adversarial examples crafted from semantic latent space. The results show that adversarial examples crafted from latent space have a high level of transferability, outperforming the transferability of adversarial examples crafted by \textbf{PGD} and FGSM.
We summarize our contributions as follows:
\begin{itemize}
\item We give explanations about the reason to noise the images in the semantic latent space. Base on a pre-trained VAE from Stable Diffusion Model, we show that adversarial examples crafted from latent space can achieve a comparable fool rate as those crafted in the pixel space, while the perturbation being more imperceptible.
\item We purpose a novel evaluation metric for adversarial examples crafted from latent space. \textbf{To the best of our knowledge, our method is the first metric specifically designed for adversarial examples crafted from latent space.}
\item We investigate the cross-model transferability of adversarial examples crafted from semantic latent space. We also investigate how the choice of loss funtions affects the transferability.
\end{itemize} \section{background} \subsection{Iterative Attack Methods}
In order to generate adversarial examples, many methods have been purposed. The most common method is to add small perturbations to the original images based on direct gradient ascent on the loss function(aka gradient-based methods). In the white box setting, many iterative optimization methods have been purposed and achieve high success rate. For examples, BIM \cite{kurakin2017adversarial}, I-FGSM \cite{kurakin2016adversarial}, MI-FGSM \cite{dong2018boosting}, PGD \cite{PGD} have all reached high performance. \subsection{Variational Auto Encoder and Stable Diffusion Models}
Variational Auto Encoder(VAE)\cite{VAE} is an encoder-decoder structure generative models for generating images. The encoder encodes image to a latent variable and the decoder decodes it back into pixel space. The sub-space that encoder encodes image into is called the latent space. As our research manipulate latent variables on latent space, the outcome of our experiments will strongly depend on a well-trained VAE that creates a proper latent space. After some pilot runs, we adopted the pre-trained VAE from Stable Diffusion Model\cite{SDM} as our VAE module. Stable Diffusion Model is Latent Diffusion Model that use a U-Net to recurrently noise and denoise the latent variable image in the latent space. \cite{SDM} purpose that the latent space created by LDMs eliminates imperceptible details of original images while maining the semantic information. The outstanding achievement of stable diffusion models strongly supports the authors' claim. Therefore, we adopt the pre-trained VAE from Stable Diffusion Model to create a strong semantic latent space for our research. \section{Related Works}
\subsection{Adversarial Attack in the latent space}
\cite{DataPoison} first purpose to generate adversarial examples by noising the latent variable created by VAE. They tends to learn a constant noise $\Delta z$ for all the latent variables. Later, an incremental work \cite{DataPoisonGen} trains a generator in the latent space to produce noised latent variables. \cite{AVAE} purpose AVAE, a model based on VAE and GAN\cite{GAN} to produce adversarial examples. And \cite{NATUREADV} purposed to search for latent variables in a latent space created by GAN and create semantical change to the image. The forementioned works are in lack of quantitive analysis of the quality of the adversarial examples and the explanation of the reason to noise the latent space.
Notably, a concurrent work \cite{Concurrent} use Latent Diffusion Model to produce noised images, which also uses the same latent space as ours. They purpose to perform 5 steps of denoising process of DDIM\cite{DDIM} on the latent variable $z=\mathcal{E}(X)$ before decoding. They tends to maximizing the loss by updating $z$, which is much similar to ours method.
However, we claim that though they take the structure into account by adding self-attention and cross-attention loss, the denoising step could still change the semantic information to a human-perceptible extent. An example is given as figure \ref{soup}. As their works manipulate the semantic meaning of the image on a high level that's visible to human, we argue that the denoising process of DDIM is too strong a perturbation to be used upon the latent variable. Thus we won't compare with their work in the experiment section, even though we reach comparable transferability as theirs.
\begin{figure}
\caption{The image on the left is the original images, and the image on the right is produced using the open-source code of \cite{Concurrent} by their default settings. As illustrated, the denoising process of DDIM has totally change the vegetable in the middle, which is a huge semantic change. The denoising process also purify the watermark on the background, which is also not expected. }
\label{soup}
\end{figure}
We admit that our basic framework for crafting adversarial examples in the latent space is not novel and has much similarity with previous work. However, the explanation for the reason to noise the latent space and the evaluation metric we purpose are novel and important contributions of our work. Our experiments also illustrate the high transferability of adversarial examples crafted from latent space, which is a novel finding. \section{Problem Formulation} We denote the target classification model as $ \mathcal{F} $ . For the VAE we used, we seperately denote the encoder and the decoder as $\mathcal{E}$ and $\mathcal{D}$. We denote the latent space as $\mathcal{Z}_{sem}$, our loss function as $\mathcal{L}$
In the noising phase, we assume to know every information about $\mathcal{F}$. For a given dataset $X$, we first encode $X $ into a latent variable $z=\mathcal{E}(X)\in \mathcal{Z}_{sem}$
The noising goal is to maximize:
$$ z' \longleftarrow \mathop{\arg\max}_{z'} \mathcal{L}(\mathcal{F}(\mathcal{D}(z'))) $$
Then the noised latent variable $z'$ is decoded back into pixel space as $X'=\mathcal{D}(z')$. The adversarial example is then generated.
\section{Why Latent Space?} Though crafting adversarial examples is a crutial topic in the field of adversarial machine learning, researches about crafting in the latent space is still relatively less explored. In this section, we will give explanations about the reasonabilityand neccessity of crafting adversarial examples in the latent space. \subsection{Comparison with Generator-Based Methods in the Pixel Space} Many researches have been done on training a noise-generator to generate adversarial examples. The most common adversarial generator model structure consists of a down-sampler, a bottle-neck module and an up-sampler. We denote the down-sampler as $\mathcal{D}$, the up-sampler as $\mathcal{U}$, the bottle-neck module as $\mathcal{B}$, then the generator can be denoted as $\mathcal{G}=\mathcal{U}\circ\mathcal{B}\circ\mathcal{D}$. If we view $\mathcal{D}$ and $\mathcal{U}$ as an encoder-decoder structure model, then $\mathcal{G}$ is a generator that produces noises in the latent space created by $\mathcal{U}$ and $\mathcal{D}$. In fact, ATGAN\cite{ATGAN} has explicitly purposed to add perturbation on the down-sampled space. Thus, we argue that the traditional generator-based methods can be viewed as a special case of our method, where the latent space is created by $\mathcal{U}$ and $\mathcal{D}$. However, as $\mathcal{U}$ and $\mathcal{D}$ is often trained under the limitation of $l_p$ norm, the latent space created by them is less semantic than a $l_p$ norm-free latent space but contains more spatial details. \subsection{Making Use of the Feature of Human Recognition System} The main difference between the latent space and the pixel space is that perturbations made on latent space cause semantic changes while those made on pixel space cause changes with spatial pattern. We argue that under the same extent of change, the Human Recognition System is more tolerant to semantic change rather than the spatial.
It's widely acknowledged that human recogizes images in a semantic-level. For example, the original image in figure \ref{catty} would be recogized by human as cat, a shoe and grass ground, along with their interactions with each other. It's shown in \cite{CLIP} that the CLIP model can encoded the semantic information of an image into a latent variable with strong semantic information, which partially proves that a feature extractor is able to extract semantic information from an image as well as human does. However, neural networks have been proved to have linear nature \cite{goodfellow} in the feature space, while human recognition system is strongly non-linear. As illustrated in figure \ref{catty}, a slightly semantically-preturbed cat would still be recogized as cat to human. But actually it is an unnatural being and should never been seen before, as no cat will have a strange pow and be without mouth. \textbf{We believe the core reason behind this scenario is that human kept a strong robustness to semantically out-of-box distribution, while neural networks are easily fooled due to their linear nature.}
\begin{figure}
\caption{An illustration of the robustness of Human Recognition System to semantically out-of-box distribution. The PGD attack is under the $l_\infty$ norm constraint with $l_\infty<16$. The noise produced by PGD as colorful stripes can be easily seen by human, while the noise produced by our method is semantically more natural. Please zoom in to see the details.}
\label{catty}
\end{figure}
However, human are much more sensitive to spatial change. We can easily recogize the spatial pattern of the noise, as to human the noise is a new semantical object that's independent to other objects in the image. Meanwhile, neural networks are equally easy to be fooled by both kind of change. Thus, it's reasonable for attackers to manipulate images by making semantical changes rather than spatial changes, as it's more covert and imperceptible to human beings.
\section{Similarity-Delta Measure($SDM$):\\ A Metric for Evaluating Adversarial Attacks on Latent Space} During practice, we found a lack of evaluation methods for adversarial attacks on the latent space. We argue that most of the existing evaluation methods are not suitable for evaluating adversarial attacks on the latent space and give our detailed analysis on existing metrics. After, we purpose a novel metric called $SDM$ to quantitively measure the quality of a latent space adversarial attack. \subsection{Analyzing Existing Evaluation Metrics}\label{analyze_metric}
Unlike the perturbation in the pixel space, the perturbation in the latent space hardly follows the $l_p$ norm constraint. Some other metrics have been used to evaluate the adversarial examples crafted from latent space. We analyze those metrics and provide reasons for why they are not suitable to evaluate adversarial attacks on latent space. \subsubsection{Deep Learning Based Metrics} The most famous deep learning based metric is Frechet Inception Distance score(FID)\cite{FID}. FID is a metric to evaluate the similarity between two set of images based on the feature extracted from the Inception model\cite{Inception}. The lower the FID score is, the more similar the two set of images are. FID is widely used in GAN\cite{GAN} and VAE\cite{VAE} to evaluate the quality of the generated images. However, we claim that FID is not suitable for evaluating adversarial examples. We take a special situation as example, where the adversarial example is crafted against the Inception model, which is also used to extract the feature in FID scoring. In this situation, the adversarial example is attacking the feature extractor itself, thereby affecting the features. Thus, the FID score will not be the actual perception distance between the adversarial example and the original image.
From a broader perspective, for other deep learning based metrics(like lpips\cite{LPIPS} ), what they actually evaluate is the distribution similarity\cite{FID} between two sets of images. However, the adversarial example is often designed to be perceptually similar to the original image, but is actually out of the original distribution. Thus from the nature of deep learning based metrics, they are unfair for adversarial examples.
\subsubsection{Traditional Metrics} For traditional metrics, the most commonly used ones are Structural Similarity Index Measure($SSIM$)\cite{SSIM} and Peak Signal-to-Noise Ratio($PSNR$)\cite{PSNR}. $SSIM$ is a metric to evaluate the similarity between two images. $PSNR$ is a metric to evaluate the quality of the image. Both of them are widely used in image processing. However, only reporting those losses is not enough, as lower $SSIM$ and higher $PSNR$ could allow more perturbation to the images, which often indicates a better attack rate. Thus a single loss without baseline is meaningless. Meanwhile, an universal baseline for such losses is hard to measure. For example, different datasets could have a different baseline of $SSIM$ and $PSNR$, which makes cross-dataset comparison unfair. Even for the same dataset, different encoder-decoder model could make the baseline different.
Particularly we give more detailed introduction about $SSIM$, as it will be used as our base metric for evaluating adversarial attacks. $SSIM$ is defined as:
$$ \operatorname{SSIM}(\mathbf{x}, \mathbf{y})=\frac{\left(2 \mu_x \mu_y+C_1\right)\left(2 \sigma_{x y}+C_2\right)}{\left(\mu_x^2+\mu_y^2+C_1\right)\left(\sigma_x^2+\sigma_y^2+C_2\right)} $$
where $\mu_x$ and $\mu_y$ are the mean of $x$ and $y$ respectively, $\sigma_x^2$ and $\sigma_y^2$ are the variance of $x$ and $y$ respectively, $\sigma_{xy}$ is the covariance of $x$ and $y$, $C_1$ and $C_2$ are two variables to stabilize the division with weak denominator. SSIM is a value between 0 and 1, where 1 means the two images are identical.
In practice, $SSIM$ is often applied to a sliding window of size $w \times w$ on both image, and takes the mean of all the $SSIM$ values as the final $SSIM$ score. We also add a gaussian filter to the sliding window to make the $SSIM$ score more robust to the noise. For implementation details please check appendix {}.
\subsection{$SDM$:Similarity-Delta Measure} \subsubsection{Similarity Functions} Though using SSIM or PSNR as metrics is not adaquate, we still believe that traditional similarity functions are good metrics to evaluate the adversarial examples and can be used for our new metric. We let our chosen similarity function as $\mathcal{S}$. For $S$ we make the following assumptions: \begin{itemize}
\item $\mathcal{S}$ is a function that takes two images as input and outputs a value between 0 and 1, where higher value means the two images are more perceptually similar.
\item $\mathcal{S}$ is symmetric, which means $\mathcal{S}(X, X') = \mathcal{S}(X', X)$
\item $\mathcal{S}(X,X')=1$ if and only if $X=X'$ \end{itemize} Many similarity functions satisfy the forementioned properties. Additionally, we assume that stronger adversarial examples lead to lower $\mathcal{S}$. And we want to find a balance between the adversarial examples' strength and perceptional similarity to the original image, while being data-independent and model-independent. Thus we propose a new metric based on SSIM, which is Similarity-Delta Measure(SDM). $SDM$ is defined under a fixed dataset $X\sim D_X$ and an encoder-decoder model $\mathcal{M}$.
Additionally, we denote $Acc$ as the accuracy of the adversarial examples crafted from $X$ and $\mathcal{S}$ as the $\mathcal{S}$ score of the adversarial examples.
$Acc_{\mathcal{M}}$ is the baseline accuracy, defined as :
$$ Acc_{\mathcal{M}}= \mathbb{E}_{X \sim \mathcal{D}_{X}}\left[\mathcal{F}(X, \mathcal{D}(\mathcal{E}(X))=labels)\right] $$
$\mathcal{S}_{\mathcal{M}}$ is the baseline $\mathcal{S}$ score on model $\mathcal{M}$, defined as:
$$ \mathcal{S}_{\mathcal{M}}= \mathbb{E}_{X \sim \mathcal{D}_{X}}\left[\mathcal{S}(X, \mathcal{D}(\mathcal{E}(X)))\right] $$
Then we define:
$$\Delta Acc=\frac{Acc_{\mathcal{M}}-Acc}{Acc_{\mathcal{M}}}$$
which indicates the fooling capability of the adversarial examples. Similarly, we define:
$$\Delta \mathcal{S}=\frac{\mathcal{S}_{\mathcal{M}-\mathcal{S}}}{\mathcal{S}_{\mathcal{M}}}$$
which indicates the decay of similarity between adversarial examples and original images.
\textbf{The $SDM$ is defined as:}
$$ SDM= \frac{\log(1-(1-\gamma)* \Delta Acc)}{\Delta \mathcal{S}+\varepsilon} $$
where $\varepsilon, \gamma $ are small numbers to prevent undefined math operations.
We set $\gamma = 1e-3$ and $\varepsilon=1e-7 $ in our experiments.
To evaluate traditional adversarial attacks, we set $\mathcal{S}_\mathcal{M}=1$. \subsection{Intuition behind the Form of SDM} As our goal is to let $SDM$ be a metrics that balance the adversarial examples' strength and perceptional similarity to the original image, it's natural to make a trade-off between $Acc$ and $\mathcal{S}$ by using a ratio of them. However, simply use $SDM=\frac{Acc}{\mathcal{S}}$ is not model and data independent, as the absolute value of $Acc$ and $\mathcal{S}$ strongly relys on the model and dataset. Thus we use the ratio of the relative value of $Acc$ and $\mathcal{S}$ to make $SDM$ model and data independent. The first version of $SDM$ is defined as:
$$ SDM=\frac{\Delta Acc}{\Delta \mathcal{S}} $$
However, from practice we observe that $\Delta S$ does not have a linear relation with $\Delta Acc$. As $\Delta Acc$ rises, $\Delta S$ rises more and more rapidly, as illustrated in figure \ref{fig:ssim_acc}.
\begin{figure}
\caption{target model is ConvNext\_Base.}
\caption{target model is ViT\_Base.}
\caption{The relation between $\Delta Acc$ and $\Delta S$ }
\label{fig:ssim_acc}
\end{figure}
It's clear that the relation between $\Delta Acc$ and $\Delta S$ is not linear. Thus we needs to find a proper function $f$ to make the relation between $f(\Delta Acc)$ and $\Delta S$ linear(or constant). Intuitionally, the attack rate is harder to increase when it is closer to 1. Thus we must pay a greater decrease in $\mathcal{S}$ as the "price". We can formulate this by a differential equation:
$$ \frac{d (\Delta Acc)}{d (\Delta S)} = f(1-\Delta Acc) $$
In this paper, we assume $f$ to be an linear function, where means $f(x)=K x$. Thus we can solve the differential equation and get:
$$ \Delta \mathcal{S}= \frac{1}{K} \log(1-\Delta Acc) $$
We notice that the value of $K$ is important, as it determines the slope of the curve of $\Delta Acc$ and $\Delta S$, which represents the strength of an adversarial attacks. If $K_1>K_2,\Delta Acc_1=\Delta Acc_2 $, we have $\Delta \mathcal{S}_1< \Delta \mathcal{S}_2 $. Thus a larger $K$ indicates that the adversarial attack is more semantically similar to the original image than other attacks under the same attack rate, which reflects the strength of the adversarial attack. Therefore, we set $SDM=K$. Then we get:
$$ SDM= K= \frac{\log(1-\Delta Acc)}{\Delta \mathcal{S}} $$
To avoid the denominator to be zero, we add a small number $\varepsilon$ to the denominator. To avoid the numerator to be undefined when $\Delta Acc=1$, we add a small number $\gamma$ to the numerator. Thus we get the final form of $SDM$:
$$ SDM= \frac{\log(1-(1-\gamma)\Delta Acc)}{\Delta \mathcal{S}+\varepsilon} $$
In practice, we found $SDM$ of this form is more stable to various attack rate, further analysis is given in section \ref{stable_sdm}. \subsection{Model-Independency and Dataset-Independency of SDM } We claim that SDM is strongly model-independent and dataset-independent. Though the absolute value of $Acc$ and $\mathcal{S}$ can differ dramatically, by comparing with the baseline of the given model and datasets, we can still get a fair cross-model and cross-dataset comparison. Below we give detailed analysis about the validity of the baseline chosen.
We claim that a well-trained encoder-decoder structure should satisfy the following properties:
(1) The encoder-decoder is trained to minimize the reconstruction loss $L_{recon}$,which should be equivalent to maximize some similarity function $\mathcal{S}'$. We assume $S'$ also satisfy the forementioned properties of similarity functions. The training goal of the encoder-decoder can be formulated as:
$$ \begin{aligned}
\mathop{\arg\min}_{\mathcal{E}, \mathcal{D}} \mathcal{L}_{\text {recon }} & = \mathop{\arg\max} _{\mathcal{E}, \mathcal{D}} \mathbb{E}_{X \sim \mathcal{D}_{X}}\left[\mathcal{S}'(X, \mathcal{D}(\mathcal{E}(X)))\right] \\
&=\mathop{\arg\max} _{\mathcal{E}, \mathcal{D}} \mathbb{E}_{X \sim \mathcal{D}_{X}}\left[\mathcal{S}'(X, X')\right] \end{aligned} $$
(2) If the encoder-decoder structure is {\em perfect}(with zero loss on any data), then the $\mathcal{S}$ score of the original image and the reconstructed image should be 1(which means they are identical). Meanwhile, any other image should have a $\mathcal{S}$ score less than 1. That can be formulated as:
$$
\begin{cases}
\mathcal{S}(X, X') = 1 & \text{if } X'=\mathcal{D}(\mathcal{E}(X))\\
\mathcal{S}(X', X') < 1 & \text{otherwise}
\end{cases} $$
(3) However, it's impossible for an encoder-decoder model to be {\em perfect} while maintaining a highly semantical latent space. But we claim that for a well-trained encoder-decoder model, the maximum $\mathcal{S}$ score could only be achieved by the image that's decoded from the encoded image of itself. That can be formulated as:
$$ \begin{aligned}
\text{Assume }\max\limits_{X' \in \mathcal{D}(\mathcal{Z}_{sem})} (\mathcal{S}(X, X')) = C \leq 1 \\
\text{Then } \mathcal{S}(X, X') = C \text{ if and only if } X'=\mathcal{D}(\mathcal{E}(X)) \end{aligned} $$
We give a intuition of why this property holds. Assume:
$$\exists z\in \mathcal{Z}_{sem},z\neq \mathcal{E}(X),\mathcal{S}(\mathcal{D}(z),X) > \mathcal{S}(\mathcal{D}(\mathcal{E}(X)),X) $$
Due to the property (1) of similarity functions, $\mathcal{S}'(\mathcal{D}(z),X)>\mathcal{S}'(\mathcal{D}(\mathcal{E}(X)),X)$ should also holds, which contradicts with property (1). Thus such $z$ could not exist.
However, it's indeed possible to have $z\in \mathcal{Z}_{sem}$ with $S(\mathcal{D}(z),X)=\mathcal{S}(\mathcal{D}(\mathcal{E}(X)),X)$, but in real world it's nearly impossible to find such $z$ that happens to be exactly identical.
Given the forementioned properties of a well-trained encoder-decoder structure, we can now validate $\mathcal{S}_{\mathcal{{M}}} $, as any perturbation on the original image should lead to a lower $\mathcal{S}$ score. Thus:
$$ \mathcal{\mathcal{M}}-\mathcal{S}+\varepsilon > 0 $$
As for $Acc$, if the accuracy of the preturbed image is higher than the baseline, then the perturbation is not adversarial. Thus for a valid adversarial attack, we have:
$$ Acc_{M}-Acc \geq 0 $$
\textbf{Thus we have some properties of the SDM score}:
\begin{itemize}
\item $SDM \geq 0$ for a well-trained encoder-decoder model and a valid adversarial attack on latent space.
\item $SDM$ is larger when adversarial are stronger and more human imperceptible.
\item $SDM$ is highly model-independent and dataset-independent. \end{itemize}
\subsection{Choice of Similarity Function:$SSIM$}\label{stable_sdm} Though $SDM$ is highly model-independent and dataset-independent, it's still sensitive to the choice of similarity function. In figure \ref{fig:ssim_acc} a strong correlation between the accuracy and $SSIM$ is shown. Thus we choose $SSIM$ as the similarity function in our experiments. We also show that use $SSIM$ as $\mathcal{S}$ brings a high level of stability to the $SDM$ metric.
To show how $SSIM$ evaluates the image, we craft two adversarial examples from a same image in the latent space, but with different extent of perturbations. The first one have a $SSIM$ score of $0.3627$, the second one have a $SSIM$ score of $0.5652$. As illustrated in figure \ref{cabbage_ssim}, we can clearly see the difference between purturbed images grows as the $SSIM$ score lowers.
\begin{figure}
\caption{Adversarial examples with different $SSIM$ scores. When $SSIM$ score is low, the perturbation is more human imperceptible.}
\label{cabbage_ssim}
\end{figure}
As illustrated in figure \ref{SDM_rb}, we show the $SDM$ score of a perturbed image with different target models, but with same encoder-decoder structure and dataset. We can see that the $SDM$ score is highly stable in a region of $\Delta Acc\in (0.6,0.995)$(denoted as $\mathcal{L}$), which is marked in green. $SDM$ score are unstable with smaller or greater $\Delta Acc$, which is marked as $red$ and $blue$ respectively. However, we argue that most adversarial attempt (as shown in \ref{fig:side:a}) should falls in $\Delta Acc \in \mathcal{L}$. Too low or too high $\Delta Acc$ means the perturbation is either too weak or over-fits the datasets. Thus we argue that $SDM$ is a stable metric for adversarial attacks.
\begin{figure}
\caption{SDM score with unstable region marked in red and blue respectively.}
\label{SDM_rb}
\end{figure} \begin{figure}
\caption{stable SDM score without sudden decay under different target models.}
\label{fig:side:a}
\end{figure} \section{Method} \subsection{Framework} We adopt the pre-trained VAE from Stable Diffusion Model as our VAE module. For a dataset $X$, we encode it into $z=\mathcal{E}(X)\in \mathcal{Z}_{sem}$, then decode it back into pixel space as $X'=\mathcal{D}(z)$. We then feed $X'$ into the target model and calculate the loss function $\mathcal{L}$, backpropagate the loss to the latent space and update the latent variable $z$ by gradient ascent. We repeat this process for $T$ step with a learning rate $lr$. The algorithm for the noising phase is shown in Algorithm \ref{alg:noise}. The sketch of our framework is illustrated in figure \ref{framework}.
\begin{algorithm}[H]
\caption{Noising Phase}
\renewcommand{\textbf{Input:}}{\textbf{Input:}}
\renewcommand{\textbf{Output:}}{\textbf{Output:}}
\label{alg:noise}
\begin{algorithmic}
\REQUIRE $X$, $\mathcal{F}$, $\mathcal{E}$, $\mathcal{D}$, $\mathcal{L}$, $T$, $lr$
\ENSURE $X'$
\STATE $z \leftarrow \mathcal{E}(X)$
\FOR{$t=1$ to $T$}
\STATE $X' \leftarrow \mathcal{D}(z)$
\STATE $loss \leftarrow \mathcal{L}(\mathcal{F}(X'))$
\STATE $z \leftarrow z + lr \cdot \nabla_z loss$
\ENDFOR
\STATE $z' \leftarrow z$
\STATE $X' \leftarrow \mathcal{D}(z')$
\end{algorithmic} \end{algorithm}
\begin{figure}
\caption{The framework of our method.}
\label{framework}
\end{figure}
\subsection{Loss Function and Evaluation Metrics} We use the most common loss function for adversarial attack, the cross entropy loss(denoted as $\mathcal{L}_{ce}$). As for evaluation metrics, we use our $SDM$ metric to evaluate the quality of our adversarial attack. We also apply $SDM$ to traditional attacking methods by setting $\mathcal{S}_\mathcal{M}=1$ and compare them with our method. \section{Experiments} \subsection{Implementation Details} We use the VAE from Stable Diffusion Model v2.0. As for dataset, we choose the validation set of ImageNet-1k. Due to our limited time and resources, in practice we randomly choose 1000 images from the validation set as our dataset. We choose several classification models as our target model including FocalNet, ConvNext, ResNet101 and ViT\_B. For each image, we first encode it into the latent space, then we do one forward, one backpropagation and do gradient ascent on the latent variable with a fixed $lr$. For each image we run 30 iterations of gradient ascent, as we find that the attack rate nearly converges after 30 epochs. For all PGD attack we use in the experiments, we set the number of iterations to 50, as we found that the loss of PGD attack converges after 50 iterations. We set $lr=5e-3$ for every attack methods. \subsection{Results}\label{results}
\subsubsection{Attacking The Target Models}\label{transfer} As found in \cite{ViTAttack}, the family of ViT\cite{VIT} differs from the traditional CNN architectures when facing adversarial attacks. So we choose models from both family and test their performance.
We also apply traditional iteration/single step methods to attack the models and test their performance. We choose PGD\cite{PGD},FGSM\cite{goodfellow} and test their performance with our method, and proves that we can reach comparable performance with traditional methods. We set $\gamma = 0.999$ and $\varepsilon=1e-7 $ in our experiments.
To quantify our adversarial examples, we use our $SDM$ metrics for evaluation. We report our top-1 accuracy and the corresponding $SDM$ score if our method and compare it to the traditional methods, as shown in table \ref{tab:transfer} and \ref{tab:sdm}.
Though our method has only achieve the best performance in terms of top-1 accuracy in the white-box setting when the target model is ViT\_B, we achieve the best $SDM$ score by about 10x compared to the traditional methods. This shows that our method is a stronger attack to the target model. However, it is not fair to compare the $SDM$ score of our method with the traditional methods, as the traditional method are not designed to maintain a high $\mathcal{S}$ like the semantic space does. But we can still see that the huge gap between the $SDM$ score of our method and the traditional method, which to some extent shows the superiority of our method. \subsubsection{Cross-Model Transferability} As the perturbation added on images is highly semantical, we expect the perturbation to be more transferable. We test our perturbation on different models but in the same dataset. The results are shown in table \ref{tab:transfer}.
For the absolute attack rate, we can see PGD attack still outperforms others. However, our method achieve a very close performance to PGD attack, while being much transferable. Our method reaches the best performance in both the average attack rate and avarage $\Delta Acc$ except for the case when our training target model is ViT\_B. However, we have the best attack rate and the best $\Delta Acc$ in this case. The results indicate our method has the best performance in transferability comparing to the traditional method. As expected, perturbation crafted in semantic latent space is much more transferable than regular pixel space-based attacks.
\begin{table}[htbp]\label{tab:transfer}
\centering
\tiny
\caption{Transferability comparison on classification with different models. Here we report the top-1 accuracy and average $\Delta Acc$, average accuracy of four target models. The higher the $\Delta Acc$ the better. For the white-box situation where the surrogate model is the same as the target model, we set the background to be gray. The best results are bolded. All the traditional iteration/single step method are within the perturbation budget of $l_\infty<10$. }
\begin{tabular}{@{}ccccccl@{}}
\toprule
& ConvNext\_base & ResNet101 & FocalNet\_base & ViT\_B & Average $\Delta Acc$ & Average Acc \\ \midrule
Clean & 0.8370 & 0.8340 & 0.8470 & 0.7410 & N/A & 0.8148 \\
PGD(ConvNext) & \cellcolor[HTML]{9B9B9B}\textbf{0.0110} & 0.5570 & 0.4070 & 0.6350 & 0.4946 & 0.5397 \\
FGSM(ConvNext) & \cellcolor[HTML]{9B9B9B}0.3980 & 0.6030 & 0.5160 & 0.6140 & 0.3409 & 0.5325 \\
\midrule
Ours(ConvNext) & \cellcolor[HTML]{9B9B9B}0.0200 & \textbf{0.3900} & \textbf{0.2590} & \textbf{0.5480} & \textbf{0.6158} & \textbf{0.3042} \\
\midrule
PGD(FocalNet) & 0.4588 & 0.6150 & \cellcolor[HTML]{C0C0C0}\textbf{0.0190} & 0.6310 & 0.4619 & 0.4309 \\
FGSM(FocalNet) & 0.5590 & 0.6380 & \cellcolor[HTML]{C0C0C0}0.4220 & 0.6260 & 0.3060 & 0.5613 \\
\midrule
Ours(FocalNet) & \textbf{0.2790} & \textbf{0.4190} & \cellcolor[HTML]{C0C0C0}0.0300 & \textbf{0.5310} & \textbf{0.6031} & \textbf{0.3148} \\
\midrule
PGD(ViT\_B) & 0.7630 & 0.7240 & 0.7660 & \cellcolor[HTML]{C0C0C0}0.0030 & 0.3279 & 0.5640 \\
FGSM(ViT\_B) & 0.6350 & 0.6110 & 0.6350 & \cellcolor[HTML]{C0C0C0}0.0630 & 0.3560 & \textbf{0.4860} \\
\midrule
Ours(ViT\_B) & 0.6690 & 0.6670 & 0.6730 & \cellcolor[HTML]{C0C0C0}\textbf{0.0020} & \textbf{0.4009} & 0.5026 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[htbp]
\caption{We report the $SDM$ score with the highest attack rate of our method and traditional methods on different models. The higher the $SDM$ score the better. The top 3 $SDM$ score is bolded.}
\label{tab:sdm}
\centering
\tiny
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
& PGD(ConvNext) & Ours(ConvNext) & PGD(FocalNet) & Ours(FocalNet) & PGD(ViT\_B) & Ours(ViT\_B) \\ \hline
SDM score & 22.48 & \textbf{153.8} & 14.78 & \textbf{149.8} & 21.68 & \textbf{225.3} \\ \hline
\end{tabular}
\end{table} \section{Future Work} In this paper we purpose a basic iteration-based framework to add perturbation on latent variables. However, we believe that more advanced update technique for latent variables can be explored. Future researches may works on more efficient and achieving ways to update latent variables, or try generator-based methods on latent space. We also show that the performance of attacks in latent space strongly depends on the quality of the latent space, which is determined by the encoder-decoder structure. Therefore, future researches may explore new encoder-decoder structure to improve the attack rate. As for our evaluation metrics $SDM$, it still faces the problem of sudden decay when attack rate is to high. Thus, future works may modify and improve the $SDM$ metric to enhance the stability. \section{Conclusions} In this paper, we explain the reasonability and neccessity to add perturbation in the semantic latent space. We also propose a basic iteration-based framework to add perturbation on latent variables on a highly semantical latent space created by a pre-trained VAE module of Stable Diffusion Model. As adversarial examples created in the latent space is hard to follow the $l_p$ norm constraint, we purpose a novel metric named $SDM$ to measure the quality of adversarial examples. To the best of our knowledge, \textbf{$SDM$ is the first metric that is specifically designed to evaluate adversarial attacks in latent space}. We also show that $SDM$ is highly model-independent and dataset-independent, while maintaining a high level of stability. We also test the cross-model transferability of perturbations crafted in latent space and proves their comparable performance with traditional methods. We believe that our $SDM$ metric can help future researches to evaluate the strength of latent spaces attack with a quantitive measure, thereby promoting the researches on latent space attack. Most importantly, we give an illustration of the potential of adversarial attacks in the semantic latent space, and \textbf{we believe that adversarial attacks in such highly semantical space can be a promising research direction in the future}.
\end{document} | arXiv |
Enhancing growth and yield of crops with nutrient-enriched organic fertilizer at wet and dry seasons in ensuring climate-smart agriculture
Taiwo B. Hammed ORCID: orcid.org/0000-0003-0350-25511,
Elizabeth O. Oloruntoba1 &
G. R. E. E. Ana1
International Journal of Recycling of Organic Waste in Agriculture volume 8, pages 81–92 (2019)Cite this article
Rapid nutrient depletion in soils is one of the major problems that affect food production and food security in Sub-Saharan Africa. Studies have linked the growth of food crops with seasonal variation and differences in weather conditions. This study was conducted to assess the effects of various organic fertilizer formulations (OFFs) on the growth and yield of selected crops (Zea mays L.; Glycine max, TX 114 and Dioscorea rotundata Poir) during rainy and dry seasons to ensuring climate-smart agriculture.
The OFFs used were plant-based (PB), animal-based (AB), rock-based (RB), organic mixture (OM-mixture of PB, AB and RB), synthetic/chemical (SC) while ordinary compost without fortification served as control. Effects of OFFs on growth parameters (number of leaves, plant height, stem girth, leaf area, and crop yield) of maize, yam and soybeans were assessed in plot experiments across the two seasons.
The RB gave highest growth performances in maize and soybean plots at both seasons when applied at 2.5 t ha−1. It also improved yam growth when applied at 2.5 t ha−1 (rainy season) and 3.0 t ha−1 (dry season) more than any other fertilizer. The largest yield of maize in the dry season was obtained from plots with PB at 2.0 t ha−1. The AB at 2.0 t ha−1 gave the largest soybean yield in the rainy season.
Organic fertilizers enriched especially with rock-based and plant-based materials have the potential to ameliorate the threat of climate change and seasonal variation to food security.
Avoid the common mistakes
The importance of agriculture to Nigeria's economy is currently at the center stage of national attention as farming is the main source of livelihood for over 70% of households in the country. In 2008, agriculture contributed 42% of the country's GDP (FMARD 2010), which was significantly higher than the 18% derived from petroleum and natural gas production. However, the country's promising agricultural potential has not been realized and low fertilizer use is a major factor contributing to the stagnant agricultural productivity in Nigeria (FMARD 2010). The compost, which is an organic fertilizer and an alternative soil amendment, is not very popular among the farmers because of its slower nutrient release potential and bulkiness. A large quantity of organic fertilizer must be applied to crops for effective results, due to low nutrient composition (Akanbi et al. 2007). Therefore, there is an urgent need to improve the quality of organic-based fertilizers for food security and environmental protection in Nigeria. However, over-dependence on expensive inorganic fertilizers may have serious environmental health hazards, such as water pollution and increased production of greenhouse gases, leading to global climate change (Arisha and Bardisi 1999) and eutrophication of water bodies that can cause algal bloom and production of toxins.
Fertilizers are broadly divided into organic (composed of enriched organic matter—plant or animal), or inorganic (composed of synthetic chemicals and/or minerals) (Heinrich 2000) types. By definition, the term 'fertilizer' refers to a soil amendment that guarantees the minimum percentages of nutrients (at least the minimum percentage of nitrogen, phosphate, and potash). An "organic fertilizer" refers to fertilizer derived from non-synthetic organic materials, including plant and animal by-products, rock powders, seaweed, inoculants, sewage sludge, animal manures, and plant residues (Benton and Jones 2012) produced through the process of drying, cooking, composting (Dadi et al. 2019), chopping, grinding, fermenting (Mario et al. 2019) or other methods (Thanaporn and Nuntavun 2019). Organic and inorganic fertilizers have been used for many centuries (Erisman et al. 2008), whereas chemically synthesized inorganic fertilizers were only widely developed during the industrial revolution. Thus, increased understanding and use of fertilizers were important parts of the pre-industrial British Agricultural Revolution and the industrial green revolution of the 20th century.
Seasonal variation and changes in weather conditions should predict the performance of microbes on bio-mineralization of organic fertilizer (Gaofei et al. 2010; John et al. 2018) and, consequently, agronomic development of crops. Knowledge of different organic fertilizer responses to seasonal variation will help in climate change resilient and climate-smart agriculture. According to Eghball (2002), compost application in excess of crop requirements can last for several years in the soil since only a fraction of nitrogen and other nutrients in compost become available in the first year after application. Previous studies could not identify the most suitable organic fertilizer for specific crops during each planting season (Dadi et al. 2019; Madhumita and Ashalata 2019; Najla et al. 2018). Zerihun and Haile (2017) tested the effect of organic and inorganic fertilizers on the yield of two soybean varieties and found out that the response of soybean varieties to applied fertilizers was significantly affected by rainfall and its geographical distribution at the two seasons. In another study, Mukhtar et al. (2010) analyzed characters of sweet potato varieties grown at varying levels of organic and inorganic fertilizer during the wet seasons of 2004 and 2005. They concluded that application of organic fertilizer increased the yield of sweet potato in both years. Specifically, this study was conducted to test the effects of different organic fertilizer formulations (OFFs) on cereal (maize), legume (soybean) and tuber (yam) crops during the two seasons (dry and rainy seasons) in terms of agronomic performances, residual soil nutrient levels and crop yield after harvesting with the aim of mitigating climate-induced drought and threat to global food security.
Description of the study area
Ibadan has been the center of administration of the old Western Region since the days of the British colonial rule, and parts of the city's ancient protective walls still stand to this day. The principal inhabitants of the city are Yoruba people. Ibadan experiences two seasons—rainy and dry. The rainy season runs from April through October, with temperature ranges from 23.1 to 27.0 °C and rainfall that ranges from 0.0 to 338.8 mm in 2005. The dry season extends from November through March. Ibadan Southwest Local Government Area (ISLGA), where the study took place, was carved out of the defunct Ibadan Municipal Government (IMG) in 1991. The Administrative Headquarter is located at Oluyole Estate. It covers a landmass of 133.5 km2 with a population density of 2401 persons per square kilometer. The 2010 estimated population for the ISLGA was projected at 320,536 people, using a growth rate of 3.2% from 2006 census. Figure 1 describes meteorological data in Ibadan in 2012 when this study was carried out.
Top left—temperature condition in Ibadan in the year 2012; top right—precipitation condition in Ibadan in the year 2012; bottom left—humidity condition in Ibadan in the year 2012; bottom right—wind speed condition in Ibadan in the year 2012
Study design and data collection procedure
A randomized complete block design (RCBD) with three replicates was conducted in plot experiments. The main plots were for three staple crops selected for the study—maize (Zea mays L.), soybean (Glycine max; TX 114) and yam (Dioscorea rotundata Poir) while five different OFFs at three levels of applications—2.0 t ha−1, 2.5 t ha−1 and 3.0 t ha−1 and control plot, applied with ordinary compost without formulation formed subplots. The compost was produced by mixing vegetable waste with cow intestinal waste in ratio 3:1, followed by wetting and turning till maturity (Hammed et al. 2011). All the organic fortifiers were also heaped up, composted and air-dried. The formulation was based on initial chemical analyses of all the fortifiers with the assumption that the OFFs should have minimum primary macronutrient values (P = 2.5% and N = 3.5%), in accordance with the national quality standard (Otu et al. 2014). The OFFs included plant-based (PB), animal/human-based (AB), rock-based (RB) and organic-based (OM) fertilizers. The PB composed of 26.07% neem, 12.50% cottonseed and 61.43% compost; AB comprised 29.40% cow blood, 11.37 bone and 59.23% compost, and Rock-based comprised 26.6% hair, 8.8% phosphate rock and 64.6% compost. Organic-based fertilizer was obtained by mixing PB, AB, and RB in the same proportion. In addition to OFFs, synthetic chemical (SC) fertilizer produced from urea and single super-phosphate (5.56% urea + 8.33% phosphorus + 86.11% compost) was also used as a chemical counterpart in this study. Urea was sourced from AFCOTT Nig. LTD, Lagos, Nigeria; it contained 45% N per 50 kg bag. The SSP was sourced from Fertilizer and Chemical LTD., Kaduna, Nigeria and contained 18% P2O5.
A 30 × 30 m2 of land along Jericho-Alesinloye road, behind the facility, was cleared for farm trials. The land was tilled to prepare maize and soybean beds and ridges for yam. A subplot was 1 × 1 m2 and in the maize and soybean subplots, each of the treatments and control subplots was planted with crops in 3 × 3 factorial design with three replicates. In the yam subplot, 2 × 3 factorial was used. A total of 72 yam tubers (each yam weight 0.55 g) were planted on the yam ridges. Thrash removed from the ground during the clearing was used as mulch. A distance of 45 cm was maintained between the crops planted. Thinning and transplanting of maize from three stands to one was carried out 2 weeks after planting and before fertilizer application. Fertilizer application to maize and soybean was done, using the ring method—3 cm deep and 5 cm away from stem 2 weeks after germination. Fertilizer was applied to yam in ring form under the mulch, a month after planting and at the first appearance of a shoot.
Soil samples for residual effect were collected at depth of 10 cm into the soil from maize, soybean and yam plots after harvesting. Grab samples were collected from each replicate, which were then pooled to form composite samples. Thinning and transplanting of maize from three stands to one was carried out 2 weeks after planting and before fertilizer application. All the measurements were taken at a week interval and the exercise continued for 11 weeks, maturity period for the crops. Standard laboratory methods as described by Motsara and Roy (2008) were followed to appraise residual nutrients and heavy metal concentration of fertilizers in the composite samples, using the Eq. 1. Agronomic data were observed viz. number of leaves, by counting; plant height, leaf area, and stem girth (in centimeters) by metric rule; and crop yield by weighing scale. Maize leaf area was calculated thus: L × B × 0.745 (Agboola 1990). Plant height was measured as the distance from the base of the plant to the height of the first tassel branch and ear height as the distance to the node bearing the upper ear (Badu-Apraku et al. 2010). The entire farm plot experiments were carried out during the two seasons—dry and rainy. The sample mean, confidence interval (at 95%), and percentage composition were computed based on the data obtained from laboratory and field trials. The data were then subjected to analysis of variance (ANOVA) and New Duncan's multiple range test (Duncan 1959) for means separation at 95% level of probability for the growth and yield parameters. Again, the Pearson's correlation coefficient between the rate of fertilizer application and agronomic data was carried out, using SPSS software version 16.
$${\text{Residual nutrient}}\;(\% ) = \frac{A - B}{N} \times 100,$$
where A is nutrients in the soil after harvesting, B is soil background nutrient level before planting and N is the nutrient composition of fertilizers.
Effect of seasonal variation on agronomic parameters of the test crops
Of all OFFs tested on the three crops, RB showed the highest effect during both seasons on all the crops. The effects of different rate of application of RB on agronomic parameters of the test crops are shown in Fig. 2. There were more growths in all the maize and yam parameters in the rainy season than those in the dry season. This could probably be due to the fact that organic fertilizer depends on soil microbes, which are living organisms for bio-mineralization, growth conditions, cultural practices, soil characteristics (Below 2001), and seasonal variation and changes in weather conditions should predict the performance of microbes and consequently, the level of bio-mineralization of organic fertilizer. According to Obiokoro (2005), climate is one of the physical factors that determines the nature of the natural vegetation, the characteristics of the soils, the crops that can be grown, and the type of farming that can be practiced in any region. A related study dealt with the response of Dioscorea alata to NPK–Ca, shows that fertilization is affected by differences in weather conditions in the two growing seasons (Hgaza et al. 2010). Maize had the highest response to the RB fertilizer when applied at the rate of 2.5 t ha−1 at both seasons, and for yam, 2.5 t ha−1 (rainy season) and 3.0 t ha−1 (dry season) were mostly effective. Conversely, soybean showed better agronomic performance in the dry season compared to that in the rainy season, especially when applied at a rate of 2.5 t ha−1. This finding is similar to a previous study conducted by Makinde and Salau (2017) who showed that the application of 2.5 t ha−1 cassava peel compost fortified with either 25 or 50 kg N ha−1 gave optimum Amaranthus growth with optimum residual soil nutrient contents.
Effect of different rate of application of RB on agronomic parameters of the test crops. PH plant height, LA leaf area, SG stem girth, NL no. of leaves
The most important climatic parameters for crop growth and yield are solar radiation, temperature, and rainfall (Ekaputa 2004). Solar radiation determines the thermal characteristics of the environment, namely net radiation, day-length or photoperiod, the air, and soil temperatures (Danjuma 2004). Soil and air temperatures affect the developmental stages more than any other factor (Ayoade 2002). Of the two, soil temperature is a better indicator of energy condition required for crop development and yield than air temperature (Song 2003). In addition, temperature and wind determine the state of soil moisture and the rate of evaporation (Okpemuoghor 2005). In order to determine the optimum microclimatic condition for crops' growth and yield, various soil surface modification systems, such as mulching and ridge construction were used in the plot experiment during this study.
Correlation matrix between agronomic parameters for maize and soybean at first and second cropping
Tables 1, 2, 3 and 4 show the correlation matrices between the agronomic parameters measured in maize and soybean during the rainy and dry seasons. There was a correlation between all agronomic parameters in maize when applied with ordinary compost during the first cropping. Very strong significant correlations were noted between PH and SG in maize plot applied with SC and PB during the first cropping with r = 0.818 and r = 0.694, respectively. In the plots applied with other formulation, no significant correlation existed; a negative correlation was even noted in SG and LA in the plot applied with RB (r = − 0.142) and between SG and PH (r = − 0.206) in OM plot. A similar situation was observed in the second cropping. However, some parameters exhibited a strong relationship in RB and AB plots and some showed a negative relationship in OM plot.
Table 1 Correlation matrix between agronomic parameters for maize at first cropping
Table 2 Correlation matrix between agronomic parameters for maize at second cropping
Table 3 Correlation matrix between agronomic parameters for soybean at first cropping
Table 4 Correlation matrix between agronomic parameters for soybean at second cropping
Correlation also existed in the soybean plots among different parameters during the two planting seasons. The observation in soybean parameter correlation was almost in the reverse direction to what obtained in the maize. Some negative correlations were observed in both rainy (NL Vs SG, r = − 0.008) and dry (SG Vs PH: r = − 0.075 and SG Vs LA: r = − 0.134) seasons in the control plots with ordinary compost. Common to both seasons, LA exhibited a positive and significant relationship with other parameters. The correlation between one parameter and others during the two planting seasons may be a clear indication that a formulation may be chosen for dual or multiple purposes. However, the observation in soybean parameter correlation was almost in the opposite direction to that obtained in the maize. This disparity might be a consequence of the specific nature of OFF application to different crops. In general, a good knowledge of correlation among agronomic parameters is required before selecting the type of OFFs to be applied to crops.
Figures 3, 4 and 5 show fresh yields of maize, soybean and yam when applied with different OFFs at the two seasons, respectively. In maize plots, formulations showed more effect on crop yield during the first cropping (rainy) than that in the second (dry) season. However, the yield in maize during the dry season far outweighed that of the rainy season when PB was applied at 2.0 t ha−1. The formulations mostly showed more effect on the soybean yield during the second (dry) season than that in the first cropping (rainy) season. The PB at rate 2.5 t ha−1 gave the highest yield of soybean at the two seasons while AB at 2.0 t ha−1 was the best OFF for soybean in the rainy season. Yam has only one season and OM (3.0 t ha−1) gave the highest yam tuber yield. This is closely followed by RB (2.0 t ha−1) and AB (2.5 t ha−1). This observation suggests seasonal specificity for OFFs. There are many research studies on the effect of organic fertilizer modification and crop yield. Loeeke et al. (2004) reported that composted manure increased corn grain yield more than fresh manure. Jayaprakash et al. (2003) conducted a field experiment to determine the effect of organic and inorganic fertilizers on the yield and yield attributes of maize under irrigated condition. Significantly highest grain yield was obtained with application of compost at 2 t ha−1, similar to that obtained for RB during the dry season in this study.
Fresh yield (mean fruit weight) of maize at first and second cropping seasons. C control, PB plant-based fertilizer, AB animal/human-based fertilizer, RB rock-based fertilizer, OM organic-based fertilizer (mixture of PB, AB, and RB), SC synthetic chemical fertilizer
Fresh yield (mean pod number) of soybean at first and second cropping seasons. C control, PB plant-based fertilizer, AB animal/human-based fertilizer, RB rock-based fertilizer, OM organic- based fertilizer (mixture of PB, AB, and RB), SC synthetic chemical fertilizer
Fresh yield (mean weight of tuber) of yam. C control, PB plant-based fertilizer, AB animal/human -based fertilizer, RB rock-based fertilizer, OM organic- based fertilizer (mixture of PB, AB, and RB), SC synthetic chemical fertilizer
The tuber yield responses to OFF application in this study are contrary to the findings of Sotomayor-Ramirez et al. (2003). The lack of tuber yield responses in their studies might be due to pest and diseases or the closeness of fertilized and non-fertilized plots as the length of roots can reach 5.5 m (O'Sullivan 2008). Organic manure can serve as an alternative practice to mineral fertilizers (Wong et al. 1999; Naeem et al. 2006) for improving soil structure (Dauda et al. 2008) and microbial biomass (Suresh et al. 2004). Therefore, the utilization of locally produced manures by vegetable production operations may increase crop yields with less use of chemical fertilizer. The use of chemical fertilizers alone to sustain high crop yield has not been quite successful due to the enhancement of soil acidity, nutrient leaching, degradation of soil physical properties, and organic matter status (Nottidge et al. 2005).
Residual potential of chemical contents of organically fortified fertilizers
As shown in Fig. 6, all the formulations showed residual nutrient potentials, though at varying levels. More quantities of OC were retained in SC plots (for yam and maize) and K (for all plots) than those in any other plot applied with other formulations. Control (ordinary compost) retained the highest levels of TN and P in the maize, yam, and soybean plots. Retention of TN and P in the maize, yam, and soybean control plots may be due to the fact that the nutrients were not in the form that could be readily absorbed by the plant roots. Additionally, high residual levels of OC and K found in SC and RB may be due to the low level of bio-mineralization. Among all the OFFs, RB was comparable to SC in terms of OC and K residual levels in maize, soybean and yam plots. In the entire main plots for maize, soybean and yam, all the formulations and control plots showed high percentage residual levels of Mn, especially when applied with SC (maize and yam) and OM (soybean) as shown in Fig. 7. Apart from the Mn, other heavy metals were almost found at zero level. The high values of Mn were due to its initial levels in the soil and compost used for fortification. Generally, organic fertilizer has a binding site to immobilize heavy metals, leading to the highest values exhibited by SC in maize and yam plots. However, the factor that was responsible for the disparity in the soybean plot is yet to be understood. Additionally, the presence of high molecular weight humic acid generally found in soil with well-decomposed organic matter reduces the bioavailability of heavy metal and its toxicity in plant (Inaba and Takenaka 2005), making them to be retained in the soil.
Residual nutrient of fertilizers in maize, soybean and yam plots (%). C control, PB plant-based fertilizer, AB animal/human-based fertilizer, RB rock-based fertilizer, OM organic-based fertilizer (mixture of PB, AB, and RB), SC synthetic chemical fertilizer
Residual heavy metal concentration of fertilizers in maize, soybean and yam plots (%). C control, PB plant-based fertilizer, AB animal/human-based fertilizer, RB rock-based fertilizer, OM organic-based fertilizer (mixture of PB, AB, and RB), SC synthetic chemical fertilizer
Seasonal variation had both negative and positive effects on the agronomic development and yield of the three crops (maize, soybean and yam) applied with different organically fortified fertilizers. Though the negative effects were more paramount especially on maize and yam during the dry season, positive effects noted for some of the fertilizers in either of the seasons should be taken as strategies to alleviate environmental stress on the crops so as to ensure climate-smart agriculture. The rate of application was another key factor that affected performances of the fertilizer on the crops. Rock-based fertilizer (RB) was generally a good promoter of maize and soybean growth at both seasons when applied at 2.5 t ha−1. Additionally, the RB fertilizer was best for yam growth when applied at 2.5 t ha−1 (rainy season) and 3.0 t ha−1 (dry season). The threat of low yield of maize in the dry season could be offset by applying PB at 2.0 t ha−1. The PB at rate 2.5 t ha−1 was good for soybean at the two seasons while AB at 2.0 t ha−1 was the best OFF for soybean in the rainy season. Fertilizers such as synthetic chemicals (SC) and plant-based (PB) had the potential to simultaneously improve two agronomic characteristics, i.e., plant height and leaf area development. Additionally, both SC and RB retained more organic carbon and potassium in the soil they were applied to. Manganese was the only heavy metal noticed in the plot with SC (maize and yam) and Organic mixture (soybean), limiting the threat of heavy metal contamination. Therefore, future studies focusing on the bioavailability of heavy metals and their toxicity to crops are recommended.
Agboola AA (1990) Organic matter and soil fertility management the humid tropics of Africa. Organic matter management and tillage in humid and sub-humid Africa. In: IBSRAM Proceedings: Bangok, Thailand, vol 10, pp 231–243
Akanbi WB, Adebayo TA, Togun OA, Adeyeye AS, Olaniran OA (2007) The use of compost extract as foliar spray nutrient source and botanical insecticide in Telfairia occidentalis. World J Agric Sci 3(5):642–652
Arisha HM, Bardisi A (1999) Effect of mineral fertilizers and organic fertilizers on growth, yield and quality of potato under sandy soil conditions. Zagazig J Agric Res 26:391–405
Ayoade JC (2002) Agro climatology. Vantage Publishers, Ibadan, pp 129–137
Badu-Apraku B, Menkir A, Ajala SO, Akinwale RO, Oyekunle M, Obeng-Antwi K (2010) Performance of tropical early-maturing maize cultivars in multiple stress environments. Can J Plant Sci 90:1–22. https://doi.org/10.1017/S0021859611000761
Below EF (2001) Nitrogen metabolism and crop productivity. In: Pessarakli M (ed) Handbook of plant and crop physiology. University of Arisona, Tucson, pp 345–401
Benton J, Jones J (2012) Inorganic chemical fertilizers and their properties in plant nutrition and soil fertility manual, 2nd edn. CRC Press, Boca Raton, pp 20–23
Dadi D, Daba G, Beyene A, Luis P, Van der Bruggen B (2019) Composting and co-composting of coffee husk and pulp with source-separated municipal solid waste: a breakthrough in valorization of coffee waste. Int J Recycl Org Waste Agric. https://doi.org/10.1007/s40093-019-0256-8
Danjuma BO (2004) Comparison of soil physico-chemical characteristics of different tillage types in North-eastern Nigeria. J Agric Environ 2(1):1–16
Dauda SN, Ajayi FA, Ndor E (2008) Growth and yield of water melon (Citrullus lanatus) as affected by poultry manure application. J Agric Soc Sci 4:121–124
Duncan RB (1959) Multiple range and multiple F-tests. Biomedics 2:1–42
Eghball B (2002) Soil properties as influenced by phosphorus- and nitrogen –based manure and compost applications. Agron J 94:128–135
Ekaputa UN (2004) Agricultural systems. UNIK Publishers, Port Harcourt, pp 126–134
Erisman JW, Sutton MA, Galloway J, Klimont Z, Winiwarter W (2008) How a century of ammonia synthesis changed the world. Nat Geosci 1(10):636E. https://doi.org/10.1038/ngeo325
Federal Ministry of Agriculture and Rural Development (FMARD). 2010. Blueprint on agriculture and rural development. A Presentation to the National Economic Management Team by the Honourable Minister of Agriculture and Rural Development, Abuja, Nigeria. http://soilsnigeria.net/Publications/Others/FertilizerPolicyforNigeria.pdf. Accessed 13 Jan 2018
Gaofei G, Zhaojun L, Fenliang F, Guixin C, Zhenan H, Yongchao L, Gaofei G, Zhaojun L, Fenliang F, Guixin C, Zhenan H, Yongchao L (2010) Soil biological activity and their seasonal variations in response to long-term application of organic and inorganic fertilizers. Plant Soil 326:31–44. https://doi.org/10.1007/s11104-009-0186-8
Hammed TB, Soyingbe AA, Adewole DO (2011) An abattoir waste water management through composting: a case study of Alesinloye waste recycling complex. Int J Int Soc Sci 6(2):67–78. https://doi.org/10.18848/1833-1882/CGP/v06i02/52024
Heinrich W (2000) Scherer "fertilizers" in Ullmann's encyclopedia of industrial chemistry. Wiley-VCH, Weinheim, pp 122–125
Hgaza VK, Diby LN, Assa A, Ake S (2010) How fertilization affects yam (Dioscorea alata L.) growth and tuber yield across the years. African J Plant Sci 4.3:053–060. https://doi.org/10.2174/1874294701105010014
Inaba S, Takenaka C (2005) Effects of dissolved organic matter on toxicity and bioavailability of copper for lettuce sprouts. Environ Int 31:603–608. https://doi.org/10.1016/j.envint.2004.10.017
Jayaprakash TC, Nagalikar VP, Pujari BT, Setty RA (2003) Effect of organic and inorganic on yield and yield attributes of maize under irrigation. Karnataka J Agric Sci 16:451–453
John F, Sean S, Bin Z, Tingting A, Jiubo P, Jie Z, Jingkuan W (2018) How soil bacterial communities with seasonal variation respond differently to long-term fertilization and plastic film mulching. Pol J Environ Stud 27(4):1483–1495. https://doi.org/10.15244/pjoes/76889
Loeeke TD, Liebman M, Cambbardella CA, Richard TL (2004) Corn response to composting and time of application of solid swine manure. Agron J 96:214–223
Madhumita G, Ashalata D (2019) Assessment of crop growth, soil properties and crop yield in an upland acidic soil with inorganic fertilizer blended with organic amendments in summer rice cropping seasons. Int J Recycl Org Waste Agric. https://doi.org/10.1007/s40093-019-0252-z
Makinde EA, Salau AW (2017) Fortified cassava peel compost amendment for Amaranthus: influence on plant growth, nutrients uptake and on soil nutrient changes. J Plant Nutr 40(5):645–655. https://doi.org/10.1080/01904167.2016.1245328
Mario AHC, Catalino JLC, Nereida RO, Joel VV, Ariadna LG, Gustavo LR (2019) Nutrient content of fermented fertilizers and its efficacy in combination with hydrogel in Zea mays L. Int J Recycl Org Waste Agric. https://doi.org/10.1007/s40093-019-0248-8
Motsara MR, Roy RN (2008) Guide to laboratory establishment for plant nutrient analysis, 19th edn. FAO fertilizer and plant nutrition bulletin, Rome, pp 42–88
Mukhtar AA, Tanimu B, Arunah UL, Babaji BA (2010) Evaluation of the agronomic characters of sweet potato varieties grown at varying levels of organic and inorganic fertilizer. World J Agric Sci 6(4):370–373
Naeem M, Iqbal J, Bakhsh MAA (2006) Comparative study of inorganic fertilizers and organic manures on yield and yield components of mungbean (Vigna radiat L.). J Agric Soc Sci 2:227–229. https://doi.org/10.5897/AJAR2017.12675
Najla D, Safaben A, Amin Z, Faouzi H, Bouthaina A, Mohandes D (2018) The challenge of using date branch waste as a peat substitute in container nursery production of lettuce (Lactuca sativa L.). Int J Recycl Org Waste Agric. https://doi.org/10.1007/s40093-018-0221-y
Nottidge DO, Ojeniyi SO, Asawalam DO (2005) Comparative effect of plant residues and NPK fertilizer on nutrient status and yield of maize (Zea mays L.) in a humid ultisol. Niger J Soil Sci 15:1–8. https://doi.org/10.4314/njss.v15i1.37439
Obiokoro OG (2005) Agro meteorology. Dunkwu Publishers, Onitsha, pp 24–30
Okpemuoghor ER (2005) Physical Environment. Dosumu Press, Lagos, pp 152–176
O'Sullivan JN (2008) Root distribution of yam (Dioscorea alata) determined by strontium tacer. Exp Agric 44:223–233. https://doi.org/10.1017/S0014479708006339
Otu WI, Idiong CI, Nsikan EB, Ekaette SU (2014) Food security and productivity of urban food crop farming households in Southern Nigeria. Agric Sci 2(3):01–12
Song AB (2003) The impact of modified physical environment on the yield of yellow and water yams in the middle belt of Nigeria. J Agric Environ 1(2):14–22
Sotomayor-Ramirez D, Gonzales-Velez A, Roman-Paoli E (2003) Yam (Dioscorea spp) response to fertilization in soils of semiarid southern coast of Puerto Rico. J Agric Univ P R 87(3):91–103
Suresh KD, Sneh G, Krishn KK, Mool CM (2004) Microbial biomass carbon and microbial activities of soils receiving chemical fertilizers and organic amendments. Arch Agron Soil Sci 50:641–647
Thanaporn P, Nuntavun R (2019) Liquid organic fertilizer production for growing vegetables under hydroponic condition. Int J Recycl Org Waste Agric. https://doi.org/10.1007/s40093-019-0252-z
Wong JWC, Ma KK, Fang KM, Cheung C (1999) Utilization of manure compost for organic farming in Hong Kong. Bio-resour Technol 67:43–46
Zerihun A, Haile D (2017) The effect of organic and inorganic fertilizers on the yield of two contrasting soybean varieties and residual nutrient effects on a subsequent finger millet crop. Agronomy 7(2):42–55. https://doi.org/10.3390/agronomy7020042
Department of Environmental Health Sciences, Faculty of Public Health, College of Medicine, University of Ibadan, P. O. Box 20593 UI HO, Ibadan, Nigeria
Taiwo B. Hammed, Elizabeth O. Oloruntoba & G. R. E. E. Ana
Taiwo B. Hammed
Elizabeth O. Oloruntoba
G. R. E. E. Ana
Correspondence to Taiwo B. Hammed.
Hammed, T.B., Oloruntoba, E.O. & Ana, G.R.E.E. Enhancing growth and yield of crops with nutrient-enriched organic fertilizer at wet and dry seasons in ensuring climate-smart agriculture. Int J Recycl Org Waste Agricult 8 (Suppl 1), 81–92 (2019). https://doi.org/10.1007/s40093-019-0274-6
Issue Date: December 2019
Organic fertilizer formulations
Natural fortifiers
Growth parameters
Seasonal variation | CommonCrawl |
\begin{document}
\author[Cai]{Mingzhong Cai} \address[Cai]{Hyperimmune Books\\ Suwanee, GA 30024\\ USA} \email{\href{mailto:[email protected]}{[email protected]}}
\author[Yiqun Liu]{Yiqun Liu} \address[Yiqun Liu]{Office of the President\\ National University of Singapore\\ Singapore 119077\\ SINGAPORE} \email{\href{mailto:[email protected]}{[email protected]}}
\author[Yong Liu]{Yong Liu} \address[Yong Liu]{School of Information Engineering\\ Nanjing Xiaozhuang University\\ CHINA} \email{\href{mailto:[email protected]}{[email protected]}}
\author[Peng]{Cheng Peng} \address[Peng]{Department of Mathematics\\ Hebei University of Technology\\ CHINA} \email{\href{mailto:[email protected]}{[email protected]}}
\author[Yang]{Yue Yang} \address[Yang]{Department of Mathematics\\ National University of Singapore\\ Singapore 119076\\ SINGAPORE} \email[Yang]{\href{mailto:[email protected]}{[email protected]}}
\subjclass[2020]{03D25}
\keywords{r.e.~degrees, minimal pair}
\thanks{Peng’s research was partially supported by NSF of China No. 12271264. Yang’s research was partially supported by NUS grant WBS: R-146-000-337-114}
\begin{abstract} Two nonzero recursively enumerable (r.e.) degrees $\mathbold{a}$ and $\mathbold{b}$ form a strong minimal pair if $\mathbold{a}\wedge \mathbold{b}=\mathbf{0}$ and $\mathbold{b}\vee \mathbold{x}\geq \mathbold{a}$ for any nonzero r.e.~degree $\mathbold{x}\leq \mathbold{a}$. We prove that there is no strong minimal pair in the r.e.~degrees. Our construction goes beyond the usual \(\mathbf{0}'''\)\nbd{}priority arguments and we give some evidence to show that it needs \(\mathbf{0}^{(4)}\)\nbd{}priority arguments. \end{abstract}
\title{On the nonexistence of a strong minimal pair} \maketitle
\section{Introduction}
In paper~\cite{Barmpalias.Cai.ea:2015}, Barmpalias, Cai, Lempp and Slaman claimed the existence of a strong minimal pair. Two nonzero recursively enumerable (r.e.) degrees $\mathbold{a}$ and $\mathbold{b}$ form a \emph{strong minimal pair} if $\mathbold{a}\wedge \mathbold{b}=\mathbf{0}$ and for any nonzero r.e.~degree $\mathbold{x}\leq \mathbold{a}$, $\mathbold{b}\vee \mathbold{x}\geq \mathbold{a}$. The notion of strong minimal pairs can also be viewed as a strengthening of the so-called ``Slaman triples''. Recall that three r.e.~degrees $\mathbold{a}, \mathbold{b}$ and $\mathbold{c}$ form a \emph{Slaman triple} if $\mathbold{a} \neq \mathbf{0}$, $\mathbold{c}\nleq \mathbold{b}$ and for any nonzero r.e.~degree $\mathbold{x}\leq \mathbold{a}$, $\mathbold{b}\vee \mathbold{x}\geq \mathbold{c}$.
In a Slaman triple formed by \(\mathbold{a}\), \(\mathbold{b}\) and \(\mathbold{c}\), it is clear that \(\mathbold{a}\) and \(\mathbold{b}\) form a minimal pair; if \(\mathbold{a}\) and \(\mathbold{c}\) coincide, then \(\mathbold{a}\) and \(\mathbold{b}\) will form a strong minimal pair.
We refer the readers to~\cite{Barmpalias.Cai.ea:2015} for the significance of this study, as well as for its ``long and twisted'' history. Their paper was devoted to show the existence of a strong minimal pair, where they gave a detailed illustration of how to combine two sets of requirements before proceeding to the long and complicated full construction. They also raised the open question whether there exists a ``two-sided'' strong minimal pair---a strong minimal pair as above but with the extra clause ``and for any nonzero r.e.~degree $\mathbold{y}\leq \mathbold{b}$, $\mathbold{a}\vee \mathbold{y}\geq \mathbold{b}$''?
While trying to settle the existence of a two-sided strong minimal pair, we encountered a serious difficulty of dealing with three sets of requirements. It turns out that their original construction cannot overcome this difficulty either. Eventually, there is another twist in the history of the problem. By employing three families of r.e.~sets, we are able to establish the \emph{nonexistence} of a strong minimal pair:
\begin{theorem} [Main]\label{thm:main}
For any r.e.~degrees $\mathbold{a}$ and $\mathbold{b}$, either $\mathbold{a}\leq \mathbold{b}$ or
there exists an r.e.~degree $\mathbold{x}\leq \mathbold{a}$ such that $\mathbold{x}\neq \mathbf{0}$
and $\mathbold{x}\vee \mathbold{b}\not\geq \mathbold{a}$. \end{theorem}
As an immediate consequence, the degrees $\mathbold{a}$ and $\mathbold{c}$ in a Slaman triple cannot coincide; in fact, $\mathbold{a}$ and $\mathbold{b}\vee\mathbold{c}$ also form a minimal pair. To see this, suppose toward a contradiction that there exists some nonzero r.e.~degree $\mathbold{x}$ below both $\mathbold{a}$ and $\mathbold{b}\vee\mathbold{c}$. Consider any nonzero r.e.~degree $\mathbold{w}$ below $\mathbold{x}$, we would have $\mathbold{b}\vee \mathbold{w}\geq \mathbold{c}$ and therefore $\mathbold{b}\vee \mathbold{w}\ge\mathbold{b}\vee \mathbold{c}\ge \mathbold{x}$, which implies that $\mathbold{x}$ and $\mathbold{b}$ form a strong minimal pair, contradicting our main theorem.
This fact echoes the gap-cogap construction used in building a Slaman triple (see, for example, Shore and Slaman \cite{Shore.Slaman:1993}), where elements are enumerated into $A$ and $B\cup C$ at alternating stages.
Besides the strong minimal pair problem itself, we are also interested in the techniques involved in the solution.
In the paper~\cite{Barmpalias.Cai.ea:2015}, a novel $c$\nbd{}outcome was introduced to handle some of the conflicts in a way that goes beyond $\mathbf{0}'''$-priority method. In this paper, we use an $(\omega+1)$\nbd{}branching tree to organize our construction, furthermore, each $(\omega+1)$\nbd{}st branch can be considered as a gateway to a parallel $\Pi_3$\nbd{}world.
A similar design has been used before, notably by Shore in~\cite{Shore:1988}. However, the conflicts between our strategies seem more severe than those in Shore's. For instance, for certain two nodes on the priority tree, one inside a $\Pi_3$\nbd{}world, the other outside, we cannot determine \emph{a priori} which node has higher priority; we have to assign priority dynamically: whichever acts first will change the environment of the other. Much more care has to be applied in showing the existence of the true path. This brings up the question of what counts as a typical $\mathbf{0}^{(4)}$\nbd{}priority argument. A $\mathbf{0}'''$\nbd{}priority argument can now be routinely presented in a framework where finite injuries happen on the true path. Our construction seems to suggest that handling the interactions inside and outside the $\Pi_3$\nbd{}world must be a feature in any framework for a $\mathbf{0}^{(4)}$\nbd{}priority argument. We hope that our work will bring us closer to a canonical framework for the $\mathbf{0}^{(4)}$\nbd{}priority argument.
\section{Toward the stage by stage construction} \subsection{Preliminaries and conventions}
A \emph{Turing functional} \(\Gamma\) is an r.e.\ set of of triples, called the \emph{axioms}, \((\sigma,n,i)\in 2^{<\omega}\times \omega\times\{0,1\}\) such that if both \((\sigma,n,i)\), \((\tau,n,j)\in \Gamma\) and \(\sigma\preceq \tau\), then \(i=j\). We write \(\Gamma(X;n)\downarrow = i\) if \(\exists l\ (X\res l,n,i)\in \Gamma\); in this case, we also define the \emph{use function} \(\gamma(X;n)\) to be the least such \(l\) plus 1. We write \(\Gamma(X;n)\uparrow\) if \(\forall l,i (X\res l, n, i)\notin \Gamma\). By assuming that Turing functionals are computing initial segments, we may further assume that the domain of a given Turing function is downward closed, and its use function is nondecreasing with respect to its arguments. For two subsets \(A\) and \(B\) of natural numbers, we write \(A\le_T B\) if there exists a Turing functional \(\Phi\) such that for each \(n\), \(A(n)=\Phi(B;n)\). Following the assumption above, whenever we write \(\Phi(B;n)=A(n)\), we mean \(\Phi(B;i)=A(i)\) for all \(i\le n\). We will use \([s]\) to indicate that all calculations are restricted to a particular stage~\(s\), and in particular, all numerical parameters occurred are \(\leq s\).
For \(\Gamma(X)=A\), the length of agreement at stage~\(s\) is defined as \[
\ell(s)=\{n\le s\mid \forall k\le n, \Gamma(X;k)[s]=A(k)[s]\}. \] We say that \(s\) is an \emph{expansionary stage} if \(\ell(s)>\ell(s^*)\) for all $s^*<s$.
We define \(\mathsf{same}(X,k,s,t)\) if and only if \[ (\forall n< k)(\forall s') [s\le s'\le t\rightarrow X(n)[s]=X(n)[s']]. \] \(\mathsf{same}(X,k,s,t)\) says that the set \(X\) up to the first \(k\) digits does not change from stage \(s\) to \(t\).
\subsection{Requirements}\label{sec:req} Given r.e.\ sets \(A\) and \(B\) with \(A\nleq_T B\), we build an r.e.\ set \(X\) satisfying the following requirements:
The first one is the permitting requirement: \begin{itemize} \item \(\mathop{\mathrm{Permit}}(X)\): \(X\leq_T A\). \end{itemize}
Fix an effective enumeration of Turing functionals \((\Gamma_e,\Delta_e)_{e\in \omega}\), we have the diagonalization requirements, here the letters $G$ and $D$ come from $\Gamma$ and $\Delta$ respectively: \begin{itemize} \item \(G_e(X)\): \(\Gamma_e(BX)\neq A\); and \item \(D_e(X)\): \(\Delta_e\neq X\). \end{itemize}
\(G_e(X)\) has three possible outcomes: Case (1), there is an \(n\) such that \(\Gamma(BX;n)\) never agrees with \(A(n)\); Case (2), there is a divergent point \(n\) such that \(\gamma(BX;n)[s]\to \infty\) as \(s\to \infty\); and Case (3), \(\Gamma_e(BX)=A\).
The strategy of \(G_e(X)\) is to check if the length of agreement between \(\Gamma(BX)\) and \(A\) goes to infinity. If the answer is no, then we win by the \(\Sigma_2\)\nbd{}outcome which is Case (1). If the answer is yes, we have a \(\Pi_2\)\nbd{}outcome indicating either Cases (2) or (3). Then we try to build a Turing functional \(\Omega\) so that \[
\Gamma_e(BX)=A \Rightarrow \Omega(B)=A. \] In a usual priority argument, if \(\Omega(B)=A\) turns out to be partial, then we shall exhibit a divergent point of \(\Gamma_e(BX)\) as in Case (2); however, in our construction we might also have the possibility of Case (3) and we shall begin to work on \emph{another} set instead of \(X\)---we take a nonuniform approach. In fact, instead of constructing a single r.e.\ set \(X\), we construct three types of candidates whose types are denoted by \(\bU\), \(\bV\) and \(\bW\). There will be a unique candidate for a \(\bU\)\nbd{}set, which will be denoted by \(U\). There will be countably many candidates each for \(\bV\)\nbd{}sets and \(\bW\)\nbd{}sets and will be denoted by \(V_\alpha\) or \(W_{\alpha,\beta}\) where the subscripts refer to their ``parent nodes''. We will argue that at least one of these candidates satisfies all requirements.
\subsection{Priority tree} Fix a list of requirements as \[
G_0(X)<D_0(X)<G_1(X)<D_1(X)<\cdots, \] where \(X\) stands for a candidate that we build. Depending on whether \(X\) is a \(\bU\)\nbd{}set, \(\bV\)\nbd{}set or \(\bW\)\nbd{}set, the strategies will be different. Besides the nodes that work directly for our requirements, we also have other auxiliary nodes.
The priority tree \(\cT\) is defined recursively as follows. The root of \(\cT\) is assigned to \(\mathop{\mathrm{Permit}}(U)\). A node \(\alpha\) assigned to \(*\) will be called a \(*\)\nbd{}node. A node \(\alpha\) is said to be \emph{in the \(\Sigma_3\)\nbd{}world} if it is not declared to be in a \emph{\(\Pi_3\)\nbd{}world} (see~\ref{it:pt C} below). For references, the first few nodes of the priority tree are given in Figure~\ref{fig:priority tree}, where we also omit some of the outcomes, particularly those with label ``\(0\)'', to save space. \begin{figure}
\caption{the priority tree}
\label{fig:priority tree}
\end{figure}
Suppose that \(\alpha\) has been assigned. \begin{enumerate}[label=(T\arabic*)]
\item Suppose that \(\alpha\) is a \(\mathop{\mathrm{Permit}}(X)\)\nbd{}node, then \(\alpha\) has a unique outgoing edge labelled \(0\) and \(\alpha\concat 0\) is assigned to \(G_0(X)\). \end{enumerate} \begin{remark*}
A \(\mathop{\mathrm{Permit}}(X)\)\nbd{}node is the place where we introduce a new candidate \(X\) and is the \emph{only} place where we enumerate numbers into \(X\).
The permitting is done in a delayed fashion, after a number gets permitted by $A$, its fate will be determined only by the next $\mathop{\mathrm{Permit}}(X)$-stage.
After introducing the set $X$, we work on \(G_0(X)\) which is the first one in its requirement list. \end{remark*}
\begin{enumerate}[resume*]
\item Suppose that \(\alpha\) is a \(G_e(X)\)\nbd{}node, then it has two outgoing edges labelled \(\infty<_L 0\), here \(<_L\) means ``to the left of''.
\begin{enumerate}[label=(T\arabic{enumi}.\arabic*)]
\item\label{it:pt GU} If \(X=U\), then \(\alpha\concat \infty\) is assigned to \(\mathop{\mathrm{Permit}}(V_\alpha)\) and \(\alpha\concat 0\) to \(D_e(U)\).
\item\label{it:pt GV} If \(X=V_{\alpha_0}\) for some \(\alpha_0\subseteq \alpha\), then \(\alpha\concat \infty\) is assigned to \(\mathop{\mathrm{Permit}}(W_{\alpha_0,\alpha})\) and \(\alpha\concat 0\) to \(D_e(V_{\alpha_0})\).
\item\label{it:pt GW} If \(X=W_{\alpha_0,\alpha_1}\) for some \(\alpha_0\subseteq\alpha_1\subseteq \alpha\),
then \(\alpha\concat \infty\) is assigned to \(R(\alpha_0,\alpha_1,\alpha)\) if \(\alpha\) is also in the \(\Sigma_3\)\nbd{}world, and to \(R^-(\alpha_0,\alpha_1,\alpha)\) if \(\alpha\) is in a \(\Pi_3\)\nbd{}world; \(\alpha\concat 0\) is assigned to \(D_e(W_{\alpha_0,\alpha_1})\).
\end{enumerate} \end{enumerate}
\begin{remark*}
A \(G_e(X)\)\nbd{}node \(\alpha\) checks if the length of agreement of \(\Gamma_e(BX)=A\) goes to infinity:
if not, it has the \(\Sigma_2\)\nbd{}outcome \(0\) and \(G_e(X)\) is satisfied easily, and we move on to the next requirement \(D_e(X)\).
One may view \(\alpha\concat 0\) as a degenerated case, as the nodes extending \(\alpha\concat 0\) can safely ignore \(\alpha\).
In the case that \(G_e(X)\)\nbd{}node has the \(\Pi_2\)\nbd{}outcome \(\infty\), we first introduce more sets to form a three element group $(U, V_{\alpha_0}, W_{\alpha_0,\alpha_1})$.
Only after we have \(X=W_{\alpha_0,\alpha_1}\) for some \(\alpha_0\) and \(\alpha_1\) with \(\alpha_0\concat \infty\subseteq \alpha_1\concat \infty\subseteq \alpha\concat \infty\),
we introduce an \(R(\alpha_0,\alpha_1,\alpha)\)\nbd{}node to detect a divergent point for
$\Gamma$\nbd{}functionals at either \(\alpha_0\), \(\alpha_1\), or \(\alpha\). An \(R^-(\alpha_0,\alpha_1,\alpha)\)\nbd{}node is a degenerate case of $R$, where we detect a divergent point for $\Gamma$\nbd{}functionals at either \(\alpha_1\) or \(\alpha\).
As one shall see, an $R^-$\nbd{}node necessarily lives in a $\Pi_3$\nbd{}world where the set $U$ is no longer considered. \end{remark*}
\begin{enumerate}[resume*]
\item Suppose that \(\alpha\) is a \(D_e(X)\)\nbd{}node, then \(\alpha\) has three outgoing edges labelled \(d<_L \infty<_L 0\).
\begin{enumerate}[label=(T\arabic{enumi}.\arabic*)]
\item\label{it:pt D-P} If \(X=U\), then \(\alpha\concat \infty\) is assigned to \(P(\eU,\eW)\) where \(E_\bU=\{\alpha\}\) and \(E_\bW\) is a finite set of nodes on the tree and will be defined later;
\(\alpha\concat 0\) and \(\alpha\concat d\) are both assigned to \(G_{e+1}(U)\).
\item\label{it:pt D-Q} If \(X=V_{\alpha_0}\), then \(\alpha\concat \infty\) is assigned to \(Q(\eV,\eW)\) where \(E_\bV=\{\alpha\}\) and \(E_\bW\) is a finite set of nodes on the tree and will be defined later;
\(\alpha\concat 0\) and \(\alpha\concat d\) are both assigned to \(G_{e+1}(V_{\alpha_0})\).
\item\label{it:pt D-CS} If \(X=W_{\alpha_0,\alpha_1}\) and \(\alpha\) is in the \(\Sigma_3\)\nbd{}world, then \(\alpha\concat\infty\) is assigned to \(C(\alpha_0,\alpha_1,\alpha)\);
if \(X=W_{\alpha_0,\alpha_1}\) and \(\alpha\) is in a \(\Pi_3\)\nbd{}world, then \(\alpha\concat\infty\) is assigned to \(S(\alpha_0,\alpha_1,\alpha)\).
In both cases, \(\alpha\concat 0\) and \(\alpha\concat d\) are assigned to \(G_{e+1}(W_{\alpha_0,\alpha_1})\).
\end{enumerate} \end{enumerate} \begin{remark*}
A \(D_e(X)\)\nbd{}node \(\alpha\) first checks if it has some \(n\) such that \(\Delta_e(n)=0\) but \(X(n)=1\).
If so, \(\alpha\) has the \(\Sigma_1\)\nbd{}outcome \(d\) and \(D_e(X)\) is satisfied in the easiest way.
If not, \(\alpha\) checks if the length of agreement between \(\Delta_e\) and \(X\) goes to infinity.
The \(\Sigma_2\)\nbd{}outcome \(0\) indicates that the length of agreement stops increasing from some stage onward; the \(\Pi_2\)\nbd{}outcome \(\infty\) indicates the other case.
Below the \(\Pi_2\)\nbd{}outcome, we do not try to diagonalize against \(\Delta_e\) immediately if \(X=W_{\alpha_0,\alpha_1}\).
The reason is that $D_e(W)$ usually showed up before some $D(U)$ or $D(V)$ nodes appear.
We are ready to attack only when we teamed up this $D_e(W)$ with either \(D(U)\) or \(D(V_{\alpha_0})\) for some \(\alpha_0\subseteq \alpha\).
The decision of choosing which set to attack and why it works are the core of our construction and will be discussed later. \end{remark*}
\begin{enumerate}[resume*]
\item Suppose that \(\alpha\) is an \(R(\alpha_0,\alpha_1,\alpha_2)\)- or an \(R^-(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, where \(\alpha_0\), \(\alpha_1\) and \(\alpha_2\) are \(G_{e_0}(U)\)-, \(G_{e_1}(V_{\alpha_0})\)- and \(G_{e_2}(W_{\alpha_0,\alpha_1})\)\nbd{}nodes for some \(e_0\), \(e_1\) and \(e_2\), respectively.
Let \(W=W_{\alpha_0,\alpha_1}\) and \(V=V_{\alpha_0}\).
\begin{enumerate}[label=(T\arabic{enumi}.\arabic*)]
\item\label{it:pt R} If \(\alpha\) is an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, then \(\alpha\) has infinitely many outgoing edges, labelled and ordered by
\[
(0,W) <_L (0,U) <_L (0,V) <_L \cdots <_L (n,W) <_L (n,U) <_L (n,V) <_L \cdots.
\]
Furthermore, for each \(n\in \omega\), \(\alpha\concat (n,W)\) is assigned to \(D_{e_2}(W)\), \(\alpha\concat (n,U)\) assigned to \(D_{e_0}(U)\) and \(\alpha\concat (n,V)\) assigned to \(D_{e_1}(V)\).
\item\label{it:pt R-} If \(\alpha\) is an \(R^-(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, then \(\alpha\) has infinitely many outgoing edges, labelled and ordered by
\[
(0,W) <_L (0,V) <_L \cdots <_L (n,W) <_L (n,V) <_L \cdots.
\]
Furthermore, for each \(n\in \omega\), \(\alpha\concat (n,W)\) is assigned to \(D_{e_2}(W)\) and \(\alpha\concat (n,V)\) assigned to \(D_{e_1}(V)\).
\end{enumerate}
\((n,W)\) is said to be a \emph{divergent outcome} for \(\alpha_2\); \((n,V)\) a divergent outcome for \(\alpha_1\);
and \((n,U)\) a divergent outcome for \(\alpha_0\) if \(\alpha\) is also an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. \end{enumerate} \begin{remark*}
A divergent outcome \((n,W)\) for \(\alpha_2\) indicates that we have some number \(m\), not necessarily \(n\) (see Section~\ref{sec:parameter} for the discussion on edge parameters),
such that \(\Gamma_{e_2}(BW;m)\uparrow\) and \(G_{e_2}(W)\) is therefore satisfied.
Extending \((n,W)\)\nbd{}outcome, we shall begin to work for the next requirement \(D_{e_2}(W)\) in the requirement list.
An \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node is necessarily in the \(\Sigma_3\)\nbd{}world while \(R^-(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node is necessarily in a \(\Pi_3\)\nbd{}world.
\end{remark*}
\begin{enumerate}[resume*]
\item\label{it:pt C} Suppose that \(\alpha\) is a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, then \(\alpha\) has infinitely many outgoing edges, labelled and ordered with order type \(\omega+1\) by
\[
(0,U) <_L \cdots <_L (n,U) <_L \cdots <_L \omega.
\]
Any extension of \(\alpha\concat \omega\) is declared to be \emph{in the \(\Pi_3\)\nbd{}world of \(\alpha\)}.
Furthermore, \((n,U)\) is said to be a divergent outcome for \(\alpha_0\) and \(\alpha\concat (n,U)\) is assigned to \(D_{e_0}(U)\), where $e_0$ comes from the index of the \(G_{e_0}(U)\)\nbd{}node \(\alpha_0\).
\(\alpha\concat \omega\) is assigned to \(S(\alpha_0,\alpha_1,\alpha_2)\). \end{enumerate}
\begin{remark*}
At a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node \(\alpha\) we try to detect a divergent point of \(\Gamma_{e_0}(BU)\).
If we find one, we continue to work on the next requirement \(D_{e_0}(U)\); otherwise, we shall not work for the candidate \(U\) anymore;
we switch to work for the candidate \(V_{\alpha_0}\) (together with some sets from $\bW$) by assigning \(\alpha\concat \omega\) to \(S(\alpha_0,\alpha_1,\alpha_2)\).
Note that by~\ref{it:pt D-CS} there will be no more \(C\)\nbd{}node extending \(\alpha\concat \omega\) and therefore there will be no nested \(\Pi_3\)\nbd{}worlds. \end{remark*}
\begin{enumerate}[resume*]
\item\label{it:pt S} Suppose that \(\alpha\) is an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, where \(\alpha_1\) is a \(G_{e_1}(V_{\alpha_0})\)\nbd{}node for some \(e_1\). Let \(V=V_{\alpha_0}\).
Then \(\alpha\) has infinitely many outgoing edges labelled and ordered by
\[
(0,V) <_L (1,V) <_L \cdots <_L (n,V) <_L \cdots.
\]
Furthermore, \((n,V)\) is said to be a divergent outcome for \(\alpha_1\) and for each \(n\in \omega\), \(\alpha\concat (n,V)\) is assigned to \(D_{e_1}(V)\). \end{enumerate} \begin{remark*}
An \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node \(\alpha\) is necessarily in the \(\Pi_3\)\nbd{}world for some \(C\)\nbd{}node \(\beta\) with \(\beta\concat \omega\subseteq \alpha\).
At \(\alpha\), we try to detect a divergent point of \(\Gamma_{e_1}(BV)\). If we find one, we shall work on the next requirement \(D_{e_1}(V)\).
If the detection fails, we shall conclude that \(A\le_T B\) toward a contradiction. \end{remark*}
\begin{enumerate}[resume*]
\item\label{it:pt P} Suppose that \(\alpha\) is a \(P(\eU,\eW)\)\nbd{}node. Then \(\alpha\) is a terminal node on the priority tree. We let \(E_\bU=\{\alpha^-\}\) as has been defined in~\ref{it:pt D-P}.
For each \(\beta\subseteq \alpha\) where \(\beta\) is a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, we enumerate \(\alpha_2\) into \(\eW\).
\item\label{it:pt Q} Suppose that \(\alpha\) is a \(Q(\eV,\eW)\)\nbd{}node. Then \(\alpha\) is a terminal node on the priority tree. We let \(E_\bV=\{\alpha^-\}\) as has been defined in~\ref{it:pt D-Q}. For each \(\beta\subseteq \alpha\) where \(\beta\) is an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, we enumerate \(\alpha_2\) into \(\eW\).
\end{enumerate} \begin{remark*}
A \(P(\eU,\eW)\)\nbd{}node \(\alpha\) will produce many potential diagonalizing witnesses for \(\beta\in E_\bU\) or \(\xi\in E_\bW\), each of whom is a \(D\)\nbd{}node.
And \(\alpha\) bets that \(A\) will permit some of the diagonalizing witnesses eventually.
If \(A\) never permits, then \(\alpha\) will demonstrate \(A\) to be recursive toward a contradiction.
Otherwise, one of the \(D\)\nbd{}node reaches a \(\Sigma_1\)\nbd{}outcome and is satisfied forever. Consequently, the $P$\nbd{}node is switched off true path.
Similar discussion applies to a \(Q\)\nbd{}node. \end{remark*}
This finishes the definition of the priority tree \(\cT\).
\begin{lemma}\label{lem:full requirement}
Let \(\rho\) be an infinite path along \(\cT\). Then there is a (unique) \(X\) such that \(\mathop{\mathrm{Permit}}(X)\), and for each \(e\), \(G_e(X)\) and \(D_e(X)\) are assigned to some node in \(\rho\).
Moreover, for this $X$, if \(\alpha\in \rho\) is assigned to \(G_e(X)\), then either
\begin{enumerate}
\item \(\alpha\concat 0 \in \rho\), or
\item \(\alpha\concat \infty \in \rho\) and there is some \(\beta\in \rho\) such that \(\beta\concat (n,X) \in \rho\) where \((n,X)\) is the divergent outcome for \(\alpha\);
\end{enumerate}
if \(\alpha\in \rho\) is assigned to \(D_e(X)\), then either \(\alpha\concat 0\) or \(\alpha\concat d\) is in \(\rho\).
\end{lemma} \begin{proof} A set \(X\) wanted by the lemma is said to be \emph{good}.
Along the infinite path \(\rho\), it is clear that if \(D_e(X)\concat 0 \in \rho\), then \(G_{e+1}(X)\in \rho\); if \(G_e(X)\concat 0\in\rho\), then \(D_e(X)\in \rho\); if \(G_e(X)\concat \infty\in \rho\) and there is some \(\beta\in \rho\) such that \(\beta\concat (n,X) \in \rho\) where \((n,X)\) is the \emph{divergent outcome} for \(G_e(X)\), then \(D_e(X)\in \rho\). Also note that \(D_e(U)\concat \infty \notin \rho\) and \(D_e(V)\concat \infty \notin \rho\) as both are assigned to either a $P$\nbd{} or a $Q$\nbd{}node, which are terminal nodes, contradicting that \(\rho\) is infinite.
Clearly the root is assigned to \(\mathop{\mathrm{Permit}}(U)\). Suppose that \(U\) is not good, then for some \(\alpha_0\in \rho\) assigned to \(G_{e_0}(U)\) for some \(e_0\), we must have that \(\alpha_0\concat \infty\in \rho\) but no divergent outcome for \(\alpha_0\) can be found in \(\rho\). Note that \(\alpha_0\concat\infty\) is assigned to \(\mathop{\mathrm{Permit}}(V_{\alpha_0})\). Suppose that \(V_{\alpha_0}\) is not good, then for some \(\alpha_1\in \rho\) assigned to \(G_{e_1}(V_{\alpha_0})\) for some \(e_1\) we must have that \(\alpha_1\concat\infty \in \rho\) but no divergent outcome for \(\alpha_1\) can be found in \(\rho\). Note that \(\alpha_1\concat\infty\) is assigned to \(\mathop{\mathrm{Permit}}(W_{\alpha_0,\alpha_1})\). We claim that \(W=W_{\alpha_0,\alpha_1}\) is good.
Case 1. Suppose \(D_{e}(W)\concat \infty \in \rho\). Then by~\ref{it:pt D-CS}, \(D_e(W)\concat \infty\) is either an \(S\)\nbd{}node or a \(C\)\nbd{}node. However, by~\ref{it:pt C}, \ref{it:pt S}, and the assumption that no divergent outcome for either \(\alpha_0\) or \(\alpha_1\) can be found in \(\rho\), we have a contradiction. Therefore if \(D_e(W)\in \rho\), then \(D_e(W)\concat 0\) or \(D_e(W)\concat d\) is in \(\rho\).
Case 2. Suppose \(G_e(W)\concat \infty \in \rho\). Then by~\ref{it:pt GW}, \(\xi=G_e(W)\concat \infty\) is either an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node or an \(R^-(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. We should have \(\xi\concat (n,W)\in \rho\). That is, if \(G_e(W)\concat \infty\in \rho\), then \(D_{e+1}(W)\in \rho\).
Hence \(W\) is good. \end{proof}
\begin{remark*} We did not formally associate explicit ``requirements'' to the type of nodes labelled $R, R^-,S, C, P$ and $Q$. If we do, the requirement associated with \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node would be: \[
(\Gamma_{e_0}(BU)=A\land \Gamma_{e_1}(BV)=A \land \Gamma_{e_2}(BW)=A) \Rightarrow A=\Phi(B), \] similarly for $R^-$ and $S$-nodes. The requirements associated with $P(E_U,E_W)$-node would be: \[ [\Delta_{e(\alpha)}=U\wedge \bigwedge_{\sigma\in E_W} \Delta_{*}=W_{\_,\_}] \Rightarrow \Theta=A, \] where $\alpha$ is the unique $D_e(U)$-node in $E_U$ and $e(\alpha)$ is the index $e$; and the missing parameters in $\Delta_{*}=W_{\_,\_}$ can be found precisely because each $\sigma\in E_W$ is necessarily a $D_{i}(W_{\alpha',\alpha''})$-node, similarly for $Q(E_V, E_W)$-nodes. The $C(\alpha_0,\alpha_1,\alpha_2)$-nodes can be better viewed as to detect if $\Gamma_{e_0}(BU)$ is total where $e_0$ is the index of the Gamma functional at $\alpha_0$, which is part of strategy to satisfy $G_{e_0}(U)$. In this sense, a $C$-node plays the analog role of a $G(X)$-node on the tree which detects if there are infinitely many expansionary stages. From these examples, it is clear that the formalism would often involve several sets and functionals whose indices can only be found in a rather complicated way, which would not provide any further insight.
On the other hand, without spilling out the requirements associated with $R, R^-,S, C, P, Q$-nodes, we can still verify the correctness of the construction as follows: First establish the existence of an (infinite) true path $\rho$; then obtain the set \(X\) given by Lemma~\ref{lem:full requirement} from \(\rho\); finally argue that requirements \(\mathop{\mathrm{Permit}}(X)\), $G_e(X)$ and $D_e(X)$, as stated in subsection 2.2, are all satisfied for this $X$. In this way, the requirements associated with $R, R^-,S, C, P, Q$-nodes are involved only indirectly. \end{remark*}
From now on, we will not pay attention to the index \(e\) in a \(G_e(X)\)\nbd{}node or a \(D_e(X)\)\nbd{}node: if \(\alpha\) is a \(G_e(X)\)\nbd{}node, we will write \(\Gamma_\alpha\) for \(\Gamma_e\) and \(\gamma_\alpha\) for \(\gamma_e\); if \(\alpha\) is a \(D_e(X)\)\nbd{}node, we will write \(\Delta_\alpha\) for \(\Delta_e\). As there will be no ambiguities, \(\xi\concat (n,U)\) will be abbreviated as \(\xi\concat n\) if \(\xi\) is a \(C\)\nbd{}node. We leave the notation of \((n,V)\)\nbd{}outcome as it is.
\begin{definition}
If \(\xi\) is a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, we write
\[
\cT[\xi, \Pi_3] = \{\beta\in \cT\mid \xi\concat \omega\subseteq \beta\}
\]
and
\[
\cT[\xi, \Sigma_3] = \{\beta\in \cT\mid (\exists n)\xi\concat n\subseteq \beta\}.
\] \end{definition} A node \(\beta\) is in a \(\Pi_3\)\nbd{}world if and only if there exists some \(C\)\nbd{}node \(\xi\) such that \(\beta\in \cT[\xi,\Pi_3]\).
We let \(\alpha^-\) denote the predecessor of \(\alpha\), and we write \(\type(\alpha)=X\) if \(\alpha\) is a \(\mathop{\mathrm{Permit}}(X)\)\nbd{}node, a \(G(X)\)\nbd{}node or a \(D(X)\)\nbd{}node.
\begin{definition}
Suppose that \(\type(\alpha)=X\). If \(X=U\), then \(\per(\alpha)\) is the root of \(\cT\);
if \(X=V_{\alpha_0}\) for some \(\alpha_0\), then \(\per(\alpha)=\alpha_0\concat \infty\);
if \(X=W_{\alpha_0,\alpha_1}\) for some \(\alpha_0,\alpha_1\), then \(\per(\alpha)=\alpha_1\concat \infty\). \end{definition}
In other words, if \(\type(\alpha)=X\), then \(\per(\alpha)\) is its ``host'' which is the \(\mathop{\mathrm{Permit}}(X)\)\nbd{}node where we introduced the set $X$.
\begin{definition} \label{def:conflict}
Let \(\alpha\) be a \(D(W)\)\nbd{}node for some \(W=W_{\alpha_0,\alpha_1}\).
An \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node \(\beta\) with \(\beta\concat (n,W)\subseteq \alpha\) for some \(n\) is said to have \emph{conflicts with} \(\alpha\)
(or equivalently, \(\alpha\) has conflicts with \(\beta\)).
Let \(\cf(\alpha)\) collects each \(\beta\subseteq \alpha\) that has conflicts with \(\alpha\). \end{definition}
\begin{remark*} The relation between \(\alpha\) and some $\beta\in \cf(\alpha)$ can be loosely stated as follows. We would like to diagonalize $\Delta$ at $\alpha$ with some witness $w$ targeting $W$. But we will see that this $w$ must be paired with some other witness $u$ or $v$ targeting $U$ or $V$ respectively. Whether $w$ is chosen (discarded) or one of $u,v$ is chosen (discarded) will depend on what happens at those $\beta\in \cf(\alpha)$. Note that \(\cf(\alpha)\) can be empty. \end{remark*}
Now we discuss some structural properties of the priority tree \(\cT\). By~\ref{it:pt C}, we have the following \begin{lemma}\label{lem:S locate C}
Let \(\alpha\) be an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, then there is a unique \(C\)\nbd{}node \(\xi\) such that \(\xi\concat\omega\subseteq \alpha\).
Moreover, \(\xi\) is a \(C(\alpha_0,\alpha_3,\alpha_4)\)\nbd{}node for some \(\alpha_3\) and \(\alpha_4\). \qed{} \end{lemma}
\begin{lemma}\label{lem:Q locate C}
Let \(\alpha\) be a \(Q(\eV,\eW)\)\nbd{}node. Then \(\eW\neq \varnothing\) if and only if there exists a unique \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node \(\xi\) for some \(\alpha_0\), \(\alpha_1\),
and \(\alpha_2\) such that \(\alpha\in \cT[\xi,\Pi_3]\). \end{lemma} \begin{proof}
\((\Leftarrow)\) Note that \(\xi\concat \omega\) is an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node and hence \(\alpha_2\in \eW\). \((\Rightarrow)\) Uniqueness is clear from the definition of priority tree as we do not have nested \(\Pi_3\)\nbd{}worlds. By~\ref{it:pt Q} and \(\eW\neq \varnothing\), there is an \(S\)\nbd{}node \(\sigma\subseteq\alpha\). Then we apply Lemma~\ref{lem:S locate C} to get the \(C\)\nbd{}node. \end{proof}
This lemma indicates where the non-uniformity comes in: those \(Q(\eV,\eW)\)\nbd{}nodes inside a \(\Pi_3\)\nbd{}world have a choice to make when comes to diagonalization, whereas the \(Q(\eV,\varnothing)\) nodes outside \(\Pi_3\)\nbd{}worlds have no choice but to enumerate elements into \(V\).
The following two lemmas are clear from the definition of \(\cT\). \begin{lemma}
Let \(\alpha\) be a \(D(X)\)\nbd{}node. Then for each \(\beta\) with \(\alpha\concat \infty \subseteq \beta\), \(\type(\beta)\neq X\). \qed{} \end{lemma}
It says that if we have not satisfied \(D(X)\) yet, we stop working on other requirements for \(X\).
\begin{lemma}
Let \(\alpha\) be a \(P(\eU,\eW)\)\nbd{}node. For \(\beta\in \eW\), \(\beta\mapsto \type(\beta)\) is injective. \qed{} \end{lemma}
We now postulate the global (or static) priority $\prec$ between the nodes on the priority tree. We use \(\alpha\prec \beta\) to denote that \(\alpha\) has higher global priority than \(\beta\). We mostly follow the left-to-right order on the priority tree.
\begin{definition} [global priority]\label{def:global priority}
Let \(\alpha,\beta\in \cT\) be such that there exists some \(\xi\) such that \(\xi\concat o_0 \subseteq \alpha\), \(\xi\concat o_1\subseteq \beta\) and \(o_0 <_L o_1\).
\begin{enumerate}
\item\label{it:prio it1} If \(\xi\) is not a \(C\)\nbd{}node, then we define \(\alpha \prec \beta\).
\item\label{it:prio it2} If \(\xi\) is a \(C\)\nbd{}node but \(o_1\) is not \(\omega\), then we define \(\alpha \prec \beta\).
\end{enumerate}
If \(\alpha\prec\beta\) or \(\beta\prec\alpha\), we say \(\alpha\) and \(\beta\) are \(\prec\)\nbd{}comparable. \end{definition}
If \(\alpha\subsetneq \beta\), we do not think that \(\alpha\) has higher priority than \(\beta\). Global priorities are similar to the usual priorities because they depend on the positions of the nodes on the tree. However,
if \(\xi\) is a \(C\)\nbd{}node and \(o_1\) is \(\omega\), then we do \emph{not} define \(\prec\) between \(\alpha\) and \(\beta\); a ``dynamic'' and ``local'' priority \(\prec_{\xi,s}\) between these two will be defined in Definition~\ref{def:local priority}. We call it ``dynamic'' because it depends on the pairs (which will be introduced in Subsection~\ref{sec:Q node}), which are not permanent and can be canceled. Moreover, even if \(\alpha\) is paired with \(\beta\), the priority between them can only be determined based on what happens on current stage.
The following lemma will be used in Lemma~\ref{lem:R con}. \begin{lemma}\label{lem:comparable nodes}
\begin{enumerate}
\item If \(\alpha\) and \(\beta\) are \(P(\eU,\eW)\)- and \(P(\eU',\eW')\)\nbd{}nodes respectively, then they are \(\prec\)\nbd{}comparable.
\item If \(\alpha\) and \(\beta\) are \(Q(\{\sigma\},\eW)\)- and \(Q(\{\tau\},\eW')\)\nbd{}nodes, respectively, with \(\type(\sigma) = \type(\tau)\), then they are \(\prec\)\nbd{}comparable.
\end{enumerate} \end{lemma}
\begin{proof} (1) holds as a \(P\)\nbd{}node never belongs to a \(\Pi_3\)\nbd{}world.
For (2), we suppose toward a contradiction that there exists a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node \(\xi\) such that \(\xi\concat n\subseteq \alpha\) and \(\xi\concat \omega \subseteq \beta\). It is clear then \(\type(\tau)=V_{\alpha_0}\neq \type(\sigma)\).
\end{proof}
\section{Parameters}\label{sec:parameter} Unlike a usual priority argument, where we might have static parameters assigned to the edges of the priority tree, our construction has dynamic parameters; each node has to update them at the end of each stage. Two kinds of the parameters are \emph{edge parameters} and \emph{pairing parameters} and will be discussed first in this section.
We assume that the readers are familiar with usual tree constructions. We now describe how a given node \(\alpha\) maintains its parameters at the end of stage~\(s\).
\noindent \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node: Suppose \(V=V_{\alpha_0}\) and \(W=W_{\alpha_0,\alpha_1}\). For each \(n\) and for each \(X\in \{U,V,W\}\), the \emph{edge parameter} \(z_{(n,X)}\) is initially set to be \(n\).
Let \((n,X)\) be the outcome such that \(\alpha\concat (n,X)\) is visited at \(s\), if any.
\begin{enumerate}
\item If \(X=W\), we update \(z_{(n,U)}=z_{(n,V)} = s\) and \(z_{(n+j,Y)}=s+j\) for each \(j\ge 1\) and each \(Y\in \{U,V,W\}\).
\item If \(X=U\) or \(X=V\), we do nothing. \end{enumerate} If \(\alpha\) is initialized at stage~\(s\), we set \(z_{(n,X)}=n\) for each \(n\) and each \(X\in \{U,V,W\}\).
\begin{remark*}
If \(z_{(n,U)}\) is stable (i.e., \(\lim_s z_{(n,U)}[s]<\infty\)),
then the divergent outcome \((n,U)\) indicates that there is some \(m\le z_{(n,U)}\) such that \(\Gamma_{\alpha_0}(BU;m)\uparrow\).
Other divergent outcomes are similar. \end{remark*}
\noindent \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node: Suppose \(V=V_{\alpha_0}\) and \(W=W_{\alpha_0,\alpha_1}\). For each \(n\), the \emph{edge parameter} \(y_{(n,V)}\) is initially set to be \(n\). Now at stage~\(s\),
let \(n\) be such that \(\alpha\concat (n,V)\) is visited at stage~\(s\). We update \(y_{(n+j,V)}=s+j\) for each \(j\ge 1\). If \(\alpha\) is initialized at stage~\(s\), we set \(y_{(n,V)}=n\).
\noindent \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node: For each \(n\), the \emph{edge parameter} \(x_{(n,U)}\) is initially set to be \(n\). Now at stage~\(s\), let \(n\) be such that \(\alpha\concat (n,U)\) is visited at stage~\(s\), we update \(x_{(n+j,U)}=s+j\) for each \(j\ge 1\). If \(\alpha\) is initialized at stage~\(s\), we set \(x_{(n,U)}=n\).
The above covers the nodes where edge parameters are needed. The next one concerns with pairing parameters.
\noindent \(Q(\eV,\eW)\)\nbd{}node: If $\eW = \varnothing$, then it needs no pairing parameters. Now we assume that \(\eW\neq \varnothing\). By Lemma~\ref{lem:Q locate C}, we let \(\xi\) be the (unique) \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node with \(\xi\concat \omega \subseteq \alpha\). The \emph{pairing parameter} \(\tp(\alpha)\) is defined to be the least number \(q\) such that \(q > y_{(n,V)}\) for each \(S\)\nbd{}node \(\beta\) and each \(n\) with \(\xi\concat \omega\subseteq \beta\concat (n,V) \subseteq \alpha\) (we assume that \(y_{(n,V)}\) has been updated because \(\beta\subseteq \alpha\)).
\(P(\eU,\eW)\)\nbd{}nodes however need no pairing parameters.
Other kinds of parameters are discussed below.
\noindent \(D(X)\)\nbd{}node: \(d\)\nbd{}outcome of \(\alpha\) can be \emph{activated} at some stage. Once it becomes activated, it remains activated forever (even if the node is initialized).
In our construction, an \(R\)- or \(R^-\)\nbd{}node builds a functional \(\Phi\), an \(S\)\nbd{}node builds a functional \(\Psi\), a \(P\)\nbd{}node builds a functional \(\Theta\), and a \(Q\)\nbd{}node builds a functional \(\Theta\) (the symbol is reused). These functionals are built locally and hence will be referred as \emph{local parameters}. Once a node is initialized or disarmed, its local parameters are discarded immediately.
A \(P(\eU,\eW)\)- or \(Q(\eV,\eW)\)\nbd{}node \(\alpha\) has to maintain a function \(\dw_\alpha\) (standing for \emph{d}iagonalizing \emph{w}itness). This \(\dw_\alpha\) is also referred as a \emph{local parameter}. A \(P\)\nbd{}node (or \(Q\)\nbd{}node, respectively) is where we attack the recursiveness of \(U\) (or \(V\), respectively) or \(W\) by enumerating a diagoalizing witness into one of them. The whole attacking process consists of four phases and \(\dw_\alpha\) is defined in the first phase:
\begin{enumerate}[fullwidth,leftmargin=*,itemindent=3em,label=(Phase\arabic*)]
\item\label{it:ph1} Prepare the diagonalizing witnesses (Subsections~\ref{sec:P node} and~\ref{sec:Q node}).
Without loss of generality, we assume that \(\alpha\) is a \(P(\eU,\eW)\)\nbd{}node. \(\alpha\) has to prepare many diagonalizing witnesses for each \(\sigma\in \eU\cup\eW\) hoping one of them gets permitted. To ensure that one of them get permitted, \(\alpha\) builds a functional \(\Theta\) so that \(\Theta=A\) in case none of them gets permitted. This would imply that \(A\) is recursive toward the contradiction. The diagonalizing witnesses are organised in the following way: for each \(a\in \omega\), \(\dw_\alpha(a)\) is a map from \(\eU\cup\eW\) to \(\omega\); for each \(\sigma\in \eU\cup\eW\), \(\dw_\alpha(a)(\sigma)\in \omega\) is the diagonalizing witness targeting the set \(\type(\sigma)\). We say that \(x=\dw_\alpha(a)(\sigma)\) is \emph{prepared} if \(\Delta_\sigma(x)=\type(\sigma)(x)=0\). Thus, enumerating \(x\) into \(\type(\sigma)\) leads to the \(\Sigma_1\)\nbd{}outcome of \(\sigma\) (and we activate \(\sigma\concat d\) in this case). Once \(\dw_\alpha(a)(\sigma)\) is prepared for each \(\sigma\in \eU\cup \eW\), we define \(\Theta(a)=A(a)\).
\item\label{it:ph2} Wait for the permission from \(A\) (Subsection~\ref{sec:permitting center}).
Suppose that \(\Theta(a)=A(a)\) has been defined and that \(a\) enters \(A\). Now \(\Theta(a)\) becomes incorrect and we have to choose (see~\ref{it:ph3}) a \(\sigma\in \eU\cup\eW\) and get ready to enumerate \(\dw_\alpha(a)(\sigma)\) into \(\type(\sigma)\).
\item\label{it:ph3} Make decision (Subsection~\ref{sec:permitting center} and~\ref{sec:R test}).
In our construction, there will be nodes setting up restraints (indirectly) on \(\type(\sigma)\) for certain \(\sigma\in \eU\cup \eW\), and our decision has to be made carefully. Once the decision is made, we do not enumerate the point immediately; it will be carried out only when \(\mathop{\mathrm{Permit}}(\type(\sigma))\)\nbd{}node is visited.
\item\label{it:ph4} Enumerate the point (Subsection~\ref{sec:permit node}).
Suppose that the chosen permission node is visited and the chosen diagonalizing point \(x=\dw_\alpha(a)(\sigma)\) has not been canceled yet, we simply enumerate it into the \(\type(\sigma)\) and activate \(\sigma\concat d\) at the same moment (and \(\alpha\) will not be visited again). \end{enumerate}
\section{Strategies and construction}\label{sec:strategies} When a node \(\alpha\) is visited, particular actions will be taken and will be described formally in the subroutine \(\visit(\alpha)\). In each of the following subsections, it describes either \(\visit(\alpha)\) or other actions. We will freely use the terms \emph{true outcome} and \emph{true path} in the usual sense in an informal remarks. Though we have a slightly modified definition for true outcome and true path (Definition~\ref{def:T star}), the intuition behind them remains the same.
\begin{definition}\label{def:announces progress} For \(\alpha\in \cT\), \(\alpha\) \emph{announces progress} if one of the following happens: \begin{enumerate}
\item it is \emph{visited};
\item it is \emph{initialized};
\item it is \emph{disarmed} (if \(\alpha\) is a \(Q\)\nbd{}node)---see Subsection~\ref{sec:disarm};
\item it \emph{receives attention} (if \(\alpha\) is a \(P\)- or \(Q\)\nbd{}node)---see Subsection~\ref{sec:permitting center};
\item it becomes \emph{activated} (if \(\alpha=\beta\concat d\) for some \(D(X)\)\nbd{}node \(\beta\))---see Subsection~\ref{sec:permit node}. \end{enumerate} \end{definition}
\begin{convention}\label{convention: init}
Whenever \(\alpha\) announces progress, each \(\beta\) of lower global priority (Definition~\ref{def:global priority}) is \emph{initialized} tacitly. If \(\alpha\) is initialized, each \(\beta\) extending \(\alpha\) is also initialized tacitly. In both cases, we say that \(\alpha\) initializes \(\beta\), or \(\beta\) is initialized by \(\alpha\). \end{convention}
\begin{remark*} The second sentence in Convention~\ref{convention: init} is to be used with~\ref{it:p2} to initialize nodes below \(\xi\concat n\) as appeared there. \end{remark*}
``Initialize'' is used in the usual sense: when a node is initialized, all local parameters are discarded. ``Disarm'' can be thought of as an analogy of ``initialize'' and specially designed for \(Q\)\nbd{}nodes (see Subsection~\ref{sec:Q node}).
\subsection{G-node}\label{sec:G node} Suppose that \(\alpha\) is a \(G(X)\)\nbd{}node for some \(X\) and the current stage is \(s\).
\noindent \(\visit(\alpha)\): \begin{enumerate}[nosep]
\item If \(s\) is an expansionary stage, then \(\visit(\alpha\concat \infty)\).
\item Otherwise, \(\visit(\alpha\concat 0)\). \end{enumerate}
\begin{remark*} The job of \(\alpha\) is to measure the length of agreements between \(\Gamma_{\alpha}(BX)\) and \(A\). If the true outcome is \(0\)\nbd{}outcome, we end up with \(\Gamma_{\alpha}(BX)\neq A\) and \(G(X)\) is therefore satisfied.
Note that having infinitely many expansionary stages does not guarantee \(\Gamma_{\alpha}(BX)=A\). In fact, there could be some divergent point which will be detected by either \(C\)-, \(R\)- (or \(R^-\)-) or \(S\)\nbd{}nodes, depending on whether \(X\) is of type \(\bU\), \(\bV\) or \(\bW\).
\end{remark*}
\subsection{D-node}\label{sec:D node} Suppose that \(\alpha\) is a \(D(X)\)\nbd{}node for some \(X\) and the current stage is \(s\).
\noindent \(\visit(\alpha)\): \begin{enumerate}[nosep]
\item If \(d\)\nbd{}outcome is activated (see Subsection~\ref{sec:permit node}), then \(\visit(\alpha\concat d)\).
\item If \(s\) is an expansionary stage, then \(\visit(\alpha\concat \infty)\).
\item Otherwise, \(\visit(\alpha\concat 0)\). \end{enumerate}
\begin{remark*} The main job of \(\alpha\) is to measure the length of agreements between \(\Delta_{\alpha}\) and \(X\). As we shall see later that once \(d\)\nbd{}outcome becomes activated, it remains activated forever. If \(d\)\nbd{}outcome is the true outcome, then we are given a point \(n\) such that \(\Delta_\alpha(n)=0\neq 1=X(n)\) and \(D(X)\) is therefore satisfied. If \(0\)\nbd{}outcome is the true outcome, then for some point \(n\), either \(\Delta_\alpha(n)\uparrow\) or \(\Delta_\alpha(n)\downarrow\neq X(n)\), which implies that \(D(X)\) is satisfied. If \(\infty\)\nbd{}outcome is the true outcome, then \(\Delta_\alpha=X\) and hence \(D(X)\) is not satisfied. In this case, the set \(X\) is not a good one in the sense of Lemma~\ref{lem:full requirement}. \end{remark*}
\subsection{P-node}\label{sec:P node} Suppose that \(\alpha\) is a \(P(\eU,\eW)\)\nbd{}node and \(\alpha\) is being visited at stage~\(s\). Let \(s^*\le s\) be the least \(\alpha\)\nbd{}stage such that \(\alpha\) never gets initialized between \(s^*\) and \(s\).
\noindent \(\visit(\alpha)\): Let \(a\ge s^*\) be the least such that \(\Theta(a)\) is not defined. \begin{enumerate}[nosep]
\item If \(\dw_\alpha(a)\) is not defined, we pick a fresh number \(x_\beta\) for each \(\beta\in \eU\cup \eW\) and define \(\dw_\alpha(a)(\beta)=x_\beta\). Stop the current stage.
\item If \(\dw_\alpha(a)\) is defined but for some \(\beta\in \eU\cup\eW\) we have \(\Delta_{\beta}(x_{\beta})\uparrow\) (i.e., \(x_\beta\) is not prepared), then we stop the current stage.
\item Otherwise, we define \(\Theta(a)=A(a)[s]\). Stop the current stage. \end{enumerate}
\begin{remark*} Basically, it just implements the~\ref{it:ph1} to prepare the diagonalizing witnesses. In (2), it means although we expect infinitely many expansionary stages, yet the length of agreements has not reached $x_{\beta}$. So we just wait. As \(P\)\nbd{}node is a terminal node, the true outcome does not matter. As we shall see later, if \(\alpha\) is on the true path, then we have \(\Theta(n)=A(n)\) for each \(n\ge s^*\); hence, \(A\) is recursive. To achieve this, whenever we have \(\Theta(a)=0\neq 1=A(a)\), we have to prevent \(\alpha\) from being visited again (i.e., switch it off the true path) by diagonalizing against some \(\beta\in \eU\cup\eW\) and activating \(\beta\concat d\). See Subsection~\ref{sec:permitting center} for details. \end{remark*}
\subsection{Q-node}\label{sec:Q node} Suppose that \(\alpha\) is a \(Q(\eV,\eW)\)\nbd{}node and \(\alpha\) is being visited at stage~\(s\). Let \(s^*\le s\) be the least \(\alpha\)\nbd{}stage such that \(\alpha\) never gets initialized or disarmed (see below) between \(s^*\) and \(s\).
\noindent \(\visit(\alpha)\): \begin{enumerate}[nosep]
\item\label{it:Q1} Suppose \(\eW\neq \varnothing\) and that \(\alpha\) is not \emph{paired}.
We let \(\xi\) be the unique \(C\)\nbd{}node with \(\xi\concat \omega\subseteq \alpha\) (Lemma~\ref{lem:Q locate C}).
We set up \((\xi\concat n, \alpha)\) as a \emph{pair} where \(n\) is the least (which always exists by the discussion in Section~\ref{sec:parameter}) such that
\begin{enumerate}
\item
\(x_{(n,U)}>\tp(\alpha)\), and
\item for each \(m\ge n\), \(\xi\concat m\) is not paired.
\end{enumerate}
Stop the current stage.
\item\label{it:Q2} Suppose \(\eW=\varnothing\) or that \(\alpha\) has been paired.
Let \(a\ge s^*\) be the least such that \(\Theta(a)\) is not defined.
\begin{enumerate}
\item Suppose that \(\dw_\alpha(a)\) is not defined, we pick a fresh number \(x_\beta\) for each \(\beta\in \eV\cup \eW\) and define \(\dw_\alpha(a)(\beta)=x_\beta\). Stop the current stage.
\item Suppose that \(\dw_\alpha(a)\) is defined but for some \(\beta\in \eV\cup\eW\) we have \(\Delta_{\beta}(x_{\beta})\uparrow\). Stop the current stage.
\item Otherwise, we define \(\Theta(a)=A(a)[s]\). Stop the current stage.
\end{enumerate} \end{enumerate}
\begin{remark*} The action of $Q$ has two parts: In~(\ref{it:Q2}), it prepares the diagonalizing witnesses like the \(P\)\nbd{}node above; in~(\ref{it:Q1}), it establishes a pair. $\Sigma_3$\nbd{} and $\Pi_3$\nbd{}worlds will interact via such pairs, based on which a ``local'' ``dynamic'' priority will be defined (see Subsection~\ref{sec:C node}, particularly Definition~\ref{def:local priority}).
Examples of pairs can be found in Figure~\ref{fig:local priority 1}, in which \((C\concat n_0,Q_0)\) and \((C\concat n_1,Q_1)\) are two pairs. \end{remark*} \begin{figure}
\caption{Examples of Pairs}
\label{fig:local priority 1}
\end{figure}
\subsection{Pair}\label{sec:a pair} Suppose that \((\xi\concat n, \alpha)\) is a pair established as in~(\ref{it:Q1}) in Subsection~\ref{sec:Q node}. We implement the following three rules: \begin{enumerate}[label=(P\arabic*)]
\item\label{it:p1} We \emph{cancel} the pair if and only if \(\alpha\) is initialized.
\item\label{it:p2} Whenever the pair is canceled, we initialize $\xi\concat n$ (and all nodes below $\xi\concat n$ by Convention~\ref{convention: init}).
\item\label{it:p3} Whenever \(\alpha\) announces progress (Definition~\ref{def:announces progress}), we initialize \(\xi\concat (n+1)\). \end{enumerate} Note that the pair is \emph{not} canceled when \(\alpha\) is disarmed (Subsection~\ref{sec:disarm}). A case of special importance shall be noted: \begin{lemma}\label{lem:cleared of P}
Let \((\xi\concat n,\alpha)\) be a pair. Whenever \((\alpha^-)\concat d\) is activated, \(\xi\concat n\) is initialized. \end{lemma}
\begin{proof}
Suppose that \((\alpha^-)\concat d\) is activated and therefore \(\alpha\) is initialized. By~\ref{it:p1}, the pair is canceled. By~\ref{it:p2}, \(\xi\concat n\) is initialized. \end{proof}
\begin{definition}\label{def:pairing priority}
For two pairs, we say that \((\xi\concat n,\alpha)\) has higher \emph{pairing priority} than \((\xi\concat m,\beta)\) if \(n<m\). (Note that \(n<m\) if and only if \(\alpha\prec\beta\).) \end{definition}
\subsection{R-node}\label{sec:R node} Suppose that \(\alpha\) is an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node or an \(R^-(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node and the current stage is \(s\). Let \(s^*<s\) be the last \(\alpha\)\nbd{}stage.
\noindent \(\visit(\alpha)\): Let \(n=0\). \begin{enumerate}[nosep]
\item Suppose \(n>s\). We stop the current stage.
\item Suppose \(\Phi(B;z_{(n,W)})\downarrow[s]\). We proceed with \(n+1\).
\item\label{it:R 3} Suppose \(\Phi(B;z_{(n,W)})\uparrow[s]\) but \(\Phi(B;z_{(n,W)})\downarrow[s^*]\).
\begin{enumerate}
\item If \(\gamma_{\alpha_2}(z_{(n,W)})[s]>\gamma_{\alpha_2}(z_{(n,W)})[s^*]\), \(\visit(\alpha\concat (n,W))\).
\item (Ignored this line if \(\alpha\) is an \(R^-\)\nbd{}node) If (a) does not happen and \(\gamma_{\alpha_0}(z_{(n,U)})[s]>\gamma_{\alpha_0}(z_{(n,U)})[s^*]\), \(\visit(\alpha\concat (n,U))\).
\item If neither (a) nor (b) happens and \(\gamma_{\alpha_1}(z_{(n,V)})[s]>\gamma_{\alpha_1}(z_{(n,V)})[s^*]\), \(\visit(\alpha\concat (n,V))\).
\item Otherwise, we define \(\Phi(B;z)=A(z)[s]\) with use \(\varphi(B;z_{(n,W)})[s^*]\) for each \(z\) with \(z_{(n-1,W)}<z\le z_{(n,W)}\) and then stop the current stage.
\end{enumerate}
\item Otherwise, for each \(z\) with \(z_{(n-1,W)}<z\le z_{(n,W)}\), we define \(\Phi(B;z)[s]=A(z)[s]\) with a fresh use \(u\), particularly
\begin{enumerate}
\item\label{it:R 4a} \(u\ge \gamma_{\alpha_2}(BW;z_{(n,W)})[s]\),
\item\label{it:R 4b} (Ignore this line if \(\alpha\) is an \(R^-\)\nbd{}node) \(u\ge \gamma_{\alpha_0}(BU;z_{(n,U)})[s]\), and
\item\label{it:R 4c} \(u\ge \gamma_{\alpha_1}(BV;z_{(n,V)})[s]\).
\end{enumerate}
Then we proceed to \(n+1\). \end{enumerate}
\begin{remark*} The $R$\nbd{}node is responsible for defining $\Phi(B; z_{(n,W)})=A(z_{(n,W)})$, whose use $\varphi(z_{(n,W)})$ is set to bound $\gamma_{\alpha_2}(BW;z_{(n,W)})[s], \gamma_{\alpha_0}(BU;z_{(n,U)})[s]$ and $\gamma_{\alpha_1}(BV;z_{(n,V)})[s]$. When $\Phi(B;z_{(n,W)})$ becomes undefined (as in~(\ref{it:R 3}) above), we track the culprit and try to demonstrate (in that order) that $\Gamma_{\alpha_2}(BW;z_{(n,W)})$, $\Gamma_{\alpha_0}(BU;z_{(n,U)})$ or $\Gamma_{\alpha_1}(BV;z_{(n,V)})$ is undefined. It is important to note that (\ref{it:R 4b}) and~(\ref{it:R 4c}) are ensured only at the stage when we define \(\Phi(B;z_{(n,W)})\); the two conditions are allowed to be violated at other stages. The reader shall find details in Subsection~\ref{sec:R strategy}.
\end{remark*}
\subsection{S-node}\label{sec:S node} Suppose that \(\alpha\) is an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node and the current stage is \(s\). Let \(s^*<s\) be the last \(\alpha\)\nbd{}stage.
\noindent \(\visit(\alpha)\): Let \(n=0\). \begin{enumerate}[nosep]
\item Suppose \(n>s\). We stop the current stage.
\item Suppose \(\Psi(B;y_{(n,V)})\downarrow[s]\). We
proceed with \(n+1\).
\item Suppose \(\Psi(B;y_{(n,V)})\uparrow[s]\) but \(\Psi(B;y_{(n,V)})\downarrow[s^*]\).
\begin{enumerate}
\item If \(\gamma_{\alpha_1}(BV;y_{(n,V)})[s]>\gamma_{\alpha_1}(BV;y_{(n,V)})[s^*]\), then \(\visit(\alpha\concat (n,V))\).
\item Otherwise, we define \(\Psi(B;y)=A(y)[s]\) with use \(\psi(B;y_{(n,V)})[s^*]\) for each \(y\) with \(y_{(n-1,V)}<y\le y_{(n,V)}\).
Then we stop the current stage.
\end{enumerate}
\item Otherwise, for each \(y\) with \(y_{(n-1,V)}<y\le y_{(n,V)}\), we define \(\Psi(B;y)=A(y)[s]\) with a fresh use \(u\), particularly \(u > \gamma_{\alpha_1}(BV;y_{(n,V)})[s]\).
Then we
proceed to \(n+1\). \end{enumerate}
\begin{remark*} The actions of $S$\nbd{}nodes are similar to those of $R$\nbd{}nodes. We define $\Psi(B;y)=A(y)$ with use $\psi(y)$ bounding $\gamma_{\alpha_1}(BV;y)$. If \((n,V)\) is the true outcome, we can conclude that \(\Psi(B;y_{(n,V)})\uparrow\), which implies \(\Gamma_{\alpha_1}(BV;y_{(n,V)})\uparrow\) and a divergent point for \(\Gamma_{\alpha_1}(BV)\) is found. Unexpectedly we might end up with \(\Gamma_{\alpha_1}(BV;y_{(n,V)})\uparrow\) but \(\Psi(B;y_{(n,V)})\downarrow\). This may happen when \(\Gamma_{\alpha_0}(BU;y_{(n,U)})\downarrow\) with \(\gamma_{\alpha_0}(BU;y_{(n,U)})<\psi(B;y_{(n,V)})\) whereas \(\visit(\alpha)\) does not say anything about \(\Gamma_{\alpha_0}(BU)\). The success of an \(S\)\nbd{}node relies on the ``totality'' (in a less strict sense) of \(\Gamma_{\alpha_0}(BU)\). Such reliance is the core of our construction and will be discussed in Subsection~\ref{sec:S strategy}.
\end{remark*}
\subsection{C-node and local dynamic priorities}\label{sec:C node} Let \(\xi\) be a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. Before we describe \(\visit(\xi)\) at stage~\(s\), we need to define the ``local'' ``dynamic'' priority $\prec_{\xi,s}$ which connects some pairs of nodes that one is in the $\Sigma_3$- and the other is in the $\Pi_3$\nbd{}worlds of $\xi$. It contrasts the global priority \(\prec\) (without subscripts) defined in Definition~\ref{def:global priority} which depends only on the priority tree. In the notation \(\prec_{\xi,s}\), the subscript \(\xi\) corresponds to the ``local'' part and \(s\) corresponds to the ``dynamic'' part.
Recall that the procedure of establishing a pair was introduced in (\ref{it:Q1}) in Subsection~\ref{sec:Q node}.
\begin{definition}[local priority]\label{def:local priority} At stage~\(s\), let \[
(\xi\concat n_0,\alpha_0), \ldots, (\xi\concat n_{k-1}, \alpha_{k-1}). \] be the list of all pairs in descending order of pairing priority (Definition~\ref{def:pairing priority}).
For \(\eta\in \cT[\xi,\Sigma_3]\) and \(\beta\in \cT[\xi,\Pi_3]\) (Note that $\eta$ and $\beta$ are not $\prec$\nbd{}comparable), we define \(\prec_{\xi,s}\) as follows: \begin{enumerate}
\item\label{it:local priority 1} \(\eta\prec_{\xi,s} \beta\) if there exists some \(i<k\) such that \(\eta\) extends \(\xi\concat m\) for some \(m\le n_i\) and \(\alpha_i\preceq \beta\),
\item\label{it:local priority 2} \(\beta\prec_{\xi,s} \eta\) if there exists some \(i<k\) such that \(\beta\preceq \alpha_i\) and \(\eta\) extends \(\xi\concat m\) for some \(n_i<m\).
\end{enumerate}
(If the list is empty, then \(\prec_{\xi,s}=\varnothing\).) \end{definition}
\begin{remark*} We will often write \(\prec_\xi\) for short as the stage~\(s\) is usually clear from context. Note also that \(\prec_\xi\) is a partial order; \(\eta\in \cT[\xi,\Sigma_3]\) and \(\beta\in \cT[\xi,\Pi_3]\) can be \(\prec_\xi\)\nbd{}incomparable. However, if \(\eta\) is a \(P\)- or \(Q\)\nbd{}node and \(\beta\) is a \(Q\)\nbd{}node that is paired, then either \(\eta\prec_\xi\beta\) or \(\beta\prec_\xi\eta\).
In Figure~\ref{fig:local priority 2}, two pairs \((C\concat n_0,Q_0)\) and \((C\concat n_1,Q_1)\) are given. An arrow \(\alpha\to\beta\) indicates \(\alpha\prec\beta\) if \(\alpha\) and \(\beta\) lie in the same row; \(\alpha\prec_\xi \beta\) if they lie in different rows: labels along the arrows refer to corresponding items in Definition~\ref{def:local priority}. For example, Item~(\ref{it:local priority 1}) of Definition~\ref{def:local priority} gives, without an explicit arrow in Figure~\ref{fig:local priority 2}, \(C\concat 0\prec_\xi \beta_1\), which follows equivalently from \(C\concat 0\prec C\concat n_0\prec_\xi Q_0\prec \beta_1\). In a similar fashion, Item~(\ref{it:local priority 2}) of Definition~\ref{def:local priority} gives \(\beta_1\prec_\xi C\concat(n_1+1)\), and we might view this as a consequence of \(\beta_1\prec Q_1\prec_\xi C\concat(n_1+1)\). Note that \(C\concat 0\) and \(\beta_0\) are not \(\prec_\xi\)\nbd{}comparable. Although \(\beta_0\) and \(C\concat n_0\) are not \(\prec_\xi\)\nbd{}comparable, we observe that whenever \(\beta_0\) announces progress, \(Q_0\) is initialized, and by~\ref{it:p1} (Subsection~\ref{sec:a pair}) the pair \((C\concat n_0,Q_0)\) is canceled and by~\ref{it:p2} \(C\concat n_0\) is therefore initialized (Lemma~\ref{lem:cleared of P} is a special case of this phenomenon). Also,~\ref{it:p3} is justified by Item~(\ref{it:local priority 2}) of Definition~\ref{def:local priority}.
\end{remark*}
\begin{figure}
\caption{An example of Local Dynamic Priority}
\label{fig:local priority 2}
\end{figure}
With the definition of local dynamic priority, we are ready to describe \(\visit(\xi)\) for the \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node \(\xi\).
\noindent \(\visit(\xi)\): Let \(s^*<s\) be the last \(\xi\)\nbd{}stage. Let \(n\) be the least (if any) such that \(\gamma_{\alpha_0}(BU;x_{(n,U)})[s]>\gamma_{\alpha_0}(BU;x_{(n,U)})[s^*]\). We are ready to visit nodes in \(\cT[\xi,\Pi_3]\). We start with \(\eta=\xi\concat \omega\). Whenever we run into \(\visit(\eta)\) for \(\eta\in \cT[\xi,\Pi_3]\), we interrupt the subroutine and do the following: \begin{enumerate}
\item If \(\abs{\eta}>s\) and \(\xi\concat n\) has not been initialized, \(\visit(\xi\concat n)\); if \(\abs{\eta}>s\) and \(\xi\concat n\) has been initialized, we stop the current stage. \emph{(After nodes in \(\cT[\xi,\Pi_3]\) has been visited, we decide if we need to visit nodes in \(\cT[\xi,\Sigma_3]\).)}
\item If \(\xi\concat n\prec_\xi \eta\), then \(\visit(\xi\concat n)\). \emph{(We stop visiting nodes in \(\cT[\xi,\Pi_3]\).)}
\item Otherwise, \(\visit(\eta)\). (\(\xi\concat n\) could potentially be initialized by~\ref{it:p1}, \ref{it:p2} or~\ref{it:p3} when certain \(\eta\) is visited.) \end{enumerate} If \(n\) does not exist, \(\visit(\xi\concat \omega)\) without interruptions.
\begin{remark*} At the $C$\nbd{}node, we are facing two parallel worlds which are connected via pairs. Instead of having two separated accessible path, we explore the two possibilities simultaneously. Whenever we see an interaction via a pair, we stay in the winner's world. If we do not see any interaction, we do not bet as either world can be the winner; indeed, we visit both worlds simultaneously.
\end{remark*}
\subsection{Permitting center}\label{sec:permitting center} The permitting center is not represented on the tree. At the beginning of a stage~\(s\), if we have \(A(a)[s-1]\neq A(a)[s]\) for some \(a\), then some \(P\)- or \(Q\)\nbd{}node \(\beta\) might have a problem because \(\Theta(a)\neq A(a)[s]\). \(\Theta(a)\) cannot be corrected and we have to prevent \(\beta\) from being visited again by diagonalizing against some ``good'' (whose existence is one of the most important parts of the verification) \(\alpha\in \eU\cup\eW\) (if \(\beta\) is a \(P(\eU,\eW)\)\nbd{}node) or \(\alpha\in \eV\cup\eW\) (if \(\beta\) is a \(Q(\eV,\eW)\)\nbd{}node) and activating \(\alpha\concat d\), which has higher global priority than \(\beta\). This section is to discuss how we choose this \(\alpha\) and this is done at the permitting center at the beginning of each stage.
\begin{remark*} All tensions among nodes on the priority tree trace back to a \(\Theta\) defined by either a \(P\)- or \(Q\)\nbd{}node and a point \(a\) such that \(\Theta(a)\neq A(a)\). \end{remark*}
Let \(\lambda\) be the root of the priority tree \(\cT\) and \(s\) be the current stage.
\noindent Permitting center: \begin{enumerate}
\item\label{it:pc 1} If there is no point \(a\) with \(A(a)[s-1]\neq A(a)[s]\), we do nothing.
\item\label{it:pc 2} (making choice) Let \(a\) be such that \(A(a)[s-1]\neq A(a)[s]\). Let \(\beta\) be the \(P(\eU,\eW)\)- or \(Q(\eV,\eW)\)\nbd{}node, if any, which has not received attention and \(\Theta(a)=0\). We list \(\eW\) as \(\xi_0\subseteq \dots \subseteq \xi_{k-1}\) where \(k=\abs{\eW}\) (the list is empty if \(\eW=\varnothing\)). We define the choice function \(\chi\) by letting
\[
\chi(\beta;a)=\xi_i
\]
for the least \(i\), if any, such that for each \(\alpha\in \cf(\xi_i)\) we have (see Subsection~\ref{sec:R test} for the definition of \(\Test\) function)
\[
\Test(\alpha; a, \dw_\beta(a)(\xi_i))=1;
\]
if such \(i\) does not exist (including the case when \(\eW=\varnothing\)),
\[
\chi(\beta;a)=\beta^-.
\]
\item\label{it:pc 3} Let \(\sigma=\chi(\beta;a)\). We \emph{send} the point \(\dw_\beta(a)(\sigma)\) to \(\per(\sigma)\), which is the \(\mathop{\mathrm{Permit}}(X)\)\nbd{}node where \(X=\type(\sigma)\).
\end{enumerate} If~(\ref{it:pc 2}) happens, we say that \(\beta\) \emph{receives attention} (on \(a\) at stage~\(s\)).
\begin{remark*} \begin{enumerate}[label=(\roman*)] \item For convenience, \(\beta\) is allowed to receive attention only once since last initialization or disarmament. \item There can exist at most one \(\beta\) as desired in~(\ref{it:pc 2}). To see this, let \(\beta_0\) and \(\beta_1\) be two \(P\)- or \(Q\)\nbd{}nodes who build \(\Theta_0\) and \(\Theta_1\) respectively. If both of them are \(P\)\nbd{}nodes (or \(Q\)\nbd{}nodes), they are \(\prec\)\nbd{}comparable; otherwise, being terminal nodes, there must exist some \(\xi\) such that \(\beta_0\in \cT[\xi,\Sigma_3]\) and \(\beta_1\in \cT[\xi,\Pi_3]\), say. By the remark below Definition~\ref{def:local priority}, they are \(\prec_\xi\)\nbd{}comparable. Therefore, whenever we visit the node of higher global (or local) priority, the one of lower global (or local) priority is initialized or disarmed.
Therefore, domains of~\(\Theta_0\) and \(\Theta_1\) must be disjoint.
\item \(\chi(\beta;a)\) chooses the node~\(\sigma\) such that we are allowed to enumerate the point \(\dw_\beta(a)(\sigma)\) into~\(X\). However, we are not enumerating it into~\(X\) immediately; we will do so when we visit \(\per(\sigma)\) provided that \(\beta\) has not been disarmed or initialized. The delay feature of the permitting is crucial to Lemma~\ref{lem:R correct}. \end{enumerate} \end{remark*}
\subsection{R-test}\label{sec:R test} Intuitively, an \(R\)-node would like to put a restraint on \(W\) while \(\Phi(B;z_{(n,W)})\downarrow\). If a \(P(\eU,\eW)\)\nbd{}node \(\beta\), for example, receives attention on \(a\), we are not free to enumerate the point \(w=\dw_\beta(a)(\sigma)\) into \(\type(\sigma)\) (where \(\sigma\in\eW\)): an \(R\)\nbd{}node \(\alpha\in \cf(\sigma)\) may still have a restraint on \(W\). To tell in which situation an \(R\)\nbd{}node \(\alpha\in \cf(\sigma)\) allows \(w\) to be enumerated in a delayed fashion, we now introduce the \(\Test\) function, which we have used without definition in Step~(\ref{it:pc 2}) of permitting center. The following definition is a bit technical and the whole picture can only be seen in the verification section, particularly Subsection~\ref{sec:R strategy}. \begin{definition}[R-test]\label{def:R test}
Let \(\alpha\) be an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. Fix numbers \(a\) and \(w=\dw_\beta(a)(\sigma)\) for some \(\beta\) and \(\sigma\). \(\beta\) is assumed to be either a \(P(\eU,\eW)\)- or a \(Q(\eV,\eW)\)\nbd{}node with \(\sigma\in \eW\) and \(\alpha\in \cf(\sigma)\).
Suppose that \(\beta\) is a \(P(\eU,\eW)\)\nbd{}node. For every $n$, we define \(\test(\alpha,n;a,w)[s]{=1}\) if one of the following holds
\begin{enumerate}[label=(R-t\arabic*)]
\item\label{it:R-t1} \(\Phi(B;z_{(n,W)})\uparrow[s]\);
\item\label{it:R-t2} \(\Phi(B;z_{(n,W)})\downarrow[s]\) and \(w>\varphi(B;z_{(n,W)})[s]\); or
\item\label{it:R-t3} \(\Phi(B;z_{(n,W)})\downarrow[s]\),
\(\mathsf{same}(U;\gamma_{\alpha_0}(BU;a)[s^*],s^*,s)\) and \(\mathsf{same}(B;\gamma_{\alpha_0}(BU;a)[s^*],s^*,s)\),
where \(s^*\le s\) is the last stage when we define \(\Phi(B;z_{(n,W)})\).
\end{enumerate}
Suppose that \(\beta\) is a \(Q(\eV,\eW)\)\nbd{}node. We define \(\test(\alpha,n;a,w)[s]=1\) if either~\ref{it:R-t1}, \ref{it:R-t2} or the following holds
\begin{enumerate}[resume*]
\item\label{it:R-t4} \(\Phi(B;z_{(n,W)})\downarrow[s]\),
\(\mathsf{same}(V;\gamma_{\alpha_1}(BV;a)[s^*],s^*,s)\) and \(\mathsf{same}(B;\gamma_{\alpha_1}(BV;a)[s^*],s^*,s)\),
where \(s^*\le s\) is the last stage when we define \(\Phi(B;z_{(n,W)})\).
\end{enumerate}
Define \(\Test(\alpha;a,w)[s]=1\) if for every \(n\) we have \(\test(\alpha,n;a,w)[s]=1\). \end{definition}
\begin{remark*} Informally, $\test(\alpha,n;a,w)[s]=1$ means that $w$ passed the test posed by $\alpha$ so that the computation $\Phi(B;z_{(n,W)})$ would not be affected by $w$ entering $W$. Items~\ref{it:R-t1} and~\ref{it:R-t2} are natural, whereas items~\ref{it:R-t3} and~\ref{it:R-t4} are toward that the computation $\Phi(B; z_{(n,W)})$ is now ``protected'' by the $U$- and $V$\nbd{}side respectively.
We could define $\test(\alpha,n;a,w)[s]=0$ if $\test(\alpha,n;a,w)[s]\neq 1$, but since we never use it, we leave it as it is. \end{remark*}
If \(\test(\alpha,n;a,w)=1\) via~\ref{it:R-t2}, then \(\test(\alpha,m;a,w)=1\) for each \(m<n\); if \(\test(\alpha,n;a,w)=1\) via~\ref{it:R-t1}, then \(\test(\alpha,m;a,w)=1\) for each \(m>n\). In fact, we have the following lemma to simplify the test, where the number \(n\) in the lemma is also easy to determine.
\begin{lemma}\label{lem:test}
At stage~\(s\), let \[
n = \max \{ m \mid \test(\alpha, m; a, w) = 1 \text{ via~\ref{it:R-t2}}\} + 1.
\]
Suppose $\test(\alpha, n; a, w) = 1$, then $\Test(\alpha; a, w) = 1$. \end{lemma}
\begin{proof}
Since \(\test(\alpha,n;a,w)=1\) via~\ref{it:R-t1} implies that \(\Test(\alpha;a,w)=1\), we assume, without loss of generality, that \(\alpha\) is a \(P\)\nbd{}node and \(\test(\alpha,n;a,w)=1\) via~\ref{it:R-t3}.
Hence we have \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;a)[s^*],s^*,s)\) and \(\mathsf{same}(B;\gamma_{\alpha_0}(BU;a)[s^*],s^*,s)\), where \(s^*\le s\) is the last stage when we define \(\Phi(B;z_{(n,W)})\).
Let \(m>n\) be such that \(\Phi(B;z_{(m,W)})\downarrow[s]\) and \(s^{**}\le s\) be the last stage when we define \(\Phi(B;z_{(m,W)})\). As \(s^{*}\le s^{**}\), therefore we have \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;a)[s^{**}],s^{**},s)\) and \(\mathsf{same}(B;\gamma_{\alpha_0}(BU;a)[s^{**}],s^{**},s)\). Hence \(\test(\alpha,m;a,w)=1\). \end{proof}
\subsection{Permit(X)-node}\label{sec:permit node} Suppose \(\alpha\) is a \(\mathop{\mathrm{Permit}}(X)\)\nbd{}node and the current stage is \(s\). Recall that at Step~(\ref{it:pc 3}) of permitting center, some point was sent to a \(\mathop{\mathrm{Permit}}(X)\)\nbd{}node.
\noindent \(\visit(\alpha)\): Let \(x=\dw_\beta(a)(\sigma)\) be the point sent by permitting center~(\ref{it:pc 3}), where \(\sigma=\chi(\beta;a)\), \(\per(\sigma)=\alpha\) and \(\type(\sigma)=X\). (We assume that \(x\) is not discarded yet). \begin{enumerate}
\item\label{it:Pi3 init Sigma3 2} We enumerate \(x\) into \(X\) and activate the \(d\)\nbd{}outcome of \(\sigma\).
\item Then \(\visit(\alpha\concat 0)\). \end{enumerate}
\begin{remark*}
Besides Convention~\ref{convention: init}, Item (1) above is also accompanied by an initialization as pointed out in Lemma~\ref{lem:cleared of P}. \end{remark*}
\subsection{Disarming process}\label{sec:disarm}
Let \(s\) be the current stage. Let \(\beta\) be a \(Q(\eV,\eW)\)\nbd{}node with \(\eW\neq \varnothing\), \(\alpha\) be an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, \(\xi\) be the unique \(C(\alpha_0,\alpha_3,\alpha_4)\)\nbd{}node with \(\xi\concat \omega \subseteq \alpha \subsetneq \alpha\concat n\subseteq \beta\), and \((\xi\concat m,\beta)\) is paired at some \(s_0 < s\).
\noindent \((\xi\concat m, \alpha\concat n,\beta)\)-DisarmingProcess: We disarm \(\beta\) immediately if one of the following events is observed at stage~\(s\): \begin{enumerate}[label=(DP-\arabic*)]
\item\label{it:DP-1} At the permitting center, some \(P\)- or \(Q\)\nbd{}node extending $\xi\concat l$ with $l \leq m$ receives attention.
\item\label{it:DP-2} At \(\mathop{\mathrm{Permit}}(U)\)\nbd{}node, both \(\lnot\,\mathsf{same}(U;\gamma_{\alpha_0}(BU;x_{(m,U)})[s_1],s_1,s)\) and \\
\(\mathsf{same}(B;\psi_\alpha(B;y_{(n,V)})[s_1],s_1,s)\) hold, where \(s_1<s\) is the last \(\alpha\)\nbd{}stage when we define \(\Psi_\alpha(B;y_{(n,V)})\).
\item\label{it:DP-3} At the node \(\xi\), the outcome to be visited is \(\xi\concat l\) for some \(l\le m\).
\item\label{it:DP-4} At the node \(\alpha\), \(\psi_\alpha(B;y_{(n,V)})\downarrow[s]<\gamma_{\alpha_0}(BU;x_{(m,U)})[s]\) and for the last \(\alpha\)\nbd{}stage~\(s_1<s\) we have either \(\psi_\alpha(B;y_{(n,V)})\uparrow[s_1]\) or \(\psi_\alpha(B;y_{(n,V)})\downarrow[s_1]\ge \gamma_{\alpha_0}(BU;x_{(m,U)})[s_1]\).
\item\label{it:DP-5} At the end of the stage when we maintain the global parameters, \(x_{(m,U)}[s-1]\neq x_{(m,U)}[s]\). \end{enumerate} If one of the events is observed, we say that \((\xi\concat m, \alpha\concat n,\beta)\)-DisarmingProcess is \emph{triggered}. The \((\xi\concat m, \alpha\concat n,\beta)\)-DisarmingProcess lasts until \((\xi\concat m,\beta)\) is canceled.
When \(\beta\) is disarmed, we discard its \(\Theta\) and \(\dw\). If there is a diagonalizing witness \(x=\dw_\beta(a)(\sigma)\) for some \(a\) and \(\sigma\in \eV\cup\eW\) having been sent to \(\per(\sigma)\) (as described in permitting center~(\ref{it:pc 3})), we also discard the diagonalizing witness \(x\).
\begin{remark*} \ref{it:DP-5} follows from~\ref{it:DP-1} and~\ref{it:DP-3} and we state it explicitly for convenience. In fact, \ref{it:DP-2} needs a bit of justification. Since~\ref{it:DP-5} implies \(x_{(m,U)}[s_1]=x_{(m,U)}[s]\), \(x_{(m,U)}\) in~\ref{it:DP-2} is legitimate.
There could be multiple disarming processes but they are working independently respecting Convention~\ref{convention: init}. The latter part of~\ref{it:DP-4} is there only to prevent one from disarming $\beta$ infinitely often if $\psi(B; y_{(n,V)})\downarrow < \gamma_{\alpha_0}(BU;x_{(m,U)})$.
\end{remark*}
\subsection{Construction}\label{sec:construction} At the beginning of stage~\(s\), we let the permitting center acts and then \(\visit(\lambda)\) where \(\lambda\) is the root of the priority tree \(\cT\). Meanwhile, the disarming processes are working in the background. We stop the current stage whenever \(\visit(\alpha)\) with \(\abs{\alpha}=s\).
This is the end of the construction.
\section{Verification}\label{sec:con and ver}
A \emph{path} of a tree is a collection of nodes such that they are pairwise comparable and are closed under initial segments. The standard way to pick up the \emph{``true path''} is to collect inductively those leftmost nodes that are visited infinitely often. In our construction, however, such nodes can be initialized infinitely often. Therefore we shall slightly modify the definition of the true path.
\begin{definition}\label{def:T star}
Let \(\cT\) be the priority tree and the construction be given as above. We define
\[
\cT^*=\{\alpha\in \cT\mid \text{\(\alpha\) is visited infinitely often and \(\alpha\) is \emph{injured} finitely often}\},
\]
where ``injured'' means ``initialized or disarmed''.
The leftmost \emph{infinite} path \(\rho\) of \(\cT^*\), if any, is the \emph{true path}. For \(\alpha\in\rho\), \(o\)\nbd{}outcome is the \emph{true outcome} if \(\alpha\concat o\in \rho\). \end{definition}
Clearly \(\lambda\in \cT^*\), \(\cT^*\) is a two-branching (the labels are not recursive) tree by Convention~\ref{convention: init}, and \(\cT^*\) is \(0''\)\nbd{}recursive. In Lemma~\ref{lem:T star path} we show that \(\cT^*\) has an infinite path. In fact, \(\cT^*\) can have multiple infinite paths, each of which is in fact as good as the true path --- we just pick one in order to discuss the index of a winning set (Lemma~\ref{lem:index}).
In Subsections~\ref{sec:R strategy} and~\ref{sec:S strategy}, we show that \(R\)- and \(S\)\nbd{}nodes can actually work as intended at each stage. Therefore nodes in \(\cT^*\) achieve their goals (Lemma~\ref{lem:T star}). In Subsection~\ref{sec:proof of main}, we argue that \(\cT^*\) has an infinite path \(\rho\) and all requirements for the set \(X\) (given by Lemma~\ref{lem:full requirement}) are satisfied.
\subsection{\texorpdfstring{Success of $R$-strategies}{success of R-strategies}}\label{sec:R strategy}
We will not distinguish between an \(R\)\nbd{}node and an \(R^{-}\)\nbd{}node in this section: the following definition also applies to an \(R^-\)\nbd{}node. Lemma~\ref{lem:R con} is strictly easier to verify (more precisely, Case~1 does not happen).
\begin{definition} [R-Condition]\label{def:R con} Let \(\alpha\) be an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. We say that \(R(\alpha,n)\)\nbd{}Condition holds at stage~\(t\) if either \begin{enumerate}
\item \(\Phi(B;z_{(n,W)})\uparrow[t]\), or
\item \(\Phi(B;z_{(n,W)})\downarrow[t]\) but \(\mathsf{same}(W,\varphi(B;z_{(n,W)})[s],s,t)\) where \(s\le t\) is the last stage when we define \(\varphi(B;z_{(n,W)})\). \end{enumerate} We say that \(R(\alpha)\)\nbd{}Condition holds if \(R(\alpha,n)\)\nbd{}Condition holds for each \(n\). \end{definition}
Whenever we define \(\Phi(B;z_{(n,W)})\) at a stage~\(s\), \(R(\alpha,n)\) holds at stage~\(s\) trivially.
The purpose of \(R\)\nbd{}Condition is to ensure the correctness of $\Phi$ defined at $R$:
\begin{lemma}[R-Correctness]\label{lem:R correct}
Let \(\alpha\) be an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, and assume that $R(\alpha)$\nbd{}Condition holds at every stage. For each \(\alpha\)\nbd{}stage~\(s\) and each \(n\le s\), if \(\Phi(B;z_{(n,W)})\downarrow[s]\), then \(\Phi(B;z_{(n,W)})\downarrow[s] = A(z_{(n,W)})[s]\). \end{lemma}
\begin{proof}
Fix an $\alpha$\nbd{}stage~\(s\) and a number $n$ such that \(\Phi(B;z_{(n,W)})\)[s] is defined.
Let \(s_0<s\) be the last $\alpha$\nbd{}stage when we defined \(\Phi(B;z_{(n,W)})\).
Therefore we have
\begin{enumerate}
\item \(\Phi(B;z_{(n,W)})[s_0]=A(z_{(n,W)})[s_0]\),
\item \(\mathsf{same}(B;\varphi(B;z_{(n,W)})[s_0],s_0,s)\), and
\item \(\varphi(B;z_{(n,W)})[s]=\varphi(B;z_{(n,W)})[s_0]>\gamma_{\alpha_2}(BW;z_{(n,W)})[s_0]\).
\end{enumerate}
Furthermore, since \(R(\alpha,n)\)\nbd{}Condition holds, we also have
\begin{enumerate}[resume]
\item \(\mathsf{same}(W;\varphi(B;z_{(n,W)})[s_0],s_0,s)\).
\end{enumerate}
Hence we have
\begin{align*}
\Phi(B;z_{(n,W)})[s] &= \Phi(B;z_{(n,W)})[s_0]\ \ \ (\mbox{by (2)})\\
&= A(z_{(n,W)})[s_0]\ \ \ (\mbox{by (1)})\\
&= \Gamma_{\alpha_2}(BW;z_{(n,W)})[s_0]\ \ (\mbox{since $s_0$ is $\alpha_2$\nbd{}expansionary})\\
&= \Gamma_{\alpha_2}(BW;z_{(n,W)})[s] \ \ \ (\mbox{by (2), (3) and (4)})\\
&= A(z_{(n,W)})[s] \ \ \ (\mbox{since $s$ is $\alpha_2$\nbd{}expansionary}). \qedhere
\end{align*} \end{proof}
By Lemma~\ref{lem:R correct}, it suffices to check that our construction ensures \(R\)\nbd{}condition.
Note that if \(\test(\alpha,n;a,w)[s]=1\) via~\ref{it:R-t1} or~\ref{it:R-t2}, enumerating \(w\) into \(W_{\alpha_0,\alpha_1}\) does not violate \(R(\alpha,n)\)\nbd{}condition; if \(\test(\alpha,n;a,w)[s]=1\) via~(R-t3), we will send \(w\) to \(\mathop{\mathrm{Permit}}(W_{\alpha_0,\alpha_1})\) and will enumerate \(w\) into \(W\) whenever \(\mathop{\mathrm{Permit}}(W_{\alpha_0,\alpha_1})\) is visited unless \(w\) is discarded.
\begin{lemma} [R-Condition Satisfaction]\label{lem:R con}
For each \(R\)\nbd{}node \(\alpha\) and each stage~\(t\), \(R(\alpha)\)\nbd{}Condition holds. \end{lemma}
\begin{proof} Let \(\alpha\) be an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, \(V=V_{\alpha_0}\) and \(W=W_{\alpha_0,\alpha_1}\). Let \(s < t\) be the last \(\alpha\)\nbd{}stage when we defined \(\varphi(B;z_{(n,W)})\). Suppose that we have \(\Phi(B;z_{(n,W)})\downarrow [t]\) but \(\lnot \mathsf{same}(W,\varphi(B;z_{(n,W)})[s],s,t)\). Then we have \begin{enumerate}
\item \(\mathsf{same}(B;\varphi(B;z_{(n,W)})[s];s,t)\),
\item \(\varphi(B;z_{(n,W)})[s]>\gamma_{\alpha_1}(BV;z_{(n,V)})[s]\),
\item \(\varphi(B;z_{(n,W)})[s]>\gamma_{\alpha_0}(BU;z_{(n,U)})[s]\), and
\item for some \(w\le \varphi(B;z_{(n,W)})[s]\), \(w\) is enumerated into \(W\), say, at some (least) stage~\(s_1\) with \(s<s_1\le t\). \end{enumerate}
We assume that at some stage~\(s_2\) with \(s<s_2\le s_1\), \(\beta\) receives attention due to~\ref{it:ph3} of the attacking process for some \(\beta\).
\textbf{Case 1.} Suppose that \(\beta\) is a \(P(\eU,\eW)\)\nbd{}node (therefore \(\alpha\) is not an \(R^-\)\nbd{}node) and \(\chi(\beta;a)=\xi\in \eW\) for some \(a\) such that \(\alpha\in \cf(\xi)\), \(\dw_\beta(a)(\xi)=w\) and \(\type(\xi)=W\). From (4) we conclude that \(\per(\xi)\) is visited at \(s_1\).
As \(w\le \varphi(B;z_{(n,W)})[s]\), we conclude that \(\beta\) extends \(\alpha\concat (m, W) \) for some \(m\le n\) and therefore \begin{enumerate}[resume]
\item \(a<z_{(n,U)}\) (by the setting of Section~\ref{sec:parameter}).
\end{enumerate} Since \(\Test(\alpha;a,w)=1\) at \(s_2\) and hence \(\test(\alpha,n;a,w)=1\), we must have~\ref{it:R-t3}, which implies \begin{enumerate}[resume]
\item \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;a)[s],s,s_2)\). \end{enumerate}
We observe that only \(P\)\nbd{}node can potentially enumerate a point into \(U\), and any two \(P\)\nbd{}nodes are always \(\prec\)\nbd{}comparable by Lemma~\ref{lem:comparable nodes}. Also, no \(P\)\nbd{}node of higher global priority than \(\beta\) receives attention between \(s_2\) and \(s_1\), because otherwise \(\beta\) would be initialized before we enumerate \(w\) into \(W\) at \(s_1\). Furthermore, any \(P\)\nbd{}node of lower global priority than \(\beta\) is initialized at \(s_2\). By (1), $\alpha\concat (n, W)$ is never visited between $s_2$ and $s_1$. Therefore, $w'$, prepared by any lower global priority $P$\nbd{}node between $s_2$ and $s_1$, must be larger than $\phi(B;z_{(n,W)})$, hence, $w' > \gamma_{\alpha_0}(BU;a)[s]$. Therefore we have \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;a),s_2,s_1)\). Together with (6), we have \begin{enumerate}[resume]
\item \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;a),s,s_1)\). \end{enumerate}
Note that \(\alpha_0\concat \infty \subseteq \mathop{\mathrm{Permit}}(W_{\alpha_0,\alpha_1})=\per(\xi)\). We have the following contradiction: \begin{eqnarray*}
0 = A(a)[s] &=& \Gamma_{\alpha_0}(BU;a)[s]\ \ \ (\mbox{since $s$ is $\alpha_0$\nbd{}expansionary})\\
&=& \Gamma_{\alpha_0}(BU;a)[s_1]\ \ \ (\mbox{by (1),(3),(5) and (7)})\\
&=& A(a)[s_1] = 1 \ \ \ (\mbox{since $s_1$ is $\alpha_0$\nbd{}expansionary}).\\ \end{eqnarray*}
\textbf{Case 2.} (This case varies only slightly.) Suppose that \(\beta\) is a \(Q(\eV,\eW)\)\nbd{}node and \(\chi(\beta;a)=\xi\in \eW\) for some \(a\) such that \(\alpha\in \cf(\xi)\), \(\dw_\beta(a)(\xi)=w\) and \(\type(\xi)=W\). For the same reason as in Case~1 we have \begin{enumerate}[resume]
\item \(a<z_{(n,V)}\). \end{enumerate}
Since \(\Test(\alpha;a,w)=1\) at \(s_2\) and hence \(\test(\alpha,n;a,w)=1\), we must have~\ref{it:R-t4}, which implies \begin{enumerate}[resume]
\item \(\mathsf{same}(V;\gamma_{\alpha_1}(BV;a)[s],s,s_2)\). \end{enumerate}
To show \(\mathsf{same}(V;\gamma_{\alpha_1}(BV;a)[s],s_2,s_1)\), we first realize that only a \(Q(\eV',\eW')\)\nbd{}node \(\beta'\) with \(\eV'=\{\tau\}\) and \(\type(\tau)=V_{\alpha_0}\) potentially enumerates a point into the set \(V=V_{\alpha_0}\). By Lemma~\ref{lem:comparable nodes}, \(\beta\) and \(\beta'\) are \(\prec\)\nbd{}comparable. Such \(Q\)\nbd{}nodes of lower global priority than \(\beta\) are initialized at \(s_2\) and those of higher priorities, if received attention before or at the beginning of stage~\(s_1\), would initialize \(\beta\). Therefore we have \(\mathsf{same}(V;\gamma_{\alpha_1}(BV;a)[s],s_2,s_1)\). Together with ($9$), we conclude that \begin{enumerate}[resume]
\item \(\mathsf{same}(V;\gamma_{\alpha_1}(BV;a)[s],s,s_1)\) \end{enumerate}
Note that \(\alpha_1\concat \infty\subseteq \mathop{\mathrm{Permit}}(W_{\alpha_0,\alpha_1})=\per(\xi)\). We have the following contradiction: \begin{align*} 0 = A(a)[s] &= \Gamma_{\alpha_1}(BV;a)[s]\ \ \ (\mbox{since $s$ is $\alpha_1$\nbd{}expansionary})\\ &= \Gamma_{\alpha_1}(BV;a)[s_1]\ \ \ (\mbox{by (1), (2), (8) and (10)})\\ &= A(a)[s_1]=1\ \ \ (\mbox{since $s_1$ is $\alpha_1$\nbd{}expansionary}). \qedhere \end{align*}
\end{proof}
\subsection{\texorpdfstring{Success of $S$-strategies}{success of S-strategies}}\label{sec:S strategy} \begin{definition} [S-Condition]\label{def:S con}
Let \(\alpha\) be an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. For each $n\in \omega$, we say that \(S(\alpha,n)\)\nbd{}Condition holds at stage~\(t\) if either
\begin{enumerate}
\item \(\Psi(B;y_{(n,V)})\uparrow[t]\), or
\item \(\Psi(B;y_{(n,V)})\downarrow[t]\) but we also have the following: (Here, \(s\le t\) is the last stage~when we defined \(\Psi(B;y_{(n,V)})\).)
\begin{enumerate}
\item\label{it:S con V} If \(\psi(B;y_{(n,V)})[s]<\gamma_{\alpha_0}(BU;y_{(n,V)})[s]\), then\\
\(\mathsf{same}(V;\psi(B;y_{(n,V)})[s],s,t)\).
\item\label{it:S con UV} If \(\psi(B;y_{(n,V)})[s] \geq \gamma_{\alpha_0}(BU;y_{(n,V)})[s]\), then either \\
\(\mathsf{same}(V;\psi(B;y_{(n,V)})[s],s,t)\) or \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;y_{(n,V)})[s],s,t)\).
\end{enumerate}
\end{enumerate}
We say that \(S(\alpha)\)\nbd{}Condition holds if \(S(\alpha,n)\)\nbd{}Condition holds for each \(n\). \end{definition}
\begin{remark*} The objective of the S-condition is to guarantee the correctness of the value of $\Psi(B; y_{(n,V)})$. This is done by ensuring $\psi(B;y)$ is bigger than either $\gamma_{\alpha_0}(BU;y_{(n,V)})[s]$ or $\gamma_{\alpha_1}(BV;y_{(n,V)})[s]$. In the case of Definition~\ref{def:S con}(\ref{it:S con V}), we have no choice but to prevent \(V\) from changing, whereas in the case of~(\ref{it:S con UV}) we are more flexible and we can ``preserve'' either $U$\nbd{}side or $V$\nbd{}side, a situation similar to the minimal pair construction.
\end{remark*}
By a similar argument as in Lemma~\ref{lem:R correct}, one can prove the following: \begin{lemma} [S-Correctness]\label{lem:S correct}
Let \(\alpha\) be an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. Assume that $S(\alpha)$\nbd{}Condition holds at every stage.
Then for each \(\alpha\)\nbd{}stage and each \(n\le s\), if \(\Psi(B;y_{(n,V)})\downarrow[s]\), then \(\Psi(B;y_{(n,V)})\downarrow[s]=A(x)[s]\). \end{lemma}
\begin{proof}
Fix an $\alpha$\nbd{}stage~\(s\) and a number $n$ such that \(\Psi(B;y_{(n,V)})[s]\) is defined.
Let \(s_0<s\) be the last $\alpha$\nbd{}stage when we defined \(\Psi(B;y_{(n,V)})\).
Therefore we have
\begin{enumerate}
\item \(\Psi(B;y_{(n,V)})[s_0]=A(y_{(n,V)})[s_0]\),
\item \(\mathsf{same}(B;\psi(B;y_{(n,V)})[s_0],s_0,s)\), and
\item \(\psi(B;y_{(n,V)})[s]=\psi(B;y_{(n,V)})[s_0]>\gamma_{\alpha_1}(BV;y_{(n,V)})[s_0]\).
\end{enumerate}
By the assumption that \(S(\alpha,n)\)\nbd{}Condition holds, either Definition~\ref{def:S con}(\ref{it:S con V}) or (\ref{it:S con UV}) holds.
Let us first prove the former case, where we have
\begin{enumerate}[resume]
\item \(\mathsf{same}(V;\psi(B;y_{(n,V)})[s_0],s_0,s)\).
\end{enumerate}
Hence we have
\begin{eqnarray*}
\Psi(B;y_{(n,V)})[s]& = & \Psi(B;y_{(n,V)})[s_0] \ \ \ (\mbox{by (2)})\\
&=& A(y_{(n,V)})[s_0] \ \ \ (\mbox{by (1)})\\
&=& \Gamma_{\alpha_1}(BV;y_{(n,V)})[s_0]\ \ (\mbox{since \(s_0\) is \(\alpha_1\)\nbd{}expansionary})\\
&=& \Gamma_{\alpha_1}(BV;y_{(n,V)})[s]\ \ \ (\mbox{by (2), (3) and (4)})\\
&=& A(y_{(n,V)})[s] \ \ \ (\mbox{since \(s\) is \(\alpha_1\)\nbd{}expansionary}).
\end{eqnarray*}
For the latter case Definition~\ref{def:S con}(\ref{it:S con UV}), we have
\begin{enumerate}[resume]
\item \(\mathsf{same}(V;\psi(B;y_{(n,V)})[s_0],s_0,s)\), or
\item \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;y_{(n,V)})[s_0],s_0,s)\).
\end{enumerate}
In the case of (5), the proof is exactly the same as above. In the case of (6), we have
\begin{align*}
\Psi(B;y_{(n,V)})[s]& =\Psi(B;y_{(n,V)})[s_0] \ \ \ (\mbox{by (2)})\\
&=A(y_{(n,V)})[s_0] \ \ \ (\mbox{by (1)})\\
&=\Gamma_{\alpha_0}(BU;y_{(n,V)})[s_0]\ \ (\mbox{since \(s_0\) is \(\alpha_0\)\nbd{}expansionary})\\
&=\Gamma_{\alpha_0}(BU;y_{(n,V)})[s]\ \ \ (\mbox{by (2), (6) and
Definition~\ref{def:S con}(\ref{it:S con UV})})\\
&=A(y_{(n,V)})[s] \ \ \ (\mbox{since \(s\) is \(\alpha_0\)\nbd{}expansionary}). \qedhere
\end{align*} \end{proof}
\begin{lemma} [S-Condition Satisfaction]\label{lem:S con}
For each \(S\)\nbd{}node \(\alpha\) and each stage~\(t\), \(S(\alpha)\)\nbd{}Condition holds. \end{lemma}
\begin{proof} Fixing an \(n\), we show that \(S(\alpha,n)\)\nbd{}Condition holds at stage~\(t\).
Suppose that \(\alpha\) is \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node, where \(\alpha_0\) is the \(G(U)\)\nbd{}node, \(\alpha_1\) is the \(G(V_{\alpha_0})\)\nbd{}node and \(\alpha_2\) is the \(D(W_{\alpha_0,\alpha_1})\)\nbd{}node. We write \(V=V_{\alpha_0}\) and \(W=W_{\alpha_0,\alpha_1}\). Let \(s\le t\) be the last \(\alpha\)\nbd{}stage when we defined \(\Psi(B;y_{(n,V)})\) (and \(\alpha\) has not been initialized between \(s\) and \(t\)). Since \(S(\alpha,n)\)\nbd{}Condition trivially holds if \(\Psi(B;y_{(n,V)})\uparrow[t]\), we assume \(\Psi(B;y_{(n,V)})\downarrow[t]\). We have
\begin{enumerate}
\item\label{it:s1} \(\psi(B;y_{(n,V)})[s]>\gamma_{\alpha_1}(BV;y_{(n,V)})[s]\) and
\item\label{it:s2} \(\mathsf{same}(B;\psi(B;y_{(n,V)})[s],s,t)\).
\end{enumerate}
As a consequence of~(\ref{it:s2}), we do not visit \(\alpha\concat (m,V)\) for \(m\le n\) between \(s\) and \(t\).
By Lemma~\ref{lem:S locate C}, there exists the unique \(C(\alpha_0,\alpha_3,\alpha_4)\)\nbd{}node \(\xi\) for some \(\alpha_3\) and \(\alpha_4\) with \(\xi\concat \omega\subseteq \alpha\). We list, if any, all paired \(Q\)\nbd{}nodes extending \(\alpha\concat (n, V)\) as
\[
\beta_0\prec \beta_1 \prec\cdots\prec\beta_{k-1}.
\]
Let \((\xi\concat n_i,\beta_i)\) be the established pair for each \(i<k\).
By the definition of pairing parameter (Section~\ref{sec:parameter} and Subsection~\ref{sec:Q node}), we have
\begin{enumerate}[resume]
\item\label{it:s3} \(y_{(n,V)} \le \tp(\beta_0) < x_{(n_0,U)}[s]<x_{(n_1,U)}[s]<\cdots < x_{(n_{k-1},U)}[s]\).
\end{enumerate}
Corresponding to clauses~(\ref{it:S con V}) and~(\ref{it:S con UV}) in \(S(\alpha,n)\)\nbd{}Condition (see Definition~\ref{def:S con}), our proof splits into two cases.
\textbf{Case 1.} Suppose \(\psi(B;y_{(n,V)})[s]<\gamma_{\alpha_0}(BU;y_{(n,V)})[s]\).
We show \(\mathsf{same}(V;\psi(B;y_{(n,V)})[s],s,t)\). Let \(s_1\) be the least stage between \(s\) and \(t\) at which some number \(a\) enters \(A\) and some \(Q\)\nbd{}node \(\beta\) receives attention.
The goal is to show that \(\beta\) will not enumerate any point less than \(\psi(B;y_{(n,V)})[s]\) into \(V\).
First of all, by the choice of \(s_1\), we have
\begin{enumerate}[resume]
\item\label{it:s4} \(\mathsf{same}(V;\psi(B;y_{(n,V)})[s],s,s_1)\).
\end{enumerate}
We may assume that \(\beta\) extends \(\alpha\concat (m,V)\) for some \(m\) as other cases are simpler. Suppose that \(w=\dw_\beta(a)(\alpha_2)\) and \(v=\dw_\beta(a)(\beta^-)<\psi(B;y_{(n,V)})[s]\) are two diagonalizing witnesses of \(\beta\).
\begin{claim}\label{cl:1}
\(m<n\).
\end{claim}
\begin{proof}[Proof of Claim~\ref{cl:1}]
If \( m > n\), the diagonalizing witnesses (particularly, \(v\)) of \(\beta\) is larger than \(\psi(B;y_{(n, V)})[s]\), which contradicts our assumption.
Now we show \(m\neq n\).
We have \(\psi(B;y_{(n,V)})[s] <\gamma_{\alpha_0}(BU;y_{(n,V)})[s] <\gamma_{\alpha_0}(BU;x_{(n_0,U)})[s]\) by the assumption of Case~1 and~(\ref{it:s3}). By \((\xi\concat n_0,\alpha\concat n,\beta_0)\)\nbd{}DisarmingProcess~\ref{it:DP-4}, \(\beta_0\) has been disarmed by $s$ (and \(\beta_i\) for \(i>0\) are also initialized).
As we do not visit $\alpha\concat (n,V)$ between $s$ and $t$, we have \(m \neq n\).
\end{proof}
By Claim~\ref{cl:1} and the definition of \(y_{(n,V)}\) (see Section~\ref{sec:parameter}), we have \(a<y_{(n,V)}\). Therefore we have \(\psi(B;y_{(n,V)})[s]>\gamma_{\alpha_1}(BV;y_{(n,V)})[s]>\gamma_{\alpha_1}(BV;a)[s]\) by~(\ref{it:s1}). By~(\ref{it:s2}) and~(\ref{it:s4}) respectively, we have
\begin{enumerate}[resume]
\item\label{it:s5} \(\mathsf{same}(B;\gamma_{\alpha_1}(BV;a)[s],s,s_1)\), and
\item\label{it:s6} \(\mathsf{same}(V;\gamma_{\alpha_1}(BV;a)[s],s,s_1)\).
\end{enumerate}
\begin{claim}\label{cl:2}
For each \(\eta\in \cf(\alpha_2)\), \(\Test(\eta;a,w)=1\).
\end{claim}
\begin{proof}[Proof of Claim~\ref{cl:2}]
Let \(l\) be such that \(\eta\concat (l,W)\subseteq\alpha_2\). It suffices to show \(\test(\eta,l;a,w)=1\) by Lemma~\ref{lem:test}.
As it is an $\alpha$\nbd{}stage, $s$ is also an $\eta\concat (l, W)$\nbd{}stage, which implies \(\Phi_\eta(B;z_{(l,W)})\uparrow[s]\).
If \(\Phi_\eta(B;z_{(l,W)})\uparrow[s_1]\), then $\test(\eta,l;a,w)[s_1] = 1$ by Definition~\ref{def:R test}~\ref{it:R-t1}. Otherwise, there exists some stage~$s^*$ such that $s < s^* < s_1$ and $s^*$ is the last stage that \(\Phi_\eta(B;z_{(l,W)})\downarrow[s^*]\). With~(\ref{it:s5}) and~(\ref{it:s6}), we conclude \(\test(\eta,l;a,w)[s_1]=1\) by Definition~\ref{def:R test} (R-t4).
\end{proof}
By Claim~\ref{cl:2} we have \(\bD(\beta,a)[s_1]\subseteq \alpha_2\). In particular, \(\bD(\beta,a)[s_1]\neq \beta^-\) and therefore \(\beta\) will not enumerate \(v\) into \(V\). Hence we have \(\mathsf{same}(V;\psi(B;y_{(n,V)})[s],s,t)\), which completes the proof of Case 1.
\textbf{Case 2.} Suppose \(\psi(B;y_{(n,V)})[s]\ge \gamma_{\alpha_0}(BU;y_{(n,V)})[s]\).
Depending on whether \(U\) or \(V\) changes first, we have two subcases.
\textbf{Case 2(a).} \(U\) changes first. We are to show \(\mathsf{same}(V;\psi(B;y_{(n,V)})[s],s,t)\). First of all, let~\(s_1\) be the least stage between \(s\) and \(t\) such that \(\mathsf{same}(V;\psi(B;y_{(n,V)}),s,s_1)\) and \(\lnot\,\mathsf{same}(U;\gamma_{\alpha_0}(BU,y_{(n,V)})[s],s,s_1)\). As we have \(y_{(n,V)} < x_{(n_0, U)}\) by~(\ref{it:s3}), we have \(\lnot\,\mathsf{same}(U;\gamma_{\alpha_0}(BU,x_{(n_0, U)})[s],s,s_1)\), which triggers \((\xi\concat n_0, \alpha\concat n,\beta_0)\)\nbd{}DisarmingProcess~\ref{it:DP-2} at \(s_1\) and therefore \(\beta_0\) is disarmed (and \(\beta_i\) for \(i>0\) are initialized).
Let \(\beta\) be a \(Q\)\nbd{}node receiving attention (say, \(a\) enters \(A\)) between \(s\) and \(t\), extending \(\alpha\concat (m,V)\) for some \(m\). By an argument similar to Claim~\ref{cl:1} and Claim~\ref{cl:2}, we conclude \(\bD(\beta,a)\subseteq\alpha_2\). Hence \(\mathsf{same}(V;\psi(B;y_{(n,V)})[s],s,t)\).
\textbf{Case 2(b).} \(V\) changes first. We are to show \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;y_{(n,V)})[s],s,t)\). Let \(s_1\) be the least stage between \(s\) and \(t\) such that \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;y_{(n,V)})[s],s,s_1)\) and \(\lnot\,\mathsf{same}(V;\psi(B;y_{(n,V)}),s,s_1)\). We first examine what happens at \(s_1\). Let \(\beta\) be the \(Q\)\nbd{}node who is responsible for this change. That is, \(\beta\) receives attention (say, \(a\) enters \(A\)) at \(s_2\le s_1\) with \(\bD(\beta,a)=\beta^-\) and therefore enumerates its diagonalizing witness \(v=\dw_\beta(a)(\beta^-)<\psi(B;y_{(n,v)})\) into \(V\) at \(s_1\).
\begin{claim}\label{cl:3}
\(\beta\) extends \(\alpha \concat (n,V)\).
\end{claim}
\begin{proof}[Proof of Claim~\ref{cl:3}]
Suppose that \(\beta\) extends \(\alpha\concat (m,V)\) for some \(m\). If \(m>n\), then we have \(v>\psi(B;y_{(n,v)})\), contradicting the assumption. If \(m<n\), by a similar argument in Claim~\ref{cl:2}, we conclude \(\bD(\beta,a)\subseteq\alpha_2\), contradicting \(\bD(\beta,a)=\beta^-\).
\end{proof}
By Claim~\ref{cl:3}, we have \(\beta=\beta_j\) for some \(j<k\). Therefore, we have
\begin{enumerate}[resume]
\item\label{it:s7} \(\gamma_{\alpha_0}(BU;x_{(n_j, U)})[s] \leq \psi(B;y_{(n,V)}) [s]\) (as otherwise, the \((\xi\concat n_j, \alpha\concat n,\beta_j)\)\nbd{}DisarmingProcess~\ref{it:DP-4} would have disarmed \(\beta_j\) by \(s\));
\item\label{it:s7b} \(\mathsf{same}(B;\gamma_{ \alpha_0}(BU; x_{(n_j,U)})[s], s, t)\) (by~(\ref{it:s2}) and~(\ref{it:s7})); and
\item\label{it:s8} all P-nodes below \(\xi\concat n_j\) are initialized at \(s_1\) by Lemma~\ref{lem:cleared of P}.
\end{enumerate}
Let \(s_3\) be the least stage between \(s\) and \(t\) such that \(b\) enters \(A\) and some P\nbd{}node \(\iota\) receives attention with diagonalizing witnesses \(w=\dw_\iota(b)(\alpha_4)\) and \(u=\dw_\iota(b)(\iota^-)<\gamma_{\alpha_0}(BU;x_{(n_j,U)})\).
Suppose that \(\iota\) extends \(\xi\concat l\) for some \(l\).
We have
\begin{claim}\label{cl:4}
\(l<n_j\) and \(s_3>s_1\).
\end{claim}
\begin{proof}[Proof of Claim~\ref{cl:4}]
If \(l> n_j\), then we have \(u>\gamma_{\alpha_0}(BU;x_{(n_j,U)})\) contradicting the assumption. If \(l\le n_j\) and \(s_3\le s_1\), then \(\beta=\beta_j\) would be disarmed by \((\xi\concat n_j, \alpha\concat n,\beta_j)\)\nbd{}DisarmingProcess~\ref{it:DP-1} at \(s_3\). If \(l=n_j\) and \(s_3>s_1\), we have a contradiction as \(\iota\) would have been initialized at \(s_1\) by~(\ref{it:s8}) (and~(\ref{it:s7b}) prevents us from visiting \(\iota\) after~\(s_1\)). Therefore, we have \(l<n_j\) and \(s_3>s_1\).
\end{proof}
By Claim~\ref{cl:4}, we have \(b<x_{(n_j,U)}\) and \(\gamma_{\alpha_0}(BU;b)[s]<\gamma_{\alpha_0}(BU;x_{(n_j,U)})[s]\). Now we have
\begin{enumerate}[resume]
\item\label{it:s9} \(\mathsf{same}(B; \gamma_{\alpha_0}(BU; b)[s], s, s_3)\) (by~(\ref{it:s7b})), and
\item\label{it:s10b} \(\mathsf{same}(U; \gamma_{\alpha_0}(BU; b)[s], s, s_3)\) (by the choice of \(s_3\)).
\end{enumerate}
From~(\ref{it:s9}) and~(\ref{it:s10b}), following the argument in Claim~\ref{cl:2}, we have \(\Test(\eta;b,w)=1\) for each \(\eta\in \cf(\alpha_4)\). Hence, we have \(\bD(\iota,b)\subseteq \alpha_4\). In particular, \(\bD(\iota,b)\neq \iota^-\) and therefore \(\iota\) will not enumerate \(u\) into \(U\). Hence we have \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;x_{(n_j,U)})[s],s,t)\). By~(\ref{it:s3}), we have \(x_{(n_j,U)}>y_{(n,V)}\). Hence we have \(\mathsf{same}(U;\gamma_{\alpha_0}(BU;y_{(n,V)})[s],s,t)\). This completes the proof of Case~2(b) and the proof of this lemma. \end{proof}
\subsection{Proof of the main theorem}\label{sec:proof of main} The goal is to show that \(\cT^*\) has an infinite path. The following notations will be convenient in the proof. For a global parameter \(x\), we say that \(x\) is \emph{stable} after stage~\(s\) if for each \(t\ge s\), \(x[t]=x[s]\). A pair \((\xi\concat n, \beta)\) is \emph{stable} after stage~\(s\) if it will not get canceled after stage~\(s\). If \(x_{(n,U)}\) is stable after \(s\), then \(\gamma(BU;x_{(n,U)})\) is \emph{stable} after \(s_0>s\) if for each \(t>s_0\), \(\gamma(BU;x_{(n,U)})[t]=\gamma(BU;x_{(n,U)})[s_0]\). \(\gamma(BV;y_{(n,V)})\) or \(\gamma(BW;z_{(n,X)})\) being \emph{stable} are defined similarly. A set \(X\res l\) is \emph{stable} after \(s_0\) if for each \(s>s_0\) we have \(\mathsf{same}(X,l,s_0,s)\).
By Lemma~\ref{lem:R correct} and~\ref{lem:S correct}, we have the following \begin{lemma}\label{lem:T star}
Let \(\alpha\in \cT^*\) be a node.
\begin{enumerate}
\item Suppose that \(\alpha\) is a \(\mathop{\mathrm{Permit}}(X)\)\nbd{}node. Then \(X\le_T A\).
\item Suppose that \(\alpha\) is a \(G_e(X)\)\nbd{}node and \(\alpha\concat 0\in T^*\). Then \(\Gamma_e(BX)\neq A\).
\item Suppose that \(\alpha\) is a \(P\)\nbd{}node. Then \(A\) is recursive.
\item Suppose that \(\alpha\) is a \(Q\)\nbd{}node. Then \(A\) is recursive.
\item Suppose that \(\alpha\) is a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node and \(\alpha\concat n\in \cT^*\). Then \(\Gamma_{\alpha_0}(BU;x_{(n,U)})\uparrow\).
\item Suppose that \(\alpha\) is an \(R(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. Let \(V=V_{\alpha_0}\) and \(W=W_{\alpha_0,\alpha_1}\). If \((n,X)\) is the leftmost outcome that is visited infinitely often, then \(\Gamma_{\alpha_i}(BX;x_{(n,X)})\uparrow\) where \(i=0,1,2\) if \(X=U,V,W\) respectively; if all outcomes are visited finitely often, then \(A\le_T B\).
\item Suppose that \(\alpha\) is an \(R^-(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. Let \(V=V_{\alpha_0}\) and \(W=W_{\alpha_0,\alpha_1}\).
If \((n,X)\) is the leftmost outcome that is visited infinitely often, then \(\Gamma_{\alpha_i}(BX;x_{(n,X)})\uparrow\) where \(i= 1,2\) if \(X= V,W\) respectively;
if all outcomes are visited finitely often, then \(A\le_T B\).
\item Suppose that \(\alpha\) is an \(S(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node. Let \(V=V_{\alpha_0}\). If \((n,V)\) is the leftmost outcome that is visited infinitely often, then \(\Gamma_{\alpha_1}(BV;y_{(n,V)})\uparrow\); if all outcomes are visited finitely often, then \(A\le_T B\).
\item Suppose that \(\alpha\) is a \(D(X)\)\nbd{}node. If the leftmost outcome that is visited infinitely often is a \(d\)\nbd{}outcome or a \(w\)\nbd{}outcome, then \(\Delta_\alpha\neq X\). \qed
\end{enumerate} \end{lemma}
\begin{lemma}\label{lem:T star C Q}
Suppose that \(\xi\) is a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node such that \(\xi\in \cT^*\) and \(\xi\concat n\notin \cT^*\) for each \(n\)\@. Suppose that \(\beta\) is a \(Q\)\nbd{}node such that \(\xi\concat \omega \subseteq \beta\) and \(\beta\) is initialized finitely often. Then \(\beta\) is disarmed finitely often. \end{lemma}
\begin{proof} Let \(V=V_{\alpha_0}\). Let \(s_0\) be the last stage when \(\beta\) is initialized (and \(\xi\) is not going to be initialized since \(\xi\in \cT^*\)). Suppose that \(\beta\) is not visited after \(s_0\), it is not paired and therefore it is not going be disarmed. Therefore we assume that \(\beta\) is visited after \(s_0\) and \((\xi\concat n,\beta)\) is the stable pair. Consider \((\xi\concat n,\alpha\concat l,\beta)\)\nbd{}DisarmingProcess for some \(S\)\nbd{}node \(\alpha\). We show that \((\xi\concat n,\alpha\concat l,\beta)\)-DisarmingProcess can be triggered finitely often.
Note first that since \(\beta\) is not initialized after \(s_0\), we are not going to visit any node \(\gamma\) extending \(\xi\concat \omega\) with \(\gamma\prec \beta\). Therefore the edge parameter \(y_{(l,V)}\) is stable for \(\alpha\).
\textbf{Case 1.} Suppose that for each \(l\le n\), \(\xi\concat l\) is visited finitely often. Then~\ref{it:DP-1} and~\ref{it:DP-3} are triggered finitely often. Since we also have \(x_{(n,U)}\) and \(\gamma_{\alpha_0}(BU;x_{(n,U)})\) stable,~\ref{it:DP-5},~\ref{it:DP-2} and~\ref{it:DP-4} are triggered finitely often.
\textbf{Case 2.} Suppose that \(l\le n\) is the least such that \(\xi\concat l\) is visited infinitely often toward a contradiction. Since \(\xi\concat l\notin\cT^*\), \(\xi\concat l\) is initialized infinitely often. Let \(k\le l\) be the least such that \(\xi\concat k\) is initialized infinitely often. As for each \(m<k\), \(\xi\concat m\) is visited finitely often, only~\ref{it:p2} or~\ref{it:p3} are possible to initialize \(\xi\concat k\) infinitely often. Suppose that~\ref{it:p2} happens after \(s_0\). If \(k<n\), then \(\beta\) would be initialized, contradicting the choice of \(s_0\); if \(k=n\), then \((\xi\concat n, \beta)\) must have been canceled, contradicting that \((\xi\concat n,\beta)\) is a stable pair. We are left with the possibility that~\ref{it:p3} happens infinitely often. We assume that \((\xi\concat (k-1),\beta')\) is the pair that triggers~\ref{it:p3} infinitely often. If \(\beta'\prec \beta\), then \(\beta'\) announcing progress would initialize \(\beta\), contradicting the choice of \(s_0\); if \(\beta'=\beta\), then \(k-1=n\), contradicting \(k\le l\le n\). \end{proof}
\begin{lemma}\label{lem:T star C alpha}
Suppose that \(\xi\) is a \(C(\alpha_0,\alpha_1,\alpha_2)\)\nbd{}node such that \(\xi\in \cT^*\) and \(\xi\concat n\notin \cT^*\) for each \(n\)\@. Suppose that \(\alpha\in \cT^*\) is a node such that \(\xi\concat \omega \subseteq \alpha\) and \(\alpha\) is not a \(Q\)\nbd{}node and \(o\) is the leftmost outcome that is visited infinitely often. Then \(\alpha\concat o\in \cT^*\)\@. \end{lemma}
\begin{proof}
Let \(s_0\) be the least stage after which \(\alpha\) is not initialized (as \(\alpha\in \cT^*\)). Let \(s_1\ge s_0\) be the least stage after which nodes (in \(\cT[\xi,\Pi_3]\)) to the left of \(\alpha\concat o\)
\begin{enumerate}
\item are never visited (by the choice of~\(o\)),
\item never become activated,
\item never receive attention (as the definitions of \(\Theta\)\nbd{}functionals of \(Q\)\nbd{}node are not going be extended),
\item are never initialized, and
\item are never disarmed (by Lemma~\ref{lem:T star C Q} and~(4) above).
\end{enumerate}
Referring to Definition~\ref{def:announces progress}, we conclude that nodes (in \(\cT[\xi,\Pi_3]\)) to the left of \(\alpha\concat o\) will not announce progress after~\(s_1\) and therefore \(\alpha\concat o\) will not be initialized after~\(s_1\).
If \(\alpha\concat o\) is not a \(Q\)\nbd{}node, then we immediately have \(\alpha\concat o\in \cT^*\);
if \(\alpha\concat o\) is a \(Q\)\nbd{}node, we use Lemma~\ref{lem:T star C Q} to conclude that \(\alpha\concat o\) is disarmed only finitely often and hence \(\alpha\concat o\in \cT^*\).
This completes the proof. \end{proof}
\begin{lemma}\label{lem:T star path}
Suppose \(A\nleq_T B\), then \(\cT^*\) has an infinite path. \end{lemma}
\begin{proof}
We recursively define a path \(\rho\subseteq \cT^*\). We enumerate the root \(\lambda\) into \(\rho\). Suppose that we have enumerated \(\alpha\) into \(\rho\). Let \(o\) be the leftmost path that is visited infinitely often (whose existence is guaranteed by Lemma~\ref{lem:T star} when applicable), then \(\alpha\concat o\in \cT^*\) (by Lemma~\ref{lem:T star C alpha} when applicable) and we enumerate \(\alpha\concat o\) into \(\rho\). \end{proof}
\begin{proof} (of Theorem~\ref{thm:main}) Let \(\rho\) be an infinite path of \(\cT^*\) given by Lemma~\ref{lem:T star path}. Let \(X\) be the set given by Lemma~\ref{lem:full requirement}. By Lemma~\ref{lem:T star}, \(\deg X\) is the desired degree. \end{proof}
\begin{remark*}
\(\cT^*\) can possibly have countably many infinite paths and each infinite path of \(\cT^*\) gives us a successful candidate for Theorem~\ref{thm:main}! Having exhibited a successful candidate \(X\), we know that all nonrecursive sets that are Turing reducible to \(X\) are also successful candidates for our Theorem~\ref{thm:main}. \end{remark*}
\section{Finding the index of a solution}\label{sec:index} \begin{theorem}\label{lem:index}
Given r.e.\ sets \(A=W_a\) and \(B=W_b\) with \(A\nleq_T B\), there is a function \(f\le_T 0^{(4)}\) such that \(0<_T W_{f(a,b)}\le_T A\) and \(A\nleq_T B\oplus W_{f(a,b)}\)\@. \end{theorem} \begin{proof}
Let \(\rho\) be the true path (Definition~\ref{def:T star}). We use \(0^{(4)}\) to decide if \(A\le_T B\) or not. If \(A\le_T B\), we define \(f(a,b)=0\). If \(A\nleq_T B\), we will use \(0^{(4)}\) to decide which set is \emph{good} (as defined in the first line of the proof of Lemma~\ref{lem:full requirement}).
First of all, the jump of \(\cT^*\) and hence \(0'''\) can compute the true path~\(\rho\).
Next, we use \(\rho'\le_T 0^{(4)}\) to decide whether there is a \(C\)\nbd{}node~\(\xi\) with \(\xi\concat \omega\in \rho\).
Note that there can be at most one such \(C\)\nbd{}node. Case (i), such \(C\)\nbd{}node~\(\xi\) exists. With~\(\xi\) as a parameter, \(\rho\) becomes \(\le_T 0''\). We start with \(\alpha=\xi\concat \omega\) and use \(\rho''\le_T 0^{(4)}\) to decide whether there are infinitely many nodes \(\beta\in \rho\) such that \(\type(\beta)=\type(\alpha)\). If yes, define \(f(a,b)\) to be the index of~\(X\); if not, we proceed to the next node in~\(\rho\) and repeat. This procedure terminates by Lemma~\ref{lem:full requirement}.
Case (ii), Such \(C\)\nbd{}node~\(\xi\) does not exist. By a similar argument, the index of the good set can be found using~\(0^{(4)}\). \end{proof}
In fact, with a very short proof, we are able to exhibit such an~\(f\) without referring to a concrete construction. \begin{proof}[Another proof.]
We write \[
\cS_{a,b}=\{e\mid 0<_T W_e\le_T W_a \land W_a\nleq_T W_b\oplus W_e\}.
\]
We have shown in Theorem~\ref{thm:main} that
\[
\forall a,b(W_a\nleq_T W_b \iff \cS_{a,b}\neq \varnothing).
\]
As \(\cS_{a,b}\le_T 0^{(4)}\) we can define the \(0^{(4)}\)\nbd{}recursive function \[
f(a,b)=\begin{cases}
0, & W_a\leq_T W_b;\\
\min \cS_{a,b}, & W_a\nleq_T W_b.
\end{cases}\qedhere
\] \end{proof}
\begin{remark*} In general, deciding whether a set like \(\cS_{a,b}\) is empty requires \(0^{(5)}\); what we have showed is that this is in fact a \(0^{(4)}\) question. \end{remark*}
Next we prove that no function $f\leq_T 0'''$ can compute the correct index. Consequently, high level of non-uniformity is necessary for the construction.
\begin{theorem}\label{thm:3} Given $f\leq_T 0'''$, there are two r.e.\ sets $A=W_a$ and $B=W_b$ with \(W_a\nleq W_b\) such that $W_{f(a,b)}\nleq_T A$ or $W_{f(a,b)}\oplus B\geq_T A$\@(i.e., \(f(a,b)\notin \cS_{a,b}\)\@). \end{theorem}
Let us first recall from~\cite{Barmpalias.Cai.ea:2015} of what can be done and what can not be done. In~\cite{Barmpalias.Cai.ea:2015}, they were trying to build two r.e.\ sets \(A=W_a\) and \(B=W_b\) so that \(A\nleq_T B\) and for all \(e\) we have \(e\notin \cS_{a,b}\). In fact, they developed a successful technique so that for any fixed \(e_0, e_1\), they could build two r.e.\ sets \(A=W_a\) and \(B=W_b\) with \(A\nleq_T B\) so that \(e_0,e_1\notin \cS_{a,b}\). The conflicts between three and more sets, however, were overlooked. Fortunately we do not need their technique here to exhibit Theorem~\ref{thm:3}. We only need the fact that for each fixed \(e\) we can build \(A=W_a\) and \(B=W_b\) with \(A\nleq_T B\) and \(e\notin \cS_{a,b}\). This construction can be done by a simple priority argument. The exact details are not critical in proving Theorem~\ref{thm:3}; we just need the construction to be \emph{simple enough}.
This paragraph serves as a reminder for readers who are interested in the construction. Fixing an index \(e\), we are building two r.e.\ sets \(A\) and \(B\) so that the following requirements for each \(i\in \omega\) are satisfied: \begin{itemize}
\item \(N_i: A\neq \Psi_i(B)\);
\item \(R_{i,e}: W_e=\Phi_i(A)\to [\exists \Gamma (A=\Gamma(BW_e)) \lor \exists \Delta(W_e=\Delta)]\). \end{itemize} The conflict is between the \(\Gamma\)\nbd{}functionals and \(N\)\nbd{}nodes. When an \(N\)\nbd{}node wants to enumerate a diagonalizing point, say \(a\), into \(A\), it enumerates the point. Then an expansionary stage of \(R\)\nbd{}node will tell us whether there is a small change in \(W_e\). If there is one, then \(\Gamma(BW_e;a)\) is undefined without a \(B\)\nbd{}change (\(N\)\nbd{}node is happy) and we redefine it with a fresh use; otherwise, \(N\)\nbd{}node loses its diagonalizing point but it can extend the definition of \(\Delta\). The construction has no more conflicts as \(W_e\) is the same set throughout the construction and hence one particular \(\Delta=W_e\) wins all \(R_{i,e}\)\nbd{}requirements for all \(i\in \omega\); this feature is indispensable for this construction.
This construction is referred as \emph{basic} construction with parameter \(e\) and the priority tree will be denoted by \(\cT_e\).
Now let us do the first warm-up to Theorem~\ref{thm:3} by assuming the given \(f\) recursive. \begin{proof}[Sketch of the proof for \(f\le_T 0\).] We are building two sets \(A\) and \(B\). By recursion theorem we know the indices \(a\) and \(b\) for \(A\) and \(B\) to be constructed. Then we do the basic construction with parameter \(f(a,b)\) to get \(A=W_a\) and \(B=W_b\) with \(W_a\nleq_T W_b\) and \(f(a,b)\notin \cS_{a,b}\). \end{proof}
As we can see from the proof, we simply apply the basic construction with the \emph{correct} parameter to construct \(A\) and \(B\). If \(f\) is recursive, we immediately compute \(f(a,b)\) for the correct parameter. However, if \(f\) is not recursive, we have to guess the value of \(f(a,b)\). Sometimes we run the basic construction with a \emph{wrong} parameter, sometimes with the correct one. We have to ensure that the basic construction with correct parameter is not interrupted by those with wrong ones: the idea is that the basic construction is fairly simple and the conflicts between basic constructions with different parameters are not fatal. Now we do the second warm-up for the double jump (we skip the warm-up for the jump).
\begin{proof}[Sketch of the proof for \(f\le_T 0''\).] Using recursion theorem we know the indices \(a\) and \(b\) for the sets we are constructing. We have to guess what \(f(a,b)\) is. Since \(f\le_T 0''\), so the relation \(f(a,b)=e\) is both \(\Sigma_3\) and \(\Pi_3\). Let \(R\) and \(S\) be recursive predicates such that \[
f(a,b)=e \iff \exists u [R(u,e) \text{ happens infinitely often}] \] and \[
f(a,b)\neq e\iff \exists u [S(u,e) \text{ happens infinitely often}], \] where \(R(u,e)=R(u,e,a,b)\) and \(S(u,e)=S(u,e,a,b)\) and we suppress the parameters \(a\) and \(b\).
Then we define the \emph{guessing tree} \(\cG\) to guide our construction. The root \(\lambda\) is assigned \(G(0)\). A \(G(e)\)\nbd{}node \(\alpha\) has \(\omega\) outcomes ordered by \(0< 1<\cdots<2u<2u+1<\cdots\). Each \(\alpha\concat (2u)\) is a terminal node and \(\alpha\concat (2u+1)\) is assigned \(G(e+1)\).
At each stage~\(s\), we begin with \(\visit(\lambda)\), where \(\visit(\alpha)\) has the following strategy:
Suppose that \(\alpha\) is an \(G(e)\)\nbd{}node. \begin{enumerate}
\item Let \(u\) be the least, if any, such that \(R(u,e)\) happens or \(S(u,e)\) happens.
\item If \(R(u,e)\) happens, \(\visit(\alpha\concat (2u))\).
\item If \(S(u,e)\) happens, \(\visit(\alpha\concat (2u+1))\). \end{enumerate}
Suppose that \(\alpha\) is \(\beta\concat (2u)\) for some \(G(e)\)\nbd{}node \(\beta\). We \(\visit(\lambda)\) for the root \(\lambda\) in \(\cT_{e}(u)\), where \(\cT_{e}(u)\) is a copy of the priority tree for the basic construction with parameter \(e\) and \(u\) is a parameter from \(\cG\). (\emph{We think of the terminal node of \(\cG\) as an entrance to the priority tree for the basic construction with corresponding parameters.})
We have the usual left-to-right initializations: if \(\beta\concat (2u)\) is initialized (because a node to the left is visited), then all nodes of \(\cT_{e}(u)\) are initialized.
The true path \(\rho\) of \(\cG\) is the leftmost path that is visited infinitely often. If \(G(e)\concat (2u+1)\in \rho\), then \(f(a,b)\neq e\); if \(G(e)\concat (2u)\in \rho\), then \(f(a,b)=e\). Therefore \(\rho\) is finite. Let \(G(e)\concat (2u)\) be the longest node of \(\rho\), then \(\cT_{e}(u)\) is visited infinitely often. Note that each node along the true path of \(\cT_{e}(u)\) is initialized still only finitely often. \end{proof}
If \(f\le_T 0''\), \(\cT_{e}(u)\) with different \(e\) and \(u\) have no conflicts other than left-to-right initializations. This will not be the case when \(f\le_T 0'''\). We present the main ideas to overcome the conflicts and to avoid overburdening the readers with technical details.
\begin{proof}[Sketch of the proof for Theorem~\ref{thm:3}] Using the limit lemma, we know \[
e=f(a,b)=\lim_{z\to \infty} g(a,b,z) \] for some \(g\le_T 0''\), therefore \(g(a,b,z)=e\) can be guessed using the same process as in the above proof.
We now define our guessing tree \(\cG\): the root \(\lambda\) is assigned \(G(0,0)\). Suppose that \(\alpha\) is assigned \(G(z,e)\), it has \(\omega\) outcomes ordered as \(0<1<\cdots<2u<2u+1<\cdots\). \(\alpha\concat (2u)\) is assigned \(G(z+1,0)\) and \(\alpha\concat (2u+1)\) is assigned \(G(z,e+1)\). If \(\rho\) is the true path along \(\cG\), we have that \(G(z,e)\concat (2u)\in \rho\) for some \(u\) if and only if \(g(a,b,z)=e\).
As we travel along the guessing tree \(\cG\) and visit a node \(G(z,e)\concat (2u)\), we better allow the basic construction with parameter \(e\) to run. But as our guessing tree \(\cG\) is more complicated and two nodes \(\alpha\subsetneq \beta\) might have guessed for different parameters \(e_\alpha\neq e_\beta\), these two basic constructions with parameters \(e_\alpha\) and \(e_\beta\) have to find ways to cooperate.
The \emph{predecessor} of \(\alpha\) is the longest node \(\beta\), if any, such that \(\beta\concat(2u)\subseteq \alpha\) for some \(u\). A \(G(z,e)\)\nbd{}node \(\alpha\) is an \emph{entrance} to the basic construction with parameter \(e\) if its predecessor is not a \(G(z-1,e)\)\nbd{}node. For \(\alpha\in \cG\), we let \[
\alpha_0\concat (2u_0)\subseteq\alpha_1\concat (2u_1)\subseteq\cdots \alpha_{k-1}\concat (2u_{k-1})\subseteq \alpha_k=\alpha \] be the sequence such that \(\alpha_0\) is an entrance for the basic construction with parameter \(e\) and \(\alpha_i\) is the predecessor of \(\alpha_{i+1}\) and suppose that \(\alpha_i\) is a \(G(z_0+i,e)\)\nbd{}node (so \(\alpha\) is a \(G(z_0+k,e)\)\nbd{}node). For each outcome \((2u)\) of \(\alpha\), we let \(\cT_{e,\alpha_0}(u_0,u_1,\dots,u_{k-1},u)\) extend \(\cT_{e,\alpha_0}(u_0,u_1,\dots,u_{k-1})\) by copying the nodes in \(\cT_e\) of length \(k\), where \(\cT_e\) is the priority tree for the basic construction with parameter \(e\). The idea is that at \(\alpha\concat (2u)\in \cG\) we are only allowed to visit a node \(\sigma\in \cT_{e,\alpha_0}(u_0,u_1,\dots,u_{k-1},u)\) with length \(k-1\): the algorithm for \(\visit(\sigma)\) is \emph{paused} whenever \(\sigma\in \cT_{e,\alpha_0}(u_0,u_1,\dots,u_{k-1},u)\) has length \(k\). We let \(\sigma_{\alpha,u}\) denote the paused node. Whenever we \(\visit(\alpha\concat (2u))\) in the guessing tree \(\cG\), we simultaneously \(\visit(\sigma_{\alpha_{k-1},u_{k-1}})\) for the node \( \sigma_{\alpha_{k-1},u_{k-1}}\in \cT_{e,\alpha_0}(u_0,u_1,\dots,u_{k-1},u)\).
The initialization is the usual left-to-right kind: if we are visiting a node to the left of \(\alpha\concat (2u)\), then the nodes in \(\cT_{e,\alpha_0}(u_0,\dots,u_{k-1},u)\setminus \cT_{e,\alpha_0}(u_0,\dots,u_{k-1})\) are initialized.
To avoid conflicts between different basic constructions, each entrance \(\alpha\) picks a fresh point \(w_u\) for each \(u\in\omega\). Whenever we \(\visit(\alpha\concat (2u))\) for some \(u\), we enumerate \(\gamma(w_u)\) into \(B\) to undefine all the \(\Gamma\)\nbd{}functionals built by some \(\beta\concat (2u')\subseteq \alpha\) for some \(u'\).
By a routine verification, one derives the following: \begin{enumerate}
\item The true path \(\rho\) of \(\cG\) is infinite and there is an infinite sequence \[
\alpha_0\concat (2u_0)\subseteq \alpha_1\concat (2u_1)\subseteq \cdots
\]
such that \(\alpha_i\in \rho\), \(\alpha_0\) is an entrance for the basic construction with parameter \(e=f(a,b)\), \(\alpha_i\) is a \(G(z_0+i,e)\)\nbd{}node for each \(i\) for some \(z_0\), and \(\alpha_i\) is the predecessor of \(\alpha_{i+1}\).
\item For each \(i\), nodes in the tree \(\cT_{e,\alpha_0}(u_0,\dots, u_i)\) are visited infinitely often and initialized finitely often.
\item The union of all \(\cT_{e,\alpha_0}(u_0,\dots, u_i)\) is a full copy of \(\cT_e\), the priority tree for the basic construction with parameter \(e\). \end{enumerate}
This completes the proof. \end{proof}
\end{document} | arXiv |
\begin{document}
\title[Accessible set functors are universal]{Accessible set functors are universal} \author{Libor Barto} \address{Charles University\\Faculty of Mathematics and Physics\\Department of Algebra\\Sokolovsk\'a 83\\186 75 Praha 8\\Czech Republic} \email{[email protected]} \thanks{ Libor Barto has received funding from the Czech Science Foundation (grant No 201/06/0664) and from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 771005). } \date{March 28, 2019} \subjclass{Primary 18B15, 18A22, 18A25; Secondary 08B05} \keywords{set functor, universal category, full embedding} \dedicatory{Dedicated to the memory of V\v era Trnkov\'a.}
\begin{abstract} It is shown that every concretizable category can be fully embedded into the category of accessible set functors and natural transformations. \end{abstract}
\maketitle
\section{Introduction}
This paper contributes to two areas in which the work of V\v era Trnkov\'a plays a major role: universal categories and set functors.
We start by providing some background to these areas and then we state the main result of this paper. The discussion is rather informal and some fine technical points are neglected. The essential concepts for this paper are formally introduced in Section~\ref{sec:prelim}.
\subsection{Universality}
One approach to measuring the complexity of a category is to characterize its full subcategories.
There is usually a natural limitation on the possible full subcategories. The following two situations are especially common: if the category in question is \emph{concretizable} (i.e., admits a faithful functor to the category of sets), then every full subcategory is concretizable as well; if the category is \emph{algebraic} (i.e., is isomorphic to a full subcategory of a category of universal algebras), then so are all of its full subcategories.
It turned out that such natural limitations are the only ones for many categories of interest~\cite{PT,T5}. An algebraic category is often \emph{alg--universal}, i.e., each category of universal algebras is isomorphic to its full subcategory. Examples include the category of graphs \footnote{If morphisms are not specified, then the most natural choice is meant. Here the morphisms are the edge--preserving mappings.} and the category of algebras with two unary operations~\cite{HP66}. Concretizable categories are sometimes even \emph{universal}, i.e., each concretizable category is among the isomorphic copies of their full subcategories. This is the case, e.g., for the category of hypergraphs~(a combination of results of Hedrl\'in and Ku\v cera, see \cite{PT}), the category of topological semigroups~\cite{T6}, or the category of topological spaces with open continuous maps as morphisms~\cite{PT}.
Although it is not immediately seen from the definition, alg--universal categories are indeed comprehensive. For instance, they contain an isomorphic copy of every \emph{small} category, i.e., a category whose object class is a set (see \cite{PT}). In particular, every partially ordered set can be represented as a set of objects with the relation ``there exists a morphism'' and, also, every group can be represented as the automorphism group of some object. The latter property of a category, \emph{group--universality}, predates the concept of alg--universality and even the notion of a category. It is a remarkable achievement of the theory that the group--universality is now usually best shown by proving the far stronger alg--universality. In fact, the development leading to this paper, discussed in Subsection~\ref{subsec:res}, is an instance of this phenomenon.
Universal categories may seem way more comprehensive than alg--universal ones. For instance, they can represent in the above sense every partially ordered class. However, a combination of results of Ku\v cera, Pultr, and Hedrl\'in proves (see, again, \cite{PT}) that the statement ``every alg--universal category is universal'' is equivalent to the set--theoretical assumption ``the class of all measurable cardinals is a set''. It follows that, in some models of set theory, the alg--universality and universality coincide. Still, an absolute proof of universality indicates, philosophically speaking, that the category is more comprehensive than any algebraic category.
\subsection{Set functors}
\emph{Set functors} are simply functors from the category $\cat{Set}$ of all sets and mappings to itself. The natural choice of morphisms between them is natural transformations. Besides being basic examples of functors (which perhaps motivated early results from the 1970s, e.g., \cite{T1,T2,T3,K1}), set functors are also among the central concepts in both Universal Coalgebra and Universal Algebra. Their appearance in the former is direct and explicit as type functors for coalgebras that model various state base systems such as streams, automata, or Kripke structures~\cite{Rut00}. The significance in the latter is a bit less direct and not so widely acknowledged in the Universal Algebra community.
One of the main focuses of Universal Algebra is the study of systems of universally quantified equations over some signature (or classes of all models of such systems --- varieties). The special attention to these objects has paid large dividends in theoretical computer science, namely in the computational complexity of constraint satisfaction problems (CSPs, for short) \footnote{A recent survey is~\cite{BKW17}. More recently, a solution to the main open problem of the area was announced independently by Bulatov~\cite{Bul17} and Zhuk~\cite{Zhu17}. Both proofs heavily use Universal Algebra.}, where to each CSP one assigns a system of equations which captures the computational complexity of that CSP, and morphisms between these systems then correspond to certain natural polynomial--time reductions. It has turned out~\cite{BOP18} that, in this context, it is enough to consider particularly simple equations, those that contain exactly one function symbol on both sides, i.e., equations of the form $t(\mbox{some variables}) = s(\mbox{some variables})$. The importance of such equations, called \emph{height one equations} in~\cite{BOP18} or \emph{flat equations} in~\cite{AGT10}, is further witnessed by a very recent development in promise CSPs~\cite{BKO}.
Returning back to set functors --- their category is equivalent to the category of systems of height one equations! \footnote{ This claim is rather sloppy; for instance, the collection of all set functors is not a class, so set functors do not form a legitimate category.}
We will now sketch one direction of this equivalence and refer the reader to a nice presentation in~\cite{AGT10} for details. Given a set functor $F$ we regard members of $FX$ as function symbols of arity $|X|$ and, for each mapping $f: X \rightarrow Y$ and $t \in FX$, we add one equation to the system in a natural way, e.g., if $X=\{x_1, x_2, x_3\}$, $Y = \{y_1,y_2\}$, $f(x_1)=f(x_2)=y_2$, $f(x_3)=y_1$, $t \in FX$, and $s = Ff(t)$, then we add the equation $t(y_2,y_2,y_1) = s(y_1,y_2)$. Natural transformations of set functors then correspond to mappings preserving height one equations.
Note that the obtained system of equations forms a proper class (except for uninteresting functors with $FX = \emptyset$ for all $X \neq \emptyset$) and involves infinitary function symbols, which is rather non-standard in Universal Algebra. However, if we restrict to \emph{accessible} set functors (to be defined in Section~\ref{sec:prelim}), then it is enough to introduce function symbols for elements of $FX$ for a sufficiently large $X$. Restricting further to \emph{finitary} functors allows us to introduce function symbols only for elements of $F(\{x_1, \dots, x_n\})$, $n \in \mathbb N$, and we are back in a standard Universal Algebraic realm.
\subsection{Universal set functors} \label{subsec:res} The original motivation for studying the universality of set functors was another open problem formulated in~\cite{BT1}: Is the category of varieties and interpretations~\cite{GT84} alg--universal? This category can be described, in the language of previous paragraphs, as the category of sets of arbitrary finitary equations, not necessarily height one. From this perspective, starting with height one equations suggests itself as a reasonable step. The first paper in this direction~\cite{BZ1} has shown, using a rather involved construction, that the category of set functors is group--universal. Then, a significantly simpler construction was carried out in~\cite{Ba2} to prove that the category of finitary set functors is alg--universal. Since this category (as well as the category of $\kappa$--accessible functors for any fixed $\kappa$) is also algebraic, the alg--universality of finitary (or $\kappa$--accessible for $\kappa \geq 7$) set functors cannot be further strengthened. Finally, the motivating problem was solved as well: the category of varieties is alg--universal~\cite{Bar07}.
This development has left the following natural question open: Is the upper bound on the arity of function symbols essential? In other words, is the category of set functors more comprehensive than the category of finitary ones (equivalently, $\kappa$-accessible for a fixed $\kappa \geq 7$)? We answer this question in affirmative by showing, in Section~\ref{sec:univ}, that the category of accessible set functors is universal. In Section~\ref{sec:concrete} we verify that this result cannot be further strengthened by proving that the category of accessible set functors is concretizable. Altogether, we get the following theorem.
\begin{theorem} \label{thm:main} A category $\cat{Z}$ is isomorphic to a full subcategory of the category of accessible set functors if and only if $\cat{Z}$ is concretizable. \end{theorem}
\section{Preliminaries} \label{sec:prelim}
\subsection{Notation}
The set $\{1,2, \dots, n\}$ is denoted $[n]$. We use the notation $A \sqcup B$ ($f \sqcup g$) for a disjoint union of sets (mappings).
Let $f: X \rightarrow Y$ be a mapping. The image of $R \subseteq X$ under $f$ is denoted by $f[R]$ and
the preimage of $S \subseteq Y$ under $f$ is denoted by $f^{-1}[S]$.
Mappings are composed from right to left, that is, $f \circ g(x) = fg(x) = f(g(x))$.
The identity mapping $X \rightarrow X$ is denoted ${\rm{id}}_X$.
\subsection{Categorical concepts} Let $\cat{M}$ and $\cat{N}$ be categories. A functor $\Phi: \cat{M} \rightarrow \cat{N}$ is \emph{faithful} (\emph{full}, respectively) if, for any two $\cat{M}$--objects $A$ and $B$, it maps the set of all $\cat{M}$--morphisms from $A$ to $B$ injectively (surjectively, respectively) into the set of all $\cat{N}$--morphisms from $\Phi A$ to $\Phi B$. A \emph{full embedding} is a full and faithful functor which is, moreover, injective on objects.
A category $\cat{M}$ is a full subcategory of $\cat{N}$ if $\cat{M}$ is obtained from $\cat{N}$ by taking some of the objects and all the morphisms between them. Note that if $\Phi: \cat{M} \rightarrow \cat{N}$ is a full embedding, then the image of $\Phi$ is a full subcategory of $\cat{N}$ which is isomorphic to $\cat{M}$.
The category of all sets and mappings is denoted $\cat{Set}$. A category $\cat{M}$ is \emph{concretizable} if there exists a faithful functor $\cat{M} \rightarrow \cat{Set}$. A category $\cat{M}$ is \emph{universal} if every concretizable category has a full embedding into $\cat{M}$.
A \emph{set functor} is a functor $\cat{Set} \rightarrow \cat{Set}$. For set functors $F$ and $G$, and a set $X$, the $X$-th component of a natural transformation $\mu: F \rightarrow G$ is denoted $\mu_X: F X \rightarrow G X$.
\subsection{Accessible set functors}
A set functor $F$ is called $\kappa$-accessible if every element of $FY$ can be accessed from an element of $FX$ with $|X| < \kappa$ in the following sense.
\begin{definition} Let $F: \cat{Set} \rightarrow \cat{Set}$ be a set functor and $\kappa$ a cardinal. $F$ is \emph{$\kappa$-accessible} if \[
FY = \bigcup \{Ff[FX] \,;\, |X| < \kappa, f: X \rightarrow Y\} \] for every set $X$.
A set functor $F$ is \emph{accessible} if it is $\kappa$-accessible for some $\kappa$, and $F$ is \emph{finitary} if it is $\aleph_0$-accessible. \end{definition}
This definition agrees with the general notion of an accessible functor, see~\cite{AP01} for several equivalent characterizations.
Strictly formally, accessible set functors do not form a legitimate category since each functor is a proper class. However, every accessible functor has a set presentation which describes the functor up to natural isomorphism. One option is to use an equational presentation as described in~\cite{AGT10}. Alternatively, it can be observed that a $\kappa$-accessible functor (as well as a natural transformation between such functors) is determined by its restriction to the set of cardinals smaller than $\kappa$. In this sense, it is legitimate to talk about the category of accessible functors and we denote this category by $\cat{AccSetFun}$.
\subsection{Topological spaces without axioms} If some known ``testing'' universal category $\cat{M}$ can be fully embedded into $\cat{N}$, then $\cat{N}$ is clearly universal as well. A particularly useful testing category proved to be ''topological spaces without axioms''. Its objects are pairs consisting of a set and a family of its ``open'' subsets, where, in contrast to topological spaces, we do not impose any restriction on the family of open sets. Morphisms, the ``continuous'' maps, are defined just like in the category of topological spaces.
\begin{definition} Objects of the category $\cat{T}$ are pairs $(A, \family{R})$, where $A$ is a set and $\family{R}$ is a family of subsets of $A$. A morphism $f: (A, \family{R}) \rightarrow (B, \family{S})$ is a mapping $f: A \rightarrow B$ such that $f^{-1}[S] \in \family{R}$ for all $S \in \family{S}$. \end{definition}
The category $\cat{T}$ is universal (Hedrl\'in, Ku\v cera, see \cite{PT}). However, we will construct a \emph{contravariant functor} from $\cat{T}$ to $\cat{AccSetFun}$, that is, a functor from the opposite category $\cat{T}^{op}$. Fortunately, it is not hard to see that the opposite category to a universal category is universal~(see Section 4.6 of~\cite{T5}).
\begin{theorem} \label{thm:topouniv} The category $\cat{T}$, as well as the opposite category $\cat{T}^{op}$, is universal. \end{theorem}
\section{Set functors are universal} \label{sec:univ}
In this section we show that the category of accessible functors is universal by constructing a contravariant functor from $\cat{T}$ to $\cat{AccSetFun}$. Let us first informally sketch the construction.
An object $(A, \family{R})$ will be sent to a set functor described by a set of height one equations in a single operation symbol $t$ of arity $|A|$. There will be one equation for each $R$: $$ t(\underbrace{x, x, \dots, x}_{R}, \underbrace{y, y, \dots, y}_{A \setminus R}) = t(\underbrace{y,y, \dots, y}_R, \underbrace{x, x, \dots, x}_{A \setminus R})\enspace. $$ This object map naturally extends to a contravariant functor and it is not hard to observe that $f: (A, \family{R}) \rightarrow (B, \family{S})$ is ``continuous'' if and only if the image of $f$ preserves height one equations.
The only substantial catch in this construction is that the above equation entails the equation where the roles of $R$ and $A \setminus R$ are swapped. For this reason we first construct an auxiliary embedding which will resolve this issue.
\subsection{Auxiliary full embedding}
Our aim now is to construct a full embedding $\Psi: \cat{T} \rightarrow \cat{T}$ whose image, denoted $\cat{K}$, has the following properties. \begin{description} \item[(K1)] For every $\cat{K}$--object $(A,\family{R})$ and $R \in \family{R}$, we have
$A \setminus R \in \family{R}$ and $R \neq \emptyset$. \item[(K2)] If $f: (A, \family{R}) \rightarrow (B, \family{S})$ is a $\cat{K}$-morphism,
then $|f[A]| > 2$. \end{description}
The construction requires a graph (symmetric, loopless, without multiply edges) on a sufficiently large even number of vertices with no nonidentical automorphisms. Let us fix one such a graph with vertex set
$V = \{v_1, v_2, \ldots, v_6\}$ and edge set $\family{G}$, where $\family{G}$ is formally handled as a family of two-element subsets of $V$. The functor $\Psi$ is defined as follows.
\begin{align*}
\Psi(A,\family{R}) &= (\overline{A}, \overline{\family{R}}) \\
{\overline{A}} &= A \sqcup V \\
\overline{\family{R}} &=
\{ \{v_i\} \,;\, i \in [6] \} \cup \{\overline{A} \setminus \{v_i\} \,;\, i \in [6]\} \cup \\
& \quad
\{ G \,;\, G \in \family{G}\} \cup \{ \overline{A} \setminus G \,;\, G \in \family{G}\} \cup \\
& \quad
\{ \{v_1, v_2, v_3\} \cup R \,;\, R \in \family{R}\} \} \cup \{\{v_4, v_5, v_6\} \cup (A \setminus R) \,;\, R \in \family{R}\} \\
\Psi f &= f \sqcup {\rm{id}}_V
\end{align*}
We first check that $\Psi f$ is always continuous.
\begin{lemma}
If $f: (A,\family{R}) \rightarrow (B, \family{S})$ is a $\cat{T}$--morphism, then
so is $\Psi f: (\overline{A},\overline{\family{R}}) \rightarrow (\overline{B},\overline{\family{S}})$
\end{lemma}
\begin{proof}
Pick $Q \in \overline{\family{S}}$. To prove that $(\Psi f)^{-1}[Q] \in \overline{\family{R}}$ we analyze the six cases in the definition of $\overline{\family{S}}$.
\begin{itemize}
\item If $Q = \{v_i\}$, $i \in [6]$, then $(\Psi f)^{-1}[\{v_i\}] = (f \sqcup {\rm{id}}_V)^{-1}[\{v_i\}] = \{v_i\} \in \overline{\family{R}}$.
\item If $Q \in \family{G}$, then $(\Psi f)^{-1}[Q] = Q \in \overline{\family{R}}$.
\item If $Q = \{v_1, v_2, v_3\} \cup S$ with $S \in \family{S}$, then $(\Psi f)^{-1}[Q] =
\{v_1, v_2, v_3\} \cup f^{-1}[S] \in \overline{\family{R}}$ since $f$ is a $\cat{T}$--morphism.
\end{itemize}
The remaining three cases are complementary and completely analogous.
\end{proof}
Clearly, $\cat{K}$ satisfies (K1) and $\Psi$ is a functor (preserves composition and identities) which is faithful and injective on objects.
It remains to prove that it is full (then (K2) follows as well).
Let
$(A,\family{R})$ and $(B, \family{S})$ be $\cat{T}$--objects and
$g: (\overline{A},\overline{\family{R}}) \rightarrow (\overline{B},\overline{\family{S}})$ be
a $\cat{T}$--morphism. We need to show that $g = \Psi f$ for a $\cat{T}$--morphism
$f: (A,\family{R}) \rightarrow (B, \family{S})$
\begin{lemma} \label{lem:one}
The function $g$ restricted to $V$ is a bijection $V \rightarrow V$ and $g[A] \subseteq B$.
\end{lemma}
\begin{proof}
Let $i \in [6]$ be arbitrary.
Since $\{v_i\} \in \overline{\family{S}}$, we must have
$g^{-1}[\{v_i\}] \in \overline{\family{R}}$. By definition of $\overline{\family{R}}$, all the members of $\overline{\family{R}}$ have a nonempty intersection with $V$, therefore
$g^{-1}[\{v_i\}] \cap V$ is nonempty. We conclude that each $v_i$ has a $g$--preimage in $V$; in other words, $g$ maps $V$ onto $V$.
Since $V$ is finite, the first claim follows.
Now we know that, for each $i \in [6]$, the set $g^{-1}[\{v_i\}] \in \overline{\family{R}}$ has a one-element intersection with $V$. The
only such elements of $\overline{\family{R}}$ are of the form $\{v_j\}$, therefore $g^{-1}[V] \subseteq V$ which is equivalent to the second claim.
\end{proof}
By Lemma~\ref{lem:one}, we can write $g$ as a disjoint union
$g = f \sqcup h$, where $f: A \rightarrow B$, and $h: V \rightarrow V$ is a bijection.
The following lemma finishes the proof that $\Psi f = g$.
\begin{lemma} \label{lem:three}
We have $h = {\rm{id}}_V$ and $f : (A, \family{R}) \rightarrow (B, \family{S})$ is a $\cat{T}$--morphism.
\end{lemma}
\begin{proof}
For every $G \in \family{G} \subseteq \overline{\family{S}}$, we have $g^{-1}[G] = h^{-1}[G] \in \overline{\family{R}}$.
Since $h$ is a bijection, the set $h^{-1}[G]$ has two elements.
The only such sets in $\overline{\family{R}}$ are the members $\family{G}$,
therefore $h^{-1}[G] \in \family{G}$. We conclude that $h^{-1}$ is a bijective endomorphism of our fixed
6-vertex graph. Since this graph is finite, every bijective endomorphism is an automorphism, hence
$h^{-1} = {\rm{id}}_V$ and the first part follows.
For every $S \in \family{S}$, we have $\{v_1, v_2, v_3\} \cup S \in \overline{\family{S}}$, therefore (by the previous paragraph)
$g^{-1}[\{v_1, v_2, v_3\} \cup S] = \{v_1, \dots, v_3\} \cup f^{-1}[S] \in \overline{\family{R}}$ and then $f^{-1}[S] \in \family{R}$, proving the second part.
\end{proof}
\subsection{Key full embedding}
In this subsection we construct a contravariant full embedding $\Phi$ from the category $\cat{K}$ (see the previous subsection) into the category $\cat{AccSetFun}$.
The set functors in the image of $\Phi$ will be quotients of (covariant) hom--functors. For a $\cat{K}$--object $\alg{A} = (A,\family{R})$ we set \[ (\Phi \alg{A}) X = \{g \,;\, g: A \rightarrow X \} /\!\!\approx_{\alg{A}} \mbox{ for any set $X$} \enspace, \] where the equivalence $\approx_{\alg{A}}$ is given by $$ g_1: A \rightarrow X \ \approx_{\alg{A}} \ g_2: A \rightarrow X \quad\mbox{ if } $$ \begin{enumerate} \item $g_1 = g_2$, or \item $g_1[A] = g_2[A]$ is a two--element set $\{x,x'\}$ and $g_1^{-1}[\{x\}] = g_2^{-1}[\{x'\}] \in \family{R}$. \end{enumerate} The definition of $\approx_{\alg{A}}$ makes sense, since $g_1^{-1}[\{x\}] = g_2^{-1}[\{x'\}] \in \family{R}$ if and only if $g_1^{-1}[\{x'\}] = g_2^{-1}[\{x\}] \in \family{R}$ as follows from the property (K1). For the same reason, $\approx_{\alg{A}}$ is an equivalence relation.
The behavior of $\Phi \alg{A}$ on a mapping $h: X \rightarrow Y$ is given by \[ ((\Phi \alg{A}) h) (g/\!\!\approx_{\alg{A}}) = hg/\!\!\approx_{\alg{A}} \mbox{ for any $g: A \rightarrow X$}\enspace. \] Clearly, $hg/\!\!\approx_{\alg{A}}$ does not depend on the choice of representative $g \in g/\!\!\approx_{\alg{A}}$.
The set functor $\Phi \alg{A}$ is $\kappa$--accessible for any $\kappa > |A|$ since, for each set $X$ and $g/\!\!\approx_{\alg{A}} \in (\Phi \alg{A})X$, we have $g/\!\!\approx_{\alg{A}} = ((\Phi \alg{A})g)({\rm{id}}_A/\!\!\approx_{\alg{A}})$.
It remains to define $\Phi$ on $\cat{K}$--morphisms. Let $\alg{A} = (A,\family{R})$ and $\alg{B} = (B,\family{S})$ be $\cat{K}$--objects, and
$f: \alg{A} \rightarrow \alg{B}$ a $\cat{K}$--morphism. The $X$-th component of the natural transformation $\Phi f: \Phi \alg{B} \rightarrow \Phi \alg{A}$ is defined by the formula \[ (\Phi f)_X (g/\!\!\approx_{\alg{B}}) = gf/\!\!\approx_{\alg{A}} \mbox{ for any $g: B \rightarrow X$} \enspace. \]
In order to see that the definition does no depend on the choice of $g \in g/\!\!\approx_{\alg{B}}$, let us consider two $\approx_{\alg{B}}$-equivalent $g_1,g_2: B \rightarrow X$. Either $g_1=g_2$, in which case $g_1f= g_2f$, or $g_1[B] = g_2[B] = \{x,x'\}$, $x \neq x'$, $g_1^{-1}[\{x\}] = g_2^{-1}[\{x'\}] \in \family{S}$. Since $f$ is a $\cat{K}$--morphism, then $f^{-1}[g_1^{-1}[\{x\}]] = (g_1f)^{-1}[\{x\}]$ and $f^{-1}[g_2^{-1}[\{x'\}]]=(g_2f)^{-1}[\{x'\}]$ are in $\family{R}$. Moreover, neither of these two subsets of $A$ is empty or equal to $A$ by the property (K1), therefore $g_1f[A] = g_2f[A] = \{x,x'\}$, hence $g_1f$ and $g_2f$ are $\approx_{\alg{A}}$-equivalent.
\begin{lemma} For any $\cat{K}$--morphism $f: \alg{A} \rightarrow \alg{B}$, the collection $\Phi f$ is a natural transformation $\Phi \alg B \rightarrow \Phi \alg A$. \end{lemma} \begin{proof} For any sets $X, Y$, mapping $h: X \rightarrow Y$, and $g/\!\!\approx_{\alg{B}} \in (\Phi \alg{B})X$ we have $$ ( (\Phi\alg{A}) h \circ (\Phi f)_X)(g/\!\!\approx_{\alg{B}}) =
((\Phi\alg{A}) h) (gf/\!\!\approx_{\alg{A}}) = [hgf]/\!\!\approx_{\alg{A}} $$ and $$ ((\Phi f)_Y \circ (\Phi\alg{B})h )(g/\!\!\approx_{\alg{B}}) =
(\Phi f)_Y(hg/\!\!\approx_{\alg{B}}) = [hgf]/\!\!\approx_{\alg{A}} \enspace. $$ Therefore $(\Phi\alg{A}) h \circ (\Phi f)_X = (\Phi f)_Y \circ (\Phi\alg{B})h$. \end{proof}
The functor $\Phi$ is injective on objects, so it remains to check that $\Phi$ is faithful and full.
\begin{lemma} $\Phi$ is faithful. \end{lemma} \begin{proof} Let $\alg{A} = (A, \family{R})$, $\alg{B} = (B, \family{S})$ be $\cat{K}$--objects and $f_1, f_2: \alg{A} \rightarrow \alg{B}$ different morphisms. Due to the property (K2), the ranges of $f_1$ and $f_2$ both contain at least three elements, so $f_1$ and $f_2$ are not $\approx_{\alg{A}}$--equivalent. But then $\Phi f_1$ and $\Phi f_2$ differ on ${\rm{id}}_B/\!\!\approx_{\alg{B}}$ since $(\Phi f_1)_B([{\rm{id}}_B]_{\alg{B}}) = f_1/\!\!\approx_{\alg{A}}$ and $(\Phi f_2)_B([{\rm{id}}_B]_{\alg{B}}) = f_2/\!\!\approx_{\alg{A}}$. \end{proof}
\begin{lemma} $\Phi$ is full. \end{lemma} \begin{proof} Let $\alg{A} = (A, \family{R}),$ $\alg{B}=(B, \family{S})$ be $\cat{K}$--objects and $\mu: \Phi \alg{B} \rightarrow \Phi \alg{A}$ a natural transformation. We must check that $\mu = \Phi f$ for some morphism $f: \alg{A} \rightarrow \alg{B}$.
Let $f: A \rightarrow B$ be any mapping such that $f/\!\!\approx_{\alg{A}} = \mu_B({\rm{id}}_B/\!\!\approx_{\alg{B}})$. From the naturality of $\mu$ we get that, for any mapping $g: B \rightarrow X$, \begin{align*} \mu_X (g/\!\!\approx_{\alg{B}}) &= (\mu_X \circ (\Phi \alg{B})g)({\rm{id}}_B/\!\!\approx_{\alg{B}}) =
((\Phi \alg{A})g \circ \mu_B)({\rm{id}}_B/\!\!\approx_{\alg{B}}) =
((\Phi \alg{A})g)(f/\!\!\approx_{\alg{A}}) \\ &=
gf/\!\!\approx_{\alg{B}}\enspace. \end{align*} Now it suffices to show that the mapping $f: A \rightarrow B$ is a $\cat{K}$--morphism $\alg{A} \rightarrow \alg{B}$, since the last equality then tells us $\mu = \Phi f$. Take an arbitrary $S \in \family{S}$ and consider the mappings $g_1,g_2: B \rightarrow [2]$ such that $g_1[S] = \{1\} = g_2[B \setminus S]$ and $g_1[B \setminus S] = \{2\} = g_2[S]$. Clearly $g_1 \approx_{\alg{B}} g_2$, therefore $\mu_2(g_1/\!\!\approx_{\alg{B}}) = \mu_2(g_2/\!\!\approx_{\alg{B}})$. Thus, using the above equality again, $g_1f \approx_{\alg{B}} g_2f$. Since $g_1f \neq g_2f$ (as $g_1(b) \neq g_2(b)$ for any $b \in B$), we get that $\family{R} \ni (g_1f)^{-1}[\{1\}] = f^{-1}[g_1^{-1}[\{1\}]] = f^{-1}[S]$ and we are done. \end{proof}
We have proved one implication in Theorem~\ref{thm:main}.
\begin{theorem} The category $\cat{AccSetFun}$ is universal. \end{theorem} \begin{proof} We have constructed two full embeddings, a covariant $\Psi: \cat{T} \rightarrow \cat{K}$ and contravariant $\Phi: \cat{K} \rightarrow \cat{AccSetFun}$. The composition $\Phi\Psi$ can be regarded as a (covariant) full embedding $\cat{T}^{op} \rightarrow \cat{AccSetFun}$. But $\cat{T}^{op}$ is universal by Theorem~\ref{thm:topouniv}, so $\cat{AccSetFun}$ is universal too. \end{proof}
\section{Accessible set functors are concretizable} \label{sec:concrete}
In this section we prove the second implication in Theorem~\ref{thm:main} by showing that $\cat{AccSetFun}$ is a concretizable category. We will use Isbell's condition for concretizability, which we now formulate.
Let $\cat{W}$ be a category and $F, G$ its objects. Let $P(F,G)$ denote the class of all pairs of $\cat{W}$--morphisms of the form $(\mu: H \rightarrow F, \nu: H \rightarrow G)$. We define an equivalence relation $\sim_{F,G}$ on $P(F,G)$ by putting $(\mu, \nu) \sim (\mu',\nu')$ if, for every pair of morphisms $(\alpha: F \rightarrow K, \beta: G \rightarrow K)$, $\alpha\mu = \beta\nu$ if and only if $\alpha\mu' = \beta\nu'$.
Isbell's condition states that the equivalence relation $\sim_{F,G}$ on $P(F,G)$ has a transversal which is a set. This condition is necessary for concretizability~\cite{Is63} and, as it turned out, is also sufficent~\cite{Fr73} (a simpler and more constructive proof is given in~\cite{Vin76}).
\begin{theorem} \label{thm:concret} A category $\cat{W}$ is concretizable if and only if, for any two $\cat{W}$--objects $F,G$, there exists a \emph{set} $Q \subseteq P(F,G)$ containing for every pair $(\mu,\nu) \in P(F,G)$ an $\sim_{F,G}$--equivalent one. \end{theorem}
We verify Isbell's condition for the category $\cat{AccSetFun}$. Fix set functors $F$ and $G$ and choose $X$ so that $F$ and $G$ are both $|X|$--accessible and $X$ is infinite.
For a pair of natural transformations $(\mu: H \rightarrow F, \nu: H \rightarrow G)$ we consider the following subsets of $FX \times GX$ and $F\emptyset \times G\emptyset$. \begin{align*} T(\mu,\nu) &= \{(\mu_X(x),\nu_X(x)) \in FX \times GX \,;\, x \in HX\} \\ T_0(\mu,\nu) &= \{(\mu_X(x),\nu_X(x)) \in F\emptyset \times G\emptyset \,;\, x \in H\emptyset\} \end{align*} We claim that, for every pair of natural transformations $(\alpha: F \rightarrow K, \beta: G \rightarrow K)$, we have $\alpha \mu = \beta \nu$ if and only if $\alpha_X(x_1) = \beta_X(x_2)$ for every $(x_1,x_2) \in T(\mu,\nu)$ and $\alpha_{\emptyset}(x_1) = \beta_{\emptyset}(x_2)$ for every $(x_1,x_2) \in T_0(\mu,\nu)$. Before proving the claim, observe that Isbell's condition is a consequence. Indeed, for each pair of subsets $R \subseteq FX \times GX$, $R_0 \subseteq F\emptyset \times G\emptyset$ we add to $Q$ a single pair $(\mu,\nu) \in P(F,G)$ with $T(\mu,\nu)=R$ and $T_0(\mu,\nu)=R_0$ provided such a pair exists. Since the claim implies that $(\mu,\nu) \sim_{F,G} (\mu',\nu')$ whenever $T(\mu,\nu)=T(\mu',\nu')$, such a set $Q$ satisfies the requirement in Theorem~\ref{thm:concret}.
One direction of the claim is obvious: if $\alpha \mu = \beta \nu$, then $\alpha_X(x_1) = \beta_X(x_2)$ for every $(x_1,x_2) \in T(\mu,\nu)$ and $\alpha_{\emptyset}(x_1) = \beta_{\emptyset}(x_2)$ for every $(x_1,x_2) \in T_0(\mu,\nu)$. Assume, conversely, that \begin{align*} (\star) \quad &\alpha_X(x_1) = \beta_X(x_2) \mbox{ for every } (x_1,x_2) \in T(\mu,\nu) \enspace \\
& \alpha_{\emptyset}(x_1) = \beta_{\emptyset}(x_2) \mbox{ for every } (x_1,x_2) \in T_0(\mu,\nu) \enspace. \end{align*} Take an arbitrary set $Y$ and an element $y \in HY$. Our aim is to prove that $\alpha_Y \mu_Y (y) = \beta_Y \nu_Y (y)$.
We distinguish two cases according to the cardinality of $Y$. The following lemma deals with the simpler case.
\begin{lemma}
If $|Y| \leq |X|$, then $\alpha_Y \mu_Y (y) = \beta_Y \nu_Y (y)$. \end{lemma}
\begin{proof} If $Y=\emptyset$, then the pair $(\mu_{\emptyset} (Hi(y)), \nu_{\emptyset}( Hi(y)))$ is in $T_0(\mu,\nu)$, so $\alpha_{\emptyset}\mu_{\emptyset} Hi (y) = \beta_{\emptyset} \nu_{\emptyset} Hi (y)$ by the second part of $(\star)$. \footnote{This is the only place where $T_0(\mu,\nu)$ is used. The case $Y = \emptyset$ was neglected in the previous version of the paper and I thank the reviewer for pointing out this mistake.}
Otherwise, take any injective mapping $i: Y \rightarrow X$ and its left inverse $k: X \rightarrow Y$ (so that $ki = {\rm{id}}_Y$). The pair $(\mu_X (Hi(y)), \nu_X( Hi(y)))$ is in $T(\mu,\nu)$, so $\alpha_X\mu_X Hi (y) = \beta_X \nu_X Hi (y)$. Since $\alpha\mu$ and $\beta\nu$ are natural transformation, we get \[ Ki \circ \alpha_Y\mu_Y (y) = \alpha_X\mu_X \circ Hi (y) = \beta_X \nu_X \circ Hi (y) = Ki \circ \beta_Y \nu_Y (y)\enspace. \] Then also \[ Kk \circ Ki \circ \alpha_Y\mu_Y (y) = Kk \circ Ki \circ \beta_Y \nu_Y (y)\enspace, \] but $Kk \circ Ki = K(ki) = K{\rm{id}}_{Y} = {\rm{id}}_{KY}$ and the claim follows. \end{proof}
In the second case, when $|Y| \geq |X|$, we obtain the following consequence of $|X|$--accessibility of $F$ and $G$.
\begin{lemma} There exist $s \in FX$, $t \in GX$ and injective mappings $i,j: X \rightarrow Y$ such that $Fi(s) = \mu_Y(y)$ and $Gj(t) = \nu_Y(y)$. \end{lemma}
\begin{proof}
As $F$ is $|X|$--accessible, there exists $X'$, $s' \in FX'$, and $f: X' \rightarrow Y$ such that $|X'| < |X|$ and $Ff(s') = \mu_Y(y)$. We factorize $f$ as $f = if'$ with $f': X' \rightarrow X$ and injective $i: X \rightarrow Y$, we set $s = Ff'(s')$, and get $Fi(s) = Fif'(s') = Ff(s') = \mu_Y(y)$. Finding $t$ and $j$ is completely analogous. \end{proof}
We fix $s,t,i,j$ satisfying the conclusion of the previous lemma and continue by finding further mappings that will help us in caluculations.
\begin{lemma} \label{lemma:aa} There exist $k,l,m: Y \rightarrow X$ and $n: X \rightarrow Y$ such that $ki = lj = {\rm{id}}_X$, $nmi=i$, and $nmj = j$. \end{lemma}
\begin{proof} The mappings $k$ and $l$ are arbitrarily chosen left inverses to $i$ and $j$, respectively.
As $X$ is infinite, the union $U =i[X] \cup j[X]$ has cardinality $|X|$, therefore there exists $m: Y \rightarrow X$ which is injective on $U$. Now we take any $n: X \rightarrow Y$ such that the restriction of $nm$ to $U$ is the identity and obtain $nmi=i$ and $nmj=j$, as required. \end{proof}
Using the mappings from Lemma~\ref{lemma:aa} we get $$ \alpha_Y\mu_Y (y) = \alpha_Y Fi (s) = \alpha_Y F(iki) (s) = \alpha_Y F(ik) Fi(s) = \alpha_Y F(ik) \mu_Y(y)\enspace. $$ Using this equality, the naturality of $\alpha$ and $\mu$, and Lemma~\ref{lemma:aa} we can continue this calculation as follows. \begin{align*} \alpha_Y F(ik) \mu_Y(y) &= \alpha_Y F(nmik) \mu_Y(y) = \alpha_Y F(nm) F(ik) \mu_Y(y) \\ &= K(nm) \alpha_Y F(ik) \mu_Y (y) = K(nm) \alpha_Y \mu_Y (y) = Kn Km \circ \alpha_Y \mu_Y (y) \\ &= Kn \circ \alpha_X \mu_X Hm (y)\enspace, \end{align*} Similarly, $$ \beta_Y\nu_Y (y) = Kn \circ \beta_X \nu_X Hm (y) $$ But $(\mu_X Hm (y),\nu_X Hm (y))$ is in $T(\mu,\nu)$, so $\alpha_X \mu_X Hm (y) = \beta_X \nu_X Hm (y)$ by the assumption $(\star)$ and we get $\alpha_Y\mu_Y (y) = \beta_Y\nu_Y (y)$, finishing the proof of the following theorem.
\begin{theorem} The category $\cat{AccSetFun}$ is concretizable. \end{theorem}
\section{Conclusion}
We have shown that the category of accessible set functors is as comprehensive as possible: it contains all concretizable categories as full subcategories. The next obvious question is how comprehensive is the collection of all set functors. Recall that this collection is not even a class and it is not clear what is a natural limitation on possible full subcollections.
Perhaps a more interesting question concerns the so called \emph{hyper--universality}. A combination of theorems by Ku\v cera~\cite{K4} and Trnkov\'a~\cite{T7} (see Section 4.7 in \cite{T5}) implies that in every universal category it is possible to find equivalences on hom--sets (which are compatible with compositions) so that the quotient category is hyper--universal, that is, it contains \emph{every} category as a full subcategory. Is there a natural hyper--universal quotient of the category of accessible set functors? Is it some kind of a homotopy equivalence on natural transformations?
Finally, let us return to Universal Algebra and finitary set functors. The main result of~\cite{Ba2} says, in the language of~\cite{BKO}, that the category of minions (with possibly infinite universes) is alg--universal. However, in the promise CSP we are so far mostly interested in minions with finite universes. How comprehensive is their category?
\end{document} | arXiv |
\begin{definition}[Definition:Integral Element of Algebra/Definition 1]
Let $A$ be a commutative ring with unity.
Let $f : A \to B$ be a commutative $A$-algebra.
Let $b\in B$.
The element $b$ is '''integral''' over $A$ {{iff}} it is a root of a monic polynomial in $A \sqbrk x$.
\end{definition} | ProofWiki |
\begin{document}
\title{Narrow operators on lattice-normed spaces}
\author{M.~Pliev}
\address{South Mathematical Institute of the Russian Academy of Sciences\\ str. Markusa 22, Vladikavkaz, 362027 Russia}
\keywords{Narrow operators, GAM-compact operators, dominated operators, lattice-normed spaces, Banach lattices}
\subjclass[2000]{Primary 46B99; Secondary 47B99.}
\begin{abstract} The aim of this article is to extend results of Maslyuchenko~O., Mykhaylyuk~V. Popov~M. about narrow operators on vector lattices. We give a new definition of a narrow operator where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every $GAM$-compact $\text{(bo)}$-norm continuous linear operator from a Banach-Kantorovich space $V$ to a Banach lattice $Y$ is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow if and only if its exact dominant is. \end{abstract}
\maketitle
\section{Introduction}
\subsection{} Today the theory of narrow operators is a very active area of Functional Analysis {\cite{B,Bi-1,Bi-2,F,Kad-2,Kad-3,Ma}}. Plichko and Popov were first $[20]$ who systematically studied this class of operators. It is worth remarking, however, that narrow operators have been studied in some particular cases by some others authors before this notion appeared. For example, Ghoussoub and Rosental $[8]$ have considered {``}norm-signed preserving operators{''} on $L_{1}[0,1]$, which are precisely the operators on $L_{1}[0,1]$ which are not narrow. On the other hand, Enflo and Starbird $[6]$ proved that if $T:L_{1}(\mu)\rightarrow L_{1}(\nu)$ is $L_{1}$-complementary singular (i.e. $T$ is invertible on no complemented subspace of $L_{1}(\mu)$isomorphic to $L_{1}(\mu)$) then $T$ is narrow. Johnson, Maurey, Schechtman and Tzafriri $[9]$ proved that every operator $T:L_{p}(\mu)\rightarrow L_{p}(\nu),1 < p<2$ which is $L_{p}$-complementary singular is narrow. Later Kadets, Shvidkov and Werner had considered narrow operators in a different context $[13]$. Flores and Ruiz considered narrow operators from a K{\"{o}}the function space $E$ $[7]$. Finally, Maslyuchenko, Mykhaylyuk and Popov have considered a general vector-lattice approach to narrow operators $[18]$.
\subsection{}
In this seminal paper [18] the authors gave a new definition of a narrow operator.
\begin{definition} \label{def:0.2}
Let $E$ be an atomless order complete vector lattice, $X$ a Banach space. A map $f:E\rightarrow X$ is called {\it narrow} if for every $x\in E_{+}$ and every $\varepsilon>0$ there exist some $y\in E$ such that $|y|=x$ and $\|f(y)\|<\varepsilon$. We say that $f$ is {\it strictly narrow} if for every $x\in E_{+}$ there exists some $y\in E$ such that $|y|=x$ and $f(y)=0$. \end{definition}
In the same paper $[18]$ another definition of a narrow operator for the case when the range space is a vector lattice, was given. Let $E,F$ be vector lattices with $E$ atomless. A linear operator $T:E\rightarrow F$ is called {\it order narrow} if for every $x\in E_{+}$ there exists a net $(x_{\alpha})$ in $E$ such that $|x_{\alpha}|=x$ for each $\alpha$ and $Tx_{\alpha}\overset{(o)}\rightarrow 0$.
\subsection{} In this paper we consider narrow operators in the framework of lattice-normed spaces. The notion of a lattice-normed space was introduced by Kantorovich in the first part of 20th century $[10]$. Later, Kusraev and his school had provided a deep theory. A detailed account the reader can find in $[15]$.
\section{Preliminaries}
The goal of this section is to introduce some basic definitions and facts. General information on vector lattices, Banach spaces and lattice-normed spaces the reader can find in the books $[1,2,15,16,17,19]$.
{\bf 1.\,1.}~Consider a vector space $V$ and a real archimedean vector lattice $E$. A map $\reduce\left\bracevert\reduce\vphantom{X} \cdot\reduce\vphantom{X}\reduce\right\bracevert\reduce:V\rightarrow E$ is a \textit{vector norm} if it satisfies the following axioms: \begin{enumerate}
\item[1)] $\reduce\left\bracevert\reduce\vphantom{X} v \reduce\vphantom{X}\reduce\right\bracevert\reduce\geq 0;$\,\, $\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce=0\Leftrightarrow v=0$;\,\,$(\forall v\in V)$.
\item[2)] $\reduce\left\bracevert\reduce\vphantom{X} v_1+v_2 \reduce\vphantom{X}\reduce\right\bracevert\reduce\leq \reduce\left\bracevert\reduce\vphantom{X} v_1\reduce\vphantom{X}\reduce\right\bracevert\reduce+\reduce\left\bracevert\reduce\vphantom{X} v_2 \reduce\vphantom{X}\reduce\right\bracevert\reduce;\,\, ( v_1,v_2\in V)$.
\item[3)] $\reduce\left\bracevert\reduce\vphantom{X}\lambda v\reduce\vphantom{X}\reduce\right\bracevert\reduce=|\lambda|\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce;\,\, (\lambda\in\Bbb{R},\,v\in V)$. \end{enumerate} A vector norm is called \textit{decomposable} if \begin{enumerate}
\item[4)] for all $e_{1},e_{2}\in E_{+}$ and $x\in V$ from $\reduce\left\bracevert\reduce\vphantom{X} x\reduce\vphantom{X}\reduce\right\bracevert\reduce=e_{1}+e_{2}$ it follows that there exist $x_{1},x_{2}\in V$ such that $x=x_{1}+x_{2}$ and $\reduce\left\bracevert\reduce\vphantom{X} x_{k}\reduce\vphantom{X}\reduce\right\bracevert\reduce=e_{k}$, $(k:=1,2)$. \end{enumerate}
A triple $(V,\reduce\left\bracevert\reduce\vphantom{X}\cdot\reduce\vphantom{X}\reduce\right\bracevert\reduce,E)$ (in brief $(V,E),(V,\reduce\left\bracevert\reduce\vphantom{X}\cdot\reduce\vphantom{X}\reduce\right\bracevert\reduce)$ or $V$ with default parameters omitted) is a \textit{lattice-normed space} if $\reduce\left\bracevert\reduce\vphantom{X}\cdot\reduce\vphantom{X}\reduce\right\bracevert\reduce$ is a $E$-valued vector norm in the vector space $V$. If the norm $\reduce\left\bracevert\reduce\vphantom{X}\cdot\reduce\vphantom{X}\reduce\right\bracevert\reduce$ is decomposable then the space $V$ itself is called decomposable. We say that a net $(v_{\alpha})_{\alpha\in\Delta}$ {\it $(bo)$-converges} to an element $v\in V$ and write $v=\omathop{\lim}{bo} v_{\alpha}$ if there exists a decreasing net $(e_{\gamma})_{\gamma\in\Gamma}$ in $E$ such that $\inf_{\gamma\in\Gamma}(e_{\gamma})=0$ and for every $\gamma\in\Gamma$ there is an index $\alpha(\gamma)\in\Delta$ such that $\reduce\left\bracevert\reduce\vphantom{X} v-v_{\alpha(\gamma)}\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq e_{\gamma}$ for all $\alpha\geq\alpha(\gamma)$. A net $(v_{\alpha})_{\alpha\in\Delta}$ is called \textit{$(bo)$-fundamental} if the net $(v_{\alpha}-v_{\beta})_{(\alpha,\beta)\in\Delta\times\Delta}$ $(bo)$-converges to zero. A lattice-normed space is called {\it $(bo)$-complete} if every $(bo)$-fundamental net $(bo)$-converges to an element of this space. Let $e$ be a positive element of a vector lattice $E$. By $[0,e]$ we denote the set $\{v\in V:\,\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq e\}$. A set $M\subset V$ is called $\text{(bo)}$-{\it bounded } if there exists $e\in E_{+}$ such that $M\subset[0,e]$. Every decomposable $(bo)$-complete lattice-normed space is called a {\it Banach-Kantorovich space} (a BKS for short).
\subsection{} Let $(V,E)$ be a lattice-normed space. A subspace $V_{0}$ of $V$ is called a $\text{(bo)}$-ideal of $V$ if for $v\in V$ and $u\in V_{0}$, from $\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq\reduce\left\bracevert\reduce\vphantom{X} u\reduce\vphantom{X}\reduce\right\bracevert\reduce$ it follows that $v\in V_{0}$. A subspace $V_{0}$ of a decomposable lattice-normed space $V$ is a $\text{(bo)}$-ideal if and only if $V_{0}=\{v\in V:\,\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce\in L\}$, where $L$ is an order ideal in $E$ [15,\,2.1.6.1]. Let $V$ be a lattice-normed space and $y,x\in V$. If $\reduce\left\bracevert\reduce\vphantom{X} x\reduce\vphantom{X}\reduce\right\bracevert\reduce\bot\reduce\left\bracevert\reduce\vphantom{X} y\reduce\vphantom{X}\reduce\right\bracevert\reduce=0$ then we call the elements $x,y$ {\it disjoint} and write $x\bot y$. The equality $x=\coprod_{i=1}^{n}x_{i}$ means that $x=\sum_{i=1}^{n}x_{i}$ and $x_{i}\bot x_{j}$ if $i\neq j$. An element $z\in V$ is called a {\it component} or a \textit{fragment} of $x\in V$ if $0\leq \reduce\left\bracevert\reduce\vphantom{X} z\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq\reduce\left\bracevert\reduce\vphantom{X} x\reduce\vphantom{X}\reduce\right\bracevert\reduce$ and $x\bot(x-z)$. Two fragments $x_{1},x_{2}$ of $x$ are called \textit{mutually complemented} or $MC$, in short, if $x=x_1+x_{2}$. The notations $z\sqsubseteq x$ means that $z$ is a fragment of $x$. According to [1,\,p.86] an element $e>0$ of a vector lattice $E$ is called an {\it atom}, whenever $0\leq f_{1}\leq e$, $0\leq f_{2}\leq e$ and $f_{1}\bot f_{2}$ imply that either $f_{1}=0$ or $f_{2}=0$. A vector lattice $E$ is atomless if there is no atom $e\in E$.
The following object will be often used in different constructions below. Let $V$ be a lattice-normed space and $x\in V$. A sequence $(x_{n})_{n=1}^{\infty}$ is called a {\it disjoint tree} on $x$ if $x_{1}=x$ and $x_{n}=x_{2n}\coprod x_{2n+1}$ for each $n\in\Bbb{N}$. It is clear that all $x_{n}$ are fragments of $x$. All lattice-normed spaces below we consider to be decomposable.
\subsection{}
Consider some important examples of lattice-normed spaces. We begin with simple extreme cases, namely vector lattices and normed spaces. If $V=E$ then the modules of an element can be taken as its lattice norm: $\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce:=|v|=v\vee(-v);\,v\in E$. Decomposability of this norm easily follows from the Riesz Decomposition Property holding in every vector lattice. If $E=\Bbb{R}$ then $V$ is a normed space.
Let $Q$ be a compact and let $X$ be a Banach space. Let $V:=C(Q,X)$ be the space of continuous vector-valued functions from $Q$ to $X$. Assign $E:=C(Q,\Bbb{R})$. Given $f\in V$, we define its lattice norm by the relation $\reduce\left\bracevert\reduce\vphantom{X} f\reduce\vphantom{X}\reduce\right\bracevert\reduce:t\mapsto\|f(t)\|_{X}\,(t\in Q)$. Then $\reduce\left\bracevert\reduce\vphantom{X}\cdot\reduce\vphantom{X}\reduce\right\bracevert\reduce$ is a decomposable norm [15,\,lemma 2.3.2].
Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure space, let $E$ be an order-dense ideal in $L_{0}(\Omega)$ and let $X$ be a Banach space. By $L_{0}(\Omega,X)$ we denote the space of (equivalence classes of) Bochner $\mu$-measurable vector functions acting from $\Omega$ to $X$. As usual, vector-functions are equivalent if they have equal values at almost all points of the set $\Omega$. If $\widetilde{f}$ is the coset of a measurable vector-function $f:\Omega\rightarrow X$ then $t\mapsto\|f(t)\|$,$(t\in\Omega)$ is a scalar measurable function whose coset is denoted by the symbol $\reduce\left\bracevert\reduce\vphantom{X}\widetilde{f}\reduce\vphantom{X}\reduce\right\bracevert\reduce\in L_{0}(\mu)$. Assign by definition $$ E(X):=\{f\in L_{0}(\mu,X):\,\reduce\left\bracevert\reduce\vphantom{X} f\reduce\vphantom{X}\reduce\right\bracevert\reduce\in E\}. $$
Then $(E(X),E)$ is a lattice-normed space with a decomposable norm [15,\,lemma 2.3.7.]. If $E$ is a Banach lattice then the lattice-normed space $E(X)$ is a Banach space with respect to the norm $|\|f|\|:=\|\|f(\cdot)\|_{X}\|_{E}$.
\subsection{}
Let $E$ be a Banach lattice and let $(V,E)$ be a lattice-normed space. By definition, $\reduce\left\bracevert\reduce\vphantom{X} x\reduce\vphantom{X}\reduce\right\bracevert\reduce\in E_{+}$ for every $x\in V$, and we can introduce some \textit{mixed norm} in $V$ by the formula $$
\||x|\|:=\|\reduce\left\bracevert\reduce\vphantom{X} x\reduce\vphantom{X}\reduce\right\bracevert\reduce\|\,\,\,(\forall\, x\in V). $$
The normed space $(V,\||\cdot|\|)$ is called a \textit{space with a mixed norm}. In view of the inequality $|\reduce\left\bracevert\reduce\vphantom{X} x\reduce\vphantom{X}\reduce\right\bracevert\reduce-\reduce\left\bracevert\reduce\vphantom{X} y\reduce\vphantom{X}\reduce\right\bracevert\reduce|\leq\reduce\left\bracevert\reduce\vphantom{X} x-y\reduce\vphantom{X}\reduce\right\bracevert\reduce$ and monotonicity of the norm in $E$, we have $$
\|\reduce\left\bracevert\reduce\vphantom{X} x\reduce\vphantom{X}\reduce\right\bracevert\reduce-\reduce\left\bracevert\reduce\vphantom{X} y\reduce\vphantom{X}\reduce\right\bracevert\reduce\|\leq\||x-y|\|\,\,\,(\forall\, x,y\in V), $$
so a vector norm is a norm continuous operator from $(V,\||\cdot|\|)$ to $E$. A lattice-normed space $(V,E)$ is called a \textit{Banach space with a mixed norm} if the normed space $(V,\||\cdot|\|)$ is complete with respect to the norm convergence.
\subsection{}
Consider lattice-normed spaces $(V,E)$ and $(W,F)$, a linear operator $T:V\rightarrow W$ and a positive operator $S\in L_{+}(E,\,F)$. If the condition $$ \reduce\left\bracevert\reduce\vphantom{X} Tv\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq S\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce;\,(\forall\, v\in V) $$ is satisfied then we say that $S$ \textit{dominates} or \textit{majorizes} $T$ or that $S$ is \textit{dominant} or {majorant} for $T$. In this case $T$ is called a \textit{dominated} or \textit{majorizable} operator. The set of all dominants of the operator $T$ is denoted by $\text{maj}(T)$. If there is the least element in $\text{maj}(T)$ with respect to the order induced by $L_{+}(E,F)$ then it is called the {\it least} or the {\it exact dominant} of $T$ and it is denoted by $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$. The set of all dominated operators from $V$ to $W$ is denoted by $M(V,W)$. Denote by $E_{0+}$ the conic hull of the set $\reduce\left\bracevert\reduce\vphantom{X} V\reduce\vphantom{X}\reduce\right\bracevert\reduce=\{\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce:\,v\in V\}$, i.e., the set of elements of the form $\sum_{k=1}^{n}\reduce\left\bracevert\reduce\vphantom{X} v_{k}\reduce\vphantom{X}\reduce\right\bracevert\reduce$, where $v_{1},\dots,v_{n}\in V$, $n\in\Bbb{N}$.
\begin{lemma}[\cite{Ku}, 4.1.2,\,4.1.5.] \label{le:1} Let $(V,E),(W,F)$ be lattice-normed spaces. Suppose $V$ is decomposable and $F$ is order complete. Then every dominated operator has the exact dominant $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$. The exact dominant of an arbitrary operator $T\in M(V,W)$ can be calculated by the following formulas: $$ \reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e) =\sup\left\{\sum\limits_{i=1}^n\reduce\left\bracevert\reduce\vphantom{X} Tv_i\reduce\vphantom{X}\reduce\right\bracevert\reduce: \sum\limits_{i=1}^n\reduce\left\bracevert\reduce\vphantom{X} v_i\reduce\vphantom{X}\reduce\right\bracevert\reduce= e, \,e\in E_{0+}\right\}; $$ $$ \reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)=\sup\{\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e_{0}):\,e_{0}\in E_{0+};\, e_{0}\leq e\} (e\in E_{+}); $$ $$ \reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)=\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e_+)+\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e_-),\, (e\in E). $$ \end{lemma}
{\bf Acknowledgment.}~I am very grateful to professor Mikhail Popov for his valuable remarks and great help. I am also grateful to the referees for their useful suggestions.
\section{Definition and some properties of narrow operators} \label{sec2}
In this section we introduce a new class of operators in lattice-normed spaces and describe some of their properties.
\begin{definition} \label{def:nar1}
Let $(V,E)$ be a lattice-normed space, $X$ a Banach space and suppose that $E$ is atomless. An operator $T:V\rightarrow X$ is called \textit{narrow}, if for every $u\in V,\,\varepsilon>0$ there exist two $MC$ fragments $u_{1},u_{2}$ of $u$ such that $\|T(u_{1}-u_{2})\|<\varepsilon$. If for every $u\in V$ there exist two $MC$ fragments $u_{1},u_{2}$ of the $u$ such that $T(u_{1}-u_{2})=0$ for then the operator $T$ is called \textit{strictly narrow}. \end{definition}
The set of all narrow operators from a lattice-normed space $(V,E)$ to a Banach space $X$ we denote by $\mathcal{N}(V,X)$.
\begin{lemma} \label{le:2} If a lattice-normed space $(V,E)$ coincides with $(E,E)$ then definitions \ref{def:0.2} and \ref{def:nar1} are equivalent. \end{lemma}
\begin{proof}
Let $X$ be a Banach space and let $T:E\rightarrow X$ be a narrow operator in accordance with Definition \ref{def:nar1}. Consider an element $e\in E_{+}$ and $\varepsilon>0$. Then there exist two $MC$ fragments $e_{1},\,e_{2}$ of $e$ such that $\|T(e_{1}-e_{2})\|<\varepsilon$. Then for $y=e_{1}-e_{2}$ one has that $\|Ty\|<\varepsilon$, that is, Definition \ref{def:0.2} for $T$ is satisfied.
Now we prove the inverse assertion. Let $T$ be a narrow operator in accordance with Definition \ref{def:0.2}. Fix any $x\in E$ and $\varepsilon>0$. Then $x=x_{+}-x_{-}$ and there exist two elements $x_{1}'$ and $x_{2}'$ such that $|x_{1}'|=x_{+},\,\|Tx_{1}'\|<\frac{\varepsilon}{2}$ and $|x_{2}'|=x_{-},\,\|Tx_{2}'\|<\frac{\varepsilon}{2}$. We consider new elements: $e_{1}:=x_{1}'\vee 0$,\, $e_{2}:=-x_{1}'\vee 0$ and $f_{1}:=x_{2}'\vee 0$,\, $f_{2}:=-x_{2}'\vee 0$. So, we have two pairs of $MC$ fragments $e_{1},e_{2}$ of $x_{+}$ and $f_{1},f_{2}$ of $x_{-}$ such that the following inequalities hold $$
\|T(e_{1}-e_{2})\|<\frac{\varepsilon}{2};\,
\|T(f_{1}-f_{2})\|<\frac{\varepsilon}{2}. $$ Then $x_{1}:=e_{1}-f_{2}$ and $x_{2}:=e_{2}-f_{1}$ are $MC$ fragments of $x$, and $$
\|T(x_{1}-x_{2})\|=\|T(e_{1}-f_{2}-e_{2}+f_{1})\|<
\|T(e_{1}-e_{2})\|+\|T(f_{1}-f_{2})\|<\varepsilon. $$ \end{proof}
Let $(V,E)$ be a lattice-normed space and let $(W,F)$ be a Banach space with a mixed norm. An operator $T:V\rightarrow W$ is called \textit{order narrow} if for every $u\in V$ there exists a net $(v_{\alpha})_{\alpha\in\Lambda}$ where every element $v_{\alpha}$ is a difference $u_{\alpha}^{1}-u_{\alpha}^{2}$ of two $MC$ fragments of $u$ such that $T(v_{\alpha})\overset{(bo)}\rightarrow 0$.
\begin{lemma} \label{le:3} Let $(V,E)$ be a lattice-normed space and let $(W,F)$ be a Banach space with a mixed norm. Then every narrow operator $T:V\rightarrow W$ is order narrow. \end{lemma}
\begin{proof}
We consider an element $u\in V$. Let $\varepsilon_{n}:=\frac{1}{2^{n}}$ and let $u_{n}^{1},u_{n}^{2}$ be $MC$ fragments~of the element $u$ and $v_{n}=(u_{n}^{1}-u_{n}^{2})$, $|\|Tv_{n}|\|\leq\varepsilon_{n}$. We set $e_{n}=\sum\limits_{k=n}^{\infty}\reduce\left\bracevert\reduce\vphantom{X} Tv_{k}\reduce\vphantom{X}\reduce\right\bracevert\reduce$, $e_{n}\in F_{+}$, $e_{n}\downarrow 0$ and the following estimate holds $\reduce\left\bracevert\reduce\vphantom{X} Tv_{n}\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq e_{n}$. Thus, $Tv_{n}\overset{(bo)}\rightarrow 0$. \end{proof}
The sets of narrow and order narrow operators coincide if a vector lattice $F$ is good enough.
\begin{lemma}\label{le:4} Let $(V,E)$ and $(W,F)$ be the same as in $3.3$ and let $F$ be a Banach lattice with order continuous norm. Then a linear operator $T:V\rightarrow W$ is order narrow if and only if $T$ is narrow. \end{lemma}
\begin{proof} Let $T$ be an order narrow operator. Then for every $u\in V$ there exist a net
$(v_{\alpha})_{\alpha\in\Lambda};\,v_{\alpha}=u_{\alpha}^{1}-u_{\alpha}^{2}$, where $u_{\alpha}^{1}$ and $u_{\alpha}^{2}$ are $MC$ fragments of $u$ and $Tv_{\alpha}\overset{(bo)}\rightarrow 0$. Fix any $\varepsilon>0$. Using the fact that the norm in $F$ is order continuous we can find $\alpha_{0}\in\Lambda$ such that $\|\reduce\left\bracevert\reduce\vphantom{X} Tv_{\alpha}\reduce\vphantom{X}\reduce\right\bracevert\reduce\|<\varepsilon$ for every $\alpha\geq\alpha_{0}$. In view of Lemma $3.3$, the converse is true. \end{proof}
The following lemma will be useful later.
\begin{lemma} \label{le:5} Let $(V,E)$ $(J,F_{1})$ and $(W,F)$ be lattice-normed spaces, $E$ an atomless vector lattice, $J$ a $\text{(bo)}$-ideal of $W$ and $F_{1}$ an order ideal of $F$. If a linear dominated operator $T:V\rightarrow J$ is order narrow then $T:V\rightarrow W$ is order narrow as well. Conversely, if a dominated linear operator $T:V\rightarrow J$ is such that $T:V\rightarrow W$ is an order narrow then so is $T:V\rightarrow J$. \end{lemma}
\begin{proof} The first part is obvious. Let $T:V\rightarrow J$ be a dominated operator such that $T:V\rightarrow W$ is order narrow. For any $v\in V$ we choose a net $(v_{\alpha})_{\alpha\in\Lambda}$ where every element $v_{\alpha}$ is a difference $u_{\alpha}^{1}-u_{\alpha}^{2}$ of two $MC$ fragments of $u$ such that $T(v_{\alpha})\overset{(bo)}\rightarrow 0$, that is, $\reduce\left\bracevert\reduce\vphantom{X} Tv_{\alpha}\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq y_{\alpha}\downarrow 0$ for some net $(y_{\alpha})_{\alpha\in\Delta}\subset F$. Using the fact that the operator $T$ is dominated and its exact dominant is $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce:E\rightarrow F_{1}$, one has that $$ \reduce\left\bracevert\reduce\vphantom{X} Tv_{\alpha}\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce=g\in F_{1}. $$ Hence, $\reduce\left\bracevert\reduce\vphantom{X} Tv_{\alpha}\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq z_{\alpha}\downarrow 0$ where $z_{\alpha}:=g\wedge y_{\alpha}$. So, we have that $(z_{\alpha})_{\alpha\in\Delta}\subset F_{1}$. Thus, the net $(Tv_{\alpha})_{\alpha\in\Delta}$ $\text{(bo)}$-converges to $0$ in $(J,F_{1})$. \end{proof}
\section{Narrow and $GAM$-compact operators} \label{sec4}
In this section we investigate connections between narrow and $GAM$-compact operators.
Let $(V,E)$ be a lattice-normed space and let $Y$ be a Banach space. A linear operator $T:V\rightarrow Y$ is called {\it $(\text{bo})$-norm continuous} whenever it sends every $\text{(bo)}$-convergent net in $V$ to a norm convergent net in $Y$. A linear operator $T:V\rightarrow Y$ is called {\it generalized $AM$-compact} or {\it $GAM$-compact} for short if for every $\text{(bo)}$-bounded set $M\subset V$ its image $T(M)$ is a relatively compact set in $Y$. Consider some examples.
{\it Example~1.}~In a particular case when $V=E$ the sets of $GAM$-compact and $AM$-compact operators from $V$ to $Y$ are equal.
{\it Example~2.}~If $E=\Bbb{R}$ then $V$ is a normed space and the sets of $GAM$-compact and compact operators from $V$ to $Y$ are equal.
{\it Example~3.}~Let $X,Y$ be Banach spaces and let $(\Omega,\Sigma,\mu)$ be a finite measure space. The space $L_{1}(\mu,X)$ is the space $\mu$-Bochner integrable functions on $\Omega$ with values in $X$, and $L_{\infty}(\mu,X)$ is the space of $X$-valued $\mu$-Bochner integrable functions on $\Omega$ that are essentially bounded. A function $g\in L_{\infty}(\mu,\mathcal{L}(X,Y))$ is said to have its essential range in the {\it uniformly compact operators} if there is a compact set $C$ in $Y$ such that $g(\omega)x\in C$ for almost all $\omega\in\Omega$ and $x\in X$, $\|x\|\leq 1$. An operator $T:L_{1}(\mu,X)\rightarrow Y$ is called {\it representable measurable kernel} if there is a bounded measurable function $g:\Omega\rightarrow\mathcal{L}(X,Y)$ such that $$ Tf=\int_{\Omega}fg\,d\mu \,\,\,\,\,\,\,\,\text{for all}\,\,f\in L_{1}(\mu,X) . $$
\begin{thm}[\cite{Ke}, Theorem~2]\label{t:1} Let $X$ be a Banach space such that $X^{\star}$ has the Radon-Nikod\'{y}m property. Then there is an isometric isomorphism between the space of compact operators $K(L_{1}(\mu,X),Y)$ and the subspace of $L_{\infty}(\mu,K(X,Y))$ consisting of those functions whose essential range is in the uniformly compact operators. In fact, $T \in K(L_{1}(\mu,X),Y)$ and $g \in L_{\infty}(\mu,K(X,Y))$ are in correspondence if and only if $$ T(f)=\int_{\Omega}g(\omega)f(\omega)\,d\mu(\omega)\,\,\, \text{for all} \,\,\,f\in L_{1}(\mu,X). $$ \end{thm}
\begin{lemma} Let $X$ be a Banach space such that $X^{\star}$ has the Radon-Nikod\'{y}m property. Then every representable measurable kernel operator $T:L_{1}(\mu,X)\rightarrow Y$ whose kernel have the essential range in the uniformly compact operators is $GAM$-compact. \end{lemma}
\begin{proof} Let $M\subset V$ be a $(\text{bo})$-bounded set. Then there exists $e\in L_{1}(\mu)$, $e\geq 0$, such that $\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq e$ for every $v\in M$. This implies that the set $M$ is norm bounded in $L_{1}(\mu, X)$ $$
\|v\|_{L_{1}(\mu, X)}=\int_{\Omega}\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce d\,\mu=
\int_{\Omega}\| v(\cdot)\|_{X} d\,\mu\leq\int_{\Omega}e d\,\mu=r $$ By Theorem $4.1$, the lemma is proved. \end{proof}
\begin{lemma} Let $(V,E)$ be a Banach space with a mixed norm and let $X$ be a Banach space. Then every $GAM$-compact operator $T:V\rightarrow X$ is norm bounded. \end{lemma}
\begin{proof}
If $T$ were unbounded then we would have a sequence $(v_{n})\subset V$ with $\|\reduce\left\bracevert\reduce\vphantom{X} v_{n}\reduce\vphantom{X}\reduce\right\bracevert\reduce\|\leq\frac{1}{2^{n}}$ and $\|Tv_{n}\|\rightarrow\infty$. The sequence $(v_{n})$ is order bounded by the element $e\in E_{+}$, $e:=\sum\limits_{n=1}^{\infty}\reduce\left\bracevert\reduce\vphantom{X} v_{n}\reduce\vphantom{X}\reduce\right\bracevert\reduce$. Hence, the sequence $(Tv_{n})$ is relatively compact, -- a contradiction. \end{proof}
\begin{cor} Let $(V,E)$ be a Banach space with a mixed norm, $E$ an order continuous Banach lattice and $X$ a Banach space. Then every $GAM$-compact operator $T:V\rightarrow X$ is $\text{(bo)}$-norm continuous. \end{cor}
The following lemma is well known [11,\,p.14].
\begin{lemma} Let $(x_{i})_{i=1}^{n}$ be a finite collection of vectors in a finite dimensional normed space $X$ and let $(\lambda_{i})_{i=1}^{n}$ be a collection of reals with $0\leq\lambda_{i}\leq 1$ for each $i$. Then there exists a collection $(\theta_{i})_{i=1}^{n}$ of numbers $\theta_{i}\in\{0,1\}$ such that $$
\Big\|\sum_{i=1}^{n}(\lambda_{i}-\theta_{i})x_{i}\Big\|\leq\frac{\text{dim}(X)}{2}\max_{i}\|x_{i}\| $$ \end{lemma}
For the rest of the section $(V,E)$ is assumed to be a Banach-Kantorovich space, $E$ is an atomless order complete vector lattice, $X$ is a Banach space and $T:V\rightarrow X$ is $\text{(bo)}$-norm continuous linear operator.
\begin{lemma}
Let $x\in V$ . Then there exist two $MC$ fragments $x_{1},x_{2}$ of $x$ such that $\|Tx_{1}\|-\|Tx_{2}\|=0$. \end{lemma}
\begin{proof}
Fix any couple $x_{1},x_{2}$ of $MC$ fragments of $x$. If $\|Tx_{1}\|-\|Tx_{2}\|=0$ then there is nothing to prove. Let $\|Tx_{1}\|-\|Tx_{2}\|>0$. Consider the partially ordered set $$
D:=\{y\sqsubseteq x_{1}:\|T(x_{1}-y)\|-\|T(x_{2}+y)\|\geq 0\} $$
where $y_{1}\leq y_{2}$ if and only if $y_{1}\sqsubseteq y_{2}$. If $B\subset D$ is a chain then $y^{\star}=\vee B\in D$ by the $\text{(bo)}$-norm continuity of $T$. By the Zorn lemma, there is a maximal element $y_{0}\in D$. Now we show $\|T(x_{1}-y_{0})\|-\|T(x_{2}+y_{0})\|=0$. Suppose on the contrary that $$
\alpha=\|T(x_{1}-y_{0})\|-\|T(x_{2}+y_{0})\|> 0. $$
Since $E$ is atomless, we can choose a fragment $0\neq y\sqsubseteq(x_{1}-y_{0})$ with $|\|Ty|\|<\frac{\alpha}{3}$. Since $y_{0}\bot y$, $y_{0}+y\sqsubseteq x_{1}$ and $$
\|T(x_{1}-y_{0}-y)\|-\|T(x_{2}+y_{0}+y)\|\geq $$ $$
\geq\|T(x_{1}-y_{0})\||-\|Ty\|-\|T(x_{2}+y_{0})\|-\|Ty\|>\frac{\alpha}{3}, $$ that contradicts the maximality of $y_{0}$. \end{proof}
\begin{lemma}
Let $v\in V$ and $(v_{n})_{n=1}^{\infty}$ be a disjoint tree on $v$. If $\|Tx_{2n}\|=\|Tx_{2n+1}\|$ for every $n\leq 1$ then $$
\lim\limits_{m\rightarrow\infty}\max\limits_{2^{m}\leq i< 2^{m+1}}\|Tv_{i}\|=0 $$ \end{lemma}
\begin{proof}
Put $\gamma_{m}=\max\limits_{2^{m}\leq i<2^{m+1}}\|Tv_{i}\|$ and $\varepsilon=\limsup\limits_{m\rightarrow\infty}\gamma_{m}$. Suppose on the contrary that $\varepsilon>0$. Then for each $n\in\Bbb{N}$ we set $$
\varepsilon_{n}=\limsup\limits_{m\rightarrow\infty}\max\limits_{2^{m}\leq i<2^{m+1},\,v_{i}\sqsubseteq v_{n}}\|Tv_{n}\|. $$ Hence, for each $m\in\Bbb{N}$ one has $$ \max\limits_{2^{m}\leq i<2^{m+1}}\varepsilon_{i}=\varepsilon.\,\,\,\,\,(\star) $$
Now we are going to construct a sequence of mutually disjoint elements $(v_{n_{j}})_{j=1}^{\infty}$ such that $\|Tv_{n_{j}}\|\leq\frac{\varepsilon}{2}$. At the first step we choose $m_{1}$ such that $\max\limits_{2^{m_{1}}\leq i<2^{m_{1}+1}}\|Tv_{i}\|\geq\frac{\varepsilon}{2}$. In accordance with $(\star)$, we choose $i_{1}$, $2^{m_{1}}\leq i_{1}<2^{m_{1}+1}$ so that $\varepsilon_{m_{1}}=\varepsilon$. Using $\|Tv_{2n}\|=\|Tv_{2n+1}\|$, we choose $n_{1}\neq i_{2}$, $2^{m_{1}}\leq i<2^{m_{1}+1}$ so that $\|Tv_{n_{1}}\|\leq\frac{\varepsilon}{2}$. At the second step we choose $m_{2}>m_{1}$ so that $$
\max\limits_{2^{m}\leq i<2^{m+1},\,v_{i}\sqsubseteq v_{i_{1}}}\|Tv_{i}\|\geq\frac{\varepsilon}{2}. $$
In accordance with $(\star)$, we choose $i_{2}$, $2^{m_{2}}\leq i_{2}<2^{m_{2}+1}$ so that $\varepsilon_{i_{2}}=\varepsilon$. Then we choose $m_{2}\neq i_{2}$, $2^{m_{2}}\leq i_{2}<2^{m_{2}+1}$ so that $\|Tv_{m_{2}}\|\geq\frac{\varepsilon}{2}$. Proceeding further, we construct the desired sequence. Indeed, $\|Tv_{m_{i}}\|\geq\frac{\varepsilon}{2}$ by the construction and mutually disjoint for $v_{m_{l}},v_{m_{j}}$, $j\neq l$ is guaranteed by the condition $m_{j}\neq i_{j}$. The elements $v_{m_{j+l}}$ are fragments of $v_{i_{j}}$ which are disjoint to $v_{m_{j}}$. \end{proof}
The following lemma indicates that operators with finite dimensional range are also narrow.
\begin{lemma} If $X$ is finite dimensional, then $T$ is narrow. \end{lemma}
\begin{proof}
Fix any $v\in V$, $\varepsilon>0$ and $\text{dim}(X)=\gamma$. Using Lemma $4.6$ we construct a disjoint tree $(v_{n})$ on $v$ with $\|Tv_{2n}\|=\|Tv_{2n+1}\|$ for all $n\in\Bbb{N}$. By lemma $4.7$ we choose $m$ such that $\gamma\alpha_{m}<\varepsilon$ where $\alpha_{m}:=\max\limits_{2^{m}\leq i<2^{m+1}}\|Tv_{i}\|$. Then using Lemma $4.5$, we choose numbers $(\lambda_{i})_{i=1}^{n}$, $\lambda_{i}\in\{0,1\}$ for $i=2^{m},\dots, 2^{m+1}-1$ so that $$
\Big\|\sum_{i=2^{m}}^{2^{m+1}-1} \Bigl(\frac{1}{2}-\lambda_{i} \Bigr)Tv_{i}\Big\|
\leq\frac{\gamma}{2}\max\limits_{2^{m}\leq i<2^{m+1}}\|Tv_{i}\|= \frac{\gamma}{2}\alpha_{m}<\frac{\varepsilon}{2}. $$
Now we consider the element $w=2 \Bigl(\sum\limits_{i=2^{m}}^{2^{m+1}-1}(\frac{1}{2}-\lambda_{i})v_{i} \Bigr)$. On the other hand, $w=\sum\limits_{i=2^{m}}^{2^{m+1}-1}u_{i}$ where $u_{i}$ are disjoint, and $u_{i}\in\{(\pm v_{i})_{i=2^{m}}^{2^{m+1}-1}\}$. Then there exist two $MC$-fragments $v_{1}$ and $v_{2}$ of $v$ such that $w=v_{1}-v_{2}$ and $\|Tw=T(v_{1}-v_{2})\|<\varepsilon$. \end{proof}
Now we are ready to prove the main result of this section.
\begin{thm} Let $(V,E)$ be a Banach-Kantorovich space, $E$ an atomless order complete vector lattice, $X$ a Banach space. Then every $GAM$-compact $\text{(bo)}$-norm continuous linear operator $T:V\rightarrow X$ is narrow. \end{thm}
\begin{proof}
It is well known that if $H$ is a relatively compact subset of $l_{\infty}(D)$ for some infinite set $D$ and $\varepsilon>0$ is an arbitrary positive number then there exists a finite rank operator $S\in l_{\infty}(D)$ such that $\|x-Sx\|\leq\varepsilon$ for every $x\in H$. So, we may consider $X$ as a subspace of some $l_{\infty}(D)$ space $$ X \hookrightarrow X^{\star\star} \hookrightarrow l_{\infty}(B_{X^{\star}})=l_{\infty}(D)=W. $$ By the notation $\hookrightarrow$ we mean isometric embedding. Fix any $v\in V$ and $\varepsilon>0$. Since $T$ is a $GAM$-compact operator, $K=\{Tu:\reduce\left\bracevert\reduce\vphantom{X} u\reduce\vphantom{X}\reduce\right\bracevert\reduce\leq\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce\}$ is relatively compact in $X$ and hence, in $W$. Then there exist a finite dimensional operator
$S\in\mathcal{L}(W)$ such that $\|x- Sx\|\leq\frac{\varepsilon}{2}$ for every $x\in K$. Then $R=S\circ T$ is a $(\text{bo})$-norm continuous finite dimensional operator. By Lemma $4.8$, there exist two $MC$ fragments $v_{1},v_{2}$ of $v$ such that $\|R(v_{1}-v_{2})\|<\frac{\varepsilon}{2}$. Thus, $$
\|T(v_{1}-v_{2})\|=\|T(v_{1}-v_{2})+S(T(v_{1}-v_{2}))-S(T(v_{1}-v_{2}))\|= $$ $$
=\|T(v_{1}-v_{2})+R(v_{1}-v_{2})-S(T(v_{1}-v_{2}))\|\leq $$ $$
\leq\|R(v_{1}-v_{2})\|+\|T(v_{1}-v_{2})-S(T(v_{1}-v_{2}))\| <\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon. $$ \end{proof}
\section{Dominated narrow operators} \label{sec5}
In this section we investigate some properties of the dominated narrow operators. Observe that for every $x,y\in L_{1}(\nu)$ the following equality holds $$
\|x-y\|=\||x|-|y|\|+\|x\|+\|y\|-\|x+y\|.\,\,\,(\star) $$
\begin{thm} Let $E,F$ be order complete vector lattices such that $E$ is atomless, $F$ an ideal of some order continuous Banach lattice and $(V,E)$ a Banach-Kantorovich space. Then every $(bo)$-continuous dominated linear operator $T:V\rightarrow F$ is order narrow if and only if $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$ is. \end{thm}
\begin{proof}
First we mention that a Banach lattice $F$ is a lattice-normed space and the vector norm $\reduce\left\bracevert\reduce\vphantom{X}\cdot\reduce\vphantom{X}\reduce\right\bracevert\reduce$ coincides with the module map $|\cdot|$. Now we prove the theorem for $F=L_{1}(\nu)$. By Lemma $3.4$, instead of order narrowness we will consider narrowness. Assume first that $T$ is narrow. Fix any $e\in E_{+}$ and $\varepsilon>0$. Let $v\in V$ and $\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce=e$. Since $$
\left\{\sum\limits_{i=1}^n|Tv_i|: \sum\limits_{i=1}^n\reduce\left\bracevert\reduce\vphantom{X} v_i\reduce\vphantom{X}\reduce\right\bracevert\reduce= e;\,\reduce\left\bracevert\reduce\vphantom{X} v_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce\bot\reduce\left\bracevert\reduce\vphantom{X} v_{j}\reduce\vphantom{X}\reduce\right\bracevert\reduce;\,i\neq j;\,n\in\Bbb{N}\right\} $$ is an increasing net, using the order continuity of $L_{1}(\nu)$, we can choose a finite collection $\{v_{1},\dots,v_{n}\}\subset V$ with $$ e=\bigsqcup\limits_{i=1}^{n}\reduce\left\bracevert\reduce\vphantom{X} v_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce $$ and $$
\Big\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)-\sum\limits_{i=1}^{n}|Tv_{i}|\,\Big\|<\varepsilon. $$ Let $v_{i}=u_{i}\coprod w_{i}$ be a decomposition into a sum of $MC$ fragments $u_{i}$ and $w_{i}$. Then we have $$
0\leq\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)-\sum\limits_{i=1}^{n}(|Tu_{i}|+|Tw_{i}|)\leq $$ $$
\leq\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)-\sum\limits_{i=1}^{n}|Tv_{i}| $$ Since $$ e=\sum\limits_{i=1}^n\reduce\left\bracevert\reduce\vphantom{X} v_i\reduce\vphantom{X}\reduce\right\bracevert\reduce=\sum\limits_{i=1}^n(\reduce\left\bracevert\reduce\vphantom{X} u_i\reduce\vphantom{X}\reduce\right\bracevert\reduce+\reduce\left\bracevert\reduce\vphantom{X} w_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce), $$ we have that $$ \reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)=\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(\sum\limits_{i=1}^n(\reduce\left\bracevert\reduce\vphantom{X} u_i\reduce\vphantom{X}\reduce\right\bracevert\reduce+\reduce\left\bracevert\reduce\vphantom{X} w_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce) =\sum\limits_{i=1}^n(\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} u_i\reduce\vphantom{X}\reduce\right\bracevert\reduce+\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} w_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce). $$
Since $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} u_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce-|Tu_{i}|$ and $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} w_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce-|Tw_{i}|$ are positive elements of $L_{1}(\nu)$ for every $i\in\{1,\dots,n\}$, the sum of their norms equals the norm of their sum. Thus, we obtain $$
\sum\limits_{i=1}^{n}(\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} u_i\reduce\vphantom{X}\reduce\right\bracevert\reduce-|Tu_{i}|\,\|+\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} w_i\reduce\vphantom{X}\reduce\right\bracevert\reduce-|Tw_{i}|\,\|)= $$ $$
=\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)-\sum\limits_{i=1}^{n}(|Tu_{i}|+|Tw_{i}|)\|\leq $$ $$
\leq\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)-\sum\limits_{i=1}^{n}|Tv_{i}|\,\|<\varepsilon. $$
For each $i=1,\dots,n$ we represent $v_{i}=u_{i}\coprod w_{i}$ so that $u_{i},w_{i}\in V$ and $\|Tw_{i}-Tu_{i}\|<\frac{\varepsilon}{n}$. Then putting $u=\coprod\limits_{i=1}^{n}u_{i}$, $w=\coprod\limits_{i=1}^{n}w_{i}$, $f_{1}=\reduce\left\bracevert\reduce\vphantom{X} u\reduce\vphantom{X}\reduce\right\bracevert\reduce$, $f_{2}=\reduce\left\bracevert\reduce\vphantom{X} w\reduce\vphantom{X}\reduce\right\bracevert\reduce$ and using inequality $(\star)$, we obtain $$
\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce f_{1}-\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce f_{2}\|\leq\sum\limits_{i=1}^{n}\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} u_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce-\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} w_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce\|\leq $$ $$
\leq\sum\limits_{i=1}^{n}\|\,|Tu_{i}|-| Tw_{i}|\,\|+
\sum\limits_{i=1}^{n}(\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} u_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce-|Tu_{i}|\,\|+
\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} w_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce-|Tw_{i}|\,\|)\leq $$ $$
\leq\sum\limits_{i=1}^{n}\| Tu_{i}- Tw_{i}\|+\varepsilon<2\varepsilon. $$ Using the arbitrariness of $e \in E_{+}$ and $\varepsilon>0$, and the fact that $f_{1}, f_{2}$ are two $MC$ fragments of $e$, we deduce that $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$ is a narrow operator.
Now let $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$ be a narrow operator, $v$ an arbitrary element of $V$, $\varepsilon>0$, $\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce=e$, $e=\coprod\limits_{i=1}^{n}\reduce\left\bracevert\reduce\vphantom{X} v_{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce$,\,$v_{i}\in V;\,\forall i\in\{1,\dots,n\}$ and again $$
\Big\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)-\sum\limits_{i=1}^{n}|Tv_{i}|\,\Big\|<\varepsilon. $$ For each $i=1,\dots,n$ we decompose $v_{i}=f^{1}_{i}\coprod f^{2}_{i}$ such that $$
\Big\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{1}^{i}\reduce\vphantom{X}\reduce\right\bracevert\reduce-\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce\Big\|<\frac{\varepsilon}{n} $$ and let $f^{1}=\coprod\limits_{i=1}^{n}f_{1}^{i}$ and $f^{2}=\coprod\limits_{i=1}^{n}f_{2}^{i}$. Taking into account that the $L_{1}$-norm of a sum of positive elements equals the sum of their norm, we obtain $$
\sum\limits_{i=1}^{n}|Tf_{i}^{j}|\leq\sum\limits_{i=1}^{n}\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{j}\reduce\vphantom{X}\reduce\right\bracevert\reduce;\,j\in\{1,2\}; $$ $$ \sum\limits_{i=1}^{n}\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce+\sum\limits_{i=1}^{n}\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce=\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e); $$ $$
\Big\|\sum\limits_{i=1}^{n}|Tf_{i}^{1}|+|Tf_{i}^{2}|\,\Big\|\leq\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce (e)\|. $$ Then, using again inequality $(\star)$, we obtain $$
\|Tf_{1}-Tf_{2}\|\leq
\sum\limits_{i=1}^{n}\|Tf_{i}^{1}-Tf_{i}^{2}\|
=\sum\limits_{i=1}^{n}\|\,|Tf_{i}^{1}|-|Tf_{i}^{2}|\,\|+ $$ $$
+\sum\limits_{i=1}^{n}\|\,|Tf_{i}^{1}|+|Tf_{i}^{2}|\,\|-\sum\limits_{i=1}^{n}\|\,|Tv_{i}|\,\|= $$ $$
=\sum\limits_{i=1}^{n}\|\,|Tf_{i}^{1}|-|Tf_{i}^{2}|+\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce-\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce+
\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce \reduce\left\bracevert\reduce\vphantom{X} f_{i}^{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce-\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce\|+ $$ $$
+\sum\limits_{i=1}^{n}\|\,|Tf_{i}^{1}|+|Tf_{i}^{2}|\,\|-\sum\limits_{i=1}^{n}\|\,|Tv_{i}|\,\|\leq $$ $$
\leq\sum\limits_{i=1}^{n}\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce-\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce\|+
\sum\limits_{i=1}^{n}\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce-|Tf_{i}^{1}|\,\|+ $$ $$
+\sum\limits_{i=1}^{n}\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} f_{i}^{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce-|Tf_{i}^{2}|\,\|+
\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)\|-\sum\limits_{i=1}^{n}\|\,|Tv_{i}|\,\| $$ Finally, using the fact that $$
\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)\|-\sum\limits_{i=1}^{n}\|\,|T v_{i}|\,\|=
\Big\|\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(e)-\sum\limits_{i=1}^{n}|T v_{i}|\,\Big\|<\varepsilon $$
we obtain that $\|Tf_{1}-Tf_{2}\|< 3\varepsilon$. Since $f_{1}$ and $f_{2}$ are $MC$ fragments of $v$, this proves that $T$ is narrow.
Now we consider the general case. Since $F$ is an ideal of some order continuous Banach lattice $H$, we have by Lemma $3.5$ that $T:V\rightarrow F$ is order narrow if and only if $ T:V\rightarrow H$ is.
We consider $T:V\rightarrow H$ and $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce:E\rightarrow H$. Fix any $v\in V$. By $E_{1}$ and $H_{1}$ we denote the principal bands in $E$ and $H$ generated by $\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce$ and $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce$ respectively. Using the fact that Boolean algebras of bands $\mathcal{B}(V)$ and $\mathcal{B}(E)$ are isomorphic [15,\,2.1.2.1], we denote $V_{1}:=\gamma(E_{1})$. Here $\gamma:\mathcal{B}(E)\rightarrow\mathcal{B}(V)$ is a boolean isomorphism. Denote by $T_{1}$ the restriction of $T$ to $V_{1}$. Operator $\reduce\left\bracevert\reduce\vphantom{X} T_{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce$ coincides with the restriction $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$ to $E_{1}$. So $H_{1}$ is an order continuous Banach lattice with weak unit $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\reduce\left\bracevert\reduce\vphantom{X} v\reduce\vphantom{X}\reduce\right\bracevert\reduce$. Then by [17,Theorem\,1.b.14] there exists a probability space $(\Omega,\Sigma,\nu)$ and an ideal $H_{2}$ of $L_{1}(\nu)$ such that $H_{1}$ is isomorphic to $H_{2}$. Let $S:H_{1}\rightarrow H_{2}$ be a lattice isomorphism. Then we set $T_{2}=S\circ T_{1}$. Moreover, $\reduce\left\bracevert\reduce\vphantom{X} T_{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce=S\circ\reduce\left\bracevert\reduce\vphantom{X} T_{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce$. By our previous consideration, $T_{2}:V_{1}\rightarrow H_{2}$ is order narrow if and only if $\reduce\left\bracevert\reduce\vphantom{X} T_{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce:E_{1}\rightarrow H_{2}$ is. Thus, we have proved that $T_{2}$ is order narrow if and only if $\reduce\left\bracevert\reduce\vphantom{X} T_{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce$ is.
Let $T:V\rightarrow H$ be order narrow. Fix an arbitrary $v\in V$. Since $T_{1}$ is order narrow, so is $T_{2}$ and hence $\reduce\left\bracevert\reduce\vphantom{X} T_{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce$. So there exists a net $(v_{\alpha})_{\alpha\in\Lambda}\subset V_{1}$ where every element $v_{\alpha}$ is a difference $u_{\alpha}^{1}-u_{\alpha}^{2}$ two $MC$ fragments of $v$ such that $\reduce\left\bracevert\reduce\vphantom{X} T_{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce(\reduce\left\bracevert\reduce\vphantom{X} v_{\alpha}\reduce\vphantom{X}\reduce\right\bracevert\reduce)\overset{(o)}\rightarrow 0$. Therefore, $\reduce\left\bracevert\reduce\vphantom{X} T_{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce(\reduce\left\bracevert\reduce\vphantom{X} v_{\alpha}\reduce\vphantom{X}\reduce\right\bracevert\reduce)\overset{(o)}\rightarrow 0$ and hence $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce(\reduce\left\bracevert\reduce\vphantom{X} v_{\alpha}\reduce\vphantom{X}\reduce\right\bracevert\reduce)\overset{(o)}\rightarrow 0$. So, $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce:E\rightarrow H$ is order narrow. \end{proof}
Let $(V,E),(W,F)$ be lattice-normed spaces and let $M(V,W)$ be the space of dominated operators from $V$ to $W$. The band generated by all lattice homomorphisms from $E$ to $F$ we denote $\mathcal{H}(E,F)$. Then, $\mathcal{H}(V,W):=\{T\in M(V,W):\,\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce\in\mathcal{H}(E,F)\}$.
\begin{thm} Let $E,F$ be order complete vector lattices such that $E$ is atomless, $F$ an ideal of some order continuous Banach lattice and let $(V,E)$ be a Banach-Kantorovich space. Then every $(bo)$-continuous dominated linear operator $T:V\rightarrow F$ is uniquely represented in the form $ T=T_{h}+T_{n}$, where $T_{h}\in \mathcal{H}(V,F)$ and $T_{n}$ is a $(bo)$-continuous order narrow operator. \end{thm}
\begin{proof} By $[15,4.2.1]$, the set $M(V,E)$ with the mapping $p: M(V,E)\rightarrow L_{+}(E,F)$ is a lattice-normed space. Here $p(T)=\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$ for every $T\in M(V,E)$. By $[15,\,4.2.6]$, the vector norm $p:M(V,F)\rightarrow L_{+}(E,F)$ is decomposable. This means that for every dominated operator $T:V\rightarrow F$ and its exact dominant $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce:E\rightarrow F$ the following statement hold: $$ \reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce=S_{1}+S_{2}\Rightarrow\exists T_{1},T_{2}\in M(V,F); $$ $$ \reduce\left\bracevert\reduce\vphantom{X} T_{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce=S_{1};\,\reduce\left\bracevert\reduce\vphantom{X} T_{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce=S_{2};\,0\leq S_{1},S_{2};\,S_{1}\bot S_{2}. $$ Fix an arbitrary $(bo)$-continuous dominated operator $T:V\rightarrow F$. By theorem $[15,4.3.2]$, every dominated operator $T:V\rightarrow F$ is $(bo)$-continuous if and only if $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce:E\rightarrow F$ is $(o)$-continuous. Hence, $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$ is $(o)$-continuous. Then by $[18,\,11.7]$ the positive $(o)$-continuous operator $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce$ is uniquely represented as a sum $\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce=S_{D}+S_{N}$ when $S_{D}$ is a $(o)$-continuous operator, $S_{D}\in\mathcal{H}(E,F)$ and $S_{N}$ is a $(o)$-continuous narrow operator. Then we obtain $$ \reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce=S_{D}+S_{N}\Rightarrow\exists T_{1},\,T_{2}\in M(V,F);\, $$ $$ \reduce\left\bracevert\reduce\vphantom{X} T_{1}\reduce\vphantom{X}\reduce\right\bracevert\reduce=S_{N};\,\reduce\left\bracevert\reduce\vphantom{X} T_{2}\reduce\vphantom{X}\reduce\right\bracevert\reduce=S_{D}. $$ Finally, by Theorem $5.1$, the proof is completed. \end{proof}
\begin{thm} Let $E(\mu)$ be a K{\"{o}}the function space over probability space $(\Omega,\Sigma,\mu)$, with atomless measure $\mu$, such that $E(\mu)'$ is order continuous, $F$ is an order continuous Banach lattice and $X$ is a Banach space. Then for every $(bo)$-continuous dominated narrow linear operator $T:E(X)\rightarrow F$ the inclusion $[0,\reduce\left\bracevert\reduce\vphantom{X} T\reduce\vphantom{X}\reduce\right\bracevert\reduce]\subset \mathcal{N}(E(X),F)$ holds. \end{thm}
\begin{proof} By theorem $[7,\,3.15]$, for every positive narrow operator $S:E(\mu)\rightarrow F$, such that $E(\mu)'$ is order continuous, the inclusion $[0,S]\subset \mathcal{N}(E(X),F)$ holds. Now by Theorem $5.1$ the proof is completed. \end{proof}
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calculate the mass of 1 atom of nitrogen
Anonymous. (Relative atomic masses: H = 1.0, N = 14.0) Reveal answer Calculate the mass of {eq}2.50 \times 10^4 {/eq} molecules of nitrogen gas. Antimony has two naturally occurring isotopes. The molar mass of water is 18.015 g/mol. The four lead isotopes have atomic masses and relative abundances of 203.973 amu (1.4%), 205.974 amu (24.1%), 206.976 amu (22.1%) and 207.977 amu (52.4%). wt. This means that a single atom of sodium weighs 23 atomic mass units (AMUs), but it also means that one mole of sodium atoms weighs 23 grams. (a) Mass of nitrogen atom = number of moles x atomic mass = 1 x 14 = 14 g (b) Mass of aluminium atom = 4 x atomic mass = 4 x 27 = 108 g Question: 1)Calculate The Mass Defect Of The Nitrogennucleus . Calculating the mass of 1 molecule of ammonia: ... ( Mass of nitrogen atom / Mass of ammonia molecule ) … Problem: The ratio of the mass of a nitrogen atom to the mass of an atom of 12C is 7:6 and the ratio of the mass of nitrogen to oxygen in N2O is 7:4. The molar mass of nitrogen is 14.0067 g/mol. is 3.0 × 10 –25 J, calculate its wavelength. 1) Calculate the molar mass. So mass of 1 mole of N2 is 28g.Therefore mass of 0.5 mole is 28/2 =14g 1-30. Mass of each atom of calcium = 6.642 x 10-26 /1 = 6.642 x 10-26 kg. A compound which contains one atom of X and two atoms of Y for each three atoms of Z is made by mixing 5. They are equal to 11 and 23, respectively. Do the same for a hydrogen chloride molecule in which the chlorine atom has an atomic mass of 34.97. 714 1561 1189 You can determine the mass percent, or how much of the total mass of ammonium carbonate is nitrogen, by first determining the mass of nitrogen and the mass of the total compound. Calculate the percentage atom economy for the reaction in Stage 7. Add the mass of Hydrogen, Nitrogen, and two Oxygens. 1. 7. Unit Conversion: In a unit conversion, the mathematical method of calculation is dimensional analysis. Find the atomic number (Z) and mass number (A). Calculate the reduced mass of a nitrogen molecule in which both nitrogen atoms have an atomic mass of 14.00. Choose your element. Calculate the atomic mass of lead. Calculate the total mass of the 7 protons and 7 neutrons in one nucleus of ""_7^14"N". Multiply the relative atomic mass by the molar mass constant. This converts atomic units to grams per mole, making the molar mass of hydrogen 1.007 grams per mole, of carbon 12.0107 grams per mole, of oxygen 15.9994 grams per mole, and of chlorine 35.453 grams per mole. Calculate the mass of the following: (i) 0.5 mole of gas (mass from mole of molecule) (ii) 0.5 mole of N atoms (mass from mole of atom) 6 moles Nitrogen to grams = 84.0402 grams. Step 2: Determine the molar mass of the element. Gram atomic mass of nitrogen = 14 g Therefore, 1 mole of nitrogen atoms contains 14 g. Calculate the mass defect for this atom in atomic mass units given the following data: mass of a proton = 1.007276 u mass of a neutron = 1.008665 u mass of an electron = 0.000549 u … Q:-Determine the empirical formula of an oxide of iron which has 69.9% iron and 30.1% dioxygen by mass. The starting point here will be the molar mass of nitrogen, which is listed as . 8 moles Nitrogen to grams = 112.0536 grams ... (iii)€€€€ The percentage yield of ammonia is the percentage, by mass, of the nitrogen and hydrogen which has been converted to ammonia. How do you calculate the mass of a single atom or molecule? 1 u = 1/12 the mass of carbon 12 by definition. Favorite Answer. The charge is 0. Calculate the maximum mass of ammonia that can be made from an excess of nitrogen and 12.0 g of hydrogen. FREE Expert Solution Show answer. This was calculated by multiplying the atomic weight of hydrogen (1.008) by two and adding the result to the weight for one oxygen (15.999). > The mass defect is the difference between the calculated mass (the sum of the masses of the protons and neutrons) and the actual mass of the nucleus. Relevance. Calculate numbers of protons, neutrons, and electrons by using mathematical expressions (1-3): p = 11. n = 23 - 11 = 12. e = 11 - 0 = 11 4. 1 moles Nitrogen to grams = 14.0067 grams. The question also states that the molar mass of nitrogen (N2) is 28.0 g/mole. So let's think : if there are 14 grams in 6,02x10^23 atoms , we just need to find this value in one atom. Let's assume that it is the atom of sodium (Na). 7 moles Nitrogen to grams = 98.0469 grams. Each mole of nitrogen contains 6.0221415 × 10^23 molecules - known as Avogadro's number. The molar mass of an element is the mass in g of one mole of the element. Q:- Atomic number is the total number of the protons that are present in the nucleus of an element's atom. (1 u is equal to 1/12 the mass of one atom of carbon-12) Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol. We must give here N2 not N because N is only atom and N2 is the molecule Molar mass : The mass of the one mole of the substance is called molar mass. Answer: Molar mass of N2 is 28(since atomic weight of nitrogen is 14and forN2=14*2). 1 mole of nitrogen atoms is equivalent to the gram atomic mass of nitrogen. Question 1: The mass of an atom of uranium-235 is observed to be 235.044 u. Answer Save. To solve this we just need a little relation , if an element has 14 U it means it has 14gram every mole of this element . If its K.E. Finding molar mass starts with units of grams per mole (g/mol). Hence 6.02 x … Percent of nitrogen: 21.2% To get the percent composition of nitrogen is ammonium sulfate, (NH_4)_2SO_4, you need to know the molar mass of the compound and that of elemental nitrogen. mass of one nitrogen atom and nitrogen molecule in kg. (the kh value for nitrogen in water is 6.1 x 10-4 m/atm.) 3 moles Nitrogen to grams = 42.0201 grams. The percentage by weight of any atom or group of atoms in a compound can be computed by dividing the total weight of the atom (or group of atoms) in the formula by the formula weight and multiplying by 100. What is the mass of a single nitrogen atom in grams? Weights of atoms and isotopes are from NIST article. Use the equation to calculate the maximum mass of magnesium oxide produced. Therefore, mass of one mole of nitrogen = 14g 100% (369 ratings) Problem Details. The mass of an electron is 9.1 × 10 –31 kg. The ratio of the mass of a nitrogen atom to the mass of an atom of {eq}_{12} {/eq}C is 7:6, and the ratio of the mass of nitrogen to oxygen in N{eq}_2 {/eq}O is 7:4. assume a total pressure of 1.0 atm and the mole fraction for nitrogen of 0.78. Weights of atoms and isotopes are from NIST article. I know that relative atomic mass of $\ce{^{12}C}$ is $12~\mathrm{u}$. Capitalize the first letter in chemical symbol and use lower case for the remaining letters: Ca, Fe, Mg, Mn, S, O, H, C, N, Na, K, Cl, Al. 1-31. You're adding the masses of uncombined protons and neutrons, 1.0073 u and 1.0087 u respectively. The atomic mass of Nitrogen is 14.00674. 1 … Gram atomic mass of an element is the amount of that element in grams whose quantity is equal to atomic mass of that element. Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). Q:-Calculate the wavelength of an electron moving with a velocity of 2.05 × 10 7 ms –1. (1 u is equal to 1/12 the mass of one atom of carbon-12) Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol. Hydrogen has an atomic mass of 1.008. Get an answer to your question Calculate the mass of nitrogen dissolved at room temperature in a 80.0 l home aquarium. 2 moles Nitrogen to grams = 28.0134 grams. But when those particles fuse together to form an atom, some of the mass is converted into energy according to E=mc^2. 2 Answers. 6. 10 years ago. The important thing to notice is that ammonium sulfate contains 2 nitrogen atoms, each belonging to one ammonium ion, NH_4^(+). Find the mass of 1 mol of oxygen atoms. 5 moles Nitrogen to grams = 70.0335 grams. Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). Please remember that you need the molar mass first when trying to find the average mass of one molecule. The atomic mass of Nitrogen is 14.0067the atomic mass of nitrogen is 14.007, so if you round off you will get 14 Therefore, gram atomic mass of nitrogen atom = 14 g ( since, atomic mass of nitrogen = 14u) Mass of one mole of an atom equal to gram atomic mass of atom . Step 1: Calculate the number of moles from the number of atoms. Take the relative atomic masses of hydrogen and nitrogen to be 1 and 14 respectively. So, the molar mass of ammonium sulfate is 132.14 g/mol and the molar mass … The mass defect is 0.108 506 u. Divide the atomic mass by the avogadro's number = 2.3258736 x 10^ -23 ~= 2.326 x 10 ^ -23 Molecular mass of nitrogen (N 2) = 14 x 2 = 28 g. 1 mole of nitrogen is 28 g = 28 x 10-3 kg. This is defined as 0.001 kilogram per mole, or 1 gram per mole. 4 moles Nitrogen to grams = 56.0268 grams. 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# Understanding the basics of linear regression
Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. It assumes that there is a linear relationship between the variables, meaning that a change in one variable is associated with a proportional change in the other variable.
The basic equation for a linear regression model is:
$$y = mx + b$$
Where:
- $y$ is the dependent variable
- $x$ is the independent variable
- $m$ is the slope of the line
- $b$ is the y-intercept
The goal of linear regression is to find the best-fitting line that minimizes the distance between the observed data points and the predicted values from the line. This line can then be used to make predictions about the dependent variable based on the values of the independent variable(s).
Linear regression can be used for both simple regression, which involves only one independent variable, and multiple regression, which involves two or more independent variables. In both cases, the goal is to estimate the coefficients (slope and intercept) that best fit the data.
Suppose we have a dataset that contains information about the number of hours studied and the corresponding test scores of a group of students. We want to use linear regression to model the relationship between the number of hours studied and the test scores.
Here is a sample of the dataset:
| Hours Studied | Test Score |
| ------------- | ---------- |
| 2 | 70 |
| 3 | 80 |
| 4 | 90 |
| 5 | 95 |
| 6 | 100 |
We can use linear regression to find the best-fitting line that represents the relationship between the number of hours studied and the test scores. This line can then be used to predict the test score for a given number of hours studied.
## Exercise
Given the dataset:
| X | Y |
| - | - |
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
| 5 | 11 |
Use linear regression to find the equation of the line that best fits the data.
### Solution
The equation of the line is:
$$y = 2x + 1$$
# Evaluating the performance of regression models
Once we have built a regression model, it is important to evaluate its performance to determine how well it fits the data and makes predictions. There are several metrics and techniques that can be used to evaluate regression models.
One commonly used metric is the coefficient of determination, also known as R-squared. R-squared measures the proportion of the variance in the dependent variable that can be explained by the independent variable(s). It ranges from 0 to 1, with a higher value indicating a better fit.
Another metric is the mean squared error (MSE), which measures the average squared difference between the predicted values and the actual values. A lower MSE indicates a better fit.
Additionally, we can use visual techniques to evaluate the model's performance. One such technique is plotting the residuals, which are the differences between the predicted values and the actual values. A random scatter of residuals around zero indicates a good fit, while a pattern in the residuals suggests that the model is not capturing all the information in the data.
Let's consider the example of predicting housing prices based on the size of the house. We have built a regression model and obtained the following metrics:
- R-squared: 0.75
- MSE: 1000
The R-squared value of 0.75 indicates that 75% of the variance in housing prices can be explained by the size of the house. The MSE value of 1000 suggests that, on average, the predicted prices are off by 1000 units.
We can also plot the residuals to visually assess the model's performance. If the residuals are randomly scattered around zero, it indicates a good fit. If there is a pattern in the residuals, it suggests that the model is not capturing all the information in the data.
## Exercise
You have built a regression model to predict the sales of a product based on advertising expenditure. The R-squared value of the model is 0.85 and the MSE is 500. Interpret these metrics.
### Solution
The R-squared value of 0.85 indicates that 85% of the variance in sales can be explained by advertising expenditure. The MSE value of 500 suggests that, on average, the predicted sales are off by 500 units.
# Incorporating multiple variables in regression models
In many cases, a single independent variable may not be sufficient to accurately predict the dependent variable. In such cases, we can incorporate multiple independent variables in the regression model.
The general equation for a multiple regression model is:
$$y = b_0 + b_1x_1 + b_2x_2 + ... + b_nx_n$$
Where:
- $y$ is the dependent variable
- $x_1, x_2, ..., x_n$ are the independent variables
- $b_0, b_1, b_2, ..., b_n$ are the coefficients
Each coefficient represents the change in the dependent variable for a one-unit change in the corresponding independent variable, holding all other independent variables constant.
To estimate the coefficients in a multiple regression model, we can use techniques such as ordinary least squares (OLS) or maximum likelihood estimation (MLE). These techniques minimize the sum of the squared differences between the observed values and the predicted values.
Suppose we want to predict the salary of an employee based on their years of experience and education level. We have collected data from a sample of employees and obtained the following results:
- Years of experience ($x_1$): 5, 10, 15, 20
- Education level ($x_2$): Bachelor's, Master's, Bachelor's, PhD
- Salary ($y$): 50000, 70000, 60000, 80000
We can incorporate both variables in the regression model to predict the salary based on years of experience and education level.
## Exercise
You have collected data on the number of hours studied, the number of practice tests taken, and the corresponding test scores of a group of students. Use multiple regression to predict the test score based on the number of hours studied and the number of practice tests taken.
### Solution
The multiple regression equation is:
$$y = b_0 + b_1x_1 + b_2x_2$$
Where:
- $y$ is the test score
- $x_1$ is the number of hours studied
- $x_2$ is the number of practice tests taken
To estimate the coefficients $b_0, b_1, b_2$, we can use techniques such as ordinary least squares (OLS) or maximum likelihood estimation (MLE).
# Utilizing predictive modeling techniques
In addition to linear regression, there are several other predictive modeling techniques that can be used to build models and make predictions. These techniques include:
1. Polynomial regression: This technique extends linear regression by including higher-order terms of the independent variables. It can capture non-linear relationships between the variables.
2. Decision trees: Decision trees are a non-parametric technique that uses a tree-like model of decisions and their possible consequences. They can handle both categorical and numerical variables.
3. Random forests: Random forests are an ensemble learning method that combines multiple decision trees to improve predictive accuracy. They can handle large datasets with high-dimensional feature spaces.
4. Support vector machines: Support vector machines are a supervised learning method that can be used for classification and regression tasks. They find the optimal hyperplane that separates the data into different classes or predicts the values of the dependent variable.
5. Neural networks: Neural networks are a set of algorithms inspired by the structure and function of the human brain. They can be used for a wide range of tasks, including regression and classification.
Each predictive modeling technique has its own strengths and weaknesses, and the choice of technique depends on the specific problem and the characteristics of the data.
Suppose we want to predict the price of a house based on its size, number of bedrooms, and location. We can use a random forest model to make predictions. The random forest model combines multiple decision trees and takes into account the interactions between the variables.
The random forest model can handle both numerical and categorical variables, making it suitable for this problem. It can also handle missing data and outliers, which are common in real-world datasets.
## Exercise
You have a dataset that contains information about the age, gender, and income of a group of individuals. You want to predict whether an individual is likely to purchase a product based on these variables. Which predictive modeling technique would you use?
### Solution
For this problem, a suitable predictive modeling technique would be a support vector machine (SVM). SVMs can handle both numerical and categorical variables, making them suitable for this problem. They can also handle binary classification tasks, which is what we want to achieve - predicting whether an individual is likely to purchase a product or not.
# Selecting the most relevant variables for regression analysis
In regression analysis, it is important to select the most relevant variables to include in the model. Including irrelevant variables can lead to overfitting and reduce the model's predictive accuracy.
There are several techniques that can be used to select the most relevant variables:
1. Forward selection: This technique starts with an empty model and adds variables one by one based on their contribution to the model's fit. The process continues until no further improvement is achieved.
2. Backward elimination: This technique starts with a model that includes all variables and removes variables one by one based on their contribution to the model's fit. The process continues until no further improvement is achieved.
3. Stepwise selection: This technique combines forward selection and backward elimination. It starts with an empty model and adds or removes variables based on their contribution to the model's fit. The process continues until no further improvement is achieved.
4. Lasso regression: Lasso regression is a regularization technique that can be used to select variables and estimate their coefficients. It adds a penalty term to the regression equation, which encourages sparsity in the coefficient estimates.
The choice of variable selection technique depends on the specific problem and the characteristics of the data. It is important to evaluate the performance of the selected variables using metrics such as R-squared and MSE.
Suppose we want to predict the price of a car based on its age, mileage, and brand. We can use forward selection to select the most relevant variables. We start with an empty model and add variables one by one based on their contribution to the model's fit. We evaluate the performance of the model at each step using metrics such as R-squared and MSE.
## Exercise
You have a dataset that contains information about the height, weight, and age of a group of individuals. You want to predict the body fat percentage based on these variables. Which variable selection technique would you use?
### Solution
For this problem, a suitable variable selection technique would be lasso regression. Lasso regression adds a penalty term to the regression equation, which encourages sparsity in the coefficient estimates. This can help select the most relevant variables and estimate their coefficients.
# Advanced regression techniques
In addition to the basic linear regression model, there are several advanced regression techniques that can be used to improve the model's predictive accuracy. These techniques include:
1. Ridge regression: Ridge regression is a regularization technique that can be used to handle multicollinearity, which occurs when there is a high correlation between the independent variables. It adds a penalty term to the regression equation, which reduces the magnitude of the coefficient estimates.
2. Elastic net regression: Elastic net regression is a combination of ridge regression and lasso regression. It can be used to handle multicollinearity and select the most relevant variables.
3. Generalized linear models: Generalized linear models extend linear regression to handle non-normal dependent variables, such as binary or count data. They use a link function to relate the linear predictor to the dependent variable.
4. Bayesian regression: Bayesian regression is a probabilistic approach that incorporates prior knowledge about the parameters of the regression model. It provides a posterior distribution for the parameters, which can be used to make predictions and estimate uncertainty.
Each advanced regression technique has its own strengths and weaknesses, and the choice of technique depends on the specific problem and the characteristics of the data.
Suppose we want to predict the probability of a customer purchasing a product based on their age, gender, and income. We can use a logistic regression model, which is a type of generalized linear model. Logistic regression models the relationship between the independent variables and the log-odds of the dependent variable.
The logistic regression model can handle binary dependent variables, making it suitable for this problem. It can also provide probabilities of the dependent variable, which can be used to make predictions.
## Exercise
You have a dataset that contains information about the number of hours studied, the number of practice tests taken, and the corresponding test scores of a group of students. You want to predict whether a student will pass the test based on these variables. Which advanced regression technique would you use?
### Solution
For this problem, a suitable advanced regression technique would be logistic regression. Logistic regression models the relationship between the independent variables and the log-odds of the dependent variable. It can handle binary dependent variables, which is what we want to achieve - predicting whether a student will pass the test or not.
# Evaluating the accuracy of predictive models
Once we have built a predictive model, it is important to evaluate its accuracy to determine how well it performs. There are several techniques that can be used to evaluate the accuracy of predictive models:
1. Cross-validation: Cross-validation is a technique that divides the dataset into multiple subsets, or folds. The model is trained on a subset of the data and tested on the remaining fold. This process is repeated multiple times, and the average performance is used as an estimate of the model's accuracy.
2. Train-test split: Train-test split is a technique that divides the dataset into a training set and a testing set. The model is trained on the training set and tested on the testing set. The performance on the testing set is used as an estimate of the model's accuracy.
3. Confusion matrix: A confusion matrix is a table that shows the performance of a classification model. It compares the predicted values with the actual values and shows the number of true positives, true negatives, false positives, and false negatives.
4. Receiver operating characteristic (ROC) curve: An ROC curve is a graphical representation of the performance of a classification model. It shows the trade-off between the true positive rate and the false positive rate for different classification thresholds.
Each evaluation technique provides different insights into the model's accuracy and can be used depending on the specific problem and the characteristics of the data.
Suppose we have built a predictive model to classify emails as spam or not spam. We have evaluated the model using cross-validation and obtained an average accuracy of 90%. This indicates that the model correctly classifies 90% of the emails.
We can also use a confusion matrix to evaluate the model's performance. The confusion matrix shows the number of true positives, true negatives, false positives, and false negatives. Based on the confusion matrix, we can calculate metrics such as precision, recall, and F1 score.
## Exercise
You have built a predictive model to classify images as cats or dogs. You have evaluated the model using train-test split and obtained an accuracy of 85%. Interpret this accuracy.
### Solution
The accuracy of 85% indicates that the model correctly classifies 85% of the images as cats or dogs.
# Incorporating non-linear relationships in regression models
In linear regression, we assume that the relationship between the independent variables and the dependent variable is linear. However, in many cases, the relationship may not be linear and may exhibit non-linear patterns. In such cases, we can incorporate non-linear relationships in regression models using various techniques.
One common technique is polynomial regression, where we introduce polynomial terms of the independent variables in the regression equation. This allows us to capture non-linear patterns by fitting a curve to the data. For example, if we have a single independent variable x, we can include x^2, x^3, and so on as additional predictors in the regression model.
Another technique is using transformation functions, such as logarithmic, exponential, or power transformations, to transform the independent variables. This can help in capturing non-linear relationships between the variables.
Additionally, we can use spline regression, which involves dividing the range of the independent variable into smaller intervals and fitting separate regression models to each interval. This allows us to capture different non-linear patterns in different parts of the data.
By incorporating non-linear relationships in regression models, we can improve the model's ability to capture complex patterns and make more accurate predictions.
Suppose we are analyzing the relationship between a person's age and their income. We suspect that the relationship may not be linear and that income may increase at a decreasing rate with age. To incorporate this non-linear relationship, we can use polynomial regression and include terms like age^2 and age^3 in the regression model. This will allow us to capture the non-linear pattern and make more accurate predictions of income based on age.
## Exercise
You are analyzing the relationship between a student's study time and their test scores. You suspect that the relationship may not be linear and that test scores may increase at an increasing rate with study time. How can you incorporate this non-linear relationship in a regression model?
### Solution
To incorporate the non-linear relationship between study time and test scores, you can use polynomial regression and include terms like study time^2 and study time^3 in the regression model. This will allow you to capture the increasing rate of test scores with study time.
# Handling missing data in regression analysis
One approach is to simply remove the observations with missing data. However, this can lead to a loss of valuable information and may introduce bias if the missing data is not random. Therefore, it is important to carefully consider the reasons for missing data and the potential impact on the analysis before deciding to remove observations.
Another approach is to impute the missing data, which means filling in the missing values with estimated values. There are various methods for imputing missing data, such as mean imputation, regression imputation, and multiple imputation. Mean imputation replaces missing values with the mean of the available data, while regression imputation uses regression models to predict the missing values based on the other variables. Multiple imputation creates multiple imputed datasets and combines the results to account for the uncertainty in the imputed values.
It is also important to consider the mechanism of missing data. Missing data can be classified as missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR). MCAR means that the missingness is unrelated to the observed and unobserved data. MAR means that the missingness is related to the observed data but not the unobserved data. MNAR means that the missingness is related to the unobserved data. The mechanism of missing data can affect the choice of imputation method and the validity of the analysis.
In summary, handling missing data in regression analysis requires careful consideration of the reasons for missing data, the mechanism of missing data, and the potential impact on the analysis. By using appropriate techniques for handling missing data, we can minimize bias and improve the accuracy of regression models.
Suppose we are analyzing the relationship between a person's income and their education level. We have data on income and education level for a sample of individuals, but some observations have missing values for income. To handle the missing data, we can use multiple imputation. We create multiple imputed datasets by imputing the missing values based on the observed data and combine the results to obtain estimates of the regression coefficients. This allows us to account for the uncertainty in the imputed values and obtain more accurate estimates of the relationship between income and education level.
## Exercise
You are analyzing the relationship between a person's age, income, and education level. The dataset contains missing values for both income and education level. How would you handle the missing data in this case?
### Solution
To handle the missing data in this case, we can use multiple imputation. We create multiple imputed datasets by imputing the missing values for income and education level based on the observed data. This allows us to account for the uncertainty in the imputed values and obtain more accurate estimates of the relationship between age, income, and education level.
# Regression model interpretation and communication
One key aspect of interpreting a regression model is understanding the coefficients. The coefficients represent the change in the dependent variable for a one-unit change in the corresponding independent variable, holding all other variables constant. It is important to consider the magnitude and direction of the coefficients to understand the relationship between the variables.
Another important aspect is assessing the statistical significance of the coefficients. This can be done by examining the p-values associated with the coefficients. A p-value less than a certain threshold (e.g., 0.05) indicates that the coefficient is statistically significant, meaning that it is unlikely to have occurred by chance. Statistical significance provides evidence that the independent variable has a meaningful impact on the dependent variable.
In addition to interpreting the coefficients, it is important to assess the overall fit of the regression model. This can be done by examining the R-squared value, which represents the proportion of the variance in the dependent variable that is explained by the independent variables. A higher R-squared value indicates a better fit of the model to the data.
Once we have interpreted the regression model, it is important to communicate the results effectively. This can be done through visualizations, such as scatter plots or line plots, that illustrate the relationship between the variables. It is also important to provide clear and concise explanations of the key findings and their implications.
In summary, interpreting and communicating regression models requires understanding the coefficients, assessing their statistical significance, evaluating the overall fit of the model, and effectively communicating the results. By following these techniques, we can gain insights from regression models and effectively communicate those insights to others.
Suppose we have built a regression model to predict housing prices based on variables such as square footage, number of bedrooms, and location. The coefficient for square footage is 100, which means that, on average, each additional square foot increases the housing price by $100, holding all other variables constant. The coefficient for number of bedrooms is 10, which means that, on average, each additional bedroom increases the housing price by $10, holding all other variables constant. Both coefficients are statistically significant with p-values less than 0.05. The R-squared value for the model is 0.80, indicating that 80% of the variance in housing prices is explained by the independent variables.
## Exercise
You have built a regression model to predict sales based on variables such as advertising expenditure, price, and seasonality. The coefficient for advertising expenditure is 0.05, which means that, on average, each additional dollar spent on advertising increases sales by 0.05 units, holding all other variables constant. The coefficient for price is -0.10, which means that, on average, each additional dollar increase in price decreases sales by 0.10 units, holding all other variables constant. Both coefficients are statistically significant with p-values less than 0.05. The R-squared value for the model is 0.70, indicating that 70% of the variance in sales is explained by the independent variables.
Based on these results, how would you interpret the relationship between advertising expenditure, price, and sales?
### Solution
The coefficient for advertising expenditure suggests that increasing advertising expenditure is associated with an increase in sales, holding all other variables constant. The coefficient for price suggests that increasing the price is associated with a decrease in sales, holding all other variables constant. Both relationships are statistically significant, meaning that they are unlikely to have occurred by chance. The R-squared value indicates that 70% of the variance in sales is explained by the independent variables, indicating a good fit of the model to the data.
# Applications of regression analysis and predictive modeling
One common application is in finance and economics. Regression analysis can be used to analyze the relationship between variables such as interest rates, inflation, and stock prices. By understanding these relationships, analysts can make predictions and inform investment decisions.
Another application is in marketing and advertising. Regression analysis can be used to analyze the impact of advertising campaigns on sales, customer behavior, and brand awareness. This information can help companies optimize their marketing strategies and allocate resources effectively.
Regression analysis is also widely used in healthcare and medical research. It can be used to analyze the relationship between variables such as patient characteristics, treatment methods, and health outcomes. This information can help researchers identify risk factors, evaluate treatment effectiveness, and improve patient care.
In addition, regression analysis and predictive modeling are used in social sciences, such as sociology and psychology. These techniques can be used to analyze the relationship between variables such as education level, income, and job satisfaction. By understanding these relationships, researchers can gain insights into social phenomena and inform policy decisions.
Furthermore, regression analysis and predictive modeling are used in environmental science and climate research. These techniques can be used to analyze the relationship between variables such as temperature, rainfall, and carbon emissions. By understanding these relationships, scientists can make predictions about future climate patterns and inform environmental policies.
Overall, regression analysis and predictive modeling have a wide range of applications in various fields. By understanding these techniques and their applications, you can apply them to solve real-world problems and make informed decisions. | Textbooks |
\begin{definition}[Definition:Polynomial in Ring Element/Definition 2]
Let $R$ be a commutative ring.
Let $S$ be a subring with unity of $R$.
Let $x\in R$.
Let $S[X]$ be the polynomial ring in one variable over $S$.
A '''polynomial in $x$ over $S$''' is an element that is in the image of the evaluation homomorphism $S[X]\to R$ at $x$.
\end{definition} | ProofWiki |
Rosehip Extract Inhibits Lipid Accumulation in White Adipose Tissue by Suppressing the Expression of Peroxisome Proliferator-activated Receptor Gamma
Nagatomo, Akifumi;Nishida, Norihisa;Matsuura, Yoichi;Shibata, Nobuhito 85
https://doi.org/10.3746/pnf.2013.18.2.085 PDF KSCI
Recent studies have shown that Rosa canina L. and tiliroside, the principal constituent of its seeds, exhibit anti-obesity and anti-diabetic activities via enhancement of fatty acid oxidation in the liver and skeletal muscle. However, the effects of rosehip, the fruit of this plant, extract (RHE), or tiliroside on lipid accumulation in adipocytes have not been analyzed. We investigated the effects of RHE and tiliroside on lipid accumulation and protein expression of key transcription factors in both in vitro and in vivo models. RHE and tiliroside inhibited lipid accumulation in a dose-dependent manner in 3T3-L1 cells. We also analyzed the inhibitory effect of RHE on white adipose tissue (WAT) in high-fat diet (HFD)-induced obesity mice model. Male C57BL/6J mice were fed HFD or HFD supplemented with 1% RHE (HFDRH) for 8 weeks. The HFDRH-fed group gained less body weight and had less visceral fat than the HFD-fed group. Liver weight was significantly lower in the HFDRH-fed group and total hepatic lipid and triglyceride (TG) content was also reduced. A significant reduction in the expression of peroxisome proliferator-activated receptor gamma (PPAR${\gamma}$) was observed in epididymal fat in the HFDRH-fed group, in comparison with controls, through Western blotting. These results suggest that downregulation of PPAR${\gamma}$ expression is involved, at least in part, in the suppressive effect of RHE on lipid accumulation in WAT.
Phosphate-Induced Rat Vascular Smooth Muscle Cell Calcification and the Implication of Zinc Deficiency in A7r5 Cell Viability
Shin, Mee-Young;Kwun, In-Sook 92
The calcification of vascular smooth muscle cells (VSMCs) is considered one of the major contributors for vascular disease. Phosphate is known as the inducer for VSMC calcification. In this study, we assessed whether phosphate affected cell viability and fetuin-A, a calcification inhibitor protein, both which are related to VSMC calcification. Also, VSMC viability by zinc level was assessed. The results showed that phosphate increased Ca and P deposition in VSMCs (A7r5 cell line, rat aorta origin). This phosphate-induced Ca and P deposition was consistent with the decreased A7r5 cell viability (P<0.05), which implies phosphate-induced calcification in A7r5 cells might be due to the decreased VSMC cell viability. As phosphate increased, the protein expression of fetuin-A protein was up-regulated. A7r5 cell viability decreased as the addition of cellular zinc level was decreased (P<0.05). The results suggested that zinc deficiency causes the decreased cell viability and it would be the future study to clarify how zinc does act for VSMC cell viability. The results suggest that the decreased VSMC viability by high P or low Zn in VSMCs may be the risk factor for vascular disease.
Effect of Phellinus baumii -Biotransformed Soybean Powder on Lipid Metabolism in Rats
Kim, Dae Ik;Kim, Kil Soo;Kang, Ji Hyuk;Kim, Hye Jeong 98
In this study, we evaluated the hypolipidemic and antioxidative effects of biotransformed soybean powder (BTS; Phellinus baumii-fermented soybean) on lipid metabolism in rats. Sprague-Dawley (SD) male rats were divided into basal diet group (BA), high fat diet group (HF), high fat diet containing 10% BTS group (10 BTS), and high fat diet containing 20% BTS group (20 BTS). Changes in the content of various isoflavones, including daidzein and genistein, within the soybean after fermentation to BTS were investigated. The levels of daidzein and genistein were $149.28{\mu}g/g$ and $364.31{\mu}g/g$, respectively. After six weeks experimental period, Food efficiency ratio in the 10 and 20 BTS group was significantly lower than the HF group (P<0.05). Total serum levels of cholesterol, triglycerides, low-density lipoprotein cholesterol, and atherogenic index ratio in the 10 or 20 BTS group were significantly lower than the HF group. The levels of alanine aminotransferase, aspartate aminotransferase and thiobarbituric acid reactive substance were significantly lower in the groups that received 10% and 20% BTS than the HF. The activities of SOD and CAT were significantly higher in the 10 and 20 BTS group than the HF group. The activity of XO in the 10 and 20 BTS group was significantly lower than in the HF group by 20% and 23%, respectively. In conclusion, these data suggest that BTS is an effective agent in improving lipid metabolism and antioxidant enzyme system.
Anti-Helicobacter pylori Properties of GutGardTM
Kim, Jae Min;Zheng, Hong Mei;Lee, Boo Yong;Lee, Woon Kyu;Lee, Don Haeng 104
Presence of Helicobacter pylori is associated with an increased risk of developing upper gastrointestinal tract diseases. Antibiotic therapy and a combination of two or three drugs have been widely used to eradicate H. pylori infections. Due to antibiotic resistant drugs, new drug resources are needed such as plants which contain antibacterial compounds. The aim of this study was to investigate the ability of GutGard$^{TM}$ to inhibit H. pylori growth both in Mongolian gerbils and C57BL/6 mouse models. Male Mongolian gerbils were infected with the bacteria by intragastric inoculation ($2{\times}10^9$ CFU/gerbil) 3 times over 5 days and then orally treated once daily 6 times/week for 8 weeks with 15, 30 and 60 mg/kg GutGard$^{TM}$. After the final administration, biopsy samples of the gastric mucosa were assayed for bacterial identification via urease, catalase and ELISA assays as well as immunohistochemistry (IHC). In the Mongolian gerbil model, IHC and ELISA assays revealed that GutGard$^{TM}$ inhibited H. pylori colonization in gastric mucosa in a dose dependent manner. The anti-H. pylori effects of GutGard$^{TM}$ in H. pylori-infected C57BL/6 mice were also examined. We found that treatment with 25 mg/kg GutGard$^{TM}$ significantly reduced H. pylori colonization in mice gastric mucosa. Our results suggest that GutGard$^{TM}$ may be useful as an agent to prevent H. pylori infection.
Safety Evaluation of Chrysanthemum indicum L. Flower Oil by Assessing Acute Oral Toxicity, Micronucleus Abnormalities, and Mutagenicity
Hwang, Eun-Sun;Kim, Gun-Hee 111
Chrysanthemum indicum is widely used to treat immune-related and infectious disorders in East Asia. C. indicum flower oil contains 1,8-cineole, germacrene D, camphor, ${\alpha}$-cadinol, camphene, pinocarvone, ${\beta}$-caryophyllene, 3-cyclohexen- 1-ol, and ${\gamma}$-curcumene. We evaluated the safety of C. indicum flower oil by conducting acute oral toxicity, bone marrow micronucleus, and bacterial reverse mutation tests. Mortality, clinical signs and gross findings of mice were measured for 15 days after the oral single gavage administration of C. indicum flower oil. There were no mortality and clinical signs of toxicity at 2,000 mg/kg body weight/day of C. indicum flower oil throughout the 15 day period. Micronucleated erythrocyte cell counts for all treated groups were not significantly different between test and control groups. Levels of 15.63~500 ${\mu}g$ C. indicum flower oil/plate did not induce mutagenicity in S. Typhimurium and E. coli, with or without the introduction of a metabolic activation system. These results indicate that ingesting C. indicum flower oil produces no acute oral toxicity, bone marrow micronucleus, and bacterial reverse mutation.
Analysis of the Prevalence and Risk Factors of Malnutrition among Hospitalized Patients in Busan
Lee, Ha-Kyung;Choi, Hee-Sun;Son, Eun-Joo;Lyu, Eun-Soon 117
This study investigated the prevalence of and risk factors for malnutrition in hospitalized patients in Busan, Republic of Korea. 944 patients (440 men and 504 women) were hospitalized in four Busan general hospitals from March through April, 2011. Nutritional status was assessed on admission by the Nutritional Risk Screening 2002. Data were collected from the electronic medical records system for the characteristics of the subjects, clinical outcomes, biochemical laboratory data, and nutrition support states. Clinical dietitians interviewed the patients using structured questionnaires involving data on weight loss and problems related to oral intakes. Malnourished patients were significantly older (P<0.001) than well-nourished patients, but the values for BMI, serum albumin, total cholesterol, TLC, hemoglobin, and hematocrit were significantly lower (P<0.001) for malnourished than for well-nourished patients. Logistic regression indicated that the main determinant factors for nutritional status were the age, length of stay, BMI, serum albumin, and total cholesterol. In order to increase therapeutic effects of hospitalized patients, clinical dietitians need to offer proper nutritional intervention based on the results of nutrition assessment and identification of malnutrition.
Hepatic Fibrosis Inhibitory Effect of Peptides Isolated from Navicula incerta on TGF-β Induced Activation of LX-2 Human Hepatic Stellate Cells
Kang, Kyong-Hwa;Qian, Zhong-Ji;Ryu, BoMi;Karadeniz, Fatih;Kim, Daekyung;Kim, Se-Kwon 124
In this study, novel peptides (NIPP-1, NIPP-2) derived from Navicula incerta (microalgae) protein hydrolysate were explored for their inhibitory effects on collagen release in hepatic fibrosis with the investigation of its underlying mechanism of action. TGF-${\beta}1$ activated fibrosis in LX-2 cells was examined in the presence or absence of purified peptides NIPP-1 and NIPP-2. Besides the mechanisms of liver cell injury, protective effects of NIPP-1 and NIPP-2 were studied to show the protective mechanism against TGF-${\beta}1$ stimulated fibrogenesis. Our results showed that the core protein of NIPP-1 peptide prevented fibril formation of type I collagen, elevated the MMP level and inhibited TIMP production in a dose-dependent manner. The treatment of NIPP-1 and NIPP-2 on TGF-${\beta}1$ induced LX-2 cells alleviated hepatic fibrosis. Moreover, ${\alpha}$-SMA, TIMPs, collagen and PDGF in the NIPP-1 treated groups were significantly decreased. Therefore, it could be suggested that NIPP-1 has potential to be used in anti-fibrosis treatment.
Influence of Extraction Method on Quality and Functionality of Broccoli Juice
Lee, Sung Gyu;Kim, Jin-Hee;Son, Min-Jung;Lee, Eun-Ju;Park, Woo-Dong;Kim, Jong-Boo;Lee, Sam-Pin;Lee, In-Seon 133
This study was performed to compare the quality and functionality of broccoli juice as affected by extraction method. Broccoli juice was extracted using method I (NUC Kuvings silent juicer), method II (NUC centrifugal juicer), and method III (NUC mixer), and the quality properties of the broccoli juices were analyzed using three different methods. Additionally, the antioxidative, anticancer, and anti-hyperglycemic activities of broccoli juice prepared by the three different methods were investigated in vitro. The broccoli juice made by method I contained the highest polyphenol and flavonoid contents at 1,226.24 mg/L and 1,018.32 mg/L, respectively. Particularly, broccoli juice prepared by method I showed higher DPPH and ABTS radical scavenging activities than those of the other samples. Additionally, broccoli juice made by method I showed the highest growth inhibitory effects against HeLa, A549, AGS, and HT-29 cancer cells. Broccoli juice prepared by method I had the highest ${\alpha}$-glucosidase inhibitory effects. These results indicate that there are important differences in chemical and functional qualities between juice extraction techniques.
Proximate Composition, Amino Acid, Mineral, and Heavy Metal Content of Dried Laver
Hwang, Eun-Sun;Ki, Kyung-Nam;Chung, Ha-Yull 139
Laver, a red algae belonging to the genus Porphyra, is one of the most widely consumed edible seaweeds. The most popular commercial dried laver species, P. tenera and P. haitanensis, were collected from Korea and China, respectively, and evaluated for proximate composition, amino acids, minerals, trace heavy metals, and color. The moisture and ash contents of P. tenera and P. haitanensis ranged from 3.66~6.74% and 8.78~9.07%, respectively; crude lipid and protein contents were 1.96~2.25% and 32.16~36.88%, respectively. Dried lavers were found to be a good source of amino acids, such as asparagine, isoleucine, leucine, and taurine, and ${\gamma}$-aminobutyric acid. K, Ca, Mg, Na, P, I, Fe, and Se minerals were selected for analysis. A clear regional variation existed in the amino acid, mineral, and trace metal contents of lavers. Regular consumption of lavers may have heath benefits because they are relatively low in fat and high in protein, and contain functional amino acids and minerals.
Monitoring and Risk Assessment of Pesticide Residues in Commercially Dried Vegetables
Seo, Young-Ho;Cho, Tae-Hee;Hong, Chae-Kyu;Kim, Mi-Sun;Cho, Sung-Ja;Park, Won-Hee;Hwang, In-Sook;Kim, Moo-Sang 145
We tested for residual pesticide levels in dried vegetables in Seoul, Korea. A total of 100 samples of 13 different types of agricultural products were analyzed by a gas chromatography-nitrogen phosphate detector (GC-NPD), an electron capture detector (GC-${\mu}ECD$), a mass spectrometry detector (GC-MSD), and a high performance liquid chromatography- ultraviolet detector (HPLC-UV). We used multi-analysis methods to analyze for 253 different pesticide types. Among the selected agricultural products, residual pesticides were detected in 11 samples, of which 2 samples (2.0%) exceeded the Korea Maximum Residue limits (MRLs). We detected pesticide residue in 6 of 9 analyzed dried pepper leaves and 1 sample exceeded the Korea MRLs. Data obtained were then used for estimating the potential health risks associated with the exposures to these pesticides. The estimated daily intakes (EDIs) range from 0.1% of the acceptable daily intake (ADI) for bifenthrin to 8.4% of the ADI for cadusafos. The most critical commodity is cadusafos in chwinamul, contributing 8.4% to the hazard index (HI). This results show that the detected pesticides could not be considered a serious public health problem. Nevertheless, an investigation into continuous monitoring is recommended.
Fatty Acid Composition and Volatile Constituents of Protaetia brevitarsis Larvae
Yeo, Hyelim;Youn, Kumju;Kim, Minji;Yun, Eun-Young;Hwang, Jae-Sam;Jeong, Woo-Sik;Jun, Mira 150
A total of 48 different volatile oils were identified form P. brevitarsis larvae by gas chromatography/mass spectrometry (GC/MS). Acids (48.67%) were detected as the major group in P. brevitarsis larvae comprising the largest proportion of the volatile compounds, followed by esters (19.84%), hydrocarbons (18.90%), alcohols (8.37%), miscellaneous (1.71%), aldehydes (1.35%) and terpenes (1.16%). The major volatile constituents were 9-hexadecenoic acid (16.75%), 6-octadecenoic acid (14.88%) and n-hexadecanoic acid (11.06%). The composition of fatty acid was also determined by GC analysis and 16 fatty acids were identified. The predominant fatty acids were oleic acid ($C_{18:1}$, 64.24%) followed by palmitic acid ($C_{16:0}$, 15.89%), palmitoleic acid ($C_{16:1}$, 10.43%) and linoleic acid ($C_{18:2}$, 4.69%) constituting more than 95% of total fatty acids. The distinguished characteristic of the fatty acid profile of P. brevitarsis larvae was the high proportion of unsaturated fatty acid (80.54% of total fatty acids) versus saturated fatty acids (19.46% of total fatty acids). Furthermore, small but significant amounts of linoleic, linolenic and ${\gamma}$-linolenic acids bestow P. brevitarsis larvae with considerable nutritional value. The novel findings of the present study provide a scientific basis for the comprehensive utilization of the insect as a nutritionally promising food source and a possibility for more effective utilization. | CommonCrawl |
RMT2015: Random Matrix Theory
RMT2012
Overview | Program | Lecturers | Organizers
Term: April 15 - April 21, 2012
Location: OIST Seaside House
Organizer: S. Hikami (Mathematical and Theoretical Physics Unit, OIST)
E. Brezin (ENS)
H. Sompolinsky (Hebrew)
J. Miller (Physics and Biology Unit, OIST)
Neural network and memory, spin glass, communication networks.
Genes analysis, co-expression of DNA, protein, bioinformatics.
Protein folding, topology, secondary structure of RNA and other biological applications.
Condensed matter application, BEC, quantum dots, quantum chaos, topological insulator, universal class.
RMT and field theory, conformal field theory, AdS/CFT.
Mathematical aspect of RMT.
Date / Time
April 14 (Sat)
Registration 1F Lobby
18:00-20:00 Dinner 3F Chura Hall
April 15 (Sun)
07:00-09:00 Breakfast 3F Chura Hall
09:00-12:00 Lecture (E. Brezin) 1F Seminar Room
12:00-14:00 Lunch 3F Chura Hall
14:00-17:00 Lecture (R. Monasson) 1F Seminar Room
April 16 (Mon)
09:00-12:00 Lecture (A. Morozov) 1F Seminar Room
14:00-17:00 Lecture (S. Ganguli) 1F Seminar Room
April 17 (Tue)
09:00-12:00 Lecture (P. Wiegmann) 1F Seminar Room
14:00-16:30 Poster Session 1F Lobby
17:00-18:00 Campus Tour OIST Campus
April 18 (Wed)
09:00-12:00 Lecture (J. Zinn-Justin) 1F Seminar Room
14:00-17:00 Lecture (H. Sompolinsky) 1F Seminar Room
18:00-20:00 Banquet 3F Chura Hall
April 19 (Thr)
09:00-12:00 Lecture (S. Cocco) 1F Seminar Room
14:00-17:00 Lecture (M. Vergassola) 1F Seminar Room
April 20 (Fri)
09:00-12:00 Lecture (J. Miller) 1F Seminar Room
14:00-18:00 Excursion Okinawa Churaumi Aquarium
Edouard Brézin
Professor (emeritus) of theoretical physics,
Ecole Normale Supérieure, Paris
Edouard Brézin has done work in quantum fieldtheory, mainly for applications in statistical physics, in particular for critical phenomena. He hasapplied field theory techniques to condensed matter problemssuch as the theory of critical wetting, localization by disorder or the study of the phase transition from anormal metal to a type II superconductor under a magnetic field. He has beeninterested in field theories with a large number of colors. This has led to arepresentation of two-dimensional quantum gravity, random fluctuating surfaces,i.e. bosonic closed string theories, in terms of random matrices. The scaling limit ofsuch models is related to integrable hierarchies such as KdV flows. He has worked on theuniversality of the correlations of eigenvalues in the local limit for random matrices and on the application of random matrices to topological properties of curves (work with S. Hikami). He is a former president of the French Academy of Sciences and a foreign member of the National Academy of Sciences (USA) and of the Royal Society (UK). He has shared the Dirac medal in 2011with J. Cardy and A. Zamolodchikov.
Recommended books, papers for the students and participants :
Random matrices (3e edition), Pure and Applied Mathematics Series 142, Elsevier (London - 2004), 688 pp. ISBN 0-12-088409-7.
* Lecture title : A brief introduction to random matrices
* Abstract : A few situations in which random matrix theory have been used in he past will be reviewed. The role of universality of the local spacing distributions will be discussed. The example of random matrices in an external matrix source will be considered in more details.
Simona Cocco
Researcher in the CNRS, Ecole Normale Superieure, Paris
Simona Cocco did her undergraduate studies in physics at the University of Rome (Italy) and did a PhD in physics and biophysics in Rome and Lyon (France). She has a position as a researcher in the CNRS in France, from 2001.
She has mainly worked on the application of statistical mechanics to biophysics. Her early works were focused on the modelling of the elastic response of a single molecule of nucleic acids (DNA, RNA), and of the unzipping experiments in which the two complementary strands of the molecule are taken apart.
More recently she has worked on inverse problems in biophysics, including: the inference of the DNA sequence from unzipping experiments, the inference of the connections between neurons from the recording of the activity of a neural population by a multi-electrode array.
* Lecture title : Inference of interactions from correlations: algorithms and applications
* Abstract :Large populations of components, such as neurons or genes, show corre-lations in their activities. A major question is to understand the underlying mechanisms responsible for those correlations. Correlations between two components can be either due a direct interaction, or to an indirect effect mediated through other components. Disentangling direct interactions from indirect correlations brings several advantages. First the network of interac- tions is generally much sparser than the one of correlations, hence a more efficient, compressed representation of the data is obtained. Secondly, interactions can be used to predict the global effect of a local perturbation on the system, such as the removal of one component or its stimulation.
We will describe how interactions can be found from correlations in the context of maximum entropy models, such as the Ising model [1]. Those models are the least constrained ones capable of reproducing the one- and two-point correlations, e.g. the average firing rates of the neurons in a multi-electrode recording and the probability that any two neurons fire simultaneously (within a, say, 20 msec time window). Finding the interactions of the Ising model given its correlations is a hard computational problem which raises several interesting, fundamental and practical questions. Among them are the consequences of bad or partial sampling. Generally only a small subset of a biological system is accessible for measurements, and over a limited
amount of time. A natural question is to ask how much the Ising model inferred from this partial and inaccurate measure of the correlations can tell us about the true, underlying interactions. We will address those questions using a recently introduced inference procedure, identifying the clusters of strongly interacting components [2], to analyze artificial data issued from Ising models with different interaction networks and neurobiological data, corresponding to multi-electrode recordings of the neural activity in the vertebrate retina or in the cortex.
The retina recordings experiments consist in extracting the retina from the eye of a salamander and putting it on a multi-electrode array. Tens of neural cells called ganglion cells, situated on the last cellular layer of the retina, can be recorded simultaneously. We will show that the Ising model is capable of correctly modeling the data and of reproducing the probability of a configuration of neural activity in a short-time window, typically of the order of 20 ms, and the higher moments of the distribution, i.e. 3-cell and higher-order correlations [3, 4]. This success is remarkable as the model does not include three cell couplings.
We will also describe experiments by G. Buzsaki [5] and F. Battaglia [6] on the in vivo recordings of the activity of a behaving rat. The experiments consists in implanting electrodes in the head of a rodent, still free to move.
The recorded area is situated in the cortex and in the hippocampus. The activity of tens of neurons (30-150) can be recorded on long time periods during which the rodent learns a task, for example to go to the left of a maze when it smells a banana clue or to the right when it smells a chocolate clue. The question raised by these experiments, that is, how the learning of a task is memorized and in which measure synaptic strength are at the basis of the memory (following the Webb's hypothesis dating back from 1949) are of fundamental importance in neuroscience.
[1] E.T. Jaynes, Phys. Rev. 106, 620630 (1957)
[2] S. Cocco, R. Monasson, Phys. Rev. Lett. 106, 090601 (2011).
[3] E. Schneidman, M. J. Berry, R. Segev, W. Bialek, Nature 440, 1007-1012
[4] S. Cocco, S. Leibler, R. Monasson, Proc. Nat. Acad. Sci. (USA) 106,
14058 (2009).
[5] S. Fujisawa, A. Amarasingham, M. T. Harrison, and G. Buzsaki, Nature
Neuroscience 11, 823-834 (2008).
[6] A. Peyrache, M. Khamassi, K. Benchenane, S. I. Wiener, and
F. P. Battaglia, Nature Neuroscience 12, 919 (2009).
Remi Monasson
Director of Research CNRS, Ecole Normale Superieure, Paris
R. Monasson got his PhD in theoretical physics at the Ecole Normale Superieure in 1993, and was a post-doc in Rome until 1995. After getting a position at CNRS, he spent two sabbatical, at the University of Chicago in 2000/2001 and at the Institute for Advanced Study, Princeton, in 2009/2011.
His research are at the crossroads between the statistical physics of disordered systems, and its interdisciplinary applications to computer science (study of phase transitions in combinatorial optimization problems with random inputs, learning in neural networks models) and biophysics (modeling of single molecule experiments, high-dimensional statistical inference).
* Lecture title :Statistical physics approaches to high-dimensional inference
* Abstract : Constant progress in experimental techniques has now made it possible to monitor the dynamics of complex systems, especially in biology. The temporal activity of a population of neurons, the expression patterns of a set of genes, and the evolution of species in an ecological system are just a few examples of dynamically evolving complex systems which can now be quantitatively recorded and made available for modeling purposes. Careful analysis of these systems is necessary to develop a quantitative systems-level understanding of biological networks and to explore how global properties arise from local interactions.
Interpreting experimental data, understanding how the components of a system interact with each other, inferring the values of those interactions from the data, and designing predictive models are formidable tasks. Difficulties hindering the modeling of complex data and the resolution of the attached inference problems include poor quality of data. Sampling may be incomplete, both from the temporal (limited acquisition frequency or recording duration) and spatial (limited access to a subpart of the system) points of view, and the data may be noisy, plagued by measurement uncertainties and/or intrinsic stochastic dynamics. There is also a trade-off between the power of models, i.e. their ability to fit the data, and the number of their defining parameters, which must be balanced to avoid overfitting. The computational cost in solving the models and inferring their parameters from the data may be great. This situation is drastically different from the usual situation encountered in the analysis of physical data, for example in condensed matter. Physical models are, generically, defined from a small number of parameters, such as the interaction strength between close spins in a ferromagnetic material or the interaction potential between two particles in a liquid. An essential property of physical systems is the indistinguishability of their components, e.g. any two electrons are identical, which make the number of parameters independent of the size of the system. In biological systems, entities such as cells exhibit individual features. Describing a collection of different entities requires a large number of parameters, extensive in the system size.
The process of inferring a large number of parameters from poor data, called high-dimensional inference, is an important goal in quantitative analysis and defines a field at the crossroads of statistical inference, machine learning, and statistical physics. In this lecture, we will concentrate on maximum-entropy models, which are currently very popular in the analysis of biological e.g. neural and protein data. These models correspond to Ising (or Potts) models in statistical physics. Briefly speaking, we want to find the interactions between a set of components from the knowledge of their correlations. This problem is at least as difficult as the direct problem of calculating the statistics of observables for a given model (whose complexity is exemplified by the notorious spin-glass problem). The inverse problem is complicated by the absence of any a priori symmetry (e.g. not defined on any a priori known lattice); the presence of noise due to poor sampling influences the estimation of observables, and thus of the model parameters.
Different approaches to solve the inverse Ising problem will be reviewed during the lecture, starting from elementary mean-field techniques to more sophisticated diagrammatic expansions and logistic regressions. An emphasis will be put on the inverse Hopfield model, useful to recast the popular principal component analysis in a controlled Bayesian framework. The relationship with random matrices and the so-called retarded learning transitions will be detailed. The approaches will be briefly illustrated on genomic data (analysis of covariations on protein families).
Alexei Morozov
Main researcher in ITEP (Institute of Theoretical and Experimental Physics, Moscow)
Main research interests:
elementary particle theory, unification models; quantum field theory, string theory; mathematical physics.
Introduction to Non-Linear Algebra, V. Dolotin, A. Morozov arXiv:hep-th/0609022
Unitary Integrals and Related Matrix Models, A.Morozov arXiv:0906.3518
Towards a proof of AGT conjecture by methods of matrix models, A.Mironov, A.Morozov, Sh.Shakirov arXiv:1011.5629, and references therein.
* Lecture title :Faces of matrix models
* Abstract : Partition functions of eigenvalue matrix models possess a number of very different descriptions:
as matrix integrals, as solutions to linear equations, as $\tau$-functions of integrable hierarchies, as result of the action of $W$-operators and of various recursions on elementary input data, as gluing of certain elementary building blocks.
All this explains the central role of such matrix models in modern mathematical physics:
they provide the basic "special functions"
to express the answers an relation between them, and they serve as a dream model of what one should try to achieve in any other field.
H. Sompolinsky
Professor of Physics, The Hebrew University
Haim Sompolinsky is a Professor of Physics and William N. Skirball Professor of Neuroscience at the Hebrew University. He is a founding member of the Interdisciplinary Center for Neural Computation (ICNC) and of the newly established Edmond and Lily Safra Center for Brain Sciences (ELSC). Sompolinsky serves as the Director of the Swartz Program for Theoretical Neuroscience at Harvard University and is an Honorary Foreign Member of the American Academy of Arts and Sciences.
Sompolinsky's early work focused on the statistical mechanics and dynamics of spin glasses. Later on, he moved to theoretical and computational neuroscience, specializing in the application of concepts and methods from statistical physics, theory of random systems, and dynamical systems theory to the study of the brain. His research includes investigations of randomness, noise and chaos in neuronal circuits, and their role in information processing, memory and learning. He has applied Random Matrix Theory to the study of spin glasses and neuronal circuits.
* Lecture title : Neuronal Circuits with Random Connectivity
* Abstract : In most neuronal systems, neurons exhibit high temporal irregularity and trial to trial variability, motivating the investigation of the conditions under which neuronal circuits exhibit chaotic dynamics and the properties of such a state. In my lectures, I will present the theory of chaos in nonlinear random neuronal circuits, developed over more than two decades. I will first describe the theory of the spectrum of the relevant random connection matrix, which governs the linear regime of the network dynamics.
I will then present the dynamic mean field theory, which describes the chaotic state of the nonlinear random network. Finally, I will discuss the nonlinear interaction between the intrinsic network dynamics and external stimuli.
* References:
1. Sommer H J, Crisanti A, Sompolinsky H, and Stein Y (1988) The Spectrum of Large Random Asymmetric Matrices. Physical Review Letters, 60: 1895-1899.
2. Rajan, K. and Abbott, L.F. (2006) Eigenvalue Spectra of Random Matrices for Neural Networks. Physical Review Letters, 97:188104.
3. Sompolinsky H, Crisanti A, and Sommers H.J (1988) Chaos in Random Neural Networks. Physical Review Letters, 61: 259-262.
4. Rajan, K, Abbott, L.F., and Sompolinsky, H. (2010) Stimulus-Dependent Suppression of Chaos in Recurrent Neural Networks, Physical Review E 82, 011903
5. Rajan K., Abbott L.F., and Sompolinsky H. (2010). Inferring stimulus selectivity from the spatial structure of neural network dynamics. Advances in Neural Information Processing, MIT Press, Cambridge MA. 23: 1975-1983
J. Zinn-Justin
Lecture title: RANDOM MATRIX AND RANDOM VECTOR THEORY: THE RENORMALIZATION GROUP APPROACH
Abstract: The realization that some ensembles of random matrices in the large size and the so-called double scaling limit could be used as toy models for quantum gravity has resulted in a tremendous expansion of random matrix theory. However, the somewhat paradoxical situation is that either models can be solved exactly or very little can be said. Since the solved models display critical points and \Red{universal properties}, it is tempting to use renormalization group (RG) ideas to reproduce universal properties, without solving models explicitly. The main ideas behind this approach are recalled here. The approach has led to encouraging results but has not yet become a universal tool as initially expected. In particular, no progress has been made for problems of quantum field theories with matrix fields. To illustrate some of the difficulties one meets, we apply in the second part of this talk the same ideas to O(N) symmetric vector models, models which can quite generally be solved in the large N limit.
P. Wiegmann
U.Chicago
Lecture title: Random Matrices, Growth models and hydrodynamic singularities.
Abstract: A broad class of non-equilibrium growth processes in two dimensions have a common law :
the velocity of the growing interface is determined by the gradient of a harmonic field (Laplacian Growth or Geometrical Growth). This kind of growth or flow is unstable, giving rise to hydrodynamic singularities which further develop into fractal singular patterns. Similar singularities occur in Random Matrix Models There a support of equilibrium measure grows with the size of the matrix according to the same law as Laplacian (or Geometrical) Growth.
In the lectures I will review this relation emphasizing a geometrical aspects of Random Matrix Theory, their hydrodynamic interpretation and the relation of growing patterns to the distribution of zeros of orthogonal polynomials.
M. Vergassola
CNRS/Institut Pasteur
*Lecture title1: Statistics of the maximum eigenvalue in random matrices
*Abstract:The statistical properties of the largest eigenvalue of a random matrix are of interest in diverse fields ranging from disordered systems and quantum mechanics to population genetics. Recent developements on the theory of large fluctuations of the first eigenvalue and some of its applications will be discussed.
*Lecture title2: Strategies of motility in living organisms
*Abstract: Challenges faced by living organisms trying to locate and move towards sources of nutrients, odors, pheromones, etc., will be discussed. Macro-organisms, such as insects and birds, lack local cues because chaotic mixing breaks up regions of high concentration into random and disconnected patches, carried by winds and currents. Thus, macroscopic animals detect patches very intermittently and have to rely on strategies more elaborate than gradient-climbing. Conversely, microorganisms, such as bacteria performing chemotaxis, can rely on local concentration cues, yet they have to cope with the stochastic nature of their microscopic world. The bacterial chemotactic response appears indeed to emerge from selective adaptation to strong fluctuations in the environments that bacterial populations experience.
Surya Ganguli
Dept. of Applied Physics, Stanford University
*Lecture title: The statistical mechanics of compressed sensing and memory through random matrices.
*Abstract:Compressed sensing (CS) is an important recent advance that shows how to reconstruct sparse high dimensional signals from surprisingly small numbers of random measurements. However, the nonlinear nature of the reconstruction process poses a challenge to understanding the performance of CS. After introducing CS, we will discuss how techniques from the statistical physics of disordered systems sheds insight into the remarkable performance of CS. We then address the seemingly unrelated question of how a neuronal network, or more generally any dissipative dynamical system, can store a memory trace for sparse temporal sequences of inputs. We show that this question is intimately related to an online, dynamical version of CS, and we discuss the properties of random networks capable of compressed sensing of time series in an online fashion.
*References:
[1] S. Ganguli, B. Huh, H. Sompolinsky, Memory Traces in Dynamical Systems, PNAS (2008)
[2] S. Ganguli and P. Latham, Feedforward to the past: the relation between neuronal connectivity, amplification, and short-term memory, Neuron (2009) 61:499-501.
[3] S. Ganguli and H. Sompolinsky, Statistical Mechanics of Compressed Sensing, Phys. Rev.Lett. (2010).
[4] S. Ganguli and H. Sompolinsky, Short-term memory in neuronal networks through dynamical compressed sensing, NIPS (2010).
[5] S. Ganguli and H. Sompolinsky, Compressed sensing, sparsity and dimensionality in neuronal information processing and data analysis, Ann. Rev. of Neurosci 35 (2012)
Vladimir Al. Osipov
Institute of Theoretical Physics, Cologne University
Zülpicher Str. 77, 50937 Cologne, Germany
Office 105 | Tel.: +49 (0) 221 470 4205 | Fax: +49 (0) 221 470 5159
Poster Title: "Ultrametric structure of the space of p-closed sequences";
Authors: Vladimir Al. Osipov, Boris Gutkin;
"The idea of p-closed sequences originated from the concept of
periodic orbits appeared in theory of quantum chaos. In the framework
of the semiclassical approach the universal spectral correlations in
the Hamiltonian systems with classical chaotic dynamics can be
attributed to the systematic correlations between actions of periodic
orbits which pass through approximately the same points of the phase
space. In the simplest way the concept is described in the following
terms. Let X and Y be two sequences of the same length n with "glued"
ends, consisting of symbols 1 and 0. Let A be a sequence of the length
p<n. The sequences X and Y are p-close (X~Y) if for any A the number
of entrances of A into X and Y is equal (could be zero). For instance,
two sequences [0010111] and [0011101] are 3-close two each other.
In our work we show that all sequences of the length n can be
distributed over clusters with respect to the naturally appearing
ultrametric distance based on the notion of p-closeness. We study the
distribution of cluster sizes in the limit of long sequences. This
problem is equivalent to the one of counting degeneracies in the
length spectrum of the de Bruijn graphs. Based on this fact, we derive
the distribution of sizes of clusters and demonstrate that in the same
limit it does not depend on n, but only on p."
Shin-ya Koyama
Professor of Department of Biomedical Engineering, Toyo University
Main interest: The theory of zeta functions.
* poster title: Quantum ergodicity of Eisenstein series in the level aspect.
* authors: Shin-ya Koyama and Sachiko Nakajima
* abstract: Quantum ergodicity is an equidistribution property of eigenfunctions for the Laplacian over a manifold as the spectrum grows. Luo and Sarnak proved it for Eisenstein series over arithmetic surfaces.
In this research we consider a famiy of arithmetic manifolds which are called the congruence surfaces of level N, and proved an equidistribution property as the level N grows with the spectrum fixed.
Zoran Ristivojevic
Postdoctoral Fellow, LPT Ecole Normale Superieure, Paris, France
I am interested in different systems where low-dimensional physics is realized, that include quantum wires, edges of quantum Hall states and cold atomic gases. I am also interested in certain aspects of some models of statistical physics that include disordered XY models.
Poster title: Some two-loop results for two XY models with disorder
Authors: Zoran Ristivojevic, P. Le Doussal, T. Giamarchi, A. Perret, A. Petkovic, G. Schehr, and K. Wiese
Abstract: We consider two disordered XY model and derive two-loop
scaling equations for them. The first model is the two-dimensional
XY model with quenched uncorrelated random symmetry-breaking
fields and it has a phase transition at a finite temperature.
Using the obtained scaling equations we calculate the amplitude of
the correlation function in the low-temperature superrough phase.
Obtained results show excellent agreement with numerical
simulation. The second model we considered has correlated disorder
in alone one direction. We calculate two correlation functions at
its Berezinskii-Kosterlitz-Thouless transition between the
Gaussian and localized phase and find the logarithmic corrections
to the naive results that could be obtained by using the
fixed-point values of the parameters.
I live in Leamington Spa, Warwickshire, England
I completed my Bachelor degree at the University of Warwick in Mathematics
I am currently at the University of Warwick studying for a Masters in Systems Biology
I will undergo two research projects this year. The first is on femtosecond spectroscopy of biomolecules and for the second I will look at EEG signals in response to visual stimuli.
My interested field of study is the application of random matrices to the study of biology
Taro Kimura
RIKEN Nishina Center Mathematical Physics Laboratory
I am a postdoctoral fellow at Mathematical Physics Laboratory,
RIKEN. A main research topic is the study of supersymmetric gauge
theory and its relation to random matrix theory. I am now exploring
the remarkable relationship between 2d conformal field theory and 4d
gauge theory, which is called the AGT relation, through their matrix
model descriptions. The matrix model is derived from the combinatorial
expression of the gauge theory partition function by considering its
asymptotic behavior. I'm also interested in other topics, lattice
gauge theory, vortex theory, quantum Hall effect, topological
insulator and so on.
* Poster Title : Matrix model from orbifold partition function
* Abstract : We investigate the matrix model associated with the
combinatorial partition function, which is derived from the instanton
counting on the four dimensional orbifold. We would like to show that
the q-deformation and its root of unity limit is relevant to this
model. It is then shown that the gauge theory consequence is extracted
by studying its asymptotic behavior. In particular Seiberg-Witten
curve is given by the spectral curve of the matrix model.
Ranjan Modak
DEPARTMENT OF PHYSICS, THEORETICAL CONDENSED MATTER,
PROFILE: I did my B.Sc from Jadavpur University,Kolkata.I have completed my M.Sc from IIT Kharagpur. Presently I am doing PhD in Department of Physics,Indian Institute of Science (IISc) ,Bangalore under the guidance of Prof. Sriram Ramaswamy and Prof. Subroto Mukerjee.
Research Interest: Thermalization of Quantum system and transport properties of integrable and non-integrable systems.
* Poster Title : Finite Size Scaling of Integrability Breaking parameter for One dimensional quantum models
* Authors : Ranjan Modak, Subroto Mukerjee, and Sriram Ramaswamy
* Abstract : We study energy level spacing statistics of 1D models of spinless fermions using numerical ex- act diagonalization. For finite-length chains,physical properties exhibit a crossover from behavior corresponding to Poisson level characteristic of integrable systems to Wigner-Dyson statistics char- acteristic of non-integrable systems. We use the Drude weight to extract the threshold value of the integrability breaking parameter and investigate its scaling with system size .The range of numer- ically accessible system sizes is not sufficient to establish the scaling with absolute certainty, but our data suggests that the threshold value decreases with increasing system size as a power law and that in the thermodynamic limit an infinitesimal value of the parameter would break integrability. We also consider a simple analytical model of random matrices that produces a power law scaling of the integrability breaking parameter with system size.
Sergio Andraus
Second-year PhD student of Physics at the U. of Tokyo, Miyashita Group
Research interests: statistical mechanics, stochastic processes and random matrix theory
email: [email protected]
* Poster Title : Dyson's model as a special case of Dunkl processes and Dunkl's intertwining operators
* Abstract :
Dyson's Brownian motion model is a family of systems in which N Brownian particles interact repulsively through a log-potential in one dimension, and it is indexed by the positive real parameter beta. Dunkl processes are stochastic processes defined as a generalization of N-dimensional Brownian motion based on a set of differential-difference operators, called Dunkl operators. These operators depend on the choice of a finite set of vectors called root system. When the A-type root system is chosen, its associated Dunkl process describes a system of Brownian particles that repel each other in one dimension and exchange positions spontaneously. An important part of the results from the theory of Dunkl operators is ob- tained through the use of the intertwining operator. This operator relates partial derivatives and Dunkl operators, but its explicit form is unknown in general.
We show that the A-type Dunkl process of parameter k equal to beta/2 under a symmetric initial condition is equivalent to Dyson's model. From this equivalence, we extract an expression for the effect of the intertwining operator on symmetric polynomials. We show that in the strong coupling limit it maps symmetric functions into a function of the sum of their variables. This allows us to study the zero-temperature limit of Dyson's model, and we show that the final configuration is proportional to a vector of the roots of the Hermite polynomials multiplied by the square root of the process time, while being independent of the initial configuration. We briefly discuss two topics of further study on the intertwining operator: its symmetric eigenfunctions and its effect on non-symmetric polynomials.
Fumihiko NAKANO
Department of Mathematics, Gakushuin University, Japan.
* Poster Title : The level statistics of one-dimensional Schroedinger operator with random decaying potential.
* Abstract : We study the level statistics problem of the one-dimensional Schr\"dingier operator with random potential decaying like $x^{-\alpha}$ at infinity. The results obtained so far is summarized as follows :
(i)(ac spectrum case)
if $\alpha > \frac 12$, the point process $\xi_L$ consisting of the rescaled eigenvalues converges to a clock process, and the fluctuation of the spacing of eigenvalues converges to Gaussian.
(ii)(critical case)
if $\alpha = \frac 12$, $\xi_L$ converges to the limit of the circular $\beta$-ensemble.
Craig Jolley
Foreign Postdoctoral Researcher
Laboratory for Systems Biology
RIKEN Center for Developmental Biology
2-2-3 Minatojima-minamimachi, Chuo-ku, Kobe-shi
Lab phone: +81-78-306-3191
e-mail: [email protected]
My current projects center on systems-biological approaches to understanding the mammalian circadian clock. The circadian clock has been extensively studied at the cellular level, and many of the molecular-level interactions are well-understood. The precise means by which the cellular-level clocks generate organism-level outputs in terms of behavior and physiology are less well-understood. This makes the circadian clock an ideal test case for organism-level systems biology, in which we are attempting to piece together a multiscale description of circadian regulatory processes at the cell, tissue, and organism scales. I'm fairly new to random matrix theory, but I hope that it might provide a way to resolve two complementary (and endemic) problems in systems biology: the "parameter problem" in large dynamical models, and the "curse of dimensionality" in high-volume data collection studies.
* Poster Title :Random Matrix Theory and Systems Biology: Some possible directions
* Abstract :Systems Biology attempts to understand the function of biological systems by leveraging two emerging technological trends. The first is the appearance of fast, inexpensive computing power which enables the study of detailed mathematical models of biological systems. The second is the development of high-throughput "omics" technologies that allow for the identification and quantification of mRNA transcripts, proteins, metabolites, and other cellular components. Taken together, these technologies provide us with an unprecedented opportunity to measure and model biological dynamics at a large scale. In both cases, however, new quantitative approaches will be required. Complex models with tens or hundreds of parameters can be fit to experimental data, but varying the values of specific parameters (often order many orders of magnitude) can have a negligible effect on the model's prediction. This makes parameter estimation by fitting a model to experimental data a precarious business. In many cases, the Hessian matrices used in model fitting have been shown to exhibit "sloppy" eigenvalue spectra, in which eigenvalues are approximately evenly-spaced (on a logarithmic axis) over many orders of magnitude. This situation is somewhat different from the matrix ensembles traditionally used in RMT, and a "sloppy universality class" has been proposed to explain these features; its features (and implications) are still not as well understood as the conventional ensembles used in RMT. High-volume data collection experiments can generate a related problem -- while it is possible to search for correlations in large data sets, the probability of finding spurious correlations between unrelated variables increases with the dimensionality of the data set. RMT can help us to reject unreliable correlations by comparing the eigenspectrum of an empirical correlation matrix with one created by assuming no correlations at all; this has been shown to be a much more reliable method for extracting meaningful correlations.
Ricky Kwok
PROFILE:I am a third year graduate student at University of California, Davis under the guidance of Craig Tracy. My research general research areas are interacting particle systems and statistical mechanics. I have previously studied Heisenberg's XXZ model and its relationship to the asymmetric simple exclusion process via unitary transformation of their matrix generators. Currently, I am working on Lieb and Liniger's model of Bose gas subject to hard wall boundary conditions for attractive particles.
Chushun Tian
Professor of Institute for Advanced Study, Tsinghua University, Beijing
Ph. D. 2005, Minnesota (supervisor: Anatoly Larkin)
Research interests: disordered systems, quantum chaos, and strongly correlated electron systems.
Jan Dahlhaus
PhD student of theoretical physics at the Lorentz Institute,
Leiden University, The Netherlands
* Poster Title :Random matrix theory of transport in topological superconductors
* Abstract :Topological superconductors are realizations of a new phase of matter that has been theoretically predicted recently and probably be found experimentally (still controversial). They are characterized by topological invariants - integer numbers that can only change when the system undergoes a quantum phase transition. The electrical transport properties of a junction between a metal and a topological superconductor can be described by a unitary scattering matrix. We investigate the situation that the junction is disordered and of irregular shape such that its electronic dynamics are chaotic. An average over different disorder realizations can then be obtained by averaging over the circular random matrix ensemble of scattering matrices that are allowed by the symmetries of the system. In our work we investigate the influence of the topological invariant on the conductance statistics of the junction.
PROFILE:Graduate student, Physics and Astronomy, UBC
My research interest is mathematical physics (operator theory, topological approach to quantum mechanics etc.)
and theoretical physics (large-scale quantum mechanics, decoherence, quantum mechanics and gravity)
* Poster Title :Decoherence and RMT
* Abstract :Decoherence is one of the most fundamental problems in quantum physics, which addresses many research area such as quantum computing, quantum biology, quantum cosmology and quantum gravity.
We will review the conventional models of environmental (ex. osccilator bath, spin bath) and intrinsic (ex. gravitational) decoherence.
Also, we will investigate the new model of decoherence using RMT and its potential relationship with quantum chaos.
Jacopo Iacovacci
PROFILE: Biophysics Student, University of Rome, 'La Sapienza'
e-mail: [email protected]
I'm a 23 student working on my master thesis in biophysics at the University of Rome 'La Sapienza'.
My primary field of study is neuroscience. I have studied the resolution of the Hopfield model for a neural network using the Random Matrix approach. In general I'm interested in the statistical mechanics applied to the study of biological systems. I'm also interested in studying quantum aspects and mechanism in living matter.
Ulisse Ferrari
PROFILE: PhD student at "La Sapienza" (Rome, Italy)
* poster title: On the critical slowing down exponents of mode coupling theory
* authors: F. Caltagirone, U. Ferrari, L. Leuzzi, G. Parisi, T. Rizzo
* abstract: An important prediction of Mode-Coupling-Theory (MCT) is the relationship between the decay exponents in the $\beta$ regime: ${\Gamma^2(1-a) \over \Gamma(1-2 a)}={\Gamma^2(1+b) \over \Gamma(1+2b)}=\lambda $.
In the original structural glass context this relationship follows from the MCT equations that are obtained making rather uncontrolled approximations. As a consequence, it is usually assumed that the relationship between the exponents is correct while $\lambda$ has to be treated like a tunable parameter. On the other hand, it is known that in some mean-field spin-glass models the dynamics is precisely described by MCT. In that context $\lambda$ can be computed exactly but, again, its computation becomes difficult when we consider more complex models including finite-dimensional ones.
In this work we unveil the physical meaning of $\lambda$ in complete generality, relating the dynamical parameter to the static Gibbs free energy and giving, thus, a ``recipe'' for its computation. In this new framework we compute the exponents $a$ and $b$ for some mean-field models and compare our results with numerical simulations or, when available, exact results obtained via purely dynamical equations.
Debayan Dey
*Position: PhD fellow (graduate student)
*PhD Advisor (PI): Prof.S.Ramakumar, Professor of Physics, Indian Institute of Science
*Address: Department of Physics, Indian Institute of Science, Bangalore 560012, India
*PhD project: Crystallography, structural bioinformatics and mathematical biology of pathogenic organisms.
* poster title: Random matrix theory and gene correlation coefficient statistics of DNA-microarray data: Application in understanding the system biology of gene regulation
* authors: Debayan Dey, S. Ramakumar
* abstract: The fundamental question in biology is to understand the mechanism by which a cell functions. The gene regulation of a cell and its interaction with environment & other cells makes a complex living organism. Gene regulation is the key process which dictates cell function and any imbalance in it results into disease. Understanding gene regulation using high throughput methods are pivotal to understand the holistic nature of gene regulatory network. But it suffers from large embedded noise within it; so a noise reduction method is very important to deduce sensible biological information which further can be experimentally tested. DNA-microarray technique provides gene expression level data for the whole cell's activity at a given time. The understanding of gene correlation matrix provided by the data is essential for biological elucidation of gene regulatory network.
Artur Święch
PROFILE: I'm studying theoretical physics on masters studies at Jagiellonian University, Krakow,. My main research interests are spectra of products of random matrices in thermodynamical limit. Currently I'm also involved in analysis of data from Collider Detector at Fermilab, i.e. events with forward rapidity gaps.
* poster title:Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices
* authors: Z. Burda, A. Jarosz, G. Livan, M. A. Nowak and A. Swiech
* abstract: We analyze spectra of products of arbitrary number of rectangular Gaussian random matrices in two cases: 1. Singular value spectra - using Free Random Variable calculus. 2. Eigenvalue spectra - using planar diagrammatics. We derive analytical expressions describing those spectra in thermodynamical limit and we propose corrections for finite sizes of matrices. The behavior of eigenvalue and singular value distributions near zero is determined to be power-law. The results are compared to numerical simulations of large random matrices.
Yutaka Shikano
PROFILE: I am the research associate professor at Institute for Molecular Science. My main research interest is the foundations of quantum mechanics, especially the quantum measurement theory, discrete time quantum walk, which is related to the quantum chaos, and the quasi-particle condensation.
* poster title: On Inhomogeneous Quantum Walks with Self-Duality
* author: Yutaka Shikano and Hosho Katsura
* abstract: We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-Andr\'e model describing the one-dimensional tight-binding model with a quasi-periodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at the origin. We also numerically study the eigenvalues of the one-step time evolution operator and find the Hofstadter butterfly spectrum which indicates the fractal nature of this class of quantum walks.
* poster title:
* author:
* abstract:
Physics and Biology Unit, OIST
Lecture title: Tails of Genome Evolution
Abstract : I describe two 'universal' features of natural genome sequences that ought to be of interest to biologists. One of them probes sequence duplication; the other, probing correlated mutation in otherwise conserved sequence, is (implicitly, but not explicitly) believed by biologists to be important. Both involve distributions of 'word' lengths that are, obviously, not Poisson.
Shinobu Hikami
Mathematical and Theoretical Physics Unit, OIST
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\begin{document}
\rule{0cm}{1cm}
\begin{center} {\Large\bf The Cauchy Operator for
Basic Hypergeometric Series} \end{center} \vskip 2mm \centerline{Vincent Y. B. Chen$^1$ and Nancy S. S. Gu$^2$ }
\begin{center} Center for Combinatorics, LPMC\\ Nankai University, Tianjin 300071\\ People's Republic of China\\
\vskip 2mm
Email: $^[email protected],
$^[email protected] \end{center}
\begin{center} {\bf Abstract} \end{center}
{\small We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ${}_2\phi_1$ transformation formula and Sears' ${}_3\phi_2$ transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it
can be considered as a generalization of the operator $T(bD_q)$. Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the $q$-analogues of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers-Szeg\"o polynomials, or the continuous big $q$-Hermite polynomials.}
\vskip 5mm
\noindent {\bf Keywords:} $q$-difference operator, the Cauchy
operator, the Askey-Wilson integral, the Askey-Roy integral, basic hypergeometric series, parameter augmentation.
\vskip 5mm \noindent{\bf AMS Subject Classification:} 05A30, 33D05, 33D15
\section{Introduction}
In an attempt to find efficient $q$-shift operators to deal with basic hypergeometric series identities in the framework of the $q$-umbral calculus \cite{Andrews, Goldman-Rota}, Chen and Liu \cite{Chen1,Chen2} introduced two $q$-exponential operators for deriving identities from their special cases. This method is called parameter augmentation. In this paper, we continue the study of parameter augmentation by defining a new operator called the Cauchy augmentation operator which is suitable for certain transformation and integral formulas.
Recall that Chen and Liu \cite{Chen1} introduced the augmentation operator \begin{equation}\label{TE} T(bD_q)=\sum_{n=0}^{\infty}\frac{(bD_q)^n}{(q;q)_n} \end{equation} as the basis of parameter augmentation which serves as a method for proving $q$-summation and integral formulas from special cases for which some parameters are set to zero.
The main idea of this paper is to introduce the Cauchy augmentation operator, or simply the Cauchy operator, \begin{equation}\label{gTE} T(a,b;D_q)=\sum_{n=0}^{\infty}\frac{(a;q)_n}{(q;q)_n}(bD_q)^n, \end{equation} which is reminiscent of the Cauchy
$q$-binomial theorem \cite[Appendix II.3]{Gasper-Rahman} \begin{equation}\label{Cauchy} \sum_{n=0}^{\infty}\frac{(a;q)_n}{(q;q)_n}z^n
=\frac{(az;q)_\infty}{(z;q)_\infty},\ \ |z|<1. \end{equation} For the same reason, the operator $T(aD_q)$ should be named the Euler operator in view of Euler's identity\cite[Appendix II.1]{Gasper-Rahman} \begin{equation}\label{Euler} \sum_{n=0}^{\infty}\frac{z^n}{(q;q)_n} =\frac{1}{(z;q)_\infty},\ \ \
\ \quad |z|<1. \end{equation}
Compared with $T(bD_q)$, the Cauchy
operator \eqref{gTE} involves two parameters. Clearly, the operator $T(bD_q)$ can be considered as a special case of the Cauchy operator \eqref{gTE} for $a=0$. In order to utilize the Cauchy operator to basic hypergeometric series, several operator identities are deduced in Section \ref{se Basic Properties}. As to the applications of the Cauchy operator, we show that many classical results on basic hypergeometric series easily fall into this framework. Heine's ${}_2\phi_1$ transformation formula \cite[Appendix III.2]{Gasper-Rahman} and Sears' ${}_3\phi_2$ transformation formula \cite[Appendix III.9]{Gasper-Rahman} can be easily obtained by the symmetric property of some parameters in two operator identities for the Cauchy operator.
In Section \ref{se Askey-Wilson Integral} and Section \ref{se Askey-Roy Integral}, we use the Cauchy operator to generalize the Askey-Wilson integral and the Askey-Roy integral. In \cite{IsmailStantonViennot}, Ismail, Stanton, and Viennot derived an integral named the Ismail-Stanton-Viennot integral which took the Askey-Wilson integral as a special case. It is easy to see that our extension of the Askey-Wilson integral is also an extension of the Ismail-Stanton-Viennot integral. In \cite{Gasper}, Gasper discovered an integral which was a generalization of the Askey-Roy integral. We observe that Gasper's formula is a special case of the formula obtained by applying the Cauchy operator directly to the Askey-Roy integral. Furthermore, we find that the Cauchy operator can be applied to Gasper's formula to derive a further extension of the Askey-Roy integral.
In Section \ref{se Bivariate Rogers-Szego Polynomials}, we present that the Cauchy operator is suitable for the study of bivariate Rogers-Szeg\"o polynomials. It can be used to derive the corresponding Mehler's and the Rogers formulas for the bivariate Rogers-Szeg\"o polynomials, which can be stated in the equivalent forms in terms of the continuous big $q$-Hermite polynomials. Mehler's formula in this case turns out to be a special case of the nonsymmetric Poisson kernel formula for the continuous big $q$-Hermite polynomials due to Askey, Rahman, and Suslov \cite{Askey-Rahman-Suslov}. Finally, in Section \ref{se Sears' Formula} and Section \ref{se q-Barnes' Lemmas}, we employ the Cauchy operator to deduce extensions of Sears' two-term summation formula \cite[Eq. (2.10.18)]{Gasper-Rahman} and the $q$-analogues of Barnes' lemmas \cite[Eqs. (4.4.3), (4.4.6)]{Gasper-Rahman}.
As usual, we follow the notation and terminology in
\cite{Gasper-Rahman}. For $|q|<1$, the $q$-shifted factorial is defined by $$(a;q)_\infty= \prod_{k=0}^{\infty}(1-aq^k) \text{\ \ and \ \ }(a;q)_n =\frac{(a;q)_\infty}{(aq^n;q)_\infty}, \text{ for } n\in \mathbb{Z}.$$
For convenience, we shall adopt the following notation for multiple $q$-shifted factorials: $$(a_1,a_2,\ldots,a_m;q)_n=(a_1;q)_n(a_2;q)_n\cdots(a_m;q)_n,$$ where $n$ is an integer or infinity.
The $q$-binomial coefficients, or the Gauss coefficients, are given by \begin{equation} {n \brack k}=\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}. \end{equation}
The (unilateral) basic hypergeometric series $_{r}\phi_s$ is defined by \begin{equation} _{r}\phi_s\left[\begin{array}{cccccc} a_1,&a_2,&\ldots,&a_r\\ b_1,&b_2,&\ldots,&b_s \end{array};q,z\right]=\sum_{k=0}^{\infty} \frac{(a_1,a_2,\ldots,a_r;q)_k}{(q,b_1,b_2,\ldots,b_s;q)_k} \left[(-1)^kq^{k\choose2}\right]^{1+s-r}z^k. \end{equation}
\section{Basic Properties}\label{se Basic Properties}
In this section, we give some basic identities involving the Cauchy
operator $T(a,b; D_q)$ and demonstrate that Heine's ${}_2\phi_1$
transformation formula and Sears' ${}_3\phi_2$ transformation formula are implied in the symmetric property of some parameters in two operator identities.
We recall that the $q$-difference operator, or Euler derivative, is defined by \begin{equation}\label{dq} D_q\{f(a)\}=\frac{f(a)-f(aq)}{a}, \end{equation} and the Leibniz rule for $D_q$ is referred to
the following identity \begin{equation}\label{Leibniz} D_q^n\{f(a)g(a)\}=\sum_{k=0}^n q^{k(k-n)}{n \brack k}D_q^k\{f(a)\}D_q^{n-k}\{g(aq^k)\}. \end{equation} The following relations are easily verified.
\begin{prop}\label{DK1}Let $k$ be a nonnegative integer. Then we have \begin{eqnarray*} D_q^k\left\{\frac{1}{(at;q)_\infty}\right\} &=&\frac{t^k}{(at;q)_\infty},\\[5pt] D_q^k\left\{(at;q)_\infty\right\} & = & (-t)^kq^{k\choose2}(atq^k;q)_\infty,\\[5pt] D_q^k\left\{\frac{(av;q)_\infty}{(at;q)_\infty}\right\} & = & t^k(v/t;q)_k\frac{(avq^k;q)_\infty}{(at;q)_\infty}.\\ \end{eqnarray*} \end{prop}
Now, we are ready to give some basic identities for the Cauchy operator $T(a,b;D_q)$. We assume that $T(a,b;D_q)$ acts on the parameter $c$. The following identity is an easy consequence of the Cauchy $q$-binomial theorem \eqref{Cauchy}.
\begin{thm}\label{gtef1} We have \begin{equation}\label{tabc} T(a,b;D_q)\left\{\frac{1}{(ct;q)_\infty}\right\}=\frac{(ab\,t;q)_\infty}{(b\,t,ct;q)_\infty}, \end{equation}
provided $|b\,t|<1$. \end{thm}
\begin{pf} By Proposition \ref{DK1}, the left hand side of \eqref{tabc} equals \[ \sum_{n=0}^{\infty}\frac{(a;q)_nb^n}{(q;q)_n} D_q^n\left\{\frac{1}{(ct;q)_\infty}\right\} =
{1\over (ct;q)_\infty} \sum_{n=0}^{\infty}
\frac{(a;q)_n(b\,t)^n}{(q;q)_n},\]
which simplifies to the right hand side of \eqref{tabc}
by the Cauchy $q$-binomial theorem \eqref{Cauchy}.\qed \end{pf}
\begin{thm}\label{gtef2} We have \begin{equation}\label{22} T(a,b;D_q)\left\{\frac{1}{(cs,ct;q)_\infty}\right\} =\frac{(ab\,t;q)_\infty}{(b\,t,cs,ct;q)_\infty} \,{}_{2}\phi_1\left[\begin{array}{cc} a,&ct\\ &ab\,t \end{array};q,bs\right], \end{equation}
provided $\max\{|bs|,|b\,t|\}<1$. \end{thm}
\begin{pf} In view of the Leibniz formula for $D_q^n$, the left hand side of \eqref{22} can be expanded as follows \begin{eqnarray*} \lefteqn{\sum_{n=0}^{\infty}\frac{(a;q)_nb^n}{(q;q)_n}\sum_{k=0}^{n}q^{k(k-n)} {n\brack k}D_q^k\left\{\frac{1}{(cs;q)_\infty}\right\}D_q^{n-k} \left\{\frac{1}{(ctq^k;q)_\infty}\right\}\nonumber}\\[6pt] &=&\sum_{n=0}^{\infty}\frac{(a;q)_nb^n}{(q;q)_n}\sum_{k=0}^{n}q^{k(k-n)}{n\brack k} \frac{s^k}{(cs;q)_\infty}\frac{(tq^k)^{n-k}}{(ctq^k;q)_\infty} \nonumber\\[6pt] &=&\frac{1}{(cs,ct;q)_\infty}\sum_{k=0}^{\infty}\frac{(ct;q)_k(bs)^k}{(q;q)_k} \sum_{n=k}^{\infty}\frac{(a;q)_n(b\,t)^{n-k}}{(q;q)_{n-k}} \nonumber\\[6pt] &=&\frac{1}{(cs,ct;q)_\infty}\sum_{k=0}^{\infty}\frac{(a,ct;q)_k(bs)^k}{(q;q)_k} \sum_{n=0}^{\infty}\frac{(aq^k;q)_n(b\,t)^{n}}{(q;q)_{n}} \quad\quad\quad\quad\quad\quad\nonumber\\[6pt] &=&\frac{(ab\,t;q)_\infty}{(b\,t,cs,ct;q)_\infty} {}_{2}\phi_1\left[\begin{array}{cc} a,&ct\\ &ab\,t \end{array};q,bs\right], \end{eqnarray*} as desired. \qed \end{pf}
Notice that when $a=0$, the ${}_2\phi_1$ series
on the right hand side of \eqref{22} can be summed by employing the Cauchy $q$-binomial theorem \eqref{Cauchy}. In this case \eqref{22} reduces to \begin{equation} T(bD_q)\left\{\frac{1}{(cs,ct;q)_\infty}\right\}= \frac{(bcst;q)_\infty}{(bs,b\,t,cs,ct;q)_\infty},\quad
\quad|bs|,|b\,t|<1, \end{equation} which was derived by Chen and Liu in \cite{Chen1}.
As an immediate consequence of the above theorem, we see that Heine's ${}_2\phi_1$ transformation formula \cite[Appendix III.2]{Gasper-Rahman} is really about the symmetry in $s$ and $t$ while applying the operator $T(a,b;q)$.
\begin{cor}[Heine's transformation]\label{Heine}We have \begin{equation}\label{sHeine} {}_{2}\phi_1\left[\begin{array}{cc} a,&b\\ &c \end{array};q,z\right]=\frac{(c/b,bz;q)_\infty} {(c,z;q)_\infty}{}_{2}\phi_1\left[\begin{array}{cc} abz/c,&b\\ &bz \end{array};q,\frac{c}{b}\right], \end{equation}
where $\max\{|z|, |c/b|\}<1$. \end{cor}
\begin{pf} The symmetry in $s$ and $t$ on the left hand side of \eqref{22} implies that \begin{equation}\label{21} \frac{(ab\,t;q)_\infty}{(b\,t,cs,ct;q)_\infty} {}_{2}\phi_1\left[\begin{array}{cc} a,&ct\\ &ab\,t \end{array};q,bs\right]=\frac{(abs;q)_\infty}{(bs,ct,cs;q)_\infty} {}_{2}\phi_1\left[\begin{array}{cc} a,&cs\\ &abs \end{array};q,b\,t\right], \end{equation}
where $\max\{|bs|,|b\,t|\}<1$.
Replacing $a, b, c, s, t$ by $b,a,a^2b/c, z/a, c/ab$ in \eqref{21}, respectively, we may easily express the above identity in the form of \eqref{sHeine}.\qed \end{pf}
\begin{rem}
A closer look at the proof of Theorem \ref{gtef2} reveals that the essence of Heine's transformation lies in the symmetry of $f$ and $g$ in Leibniz's formula \eqref{Leibniz}. \end{rem}
We should note that we must be cautious about the convergence conditions while utilizing the Cauchy operator. In general, it would be safe to apply the Cauchy operator if the resulting series is convergent. However, it is possible that from a convergent series one may obtain a divergent series after employing the Cauchy
operator. For example, let us consider Corollary \ref{Heine}. The resulting series \eqref{21} can be obtained by applying the Cauchy operator $T(a,b;D_q)$ to $1/(cs,ct;q)_\infty$
which is convergent for all $t$. However, the resulting series on the left hand side of \eqref{21} is not convergent for $|t|>1/|b|$.
Combining Theorem \ref{gtef1} and the Leibniz rule \eqref{Leibniz}, we obtain the following identity which implies Theorem \ref{gtef2} by setting $v=0$. Sears' ${}_3\phi_2$ transformation formula \cite[Appendix III.9]{Gasper-Rahman} is also a consequence of Theorem \ref{gtef3}.
\begin{thm}We have\label{gtef3} \begin{eqnarray}\label{tab} T(a,b;D_q)\left\{\frac{(cv;q)_\infty}{(cs,ct;q)_\infty}\right\} =\frac{(abs,cv;q)_\infty}{(bs,cs,ct;q)_\infty}{}_{3}\phi_2\left[\begin{array}{ccc} a,&cs,&v/t\\ &abs,&cv \end{array};q,b\,t\right], \end{eqnarray}
provided $\max\{|bs|,|b\,t|\}<1$. \end{thm}
\begin{pf} In light of Leibniz's formula, the left hand side of \eqref{tab} equals \begin{eqnarray} \lefteqn{\sum_{n=0}^{\infty}\frac{(a;q)_nb^n}{(q;q)_n} D_q^n\left\{\frac{(cv;q)_\infty}{(cs,ct;q)_\infty}\right\}\nonumber} \\[6pt] &=&\sum_{n=0}^{\infty}\frac{(a;q)_nb^n}{(q;q)_n} \sum_{k=0}^{n}q^{k(k-n)} {n\brack k}D_q^k\left\{\frac{(cv;q)_\infty}{(ct;q)_\infty}\right\} D_q^{n-k}\left\{\frac{1}{(csq^k;q)_\infty}\right\}\nonumber\\[6pt] &=&\sum_{n=0}^{\infty}\frac{(a;q)_nb^n}{(q;q)_n} \sum_{k=0}^{n}q^{k(k-n)}{n\brack k} \frac{t^k(v/t;q)_k(cvq^k;q)_\infty}{(ct;q)_\infty} D_q^{n-k}\left\{\frac{1}{(csq^k;q)_\infty}\right\}\nonumber\\[6pt] &=&\sum_{k=0}^{\infty}\frac{(v/t;q)_k(cvq^k;q)_\infty{t}^k} {(q;q)_k(ct;q)_\infty} \sum_{n=k}^{\infty}\frac{b^nq^{k(k-n)}(a;q)_n}{(q;q)_{n-k}} D_q^{n-k}\left\{\frac{1}{(csq^k;q)_\infty}\right\}\nonumber\\[6pt] &=&\sum_{k=0}^{\infty}\frac{(a,v/t;q)_k(cvq^k;q)_\infty(b\,t)^k} {(q;q)_k(ct;q)_\infty} \sum_{n=0}^{\infty}\frac{(bq^{-k})^n(aq^k;q)_n}{(q;q)_n} D_q^{n}\left\{\frac{1}{(csq^k;q)_\infty}\right\}\nonumber\\[6pt] &=&\sum_{k=0}^{\infty}\frac{(a,v/t;q)_k(cvq^k;q)_\infty(b\,t)^k}{(q;q)_k(ct;q)_\infty} T(aq^k,bq^{-k};D_q)\left\{\frac{1}{(csq^k;q)_\infty}\right\} \nonumber. \end{eqnarray} By Theorem \ref{gtef1}, the above sum equals \begin{eqnarray} \lefteqn{\sum_{k=0}^{\infty}\frac{(a,v/t;q)_k(cvq^k;q)_\infty(b\,t)^k}{(q;q)_k(ct;q)_\infty} \frac{(absq^k;q)_\infty}{(bs,csq^k;q)_\infty}\nonumber}\\[6pt] &=&\frac{(cv;q)_\infty}{(cs,ct;q)_\infty}\sum_{k=0}^{\infty} \frac{(a,cs,v/t;q)_k(b\,t)^k}{(q,cv;q)_k} \frac{(absq^k;q)_\infty}{(bs;q)_\infty}\nonumber\\[6pt] &=&\frac{(abs,cv;q)_\infty} {(bs,cs,ct;q)_\infty}{}_{3}\phi_2\left[\begin{array}{ccc} a,&cs,&v/t\\ &abs,&cv \label{33} \end{array};q,b\,t\right], \end{eqnarray} as desired. \qed \end{pf}
\begin{cor}[Sears' transformation]We have \begin{equation} {}_{3}\phi_2\left[\begin{array}{ccc} a,&b,&c\\ &d,&e \end{array};q,\frac{de}{abc}\right]= \frac{(e/a,de/bc;q)_\infty}{(e,de/abc;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} a,&d/b,&d/c\\ &d,&de/bc \end{array};q,\frac{e}{a}\right],\label{hall} \end{equation}
where $\max\{|de/abc|,|e/a|\}<1$. \end{cor}
\begin{pf} Based on the symmetric property of the parameters $s$ and $t$ on the left hand side of \eqref{tab}, we find that \begin{equation*} \frac{(abs,cv;q)_\infty}{(bs,cs,ct;q)_\infty}{}_{3}\phi_2\left[\begin{array}{ccc} a,&cs,&v/t\\ &abs,&cv \end{array};q,b\,t\right]=\frac{(ab\,t,cv;q)_\infty}{(b\,t,ct,cs;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} a,&ct,&v/s\\ &ab\,t,&cv \end{array};q,bs\right], \end{equation*}
where $\max\{|bs|,|b\,t|\}<1$.
Making the substitutions $c \rightarrow ab^2/e$, $v \rightarrow de/ab^2$, $s \rightarrow e/ab$, and $t\rightarrow de/ab^2c$, we get the desired formula.\qed \end{pf}
We see that the essence of Sears' transformation also lies in the symmetry of $s$ and $t$ in the application of Leibniz rule.
\section{An Extension of the Askey-Wilson Integral} \label{se Askey-Wilson Integral}
The Askey-Wilson integral \cite{AskeyWilson} is a significant extension of the beta integral. Chen and Liu \cite{Chen1} presented a treatment of the Askey-Wilson integral via parameter augmentation. They first got the usual Askey-Wilson integral with one parameter by the orthogonality relation obtained from the Cauchy $q$-binomial theorem \eqref{Cauchy} and the Jacobi triple product identity \cite[Appendix II.28]{Gasper-Rahman}, and then they applied the operator $T(bD_q)$ three times to deduce the Askey-Wilson integral involving four parameters \cite{Askey,IsmailStantonViennot, IsmailStanton,Kalnins,Rahman,WilfZeilberger} \begin{eqnarray} &&\int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta};q)_\infty d\theta} {(ae^{i\theta},ae^{-i\theta},be^{i\theta},be^{-i\theta}, ce^{i\theta},ce^{-i\theta},de^{i\theta},de^{-i\theta};q)_\infty}\nonumber\\[6pt] &&=\frac{2\pi(abcd;q)_\infty}{(q,ab,ac,ad,bc,bd,cd;q)_\infty},\label{Askeywilson} \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|\}<1$.
In this section, we derive an extension of the Askey-Wilson integral \eqref{Askeywilson} which contains the following Ismail-Stanton-Viennot's integral \cite{IsmailStantonViennot} as a special case: \begin{eqnarray}\label{ISV} \lefteqn{\int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta};q)_\infty d\theta} {(ae^{i\theta},ae^{-i\theta},be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta}, de^{i\theta},de^{-i\theta},ge^{i\theta},ge^{-i\theta};q)_\infty}}\nonumber\\[6pt] &=&\frac{2\pi(abcg,abcd;q)_\infty}{(q,ab,ac,ad,ag,bc,bd,bg,cd,cg;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} ab,&ac,&bc\\ &abcg,&abcd \end{array};q,dg\right], \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|,|g|\}<1$.
\begin{thm}[Extension of the Askey-Wilson integral]We have \label{ThmAskeyWilson} \begin{eqnarray}\label{exAskeywilson} \lefteqn{\int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta},fge^{i\theta};q)_\infty} {(ae^{i\theta},ae^{-i\theta},be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta}, de^{i\theta},de^{-i\theta},ge^{i\theta};q)_\infty}}\nonumber\\[6pt] &&\quad\quad\quad\times{}_{3}\phi_2\left[\begin{array}{ccc} f,&ae^{i\theta},&be^{i\theta}\\ &fge^{i\theta},&ab \end{array};q,ge^{-i\theta}\right]d\theta\nonumber\\[6pt] &=&\frac{2\pi(cfg,abcd;q)_\infty}{(q,ab,ac,ad,bc,bd,cd,cg;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} f,&ac,&bc\\ &cfg,&abcd \end{array};q,dg\right], \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|,|g|\}<1$. \end{thm}
\begin{pf} The Askey-Wilson integral \eqref{Askeywilson} can be written as \begin{eqnarray}\label{integral} &&\int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta};q)_\infty} {(be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta},de^{i\theta},de^{-i\theta};q)_\infty} \frac{(ab;q)_\infty}{(ae^{i\theta},ae^{-i\theta};q)_\infty}d\theta\nonumber\\[6pt] &&\qquad =\frac{2\pi}{(q,bc,bd,cd;q)_\infty}\frac{(abcd;q)_\infty}{(ac,ad;q)_\infty}. \end{eqnarray}
Before applying the Cauchy operator to an integral, it is necessary to show that the Cauchy operator commutes with the integral. This fact is implicit in the literature. Since this commutation relation depends on some technical conditions in connection with the integrands, here we present a complete proof.
First, it can be easily verified that the $q$-difference operator $D_q$ commutes with the integral. By the definition of $D_q$ \eqref{dq}, it is clear that \begin{equation} D_q\left\{\int_C f(\theta,a){d}\theta\right\} = \int_C D_q \left\{f(\theta,a)\right\}{d}\theta. \end{equation} Consequently, the operator $D_q^n$ commutes with the integral. Given a Cauchy operator $T(f,g;D_q)$, we proceed to prove that it commutes with the integral. From the well-known fact that, for a sequence of continuous functions $u_n(\theta)$ on a curve $C$, the sum commutes with the integral in \[ \sum_{n=0}^{\infty}\int_{C}u_n(\theta){d}\theta\]
provided that $\sum_{n=0}^{\infty}u_n(\theta)$ is uniformly convergent. It is sufficient to check the convergence condition for the continuity is obvious. This can be done with the aid of the Weierstrass M-Test \cite{Arfken}. Using the Cauchy operator $T(f,g;D_q)$ to the left hand side of \eqref{integral}, we find that \begin{eqnarray} \lefteqn{T(f,g;D_q)\left\{\int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta};q)_\infty} {(be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta},de^{i\theta},de^{-i\theta};q)_\infty} \frac{(ab;q)_\infty}{(ae^{i\theta},ae^{-i\theta};q)_\infty}{d}\theta\right\} }\label{44}\\[6pt] &=& \sum_{n=0}^{\infty}\frac{(f;q)_{n}}{(q;q)_{n}}(gD_q)^n \int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta};q)_\infty} {(be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta},de^{i\theta},de^{-i\theta};q)_\infty} \frac{(ab;q)_\infty}{(ae^{i\theta},ae^{-i\theta};q)_\infty}{d}\theta\nonumber\\[6pt] &=& \sum_{n=0}^{\infty}\int_{0}^{\pi} \frac{(e^{2i\theta},e^{-2i\theta};q)_\infty} {(be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta},de^{i\theta},de^{-i\theta};q)_\infty} \frac{(f;q)_{n}g^n}{(q;q)_{n}}D_q^n\left\{\frac{(ab;q)_\infty} {(ae^{i\theta},ae^{-i\theta};q)_\infty}\right\}{d}\theta.\nonumber \end{eqnarray} Let $U_n(\theta)$ denote the integrand in the last line of the above equation. We make the assumption $0 < q < 1$ so that, for $0 \leq \theta \leq \pi$, \begin{equation}
|(|x|;q)_{\infty}| \leq |(xe^{\pm i\theta};q)_{\infty}| \leq
(-|x|;q)_{\infty} \end{equation} and \begin{equation}
|(e^{\pm 2i\theta};q)_\infty| \leq (-1;q)_{\infty}. \end{equation} Now we rewrite the series $\sum_{n=0}^{\infty}U_n(\theta)$ into another form $\sum_{n=0}^{\infty}V_n(\theta)$ in order to prove its uniform convergence. In the proof of Theorem \ref{gtef3}, one sees that the
absolute convergence of the ${}_3\phi_2$ series under the condition
$|bs|,\,|b\,t|<1$
implies the absolute convergence of the sum $$\sum_{n=0}^{\infty}\frac{(a;q)_nb^n}{(q;q)_n} D_q^n\left\{\frac{(cv;q)_\infty}{(cs,ct;q)_\infty}\right\}.$$
Therefore, under the condition $|g|<1$, it follows from Theorem \ref{gtef3} that \begin{eqnarray} \lefteqn{\sum_{n=0}^{\infty}\frac{(f;q)_{n}g^n}{(q;q)_{n}} D_q^n\left\{\frac{(ab;q)_\infty}{(ae^{i\theta}, ae^{-i\theta};q)_\infty}\right\}}\nonumber\\[6pt] &=&\frac{(fge^{i\theta},ab;q)_\infty}{(ge^{i\theta}, ae^{i\theta},ae^{-i\theta};q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} f,&ae^{i\theta},&be^{i\theta}\\ &fge^{i\theta},&ab \end{array};q,ge^{-i\theta}\right]. \end{eqnarray} Hence \begin{eqnarray} \sum_{n=0}^{\infty}U_n(\theta)&=&\frac{(e^{2i\theta},e^{-2i\theta},fge^{i\theta},ab;q)_\infty} {(ae^{i\theta},ae^{-i\theta},be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta}, de^{i\theta},de^{-i\theta},ge^{i\theta};q)_\infty}\nonumber\\[6pt] &&\quad \times {}_{3}\phi_2\left[\begin{array}{ccc} f,&ae^{i\theta},&be^{i\theta}\\ &fge^{i\theta},&ab \end{array};q,ge^{-i\theta}\right]\nonumber\\[6pt] &=&\frac{(e^{2i\theta},e^{-2i\theta},fge^{i\theta},ab;q)_\infty} {(ae^{i\theta},ae^{-i\theta},be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta}, de^{i\theta},de^{-i\theta},ge^{i\theta};q)_\infty}\nonumber\\[6pt] &&\quad \times\sum_{n=0}^{\infty}\frac{(f,ae^{i\theta},be^{i\theta};q)_n} {(q,fge^{i\theta},ab;q)_n} \left(ge^{-i\theta}\right)^n. \end{eqnarray} Now, let \begin{eqnarray} V_n(\theta)&=&\frac{(e^{2i\theta},e^{-2i\theta},fge^{i\theta},ab;q)_\infty} {(ae^{i\theta},ae^{-i\theta},be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta}, de^{i\theta},de^{-i\theta},ge^{i\theta};q)_\infty}\nonumber\\[6pt] &&\quad \times\frac{(f,ae^{i\theta},be^{i\theta};q)_n}{(q,fge^{i\theta},ab;q)_n} \left(ge^{-i\theta}\right)^n. \end{eqnarray}
By the Weierstrass M-Test, it remains to find a convergent series $\sum_{n=0}^{\infty}M_n$, where $M_n$ is independent of $\theta$, such that $|V_n(\theta)|\leq M_n$. For
$\max\{|a|,|b|,|c|,|d|,|g|\}<1$, we may choose \begin{equation} M_n=\left(\frac{(-1;q)_\infty}
{(|a|,|b|,|c|,|d|;q)_\infty}\right)^2\frac{(-|fg|,ab;q)_\infty}
{(|g|;q)_\infty}\frac{(-|f|,-|a|,-|b|;q)_{n}|g|^n}{|(q,|fg|,ab;q)_{n}|}. \end{equation} It is easy to see that $\sum_{n=0}^{\infty}M_n$ is convergent when
$|g|<1$. It follows that the Cauchy operator commutes with the integral in \eqref{44}, so \eqref{44} can be written as \begin{eqnarray*} \lefteqn{ \int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta};q)_\infty} {(be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta},de^{i\theta},de^{-i\theta};q)_\infty} \sum_{n=0}^{\infty}\frac{(f;q)_{n}g^n}{(q;q)_{n}} D_q^n\left\{\frac{(ab;q)_\infty}{(ae^{i\theta},ae^{-i\theta};q)_\infty}\right\}{d}\theta} \nonumber \\[6pt] &=& \int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta};q)_\infty} {(be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta},de^{i\theta},de^{-i\theta};q)_\infty} T(f,g;D_q)\left\{\frac{(ab;q)_\infty} {(ae^{i\theta},ae^{-i\theta};q)_\infty}\right\}{d}\theta. \end{eqnarray*}
Finally, we may come to the general condition $|q|<1$ by the argument of analytic continuation. Hence, under the condition
$\max\{|a|,|b|,|c|,|d|,|g|\}<1$, we have shown that it is valid to exchange the Cauchy operator and the integral when we apply the Cauchy operator to \eqref{integral}.
Now, applying $T(f,g;D_q)$ to \eqref{integral} with respect to the parameter $a$ gives \begin{eqnarray} \lefteqn{\int_{0}^{\pi}\frac{(e^{2i\theta},e^{-2i\theta};q)_\infty} {(be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta}, de^{i\theta},de^{-i\theta};q)_\infty} \frac{(fge^{i\theta},ab;q)_\infty}{(ge^{i\theta}, ae^{i\theta},ae^{-i\theta};q)_\infty}}\nonumber\\[6pt] &&\quad\quad\quad\times{}_{3}\phi_2\left[\begin{array}{ccc} f,&ae^{i\theta},&be^{i\theta}\\[6pt] &fge^{i\theta},&ab \end{array};q,ge^{-i\theta}\right]{d}\theta\nonumber\\[6pt] &=&\frac{2\pi}{(q,bc,bd,cd;q)_\infty}\frac{(cfg,abcd;q)_\infty} {(cg,ac,ad;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} f,&ac,&bc\\ &cfg,&abcd \end{array};q,dg\right], \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|,|g|\}<1$. This implies the desired formula. The proof is completed. \qed \end{pf}
In fact, the above proof also implies the convergence of the integral in Theorem \ref{ThmAskeyWilson}. Once it has been shown that the sum commutes with the integral, one sees that the integral obtained from exchanging the sum and the integral is convergent.
Setting $f=ab$ in \eqref{exAskeywilson}, by the $q$-Gauss sum \cite[Appendix II.8]{Gasper-Rahman}: \begin{equation} {}_{2}\phi_1\left[\begin{array}{cc} a,&b\\ &c \end{array};q,\frac{c}{ab}\right]=\frac{(c/a,c/b;q)_\infty}
{(c,c/ab;q)_\infty},\ \ |c/ab|<1,\label{gauss} \end{equation}
we arrive at the Ismail-Stanton-Viennot integral \eqref{ISV}.
Setting $f=abcd$ in \eqref{exAskeywilson}, by means of the $q$-Gauss sum \eqref{gauss} we find the following formula which we have not seen in the literature.
\begin{cor} We have \begin{eqnarray} \lefteqn{\int_{0}^{\pi} \frac{(e^{2i\theta},e^{-2i\theta},abcdge^{i\theta};q)_\infty} {(ae^{i\theta},ae^{-i\theta},be^{i\theta}, be^{-i\theta},ce^{i\theta},ce^{-i\theta}, de^{i\theta},de^{-i\theta},ge^{i\theta};q)_\infty}}\nonumber\\[6pt] &&\quad\quad\times{}_{3}\phi_2\left[\begin{array}{ccc} abcd,&ae^{i\theta},&be^{i\theta}\\ &abcdge^{i\theta},&ab \end{array};q,ge^{-i\theta}\right]{d}\theta\nonumber\\[6pt] &=&\frac{2\pi(abcd,acdg,bcdg;q)_\infty}{(q,ab,ac,ad,bc,bd,cd,cg,dg;q)_\infty}, \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|,|g|\}<1$. \end{cor}
\section{A Further Extension of the Askey-Roy Integral} \label{se Askey-Roy Integral}
Askey and Roy \cite{Askey-Roy} used Ramanujan's ${}_1\psi_1$ summation formula \cite[Appendix II.29]{Gasper-Rahman} to derive the following integral formula: \begin{eqnarray}\label{Askey-Roy} \lefteqn{\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(\rho e^{i\theta}/d,qde^{-i\theta}/\rho,\rho ce^{-i\theta},qe^{i\theta}/c\rho;q)_\infty} {(ae^{i\theta},be^{i\theta},ce^{-i\theta}, de^{-i\theta};q)_\infty}{d}\theta \nonumber}\\[6pt] &&=\frac{(abcd,\rho c/d,dq/\rho c,\rho,q/\rho;q)_\infty}{(q,ac,ad,bc,bd;q)_\infty}, \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|\}<1$ and $cd\rho\neq0$, which is called the Askey-Roy integral.
In \cite{Gasper}, Gasper discovered an integral formula \begin{eqnarray} \lefteqn{\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(\rho e^{i\theta}/d,qde^{-i\theta}/\rho,\rho ce^{-i\theta},qe^{i\theta}/c\rho,abcdfe^{i\theta};q)_\infty} {(ae^{i\theta},be^{i\theta},fe^{i\theta},ce^{-i\theta},de^{-i\theta};q)_\infty} {d}\theta}\nonumber\\[6pt] &=&\frac{(abcd,\rho c/d,dq/\rho c,\rho,q/\rho,bcdf,acdf;q)_\infty} {(q,ac,ad,bc,bd,cf,df;q)_\infty},\label{Gasper} \end{eqnarray}
provided $\max\{|a|,|b|,|c|,|d|,|f|\}<1$ and $cd\rho\neq0$, which is an extension of the Askey-Roy integral. Note that Rahman and Suslov \cite{RahmanSuslov} found a proof of Gasper's formula \eqref{Gasper} based on the technique of iteration with respect to the parameters of $\rho(s)$ in the integral $$\int_{C}\rho(s)q^{-s}{d}s,$$ where $\rho(s)$ is the solution of a Pearson-type first-order difference equation.
In this section, we first derive an extension of the Askey-Roy integral by applying the Cauchy operator. We see that Gasper's formula \eqref{Gasper} is a special case of this extension \eqref{askey}. Moreover, a further extension of the Askey-Roy integral can be obtained by taking the action of the Cauchy
operator on Gasper's formula.
\begin{thm}\label{t-Roy} We have \begin{eqnarray}\label{Roy} \lefteqn{\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(\rho e^{i\theta}/d,qde^{-i\theta}/\rho,\rho ce^{-i\theta},qe^{i\theta}/c\rho,abcdfe^{i\theta},ghe^{i\theta};q)_\infty} {(ae^{i\theta},be^{i\theta},fe^{i\theta},he^{i\theta}, ce^{-i\theta},de^{-i\theta};q)_\infty}}\nonumber\\[6pt] &&\times{}_{3}\phi_2\left[\begin{array}{ccc} g,&ae^{i\theta},&fe^{i\theta}\\ &ghe^{i\theta},&abcdfe^{i\theta} \end{array};q,bcdh\right] {d} \theta\nonumber\\[6pt] &=&\frac{(abcd,\rho c/d,dq/\rho c,\rho,q/\rho,bcdf,acdf,cgh;q)_\infty} {(q,ac,ad,bc,bd,cf,ch,df;q)_\infty}\nonumber\\[6pt] &&\quad \times{}_{3}\phi_2\left[\begin{array}{ccc} g,&ac,&cf\\ &cgh,&acdf \end{array};q,dh\right], \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|,|f|,|h|\}<1$ and $cd\rho\neq0$. \end{thm}
\begin{pf} As in the proof of the extension of the Askey-Wilson integral, we can show that the Cauchy operator also commutes with the Aksey-Roy integral. So we may apply the Cauchy operator $T(f,g;D_q)$ to both sides of the Askey-Roy integral \eqref{Askey-Roy} with respect to the parameter $a$. It follows that \begin{eqnarray}\label{askey} \lefteqn{\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(\rho e^{i\theta}/d,qde^{-i\theta}/\rho,\rho ce^{-i\theta},qe^{i\theta}/c\rho,fge^{i\theta};q)_\infty} {(ae^{i\theta},be^{i\theta},ce^{-i\theta},de^{-i\theta},ge^{i\theta};q)_\infty} {d}\theta}\nonumber\\[6pt] &=&\frac{(abcd,cfg,\rho c/d,dq/\rho c,\rho,q/\rho;q)_\infty}{(q,ac,ad,bc,bd,cg;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} f,&ac,&bc\\ &cfg,&abcd \end{array};q,dg\right], \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|,|g|\}<1$ and $cd\rho\neq0$. \end{pf}
Putting $f= abcd$ and $g= f$ in \eqref{askey}, by the $q$-Gauss sum \eqref{gauss}, we get the formula \eqref{Gasper} due to Gasper.
In order to apply the Cauchy operator to Gasper's formula \eqref{Gasper}, we rewrite it as \begin{eqnarray}\label{Gasperch} &&\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(\rho e^{i\theta}/d,qde^{-i\theta}/\rho,\rho ce^{-i\theta},qe^{i\theta}/c\rho;q)_\infty} {(be^{i\theta},fe^{i\theta},ce^{-i\theta},de^{-i\theta};q)_\infty} \frac{(abcdfe^{i\theta};q)_\infty}{(ae^{i\theta},abcd;q)_\infty} {d}\theta \nonumber\\[6pt] &&\quad=\frac{(\rho c/d,dq/\rho c,\rho,q/\rho,bcdf;q)_\infty}{(q,bc,bd,cf,df;q)_\infty} \frac{(acdf;q)_\infty}{(ac,ad;q)_\infty}. \end{eqnarray} The proof is thus completed by employing the operator $T(g,h;D_q)$ with respect to the parameter $a$ to the above identity.\qed
Replacing $a$, $g$ by $g$, $cdfg$, respectively, and then taking $h=a$ in \eqref{Roy}, we are led to the following identity due to
Zhang and Wang \cite{Zhang-Wang}.
\begin{cor} We have \begin{eqnarray} &&\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(\rho e^{i\theta}/d, qde^{-i\theta}/\rho, \rho ce^{-i\theta}, qe^{i\theta}/c\rho, abcdfge^{i\theta}, bcdfge^{i\theta};q)_{\infty}}{(ae^{i\theta}, be^{i\theta}, fe^{i\theta}, ge^{i\theta}, ce^{-i\theta}, de^{-i\theta};q)_{\infty}} \nonumber \\[6pt] &&\quad\quad\quad\quad\times {}_{3}\phi_2\left[\begin{array}{ccc} fe^{i\theta},&ge^{i\theta},&gcdf\\ &acdfge^{i\theta},&bcdfge^{i\theta} \end{array};q,abcd\right]{d}\theta \nonumber \\[6pt] &&\quad\quad\quad=\frac{(\rho c/d, dq/\rho c, \rho, q/\rho, acdf, acdg, bcdf, bcdg, cdfg;q)_{\infty}}{(q, ac, ad, bc, bd, cf, df, cg, dg;q)_{\infty}}, \end{eqnarray}
where $\max\{|a|,|b|,|c|,|d|,|f|,|g|\}<1$ and $cd\rho\neq0$. \end{cor}
\section{The Bivariate Rogers-Szeg\"o Polynomials}\label{se Bivariate Rogers-Szego Polynomials}
In this section, we show that Mehler's formula and the Rogers formula for the bivariate Rogers-Szeg\"o polynomials can be easily derived from the application of the Cauchy operator. The bivariate Rogers-Szeg\"{o} polynomials are closely related to the continuous big $q$-Hermite polynomials. However, it seems that the following form of the bivariate Rogers-Szeg\"o polynomials are
introduced by Chen, Fu and Zhang \cite{ChenFuZhang}, as defined by \begin{equation}
h_n(x,y|q)=\sum_{k=0}^{n}{n \brack k}P_k(x,y), \end{equation} where the Cauchy polynomials are given by \[ P_k(x,y)=x^k(y/x;q)_k=(x-y)(x-qy)\cdots (x-q^{n-1}y),\]
which naturally arise in the $q$-umbral calculus. Setting $y=0$, the polynomials $h_n(x,y|q)$ reduce to the classical Rogers-Szeg\"{o}
polynomials $h_n(x|q)$ defined by \begin{equation}
h_n(x|q)=\sum_{k=0}^{n}{n \brack k}x^k. \end{equation}
It should be noted that Mehler's formula for the bivariate Rogers-Szeg\"o polynomials is due to Askey, Rahman, and Suslov \cite[Eq. (14.14)]{Askey-Rahman-Suslov}. They obtained the nonsymmetric Poisson kernel formula for the continuous big
$q$-Hermite polynomials, often denoted by $H_n(x;a|q)$. The formula of Askey, Rahman, and Suslov can be easily formulated in terms of
$h_n(x,y|q)$. Recently, Chen, Saad, and Sun presented an approach to Mehler's formula and the Rogers formula for $h_n(x,y|q)$ by using the homogeneous difference operator $D_{xy}$ introduced by Chen, Fu, and Zhang. As will be seen, the Cauchy operator turns out to be more efficient
compared with the techniques used in \cite{Chen-Saad-Sun}.
We recall that the generating function of the bivariate Rogers-Szeg\"{o} polynomials \begin{equation}\label{gf}
\sum_{n=0}^{\infty}h_n(x,y|q)\frac{t^n}{(q;q)_n}=\frac{(yt;q)_\infty} {(t,xt;q)_\infty}, \end{equation}
where $\max\{|x|,|xt|<1\}$, can be derived from the Euler identity \eqref{Euler}
using the Cauchy operator.
A direct calculation shows that \begin{eqnarray}\label{DK2} D_q^k\left\{a^n\right\} & = & \left\{\begin{array}{ll} a^{n-k}(q^{n-k+1};q)_k,& 0\leq k\leq n,\\[6pt] 0, &k > n.\end{array}\right. \end{eqnarray}
From the identity \eqref{DK2}, we can
easily establish the following lemma.
\begin{lem} \label{m-l} We have \begin{equation} T(a,b;D_q)\left\{c^n\right\}=\sum_{k=0}^{n}{n\brack k}(a;q)_kb^kc^{n-k}. \end{equation} \end{lem}
Applying $T(a,b;D_q)$ to the Euler identity \eqref{Euler} with respect to the parameter $z$, we get \begin{equation} \sum_{n=0}^{\infty}\frac{z^n}{(q;q)_n}\sum_{k=0}^{n}{n\brack k}(a;q)_k\left(\frac{b}{z}\right)^k=\frac{(ab;q)_\infty} {(b,z;q)_\infty},\label{rogers-szego} \end{equation} which leads to \eqref{gf} by suitable substitutions.
The reason that we employ the Cauchy operator to deal with the bivariate Rogers-Szeg\"o polynomials is based on the following fact \begin{equation}
h_n(x,y|q)=\lim_{c \rightarrow 1}T(y/x,x;D_q)\left\{c^n\right\}. \end{equation}
We are ready to describe how one can employ the Cauchy operator to derive Mehler's formula and the Rogers formula for $h_n(x,y|q)$.
\begin{thm}[Mehler's formula for $h_n(x,y|q)$] We have \begin{equation}\label{gemehler}
\sum_{n=0}^{\infty}h_n(x,y|q)h_n(u,v|q) \frac{t^n}{(q;q)_n}=\frac{(ty,tv;q)_\infty}{(t,tu,tx;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} t,&y/x,&v/u\\ &ty,&tv \end{array};q,tux\right], \end{equation}
where $\max\{|t|,|tu|,|tx|,|tux|\}<1$. \end{thm}
\begin{pf} By Lemma \ref{m-l}, the left hand side of (\ref{gemehler}) can be written as \begin{eqnarray*}
\lefteqn{ \sum_{n=0}^{\infty}h_n(x,y|q)\lim_{c \rightarrow 1}T(v/u,u;D_q)\left\{c^n\right\} \frac{t^n}{(q;q)_n}\nonumber}\\[6pt] & = &
\lim_{c \rightarrow 1}T(v/u,u;D_q)\left\{
\sum_{n=0}^{\infty}h_n(x,y|q)\frac{(ct)^n}{(q;q)_n}\right\}\nonumber. \end{eqnarray*} In view of the generating function (\ref{gf}), the above sum equals \begin{eqnarray} \lefteqn{\lim_{c \rightarrow 1}T(v/u,u;D_q)\left\{\frac{(cty;q)_\infty} {(ct,ctx;q)_\infty}\right\}\nonumber} \\[6pt] &=&\lim_{c \rightarrow 1}\left(\frac{(tv,cty;q)_\infty}{(tu,ct,ctx;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} v/u,&ct,&y/x\\ &tv,&cty \end{array};q,tux\right]\right)\nonumber\\[6pt] &=&\frac{(ty,tv;q)_\infty}{(t,tu,tx;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} t,&y/x,&v/u\\ &ty,&tv \end{array};q,tux\right], \end{eqnarray}
where $\max\{|t|,|tu|,|xt|,|tux|\}<1$. This completes the proof. \qed \end{pf}
We see that \eqref{gemehler} is equivalent to \cite[Eq. (2.1)]{Chen-Saad-Sun} in terms of Sears' transformation formula \eqref{hall}. Setting $y=0$ and $v=0$ in \eqref{gemehler} and employing the Cauchy $q$-binomial theorem \eqref{Cauchy}, we obtain Mehler's formula \cite{Chen1, IsmailStanton, Rogers1893, stanton2000} for the Rogers-Szeg\"{o} polynomials.
\begin{cor}We have \begin{equation}\label{mehler}
\sum_{n=0}^{\infty}h_n(x|q)h_n(u|q) \frac{t^n}{(q;q)_n}=\frac{(t^2ux;q)_\infty}{(t,tu,tx,tux;q)_\infty}, \end{equation}
where $\max\{|t|,|tu|,|tx|,|tux|\}<1$. \end{cor}
\begin{thm}[The Rogers formula for $h_n(x,y|q)$]We have \begin{equation}
\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}h_{m+n}(x,y|q) \frac{t^n}{(q;q)_n}\frac{s^m}{(q;q)_m}=\frac{(ty;q)_\infty}{(s,t,tx;q)_\infty} {}_{2}\phi_1\left[\begin{array}{cc} t,&y/x\\ &ty \end{array};q,sx\right],\label{homo2} \end{equation}
where $\max\{|s|,|t|,|sx|,|tx|\}<1$. \end{thm}
\begin{pf} Using Lemma \ref{m-l}, the left hand side of \eqref{homo2} equals \begin{eqnarray} \lefteqn{\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \lim_{c \rightarrow 1}T(y/x,x;D_q)\left\{c^{m+n}\right\}\frac{t^n}{(q;q)_n}\frac{s^m} {(q;q)_m}}\nonumber\\[6pt] &=&\lim_{c \rightarrow 1}T(y/x,x;D_q)\left\{\sum_{n=0}^{\infty}\frac{(ct)^n}{(q;q)_n} \sum_{m=0}^{\infty}\frac{(cs)^m}{(q;q)_m}\right\} \end{eqnarray} \begin{eqnarray} &=&\lim_{c \rightarrow 1}T(y/x,x;D_q)\left\{\frac{1}{(cs,ct;q)_\infty}\right\} \qquad\qquad\qquad\nonumber\\[6pt] &=&\frac{(ty;q)_\infty}{(s,t,tx;q)_\infty} {}_{2}\phi_1\left[\begin{array}{cc} t,&y/x\\ &ty \end{array};q,sx\right], \end{eqnarray}
where $\max\{|s|,|t|,|sx|,|tx|\}<1$.\qed \end{pf}
Note that \eqref{homo2} is equivalent to \cite[Eq. (3.1)]{Chen-Saad-Sun} in terms of Heine's transformation formula \cite[Appendix III.1]{Gasper-Rahman}. Setting $y=0$ in \eqref{homo2}, by the Cauchy $q$-binomial theorem \eqref{Cauchy} we get the Rogers formula \cite{Chen1, Rogers1893, Rogers1} for the Rogers-Szeg\"{o} polynomials.
\begin{cor} We have \begin{equation}\label{Rogers}
\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}h_{m+n}(x|q) \frac{t^n}{(q;q)_n}\frac{s^m}{(q;q)_m}=\frac{(stx;q)_\infty} {(s,sx,t,tx;q)_\infty}, \end{equation}
where $\max\{|s|,|t|,|sx|,|tx|\}<1$. \end{cor}
\section{An Extension of Sears' Formula}\label{se Sears' Formula}
In this section, we give an extension of the Sears two-term summation formula \cite[Eq. (2.10.18)]{Gasper-Rahman}: \begin{eqnarray}\label{sears} \lefteqn{\int_{c}^{d}\frac{(qt/c,qt/d,abcdet;q)_\infty} {(at,bt,et;q)_\infty}{d}_qt\nonumber}\\ &=&\frac{d(1-q)(q,dq/c,c/d,abcd,bcde,acde;q)_\infty} {(ac,ad,bc,bd,ce,de;q)_\infty}, \end{eqnarray}
where $\max\{|ce|,|de|\}<1$.
From the Cauchy operator, we deduce the following extension of (\ref{sears}).
\begin{thm} We have \begin{eqnarray}\label{tSears} \lefteqn{\int_{c}^{d}\frac{(qt/c,qt/d,abcdet,fgt;q)_\infty} {(at,bt,et,gt;q)_\infty} {}_{3}\phi_2\left[\begin{array}{ccc} f,&at,&et\\ &fgt,&abcdet \end{array};q,bcdg\right] {d}_qt}\nonumber\\[6pt] &=&\frac{d(1-q)(q,dq/c,c/d,abcd,bcde,acde,cfg;q)_\infty} {(ac,ad,bc,bd,ce,cg,de;q)_\infty}\nonumber\\[6pt] &&\times{}_{3}\phi_2\left[\begin{array}{ccc} f,&ac,&ce\\ &cfg,&acde \end{array};q,dg\right], \end{eqnarray}
where $\max\{|bcdg|,|ce|,|cg|,|de|,|dg|\}<1$. \end{thm}
\begin{pf} We may rewrite \eqref{sears} as \begin{eqnarray} \lefteqn{\int_{c}^{d}\frac{(qt/c,qt/d;q)_\infty} {(bt,et;q)_\infty}\frac{(abcdet;q)_\infty}{(at,abcd;q)_\infty} \text{d}_qt}\nonumber\\[6pt] &=&\frac{d(1-q)(q,dq/c,c/d,bcde;q)_\infty} {(bc,bd,ce,de;q)_\infty}\frac{(acde;q)_\infty}{(ac,ad;q)_\infty}. \end{eqnarray} Applying the operator $T(f,g;D_q)$ with respect to the parameter $a$, we obtain \eqref{tSears}.\qed \end{pf}
As far as the convergence is concerned, the above integral is of the following form \begin{equation}\label{abn} \sum_{n=0}^{\infty}A(n)\sum_{k=0}^{\infty}B(n,k). \end{equation} To ensure that the series (\ref{abn}) converges absolutely, we assume that the following two conditions are satisfied: \begin{enumerate} \item $\sum_{k=0}^{\infty}B(n,k)$ converges to $C(n)$, and $C(n)$
has a nonzero limit as $n\rightarrow \infty$. \item $\lim\limits_{n \rightarrow
\infty}|\frac{A(n)}{A(n-1)}|<1$. \end{enumerate}
It is easy to see that under the above assumptions, (\ref{abn}) converges absolutely, since \[ \lim_{n \rightarrow
\infty}\left|\frac{A(n)C(n)}{A(n-1)C(n-1)}\right| = \lim_{n
\rightarrow \infty}\left|\frac{A(n)}{A(n-1)}\right|<1 . \]
It is easy to verify the double summations in \eqref{tSears} satisfy the two assumptions of \eqref{abn}, so the convergence is guaranteed.
\section{Extensions of $q$-Barnes' Lemmas}\label{se q-Barnes' Lemmas}
In this section, we obtain extensions of the $q$-analogues of Barnes' lemmas. Barnes' first lemma \cite{Barne1} is an integral analogue of Gauss'
${}_2F_1$ summation formula.
Askey and Roy \cite{Askey-Roy} pointed out that Barnes' first lemma is also an extension of the beta integral. Meanwhile, Barnes' second lemma \cite{Barne2} is an integral analogue of Saalsch\"{u}tz's formula.
The following
$q$-analogue of Barnes' first lemma is due to Watson, see \cite[Eq. (4.4.3)]{Gasper-Rahman}: \begin{eqnarray}\label{barnes1} &&\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{(q^{1-c+s},q^{1-d+s};q)_\infty}{(q^{a+s},q^{b+s};q)_\infty} \frac{\pi q^s {d}s}{\sin \pi(c-s) \sin\pi(d-s)} \nonumber \\[6pt] &&\quad\quad= \frac{q^c}{\sin \pi(c-d)} \frac{(q,q^{1+c-d},q^{d-c},q^{a+b+c+d};q)_{\infty}} {(q^{a+c},q^{a+d},q^{b+c},q^{b+d};q)_{\infty}}. \end{eqnarray} The $q$-analogue of Barnes' second lemma is due to Agarwal, see \cite{Agarwal} and \cite[Eq. (4.4.6)]{Gasper-Rahman}: \begin{eqnarray}\label{barnes2} &&\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{(q^{1+s},q^{d+s},q^{1+a+b+c+s-d};q)_\infty} {(q^{a+s},q^{b+s},q^{c+s};q)_\infty}\ \frac{\pi q^s {d}s}{\sin \pi s \sin\pi(d+s)} \nonumber \\[6pt] &&\quad\quad=\csc \pi d\ \frac{(q,q^d,q^{1-d},q^{1+b+c-d},q^{1+a+c-d},q^{1+a+b-d};q)_{\infty}} {(q^a,q^b,q^c,q^{1+a-d}, q^{1+b-d},q^{1+c-d};q)_{\infty}}, \end{eqnarray} where $\text{Re}\{s\log q-\log(\sin\pi s\sin\pi(d+s))\}<0$ for large
$|s|$. Throughout this section, the contour of integration always ranges from $-i\infty$ to $i\infty$ so that the increasing sequences of poles of integrand lie to the right and the decreasing sequences of poles lie to the left of the contour, see \cite[p. 119]{Gasper-Rahman}. In order to ensure that the Cauchy operator commutes with the integral, we assume that $q=e^{-\omega},\ \omega>0$.
We obtain the following extension of Watson's $q$-analogue of Barnes' first lemma.
\begin{thm} We have \begin{eqnarray}\label{barnes1-1} \lefteqn{\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{(q^{1-c+s},q^{1-d+s},q^{e+f+s};q)_\infty} {(q^{a+s},q^{b+s},q^{f+s};q)_\infty} \frac{\pi q^s {d}s}{\sin \pi(c-s) \sin\pi(d-s)}} \nonumber \\[6pt] &=&\frac{q^c}{\sin \pi(c-d)} \frac{(q,q^{1+c-d},q^{d-c},q^{a+b+c+d},q^{c+e+f};q)_{\infty}} {(q^{a+c},q^{a+d},q^{b+c},q^{b+d},q^{c+f};q)_{\infty}}\nonumber\\[6pt] &&\times{}_{3}\phi_2\left[\begin{array}{ccc} q^e,&q^{a+c},&q^{b+c}\\ &q^{c+e+f},&q^{a+b+c+d} \end{array};q,q^{d+f}\right], \end{eqnarray}
where $\max\{|q^f|,|q^{c+f}|,|q^{d+f}|\}<1$. \end{thm} \begin{pf} Applying the operator $T(q^e,q^f;D_q)$ to \eqref{barnes1} with respect to the parameter $q^a$, we arrive at \eqref{barnes1-1}. \qed \end{pf}
Let us consider the special case when $e=a+b+c+d$.
The ${}_{3}\phi_2$ sum on the right hand side of \eqref{barnes1-1} turns out to be a ${}_{2}\phi_1$ sum and can be summed by the $q$-Gauss formula \eqref{gauss}. Hence we get the following formula derived by Liu \cite{Liu}, which is also an extension of $q$-Barnes' first Lemma.
\begin{cor}\label{barnes-cor1} We have \begin{eqnarray}\label{barnes1-3} \lefteqn{\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{(q^{1-c+s},q^{1-d+s},q^{a+b+c+d+f+s};q)_\infty} {(q^{a+s},q^{b+s},q^{f+s};q)_\infty} \frac{\pi q^s {d}s}{\sin \pi(c-s) \sin\pi(d-s)}} \nonumber \\[6pt] &=&\frac{q^c}{\sin \pi(c-d)} \frac{(q,q^{1+c-d},q^{d-c},q^{a+b+c+d},q^{a+c+d+f},q^{b+c+d+f};q)_{\infty}} {(q^{a+c},q^{a+d},q^{b+c},q^{b+d},q^{c+f},q^{d+f};q)_{\infty}}, \end{eqnarray}
where $\max\{|q^f|,|q^{c+f}|,|q^{d+f}|\}<1$. \end{cor}
Clearly, \eqref{barnes1-3} becomes $q$-Barnes' first Lemma \eqref{barnes1} for $f\rightarrow \infty$. Based on Corollary \ref{barnes-cor1}, employing the Cauchy
operator again, we derive the following further extension of $q$-Barnes' first Lemma.
\begin{thm} We have \begin{eqnarray}\label{barnes1-4} \lefteqn{\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{(q^{1-c+s},q^{1-d+s},q^{a+b+c+d+f+s},q^{e+g+s};q)_\infty} {(q^{a+s},q^{b+s},q^{f+s},q^{g+s};q)_\infty} \frac{\pi q^s }{\sin \pi(c-s) \sin\pi(d-s)}} \nonumber \\[6pt] &&\quad\times{}_{3}\phi_2\left[\begin{array}{ccc} q^e,&q^{a+s},&q^{b+s}\\ &q^{e+g+s},&q^{a+b+c+d+f+s} \end{array};q,q^{c+d+f+g}\right]{d}s\nonumber\\[6pt] &=&\frac{q^c}{\sin \pi(c-d)} \frac{(q,q^{1+c-d},q^{d-c},q^{a+b+c+d},q^{a+c+d+f}, q^{b+c+d+f},q^{c+e+g};q)_{\infty}} {(q^{a+c},q^{a+d},q^{b+c},q^{b+d},q^{c+f}, q^{c+g},q^{d+f};q)_{\infty}}\nonumber\\[6pt] &&\quad\quad\times{}_{3}\phi_2\left[\begin{array}{ccc} q^e,&q^{a+c},&q^{b+c}\\ &q^{c+e+g},&q^{a+b+c+d} \end{array};q,q^{d+g}\right], \end{eqnarray} where
$\max\{|q^f|,|q^g|,|q^{c+f}|,|q^{c+g}|,|q^{d+f}|,|q^{d+g}|,|q^{c+d+f+g}|\}<1$. \end{thm}
We conclude this paper with the following extension of
Agarwal's $q$-analogue of Barnes' second lemma. The proof
is omitted.
\begin{thm} We have \begin{eqnarray}\label{barnes2-1} \lefteqn{\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{(q^{1+s},q^{d+s},q^{1+a+b+c+s-d},q^{e+f+s};q)_\infty} {(q^{a+s},q^{b+s},q^{c+s},q^{f+s};q)_\infty}\ \frac{\pi q^s }{\sin \pi s \sin\pi(d+s)} }\nonumber \\[6pt] &&\times{}_{3}\phi_2\left[\begin{array}{ccc} q^e,&q^{a+s},&q^{b+s}\\ &q^{e+f+s},&q^{1+a+b+c+s-d} \end{array};q,q^{1+c+f-d}\right]{d}s\nonumber\\[6pt] &=&\csc \pi d\ \frac{(q,q^d,q^{1-d},q^{1+b+c-d},q^{1+a+c-d},q^{1+a+b-d},q^{e+f};q)_{\infty}} {(q^a,q^b,q^c,q^{f},q^{1+a-d}, q^{1+b-d},q^{1+c-d};q)_{\infty}}\nonumber\\[6pt] &&\times{}_{3}\phi_2\left[\begin{array}{ccc} q^a,&q^{b},&q^{e}\\ &q^{e+f},&q^{1+a+b-d} \end{array};q,q^{1+f-d}\right], \end{eqnarray}
where $\max\{|q^f|,|q^{1+f-d}|,|q^{1+c+f-d}|\}<1$ and $\text{Re}\{s\log q-\log(\sin\pi s\sin\pi(d+s))\}<0$ for large
$|s|$. \end{thm}
\noindent{\bf Acknowledgments.} We would like to thank the referee and Lisa H. Sun for helpful comments. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the Ministry of Science and Technology and the National Science Foundation of China.
\end{document} | arXiv |
\begin{definition}[Definition:Perfect Square Dissection]
A '''perfect square dissection''' is a dissection of an integer square into a number of smaller integer squares all of different sizes.
\end{definition} | ProofWiki |
\begin{document}
\title {Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems
}
\author[1]{Yoshiyuki Sekiguchi\thanks{2-1-6, Etchujima, Koto, Tokyo 135-8533, JAPAN. [email protected]}} \author[2]{Hayato Waki\thanks{744 Motooka, Nishi-ku, Fukuoka 819-0395, JAPAN. [email protected]}} \affil[1]{Tokyo University of Marine Science and Technology} \affil[2]{Institute of Mathematics for Industry, Kyushu University}
\date{}
\maketitle
\begin{abstract} We consider sensitivity of a semidefinite program under
perturbations
in the case that the primal problem is strictly feasible and the dual problem is weakly feasible.
When the coefficient matrices are perturbed,
the optimal values can change discontinuously
as explained in concrete examples.
We show that the
optimal value of such a semidefinite program changes continuously under conditions involving the behavior of the minimal faces of the perturbed dual problems.
In addition, we determine what kinds of perturbations keep the minimal
faces invariant, by using the reducing certificates, which are produced in facial reduction. Our results allow us to classify the behavior of the minimal face of a semidefinite program obtained from a control problem. \end{abstract}
\begin{keywords} Semidefinite programming, sensitivity, facial reduction,
minimal face, H-infinity feedback control problem \end{keywords} \begin{classification} 90C31, 90C22, 90c51, 93D15 \end{classification}
\section{Introduction} \label{section:intro}
A \textit{semidefinite program} is the problem of maximizing a linear function subject to the constraint that an affine combination of matrices is positive semidefinite, where the constraint is called a linear matrix inequality.
Semidefinite programs have various applications, such as discrete optimization, polynomial optimization and control problems (e.g. \cite{Anjos12,Scherer06} ). If the feasible sets of a semidefinite program and the dual problem satisfy the constraint qualifications, both of which are called \textit{strict feasibility}, then interior point methods compute an approximation to an exact solution efficiently; see, e.g., \cite{deKlerk02,Tuncel10}.
With a lack of strict feasibility, interior point methods are numerically unstable and often give wrong optimal values [5-7].
To avoid such numerical instability, we can use the technique called \textit{facial reduction}, which finds the minimal face among the faces of the positive semidefinite cone containing a feasible set. Such a face is called the \textit{minimal face of a semidefinite program} [8-11].
Then, we obtain a semidefinite program that satisfies strict feasibility and has the same optimal value as the original problem. For applications of facial reduction, see the monograph \cite{Drusvyatskiy17} and the references therein.
The first contribution of this paper is to provide sufficient conditions for continuity of the optimal value under perturbations, in the case that the primal problem is strictly feasible and the dual problem is feasible but not strictly feasible (Theorem \ref{thm:main2}). In that case, if we perturb the constant matrix in the constraint of the primal problem, then it can be shown from the general theory of convex analysis that the optimal value changes continuously \cite[Corollary 7.5.1]{Rockafellar70}. For more detailed analysis, see \cite{Cheung14}. However if we also perturb the coefficient matrices of the variables, then the optimal value may change discontinuously (Example \ref{ex:notrank}, \ref{ex:notface}). Here one of the keys to the phenomenon is the behavior of the minimal face of the dual problem under the perturbation. By using concrete examples, we argue that our sufficient conditions are hard to remove.
In the case that both of the primal and dual problems are strictly feasible, continuity of the optimal value can be shown by Gol'{\v s}hte{\u \i}n \cite[Theorem 17]{Golshtein72}. Moreover if perturbations are restricted on the constant matrices in the constraint of a semidefinite program, several authors have studied stability of optimal solutions; see, e.g., [16-18].
Perturbation analysis of general nonlinear programming has been studied thoroughly by Bonnans and Shapiro \cite{Bonnans00}.
The second contribution is to obtain sufficient conditions for the perturbations to keep the minimal face invariant (Proposition \ref{prop:invariant1}, \ref{prop:invariant3}). If the minimal face does not change under a perturbation, then one of the conditions in Theorem \ref{thm:main2} is satisfied. These results give a new insight to perturbation analysis of semidefinite programs. Here we use \textit{reducing certificates}, which are generated by facial reduction to find the minimal face \cite{Pataki13}. We remark that reducing certificates are often obtained without solving semidefinite programs if the problems are generated from combinatorial optimization problems, matrix completion problems, sums of squares problems \cite{Drusvyatskiy17} or $H_\infty$ control problems \cite{Waki15}. Using these conditions, we investigate a semidefinite program generated from an $H_\infty$ state feedback control problem.
The organization of this paper is as follows: preliminaries on semidefinite programs and facial reduction are given in Section \ref{preliminary}. In Section \ref{sec:main}, we show the main result on continuity of the optimal value of a semidefinite program. In Section \ref{section:face_invariant}, we give sufficient conditions on the perturbations under which the minimal face does not change. We devote Section \ref{sec:example} to applications to a control problem and numerical experiements. The conclusions are given in Section \ref{sec:conclusion}.
\section{Preliminaries on
Semidefinite Program and Facial Reduction}\label{preliminary}
\subsection{\textbf{Semidefinite Program}}
Let $\mathbb{S}^n$, $\mathbb{S}^n_+$ and $\mathbb{S}^n_{++}$ be the sets of $n\times n$ symmetric matrices, positive semidefinite matrices and positive definite matrices, respectively. In this paper, the primal semidefinite program (SDP) \eqref{P} and its dual \eqref{D} are formulated as follows: \begin{align} \label{P} &\quad \sup_{y, Z}\left\{ b^Ty : A_0 - \sum_{k=1}^m y_k A_k =Z, \ y\in \Bbb{R}^m,\ Z \in\Bbb{S}^n_+\right\}, \tag{$P$}\\
\label{D} &\quad\inf_{X}\left\{A_0 \bullet X : A_k \bullet X = b_k \ (k \in [m]), X \in\Bbb{S}^n_+\right\}, \tag{$D$} \end{align} where $A_0, A_1, \ldots, A_m\in\mathbb{S}^n$, $b\in\Bbb{R}^m$, $[m] := \{1, \ldots, m\}$, and the inner product $A\bullet B$ is defined by $\sum_{i, j=1}^n A_{ij}B_{ij}$ for $A, B\in\mathbb{S}^n$.
Problem \eqref{P} is said to be {\itshape strictly feasible} if there exists a feasible solution $(y, Z)$ in \eqref{P} such that $Z\in\mathbb{S}^n_{++}$. Problem \eqref{D} is said to be {\itshape strictly feasible} if there exists a feasible solution $X$ in \eqref{D} such that $X\in\mathbb{S}^n_{++}$. We say that \eqref{P} (resp. \eqref{D}) is {\itshape weakly feasible}, if \eqref{P} (resp. \eqref{D}) is feasible but not strictly feasible.
Throughout this paper, we deal with only the case where both \eqref{P} and its dual \eqref{D} are feasible. We say that \eqref{P} is {\itshape nonsingular} if both \eqref{P} and \eqref{D} are strictly feasible and the coefficient matrices $A_1, \ldots, A_m$ are linearly independent. We say that \eqref{P} is {\itshape singular} if the coefficient matrices are linear dependent or at least one of \eqref{P} and \eqref{D} is weakly feasible.
\subsection{\textbf{Facial Reduction for SDP}} The definition of a face of a general convex set is provided in \cite{Rockafellar70}. The following lemma provides results on a facial structure of $\Bbb{S}^n_+$, e.g. \cite{Borwein81b,Pataki13}.
\begin{lemma}\label{facest} \begin{enumerate} \item\label{F1} Any face of $\Bbb{S}^n_+$ is either the
empty set, $\{O_{n\times n}\}$,
$\Bbb{S}^n_+$, or \begin{align*} &\left\{ Q\begin{pmatrix} O_{(n-r)\times (n-r)} & O_{(n-r)\times r}\\ O_{r\times (n-r)} & M \end{pmatrix}Q^T : M\in\Bbb{S}^r_+ \right\}, \end{align*} where $Q$ is an $n\times n$ nonsingular matrix.
\item\label{F2} The set $\Bbb{S}^n_+ + F^{\perp}$ is closed for all faces $F$ of $\Bbb{S}^n_+$, where $F^{\perp}$ stands
for the set $\{Z\in\Bbb{S}^n : Z\bullet X = 0 \ (\forall X\in
F)\}$. \end{enumerate} \end{lemma} We call $Q$ in Part \ref{F1} of Lemma \ref{facest} the \textit{nonsingular matrix associated to the face}.
It follows from this property that for any $U\in\mathbb{S}^n_+$, the set $\Bbb{S}^n_+\cap\{U\}^{\perp}$ is a face of $\Bbb{S}^n_+$,
where
$\{U\}^{\perp} = \{ X\in\Bbb{S}^n : X\bullet U = 0\}$. The property given in Part \ref{F2} of Lemma \ref{facest}, which is called niceness,
implies that $F^* = \Bbb{S}^n_+ + F^{\perp}$ for all faces $F$ of
$\Bbb{S}^n_+$. Here $F^*$ is the dual cone of $F$, i.e. $F^*=\{Z\in\Bbb{S}^n : Z\bullet X \ge 0 \ (\forall X\in F)\}$.
We define the minimal face of \eqref{D} and introduce facial reduction for \eqref{D}. The minimal face of \eqref{D} is defined as the intersection of all faces of $\mathbb{S}^n_{+}$ that contain the feasible region of \eqref{D}. We denote the minimal face by $F_{\min}$. The following result on the minimal face is obtained by \cite{Pataki13} and Part \ref{F2} in Lemma \ref{facest}.
\begin{lemma}
\cite[\textrm{SDP version of Section 28.2.6 and Lemma 28.4}]{Pataki13}\label{freduction} Assume that \eqref{P} and \eqref{D} are feasible. Let $F$ be a face
of $\Bbb{S}^n_+$ that contains $F_{\min}$ and $\mathop\mathrm{rint} F$ be its
relative interior. Then
the following are equivalent; \begin{enumerate}
\item $F\neq F_{\min};$
\item
\label{lemma:freduction_cond}
There exists $(y, U, V)\in\Bbb{R}^m\times \Bbb{S}^n_+\times
F^{\perp}$ such that
\begin{align}
\label{certificate} b^Ty &= 0,\quad -\sum_{k\in [m]}y_kA_k = U+V \mbox{ and } U+V \not\in F^{\perp};
\end{align}
\item $\displaystyle \left\{X\in\mathop\mathrm{rint} F : A_k\bullet X = b_k \
(k\in [m]) \right\}=\emptyset.$ \end{enumerate}
If $U$ satisfies the system in \ref{lemma:freduction_cond},
then we have $F_{\min}\subseteq F\cap \{U\}^{\perp}\subsetneq F$.
\end{lemma} We call the above system \eqref{certificate} the \textit{discriminant system} of the facial reduction for \eqref{D}, and a solution $(y, U, V)$ a \textit{reducing certificate}.
The facial reduction for SDP in e.g. \cite{Pataki13,Waki13} is a procedure based on Lemma \ref{freduction}.
It generates a sequence
$\{F_i\}_{i=0}^s$ of faces of $\Bbb{S}^n_+$
such that
\begin{align*}
F_0 = \Bbb{S}^n_+,\ F_{i} &= F_{i-1}\cap\{U^i\}^\perp \ (i=1, \ldots, s) \mbox{ and } F_s =F_{\min}.
\end{align*} Therefore, the iterative process can be expressed as
\begin{align*}
\Bbb{S}_+^n &= F_0 \overset{(y^1, U^1, V^1)}{\longrightarrow} F_1
\overset{(y^2, U^2, V^2)}{\longrightarrow} F_2
\overset{(y^3, U^3, V^3)}{\longrightarrow}
\cdots
\overset{(y^s, U^s, V^s)}{\longrightarrow} F_s=F_{\min},
\end{align*}
where we call $\{(y^i, U^i, V^i)\}_{i=1}^s$ a \textit{facial reduction sequence} for \eqref{D}.
Here we note that $U^i,V^i$ need to satisfy $U^i+V^i \notin F_{i-1}^\perp$.
Examples of facial reduction for SDP can be seen in e.g. \cite[Example 28.3]{Pataki13} and \cite[Example 3.1]{Waki13}.
If the discriminant system \eqref{certificate} has multiple solutions, then we have flexibility in choosing a facial reduction sequence for \eqref{D}. Cheung and Wolkowicz \cite[Proposition B.1]{Cheung14} prove that any two facial reduction sequences must be of the same length
when a reducing certificate $(y,U,V)$ is selected at each iteration so that $U$ has the maximal rank. The length is called the degree of singularity for \eqref{D}.
The degree of singularity is used in \cite{Cheung14} for the sensitivity analysis of SDPs and in \cite{Sturm00b} for the error bounds.
Although we deal with only the feasible SDPs in the present paper, we introduce a study on the infeasibility briefly. Infeasibility of SDP has two types as well as feasibility, i.e. strong infeasibility and weak infeasibility. The authors in \cite{Liu18,Lourenco16} discuss a characterization of infeasibility by facial reduction.
\section{Main Result}\label{sec:main} \subsection{\textbf{Stability of Singular Semidefinite Programs}}\label{SlaterFailsD} We define the perturbed problems for \eqref{P} by \begin{align} \label{Pt} & \sup_{y, Z}\left\{ b(t)^Ty : \sum_{k \in [m]} y_k A_k(t) + Z = A_0(t),\
y\in \Bbb{R}^m,\ Z \in\Bbb{S}^n_+\right\}, \tag{$P_t$}\\ \label{Dt} &\inf_{X}\left\{A_0(t) \bullet X : A_k(t) \bullet X = b_k(t) \ (k \in [m]),\ X \in\Bbb{S}^n_+\right\}, \tag{\text{$D_t$}}
\end{align} where $t\ge 0$, $A_k(t)\in \Bbb{S}^n,\ b(t)\in \Bbb{R}^m$ are continuous at $t=0$, and $A_k(0) = A_k,\ b(0) = b$.
In this subsection, the following conditions are imposed on the initial SDP:
\begin{condition}
\label{cond:singular}
\hspace{1ex}
\begin{enumerate}[label=(C\arabic*)]
\item\label{C1} \eqref{D} is feasible, and \eqref{P} is strictly feasible;
\item\label{C2} $A_1, \ldots, A_m$ are linearly independent.
\end{enumerate}
\end{condition}
Then, by applying the facial reduction to \eqref{D}, there exist a nonsingular matrix $Q$ and $r\in \Bbb{N}$ such that {\small \begin{align}
&\inf_{X_3}\left\{Q^TA_0Q \bullet \begin{pmatrix}
O & O \\
O & X_3
\end{pmatrix} :
\begin{array}{l}
Q^TA_kQ \bullet
\begin{pmatrix}
O & O \\
O & X_3
\end{pmatrix} = b_k \ (k \in [m]),\
X_3 \in \Bbb{S}^r_+
\end{array}
\right\} \label{FD}\tag{\text{$F(D)_0$}} \end{align} }
has the same optimal value as \eqref{D}, and \ref{FD} is strictly feasible due to Lemma \ref{freduction}.
Here, for $n\times n$ matrix $M$, we denote by $M_3$ the right bottom
block of the partitioning \begin{equation}
M = \begin{pmatrix}
M_1 & M_2^T \\
M_2 & M_3
\end{pmatrix},
\label{eq:partition} \end{equation} where the partitioning is uniquely determined by Lemma \ref{facest} for the minimal face of \eqref{D} with $M_1 \in \Bbb{S}^{n-r}, M_2 \in \Bbb{R}^{r\times (n-r)}, M_3\in \Bbb{S}^r$.
We call $M_3$ the \textit{third block} of $M$ associated to the minimal face of \eqref{D}. Then we can rewrite \ref{FD} as follows: \begin{align} \label{FD2}
&\inf_{X}\left\{ (Q^TA_0Q)_3 \bullet X :
(Q^TA_kQ)_3 \bullet X = b_k \ (k\in [m]), X \in\Bbb{S}^r_+
\right\}. \tag{$F(D)$}
\end{align} For $A=(a_{ij})_{1\le i, j\le n}\in\Bbb{S}^n$, we define $\mathop\mathrm{vec}(A)$ as the vectorization of $A$, i.e., \[ \mathop\mathrm{vec}(A) = (a_{11}, a_{12}, \ldots, a_{1n}, a_{21}, a_{22}, \ldots, a_{n1}, \ldots, a_{nn})^T. \] Let $r(A_1, \ldots, A_m)$ be the rank of the matrix $(\mathop\mathrm{vec}(A_1), \ldots, \mathop\mathrm{vec}(A_m))$.
The following theorem is the main result of this paper.
\begin{theorem}
\label{thm:main2}
Under Condition \ref{cond:singular}, suppose that the minimal face $F_{\min}$ of \eqref{D} can be written as
\[
F_{\min} = \left\{
Q
\begin{pmatrix}
O_{(n-r)\times(n-r)} & O_{(n-r)\times r} \\
O_{r\times (n-r)} & X
\end{pmatrix}Q^T : X \in\Bbb{S}^r_+\right\}
\]
for some nonsingular matrix $Q\in\Bbb{R}^{n\times n}$ and $r\in\Bbb{N}$.
In addition, we suppose that the set $\{(A_0(t), \ldots, A_m(t), b(t)) : 0 \leq t \leq \delta\}$ satisfies the following assumptions for some $\delta>0$: \begin{enumerate}
\item\label{thm:sing:cond1} \eqref{Dt} is feasible for each $t\in [0,\delta]$;
\item\label{thm:sing:cond2} For each $t\in [0,\delta]$,
there exists a nonsingular matrix $Q(t)$ such that
$\displaystyle\lim_{t\to 0}Q(t) = Q$, and the minimal face of \eqref{Dt} can
be written as
\[
\left\{
Q(t)
\begin{pmatrix}
O_{(n-r)\times(n-r)} & O_{(n-r)\times r} \\
O_{r\times (n-r)} & X
\end{pmatrix}Q(t)^T : X \in\Bbb{S}^r_+\right\};
\]
\item\label{thm:sing:cond3} For each $t\in [0,\delta]$, we have \[
r \left((Q(t)^TA_1(t)Q(t))_3, \ldots,
(Q(t)^TA_m(t)Q(t))_3 \right) \\
= r \left( (Q^TA_1Q)_3, \ldots, (Q^TA_mQ)_3
\right), \] where $M_3$ is the third block of $M\in \Bbb{S}^n$ associated with
the minimal face of \eqref{D}. \end{enumerate} Then the optimal value of \eqref{Dt} varies continuously at $t = 0$.
\end{theorem}
The following is an immediate corollary.
\begin{corollary}
\label{cor:cor1}
Under Condition \ref{cond:singular}, suppose that there exists $\delta>0$ such that
\eqref{Dt} has a nonempty feasible set
and the same minimal face as \eqref{D}, and
\[ r \left( (Q^TA_1(t)Q)_3, \ldots,
(Q^TA_m(t)Q)_3 \right)
= r \left( (Q^TA_1Q)_3, \ldots, (Q^TA_mQ)_3
\right)
\]
for $t\in [0,\delta]$. Then the optimal value of \eqref{Dt} varies continuously at $t = 0$.
\end{corollary}
Before proceeding to the proof, we investigate examples and show that the rank condition or the condition on the face can not be removed from Theorem \ref{thm:main2} and Corollary \ref{cor:cor1}.
\begin{example}
\label{ex:notrank}
The following example satisfies the condition on the face
but does not satisfy the rank condition.
We set $b = (0, 2, 2)^T$ and \[
A_0 = \left(
\begin{smallmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1
\end{smallmatrix}
\right),\
A_1 = \left(\begin{smallmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{smallmatrix}\right),\
A_2 = \left(\begin{smallmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{smallmatrix}
\right),
A_3 = \left(
\begin{smallmatrix}
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{smallmatrix}
\right) \]
in \eqref{P} and \eqref{D}. Then $A_1, A_2, A_3$ are linearly
independent, $(P)$ is strictly feasible, and $(D)$ is weakly
feasible. The optimal value is $0$
and an optimal pair is
$X = \left(\begin{smallmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 0
\end{smallmatrix}\right),\ y = (0, 0, 0),\
Z = \left(\begin{smallmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{smallmatrix}\right)$. The minimal face of \eqref{D} is \[F_{\min} =\left\{\left(\begin{smallmatrix}
O_{2\times 2} & O_{2\times 2} \\
O_{2\times 2} & X_3
\end{smallmatrix}\right)\in \Bbb{S}^4_+:
X_3 \in \Bbb{S}^2_+ \right\}.\]
If we perturb the matrices as \[
A_i(t) = A_i\ (i=0,1, 2), A_3(t) =
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 1 + t & 0\\
0 & 0 & 0 & 1 - t
\end{smallmatrix}\right), \]
then \eqref{Dt} remains feasible for each $t>0$.
In fact the feasible points of \eqref{Dt} can be
written as $X = \left(\begin{smallmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & \alpha \\
0 & 0 & \alpha & 1
\end{smallmatrix}\right)\
(-1\leq\alpha\leq 1)$.
Thus the minimal face of \eqref{Dt} is equal to $F_{\min}$ for
each $t>0$.
Now the dimension of the span of the third blocks of
the matrices $A_1, A_2, A_3$ is $1$, while that of $A_1(t),
A_2(t), A_3(t)$ is $2$ for each $t>0$.
The optimal value of \eqref{Dt} is $1$
and the optimal pairs are
$X = \left(\begin{smallmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & \alpha \\
0 & 0 & \alpha& 1
\end{smallmatrix}\right)\
(-1\leq\alpha\leq 1),\
y = \left(\beta, \frac{1+t}{2t}, -\frac{1}{2t}\right),\
Z = \left(\begin{smallmatrix}
-\beta & \frac{1}{2t} & 0 & 0 \\
\frac{1}{2t} & -\beta & 0 & 0 \\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{smallmatrix}\right)\ \left(\beta\leq -\frac{1}{2t}\right)$ for each $t>0$.
Thus the optimal value changes discontinuously at $t=0$.
\end{example}
\begin{example}
\label{ex:notface}
The following example satisfies the rank condition but
does not satisfy the condition on the face.
We set $b = (2, 2, 2, 0)^T$ and \[
A_0 = \left(
\begin{smallmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1
\end{smallmatrix}
\right),
A_1 = \left(
\begin{smallmatrix}
0 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{smallmatrix}
\right),
A_2 = \left(\begin{smallmatrix}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{smallmatrix}
\right),
A_3 = \left(
\begin{smallmatrix}
1 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{smallmatrix}
\right),
A_4 = \left(
\begin{smallmatrix}
0 & 0 & 1\\
0 & 0 & 0\\
1 & 0 & 0
\end{smallmatrix}
\right) \]
in \eqref{P} and \eqref{D}.
Then $A_1, \ldots, A_4$ are linearly independent, \eqref{P} is strictly
feasible, and
\eqref{D} is weakly feasible.
The optimal value is $\frac{1}{2}$, and
the optimal pairs are
$X=\left(
\begin{smallmatrix}
0 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & \frac{1}{2}
\end{smallmatrix}\right),\
y=\left(-\frac{1}{4}, \frac{1}{2}-y_3,y_3,0\right),
Z=\left(
\begin{smallmatrix}
-y_3 & 0 & 0\\
0 & \frac{1}{4} & -\frac{1}{2} \\
0 & -\frac{1}{2} & 1
\end{smallmatrix}\right),\ y_3\leq0$.
The minimal face of \eqref{D} is \[F_{\min} =\left\{\left(\begin{smallmatrix}
0 & O_{1\times 2} \\
O_{2\times 1} & X_3
\end{smallmatrix}\right)\in \Bbb{S}^3_+:
X_3 \in \Bbb{S}^2_+ \right\}.\]
If we perturb the matrices as
\begin{align*}
& A_i(t) = A_i\ (i=0, 1,2), A_3(t) =
\left(\begin{smallmatrix}
1 & 0 & 0\\
0 & 0 & 1-t^2\\
0 & 1-t^2 & 0
\end{smallmatrix}\right),
A_4(t) =
\left(\begin{smallmatrix}
0 & 0 & 1\\
0 & -2t & 0\\
1 & 0 & 0
\end{smallmatrix}\right),
\end{align*}
then \eqref{Dt} is strictly feasible for each $t>0$.
In fact,
$X=\left(
\begin{smallmatrix}
2t^2 & 0 & 2t\\
0 & 2 & 1\\
2t & 1 & 3
\end{smallmatrix}\right)$ are strict feasible points
of \eqref{Dt}.
Thus the minimal face of \eqref{Dt} is $\mathbb{S}^3_+$ for each $t>0$.
Since the span of the third blocks of the matrices $A_1(t),\ldots, A_4(t)$ has the same basis
as that of $A_1,\ldots, A_4 $ for each $t>0$, the rank condition is satisfied.
However the optimal value of \eqref{Dt} is $2$ with $X=\left(
\begin{smallmatrix}
2t^2 & t & 2t\\
t & 2 & 1\\
2t & 1 & 2
\end{smallmatrix}\right),\
y=\left(2, \frac{1}{t^2} - 1,-\frac{1}{t^2},\frac{1}{t}\right),
Z=\left(
\begin{smallmatrix}
\frac{1}{t^2} & 0 & -\frac{1}{t}\\
0 & 0 & 0\\
-\frac{1}{t} & 0 & 1
\end{smallmatrix}\right) $
being the unique optimal pair for each $t>0$. Thus the optimal value changes discontinuously at $t=0$.
\end{example}
\begin{example}\label{ex:theorem}
Consider the same SDP as in Example \ref{ex:notface}.
If we perturb the matrices as \[
A_i(t) = A_i\ (i=0, 1,4), A_2(t) =
\left(\begin{smallmatrix}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & t
\end{smallmatrix}\right),\
A_3(t) = \left(\begin{smallmatrix}
1 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & t
\end{smallmatrix}\right), \]
the minimal face of each perturbed problem is equal to
$F_{\min}$ in Example \ref{ex:notface}.
Here the condition on the face and the rank condition are satisfied for
sufficiently small $t>0$. Thus Theorem \ref{thm:main2} guarantees the continuity of
the optimal value.
In fact, the optimal value of \eqref{Dt} is $\frac{ 2t+4 - 4\,\sqrt{t+1}}{t^2}$
and converges to $\frac{1}{2}$ as $t\to 0$.
The optimal pairs are {\small
\begin{align*}
& X = \left(\begin{smallmatrix}
0&0&0\cr 0&2&{{2\,\sqrt{t+1}-2}\over{t}}\cr 0&{{2\,\sqrt{t
+1}-2}\over{t}}&{{ 2t+4 - 4\,\sqrt{t+1}}\over{t^2}}\cr
\end{smallmatrix}\right),\
y = \textstyle{\left(-\frac{\sqrt{t+1}\,\left(t+2\right)-2\,t-2}{t^3+t^2},
\beta, -\frac{\sqrt{t+1}+\beta\,t^2+\left(
\beta-1\right)\,t-1}{t^2+t}, 0
\right)},\\ & Z = \left(\begin{smallmatrix}
{{\sqrt{t+1}+ \beta\,t^2+\left(\beta-1
\right)\,t-1}\over{t^2+t}}&0&0\cr 0&{{\sqrt{t+1}\,\left(t+2\right)-2
\,t-2}\over{t^3+t^2}}&{{\sqrt{t+1}-t-1}\over{t^2+t}}\cr 0&{{\sqrt{t+
1}-t-1}\over{t^2+t}}&{{1}\over{\sqrt{t+1}}}\cr
\end{smallmatrix}\right) \end{align*} }
for all $\beta$ such that $(1,1)$st element of $Z$ is nonnegative.
\end{example}
\begin{proof}[Proof of Theorem \ref{thm:main2}.]
By the assumptions \ref{thm:sing:cond1} and \ref{thm:sing:cond2} in Theorem \ref{thm:main2},
the optimal value of \eqref{Dt} is equal to \[
\inf_{X}\left\{ (Q(t)^TA_0(t)Q(t))_3 \bullet X : \right.\\
\left. (Q(t)^TA_k(t)Q(t))_3 \bullet X = b_k(t)\ (k\in [m]),
X \in\Bbb{S}^r_+
\right\}\label{FDt}\tag{\text{$F(D_t)$}}, \]
and \ref{FDt} has a nonempty feasible set for each $t\in [0,\delta]$.
Thus if continuity of
the optimal value of \ref{FDt} at $t = 0$ is shown,
then that of the optimal value of \eqref{Dt} is also shown.
For each $t\in [0,\delta]$, we have that the dual of \ref{FDt} is
\begin{align}
& \sup_{y, Z}\left\{b(t)^Ty :
\displaystyle\sum_{k\in [m]} y_k(Q(t)^TA_k(t)Q(t))_3 + Z = (Q(t)^TA_0Q(t))_3,
Z\in\Bbb{S}^r_+
\right\}. \label{FDtd}\tag{\text{$F(D_t)^{\prime}$}} \end{align}
Then \ref{FDt} has the same optimal value as \ref{FDtd}
because \ref{FDt} and \ref{FDtd} are
strictly feasible.
In fact, strict feasibility of \ref{FDt} follows from the properties of facial reduction.
Since, for a strictly feasible point $(\tilde{y}, \tilde{Z})$ of \eqref{Pt}, $(\tilde{y}, (Q(t)^T\tilde{Z}Q(t))_3)$ is also a strictly feasible point of \ref{FDtd}, and hence \ref{FDtd} is strictly feasible.
Therefore, the proof is done by showing Theorem \ref{thm:contopt_rank}. \hspace{\fill} \qed
\end{proof}
\begin{theorem}
\label{thm:contopt_rank}
If both \eqref{P} and \eqref{D} are strictly feasible,
\eqref{Dt} is feasible, and
$r\left(A_1(t),\ldots, A_m(t)\right)
= r \left( A_1,\ldots, A_m \right)$ for each sufficiently
small $t>0$,
then the optimal value of \eqref{Dt} varies continuously at $t=0$.
\end{theorem}
We will prove Theorem \ref{thm:contopt_rank} in Subsection
\ref{subsection:Stab}.
\begin{remark}
The coefficient matrices $A_1 ,\ldots, A_m$ in \eqref{P} are usually assumed to be linearly independent in the literature.
However the coefficient matrices in
\ref{FD2}
can be linearly dependent even if the initial SDP has linearly independent constraints. In fact, the coefficient matrices of the reduced SDPs are linearly dependent in Examples \ref{ex:notrank}, \ref{ex:notface} and \ref{ex:theorem}. Thus we need to consider SDPs with linearly dependent coefficient matrices in Theorem \ref{thm:contopt_rank}.
\end{remark}
As in Example \ref{ex:notrank} and \ref{ex:notface},
if $r\left(A_1(t),\ldots, A_m(t)\right)=r \left( A_1,\ldots, A_m \right)$ and \eqref{D} is weakly feasible,
then the optimal value of \eqref{Dt} can vary
discontinuously.
We present an additional example and show that
the feasibility condition on \eqref{Dt} or the rank condition
can not be removed from Theorem \ref{thm:contopt_rank}.
\begin{example}
\label{ex:perturb_infeasible}
In \eqref{P} and \eqref{D}, we set $b=(2, 2)^T$,
\[
A_0 = \begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix},\
A_1 = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},
A_2 = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}.
\]
Then \eqref{P} and \eqref{D} are strictly feasible.
The optimal value is $0$, and the optimal pairs are $X = \left(\begin{smallmatrix}
2 & 0 \\
0 & 0
\end{smallmatrix}\right),\
y = \left(\alpha, -\alpha\right),\
Z = \left(\begin{smallmatrix}
0 & 0 \\
0 & 1
\end{smallmatrix}\right)$ for any $\alpha \in \Bbb{R}$.
However, if we take
$A_2(t) = \left(\begin{smallmatrix}
1 + t & 0\\
0 & 1 + t
\end{smallmatrix}\right)$,
then $r(A_1(t), A_2(t)) = r(A_1, A_2)
= 1$ but
\eqref{Dt} is infeasible.
Therefore feasibility of \eqref{Dt} can not be derived from the rank
condition and needs to be assumed.
On the other hand, if we take
$A_2(t) = \left(\begin{smallmatrix}
1 + t & 0 \\
0 & 1 - t
\end{smallmatrix}\right)$, then
\eqref{Dt} is feasible and
$r(A_1(t), A_2(t)) = 2$ for all $t>0$.
The optimal value is $1$, and the optimal pair
is $X = \left(\begin{smallmatrix}
1 & \beta \\
\beta & 1
\end{smallmatrix}\right)\ (-1\leq \beta \leq 1),\
y = \left(\frac{1+t}{2t},-\frac{1}{2t}\right),\
Z = \left(\begin{smallmatrix}
0 & 0 \\
0 & 0
\end{smallmatrix}\right)
$. Thus the optimal value varies discontinuously at $t=0$ without the rank condition.
\end{example}
\subsection{\textbf{Proof of Theorem \ref{thm:contopt_rank}}} \label{subsection:Stab}
First, we recall an existence theorem for optimal solutions to an SDP with a focus on the linear independence of the coefficient matrices.
\begin{theorem}
\cite[Theorem 4.1 and Corollary 4.1]{Todd01}
\label{thm:Todd01}
Suppose \eqref{P} is strictly feasible and \eqref{D} is feasible.
Then \eqref{D} has a nonempty compact optimal set
and the same optimal value as \eqref{P}. Also, suppose that \eqref{P} is feasible and \eqref{D} is strictly
feasible. If the coefficient matrices $A_1, \ldots, A_m$
are linearly independent, then \eqref{P} has a nonempty compact optimal set and the same optimal value as \eqref{D}.
\end{theorem}
\begin{remark}
\label{remark:Todd01}
\begin{enumerate}
\item Suppose that
\eqref{P} is feasible and \eqref{D} is strictly feasible.
However, we do not assume that the coefficient matrices $A_1,\ldots, A_m$
are linearly independent.
Then easy arguments show that \eqref{P} has a nonempty optimal set and the same optimal value as \eqref{D}.
Here we lost the compactness of the optimal set of \eqref{P}.
\item The set of the optimal
solutions $(y, Z)$ of \eqref{P} is unbounded
when the matrices $A_1, \ldots, A_m$ are linearly dependent.
However Lemma
\ref{lemma:unif_bdd} bellow tells that the image of the optimal solutions under the projection $(y,Z)\mapsto Z$ is bounded if \eqref{P} and \eqref{D} are strictly feasible.
\end{enumerate}
\end{remark}
We note that we do not assume the linear independence of the coefficient matrices $A_1,\ldots, A_m$ in the following arguments.
We will use the symbol $S(t) =(\mathop\mathrm{vec}\left(A_1(t)\right), \ldots, \mathop\mathrm{vec}\left(A_m(t)\right))\in\Bbb{R}^{n^2\times m}$ and the symbol $(S(t)^T)^\dagger$ for the Moor-Penrose generalized inverse of $S(t)^T$ \cite{HornJohnson13}.
\begin{lemma}
\label{thm:contP}
Suppose $X_0$ is a strictly feasible point of \eqref{D}.
If \eqref{Dt} is feasible and\\
$r\left(A_1(t),\ldots, A_m(t)\right)
= r\left(A_1,\ldots, A_m\right)$ for each $t\in [0,\delta]$,
then there exist strictly feasible points $X_t$ of
\eqref{Dt} for all sufficiently small $t>0$
such that $X_t \to X_0$ as $t \to 0$.
\end{lemma}
\begin{proof}
We can write the equality constraints of \eqref{D} and \eqref{Dt} by $S(0)^T\mathop\mathrm{vec}(X) = b$ and $S(t)^T\mathop\mathrm{vec}(X) = b(t)$, respectively.
Note that $A_k(0) = A_k\ (k\in [m]),\ b = b(0)$. We set
\begin{align*} \mathop\mathrm{vec}(X_0) &= (I - (S(0)^T)^\dagger S(0)^T)\mathop\mathrm{vec}(X_0) + (S(0)^T)^\dagger b(0) \mbox{ and } \\\mathop\mathrm{vec}(X_t) &=
(I - (S(t)^T)^\dagger
S(t)^T)\mathop\mathrm{vec}(X_0)
+ (S(t)^T)^\dagger b(t). \end{align*} Then we can check that $S(0)^T\mathop\mathrm{vec}(X_0) = b$ and $S(t)^T\mathop\mathrm{vec}(X_t) = b(t)$, by using the fact that $S(t)^T(S(t)^T)^\dagger v = v$ if and only if $v\in \mathop\mathrm{Im} S(t)^T$. Since we have $\mbox{rank}(S(t)) = r (A_1(t), \ldots, A_m(t))$ for all $t\ge 0$, it follows from the assumption on the rank and \cite[Theorem
5.2]{Stewart69} that $(S(t)^T)^\dagger \to (S(0)^T)^\dagger$ as $t\to 0$. Therefore $X_t\to X_0$ as $t\to 0$. \hspace{\fill} \qed
\end{proof}
\begin{remark}
\label{remark:s_feasible_P}
Unlike \eqref{Dt},
we can easily prove that \eqref{Pt} have strictly feasible points $(y_t,
Z_t)$ for all sufficiently small $t\geq0$ without assuming the rank condition.
If \eqref{P} is strictly feasible, there exists $y_0\in \Bbb{R}^m$ such that $A_0 - \sum_k y_{0,k} A_k \in \Bbb{S}_{++}^n$. Then we have that $Z_t:=A_0(t) - \sum_k y_{0,k} A_k(t)\in\Bbb{S}_{++}^n$ for all sufficiently small $t\geq0$. For each $t>0$, $(y_0, Z_t)$ is a strictly feasible point of \eqref{Pt} and converges to a strict feasible point of \eqref{P}.
\end{remark}
Let $\mathcal{U}(t)$ be the set of optimal solutions of \eqref{Dt}, and \[
\mathcal{V}(t) = \{Z\in \Bbb{S}^n : (y,Z) \text{ is optimal to \eqref{Pt}}
\ \text{for some $y\in \Bbb{R}^m$}\}. \]
\begin{lemma}
\label{lemma:unif_bdd}
Suppose that \eqref{P} is strictly feasible.
If there exist strictly feasible points $X_t$ of \eqref{Dt}
for all sufficiently small
$t\geq0$ such that $X_t \to X_0$ as $t \to 0$,
then both sets $\mathcal{U}(t)$ and $\mathcal{V}(t)$ are nonempty and uniformly bounded;
i.e., there exist $\delta>0$ and compact sets $C_1, C_2$ such that
\[
\mathcal{U}(t) \subset C_1,\ \mathcal{V}(t) \subset C_2\quad (0 \leq t \leq \delta).
\]
\end{lemma}
\begin{proof}
Since \eqref{Dt} and \eqref{Pt} have strictly feasible points, Remark
\ref{remark:Todd01} ensures that they have the same optimal
value and that $\mathcal{U}(t)$ and $\mathcal{V}(t)$ are nonempty for all sufficiently small $t\ge 0$.
For a strictly feasible point $(y_0, Z_0)$ of \eqref{P},
we set
$ y_t = y_0$ and $Z_t = A_0(t) - \sum_ky_{0,k}A_k(t)$.
Then $(y_t, Z_t)$ is a strictly feasible point of
\eqref{Pt} for each small $t\geq 0$ as explained in Remark \ref{remark:s_feasible_P}.
Let $X$ and $(y, Z)$ be arbitrary optimal solutions to \eqref{Dt} and
\eqref{Pt} respectively.
Since $X_t$ and $(y_t, Z_t)$ are feasible points, we have
\[
A_k(t) \bullet (X - X_t) = 0,\ \sum_{k \in [m]} (y_k - y_{t,k}) A_k(t)
+ Z - Z_t = 0.
\]
Then it follows that
$(X - X_t) \bullet (Z - Z_t) = 0$
and hence that
$X \bullet Z_t + X_t \bullet Z = X_t \bullet Z_t$.
Moreover, positive semidefiniteness of $X_t$ and $Z$ guarantees that
$X \bullet Z_t \leq X_t \bullet Z_t$.
Thus, by positive definiteness
of $Z_t$, there exists $\epsilon>0$
such that for all sufficiently small $t>0$, we have
\[
\|X\| \leq \frac{X_t \bullet Z_t}{\lambda_{\min}(Z_t)}
< \frac{X_0 \bullet Z_0 + \epsilon}{\lambda_{\min}(Z_0) - \epsilon},
\]
where $\lambda_{\min}(M)$ is the smallest eigenvalue of a
matrix $M$.
Therefore, $\mathcal{U}(t)$ is uniformly bounded for all sufficiently small $t>0$.
Similar arguments are applied to $\mathcal{V}(t)$. \hspace{\fill} \qed
\end{proof}
The following lemma is well-known, and the proof is omitted.
\begin{lemma}
\label{lemma:minimax}
Suppose that \eqref{D} has the same optimal value as \eqref{P} and that both
of \eqref{D} and \eqref{P} have optimal solutions. We define the function $L :
\Bbb{S}^n\times\Bbb{R}^m \to \Bbb{R}$ as follows:
\[
L(X, y) = A_0 \bullet X + \sum_{k\in [m]} y_k(b_k - A_k
\bullet X).
\]
Then $\widetilde{X}$ and $(\tilde{y}, A_0 - \sum_k \tilde{y}_kA_k)$ are optimal solutions of
\eqref{D} and \eqref{P} respectively if and only if $(\widetilde{X}, \tilde{y})\in\Bbb{S}^n_+\times \Bbb{R}^m$ satisfies
\[
L(\widetilde{X}, y) \leq L(\widetilde{X}, \tilde{y}) \leq L(X, \tilde{y}),\ \forall (X,
y) \in \Bbb{S}^n_+ \times \Bbb{R}^m.
\]
\end{lemma}
\begin{lemma}
\label{lemma:pinverse}
Let $S$ be a matrix $\left(\mathop\mathrm{vec}(A_1)\ \ldots\ \mathop\mathrm{vec}(A_m)\right) \in \Bbb{R}^{n^2\times m}$.
If $(\tilde{y},\widetilde{Z})$ is an optimal solution to \eqref{P},
then $(y_*, \widetilde{Z})$ is also an optimal solution to \eqref{P}, where $y_* = S^\dagger (\mathop\mathrm{vec}(A_0) - \mathop\mathrm{vec}(\widetilde{Z}))$.
\end{lemma}
\begin{proof}
By feasibility of $(\tilde{y}, \widetilde{Z})$, we have $S \tilde{y} = \mathop\mathrm{vec}(A_0) - \mathop\mathrm{vec}(\widetilde{Z})$.
Since $SS^\dagger v = v$ if and only if $v \in \mathop\mathrm{Im} S$,
we see that $Sy_* = \mathop\mathrm{vec}(A_0) - \mathop\mathrm{vec}(\widetilde{Z})$.
Then we obtain $y_* \in \tilde{y} + \ker S$.
Here we have $\ker S \subset (\mathop\mathrm{Span}\{b\})^\perp$
since otherwise the optimal value of \eqref{P} is infinity and hence this contradicts finiteness of the optimal value.
Thus $b^Ty_* = b^T\tilde{y}$, and therefore, $(y_*, \widetilde{Z})$ is optimal. \hspace{\fill} \qed
\end{proof}
Lemma \ref{thm:stability} plays an essential role in the proof of Theorem \ref{thm:contopt_rank}. Lemma \ref{lemma:unif_bdd} and \ref{thm:stability} ensure outer semicontinuity of the set-valued map $t\mapsto \mathcal{U}(t)\times \mathcal{V}(t)$; see \cite[Section 5.B]{RockWets98}. In the following, $\Bbb{B}$ denotes the closed unit ball in $\Bbb{S}^n$. We define, for $X\in \Bbb{S}^n$ and $C\subset \Bbb{S}^n$, \[
d(X,C) = \inf\{\|X - Y\|:Y \in C\}. \]
\begin{lemma}
\label{thm:stability}
Suppose that \eqref{P} is strictly feasible.
If there exist strictly feasible points $X_t$ of \eqref{Dt}
for all sufficiently small $t \geq 0$ such that $X_t \to X_0$ as $t\to 0$, then
for any $\epsilon>0$, there exists $\eta>0$ such that
\[
\mathcal{U}(t) \subset \mathcal{U}(0) + \epsilon \Bbb{B},\ \mathcal{V}(t) \subset \mathcal{V}(0) + \epsilon \Bbb{B}
\quad (0 \leq t \leq \eta).
\]
\end{lemma}
\begin{proof} By Remark \ref{remark:Todd01},
\eqref{Dt} and \eqref{Pt} have optimal solutions and the
same optimal value. Suppose that the conclusion is false.
Then there exist $\epsilon>0$, $\{t_j\}$, $X(t_j)\in
\mathcal{U}(t_j)$ and $Z(t_j)\in \mathcal{V}(t_j)$ such that
$t_j \to 0$ and \begin{equation}
d\left(X(t_j), \mathcal{U}(0)\right) \geq \epsilon,\ d\left(Z(t_j), \mathcal{V}(0)\right) \geq
\epsilon,
\label{eq:contra} \end{equation}
for all $j$. Recall that
$S(t)$ denotes the matrix $\left(
\mathop\mathrm{vec}(A_1(t)) \cdots \mathop\mathrm{vec}(A_m(t))
\right)$. Let
$y(t_j) = S(t_j)^\dagger (\mathop\mathrm{vec}(A_0) - \mathop\mathrm{vec}(Z(t_j)))$.
Then, Lemma \ref{lemma:pinverse} implies that
the feasible solution $(y(t_j), Z(t_j))$ is optimal for $(P_{t_j})$ for each $j$.
We define
\[
L(X, y,t) = A_0(t) \bullet X + \sum_{k\in [m]} y_k(b_k(t) - A_k(t)
\bullet X).
\]
Then, we note that $L(X, y, 0)$ is equal to $L(X, y)$ defined in Lemma \ref{lemma:minimax}.
By Lemma \ref{lemma:minimax},
we have
\[
L(X(t_j), y, t_j) \leq L(X(t_j), y(t_j), t_j) \leq L(X, y(t_j), t_j),\ \forall (X,y) \in \Bbb{S}_+^n \times \Bbb{R}^m.
\]
Since Lemma \ref{lemma:unif_bdd} ensures that $\{(X(t_j), Z(t_j))\}$ is uniformly bounded,
we may assume that
\[
(X(t_j), y(t_j), Z(t_j)) \to (\widetilde{X}, \tilde{y}, \widetilde{Z})
\]
as $j \to \infty$ for some $(\widetilde{X}, \tilde{y}, \widetilde{Z})$.
Thus we have \[
L(\widetilde{X}, y, 0) \leq L(\widetilde{X},\tilde{y}, 0) \leq L(X, \tilde{y}, 0),\ \forall (X,y) \in \Bbb{S}_+^n \times \Bbb{R}^m. \] By applying Lemma \ref{lemma:minimax} again, $\widetilde{X}$ and $(\tilde{y}, \widetilde{Z})$ are
optimal for \eqref{P} and \eqref{D} respectively. This contradicts the inequalities \eqref{eq:contra}. \hspace{\fill} \qed
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:contopt_rank}.]
By Lemma \ref{thm:contP} and \ref{thm:stability},
we have that for any $\epsilon>0$
and $X(t) \in \mathcal{U}(t)$, there
exist $\eta>0$ and $\widetilde{X}^t \in \mathcal{U}(0)$ such
that for $t\in [0, \eta]$,
\[
|A_0(t) \bullet X(t) - A_0 \bullet \widetilde{X}^t|
\leq k_1\|X(t) - \widetilde{X}^t\| + k_2\|A_0(t) - A_0(0)\|< \epsilon
\]
for some $k_1, k_2>0$. This completes the proof of Theorem \ref{thm:contopt_rank}. \hspace{\fill} \qed
\end{proof}
\begin{corollary}
\label{thm:contopt}
If both \eqref{P} and \eqref{D} are strictly feasible and
$A_1,\ldots, A_m$ are linearly independent, then the optimal value of \eqref{Dt} varies
continuously at $t=0$.
\end{corollary} \begin{proof}
By strict feasibility and the linear independence condition, for all sufficiently small $t>0$,
\eqref{Pt} and \eqref{Dt} are feasible,
and the rank condition is satisfied.
\hspace*{\fill} \qed \end{proof}
\section{Behavior of a Minimal Face under Perturbations} \label{section:face_invariant}
In this section, the behavior of a minimal face under perturbations is investigated. In particular, we give criteria for perturbations to keep the minimal face invariant.
We slightly simplify the situations and consider the following perturbed problem: \begin{align}\label{Dt4} &\inf_X\left\{ A_0 \bullet X : (A_k + E_k(t)) \bullet X = b_k \ (k\in [m]), X \in\Bbb{S}^n_+\right\}, \tag{\text{$D_t$}} \end{align} where $E_k(t) = A_k(t) - A_k$ for all $k\in [m]$. We note $E_k(t) \to 0$ as $t\to 0$ since we assume that $A_k(t)$ are continuous at $t=0$ and $A_k(0) = A_k$. Throughout this section, we assume the following conditions:
\begin{condition}
\label{cond:perturb}
\hspace{1ex}
\begin{enumerate}
\item\label{D1} \eqref{D} is feasible, and \eqref{P} is strictly feasible;
\item\label{D2} $A_1, \dots, A_m$ are linearly independent;
\item\label{D3} \eqref{Dt4} is feasible for each sufficiently small $t>0$.
\end{enumerate}
\end{condition}
We say that $\{\eqref{Dt4}\}_{t\geq 0}$ satisfies the \textit{rank condition} if there exist an associated nonsingular matrix $Q$ to the minimal face of
\eqref{D} and $\delta>0$ such that for all $t\in [0, \delta]$, \[
r \left( (Q^T(A_1 \!+\! E_1(t))Q)_3, \ldots,
(Q^T(A_m\!+\!E_m(t))Q)_3 \right) \\
= r \left( (Q^TA_1Q)_3, \ldots, (Q^TA_mQ)_3
\right), \] where the submatrix $M_3$ for $M\in \Bbb{S}^n$ is determined by the minimal face of \eqref{D} as in \eqref{eq:partition}. Here we note that $Q$ in the left hand side does not depend on $t$. We start with the following lemma.
\begin{lemma}
\label{lemma:rank_cond}
Let $F_{\min}$ and $F^t_{\min}$ be the minimal faces of \eqref{D} and
\eqref{Dt4} respectively.
Suppose $\{\eqref{Dt4}\}_{t\geq 0}$ satisfies the rank condition.
If there exists $\delta>0$ such that $F^t_{\min}\subset F_{\min}$ for all $t\in [0,\delta]$,
we have $F^t_{\min} = F_{\min}$ for all sufficiently small $t> 0$.
\end{lemma} \begin{proof}
By Lemma \ref{freduction},
the reduced problem \ref{FD2} of \eqref{D} has a strictly feasible
point which solves
$
\left(Q^T A_k Q\right)_3 \bullet X = b_k \ (k\in [m]), X\in\Bbb{S}^r_{++}
$ for some $r>0$,
where $Q$ is an associated nonsingular matrix to the minimal
face $F_{\min}$ of \eqref{D}.
For each $t \in [0, \delta]$, feasibility of \eqref{Dt4} and
$F^t_{\min}\subset F_{\min}$ imply that there exists $\tilde{X}\in F_{\min}$ such that $\left(A_k + E_k(t)\right) \bullet \tilde{X} = b_k \ (k\in [m])$. It follows from the representation of $F_{\min}$ with $Q$
that
\[
\left(Q^T \left(A_k + E_k(t)\right) Q\right)_3 \bullet X = b_k \
(k\in [m]),\
X\in\Bbb{S}^r_{+}
\]
is feasible. Consider the following problem obtained by perturbing \ref{FD2}:
\begin{align}\label{pertFD}
\inf_{X}\left\{\left(Q^TA_0Q\right)_3 \bullet X:
\left(Q^T \left(A_k + E_k(t)\right) Q\right)_3 \bullet X = b_k \
(k\in [m]),\
X\in\Bbb{S}^r_{+}
\right\}.
\end{align}
Here, \ref{FD2} has a strictly feasible point, \eqref{pertFD} is feasible, and the rank condition is satisfied. Thus Lemma \ref{thm:contP} implies
that for each sufficiently small $t>0$, \eqref{pertFD} has a strictly feasible point. It means that $\{X \in \mathop\mathrm{rint} F_{\min}: (A_k + E_k(t))\bullet X = b_k\ (k\in [m])\}\neq \emptyset$
for each sufficiently small $t>0$.
Since $F_{\min}$ is a face of $\Bbb{S}^n_+$
containing $F_{\min}^t$, we have $F_{\min}=F_{\min}^t$
by Lemma \ref{freduction}. \hspace{\fill} \qed \end{proof}
\begin{example}
Lemma \ref{lemma:rank_cond} does not hold without the assumption
$F^t_{\min} \subset F_{\min}$.
In Example \ref{ex:notface},
the perturbation is
of the same type as this section is considering.
Condition \ref{cond:perturb} and the rank condition are satisfied,
but the minimal faces $F^t_{\min}$ of \eqref{Dt4} are not equal to
$F_{\min}$.
Here $F^t_{\min}$ are not included in $F_{\min}$.
\end{example}
We first give simple sufficient conditions that can be shown easily.
\begin{proposition}
\label{prop:invariant1}
For a facial reduction sequence $\{(\hat{y}^i, \widehat{U}^i, \widehat{V}^i)\}_{i=1}^s$
of \eqref{D},
let the minimal face of \eqref{D} be $F_{\min}$
and $\hat{K} = \{k: \hat{y}_k^i = 0 \ (\forall i=1, \ldots, s)\}$.
Suppose that $\{\eqref{Dt4}\}_{t\geq 0}$ satisfies the rank
condition
and $E_k(t) = O_{n\times n} \ (k \notin \hat{K})$.
Then the minimal faces of \eqref{Dt4} are equal to $F_{\min}$
for all sufficiently small $t>0$.
\end{proposition}
\begin{proof}
Let $\{F_i\}_{i=1}^s$ be the sequence of faces generated by the facial reduction sequence $\{(\hat{y}^i, \widehat{U}^i, \widehat{V}^i)\}_{i=1}^s$
of \eqref{D}.
Since $E_k(t)=O_{n\times n} $ for all $k\not\in\hat{K}$, we have
\[
-\sum_{k\in [m]}\hat{y}^i_k (A_k + E_k(t))
= -\sum_{k\in [m]}\hat{y}^i_k A_k = \widehat{U}^i + \widehat{V}^i
\]
for $i = 1, \ldots, s$.
Thus $\{(\hat{y}^i, \widehat{U}^i, \widehat{V}^i)\}_{i=1}^s$ is a facial reduction sequence of \eqref{Dt4} up to the $s$-th iteration.
It is summarized as
\[ \eqref{Dt4}\quad \Bbb{S}_+^n \overset{(\hat{y}^1, \widehat{U}^1, \widehat{V}^1)}{\longrightarrow} F_1
\overset{(\hat{y}^2, \widehat{U}^2, \widehat{V}^2)}{\longrightarrow} F_2
\overset{(\hat{y}^3, \widehat{U}^3, \widehat{V}^3)}{\longrightarrow}
\cdots
\overset{(\hat{y}^s, \widehat{U}^s, \widehat{V}^s)}{\longrightarrow} F_s = F_{\min}. \]
Thus the minimal faces of \eqref{Dt4} are contained in $F_{\min}$.
In addition, since $\{\eqref{Dt4}\}_{t\geq 0}$ satisfies the rank condition,
it follows from Lemma \ref{lemma:rank_cond} that
the minimal faces of \eqref{Dt4} are equal to $F_{\min}$ for sufficiently small $t>0$. \hspace{\fill} \qed
\end{proof}
Next, we will use the positive eigenvectors of reducing certificates to give sufficient conditions for the minimal face to be invariant under a peturbation.
\begin{proposition}
\label{prop:invariant3}
Let $\{(\hat{y}^i, \widehat{U}^i, \widehat{V}^i)\}_{i=1}^s$ be a facial reduction sequence of
\eqref{D},
$F_0 = \Bbb{S}_+^n$ and $F_1,\ldots, F_s$ be the generated faces.
In addition, let
\[
L_i = \mathop\mathrm{Span}\{qq^T \colon q \text{ is an eigenvector of $\widehat{U}^i$} \\
\text{associated with a positive eigenvalue}\}.
\]
Suppose that $\{\eqref{Dt4}\}_{t\geq 0}$ satisfies the rank condition, and
for each $i = 1, \ldots, s$,
\[
\sum_{k\in [m]} \hat{y}_k^i E_k(t) + v^i(t)
\in
L_i
\]
for some $v^i(t) \in F_{i-1}^\perp$ with $v^i(t) \to O_{n\times n}$ as $t\to 0$.
Then \eqref{Dt4} have the same minimal face as \eqref{D} for
all sufficiently
small $t>0$.
\end{proposition}
Before proceeding to the proof, we present an example and a remark.
\begin{example}
\label{ex:invariant}
The SDP in Example $\ref{ex:notface}$ has
a facial reduction sequence consisting of
only one certificate
$(\hat{y}, \widehat{U}, \widehat{V})=\left((0,1,-1,0)^T,\left(\begin{smallmatrix}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{smallmatrix}\right),
O_{3\times 3}\right)$, and hence
$L_1 = \mathop\mathrm{Span}\{\widehat{U}\}$.
If we perturb the matrices as
{\small
\[
A_1(t) = \left(
\begin{smallmatrix}
3t & 4t & 5t\\
4t & 1 & 0\\
5t & 0 & t
\end{smallmatrix}
\right),
A_2(t) = \left(\begin{smallmatrix}
0 & 3t & 2t\\
3t & 0 & 1\\
2t & 1 & t
\end{smallmatrix}
\right),
A_3(t) = \left(
\begin{smallmatrix}
1+4t & 3t & 2t\\
3t & 0 & 1\\
2t & 1 & t
\end{smallmatrix}
\right),
A_4(t) = \left(
\begin{smallmatrix}
2t & 5t & 1 + 3t\\
5t & t & -t\\
1+3t & -t & 0
\end{smallmatrix}
\right),
\]}
then the corresponding $E_i(t)$ satisfy
$\sum_{k=1}^4 \hat{y}_k E_k(t) = -4t\widehat{U} \in L_1$.
Thus the conditions of Proposition $\ref{prop:invariant3}$ are satisfied, and
the minimal face is invariant under the perturbation
for sufficiently small $t>0$.
In fact, since the minimal face of \eqref{Dt4} is contained in $\Bbb{S}^n_+\cap \{\widehat{U}\}^\perp$ and
$\left(\begin{smallmatrix}0 & 0 & 0\\
0 & 2-t & 1-t/2\\
0 & 1-t/2 & 1\end{smallmatrix}\right)$
is a feasible point, the minimal face of \eqref{Dt4} is equal to $F_{\min}$ in Example \ref{ex:notrank}.
More generally, to apply Proposition \ref{prop:invariant3},
it suffices that we choose $E_i(t)$ such that \eqref{Dt4} are feasible, the rank condition holds, and $E_2(t) - E_3(t) = \alpha_t\widehat{U}$ for some $\alpha_t \in \Bbb{R}$.
\end{example}
\begin{remark}
In particular,
the inclusion in Proposition
\ref{prop:invariant3} holds if we have
\[
-\sum_{k\in [m]} \hat{y}_k^i E_k(t) \in
\alpha^i(t)\widehat{U}^i + F_{i-1}^\perp,
\]
with $\alpha^i(t)\to 0$ as $t \to 0$ for each $i=1, \ldots, s$.
\end{remark}
\begin{proof}[Proof of Proposition $\ref{prop:invariant3}$]
Since $\{(\hat{y}^i,\widehat{U}^i,\widehat{V}^i)\}_{i=1}^s$ is a facial reduction sequence of \eqref{D}, we have that $\widehat{U}^i\in \Bbb{S}_+^n$, $\widehat{V}^i
\in F_{i-1}^\perp$ and that
$-\sum_k\hat{y}^i_kA_k = \widehat{U}^i + \widehat{V}^i \notin F_{i-1}^\perp$ for each $i=1,\ldots, s$.
Let us fix $i$.
Let $\{q_l\}$ be the set of the eigenvectors of $\widehat{U}^i$
that are associated with positive eigenvalues, orthogonal to
each other, and $\|q_l\| = 1$. Then every matrix in $L_i$ can be written
as a linear combination of $q_l{q_l}^T$.
By the assumption,
there exist
$\alpha_{l}(t)\in \Bbb{R}$ and $v(t)\in F_{i-1}^\perp$
such that $-\sum_k \hat{y}_k^i E_k(t) = \sum_l\alpha_{l}(t)q_l{q_l}^T +
v(t)$ and $v(t)\to O_{n\times n}$.
Since $\sum_k \hat{y}_k^i E_k(t) \to O_{n
\times n}$
as $t\to 0$
and $\{q_lq_l^T\}$ is linearly independent,
we have $\alpha_{l}(t)\to 0$ for each $l$.
We set
\[
U^i = \widehat{U}^i + \sum_l\alpha_{l}(t)q_l{q_l}^T,\quad
V^i = \widehat{V}^i + v(t)
\]
Then $V^i \in F_{i-1}^\perp$.
Since $\widehat{U}^i$ can be written as $\sum_l\lambda_lq_l{q_l}^T$,
where $\lambda_l$ is the positive eigenvalue of $\widehat{U}^i$
corresponding to $q_l$,
we see that $U^i\in \Bbb{S}_+^n$ for all sufficiently small $t>0$.
Thus we have
\[
-\sum_k \hat{y}_k^i \left( A_k + E_k(t) \right) = \widehat{U}^i + \widehat{V}^i + \sum_\ell \alpha_\ell(t)q_\ell q_\ell^T + v(t) = U^i + V^i.
\]
Since $\widehat{U}^i + \widehat{V}^i \notin F_{i-1}^\perp$ by the definition of the facial reduction sequence and
$F_{i-1}^\perp$ is closed,
we also have $U^i + V^i \notin F_{i-1}^\perp$ for all
sufficiently small $t>0$.
In addition, we obtain that \begin{align*}
F_{i-1} \cap \left\{U^i\right\}^\perp
& = F_{i-1} \cap \Big\{ \widehat{U}^i + \sum_l\alpha_{l}(t)
q_{l}q_{l}^T\Big\}^\perp\\
& = F_{i-1} \cap \Big\{ \sum_l(\lambda_l + \alpha_{l}(t)
)q_{l}q_{l}^T\Big\}^\perp
= F_{i-1} \cap \Big\{\widehat{U}^i\Big\}^\perp = F_i. \end{align*}
Therefore, we have shown that $\{U^i\}_{i=1}^s$ also generates
the faces $F_1,\dots,F_s$
and that $(\hat{y}^i,U^i,V^i)$ is a reducing certificate of \eqref{Dt4} at the $i$-th iteration for each $i=1, \ldots, s$.
Thus $F_s$ contains
the minimal face of \eqref{Dt4} for each sufficiently small $t>0$.
In addition, since $\{\eqref{Dt4}\}_{t\geq 0}$ satisfies the rank condition,
Lemma
\ref{lemma:rank_cond} implies that
the minimal face of \eqref{Dt4} is equal to $F_s$ for each sufficiently small $t>0$. \hspace*{\fill} \qed
\end{proof}
\section{Application to a Control Problem}\label{sec:example}
\subsection{\textbf{A Singular SDP in $H_\infty$ State Feedback Control Problem}}\label{subsec:example} We present a singular SDP arising from $H_\infty$ state feedback control problem. The $H_{\infty}$ control problem is one of the most successful applications of SDP and is the problem for designing a controller that achieves stabilization with some guaranteed performance based on the $H_\infty$ norm. In particular, the $H_{\infty}$ state feedback control problem is a special case of the $H_{\infty}$ control problem. See, e.g., \cite{Iwasaki94,Scherer06} for the detail on the SDP formulation.
In this section, we deal with the following SDP problem: \begin{align} \label{LMI} &\sup\left\{ -\gamma: \begin{pmatrix}
-\He(AY_1 + B_2Y_2) & & \\
-C_1Y_1 - D_{12}Y_2 & \gamma I_2 & \\
-B_1^T & -D_{11}^T & \gamma I_{2}
\end{pmatrix}\in \Bbb{S}_+^6, \begin{array}{l}
Y_1\in \Bbb{S}_+^2, \\
Y_2\in \Bbb{R}^{2\times 2},\\
\gamma\in\Bbb{R}
\end{array}\right\},\tag{P0} \end{align} where $\He(X) = X+ X^T$ for $X\in\Bbb{R}^{n\times n}$ and the blanks in the matrices stand for the transpose of the lower triangular block part. Also, the matrices $A$, $B_1$, $B_2$, $C_{1}$, $D_{11}$ and $D_{12}$ are defined as follows: \begin{equation}
\left(
\begin{array}{c|c|c}
A & B_1 & B_2 \\
\hline
C_1 & D_{11} & D_{12}
\end{array}\right)
= \left(
\begin{array}{cc|cc|c}
-1 & -1 & -1 & -1 & 0 \\
1 & 0 & -1 & 0 & 1 \\
\hline
2 & -1 & -1 & 0 & 2 \\
-1 & 2 & -1 & 0 & -1 \end{array}
\right). \label{eq:control_matrix} \end{equation} Its dual can be formulated as follows: \begin{align} \label{DUAL} &\inf\left\{
-\begin{pmatrix} O&&\\ O&O&\\ B_1^T&D_{11}^T &O \end{pmatrix} \bullet Z : \begin{array}{l} \He(A^TZ_{11}+C_1^TZ_{21})\in\mathbb{S}^{2}_+, \\ I_p\bullet Z_{22} + I_{m_1}\bullet Z_{33} = 1, \\ B_2^TZ_{11}+D_{12}^TZ_{21}=O, \\
Z=\left(Z_{ij}\right)_{1\le i, j\le 3}\in\mathbb{S}^{ 6}_+ \end{array} \right\}. \end{align}
To adjust our SDP problem of interest to the form of \eqref{P}, we define the coefficient matrices $A_k$ \ $(k\in [6]\cup\{0\})$ and vector $b$ by $ A_k = \left(\begin{smallmatrix} A_{k, 1} & O\\ O & A_{k, 2} \end{smallmatrix}\right)$ and $b = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix}^T$, where $A_{k, 1}\in\mathbb{S}^6$ and $A_{k, 2}\in\mathbb{S}^2$ for all $k$. Furthermore, we rewrite variables $Y_1$, $Y_2$, $\gamma$ as follows: \[ Y_1 = \begin{pmatrix} y_1 & y_2 \\ y_2 & y_3 \end{pmatrix}, Y_2 = \begin{pmatrix} y_4 & y_5 \end{pmatrix}, y_6 = \gamma. \]
It follows from \cite[Theorems 3.3 and 3.5]{Waki15} that \eqref{LMI} is strictly feasible but its dual problem is weakly feasible.
Thus we can say that \eqref{LMI} is singular.
We compare computational results on \eqref{LMI} with the following three perturbed SDPs for \eqref{LMI}: For $\epsilon = $1.0e-16, \begin{enumerate}[label=(P\arabic*),leftmargin=*]
\item SDP obtained by perturbing the $(2, 2)$nd element of $A_{5, 1}$ into $-2(1+\epsilon)$,\label{p1} \item SDP obtained by perturbing the $(2, 3)$rd and $(3, 2)$nd elements in $A_{5, 1}$ into $-2(1+\epsilon)$, and \label{p2} \item SDP obtained by perturbing the $(2, 4)$th and $(4, 2)$nd elements of $A_{5, 1}$ into $1+\epsilon$. \label{p3} \end{enumerate}
We apply SDPA-GMP \cite{SDPA} to solve \eqref{LMI} to \ref{p3} with stopping tolerances $\delta$ ($\delta$=1.0e-10, 1.0e-30 and 1.0e-50) and set the floating point computation to approximately 300 significant digits. We set {\ttfamily maxIteration} = 10000 and {\ttfamily betaStar} = {\ttfamily betaBar} = {\ttfamily gammaStar} = 0.5 for parameters of SDPA-GMP. See \cite{SDPA} for more details on parameters. Table \ref{results} shows the numerical results. We observe the following:
\begin{table}[htbp] \caption{Computed values for \eqref{LMI}, its perturbed problems \ref{p1}, \ref{p2} and \ref{p3}} \centering
\begin{tabular}{|c|c|c|c|} \hline & $\delta=$1.0e-10 &$\delta=$1.0e-30 &$\delta=$1.0e-50 \\ \hline \eqref{LMI} &-2.2360679775444764 &-2.2360679774997897 &-2.2360679774997897 \\ \ref{p1} &-2.2360072694172072 &-2.1078335768712432 & -1.4142135623730950\\ \ref{p2} &-2.2360072694172055 &-2.0000000000000000 & -2.0000000000000000\\ \ref{p3} &-2.2360072665294605 &-1.4142135623730950 &-1.4142135623730950\\ \hline \end{tabular} \label{results} \end{table}
\begin{itemize} \item The computed values of \eqref{LMI} are almost same
for all $\delta$, whereas the values for perturbed problems \ref{p1}, \ref{p2} and \ref{p3} are different. These significant differences imply that one needs to choose suitable
tolerances $\delta$ in order to use the floating point computation
with longer significant digits for singular SDPs.
\item We can verify that the optimal value of \eqref{LMI} is $-\sqrt{5}$, while the
optimal value of the perturbed problem \ref{p1} is $-\sqrt{2}$.
These differences show that a small perturbation of coefficient matrices $A_k$ in \eqref{LMI} may yield a significant change of the optimal value of \eqref{LMI}. \end{itemize}
\subsection{\textbf{Behavior of Minimal Faces under Perturbations for Our Example}}\label{example:invariant}
We show that matrix-wise perturbations make the minimal face of the dual problem of
\eqref{LMI} invariant or full-dimensional, i.e., $\Bbb{S}^8_+$.
Let $A = \left(\begin{smallmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{smallmatrix}\right)$,
$B_2 = \left(\begin{smallmatrix}
b_1 \\
b_2
\end{smallmatrix}\right)$,
$C_1 = \left(\begin{smallmatrix}
c_{11} & c_{12} \\
c_{21} & c_{22}
\end{smallmatrix}\right)$,
$D_{12} = \left(\begin{smallmatrix}
d_{11}\\
d_{21}
\end{smallmatrix}\right)$, and let $B_1$ and $D_{11}$ be the
same matrices as in \eqref{eq:control_matrix}. Then the first
constraint in \eqref{LMI}
means that {\footnotesize
\[
\begin{pmatrix}
-2a_{11}y_1 - 2a_{12}y_2- 2b_1y_4 & \\
-a_{21}y_1 - (a_{11} + a_{22})y_2
- a_{12} y_3 - b_2 y_4 - b_1 y_5
& - 2a_{21} y_2 - 2a_{22} y_3 - 2b_2 y_5 & \\
-c_{11}y_1 - c_{12}y_2 - d_1y_4 & -c_{11}y_2 - c_{12}y_3 - d_1y_5
& y_6 \\
-c_{21}y_1 - c_{22}y_2 - d_2y_4 & -c_{21}y_2 - c_{22}y_3 - d_2 y_5
& 0 & y_6 \\
1 & 1 & 1 & 1 & y_6 \\
1 & 0 & 0 & 0 & 0 & y_6
\end{pmatrix}
\]
}
is contained in $\Bbb{S}_+^6$.
The related part with $a_{11}$ in the above matrix can be extracted as
\begin{align*} a_{11}
\left(\begin{smallmatrix}
-2y_1 & -y_2 & 0 & 0 & 0 & 0 \\
-y_2 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{smallmatrix}\right)&=a_{11}y_1\left(\begin{smallmatrix}
-2 &0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{smallmatrix}\right) +a_{11}y_2 \left(\begin{smallmatrix}
0 & -1 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{smallmatrix}\right)\\
&=:a_{11}(y_1 E_{1,1} + y_2 E_{2,1}).
\end{align*}
Since a perturbation on $a_{11}$ affects the coefficient matrices of $y_1$ and $y_2$, the corresponding perturbing matrices are
$E_1(t)=\left(\begin{smallmatrix}
tE_{1,1} & O \\
O&O_{2\times 2}\end{smallmatrix}\right)$, $E_2(t) =\left(\begin{smallmatrix}
tE_{2,1} & O \\
O&O_{2\times 2}\end{smallmatrix}\right)$
and $E_k(t) = O_{8\times 8}\ (k =
3, \ldots, 6)$.
We remark that we need to consider block matrices with two blocks for the perturbation because the coefficient matrices for $y_1$ also appear in the constraint $Y_1 \in \Bbb{S}^2_+$ of \eqref{LMI}.
Consider the problem \eqref{Dt4} perturbed with $\{E_k(t)\}$.
Then one can verify that the length of the facial reduction sequence
for \eqref{Dt4} is one and that it is $\{(y, U, V)\}$, where
\begin{align}\label{eq:ex1} &\left\{ \begin{array}{l}
y = (1, 0, 0, -1, 0, 0)^T, U = \begin{pmatrix}
U_1 & O\\
O& U_2
\end{pmatrix}, V = \begin{pmatrix}
V_1 & O\\
O& V_2
\end{pmatrix},\\
V_1 =O_{6\times 6},
V_2=O_{2\times 2}, U_1= \begin{pmatrix}
1 & 0^T\\ 0 & O_{5\times 5} \end{pmatrix}, U_2 = \begin{pmatrix}
1 & 0\\
0 & 0
\end{pmatrix}.
\end{array}
\right. \end{align}
Let $e_1\in\Bbb{R}^6$ and $f_1\in\Bbb{R}^2$ be the unit vectors whose first entry is $1$ and others are
zero. Then the positive eigenvalues of $U$ are $2, 1$, and
the associated eigenvectors
are $(e_1, 0_2^T)^T$, $(0_6^T, f_1^T)^T$ respectively. Here $0_p$ is the $p$-dimensional zero vector for a given positive integer $p$.
Since we have that
\[
-\left( 1 \cdot E_1(t) + 0 \cdot E_2(t) \right)
\in\mathop\mathrm{Span}\left\{\left(\begin{smallmatrix}e_1{e_1}^T & O \\ O & O_{2\times 2}\end{smallmatrix}\right), \left(\begin{smallmatrix} O_{6\times 6} & O \\ O & f_1{f_1}^T\end{smallmatrix}\right)
\right\}
\]
and that $\{\eqref{Dt4}\}_{t\geq 0}$ satisfies the rank condition,
Proposition \ref{prop:invariant3} implies that this perturbation does not
change the minimal face of the dual problem.
We can apply similar arguments to see behavior of the minimal face of the dual problem for the other perturbations and observe the followings: \begin{itemize} \item The minimal face is invariant under the matrix-wise perturbation with respect to $a_{11}$, $a_{12}$, $a_{22}$, $c_{12}$, $c_{22}$ and $b_1$. The optimal value of \eqref{Dt4} changes continuously at $t = 0$ due to Theorem 3.1. \item The other perturbations, i.e. $a_{21}$, $c_{11}$, $c_{21}$, $b_2$ $d_1$ and $d_2$, make the minimal face
of the dual problem to be $\Bbb{S}^8_+$, which implies that the perturbed problem is strictly
feasible. However, we have numerically confirmed that the optimal value of \eqref{Dt4} also varies continuously in this case. It is a future study to find other conditions that ensure the continuity of the optimal value under any matrix-wise perturbations.
\item Hence if we perturb matrices $A$, $B_2$, $C_1$ and
$D_{12}$ in the structured form, the minimal face may be
different,
but can not be smaller. \end{itemize}
\section{Conclusions}\label{sec:conclusion} We consider perturbations of the coefficient matrices of a semidefinite program, in the case that the primal problem is strictly feasible and the dual problem is weakly feasible. We give sufficient conditions for continuity of the optimal value. These conditions involve the behavior of the minimal faces of the perturbed dual problems and the submatrices of the coefficient matrices associated with the minimal faces. By using examples, it is argued that these conditions are hard to remove. We further obtain sufficient conditions for the perturbations to keep the minimal face invariant. A facial reduction sequence, which is obtained in the process of facial reduction, plays the central role. Then our results are applied to a semidefinite program obtained from an $H_\infty$ control problem. By presenting numerical experiments with interior point methods, we also discuss the importance of computations with arbitrary precision arithmetic, together with an appropriate parameter for the stopping criteria, in order to obtain an approximation to the optimal value of a singular semidefinite program.
In the future work, it is worth considering to use a facial reduction sequence to analyze other properties of a semidefinite program. In addition, it may be interesting to find combinatorial structures in the elements of perturbing matrices that preserve the minimal face of a semidefinite program.
\end{document} | arXiv |
\begin{document}
\newcommand{\comment}[1]{} \comment{ Article-id: 0904.4324, Article password: v538r
Spherical and Whittaker Functions via DAHA Ivan Cherednik, Xiaoguang Ma
This work grew out of the lectures given by the first author at Harvard in February and March, 2009. A draft of the lecture notes was prepared by the second author, and then expanded and made into their final form by the first author. It begins with an introduction to the classical p-adic theory of the Macdonald, Matsumoto and Whittaker functions. Its major directions are as follows:
1) extending the theory of DAHA to arbitrary levels; 2) the affine Satake map and Hall functions via DAHA; 3) the spinor Dunkl operators for the Q-Toda operators; 4) applications to nil-DAHA and Q-Whittaker functions; 5) the technique of spinors in the differential theory.
The latest variant is essentially mathematically equivalent to variant 5, though there the were some improvements, extensive editing and adding some justifications.
}
\title[Spherical and Whittaker functions via DAHA] {\footnotesize A new take on spherical, Whittaker and Bessel functions\\ {\tiny (Spherical and Whittaker functions via DAHA I,II)}}
\author[Ivan Cherednik]{Ivan Cherednik $^\dag$} \author[Xiaoguang Ma]{Xiaoguang Ma}
\begin{abstract} This paper begins with an exposition of the classical p-adic theory of the Macdonald, Matsumoto and Whittaker functions aimed at the affine generalizations. The major directions are the theory of DAHA for arbitrary levels and the affine Satake map and Hall functions via DAHA. The key result is the proportionality of the two different formulas for the affine symmetrizer, the Satake-type formula and that based on the polynomial representation of DAHA. The latter approach results in two important formulas for the affine symmetrizer generalizing the relations between the Kac-Moody characters and Demazure characters.
The second part of this paper is focused on the spinor (nonsymmetric) Whittaker functions in the rank one, related q-Toda-Dunkl operators, and other aspects of the spinor construction, including one-dimensional Bessel functions, and the isomorphism between the affine Knizhnik-Zamolodchikov equation and the Quantum Many-Body problem (the Heckman-Opdam system). \end{abstract}
\date{October 24, 2012}
\address {\vskip -0.5cm (I.Cherednik) Department of Mathematics, UNC Chapel Hill, North Carolina 27599, USA\\ [email protected]} \address {\vskip -0.7cm (X. Ma) 632 CAB, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA\\ [email protected]}
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\thanks{$^\dag$ Partially supported by NSF grant DMS--0800642}
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\setcounter{section}{-1} \setcounter{equation}{0} \section{\sc{Introduction}}
\noindent This work grew out of the lectures given by the first author at Harvard in February and March, 2009. A draft of the lecture notes was prepared by the second author, and then expanded and made into their final form by the first author. It will be published by Selecta Mathematica in two parts ``Spherical and Whittaker functions via DAHA, I,II," essentially corresponding to Sections 0,1,2,3 and Sections 4,5,6,7 of this preprint, which is a somewhat extended version of its previous variant (posted in 2009). These two parts are related but relatively independent; the theory of spherical functions is the main unifying theme.
\subsection{{\bf Objectives and main results}} The first aim of the first part of this work is to connect DAHA with the theory of affine Hall functions using the approach to the classical Hall polynomials ($=\, p$\~adic spherical functions) via the Matsumoto $p$\~adic functions, an important special case of the theory of nonsymmetric Macdonald polynomials.
It is closely connected to the second major direction of this work, which is the nonsymmetric Whittaker theory. Classical Whittaker functions are already nonsymmetric, so we need a new theory of spinors (generally, $W$\~spinors) to achieve this; some instances already appeared in the related harmonic analysis.
Dunkl-$q$-Toda operators and their eigenfunctions, the spinor $q$\~Whit\-taker functions, are introduced and studied for $A_1$ in the second part of this paper. The $p$\~adic limits of these function are well defined and result in {\em new} Matsumoto-type (``nonsymmetric") $p$\~adic Whittaker functions; see Sections \ref{sect:Whitf}. The definition can be given for any root systems.
Another (actually related) possible output of this project could be the theory of nonsymmetric counterparts of the affine Hall functions and the corresponding Satake map, including their connections with the DAHA elliptic-type representations from \cite{C5}; cf. Section \ref{sec: p-adic-e}. This work is in progress.
More specifically, the results of this work (both parts) can be grouped as follows.
(1) The theory of DAHA modules of arbitrary levels $l$ (not only $l=0,1$ as in \cite{C101}), which technically means that its polynomial representation can be multiplied by any powers of the Gaussian.
(2) The affine Satake isomorphism and affine Hall functions via DAHA; the latter functions attract growing attention, though not much is known so far for arbitrary $q$ and $t$, the DAHA parameters.
(3) Establishing connections with the theory of Kac-Moody characters, the $t\to\infty$ limits of the affine Hall functions, and the level one Demazure characters.
(4) The theory of coinvariants of DAHA, their relations to the bilinear symmetric invariant forms on DAHA of higher levels, and the corresponding spaces of Looijenga functions.
(5) Revisiting classical $p$\~adic theory of the Satake-Macdonald, Matsumoto and Whittaker functions with the focus on the Matsumoto functions and aiming at the DAHA generalizations.
(6) The study of new spinor Dunkl operators serving the $q$\~Toda operators and the $q$\~Whittaker functions, the related theory of the nil-DAHA and the spinor Whittaker functions.
(7) Developing the technique of $W$\~spinors, including the differential theory and its application to the Bessel functions, symmetric and nonsymmetric, and the AKZ$\leftrightarrow$QMBP isomorphism theorem.
\subsubsection{\sf Affine Satake isomorphisms} Among the main topics we consider, are the {\em DAHA-Satake map}, which is the infinite symmetrizer on the affine Hecke subalgebra, and its relation to the {\em affine Satake map} (and related constructions) defined by the formulas used in \cite{Ka,FGT,BK}. The latter map is directly connected with the theory of Jackson integration developed in \cite{C1,C5,Sto}, which provides exact formulas at levels $0,1$; see also \cite{FGT}, Section 12.7 ``Lattice-hypergeometric sums." Interestingly, the DAHA-Satake map and the affine Satake map have different convergence ranges. The latter is well defined for any nonzero $t$, the former only as $\Re k < -1/h$ for $t=q^k$ and the Coxeter number $h$;
$|q|<1$ in the paper. When both converge, they are proportional to each other.
The affine Satake series becomes essentially the {\em Weyl-Kac character formula} in the limit $t\to \infty$. On the other hand, the DAHA-Satake map appeared to be related to the {\em Demazure characters}, due to the main proportionality theorem and the $Y$\~formulas from Theorems \ref{YLEFT},\ref{YLEFTNEW}. We note that $t$\~counterparts of the Kac-Moody string functions (and related matters) are not discussed in this paper; see \cite{FGT,Vi}. Also, what seems promising to us is the study of the monodromy of the affine Hall functions (generalizing the classical theorem due to Kac and Peterson); we hope to consider this problem in other works.
Concerning the algebraic theory of DAHA, the Satake map and affine Hall functions are closely related to {\em DAHA coinvariants\,}, which, in turn, are directly connected with the symmetric invariant bilinear forms on DAHA of levels $l\ge 0$. The bilinear forms of level $0$ and $1$ are exactly the key inner products from \cite{C101} and other works of the first author. For arbitrary levels $l>0$, the space of DAHA coinvariants is isomorphic to the corresponding Looijenga space. Various applications of the DAHA coinvariants are expected in mathematics and physics.
\subsubsection{\sf Spinor Whittaker functions} The focus of the second part of this work is on the nonsymmetric Whittaker theory for $A_1$. The classical Whittaker functions are already nonsymmetric, so we need a new theory of spinors (generally $W$\~spinors) to achieve this; some its instances already appeared in the related harmonic analysis (we will discuss this).
The construction of the {\em spinor-Dunkl operators} for the $q$\~Toda operators (also called chains) is an important and unexpected development in this classical field. It can be presented as an isomorphism between the standard polynomial representation of {\em nil-DAHA} and the spinor-polynomial representation of its dual. The reproducing kernel of this isomorphism is the {\em spinor nonsymmetric Whittaker function}, which was mentioned in \cite{ChW} as a possible major continuation of the theory of $q$\~Whittaker functions. We note that the definition of the difference (relativistic) Toda chain in the case of $A_n$ in the classical and quantum variants is essentially due to Ruijsenaars; see \cite{Rui} for a review.
In this paper the formula for the {\em nonsymmetric} Whittaker function is discussed for $A_1$ only. See \cite{ChW} for the theory of {\em global symmetric $q$\~Whittaker functions}, which are closely connected with the theory of affine flag varieties and Givental-Lee theory. They may have other applications too; see \cite{GLO}. Technically, the introduction of {\em nonsymmetric} Whittaker functions is an important step for using DAHA methods at their full potential.
It is important that the same limit $t\to \infty$ serves the $q$\~Whittaker functions and the passage to Kac-Moody theory. However, this limit must be calibrated in a very special way in the Whittaker case following the Ruijsenaars procedure (see \cite{Et1} and \cite{ChW}). As a matter of fact, obtaining the Kac-Moody characters is also not immediate from DAHA; the affine Satake map is needed here, the major theme of the first part of this work. The $q$\~Hermite polynomials emerge in the limit $t\to\infty$ for both, $q$\~Whittaker and Kac-Moody theories. They play an important role in our analysis. The resulting connection between Kac-Moody theory and $q$\~Whittaker theory is expected to be related to the geometric quantum Langlands program.
\subsection{{\bf Dunkl operators via DAHA}} To put this paper into perspective, let us briefly outline the (current) status of DAHA theory from the viewpoint of the constructions of the Dunkl operators. The families of the Dunkl operators are essentially in one-to-one correspondence with the constructions of DAHA ``polynomial representations". The latter are generally those induced from the affine Hecke subalgebras of DAHA, their variants and degenerations. Not all of them are really polynomial; {\em Fock representations} may be a better name.
Such approach to reviewing applications of DAHA is of course simplified, but maybe not too much. For instance, if the polynomial representation is known and well studied, then we know a lot about the corresponding DAHA. It gives the PBW theorem, the zeros of the corresponding Bernstein-Sato polynomial, the definition of the localization functor, the construction of the corresponding spherical function and more of these.
The {\em spinor-polynomial} representation needed for the $q$\~Toda-Dunkl operators appeared of a new type (not exactly induced from AHA), which reflects interesting new features of nil-DAHA. To explain it, let us begin with the list of major families of Dunkl operators.
\subsubsection{\sf Main families of Dunkl operators} We will stick to the crystallographic case; there are important developments for the groups generated by complex reflections and those generated by symplectic ones (though the latter generally do not result in Dunkl-type operators). With this reservation, the list of major known families of Dunkl operators and corresponding polynomial representations is as follows.
(a) The rational-differential operators due to Charles Dunkl; {\em rational DAHA\,} is self-dual and its theory (including the polynomial representation) is the most developed now.
(b) Differential-trigonometric and difference-rational polynomial representations of {\em degenerate DAHA\,}; they are connected by the generalized Harish-Chandra transform.
(c) Macdonald theory and $q,t$\~DAHA, corresponding to the difference-trigonometric polynomial representation and the corresponding Dunkl operators; it is self-dual as in the rational case.
(d) Differential-elliptic representation of degenerate DAHA and the difference-elliptic representation of $q,t$\~DAHA \cite{Ch12,Ch13}; their dual counterparts have not been studied so far.
(e) The specializations of the representations from $(b)$ in the theory of Yang-type systems of spin-particles. The references are \cite{Ug} and \cite{EOS}; degenerate DAHA governs their theory.
Let us mention that the families from $(d)$ were introduced in \cite{Ch12} and \cite{Ch13}, but there is no reasonably complete theory of these representations so far. They are connected with the {\em affine Hall functions\,}, the major theme of the first part of the paper.
\subsubsection{\sf The Toda-Whittaker case} The nonsymmetric $q$\~Whittaker functions are eigenfunctions of new {\em spinor\,} Dunkl operators defined using {\em nil-DAHA}, which adds a new dimension to the list above. The $q$\~Toda-Dunkl operators do require the spinors; they are different from those of the induced type defined in \cite{Ch8} (and their degenerations).
The usual (``symmetric") $q$\~Whittaker functions have various applications, exceeding those of the difference spherical functions. One of the reasons is that the coefficients of the $q$\~Whittaker functions are $q$\~integers.
There is a limiting procedure due to Ruijsenaars that connects the $q$\~Toda operators and the difference QMBP; see \cite{Rui, Et1}. It must be significantly modified in the nonsymmetric case using the spinor setting and eventually leads to the {\em spinor polynomial representation}, an irreducible module of {\em nil-DAHA\,} of a new kind.
To be more exact, the latter representation is a counterpart of the polynomial representation multiplied by the Gaussian. Its nil-Fourier-dual equals the Gaussian times the standard polynomial representation of nil-DAHA. The map intertwining these two representations is given in terms of the {\em nonsymmetric spinor global $q$\~Whittaker function\,}. The construction is a general one, but we will stick to the $A_1$\~case in this work.
\subsection{{\bf The technique of spinors}} It is an important tool in the QMBP (the Heckman-Opdam eigenvalue problem) and DAHA theory. The main objective of the spinors is to address the problem that the Dunkl operators are not local; they become local in the space of spinors. Another (related) purpose of this technique is to incorporate into DAHA theory all solution, not only $W$\~invariant, of the QMBP, its generalizations and variants. Solving QMBP in the class of all functions has interesting algebraic and analytic aspects. We will not try to review them here.
As far as we know, this technique was used explicitly for the first time in \cite{C13}, when proving the so-called Matsuo- Cherednik isomorphism theorem. This theorem establishes an equivalence of the affine Knizhnik-Zamolodchikov equation, AKZ, in the modules of the degenerate Hecke algebra induced from (dominant) characters and the corresponding Heckman-Opdam system (QMBP). See Chapter 1 of \cite{C101}, \cite{O2} and Section \ref{sect:AKZ} below.
Using the technique of spinors systematically (see Section \ref{sect:AKZ}) makes the proof from \cite{C13} entirely algebraic and establishes its direct connection with the proof suggested (several years later) in \cite{O2}; compare Lemma 3.2 there with Theorem \ref{SPINDUN} below. The approach from \cite{O2} is actually very close to the justification of this theorem in our paper. Mathematically, Opdam's proof was essentially equivalent to the one from \cite{C13}, but this was done in \cite{O2} entirely algebraically; the spinors in their algebraic variant were certainly present there.
We note the technique of spinors (combined with the explicit calculation of the AKZ-monodromy) was actually used in \cite{C13} to obtain the {\em nonsymmetric spherical function\,}, called the $G$\~function in \cite{O2}.
Generally speaking, there is nothing very new about the definition of spinors, $W$\~spinors to be more exact. They are simply sets of functions $\{f_w\}$ numbered by the elements from the Weyl group $W$ with the action of $W$ on the indices. The principle spinors are in the form $\{w^{-1}(f),\,w\in W\}$ for a global function $f$; generally $f_w$ are absolutely independent functions. For instance, the {\em real spinors} are functions on the disjoint union of all Weyl chambers, collected (using $W$) in the fundamental Weyl chamber. It is not surprising that they appeared in various contexts before.
\subsubsection{\sf Connections to AKZ} The Matsuo proof of the relation between AKZ and QMBP from paper \cite{Mats} was a direct algebraic verification. The Grothendieck-type notion of the monodromy {\em without a fixed point} used in \cite{C13} made the proof very short and entirely conceptual; also, this paper was written for the vector-valued solutions and included the rational QMBP. Using this approach, such an equivalence was extended to the difference and elliptic cases. In the difference theory, this map can be an embedding of the spaces of solutions (not an isomorphism); see \cite{Ch8}, which was finalized in \cite{Sto2}.
The definition of the elliptic QMBP requires the trivial central charge condition, which is $l=-kh$ for the Coxeter number $h$ (where $t=q^k$); then the equivalence will hold too. Apart from the elliptic case, the isomorphism theorems from \cite{C13} and \cite{C101} (Chapter 1) can be stated as follows.
\newtheorem*{keytheorem}{Theorem}
\begin{keytheorem}[{\sf AKZ$\to$Dunkl$\to$QMBP}] Given an arbitrary weight $\lambda$, the space of AKZ-solutions in the induced module $I_\lambda$ of the (degenerate) affine Hecke algebra can be identified with the $\lambda$\~eigenspace of Dunkl operators in the corresponding DAHA {\sf spinor} representation. Then the latter eigenspace can be mapped to the space of all, not necessarily symmetric, solutions of the corresponding QMBP. For generic $\lambda$, this map is an isomorphism (an embedding in the difference setting). \end{keytheorem}
The spinors needed here are {\em complex}, defined in the domain $U=\{z\}$ such that $\Im(z)$ belongs to the corresponding fundamental Weyl chamber. They can be interpreted as functions in the disjoint union $\cup_{w\in W} w(U)$; then the principle spinors are global analytic functions. Using $W$, we can gather these functions in $U$. Only functions in $U$ emerge in the spinor theory of the Dunkl-type eigenvalue problem, including the spinor integration and related inner products.
\subsubsection{\sf On the localization functor} This construction is connected with the {\em localization functor}, one of the most powerful tools in the theory of DAHA. See \cite{GGOR} and \cite{VV1}. The localization construction assigns a local system to a module of DAHA (from a proper category); the case of induced representations is related to AKZ and paper \cite{C13} as follows.
The starting point of the latter paper was the AKZ with values in an arbitrary finite-dimensional module $V$ of AHA (or degenerate AHA). Then the spinor Dunkl operators were defined for these AKZ via the monodromy representation. Combining these Dunkl operators with the operators of multiplication by functions supplies the space of $V$\~valued analytic functions with the DAHA action.
The relation of the spinor Dunkl operators to the monodromy of AKZ is of independent interest. The {\em monodromy cocycle} on $W$ from \cite{C13} (see also \cite{C101}, Chapter 1) can be expressed in terms of the (usual) monodromy homomorphism of the braid group. This establishes a link to the localization functor.
We note that the construction AKZ$\to$Dunkl$\to$QMBP was aimed at applications to the corresponding eigenvalue problems and was done only within the class of induced modules; the projective modules are of key importance for the theory of the localization functor.
\subsubsection{\sf The setting of the work} We mainly use the standard affine root systems in contrast to the {\em twisted} affine root systems considered in \cite{C101} and many papers on DAHA. The standard (untwisted) ``affinization" is (presumably) exactly the one compatible with the quantum Langlands duality. For instance, the {\em untwisted} affine exponents from \cite{C103}, describing the reducibility of the polynomial representation, obey the quantum Langlands-type duality for the modular transformation $q\mapsto \widehat{q}$. This kind of duality does not hold in the {\em twisted case} (at least, we do not know how to formulate it). On the other hand, the twisted affinization has obvious merits (versus the standard setting) for the theory of Gaussians. This is parallel to the advantages of the twisted case for level-one character formulas in Kac-Moody theory.
Due to the standard (untwisted) setting, we need to state some of the results of this paper, especially where the Gaussians are involved, only for the simply-laced root systems. We hope to consider the corresponding {\em twisted} case in other publications. Using $t$ in this paper is relaxed as well; we simply treat it as a single parameter. Generally, $t$ (or $k$) are supposed to depend on the length of the corresponding root. In the second part of this work, we present some constructions only in the $A_1$\~case, where practically everything can be calculated explicitly. However, the major results of this paper can be transferred to (or expected to hold for) arbitrary root systems.
The readers familiar with AHA and classical $p$\~adic theory can go directly to the double affine generalizations, though the introduction of the Macdonald's $p$\~adic spherical functions as symmetrizations of Matsumoto functions, which are essentially delta functions, is not quite standard (even for specialists).
\subsection{{\bf Acknowledgements}} \subsubsection{\sf Harvard lectures} The paper is based on a series of lectures delivered by the first author at Harvard (February-March 2009); he is responsible for the scientific contents of this paper.
It was a somewhat unusual series, a sort of reporting the current research activities on weekly basis. The output of these lectures appeared better than the lecturer expected (hopefully, for the listeners too). The initial TeX files of the lectures were prepared by Xiaoguang Ma.
Extensive usage of examples and exposition of the classical topics are an organic part of the design of this work. However, the focus is on general approaches and new results. Almost all examples and direct verifications are needed to prepare affine and spinor generalizations, the main purpose of this work.
\subsubsection{\sf Special thanks} My special thanks go to Dennis Gaitsgory and Pavel Etingof for participating in these lectures, shaping their direction and contents, and for various important discussions.
Alexander Braverman and David Kazhdan significantly influenced the key topics of this series of lectures and the papers. David Kazhdan greatly helped in improving the theory of the affine symmetrizers presented in this work. I am grateful to Alexander Braverman, Ian Grojnowski and Manish Patnaik for discussions concerning the affine Hall functions. Talks to Victor Kac and Boris Feigin were helpful in establishing the connections with Kac-Moody theory. I am thankful to Roman Bezrukavnikov, Michael Finkelberg and Victor Ostrik for discussions of the affine flag varieties and quantum groups. I thank Pavel Etingof, Eric Opdam, Simon Ruijsenaars, Jasper Stokman, the referee, and Ann Kostant for valuable comments.
The work is partially based on my notes on spinors (reported at the University Paris VI in 2004 and at RIMS in 2005) and on the DAHA approach to the decomposition of the regular representation of AHA (see \cite{ChL},\cite{HO2})
reported at CIRM (2006), MIT (2007) and at the University of Amsterdam (2008). Working on these papers continued at RIMS (Kyoto University, 2009) and completed at the Hebrew University (2012). I am very grateful for the invitations.
Quite a few topics were stimulated by my talks to physicists; special thanks to Anton Gerasimov who introduced me to the brave new world of $q$\~Whittaker functions.
I am grateful to the many people, mathematicians and physicists, I talked to on these and related matters at Harvard, MIT, RIMS, the Hebrew University, and in many other places.
{--Ivan Cherednik}
\setcounter{equation}{0} \section{\sc{P-adic theory revisited}} \noindent The area of affine Hecke algebras, AHA, and spherical functions is vast. The classical $\mathfrak{p}$\~adic spherical functions were subject to various generalizations. It is most important to note that they are limits as $q\to 0$ of the {\em symmetric Macdonald polynomials}, due to Ian Macdonald. Similarly, the limits of the {\em nonsymmetric Macdonald polynomials} are the Matsumoto spherical functions, key to our approach. The DAHA methods help a lot in clarifying the algebraic aspects of their theory. See Section 2.11 from Chapter 2 in \cite{C101} (and references therein) and \cite{O4}; see also \cite{Ion2,O5}.
The purpose of this section is revisiting the $\mathfrak{p}$\~adic theory from the viewpoint of DAHA, which aims at establishing connections with the affine Hall functions and $q$\~Whittaker functions.
\subsection{{\bf Affine Weyl group}} \label{sect:affineweyl} \subsubsection{\sf Root systems} Concerning the classical theory of root systems and Weyl groups, the standard references are \cite{Bo,Hu}; if these sources are insufficient, then see \cite{C101}.
In this paper $R=\{\alpha\}\subset \mathbb{R}^{n}$ is a simple reduced root system with respect to a nondegenerate symmetric bilinear form $(,)$ on $\mathbb{R}^{n}$. Let $\{\alpha_{i}\}_{i=1}^{n}\subset R$ be the set of simple roots and let $R_{+}$ (or $R_{-}$) be the set of positive (or negative) roots. The coroots are denoted by $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$; $W$ is the Weyl group generated by $s_\alpha$.
Let $Q=\bigoplus_{i=1}^{n}\mathbb{Z}\alpha_{i}$, $P=\bigoplus_{i=1}^{n}\mathbb{Z}\omega_i$, correspondingly, let $Q^{\vee}=\bigoplus_{i=1}^{n}\mathbb{Z}\alpha_{i}^{\vee}$ be the coroot lattice and $P^{\vee}=\bigoplus_{i=1}^{n}\mathbb{Z} \omega^{\vee}_{i}$ the coweight lattice, where $\{\omega_{i}^{\vee}\}$ are the fundamental coweights, i.e., $(\omega^{\vee}_{i}, \alpha_{j})=\delta_{ij}$. Replacing $\mathbb{Z}$ by $\mathbb{Z}_{+}={\mathbb Z}_{\ge 0}$, we obtain $Q_+,Q^{\vee}_{+}$ and $P_+,P^{\vee}_{+}$.
The maximal positive root will be denoted by $\theta$, and the bilinear form will be normalized by the condition $(\theta,\theta)=2$; also $\rho\stackrel{\,\mathbf{def}}{= \kern-3pt =}\dfrac{1}{2}\sum_{\alpha\in R_{+}}\alpha$. Due to this normalization, \begin{align*} &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &&Q& &\subset& &P& &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &&\cup& && &\cup& &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &&Q^\vee& &\subset& &P^\vee.& &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \end{align*}
We stick to reduced root systems in this paper, sometimes even to the $A-D-E$ systems. Almost all results in the theory of DAHA and related Macdonald polynomials for reduced root systems were transferred to the case of $C^\vee C$, the ultimate nonreduced system, and to the corresponding theory of Koornwinder polynomials.
\subsubsection{\sf Affine root systems} The vectors $\widetilde{\alpha}= [\alpha,j]\in \mathbb{R}^{n}\times\mathbb{R}$, where $\alpha\in R$ and $j\in \mathbb{Z}$, form the {\em standard affine root system} $\widetilde{R}$. The set of positive affine roots is
$\widetilde{R}_{+}=\{[\alpha,j]\,|\,j\in \mathbb{Z}_{>0}\}\cup
\{[\alpha,0]\,|\,\alpha\in R_{+}\}$. Define $\alpha_{0}=[-\theta, 1]$, where $\theta$ is the maximal positive root in $R$. We will identify $\alpha\in R$ with $\widetilde{\alpha}=[\alpha,0]\in \widetilde{R}$. The affine simple roots $\{\alpha_i, 0\le i\le n\}$ form the extended (also called affine) Dynkin diagram $Dyn^{\tt aff} \supset Dyn=\{\alpha_i,1\le i\le n\}$.
For an arbitrary affine root $\widetilde{\alpha}=[\alpha,j]$ and $\widetilde{z}=[z, \zeta]\in {\mathbb R}^{n+1}$, the corresponding reflection is defined as follows: $$ s_{\widetilde{\alpha}}(\tilde z) =\tilde z-2\frac{(z, \alpha)}{(\alpha,\alpha)}\,\widetilde{\alpha} =\tilde z-(z,\alpha^{\vee})\,\widetilde{\alpha}. $$ We set $s_i=s_{\alpha_i}$ for $i=0,\ldots,n$. The affine Weyl group $\widetilde{W}$ is generated by
$\{s_{\widetilde{\alpha}}\,|\,\widetilde{\alpha}\in \widetilde{R}_{+}\}$;\, $\{s_i\}$ for $i\ge 0$ are sufficient.
\begin{theorem} We have an isomorphism $$\widetilde{W}\cong W\ltimes Q^{\vee},$$ where the translation $\alpha^\vee\in Q^\vee$ is naturally identified with the composition $s_{[-\alpha,1]}s_{\alpha}\in \widetilde W$. In terms of the action in ${\mathbb R}^{n+1}\ni \tilde z$, one has $b(\tilde z)=[z,\zeta-(b,z)]$ for $\tilde z=[z,\zeta],\, b\in Q^\vee;$ notice the sign of $(b,z)$.\phantom{1}
$\qed$ \end{theorem}
Define the {\em extended affine Weyl group} to be $\widehat{W}=W\ltimes P^{\vee}$ acting on ${\mathbb R}^{n+1}$ via the last formula from the theorem with $b\in P^\vee$. Then $\widetilde{W}\subset \widehat{W}$. Moreover, we have the following theorem. Let $\operatorname{Aut}=\operatorname{Aut}(Dyn^{\tt aff})$\, and
$O\stackrel{\,\mathbf{def}}{= \kern-3pt =} \{r\}$ for $\operatorname{Aut}(\alpha_0)=\{\alpha_r\}$, i.e., $O$ is formed by the indices of the simple roots from the Aut\~orbit $\operatorname{Aut}(\alpha_0)$ of $\alpha_0$.
\begin{theorem} (i) The group $\widetilde{W}$ is a normal subgroup of $\widehat{W}$ and $\widehat{W}/\widetilde{W}=P^{\vee}/Q^{\vee}$. The latter group can be identified with the group $\Pi=\{\pi_r\}$ of the elements of $\widehat{W}$ permuting simple affine roots under their action in ${\mathbb R}^{n+1}$. It is a normal commutative subgroup of $\operatorname{Aut}$; the quotient $\operatorname{Aut}/\Pi$ is isomorphic to the group $A_0=\operatorname{Aut}(Dyn)$ of the automorphisms preserving $\alpha_0$.
(ii) The indices $r\in O^*\stackrel{\,\mathbf{def}}{= \kern-3pt =} O\setminus \{0\}$ are exactly those for the minuscule coweights $\omega_r^\vee$ satisfying the inequalities $(\alpha,\omega_r^\vee)\le 1$ for all $\alpha\in R_+$. The elements $\pi_r\in \Pi$ are uniquely determined by the relations $\pi_r(\alpha_0)=\alpha_r$ ($\pi_0=$id). An arbitrary element $\widehat{w}\in \widehat{W}$ can be uniquely represented as $\widehat{w}=\pi_r\widetilde w$ for $\widetilde w\in \widetilde W$.\phantom{1}
$\qed$ \end{theorem}
It is not difficult to calculate $\pi_r$ explicitly (see \cite{C101}): \begin{align}\label{piromr} &\pi_r=\omega_r^\vee\, u_r^{-1} \hbox{\ \,for minuscule\ \,} \omega_r^\vee\in P_+^\vee\subset \widehat{W},\ u_r=w_0w_0^{(r)}, \end{align} where $w_0^{(r)}$ is the element of maximal length in the centralizer of $\omega_r^\vee$ in $W$ for $r\in O^*$, $w_0$ is the element of maximal length in $W$. Equivalently, $u_r$ is of minimal possible length such that $u_r(\omega_r) \in P_-=-P_+$ (see the next section). Note that $\pi_{r}s_{i}\pi_{r}^{-1}=s_{j}$ if $\pi_{r}(\alpha_{i})=\alpha_{j}$, $0\leq i\leq n$.
\subsubsection{\sf The length function}\label{sect:length} Any element $\widehat{w}\in \widehat{W}$ can be written as $\widehat{w}=\pi_{r}\widetilde{w}$ for $\pi_{r}\in \Pi$ and $\widetilde{w}\in \widetilde{W}$. The length $l(\widehat{w})$ is defined to be the length of the {\em reduced decomposition} $\widetilde{w}=s_{i_{l}}\cdots s_{i_{1}}$ (i.e., with minimal possible $l$) in terms of the simple reflections $s_{i}$. Thus, by definition, $l(\pi_{r})=0$.
This is the standard {\em group-theoretical} definition. There are two other (equivalent) definitions of the length for the crystallographic groups, {\em combinatorial} and {\em geometric}. Namely, the length $l(\widehat{w})$ is the cardinality
$|\tilde R_+\cap \widehat{w}^{-1}(\tilde R_-)|$ and can also be interpreted as the ``distance" from the standard affine Weyl chamber to its image under $w$. Both definitions readily give that $l(\pi_{r})=0$; indeed, $\pi_r$ sends positive roots $\tilde{\alpha}$ to positive roots and (therefore) leaves the standard affine Weyl chamber invariant.
Either the combinatorial or the geometric definition can be used to check that $l(w(b))=2(\rho,b)$ for arbitrary $b\in P^\vee_+$ and $w\in W$.
All three approaches to the length function are important in the combinatorial theory of affine Weyl groups, which is far from being simple and completed.
\subsubsection{\sf Twisted affinization}\label{sect:thinaff} There is another affine extension $R^\nu$ of $R$, convenient in quite a few constructions (especially, when the DAHA Fourier transform and the Gaussians are studied). This is the setting in \cite{C101} and in quite a few of author's papers. This extension is defined for the maximal {\em short} root ${\vartheta}$ instead of the maximal root $\theta$. Accordingly, $(\alpha,\alpha)=2\,$ for short roots and affine roots are introduced as $\widetilde{\alpha}=[\alpha,\nu_\alpha\, j]$ for $\nu_\alpha\stackrel{\,\mathbf{def}}{= \kern-3pt =}\frac{(\alpha,\alpha)}{2}$ ($=1,2,3$). Adding $\alpha_0=[-{\vartheta},1]$ for such ${\vartheta}$ to $\{\alpha_i,i>0\}$, the resulting diagram is the extended Dynkin diagram $(Dyn^\vee)^{\tt aff}$ for $R^\vee$ where all the arrows are reversed. On can simply set $\tilde R^\nu\stackrel{\,\mathbf{def}}{= \kern-3pt =}((R^\vee)^{\tt aff})^\vee$,
where the form in $R^\vee$ is normalized by the (usual) condition $(\alpha^\vee,\alpha^\vee)=2$ for long $\alpha^\vee$, which makes ${\vartheta}$ the maximal root in $R^\vee$. The second check in $((R^\vee)^{\tt aff})^\vee$ is applied to the {\em affine} roots. The formula $s_{[-\alpha,\nu_\alpha]}s_{\alpha}=\alpha\,$ naturally results in unchecked $Q,P$ in the {\em twisted affine Weyl group}: $$\ \hbox{ for } \ R^\nu\,:\, \ \widetilde{W}\cong W\ltimes Q,\ \widehat{W}\cong W\ltimes P. $$ In $\mathfrak{p}$-adic theory, the twisted Chevalley group is a {\em form} of the split group for a proper Galois extension of the starting field.
The appearance of $Q,P$ in $\widetilde W,\widehat{W}$ results in the invariance of the corresponding DAHA with respect to the Fourier transform and other basic automorphisms. This is the main reason why the book \cite{C101} is mainly written in such a ``self-dual" setting. Due to the special choice of the normalization, $Q\subset Q^\vee$ in this case; recall that $({\vartheta},{\vartheta})=2$. The term ``twisted" matches similar terminology in Kac-Moody theory.
\subsection{{\bf AHA and spherical functions}} \subsubsection{\sf Affine Hecke algebras} The affine Hecke algebra $\mathcal{H}$ is generated by $T_{0}, T_{1}, \ldots, T_{n}$ and the group $\Pi=\{\pi_{r}\}$ with the relations: \begin{align} &\underbrace{T_{i}T_{j}T_{i}\ldots}_{m_{ij} \text{ times}}= \underbrace{T_{j}T_{i}T_{j}\ldots}_{m_{ij} \text{ times}}\,,\notag\\ &(T_{i}-t^{1/2})(T_{i}+t^{-1/2})=0,\label{heckerel}\\ &\pi_{r}T_{i}\pi_{r}^{-1}=T_{\pi_{r}(i)}.\notag \end{align} where $\pi_r(i)$ is the suffix of the simple root $\pi_r(\alpha_i)$; $m_{ij}$ is the number of edges between vertex $i$ and vertex $j$ in the affine Dynkin diagram $Dyn^{\tt aff}$ and $t$ is a formal parameter (later, mainly a nonzero number).
{\bf Comment.\ } The above definition gives the affine Hecke algebra with {\em equal parameters}. More systematically, we can introduce a family of formal
parameters $\{t_{\alpha}\}$ depending only on $|\alpha|$, setting $t_{i}=t_{\alpha_i}$ for $0\le i\le n$. Replacing relations (\ref{heckerel}) by the relations $(T_{i}-t_{i}^{1/2})(T_{i}+t_{i}^{-1/2})=0$, we come to the definition of the affine Hecke algebra standard in (modern) geometric and/or algebraic theory (in the case of {\em unequal parameters}).
The formulas below can be readily adjusted to this setting, namely, $t_i$ must be used for $T_i$ and the subscript $\alpha$ must be added to $t$ in the formulas involving $Y_{\alpha^\vee}$. In DAHA theory, the same must be done for $X_\alpha$; also, the relation $t=q^k$ below will become $t_\alpha=q^{k_\alpha}$. If $\tilde R^\nu$ is used instead of $\tilde R$, with $Y_\alpha$ instead of $Y_{\alpha^\vee}$, then $q$ must be also replaced by $q_\alpha=q^{\nu_\alpha}$ in the formulas; accordingly, $t_\alpha=q_\alpha^{k_\alpha}$. \phantom{1}
$\qed$
For any element $\widehat{w}\in \widehat{W}$, define $T_{\widehat{w}}=\pi_{r}T_{i_{l}}\cdots T_{i_{1}}$, where $\widehat{w}=\pi_{r}s_{i_{l}}\cdots s_{i_{1}}$ is a reduced representation of $\widehat{w}$. The definition of $T_{\widehat{w}}$ does not depend on the choice of the reduced decomposition.
Setting $Y_b=T_b$ for $b\in P^{\vee}_{+}\subset \widehat{W}$, one has $Y_{b}Y_{c}=Y_{c}Y_{b}$ for such (dominant) $b,c$; use that $l(b)=2(\rho,b)$ for dominant $b$. For any $a\in P^{\vee}$, we set $Y_{a}\stackrel{\,\mathbf{def}}{= \kern-3pt =} Y_{b}Y_{c}^{-1}$, where $a=b-c$ with some $b,c\in P^{\vee}_{+}$; the commutativity guarantees that $Y_a$ depends only on $a$. This definition is due to Bernstein, Zelevinsky, and Lusztig, see, e.g., \cite{L}.
Let $\mathscr{Y}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \mathbb{C}[Y_{\omega_{i}^{\vee}}^{\pm}] \subset \mathcal{H}$. Then $$\mathcal{H}=\langle\mathscr{Y}, T_{1}, \ldots, T_{n}\rangle.$$ Indeed, $T_0=Y_\theta T_{s_\theta}^{-1}$ and $\pi_r= Y_{\omega_r^\vee}T_{u_r}^{-1}$ (see (\ref{piromr})).
\begin{theorem}\label{PBW} (i) An arbitrary element $H\in \mathcal{H}$ can be uniquely represented as $H=\sum c_{b,w}\,Y_b\,T_w$ (a finite sum) for $b\in P^\vee, w\in W$, which is called the {\sf PBW Theorem}.
(ii) The subalgebra $\mathscr{Y}^W$ of $W$\~invariant $Y$\~polynomials is the center of $\mathcal{H}$ (the {\sf Bernstein Lemma}); here $w(Y_b)=Y_{w(b)}$, see also Lemma \ref{Berns}. \end{theorem}
\subsubsection{\sf Matsumoto functions} Let $\mathbf{H}=\mathcal{H}_{\tt nonaff}$ be the Hecke algebra associated with the nonaffine root system $R$, i.e., generated by $T_i$ for $1\le i\le n$. We define the $t$\~symmetrizer by the formula $$ \mathscr{P}_{+}=\frac{\sum_{w\in W}t^{l(w)/2}T_{w}} {\sum_{w\in W}t^{l(w)}}\in \mathbf{H}\,. $$ One checks directly or using (\ref{tidelta}) below that $$ \frac{(1+t^{1/2}T_{i})\mathscr{P}_{+}}{1+t}= \mathscr{P}_{+},\ 1\le i\le n. $$
The following renormalization $\delta_{\widehat{w}}=t^{-l(\widehat{w})/2}T_{\widehat{w}}$ of $T_{\widehat{w}}$ (any $\widehat{w}\in \widehat{W}$) is convenient to establish the connection with $\mathfrak{p}$-adic theory. Then\, \begin{equation}\label{tidelta} T_{i}\delta_{\widehat{w}}=\left\{\begin{array}{ccc}t^{1/2} \delta_{s_i\widehat{w}}, & & \text{ if }l(s_{i}\widehat{w})=l(\widehat{w})+1; \\ t^{-1/2}\delta_{s_i\widehat{w}}+(t^{1/2}-t^{-1/2})\delta_{\widehat{w}}, & & \text{ otherwise.}\end{array}\right. \end{equation}
Let $\Delta=\bigoplus_{\widehat{w}\in\widehat{W}}\mathbb{C}\delta_{\widehat{w}}$ be the (left) regular representation of $\mathcal{H}$. Its {\em spherical submodule} is defined as follows: $$\Delta^{\sharp}=\Delta\mathscr{P}_{+}\cong \mathscr{Y}\mathscr{P}_{+}.$$ Identification with the Laurent $Y$\~polynomials is based on claim $(i)$ (PBW) of Theorem \ref{PBW}.
From now on $\Delta^{\sharp}$ will be identified with $\mathscr{Y}$, i.e., $1\in \mathscr{Y}$ will be actually $\mathscr{P}_+$. By $\delta^{\sharp}_{\widehat{w}}$, we denote the image of $\delta_{\widehat{w}}$ in $\Delta^{\sharp}$; explicitly,\, $\delta_{\widehat{w}}^{\sharp}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \delta_{\widehat{w}}\mathscr{P}_{+}$.
The {\em Matsumoto functions} \cite{Mat}, also called nonsymmetric $\mathfrak{p}$-adic spherical functions, are defined (in this approach) to be $$\varepsilon_{b}=\delta_{b}^{\sharp}, \quad\forall\, b\in P^{\vee},$$ i.e., we simply restrict $\delta^{\sharp}$ to $P^\vee$ here. From this definition, $\varepsilon_{b}=t^{-(b,\rho)}Y_{b}$ for any $b\in P_{+}^{\vee}$. Representing (calculating) $\varepsilon_{b}$ as a Laurent polynomial in terms of $Y$ for any $b\in P^{\vee}$ is of fundamental importance.
\subsubsection{\sf The rank-one case}\label{sec:rank1} In the $A_{1}$ case, we can set $\omega=\omega_1^{\vee}=\omega^\vee$; then $\alpha=\alpha_1=2\omega$ and $\rho=\omega$. The extended affine Weyl group $\widehat{W}$ is generated by $\pi=\pi_1$ and the reflection $s=s_{\alpha}$. As an element of $\widehat{W}$, $\omega=\pi s$. Let $T=T_{1}\in \mathcal{H}$; then $Y=Y_{\omega}=\pi T $.
The affine Hecke algebra can be written as $\mathcal{H}=\langle Y, T\rangle$ subject to $T^{-1}YT^{-1}=Y^{-1}$ and $(T-t^{1/2})(T+t^{-1/2})=0$. The first of these relations is equivalent to $\pi^2=1$ for $\pi$ introduced as $YT^{-1}$.
The symmetrizer is $$\mathscr{P}_{+}=\frac{1+t^{1/2}T}{1+t}.$$
For any $m\in \mathbb{Z}$, let $\delta_{m}=\delta_{m\omega}$ and $\varepsilon_{m}=\delta_{m\omega}^{\sharp}
=t^{-|m|/2}T_{m\omega}\mathscr{P}_{+}$. Then we have for $m\geq 0$, \begin{align} &T\varepsilon_{m}\ =\ t^{1/2}\varepsilon_{-m},\label{Tvepm}\\ &T\varepsilon_{-m}\ =\ t^{-1/2} \varepsilon_{-m}+(t^{1/2}-t^{-1/2})\varepsilon_{m}.\label{T-vepm} \end{align} Similarly, for $m\geq 0$, \begin{align*} &T^{-1}\varepsilon_{-m}\ =\ t^{-1/2}\varepsilon_{m},\\ &T^{-1}\varepsilon_{m}\ =\ (T-(t^{1/2}-t^{-1/2}))\varepsilon_{m}=t^{1/2} \varepsilon_{-m}-(t^{1/2}-t^{-1/2})\varepsilon_{m}. \end{align*}
\begin{lemma}\label{LEMPI} For any $m\in {\mathbb Z}$, $\pi\varepsilon_{m}=\varepsilon_{1-m}$. \end{lemma} {\em Proof.} Since $\pi^{2}=1$, it suffices to calculate $\pi \varepsilon_{-m}$ for $m\ge 0$. Using that $Y\varepsilon_m=t^{1/2}\varepsilon_{m+1}$ (it results from the definition of $\varepsilon$ for such $m$), $$\pi \varepsilon_{-m}=YT^{-1}\varepsilon_{-m}= t^{-1/2}Y\varepsilon_{m}= \varepsilon_{1+m}.$$ \vskip -0.7cm \phantom{1}
$\qed$
Let us apply the lemma to write down the action of $Y^{\pm1}$ on $\varepsilon_{m},\varepsilon_{-m}$ for $m\ge 0$:
\begin{eqnarray} Y\varepsilon_{m} &=&t^{1/2}\varepsilon_{m+1}\label{yepm},\\ Y\varepsilon_{-m} &=&t^{-1/2}\varepsilon_{-m+1}+(t^{1/2}-t^{-1/2})\varepsilon_{m+1} \label{yep-m},\\ Y^{-1}\varepsilon_{m+1} &=&t^{-1/2}\varepsilon_{m}\label{y-epm},\\ Y^{-1}\varepsilon_{-m} &=&t^{1/2}\varepsilon_{-m-1}-(t^{1/2}-t^{-1/2})\varepsilon_{m+1}. \label{y-ep-m} \end{eqnarray} Note that (\ref{yepm})and (\ref{yep-m}) overlap at $m=0$, as well as (\ref{Tvepm})and (\ref{T-vepm}).
The formulas for the action of $Y$ and $Y^{-1}$ are called {\em nonsymmetric Pieri rules}; they are {\em obviously} sufficient to calculate the $\varepsilon$\~functions (which holds in any ranks). However, the {\em technique of intertwiners} is generally more efficient for calculating the $\varepsilon$\~polynomials and their variants than direct usage of the Pieri formulas (see, e.g., \cite{C101}). In this particular example, formula (\ref{Tvepm}) is such an intertwiner. It is sufficient indeed: \begin{eqnarray}\label{vepformulas} \varepsilon_m & = & t^{-\frac{m}{2}}Y^m \ \hbox{ for } \ m\ge 0 \hbox{\ \, implies \ that}\\ \varepsilon_{-m} &=& t^{-\frac{1}{2}}T\varepsilon_m=t^{-\frac{m+1}{2}}T(Y^m)\notag\\ &=&t^{-\frac{m+1}{2}}(t^{\frac{1}{2}}Y^{-m}+ (t^{\frac{1}{2}}-t^{-\frac{1}{2}})\frac{Y^{-m}-Y^{m}} {Y^{-2}-1}).\notag \end{eqnarray}
We are now ready to introduce the {\em $\mathfrak{p}$-adic spherical functions}. In this (algebraic) approach, they are \begin{equation*} \varphi_{m}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\frac{1+t^{1/2}T}{1+t}\,\varepsilon_{m},\ m\ge 0. \end{equation*} Using formulas (\ref{yepm}), (\ref{y-epm}) and the commutativity of $Y+Y^{-1}$ with $T$ (check it directly or see below), we establish the {\em symmetric Pieri rules}: \begin{align} &(Y+Y^{-1})\varphi_m\ =\ t^{1/2}\varphi_{m+1}+t^{-1/2}\varphi_{m-1} \hbox{\ as\ } m>0,\notag\\ &(Y+Y^{-1})\varphi_0\ =\ (t^{1/2}+t^{-1/2})\varphi_{1}. \label{pieri} \end{align}
Note that the latter relation follows from the former if one formally imposes the periodicity condition $\varphi_{-1}=\varphi_1$. By construction, $\varphi_0=1$; all other functions can be calculated using the Pieri rules. All $\varphi_{i}$'s are invariant under $s:\,Y\mapsto Y^{-1}$ due to the commutativity $[Y+Y^{-1},T]=0$.
The first three $\varphi_{m}$'s are as follows: \begin{eqnarray*} \varphi_{0}=1,\quad \varphi_{1}=\frac{Y+Y^{-1}}{t^{1/2}+t^{-1/2}}, \quad \varphi_{2}=\frac{(Y+Y^{-1})^{2}}{1+t}-t^{-1}. \end{eqnarray*}
For the system $A_1$, the symmetric Pieri rules look simpler than their $\varepsilon$\~counterparts, but this is exactly the other way around in higher ranks. Generally, there are no good formulas for the action of $W$\~orbitsums in the form $\sum_w Y_{w(b)}$ on the spherical functions (see (\ref{pieri})) except for the minuscule $b=\omega_r^\vee$ and $b=\theta$. Theoretically, the Pieri formulas are sufficient to calculate all $\varphi$\~polynomials, but this can be used mainly for $A_n$ and in some cases of small ranks. The nonsymmetric formulas of type (\ref{yepm}--\ref{y-ep-m}) exist (and are reasonably convenient to deal with) for arbitrary root systems.
\subsection{{\bf Spherical functions as Hall polynomials}} \label{sec:Sph via Hall} \subsubsection{\sf Macdonald's formula} In general (for any root system $R$ as above), we can define the {\em spherical function} as follows: \begin{eqnarray*} \varphi_{b}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\mathscr{P}_{+}\varepsilon_{b} = t^{-(\rho,b)}\mathscr{P}_{+}Y_{b}\mathscr{P}_{+}\in \mathscr{Y},\ b\in P^{\vee}_+. \end{eqnarray*} They become $W$\~invariant $Y$\~polynomials upon the identification of $\Delta^{\sharp}$ and $\mathscr{Y}$ (the Bernstein Lemma), where $w(Y_b)\stackrel{\,\mathbf{def}}{= \kern-3pt =} Y_{w(b)}$ for $w\in W$. Their ($\mathfrak{p}$-adic) theory was developed by Satake, Macdonald and others; we will mainly call them the {\em Macdonald spherical functions}. Macdonald established the following fundamental fact.
\begin{theorem} Let $P(t)$ be the Poincar\'e polynomial, namely, $P(t)=\sum_{w\in W}t^{l(w)}$. Then \begin{eqnarray}\label{macdvarphi} \varphi_{b}(Y)=\frac{t^{-(\rho,b)}}{P(t^{-1})} \sum_{w\in W}Y_{w(b)}\prod_{\alpha\in R_{+}} \frac{1-t^{-1}Y_{w(\alpha^{\vee})}^{-1}} {1-Y_{w(\alpha^{\vee})}^{-1}}. \end{eqnarray}\phantom{1}
$\qed$ \end{theorem}
The summation on the right-hand side is proportional to the {\em Hall-Littlewood polynomial} associated with $b\in P_+^\vee$. The potential poles (due to the denominators) will cancel each other, so it is really a Laurent $Y$\~polynomial. It can be readily deduced from the fact that all anti-symmetric polynomials in $\mathscr{Y}$ are divisible by the {\em discriminant}, the common denominator on the right-hand side. The proof of this theorem will be given in the next section.
In the case of $A_{1}$, we obtain \begin{align} \varphi_{m}=&\frac{t^{-m/2}}{1+t^{-1}} \left(\frac{Y^{m}-t^{-1}Y^{m-2}-Y^{-m-2}+t^{-1}Y^{-m}} {1-Y^{-2}}\right)\notag\\ =&\frac{t^{-m/2}}{1+t^{-1}} \left(\frac{(Y^{m+1}-Y^{-m-1})-t^{-1}(Y^{m-1}-Y^{1-m})} {Y-Y^{-1}}\right),\label{vphiformula} \end{align} which matches our calculations above based on the Pieri rules. Compare with the ``nonsymmetric" formulas (\ref{vepformulas}). Macdonald established his formula by calculating the Satake $\mathfrak{p}$\~adic integral representing the spherical function (see below).
One can try to use the Pieri rules to justify the theorem, but as we noted above, reasonably simple explicit formulas exist only for $A_n$ and in some cases of small ranks. There is another, much more direct approach (any root systems), which can be generalized to DAHA theory. We will switch to it after the following remarks clarifying the $\mathfrak{p}$\~adic origins of the Pieri rules, to be continued in Section \ref{sect:S-Mtheory} on the classical $\mathfrak{p}$\~adic theory of spherical functions.
\subsubsection{\sf Comments on Pieri rules} Formulas (\ref{pieri}) match the classical arithmetical definition of the (one-dimensional) Hecke operator. Let $t$ be the cardinality of the residue field of a $\mathfrak{p}$\~adic field $K$ ($t=p$ for ${\mathbb Q}_p$). The {\em Bruhat-Tits building} of type $A_1$ is a {\em tree} with $t+1$ edges from each vertex; the {\em vertices} $\{v\}$ correspond to the maximal parahoric subgroups of $G=PGL_2(K)$, which are (all) conjugated to $U=PGL_2(\mathcal{O})\subset G=PGL_2(K)$ for the ring of integers $\mathcal{O}\subset K$. Two vertices are connected by an {\em edge} if their intersection is an Iwahori subgroup, i.e., is conjugated to $B=\{g\in U\,\mid\, g_{21}\in \mathfrak{p}\}$ for the maximal ideal $\mathfrak{p}\subset \mathcal{O}$. The group $G$ naturally acts on this tree by conjugation. Identifying the vertices with the cosets of $G/U$, the action of $G$ becomes left regular.
Let $d(v)$ be the distance (in the tree) of the vertex $v$ from the origin $o$, which corresponds to $U$. The functions $f(m)$ on this tree depending only on the distance $m=d(v)\ge 0$ are exactly the functions on $G/\!/\,U=U\backslash G/U$. The figure is as follows ($t=p=3$): \begin{equation*} \xy <1cm,0cm>: (0,1)*=0{\bullet}="-" ; (1,1)*=0{\bullet}="*", **@{-}; (2,1)*=0{\bullet}="*", **@{-}; (3,1)*=0{\bullet}="*", **@{.}; (5,1)*=0{\bullet}="*", **@{-}; (4,1)*=0{\bullet}="*", **@{-}; (5,0)*=0{\bullet}="*", **@{-}, (4,1); (5,2)*=0{\bullet}="*", **@{-}, (2,1); (0,0.6)*+{^0}; (1,0.6)*+{^1}; (2,0.6)*+{^2}; (3,0.6)*+{^{m-1}}; (4,0.6)*+{^m}; (5.2,1.6)*+{^{m+1}}; (5.2,-0.4)*+{^{m+1}}; (5.2,0.6)*+{^{m+1}}; \endxy \end{equation*}
The classical {\em Hecke operator} is the (radial) Laplace operator $\Delta$ on this tree, the averaging over the neighbors. Explicitly, $$ \Delta f(m)=\frac{tf(m+1)+f(m-1)}{t+1} \ \hbox{ for } \ m>0, \ \, \Delta f (0)=\frac{f(1)}{t+1}. $$ Thus (\ref{pieri}) is exactly the eigenvalue problem for $(t^{1/2}+t^{-1/2})\Delta$ with the eigenvalue $Y+Y^{-1}$, where $Y$ is treated as a free parameter.
For arbitrary Chevalley groups, a combinatorial definition of the Laplace-type operator and its higher analogs in terms of the Bruhat-Tits buildings becomes involved. The case of $A_n$ was considered by Drinfeld.
The Bruhat-Tits building is equally useful in the theory of {\em Whittaker functions}. There is a unique infinite path from the origin such that the elements of the {\em unipotent} subgroup $N\subset G$ preserve its {\em direction to infinity}; only the direction, any finite number of vertices can be ignored. Let us extend this path to a {\em road}, infinite in both directions. Then any vertex can be mapped onto this road (identified with $N\backslash G/U$) using $N$; its image is unique. The Whittaker function can be interpreted as a function on this road, nonzero only on the original (positive) path; see Section \ref{sect:Whitf} below for more detail.
\subsubsection{\sf The major limits}\label{sect:thelimits} Let us switch from the normalization we used (compatible with the $\mathfrak{p}$\~adic Hecke operators), to the one more convenient algebraically. Namely, we set $\widetilde{\varphi}_{m}\stackrel{\,\mathbf{def}}{= \kern-3pt =} t^{m/2}\varphi_{m}$, which readily simplifies the (symmetric) Pieri rules: $$ (Y+Y^{-1})\widetilde{\varphi}_{m}= \widetilde{\varphi}_{m+1}+\widetilde{\varphi}_{m-1}. $$ This recurrence has the following elementary solutions for $m\ge 0$.
1) The monomial symmetric functions (divided by $2$): $$\mathcal{M}_{m}=(Y^{m}+Y^{-m})/2. $$
2) The classical Schur functions $\chi_{m}$: $$ \chi_{m}=\frac{Y^{m+1}-Y^{-m-1}}{Y-Y^{-1}}. $$
3) The renormalized Macdonald spherical functions: $$ \widetilde{\varphi}_{m}=\frac{1}{1+t^{-1}}\cdot \frac{Y^{m+1}-Y^{-m-1}-t^{-1}(Y^{m-1}-Y^{1-m})}{Y-Y^{-1}}. $$
All three sequences begin with $1$ at $m=0$. They are different due to the {\em boundary conditions\,} at $m=-1$\,: $$ 1)\,\mathcal{M}_{-1}=\mathcal{M}_1,\ \ \,2)\,\chi_{-1}=0,\ \ \, 3)\,\widetilde{\varphi}_{-1}= \widetilde{\varphi}_{1}t^{-1}. $$ The first two cases are limits of the third one: \begin{equation*} \xymatrix{ -\chi_{m-2} & \widetilde{\varphi}_{m}\ar[l]_{\quad t\to 0} \ar[d]_{t\to 1}\ar[r]^{t\to \infty} &\chi_{m}\\ & \mathcal{M}_{m}}. \end{equation*} The limit $t\to \infty$ is actually the degeneration of the Macdonald spherical functions to the Whittaker functions; see Section \ref{sect:Whitf}.
\subsubsection{\sf The nonsymmetric case} The Matsumoto spherical functions are right $U$\~invariant and left Iwahori\~invariant, so they can be naturally identified with the functions depending on the distances from the origin $\,o\,$ in the following two halves of the Bruhat-Tits building\,:\\ ($+$) the paths from $\,o\,$ through the {\em nonaffine} neighbors of $\,o\,$ ($t$ of them),\\ ($-$) the paths from $\,o\,$ through the {\em affine} neighbor $\,\widehat{o}\,$ of $\,o\,$ (only one).
The elements of $B\subset G$ are exactly those preserving $\,o\,$ and the edge between $\,o\,$ and $\,\widehat{o}\,$. We will measure the distance using negative numbers in the second half ($-$). Then the functions on $B\backslash G/U$ become $f(m)$ for $m\in {\mathbb Z}$, where $m=d'(v)\in Z$ for the new distance (may be negative).
Check that $d'(v)$ is the only invariant of the vertex under the action of the Iwahori subgroup and interpret combinatorially formulas (\ref{yepm},\ref{yep-m}) in terms of $m=d'(v)$.
Let us switch in (\ref{vepformulas}) to
$\widetilde{\varepsilon}_m=t^{|m|/2}\varepsilon_m$. Then \begin{align}\label{veptilde} &\widetilde{\varepsilon}_m=t^{m/2}\varepsilon_m=Y^m,\ \widetilde{\varepsilon}_{-m}=Y^{-m}+ (1-t^{-1})\frac{Y^{-m}-Y^{m}} {Y^{-2}-1}, \end{align} where $m\ge 0$. There is no dependence on $t$ for nonnegative indices (so the corresponding limits are trivial). The graph of the limits for $-m\,(m>0)\,$ reads as follows: \begin{equation*} \xymatrix{ \infty & \widetilde{\varepsilon}_{-m}\ar[l]_{\quad t\to 0\quad} \ar[d]_{t\to 1}\ar[r]^{t\to \infty} &\chi_m\\ & Y^{-m}}. \end{equation*}
\subsubsection{\sf Proof of Macdonald's formula} Recall that the affine Hecke algebra $\mathcal{H}$ in the $T$-$Y$\~presentation is generated by the elements $T_{1}, \ldots, T_{n}$ and $Y_{b}$ for $b\in P^{\vee}$. The defining relations between $T_{i}$'s and $Y_{b}$'s are: \begin{align}\label{tyhecke} &T_{i}^{-1}Y_{b}T_{i}^{-1}=Y_{b}Y^{-1}_{\alpha_{i}},\ \text{ if }(b, \alpha_{i})=1,\\ &T_{i}Y_{b}=Y_{b}T_{i}, \text{ if }(b, \alpha_{i})=0,\ i>0. \label{t-yhecke} \end{align} The connection with the original definition is as follows: \begin{align*} T_{0}=Y_{\theta}T_{s_{\theta}}^{-1}, \ \pi_{r}=Y_{\omega^{\vee}_{r}}T_{u_{r}}^{-1}, \end{align*} where $u_r$ are from (\ref{piromr}).
Formulas (\ref{tyhecke}),(\ref{t-yhecke}) are actually the relations of the orbifold braid group of ${\mathbb C}^*/W$. Using the quadratic relations, \begin{equation}{\label{lus}} T_{i}Y_{b}-Y_{s_{i}(b)}T_{i}=(t^{1/2}-t^{-1/2}) \frac{Y_{s_{i}(b)}-Y_{b}}{Y^{-1}_{\alpha_{i}^{\vee}}-1},\, i>0. \end{equation} These formulas are due to Lusztig (see e.g., \cite{L}).
\begin{lemma}\label{Berns} The center of the affine Hecke algebra is $$Z(\mathcal{H})=\mathscr{Y}^{W}=\mathbb{C}[Y_{b}]^{W}.$$ \end{lemma} {\em Proof.} By regarding both sides of \eqref{lus} as operators on $\mathscr{Y}\ni f(Y)$, we have \begin{equation}\label{lusoper} T_{i}(f)=t^{1/2}s_i(f)+(t^{1/2}-t^{-1/2}) \frac{s_{i}(f)-f}{Y^{-1}_{\alpha_{i}^{\vee}}-1}. \end{equation} Thus $T_{i}(f)=t^{1/2}f$ for all $i>0$ are equivalent to the relations $s_{i}(f)=f$ for all $i>0$, which means that $f\in \mathscr{Y}^{W}$. \phantom{1}
$\qed$
\begin{theorem}[{\sf Operator Macdonald Formula}]{\label{MAC}} Let $$\widetilde{M}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \prod_{\alpha\in R_+}\frac{1-t^{-1}Y_{\alpha^{\vee}}^{-1}} {1-Y_{\alpha^{\vee}}^{-1}}.$$ Then we have the following identity of operators acting in $\mathscr{Y}$ \begin{equation}{\label{mac1}} P(t^{-1})\mathscr{P}_{+}=(\sum_{w\in W}w)\circ \widetilde{M}, \end{equation} Using the definition of $\mathscr{P}_{+}$, \begin{equation}{\label{mac2}} \sum_{w\in W}T_{w}^{-1}t^{-l(w)/2}=(\sum_{w\in W}w)\circ \widetilde{M}. \end{equation} Equivalently, (\ref{mac1}) holds in the (abstract) algebra \,$\mathcal{B}$\, of operators generated by $W\ni w$ and rational functions in terms of \,$\{Y_b\}$ subject to the relations \ $w Y_b w^{-1}=Y_{w(b)}$\, ($w\in W,b\in P$). \end{theorem}
{\em Proof.} The equivalence of \eqref{mac1} and \eqref{mac2} is due to \begin{equation*} \mathscr{P}_{+}=\frac{\sum_{w\in W}t^{l(w)/2}T_{w}} {\sum_{w\in W}t^{l(w)}} =\frac{\sum_{w\in W}t^{-l(w)/2}T^{-1}_{w}} {\sum_{w\in W}t^{-l(w)}}. \end{equation*} Indeed, both operators are divisible by $1+t^{1/2}T_i$ on the right and on the left for any $i>0$, and act identically on $1\in \mathscr{Y}$ (which provides the exact normalization factors).
Following \cite{Ch10} (upon the affine degeneration), let us introduce the following involution acting on the operators from the algebra $\mathcal{B}$, \begin{align} \iota: Y_{b}\mapsto Y_{b},\ t^{1/2}\mapsto -t^{-1/2}, \ s_{i}\mapsto -s_{i}. \end{align} Applying this involution to the operator from (\ref{lusoper}), \begin{equation*} T_{i}=t^{1/2}s_{i}+\frac{t^{1/2}-t^{-1/2}} {Y_{\alpha_{i}^{\vee}}^{-1}-1}(s_{i}-1), \end{equation*} one readily obtains \begin{equation*} T_{i}^\iota\ =\ t^{-1/2}s_{i}- \frac{t^{1/2}-t^{-1/2}}{Y_{\alpha_{i}^{\vee}}^{-1}-1} (s_{i}+1). \end{equation*}
The $q\to 0$ limit of the $\mu$\~function\, from the DAHA theory is $$ M\stackrel{\,\mathbf{def}}{= \kern-3pt =} \prod_{\alpha\in R_+}\frac{1-Y_{\alpha^{\vee}}^{-1}} {1-tY_{\alpha^{\vee}}^{-1}}. $$ This function is {\em equivalent} ($\leftrightarrows$) to $\widetilde{M}$ in the following sense: they coincide up to a $W$\~invariant factor. Indeed, $$ \widetilde{M}\leftrightarrows \widetilde{M}'\stackrel{\,\mathbf{def}}{= \kern-3pt =}\prod_{\alpha\in R_+}\frac{1-Y_{\alpha^{\vee}}} {1-t^{-1}Y_{\alpha^{\vee}}}\leftrightarrows M .$$
\begin{lemma}\label{IOTAM} $MT_{i}M^{-1}=T_{i}^{\iota}$ for $i=1,\ldots,n$ (see \cite{Ch10}). \phantom{1}
$\qed$\end{lemma}
\begin{lemma}\label{TLEFTRIGHT} For $i\ge 1$,
$T_{i}\,+t^{-1/2}=(s_{i}+1)\cdot F_i$ for a rational function $F_i(Y)$,
$T_{i}^{\iota}+t^{-1/2}=G_i\cdot(s_{i}+1)$ for a rational function $G_i(Y)$. \phantom{1}
$\qed$\end{lemma}
Returning to the proof of the theorem, $\mathscr{P}_{+}\circ \widetilde{M}^{-1}\leftrightarrows \mathscr{P}_{+}\circ M^{-1}$, and these operators are divisible by $(1+t^{1/2}T_i)$ on the left and by $(1+t^{1/2}T_i^\iota)$ on the right. The left divisibility is straight from that of $\mathscr{P}_{+}$; the right divisibility results from Lemma \ref{IOTAM}.
Using Lemma \ref{TLEFTRIGHT}, we obtain that $\mathscr{P}_{+}\circ \widetilde{M}^{-1}$ is divisible on the right and on the left by $(s_i+1)$. Thus it commutes with the operators of multiplication by functions from $\mathscr{Y}^W$ and must be in the form $G(Y)\circ\sum_{w\in W}w$ for a $W$\~invariant (rational) function $G(Y)$. Hence, $G=P(t^{-1})^{-1}$ due to $\sum_{w\in W}w(\widetilde{M})=P(t^{-1})$. The latter is an immediate corollary of the divisibility of antisymmetric Laurent polynomials by the discriminant; see \cite{Bo} and \cite{Hu}, formula (35), Section 3.20. \phantom{1}
$\qed$
The operator Macdonald formula is actually from \cite{Ma5}, formula (5.5.14). We deduced this from Lemma \ref{IOTAM}; Macdonald checks the divisibility of the operator $(\sum_{w\in W}w)\circ \widetilde{M}$ by $1+t^{1/2}T_i$ on the left and on the right directly. Then he equates the leading terms in (\ref{mac1}), the coefficients of the longest element $w_0\in W$. Note that his last step cannot be used in DAHA theory (the longest element does not exist in $\widehat{W}$). We think that the interpretation of $M$ and $\mu$ from \cite{Ch10} as intertwiners between the symmetric and antisymmetric polynomial representations clarifies well their appearance in this context.
\subsection{{\bf Satake-Macdonald theory}}\label{sect:S-Mtheory} \subsubsection{\sf Chevalley groups}\label{sect:Cheva} Let $K$ be a $\mathfrak{p}$\~adic field and $\mathcal{O}\subset K$ the valuation ring in $K$ with the (unique) prime ideal $\mathfrak{p}=(\varpi)$ for the uniformizing element $\varpi$.
We set $t=|k|$, where $k$ is the residue field $\mathcal{O}/(\varpi)$.
For an irreducible reduced root system $R$ as above and the coweight lattice $P^{\vee}$, the Lie algebra $\mathfrak{g}_{K}$ is defined as the $\mathfrak{g}\otimes K$ for the Lie algebra $\mathfrak{g}$ defined over ${\mathbb Z}$ as the span of $\{x_{\alpha}, h_{b}\}$ for $\alpha\in R, b\in P^{\vee}$ subject to the relations \begin{align*} &[h_a,h_b]=0,\ [h_{b},x_{\alpha}]=(b, \alpha)x_{\alpha}, \ [x_{\alpha}, x_{-\alpha}]=h_{\alpha^\vee}, \\ &[x_{\alpha},x_{\beta}]=N_{\alpha,\beta}x_{\alpha+\beta}\ \text{ if }\alpha+\beta\in R,\, \text{ otherwise } 0. \end{align*} Accordingly, $\mathfrak{g}_{\mathcal{O}}=\mathfrak{g}\otimes \mathcal{O}$. The integers $N_{\alpha,\beta}$ can be chosen here uniquely up to signs; we will omit their discussion.
The unipotent groups $X_{\alpha}$ are defined for $\alpha\in R$ as ``exponents" of $Kx_{\alpha}$; $H$ is the $K$\~torus corresponding to $P^\vee$. By construction, these groups act on $\mathfrak{g}_K$. We will also need the group lattice formed by the elements $\varpi^b\in H$ for $b\in P^\vee$ defined as follows: $$ \varpi^{b}(x_{\alpha})=\varpi^{(b,\alpha)}x_{\alpha},\, \forall\, \alpha\in R. $$ Finally, the (split) {\em Chevalley group} $G$ is the span of $X_\alpha$ for all $\alpha\in R$ and $H$. The standard {\em unipotent subgroup} $N$ is the group span of $X_\alpha$ for $\alpha\in R_+$. The {\em maximal parahoric subgroup} $U$ is the centralizer of $\mathfrak{g}_{\mathcal{O}}$ in $G$. Note that $P^\vee$ is used here; if it is replaced by $Q^\vee$, then the corresponding group is the group of $K$\~points of the connected simply connected split algebraic group associated with $R$.
We have the Cartan decomposition of $G$ \begin{equation} G=UH_{+}U=\bigcup_{b\in P^{\vee}_{+}} U\varpi^{b}U, \end{equation} and the Iwasawa decomposition \begin{equation}{\label{Iwasawa}} G=UHN=\bigcup_{b\in P^{\vee}} U\varpi^{b}N\,; \end{equation} the unions are disjoint.
As an exercise, introduce the Chevalley group corresponding to the {\em twisted affinization} $\tilde R^\nu$ of $R$ considered in Section \ref{sect:thinaff}. Using algebraic groups, it will be a group of $K$\~points of a nonsplit group over $K$, which splits over certain ramified extension of $K$.
\subsubsection{\sf The Satake integral} Let $L(G,U)$ be the space of complex valued functions $f$ on $G$, compactly supported, satisfying the bi-$U$-invariance condition: \begin{equation*} f(u_{1}x u_{2})=f(x) \text{ for all } x\in G, \text{ and any } u_{1}, u_{2}\in U. \end{equation*}
This is a ring; the product of two functions $f,g\in L(G,U)$ is defined by the {\em convolution\,} \begin{equation*} f\ast g(x)=\int_{G}f(xy^{-1})g(y)dy, \end{equation*} where $dy$ is the Haar measure on $G$ normalized by $\int_{U}dy=1$. Moreover, it is a commutative ring (use the ``$-1$"\~automorphism of $R$ and $R^\vee$ extended to $G$).
The {\em zonal spherical function} on $G$ relative to $U$ is a continuous bi-$U$-invariant complex-valued function $\Phi$ on $G$ satisfying the following condition: \begin{align} \label{sphdef} &\Phi\ast f =c_{f}\Phi \hbox{\ \, for\ any\ \,} f\in L(G,U), \end{align} and for the constants $c_{f}$ depending on $f$. In other words, $\Phi$ is a common eigenfunction of all the convolution operators with the elements $f\in L(G,U)$; then $c_{f}$ are the corresponding eigenvalues. The normalization is $\Phi(1)=1$.
Satake (following Harish-Chandra) found that an {\em arbitrary} zonal spherical function can be uniformly described in terms of the vector $\lambda\in {\mathbb C}\otimes_{{\mathbb Z}}P\cong {\mathbb C}^n$. Using the Iwasawa decomposition \eqref{Iwasawa}, let us define the projection map onto $P^\vee$ \begin{align} &\mathrm{pr}:G\ \to\ P^{\vee}, \ x\in U\varpi^{b}N\ \mapsto \ b. \end{align} Using this map, the zonal spherical functions are given as follows: \begin{equation}\label{satake} \Phi_{\lambda}(x)=\int_{U}t^{(\mathrm{pr} (x^{-1}u),\rho-\lambda)}du \end{equation} for the Haar measure restricted to $U$.
Macdonald calculated this integral in \cite{Ma1} using the combinatorics of $U$. This was not too simple; see his Madras lectures \cite{Ma2} (the lectures also include relations to the real theory, positivity matters and other issues). It suffices to evaluate $\Phi_{\lambda}$ at $\varpi^b$. His formula reads as \begin{align}\label{macdformula} &\Phi_\lambda (\varpi^b)=\frac{1}{P(t^{-1})} \sum_{w\in W}t^{(b,w(\lambda)-\rho)}\,\prod_{\alpha\in R_+} \frac{1-t^{-1-(\alpha^\vee,w(\lambda))}}{1-t^{-(\alpha^\vee,w(\lambda))}}. \end{align} Connecting $\mathfrak{p}$\~adic theory and our algebraic approach can be achieved by replacing $Y_{b}$ by $t^{(b,\lambda)}$, namely, \begin{align*} &\varphi_b(Y)\ =\ \Phi_{\lambda}(\varpi^b)\,\bigl [t^{(b,\lambda)}\mapsto Y_b \bigr]. \end{align*} Recall that in (\ref{macdvarphi}), \begin{eqnarray*} \varphi_{b}(Y)=\frac{t^{-(\rho,b)}}{P(t^{-1})} \sum_{w\in W}Y_{w(b)}\prod_{\alpha\in R_+} \frac{1-t^{-1}Y_{w(\alpha^{\vee})}^{-1}} {1-Y_{w(\alpha^{\vee})}^{-1}}. \end{eqnarray*}
\subsubsection{\sf The universality principle} The approach via the Matsumoto spherical functions establishes a bridge between the algebraic theory above and the $\mathfrak{p}$\~adic theory, and {\em proves} (\ref{macdformula}) without taking a single $\mathfrak{p}$\~adic integral.
The coincidence of these two theories, algebraic and $\mathfrak{p}$\~adic, can be also seen by observing that the defining relations from (\ref{sphdef}) are nothing but the Pieri rules in the algebraic theory. However this is with the reservation that the (symmetric) Pieri rules are generally not explicit.
One can also use the following {\em universality principle}.
Formula (\ref{sphdef}) ensures that there exists a family of {\em pairwise commutative} {\em difference} operators in terms of $b$; they are convolutions with different $f\in L(G,U)$. It is not necessary to know exactly how the convolution is defined; it can be of any origin, say, from geometric theories. Provided there exist such operators (differential or difference) and certain natural {\em symmetries}, such a family is essentially unique for a given root system.
This claim can be made rigorous if more information on the structure of difference or differential operators under consideration is available.
The key point is that we have very few such families in mathematics (subject to certain symmetries and boundary conditions). Cf. the discussion in Section \ref{sect:thelimits}. Major examples come from the theory of Macdonald polynomials and DAHA, from their counterparts, generalizations and degenerations. In physics, the same phenomenon is the universality of the quantum many body problem.
Thus, one can expect {\em a priori} (or even conclude rigorously) that $\mathfrak{p}$\~adic spherical functions must be proper specializations of the Macdonald polynomials. In our case, specialization of the general $q,t$\~theory is by letting $q\to 0$ and under minor renormalizations. The justification of this connection is straightforward if the algebraic approach via the Matsumoto functions is used. However, it is not obvious at all if the spherical functions and the operators are defined $\mathfrak{p}$\~adically, via the convolution on $G$.
\subsubsection{\sf Whittaker functions}\label{sect:Whitf} The universality principle discussed above works equally well for the Whittaker functions. We introduce them following \cite{CS} with some simplifications; see also \cite{Shi} for the $GL_n$\~case. The notation is from Section \ref{sect:Cheva}.
The theory of $q$\~Whittaker functions will be discussed in the second part of this work, including the nonsymmetric (spinor) functions. A natural challenge is to define the Matsumoto-type (``nonsymmetric") p-adic Whittaker functions (which can be only spinor ones); their definition is outlined below.
The unramified $\mathfrak{p}$\~adic Whittaker function $\mathcal{W}$ is introduced for an additive character $\psi$, the product of the ($K$\~additive) characters $\psi_i: K\to K/\mathcal{O} \to {\mathbb C}^*\ (i=1,\ldots,n)$; each $\psi_i$ must be nontrivial on $\varpi^{-1}\mathcal{O}/\mathcal{O}$. This can be naturally extended to a character of the group $N$ (vanishing on $X_\alpha$ for nonsimple roots $\alpha>0$).
For an algebra homomorphism $\chi:\,L(G,U)\to {\mathbb C}$, there is a unique function $\mathcal{W}_{\chi}$ on $G$ such that $\mathcal{W}_{\chi}(1)=1$, \begin{align} \label{whit} &\mathcal{W}_\chi(ngu)=\psi(n)\,\mathcal{W}_\chi(g) \ \hbox{ for } \ n\in N,\, u\in U,\, g\in G, \\ &\ \hbox{ and } \ \mathcal{W}_\chi\ast f =\chi(f)\,\mathcal{W} \hbox{\ \, for\ any\ \,} f\in L(G,U). \notag \end{align}
Similar to the spherical function $\Phi$, it suffices to know the values $\mathcal{W}_{\chi}(\varpi^b)$ for $b\in P^\vee$. However, $\mathcal{W}_{\chi}(\varpi^b)$ is {\em not} a $W$\~invariant function of $b$. Moreover, $\mathcal{W}_{\chi}(\varpi^b)=0$ unless $b\in P^\vee_+$ (anti-dominant in Lemma 5.1 from \cite{CS}).
The universality principle is actually sufficient to conclude/expect that, up to a certain renormalization, $\mathcal{W}_{\chi}(\varpi^b)$ {\em does not depend on $t$} (a surprising fact!) and that it is a classical finite-dimensional character of the Langlands dual group of $G$. Here the corresponding dominant weight is $b$ and $\chi$ must be treated as the argument. See Theorem 5.4 from \cite{CS} and \cite{Shi} for the precise statements.
The fact that $\mathcal{W}_{\chi}(\varpi^b)$ vanishes for $b\not\in P^\vee_+$ is the key here.
It provides the boundary condition sufficient to identify the Whittaker functions with the characters (practically without calculations). Cf. Section \ref{sect:thelimits}, case (2). A counterpart of this property in the theory of real and complex Whittaker functions is a certain decay condition; see \cite{ChW} for the $q$\~Whittaker functions.
Let us demonstrate the mechanism of this vanishing property in the case of $GL_2(K)$. Using the first relation from the definition of $\mathcal{W}=\mathcal{W}_\chi$,
\begin{align*} &\psi(\varpi^{-1}) \mathcal{W}\Biggl(\left(\begin{array}{cc} \varpi^n & 0 \\mathbf{0} & \varpi^{n+1} \\ \end{array}\right)\Biggr) =\mathcal{W}\Biggl( \left(\begin{array}{cc} 1& \varpi^{-1} \\mathbf{0} & 1 \\ \end{array}\right) \left(\begin{array}{cc} \varpi^n & 0 \\mathbf{0} & \varpi^{n+1} \\ \end{array}\right) \Biggr)\\ &=\mathcal{W}\Biggl( \left(\begin{array}{cc} \varpi^n& \varpi^n \\mathbf{0} & \varpi^{n+1} \\ \end{array}\right) \Biggr) =\mathcal{W}\Biggl( \left(\begin{array}{cc} \varpi^n& \varpi^n \\mathbf{0} & \varpi^{n+1} \\ \end{array}\right) \left(\begin{array}{cc} 1& -1 \\mathbf{0} & 1 \\ \end{array}\right) \Biggr)\\ &=\mathcal{W}\Biggl(\left(\begin{array}{cc} \varpi^n & 0 \\mathbf{0} & \varpi^{n+1} \\ \end{array}\right)\Biggr) =0 \hbox{\ \, due to \ \,} \psi(\varpi^{-1})\neq 1. \end{align*} At the level of formulas, $\mathcal{W}_{\chi}(\varpi^b)$ for $\lambda=\chi$ is the limit $t\to \infty$ of the $\mathfrak{p}$\~adic spherical function from (\ref{macdformula}); see Section \ref{sect:thelimits} for the demonstration in the $A_1$\~case.
Generalizing (any root systems), we claim that the $\mathfrak{p}$\~adic Whittaker functions can be obtained as limits of the properly normalized spherical functions when the cardinality of the residue field $k$ tends to $\infty$. I.e., we replace the starting $\mathfrak{p}$\~adic field by (the completion of) its maximal unramified extension; the limiting procedure can be correctly defined. It results in the switch from the affine Hecke algebra to the affine nil-Hecke algebra. The Matsumoto functions go to new {\em spinor-Whittaker functions} in this limit.
Let us make this explicit for $A_1$. The quadratic relation becomes $T(T-1)=0$ in such a limit. Correspondingly, $T^{-1}$ in the formulas must be replaced by $T'\stackrel{\,\mathbf{def}}{= \kern-3pt =} T-1$. For instance, the relation $TYT=Y^{-1}$ now becomes $T' Y=Y^{-1} T$; more generally, \begin{equation*} T Y^n-Y^{-n}T= \frac{Y^{-n}-Y^n}{Y^{-2}-1} \ \hbox{ for } \ n\in {\mathbb Z}. \end{equation*} Cf. Section \ref{sec:rank1} above.
The definition of the Matsu\-mo\-to- Whittaker function remains $T_{\widehat{w}}\mathscr{P}_+$ for $\widehat{w}\in \widehat{W}$ and for the symmetrizer $\mathscr{P}_+$, which is now simply $T$ (for $A_1$). Following (\ref{vepformulas}), they must be expressed in terms of $Y^{\pm 1}$. Setting $\psi_{-n}=Y^n T$ for $n\ge 0$, the nil-counterpart of $(T\pi)^n \mathscr{P}_+\,$ is \begin{align*} &\psi_n\, =\, T Y^n T= Y^{-n} T+ \frac{Y^{-n}-Y^n}{Y^{-2}-1}T =(\sum_{m=0}^n Y^{n-2m})T \ \hbox{ for } \ n\ge 0. \end{align*}
Thus $\psi_n=\{\,Y^{|n|}\hbox{ for } n\le 0,\ (Y^{n+1}-Y^{-n-1})/(Y-Y^{-1})\hbox{ for } n> 0\,\}.$
The identities $T\psi_{-n}=\psi_n=T\psi_{n}\, (n\ge 0)$ are directly connected with the theory of the second part of this work. The symmetrization of the spinor Whittaker function (here applying $T$) must be the {\em diagonal spinor} (under the symmetry $n\to -n$) constructed from the ``symmetric" Whittaker function.
The connection to the {\em spinor $q$\~Whittaker function} from the second part of this work is direct. Namely, it is the limit $t\to 0$ where $\Lambda$ is replaced by $Y$. Recall that $t$, the cardinality of the residue field, changes to $t^{-1}$ in the $q,t$\~theory. The theory of the spinor $q$\~Whittaker functions for arbitrary root systems is in progress, including the $\mathfrak{p}$\~adic applications.
\setcounter{equation}{0} \section{\sc{Double affine generalizations}}{\label{sect:daha}} \subsection{{\bf Double affine Hecke algebra}}
We continue to use the notations from Section \ref{sect:affineweyl}. Let $\widehat{P}=\{\widehat{a}=[a,j]\,|\,a\in P, j\in \mathbb{Z}\}\subset \mathbb{R}^{n}\times \mathbb{R}$ be the {\em affine weight lattice}. Correspondingly, let $X_{[a,j]}\stackrel{\,\mathbf{def}}{= \kern-3pt =} X_a q^j$ for pairwise commutative $X_a$ ($X_{a+b}=X_aX_b$) and a parameter $q$ (later, a nonzero number). Setting $X_j=X_{\omega_j}$ for $j=1,\ldots,n$ (they are algebraically independent)\,: $$X_{a}=\prod_{j=1}^{n}X_{j}^{l_j}, \hbox{\ \, where\,\ } l_j=(a,\alpha_j^\vee) \text{\ \,due to \ \,} a=\sum_{j=1}^{n}l_{j}\omega_{j}. $$
Recall the definition of the action of the extended affine Weyl group $\widehat{W}=W\ltimes P^{\vee}$ in ${\mathbb R}^{n+1}$: $$ b[z,\xi]=[z, \xi-(b,z)]\, (b\in P^{\vee}),\ w[z,\xi]=[w(z),\xi]\, (w\in W). $$ Accordingly, we set $\widehat{w}(X_{\widehat{a}})\stackrel{\,\mathbf{def}}{= \kern-3pt =} X_{\widehat{w}(\widehat{a})}.$
This action is dual to the {\em standard affine action} of $\widehat{W}\ni \widehat{w}$ in ${\mathbb R}^n\ni x$ via the translations defined as $wb(x)=w(x+b)$ for $w\in W,\, b\in P^\vee$. In the space of functions of $x$, this reads as $\widehat{w}(f)(x)=f(\widehat{w}^{-1}(x))$ (notice the sign). Applying $\widehat{w}=wb\in \widehat{W}$ to $\, X_a\stackrel{\,\mathbf{def}}{= \kern-3pt =} q^{x_a}\,$ for $x_a\stackrel{\,\mathbf{def}}{= \kern-3pt =} (x,a)$, one has \begin{align}\label{xbaction} &\widehat{w}(X_a)=q^{(w^{-1}x-b,a)}=q^{(x,w(a)-(b,a))}=X_{[w(a),-(b,a)]}= X_{\widehat{w}(a)}. \end{align}
The {\em double affine Hecke algebra} (DAHA), denoted by $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$, is defined over the ring of constants ${\mathbb Z}[q^{\pm 1/m},t^{\pm 1/2}]$ for $m\in Z_+$ such that $(P,P^\vee)= \frac{1}{m}{\mathbb Z}$. In this paper we will mainly consider DAHA over the field ${\mathbb C}_{q,t}\stackrel{\,\mathbf{def}}{= \kern-3pt =} {\mathbb C}(q^{1/m},t^{1/2}).$ This algebra is generated by the affine Hecke algebra $\mathcal{H}= \langle T_{i}, i=0,\ldots,n, \Pi\rangle$ defined above and pairwise commutative elements $\{X_{a}, a\in P\}$ subject to the following {\em cross-relations}: \begin{align} &T_{i}X_{a}T_{i}=X_{a}X_{\alpha_{i}}^{-1} \text{\ if\ } (a,\alpha_{i}^{\vee})=1, \notag\\ &T_{i}X_{a}=X_{a}T_{i} \text{\ if\ } (a, \alpha_{i}^{\vee})=0,\label{dahadef} \\ &\pi_{r}X_{b}\pi_{r}^{-1}=X_{\pi_{r}(b)},\notag \end{align} where $r\in O$ is from the orbit $O$ of $\alpha_{0}$ in $Dyn^{\tt aff}$; see (\ref{heckerel}).
Recall that the $Y_{b}$ for $b\in P^\vee$ from (\ref{tyhecke}) satisfy the {\em dual} cross-relations: \begin{align*} &T_{i}Y_{b}T_{i}=Y_{b}Y_{\alpha_{i}^\vee}^{-1},\ \text{ if }(b,\alpha_{i})=1,\\ &T_{i}Y_{b}=Y_{b}T_{i}, \text{ if } (b, \alpha_{i})=0. \end{align*} Using $Y_b$ instead of $\{\pi_r,T_0\}$, $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}=\langle X_{a}\,(a\in P),\, Y_{b}\,(b\in P^\vee), T_{1}, \ldots, T_{n}\rangle$.
\subsubsection{\sf The PBW Theorem} An important fact is the PBW Theorem (actually, there are $6$ of them depending on the ordering of $X,T,Y$): \begin{theorem}[{\sf PBW for DAHA}]\label{PBWDAHA} Every element in $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ can be uniquely written in the form \begin{equation}\label{pbwcdaha} \sum_{a,w,b}C_{a,w,b}\,X_{a}T_{w}Y_b \text{\ \, for \,} C_{a,w,b}\in {\mathbb C}_{q,t},\ a\in P,\ w\in W,\ b\in P^\vee. \end{equation} \vskip -1cm\phantom{1}
$\qed$ \end{theorem}
The theorem readily results in the definition of the {\em polynomial representation} of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ in $\mathscr{X}\stackrel{\,\mathbf{def}}{= \kern-3pt =}{\mathbb C}_{q, t}[X_b]={\mathbb C}_{q,t}[X_{\omega_i}]$;
the ring ${\mathbb Z}[q^{\pm 1/m},t^{\pm 1/2}]$ is sufficient in its definition.
Using Theorem \ref{PBWDAHA}, we can identify $\mathscr{X}$ with the induced representation Ind$_\mathcal{H}^{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}\,{\mathbb C}_+$, where ${\mathbb C}_+$ is the one-dimensional module of $\mathcal{H}$ such that $T_{\widehat{w}}\mapsto t^{l(\widehat{w})/2}$.
The generators $X_{b}$ act by multiplication; $T_i (i\ge 0)$ and $\pi_r (r\in O^*)$ act in $\mathscr{X}$ as follows: \begin{eqnarray}\label{pitpolyn} &\pi_{r}\mapsto \pi_{r},\ \, T_{i}\mapsto t^{1/2}s_{i}+\dfrac{t^{1/2}-t^{-1/2}} {X_{\alpha_{i}}-1}(s_{i}-1). \end{eqnarray} Here $s_{0}(X_{b})=X_{b}X_{\theta}^{-(b, \theta)}q^{(b, \theta)}$.
{\bf Comment.\ } If one begins with formulas (\ref{pitpolyn}), then the DAHA relations for these operators are not difficult to check directly. This approach gives the PBW Theorem for $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ (the polynomial representation is faithful if $q$ is not a root of unity). In the affine case the deduction of the PBW Theorem from the (nonaffine) formulas (\ref{pitpolyn}), checked directly, is actually due to Lusztig (in one of his first papers on AHA). Kato interpreted these formulas as those in Ind$_{\mathbf{H}}^\mathcal{H}{\mathbb C}_+$ for nonaffine $\mathbf{H}$ and the plus-representation ${\mathbb C}_+$ (but then you need to use the PBW Theorem).
In the DAHA case the best way to obtain the PBW Theorem is by defining the representation $\mathscr{X}$ via the formulas from (\ref{pitpolyn})and checking that it is faithful for generic $q$. There is no problem to order $X,Y,T$ as in (\ref{pbwcdaha}) for any $q,t\in {\mathbb C}^*$ using the DAHA relations, so the polynomial representation for generic $q$ (when this representation is faithful) provides the uniqueness of such expansions (which is the key) for all $q$.
\subsubsection{\sf The mu-functions} We set \begin{equation}\label{mutildemu} \mu(X;q,t)=\prod_{\widetilde{\alpha}>0}\frac{1-X_{\widetilde{\alpha}}} {1-tX_{\widetilde{\alpha}}},\ \ \widetilde{\mu}(X;q,t)= \prod_{\widetilde{\alpha}>0}\frac{1-t^{-1}X_{\widetilde{\alpha}}} {1-X_{\widetilde{\alpha}}}. \end{equation} Following Section \ref{sect:length}, \begin{align}\label{Lahw} &\Lambda(\widehat{w})\,\stackrel{\,\mathbf{def}}{= \kern-3pt =}\,\widetilde{R}_{+}\cap \widehat{w}^{-1}(\widetilde{R}_{-})
=\{\tilde{\alpha}>0\,|\,\widehat{w}(\tilde{\alpha})<0\} \ \hbox{ for } \ \widehat{w}\in \widehat{W} \end{align} consists of $l(\widehat{w})$ positive roots. The following are the key relations for the functions $\mu,\widetilde{\mu}$: \begin{align}\label{murelations} &\frac{\widehat{w}^{-1}(\mu)}{\mu}= \frac{\widehat{w}^{-1}(\widetilde{\mu})}{\widetilde{\mu}}= \prod_{\widetilde{\alpha}\in \Lambda(\widehat{w})} \frac{1-t^{-1}X_{\widetilde{\alpha}}^{-1}}{1-X_{\widetilde{\alpha}}^{-1}}\cdot \frac{1-X_{\widetilde{\alpha}}}{1-t^{-1}X_{\widetilde{\alpha}}}\\ =&\prod_{\widetilde{\alpha}\in \Lambda(\widehat{w})} \frac{1-t^{-1}X_{\widetilde{\alpha}}^{-1}}{1-t^{-1}X_{\widetilde{\alpha}}}\cdot \frac{1-X_{\widetilde{\alpha}}}{\ \,1-X_{\widetilde{\alpha}}^{-1}}= \prod_{\widetilde{\alpha}\in \Lambda(\widehat{w})} \frac{t^{-1}-X_{\widetilde{\alpha}}}{1-t^{-1}X_{\widetilde{\alpha}}}.\notag \end{align} We see that $\mu/\widetilde{\mu}$ is (formally) a $\widehat{W}$\~invariant function. Note that both functions, $\mu$ and $\widetilde{\mu}$, are invariant under the action of $\Pi=\{\pi_r,r\in O\}$.
We will need the formula for the constant term ct$(t)$ of $\mu$ (the coefficient of $X^0$): \begin{align}\label{consterm} &\hbox{ct}(t)=\prod_{\alpha\in R_{+}}\prod_{i=1}^{\infty} \frac{(1-t^{(\alpha,\rho^{\vee})}q^{i})^{2}} {(1-t^{(\alpha,\rho^{\vee})+1}q^{i})(1-t^{(\alpha,\rho^{\vee})-1}q^{i})}. \end{align} It will be treated as an element in $\mathbb{C}[t][[q]]$; we will use this formula mainly for $t^{-1}$ instead of $t$.
\subsection{{\bf Affine symmetrizers}}
\subsubsection{\sf The hat-symmetrizers} Let us introduce formally the infinite counterpart of the $P$\~symmetrizer as follows: \begin{equation}\label{defhatP} \widehat{\mathscr{P}}_{+} =\sum_{\widehat{w}\in \widehat{W}}t^{-l(\widehat{w})/2} T^{-1}_{\widehat{w}}/\widehat{P}(t^{-1}) \ \hbox{ for } \ \widehat{P}(t)=\sum_{\widehat{w}\in \widehat{W}} t^{l(\widehat{w})}, \end{equation} the affine Poincar\'e series, which is a rational function of $t$. Here and below $\widehat{P}(t^{-1})^{-1}$ is expanded with respect to $t^{-1}$. We also set
$$ \widehat{\mathscr{P}}\,'_+\stackrel{\,\mathbf{def}}{= \kern-3pt =} \sum_{\widehat{w}\in \widehat{W}}t^{-l(\widehat{w})/2} T^{-1}_{\widehat{w}},\ \ \widehat{\mathscr{S}}\,'_+\stackrel{\,\mathbf{def}}{= \kern-3pt =}\sum_{\widehat{w}\in \widehat{W}}\widehat{w},\ \ \widehat{\mathscr{I}}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \widehat{\mathscr{S}}\,'_+ \circ \widetilde{\mu}\,. $$ All constructions below can be extended to the {\em minus-symmetrizers} (generally, to arbitrary characters of the affine Hecke algebra), but we will stick to the plus-case in this paper.
We understand these operators in this paper mainly (but not always) as follows. Let us move all $\widehat{w}\in \widehat{W}$ in the series for $\widehat{\mathscr{S}}\,'_+ \circ \widetilde{\mu}$ to the right and expand the coefficients in terms of $X_{\alpha_i}$ for $i=0,\ldots,n$. Such expansions will contain only nonnegative powers of $q$. Similarly, $t^{-l(\hat{u})/2}T_{\hat{u}}^{-1}$ are understood as operators in the polynomial representation, where we move all $\widehat{w}$ to the right. The resulting coefficients will be {\em infinite} sums in terms of $X_{\alpha_i}\,(i\ge 0)$ by construction, to be analyzed in the next theorem, which extends Theorem \ref{MAC} to the affine case.
\subsubsection{\sf The kernel and the image} \begin{theorem}\label{KERIMAGE} (i) The coefficients of $\widehat{w}$ in the above representations of $\widehat{\mathscr{S}}\,'_+ \circ \widetilde{\mu}$ and $\widehat{\mathscr{P}}\,'_+$ will contain only nonpositive powers of $t$. These coefficients are well defined as formal series in terms of $X_{\alpha_i}$ for
$i\ge 0$ and $t^{-1}$. Moreover, provided that $|q|<1$ and $|t|>1$, the coefficients of individual $X_a \widehat{w}\,(a\in Q\subset P, \widehat{w}\in \widehat{W})\,$ will converge as series in terms of $q,t^{-1}$.
(ii) Letting $\mathcal{A}=\widehat{\mathscr{P}}\,'_+$ or $\mathcal{A}=\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$, the following annihilation properties hold: \begin{align}\label{annihil} (\widehat{w}-1)\mathcal{A}\ =\ 0\ =\ &(t^{-\frac{l(\widehat{w})}{2}}T_{\widehat{w}}-1)\mathcal{A} \notag\\ \ =\ 0\ =\ &\mathcal{A}(t^{-\frac{l(\widehat{w})}{2}}T_{\widehat{w}}-1). \end{align} The products in (\ref{annihil}) must be transformed in the same way as $\widehat{\mathscr{S}}\,'_+ \circ \widetilde{\mu}$ and $\widehat{\mathscr{P}}\,'_+$. Namely, all $\{T_{\widehat{w}}^{\pm 1}\}$ must be expressed via $\{\widehat{w}\}$ using (\ref{pitpolyn})\,; then all $\widehat{w}$ must be moved to the right and, finally, the resulting coefficients of $\widehat{w}$ must be expanded as series from ${\mathbb Z}[[t^{-1/2},X_{\alpha_i},i\ge 0]]$.
(iii) The right multiplication by \, $(t^{-\frac{l(\widehat{w})}{2}}T_{\widehat{w}}-1)$\, is well defined for any series $\mathcal{C}=\sum_{\hat{u}}\, C_{\hat{u}}\, \hat{u}$ with the coefficients in ${\mathbb Z}[[t^{-1/2},X_{\alpha_i},i\ge 0]]$ or its localization by $t$. Namely, given $\widehat{w}\in \widehat{W}$, \begin{align*} &\mathcal{C}\,(t^{-\frac{l(\widehat{w})}{2}}T_{\widehat{w}}-1)\ =\ \sum_{\hat{u},\hat{v}} \,C_{\hat{u}} B^{\hat{u}}_{\hat{v}}\,\hat{u} \hat{v} \ \hbox{ for } \ \\ &\hat{u} \,(t^{-\frac{l(\widehat{w})}{2}}T_{\widehat{w}}-1)\ =\ \sum_{\hat{v}}\, B^{\hat{u}}_{\hat{v}}\,\hat{u} \hat{v},\ \, B^{\hat{u}}_{\hat{v}}\in {\mathbb Z}[[t^{-1/2},X_{\alpha_i},i\ge 0]], \end{align*} where $\hat{v}$ are taken from the (finite) Bruhat set of the element $\widehat{w}$. \end{theorem} {\em Proof.} To check $(i)$ for $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$, let us divide it by $\widetilde{\mu}$ on the left. Then, using (\ref{murelations}), \begin{eqnarray}\label{tmudivided} \widetilde{\mu}^{-1}\circ\widehat{\mathscr{S}}\,'_{+}\circ\widetilde{\mu} =\sum_{\widehat{w}\in \widehat{W}}\prod_{\widetilde{\alpha}\in \Lambda(\widehat{w})} \frac{t^{-1}-X_{\widetilde{\alpha}}}{1-t^{-1}X_{\widetilde{\alpha}}}\circ \widehat{w}^{-1}, \end{eqnarray} which can be readily expanded in terms of $t^{-1}$. Multiplying (\ref{tmudivided}) by the expansion of $\widetilde{\mu}$ in terms of $X_{\alpha_i}$ for $i\ge 0$, we obtain the required.
Only the nonnegative powers of $t^{-1}$ appear in the expressions of $t^{-l(\widehat{w})/2}T_{\widehat{w}}^{-1}$ and $\widehat{\mathscr{P}}\,'_+$. Indeed, using (\ref{pitpolyn}), \begin{align*} t^{-1/2}T_{i}^{-1}\ &=\ t^{-1/2}(t^{-1/2}s_{i}+\dfrac{t^{-1/2}-t^{1/2}} {X_{\alpha_{i}}^{-1}-1}(s_{i}-1))\\ &=\ t^{-1}s_{i}+\dfrac{(t^{-1}-1)X_{\alpha_i}} {1-X_{\alpha_{i}}}(s_{i}-1). \end{align*} The $\widehat{w}$\~coefficients of $\widehat{\mathscr{P}}\,'_+$ are infinite sums, well defined due to part $(e)$ of Lemma \ref{LEMThatw} below.
We note that the operators $\widehat{\mathscr{S}}\,'_+ \circ \widetilde{\mu}$ and $\widehat{\mathscr{P}}\,'_+$ will be used later in concrete spaces; then their coefficients will be treated as (meromorphic) functions of $X,q,t$.
The convergence of the coefficients of
$\widehat{\mathscr{P}}\,'_+$ subject to $|q|<1<|t|$ is part of Theorem \ref{PSBULLET}. It can be also obtained from Theorem \ref{LEVZERO}; see an outline of its proof in Section \ref{sect: Coefficient}. The sharp estimate
is actually $|t|>q^{1/h}$ (see below).
Let $\iota$ be the involution, not an anti-involution, in $\mathscr{X}$ or acting in a proper localization of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ given by \begin{equation*} \iota: s_{i}\mapsto -s_{i}\, (i\ge 0),\, \pi_r\to\pi_r,\, \ X_{a}\mapsto X_{a},\, q\mapsto q, \, t^{1/2}\mapsto -t^{-1/2}. \end{equation*}
We have the following two lemmas extending the corresponding nonaffine Lemmas \ref{IOTAM} and \ref{TLEFTRIGHT} (used for verifying the Macdonald formula).
\begin{lemma}\label{IOTLEM} $\mu T_{i}\mu^{-1}=T_{i}^{\iota}$,\, for\, $i=0, \ldots, n$\, (see \cite{Ch10}). \phantom{1}
$\qed$ \end{lemma} \begin{lemma} For $i\geq 0$,
$t^{1/2}T_{i}+1=(s_{i}+1)\cdot F_i$ for a rational function $F_i$,
$t^{1/2}T_{i}^{\iota}+1=G_i\cdot(s_{i}+1)$ for a rational function $G_i$. \phantom{1}
$\qed$ \end{lemma} Note that the automorphism $\,H\mapsto \mu^\iota H^\iota (\mu^\iota)^{-1}\,$ acts trivially on the element $T_i(i\ge 0),\,X_a,\,Y_a,\,q$, changing only $t$.
These lemmas are sufficient to establish $(ii)$. Claim $(iii)$ is straightforward. \phantom{1}
$\qed$
\subsubsection{\sf Employing the E-polynomials} From now on we will frequently represent $t$ in the form $t=q^k$. Given $a\in P$, $u_{a}$ will be the element of minimal possible length in $W$ such that $u_{a}(a)\in P_{-}$. We set \begin{align}\label{pibdef} &a_-\stackrel{\,\mathbf{def}}{= \kern-3pt =} u_a(a)\in P_-,\ \, \pi_a\stackrel{\,\mathbf{def}}{= \kern-3pt =} au_a^{-1}. \end{align} Here $l(\pi_a w)=l(\pi_a)+l(w)$ for an arbitrary $w\in W$, which is the defining property of $\{\pi_a\}$.
The Macdonald polynomials $E_{a}$, $a\in P$ are $Y$\~eigenvectors: \begin{align}\label{epolyno} &Y_{b}^{-1}(E_{a})\,=\,q^{(b, a_{\sharp})}E_{a}, \ b\in P^\vee,\ a_{\sharp}=a-k u_{a}^{-1}(\rho), \end{align} which fix them uniquely up to proportionality for generic $k$. The standard normalization condition is $E_a=X_a+$(lower terms); see books \cite{Ma4,C101}. Note that $u_0=$id and $0_\sharp=-k\rho$. More generally, $u_{a}=\,$id for $a\in P_{-}$ and $Y_{b}^{-1}(E_{a})=q^{(b, a-k \rho)}E_{a}$ for such $a$ and any $b\in P^\vee$.
These polynomials were introduced by Heckman and Opdam in the differential setting, then by Macdonald for $t=q^k$ for integers $k$ and then in \cite{C4} in complete generality (in the reduced case). They are orthogonal Laurent polynomials with respect to the inner product $$ \hbox{Constant Term\,}(fg^*\mu) \ \hbox{ for } \ f,g\in \mathscr{X},\ q^*=q^{-1},t^*=t^{-1}, X_b^*=X_b^{-1}. $$ See \cite{Ma4,C101} and also \cite{OS}; the latter contains historic remarks and references including the important $C^\vee C$\~case, which we do not discuss here. The symmetric Macdonald polynomials for the classical root systems were defined (and used) for the first time by Kevin Kadell.
Among quite a few properties of the $E$\~polynomials, let us mention the nonsymmetric Macdonald conjectures, namely, the norm-formula, the duality-evaluation formula and the Pieri rules. They are now established in an entirely conceptual way (see \cite{C101} and \cite{C103}); these properties can be deduced from the self-duality of DAHA practically without calculations.
In a sense the duality claim is the starting (and the simplest) in this chain of properties and the constant term formula is the endpoint. The nonsymmetric Pieri rules do not belong to the standard list of Macdonald's conjectures, but they are the key to connect the duality with the evaluation and norm formulas. We note that their proof in \cite{C4} goes via the reduction to the roots of unity.
The symmetric (usual) Macdonald conjectures can be deduced from the nonsymmetric ones or can be obtained directly from the DAHA theory upon symmetrization. The key feature of the nonsymmetric theory, which has no symmetric counterpart, is the technique of intertwiners. It simplifies dealing with the $E$\~polynomials significantly vs. the symmetric theory (the $P$\~polynomials).
We note that \cite{C101} and other works of the first author are mainly written for the {\em twisted affinization} $\tilde R^\nu$ (in the reduced case). A natural notation is $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(\tilde R^\nu;\tilde R^\nu)$, which means that the $X$\~generators and $Y$\~generators are labeled by same lattice $P$. Then the $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ from this paper must be denoted by $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(\tilde R;\widetilde{R^\vee})$.
The technique of intertwiners can be transferred to $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(\tilde R;\widetilde{R^\vee})$ (which is the setting of this paper). The norm and evaluation formulas for $\tilde R^\vee$ hold for $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(\tilde R;\widetilde{R^\vee})$ upon natural modifications of the formulas. For instance, the evaluation formula for $E_{a}(t^{-\rho^\vee})$ can be obtained from the one in \cite{C101} or from the Main Theorem of \cite{C4} (formula (5.4)) by the following transformations:
(a) adding check to $\rho$,\ (b) replacing $q_\alpha$ by $q$,\ and\ (c) setting $t_\alpha=q^{k_\alpha}$.
Explicitly, for $b\in P$, \begin{align} &E_{b}(t^{-\rho^\vee}) \ =\ t^{(\rho^\vee,\,b_-)} \prod_{[\alpha,j]\in \Lambda'(\pi_b)} \Bigl( \frac{ 1- q^{j}t^{1+(\rho^\vee,\,\alpha)}
}{ 1- q^{j}t^{(\rho^\vee,\,\alpha)} } \Bigr),\ \hbox{ where } \ \label{evaluform}\\ &\Lambda'(\pi_b)\ =\
\{[\alpha,j]\ |\ [-\alpha,\nu_\alpha j]\in \Lambda(\pi_b)\} \ \hbox{ for } \ \pi_b\stackrel{\,\mathbf{def}}{= \kern-3pt =} bu_b^{-1}, \notag \end{align} and we use the elements $u_b,\pi_b$ from (\ref{epolyno}),(\ref{pibdef}). The same transformation must be performed with the norm-formula (5.5) from \cite{C4}.
{\bf Comment.\ } We note that the DAHA of {\em untwisted} type $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(\tilde R;\widetilde{R^\vee})$ are expected to satisfy the quantum Langlands duality (see \cite{C103}). Trying to help the readers interested in this setting, let us discuss briefly the changes with the key DAHA\~automorphisms from \cite{C101} needed in the untwisted case. The $\sigma$ from \cite{C101} (coinciding with $\omega^{-1}$ from \cite{C4}) maps now $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(\tilde R;\widetilde{R^\vee})$ to $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(\widetilde{R^\vee},\tilde R)$. The automorphism $\tau_+$ acts in the former, $\tau_-$ in the latter. One has $$\sigma \tau_+^{-1}=\tau_-\sigma,\ \sigma \tau_+=\tau_-^{-1}\sigma.$$ There are unsettled questions with the difference Mehta-Macdonald formulas from \cite{C5} in the untwisted case; they will be partially addressed when discussing the affine Hall functions of level one. \phantom{1}
$\qed$
\subsubsection{\sf Convergence at level zero} Let us begin with the remark that the summation formula for ct$(t)$ from \cite{Ma3} was interpreted in \cite{C1} as the Jackson integration version of the {\em constant term conjecture}. It was generalized there to the Jackson-type norm formulas for arbitrary $E$\~polynomials. The relation of \cite{C1} to the present paper is direct; the definition of the Jackson integral of $f(X)$ from \cite{C1} is nothing but $$ \widehat{\mathscr{S}}\,'_+(\widetilde{\mu} f(X))\,[X\mapsto q^\xi] \ \hbox{ for } \ \xi\in {\mathbb C}^n; $$ the vector $\xi$ (arbitrary) is called the origin of the Jackson integral, which is a summation. The following theorem is a particular case of the Jackson norm-formulas from \cite{C1}, Proposition 5.7.
\begin{theorem}\label{THMSE}
For $|q|<1$, $t=q^k$ and $a\in P$ such that $E_{a'}$ are well defined for all $a'\in W(a)$, the sums $\widehat{\mathscr{S}}\,'_{+} (\widetilde{\mu}E_{a'})$ absolutely converge if and only if\, $\Re(2k\rho+a_{+},\omega_i)<0$ for all $i=1,\ldots,n$. Here $\{a_-\}=W(a)\cap P_-$, $a_+=w_0(a_-)$ for the element $w_0$ of maximal length in $W$, $\Re$ denotes the real part. Under this condition, $\widehat{\mathscr{S}}\,'_{+} (\widetilde{\mu}E_{a'})=0$ for $a\neq 0$ and all $a'\in W(a)$. \phantom{1}
$\qed$ \end{theorem} To give some examples, the (absolute) convergence range for $a=\rho= \alpha_1+\alpha_2$ in the case of $A_2$ is $\{\Re k>-1/2\}$; it becomes $\{\Re k>-1/3\}$ for $a=\omega_1=\omega_1^\vee =(2\alpha_1+\alpha_2)/3$.
We continue to assume that $k$ is generic (we will need this to employ the $E$\~polynomials). Considering generic $k$ in Theorem \ref{THMSE} and in a similar convergence statement is sufficient for us. Indeed, the inequalities for $\Re k$ that provide the convergence (in a given finite-dimensional subspace of $\mathscr{X}$) for all but finitely many special $k$ hold automatically for such special values. The convergence can be better at such special values, but no worse than at generic $k$, which is sufficient in what will follow.
\begin{theorem}\label{LEVZERO} The sum $\widehat{\mathscr{P}}'_{+}(E_{a'}) =\sum_{\widehat{w}\in \widehat{W}}t^{-l(\widehat{w})/2} T_{\widehat{w}}^{-1}(E_{a'})$ absolutely converges for any $a'\in W(a)$ if and only the following its sub-sum converges absolutely: $\sum_{b\in P^\vee_{+}}t^{-(\rho, b)}Y_{b}^{-1}(E_{a_-})$. Using (\ref{epolyno}), this readily results in the same condition as from the previous theorem, namely, $\Re(2k\rho+a_{+},\omega_i)<0$ for all $i=1,\ldots,n$. Provided the convergence, \begin{align}\label{propor} &\widehat{\mathscr{P}}\,'_+ =ct(t^{-1})\widehat{\mathscr{S}}\,'_+ \circ \widetilde{\mu} \hbox{\ \, as operators acting in\ \,} \mathscr{X}, \end{align} where ct$(t^{-1})$ is the constant term of $\mu(X;q,t^{-1})$: $$\hbox{ct}(t^{-1})=\prod_{\alpha\in R_{+}}\prod_{i=1}^{\infty} \frac{(1-t^{-(\alpha,\rho^{\vee})}q^{i})^{2}} {(1-t^{-(\alpha,\rho^{\vee})-1}q^{i}) (1-t^{-(\alpha,\rho^{\vee})+1}q^{i})} \in \mathbb{C}[t^{-1}][[q]].$$ \end{theorem} {\em Proof.} Let us begin with establishing the proportionality claim from (\ref{propor}) assuming the convergence. Copying the affine case, $\widehat{\mathscr{P}}\,'_{+}\circ\,\widetilde{\mu}^{-1}$ is divisible by $(t^{1/2}T_{i}+1)$ on the left and by $(t^{1/2}T_{i}^{\iota}+1)$ on the right. Hence it is divisible by $(s_{i}+1)$ on the left and on the right. Therefore $$ \widehat{\mathscr{P}}\,'_{+}\circ\,\widetilde{\mu}^{-1}\ =\ G(X)\sum_{\widehat{w}\in \widehat{W}}\widehat{w}\ =\ G(X)\cdot \widehat{\mathscr{S}}\,'_+ $$ for a certain $\widehat{W}$\~invariant function $G(X)$. Using \cite{Ma3}, $G=ct(t^{-1})$.
More directly, we can check that $\widehat{\mathscr{P}}\,'_{+}(E_a)=0$ for any $a\in P\setminus \{0\}$; combining this with Theorem \ref{THMSE} we readily establish the required proportionality.
The operator $\widehat{\mathscr{S}}\,'_+$ of course diverges in (the whole) $\mathscr{X}$, so we must apply the argument above as follows. Given $N\in {\mathbb N}$, formulas (\ref{annihil}) guarantee that the images and the kernels of $\widehat{\mathscr{P}}\,'_+$ and $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ coincide upon acting in the linear spaces $V_N=\oplus_{(\rho,a_+)<N} {\mathbb C} X_a$, provided that
$\Re k<0$ and $|\Re k|$ is sufficiently large (depending on $N$). Thus these operators are proportional in every $V_N$ and the coefficient of proportionality (a constant) does not depend on $N$.
The convergence analysis for $\widehat{\mathscr{P}}\,'_+$ in $\mathscr{X}$ is different from that for $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$. First, it suffices to assume that $a\in P_-$, using the standard relations between the polynomials $E_{a\,'}$ for $a'\in W(a)$. Second, we observe that the convergence is the worst for terms $Y_b^{-1}(E_a)$ with $b\in P_+^\vee$ and $a\in P_-$. Thus, we need to analyze \begin{eqnarray*} \sum_{b\in P^\vee_{+}}t^{-(\rho, b)}Y_{b}^{-1}(E_{a}) &=&\sum_{b\in P_{+}}q^{(b, a-2k\rho)}E_{a}; \end{eqnarray*} this sum converges absolutely if and only if \, $\Re(2k\rho+a_{+})\in {\mathbb R}_{>0}Q_+$. The completion of this argument is based on the following theorem.
\subsubsection{\sf Y-formulas for P-hat} \label{sect:RatPhat} Recall that $\widehat{\mathscr{P}}\,'_+$ is the plus-symmetrizer without the exact projector normalization, i.e., without the division by $\widehat{P}(t^{-1})$. By $P(t)$, we denote the {\em nonaffine} Poincar\'e polynomial. For a subset $\mathbf{I}\subset \{1,2,\ldots,n\}$, the Poincar\'e polynomial of the root subsystem $R_{\mathbf{I}}\subset R$ generated by the simple roots $\{\alpha_i\,\mid\, i\in \mathbf{I}\}$ will be denoted by $P_{\mathbf{I}}(t)$. It is $1$ if $\mathbf{I}=\emptyset$.
\begin{theorem}\label{P+FORMULA} The symmetrizer $\widehat{\mathscr{P}}\,'_+$ can be presented as the following summation over all subsets $\mathbf{I}\subset \{1,2,\ldots,n\}$ including the empty set and $\mathbf{I}=\{1,\ldots,n\}$\,: \begin{align}\label{hatPrat} &\widehat{\mathscr{P}}\,'_+ = P(t^{-1})\mathscr{P}_+\Bigl(\, \sum_{\mathbf{I}}\frac{P(t)}{P_{\mathbf{I}}(t)} \prod_{i\not\in \mathbf{I}} \frac{\,t^{-(\omega_i^\vee,\,\rho)}\,Y_{\omega_i^\vee}^{-1}} {1-t^{-(\omega_i^\vee,\,\rho)}\,Y_{\omega_i^\vee}^{-1}}\,\Bigr) \mathscr{P}_+, \end{align} which is understood coefficient-wise upon the expansion of the rational expressions in the products in terms of $t^{-1}$ (a set of identities in $\mathcal{H}_Y$). \end{theorem} {\em Proof.} We employ the key property of the elements $\pi_b$ from (\ref{pibdef}), namely, the equality $l(\pi_b w)=l(\pi_b)+l(w)$ for any $w\in W$. Since $\pi_b=b u_b^{-1}$, one has $\pi_b w=u_b^{-1}\,b_-\,w$. The element $u=u_b$ can be arbitrary such that its length is minimal possible for a given $b=u^{-1}(b_-)$, i.e., minimal in the coset $Z(b_-)u$ for the centralizer $Z(b_-)$ of $b_-$ in $W$. It results in (\ref{hatPrat}).\phantom{1}
$\qed$
Note that formula (\ref{hatPrat}) gives a {\em rational} expression for the affine Poincar\'e series $\widehat{P}(t^{-1})$ $=\widehat{\mathscr{P}}\,'_+(1)$. Provided that $\widehat{P}(t^{-1})\neq 0$, the theorem gives a universal map {\em onto} the space of $Y$-{\em spherical vectors} $$ \{\, v\,\mid\, T_{\widehat{w}}(v)=t^{l(\widehat{w})/2)}v \ \hbox{ for } \ \widehat{w}\in \widehat{W}\,\}, $$ which is applicable to $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\,$\~modules that are unions of {\em finite-dimensional} $Y$\~invariant subspaces, including $\mathscr{X}$. Theorem \ref{SYMRANK1} can be readily extended to arbitrary one-dimensional characters of $\mathcal{H}_Y$; the case of the {\em affine minus-symmetrizer}, corresponding to $\{T_{\widehat{w}}\mapsto (-t^{-1/2})^{l(\widehat{w})}\}$, is of importance.
The right-hand side of formula (\ref{hatPrat}) is a rational function and can be used as such without the $t^{-1}$\~expansion. However, one has to ensure that the denominators in (\ref{hatPrat}) are nonzero. For instance, this formula can be used in the (whole) polynomial representation $\mathscr{X}$ for $A_1$ with any $q,t$ unless $t^2\in q^{-1-{\mathbb Z}_+}$ and for $A_2$ unless $t^6\in q^{-1-{\mathbb Z}_+}$ or $t^3\in q^{1+{\mathbb Z}_+}$. It is under the assumption that $q$ is not a root of unity and $\widehat{P}(t^{-1})\neq 0$. At roots of unity, this formula can be applied only in certain quotients of $\mathscr{X}$.
Formula (\ref{hatPrat}) is the subject of Theorem \ref{SYMRANK1} in the case of $A_1$. For $A_2$, it reads as follows: \begin{align*} &\widehat{\mathscr{P}}\,'_+\ =\ P(t)P(t^{-1})\mathscr{P}_+\Bigl(\, \frac{\,t^{-2}\,Y_{\omega_1+\omega_2}^{-1}} {(1-t^{-1}\,Y_{\omega_1}^{-1})(1-t^{-1}\,Y_{\omega_2}^{-1})}\\ &+\, \frac{1}{1+t}\, \bigl(\frac{\,t^{-1}\,Y_{\omega_1}^{-1}} {1-t^{-1}\,Y_{\omega_1}^{-1}}\, +\frac{\,t^{-1}\,Y_{\omega_2}^{-1}} {1-t^{-1}\,Y_{\omega_2}^{-1}}\bigr)\, +\,\frac{1}{(1+t)(1+t+t^2)}\, \Bigr)\mathscr{P}_+. \end{align*} Here $\rho=\alpha_1+\alpha_2$ and $(\rho,\omega_i)=1$ for $i=1,2$; $P(t)= (1+t)(1+t+t^2)$. Recall that $\omega_i=\omega_i^\vee.$ Applying this formula to $1\in \mathscr{X}$ and using that $t^{-1}\,Y_{\omega_i}^{-1}(1)$$=t^{-2}$, the resulting series is the $t^{-1}$\~expansion of $\widehat{P}(t^{-1})$; we arrive at the formula $\widehat{P}(t^{-1})=3(1-t^{-3})/(1-t^{-1})^3$.
The expression on the right-hand side of (\ref{hatPrat}) treated as an element in the localization of affine Hecke subalgebra $\mathcal{H}_Y=\langle T_{\widehat{w}},\widehat{w}\in \widehat{W}\rangle$ must be {\em identically} zero. Indeed, no affine symmetrizer exists in $\mathcal{H}_Y$ or its localizations unless completions are allowed. Similarly, this expression becomes identically zero when applied in $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\~modules that are unions of finite-dimensional $\mathcal{H}_Y$\~modules containing {\em no} $Y$\~spherical vectors. This is the key point of the following theorem; we mention that the $A_1$\~case is considered in full detail in Theorem \ref{SYMRANK1Y} below.
\begin{theorem}\label{YLEFT} Given a set of representatives $\mathbf b=\{b^1,\ldots,b^p\}\subset P_+^\vee$ for the group $\Pi=P^\vee/Q^\vee$ (of cardinality $p$), let \begin{align}\label{SiYgen} &\widetilde{\Sigma}_{\mathbf b}= \prod_{\alpha\in R_+}\frac{(1-tY_{\alpha^\vee}^{-1})} {(1-Y_{\alpha^\vee}^{-1})}\, \frac{\sum_{j=1}^p t^{-(b^j,\rho)}Y_{b^j}} {\prod_{i=1}^n (1-tY_{\alpha_i^\vee}^{-1})}\,,\\ &\overline{\Sigma}_{\mathbf b}=\frac {\prod_{\alpha\in R_+\setminus\{\alpha_1,\, \ldots,\,\alpha_n\}}\,(1-t^{1-(\alpha^\vee,\rho)})} {\prod_{\alpha>0}(1-t^{-(\alpha^\vee,\rho)})} \sum_{j=1}^p\, t^{-(b^j,\rho)}Y_{b^j}. \label{SiYgenbar} \end{align} We consider $\widehat{\mathscr{P}}\,'_+$ as a standard formal series $\sum_{\widehat{w}}C_{\widehat{w}}\widehat{w}$ provided the convergence of the coefficients as formal series or point-wise or as an operator acting in any representations of $\mathcal{H}_Y$ where it is well defined. If $t$ is treated as a number, $\widehat{P}(t^{-1})$ is supposed to be invertible.
Let $b^j\to\infty$, which means that $(b^j,\alpha_i)\to \infty$ for all $1\le j\le p,$\, $i>0$. We also assume that \begin{align}\label{tYvanish} &\lim_{b^j\to\infty} t^{-(b^j,\rho)}Y_{w(b^j)}\mathscr{P}\,'_+= \left\{\begin{array}{c}
\hbox{\, exists\, for\, all \,\,} w\in W\\
\hbox{\, equals zero for\ } w\neq \hbox{\sf id} \end{array}\right\} \end{align} coefficient-wise in the standard $\widehat{w}$\~expansions
(provided then that $|q|$ is sufficiently small if the coefficients are treated as meromorphic functions) or element-wise in a given $\mathcal{H}_Y$\~module. Then \begin{align}\label{PSiY} &\widehat{\mathscr{P}}\,'_+\, =\, \lim_{\mathbf b\to\infty} \widetilde{\Sigma}_{b}\,\mathscr{P}\,'_+\,=\, \lim_{\mathbf b\to\infty} \overline{\Sigma}_{b}\,\mathscr{P}\,'_+\ \hbox{ for } \ \mathscr{P}\,'_+\, =\, P(t^{-1})\,\mathscr{P}_{\,+}. \end{align} In the one-dimensional representation of $\mathcal{H}_Y$ corresponding to ``$+$", (\ref{PSiY}) results in formula (5.9) from \cite{Ma3} for the affine Poincar\'e series $\widehat{P}(t)$ in terms of the degrees $d_i$: \begin{align}\label{rataffpoin}
\widehat{P}(t) =\frac{|\Pi|}{(1-t)^{n}} \,\prod_{i=1}^n\frac{1-t^{d_i}}{1-t^{d_i-1}},\hbox{\ where\ } P(t)=\frac{\prod_{i=1}^n (1-t^{d_i})}{(1-t)^n}. \end{align} \end{theorem} {\em Sketch of the proof.} \comment{ In the limit ${\mathbf b}\to\infty$, the convergence of $\widehat{\mathscr{P}}\,'_+$ implies that \begin{align}\label{tYvanish} &t^{-l(a)/2}Y_a\to 0\ \hbox{ for } \ a\not\in P^\vee_+, \ \hbox{ where } \ l(a)=l(a_+)=2(a_+,\rho); \end{align} we omit the justification. } Relation (\ref{tYvanish}) implies that $\overline{\Sigma}_{b}\,\mathscr{P}\,'_+\ $ from (\ref{PSiY}) converges to the affine symmetrizer up to proportionality, i.e., satisfies the invariance properties upon multiplication by $T_{\widehat{w}}$ ($\widehat{w}\in \widehat{W}$) on the right and on the left. It is obvious when $T_{\widehat{w}}=Y_a(a\in P_+^\vee)$, which is sufficient. Cf. Theorem \ref{SYMRANK1Y} below for $A_1$.
A straightforward calculation of the coefficient of proportionality results in the first equality in (\ref{PSiY}). It readily gives that $\widetilde{\Sigma}_{\mathbf b}\mathscr{P}\,'_+$ and $\overline{\Sigma}_{\mathbf b}\mathscr{P}\,'_+$ must coincide in the limit provided the convergence of the latter expression. Indeed, the multiplication or division by the ratio $(1-CY_{\alpha^\vee})/(1-Ct^{(\alpha^\vee,\rho)})$ will not change $\widetilde{\Sigma}_{\mathbf b}$ in the limit for a sufficiently general constant $C$.
As noted above, the first equality in (\ref{PSiY}) can be deduced directly from relation (\ref{hatPrat}); let us outline the main steps.
We introduce the truncation $\Upsilon_{\mathbf b}$ of the $Y$\~expression between the two $\mathscr{P}_+$ in formula (\ref{hatPrat}) as follows. Upon the $Y^{-1}$\~expansion, only the monomials $Y_a^{-1}$ subject to $b^j- Q_+\ni a\in P_+$ for $b^j=a\mod Q$ will be kept. Let $b^j=\sum_{i=1}^n r_i^j \alpha_i^\vee$; recall that $(b^j,\alpha_i)\to \infty$, so the whole $\widehat{\mathscr{P}}\,'_+\,$ will be obtained in this limit. The {\em finite} sum \begin{align}\label{Upsilonb} \Upsilon_{\mathbf b}^\flat\,\stackrel{\,\mathbf{def}}{= \kern-3pt =}\,P(t)\sum_{j=1}^p \prod_{i=1}^n \frac{(1-t^{-1-r_i^j(\alpha_i^\vee,\,\rho)}\, Y_{\alpha_i^\vee}^{-r_i^j-1})} {(1-t^{-(\alpha_i^\vee,\,\rho)}\,Y_{\omega_i^\vee}^{-1})} \end{align} contains all such $Y_a^{-1}$, i.e., contains $\Upsilon_{\mathbf b}$, but there will be extra (nondominant) terms there with $a\not\in P_+^\vee$.
We are going now to use nonaffine formulas (\ref{mac1}) and (\ref{mac2})\,: \begin{equation}{\label{mac12}} P(t^{-1})\mathscr{P}_{+}=(\sum_{w\in W}w)\circ \widetilde{M}\ \hbox{ for } \ \widetilde{M}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \prod_{\alpha\in R_+}\frac{1-t^{-1}Y_{\alpha^{\vee}}^{-1}} {1-Y_{\alpha^{\vee}}^{-1}}. \end{equation}
Due to these formulas combined with the vanishing property from (\ref{tYvanish}), the contributions of $Y_a^{-1}$ in (\ref{Upsilonb}) with $(a,\rho)\ll (b^j,\rho)$ for $a\in b^j-Q_+$ tend to zero in the limit. Thus the nondominant terms can be disregarded in $\Upsilon_{\mathbf b}^\flat$. Moreover, it suffices to consider only $\mathbf{I}=\emptyset$ in Theorem \ref{P+FORMULA} in the limit upon applying the operator from (\ref{mac12}).
Similarly, the numerator in formula (\ref{Upsilonb}) can be actually reduced to $P(t)\,\bigl(1+(-1)^n\prod_{i=1}^n t^{-1-r_i^j(\alpha_i^\vee,\,\rho)}\,Y_{\alpha_i^\vee}^{-r_i^j-1}\bigr)$. Using that (\ref{hatPrat}) is zero in localizations of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\,$, $ \mathscr{P}_{+}\,\Upsilon^\flat_{\infty}\, \mathscr{P}_+=0 \ \hbox{ for } \ \Upsilon^\flat_{\infty} $ for $\Upsilon_{\infty}^{\flat}$ given by (\ref{Upsilonb}) upon making the numerators $1$, i.e., by deleting the terms that contain any $r_i^j$. This identity can be obtained directly from (\ref{mac12}); use the divisibility of the anti-invariant Laurent polynomials by the discriminant.
This makes it possible to switch to $Y_{b^j+a}^{-1}$ with $a\in Q_+^\vee$ in the limit; the terms here apart from the initial truncation will not contribute to the limit. Therefore $\Upsilon_{\mathbf b}^{\flat}$ can be replaced by \begin{align*} \Upsilon^\sharp_b\,\stackrel{\,\mathbf{def}}{= \kern-3pt =}\, P(t)&\,(-1)^n\,\sum_{j=1}^p\,t^{-(b^j,\rho)}\,Y_{b^j}^{-1} \,\prod_{i=1}^n \frac {t^{-(\alpha_i^\vee,\rho)}Y_{\alpha_i^\vee}^{-1}} {(1-t^{-1}Y_{\alpha_i^\vee}^{-1})},\\ \hbox{and\ } & \widehat{\mathscr{P}}\,'_+\ =\ \lim_{\mathbf b\to\infty} \sum_{w\in W}\, w\bigl(\Upsilon^\sharp_b\widetilde{M}\bigr)\mathscr{P}_+. \end{align*}
Using the vanishing condition from (\ref{tYvanish}) once again, we see that only $w=w_0$ here really contributes to $\widehat{\mathscr{P}}\,'_+$ in the limit. Let us substitute $b\mapsto -w_0(b)\,$ in the resulting expression. Then $\widehat{\mathscr{P}}\,'_+$ becomes the limit of \begin{align*} (-1)^n\,\sum_{j=1}^p\frac {t^{-(b^j+\rho^\vee,\rho)}Y_{b^j+\rho^\vee}} {\prod_{i=1}^n(1-t^{-(\alpha_i^\vee,\rho)}Y_{\alpha_i^\vee})} \prod_{\alpha\in R_+}\frac {(1-t^{-1}Y_{\alpha^\vee})} {(1-Y_{\alpha^\vee})}\, P(t)\mathscr{P}_+, \end{align*} where we use that $\rho^\vee=\sum_{i=1}^n \omega_i^\vee$. Rewriting the latter formula in terms of $Y_{\alpha^\vee}^{-1}$, we finalize (\ref{PSiY}).
Applying (\ref{PSiY}) to $1$ in the standard one-dimensional representation of $\mathcal{H}_Y$, one arrives at (\ref{rataffpoin}). Indeed, $Y_a$ become $t^{(a,\rho)}$ upon this evaluation and $\mathscr{P}(1)=1$. This formula is due to Matsumoto and Macdonald; see formula (5.9) from \cite{Ma3}. \phantom{1}
$\qed$
The conditions from (\ref{tYvanish}) hold coefficient-wise via the action of $Y_b$ in the polynomial representation followed by the standard expansion $Y_b= \sum _{\widehat{w}\in \widehat{W}} C_{\widehat{w}}\widehat{w}$ and in the representations $\mathscr{X}q^{\,lx^{2}/2}$ for $l>0$. See Theorem \ref{YLEFTNEW} below; the standard expansions of $Y_{w(b)}$ are discussed there in detail.
Formula (\ref{SiYgen}) for $\overline{\Sigma}_b$ coincides with formula (\ref{YSiabsbar}) for $\overline{\Sigma}_M$ below in the case of $A_1$. One needs to set $b^1=M\omega, b^2=(M-1)\omega$ for $\omega=\omega_1$. We note that $b^1$ and $b^2$ can be taken arbitrary (approaching infinity); the $\widehat{w}$\~expansions of $Y_{b^j}$ in (\ref{YSiabsbar}) are for two disjoint sets of $\widehat{w}$, for $j=1$ and $j=2$.
The vanishing condition from (\ref{tYvanish}) becomes (\ref{YMzero}) for $A_1$ and always holds provided the existence of $\widehat{\Sigma}_\infty^+$ in Theorem \ref{SYMRANK1Y}.
{\bf Comment.\ } In the Kac-Moody limit $t\to\infty$, (\ref{SiYgenbar}) combined with the proportionality claim from (\ref{propor}) give a presentation of the Kac-Moody characters as limits of the (affine) Demazure characters. The latter are directly related to the operators $T^\infty_{\widehat{w}}=\lim_{t\to\infty}t^{-l(\widehat{w})/2}T_{\widehat{w}}$. Namely, the corresponding Demazure characters are proportional to $q^{-l\frac{x^2}{2}}\, T^\infty_{\widehat{w}}(X_{-a}\,q^{l\frac{x^2}{2}})$ upon the substitution $X_b\mapsto e^{-b}$. Here $a$ are affine $l$\~dominant weights, i.e., $a\in P_+$ and $(a,\theta)\le l$. For $\widehat{w}=b\in P_+$ as $b\to\infty$, they approach $\prod_{i=1}^\infty \frac{1}{(1-q^i)^n}\widehat{\chi}_a^{(l)}$; see (\ref{Kac-Moody}) below.
Here it is not necessary to stick to the affine dominant weights $a$ of level $l$. One can define the Kac-Moody characters formally for arbitrary $a\in P$ using the Kac-Weyl formula. The proportionality claim (\ref{propor}) itself provides that the Kac-Moody characters are {\em sums} of properly normalized Demazure characters, which is connected with the (infinite-dimensional) Demazure modules associated with the opposite Borel subalgebra (to that used for the highest vectors).
For arbitrary $t$, (\ref{SiYgenbar}) states that the corresponding affine Hall functions from (\ref{affineHall}) are limits of the {\em Demazure $t$\~characters} for $a\in P_+$ defined (formally) as $q^{-l\frac{x^2}{2}}\, \overline{\Sigma} \mathscr{P}_+'(X_{-a}\,q^{l\frac{x^2}{2}})$, where actually we do not need $\mathscr{P}_+'$ (see below). The summation formula also holds and is equally important.
\subsubsection{\sf Coefficient-wise proportionality} \label{sect: Coefficient} Theorem \ref{LEVZERO} is sufficient to claim the existence of the coefficients of the operator $\widehat{\mathscr{P}}\,'_+ $ as meromorphic functions and the coefficient-wise proportionality from (\ref{propor}). We will outline here an analytic version of this approach based on a natural analytic extension of the polynomial representation.
\begin{theorem} \label{TQ1H}
Let $|t|>q^{1/h}$ for the Coxeter number $h=(\theta,\rho)+1$. Expanding $\widehat{\mathscr{P}}\,'_+=\sum_{\widehat{w}\in \widehat{W}} F_{\widehat{w}}(X)\,\widehat{w}$, the coefficients $F_{\widehat{w}}$ converge absolutely and to an analytic function on any given compact subsets in $\{0\neq X_\alpha\not\in q^{{\mathbb Z}}, \alpha\in R\}$
for sufficiently small $|q|$ depending on this subset. Moreover, $F_{\widehat{w}}$ coincide with the corresponding coefficients of\, $ct(t^{-1})\,\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ in this range; for instance, $F_{\hbox{\tiny id}}=ct(t^{-1})\,\widetilde{\mu}(X;q,t)$.
\end{theorem}
The proof of this theorem, including the proportionality claim and the sharp estimate of the radius of convergence with respect to $t$ of the coefficients of $\widehat{w}$, results from Theorem \ref{PSBULLET} below, based on the representations of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ in the space of delta functions. Also, the existence of $\{F_{\widehat{w}}\}$ as meromorphic functions can be obtained using direct estimates for the coefficients of operators $Y_b$\,; see Lemma \ref{LEMThatw} below and Theorem \ref{YEXPA} in the case of $A_1$. Nevertheless, it is quite natural to try to deduce the convergence and proportionality directly from the properties of $\widehat{\mathscr{P}}\,'_+$, considered as an operator acting in the polynomial representation and its extensions.
Let us outline here an approach to the coefficient-wise existence and the proportionality utilizing the following analytic modification of Theorem \ref{LEVZERO}. As a matter fact, the approach from Theorem \ref{PSBULLET} (entirely algebraic) is very similar to the following considerations.
We will assume in the sketch below that $|t|>1$. When dealing with the affine symmetrizers analytically, it is convenient to replace $\mathscr{X}$ by the union of Paley\~Wiener type spaces $\mathscr{P\!W\!}_{M}(\mathcal{U})$ of analytic functions in a given $\widehat{W}$\~invariant domain ${\mathbb R}^n\subset \mathcal{U}\subset{\mathbb C}^n$. Here $M\in {\mathbb Z}_+$ and the growth condition is as follows: $$ f(x)\in \mathscr{P\!W\!}_{M}(\mathcal{U})\,\Rightarrow\, {}^{bw}f(x)<C_{x}(M)\, q^{-M(b_+,\rho)},\ b\in P^\vee,\, w\in W, $$ for a constant $C_{x}(M)$ continuously depending on $x\in \mathcal{U}$. For $M=0$, this space includes $1$ and all $\widehat{W}$\~invariant functions analytic in $\mathcal{U}$, for instance, the images of $\widehat{\mathscr{P}}\,'_+$ and $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$. These two operators act in $\mathscr{P\!W\!}_{M}(\mathcal{U})$ for sufficiently large negative $\,\Re k\,$, depending on $M$, and for sufficiently small $\mathcal{U}$ containing ${\mathbb R}^n$.
The kernels and images of these operators in $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\~invariant subspaces of $\cup_{M\ge 0}\mathscr{P\!W\!}_{M}(\mathcal{U})$ coincide and Theorem \ref{KERIMAGE} (in an analytic variant) implies the proportionality \begin{align}\label{proporan} &\widehat{\mathscr{P}}\,'_+= ct(t^{-1})\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu} \hbox{\ \, provided the convergence.\,\ } \end{align}
To extract and then equate the coefficients of the operators under consideration, we need certain modifications of delta functions in the space $\mathscr{P\!W\!}_{0}(\mathcal{U})$ in a sufficiently small neighborhood $\mathcal{U}$ of $0\in{\mathbb R}^n$. Let $\mathcal{A}=\sum_{\widehat{w}\in\widehat{W}}F_{\widehat{w}}(X)\widehat{w}$, assuming that this operator is convergent with the coefficients analytic in $\mathcal{U}$ and satisfies the conditions from (\ref{annihil}). It suffices to know $\widetilde{F}_b\stackrel{\,\mathbf{def}}{= \kern-3pt =} \sum_{w\in W}F_{bw}(X)$ for $b\in P^\vee$; expand $\mathcal{A}$ in terms of $bT_{w}$ for $\widehat{w}=bw$ to see it (use that $q,t$ are generic).
Let us extract from $\mathcal{A}$ the value of the coefficient $\widetilde{F}_0$ at $x=0$. Recall the notation $X=q^x$, $x_\alpha=(\alpha,x)$. The following {\em probe function} from $\mathscr{P\!W\!}_{0}(\mathcal{U})$ can be used, a substitute for the delta function at zero:\, $$ \zeta_N(x)=-\prod_{\alpha\in R_+} \frac{(\exp({N\pi \imath x_\alpha)-\exp(-N\pi \imath x_\alpha}))^2} {(\exp(N\pi x_\alpha)-\exp(-N\pi x_\alpha))^2}, $$
where $N\in {\mathbb N}$, $\imath^2=-1$. This function is of order $1+O(|x|^2/N)$ near $x=0$ and of order
$O\bigl(\,|x-b|^2\cdot\frac{\exp(-CN)}{N}\,\bigr)$ for $x\approx b\in P^\vee\setminus 0$ for some constant $C>0$. Obviously, $\mathcal{A}(\zeta_N)(x=0)=\widetilde{F}_0(x=0)$, and we recover the value of $\widetilde{F}$ at $x=0$.
Using the function $\sum_w \zeta_N(w(x)-x_0)$ in the same manner, we can find the values $\widetilde{F}_0(x=x_0)$ for any given $x_0$ in a sufficiently small neighborhood of $x=0$. This gives the function $\widetilde{F}_0$ in $\mathcal{U}$ pointwise in terms of the action of $\mathcal{A}$ in $\mathscr{P\!W\!}_{0}(\mathcal{U})$. Alternatively, recovering $\widetilde{F}_0(x)$ for small $x$ can be achieved by tending $N$ to $\infty$ (we will omit details).
The same approach can be used for extracting any $\widetilde{F}_b$ from $\mathcal{A}$ upon applying the translations by $b\in P^\vee$ to the argument $x$ in the probe function (which fix its numerator).
This is of course based on the existence of $\mathcal{A}$ when applied to $\zeta_N$ in a neighborhood of $x=0$. The numerator of $\zeta_N$ is a {\em pseudo-constant}, a $\widehat{W}$\~invariant function. Thus, the rate of convergence depends only on the denominator and the convergence of the operators $\widehat{\mathscr{P}}\,'_+$ and $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$\, applied to $\zeta_N$ is no worse than that for constants (or pseudo-constants). Actually, it is better than this; it holds for small {\em positive} $\Re k$ too (presumably, the inequality $\Re k<1/h$ is sufficient here).
As a matter of fact, we need to know here the convergence only for large negative $\Re k$ (for recovering the coefficients), a weaker fact. Indeed, the coefficients of $\widehat{\mathscr{P}}\,'_+$ and $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$\, are meromorphic functions in $k$ (provided the convergence). If the proportionality of these operators is known for $\Re k \ll 0$, then it holds coefficient-wise. Thus, the coefficient-wise existence and proportionality require only Theorem \ref{propor} extended analytically to the functions similar to $\zeta_N$; the proportionality factor will be automatically $ct(t^{-1})$. Theorem \ref{PSBULLET} below is an algebraic variant of this approach.
\subsection{{\bf Affine Hall functions}} \subsubsection{\sf Main definition} The above considerations were for the $0$\~level case of the general theory of {\em affine Hall functions of arbitrary levels}, which will be the subject of this section.
We continue to assume that $|q|<1$.
Expressing $X_{a}=q^{x_{a}}=q^{(x,a)}$, let us introduce the {\em $l$\~Gaussian} as $q^{\,l\,x^2/2}$ for $x^{2}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\sum_{i=1}^{n}x_{\omega_{i}}x_{\alpha_{i}^{\vee}}.$ In the case of $A_{2}$, for example, we have $\alpha_{1}=\alpha_1^\vee=2\omega_{1}-\omega_{2}$, $\alpha_{2}=\alpha_2^\vee=2\omega_{2}-\omega_{1}$ and $$\frac{x^{2}}{2}= \frac{x_{1}(2x_{1}-x_{2})}{2}+\frac{x_{2}(2x_{2}-x_{1})}{2}= x_{1}^{2}-x_{1}x_{2}+x_{2}^{2}.$$ One readily checks that $$ \widehat{w}(q^{\,lx^2/2})= q^{\,lb^2/2} X_{lb}^{-1} q^{\,lx^2/2} \ \hbox{ for } \ \widehat{w}=bw,\, b\in P^\vee, w\in W. $$ These formulas are actually the defining relations of the Gaussian in what will follow. Recall that $\,bw(X_a)=q^{-(b,w(a))}X_a\,$ for $\,a\in P$.
To simplify notations, we set \begin{align}\label{Jproj} \widehat{\mathscr{I}}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \widehat{\mathscr{S}}\,'_{+}\circ\widetilde{\mu},\ \ \mathscr{I} \hbox{\ stays here for ``integration"}. \end{align}
The {\em Hall functions of level $l>0$} are defined as \begin{align}\label{affineHall} &H_{a}^{(l)}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\widehat{\mathscr{I}}(X_{a}q^{\,lx^{2}/2}), \ a\in P,\ \,\mathscr{H}_{l}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \widehat{\mathscr{I}}(\mathscr{X}q^{\,lx^{2}/2}). \end{align} Thanks to the presence of the Gaussian, $q^{\,-lx^2/2}\,H_a^{(l)}$ are absolutely convergent series in terms of $X_b$ ($b\in P$) for all $x$ and $t$ (no poles due to the denominator of $\widetilde{\mu}$ will occur). This is known and can be readily checked using $\widehat{\mathscr{P}}_+^{\,'\,}$, which preserves the Laurent polynomials. Indeed, the residues at (potential) poles of $H_a^{(l)}$ are meromorphic functions in terms of $q,t$; however they must vanish for sufficiently general $t$ due to (\ref{propor}) or (\ref{proporan}), the proportionality.
The absolute convergence actually holds here for any $l\in {\mathbb C}$ such that $\Re l>0$, but then we will not be able to represent the funcuions $q^{-\,lx^{2}/2}\,H_{a}^{(l)}$ as Laurent series. Also, singularities in $x$ can appear for nonintegral $\,l\,$ at {\em nonreal poles} of $\widetilde{\mu}(q^x)$, which are as follows: \begin{align}\label{polesmu} \bigl\{x\,\mid\, (x,\alpha)+j\in 2\pi \imath\, \log(q)\, \{P^\vee\setminus 0\},\ [\alpha,j]\in \tilde R_+\bigr\}, \end{align} where $\imath$ is the imaginary unit. There will be no singularities in a sufficiently small neighborhood of ${\mathbb R}^n\subset {\mathbb C}^n$ for nonintegral levels.
Note that for any $\widehat{W}$\~invariant function $f$, called a {\em pseudo-constant}, \begin{equation}\label{PSonf} \widehat{\mathscr{P}}\,'_{+}(f)\,=\,\widehat{P}(t^{-1})f\,=\, \hbox{ct}(t^{-1}) \widehat{\mathscr{I}}(f), \end{equation} where we need to assume that $\Re k<0$ to ensure convergence. Here $\widehat{P}(t)$ is the affine Poincar\'e series.
The coefficient of proportionality is the same as in (\ref{propor}) because the action of our operators on any pseudo-constants $f$ is no different from the action on $1\in \mathscr{X}$. For instance, (\ref{PSonf}) holds for functions from $\mathscr{H}_l$ provided that $\Re k<0$.
{\bf Comment.\ } The proportionality from (\ref{propor}) cannot hold for all $k$; otherwise $H_a^{(l)}$ would vanishes {\em identically} for all $a\in P$ at the poles of $\hbox{ct}(t^{-1})$, which is not the case. For instance, $\mathscr{H}_l$ must be $\{0\}$ as $t=q^{1/h}$ for the Coxeter number $h$ if $\mathscr{P}\,'_+$ is well defined at this point, which happens only for $l=1$. Indeed, the proportionality always holds when both operators are well defined.
We claim that for any (integral) $l>0$, the space $\mathscr{H}_{l}$ is always smaller than the corresponding Looijenga space (see the definition below) at $t=q^{1/h}$ and at other zeros of $H_{a}^{(l=1)}$ from part $(ii)$ of the next Theorem \ref{HALLONE} (the simply-laced case). However it is generally nonzero. The justification of this and similar facts is based on diminishing the level due to formula (\ref{PSonf}).
Numerical calculations of the space ${\mathscr H}_{l}$ $=\widehat{\mathscr{I}}(\mathscr{X})$ for $A_1,A_2,B_2$ show that this space is really nonzero at $t=q^{1/h}$, i.e., that, generally, $\widehat{\mathscr{P}}\,'_+$ cannot be continued analytically to $\Re k\ge 1/h$. The latter inequality seems sharp for $l>1$, namely, the convergence of $\widehat{\mathscr{P}}\,'_+$ and (its corollary) the vanishing property $\mathscr{H}_l(k=1/h)=\{0\}$ are not expected to hold for $l=1\pm \varepsilon$ for arbitrarily small $\varepsilon>0$. Only integral $l$ are considered in this paper, but the definition of the corresponding spaces for any complex $l$ with $\Re l>0$ is straightforward. \phantom{1}
$\qed$
\comment{ {\em DAHA-Matsumoto polynomials}. Let us connect this definition with the approach from Section \ref{sec:Sph via Hall} based on the Matsumoto (nonsymmetric) spherical functions, which were introduced there in an entirely algebraic way. We define the {\em DAHA-Matsumoto polynomials} $H\!\!\!E_{a}^{(l)}\in \mathscr{X} q^{\,lx^2/2}\,$ of level $l\ge 0$ by the relation: \begin{align}\label{affineHallns} &H\!\!\!E_{a}^{(l)}\widehat{\mathscr{P}}_+ \ =\ \sigma^{-1}(T_{a})\,q^{\,lx^{2}/2}\, \widehat{\mathscr{P}}_+ \,\in\, \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\, q^{\,lx^2/2}\widehat{\mathscr{P}}_+ . \end{align} Here $\tau_{\pm}$ and $\sigma=\tau_+\tau_-^{-1}\tau_+$ are the standard automorphisms of DAHA; see \cite{C101}. We will use that $\sigma^{-1}(Y_a)=X_a^{-1}$ for $a\in P$.
Because of this definition, we are supposed to add formally all Gaussians $q^{\,lx^2/2}\, (l\in {\mathbb Z}_+)$ to $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ with the relations $q^{\,lx^2/2} H=\tau_+^l(H) q^{\,lx^2/2}$ for $H\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$, which allow moving $q^{\,lx^2/2}$ to the right. However, such an extension is not really necessary; the strict meaning of (\ref{affineHallns}) is as follows.
First, we apply the DAHA-PBW theorem in terms of the basis $X_b\, (b\in P)$, $T_{w}\,(w\in W)$ and $\tau_+^l(Y_b)\, (b\in P)$ to the element $\sigma^{-1}(T_{a})$. Second, we replace the elements $\tau_+^l(T_{\widehat{w}})$ (placed on the right) by $t^{l(\widehat{w})/2}$; the result will be a {\em polynomial} in terms of $X_b$. Then $H\!\!\!E_{a}^{(l)}$ will be this polynomial upon its multiplication by $q^{\,lx^2/2}$.
For example, $H\!\!\!E_{a}^{(l)}=X_a^{-1} q^{\,lx^2/2}$ for $a\in P_+$, since $T_a=Y_a$ for such $a$. It readily results in \begin{align}\label{hnonsym-sym} \widehat{\mathscr{P}}\,'_+ (H\!\!\!E_{a}^{(l)})\ = \ ct(t^{-1}) H_a^{(l)} \ \hbox{ for } \ a\in P_+ \end{align} due to $\widehat{\mathscr{P}}\,'=ct(t^{-1})\,\widehat{\mathscr{I}}$. It of course includes the analysis of the coefficient-wise convergence of the operator $\widehat{\mathscr{P}}\,'$ and when it acts in $\mathscr{X} q^{\,lx^2/2}$ from \cite{ChNT} and this work.
We think that such an interpretation of our proportionality theorem (it was not stated in \cite{ChNT}, v5) can be of importance. A statement similar to (\ref{hnonsym-sym}) was known to Alexander Braverman (unpublished), with a reservation that only dominant weights $a$ subject to $(a,\theta)\le l$ appear in his approach. He did not define the nonsymmetric affine Hall polynomials and we do not know how he justified the proportionality theorem (presumably, using the theory from \cite{ChNT}). The convergence of $\widehat{\mathscr{P}}\,'$ is of course the key here.
We are very thankful to him for triggering (\ref{hnonsym-sym}). As it was explained above, the extension of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ by the powers of the Gaussians is not necessary here and in (\ref{affineHallns}); the PBW theorem twisted by $\tau_+^l$ is the main algebraic ingredient of the construction. }
\subsubsection{\sf Discussion, some references} The formula for the affine Satake-type operator $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ was considered by several specialists as a ``natural" extension of the Macdonald $\mathfrak{p}$\~formula, including certain geometric aspects and applications.
The main reference is \cite{Ka}; see also \cite{FGT,BK}. Equivalent definitions of the affine Hall-Littlewood functions were suggested by several authors (not always published), for instance, by Feigin and Grojnowski; let us also mention Garland's works.
Independently, the affine Hall functions of level one were explicitly calculated in \cite{C5} in the context of Jackson integrals (see also \cite{Sto}). The paper \cite{FGT} contains an important interpretation of the affine Hall functions via the Dolbeault cohomology of the {\em affine Grassmannian} and related flag varieties. The appearance of the $ct(t^{-1})$ in the formulas is interpreted there as the ``failure of the Hodge decomposition." See also Section 12.7 in \cite{FGT} concerning the level-one formulas.
The definition of $\widehat{\mathscr{P}}\,'_+$ is straight; it belongs to a completion of the corresponding affine Hecke algebra. It becomes really interesting when acting in DAHA modules; this theory is new.
Both operators, $\widehat{\mathscr{I}}\,=\,\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$\, and $\widehat{\mathscr{P}}\,'_+$, are proportional whenever the operator $\widehat{\mathscr{P}}\,'_+$ exists (see Theorem \ref{HALLONE}). They complement each other in the following sense.
The convergence of the $\widehat{\mathscr{I}}\,$ for $l>0$ is better and much simpler to manage than that of $\widehat{\mathscr{P}}\,'_+$. However, the latter operator acts naturally in DAHA modules and, importantly, does not require a priori knowledge of the $\mu$\~function; for instance, this provides an alternative way to supply the polynomial and similar representations with inner products. Accordingly, this operator
has no singularities (at the denominator of $\widetilde{\mu}$). Also, $\widehat{\mathscr{P}}\,'_+\,$ is an exact DAHA-version of the classical Satake isomorphism in the AHA theory and it is closely connected with the theory of Demazure characters. Let us comment on the latter.
Under the limit $t\to \infty$, the operator $\widehat{\mathscr{I}}\,$ is directly connected with the {\em Weyl-Kac formula} for Kac-Moody characters; the functions $ct(t^{-1})H_{-b}^{(l)}$ tend to the corresponding characters for the affine dominant weights $b$. Theorem \ref{YLEFT} generalizes the presentation of the corresponding Kac-Moody character as an inductive limit of the {\em Demazure characters}\,.
The proportionality itself is an operator $t$\~variant of the presentation of the Kac-Moody characters as {\em sums} of properly normalized Demazure characters associated with the Demazure modules the Borel subalgebra opposite to the one used for the highest vectors.
\subsubsection{\sf Proportionality for \texorpdfstring{$l>0$}{level>0}} Let us begin with the level-one case. Then we have a reasonably complete theory from \cite{C5} (see also \cite{C101}) and \cite{Sto} devoted to the $C^\vee C$\~case. Let us mention \cite{Vi}, where the level-one case is addressed in the simply-laced case. Theorem 2 there is a special case of Theorem 7.1 from \cite{C5} (for simply-laced root systems). The relation of Theorem 2 to the difference Mehta-Macdonald formulas from \cite{C5} {\em in the compact case} is discussed in \cite{Vi}. The compact case is that based on the constant term inner product (more generally, on the imaginary integration). However, it is the {\em noncompact case}, namely the Jackson integration formula from \cite{C5} (not mentioned in \cite{Vi}), that is {\em directly} connected with the affine Hall functions of level one.
Works \cite{C5,C101} were written in the self-dual setting, i.e., for the {\em twisted} affine root system $\tilde R^\nu$, where the same lattice $P$ is used in $\widehat{W}$ and for $X_a$ (and $E_a$). Accordingly, the operator $T_0$ changes to the one with $\alpha_0=[-{\vartheta},1]$ for the maximal {\em short} root ${\vartheta}$. Restricting ourselves to the simply-laced case, the results from \cite{C5} on the Mehta-Macdonald formulas in the context of Jackson integration can be formulated as follows. Recall that $\alpha^\vee=\alpha$, $\omega_i^\vee=\omega_i$ in this case due to the normalization $(\alpha,\alpha)=2$ for $\alpha\in R$.
\begin{theorem}\label{HALLONE} Let $R$ be a simply-laced root system. We set $\gamma(x)\stackrel{\,\mathbf{def}}{= \kern-3pt =}
|W|^{-1}\,\sum_{\widehat{w}\in \widehat{W}}\widehat{w}(q^{x^{2}/2})$
$=q^{x^2/2}\,\sum_{b\in P}\,X_{b}\,q^{b^{2}/2}$ for the order $|W|$ of the nonaffine Weyl group $W$. Let $X_b(q^a)\stackrel{\,\mathbf{def}}{= \kern-3pt =} q^{(b,a)}$, $\widehat{P}(t^{-1})$ is from (\ref{rataffpoin}). The level will be $l=1$.
(i) The series $\widehat{\mathscr{P}}\,'_+$ considered as an operator in $\mathscr{X}q^{x^2/2}$ converges element-wise for all $t\in {\mathbb C}^*$. The proportionality relation $$ \widehat{\mathscr{I}}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \widehat{\mathscr{S}}\,'_+ \circ \widetilde{\mu}\ =\ \ct(t^{-1})^{-1}\widehat{\mathscr{P}}\,'_+ $$ holds for any $t\neq 0$ as well; cf. (\ref{propor}).
(ii) Assuming that $E_a$ is well defined, \begin{align}\label{levonehall} &\widehat{\mathscr{S}}\,'_{+}(\widetilde{\mu}\,E_{a}q^{x^{2}/2})= \frac{\widehat{P}(t^{-1})}{\hbox{ct}(t^{-1})} \widehat{\mathscr{P}}_{+}(E_{a}q^{x^{2}/2})\\ &=E_{a}(q^{-k\rho})\, q^{-a^{2}/2-k(a_{+},\rho)} \cdot\prod_{\alpha\in R_{+}} \prod_{j=0}^{\infty}\frac{1-t^{-1-(\rho,\alpha)}q^{j}} {1-t^{-(\rho,\alpha)}q^{j}}\cdot \gamma(x).\notag \end{align}
(iii) If $t$ is not a root of unity, then the linear map $\widehat{\mathscr{P}}\,'_{+}$ is identically zero in $\mathscr{X}q^{x^{2}/2}$ if and only if $t^{m_i}=q^j$ for $j\in {\mathbb N}$ (for instance, for $t=q$). Here $\{m_1,m_2,\ldots,m_n\}$ are the exponents of $R$; $m_i=d_i-1$ for the degrees $\{d_i\}$. The map $\widehat{\mathscr{I}}$ is identically zero on $\mathscr{X}q^{x^{2}/2}$ if and only if $t^{d_i}=q^j$ for $j\in {\mathbb N}$ and $j/d_{i}\not\in {\mathbb N}$ (for instance, this map vanishes identically at $t=q^{1/h}$, where $h=(\theta,\rho)+1$ is the Coxeter number). \end{theorem}
{\em Sketch of the proof.} The existence of $\widehat{\mathscr{P}}\,'_+$ for all $k\in {\mathbb C}$ and the corresponding extension of the proportionality from $(\ref{propor})$ is due to the fact that the image of this operator is one-dimensional for generic $k$ and therefore proportional to $\gamma(x)$. The best way to proceed here is via the level-one variant of Theorem \ref{PSBULLET}, namely, by considering the inner product \begin{align*} &\langle f,g\rangle_1\ =\ \bigl(\widehat{\mathscr{P}}\,'_+(fg q^{x^2/2})\bigr)(id)\,. \end{align*}
Paper \cite{C5} contains the formula for $\widehat{\mathscr{S}}\,'_{+}(\widetilde{\mu}\,E_{a}q^{x^{2}/2})$ from (\ref{levonehall}). To check $(iii)$, use the explicit formula for ct$(t^{-1})$ and the fact that all $E_a$ are well defined with nonzero $E_a(q^{-k\rho})$ for positive $\Re k$. \phantom{1}
$\qed$
{\bf Comment.\ } The levels $0$ and $1$ are exceptional from the viewpoint of convergence. For $l=0$, the convergence of both, $\widehat{\mathscr{I}}$ and $\widehat{\mathscr{P}}\,'_+$, is (naturally) significantly worse than the convergence in the presence of the Gaussian. For $l=1$, $\widehat{\mathscr{P}}\,'_+$ converges much better than for (integral) $l>1$ due to the fact that its image is one-dimensional. Recall that $\widehat{\mathscr{I}}$ always converges for $l>0$.
\phantom{1}
$\qed$
\begin{theorem}\label{GENPROPR} We continue to assume that $R$ is simply-laced, but $l$ can be an arbitrary complex number now such that $\Re l>0$. If $l\not\in {\mathbb Z}$, then we need to avoid the {\sf\, nonreal singularities\,} of the function $\widetilde{\mu}(X;q,t)$; see (\ref{mutildemu}) and (\ref{polesmu}). Restricting the functions to a sufficiently small neighborhood of $x=0$ is sufficient. Considering $\widehat{\mathscr{I}}$ and $\widehat{\mathscr{P}}\,'_+$ as operators acting in the space $\mathscr{X}q^{\,lx^{2}/2}$, the former operator converges absolutely element-wise for any $k$ and the latter converges absolutely as $\Re k <1/h$ for the Coxeter number $h$. Under the condition $\Re k <1/h$, the proportionality holds: $\ct(t^{-1})\widehat{\mathscr{I}}\ =\ \widehat{\mathscr{P}}\,'_+.$ \end{theorem} {\em Proof.} The convergence and proportionality here can be deduced from the corresponding coefficient-wise claims from Theorem \ref{TQ1H} in Section \ref{sect: Coefficient}. See also Lemma \ref{LEMThatw} concerning the convergence. The estimates in $(e)$ there and the fact that the growth of the coefficients of $\widehat{\mathscr{P}}\,'_+$ is no greater than exponential are sufficient for the convergence due to the presence of the Gaussian. Theorem \ref{YEXPA} below provides sharp estimates for the coefficients of $Y$\~operators in the case of $A_1$.
For $\Re k<0$, the absolute convergence of $\widehat{\mathscr{P}}\,'_+$ and, therefore, the proportionality follow from the convergence of this operator in the space $\mathscr{PW}_0(\mathcal{U})$ there. The estimates from Theorem \ref{LEVZERO} (the level zero case) can be almost directly used for such $k$ as well; the convergence will be no worse than it was for $a=0$ in this theorem. \phantom{1}
$\qed$
{\bf Comment.\ } Let us mention the symmetrizer $ \sum_{\widehat{w}}t^{l(\widehat{w})/2}T_{\widehat{w}}$, with $t,T$ instead of $t^{-1},T^{-1}$. Its convergence range in the space $\mathscr{X}q^{-lx^2/2}$ is $\Re k>-1/h$ (unless $l=0,1$), i.e., negating the range for $\widehat{\mathscr{P}}\,'_+$ (acting in
$\mathscr{X}q^{+lx^2/2}$). We continue to assume that $|q|<1$.
This symmetrizer corresponds to the theory of {\em imaginary integration}. Applying it to $\mathscr{X}q^{+lx^2/2}$ with positive $\Re l$ is possible too provided that $\Re k>0$, however the result will be zero identically. \phantom{1}
$\qed$
\subsubsection{\sf The Looijenga spaces}{\label{sect:loo}} For positive integral levels $l>0$, let us introduce the {\em Looijenga space} \begin{equation*} \mathcal{L}_{l}= \{\sum_{\widehat{w}\in \widehat{W}}\widehat{w}(X_{a}q^{lx^{2}/2}),\ a\in P\}. \end{equation*}
It can be identified with the space Funct$\,(P/lP^\vee)^{\Pi W}$ formed by the $\Pi W$\~invariant functions on the set $P/lP^\vee$. Recall that $P^\vee\subset P$ due to the normalization $(\theta,\theta)=2$. The action of $W$ is natural. The action of the group $\Pi=\{\pi_r=\omega_r u_r^{-1}\,\mid\, r\in O\}$
is as follows.
Let us identify the space Funct$\,(P/lP^\vee)^{W}$ with the space Funct$\,(\mathcal{C}_l)$, defined for the set
$\mathcal{C}_l\stackrel{\,\mathbf{def}}{= \kern-3pt =}\{b\in P_+\,|\, (b,\theta)\le l\}$. The group $\Pi$ naturally acts on the {\,\em set\,} $\mathcal{C}_l$ through its action on the {\em closed fundamental affine Weyl
chamber} $\{x\in {\mathbb R}_+\cdot P_+\,|\, (x,\theta)\le 1\}$ ``multiplied" by $l$. More algebraically, we can identifying $\Pi$ with the group $\{(l\omega_r) u_r\,\mid\, r\in O\}$ and consider the affine action of the latter on the points of the set $\mathcal{C}_l$. Then $\mathcal{L}_{l}$ becomes isomorphic to Funct$\,(\mathcal{C}_l)^\Pi$.
For instance,
the permutation induced by $\pi_1\in \Pi$ on $\mathcal{C}_2$ in the case of $A_2$ reads as follows: \begin{align*} \mathcal{C}_2=&\{0,\omega_1,\omega_2,\omega_1+\omega_2, 2\omega_1,2\omega_2\}\\ \pi_1(\mathcal{C}_2)=&\{2\omega_1,\omega_1+\omega_2,\omega_1,\omega_2,2\omega_2,0\}. \end{align*}
Thus dim\,$\mathcal{L}_2=6/|\Pi|=2$ in this example. Only the sets $\mathcal{C}_{3p}$ contain a (unique) $\Pi$\~invariant point, which is $p(\omega_1+\omega_2)$. The general dimension formula for $A_2$ $(l>0)$ is $$ \hbox{dim}\,\mathcal{L}_{l}=(\frac{(l+2)(l+1)}{2} + \delta_l)/3 \ \hbox{ for } \ \delta_{3p}=2, \delta_{3p\pm 1}=0. $$
For $A_1$, $\hbox{dim}\,\mathcal{L}_{l}=1+[l/2]$, where $[\cdot ]$ is the integer part. Indeed, $\pi_1$ transposes $0$ and $l\omega_1$ in this case and has a fixed point if and only if $l$ is even.
\begin{theorem} The space $\mathscr{H}_{l}= \widehat{\mathscr{I}}(\mathscr{X}q^{lx^{2}/2})$\, belongs to $\mathcal{L}_{l}.$ For generic $k$, for instance, provided that $\Re k<0$, this space coincides with $\mathcal{L}_l$. \end{theorem} {\em Proof}. The surjectivity of the map $\widehat{\mathscr{I}}: \mathscr{X}q^{lx^{2}/2}\to$ $\mathcal{L}_{l}$ for generic $k$ is straightforward; adding $\widetilde{\mu}$ does not change the image. One can also use that this map is zero on $\mathcal{J}_{l}(\mathscr{X})q^{lx^{2}/2}$ (see below) and apply Theorem \ref{THMDIMCOINV}.\phantom{1}
$\qed$
Note that the group of the automorphisms of the {\em nonaffine} Dynkin diagram acts in Funct$\,(\mathcal{C}_l)^\Pi$. This action commutes with the action of this groups on $\mathscr{X}$ under the map $\widehat{\mathscr{I}}$, since the Gaussian is invariant with respect to these automorphisms. For instance, \begin{align}\label{varsigmaf} (H^{(l)}_{a})^{\varsigma}=H^{(l)}_{\varsigma(a)} \ \hbox{ for } \ \varsigma(a)=-w_{{}_0}(a),\ X^\varsigma_a=X_{\varsigma(a)}. \end{align}
\subsection{{\bf DAHA coinvariants}}\label{sec:coinvariants} \subsubsection{\sf Polynomial coinvariants}\label{sec:pcoinv} We will introduce the coinvariants only in the context of the polynomial representation. The space of {\em coinvariants of level $l$} is $\mathscr{X}/\mathcal{J}_{l}(\mathscr{X})$ for the subspace $$ \mathcal{J}_{l}(\mathscr{X})\stackrel{\,\mathbf{def}}{= \kern-3pt =}\langle\, q^{-lx^{2}/2}\, T_{\widehat{w}}\,q^{\,lx^{2}/2}(X_a)-
t^{l(\widehat{w})/2}X_a\,|\, \widehat{w}\in \widehat{W}, \, a\in P\,\rangle\subset \mathscr{X}. $$ We note that taking only finitely many $X_a$ is sufficient in this definition (and all $\widehat{w}$). For instance, it suffices to make $a=0$ if the quotient is one-dimensional (say, when $l=1$ in the simply-laced case).
By construction, $\mathcal{J}_{l}(\mathscr{X})q^{\,lx^{2}/2}$ belongs to the kernel of the map $\widehat{\mathscr{I}}$. Denoting the map $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\ni A\mapsto q^{x^{2}/2}A\,q^{-x^{2}/2}$ by $\tau$ (it is an automorphism of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$), $\mathcal{J}_{l}(\mathscr{X})= \tau^{-l}(\mathcal{J}_{0}(\mathscr{X}))$.
We claim that the dimension of $\mathscr{X}/\mathcal{J}_{l}(\mathscr{X})$ {\em always} coincides with that of the Looijenga space (defined above). The dimension of the space of coinvariants can be calculated without any reference to the Looijenga space.
\begin{theorem}\label{THMDIMCOINV} For any $q,t\in {\mathbb C}^*$ and $l>0$, \begin{align*} {\hbox{\rm dim}}_{\mathbb C}\,(\mathscr{X}/\mathcal{J}_{l}(\mathscr{X})) \,=\,{\hbox{\rm dim}}_{\mathbb C}\,( \text{Funct\,}(\mathcal{C}_l)^{\Pi}). \end{align*} \end{theorem} {\em Sketch of the proof.} We use the PBW theorem to establish the inequality \begin{align}\label{dimineqality} {\hbox{\rm dim}}_{\mathbb C}\,(\mathscr{X}/\mathcal{J}_{l}(\mathscr{X})) \,\leq\,{\hbox{\rm dim}}_{\mathbb C}\,( \text{Funct\,} (\mathcal{C}_l)^{\Pi}). \end{align} Let $k\to 0$ $(t=q^k\to 1)$. Then $T_{\widehat{w}}\to \widehat{w}$ and $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(t=1)$ becomes the classical Weyl algebra generated by $X_a$ and $Y_b$ extended by $W$. The dimension can be readily calculated at $k=0$; it equals ${\hbox{\rm dim}}_{\mathbb C}\,( \text{Funct\,}(\{b\in P_{+}, (b,\theta)\leq l\})^{\Pi})$. Due to (\ref{dimineqality}), this dimension must remain the same for all $q,t$.\phantom{1}
$\qed$
\subsubsection{\sf The B-case}\label{sect:THEBCASE} Avoiding the non-simply-laced root systems in Theorem \ref{HALLONE} is not only a technicality. The dimension of $\mathcal{L}_{1}$ is greater than one if $P\neq P^\vee$, so it is not true (generally) that all level-one Hall functions are proportional to $\gamma(x)$, as stated in this theorem. However for $B_n$, there is the following possibility to make the image really one-dimensional (for $l=1$).
We use that $Q=P^\vee$ in this case and consider $\mathscr{X}\,'={\mathbb C}_{q,t}[X_a,a\in Q]$ instead of the complete polynomial representation $\mathscr{X}$. The space $\mathscr{X}\,'$ is a module over the {\em little DAHA} (in the terminology from \cite{C101}), which is generated by $\mathscr{X}\,'$ and the same $\{T_{\widehat{w}},\widehat{w}\in \widehat{W}\}$; all the considerations above hold under this restriction. The corresponding level-one Looijenga space will be isomorphic to Funct$\,(Q/lQ^\vee)^{\,W}$, i.e., will be of dimension one as $l=1$. The formula (\ref{levonehall}) holds if $\rho$ is replaced by $\rho^\vee$ and $a\in Q$.
Generally, if there is any DAHA-submodule $\mathscr{X}\,'$, then, automatically, $$\widehat{\mathscr{I}}(\mathscr{X}\,'\,q^{lx^{2}/2}) \subset \{\sum_{\widehat{w}\in \widehat{W}}\widehat{w}(G(X)\,q^{lx^{2}/2}),\, G(X)\in \mathscr{X}\,'\} \hbox{\ \, for\ any\ } l>0. $$
\subsubsection{\sf Levels 0 and 1} Let us consider the (simplest) cases when the space of coinvariants is one-dimensional. \begin{theorem}\label{thm:coinv} In the level-zero case, provided that the space of $Y$\~eigenvectors with the eigenvalue $t^{\rho}$ (i.e., containing $E_{0}=1$) is one dimensional in $\mathscr{X}$, $$ {\hbox{\rm dim}}_{\mathbb C}\,(\mathscr{X}/\mathcal{J}_0(\mathscr{X}))=1\ \, \hbox{and\ \,} \oplus_{\,q^\lambda\,\neq\, t^{\rho}\,}\mathbb{C}\mathscr{X}_{\lambda}= \mathcal{J}_0(\mathscr{X}), $$
where $\mathscr{X}_\lambda=\{f\in \mathscr{X}\,|\, (Y_a-q^{(\lambda,a)})^N (f)=0\}$ for sufficiently large $N$; we identify $q^\lambda$ if they give coinciding $Y$\~eigenvalues. This dimension is one for $l=1$ as well in the simply-laced case; then $q,t$ can be {\sf arbitrary} nonzero. \end{theorem} {\em Proof.} If the nonsymmetric Macdonald polynomials $E_a$ are well defined, then they form a basis for $\mathscr{X}$. Otherwise, use the generalized $Y$\~eigenvectors in the following reasoning. Recall that the action of $Y_{b}$ is given by $Y_{b}^{-1}(E_{a})=q^{(a_{\sharp},b)}E_{a}$ for $a\in P,\, b\in P^\vee$. So for any $a\in P$ such that $q^{(a_{\sharp},b)}\neq q^{k(\rho,b)}$, we have $E_{a}\in \mathcal{J}_0(\mathscr{X})$. Then $E_{0}=1$ is of multiplicity one in $\mathscr{X}$ and ${\hbox{\rm dim}}_{\mathbb C}\,(\mathscr{X}/\mathcal{J}(\mathscr{X}))=1$.
In the case $l=1$, we use that $\tau_-\tau_+^{-1}(Y_b)=$ $\tau_-\tau_+^{-1}\tau_-(Y_b)= \sigma^{-1}(Y_b)=X_b^{-1}$ and apply $\tau_-^{-1}$ to the triple $\{\,\{X_a\}, \{T_w\}, \{Y_b\}\,\}$, satisfying the PBW theorem. See \cite{C101} for the definitions of $\tau_{\pm}, \sigma$ and also see Lemma \ref{lem:basis} below for the case of $A_1$. \phantom{1}
$\qed$
\subsection{{\bf Kac-Moody limit}} The limiting case $t\to\infty$ ($k\to -\infty$) is important. Then the Hall function $\widetilde{H}_{a}^{(l)}$ for a weight $a\in P_+$ subject to $(a,\theta)\le l$ becomes proportional to the character of the corresponding integrable Kac-Moody module. The level $\,l\in {\mathbb N}\,$ equals the action of the central element $c$ in the standard normalization; we consider here only the case of standard (split) Kac-Moody algebras.
Notice that we use the extended affine Weyl group $\widehat{W}$ with $P^\vee$ instead of $Q^\vee$ (usual in Kac-Moody theory) and that, in our approach, the weights $a\in P$ are not supposed to be $l$\~dominant. The Hall functions can be defined for any $a$, but their interpretation as characters of integrable modules of level $\,l\,$ in the limit does require $a\in P_+$ and the inequality $(a,\theta)\le l$. This connection with the Kac-Moody characters is known; see e.g., \cite{Vi}. Let us discuss this in detail.
\subsubsection{\sf Explicit formulas} From (\ref{mutildemu}) and (\ref{consterm}), \begin{align}\label{tildemulim} &\widetilde{\mu}(t\to\infty) = \prod_{\widetilde{\alpha}>0}\frac{1} {1-X_{\widetilde{\alpha}}},\ \lim_{t\to\infty}\hbox{ct}(t^{-1})=\prod_{i=1}^\infty\frac{1} {(1-q^{i})^n}. \end{align}
Also, $\widehat{P}(t^{-1})\to |\Pi|$ as $t\to\infty$\, for\, $\Pi=P^\vee/Q^\vee$. Setting $$ \widehat{\chi}_a ^{(l)} \stackrel{\,\mathbf{def}}{= \kern-3pt =} q^{-l\frac{x^2}{2}}\, \lim_{t\to \infty}\widetilde{H}_{-a}^{(l)} \ \hbox{ for } \ a\in P \hbox{\ \,(notice\ $-a$)}, $$ \begin{align}\label{Kac-Moody} \widehat{\chi}_a^{(l)}\ &=\ q^{-l\frac{x^2}{2}} \sum_{\widehat{w}\in \widehat{W}}\, \widehat{w}(X_{a}^{-1}\,\widetilde{\mu}(t\to\infty)\,q^{l\frac{x^2}{2}})\\ &=\ \bigl(\sum_{\widehat{w}=bw} (-1)^{l(\widehat{w})} X_{\widehat{w}(\widehat{\rho}+a)-\widehat{\rho}+lb}^{-1}\ q^{\,lb^2/2}\bigr)/ \prod_{\tilde{\alpha}\in \tilde R_+}(1-X_{\tilde{\alpha}}). \notag \end{align} Here the summation is over all $b\in P^\vee, w\in W$ and we set (symbolically) $\widehat{\rho}=\frac{1}{2} \sum_{\tilde{\alpha}\in \tilde R_+}\tilde{\alpha}$ (as for Kac-Moody algebras). What we really need is the relation $$\sum_{\tilde{\alpha}\in \Lambda(\widehat{w}^{-1})}\tilde{\alpha}\ =\ \widehat{\rho}-\widehat{w}(\widehat{\rho}) $$ for the sets $\Lambda(\widehat{w}^{-1})$ defined in (\ref{Lahw}); note $\widehat{w}^{-1}$ here. Using the level-zero and level-one formulas for $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$, \begin{align}\label{thetaprod} \prod_{\tilde{\alpha}\in \tilde R_+}(1-X_{\tilde{\alpha}})&= \frac{\sum_{\widehat{w}=bw} (-1)^{l(\widehat{w})} X_{\widehat{\rho}-\widehat{w}(\widehat{\rho})}}
{|\Pi|\,\prod_{j=1}^\infty (1-q^j)^n}\\ &=\frac{\sum_{\widehat{w}=bw} (-1)^{l(\widehat{w})} X_{\widehat{\rho}-\widehat{w}(\widehat{\rho})-b}\ q^{\,b^2/2}}{\sum_{b\in P}\,X_{b}\,q^{\,b^2/2}}. \label{thetaprodone} \end{align} Formula (\ref{thetaprodone}) is stated here in the simply-laced case as in (\ref{levonehall}). One can readily adjust this formula to the setting of \cite{C101}, i.e., to the case of {\em twisted} $\tilde R^\nu$\~affinization (then an arbitrary reduced nonaffine $R$ can be used).
These two formulas are the denominator identity and the level-one formula due to Kac. See
Theorem 10.4, Lemma 12.7 and (12.13.6) from \cite{Kac}. We conclude that $ct(t^{-1})|_{t\to \infty}\,\widehat{\chi}_a^{(l)}$ is the character of the corresponding Kac-Moody integrable module of level $l$ provided that $a\in P_+$ and $(a,\theta)\le l$; this is upon the substitution $X_{b}\mapsto e^{-b}$.
Let us provide the first few terms of the numerators of these formulas in the case of $A_1$: \begin{align*} & \sum_{\widehat{w}=bw} (-1)^{l(\widehat{w})} X_{\widehat{\rho}-\widehat{w}(\widehat{\rho})-lb}\ q^{\,lb^2/2}\mod (q^2)\\ =&\left\{\begin{array}{ccc} 1-X^2+q^{1/4}(X^{-1}-X^3-qX^{-3}+qX^5) &\ \hbox{ for } \ l=1,& \\ 2(1-X^2+qX^{4}-qX^{-2}) &\ \hbox{ for } \ l=0,& \end{array}\right. \end{align*} where $X=X_{\omega_1}$. Compare this with the left-hand side of (\ref{thetaprod}) and (\ref{thetaprodone}) multiplied by the corresponding denominators (here the calculations are direct).
In our approach, there are no clear reasons to stick here to {\em affine $l$\~dominant} weights, i.e., to $a\in P_+$ subject to $(a,\theta)\le l$. Apart from the weights of integrable modules, i.e., for arbitrary $a\in P$, the following level-one formulas in terms of the polynomials $\widetilde{E}_a\stackrel{\,\mathbf{def}}{= \kern-3pt =} E_a(t\to \infty)$ are worth mentioning: \begin{align}\label{Kac-M-Herm} \widehat{\mathscr{S}}_+'\bigl( \widetilde{\mu}(t\to\infty)\, \widetilde{E}_a\,q^{x^2/2}\bigr)= \left\{\begin{array}{ccc}q^{-a^2/2}\gamma(x), & & \hbox{if\ } a\in P_-, \\mathbf{0}, & & \hbox{otherwise}. \end{array}\right. \end{align}
We use formula (\ref{levonehall}). The polynomials
$\widetilde{E}_a$ are closely connected with the $q$\~Hermite polynomials $E_a(t\to 0)$ studied in \cite{ChW} (and which play the key role in the theory of $q$\~Whittaker functions).
{\bf Comment.\ } Let us consider briefly the limit $t\to 0$ ($k\to \infty$). Then the series $\widetilde{\mu}^{-1}\circ \widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ can also be interpreted via the Kac-Moody characters. Due to (\ref{murelations}), \begin{align*} &q^{-l\frac{x^2}{2}}\, \lim_{t\to 0}\widetilde{H}_{a}^{(l)} \ =\ \frac{\sum_{\widehat{w}=bw} (-1)^{l(\widehat{w})} X_{\widehat{w}(\widehat{\rho}+a)-\widehat{\rho}-lb}\ q^{\,lb^2/2}}{ \prod_{\tilde{\alpha}\in \tilde R_+}(1-X_{\tilde{\alpha}})}. \end{align*}
\subsubsection{\sf Match at level one} We note that (12.13.6) from Kac's book is stated in the simply-laced case, which matches the setting we use for formulas (\ref{levonehall}) and (\ref{thetaprodone}). Calculating the level-one characters in the cases $B_n, F_4, G_2$ is due to Kac and Peterson. As for the $k$\~case (i.e., when $t$ is added), we explained in Section \ref{sect:THEBCASE} how to proceed in the $B$\~case for the lattice $Q^\vee$. The root systems $F_4$ and $G_2$ with $k$ also seem doable.
The most difficult case in the theory of level-one Kac-Moody characters is $C_n$ (managed by Kac and Wakimoto); it seems exactly parallel to the problem with explicit formulas for the affine Hall functions of type $C_n$ for $l=1$ (untwisted). The paper \cite{Sto} devoted to the $C^\vee C$ may contain the methods and results sufficient to manage this case.
The above discussion and considerations of this section are in the {\em untwisted} case. The formulas for the {\em twisted Kac-Moody characters} are known for any root systems. The twisted KM\~characters correspond (with some reservations) to our using $\tilde R^\nu$, the twisted affinization from Section \ref{sect:thinaff}. Similar to Kac-Moody theory, the level-one formulas with $k$ were obtained (uniformly) in \cite{C5} for {\em any} reduced root systems.
It is worth mentioning that the classification of Kac-Moody algebras is {\em not} the same as that for DAHA (which continues the classical classification of symmetric spaces). However, when they intersect, it seems that there is almost an exact match between the problems arising in the theory of Kac-Moody characters and those for the affine Hall functions (with $k$). At least, this is so in the level-one case. Recall that the affine Hall functions belong to the same Looijenga space as the Kac-Moody characters. We do not discuss explicit formulas for $l>1$, where not much is actually known; see \cite{Vi}.
Let us mention that in the level-one case, the affine Demazure characters are directly connected with the nonsymmetric $q$\~Hermite polynomials $E_a(t\to 0)$ (see above). They become $W$\~invariant for $a\in P_-$ and their coefficients in this case are given in terms of the $q$\~Kostka numbers (see \cite{San},\cite{Ion1}).
We are grateful to Victor Kac who helped us establish the correspondence between the two theories, the classical KM theory and the one for arbitrary $k$. We thank Boris Feigin for a helpful discussion. As a matter of fact, we introduce in this paper certain $t$\~deformations of the Demazure characters, but our definition is of a technical nature and we do not now how far this can go.
\subsection{{\bf Shapovalov forms}} We will begin with a very general approach to constructing inner products (in functional analysis, known as $GNS$ construction). Let $\mathcal{F}$ be a cyclic $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\~module, i.e., $\mathcal{F}=\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(vac)$ for some $vac\in \mathcal{F}$. Actually $\mathcal{F}$ can be absolutely arbitrary in the following (formal) considerations, but we prefer to restrict ourselves to cyclic modules here. We assume that $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ and $\mathcal{F}$ are defined over a field $\widetilde{{\mathbb C}}$. It can be ${\mathbb C}_{q,t}$, the definition field for the polynomial representation of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$, or its extension by the parameters of $\mathcal{F}$ (treated as independent variables). If $q,t$ and the parameters of $\mathcal{F}$ are considered as nonzero complex numbers, then $\widetilde{{\mathbb C}}={\mathbb C}$.
\subsubsection{\sf Symmetric J-coinvariants}
We set $\mathcal{J}=\{A\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\,|\,A(vac)=0\}$ (a left ideal). Then $\mathcal{F}\cong \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}/\mathcal{J}$. {\em Any} form on $\mathcal{F}$ which is symmetric and $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\~invariant with respect to a given anti-involution $\star$ can be obtained as follows.
Let $\star$ be an anti-involution $\star$ on $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\, ($\star^2=1$ is required because the form must be symmetric) and let $\varrho: \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\to \widetilde{{\mathbb C}}$ be a functional on $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ such that $\varrho(A^{\star})=\varrho(A)$ and $\varrho(\mathcal{J})=0$. Automatically, we have that $\varrho(\mathcal{J}^{\star})=0$ ($\mathcal{J}^{\star}$ is a right ideal in $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$). Since $\varrho(\mathcal{J}+\mathcal{J}^{\star})=0$, it comes from a functional $\varrho': \mathcal{F}\to \mathcal{F}/\mathcal{J}^{\star}(\mathcal{F})\to \widetilde{{\mathbb C}}$.
Then the form on $\mathcal{F}$ is introduced as follows: $$\langle f, g\rangle\stackrel{\,\mathbf{def}}{= \kern-3pt =}\varrho(\bar{f}^{\star}\,\bar{g})= \varrho'(f^{*}\,g), \, f,g\in \mathcal{F},$$ where we lift $f,g$ to $\bar{f},\bar{g}\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ and set $f^{*}=\bar{f}^{\star}(vac)$.
This form $\langle\,,\,\rangle$ is obviously symmetric and $\star$\~invariant: $$\langle A(f), g\rangle=\langle f, A^{\star}(g)\rangle, \text{ where } f,g\in \mathcal{F}, A\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}. $$
To describe all such forms, let us introduce the space \begin{align}\label{starcoinv} &\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}/(\mathcal{J}+\mathcal{J}^{\star})=\mathcal{F}/\mathcal{J}^{\star}(\mathcal{F}). \end{align} and its dual $\operatorname{Hom}_{\widetilde{{\mathbb C}}}(\mathcal{F}/\mathcal{J}^{\star}(\mathcal{F}),\widetilde{{\mathbb C}})$. Both have a natural action of $\star$ and are direct sums of $\pm 1$\~eigenspaces.
The subspace of $\star$\,\~invariant elements of $\operatorname{Hom}_{\widetilde{{\mathbb C}}}(\mathcal{F}/\mathcal{J}^{\star}(\mathcal{F}),\widetilde{{\mathbb C}})$ will be called the {\em space of $\star$\,\~symmetric $\mathcal{J}$\~coinvariants}. We will always assume that $1^\star=1$, correspondingly, $vac^\star=vac$.
The $\pm 1$\~eigenvectors of $\star$ from $\operatorname{Hom}_{\widetilde{{\mathbb C}}}(\mathcal{F}/\mathcal{J}^{\star}(\mathcal{F}),\widetilde{{\mathbb C}})$ lead to either $\star$\,\~invariant forms or to $\star$\,\~anti-invariant ones, respectively. In the examples we consider, the action of $\star$ is trivial in the whole space from (\ref{starcoinv}) and its dual, but the minus-sign (equally interesting) may occur as well.
Let us discuss basic examples.
\subsubsection{\sf Shapovalov pairs} We call the nonzero form $\langle\,,\,\rangle$ a {\em Shapovalov form} if $$\hbox{dim}_{\widetilde{{\mathbb C}}} \bigl(\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}/(\mathcal{J}+\mathcal{J}^{\star})\bigr)=1= \hbox{dim}_{\widetilde{{\mathbb C}}}\bigl(\mathcal{F}/\mathcal{J}^{\star}(\mathcal{F})\bigr), $$ and therefore this form is a unique symmetric $\star$\,\~invariant form in $\mathcal{F}$ up to proportionality. Accordingly, $\{\mathcal{J},\star\}$ is called a {\em Shapovalov pair}.
This terminology may be somewhat misleading. The anti-involutions we are going to consider generally have little to do with those in Lie theory; the connection with the Heisenberg and Weyl algebras is significantly more direct. However, our usage of the PBW theorem is really similar to the original Shapovalov construction.
Given a Shapovalov pair $\{\mathcal{J},\star\}$, finding $\langle f,g\rangle$ is purely algebraic problem directly related to the PBW theorem. For instance, $\langle f,g\rangle$ always depends rationally on the parameters $t,q$ of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$. It is valuable, since the forms given by integrals (or similar) are generally well defined only for some $t,q$. Their meromorphic continuation to other values of $q,t$ can be involved.
{\bf Comment.\ } We follow in this section unpublished notes by the first author devoted to the Arthur-Heckman-Opdam formulas \cite{HO2} in the theory of the spectral decomposition of AHA (due to Lusztig and many others). This approach is based on a relatively direct (without geometry) meromorphic continuation of the corresponding Plancherel formula and ``picking the residues".
The DAHA version of this decomposition is completed (by now) only for $A_n$ (unpublished). The best reference we can give so far is \cite{ChL}. The main theorem is that the Shapovalov form coincides with the analytic continuation of the corresponding inner product defined in terms of the standard integration over $i{\mathbb R}^n$ subject to $\Re k>0$. A direct analytic continuation of the latter to negative $\Re k$ appeared a certain generalization of the ``picking the residues" in AHA theory. In contrast to the Arthur-Heckman-Opdam method \cite{HO2}, the result of this procedure is known {\em a priori}. It is the Shapovalov form, which is defined entirely algebraically, and is rational or even regular in terms of $t$; see Theorem \ref{Shapind} below. \phantom{1}
$\qed$
The case of the standard form associated with the anti-involution $\ast$ of the polynomial representation, sending $t,q,X_a,Y_b,T_i$ to their inverses, was considered in \cite{C101},Proposition 3.3.2. The rational dependence of the corresponding inner products in terms of $q,t$ was deduced there from the uniqueness of such a form up to proportionality. A similar approach was applied to the anti-involution $\phi$ (governing the duality) in \cite{C101} and to the bilinear invariant forms involving the $q$\~Gaussians (generalizations of the Mehta-Macdonald integrals). See (\ref{diamondef}) below.
\subsubsection{\sf Y-induced modules} Let us discuss the Shapovalov forms for the $Y$\~induced modules $\mathcal{F}=\mathcal{I}_\lambda$, where $\lambda\in \widetilde{{\mathbb C}}^n$. By definition, $\mathcal{I}_\lambda$ is a free $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\~module over $\widetilde{{\mathbb C}}$ generated by $vac$ with the defining relations $Y_b(vac)=q^{(\lambda,b)}\,vac$. It belongs to the category $\mathcal{O}$ with respect to the action of $Y$\~elements, i.e., it can be represented as a direct sum of the {\em finite-dimensional\,} spaces of generalized $Y$\~eigenvectors. For the sake of definiteness, let us assume that $T_i^{\star}=T_i$ for $i=1,\ldots,n$. Then the corresponding $\varrho$ satisfies the following: \begin{align}\label{Shapo} \varrho(Y_a^{\star}T_w Y_b)=q^{(\lambda,a+b)}\varrho(T_w),\ \varrho(T_w)=\varrho(T_{w^{-1}}) \ \hbox{ for } \ w\in W. \end{align} The latter relation simply means that $\varrho$ is a trace functional on the nonaffine Hecke algebra $\mathbf{H}$.
We call the anti-involution $\,\star\, $ of {\em strong Shapovalov type} with respect to $\mathscr{Y}$ if $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ satisfies the PBW condition for $\mathscr{Y}$, $\mathbf{H}$ and $\mathscr{Y}^{\star}$ (replacing $\mathscr{X}$). Namely, if an arbitrary $A\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ can be uniquely represented as $c_{awb}\,Y_a^{\star}T_w Y_b$ for $a,b\in P^\vee$ and $w\in W$. Then the conditions from (\ref{Shapo}) determine $\rho$ completely. We see that the simply-laced root systems are generally needed here, unless in the twisted (self-dual) setting for the affine root system $\tilde R^\nu$, as in \cite{C101}. Note that the definition of strong Shapovalov anti-involutions depends only on $\star$ and $\mathscr{Y}$, not on the module $\mathcal{I}_\lambda$ ($\lambda$ can be arbitrary).
An important example of the {\em weak\,} (not strong) Shapovalov anti-involution in $\mathcal{I}_{\lambda}$ is when $\mathscr{Y}^{\star}=\mathscr{Y}$, i.e., $\mathscr{Y}$ is a {\em normal subalgebra} with respect to $\star$. Then the Shapovalov condition holds for $\mathcal{I}_\lambda$ provided that the {\em generalized\,} $Y$\~eigenspace containing $vac$ is one-dimensional in $\mathcal{I}_\lambda$. Indeed, the linear span of the spaces $(Y_a-q^{(a,\lambda)})\mathcal{I}_\lambda\subset$Ker$(\varrho)$ is of codimension one in $\mathcal{I}_\lambda$ in this case. Here $\star$ can be arbitrary, provided $\mathscr{Y}$ is normal.
There are actually only a few {\em strong} Shapovalov anti-involutions in DAHA theory, essentially the examples (1) and (3) considered below (for the subalgebra $\mathscr{Y}$). They play a significant role. The corresponding PBW property holds for any (nonzero) $q$ and $t$ for these anti-involutions.
The following rationality theorem clarifies the importance of the Shapovalov property in both, the weak and strong variants. The first generally guarantees rational dependence of the inner products on the parameters (including $q,t$); the second provides regular dependence.
We follow Proposition 3.3.2 from \cite{C101}. Let the algebra $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$, the representation $\mathcal{F}$, and the functional $\rho$ be defined over the same field $\widetilde{{\mathbb C}}$. For instance, the field of rationals ${\mathbb C}(q^{1/m},t^{1/2})$ can be taken for the polynomial representation (generally this field is supposed to contain the parameters of the module $\mathcal{F}$).
\begin{theorem}\label{Shapind} (i) A form $\langle\,,\,\rangle$ on $\mathcal{F}$ corresponding to a Shapovalov pair $\{\mathcal{J},\star\}$ is a unique symmetric $\star$\,\~invariant form in $\mathcal{F}$ up to proportionality; let us normalize it by the condition $\langle 1,1\rangle=1$. Then given $f,g\in \mathcal{F}$, their inner product $\langle f,g\rangle$ belongs to the field $\widetilde{{\mathbb C}}$ (which may include the parameters of $\mathcal{F}$).
(ii) Assuming that $\star$ satisfies the strong Shapovalov property for any nonzero $q$ and $t$, let $f,g$ be taken from $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}_{int}(vac)$, where \begin{align}\label{HHinteg} \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}_{int}={\mathbb C}[\,q^{\pm 1/m},t^{\pm 1/2}\,]\,[X_a,Y_b,T_w]_ {\hbox{\tiny nc}}\ \subset\ \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}. \end{align} The ring of coefficients here is the standard ${\mathbb C}$\~algebra necessary for the defining DAHA relations and by $[\ ]_{\hbox{\tiny nc}}$ we mean the noncommutative algebraic span. Then the inner product $\langle f,g\rangle$ is well defined for any nonzero $q,t$. In other words, if the PBW property holds for $\mathscr{Y}$,\, $\mathbf{H}$ and $\mathscr{Y}^{\star}$, then the corresponding form is regular in terms of $q^{\pm 1/m},t^{\pm 1/2}$. \end{theorem}
\subsubsection{\sf The polynomial case}{\label{sect:poly}} Let us discuss the Shapovalov condition for an arbitrary anti-involution $\star$, fixing $T_i$ for $i>0$, combined with the polynomial representation $\mathscr{X}$. This representation is a quotient of $\mathcal{I}_{\lambda}$ for $\lambda=k\rho$; the vacuum element (the cyclic generator of $\mathcal{I}_{k\rho}$) becomes $1\in \mathscr{X}$. One has $$\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}/(\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\mathcal{J}+\mathcal{J}^{\star}\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}) \cong \mathscr{X}/\mathcal{J}^{\star}(\mathscr{X})$$ for the left ideal $\mathcal{J}$ linearly generated by the spaces $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(T_{\widehat{w}}-t^{l(w)/2})$. This results in\ $ \varrho(Y_a^{\star}T_w Y_b)=t^{(\rho,a+b)+l(w)/2}.\ $
Chapter 3 of \cite{C101} is actually the theory of the following three anti-involutions and the corresponding symmetric forms: \begin{align}\label{diamondef} (1)&\ \ \varphi: X_a\leftrightarrow Y_a^{-1}, T_w\mapsto T_{w^{-1}},\notag \\ (2)&\ \ \Diamond: X_a\mapsto T_{w_0}^{-1}X_{-w_0(a)}T_{w_0}, Y_b\mapsto Y_b, T_w\mapsto T_{w^{-1}},\\ (3)&\ \ \Diamond_1= q^{-x^2/2}\circ\Diamond\,\circ q^{x^2/2}: Y_a\mapsto q^{-x^2/2}Y_a q^{x^2/2}.\notag \end{align} We assume that $R$ is simply-laced in (1) (it is arbitrary in \cite{C101} because $\tilde R^\nu$ is considered there). Let us provide some details.
(1) This anti-involution controls the duality and evaluation conjectures and is related to the Fourier transform. The Shapovalov property for $\varphi$ is {\em exactly} the PBW Theorem (any $q,t$). The corresponding form is well defined for any $q,t$ and the study of its radical is an important tool in the theory of the polynomial representation of DAHA.
(2) The second anti-involution governs the inner product in $\mathscr{X}$ (without conjugating $q,t$); $\Diamond$ is of Shapovalov type only for generic $k$ (and there is no immediate relation to the PBW theorem). So it is {\em weak\,}. The corresponding bilinear form is the key in the DAHA harmonic analysis, including the Plancherel formula for $\mathscr{X}$ and its Fourier image, the representation of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ in delta functions.
(3) The third appears in the difference Mehta-Macdonald formulas and is used to prove that the Fourier transform of the DAHA module $\mathscr{X}q^{-x^2/2}$ is $\mathscr{X}q^{+x^2/2}$. The strong Shapovalov property holds here, so the form is well defined for any $q,t$. The radical of the corresponding pairing is closely related to that in (1) (they coincide in the rational theory).
\subsection{{\bf Using induced modules}} \subsubsection{\sf Level-zero forms}{\label{sect:form}} Let us consider the coinvariants in the case $l=0$ via the affine symmetrizer $\widehat{\mathscr{P}}_+$. The $\widehat{\mathscr{P}}$\~symmetrizer is more convenient here than $\widehat{\mathscr{I}}$. The definition is in (\ref{defhatP}); we will also use the rational formula of Theorem \ref{P+FORMULA}, which gives a $t$\~meromorphic continuation of this operator when acting in $\mathscr{X}$.
Recall that $ \,\widehat{\mathscr{P}}_{+}(f)\,=\,\widehat{\mathscr{P}}\,'_{+}(f)/ \widehat{P}(t^{-1}),\, $ where $\widehat{P}(t)$ is the affine Poincar\'e series; see (\ref{defhatP}). We continue using the notation $\mathcal{J}\subset \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ for the ideal such that $\mathscr{X}=\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}/\mathcal{J}$; it is the linear span of subspaces \begin{equation*} \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}(T_{\widehat{w}}-t^{l(\widehat{w})/2}) \ \hbox{ for } \ \widehat{w}\in \widehat{W}. \end{equation*}
For the anti-involution $\Diamond$ in (\ref{diamondef}), the functional \begin{align*} \varrho_+:\, \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$} \, \to\, {\mathbb C}_{q,t}\hbox{\ \,sending\ \,} \ A\, \mapsto\, \widehat{\mathscr{P}}_+A(1) \end{align*} satisfies the $\Diamond$\~invariance property $ \varrho_+(\mathcal{J}^{\Diamond}+\mathcal{J})=0. $ Indeed, $$ \varrho_+(f)=\widehat{\mathscr{P}}_+(f),\ \varrho_+((T_{\widehat{w}}^{\Diamond}-t^{l(\widehat{w})/2})f)=0 \ \hbox{ for } \ f\in\mathscr{X}, $$ since $\Diamond$ preserves $\mathcal{H}=\langle T_{\widehat{w}}\rangle$. Thus, $\varrho_+$ can be used to construct a symmetric form on $\mathscr{X}$ corresponding to the anti-involution $\Diamond$.
This argument is of course {\em formal}; one needs to address the existence of $\widehat{\mathscr{P}}_+(f)$. Theorem \ref{P+FORMULA} provides the existence of $\widehat{\mathscr{P}}_+$ if there are no $Y_{\omega_i^\vee}$\~eigenvectors in $\mathscr{X}$ with the eigenvalue $t^{-(\rho,\omega_i^\vee)}$ for $i=1,2,\ldots,n$.
{\bf Comment.\ } The rational formula for $\widehat{\mathscr{P}}_+(f)$ from Theorem \ref{P+FORMULA} cannot be used in (the whole) $\mathscr{X}$ if $q$ is a root of unity even if $t$ is sufficiently general. Indeed, recall that the $Y$\~eigenvalue of $1\in \mathscr{X}$ is $t^{\rho}$. For generic $q$, the parameter $t$ can be an $N$-th root of unity for sufficiently large $N$. The latter is needed to avoid the zeros of $\widehat{P}(t^{-1})$. \phantom{1}
$\qed$
Under these conditions, the space of $\{\varrho_+\}$ is one-dimensional and $\widehat{\mathscr{P}}_{+}$ becomes a {\em universal}\, $\Diamond$\~coinvariant, which leads to the following construction. Recall that $\varsigma(a)=-w_{{}_0}(a),\ X^\varsigma_a=X_{\varsigma(a)}$; see (\ref{varsigmaf}).
\begin{theorem}\label{DIAZERO} (i) Let us assume that $\mathscr{X}$ has a nonzero symmetric form $\langle f,g \rangle$ with the anti-involution $\Diamond$ normalized by $\langle 1,1\rangle=1$. Given any $f,g\in \mathscr{X}$, $\langle f,g\rangle$ is a rational function in terms of $q,t$. Provided that $\Re k<0$
and $|\Re k|$ is sufficiently large (depending on $f,g$), \begin{align}\label{psymformula} \langle f,g\rangle= t^{-l(w_0)/2}\,\widehat{\mathscr{P}}_{+}(f T_{w_0}(g^\varsigma)). \end{align}
(ii) Let $\widehat{P}(t^{-1})\neq 0$ for the affine Poincar\'e series expressed as in (\ref{rataffpoin}), $\mathcal{F}$ be a $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\~quotient of $\mathscr{X}$ such that it has no $Y_{\omega_i^\vee}$\~eigenvectors with the eigenvalue $t^{-(\rho,\omega_i^\vee)}$. for any $i=1,2,\ldots,n.$ Using the rational presentation for $\widehat{\mathscr{P}}\,'_{+}$ from Theorem \ref{P+FORMULA}, formula (\ref{psymformula}) supplies $\mathcal{F}$ with a bilinear symmetric form associated with the anti-involution $\Diamond$ and satisfying $\lan1,1\rangle=1$.\phantom{1}
$\qed$ \end{theorem}
Compare with Proposition 3.3.2 from \cite{C101} and with Theorem \ref{Shapind} above.
\subsubsection{\sf X-induced modules} A modification of formula (\ref{psymformula}) can be used in $X$\~induced $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$- modules. They are defined as universal $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\~modules $\mathcal{I}^X_\xi$ generated by $v$ subject to $X_a(v)=q^{(\xi,a)}v$ for $\xi\in {\mathbb C}^n$, $a\in P$. If $\xi$ is generic, then the module $\mathcal{I}^X_\xi$ is $X$\~semisimple and can be identified with the {\em delta-representation} of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ in the space $$ \Delta_\xi\stackrel{\,\mathbf{def}}{= \kern-3pt =} \oplus_{\widehat{w}\in \widehat{W}}\,{\mathbb C}_{q,t}\,\chi_{\widehat{w}} $$ in terms of the {\em characteristic functions} $\chi_{\widehat{w}}$ defined as follows: $$ \chi_{\widehat{w}}(\hat{u})= \delta_{\widehat{w},\hat{u}}\,,\ \, \chi_{\widehat{w}}\chi_{\hat{u}} = \delta_{\widehat{w},\hat{u}}\,\chi_{\widehat{w}} \hbox{\ \ for the Kronecker delta }. $$ The action of the $X$\~operators is via their evaluations at $\{q^{\widehat{w}(\xi)}\}$: \begin{align*} &X_a(\chi_{\widehat{w}})\stackrel{\,\mathbf{def}}{= \kern-3pt =} X_a(\widehat{w})\chi_{\widehat{w}} \ \hbox{ for } \ a\in P, \widehat{w}\in \widehat{W},\\ &X_a(bw)\stackrel{\,\mathbf{def}}{= \kern-3pt =} X_a(q^{b+ w(\xi)})\, =\, q^{(a,b)}X_{w^{-1}(a)}(q^{\xi}).\notag
\end{align*}
The group $\widehat{W}$ acts on the characteristic functions through their indices:\, $\hat{u}(\chi_{\widehat{w}})=\chi_{\hat{u}\widehat{w}}$ \, for $\hat{u},\widehat{w}\in \widehat{W}$. Accordingly, \begin{align*} T_i(\chi_{\widehat{w}})\,&=\, \frac{t^{1/2}X_{\alpha_i}^{-1}(q^{w(\xi)}) q^{-(\alpha_i,b)} - t^{-1/2}}{ {X_{\alpha_i}^{-1}(q^{w(\xi)})q^{-(\alpha_i,b)} - 1} }\, \chi_{s_i\widehat{w}}\notag\\ &-\,\frac{t^{1/2}-t^{-1/2}}{ {X_{\alpha_i}(q^{w(\xi)})q^{(\alpha_i,b)} - 1} }\, \chi_{\widehat{w}} \ \hbox{ for } \ \widehat{w}=bw\in \widehat{W}, \notag\\ \pi_r(\chi_{\widehat{w}})\,=\, &\chi_{\pi_r\widehat{w}}, \ \hbox{ where } \ \pi_r\in \Pi,\, 0\le i\le n,\, X_{\alpha_0}=qX_{\theta}^{-1}.
\end{align*}
The $X$\~weight $q^\xi$ is assumed generic in this formula and below. We follow Section 3.4.2, ``Discretization", from \cite{C101}.
The {\em delta functions\,} are defined as $ \delta_{\widehat{w}}(\hat{u})\ =\ \mu_\bullet(\widehat{w})^{-1}\chi_{\widehat{w}} $ for $\mu_\bullet(\widehat{w})\stackrel{\,\mathbf{def}}{= \kern-3pt =}\mu(\widehat{w})/\mu(\hbox{id})$, the measure function in the following inner product: \begin{align} &\langle f,g\rangle_\bullet = \sum_{\widehat{w}\in \widehat{W}} \mu_\bullet(\widehat{w}) f(\widehat{w})\ g(\widehat{w})\ =\ \langle g,f\rangle_\bullet\,. \label{innerdelnew} \end{align} Here $f,g$ are finite or infinite (provided the convergence) linear combinations of the characteristic functions considered as functions on $\widehat{W}$. By construction, $\langle \chi_{\hat{u}},\delta_{\widehat{w}}\rangle_{\bullet}= \delta_{\hat{u},\widehat{w}}$ for $\hat{u},\widehat{w}\in \widehat{W}$ and Kronecker's $\delta_{\hat{u},\widehat{w}}$.
The values $\mu_\bullet(\widehat{w})$ are given by formulas in (\ref{murelations}); replace in this formula $X$ by $q^\xi$ and $\widehat{w}$ by $\widehat{w}^{-1}$. We see that (\ref{innerdelnew}) is directly connected with the affine symmetrizer $\widehat{\mathscr{S}}\,'_+\circ\widetilde{\mu}$: \begin{align}\label{bullettildes} &\langle f,g\rangle_\bullet\ =\ \widetilde{\mu}^{-1} \widehat{\mathscr{S}}\,'_+(\widetilde{\mu}fg)(\hbox{id}); \end{align} recall that $F(X)(\hbox{id})=F(q^\xi)$ for functions $F$ of $X$ and $\chi_{\widehat{w}}(\hbox{id})=\delta_{\widehat{w},\hbox{\tiny id}}$.
The anti-involution of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ associated with $\langle\,,\,\rangle_\bullet$ is \begin{align}\label{diambul} &\Diamond_\bullet:\,T_i\mapsto T_i(i\ge 0),\ \,X_a\mapsto X_a (a\in P),\ \, \Pi\ni\pi_r\mapsto \pi_r^{-1}. \end{align} See Section 3.2.2 from \cite{C101} and formula (3.9.4) from Section 3.9.1; compare with the definition of $\Diamond$ from (\ref{diamondef}). The (ideal of the) module $\Delta_\xi$ and the anti-involution $\Diamond_\bullet$ satisfy the nonstrong Shapovalov property (for generic $q^\xi$).
\subsubsection{\sf Theorems \ref{TQ1H}, \ref{GENPROPR} revisited} \label{sec:appl-to-conv} The Shapovalov property of $\Delta_\xi$ and $\Diamond_\bullet$ guarantees that this module has a unique up to proportionality bilinear form associated with $\Diamond_\bullet$ (for sufficiently general $\xi$). Using $\widehat{\mathscr{P}}\,'_+$ instead of $\widehat{\mathscr{S}}\,'_+\circ\widetilde{\mu}$ in (\ref{bullettildes}), one can establish the coefficient-wise proportionality of these operators. A direct usage of the divisibility argument as in Theorem \ref{KERIMAGE} can be now avoided;\, though it is of course present in this approach. The justification goes as follows.
\begin{theorem}\label{PSBULLET} Let $\widehat{\mathscr{P}}\,'_+=\sum_{\widehat{w}\in \widehat{W}}C_{\widehat{w}}\widehat{w}$ be the expansion from Theorem \ref{KERIMAGE},$(i)$ (see also Lemma \ref{LEMThatw} below). We set $$\widehat{\mathscr{P}}_+^\circledast\,=\, \sum_{\widehat{w}\in \widehat{W}}\,C_{\widehat{w}}^\circledast\widehat{w} \ \hbox{ for } \ C_{\widehat{w}}^\circledast\stackrel{\,\mathbf{def}}{= \kern-3pt =} C_{\widehat{w}}/C_{\hbox{\tiny id}}, $$ where $C_{\widehat{w}},C_{\widehat{w}}^\circledast\,\in\, {\mathbb Z}[[t^{-1/2},\,X_{\alpha_i},\,i\ge 0]]$. Then for $f,g\in \Delta_\xi$ (with the coefficient-wise multiplication), $$ \bigl(\widehat{\mathscr{P}}_+^\circledast(fg)\bigr)(\hbox{id})\, =\,\langle f,g\rangle_\bullet\,=\, \bigl(\widetilde{\mu}^{-1}\widehat{\mathscr{S}}\,'_+(\widetilde{\mu}fg)\bigr) (\hbox{id})\,, $$ where the values are in the algebra ${\mathbb C}[[t^{-1/2},\,q^{(\xi,\alpha_i)}\, ,i\ge 0]]$. In particular, $\,C_{\widehat{w}}^\circledast(q^\xi)=\mu_\bullet(\widehat{w}^{-1})\,$ for any $\widehat{w}\in \widehat{W}$ (when $f=\chi_{\widehat{w}^{-1}}=g$ are taken). Thus, $\widehat{\mathscr{P}}_+\,'$ and $\widehat{\mathscr{S}}\,'_+\circ\widetilde{\mu}$ are proportional to each other, which readily results in the proportionality claim from (\ref{propor}) or (\ref{proporan}). \end{theorem}
{\em Proof.} We define the inner product $\langle f,g\rangle'\,$ for $f,g\in \Delta_\xi$ using a direct counterpart of (\ref{bullettildes}):
\begin{align}\label{bullettildep} &\langle f,g\rangle'\ =\ \bigl(\widehat{\mathscr{P}}^\circledast_+(fg)\bigr)(id)\,; \end{align} cf. formula (\ref{psymformula}). Here $\widehat{\mathscr{P}}\,'_+$ is considered as in its original definition from Theorem \ref{KERIMAGE}, i.e., with the coefficients $C_{\widehat{w}}\in {\mathbb Z}[[t^{-1/2},\,X_{\alpha_i}\, ,i\ge 0]]$ in its decomposition $\sum_{\widehat{w}}C_{\widehat{w}}\widehat{w}$. Then this series is applied to $fg\in \Delta_\xi$ and finally the coefficient of $\delta_{id}=\chi_{id}$ has to be considered. The output will be a finite linear combination of proper $C_{\widehat{w}}$ evaluated at $q^\xi$, i.e., an element of ${\mathbb Z}[[t^{-1/2},\,q^{(\xi,\alpha_i)}\, ,i\ge 0]]$. We assume here that the coefficients of the expansion of $fg$ in terms of $\chi_{\widehat{w}}$ are from ${\mathbb Z}$ or from this algebra.
Due to formula (\ref{annihil}), \begin{align}\label{Pa-right} \widehat{\mathscr{P}}\,'_+\,T_{\widehat{w}}\ =\ t^{\frac{l(\widehat{w})}{2}}\, \widehat{\mathscr{P}}\,'_+\,. \end{align} Note that $T_{\widehat{w}}$ are placed here on the right, which is covered by part $(iii)$ of Theorem \ref{KERIMAGE}.
Relation (\ref{Pa-right}) provides that all images $\widehat{\mathscr{P}}\,'_+(\delta_{\widehat{w}})$ are proportional to each other with certain constant coefficients of proportionality. Thus, taking the evaluation at any $\hat{u}$ instead of $id$ in (\ref{bullettildep}) will not change this bilinear form up to proportionality.
These relations are sufficient to conclude that the form $\langle f,g\rangle'$ satisfies all properties of the form from (\ref{bullettildes}). It can be checked directly, but this can be avoided since we already know that the vanishing conditions from (\ref{annihil}) are the same for $\widehat{\mathscr{P}}\,'_+$ and for $\widehat{\mathscr{S}}\,'_+\circ\widetilde{\mu}$.
We see that $\langle f,g\rangle'$ for $f,g\in \Delta_\xi$ is $\langle\,,\,\rangle_\bullet$ times a constant, which may, generally speaking, depend on $\xi$. This constant is easy to find by taking $f=1=g$. Here $1=\sum_{\widehat{w}}\chi_{\widehat{w}}$ is an infinite sum in $\Delta_\xi$, but the limits $\langle 1,1 \rangle_\bullet$ and $\langle 1,1 \rangle'$ are well defined. \phantom{1}
$\qed$
Theorem \ref {PSBULLET} establishes that the coefficients of the expansion of $\widehat{\mathscr{P}}\,'_+\,$ (in its initial definition from Theorem \ref{KERIMAGE}) are actually those of $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}\,$ up to a general (functional) coefficient of proportionality. The coefficient of proportionality is immediate; it is $ct(t^{-1})$ due to Macdonald.
Moreover, there is a common radius of convergence of the coefficients of $\widehat{\mathscr{P}}\,'_+$ with respect to $t^{-1}$, which depends only on the ``first appearance" of the singularities in $ct(t^{-1})$ and readily results in the estimate $\Re k <1/h$,
equivalently, $|t|\,>q^{1/h}\,.$ For such $t$ and $|q|<1$, $$ \widehat{\mathscr{P}}\,'_+= ct(t^{-1})\,\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}\,, $$ which finalizes Theorem \ref{TQ1H}. We use that the coefficients of $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ are well defined for any $t$.
{\em Theorem \ref{GENPROPR}.} Similarly, the operator $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$
converges for $|q|<1$ (any $t$) when applied to the functions from the spaces $\mathscr{X}q^{\,lx^{2}/2}$ for the levels $0<l\in {\mathbb Z}_+$. It is apart from potential $X$\~singularities, which are actually not present due to cancelations of residues (see below).
The operator $\widehat{\mathscr{P}}\,'_+$ acts there too; by construction, its images are certain series multiplied by $q^{\,lx^{2}/2}$. The coefficient-wise proportionality provides that these images are actually
expansions of meromorphic $X$\~functions when $|t|\,>q^{1/h}\,.$ This finalizes Theorem \ref{GENPROPR}.
As a byproduct, we obtain that $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ has no singularities when acting in $\mathscr{X}q^{\,lx^{2}/2} \, (0<l\in {\mathbb Z}_+)$. A direct justification of this (known) fact is by establishing the cancelation of singularities, which is not needed now due to the proportionality theorem. Indeed, it suffices
to assume here that $|q|$ is small. Then $\widehat{\mathscr{P}}\,'_+$ converges and has no singularities because it is defined in terms of the divided differences, which preserve Laurent polynomials.
We note that given $t\in {\mathbb C}^*$ and $ b\in P$, it is not too difficult to check directly that $\widehat{\mathscr{P}}\,'_+(X_b q^{\,lx^{2}/2}) $
is an analytic function for $|q|<1$ and sufficiently large $|t|$.
To conclude the convergence and proportionality matters, let us emphasize that there are two major approaches to the analysis of $\widehat{\mathscr{P}}\,'_+$. The first is based on its $Y$-{\em rational} presentation from formula (\ref{hatPrat}), which, for instance, results in Theorem \ref{YLEFT}. The second is Theorem \ref {PSBULLET} (and its predecessors), which equates this operator with $ct(t^{-1})\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ and then the theory of the latter can be used.
\subsubsection{\sf Application to Theorem \ref{YLEFT}} A similar approach can be used to finalize Theorem \ref{YLEFT}. We will prove here that the convergence and vanishing assumptions in this theorem hold when $\, \widehat{\mathscr{P}}\,'_+$,\, $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}\,,$ $\Sigma_{\infty}$\, and other operators involved are understood coefficient-wise.
The coefficients will be treated as the elements of the algebra \begin{align}\label{ZX-prime} &{\mathbb Z}\!X_+\stackrel{\,\mathbf{def}}{= \kern-3pt =} {\mathbb Z}[[t^{-1},\,X_{\alpha_i}\, ,i\ge 0]]\,;
\end{align} recall that it contains all positive powers of $q$. We will use $\widehat{P}(t)$, the affine Poincar\'e series from (\ref{rataffpoin}). By $b\to \infty$, we mean that $b\in P_+$ and $(b,\alpha_i)\to \infty$ for all $i>0$. Similarly,\, $\mathbf b\to\infty$ for a set $\mathbf b=\{b^j\}$ if and only if $b^j\to \infty$ for all $j$.
\begin{theorem}\label{YLEFTNEW}
Given a system of representatives $\mathbf b=\{b^1,\ldots,b^p\}\subset P_+^\vee$ for the group $\Pi=P^\vee/Q^\vee$ (of cardinality $p$), we set \begin{align}\label{SiYgennew} &\Sigma_{\mathbf b}\, =\,
\frac{1}{|\Pi|}\sum_{j=1}^p\, t^{-(b^j,\rho)}Y_{b^j},
\ \,\Sigma_\infty\, =\, \frac{1}{|\Pi|}\lim_{\mathbf b\to\infty}\, \sum_{j=1}^p t^{-(b^j,\rho)}\,Y_{b^j}. \end{align}
(i) Given $W\ni u\neq \hbox{id}\,$ and $\widehat{w}\in \widehat{W}$, there exists a constant $\delta=\delta_{\widehat{w}}>0$ such that for all $\mathbf b$ sufficiently close to $\infty$, \begin{align}\label{tYlimdefw} &C_{\widehat{w}}^u\,\in \, q^v \,{\mathbb Z}\!X_+ \ \hbox{ for } \ v> \delta\,(\sum_{j=1}^p(b^j,\rho)) \hbox{,\ \, where } \notag \\
&\Sigma_{\infty}^u\, \stackrel{\,\mathbf{def}}{= \kern-3pt =}\, \frac{1}{|\Pi|} \sum_{j=1}^p \lim_{\mathbf b\to\infty} t^{-(b^j,\rho)}Y_{u(b^j)}\,=\sum_{\widehat{w}\in \widehat{W}} C^u_{\widehat{w}} \widehat{w}. \end{align}
(ii) The limit $\Sigma_\infty\,$ exists as a series $\sum_{\widehat{w}\in \widehat{W}} C_{\widehat{w}}\,\widehat{w}$ with the coefficients $C_{\widehat{w}}=C_{\widehat{w}}^{\hbox{\tiny id}}$ in the algebra ${\mathbb Z}\!X_+$. Recall that we replace $Y_b$ by the corresponding operators acting in the polynomial representation, move $\widehat{w}$ to the right and then expand the resulting $X$\~rational coefficients of $\widehat{w}$ in terms of $X_{\alpha_i} (i\ge 0)$.
Given $\widehat{w}\in \widehat{W}$ and a compact subset belonging to $\{0\neq X_\alpha\not\in q^{{\mathbb Z}}, \alpha\in R\}$,
the coefficients $C_{\widehat{w}}^u$ converges uniformly in this subset provided that $|t|>1$ and $|q|$ is sufficiently small (depending on $|t|$ and this subset); moreover, $C_{\widehat{w}}^u\to 0$ for $u\neq 0.$
(iii) Treating the $C$\~coefficients as elements from ${\mathbb Z}\!X_+$, \begin{align}\label{tYliminv} &\Sigma_\infty Y_a\ =\ \ t^{(a,\rho)}\,\Sigma_\infty \ \ \hbox{ for } \ \, a\in P,\notag\\ &\Sigma_\infty T_{\widehat{w}}\ =\ t^{l(\widehat{w})/2}\,\Sigma_\infty \ \hbox{ for } \ \widehat{w}\in \widehat{W}. \end{align} These identities formally result in \begin{align}\label{tYliminvS} \Sigma_\infty\ =\ \widehat{\mathscr{P}}_+\ =\ (ct(t^{-1})/\widehat{P}(t^{-1}))\, \widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}\,. \end{align}
(iv) We continue (ii) and (iii). For $0\le |q|<1$ and $X$ from a given compact subset of $\{0\neq X_\alpha\not\in q^{{\mathbb Z}}, \alpha\in R\}$, the condition
$|t|>1$ is sufficient for the uniform point-wise convergence of the coefficients $C_{\widehat{w}}^u$ of $\Sigma_\infty^u$; the convergence is to $0$ for $u\neq$id.
Correspondingly, the coefficients of the $\widehat{w}$\~expansions in the identities from (\ref{tYliminvS}) coincide
point-wise provided that $0\le |q|, |t|^{-1}<1$ subject to $\{0\neq X_\alpha\not\in q^{{\mathbb Z}}, \alpha\in R\}$. \end{theorem} {\em Proof.} We will use the following presentation of $T_{\hat{u}}$ ($\hat{u}\in \widehat{W}$) acting in $\mathscr{X}$, which is especially convenient for $Y_b=T_b\, (b\in P_+)$. Let \begin{align}\label{Gtaldef} &G_{\tilde{\alpha}} \ \stackrel{\,\mathbf{def}}{= \kern-3pt =}\ 1+\frac{1-t^{-1}} {X_{\tilde{\alpha}}^{-1}-1}\,(1-s_{\tilde{\alpha}})\ =\ \frac{X_{\tilde{\alpha}}^{-1}-t^{-1}}{X_{\tilde{\alpha}}^{-1}-1}- \frac{1-t^{-1}}{X_{\tilde{\alpha}}^{-1}-1}\,s_{\tilde{\alpha}},\\ &G'_{\tilde{\alpha}}\ \stackrel{\,\mathbf{def}}{= \kern-3pt =}\ G_{\tilde{\alpha}}(X\mapsto X^{-1})\ =\ G_{-\tilde{\alpha}}\ =\ \frac{X_{\tilde{\alpha}}-t^{-1}}{X_{\tilde{\alpha}}-1}- \frac{1-t^{-1}}{X_{\tilde{\alpha}}-1}\,s_{\tilde{\alpha}}\notag \end{align} for $\tilde{\alpha}\in \widetilde{R}$;\ recall that $X_{\tilde{\alpha}}=X_{\alpha}q^j$\, for $\tilde{\alpha}=[\alpha,j]$.
Given a reduced decomposition $b= \pi_r s_{j_l}\cdots s_{j_1}$ for $\ l=l(b)$ and $r\in O$, one has \begin{align}\label{prodforgb} &t^{-(\rho,b)}Y_b\ =\ b\, G_{\tilde{\alpha}^{\,l}}\cdots G_{\tilde{\alpha}^1}= \ G'_{\tilde{\beta}^{\,l}}\cdots G'_{\tilde{\beta}^1}\,b,\\ &\hbox{for\ \ } \tilde{\alpha}^1=\alpha_{j_1}, \tilde{\alpha}^2=s_{j_1}(\alpha_{j_2}),\,\ldots\,,\, \tilde{\beta}^r=-b(\tilde{\alpha}^{\,r})\in \tilde R_+. \notag \end{align} The set $\{\tilde{\alpha}^1,\tilde{\alpha}^2,\, \ldots\,\}=\Lambda(b)\subset \tilde R_+$ is the $\Lambda$\~set defined in (\ref{Lahw}). Note that $b s_{\tilde{\alpha}^l}\,\cdots\, s_{\tilde{\alpha}^1}=$ $\pi_r.$ See e.g., \cite{Ch13}, where the main construction is actually close to the limits we consider here.
Here and below we need some basic properties of the $\Lambda$\~sets and the Bruhat ordering in $\widehat{W}$.
Only $X_{\tilde{\beta}}$ with positive $\tilde{\beta}$ are present in the terms of the $G'$\~product from (\ref{prodforgb}), but negative roots do appear when calculating its (right) $\widehat{w}$\~expansion when the terms with $s_{\tilde{\alpha}}$ are taken from the corresponding binomials. The resulting $\{\widehat{w}\}$ will form the Bruhat set for $b$ (pure $b$ is obtained if no single $s_{\tilde{\alpha}}$ is taken).
As always in this paper, we expand the resulting $X$\~rational coefficients in terms of $X_{\alpha_i}$ $(i\ge 0)$ and can readily check that they are actually from ${\mathbb Z}\!X_+$ for any given $\hat{u}\in \widehat{W}$. The problem is to justify the existence of these coefficiients when $l(\hat{u})\to \infty$.
Claim $(ii)$ is the key in this theorem; it will be deduced from Theorem \ref{YLEFT}, where $\Sigma_{\infty\,}^+$ was obtained from $\widehat{\mathscr{P}}\,'_+$ assuming $(i)$. The fact that the $C$\~coefficients of $\Sigma_{\infty\,}$ converge cannot be justified at the moment directly from (\ref{prodforgb}), at least for arbitrary root systems. Potentially, there can be terms in the resulting summation destroying the convergence; they cancel each other, which follows from Theorem \ref{YLEFT}.
{\em Claim $(i)$.\,} We need the following modification of (\ref{prodforgb}).
Given a reduced decomposition $\hat{u}=\pi_r s_{j_l}\cdots s_{j_1}$, let \begin{align}\label{prodforg} &t^{-l(\hat{u})/2}T_{\hat{u}}^{-1}\ = \widetilde{G}_{\tilde{\alpha}^1}\cdots \widetilde{G}_{\tilde{\alpha}^l}\, \hat{u}^{-1},\\ &\widetilde{G}_{\tilde{\alpha}}= \frac{1-t^{-1}X_{\tilde{\alpha}}^{-1}}{1-X_{\tilde{\alpha}}^{-1}}+ \frac{1-t^{-1}}{1-X_{\tilde{\alpha}}^{-1}}s_{\tilde{\alpha}}.\notag \end{align} Note that $\tilde{\alpha}^1=\alpha_{j_1}$, $\tilde{\alpha}^2=s_{j_1}(\alpha_{j_2})$, $\tilde{\alpha}^3=s_{j_1}s_{j_2}(\alpha_{j_3})$ and so on constitute the set $\Lambda(\hat{u})$.
First, it is simple to calculate the ``greatest" $C$\~coefficient in this expression, which is that of $\hat{u}^{-1}$. It can be obtained only by picking the terms without $\,s_{\tilde{\alpha}}\,$ from all binomials in (\ref{prodforg}). Thus, \begin{align}\label{cminbt} C_{\hat{u}^{-1}}\ =\ \prod_{[\alpha,j]\in \Lambda(\hat{u})} \frac{t^{-1}-q^j X_{\alpha}}{1-q^j X_{\alpha}}\ =\ t^{-l(\hat{u})}\!\!\prod_{[\tilde{\alpha}]\in \Lambda(\hat{u})} \frac{1-t X_{\tilde{\alpha}}}{1-X_{\tilde{\alpha}}}\,. \end{align} It readily converges to zero in the sense of $(i)$, i.e., it will become divisible in ${\mathbb Z}\!X_+$ by growing positive powers either of $t^{-1}$ or of $q$ as $l(\hat{u})\to \infty$. Recall that we expand the denominators here and in any other products in terms of nonnegative powers of $X_{\alpha_i}$ for $i\ge 0$.
The case of fixed (bounded) $\widehat{w}$ as $l(\hat{u})\to \infty$ is, in a sense, opposite to this example. The following lemma addresses it.
\subsubsection{\sf Combinatorics of C-coefficients} Let us examine the individual products contributing to the coefficients $C_{\widehat{w}}$ in the standard decomposition $$ t^{-l(\hat{u})/2}T_{\hat{u}}^{-1}=\sum_{\widehat{w}\in \widehat{W}} C_{\widehat{w}} \widehat{w}. $$
\begin{lemma}\label{LEMThatw} (a) There exists a constant $\delta_{\hbox{\tiny total}}\,>\,0$ such that for any $\hat{u}\in \widehat{W} \ni\widehat{w} $ all such individual products belong to $$ (q^v+t^{-v})\,{\mathbb Z}\!X_+ \ \hbox{ for } \ v\,>\,\delta_{\hbox{\tiny total}}\ l(\hat{u}). $$
(b) Given $\,\widehat{w}\in \widehat{W}$, there exists a constant $\delta_{\widehat{w}}>0$ such that for any $\hat{u}\in P_+$, the corresponding individual products from (a) belong to $$ q^v\,{\mathbb Z}\!X_+ \ \hbox{ for } \ v\,>\,\delta_{\widehat{w}}\, l(\hat{u}). $$
(c) Given \ id\ $\neq u\in W$, the same holds for the standard decomposition of \ $t^{-(\rho,b)}Y_{u(b)}^{-1}\,$, where $b\in P_+$ and $l(\hat{u})$ is replaced by $l(b)=2(\rho,b)$;\, we assume that $b\to\infty$, i.e., $(b,\alpha_i)\to\infty$ for all $i>0$.
(d) Claims from (b,c) hold when the algebra ${\mathbb Z}\!X_+$ from (\ref{ZX-prime}) is changed to algebra ${\mathbb Z}\!X'_+\stackrel{\,\mathbf{def}}{= \kern-3pt =} {\mathbb Z}[[\,t'\, ,\ X_{\alpha_i}\, ,i\ge 0\,]] \ \hbox{ for } \ t'\stackrel{\,\mathbf{def}}{= \kern-3pt =} 1-t^{-1}$, i.e., when we expand the coefficients at the point $\,t=1\,$ instead of $\,t=0$.
(e) The $C$\~coefficients of the decomposition $\widehat{\mathscr{P}}\,'_+\ =\ \sum_{\widehat{w}\in \widehat{W}}\,C_{\widehat{w}}\,\widehat{w} $ are well-defined elements of ${\mathbb Z}\!X_+$ or ${\mathbb Z}\!X'$. Moreover, they belong to $$ \sum_{j=0}^\infty q^j\, \Bigl(\,{\mathbb Z}[[\,X_{\alpha_i}\, ,i\ge 0\,]]\,[t^{-1}]\,\Bigr)\ \subset\ {\mathbb Z}\!X_+\,\cap\,{\mathbb Z}\!X_+'. $$ \end{lemma} {\em Proof.} The smallest possible $\widehat{w}$ that can be obtained from $\hat{u}$ is when we always pick the terms with $\,s_{\tilde{\alpha}}\,$ from the binomials in the product (\ref{prodforg}) for $\,t^{-l(\hat{u})/2}T_{\hat{u}}^{-1}\,.$ It will contribute to the $C$\~coefficient of $\pi_r^{-1}$ (maximally distant from $\widehat{w}^{-1}$). There can be of course other products that contribute to $C_{\pi_r^{-1}}\,$; their number grows exponentially in terms of $l(\hat{u})$. The corresponding product is as follows: \begin{align}\label{prodforpi} \prod_{r=1}^l \frac{1-t^{-1}}{1-X_{\alpha_{j_r}}^{-1}}\ &=\ (-q/X_\theta)^{l_0}\frac{(1-t^{-1})^l}{(1-q/X_{\theta})^{l_0}} \,\prod_{j_r\neq 0}\frac{1}{1-X_{\alpha_{j_r}}^{-1}}\notag\\ &=\ (-1)^l\,\frac{(1-t^{-1})^l(q/X_\theta)^{l_0}}{(1-q/X_{\theta})^{l_0}} \,\prod_{j_r\neq 0}\frac{X_{\alpha_{j_r}}}{1-X_{\alpha_{j_r}}}, \end{align} where $l_0$ is the number of indices $j_r=0\ (1\le r\le l)$ in the reduced decomposition of $\hat{u}$; recall that $\alpha_j$ are simple roots.
Obviously, this product satisfies $(a)$. Moreover, its minimal power of $q$ will grow linearly with respect to $l(\hat{u})$. We use that for any given $i\ge 0$, the number of $j_r$ such that $j_r=i$ must grow linearly with $l(\widehat{w})$. Indeed, if in a certain connected portion of the reduced decomposition of $\hat{u}$, the simple reflection $s_i$ is missing, then this portion comes from a finite Weyl subgroup of $\widehat{W}$ (for instance, $W$ for $i=0$). Therefore, the maximal possible length of such a segment is bounded by the length $l(w_0)$ of the element $w_0$ of the maximal length in $W$.
We conclude that the number of $X_{\theta}^{-1}$ from $(q/X_{\alpha_0})^{l_0}$ that we can terminate using $X_{\alpha_{j_r}}$ will grow linearly together with $l(\hat{u})$. This will release the power of $q$ growing linearly with respect to $l(\hat{u})$.
{\em Omitting some $s_{\tilde{\alpha}}$.\,} Let us take now one $\widetilde{G}_{\tilde{\alpha}^p}$ for some $p$ in (\ref{prodforg}) and pick the term there without $s_{\tilde{\alpha}}$ for $\tilde{\alpha}=\tilde{\alpha}^p$; anything else remains unchanged. The corresponding contribution will be to the coefficient $C_{\widehat{w}}$ for $\widehat{w}= s_{j_p}\pi_r^{-1}$. It is a pure product equal to \begin{align}\label{prodforpi1} &\Bigl(\,\prod_{r=1}^{p-1} \frac{1-t^{-1}}{1-X_{\alpha_{j_r}}^{-1}}\,\Bigr)\, \frac{1-t^{-1}X_{\alpha_{j_p}}^{-1}}{1-X_{\alpha_{j_p}}^{-1}}\, \Bigl(\,\prod_{r=p+1}^{l} \frac{1-t^{-1}}{1-X_{\beta_{r}}^{-1}}\,\Bigr),\notag\\ &\ \ \ \ \ \ \ \hbox{\ where\ } \beta_{r}\ =\ s_{j_p}(\alpha_{j_r})\, \ \hbox{ for } \ r>p\,. \end{align} Unless $j_p=0$, the estimate of the $q$\~power is completely parallel to that for (\ref{prodforpi}).
If $j_p=0$, then the indices $\{j_r=0\}$ {\em after} $j_p$ will not contribute any longer to the total power of $q$, since $s_0(\alpha_0)=-\alpha_0$. However, $s_0(\alpha_i)=\alpha_0+\alpha_i$ for simple $\,\alpha_i\,(i>0)\,$ neighboring to $\alpha_0$ in the completed Dynkin diagram. The number of such indices $\,i\,$ in the reduced decomposition of $\hat{u}$ will tend to infinity together with $l(\hat{u})\to\infty$. The corresponding $X_{\alpha_i-\theta}$ in the numerator can be terminated by using nonaffine $X_{\alpha}$ with $\alpha>0$ exactly in the same way as it was done in (\ref{prodforpi}) for $X_{-\theta}$. The released $q$ will provide the required growth of the total power of $q$.
If there are two places $p<p'$ where the terms without $s_{\tilde{\alpha}}$ are taken, then the resulting product will contribute to $C_{\widehat{w}}$ for $\widehat{w}=s_{j_p}s_{j_{p'}}\pi_r^{-1}\,;\,$ it reads \begin{align}\label{prodforpi2} &\Bigl(\,\prod_{r=1}^{p-1} \frac{1-t^{-1}}{1-X_{\alpha_{j_r}}^{-1}}\,\Bigr)\, \frac{1-t^{-1}X_{\alpha_{j_p}}^{-1}}{1-X_{\alpha_{j_p}}^{-1}}\, \Bigl(\,\prod_{r=p+1}^{p'-1} \frac{1-t^{-1}}{1-X_{\beta_{r}}^{-1}}\,\Bigr)\\ &\ \ \times\,\frac{1-t^{-1}X_{\beta_{p'}}^{-1}}{1-X_{\beta_{p'}}^{-1}}\, \Bigl(\,\prod_{r=p'+1}^{l} \frac{1-t^{-1}}{1-X_{\beta_r}^{-1}}\,\Bigr)\,, \ \hbox{ where } \ \notag\\ \beta_{r}=&s_{j_p}(\alpha_{j_r}) \hbox{\, if \,} p'\ge r>p\,,\ \beta_{r}=s_{j_p}s_{j_{p'}}(\alpha_{j_r}) \hbox{\, if \,} r>p'\,.\notag \end{align} For the sake of uniformity, we will replace here the remaining $\alpha_{j_r}$ by $\beta_r$ as well, setting $\beta_r=\alpha_{i_r}$ for $p\ge r >0$.
{\em Minimal $q$\~powers\,.} The analysis of the minimal power of $\,q\,$ remains essentially the same in the case of two $p$. The number of the indices $r>p\,$ with {\em affine negative\,} $\beta_r\,$ (they do not contribute to the minimal power of $q$) {\em approximately}, i.e., in the limit $l(\hat{u})\to\infty$\,, is no smaller than the number of positive affine $\beta_r$ (which do contribute). It can be readily generalized to any number of indices $\,p\,$ such that the corresponding terms without $s_{\tilde{\alpha}}$ are taken. Let $\widehat{w}$ will be the corresponding index of the $C$\~coefficient; $\widehat{w}=s_{j_p}s_{j_{p'}}\pi_r^{-1}$ for two $p$.
We can assume that $\widehat{w}(\alpha_0)<0$ for $\widehat{w}\in \widehat{W}$. Indeed, if the reduced decomposition of $\hat{u}\,$ grows to infinity after $\{p\}$, then we can assume that $\widehat{w}(\alpha_0)<0$ for $\widehat{w}\in \widehat{W}$; otherwise such $\widehat{w}$ will not change the positivity of $X_{\alpha_0}$ after $\{p\}$. If such a growing interval in the reduced decomposition of $\hat{u}$ occurs between some $p$, we can diminish $\{p\}$ to end this sequence before this interval. The positivity of the terms before $\{p\}$ remains unchanged, which provides the required power of $q$ if such growing interval occurs before $\{p\}$.
Representing $\widehat{w}=v a$ for $v\in W,\,a\in P$, the condition $\widehat{w}(\alpha_0)<0$ can happen only if $(a,\theta)<0$. Indeed, $\widehat{w}(\alpha_0)=[-v(\theta),1+(a,\theta)]$. However, $\theta$ is a sum of simple roots with positive coefficients. Thus $(a,\alpha_j)=-d<0$ for at least one $j>0$ and $\widehat{w}(\alpha_j)=[v(\alpha_j),d]>0$.
Finally, $d$ here will tend to $\infty$ together with $l(\hat{u})$ because, as we already used, the number of $s_j$ (for any given $j>0$) in the reduced decomposition of $\hat{u}$ grows linearly with respect to $l(\hat{u})$.
{\em Omitting many $p$\,.} Let $\mathbf p=\{\cdots > p''>p'>p\,\}\,$ be the sequence of the terms (binomials) where we omit the corresponding $s_{j_p}\,$. If $\mathbf p=\Lambda(\hat{u})$, then we arrive at (\ref{cminbt}); however, now we are interested in the case when the corresponding $\widehat{w}\,$ remains bounded.
One has $$ \beta_p=\alpha_{j_p},\ \beta_{p'}=s_{j_p}(\alpha_{j_{p'}}),\ \beta_{p''}=s_{j_p}s_{j_{p'}}(\alpha_{j_{p''}}) \hbox{\,,\ and so on}. $$
The corresponding product will contribute to the coefficient $C_{\widehat{w}}$ for $\widehat{w}=s_{j_{p}}s_{j_{p'}}s_{j_{p''}}\,\cdots\,\pi_r^{-1}$. The set $\Lambda(\widehat{w}^{-1})$ is very explicit; it is obtained from the set $\mathbf{\beta}=\{\,\ldots \beta_{p''},\beta_{p'},\beta_{p}\,\}\,$ by removing all pairs in this set in the form $\,\{\tilde{\alpha},-\tilde{\alpha}\}\,.$
The main problem we have to address is that $\widehat{w}$ can be small for arbitrarily large $\hat{u}$. This is actually the key point of the justification of existence of $\widehat{\mathscr{P}}^{\,'}_+$ and other operators under consideration. Given $\widehat{w}$, the number of contributions to $C_{\widehat{w}}$ of this kind will go to $\infty$ together with $l(\hat{u})$.
For instance, one can take $\,\hat{u}=(-b_+)w_0 b_+\,$ for any $b_+\in P_+$ such that $-b_+=w_0(b_+)$. Then the corresponding length will be $l(\hat{u})=2\,l(b_+)+l(w_0)$, but if we delete $w_0$ it will drop to zero. Actually, this is a typical example. The corresponding product will be that from (\ref{prodforpi2}) with only one group in the parentheses, corresponding to $w_0$, and with the products before and after it (without the parentheses) corresponding to $b_+$ and $(-b_+).$
Assuming that $|\mathbf p|$ is large and $l(\widehat{w})$ is bounded by a certain constant, almost all elements of $\Lambda(\widehat{w}^{-1})$ will appear in the pairs $\{\tilde{\alpha},-\tilde{\alpha}\}$ for $\tilde{\alpha}=[\alpha,j]>0.$ Any such a pair will contribute either $t^{-1}$ or at least $q^{j-1}$ to the resulting product. Indeed, the product of the corresponding quantities will be (before the expansion in terms $X_{\tilde{\beta}}$ with $\tilde{\beta}>0$) $$ \frac{(1-t^{-1}X_{\tilde{\alpha}})(t^{-1}-X_{\tilde{\alpha}})}{(1-X_{\tilde{\alpha}})^2}\,. $$ See (\ref{prodforpi2}), the terms there without the parentheses. This concludes $(a)$, but the resulting powers of $q$ can be estimated better than needed in $(a)$, which is part $(b)$ of the lemma. Note that this argument is not applicable to $(d)$, though the estimates for the $q$\~powers below hold in this case.
{\em Part (b).} Since the original decomposition of $\hat{u}$ was reduced, there will be $\alpha_{j_r}$ for $r$ with positive $\beta_r$ between some of $p$ with their affine components approaching $\infty$. Indeed, if all these affine components remain bounded, then $l(\widehat{w})\to \infty$ together with $l(\hat{u})$.
For instance, in the example of $\,\hat{u}=(-b_+)w_0 b_+\,$ with $b_+\in P_+\,$, the element $(-b_+)$ must be on the left and $b_+$ on the right to ensure that the corresponding combined decomposition is reduced. Thus $b_+^{-1}(\alpha_i)=[\alpha_i, (b_+,\alpha_i)]$ and there must be growing positive affine components at least for some $i>0$.
To demonstrate the essence of our estimates in the case of large
$|\mathbf b|$ with bounded $l(\widehat{w})$, let us insert here any $v\in W$ instead of $w_0$. For $c=v(b), \,b\in P_+\,$, the length of $\hat{u}=c\,v\, b=v\cdot(2b)$ is $l(c)+l(v)+l(b)$. We pick the terms with $s_{\tilde{\alpha}}$ only from $v$ here (the corresponding portion of the reduced decomposition of $\hat{u}$) assuming that $v(b)+b$ is bounded. The set $\Lambda(v)$ is formed by certain positive linear combinations of $\alpha_{i_r}$ with {\em nonnegative} integral coefficients for the indices $I_v=\{i_r\}$ from a given reduced decomposition $v=s_{i_m}\cdots s_{i_1}$.
One has $b^{-1}(\Lambda(v))\in \tilde R_+$. If all scalar products $(b,\alpha_i)$ here are no greater than a certain constant for all $i\in I_v$, then there must exist an index $0<k\not\in I_v$ such that $M=(b,\alpha_k)$ is positive and large compared to $(b,\rho)$. Representing $b=b'+b''$ for $b'=M\omega_k$, we can assume that $v(b')=b'$; otherwise $M$ will contribute to the power of $q$ (the terms inside the parentheses) and we will obtain the required growth. However the relation $v(b')=b'$ readily contradicts the assumption that $v(b)+b$ is bounded.
{\em The general case.} We need to examine the products in the form $\hat{u}=\hat{u}^*\,\hat{v}\,\hat{u}'$, where $\hat{v}\in \widehat{W} \ni \hat{u}',\hat{u}^*$, such that $l(\hat{u}^*\,\hat{v}\,\hat{u}')=l(\hat{u}^*)+l(\hat{v})+l(\hat{u}')$ and the set $\Lambda(\hat{u}^*\,\hat{u}')$ remains bounded as $l(\hat{u}^*\,\hat{v}\,\hat{u}')\to \infty.$
It is a generalization of the example considered above, where $\hat{v}$ substitutes for $v$ and $\hat{u}'$ replaces $b\in P_+$.
Let $I_{\hat{v}}=\{i_r\}$ for a given reduced decomposition $\hat{v}=s_{i_m}\cdots s_{i_1}\,$. We set $\,\hat{u}=a u\, (a\in P, u\in W)$, $a=\sum_{i=1}^nM_i \omega_i$ and $a=a'+a''$ for $a'=\sum_k M_k\omega_k$ for the set
$K=\{k\}$ of all indices $i$ such that $|M_i|\to \infty$. Using that $a^{-1}(\Lambda(\hat{v}))\in \tilde R_+$, we can check by induction that $K\cap I_{\hat{v}}=\emptyset$ unless the power of $q$ tends to infinity together with $l(\hat{u}^*\,\hat{v}\,\hat{u}')$ (the fact we need to establish). Let us demonstrate it.
Indeed, for the first appearance of $i$ in the sequence $I_{\hat{v}}$, the root $(a')^{-1}(\alpha_i)$ and the inner product $(a',\alpha_i)$ must be nonnegative to ensure the positivity of $(\hat{u}')^{-1}(\alpha_i)\in \Lambda(\hat{v}\,\hat{u}')$. This holds assuming that we already know that $(a',\alpha_{i'})=0$ for all previous $i'$ in $I_{\hat{v}}$. The corresponding contribution to the resulting power of $q$ will be $(a',\alpha_i)\to \infty$ as $l(\hat{u}^*\,\hat{v}\,\hat{u}')\to \infty$ if $(a',\alpha_{i})>0$. Therefore, $(a',\alpha_i)=0$. Check that here $\alpha_i$ can be allowed to be $\alpha_0$. Thus we conclude that $\hat{v}(a')=a'$.
To finalize this reasoning (and part $(b)$), one can assume that $\hat{u}^*$ in $\hat{u}^*\,\hat{v}\,\hat{u}'$ can be represented as $\hat{u}^*=\widehat{w}^*\, b^*$ for $b^*\in P, \widehat{w}^*\in\widehat{W}$ such that $l(\hat{u}^*)=l(\widehat{w}^*)+l(b^*)$ and the sum $b^*+a'$ remains bounded in the limit. However then $b^*\hat{v} a'$ cannot be reduced. This contradiction shows that the powers of $q$ in the products under consideration (contributing to $C_{\hat{u}^*\hat{u}'}$ for bounded $\hat{u}^*\hat{u}'$) go to infinity as $l(\hat{u}^*\,\hat{v}\,\hat{u}')\to \infty$.
{\bf Comment.\ } We note that continuing the (combinatorial) analysis of the products contributing to the coefficients $C_{\widehat{w}}$ in the decomposition of $t^{-l(\hat{u})/2}T_{\hat{u}}^{-1}$ for fixed $\widehat{w}$ and growing $\hat{u}\in \widehat{W}$, one can eventually
arrive at the sharp convergence range $|t|>q^{1/h}$ for the $C$\~coefficients of the operator $\widehat{\mathscr{P}}^{\,'}_+$; see $(e)$. We will not demonstrate this here, since it formally results from
the proportionality of this operator with $\widehat{\mathscr{I}}$, where sufficiently small $|q|$ and $|t|^{-1}$ are sufficient (the coefficients of $\widehat{\mathscr{I}}$ are explicit). Let us mention that the Coxeter number $h$ appears in our considerations due to counting the ``density" of $s_{0}$ in the reduced decomposition of $\hat{u}$ and in similar estimates. \phantom{1}
$\qed$
{\em Part (c).\,} Let us apply the $q$\~estimates from $(a,b)$ to $Y_{u(b)}$ for id$\neq u\in W$, $b\in P_+$. The above analysis of the expansion $t^{-(\rho,b)}Y_b^{-1}=\sum_{\widehat{w}\in \widehat{W}} C_{\widehat{w}}\,\widehat{w}$\ for $b\in P_+$\, corresponds to $u=w_0$ and can be readily extended to arbitrary such $u$. The main step is as follows.
Let $c=u(b)$ for $b\in P_+$. Then $l(c)=l(b)$ and the representation $c= b'-b''$ for $b',b''\in P_+$ results in the following. Setting $c=u(b)=\pi_r s_{j_l}\,\cdots\, s_{j_1}\ (l=l(c))\,$, \begin{align}\label{prodforgc} &Y_c\ =\ \pi_r T_{j_l}^{\epsilon_l}\,\cdots T_{j_1}^{\epsilon_1}\,, \ \hbox{ where } \ \,\epsilon_j=\pm 1;\hbox{\ \, correspondingly\,,}\notag\\ &t^{-(\rho,b)}Y_c = c\,\widehat{G}_{\tilde{\alpha}^l}\,\cdots\, \widehat{G}_{\tilde{\alpha}^1} \ \hbox{ for } \ \tilde{\alpha}^1=\alpha_{j_1}, \tilde{\alpha}^2=s_{j_1}(\alpha_{j_2}),\,\ldots\,,\\ &\hbox{where\, }\widehat{G}_{\tilde{\alpha}^j}=G_{\tilde{\alpha}^j}\, \hbox{ for }\epsilon_j=+1\, \hbox{ and } \widehat{G}_{\tilde{\alpha}^j}=\widetilde{G}_{\tilde{\alpha}^j} \hbox{\, otherwise}.\notag \end{align}
Then we move $\,c\,$ to the right and focus on the terms $\widetilde{G}_{\tilde{\alpha}^j}$ for $\epsilon_j=-1$; the condition $b\to \infty$ guarantees complete analogy with the considerations for $c=-b\, (b\in P_+)$.
{\em Parts (d,e).} The estimates for the power of $q$, and therefore the claims from $(b)$ and $(c)$ hold when we replace $t^{-1}$ by $1-t'$ for $t'=1-t^{-1}$ and analyze the resulting expressions. This is claimed in $(d)$. It provides the existence of the coefficients of $\widehat{\mathscr{P}}^{\,'}_+$ and those of $\Sigma_{\infty}$ as formal series in terms of $q$ an $t'$.
Part $(e)$ follows from $(b)$ and the observation (obvious from its justification) that the power of $t^{-1}$ is bounded in the products contributing to $C_{\widehat{w}}$ unless they are divisible in ${\mathbb Z}\!X_+$ by powers of $q$ approaching $\infty\,$. \phantom{1}
$\qed$
\subsubsection{\sf Back to Theorem \ref{YLEFTNEW}} {\em Claim $(ii)$.} Combining $(i)$ and Theorem \ref{YLEFT}, we obtain that $\Sigma_\infty^+=\Sigma_\infty\,\mathscr{P}_+$ exists as a series with coefficients in ${\mathbb Z}\!X_+$. A justification of this fact based directly on (\ref{prodforgb}) is not known at the moment (at least for arbitrary root systems).
Recall that the connection of $\widehat{\mathscr{P}}\,'_+$ and $\Sigma_{\infty\,}^+$ is a sequence of algebraic manipulations based on the fact that $\widehat{\mathscr{P}}\,'_+$ treated as a rational function is identically zero. See (\ref{hatPrat}) and also Theorem \ref{SYMRANK1} below (the case of $A_1$).
Since $\Sigma_{\infty\,}$ is given in terms of $b^j$ representing all elements in $\Pi$, this formally results in the existence of $(\Sigma_{\infty\,}^{\pi})^+=\Sigma_{\infty\,}^{\pi}\mathscr{P}_+$ for any $\pi=\pi_r\in \Pi$, where \begin{align*} \Sigma_{\infty\,}^{\pi}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \lim_{b\to \infty}\, t^{-(\rho,b)}Y_{b} \ \hbox{ for } \ b \hbox{\ \ such that\ \ } b-\omega_r\in Q. \end{align*}
We will use the nonaffine $(i>0)$ intertwining operators; cf. Section \ref{sect:intertw}. One has \begin{align}\label{y-intert} &\Phi_i Y_b=Y_{s_i(b)}\Phi_i\, \ \hbox{ for } \ \, \Phi_i\stackrel{\,\mathbf{def}}{= \kern-3pt =} \frac{T_i+(t^{1/2}-t^{-1/2})/(Y_{\alpha_i}^{-1}-1)} {t^{1/2}+(t^{1/2}-t^{-1/2})/(t^{-(\rho,\alpha_i)}-1)},\\ &\mathscr{P}_+\ =\ \sum_{u\in W} \Phi_{u}, \ \hbox{ where } \ \, \Phi_{uv}=\Phi_{u}\Phi_{v}\, (u,v\in W), \ \, \Phi_i=\Phi_{s_i}. \label{phivias} \end{align} Also, $\mathscr{P}_+\,$ is divisible by $\sum_{u\in W} u$ on the left; see (\ref{mac1}).
Applying $(i)$, \begin{align*} (\Sigma_{\infty\,}^{\pi})^+=(\widetilde{\Sigma}_{\infty\,}^{\pi})^+\ \ \hbox{ for } \ \widetilde{\Sigma}_{\infty\,}^{\pi}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \lim_{b\to \infty}\, t^{-(\rho,b)}(\sum_{u\in W}Y_{u(b)}), \ \,b\in \omega_r+ Q. \end{align*} Moreover, representing $\mathscr{P}_+$ via the $Y$\~intertwiners, we can place it on the left: \begin{align}\label{sipinftyx} (\Sigma_{\infty\,}^{\pi})^+\ =\ {}^+\widetilde{\Sigma}_{\infty\,}^{\pi}\,\stackrel{\,\mathbf{def}}{= \kern-3pt =}\, \mathscr{P}_+\, \lim_{b\to \infty}\, t^{-(\rho,b)}(\sum_{u\in W}Y_{u(b)}) \end{align} for $b$ from $\omega_r+Q$.
Since $\mathscr{P}_+\,$ is divisible by $\,\sum_{u\in W} u\,$ on the left, the $C$\~coefficients of ${}^+\widetilde{\Sigma}_{\infty\,}^{\pi}$ must satisfy the $W$\~invariance relations $C_{u\widehat{w}}=C_{\widehat{w}}$ for $u\in W$ and $\widehat{w}\in \widehat{W}$. The $C$\~coefficients of sums $\sum_{u\in W}Y_{u(b)}$ have the same invariance condition up to the terms from $q^N{\mathbb Z}\!X_+$ for $N$ growing together with $(\rho,b)$. Use the $Y$\~intertwiners and part $(i)$ (see also part $(iii)$ below).
However, $\mathscr{P}_+\,$ for generic $t$ has no kernel when acting in the space of $W$\~invariant delta function defined as follows: $\delta_a(X_{w(c)})=q^{(a,c)}$ for $a,c\in P_+, \, w\in W$. Therefore, the $C$\~coefficients of $\sum_{u\in W}Y_{u(b)}$ can be uniquely recovered from those of $\mathscr{P}_+\,\sum{u\in W}Y_{u(b)}$ modulo $q^N{\mathbb Z}\!X_+$. This provides the existence of $\Sigma_{\infty\,}^{\pi}$ and justifies the first part of $(ii)$.
{\em Claim $(iii)$}. Actually, we have already used the main arguments needed here in $(ii)$ above. Nevertheless, let us see how the proportionality claims can be obtained directly from the existence of $\Sigma_{\infty\,}$.
The first of the formulas from (\ref{tYliminv}) results from $(ii)$. Let us demonstrate that the second follows directly from $(i)$. Using (\ref{y-intert}), $\Sigma_\infty \Phi_i = \Phi_i \Sigma^{s_i}_\infty=0$ upon the $\widehat{w}$\~expansions with $C_{\widehat{w}}$ treated as a formal series with the coefficients in ${\mathbb Z}\!X_+$ (or
point-wise for sufficiently small $|q|$). Therefore $$ \Sigma_\infty T_i=-\frac{t^{1/2}-t^{-1/2}}{t^{-1}-1} \Sigma_\infty =t^{1/2}\Sigma_\infty. $$
These formulas show that we can use $\Sigma_\infty$ exactly in the same way as $\widehat{\mathscr{P}}\,'_+\,$ in Theorem \ref{PSBULLET}, i.e., it can be used to define the corresponding form $\langle f, g\rangle$ in $\Delta_\xi$. The uniqueness of such a bilinear form in $\Delta_\xi$ up to proportionality results in the coefficient-wise proportionality of $\Sigma_\infty$, $\widehat{\mathscr{P}}\,'_+\,$ and $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}\,$. Upon evaluation at $1$, we see that $\Sigma_\infty=\widehat{\mathscr{P}}_+\,.$
{\em Point-wise convergence; (ii) and (iv).} The considerations from $(i)$ can be equally used for the point-wise convergence to zero of the $C$\~coefficients of $t^{-(\rho,b)}Y_b^{-1}$ in the limit $b\to\infty$ and, more generally, the coefficients of $\Sigma^u_{\infty}$ for id$\neq u\in W.$
If $\widehat{w}$ is fixed, then the minimal common power of $q$ in the expansion of $C_{\widehat{w}}$ will grow linearly together with $(\rho,b)$. Therefore the sum of absolute values of all coefficients of $C_{\widehat{w}}$ expanded in terms of the powers of $t'$ and $X_{\alpha_i}\,(i\ge 0)$
can grow no greater than exponentially. Thus the functional convergence of the series for $C_{\widehat{w}}$ to zero can be achieved
by making $\,|q|\,$ sufficiently small, depending on $t$ and the compact set were $X$ is taken, naturally apart from the singularities. This can be readily extended to any $u\neq$id.
{\bf Comment.\ } We note that a direct justification of the functional (point-wise) convergence to $0$ in (\ref{prodforpi}) for the whole $C_{\pi_r^{-1}}$ and for any $C_{\widehat{w}}$ is doable as well (without the $X,q,t^{-1}$\~expansions), though it follows essentially the same lines. Let us also mention that the statements from $(ii)$ are discussed in detail in the case of $A_1$ in (\ref{YexpB}); see the first formula there and Theorem \ref{SIMINV}. \phantom{1}
$\qed$
Similar estimates show that the coefficients of $\Sigma_{\infty}\,$ exist as analytic functions
for sufficiently small $|q|,|t'|$. Thus (\ref{tYliminvS}) holds for such $q,t$, where the $C$\~coefficients are treated analytically (in this range), which is the first part of $(iv)$.
{\em Sharp estimates.\,} The exact estimates from $(iv)$ including the coincidence statements formally follow from its first part, which is for sufficiently
small $|q|$ and $|t'|$. The following actually repeats the argument that have been already used for the sharp estimates of the convergence of the coefficients of $\widehat{\mathscr{P}}\,'_+\,$.
Let $|q|<1$. The coefficients of $(ct(t^{-1})/\widehat{P}(t^{-1}))\, \widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ are very explicit and the first singularity with respect to $t$ is at a certain root of unity due to the zeros of $\widehat{P}(t^{-1}))$. Due to (\ref{tYliminvS}), this gives that the radius of convergence is
$|t|^{-1}<1$ for all coefficients of $\Sigma_\infty\ $. Note that this is different from the answer obtained for $\widehat{\mathscr{P}}\,'_+\,$ due to the presence of the Poincar\'e series in the coefficient of proportionality.
This concludes the justification of Theorem \ref{YLEFTNEW}. \phantom{1}
$\qed$
{\bf Comment.\ } Let us discuss very briefly the case $|t|<1$. We set $\Sigma^+_{\mathbf b}\,=\,
(1/|\Pi|)\sum_{j=1}^p\, (-t)^{(b^j,\rho)}Y_{b^j}\,$ instead of that in (\ref{SiYgennew}). Then $\Sigma^+_{\infty}$ defined as $ \lim_{{\mathbf b} \to \infty}\Sigma^+_{\mathbf b}\,$ will be proportional to \,$\sum_{\widehat{w}\in \widehat{W}} (-t)^{l(\widehat{w})/2}T_{\widehat{w}}^{-1}$\, and to $\mu(X; q,t)^{-1}\circ \sum_{\widehat{w}\in \widehat{W}} (-1)^{l(\widehat{w})}\,\widehat{w}$ for
$\mu$ from (\ref{mutildemu}), the $|t|<1$ counterpart of $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$. This makes the standard {\em right} decomposition $\Sigma^+_{\infty}=\sum C_{\widehat{w}}^+\widehat{w}$ naturally much simpler than that for $\Sigma_{\infty}$. We note that the exact coefficients of proportionality can be obtained from
the theory $|t|>1$ using the DAHA involution (not an anti-involution) sending $t^{1/2}\mapsto -t^{-1/2}$ and fixing $q$ and the generators of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$. It coincides with $\,H\mapsto \mu^{-1}\, H^\iota \,\mu\,$ for the involution $\iota\,$ used in Lemma \ref{IOTLEM}, when acting on operators in a proper completion of the polynomial representation. It results in $$ \Sigma^+_\infty\ \,=\ \, (ct(t)/\widehat{P}(t))\, \mu(X; q,t)^{-1}\circ \sum_{\widehat{w}\in \widehat{W}} (-1)^{l(\widehat{w})}\,\widehat{w}. $$ Compare with (\ref{YexpA}) in the case of $A_1$. \phantom{1}
$\qed$
\subsubsection{\sf Higher levels} Conjugating $\Diamond$ from (\ref{diamondef}) by $q^{\,lx^2/2}$ for an integer $l\ge 0$, one obtains the following anti-involution: \begin{eqnarray*} &\Diamond_l:\, T_{i}\, \mapsto\, T_{i},\ (i>0), Y_{b}\, \mapsto\, q^{-lx^2/2}\,Y_{b}\,q^{lx^2/2}, \ (b\in P^{\vee}), \\ &X_{a}\, \mapsto\, T_{w_{0}}^{-1}X_{a^{\varsigma}}T_{w_{0}},\ X_{a^{\varsigma}}=\varsigma(X_{a})=X_{-w_{0}(a)}, \ a\in P. \end{eqnarray*} The formulas for $T_0$ and $\pi_r$ can also be calculated but they are not that direct. Let us discuss the invariant forms corresponding to $\Diamond_l$ for $l>0$. The $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\~module will be the polynomial representation $\mathscr{X}$.
We use that $\widehat{\mathscr{I}}$ identifies the space of coinvariants $\mathscr{X}/\mathcal{J}_{l}(\mathscr{X})$, from Section \ref{sec:coinvariants} with the Looijenga space $\mathcal{L}_l\, (l\in {\mathbb N})$ for generic $k$. Recall that $\mathcal{J}_{l}(\mathscr{X})$ is the span of linear spaces $$q^{-lx^{2}/2}\,(T_{\widehat{w}}- t^{l(\widehat{w})/2})(\mathscr{X}q^{\,lx^{2}/2}) \ \hbox{ for } \ \widehat{w}\in \widehat{W}. $$ Thus this is exactly the space of $\Diamond_l$\~coinvariants from (\ref{starcoinv}): \begin{align*} &\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}/(\mathcal{J}+\mathcal{J}^{\Diamond_l})= \mathscr{X}/\mathcal{J}^{\Diamond_l}(\mathscr{X}),\ \mathcal{J}=\,\hbox{Ker\,}(\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\ni A\mapsto A(1)\in \mathscr{X}); \end{align*} the subspaces $\mathcal{J}_l\subset \mathscr{X}$ from Section \ref{sec:pcoinv} and $\mathcal{J}^{\Diamond_l}$ coincide.
The action of $\Diamond_l$ is trivial in this quotient; use the limit $t\to 1$ to see this. Therefore every functional on this space can be used to construct a form associated with $\Diamond_l$, and every such a form can be obtained in this way. Using $\widehat{\mathscr{I}}$, we come to the following extension of Theorem \ref{DIAZERO} from $l=0$ to $l>0$.
\begin{theorem}\label{DIAGAU} Let us assume that $\mathscr{X}$ has a nonzero symmetric form $\langle f,g \rangle$ corresponding to the anti-involution $\Diamond_l$ and normalized by $\langle 1,1\rangle=1$. Provided that $\widehat{\mathscr{I}}(\mathscr{X})=\mathcal{L}_l$, this form can be represented as follows: $$\langle f,g\rangle= \psi(\widehat{\mathscr{I}}(f T_{w_0}(g^\varsigma)\,q^{l x^2/2})) $$ for a proper linear functional $\psi: \mathcal{L}_l\to {\mathbb C}$. When $l=1$, the resulting symmetric form satisfies the Shapovalov property; the corresponding anti-involution is of strong type with respect to $\mathscr{Y}$. \phantom{1}
$\qed$ \end{theorem}
{\bf Analytic theories.}
\ Let us take a function $\phi(x)$ such that $\phi\, q^{-l x^2/2}$ is $\widehat{W}$\~invariant, for instance $\phi=q^{-l x^2/2}$. Then the form \begin{equation}\label{diaform} \langle f,g\rangle_\phi= t^{-l(w_{0})/2}\int f T_{w_{0}}(g^{\varsigma})\phi\mu',\, \mu'=\mu (l<0),\, \mu'=\widetilde{\mu} (l>0). \end{equation} is symmetric and is served by $\Diamond_{l}$ for the following major choices of the integration (``theories"):
(a) imaginary integration $\int_{e+\imath {\mathbb R}^n}$ for $e\in {\mathbb R}^n$ subject to $l< 0$,
(b) real integration $\sum_{w\in W}\int_{w(e)+{\mathbb R}^n}$ for $\,e\not\in {\mathbb R}^n\,$, where $l>0$,
(c) Jackson integration $\int_\xi f\,=\, \sum_{\widehat{w}\in \widehat{W}}\,f(q^{\widehat{w}(\xi)})$, where $l>0$.
The function $\phi$ must be analytic in a neighborhood of the integration contour for $(a)$ and everywhere for $(b)$ to ensure that the integral does not depend on the choice of $e\in {\mathbb R}^n$. Finding the kernels of the linear map $\phi\mapsto \langle\, \cdot\,,\,\cdot\,\rangle_\phi$ is an interesting problem; the dimension of its image equals dim\,$\mathcal{L}_l$.
Establishing connections between these theories is fundamental in harmonic analysis. Relating them to {\em algebraic} Shapovalov-type inner products is equally important. The latter inner products do not involve integrations and are well defined for all or almost all $q,t$. This problem is directly linked to the DAHA-generalization of the Arthur-Heckman-Opdam approach from \cite{HO2}, which can be stated as the problem of {\em finding presentations of algebraically defined inner products in DAHA\~modules} (Shapovalov-type ones) {\em in terms of the integrations} (with respect to the affine residual subtori).
\setcounter{equation}{0} \section{\sc{The rank-one case}} \subsection{{\bf Polynomial representation}} \subsubsection{\sf Basic definitions} Let us consider the root system $A_{1}$. Following Section \ref{sec:rank1}, $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ is generated by $Y=Y_{\omega_1},T=T_1,X=X_{\omega_1}$ subject to the quadratic relation $(T-t^{1/2})(T+t^{-1/2})=0$ and the cross-relations: \begin{align}\label{dahaone} &TXT=X^{-1},\ T^{-1}YT^{-1}=Y^{-1},\ Y^{-1}X^{-1}YXT^2q^{1/2}=1. \end{align} Using $\pi\stackrel{\,\mathbf{def}}{= \kern-3pt =} YT^{-1}$, the second relation becomes $\pi^2=1$. The field of definition will be ${\mathbb C}(q^{1/4},t^{1/2})$, though ${\mathbb Z}[q^{\pm 1/2},t^{\pm 1/2}]$ is sufficient for many constructions; actually $q^{\pm 1/4}$ will be needed only in the automorphisms $\tau_{\pm}$ below. We will frequently treat $q,t$ as numbers; then the field of definition will be ${\mathbb C}$.
The following map can be extended to an anti-involution on $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$\, $\varphi: X \leftrightarrow Y^{-1}, T\to T$. The first two relations in (\ref{dahaone}) are obviously fixed by $\varphi$; as for the third, check that $\varphi(Y^{-1}X^{-1}YX)=Y^{-1}X^{-1}YX$.
The following DAHA automorphism is of key importance in this paper: $$ \tau_+(X)=X,\ \tau_+(T)=T,\ \tau_+(Y)=q^{-1/4}XY,\ \tau_+(\pi)=q^{-1/4}X\pi, $$ which can be interpreted as conjugation by the Gaussian $q^{x^2}$ for $X=q^x$. Check that $T^{-1}YT^{-1}=Y^{-1}$ is transformed to $Y^{-1}X^{-1}YXT^2q^{1/2}=1$ under $\tau_+$. Applying $\varphi$ we obtain an automorphism $\tau_-=\varphi\tau_+\varphi$\,: $$ \tau_-(Y)=Y,\ \tau_-(T)=T,\ \tau_-(X)=q^{1/4}YX. $$
The Fourier transform corresponds to the following automorphism of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ (it is not an involution)\,: \begin{align} \sigma(X)= Y^{-1},\ \sigma(T)=T,\ &\sigma(Y)=q^{-1/2}Y^{-1}XY=XT^2,\ \sigma(\pi)=XT, \notag\\ \sigma\ &=\ \tau_+\tau_-^{-1}\tau_+\ =\ \tau_-^{-1}\tau_+\tau_-^{-1}. \label{tautautau} \end{align} Check that $\,\sigma\tau_+=\tau_-^{-1}\sigma,\ \, \sigma\tau_+^{-1}=\tau_-\sigma.$
The polynomial representation is defined as $\mathscr{X}={\mathbb C}_{q,t}[X^{\pm 1}]$ over the field ${\mathbb C}_{q,t}={\mathbb C}(q^{1/4},t^{1/2})$ with $X$ acting by the multiplication. The formulas for the other generators are \begin{eqnarray*} T=t^{1/2}s+\frac{t^{1/2}-t^{-1/2}}{X^{2}-1} \circ (s-1),\ \, Y=\pi T \end{eqnarray*} in terms of the multiplicative reflection $s(X^n)=X^{-n}$ and $\pi(X^n)=q^{n/2}X^{-n}$ for $n\in{\mathbb Z}$.
The Gaussian $q^{x^2}$ is an element of a completion of $\mathscr{X}$. However the conjugation $A\mapsto q^{x^2}\,A\,q^{-x^2}$ for $A\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ preserves $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ and coincides with $\tau_+$. To see this use that $$ Y=\omega\circ(t^{1/2}+\frac{t^{1/2}-t^{-1/2}}{X^{-2}-1} \circ (1-s)). $$ Recall that $X=q^x$ and \begin{align*} &s(x)=-x,\ \omega(f(x))=f(x-1/2),\ \pi=\omega s,\ \pi(x)=1/2-x,\\ &\omega(q^{x^2})\,=\,q^{1/4}X^{-1}q^{x^2},\ \, Y(q^{-x^2})\,=\,\omega(q^{-x^2})\,=\,q^{-1/4}X q^{-x^2}. \end{align*}
It is important that $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ at $t=1$ becomes the Weyl algebra defined as the span $\langle X,Y\rangle /(Y^{-1}X^{-1}YXq^{1/2}=1)$ extended by the inversion $s=T(t=1)$ sending $X\mapsto X^{-1}$ and $Y\mapsto Y^{-1}$.
\subsubsection{\sf The E-polynomials}{\label{sect:macpoly}} Let us assume that $k$ is generic; we set $t=q^k$. The definition is as follows: \begin{align}\label{nonsymp} YE_{n}=q^{-n_{\sharp}}E_{n}\ \hbox{ for } \ n\in Z,\ \,E_n\in \mathscr{X}, &&&&\\ n_{\sharp}=\left\{\begin{array}{ccc}\frac{n+k}{2} & & n>0, \\\frac{n-k}{2} & & n\le 0,\end{array}\right\}, \text{\, note that }\, 0_{\sharp}=-\frac{k}{2}.\label{nonsymp1} \end{align} The normalization is $E_{n}=X^{n}+\text{ ``lower terms'' },$
where by ``lower terms'', we mean polynomials in terms of $X^{\pm m}$ as $|m|<n$
and, additionally, $X^{|n|}$ for negative $n$. It gives a filtration in $\mathscr{X}$ with the consecutive quotients of dimension $1$. Check that $Y$ preserves it, which justifies that $Y$ is diagonalizable in $\mathscr{X}$ and readily provides the formulas for the eigenvalues from (\ref{nonsymp}),(\ref{nonsymp1}).
The $E_{n} (n\in {\mathbb Z})$ are called {\em nonsymmetric Macdonald polynomials} or simply $E$\~polynomials. Obviously, $E_{0}=1, E_{1}=X$.
\subsubsection{\sf The intertwiners}\label{sect:intertw} The first intertwiner comes from AHA theory: $$ \Phi\stackrel{\,\mathbf{def}}{= \kern-3pt =} T+\frac{t^{1/2}-t^{-1/2}}{Y^{-2}-1}\,:\ \Phi Y=Y^{-1}\Phi. $$ The second is $\Pi\stackrel{\,\mathbf{def}}{= \kern-3pt =} q^{1/4}\tau_+(\pi)$; obviously, $\Pi^2=q^{1/2}$. Explicitly, $$ \Pi=X\pi=q^{1/2}\pi X^{-1}\,:\ \Pi Y=q^{-1/2}Y^{-1}\Pi. $$ Use that $\phi(\Pi)=\Pi$ to deduce the latter relation from $\Pi X\Pi^{-1}=q^{1/2}X^{-1}$. The $\Pi$\~type intertwiner is due to Knop and Sahi for $A_n$ (the case of arbitrary reduced systems was considered in \cite{C1}). Since $\Phi,\Pi$ ``intertwine" $\mathscr{Y}$, they can be used for generating the $E$\~polynomials. Namely, \begin{align}{\label{signE}} &E_{n+1}=q^{n/2}\Pi (E_{-n}) \ \hbox{ for } \ n\ge 0,\\ &E_{-n}=t^{1/2}(T+\frac{t^{1/2}-t^{-1/2}}{q^{2n_{\sharp}}-1})E_{n}. \label{interphi} \end{align} Beginning with $E_0=1$, one can readily construct the whole family of $E$\~polynomials. For instance, \begin{eqnarray*} T(X)&=&t^{1/2}X^{-1}+\frac{(t^{1/2}-t^{-1/2})(X^{-1}-X)}{X^{2}-1}\\ &=&t^{1/2}X^{-1}-(t^{1/2}-t^{-1/2})X^{-1}\ =\ t^{-1/2}X^{-1}, \\ E_{-1}&=&t^{1/2}(T+\frac{t^{1/2}-t^{-1/2}}{qt-1})E_{1} =X^{-1}+\frac{1-t}{1-t q}X. \end{eqnarray*} Using $\Pi$, \begin{align*} &E_{2} = q^{1/2}\Pi E_{-1}= X^{2}+q\frac{1-t}{1-tq}. \end{align*} Applying $\Phi$ and then $\Pi$, \begin{align*} &E_{-2} = X^{-2}+\frac{1-t}{1-tq^{2}}X^{2} +\frac{(1-t)(1-q^{2})}{(1-tq^{2})(1-q)},\\ &E_{3}=X^{3}+q^2\frac{1-t}{1-tq^{2}}X^{-1}+q\frac{(1-t)(1-q^{2})} {(1-tq)(1-q)}X. \end{align*} It is not difficult to find the general formula. See e.g., (6.2.7) from \cite{Ma4} for integral $k$. However, recalculating these formulas from integral $k$ to generic $k$ is not too simple; we will provide the exact formulas for the $E$\~polynomials below (in the form we need).
\subsubsection{\sf The E-Pieri rules} For any $n\in {\mathbb Z}$, we have the {\em evaluation formula} \begin{eqnarray*}
E_{n}(t^{-1/2})=t^{-|n|/2}\prod_{0<j<|\widetilde{n}|} \frac{1-q^{j}t^{2}}{1-q^{j}t}, \end{eqnarray*}
where $|\widetilde{n}|=|n|+1$ if $n\leq 0$ and
$|\widetilde{n}|=|n|$ if $n>0$.
It is used to introduce the {\em nonsymmetric spherical polynomials} $$\mathcal{E}_{n}=\frac{E_{n}}{E_{n}(t^{-1/2})}.$$ This normalization is important in many constructions due to the {\em duality formula} $\mathcal{E}_m(q^{n_\sharp})=\mathcal{E}_n(q^{m_\sharp})$. The Pieri rules are the simplest for the $E$\~spherical polynomials: \begin{eqnarray}\label{pierie} X\mathcal{E}_{n} =\frac{t^{-1/2\pm 1}q^{-n}-t^{1/2}} {t^{\pm1}q^{-n}-1}\mathcal{E}_{n+1} +\frac{t^{1/2}-t^{-1/2}}{t^{\pm1}q^{-n}-1}\mathcal{E}_{1-n}. \end{eqnarray} Here the sign is $\pm=+$\, if\, $n\leq 0$\ and $\pm=-$ if $n>0$. These formulas give an alternative approach to constructing the $E$\~polynomials and establishing their connections with other theories, for instance, with $\mathfrak{p}$\~adic Matsumoto functions.
\subsubsection{\sf Rogers' polynomials}{\label{sect:Rogers}} Let us introduce the {\em Rogers polynomials} for $n\ge 0$: $$ P_{n}=(1+t^{1/2}T)\bigl(E_{n}\bigr)=(1+s) \bigl(\frac{t-X^2}{1-X^2}E_n\bigr)=E_{-n}+\frac{t-tq^n}{1-tq^n}E_n, $$ $P_n=X^n+X^{-n}+$``lower terms", where the latter are $X^m+X^{-m}$ for $0\le m<n$. They are eigenfunctions of the following well-known operator \begin{align}\label{Lopertaor} \mathcal{L}=\frac{t^{1/2}X-t^{-1/2}X^{-1}}{X-X^{-1}}\Gamma+ \frac{t^{1/2}X^{-1}-t^{-1/2}X}{X^{-1}-X^{1}}\Gamma^{-1}, \end{align} where we set $\Gamma(f(x))=f(x+1/2),\, \Gamma(X)=q^{1/2}X$, i.e., $\Gamma$ acts as $-\omega$ in $\mathscr{X}$. This operator is the restriction of the operator $Y+Y^{-1}$ to symmetric polynomials, which is the key point of the DAHA approach to the theory of the Macdonald polynomials.
The exact eigenvalues are as follows: \begin{equation}{\label{eqn:LP}} \mathcal{L}(P_n)\ = \ (q^{n/2}t^{1/2}+q^{-n/2}t^{-1/2})\,P_n,\ n\ge 0. \end{equation} The evaluation formula reads $$ P_n(t^{\pm1/2})=t^{-n/2}\prod_{0\le j\le n-1} \frac{1-q^{j}t^{2}}{1-q^{j}t}. $$ The spherical $P$\~polynomials $\mathcal{P}_n\stackrel{\,\mathbf{def}}{= \kern-3pt =} P_n/P_n(t^{1/2})$ satisfy the duality $\mathcal{P}_n(t^{1/2}q^{m/2})=\mathcal{P}_m(t^{1/2}q^{n/2})$.
\subsubsection{\sf Explicit formulas}{\label{sect:formulas}} Let us begin with the well-known formulas for the Rogers polynomials ($n\ge 0$): \begin{align}\label{exactp} &P_{n}=X^{n}+X^{-n}+\sum_{j=1}^{[n/2]}M_{n-2j}\prod_{i=0}^{j-1} \frac{(1-q^{n-i})}{(1-q^{1+i})}\,\frac{(1-tq^{i})\ } {(1-tq^{n-i-1})}, \end{align} where $M_{n}=X^{n}+X^{-n}\, (n>0)$ and $M_{0}=1$.
The formulas for the $E$\~polynomials are as follows ($n>0$): \begin{align}\label{exacte-} E_{-n}&=X^{-n}+X^n\frac{1-t}{1-tq^n}+ \sum_{j=1}^{[n/2]}X^{2j-n}\,\prod_{i=0}^{j-1} \frac{(1-q^{n-i})}{(1-q^{1+i})}\,\frac{(1-tq^{i})}{(1-tq^{n-i})} \notag\\ &+\sum_{j=1}^{[(n-1)/2]}X^{n-2j}\, \frac{(1-tq^{j})}{(1-tq^{n-j})}\prod_{i=0}^{j-1} \frac{(1-q^{n-i})}{(1-q^{1+i})}\,\frac{(1-tq^{i})}{(1-tq^{n-i})}, \end{align} \begin{align}\label{exacte+} E_{n}=X^{n}+& \sum_{j=1}^{[n/2]}X^{2j-n}\,q^{n-j}\, \frac{(1-q^{j})}{(1-q^{n-j})}\prod_{i=0}^{j-1} \frac{(1-q^{n-i-1})}{(1-q^{1+i})}\, \frac{(1-tq^{i}\ ) }{(1-tq^{n-i-1})} \notag\\ &+\sum_{j=1}^{[(n-1)/2]}X^{n-2j}\,q^j\, \prod_{i=0}^{j-1}\frac{(1-q^{n-i-1})}{(1-q^{1+i})}\, \frac{(1-tq^{i}\ ) }{(1-tq^{n-i-1})}. \end{align}
\subsection{{\bf The p-adic limit}} Let us ``separate" $t$ and $q$; they will not be connected any longer by the relation $t=q^k$ in this section.
\subsubsection{\sf The limits of P-polynomials} We will begin with the symmetric case. Formula (\ref{exactp}) readily gives that \begin{align*} P^{0}_{n}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\lim_{q\to 0}P_{n}&=X^{n}+X^{-n}+ \sum_{j=1}^{[n/2]}M_{n-2j} \prod_{i=0}^{j-1}(X^{n-2j}+X^{2j-n})(1-t)\\ &=X^{n}+X^{-n}+(1-t)\chi_{n-2} =\chi_{n}-t\chi_{n-2} \end{align*} for the monomial symmetric functions $M_n$ and the classical characters $\chi_n=(X^{n+1}-X^{-n-1})/ (X-X^{-1})$. In the spherical normalization, $\mathcal{P}_n=P_n/P_n(t^{1/2})$, where $P^0_n(t^{1/2})=t^{-n/2}(1+t)$. One has $$ \mathcal{P}_{n}^{0}=(\chi_{n}-t\chi_{n-2})\frac{t^{n/2}}{1+t}. $$ By letting $t\to t^{-1}$ and $X\to Y$, we obtain that $\mathcal{P}^{0}_{n}$ coincides with the spherical function $\varphi_{n}$.
Let us obtain this fact directly from the definition of the Rogers polynomials $P_{n}$ in terms of the operator $\mathcal{L}$: \begin{align*} \left(\frac{t^{1/2}X-t^{-1/2}X^{-1}}{X-X^{-1}}\Gamma+ \frac{t^{1/2}X^{-1}-t^{-1/2}X}{X^{-1}-X^{1}}\Gamma^{-1}\right) &P_{n}\\ =\ (q^{n/2}t^{1/2}+q^{-n/2}t^{-1/2})&P_{n}\,; \end{align*} see Section \ref{sect:Rogers}.
Recall that $\Gamma(X^{m})=q^{m/2}X^{m}\,$ for any $\,m\in {\mathbb Z}\,$. It gives that \begin{align}\label{limgax} &\lim_{q\to 0}q^{n/2}\Gamma^{\pm 1}(X^{\pm m})=0
\ \hbox{ for } \ |m|\le n \hbox{\ \ unless\ }\notag\\ &\lim_{q\to 0}q^{n/2}\Gamma(X^{-n})=X^{-n},\ \lim_{q\to 0}q^{n/2}\Gamma^{-1}(X^{n})=X^{n} \ \hbox{ for } \ n\ge 0. \end{align} Therefore \begin{align*} t^{-1/2}P_{n}^{0}=\frac{t^{1/2}X-t^{-1/2}X^{-1}} {X-X^{-1}}X^{-n} +\frac{t^{1/2}X^{-1}-t^{-1/2}X}{X^{-1}-X}X^{n}. \end{align*} Using that $P^0_n(t^{1/2})=t^{-n/2}(1+t)$, we obtain that \begin{align*} \mathcal{P}_{n}^{0}=\left(\frac{tX^{2}-1}{X^{2}-1}X^{-n} +\frac{tX^{-2}-1}{X^{-2}-1}X^{n}\right)\frac{t^{n/2}}{1+t}, \end{align*} which is exactly the Macdonald summation formula (\ref{vphiformula}) under the substitution $X\mapsto Y,t\mapsto t^{-1}$: \begin{align}\label{Macdsummat} \varphi_{n}\ =\ \frac{t^{-n/2}}{1+t^{-1}} \left(\frac{1-t^{-1}Y^{-2}}{1-Y^{-2}}Y^{-n} +\frac{1-t^{-1}Y^{2}}{1-Y^{2}}Y^{n}\right). \end{align} We see that the right-hand side of (\ref{Macdsummat}) is actually the limit of the operator $\mathcal{L}$; this is a general fact (true for any root systems).
\subsubsection{\sf The limits of E-polynomials}\label{sec: p-adic-e}
We mainly follow \cite{C101}, however, with certain technical modifications. \begin{theorem} The limit $\mathcal{E}^0_n(X)= \lim_{q\to 0}\mathcal{E}_{n}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \mathcal{E}_{n}^{0}$ exists. The Matsumoto functions $\varepsilon_{n}$ from (\ref{vepformulas}) are connected with $\mathcal{E}^0_n$ as follows: $$ \varepsilon_{n}= \mathcal{E}_{n}^{0}(t\to t^{-1}, X\to Y). $$ \end{theorem} {\em Proof.}
First, $\lim_{q\to 0}E_{n}(t^{-1/2})=t^{-|n|/2}$. For $n>0$, we have \begin{eqnarray*} X\mathcal{E}_{n}^{0} & = & t^{-1/2}\mathcal{E}_{n+1}^{0},\\ X\mathcal{E}_{-n}^{0} & = & t^{1/2}\mathcal{E}_{-n+1}^{0} -(t^{1/2}-t^{-1/2})\mathcal{E}_{n+1}^{0}. \end{eqnarray*} These are exactly the Pieri relations for the Matsumoto functions from (\ref{yepm}--\ref{y-epm}) upon the substitution $Y\mapsto X,t\mapsto t^{-1}$.\phantom{1}
$\qed$
We know from (\ref{vepformulas}) that for $n\ge 0$, \begin{align}\label{vepnfor} &\varepsilon_n = t^{-\frac{n}{2}}Y^n,\ \varepsilon_{-n} =t^{-\frac{n+1}{2}}(t^{\frac{1}{2}}Y^{-n}+ (t^{\frac{1}{2}}-t^{-\frac{1}{2}})\frac{Y^{-n}-Y^{n}} {Y^{-2}-1}). \end{align} Obtaining these formulas directly from (\ref{exacte-}) and (\ref{exacte+}) is of some interest.
Let us show how to use here the $Y$\~operator, namely, the formulas for the action of $Y$ and $Y^{-1}$, correspondingly, on $E_n (n>0)$ and $E_{-n}(n\ge 0)$. One can present (\ref{nonsymp}) as follows: \begin{align*} &\frac{1-qtX^{-2}}{1-qX^{-2}}\,q^{n/2}\Gamma^{-1}(E_{n})+ \frac{1-t}{1-q X^{-2}}\,s(q^{n/2}\Gamma(E_{n}))\,=\, E_n \ (n>0),\\ &(\frac{1-t^{-1}X^{-2}}{1-X^{-2}}- \frac{1-t^{-1}}{1-X^{-2}}\,s)\,(q^{-n/2}\Gamma(E_{-n}))\,=\, E_{-n} \ \hbox{ for } \ n\ge 0. \end{align*} Setting $E_m^0=E_m(X; q=0)$ and applying (\ref{limgax}), \begin{align*} &E_{n}^0=X^n \ \hbox{ and } \ E_{-n}^0= (\frac{1-t^{-1}X^{-2}}{1-X^{-2}}- \frac{1-t^{-1}}{1-X^{-2}}\,s)(X^{-n}) \ \hbox{ for } \ n\ge 0, \end{align*} where in the first formula we use that $q^{n/2}\Gamma(E_{n})=0$ in the limit $n\to\infty\,$ because $E_{n>0}$ does not contain $X^{-n}$. Switching from $E_m^0$ to $\mathcal{E}_m^0$ and then to $\varepsilon_m$ (any $m\in {\mathbb Z}$), we arrive at (\ref{vepnfor}).
{\bf Comment.\ } We expect that a similar connection holds between the {\em difference-elliptic\,} symmetric Macdonald-type Looijenga functions (which can be called elliptic $P$\~functions) and the affine Hall functions. There is no general theory of such Macdonald-Looijenga functions so far; the paper \cite{Ch13} dealt with the difference-elliptic theory only at the level of operators. Their diagonalization was not performed there. The elliptic Ruijsenaars operators and their generalizations to arbitrary root systems are really connected with the affine symmetrizers in the corresponding limit.
Paper \cite{Ch13} indicates that the corresponding {\em nonsymmetric Hall polynomials\,} can be obtained from the nonsymmetric elliptic Macdonald-Looijenga functions, {\em elliptic $E$\~functions}, for the direct counterpart of the limit $q\to 0$. The nonsymmetric elliptic $E$\~functions are actually simpler to define than the $P$\~functions. The limit $q\to 0$ is well defined for the properly normalized difference-elliptic $Y$ operators from \cite{Ch13}, which provides certain nonsymmetric variant of the DAHA symmetrizer considered in this paper; it is in progress. A connection is expected with \cite{EFMV}.
Note that there are other theories of elliptic orthogonal polynomials. The most advanced theory we know is \cite{Ra}; however, this seems not what is needed here.
\subsection{{\bf Coinvariants and symmetrizers}} \subsubsection{\sf DAHA coinvariants} Let us prove Theorem \ref{thm:coinv} for the level $l=1$ in the case of $A_1$. \begin{theorem} For any $q,t=q^k$, ${\hbox{\rm dim}}_{\mathbb C}\,(\mathscr{X}/\mathcal{J}_{1}(\mathscr{X}))=1$. \end{theorem} {\em Proof.} Let $\varrho: \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\to \mathbb{C}$ be a functional on $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ such that \begin{align}\label{eqn:varrho1} \varrho(\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\cdot(T_{\widehat{w}}-t^{l(\widehat{w})/2}))\,=\,0 \text{\ \,and\ }\\ \label{eqn:varrho2} \varrho(\tau_+^{-1}(T_{\widehat{w}}-t^{l(\widehat{w})/2})\cdot \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$})\,=\,0 \end{align} for all $\widehat{w}\in \widehat{W}=W{\symbol n} P^\vee=\mathbf{S}_2{\symbol n} {\mathbb Z}\omega$. \begin{lemma} \label{lem:basis} An arbitrary $A\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ can be uniquely represented as \begin{align*} A=\sum c_{n,\varepsilon,m}\tau_+^{-1}(Y^{n})T^{\varepsilon}Y^{m}, \end{align*} where $\varepsilon=0$ or $1$,\ $m, n$ are integers and $c_{n,\varepsilon,m}$ are constants. \end{lemma} {\em Proof of Lemma \ref{lem:basis}.} One has $\tau_-\tau_+^{-1}(Y)=$ $\tau_-\tau_+^{-1}\tau_-(Y)= \sigma^{-1}(Y)=X^{-1}$. Applying $\tau_-^{-1}$ to $X^{-n}$, $T^{\varepsilon}$ and $Y^m$, we obtain that the elements $\{\tau_+^{-1}(Y^{n})T^{\varepsilon}Y^{m}\}$ form a PBW basis for $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$. \phantom{1}
$\qed$
Now, for $A\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$, relations \eqref{eqn:varrho1} and \eqref{eqn:varrho2} give that \begin{align*} \varrho(\tau_+^{-1}(Y^{n})A)\,=\, t^{n/2}\varrho(A) \text{\ and \ } \varrho(AT_{\widehat{w}})\,=\,t^{l(\widehat{w})/2}\varrho(A). \end{align*} Representing $A$ as in Lemma \ref{lem:basis}, \begin{align*} \varrho(A)=\sum c_{n,\varepsilon,m} \varrho(\tau_+^{-1}(Y^{n})T^{\varepsilon}Y^{-m})\,= \sum c_{n,\varepsilon,m}\,t^{n/2+\varepsilon/2-m/2}. \end{align*} Thus ${\hbox{\rm dim}}_{\mathbb C}\,(\mathscr{X}/\mathcal{J}_{1}(\mathscr{X}))=1$. \phantom{1}
$\qed$
{\bf Comment.\ } A similar argument can be employed for arbitrary simply-laced root systems (or if the twisted setting is used). A counterpart of Lemma \ref{lem:basis} is the claim that an arbitrary $A\in \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ can be uniquely represented as \begin{align*} A=\sum c_{b,w,a}\tau_+^{-1}(Y_{b})T_{w}Y_{a}, \end{align*} where $w\in W$,\, $a,b\in P$\, and $c_{b,w,a}$ are constants.
For any level $l>0$, $\tau_+^{-l}(Y)=q^{-l/4}X^{-l}Y$. Calculating the space of coinvariants generally requires knowing $\tau_+^{-l}(Y^m)$. The latter can be computed using the relation $Y^{-1}X^{-1}YXT^2q^{1/2}=1$, but explicit formulas are involved. Nevertheless, they are sufficient for finding the dimension of the space of coinvariants (for arbitrary simply-laced root systems as well).
\subsubsection{\sf P-hat in rank one}\label{sect:symmrank1} Let us discuss the rank-one version of Theorem \ref{P+FORMULA}. The explicit list of the elements $\widehat{w}\in \widehat{W}$ (there are four types) and the corresponding $T_{\widehat{w}}'\stackrel{\,\mathbf{def}}{= \kern-3pt =} t^{-l(w)/2}T_{\widehat{w}}^{-1}$, presented in terms of $Y,T$, is as follows: \begin{align*} &1)\ \widehat{w}= &m\omega\cdot s\, &(m>0),\ \ l(\widehat{w})=&m-1&,\ \ T_{\widehat{w}}'= &t^{-\frac{m-1}{2}}TY^{-m},\\ &2) &m\omega\, &(m>0),\, &m&,\, &t^{-\frac{m}{2}}Y^{-m},\\ &3) &-m\omega\, &(m\ge 0),\, &m&,\, &t^{-\frac{m}{2}}TY^{-m}T^{-1},\\ &4) &(-m\omega)\cdot s\, &(m\ge 0),\, &m+1&,\, &t^{-\frac{m+1}{2}}Y^{-m}T^{-1}. \end{align*} Note that we use a presentation that is somewhat different from the one used in the justification of Theorem \ref{P+FORMULA}.
\begin{theorem}\label{SYMRANK1} The affine symmetrizer $\widehat{\mathscr{P}}\,'_+$ (the prime here indicates that there is no division by $\widehat{P}(t^{-1})$) can be expressed as follows: \begin{align}\label{hatprank1} &\widehat{\mathscr{P}}\,'_+ = (1+t^{\frac{1}{2}}T)\,\Bigl(\, \frac{\,t^{-\frac{1}{2}}\,Y^{-1}} {1-t^{-\frac{1}{2}}\,Y^{-1}}\, (1+t^{-\frac{1}{2}}T^{-1})+t^{-\frac{1}{2}}T^{-1}\,\Bigr). \end{align} In particular, $\widehat{\mathscr{P}}\,'_+(1)=2\,\frac{1+t^{-1}}{1-t^{-1}}=
\widehat{P}(t^{-1})=2+\sum_{m=1}^{\infty}4t^{-m}$ for $|t|>1$. \phantom{1}
$\qed$\end{theorem}
The formula for $\widehat{\mathscr{P}}\,'_+$ from the theorem in terms of $t^{-1/2}$ is exactly the definition of the $P$\~hat symmetrizer upon using (1,2,3,4) above, as well as its particular case, the sum $2+\sum_{m=1}^{\infty}4t^{-m}$. Note that $ (1+t^{1/2}T)t^{-1/2}T^{-1}=1+t^{-1/2}T^{-1}. $
As remarked in Section \ref{sect:RatPhat} concerning formula (\ref{hatPrat}), the right-hand side of (\ref{hatprank1}) becomes {\em identically} zero when treated as an element of a proper localization of the affine Hecke algebra $\mathcal{H}_Y=\langle T,Y^{\pm 1}\rangle$. Indeed, this expression can be only zero because the localization is not sufficient to construct such an affine symmetrizer in $\mathcal{H}_Y$ (a completion is needed). One can deduce that (\ref{hatprank1}) vanishes directly from the relation \begin{align}\label{aharat} Tf(Y)-f(Y^{-1})T=\frac{t^{1/2}-t^{-1/2}}{Y^{-2}-1} (f(Y^{-1})-f(Y)) \end{align} extended to rational functions $f(Y)$; let us demonstrate it. First, the extension of (\ref{aharat}) to rational functions is straightforward since an arbitrary rational function in terms of $Y$ can be represented as a Laurent polynomial divided by a {\em $W$\~invariant} Laurent polynomial, commuting with $T$.
Let $1^+\stackrel{\,\mathbf{def}}{= \kern-3pt =} (1+t^{-1/2}T^{-1})$, \begin{align}\label{UUplus} U\stackrel{\,\mathbf{def}}{= \kern-3pt =} \frac{t^{-1/2}\,Y^{-1}}{ 1-t^{-1/2}\,Y^{-1}},\ U^+=U(1+t^{-1/2}T^{-1}). \end{align} Then \begin{align*} & TU^+=-\frac{t^{-1/2}}{1-t^{-1/2}Y^{-1}}(1+t^{-1/2}T^{-1})= -t^{-1/2}U^+ - t^{-1/2}1^+\,; \end{align*} therefore, $(1+t^{1/2}T)U^+ +1^+=0$, which is exactly vanishing the right-hand side of (\ref{hatprank1}).
This identity is the key point of the formula for $\widehat{\mathscr{P}}\,'_+$ as a limit of the powers of $Y$. Let us discuss in detail the corresponding deduction of Theorem \ref{YLEFT} from Theorem \ref{P+FORMULA} in the case of $A_1$.
For integers $M>0$, we introduce the {\em truncated symmetrizers} \begin{align}\label{hatprank1M} &\widehat{\mathscr{P}}\,'_M = (1+t^{\frac{1}{2}}T)\,\Bigl(\, \,\sum_{j=1}^M\,t^{-\frac{j}{2}}\,Y^{-j} (1+t^{-\frac{1}{2}}T^{-1})\Bigr)+ 1+t^{-\frac{1}{2}}T^{-1}. \end{align} \begin{theorem}\label{SYMRANK1Y} (i) Moving $T$ in $\widehat{\mathscr{P}}\,'$ via (\ref{aharat}), one arrives at identities in $\mathcal{H}_Y$: \begin{align}\label{hatsirank1} &\widehat{\mathscr{P}}\,'_M= \widehat{\Sigma}^+_M\stackrel{\,\mathbf{def}}{= \kern-3pt =}\,\widehat{\Sigma}_M\,(1+t^{-1/2}T^{-1}), \ \hbox{ for } \ \\ &\widehat{\Sigma}_M \stackrel{\,\mathbf{def}}{= \kern-3pt =} t^{-[\frac{M}{2}]}+ \sum_{j=1}^M\, t^{-[\frac{M-j}{2}]-\frac{j}{2}}\,(Y^j+Y^{-j}), \notag \end{align} where $[a/b]$ is the integer part.
(ii) The operator $\widehat{\mathscr{P}}\,'_+$ is well defined if and only if the limit $\widehat{\Sigma}_\infty^+=\lim_{M\to\infty} \widehat{\Sigma}^+_M$ exists; then these operators coincide. The existence of $\widehat{\Sigma}_\infty^+$ formally results in the following condition: \begin{align}\label{YMzero} &\lim_{M\to\infty} t^{-M/2}(Y^{-M})^+=0. \end{align} In its turn, (\ref{YMzero}) ensures that $\widehat{\Sigma}_\infty^+$ is an {\sf affine symmetrizer} if it exists, i.e., satisfies the symmetries \begin{align}\label{YSiabs} &Y\,\widehat{\Sigma}^+_\infty= \widehat{\Sigma}^+_\infty\, Y= t^{\frac{1}{2}}\,\widehat{\Sigma}^+_\infty =T\,\widehat{\Sigma}^+_\infty = \widehat{\Sigma}^+_\infty\, T. \end{align}
(iii) Finally, we claim that the existence of $\widehat{\mathscr{P}}\,'_+$, for instance its coefficient-wise convergence in the $\widehat{w}$\~decomposition, results in the identity \begin{align} &\widehat{\mathscr{P}}\,'_+= \lim_{M\to\infty}\overline{\Sigma}^+_M \,\ \hbox{ for } \ \, \overline{\Sigma}_M\stackrel{\,\mathbf{def}}{= \kern-3pt =}\frac {t^{-\frac{M}{2}}\,Y^M+t^{-\frac{M-1}{2}}\,Y^{M-1}}{1-t^{-1}}, \label{YSiabsbar} \end{align} which includes the existence (convergence) of $\overline{\Sigma}^+_{\infty\,}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \lim_{M\to\infty}\overline{\Sigma}^+_M$ in the same sense as that for $\widehat{\mathscr{P}}\,'_+$. \end{theorem}
\subsubsection{\sf Proof of Theorem \ref{SYMRANK1Y}} Let us prove this theorem (and Theorem \ref{YLEFT} for $A_1$); (\ref{YSiabsbar}) is its main part, called {\em the sigma-formula}. The following was outlined in Theorem \ref{YLEFT} for arbitrary root systems.
Only the $t$\~powers $t^{-M/2}$ and $t^{(1-M)/2}$ appear in the formula for $\widehat{\Sigma}_M$\,: \begin{align*} \widehat{\Sigma}_M\ =\ &t^{-\frac{M}{2}}(Y^M+Y^{-M})+ t^{\frac{1-M}{2}}(Y^{M-1}+Y^{1-M})\\ +\ &t^{-\frac{M}{2}}(Y^{M-2}+Y^{2-M})+\,\ldots\, +t^{-[\frac{M}{2}]}. \end{align*} For instance, in the case of even $M$, $$ \widehat{\Sigma}_M(1)=\sum_{j=2l} t^{-M/2}(t^{j/2}+t^{-j/2})+ \sum_{j=2l-1}t^{-M/2+1/2}(t^{j/2}+t^{-j/2}) $$ for $l=1,2\ldots,M/2$. The resulting $t^{-1}$\~series is $2+2t^{-1}+2t^{-2}+\ldots$\,; we obtain that $$ \lim_{M\to\infty}\widehat{\Sigma}_M\cdot(1+t^{-1/2}T^{-1})(1)= 2\,\frac{1+t^{-1}}{1-t^{-1}}=\widehat{P}(t^{-1})
\ \hbox{ for } \ |t|>1. $$
Let us check (\ref{hatsirank1}); we use the truncation $U_M=\sum_{j=1}^M t^{-j/2}Y^{-j}$ of the series $U$ introduced in (\ref{UUplus}) and set $U_M^+=U_M\,(1+t^{-1/2}T^{-1})$ for $U_M$ and other operators. Then \begin{align*} \widehat{\mathscr{P}}\,'_M-1^+\ =\ &(1+t^{\frac{1}{2}}T)\,U_M^+\\ =\ &\,U_M^+ + ts_{{}_Y}(U_M)^+ + \frac{t-1}{Y^{-2}-1}(ts_{{}_Y}(U_M) -U_M)^+\\ =\ \sum_{j=1}^M t^{-\frac{j}{2}}&\Bigl( (Y^{-j}+tY^{j}) + (1-t)\bigl(Y^j+Y^{j-2}+\ldots+Y^{2-j}\bigr)\Bigr)^+ \end{align*} for $s_{{}_Y}(Y^j)=Y^{-j}$. Collecting the terms with $Y^{\pm i}$, we obtain that \begin{align*} \widehat{\mathscr{P}}\,'_M\ =\ \sum_{i=1}^M &\Bigl(\bigl(\frac{1-t}{1-t^{-1}}\, t^{-\frac{i}{2}}(1-t^{-1-[\frac{M-i}{2}]})+t^{1-\frac{i}{2}} \bigr)\,Y^i\Bigr)^+\\ +\sum_{i=0}^{M-2} &\Bigl(\bigl(\frac{1-t}{1-t^{-1}}\, t^{-1-\frac{i}{2}}(1-t^{-[\frac{M-i}{2}]})+t^{-\frac{i}{2}} \bigr)\,Y^{-i}\Bigr)^+ \\ +\ &\Bigl( t^{-\frac{M}{2}}Y^{-M}\Bigr)^+ +\Bigl(t^{\frac{1}{2}-\frac{M}{2}}Y^{1-M} \Bigr)^+, \end{align*} where the last term is present only for $M\ge 2$. For $M=1$: $$\widehat{\mathscr{P}}\,'_M=1^+ +(1+t^{1/2}T)(t^{-1/2}Y^{-1})^+= 1^+ +t^{-1/2}(Y+Y^{-1})^+,$$ which immediately follows from (\ref{dahaone}).
As we have already checked, this sum becomes identically zero as $M\to \infty$. Therefore significant algebraic simplifications are granted; only the terms containing $M$ will contribute.
Finally, \begin{align*} \widehat{\mathscr{P}}\,'_M= \Bigl( \sum_{i=1}^M &t^{-\frac{i}{2}-[\frac{M-i}{2}]})Y^i+ \sum_{i=0}^{M-2}t^{-\frac{i}{2}-[\frac{M-i}{2}]})Y^{-i}\\ +\,&t^{-\frac{M}{2}}Y^{-M}+t^{\frac{1}{2}-\frac{M}{2}} Y^{1-M}\Bigr)^+, \end{align*} which can be readily transformed to formula (\ref{hatsirank1}). Claim $(i)$ is checked.
{\em Claim (ii).} Let us demonstrate that \begin{align}\label{YSiplus} t^{-\frac{1}{2}}Y\,\widehat{\Sigma}^+_\infty\ =\ \widehat{\Sigma}^+_\infty\ =\ t^{-\frac{1}{2}}T\,\widehat{\Sigma}^+_\infty\,. \end{align} The second of these formulas is an immediate corollary of the $s_{{}_Y}$\~invariance of $\widehat{\Sigma}^+_\infty$.
Provided the convergence of $\widehat{\mathscr{P}}\,'_+$ or (equivalently) $\widehat{\Sigma}^+_{\infty\,}$, the first relation from (\ref{YSiplus}) is formally equivalent to the condition \begin{align}\label{limYM} &\lim_{M\to \infty} t^{-M/2}(Y^{-M})^+=0. \end{align} Indeed, if $\widehat{\Sigma}^+_M$ converges, then so does \begin{align*} t^{-1/2}Y\,\widehat{\Sigma}^+_M\ =\ \widehat{\Sigma}^+_{M+1}- (t^{-(M+1)/2}Y^{-M-1} +t^{-M/2}Y^{-M})^+. \end{align*} Thus the condition $(t^{-(M+1)/2}Y^{-M-1}-t^{-M/2}Y^{-M})^+\to 0$ as $M\to \infty$ is necessary for the existence of $\widehat{\Sigma}^+_\infty$. This condition holds if and only if it is satisfied for each of the two terms separately, which is (\ref{limYM}). We conclude that the existence of $\widehat{\Sigma}^+_\infty$ results in (\ref{limYM}) and the latter, in its turn, gives the $t^{-\frac{1}{2}}Y$\~invariance condition from (\ref{YSiplus}).
Then $$ (1+t^{-1})\widehat{\Sigma}^+_\infty\ =\lim_{M\to\infty}\ (1+t^{-1/2}T^{-1})\widehat{\Sigma}_M (1+t^{-1/2}T^{-1}), $$ and we see that $\widehat{\Sigma}^+_\infty$ is invariant under the action of the {\em anti-involution} of $\mathcal{H}_Y$ sending $Y\mapsto Y$ and $T\mapsto T$ (and fixing $t,q$). Applying this anti-involution to (\ref{YSiplus}), we arrive at the counterpart of these relations with $\widehat{\Sigma}^+_\infty\ $ placed on the left and $Y,T$ on the right.
{\em Claim (iii).} Finally, relation (\ref{limYM}) readily results in (\ref{YSiabsbar}). \phantom{1}
$\qed$
{\bf Comment.\ }
It is worth mentioning that the sigma formula for $\widehat{\mathscr{P}}\,'_+$ makes it possible to calculate its $C$\~coefficients {\em directly}\, and establish the proportionality with $\widehat{\mathscr{S}}\,'_+\circ \widetilde{\mu}$ in the most explicit way.
Theorem \ref{YLEFTNEW}, which is a continuation of Theorem \ref {YLEFT}, establishes that the right multiplication of $\overline{\Sigma}_M$ by $(1+t^{-1/2}T^{-1})$ (the notation was $\overline{\Sigma}_M^+$) is actually not necessary in (\ref{YSiabsbar}). The following holds: \begin{align} &\widehat{\mathscr{P}}\,'_+\ =\ \lim_{M\to\infty}\overline{\Sigma}_M. \label{YSiabsbar1} \end{align} The coefficients here can be treated as formal series in terms of $X_{\alpha_1}=X^2,$ $X_{\alpha_0}=q X^{-2},$ $t^{-1}$
or as functions provided that $|t|>1>|q|$. One needs to check that $\lim_{M\to \infty} t^{-M/2}\,Y^{-M}\,=0\,$ without ${}^+$ as in (\ref{limYM}); this formally results in $$ \overline{\Sigma}^+_{\infty\,}= (1+t^{-1})\overline{\Sigma}_{\infty\,}. $$ The convergence in the algebraic variant means here that $t^{-M/2}\,Y^{-M}$ is getting divisible by powers of
$q$ growing together with $M.$ See Theorem \ref{YLEFTNEW} and below, the second formula in (\ref{YexpB}) and Theorem \ref{SIMINV}.
{\bf Comment.\ } We note that under the Kac-Moody limit $t\to\infty$, formula (\ref{YSiabsbar}) leads to a presentation of the Kac-Moody characters introduced for affine dominant weights as inductive limits of the corresponding Demazure characters. It can be used of course for arbitrary weights, not necessarily dominant, or even for arbitrary functions provided the convergence, which is an interesting development of this classical direction. Actually $$ \overline{\Sigma}_M\stackrel{\,\mathbf{def}}{= \kern-3pt =}\frac {t^{-\frac{M}{2}}\,Y^M+t^{-\frac{M-1}{2}}\,Y^{M-1}}{1-t^{-1}}, $$ applied to $q^{l x^2/4}$, and its generalization via $\overline{\Sigma}_{\mathbf b}$ from Theorem \ref{YLEFT} can be considered as certain $q,t$\~Demazure characters.
\phantom{1}
$\qed$
\subsubsection{\sf Stabilization of Y-powers} Let us provide explicit analysis of the limits of the powers $Y$\~operators, including the coefficient-wise convergence of $\overline{\Sigma}_\infty$ and $\widehat{\mathscr{P}}\,'_+$; see Theorem \ref{SYMRANK1Y}. Recall that we expand operators in the form $\sum_{\widehat{w}}C_{\widehat{w}}\widehat{w}$, where $C_{\widehat{w}}$ can be considered as formal series or functions of $X$. Let us treat them as (meromorphic) functions.
Note that if we know that the $C$\~coefficients are meromorphic functions, this does not guarantees that this operator converges in the corresponding space. For instance, when acting in the polynomial representation $\mathscr{X}$, it is well defined at
a given Laurent polynomials $P(X)$ only for sufficiently large negative $\Re k$ (depending on $P$), which is significantly worse than the condition $|qt^{-2}|<1$ (necessary and) sufficient for the coefficient-wise convergence of $\widehat{\mathscr{P}}\,'_+$.
In contrast to the case $l=0$, the convergence of $\widehat{\mathscr{P}}\,'_+$ in the spaces $\mathscr{X}q^{lx^2}$ for $l>0$ is equivalent to the existence of the corresponding $\{C_{\widehat{w}}\}$ (considered in the next theorem). It is with a reservation concerning $l=1$, where the operator $\widehat{\mathscr{P}}\,'_+$ is well defined for any $t$. This fact is not very surprising due to the presence of the Gaussians; the growth of the $C_{\widehat{w}}$\~coefficients is no greater than exponential in terms of $l(\widehat{w})$.
The following theorem is directly related to Theorems \ref{TQ1H} and \ref{GENPROPR}.
\begin{theorem} \label{YEXPA} Continuing to assume that $|q|<1$, we represent: \begin{align}\label{YexpA}
\ \hbox{ for } \ |t|<1\,:\ \,t^{\frac{m}{2}}q^{-\frac{m}{2}}Y^{-m}&= \sum_{\widehat{w}\in \widehat{W}}A^{(-m)}_{\widehat{w}}(X)\,\widehat{w} \\ \ \hbox{ and } \ \ \,t^{\frac{m}{2}}Y^{m}&= \sum_{\widehat{w}\in \widehat{W}}A^{(m)}_{\widehat{w}}(X)\,\widehat{w} \,,\notag\\ \label{YexpB}
\ \hbox{ for } \ |t|>1\,:\ t^{-\frac{m}{2}}q^{-\frac{m}{2}}Y^{-m}&= \sum_{\widehat{w}\in \widehat{W}}B^{(-m)}_{\widehat{w}}(X)\,\widehat{w} \\ \ \hbox{ and } \ \ t^{-\frac{m}{2}}Y^{m}&= \sum_{\widehat{w}\in \widehat{W}}B^{(m)}_{\widehat{w}}(X)\,\widehat{w} \,,\notag \end{align} where $m\in {\mathbb Z}_+$. These are just algebraic expansions in the polynomial representations; the sums are finite.
The claim is that, given $\widehat{w}\in \widehat{W}$, the limits $A^{\pm\infty}_{\widehat{w}}=\lim_{m\to\infty} A^{(\pm m)}_{\widehat{w}}$ and $B^{\pm\infty}_{\widehat{w}}=\lim_{m\to\infty} B^{(\pm m)}_{\widehat{w}}$ exist and are meromorphic functions in terms of $X^2$ analytic apart from $0\neq X^2 \not\in q^{{\mathbb Z}}$. \phantom{1}
$\qed$\end{theorem}
Using the second formula, we see that the operator
$t^{-m/2}Y^{-m}$ for $|t|>1$ has the coefficients tending to zero as $m\to\infty$. Indeed, given $\widehat{w}\in \widehat{W}$, the coefficient $B^{(-m)}_{\widehat{w}}(X)$ behaves as $q^{m/2}B^{-\infty}_{\widehat{w}}(X)$ in the limit of large $m>0$. Similarly, $t^{-m/2}Y^{-m}$ has the $A$\~coefficients (for
$|t|<1$) convergent to zero as $m\to\infty$ if
$|qt^{-2}|<1$.
It readily results in formula (\ref{limYM}) needed above. We obtain that the $C_{\widehat{w}}$\~coefficients of $\widehat{\mathscr{P}}\,'_+$ are meromorphic functions
when $|qt^{-2}|<1$. Here one can use (\ref{YSiabsbar}) or directly (\ref{hatprank1}).
{\bf Comment.\ } Note that the case $|t|=1$ is not covered by Theorem \ref{YEXPA}. In this case, the $A,B$\~coefficients remain bounded for large $-m$, which is sufficient for the application to $\widehat{\mathscr{P}}\,'_+$. \phantom{1}
$\qed$
The theorem is closely connected with the action of $Y^{\pm m}$ in the polynomial representation. For instance, the first line of (\ref{YexpB}) is related to the fact that for any given $n\in {\mathbb Z}_+$,\,
$\lim_{m\to \infty}t^{-m/2}Y^{-m}(X^{\pm n})= 0$, provided that $|q|<1$ and $|tq^{n/2}|>1$, i.e., for sufficiently large $t$, exactly, when
$|t|>|q|^{-n/2}$. This fact was actually used in Theorem \ref{LEVZERO} (for arbitrary root systems). It can be readily checked by expressing $X^{\pm n}$ in terms of the $E$\~polynomials. For the latter, \begin{align*} &t^{-m/2}Y^{-m}(E_{-n})=t^{-m}q^{-mn/2}E_{-n}= (t q^{n/2})^{-m} E_{-n} \ \hbox{ for } \ n\ge 0, \\ &t^{-m/2}Y^{-m}(E_{n})\ =\ q^{mn/2}E_{n} \ \hbox{ for } \ n>0. \end{align*}
\subsubsection{\sf More on stabilization} The expansions from (\ref{YexpB}) for the $B$\~coefficients and the relations from (\ref{YSiabs}) are of clear algebraic nature. Let us demonstrate it. The expansion of the operators in the following theorem will be considered in the polynomial representation as above, however we will now treat their coefficients as formal $q$\~series.
\begin{theorem} \label{SIMINV} (i) The $C$\~coefficients in the expansion $t^{-m/2}Y^{-m}=\sum_{\widehat{w}\in \widehat{W}}C^{(-m)}_{\widehat{w}}\widehat{w}$ \, for $m\ge 0$\, are from the ring $$ \mathbb{X}\ =\ {\mathbb Z}[t^{-1},q^{1/2},X^{\pm 2},(1-q^l X^{\pm 2r})^{-1})], $$ where $l,r\in {\mathbb Z}_+,\, r>0,\, l>0$ for $-2r$. Moreover, the coefficient $C^{(-m)}_{\,w\cdot b\,}$, where $w=1,s$ and $b=\pm n$ for $m\ge n\ge 0$, belongs to $q^{(m-n)/2}\,\mathbb{X}\subset\mathbb{X}$.
(ii) In particular, the coefficients of $w\cdot(\pm n)$ in the $\widehat{w}$\~expansions of \begin{align}\label{YSiprime} &t^{-\frac{1}{2}}Y\,\widehat{\Sigma}^+_M-\widehat{\Sigma}^+_M \ \hbox{ and } \ t^{-\frac{1}{2}}T\,\widehat{\Sigma}^+_M-\widehat{\Sigma}^+_M \end{align} belong to the ideal\, $q^{(M-n)/2}\,\mathbb{X}$\, for $0\le n\le M$. If $n$ is fixed and $M\to\infty$, these coefficients tend to zero with respect to the system of ideals $q^{m}\,\mathbb{X}$ for $m\to\infty$.\phantom{1}
$\qed$ \end{theorem}
This theorem is a refined $A_1$\~version of the corresponding (algebraic) part $(ii)$ of Theorem \ref{YLEFTNEW}, which established that given $\widehat{w}\in \widehat{W}$, the coefficients $C_{\widehat{w}}$, counterparts of the coefficients $C^{(-m)}_{\widehat{w}}$, are divisible by powers of $q$ growing linearly in the limit $\mathbf b \to \infty$, corresponding to $m\to \infty$.
{\em Exact formulas as $t=0$ and $t\to \infty$}. Let us provide the first several exact formulas for the coefficients $A_{\widehat{w}}^{(\pm m)}$ and $B_{\widehat{w}}^{(\pm m)}$ from Theorem \ref{YEXPA} in the corresponding limits $t\to 0$ and $t\to \infty$. We will consider only the case of even powers. Then $\widehat{w}$ that appear in the formulas can be represented as $$ [n;\epsilon]\,\stackrel{\,\mathbf{def}}{= \kern-3pt =}\, \Gamma^{2n}\, s^\epsilon\,=\,(-2n\omega)\, s^\epsilon\, \ \hbox{ for } \ n\in {\mathbb Z},\ \epsilon=0,1. $$ When $m\ge 1$, the range of nonzero terms in the $A,B$\~coefficients is \begin{align}\label{ab-range} &-m\le n\le m-1, \ \hbox{ where } \ \epsilon=0,1 \ \hbox{ for } \ A_{[n;\,\epsilon]}^{(+2m)}, B_{[n;\,\epsilon]}^{(+2m)},\\ &-m\le n\le m,\ \epsilon=1 \hbox{\, if\, }\ n=-m,\, \ \hbox{ for } \ A_{[n;\,\epsilon]}^{(-2m)}, B_{[n;\,\epsilon]}^{(-2m)}.\notag \end{align} The elements $\widehat{w}$ not in the form $[n;\epsilon]$ will not contribute.
We set $\bar{A},\bar{B}$ for the limits of these coefficients respectively for $t\to 0$ and $t\to \infty$. The formulas below (they are known and are ``pure" products for any coefficients) are of importance when analyzing the relations to the Demazure characters in Kac-Moody theory and, hopefully, for the study of the $t$\~deformations of the Demazure characters.
First, \begin{align}\label{ab-zero} &\bar{A}_{[0;0]}^{(\pm 2m)}\ =\ \bar{B}_{[0;0]}^{(\pm 2m)}\ =\ C_0^m\,\stackrel{\,\mathbf{def}}{= \kern-3pt =}\, \prod_{i=2}^{2m-1}(1-q^i)\\ \times\bigl(&\,\prod_{i=2}^{2[\frac{m}{2}]}(1-q^i)\, \prod_{i=1}^{2[\frac{m+1}{2}]-1}(1-q^i)\, \prod_{i=0}^{m-1} (1-q^{i+1}X^{-2})(1-q^i X^2)\,\bigr)^{-1}\notag \end{align} for $m>1$ and with $C_0^1\stackrel{\,\mathbf{def}}{= \kern-3pt =} 1$ as $m=1$ (any signs of $\pm 2m$); here $[m/2]$ is the integer part of $m/2$.
Second, the case of the reflection, \begin{align}\label{ab-min} &\bar{A}_{[0;1]}^{(2m)}\ =\ -C_0^m,\ \ \ \ \, \bar{A}_{[0;1]}^{(-2m)}\ =\ -X^2 C_0^m \, \frac{1-q^m X^{-2}}{1-q^m X^2},\\ &\bar{B}_{[0;1]}^{(2m)}\ =\ -X^2 C_0^m ,\ \bar{B}_{[0;1]}^{(-2m)}\ =\ -X^4 C_0^m \, \frac{1-q^m X^{-2}}{1-q^m X^2}. \notag \end{align}
Then for $\Gamma^2$ (an element of length $2$)\,: \begin{align}\label{abc10} &\bar{A}_{[1;0]}^{(2m)}=C_0^m \frac{(1-q^{m-1})}{(1-q^{m+1})} \frac{(1-q^m X^{-2})}{(1-q^m X^2)},\\ &\bar{A}_{[1;0]}^{(-2m)}=q^{-1} C_0^m \frac{1-q^m X^{-2}}{1-q^m X^2}, \notag\\ &\bar{B}_{[1;0]}^{(2m)}=q X^4 C_0^m \frac{(1-q^{m-1})}{(1-q^{m+1})} \frac{(1-q^m X^{-2})}{(1-q^m X^2)},\notag\\ &\bar{B}_{[1;0]}^{(-2m)}=X^4 C_0^m \frac{1-q^m X^{-2}}{1-q^m X^2}, \notag \end{align} and for $s \Gamma^2$ (its length is $1$): \begin{align}\label{abc-10} &\bar{A}_{[-1;1]}^{(2m)}=-C_0^m \frac{1-q^{m-1}X^2}{1-q^{m+1}X^{-2}},\ \ \bar{A}_{[-1;1]}^{(-2m)}=-q^{-1}X^2 C_0^m ,\\ &\bar{B}_{[-1;1]}^{(2m)}=-q X^{-2} C_0^m \frac{1-q^{m-1}X^2}{1-q^{m+1}X^{-2}},\ \ \bar{B}_{[-1;1]}^{(-2m)}=-C_0^m. \notag \end{align}
Finally, $\Gamma^2 s$ (an element of length $3$), \begin{align}\label{abc11} &\bar{A}_{[1;1]}^{(2m)}=-C_0^m \frac{(1-q^{m-1})}{(1-q^{m+1})} \frac{(1-q^{m}X^{-2})}{(1-q^{m}X^{2})},\\ &\bar{A}_{[1;1]}^{(-2m)}=-q X^2 C_0^m \, \frac{(1-q^{m-1})}{(1-q^{m+1})} \frac{(1-q^{m-1}X^{-2})(1-q^{m}X^{-2})} {(1-q^{m}X^{2})(1-q^{m+1}X^{2})}\notag,\\ &\bar{B}_{[1;1]}^{(2m)}=-q^3 X^{6} C_0^m \frac{(1-q^{m-1})}{(1-q^{m+1})} \frac{(1-q^{m-1}X^{-2})}{(1-q^{m+1}X^2)},\notag\\ &\bar{B}_{[1;1]}^{(-2m)}=-q^4 X^{8}C_0^m \frac{(1-q^{m-1})}{(1-q^{m+1})} \frac{(1-q^{m-1}X^{-2})(1-q^{m}X^{-2})} {(1-q^{m}X^{2})(1-q^{m+1}X^{2})}. \notag \end{align}
\setcounter{equation}{0} \section{\sc{Spinor Whittaker function}} \subsection{{\bf Q-Hermite polynomials}} We will begin with the limiting procedures connecting $q$\~Toda theory with the difference QMBP.
\subsubsection{\sf The Ruijsenaars limit} \label{sect:whitlim} Recall the definition of the $L$\~operator from (\ref{Lopertaor}) : \begin{align}\label{Lopertaorx} \mathcal{L}=\frac{t^{1/2}X-t^{-1/2}X^{-1}}{X-X^{-1}}\Gamma+ \frac{t^{1/2}X^{-1}-t^{-1/2}X}{X^{-1}-X^{1}}\Gamma^{-1}, \end{align} where we set $\Gamma(f(x))=f(x+1/2),\, \Gamma(X^n)=q^{n/2}X$ for $X=q^x$. It is symmetric with respect to the action of $s\,:\, X\mapsto X^{-1},\, \Gamma\mapsto \Gamma^{-1}$.
This operator preserves the space of {\em symmetric} Laurent polynomials. The space of all Laurent polynomials will be denoted by $\mathscr{X}={\mathbb C}_{q,t}[X^{\pm}]$, where the field of definition is ${\mathbb C}_{q,t}\stackrel{\,\mathbf{def}}{= \kern-3pt =}{\mathbb C}(q^{1/2},t^{1/2})$.
The Rogers polynomials $P_n\in \mathscr{X}\, (n\ge 0)$ are the eigenfunctions of $\mathcal{L}$ normalized by the conditions $P_n=X^n+X^{-n}+$``lower terms". The eigenvalues are as follows (see (\ref{eqn:LP}))\,: \begin{equation}{\label{eqn:LPx}} \mathcal{L}(P_n)\ = \ (q^{n/2}t^{1/2}+q^{-n/2}t^{-1/2})\,P_n,\ n\ge 0. \end{equation}
In this section, $|q|<1$ and $t=q^k$ for $k\in {\mathbb C}$. We will use the difference operator $\Gamma_k(X^n)\stackrel{\,\mathbf{def}}{= \kern-3pt =} t^{k/2}X^n$,
Following Ruijsenaars, Etingof demonstrates in \cite{Et1} that $$ \lim_{k\to -\infty}q^{-kx}\Gamma_{k}\,\mathcal{L}\,\Gamma_{-k}q^{kx} $$ becomes the so-called $q$\~Toda (difference) operator. To be exact, they considered the case of $A_n$. The {\em difference} Toda operators of type $A_n$ are due to Ruijsenaars too; see e.g., \cite{Rui}. Inozemtsev extended Ruijsenaars' limiting procedure to the case of differential {\em periodic} Toda lattice (which we do not consider here).
The $A_n$ is exceptional because all fundamental weights are minuscule and the formulas for the Macdonald-Ruijsenaars difference QMBP operators are explicit. The justification of this limiting procedure in the case of arbitrary (reduced) root systems (conjectured by Etingof) was obtained in \cite{ChW}; one can employ the Dunkl operators in Macdonald theory or use directly the formula for the {\em global} $q$\~Whittaker function from \cite{ChW}. It is worth mentioning that the classical integrability (at the level of the Poisson brackets) of QMBP and the classical Toda chain is significantly simpler than that of its quantum (operator) generalization.
Following \cite{ChW}, we tend $k$ to $\infty$ ($t\to 0$) in this section. Let \begin{align*} \hbox{\ae}(\mathcal{L})&\stackrel{\,\mathbf{def}}{= \kern-3pt =} q^{kx}\Gamma_{k}^{-1}\,\mathcal{L}\,\Gamma_{k}q^{-kx},\ R\!E(\mathcal{L})\stackrel{\,\mathbf{def}}{= \kern-3pt =} \lim_{k\to \infty} \hbox{\ae}(\mathcal{L}), \end{align*} where the second limit is the {\em Ruijsenaars-Etingof procedure}. At the level of functions $F(X)$: $$R\!E(F)=\lim_{k\to \infty}q^{kx}\,F(q^{-k/2}X) =\lim_{k\to \infty}q^{kx}\Gamma_{k}^{-1}(F). $$ Generally, the $R\!E$ procedure requires very specific functions $F$ to be well defined. Formally, if $\mathcal{L}(\Phi)=(\Lambda+\Lambda^{-1})\Phi$, then $$ R\!E(\mathcal{L})(\mathcal{W}) =(\Lambda+\Lambda^{-1})\mathcal{W} \ \hbox{ for } \ \mathcal{W}=R\!E(\Phi) \hbox{\, provided its existence}. $$ At the level of operators, \begin{align}\label{whitoper} \hbox{\ae}(\mathcal{L}) &=\,\frac{X-X^{-1}}{t^{-1/2}X-t^{1/2}X^{-1}}t^{-1/2}\Gamma +\frac{tX^{-1}-t^{-1}X}{t^{1/2}X^{-1}-t^{-1/2}X}t^{1/2}\Gamma \notag\\ &=\frac{X-X^{-1}}{X-tX^{-1}}\Gamma +\frac{t^{2}X^{-1}-X}{tX^{-1}-X}\Gamma^{-1}. \end{align} Therefore \begin{align}\label{qToda} R\!E(\mathcal{L}) =\frac{X-X^{-1}}{X}\Gamma+\Gamma^{-1}= (1-X^{-2})\Gamma+\Gamma^{-1}, \end{align} where the latter is the {\em $q$\~Toda operator}.
One of the main results of \cite{ChW} states that the $R\!E$\~image of the {\em global $q,t$\~spherical function} (arbitrary reduced root systems; see the definition there) is as follows: \begin{align}\label{Whitsym} \mathcal{W}_{q}(X,\Lambda)\,=\, \sum_{m=0}^{\infty}q^{m^{2}/4}\,X^m \overline{P}_{m}(\Lambda)\, \prod_{s=1}^{m}\frac{1}{1-q^{s}}\,q^{x^{2}}\,q^{\lambda^{2}}, \end{align} where $\prod_{s=1}^0=1$, $\Lambda=q^\lambda$ as for $X$, $\overline{P}_m$ are the symmetric $q$\~Hermite polynomials, defined as the specializations of $P_m$ at $t=0$.
The existence of $\{\overline{P}_m\}$ can be readily deduced from the explicit formulas from the previous part of the paper. It will be discussed systematically (from scratch) below.
One of the key properties of $\mathcal{W}_{q}(X,\Lambda)$ is the Shintani-type formula; see \cite{ChW}. Setting $ \widetilde{\mathcal{W}}_q(X,\Lambda) \stackrel{\,\mathbf{def}}{= \kern-3pt =} \sum_{m=0}^\infty q^{\frac{m^2}{4}} \frac{X^m\ \overline{P}_m(\Lambda) } {\prod_{s=1}^{m} (1-q^{s}) }\,$ one has:\, $\widetilde{\mathcal{W}}_q(q^{n/2},\Lambda)=0$\, for\, $n>0$\, and \begin{align}\label{shintqa1} &q^{n^2/4}\,\widetilde{\mathcal{W}}_q(q^{-n/2},\Lambda)\ =\ \theta(\Lambda)\overline{P}_n(\Lambda) \prod_{j=1}^{\infty}\Bigl( \frac{1}{1-q^j }\Bigr), \end{align} where $\,n\ge 0, \ \theta(\Lambda)\, =\, \sum_{j=-\infty}^\infty q^{j^2/4}\Lambda^j.$
\subsubsection{\sf Nonsymmetric polynomials} \label{sect:QHermite} We will use the $E$\~polynomials $E_{a}\in \mathscr{X}$ from the previous part of the paper, which are the eigenfunctions of the difference Dunkl operator $$ Y\stackrel{\,\mathbf{def}}{= \kern-3pt =}\Gamma^{-1}\circ(t^{1/2}+\frac{t^{1/2}-t^{-1/2}}{X^{-2}-1} \circ (1-s)). $$ Namely, see (\ref{nonsymp}) above, \begin{align}\label{nonsympx} YE_{n}=q^{-n_{\sharp}}E_{n}\ \hbox{ for } \ n\in Z,&&&&\\ n_{\sharp}=\left\{\begin{array}{ccc}\frac{n+k}{2} & & n>0, \\\frac{n-k}{2} & & n\le 0,\end{array}\right\}, \text{\, note that }\, 0_{\sharp}=-\frac{k}{2}. \end{align} The normalization is $E_{n}=X^{n}+\text{ ``lower terms'' },$
where by ``lower terms'', we mean polynomials in terms of $X^{\pm m}$ as $|m|<n$
and, additionally, $X^{|n|}$ for negative $n$.
Let us define their two limits: $$ \widetilde{E}_{a}=\lim_{t\to\infty}E_{a} \ \hbox{ and } \ \overline{E}_{a}=\lim_{t\to 0}E_{a}. $$ Both limits exist (use the explicit formulas or the intertwining operators from the previous part of the paper) and are closely connected to each other. The following theorem provides the connection.
\begin{theorem} For $n\geq 0$, \begin{align}\label{tildee} \widetilde{E}_{-n}\, =\,\bigl(q^{\frac{n}{2}}\overline{E}_{-n}(Xq^{\frac{1}{2}})\bigr)
\Big|_{q\to q^{-1}},\ \widetilde{E}_{n}\, =\,\bigl(q^{-\frac{n}{2}}\overline{E}_{n}(Xq^{\frac{1}{2}})\bigr)
\Big|_{q\to q^{-1}}. \end{align} \end{theorem}
\phantom{1}
$\qed$
The polynomials $\overline{E}_{a}$ are called {\em nonsymmetric (continuous) $q$\~Hermite polynomials} (see \cite{ChW} and references therein). Upon the substitution $X\mapsto X^{-1}$, the polynomials $\overline{E}_a$ are directly connected with the Demazure characters of level-one Kac-Moody integrable modules; see \cite{San} for the $GL_n$\~case. Generally it holds only for the twisted affinization; see \cite{Ion1}. These polynomials also appear naturally in formulas $\widehat{\chi}_{a}^{(l=1)}$ from (\ref{Kac-Moody}), when the latter are used for arbitrary $a\in P$; see also (\ref{Kac-M-Herm}) there.
More systematically, let us define \begin{align*} \overline{T}\,\stackrel{\,\mathbf{def}}{= \kern-3pt =}\, \lim_{t\to 0}t^{1/2}T= \frac{1}{1-X^{2}}\circ (s-1),\ \overline{T}(\overline{T}+1)=0. \end{align*} Using intertwiners, $\overline{E_0}=1$, \begin{align*} \overline{E}_{1+n}\,=\,q^{n/2}\Pi \overline{E}_{-n},\\ \overline{E}_{-n}\,=\,(\overline{T}+1)\overline{E}_{n} \end{align*} for $n\ge 0$; the raising operator $\Pi\stackrel{\,\mathbf{def}}{= \kern-3pt =} X\pi$ was discussed in Section \ref{sect:intertw}.
From the divisibility condition $\overline{T}+1=(s+1)\cdot\{\,\}$, we obtain that $\overline{E}_{-n}$ is symmetric ($s$\~invariant) and $\overline{P}_{n}=\overline{E}_{-n}$ for $n\ge 0.$
Explicitly, \begin{align*} \overline{E}_{-n-1}\,=\,((\overline{T}+1)\Pi q^{n/2}) \overline{E}_{-n},\\ (\overline{T}+1)\Pi \,=\,\frac{X^{2}\Gamma^{-1}-X^{-2}\Gamma}{X-X^{-1}}. \end{align*}
The bar-Pieri rules read as follows: \begin{align}\label{pienilp+} &X^{-1}\overline{E}_{-n}=\overline{E}_{-n-1}-\overline{E}_{n+1} \ (n\ge 0),\\ &X^{-1}\overline{E}_n=(1-q^{n-1})\overline{E}_{n-1}+q^{n-1} \overline{E}_{1-n}\ (n\ge 1),\notag\\ \label{pienilp-} &X\overline{E}_{-n}=(1-q^n)\overline{E}_{1-n}+\overline{E}_{n+1} \ (n\ge 0),\\ &X\overline{E}_n=\overline{E}_{n+1}-q^{n} \overline{E}_{1-n}\ (n\ge 1).\notag \end{align}
Let $\overline{Y}=\pi \overline{T}=\lim_{t\to 0}t^{1/2}Y$. Recall that \begin{eqnarray*} YE_{n}=\left\{\begin{array}{ccc}t^{-1/2}q^{-n/2}E_n, & & n>0, \\ t^{1/2}q^{n/2}E_{n}, & & n\leq 0.\end{array}\right. \end{eqnarray*} In the limit, \begin{eqnarray}\label{Ynilp+} \overline{Y}\,\overline{E}_{n}
=\left\{\begin{array}{ccc}q^{-|n|/2}\overline{E}_{n}, & & n>0, \\ 0, & & n\le 0.\end{array}\right. \end{eqnarray} Since $\overline{Y}$ is not invertible, we need to introduce $$ \overline{Y}\,'=\lim_{t\to 0}t^{1/2}Y^{-1}= \lim_{t\to 0}t^{1/2}T^{-1}\pi=\overline{T}\,'\pi $$ for $\overline{T}\,'=\overline{T}+1$. Then $\overline{Y}\,\overline{Y}\,'=0=\overline{Y}\,'\,\overline{Y}$ and \begin{eqnarray}\label{Ynilp-} \overline{Y}\,'\,\overline{E}_{n}
=\left\{\begin{array}{ccc}q^{-|n|/2}\overline{E}_{n}, & & n\le 0, \\ 0, & & n>0.\end{array}\right. \end{eqnarray} Finally, see \eqref{eqn:LP}, \begin{align*} \overline{\mathcal{L}}=\lim_{t\to 0}t^{1/2}\mathcal{L}=\overline{Y}\,' +\overline{Y} \,=\,\frac{1}{1-X^{2}}\Gamma+\frac{1}{1-X^{-2}}\Gamma^{-1} \end{align*} and $\overline{\mathcal{L}}\,\overline{P}_{n}= q^{-n/2}\overline{P}_{n},\, n\ge 0$; recall that $\overline{P}_n=\overline{E}_{-n}$.
\subsubsection{\sf Nil-DAHA} We come to the following definition of {\em nil-DAHA\,} (which can be readily adjusted to any reduced root systems).
\begin{theorem}\label{TWONIL} (i) {\sf Nil-DAHA}\, $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_+$ is generated by $T,\pi_+, X^{\pm 1}$ over the ring ${\mathbb C}[q^{\pm 1/4}]$ with the defining relations $T(T+1)=0$, \begin{align}\label{nildahax} \pi_+^2=1,\ \pi_+ X\pi_+=q^{1/2}X^{-1}, \ TX-X^{-1}T=X^{-1}, \end{align} resulting in $X^{-1}=XT-TX^{-1}$. Setting $\,Y\stackrel{\,\mathbf{def}}{= \kern-3pt =}\pi_+ T\,$ and $\,Y'\stackrel{\,\mathbf{def}}{= \kern-3pt =} T'\pi_+\,$ for $\,T'\stackrel{\,\mathbf{def}}{= \kern-3pt =} (T+1)\,$, the relation $TT'=0$ gives that (\ref{nildahax}) gives that $\,TY-Y'T=-Y,\ TY'=0=YT'$,\, which results in \, $TY'-YT=Y$.
(ii) Similarly, one can define $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-={\mathbb C}[q^{\pm 1/4}]\langle T,\pi_-, Y^{\pm 1}\rangle$ subject to $T(T+1)=0$\, and \begin{align}\label{nildahay} &\pi_-^2=1,\ \pi_- Y\pi_-=q^{-1/2}Y^{-1},\\ &TY-Y^{-1}T=-Y\, \Rightarrow\, YT-TY^{-1}=-Y.\notag \end{align} Setting $X\stackrel{\,\mathbf{def}}{= \kern-3pt =}\pi_- T',\, X'\stackrel{\,\mathbf{def}}{= \kern-3pt =} T\pi_-,\, T'=T+1,$\ one has $$ TX-X'T=X',\ T'X'=0=XT\, \Rightarrow\, TX'-XT=-X'. $$
(iii) The algebra $\,\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-\,$ is the image of the algebra $\,\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_+\,$ under the anti-isomorphism $$ \varphi: T\mapsto T,\, \pi_+\mapsto\pi_-,\, X\mapsto Y^{-1}. $$ Correspondingly, $\varphi: Y\mapsto X', Y'\mapsto X$. There is also an isomorphism $\sigma:$ $\,\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_+\,\to\,\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-\,$ sending \begin{align*} &\sigma: T\mapsto T,\ \,X\mapsto Y,\ \,\pi_+\mapsto \pi_-,\\ &\sigma:\, Y\mapsto \pi_-T,\ \, Y'\mapsto T'\pi_-.
\end{align*}
(iv) The automorphism $\tau_+$ fixing $T,X$ and sending $Y\mapsto q^{-1/4}XY$ acts in $\,\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_+\,$. Correspondingly, $\tau_-\stackrel{\,\mathbf{def}}{= \kern-3pt =}\varphi\tau_+\varphi^{-1}$ acts in $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-$ preserving $T,Y$ and sending \, $X\mapsto q^{1/4}YX$. One has the relations \begin{align}\label{sitausi} &\sigma\tau_+\ =\ \tau_-^{-1}\sigma, \ \, \sigma\tau_+^{-1}\ =\ \tau_-\sigma, \end{align} matching the identity from (\ref{tautautau}) in the generic case. \phantom{1}
$\qed$ \end{theorem} Both algebras $\,\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_{\pm}\,$ satisfy the PBW Theorem, so $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ is their {\em flat} deformation. The formulas above give an explicit description of the {\em bar-polynomial\,} representation of $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_+$ in $\mathscr{X}= {\mathbb C}_{q}[X^{\pm 1}]$; recall that $T,\pi_+, X^{\pm 1},Y,Y'$ are mapped to the operators $\overline{T},\pi, X^{\pm 1}, \overline{Y},\overline{Y}\,'$. It holds even if $q$ is a root of unity, including the construction of the $q$\~Hermite polynomials (use the intertwiners).
A surprising fact is that the construction of nonsymmetric Whittaker functions naturally leads to a module over $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-\,$ that is similar to $\mathscr{X}$ as a vector space but has a very different module structure. We will call it later the {\em hat-polynomial\,} representation; this will require using the {\em spinors}, to be discussed next.
{\bf Comment.\ } Let us mention the relation of nil-DAHA $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_+\,$ to the $T$\~equivariant $K_T(\mathcal{B})$ for affine flag varieties $\mathcal{B}$ from \cite{KK} and the Demazure-type operators on this (commutative) ring considered in this paper. Here $T$ is the maximal torus in the Lie group $G$ constructed by the root system $R$.
The exact $K$\~theoretic interpretation of DAHA was obtained in \cite{GG} (see also \cite{GKV}). Namely, $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ is essentially $K^{T\times {\mathbb C}^*}(\Lambda)$ for a certain canonical Lagrangian subspace $\Lambda\subset \mathcal{T}^*(\mathcal{B}\times\mathcal{B})$, that is the Grothendieck group of the (derived) category of $T\times {\mathbb C}^*$\~equivariant coherent sheaves on $\Lambda$.
This interpretation is for arbitrary $q,t$. Switching from $\mathcal{B}$ in \cite{KK} to $\Lambda\subset \mathcal{T}^*(\mathcal{B}\times\mathcal{B})$ is important because it gives the definition of convolution and, therefore, supplies $K^{T\times C^*}(\Lambda)$ with a structure of algebra (isomorphic to $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$). We note that the Gaussians were added to the definition of DAHA in \cite{GG}. We prefer not to consider the Gaussians as part of the definition of DAHA, treating them as outer automorphisms of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}\,$, as in the classical theory of Heisenberg-Weyl algebras and metaplectic representations.
\subsection{{\bf Nonsymmetric Q-Toda theory}} The problem of finding Dunkl operators for the {\em $q$\~Toda operator} from (\ref{qToda}) seems not well defined since the Toda operators are not symmetric. Nevertheless, it has a solution ( below). It provides a spinor variant of the representation $\mathcal{L}=Y+Y^{-1}$ (upon the restriction to the symmetric functions) for $\mathcal{L}$ from (\ref{Lopertaorx}).
The spinor-Dunkl operators make it possible to use DAHA methods at their full potential algebraically and in the theory of the $q$\~Whittaker functions. The construction can be extended to arbitrary root systems (in progress). We will begin with the introduction of the spinors.
\subsubsection{\sf The spinors}\label{sect:spinors} Generally, the $W$\~spinors are needed in DAHA theory as discussed in the introduction. In the $A_1$\~case, we will call them simply {\em spinors}. In this case, they are really connected with spinors from the theory of the Dirac operator (and with super-algebras). Under the rational degeneration, the Dunkl operator for $A_1$ becomes the square root of the (radial part of the) Laplace operator, i.e., the {\em Dirac operator}. However, this relation (and using super-variables) is a special feature of the root system $A_1$.
For practical calculations with spinors, the language of ${\mathbb Z}_2$\~graded algebras can be used in the $A_1$\~case (see the differential theory below). However, we prefer to proceed here in a way that does not rely on the special symmetry of the $A_1$\~case and can be transferred to $W$\~spinors for arbitrary root systems.
The {\em spinors\,} are simply pairs $\{f_1,f_2\}$ of elements (functions) from a space $\mathcal{F}$ with an action of $s$; the addition or multiplication (if applicable) of spinors is componentwise. The space of spinors will be denoted by $\widehat{\mathcal{F}}$.
The involution $s$ on spinors is defined as follows $s\{f_1,f_2\}=\{f_2,f_1\}$, so this does not involve the action of $s$ in $\mathcal{F}$. There is a ``natural" embedding $\rho:\mathcal{F}\to \widehat{\mathcal{F}}$ mapping $f\mapsto f^\rho=\{f,s(f)\}$ and the diagonal embedding $\delta: \mathcal{F}\to \widehat{\mathcal{F}}$ sending $f\mapsto f^\delta=\{f,f\}$. Accordingly, for an arbitrary operator $A$ acting in $\mathcal{F}$, $A^\rho=\{A,s(A)\}, A^\delta=\{A,A\}$. The images $f^\rho$ of $f\in\mathcal{F}$ are called {\em functions} (in contrast to {\em spinors}) or {\em principle spinors} (like for adeles).
For instance, for $\mathcal{F}=\mathscr{X}$, \begin{align*} &X^\rho:\{f_1,f_2\}\mapsto\{Xf_1,X^{-1}f_2\},\ &&\Gamma^\rho:\{f_1,f_2\}\mapsto\{\Gamma(f_1), \Gamma^{-1}(f_2)\},\\ &X^\delta:\{f_1,f_2\}\mapsto\{Xf_1,Xf_2\},\ &&\Gamma^\delta:\{f_1,f_2\}\mapsto\{\Gamma(f_1),\Gamma(f_2)\}, \end{align*} where, recall, $\Gamma(X)=q^{1/2}X$. We simply put $$ X^\rho=\{X,X^{-1}\},\ \Gamma^\rho=\{\Gamma,\Gamma^{-1}\},\ X^\delta=\{X,X\},\ \Gamma^\delta=\{\Gamma,\Gamma\}. $$ Obviously, $s^\rho=s=s^\delta$.
If a function $f\in \mathcal{F}$ or an operator $A$ acting in $\mathcal{F}$ have no super-index $\delta$, then they will be treated as $f^\rho, A^\rho$. I.e., by default, functions and operators are embedded into $\widehat{\mathcal{F}}$ and the algebra of spinor operators using $\rho$.
If the operator $A$ is explicitly expressed as $\{A_1,A_2\}$, then $A_1$ and $A_2$ must be applied to the corresponding components of $f=\{f_1,f_2\}$. In the calculations below, $A_i$ will be allowed to contain $s$ placed on the right, i.e., in the form $A_i=A_i'\cdot s$, where $A_i'$ contains no $s$\,. The latter can be always achieved by using the commutation relations between $s$ and $X, \Gamma$. Then the component $i$ of $Af$ will be (by definition) $A_i'(f_{3-i})$. I.e., $s$ placed on the right {\em inside a spinor component of the operator} will mean the switch to the other component ($i\mapsto 3-i$) before applying the rest of the operator, which is $A_i'$.
For instance, $\{\Gamma s,\,s-1\}(\{f_1,f_2\})=$ $\{\Gamma(f_2), f_1-f_2\}$.
We will frequently use the vertical mode for spinors: $$ \{f_1,f_2\}= \left\{\begin{array}{c}f_1 \\mathcal{F}_2\end{array}\right\},\ \{A_1,A_2\}= \left\{\begin{array}{c}A_1 \\A_2\end{array}\right\}. $$
\comment{ \begin{align*} &X^\rho:\left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\} \mapsto\left\{\begin{array}{c}Xf_1 \\X^{-1}f_2 \end{array}\right\} &&\Gamma^\rho:\left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\} \mapsto\left\{\begin{array}{c}\Gamma(f_1) \\\Gamma^{-1}(f_2) \end{array}\right\} \\ &X^\delta:\left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\} \mapsto\left\{\begin{array}{c}Xf_1 \\Xf_2 \end{array}\right\},\ &&\Gamma^\delta:\left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\} \mapsto\left\{\begin{array}{c}\Gamma(f_{1}) \\ \Gamma(f_{2})\end{array}\right\}, \end{align*} }
\subsubsection{\sf Q-Toda via DAHA} The $q$\~Toda {\em spinor\,} operator is the following {\em symmetric\,} (i.e., $s$\~invariant) difference {\em spinor\,} operator \begin{align}\label{Lspin} \widehat{\mathcal{L}}\ =\ \{\Gamma^{-1}+(1-X^{-2})\Gamma,\,\Gamma^{-1}+(1-X^{-2})\Gamma\}. \end{align} Its first component is the operator $R\!E(\mathcal{L})$ from Section \ref{sect:whitlim}; we will use the notation and definitions from this section.
We claim that $\widehat{\mathcal{L}}$ can be represented in the form $\widehat{Y}+\widehat{Y}^{-1}$ upon the restriction to {\em symmetric spinors}, i.e., to $\{f,f\}\in \widehat{\mathcal{F}}$. The construction of the {\em spinor-difference Dunkl operator} $\widehat{Y}$ goes as follows.
Let us introduce the following map on the operators in terms of $X,\Gamma$ and $s$ with values in spinor operators: \begin{align}\label{aede} &\hbox{\ae}^\delta:\, X\mapsto \widetilde{t}^{\ -1/2}X,\, \Gamma \mapsto \widetilde{t}^{\ -1/2}\Gamma,\, s\mapsto s \end{align} for the {\em spinor constant\,} $\widetilde{t}^{\ 1/2}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\{t^{1/2},t^{-1/2}\}$. Spinor constants are constant diagonal matrices; they commute with $\Gamma$ and $X$ but not with $s$ unless they are scalar. The {\em spinor $R\!E$-construction\,} is $$ R\!E^\delta:\ A\mapsto \lim_{t\to 0}\, \hbox{\ae}^\delta(A). $$ It is of course very different from the procedure $R\!E^\rho$ from Section \ref{sect:whitlim}. The spinor-Dunkl operators are $\widehat{Y}=R\!E^\delta(Y),\, \widehat{Y}\,'= R\!E^\delta(Y^{-1})$. They are inverse to each other: $\widehat{Y}\widehat{Y}\,'=1$.
\begin{theorem}\label{MainToda} The map \begin{align*} &Y^{\pm 1}\mapsto \widehat{Y}^{\pm 1},\ \pi_-\mapsto R\!E^\delta(XT),\\ &T\mapsto \widehat{T}=R\!E^\delta(t^{1/2}T),\ T'\mapsto \widehat{T}'=R\!E^\delta(t^{1/2}T^{-1})\, \end{align*} can be extended to a representation of the algebra $\,\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-\,$ in the space $\,\widehat{\mathscr{X}}\,$ of spinors over $\mathscr{X}={\mathbb C}[q^{\pm 1/4}][X^{\pm 1}]$. Correspondingly, \begin{align*} &X\mapsto R\!E^\delta(t^{1/2}X)=R\!E^\delta(\pi_-)\circ \widehat{T}',\\ &X'\mapsto R\!E^\delta(t^{1/2}X^{-1})=\widehat{T} \circ R\!E^\delta(\pi_-). \end{align*} The commutativity of\, $T$ and\, $Y+Y^{-1}$\, in $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-$ results in the $s$\~invariance of $\ \widehat{Y}+\widehat{Y}^{-1}\,$ and the $s$\~invariance of this operator upon its restriction to the space of $s$\~invariant spinors, which is the one from (\ref{Lspin}).\phantom{1}
$\qed$ \end{theorem}
It is clear from the construction that all hat-operators preserve the space of spinors for Laurent polynomials in terms of $X^{\pm 1}$. We will give below explicit formulas. Upon multiplication by the Gaussian, this $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-$\~module contains an irreducible submodule, the {\em spinor polynomial representation}, isomorphic to the Fourier image of the bar-polynomial representation times the Gaussian; see Section \ref{sect:QHermite}, formula (\ref{sitausi}) and Theorem \ref{SPIN-polyn} below. The reproducing kernel of the isomorphism between these two modules inducing $\sigma:\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_+\to$ $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-$ at the operator level is given by the {\em nonsymmetric $q$\~Whittaker function}; its existence was conjectured in \cite{ChW}.
\subsubsection{\sf Spinor-Dunkl operators} Let us calculate explicitly the operator $\widehat{Y}=R\!E^{\delta}(Y)= \lim_{t\to 0}\hbox{\ae}^\delta(Y)$.
Recall that $s$ placed on the right {\em inside a spinor component of the operator} always mean the switch to the other component before applying the rest of the operator in this component. \comment{ For instance (see below), \begin{align*} &\left\{\begin{array}{c} \Gamma^{-1}\cdot(1-s) \\ \Gamma-\Gamma\cdot X^{-2}\cdot (1-s)\end{array}\right\}\{f_1,f_2\} \\ &\ =\ \{\Gamma^{-1}(f_1-f_2), \Gamma(f_2)-\Gamma(X^{-2}(f_2-f_1))\}. \end{align*} }
Using formulas (\ref{aede}): \begin{eqnarray*} \hbox{\ae}^\delta(Y) &=&s\cdot(\widetilde{t}^{\ -1/2}\Gamma)\cdot \left(t^{1/2}s+\frac{t^{1/2}-t^{-1/2}} {\widetilde{t}^{-1}X^{2}-1}\cdot (s-1)\right)\\ &=&t^{1/2}\widetilde{t}^{\ 1/2}\Gamma^{-1} +\widetilde{t}^{\ 1/2}\Gamma^{-1}\cdot\frac{t^{1/2}-t^{-1/2}} {\widetilde{t}X^{-2}-1}\cdot(1-s)\\ &=&\left\{\begin{array}{c} t\Gamma^{-1}+\Gamma^{-1}\frac{t-1}{tX^{-2}-1}\cdot(1-s) \\ \Gamma+\Gamma\frac{1-t^{-1}}{t^{-1}X^{2}-1}\cdot(1-s) \end{array}\right\}\\ \xrightarrow{t\to 0}\ \widehat{Y}&=& \left\{\begin{array}{c} \Gamma^{-1}\cdot(1-s) \\ \Gamma-\Gamma\cdot X^{-2}\cdot (1-s)\end{array}\right\}. \end{eqnarray*} Recall that $\widetilde{t}^{\ 1/2}= \{t^{1/2},t^{-1/2}\}$. A little bit more involved calculation is needed for $\widehat{Y}\,'=R\!E^{\delta}(Y^{-1})$:
\begin{eqnarray*} \hbox{\ae}^\delta(Y^{-1}) &=&\left(t^{-1/2}s+\frac{t^{-1/2}-t^{1/2}}{\widetilde{t}X^{-2}-1} \cdot (s-1)\right)\cdot (\widetilde{t}^{\ 1/2}\Gamma^{-1}s)\\ &=&\left(\frac{t^{\ -1/2}\widetilde{t}X^{-2}-t^{1/2}} {\widetilde{t}X^{-2}-1}\cdot s -\frac{t^{\ -1/2}-t^{1/2}}{\widetilde{t}X^{-2}-1}\right) \cdot(\widetilde{t}^{\ 1/2}\Gamma^{-1}s)\\ &=&\frac{t^{\ -1/2}\widetilde{t}X^{-2}-t^{1/2}} {\widetilde{t}X^{-2}-1}\widetilde{t}^{\ -1/2}\Gamma -\frac{t^{-1/2}-t^{1/2}}{\widetilde{t}X^{-2}-1} \widetilde{t}^{\ 1/2}\Gamma^{-1}s\\ &=&\left\{\begin{array}{c} \frac{X^{-2}-1}{tX^{-2}-1}\Gamma-\frac{1-t}{tX^{-2}-1} \Gamma^{-1}s\\ \frac{t^{-1}X^{2}-t}{t^{-1}X^{2}-1} \Gamma^{-1}-\frac{t^{-1}-1}{t^{\,-1}X^{2}-1}\Gamma s \end{array}\right\}\xrightarrow{t\to 0}\\ \widehat{Y}\,'&=& \left\{\begin{array}{c} (1-X^{-2})\Gamma+\Gamma^{-1}s \\ \Gamma^{-1}-\frac{1}{X^{2}}\Gamma s\end{array}\right\}\\ &=& \left\{\begin{array}{c}1-X^{-2} \\1\end{array}\right\} \Gamma+\left\{\begin{array}{c}1 \\-X^{2}\end{array}\right\} \Gamma^{-1}s. \end{eqnarray*} Automatically, $\widehat{Y}\widehat{Y}\,'=1$, since these operators were obtained by the $R\!E^{\delta}$\~construction. Now, as we claimed, \begin{eqnarray*} &&R\!E^{\delta}(Y+Y^{-1})=\lim_{t\to 0}\hbox{\ae}^\delta(Y+Y^{-1})\\ &=&\left\{\begin{array}{c}\Gamma^{-1}(1-s)+(1-X^{-2}) \Gamma+\Gamma^{-1}s
\\\Gamma-\Gamma\frac{1}{X^{2}}(1-s)+\Gamma^{-1}
-\frac{1}{X^{2}}\Gamma s\end{array}\right\}\\ &=&\left\{\begin{array}{c}\Gamma^{-1}+(1-X^{-2})\Gamma \\ \Gamma^{-1}+(1-X^{-2})\Gamma\end{array}\right\} \bigl(\text{ mod }\bigl(\cdot\bigr)(s-1)\bigr). \end{eqnarray*}
For $X$ and $X^{-1}$, we have \begin{align}{\label{eqn:hatX}} \widehat{X}\!=\!R\!E^{\delta}(t^{1/2}X) =\lim_{t\to 0}\hbox{\ae}^\delta(t^{1/2}X) =\lim_{t\to 0}t^{1/2}\widetilde{t}^{\ -1/2}X =\left\{\begin{array}{c}X \\mathbf{0}\end{array}\right\},\\ \widehat{X}\,'\!=\!R\!E^{\delta}(t^{1/2}X^{-1}) =\lim_{t\to 0}\hbox{\ae}^\delta(t^{1/2}X^{-1}) =\lim_{t\to 0}t^{1/2}\widetilde{t}^{\ -1/2}X^{-1} =\left\{\begin{array}{c}0 \\X\end{array}\right\}.\notag \end{align} Obviously, $\widehat{X}\widehat{X}\,'=0$. Next, \begin{align*} &\widehat{T}=R\!E^{\delta}(t^{1/2}T)= \lim_{t\to 0}\hbox{\ae}^\delta(t^{1/2}T) =\left\{\begin{array}{c}0 \\mathcal{S}-1\end{array}\right\},\\ &\widehat{T}\,'=R\!E^{\delta}(t^{1/2}T^{-1})=\lim_{t\to 0} \hbox{\ae}^\delta(t^{1/2}T^{-1}) =\left\{\begin{array}{c} 1 \\ s \end{array}\right\}. \end{align*}
It is instructional to check the following relations using the explicit formulas we obtained (they of course follow from Theorem \ref{MainToda}): \begin{align} &\widehat{T}'=\widehat{T}+1,\ \widehat{T}\widehat{T}\,'\,=\,0\,=\,\widehat{T}\,'\widehat{T},\ \, \widehat{T}\,'\widehat{X}\,'=0=\widehat{X}\widehat{T}, {\label{eqn:TY1}}\\ &\widehat{T}\widehat{Y}-\widehat{Y}^{-1}\widehat{T}\,=\,-\widehat{Y}, \ \widehat{T}\widehat{Y}^{-1}-\widehat{Y}\widehat{T}\,=\,\widehat{Y}, {\label{eqn:TY2}}\\ &\widehat{T}\widehat{X}-\widehat{X}\,'\widehat{T}=\widehat{X}\,',\ \widehat{T}\widehat{X}\,'-\widehat{X}\widehat{T}=-\widehat{X}\,', \ \widehat{X}+\widehat{X'}=X^{\delta}.{\label{eqn:TX}} \end{align} Relations \eqref{eqn:TY2} imply that \begin{align}\label{hatTYcom} \widehat{T}(\widehat{Y}+\widehat{Y}^{-1})=(\widehat{Y}+\widehat{Y}^{-1})\widehat{T}. \end{align} It proves that the spinor operator $\widehat{Y}+\widehat{Y}^{-1}$ is symmetric (recall that $\widehat{Y}\,'=\widehat{Y}^{-1}$). Indeed, applying (\ref{hatTYcom}) to a symmetric spinor $\{f,f\}$, let $(\widehat{Y}+\widehat{Y}^{-1})(\{f,f\})=\{g_1,g_2\}$. Then $\widehat{T}(\{g_1,g_2\})=0$, which is possible if and only if $g_1=g_2$.
\subsubsection{\sf Using the components} Explicitly, the action of $\widehat{Y}$ and $\widehat{Y}\,'$ on the spinors is as follows: \begin{align*} &\widehat{Y}(\left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\}) =\left\{\begin{array}{c}\Gamma^{-1}(f_{1}-f_{2}) \\ \Gamma(f_{2})-\Gamma(\frac{f_{2}-f_{1}}{X^{2}}) \end{array}\right\}, \\ &\widehat{Y}\,'(\left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\}) =\left\{\begin{array}{c}(1-X^{-2}) \Gamma(f_{1})+\Gamma^{-1}(f_{2}) \\ \Gamma^{-1}(f_{2})-\frac{1}{X^{2}}\Gamma(f_{1}) \end{array}\right\}. \end{align*} It is simple but not immediate to check the relation $\widehat{Y}\widehat{Y}\,'=$\,id and other identities for $\widehat{Y}^{\pm1}$ using the component formulas. The explicit formulas for $\widehat{T}$ and $\widehat{T}\,'$ are: \begin{align}\label{Tspinors} &\widehat{T}(\left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\}) =\left\{\begin{array}{c}0\\mathcal{F}_{1}-f_{2} \end{array}\right\}, &\widehat{T}\,'(\left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\}) =\left\{\begin{array}{c}f_{1}\\ f_{1}\end{array}\right\}. \end{align} It readily gives \eqref{eqn:TY1}, \eqref{eqn:TY2}.
Generally, there is no need to establish and check the formulas for $\widehat{X}$ and $\widehat{X}\,'$ (although they are simple). From Theorem \ref{MainToda}, \begin{align*} &\widehat{X}\ =\ R\!E^\delta(\pi_-)\cdot \widehat{T}\,',\ \widehat{X}\,'\ =\ \widehat{T} \cdot R\!E^\delta(\pi_-). \end{align*} Thus we need only to know $\widehat{\pi}\stackrel{\,\mathbf{def}}{= \kern-3pt =} R\!E^\delta(\pi_-)$, where $\pi_-=XT$. We have \begin{eqnarray*} \hbox{\ae}^\delta(XT) &=&(\widetilde{t}^{-1/2}X)(t^{1/2}s +\frac{t^{1/2}-t^{-1/2}}{\widetilde{t}^{-1}X^{2}-1}(s-1))\\ &=&\widetilde{t}^{-1/2}t^{1/2}Xs +\frac{X(\widetilde{t}^{-1/2}t^{1/2}-\widetilde{t}^{-1/2}t^{-1/2})} {\widetilde{t}^{-1}X^{2}-1}(s-1)\\ &=&\left\{\begin{array}{c}Xs \\tX^{-1}s\end{array}\right\} +\left\{\begin{array}{c}\frac{X(1-t^{-1})} {t^{-1}X^{2}-1}(s-1) \\ \frac{X^{-1}(t-1)}{tX^{-2}-1}(s-1)\end{array}\right\}. \end{eqnarray*} Taking the limit $t\to 0$, \begin{align*} \widehat{\pi}=\left\{\begin{array}{c}Xs \\mathbf{0}\end{array}\right\} +\left\{\begin{array}{c}-X^{-1}(s-1) \\X^{-1}(s-1) \end{array}\right\} =\left\{\begin{array}{c}Xs-X^{-1}(s-1) \\X^{-1}(s-1) \end{array}\right\}. \end{align*} Using the components, \begin{align}\label{picompo} \widehat{\pi}:\ \left\{\begin{array}{c}f_{1} \\mathcal{F}_{2} \end{array}\right\}\mapsto \left\{\begin{array}{c}Xf_{2}+\frac{f_{1}-f_{2}}{X} \\ \frac{f_{1}-f_{2}}{X}\end{array}\right\}. \end{align} Check directly that $\widehat{\pi}^{2}=id$.
This formula completes the ``component presentation" of the {\em hat-module} of $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-$ from Theorem \ref{MainToda}: $$ T, \pi_-, Y\ \mapsto\
\widehat{T}, \widehat{\pi}, \widehat{Y}. $$
The extension of this Theorem to arbitrary (reduced) root systems is straightforward as well as the justification; we will address this (and the applications) in further paper(s). The formulas for the $\overline{Y}$\~operators are of course getting more involved. The justifications in the spinor $q$\~Toda theory (including global Whittaker functions) are entirely based on DAHA theory. We calculate and check practically everything explicitly in this work mainly to demonstrate the practical aspects of the technique of spinors (and because of novelty of this topic).
\subsubsection{\sf Spinor Whittaker function} Let us apply the procedure $R\!E^\delta$ to the {\em global difference spherical function} $\mathcal{E}_q(x,\lambda)$ from \cite{C5}, Section 5 (upon the specialization to the case of $A_1$). We do not give here its exact definition and do not discuss the details of the procedure. Actually, the only point that requires comments is using the conjugated $E$\~polynomials, $E_b^*$ in the formula for $\mathcal{E}_q$ in \cite{C5}. Generally, the relation of $\{E_b\}$ and its conjugates is via the action of $T_{w_0}$; compare with Theorem \ref{tildee} in the case of $A_1$.
We arrive at the following spinor nonsymmetric generalization of the function $\mathcal{W}_q$ from (\ref{Whitsym}) above: \begin{align} \Omega(X,\Lambda)= q^{x^2}q^{\lambda^2}\,\Bigl(1+ \sum_{m=1}^{\infty} q^{m^{2}/4}\, \bigl(&\frac{\overline{E}_{-m}(\Lambda)} {\prod_{s=1}^{m} (1-q^{s})} \left\{\begin{array}{c}X^m \\ q^mX^m\end{array}\right\}\notag\\ +\,&\frac{\overline{E}_{m}(\Lambda)} {\prod_{s=1}^{m-1} (1-q^{s})} \left\{\begin{array}{c} 0 \\ X^m\end{array}\right\} \bigr)\Bigr).\label{spinwhito} \end{align} Using the Pieri rules from (\ref{pienilp-}), we can present it as follows: \begin{align}\label{spinwhit} \Omega= q^{x^2}q^{\lambda^2}\, \sum_{m=0}^{\infty} \frac{q^{m^{2}/4}} {\prod_{s=1}^{m} (1-q^{s})} \left\{\begin{array}{c}X^m \overline{E}_{-m}(\Lambda) \\ X^m \Lambda^{-1}\overline{E}_{m+1}(\Lambda)\end{array}\right\}. \end{align} Either of these two presentations readily gives that the spinor- symmetrization of $\Omega$ is $\{\mathcal{W},\,\mathcal{W}\}$. We need to apply the symmetrizer $\mathscr{P}'=T'=T+1$ to $\Omega$, equivalently, duplicate its first component; see (\ref{Tspinors}). Note that $\Lambda$ is a (nonspinor) variable.
The spinor $\Omega$ intertwines the bar-representation of $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_+$ and the hat-representation of $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-$. Namely, \begin{align} &\widehat{Y}(\Omega)=\Lambda^{-1}(\Omega),\ \widehat{X}(\Omega)=\overline{Y}\,'_\Lambda (\Omega),\ \widehat{X}\,'(\Omega)=\overline{Y}_\Lambda (\Omega),\notag\\ &\widehat{\pi}(\Omega)\ =\ \pi_\Lambda(\Omega),\ \, \widehat{T}(\Omega)\ =\ \overline{T}_\Lambda(\Omega), \label{Omegainter} \end{align} where $\overline{Y}\,'_\Lambda,\, \overline{Y}_\Lambda,\,\pi_\Lambda,\, \overline{T}_\Lambda$\, act on the argument $\Lambda$; the other operators are $X$\~operators. These (and other related identities) follow from the general theory for any reduced root systems (at least in the twisted case). However, in the rank-one case (and for $A_n$), one can use the Pieri rules from (\ref{pienilp+}),(\ref{pienilp-}) and formulas (\ref{Ynilp+}), (\ref{Ynilp-}) for the direct verification.
Let us calculate $\overline{Y}\,'_\Lambda(\Omega)$. First, $\overline{Y}\,'_\Lambda(\overline{E}_n(\Lambda))=0$ for $n>0$. Second, $q^{\,-\lambda^2}\,\overline{Y}\,'_\Lambda\,q^{\lambda^2}$ $=q^{-1/4}\,\overline{Y}\,'_\Lambda\cdot\Lambda$. For instance, $$ \overline{Y}\,'_\Lambda(q^{\lambda^2})=q^{-1/4} \overline{Y}\,'_\Lambda(\Lambda)\,q^{\lambda^2}= \overline{Y}\,'_\Lambda(\overline{E}_1(\Lambda))\,q^{\lambda^2}=0. $$ We see that the second spinor component of $\overline{Y}\,'_\Lambda(\Omega)$ vanishes, as it is supposed to be because the second component of $\widehat{X}(\Omega)$ is obviously zero.
The first component reads as follows: \begin{align*} \overline{Y}\,'_\Lambda(\Omega)&= q^{x^2}q^{\lambda^2}\, \sum_{m=0}^{\infty} \frac{q^{m^{2}/4-1/4}\,X^m\, \overline{Y}\,'_\Lambda(\Lambda \overline{E}_{-m}) }{\prod_{s=1}^{m} (1-q^{s})}\\ &= q^{x^2}q^{\lambda^2}\, \sum_{m=0}^{\infty} \frac{q^{m^{2}/4-1/4-m/2+1/2}\,X^m\, (1-q^m)\overline{E}_{1-m} }{\prod_{s=1}^{m} (1-q^{s})}\\ &=q^{x^2}q^{\lambda^2}\, X\,\sum_{m=1}^{\infty} \frac{q^{(m-1)^{2}/4}\,X^{m-1}\, \overline{E}_{1-m}) }{\prod_{s=1}^{m-1} (1-q^{s})},\\ \end{align*} which coincides with the first component of $\widehat{X}(\Omega)$ (its second component is zero). We have used here the nil-Pieri formula: $$\Lambda\overline{E}_{-n}= (1-q^n)\overline{E}_{1-n}+\overline{E}_{n+1} \ \hbox{ for } \ n>0; $$ the second term, $\overline{E}_{n+1}$, does not contribute to the final formula, since $\overline{Y}\,'(\overline{E}_{n+1})=0$.
The (key) relation $\widehat{Y}(\Omega)=\Lambda^{-1}(\Omega)$ can be verified directly in a similar manner.
First, $q^{-x^2}\,\widehat{Y}\,q^{x^2}=q^{1/4}X^{-1}\widehat{Y}$. Therefore \begin{align}\label{hatYga} q^{-x^2}\,\widehat{Y}\,q^{x^2}\left\{\begin{array}{c}f_1\\ f_2\end{array}\right\}=q^{1/4} \left\{\begin{array}{c} X^{-1}\Gamma^{-1}(f_1-f_2) \\ X\Gamma(f_2)+q^{-1}X^{-1}\Gamma(f_1-f_2)\end{array}\right\}. \end{align} Second, $\mathcal{F}_m\stackrel{\,\mathbf{def}}{= \kern-3pt =} \overline{E}_{-m}(\Lambda)- \Lambda^{-1}\overline{E}_{m+1}(\Lambda)= (1-q^m)\Lambda^{-1}E_{1-m}(\Lambda)$ (the Pieri rules). Now,\ \, $\Lambda^{-1}q^{-x^2}q^{-\lambda^2}\,\widehat{Y}(\Omega)=$ \begin{align*} &\Lambda^{-1}\sum_{m=0}^{\infty} \frac{q^{m^{2}/4+1/4}} {\prod_{s=1}^{m} (1-q^{s})} \left\{\begin{array}{c} q^{-\frac{m}{2}}X^{m-1}\mathcal{F}_m \\ q^{\frac{m}{2}}X^{m+1}\Lambda^{-1}\overline{E}_{m+1}(\Lambda) +q^{\frac{m}{2}-1}X^{m-1}\mathcal{F}_m \end{array}\right\}\\ =&\sum_{m=0}^{\infty} \frac{q^{m^{2}/4+1/4}} {\prod_{s=1}^{m} (1-q^{s})} \left\{\begin{array}{c} q^{-\frac{m}{2}}(1-q^m)X^{m-1} \overline{E}_{1-m} \\ q^{\frac{m}{2}}X^{m+1}\overline{E}_{m+1} +q^{\frac{m}{2}-1}(1-q^m)X^{m-1}\overline{E}_{1-m} \end{array}\right\}. \end{align*} Collecting the terms with $(1-q^m)$, we obtain that \begin{align*} \widehat{Y}(\Omega)= &\Lambda^{-1}q^{x^2}q^{\lambda^2}\, \sum_{m=1}^{\infty} \frac{q^{(m-1)^{2}/4}} {\prod_{s=1}^{m-1} (1-q^{s})} \left\{\begin{array}{c} X^{m-1}\overline{E}_{1-m}(\Lambda) \\ q^{m-1}X^{m-1}\overline{E}_{1-m}(\Lambda) \end{array}\right\}\\ + &\Lambda^{-1}q^{x^2}q^{\lambda^2} \sum_{m=0}^{\infty} \frac{q^{(m+1)^2/4}} {\prod_{s=1}^{m} (1-q^{s})} \left\{\begin{array}{c} 0\\ X^{m+1}\overline{E}_{m+1}(\Lambda) \end{array}\right\}, \end{align*} i.e., exactly the presentation from (\ref{spinwhito}) multiplied by $\Lambda^{-1}$.
Formulas (\ref{hatYga}), (\ref{picompo}) and (\ref{Tspinors}) result in the definition of the {\em spinor-polynomial} representation: $$ \mathscr{X}_{spin}={\mathbb C}\oplus\ \bigl (\oplus_{m=1}^{\ \infty} ({\mathbb C}\{X^m,0\}\oplus {\mathbb C}\{0,X^m\})\bigr). $$ \begin{theorem}\label{SPIN-polyn} The space $\mathscr{X}_{spin}$ is an irreducible $\overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-$\~submodule of the space of spinors over ${\mathbb C}[X^{\pm1}]$ supplied with the twisted action: $$ \overline{\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}}_-\ni A\mapsto q^{-x^2}\,\widehat{A}\,q^{x^2}. $$ More explicitly, $\mathscr{X}_{spin}$ is invariant and irreducible under the action of operators $\widehat{T}, \widehat{\pi}$ and $q^{-x^2}\,\widehat{Y}\,q^{x^2}$. \phantom{1}
$\qed$ \end{theorem}
The general theory of spinor nonsymmetric Whittaker functions will be published elsewhere. Let us now consider the technique of spinors in the differential setting.
\setcounter{equation}{0} \section{\sc{Differential theory}} \subsection{{\bf The degenerate case}} \subsubsection{\sf Degenerate DAHA} Let us begin with the definition of {\em degenerate double affine Hecke algebra} for an arbitrary (reduced) root system $R$. Recall that $\widehat{W}=W\ltimes P^\vee$ for the coweight lattice $P^{\vee}$.
\begin{definition}\label{dDAHA} The {\sf degenerate double affine Hecke algebra} $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}'$ is generated by $\widehat{W}$ (with the corresponding group relations) and pairwise commutative elements $y_{b}$, $b\in P$ satisfying the following relations: \begin{align}\label{ddaha} &s_{i}y_{b}-y_{s_{i}(b)}s_{i}\,=\,-k(b, \alpha_{i}^\vee) \text{\ \,for\ \,}i\geq 1, \\ &s_{0}y_{b}-y_{s_{0}(b)}s_{i}\,=\,k(b, \theta) \text{\ and \ } \pi_{r}y_{b}=y_{\pi_{r}(b)}\pi_{r},\notag \end{align} where $y_{[b,j]}=y_b+j$, $y_{b+c}=y_b+y_c$. \end{definition} Note that in contrast to the definition of DAHA from (\ref{dahadef}), $y_b$ are labeled by $b\in P$ (not by $P^\vee$). It is convenient because $X_a$ (to be introduced later) will be naturally labeled by $a\in P^\vee$.
Due to the additive dependence of $y_b$ of $b$, the exact choice ($P$ or $P^\vee$) is not too important here; one can even take $b\in {\mathbb C}^n$. Similarly, changing $(b, \alpha_{i}^\vee)$ to $(b, \alpha_{i})$ will simply re-scale the $k$\~parameters. However, the exact choice of the lattice is important to ensure the compatibility of this definition with the limit $q\to 1$ from $q,t$\~DAHA (see below). The operators $X_a$ will be (translations by) $a\in P^\vee$ considered as elements of $\widehat{W}\subset \hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}'$. The PBW Theorem holds for $\{X_a,y_b,W\}$.
This algebra was introduced for the first time as the limit $q\to 1$ of $q,t$\~DAHA; see \cite{C101}, Chapter 2, Section ``Degenerate DAHA." There is another approach to its definition via the compatibility and $\widehat{W}$\~equivariance of the {\em affine infinite Knizhnik-Zamolodchikov equation} from \cite{C1,Ch12}. It can be called ``elliptic AKZ" (though no elliptic functions are used in its definition) because this system of equations at critical level is equivalent to the eigenvalue problem for the elliptic deformation of the Heckman-Opdam operators. The latter is due to Olshanetsky -Perelomov for $A_n$\,,\,Ochiai -Oshima -Sekiguchi for the classical root systems, and from \cite{Ch12} for any (reduced) root systems.
Let us consider the $A_{1}$\~case. Then $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}'$ will be generated by $s, \pi, y$ with the following defining relations: \begin{align*} s^2=1,\ sy+ys\,=\,-k,\ \pi y\,=\,(\frac{1}{2}-y)\pi. \end{align*} Recall that we set $s=s_1$,\, $\omega=\omega_1$,\, $\pi=\omega s$,\, $y=y_{\omega}$; for instance, $\pi(\omega)=[-\omega,\frac{1}{2}]$.
Letting $X=\pi s$, one has that $sXs\,=\,X^{-1}$, $(Xs)y\,=\,(\frac{1}{2}-y)(Xs)$ and finally \begin{align*} \ X(-k-ys)\,=\,(\frac{1}{2}-y)Xs \ \Rightarrow\ [y, X]\,=\,\frac{1}{2}X+kXs. \end{align*} Similar to DAHA, $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}'$ can be represented as $\langle y, s, X^{\pm1}\rangle$ subject to the relations: \begin{align}\label{trigA1} sXs\,=\,X^{-1},\ sy+ys\,=\,-k,\ s^{2}\,=\,1,\ [y, X]\,=\,\frac{1}{2}X+kXs. \end{align}
This algebra can be obtained as the limit (``degeneration") of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ from (\ref{dahaone}) as follows. We set $q=\exp(h)$, $t=q^{k}=\exp(hk)$. Let $Y=\exp(-hy)$,\, $X=X$\, and\, $T=s+\frac{hk}{2}$. Note that now $X$ comes from the multiplication operator (not from translations). The letter relation is necessary to ensure that the quadratic relation holds modulo $\,(h^2)\,$. Indeed, then \begin{align*} T^{2}\ =\ 1+hks\ =\ (t^{1/2}-t^{-1/2})T+1\mod (h^2). \end{align*} Check that the coefficient of $h$ in $TY^{-1}T=Y$ readily results in the relation $sys+ks=-y$.
\subsubsection{\sf Polynomial representation} Continuing with the $A_1$\~case, $X$ and $s$ remain the same as in the $q,t$\~case, however, now we set $X=e^{x}$. The generator $y$ is mapped to the differential operator \begin{align}\label{yformula} y=\frac{1}{2}\frac{d}{dx}+\frac{k}{1-X^{2}}(1-s)-\frac{k}{2}\,, \end{align} called the trigonometric Dunkl or Cherednik-Dunkl operator. It is simple to check directly that $sys+y=-ks$\, and that \begin{align*} [y,X]\,=\,\frac{1}{2}X+\frac{k}{1-X^{-2}}(Xs-X^{-1}s)\,=\, \frac{1}{2}X+kXs. \end{align*}
The constant $-k/2$ in formula (\ref{yformula}) automatically results from the limiting procedure. However, its appearance here can be clarified without any reference to DAHA or degenerate DAHA. \begin{lemma}\label{deltak} Let $\Delta_{k}\stackrel{\,\mathbf{def}}{= \kern-3pt =} (e^{x}-e^{-x})^{k}$. Then \begin{align*} \widetilde{y}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\Delta_{k}\,y\,\Delta_{k}^{-1}= \frac{1}{2}\frac{d}{dx}-\frac{k}{1-X^{-2}}s. \end{align*} \end{lemma} {\em Proof.} Indeed, we have \begin{eqnarray*} \Delta_{k}\,y\,\Delta_{k}^{-1} &=& \frac{1}{2}\frac{d}{dx}- \frac{k}{2}\,\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} +\frac{k}{1-X^{-2}}(1-s)-\frac{k}{2}\\ &=& \frac{1}{2}\frac{d}{dx}+ \frac{k}{2}\,\bigl(1-\frac{2e^{x}}{e^{x}-e^{-x}}\bigr) +\frac{k}{1-X^{-2}}(1-s)-\frac{k}{2}\\ &=&\frac{1}{2}\frac{d}{dx}-\frac{k}{1-X^{-2}}s. \end{eqnarray*}
\phantom{1}
$\qed$
Thus the constant $-k/2$\, is necessary to make the conjugation of the trigonometric Dunkl operator by $\Delta_k$ with pure $s$ (but then the Laurent polynomials will not be preserved). We mention that the trigonometric Dunkl operators were introduced in \cite{C13} in terms of $(c-s)$ for an arbitrary constant $c$ (including $c=0$) and in the matrix setting. We see that the constant $c$ can be changed using conjugations by powers of the discriminant.
{\bf Comment.\ } For complex $k$, we need to take the function
$|e^{x}-e^{-x}|^{k}$ in the lemma (to avoid problems with complex powers). However, the claim of the lemma is entirely algebraic. The best way to proceed here algebraically is to conjugate by the {\em even spinor} $$\{(e^{x}-e^{-x})^{k}, (e^{x}-e^{-x})^{k}\}$$ for any branch of $(e^{x}-e^{-x})^{k}$. It is the first appearance of spinors in this part of the paper. \phantom{1}
$\qed$
\subsubsection{\sf The self-adjointness} Let us first establish the connection of the trigonometric Dunkl operator to the $k$\~deformation of Harish- Chandra theory of the radial parts of Laplace operators on symmetric spaces. One has
$$L\,'\stackrel{\,\mathbf{def}}{= \kern-3pt =} 2y^{2}|_{\mathrm{sym}} =\frac{1}{2}\frac{d^{2}}{dx^{2}}+ k\frac{(1+e^{-2x})}{(1-e^{-2x})}\frac{d}{dx}+ \frac{k^{2}}{2}.$$
The restriction $\,|_{\mathrm{sym}}\,$ to symmetric (even) functions simply means that we move all $s$ to the right and then delete them.
In Harish-Chandra theory, $k$ is one-half of the {\em root multiplicity} of the restricted root system corresponding to the symmetric space. For instance, $k=1$ in the so-called group case. Let us mention the contributions of Koornwinder, Calogero, Sutherland, Heckman, Opdam and van den Ban to developing the theory for arbitrary $k$. See e.g, \cite{HO1} (we do not need anything beyond the results of this paper in this section).
Lemma \ref{deltak} readily gives that \begin{eqnarray*} \widetilde{L}\,'\stackrel{\,\mathbf{def}}{= \kern-3pt =}\Delta_{k}\,L\,'\,\Delta_{k}^{-1} \,=\,\frac{1}{2}\frac{d^{2}}{dx^{2}}+\frac{2k(1-k)} {(e^{x}-e^{-x})^{2}}. \end{eqnarray*}
Now let us discuss the inner product. We set formally: $$\langle f, g\rangle\,\stackrel{\,\mathbf{def}}{= \kern-3pt =}\,\int f(x)g(-x)\Delta_{k}^{2}dx. $$ For instance, the integration here can be taken over ${\mathbb R}$;
then $\Delta_{k}^{2}$ must be understood as $|e^x-e^{-x}|^{2k}$; the functions $f,g$ must be chosen to ensure the convergence.
The anti-involution ${}^+$ (formally) corresponding to the ``free" inner product \,$\int f(x)g(-x)dx$\, acts as follows: \begin{align*} x^{+}\,=\,x, \,\, (\frac{d}{dx})^{+}=\frac{d}{dx}. \end{align*} Then the anti-involution $A^{\Diamond}=\Delta_{k}^{-2}\,A^{+}\,\Delta_{k}^2$ serves $\langle f,g\rangle$.
\begin{lemma} One has \begin{align*} X^{\Diamond}\,=\,X^{-1},\ y^{\Diamond}\,=\,y,\ s^{\Diamond}\,=\,s, \end{align*} which implies that $(L\,')^{\Diamond}=L\,'$. \end{lemma} {\em Proof.} One can check the self-adjointness of $y$ and $L'$ directly. However, the best way is via Lemma \ref{deltak} (first, for $y$ and, second, for $L'$). Using that $\widetilde{y}^{+}=\widetilde{y}$, one obtains that \begin{eqnarray*} &y^{\Diamond}\, =\,\Delta_{k}^{-2}\,(\Delta_{k}^{-1}\,\widetilde{y}\, \Delta_{k}^{\,})^{\,+}\,\Delta_{k}^{2} \,=\,\Delta_{k}^{-2}\,(\Delta_{k}\,\widetilde{y}\, \Delta_{k}^{-1})\,\Delta_{k}^{2} \,=\,\Delta_{k}^{-1}\,\widetilde{y}\,\Delta_{k}=y.& \end{eqnarray*}
\subsubsection{\sf The Ruijsenaars limit} The procedure is as follows. We begin with $\widetilde{L}\,' =\frac{1}{2}\frac{d^{2}}{dx^{2}}+\frac{2k(1-k)} {(e^{x}-e^{-x})^{2}}$, replace $x$ by $x+M$ and connect $M$ with $k$ by the relation $k(1-k)=e^{2M}$. Finally, we set $\Re M\to+\infty$. Then the resulting operator will be $\frac{1}{2}\frac{d^{2}}{dx^{2}}+2e^{-x},$ the {\em Toda operator}.
Applying this method to arbitrary root systems, one obtains a system of pairwise commutative Toda operators. In contrast to $L\,'$, these operators are {\em not} $W$\~invariant. The (real) {\em Whittaker function} is their eigenfunction. Given a weight (the set of eigenvalues), the dimension of the corresponding space of all eigenfunctions is
$|W|$. The ``true" Whittaker function belongs to this space and can be fixed uniquely there using certain decay conditions.
Let us give a reference to paper \cite{Shim}, where this procedure was applied to the Heckman-Opdam functions from \cite{HO1}; their limits are, indeed, the {\em true} Whittaker ones.
Note that $k$ must be arbitrary in QMBP for the Ruijsenaars- Etingof procedure. It is impossible to obtain the Whittaker function directly from the classical Harish-Chandra spherical function (which is for very special $k$). It is somewhat different from $\mathfrak{p}$\~adic theory, where the passage from the Satake-Macdonald spherical function to the $\mathfrak{p}$\~adic Whittaker function can be established via switching to the maximal unramified extension from a given $\mathfrak{p}$\~adic field.
\subsection{{\bf Dunkl operator and Bessel function}} Let $X=e^{\varepsilon x}$ with $\varepsilon>0$. Then the trigonometric Dunkl operator $y$ becomes \begin{align*} \frac{1}{2\varepsilon} \frac{d}{dx}+\frac{k}{2\varepsilon x}(1-s)-\frac{k}{2}+ o(\varepsilon). \end{align*} Letting $\varepsilon\to 0$, \begin{align*} \varepsilon y\to \frac{1}{2}\frac{d}{dx}+ \frac{k}{2x}(1-s). \end{align*} We will use the same letter $y$ on the right-hand side. However, the {\em Dunkl operator}\, will be more convenient: $$ \mathscr{D}\stackrel{\,\mathbf{def}}{= \kern-3pt =} 2y=\frac{d}{dx}+\frac{k}{x}(1-s). $$
This definition is due to Charles Dunkl \cite{Du}, who introduce Dunkl (rational) operators for arbitrary root systems and also for some groups generated by complex reflections.
\subsubsection{\sf Rational DAHA} \begin{definition} The {\sf rational double affine Hecke algebra} $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}''$ is generated by $x, y, s$ with the following relations: \begin{align*} &sxs\,=\,-x, \,\, sys\,=\,-y, \,\, s^{2}\,=\,1, \,\, [y,x]=\frac{1}{2}+ks. \end{align*} \end{definition} It is the limit of the relations from (\ref{trigA1}). An abstract (and very general) variant of this definition is actually due to Drinfeld \cite{Dr} (though he did not consider its polynomial representation).
The assignment $x\to x$, $y\to \mathscr{D}/2$, $s\to s$ defines the {\em polynomial representation} of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}''$ in $\mathbb{C}[x]$. It is an induced module from the character of the subalgebra generated by $y,s$ sending $y$ to $y(1)=0$ and $s$ to $s(1)=1$. The PBW Theorem is almost immediate in the rational setting (it also follows from the existence of the polynomial representation).
Upon the symmetrization of $\mathscr{D}^2$, we obtain the key operator in the classical theory of Bessel functions: \begin{align*}
L\stackrel{\,\mathbf{def}}{= \kern-3pt =}\mathscr{D}^{2}|_{\mathrm{sym}} =\frac{d^{2}}{dx^{2}}+\frac{2k}{x}\frac{d}{dx}. \end{align*}
\begin{lemma}\label{LEMDIAM} (i) One has $$ x^{k}\cdot\mathscr{D}\cdot x^{-k}= \widetilde{\mathscr{D}}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\frac{d}{dx}-\frac{k}{x}s,\ \ x^{k}\cdot L\cdot x^{-k}=\widetilde{L}\stackrel{\,\mathbf{def}}{= \kern-3pt =}\frac{d^{2}} {dx^{2}}+\frac{k(1-k)}{x^{2}}. $$
(ii) Let $A^{\Diamond}=x^{-2k}\cdot A^{*}\cdot x^{2k}$, where the anti-involution $*$ is as follows: $$x^{*}\,=\,x, \quad (\frac{d}{dx})^{*}\,=\,-\frac{d}{dx}; $$ the anti-involution $\,\Diamond\,$ formally serves the bilinear symmetric form $\langle f, g\rangle$ $=\int f(x)g(x)x^{2k}dx\,.$ One has that\, $\mathscr{D}^{\Diamond}=-\mathscr{D}$\,, and\, $L^{\Diamond}=L$\,.\phantom{1}
$\qed$ \end{lemma}
\subsubsection{\sf Bessel functions} Assuming that $\lambda\neq 0$, an arbitrary solution $\varphi_{\lambda}^{(k)}$ of the eigenvalue problem \begin{align}\label{Lvarphi} L \varphi_{\lambda}^{(k)}=4\lambda^{2}\varphi_{\lambda}^{(k)} \end{align} analytic in a neighborhood of $x=0$ can be represented as $$\varphi_{\lambda}^{(k)}(x)=\varphi^{(k)}(x\lambda). $$ Here $\varphi^{(k)}$ can be readily calculated: \begin{eqnarray}\label{varphiform} \varphi^{(k)}(t)=\sum_{m=0}^{\infty} \frac{t^{2m}\Gamma(k+1/2)}{m!\Gamma(k+n+1/2)} \end{eqnarray} for the Gamma\~function, satisfying $\Gamma(x+1)\,=\,x\Gamma(x)$, $\Gamma(1)=1$. The parameter $k$ is arbitrary here provided that $k\neq -1/2-m$ for $m\in {\mathbb Z}_+$. The function $\varphi^{(k)}(t)$ is a variant of the Bessel $J$\~function.
See \cite{O3} (and references therein) for the theory of multi-dimensional Bessel functions.
Notice that \begin{eqnarray*} \varphi^{(k)}(t)\xrightarrow{k\to 0} \sum_{m=0}^{\infty}\frac{(2t)^{2m}}{(2m)!} =\frac{e^{2t}+e^{-2t}}{2}, \end{eqnarray*} due to the relations: \begin{align*} \Gamma(n+1)\Gamma(n+\frac{1}{2})\,=\,2^{-2n}(2n)!\sqrt{\pi}, \ \, \Gamma(\frac{1}{2})\,=\,\sqrt{\pi}. \end{align*}
Using the passage to the Sturm-Louiville operator $\widetilde{L}$, we can control the growth of $\varphi_{\lambda}^{(k)}$ at infinity.
\begin{lemma} The differential equation $L \varphi =4\lambda^{2}\varphi$ has the following two fundamental solutions for real $x$. If $\lambda=0$, then $1$ and $x^{1-2k}$ can be taken. If $\lambda\neq 0$, the asymptotic behavior can be used to fix them: \begin{align*} \varphi_{\lambda}^{\pm}= x^{-k}e^{\pm 2\lambda x}(1+o(1)) \hbox{\ as\ } x\to +\infty. \end{align*} Any solution $\varphi$ is a linear combination of these two. In particular, the growth of any solution as $x \to\pm\infty$ is no greater than exponential, namely, $O(\,x^{-\Re k} e^{\pm 2x\Re \lambda})$ for $\lambda\neq 0$. \phantom{1}
$\qed$ \end{lemma} We will use this lemma only for justifying that the Gauss-Bessel integrals we will need below are well defined. The following is the classical formula; see Introduction and Chapter 1 from \cite{C101} for a more comprehensive exposition.
\subsubsection{\sf Hankel transform} \begin{theorem}{\label{thm:master}} \begin{align*} \int^{+\infty}_{-\infty}\, \varphi_{\lambda}^{(k)}(x)\,
\varphi_{\mu}^{(k)}(x)\,e^{-x^{2}}|x|^{2k} dx=\Gamma(k+\frac{1}{2})\,\varphi_{\mu}^{(k)} (\lambda)\,e^{\lambda^{2}+\mu^{2}}, \end{align*} where $\Re k>-\frac{1}{2}$. The normalization is given by the Euler integral: \begin{align*}
\int^{+\infty}_{-\infty}e^{-x^{2}}|x|^{2k} dx=\Gamma(k+\frac{1}{2}). \end{align*} Here one can set $\int^{+\infty}_{-\infty}=2\int^{+\infty}_{0}$, since all functions are even. \phantom{1}
$\qed$ \end{theorem}
In order to prove Theorem \ref{thm:master}, we need the following definition. \begin{definition} The {\sf Hankel transform} for even functions $f$ is given by \begin{align}\label{mastersym} \mathbb{H}f(\lambda)=\frac{1}{\Gamma(k+\frac{1}{2})}
\int_{\mathbb{R}}f(x)\varphi_{\lambda}^{(k)}(x)|x|^{2k}dx \end{align} in proper functional spaces. \end{definition}
\subsubsection{\sf Its properties} Let us denote the operator $L$ acting in the $\lambda$\~space by $L_\lambda$; $L$ without the suffix $\lambda$ will continue to be the operator above in terms $x$. Recall that the operator $L$ depends on $k$; we will sometimes denote it by $L^{(k)}$.
\begin{lemma} For any functional spaces (not only for even functions), provided $L$ and $\mathbb{H}$ are well defined there, \begin{enumerate} \item [(a)] $\mathbb{H}(L)=4\lambda^{2}$, $\mathbb{H}(4x^{2})=L_\lambda$; \item [(b)] $e^{-x^{2}}\,L\,e^{x^{2}}=L+4x^{2}+[L,x^{2}]$. \end{enumerate} \end{lemma} {\em Proof.} Claim $(a)$ is based on the $x\!\leftrightarrow\! \lambda$\~symmetry of $\varphi_{\lambda}^{(k)}(x)$ and on the self-adjointness of the operators $L$ and $x^2$ with respect to the measure we consider.
Checking $(b)$ is direct. One can also use the following important connection with the theory of $\mathfrak{sl}(2)$. Setting $$ e=x^{2},\ f=-\frac{L}{4},\ h=[e,f]=x\frac{d}{dx}+\frac{1}{2}+k, $$ we obtain a representation of this Lie algebra. Then $e^{-x^{2}}Le^{x^{2}}$ can be interpreted and calculated using the adjoint action of $SL_2$. It must be {\em a priori} a linear combination of $e,f,h$; the exact formula is simple. Note that the Hankel transformation becomes the group element $s\in SL_2$ in this interpretation. \phantom{1}
$\qed$
{\em Proof of theorem \ref{thm:master}}. Let $\widehat{\varphi}_{\mu}^{(k)}(\lambda) \stackrel{\,\mathbf{def}}{= \kern-3pt =} e^{-\lambda^{2}}\mathbb{H}(\varphi^{(k)}_{\mu}(x)e^{-x^{2}})$. Due to the lemma, $\widehat{\varphi}_{\mu}^{(k)}(\lambda)$ satisfies $L^{(k)}_{\lambda}\widehat{\varphi}_{\mu}^{(k)}= 4\mu^{2}\widehat{\varphi}_{\mu}^{(k)}$. However, this solution is unique up to proportionality in the class of even analytic functions in a neighborhood of $x=0$. Thus $\widehat{\varphi}_{\mu}^{(k)}(\lambda)=C_\mu \varphi_{\mu}^{(k)}(\lambda)$. It gives (\ref{mastersym}) up to proportionality. Using the $\lambda\!\leftrightarrow\!\mu$\~symmetry on the left-hand side of this formula and the same symmetry of $\varphi_{\mu}^{(k)}(\lambda)$, we obtain that $C_\mu=Ce^{\mu^2}$ for an absolute constant $C$, which can be readily determined. \phantom{1}
$\qed$
\subsubsection{\sf Tilde-Bessel functions} Let us try to apply the master formula to other solutions of the eigenvalue problem (\ref{Lvarphi}). We will manipulate algebraically for some time, without exact analytic justifications. The proof above looks very algebraic; we even did not use that $\varphi_{\lambda}^{(k)}(x)$ is even.
For $\lambda\neq 0$, there exists another solution $\widetilde{\varphi}_{\lambda}^{(k)}(x)= (x\lambda)^{1-2k}\varphi_{\lambda}^{(1-k)}(x)$ of (\ref{Lvarphi}). If $\lambda=0$, let $\widetilde{\varphi}_{\lambda}^{(k)}(x)\stackrel{\,\mathbf{def}}{= \kern-3pt =} x^{1-2k}$. We need to assume that $\Re (k)<1/2$ to avoid the singularity at $0$ in these solutions.
Applying the reasoning above (formally), we obtain that \begin{align}\label{genmaster} \mathbb{H}(\widetilde{\varphi}_{\mu}^{k}e^{-x^{2}}) =\breve{\varphi}_{\mu}^{(k)}(\lambda) e^{\lambda^{2}+\mu^{2}} \end{align} for a certain solution $\breve{\varphi}_{\mu}^{(k)}$ of the same eigenvalue problem, a linear combination of $\varphi_{\mu}^{(k)}$ and $\widetilde{\varphi}_{\mu}^{(k)}$.
If we assume here that $0<\Re(k)<1/2$ and set $\mu=0$, then $\widetilde{\varphi}_{\mu}^{(k)}(0)=0$. Upon obvious cancelations, we come to the following brand new identity in the theory of Bessel functions: $$ \int_{-\infty}^{+\infty}
\varphi_{\lambda}^{(k)}(x)|x|e^{-x^{2}}dx=e^{\lambda^{2}}. $$ {\em Unfortunately this formula is wrong}. Let us explain why.
Informally this is wrong simply because no {\em new} identities of such a kind can be expected in the very classical field of Bessel functions and Hankel transform. The exact mathematical reason for this failure is as follows. The integration by parts, necessary for the self-adjointness claim, requires the convergence at $0$ of the {\em first two derivatives} of the functions involved. The existence of the starting and the final integral can be insufficient; one need to justify the convergence of all intermediate integrals as well.
The following analytic constraints make claim $(ii)$ of Lemma \ref{LEMDIAM} rigorous. These conditions are not exactly sharp, but sufficient for us.
Provided that $f,g\in C^{2}(\mathbb{R}_{+})$ and
$f(x)|x|^{k}$, $g(x)|x|^{k}$ are absolutely integrable, \begin{align*}
\int_{-\infty}^{+\infty}L(f)g|x|^{2k}dx=
\int_{-\infty}^{+\infty}fL(g)|x|^{2k}dx. \end{align*}
\subsubsection{\sf Complex analytic theory} The deduction above of (\ref{genmaster}) from the properties of the Hankel transform is of course formally correct; this simply gives nothing new in the case of real integration due to the divergence at $0$ of the derivatives of the tilde-solution. The Laplace integration, was design exactly to avoid the divergences of this kind. Let us first re-establish the usual master formula in the Laplace setting. \begin{theorem} For all $k\in \mathbb{C}$ such that $k\neq -\frac{1}{2}-m$, $m\in \mathbb{Z}_{+}$, \begin{align*} \int_{i\varepsilon +\mathbb{R}} \varphi_{\lambda}^{(k})(x) \varphi_{\mu}^{(k)}(x)e^{-x^{2}}(-x^{2})^{k}dx =\frac{\pi}{\Gamma(\frac{1}{2}-k)} \varphi_{\lambda}^{(k)}(\mu)e^{\lambda^{2}+\mu^{2}}. \end{align*} Here $\varepsilon>0$; the condition $k\neq -\frac{1}{2}-m$ is necessary for the existence of $\varphi_{\lambda}^{(k)}(x)$.\phantom{1}
$\qed$ \end{theorem}
For any complex number $k$, the function $(-x^{2})^{k}$ is defined as the function $\exp(k\log(-x^2))$ continued along the integration path $x\in i\varepsilon+{\mathbb R}$ for the usual branch of $\log$ with the cutoff at ${\mathbb R}_-$. Using $(-x^{2})^{k}$ is quite standard in classical works on $\Gamma$ and related functions.
Due to the Gamma-term on the right-hand side, this integral must be zero at $k=\frac{1}{2}+m$, $m\in \mathbb{Z}_{+}$. It is simple to demonstrate directly. Indeed, $$(-x^{2})^{1/2}=-ix \hbox{\ \, along the path\,\ } i\varepsilon+{\mathbb R}\,; $$ check the point $x=i\varepsilon$ using that $(\varepsilon^2)^{1/2}=\varepsilon$. The integrand is analytic at zero for such $k$, so we can tend $\varepsilon\to 0$. However the integrand is an odd function on ${\mathbb R}$ and, therefore, \begin{align*} \int_{i\varepsilon +\mathbb{R}} \varphi_{\lambda}^{(k)}(x) \varphi_{\mu}^{(k)}(x)e^{-x^{2}}(-ix)^{2m+1}dx=0. \end{align*}
Similarly, for $\widetilde{\varphi}_{\lambda}(x) \stackrel{\,\mathbf{def}}{= \kern-3pt =}(-\lambda^{2})^{1/2-k}(-x^{2})^{1/2-k} \varphi_{\lambda}^{(1-k)}(x)$, which is the complex analytic variant of the tilde-solution considered above, \begin{eqnarray*} &&\int_{i\varepsilon +\mathbb{R}} \varphi_{\lambda}^{(k)}(x) \widetilde{\varphi}^{(k)}_{\mu}(x)(-x^{2})^{(k)} e^{-x^{2}}dx\\ &=&\int_{i\varepsilon +\mathbb{R}}\varphi_{\lambda}^{(k)}(x) \varphi^{(1-k)}_{\mu}(x)(-x^{2})^{1/2}dx\\ &=&\int_{\mathbb{R}}\varphi_{\lambda}^{(k)}(x) \varphi^{(1-k)}_{\mu}(x)(-ix)dx\ =\ 0. \end{eqnarray*} Thus the standard solution $\varphi_{\lambda}^{(k)}(x)$ and the complex-analytic tilde-solution are orthogonal to each other in the master formula.
It is straightforward to calculate the master formula for the tilde-solutions $\widetilde{\varphi}^{(k)}_{\lambda}(x), \widetilde{\varphi}^{(k)}_{\mu}(x)$ coupled together in the Gauss-Bessel integral. We will provide the corresponding formulas below when doing the nonsymmetric master formula.
\setcounter{equation}{0} \section{\sc{Spinor eigenfunctions}} We will begin with the eigenvalue problem for the Dunkl operator. The latter is not a differential operator, but it shares some (but not all) properties with the first order {\em differential} operators. \begin{lemma}\label{DUNEIG} (i) The eigenvalue problem \begin{align}\label{Duneigen} \mathscr{D}\psi=2\lambda\psi, \text{ for } \mathscr{D}=\frac{d}{dx}+\frac{k}{x}(1-s) \end{align} has a unique analytic at $0$ solution $\psi=\psi_\lambda^{(k)}(x)$ satisfying $\psi(0)=1$ if and only if $k\not \in -1/2-{\mathbb Z}_+.$
(ii) Namely, it is $\psi=1$ for $\lambda=0$ and $ \psi(x)=\psi^{(k)}(\lambda x) $ for $$ \psi^{(k)}(t)= \varphi^{(k)}(t)+\frac{1}{2}(\varphi^{(k)})'(t) $$ in terms of $\varphi^{(k)}(t)$ from (\ref{varphiform}).
(iii) When $\lambda=0$ and $k=-\frac{1}{2}-m$, the space of analytic solutions is generated by $\psi=1$ and $\psi=x^{2m+1}$. When $\lambda\neq 0$ for the same $k$, the analytic solution $\psi$ exists and is unique up to proportionality, but vanishes at \, $0$. \phantom{1}
$\qed$ \end{lemma} The fact that the dimension of the space of solutions of (\ref{Duneigen}) can be $2$ (for special values of the parameters) requires attention and will eventually lead us to the spinor extension of the space of functions.
\subsection{{\bf Nonsymmetric master formula}} For $k\neq -1/2-m$, $m\in {\mathbb Z}_{+}$ and the function $\psi_{\lambda}^{(k)}(x)=\psi^{(k)}(\lambda x)$ from Lemma \ref{DUNEIG}, the following holds. \begin{theorem}\label{NONSYMPSI} (i) For $\Re k>-1/2$, \begin{align*} \int_{\mathbb{R}}
\psi_{\lambda}^{(k)}(x)\psi_{\mu}^{(k)}(x)e^{-x^{2}}|x|^{2k}dx =\Gamma(k+\frac{1}{2}) \psi_{\lambda}(\mu)^{(k)}e^{\lambda^{2}+\mu^{2}}. \end{align*}
(ii) Denote $\int_{{\mathbb R}}^{\varepsilon}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \frac{1}{2}(\int_{i\varepsilon +\mathbb{R}}+ \int_{-i\varepsilon +\mathbb{R}})$, then \begin{align*} \int^{\varepsilon}_{\mathbb{R}} \psi_{\lambda}^{(k)}(x)\psi_{\mu}^{(k)}(x)e^{-x^{2}} (-x^{2})^{k}dx =\frac{\pi}{\Gamma(\frac{1}{2}-k)} \psi_{\lambda}^{(k)}(\mu)e^{\lambda^{2}+\mu^{2}}. \end{align*} \end{theorem} {\em Proof.} As in the symmetric theory, the formula readily results from the basic facts concerning the {\em nonsymmetric Hankel transform}. The (general) definition of this transform is due to Dunkl \cite{Du2}. Its one-dimensional version can be found in Hermite's works, but this was used only marginally in the classical theory. This transform is given by \begin{align}\label{masternon} \mathbb{H}_{ns}f(\lambda)=\frac{1}{\Gamma(k+\frac{1}{2})}
\int_{\mathbb{R}}f(x)\psi_{\lambda}^{(k)}(x)|x|^{2k}dx, \end{align} provided the existence. Its theory is actually simpler than that of the classical symmetric Hankel transform (at least the algebraic aspects). We use the notation $\mathscr{D}_\lambda$ for the Dunkl operator acting in the $\lambda$\~space.
The following analytic conditions for the functions $f,g$ and their derivatives $f',g'$ are sufficient to ensure that \begin{align}\label{intRD}
&\int_{{\mathbb R}}\,\mathscr{D}(f) g |x|^{2k}dx\ =\
-\int_{{\mathbb R}}\,f\mathscr{D}(g) |x|^{2k}dx: \end{align}
(1) $f(x), g(x)$ are continuous and $f'(x), g'(x)$ exist in $\mathbb{R}\setminus 0$;
(2) the function $f(x)g(x)|x|^{2k}$ is integrable and continuous at $0$;
(3) $f(x)g(x)|x|^{2k-1}$, $f'(x)g(x)|x|^{2k}$,
$f(x)g'(x)|x|^{2k}$, $f(x)g(-x)|x|^{2k}$
\ \ \ \ \ are integrable at zero.
For the integration $\int_{\mathbb{R}}^{\varepsilon},$ only the integrability at infinity is needed for (\ref{intRD}). The theorem readily follows from the following lemma.
\begin{lemma} For $f$ as above and provided the existence of $\mathbb{H}_{ns}$, \begin{enumerate} \item [(a)] $\mathbb{H}_{ns}(\mathscr{D})=2\lambda$, $\mathbb{H}_{ns}(2x)=\mathscr{D}_\lambda$; \item [(b)] $e^{-x^{2}}\,\mathscr{D}\,e^{x^{2}}=\mathscr{D}+2x$\,, \end{enumerate} where the integration in (\ref{masternon}) can be either $\int_{\mathbb{R}}$ or $\int_{\mathbb{R}}^{\varepsilon}.$\phantom{1}
$\qed$ \end{lemma}
{\bf Comment.\ } Similar to the symmetric case, the integrals from Theorem \ref{NONSYMPSI} in the complex case are identically zero as $k\in 1/2+{\mathbb Z}_+$. It corresponds to the vanishing condition of the inner products associated with level-one coinvariants from
Theorem \ref{HALLONE}. See also formula (\ref{diaform}) (the real case $(b)$ there).
The affine symmetrizer $\widehat{\mathscr{I}}\,$ from
(\ref{Jproj})\, is a $q,t$\~Jackson counterpart of the integration $\int_{i\varepsilon +{\mathbb R}}f(x) (-x^2)^k dx$. The zeros of the inner product $\widehat{\mathscr{I}}\,(f\, T(g))$ for $A_1$ are exactly in the set $1/2+{\mathbb Z}_+$.
\subsubsection{\sf Using spinors} The theory of the nonsymmetric tilde-solutions requires the technique of spinors (already used above). They are pairs $f=\{f_{1}, f_{2}\}$ of functions defined in an open set $U$ in ${\mathbb R}$ or ${\mathbb C}$. {\em Real spinor} are defined for $U=\{x\in {\mathbb R},\, x>0\}$; {\em complex spinors} are defined for the set $U=\{x\in {\mathbb C},\, \Im x>0\}$. The operators act naturally on spinors; see Section \ref{sect:spinors}. For instance, $$ s\{f_{1}, f_{2}\}=\{f_{2}, f_{1}\},\ x\{f_{1}, f_{2}\}= \{xf_{1}, -xf_{2}\},\ \{f_{1}, f_{2}\}'=\{f_{1}', -f_{2}'\}, $$ where here and below $f'\stackrel{\,\mathbf{def}}{= \kern-3pt =} df/dx$.
The {\em super-presentation} of a spinor $f$ is defined to be \begin{align*} f=\llbracket f^{0}, f^{1}\rrbracket, \text{ where } f^{0}=\frac{f_{1}(x)+f_{2}(x)}{2}, \, f^{1}=\frac{f_{1}(x)-f_{2}(x)}{2}. \end{align*}
For any two spinors, $f=\{f_{1}, f_{2}\}$, $g=\{g_{1}, g_{2}\}$, their product is given by $f\cdot g=\{f_{1}g_{1}, f_{2}g_{2}\}$. In the super-presentation: \begin{align*} f\cdot g=\llbracket f^{0}g^{0}+f^{1}g^{1}, f^{0}g^{1}+f^{1}g^{0}\rrbracket. \end{align*} It is the standard stuff about ${\mathbb Z}_2$\~graded algebras.
A spinor $f=\{f_{1}, f_{2}\}$ is called a {\em principal spinor (function)} if the following holds. There must exist an open {\em connected\,} set $\widetilde{U}$ and a function $\widetilde{f}$ on $\widetilde{U}$ such that $U$, $U^{s}\stackrel{\,\mathbf{def}}{= \kern-3pt =} s(U)\subset \widetilde{U}$ and
$f_{1}=\widetilde{f}|_{U}$, $f_{2}=s(\widetilde{f})|_{U}$.
The differentiation of spinors $\frac{d}{dx}$ is an odd operator defined by $$ \frac{d}{dx}\llbracket f^{0}, f^{1}\rrbracket =\llbracket \frac{d}{dx}f^{1}, \frac{d}{dx}f^{0}\rrbracket. $$ The spinor integration is given by $$
\int_\gamma \llbracket f^{0}, \ f^{1}\rrbracket\stackrel{\,\mathbf{def}}{= \kern-3pt =} \int_{\gamma}f^{0}, $$ where $\gamma\subset U$ is a path in the set $U$.
\subsubsection{\sf Spinor Bessel functions} The Dunkl spinor eigenvalue problem is \begin{align}\label{dunspineig} \mathscr{D}(\psi)= \llbracket(\psi^{1})'+\frac{2k\psi^{1}}{x},(\psi^{0})'\rrbracket =\llbracket 2\lambda \psi^{0}, 2\lambda\psi^{1}\rrbracket. \end{align} In the standard representation $\{\psi_1,\psi_2\}$, it reads as follows: \begin{align*} \mathscr{D}(\psi)= \{\,\psi_1'+\frac{k(\psi_1-\psi_2)}{x}, -\psi_2'-\frac{k(\psi_2-\psi_1)}{x}\,\}= \{\,2\lambda \psi_1, 2\lambda\psi_2\,\}. \end{align*}
\begin{lemma}\label{DUNSOLUT} The space of solutions of the eigenvalue problem (\ref{dunspineig}) is always two-dimensional. There are three cases: \begin{enumerate} \item if $\lambda\neq 0$, then all the solutions are in the form $\psi=\llbracket \varphi, \frac{\varphi'}{2\lambda}\rrbracket$ for $\varphi$ satisfying $L\varphi=4\lambda^{2}\varphi$, and only one of them (up to proportionality) is a function (i.e., a principle spinor); \item if $\lambda=0$ and $k\not\in -1/2-{\mathbb Z}_+$ then $\psi=1$ is a solution and also there is an odd spinor solution $\chi_{k}$, given by
$\chi_{k}=\llbracket 0, |x|^{-2k}\rrbracket$ in the real case and $\chi_{k}=\llbracket 0, (-x^{2})^{-k}\rrbracket$ in the complex case; \item when $\lambda=0$ and $k= -1/2-m$ for $m\in {\mathbb Z}_+$, then the solutions are $1$ and $x^{2m+1}$, i.e., both are principle spinors (functions).
\phantom{1}
$\qed$ \end{enumerate} \end{lemma}
{\em Nonsymmetric tilde-solutions.} For $k\notin 1/2+\mathbb{Z}_{+}$, the spinor \begin{align*} \widetilde{\psi}_{\lambda}^{(k)}=\chi_{k}(x)\chi_{k}(\lambda) \psi_{\lambda}^{(-k)}(x) \end{align*} satisfies (\ref{dunspineig}). Actually it is a {\em bi-spinor}, in terms of $x$ and $\lambda$; we will skip the formal definition.
Let us incorporate the tilde-solution into the master formula. We need to redefine the inner product. Let \begin{align*} x^{2k}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \left\{\begin{array}{ccc}
\llbracket\ |x|^{2k},\,0\ \rrbracket, & & \hbox{\ real\ case;} \\ \llbracket\, (-x^{2})^{k},\,0\ \rrbracket, & & \hbox{\ complex\ case.}\end{array}\right. \end{align*} I.e., both are even spinors (functions, if $k\in {\mathbb Z}$). Note that $\chi_{k}(x)x^{2k}=\llbracket 0,1\rrbracket$ is an odd constant (a spinor of course). The integration will be \begin{eqnarray*} \int f(x) & \stackrel{\,\mathbf{def}}{= \kern-3pt =} & 2\int_{0}^{+\infty}f^{0}(x)dx \text{\ \,in the real case; }\\ \int f(x) & \stackrel{\,\mathbf{def}}{= \kern-3pt =} & \int_{i\varepsilon+\mathbb{R}}f^{0}(x)dx \text{\ \, in the complex case. } \end{eqnarray*}
Let us check that the $\psi$\~solution and the $\widetilde{\psi}$\~solution are orthogonal to each other in the master formula. Similar to the symmetric case, we have the divergence problem with the integration by parts, so only the complex case will be considered. Then the integral \begin{align}{\label{eqn:complexint}} \int \psi^{(k)}_{\lambda}(x) \widetilde{\psi}^{(k)}_{\mu}(x)e^{-x^{2}}x^{2k}
\end{align} is proportional to \begin{align*} I=\int_{i\varepsilon+{\mathbb R}}e^{-x^{2}} (\psi_{\lambda}^{(k)}\psi_{\mu}^{(-k)} \cdot\llbracket 0,1\rrbracket)^{0}\,dx= \int_{i\varepsilon+{\mathbb R}}e^{-x^{2}} (\psi_{\lambda}^{(k)}\psi_{\mu}^{(-k)})^1\,dx. \end{align*} However, $e^{-x^2}\psi_{\lambda}^{(k)}(x)\psi_{\mu}^{(-k)}(x)$ is a principal spinor, i.e., a restriction of an analytic function $F$. Therefore the component $F^1$ is an odd function on ${\mathbb R}$. Letting $\varepsilon\to 0$ in the integration path, we conclude that $I=0$.
\subsubsection{\sf Tilde master formulas} Let us list explicitly the Gauss-Bessel integrals for the tilde-solutions. \begin{theorem} In the real case, \begin{align*} 2\int^{+\infty}_{0} (\widetilde{\psi}_{\lambda}^{(k)}
\widetilde{\psi}_{\mu}^{(k)})^{0}e^{-x^{2}}|x|^{2k}dx =\widetilde{\psi}_{\lambda}^{(k)}(\mu) e^{\lambda^{2}+\mu^{2}}\Gamma(\frac{1}{2}-k) \text{\ \,for\ \, }\Re k<\frac{1}{2}. \end{align*} In the complex case, \begin{align*} \int_{i\varepsilon+\mathbb{R}} (\widetilde{\psi}_{\lambda}^{(k)}\widetilde{\psi}_{\mu}^{(k)})^{0} e^{-x^{2}}(-x^{2})^{k}dx =\frac{\pi}{\Gamma(\frac{1}{2}+k)}\widetilde{\psi}_{\lambda}^{(k)} (\mu)e^{\lambda^{2}+\mu^{2}} \text{ as }k\notin \frac{1}{2}+\mathbb{Z}_{+}; \end{align*} this integral is zero when $k=-1/2-m$ for $m\in \mathbb{Z}_{+}$. \phantom{1}
$\qed$ \end{theorem} We note that the spinors we integrate and those in the right-hand side are actually {\em bi-spinors}, i.e., spinors in terms of $x$ and spinors in terms of $\lambda,\mu$. the formal definitions are straightforward. It suffices here to use directly the definition: $\widetilde{\psi}_{\lambda}^{(k)}(\mu)= \chi_{k}(\lambda)\chi_{k}(\mu) \psi_{\lambda}^{(-k)}(\mu)$.
Let us also provide the symmetric tilde-formulas (no spinors are needed): \begin{align*} 2\int^{+\infty}_{0} \widetilde{\varphi}_{\lambda}^{(k)}
\widetilde{\varphi}_{\mu}^{(k)}\,e^{-x^{2}}|x|^{2k}\,dx\ =\ \Gamma(\frac{3}{2}-k)\,\widetilde{\varphi}_{\mu}^{(k)} (\lambda)\,e^{\lambda^{2}+\mu^{2}},\ \Re k<\frac{3}{2}\,, \end{align*} \begin{align*} \int_{i\varepsilon+\mathbb{R}}\widetilde{\varphi}_{\lambda}^{(k)} \widetilde{\varphi}_{\mu}^{(k)}\,e^{-x^{2}}(-x^{2})^{k}\,dx =\frac{\pi}{\Gamma(-\frac{1}{2}+k)}\, \widetilde{\varphi}_{\mu}^{(k)}(\lambda)\, e^{\lambda^{2}+\mu^{2}},\ k\notin \frac{3}{2}+\mathbb{Z}_{+}, \end{align*} and the latter integral is zero at $k=1/2-m$ for $m\in \mathbb{Z}_{+}$.
An obvious problem is in extending the nonsymmetric master formula to all spinor solutions for arbitrary root systems. One cannot expect the formulas to be so simple as for $A_1$, because the Weyl groups $W$ have irreducible representations of higher dimensions. We do not have the general formulas at the moment. Similar questions can be posted for arbitrary, not necessarily symmetric, solutions of the $L$\~eigenvalue problems in arbitrary ranks, when no spinors are needed.
We mention that the orthogonality relations for $\psi$ coupled with $\widetilde{\psi}$ can be extended to the trigonometric- differential and trigonometric- difference settings (any root systems), provided we have the $Y$\~semisimplicity. Hopefully this can be sufficient to manage the rational case.
\subsection{{\bf Affine KZ equations}}\label{sect:AKZ} \subsubsection{\sf Degenerate AHA and AKZ} Let $R$ be an arbitrary (reduced) root system, $R^\vee$ its dual, $P$ and $P^\vee$ the corresponding weight and coweight lattices. We set $z_{a}=(z,a)$ for $z\in {\mathbb C}^n$ and define the differentiation $\partial_{b}z_{a}\stackrel{\,\mathbf{def}}{= \kern-3pt =} (b,a)$ for arbitrary vectors $a,b$ (to be used mainly for $b\in P$, $a\in P^\vee$). Let ${}^wf(z)=f(w^{-1}(z))$ for $w\in W$, $s_{\alpha}$ be the reflections corresponding to the roots $\alpha$ and $\{y_{b}\}$ pairwise commutative elements satisfying $y_{a+b}=y_a+y_b$ for $a,b\in P$.
We will follow Definition \ref{dDAHA} of degenerate DAHA restricted to the AHA case, i.e., consider only nonaffine reflections $s_i$; also $-k$ will be replaced by $k$. The relations of {\em degenerate AHA}, due to Drinfeld for $GL_n$ \cite{Dr}
and Lusztig \cite{L}, are \begin{align}\label{ddahanon} &s_{i}y_{b}-y_{s_{i}(b)}s_{i}\,=\,k(b, \alpha_{i}^\vee), \text{ for }i\geq 1. \end{align} The corresponding algebra will be denoted by $\mathcal{H}'$.
Let $\Phi\,$ be a function of $z\,$ taking its values in the abstract algebraic span $\langle s_{\alpha}, y_{b}\rangle$.
The {\em affine Knizhnik-Zamolodchikov equation}, AKZ, is the following system of differential equations \begin{align}{\label{eqn:AKZ}} \partial_{b}(\Phi) =\left(\sum_{\alpha\in R_{+}^\vee}\frac{k(b,\alpha) s_{\alpha}}{e^{z_{\alpha}}-1}+y_{b}\right)\Phi, \text{ where }b\in P. \end{align} Actually, $b$ can be arbitrary complex vectors here and below.
\begin{theorem} The AKZ is self-consistent and $W$\~equivariant if and only if the elements $s_{\alpha}\,$ and $y_{b}\,$ satisfy the relations from (\ref{ddahanon}). The equivariance here means that if $\Phi\,$ is a solution of AKZ, then so is $\,w({}^w\Phi(z))=w(\Phi(w^{-1}(z)))\,$.\phantom{1}
$\qed$ \end{theorem}
The definition of AKZ and this theorem were the starting point of DAHA theory; here and below see Chapter 1 of \cite{C101}. The following construction is basically from \cite{C13}, but using the technique of spinors consistently makes it entirely algebraic (and essentially coinciding with that from \cite{O2}).
In \cite{C13} and other first author's papers, the values of AKZ were considered in $\mathcal{H}'$\~modules induced from arbitrary finite-dimensional representations of $W$ or induced from the characters of the polynomial algebra ${\mathbb C}[y]={\mathbb C}[y_b,\, b\in P]$. In this paper we will stick to the modules induced from ${\mathbb C}[y]$.
\subsubsection{\sf Spinor Dunkl operators} The Dunkl operators will be needed here in the following form: \begin{align}{\label{eqn:Duntrig}} \mathscr{D}_b^0 =\partial_b-\sum_{\alpha\in R_{+}^\vee}\frac{k(b,\alpha) \sigma_{\alpha}}{e^{z_{\alpha}}-1},\ \hbox{ where } \ \sigma_\alpha(z_a)=z_{s_\alpha(a)}. \end{align} Here $\sigma$ stays for the action on the argument of functions: $\sigma_u(f)(z)=f(u^{-1}z)$,\, $u\in W$. The relation to AKZ is established via the {\em spinor Dunkl operators\,} defined as a natural extension of (\ref{eqn:Duntrig}) to the space of $W$\~spinors.
The {\em spinors} are collections $\widehat{\psi}=\{\psi_w,\, w\in W\}$ of (arbitrary) scalar functions with component-wise addition, multiplication and the differentiations by $\partial_b$. The action $\sigma_u$ for $u\in W$ is through permutations of the indices: $$ \sigma_u(\widehat{\psi})=\{\psi_{u^{-1}w},w\in W\}. $$ Note the sign of $u^{-1}$, which ensures that we really have a representation of $W$; the spinors are actually {\em functions} on $W\times {\mathbb C}^n$ so $u^{-1}$ (the dualization) is necessary. This definition matches the action of $W$ on functions $f$ of $z$, which will be considered as {\em principle spinors} under the embedding $$ f\mapsto f^\rho\stackrel{\,\mathbf{def}}{= \kern-3pt =}\{f_w={}^{w^{-1}}f,\,w\in W\}. $$ Indeed, we have the commutativity \, $(\sigma_u(f))^\rho=\sigma_u(f^\rho)$. The definition of $\rho$ can be naturally extended to the operators acting on functions.
For instance, the function $z_\alpha$ becomes the spinor $\{z_{w^{-1}(\alpha)},w\in W\}$ under this embedding; also, $(\partial_b)^\rho=\{\partial_{w^{-1}(b)},w\in W\}$.
\begin{theorem}\label{SPINDUN} For a solution $\Phi$ of the AKZ with values in $\mathcal{H}'$, let us define the spinor $\widehat{\Psi}=\{w(\Phi),w\in W\}$ for the action of $w\in W$ in $\mathcal{H}'$ by left multiplications. Then $\widehat{\Psi}$ satisfies the following {\sf spinor Dunkl eigenvalue problem}: \begin{align}\label{Dspineigen} \mathscr{D}_b^0(\widehat{\Psi})=y_b\widehat{\Psi},\ b\in P. \end{align} \end{theorem} {\it Proof.} The $W$\~equivariance of AKZ readily establishes the equivalence of this theorem with the previous one. Explicitly, $\sigma_\alpha(\widehat{\Psi})=\{s_\alpha w(\Phi),w\in W\}$ and the relations for the component $w=u$ of $\widehat{\Psi}$ read as follows: $$ \partial_{u^{-1}(b)}\,u(\Phi)= \sum_{\alpha\in R_{+}^\vee}\frac{k(b,\alpha) s_{\alpha}\,u(\Phi)}{\exp(z_{u^{-1}(\alpha)})-1}+y_b\, u(\Phi), \ b\in P. $$ This can be recalculated to the same AKZ system for $\Phi$ due to the $W$\~equivariance. \phantom{1}
$\qed$
{\bf Comment.\ } In \cite{C13}, an analytic variant of this construction was used. The algebraic formalization of the argument from \cite{C13} can be found in Lemma 3.2 from \cite{O2}; the proof above is very similar to that in \cite{O2}. This ``algebraization" can be readily extended to the difference and elliptic theories (considered in \cite{C101} and previous first author's works). From the viewpoint of the applications to the isomorphism theorems, both approaches are equivalent.
As far as the reduction of AKZ to the Dunkl eigenvalue problem is concerned, arbitrary modules of $\mathcal{H}'$ were considered (not only induced) in \cite{C13}. The Dunkl operators there were given in terms of the action of $W$ via the monodromy of AKZ (see below). Treating formally the corresponding $W$\~orbits as {\em spinors}, one makes the construction entirely algebraic (as in Theorem \ref{SPINDUN} and in \cite{O2}).
It is important that the monodromy can be calculated {\em explicitly} for the asymptotically free solutions of AKZ. For instance, these explicit formulas were used in Theorem 4.3 from \cite{C13} to solve the {\em real\,} (nonspinor) Dunkl eigenvalue problem via AKZ in {\em functions} (not only in {\em spinors}). The solution found in \cite{C13} using the monodromy approach is the $G$\~function that was introduced (later) and played the key role in paper \cite{O2}. \phantom{1}
$\qed$
\subsubsection{\sf The isomorphism theorem} Let us apply Theorem \ref{SPINDUN} to {\em induced representations}. Given a one-dimensional representation ${\mathbb C}_{\lambda}={\mathbb C} v_\circ$ of ${\mathbb C}[y]\,$ defined by $y_{b}(v_\circ)=\lambda_b v_\circ$ for $\lambda_b=(\lambda, b)$, where $\lambda\in {\mathbb C}^n$,\, let $ \,I_{\lambda}=\operatorname{Ind}^{\mathcal{H}'}_{{\mathbb C}[y]}{\mathbb C}_\lambda\, $ be the $\mathcal{H}'$\~module induced from ${\mathbb C}_\lambda$.
We note that if the space of eigenvectors (pure, not generalized) for the eigenvalue $\lambda$ is one-dimensional in $I_\lambda$, then there exists a rational expression in terms of $y_b$ serving as a projector of $I_\lambda$ onto ${\mathbb C} v_\circ\subset I_\lambda$.
Let $I_\lambda^*$ be $Hom(I_\lambda,{\mathbb C})$ supplied with the natural action of $\mathcal{H}'$ via the canonical {\em anti-involution} of $\mathcal{H}'$ preserving the generators $s_i, y_b$ (reversing the order in products). We use here that the relations in the degenerate affine Hecke algebra are self-dual.
Next, we define the linear functional \,$\varpi: f\mapsto f(v_\circ)$\, on \,$I_\lambda^*\ni f$\, satisfying the conditions\, $ \varpi((y_b-\lambda_b)I_\lambda^*)=0 \ \hbox{ for } \ b\in P.\, $ Assuming, that the space of $\lambda$\~eigenvectors in $I_\lambda$ is one-dimensional, these conditions determine $\varpi$ uniquely up to proportionality.
The functional $\varpi$ is nonzero on any nonzero $\mathcal{H}'$\~submodule $V^*\subset I_\lambda^*$, since $I_\lambda$ is cyclic generated by $v_\circ$. Indeed, if $\varpi(f)=0$ for all $f\in V^*$, then $f(\mathcal{H}'v_\circ)=0=f(I_\lambda)$ for all such $f$.
Let $U_{0}\subset \mathbb{C}^{n}$ be a open neighborhood of $0$ in ${\mathbb C}^n$; we set $U_{0}'=\cap_{{}_{w\in W}}\,w(U_{0})$. We assume that $U_{0}$ satisfies the following properties (necessary for the monodromy interpretation below): \begin{enumerate} \item $U_{0}$ does not contain any zeros of $\prod_{\alpha\in R_{+}^\vee}(e^{z_{\alpha}}-1)$; \item $U_{0}$ is simply connected and $U_{0}'/W$ is connected; \end{enumerate} $U_0^\star$ will be one of the connected components of $U_0'$
(the latter set is a disjoint union of $|W|$ connected open sets).
By $\,Sol^{\lambda}_{AKZ}(U_{0})\,$, we denote the space of $I_{\lambda}^*$\~valued analytic solutions $\phi$ of the AKZ equation in $U_0$.
Let $\,Sol_{\mathscr{D}}^{\lambda}(U^\star_{0})\,$ be the space of $W$-{\em spinor solutions} $\widehat{\psi}$ in $U^\star_{0}$ of the {\em scalar} eigenvalue problem \begin{align}\label{Dspineig} \mathscr{D}_b^0(\widehat{\psi})=\lambda_b\widehat{\psi},\ b\in P. \end{align} The spinors here are collections $\widehat{\psi}=\{\psi_w,\, w\in W\}$ of (arbitrary) scalar analytic functions in $U^\star_{0}$.
\begin{theorem}\label{ISOTHMFULL} The dimension of the space
$Sol_{\mathscr{D}}^{\lambda}(U^\star_{0})$ equals the cardinality $|W|$ of $W$. There is an isomorphism \begin{align}\label{solakzD} \eta:\, Sol^{\lambda}_{AKZ}(U_{0})\ni\phi\,\mapsto\, \{\varpi(w(\phi))\downarrow_{U_0^\star},\, w\in W\}\,\in\, Sol_{\mathscr{D}}^{\lambda}(U^\star_{0}) \end{align} for the action of $w\in W$ on the values of $\phi$, which are from $I_\lambda$. \end{theorem} {\em Proof}. The claim that $\eta$ is a map between the required spaces of solutions follows from Theorem \ref{SPINDUN}. Due to the coincidence of the dimensions of the spaces in (\ref{solakzD}), we need only to check that $\eta$ is injective. As in \cite{C13}, this follows from the fact that $\varpi$ is nonzero on any $\mathcal{H}'$\~submodule of $I_{\lambda}^*$. Note that the construction of $\eta$ is entirely algebraic, so it suffices to assume that $\phi$ is defined in the same open set $U_0^\star$ as in the statement of the theorem.
\subsubsection{\sf The monodromy interpretation} Let $\Phi(z)$ be an invertible matrix solution of AKZ in $U_0$ with values in $\hbox{Aut}(I_\lambda^*)$. For any $w\in W$, let us define the {\em monodromy matrix} $\mathcal{T}_{w}$ by \begin{align*} w(\Phi(z))=\Phi(w(z))\mathcal{T}_{w}. \end{align*} Here $\Phi(w(z))$ is well defined in $U_0\cap w^{-1}(U_0)$, so is $\mathcal{T}_{w}$. The matrix solution $\Phi$ is nothing but a choice of the basis of fundamental solutions in $Sol^{\lambda}_{AKZ}(U_{0})$ (its columns). Changing the basis conjugates all $\mathcal{T}_w$ by a constant invertible matrix. The matrix-valued functions $\mathcal{T}_{w}$ have the following properties: \begin{enumerate} \item[(a)] $\mathcal{T}_{w}$ are defined in $U_{0}'$ and are locally constant; \item[(b)] $\mathcal{T}_{uw}={}^{w^{-1}}\mathcal{T}_{u} \mathcal{T}_{w} =\mathcal{T}_{u}(w(z))\mathcal{T}_{w}(z)$ for $u,w\in W$. \end{enumerate} For each $w\in W$, let us define its {\em $\sigma'$-action}\,: $$\sigma'_{w}(F)={}^{w}F\mathcal{T}_{w^{-1}}= F(w^{-1}(z))\mathcal{T}_{w^{-1}}(z). $$ Then $\sigma'_{1}=1$, $\sigma'_{uw}=\sigma'_{u}\sigma'_{w}$ and $\sigma'_{w}\partial_{a}=\partial_{w(a)}\sigma'_{w}$ for $u,w\in W$ and $a\in P$. We naturally set $\sigma'_{\alpha}=\sigma'_{s_\alpha}$ and $\sigma'_i=\sigma'_{\alpha_i}$. Here $F$ can be an arbitrary function in $U_0'$ with values in $\hbox{Aut}(I_\lambda^*)$.
Introducing $$ \mathscr{D}_{b}'\stackrel{\,\mathbf{def}}{= \kern-3pt =} \partial_{b}- k\sum_{\alpha\in R_{+}^\vee}\frac{(\alpha,b)\sigma'_{\alpha}} {e^{z_{\alpha}}-1}, $$ one readily obtains that \begin{align}\label{abphi} y_{b}\Phi=\left(\partial_{b}-k \sum_{\alpha\in R_{+}^\vee}\frac{(\alpha,b)\sigma'_{\alpha}} {e^{z_{\alpha}}-1}\right)\Phi=\mathscr{D}_{b}'\Phi. \end{align} We simply employ the definition of $\sigma'$ here. The action of $\mathscr{D}_{b}'$ is given in terms of the $W$\~action on $z$ and the {\em right} multiplications by matrices $\mathcal{T}_{s_\alpha}$. So this action commutes with $y_b$, which are {\em left} multiplications by constant matrices. Therefore we can apply the functional $\varpi$ to $\Phi$ in (\ref{abphi}), which gives that (\ref{abphi}) holds for $\varpi(\Phi)$. The spinor $\widehat{\Psi}$ from Theorem \ref{SPINDUN} is nothing but $\{\Psi_w=\sigma'_{w^{-1}}(\Phi)\downarrow U_0^\star, w\in W\}$.
\comment{ (with the natural action of $W$ on the indices $w$ of $\Psi_w$). Then we claim that \begin{align}\label{spinpsi} \widehat{\psi}=\{\psi_w,\,w\in W\}\stackrel{\,\mathbf{def}}{= \kern-3pt =} \varpi(\widehat{\Psi})\,=\,\{\varpi(\Psi_w)\} \end{align} solves the eigenvalue (scalar) problem (\ref{Dspineig}) in the space of spinors for the connected component $U_0^\star$. If $\phi$ is the first column of $\Phi$, then $\psi_1=\psi_{id}$ is $\varpi(\phi)$ upon the restriction to $U_0^\star$. Thus the map claimed in the theorem is $$ \phi\mapsto \widehat{\psi},\ \psi_1=\varpi(\phi)\downarrow_{U_0^\star}. $$ Following \cite{C13} and using that $\varpi$ is nonzero on any $\mathcal{H}'$\~submodule of $I_{\lambda}^*$, we prove the theorem.\phantom{1}
$\qed$ }
\subsubsection{\sf Connection to QMBP} Continuing this construction, one can combine the isomorphism we found with the symmetrization map, which acts from $Sol_{\mathscr{D}}^{\lambda}(U_{0}^\star)$ to the space of solutions of the Heckman-Opdam system (QMBP) in $U_0^\star$ corresponding to $\lambda$. To be exact, the map from $Sol^{\lambda}_{AKZ}(U_{0})$ to $Sol^{\lambda}_{QMBP}(U_{0})$ is the projection of the space of values onto the one-dimensional subspace of $W$\~invariants inside $I_{\lambda}^*$. It gives the Matsuo- Cherednik isomorphism theorem from \cite{Mats,C13} (the proof follows \cite{C13}). The spinors do not appear in the construction of this map and the statement of the theorem; however, they provide {\em the} best way to verify it (and dramatically reduce the proof from \cite{Mats}).
We note that the relation of the Dunkl-spinor eigenvalue problem above to QMBP is actually very similar to Lemma \ref{DUNSOLUT}, which addresses solving the Dunkl eigenvalue problem in {\em spinors}. Let us mention Corollary 3.4 from \cite{O2}, where a similar extension of the Dunkl eigenvalue problem was considered.
Certain conditions on the module $I_\lambda$ are necessary to ensure the {\em isomorphism} with QMBP. Namely, this module must be assumed {\em spherical\,}, $\mathcal{H}'$\~generated by $\sum_{w\in W} w(v_\circ)$, correspondingly, $I_\lambda^*$ will be {\em co-spherical\,}. See \cite{C13} and \cite{C101}.
{\bf Comment.\ } There are relations to the localization functor from \cite{GGOR,VV1}. The later is, very briefly, taking the monodromy representation of the local systems analogous to AKZ (in more general modules). Starting with certain rational or degenerate DAHA modules, the monodromy results in the representations of nonaffine (affine) $t$\~Hecke algebras.
The monodromy is important in our approach too (the cocycle $\{\mathcal{T}_w\}$ does contain $t$). The actual output of our approach is a complete system of eigenfunctions of Dunkl operators in the corresponding $y$\~eigenspaces of the initial $\mathcal{H}'$\~module, the $G$\~function in the terminology from \cite{O2}. Algebraically, the Dunkl operators and the operators of multiplication by the (trigonometric) coordinates generate the corresponding DAHA module.
The localization functor is understood completely (so far) only in the rational case and in the differential -trigonometric case (corresponding to the setting of this section); see \cite{GGOR,VV1}. Our construction and the isomorphism theorems hold for all known families of AKZ and Dunkl operators (including the elliptic theories). See \cite{C13}, \cite{Ch12}, Chapter 1 from \cite{C101} and \cite{Sto2}. The exact connection is still not clarified.
\setcounter{equation}{0} \section{\sc{Conclusion}} To try to connect better the topics of this work and to put it into perspective, we will touch upon the relations of DAHA, mainly the $q$\~Whittaker functions, to the geometric quantum Langlands program, though not much is known in this direction. The relation of the Verlinde algebras to the Lusztig category of the representations of quantum groups from \cite{L1} is of key importance here; this is the main focus of this section.
We will not try to review the applications (known and expected) of the {\em ``symmetric" global $q$\~Whittaker functions}, including the Shintani -Casselman -Shalika formula, the relations to Givental-Lee theory and possible applications in physics. See \cite{ChW} and \cite{GLO} for a discussion. Generally, the (coefficients of) $q$\~Whittaker functions are expected to contain a lot of information about quantum $K$\~theory and $IC$\~theory of affine flag varieties. Givental-Lee theory deals with quantum $K$\~theory of the flag variety.
We are very thankful to Roman Bezrukavnikov, Alexander Braverman, Dennis Gaitsgory, Michael Finkelberg, David Kazhdan, Victor Ostrik for various discussions on quantum groups, affine Grassmannians, quantum Langlands program and neighboring topics (though they do not always agree with what will follow).
\subsection{{\bf Verlinde algebras and QG}} The relations to DAHA are expected upon applying $K_0$ (the Grothendieck group) to the categories used in the quantum geometric Langlands program and related directions. Then these categories become commutative rings with inner products and sometimes with a projective action of $PSL_2({\mathbb Z})$. Generally, the number of simple objects must be finite for the latter action to exist.
As it was pointed out in Section ``Abstract Verlinde Algebras" from \cite{C101}, such rings (even if some of these structures are missing) are very exceptional. For instance, one can formally prove counterparts of the Macdonald conjectures (the norm formulas and the evaluation-duality formulas) in the abstract Verlinde-type setting, establish Pieri rules and do more; cf. \cite{C103}.
It is unlikely that there are many commutative rings with such rich structures. The major candidates are quotients of the polynomial and various similar representations of DAHA, including infinite-dimensional ones and the corresponding (commutative) algebras of the $W$\~invariants.
\subsubsection{\sf Quantum groups} The expected connections to the Langlands program and related projects are grouped around the following.
\newtheorem*{keyconjecture}{Conjecture}
\begin{keyconjecture} \label{CONJCH} The commutative algebra $K_0(Rep_q\, G)$ for the category $Rep_q\, G\,$ of finite-dimensional representations of Lusztig's quantum group can be canonically identified with the algebra $\mathscr{X}^W$ of $W$\~invariants of the polynomial representation $\mathscr{X}$ of DAHA at $t=q$, defined for the corresponding root system. It includes the roots of unity $q$. Then sub-quotients of the $\mathscr{X}^W$ under the action of the subalgebra of invariants of $\hbox{${\mathcal H}$\kern-5.2pt${\mathcal H}$}$ (the elements commuting with $T_w$ for $w \in W$) correspond to categorical sub-quotients of $Rep_q\, G\,$. Such sub-quotient of $Rep_q\, G\,$ has the structure of modular category if $PSL(2,{\mathbb Z})$ acts projectively in the corresponding sub-quotient of $\mathscr{X}^W$. \end{keyconjecture}
For generic $q\,$, the simple objects correspond to the classical finite-dimensional characters, which are eigenfunctions of the $W$\~invariant $Y$\~operators. The most interesting here is the case of roots of unity, when $\mathscr{X}$ and $\mathscr{X}^W$ become reducible.
For $q=e^{2\pi \imath /N}$, the algebra of $W$\~invariants of the nonzero (canonical) irreducible quotient of $\mathscr{X}$ can be naturally identified with the {\em Verlinde algebra} in the special case $k=1\,(t=q)$; see \cite{C101}, Section 0.4. The projective DAHA-action of the $PSL(2,{\mathbb Z})$ leads to the Verlinde $T,S$\~operators.
The Verlinde algebra was originally defined in terms of integrable Kac-Moody modules with the {\em fusion} directly related to the {\em conformal field theory}. Equivalently, it is isomorphic to the quotient of $K_0(Rep_q\, G)$ by the modules of zero $q$\~dimension, i.e., $K_0$ of the so-called {\em reduced category\,}. The equivalence of these two approaches at roots of unity is due to Finkelberg \cite{Fi} (\cite{KL2} apart from the roots of unity). It confirms the conjecture for the {\em perfect\,} quotients of $\mathscr{X}$.
The categorical sub-quotients in the conjecture generally cannot be expected to be tensor categories for Lusztig's big quantum group unless in some special cases, including the reduced category. The first author is grateful to Michael Finkelberg and Victor Ostrik for clarifying discussions on these matters.
The next case after the reduced category (actually the key) is the so-called {\em parallelogram quotient} of $Rep_q\, G$. It is the category of representations of the {\em small quantum group\,} \cite{AG}, which attracts a lot of attention now. We expect that its $K_0$ corresponds to the algebra of $W$\~invariants of the {\em DAHA parallelogram module} under the same relation $t=q$. The latter is defined for $A_1$ as $$ V^{-2}={\mathbb C}[X,X^{-1}]/(X^{2N}+X^{-2N}-2)= {\mathbb C}[X,X^{-1}]/(X^{N}-X^{-N})^2 $$ in the notation from \cite{C101} Section 2.9.3;\, its dimension is $4N$.
Let us discuss the rank-one case in greater detail.
\subsubsection{\sf The rank-one case} The {\em perfect} quotient of $V^{-2}$ for $q=e^{2\pi \imath/N}$ and integral $0\le k<N/2$ will be denoted by $V_{2N-4k}$; its dimension is $2N-4k$. Here one can consider half-integral $k$ too (we will not discuss it). Let $V_{2N+4k}\,$ be the kernel of the natural map $V^{-2}\to V_{2N-4k}$.
Both are irreducible DAHA modules with the projective $PSL(2,{\mathbb Z})$-action. They are commutative algebras because so is $\mathscr{X}$; $V_{2N-4k}$ is semisimple, but $V_{2N+4k}$ for $k>0$ is not. The action of $X$ in the latter has $\,4k\,$ Jordan $2$\~blocks ($2$\~dimensional blocks) with pairwise distinct eigenvalues and $\,2N-4k\,$ simple eigenvectors. Due to the projective $PSL(2,{\mathbb Z})$\~action (we need $\sigma$), the Jordan decomposition must be of the same type for $Y$ instead of $X$.
The Jordan decomposition of $Y$ in the whole $V^{-2}$ is different from that of $X$. Namely, $Y$ has $\,4k\,$ Jordan $2$\~blocks and the rest of it is semisimple (all eigenvalues are of multiplicity $2$). The decomposition of $X$ in $V^{-2}$ obviously consists of the $2$\~blocks only (see its definition); their number is $2N$. Hence, there can be no projective action of $PSL(2,{\mathbb Z})$ in $V^{-2}$ extending that in $V_{2N\pm 4k}$.
Upon taking the $W$\~invariants,\, ${\hbox{\rm dim}}_{\mathbb C}\, (V^{-2})^W=2N\,,$ $$ {\hbox{\rm dim}}_{\mathbb C}\, V_{2N-4k}^W\, =\, N-2k+1,\ \ \hbox{ and } \ {\hbox{\rm dim}}_{\mathbb C}\, V_{2N+4k}^W\, =\, N+2k-1. $$ The latter two algebras are projective $PSL(2,{\mathbb Z})$\~invariant because the generator $T$ is fixed under this action.
Let us discuss the case $k=1$ in more detail. One has $$ V_{2N-4}= {\mathbb C}[X,X^{-1}]/(F) \ \hbox{ for } \ F=\frac{X^{2N}-1}{(X^2-1)(X^2-q)}. $$ For instance, $F=(X-q)(X+q)$ for (the minimal possible) $N=3$ and the Verlinde algebra is ${\mathbb C}[Z]/(Z^2-1)$ for $Z=X+X^{-1}$ in this case; $\,q=\exp(2\pi \imath/3)$.
Importantly, $Y+Y^{-1}$ acts semisimply in the invariants of the polynomial representation for $k=1$. It is due to the fact that the $(Y+Y^{-1})$\~eigenvectors in ${\mathbb C}[X+X^{-1}]$ do not depend on $q$ when $t=q$ and are proportional to the $SL(2)$\~Schur functions (it holds for any root systems). Accordingly, $(V^{-2})^W$ and $V_{2N+4}^W$ are $(Y+Y^{-1})$\~semisimple in this case. The spectrum of $Y+Y^{-1}$ in $(V^{-2})^W$ is $\{q^{i/2}+q^{-i/2}, 1\le i\le 2N\}$ for $q^{1/2}= e^{\pi\imath/N}$; thus, $\,2,-2\,$ are simple eigenvalues and the others are of multiplicity $2$.
The operator $X+X^{-1}$ in $(V^{-2})^W$ is {\,\em not\,} semisimple even for $k=1$. Namely, $2,-2$ are its simple eigenvalues, but $q^{i/2}+q^{-i/2}$ correspond to the Jordan $2$\~blocks for $1\le i <N$. Since this is different from the Jordan decomposition of $Y+Y^{-1}$ in this space, we conclude that there can be no projective action of $PSL(2,{\mathbb Z})$ in $(V^{-2})^W$ for $k=1$ ($N\ge 3$). If the conjecture above holds, then no such an action can be expected in the parallelogram quotient of $Rep_q\, G$ at roots of unity extending that in the Verlinde algebra; so it cannot be a {\em modular category}.
It is likely that the irreducible constituents of the parallelogram DAHA modules for integral $k$ are always projective $PSL(2,{\mathbb Z})$\~modules, but this is known only for $A_1$; it may be connected with \cite{Lyu}. The parallelogram module, as the whole, has no natural (projective) $PSL(2,{\mathbb Z})$\~structure (only $\tau_-$ acts there).
As a related direction, we would like to mention that Tipunin and others successfully calculated certain generalized Verlinde algebras of nonsemisimple type using the {\em logarithmic conformal theory}; see e.g., \cite{MT}. They obtained exactly the ones described in \cite{C101}, Proposition 2.9.6 (upon taking the $W$\~invariants). Technically, the (canonical) irreducible quotient of the polynomial representation becomes nonsemisimple for integral $N>k>N/2$ ; it can be identified with $V_{2N+4k}$ considered above for $0<k<N/2$.
Let us also note that the limit of the minimal models as $c\to 1$ is important in physics applications; the corresponding infinite-dimensional Verlinde-type algebra is likely to be the polynomial DAHA representation itself.
\subsection{{\bf Expected developments}} \subsubsection{\sf Approaching the conjecture} The most conceptual reason for the conjecture above is a very close relation of DAHA (almost at the level of its definition) to $K$\~theory of affine flag varieties. However, there are other aspects too.
{\em KZ equations.} The affine Knizhnik-Zamolodchikov equations and the so-called $r$\~matrix KZ (see \cite{C101}, Section 1.5) can be employed here. These KZ are directly connected with the {\em coinvariants} and the {\em $\tau$\~function} for {\em factorizable Kac-Moody algebras} associated with $r$\~matrices (introduced in the first author's works). Generally, the approach based on the $KZ$ equation is of key importance in \cite{KL2}, \cite{Fi} and in \cite{Ga}, so this technique is certainly relevant for the conjecture.
{\em Nonsymmetric theory.} DAHA gives the most in the nonsymmetric setting, when we switch from the $W$\~invariant polynomials to the whole polynomial representation. However, we do not know much about the geometric meaning of the {\em nonsymmetric Macdonald polynomials}. There are two major general facts here. They are connected with the Matsumoto spherical functions and with the level-one Demazure characters; these examples are degenerate but nevertheless important.
Generally, taking the $W$\~invariants in DAHA-modules seems really necessary to relate them to Lie-Kac-Moody theory. The technique of spinors, which establishes a connection of DAHA to non-$W$\~invariant sections of local systems like QMBP (the Heckman-Opdam system), could be a bridge from the nonsymmetric theory to geometry.
{\em Finite-dimensional modules.} It is worth mentioning that the specialization $t=q$ used in this conjecture does not seem the only one related to $Rep_q\, G$. Let us restrict ourselves to the spherical case, which means that we will consider only the quotients of the polynomial representation $\mathscr{X}$. Then such modules will be commutative algebras and the corresponding categories, if any, can be expected monoidal.
Important generalizations of Verlinde algebras can be obtained when the polynomial representation $\mathscr{X}$ and its nonzero irreducible quotient are considered for the following DAHA parameters:
(a)\, $t=q^k$ for {\em \, singular\,} rational $k=-\frac{s}{d}<0\,$ and any unimodular $q$,\,
(b)\, $t\in {\mathbb C}$ but $q$ is a root of unity (a variant of the parallelogram case),
(c)\, and when $q$ is a root of unity under the limits $\,t\to 0$ or $\,t\to \infty\,$,\\ although not all structures are present in these three cases.
Only the integrality of the structural constants of the Verlinde algebra will be missing in $(a)$ (since $q$ is not a root of unity); the positivity of the Verlinde inner product will hold for sufficiently small $\arg(q)$.
More significantly, there will be no projective action of $PSL(2,{\mathbb Z})$ in the cases $(b,c)$.
The limits from $(c)$ (which are actually particular cases of $(b)$) are very interesting because of possible (no exact confirmations so far) relations to the following.
\subsubsection{\sf Toward Langlands program} The (local) quantum geometric Langlands program will be discussed here very introductory. Let $G$ be the simply connected Lie group over ${\mathbb C}$ corresponding to a given root system $R$,\, ${}^L G$ its Langlands dual (though we mainly stick to the simply-laced $R$ in this work).
The global ``symmetric" $q$\~Whittaker function can be interpreted as the Fourier transform of
$K_0(Rep_q\, {}^LG)$ for generic $q$; we actually need $|q|<1$ here to ensure the convergence. The challenge is to connect it with the category $\hbox{Whit}^c$ (see below) and the Gaitsgory-Lurie transform $$ K_0(Rep_q\, {}^LG)\ \to\ \hbox{Whit}^c(Gr_G), $$ a complicated functor between the corresponding $2$-categories. Such connection seems almost inevitable if this transform has something to do with the $q$\~Toda operators, which is exactly the key question.
The images of the simple objects of $K_0(Rep_q {}^LG)$ in $K_0(\hbox{Whit}^c(Gr_G))$ under the Gaitsgory-Lurie transform are of major importance; for many applications, knowing them is quite sufficient. The problem is that this map cannot be fixed uniquely at the level of $K_0$ without using involved categorical (or other?) methods.
Assuming that ${}^L\!\mathscr{X}^W$, where ${}^L\!\mathscr{X}$ is the Langlands dual of $\mathscr{X}$, is a substitute for $K_0(Rep_q {}^LG)$ (the conjecture), its limits $t\to 0$ or $t\to\infty$ could be equally relevant here for generic $q$ (they are connected with each other). Then the Fourier transform of $\, \lim_{t\to 0}\,{}^L\!\mathscr{X}^W$ could be, hopefully, a DAHA counterpart of $K_0$ of the category $$ \hbox{Whit}^c(Gr_G)\ =\ D\,\hbox{mod}\,^c ( G((z))/G_0)^{N((z))}, $$ where $N\subset G$ is the standard unipotent subgroup and an unramified character on $N((z))$ is needed here to define the equivariant modules.
Without going into detail, it is a category of $N((z))$\~equivariant $c$\~twisted $D$\~modules on the affine flag variety $Gr_G$, which is the group $G((z))$ of (formal) meromorphic loops divided by the group $G_0=G[[z]]$ of holomorphic ones.
The category $\hbox{Whit}^c(Gr_G)$ was proven by Gaitsgory in \cite{Ga} to be equivalent (for generic $q$ and under some technical restrictions) to $Rep_q\, {}^LG$ for $q=e^{\pi c}$, which was conjectured by Lurie.
{\em $Q$\~Toda system as Hitchin system.} The Fourier image of $\lim_{t\to 0}\,{}^L\!\mathscr{X}$ twisted by the Gaussian is the spinor polynomial representation of nil-DAHA from Theorems \ref{TWONIL}, called there the {\em hat-representation} (see also Theorem \ref{MainToda}). So the $W$\~invariant part of the hat-representation may be a candidate for $K_0(\hbox{Whit}^c(Gr_G)$.
A certain indirect confirmation is the relation of $\hbox{Whit}^c(Gr_G)$ to the $W$\~algebras and their Verlinde algebras, which, in their turn, are connected with the DAHA-Verlinde algebras.
If one replaces the {\em Hitchin system} in the geometric Langlands duality by the $q$\~Toda eigenvalue problem, then the ``symmetric" (non-spinor) $q$\~Whittaker function will become the reproducing kernel of the corresponding Fourier transform. For any fixed set of eigenvalues, the corresponding $q$\~Toda eigenvalue problem can be interpreted as a $D$\~module very similar to those in the category $\hbox{Whit}^c(Gr_G)$ (upon the switch from quantum groups to Kac-Moody theory). The exact relation of this approach to the quantum Langlands program is not established so far.
\subsubsection{\sf Affine flag varieties etc.} Another source of inspiration could be Theorem 3 from \cite{BF}, which may be more directly connected with $q$\~Whittaker functions than the Gaitsgory-Lurie transform. In its $K$\~theoretic variant (a conjecture), it looks related to the Fourier duality we establish between nil-DAHA from Theorem \ref{TWONIL}.
If such a connection really exists, then it could result in the $K$\~theoretical interpretation of the {\em spinor $q$\~Whittaker function} from (\ref{spinwhit}). It provides the duality between the spinor hat-representation and the bar-representation. The latter has a clear $K$\~theoretic meaning; thus the former can be of geometric nature too.
Also, we expect the modular {\em translation functor\,} and the so-called {\em wall-crossing\,} to be related to the DAHA {\em intertwiners\,} and, more specifically, to the analytic continuation of the asymptotic expansions of the global $q$\~functions from one asymptotic sector to another.
The mod $p\,$ methods were already used for DAHA; this is a powerful tool. The wall-crossing is expected to be connected with the theory of nil-DAHA; its relation to global functions is not based on any solid evidence at the moment.
Let us outline a possible approach to geometric theory of global functions based on their asymptotic expansions. The definition of these functions and the existence of their limits at infinity are from \cite{C5}, \cite{ChW}; let us also mention Stokman's definition of the global functions for $C^\vee C$ and his recent results on the difference Harish-Chandra theory.
{\em Global functions geometrically.} A complete description of the asym\~ptotic expansions of a global function, namely, inside the asymptotic sectors, then at their walls, then at the walls of walls and so on, called the {\em resonance conditions\,}, would fix it uniquely as an analytic function without any reference to the Macdonald or $q$\~Toda operators.
Generally, the continuation of the functions/sections from their natural domains to the boundary requires involved tools (like intersection cohomology). {\em Global functions\,} are automatically such continuations of their asymptotic expansions, so they are expected to be canonical in every possible sense. In their definition, we use that the polynomial representation multiplied by the Gaussian is self-dual with respect to the DAHA-Fourier transforms; the global functions are the corresponding reproducing kernels. It provides a conceptual explanation of their remarkable algebraic and analytic properties.
The resonance theory of global $q,t$\~spherical and $q$\~Whittaker functions, a continuation of the program due to Harish-Chandra, Casselman \cite{Ca} and others, is in progress. The first development here was the Harish-Chandra theory of asymptotic decomposition (the first author and Stokman), including the representation of a global function as a weighted $W$\~summation of its asymptotic expansions.
{\em Associators and dilogarithm.\,} We note that DAHA can be applied to catch certain categorical structures beyond $K_0$. Generally, changing the asymptotic sectors of the Knizhnik-Zamolodchikov equations gives the {\em associators\,} due to Drinfeld. In DAHA theory we restrict ourselves to the (various) AKZ equations. In several examples, these associators correspond to different choices of maximal commutative subalgebras in AHA or DAHA and can be calculated.
The resulting $q,t$\~pentagon-type relations in the limit $t\to 1$ may be connected with \cite{FG}. It is certainly connected with the theory of asymptotic decomposition of the global functions outlined above. It is worth mentioning that the well-known pentagon relation for the quantum {\em dilogarithm}, which is nothing but the $q$\~Gamma function, does not play any significant role in DAHA theory so far, though there are recent developments in this direction in the theory of nil-DAHA. Adding dilogarithms to DAHA would be an important development.
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On This Day in Math - March 31
God runs electromagnetics on Monday, Wednesday, and Friday by the wave theory, and the devil runs it by quantum theory on Tuesday, Thursday, and Saturday.
~Sir Lawrence Bragg
The 91st day of the year; 10n + 91 and 10n + 93 are twin primes for n = 1, 2, 3 and 4. (For n less than ten, one of these expressions is prime for some other values of n, which?)
91 and it's reversal 19 are related to Ramanujan's Taxi-cab number, 1729 = 19x91, a palindrome product. Note that the sum of the digits of 1729 are 19.
91 is : The sum of thirteen consecutive integers = 1 + 2 + 3 + ... + 11 + 12 + 13
The sum of one of each US coin less than a Silver Dollar is 91 cents.
and of six consecutive squares= 12 + 22 + 32 + 42 + 52 + 62
two consecutive cubes = 33 + 43 and the difference of two consecutive cubes = 63 - 53
In 1851, Leon Foucault demonstrated his pendulum experiment at the Pantheon of Paris at the request of Napoleon Bonaparte, who had been informed of Foucault's recent discovery on 6 Jan 1851. He had installed a pendulum in his cellar in the Arras Street of Paris. It was made from 2 m (6.5-ft) long wire supporting a 5-kg weight. He observed a small movement of the oscillation plane of the pendulum - showing that the Earth was rotating underneath the swinging pendulum. A month later, he repeated the experiment at the observatory of Paris, with a 11-m pendulum which gave longer swings and a more clearly visible deviation. His March demonstration at the Pantheon used a 28-kg sphere on a 67-m (220-ft) wire. *TIS (The first date of this demonstration seems to have been on March 28,
1854 The University of Konigsberg awarded Weierstrass an honorary doctorate. Previously he was a Gymnasium teacher without a university degree. *VFR The award was the result of the attention his 1854 paper, Zur Theorie der Abelschen Functionen, which appeared in Crelle's Journal. This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series. With this paper Weierstrass burst from obscurity.*SAU
1899 The EIFFEL TOWER, was built in 26 months and opened in Mar 1889 for the Universal Exposition. it is 320.75 m (1051 ft) high and only weighs 7000 tons – less than the air around it! The tower was inaugurated on 31 March 1889, and opened on 6 May. *VFR
1959 Sof'ja Janovskaja became the first chairperson of the newly created department of mathematical logic at the Moscow State University. *Women of Mathematics
1918 Daylight Savings Time for the USA first applied. Standard time was adopted throughout the United States. 'An Act to preserve daylight and provide standard time for the United States' was enacted on March 19, 1918. It both established standard time zones and set summer DST to begin on March 31, 1918. *WebExhibits.org
I understand that at least three states are trying to repeal daylight savings in their states as of 2014.
In 1921, Professor Albert Einstein arrived in New York to give a lecture on his new theory of relativity. *TIS
1936 The Last day of service of the US Post Office in Eight, West Va. (It Seems the PO in nearby Six, W. Va lasted a little longer, but I can't find it now in Post Office Listings)
1939 Harvard and IBM Agree to Build The Mark I "Giant Brain":
Harvard and IBM sign an agreement to build the Mark I, also known as the IBM Automatic Sequence Controlled Calculator (ASCC). Project leader Howard Aiken developed the original concept of the machine: a series of switches, relays, rotating shafts and clutches. The Mark I weighed about five tons and contained more than 750,000 components. It read instructions from paper tape and data from punch cards.*CHM
1981 Time (p. 51) reported that Educational Testing Service had to change the scores on 250,000 PSAT and 19,000 SAT papers because a student had successfully challenged a mathematical question about polyhedrons with no right answer. Mathematics Magazine 54 (1981), pp 152 and 277. *VFR Daniel Lowen, 17, a junior at Cocoa Beach High School in Florida was the first to call the ETS attention to their error. The problem involved putting two pyramids together and determining the number of faces on the new figure. The ETS had failed to allow for the fact that when two faces are joined, other faces meeting at the edges of the union might meld into one face.
1984 Science News reports that Persi Diaconis, a statistician at Stanford, can do a perfect riffle shuffle eight times in a row, thereby returning the 52-card deck to its original order. He has also proved that seven ordinary shuffles is enough to randomize a deck of cards. *VFR
1993 The birth of Spamming, A bug in a program written by Richard Depew sends an article to 200 newsgroups simultaneously. The term spamming is coined by Joel Furr (a writer and software trainer notable as a Usenet personality in the early and mid 1990s.) to describe the incident. *Wik
2011 The first ever "On This Day in Math"... thanks to hundreds of you for all the help.
1596 René Descartes (31 March 1596 in La Haye (now Descartes),Touraine, France
- 11 Feb 1650 in Stockholm, Sweden)was a French philosopher whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry. His work had a great influence on both mathematicians and philosophers. La Géométrie is by far the most important part of this work. Scott summarises the importance of this work in four points:
He makes the first step towards a theory of invariants, which at later stages derelativises the system of reference and removes arbitrariness.
Algebra makes it possible to recognise the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all.
Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method.
Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all.
*SAU His lifelong habit of laying abed till noon was interrupted by Descartes' new employer, the athletic, nineteen-year-old Queen Christiana of Sweden, who insisted he tutor her in philosophy in an unheated library early in the morning. This change of lifestyle caused the illness that killed him. [Eves, Circles, 177◦]*VFR
1730 – Étienne Bézout (31 March 1730 in Nemours, France - 27 Sept 1783 in Basses-Loges (near Fontainbleau), France) His most famous and well used book "including an incorrect proof that the quintic was solvable by radicals. In the early nineteenth century some of his in influential textbooks were translated into English. One translator, John Farrah, used
them to teach calculus at Harvard." *VFR
Bezout's theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his principia, where he claims that two curves have a number of intersection points given by the product of their degrees. The theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques. Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions. From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold. This led to a sentiment, expressed by certain authors, that his proof was neither correct nor the first proof to be given.
1795 Louis Paul Emile Richard (31 March 1795 in Rennes, France - 11 March 1849 in Paris, France) Richard perhaps attained his greatest fame as the teacher of Galois and his report on him which stated, "This student works only in the highest realms of mathematics.... "
It is well known. However, he also taught several other mathematicians whose biographies are included in this archive including Le Verrier, Serret and Hermite. He fully realised the significance of Galois' work and so, fifteen years after he left the college, he gave Galois' student exercises to Hermite so that a record of his school-work might be preserved. It is probably fair to say that Richard chose to give them to Hermite since in many ways he saw him as being similar to Galois. Under Richard's guidance, Hermite read papers by Euler, Gauss and Lagrange rather than work for his formal examinations, and he published two mathematics papers while a student at Louis-le-Grand.
Despite being encouraged by his friends to publish books based on the material that he taught so successfully, Richard did not wish to do so and so published nothing. This is indeed rather unfortunate since it would now be very interesting to read textbooks written by the teacher of so many world-class mathematicians.*SAU
1806 Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, and was listed by Alexander Macfarlane as one of ten leading 19th-century British mathematicians. Kirkman's schoolgirl problem, an existence theorem for Steiner triple systems that founded the field of combinatorial design theory, is named after him.
Kirkman's first mathematical publication was in the Cambridge and Dublin Mathematical Journal in 1846, on a problem involving Steiner triple systems that had been published two years earlier in the Lady's and Gentleman's Diary by Wesley S. B. Woolhouse. Despite Kirkman's and Woolhouse's contributions to the problem, Steiner triple systems were named after Jakob Steiner who wrote a later paper in 1853. Kirkman's second research paper paper, in 1848, concerned hypercomplex numbers.
In 1850, Kirkman observed that his 1846 solution to Woolhouse's problem had an additional property, which he set out as a puzzle in the Lady's and Gentleman's Diary:
Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast.
This problem became known as Kirkman's schoolgirl problem, subsequently to become Kirkman's most famous result. He published several additional works on combinatorial design theory in later years. Kirkman also studied the Pascal lines determined by the intersection points of opposite sides of a hexagon inscribed within a conic section. Any six points on a conic may be joined into a hexagon in 60 different ways, forming 60 different Pascal lines. Extending previous work of Steiner, Kirkman showed that these lines intersect in triples to form 60 points (now known as the Kirkman points), so that each line contains three of the points and each point lies on three of the lines. *Wik
1847 – Yegor Ivanovich Zolotarev, (March 31, 1847, Saint Petersburg – July 19, 1878, Saint Petersburg) In 1874, Zolotarev become a member of the university staff as a lecturer and in the same year he defended his doctoral thesis "Theory of Complex Numbers with an Application to Integral Calculus". The problem Zolotarev solved there was based on a problem Chebyshev had posed earlier. His steep career ended abruptly with his early death. He was on his way to his dacha when he was run over by a train in the Tsarskoe Selo station. On July 19, 1878 he died from blood poisoning. *Wik
1890 Sir William Lawrence Bragg (31 Mar 1890; 1 Jul 1971 at age 81) was an Australian-English physicist and X-ray crystallographer who at the early age of 25, shared the Nobel Prize for Physics in 1915 (with his father, Sir William Bragg). Lawrence Bragg formulated the Bragg law of X-ray diffraction, which is basic for the determination of crystal structure: nλ = 2dsinθ which relates the wavelength of x-rays, λ, the angle of incidence on a crystal, θ, and the spacing of crystal planes, d, for x-ray diffraction, where n is an integer (1, 2, 3, etc.). Together, the Braggs worked out the crystal structures of a number of substances. Early in this work, they showed that sodium chloride does not have individual molecules in the solid, but is an array of sodium and chloride ions. *TIS
1906 Shin'ichiro Tomonaga (31 Mar 1906; 8 Jul 1979 at age 73)Japanese physicist who shared the Nobel Prize for Physics in 1965 (with Richard P. Feynman and Julian S. Schwinger of the U.S.) for independently developing basic principles of quantum electrodynamics. He was one of the first to apply quantum theory to subatomic particles with very high energies. Tomonaga began with an analysis of intermediate coupling - the idea that interactions between two particles take place through the exchange of a third (virtual particle), like one ship affecting another by firing a cannonball. He used this concept to develop a quantum field theory (1941-43) that was consistent with the theory of special relativity. WW II delayed news of his work. Meanwhile, Feynman and Schwinger published their own independent solutions.*TIS
1624 Joao Baptista Lavanha (1550 in Portugal - 31 March 1624 in Madrid, Spain)Lavanha is said to have studied in Rome. He was appointed by Philip II of Spain to be professor of mathematics in Madrid in 1582.
Philip had sent the Duke of Alba with an army to conquer Portugal in 1580 and soon realized that Portugal was more advanced in studies of navigation than Spain. In an attempt to correct this, Philip founded an Academy of Mathematics in Madrid with Lavanha as its first professor.
From 1587 Lavanha became chief engineer to Philip II. He was appointed cosmographer to the king in 1596 and about the same time he moved to Lisbon where he taught mathematics to sailors and navigators.
Lavanha is best known for his contributions to navigation. His book Regimento nautico gives rules for determining latitude and tables of declination of the Sun. He also worked on maps, producing some interesting new ideas. He produced a map of Aragon in about 1615. Among his publications was a translation of Euclid.
Lavanha also studied instruments used in navigation, constructing astrolabes, quadrants and compasses. *SAU
1726/7 Isaac Newton (25 December 1642 – 20 March 1727 [NS: 4 January 1643 – 31 March 1727) English physicist and mathematician, who made seminal discoveries in several areas of science, and was the leading scientist of his era. His study of optics included using a prism to show white light could be split into a spectrum of colors. The statement of his three laws of motion are fundamental in the study of mechanics. He was the first to describe the moon as falling (in a circle around the earth) under the same influence of gravity as a falling apple, embodied in his law of universal gravitation. As a mathematician, he devised infinitesimal calculus to make the calculations needed in his studies, which he published in Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy, 1687)*TIS
Newton died intestate. Immediately his relatives began to quarrel over the division of his estate, which amounted to a considerable fortune. Thomas Pellet examined Newton's manuscript holdings in hopes of turning a quick profit. His "thick clumsy annotations 'Not fit to be printed,' now seem at once pitiful and ludicrous." See Whiteside, Newton Works, I, xvii ff for details. *VFR
1776 John Bird (1709 – March 31, 1776), the well known mathematical instrument maker, was born at Bishop Auckland. He worked in London for Jonathan Sisson, and by 1745 he had his own business in the Strand. Bird was commissioned to make a brass quadrant 8 feet across for the Royal Observatory at Greenwich, where it is still preserved. Soon after, duplicates were ordered for France, Spain and Russia.
Bird supplied the astronomer James Bradley with further instruments of such quality that the commissioners of longitude paid him £500 (a huge sum) on condition that he take on a 7-year apprentice and produce in writing upon oath, a full account of his working methods. This was the origin of Bird's two treatises The Method of Dividing Mathematical Instruments (1767) and The Method of Constructing Mural Quadrants (1768). Both had a foreword from the astronomer-royal Nevil Maskelyne. When the Houses of Parliament burned down in 1834, the standard yards of 1758 and 1760, both constructed by Bird, were destroyed.
Bird was an early influence in the life of Jerimiah Dixon, and in all probability it was he who recommended Dixon as a suitable companion to accompany Mason. *Wik
1841 George Green (14 Jul 1793, 31 Mar 1841 at age 47) was an English mathematician, born near Nottingham, who was first to attempt to formulate a mathematical theory of electricity and magnetism. He was a baker while, remarkably, he became a self-taught mathematician. He became an undergraduate at Cambridge in October 1833 at the age of 40. Lord Kelvin (William Thomson) subsequently saw, was excited by the Essay. Through Thomson, Maxwell, and others, the general mathematical theory of potential developed by an obscure, self-taught miller's son heralded the beginning of modern mathematical theories of electricity.*TIS His most famous work, An Essay on the Application of Mathematical Analysis to the Theory of Electricity and Magnetism was published, by subscription, in March 1828. Most of the fifty-two subscribers were friends and patrons. The work lay unnoticed until William Thomson rediscovered it and showed it to Liouville and Sturm in Paris in 1845. The Theory of Potential it developed led to the modern mathematical theory of electicity. *VFR
1997 Friedrich (Hermann) Hund (4 Feb 1896 - 31 Mar 1997) was a German physicist known for his work on the electronic structure of atoms and molecules. He introduced a method of using molecular orbitals to determine the electronic structure of molecules and chemical bond formation. His empirical Hund's Rules (1925) for atomic spectra determine the lowest energy level for two electrons having the same n and l quantum numbers in a many-electron atom. The lowest energy state has the maximum multiplicity consistent with the Pauli exclusion principle. The lowest energy state has the maximum total electron orbital angular momentum quantum number, consistent with rule. They are explained by the quantum theory of atoms by calculations involving the repulsion between two electrons. *TIS
Harold Scott MacDonald Coxeter (9 Feb 1907 in London, England - 31 March 2003 in Toronto, Canada) graduated from Cambridge and worked most of his life in Canada. His work was mainly in geometry. In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics. Among his most famous geometry books are The real projective plane (1955), Introduction to geometry (1961), Regular polytopes (1963), Non-euclidean geometry (1965) and, written jointly with S L Greitzer, Geometry revisited (1967). He also published a famous work on group presentations, which was written jointly with his first doctoral student W O J Moser, Generators and relations for discrete groups.
His 12 books and 167 published articles cover more than mathematical research. Coxeter met Escher in 1954 and the two became lifelong friends. Another friend, R Buckminister Fuller, used Coxeter's ideas in his architecture. In 1938 Coxeter revised and updated Rouse Ball's Mathematical recreations and essays, a book which Rouse Ball first published in 1892. *SAU
*WM = Women of Mathematics, Grinstein & Campbel
Jaime Escalante *Wik
A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.
~Stefan Banach
The 90th day of the year; 90 is the only number that is the sum of its digits plus the sum of the squares of its digits. (Is there any interesting distinction to the rest of the numbers for which this sum is more (or less) than the original number?)
\( \frac{90^3 - 1}{90 - 1} \) is a Mersenne prime.
90 is the smallest number having 6 representations as a sum of four positive squares
90 is the number of degrees in a right angle. Moreover, as a compass direction, 90 degrees corresponds to east.
Which reminds me of a fun math joke:"The number you have dialed is imaginary. Please rotate you phone by 90 degrees and dial again."
And 90 is the sum of the first 9 consecutive even numbers,
the sum of consecutive integers in two different ways,
the sum of two consecutive primes,
and of six consecutive primes,
and the sum of five consecutive squares.
(all proofs left to the reader.)
In 239, B.C., was the first recorded perihelion passage of Halley's Comet by Chinese astronomers in the Shih Chi and Wen Hsien Thung Khao chronicles. Its highly elliptical, 75-year orbit carries it out well beyond the orbit of Neptune and well inside the orbits of Earth and Venus when it swings in around the Sun, traveling in the opposite direction from the revolution of the planets. It was the first comet that was recognized as being periodic. An Englishman, Edmond Halley predicted in 1705 that the comet that appeared over London in 1682 would reappear again in 1759, and that it was the same comet that appeared in 1607 and 1531. When the comet did in fact reappear again in 1759, as correctly predicted, it was named (posthumously) after Halley. *TIS
Comets have been observed and recorded in China since the Shang Dynasty (1600-1046 BC). The set of comet illustrations shown below is from a silk book written during the western Han period.
* Marilyn Shea,umf.maine.edu
1612 The Jesuit astronomer Christoph Scheiner thought he had discovered a 5th Jupiter moon He was mistaken. *Thony Christie, @rmathematicus
In 1791, after a proposal by the Académie des sciences (Borda, Lagrange, Laplace, Monge and Condorcet), the French National Assembly finally chose that a metre would be a 1/10 000 000 of the distance between the north pole and the equator. *TIS (although at the time, this distance was not known. To determine the distance from the North Pole to the equator it was assumed that a portion of a meridian could be measured accurately and the whole distance could then be estimated from this sample. The meridian chosen went from Barcelona in Spain, to Dunquerque in France; this choice was an early example of the intended international nature of the metric system. Two astronomers, Borda and Méchain, were appointed to carry out the measurement. )
1796 The nineteen year old Gauss began his scientific diary with his construction of the regular 17-gon. The Greeks had ruler-and-compass constructions for the regular polygons with 3, 4, 5 and 15 sides, and for all others obtainable from these by doubling the number of sides. Here the problem rested until Gauss completely solved it: A regular n-gon is constructable IFF n is a product of a power of 2 and one or more distinct Fermat primes, i.e., primes of the form 22n +1. This discovery led Gauss to devote his life to mathematics rather than philology. *VFR Gauss told his close friend Bolyai that the regular 17-gon should adorn his tombstone, but this was not done. There is a 17 pointed star on the base of a monument to him in Brunswick because the stonemason felt everyone would mistake the 17-gon for a circle. Gauss gave the tablet on which he had made the discovery to Bolyai, along with a pipe, as a souvenir. (I have been unable to find any later trace of the pipe or tablet, but if anyone has knowledge of the I would appreciate any information.)
*Genial Gauss Gottingen
1858 Pencil with attached eraser patented. It has benefited generations of mathematics students. The first patent for attaching an eraser to a pencil was issued to a man from Philadelphia named Hyman Lipman. This patent was later held to be invalid because it was merely the combination of two things, without a new use. I found a note at about.com that said that "Before rubber, breadcrumbs had been used to erase pencil marks."
1867 The U. S. purchases Alaska from Russia for $7,200,000 in gold. The most prominent American mathematician of the time, Benjamin Peirce, then superintendent of the Coast Survey, played a role in the acquisition by sending out a reconnaissance party whose reports were important aids to proponents of the purchase. *VFR
1951 UNIVAC I turned over to Census Bureau. During ENIAC project, Mauchly met with several Census Bureau officials to discuss non-military applications for electronic computing devices. In 1946, with ENIAC completed, Mauchly and Eckert were able to secure a study contract from the National Bureau of Standards (NBS) to begin work on a computer designed for use by the Census Bureau. This study, originally scheduled for six months, took about a year to complete. The final result were specifications for the Universal Automatic Computer (UNIVAC).
UNIVAC was, effectively, an updated version of ENIAC. Data could be input using magnetic computer tape (and, by the early 1950's, punch cards). It was tabulated using vacuum tubes and state-of-the-art circuits then either printed out or stored on more magnetic tape.
Mauchly and Eckert began building UNIVAC I in 1948 and delivered the completed machine to the Census Bureau in March 1951. The computer was used to tabulate part of the 1950 population census and the entire 1954 economic census. Throughout the 1950's, UNIVAC also played a key role in several monthly economic surveys. The computer excelled at working with the repetitive but intricate mathematics involved in weighting and sampling for these surveys.
UNIVAC I, as the first successful civilian computer, was a key part of the dawn of the computer age *US CENSUS Bureau Web page
In 1953, Albert Einstein announced his revised unified field theory.*TIS
1985 M.I.T. computer science graduate students Robert W. Baldwin and Alan T. Sherman successfully decode a cipher consisting of a series of numbers separated by commas. They failed to share in the $116,000 prize offered by Decipher Inc. since they misread the contest rules—the contest ended the previous evening. [Burlington Free Press, 5 April 1985.]
2010 A Blue moon - The second full moon of the month of March. The next month with a blue moon will be in 2012: August 2, August 31
1862 Leonard James Rogers (30 March 1862, 12 Sept 1933) Rogers was a man of extraordinary gifts in many fields, and everything he did, he did well. Besides his mathematics and music he had many interests; he was a born linguist and phonetician, a wonderful mimic who delighted to talk broad Yorkshire, a first-class skater, and a maker of rock gardens. He did things well because he liked doing them. Music was the first necessity in his intellectual life, and after that came mathematics. He had very little ambition or desire for recognition.
Rogers is now remembered for a remarkable set of identities which are special cases of results which he had published in 1894. Such names as Rogers-Ramanujan identities, Rogers-Ramanujan continued fractions and Rogers transformations are known in the theory of partitions, combinatorics and hypergeometric series. *SAU
1864 Helen Abbot Merrill born in Llewellyn Park, Orange, New Jersey. She graduated from Wellesley College in 1886, taught school for several years and then returned to teach at Wellesley from 1893 until her retirement in 1932. She studied function theory with Heinrich Maschke at Chicago, descriptive geometry with G. F. Shilling at G¨ottingen, and function theory with James Pierpont at Yale, where she received her Ph.D. in 1903. She wrote a popular book about mathematics, Mathematical Excursions (1933), that has been reprinted by Dover.*WM
A rare (and a little pricey) collectors favorite
1879 Bernhard Voldemar Schmidt (30 Mar 1879, 1 Dec 1935) Astronomer and optical instrument maker who invented the telescope named for him. In 1929, he devised a new mirror system for reflecting telescopes which overcame previous problems of aberration of the image. He used a vacuum to suck the glass into a mold, polishing it flat, then allowing in to spring back into shape. The Schmidt telescope is now widely used in astronomy to photograph large sections of the sky because of its large field of view and its fine image definition. He lost his arm as a child while experimenting with explosives. Schmidt spent the last year of his life in a mental hospital.*TIS
1886 Stanisław Leśniewski (March 30, 1886, Serpukhov – May 13, 1939, Warsaw) was a Polish mathematician, philosopher and logician. Leśniewski belonged to the first generation of the Lwów-Warsaw School of logic founded by Kazimierz Twardowski. Together with Alfred Tarski and Jan Łukasiewicz, he formed the troika which made the University of Warsaw, during the Interbellum, perhaps the most important research center in the world for formal logic. *Wik
1892 Stefan Banach (30 Mar 1892, 31 Aug 1945) Polish mathematician who founded modern functional analysis and helped develop the theory of topological vector spaces. In addition, he contributed to measure theory, integration, the theory of sets, and orthogonal series. In his dissertation, written in 1920, he defined axiomatically what today is called a Banach space. The idea was introduced by others at about the same time (for example Wiener introduced the notion but did not develop the theory). The name 'Banach space' was coined by Fréchet. Banach algebras were also named after him. The importance of Banach's contribution is that he developed a systematic theory of functional analysis, where before there had only been isolated results which were later seen to fit into the new theory. *TIS
His doctoral dissertation, which was published in Fundamenta Mathematicae in 1922, marks the birth of functional analysis. *VFR
1921 Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician who made contributions in combinatorics, graph theory, number theory but mostly in probability theory. He proved, using the large sieve, that there is a number K such that every even number is the sum of a prime number and a number that can be written as the product of at most K primes. See also Goldbach conjecture.
In information theory, he introduced the spectrum of Rényi entropies of order α, giving an important generalisation of the Shannon entropy and the Kullback-Leibler divergence. The Rényi entropies give a spectrum of useful diversity indices, and lead to a spectrum of fractal dimensions. The Rényi–Ulam game is a guessing game where some of the answers may be wrong.
He wrote 32 joint papers with Paul Erdős, the most well-known of which are his papers introducing the Erdős–Rényi model of random graphs. Rényi, who was addicted to coffee, invented the quote: "A mathematician is a device for turning coffee into theorems.", which is generally ascribed to Erdős. The sentence was originally in German, being a wordplay on the double meaning of the word Satz (theorem or residue of coffee). *Wik
1929 Ilya Piatetski-Shapiro (30 March 1929 – 21 February 2009) During a career that spanned 60 years he made major contributions to applied science as well as theoretical mathematics. In the last forty years his research focused on pure mathematics; in particular, analytic number theory, group representations and algebraic geometry. His main contribution and impact was in the area of automorphic forms and L-functions.
For the last 30 years of his life he suffered from Parkinson's disease. However, with the help of his wife Edith, he was able to continue to work and do mathematics at the highest level, even when he was barely able to walk and speak.*Wik
1559 Adam Ries (23 Dec 1492 in Staffelstein (near Bamberg), Upper Franconia (now Germany) - 30 March 1559 in Annaberg, Saxony (now Annaberg-Buchholz, Germany) Ries's income came mainly from his arithmetic textbooks. The first of these was Rechnung auff der linihen written while he was in Erfurt and printed in that city in 1518 by Mathes Maler. The book was intended to teach people how to use a calculating board similar to an abacus. This type of device is described by the Money Museum,
Four horizontal and five vertical lines were painted or carved on the calculating boards to represent the decimal values in ascending order. The arithmetical sums were worked out with the help of coin-like counters. They were placed on the respective lines according to the values of the numbers and then, depending on the calculation, these were moved, removed or added to the lines until the final result could be read off. No numbers were printed on the counters; they amounted to as much as the line on which they were placed.
No copy of the first edition of this book has survived, the earliest that we have is the second of the four editions which was published in 1525.
Dirk Struik writes,
Adam Ries has remained in German memory because of his Rechenbücher -schoolbooks on arithmetic, popular for a century and a half. It is less known that he also wrote an algebra, called the Cosz, but this work has remained in manuscript form. Three of these manuscripts were bound together in 1664 by the Dresden Rechenmeister Martin Kupffer. They were thought to be lost until they were found in 1855, and are now kept at the Erzgebirgsmuseum Annaberg-Buchholz, Annaberg being the Saxonian mining town where Ries lived as a respected citizen and teacher for many years until his death. The impressive folio facsimile, published on the occasion of the 500th birthday of Ries, contains three manuscripts: Cosz I (pp. 1-325) was finished in 1524, Cosz II (pp. 329-499) was written between 1545 and 1550 ...
*SAU
Thony Christie pointed out to me that the German Wikipedia gives his date of death as April 2. He also has confirmed that the phrase "das macht nach Adam Ries" (That's according to Adam Ries) is still used in Germany to indicate something is done correctly, sort of like the American idiom, "according to Hoyle."
And here is the amazing story of how he was billed for his television license over 450 years after his death.
1832 Stephen Groombridge (7 Jan 1755; 30 Mar 1832) English astronomer and merchant, who compiled the Catalogue of Circumpolar Stars (corrected edition published 1838), often known as the Groombridge Catalog. For ten years, from 1806, he made observations using a transit circle, followed by another 10 years adjusting the data to correct for refraction, instrument error and clock error. He retired from the West Indian trade in 1815 to devote full time to the project. He was a founder of the Astronomical Society (1820). His work was continued by others when he was struck (1827) with a "severe attack of paralysis" from which he never fully recovered. The catalog eventually listed 4,243 stars situated within 50° of the North Pole and having apparent magnitudes greater than 9. Editions of the catalog were published posthumously. The 1833 edition was withdrawn due to errors, and corrected in 1838 by A Catalog of Circumpolar Stars, Reduced to January 1, 1810, edited by G. Biddell Airy. *TIS
1914 John Henry Poynting (9 Sep 1852; 30 Mar 1914)British physicist who introduced a theorem (1884-85) that assigns a value to the rate of flow of electromagnetic energy known as the Poynting vector, introduced in his paper On the Transfer of Energy in the Electromagnetic Field (1884). In this he showed that the flow of energy at a point can be expressed by a simple formula in terms of the electric and magnetic forces at that point. He determined the mean density of the Earth (1891) and made a determination of the gravitational constant (1893) using accurate torsion balances. He was also the first to suggest, in 1903, the existence of the effect of radiation from the Sun that causes smaller particles in orbit about the Sun to spiral close and eventually plunge in.*TIS
1944 Sir Charles Vernon Boys (15 Mar 1855; 30 Mar 1944 at age 88) English physicist and inventor of sensitive instruments. He graduated in mining and metallurgy, self-taught in a wide knowledge of geometrical methods. In 1881, he invented the integraph, a machine for drawing the antiderivative of a function. Boys is known particularly for his utilization of the torsion of quartz fibres in the measurement of minute forces, enabling him to elaborate (1895) on Henry Cavendish's experiment to improve the values obtained for the Newtonian gravitational constant. He also invented an improved automatic recording calorimeter for testing manufactured gas (1905) and high-speed cameras to photograph rapidly moving objects, such as bullets and lightning discharges. Upon retirement in 1939, he grew weeds.*TIS
1954 Fritz Wolfgang London (7 Mar 1900; 30 Mar 1954 at age 53) German-American physicist who, with Walter Heitler, devised the first quantum mechanical treatment of the hydrogen molecule, while working with Erwin Schrödinger at the University of Zurich. In a seminal paper (1927), they developed a wave equation for the hydrogen molecule with which it was possible to calculate approximate values of the molecule's ionization potential, heat of dissociation, and other constants. These predicted values were reasonably consistent with empirical values obtained by spectroscopic and chemical means. This theory of the chemical binding of homopolar molecules is considered one of the most important advances in modern chemistry. The approach is later called the valence-bond theory. *TIS
1995 John Lighton Synge (March 23, 1897–March 30, 1995) was an Irish mathematician and physicist. Synge made outstanding contributions to different fields of work including classical mechanics, general mechanics and geometrical optics, gas dynamics, hydrodynamics, elasticity, electrical networks, mathematical methods, differential geometry, and Einstein's theory of relativity. He studied an extensive range of mathematical physics problems, but his best known work revolved around using geometrical methods in general relativity.
He was one of the first physicists to seriously study the interior of a black hole, and is sometimes credited with anticipating the discovery of the structure of the Schwarzschild vacuum (a black hole).
He also created the game of Vish in which players compete to find circularity (vicious circles) in dictionary definitions. *Wik
2000 George Keith Batchelor FRS (8 March 1920 – 30 March 2000) was an Australian applied mathematician and fluid dynamicist. He was for many years the Professor of Applied Mathematics in the University of Cambridge, and was founding head of the Department of Applied Mathematics and Theoretical Physics (DAMTP). In 1956 he founded the influential Journal of Fluid Mechanics which he edited for some forty years. Prior to Cambridge he studied in Melbourne High School.
As an applied mathematician (and for some years at Cambridge a co-worker with Sir Geoffrey Taylor in the field of turbulent flow), he was a keen advocate of the need for physical understanding and sound experimental basis.
His An Introduction to Fluid Dynamics (CUP, 1967) is still considered a classic of the subject, and has been re-issued in the Cambridge Mathematical Library series, following strong current demand. Unusual for an 'elementary' textbook of that era, it presented a treatment in which the properties of a real viscous fluid were fully emphasized. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1959.*Wik
2010 Jaime Alfonso Escalante Gutierrez (December 31, 1930 — March 30, 2010) was a Bolivian educator well-known for teaching students calculus from 1974 to 1991 at Garfield High School, East Los Angeles, California. Escalante was the subject of the 1988 film Stand and Deliver, in which he is portrayed by Edward James Olmos.*Wik
von Mädler Map of Moon, *www.RareMaps.com
Natural selection is a mechanism for generating an exceedingly high degree of improbability.
~R. A. Fisher
The 89th day of the year; 89 is the fifth Fibonacci prime and the reciprocal of 89 starts out 0.011235... (generating the first five Fibonacci numbers) *Prime Curios
and 89 can be expressed by the first 5 integers raised to the first 5 Fibonacci numbers: 11 + 25 + 33 + 41+ 52
If you write any integer and sum the square of the digits, and repeat, eventually you get either 1, or 89
(ex: 16; \( 1^2 + 6^2 = 37; 3^2 + 7^2 = 58; 5^2 + 8^2 = 89 \)
An Armstrong (or Pluperfect digital invariant) number is a number that is the sum of its own digits each raised to the power of the number of digits. For example, 371 is an Armstrong number since \(3^3+7^3+1^3 = 371\). There are exactly 89 such numbers, including two with 39 digits. (115,132,219,018,763,992,565,095,597,973,522,401 is the largest) (Armstrong numbers are named for Michael F. Armstrong who named them for himself as part of an assignment to his class in Fortran Programming at the University of Rochester \)
89 is a numeric ambigram (a number that rotates to form a different number), and is the sum of four strobogrammatic numbers (rotate and stay the same) , 1+8+11+69 = 89.
1796 Gauss achieved the construction of the 17-gon and a week later he would obtain his first proof of the quadratic reciprocity law. These two accomplishments mark the emergence from the ingenious manipulations of his youth, to the polished proofs of the mature mathematician. *Merzbach, An Early Version of Guass' Disquisitiones Arithmeticae, Mathematical Perspectives Academic Press 1981
first image obtained by NASA's Dawn spacecraft .
In 1807, Vesta 4, the only asteroid visible to the naked eye, thus the brightest on record, was first observed by the amateur astronomer Heinrich Wilhelm Olbers from Bremen. Vesta is a main belt asteroid with a diameter of 525-km and a rotation period of 5.34 hours. Pictures taken by the Hubble Space Telescope in 1995 show Vesta's complex surface, with a geology similar to that of terrestrial worlds - such as Earth or Mars - a surprisingly diverse world with an exposed mantle, ancient lava flows and impact basins. Though no bigger than the state of Arizona, it once had a molten interior. This contradicts conventional ideas that asteroids essentially are cold, rocky fragments left behind from the early days of planetary formation. *TIS Since the discovery of Ceres in 1801, and the asteroid Pallas in 1802, he had corresponded and became close friends with Gauss. For that reason he allowed Gauss to name the new "planet".
1933 Italy issued the world's first postage stamp portraying Galileo. [Scott #D16] *VFR
Galileo Galilei (1564–1642) made his first appearance on this stamp in 1933 for use in pneumatic postal systems (hence the wording "Posta Pneumatica" on the stamp). Pneumatic post involved placing letters in canisters which were then shot along pipes by compressed air from one Post Office to another. Pneumatic postal systems were set up in several European and American cities, including Rome, Naples, and Milan. Italy was the only country to issue stamps specifically for pneumatic postal use. Two of the designs showed Galileo – this one and a modified version with different face value and colour issued in 1945. The portrait is based on one by Justus Sustermans painted in 1636 when Galileo was aged 72. *Ian Ridpath, World's Oldest Astro Stamps page.
1989 Pixar Wins Academy Award for "Tin Toy":
Pixar wins an Academy Award for "Tin Toy," the first entirely computer-animated work to win in the best animated short film category. Pixar, now a division of Disney, continued its success with a string of shorts and the first entirely computer-animated feature-length film, the best-selling "Toy Story." *CHM
2012 Buzz Lightyear that flew in space joins Smithsonian collection. Launched May 31, 2008, aboard the space shuttle Discovery with mission STS-124 and returned on Discovery 15 months later with STS-128, the 12-inch action figure is the longest-serving toy in space. Disney Parks partnered with NASA to send Buzz Lightyear to the International Space Station and create interactive games, educational worksheets and special messages encouraging students to pursue careers in science, technology, engineering and mathematics (STEM). The action figure will go on display in the museum's "Moving Beyond Earth" gallery in the summer. The Toy Story character became part of the National Air and Space Museum's popular culture collection. *http://airandspace.si.edu [I still have a Buzz Lightyear toy on my book case given to me by some students because I used to use his trademark quote in (my very questionable) Latin, "ad infinitum, et ultra." ]
1794 Johann Heinrich von Mädler (29 May 1794, 14 Mar 1874 at age 79) German astronomer who (with Wilhelm Beer) published the most complete map of the Moon of the time, Mappa Selenographica, 4 vol. (1834-36). It was the first lunar map to be divided into quadrants, and it remained unsurpassed in its detail until J.F. Julius Schmidt's map of 1878. Mädler and Beer also published the first systematic chart of the surface features of the planet Mars (1830). *TIS
1825 Francesco Faà di Bruno (29 March 1825–27 March 1888) was an Italian mathematician and priest, born at Alessandria. He was of noble birth, and held, at one time, the rank of captain-of-staff in the Sardinian Army. He is the eponym of Faà di Bruno's formula. In 1988 he was beatified by Pope John Paul II. Today, he is best known for Faà di Bruno's formula on derivatives of composite functions, although it is now certain that the priority in its discovery and use is of Louis François Antoine Arbogast: Faà di Bruno should be only credited for the determinant form of this formula. However, his work is mainly related to elimination theory and to the theory of elliptic functions.
He was the author of about forty original articles published in the "Journal de Mathématiques" (edited by Joseph Liouville), Crelle's Journal, "American Journal of Mathematics" (Johns Hopkins University), "Annali di Tortolini", "Les Mondes", "Comptes rendus de l'Académie des sciences", etc.*Wik
1830 Thomas Bond Sprague (29 March 1830 in London, England - 29 Nov 1920 in Edinburgh, Scotland) studied at Cambridge and went on to become the most important actuary of the late 19th Century. He wrote more than 100 papers including many in the Proceedings of the EMS. *SAU
1873 Tullio Levi-Civita (29 Mar 1873, 29 Dec 1941) Italian mathematician who was one of the founders of absolute differential calculus (tensor analysis) which had applications to the theory of relativity. In 1887, he published a famous paper in which he developed the calculus of tensors. In 1900 he published, jointly with Ricci, the theory of tensors Méthodes de calcul differential absolu et leures applications in a form which was used by Einstein 15 years later. Weyl also used Levi-Civita's ideas to produce a unified theory of gravitation and electromagnetism. In addition to the important contributions his work made in the theory of relativity, Levi-Civita produced a series of papers treating elegantly the problem of a static gravitational field. *TIS
1890 Sir Harold Spencer Jones (29 Mar 1890, 3 Nov 1960) English astronomer who was 10th astronomer royal of England (1933–55). His work was devoted to fundamental positional astronomy. While HM Astronomer at the Cape of Good Hope, he worked on poper motions and parallaxes. Later he showed that small residuals in the apparent motions of the planets are due to the irregular rotation of the earth. He led in the worldwide effort to determine the distance to the sun by triangulating the distance of the asteroid Eros when it passed near the earth in 1930-31. Spencer Jones also improved timekeeping and knowledge of the Earth's rotation. After WW II he supervised the move of the Royal Observatory to Herstmonceux, where it was renamed the Royal Greenwich Observatory.*TIS
1893 Jason John Nassau (29 March 1893 in Smyrna, (now Izmir) Turkey - 11 May 1965 in Cleveland, Ohio, USA) was an American astronomer.
He performed his doctoral studies at Syracuse, and gained his Ph.D. mathematics in 1920. (His thesis was Some Theorems in Alternants.) He then became an assistant professor at the Case Institute of Technology in 1921, teaching astronomy. He continued to instruct at that institution, becoming the University's first chair of astronomy from 1924 until 1959 and chairman of the graduate division from 1936 until 1940. After 1959 he was professor emeritus.
From 1924 until 1959 he was also the director of the Case Western Reserve University (CWRU) Warner and Swasey Observatory in Cleveland, Ohio. He was a pioneer in the study of galactic structure. He also discovered a new star cluster, co-discovered 2 novae in 1961, and developed a technique of studying the distribution of red (M-class or cooler) stars.*Wik
1896 Wilhelm Friedrich Ackermann (29 March 1896 – 24 December 1962) was a German mathematician best known for the Ackermann function, an important example in the theory of computation.*Wik
1912 Martin Eichler (29 March 1912 – 7 October 1992) was a German number theorist. He received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936.
Eichler once stated that there were five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. He is linked with Goro Shimura in the development of a method to construct elliptic curves from certain modular forms. The converse notion that every elliptic curve has a corresponding modular form would later be the key to the proof of Fermat's last theorem.*Wik
1912 Caius Jacob (29 March 1912 , Arad - 6 February 1992 , Bucharest ) was a Romanian mathematician and member of the Romanian Academy. He made contributions in the fields of fluid mechanics and mathematical analysis , in particular vigilance in plane movements of incompressible fluids, speeds of movement at subsonic and supersonic , approximate solutions in gas dynamics and the old problem of potential theory. His most important publishing was Mathematical introduction to the mechanics of fluids. *Wik
1941 Joseph Hooton Taylor, Jr. (March 29, 1941, ) is an American astrophysicist and Nobel Prize in Physics laureate for his discovery with Russell Alan Hulse of a "new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation." *Wi
1772 Emanuel Swedenborg (29 Jan 1688; 29 Mar 1772) Swedish scientist, philosopher and theologian. While young, he studied mathematics and the natural sciences in England and Europe. From Swedenborg's inventive and mechanical genius came his method of finding terrestrial longitude by the Moon, new methods of constructing docks and even tentative suggestions for the submarine and the airplane. Back in Sweden, he started (1715) that country's first scientific journal, Daedalus Hyperboreus. His book on algebra was the first in the Swedish language, and in 1721 he published a work on chemistry and physics. Swedenborg devoted 30 years to improving Sweden's metal-mining industries, while still publishing on cosmology, corpuscular philosophy, mathematics, and human sensory perceptions. *TIS
1806 John Thomas Graves (4 December 1806, Dublin, Ireland–29 March 1870, Cheltenham, England) was an Irish jurist and mathematician. He was a friend of William Rowan Hamilton, and is credited both with inspiring Hamilton to discover the quaternions and with personally discovering the octonions, which he called the octaves. He was the brother of both the mathematician Charles Graves and the writer and clergyman Robert Perceval Graves.
In his twentieth year (1826) Graves engaged in researches on the exponential function and the complex logarithm; they were printed in the Philosophical Transactions for 1829 under the title An Attempt to Rectify the Inaccuracy of some Logarithmic Formulæ. M. Vincent of Lille claimed to have arrived in 1825 at similar results, which, however, were not published by him till 1832. The conclusions announced by Graves were not at first accepted by George Peacock, who referred to them in his Report on Algebra, nor by Sir John Herschel. Graves communicated to the British Association in 1834 (Report for that year) on his discovery, and in the same report is a supporting paper by Hamilton, On Conjugate Functions or Algebraic Couples, as tending to illustrate generally the Doctrine of Imaginary Quantities, and as confirming the Results of Mr. Graves respecting the existence of Two independent Integers in the complete expression of an Imaginary Logarithm. It was an anticipation, as far as publication was concerned, of an extended memoir, which had been read by Hamilton before the Royal Irish Academy on 24 November 1833, On Conjugate Functions or Algebraic Couples, and subsequently published in the seventeenth volume of the Transactions of the Royal Irish Academy. To this memoir were prefixed A Preliminary and Elementary Essay on Algebra as the Science of Pure Time, and some General Introductory Remarks. In the concluding paragraphs of each of these three papers Hamilton acknowledges that it was "in reflecting on the important symbolical results of Mr. Graves respecting imaginary logarithms, and in attempting to explain to himself the theoretical meaning of those remarkable symbolisms", that he was conducted to "the theory of conjugate functions, which, leading on to a theory of triplets and sets of moments, steps, and numbers" were foundational for his own work, culminating in the discovery of quaternions.
For many years Graves and Hamilton maintained a correspondence on the interpretation of imaginaries. In 1843 Hamilton discovered the quaternions, and it was to Graves that he made on 17 October his first written communication of the discovery. In his preface to the Lectures on Quaternions and in a prefatory letter to a communication to the Philosophical Magazine for December 1844 are acknowledgments of his indebtedness to Graves for stimulus and suggestion. After the discovery of quaternions, Graves employed himself in extending to eight squares Euler's four-square identity, and went on to conceive a theory of "octaves" (now called octonions) analogous to Hamilton's theory of quaternions, introducing four imaginaries additional to Hamilton's i, j and k, and conforming to "the law of the modulus".
Graves devised also a pure-triplet system founded on the roots of positive unity, simultaneously with his brother Charles Graves, the bishop of Limerick. He afterwards stimulated Hamilton to the study of polyhedra, and was told of the discovery of the icosian calculus. *Wik
1873 Francesco Zantedeschi (born 1797, 29 Mar 1873) Italian priest and physicist, who published papers (1829, 1830) on the production of electric currents in closed circuits by the approach and withdrawal of a magnet, preceding Faraday's classic experiment of 1831. Studying the solar spectrum, Zantedeschi was among the first to recognize the marked absorption by the atmosphere of the red, yellow, and green light. Though not confirmed, he also thought he detected a magnetic action on steel needles by ultra-violet light (1838), at least suspecting a connection between light and magnetism many years before Clerk-Maxwell's announcement (1867) of the electromagnetic theory of light. He experimented on the repulsion of flames by a strong magnetic field.*TIS
1912 Robert Falcon Scott, (6 June 1868 - 29 March 1912) was a Royal Navy officer and explorer who led two expeditions to the Antarctic regions: the Discovery Expedition, 1901–04, and the ill-fated Terra Nova Expedition, 1910–13. During this second venture, Scott led a party of five which reached the South Pole on 17 January 1912, only to find that they had been preceded by Roald Amundsen's Norwegian expedition. On their return journey, Scott and his four comrades all died from a combination of exhaustion, starvation and extreme cold. *Wik
1944 Grace Chisholm Young (née Chisholm; 15 March 1868 – 29 March 1944) was an English mathematician. She was educated at Girton College, Cambridge, England and continued her studies at Göttingen University in Germany. Her early writings were published under the name of her husband, William Henry Young, and they collaborated on mathematical work throughout their lives. For her work on calculus (1914–16), she was awarded the Gamble Prize.
Her son, Laurence Chisholm Young, was also a prominent mathematician. One of her living granddaughters, Sylvia Wiegand (daughter of Laurence), is also a mathematician (and a past president of the Association for Women in Mathematics.)*Wik
1980 William Gemmell Cochran (15 July 1909, Rutherglen – 29 March 1980, Orleans, Massachusetts)In 1934 R A Fisher left Rothamsted Experimental Station to accept the Galton chair at University College, London and Frank Yates became head at Rothamsted. Cochran was offered the vacant post but he had not finished his doctoral course at Cambridge. Yates later wrote:-
... it was a measure of good sense that he accepted my argument that a PhD, even from Cambridge, was little evidence of research ability, and that Cambridge had at that time little to teach him in statistics that could not be much better learnt from practical work in a research institute.
Cochran accepted the post at Rothamsted where he worked for 5 years on experimental designs and sample survey techniques. During this time he worked closely with Yates. At this time he also had the chance to work with Fisher who was a frequent visitor at Rothamsted.
Cochran visited Iowa Statistical Laboratory in 1938, then he accepted a statistics post there in 1939. His task was to develop the graduate programe in statistics within the Mathematics Department. In 1943 he joined Wilks research team at Princeton.
At Princeton he was involved in war work examining probabilities of hits in naval warfare. By 1945 he was working on bombing raid strategies.
He joined the newly created North Carolina Institute of Statistics in 1946, again to develop the graduate programe in statistics. From 1949 until 1957 he was at Johns Hopkins University in the chair of biostatistics. Here he was more involved in medical applications of statistics rather than the agricultural application he had studied earlier.
From 1957 until he retired in 1976 Cochran was at Harvard. His initial task was to help set up a statistics department, something which he had a great deal of experience with by this time. He had almost become a professional at starting statistics within universities in the USA. *SAU
1983 Sir Maurice George Kendall, FBA (6 September 1907 – 29 March 1983) was a British statistician, widely known for his contribution to statistics. The Kendall tau rank correlation is named after him.*Wik He was involved in developing one of the first mechanical devices to produce (pseudo-) random digits, eventually leading to a 100,000-random-digit set commonly used until RAND's (once well-known) "A Million Random Digits With 100,000 Normal Deviates" in 1955.
Kendall was Professor of Statistics at the London School of Economics from 1949 to 1961. His main work in statistics involved k-statistics, time series, and rank-correlation methods, including developing the Kendall's tau stat, which eventually led to a monograph on Rank Correlation in 1948. He was also involved in several large sample-survey projects. For many, what Kendall is best known for is his set of books titled The Advanced Theory of Statistics (ATS), with Volume I first appearing in 1943 and Volume II in 1946. Kendall later completed a
rewriting of ATS, which appeared in three volumes in 1966, which were updated by collaborator Alan Stuart and Keith Ord after Kendall's death, appearing now as "Kendall's Advanced Theory of Statistics". *David Bee
1999 Boris A. Kordemsky ( 23 May 1907 – 29 March, 1999) was a Russian mathematician and educator. He is best known for his popular science books and mathematical puzzles. He is the author of over 70 books and popular mathematics articles.
Kordemsky received Ph.D. in education in 1956 and taught mathematics at several Moscow colleges.
He is probably the best-selling author of math puzzle books in the history of the world. Just one of his books, Matematicheskaya Smekalka (or, Mathematical Quick-Wits), sold more than a million copies in the Soviet Union/Russia alone, and it has been translated into many languages. By exciting millions of people in mathematical problems over five decades, he influenced generations of solvers both at home and abroad. *Age of Puzzles, by Will Shortz and Serhiy Grabarchuk (mostly)
1908 John Bardeen (23 May 1908; 30 Jan 1991 at age 82) American physicist who was cowinner of the Nobel Prize for Physics in both 1956 and 1972. He shared the 1956 prize with William B. Shockley and Walter H. Brattain for their joint invention of the transistor. With Leon N. Cooper and John R. Schrieffer he was awarded the 1972 prize for development of the theory of superconductors, usually called the BCS-theory (after the initials of their names). *TIS
*George W. Hart, Sculpture
`The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps ...'.
Alexandre Grothendieck in a letter in 1982 to Ronald Brown
The 88th day of the year; 882 = 7744, it is one of only 5 numbers known whose square has no isolated digits. (Can you find the others?) [Thanks to Danny Whittaker @nemoyatpeace for a correction on this.]
There are only 88 narcissistic numbers in base ten, (an n-digit number that is the sum of the nth power of its digits, 153=13 + 53 + 33
88 is also a chance to introduce a new word (new for me). 88 is strobogrammatic, a number that is the same when it is rotated 180o about its center... 69 is another example. If they make a different number when rotated, they are called invertible (109 becomes 601 for example). *Prime Curios
And with millions (billions?) of stars in the sky, did you ever wonder how many constellations there are? Well, according to the Internationals Astronomical Union, there are 88.
Currently, 14 men and women, 9 birds, two insects, 19 land animals, 10 water creatures, two centaurs, one head of hair, a serpent, a dragon, a flying horse, a river and 29 inanimate objects are represented in the night sky (the total comes to more than 88 because some constellations include more than one creature.)
And if you chat with Chinese friends, the cool way to say bye-bye is with 88, from Mandarin for 88, "bā ba".
Not too far from my home near Possum Trot, Ky, there is a little place called Eighty-eight, Kentucky. One strory of the naming (there could be as many as 88 of them) is that the town was named in 1860 by Dabnie Nunnally, the community's first postmaster. He had little faith in the legibility of his handwriting, and thought that using numbers would solve the problem. He then reached into his pocket and came up with 88 cents.
In the 1948 presidential election, the community reported 88 votes for Truman and 88 votes for Dewey, which earned it a spot in Ripley's Believe It or Not.
And expanding the "88 is strobogrammatic" theme, INDER JEET TANEJA came up with this beautiful magic square with a constant of 88 that was used in a stamp series in Macao in 2014 and 2015. This image shows the reflections both horizontally and vertically, as well as the 180 degree rotation, each is a magic square.
The stamps had denominations of 1 through 9 pataca and when two sheets were printed you could do your own Luo Shu magic square with the denominations. The Luo Shu itself was featured on the 12 pataca stamp.
In 1747, the fascination with electricity upon reaching the American colonies was the subject of Benjamin Franklin's first of the famous series of letters in which he described his experiments on electricity to Peter Collinson, Esq., of London. He thanked Collison for his "kind present of an electric tube with directions for using it" with which he and others did electrical experiments. "For my own part I never was before engaged in any study that so totally engrossed my attention and my time as this has lately done; for what with making experiments when I can be alone, and repeating them to my friends and acquaintances, who, from the novelty of the thing, come continually in crowds to see them, I have, during some months past, had little leisure for anything else."*TIS
1764 In a second trial of John Harrison's marine timekeeper, his son William departed for Barbados aboard the Tartar. As with the first trial, William used H4 to predict the ship's arrival at Madeira with extraordinary accuracy. The watch's error was computed to be 39.2 seconds over a voyage of 47 days, three times better than required to win the maximum reward of £20,000. *Royal Museum Greenwich
1802 Olbers, while observing the constellation Virgo, had observed a "star" of the seventh-magnitude not found on the star charts. Over the following week he would observe the motion and determined that it was a planet. In early April he sent the data to Gauss to compute the orbit. On the 18th of April, Gauss computed the orbit in only three hours, placing the orbit between Mars and Jupitor. Olbers named the new planetoid Pallas, and predicted there would be others found in the same area. John Herschel dismissed this speculation as "dreams in which astronomers... indulge" but over 1000 such planetoids have been observed. *Dunnington, Gray, & Dohse; Carl Friedrich Gauss: Titan of Science
1809 Gauss finished work on his Theoria Motus. It explains his methods of computing planetary orbits using least squares. [Springer's 1985 Statistics Calendar] *VFR
In 1946, the Census Bureau and the National Bureau of Standards met to discuss the purchase of a computer. The agencies agreed to buy UNIVAC, the world's first general all-purpose business computer, from Presper Eckert and John Mauchly for a mere $225,000. Unfortunately, UNIVAC cost far more than that to develop. Eckert and Mauchly's venture floundered as the company continued to build and program UNIVACs for far less than the development cost. Eventually, the company was purchased by Remington Rand. *TIS
1949 The phrase "Big Bang" is created. Shortly after 6:30 am GMT on BBC's The Third Program, Fred Hoyle used the term in describing theories that contrasted with his own "continuous creation" model for the Universe. "...based on a theory that all the matter in the universe was created in one big bang ... ". *Mario Livio, Brilliant Blunders
1959 Germany issued a stamp commemorating the 400th anniversary of the death of Adam Riese [Scott #799] *VFR I understand that the German expression "nach Adam Riese", is still used today. It means "according to Adam Ries" and it is used in saying something is exactly correct.
In 2006, a substantial "lost" book of manuscripts by Robert Hooke in his own handwriting was bought for the Royal Society by donations of nearly £1 million. The book was just minutes before going on the auction block when a last-minute purchase agreement was made and kept the precious document in Britain. Hooke is now often overlooked, except for his law of elasticity, although in his time, he was a prolific English scientist and contributed greatly to planning the rebuilding of London after the Great Fire of 1666. The document of more than 520 pages of manuscripts included the minutes of the Royal Society from 1661-82. It had been found in a cupboard in a private house by an antiques expert there to value other items. *TIS
BIRTHS 1847 Gyula Farkas (28 March 1847 in Sárosd, Fejér County, Hungary - 27 Dec 1930 in Pestszentlorinc, Hungary) He is remembered for Farkas theorem which is used in linear programming and also for his work on linear inequalities. In 1881 Gyula Farkas published a paper on Farkas Bolyai's iterative solution to the trinomial equation, making a careful study of the convergence of the algorithm. In a paper published three years later, Farkas examined the convergence of more general iterative methods. He also made major contributions to applied mathematics and physics, particularly in the areas of mechanical equilibrium, thermodynamics, and electrodynamics.*SAU
1923 Israel Nathan Herstein (March 28, 1923, Lublin, Poland – February 9, 1988, Chicago, Illinois) was a mathematician, appointed as professor at the University of Chicago in 1951. He worked on a variety of areas of algebra, including ring theory, with over 100 research papers and over a dozen books.
He is known for his lucid style of writing, as exemplified by the classic and widely influential Topics in Algebra, an undergraduate introduction to abstract algebra that was published in 1964, which dominated the field for 20 years. A more advanced classic text is his Noncommutative Rings in the Carus Mathematical Monographs series. His primary interest was in noncommutative ring theory, but he also wrote papers on finite groups, linear algebra, and mathematical economics.*Wik
1928 Alexander Grothendieck (28 Mar 1928-13 November 2014) In 1966 he won a Fields Medal for his work in algebraic geometry. He introduced the idea of K-theory and revolutionized homological algebra. Within algebraic geometry itself, his theory of schemes is used in technical work. His generalization of the classical Riemann-Roch theorem started the study of algebraic and topological K-theory. His construction of new cohomology theories has left consequences for algebraic number theory, algebraic topology, and representation theory. His creation of topos theory has appeared in set theory and logic.
One of his results is the discovery of the first arithmetic Weil cohomology theory: the ℓ-adic étale cohomology. This result opened the way for a proof of the Weil conjectures, ultimately completed by his student Pierre Deligne. To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with applications to the Langlands program.
Grothendieck influenced generations of mathematicians after his departure from mathematics. His emphasis on the role of universal properties brought category theory into the mainstream as an organizing principle. His notion of abelian category is now the basic object of study in homological algebra. His conjectural theory of motives has been behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration. *Wik
1678 Claude François Milliet Dechales (1621 in Chambéry, France - 28 March 1678 in Turin, Italy) Dechales is best remembered for Cursus seu mundus mathematicus published in Lyons in 1674, a complete course of mathematics. Topics covered in this wide ranging work included practical geometry, mechanics, statics, magnetism and optics as well as topics outwith the usual topics of mathematics such as geography, architecture, astronomy, natural philosophy and music. In 1678 he published in Lausanne his edition of Euclid, The Elements of Euclid Explained in a New but Most Easy Method: Together with the Use of Every Proposition through All Parts of the Mathematics, written in French by That Most Excellent Mathematician, F Claude Francis Milliet Dechales of the Society of Jesus. This work covers Books 1 to 6, together with Books 11 and 12, of Euclid's Elements. A second edition was published in 1683, then an edition revised by Ozanam was published in Paris in 1753. An English translation was published in London by M Gillyflower and W Freeman, the translation being by Reeve Williams. A second edition of this English translation appeared in 1696. Schaap writes, "Dechales's separate edition of Euclid, long a favourite in France and elsewhere on the Continent, never became popular in England." *SAU
1794 Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet (17 September 1743 – 28 March 1794), known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election. Unlike many of his contemporaries, he advocated a liberal economy, free and equal public education, constitutionalism, and equal rights for women and people of all races. His ideas and writings were said to embody the ideals of the Age of Enlightenment and rationalism, and remain influential to this day. He died a mysterious death in prison after a period of being a fugitive from French Revolutionary authorities.*Wik
Condorcet committed suicide by poisoning while in jail so that the republican terrorists could not take him to Paris. *VFR (The St Andrews site has the date of his death one day later.)
1840 Simon Antoine Jean Lhuilier (24 April 1750 in Geneva, Switzerland - 28 March 1840 in Geneva, Switzerland) His work on Euler's polyhedra formula, and exceptions to that formula, were important in the development of topology. Lhuilier also corrected Euler's solution of the Königsberg bridge problem. He also wrote four important articles on probability during the years 1796 and 1797. His most famous pupil was Charles-François Sturm who studied under Lhuilier during the last few years of his career in Geneva. *SAU
1850 Bernt Michael Holmboe (23 March 1795 – 28 March 1850) was a Norwegian mathematician. Holmboe was hired as a mathematics teacher at the Christiania Cathedral School in 1818, where he met the future renowned mathematician Niels Henrik Abel. Holmboe's lasting impact on mathematics worldwide has been said to be his tutoring of Abel, both in school and privately. The two became friends and remained so until Abel's early death. Holmboe moved to the Royal Frederick University in 1826, where he worked until his own death in 1850.
Holmboe's significant impact on mathematics in the fledgling Norway was his textbook in two volumes for secondary schools. It was widely used, but faced competition from Christopher Hansteen's alternative offering, sparking what may have been Norway's first debate about school textbooks. *Wik
1874 Peter Andreas Hansen (8 Dec 1795; 28 Mar 1874) Danish astronomer whose most important work was the improvement of the theories and tables of the orbits of the principal bodies in the solar system. At Altona observatory he assisted in measuring the arc of meridian (1821). He became the director (1825) of Seeberg observatory, which was removed to Gotha in a new observatory built for him (1857). He worked on theoretical geodesy, optics, and the theory of probability. The work in celestial mechanics for which he is best known are his theories of motion for comets, minor planets, moon and his lunar tables (1857) which were in use until 1923. He published his lunar theory in Fundamenta ("Foundation") in 1838, and Darlegung ("Explanation") in 1862-64.*TIS
1950 Ernst David Hellinger (0 Sept 1883 in Striegau, Silesia, Germany (now Strzegom, Poland) - 28 March 1950 in Chicago, Illinois, USA) introduced a new type of integral: the Hellinger integral . Jointly with Hilbert he produced an important theory of forms. From 1907 to 1909 he was an assistant at Göttingen and, during this time, he ".. edited Hilbert's lecture notes and Felix Klein's influential Elementarmathematik vom höheren Standpunkte aus (Berlin, 1925) which was translated into English (New York, 1932). Years later the story is told that,
Shortly after his arrival at Northwestern, one of the professors in describing Northwest's mathematics program to him remarked that in the honours course Felix Klein's 'Elementary mathematics from an advanced standpoint' was used as a text and "perhaps Hellinger was familiar with it". At this Hellinger ... replied "familiar with it, I wrote it!".
Charles Minard's Napoleon's March,
Modern science, as training the mind to an exact and impartial analysis of facts, is an education specially fitted to promote citizenship.
~Karl Pearson
The 87th day of the year; the sum of the squares of the first four primes is 87. \(87 = 2^2 + 3^2 + 5^2 + 7^2 \)
87 = 3 * 29, \(87^2 + 3^2 + 29^2 and 87^2 - 3^2 - 29^2 \)are both primes
Among Australian cricket players, it seems, 87 is an unlucky score and is referred to as "the devil's number", supposedly because it is 13 runs short of 100.
And 87 is, of course, the number of years between the signing of the U.S. Declaration of Independence and the Battle of Gettysburg, immortalized in Abraham Lincoln's Gettysburg Address with the phrase "fourscore and seven years ago..."
In 1827, Charles Darwin, aged 18, submitted his first report of an original scientific discovery to the Plinian Society in Edinburgh, Scotland. Darwin had discovered several things about the biology of tiny marine organisms found along the Scottish coast. *TIS
1921 On the morning of Easter Sunday, Otto Loewi awoke with the memory he had had an important dream during the night and written down some notes, but when he tried to retrieve them, the writing was hopelessly illegible. After trying to recall the dream all day, he retired early in the evening and eventually the dream came again. The dream was about a way to determine if transmissions between nerve cells was chemical or not. He immediately got out of bed and went to his laboratory. With a single experiment on a frog's heart he confirmed his own thesis of seventeen years before, that the transfer was indeed a chemical process. *Michael Brooks, Free Radicals (pg 24-25)
1958 The first national high-school mathematics competition in the U.S. was held. Since 1983 it has been known as the American High School Mathematics Examination (AHSME). [The College Mathematics Journal, 16 (1985), p. 331] *VFR
1976 20-Year Old Bill Gates Gives Opening Address to Hobbyists:
Bill Gates gives the opening address at the First Annual World Altair Computer Convention in Albuquerque, N.M. MITS, the company that developed the Altair, had set up shop in the southwestern city to develop its kit computer, which was a hit among hobbyists after it graced the cover of "Popular Mechanics" magazine. Gates, then a 20-year-old erstwhile Harvard student, had helped develop the form of BASIC sold with the Altair. *CHM
1781 Charles Joseph Minard (27 Mar 1781; 24 Oct 1870 at age 89) French civil engineer who made significant contributions to the graphical representations of data. His best-known work, Carte figurative des pertes successives en hommes de l'Armee Français dans la campagne de Russe 1812-1813, dramatically displays the number of Napoleon's soldiers by the width of an ever-reducing band drawn across a map from France to Moscow. At its origin, a wide band shows 442,000 soldiers left France, narrowing across several hundred miles to 100,000 men reaching Moscow. With a parallel temperature graph displaying deadly frigid Russian winter temperatures along the way, the band shrinks during the retreat to a pathetic thin trickle of 10,000 survivors returning to their homeland. *TIS
1824 Johann Wilhelm Hittorf (27 Mar 1824, 28 Nov 1914) German physicist who was a pioneer in electrochemical research. His early investigations were on the allotropes (different physical forms) of phosphorus and selenium. He was the first to compute the electricity- carrying capacity of charged atoms and molecules (ions), an important factor in understanding electrochemical reactions. He investigated the migration of ions during electrolysis (1853-59), developed expressions for and measured transport numbers. In 1869, he published his laws governing the migration of ions. For his studies of electrical phenomena in rarefied gases, the Hittorf tube has been named for him. Hittorf determined a number of properties of cathode rays, including (before Crookes) the deflection of the rays by a magnet. *TIS
1845 Wilhelm Conrad Röntgen (27 Mar 1845 - 10 Feb 1923 at age 77) was a German physicist who discovered the highly penetrating form of radiation that became known as X-rays on 8 Nov 1895. He received the first Nobel Prize for Physics (1901), "in recognition of the extraordinary services he has rendered by the discovery of the remarkable rays subsequently named after him." This high-energy radiation, though first called Röngen rays, became known as X-rays. His discovery initiated revolutionary improvements in making medical diagnoses and enabled many new advances in modern physics. *TIS "In 1901 he became the first physicist to receive a Nobel prize." *VFR
1855 Sir Alfred Ewing (27 Mar 1855, 7 Jan 1935) was a Scottish physicist who discovered and named hysteresis (1881), the resistance of magnetic materials to change in magnetic force. Ewing was born and educated in Dundee and studied engineering on a scholarship at Edinburgh University. He helped Sir William Thomson, later Lord Kelvin in a cable laying project. In 1878 he became professor of Mechanical Engineering and Physics at Tokyo University, where he devised instruments for measuring earthquakes. In 1903 he moved to the Admiralty as head of education and training, where during WW I, he and his staff took on the task of deciphering coded messages. *TIS
1857 Karl Pearson (27 Mar 1857; 27 Apr 1936 at age 79) English mathematician who was one of the founders of modern statistics. His lectures as professor of geometry evolved into The Grammar of Science (1892), his most widely read book and a classic in the philosophy of science. Stimulated by the evolutionary writings of Francis Galton and a personal friendship with Walter F.R. Weldon, Pearson became immersed in the problem of applying statistics to biological problems of heredity and evolution. The methods he developed are essential to every serious application of statistics. From 1893 to 1912 he wrote a series of 18 papers entitled Mathematical Contributions to the Theory of Evolution, which contained much of his most valuable work, including the chi-square test of statistical significance. *TIS
1897 Douglas Rayner Hartree PhD, FRS (27 March 1897 – 12 February 1958) was an English mathematician and physicist most famous for the development of numerical analysis and its application to the Hartree-Fock equations of atomic physics and the construction of the meccano differential analyser. *Wik
1905 László Kalmár (27 March 1905 in Edde (N of Kaposvar), Hungary - 2 Aug 1976 in Mátraháza, Hungary) worked on mathematical logic and theoretical computer science. He was ackowledged as the leader of Hungarian mathematical logic. *SAU
1928 Alexander Grothendieck In 1966 he won a Fields Medal for his work in algebraic geometry. He introduced the idea of K-theory and revolutionized homological algebra. *VFR
1850 Wilhelm Beer (4 Jan 1797, 27 Mar 1850 at age 53) German banker and amateur astronomer who owned a fine Fraunhofer refractor which he used in his own a private observatory. He worked jointly with Johann Heinrich von Mädler, to produce the first large-scale moon map to be based on precise micrometric measurements. Their four-year effort was published as Mappa Selenographica (1836). This fine lithographed map provided the most complete details of the Moon's surface in the first half of the 19th century. It was the first lunar map divided in quadrants, and recorded the Moon's face in great detail detail. It was drawn to a scale of scale of just over 38 inches to the moon's diameter. Mädler originated a convention for naming minor craters with Roman letters appended to the name of the nearest large crater (ex. Egede A,B, and C).
1923 Sir James Dewar (20 Sep 1842; 27 Mar 1923) British chemist and physicist. Blurring the line between physics and chemistry, he advanced the research frontier in several fields at the turn of the century, and gave dazzling lectures. His study of low-temperature phenomena entailed making an insulating double-walled flask of his own design by creating a vacuum between the two silvered layers of steel or glass (1892). This Dewar flask that has been named for him led to the domestic Thermos bottle. In June 1897, The Scientific American reported that "Dewar has just succeeded in liquefying fluorine gas at a temperature of -185 degrees C." He obtained liquid hydrogen in 1898. Dewar also invented cordite, the first smokeless powder.*TIS
1925 Carl Gottfried Neumann,(7 May 1832 in Königsberg, Germany (now Kaliningrad, Russia) - 27 March 1925 in Leipzig, Germany) He worked on a wide range of topics in applied mathematics such as mathematical physics, potential theory and electrodynamics. He also made important pure mathematical contributions. He studied the order of connectivity of Riemann surfaces.
During the 1860s Neumann wrote papers on the Dirichlet principle and the 'logarithmic potential', a term he coined. In 1890 Émile Picard used Neumann's results to develop his method of successive approximation which he used to give existence proofs for the solutions of partial differential equations.*SAU
1929 Samuil Shatunovsky (25 March 1859 – 27 March 1929) was a Russian mathematician. focused on several topics in mathematical analysis and algebra, such as group theory, number theory and geometry. Independently from Hilbert, he developed a similar axiomatic theory and applied it in geometry, algebra, Galois theory and analysis.[1] However, most of his activity was devoted to teaching at Odessa University and writing associated books and study materials.*Wik
1972 Maurits Cornelius Escher (17 June 1898 in Leeuwarden, Netherlands - 27 March 1972 in Laren, Netherlands) an artist whose works have included a considerable mathematical content. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture, and tessellations. *Wik
On This Day in Math - March 9
On This Day in Math -March 6 | CommonCrawl |
\begin{definition}[Definition:Interior (Topology)/Definition 2]
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
The '''interior''' of $H$ is defined as the largest open set of $T$ which is contained in $H$.
\end{definition} | ProofWiki |
\begin{document}
\title{Affine Monotonic and Risk-Sensitive Models in Dynamic Programming}
\author{Dimitri~P.~Bertsekas \thanks{} \thanks{D.\ P.\ Bertsekas is with the Laboratory for Information and Decision Systems (LIDS), M.I.T. Email: [email protected]. Address: 77 Massachusetts Ave.,\ Rm 32-D660, Cambridge, MA 02139, USA.} }
\markboth{Report~LIDS-3204,~June 2016~ (Revised,~November~2017)} {Bertsekas: Affine Monotonic and Risk-Sensitive Models in Dynamic Programming}
\maketitle
\begin{abstract} In this paper we consider a broad class of infinite horizon discrete-time optimal control models that involve a nonnegative cost function and an affine mapping in their dynamic programming equation. They include as special cases several classical models such as stochastic undiscounted nonnegative cost problems, stochastic multiplicative cost problems, and risk-sensitive problems with exponential cost. We focus on the case where the state space is finite and the control space has some compactness properties. We assume that the affine mapping has a semicontractive character, whereby for some policies it is a contraction, while for others it is not. In one line of analysis, we impose assumptions guaranteeing that the noncontractive policies cannot be optimal. Under these assumptions, we prove strong results that resemble those for discounted Markovian decision problems, such as the uniqueness of solution of Bellman's equation, and the validity of forms of value and policy iteration. In the absence of these assumptions, the results are weaker and unusual in character: the optimal cost function need not be a solution of Bellman's equation, and an optimal policy may not be found by value or policy iteration. Instead the optimal cost function over just the contractive policies is the largest solution of Bellman's equation, and can be computed by a variety of algorithms. \end{abstract}
\begin{IEEEkeywords} Dynamic programming, Markov decision processes, stochastic shortest paths, risk sensitive control. \end{IEEEkeywords}
\section{Introduction} \label{sec-intro}
We consider an infinite horizon optimal control model, characterized by an affine and monotone abstract mapping that underlies the associated Bellman equation of dynamic programming (DP for short). This model was formulated with some analysis in the author's monograph [1] as a special case of abstract DP. In the present paper we will provide a deeper analysis and more effective algorithms for the finite-state version of the model, under considerably weaker assumptions.
To relate our analysis with the existing literature, we note that DP models specified by an abstract mapping defining the corresponding Bellman equation have a long history. Models where this mapping is a sup-norm contraction over the space of bounded cost functions were introduced by Denardo [2]; see also Denardo and Mitten [3]. Their main area of application is discounted DP models of various types. Noncontractive models, where the abstract mapping is not a contraction of any kind, but is instead monotone, were considered by Bertsekas [4], [5] (see also Bertsekas and Shreve [6], Ch.\ 5). Among others, these models cover the important cases of positive and negative (reward) DP problems of Blackwell [7] and Strauch [8], respectively. Extensions of the analysis of [5] were given by Verdu and Poor [9], which considered additional structure that allows the development of backward and forward value iterations, and in the thesis by Szepesvari [10], [11], which introduced non-Markovian policies into the abstract DP framework. The model of [5] was also used to develop asynchronous value iteration methods for abstract contractive and noncontractive DP models; see [12], 13]. Moreover, there have been extensions of the theory to asynchronous policy iteration algorithms and approximate DP by Bertsekas and Yu ([13], [14], [15]).
A type of abstract DP model, called {\it semicontractive\/}, was introduced in the monograph [1]. In this model, the abstract DP mapping corresponding to some policies has a regularity/contraction-like property, but the mapping of others does not. A prominent example is the stochastic shortest path problem (SSP for short), a Markovian decision problem where we aim to drive the state of a finite-state Markov chain to a cost-free and absorbing termination state at minimum expected cost. In SSP problems, the contractive policies are the so-called {\it proper} policies, which are the ones that lead to the termination state with probability 1. The SSP problem, originally introduced by Eaton and Zadeh [16], has been discussed under a variety of assumptions, in many sources, including the books [17], [18], [19], [20], [21], [22], [23], [24], and [12], where it is sometimes referred to by other names such as ``first passage problem" and ``transient programming problem." It has found a wide range of applications in regulation, path planning, robotics, and other contexts.
In this paper we focus on a different subclass of semicontractive models, called {\it affine monotonic\/}, where the abstract mapping associated with a stationary policy is affine and maps nonnegative functions to nonnegative functions. These models include as special cases stochastic undiscounted nonnegative cost problems (including SSP problems with nonnegative cost per stage), and multiplicative cost problems, such as problems with exponentiated cost. A key idea in our analysis is to use the notion of a {\it contractive policy} (one whose affine mapping involves a matrix with eigenvalues lying strictly within the unit circle). This notion is analogous to the one of a proper policy in SSP problems and is used in similar ways.
Our analytical focus is on the validity and the uniqueness of solution of Bellman's equation, and the convergence of (possibly asynchronous) forms of value and policy iteration. Our results are analogous to those obtained for SSP problems by Bertsekas and Tsitsiklis [25], and Bertsekas and Yu [26]. As in the case of [25], under favorable assumptions where noncontractive policies cannot be optimal, we show that the optimal cost function is the unique solution of Bellman's equation, and we derive strong algorithmic results. As in the case of [26], we consider more general situations where the optimal cost function need not be a solution of Bellman's equation, and an optimal policy may not be found by value or policy iteration. To address such anomalies, we focus attention on the optimal cost function over just the contractive policies, and we show that it is the largest solution of Bellman's equation and that it is the natural limit of value iteration.
However, there are some substantial differences from the analyses of [25] and [26]. The framework of the present paper is broader than SSP and includes in particular multiplicative cost problems. Moreover, some of the assumptions are different and necessitate a different line of analysis; for example there is no counterpart of the assumption that the optimal cost function is real-valued, which is fundamental in the analysis of [26]. As an indication, we note that deterministic shortest path problems with negative cost cycles can be readily treated within the framework of the present paper, but cannot be analyzed as SSP problems within the standard framework of [25] and the weaker framework of [26] because their optimal shortest path length is equal to $-\infty$ for some initial states.
In this paper, we also pay special attention to exponential cost problems, extending significantly some of the classical results of Denardo and Rothblum [27], and the more recent results of Patek [28]. Both of these papers impose assumptions that guarantee that the optimal cost function is the unique solution of Bellman's equation, whereas our assumptions are much weaker. The paper by Denardo and Rothblum [27] also assumes a finite control space in order to bring to bear a line of analysis based on linear programming (see also the discussion of Section II).
The paper is organized as follows. In Section II we introduce the affine monotonic model, and we show that it contains as a special case multiplicative and exponential cost models. We also introduce contractive policies and related assumptions. In Section III we address the core analytical questions relating to Bellman's equation and its solution, and we obtain favorable results under the assumption that all noncontractive policies have infinite cost starting from some initial state. In Section IV we remove this latter assumption, and we show favorable results relating to a restricted problem whereby we optimize over the contractive policies only. Algorithms such as value iteration, policy iteration, and linear programming are discussed somewhat briefly in this paper, since their analysis follows to a great extent established paths for semicontractive abstract DP models [1].
Regarding notation, we denote by $\Re^n$ the standard Euclidean space of vectors $J=\big(J(1),\ldots,J(n)\big)$ with real-valued components, and we denote by $\Re$ the real line. We denote by $\Re^n_+$ the set of vectors with nonnegative real-valued components, $$\Re^n_+=\big\{J\mid 0\le J(i)<\infty,\ i=1,\ldots,n\big\},$$ and by ${\cal E}_+^n$ the set of vectors with nonnegative extended real-valued components, $${\cal E}_+^n=\big\{J\mid 0\le J(i)\le\infty,\ i=1,\ldots,n\big\}.$$ Inequalities with vectors are meant to be componentwise, i.e., $J\le J'$ means that $J(i)\le J'(i)$ for all $i$. Similarly, in the absence of an explicit statement to the contrary, operations on vectors, such as $\lim$, $\limsup$, and $\inf$, are meant to be componentwise.
\section{Problem Formulation}
We consider a finite state space $X=\{1,\ldots,n\}$ and a (possibly infinite) control constraint set $U(i)$ for each state $i$. Let ${\cal M}$ denote the set of all functions $\m=\big(\m(1),\ldots,\m(n)\big)$ such that $\m(i)\in U(i)$ for each $i=1,\ldots,n$. By a {\it policy} we mean a sequence of the form $\pi=\{\mu_0,\mu_1,\ldots\}$, where $\mu_k\in{\cal M}$ for all $k=0,1,\ldots$. By a {\it stationary policy} we mean a policy of the form $\{\m,\m,\ldots\}$. For convenience we also refer to any $\m\in{\cal M}$ as a ``policy" and use it in place of the stationary policy $\{\m,\m,\ldots\}$, when confusion cannot arise.
We introduce for each $\m\in{\cal M}$ a mapping $T_\m:{\cal E}_+^n\mapsto{\cal E}_+^n$ given by \begin{equation} \label{eq-mumap} T_\m J=b_\m+A_\m J, \end{equation} where $b_\m$ is a vector of $\Re^n$ with components $b\big(i,\m(i)\big)$, $i=1,\ldots,n$, and $A_\m$ is an $n\times n$ matrix with components $A_{ij}\big(\m(i)\big)$, $i,j=1,\ldots,n$. We assume that $b(i,u)$ and $A_{ij}(u)$ are nonnegative, \begin{equation} \label{eq-affmonnoneg} b(i,u)\ge0,\quad A_{ij}(u)\ge0,\quad \forall\ i,j=1,\ldots,n,\ u\in U(i). \end{equation} We define the mapping $T:{\cal E}_+^n\mapsto{\cal E}_+^n$, where for each $J\in {\cal E}_+^n$, $TJ$ is the vector of ${\cal E}_+^n$ with components \begin{align} \label{eq-tmapaffmon} (TJ)(i)=\inf_{u\in U(i)}&\left[b(i,u)+\sum_{j=1}^nA_{ij}(u)J(j)\right],\notag\\ &\qquad \qquad \qquad i=1,\ldots,n. \end{align} Note that since the value of the expression in braces on the right depends on $\m$ only through $\m(i)$, which is just restricted to be in $U(i)$, we have $$(TJ)(i)=\inf_{\m\in{\cal M}}(T_{\m} J)(i),\qquad i=1,\ldots,n,$$ so that $(TJ)(i)\le (T_{\m} J)(i)$ for all $i$ and $\m\in{\cal M}$.
We now define a DP-like optimization problem that involves the mappings $T_\m$. We introduce a special vector $\bar J\in\Re^n_+$, and we define the cost function of a policy $\pi=\{\m_0,\m_1,\ldots\}$ in terms of the composition of the mappings $T_{\m_k}$, $k=0,1,\ldots$, by $$J_\pi(i)=\limsup_{N\to\infty}\,(T_{\m_0}\cdots T_{\m_{N-1}} \bar J)(i),\qquad i=1,\ldots,n.$$ The cost function of a stationary policy $\m$, is written as $$J_\m(i)=\limsup_{N\to\infty}\,(T_{\m}^N \bar J)(i),\qquad i=1,\ldots,n.$$ (We use $\limsup$ because we are not assured that the limit exists; our analysis and results remain essentially unchanged if $\limsup$ is replaced by $\liminf$.) In contractive abstract DP models, $T_\m$ is assumed to be a contraction for all $\m\in{\cal M}$, in which case $J_\m$ is the unique fixed point of $T_\m$ and does not depend on the choice of $\bar J$. Here we will not be making such an assumption, and the choice of $\bar J$ may affect profoundly the character of the problem. For example, in SSP and other additive cost Markovian decision problems $\bar J$ is the zero function, $\bar J(i)\equiv0$, while in multiplicative cost models $\bar J$ is the unit function, $\bar J(i)\equiv1$, as we will discuss shortly. Also in SSP problems $A_\m$ is a substochastic matrix for all $\m\in{\cal M}$, while in other problems $A_\m$ can have components or row sums that are larger and as well as smaller than 1.
We define the optimal cost function $J^*$ by $$J^*(i)=\inf_{\pi\in\Pi}J_\pi(i),\qquad i=1,\ldots,n,$$ where $\Pi$ denotes the set of all policies. We wish to find $J^*$ and a policy $\pi^*\in\Pi$ that is optimal, i.e., $J_{\pi^*}=J^*$. The analysis of affine monotonic problems revolves around the equation $J=TJ$, or equivalently \begin{equation} \label{eq-beaffmon} J(i)=\inf_{u\in U(i)}\left[b(i,u)+\sum_{j=1}^nA_{ij}(u)J(j)\right],\quad j=1,\ldots,n. \end{equation} This is the analog of the classical infinite horizon DP equation and it is referred to as {\it Bellman's equation\/}. We are interested in solutions of this equation (i.e., fixed points of $T$) within ${\cal E}_+^n$ and within $\Re^n_+$. Usually in DP models one expects that $J^*$ solves Bellman's equation, while optimal stationary policies can be obtained by minimization over $U(i)$ in its right-hand side. However, this is not true in general, as we will show in Section IV.
Affine monotonic models appear in several contexts. In particular, finite-state sequential stochastic control problems (including SSP problems) with nonnegative cost per stage (see, e.g., [12], Chapter 3, and Section IV) are special cases where $\bar J$ is the identically zero function [$\bar J(i)\equiv0$]. Also, discounted problems involving state and control-dependent discount factors (for example semi-Markov problems, cf.\ Section 1.4 of [12], or Chapter 11 of [29]) are special cases, with the discount factors being absorbed within the scalars $A_{ij}(u)$. In all of these cases, $A_\m$ is a substochastic matrix. There are also other special cases, where $A_\m$ is not substochastic. They correspond to interesting classes of practical problems, including SSP-type problems with a multiplicative or an exponential (rather than additive) cost function, which we proceed to discuss.
\subsection{Multiplicative and Exponential Cost SSP Problems}
We will describe a type of SSP problem, where the cost function of a policy accumulates over time multiplicatively, rather than additively, up to the termination state. The special case where the cost from a given state is the expected value of the exponential of the length of the path from the state up to termination was studied by Denardo and Rothblum [27], and Patek [28]. We are not aware of a study of the multiplicative cost version for problems where a cost-free and absorbing termination state plays a major role (the paper by Rothblum [30] deals with multiplicative cost problems but focuses on the average cost case).
Let us introduce in addition to the states $i=1,\ldots,n$, a cost-free and absorbing state $t$. There are probabilistic state transitions among the states $i=1,\ldots,n$, up to the first time a transition to state $t$ occurs, in which case the state transitions terminate. We denote by $p_{it}(u)$ and $p_{ij}(u)$ the probabilities of transition from $i$ to $t$ and to $j$ under $u$, respectively, so that $$p_{it}(u)+\sum_{j=1}^np_{ij}(u)=1,\qquad i=1,\ldots,n,\ u\in U(i).$$ Next we introduce nonnegative scalars $h(i,u,t)$ and $h(i,u,j)$, $$h(i,u,t)\ge 0,\quad h(i,u,j)\ge0,\quad \forall\ i,j=1,\ldots,n,\ u\in U(i),$$ and we consider the affine monotonic problem where the scalars $A_{ij}(u)$ and $b(i,u)$ are defined by \begin{equation} \label{eq-hmultit} A_{ij}(u)=p_{ij}(u)h(i,u,j),\qquad i,j=1,\ldots,n,\ u\in U(i), \end{equation} and \begin{equation} \label{eq-hmultio} b(i,u)=p_{it}(u)h(i,u,t),\quad i=1,\ldots,n,\ u\in U(i), \end{equation} and the vector $\bar J$ is the unit vector, $$\bar J(i)=1,\qquad i=1,\ldots,n.$$ The cost function of this problem has a multiplicative character as we show next.
Indeed, with the preceding definitions of $A_{ij}(u)$, $b(i,u)$, and $\bar J$, we will prove that the expression for the cost function of a policy $\pi=\{\m_0,\m_1,\ldots\}$, $$J_\pi(x_0)=\limsup_{N\to\infty}\,(T_{\m_0}\cdots T_{\m_{N-1}} \bar J)(x_0),\qquad x_0=1,\ldots,n,$$ can be written in the multiplicative form \begin{equation} \label{eq-infmulticost} J_\pi(x_0)=\limsup_{N\to\infty}\,E\left\{\prod_{k=0}^{N-1}h\big(x_k,\m_k(x_k),x_{k+1}\big)\right\}, \end{equation} where: \begin{itemize} \item [(a)] $\{x_0,x_1,\ldots\}$ is the random state trajectory generated starting from $x_0$ and using $\pi$. \item [(b)] The expected value is with respect to the probability distribution of that trajectory. \item [(c)] We use the notation $$h\big(x_k,\m_k(x_k),x_{k+1}\big)=1,\qquad \hbox{if }x_k=x_{k+1}=t,$$ (so that the multiplicative cost accumulation stops once the state reaches $t$). \end{itemize} \par\noindent Thus, we claim that {\it $J_\pi(x_0)$ can be viewed as the expected value of cost accumulated multiplicatively, starting from $x_0$ up to reaching the termination state $t$ (or indefinitely accumulated multiplicatively, if $t$ is never reached)\/}.
To verify the formula \eqref{eq-infmulticost} for $J_\pi$, we use the definition $T_\m J=b_\m+A_\m J,$ to show by induction that for every $\pi=\{\m_0,\m_1,\ldots\}$, we have \begin{align} \label{eq-finhorcost} T_{\m_0}\cdots T_{\m_{N-1}} \bar J&=A_{\m_0}\cdots A_{\m_{N-1}}\bar J\notag\\ &\qquad +b_{\m_0}+\sum_{k=1}^{N-1}A_{\m_0}\cdots A_{\m_{k-1}}b_{\m_k}. \end{align} We then interpret the $n$ components of each vector on the right as conditional expected values of the expression \begin{equation} \label{eq-multicost} \prod_{k=0}^{N-1}h\big(x_k,\m_k(x_k),x_{k+1}\big) \end{equation} multiplied with the appropriate conditional probability. In particular: \begin{itemize} \item [(a)] The $i$th component of the vector $A_{\m_0}\cdots A_{\m_{N-1}}\bar J$ in Eq.\ \eqref{eq-finhorcost} is the conditional expected value of the expression \eqref{eq-multicost}, given that $x_0=i$ and $x_N\ne t$, multiplied with the conditional probability that $x_N\ne t$, given that $x_0=i$. \item [(b)] The $i$th component of the vector $b_{\m_0}$ in Eq.\ \eqref{eq-finhorcost} is the conditional expected value of the expression \eqref{eq-multicost}, given that $x_0=i$ and $x_1=t$, multiplied with the conditional probability that $x_1= t$, given that $x_0=i$. \item [(c)] The $i$th component of the vector $A_{\m_0}\cdots A_{\m_{k-1}}b_{\m_k}$ in Eq.\ \eqref{eq-finhorcost} is the conditional expected value of the expression \eqref{eq-multicost}, given that $x_0=i$, $x_1,\ldots,x_{k-1}\ne t$, and $x_k=t$, multiplied with the conditional probability that $x_1,\ldots,$ $x_{k-1}\ne t$, and $x_k=t$, given that $x_0=i$. \end{itemize} \par\noindent By adding these conditional probability expressions, we obtain the $i$th component of the unconditional expected value $$E\left\{\prod_{k=0}^{N-1}h\big(x_k,\m_k(x_k),x_{k+1}\big)\right\},$$ thus verifying the formula \eqref{eq-infmulticost}.
A special case of multiplicative cost problem is the {\it risk-sensitive SSP problem with exponential cost function\/}, where for all $i=1,\ldots,n,$ and $u\in U(i)$, \begin{equation} \label{eq-hexponspec} h(i,u,j)=\hbox{exp}\big({g(i,u,j)}\big),\quad j=1,\ldots,n, t, \end{equation} and the function $g$ can take both positive and negative values. The Bellman equation for this problem is \begin{align} \label{eq-hexpon} J(i)=&\inf_{u\in U(i)}\Bigg[p_{it}(u)\hbox{exp}\big({g(i,u,t)}\big)\notag\\ &\ \ \ +\sum_{j=1}^np_{ij}(u)\hbox{exp}\big({g(i,u,j)}\big)J(j)\Bigg],\quad i=1,\ldots,n. \end{align} Based on Eq.\ \eqref{eq-infmulticost}, we have that $J_\pi(x_0)$ is the limit superior of the expected value of the exponential of the $N$-step additive finite horizon cost up to termination, i.e., $$\sum_{k=0}^{\bar k}g\big(x_k,\m_k(x_k),x_{k+1}\big),$$
where $\bar k$ is equal to the first index prior to $N-1$ such that $x_{\bar k+1}=t$, or is equal to $N-1$ if there is no such index. The use of the exponential introduces risk aversion, by assigning a strictly convex increasing penalty for large rather than small cost of a trajectory up to termination (and hence a preference for small variance of the additive cost up to termination).
In the cases where $0\le g$ or $g\le 0$, we also have $\bar J\le T\bar J$ and $T\bar J\le \bar J$, respectively, corresponding to a monotone increasing and a monotone decreasing problem, in the terminology of [1]. Both of these problems admit a favorable analysis, highlighted by the fact that $J^*$ is a fixed point of $T$ (see [1], Chapter 4).
The case where $g$ can take both positive and negative values is more challenging, and is the focus of this paper. We will consider two cases, discussed in Sections III and IV of this paper, respectively. Under the assumptions of Section III, $J^*$ is shown to be the unique fixed point of $T$ within $\Re^n_+$. Under the assumptions of Section IV, it may happen that $J^*$ is not a fixed point of $T$ (see Example \vskip1pt\pn{examplebelcounterexp} that follows). Denardo and Rothblum [27] and Patek [28] consider only the more benign Section III case, for which $J^*$ is a fixed point of $T$. Also, the approach of [27] is very different from ours: it relies on linear programming ideas, and for this reason it requires a finite control constraint set and cannot be readily adapted to an infinite control space. The approach of [28] is closer to ours in that it also descends from the paper [25]. It allows for an infinite control space under a compactness assumption that is similar to our Assumption 2.2 of the next section, but it also requires that $g(i,u,j)>0$ for all $i,u,j$, so it deals only with a monotone increasing case where $T_\m \bar J\ge \bar J$ for all $\m\in{\cal M}$.
The deterministic version of the exponential cost problem where for each $u\in U(i)$, only one of the transition probabilities $p_{it}(u),p_{i1}(u),\ldots,p_{in}(u)$ is equal to 1 and all others are equal to 0, is mathematically equivalent to the classical deterministic shortest path problem (since minimizing the exponential of a deterministic expression is equivalent to minimizing that expression). For this problem a standard assumption is that there are no cycles that have negative total length to ensure that the shortest path length is finite. However, it is interesting that this assumption is not required in the present paper: when there are paths that travel perpetually around a negative length cycle we simply have $J^*(i)=0$ for all states $i$ on the cycle, which is permissible within our context.
\subsection{Assumptions on Policies - Contractive Policies}
\par\noindent We now introduce a special property of policies which is central for the purposes of this paper. We say that a given stationary policy $\m$ is {\it contractive if $A_\m^N\to0$ as $N\to\infty$\/}. Equivalently, $\m$ is contractive if all the eigenvalues of $A_\m$ lie strictly within the unit circle. Otherwise, $\m$ is called {\it noncontractive\/}. It follows from a classical result that a policy $\m$ is contractive if and only if $T_\m$ is a contraction with respect to some norm. Because $A_\m\ge0$, a stronger assertion can be made: $\m$ is contractive if and only if $A_\m$ is a contraction with respect to some weighted sup-norm (see e.g., the discussion in [22], Ch.\ 2, Cor.\ 6.2, or [12], Section 1.5.1). In the special case of SSP problems with additive cost function (all matrices $A_\m$ are substochastic), the contractive policies coincide with the {\it proper} policies, i.e., the ones that lead to the termination state with probability 1, starting from every state.
A particularly favorable situation for an SSP problem arises when all policies are proper, in which case all the mappings $T$ and $T_\m$ are contractions with respect to some common weighted sup norm. This result was shown in the paper by Veinott [31], where it was attributed to A.\ J.\ Hoffman. Alternative proofs of this contraction property are given in Bertsekas and Tsitsiklis, [22], p.\ 325 and [32], Prop.\ 2.2, Tseng [33], and Littman [34]. The proofs of [32] and [34] are essentially identical, and easily generalize to the context of the present paper. In particular, it can be shown that if all policies are contractive, all the mappings $T$ and $T_\m$ are contractions with respect to some common weighted sup norm. However, we will not prove or use this fact in this paper.
Let us derive an expression for the cost function of contractive and noncontractive policies. By repeatedly applying the mapping $T$ to both sides of the equation $T_\m J=b_\m+A_\m J$, we have $$T^N_\m J=A_\m^N J+\sum_{k=0}^{N-1} A_\m^k b_\m,\qquad \forall\ J\in \Re^n,\ N=1,2,\ldots,$$ and hence \begin{equation} \label{eq-infhorcost} J_\m=\limsup_{N\to\infty}T_\m^N\bar J=\limsup_{N\to\infty}A_\m^N \bar J+\sum_{k=0}^{\infty} A_\m^k b_\m. \end{equation} From these expressions, it follows that if $\m$ is contractive, the initial function $\bar J$ in the definition of $J_\m$ does not matter, and we have \begin{align*} J_\m&=\limsup_{N\to\infty}T_\m^NJ\notag\\ &=\limsup_{N\to\infty}\sum_{k=0}^{N-1} A_\m^k b_\m,\quad \forall\ \m\hbox{: contractive},\ J\in\Re^n. \end{align*} Moreover, since for a contractive $\m$, $T_\m$ is a contraction with respect to a weighted sup-norm, the $\lim\sup$ above can be replaced by $\lim$, so that \begin{equation} \label{eq-contractiveexpr} J_\m=\sum_{k=0}^{\infty} A_\m^k b_\m=(I-A_\m)^{-1}b_\m,\qquad \forall\ \m\hbox{: contractive}. \end{equation} Thus if $\m$ is contractive, $J_\m$ is real-valued as well as nonnegative, i.e., $J_\m\in\Re^n_+$. If $\m$ is noncontractive, we have $J_\m\in{\cal E}_+^n$ and it is possible that $J_\m(i)=\infty$ for some states $i$. We will assume throughout the paper the following.
\begin{assumption}\label{assumptiontwoone} There exists at least one contractive policy. \end{assumption}
The analysis of finite-state SSP problems typically assumes that the control space is either finite, or satisfies a compactness and continuity condition introduced in [25]. The following is a similar condition, and will be in effect throughout the paper.
\begin{assumption}[Compactness and Continuity]\label{assumptiontwotwo} The control space $U$ is a metric space, and $p_{ij}(\cdot)$ and $b(i,\cdot)$ are continuous functions of $u$ over $U(i)$, for all $i$ and $j$. Moreover, for each state $i$, the sets
$$\left\{u\in U(i)\ \Big|\ b(i,u)+\sum_{j=1}^nA_{ij}(u)J(j)\le \l\right\}$$ are compact subsets of $U$ for all scalars $\l\in\Re$ and $J\in \Re^n_+$. \end{assumption}
The preceding assumption is satisfied if the control space $U$ is finite. One way to see this is to simply identify each $u\in U$ with a distinct integer from the real line. Another interesting case where the assumption is satisfied is when for all $i$, $U(i)$ is a compact subset of the metric space $U$, and the functions $b(i,\cdot)$ and $A_{ij}(\cdot)$ are continuous functions of $u$ over $U(i)$.
An advantage of allowing $U(i)$ to be infinite and compact is that it makes possible the use of randomized policies for problems where there is a {\it finite} set of feasible actions at each state $i$, call it $C(i)$. We may then specify $U(i)$ to be the set of all probability distributions over $C(i)$, which is a compact subset of a Euclidean space. In this way, our results apply to finite-state and finite-action problems where randomization is allowed, and $J^*$ is the optimal cost function over all randomized nonstationary policies. Note, however, that the optimal cost function may change when randomized policies are introduced in this way. Basically, for our purposes, optimization over nonrandomized and over randomized policies over finite action sets $C(i)$ are two different problems, both of which are interesting and can be addressed with the methodology of this paper. However, when the sets $C(i)$ are infinite, a different and mathematically more sophisticated framework is required in order to allow randomized policies. The reason is that randomized policies over the infinite action sets $C(i)$ must obey measurability restrictions, such as universal measurability; see Bertsekas and Shreve [6], and Yu and Bertsekas [35].
The compactness and continuity part of the preceding assumption guarantees some important properties of the mapping $T$. These are summarized in the following proposition.
\begin{proposition}\label{prp-propcompactimpl} Let Assumptions \vskip1pt\pn{assumptiontwoone} and \vskip1pt\pn{assumptiontwotwo} hold. \begin{itemize} \item [(a)] The set of $u\in U(i)$ that minimize the expression \begin{equation} \label{eq-minimexpr} b(i,u)+\sum_{j=1}^nA_{ij}(u)J(j) \end{equation} is nonempty and compact. \item [(b)] Let $J_0$ be the zero vector in $\Re^n$ [$J_0(i)\equiv0$]. The sequence $\{T^kJ_0\}$ is monotonically nondecreasing and converges to a limit $\tilde J\in \Re^n_+$ that satisfies $\tilde J\le J^*$ and $\tilde J=T \tilde J$. \end{itemize} \end{proposition}
\proof (a) The set of $u\in U(i)$ that minimize the expression in Eq.\ \eqref{eq-minimexpr} is the intersection $\cap_{m=1}^\infty U_m$ of the nested sequence of sets \begin{align*}
U_m=\left\{u\in U(i)\ \Big|\ b(i,u)+\sum_{j=1}^nA_{ij}(u)J(j)\le\l_m\right\},\\ m=1,2,\ldots, \end{align*} where $\{\l_m\}$ is a monotonically decreasing sequence such that $$\l_m\downarrow \inf_{u\in U(i)}\left[b(i,u)+\sum_{j=1}^nA_{ij}(u)J(j)\right].$$ Each set $U_m$ is nonempty, and by Assumption \vskip1pt\pn{assumptiontwotwo}, it is compact, so the intersection is nonempty and compact.
\par\noindent (b) By the nonnegativity of $b(i,u)$ and $A_{ij}(u)$, we have $J_0\le TJ_0$, which by the monotonicity of $T$ implies that $\{T^kJ_0\}$ is monotonically nondecreasing to a limit $\tilde J\in{\cal E}_+^n$, and we have \begin{equation} \label{eq-incrsect} J_0\le TJ_0 \le \cdots\le T^kJ_0\le \cdots\le \tilde J. \end{equation} For all policies $\pi=\{\m_0,\m_1,\ldots\}$, we have $T^kJ_0\le T^k\bar J\le T_{\m_0}\cdots T_{\m_{k-1}}\bar J$, so by taking the limit as $k\to\infty$, we obtain $\tilde J\le J_\pi$, and by taking the infimum over $\pi$, it follows that $\tilde J\le J^*$. By Assumption \vskip1pt\pn{assumptiontwoone}, there exists at least one contractive policy $\m$, for which $J_\m$ is real-valued [cf.\ Eq.\ \eqref{eq-contractiveexpr}], so $J^*\in\Re^n_+$. It follows that the sequence $\{T^kJ_0\}$ consists of vectors in $\Re^n_+$, while $\tilde J\in\Re^n_+$.
By applying $T$ to both sides of Eq.\ \eqref{eq-incrsect}, we obtain \begin{align*} (T^{k+1}J_0)(i) &= \inf_{u\in U(i)} \left[b(i,u)+\sum_{j=1}^nA_{ij}(u)(T^kJ_0)(j)\right]\\ &\le (T\tilde J)(i), \end{align*} and by taking the limit as $k\to\infty$, it follows that $\tilde J\le T\tilde J.$ Assume to arrive at a contradiction that there exists a state $\tilde i$ such that \begin{equation} \label{eq-onetwethsf} \tilde J(\tilde i)< (T\tilde J)(\tilde i). \end{equation} Consider the sets $$U_k(\tilde i)= \left\{ u\in U(\tilde i)\
\Big|\ b(\tilde i,u) + \sum_{j=1}^nA_{\tilde ij}(u)(T^kJ_0)(j)\le \tilde J(\tilde i)\right\}$$ for $k\ge0$. It follows from Assumption \vskip1pt\pn{assumptiontwotwo} and Eq.\ \eqref{eq-incrsect} that $\big\{U_k(\tilde i)\big\}$ is a nested sequence of compact sets. Let also $u_k$ be a control attaining the minimum in $$\min_{u\in U(\tilde i)}\left[b(\tilde i,u)+\sum_{j=1}^nA_{\tilde ij}(u)(T^kJ_0)(j)\right];$$ [such a control exists by part (a)]. From Eq.\ \eqref{eq-incrsect}, it follows that for all $m\ge k$, \begin{align*} b(\tilde i,u_m)&+ \sum_{j=1}^nA_{\tilde ij}(u_m)(T^kJ_0)(j) \\ &\le b(\tilde i,u_m) + \sum_{j=1}^nA_{\tilde ij}(u_m)(T^mJ_0)(j)\\ &\le \tilde J(\tilde i). \end{align*} Therefore $\{ u_m\}_{m=k}^\infty \subset U_k(\tilde i)$, and since $U_k(\tilde i)$ is compact, all the limit points of $\{ u_m\}_{m=k}^\infty$ belong to $U_k(\tilde i)$ and at least one such limit point exists. Hence the same is true of the limit points of the entire sequence $\{ u_m\}_{m=0}^\infty$. It follows that if $\tilde u$ is a limit point of $\{ u_m\}_{m=0}^\infty$ then $$\tilde u\in \cap_{k=0}^\infty U_k(\tilde i).$$ This implies that for all $k\ge 0$ $$(T^{k+1}J_0)(\tilde i)\le b(\tilde i,\tilde u) + \sum_{j=1}^nA_{\tilde ij}(\tilde u)(T^kJ_0)(j)\le \tilde J(\tilde i).$$ By taking the limit in this relation as $k\to\infty$, we obtain $$\tilde J(\tilde i) = b(\tilde i,\tilde u) + \sum_{j=1}^nA_{\tilde ij}(\tilde u)\tilde J(j).$$ Since the right-hand side is greater than or equal to $(T\tilde J)(\tilde i)$, Eq.\ \eqref{eq-onetwethsf} is contradicted, implying that $\tilde J =T\tilde J$. \qed
\section{Case of Infinite Cost Noncontractive Policies}
\par\noindent We now turn to questions relating to Bellman's equation, the convergence of value iteration (VI for short) and policy iteration (PI for short), as well as conditions for optimality of a stationary policy. In this section we will use the following assumption, which parallels the central assumption of [25] for SSP problems. We will not need this assumption in Section IV.
\begin{assumption}[Infinite Cost Condition] \label{assumptionthreeone} For every noncontractive policy $\m$, there is at least one state such that the corresponding component of the vector $\sum_{k=0}^\infty A_\m^k b_\m$ is equal to $\infty$. \end{assumption}
Note that the preceding assumption guarantees that for every noncontractive policy $\m$, we have $J_\m(i)=\infty$ for at least one state $i$ [cf.\ Eq.\ \eqref{eq-infhorcost}]. The reverse is not true, however: $J_\m(i)=\infty$ does not imply that the $i$th component of $\sum_{k=0}^\infty A_\m^k b_\m$ is equal to $\infty$, since there is the possibility that $A_\m^N\bar J$ may become unbounded as $N\to\infty$ [cf.\ Eq.\ \eqref{eq-infhorcost}].
Under Assumptions \vskip1pt\pn{assumptiontwoone}, \vskip1pt\pn{assumptiontwotwo}, and \vskip1pt\pn{assumptionthreeone}, we will now derive results that closely parallel the standard results of [25] for additive cost SSP problems. We have the following characterization of contractive policies.
\begin{proposition}[Properties of Contractive Policies]\label{prp-proptto} Let Assumption \vskip1pt\pn{assumptionthreeone} hold. \begin{itemize} \item [(a)] For a contractive policy $\m$, the associated cost vector $J_\m$ satisfies $$\lim_{k\to\infty}(T_\m^k J)(i)=J_\m(i),\qquad i=1,\ldots,n,$$ for every vector $J\in\Re^n$. Furthermore, we have $J_\m=T_\m J_\m,$
and $J_\m$ is the unique solution of this equation within $\Re^n$. \item [(b)] A stationary policy $\m$ is contractive if and only if it satisfies $J\ge T_\m J$ for some vector $J\in\Re^n_+$. \end{itemize} \end{proposition}
\proof (a) Follows from Eqs.\ \eqref{eq-finhorcost} and \eqref{eq-contractiveexpr}, and by writing the equation $J_\m=T_\m J_\m$ as $(I-A_\m) J_\m=b_\m$.
\par\noindent(b) If $\m$ is contractive, by part (a) we have $J\ge T_\m J$ for $J=J_\m\in \Re^n_+$. Conversely, let $J$ be a vector in $\Re^n_+$ with $J\ge T_\m J$. Then the monotonicity of $T_\m$ and Eq.\ \eqref{eq-finhorcost} imply that for all $N$ we have
$$J\ge T_\m^N J=A_\m^NJ+\sum_{k=0}^{N-1}A_\m^k b_\m\ge \sum_{k=0}^{N-1}A_\m^k b_\m\ge0.$$ It follows that the vector $\sum_{k=0}^{\infty}A_\m^k b_\m$ is real-valued so that, by Assumption \vskip1pt\pn{assumptionthreeone}, $\m$ cannot be noncontractive. \qed
The following proposition is our main result under Assumption \vskip1pt\pn{assumptionthreeone}. It parallels Prop.\ 3 of [25] (see also Section 3.2 of [12]). In addition to the fixed point property of $J^*$ and the convergence of the VI sequence $\{T^kJ\}$ to $J^*$ starting from any $J\in\Re^n_+$, it shows the validity of the PI algorithm. The latter algorithm generates a sequence $\{\m^k\}$ starting from any contractive policy $\m^0$. Its typical iteration consists of a computation of $J_{\m^k}$ using the policy evaluation equation $J_{\m^k}=T_{\m^k}J_{\m^k}$, followed by the policy improvement operation $T_{\m^{k+1}}J_{\m^k}=TJ_{\m^k}$.
\begin{proposition}[Bellman's Equation, Policy Iteration, Value Iteration, and Optimality Conditions]\label{prp-propttt} Let Assumptions \vskip1pt\pn{assumptiontwoone}, \vskip1pt\pn{assumptiontwotwo}, and \vskip1pt\pn{assumptionthreeone}\ hold. \begin{itemize} \item [(a)] The optimal cost vector $J^* $ satisfies the Bellman equation $J =T J$. Moreover, $J^* $ is the unique solution of this equation within $\Re^n_+$. \item [(b)] Starting with any contractive policy $\m^0$, the sequence $\{\m^k\}$ generated by the PI algorithm consists of contractive policies, and any limit point of this sequence is a contractive optimal policy. \item [(c)] We have $$\lim_{k\to\infty}(T^kJ)(i)=J^* (i),\qquad i=1,\ldots,n,$$ for every vector $J\in\Re^n_+$. \item [(d)] A stationary policy $\m$ is optimal if and only if
$T_\m J^* =TJ^* .$ \item [(e)] For a vector $J\in \Re^n_+$, if $J\le TJ$ then $J\le J^* $, and if $J\ge TJ$ then $J\ge J^* $.\end{itemize} \end{proposition}
\proof (a), (b) From Prop.\ \vskip1pt\pn{prp-propcompactimpl}(b), $T$ has as fixed point the vector $\tilde J\in\Re^n_+$, the limit of the sequence $\{T^kJ_0\}$, where $J_0$ is the identically zero vector [$J_0(i)\equiv0$]. We will show parts (a) and (b) simultaneously and in stages. First we will show that $\tilde J$ is the unique fixed point of $T$ within $\Re^n_+$. Then we will show that the PI algorithm, starting from any contractive policy, generates in the limit a contractive policy $\overline \m$ such that $J_{\overline\m}=\tilde J$. Finally we will show that $J_{\overline\m}=J^*$.
Indeed, if $J$ and $J'$ are two fixed points, then we select $\m$ and $\m'$ such that $J=TJ =T_\m J$ and $J'=TJ' =T_{\m'}J'$; this is possible because of Prop.\ \vskip1pt\pn{prp-propcompactimpl}(a). By Prop.\ \vskip1pt\pn{prp-proptto}(b), we have that $\m$ and $\m'$ are contractive, and by Prop.\ \vskip1pt\pn{prp-proptto}(a) we obtain $J=J_\m$ and $J'=J_{\m'}$. We also have $J=T^kJ\le T_{\m'}^kJ$ for all $k\ge1$, and by Prop.\ \vskip1pt\pn{prp-proptto}(a), it follows that $J\le\lim_{k\to\infty}T_{\m'}^k J=J_{\m'}=J'$. Similarly, $J'\le J$, showing that $J=J'$. Thus $T$ has $\tilde J$ as its unique fixed point within $\Re^n_+$.
We next turn to the PI algorithm. Let $\m$ be a contractive policy (there exists one by Assumption \vskip1pt\pn{assumptiontwoone}). Choose $\m'$ such that $$T_{\m'} J_\m=TJ_\m.$$
Then we have $J_\m=T_\m J_\m\ge T_{\m'}J_\m$. By Prop.\ \vskip1pt\pn{prp-proptto}(b), $\m'$ is contractive, and using the monotonicity of $T_{\m'}$ and Prop.\ \vskip1pt\pn{prp-proptto}(a), we obtain \begin{equation} \label{eq-ones} J_\m\ge\lim_{k\to\infty}T_{\m'}^k J_\m=J_{\m'}. \end{equation} Continuing in the same manner, we construct a sequence $\{\m^k\}$ such that each $\m^k$ is contractive and \begin{equation} \label{eq-onese} J_{\m^k}\ge T_{\m^{k+1}}J_{\m^k} =TJ_{\m^k} \ge J_{\m^{k+1}},\qquad k=0,1,\ldots \end{equation} The sequence $\{J_{\m^k}\}$ is real-valued, nonincreasing, and nonnegative so it converges to some $J_\infty\in\Re^n_+$.
We claim that the sequence of vectors $\m^k=\big(\m^k(1),\ldots,\m^k(n)\big)$ has a limit point $\big(\overline \m(1),\ldots,\overline\m(n)\big)$, with $\overline\m$ being a feasible policy. Indeed, using Eq.\ \eqref{eq-onese} and the fact $J_\infty\le J_{\m^{k-1}}$, we have for all $k=1,2,\ldots,$ $$T_{\m^{k}}J_\infty\le T_{\m^{k}}J_{\m^{k-1}}=TJ_{\m^{k-1}}\le T_{\m^{k-1}}J_{\m^{k-1}}=J_{\m^{k-1}}\le J_{\m^0},$$ so $\m^{k}(i)$ belongs to the set
$$\hat U(i)=\left\{u\in U(i)\ \Big|\ b(i,u)+\sum_{j=1}^nA_{ij}(u)J_\infty(j)\le J_{\m^0}(i)\right\},$$ which is compact by Assumption \vskip1pt\pn{assumptiontwotwo}. Hence the sequence $\{\m^k\}$ belongs to the compact set $\hat U(1)\times\cdots\times \hat U(n)$, and has a limit point $\overline\m$, which is a feasible policy. In what follows, without loss of generality, we assume that the entire sequence $\{\m^k\}$ converges to $\overline\m.$
Since $J_{\m^k}\downarrow J_\infty\in \Re^n_+$ and $\m^k\to\overline \m$, by taking limit as $k\to\infty$ in Eq.\ \eqref{eq-onese}, and using the continuity part of Assumption \vskip1pt\pn{assumptiontwotwo}, we obtain $J_\infty=T_{\overline\m} J_\infty$. It follows from Prop.\ \vskip1pt\pn{prp-proptto}(b) that $\overline\m$ is contractive, and that $J_{\overline\m}$ is equal to $J_\infty$. To show that $J_{\overline\m}$ is a fixed point of $T$, we note that from the right side of Eq.\ \eqref{eq-onese}, we have for all policies $\m$, $T_\m J_{\m^k} \ge J_{\m^{k+1}}$, which by taking limit as $k\to\infty$ yields $T_\m J_{\overline\m}\ge J_{\overline\m}$. By taking minimum over $\m$, we obtain $T J_{\overline\m}\ge J_{\overline\m}$. Combining this with the relation $J_{\overline\m}=T_{\overline\m} J_{\overline\m}\ge TJ_{\overline\m}$, it follows that $J_{\overline\m}= TJ_{\overline\m}$. Thus $J_{\overline\m}$ is equal to the unique fixed point $\tilde J$ of $T$ within $\Re^n_+$.
We will now conclude the proof by showing that $J_{\overline\m}$ is equal to the optimal cost vector $J^* $ (which also implies the optimality of the policy $\overline\m$, obtained from the PI algorithm starting from a contractive policy). By Prop.\ \vskip1pt\pn{prp-propcompactimpl}(b), the sequence $T^kJ_0$ converges monotonically to $\tilde J$, which is equal to $J_{\overline\m}$. Also, for every policy $\pi=\{\m_0,\m_1,\ldots\}$, we have $$T^kJ_0\le T^k\bar J\le T_{\m_0}\cdots T_{\m_{k-1}}\bar J,\qquad k=0,1,\ldots,$$ and by taking the limit as $k\to\infty$, we obtain $J_{\overline \m}=\tilde J=\lim_{k\to\infty}T^k J_0\le J_\pi$ for all $\pi$, showing that $J_{\overline\m}=J^*$. Thus $J^*$ is the unique fixed point of $T$ within $\Re^n_+$, and $\overline \m$ is an optimal policy.
\par\noindent (c) From the preceding proof, we have that $T^kJ_0\to J^*$, which implies that \begin{equation} \label{eq-convbelow} \lim_{k\to\infty}T^kJ=J^*,\qquad \forall\ J\in\Re^n_+\hbox{ with }J\le J^*. \end{equation} Also, for any $J\in\Re^n_+$ with $J\ge J^*$, we have $$T_{\overline\m}^kJ\ge T^kJ\ge T^kJ^*=J^*=J_{\overline\m},$$ where $\overline\m$ is the contractive optimal policy obtained by PI in the proof of part (b). By taking the limit as $k\to\infty$ and using the fact $T_{\overline\m}^kJ\to J_{\overline\m}$ (which follows from the contractiveness of $\overline\m$), we obtain \begin{equation} \label{eq-convabove} \lim_{k\to\infty}T^kJ=J^*,\qquad \forall\ J\in\Re^n_+\hbox{ with }J\ge J^*. \end{equation} Finally, given any $J\in\Re^n_+$, we have from Eqs.\ \eqref{eq-convbelow} and \eqref{eq-convabove}, $$\lim_{k\to\infty}T^k\big(\min\{J,J^*\}\big)=J^*,\quad \lim_{k\to\infty}T^k\big(\max\{J,J^*\}\big)=J^*,$$ and since $J$ lies between $\min\{J,J^*\}$ and $\max\{J,J^*\}$, it follows that $T^kJ\toJ^*$.
\par\noindent(d) If $\m$ is optimal, then $J_\m=J^* $ and since by part (a) $J^*$ is real-valued, $\m$ is contractive. Therefore, by Prop.\ \vskip1pt\pn{prp-proptto}(a), $$T_{\m}J^* =T_{\m}J_{\m}=J_{\m}=J^* =TJ^* .$$ Conversely, if $J^* =TJ^* =T_{\m}J^* $, it follows from Prop.\ \vskip1pt\pn{prp-proptto}(b) that $\m$ is contractive, and by using Prop.\ \vskip1pt\pn{prp-proptto}(a), we obtain $J^* =J_\m$. Therefore $\m$ is optimal. The existence of an optimal policy follows from part (b).
\par\noindent(e) If $J\in \Re^n_+$ and $J\le TJ$, by repeatedly applying $T$ to both sides and using the monotonicity of $T$, we obtain $J\le T^kJ$ for all $k.$ Taking the limit as $k\to\infty$ and using the fact $T^kJ\to J^* $ [cf.\ part (c)], we obtain $J\le J^* $. The proof that $J\ge J^* $ if $J\ge TJ$ is similar.
\qed
\subsection{Computational Methods}
Proposition \vskip1pt\pn{prp-propttt}(b) shows the validity of PI when starting from a contractive policy. This is similar to the case of additive cost SSP, where PI is known to converge starting from a proper policy (cf.\ the proof of Prop.\ 3 of [25]).
There is also an asynchronous version of the PI algorithm proposed for discounted and SSP models by Bertsekas and Yu [14], [15], which does not require an initial contractive policy and admits an asynchronous implementation. This algorithm extends straightforwardly to the affine monotonic model of this paper under Assumptions 2.1, 2.2, and 3.1 (see [1], Section 3.3.2, for a description of this extension to abstract DP models).
Proposition \vskip1pt\pn{prp-propttt}(c) establishes the validity of the VI algorithm that generates the sequence $\{T^kJ\}$, starting from any initial $J\in \Re^n_+$. An asynchronous version of this algorithm is also valid; see the discussion of Section 3.3.1 of [1].
Finally, Prop.\ \vskip1pt\pn{prp-propttt}(e) shows it is possible to compute $J^*$ as the unique solution of the problem of maximizing $\sum_{i=1}^n \b_iJ(i)$ over all $J=\big(J(1),\ldots,J(n)\big)$ such that $J\le TJ$, where $\b_1,\ldots,\b_n$ are any positive scalars. This problem can be written as \begin{align} \label{eq-lpproblem} \hbox{\rm maximize}\quad &\sum_{i=1}^n \b_i J(i)\cr \hbox{\rm subject to\ \ } &J(i)\le b(i,u) + \sum_{j=1}^n A_{ij}(u)J(j),\notag\\ &\quad \quad \quad i =1,\ldots,n, \quad u\in U(i), \end{align} and it is a linear program if each $U(i)$ is a finite set.
\section{Case of Finite Cost noncontractive Policies}
\par\noindent We will now eliminate Assumption \vskip1pt\pn{assumptionthreeone}, thus allowing noncontractive policies with real-valued cost functions. We will prove results that are weaker yet useful and substantial. An important notion in this regard is the optimal cost that can be achieved with contractive policies only, i.e., the vector $\hat J$ with components given by \begin{equation} \label{eq-optcontractive} \hat J(i)=\inf_{\m:\, \hbox{contractive}}J_\m(i),\qquad i=1,\ldots,n. \end{equation} We will show that $\hat J$ is a solution of Bellman's equation, while $J^*$ need not be. To this end, we give an important property of noncontractive policies in the following proposition.
\begin{proposition}\label{prp-propnoncontractive} If $\m$ is a noncontractive policy and all the components of $b_\m$ are strictly positive, then there exists at least one state $i$ such that the corresponding component of the vector $\sum_{k=0}^{\infty} A_\m^k b_\m$ is $\infty$. \end{proposition}
\proof According to the Perron-Frobenius Theorem, the nonnegative matrix $A_\m$ has a real eigenvalue $\l$, which is equal to its spectral radius, and an associated nonnegative eigenvector $\xi\ne0$ (see e.g., [22], Chapter 2, Prop.\ 6.6). Choose $\gamma>0$ to be such that $b_\m\ge \gamma\xi$, so that $$\sum_{k=0}^{\infty} A_\m^k b_\m\ge \gamma\sum_{k=0}^{\infty} A_\m^k \xi=\gamma\left(\sum_{k=0}^{\infty} \l^k\right) \xi.$$ Since some component of $\xi$ is positive while $\l\ge1$ (since $\m$ is noncontractive), the corresponding component of the infinite sum on the right is infinite, and the same is true for the corresponding component of the vector $\sum_{k=0}^{\infty} A_\m^k b_\m$ on the left. \qed
\subsection{The $\delta$-Perturbed Problem}
We now introduce a perturbation line of analysis, also used in [1] and [26], whereby we add a constant $\delta>0$ to all components of $b_\m$, thus obtaining what we call the {\it $\delta$-perturbed affine monotonic model\/}. We denote by $J_{\m,\delta}$ and $J^*_\delta$ the cost function of $\m$ and the optimal cost function of the $\delta$-perturbed model, respectively. We have the following proposition.
\begin{proposition}\label{prp-propdeltaoptam} Let Assumptions \vskip1pt\pn{assumptiontwoone} and \vskip1pt\pn{assumptiontwotwo} hold. Then for each $\delta>0$: \begin{itemize} \item [(a)] $J^*_\delta$ is the unique solution within $\Re^n_+$ of the equation $$J(i)=(TJ)(i)+\delta,\qquad i=1,\ldots,n.$$ \item [(b)] A policy $\m$ is optimal for the $\delta$-perturbed problem (i.e., $J_{\m,\delta}=J^*_\delta$) if and only if $T_{\m}J^*_\delta =TJ^*_\delta$. Moreover, for the $\delta$-perturbed problem, all optimal policies are contractive and there exists at least one contractive policy that is optimal. \item [(c)] The optimal cost function over contractive policies $\hat J$ [cf.\ Eq.\ \eqref{eq-optcontractive}] satisfies $$\hat J(i)=\lim_{\delta\downarrow0}J^*_{\delta}(i),\qquad i=1,\ldots,n.$$ \item [(d)] If the control constraint set $U(i)$ is finite for all states $i=1,\ldots,n$, there exists a contractive policy $\hat \m$ that attains the minimum over all contractive policies, i.e., $J_{\hat \m}=\hat J$. \end{itemize} \end{proposition}
\proof (a), (b) By Prop.\ \vskip1pt\pn{prp-propnoncontractive}, we have that Assumption \vskip1pt\pn{assumptionthreeone} holds for the $\delta$-perturbed problem. The results follow by applying Prop.\ \vskip1pt\pn{prp-propttt} [the equation of part (a) is Bellman's equation for the $\delta$-perturbed problem].
\par\noindent (c) For an optimal contractive policy $\m^*_\delta$ of the $\delta$-perturbed problem [cf.\ part (b)], we have for all $\m'$ that are contractive $$\hat J=\inf_{\m:\, \hbox{contractive}}J_\m\le J_{\m^*_\delta}\le J_{\m^*_\delta,\delta}= J^*_\delta\le J_{\m',\delta}. $$ Since for every contractive policy $\m'$, we have $\lim_{\delta\downarrow0}J_{\m',\delta}=J_{\m'},$ it follows that $$\hat J\le \lim_{\delta\downarrow0}J_{\m^*_\delta}\le J_{\m'},\qquad \forall\ \m': \hbox{contractive}. $$ By taking the infimum over all $\m'$ that are contractive, the result follows.
\par\noindent (d) Let $\{\delta_k\}$ be a positive sequence with $\delta_k\downarrow0$, and consider a corresponding sequence $\{\m_{k}\}$ of optimal contractive policies for the $\delta_k$-perturbed problems. Since the set of contractive policies is finite, some policy $\hat \m$ will be repeated infinitely often within the sequence $\{\m_{k}\}$, and since $\{J^*_{\delta_k}\}$ is monotonically nonincreasing, we will have $$\hat J\le J_{\hat \m}\le J^*_{\delta_k},$$ for all $k$ sufficiently large. Since by part (c), $J^*_{\delta_k}\downarrow\hat J$, it follows that $J_{\hat \m}=\hat J$.
\qed
\subsection{Main Results}
We now show that $\hat J$ is the largest fixed point of $T$ within $\Re^n_+$. This is the subject of the next proposition, which also provides a convergence result for VI as well as an optimality condition; see Fig.\ \vskip1pt\pn{fig-Beleq}.
\begin{proposition}[Bellman's Equation, Value Iteration, and Optimality Conditions]\label{prp-propfixedviaff} Let Assumptions \vskip1pt\pn{assumptiontwoone} and \vskip1pt\pn{assumptiontwotwo} hold. Then: \begin{itemize} \item [(a)] The optimal cost function over contractive policies, $\hat J$, is the largest fixed point of $T$ within $\Re^n_+$, i.e., $\hat J$ is a fixed point that belongs to $\Re^n_+$, and if $J'\in\Re^n_+$ is another fixed point, we have $J'\le \hat J$. \item [(b)] We have $T^kJ\to \hat J$ for every
$J\in \Re^n_+$ with $J\ge \hat J$. \item [(c)] Let $\m$ be a contractive policy. Then $\m$ is optimal within the class of contractive policies (i.e., $J_\m=\hat J$) if and only if $T_{\m} \hat J =T\hat J$. \end{itemize} \end{proposition}
\proof (a), (b) For all contractive $\m$, we have $J_\m=T_\m J_\m\ge T_\m\hat J\ge T\hat J.$ Taking the infimum over contractive $\m$, we obtain $\hat J\ge T\hat J$. Conversely, for all $\delta>0$ and $\m\in{\cal M}$, we have $$J^*_\delta=TJ^*_\delta+\delta e\le T_\m J^*_\delta+\delta e.$$ Taking limit as $\delta\downarrow0$, and using Prop.\ \vskip1pt\pn{prp-propdeltaoptam}(c), we obtain $\hat J\le T_\m\hat J$ for all $\m\in{\cal M}$. Taking infimum over $\m\in{\cal M}$, it follows that $\hat J\le T\hat J$. Thus $\hat J$ is a fixed point of $T$.
For all $J\in\Re^n$ with $J\ge \hat J$ and contractive $\m$, we have by using the relation $\hat J=T\hat J$ just shown, $$\hat J=\lim_{k\to\infty}T^k\hat J\le \lim_{k\to\infty}T^k J\le \lim_{k\to\infty}T_\m^k J=J_\m.$$ Taking the infimum over all contractive $\m$, we obtain $$\hat J\le \lim_{k\to\infty}T^k J\le \hat J,\qquad \forall\ J\ge \hat J.$$ This proves that $T^kJ\to\hat J$ starting from any $J\in \Re^n_+$ with $J\ge \hat J$. Finally, let $J'\in\Re^n_+$ be another fixed point of $T$, and let $J\in \Re^n_+$ be such that $J\ge \hat J$ and $J\ge J'$. Then $T^kJ\to\hat J$, while $T^kJ\ge T^kJ'=J'$. It follows that $\hat J\ge J'$.
\par\noindent (c) If $\m$ is a contractive policy with $J_\m=\hat J$, we have $\hat J=J_{\m}=T_{\m}J_{\m}=T_{\m}\hat J,$ so, using also the relation $\hat J=T\hat J$ [cf.\ part (a)], we obtain $T_{\m}\hat J=T\hat J$.
Conversely, if $\m$ satisfies $T_{\m}\hat J=T\hat J$, then from part (a), we have $T_\m \hat J=\hat J$ and hence $\lim_{k\to\infty}T_\m^k\hat J = \hat J$. Since $\m$ is contractive, we obtain $J_\m=\lim_{k\to\infty}T_\m^k\hat J$, so $J_\m=\hat J$. \qed
\begin{figure}
\caption{Schematic illustration of Prop.\ \vskip1pt\pn{prp-propfixedviaff} for a problem with two states. The optimal cost function over contractive policies, $\skew6\hat J$, is the largest solution of Bellman's equation, while VI converges to $\skew6\hat J$ starting from $J\ge \skew6\hat J$. }
\label{fig-Beleq}
\end{figure}
Note that it is possible that there exists a noncontractive policy $\m$ that is strictly suboptimal and yet satisfies the optimality condition $T_\m J^* =TJ^*$ (there are simple deterministic shortest path examples with a zero length cycle that can be used to show this; see [1], Section 3.1.2). Thus contractiveness of $\m$ is an essential assumption in Prop.\ \vskip1pt\pn{prp-propfixedviaff}(c).
The following proposition shows that starting from any $J\ge \hat J$, the convergence rate of VI to $\hat J$ is linear. The proposition also provides a corresponding error bound. The proof is very similar to a corresponding result of [26] and will not be given.
\begin{proposition}[Convergence Rate of VI]\label{prp-propvirateconvam} Let Assumptions \vskip1pt\pn{assumptiontwoone} and \vskip1pt\pn{assumptiontwotwo} hold, and assume that there exists a contractive policy $\hat \m$ that is optimal within the class of contractive policies, i.e., $J_{\hat \m}=\hat J$. Then
$$\big\|TJ-\hat J\|_v\le \b\|J-\hat J\|_v,\qquad \forall\ J\ge \hat J,$$
where $\|\cdot\|_v$ is a weighted sup-norm for which $T_{\m^*}$ is a contraction and $\b$ is the corresponding modulus of contraction. Moreover, we have
$$\|J-\hat J\|_v\le {1\over 1-\b}\max_{i=1,\ldots,n}{J(i)-(TJ)(i)\over v(i)},\qquad \forall\ J\ge \hat J.$$\end{proposition}
We note that if $U(i)$ is infinite it is possible that $\hat J=J^*$, but the only optimal policy is noncontractive, even if the compactness Assumption \vskip1pt\pn{assumptiontwotwo} holds. This is shown in the following example, which is adapted from the paper [26] (Example 2.1).
\begin{example}[A Counterexample on the Existence of an Optimal Contractive Policy] \label{examplecounterexp} Consider an exponential cost SSP problem with a single state 1 in addition to the termination state $t$; cf.\ Fig.\ \vskip1pt\pn{fig-exp-cost}. At state 1 we must choose $u\in[0,1]$. Then, we terminate at no cost [$g(1,u,t)=0$ in Eq.\ \eqref{eq-hexponspec}] with probability $u$, and we stay at state 1 at cost $-u$ [i.e., $g(1,u,1)=-u$ in Eq.\ \eqref{eq-hexponspec}] with probability $1-u$. We have $b(i,u)=u\exp{(0)}$ and $A_{11}(u)=(1-u)\exp{(-u)}$, so that $$H(1,u,J)=u+(1-u)\exp{(-u)}J.$$ Here there is a unique noncontractive policy $\m'$: it chooses $u=0$ at state 1, and has cost $J_{\m'}(1)=1$. Every policy $\m$ with $\m(1)\in (0,1]$ is contractive, and $J_\m$ can be obtained by solving the equation $J_\m=T_\m J_\m$, i.e., $$J_\m(1)=\m(1)+\big(1-\m(1)\big)\exp{\big(-\m(1)\big)}J_\m(1).$$ We thus obtain $$J_\m(1)={\m(1)\over \big(1-\m(1)\big)\exp{\big(-\m(1)\big)}}.$$ It can be seen that $\skew5\hat J(1)=J^*(1)=0$, but there exists no optimal policy, and no optimal policy within the class of contractive policies. \end{example}
\begin{figure}
\caption{The exponential cost SSP problem with a single state of Example \vskip1pt\pn{examplecounterexp}.}
\label{fig-exp-cost}
\end{figure}
Let us also show that generally, under Assumptions \vskip1pt\pn{assumptiontwoone} and \vskip1pt\pn{assumptiontwotwo}, $J^*$ need not be a fixed point of $T$. The following is a straightforward adaptation of Example 2.2 of [26].
\begin{example}[An Exponential Cost SSP Problem Where $J^*$ is not a Fixed Point of $T$] \label{examplebelcounterexp} Consider the exponential cost SSP problem of Fig.\ \vskip1pt\pn{fig-belcounterex1s}, involving a noncontractive policy $\m$ whose transitions are marked by solid lines in the figure and form the two zero length cycles shown. All the transitions under $\m$ are deterministic, except at state 1 where the successor state is 2 or 5 with equal probability $1/2$. We assume that the cost of the policy for a given state is the expected value of the exponential of the finite horizon path length. We first calculate $J_\m(1)$. Let $g_k$ denote the cost incurred at time $k$, starting at state 1, and let $s_N(1)=\sum_{k=0}^{N-1} g_k$ denote the $N$-step accumulation of $g_k$ starting from state 1. We have $$s_N(1)=0\qquad \hbox{if $N=1$ or $N=4+3t$, $t=0,1,\ldots$},$$ and \begin{align*} s_N(1)=1\ \hbox{or $s_N(1)=-1$ with probability 1/2 each}\\ \qquad \qquad \hbox{if $N=2+3t$ or $N=3+3t$, $t=0,1,\ldots$.} \end{align*} Thus $$J_\m(1)=\limsup_{N\to\infty}E\left\{e^{s_N(1)}\right\}={1\over 2} (e^1+e^{-1}).$$ On the other hand, a similar (but simpler) calculation shows that $$J_\m(2)=J_\m(5)=e^1,$$ (the $N$-step accumulation of $g_k$ undergoes a cycle $\{1,-1,0,1,-1,0,\ldots\}$ as $N$ increases starting from state 2, and undergoes a cycle $\{-1,1,0,-1,1,0,\ldots\}$ as $N$ increases starting from state 5). Thus the Bellman equation at state 1, $$J_\m(1)={1\over 2} \big(J_\m(2)+J_\m(5)\big),$$ is not satisfied, and $J_\m$ is not a fixed point of $T_\m$. If for $i=1,4,7,$ we have transitions (shown with broken lines) that lead from $i$ to $t$ with a cost $c>2$, the corresponding contractive policy is strictly suboptimal, so that $\m$ is optimal, but $J_\m=J^*$ is not a fixed point of $T$. \end{example}
\begin{figure}
\caption{An example of a noncontractive policy $\m$, where $J_\m$ is not a fixed point of $T_\m$. All transitions under $\m$ are shown with solid lines. These transitions are deterministic, except at state 1 where the next state is 2 or 5 with equal probability $1/2$. There are additional transitions from nodes 1, 4, and 7 to the destination (shown with broken lines) with cost $c>2$, which create a suboptimal contractive policy. We have $J^*=J_\m$ and $J^*$ is not a fixed point of $T$.}
\label{fig-belcounterex1s}
\end{figure}
\subsection{Computational Methods}
\par\noindent Regarding computational methods, Prop.\ \vskip1pt\pn{prp-propfixedviaff}(b) establishes the validity of the VI algorithm that generates the sequence $\{T^kJ\}$ and converges to $\hat J$, starting from any initial $J\in \Re^n_+$ with $J\ge \hat J$. Moreover, Prop.\ \vskip1pt\pn{prp-propvirateconvam} yields a linear rate of convergence result for this VI algorithm, assuming that there exists a contractive policy $\hat \m$ that is optimal within the class of contractive policies. Convergence to $\hat J$ starting from within the region $\{J\mid 0\le J\le \hat J\}$ cannot be guaranteed, since there may be fixed points other than $\hat J$ within that region. There are also PI algorithms that converge to $\hat J$. As an example, we note a PI algorithm with perturbations for abstract DP problems developed in Section 3.3.3 of [1], which can be readily adapted to affine monotonic problems. Finally, it is possible to compute $\hat J$ by solving a linear programming problem, in the case where the control space $U$ is finite, by using the following proposition.
\begin{proposition}\label{prp-proppertlp} Let Assumptions \vskip1pt\pn{assumptiontwoone} and \vskip1pt\pn{assumptiontwotwo} hold. Then if a vector $J\in \Re^n$ satisfies $J\le TJ$, it also satisfies $J\le \hat J $. \end{proposition}
\proof Let $J\le TJ$ and $\delta>0$. We have $J\le TJ+\delta e=T_\delta J$, and hence $J\le T^k_\delta J$ for all $k$. Since the infinite cost conditions hold for the $\delta$-perturbed problem, it follows that $T^k_\delta J\to J^*_\delta$, so $J\le J^*_\delta$. By taking $\delta\downarrow 0$ and using Prop.\ \vskip1pt\pn{prp-propdeltaoptam}(c), it follows that $J\le \hat J$. \qed
The preceding proposition shows that $\hat J$ is the unique solution of the problem of maximizing $\sum_{i=1}^n \b_iJ(i)$ over all $J=\big(J(1),\ldots,J(n)\big)$ such that $J\le TJ$, where $\b_1,\ldots,\b_n$ are any positive scalars, i.e., the problem of Eq.\ \eqref{eq-lpproblem}. This problem is a linear program if each $U(i)$ is a finite set.
\vskip-1.5pc
\section{Concluding Remarks}
In this paper we have expanded the SSP methodology to affine monotonic models that are characterized by an affine mapping from the set of nonnegative functions to itself. These models include among others, multiplicative and risk-averse exponentiated cost models. We have used the conceptual framework of semicontractive DP, based on the notion of a contractive policy, which generalizes the notion of a proper policy in SSP. We have provided extensions of the basic analytical and algorithmic results of SSP problems, and we have illustrated their exceptional behavior within our broader context.
Another case of affine monotonic model that we have not considered, is the one obtained when $\bar J\le0$ and $$b(i,u)\le0,\qquad A_{ij}(u)\ge0,\qquad \forall\ i,j=1,\ldots,n,\ u\in U(i),$$ so that $T_\m$ maps the space of nonpositive functions into itself. This case has different character from the case $\bar J\ge0$ and $b(i,u)\ge0$ of this paper, in analogy with the well-known differences in structure between stochastic optimal control problems with nonpositive and nonnegative cost per stage.
\vskip-1.5pc
\def\vskip1pt\pn{\vskip1pt\par\noindent}
\def\ref{\vskip1pt\pn}
\vskip1pt\pn[1] Bertsekas, D.\ P., 2013.\ Abstract Dynamic Programming, Athena Scientific, Belmont, MA.
\vskip1pt\pn [2] Denardo, E.\ V., 1967.\ ``Contraction Mappings in the Theory Underlying Dynamic Programming," SIAM Review, Vol.\ 9, pp.\ 165-177.
\vskip1pt\pn[3] Denardo, E.\ V., and Mitten, L.\ G., 1967.\ ``Elements of Sequential Decision Processes," J.\ Indust.\ Engrg., Vol.\ 18, pp.\ 106-112.
\vskip1pt\pn [4] Bertsekas, D.\ P., 1975.\ ``Monotone Mappings in Dynamic Programming," 1975 IEEE Conference on Decision and Control, pp.\ 20-25.
\vskip1pt\pn [5] Bertsekas, D.\ P., 1977.\ ``Monotone Mappings with Application in Dynamic Programming," SIAM J.\ on Control and Opt., Vol.\ 15, pp.\ 438-464.
\vskip1pt\pn [6] Bertsekas, D.\ P., and Shreve, S.\ E., 1978.\ Stochastic Optimal Control: The Discrete Time Case, Academic Press, N.\ Y.; may be downloaded from http://web.mit.edu/dimitrib/www/home.html
\vskip1pt\pn [7] Blackwell, D., 1965.\ ``Positive Dynamic Programming," Proc.\ Fifth Berkeley Symposium Math.\ Statistics and Probability, pp.\ 415-418.
\vskip1pt\pn [8] Strauch, R., 1966.\ ``Negative Dynamic Programming," Ann.\ Math.\ Statist., Vol.\ 37, pp.\ 871-890.
\vskip1pt\pn[9] Verdu, S., and Poor, H.\ V., 1984.\ ``Backward, Forward, and Back\-ward-Forward Dynamic Programming Models under Commutativity Conditions," Proc.\ 1984 IEEE Decision and Control Conference, Las Vegas, NE, pp.\ 1081-1086.
\vskip1pt\pn[10] Szepesvari, C., 1998.\ Static and Dynamic Aspects of Optimal Sequential Decision Making, Ph.D.\ Thesis, Bolyai Institute of Mathematics, Hungary.
\vskip1pt\pn[11] Szepesvari, C., 1998.\ ``Non-Markovian Policies in Sequential Decision Problems," Acta Cybernetica, Vol.\ 13, pp.\ 305-318.
\vskip1pt\pn[12] Bertsekas, D.\ P., 2012.\ Dynamic Programming and Optimal Control, Vol.\ II, 4th Edition: Approximate Dynamic Programming, Athena Scientific, Belmont, MA.
\vskip1pt\pn[13] Bertsekas, D.\ P., and Yu, H., 2010.\ ``Asynchronous Distributed Policy Iteration in Dynamic Programming," Proc.\ of Allerton Conf.\ on Com., Control and Comp., Allerton Park, Ill, pp.\ 1368-1374.
\ref[14] Bertsekas, D.\ P., and Yu, H., 2012.\ ``Q-Learning and Enhanced Policy Iteration in Discounted Dynamic Programming," Math.\ of OR, Vol.\ 37, pp.\ 66-94.
\vskip1pt\pn[15] Yu, H., and Bertsekas, D.\ P., 2013.\ ``Q-Learning and Policy Iteration Algorithms for Stochastic Shortest Path Problems," Annals of Operations Research, Vol.\ 208, pp.\ 95-132.
\vskip1pt\pn [16] Eaton, J.\ H., and Zadeh, L.\ A., 1962.\ ``Optimal Pursuit Strategies in Discrete State Probabilistic Systems," Trans.\ ASME Ser.\ D.\ J.\ Basic Eng., Vol.\ 84, pp.23-29.
\vskip1pt\pn[17] Pallu de la Barriere, R., 1967.\ Optimal Control Theory, Saunders, Phila; republished by Dover, N. Y., 1980.
\vskip1pt\pn [18] Derman, C., 1970.\ Finite State Markovian Decision Processes, Academic Press, N.\ Y.
\vskip1pt\pn [19] Whittle, P., 1982.\ Optimization Over Time, Wiley, N.\ Y., Vol.\ 1, 1982, Vol.\ 2, 1983.
\vskip1pt\pn[20] Kallenberg, L.\ C.\ M.\ 1983.\ Linear Programming and Finite Markovian Control Problems, MC Tracts 148, Amterdam.
\vskip1pt\pn[21] Bertsekas, D.\ P., 1987.\ Dynamic Programming: Deterministic and Stochastic Models, Prentice-Hall, Englewood Cliffs, N.\ J.
\ref[22] Bertsekas, D.\ P., and Tsitsiklis, J.\ N., 1989.\ Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Englewood Cliffs, N.\ J.
\vskip1pt\pn[23] Altman, E., 1999.\ Constrained Markov Decision Processes, CRC Press, Boca Raton, FL.
\vskip1pt\pn[24] Hernandez-Lerma, O., and Lasserre, J.\ B., 1999.\ Further Topics on Discrete-Time Markov Control Processes, Springer, N.\ Y.
\vskip1pt\pn [25] Bertsekas, D.\ P., and Tsitsiklis, J.\ N., 1991.\ ``An Analysis of Stochastic Shortest Path Problems," Math.\ of OR, Vol.\ 16, pp.\ 580-595.
\vskip1pt\pn[26] Bertsekas, D.\ P., and Yu, H., 2016.\ ``Stochastic Shortest Path Problems Under Weak Conditions," Lab.\ for Information and Decision Systems Report LIDS-2909, August 2013, revised March 2015 and January 2016.
\vskip1pt\pn[27] Denardo, E.\ V., and Rothblum, U.\ G., 1979.\ ``Optimal Stopping, Exponential Utility, and Linear Programming," Math.\ Programming, Vol.\ 16, pp.\ 228-244.
\vskip1pt\pn[28] Patek, S.\ D., 2001.\ ``On Terminating Markov Decision Processes with a Risk Averse Objective Function," Automatica, Vol.\ 37, pp.\ 1379-1386.
\vskip1pt\pn [29] Puterman, M.\ L., 1994.\ Markov Decision Processes: Discrete Stochastic Dynamic Programming, J.\ Wiley, N.\ Y.
\vskip1pt\pn[30] Rothblum, U.\ G., 1984.\ ``Multiplicative Markov Decision Chains," Math.\ of OR, Vol.\ 9, pp.\ 6-24.
\vskip1pt\pn [31] Veinott, A.\ F., Jr., 1969.\ ``Discrete Dynamic Programming with Sensitive Discount Optimality Criteria," Ann.\ Math.\ Statist., Vol.\ 40, pp.\ 1635-1660.
\vskip1pt\pn [32] Bertsekas, D.\ P., and Tsitsiklis, J.\ N., 1996.\ Neuro-Dynamic Programming, Athena Scientific, Belmont, MA.
\vskip1pt\pn[33] Tseng, P., 1990.\ ``Solving $H$-Horizon, Stationary Markov Decision Problems in Time Proportional to $\log (H)$,'' Operations Research Letters, Vol.\ 9, pp.\ 287-297.
\vskip1pt\pn[34] Littman, M.\ L., 1996.\ Algorithms for Sequential Decision Making, Ph.D.\ thesis, Brown University, Providence, R.\ I.
\vskip1pt\pn[35] Yu, H., and Bertsekas, D.\ P., 2013.\ ``A Mixed Value and Policy Iteration Method for Stochastic Control with Universally Measurable Policies," arXiv preprint arXiv:1308.3814, to appear in Math.\ of OR.
\begin{IEEEbiography}{Dimitri P.\ Bertsekas} studied engineering at the National Technical University of Athens, Greece, obtained his MS in electrical engineering at the George Washington University, Wash. DC in 1969, and his Ph.D. in system science in 1971 at the Massachusetts Institute of Technology.
Dr. Bertsekas has held faculty positions with the Engineering-Economic Systems Dept., Stanford University (1971-1974) and the Electrical Engineering Dept. of the University of Illinois, Urbana (1974-1979). Since 1979 he has been teaching at the Electrical Engineering and Computer Science Department of the Massachusetts Institute of Technology (M.I.T.), where he is currently McAfee Professor of Engineering. He consults regularly with private industry and has held editorial positions in several journals. His research at M.I.T. spans several fields, including optimization, control, large-scale computation, and data communication networks, and is closely tied to his teaching and book authoring activities. He has written numerous research papers, and fourteen books, several of which are used as textbooks in MIT classes.
Professor Bertsekas was awarded the INFORMS 1997 Prize for Research Excellence in the Interface Between Operations Research and Computer Science for his book ``Neuro-Dynamic Programming" (co-authored with John Tsitsiklis), the 2001 ACC John R. Ragazzini Education Award, the 2009 INFORMS Expository Writing Award, the 2014 ACC Richard E. Bellman Control Heritage Award, the 2014 Khachiyan Prize, and the SIAM/MOS 2015 George B. Dantzig Prize. In 2001, he was elected to the United States National Academy of Engineering.
Dr. Bertsekas' recent books are ``Dynamic Programming and Optimal Control: 4th Edition" (2017), ``Noninear Programming: 3rd Edition" (2016), ``Convex Optimization Algorithms" (2015), and ``Abstract Dynamic Programming" (2013) all published by Athena Scientific. \end{IEEEbiography}
\end{document}
\end{document} | arXiv |
Volume element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
$\mathrm {d} V=\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}$
where the $u_{i}$ are the coordinates, so that the volume of any set $B$ can be computed by
$\operatorname {Volume} (B)=\int _{B}\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}.$
For example, in spherical coordinates $\mathrm {d} V=u_{1}^{2}\sin u_{2}\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}$, and so $\rho =u_{1}^{2}\sin u_{2}$.
The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a kind of measure on a manifold. On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form. On a non-orientable manifold, the volume element is typically the absolute value of a (locally defined) volume form: it defines a 1-density.
Volume element in Euclidean space
In Euclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates
$\mathrm {d} V=\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.$
In different coordinate systems of the form $x=x(u_{1},u_{2},u_{3})$, $y=y(u_{1},u_{2},u_{3})$, $z=z(u_{1},u_{2},u_{3})$, the volume element changes by the Jacobian (determinant) of the coordinate change:
$\mathrm {d} V=\left|{\frac {\partial (x,y,z)}{\partial (u_{1},u_{2},u_{3})}}\right|\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}.$
For example, in spherical coordinates (mathematical convention)
${\begin{aligned}x&=\rho \cos \theta \sin \phi \\y&=\rho \sin \theta \sin \phi \\z&=\rho \cos \phi \end{aligned}}$
the Jacobian determinant is
$\left|{\frac {\partial (x,y,z)}{\partial (\rho ,\theta ,\phi )}}\right|=\rho ^{2}\sin \phi $
so that
$\mathrm {d} V=\rho ^{2}\sin \phi \,\mathrm {d} \rho \,\mathrm {d} \theta \,\mathrm {d} \phi .$
This can be seen as a special case of the fact that differential forms transform through a pullback $F^{*}$ as
$F^{*}(u\;dy^{1}\wedge \cdots \wedge dy^{n})=(u\circ F)\det \left({\frac {\partial F^{j}}{\partial x^{i}}}\right)\mathrm {d} x^{1}\wedge \cdots \wedge \mathrm {d} x^{n}$
Volume element of a linear subspace
Consider the linear subspace of the n-dimensional Euclidean space Rn that is spanned by a collection of linearly independent vectors
$X_{1},\dots ,X_{k}.$
To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the $X_{i}$ is the square root of the determinant of the Gramian matrix of the $X_{i}$:
${\sqrt {\det(X_{i}\cdot X_{j})_{i,j=1\dots k}}}.$
Any point p in the subspace can be given coordinates $(u_{1},u_{2},\dots ,u_{k})$ such that
$p=u_{1}X_{1}+\cdots +u_{k}X_{k}.$
At a point p, if we form a small parallelepiped with sides $\mathrm {d} u_{i}$, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix
${\sqrt {\det \left((du_{i}X_{i})\cdot (du_{j}X_{j})\right)_{i,j=1\dots k}}}={\sqrt {\det(X_{i}\cdot X_{j})_{i,j=1\dots k}}}\;\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\cdots \,\mathrm {d} u_{k}.$
This therefore defines the volume form in the linear subspace.
Volume element of manifolds
See also: Riemannian volume form
On an oriented Riemannian manifold of dimension n, the volume element is a volume form equal to the Hodge dual of the unit constant function, $f(x)=1$:
$\omega =\star 1.$
Equivalently, the volume element is precisely the Levi-Civita tensor $\epsilon $.[1] In coordinates,
$\omega =\epsilon ={\sqrt {\left|\det g\right|}}\,\mathrm {d} x^{1}\wedge \cdots \wedge \mathrm {d} x^{n}$
where $\det g$ is the determinant of the metric tensor g written in the coordinate system.
Area element of a surface
A simple example of a volume element can be explored by considering a two-dimensional surface embedded in n-dimensional Euclidean space. Such a volume element is sometimes called an area element. Consider a subset $U\subset \mathbb {R} ^{2}$ and a mapping function
$\varphi :U\to \mathbb {R} ^{n}$
thus defining a surface embedded in $\mathbb {R} ^{n}$. In two dimensions, volume is just area, and a volume element gives a way to determine the area of parts of the surface. Thus a volume element is an expression of the form
$f(u_{1},u_{2})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}$
that allows one to compute the area of a set B lying on the surface by computing the integral
$\operatorname {Area} (B)=\int _{B}f(u_{1},u_{2})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}.$
Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix of the mapping is
$\lambda _{ij}={\frac {\partial \varphi _{i}}{\partial u_{j}}}$
with index i running from 1 to n, and j running from 1 to 2. The Euclidean metric in the n-dimensional space induces a metric $g=\lambda ^{T}\lambda $ on the set U, with matrix elements
$g_{ij}=\sum _{k=1}^{n}\lambda _{ki}\lambda _{kj}=\sum _{k=1}^{n}{\frac {\partial \varphi _{k}}{\partial u_{i}}}{\frac {\partial \varphi _{k}}{\partial u_{j}}}.$
The determinant of the metric is given by
$\det g=\left|{\frac {\partial \varphi }{\partial u_{1}}}\wedge {\frac {\partial \varphi }{\partial u_{2}}}\right|^{2}=\det(\lambda ^{T}\lambda )$
For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.
Now consider a change of coordinates on U, given by a diffeomorphism
$f\colon U\to U,$
so that the coordinates $(u_{1},u_{2})$ are given in terms of $(v_{1},v_{2})$ by $(u_{1},u_{2})=f(v_{1},v_{2})$. The Jacobian matrix of this transformation is given by
$F_{ij}={\frac {\partial f_{i}}{\partial v_{j}}}.$
In the new coordinates, we have
${\frac {\partial \varphi _{i}}{\partial v_{j}}}=\sum _{k=1}^{2}{\frac {\partial \varphi _{i}}{\partial u_{k}}}{\frac {\partial f_{k}}{\partial v_{j}}}$
and so the metric transforms as
${\tilde {g}}=F^{T}gF$
where ${\tilde {g}}$ is the pullback metric in the v coordinate system. The determinant is
$\det {\tilde {g}}=\det g\left(\det F\right)^{2}.$
Given the above construction, it should now be straightforward to understand how the volume element is invariant under an orientation-preserving change of coordinates.
In two dimensions, the volume is just the area. The area of a subset $B\subset U$ is given by the integral
${\begin{aligned}{\mbox{Area}}(B)&=\iint _{B}{\sqrt {\det g}}\;\mathrm {d} u_{1}\;\mathrm {d} u_{2}\\&=\iint _{B}{\sqrt {\det g}}\left|\det F\right|\;\mathrm {d} v_{1}\;\mathrm {d} v_{2}\\&=\iint _{B}{\sqrt {\det {\tilde {g}}}}\;\mathrm {d} v_{1}\;\mathrm {d} v_{2}.\end{aligned}}$
Thus, in either coordinate system, the volume element takes the same expression: the expression of the volume element is invariant under a change of coordinates.
Note that there was nothing particular to two dimensions in the above presentation; the above trivially generalizes to arbitrary dimensions.
Example: Sphere
For example, consider the sphere with radius r centered at the origin in R3. This can be parametrized using spherical coordinates with the map
$\phi (u_{1},u_{2})=(r\cos u_{1}\sin u_{2},r\sin u_{1}\sin u_{2},r\cos u_{2}).$
Then
$g={\begin{pmatrix}r^{2}\sin ^{2}u_{2}&0\\0&r^{2}\end{pmatrix}},$
and the area element is
$\omega ={\sqrt {\det g}}\;\mathrm {d} u_{1}\mathrm {d} u_{2}=r^{2}\sin u_{2}\,\mathrm {d} u_{1}\mathrm {d} u_{2}.$
See also
• Cylindrical coordinate system § Line and volume elements
• Spherical coordinate system § Integration and differentiation in spherical coordinates
• Surface integral
• Volume integral
References
• Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp. xii+510, ISBN 978-3-540-15279-8
1. Carroll, Sean. Spacetime and Geometry. Addison Wesley, 2004, p. 90
| Wikipedia |
\begin{document}
\title{A value distribution result related to Hayman's alternative} \author[K. S. Charak]{Kuldeep Singh Charak} \address{ \begin{tabular}{lll} &Kuldeep Singh Charak\\ &Department of Mathematics\\ &University of Jammu\\ &Jammu-180 006\\ &India\\ \end{tabular}} \email{[email protected]}
\author[A. Singh]{Anil Singh} \address{ \begin{tabular}{lll} &Anil Singh\\ &Department of Mathematics\\ &University of Jammu\\ &Jammu-180 006\\ &India \end{tabular}} \email{[email protected] }
\begin{abstract} Motivated by Bloch's Principle, we prove a value distribution result for meromorphic functions which is related to Hayman's alternative in certain sense. \end{abstract}
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} \footnotetext{2010 {\it Mathematics Subject Classification}. 30D35, 30D45.} \footnotetext{{\it Keywords and phrases}. Meromorphic function, Value distribution theory, Normal families, Bloch's principle.}
\maketitle
\section{Introduction and Main Result} The reader is assumed to be familiar with the standard notations of Nevanlinna value distribution theory of meromorphic functions (one may refer to \cite{ cherry, Hayman-1}) such as $T\left(r,f\right)$, $m\left(r,f\right)$, $N\left(r,f\right),$ etc. We shall denote the class of all meromorphic functions on a domain $D$ in $\ensuremath{\mathbb{C}}$ by $\mathcal{M}(D)$ and we shall write, `$\left\langle f,D\right\rangle\in\mathcal{P}$' for `$f\in \mathcal{M}(D)$ satisfies the property $\mathcal{P}$ on $D$' .
We say that $\phi \in \mathcal{M}(\ensuremath{\mathbb{C}})$ is a small function of $f \in \mathcal{M}(\ensuremath{\mathbb{C}})$ if $T(r,\phi)=S(r,f)$ as $r\to \infty$ possibly outside a set of $r$ of finite linear measure.
W.K. Hayman proved the following `Picard type' theorem, also known as Hayman's alternative:
\begin{theorem}\label{alternative} \cite{Hayman-2} Let $f\in \mathcal{M}(\ensuremath{\mathbb{C}})$ and let $l\geq 1.$ Suppose that $f(z)\neq 0, \mbox{ and } f^{(l)}(z)-1\neq 0$ for all $z\in\mathbb{C}.$ Then $f$ is constant. \end{theorem}
A subfamily $\mathcal{F}$ of $\mathcal{M}(D)$ is said to be normal in $D$ if every sequence of members of $\mathcal{F}$ contains a subsequence that converges locally uniformly (w.r.t. the spherical metric) in $D$. Recall Bloch's Principle (see\cite{rubel,Schiff}): {\it A subfamily $\mathcal{F}$ of $\mathcal{M}(D)$ with $\left\langle f,D\right\rangle\in\mathcal{P}$ for each $f\in \mathcal{F}$ is likely to be normal on $D$ if $\mathcal{P}$ reduces every $f\in \mathcal{M}(\ensuremath{\mathbb{C}})$ to a constant}. Neither Bloch's Principle nor its converse is true (see \cite{charak1,charak2,charak3,lahiri,rubel}).
According to Bloch's Principle, to every `Picard type' theorem there corresponds a normality criterion. A normality criterion corresponding to Theorem \ref{alternative} was proved by Y.Gu as follows:
\begin{theorem}\label{Gu} \cite{Gu} Let $\mathcal{F} \subseteq \mathcal{M}(D)$ and let $l\geq 1.$ Suppose that $f(z)\neq 0, \mbox{ and }f^{(l)}(z)-1\neq 0$ for all $z\in D$ and $f\in\mathcal{F}.$ Then $\mathcal{F}$ is normal in $D.$ \end{theorem}
The constants $0 \mbox{ and }1$ in Theorem \ref{alternative} and Theorem \ref{Gu} can be replaced by arbitrary constants $a \mbox{ and }b\neq 0$: \begin{theorem} \cite{Hayman-2}\label{alternative1} Let $f \in \mathcal{M}(D)$ and let $l\geq 1.$ Suppose that $f(z)\neq a, \mbox{ and }f^{(l)}(z)-b\neq 0$ for all $z\in\mathbb{C},$ where $a,b\in\mathbb{C},~b\neq 0.$ Then $f$ is constant. \end{theorem} \begin{theorem}\cite{Gu}\label{Gu1} Let $\mathcal{F} \subseteq \mathcal{M}(D)$ and let $l\geq 1.$ Suppose that $f(z)\neq a, \mbox{ and }f^{(l)}(z)-b\neq 0$ for all $z\in D,$ $f\in\mathcal{F}$, where $a,b\in\mathbb{C}$ with $~b\neq 0.$ Then $\mathcal{F}$ is normal in $D.$ \end{theorem} Note that if $l\geq 1$ and $b\in\mathbb{C}\setminus\left\{0\right\}$, then there is a polynomial $P(z)$ such that $P^{(l)}(z)=b.$ Using this observation, Theorem \ref{alternative1} and Theorem \ref{Gu1} can be restated as: \begin{theorem}\label{alternative2} Suppose that $P(z)$ is a polynomial of degree $l\geq 1$ and $a\in\mathbb{C}.$ If $f\in \mathcal{M}(\ensuremath{\mathbb{C}})$ is such that $f(z)\neq a \mbox{ and }\left(f(z)-P(z)\right)^{(l)}\neq 0,$ then $f$ is constant. \end{theorem} \begin{theorem}\label{Gu2}Suppose that $P(z)$ is a polynomial of degree $l\geq 1$ and $a\in\mathbb{C}.$ If $\mathcal{F} \subseteq \mathcal M(D)$ is such that each $f\in\mathcal{F}$ satisfies: $$f(z)\neq a\mbox{ and } \left(f(z)-P(z)\right)^{(l)}\neq 0,$$ then $\mathcal{F}$ is normal in $D.$ \end{theorem} \begin{remark}\label{remark} {\it Put $g=f-P\mbox{ and } R=Q-P$, where $P\mbox{ and }Q$ are polynomials with $deg(P-Q)=deg(Q)=l$ and $Q$ is non-constant. If $f(z)-P(z)\neq 0$ and $\left(f(z)-Q(z)\right)^{(l)}\neq 0,$ then by using Theorem \ref{alternative2}, we find that $f(z)=P(z)+c$, for some constant $c\neq 0.$} \end{remark} Remark \ref{remark} shows that Theorem \ref{alternative1} does not hold if $a$ is replaced by some non-constant function.
\begin{remark} {\it Suppose $f$ is an entire function such that $f-g$ has only finitely many zeros in the plane, where $g$ is some non-constant entire function. Further, let $$F(z)=\sum_{k=1}^{n}a_k(z)\left(f-g\right)^{(k)}$$ omits $1,$ where $a_k(z)$ are small functions of $f.$ Then by using Theorem 3.2 in \cite{Hayman-1}, we find that
$f(z)=g(z)+p(z)$, for some polynomial $p(z).$ Indeed, $$T(r,f-g)< \overline{N}(r,f-g)+N(r, {1\over f-g})+\overline{N}\left(r,{1\over F-1}\right)-N_0\left(r,{1\over F'}\right)+S(r,f),$$ where $N_0\left(r,{1\over F'}\right)$ is the counting function of the zeros of $F'$ which are not zeros of $F-1.$
Since $f-g$ is entire and has only finitely many zeros, it follows that
\begin{flalign*} T(r,f-g)< \overline{N}\left(r,{1\over F-1}\right)-N_0\left(r,{1\over F'}\right)+S(r,f)\\ \Rightarrow T(r,f-g)< \overline{N}\left(r,{1\over F-1}\right)+S(r,f). \end{flalign*} If $f-g$ is transcendental, then $F(z)=1$ must have infinitely many roots, which is a contradiction and hence $f-g$ must be a polynomial, say $p(z);$ that is, $f(z)=g(z)+p(z).$ } \end{remark}
Let $P\mbox{ and }Q$ be polynomials with $1\leq deg(P)<deg(Q)=l$ and $\mathcal{P}$ be the property defined as follows:
$``\left\langle f,D\right\rangle\in\mathcal{P}\Leftrightarrow f-P\neq 0\mbox{ and }\left(f-Q\right)^{(l)}\neq 0 \mbox{ on }D$''. That is, $f$ satisfies the property $\mathcal{P}$ if, and only if $f-P\mbox{ and }\left(f-Q\right)^{(l)}$ have no zeros in $D.$ With this $\mathcal{P}$, Theorem \ref{Gu2} immediately yields:
\begin{theorem}\label{Gu3} The family $\mathcal{F}:=\left\{f\in\mathcal{M}\left(D\right):\left\langle f,D\right\rangle\in\mathcal{P}\right\}$ is normal in $D.$
\end{theorem} Note that Remark \ref{remark} and Theorem \ref{Gu3} provide a counterexample to the converse of Bloch's Principle.
W. Schwick generalized Theorem \ref{Gu} :
\begin{theorem}\label{schwick} \cite{schiwick} Let $g\not\equiv 0$ be in $\mathcal{M}(D)$ and let $l\in\mathbb{N}.$ Let $\mathcal{F} \subseteq \mathcal{M}(D)$ be such that $f\neq 0,$ $f^{(l)}\neq g,$ and $f\mbox{ and }g$ have no common poles for each $f\in\mathcal{F}.$ Then $\mathcal{F}$ is normal in $D.$ \end{theorem}
According to the converse of Bloch's Principle, one may find a `Picard type' theorem corresponding to Theorem \ref{schwick}, and this is the purpose of this paper. In fact, we prove the following value distribution result corresponding to Theorem \ref{schwick} which is related to Hayman's alternative in certain sense:
\begin{theorem}\label{MTA} Suppose that $f\in \mathcal{M}(\ensuremath{\mathbb{C}})$ is transcendental and $\phi$ is a small function of $f$ such that $f\mbox{ and }\phi$ have no common poles. Let $l\in\mathbb{N}$ and $\psi(z)=f^{(l)}(z)$. If $f(z)\neq 0\mbox{ and } \psi(z)\neq \phi(z)$ for all $z\in\mathbb{C},$ then
$\psi'(z)= \phi(z)$ and $\psi'(z)= \phi'(z)$ have infinitely many solutions.
\end{theorem}
\section{ Proof of Theorem \ref{MTA}}\label{maintheorem} Since the proof of Theorem \ref{MTA} is based on Milloux techniques (see \cite{Hayman-1} p.60), we need to prove some key lemmas for the proof of Theorem \ref{MTA}. Throughout this paper, we shall denote $f^{(l)}(z)$ by $\psi(z)$, where $l\in\mathbb{N}$.
\begin{lemma}\label{theorem 3.2} Let $f \in \mathcal{M}(\ensuremath{\mathbb{C}})$ and let $\phi$ be a small function of $f.$ Then for $r\to\infty$ outside a set of finite linear measure, \begin{equation}\label{2.1}T\left(r,f\right)\leq\bar{N}\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^0_2\left(r,\psi\right)+S(r,f), \end{equation} where $N^0_2\left(r,\psi\right)=N\left(r,\frac{1}{\psi'}\right)-N_0\left(r,\frac{1}{\psi'}\right)$ and $N_0\left(r,\frac{1}{\psi'}\right)$ is the counting function of zeros of $\psi'$ which are not zeros of $\psi$. \end{lemma} \begin{proof} By the second fundamental theorem of Nevanlinna for three small functions (see \cite{Hayman-1} Theorem. 2.5, also see\cite{cherry} Theorem. 5.9.1) with $a_1=0,a_2=\infty\mbox{ and } a_3=\phi$, we have \begin{equation}\label{one} T(r,f)\leq \bar{N}\left(r,\frac{1}{f}\right)+\bar{N}\left(r,{f}\right)+\bar{N}\left(r,\frac{1}{f-\phi}\right)+S(r,f). \end{equation} as $r\to\infty$ outside a set of $r$ of finite linear measure.
Since
$\bar{N}\left(r,\frac{1}{f}\right)={N}\left(r,\frac{1}{f}\right)-{N}\left(r,\frac{1}{f'}\right)+{N}_0\left(r,\frac{1}{f'}\right),$\\ $~~~~\bar{N}\left(r,\frac{1}{f-\phi}\right)={N}\left(r,\frac{1}{f-\phi}\right)-{N}\left(r,\frac{1}{f'-\phi '}\right)+{N}_0\left(r,\frac{1}{f'-\phi '}\right)$\\
and
$\bar{N}\left(r,f\right)=N\left(r,f'\right)-N\left(r,f\right),$\\ therefore (\ref{one}) yeilds (after adding $m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{f-\phi}\right)$ to both sides) \begin{flalign*} T\left(r,f\right)+m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{f-\phi}\right) & \leq m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{f-\phi}\right)\\ &+N\left(r,\frac{1}{f}\right)+N\left(r,\frac{1}{f-\phi}\right) +N\left(r,{f'}\right)-N\left(r,{f}\right) &&\\ &-N\left(r,\frac{1}{f'}\right)+N_0\left(r,\frac{1}{f'}\right) +N_0\left(r,\frac{1}{f'-\phi '}\right)&&\\ &-N\left(r,\frac{1}{f'-\phi '}\right)+S(r,f) && \end{flalign*}
\begin{flalign}\label{eq:2MT} \Rightarrow m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{f-\phi}\right)+m\left(r,{f}\right) & \leq 2T\left(r,f\right)-N_1\left(r,f\right) +N_0\left(r,\frac{1}{f'}\right) \nonumber\\
& +N_0\left(r,\frac{1}{f'-\phi '}\right)-N\left(r,\frac{1}{f'-\phi '}\right)+S(r,f),&& \end{flalign}
where $N_1\left(r,f\right)=N\left(r,\frac{1}{f'}\right)+2N\left(r,f\right)-N\left(r,f'\right).$
Applying (\ref{eq:2MT}) to $\psi=f^{(l)} \mbox{ and put }g=\psi-\phi, $ we have \begin{flalign}\label{eq:3MT} m\left(r,\frac{1}{\psi}\right)+m\left(r,\frac{1}{g}\right)+m\left(r,{\psi}\right) &\leq 2T\left(r,\psi\right)-N_1\left(r,\psi\right)+N_0\left(r,\frac{1}{\psi'}\right) \nonumber\\
&+N_0\left(r,\frac{1}{g '}\right)-N\left(r,\frac{1}{g'}\right)+S(r,\psi) && \end{flalign} as $r\to\infty$ outside a set of $r$ of finite linear measure.\\
Since
$N\left(r,\psi'\right)-N\left(r,\psi\right)=\bar{N}\left(r,f\right)$\\
and
$N\left(r,\frac{1}{g}\right)-N\left(r,\frac{1}{g'}\right)=\bar{N}\left(r,\frac{1}{g}\right)-N_0\left(r,\frac{1}{g'}\right)$
and using the first fundamental theorem of Nevanlinna, we have \begin{flalign*}\label{eq:4MT} 2T\left(r,\psi\right)-N_1\left(r,\psi\right)& =m\left(r,{\psi}\right)+m\left(r,\frac{1}{g}\right)+N\left(r,\psi\right)+N\left(r,\frac{1}{g}\right)\\ &-N\left(r,\frac{1}{\psi '}\right)-2N\left(r,\psi\right)+N\left(r,\psi '\right) +S(r,f) &&\\ &=m\left(r,{\psi}\right)+m\left(r,\frac{1}{g}\right) + N\left(r,\frac{1}{g}\right)-N\left(r,\frac{1}{\psi'}\right) &&\\ &+\bar{N}\left(r,f\right)+S(r,f)&& \end{flalign*} and hence (\ref{eq:3MT}) reduces to \begin{equation}\label{eq:proximity}
m\left(r,\frac{1}{\psi}\right)\leq \bar{N}\left(r,\frac{1}{g}\right)-N\left(r,\frac{1}{\psi'}\right)+\bar{N}\left(r,f\right)+N_0\left(r,\frac{1}{\psi'}\right)+S(r,f).
\end{equation}
Also
\begin{flalign}\label{eq:10MT}
T\left(r,f\right) &=T\left(r,\frac{1}{f}\right)+O(1)\nonumber\\
&=m\left(r,\frac{\psi}{f\psi}\right)+N\left(r,\frac{1}{f}\right)+O(1)&&\nonumber\\
&\leq m\left(r,\frac{\psi}{f}\right)+m\left(r,\frac{1}{\psi}\right)+N\left(r,\frac{1}{f}\right)+O(1)&&\nonumber\\
&=m\left(r,\frac{1}{\psi}\right)+N\left(r,\frac{1}{f}\right)+S(r,f).&&
\end{flalign} Now by using (\ref{eq:proximity}) in (\ref{eq:10MT}), we get \begin{flalign}\label{eq:11MT} T\left(r,f\right) &\leq \bar{N}\left(r,f\right)+N\left(r,\frac{1}{f} \right) +\bar{N}\left(r,\frac{1}{g}\right) -N\left(r,\frac{1}{\psi'}\right)+N_0\left(r,\frac{1}{\psi'}\right)+S(r,f) \nonumber\\ &=\bar{N}\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^0_2\left(r,{\psi}\right)+S(r,f)&& \end{flalign} as $r\to\infty$ outside a set of $r$ of finite linear measure, where $$N^0_2\left(r,\psi\right)=N\left(r,\frac{1}{\psi'}\right)-N_0\left(r,\frac{1}{\psi'}\right)=N\left(r,\frac{1}{\psi}\right)-\bar{N}\left(r,\frac{1}{\psi}\right)$$ counts only repeated zeros of $\psi$ with multiplicity reduced by $1$. \end{proof}
\begin{lemma}\label{lemma3.1A}
Let $f\in\mathcal{M}\left(\mathbb{C}\right)$ and let $\phi$ be a small function of $f$ such that $f \mbox{ and }\phi$ have no common poles. Then \begin{equation} lN_1\left(r,f\right)\leq \bar{N_2}\left(r,f\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\frac{1}{\psi'-\phi}\right)+S(r,f), \end{equation}
where $N_1\left(r,f\right)$ is the counting function of simple poles of $f$and $\bar{N_2}\left(r,f\right)$ is the counting function of multiple poles of $f$ counted once. \end{lemma} The proof of Lemma \ref{lemma3.1A} is carried out by following the proof of Lemma 3.1 in \cite{Hayman-1} with certain modifications.
{\bf Proof of Lemma \ref{lemma3.1A}.}
Put $$G(z)={\left\{\psi'(z)-\phi(z)\right\}^{l+1}\over {\left\{ \phi(z)-\psi(z)\right\}^{l+2}}}.$$
If $z_0$ is a simple pole of $f(z),$ then near $z_0$ we have $$f(z)={a\over z-z_0}+O(1),$$ and $$\psi(z)={(-1)^{l}a (l!) \over \left(z-z_0\right)^{l+1}}+O(1)$$
near $z_0$, and hence $$ \psi'={(-1)^{l+1}a(l+1)! \over \left(z-z_0\right)^{l+2}}\left\{1+O(z-z_0)^{l+2}\right\}.$$
Since $f$ and $\phi$ have no common poles, therefore near $z_0,$ we have $$G(z)={(-1)^{l+1}(l+1)^{l+1} \over {al!}}\left\{1+O(z-z_0)^{l+1}\right\}$$
which implies that $ G(z_0)\neq 0,\infty,$ and $G'(z)$ has a zero of order atleast $l$ at $z_0$ and so \begin{eqnarray}\label{eq:1L} l N_1\left(r,f\right)\leq N_0\left(r,\frac{1}{G'}\right). \end{eqnarray}
Applying Jensen's formula to ${G'/{G}}$ we get \begin{equation}\label{eq:2L} N\left(r,\frac{G}{G'}\right)-N\left(r,\frac{G'}{G}\right)=m\left(r,\frac{G'}{G}\right)-m\left(r,\frac{G}{G'}\right)+O(1). \end{equation} Since the only zeros of ${G'}/{G}$ are the zeros of $G'$ which are not zeros of $G,$ we have \begin{eqnarray}\label{eq:3L} N\left(r,\frac{G}{G'}\right)=N_0\left(r,\frac{1}{G'}\right). \end{eqnarray}
Also, ${G'}/{G}$ has only simple poles at the zeros and poles of $G,$ so \begin{eqnarray}\label{eq:4L}
N\left(r,\frac{G'}{G}\right)=\bar{N}\left(r, G\right)+\bar{N}\left(r,\frac{1}{G}\right). \end{eqnarray}
Using (\ref{eq:3L}) and (\ref{eq:4L}) in (\ref{eq:2L}), we obtain \begin{flalign}\label{eq:5L} N_0\left(r,\frac{1}{G'}\right)-\bar{N}\left(r,\frac{1}{G}\right)-\bar{N}\left(r,{G}\right) =m\left(r,\frac{G'}{G}\right)-m\left(r,\frac{G}{G'}\right)+O(1). \end{flalign} Now, from (\ref{eq:1L}), (\ref{eq:2L}) and (\ref{eq:5L}), we have \begin{flalign}\label{eq:simple pole}
lN_1\left(r,f\right) &\leq N_0\left(r,\frac{1}{G'}\right)\nonumber\\ &= \bar{N}\left(r,\frac{1}{G}\right)+\bar{N}\left(r,{G}\right)+m\left(r,\frac{G'}{G}\right)-m\left(r,\frac{G}{G'}\right)+O(1)&&\nonumber\\ &\leq \bar{N}\left(r,\frac{1}{G}\right)+\bar{N}\left(r,{G}\right)+m\left(r,\frac{G'}{G}\right)+O(1).&& \end{flalign}
Let $z_0$ be a pole of $\psi(z)-\phi(z)$ of order $m$, say. Then near $z_0$ $$\psi(z)-\phi(z)={s_0(z)\over\left(z-z_0\right)^m}$$ for some function $s_0(z)$ analytic in a neighborhood of $z_0 \mbox{ such that } s_0(z_0)\neq 0.$
Now, there are two cases:
\begin{itemize}
\item[Case 1:] $z_0$ is a pole of $f(z)$. Then $m=k+l $, where $k>1$ is the multiplicity of $z_0$ as a pole of $f$. Since $z_0$ is not a pole of $\phi,$ we see that $z_0$ is a pole of $\psi'(z)-\phi(z)$ of multiplicity $m+1.$ Therefore, near $z_0$
$$G(z)=t_0(z)\left(z-z_0\right)^{k-1},$$
for some function $t_0(z)$ analytic in a neighborhood of $z_0 \mbox{ such that } t_0(z_0)\neq 0.$ So $z_0$ is a zero of order $k-1$ of $G(z)$.
\item[Case 2:] $z_0$ is a pole of $\phi(z).$ Then $z_0$ is also a pole of $\psi'(z)-\phi(z)$ of multiplicity $m.$ Therefore, near $z_0$
$$G(z)=t_1(z)\left(z-z_0\right)^{m},$$
for some function $t_1(z)$ analytic in a neighborhood of $z_0 \mbox{ such that } t_1(z_0)\neq 0.$ This shows that $z_0$ is a pole of $G(z)$ of the same multiplicity as that of $\phi(z).$ \end{itemize} Similarly, looking at the poles of $\psi'(z)-\phi(z),$ we obtain the same conclusion as in the case of poles of $\psi(z)-\phi(z).$
Next, correspnding to the zeros of $\psi(z)-\phi(z)$ and $\psi'(z)-\phi(z),$ we have the following three cases:
\begin{itemize}
\item[Case 1:] $z_0$ is a zero of $\psi(z)-\phi(z)$ but it is not a zero of $\psi'(z)-\phi(z).$ Then $z_0$ is a pole of $G(z)$.
\item[Case 2:] $z_0$ is zero of $\psi'(z)-\phi(z)$ but it is not a zero of $\psi(z)-\phi(z).$ Then $z_0$ is a zero of $G(z)$.
\item[Case 3:] $z_0$ is a common zero of $\psi'(z)-\phi(z) \mbox{ and }\psi(z)-\phi(z)$. Let $j$ and $k$ be the multiplicities of $z_0$ as a zero of $\psi'(z)-\phi(z) \mbox{ and }\psi(z)-\phi(z)$, respectively. Then near $z_0$,
$$G(z)=t_2(z)\left(z-z_0\right)^{(l+1)j-(l+2)k}$$
for some function $t_2(z)$ analytic in a neighborhood of $z_0 \mbox{ such that }t_2(z_0)\neq 0$.
Thus $z_0$ is a pole of $G(z)$ if $k>\frac{l+1}{l+2}j$ and $z_0$ is a zero of $G(z)$ if $k<\frac{l+1}{l+2}j$. \end{itemize}
Let $N(r,\frac{1}{f},\frac{1}{g^0})$ be the counting function of zeros of $f$ which are not zeros of $g$ , $N(r,\frac{1}{f},\frac{1}{g})$ be the counting function corresponding to the common zeros of $f\mbox{ and }g$ and $N^{(\alpha)}(r,\frac{1}{f},\frac{1}{g})$ be the counting function corresponding to the common zeros of $f$ and $g$, such that the $m(f, z_0)>\alpha m(g,z_0)$, where by $m(f,z_0)$ we denote the multiplicity of $z_0$ as a zero of $f$. With these notations and the preceding arguments , we find that
\begin{flalign}\label{eq:poles}
\bar{N}\left(r,\frac{1}{G}\right) &\leq \bar{N}_2(r,f)+\bar{N}(r,\phi) + \bar{N}\left(r,\frac{1}{\psi'-\phi},\frac{1}{(\psi-\phi)^0}\right) \nonumber \\ &+\bar{N}^{\left(\frac{l+2}{l+1}\right)}\left(r,\frac{1}{\psi'-\phi},\frac{1}{\psi-\phi}\right) && \end{flalign} and \begin{flalign}\label{eq:zeros} \bar{N}(r,G) &\leq \bar{N}\left(r,\frac{1}{\psi-\phi},\frac{1}{(\psi'-\phi)^0}\right)+\bar{N}^{\left(\frac{l+1}{l+2}\right)}\left(r,\frac{1}{\psi-\phi},\frac{1}{\psi'-\phi}\right). \end{flalign} Note that \begin{flalign*}
&\bar{N}\left(r,\frac{1}{\psi-\phi},\frac{1}{(\psi'-\phi)^0}\right)+\bar{N}^{\left(\frac{l+1}{l+2}\right)}\left(r,\frac{1}{\psi-\phi},\frac{1}{\psi'-\phi}\right) &&\\ &+\bar{N}\left(r,\frac{1}{\psi'-\phi},\frac{1}{(\psi-\phi)^0}\right)+\bar{N}^{\left(\frac{l+2}{l+1}\right)}\left(r,\frac{1}{\psi'-\phi},\frac{1}{\psi-\phi}\right) && \\ &\leq \bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\frac{1}{\psi'-\phi}\right).&& \end{flalign*} Therefore, using (\ref{eq:poles}) and (\ref{eq:zeros}) in (\ref{eq:simple pole}), we get \begin{flalign}\label{eq:I1}
lN_1\left(r,f\right) & \leq \bar{N}_2(r,f)+\bar{N}(r,\phi) + \bar{N}\left(r,\frac{1}{\psi-\phi}\right)
+\bar{N}\left(r,\frac{1}{\psi'-\phi}\right)+m\left(r,\frac{G'}{G}\right)+O(1). && \end{flalign} Since $T(r,\phi)=S(r,f)$ and $S(r,\psi)=S(r,f)$, by Theorem 3.1 in \cite{Hayman-1}, we have $$m\left(r,\frac{G'}{G}\right)=S(r,f).$$
Thus from (\ref{eq:I1}), it follows that
\begin{flalign}
lN_1\left(r,f\right) & \leq \bar{N}_2(r,f) + \bar{N}\left(r,\frac{1}{\psi-\phi}\right)
+\bar{N}\left(r,\frac{1}{\psi'-\phi}\right)+S(r,f). && \end{flalign} \qed
\begin{lemma}\label{22feb} Let $f\in\mathcal{M}\left(\mathbb{C}\right)$ and let $\phi$ be a small function of $f$ such that $f \mbox{ and }\phi$ have no common poles. Then \begin{equation} lN_1\left(r,f\right)\leq \bar{N_2}\left(r,f\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+ N_0\left(r,\frac{1}{\psi'-\phi'}\right)+S(r,f), \end{equation}
where $N_1\left(r,f\right)$ is the counting function of simple poles of $f$, $\bar{N_2}\left(r,f\right)$ is the counting function of multiple poles of $f$ counted once and $N_0\left(r,\frac{1}{\psi'-\phi'}\right)$ is the counting function of zeros of $\psi'-\phi'$ which are not repeated zeros of $\psi-\phi$. \end{lemma} \begin{proof}
Define $$G(z)={\left\{\psi'(z)-\phi'(z)\right\}^{l+1}\over {\left\{ \phi(z)-\psi(z)\right\}^{l+2}}}.$$ Then as in the proof of Lemma \ref{lemma3.1A} above, we
again arrive at (\ref{eq:simple pole}).
Next, to find the distribution of poles and zeros of $G(z)$, we proceed as follows;
Put, $$h(z)=\psi(z)-\phi(z).$$
If $z_0$ is a pole of $h(z)$ of order $m,$ then near $z_0,$ $$h(z)={s(z)\over\left(z-z_0\right)^m}\mbox{ and }h'(z)={t(z)\over\left(z-z_0\right)^{m+1}}~,$$
where $s(z)\mbox{ and }t(z)$ are functions analytic in a neighborhood of $z_0$ and both have no zeros at $z_0.$ So, \begin{eqnarray}\label{eq:6LA} G(z)={w(z)\over\left(z-z_0\right)^{l+1-m}} \end{eqnarray}
for some function $w(z)$ analytic in a neighborhood of $z_0 \mbox{ such that }w(z_0)\neq 0.$
Next if $z_0$ is a zero of $h(z),$ then near $z_0$, $h(z)={l(z)\left(z-z_0\right)^m}$ and so \begin{eqnarray}\label{eq:7LA} G(z)={m(z)\over\left(z-z_0\right)^{l+1+m}} \end{eqnarray} where $l(z)\mbox{ and }m(z)$ are functions analytic in a neighborhood of $z_0$ and both have no zeros at $z_0.$
From (\ref{eq:7LA}) and (\ref{eq:6LA}) we see that the only poles of $G(z)$ occur at
\begin{itemize}
\item[(i)] the roots of $h(z)=0$ and
\item[(ii)] the poles of $\phi(z)$ of multiplicity less than $l+1$.
\end{itemize}
Therefore, \begin{eqnarray}\label{eq:8LA} \bar{N}\left(r,G\right)\leq\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\phi\right)-\bar{N}_{l+1}\left(r,\phi\right), \end{eqnarray}
where $\bar{N}_{k}\left(r,\phi\right)$ is the counting function of poles of $\phi(z)$ which have multiplicity atleast $k$, each pole is counted once.
Since the zeros of $G(z)$ occur at
\begin{itemize}
\item[(i)] the roots of $h'(z)=0$ which are not the roots of $h(z)=0$
\item[(ii)] multiple poles of $f(z)$ and
\item[(iii)] poles of $\phi$ of multiplicity greater than $l+1$, \end{itemize}
therefore, \begin{eqnarray}\label{eq:9LA} \bar{N}\left(r,\frac{1}{G}\right)\leq \bar{N}_0\left(r,\frac{1}{\psi'-\phi'}\right)+\bar{N}_2\left(r,f\right)+\bar{N}_{l+1}\left(r,\phi\right). \end{eqnarray}
Adding (\ref{eq:8LA}) and (\ref{eq:9LA}), we have; \begin{flalign}\label{eq:10LA} \bar{N}\left(r,G\right)+\bar{N}\left(r,\frac{1}{G}\right) \leq \bar{N_2}\left(r,f\right)+\bar{N}\left(r,\phi\right) +\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+N_0\left(r,\frac{1}{\psi'-\phi'}\right). \end{flalign} Since $T(r,\phi)=S(r,f)$ and $S(r,\psi)=S(r,f)$, by Theorem $3.1$ in \cite{Hayman-1}, we have $$m\left(r,\frac{G'}{G}\right)=S(r,f) .$$
Thus, from (\ref{eq:simple pole}) and (\ref{eq:10LA}), it follows that
\begin{equation}\label{eq:12LA}
lN_1\left(r,f\right)\leq \bar{N_2}\left(r,f\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+N_0\left(r,\frac{1}{\psi'-\phi'}\right)+ S(r,f).
\end{equation} \end{proof}
\begin{lemma}\label{final lemmaA} Let $f, \psi \mbox{ and }\phi$ be as in Lemma \ref{lemma3.1A}. Then \begin{flalign*}
(a)~ T\left(r,f\right)&\leq \left(2+\frac{1}{l}\right)N\left(r,\frac{1}{f}\right)+\left(2+\frac{2}{l}\right)\bar{N}\left(r,\frac{1}{\psi-\phi}\right)\\ &+\frac{1}{l}\bar{N}\left(r,\frac{1}{\psi'-\phi}\right)-\left(2+\frac{1}{l}\right)N^0_2\left(r,\psi\right)+S(r,f).&& \end{flalign*}
\begin{flalign*} (b)~ T\left(r,f\right)&\leq \left(2+\frac{1}{l}\right)N\left(r,\frac{1}{f}\right)+\left(2+\frac{2}{l}\right)\bar{N}\left(r,\frac{1}{\psi-\phi}\right)\\ &+\frac{1}{l}N_0\left(r,\frac{1}{\psi'-\phi'}\right)-\left(2+\frac{1}{l}\right)N^0_2\left(r,\psi\right)+S(r,f).&& \end{flalign*} \end{lemma} \begin{proof} By Lemma \ref{theorem 3.2}, we have \begin{flalign}\label{eq:1FL}
N_1\left(r,f\right)+2\bar{N_2}\left(r,f\right)&\leq N\left(r,f\right) \nonumber\\ &\leq T\left(r,f\right) &&\nonumber\\ &\leq\bar{N}\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^0_2\left(r,\psi\right)+S(r,f).&& \end{flalign} Since $\bar{N}\left(r,f\right)=N_1\left(r,f\right)+\bar{N_2}\left(r,f\right),$ from (\ref{eq:1FL}) we have \begin{eqnarray}\label{eq:2FL} \bar{N_2}\left(r,f\right)\leq N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^0_2\left(r,\psi\right)+S(r,f) . \end{eqnarray}
Using (\ref{eq:2FL}) in Lemma \ref{lemma3.1A}, we obtain \begin{flalign}\label{eq:3FL} N_1\left(r,f\right) &\leq \frac{1}{l} N\left(r,\frac{1}{f}\right)+\frac{2}{l}\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-\frac{1}{l}N^0_2\left(r,\psi\right)\nonumber\\ &+\frac{1}{l}\bar{N}\left(r,\frac{1}{\psi'-\phi}\right)+ S(r,f).&& \end{flalign}
Now from (\ref{eq:2FL}) and (\ref{eq:3FL}) it follows that \begin{flalign}\label{eq:4FL} \bar{N}\left(r,f\right)&= N_1\left(r,f\right)+\bar{N_2}\left(r,f\right) \nonumber\\ &\leq N_1\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^0_2\left(r,\psi\right)+S(r,f) &&\nonumber\\ &\leq \left(1+\frac{1}{l} \right)N\left(r,\frac{1}{f}\right)+\left(1+\frac{2}{l}\right)\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-\left(1+\frac{1}{l}\right)N^0_2\left(r,\psi\right) &&\nonumber\\ &+\frac{1}{l}\bar{N}\left(r,\phi\right)+\frac{1}{l}\bar{N}\left(r,\frac{1}{\psi'-\phi}\right)+S(r,f).&&
\end{flalign}
Now in view of (\ref{eq:4FL}), Lemma \ref{theorem 3.2} yields \begin{flalign*} T\left(r,f\right)&\leq \left(2+\frac{1}{l}\right)N\left(r,\frac{1}{f}\right)+\left(2+\frac{2}{l}\right)\bar{N}\left(r,\frac{1}{\psi-\phi}\right)\\ &+\frac{1}{l}\bar{N}\left(r,\frac{1}{\psi'-\phi}\right)-\left(2+\frac{1}{l}\right)N^0_2\left(r,\psi\right)+S(r,f),&& \end{flalign*} which proves $(a)$.
The conclusion $(b)$ follows by using Lemma \ref{22feb} instead of Lemma \ref{lemma3.1A} in the proof of $(a)$, above.
\end{proof}
\textbf{Proof of Theorem \ref{MTA}: } Since $N^0_2\left(r,\psi\right)\geq 0$, by Lemma \ref{final lemmaA}$(a)$ we have \begin{flalign}\label{eq:1} T\left(r,f\right)&\leq 3N\left(r,\frac{1}{f}\right)+4\bar{N}\left(r,\frac{1}{\psi-\phi}\right)
+\bar{N}\left(r,\frac{1}{\psi'-\phi}\right)+S(r,f). \end{flalign}
Since $f\mbox{ and } \psi-\phi$ have no zeros, $N\left(r,\frac{1}{f}\right)=0 \mbox{ and }\bar{N}\left(r,\frac{1}{\psi-\phi}\right)=0.$ Therefore, (\ref{eq:1}) reduces to \begin{eqnarray}\label{eq:4} T\left(r,f\right)\leq \bar{N}\left(r,\frac{1}{\psi'-\phi}\right)+S(r,f). \end{eqnarray} Since $f\in\mathcal{M}\left(\mathbb{C}\right)$ is transcendental, (\ref{eq:4}) implies that $\psi'(z)=\phi(z)$ has infinitely many solutions.\\
Similarly, Lemma \ref{final lemmaA}$(b)$ implies that $\psi'(z)=\phi'(z)$ has infinitely many solutions. ~~~~~~~~~~~~~~~~~~~~~~~~~$\qed$
\end{document} | arXiv |
\begin{definition}[Definition:Lipschitz Continuity/Real Function]
Let $A \subseteq \R$.
Let $f: A \to \R$ be a real function.
Let $I \subseteq A$ be a real interval on which:
:$\exists K \in \R_{\ge 0}: \forall x, y \in I: \size {\map f x - \map f y} \le K \size {x - y}$
Then $f$ is '''Lipschitz continuous on $I$'''.
The constant $K$ is known as '''a Lipschitz constant for $f$'''.
\end{definition} | ProofWiki |
Douady rabbit
The Douady rabbit is any of various particular filled Julia sets whose parameter is near the center of a period 3 bud of the Mandelbrot set for complex quadratic map. It is named after French mathematician Adrien Douady.
Formula
The rabbit is generated by iterating the Mandelbrot set map $z=z^{2}+c$ on the complex plane with $c$ fixed to lie in the period three bulb off the main cardiod and $z$ ranging over the plane. The pixels in the image are then colored to show whether for a particular value of $z$ the iteration converged or diverged.
Variants
The Twisted rabbit[1] is the composition of the rabbit polynomial with nth powers of the Dehn twists about its ears.[2]
Corabbit is symmetrical image of rabbit. Here parameter $c\approx -0.1226-0.7449i$. It is one of 2 other polynomials inducing the same permutation of their post-critical set are the rabbit.
3D
The Julia set has no direct analog in 3D.
4D
Quaternion Julia set with parameters c = −0,123 + 0.745i and a cross section in the XY plane. The "Douady Rabbit" Julia set is visible in the cross section.
Embedded
A small "embedded" homeomorphic copy of rabbit in the center of a Julia set[3]
Fat
The fat rabbit or chubby rabbit has c at the root of 1/3-limb of the Mandelbrot set. It has a parabolic fixed point with 3 petals.[4]
• fat rabbit
• parabolic chessboard
n-th eared
• period 4 bulb rabbit = Three-Eared Rabbit
• period 5 bulb rabbit = Four-Eared Rabbit
In general, the rabbit for the period-(n+1) bulb off the main cardiod will have n ears[5]
Perturbed
Perturbed rabbit[6]
• Perturbed Rabbit
• Perturbed Rabbit
• Perturbed rabbit zoom
Forms of the complex quadratic map
There are two common forms for the complex quadratic map ${\mathcal {M}}$. The first, also called the complex logistic map, is written as
$z_{n+1}={\mathcal {M}}z_{n}=\gamma z_{n}\left(1-z_{n}\right),$
where $z$ is a complex variable and $\gamma $ is a complex parameter. The second common form is
$w_{n+1}={\mathcal {M}}w_{n}=w_{n}^{2}-\mu .$
Here $w$ is a complex variable and $\mu $ is a complex parameter. The variables $z$ and $w$ are related by the equation
$z=-{\frac {w}{\gamma }}+{\frac {1}{2}},$
and the parameters $\gamma $ and $\mu $ are related by the equations
$\mu =\left({\frac {\gamma -1}{2}}\right)^{2}-{\frac {1}{4}}\quad ,\quad \gamma =1\pm {\sqrt {1+4\mu }}.$
Note that $\mu $ is invariant under the substitution $\gamma \to 2-\gamma $.
Mandelbrot and filled Julia sets
There are two planes associated with ${\mathcal {M}}$. One of these, the $z$ (or $w$) plane, will be called the mapping plane, since ${\mathcal {M}}$ sends this plane into itself. The other, the $\gamma $ (or $\mu $) plane, will be called the control plane.
The nature of what happens in the mapping plane under repeated application of ${\mathcal {M}}$ depends on where $\gamma $ (or $\mu $) is in the control plane. The filled Julia set consists of all points in the mapping plane whose images remain bounded under indefinitely repeated applications of ${\mathcal {M}}$. The Mandelbrot set consists of those points in the control plane such that the associated filled Julia set in the mapping plane is connected.
Figure 1 shows the Mandelbrot set when $\gamma $ is the control parameter, and Figure 2 shows the Mandelbrot set when $\mu $ is the control parameter. Since $z$ and $w$ are affine transformations of one another (a linear transformation plus a translation), the filled Julia sets look much the same in either the $z$ or $w$ planes.
Figure 1: The Mandelbrot set in the $\gamma $ plane.
Figure 2: The Mandelbrot set in the $\mu $ plane.
The Douady rabbit
The Douady rabbit is most easily described in terms of the Mandelbrot set as shown in Figure 1 (above). In this figure, the Mandelbrot set, at least when viewed from a distance, appears as two back-to-back unit discs with sprouts. Consider the sprouts at the one- and five-o'clock positions on the right disk or the sprouts at the seven- and eleven-o'clock positions on the left disk. When $\gamma $ is within one of these four sprouts, the associated filled Julia set in the mapping plane is a Douady rabbit. For these values of $\gamma $, it can be shown that ${\mathcal {M}}$ has $z=0$ and one other point as unstable (repelling) fixed points, and $z=\infty $ as an attracting fixed point. Moreover, the map ${\mathcal {M}}^{3}$ has three attracting fixed points. Douady's rabbit consists of the three attracting fixed points $z_{1}$, $z_{2}$, and $z_{3}$ and their basins of attraction.
For example, Figure 3 shows Douady's rabbit in the $z$ plane when $\gamma =\gamma _{D}=2.55268-0.959456i$, a point in the five-o'clock sprout of the right disk. For this value of $\gamma $, the map ${\mathcal {M}}$ has the repelling fixed points $z=0$ and $z=.656747-.129015i$. The three attracting fixed points of ${\mathcal {M}}^{3}$ (also called period-three fixed points) have the locations
$z_{1}=0.499997032420304-(1.221880225696050\times 10^{-6})i{\;}{\;}{\mathrm {(red)} },$
$z_{2}=0.638169999974373-(0.239864000011495)i{\;}{\;}{\mathrm {(green)} },$
$z_{3}=0.799901291393262-(0.107547238170383)i{\;}{\;}{\mathrm {(yellow)} }.$
The red, green, and yellow points lie in the basins $B(z_{1})$, $B(z_{2})$, and $B(z_{3})$ of ${\mathcal {M}}^{3}$, respectively. The white points lie in the basin $B(\infty )$ of ${\mathcal {M}}$.
The action of ${\mathcal {M}}$ on these fixed points is given by the relations
${\mathcal {M}}z_{1}=z_{2},$
${\mathcal {M}}z_{2}=z_{3},$
${\mathcal {M}}z_{3}=z_{1}.$
Corresponding to these relations there are the results
${\mathcal {M}}B(z_{1})=B(z_{2}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {red} }\subseteq {\mathrm {green} },$
${\mathcal {M}}B(z_{2})=B(z_{3}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {green} }\subseteq {\mathrm {yellow} },$
${\mathcal {M}}B(z_{3})=B(z_{1}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {yellow} }\subseteq {\mathrm {red} }.$
As a second example, Figure 4 shows a Douady rabbit when $\gamma =2-\gamma _{D}=-.55268+.959456i$, a point in the eleven-o'clock sprout on the left disk. (As noted earlier, $\mu $ is invariant under this transformation.) The rabbit now sits more symmetrically in the plane. The period-three fixed points then are located at
$z_{1}=0.500003730675024+(6.968273875812428\times 10^{-6})i{\;}{\;}({\mathrm {red} }),$
$z_{2}=-0.138169999969259+(0.239864000061970)i{\;}{\;}({\mathrm {green} }),$
$z_{3}=-0.238618870661709-(0.264884797354373)i{\;}{\;}({\mathrm {yellow} }),$
The repelling fixed points of ${\mathcal {M}}$ itself are located at $z=0$ and $z=1.450795+0.7825835i$. The three major lobes on the left, which contain the period-three fixed points $z_{1}$,$z_{2}$, and $z_{3}$, meet at the fixed point $z=0$, and their counterparts on the right meet at the point $z=1$. It can be shown that the effect of ${\mathcal {M}}$ on points near the origin consists of a counterclockwise rotation about the origin of $\arg(\gamma )$, or very nearly $120^{\circ }$, followed by scaling (dilation) by a factor of $|\gamma |=1.1072538$.
Twisted rabbit problem
In the early 1980s, Hubbard posed the so-called twisted rabbit problem, a polynomial classification problem. The goal is to determine Thurston equivalence types of functions of complex numbers that usually are not given by a formula (these are called topological polynomials)."[7]
• given a topological quadratic whose branch point is periodic with period three, to which quadratic polynomial is it Thurston equivalent?
• determining the equivalence class of “twisted rabbits”, i.e. composita of the “rabbit” polynomial with nth powers of the Dehn twists about its ears.
It was originally solved by Laurent Bartholdi and Volodymyr Nekrashevych[8] using iterated monodromy groups.
The generalization of the twisted rabbit problem to the case where the number of post-critical points is arbitrarily large is also solved.[9]
Gallery
• Gray levels indicate the speed of convergence to infinity or to the attractive cycle
• boundaries of level sets
• binary decomposition
• with spine
• with external rays
See also
• Dragon curve
• Herman ring
• Siegel disc
References
1. "A Geometric Solution to the Twisted Rabbit Problem by Jim Belk, University of St Andrews" (PDF). Archived (PDF) from the original on 2022-11-01. Retrieved 2022-05-03.
2. "Thurston equivalence of topological polynomials by Laurent Bartholdi, Volodymyr Nekrashevych". Archived from the original on 2017-11-14. Retrieved 2018-01-26.
3. "Period-n Rabbit Renormalization. "Rabbit's show" by Evgeny Demidov". Archived from the original on 2022-05-03. Retrieved 2022-05-03.
4. Note on dynamically stable perturbations of parabolics by Tomoki Kawahira Archived October 2, 2006, at the Wayback Machine
5. "Twisted Three-Eared Rabbits: Identifying Topological Quadratics Up To Thurston Equivalence by Adam Chodof" (PDF). Archived (PDF) from the original on 2022-05-03. Retrieved 2022-05-03.
6. "Recent Research Papers (Only since 1999) Robert L. Devaney: Rabbits, Basilicas, and Other Julia Sets Wrapped in Sierpinski Carpets". Archived from the original on 2019-10-23. Retrieved 2020-04-07.
7. "Polynomials, dynamics, and trees by Becca Winarski" (PDF). Archived (PDF) from the original on 2022-11-01. Retrieved 2022-05-08.
8. "Thurston equivalence of topological polynomials by Laurent Bartholdi, Volodymyr Nekrashevych". Archived from the original on 2022-05-08. Retrieved 2022-05-08.
9. "RECOGNIZING TOPOLOGICAL POLYNOMIALS BY LIFTING TREES by JAMES BELK, JUSTIN LANIER, DAN MARGALIT, AND REBECCA R. WINARSKI". Archived from the original on 2022-05-08. Retrieved 2022-05-08.
External links
• Weisstein, Eric W. "Douady Rabbit Fractal". MathWorld.
• Dragt, A. "Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics".
• Adrien Douady: La dynamique du lapin (1996) - video on the YouTube
This article incorporates material from Douady Rabbit on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
| Wikipedia |
Shannon sampling theorem
Every signal function $f(t)$ that is band-limited to $[-\pi W,\pi W]$ for some $W>0$ can be completely reconstructed from its sample values $f(k/W)$, taken at nodes $k/W$, $k\in Z$, equally spaced apart on the real axis $\textbf{R}$, in the form
\begin{equation}f(t)=\sum_{k=-\infty}^{\infty}f\bigg(\frac{k}{W}\bigg)\frac{\sin\pi(Wt-k)}{\pi(Wt-k)},\end{equation}
the series being absolutely and uniformly convergent on $\textbf{R}$. The latter series will be denoted by $(S_Wf)(t)$. Here, band-limited means that $f$ contains no frequencies higher than $\pi W$ or, in mathematical terms, that $f$ is continuous, square-integrable on $\textbf{R}$ (i.e., of finite energy) and that its $L_2(\textbf{R})$-Fourier transform $\hat{f}(v)=(1/\sqrt{2\pi})\int_{-\infty}^{\infty}f(u)e^{-iuv}du$ vanishes outside $[-\pi W,\pi W]$ (see [a13], [a10], p. 51, [a8], [a5], [a14]).
The theorem is also associated with the names of E.T. Whittaker, K. Ogura, V.A. Kotel'nikov, H.P. Raabe, and I. Someya.
Band-limited signals suffer under severe restrictions, since, by the Paley–Wiener theorem, they can be extended to an entire function on the whole complex plane $\textbf{C}$. Thus, they cannot be simultaneously time-limited. However, both situations are covered by the so-called "approximate sampling" theorem, which is valid for not necessarily band-limited signals. It is due to J.R. Brown [a2] and states that if $f\in L_2(\textbf{R})\cap C(\textbf{R})$ and $\hat{f}\in L_1(\textbf{R})$, then for the aliasing error $(R_Wf)(t)=f(t)-(S_Wf)(t)$ one has the estimate
\begin{equation}\sup_{t\in R}|R_Wf(t)|\leq \sqrt{\frac{2}{\pi}}\int_{|v|>\pi W}|\hat{f}(v)|dv,\end{equation}
so that $\lim_{W\to \infty}(S_Wf)(t)=f(t)$ uniformly in $t\in R$. In particular, if $f$ is band-limited to $[-\pi W,\pi W]$, then $f(t)=(S_Wf)(t)$ for $t\in \textbf{R}$.
In essence, the sampling theorem is equivalent (in the sense that each can be deduced from the others) to five fundamental theorems in four different fields of mathematics. In fact, for band-limited functions the sampling theorem (including sampling of derivatives) is equivalent to the famous Poisson summation formula (Fourier analysis) and the Cauchy integral formula (complex analysis, cf. Cauchy integral theorem). Further, the approximate sampling theorem is equivalent to the general Poisson summation formula, the Euler–MacLaurin formula, the Abel–Plana summation formula (numerical mathematics), and to the basic functional equation for the Riemann zeta-function (number theory). The Poisson summation formula can also be interpreted as a trace formula [a15], p. 48. Two final connections are that the series $(S_Wf)(t)$ can also be regarded as a limiting case of the Lagrange interpolation formula (as the number of nodes tends to infinity), while the Gauss summation formula of special function theory is a particular case of Shannon's theorem. See, e.g., [a7], [a4] and the references therein.
There are several types of errors that may influence the accuracy of the reconstruction of a signal function from its sample values; namely, the truncation error, which arises if only a finite number of samples is taken into account, the amplitude error, also called quantization error, which arises when instead of the exact values $f(k/W)$ only approximate values are available, and the time jitter error, which arises when the sample points are not met correctly but might differ by some $\delta$. All such errors (including the aliasing error) can occur in combination (see, e.g., [a8]).
Sampling theory can be put in an abstract setting, in which the band-limited function $f$ is represented by a sampling series of the form $f(t)=\sum_kf(\lambda_k)S_k(t)$, where $\{\lambda_k\}$ is a discrete subset of the domain of $f$ and $\{S_k\}$ is a set of appropriate "reconstruction functions" forming a basis or frame for a suitable function space (most often, a Hilbert space with reproducing kernel; cf. also Hilbert space). The methods give some unification of the approaches, and facilitate connections with important principles of sampling theory, including the Nyquist–Landau minimal rate for stable sampling, and sets of stable sampling, interpolation or uniqueness. The approach is flexible, and applies to multi-band (i.e., the support of the Fourier transform is a union of several disjoint intervals), multi-channel (i.e., the samples are not all taken from $f$ itself, but also from a transformed version of $f$), and multi-dimensional sampling. See [a10].
The first study of sampling and reconstruction in an abstract harmonic analysis setting is due to I. Kluvánek (1965). He replaced the time domain $\textbf{R}$ by a general locally compact Abelian group $G$, and the frequence domain, $\textbf{R}$ again, by the dual group $\hat{G}$. Instead of sampling at regularly distributed points $\{k/W\}\subset \textbf{R}$, a function defined on $G$ is now sampled at the points of a discrete subgroup of $G$. See [a1].
When the Fourier transform is replaced by the Mellin transform, with $G=\textbf{R}_+$, one is led to the exponential sampling theorem of N. Ostrowski and others (1981); see [a6].
[a1] M.G. Beaty, M.M. Dodson, "Abstract harmonic analysis and the sampling theorem" J.R. Higgins (ed.) R.L. Stens (ed.) , Sampling Theory in Fourier and Signal Analysis: Advanced Topics , Clarendon Press (1999)
[a2] J.L. Brown Jr., "On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem" J. Math. Anal. Appl. , 18 (1967) pp. 75–84
[a3] P.L. Butzer, "A survey of the Whittaker-Shannon sampling theorem and some of its extensions" J. Math. Research Exp. , 3 (1983) pp. 185–212
[a4] P.L. Butzer, M. Hauss, "Applications of sampling theory to combinatorial analysis, Stirling numbers, special functions and the Riemann zeta function" J.R. Higgins (ed.) R.L. Stens (ed.) , Sampling Theory in Fourier and Signal Analysis: Advanced Topics , Clarendon Press (1999)
[a5] P.L. Butzer, J.R. Higgins, R.L. Stens, "Sampling theory of signal analysis 1950-1995" J.P. Pier (ed.) , Development of Mathematics 1950-2000 , Birkhäuser (to appear)
[a6] P.L. Butzer, S. Jansche, "The exponential sampling theorem of signal analysis" Atti Sem. Fis. Univ. Modena , 46 (1998) pp. 99–122 (C. Bardaro and others (eds.): Conf. in Honour of C. Vinti (Perugia, Oct. 1996))
[a7] P.L. Butzer, G. Nasri-Roudsari, "Kramer's sampling theorem in signal analysis and its role in mathematics" J.M. Blackedge (ed.) , Image Processing: Math. Methods and Appl. , Inst. Math. Appl. New Ser. , 61 , Clarendon Press (1997) pp. 49–95
[a8] P.L. Butzer, W. Splettstösser, R.L. Stens, "The sampling theorem and linera predeiction in signal analysis" Jahresber. Deutsch. Math. Ver. , 90 (1988) pp. 1–70
[a9] J.R. Higgins, "Five short stories about the cardinal series" Bull. Amer. Math. Soc. , 12 (1985) pp. 45–89
[a10] J.R. Higgins, "Sampling theory in Fourier and signal analysis: Foundations" , Clarendon Press (1996)
[a11] "Sampling theory in Fourier and signal analysis: Advanced topics" J.R. Higgins (ed.) R.L. Stens (ed.) , Clarendon Press (1999)
[a12] A.J. Jerri, "The Shannon sampling theorem: its various extensions and applications: A tutorial review" Proc. IEEE , 65 (1977) pp. 1565–1589
[a13] "Claude Elwood Shannon: Collected papers" N.J.A. Sloane (ed.) A.D. Wyer (ed.) , IEEE (1993)
[a14] A.I. Zayed, "Advances in Shannon's sampling theory" , CRC (1993)
[a15] A. Terras, "Harmonic analysis on symmetric spaces and applications" , 1 , Springer (1985)
Shannon sampling theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shannon_sampling_theorem&oldid=51082
This article was adapted from an original article by P.L. Butzer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Shannon_sampling_theorem&oldid=51082" | CommonCrawl |
\begin{document}
\title{\LARGE {\bf Optimal Insurance with Limited Commitment in a Finite Horizon\footnote{
Junkee Jeon gratefully acknowledges the support of the National Research Foundation of Korea (NRF) grant funded by the Korea government (Grant No. NRF-2017R1C1B1001811). Hyeng Kuen Koo gratefully acknowledges the support of the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (Grant No. NRF-2016R1A2B4008240). Kyunghyun Park is supported by NRF Global Ph.D Fellowship (2016H1A2A1908911). }}}
\author{
Junkee Jeon \footnote{E-mail: {\tt [email protected]}\;Department of Applied Mathematics, Kyung Hee University, Korea.}
\and
Hyeng Keun Koo\footnote{E-mail: {\tt [email protected]}\;Department of Financial Engineering, Ajou University, Korea.}
\and
Kyunghyun Park\footnote{ E-mail: {\tt [email protected]}\;Department of Mathematical Sciences, Seoul National University, Korea.}
}
\date{\today}
\maketitle \pagestyle{plain} \pagenumbering{arabic}
\abstract{We study a finite horizon optimal contracting problem with limited commitment. A risk-neutral principal enters into an insurance contract with a risk-averse agent who receives a stochastic income stream and is unable to make any commitment. This problem involves an infinite number of constraints at all times and at each state of the world. \citet{MJ} have developed a dual approach to the problem by considering a Lagrangian and derived a Hamilton-Jacobi-Bellman equation in an infinite horizon. We consider a similar Lagrangian in a finite horizon, but transform the dual problem into an infinite series of optimal stopping problems. For each optimal stopping problem we provide an analytic solution by providing an integral equation representation for the free boundary. We provide a verification theorem that the value function of the original principal's problem is the Legender-Fenchel transform of the integral of the value functions of the optimal stopping problems. We also provide numerical simulations results of the optimal contracting strategies.}
{\em Keywords} : Optimal contract, Limited commitment, Principal-Agent problem, Optimal stopping problem, Variational inequality, Singular control problem \\
\section{Introduction}
In this paper we investigate a contracting problem with limited commitment in continuous time. A risk-neutral principal enters into an insurance contract with a risk-averse agent who receives a stream of random income. The principal, typically a large institutional agent such as a government agency or a financial institution, is able to make firm commitment to keep the contract for reputational or for other reasons. {However, the} agent{,} who may be an individual or a small country{,} is not able to make such commitment. If both parties were able to make firm commitment, the contract would result in a classical outcome {of} full insurance{, where} the principal absorbs all the risk in the agent's income and provides a constant stream of income to the agent. In the limited commitment case{,} the full insurance outcome is not attainable {and} the principal provides only partial insurance to the agent.
There has been intensive economic research on optimal contract {s} with limited commitment (see e.g.{,} \citet{EG}, \citet{TW}, \citet{KL}). Typically{,} the authors in the literature show that credit limits are used as a mechanism to enforce the contract and try to characterize the credit limits. They also show qualitative features of the contract. Recently, \citet{GZ} and \citet{MJ} {proposed} formulating the problem in continuous time and obtaining a closed form characterization of the optimal contract. The closed-form outcome exhibits the following feature of the optimal contract: the optimal contract starts with a payment much lower than in the full commitment case and ratchets up whenever the income process hits a new high level. They also show that the contract outcome can be enforced by providing unlimited insurance through futures contracts and {optimal} imposing credit limits.
We consider the continuous-time contracting problem with a finite contracting period. The infinite horizon models studied by \citet{GZ} and \citet{MJ} do not capture an important feature of {real-world} contracts: contracts mostly have finite maturity dates. This is the motivation of our paper.
We use the dual approach which allows us to use the Lagrangian method to solve the optimization problem. {To} apply th {is} approach, we need to transform an infinite number of constraints into one constraint. We utilize a method proposed by \citet{HP} or \citet{MJ}. We construct the Lagrangian and define the dual problem by using the state variable {, i.e.,} which is the agent's income process{,} and a costate variable{,} {i.e.,} the cumulative Lagrange multiplier process arising from the dynamic participation constraints. We solve the dual problem by transforming the problem into a series of optimal stopping problems. The optimal stopping problems are, in a formal sense, equivalent to those of early exercise of American options. Then, we can apply well-developed techniques to the {latter} contracting problem. In particular, we apply the integral representation of an American option value (see e.g. \citet{K}, \citet{J}, \citet{CJM}, \citet{Det}) and derive the optimal contract in analytic form. {To} obtain a concrete solution, we apply the recursive integration method proposed by \citet{Huang} to solve numerically the integral equation. \\
\noindent {\it Contributions.} Our contributions are as follows. First, we provide an analytic solution to the optimal contracting problem with limited commitment with a finite contracting period. Second, we make a technical contribution, providing a connection between the contracting problem and the irreversible investment problem involving real options studied by \citet{DixitPindyck}. \citet{MJ} establish the duality theorem and provide a dynamic programming characterization of the dual problem. However, it is difficult to apply their method to the problem with a finite period, since one has to consider a Hamilton-Jacobi-Bellman(HJB) equation with a gradient constraint involving three variables {:} time, income, and the agent's continuation value. Moreover, it is not easy to find a solution to the HJB equation. We overcome th {is} difficulty by considering a transformed problem, which is similar in its formal structure to an irreversible incremental problem and equivalent to an infinite series of optimal stopping problems, {so that it} essentially becomes a single optimal stopping problem in our model.\\
\noindent{\it Related literature.} In addition to the research mentioned above, there is extensive literature on the contracting problem with limited commitment. \citet{KL} and \citet{AJ2000} investigate asset pricing based on the model with limited commitment. \citet{Zhang} provides a solution to a long-term contracting problem in discrete time using a stopping-time approach. \citet{BWY} shows that there is equivalence between the household problem and the contracting problem. \citet{AH2014}, \citet{AL}, \citet{AKL}, and \citet{BWY} study optimal contracting between investors and an entrepreneur in a continuous-time model. Our paper, however, is different from the papers in the literature in the following aspects. First, the contract maturity is finite in our continuous-time model, whereas most papers consider either discrete-time long-horizon or continuous-time infinite-horizon models. Second, most authors use dynamic programming{,} and \citet{MJ} use {in particular} the dual approach, whereas we also use the dual approach {as well as} transform {ing the original problem} into optimal stopping problems. The duality approach has been used to study continuous-time portfolio selection problems (see e.g., \citet{CoxH}, \citet{KLS}). \citet{Jeon} also apply the dual approach and the transformation to study an optimal consumption and portfolio selection problem in which an economic agent does not tolerate a decline in standard of living. \\
\noindent {\it Organization.} The rest of paper is organized as follows. Section \ref{sec:2} explains the model of the optimal contracting problem with limited commitment in a finite-horizon framework. In Section \ref{sec:3}, we state our optimization problem. By constructing the Lagrangian of the optimization problem, we define the dual problem. {In {S}ection \ref{sec:B}, we analyse the variational inequality arising from the dual problem. Section \ref{sec:duality} establishes the duality theorem and provides the analytic representation of the optimal strategies.} In {S}ection \ref{sec:4}, we provide numerical simulation results. In Section \ref{sec:5} we draw conclusions.
\section{Economic model}\label{sec:2}
We extend a contracting model with limited commitment in a continuous-time framework studied in \citet{GZ}, {and} \citet{MJ}. The crucial difference between the existing models and ours is that our model is set up in the finite horizon to study how the horizon affect {s} the contract.
The agent receives an income stream $Y_t$ which is an $\mathcal{F}_t$-adapted process. We consider the following geometric Brownian motion for the income process: $$ dY_t= \mu Y_t dt+ \sigma Y_t dB_t,\;\;\;\;\;Y_0=y, $$ where $\mu>0, \sigma>0$ is constant and $\{B_t\}_{t=0}^{T}$ is a standard Brownian motion on the underlying probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and $\{\mathcal{F}_t\}_{t=0}^{T}$ is the augmentation under $\mathbb{P}$ of the natural filtration generated by the standard Brownian motion $\{B_t\}_{t=0}^{T}$. The agent's income process $Y$ is publicly observable by both the principal and the agent.
The agent is risk-averse with subject discount rate $\rho>0$. The contract horizon is $[0,T]$. At time $0$, the principal offers {a} contract $(C,T)$ to the agent. Then the instantaneous payment or consumption of the agent $C=\{C_t\}_{t=0}^{T}$ is a non-negative $\mathcal{F}_t$-adapted process satisfying $$ \mathbb{E}\left[\int_0^T e^{-rt}C_t dt\right] < \infty. $$ Note that the contract is dependent on the {entire} history of {the} $Y_t$ process in our model.
We assume that the agent has no access to the financial market. The agent's utility at time $0$ is defined by $$ U_0^a(C) \equiv \mathbb{E}\left[\int_0^T e^{-\rho t} u(C_t) dt \right]{,} $$ where $u(\cdot)$ is a continuously differentiable, strictly concave, and strictly increasing function. Thus, the agent's continuation utility at date $t$ {is} given by $$ U_t^a(C) \equiv \mathbb{E}_t \left[ \int_t^{T} e^{-\rho(s-t)} u(C_s)ds \right]{,} $$ where $\mathbb{E}_t[\cdot]=\mathbb{E}[\cdot \mid \mathcal{F}_t]$ is the conditional expectation at time $t$ on the filtration $\mathcal{F}_t$.
We assume that the principal can freely access the financial market with constant risk-free rate $r>0$ and can derive utility according to $$ U^p(y,C) \equiv \mathbb{E}\left[\int_0^Te^{-rt}(Y_t-C_t)dt\right]. $$ Without loss of generality{,} we assume $0<r\le \rho$. This also means that the principal is more patient than or equally patient with the agent.
{To obtain} explicit solutions, we assume that the agent {uses} the the constant relative risk aversion (CRRA) utility function as a representative utility function. \begin{as}\label{as:1}
\begin{eqnarray}\label{eq:CRRA}
u(c)\equiv \frac{c^{1-\gamma}}{1-\gamma},~~~~~~\gamma>0,\gamma \neq 1{,}
\end{eqnarray}
where $\gamma$ is the agent's coefficient of relative risk aversion. \end{as} The agent has a limited commitment. He/she can walk away from the contract and take an outside value at any time after signing the contract. The outside value (or autarky value) $U_d$ at time $t$ is given by \begin{eqnarray*}
\begin{split}
U_d(t,Y_t) \equiv& \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds\right]
=\dfrac{{(Y_t)}^{1-\gamma}}{1-\gamma}\dfrac{1-e^{-\hat{\rho}(T-t)}}{\hat{\rho}},
\end{split} \end{eqnarray*} where $\hat{\rho}$ is defined by $$ \hat{\rho}\equiv \rho-(1-\gamma)\mu+\frac{1}{2}\gamma(1-\gamma)\sigma^2. $$ We make the following assumption to ensure a finite outside value for sufficiently large $T$. \begin{as}
$$\hat{\rho}>0.$$ \end{as}
To ensure that the agent does not walk away, we impose the following dynamic participation constraint: \begin{eqnarray}\label{eq:EN2} U_t^a(C) \geq U_d(t,Y_t),\;\;\;\;\forall t \in[0,T]. \end{eqnarray}\ We also impose the promise keeping constraint (or individual rationality constraint): \begin{eqnarray}\label{eq:EN1} w_0=U_0^a(C){,} \end{eqnarray} where $w_0$ is an initial promised value to the agent. \begin{as}\label{as:promise_value}~
We assume that the initial promised value $w$ satisfies the following inequality:
$$
w \ge U_d(0,y).
$$ \end{as} Lastly, the continuation utility of the principal at date $t$ is defined by $$ U_t^p({Y},C) \equiv \mathbb{E}_t\left[\int_t^Te^{-r(s-t)}(Y_s-C_s)ds\right]. $$ We call a consumption plan $\{C_s\}_{s=t}^{T}$ \textit{enforceable} at time $t\in[0,T]$ if the following conditions hold. \begin{itemize}
\item[(i)](Integrability condition)
\begin{eqnarray}\label{eq:int}
\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}C_s ds\right] < \infty.
\end{eqnarray}
\item[(ii)](Participation constraint)
\begin{eqnarray}\label{eq:parti}
U_s^a(C) \geq U_d(s,Y_s),~~~\forall s \in[t,T].
\end{eqnarray} \end{itemize} Let $\Gamma_t(y,w)$ be the set of all \textit{enforceable} consumption plans at time $t$.
For some technical aspects, we make the following assumption: \begin{as}\label{thm:as1}
\begin{eqnarray}
\begin{aligned}
\hat{r}\equiv r-\mu >0,\;\;K\equiv r+\frac{\rho-r}{\gamma}~>0,~~{\textrm{and}}~~\mu>\frac{\sigma^2}{2}.
\end{aligned}
\end{eqnarray} \end{as}
\section{ Optimization Problem}\label{sec:3}
\subsection{First-Best Allocation} We first consider the first-best allocation contracting problem without participation constraint \eqref{eq:EN2} as the first-best benchmark. We use the agent's continuation value $w_t\equiv U_t^a(C)$ as a state variable. \begin{pr}[First-Best Problem]\label{pro:FB} ~\\
For given $w_t=w,\;{Y_t=y}$, the principal's problem is to maximize
\begin{equation*}
V^{FB}(t, y, w)= \sup_{\Gamma_t'(y,w)}U_t^p(y,C),
\end{equation*}
where $\Gamma'_t(y,w)$ is the set of all consumption plans $\{C_s\}_{s=t}^T$ satisfying
the integrability condition \eqref{eq:int}. \end{pr} Note that the actual principal's problem for first-best case is to find $V^{FB}(0,y,w)$ at $t = 0$. However, the value function of Problem \ref{pro:FB} is well-defined for any $t > 0$ by the dynamic programming principle (\citet{Bellman}) since the principal fully commits to the contract until $T$.
For Lagrangian multiplier $\lambda^{*}>0$, let us consider the {following} Lagrangian for {the} first-best case: \begin{eqnarray*}
\begin{split}
{\bf L}_{FB}\equiv&\;\mathbb{E}_{t}\left[\int_{t}^{T}e^{-r(s-t)}(Y_s-C_s)ds\right]+\lambda^{*}\left(\mathbb{E}_{t}\left[\int_{t}^{T}e^{-\rho(s-t)}u(C_s)ds\right]-w\right).
\end{split} \end{eqnarray*} Define the \textit{convex dual function} of $u$ as follows
\begin{eqnarray}\label{eq:DUAL} \tilde{u}(z) \equiv \max_{c>0} \left\{zu(c)-c\right\},~~~\textrm{for}~~z>0. \end{eqnarray} In the case of the CRRA utility function, the dual function is derived as follows:
\begin{eqnarray}\label{eq:CRRA_D} \tilde{u}(z) =\frac{\gamma}{1-\gamma}z^{\frac{1}{\gamma}}. \end{eqnarray} Here, the first-best consumption is given by \begin{equation}\label{FB} C_{s}^{FB}=(u')^{-1}\left({e^{(\rho-r)(s-t)}}/\lambda^{*}\right),\;\;\;\;s\in[t,T]. \end{equation} From Lagrangian ${\bf L}_{FB}$, we can define the dual value function {$\tilde{V}^{FB}(t,\lambda^{*},y)$} as follows: \begin{eqnarray*}
\begin{split}
\tilde{V}^{FB}(t,\lambda^{*},y)=&\;\mathbb{E}_{t}\left[\int_{t}^{T}e^{-r(s-t)}\tilde{u}\left(e^{-(\rho-r)(s-t)}\lambda^{*}\right)ds\right] +\mathbb{E}_{t}\left[\int_{t}^{T}e^{-r(s-t)}Y_s ds\right].
\end{split} \end{eqnarray*} Then, we can deduce the following duality-relationship: $$ V^{FB}(t,y,w)=\inf_{\lambda^{*}>0}\left(\tilde{V}^{FB}(t,\lambda^{*},y)-w\lambda^{*}\right). $$ By a first-order condition, we can obtain $$ w= \dfrac{1-e^{-K(T-t)}}{K} \dfrac{1}{1-\gamma}(\lambda^{*})^{\frac{1}{\gamma}-1}. $$ From \eqref{FB}, \begin{eqnarray} \begin{split}\label{eq:FB_C} C^{FB}_{s}=&e^{-\frac{(\rho-r)}{\gamma}(s-t)}\left(\dfrac{K(1-\gamma)w}{{1-e^{-K(T-t)}}}\right)^{\frac{1}{1-\gamma}},\qquad s\in[t,T]. \end{split} \end{eqnarray} For the first-best consumption $C^{FB}$, the following equality holds: $$ U_0^a(C^{FB})=w. $$ This means that the risk-neutral principal bears all uncertainty and fully insures the risk-averse agents.
{Since $0<r\le \rho$,} we can easily confirm that the first-best consumption process $C^{FB}$ is non-increasing function over time.
\subsection{Limited Commitment} We now write down the optimal contract problem with limited commitment between the principal and the agent. \begin{pr}[Primal problem]\label{pr:main}~
Given $w_t=w$ and {$Y_t=y$}, we consider the following maximization problem:
$$
V(t,y,w) \equiv \sup_{C \in \Gamma_t(y,w)}U_t^p(y,C).
$$ \end{pr} To obtain the solution of Problem \ref{pr:main}, we will construct a Lagrangian for Problem \ref{pr:main}. The key is to write the part of the Lagrangian corresponding to the participation constraint \eqref{eq:parti}, which should hold at every $s\in[t,T]$ and thus is comprised of infinite individual constraints. By utilizing the similar method proposed by \citet{HP}, \citet{MJ} {wrote} an infinite number of constraints as an integral of the constraints. We write down the Lagrangian as follows:
\begin{eqnarray}\label{eq:LGR} \begin{aligned} {\bf L} \equiv & \mathbb{E}_t\left[ \int_t^T e^{-r(s-t)} (Y_s-C_s)ds \right] + \lambda \left(\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)} u(C_s)ds\right]-w\right)\\ +&\mathbb{E}_t\left[\int_t^T e^{-r(s-t)} \eta_s \left(\int_s^T e^{-\rho(\xi-s)}(u(C_\xi)-u(Y_\xi))d\xi \right)ds \right], \end{aligned} \end{eqnarray} where $\lambda >0$ is the Lagrange multiplier associated with the promise-keeping constraint (\ref{eq:EN1}) at each time $t\geq 0$ and $e^{-r(s-t)}\eta_s \geq 0 $ is the Lagrange multiplier associated with the participation constraint (\ref{eq:EN2}) at each time $s \in [t,T]$.\\ Using integration by parts, the third term of right-hand side in (\ref{eq:LGR}) can be given as follows:
\begin{eqnarray}\label{eq:IB} \begin{aligned} &\mathbb{E}_t\left[\int_t^T e^{-r(s-t)} \eta_s \left(\int_s^T e^{-\rho(\xi-s)}(u(C_\xi)-u(Y_\xi))d\xi \right)ds \right]\\ =&\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)} \cdot e^{(\rho-r)(s-t)} \eta_s \left(\int_s^T e^{-\rho(\xi-s)}(u(C_\xi)-u(Y_\xi))d\xi \right)ds \right]\\ =&\mathbb{E}_t\left[\int_t^T d\left(\int_t^s e^{(\rho-r)(\xi-t)} \eta_\xi d\xi\right)\left(\int_s^T e^{-\rho(\xi-t)}(u(C_\xi)-u(Y_\xi))d\xi \right)ds \right]\\ =&\mathbb{E}_t\left[\int_t^T\left(\int_t^se^{(\rho-r)(\xi-t)}\eta_\xi d\xi\right)e^{-\rho(s-t)}(u(C_s)-u(Y_s))ds\right]. \end{aligned} \end{eqnarray} Plugging th {e} equation \eqref{eq:IB} into the Lagrangian \eqref{eq:LGR}, we obtain
\begin{eqnarray}\label{eq:LGR2} \begin{aligned} {\bf L} = & \mathbb{E}_t\left[ \int_t^T e^{-r(s-t)} (Y_s-C_s)ds \right] +\lambda \mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds\right]\\ +&\mathbb{E}_t\left[\int_t^T\left(\int_t^se^{(\rho-r)(\xi-t)}\eta_\xi d\xi+\lambda \right)e^{-\rho(s-t)}(u(C_s)-u(Y_s))ds\right]-\lambda w. \end{aligned} \end{eqnarray} We define a \textit{costate process} $\{X_s\}_{s=t}^T$ as the cumulative amounts of the Lagrangian multipliers,
\begin{eqnarray}\label{eq:ND} X_s \equiv \int_t^s e^{(\rho-r)(\xi-t)}\eta_\xi d\xi + \lambda, \;\;\;\;X_t=\lambda. \end{eqnarray} This process is non-decreasing, continuous, and satisfies $$ dX_s=e^{(\rho-r)(s-t)}\eta_s ds,\;\;\;\;s\in[t,T]. $$ Then we can write down
\begin{eqnarray}\label{eq:LGR3} \begin{aligned} {\bf L} =& \mathbb{E}_t\left[ \int_t^T e^{-r(s-t)} (Y_s-C_s)ds \right] +\lambda \mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds\right]\\ +&\mathbb{E}_t\left[\int_t^T X_s e^{-\rho(s-t)}(u(C_s)-u(Y_s))ds\right]-\lambda w.\\ \end{aligned} \end{eqnarray} For every enforceable consumption plan $\{C_s\}_{s=t}^{T}$, we can deduce that \begin{eqnarray} \mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(Y_s-C_s)ds\right] \le \sup_{\{C_t\}}{\bf L}. \end{eqnarray} To derive the dual problem, we first choose the consumption at each time to obtain the maximum of the Lagrangian \eqref{eq:LGR3}, which takes the following form:
\begin{eqnarray}\label{eq:LGR4} \begin{aligned} {\bf L}(X) =& \sup_{\{C_t\}} \left\{ \mathbb{E}_t \left[\int_t^T e^{-r(s-t)}\Big(e^{-(\rho-r)(s-t)}X_su(C_s)-C_s\Big)ds\right]\right.\\ +&\left.\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}Y_s-e^{-\rho(s-t)}X_su(Y_s) ds\right]+\lambda \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right]-\lambda w \right\}.\\ \end{aligned} \end{eqnarray} By the first-order condition for a consumption plan $\{C_s\}_{s=t}^T$ in (\ref{eq:LGR4}), the optimal consumption $\{C^*_s\}_{s=t}^T$ is given as:
\begin{eqnarray}\label{eq:optimal_C} C^*_s = (u^{\prime})^{-1}\left(\dfrac{e^{(\rho-r)(s-t)}}{X_s}\right)=(e^{-(\rho-r)(s-t)}X_s)^{\frac{1}{\gamma}},\;\;\;\;s\in[t,T]. \end{eqnarray} Then {\bf L}$(X)$ can be given by
\begin{eqnarray}\label{eq:LGR5} \begin{aligned} {\bf L}(X) =& \mathbb{E}_t \left[\int_t^T e^{-r(s-t)} \Big( \tilde{u}(e^{-(\rho-r)(s-t)}X_s) +Y_s - e^{-(\rho-r)(s-t)}X_s u(Y_s)\Big)ds\right] \\ +&\lambda \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right]-\lambda w. \\ \end{aligned} \end{eqnarray} To ensure that the Lagrangian \eqref{eq:LGR5} is finite, we assume the following integrability conditions: \begin{eqnarray}\label{eq:integrability} \begin{split}
&{\mathbb{E}_t\left[\int_{t}^{T}e^{-\rho (s-t)}|u(Y_s)|X_s ds \right]<\infty,}\\
&\mathbb{E}_t\left[\int_{t}^{T}e^{-r (s-t)} |\tilde{u}(X_s e^{-(\rho-r)(s-t)})|ds\right]<\infty. \end{split} \end{eqnarray} Let us define \begin{eqnarray} \begin{split}\label{eq:fun_cal_J} \mathcal{J}(t,\lambda,y,X)\equiv& \mathbb{E}_t\left[\int_t^T e^{-r(s-t)} \Big( \tilde{u}(e^{-(\rho-r)(s-t)}X_s) +Y_s - e^{-(\rho-r)(s-t)}X_s u(Y_s)\Big)ds \right] \\ +&\lambda \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right]. \end{split} \end{eqnarray} {By the definition of the Lagrangian ${\bf L}(X)$ and the function $\mathcal{J}$} defined in \eqref{eq:fun_cal_J}, \begin{eqnarray*}
\begin{split}
\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(Y_s-C_s)ds\right]\le \mathcal{J}(t,\lambda,y,X)-\lambda w.
\end{split} \end{eqnarray*} {Then, it is clear that} \begin{eqnarray*}
\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(Y_s-C_s)ds\right] \le \inf_{\lambda>0, X\in \mathcal{ND}(\lambda)}\left[\mathcal{J}(t,\lambda,y,X)-\lambda w\right]{,} \end{eqnarray*} where $\mathcal{ND}(\lambda)$ denotes the set of all positive non-decreasing, adaptable with respect to $\mathcal{F}$, right-continuous processes $X$ with left-limits (RCLL) and starting at $X_t=\lambda$, satisfying \eqref{eq:integrability}.
Hence, the value function $V(t,y,w)$, the maximized utility value of the principal, satisfies the following inequality: \begin{eqnarray} \begin{split}\label{eq:du1} V(t,y,w)=\sup_{C \in \Gamma_t(y,w)}\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(Y_s-C_s)ds\right]\le \inf_{\lambda>0, X\in \mathcal{ND}(\lambda)}\left[\mathcal{J}(t,\lambda,y,X)-\lambda w\right]. \end{split} \end{eqnarray} We will show in Theorem \ref{thm:main} that the maximized value is indeed equal to the right-hand side of the inequality in \eqref{eq:du1} with the infimum being replaced by the minimum, i.e., \begin{eqnarray} \begin{split}\label{eq:du2} V(t,y,w)=\min_{\lambda>0, X\in \mathcal{ND}(\lambda)}\left[\mathcal{J}(t,\lambda,y,X)-\lambda w\right]= \min_{\lambda>0}\left[\min_{X\in \mathcal{ND}(\lambda)}\mathcal{J}(t,\lambda,y,X)-\lambda w\right]. \end{split} \end{eqnarray} That is, we should choose the process $X$ to minimize ${\bf L}(X)$:
\begin{eqnarray}\label{eq:minimize_L} \inf_{X\in \mathcal{ND}(\lambda)}{\bf L}(X). \end{eqnarray} We now study the minimization problem inside the bracket of the right-hand side of the last equality in \eqref{eq:du2}, which we will call {as} the dual problem of Problem \ref{pr:main}. \begin{pr}[Dual problem]\label{pr:dual}~
Given $\lambda>0$, consider the minimization problem:
\begin{eqnarray}\label{eq:dual_value}
\begin{split}
J(t,\lambda,y)=&\inf_{X \in \mathcal{ND}(\lambda)}\mathcal{J}(t,\lambda,y,X).
\end{split}
\end{eqnarray} \end{pr}
To obtain the optimal costate process $\{X_s^*\}_{s=t}^T$, we transform Problem \ref{pr:dual} to the \textit{optimal stopping problem}. This transformation is due to the existence of one-to-one correspondence between the set of all costate processes $\{X_s\}_{s=t}^T\in\mathcal{ND}(\lambda)$ and the set of all infinite series of $\mathcal{F}$-stopping times $\{\tau(x)\}_{x\ge\lambda}$ taking values in $[t,T]$ which is non-decreasing and left-continuous with right limits as a function of $x$.
The correspondence is given by $$ \tau(x)=\inf\{s\ge t \mid X_s \ge x\}\wedge T. $$ The problem of choosing a non-decreasing process $\{X_s\}_{s=t}^T$ is similar to an irreversible incremental investment problem studied by \citet{Pindyck88} and \citet{DixitPindyck}. They consider the capacity expansion decision of a firm as a series of optimal stopping problems: for each level of capacity, there is {a corresponding} stopping problem where the firm chooses the optimal time to expand its capacity to {an appropriate} level. Based on this idea, we transform Problem \ref{pr:dual} into a series of optimal stopping problems in the following lemma. \begin{lem}\label{thm:lem1}~
We can write the dual value function as follows:
\begin{eqnarray}\label{eq:dual_ex1}
J(t,\lambda,y)= -y^{1-\gamma} \int_{\lambda}^{\infty} \left( \sup_{\tau(x)\in[t,T]}\mathbb{E}_t^{\mathbb{Q}} \left[ e^{-\hat{\rho}(\tau(x)-t)} h(\tau(x),x\mathcal{H}_{\tau(x)}) \right]\right)dx + J_0(t,\lambda,y){,}
\end{eqnarray}
where
\begin{eqnarray}
\begin{aligned}
\mathcal{H}_s &\equiv e^{-(\rho-r)(s-t)}Y_s^{-\gamma},\\
h(t,z)&\equiv\frac{1}{1-\gamma}\left(\frac{1-e^{-\hat{\rho}(T-t)}}{\hat{\rho}}-\frac{1-e^{-K(T-t)}}{K}z^{\frac{1}{\gamma}-1}\right),\\
{J}_0(t,\lambda,y)&\equiv\frac{\gamma}{1-\gamma}\frac{1-e^{-K(T-t)}}{K}\lambda^{\frac{1}{\gamma}}+\frac{1-e^{-\hat{r}(T-t)}}{\hat{r}}y .\\
\end{aligned}
\end{eqnarray}
and the measure $\mathbb{Q}$ is defined in the proof. $\mathbb{E}^{\mathbb{Q}}[\cdot]$ is the expectation with respect to measure $\mathbb{Q}$. The corresponding standard Brownian motion $B_s^{\mathbb{Q}}$ is defined as $$ B_s^{\mathbb{Q}}\equiv B_s-(1-\gamma)\sigma s,\;s\in[t,T].$$ \end{lem} \noindent{\bf Proof.} Define a function $f$ as follows $$ f(x)=e^{-r(s-t)}\left(\tilde{u}(e^{-(\rho-r)(s-t)}x) +Y_s - e^{-(\rho-r)(s-t)}x \cdot u(Y_s) \right). $$ By assigning (\ref{eq:CRRA}) and (\ref{eq:CRRA_D}) to $f$, \begin{eqnarray} f(x)=e^{-r(s-t)}\left(\frac{\gamma}{1-\gamma}(e^{-(\rho-r)(s-t)}x)^{\frac{1}{\gamma}} +Y_s - e^{-(\rho-r)(s-t)}x \frac{1}{1-\gamma}Y_s^{1-\gamma} \right) \end{eqnarray} and \begin{eqnarray}\label{eq:fprime} f^{\prime}(x)=\frac{e^{-\rho(s-t)}}{1-\gamma}\left((e^{-(\rho-r)(s-t)}x)^{\frac{1}{\gamma}-1} - Y_s^{1-\gamma} \right). \end{eqnarray} Then,
\begin{eqnarray}\label{eq:ST} \begin{aligned} &J(t,\lambda,y)\\ =& \inf_{X_s \in \mathcal{ND}(\lambda)} \mathbb{E}_t \left[\int_t^T f(X_s)ds \right]+\lambda\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds\right]\\ =& \inf_{X_s \in \mathcal{ND}(\lambda)} \mathbb{E}_t \left[\int_t^T \left(\int_{X_t}^{X_s} f^{\prime}(x)dx+f(X_t)\right)ds\right]+\lambda\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds\right]\\ =& \inf_{X_s \in \mathcal{ND}(\lambda)} \mathbb{E}_t \left[\int_t^T \left(\int_{\lambda}^{\infty} f^{\prime}(x)\cdot {\bf 1}_{\{x\leq X_s\}}dx\right)ds\right]+\mathbb{E}_t\left[\int_t^T f(\lambda) ds \right]+\lambda\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds\right]\\ =& \inf_{X_s \in \mathcal{ND}(\lambda)} \mathbb{E}_t \left[\int_t^T \left(\int_{\lambda}^{\infty} f^{\prime}(x)\cdot {\bf 1}_{\{x\leq X_s\}}dx\right)ds\right]+\lambda\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds\right]\\ +&\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}\left(\tilde{u}(e^{-(\rho-r)(s-t)}\lambda) +Y_s - e^{-(\rho-r)(s-t)}\lambda \cdot u(Y_s) \right) ds \right]\\ =&\inf_{X_s \in \mathcal{ND}(\lambda)} \mathbb{E}_t \left[\int_t^T \left(\int_{\lambda}^{\infty} f^{\prime}(x)\cdot {\bf 1}_{\{w\leq X_s\}}dx\right)ds\right] + \mathbb{E}_t\left[\int_t^T e^{-r(s-t)}\left(\tilde{u}(e^{-(\rho-r)(s-t)}\lambda) +Y_s \right) ds \right]\\ =&\inf_{\tau(x) \in [t,T]} \mathbb{E}_t \left[\int_{\lambda}^{\infty} \left(\int_{t}^{T} f^{\prime}(x)\cdot {\bf 1}_{\{s> \tau(x)\}}ds\right)dx\right] + \mathbb{E}_t\left[\int_t^T e^{-r(s-t)}\left(\tilde{u}(e^{-(\rho-r)(s-t)}\lambda) +Y_s \right) ds \right]\\ =&\int_{\lambda}^{\infty} \left(\inf_{\tau(x) \in [t,T]} \mathbb{E}_t \left[\int_{\tau(x)}^{T} f^{\prime}(x)ds\right]\right)dx + \mathbb{E}_t\left[\int_t^T e^{-r(s-t)}\left(\tilde{u}(e^{-(\rho-r)(s-t)}\lambda) +Y_s \right) ds \right].\\ \end{aligned} \end{eqnarray} where a stopping time $\tau(x)$ is defined by \begin{eqnarray}\label{eq:ST1}
\tau(x) = \inf\left\{s\geq t | X_s \geq x\right\}\wedge T,\;\;\;\forall x\ge\lambda. \end{eqnarray} Note that Fubini's theorem implies that $$ \inf_{X_s \in \mathcal{ND}(\lambda)} \mathbb{E}_t \left[\int_t^T \left(\int_{\lambda}^{\infty} f^{\prime}(x)\cdot {\bf 1}_{\{x\leq X_s\}}dx\right)ds\right]=\inf_{\tau(x) \in [t,T]} \mathbb{E}_t \left[\int_{\lambda}^{\infty} \left(\int_{t}^{T} f^{\prime}(x)\cdot {\bf 1}_{\{s> \tau(x)\}}ds\right)dx\right]. $$ Define a exponential martingale process $Z_s^t$ and corresponding probability measure $\mathbb{Q}$ for each $s \in [t,T]$, $$ Z_s^t=\exp\left\{-\frac{1}{2}(1-\gamma)^2\sigma^2(s-t) + (1-\gamma)\sigma (B_s-B_t) \right\},~~\textrm{and}~~~\frac{d\mathbb{Q}}{d\mathbb{P}}=Z_s^t, $$ {respectively}.
Girsanov's theorem implies that $$ dB_s^{\mathbb{Q}}=dB_s - (1-\gamma)\sigma ds,\;\;\;\;s\in[t,T] $$ is a standard Brownian motion under the measure $\mathbb{Q}$.
Since $Y_s = Y_t e^{(\mu-\frac{1}{2}\sigma^2)(s-t)+\sigma(B_s-B_t)}$ and $Y_t=y$, let us define $Y_s^t \equiv e^{(\mu-\frac{1}{2}\sigma^2)(s-t)+\sigma(B_s-B_t)}$. By using (\ref{eq:fprime}), the first term of the last equation in (\ref{eq:ST}) can be derived as follows.
\begin{footnotesize}
\begin{eqnarray}\label{eq:ST_1}
\begin{aligned}
&\inf_{\tau(x) \in [t,T]}\mathbb{E}_t\left[\int_{\tau(x)}^T f^{\prime}(x)ds\right]\\
=&\inf_{\tau(x) \in [t,T]}\mathbb{E}_t\left[\int_{\tau(x)}^T \frac{e^{-\rho(s-t)}}{1-\gamma}\left((e^{-(\rho-r)(s-t)}x)^{\frac{1}{\gamma}-1} - Y_s^{1-\gamma} \right) ds \right]\\
=&\inf_{\tau(x) \in [t,T]}\frac{1}{1-\gamma}\mathbb{E}_t \left[e^{-\rho(\tau(x)-t)}\mathbb{E}_{\tau(x)}\left[\int_{\tau(x)}^T e^{-\rho(s-\tau(x))}\left((e^{-(\rho-r)(s-t)}x)^{\frac{1}{\gamma}-1}- Y_s^{1-\gamma}\right)ds \right]\right]\\
=&\inf_{\tau(x) \in [t,T]}\frac{y^{1-\gamma}}{1-\gamma}\mathbb{E}_t \left[ e^{-\rho(\tau(x)-t)} (Y^t_{\tau(x)})^{1-\gamma}\mathbb{E}_{\tau(x)}\left[\int_{\tau(x)}^T e^{-\rho(s-\tau(x))} (Y_s^{\tau(x)})^{1-\gamma} \left(\left(\frac{e^{-(\rho-r)(s-t)}x}{(Y_s)^{\gamma}}\right)^{\frac{1}{\gamma}-1}- 1\right)ds \right]\right]\\
=&\inf_{\tau(x) \in [t,T]}\frac{y^{1-\gamma}}{1-\gamma}\mathbb{E}_t \left[ e^{-\hat{\rho}(\tau(x)-t)} Z^t_{\tau(x)}\mathbb{E}_{\tau(x)}\left[\int_{\tau(x)}^T e^{-\hat{\rho}(s-\tau(x))}Z_s^{\tau(x)}\left(\left(\frac{e^{-(\rho-r)(s-t)}x}{(Y_s)^{\gamma}}\right)^{\frac{1}{\gamma}-1}- 1\right)ds \right]\right]\\
=&\inf_{\tau(x) \in [t,T]}\frac{y^{1-\gamma}}{1-\gamma}\mathbb{E}_t^{\mathbb{Q}} \left[ e^{-\hat{\rho}(\tau(x)-t)} \mathbb{E}_{\tau(x)}^{\mathbb{Q}}\left[\int_{\tau(x)}^T e^{-\hat{\rho}(s-\tau(x))}\left(\left(\frac{e^{-(\rho-r)(s-t)}x}{(Y_s)^{\gamma}}\right)^{\frac{1}{\gamma}-1}- 1\right)ds \right]\right]\\
=&\inf_{\tau(x) \in [t,T]}\frac{y^{1-\gamma}}{1-\gamma}\mathbb{E}_t^{\mathbb{Q}} \left[ e^{-\hat{\rho}(\tau(x)-t)} \mathbb{E}_{\tau(x)}^{\mathbb{Q}}\left[\int_{\tau(x)}^T\left( e^{-\hat{\rho}(s-\tau(x))} \left(x\mathcal{H}_{s}\right)^{\frac{1}{\gamma}-1}\cdot e^{-(\rho-r)(s-\tau(x))\left(\frac{1}{\gamma}-1\right)}(Y_s^{\tau(x)})^{\gamma-1}\right.\right.\right.\\ -&\left.\left.\left.e^{-\hat{\rho}(s-\tau(x))} \right)ds \right]\right]\\
=& \inf_{\tau(x) \in [t,T]}\frac{y^{1-\gamma}}{1-\gamma}\mathbb{E}_t^{\mathbb{Q}} \left[ e^{-\hat{\rho}(\tau(x)-t)}\left(\frac{1-e^{-K(T-\tau(x))}}{K}(x\mathcal{H}_{\tau(x)})^{\frac{1}{\gamma}-1} - \frac{1-e^{-\hat{\rho}(T-\tau(x))}}{\hat{\rho}} \right) \right].
\end{aligned}
\end{eqnarray} \end{footnotesize} \begin{rem}\label{thm:Qprocess}~ Under the measure $\mathbb{Q}$,
\begin{eqnarray}\label{eq:Hprocess}
\begin{aligned}
&dY_s=(\mu+(1-\gamma)\sigma^2)Y_sds + \sigma Y_s dB_s^{\mathbb{Q}},\\
&d\mathcal{H}_s=(\hat{r}-\hat{\rho}+\sigma^2\gamma^2)\mathcal{H}_sds-\gamma\sigma \mathcal{H}_sdB_s^{\mathbb{Q}}.
\end{aligned}
\end{eqnarray}
The process $\mathcal{H}_s$ can be easily derived by It{\^o}'s lemma. We will use this process to prove variational inequality(VI) later. \end{rem} The other term of last equation in (\ref{eq:ST}) can be directly derived. \begin{eqnarray}\label{eq:ST_2} \begin{aligned} &\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}\left(\tilde{u}(e^{-(\rho-r)(s-t)}\lambda) +Y_s \right) ds \right]=\frac{\gamma}{1-\gamma} \frac{1-e^{-K(T-t)}}{K}\lambda^{\frac{1}{\gamma}}+\frac{1-e^{-\hat{r}(T-t)}}{\hat{r}}y. \end{aligned} \end{eqnarray} \\ By (\ref{eq:ST_1}) and (\ref{eq:ST_2}), \begin{footnotesize}
\begin{eqnarray}
\begin{aligned}
&J(t,\lambda,y)\\
=& \int_{\lambda}^{\infty} \left( \inf_{\tau(x) \in [t,T]}\frac{y^{1-\gamma}}{1-\gamma}\mathbb{E}_t^{\mathbb{Q}} \left[ e^{-\hat{\rho}(\tau(x)-t)}\left(\frac{1-e^{-K(T-\tau(x))}}{K}(x\mathcal{H}_{\tau(x)})^{\frac{1}{\gamma}-1} - \frac{1-e^{-\hat{\rho}(T-\tau(x))}}{\hat{\rho}} \right) \right]\right)dx\\
+&\frac{\gamma}{1-\gamma}\left(\frac{1-e^{-K(T-t)}}{K}\lambda^{\frac{1}{\gamma}}+\frac{1-e^{-\hat{r}(T-t)}}{\hat{r}}y \right)\\
=&-y^{1-\gamma} \int_{\lambda}^{\infty} \left( \sup_{\tau(x) \in [t,T]}\mathbb{E}_t^{\mathbb{Q}} \left[ e^{-\hat{\rho}(\tau(x)-t)}\frac{1}{1-\gamma}\left(\frac{1-e^{-\hat{\rho}(T-\tau(x))}}{\hat{\rho}}-\frac{1-e^{-K(T-\tau(x))}}{K}(x\mathcal{H}_{\tau(x)})^{\frac{1}{\gamma}-1} \right) \right]\right)dx\\
+&{ \frac{\gamma}{1-\gamma}\left(\frac{1-e^{-K(T-t)}}{K}\lambda^{\frac{1}{\gamma}}+\frac{1-e^{-\hat{r}(T-t)}}{\hat{r}}y \right)}.\\
\end{aligned}
\end{eqnarray} \end{footnotesize} This ends the proof of Lemma \ref{thm:lem1}.
$\Box$
In Lemma \ref{thm:lem1}, the dual problem can be solved by converting it to the infinite series of optimal stopping time problems. Since the underlying process $(\mathcal{H}_s)_{s=t}^{T}$ has {an} exponential form, however, the multiplicative factor $x$ can be absorbed into the initial condition problem. Thus, the problems can be {combined} {in}to a single problem{,} as shown below. \begin{pr}[Optimal stopping problem]\label{pr:OS}~
We consider the following optimal stopping problem:
$$
g(t,z)=\sup_{\tau \in \mathcal{S}(t,T)} \mathbb{E}^{\mathbb{Q}}\left[e^{-\hat{\rho}(\tau-t)}h(\tau,\mathcal{H}_{\tau}) \bigm| \mathcal{H}_t=z\right]{,}
$$
where $\mathcal{S}(t,T)$ denotes the set of all stopping times of the filtration $\mathcal{F}$ taking values in $[t,T]$, and
$$
h(t,z)=\frac{1}{1-\gamma}\left(\frac{1-e^{-\hat{\rho}(T-t)}}{\hat{\rho}}-\frac{1-e^{-K(T-t)}}{K}z^{\frac{1}{\gamma}-1}\right)
$$
and $(\mathcal{H}_s)_{s\geq t}$ satisfies the following stochastic diffusion process:
$$
d\mathcal{H}_s=(\hat{r}-\hat{\rho}+\sigma^2\gamma^2)\mathcal{H}_sds-\gamma\sigma \mathcal{H}_sdB_s^{\mathbb{Q}}.
$$ \end{pr} Notice that Problem \ref{pr:OS} is equivalent to that of finding the optimal exercise time of an American option written on the underlying process $\{\mathcal{H}_s\}_{s=t}^{T}$ with payoff equal to $h(\tau,\mathcal{H}_\tau)$ at the time of exercise time $\tau$. The exercise time is characterized as the first time for the underlying process to hit the early exercise boundary(or free boundary), and thus the problem is to derive the early exercise boundary. \citet{K}, \citet{CJM}, and \citet{Det} provide an integral equation representation of the American option value from which one can derive a functional equation for the early exercise boundary.
According to the standard technique {for} the optimal stopping problem, $g(T,z)$ can be derived from the following variational inequality(VI). (See Chapter 2 of \citet{KS} or \citet{YK}): \begin{eqnarray}\label{eq:VI1} \begin{split} \left\{ \begin{array}{l} -\partial_t g-{\cal L} g=0,~~ \mbox{if}~~g(t,z)>h(t,z)~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ -\partial_t g-{\cal L} g\geq 0,~~ \mbox{if}~~g(t,z)=h(t,z) ~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ g(T,z)=h(T,z),\quad \forall \;z\in(0,+\infty){,} \end{array} \right. \end{split} \end{eqnarray} where $\mathcal{M}_T=[0,T)\times(0,+\infty)$ and the operator $\mathcal{L}$ is generated by the process $\mathcal{H}_s$: $$ \mathcal{L}= \frac{\gamma^2\sigma^2}{2}z^2 \partial_{zz}+(\hat{r}-\hat{\rho}+\sigma^2\gamma^2)z\partial_z-\hat{\rho}. $$
\section{Analysis of variational inequality arising from Problem \ref{pr:OS}}\label{sec:B}
{In this section we provide a complete self-contained derivation of the solution to VI \eqref{eq:VI1} arising from Problem \ref{pr:OS} by borrowing the ideas and proofs in \citet{YK}.}
For convenience of proof, we substitute the above VI \eqref{eq:VI1} for the following to make the lower obstacle 0. $$ Q(t,z)=g(t,z)-h(t,z). $$ Then, the (\ref{eq:VI1}) can be converted to \begin{eqnarray}\label{eq:VI2} \begin{split} \left\{ \begin{array}{l} -\partial_t Q-{\cal L} Q=\dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~ \mbox{if}~~Q(t,z)>0~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ -\partial_t Q-{\cal L} Q\geq \dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~ \mbox{if}~~Q(t,z)=0 ~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ Q(T,z)=0,\quad \forall \;z\in(0,+\infty). \end{array} \right. \end{split} \end{eqnarray} Now, we will prove the existence and uniqueness of $W^{1,2}_{p,loc}$ solution {$(p\geq1)$} to VI \eqref{eq:VI2} and describe properties of the solution. \begin{lem}\label{thm:lemQ}~
VI (\ref{eq:VI2}) has a unique strong solution $Q$ satisfying the following properties:\\
\noindent 1. $Q \in W^{1,2}_{p,loc}(\mathcal{M}_T) \cap C(\widetilde{\mathcal{M_T}})$ for any $p \geq 1 $ and $\partial_z Q \in C(\widetilde{\mathcal{M}_T})$, where $\widetilde{\mathcal{M}}_T=[0,T]\times (0,+\infty)$.\\
\noindent 2. $\partial_zQ \geq 0$ a.e. in $\widetilde{\mathcal{M}_T}$ and $\partial_t Q \leq 0$ a.e. in $\widetilde{\mathcal{M}_T}$. \end{lem} \noindent{\bf Proof.} 1. To use the Theorem of \cite{F2}, replace the PDE operator $\mathcal{L}$ with a non-degenerate parabolic equation.
Define $$ \zeta=\log z,~~~~ \bar{Q}(t,\zeta)=Q(t,z). $$ Then $\bar{Q}(t,\zeta)$ satisfies \begin{eqnarray}\label{eq:VI3} \begin{split} \left\{ \begin{array}{l} -\partial_t \bar{Q}-\bar{{\cal L}} \bar{Q}=\dfrac{1}{1-\gamma}\left(e^{\zeta(\frac{1}{\gamma}-1)}-1\right),~~ \mbox{if}~~\bar{Q}(t,\zeta)>0~~\textrm{and}~~{(t,e^\zeta)\in \mathcal{M}_T}
\\ -\partial_t \bar{Q}-\bar{{\cal L}} \bar{Q}\geq \dfrac{1}{1-\gamma}\left(e^{\zeta(\frac{1}{\gamma}-1)}-1\right),~~ \mbox{if}~~\bar{Q}(t,\zeta)=0 ~~\textrm{and}~~{(t,e^\zeta)\in \mathcal{M}_T}
\\ \bar{Q}(T,\zeta)=0,\quad \forall \;\zeta\in(0,+\infty). \end{array} \right. \end{split} \end{eqnarray} where $$ \bar{{\cal L}}= \frac{\gamma^2\sigma^2}{2} \partial_{\zeta\zeta}+(\hat{r}-\hat{\rho}+\frac{\sigma^2\gamma^2}{2})\partial_\zeta-\hat{\rho}. $$ Since the inhomogeneous term `$\dfrac{1}{1-\gamma}\left(e^{\zeta(\frac{1}{\gamma}-1)}-1\right)$', the lower obstacle `0' and the terminal value `0' are all smooth functions, we can easily show that VI(\ref{eq:VI3}) has a unique solution satisfying $\bar{Q} \in W^{1,2}_{p,loc}(\mathcal{M}_T) \cap C(\widetilde{\mathcal{M_T}})$ for any $p \geq 1 $ and $\partial_\zeta \bar{Q} \in C(\widetilde{\mathcal{M}_T})$(See \cite{F2}).\\
2. Let us denote $\widetilde{Q}(t,z)=Q(t,\eta z)$ for any $\eta >1$. Then $\widetilde{Q}$ satisfies following Variational Inequality: \begin{eqnarray}\label{eq:VI4} \begin{split} \left\{ \begin{array}{l} -\partial_t \widetilde{Q}-{\cal L} \widetilde{Q}=\dfrac{1}{1-\gamma}\left((\eta z)^{\frac{1}{\gamma}-1}-1\right),~~ \mbox{if}~~\widetilde{Q}(t,z)>0~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ -\partial_t \widetilde{Q}-{\cal L} \widetilde{Q}\geq \dfrac{1}{1-\gamma}\left((\eta z)^{\frac{1}{\gamma}-1}-1\right),~~ \mbox{if}~~\widetilde{Q}(t,z)=0 ~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ \widetilde{Q}(T,z)=0\quad \forall \;z\in(0,+\infty). \end{array} \right. \end{split} \end{eqnarray} For any $\eta >1$, we can easily check that the inhomogeneous term of $\widetilde{Q}$ is greater than $Q$: $$ \frac{1}{1-\gamma}\left((\eta z)^{\frac{1}{\gamma}-1}-1\right) > \frac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~~~~~\textrm{for}~\forall~\gamma>0 (\gamma \neq 1). $$ In addition, since the terminal value of $Q$ and $\widetilde{Q}$ are the same, the comparison theory for VI implies that $\widetilde{Q}(t,z)=Q(t,\eta z) \geq Q(t,z)$ for any $\eta >1$ and $(t,z) \in \mathcal{M}_T$. So we obtain $\partial_z Q \geq 0$ in $\mathcal{M}_T$.\\ Define $\hat{Q}(t,z)=Q(t-\delta,z)$ with $\delta>0$ being sufficiently small. Then $\hat{Q}$ follows: \begin{eqnarray}\label{eq:VI5} \begin{split} \left\{ \begin{array}{l} -\partial_t \hat{Q}-{\cal L} \hat{Q}=\dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~ \mbox{if}~~\hat{Q}(t,z)>0~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ -\partial_t \hat{Q}-{\cal L} \hat{Q}\geq \dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~ \mbox{if}~~\hat{Q}(t,z)=0 ~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ \hat{Q}(T,z)=0\quad \forall \;z\in(0,+\infty). \end{array} \right. \end{split} \end{eqnarray} Since $\hat{Q}(T,z)(=Q(T-\delta,z)) \geq Q(T,z)$ for any $\delta>0,z>0$, the following can be deduced by the comparison theory for the same reason: $\hat{Q}(t,z)=Q(t-\delta,z) \leq Q(t,z)$. Therefore we can prove that $\partial_t Q \leq 0$, a.e.
$\Box$
We can define two regions derived from the Variational Inequality (\ref{eq:VI2}): \begin{eqnarray}\label{eq:def_region1}
\Omega_1=\left\{(t,z)~ |~ Q(t,z)=0 \right\},~~~\Omega_2=\left\{(t,z)~ |~ Q(t,z)>0 \right\}. \end{eqnarray} Since $\partial_z Q \ge 0$, we can define the free boundary $z^{\star}(t)$ as follows: \begin{eqnarray}
z^{\star}(t)=\inf \left\{ z\geq 0 ~|~ Q(t,z)>0 \right\},~~~~\forall t\in[0,T). \end{eqnarray} The two regions can be defined according to the boundary as follows. \begin{eqnarray} \begin{aligned}\label{eq:def_region2}
&\Omega_1=\left\{(t,z)~ |~ 0<z\leq z^{\star}(t),~ t \in [0,T] \right\},\\
&\Omega_2=\left\{(t,z)~ |~ z> z^{\star}(t),~ t \in [0,T] \right\}. \end{aligned} \end{eqnarray} \begin{lem}\label{lem-cinfinity}~The free boundary $z^{\star}$ is smooth, i.e., $z^{\star}(t)\in C[0,T]\cap C^{\infty}([0,T))$. Moreover, the solution $Q\equiv 0$ in $\Omega_1$,\;and $Q\in C^{\infty}\left(\{(t,z)\mid z \ge z^{\star}(t),\;t\in[0,T]\}\right)$, and $\partial_{t} Q \in C(\widetilde{\mathcal{M}}_{T})$. \end{lem} \noindent{\bf Proof.} By Lemma~\ref{thm:lemQ}, we obtain $\partial_{t} Q \le 0$ \rm{a.e.} in $\widetilde{\mathcal{M}}_{T}$. Moreover, the coefficient functions in the operator $\mathcal{L}$, the lower obstacle function, the terminal function, and the non-homogeneous term $\frac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right)$ are all smooth. Therefore, the regularity results in the lemma follow from Theorem 3.1 in ~\citet{F2}.
$\Box$
Consider the following function $Q_{\infty}$ as follow: \begin{eqnarray}\label{eq:INFQ} \begin{split} Q_{\infty}(t,z)= \left\{ \begin{array}{l} \left(-\dfrac{1}{\gamma}\dfrac{1}{K}\dfrac{1}{\alpha_-}(z_{\infty})^{\frac{1}{\gamma}-1-\alpha_-}\right)z^{\alpha_-}+\dfrac{1}{1-\gamma}\left(\dfrac{1}{K}z^{\frac{1}{\gamma}-1}-\dfrac{1}{\hat{\rho}}\right),~~ \mbox{if}~~(t,z)\in\Omega_2^{\infty},~~
\\ 0,\;\;\;\;\;\;\;\;\;\;\;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \mbox{if}~~(t,z)\in\Omega_1^{\infty},
\end{array} \right. \end{split} \end{eqnarray} where \begin{eqnarray}\label{eq:INFZ} \begin{aligned}
&\Omega_1^{\infty}=\left\{(t,z)~ |~ 0<z\leq z_{\infty},~ t \in [0,T] \right\},~~\Omega_2^{\infty}=\left\{(t,z)~ |~ z> z_{\infty},~ t \in [0,T] \right\} \end{aligned} \end{eqnarray} and \begin{eqnarray} z_{\infty}=\left(\frac{\hat{\rho}(\alpha_- \gamma + \gamma -1)}{K \alpha_- \gamma}\right)^{\frac{\gamma}{\gamma-1}}. \end{eqnarray} $\alpha_+$ and $\alpha_-$ are the positive and negative root of the following quadratic equation $f(\alpha)=0$, respectively: \begin{eqnarray}\label{eq:f1} f(\alpha)=\frac{\gamma^2 \sigma^2}{2}\alpha^2+(\hat{r}-\hat{\rho}+\frac{\gamma^2\sigma^2}{2})\alpha-\hat{\rho}=0. \end{eqnarray} It is easy to confirm that $$ f(-1)=-\hat{r}=\mu-r<0,\;\;\mbox{and}\;\alpha_- < -1. $$ \begin{lem}\label{thm:lem2}~The solution $Q$ satisfy the following statements:\\
\noindent 1.~$Q(t,z)\equiv0 $ in $\Omega_1^\infty$.\\
\noindent 2. $Q(t,z)>0$ in $[0,T) \times (z^T,\infty)$, where $z^T=1$. \end{lem} \noindent{\bf Proof.}\\ {1.} It is easy to check that $$ Q_{\infty} \in W^{1,2}_{p,loc}(\mathcal{M}_T) \cap C(\widetilde{\mathcal{M_T}})~~\textrm{for}~~p\geq 1 . $$ Since $\alpha_-<-1$, we deduce $$
\frac{\partial Q_{\infty}}{\partial z}=\frac{1}{\gamma K}z^{\frac{1}{\gamma}-2}\left(1-\left(\frac{z}{z_\infty}\right)^{\alpha_-+1-\frac{1}{\gamma}}\right)>0,~~\textrm{in}~~\Omega_2^{\infty}=\left\{(t,z)~ |~ z> z_{\infty},~ t \in [0,T] \right\}. $$ We can see that $Q_{\infty}$ satisfies the following equality. \begin{eqnarray}\label{eq:VI_infty} \begin{split} -\partial_t Q_{\infty}-{\cal L} Q_{\infty}= \left\{ \begin{array}{l} \dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~
\mbox{in}~~\Omega_2^{\infty}=\left\{(t,z)~ |~ z> z_{\infty},~ t \in [0,T] \right\},
\\ 0,\;\;\;~~~~~~~~~~~~~~~~~~~~~~
\mbox{in}~~\Omega_1^{\infty}=\left\{(t,z)~ |~ 0<z\leq z_{\infty},~ t \in [0,T] \right\}.
\\ \end{array} \right. \end{split} \end{eqnarray} Prior to proving $1.$ of Lemma \ref{thm:lem2} , we can first show the following inequality: \begin{eqnarray}\label{eq:Z1} z_\infty < z^T ~~~~\Longleftrightarrow~~~~\left(\frac{\hat{\rho}(\alpha_- \gamma + \gamma -1)}{K \alpha_- \gamma}\right)^{\frac{\gamma}{\gamma-1}}<1. \end{eqnarray} If $0<\gamma<1$, the above inequality is equivalent to $$ \hat{\rho}(\alpha_-\gamma+\gamma-1) < K\alpha_-\gamma ~~~~\Longleftrightarrow~~~~(\hat{\rho}-K)\gamma \alpha_- < \hat{\rho}(1-\gamma). $$ By Assumption \ref{thm:as1}, $$ \hat{\rho}-K=(\rho-r)\left(1-\frac{1}{\gamma} \right)-(1-\gamma)\mu +\gamma(1-\gamma)\frac{\sigma^2}{2} <0. $$ It is enough to show $$ \alpha_->\frac{\hat{\rho}(1-\gamma)}{\gamma(\hat{\rho}-K)}. $$ Since $\alpha_-$ is negative root of quadratic equation $f(\alpha)=0$ in \eqref{eq:f1}, we need to show $f\left(\frac{\hat{\rho}(1-\gamma)}{\gamma(\hat{\rho}-K)}\right)>0$. By simple calculations, $$ f\left(\frac{\hat{\rho}(1-\gamma)}{\gamma(\hat{\rho}-K)}\right)=\frac{K \hat{\rho} \sigma^2 (1-\gamma)^2}{2\gamma^2(\hat{\rho}-K)^2}>0. $$ The following $\gamma>1$ case is also the same as the previously proven case $0<\gamma<1$ and thus is omitted here.
By using (\ref{eq:VI_infty}) and the inequality $z_\infty < z^T$ in (\ref{eq:Z1}), $$ -\partial_t Q_{\infty}-{\cal L} Q_{\infty} \geq \frac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right)~~~\forall ~(t,z) \in \mathcal{M}_T $$ Note that $\frac{\partial Q_{\infty}}{\partial z}(t,z_\infty)=Q_{\infty}(t,z_\infty)=0$ and $\frac{\partial Q_{\infty}}{\partial z}(t,z)>0$ in $\Omega_2$. Therefore $Q_{\infty}(t,z)>0$ in $\Omega_2$ and $Q_{\infty}$ satisfies the following VI: \begin{eqnarray}\label{eq:VI_infty2} \begin{split} \left\{ \begin{array}{l} -\partial_t Q_{\infty}-{\cal L} Q_{\infty}=\dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~ \mbox{in}~~Q_{\infty}>0~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ -\partial_t Q_{\infty}-{\cal L} Q_{\infty} \geq \dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~ \mbox{in}~~Q_{\infty}=0~~\textrm{and}~~(t,z)\in \mathcal{M}_T,
\\ Q_{\infty}(t,z)\geq 0,~~~~\forall~~(t,z)\in \mathcal{M}_T. \end{array} \right. \end{split} \end{eqnarray} By the comparison principle for VI, we have $$ Q(t,z)\leq Q_{\infty}(t,z),~~~~~\forall~ (t,z)~~ \textrm{in}~~ \mathcal{M}_T. $$ So we can get the first result of this lemma $Q=0$~~in~$\Omega_1^\infty$. $$
0\leq Q(t,z) \leq Q_{\infty}(t,z)\leq0,~~\textrm{in}~~\Omega_1^\infty=\left\{(t,z)~ |~ 0<z\leq z_{\infty},~ t \in [0,T] \right\}. $$ Furthermore, this means that $$
\Omega_1=\left\{(t,z)~ |~ 0<z\leq z^\star(t),~ t \in [0,T] \right\}~~\supseteq~~\Omega_1^\infty=\left\{(t,z)~ |~ 0<z\leq z_{\infty},~ t \in [0,T] \right\}. $$ Thus, we obtain that $z^{\star}(t)\geq z_\infty$ in $[0,T]$.\\
\noindent {2.} Finally, let us prove the second property of this Lemma. VI \eqref{eq:VI2} implies that $$ Q=0,~~~ -\partial_t Q-{\cal L} Q-\frac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right) \geq 0,~~~~\textrm{in}~~\Omega_1. $$
This leads to $\Omega_1 \subseteq [0,T]\times (0,z^T]$. Hence, it is obvious that $\Omega_2=\left\{(t,z)~ |~ Q(t,z)>0 \right\} \supseteq [0,T)\times(z^T,\infty)$. Thus, $Q(t,z)>0$ in $[0,T) \times (z^T,\infty)$. Moreover, $$ z^{\star}(t)\leq z^T(=1)~~\textrm{in}~~[0,T]. $$ This is the end of the lemma.
$\Box$
\begin{lem}\label{thm:FB}~
The free boundary $z^\star(t)$, $t \in [0,T]$ is strictly increasing with the terminal point $z^{\star}(T)=\displaystyle\lim_{t\rightarrow T^-}z^{\star}(t)=z^T$. And $z_{\infty}<z^{\star}(t)<z^T$, ~~$\forall t \in [0,T)$. \end{lem} \noindent{\bf Proof.} By Lemma \ref{thm:lemQ}, $$ \partial_z Q \geq 0,~\partial_t Q \leq 0 ~~\textrm{a.e. in}~~\mathcal{M}_T,~~~~Q\in C(\widetilde{\mathcal{M}}_T). $$ For any fixed $t_0 \in [0,T)$, $$ 0\leq Q(t,z) \leq Q(t,z^{\star}(t_0)) \leq Q(t_0,z^{\star}(t_0))=0,~~~~~\forall t \in (t_0,T]~~ \textrm{and}~~ \forall z \in (0,z^\star(t_0)]. $$ By the definition of the free boundary $z^{\star}(t)$, $$ z^{\star}(t) \geq z^{\star}(t_0),~~~~0\leq t_0 \leq t <T. $$ The above {inequality} shows that $z^\star(t)$ is just increasing function with respect to $t$. (We still need to show the strict increasing property of $z^\star(t)$).\\ Since $z^\star(t)$ is increasing, the $z^\star(T)=\displaystyle\lim_{t\rightarrow T-}z^\star(t)$ exists. We know that $z^\star(t)\leq z^T$ in $\forall t \in[0,T]$. It is therefore sufficient to show that $z^\star(T)\geq z^T$. Otherwise, there exists interval $(z^\star(T),z^T)$ such that $[0,T) \times (z^\star(T),z^T) \subset \Omega_2$. So, we have shown that \begin{eqnarray*}
\begin{split}
\left\{
\begin{array}{l}
-\partial_t Q-{\cal L} Q=\dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),~~
\mbox{in}~~\Omega_2
\\
Q(T,z)= 0,~~~~\forall~z \geq z^\star(T),~~~~~~Q(t,z^\star(t))=0~~~\forall~t\in[0,T].
\end{array}
\right.
\end{split} \end{eqnarray*} By using the above fact, we can deduce that at time $T$ $$ \partial_t Q(T,z)=-{\cal L} Q(T,z)-\frac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right)=-\frac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right)>0~~~~\forall~z\in(z^\star(T),z^T). $$ However, it is easy to see that the above inequality is inconsistent with the results of Lemma \ref{thm:lemQ}.
Finally, we show that $z^\star(t)$ is strictly increasing. Otherwise, there exists $t_1,t_2$ and $z_1$ such that $z^\star(t)=z_1$ for all $t \in[t_2,t_1]$ where $0\leq t_2 < t_1 \leq T$ and $z_1 \in [z_{\infty},z^T]$. Then, it is clear that $Q(t,z)=0$ for all $(t,z)\in[t_2,t_1]\times(0,z_1)$. Because of the continuity at the free boundary of $\partial_zQ$, $\partial_z Q (t,z_1)=0$ for all $t\in[t_2,t_1]$. Therefore, we obtain that $\partial_tQ(t,z_1)=\partial_z\partial_zQ(t,z_1)=0$ for all $t\in[t_2,t_1]$. In domain $[t_2,t_1)\times(z_1,\infty)$, $\partial_tQ$ satisfies \begin{eqnarray*}
\begin{split}
\left\{
\begin{array}{l}
-\partial_t\partial_t Q-{\cal L} \partial_t Q=0,~\partial_t Q \leq 0,~~
\mbox{in}~~[t_2,t_1)\times(z_1,\infty),
\\
\partial_tQ(t,z_1)=0,~~~~\forall~t\in(t_2,t_1).
\end{array}
\right.
\end{split} \end{eqnarray*} According to Hopf's boundary point lemma (See \cite{Liu}), we obtain that $\partial_z(\partial_tQ)<0$, which contradicts the $\partial_z\partial_tQ(t,z_1)=0$ in $t\in[t_2,t_1]$. So, the free boundary $z^{\star}(t)$ is strictly increasing. Thus, We conclude that $z_{\infty}<z^{\star}(t)<z^T$ for all $t\in[0,T)$
$\Box$
\begin{lem}\label{thm:estimate}
~For $(t,z)\in\mathcal{M}_T$,
\begin{eqnarray}
0\le \partial_z Q(t,z) \le \dfrac{1}{\gamma}\cdot\dfrac{1-e^{-K(T-t)}}{K}z^{\frac{1}{\gamma}-2}.
\end{eqnarray} \end{lem} \noindent{\bf Proof.} From VI \eqref{eq:VI2}, $Q(t,z)$ satisfies \begin{eqnarray*}
\begin{split}
\begin{cases}
&-\partial_t Q -\mathcal{L} Q =\dfrac{1}{1-\gamma}\left(z^{\frac{1}{\gamma}-1}-1\right),\;\;\;\mbox{in}\;\Omega_2,\\
&Q(T,z)=0,\;\;\forall\;z\ge z^{\star}(T);\qquad Q(t,z^{\star}(t))=0,\;\;\forall\;t\in[0,T].
\end{cases}
\end{split} \end{eqnarray*} Since $\mathcal{L}(z\partial_z Q)=z\partial_z(\mathcal{L}Q)$ and $\partial_zQ(t,z^{\star}(t))=0$, we have \begin{eqnarray*}
\begin{split}
\begin{cases}
&-\partial_t (z\partial_z Q) -\mathcal{L} (z\partial_zQ) =\dfrac{1}{\gamma}z^{\frac{1}{\gamma}-1},\;\;\;\mbox{in}\;\Omega_2,\\
&(z\partial_z Q)(T,z)=0,\;\;\forall\;z\ge z^{\star}(T);\qquad (z\partial_zQ)(t,z^{\star}(t))=0,\;\;\forall\;t\in[0,T].
\end{cases}
\end{split} \end{eqnarray*} Let us temporarily denote $$ Q_1(t,z)=\dfrac{1}{\gamma}\cdot\dfrac{1-e^{-K(T-t)}}{K}z^{\frac{1}{\gamma}-1}. $$ Then, $Q_1(t,z)$ satisfies \begin{eqnarray*}
\begin{split}
\begin{cases}
&-\partial_t Q_1 -\mathcal{L} Q_1 =\dfrac{1}{\gamma}z^{\frac{1}{\gamma}-1},\;\;\;\mbox{in}\;\Omega_2,\\
&Q_1(T,z)=0,\;\;\forall\;z\ge z^{\star}(T);\; Q_1(t,z^{\star}(t))=\dfrac{1}{\gamma}\cdot\dfrac{1-e^{-K(T-t)}}{K}(z^{\star}(t))^{\frac{1}{\gamma}-1},\;\;\forall\;t\in[0,T].
\end{cases}
\end{split} \end{eqnarray*} By the comparison principle for PDEs(see \citet{Lieberman}), $$ z\partial_z Q(t,z)\le Q_1(t,z). $$ From Lemma \ref{thm:lemQ}, we can conclude $$ 0\le \partial_zQ(t,z) \le \dfrac{1}{\gamma}\cdot\dfrac{1-e^{-K(T-t)}}{K}z^{\frac{1}{\gamma}-2}. $$
$\Box$
{We provide the integral equation representation of $Q(t,z)$ in the following lemma.} \begin{lem}\label{lem:integral_Q}~
In the region $\Omega_2$, the value function $Q(t,z)$ has the following integral equation representation:
\begin{eqnarray*}
\begin{split}
Q(t,z)=&\dfrac{z^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(d^{\gamma}\left(s-t,\frac{z}{z^{\star}(s)}\right)\right)ds\\-&\dfrac{1}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(d^1\left(s-t,\frac{z}{z^{\star}(s)}\right)\right)ds,
\end{split}
\end{eqnarray*}
where
$$
d^1(t,z)=\dfrac{\log{z}+(\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma\sigma)^2)t}{\gamma\sigma\sqrt{t}},\;\;d^\gamma(t,z)=\dfrac{\log{z}+(\hat{r}-\hat{\rho}-\frac{1}{2}(\gamma\sigma)^2+\frac{1}{\gamma}(\gamma\sigma)^2)t}{\gamma\sigma\sqrt{t}},
$$
and $\mathcal{N}(\cdot)$ is a standard normal distribution function.
Moreover, the free boundary $z^{\star}(t)$ satisfies the following integral equation:
\begin{eqnarray*}
\begin{split}
0=&\dfrac{(z^{\star}(t))^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(d^{\gamma}\left(s-t,\frac{z^{\star}(t)}{z^{\star}(s)}\right)\right)ds\\-&\dfrac{1}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(d^1\left(s-t,\frac{z^{\star}(t)}{z^{\star}(s)}\right)\right)ds.
\end{split}
\end{eqnarray*} \end{lem} \noindent{\bf Proof.} From Lemma \ref{thm:lemQ}, $$ Q \in W^{1,2}_{p,loc}(\mathcal{M}_T) \cap C(\widetilde{\mathcal{M_T}}),~~~ {p \geq 1 }. $$ By applying It\'o lemma to $e^{-\hat{\rho} s}Q(s.\mathcal{H}_s)$ (see \citet{Krylov}), \begin{eqnarray}\begin{split}\label{ito_Q} \int_{t}^{T}d\left(e^{-{\hat{\rho}}s}Q(s,\mathcal{H}_s)\right)=\int_{t}^{T}e^{-{\hat{\rho}}s}\left(\dfrac{\partial Q}{\partial s} + \mathcal{L}Q\right)ds - {\gamma \sigma}\int_{t}^{T} e^{-{\hat{\rho}}s}\mathcal{H}_s \dfrac{\partial Q}{\partial z}dB^{\mathbb{Q}}_{s}. \end{split}\end{eqnarray} By Lemma \ref{thm:estimate}, \begin{eqnarray*}
\begin{split}
\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\left(\gamma\sigma e^{-\hat{\rho}t}\mathcal{H}_t\dfrac{\partial Q}{\partial z}\right)^2dt\right]\le \dfrac{\sigma^2}{K^2} \mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\left(e^{-\hat{\rho}t}{\mathcal{H}_t}^{\frac{\gamma-1}{\gamma}}\right)^2 dt\right] <\infty.
\end{split} \end{eqnarray*} This implies that $$ {\gamma \sigma}\int_{t}^{T} e^{-{\hat{\rho}}s}\mathcal{H}_s \dfrac{\partial Q}{\partial z}dB^{\mathbb{Q}}_{s} $$ is a martingale under $\mathbb{Q}$ measure and $$ \mathbb{E}_t^{\mathbb{Q}}\left[ {\gamma \sigma}\int_{t}^{T} e^{-{\hat{\rho}}s}\mathcal{H}_s \dfrac{\partial Q}{\partial z}dB^{\mathbb{Q}}_{s}\right]=0. $$ (see Chapter 3 in \citet{OS})
By taking expectation to both-side of the equation \eqref{ito_Q}, \begin{eqnarray*}
\begin{split}
Q(t,z)=&\mathbb{E}_t^{\mathbb{Q}}\left[e^{-\hat{\rho}(T-t)}Q(T,\mathcal{H}_T)\right]-\mathbb{E}_t^{\mathbb{Q}}\left[\int_{t}^{T}e^{-{\hat{\rho}}(s-t)}\left(\dfrac{\partial Q}{\partial s} + \mathcal{L}Q\right)ds\right]\\
=&-\mathbb{E}^{\mathbb{Q}}_t\left[\int_{t}^{T}e^{-{\hat{\rho}}(s-t)}\left(\dfrac{\partial Q}{\partial s} + \mathcal{L}Q\right){\bf 1}_{\{(s,\mathcal{H}_s)\in{\Omega_2}\}}ds\right]\\
=&-\mathbb{E}^{\mathbb{Q}}_t\left[\int_{t}^{T}e^{-{\hat{\rho}}(s-t)}\left(\dfrac{\partial Q}{\partial s} + \mathcal{L}Q\right){\bf 1}_{\{\mathcal{H}_s\ge z^{\star}(s)\}}ds\right]\\
=&\dfrac{1}{1-\gamma}\mathbb{E}_t^{\mathbb{Q}}\left[\int_{t}^{T}e^{-{\hat{\rho}}(s-t)}\left({\mathcal{H}_s}^{\frac{1-\gamma}{\gamma}}-1\right){\bf 1}_{\{\mathcal{H}_s\ge z^{\star}(s)\}}ds\right]\\
=&\dfrac{1}{1-\gamma}\mathbb{E}_t^{\mathbb{Q}}\left[\int_{t}^{T}e^{-{\hat{\rho}}(s-t)}{\mathcal{H}_s}^{\frac{1-\gamma}{\gamma}}{\bf 1}_{\{\mathcal{H}_s\ge z^{\star}(s)\}}ds\right]-\dfrac{1}{1-\gamma}\mathbb{E}^{\mathbb{Q}}_t\left[\int_{t}^{T}e^{-{\hat{\rho}}(s-t)}{\bf 1}_{\{\mathcal{H}_s\ge z^{\star}(s)\}}ds\right].
\end{split} \end{eqnarray*} Since $$ \dfrac{d\mathbb{P}}{d\mathbb{Q}}=\exp{\left\{-\dfrac{1}{2}(1-\gamma)^2\sigma^2(s-t)-(1-\gamma)\sigma(B_s^{\mathbb{Q}}-B_t^{\mathbb{Q}})\right\}} $$ and $B_s=B_t^{\mathbb{Q}}+(1-\gamma)\sigma s,\;\;\mbox{for}\;s\in[t,T]$, \begin{eqnarray*}
\begin{split}
&\mathbb{E}^{\mathbb{Q}}_t\left[\int_{t}^{T}e^{-{\hat{\rho}}(s-t)}{\mathcal{H}_s}^{\frac{1-\gamma}{\gamma}}{\bf 1}_{\{\mathcal{H}_s\ge z^{\star}(s)\}}ds\right]\\=&z^{\frac{1}{\gamma}-1}\mathbb{E}_t\left[\int_{t}^{T}e^{-K(s-t)}{\bf 1}_{\{\mathcal{H}_s\ge z^{\star}(s)\}}ds\right]\\
=&z^{\frac{1}{\gamma}-1}\int_{t}^{T}e^{-K(s-t)}\mathbb{P}(\mathcal{H}_s\ge z^{\star}(s))ds\\
=&z^{\frac{1}{\gamma}-1}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(\dfrac{\log{\frac{z}{z^{\star}(s)}}+(\hat{r}-\hat{\rho}-\frac{1}{2}(\gamma\sigma)^2+\frac{1}{\gamma}(\gamma\sigma)^2)(s-t)}{\gamma\sigma\sqrt{s-t}}\right)ds.
\end{split} \end{eqnarray*} Similarly, \begin{eqnarray*}
\begin{split}
&\mathbb{E}_t^{\mathbb{Q}}\left[\int_{t}^{T}e^{-{\hat{\rho}}(s-t)}{\bf 1}_{\{\mathcal{H}_s\ge z^{\star}(s)\}}ds\right]\\=&\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathbb{Q}(\mathcal{H}_s\ge z^{\star}(s))ds\\
=&\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(\dfrac{\log{\frac{z}{z^{\star}(s)}}+(\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma\sigma)^2)(s-t)}{\gamma\sigma\sqrt{s-t}}\right)ds.
\end{split} \end{eqnarray*} Thus, \begin{eqnarray*}
\begin{split}
Q(t,z)=&\dfrac{z^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(\dfrac{\log{\frac{z}{z^{\star}(s)}}+(\hat{r}-\hat{\rho}-\frac{1}{2}(\gamma\sigma)^2+\frac{1}{\gamma}(\gamma\sigma)^2)(s-t)}{\gamma\sigma\sqrt{s-t}}\right)ds\\
-&\frac{1}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(\dfrac{\log{\frac{z}{z^{\star}(s)}}+(\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma\sigma)^2)(s-t)}{\gamma\sigma\sqrt{s-t}}\right)ds.
\end{split} \end{eqnarray*} By the smooth-pasting condition (Lemma \ref{lem-cinfinity}), \begin{eqnarray*}
\begin{split}
0=&(z^{\star}(t))^{\frac{1}{\gamma}-1}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(\dfrac{\log{\frac{z^{\star}(t)}{z^{\star}(s)}}+(\hat{r}-\hat{\rho}-\frac{1}{2}(\gamma\sigma)^2+\frac{1}{\gamma}(\gamma\sigma)^2)(s-t)}{\gamma\sigma\sqrt{s-t}}\right)ds\\
-&\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(\dfrac{\log{\frac{z^{\star}(t)}{z^{\star}(s)}}+(\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma\sigma)^2)(s-t)}{\gamma\sigma\sqrt{s-t}}\right)ds.
\end{split} \end{eqnarray*}
$\Box$
{We will show that the value function $Q(t,z)$ converges when the time-to-maturity goes to infinity. We will need the following lemma. } \begin{lem} ~For arbitrary $c>0$ and $d \in\mathbb{R}$, \label{lem:lemcd}
\begin{eqnarray*}
\int_0^{\infty}e^{-c\xi}\mathcal{N}(d\sqrt{\xi})d\xi = \frac{1}{2c}\left(1+\frac{d}{\sqrt{d^2+2c}}\right).
\end{eqnarray*} \end{lem} \noindent{\bf Proof.} By integration by parts, \begin{eqnarray} \int_0^{\infty}e^{-c\xi}\mathcal{N}(d\sqrt{\xi})d\xi =\left[-\frac{1}{c} e^{-c \xi}\mathcal{N}(d\sqrt{\xi})\right]_{\xi=0}^{\xi=\infty} + \frac{d}{2c \sqrt{2\pi}}\int_0^{\infty} e^{-c\xi -\frac{d^2}{2}\xi}\frac{1}{\sqrt{\xi}}d\xi. \end{eqnarray} By \citet{AS} (p.304, equation (7.4.33)), for any $a,b \in \mathbb{R}$, \begin{eqnarray*}
\int_0^{\infty}\exp\left\{ -a^2x^2-\frac{b^2}{x^2} \right\}dx = \frac{\sqrt{\pi}}{2|a|}e^{-2|a||b|}. \end{eqnarray*} Therefore, \begin{eqnarray*}
\int_0^{\infty} e^{-(c +\frac{d^2}{2})\xi}\frac{1}{\sqrt{\xi}}d\xi=\frac{\sqrt{\pi}}{\sqrt{c+\frac{d^2}{2}}}. \end{eqnarray*}
$\Box$
{We now provide the convergence of $Q(t,z)$ in the following lemma.} \begin{lem}\label{lem:inf}
\begin{eqnarray*}
\begin{split}
\lim_{T-t \to \infty} Q(t,z)=Q_{\infty}(z)\;\;\;\mbox{and}\;\;\lim_{T-t\to \infty}z^{\star}(t)=z_\infty,
\end{split}
\end{eqnarray*}
where $Q_{\infty}(z)$ and $z_\infty$ are defined in \eqref{eq:INFQ} and \eqref{eq:INFZ}, respectively. \end{lem} \noindent{\bf Proof.} \noindent Let $\widetilde{z}_{\star}$ define such that $\widetilde{z}_{\star}(T-t) \equiv z^{\star}(t)$, then \begin{eqnarray*}
z_\infty = \lim_{T-t \rightarrow \infty} z^{\star}(t)=\lim_{T-t \rightarrow \infty} \widetilde{z}_{\star}(T-t). \end{eqnarray*} \noindent From the integral equation of $z^{\star}$ in Lemma \ref{lem:integral_Q} and $T-t \rightarrow \infty$, \begin{eqnarray} \label{eq:yinfty} \begin{split} 0&= {z_{\infty}}^{\frac{\gamma-1}{\gamma}} \int_0^\infty e^{-K\xi} \mathcal{N}\left(\frac{\hat{r}-\hat{\rho}-\frac{1}{2}(\gamma \sigma)^2+\frac{1}{\gamma}(\gamma \sigma)^2 }{\gamma \sigma}\sqrt{\xi}\right) d\xi\\ &~-\int_0^\infty e^{-\hat{\rho}\xi} \mathcal{N}\left(\frac{\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma \sigma)^2 }{\gamma \sigma}\sqrt{\xi}\right) d\xi.\\ \end{split} \end{eqnarray} By applying Lemma \ref{lem:lemcd} and direct computation, we can get $z_\infty$ as follows: \begin{eqnarray*}
z_{\infty}=\left(\frac{\hat{\rho}(\alpha_- \gamma + \gamma -1)}{K \alpha_- \gamma}\right)^{\frac{\gamma}{\gamma-1}}. \end{eqnarray*} Consider the integral equation representation of $Q(t,y)$ in Lemma \ref{lem:integral_Q}, then \begin{eqnarray*}
\begin{aligned}
Q_{\infty}(z) \equiv &\lim_{T-t \rightarrow \infty}Q(t,z)\\
=&\lim_{T-t \rightarrow \infty}\dfrac{z^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{0}^{T-t}e^{-K\xi}\mathcal{N}\left(d^{\gamma}\left(\xi,\frac{z}{\widetilde{z}^{\star}(T-t-\xi)}\right)\right)d\xi\\
&-\lim_{T-t \rightarrow \infty} \dfrac{1}{1-\gamma}\int_{0}^{T-t}e^{-\hat{\rho}\xi}\mathcal{N}\left(d^1\left(\xi,\frac{z}{\widetilde{z}^{\star}(T-t-\xi)}\right)\right)d\xi\\
=& \dfrac{z^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{0}^{\infty}e^{-K\xi}\mathcal{N}\left(d^{\gamma}\left(\xi,\frac{z}{z_\infty}\right)\right)d\xi - \dfrac{1}{1-\gamma}\int_{0}^{\infty}e^{-\hat{\rho}\xi}\mathcal{N}\left(d^1\left(\xi,\frac{z}{z_\infty}\right)\right)d\xi \\
=&\dfrac{z^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{0}^{\infty}e^{-K\xi}\mathcal{N}\left(\left(\dfrac{\log\frac{z}{z_\infty}}{\gamma \sigma}\right) \xi^{-\frac{1}{2}}+\left(\dfrac{\hat{r}-\hat{\rho}-\frac{1}{2}(\gamma \sigma)^2+\frac{1}{\gamma}(\gamma \sigma)^2}{\gamma \sigma}\right)\xi^{\frac{1}{2}}\right)d\xi \\
&-\dfrac{1}{1-\gamma}\int_{0}^{\infty}e^{-\hat{\rho}\xi}\mathcal{N}\left(\left(\dfrac{\log\frac{z}{z_\infty}}{\gamma \sigma}\right)\xi^{-\frac{1}{2}}+\left(\dfrac{\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma \sigma)^2}{\gamma \sigma}\right)\xi^{\frac{1}{2}} \right)d\xi.
\end{aligned} \end{eqnarray*} The exact solution of the above integral equation can be easily calculated by integration by parts and some characteristic solutions.
First, consider the following integral equation: \begin{eqnarray*}
\int_{0}^{\infty}e^{-K\xi}\mathcal{N}\left(\left(\dfrac{\log\frac{z}{z_\infty}}{\gamma \sigma}\right) \xi^{-\frac{1}{2}}+\left(\dfrac{\hat{r}-\hat{\rho}-\frac{1}{2}(\gamma \sigma)^2+\frac{1}{\gamma}(\gamma \sigma)^2}{\gamma \sigma}\right)\xi^{\frac{1}{2}}\right)d\xi. \end{eqnarray*} Let \begin{eqnarray} c_1=\left(\dfrac{\log\frac{z}{z_\infty}}{\gamma \sigma}\right),\;\;\mbox{and}\;\;c_2=\left(\dfrac{\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma \sigma)^2}{\gamma \sigma}\right). \end{eqnarray} Since we consider the value of $Q$ in the $\Omega_2$, $z\geq z^\star(\xi+t) \geq z_\infty$ implies $c_1>0$.\\ By integration by parts, \begin{eqnarray*}
\begin{aligned}
&\int_0^\infty e^{-K \xi} \mathcal{N}\left(c_1 \xi^{-\frac{1}{2}}+c_2 \xi^{\frac{1}{2}} \right)\\
=& \left.-\frac{1}{K}e^{-K\xi}\mathcal{N}\left(c_1 \xi^{-\frac{1}{2}}+c_2\xi^{\frac{1}{2}}\right)\right]_{\xi=0}^\infty\\
&+\frac{1}{K}\frac{1}{\sqrt{2\pi}}\int_0^\infty e^{-K\xi}\cdot e^{\frac{1}{2}(c_1 \xi^{-\frac{1}{2}}+c_2 \xi^{\frac{1}{2}})^2}\cdot \left(-\frac{c_1}{2}\xi^{-\frac{3}{2}}+\frac{c_2}{2}\xi^{-\frac{1}{2}}\right)d\xi\\
=&\frac{1}{K}+\frac{e^{-c_1c_2}}{K \sqrt{2\pi}}\int_0^\infty e^{-\left(
\frac{c_1}{\sqrt{2}}\right)^2x^{-2} -\left(\frac{\sqrt{c_2^2+2K}}{\sqrt{2}}\right)^2x^2}(-c_1x^{-2}+c_2)dx.~~~~(\xi=x^2)\\
\end{aligned} \end{eqnarray*} Since \citet{AS} (p.304, equation (7.4.33)), \begin{eqnarray*}
\begin{split}
&\int_0^{\infty} \exp \left\{-a^2x^2-\frac{b^2}{x^2}\right\}dx = \frac{\sqrt{\pi}}{2|a|}e^{-2|a||b|},\\
&\int_0^{\infty} \frac{1}{x^2} \exp \left\{-a^2x^2-\frac{b^2}{x^2}\right\}dx = \frac{\sqrt{\pi}}{2|b|}e^{-2|a||b|}.\\
\end{split} \end{eqnarray*} By using above equations , we can get \begin{eqnarray}\label{eq:INF1} \begin{aligned} \int_0^\infty e^{-K \xi} \mathcal{N}\left(c_1 \xi^{-\frac{1}{2}}+c_2 \xi^{\frac{1}{2}} \right)=&\frac{1}{K}-\frac{1}{2K}\left(1-\frac{c_2}{\sqrt{c_2^2 + 2K}}\right)e^{-c_1(\sqrt{c_2^2 + 2K}+c_2)}\\ =&\frac{1}{K}\left(1-\left(\frac{\alpha_++\left(\tfrac{\gamma-1}{\gamma}\right)}{\alpha_+-\alpha_-}\right)\cdot \left(\frac{z}{z_\infty}\right)^{\alpha_-+\left(\tfrac{\gamma-1}{\gamma}\right)}\right), \end{aligned} \end{eqnarray} where \begin{eqnarray*}
\begin{split}
&\sqrt{c_2^2+ 2K} = \frac{\gamma \sigma(\alpha_+-\alpha_-)}{2},\;\;\sqrt{c_2^2+ 2K}+c_2 = -\gamma \sigma(\alpha_-+\left(\tfrac{\gamma-1}{\gamma}\right)),\\
&\sqrt{c_2^2+ 2K}-c_2 = \gamma \sigma (\alpha_++\left(\tfrac{\gamma-1}{\gamma}\right)).
\end{split} \end{eqnarray*} Similarly, we can derive \begin{eqnarray}\label{eq:INF2} \begin{aligned} &\int_{0}^{\infty}e^{-\hat{\rho}\xi}\mathcal{N}\left(\left(\dfrac{\log\frac{z}{z_\infty}}{\gamma \sigma}\right)\xi^{-\frac{1}{2}}+\left(\dfrac{\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma \sigma)^2}{\gamma \sigma}\right)\xi^{\frac{1}{2}} \right)d\xi =\frac{1}{\hat{\rho}}\left(1-\left(\frac{\alpha_+}{\alpha_+-\alpha_-}\right)\left(\frac{z}{z_\infty}\right)^{\alpha_-}\right). \end{aligned} \end{eqnarray} By the equation (\ref{eq:INF1}) and (\ref{eq:INF2}), we can get $Q_{\infty}$ as follows: \begin{eqnarray*}
\begin{aligned}
Q_{\infty}(z)=\left(-\dfrac{1}{\gamma}\dfrac{1}{K}\dfrac{1}{\alpha_-}(z_{\infty})^{\frac{1}{\gamma}-1-\alpha_-}\right)z^{\alpha_-}+\dfrac{1}{1-\gamma}\left(\dfrac{1}{K}z^{\frac{1}{\gamma}-1}-\dfrac{1}{\hat{\rho}}\right),~~~~~
\mbox{in}~~~\Omega_2^{\infty}.
\end{aligned} \end{eqnarray*}
$\Box$
\section{Duality Theorem and Optimal Strategies}\label{sec:duality} By analyzing the variational inequality \eqref{eq:VI1} in Section \ref{sec:B}, we derive the following integral equation representation of value function of $g(t,z)$ of Problem \ref{pr:OS}: \begin{eqnarray*}
\begin{split}
g(t,z)=&\dfrac{1}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1\left(s-t,\frac{z}{z^{\star}(s)}\right)\right)ds\\-&\dfrac{z^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(-d^{\gamma}\left(s-t,\frac{z}{z^{\star}(s)}\right)\right)ds,
\end{split} \end{eqnarray*} where $z^{\star}(t)$ is the free boundary of Problem \ref{pr:OS}, $\mathcal{N}(\cdot)$ is a standard normal distribution function, and $$ d^1(t,z)=\dfrac{\log{z}+(\hat{r}-\hat{\rho}+\frac{1}{2}(\gamma\sigma)^2)t}{\gamma\sigma\sqrt{t}},\;\;d^\gamma(t,z)=\dfrac{\log{z}+(\hat{r}-\hat{\rho}-\frac{1}{2}(\gamma\sigma)^2+\frac{1}{\gamma}(\gamma\sigma)^2)t}{\gamma\sigma\sqrt{t}}. $$ Also, in terms of free boundary $z^{\star}(t)$, we can define the jump region {\bf JR} and the no-jump region {\bf NR} as follows: $$ {\bf JR}=\{(t,\lambda,y) \mid \lambda \le z^{\star}(t) y^\gamma\}\;\;\;\mbox{and}\;\;\;{\bf NR}=\{(t,\lambda,y)\mid \lambda > z^{\star}(t)y^\gamma \}. $$ As seen in Figure \ref{fig:1}, the free boundary $z^{\star}(t)$ partitions the $(t,z)$-region into the jump region and no-jump region. \begin{figure}
\caption{ The jump-region and the no-jump region in $(t,z)$-domain.}
\label{fig:1}
\end{figure} \begin{rem}~ In terms of te regions $\Omega_1$ and $\Omega_2$ defined in Section \ref{sec:B},
$$
{\bf JR}=\{(t,\lambda,y)\mid (t,\dfrac{\lambda}{y^\gamma})\in \Omega_1\}\;\;\;\textrm{and}\;\;\;{\bf NR}=\{(t,\lambda,y)\mid (t,\dfrac{\lambda}{y^\gamma})\in \Omega_2\}.
$$ \end{rem}
If initially $(t,\lambda,y)\in {\bf JR}${,} then $X$ should {increase} immediately, such that $\dfrac{\lambda}{y^\gamma}$ reaches the free boundary $z^{\star}(t)$. That is, the principal should increase the agent's consumption process. On the other hand, if $(t, \lambda,y)\in {\bf NR}$, $X$ must stay constant and this implies that the principal does not adjust the agent's consumption process. Thus, we call {\bf JR} and {\bf NR} the jump region and the no-jump region, respectively.
Moreover, the free boundary $z^{\star}(t)$ satisfies the following integral equation: \begin{eqnarray} \begin{split}\label{eq:integral} 0=&\dfrac{(z^{\star}(t))^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(d^{\gamma}\left(s-t,\frac{z^{\star}(t)}{z^{\star}(s)}\right)\right)ds\\-&\dfrac{1}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1\left(s-t,\frac{z^{\star}(t)}{z^{\star}(s)}\right)\right)ds. \end{split} \end{eqnarray} Then, we can directly obtain the following lemma. \begin{lem}~\label{lem:time}
The optimal stopping time $\tau^*$ for Problem \ref{pr:OS} is given by
$$
\tau^* \equiv \inf \left\{s \ge t \bigg| \mathcal{H}_s \le z^{\star}(t) \right\}\wedge T,
$$
where $z^{\star}(t)$ satisfies the integral equation \eqref{eq:integral}. \end{lem} By using Lemma \ref{thm:lem1} and Lemma \ref{lem:time}, we provide a solution to Problem \ref{pr:dual} in the following proposition. \begin{pro}~\label{pro:dual}
\begin{itemize}
\item[(a)] The infinite series of optimal stopping times $\{\tau^*(x)\}_{x\ge \lambda}$ in Lemma \ref{thm:lem1} is given by
$$
\tau^*(x) = \inf \{s\ge t \mid x\mathcal{H}_s \le z^{\star}(t)\} \wedge T.
$$
\item[(b)] The dual value function is given by
\begin{eqnarray*}
\begin{split}
J(t,\lambda,y)=&-y^{1-\gamma}\int_{\lambda}^{\infty}g(t,\dfrac{x}{y^{\gamma}})dx + J_0(t,\lambda,y)\\
=&-\dfrac{y^{1-\gamma}}{1-\gamma}\int_{\lambda}^{\infty}\left[\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^\gamma(s-t,\frac{x}{ z^{\star}(s)y^\gamma})\right)ds\right.\\&\left.-\left(\dfrac{x}{y^\gamma}\right)^{\frac{1}{\gamma}-1}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1(s-t,\frac{x}{z^{\star}(s)y^\gamma})\right)ds\right]dx\\
&+\dfrac{\gamma}{1-\gamma}\dfrac{1-e^{-K(T-t)}}{K}\lambda^{\frac{1}{\gamma}}+\dfrac{1-e^{-\hat{r}(T-t)}}{\hat{r}}y.
\end{split}
\end{eqnarray*}
\end{itemize} \end{pro}
\begin{rem}~ It is easy to see that the infinite series of optimal stopping times $\{\tau^*(x)\}_{x\ge \lambda}$ defined in Proposition \ref{pro:dual} is non-decreasing, left continuous with right limits as function of $x$. Thus, the optimal stopping problem in \eqref{eq:dual_ex1} can be expressed as follows:
\begin{eqnarray}
\sup_{\tau^*(x)\in[t,T]}\mathbb{E}_t^{\mathbb{Q}} \left[ e^{-\hat{\rho}(\tau^*(x)-t)} h(\tau^*(x),x\mathcal{H}_{\tau^*(x)}) \right]=g(t,xH_t).
\end{eqnarray} \end{rem} Proposition \ref{pro:dual} (a) characterizes the optimal time to adjust the process $X$ as we discussed earlier. Proposition \ref{pro:dual} (b) provides the dual value function by using the time-varying function $z^{\star}(t)$ determining the free boundary for the optimal stopping problem.
By Proposition \ref{pro:dual}, \begin{eqnarray}\label{eq:partial_J} \partial_\lambda J(t,\lambda,y)&=y^{1-\gamma}g\left(t,\dfrac{\lambda}{y^{\gamma}}\right)+\partial_\lambda J_0(t,\lambda,y). \end{eqnarray} From \eqref{eq:def_region1} and \eqref{eq:def_region2} in Section \ref{sec:B}, \begin{eqnarray*}
\begin{split}
\partial_\lambda J(t,\lambda,y)&> y^{1-\gamma}h\left(t,\dfrac{\lambda}{y^\gamma}\right)+\partial_\lambda J_0\left(t,\lambda,y\right)=U_d(t,y),\;\;\;\mbox{for}\;\;\lambda>y^{\gamma}z^{\star}(t),\\
\partial_\lambda J(t,\lambda,y)&= y^{1-\gamma}h\left(t,\dfrac{\lambda}{y^\gamma}\right)+\partial_\lambda J_0(t,\lambda,y)=U_d(t,y),\;\;\;\mbox{for}\;\;\lambda\le y^{\gamma}z^{\star}(t).\\
\end{split} \end{eqnarray*} Hence, we can rewrite \begin{eqnarray*}
\begin{split}
{\bf JR}&=\{(t,\lambda,y) \mid \partial_{\lambda}J(t,\lambda,y)=U_d(t,y) \},\\
{\bf NR}&=\{(t,\lambda,y) \mid \partial_{\lambda}J(t,\lambda,y)>U_d(t,y) \}.
\end{split} \end{eqnarray*} Now we can state the following corollary. \begin{cor}
~The dual value function $J(t,\lambda,y)$ can be rewritten by
\begin{itemize}
\item[(a)] In the no-jump region {\bf NR},
\begin{eqnarray*}
\begin{split}
J(t,\lambda,y)=&-\dfrac{y^{1-\gamma}}{1-\gamma}\int_{\lambda}^{\infty}\left[\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1\left(s-t,\frac{x}{ z^{\star}(s)y^\gamma}\right)\right)ds\right.\\&\left.-\left(\dfrac{x}{y^\gamma}\right)^{\frac{1}{\gamma}-1}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^\gamma \left(s-t,\frac{x}{z^{\star}(s)y^\gamma}\right)\right)ds\right]dx\\
&+\dfrac{\gamma}{1-\gamma}\cdot \dfrac{1-e^{-K(T-t)}}{K}\lambda^{\frac{1}{\gamma}}+\dfrac{1-e^{-\hat{r}(T-t)}}{\hat{r}}y.
\end{split}
\end{eqnarray*}
\item[(b)] {In} the jump-region {\bf JR},
\begin{eqnarray*}
\begin{split}
J(t,\lambda,y)=J(t,z^{\star}(t)y^{\gamma},y) + (\lambda-z^{\star}(t)y^{\gamma})U_d(t,y).
\end{split}
\end{eqnarray*}
\end{itemize} \end{cor} By applying a standard method of singular control problem developed by \citet{DN} or \citet{FS} to Problem \ref{pr:dual}, the dual value function $J(t,\lambda,y)$ satisfies the certain Hamilton-Jacobi-Bellman(HJB) equation. In fact, \citet{MJ} study the infinite-horizon problem by solving the linear Hamilton-Jacobi-Bellman equation. The following proposition provides that our derived dual value function $J(t,\lambda,y)$ satisfies the associated HJB equation. \begin{pro}\label{pro:dual_HJB}~
The dual value function $J(t,\lambda,y)$ satisfies the following HJB equation:
\begin{eqnarray*}
\begin{split}
\begin{cases}
&\min\left\{\partial_t J+ \mathcal{A}J+y+\tilde{u}(\lambda),\;\partial_\lambda J-U_d(t,y)\right\}=0,\;\;\;(t,\lambda,y)\in[0,T]\times\mathbb{R}_{+}\times\mathbb{R}_+,\\
&J(T,\lambda,y)=0,
\end{cases}
\end{split}
\end{eqnarray*}
where
$$
\mathcal{A}=\frac{\sigma^2}{2}y^2 \partial_{yy}+\mu y\partial_y+(r-\rho)\lambda\partial_{\lambda} -r.
$$ \end{pro} \noindent{\bf Proof.} It is sufficient to $$ \partial_t J+ \mathcal{A}J+y+\tilde{u}(\lambda)=0 \;\;\;\mbox{in}\;\; {\bf NR}, $$ and $$ \partial_t J+\mathcal{A}J+y+\tilde{u}(\lambda)\ge 0 \;\;\;\mbox{in}\;\; {\bf JR}. $$ From the representation of $J(t,\lambda,y)$ in Proposition \ref{pro:dual}, we have \begin{eqnarray}\label{eq:DER_J} \begin{split} \partial_tJ&=y^{1-\gamma}\int_\lambda^\infty -\partial_tg\left(t,\dfrac{x}{y^{\gamma}}\right)dx -\frac{\gamma}{1-\gamma}e^{-K(T-t)}\lambda^{\frac{1}{\gamma}}-e^{\hat{r}(T-t)}y\\ y \partial_y J &= y^{1-\gamma}\int_\lambda^\infty \Big[-(1-\gamma) g\left(t,\dfrac{x}{y^{\gamma}}\right) + \gamma \dfrac{x}{y^{\gamma}}\partial_z g\left(t,\dfrac{x}{y^{\gamma}}\right)\Big]dx\\ y^2 \partial_{yy}J &=y^{1-\gamma}\int_\lambda^\infty \Big[\gamma(1-\gamma)g\left(t,\dfrac{x}{y^{\gamma}}\right)-\left((\gamma^2-\gamma)+ 2\gamma^2 \right)\dfrac{x}{y^{\gamma}}\partial_z g\left(t,\dfrac{x}{y^{\gamma}}\right) \Big.\\ &\Big.-\gamma^2 \left(\dfrac{x}{y^{\gamma}}\right)^2\partial_{zz} g\left(t,\dfrac{x}{y^{\gamma}}\right)\Big]dx\\ \lambda \partial_\lambda J&=y^{1-\gamma}\lambda g\left(t,\frac{\lambda}{y^\gamma}\right)+\frac{1}{1-\gamma}\frac{1-e^{-K(T-t)}}{K}\lambda^{\frac{1}{\gamma}}. \end{split} \end{eqnarray} Since the value function $g(t,z)$ satisfies the variational inequality \eqref{eq:VI1}, \begin{eqnarray}\label{eq:NR_G} -\partial_t g(t,\dfrac{x}{y^{\gamma}})-\mathcal{L}g(t,\dfrac{x}{y^\gamma})=0,~~~\textrm{for}\;\;(x,y)\in{\bf NR}. \end{eqnarray} By using (\ref{eq:DER_J}) and (\ref{eq:NR_G}), $\mathcal{A}J(t,\lambda,y)$ can be derived as follows: \begin{eqnarray} \begin{aligned} &\partial_t J(t,\lambda,y)+\mathcal{A}J(t,\lambda,y)\\ =&y^{1-\gamma}\int_\lambda^\infty \Big[-\partial_t g-\frac{\gamma^2\sigma^2}{2}\left(\dfrac{x}{y^{\gamma}}\right)^2\partial_{zz}g-(\hat{r}-\hat{\rho}+\gamma^2\sigma^2+(\rho-r))\dfrac{x}{y^{\gamma}}\partial_z g+(\hat{\rho}-(\rho-r))g \Big]dx\\ &-y^{1-\gamma}(\rho-r)\lambda g\left(t,\frac{\lambda}{y^\gamma}\right)-\frac{\gamma}{1-\gamma}\lambda^{\frac{1}{\gamma}}-y\\ =&y^{1-\gamma}\int_\lambda^\infty \Big[ -\partial_t g-\mathcal{L}g(t,\dfrac{x}{y^{\gamma}})-(\rho-r)\Big(\dfrac{x}{y^{\gamma}}\partial_z g(t,\dfrac{x}{y^{\gamma}})+g(t,\dfrac{x}{y^{\gamma}}) \Big)\Big]dx\\ &-y^{1-\gamma}(\rho-r)\lambda g\left(t,\frac{\lambda}{y^\gamma}\right)-\frac{\gamma}{1-\gamma}\lambda^{\frac{1}{\gamma}}-y\\ =& -y^{1-\gamma}(\rho-r)\left(\int_\lambda^\infty \left[\frac{x}{y^\gamma}\partial_zg\left(t,\frac{x}{y^\gamma}\right)+g\left(t,\frac{x}{y^{\gamma}}\right)\right]dx+\lambda g\left(t,\frac{\lambda}{y^\gamma}\right)\right)-\frac{\gamma}{1-\gamma}\lambda^{\frac{1}{\gamma}}-y\\ =& -y^{1-\gamma}(\rho-r)\left( \left[xg(t,\dfrac{x}{y^\gamma})\right]_{x=\lambda}^{\infty}+\lambda g\left(t,\frac{\lambda}{y^\gamma}\right)\right)-\frac{\gamma}{1-\gamma}\lambda^{\frac{1}{\gamma}}-y. \end{aligned} \end{eqnarray} By Lemma \ref{lem:integral_Q} and $g(t,z)=Q(t,z)+h(t,z)$, we have \begin{eqnarray*}
\begin{split}
g(t,z)=&\dfrac{z^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(-d^{\gamma}\left(s-t,\frac{z}{z^{\star}(s)}\right)\right)ds\\-&\dfrac{1}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1\left(s-t,\frac{z}{z^{\star}(s)}\right)\right)ds,
\end{split} \end{eqnarray*} Since $\mathcal{N}\left(-d^{\gamma}\left(s-t,\frac{z}{z^{\star}(s)}\right)\right)$ exponentially converge to $0$ {as} $z$ goes to infinity, it is easy to show that $$ \lim_{z\to \infty} zg(t,z)=0. $$ This implies that \begin{eqnarray*}
\begin{split}
\partial_t J(t,\lambda,y)+\mathcal{A}J(t,\lambda,y)=&-y^{1-\gamma}(\rho-r)\left( \left[xg(t,\dfrac{x}{y^\gamma})\right]_{x=\lambda}^{\infty}+\lambda g\left(t,\frac{\lambda}{y^\gamma}\right)\right)-\frac{\gamma}{1-\gamma}\lambda^{\frac{1}{\gamma}}-y\\
=&-\frac{\gamma}{1-\gamma}\lambda^{\frac{1}{\gamma}}-y.
\end{split} \end{eqnarray*} and \begin{eqnarray*}
\partial_t J(t,\lambda,y)+\mathcal{A} J(t,\lambda,y)+y+\tilde{u}(\lambda)=0. \end{eqnarray*} Since \begin{eqnarray}\label{eq:JR_G} -\partial_t g(t,\dfrac{x}{y^{\gamma}})-\mathcal{L}g(t,\dfrac{x}{y^\gamma})\ge 0~~~\textrm{for}\;\;(x,y)\in{\bf JR}, \end{eqnarray} we can similarly show that \begin{eqnarray*}
\partial_t J(t,\lambda,y)+\mathcal{A} J(t,\lambda,y)+y+\tilde{u}(\lambda)\ge 0\;\;\mbox{in}\;\;{\bf JR}. \end{eqnarray*}
$\Box$
We will now state and prove the main theorem of this paper. \begin{thm}~\label{thm:main}
\begin{itemize}
\item[(a)] For given $w$ satisfying Assumption \ref{as:promise_value}, the value function $V(t,w,y)$ of Problem \ref{pr:main} and the dual value function $J(t,\lambda,y)$ derived in Proposition \ref{pro:dual} satisfy the following duality relationship {:}
\begin{eqnarray}
\begin{split}\label{eq:duality}
V(t,w,y)=\min_{\lambda >0}\left(J(t,\lambda,y)-\lambda w\right).
\end{split}
\end{eqnarray}
There exists a unique solution $\lambda^*$ with $(t,\lambda^*,y)\in {\bf NR}$ for the minimization problem \eqref{eq:duality}.
\item[(b)] For $s\in[t,T]$, the optimal costate process $X_s^*$ is given by
\begin{equation}
X_s^*=\max\left(\lambda^*, \sup_{t\le \xi \le s}e^{(\rho-r)(\xi-t)}Y_{\xi}^\gamma z^{\star}(\xi)\right),
\end{equation}
where $\lambda^*$ is the unique solution to the minimization problem \eqref{eq:duality} in (a). Moreover, the optimal costate process $\{X_s^*\}_{s=t}^{T}$ satisfies the integrability condition \eqref{eq:integrability}.
\item[(c)] For $s\in[t,T]$, the optimal consumption plan $c_s^*$ and continuation value $w_t^*$ are, respectively, given by
\begin{eqnarray*}
\begin{split}
C_s^*=& \left(\lambda_s^*\right)^{\frac{1}{\gamma}},\\
w_s^*=& \dfrac{{(\lambda_s^*)}^{1-\gamma}}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1\left(s-t,\dfrac{\lambda_s^*}{y_s^{\gamma}z^{\star}(t)}\right)\right)ds\\+&\dfrac{(\lambda_s^*)^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(d^{\gamma}\left(s-t,\dfrac{\lambda_s^*}{y_s^\gamma z^\star(s)}\right)\right)ds
\end{split}
\end{eqnarray*}
with $\lambda_s^*\equiv e^{-(\rho-r)(s-t)}X_s^*$.
\end{itemize} \end{thm} \noindent{\bf Proof.} We will prove the duality relationship in the theorem in the following steps.\\
\noindent {\bf (Step 1)} First, we will show that the dual value function $J(t,\lambda,y)$ is strictly convex in $\lambda$ in {\bf NR}.\\
\noindent{\bf Proof of {\bf (Step 1)}} Consider $\lambda^1,\lambda^2>0\;(\lambda^1\neq \lambda^2)$ with $$ (t,\lambda^1,y),\;(t,\lambda^2,y)\in{\bf NR}. $$ Let $\lambda^3=\alpha \lambda^1 + (1-\alpha)\lambda^2$ with $\alpha\in(0,1)$. Then, clearly $$ (t,\lambda^3,y)\in{\bf NR}. $$ Also, let $X^{*,j}$ be the optimal process for the minimization problem {\eqref{eq:dual_value}} with $\{X_s^{*,j}\}_{s=t}^{T}\in\mathcal{ND}(\lambda^{j})$, $j=1,2,3$. In other words, for $j=1,2,3$, \begin{eqnarray*}
\begin{split}
J(t,\lambda^{j},y)=& \inf_{X \in \mathcal{ND}(\lambda^j)} \left\{\mathbb{E}\left[\int_t^T e^{-r(s-t)} \Big( \tilde{u}(e^{-(\rho-r)(s-t)}X_s) +Y_s - e^{-(\rho-r)(s-t)}X_s u(Y_s)\Big)ds \right] \right. \\
+&\left. \lambda^j \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right] \right\}\\
=&\mathbb{E}\left[\int_t^T e^{-r(s-t)} \Big( \tilde{u}(e^{-(\rho-r)(s-t)}X_s^{*,j}) +Y_s - e^{-(\rho-r)(s-t)}X_s^{*,j} u(Y_s)\Big)ds \right] \\
+&\lambda^j \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right].
\end{split} \end{eqnarray*} Then, \begin{eqnarray*}
\begin{split}
&\alpha J(t,\lambda^{1},y)+(1-\alpha)J(t,\lambda^2,y)\\
=&\mathbb{E}\left[\int_t^T e^{-r(s-t)} \Big( \alpha\tilde{u}(e^{-(\rho-r)(s-t)}X_s^{*,1}) +(1-\alpha) \tilde{u}(e^{-(\rho-r)(s-t)}X_s^{*,2})\right.\\+&\left.Y_s - e^{-(\rho-r)(s-t)}(\alpha X_s^{*,1}+(1-\alpha)X_s^{*,2}) u(Y_s)\Big)ds \right]
+\lambda^3 \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right]
\end{split} \end{eqnarray*} By (b) in Proposition \ref{pro:dual}, the optimal processes $X^{*,j}$($j=1,2,3$) are given by \begin{equation*} X_s^{*,j}=\max\left(\lambda^{j}, \sup_{t\le \xi \le s}e^{(\rho-r)(s-t)}Y_{\xi}^\gamma z^{\star}(\xi)\right). \end{equation*} Since $\lambda_1\neq\lambda_2$ and $X_t^{*,j}=\lambda^{j}$, we can deduce that $$ X_{s}^{*,1}\neq X_{s}^{*,2}\;\mbox{a.s.} $$ Thus, by strict convexity of $\tilde{u}(\cdot)$, $$ \alpha\tilde{u}(e^{-(\rho-r)(s-t)}X_s^{*,1}) +(1-\alpha) \tilde{u}(e^{-(\rho-r)(s-t)}X_s^{*,2})>\tilde{u}(e^{-(\rho-r)(s-t)}(\alpha X_s^{*,1}+(1-\alpha)X_s^{*,2})). $$ Let us temporarily denote $\bar{X}^{*}=(\alpha X_s^{*,1}+(1-\alpha)X_s^{*,2})$. Then, \begin{eqnarray*}
\begin{split}
&\alpha J(t,\lambda^{1},y)+(1-\alpha)J(t,\lambda^2,y)\\
>&\mathbb{E}\left[\int_t^T e^{-r(s-t)} \Big( \tilde{u}(e^{-(\rho-r)(s-t)}\bar{X}_s^{*}) +Y_s - e^{-(\rho-r)(s-t)}\bar{X}_s^{*}u(Y_s)\Big)ds \right] \\
+&\lambda^3 \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right]\\
\ge& \inf_{X \in \mathcal{ND}(\lambda^3)} \left\{\mathbb{E}\left[\int_t^T e^{-r(s-t)} \Big( \tilde{u}(e^{-(\rho-r)(s-t)}X_s) +Y_s - e^{-(\rho-r)(s-t)}X_s u(Y_s)\Big)ds \right] \right. \\
+&\left. \lambda^3 \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right] \right\}\\
=&\mathbb{E}\left[\int_t^T e^{-r(s-t)} \Big( \tilde{u}(e^{-(\rho-r)(s-t)}X_s^{*,3}) +Y_s - e^{-(\rho-r)(s-t)}X_s^{*,3} u(Y_s)\Big)ds \right] \\
+&\lambda^3 \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right]=J(t,\lambda^3,y).
\end{split} \end{eqnarray*} This implies that $J(t,\lambda,y)$ is strictly convex in $\lambda$ in ${\bf NR}$.
$\Box$
Now, we rewrite the Lagrangian {\bf L} in \eqref{eq:LGR5} as \begin{eqnarray}\label{eq:LGR6} \begin{aligned} {\bf L}(t,\lambda,y,X) =& \mathbb{E}_t \left[\int_t^T e^{-r(s-t))} \Big( \tilde{u}(e^{-(\rho-r)(s-t)}X_s) +Y_s - e^{-(\rho-r)(s-t)}X_s u(Y_s)\Big)ds\right] \\ +&\lambda \mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right]-\lambda w. \\ \end{aligned} \end{eqnarray} with $X_t=\lambda, Y_t=y$. For simplicity, let ${\bf L}(t,\lambda,y,X)={\bf L}(\lambda,X)$. \\
\noindent {\bf (Step 2)} For every enforceable plan $C\in\Gamma(t,y,w)$, every $x>0$, and every $X\in\mathcal{ND}(\lambda)$, the following inequality is established: \begin{eqnarray*}
\begin{split}
L(\lambda,X)\ge U_t^P(y,C)=\mathbb{E}_t\left[\int_{t}^T e^{-r(s-t)}(Y_s-C_s)ds\right]
\end{split} \end{eqnarray*} The equality holds if and only if for all $s\in[t,T]$, $$ X_s e^{-(\rho-r)(s-t)}u'(C_s)-1=0,\;\;\int_{s}^{T}e^{-\rho(\xi-s)}(U_\xi^a(C)-U_d(\xi,Y_\xi))dX_{\xi}=0. $$ This leads to the following {\it weakly duality} relationship: \begin{eqnarray} V(t,w,y)\le \inf_{\lambda>0} \left( J(t,\lambda,y)-\lambda w \right). \end{eqnarray} \noindent{\bf Proof of {\bf (Step 2)}} By the definition of $\tilde{u}(\cdot)$, $$ \tilde{u}(e^{-(\rho-r)(s-t)}X_s) \ge X_s e^{-(\rho-r)(s-t)}u(C_s)-C_s. $$ This leads to \begin{eqnarray*}
\begin{split}
&{\bf L}(\lambda,X)\\\ge&\mathbb{E}_t \left[\int_t^T e^{-r(s-t)} \left(Y_s-C_s\right)ds\right] +\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}\left(X_s e^{-(\rho-r)(s-t)}(u(C_s)-u(Y_s)\right)ds\right]\\
+&\lambda \left(\mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(Y_s)ds \right]-w\right)\\
=&\mathbb{E}_t\left[\int_{t}^{T} e^{-r(s-t)}(Y_s-C_s)ds\right]+\mathbb{E}_t\left[\int_{t}^{T}e^{-\rho(s-t)}\int_{s}^{T}e^{-\rho(\xi-s)}\left(u(C_{\xi})-u(Y_{\xi})\right)d\xi \cdot dX_s\right]\\
+&\lambda \left(\mathbb{E}_t \left[\int_t^T e^{-\rho(s-t)}u(C_s)ds \right]-w\right)\\
\ge&\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(Y_s-C_s)ds\right],
\end{split} \end{eqnarray*} where the middle equation follows from integration by parts, and the last inequality follows from the fact $C$ is enforceable and $X$ is in $\mathcal{ND}(\lambda)$.
Clearly, the equality holds if and only if $$ X_s e^{-(\rho-r)(s-t)}u'(C_s)-1=0,\;\;\int_{s}^{T}e^{-\rho(\xi-s)}(U_\xi^a(C)-U_d(\xi,Y_\xi))dX_{\xi}=0. $$ Moreover, we can immediately obtain \begin{eqnarray*}
V(t,w,y)\le \inf_{\lambda>0} \left( J(t,\lambda,y)-\lambda w \right). \end{eqnarray*}
$\Box$
\noindent {\bf (Step 3)} The consumption plan $C^{*}$ defined in \eqref{eq:optimal_C} is enforceable and optimal.\\
\noindent{\bf Proof of {\bf (Step 3)}} Let us assume that $(X_s^*)_{s=t}^{T}\in\mathcal{ND}(\lambda^*)$ minimize the Lagrangian {\bf L} in \eqref{eq:minimize_L}. Under this assumption we first prove that $(c_s^*)_{s=t}^{T}$ is enforceable and optimal. Then we will show the existence and uniqueness of $\lambda^*$ and derive $(X_s^*)_{s=t}^{T}$. Finally, we derive the optimal continuation process.
Since $(X_s^*)_{s=t}^{T}\in\mathcal{ND}(\lambda^*)$, we know that $$
\mathbb{E}_t\left[\int_{t}^{T}e^{-r(s-t)}|\tilde{u}(X_s^* e^{-(\rho-r)(s-t)})|\right]<\infty. $$ From this, it is easy to check that $$ \mathbb{E}_t\left[\int_{t}^T e^{-r(s-t)}C_s^*ds\right]<+\infty. $$ For sufficiently small $\delta>0$ and $h\in(0,\delta)$, we consider $$ \lambda^h \equiv \lambda^* + h,~~~~X_s^h \equiv X_s^* + h,~~~ \textrm{and}~~~X_s^\delta \equiv X_s^* + \delta $$ Then \begin{eqnarray}\label{eq:enforceable1} \begin{split}
&\mathbb{E}_t\left[\int_{t}^{T}e^{-r(s-t)}|\tilde{u}((X_s^*+\delta) e^{-(\rho-r)(s-t)})|ds\right]\\
&<\left|\dfrac{\gamma}{1-\gamma}\right|\mathbb{E}_t\left[\int_{t}^{T}e^{-r(s-t)}\left(e^{-(\rho-r)(s-t)}(X_s^*(1+\delta/X_t^*))\right)^{\frac{1}{\gamma}}ds\right]\\
&<(1+\delta/\lambda)^{\frac{1}{\gamma}}\mathbb{E}_t\left[\int_{t}^{T}e^{-r(s-t)}|\tilde{u}(X_s^* e^{-(\rho-r)(s-t)})|\right]<\infty. \end{split} \end{eqnarray} Similarly, \begin{eqnarray}\label{eq:enforceable15} \begin{split}
\mathbb{E}_t\left[\int_{t}^{T}e^{-r(s-t)}|\tilde{u}((X_s^*(1\pm\delta))e^{-(\rho-r)(s-t)})|ds\right]<\infty. \end{split} \end{eqnarray} Since $$
\mathbb{E}_t\left[\int_{t}^{T}e^{-\rho(s-t)}|U_d(Y_s)|X_s^* ds\right]<\infty, $$ clearly, we can have $$
\mathbb{E}_t\left[\int_{t}^{T}e^{-\rho(s-t)}|U_d(Y_s)|(X_s+\delta) ds\right]<\infty. $$ {The convexity of $\tilde{u}(\cdot)$} implies \begin{eqnarray} \begin{split}\label{eq:enforceable2} e^{-(\rho-r)(s-t)}u(C_s^*)\le& \dfrac{\tilde{u}(X_s^h e^{-(\rho-r)(s-t)})-\tilde{u}(X_s^* e^{-(\rho-r)(s-t)})}{h}\\\le& \dfrac{\tilde{u}(X_s^\delta e^{-(\rho-r)(s-t)})-\tilde{u}(X_s^* e^{-(\rho-r)(s-t)})}{\delta} \end{split} \end{eqnarray} Thus, \begin{eqnarray}\label{eq:enforceable3} \begin{split}
&\mathbb{E}\left[\int_t^{T}e^{-\rho(s-t)}|u(C_s^*)|ds\right]\\
\le&\mathbb{E}\left[\int_t^{T}e^{-r(s-t)}|u(C_s^*)|\dfrac{e^{-(\rho-r)(s-t)}X_s^*}{\lambda^*}ds\right]\\
\le &\dfrac{1}{\lambda^*}\mathbb{E}\left[\int_t^{T}e^{-r(s-t)}\left(|\tilde{u}(X_s^*e^{-(\rho-r)(s-t)})|+C_s^*\right)ds\right]<\infty. \end{split} \end{eqnarray} \eqref{eq:enforceable1},\eqref{eq:enforceable2}, and \eqref{eq:enforceable3} imply $X^h \in \mathcal{ND}(\lambda^* + h)$.
Since $(X_s^*)_{s=t}^{T}\in\mathcal{ND}(\lambda^*)$ minimizes the Lagrangian {\bf L}, $$ {\bf L}(\lambda^h, X^h) \ge {\bf L}(\lambda^*, X^*). $$ Hence, $$ \lim_{h \downarrow 0} \dfrac{{\bf L}(\lambda^h,X^h)-{\bf L}(\lambda^*,X^*)}{h}\ge 0. $$ or, equivalently, $$ \lim_{h \downarrow 0}\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}\dfrac{\tilde{u}(X_s^h e^{-(\rho-r)(s-t)})-\tilde{u}(X_s^* e^{-(\rho-r)(s-t)})}{h}ds\right] -w \ge 0. $$ The Dominated Convergence Theorem implies \begin{eqnarray*}
\begin{split}
\lim_{h \downarrow 0}\;&\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}\dfrac{\tilde{u}(X_s^h e^{-(\rho-r)(s-t)})-\tilde{u}(X_s^* e^{-(\rho-r)(s-t)})}{h}ds\right] -w\\
=&\mathbb{E}_t\left[\lim_{h \downarrow 0}\int_t^T e^{-r(s-t)}\dfrac{\tilde{u}(X_s^h e^{-(\rho-r)(s-t)})-\tilde{u}(X_s^* e^{-(\rho-r)(s-t)})}{h}ds\right] -w \\
=&\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}u(C_s^*)ds\right]-w \ge 0.
\end{split} \end{eqnarray*} Thus, $C^*$ satisfies the promise-keeping constraints.
Similar to \citet{MJ}, define $X^h(w,\xi)\equiv X^*(t,\xi)+h{\bf 1}_{A\times(s,T]}(w,\xi)$ for $h\in(0,\delta)$, $s\in[t,T]$ and $A\in\mathcal{F}_s$. Note $X_t^h=X_t^*=\lambda^*$.
By a similar argument, we can obtain $$ X^{h}\in\mathcal{ND}(\lambda^*). $$ Since $$ \lim_{h \downarrow 0}\dfrac{{\bf L}(\lambda^*,X^h)-{\bf L}(\lambda^*,X^*)}{h}\ge 0. $$ This leads to $$ \mathbb{E}_t\left[{\bf 1}_{A}\int_{s}^{T}e^{-\rho(\xi-t)}U(C_\xi^*)d\xi\right]\ge \mathbb{E}_t\left[{\bf 1}_{A}e^{-\rho(s-t)}U_d(s,Y_s)\right]. $$ Since $A$ was an arbitrary set in $\mathcal{F}_s$, we deduce that \begin{eqnarray*}
\begin{split}
U_s(C)=\mathbb{E}_s\left[\int_{s}^{T}e^{-\rho(\xi-s)}U(C_\xi^*)d\xi\right]\ge U_d(s,Y_s),
\end{split} \end{eqnarray*} for any $s\in[t,T]$.
Therefore, the consumption plan $C^*$ is enforceable. Now we will show that $C^*$ is optimal.
For $h\in(0,\delta)$, let us consider $$ X^{\pm h}\equiv X^*(1\pm h). $$ By the following convexity of $\tilde{u}(\cdot)$, \begin{eqnarray*}
\begin{split}
\dfrac{\tilde{u}(X_s^{-\delta}e^{-(\rho-r)(s-t)})-\tilde{u}(X_s^{*}e^{-(\rho-r)(s-t)})}{-\delta}
\le&\dfrac{\tilde{u}(X_s^{\pm h}e^{-(\rho-r)(s-t)})-\tilde{u}(X_s^{*}e^{-(\rho-r)(s-t)})}{\pm h}\\
\le&\dfrac{\tilde{u}(X_s^{\delta}e^{-(\rho-r)(s-t)})-\tilde{u}(X_s^{*}e^{-(\rho-r)(s-t)})}{\delta}
\end{split} \end{eqnarray*} and the condition \eqref{eq:enforceable15}, we can deduce $X^{\pm h}\in\mathcal{ND}(\lambda^*(1+h))$.
Since ${\bf L}(\lambda^*(1\pm h),X^{\pm h})\ge {\bf L}(\lambda^*,X^{*})$, $$ \lim_{h\downarrow 0}\dfrac{{\bf L}(\lambda^*(1+h),X^{h})-{\bf L}(\lambda^*,X^{*})}{h}\ge 0,\;\;\lim_{h\uparrow 0}\dfrac{{\bf L}(\lambda^*(1-h),X^{-h})-{\bf L}(\lambda^*,X^{*})}{-h}\ge 0 $$ By the Dominated Convergence theorem and integration by parts, $$ \lambda^*\left(U_t(C^*)-w\right)+\mathbb{E}_t\left[\int_{t}^{T}e^{-\rho(s-t)}\int_{s}^{T}e^{-\rho(\xi-s)}\left(u(C_{\xi}^*)-u(Y_{\xi})\right)d\xi \cdot dX_s\right]=0. $$ Thus, the promise keeping constraint and the participation constraint must hold with equality for the consumption plan $C^*$.
Since $$ U_t^P(y,C^*)\le\sup_{C \in \Gamma(t,y,w)}U_t^P(t,y,C)\le \inf_{X \in \mathcal{ND}(\lambda)}{\bf L}(t,\lambda,y,X)-\lambda w \le{\bf L}(t,\lambda^*,X^*)-\lambda^* w, $$ we conclude that the consumption plan $C^*$ is optimal and the following duality relationship holds: $$ V(t,w,y)=\inf_{\lambda>0}\left\{J(t,\lambda,y)-\lambda w\right\}. $$
$\Box$
\noindent {\bf (Step 4)} Determination of $\lambda^*$ in the duality relationship \eqref{eq:duality} and the optimal process $(X_s^*)_{s=t}^{T}$.\\
By Lemma \ref{lem:integral_Q} and \eqref{eq:partial_J}, \begin{eqnarray*}
\begin{split}
\partial_{\lambda}J(t,\lambda,y)=&y^{1-\gamma}g(t,\dfrac{\lambda}{y^{\gamma}})+\partial_{\lambda}J_0(t,\lambda,y)\\
=&\dfrac{y^{1-\gamma}}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1(s-t,\dfrac{\lambda}{y^{\gamma}z^{\star}(s)})\right)ds\\~+&\dfrac{\lambda^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(d^{\gamma}(s-t,\dfrac{\lambda}{y^\gamma z^\star(s)})\right)ds.\\
\end{split} \end{eqnarray*} and we deduce that \begin{eqnarray*}
\begin{split}
\lim_{\lambda \to \infty}\partial_{\lambda}J(t,\lambda,y)=\begin{cases}
\infty\qquad&\mbox{if}\;\;0<\gamma<1,\\
0\qquad&\mbox{if}\;\;1<\gamma.
\end{cases}
\end{split} \end{eqnarray*} Moreover, by Lemma \ref{lem-cinfinity} or the value matching condition for $g(t,z)$, $$ \partial_{\lambda}J(t,z^{\star}(t)y^\gamma,y)=U_d(t,y). $$ By {\bf (Step 1)}, $J(t,\lambda,y)$ is strictly convex in $\lambda$ in {\bf NR}, i.e., $$ \partial_{\lambda\lambda}J(t,\lambda,y)>0 \;\;\mbox{in}\;\;{\bf NR}. $$ This implies that $\partial_{\lambda}J(t,\lambda,y)$ is strictly increasing in $\lambda$ in {\bf NR}.
Thus, for given $w$ satisfying Assumption \ref{as:promise_value}, there exists a unique $\lambda^*$ such that $$ w=\partial_{\lambda}J(t, {\lambda^*},y) $$ and $(t,\lambda^*,y)\in{\bf NR}$.
Now, we will determine the optimal shadow process $(X_s^*)_{s=t}^T\in\mathcal{ND}(\lambda^*)$.
{Since the optimal stopping time $\tau$ in Problem \ref{pr:OS} is given by
\begin{eqnarray*}
\tau = \inf\{s\mid 1\le {e^{(\rho-r)(s-t)}}z^\star(s){Y_s^\gamma}\},
\end{eqnarray*}
the optimal stopping time $\tau(x)$ in Lemma \ref{thm:lem1} can be written by
\begin{eqnarray*}
\tau(x) = \inf\{s\mid x\le {e^{(\rho-r)(s-t)}}z^\star(s){Y_s^\gamma}\}.
\end{eqnarray*} } Since $$ \{X_s^*<x\}=\{s<\tau(x)\}=\left\{\max_{t\le \theta \le s}e^{(\rho-r)(\theta-t)}z^{\star}(\theta)Y_\theta^\gamma <x\right\}, $$ the optimal process $(X_s^*)_{s=t}^T$ can be expressed explicitly as \begin{eqnarray*}
X_s^*=\max\left(\lambda^*, \max_{t\le \theta \le s}e^{(\rho-r)(\theta-t)}z^{\star}(\theta)Y_\theta^\gamma \right). \end{eqnarray*}
By defining $\lambda_s^*=e^{-(\rho-r)(s-t)}X_s^*$, since Problem \ref{pro:dual} is {\it time consistent}, we can deduce that dual value function $J(s,\lambda_s^*,Y_s)$ at time $s\in[t,T]$ is given by \begin{eqnarray*}
\begin{split}
J(s,\lambda_s^*,Y_s)=&-\dfrac{Y_s^{1-\gamma}}{1-\gamma}\int_{\lambda_s^*}^{\infty}\left[\int_{s}^{T}e^{-\hat{\rho}(\xi-s)}\mathcal{N}\left(-d^1(\xi-s,\frac{x}{ z^{\star}(\xi)Y_s^\gamma})\right)d\xi\right.\\&\left.-\left(\dfrac{x}{Y_s^\gamma}\right)^{\frac{1}{\gamma}-1}\int_{s}^{T}e^{-\hat{\rho}(\xi-s)}\mathcal{N}\left(-d^1(\xi-s,\frac{x}{z^{\star}(\xi)Y_s^\gamma})\right)d\xi\right]dx\\
&+\dfrac{\gamma}{1-\gamma}\left(\dfrac{1-e^{-K(T-s)}}{K}(\lambda_s^*)^{\frac{1}{\gamma}}+\dfrac{1-e^{-\hat{r}(T-s)}}{\hat{r}}Y_s\right).
\end{split} \end{eqnarray*} and satisfies the duality-relationship: \begin{eqnarray*}
{V(s,Y_s,w_s)=\inf_{\lambda>0}\left\{J(s,\lambda,Y_s) - w_s \lambda\right\}=J(s,\lambda_s^*,Y_s) - w_s \lambda_s^*.} \end{eqnarray*} By the first-order condition, we obtain \begin{eqnarray*}
\begin{split}
w_s =&\partial_{\lambda}J(s,\lambda_s^*,Y_s)\\
=&\dfrac{Y_s^{1-\gamma}}{1-\gamma}\int_{t}^{T}e^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1(s-t,\dfrac{\lambda_s^*}{Y_s^{\gamma}z^{\star}(s)})\right)ds\\+&\dfrac{(\lambda_s^*)^{\frac{1}{\gamma}-1}}{1-\gamma}\int_{t}^{T}e^{-K(s-t)}\mathcal{N}\left(d^{\gamma}(s-t,\dfrac{\lambda_s^*}{Y_s^\gamma z^\star(s)})\right)ds.
\end{split} \end{eqnarray*} \noindent {\bf (Step 5)} The optimal process $(X_s^*)_{s=t}^T$ satisfies the integrability condition \eqref{eq:integrability}.\\
\noindent{\bf Proof of {\bf (Step 5)}} It is enough to show that $$ \mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(e^{-(\rho-r)(s-t)}X_s^*)^{\frac{1}{\gamma}} ds\right]<\infty\;\;\;\mbox{and}\;\;\;\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}Y_s^{1-\gamma}X_s^*ds\right]<\infty $$ First, we will utilize the idea in Lemma \ref{thm:lem1}. \begin{eqnarray*}
\begin{split}
&\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(e^{-(\rho-r)(s-t)}X_s^*)^{\frac{1}{\gamma}} ds\right]\\
=&\mathbb{E}_t\left[\int_t^T e^{-K(s-t)}(X_s^*)^\frac{1}{\gamma}ds\right]\\
=&\mathbb{E}_t\left[\int_t^T e^{-K(s-t)}\left(\int_{X_t^*}^{X_s^*}\frac{1}{\gamma}x^{\frac{1}{\gamma}-1}dx + X_t^*\right)ds\right]\\
=&\dfrac{1-e^{-K(T-t)}}{K}\lambda^* + \dfrac{1}{\gamma}\mathbb{E}_t\left[\int_t^T e^{-K(s-t)}\left(\int_{\lambda^*}^{\infty}x^{\frac{1}{\gamma}-1}{\bf 1}_{\{X_s^*\ge x\}}dx\right)ds\right]\\
=&\dfrac{1-e^{-K(T-t)}}{K}\lambda^* +\dfrac{1}{\gamma}\int_{\lambda^*}^{\infty}\int_t^T e^{-K(s-t)}x^{\frac{1}{\gamma}-1}\mathbb{E}_t\left[{\bf 1}_{\{s\ge \tau(x)^*\}}\right]ds dx,
\end{split} \end{eqnarray*} where the last equality is obtained from Fubini's theorem.
By Proposition \ref{pro:dual}, we can obtain $$ \mathbb{P}(s\ge \tau(x)^*)=\mathbb{P}(x\mathcal{H}_s \le z^\star(s)). $$ Since $$ d\mathcal{H}_s = (\hat{r}-\hat{\rho}+\gamma\sigma^2)\mathcal{H}_s ds -\gamma\sigma\mathcal{H}_s dB_s $$ under the measure $\mathbb{P}$ with $\mathcal{H}_t= 1/y^\gamma$, we can easily obtain that $$ \mathbb{P}(s\ge \tau(x)^*)=\mathcal{N}\left(-d^\gamma(s-t,\dfrac{x}{z^\star(s)y^\gamma})\right). $$ For $\widetilde{z}^\star(s)=z^\star(T-s)$, $\xi=s-t$ and $\tau=T-t$, \begin{eqnarray}\label{eq:integral1} \begin{split} &\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(e^{-(\rho-r)(s-t)}X_s^*)^{\frac{1}{\gamma}} ds\right]\\ =&\dfrac{1-e^{-K(T-t)}}{K}\lambda^* + \dfrac{1}{\gamma}\int_{\lambda^*}^{\infty}\int_0^\tau e^{-K\xi}x^{\frac{1}{\gamma}-1}\mathcal{N}\left(-d^\gamma(\xi,\dfrac{x}{\widetilde{z}^\star(\tau-\xi)y^\gamma})\right)d\xi dx\\ < &\dfrac{\lambda^*}{K} + \dfrac{1}{\gamma}\int_{\lambda^*}^{\infty}\lim_{\tau \to \infty}\left[\int_0^\tau e^{-K\xi}x^{\frac{1}{\gamma}-1}\mathcal{N}\left(-d^\gamma(\xi,\dfrac{x}{\widetilde{z}^\star(\tau-\xi)y^\gamma})\right)d\xi \right]dx\\ =&\dfrac{\lambda^*}{K} + \dfrac{1}{\gamma}\int_{\lambda^*}^{\infty}x^{\frac{1}{\gamma}-1}\int_0^\infty e^{-K\xi}\mathcal{N}\left(-d^\gamma(\xi,\dfrac{x}{z_\infty y^\gamma})\right)d\xi dx. \end{split} \end{eqnarray} By using the result in Proof of Lemma \ref{lem:inf}, we can derive \begin{eqnarray} \begin{split}\label{eq:integral2} \int_0^\infty e^{-K\xi}\mathcal{N}\left(-d^\gamma(\xi,\dfrac{x}{z_\infty y^\gamma})\right)d\xi=\frac{1}{K}\left(\frac{\alpha_++\left(\frac{\gamma-1}{\gamma}\right)}{\alpha_+-\alpha_-}\right)\cdot \left(\frac{x}{z_\infty y^\gamma}\right)^{\alpha_-+\left(\frac{\gamma-1}{\gamma}\right)} \end{split} \end{eqnarray} By \eqref{eq:integral1} and \eqref{eq:integral2}, we deduce \begin{eqnarray*}
\begin{split}
&\mathbb{E}_t\left[\int_t^T e^{-r(s-t)}(e^{-(\rho-r)(s-t)}X_s^*)^{\frac{1}{\gamma}} ds\right]\\
<&\dfrac{\lambda^*}{K}+\dfrac{1}{\gamma}\dfrac{1}{K}\left(\frac{\alpha_++\left(\frac{\gamma-1}{\gamma}\right)}{\alpha_+-\alpha_-}\right)(z_\infty y^\gamma)^{\frac{1}{\gamma}-1-\alpha_-}\int_{\lambda^*}^{\infty}x^{\alpha_-}dx<\infty.
\end{split} \end{eqnarray*} Similarly, \begin{eqnarray*}
\begin{split}
&\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}Y_s^{1-\gamma}X_s^*ds\right]\\
=&\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}Y_s^{1-\gamma}\left(\int_{X_t^*}^{X_s^*}1dx + X_t^*\right)\right]\\
=&\dfrac{1-e^{-\hat{\rho}(T-t)}}{\hat{\rho}}y^{1-\gamma}\lambda^* + \mathbb{E}_t \left[\int_t^T e^{-{\rho}(s-t)}Y_s^{1-\gamma}\left(\int_{X_t^*}^{X_s^*}1dx\right) ds\right]\\
=&\dfrac{1-e^{-\hat{\rho}(T-t)}}{\hat{\rho}}y^{1-\gamma}\lambda^*+y^{1-\gamma}\mathbb{E}^{\mathbb{Q}}\left[\int_t^T e^{-\hat{\rho}(s-t)}\left(\int_{\lambda^*}^{\infty}{\bf 1}_{\{X_s^* \ge x \}}\right)ds\right]\\
=&\dfrac{1-e^{-\hat{\rho}(T-t)}}{\hat{\rho}}y^{1-\gamma}\lambda^* + y^{1-\gamma}\int_{\lambda^*}^{\infty}\int_t^Te^{-\hat{\rho}(s-t)}\mathbb{E}^{\mathbb{Q}}\left[{\bf 1}_{\{s\ge \tau(x)^* \}}\right]ds dx,
\end{split} \end{eqnarray*} where the measure $\mathbb{Q}$ is defined in Lemma \ref{thm:lem1}.
Since $$ d\mathcal{H}_s = (\hat{r}-\hat{\rho}+\gamma^2\sigma^2)\mathcal{H}_s ds -\gamma\sigma\mathcal{H}_s dB_s^{\mathbb{Q}}, $$ we have $$ \mathbb{Q}(s\ge \tau(x)^*)=\mathbb{Q}(x\mathcal{H}_s \le z^{\star}(t))=\mathcal{N}\left(-d^1(s-t,\dfrac{x}{z^\star(s)y^\gamma})\right). $$ By using the result in Proof of Lemma \ref{lem:inf}, we deduce that \begin{eqnarray*}
\begin{split}
&\mathbb{E}_t\left[\int_t^T e^{-\rho(s-t)}Y_s^{1-\gamma}X_s^*ds\right]\\
=&\dfrac{1-e^{-\hat{\rho}(T-t)}}{\hat{\rho}}y^{1-\gamma}\lambda^* + y^{1-\gamma}\int_{\lambda^*}^{\infty}\int_t^Te^{-\hat{\rho}(s-t)}\mathcal{N}\left(-d^1(s-t,\dfrac{x}{z^\star(s)y^\gamma})\right)ds dx\\
<&\dfrac{y^{1-\gamma}\lambda^*}{\hat{\rho}}+y^{1-\gamma}\int_{\lambda^*}^{\infty}\int_0^{\infty}e^{-\hat{\rho}\xi}\mathcal{N}\left(-d^1(\xi,\dfrac{x}{z_\infty y^\gamma})\right)d\xi dx\\
=&\dfrac{y^{1-\gamma}\lambda^*}{\hat{\rho}}+\dfrac{1}{\hat{\rho}}\left(\dfrac{\alpha_+}{\alpha_+ - \alpha_-}\right)(z_\infty)^{-\alpha_-}y^{1-\gamma-\alpha_-\gamma}\int_{\lambda^*}^\infty x^{\alpha_-}dx <\infty.
\end{split} \end{eqnarray*} Thus, we can conclude that the optimal process $(X_s^*)_{s=t}^T$ satisfies the integrability condition \eqref{eq:integrability}.
$\Box$
From {\bf (Step 1)}--{\bf (Step 5)}, we have just proved Theorem \ref{thm:main}.
$\Box$
We will now show the convergence of the optimal strategies for a finite horizon problem to those for the infinite horizon problem. \begin{thm}\label{thm:infinite}~
For $t\ge 0$, as time to maturity goes to infinity, i.e., $T-t\to \infty$, the agent's optimal consumption $C_{t,\infty}^*$ and the optimal promised value $w_{t,\infty}^*$ at time $t$ are given by
\begin{eqnarray*}
\begin{split}
C_{t,\infty}^*=&(\lambda_{t,\infty}^*)^{\frac{1}{\gamma}},\\
w_{t,\infty}^*=&-\dfrac{1}{\gamma}\dfrac{1}{K}\dfrac{1}{\alpha_-}(z_\infty Y_t^\gamma)^{\frac{1}{\gamma}-1-\alpha_-}(\lambda_{t,\infty}^*)^{\alpha_-} + \dfrac{1}{1-\gamma}\dfrac{1}{K}(\lambda_{t,\infty}^*)^{\frac{1}{\gamma}-1},
\end{split}
\end{eqnarray*}
where $\alpha_-$ and $z_\infty$ are defined in \eqref{eq:f1} and \eqref{eq:INFZ} in Section \ref{sec:B}, respectively, $\lambda_{t,\infty}^*=e^{-(\rho-r)t}X_{t,\infty}^*$, and
$$
X_{t,\infty}^*=\max\left(\lambda_{\infty}^*, z_\infty\sup_{0\le \xi \le t}e^{(\rho-r)\xi}Y_{\xi}^\gamma \right).
$$
Moreover, $\lambda_{\infty}^*$ is a unique solution to the following algebraic equation:
\begin{eqnarray*}
\begin{split}
w=&-\dfrac{1}{\gamma}\dfrac{1}{K}\dfrac{1}{\alpha_-}(z_\infty y^\gamma)^{\frac{1}{\gamma}-1-\alpha_-}(\lambda_{\infty}^*)^{\alpha_-} + \dfrac{1}{1-\gamma}\dfrac{1}{K}(\lambda_{\infty}^*)^{\frac{1}{\gamma}-1}.
\end{split}
\end{eqnarray*} \end{thm} \noindent{\bf Proof.} By Lemma \ref{lem:inf}, $$ \lim_{T-t \to \infty} Q(t,z)=Q_\infty (z)\;\;\;\mbox{and}\;\;\;\lim_{T-t\to \infty}z^\star(t)=z_\infty, $$ where $Q_\infty(z)$ and $z_\infty$ are defined in \eqref{eq:INFQ} and \eqref{eq:INFZ}, respectively.
Thus, we have \begin{eqnarray*}
\begin{split}
g_\infty(z)\equiv \lim_{T-t \rightarrow \infty} g(t,z)=Q_\infty(z)+\dfrac{1}{1-\gamma}\left(\dfrac{1}{\hat{\rho}}-\dfrac{1}{K}z^{\frac{1}{\gamma}-1}\right),
\end{split} \end{eqnarray*} and \begin{eqnarray*}
\begin{split}
J_\infty(y,w)=\lim_{T-t \rightarrow \infty}J(t,y,w)=-y^{1-\gamma}\int_{\lambda}^{\infty}g_\infty\left(\dfrac{x}{y^\gamma}\right)dx +\dfrac{\gamma}{1-\gamma}\dfrac{1}{K}\lambda^{\frac{1}{\gamma}} +\dfrac{1}{\hat{r}}y.
\end{split} \end{eqnarray*} From Theorem \ref{thm:main}, we can derive \begin{eqnarray*}
\begin{split}
C_{t,\infty}^*=&(\lambda_{t,\infty}^*)^{\frac{1}{\gamma}}\\
w_{t,\infty}^*=&\partial_\lambda J_\infty(\lambda_{t,\infty}^*Y_t^\gamma)\\
=&-\dfrac{1}{\gamma}\dfrac{1}{K}\dfrac{1}{\alpha_-}(z_\infty Y_t^\gamma)^{\frac{1}{\gamma}-1-\alpha_-}(\lambda_{t,\infty}^*)^{\alpha_-} + \dfrac{1}{1-\gamma}\dfrac{1}{K}(\lambda_{t,\infty}^*)^{\frac{1}{\gamma}-1}.
\end{split} \end{eqnarray*} where $\lambda_{t,\infty}^*=e^{-(\rho-r)t}X_{t,\infty}^*$ and $$ X_{t,\infty}^*=\max\left(\lambda_{\infty}^*, z_\infty\sup_{0\le \xi \le t}e^{(\rho-r)\xi}Y_{\xi}^\gamma \right). $$ Here, $\lambda_{\infty}^*$ is a unique solution to the following algebraic equation: \begin{eqnarray*}
\begin{split}
w=&-\dfrac{1}{\gamma}\dfrac{1}{K}\dfrac{1}{\alpha_-}(z_\infty y^\gamma)^{\frac{1}{\gamma}-1-\alpha_-}(\lambda_{\infty}^*)^{\alpha_-} + \dfrac{1}{1-\gamma}\dfrac{1}{K}(\lambda_{\infty}^*)^{\frac{1}{\gamma}-1}.
\end{split} \end{eqnarray*} For this infinite horizon case, the uniqueness of $\lambda_{\infty}^*$ was proven by \cite{MJ}.
$\Box$
\begin{rem}~
For $\alpha=1-\gamma$ and $\beta=1-\gamma-\gamma\alpha_-$, the results in Theorem \ref{thm:infinite} {are} consistent {with} those of Section 2 in \citet{MJ}. \end{rem}
\section{Numerical Illustrations}\label{sec:4} In this section, we illustrate numerical simulation results for optimal contracting policies. The optimal contracting policies stated in Theorem \ref{thm:main} is not fully explicit, since it requires solving the integral equation \eqref{eq:integral} for the free boundary $z^{\star}(t)$. Thus, we should solve for the free boundary by numerical methods. Thus, we solve the integral equation \eqref{eq:integral} by the recursive integration method proposed by \citet{Huang}. \begin{figure}
\caption{ Simulation of the optimal consumption $C^*$, the optimal process $X^*$ and the regulated process $\lambda^*/Y^\gamma$. The parameter values are as follows: $\rho=0.04,\;r=0.04,\;\mu=0.02\;\sigma=0.1,\;\gamma=3,\;w=-5$, and $T=30$.}
\label{fig00b}
\label{fig00b}
\label{fig00a}
\label{fig:2}
\end{figure}
Figure \ref{fig:2} presents simulation paths of the optimal consumption $C^*$, the optimal costate process $X^*$, and the regulated process ${\lambda^*}/{Y^\gamma}$. The optimal costate process $X_s^*$ ensures that $(s,{\lambda_s^*}/{Y_s^\gamma})$ will never leave the no-jump region for $s\in[t,T]$ as shown in Figure \ref{fig:2} (a). Whenever $\lambda^*/Y^\gamma$ low enough to hit the free boundary $z^\star$, the non-decreasing process $X^*$ increases in Figure \ref{fig:2} (b). The reason is that the optimal costate process $X^*$ have the property that it increases only when $\lambda^*/Y^\gamma$ hits the free boundary, at which time the participation constraints \eqref{eq:parti} bind. This leads to increase {s in} the optimal consumption $C^*$. In other words, if income process $Y$ increases enough and thus $\lambda^*/Y^\gamma$ decreases and hits the free boundary, the principal should increase the agent's consumption $C^*$ so that the agent does not walk away from the contract{,} as shown in Figure \ref{fig:2} (c). \begin{figure}
\caption{ Simulation of optimal consumption $C^*$. The parameter values are as follows: $\rho=0.07,\;r=0.04,\;\mu=0.02\;\sigma=0.1,\;\gamma=3\;w=-5$ and $T=30$.}
\label{fig01a}
\label{fig:3}
\end{figure}
Figure \ref{fig:3} which is representative of the case where $\rho>r${,} show {ing} a simulation path of optimal consumption allocation. In the case of $\rho>r$, however, the optimal consumption $C^*$ gradually decreases even when the regulated process $\lambda^*/Y^\gamma$ does not hit the free boundary. \begin{figure}
\caption{ Simulation of optimal consumption $C^*$. The parameter values are as follows: $\rho=0.07,\;r=0.04,\;\mu=0.02\;\sigma=0.1,\;\gamma=3\;w=-5$ and $T=30$.}
\label{fig01a}
\label{fig01a}
\label{fig:4}
\end{figure}
Figure \ref{fig:4} shows the comparison results of the optimal consumption $C^*$ and the first-best consumption $C^{FB}$ in two different scenarios. Two scenarios correspond to two different sample paths of the income process generated with the same parameter values. In the first scenario (scenario 1), the income process steadily increases whereas the income process tends to decrease in the second scenario (scenario 2). Since the first-best consumption defined in \eqref{eq:FB_C} is a deterministic function, $C^{FB}$ is the same in both scenarios. In contrast to the first-best allocation, the optimal consumption $C^*$ in limited commitments depends on {the entire} history of income process $Y_t$. In other words, the optimal consumption $C^*$ also increases as income steadily increases in the scenario 1. That is, although $C^* <C^{FB}$ at the beginning, as time passes, the optimal consumption $C^*$ exceeds the first-best allocation $C^{FB}${,} as shown in Figure \ref{fig:4} (a). In scenario 2, however, since the income process $Y$ gradually decreases, the optimal consumption $C^*$ does not exceed first-best allocation $C^{FB}$ until maturity $T$.
\section{Concluding Remarks}\label{sec:5}
In this paper we study the optimal contracting problem with limited commitment between a risk-neutral principal and a risk-averse agent in the finite horizon. We establish the duality relationship and transform the dual problem into an infinite series of optimal stopping problems, which essentially becomes a single optimal stopping problem. This is similar in its formal structure to an irreversible incremental investment problem studied in \citet{Pindyck88} {and} \citet{DixitPindyck}. {B}ased on partial differential equation theory, we characterize the variational inequality arising from the optimal stopping problem. We also derive an integral equation representation of optimal strategy and provide numerical results {for} the optimal strategy. \section*{Acknowledgments.}
Junkee Jeon gratefully acknowledges the support of the National Research Foundation of Korea (NRF) grant funded by the Korea government (Grant No. NRF-2017R1C1B1001811). Hyeng Kuen Koo gratefully acknowledges the support of the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (Grant No. NRF-2016R1A2B4008240). Kyunghyun Park is supported by NRF Global Ph.D Fellowship (2016H1A2A1908911).
\end{document} | arXiv |
\begin{document}
\begin{abstract}
We construct an infinite family of smoothly slice knots that we prove are topologically doubly slice. Using the correction terms coming from Heegaard Floer homology, we show that none of these knots is smoothly doubly slice. We use these knots to show that the subgroup of the double concordance group consisting of smoothly slice, topologically doubly slice knots is infinitely generated. As a corollary, we produce an infinite collection of rational homology 3--spheres that embed in $S^4$ topologically, but not smoothly. \end{abstract}
\title{Distinguishing topologically and smoothly doubly slice knots}
\rhead{\thepage} \lhead{\author} \thispagestyle{empty}
\raggedbottom
\pagenumbering{arabic}
\setcounter{section}{0}
\section{introduction}\label{section:introduction}
A knot $K$ in $S^3$ is called \emph{smoothly doubly slice} if there exists a smoothly embedded, unknotted 2--sphere $\kappa$ in $S^4$ such that $\kappa\cap S^3=K$. Analogously, $K$ is called \emph{topologically doubly slice} if $\kappa$ is topologically locally flat. The question of which slice knots are doubly slice was first posed by Fox in 1961 \cite{fox:problems}, and Zeeman showed that $K\#(-K)$ is always doubly slice \cite{zeeman}. Work of Sumners encapsulates what was known up to about 1970 \cite{sumners:invertible}. In particular, he gave necessary algebraic conditions for a knot to be doubly slice and proved that $9_{46}$ is the only doubly slice knot up to 9 crossings. Although his proof that $9_{46}$ is doubly slice is (necessarily) geometric in nature, his obstruction methods are actually purely algebraic. He showed that $9_{46}$ is the only knot up to 9 crossings that is algebraically doubly slice. A knot $K$ is called \emph{algebraically doubly slice} if there exists an invertible $\mathbb{Z}$--valued matrix $P$ such that $$PA_KP^\tau = \begin{bmatrix}
0 & B_1 \\
B_2 & 0
\end{bmatrix},$$ where $A_K$ is a Seifert matrix for $K$, and $B_1$ and $B_2$ are square matrices of equal dimension. Matrices of this form are often called \emph{hyperbolic}, and have been studied by Levine \cite{levine:hyperbolic}. We remark that all these concepts generalize to higher dimensions (see, for example \cite{sumners:invertible}), but we will restrict our attention to the classical dimension.
Since the work of Sumners, there have been three major geometric developments in the theory, all in the topologically locally flat category. In what follows, we will take `slice' and `doubly slice' to mean `topologically slice' and `topologically doubly slice' and clarify the category when necessary or helpful.
First, in 1983, Gilmer-Livingston showed, using Casson-Gordon invariants, that there exist slice knots that are algebraically doubly slice, but not doubly slice \cite{gilmer-livingston:embedding}. Second, about 10 years ago, Kim \cite{kim:new} extended the bi-filtration technology introduced by Cochran-Orr-Teichner in \cite{COT} to the class of topologically doubly slice knots. At the same time, Friedl \cite{friedl:eta} showed that certain $\eta$--invariants coming from metabelian representations $\pi_1(M_K)\to U(k)$, where $M_K$ denotes 0--surgery on $K$, can be used to obstruct double sliceness.
In this paper, the invariants used are the correction terms coming from Heegaard Floer homology (see \cite{oz-sz:absolute}). These are smooth manifold invariants, so they are well suited to distinguish the smooth and topologically locally flat categories. A second property these invariants enjoy is the fact that, while they can be used to obstruct smooth sliceness, they do not completely vanish for smoothly slice knots, as do invariants such as the signature, $\tau$--invariant, or $s$--invariant. In other words, they encode enough information to distinguish smooth double sliceness and smooth sliceness. The main result of the present paper is the following.
\begin{lettertheorem}\label{thm:main} There exists an infinite family of smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. \end{lettertheorem}
Recall that two knots $K_0$ and $K_1$ are said to be \emph{concordant} if $K_1\#(-K_2)$ is slice (where $-K$ denotes the mirror reverse of $K$) or, equivalently, if there exists a properly embedded cylinder $C\subset S^3\times I$ such that $C\cap S^3\times\{i\}=K_i$ for $i=0,1$. If $K_0$ and $K_1$ are concordant, we write $K_0\sim K_1$. Concordance can be studied in either the smooth or the topologically locally flat categories and induces (different) equivalence relations therein. Let $\mathcal C$ denote the set of knots in $S^3$ up to smooth concordance. Under connected sum, $\mathcal C$ inherits an abelian group structure and is called the smooth \emph{concordance group}. Similarly, one can define the topological concordance group $\mathcal C^{top}$ and the algebraic concordance group $\mathcal G$. There exist surjective homomorphisms $$\mathcal C\stackrel{\psi}{\longrightarrow} \mathcal C^{top}\stackrel{\phi}{\longrightarrow}\mathcal G.$$
These groups have received a large amount of attention, and many interesting theorems and examples have expanded our understanding of their nature; however, there remain many open problems. For example, it is still not known whether or not $\mathcal C$ and $\mathcal C^{top}$ contain elements of finite order greater than two. On the other hand, Levine \cite{levine:invariants,levine:groups} proved that $$\mathcal G\cong \mathbb{Z}^\infty\oplus\mathbb{Z}_2^\infty\oplus\mathbb{Z}^\infty_4.$$ For an excellent survey, see \cite{livingston:survey}.
It would be natural to define $K_0$ and $K_1$ to be doubly concordant if $K_0\#(-K_1)$ is doubly slice. However, it is not known whether this gives an equivalence relation. The issue is the following unsolved problem.
\begin{question}\label{question:cancelation} Suppose that $K$ is doubly slice and that $J\#K$ is doubly slice. Then, must $J$ be doubly slice? \end{question}
Without an affirmative answer to Question \ref{question:cancelation}, one cannot prove transitivity of the desired equivalence relation. Following \cite{stoltzfus:double}, we say that $J$ is \emph{stably doubly slice} if $J\#K$ is doubly slice for some doubly slice knot $K$. Then, Question \ref{question:cancelation} is simply asking whether or not there exist stably doubly slice knots that are not doubly slice. Because of these difficulties, we must adopt a different definition of doubly concordant.
Recall that two knots $K_0$ and $K_1$ are concordant if and only if there exist two slice knots $J_0$ and $J_1$ such that $K_0\#J_0=K_1\#J_1$. This follows from the more common definition of concordant by realizing that the analogue of Question \ref{question:cancelation} for slice knots is true: If $K$ is slice and $J\#K$ is slice, then $J$ is slice. With this in mind, we adopt the following definition.
\begin{definition} Two knots $K_0$ and $K_1$ are smoothly \emph{doubly concordant} if there exist smoothly doubly slice knots $J_0$ and $J_1$ such that $K_0\#J_0=K_1\#J_1$. We write $K_0\stackrel{\mathcal D}{\sim}K_1$. \end{definition}
It is straightforward to verify that $\stackrel{\mathcal D}{\sim}$ is an equivalence relation. We let $\mathcal C_\mathcal D$ denote the set of knots in $S^3$ modulo this relation, which inherits an abelian group structure under connected sum and is called the smooth \emph{double concordance group}. Analogously, we can define the topological double concordance group $\mathcal C_\mathcal D^{top}$ and the algebraic double concordance group $\mathcal G_\mathcal D$, and we have surjective homomorphisms $$\mathcal C_\mathcal D\stackrel{\psi_\mathcal D}{\longrightarrow} \mathcal C^{top}_\mathcal D\stackrel{\phi_\mathcal D}{\longrightarrow}\mathcal G_\mathcal D.$$
The study of these structures is complicated by Question \ref{question:cancelation}. In Subsection \ref{subsec:question}, we show that, under certain conditions, if $K$ is smoothly stably doubly slice, then the correction terms of $\Sigma_2(K)$ must vanish in the same way as when $K$ is smoothly doubly slice. In this light, one consequence of Theorem \ref{thm:main} is that $\mathcal T_\mathcal D\not=0$, where $\mathcal T_\mathcal D = \ker(\psi_\mathcal D)$.
In \cite{GRS}, Grigsby, Ruberman, and Strle (building on work of Jabuka and Naik \cite{jabuka-naik:order}) defined invariants that can be used to obstruct a knot from having finite order in $\mathcal C$. After a slight modification, we show that similar invariants can be applied to $\mathcal C_\mathcal D$. After restricting our attention to a certain subfamily of the knots from Theorem \ref{thm:main}, we are able to show the following.
\begin{lettertheorem}\label{thm:main2} There is an infinitely generated subgroup $\mathcal S$ inside $\mathcal T_\mathcal D$, generated by smoothly slice knots whose order in $\mathcal C_\mathcal D$ is at least three. \end{lettertheorem}
One would like to say that the knots in $\mathcal S$ have infinite order in $\mathcal C_\mathcal D$. Unfortunately, due to Question \ref{question:cancelation}, we can only obstruct order one and order two.
\begin{letterconjecture}\label{conj:infty} The subgroup $\mathcal S\subset \mathcal T_\mathcal D$ is isomorphic to $\mathbb{Z}^\infty$. \end{letterconjecture}
We have the following corollary to Theorem \ref{thm:main}.
\begin{lettercorollary}\label{coro:embed} There exists an infinite family of rational homology 3--spheres that embed in $S^4$ topologically, but not smoothly. \end{lettercorollary}
Note that these manifolds are not integral homology spheres. An affirmative answer to Question \ref{question:cancelation} would imply Conjecture \ref{conj:infty}. If the Conjecture \ref{conj:infty} is false, then there are knots in $\mathcal S$ whose branched double covers do not smoothly embed in $S^4$, but do stably embed smoothly in $S^4$. See \cite{budney-burton:embeddings} for a survey concerning 3--manifold embeddings in $S^4$.
\subsection*{Organization} In Section \ref{section:background}, we give a brief outline of the proofs of Theorems \ref{thm:main} and \ref{thm:main2} and give a background overview of the relevant theories. In Section \ref{section:geometry}, we give the construction of the pertinent family of knots and prove that they are topologically doubly slice. We also introduce and discuss the 3--manifolds and 4--dimensional cobordisms that are used in the proof of Theorem \ref{thm:main}, discuss the sub-family of knots used to prove Theorem \ref{thm:main2}, and address the subtlety of Question \ref{question:cancelation}. In Section \ref{section:HF}, we recall the pertinent aspects of Heegaard Floer theory. In Section \ref{section:calculations}, we perform the calculations necessary to prove that the knots are not smoothly doubly slice. In Section \ref{section:infinite_order}, we use invariants introduced by Grigsby, Ruberman, and Strle to prove Theorem \ref{thm:main2}. The proofs of the main theorems rely on calculations of the knot Floer complexes for certain torus knots and the positive, untwisted Whitehead double of the right-handed trefoil. These facts, some of which are found in \cite{HKL}, are presented in Appendix \ref{appendix}.
\begin{figure}
\caption{One member of the family $\mathcal K_p$; here, $p=5$.}
\label{fig:InfectedKnotSmall}
\end{figure}
\section{Background and outline of proof}\label{section:background}
In Section \ref{section:geometry}, we construct the knots $\mathcal K_p$ for odd primes $p$, and prove that they are topologically doubly slice. (See Figure \ref{fig:InfectedKnotSmall} for an example.)
The most difficult task of this paper is showing that the $\mathcal K_p$ are not smoothly doubly slice. This is accomplished by studying the double covers of $S^3$ branched along these knots. If $K$ is a smoothly doubly slice knot, then it is the intersection of a smoothly unknotted 2--sphere $\kappa\subset S^4$ with the standard $S^3\subset S^4$. So we have $(S^3,K)\subset (S^4,\kappa)$, where the first pair sits as the equator of the second. Taking the branched double cover, we get $(\Sigma_2(K),K)\subset (S^4,\kappa)$. This gives a smooth embedding of the branched double cover $\Sigma_2(K)$ of $K$ into $S^4$. We have proved the following proposition, which first appeared in \cite{gilmer-livingston:embedding}.
\begin{proposition} If $K$ is a smoothly doubly slice knot, then $\Sigma_2(K)$ embeds smoothly into $S^4$. \end{proposition}
Thus, we can prove that a knot is not smoothly doubly slice by showing that its branched double cover does not embed smoothly in $S^4$. To do this, we make use of the correction terms coming from Heegaard Floer homology. For more details, see Section \ref{section:HF}. For now, let $M$ denote a closed 3--manifold, and let $\frak s\in\text{Spin}^c(M)$. Let $d(M,\frak s)$ denote the correction term associated to the pair $(M,\frak s)$. The main tool in this paper is the following theorem, which also appears in \cite{donald:embedding} and \cite{gilmer-livingston:embedding} in one form or another.
\begin{theorem}\label{thm:hyp_corr_terms} Let $M$ be a rational homology 3--sphere that embeds smoothly in $S^4$. Then $H_1(M)= G_1\oplus G_2$ with $G_1\cong G_2$. Furthermore, there is an identification $\text{Spin}^c(M)\cong H^2(M;\mathbb{Z})\cong H_1(M)$ such that $$d(M,\frak s)=0\hspace{.25in} \forall \frak s\in G_1\cup G_2.$$
In other words, if $|H_1(M)|=n^2$, then at least $2n-1$ of the $n^2$ correction terms associated to $M$ must vanish. \end{theorem}
\begin{proof} Since $M$ embeds smoothly in $S^4$, we get a decomposition $S^4 = U_1\cup_MU_2$, where $U_i$ is a rational homology 4--ball for $i=1,2$. Let $G_i = H_1(U_i)$ for $i=1,2$. By analyzing the Mayer-Vietoris sequence induced by this decomposition, we see that $H_1(M)\cong H_1(U_1)\oplus H_1(U_2)=G_1\oplus G_2$. The proof that $G_1\cong G_2$ is due to Hantzsche \cite{hantzche}, and is as follows. By analyzing the relative sequence for $(S^4,U_1)$, we see that $H_1(U_1)\cong H_2(S^4,U_1)$. By excision, $H_2(S^4,U_1)\cong H_2(U_2,M)$, and by Lefschetz duality, $H_2(U_2,M)\cong H^2(U_2)$. Finally, by the universal coefficients theorem, $H^2(U_2)\cong H_1(U_2)$ (since $H_1(U_2)$ and $H_2(U_2)$ are both torsion).
Now consider the dual isomorphism $G_1\oplus G_2\cong H^2(M)$, whose restrictions to $G_i$ are induced by the inclusion $M\hookrightarrow U_i$ for $i=1,2$. Elements in $H^2(M)$ that are in the image of this inclusion from $G_i$ correspond to $\text{Spin}^c$ structures on $M$ that extend to $\text{Spin}^c$ structures over $U_i$ for $i=1,2$. However, for any 3--manifold $Y$ and $\frak s\in\text{Spin}^c(Y)$, we have that $d(Y,\frak s) = 0$ whenever $(Y,\frak s) = \partial (W,\frak t)$, where $W$ is a rational homology 4--ball and $\frak t$ extends $\frak s$ (see \cite{oz-sz:absolute}).
If follows that $d(M,\frak s)=0$ for any $\frak s\in G_1\cup G_2$, which is a set of cardinality $2n-1$. \end{proof}
Let $\mathcal Z_p$ denote the double cover of $S^3$ branched along the knot $\mathcal K_p$. In Section \ref{section:calculations}, we make use of the surgery exact triangle to relate the Heegaard Floer homology of $\mathcal Z_p$ to that of simpler manifolds (manifolds obtained as surgery on knots in $S^3$, to be precise). Using this set-up, we show in Corollary \ref{coro:terms} that only $2p-3$ of the $p^2$ correction terms associated to $\mathcal Z_p$ vanish. By Theorem \ref{thm:hyp_corr_terms}, this implies Theorem \ref{thm:main}, as well as Corollary \ref{coro:embed}.
Of course, the statement that at least $2n-1$ of the $n^2$ correction terms must vanish does not use the full strength of Theorem \ref{thm:hyp_corr_terms}, since it makes no use of the group structure of the correction terms. Jabuka and Naik \cite{jabuka-naik:order} used this group structure to prove that many low crossing knots (whose concordance order was unknown) are not order 4 in $\mathcal C$. Grigsby, Ruberman, and Strle investigated this concept further in \cite{GRS}, and introduced knot invariants that can be used to obstruct finite concordance order among knots. We refine one set of these invariants so that they can be used to obstruct order one and order two in the double concordance group, and use them to prove that a family related to the $\mathcal K_p$ generates an infinite rank subgroup in $\mathcal C_\mathcal D$ (see Section \ref{section:infinite_order}). This proves Theorem \ref{thm:main2}.
\section{Geometric considerations}\label{section:geometry}
In this section, we use the method of infection to construct the knots $\mathcal K_p$ and $\mathcal K_{p,k}$. We then describe a sufficient condition for a knot to be doubly slice and use it to prove that these knots are topologically doubly slice. Next, we introduce the 3--manifolds triad that will be used in Section \ref{section:calculations}, and describe the 4--dimensional cobordisms relating them. Finally, we address Question \ref{question:cancelation}.
\subsection{Infection and the knots $\mathcal K_p$}\label{subsec:infection}\
Let $\vec\eta = (\eta_1, \ldots, \eta_n)$ be an $n$--component unlink in $S^3$, and choose an open tubular neighborhood $N_i$ of each $\eta_i$ such that $\overline N_i\cap \overline N_j=\emptyset$ for $i\not=j$. Let $E = S^3 - \cup_{i=1}^n N_i$. Next, consider a collection of knots $\vec J=(J_1,\ldots, J_n)$, and let $E_{J_i}$ denote the exterior of $J_i$. Let $M$ be the manifold obtained by gluing $E_{J_i}$ to $E$ along $\partial N_i$ such that the meridian and longitude of $\eta_i$ are identified with the longitude and meridian, respectively, of $J_i$. This choice of gluing ensures that $M$ is diffeomorphic to $S^3$.
Let $K\subset E$, and let $f:E\to M$ be the natural inclusion. Then the knot $K_{\vec\eta}(\vec J) = f(K)$ is the result of \emph{infection} on $K$ by $\vec J$ along $\vec \eta$. In the case when $\vec\eta$ is a knot, we simply write $I_\eta(J)$. See Figure \ref{fig:InfectedKnots}. The construction, as given, dates back at least as far as \cite{gilmer:slice}.
\begin{example}\ \begin{enumerate} \item If $n=1$, we recover the satellite construction. In particular, if $\eta$ is chosen to be a meridian of $K$, then infection of $K$ by $J$ along $\eta$ is simply $K\#J$. \item If $K\cup\eta$ is the positive Whitehead link (see Figure \ref{fig:WhiteheadAndTrefoil} (b)), then infection of $K$ by $J$ along $\eta$ is the positive, untwisted Whitehead double of $J$, which we denote by $Wh^+(J,0)$. For example, if $J$ is the right-handed trefoil, then $Wh^+(J,0)$ is shown in Figure \ref{fig:WhiteheadAndTrefoil} (c). \end{enumerate} \end{example}
\begin{figure}
\caption{(a) The right-handed trefoil, (b) the positive Whitehead link, and (c) the positive, untwisted Whitehead double of the right-handed trefoil.}
\label{fig:WhiteheadAndTrefoil}
\end{figure}
Throughout, we will denote the $(p,q)$--torus knot by $T_{p,q}$ for $2\leq p<|q|$ (see Figure \ref{fig:TorusKnot}).
Let $I_{J,p}$ denote the knot obtained by infecting $T_{2,p}\#(T_{2,-p})$ with $J$ along $\eta$ (see Figure \ref{fig:InfectedKnots}). \begin{figure}
\caption{An example of the torus knot $T_{p,p+1}$; here $p=5$.}
\label{fig:TorusKnot}
\end{figure} Let $D$ be the positive, untwisted Whitehead double of the right handed trefoil, and let $\mathcal K_p = I_{D,p}$ for $p$ an odd prime (see Figures \ref{fig:InfectedKnotSmall} and \ref{fig:InfectedKnots}(b)). Let $\mathcal K_{p,k} =I_{\#_kD,p}$, and note that $\mathcal K_{p,1}=\mathcal K_p$. The rest of the paper will be devoted to proving that these knots are topologically doubly slice, but not smoothly doubly slice.
\begin{figure}
\caption{(a) The knot $T_{2,p}\#T_{2,-p}$ along with the infection curve $\eta$. (b) Two descriptions of the result of infecting $T_{2,p}\#T_{2,-p}$ with some knot $J$ along $\eta$. Here, $p=5$.}
\label{fig:InfectedKnots}
\end{figure}
\subsection{A sufficient condition for double sliceness}\label{subset:first}\
In this subsection we will present a sufficient condition for a knot $K$ to be doubly slice that applies when $K$ is obtained by a certain type of infection. We remark that Donald \cite{donald:embedding} gives a different sufficient condition: one which involves systems of ribbon bands for $K$.
Our criterion will make use of some well-known facts about topologically locally flat surfaces in 4--manifolds that result from the work of Freedman and Quinn \cite{freedman:4manifolds,freedman-quinn}.
\begin{theorem}\label{thm:freedman}\ \begin{enumerate} \item Let $K$ be a knot in $S^3$ with Alexander polynomial $\Delta_K=1$. Then, there exists a topologically locally flat disk $D$ properly embedded in $B^4$ with $\partial D=K$ and $\pi_1(B^4-D)\cong\mathbb{Z}$.
\item Let $\kappa$ be a topologically locally flat 2--knot in $S^4$ with $\pi_1(S^4-\kappa)\cong\mathbb{Z}$. Then, there exists an embedded 3--ball $B\subset S^4$ with $\partial B=\kappa$.
\end{enumerate} \end{theorem}
There is a simple corollary to this theorem that will be useful below (cf. \cite{kirby-melvin,gordon-sumners}).
\begin{corollary}\label{coro:freedman} Let $K$ be a knot in $S^3$ with $\Delta_K=1$. Then, $K$ is topologically doubly slice. \end{corollary}
\begin{proof} By Theorem \ref{thm:freedman}, we know that $K$ bounds a topological disk $D\subset B^4$ whose complement has fundamental group $\mathbb{Z}$. Moreover, we have $\pi_1(S^3-K)\to\pi_1(B^4-D)\cong \mathbb{Z}$ is surjective. If we double the pair $(B^4, D)$ along the boundary $(S^3,K)$, then we get $(S^4, \kappa)$, where $\kappa$ is a topological 2--knot. It follows that $\pi_1(S^4-\kappa)\cong \mathbb{Z}$ by van Kampen's theorem (this uses the surjectivity). Thus, $\kappa$ is topologically unknotted with $\kappa\cap S^3=K$, so $K$ is topologically doubly slice. \end{proof}
\begin{proposition}\label{prop:topo_infection} Let $K$ be a topologically doubly slice knot and let $K'=I_{\vec\eta}(\vec J)$ be the result of infecting $K$ with the knots $J_i$, each of which is topologically doubly slice. Then $K'$ is topologically doubly slice. \end{proposition}
\begin{proof} We can isotope the link $K\cup\vec\eta$ so that the $\eta_i$ span small, disjoint disks $D_i$ for $i=1,\ldots, n$, which $K$ intersects transversely in $m_i$ points. Because $K$ is doubly slice, there is an unknotted 2--sphere $\kappa\subset S^4$ such that $\kappa\cap (S^3\times[-1,1]) = K\times[-1,1]$. Let $D_i\times I$ denote a a thickening of $D_i$ in $S^3$, so $(D_i\times I, K\times I)$ is a trivial $m_i$--strand tangle. From each $D_i\times I\times[-1,1]$, we will remove the interior of a small 4--ball $B_i$ such such that $B_i\cap (K\times[-1,1])$ is a disjoint collection of $m_i$ parallel disks and $B_i\cap (S^3,K)$ is a trivial tangle of $m_i$ strands. Let $m=\sum_{i=1}^nm_i$. Let $\overline B$ be the result of this removal, i.e., to form $\overline B$ we have removed $n$ 4--balls from $S^4$ and and $m$ 2--disks from $\kappa$ to form a punctured manifold pair.
Now, let $J_i$ be one of the topologically doubly slice knots that will be used in the infection. Let $\mathcal J_i$ be an unknotted 2--sphere in $S^4$ such that $\mathcal J_i\cap(S^3\times[-1,1])=J_i\times[-1,1]$. Let $\lambda_i$ denote the disjoint union of $m_i$ parallel copies of $\mathcal J_i$. Then, $\lambda_i\cap S^3$ is the $(m_i,0)$--cable $C_i$ of $J_i$, and $\lambda_i\cap(S^3\times[-1,1])=C_i\times[-1,1]$.
We can assume that the parallel copies of $\mathcal J_i$ are close enough so that there is a small 4--ball $B'_i\subset S^3\times [-1,1]$ such that $B_i'\cap(C_i\times I)$ is a collection of $m_i$ parallel disks and $B_i'\cap(S^3,C_i)$ is a trivial tangle of $m_i$ strands. Form $\overline B_i$ by removing the interior of $B_i'$. Then $\overline B_i$ is a 4--ball that contains $m_i$ parallel, topologically unknotted disks that intersect the $B^3$ cross-section of $B^4$ in the tangle $(B^3,C_i)$, i.e., a 3--ball containing $m_i$ arcs that are tied in $C_i$.
Finally, we will re-form $S^4$ from $\overline B$ by gluing in $\overline B_i$ along $\partial B_i\subset \overline B$. This has the effect of replacing each parallel set of $m_i$ topological disks that we removed from $\kappa$ with a parallel set of $m_i$ topological disks. Since $\kappa$ was originally topologically unknotted, this new 2--sphere $\kappa'$ is clearly topologically unknotted. Furthermore, for each $i$, we removed from $(S^3, K)$ a trivial tangle of $m_i$ strands. We have now replaced that tangle with the $(B^3, C_i)$ tangle described above. The result of this is to tie the $m_i$ strands in the knot $C_i$. This is precisely the effect of infection of $K$ with $J_i$ along $\eta_i$. In other words, $\kappa'$ is a topologically unknotted 2--sphere with $\kappa'\cap S^3 = I_{\vec\eta}(\vec J)=K'$. It follows that $K'$ is topologically doubly slice. \end{proof}
We remark that the conclusion of Proposition \ref{prop:topo_infection} holds if $K$ is smoothly doubly slice and that an analogous proposition holds in the smooth category. We can apply the previous proposition to the knots $\mathcal K_{p,k}$, proving that the knots referenced in Theorems \ref{thm:main} and \ref{thm:main2} are topologically doubly slice.
\begin{corollary}\label{coro:doubly_slice} The knots $\mathcal K_{p,k}$ are topologically doubly slice and smoothly slice. \end{corollary}
\begin{proof} Let $K=T_{2,p}\#T_{2,-p}$, let $J=\#_kD$, and let $\eta$ be as shown in Figure \ref{fig:InfectedKnots}. Then, $\mathcal K_{p,k} = K_\eta(J)$, with $K$ smoothly doubly slice (by Zeeman \cite{zeeman}) and $J$ topologically doubly slice (by Corollary \ref{coro:freedman}, since $\Delta_J=1$). Thus, by Proposition \ref{prop:topo_infection}, $\mathcal K_{p,k}$ is topologically doubly slice.
To see that $\mathcal K_{p,k}$ is smoothly slice, consider it as the boundary of a punctured Klein bottle, as in Figure \ref{fig:DoubleBranchedCover}(a). This punctured Klein bottle is formed by attaching two bands to a disk. In this case, the right most band is unknotted and untwisted. It follows that we can push the interior of the punctured Klein bottle into the 4--ball and surger it along the core of this band. The result is a smooth, properly embedded disk in the 4--ball with boundary $\mathcal K_{p,k}$. \end{proof}
\subsection{Relevant 3--manifolds and 4--dimensional cobordisms}\label{subsec:triad}\
Let $I_{J,n}$ be the infected knot described above, and let $Z_{J,n}$ be the double-cover of $S^3$ branched along $I_{J,n}$. In \cite{akbulut-kirby}, Akbulut and Kirby described how to get a surgery diagram for the double-cover of $B^4$ branched along a surface bounded by a knot. Applying this technique, we see that $Z_{J,n} = S^3_{n,-n}((J\#J)_{(2,0)})$, i.e., surgery on the (2,0)--cable of $J\#J$ with surgery coefficients $n$ and $-n$ (see Figure \ref{fig:DoubleBranchedCover}). Note that throughout this paper, $J$ will be a reversible knot, so $J^r=J$.
\begin{figure}
\caption{(a) The knot $I_{J,n}$, shown as the boundary of a punctured Klein bottle. The boxes indicate $n$ positive half-twists. (b) Two descriptions of the resulting branched double cover, $Z_{J,n}$, which are related by a handleslide.}
\label{fig:DoubleBranchedCover}
\end{figure}
Let $X=S^3_n(J\#J)$, and let $K\subset X$ be the null-homologous knot shown in Figure \ref{fig:KnotInY}. If we think of $X$ as $n$--surgery on one component of the (2,0)--cable of $J\#J$, then $K$ is the image (in the surgery manifold) of the second component of the (2,0)--cable. Since $K$ is a longitudinal push-off of $J\#J$ in $S^3$, it bounds, in $S^3$, a Seifert surface $F$ with $g(F) = g(J\#J)$. Since $F$ is disjoint from $J\#J$, we see that $F$ is a Seifert surface for $K$ in $X$, as well. Thus, $K$ is null-homologous in $X$. With respect to the Seifert framing of $K$ in $X$, we have $X_{-n}(K) = Z_{J,n}$.
\begin{figure}
\caption{Two equivalent views of the null-homologous knot $K$ in $X=S^3_n(J\#J^r)$. Note that the Seifert framing on $K$ is different in these two descriptions. Compare with Figure \ref{fig:DoubleBranchedCover} to see that $Z$ is obtained by surgery on $K$.}
\label{fig:KnotInY}
\end{figure}
Now, let $Y = X_{-n-1}(K)$. After performing a handle-slide and blowing down (see Figure \ref{fig:BlowdownY}), we see that $Y = S^3_{n^2+n}(J\#J\#T_{n,n+1})$. These three manifolds, $X, Y,$ and $Z=Z_{J,n}$ form a triad:
\begin{figure}
\caption{The manifold $Y$ is obtained as $(-1)$--surgery on $K$ in $X$. After a blowdown, $Y$ can be realized by $(n^2+n)$--surgery on $J\#J\#T_{n,n+1}$.}
\label{fig:BlowdownY}
\end{figure}
$$\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=4em,column sep=4em,minimum width=2em] {
X& \ & Z \\
\ & Y & \ \\};
\path[-stealth]
(m-1-1) edge node [pos=.6,left] {$W_1\ $} (m-2-2)
(m-2-2) edge node [pos=.4,right] {$\ W_2$} (m-1-3)
(m-1-3) edge node [above] {$W_3$} (m-1-1); \end{tikzpicture}$$
Now, since $-\overline{W_3}$ is the cobordism from $X$ to $Z$ corresponding to attaching a $(-n)$--framed 2--handle along $K$ in $X$, we have that $H_2(-\overline{W_3})\cong\mathbb{Z}$ is generated by the class $S_3 = F\cup D^2$ (i.e., the genus $g$ Seifert surface for $K$, capped off with the core disk of the 2--handle), and $[S_3]\cdot [S_3] = -n$ in $-\overline{W_3}$. Therefore, $W_3$ is a positive definite cobordism whose second homology is generated by a surface of genus $g(J\#J)$ with self-intersection $n$.
Similarly, $W_1$ is formed by attaching a $(-n-1)$--framed 2--handle to $X$ along $K$. The result is that $W_1$ is a negative definite cobordism whose second homology is generated by a class $[S_1]$, where $S_1$ is a surface of genus $g=g(K)$ with self-intersection $-n-1$. Note also that $H^2(W_1)\cong \mathbb{Z}_n\oplus\mathbb{Z}$. The map from $H^2(W_1)\to H^2(X)$ induced by restricting to $X$ is realized by projection onto the first component: $\mathbb{Z}_n\oplus\mathbb{Z}\to\mathbb{Z}_n$, while the corresponding map from $H^2(W_1)\to H^2(Y)$ is reduction modulo $n+1$ of the second component and the identity on the first: $\mathbb{Z}_n\oplus\mathbb{Z}\to\mathbb{Z}_n\oplus\mathbb{Z}_{n+1}$.
Finally, $W_2$ is obtained by attaching a $(-1)$--framed 2-handle along the meridian $\mu$ shown in Figure \ref{fig:MeridinalBlowdown}. In fact, $\mu$ is rationally null-homologous, and bounds a rational Seifert surface, $S_2$. It turns out that this surface has self-intersection $-n^2-n$ and $[S_2]$ generates the second homology of $W_2$, so $W_2$ is negative definite. Note also that $H^2(W_2)\cong \mathbb{Z}_n\oplus\mathbb{Z}$. The map from $H^2(W_2)\to H^2(Y)$ induced by restricting to $Y$ is realized by reduction modulo $n+1$ of the second component and the identity on the first: $\mathbb{Z}_n\oplus\mathbb{Z}\to\mathbb{Z}_n\oplus\mathbb{Z}_{n+1}$, while the corresponding map from $H^2(W_2)\to H^2(Z)$ is reduction modulo $n$ of the second component and the identity on the first: $\mathbb{Z}_n\oplus\mathbb{Z}\to\mathbb{Z}_n\oplus\mathbb{Z}_n$.
\begin{figure}
\caption{(a) The manifold $Y'$ shown with the rationally null-homologous meridian $\mu$. (b) The manifold $Z$, obtained by $(-1)$--surgery on $\mu$.}
\label{fig:MeridinalBlowdown}
\end{figure}
Let us see why the capped off rational Seifert surface has self-intersection $-n^2-n$. We are performing $(-1)$--surgery on a meridian, $\mu$, to one component of the framed link giving $Y$. The effect of this surgery is to attach a 0--framed disk to every $(-1,1)$--curve on $\partial N(\mu)$. If we select $n+1$ of these curves, we get the torus link $T_{n+1,n+1}$. Since this is an $(n+1)$--component link and each component is a meridian, it is homologous to $(n+1)\cdot\mu=0$. So this $T_{n+1,n+1}$ bounds an orientable surface in $Y$. If we attach 0--framed 2--handles to each component, it is easy to see that the intersection among these disks is simply given by the total linking of the components of $T_{n+1,n+1}$. Let $S_2$ be the surface obtained by capping off the $n+1$ boundary components of this orientable surface with these 0--framed disks. Then, $S_2\cdot S_2 = -n(n+1)$.
The following example will be pertinent to our calculations in Section \ref{section:calculations}.
\begin{example}\label{ex:triad_spaces} If $J$ is the unknot, then \begin{eqnarray*} X &=&L(n,1), \\ Y &=& S^3_{n^2+n}(T_{n,n+1}) = L(n,1)\#L(n+1,-1), \text{ and } \\ Z &=&L(n,1)\#L(n,-1). \end{eqnarray*}
In general, \begin{eqnarray*} X & = & S^3_n(J\#J), \\ Y & = & S^3_{n^2+n}(J\#J\#T_{n,n+1}), \text{ and }\hspace{.75in}\hspace{.4in} \\ Z & = & S^3_{n,-n}((J\#J)_{(2,0)}). \end{eqnarray*}
\end{example}
\subsection{Enumerating $\text{Spin}^c$ structures}\label{subsec:enumerate}\
This nice homological set-up gives us natural enumerations of the $\text{Spin}^c$ structures on the manifolds in question. Since $X$ is surgery on a knot in $S^3$, there is an enumeration of $\text{Spin}^c(X)$ by $i\in\mathbb{Z}_n$. Let $\frak s_i\in\text{Spin}^c(X)$ for some $i\in\mathbb{Z}_n$.
Let $[\frak s_i,\frak s_j]\in\text{Spin}^c(Y)$ denote the $\text{Spin}^c$ structure on $Y$ that is cobordant to $\frak s_i$ via a $\text{Spin}^c$ structure $[\frak s_i, \frak t_m]$ with $$\langle c_1([\frak s_i,\frak t_m]),[S_1]\rangle = 2m+n,$$ where $m\in\mathbb{Z}$ is any integer satisfying $m\equiv j\pmod{n+1}$.
Let $[\frak s_i, \frak s_k]\in\text{Spin}^c(Z)$ denote the $\text{Spin}^c$ structure that is cobordant to $[\frak s_i,\frak s_j]$ via $[\frak s_i,\frak r_m]$ with $$\langle c_1([\frak s_i,\frak r_m]),[S_2]\rangle = 2m+n(n+1),$$ where $m\in\mathbb{Z}$ is any integer satisfying $m\equiv j \pmod{n+1}$ and $m\equiv k\pmod n$.
A key feature of this set-up is that we are given affine identifications: \begin{eqnarray*} \text{Spin}^c(X) & \cong & \mathbb{Z}_n \\ \text{Spin}^c(Y) & \cong & \mathbb{Z}_n\oplus\mathbb{Z}_{n+1} \\ \text{Spin}^c(Z) & \cong & \mathbb{Z}_n\oplus\mathbb{Z}_n, \end{eqnarray*} the first and third of which take the unique spin structure to the identity element.
\subsection{Remarks about surgery coefficients}\label{subsec:coefficients}\
In what follows, we will use Heegaard Floer theory to study the manifolds described above. In general, when studying the Heegaard Floer homology of surgeries on knots, calculations become much simpler when dealing with large surgery coefficients. For example, Theorem \ref{thm:surgery_formula}, which we will use extensively, requires that the surgery coefficient be positive and at least $2g-1$, where $g$ is the genus of the knot that is being surgered. The purpose of this subsection is to show that this criterion is met in what follows and to examine the knots $\mathcal K_{p,k_p}$, which will be used in Section \ref{section:infinite_order} to prove Theorem \ref{thm:main2}.
Let $I_{J,p}$ be the knot formed by infecting $T_{2,p}\#T_{2,-p}$ with $J$ along $\eta$, as shown in Figure \ref{fig:InfectedKnots}. Consider $J=\#_kD$, which is a knot of genus $k$. In order to apply Theorem \ref{thm:surgery_formula} to the manifold $X=S^3_p(J\#J)$, we must have $p\geq 2g(J\#J)-1=4k-1$. In order to apply Theorem \ref{thm:surgery_formula} to the manifold $Y=S^3_{p^2+p}(J\#J\#T_{p,p+1})$, we must have $$p^2+p\geq 2g(J\#J\#T_{p,p+1})-1=2\left(2k + \frac{p(p-1)}{2}\right)-1.$$ So, we must have $p\geq \frac{4k-1}{2}$, i.e., $k\leq \frac{2p+1}{4}$. In Section \ref{section:infinite_order}, it will be necessary for us to consider knots where $k\geq\frac{p+5}{12}$. Let $k_p = \lceil\frac{p+6}{12}\rceil$, and define $\mathcal K_{p,k_p} = I_{\#_{k_p}D,p}$. Then, the manifolds associated to $\mathcal K_{p,k_p}$ are surgeries of appropriately large coefficient and $k_p$ is large enough to satisfy the conditions in Section \ref{section:infinite_order}: $$ 4k_p-1 \leq 4\left[\frac{p+6}{12}+1\right]-1 = \frac{p+18}{3}-1 \leq p, $$ and $$ \frac{4k_p-1}{2}\leq \frac{4\left[\frac{p+6}{12}+1\right]-1}{2} = \frac{p+15}{6}\leq p .$$ These inequalities will be satisfied for large $p$, and for small $p$ it is easy to see that the condition on $k_p$ can be relaxed. It should be noted that there are, in general, many values of $k$ that will suffice for each value of $p$, we have simply chosen one that will work for all large values of $p$.
\subsection{Linking forms and Question \ref{question:cancelation}}\label{subsec:question}\
A knot $K\subset S^3$ is called \emph{stably doubly slice} if there exists a doubly slice knot $J$ such that $K\#J$ is doubly slice. Question \ref{question:cancelation} can be rephrased to ask whether there exist stably doubly slice knots that are not doubly slice. In this subsection we show that the correction terms could possibly detect the difference between smoothly doubly slice knots and smoothly stably doubly slice knots.
Analogously, we say that a 3--manifold $M$ \emph{stably embeds} smoothly in $S^4$ if there is a 3--manifold $N$ that embeds smoothly in $S^4$ such that $M\#N$ embeds smoothly in $S^4$. It is not known if such an $M$ must itself embed in $S^4$.
Give a finite abelian group $G$, a \emph{linking form} on $G$ is a non-degenerate, symmetric, bilinear form $\lambda:G\times G\to\mathbb{Q}/\mathbb{Z}$. For every rational homology 3--sphere $M$ there is a linking form $\lambda:H_1(M)\times H_1(M)\to\mathbb{Q}/\mathbb{Z}$ defined by Poincar\'e duality.
Now we will consider \emph{linking triples} $(G,\lambda, f)$, where $G$ is a finite abelian group, $\lambda$ is a linking form on $G$, and $f:G\to\mathbb{Q}$ is a function (not necessarily a homomorphism). Such a triple is called \emph{metabolic} if there is a subgroup $G_1<G$ with $|G_1|^2 =|G|$ such that $\lambda|_{G_1}\equiv 0$ and $f(G_1)=0$. The triple is called \emph{hyperbolic} if $G=G_1\oplus G_2$ with $G_1\cong G_2$ such that $\lambda|_{G_i}\equiv 0$ and $f(G_i)=0$ for $i=1,2$. Note that the set of linking triples has an additive structure given by orthogonal sum.
\begin{lemma}\label{lemma:hyp_cancel} Let $(A,\mu,f)$ and $(B,\nu,g)$ be linking triples. If $(A,\mu,f)$ and $(A\oplus B, \mu\oplus\nu,f\oplus g)$ are both hyperbolic, then $(B,\nu,g)$ is metabolic. \end{lemma}
Though we will use the hypotheses that $(A,\mu,f)$ and $(A\oplus B, \mu\oplus\nu,f\oplus g)$ are hyperbolic, the result hold if these objects are merely metabolic. The following proof is, in essence, due to Kervaire \cite{kervaire} (cf. \cite{gilmer:slice}).
\begin{proof} Let $A=A_0\oplus A_1$ and $A\oplus B = L\oplus M$ be the hyperbolic splittings of $A$ and $A\oplus B$. Let $L_i=L\cap(A_i\oplus B)$ and $M_i=M\cap(A_i\oplus B)$ for $i=0,1$. Let $B_i^L$ and $B_i^M$ be the projections of $L_i$ and $M_i$ onto $B$, respectively. From now on, we will restrict our attention to $B_0^L$.
Let $b,b'\in B_0^L$. Then there exist $a,a'\in A_0$ such that $a\oplus b,a'\oplus b'\in L$. Then, $$\nu(b,b') = \mu(a,a')+\nu(b,b') = \mu\oplus\nu(a\oplus b,a'\oplus b') = 0,$$ and $$ g(b) = f(a) + g(b) = f\oplus g(a\oplus b) = 0.$$
Thus, the restrictions of $\nu$ and $g$ to the $B_0^L$ vanish. Next we show that $|B_0^L|^2=|B|$. Consider the following two short exact sequences: $$ 0 \longrightarrow L_0 \longrightarrow L \stackrel{\pi_{A_1}}{\longrightarrow} L_{A_1} \longrightarrow 0,$$ where $\pi_{A_1}:A\oplus B\to A_1$ is projection onto $A_1<A$, and $$ 0 \longrightarrow L\cap(A_0\oplus0) \longrightarrow L_0 \stackrel{\pi_{B}}{\longrightarrow} B_0^L \longrightarrow 0,$$ where $\pi_B:A\oplus B$ is projection onto $B$.
Next, we claim that $|L\cap(A_0\oplus 0)|\cdot|A_0|\cdot|L_{A_1}|\leq |A|$. Assuming this, we see that
$$|B_0^L| = \frac{|L_0|}{|L\cap(A_0\oplus 0)|}=\frac{|L|}{|L\cap(A_0\oplus 0)|\cdot|L_{A_1}|}\geq \frac{|L|\cdot|A_0|}{|A|} = |B|^{1/2}.$$
Because $B_0^L$ is isotropic and $\nu$ is non degenerate, we have that $|B_0^L|^2=|B|$, as desired. To justify claim assumed above, we will prove that $L\cap(A_0\oplus0)$ is orthogonal to $A_0\oplus L_{A_1}$ under $\mu$. Clearly, $L\cap(A_0\oplus 0)\perp A_0$. Let $u\in L\cap(A_0\oplus0)$ and $w\in L_{A_1}$. Then there exists $v\oplus x\in L_0$ such that $(v+w)\oplus x\in L$. Then, $$\mu(u,w) = \mu(u,w) + \mu(u,v) = \mu(u,w+v) + \nu(0,x) = \mu\oplus\nu(u\oplus0,(w+v)\oplus x) = 0.$$
This shows that $B_0^L$ is a metabolizing summand of $B$. The same is true for $B_1^L, B_0^M,$ and $B_1^M$.\end{proof}
Note that the four metabolizers produced in the proof above are all isomorphic. This follows from the classification of linking forms, specifically the fact that a linking form splits over the homogeneous $p$--group components of the group \cite{wall:linking}. Because of this, we could have performed the above analysis one homogeneous $p$--group component at a time, each of which would split via $L$ and $M$.
Next, we investigate how these metabolizers sit inside $A$ and $B$. Suppose that $A$ and $B$ are homogeneous $p$--groups with a common exponent and have ranks $2r$ and $2s$, respectively. Without loss of generality, we can write $$L = \langle (a_1, b_1), \ldots, (a_t,b_t), (0,b_{t+1}), \ldots, (0, b_{t+l}), (a_{t+l+1},0),\ldots, (a_{r+s}, 0),$$ where the $b_i$ are linearly independent, and the $a_j$ are linearly independent. Let $t'=r-l$. Without loss of generality, we can assume that $a_1, \ldots, a_{t'}\in A_0$ and $a_{t'+1},\ldots, a_{2t'}\in A_1$ (by consideration of the ranks of $B_0^L$ and $B_1^L$). Since $\nu$ is non-degenerate, we can assume that, for $0\leq i\leq t'$ and $t'+1\leq j\leq 2t'$, $\nu(b_i, b_j)\not=0$ if and only if $j=t'+i$ (perform change of bases within these rank $t'$ summands). Note that $B/\langle b\rangle^\perp$ has rank one for each $b\in B$.
Clearly, $t+l\leq 2s$, and, in fact, we have that $t$ is even with $t/2+l = r$, i.e., $t=2t'$. This claim follows from the $\nu$ being non-degenerate; if $t/2<r-l$, there is an element $(a_{2t'+1}, b_{2t'+1})$ that an be assumed to have the property that $\nu(b_{2t'+1}, b_i)=0$ for all $0\leq i\leq t'$. However, if this were the case, then $\langle b_1,\ldots, b_{t'},b_{2t'+1}, b_{t+1}, \ldots, b_{t+l}\rangle$ would have rank $r+1$ and be isotropic, a contradiction.
It follows that each $a_i$ for $0\leq i\leq t$ is in either $A_0$ or $A_1$. Together with a similar argument for $M$, we get that $\pi_B(L_0+L_1+M_0+M_1) =B$. In particular, $B = B_0^L+B_1^L+B_0^M+B_1^M$. We can use this to prove a simple corollary.
\begin{corollary}\label{coro:rank4} Let $(A,\mu,f)$ and $(B,\nu,g)$ be linking triples. If $(A,\mu,f)$ and $(A\oplus B, \mu\oplus\nu,f\oplus g)$ are both hyperbolic, and if each homogeneous $p$--group component of $B$ is at most rank 4, then $(B,\nu,g)$ is hyperbolic. \end{corollary}
\begin{proof} Since $B$ is rank 4 and spanned by four metabolizers of rank 2 (by the comments above), either some pair of the metabolizers are disjoint, or there is an element $b$ common to each of the four metabolizers. However, the latter case implies that $(0,b)\in L\cap M$, a contradiction. Thus, there is a pair giving a hyperbolic splitting of $(B,\nu,g)$. If $B$ is rank 2, a similar argument works. \end{proof}
Next, we give a counterexample that shows that Corollary \ref{coro:rank4} is as strong as possible, in some sense.
\begin{example}\label{ex:rank6} Let $A\cong\mathbb{Z}_p^6=\langle z_1, w_1, z_2, w_2, z_3, w_4\rangle$ and let $B\cong\mathbb{Z}_p^6=\langle x_1, y_1, x_2, y_2, x_3, y_4\rangle$. Let $A_0 = \langle z_1, z_2, z_3\rangle$ and $A_1 = \langle w_1, w_2, w_3\rangle$. With respect to these bases, let $\mu$ and $\nu$ be linking forms given by $$\bigoplus_3\begin{pmatrix} 0 & -2/p \\ -2/p & 0 \end{pmatrix} \text{ and } \bigoplus_3\begin{pmatrix} 0 & 2/p \\ 2/p & 0 \end{pmatrix},$$ respectively. Consider the splitting $A\oplus B=L\oplus M$, where $$L =\langle (z_1, x_1), (z_2, x_2), (w_1, y_1), (w_2, y_2), (0, x_3), (w_3, 0)\rangle,$$ and $$M =\langle (z_1, y_2), (z_3,x_1), (w_1, x_2), (w_3, y_1), (0, y_3), (w_2, 0)\rangle.$$ It is straightforward to check that $L\cap M=0$ and that $L+M=A\oplus B$. Furthermore, it is obvious that $\mu\oplus\nu$ vanishes on both $L$ and $M$. Next, notice that \begin{eqnarray*} B_0^L & = & \langle x_1, x_2, x_3\rangle \\ B_1^L & = & \langle y_1, y_2, x_3\rangle \\ B_0^M& = & \langle x_1, y_2, y_3\rangle \\ B_1^M & = & \langle y_1, x_2, y_3\rangle. \end{eqnarray*} No pair of these metabolizers is disjoint. Define $g:B\to\mathbb{Q}$ by $$g(b) = \begin{cases} 0 & \text{ if $b\in B_0^L\cup B_1^L\cup B_0^M\cup B_1^M$}, \\ 1 & \text{ otherwise} \end{cases}.$$ Define $f:B\to\mathbb{Q}$ by $$f(a) = \begin{cases} -g(b_a) & \text{ if $a\not\in A_0\cup A_1$}, \\ 0 & \text{ if $a\in A_0\cup A_1$} \end{cases},$$ Where $a\mapsto b_a$ its the isomorphism from $A$ to $B$ that sends the $z_i$ to the $x_i$ and the $w_i$ to the $y_i$.
With this set up, it is clear that $(A,\mu, f)$ is hyperbolic and that $g:B\to \mathbb{Q}$ is not hyperbolic. It remains to show that $f\oplus g:A\oplus B\to\mathbb{Q}$ vanishes on $L$ and $M$. This will imply that $(A\oplus B, \mu\oplus\nu, f\oplus g)$ is hyperbolic, thus exemplifying the necessity of the rank restriction in Corollary \ref{coro:rank4}.
Let $l\in L\cup M$ with $l=(a,b)$. It suffices to check that $f(a)=0$ if $b$ is in one of the metabolizers listed above and that $a\not\in A_0\cup A_1$ if $b$ is not in one of these metabolizers. It is straightforward to check that these criteria are met.
\end{example}
Let $K\subset S^3$, and let $\mathcal A=(A, \mu,f)$ be the linking triple associated to $\Sigma_2(K)$, i.e., $A=H_1(\Sigma_2(K))$, $\mu$ is the linking from on $A$, and $f(a) = d(\Sigma_2(K),\frak s_a)$, where $\frak s_a$ is the $\text{Spin}^c$ structure corresponding to $a\in H_1(\Sigma_2(K))$. Let $\mathcal A_{p^k}$ denote the restriction of this triple to the homogeneous $p^k$--group component of $A$. We have shown the following.
\begin{proposition}\label{prop:stable_terms}
Let $K\subset S^3$ and let $\mathcal A$ be the associated linking triple. Suppose that $\det(K) = |A| = p_1^{k_1}\cdots p_n^{k_n}$. \begin{enumerate} \item If $K$ is smoothly doubly slice, then $\mathcal A$ is hyperbolic. \item If $K$ is smoothly stably doubly slice, then $\mathcal A_{p_i^{k_i}}$ is hyperbolic whenever $k_i\leq 4$. \end{enumerate} \end{proposition}
Note that (1) is a restatement of Theorem \ref{thm:hyp_corr_terms}. We will use this result in Sections \ref{section:calculations} and \ref{section:infinite_order} to help prove Theorems \ref{thm:main} and \ref{thm:main2}.
\section{Heegaard Floer homology}\label{section:HF}
Below, we collect some basic facts about the suite of invariants known as Heegaard Floer homology. For complete details, see (for example) \cite{oz-sz:absolute,oz-sz:knots,oz-sz:3-manifolds_1}. Throughout, let $\mathbb{F}$ denote the field with two elements.
\subsection{3--manifold invariants}\
Let $M$ be a closed 3--manifold, and let $\frak s\in\text{Spin}^c(M)$ be a torsion $\text{Spin}^c$ structure on $M$. Heegaard Floer homology theory associates to $(M,\frak s)$ a $\mathbb{Z}$--filtered, $\mathbb{Q}$--graded chain complex $CF^\infty$, which is well-defined up to filtered chain homotopy equivalence. This complex is a free, finitely generated $\mathbb{F}[U,U^{-1}]$--module. The action of $U$ lowers the filtration level by one, and lowers the grading by two. Henceforth, if $C$ is any filtered, graded chain complex, then $C_{\{i\leq n\}}$ denotes the subcomplex consisting of elements of filtration level at most $n$.
Denote the associated homology group by $HF^\infty(M,\frak s)$. If $M$ is a rational homology 3--sphere, it turns out that these groups are uninteresting. Let $\mathcal T^\infty = \mathbb{F}[U,U^{-1}]$. Then, for any rational homology 3--sphere $M$ and any $\frak s\in\text{Spin}^c(M)$, we get $HF^\infty(M,\frak s)\cong\mathcal T^\infty$. This means that any interesting information about $(M,\frak s)$ must be stored at the chain complex level.
Indeed, there are associated sub- and quotient-complexes: $$CF^-(M,\frak s) = CF^\infty(M,\frak s)_{\{i<0\}},$$ $$CF^+(M,\frak s) = CF^\infty(M,\frak s)/CF^-(M,\frak s),$$ and $$\widehat{CF}(M,\frak s) = CF^\infty(M,\frak s)_{\{i\leq0\}}/CF^-(M,\frak s).$$
The corresponding homology groups, $HF^-(M,\frak s)$, $HF^+(M,\frak s)$ , and $\widehat{HF}(M,\frak s)$ turn out to be very powerful 3--manifold invariants. These groups are related by two important long exact sequences: $$ \cdots \longrightarrow HF^-(M,\frak s) \stackrel{\iota}{\longrightarrow} HF^\infty(M,\frak s) \stackrel{\pi}{\longrightarrow} HF^+(M,\frak s) \longrightarrow\cdots$$ and $$ \cdots \longrightarrow \widehat{HF}(M,\frak s) \stackrel{\hat\iota}{\longrightarrow} HF^+(M,\frak s) \stackrel{U}{\longrightarrow} HF^+(M,\frak s) \longrightarrow\cdots.$$
Note that $\widehat{HF}(M,\frak s)$ is a finitely generated $\mathbb{F}$--vector space. Define $$HF_{red}(M,\frak s) = HF^+(M,\frak s)/\text{Im}(\pi).$$ Let $\mathcal T^+ = \mathbb{F}[U, U^{-1}]/U\cdot\mathbb{F}[U]$. If $M$ is a rational homology 3--sphere, we have the following decomposition: $$HF^+(M,\frak s) = \mathcal T^+\oplus HF_{red}(M,\frak s).$$ It turns out that the grading of the element of lowest grading living in $\mathcal T^+$, which we call the tower part of $HF^+(M,\frak s)$, is an interesting invariant called the \emph{correction term}.
\begin{definition} The \emph{correction term} (or \emph{$d$--invariant}) of $(M,\frak s)$ is denoted $d(M,\frak s)$ and is given by $$\min\{gr(\pi(\alpha)):\alpha\in HF^\infty(M,\frak s)\}.$$ \end{definition}
The correction term enjoys a number of nice properties, including the fact that $d$ is a $\text{Spin}^c$ rational homology cobordism invariant (see \cite{oz-sz:absolute}): \begin{enumerate} \item $d(M_1\#M_2, \frak s_1\#\frak s_2) = d(M_1,\frak s_1)+d(M_2,\frak s_2)$, \item $d(-M,\frak s) = -d(M,\frak s)$, where $-M$ denotes $M$ with the opposite orientation, and
\item $d(M,\frak s) = 0$ whenever $(M,\frak s) = \partial (W,\frak t)$, where $W$ is a rational-homology 4--ball, and $\frak t|_{\partial W} = \frak s$. \end{enumerate} This last property is key in the proof of Theorem \ref{thm:hyp_corr_terms}.
As mentioned above, there are affine identifications $\text{Spin}^c(M)\cong H^2(M;\mathbb{Z})$, so a rational homology 3--sphere $M$ will have $|H^2(M)|$ correction terms. We will denote the collection of correction terms associated to $M$ by $\mathcal D(M)$. When possible, the group structure of $H^2(M)$ will be implicit in our presentation of $\mathcal D(M)$. For example, in \cite{oz-sz:absolute} a formula for the correction terms of lens spaces is given. In particular,
\begin{equation}\label{eqn:lens_corr_terms} d(L(p,1),i) = \frac{p-(2i-p)^2}{4p}. \end{equation}
\begin{example}\label{ex:lens_corr_terms} Consider the case from Section \ref{section:geometry} when $J$ is unknotted and $n=5$. Then, $Y=L(5,1)$, and Equation \ref{eqn:lens_corr_terms} tells us that, $$\mathcal D(L(5,1)) = \{1,1/5,-1/5,-1/5,1/5\}.$$ By the additivity of the correction terms, we have the following: $$\mathcal D(L(5,1)\#L(5,-1)) = \left\{ \begin{array}{ccccc} 0 & 4/5 & 6/5 & 6/5 & 4/5 \\ -4/5 & 0 & 2/5 & 2/5 & 0 \\ -6/5 & -2/5 & 0 & 0 & -2/5 \\ -6/5 & -2/5 & 0 & 0 & -2/5 \\ -4/5 & 0 & 2/5 & 2/5 & 0 \end{array} \right\}.$$
Note that implicit in the presentation matrix is the affine identification $\text{Spin}^c(L(n,1)\#L(n,-1))\cong\mathbb{Z}_5\oplus\mathbb{Z}_5$ given by $[\frak s_i,\frak s_j]\sim (i,j)$. For example, the correction terms vanish on all elements of the subgroups generated by $(1,1)$ and $(1,4)$ in $\mathbb{Z}_5\oplus\mathbb{Z}_5$.
It will sometimes be helpful to write such collections as follows: $$\mathcal D(L(5,1)) = \{-1/5, 1/5,1,1/5,-1/5\}.$$ and $$\mathcal D(L(5,1)\#L(5,-1)) = \left\{ \begin{array}{ccccc} 0 & -2/5 & -6/5 & -2/5 & 0 \\ 2/5 & 0 & -4/5 & 0 & 2/5 \\ 6/5 & 4/5 & 0 & 4/5 & 6/5 \\
2/5 & 0 & -4/5 & 0 & 2/5 \\ 0 & -2/5 & -6/5 & -2/5 & 0 \\ \end{array} \right\}.$$ The only difference here, is that we have centered our indexing set about zero, using $$\{-(p-1)/2, -(p-3)/2,\ldots,-1,0,1,\ldots (p-3)/2,(p-1)/2\}$$ to index $\mathbb{Z}_p$ instead of \{$0,1,2,\ldots, p-1\}$. \end{example}
\subsection{The surgery exact triangle and 4--dimensional cobordisms}\label{subsec:cobordism}\ A $\text{Spin}^c$--cobordism between two $\text{Spin}^c$ 3--manifolds induces certain maps between the Heegaard Floer homology groups associate to the two manifolds. We now turn our attention to some aspects of these induced maps.
Let $M$ be a rational homology 3--sphere, and let $K$ be a null-homologous knot in $M$. Let $M_0$ be the result of $N$--surgery on $K$, and let $M_1$ be the result of $(N+1)$--surgery on $K$. This is a special case of a broader context in which the triple $(M, M_0, M_1)$ is called a \emph{triad}. For a discussion relevant to this subsection, see \cite{oz-sz:lectures}. Implicit in this set up is a triple of cobordisms obtained by 2--handle addition (cf. Subsection \ref{subsec:triad}).
$$\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=4em,column sep=4em,minimum width=2em] {
M& \ & M_1 \\
\ & M_0 & \ \\};
\path[-stealth]
(m-1-1) edge node [pos=.6,left] {$W_1\ $} (m-2-2)
(m-2-2) edge node [pos=.4,right] {$\ W_2$} (m-1-3)
(m-1-3) edge node [above] {$W_3$} (m-1-1); \end{tikzpicture}$$
\begin{theorem} Let $(M, M_0, M_1)$ be a triad, then there exist exact triangles relating their Heegaard Floer homologies: $$\begin{array}{cc} \begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=2.8em,column sep=1.25em,minimum width=1em] {
\widehat{HF}(M)& \ & \widehat{HF}(M_1) \\
\ & \widehat{HF}(M_0) & \ \\};
\path[-stealth]
(m-1-1) edge node [pos=.6,left] {$\widehat F_1\ $} (m-2-2)
(m-2-2) edge node [pos=.4,right] {$\ \widehat F_2$} (m-1-3)
(m-1-3) edge node [above] {$\widehat F_3$} (m-1-1); \end{tikzpicture} & \begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=1.25em,minimum width=1em] {
HF^+(M)& \ & HF^+(M_1) \\
\ & HF^+(M_0) & \ \\};
\path[-stealth]
(m-1-1) edge node [pos=.6,left] {$F^+_1\ $} (m-2-2)
(m-2-2) edge node [pos=.4,right] {$\ F^+_2$} (m-1-3)
(m-1-3) edge node [above] {$F^+_3$} (m-1-1); \end{tikzpicture} \end{array}$$ These maps are induced by the 2--handle cobordisms relating the triad. \end{theorem}
Moreover, the grading shifts associated to these induced maps are given by the following formula: $$gr(F^\circ_i) = gr(F^\circ_i(x))-gr(x) = \frac{(c_1(\frak t))^2 -2\chi(W_i)-3\sigma(W_i)}{4}.$$
This set-up can be applied to the 3--manifolds and 4--dimensional cobordisms introduced in Section \ref{section:geometry}. Below, we will use these exact triangles to understand the Heegaard Floer homology of $Z$ (i.e., $M_1$) via the Heegaard Floer homology of $X$ and $Y$ (i.e., $M$ and $M_0$), which are more tractable, since they are each realized by surgery on knots in $S^3$.
In addition to this nice set-up, we have two important theorems about the behavior of these maps on certain types of cobordisms.
\begin{theorem}[\cite{oz-sz:absolute}]\label{thm:infty_isom} Let $W$ be a cobordism between rational homology 3--manifolds obtained by surgery on a knot such that $b_2^+(W)=0$. Then $F^\infty_{W,\frak t}$ is an isomorphism for all $\frak t\in\text{Spin}^c(W)$. \end{theorem}
The following theorem is implicit in the work of Ozsv\'ath and Szab\'o \cite{oz-sz:absolute}, and can also be found in \cite{lisca-stipsicz}.
\begin{theorem}\label{thm:hat_vanish} Let $W$ be a cobordism induced by attaching a 2--handle to a rational homology 3--sphere, and let $\frak t\in\text{Spin}^c(W)$. Suppose that $W$ contains a smoothly embedded, closed, orientable surface $\Sigma$ with $g(\Sigma)>0$ such that
$$\Sigma\cdot\Sigma\geq 0 \text{ and } |\langle c_1(\frak t),[\Sigma]\rangle| + \Sigma\cdot\Sigma > 2g(\Sigma) - 2.$$ Then $\widehat F_{W,\frak t}$ is zero. \end{theorem}
For example, if the 2--handle attachment occurs along a knot with large positive framing relative to its genus, the induced map $\widehat F_{W,\frak t}$ will vanish for all $\frak t\in\text{Spin}^c(W)$.
\subsection{Knot complexes}\
A rationally null-homologous knot $K$ in $M$ induces a second filtration on $CF^\infty(M,\frak s)$, which thus becomes a $\mathbb{Z}\oplus\mathbb{Z}$--filtered, $\mathbb{Q}$--graded complex, and is denoted $CFK^\infty(M,K,\frak s)$. The action of $U$ lowers both filtrations by one, and lowers the grading by two. For our purposes, the most important aspect of this complex is that it can be used to determine the Heegaard Floer homology of surgeries on $K$.
For a positive integer $p$, let $\frak s_m$ denote the element of $\text{Spin}^c(S^3_p(K))$ which is $\text{Spin}^c$ cobordant to the unique $\text{Spin}^c$ structure on $S^3$ via an element $\frak t_m\in\text{Spin}^c(W)$ (where $W$ is the 2--handle cobordism induced by $p$--surgery) satisfying $$\langle c_i(\frak t_m, [S]\rangle +p = 2m,$$ where $S$ denotes a Seifert surface for $K$, capped off with the core of the 2--handle. Then the following theorem is stated as in \cite{HLR}, but is originally proved in \cite{oz-sz:knots}.
\begin{theorem}\label{thm:surgery_formula}
Let $K$ be a knot in $S^3$, and suppose that $g(K)=g$. Let $p\geq 2g-1$. Then for all $m$ satisfying $|m|\leq \frac{1}{2}(p-1)$, there is a chain homotopy equivalence of graded complexes over $\mathbb{F}[U]$: $$CF_k^+(S^3_p(K),\frak s_m)\simeq CFK^\infty_l(M,K,\frak s)_{\{\max(i,j-m)\geq 0\}},$$ where $$k=l+\frac{p-(2m-p)^2}{2p}.$$ \end{theorem}
Equation \ref{eqn:lens_corr_terms} can be viewed a special case of this (i.e., when $K$ is the unknot). We make extensive use of this theorem in the calculations required by the proof in Section \ref{section:calculations}, which can be found in Appendix \ref{appendix}.
One corollary of this set-up is that the correction terms of manifolds obtained by surgery on knots can be compared to those of lens spaces. We refer the reader to \cite{ni-wu:cosmetic,ni-wu:rational} for a nice development. In short, by considering $CFK^\infty(S^3,K)$, one can define two sequences of nonnegative integers $V_k, H_k$ for $k\in\mathbb{Z}$ satisfying $$V_k = H_{-k},\hspace{.25in} V_k\geq V_{k+1}\geq V_k-1, \hspace{.25in} V_{g(K)}=0.$$ It turns out that the correction terms of surgeries on $K$ are determined by these integers.
\begin{theorem}\label{thm:ni-wu} Let $K$ be a knot in $S^3$. Then, $$d(S^3_p(K),i) = d(L(p,1),i) - 2\max\{V_i,H_{i-p}\}.$$ \end{theorem}
\section{Proof of Theorem \ref{thm:main}}\label{section:calculations}
In this section, we prove the following proposition, whose corollary, together with Corollary \ref{coro:doubly_slice}, implies Theorem \ref{thm:main}. Recall the geometric set-up from Section \ref{section:geometry}. In particular, let $\mathcal Z_{p,k_p}$ be the double branched cover of the knot $\mathcal K_{p,k_p}$.
\begin{proposition}\label{prop:X_corr_terms} The difference $\mathcal D(L(p,1)\#L(p,-1)) - \mathcal D(\mathcal Z_{p,k_p})$ is given by the following matrix $\mathcal M$. $$ { \left[ \setlength{\arraycolsep}{3.5pt} \begin{array}{cccccccccccccccccccccc} 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0& \cdots & 0 \\ \vdots & & \vdots &\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots && \vdots& \vdots & & \vdots \\ 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0& \cdots & 0 \\ 2 & \cdots & \textcolor{Black}{2} & 0 & \textcolor{Black}{0} & 0 & 0 & 0 &0 & 0 & 2 & 2 & 2 & 2 & \cdots & 2 & 2 & \cdots & 2 \\ 2 & \cdots & 2 & \textcolor{Black}{2} & 0 & \textcolor{Black}{0} & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & \cdots & 2 & 2 & \cdots & 2 \\ 4 & \cdots & 4 & 4 & \textcolor{Black}{2} & 0 & \textcolor{Black}{0} & 0 & 0 & 0 & 4 & 4 & 4 & 4 & \cdots & 4 & 4 & \cdots & 4 \\ 4 & \cdots & 4 & 4 & 4 & \textcolor{Black}{2} & 0 & \textcolor{Black}{0} & 0 & 0 & 4 & 4 & 4 & 4 & \cdots & 4 & 4 & \cdots & 4\\ \vdots & & \vdots & \vdots & & & \textcolor{Black}{\ddots} & \ddots & \textcolor{Black}{\ddots} & & \vdots & \vdots & \vdots &\vdots & & \vdots & \vdots& & \vdots \\ 2k_p & \cdots & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & \textcolor{Black}{2} & 0 & \textcolor{Black}{0} & 2k_p & 2k_p & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & \cdots & 2k_p \\ 2k_p & \cdots & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & 2k_p & \textcolor{Black}{2} & 0 & \textcolor{Black}{2} & 2k_p & 2k_p &2k_p & \cdots & 2k_p & 2k_p & \cdots & 2k_p \\ 2k_p & \cdots & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & 2k_p & 2k_p & \textcolor{Black}{0} & 0 & \textcolor{Black}{2} & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & \cdots & 2k_p \\ \vdots & & \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & & \textcolor{Black}{\ddots} & \ddots & \textcolor{Black}{\ddots} & & & \vdots & \vdots& & \vdots \\ 4 & \cdots & 4 & 4 & \cdots & 4 & 4 & 4 & 4 & 0 & 0 & \textcolor{Black}{0} & 0 & \textcolor{Black}{2} & 4 & 4 & 4 & \cdots & 4 \\ 4 & \cdots & 4 & 4 & \cdots& 4 & 4 & 4 &4 & 0 & 0 & 0 & \textcolor{Black}{0} & 0 & \textcolor{Black}{2} & 4 & 4 & \cdots & 4 \\ 2 & \cdots & 2 & 2 & \cdots& 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & \textcolor{Black}{0} & 0 & \textcolor{Black}{2} & 2 & \cdots & 2 \\ 2 & \cdots & 2 & 2 & \cdots & 2 & 2 & 2 & 2& 0 & 0& 0 & 0 & 0 & \textcolor{Black}{0} & 0 & \textcolor{Black}{2} & \cdots & 2 \\ 0 & \cdots & 0 & 0 & \cdots & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & \cdots & 0 \\ \vdots & & \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots& \vdots & \vdots & \vdots & & \vdots \\ 0 & \cdots & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 \\ \end{array} \right]. } $$ \end{proposition}
This matrix presentation makes use of the affine identification $\text{Spin}^c(\mathcal Z_{p,k_p})\cong H^2(\mathcal Z_{p,k_p})\cong\mathbb{Z}_p\oplus\mathbb{Z}_p$, where $(i,j)\in\mathbb{Z}_p\oplus\mathbb{Z}_p$ is such that $|i|,|j|\leq \frac{p-1}{2}$. There is an indeterminacy present that must be discussed. In Appendix \ref{appendix}, the calculation of the correction terms for $Y =S^3_{p^2+p}(J\#J\#T_{p,p+1})$ (with $J=\#_{k_p}D$) is done in a way that forgets the explicit identification of $\text{Spin}^c(Y)\cong H^2(Y)\cong\mathbb{Z}_p\oplus\mathbb{Z}_{p+1}$. Thus, we lose track of the difference between $j$ and $-j$ in $\mathbb{Z}_{n+1}$ and between $i$ and $-i$ in $\mathbb{Z}_n$. As a consequence, when regarding the matrix above, we must consider it only up to horizontal reflection about the central column and vertical reflection about the central row. This indeterminacy is inconsequential in what follows. In particular, the following corollary holds.
\begin{corollary}\label{coro:terms} The manifold $\mathcal Z_{p,k_p}$ has precisely $2p-2k_p-1$ vanishing correction terms. Therefore, $\mathcal K_{p,k_p}$ is not smoothly doubly slice. Moreover, the $\mathcal K_{p,k_p}$ are nontrivial in $\mathcal C_\mathcal D$. \end{corollary}
\begin{proof} The $(p\times p)$--matrix for $\mathcal D(L(p,1)\#L(p,-1))$ has zeros along the two (orthogonal) diagonals and non-integer rational numbers elsewhere. The corresponding entries in the matrix for $\mathcal D(Z_{p,k_p})$ are lowered by even integers, corresponding to the matrix $\mathcal M$ by Proposition \ref{prop:X_corr_terms}. It is easy to see that precisely $2k_p$ of the $2p-1$ vanishing entries in $\mathcal D(L(p,1)\#L(p,-1))$ will be lowered by a nonzero amount. These changes correspond to the non-zero entries of $\mathcal M$ on the cross-diagonal. Since all entries are changed by an even integer, no new zeros will be created. Therefore, $\mathcal Z_{p,k_p}$ has precisely $2p-2k_p-1$ vanishing correction terms. By Theorem \ref{thm:hyp_corr_terms}, this implies that the $\mathcal K_{p,k_p}$ are not smoothly doubly slice. In fact, by Proposition \ref{prop:stable_terms}, this implies that each $\mathcal K_{p,k_p}$ is not even smoothly stably doubly slice, since $\det(\mathcal K_{p,k_p}) = p^2$. Therefore, each $\mathcal K_{p,k_p}$ represents a nontrivial element in $\mathcal C_\mathcal D$. \end{proof}
\subsection{Notation and set-up}\
Let $X=S^3_n(K)$, and let $[\frak s_i]\in\text{Spin}^c(X)$ be the enumeration of $\text{Spin}^c(X)$ introduced in Subsection \ref{subsec:enumerate}. Then we have the following decomposition: $$HF^\infty(X) = \bigoplus_{i=0}^{n-1} HF^\infty(X,\frak s_i)=\bigoplus_{i=0}^{n-1}\mathcal T^\infty_i(X).$$ Note that here and throughout, subscripts will correspond to the labelings of $\text{Spin}^c$ structures on the manifolds. Theorem \ref{thm:surgery_formula} implies that, for any $x\in\mathcal T^\infty_i(X)$, $$gr(x)\equiv d(L(n,1),i) \pmod 2$$ for all $i\in\mathbb{Z}_n$. Let $\bar x^\infty_i$ denote the element in $\mathcal T^\infty_i(X)$ such that $$gr(\bar x^\infty_i)=d(L(n,1),i).$$
Let $Y = X_{-n-1}(K)$ for a null homologous knot $K$ in $X$, and let $[\frak s_i,\frak s_j]\in\text{Spin}^c(Y)$ be the enumeration of $\text{Spin}^c(Y)$, as in Subsection \ref{subsec:enumerate}. This gives the following decomposition: $$HF^\infty(Y) = \bigoplus_{i=0}^{n-1}\bigoplus_{j=0}^nHF^\infty(Y,[\frak s_i, \frak s_j]) = \bigoplus_{i=0}^{n-1}\bigoplus_{j=0}^n\mathcal T^\infty_{i,j}(Y).$$
Let $F^\infty_{W_1,[\frak s_i,\frak t_m]}:HF^\infty(X,\frak s_i)\to HF^\infty(Y, [\frak s_i,\frak s_j])$ be the map induced by $(W_1,[\frak s_i,\frak t_m])$, as in Subsection \ref{subsec:cobordism}. Since $W_1$ is negative definite, we can conclude (see \cite{oz-sz:absolute}) that $F_{W_1,\frak t}^\infty$ is an isomorphism for all $\frak t\in\text{Spin}^c(W_1)$. Furthermore, $$gr\left(F^\infty_{W_1,[\frak s_i,\frak t_m]}\right)= \frac{(n+1)-(2m+(n+1))^2}{4(n+1)}$$ for each $i\in\mathbb{Z}_n$. In general, if $F$ is any graded map between graded abelian groups, we denote the grading shift of $F$ by $gr(F)$.
\begin{lemma}\label{lemma:mod2} For all $i\in\mathbb{Z}_n$ and $j\in\mathbb{Z}_{n+1}$, let $y$ be any element in $\mathcal T_{i,j}^\infty(Y)$, then $$gr(y)\equiv gr(L(n,1),i)-gr(L(n+1,1),j) \pmod 2.$$ \end{lemma}
\begin{proof} The fact that $F^\infty_{W_1,[\frak s_i,\frak t_{m}]}$ is an isomorphism, and the labeling of $\text{Spin}^c$ structures, implies that $F^\infty_{W_1,[\frak s_i,\frak t_m]}(x_i^\infty)\subset\mathcal T^\infty_{i,j}$ if and only if $m\equiv j\pmod{n+1}$. Let $m=-j$, then, since all elements in $\mathcal T^\infty_{i,j}$ can be obtained from each other by translation by $U$, \begin{eqnarray*} gr(y) & \equiv & gr\left(F^\infty_{W_1,[\frak s_i,\frak t_{-j}]}(\bar x_i^\infty)\right) \pmod 2 \\ & \equiv & gr(\bar x_i^\infty) + gr(F^\infty_{W_1,[\frak s_i,\frak t_{-j}]}) \\ & \equiv & d(L(n,1),i)-d(L(n+1,1),j) \end{eqnarray*} \end{proof}
Let $\bar y^\infty_{i,j}$ denote the element in $\mathcal T^\infty_{i,j}(Y)$ satisfying $$gr(\bar y_{i,j}^\infty) = d(L(n,1),i)-d(L(n+1,1),j).$$ Using this notation, we gain a precise understanding of the map $$ F^\infty_1 = \sum_{\frak t\in\text{Spin}^c(W_1)}F^\infty_{W_1,\frak t},$$ given by the following lemma. Note that $F_1^\infty$ is not a well-defined map to $HF^\infty(Y)$, since its image will generally consist of infinite sums of elements in $HF^\infty(Y)$. The important fact for us is that all but finitely many of the terms will have coefficients that are large powers of $U$.
\begin{lemma}\label{lemma:F1_infty} Let the $\bar x_i^\infty$ and $\bar y_{i,j}^\infty$ be defined as above. Then, for all $i\in\mathbb{Z}_n$, $$F^\infty_1(\bar x_i^\infty) = (\bar y^\infty_{i,1} + \bar y_{i,2}^\infty + \cdots + \bar y_{i,n}^\infty) + U(\bar y_{i,1}^\infty+\bar y_{i,n}^\infty) + U^2(\bar y_{i,2}^\infty + \bar y_{i,n-1}^\infty)+\cdots,$$ where the expression continues indefinitely with increasing positive powers of $U$ as coefficients. \end{lemma}
\begin{proof} The proof of this lemma is a simple examination of $gr(F^\infty_{W_1,[\frak s_i,\frak t_m]})$ as $m$ varies over the integers. The powers of $U$ in the tail follow a growth pattern that depends quadratically on $n$ in a simple way, but will not be relevant in what follows. \end{proof}
Continuing, let $Z$ be obtained from $Y$ by blowing down a meridian, as in Subsection \ref{subsec:triad}, let $W_2$ be the induced cobordism, and let $[\frak s_i,\frak s_k]\in\text{Spin}^c(Z)$ be the enumeration of $\text{Spin}^c(Z)$, as in Subsection \ref{subsec:enumerate}. This gives the following decomposition: $$HF^\infty(Z) = \bigoplus_{i=0}^{n-1}\bigoplus_{k=0}^{n-1}HF^\infty(Z,[\frak s_i, \frak s_k]) = \bigoplus_{i=0}^{n-1}\bigoplus_{k=0}^{n-1}\mathcal T^\infty_{i,k}(Z).$$
Let $F^\infty_{W_2,[\frak s_i,\frak r_m]}:HF^\infty(Y,[\frak s_i,\frak s_j])\to HF^\infty(Z, [\frak s_i,\frak s_k])$ be the map induced by $(W_2,[\frak s_i, \frak r_m])$. Since $W_2$ is negative definite, we can conclude that $F_{W_2,\frak r}^\infty$ is an isomorphism for all $\frak r\in\text{Spin}^c(W_2)$. Furthermore, $$gr\left(F^\infty_{W_2,[\frak s_i,\frak r_m]}\right) = \frac{n(n+1)-(2m+n(n+1))^2}{4n(n+1)}$$ for each $i\in\mathbb{Z}_n$ and $j\in\mathbb{Z}_{n+1}$.
\begin{lemma} Let $z$ be any element in $\mathcal T_{i,k}^\infty(Z)$, then $$gr(z)\equiv gr(L(n,1),i)-gr(L(n,1),k) \pmod 2$$ for all $i\in\mathbb{Z}_n$ and $k\in\mathbb{Z}_{n}$. \end{lemma}
\begin{proof} This proof is identical to that of Lemma \ref{lemma:mod2}. \end{proof}
Let $\bar z^\infty_{i,k}$ denote the element in $\mathcal T^\infty_{i,k}(Z)$ satisfying $$gr(\bar z_{i,k}^\infty) = d(L(n,1),i)-d(L(n,1),j).$$ Using this notation, we gain a precise understanding of the map $$ F^\infty_2 = \sum_{\frak r\in\text{Spin}^c(W_2)}F^\infty_{W_2,\frak r},$$ in an analogous way to Lemma \ref{lemma:F1_infty}. From this point on, we will index $H_1(X)\cong\mathbb{Z}_n,H_1(Y)\cong\mathbb{Z}_n\oplus\mathbb{Z}_{n+1}$, and $H_1(Z)\cong\mathbb{Z}_n\oplus\mathbb{Z}_n$ by $i,(i,j)$, and $(i,k)$ (respectively), such that $-\frac{n-1}{2}\leq i,k\leq \frac{n-1}{2}$ and $-\frac{n+1}{2}\leq j\leq \frac{n-1}{2}$.
\begin{lemma}\label{lemma:F2_infty} Let the $\bar y_{i,j}^\infty$ and $\bar z_{i,k}^\infty$ be defined as above. Then, for all $i\in\mathbb{Z}_n$, \begin{eqnarray*} F^\infty_2(\bar y_{i,0}^\infty) & = & \bar z^\infty_{i,0} + U(\bar z_{i,1}^\infty+\bar z_{i,n-1}^\infty) + U^5(\bar z_{i,2}^\infty+ \bar z_{i,n-2}^\infty)+\cdots, \\ F^\infty_2(\bar y_{i,j}^\infty) & = & \bar z^\infty_{i,j-1}+ \bar z^\infty_{i,j} + U(\bar z_{i,j+1}^\infty) + U^3(\bar z_{i,j-2}^\infty)+ \cdots, \end{eqnarray*}
if $|j|=1$, and \begin{eqnarray*} \hspace{.7in} F^\infty_2(\bar y_{i,j}^\infty) & = & \bar z^\infty_{i,j-1}+ \bar z^\infty_{i,j} + U(\bar z_{i,j-2}^\infty+ \bar z_{i,j+1}^\infty) + U^3(\bar z_{i,j-3}^\infty+ \bar z_{i,j+2}^\infty)+\cdots, \end{eqnarray*}
for $|j|>1$. The expressions continue indefinitely with increasing positive powers of $U$ as coefficients. \end{lemma}
\begin{proof} This proof is the same as that of Lemma \ref{lemma:F1_infty}. \end{proof}
Let $\pi:HF^\infty(M,\frak s)\to HF^+(M,\frak s)$, be the natural projection map. Let $\bar x_i^+ = \pi(\bar x_i^\infty)$, and define $\bar y_{i,j}^+$ and $\bar z_{i,k}^+$ similarly. Analogous to the discussion above, we have the following decomposition: $$HF^+(X)/HF_{red}(X) = \bigoplus_{i=-\frac{n-1}{2}}^{\frac{n-1}{2}}\mathcal T_i^+(X),$$ as well as similar decompositions corresponding to $Y$ and $Z$. Note that we are not claiming that $\bar x_i^+$ is nonzero in $\mathcal T_i^+(X)$. Similarly, it may be that $\bar y_{i,j}^+$ and the $\bar z_{i,k}^+$ vanish. Define $$F_1^+ = \sum_{\frak t\in\text{Spin}^c(W_1)}F^+_{W_1,\frak t},$$ and $$F_2^+ = \sum_{\frak r\in\text{Spin}^c(W_2)}F^+_{W_2,\frak r}.$$
\subsection{Proof of Proposition \ref{prop:X_corr_terms}}\
With this notational set-up, we recall that the triad $(X,Y,Z)$ introduced in Section \ref{section:geometry} induces certain long exact sequence (discussed in Section \ref{section:HF}), which will be used below in the proof of Proposition \ref{prop:X_corr_terms}.
Let $J=\#_{k_p}D$, so $X=S^3_p(J\#J)$, $Y = S^3_{p^2+p}(J\#J\#T_{p,p+1})$, and $Z=\mathcal Z_{p,k_p}=\Sigma_2(I_{\#_{k_p}D,p})$. The calculations made in Appendix \ref{appendix} give us the correction terms for $X$ and $Y$. In particular, Lemma \ref{lemma:Y_corr_terms} tells us that $\mathcal D(L(p,1)) - \mathcal D(X)$ is given by $$\vec w = \{0,\ldots, 0,2,2,4,4,\ldots,2k_p-2,2k_p-2,2k_p,2k_p,2k_p,2k_p-2,2k_p-2,\ldots,4,4,2,2,0,\ldots, 0\},$$ where $2w_i$ is the value of the $i^\text{th}$ coordinate of $\vec w$ for $i\in\mathbb{Z}$ with our symmetric labeling.
Let $x_i^\infty = U^{w_i}\bar x_i^\infty$, and let $\pi(x_i^\infty) = x_i^+$. It follows that $x_i^+$ is the element of lowest grading in $\mathcal T_i^+(X)$, i.e., $gr(x_i^+) = d(X,\frak s_i)$. Similarly, by Corollary \ref{coro:Y'_corr_terms}, $\mathcal D(L(n,1)\#L(n+1,-1)) - \mathcal D(Y)$ is given by the matrix $\mathcal M = (2m_{i,j})$, which has the following form. $$ { \left[ \setlength{\arraycolsep}{2.3pt} \begin{array}{cccccccccccccccccccccc} 0 & \cdots & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0& 0& \cdots & 0 \\ \vdots & & \vdots & \vdots &\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots && \vdots& \vdots & \vdots& & \vdots \\ 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0& 0& \cdots & 0 \\ 2 & \cdots & 2 & 2 & 0 & 0 & 0 & 0 & 0 &0 & 0 & 2 & 2 & 2 & 2 & \cdots & 2 & 2 & 2& \cdots & 2 \\ 2 & \cdots & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & \cdots & 2 & 2 & 2& \cdots & 2 \\ 4 & \cdots & 4 & 4 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 4 & 4 & \cdots & 4 & 4 & 4& \cdots & 4 \\ 4 & \cdots & 4 & 4 & 4 & 4 & 2 & 0 & 0 & 0 & 0 & 4 & 4 & 4 & 4 & \cdots & 4 & 4 & 4& \cdots & 4\\ \vdots & & \vdots & \vdots & \vdots & & & \ddots & \ddots & \ddots & & \vdots & \vdots & \vdots &\vdots & & \vdots & \vdots& \vdots& & \vdots \\ 2k_p & \cdots & 2k_p & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & 2 & 0 & 0 & 2k_p & 2k_p & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & 2k_p & \cdots & 2k_p \\ 2k_p & \cdots & 2k_p & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & 2k_p & 2 & 0 & 2 & 2k_p & 2k_p &2k_p & \cdots & 2k_p & 2k_p & 2k_p& \cdots & 2k_p \\ 2k_p & \cdots & 2k_p & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & 2k_p & 2k_p & 0 & 0 & 2 & 2k_p & 2k_p & \cdots & 2k_p & 2k_p & 2k_p& \cdots & 2k_p \\ \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & & \ddots & \ddots & \ddots & & & \vdots & \vdots& \vdots& & \vdots \\ 4 & \cdots & 4 & 4 & 4 & \cdots & 4 & 4 & 4 & 4 & 0 & 0 & 0 & 0 & 2 & 4 & 4 & 4 & 4& \cdots & 4 \\ 4 & \cdots & 4 & 4 & 4 & \cdots& 4 & 4 & 4 &4 & 0 & 0 & 0 & 0 & 0 & 2 & 4 & 4 & 4& \cdots & 4 \\ 2 & \cdots & 2 & 2 & 2 & \cdots& 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2& \cdots & 2 \\ 2 & \cdots & 2 & 2 & 2 & \cdots & 2 & 2 & 2 & 2& 0 & 0& 0 & 0 & 0 & 0 & 0 & 2 & 2& \cdots & 2 \\ 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0& \cdots & 0 \\ \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots& \vdots & \vdots & \vdots & \vdots& & \vdots \\ 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0& \cdots & 0 \\ \end{array} \right] } $$ Note that the values in the $i^\text{th}$ row of $\mathcal M$ are bounded above by $2w_i$. (Remember that the rows are labeled by $\mathbb{Z}_n$ symmetrically about zero, and the columns are labeled by $\mathbb{Z}_{n+1}$ by $j\in[-\frac{n+1}{2},\frac{n-1}{2}]$). We remark again that the calculation given in the proof of Corollary \ref{coro:Y'_corr_terms} introduces an indeterminacy regarding our presentation of the correction terms. Namely, we cannot distinguish between $i$ and $-i$ and $j$ and $-j$ in the present labeling. This indeterminacy is merely notational and will not affect the results.
Let $y_{i,j}^\infty = U^{m_{i,j}}\bar y_{i,j}^\infty$, and let $y_{i,j}^+ = \pi(y_{i,j}^\infty)$. It follows that $y_{i,j}^+$ is the element of lowest grading in $\mathcal T_{i,j}^+(Y)$, i.e., $gr(y_{i,j}^+) = d(Y,[\frak s_i, \frak s_j])$. With this notational set-up, we can prove the following lemma about the map $F_1^+:HF^+(X)\to HF^+(Y)$.
\begin{lemma} Let $x_i^+\in\mathcal T_i(X)$ and $y_{i,j}^+\in\mathcal T_{i,j}^+(Y)$ be elements of lowest grading in their respective towers. Then, $$F_1^+(x_i^+) = \sum_{j\in\mathcal I_i}y_{i,j}^+,$$ where $\mathcal I_i = \{j\not=0\ :\ m_{i,j}=w_i\}$. \end{lemma}
\begin{proof} By Lemma \ref{lemma:F1_infty} we have that $$F_1^\infty(\bar x_i^\infty) = \sum_{j\not=0}\bar y_{i,j}^\infty + \mathcal U(\bar y^\infty_{i,j}),$$ where $\mathcal U(\bar y_{i,j}^\infty)$ represents the terms that are positive $U$--translates of the $\bar y_{i,j}^\infty$. By $U$--equivariance, we have $$F_1^\infty(x_i^\infty) = U^{w_i}F_1^\infty(\bar x_i^\infty) = \sum_{j\not=0}U^{w_i}\bar y_{i,j}^\infty + U^{w_i}\mathcal U(\bar y_{i,j}^\infty).$$ Since $F_1$ commutes with the natural projection $\pi$ (which is $U$--equivariant), we see that $$F_1^+(x_i^+) = \pi(F_1^\infty(x_i^\infty)) = \sum_{j\not=0}U^{w_i}\pi(\bar y^\infty_{i,j}) = \sum_{j\not=0}U^{w_i}\bar y^+_{i,j},$$ where the tail has vanished, by $U$--equivariance. By definition, we have $U^{w_i}\bar y_{i,j}^+ = U^{w_i-m_{i,j}}y_{ij}^+$, and this term will be nonzero if and only if $w_i\leq m_{i,j}$. This can only happen if $w_i = m_{i,j}$, since, as we noticed above, $m_{i,j}\leq w_i$. \end{proof}
Note that $|\mathcal I_i|\geq\frac{p+1}{2}$ for each $i$; so, in particular, $F_1^+(x_i^+)$ is a linear combination of at least $\frac{p+1}{2}$ terms for each $i$. One consequence of this is that $y_{i,j}^+$ is not in the image of $F_1^+$ for any $i,j$.
Let $z_{i,j}^+$ denote the element of lowest grading in $\mathcal T_{i,j}^+(Z)$. We know by $U$--equivariance that $$F_{W_2,[\frak s_i,\frak s_k]}(y_{i,j}^+) = U^{c_{i,k}}z_{i,k}^+$$ for some nonnegative integer $c_{i,k}$. If we can show that $c_{i,j}=0$ for all $i,k$, we will have proved Proposition \ref{prop:X_corr_terms}, because we will have shown that $z_{i,k}^\infty = U^{m_{i,k}}\bar z_{i,k}^\infty$. This is accomplished by the following lemma. Recall the natural inclusion map $\hat\iota:\widehat{HF}(Z)\to HF^+(Z)$.
\begin{lemma} Let $z_{i,k}^+$ be the element of lowest grading in $\mathcal T_{i,k}^+(Z)$, and let $y_{i,k}^+$ be the element of lowest grading in $\mathcal T_{i,k}^+(Y)$. Then, $$gr(z_{i,k}^+) = gr\left(F_{W_2,[\frak s_i,\frak t_k]}^+(y_{i,k}^+)\right).$$ \end{lemma}
\begin{proof} Let $\hat z\in\widehat{HF}(Z)$ such that $\hat\iota(\hat z) = z_{i,k}^+$. By Theorem \ref{thm:hat_vanish}, we know that $\widehat F_3(\hat z)=0$. (Recall that $-\overline{W_3}$ is induced by $(-p)$--surgery on a knot of genus $2k_p$ with $p>4p_k-1$, see Subsection \ref{subsec:coefficients}.) By the exactness at $\widehat{HF}(X)$, there exists some $\hat y\in\widehat{HF}(Y')$ such that $\widehat F_2(\hat y) = \hat z$. Now, $\hat y$ may not be homogeneous, so write $\hat y = \sum_a\hat y_a$, where each $\hat y_a$ is homogeneous and in $\widehat{HF}(Y',[\frak s_i,\frak s_{j_a}])$. By Lemma \ref{lemma:reduced_bound}, we know that for each $a$, $gr(\hat y_a) \leq gr(y_{i,j_a}^+)$. So, we have
\begin{eqnarray*} gr\left(\widehat F_{W_2,[\frak s_i,\frak t_{m_a}]}(\hat y_a)\right) & = & gr(\hat y_a) + gr\left(\widehat F_{W_2,[\frak s_i,\frak t_{m_a}]}\right) \\ & \leq & gr(y_{i,j_a}^+) + gr\left(F^+_{W_2,[\frak s_i,\frak t_{m_a}]}\right) \\ & = & gr\left( F_{W^+_2,[\frak s_i,\frak t_{m_a}]}(y_{i,j_a}^+)\right) \\ & \leq & gr\left( F_{W^+_2,[\frak s_i,\frak t_k]}(y_{i,k}^+)\right), \\ \end{eqnarray*}
where the last inequality follows from the fact that $gr\left(F_{W^+_2,[\frak s_i,\frak t_m]}(y_{i,j}^+)\right)$ is maximized when by $j$ with $|j|= k$. (Note that $j_a\equiv k\pmod{p}$.) Since $$gr(\hat z) \leq \max_a gr\left(\widehat F_{W_2,[\frak s_i,\frak t_{m_a}]}(\hat y_a)\right),$$ we have $$gr(\hat z) \leq gr\left( F_{W^+_2,[\frak s_i,\frak t_k]}(y_{i,k}^+)\right).$$ This implies the desired equality once we recall that $$gr(\hat z) = gr(z_{i,k}^+) \geq gr\left( F_{W^+_2,[\frak s_i,\frak t_k]}(y_{i,k}^+)\right),$$ by $U$--equivariance. \end{proof}
\section{Proof of Theorem \ref{thm:main2}}\label{section:infinite_order}
In this section, we give a reformulation of one of the invariants introduced in \cite{GRS} for the study of double concordance of knots and use it to find an infinitely generated subgroup in $\ker\psi_\mathcal D$.
Let $A$ be a finite abelian group, so $A$ can be written as the product of cyclic groups. Let $r_{p,k}(A)$ denote the number of copies of $\mathbb{Z}_{p^k}$ in the decomposition of $A$. Let $r_p(A) = \sum_{k=1}^\infty r_{p,k}(A)$. In other words, any generating set for $A$ must contain at least $r_p(A)$ elements of order $p^k$ for some $k\in\mathbb{N}$.
The following definition differs from \cite{GRS} only in the use of $r_p(A)$.
\begin{definition} Let $K$ be a knot in $S^3$ and let $p\in\mathbb{N}$ be a positive prime. Let $M=\Sigma_2(K)$. Fix an affine identification between $\text{Spin}^c(M)$ and $A=H^2(M;\mathbb{Z})$ such that the unique spin structure $\frak s_0$ gets identified with zero in $A$. Let $\mathcal G_p$ denote the collection of all subgroups of $A$ of order $p$. Define
$$\frak D_p(K) = \min\left\{ \left|\sum_{H\in\mathcal G_p}n_HS_H(d(M))\right|\ :\ \begin{array}{c} n_H \geq 0 \text{ for all $H$}, \\ \text{ at least $r_p(A)$ of the $n_H$ are nonzero}\end{array}\right\}$$ if $p$ divides $\det(K)$ and $$\frak D_p(K) =0$$ otherwise, where $S_H(d(M)) = \sum_{h\in H}d(M,h)$. \end{definition}
The proof of the following theorem is essentially given in \cite{GRS}, but is formulated here for double concordance.
\begin{theorem}\label{thm:GRS} Let $K\subset S^3$ be a knot and $p\in\mathbb{N}$ a positive prime. If there is a positive $n\in\mathbb{N}$ such that $\#_nK$ is smoothly doubly slice, then $\frak D_p(K)=0$. \end{theorem}
\begin{proof} Suppose that $J=\#_nK$ is smoothly doubly slice. Let $N =\Sigma_2(J) = \#_n\Sigma_2(K)$. The identification of $\text{Spin}^c(\Sigma_2(K))$ with $A$ gives an identification of $\text{Spin}^c(N)$ with $A^n$. By Theorem \ref{thm:hyp_corr_terms}, there exists subgroups $G$ and $H$ in $A^n$ such that $G\oplus H=A^n$ and $G\cong H$.
Assume that $p$ divides $\det(K)$, and let $r=r_p(A)$. Projection onto the first coordinate $\pi:A^n\to A$ is onto, so $\pi(G)+\pi(H) = A$. Let $a_1, \ldots, a_r$ be linearly independent generators of $A$ of $p$--power order such that $\pi^{-1}(a_i)\cap(G\cup H)$ is nonempty. Let $g_i'\in \pi^{-1}(a_i)\cap(G\cup H)$, then $|g_i'| = p^{k_i}q$ for some positive $q\in\mathbb{Z}$ relatively prime to $p$. Let $g_i = qp^{k_i-1}g_i'$. Then $\{g_1, \ldots, g_r\}$ is a collection elements of order $p$ in $G\cup H$. Furthermore, the elements of $\{\pi(g_1), \ldots, \pi(g_r)\}$ are linearly independent in $A$, so, as elements of $\mathcal G_p$, $\langle g_i\rangle = \langle g_j\rangle$ if and only if $i=j$. Write $g_i = (g_i^1, \ldots, g_i^n)$ for $i=1, \ldots, r$.
By Theorem \ref{thm:hyp_corr_terms}, $d(N, x) = 0$ for all $x\in G\cup H$. Let $f:A\to\mathbb{Q}$ be given by $f(x) = d(N,x)$, and let $f^{(n)}:A^n\to\mathbb{Q}$ be given by $f(x_1,\ldots,x_n) = f(x_1)+\cdots+f(x_n)$. Since $\langle g_i\rangle<G\cup H$, we have \begin{eqnarray*} \sum_{m=0}^{p-1}f^{(n)}(mg_i)= 0 & \implies & \sum_{m=0}^{p-1}\sum_{j=1}^nf(mg_i^j)= 0 \\ & \implies & \sum_{j=1}^n\sum_{m=0}^{p-1}f(mg_i^j)= 0 \\ & \implies & \sum_{j=1}^nS_{\langle g_i^j\rangle}(f)= 0 \\ & \implies & \sum_{j=1}^nS_{\langle g_i^j\rangle}(d(N))= 0 \end{eqnarray*} Since $\langle g_i^j\rangle\in \mathcal G_p$ for each $j$, $$\sum_{j=1}^nS_{\langle g_i^j\rangle}(d(N)) = \sum_{H\in\mathcal G_p}n_HS_H(d(N)),$$ with at least one $n_H$ nonzero (since at least $g_i^1$ is nontrivial). For each $j=1, \ldots, r$, we get a similar linear combination, and, since the $g_j^1$ are independent, each linear combination is nontrivial on a distinct element of $\mathcal G_p$. Summing, we get $$\sum_{i=1}^r\sum_{j=1}^nS_{\langle g_i^j\rangle}(d(N)) = \sum_{H\in\mathcal G_p}n_HS_H(d(N)),$$ where at least $r$ of the $n_H$ are nonzero. It follows that $\mathcal D_p(K)=0$, as desired.
\end{proof}
To prove Theorem \ref{thm:main2}, we will need to understand $S_G(f)$ for each subgroup $G$ of $\mathbb{Z}_p\oplus\mathbb{Z}_p$. Let $G_\star =\langle(1,1)\rangle$ and let $G_a = \langle (a,a+1)\rangle$ for $a\in\mathbb{Z}_p$. Then, together, $G_\star$ and the $G_a$ represent the $p+1$ distinct order $p$ subgroups of $\mathbb{Z}_p\oplus\mathbb{Z}_p$.
First let us consider $Z=L(p,1)\#L(p,-1)$ for a positive prime $p$. We saw in Subsection \ref{subsec:enumerate} that we have an affine identification $[\frak s_i,\frak s_j]\sim (i,j)$ between $\text{Spin}^c(Z)$ and $Z_p\oplus\mathbb{Z}_p$.
Let $f:H_1(Z)\to\mathbb{Q}$ be given by $f(x)=d(Z,[\frak s_i,\frak s_j])$, where $[\frak s_i,\frak s_j]\sim x$ is the given affine identification. It is possible to check using Equation \ref{eqn:lens_corr_terms} that $$S_a^{\text{lens}}=S_{G_a}(f) = \begin{cases} \frac{(p-1)(p+1)}{6} & \text{ if $a=0$,} \\ -\frac{(p-1)(p+1)}{6} & \text{ if $a=p-1$,} \\ 0 & \text{ if $a=\star$,}\\ 0 & \text{ otherwise.} \end{cases} $$
Furthermore, by Proposition \ref{prop:X_corr_terms}, we know that $$\mathcal D(L(n,1)\#L(n,-1)) - \mathcal D(\mathcal Z_{p,k_p})$$ is given by $\mathcal M$.
Let $S'_G = \sum_{g\in G}\mathcal M_g$, where $\mathcal M_g=\mathcal M_{i,j}$, if $g=(i,j)\in\mathbb{Z}_p$. Then, we see that
$$S'_{G_a} = \begin{cases} 2k(p-3)+4 & \text{ if $a=0$,} \\ 0 & \text{ if $a=p-1$,} \\ 0 & \text{ if $a=\star$,}\\ \text{(large positive number)} & \text{ otherwise.} \end{cases} $$
It follows that the pertinent sums for $\mathcal Z_{p,k_p}$ are given by $S^{\mathcal Z_{p,k_p}}_{G_a}(f) = S_a^{\text{lens}} - S'_{G_a}$. So, $$S^{\mathcal Z_{p,k_p}}_{G_a} = \begin{cases} \frac{(p-1)(p+1)}{6}-(2k(p-3)+4) & \text{ if $a=0$,} \\ -\frac{(p-1)(p+1)}{6} & \text{ if $a=p-1$,} \\ 0 & \text{ if $a=\star$,}\\ \text{(large negative number)} & \text{ otherwise.} \end{cases} $$
The upshot is that $S_{G_a}^{\mathcal Z_{p,k_p}}$ will be strictly negative for all $a\not=\star$ if and only if $$\frac{(p-1)(p+1)}{6}-(2k(p-3)+4)<0.$$ The left side will be negative if $k\geq \frac{p+5}{12}$. As we saw above in Subsection \ref{subsec:coefficients}, we will let $k_p=\left\lceil{\frac{p+6}{12}}\right\rceil$, which will satisfy this condition. Now we can prove the following, recalling our set-up from Section \ref{section:geometry}.
\begin{proposition} Let $\mathcal K_{p,k_p} = I_{\#_{2k_p}J,p}$, where $J$ is $T_{2,3}$ or $D$, and where $k_p=\left\lceil{\frac{p+6}{12}}\right\rceil$. Then, \begin{enumerate} \item No knot in the span (under connected sum) of the $\mathcal K_{p,k_p}$ is smoothly doubly slice. \item Each of the $\mathcal K_{p,k_p}$ has order greater than two in $\mathcal C_{\mathcal D}$. \item The collection $\{\mathcal K_{p,k_p}\}$ forms a basis for an infinitely generated subgroup of $\mathcal C_\mathcal D$. \end{enumerate} \end{proposition}
Note that this is independent of the indeterminacies $i\leftrightarrow -i$ and $j\leftrightarrow -j$ discussed earlier. Notice also that Example \ref{ex:rank6} illustrates why we cannot claim that the $\mathcal K_{p,k_p}$ have infinite order in $\mathcal C_{\mathcal D}$.
\begin{proof} By Corollary \ref{coro:terms}, we know that each of these knots is nontrivial in $\mathcal C_\mathcal D$. The discussion preceding this proposition shows that the Grigsby-Ruberman-Strle invariant $\mathcal D_p$ is nonzero for $\mathcal K_{p,k_p}$. This follows because, for these knots, $S^{\mathcal Z_{p,k_p}}_{G_a}$ is nonnegative for only one subgroup of $\mathbb{Z}_p\oplus\mathbb{Z}_p$. Since the condition on $\mathcal D_p$ states that $n_G$ must be nonzero for at least two distinct subgroups $G$, the sum $\sum_{G\in\mathcal G_p}n_GS_G(M)$ will always be nonzero. By Theorem \ref{thm:GRS}, this shows that $\#_a\mathcal K_{p,k_p}$ is not doubly slice for all $a\in \mathbb{N}$. By Proposition \ref{prop:stable_terms}, $\mathcal K_{p,k_p}\#\mathcal K_{p,k_p}$ is nontrivial in $\mathcal C_\mathcal D$, since $\mathcal A_p$ is rank 4 and not hyperbolic for these knots.
Suppose that $$\mathcal K = \left(\#_{n_{p_1}}\mathcal K_{p_1,k_{p_1}}\right)\#\left(\#_{n_{p_2}}\mathcal K_{p_2,k_{p_2}}\right)\#\cdots\#\left(\#_{n_{p_m}}\mathcal K_{p_m,k_{p_m}}\right).$$ Since the $p_i$ are all distinct primes, we get that $\mathcal D_{p_i}(K) = \mathcal D_{p_i}(\#_{n_i}\mathcal K_{p_i,k_{p_i}})$. It is easy to see that, for the knots in question, $\mathcal D_{p_i}(\#_{n_i}\mathcal K_{p_i,k_{p_i}}) \not=0$, since $S^{\mathcal Z_{p_i,k_{p_i}}}_{G_a}$ is always nonpositive and strictly negative away from a single metabolizer. It follows that $\mathcal D_{p_i}(\mathcal K)\not=0$. This proves that $K$ is not doubly slice, and if any of the $n_{p_i}$ are less than 3, then $\mathcal K$ is nontrivial in $\mathcal C_\mathcal D$. \end{proof}
By Corollary \ref{coro:doubly_slice}, each member of $\{\mathcal K_{p,k_p}\}$ is topologically doubly slice. It follows that these knots generate an infinitely generated subgroup of $\ker\psi_\mathcal D$ that consists of knots that are not smoothly doubly slice. This proves Theorem \ref{thm:main2}.
\appendix
\section{Assorted knot Floer complex calculations}\label{appendix}
The goal of this appendix is to perform the correction term calculations required by the proof in Section \ref{section:calculations}. Throughout, we will let $J = \#_mK$ be the connected sum of $m$ copies of $K$, where $K$ will always be one of three knots: the unknot; the right-handed trefoil $T_{2,3}$; or the positive, untwisted Whitehead double of the right-handed trefoil $D$. Let $X = S^3_n(J\#J)$ and $Y=S^3_{n^2+n}(J\#J\#T_{n,n+1})$; throughout, $n$ will be a positive odd number. The following facts are collected from two theorems of Hedden, Kim, and Livingston \cite[Proposition 6.1, Theorem B.1]{HKL}, and are the basis what follows. We will work with coefficients in $\mathbb{F}_2$ throughout.
\begin{theorem}[\cite{HKL}]\label{thm:HKL}\ \begin{enumerate} \item The chain complex $CFK^\infty(S^3, D)$ is filtered chain homotopy equivalent to $CFK^\infty(S^3, T_{2,3})\oplus \mathcal A$, where $\mathcal A$ is an acyclic subcomplex. \item The chain complex $CFK^\infty(S^3,\#_mT_{2,3})\simeq CFK^\infty(S^3, T_{2,3})^{\otimes m}$ is filtered chain homotopy equivalent to $CFK^\infty(S^3, T_{2,2m+1})\oplus\mathcal A'$, where $\mathcal A'$ is an acyclic subcomplex.
\end{enumerate} \end{theorem}
First, we calculate the correction terms for $X$ when $m=2k$, the case relevant to our discussion. Recall that the affine identification $\text{Spin}^c(X)\cong \mathbb{Z}_n$ gives rise to a natural indexing of $\frak s_i\in\text{Spin}^c(X)$, where $|i|\leq (n-1)/2$. This symmetry of this indexing is advantageous, and will be used here. Let $\mathcal D(X)$ denote the collection of correction terms associated to $X$, i.e., $$\mathcal D(X) = \{d(X,\frak s_{\frac{-n+1}{2}}), d(X,\frak s_{\frac{-n+3}{2}}),\ldots, d(X,\frak s_{-1}), d(X,\frak s_0), d(X,\frak s_1),\ldots, d(X,\frak s_{\frac{n-3}{2}}), d(X,\frak s_{\frac{n-1}{2}})\}.$$
\begin{lemma}\label{lemma:Y_corr_terms} Let $X=S^3_n(\#_{2k}K)$, where $K$ is $T_{2,3}$ or $Wh^+(T_{2,3},0)$. Then, $ \mathcal D(L(n,1)) - \mathcal D(X)$ is given by $$ \{0,\ldots, 0,2,2,4,4,\ldots,2k-2,2k-2,2k,2k,2k,2k-2,2k-2,\ldots,4,4,2,2,0,\ldots, 0\}.$$ \end{lemma}
Of course, if $K$ is the unknot, then $\mathcal D(X) = \mathcal D(L(n,1))$.
\begin{proof} By combining parts (1) and (2) of Theorem \ref{thm:HKL}, we get that $$CFK^\infty(S^3, \#_mD)\simeq CFK^\infty(S^3, \#_mT_{2,3})\oplus\mathcal A'' \simeq CFK^\infty(S^3, T_{2,2m+1})\oplus\mathcal A'''.$$
The acyclic pieces can contribute to the homology of $HF^+(X)$, but these contributions are confined to $HF_{red}(X)$ and will not affect the correction term calculations. It follows that $d(X,\frak s_i) = d(S^3_n(T_{2,2m+1}),\frak s_i)$ for all $|i|\leq (n-1)/2$.
The complex $C=CFK^\infty(S^3, T_{2,2m+1})$ can be easily obtained from the Alexander polynomial $\Delta_{T_{2,2m+1}}(t)$, since $T_{2,2m+1}$ is an alternating $L$-space knot \cite{oz-sz:alternating}, and is shown in Figure \ref{fig:AltTorusComplexes}. Its basic building block (which we call a \emph{germ}) can be seen in Figure \ref{fig:AltTorusComplexes}. One way to characterize which piece of the total complex is the germ $G$ is to say that $G$ is contained in the first $(i,j)$--quadrant, but $UG$ is not. The total complex is obtained by taking $\mathbb{Z}$ copies of $G$, which are related by $U$--translation, i.e., $C = \sqcup_{z\in\mathbb{Z}}U^zG$.
There is a simple way to calculate $V_l=V_l(T_{2,2m+1})$ in this case \cite{ni-wu:cosmetic, ni-wu:rational}. Consider the subcomplex $C_{\{\max(i,j-l)\geq 0\}}$. Then, $$V_l = \max\{z\ :\ U^zG\cap C_{\{\max(i,j-l)\geq 0\}}=\emptyset\}.$$ See, for example, Figure \ref{fig:VsAltTorus}. With this in mind, it is now easy to see that $$\{V_l(T_{2,2m+1})\}_{l\geq 0} = \begin{cases} \{k,k,k-1,k-1,\ldots, 2,2,1,1,0,\ldots\} & \text{ if $m=2k$}, \\ \{k,k-1,k-1,\ldots, 2,2,1,1,0,\ldots\} & \text{ if $m=2k+1$}, \\ \end{cases} $$ where each value less than $k$ appears twice in each list, and the infinite tails each consists of zeros. Simply start with the shaded box in the third quadrant, as in Figure \ref{fig:VsAltTorus}(b), and notice how the homology changes as the box is moved vertically upward. If we recall that $H_{l} = V_{-l}$, then Theorem \ref{thm:ni-wu} completes the proof. \end{proof}
\begin{figure}
\caption{Portions of the complex $CFK^\infty(S^3, T_{2,2m+1})$ is shown above for (a) $m=4$ and (b) $m=6$. The germ of each complex is shown hollow in red.}
\label{fig:AltTorusComplexes}
\end{figure}
\begin{figure}
\caption{(a) The calculation showing that $V_1(T_{2,9})=2$. (b) The calculation showing that $V_0(T_{2,13})=3$.}
\label{fig:VsAltTorus}
\end{figure}
Let $L_k$ denote the list of even integers given in Lemma \ref{lemma:Y_corr_terms}, but with each value halved, and consider the bijection between $L_k$ and $\mathbb{Z}$ where the central $k$ corresponds to zero and the values to the left and right correspond to the negative and positive integers, respectively. Let $L_k^t$ denote a truncated version of $L_k$ where the value of any term in $L_k^t$ corresponding to an integer less than $-t$ is set to zero. Let $L_k^t(x)$ represent the element of $L_k^t$ corresponding to $x\in\mathbb{Z}$. For example, $$ \begin{array}{ccc} L_3 & = & \{\ldots,0,0,1,1,2,2,3,3,3,2,2,1,1,0,0\ldots\}, \\ L_3^1 & = & \{\ldots,0,0,0,0,0,0,3,3,3,2,2,1,1,0,0\ldots\}, \\ L_3^3 & = & \{\ldots,0,0,0,0,2,2,3,3,3,2,2,1,1,0,0\ldots\}, \\ \end{array} $$ and $L_3^1(-1)=3$. We will make use of these truncated lists later.
\begin{figure}
\caption{The complexes (a) $CFK^\infty(S^3,T_{2,3})$ and (b) $CFK^\infty(S^3,D)$ are shown with gradings; the three chains adjacent to the star have gradings $-2,-2$, and $-1$.}
\label{fig:TrefoilComplexes}
\end{figure}
Our next task is to give a calculation for the correction terms of $Y$. To start, consider the case when $K$ is the unknot, so $Y = S^3_{n^2+n}(T_{n,n+1})$.
\begin{lemma}\label{lemma:Vs_Torus} Let $n=2d+1$ for $d\in\mathbb{N}$. Then, $\{V_l(T_{n,n+1})\}_{l\geq 0}$ is given by $$ {\SMALL \begin{array}{cccccccccc} \{ & Tr(d), & Tr(d), & \ldots & Tr(d), & Tr(d)-1, & Tr(d) - 2, & \ldots, & Tr(d-1)+2, & Tr(d-1) + 1, \\
& Tr(d-1), & Tr(d-1), & \ldots & Tr(d-1), & Tr(d-1)-1, & Tr(d-1) - 2, & \ldots, & Tr(d-2)+2, & Tr(d-2) + 1, \\
& Tr(d-2), & Tr(d-2), & \ldots & Tr(d-2), & Tr(d-2)-1, & Tr(d-2) - 2, & \ldots, & Tr(d-3)+2, & Tr(d-3) + 1, \\
& & & & & \vdots & & & & \\
& 3, & 3, & 3, & \ldots, & & \ldots, & 3, & 3, & 2, \\
& 1, & 1, & 1, & \ldots, & & \ldots, & 1, & 1, & 1, \\
& 0, & 0, & 0, & \ldots, & \}, && & & \\ \end{array} } $$ where $Tr(k)$ denotes the $k^\text{th}$ triangular number. \end{lemma}
To clarify, the above list has been displayed so as to make the pattern of its elements more clear. On the $i^\text{th}$ line, the value $Tr(d-i+1)$ appears $d+i+1$ times, followed by sequential decreases by 1, until the next triangular number is hit, which begins a new line. The tail of the list is all zeros. We will refer to the first appearance of each triangular number (i.e., the first element of each line) as a \emph{pivot}. These pivots occur when $l$ is a multiple of $n$ and correspond to the cycles in the germ $G$ of $CFK^\infty(S^3, T_{n,n+1})$ (see Figure \ref{fig:TorusKnotComplex}).
\begin{figure}
\caption{(a) The complex $CFK^\infty(S^3,T_{n,n+1})$; here $n=5$. The calculation showing that $V_4(T_{5,6}) = 2$.}
\label{fig:TorusKnotComplex}
\end{figure}
\begin{proof} As noted in \cite{HKL}, $C=CFK^\infty(S^3, T_{n,n+1})$ has germ $G$ as shown in red in Figure \ref{fig:TorusKnotComplex}(a). The total complex is obtained by taking $\mathbb{Z}$ copies of this germ, which are related by $U$--translation, i.e., $C = \cup_{z\in\mathbb{Z}}U^zG$. As in the proof of Lemma \ref{lemma:Y_corr_terms}, the $V_l = V_l(T_{n,n+1})$ are given by $$V_l = \max\{z\ :\ U^zG\cap C_{\{\max(i,j-l)\geq 0\}} = \emptyset\}.$$ See for example, Figure \ref{fig:TorusKnotComplex}(b). Putting all this together, it is easy to see that $\{V_l\}_{l\geq 0}$ is as claimed. \end{proof}
Lemma \ref{lemma:Vs_Torus} gives us a basis to understand the correction terms for surgeries on $J\#J\#T_{n,n+1}$. To continue, we need to understand how the knot chain complex for $T_{n,n+1}$ changes under connected sum with $K$.
\begin{lemma}\label{lemma:Complex_Sum} After a filtration-preserving change of basis, $CFK^\infty(S^3,J\#J\#T_{n,n+1}) = C_{sum}\oplus\mathcal A$, where a germ for $C_{sum}$ is made up of the the characteristic pieces shown in Figure \ref{fig:SumComplex}, and $\mathcal A$ is an acyclic subcomplex. \end{lemma}
\begin{figure}
\caption{The three filtered chain homotopy equivalences used in the proof of Lemma \ref{lemma:Complex_Sum}. Each is obtained by a filtration-preserving change of basis. Note that in (a) and (b), if the off-diagonal group overlaps with either of the other two groups, then the result has a slightly different form, but is qualitatively the same. Also, compare with Figures \ref{fig:SumComplex}(a) and \ref{fig:SumComplex}(b). Here, each group of ``stairs'' represents a copy of $CFK^\infty(S^3, T_{2,2m+1})$; here $m=3$. The groups of stairs are referred to as the $a$--, $b$--, and $c$--groups.}
\label{fig:PictureProofs}
\end{figure}
\begin{proof} Recall that $CFK^\infty(S^3, J\#J)\simeq CFK^\infty(S^3, T_{2,2m+1})\oplus \mathcal A$, by Theorem \ref{thm:HKL}. It follows that $CFK^\infty(S^3, J\#J\#T_{n,n+1})\simeq CFK^\infty(S^3, T_{n,n+1})\otimes CFK^\infty(S^3, T_{2,2m+1})\oplus\mathcal A$. (Here, we use $\mathcal A$ to represent potentially different acyclic subcomplexes.) To see that this tensor product has the desired form, we need three pictorial lemmas, shown in Figure \ref{fig:PictureProofs}. All three parts show a chain homotopy equivalence achieved via a filtration preserving change of basis. Consider Figure \ref{fig:PictureProofs}(a), and denote the chains by $a_1, \ldots, a_{2m+1}$, $b_1,\ldots, b_{2m+1}$, and $c_1, \ldots, c_{2m+1}$ (i.e., $b_i\mapsto a_i+c_i$). The double arrows mean that there should be an arrow between each pair of vertically or horizontally aligned chains. The pertinent change of basis is: $$b_k\mapsto b_k+\sum_{\substack{i<k, \\ i \text{ even}, \\ n_j(b_k)>n_j(c_i)}}c_i + \sum_{\substack{j>k, \\ j \text{ even} \\ n_i(b_k)>n_i(a_i)}}a_j,$$ for odd $k$. (Note that the indexing variables $i,j$, and $k$ used here are not related to the uses of $i,j,$ and $k$ used elsewhere; in particular, $i$ and $n_i$ are not related here.) The third condition on each summation guarantees that this change of basis is filtration preserving. Note that any vertical arrows hitting the $a$--group or horizontal arrows hitting the $c$--group are unaffected, since the chains in these groups are unchanged. The result is that the only arrows from the $b$--group to either the $a$--group or the $c$--group will go from $b_k$ with odd $k$ to $a_i$ and $c_j$ with odd $i$ and odd $j$. Parts (a) and (b) of Figure \ref{fig:SumComplex} show the possible results of this local change of basis on the germ of $CFK^\infty(S^3, J\#J\#_{n,n+1})$. In (a), the $a$ and $b$ pieces overlap; in (b) they do not.
Next, Figure \ref{fig:PictureProofs}(b) corresponds to the filtration preserving change of basis given by $$c_k\mapsto c_k+\sum_{\substack{j>k, \\ i \text{ even}, \\ n_j(b_j)>n_j(c_k)}}b_j,$$ for odd $k$, and $$a_k\mapsto a_k + \sum_{\substack{i<k, \\ i \text{ even} \\ n_i(b_i)>n_i(a_k)}}b_i,$$ for odd $k$.
Finally, Figure \ref{fig:PictureProofs}(c) corresponds to the filtration preserving change of basis given by $$a_k\mapsto a_k+c_1,$$ for odd $k\geq 3$, $$a_k\mapsto a_k + b_{k-1} + \sum_{\substack{i<k, \\ i \text{ even}}}c_i,$$ for even $k$, and $$b_k\mapsto b_k + \sum_{\substack{i<k, \\ i \text{ even}}}c_i,$$ for odd $k>2$. See also Figure \ref{fig:SumComplex}(c).
By applying these three types of change of basis, we see that $CFK^\infty(S^3, J\#J\#T_{n,n+1}) = C_{sum}\oplus\mathcal A$, where the characteristic pieces of $C_{sum}$ are shown in Figure \ref{fig:SumComplex}. In other words, connected summing $T_{n,n+1}$ with $J\#J$ introduces a stepping pattern at every joint (i.e., the cycles) of the germ of $CFK^\infty(S^3, T_{n,n+1})$, except the first and last, where the germ simply extends by $m$. \end{proof}
\begin{figure}\label{fig:SumComplex}
\end{figure}
\begin{lemma}\label{lemma:Vs_Sum}
Let $K$ be $T_{2,3}$ or $D$ and let $J=\#_{k}K$. Then, for $|l|\leq \frac{n^2+n}{2}$, $$V_l(J\#J\#T_{n,n+1}) - V_l(T_{n,n+1}) = \begin{cases} L_k^{t(l)}(\overline l) & \text{ if \ $0\leq l\leq \frac{n(n-1)}{2}$}, \\ 1 & \text{ if \ $\frac{n(n-1)}{2}\leq l \leq \frac{n(n-1)}{2}+k$}, \\ 0 & \text{ if \ $\frac{n(n-1)}{2}+k<l$}, \end{cases} $$
where $\overline l\in[\frac{-n+1}{2},\frac{n-1}{2}]$ is the mod $n$ reduction of $l$, and $t(l) = \frac{n-3}{2} - a$ if $|l|\in[an-\frac{n-1}{2},an+\frac{n+1}{2})$. \end{lemma}
\begin{proof}
Lemma \ref{lemma:Complex_Sum} tells us that the germ of $CFK^\infty(S^3, J\#J\#T_{n,n+1})$ is given locally as in Figure \ref{fig:SumComplex}. Forming the connected sum changes the complex for $T_{n,n+1}$ by introducing a stepping pattern at each joint. It is straightforward to see the effect of this on the $V_l(T_{n,n+1})$. For each joint, one simply superimposes a copy of $L_k^t$ over $\{V_l(T_{n,n+1})\}_{\geq 0}$, with $L_k(0)$ centered over the $l$ corresponding to the joint. If $C_{sum}$ looks locally like Figure \ref{fig:SumComplex}(a) at the joint (i.e., there is some overlap), then we use $L^t_k$ where $t-1$ is the amount of overlap (in Figure \ref{fig:SumComplex}(a), the overlap shown is 3). If $C_{sum}$ looks locally like Figure \ref{fig:SumComplex}(b) at the joint (i.e., there is no overlap), then $L^t_k = L_k$ (i.e., there is no truncation).
To clarify, $\{V_l(T_{n,n+1})\}_{\geq 0}$ is shown below, with the pivots highlighted in red. These pivots correspond to the joints of $CFK^\infty(S^3, T_{n,n+1})$. In $C_{sum}$, each such joint has been tensored with the germ for $CFK^\infty(S^3, T_{2,2m+1})$ and looks locally as in Figure \ref{fig:SumComplex}. By considering these local pictures, we can see that $V_l=V_l(J\#J\#T_{n,n_1})$ will have the value claimed, because the introduction of the stepping pattern corresponds precisely to adding $L_k^{t(l)}(\overline l)$ to $V_l$.
$$ {\SMALL \begin{array}{cccccccccc} \{ & \textcolor{Red}{Tr(d)}, & Tr(d), & \ldots & Tr(d), & Tr(d)-1, & Tr(d) - 2, & \ldots, & Tr(d-1)+2, & Tr(d-1) + 1, \\
& \textcolor{Red}{Tr(d-1)}, & Tr(d-1), & \ldots & Tr(d-1), & Tr(d-1)-1, & Tr(d-1) - 2, & \ldots, & Tr(d-2)+2, & Tr(d-2) + 1, \\
& \textcolor{Red}{Tr(d-2)}, & Tr(d-2), & \ldots & Tr(d-2), & Tr(d-2)-1, & Tr(d-2) - 2, & \ldots, & Tr(d-3)+2, & Tr(d-3) + 1, \\
& & & & & \vdots & & & & \\
& \textcolor{Red}{3}, & 3, & 3, & \ldots, & & \ldots, & 3, & 3, & 2, \\
& \textcolor{Red}{1}, & 1, & 1, & \ldots, & & \ldots, & 1, & 1, & 1, \\
& \textcolor{Red}{0}, & 0, & 0, & \ldots, & \}, && & & \\ \end{array} } $$ \end{proof}
Now that we have calculated $\{V_l(J\#J\#T_{n,n+1})\}_{l\geq 0}$, it is straightforward to calculate the correction terms for $Y$.
\begin{corollary}\label{coro:Y'_corr_terms} Let $K$ be $T_{2,3}$ or $D$, let $J=\#_kK$, and let $Y = S^3_{n^2+n}(J\#J\#T_{n,n+1})$. Then, $$\mathcal D(L(n,1)\#L(n+1,-1))-\mathcal D(Y) $$ is given by $$ {\small \left[ \setlength{\arraycolsep}{4pt} \begin{array}{cccccccccccccccccccccc} 0 & \cdots & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0& 0& \cdots & 0 \\ \vdots & & \vdots & \vdots &\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots && \vdots& \vdots & \vdots& & \vdots \\ 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0& 0& \cdots & 0 \\ 2 & \cdots & 2 & \textcolor{Red}{2} & 0 & \textcolor{Red}{0} & 0 & 0 & 0 &0 & 0 & 2 & 2 & 2 & 2 & \cdots & 2 & 2 & 2& \cdots & 2 \\ 2 & \cdots & 2 & 2 & \textcolor{Red}{2} & 0 & \textcolor{Red}{0} & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & \cdots & 2 & 2 & 2& \cdots & 2 \\ 4 & \cdots & 4 & 4 & 4 & \textcolor{Red}{2} & 0 & \textcolor{Red}{0} & 0 & 0 & 0 & 4 & 4 & 4 & 4 & \cdots & 4 & 4 & 4& \cdots & 4 \\ 4 & \cdots & 4 & 4 & 4 & 4 & \textcolor{Red}{2} & 0 & \textcolor{Red}{0} & 0 & 0 & 4 & 4 & 4 & 4 & \cdots & 4 & 4 & 4& \cdots & 4\\ \vdots & & \vdots & \vdots & \vdots & & & \textcolor{Red}{\ddots} & \ddots & \textcolor{Red}{\ddots} & & \vdots & \vdots & \vdots &\vdots & & \vdots & \vdots& \vdots& & \vdots \\ 2k & \cdots & 2k & 2k & 2k & \cdots & 2k & 2k & \textcolor{Red}{2} & 0 & \textcolor{Red}{0} & 2k & 2k & 2k & 2k & \cdots & 2k & 2k & 2k & \cdots & 2k \\ 2k & \cdots & 2k & 2k & 2k & \cdots & 2k & 2k & 2k & \textcolor{Red}{2} & 0 & \textcolor{Red}{2} & 2k & 2k &2k & \cdots & 2k & 2k & 2k& \cdots & 2k \\ 2k & \cdots & 2k & 2k & 2k & \cdots & 2k & 2k & 2k & 2k & \textcolor{Red}{0} & 0 & \textcolor{Red}{2} & 2k & 2k & \cdots & 2k & 2k & 2k& \cdots & 2k \\ \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & & \textcolor{Red}{\ddots} & \ddots & \textcolor{Red}{\ddots} & & & \vdots & \vdots& \vdots& & \vdots \\ 4 & \cdots & 4 & 4 & 4 & \cdots & 4 & 4 & 4 & 4 & 0 & 0 & \textcolor{Red}{0} & 0 & \textcolor{Red}{2} & 4 & 4 & 4 & 4& \cdots & 4 \\ 4 & \cdots & 4 & 4 & 4 & \cdots& 4 & 4 & 4 &4 & 0 & 0 & 0 & \textcolor{Red}{0} & 0 & \textcolor{Red}{2} & 4 & 4 & 4& \cdots & 4 \\ 2 & \cdots & 2 & 2 & 2 & \cdots& 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & \textcolor{Red}{0} & 0 & \textcolor{Red}{2} & 2 & 2& \cdots & 2 \\ 2 & \cdots & 2 & 2 & 2 & \cdots & 2 & 2 & 2 & 2& 0 & 0& 0 & 0 & 0 & \textcolor{Red}{0} & 0 & \textcolor{Red}{2} & 2& \cdots & 2 \\ 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0& \cdots & 0 \\ \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots& \vdots & \vdots & \vdots & \vdots& & \vdots \\ 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0& \cdots & 0 \\ \end{array} \right], } $$ where the rows are indexed by $i\in[\frac{-n+1}{2},\frac{n-1}{2}]$ and the columns are indexed by $j\in[\frac{-n}{2},\frac{n-2}{2}]$. \end{corollary} This matrix, $\mathcal M$ is not as complicated as it looks. It is a $(n\times(n+1))$--matrix, with zeros along the right off-diagonal. If the first column is removed, it is rotationally symmetric. Think of $\mathcal M$ as the union of its diagonals. Every diagonal (other than the middle three) is simply a copy of (twice) $L_k^t$, where $t$ is the displacement (left or right) from the center three diagonals. For example, if we consider the diagonal directly to the left of the middle three diagonals, we see a copy of (twice) $L_k^1$ that emanates towards the northwest.
\begin{proof} The matrix presentation for $ \mathcal D(L(n,1)\#L(n+1,-1)) - \mathcal D(Y) $ follows from Lemma \ref{lemma:Vs_Sum} in light of the following remark. Knowing $V_l(J\#J\#T_{n,n+1})$ allows us to calculate $d(Y,\frak s_l)$, but in the matrix above, we have given $d(Y, [\frak s_i, \frak s_j])$, which uses the enumeration of $\text{Spin}^c$ structures introduced in Subsection \ref{subsec:enumerate}. This correspondence is given by $l = \frac{n(n+1)}{2} +(n+1)i - nj$, where $l,i$, and $j$ are all taken to be centered about zero. This identification maps the subgroup generated by $n+1\in\mathbb{Z}_{n^2+n}$ to the subgroup generated by $(1,1)\in\mathbb{Z}_n\oplus\mathbb{Z}_{n+1}$. \end{proof}
There is a slight issue related to our choice of identification. In fact, there are four different identifications we could have chosen, each of which is related to the others by negating $i$, $j$, or both. The following lemma proves that these are the only four identifications that we should concern ourselves with, since any other identification will not preserve the equivalence class of the correction terms modulo 2.
Even if we are content with only these four identifications, we need to note that a different choice of identification changes our labeling of the correction terms. We have introduced an indeterminacy in which we cannot distinguish $i$ from $-i$ or $j$ from $-j$ in our labelings. Fortunately, this does not affect the proofs of Theorem \ref{thm:main} and \ref{thm:main2}.
\begin{lemma} Let $l = \frac{n(n+1)}{2} + (n+1)i - nj$. Then, $$d(S^3_{n^2+n}(J\#J\#T_{n,n+1}),l) \equiv d(L(n,1),i) - d(L(n+1,1),j) \pmod 2.$$ \end{lemma}
\begin{proof} By the integer surgery formula, we see that $d(S^3_{n^2+n}(T_{n,n+1}),l) \equiv d(L(n^2+n,1),l) \pmod 2$. Let $k=j-i$, then it is straightforward to show that $$\frac{(2l-n(n+1))^2-n(n+1)}{4n(n+1)} - \frac{(2i-n)^2-n}{4n}-\frac{(2j-(n+1))^2-(n+1)}{4(n+1)} = k^2-k.$$
From this, it follows that $d(L(n^2+n,1),l)\equiv d(L(n,1),i)-d(L(n+1,1),j) \pmod 2$ and that $$d(S^3_{n^2+n}(T_{n,n+1}),l) \equiv d(L(n,1),i)-d(L(n+1,1),j) \pmod 2.$$
Furthermore, it is easy to see that $d(L(p,1),a)\equiv d(L(p,1),b) \pmod 2$ if and only if $b=p-a$, assuming $0\leq a<b<p$. Thus, for any $Y$ obtained by surgery on a knot in $S^3$, we know that $d(Y,a) = d(Y,b)$ if and only if $b=a$ or $p-a$. In other words, the equality of two correction terms is determined by their mod 2 equivalence class for such 3--manifolds. This implies that $$d(S^3_{n^2+n}(T_{n,n+1}),l) = d(L(n,1),i) - d(L(n+1,1),j).$$ To complete the proof, we simply note that $V_l(J\#J\#T_{n,n+1}) \equiv V_l(T_{n,n+1}) \pmod 2$, by Lemma \ref{lemma:Vs_Sum}. \end{proof}
\begin{figure}
\caption{The filtered chain homotopy equivalence shown above is a straightforward exercise.}
\label{fig:TensorLemma}
\end{figure}
\begin{figure}
\caption{Germs for the total complexes of (a) $CFK^\infty(S^3, \#_mT_{2,3})$ and (b) $CFK^\infty(S^3, \#_mD)$. Note that each square above is meant to represent a multitude of overlaid squares. In (a), gradings of overlaid squares match up and are as shown. In (b), gradings in overlaid squares may be lower than shown. (See text.) }
\label{fig:TorsionComplexes}
\end{figure}
Descriptions of the germs for the total complexes $CFK^\infty(S^3, \#_mT_{2,3})$ and $CFK^\infty(S^3, \#_mD)$ are given in Figure \ref{fig:TorsionComplexes}. These presentation follow from Theorem \ref{thm:HKL}, the pictorial lemma shown in Figure \ref{fig:TensorLemma}, and induction. The proof of this pictorial lemma is straightforward. Note that in Figure \ref{fig:TorsionComplexes} each acyclic square shown is meant to represent a multitude of overlying acyclic squares, but the gradings are controlled.
Here is how the gradings behave in Figure \ref{fig:TorsionComplexes}. In (a), the gradings are as shown, and overlaid squares have the same gradings as the representative shown. In (b), for each collection of overlaid squares, the maximally graded representative is shown, and there are $m+1$ different possible gradings. For example, consider the bottom-left square in the right part of (b). This square represents many squares, each of which has as its bottom-left corner a chain with grading in $\{-m,-(m+1),-(m+2),\ldots, -2m\}$.
We are now prepared to prove one final property about $HF^+(Y')$, which we will need in order to complete the proof in Section \ref{section:calculations}.
\begin{lemma}\label{lemma:reduced_bound} Let $Y = S^3_{n^2+n}(J\#J\#T_{n,n+1})$, and let $\xi\in HF_{red}(Y,[\frak s_i,\frak s_j])$. Then $$gr(\xi) \leq gr\left(\mathcal T_{i,j}^+(Y)\right).$$ \end{lemma}
\begin{proof} Let $C = CFK^\infty(S^3, J\#J\#T_{n,n+1})$, let $C^1 = CFK^\infty(S^3,J\#J)$, and let $C^2 = CFK^\infty(S^3, T_{n,n+1})$, so $C = C^1\otimes C^2$. Let $G, G^1$, and $G^2$ be the germs for $C, C^1$, and $C^2$, respectively. Let $\xi\in HF_{red}(Y,[\frak s_i,\frak s_j])$, and let $c\in C$ be any chain such that $[c]=\xi$.
Let $c'\in C$ be any chain such that $[c']$ is the element of lowest grading in $\mathcal T_{i,j}(Y)$, so $gr(c') = gr(\mathcal T_{i,j}(Y))$. Let $G' = U^{z'}G$ be the germ containing $c'$, where $z'\in\mathbb{Z}$. Any chain in $\cup_{e> 0}U^eG'$ that is not homologous to a $U$--translate of $c'$ is not a cycle. To see this, simply observe that $H_*(\cup_{e> 0}U^eG') \cong \mathcal T^+$, and is generated by $U$--translates of $[c']$.
Suppose that $c\in U^zG$ for some $z\in\mathbb{Z}$. Since $c$ is a cycle and not homologous to a $U$--translate of $c'$, we see that $z\geq z'$. Let $c'' = U^{z'-z}c$, so $c''\in G'$, and let $c'' = c^1\otimes c^2$ with $c^1\in G^1$ and $c^2\in U^{z'}G^2$. Note that $$0 = \partial c'' = \partial(c^1\otimes c^2) = \partial c^1\otimes c^2 + c^1\otimes\partial c^2.$$ It follows that $c^1$ and $c^2$ are both cycles
By considering Figure \ref{fig:TorsionComplexes}, we see that any cycle in $G^1$ has nonpositive grading. Furthermore, any cycle in $U^{z'}G^2$ has grading $-2z'$. Let $c' = c^3\otimes c^4$, where $c^3\in G^1$ and $c^4\in U^{z'}G^2$. Since $[c']$ is the element of lowest grading in $\mathcal T_{i,j}(Y)$ it follows that $gr(c^3)=0$ and $gr(c^4)=-2z'$.
It follows that $gr(c)\leq gr(c'')\leq gr(c')$, as desired.
\end{proof}
\end{document} | arXiv |
Find the number of real solutions of the equation
\[\frac{4x}{x^2 + x + 3} + \frac{5x}{x^2 - 5x + 3} = -\frac{3}{2}.\]
Let $y = x^2 + x + 3.$ Then we can write the given equation as
\[\frac{4x}{y} + \frac{5x}{y - 6x} + \frac{3}{2} = 0.\]Multiplying everything by $2y(y - 6x),$ we get
\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\]Expanding, we get $3y^2 - 48x^2 = 0,$ so $y^2 - 16x^2 = (y - 4x)(y + 4x) = 0.$ Thus, $y = 4x$ or $y = -4x.$
If $y = 4x,$ then $x^2 + x + 3 = 4x,$ so $x^2 - 3x + 3 = 0.$ This quadratic has no real solutions.
If $y = -4x,$ then $x^2 + x + 3 = -4x,$ so $x^2 + 5x + 3 = 0.$ This quadratic has two real solutions, giving us a total of $\boxed{2}$ real solutions. | Math Dataset |
\begin{definition}[Definition:Angle/Unit/Radian]
The '''radian''' is a measure of plane angles symbolized either by the word $\radians$ or without any unit.
'''Radians''' are pure numbers, as they are ratios of lengths. The addition of $\radians$ is merely for clarification.
$1 \radians$ is the angle subtended at the center of a circle by an arc whose length is equal to the radius:
:360px
\end{definition} | ProofWiki |
Arthritis Research & Therapy
Box 1 Dosage of resistance exercise, individualized according to each participant's resources
Resistance exercise improves muscle strength, health status and pain intensity in fibromyalgia—a randomized controlled trial
Anette Larsson†1, 2Email author,
Annie Palstam†1, 2,
Monika Löfgren3,
Malin Ernberg4,
Jan Bjersing5,
Indre Bileviciute-Ljungar3,
Björn Gerdle6, 7,
Eva Kosek8 and
Kaisa Mannerkorpi1, 2, 9
Arthritis Research & Therapy201517:161
© Larsson et al. 2015
Received: 3 February 2015
Fibromyalgia (FM) is characterized by persistent widespread pain, increased pain sensitivity and tenderness. Muscle strength in women with FM is reduced compared to healthy women. The aim of this study was to examine the effects of a progressive resistance exercise program on muscle strength, health status, and current pain intensity in women with FM.
A total of 130 women with FM (age 22–64 years, symptom duration 0–35 years) were included in this assessor-blinded randomized controlled multi-center trial examining the effects of progressive resistance group exercise compared with an active control group. A person-centred model of exercise was used to support the participants' self-confidence for management of exercise because of known risks of activity-induced pain in FM. The intervention was performed twice a week for 15 weeks and was supervised by experienced physiotherapists. Primary outcome measure was isometric knee-extension force (Steve Strong®), secondary outcome measures were health status (FIQ total score), current pain intensity (VAS), 6MWT, isometric elbow-flexion force, hand-grip force, health related quality of life, pain disability, pain acceptance, fear avoidance beliefs, and patient global impression of change (PGIC). Outcomes were assessed at baseline and immediately after the intervention. Long-term follow up comprised the self-reported questionnaires only and was conducted after 13–18 months. Between-group and within-group differences were calculated using non-parametric statistics.
Significant improvements were found for isometric knee-extension force (p = 0.010), health status (p = 0.038), current pain intensity (p = 0.033), 6MWT (p = 0.003), isometric elbow flexion force (p = 0.02), pain disability (p = 0.005), and pain acceptance (p = 0.043) in the resistance exercise group (n = 56) when compared to the control group (n = 49). PGIC differed significantly (p = 0.001) in favor of the resistance exercise group at post-treatment examinations. No significant differences between the resistance exercise group and the active control group were found regarding change in self-reported questionnaires from baseline to 13–18 months.
Person-centered progressive resistance exercise was found to be a feasible mode of exercise for women with FM, improving muscle strength, health status, and current pain intensity when assessed immediately after the intervention.
ClinicalTrials.gov identification number: NCT01226784, Oct 21, 2010.
Fibromyalgia Impact Questionnaire
Active Control Group
Pain Acceptance
Pain Disability Index
Musculoskeletal pain has a negative impact on quality of life and work capacity [1] for individuals and entails large costs for society due to long-term sickness absence [2, 3]. Fibromyalgia (FM) affects approximately 1–3 % of the general population, and it is more common among women and in older age [4, 5]. FM is characterized by persistent widespread pain, increased pain sensitivity and tenderness [6] and is associated with impaired physical capacity [7–9], activity limitations [10], fatigue, and distress [6, 11].
The pain in FM is attributed to amplification of nociceptive input due to central sensitization and impaired central pain inhibition [12, 13]. Hypothetically, physical deconditioning leads to enhanced muscle ischemia, increasing peripheral sensitization and thus contributing to the central sensitization [14]. Although the precise etiology of FM is not known, physical deconditioning is believed to contribute to the development of FM [15]. Muscle strength in women with FM has been reported to be reduced by an average of 39 % compared to healthy women [16]. Possible physiological explanations for the reduced strength include structural changes in muscle fibers [17], altered neuromuscular control mechanisms [18], impaired blood circulation [19], and disturbances in regulation of growth and energy metabolism [20]. However, no differences in neuromuscular coordination was found in a study comparing exercising women with FM to sedentary controls, and both groups improved their motor unit activity during a resistance exercise program [21]. This indicates that a reason for reduced muscle strength in FM might be a low amount of physical activity at such a level that is required to maintain or improve muscle strength. Also a recent study using accelerometers indicates that the amount of physical activity at moderate and vigorous level is low among patients with FM when compared to healthy individuals [22]. Activity-induced pain is a common feature in FM [23] and might be a reason why the patients avoid activities and exercise, which could increase pain. Current guidelines for patients with FM include recommendations for aerobic exercise, such as brisk walking and cycling, as an important part of long-term management of FM [24, 25], as in several studies this mode of exercise has been shown to improve general health and physical function in patients with FM [26, 27]. Although muscle deconditioning is known to increase the susceptibility to microtrauma related to mechanical strain during physical activities [28], few studies have evaluated the effects of resistance exercise designed to improve muscle strength in FM [15]. However they have documented promising effects of resistance exercise on muscle strength, health status and pain, but the paucity of studies implies a low quality of evidence [15]. Meta-analyses in a Cochrane report of resistance exercise are based on one to three trials [15], warranting further research to improve confidence for estimated effects of resistance exercise for patients with FM [15].
A possible reason for the paucity of studies evaluating the effects of resistance exercise in FM is the risk of increased pain during isometric muscle contraction [23]. However exercise-induced pain during progressive resistance exercise might be avoided by gradual introduction to heavier loads [29]. Furthermore a theory of person-centeredness, which emphasizes active involvement of the patient in planning the treatment, is suggested to enhance the patient's ability to manage health problems [30]. In the present study, the details of the exercise program were planned together with each patient, using the principles of person-centeredness, to support each participant's ability to manage the exercise and the progress of it.
Relaxation therapy was chosen as an active control intervention as it is often integrated in multidisciplinary rehabilitation for patients with FM [24], rather than to control for treatment as usual. There is little evidence of the effects of relaxation therapy as an isolated therapy in FM [31], but relaxation therapy is assumed to improve overall wellbeing, thus providing a meaningful and inspiring therapy to control for the natural course and some aspects of attention and expectations. Patients recruited to the study were informed that the aim of the study was to compare these two treatment modalities. The aim of this study was to examine the effects of a progressive resistance exercise program using a person-centered approach, on muscle strength, health status, and current pain intensity in women with FM. Relaxation therapy was selected as an active control intervention.
An assessor-blinded randomized controlled multicenter trial examined the effects of progressive resistance group exercise compared with an active control group. The trial was registered with ClinicalTrials.gov identification number: NCT01226784.
The recruitment started in 2010 and data collection was completed at all sites (Gothenburg, Stockholm, and Linköping) in 2013. Inclusion criteria were women aged 20–65 years, meeting the American College of Rheumatology (ACR) 1990 classification criteria for FM [6]. Comorbidity as an exclusion criterion was defined by anamnesis. Exclusion criteria were high blood pressure (>160/90 mmHg), osteoarthritis (OA) in hip or knee, confirmed by radiological findings and affecting activities of daily life such as stair climbing or walking, other severe somatic or psychiatric disorders, other dominating causes of pain than FM, high consumption of alcohol (alcohol use disorders identification test (AUDIT) score >6) [32], participation in a rehabilitation program within the past year, regular resistance exercise or relaxation exercise twice a week or more, inability to understand or speak Swedish, and not being able to refrain from analgesics, non-steroidal anti-inflammatory drugs (NSAID) or hypnotic drugs for 48 hours prior to examinations.
Women with FM were recruited by newspaper advertisement in the local newspapers of three cities in Sweden (Gothenburg, Stockholm, and Linköping). A total of 402 women with FM who notified their interest for participation in the study were telephone screened for possible eligibility and informed about the study procedure. Out of these, 177 women who were interested in participation were referred for medical examination for further enrollment, while 225 were not eligible for enrollment (for details see Fig. 1). The 177 women were screened for eligibility by an experienced physician to verify ACR 1990 criteria for FM by means of a standardized interview and palpation of tender points [6]. A total of 47 women were found not eligible due to not meeting the inclusion criteria (n = 28), or declining participation (n = 19). One-hundred and thirty women with FM fulfilled the inclusion criteria. They were given written and oral information and were referred for baseline examinations (Fig. 1). Informed written consent was obtained from all participants before the baseline examination. After completing baseline examinations, the participants were randomized and informed of group allocation. An appointment for an individual introductory meeting with the specific physiotherapist guiding each intervention was scheduled with each participant. The study was approved for all sites by the Regional ethics committee in Stockholm (2010/1121-31/3).
Consolidated Standards of Reporting Trials (CONSORT) flow diagram of the progress of the two groups of the randomized trial. FM fibromyalgia
Randomization was conducted separately for each site in blocks of six subjects by a computer generated sequence [33]. For each participant the treatment was concealed in sequentially numbered, sealed, opaque envelopes. Randomization and concealment was done by a person not involved in the examinations or treatments (ME). When a participant had been included the envelope was opened together with each participant, after which she was informed about the group to which she had been allocated.
Rationale for the resistance exercise program: the main goal was to improve muscle strength and health status by progressive resistance exercise, but without risking increased pain while loading the muscles. It was unclear how many participants would be able to manage exercise at higher loads. Exercise of large muscle groups, preferably muscles in the lower extremities were chosen, as risk of activity-induced pain was anticipated to be higher when loading muscles of upper extremities. Exercises improving core stability and power were included in the program.
Person-centered intervention: the resistance exercise program was performed twice a week for 15 weeks and was supervised by experienced physiotherapists. It was conducted at physiotherapy premises and at a local gym at four different sites in groups comprising five to seven participants to promote interaction between participants and to facilitate physiotherapeutic guidance. The intervention was preceded by an individual introductory meeting. The meeting was commenced with a dialogue between the participant and the physiotherapist about the participant's earlier experiences and thoughts of exercise, which could potentially be an obstacle for her ability to exercise despite her explicit intention to do so. The introductory meeting also included exercise instructions, testing and adjustment of loads and modifications of specific exercises according to individual conditions and according to self-efficacy principles [34] of each participant's confidence in their ability to perform each exercise and to manage specific loads. The meeting resulted in a written protocol with descriptions of specific exercises and loads, which was used by each participant as an exercise program at each exercise session. To promote the participant's sense of control, and to avoid possible negative effects related to exercise, the exercise was initiated at low loads, and possibilities for progressions of loads were evaluated every 3−4 weeks in dialogue between the physiotherapist and participant (Box 1). When the participant was not ready to increase exercise loads, she continued exercising on the same loads until she was ready to do so. This mode of exercise was anticipated to increase exercise self-efficacy, enhance the ability to choose the proper level of exercise and better manage symptoms.
Estimation of one repetition maximum (1RM) was made by submaximal ratings of perceived exertion for health and safety reasons [35]. The participants were asked to perform their maximum number of repetitions until perceived exhaustion at an individually adjusted, given resistance. 1RM was based on the number of repetitions performed.
Active control group: the relaxation therapy was performed twice a week for 15 weeks and was guided by experienced physiotherapists. It was conducted at physiotherapy premises at four different sites in groups comprising five to eight participants and was preceded by an individual introductory meeting at the premises, which included instructions and allowed for preparations and modifications of practical matter such as positioning and the use of mattresses and pillows to reach a good level of comfort. The relaxation therapy was performed as autogenic training [31], which refers to a series of mental exercises including relaxation and autosuggestion. The physiotherapist guided the participants through their bodies, during approximately 25 minutes, by focusing their minds on the bodily experience of relaxation and letting the body part in focus rest on the ground. This was repeated for each specific body-part, aiming at feeling as relaxed as possible in the whole of the body at the end of the session. After each session the participants were invited to share experiences and ask each other and the physiotherapist questions, and continued thereafter with the stretching exercises.
Outcomes were assessed at baseline and at post-treatment examination after 15 weeks. Follow up was conducted 13–18 months after the baseline and only included self-reported questionnaires. All participants were invited to a post-treatment examination according to an intention-to-treat design. Baseline examinations and examinations after 15 weeks of intervention included serum samples for later analysis (not analyzed in this study), self-reported questionnaires, performance-based tests of muscle strength and physical capacity and assessment of current pain intensity. Background data were gathered using a standardized interview and included age, symptom duration, tender points, body mass index (BMI), level of leisure time physical activity (LTPAI) [36], pharmacological treatment, education, family status, country of birth, work status, and sick leave (Table 1). Examinations were conducted at physiotherapy premises by physiotherapists who were blinded to group allocation. Baseline and post-treatment examinations were performed by the same physiotherapists. The 13–18 month follow up comprised self-reported questionnaires only and was sent to the participants by mail. The participants who did not return the questionnaires in a reasonable time were reminded by telephone. After three reminders, participants that had not returned their questionnaires were regarded as missing. The participants that were already lost at post-treatment examinations were regarded as missing and were not contacted for follow up at 13–18 months.
Characteristics of the study population
Resistance exercise (experimental)
Relaxation therapy (control)
(n = 67)
Mean (SD)
Median (min; max)
Age, years
51 (25; 64)
Symptom duration, years
9 (0; 35)
Tender points, number
LTPAI, h
Number (%)
Pharmacologigal treatment
NSAID paracetamol
53 (79 %)
Opioids for mild to moderate pain
Anticonvulsives
4 (6 %)
≤9 years
8 (12 %)
Living with an adult
Sick leave/disability pension
Missing values, LTPAI: n = 1. NSAID non-steroidal anti-inflammatory drugs, LTPAI leisure time physical activity instrument
Resistance exercise was the core element of the intervention, but the program also involved other interacting components, such as the partnership between the physiotherapist and the participant when planning and progressing the exercise program. It was anticipated that both the voluntary activity of the participant and the severity of her health problems would influence the outcomes of the intervention. The prevalent evidence of the benefits of resistance exercise in FM is of low quality [15], and therefore muscle strength was selected as the primary outcome. Pain intensity, physical capacity, health status and other variables associated with health problems in chronic pain were selected for secondary outcomes, as exercising was also thought to impact on these variables.
The primary outcome was isometric knee-extension force (N) measured with a dynamometer (Steve Strong®: Stig Starke HBI, Göteborg, Sweden) using a standard protocol. The participant was in a fixed seated position with back support, knee and hip in 90 ° of flexion and legs hanging freely. A non-elastic strap was placed around the ankle and attached to a pressure transducer with an amplifier. The subjects were instructed and verbally encouraged to pull the ankle strap with maximal force for 5 seconds. Three trials were performed for each test and there was a one minute rest between each trial. The best performance out of three trials was recorded. A mean value from the right and left leg was calculated. The instrument has been used in previous studies of physical performance [37, 38] and has been reported to show satisfactory test-retest reliability for patients with a chronic condition [37].
Secondary outcomes were: the fibromyalgia impact questionnaire (FIQ) is a disease-specific self-reported questionnaire that comprises ten subscales of disabilities and symptoms ranging from 0 to 100. The total score is the mean of ten subscales. A higher score indicates a lower health status [39]. This instrument has shown good sensitivity in demonstrating therapeutic change [40]; current pain intensity (VAS), rated on a plastic 0-100 visual analogue scale with a moveable cursor along a line and anchors at the extremes. The participant was asked to rate her current pain intensity ranging from no pain at all to the worst imaginable pain; the six-minute walk test (6MWT), a performance-based test that measures total walking distance (m) during a period of 6 minutes [41]; maximal isometric elbow flexion force (kg) in both arms, was measured one by one using a dynamometer (Isobex®: Medical Device Solutions AG, Oberburg, Switzerland). The participant was in a seated position without back support, with the legs stretched out in front. The upper arm was aligned with the trunk and the elbow was placed in 90° of flexion. The maximum strength obtained during a period of 5 seconds was recorded and used in the present study [42]. Three trials were performed for each test and there was a one minute rest between each trial. The best performance out of three trials was recorded. A mean value from the right and left arm was calculated; hand grip force (N) bilaterally registered using Grippit® (AB Detektor, Göteborg, Sweden). The mean force over a set period of time (10 seconds) was recorded [43]. Two trials were performed for each test and there was a one minute rest between each trial. The best performance out of two trials was recorded. A mean value of the right and left hand was calculated and used in the present study; Short Form Health Survey (SF-36), a generic instrument assessing health related quality of life [44]. A higher score indicates better health. The subscales that build two composite scores, the physical component scale (PCS) and the mental component scale (MCS), were used in this study; the pain disability index (PDI), an instrument for measuring the impact that pain has on the ability of a person to participate in essential life activities on a scale from 0 to 70. The higher the index is, the greater the person's disability due to pain [45, 46]; the chronic pain acceptance questionnaire (CPAQ), which assesses the degree of pain-related acceptance. It consists of 20 items ranging from 0 (never true) to 6 (always true). A higher score indicates a higher level of acceptance. The total score (0–120) is presented in this study [47]; the fear avoidance beliefs questionnaire (FABQ), a questionnaire with two sub-scales that assess the extent to which fear and avoidance affect work beliefs (7 items range 0–42) and physical beliefs (4 items 0–24) in patients with chronic pain. A higher score represents greater fear avoidance beliefs [48]; and the patient global impression of change (PGIC), a numeric scale ranging from 1 (very much improved) to 7 (very much worse), where a lower score indicates greater improvement. This instrument assesses perceived global impression of change from the patient's perspective [49]. The PGIC was measured at post-treatment examinations and at 13–18 month follow up.
Data were computerized and analyzed using the Statistical Package Software for the Social Sciences (SPSS version 22.0, Chicago, IL, USA). Descriptive data are presented as mean, SD, median (min; max) for continuous variables or the number (n) and percentage (%) for categorical variables. For comparison between two groups, the Mann-Whitney U test was used for continuous variables, Fisher's exact test was used for dichotomous variables, and the Mantel Haenzel test was used for ordinal categorical variables. The Wilcoxon signed rank test was used for comparison between baseline and post-test within groups for continuous variables. Spearman correlation analysis was used for analyzing correlations between the PGIC and outcomes. To control possible type I errors, the upper limit of the expected number of false significant results for the analyses was calculated by the following formula:
$$ \alpha /1\hbox{--} \alpha \times \left(\mathrm{Number}\ \mathrm{of}\ \mathrm{tests}\ \hbox{--}\ \mathrm{Number}\ \mathrm{of}\ \mathrm{significant}\ \mathrm{tests}\right), $$
where α is the significance level. All significance tests were two-sided and conducted at the 5 % significance level. Outcomes were analyzed according to the intention-to-treat design, implying that all participants were invited to post-treatment examination, whether they had participated in the intervention or not. Only measured values were included in analyses of changes over time between the two groups and within the groups implying that cases missing were not included in the analysis. Effect size was calculated for variables showing a significant change. Effect size for between-group analyses was calculated by dividing the mean difference between the post-treatment score and baseline score in the intervention group and in the control group by the pooled SD for difference. Effect sizes from 0.20 to <0.50 were regarded as small, while effect sizes from 0.50 to <0.80 were regarded as moderate [50]. No previous data were found for isometric knee-extension force in FM using the selected dynamometer (the primary outcome), but the same methodology was applied in a study of women with chronic disease. Their isometric knee-extension force was 263 N, SD 100 [37]. Based on that report, 59 participants per group would be satisfactory to detect a 20 % difference with 80 % power when the significance level was set to 5 %.
There were no significant baseline differences between the resistance exercise group (n = 67) and the active control group (n = 63) in background data (Table 1), or the primary outcome or secondary outcomes.
Intention-to-treat analysis
All participants were invited to a post-treatment examination according to the intent-to-treat design and 81 % of the total sample completed the test, 56 (84 %) belonging to the resistance exercise group and 49 (76 %) in the active control group (Fig. 1). A total of 17 participants (25 %) in the resistance exercise group, and 20 (32 %) in the active control group discontinued the intervention for various reasons (Fig. 1). No significant differences were found when comparing the baseline characteristics of the women who completed and the women who failed to complete the post-treatment examinations. Adverse effects were reported by five participants, all in the resistance exercise group, who chose to discontinue the intervention due to increased pain (Fig. 1), but two of these participants completed post-treatment examinations. Mean attendance rate at the resistance exercise sessions was 71 % and 64 % at the relaxation therapy sessions (range 0 to 100 % in both groups).
Exercise loads
A total of 42 participants (62.7 %) in the resistance exercise group reached exercise loads of 80 % of 1RM while 7 participants (10.4 %) reached exercise loads of 60 % of 1RM. The women in the resistance exercise group who managed to reach exercise loads of 80 % of 1RM (n = 42, 63 %) showed significantly better physical capacity represented by 6MWT (p = 0.040) and health status represented by FIQ total score (p = 0.029) at baseline than the women in the resistance exercise group who did not reach exercise levels of 80 % of 1RM (Table 2).
Comparison of baseline values for participants in the resistance exercise group who reached exercise loads of 80 % and participants in the resistance exercise group who reached exercise loads up to 60 %
Baseline values of participants reaching loads of 80 %
Baseline values of participants reaching loads ≤60 %
Comparison of baseline values between groups
51.7 (8.1)
10 (0; 35)
Tender point count, number
4 (0;18)
Primary outcome
Isometric knee-extension force, N
347.3 (106.2)
344 (114;643)
Secondary outcomes
FIQ total score (0-100)
55.2 (31; 81)
Current pain intensity, visual analog scale
48.8 (27.19)
50 (5;100)
6MWT, m
573.7 (70.3)
570 (376; 766)
Isometric elbow-flexion force, kg
Isometric hand-grip force, N
162 (34; 319)
SF36 PCS, 0–100
SF36 MCS, 0–100
PDI, 0–70
CPAQ total, 0–120
59 (19; 106)
FABQphysical, 0–24
FABQwork, 0–42
6MWT 6 minute walk test, SF36 PCS Short Form 36 physical component scale, SF36 MCS Short Form 36 mental component scale, CPAQ chronic pain acceptance questionnaire, PDI pain disability index, FABQ fear avoidance beliefs questionnaire
Type I error: the between-group analyses comprised a total of 12 statistical analyses, with 7 significant values at the significance level 0.05, and the upper level of the number of false significant results was 0.26, which indicates that 0–1 of the significant results observed might be false.
Significantly greater improvement (p = 0.010) was found for isometric knee-extension force in favor of the resistance exercise group as compared to the active control group (Table 3). The effect size of change in isometric knee-extension force for the intervention group compared with the active control group was 0.55 (i.e., a moderate effect size). A total of 30 % of the participants fulfilling the resistance exercise program (n = 49) improved their isometric knee-extension force by 20 % or more, while the changes ranged from 51 % to 126 % on the individual level, implying large variation among the participants.
Between-group analysis and within-group analysis of the primary and secondary outcomes
Between-group analysis of change
Post test
Post-test- baseline
Within-group analysis
Δ (SD)
Δ (min; max)
−8.8 (70.0)
31 (−157; 178)
−10 (−222; 132)
FIQ total, 0–100
−3 (−51; 20)
1 (−34; 25)
Current pain intensity, VAS
−11.5 (25.1)
50 (5; 100)
−13 (−83; 48)
1 (−125; 101)
Isometric elbow flexion force, kg
2 (−5; 12)
Hand-grip force, N
14 (−32; 158)
9 (−101; 98)
−0.4 (9.5)
CPAQ, 0–120
−1 (−19;19)
0 (−18;10)
16.67 (12.5)
Missing values at baseline: Resistance exercise group, SF36 PCS and SF36 MCS, n = 1; FABQwork, n = 6, FABQphysical, n = 1; Relaxation therapy group, CPAQ, n = 1; FABQwork, n = 8. Missing post-test values: Resistance exercise group, SF36 MCS and PCS, n = 3; FABQwork, n = 7; FABQphysical, n = 2; Relaxation therapy group, FIQtotal, n = 1; SF36 PCS and MCS, n = 2; FABQwork, n = 9. Significant p values are shown in bold text. 6MWT six-minute walk test, FIQ fibromyalgia impact questionnaire, VAS visual analog scale, SF36 short-form 36, PDI pain disability index, CPAQ chronic pain acceptance questionnaire, FABQ fear avoidance beliefs questionnaire
No significant baseline differences (p = 0.51) were found for isometric knee-extension force between the three sites. Mean difference for change in isometric knee-extension force from baseline to post-treatment examinations between the two groups was 43.0 N (standard error (SE) 22.5), 40.1 N (SE 24.6), and 35.2 N (SE 23.1) at each site respectively.
Significantly greater improvement was observed in health status (FIQ total score) (p = 0.038) in the resistance exercise group compared to the active control group (Table 3). The effect size of the change in the FIQ total score for the intervention group compared with the active control group was 0.41. Significantly greater improvement was observed in current pain intensity (VAS) (p = 0.033) in the resistance exercise group compared to the active control group (Table 3). The effect size of the change in current pain intensity for the intervention group compared with the active control group was 0.46. Significantly greater improvement was observed in the 6MWT (p = 0.003) in the resistance exercise group compared to the active control group (Table 3). The effect size of the change in the 6MWT for the intervention group compared with the active control group was 0.45. Significantly greater improvement was observed in isometric elbow-flexion force (p = 0.020) in the resistance exercise group compared with the active control group (Table 3). The effect size of the change in isometric elbow flexion force for the intervention group compared with the active control group was 0.36. There was no significant difference between groups in hand-grip force; both the exercise intervention group and the active control group improved their strength significantly (p <0.001, p = 0.013) (Table 3).
Significant improvements were observed in the health related quality of life (SF-36 PCS and MCS) (p = 0.007) within the resistance exercise group, reflecting what is considered to be a clinically important difference [51]. No significant differences were found when comparing the resistance exercise group with the active control group (Table 3). Significantly greater improvement in pain disability represented by the PDI (p = 0.005) was observed in the resistance exercise group compared to the active control group (Table 3). The effect size of the change in the PDI for the intervention group compared with the active control group was 0.53. Significantly greater improvement in pain acceptance represented by the CPAQ (p = 0.043) was observed in the resistance exercise group as compared to the active control group (Table 3). The effect size of the change in the CPAQ for the intervention group compared with the active control group was 0.45. No differences within groups or between groups were found for fear avoidance beliefs represented by the FABQ (Table 3).
PGIC
PGIC differed significantly (p = 0.001) in favor of the resistance exercise group as compared with the active control group at post-treatment examinations. A total of 62.5 % of the participants in the resistance exercise group and 32.7 % in the active control group reported improvement in symptoms. PGIC ratings correlated significantly with improvements in current pain intensity (VAS) (r s 0.38, p = 0.004) and SF-36 PCS (r s 0.54, p <0.001) in the resistance exercise group.
There were no significant differences in the level of leisure time physical activity (LTPAI) (p = 0.74) between the resistance exercise group (5.6 h, SD 5.1) and the active control group (5.9 h, SD 5.2) at baseline. The level of moderate to vigorous physical activity at baseline in the resistance exercise group was 2.4 h (SD 2.6) and in the active control group 2.2 h (SD 2.1) (p = 0.89). During the intervention period, the level of physical activity increased significantly (p <0.001) in the resistance exercise group (2.3 h, SD 4.8) compared to the active control group (−0.1 h, SD 4.8). Moderate to vigorous physical activity increased significantly more (p = 0.003) in the resistance exercise group (1.8 h, SD 3.0) compared to the active control group (0.4 h, SD 2.6).
Follow up at 13–18 months
A total of 91 (70 %) participants completed the follow up, 48 (72 %) in the resistance exercise group and 43 (68 %) in the active control group, respectively (Fig. 1). No significant differences between the resistance exercise group and the active control group were found at follow up after 13–18 months when compared to baseline measures of these outcomes (Table 4). The only significant within-group improvement at follow up in the resistance exercise group was for pain acceptance (CPAQ) (p = 0.044) (Table 4).
Between-group analysis and within-group analysis of outcomes assessed at follow up after 13–18 months
Between group analysis for change
13–18 month follow up
Follow up- baseline
Follow up-baseline
−1 (−14; 5)
Missing values at baseline: Resistance exercise group, SF36 PCS and SF36 MCS, n = 1; FABQwork, n = 6; FABQphysical, n = 1. Relaxation therapy group, CPAQ, n = 1; FABQwork, n = 8. Missing values at 13–18 month follow up, Resistance exercise group, current pain intensity, n = 2; FABQwork, n = 5; SF36 PCS and SF36 MCS, n = 1. Relaxation therapy group, FABQwork, n = 7; FABQphysical, n = 1; SF36 PCS and SF36 MCS, n = 1. Significant p values are shown in bold text. FIQ fibromyalgia impact questionnaire, VAS visual analog scale, SF36 short-form 36, PDI pain disability index, CPAQ chronic pain acceptance questionnaire, FABQ fear avoidance beliefs questionnaire
There was no significant difference (p = 0.07) in the level of leisure time physical activity (LTPAI) from baseline to follow up between the resistance exercise group (−0.4, SD 4.9) and the active control group (−1.4, SD 2.6), as both groups had slightly decreased their total physical activity. However, at the same time, both groups had increased the level of moderate to vigorous physical activity (p = 0.74), represented by 0.8 h (SD 4.5) in the resistance exercise group and 0.9 h (SD 3.1) in the active control group.
The main findings of this study were significant improvements in isometric knee-extension force, current pain intensity, and other aspects of health in the resistance exercise group compared to the active control group. These results were supported by significant within-group improvements in the resistance exercise group. The improvement of the knee-extension force is in line with a previous study of resistance exercise in women with FM [21]. The effect size of change in the isometric knee-extension force indicated a moderate improvement in the resistance exercise group as compared with the control group. The mean improvement in isometric knee-extension force was smaller in our sample than in the two previous studies [21, 52], and one reason for this might be the longer intervention period of these studies which was 21 weeks compared to 15 weeks in our study [21, 52]. Other reasons for the difference in improvement may be related to differences in the measurement equipment, population characteristics, disease severity and exercise parameters. The significant between-group differences found in elbow-flexion force in favor of the resistance exercise group were supported by significant within-group improvements in the resistance exercise group. To our knowledge this is the first resistance exercise study showing that women with FM can improve their biceps strength by resistance exercise. Elbow-flexion force and hand-grip force had also improved significantly in the active control group. A reason for the improvement might be reduced tension in upper-extremity muscles. A previous study showed that patients with FM displayed higher levels of unnecessary tension in shoulder flexors and also had reduced strength during dynamic activities compared to pain-free controls [53]. Our interpretation is that the relaxation therapy resulted in lower tension in the shoulder muscles and thus increased the hand-grip force. Also, walking capacity, measured with 6MWT improved with resistance exercise which is in line with previous reports of resistance exercise in women with FM [54].
The mean improvement in current pain intensity in the resistance exercise group represented an improvement of 23 %, which is considered a clinically important difference, as a reduction of 15 % represents a minimal clinically important difference [55]. The improvements in pain intensity are in line with reports from previous studies of improvements in pain following resistance exercise in FM [21, 54, 56, 57]. Also, the improvements in FIQ total score support previous findings in studies of women with FM engaging in resistance exercise [56, 57].
PGIC differed significantly in favor of the resistance exercise group. We found that PGIC correlated with improvements in current pain intensity and SF-36 physical component score, which implies that the participants' overall impressions of change reflect clinically important improvements in disease-related health problems. These findings are in line with a previous report on FM from OMERACT [58]. Notable is that the improvements in SF36 scores in the resistance exercise group can be regarded as clinically important differences [51]. Significantly improved pain disability (assessed by the PDI) with a moderate effect-size for change indicated improvement in participation in everyday life activities, which reflects that the intervention focusing on enhancement in self-confidence and pain management during the exercise sessions was successful. Further, significantly improved pain acceptance (assessed by the CPAQ), was found in the resistance exercise group compared with the active control group. Acceptance of pain is assumed to be associated with less disability and better functioning in patients with chronic pain [47], and the results of this study indicate that pain acceptance can be improved when engaging in exercise.
The mean attendance rate was 71 % at the resistance exercise sessions and 64 % at the relaxation therapy sessions, which is regarded as a satisfactory rate in patients with severe health problems. The progression of the resistance exercise program proved to be a successful mode of exercise for most participants, as the majority tolerated the exercise well and few participants experienced aggravated symptoms.
Pain acceptance (CPAQ) was the only significant improvement found at follow up after 13–18 months in the resistance exercise group, which implies that the intervention promoted a process of pain acceptance that has long-term effects. However this finding should be interpreted with caution due to the fact that multiple comparisons were conducted. A probable reason for the lack of other long-term effects is that physical activity levels declined to baseline levels after the end of the intervention period. This implies that the participants had difficulties with maintaining regular resistance exercise without supervision. Some of the reasons given by participants for not continuing exercising were expensive gym membership, need for continued supervision and guidance, and difficulties in prioritizing exercise in daily life. A similar lack of long-lasting effects and difficulties among women with FM to maintain their levels of resistance exercise after the end of intervention have previously been reported [57]. A longer period of guidance and support might increase the prospects for long-lasting effects [59].
In this study, 81 % of the study population completed post-treatment examination, indicating satisfactory compliance. Five participants (7 %) in the resistance exercise group reported adverse effects and three of these did not complete the post-treatment examination. Adverse effects in this study are in line with a previous study of resistance exercise in women with FM [56]. In this study, 63 % of the participants managed to attain exercise loads of 80 % of 1RM. At baseline, these participants presented with better physical capacity in terms of 6MWT, and health status in terms of FIQ total score than those who did not reach loads of 80 %, implying that personal instructions and progression of exercise loads need to be adjusted to the participants' physical resources and health status.
Muscle-strengthening activity, such as resistance exercise, at least twice a week is recommended for preventing age-related loss of muscle mass, impaired physical function [60] and the development of degenerative age-related chronic conditions [61] in older adults in the general population. The prevention of loss of muscle mass and physical function due to aging could be argued as even more important in this population given the impaired muscle strength [8, 9] and reduced levels of physical activity [22] previously shown in women with FM. Further, the benefits of regular progressive resistance exercise on muscle strength, pain, health status and participation in daily life activities shown in this study implies that resistance exercise can be recommended as a safe and effective option for exercise, which warrants inclusion in the management of FM.
The positive results of the study showed that a supervised resistance exercise program based on person-centered principles with individually adjusted loads and progression according to each participant's resources is feasible and successful. This program can be recommended to the general population of women with FM as the characteristics of our study sample in terms of tender points, pain intensity levels and FIQ scores appear to be representative of women with FM in general [54, 56, 62]. However, strategies to support long-term regular exercise should be developed to ensure longstanding health effects.
The recruitment procedure i.e., newspaper advertisement may have resulted in recruitment of participants who were motivated to exercise and this could bias the results. To minimize this risk the advert was designed to recruit participants to both interventions so none of the participants would know in advance which intervention was the active intervention or the control intervention. Unlike pharmacological studies, which are easily blinded, behavioral and physical treatment requiring the active participation of patients is virtually impossible to blind or make inert.
Person-centered progressive resistance exercise was shown to be a feasible mode of exercise for women with FM, improving muscle function, health status, current pain intensity, pain management and participation in activities of daily life. At long-term follow up the effects had declined to baseline levels, implying that a longer period of guidance and support is recommended to increase the possibilities of maintaining regular exercise habits.
Frequency: 2 times per week for 15 weeks, supervised by physiotherapists
Exercise sessions: 10 minute warm up, 50 minute standardized protocol including: leg-press, knee-extension and knee-flexion using weight machine, biceps curl and hand grip strength using free weights, heel raise and core stability using body weight and 10 minutes of stretching exercises. Exercises for explosive strength were added to the protocol five weeks, and eight weeks into the intervention with rapid heel-raises and explosive knee-extensions respectively.
Progression: baseline: 40 % of 1 RM, with 15–20 repetitions in 1–2 sets. Three to four weeks: 60 % of 1 RM, with 10–12 repetitions in 1–2 sets. 6–8 weeks: 80 % of 1RM, performed with 5–8 repetitions in 1–2 sets.
Between each set there was a 1 minute recovery.
Anette Larsson and Annie Palstam contributed equally to this work.
1RM:
one repetition maximum
6MWT:
six minute walk test
ACR:
CPAQ:
chronic pain acceptance questionnaire
FIQ:
FM:
LTPAI:
leisure time physical activity
NSAID:
PGIC:
patient global impression of change
SF36:
Short Form 36
SF36-MCS:
Short Form 36-mental component scale
SF36-PCS:
Short Form 36-physical component scale
visual analog scale
We would like to thank all colleagues that performed examinations and supervised the groups in Gothenburg, Alingsås, Linköping, and Stockholm. We would like to thank City Gym in Gothenburg for letting us use the facilities. The study was supported by the Swedish Rheumatism Association, the Swedish Research Council, the Health and Medical Care Executive Board of Västra Götaland Region, ALF-LUA at Sahlgrenska University Hospital, Stockholm County Council (ALF), The Norrbacka-Eugenia foundation, and Gothenburg Center for Person Centered Care (GPCC). The authors declare no conflicts of interest.
The authors of this multicenter study have contributed as follows and their research site is also specified: AL and AP (Gothenburg site) worked with study design, data analysis, interpretation of data and manuscript writing; ML (Stockholm site) worked with acquisition; ME (Stockholm site) participated in study conception and study design; JB (Gothenburg site) worked with acquisition and interpretation of data; IB-L (Stockholm site) participated in study design and acquisition; BG (Linköping site) worked with study conception, study design, and acquisition; EK (Stockholm site) worked with study conception, study design, and acquisition; KM (Gothenburg site) worked with study conception, study design, acquisition, analysis, and interpretation of data. All the authors were involved in the drafting of the article and revising it critically for important intellectual content. All the authors approved the final version of the article.
Institute of Medicine, Department of Rheumatology and Inflammation research, Sahlgrenska Academy, University of Gothenburg, Guldhedsgatan Box 480, 405 30 Göteborg, Sweden
University of Gothenburg Centre for Person Centered Care (GPCC), Göteborg, Sweden
Department of Clinical Sciences, Danderyd Hospital, Karolinska Institute, Stockholm, Sweden
Department of Dental Medicine, Karolinska Institute, Stockholm, Sweden
Sahlgrenska University Hospital, Rheumatology, Göteborg, Sweden
Department of Pain and Rehabilitation Center, Linköping University, Linköping, Sweden
Department of Medical and Health Sciences, Linköping University, Linköping, Sweden
Department of Clinical Neuroscience, Karolinska Institute, Stockholm, Sweden
Institute of Neuroscience and Physiology, Section of Health and Rehabilitation, Physiotherapy, Sahlgrenska Academy, University of Gothenburg, Göteborg, Sweden
McDonald M, daCosta DiBonaventura M, Ullman S. Musculoskeletal pain in the workforce: the effects of back, arthritis, and fibromyalgia pain on quality of life and work productivity. J Occup Environ Med. 2011;53:765–70.PubMedView ArticleGoogle Scholar
Fjell Y, Alexanderson K, Karlqvist L, Bildt C. Self-reported musculoskeletal pain and working conditions among employees in the Swedish public sector. Work. 2007;28:33–46.PubMedGoogle Scholar
Annemans L, Le Lay K, Taïeb C. Societal and patient burden of fibromyalgia syndrome. Pharmacoeconomics. 2009;27:547–59.PubMedView ArticleGoogle Scholar
Gran JT. The epidemiology of chronic generalized musculoskeletal pain. Best Pract Res Clin Rheumatol. 2003;17:547–61.PubMedView ArticleGoogle Scholar
Lindell L, Bergman S, Petersson IF, Jacobsson LT, Herrstrom P. Prevalence of fibromyalgia and chronic widespread pain. Scand J Prim Health Care. 2000;18:149–53.PubMedView ArticleGoogle Scholar
Wolfe F, Smythe HA, Yunus MB, Bennett RM, Bombardier C, Goldenberg DL, et al. The American College of Rheumatology 1990 Criteria for the Classification of Fibromyalgia. Report of the Multicenter Criteria Committee. Arthritis Rheum. 1990;33:160–72.PubMedView ArticleGoogle Scholar
Mannerkorpi K, Burckhardt CS, Bjelle A. Physical performance characteristics of women with fibromyalgia. Arthritis Care Res. 1994;7:123–9.PubMedView ArticleGoogle Scholar
Henriksen M, Lund H, Christensen R, Jespersen A, Dreyer L, Bennett RM, et al. Relationships between the fibromyalgia impact questionnaire, tender point count, and muscle strength in female patients with fibromyalgia: a cohort study. Arthritis Rheum. 2009;61:732–9. doi:10.1002/art.24512.PubMedView ArticleGoogle Scholar
Góes SM, Leite N, Shay BL, Homann D, Stefanello JMF, Rodacki ALF. Functional capacity, muscle strength and falls in women with fibromyalgia. Clin Biomech. 2012;27:578–83.View ArticleGoogle Scholar
Henriksson C, Gundmark I, Bengtsson A, Ek AC. Living with fibromyalgia: Consequences for everyday life. Clin J Pain. 1992;8:138–44.PubMedView ArticleGoogle Scholar
Wolfe F, Ross K, Anderson J, Russell IJ, Hebert L. The prevalence and characteristics of fibromyalgia in the general population. Arthritis Rheum. 1995;38:19–28.PubMedView ArticleGoogle Scholar
Lannersten L, Kosek E. Dysfunction of endogenous pain inhibition during exercise with painful muscles in patients with shoulder myalgia and fibromyalgia. Pain. 2010;151:77–86.PubMedView ArticleGoogle Scholar
Jensen KB, Kosek E, Petzke F, Carville S, Fransson P, Marcus H, et al. Evidence of dysfunctional pain inhibition in Fibromyalgia reflected in rACC during provoked pain. Pain. 2009;144:95–100.PubMedView ArticleGoogle Scholar
Bennett RM. Emerging concept in the neurobiology of chronic pain: evidence of abnormal sensory processing in fibromyalgia. Mayo Clinic Proceedings Elsevier. 1999;74:385–98.View ArticleGoogle Scholar
Busch AJ, Webber SC, Richards RS, Bidonde J, Schachter CL, Schafer LA, et al. Resistance exercise training for fibromyalgia. Cochrane Database Syst Rev. 2013;12:CD010884. doi:10.1002/14651858.cd010884.PubMedGoogle Scholar
Maquet D, Croisier JL, Renard C, Crielaard JM. Muscle performance in patients with fibromyalgia. Joint Bone Spine. 2002;69:293–9.PubMedView ArticleGoogle Scholar
Bengtsson A. The muscle in fibromyalgia. Rheumatology (Oxford). 2002;41:721–4.View ArticleGoogle Scholar
Gerdle B, Grönlund C, Karlsson SJ, Holtermann A, Roeleveld K. Altered neuromuscular control mechanisms of the trapezius muscle in fibromyalgia. BMC Musculoskelet Disord. 2010;11:42.PubMed CentralPubMedView ArticleGoogle Scholar
Elvin A, Siösteen AK, Nilsson A, Kosek E. Decreased muscle blood flow in fibromyalgia patients during standardised muscle exercise: a contrast media enhanced colour Doppler study. Eur J Pain. 2006;10:137.PubMedView ArticleGoogle Scholar
Bennett RM. Adult growth hormone deficiency in patients with fibromyalgia. Curr Rheumatol Rep. 2002;4:306–12.PubMedView ArticleGoogle Scholar
Hakkinen A, Hakkinen K, Hannonen P, Alen M. Strength training induced adaptations in neuromuscular function of premenopausal women with fibromyalgia: comparison with healthy women. Ann Rheum Dis. 2001;60:21–6.PubMed CentralPubMedView ArticleGoogle Scholar
McLoughlin MJ, Colbert LH, Stegner AJ, Cook DB. Are women with fibromyalgia less physically active than healthy women? Med Sci Sports Exerc. 2011;43:905–12. doi:10.1249/MSS.0b013e3181fca1ea.PubMedView ArticleGoogle Scholar
Staud R, Robinson ME, Price DD. Isometric exercise has opposite effects on central pain mechanisms in fibromyalgia patients compared to normal controls. Pain. 2005;118:176–84.PubMedView ArticleGoogle Scholar
Carville SF, Arendt-Nielsen S, Bliddal H, Blotman F, Branco JC, Buskila D, et al. EULAR evidence-based recommendations for the management of fibromyalgia syndrome. Ann Rheum Dis. 2008;67:536–41. doi:10.1136/ard.2007.071522.PubMedView ArticleGoogle Scholar
Häuser W, Thieme K, Turk DC. Guidelines on the management of fibromyalgia syndrome–a systematic review. Eur J Pain. 2010;14:5–10.PubMedView ArticleGoogle Scholar
Busch AJ, Barber KA, Overend TJ, Peloso PM, Schachter CL. Exercise for treating fibromyalgia syndrome. Cochrane Database Syst Rev. 2007;4:CD003786. doi:10.1002/14651858.CD003786.pub2.PubMedGoogle Scholar
Busch AJ, Schachter CL, Overend TJ, Peloso PM, Barber KA. Exercise for fibromyalgia: a systematic review. J Rheumatol. 2008;35:1130–44.PubMedGoogle Scholar
Bennett RM. The origin of myopain: An integrated hypothesis of focal muscle changes and sleep disturbance in patients with the fibromyalgia syndrome. J Musculoskelet Pain. 1993;1:95–112.View ArticleGoogle Scholar
Kristensen J, Franklyn-Miller A. Resistance training in musculoskeletal rehabilitation: a systematic review. Br J Sports Med. 2012;46:719–26. doi:10.1136/bjsm.2010.079376.PubMedView ArticleGoogle Scholar
Ekman I, Swedberg K, Taft C, Lindseth A, Norberg A, Brink E, et al. Person-centered care—Ready for prime time. Eur J Cardiovasc Nurs. 2011;10:248–51.PubMedView ArticleGoogle Scholar
Meeus M, Nijs J, Vanderheiden T, Baert I, Descheemaeker F, Struyf F. The effect of relaxation therapy on autonomic functioning, symptoms and daily functioning, in patients with chronic fatigue syndrome or fibromyalgia: a systematic review. Clin Rehabil. 2014;29:221–33.PubMedView ArticleGoogle Scholar
Saunders JB, Aasland OG, Babor TF, de la Fuente JR, Grant M. Development of the alcohol use disorders identification test (AUDIT): WHO collaborative project on early detection of persons with harmful alcohol consumption-II. Addiction. 1993;88:791–804.PubMedView ArticleGoogle Scholar
Randomization. www.randomization.com.
Bandura A. Self-efficacy: The exercise of control. New York: Freeman; 1997.Google Scholar
Fleck SJ, Kraemer WJ. Designing resistance training programs. 3rd ed. Champaign: Human Kinetics; 2004.Google Scholar
Mannerkorpi K, Hernelid C. Leisure Time Physical Activity Instrument and Physical Activity at Home and Work Instrument. Development, face validity, construct validity and test-retest reliability for subjects with fibromyalgia. Disabil Rehabil. 2005;27:695–701. doi:10.1080/09638280400009063.PubMedView ArticleGoogle Scholar
Brodin E, Ljungman S, Sunnerhagen KS. Rising from a chair: a simple screening test for physical function in predialysis patients. Scand J Urol Nephrol. 2008;42:293–300. doi:10.1080/00365590701797556.PubMedView ArticleGoogle Scholar
Schaufelberger M, Eriksson B, Grimby G, Held P, Swedberg K. Skeletal muscle alterations in patients with chronic heart failure. Eur Heart J. 1997;18:971–80.PubMedView ArticleGoogle Scholar
Hedin PJ, Hamne M, Burckhardt CS, Engstrom-Laurent A. The Fibromyalgia Impact Questionnaire, a Swedish translation of a new tool for evaluation of the fibromyalgia patient. Scand J Rheumatol. 1995;24:69–75.PubMedView ArticleGoogle Scholar
Bennett R. The Fibromyalgia Impact Questionnaire (FIQ): a review of its development, current version, operating characteristics and uses. Clin Exp Rheumatol. 2005;23:S154–62.PubMedGoogle Scholar
Mannerkorpi K, Svantesson U, Carlsson J, Ekdahl C. Tests of functional limitations in fibromyalgia syndrome: a reliability study. Arthritis Care Res. 1999;12:193–9.PubMedView ArticleGoogle Scholar
Leggin BG, Neuman RM, Iannotti JP, Williams GR, Thompson EC. Intrarater and interrater reliability of three isometric dynamometers in assessing shoulder strength. J Shoulder Elbow Surg. 1996;5:18–24.PubMedView ArticleGoogle Scholar
Nordenskiold UM, Grimby G. Grip force in patients with rheumatoid arthritis and fibromyalgia and in healthy subjects: A study with the Grippit instrument. Scand J Rheumatol. 1993;22:14–9.PubMedView ArticleGoogle Scholar
Ware Jr JE, Sherbourne CD. The MOS 36-item short-form health survey (SF-36). I. Conceptual framework and item selection. Med Care. 1992;30:473–83.PubMedView ArticleGoogle Scholar
Tait RC, Pollard CA, Margolis RB, Duckro PN, Krause SJ. The Pain Disability Index: psychometric and validity data. Arch Phys Med Rehabil. 1987;68:438–41.PubMedGoogle Scholar
Tait RC, Chibnall JT, Krause S. The Pain Disability Index: psychometric properties. Pain. 1990;40:171–82.PubMedView ArticleGoogle Scholar
McCracken LM, Vowles KE, Eccleston C. Acceptance of chronic pain: component analysis and a revised assessment method. Pain. 2004;107:159–66.PubMedView ArticleGoogle Scholar
Waddell G, Newton M, Henderson I, Somerville D, Main CJ. A Fear-Avoidance Beliefs Questionnaire (FABQ) and the role of fear-avoidance beliefs in chronic low back pain and disability. Pain. 1993;52:157–68.PubMedView ArticleGoogle Scholar
Hurst H, Bolton J. Assessing the clinical significance of change scores recorded on subjective outcome measures. J Manipulative Physiol Ther. 2004;27:26–35. doi:10.1016/j.jmpt.2003.11.003.PubMedView ArticleGoogle Scholar
Fayers P, Machin D. Quality of life: the assessment, analysis and interpretation of patient-reported outcomes. John Wiley & Sons; 2007. The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, EnglandGoogle Scholar
Lubeck DP. Patient-reported outcomes and their role in the assessment of rheumatoid arthritis. Pharmacoeconomics. 2004;22:27–38.PubMedView ArticleGoogle Scholar
Valkeinen H, Alen M, Hannonen P, Hakkinen A, Airaksinen O, Hakkinen K. Changes in knee extension and flexion force, EMG and functional capacity during strength training in older females with fibromyalgia and healthy controls. Rheumatology (Oxford). 2004;43:225–8. doi:10.1093/rheumatology/keh027.View ArticleGoogle Scholar
Elert J, Kendall SA, Larsson B, Månsson B, Gerdle B. Chronic pain and difficulty in relaxing postural muscles in patients with fibromyalgia and chronic whiplash associated disorders. J Rheumatol. 2001;28:1361–8.PubMedGoogle Scholar
Bircan C, Karasel SA, Akgun B, El O, Alper S. Effects of muscle strengthening versus aerobic exercise program in fibromyalgia. Rheumatol Int. 2008;28:527–32. doi:10.1007/s00296-007-0484-5.PubMedView ArticleGoogle Scholar
Salaffi F, Stancati A, Alberto Silvestri C, Ciapetti A, Grassi W. Minimal clinically important changes in chronic musculoskeletal pain intensity measured on a numerical rating scale. Eur J Pain. 2004;8:283–91.PubMedView ArticleGoogle Scholar
Jones KD, Burckhardt CS, Clark SR, Bennett RM, Potempa KM. A randomized controlled trial of muscle strengthening versus flexibility training in fibromyalgia. J Rheumatol. 2002;29:1041–8.PubMedGoogle Scholar
Kayo AH, Peccin MS, Sanches CM, Trevisani VF. Effectiveness of physical activity in reducing pain in patients with fibromyalgia: a blinded randomized clinical trial. Rheumatol Int. (2012)32:2285-2292. doi:10.1007/s00296-011-1958-z.
Mease P, Arnold LM, Choy EH, Clauw DJ, Crofford LJ, Glass JM, et al. Fibromyalgia syndrome module at OMERACT 9: domain construct. J Rheumatol. 2009;36:2318–29.PubMed CentralPubMedView ArticleGoogle Scholar
Brodin N, Eurenius E, Jensen I, Nisell R, Opava CH. Coaching patients with early rheumatoid arthritis to healthy physical activity: a multicenter, randomized, controlled study. Arthritis Care Res. 2008;59:325–31.View ArticleGoogle Scholar
Nelson ME, Rejeski WJ, Blair SN, Duncan PW, Judge JO, King AC, et al. Physical activity and public health in older adults: recommendation from the American College of Sports Medicine and the American Heart Association. Circulation. 2007;116:1094.PubMedView ArticleGoogle Scholar
Chodzko-Zajko WJ, Proctor DN, Fiatarone Singh MA, Minson CT, Nigg CR, Salem GJ, et al. American College of Sports Medicine position stand: Exercise and physical activity for older adults. Med Sci Sports Exerc. 2009;41:1510–30. doi:10.1249/MSS.0b013e3181a0c95c.PubMedView ArticleGoogle Scholar
Mannerkorpi K, Nordeman L, Cider Å, Jonsson G. Does moderate-to-high intensity Nordic walking improve functional capacity and pain in fibromyalgia? A prospective randomized controlled trial. Arthritis Res Ther. 2010;12:R189.PubMed CentralPubMedView ArticleGoogle Scholar
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Super-Poincaré algebra
In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symmetries), and are Lie superalgebras. Thus a super-Poincaré algebra is a Z2-graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part.
Informal sketch
The Poincaré algebra describes the isometries of Minkowski spacetime. From the representation theory of the Lorentz group, it is known that the Lorentz group admits two inequivalent complex spinor representations, dubbed $2$ and ${\overline {2}}$.[nb 1] Taking their tensor product, one obtains $2\otimes {\overline {2}}=3\oplus 1$; such decompositions of tensor products of representations into direct sums is given by the Littlewood–Richardson rule.
Normally, one treats such a decomposition as relating to specific particles: so, for example, the pion, which is a chiral vector particle, is composed of a quark-anti-quark pair. However, one could also identify $3\oplus 1$ with Minkowski spacetime itself. This leads to a natural question: if Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry be extended to the fundamental representation? Well, it can: this is exactly the super-Poincaré algebra. There is a corresponding experimental question: if we live in the adjoint representation, then where is the fundamental representation hiding? This is the program of supersymmetry, which has not been found experimentally.
History
The super-Poincaré algebra was first proposed in the context of the Haag–Łopuszański–Sohnius theorem, as a means of avoiding the conclusions of the Coleman–Mandula theorem. That is, the Coleman–Mandula theorem is a no-go theorem that states that the Poincaré algebra cannot be extended with additional symmetries that might describe the internal symmetries of the observed physical particle spectrum. However, the Coleman–Mandula theorem assumed that the algebra extension would be by means of a commutator; this assumption, and thus the theorem, can be avoided by considering the anti-commutator, that is, by employing anti-commuting Grassmann numbers. The proposal was to consider a supersymmetry algebra, defined as the semidirect product of a central extension of the super-Poincaré algebra by a compact Lie algebra of internal symmetries.
Definition
The simplest supersymmetric extension of the Poincaré algebra contains two Weyl spinors with the following anti-commutation relation:
$\{Q_{\alpha },{\bar {Q}}_{\dot {\beta }}\}=2{\sigma ^{\mu }}_{\alpha {\dot {\beta }}}P_{\mu }$
and all other anti-commutation relations between the Qs and Ps vanish.[1] The operators $Q_{\alpha },{\bar {Q}}_{\dot {\alpha }}$ are known as supercharges. In the above expression $P_{\mu }$ are the generators of translation and $\sigma ^{\mu }$ are the Pauli matrices. The index $\alpha $ runs over the values $\alpha =1,2.$ A dot is used over the index ${\dot {\beta }}$ to remind that this index transforms according to the inequivalent conjugate spinor representation; one must never accidentally contract these two types of indexes. The Pauli matrices can be considered to be a direct manifestation of the Littlewood–Richardson rule mentioned before: they indicate how the tensor product $2\otimes {\overline {2}}$ of the two spinors can be re-expressed as a vector. The index $\mu $ of course ranges over the space-time dimensions $\mu =0,1,2,3.$
It is convenient to work with Dirac spinors instead of Weyl spinors; a Dirac spinor can be thought of as an element of $2\oplus {\overline {2}}$; it has four components. The Dirac matrices are thus also four-dimensional, and can be expressed as direct sums of the Pauli matrices. The tensor product then gives an algebraic relation to the Minkowski metric $g^{\mu \nu }$ which is expressed as:
$\{\gamma ^{\mu },\gamma ^{\nu }\}=2g^{\mu \nu }$
and
$\sigma ^{\mu \nu }={\frac {i}{2}}\left[\gamma ^{\mu },\gamma ^{\nu }\right]$
This then gives the full algebra[2]
${\begin{aligned}\left[M^{\mu \nu },Q_{\alpha }\right]&={\frac {1}{2}}(\sigma ^{\mu \nu })_{\alpha }^{\;\;\beta }Q_{\beta }\\\left[Q_{\alpha },P^{\mu }\right]&=0\\\{Q_{\alpha },{\bar {Q}}_{\dot {\beta }}\}&=2(\sigma ^{\mu })_{\alpha {\dot {\beta }}}P_{\mu }\\\end{aligned}}$
which are to be combined with the normal Poincaré algebra. It is a closed algebra, since all Jacobi identities are satisfied and can have since explicit matrix representations. Following this line of reasoning will lead to supergravity.
Extended supersymmetry
It is possible to add more supercharges. That is, we fix a number which by convention is labelled ${\mathcal {N}}$, and define supercharges $Q_{\alpha }^{I},{\bar {Q}}_{\dot {\alpha }}^{I}$ with $I=1,\cdots ,{\mathcal {N}}.$
These can be thought of as many copies of the original supercharges, and hence satisfy
$[M^{\mu \nu },Q_{\alpha }^{I}]=(\sigma ^{\mu \nu })_{\alpha }{}^{\beta }Q_{\beta }^{I}$
$[P^{\mu },Q_{\alpha }^{I}]=0$
and
$\{Q_{\alpha }^{I},{\bar {Q}}_{\dot {\alpha }}^{J}\}=2\sigma _{\alpha {\dot {\alpha }}}^{\mu }P_{\mu }\delta ^{IJ}$
but can also satisfy
$\{Q_{\alpha }^{I},Q_{\beta }^{J}\}=\epsilon _{\alpha \beta }Z^{IJ}$
and
$\{{\bar {Q}}_{\dot {\alpha }}^{I},{\bar {Q}}_{\dot {\beta }}^{J}\}=\epsilon _{{\dot {\alpha }}{\dot {\beta }}}Z^{\dagger IJ}$
where $Z^{IJ}=-Z^{JI}$ is the central charge.
Super-Poincaré group and superspace
Just as the Poincaré algebra generates the Poincaré group of isometries of Minkowski space, the super-Poincaré algebra, an example of a Lie super-algebra, generates what is known as a supergroup. This can be used to define superspace with ${\mathcal {N}}$ supercharges: these are the right cosets of the Lorentz group within the ${\mathcal {N}}$ super-Poincaré group.
Just as $P_{\mu }$ has the interpretation as being the generator of spacetime translations, the charges $Q_{\alpha }^{I},{\bar {Q}}_{\dot {\alpha }}^{I}$, with $I=1,\cdots ,{\mathcal {N}}$, have the interpretation as generators of superspace translations in the 'spin coordinates' of superspace. That is, we can view superspace as the direct sum of Minkowski space with 'spin dimensions' labelled by coordinates $\theta _{\alpha }^{I},{\bar {\theta }}^{I{\dot {\alpha }}}$. The supercharge $Q_{\alpha }^{I}$ generates translations in the direction labelled by the coordinate $\theta _{\alpha }^{I}.$ By counting, there are $4{\mathcal {N}}$ spin dimensions.
Notation for superspace
The superspace consisting of Minkowski space with ${\mathcal {N}}$ supercharges is therefore labelled $\mathbb {R} ^{1,3|4{\mathcal {N}}}$ or sometimes simply $\mathbb {R} ^{4|4{\mathcal {N}}}$.
SUSY in 3 + 1 Minkowski spacetime
In (3 + 1) Minkowski spacetime, the Haag–Łopuszański–Sohnius theorem states that the SUSY algebra with N spinor generators is as follows.
The even part of the star Lie superalgebra is the direct sum of the Poincaré algebra and a reductive Lie algebra B (such that its self-adjoint part is the tangent space of a real compact Lie group). The odd part of the algebra would be
$\left({\frac {1}{2}},0\right)\otimes V\oplus \left(0,{\frac {1}{2}}\right)\otimes V^{*}$
where $(1/2,0)$ and $(0,1/2)$ are specific representations of the Poincaré algebra. (Compared to the notation used earlier in the article, these correspond ${\overline {2}}\oplus 1$ and $1\oplus 2$, respectively, also see the footnote where the previous notation was introduced). Both components are conjugate to each other under the * conjugation. V is an N-dimensional complex representation of B and V* is its dual representation. The Lie bracket for the odd part is given by a symmetric equivariant pairing {.,.} on the odd part with values in the even part. In particular, its reduced intertwiner from $\left[\left({\frac {1}{2}},0\right)\otimes V\right]\otimes \left[\left(0,{\frac {1}{2}}\right)\otimes V^{*}\right]$ to the ideal of the Poincaré algebra generated by translations is given as the product of a nonzero intertwiner from $\left({\frac {1}{2}},0\right)\otimes \left(0,{\frac {1}{2}}\right)$ to (1/2,1/2) by the "contraction intertwiner" from $V\otimes V^{*}$ to the trivial representation. On the other hand, its reduced intertwiner from $\left[\left({\frac {1}{2}},0\right)\otimes V\right]\otimes \left[\left({\frac {1}{2}},0\right)\otimes V\right]$ is the product of a (antisymmetric) intertwiner from $\left({\frac {1}{2}},0\right)\otimes \left({\frac {1}{2}},0\right)$ to (0,0) and an antisymmetric intertwiner A from $N^{2}$ to B. Conjugate it to get the corresponding case for the other half.
N = 1
B is now ${\mathfrak {u}}(1)$ (called R-symmetry) and V is the 1D representation of ${\mathfrak {u}}(1)$ with charge 1. A (the intertwiner defined above) would have to be zero since it is antisymmetric.
Actually, there are two versions of N=1 SUSY, one without the ${\mathfrak {u}}(1)$ (i.e. B is zero-dimensional) and the other with ${\mathfrak {u}}(1)$.
N = 2
B is now ${\mathfrak {su}}(2)\oplus {\mathfrak {u}}(1)$ and V is the 2D doublet representation of ${\mathfrak {su}}(2)$ with a zero ${\mathfrak {u}}(1)$ charge. Now, A is a nonzero intertwiner to the ${\mathfrak {u}}(1)$ part of B.
Alternatively, V could be a 2D doublet with a nonzero ${\mathfrak {u}}(1)$ charge. In this case, A would have to be zero.
Yet another possibility would be to let B be ${\mathfrak {u}}(1)_{A}\oplus {\mathfrak {u}}(1)_{B}\oplus {\mathfrak {u}}(1)_{C}$. V is invariant under ${\mathfrak {u}}(1)_{B}$ and ${\mathfrak {u}}(1)_{C}$ and decomposes into a 1D rep with ${\mathfrak {u}}(1)_{A}$ charge 1 and another 1D rep with charge -1. The intertwiner A would be complex with the real part mapping to ${\mathfrak {u}}(1)_{B}$ and the imaginary part mapping to ${\mathfrak {u}}(1)_{C}$.
Or we could have B being ${\mathfrak {su}}(2)\oplus {\mathfrak {u}}(1)_{A}\oplus {\mathfrak {u}}(1)_{B}$ with V being the doublet rep of ${\mathfrak {su}}(2)$ with zero ${\mathfrak {u}}(1)$ charges and A being a complex intertwiner with the real part mapping to ${\mathfrak {u}}(1)_{A}$ and the imaginary part to ${\mathfrak {u}}(1)_{B}$.
This doesn't even exhaust all the possibilities. We see that there is more than one N = 2 supersymmetry; likewise, the SUSYs for N > 2 are also not unique (in fact, it only gets worse).
N = 3
It is theoretically allowed, but the multiplet structure becomes automatically the same with that of an N=4 supersymmetric theory. So it is less often discussed compared to N=1,2,4 version.
N = 4
This is the maximal number of supersymmetries in a theory without gravity.
N = 8
This is the maximal number of supersymmetries in any supersymmetric theory. Beyond ${\mathcal {N}}=8$, any massless supermultiplet contains a sector with helicity $\lambda $ such that $|\lambda |>2$. Such theories on Minkowski space must be free (non-interacting).
SUSY in various dimensions
In 0 + 1, 2 + 1, 3 + 1, 4 + 1, 6 + 1, 7 + 1, 8 + 1, and 10 + 1 dimensions, a SUSY algebra is classified by a positive integer N.
In 1 + 1, 5 + 1 and 9 + 1 dimensions, a SUSY algebra is classified by two nonnegative integers (M, N), at least one of which is nonzero. M represents the number of left-handed SUSYs and N represents the number of right-handed SUSYs.
The reason of this has to do with the reality conditions of the spinors.
Hereafter d = 9 means d = 8 + 1 in Minkowski signature, etc. The structure of supersymmetry algebra is mainly determined by the number of the fermionic generators, that is the number N times the real dimension of the spinor in d dimensions. It is because one can obtain a supersymmetry algebra of lower dimension easily from that of higher dimensionality by the use of dimensional reduction.
Upper bound on dimension of supersymmetric theories
The maximum allowed dimension of theories with supersymmetry is $d=11=10+1$, which admits a unique theory called 11-dimensional supergravity which is the low-energy limit of M-theory. This incorporates supergravity: without supergravity, the maximum allowed dimension is $d=10=9+1$.[3]
d = 11
The only example is the N = 1 supersymmetry with 32 supercharges.
d = 10
From d = 11, N = 1 SUSY, one obtains N = (1, 1) nonchiral SUSY algebra, which is also called the type IIA supersymmetry. There is also N = (2, 0) SUSY algebra, which is called the type IIB supersymmetry. Both of them have 32 supercharges.
N = (1, 0) SUSY algebra with 16 supercharges is the minimal susy algebra in 10 dimensions. It is also called the type I supersymmetry. Type IIA / IIB / I superstring theory has the SUSY algebra of the corresponding name. The supersymmetry algebra for the heterotic superstrings is that of type I.
Remarks
1. The barred representations are conjugate linear while the unbarred ones are complex linear. The numeral refers to the dimension of the representation space. Another more common notation is to write (1⁄2, 0) and (0, 1⁄2) respectively for these representations. The general irreducible representation is then (m, n), where m, n are half-integral and correspond physically to the spin content of the representation, which ranges from |m + n| to |m − n| in integer steps, each spin occurring exactly once.
Notes
1. Aitchison 2005
2. van Nieuwenhuizen 1981, p. 274
3. Tong, David. "Supersymmetry". www.damtp.cam.ac.uk. Retrieved 3 April 2023.
References
• Aitchison, Ian J R (2005). "Supersymmetry and the MSSM: An Elementary Introduction". arXiv:hep-ph/0505105.
• Gol'fand, Y. A.; Likhtman, E. P. (1971). "Extension of the algebra of the Poincare group generators and violation of P invariance". JETP Lett. 13: 323–326. Bibcode:1971JETPL..13..323G.
• van Nieuwenhuizen, P. (1981). "Supergravity". Phys. Rep. 68 (4): 189–398. Bibcode:1981PhR....68..189V. doi:10.1016/0370-1573(81)90157-5.
• Volkov, D. V.; Akulov, V. P. (1972). "Possible Universal Neutrino Interaction". JETP Lett. 16 (11): 621 pp.
• Volkov, D. V.; Akulov, V. P. (1973). "Is the neutrino a goldstone particle". Phys. Lett. B. 46 (1): 109–110. Bibcode:1973PhLB...46..109V. doi:10.1016/0370-2693(73)90490-5.
• Weinberg, Steven (2000). Supersymmetry. The Quantum Theory of Fields. Vol. 3 (1st ed.). Cambridge: Cambridge University Press. ISBN 978-0521670555.
• Wess, J.; Zumino, B. (1974). "Supergauge transformations in four dimensions". Nuclear Physics B. 70 (1): 39–50. Bibcode:1974NuPhB..70...39W. doi:10.1016/0550-3213(74)90355-1.
Supersymmetry
General topics
• Supersymmetry
• Supersymmetric gauge theory
• Supersymmetric quantum mechanics
• Supergravity
• Superstring theory
• Super vector space
• Supergeometry
Supermathematics
• Superalgebra
• Lie superalgebra
• Super-Poincaré algebra
• Superconformal algebra
• Supersymmetry algebra
• Supergroup
• Superspace
• Harmonic superspace
• Super Minkowski space
• Supermanifold
Concepts
• Supercharge
• R-symmetry
• Supermultiplet
• Short supermultiplet
• BPS state
• Superpotential
• D-term
• FI D-term
• F-term
• Moduli space
• Supersymmetry breaking
• Konishi anomaly
• Seiberg duality
• Seiberg–Witten theory
• Witten index
• Wess–Zumino gauge
• Localization
• Mu problem
• Little hierarchy problem
• Electric–magnetic duality
Theorems
• Coleman–Mandula
• Haag–Łopuszański–Sohnius
• Nonrenormalization
Field theories
• Wess–Zumino
• N = 1 super Yang–Mills
• N = 4 super Yang–Mills
• Super QCD
• MSSM
• NMSSM
• 6D (2,0) superconformal
• ABJM superconformal
Supergravity
• Pure 4D N = 1 supergravity
• N = 8 supergravity
• Higher dimensional
• Gauged supergravity
Superpartners
• Axino
• Chargino
• Gaugino
• Goldstino
• Graviphoton
• Graviscalar
• Higgsino
• LSP
• Neutralino
• R-hadron
• Sfermion
• Sgoldstino
• Stop squark
• Superghost
Researchers
• Affleck
• Bagger
• Batchelor
• Berezin
• Dine
• Fayet
• Gates
• Golfand
• Iliopoulos
• Montonen
• Olive
• Salam
• Seiberg
• Siegel
• Roček
• Rogers
• Wess
• Witten
• Zumino
| Wikipedia |
# Activation functions and their role in neural networks
Activation functions are fundamental components of neural networks. They introduce non-linearity into the network, allowing it to learn complex patterns and relationships. Commonly used activation functions include sigmoid, ReLU (Rectified Linear Unit), and tanh.
Sigmoid function:
The sigmoid function is defined as:
$$\sigma(z) = \frac{1}{1 + e^{-z}}$$
It maps any input to a value between 0 and 1, which is useful for binary classification tasks.
ReLU function:
The ReLU function is defined as:
$$ReLU(z) = max(0, z)$$
It is simple to compute and helps prevent the vanishing gradient problem, which can occur in deep networks.
Tanh function:
The tanh function is defined as:
$$tanh(z) = \frac{e^z - e^{-z}}{e^z + e^{-z}}$$
It maps any input to a value between -1 and 1, making it useful for regression tasks.
## Exercise
Consider the following activation functions:
1. Sigmoid
2. ReLU
3. Tanh
Which of these functions is typically used for binary classification tasks?
A. Sigmoid
B. ReLU
C. Tanh
The correct answer is A. Sigmoid.
# Memory processing and its relation to neural networks
Memory processing is a crucial aspect of neural networks, as it allows the network to store and retrieve information. This is particularly important for tasks involving sequence processing, such as natural language processing.
Long Short-Term Memory (LSTM) networks are a type of recurrent neural network that are especially effective at handling sequence data. LSTMs have a special type of memory cell called a "memory cell," which allows them to capture and store information over long sequences.
Gated Recurrent Unit (GRU) networks are another type of recurrent neural network that can effectively handle sequence data. GRUs have a more streamlined architecture compared to LSTMs, making them computationally more efficient.
# Natural language processing and its application in neural networks
Natural language processing (NLP) is a field that focuses on enabling computers to understand, interpret, and generate human language. Neural networks have been instrumental in advancing NLP techniques, particularly through the use of word embeddings and recurrent neural networks.
Word embeddings are a way to represent words as vectors in a high-dimensional space. They allow neural networks to capture the semantic and syntactic properties of words, enabling more accurate and nuanced language processing.
Recurrent neural networks, such as LSTMs and GRUs, have been used to process sequence data in NLP tasks, such as sentiment analysis, machine translation, and text summarization. These networks can capture the context and structure of sentences, making them more effective than traditional bag-of-words models.
# Foundational neural network architectures
There are several foundational neural network architectures that have been used to build more complex models. Some of the most important include:
1. Feedforward Neural Networks (FNNs): These are the simplest type of neural networks, where the information flows in one direction from the input layer to the output layer.
2. Convolutional Neural Networks (CNNs): These are designed for processing grid-like data, such as images. They have a series of convolutional layers that learn local patterns in the data.
3. Recurrent Neural Networks (RNNs): These are designed for processing sequence data. They have a loop that allows information to flow repeatedly within the network, capturing long-range dependencies.
4. Transformer Networks: These are based on the self-attention mechanism, which allows the network to weigh the importance of different parts of the input sequence. They have become the foundation for many state-of-the-art NLP models.
# Spreading activation and its role in neural network performance
Spreading activation is a phenomenon in neural networks where the activation of a neuron can influence the activation of other neurons, even if they are not directly connected. This can lead to improved performance and faster convergence of the network.
One way to achieve spreading activation is through the use of local connectivity patterns, where neurons in the same region of the network are more likely to be connected. This can help the network learn more complex patterns and relationships.
Another way to achieve spreading activation is through the use of lateral connections, where neurons in different layers of the network are connected. This can help the network learn from information at multiple levels and improve its overall performance.
# Deep learning and its impact on neural networks
Deep learning refers to the use of deep neural networks, which have many layers of neurons. These networks can learn complex patterns and representations from large amounts of data, making them highly effective for a wide range of tasks.
Deep learning has had a transformative impact on the field of neural networks. It has led to advances in areas such as computer vision, natural language processing, and speech recognition. It has also enabled the development of more sophisticated models, such as generative adversarial networks (GANs) and transformer networks.
# Examples of neural network applications in memory and language processing
Neural networks have been applied to a wide range of memory and language processing tasks. Some examples include:
1. Image recognition: Neural networks have been used to classify and recognize objects in images, such as determining whether an image contains a cat or a dog.
2. Natural language understanding: Neural networks have been used to parse and understand the meaning of text, such as sentiment analysis and machine translation.
3. Speech recognition: Neural networks have been used to recognize and transcribe spoken words, enabling more accurate and efficient speech-to-text conversion.
4. Recommender systems: Neural networks have been used to recommend items to users based on their preferences and behavior, such as suggesting movies or products to a user.
# Challenges and future directions in neural network research for memory and language processing
Despite the impressive advances in neural network research for memory and language processing, there are still several challenges that need to be addressed. These include:
1. Interpretability: Neural networks can be difficult to interpret and understand, making it challenging to explain why they make certain decisions.
2. Generalization: Neural networks can sometimes struggle to generalize well to new data, leading to overfitting and poor performance on unseen data.
3. Fairness and bias: Neural networks can inadvertently learn and perpetuate biases present in the training data, leading to unfair or discriminatory behavior.
4. Computational efficiency: Deep neural networks can be computationally expensive to train and evaluate, limiting their practical application in some domains.
In the future, researchers will likely continue to explore ways to address these challenges and push the boundaries of neural network research for memory and language processing. This may involve developing more interpretable and explainable models, improving generalization capabilities, reducing biases, and improving computational efficiency.
# Conclusion: The importance of neural networks in advancing our understanding of memory and language processing
Neural networks have revolutionized the field of memory and language processing, enabling us to build more effective and powerful models for a wide range of tasks. Their ability to learn complex patterns and representations from data has led to significant advances in areas such as computer vision, natural language understanding, and speech recognition.
However, there are still several challenges that need to be addressed in order to fully realize the potential of neural networks in memory and language processing. Researchers will continue to explore ways to address these challenges and push the boundaries of neural network research in this domain.
In conclusion, neural networks have become an indispensable tool for advancing our understanding of memory and language processing. Their ability to learn complex patterns and representations from data has led to significant advances in various fields, and their impact is likely to continue to grow in the coming years. | Textbooks |
Sieve of Pritchard
In mathematics, the sieve of Pritchard is an algorithm for finding all prime numbers up to a specified bound. Like the ancient sieve of Eratosthenes, it has a simple conceptual basis in number theory.[1] It is especially suited to quick hand computation for small bounds.
Whereas the sieve of Eratosthenes marks off each non-prime for each of its prime factors, the sieve of Pritchard avoids considering almost all non-prime numbers by building progressively larger wheels, which represent the pattern of numbers not divisible by any of the primes processed thus far. It thereby achieves a better asymptotic complexity, and was the first sieve with a running time sublinear in the specified bound. Its asymptotic running-time has not been improved on, and it deletes fewer composites than any other known sieve. It was created in 1979 by Paul Pritchard.[2]
Since Pritchard has created a number of other sieve algorithms for finding prime numbers,[3][4][5] the sieve of Pritchard is sometimes singled out by being called the wheel sieve (by Pritchard himself[1]) or the dynamic wheel sieve.[6]
Overview
A prime number is a natural number that has no natural number divisors other than the number $1$ and itself.
To find all the prime numbers less than or equal to a given integer $N$, a sieve algorithm examines a set of candidates in the range $2,3,...,N$, and eliminates those that are not prime, leaving the primes at the end. The sieve of Eratosthenes examines all of the range, first removing all multiples of the first prime $2$, then of the next prime $3$, and so on. The sieve of Pritchard instead examines a subset of the range consisting of numbers that occur on successive wheels, which represent the pattern of numbers left after each successive prime is processed by the sieve of Eratosthenes.
For $i>0$ the $i$'th wheel $W_{i}$ represents this pattern. It is the set of numbers between $1$ and the product $P_{i}=p_{1}*p_{2}*...*p_{i}$ of the first $i$ prime numbers that are not divisible by any of these prime numbers (and is said to have an associated length $P_{i}$). This is because adding $P_{i}$ to a number doesn't change whether or not it is divisible by one of the first $i$ prime numbers, since the remainder on division by any one of these primes is unchanged.
So $W_{1}=\{1\}$ with length $P_{1}=2$ represents the pattern of odd numbers; $W_{2}=\{1,5\}$ with length $P_{2}=6$ represents the pattern of numbers not divisible by $2$ or $3$; etc. Wheels are so-called because $W_{i}$ can be usefully visualized as a circle of circumference $P_{i}$ with its members marked at their corresponding distances from an origin. Then rolling the wheel along the number line marks points corresponding to successive numbers not divisible by one of the first $i$ prime numbers. The animation shows $W_{2}$ being rolled up to 30.
It's useful to define $W_{i}\rightarrow n$ for $n>0$ to be the result of rolling $W_{i}$ up to $n$. Then the animation generates $W_{2}\rightarrow 30=\{1,5,7,11,13,17,19,23,25,29\}$. Note that up to $5^{2}-1=24$, this consists only of $1$ and the primes between $5$ and $25$.
The sieve of Pritchard is derived from the observation[1] that this holds generally: for all $i>0$, the values in $W_{i}\rightarrow {(p_{i+1}^{2}-1)}$ are $1$ and the primes between $p_{i+1}$ and $p_{i+1}^{2}$. It even holds for $i=0$, where the wheel has length $1$ and contains just $1$ (representing all the natural numbers). So the sieve of Pritchard starts with the trivial wheel $W_{0}$ and builds successive wheels until the square of the wheel's first member after $1$ is at least $N$. Wheels grow very quickly, but only their values up to $N$ are needed and generated.
It remains to find a method for generating the next wheel. Note in the animation that $W_{3}=\{1,5,7,11,13,17,19,23,25,29\}-\{5*1,5*5\}$ can be obtained by rolling $W_{2}$ up to $30$ and then removing $5$ times each member of $W_{2}$. This also holds generally: for all $i\geq 0$, $W_{i+1}=(W_{i}\rightarrow P_{i+1})-\{p_{i+1}*w|w\in W_{i}\}$.[1] Rolling $W_{i}$ past $P_{i}$ just adds values to $W_{i}$, so the current wheel is first extended by getting each successive member starting with $w=1$, adding $P_{i}$ to it, and inserting the result in the set. Then the multiples of $p_{i+1}$ are deleted. Care must be taken to avoid a number being deleted that itself needs to be multiplied by $p_{i+1}$. The sieve of Pritchard as originally presented[2] does so by first skipping past successive members until finding the maximum one needed, and then doing the deletions in reverse order by working back through the set. This is the method used in the first animation above. A simpler approach is just to gather the multiples of $p_{i+1}$ in a list, and then delete them.[7] Another approach is given by Gries and Misra.[8]
If the main loop terminates with a wheel whose length is less than $N$, it is extended up to $N$ to generate the remaining primes.
The algorithm, for finding all primes up to N, is therefore as follows:
1. Start with a set W={1} and length=1 representing wheel 0, and prime p=2.
2. As long as p2 <= N, do the following
1. if length < N then
1. extend W by repeatedly getting successive members w of W starting with 1 and inserting length+w into W as long as it doesn't exceed p*length or N;
2. increase length to the minimum of p*length and N.
2. repeatedly delete p times each member of W by first finding the largest <= length and then working backwards.
3. note the prime p, then set p to the next member of W after 1 (or 3 if p was 2).
3. if length < N then extend W to N by repeatedly getting successive members w of W starting with 1 and inserting length+w into W as long as it doesn't exceed N;
4. On termination, the rest of the primes up to N are the members of W after 1.
Example
To find all the prime numbers less than or equal to 150, proceed as follows.
Start with wheel 0 with length 1, representing all natural numbers 1, 2, 3...:
1
The first number after 1 for wheel 0 (when rolled) is 2; note it as a prime. Now form wheel 1 with length 2x1=2 by first extending wheel 0 up to 2 and then deleting 2 times each number in wheel 0, to get:
1 2
The first number after 1 for wheel 1 (when rolled) is 3; note it as a prime. Now form wheel 2 with length 3x2=6 by first extending wheel 1 up to 6 and then deleting 3 times each number in wheel 1, to get
1 2 3 5
The first number after 1 for wheel 2 is 5; note it as a prime. Now form wheel 3 with length 5x6=30 by first extending wheel 2 up to 30 and then deleting 5 times each number in wheel 2 (in reverse order!), to get
1 2 3 5 7 11 13 17 19 23 25 29
The first number after 1 for wheel 3 is 7; note it as a prime. Now wheel 4 has length 7x30=210, so we only extend wheel 3 up to our limit 150. (No further extending will be done now that the limit has been reached.) We then delete 7 times each number in wheel 3 until we exceed our limit 150, to get the elements in wheel 4 up to 150:
1 2 3 5 7 11 13 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97 101 103 107 109 113 119 121 127 131 133 137 139 143 149
The first number after 1 for this partial wheel 4 is 11; note it as a prime. Since we've finished with rolling, we delete 11 times each number in the partial wheel 4 until we exceed our limit 150, to get the elements in wheel 5 up to 150:
1 2 3 5 7 11 13 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97 101 103 107 109 113 119 121 127 131 133 137 139 143 149
The first number after 1 for this partial wheel 5 is 13. Since 13 squared is at least our limit 150, we stop. The remaining numbers (other than 1) are the rest of the primes up to our limit 150.
Just 8 composite numbers are removed, once each. The rest of the numbers considered (other than 1) are prime. In comparison, the natural version of Eratosthenes sieve (stopping at the same point) removes composite numbers 184 times.
Pseudocode
The sieve of Pritchard can be expressed in pseudocode, as follows:[1]
algorithm Sieve of Pritchard is
input: an integer N >= 2.
output: the set of prime numbers in {1,2,...,N}.
let W and Pr be sets of integer values, and all other variables integer values.
k, W, length, p, Pr := 1, {1}, 2, 3, {2};
{invariant: p = pk+1 and W = Wk $\cap $ {1,2,...,N} and length = minimum of Pk,N and Pr = the primes up to pk}
while p2 <= N do
if (length < N) then
Extend W,length to minimum of p*length,N;
Delete multiples of p from W;
Insert p into Pr;
k, p := k+1, next(W, 1)
if (length < N) then
Extend W,length to N;
return Pr $\cup $ W - {1};
where next(W, w) is the next value in the ordered set W after w.
procedure Extend W,length to n is
{in: W = Wk and length = Pk and n > length}
{out: W = Wk$\rightarrow $n and length = n}
integer w, x;
w, x := 1, length+1;
while x <= n do
Insert x into W;
w := next(W,w);
x := length + w;
length := n;
procedure Delete multiples of p from W,length is
integer w;
w := p;
while p*w <= length do
w := next(W,w);
while w > 1 do
w := prev(W,w);
Remove p*w from W;
where prev(W, w) is the previous value in the ordered set W before w. The algorithm can be initialized with $W_{0}$ instead of $W_{1}$ at the minor complicaion of making next(W,1) a special case when k = 0.
This abstract algorithm uses ordered sets supporting the operations of insertion of a value greater than the maximum, deletion of a member, getting the next value after a member, and getting the previous value before a member. Using one of Mertens' theorems (the third) it can be shown to use $O(N/\log \log N)$ of these operations and additions and multiplications.[2]
Implementation
An array-based doubly-linked list s can be used to implement the ordered set W, with s[w] storing next(W,w) and s[w-1] storing prev(W,w). This permits each abstract operation to be implemented in a small number of operations. (The array can also be used to store the set Pr "for free".) Therefore the time complexity of the sieve of Pritchard to calculate the primes up to $N$ in the random access machine model is $O(N/\log \log N)$ operations on words of size $O(\log N)$. Pritchard also shows how multiplications can be eliminated by using very small multiplication tables,[2] so the bit complexity is $O(N\log N/\log \log N)$ bit operations.
In the same model, the space complexity is $O(N)$ words, i.e., $O(N\log N)$ bits. The sieve of Eratosthenes requires only 1 bit for each candidate in the range 2 through $N$, so its space complexity is lower at $O(N)$ bits. Note that space needed for the primes is not counted, since they can printed or written to external storage as they are found. Pritchard[2] presented a variant of his sieve that requires only $O(N/\log \log N)$ bits without compromising the sublinear time complexity, making it asymptotically superior to the natural version of the sieve of Eratostheses in both time and space.
However, the sieve of Eratostheses can be optimized to require much less memory by operating on successive segments of the natural numbers.[9] Its space complexity can be reduced to $O({\sqrt {N}})$ bits without increasing its time complexity[3] This means that in practice it can be used for much larger limits $N$ than would otherwise fit in memory, and also take advantage of fast cache memory. For maximum speed it is also optimized using a small wheel to avoid sieving with the first few primes (although this does not change its asymptotic time complexity). Therefore the sieve of Pritchard is not competitive as a practical sieve over sufficiently large ranges.
Geometric model
At the heart of the sieve of Pritchard is an algorithm for building successive wheels. It has a simple geometric model as follows:
1. Start with a circle of circumference 1 with a mark at 1
2. To generate the next wheel:
1. Go around the wheel and find (the distance to) the first mark after 1; call it p
2. Create a new circle with p times the circumference of the current wheel
3. Roll the current wheel around the new circle, marking it where a mark touches it
4. Magnify the current wheel by p and remove the marks that coincide
Note that for the first 2 iterations it is necessary to continue round the circle until 1 is reached again.
The first circle represents $W_{0}=\{1\}$, and successive circles represent wheels $W_{1},W_{2},...$. The animation on the right shows this model in action up to $W_{3}$.
It is apparent from the model that wheels are symmetric. This is because $P_{k}-w$ is not divisible by one of the first $k$ primes if and only if $w$ is not so divisible. It is possible to exploit this to avoid processing some composites, but at the cost of a more complex algorithm.
Related sieves
Once the wheel in the sieve of Pritchard reaches its maximum size, the remaining operations are equivalent to those performed by Euler's sieve.
The sieve of Pritchard is unique in conflating the set of prime candidates with a dynamic wheel used to speed up the sifting process. But a separate static wheel (as frequently used to speed up the sieve of Eratosthenes) can give an $O(\log \log N)$ speedup to the latter, or to linear sieves, provided it is large enough (as a function of $N$). Examples are the use of the largest wheel of length not exceeding ${\sqrt {N}}/log^{2}N$ to get a version of the sieve of Eratosthenes that takes $O(N)$ additions and requires only $O({\sqrt {N}}/\log \log N)$ bits,[3] and the speedup of the naturally linear sieve of Atkin to get a sublinear optimized version.
Bengalloun found a linear smoothly incremental sieve,[10] i.e., one that (in theory) can run indefinitely and takes a bounded number of operations to increment the current bound $N$. He also showed how to make it sublinear by adapting the sieve of Pritchard to incrementally build the next dynamic wheel while the current one is being used. Pritchard[5] showed how to avoid multiplications, thereby obtaining the same asymptotic bit-complexity as the sieve of Pritchard.
Runciman provides a functional algorithm[11] inspired by the sieve of Pritchard.
See also
• Sieve of Eratosthenes
• Sieve of Atkin
• Sieve theory
References
1. Pritchard, Paul (1982). "Explaining the Wheel Sieve". Acta Informatica. 17 (4): 477–485. doi:10.1007/BF00264164. S2CID 122592488.
2. Pritchard, Paul (1981). "A Sublinear Additive Sieve for Finding Prime Numbers". Communications of the ACM. 24 (1): 18–23. doi:10.1145/358527.358540. S2CID 16526704.
3. Pritchard, Paul (1983). "Fast Compact Prime Number Sieves (Among Others)". Journal of Algorithms. 4 (4): 332–344. doi:10.1016/0196-6774(83)90014-7. hdl:1813/6313. S2CID 1068851.
4. Pritchard, Paul (1987). "Linear prime-number sieves: A family tree". Science of Computer Programming. 9 (1): 17–35. doi:10.1016/0167-6423(87)90024-4. S2CID 44111749.
5. Pritchard, Paul (1980). "On the prime example of programming". Language Design and Programming Methodology. Lecture Notes in Computer Science. Vol. 877. pp. 280–288. CiteSeerX 10.1.1.52.835. doi:10.1007/3-540-09745-7_5. ISBN 978-3-540-09745-7. S2CID 9214019.
6. Dunten, Brian; Jones, Julie; Sorenson, Jonathan (1996). "A Space-Efficient Fast Prime Number Sieve". Information Processing Letters. 59 (2): 79–84. CiteSeerX 10.1.1.31.3936. doi:10.1016/0020-0190(96)00099-3. S2CID 9385950.
7. Mairson, Harry G. (1977). "Some new upper bounds on the generation of prime numbers". Communications of the ACM. 20 (9): 664–669. doi:10.1145/359810.359838. S2CID 20118576.
8. Gries, David; Misra, Jayadev (1978). "A linear sieve algorithm for finding prime numbers". Communications of the ACM. 21 (12): 999–1003. doi:10.1145/359657.359660. hdl:1813/6407. S2CID 11990373.
9. Bays, Carter; Hudson, Richard H. (1977). "The segmented sieve of Eratosthenes and primes in arithmetic progressions to 1012". BIT. 17 (2): 121–127. doi:10.1007/BF01932283. S2CID 122592488.
10. Bengelloun, S. A. (2004). "An incremental primal sieve". Acta Informatica. 23 (2): 119–125. doi:10.1007/BF00289493. S2CID 20118576.
11. Runciman, C. (1997). "Lazy Wheel Sieves and Spirals of Primes" (PDF). Journal of Functional Programming. 7 (2): 219–225. doi:10.1017/S0956796897002670. S2CID 2422563.
Number-theoretic algorithms
Primality tests
• AKS
• APR
• Baillie–PSW
• Elliptic curve
• Pocklington
• Fermat
• Lucas
• Lucas–Lehmer
• Lucas–Lehmer–Riesel
• Proth's theorem
• Pépin's
• Quadratic Frobenius
• Solovay–Strassen
• Miller–Rabin
Prime-generating
• Sieve of Atkin
• Sieve of Eratosthenes
• Sieve of Pritchard
• Sieve of Sundaram
• Wheel factorization
Integer factorization
• Continued fraction (CFRAC)
• Dixon's
• Lenstra elliptic curve (ECM)
• Euler's
• Pollard's rho
• p − 1
• p + 1
• Quadratic sieve (QS)
• General number field sieve (GNFS)
• Special number field sieve (SNFS)
• Rational sieve
• Fermat's
• Shanks's square forms
• Trial division
• Shor's
Multiplication
• Ancient Egyptian
• Long
• Karatsuba
• Toom–Cook
• Schönhage–Strassen
• Fürer's
Euclidean division
• Binary
• Chunking
• Fourier
• Goldschmidt
• Newton-Raphson
• Long
• Short
• SRT
Discrete logarithm
• Baby-step giant-step
• Pollard rho
• Pollard kangaroo
• Pohlig–Hellman
• Index calculus
• Function field sieve
Greatest common divisor
• Binary
• Euclidean
• Extended Euclidean
• Lehmer's
Modular square root
• Cipolla
• Pocklington's
• Tonelli–Shanks
• Berlekamp
• Kunerth
Other algorithms
• Chakravala
• Cornacchia
• Exponentiation by squaring
• Integer square root
• Integer relation (LLL; KZ)
• Modular exponentiation
• Montgomery reduction
• Schoof
• Trachtenberg system
• Italics indicate that algorithm is for numbers of special forms
| Wikipedia |
List of things named after Jean-Pierre Serre
These are the things named after Jean-Pierre Serre, a French mathematician.
• Bass–Serre theory
• Serre class
• Quillen–Suslin theorem (sometimes known as "Serre's Conjecture" or "Serre's problem")
• Serre's Conjecture concerning Galois representations
• Serre's "Conjecture II" concerning linear algebraic groups
• Serre's criterion (there are several of them.)
• Serre duality
• Serre–Grothendieck–Verdier duality
• Serre's FAC
• Serre fibration
• Serre's C-theory
• Serre's inequality on height
• Serre group
• Serre's modularity conjecture
• Serre's multiplicity conjectures
• Serre's open image theorem
• Serre's property FA
• Serre relations
• Serre subcategory
• Serre functor
• Serre spectral sequence
• Lyndon–Hochschild–Serre spectral sequence
• Serre–Swan theorem
• Serre–Tate theorem
• Serre's theorem in group cohomology[1]
• Serre's theorem on affineness
• Serre twist sheaf
• Serre's vanishing theorem
• Thin set in the sense of Serre
See also
• Serre conjecture (disambiguation)
Notes
1. "Serre theorem in group cohomology - Encyclopedia of Mathematics".
| Wikipedia |
Topological tic-tac-toe
Alex Bolton plays noughts and crosses on unusual surfaces
by Alex Bolton. Published on 18 October 2018.
Tic-tac-toe also known as noughts and crosses) is a classic game known for its simplicity, and has been popular since ancient times. You and a friend (or enemy!) take turns to mark the squares of a $3 \times 3$ grid. The winner is the first to get three of their symbols in a row (horizontal, vertical or diagonal). A winning game for X is shown below right.
A winning game for X
It's not difficult to work out how to play optimally on a standard board, where you're guaranteed to at least draw with your opponent. However, what if you're not restricted to the standard two-dimensional square grid? How would you play then?
To present a fresh challenge and to make tic-tac-toe exciting again, here is a collection of puzzles where you will be swapping your standard square board with one on various topological surfaces. Have fun!
The cylinder
The first new board to consider is the cylinder. To form a cylinder as in the diagram below, imagine that the board is wrapped around so that the left and right edges of the board are connected to each other, like a piece of paper that has been rolled into a tube. Matching edges are denoted with a $\uparrow$.
In all the subsequent puzzles, it is X's turn to play, and it is possible for X to win in some number of moves, even if O plays optimally.
Puzzle 1: cylinders
How does X win both of these games on cylindrical boards? It is possible to win the first game in one move.
The first game can be won in one move. A demonstration of this, along with the folded cylindrical board, is shown below.
Bending the board into a cylinder. The winning move for the first puzzle is shown in red and the winning line is marked in blue.
The Möbius strip
A Möbius strip. Image: David Benbennick, CC BY-SA 3.0
How about a Möbius strip? Imagine that the right edge of the board is wrapped around the back of the board and given a half turn, so that it connects to the left edge of the board. The half turn means that top and bottom become flipped as we move off the left or right edge of the board. The edges with a half turn are denoted by a $\uparrow$ and a $\downarrow$. (If you want to actually construct this board, draw each grid square as a wide rectangle so that the grid is wide enough to wrap around with a half twist.)
The figure on below shows adjacent squares in the new board by making copies of the Möbius strip board, and puzzle 2 gives two puzzles on the Möbius strip board.
A figure showing which grid squares are adjacent on a Möbius strip board. For example, if you go one place right from the top-right square 3, you will go to the bottom-left square 7.
Puzzle 2: Möbius strips
How does X win both of these games on Möbius strip boards? It is possible to win the first game in one move.
The torus
There's no need to limit ourselves to only connecting the left and right edges—we can also connect the top and bottom edges. If we return to the cylinder formed by wrapping the right edge of the board round to meet the left edge, we can now connect the top and bottom edges together to form a torus. Now we have two pairs of connected edges, denoted by $\rightarrow$ and $\uparrow$.
For tic-tac-toe purposes, the cylindrical and toroidal boards are identical, and this holds even if we change the game to be `make a line of length $n$ on an $n \times n$ board' for any $n$. However, for puzzle 3, you need to find a line of length 3 on a $4 \times 4$ board.
Puzzle 3: torus puzzle
What should X do to make three in a row on a torus?
On the torus, we can think of any row (or column) as being the central one, so it's easier to prove facts about the torus than for other boards. Have a go at the following challenges.
Puzzle 4: more torus puzzles
Can you show that making any line of length $n$ on an $n \times n$ cylindrical board is also a line of length $n$ on an $n \times n$ toroidal board and vice versa (so a game on a torus is equivalent to a game on a cylinder)?
Are any starting positions better than others on a $3 \times 3$ torus?
Is it possible for the game on a $3 \times 3$ toroidal board to end in a draw, with neither player getting 3 in a row (the previous question gives you a shortcut to solving this one)?
The Klein bottle
A figure showing which grid squares are adjacent on Klein bottle board.
Now we will think about playing on a Klein bottle, a shape that cannot be constructed in three dimensions without it intersecting with itself. Fold the top edge of the board over to touch the bottom edge, and connect the left and right edges with a half twist like for the Möbius strip.
The figure to the right shows which squares are adjacent on the Klein bottle board, and puzzle 5 gives two puzzle.
Puzzle 5: Klein bottle puzzles
How does X win both of these games on Klein bottle boards? It is possible to win the first game in one move.
The projective plane
Another shape that it is not possible to construct in three dimensions without the shape intersecting itself is the projective plane. This is a shape where the top and bottom edges are connected by a half twist, as are the left and right edges. To imagine how it would be made, think about connecting the single edge of the Möbius strip to itself. (You'll have to be very dextrous to actually create this from paper!) Puzzle 6 gives two puzzles using the projective plane. Top tip: construct an adjacency map like the one for the Klein bottle.
Puzzle 6: projective plane puzzles
How does X win both of these games on projective plane boards?
(Note: I'm assuming that a valid line comprises three distinct squares, so a single square does not appear more than once in a valid line.)
That brings us to the end of our foray into topological tic-tac-toe. I hope you enjoyed these mind-bending puzzles! The inspiration for this article came from Across The Board by John Watkins, the most complete book on chessboard and other grid puzzles. I would recommend this book for some further interesting puzzles, eg how many queens are needed so that every square on a chessboard is targeted or occupied by one of the queens? The author gives an interesting history of chess problems and shows that they have inspired important advances in maths.
For solutions to the puzzles above, see this article.
Alexander Bolton
Alexander Bolton completed a maths PhD at Imperial College London, applying Bayesian statistics to cyber security problems. He now works as a quantitative analyst at G-Research.
+ More articles by Alexander
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\begin{document}
\begin{center} {\large \bf Repetition in Colored Sequences of Balls}\\ Jeremy M. Dover \end{center} \begin{abstract} In responding to a question on Math Stackexchange, the author~\cite{2109013} formulated the problem of determining the number of sequences of balls colored in most $n$ colors where some fixed numbers of balls share colors with other balls in the sequence. In this paper, we formulate the problem more formally, and solve several variations. \end{abstract}
\section{The setup} Suppose you have balls of $n$ different colors, with an ample supply of each color, and you wish to create a sequence of $k$ balls. We assume that balls of the same color are indistinguishable, but positions in the sequence are distinguishable. A very natural question is to ask how many such sequences contain exactly $m$ balls which are the same color as another ball. Conceptually, this problem is easily solved in terms of sums of multinomial coefficents, but the challenge is figuring out which selections of colors to sum over. To break the problem down more effectively, we pose a more specific problem: how many ways are there to color a sequence of $k$ balls with at most $n$ colors such that exactly $m$ balls are the same color as another ball, and exactly $\lambda$ colors are repeated. We denote this number as $Z(k,n,m,\lambda)$.
First, note some constraints, added to the obvious constraints that all of $k$, $n$, $m$, and $\lambda$ must be non-negative integers: \begin{itemize} \item If $k>n$, then there must be at least $k-n+1$ balls that match some other ball, since at most $n$ balls are the first instance of their color in the sequence, meaning the remaining $k-n$ balls must match one of these, and at least one of these first instances must be matched. Hence $Z(k,n,m,\lambda) = 0$ for all $m<k-n+1$. \item If exactly $m$ balls match some other ball, then at most $\lfloor \frac{m}{2} \rfloor$ colors can be repeated. Hence $Z(k,n,m,\lambda) = 0$ for $\lambda > \lfloor \frac{m}{2} \rfloor$. \item If exactly $m$ balls match some other color, then $k-m$ balls do not match any other color. Hence the number of colors not repeated in the sequence, $n - \lambda$, must be at least $k-m$. Hence $Z(k,n,m,\lambda) = 0$ for all $\lambda > n-k+m$. \item If $m=0$, then $\lambda$ is necessarily 0 as well, and we have $Z(k,n,0,0) = \frac{k!}{(k-n)!}$ for all $n \le k$, and $Z(k,n,0,\lambda) = 0$ for all other choices of $k$, $n$, and $\lambda$. \item We cannot have exactly one ball matching, so $Z(k,n,1,\lambda) = 0$ for all $k$, $n$, and $\lambda$ \end{itemize}
Given that we have values for $k$, $n$, $m$, and $\lambda$ which meet the constraints above, the count is fairly straightforward. First, select the $\lambda$ colors to be repeated, which can be done in $n \choose \lambda$ ways. Next, select the $m$ balls in the sequence which are going to match other positions, which can be done in $k \choose m$ ways. Now color the $k-m$ balls which need to be uniquely colored with the remaining $n-\lambda$ colors, which can be done in $\frac{(n-\lambda)!}{(n-\lambda+m-k)!}$ ways.
The final component is to color the $m$ balls that will match with the $\lambda$ repeated colors such that each color is used at least twice. Thinking of the color assignment as a function, this function is doubly-surjective, that is, each color is the image of at least two balls in the domain. Doubly-surjective functions have been studied by Walsh~\cite{walsh}, who defines $s(m,\lambda)$ to be the number of doubly-surjective functions from a set of $m$ elements into a set of $\lambda$ elements. Walsh calculates the following formula: $$s(m,\lambda) = \sum_{j=0}^\lambda (-1)^j{\lambda \choose j}\sum_{i=0}^j {j \choose i}\frac{m!}{(m-i)!}(\lambda-j)^{m-i}$$
With this notation, we have $$Z(k,n,m,\lambda) = {n \choose \lambda}{k \choose m}\frac{(n-\lambda)!}{(n-\lambda+m-k)!}s(m,\lambda)$$ where $m\ge k-n+1$ and $\lambda \le {\rm min}\{\lfloor \frac{m}{2} \rfloor, n-k+m\}$.
\section{Counting matching balls} With the formula in the previous section, we have the building blocks to solve several similar, but slightly different problems.
{\em Problem 1: How many sequences of $k$ balls colored in at most $n$ colors contain exactly $m \ge 2$ balls with a color matching some other ball?}
From our previous analysis, any sequence satisfying the conditions of the problem must have $\lambda$ colors which are repeated, where $1 \le \lambda \le {\rm min}\{\lfloor \frac{m}{2} \rfloor, n-k+m\}$. Therefore the solution to this problem is simply
$$\sum_{\lambda=0}^{{\rm min}\{\lfloor \frac{m}{2} \rfloor, n-k+m\}} Z(k,n,m,\lambda)$$
As an example, we can count the sequences of 5 balls colored in at most three colors such that exactly 4 balls match some other color. In this case we could have $\lambda=1$ or $\lambda=2$. If $\lambda=1$ we must have four balls of one color, which can be chosen in three ways, and one ball of a differing color, which can be chosen in two ways. Once the balls are chosen, there are five ways to put them in sequence, namely $AAAAB, AAABA, \ldots, BAAAA$. There are 30 such sequences.
The case for $\lambda=2$ is slightly more complex. Since four balls match and two colors are repeated, we must have two balls of each of two colors, and one of the third color; these choices can be made in three ways, since all colors are effectively assigned once the singleton color is picked. Once the balls are picked, there are five ways to place the singleton, then ${4 \choose 2} = 6$ ways to place the first matching pair. This yields $3 \times 5 \times 6 = 90$ possible sequences of this form. Summing with the previous case gives 120 possible solutions, which matches the formula calculation.
{\em Problem 2: How many sequences of balls colored in at most $n$ colors contain exactly $m \ge 2$ balls with a color matching some other ball?}
This problem is similar to the previous, but note that the length constraint has been removed. For fixed $m$ and $n$, the only sequences colored with at most $n$ colors and having exactly $m$ balls matching another ball must have length between $m$ and $m+n-1$, using our constraints above. This can be answered via summing over the appropriate answers to the previous problem:
$$\sum_{k=m}^{m+n-1}\sum_{\lambda=0}^{{\rm min}\{\lfloor \frac{m}{2} \rfloor, n-k+m\}} Z(k,n,m,\lambda)$$
\section{Counting repeats}
Conceptually, when counting matches in the previous problem, we assume that the entire sequence has already been created, and we look at each ball of the sequence and can determine if some other ball has a matching color, no matter where in the sequence the match arises. Instead, suppose we are drawing the balls sequentially, and must make a ``matching'' decision when the ball is drawn; thus we are counting the number of balls in the sequence whose color has previously been seen in the sequence. This produces a subtle difference in our counting, because we do not go back and retrospectively count the first occurrence of a color that will later match.
To see the difference, consider the sequence $AABBCCDDDD$. This sequence has 10 balls and four colors, and all 10 balls match some other color. However, only 6 balls match a color previously seen in the sequence, since the first instance of each of the four colors is not counted.
{\em Problem 3: How many sequences of $k$ balls colored in at most $n$ colors contain exactly $\mu \ge 1$ balls whose color appears earlier in the sequence?}
As before, there are some constraints on values for $k$, $n$ and $\mu$ for this problem to make sense, beyond the obvious non-negativity constraints. First note that $\mu \le k-1$, since the first ball of a sequence will never be counted, though all of the remaining balls may be. In addition, only the first instance of a color in a sequence is not counted as a repeat, and there are at most $n$ such first instances, so we must have $k \le n+\mu$.
Using the same trick as before, let $\lambda$ be the number of colors which occur at least twice in a particular sequence. Clearly, we must have $1 \le \lambda \le \mu$, and any sequence with exactly $\mu$ balls whose color appears previously in the sequence has exactly $\mu+\lambda$ balls whose color matches some other in the sequence ($\mu$ repeats, plus the original occurrences of each of $\lambda$ colors). Therefore we can use the numbers $Z(k,n,m,\lambda)$ to solve this problem as well, with the answer being $$\sum_{\lambda=0}^\mu Z(k,n,\mu+\lambda,\lambda)$$
{\em Problem 4: How many sequences of balls colored in at most $n$ colors contain exactly $\mu \ge 1$ balls whose color appears earlier in the sequence?}
Like the problems in the previous section, for fixed $\mu$ and $n$ there only exist sequences of balls colored with at most $n$ colors and having exactlu $\mu$ balls match a color previously seen for lengths $k$ satisfying $\mu +1 \le k \le n+\mu$. As before, we can simply sum the corresponding values for the individual lengths to obtain the answer:
$$\sum_{k=\mu+1}^{n+\mu}\sum_{\lambda=0}^\mu Z(k,n,\mu+\lambda,\lambda)$$
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\end{document} | arXiv |
\begin{document}
\begin{frontmatter}
\title{Partial Frames, Their Free Frames\\ and Their Congruence Frames}
\author{Anneliese Schauerte\thanksref{myemail}}
\author{John Frith\thanksref{coemail}}
\thanks[myemail]{Email: \href{mailto:[email protected]} {\texttt{\normalshape [email protected]}}}
\address[b]{Department of Mathematics and Applied Mathematics\\University of Cape Town\\
Cape Town, South Africa}
\thanks[coemail]{Email: \href{mailto:[email protected]} {\texttt{\normalshape
[email protected]}}} \begin{abstract}
The context of this work is that of partial frames; these are meet-semilattices where not all subsets need have joins. A selection function, $\mathcal{S}$, specifies, for all meet-semilattices, certain subsets under consideration, which we call the ``designated'' ones; an $\calS$-frame{} then must have joins of (at least) all such subsets and binary meet must distribute over these. A small collection of axioms suffices to specify our selection functions; these axioms are sufficiently general to include as examples of partial frames, bounded distributive lattices, $\sigma$-frames, $\kappa$-frames and frames.
We consider right and left adjoints of $\calS$-frame{} maps, as a prelude to the introduction of closed and open maps.
Then we look at what might be an appropriate notion of Booleanness for partial frames. The obvious candidate is the condition that every element be complemented; this concept is indeed of interest, but we pose three further conditions which, in the frame setting, are all equivalent to it. However, in the context of partial frames, the four conditions are distinct. In investigating these, we make essential use of the free frame over a partial frame and the congruence frame of a partial frame.
We compare congruences of a partial frame, technically called $\sels$-congruence{s}, with the frame congruences of its free frame. We provide a natural transformation for the situation and also consider right adjoints of the frame maps in question. We characterize the case where the two congruence frames are isomorphic and provide examples which illuminate the possible different behaviour of the two.
We conclude with a characterization of closedness and openness for the embedding of a partial frame into its free fame, and into its congruence frame. \end{abstract} \begin{keyword}
frame, partial frame, $\cal{S}$-frame, $\kappa$-frame, $\sigma$-frame, free frame over partial frame, congruence frame, Boolean algebra, closed map, open map \end{keyword} \end{frontmatter} \section{Introduction}\label{intro}
Partial frames are meet-semilattices where, in contrast with frames, not all subsets need have joins. A selection function, $\mathcal{S}$, specifies, for all meet-semilattices, certain subsets under consideration, which we call the ``designated'' ones; an $\calS$-frame{} then must have joins of (at least) all such subsets and binary meet must distribute over these. A small collection of axioms suffices to specify our selection functions; these axioms are sufficiently general to include as examples of partial frames, bounded distributive lattices, $\sigma$-frames, $\kappa$-frames and frames.
We consider the classical notions of right and left adjoints, for $\calS$-frame{} maps. Unlike the situation for full frames, such maps need not have right adjoints. This is a prelude to the introduction of closed and open maps, and a discussion of their properties.
What is an appropriate notion of Booleanness for partial frames? The obvious answer is that the partial frame should have every element complemented; this concept is indeed of interest, but we pose three further conditions which, in the frame setting, are all equivalent to it. However, in the context of partial frames, the four conditions are distinct. In investigating these, we make essential use of the free frame over a partial frame and the congruence frame of a partial frame.
We compare congruences of a partial frame, technically called $\sels$-congruence{s}, with the frame congruences of its free frame. We provide a natural transformation for the situation and also consider right adjoints of the frame maps in question. We characterize the case where the two congruence frames are isomorphic and provide examples which illuminate the possible different behaviour of the two.
We conclude with a characterization of closedness and openness for the embedding of a partial frame into its free fame, and into its congruence frame.
Since this document is intended as an extended abstract, proofs are omitted. \section{Background}
This background section is taken largely from \cite{jfasBoolVar}. See \cite{picpulframesbook} and \cite{jo} as references for frame theory; see \cite{BBCrag} and \cite{bbnotes} for $\sigma$-frames; see \cite{madden} and \cite{manpap} for $\kappa$-frames; see \cite{maclane} and \cite{ahs} for general category theory.
\vskip5pt\noindent The basics of our approach to partial frames can be found in \cite{jfascgasa1}, \cite{jfascgasa2} and \cite{jfas_hcozforqm}. Our papers with a more topological flavour are \cite{jfascplns}, \cite{jfasslovak}, \cite{jfasonepoint}, \cite{jfaspartframfilt} and \cite{jfascompreflpart}. Our papers with a more algebraic flavour are \cite{jfascov}, \cite{jfasmadden} and \cite{jfassemilatticescong}. Crucial for this paper is \cite{jfasheyt}. We are indebted to earlier work by other authors in this field: see \cite{paseka}, \cite{zhaonuclei}, \cite{zhao} and \cite{zenk}. For those interested in a comparison of the various approaches, see \cite{jfascgasa2}. \vskip5pt\noindent A \textit{meet-semilattice} is a partially ordered set in which all finite subsets have a meet. In particular, we regard the empty set as finite, so a meet-semilattice comes equipped with a top element, which we denote by $1$. We do not insist that a meet-semilattice should have a bottom element, which, if it exists, we denote by $0$. A function between meet-semilattices $f:L\to M$ is a \textit{meet-semilattice map} if it preserves finite meets, as well as the top element. A \textit{sub meet-semilattice{}} is a subset for which the inclusion map is a meet-semilattice{} map.
The essential idea for a \textit{partial frame} is that it should be ``frame-like'' but that not all joins need exist; only certain joins have guaranteed existence and binary meets should distribute over these joins. The guaranteed joins are specified in a global way on the category of meet-semilattice{s} by specifying what is called a selection function; the details are given below. \begin{definition}\label{AB} A \textit{selection function} is a rule, which we usually denote by $\mathcal{S}$, which assigns to each meet-semilattice $A$ a collection $\mathcal{S} A$ of subsets of $A$, such that the following conditions hold (for all meet-semilattices $A$ and $B$): \begin{enumerate} \item[] \begin{enumerate} \item[(S1)] For all $x\in A$, $\{x\}\in\mathcal{S} A$. \item[(S2)] If $G, H\in\mathcal{S} A$ then $\{x\wedge y:x\in G,y\in H\}\in\mathcal{S} A$. \item[(S2)$'$] If $G,H\in\mathcal{S} A$ then $\{x\vee y:x\in G,y\in H\}\in\mathcal{S} A$. \item[(S3)] If $G\in\mathcal{S} A$ and, for all $x\in G$, $x=\bigvee H_x$ for some $H_x\in \mathcal{S} A$, then $$\bigcup\limits_{x\in G}H_x\in\mathcal{S} A.$$ \item[(S4)] For any meet-semilattice map $f:A\to B$, $$\mathcal{S}(f[A])=\{f[G]:G\in\mathcal{S} A\}\subseteq \mathcal{S} B.$$ \item[(SSub)] For any sub meet-semilattice{} $B$ of meet-semilattice{} $A$, if $G\subseteq B$ and $G\in\mathcal{S} A $, then $G\in\mathcal{S} B$. \item[(SFin)] If $F$ is a finite subset of $A$, then $F\in\mathcal{S} A$. \item[(SCov)] If $G\subseteq H$ and $H\in\mathcal{S} A$ with $\bigvee H=1$ then $G\in\mathcal{S} A$. (Such $H$ are called $\mathcal{S}$-\textit{covers}.) \item[(SRef)] Let $X,Y\subseteq A$. If $X\leq Y$ with $X\in\mathcal{S} A$ there is a $C\in \mathcal{S} A$ such that $X\leq C\subseteq Y$. (By $X\leq Y$ we mean, as usual, that for each $x\in X$ there exists $y\in Y$ such that $x\leq y$.) \end{enumerate} \end{enumerate} \end{definition} Of course (SFin) implies (S1) but there are situations where we do not impose (SFin) but insist on (S1). Note that we always have $\emptyset\in\mathcal{S} A$. Once a selection function, $\mathcal{S}$, has been fixed, we speak informally of the members of $\mathcal{S} A$ as the \textit{designated} subsets of $A$.
\begin{definition} An \textit{$\calS$-frame} $L$ is a meet-semilattice{} in which every designated subset has a join and for any such designated subset $B$ of $L$ and any $a\in L$, $$a\wedge\bigvee B=\bjoinl{b\in B}a\wedge b.$$Of course such an $\calS$-frame{} has both a top and a bottom element which we denote by $1$ and $0$ respectively.\\ A meet-semilattice{} map $f:L\to M$, where $L$ and $M$ are $\calS$-frame{s}, is an \textit{$\calS$-frame{} map} if $f(\bigvee B)=\bjoinl{b\in B}f(b)$ for any designated subset $B$ of $L$. In particular such an $f$ preserves the top and bottom element.\\ A \textit{sub $\calS$-frame{}} $T$ of an $\calS$-frame{} $L$ is a subset of $L$ such that the inclusion map $i:T\to L$ is an $\calS$-frame{} map. \\ The category $\sels$\textbf{Frm}{} has objects $\calS$-frame{s} and arrows $\calS$-frame{} maps. \end{definition}
We use the terms ``partial frame'' and ``$\calS$-frame'' interchangeably, especially if no confusion about the selection function is likely. We also use the term \textit{full frame} in situations where we wish to emphasize that all joins exist.
\begin{note} Here are some examples of different selection functions and their corresponding $\calS$-frame{s}. \begin{itemize} \item[1.] In the case that all joins are specified, we are of course considering the notion of a frame. \item[2.] In the case that (at most) countable joins are specified, we have the notion of a $\sigma$-frame. \item[3.] In the case that joins of subsets with cardinality less than some (regular) cardinal $\kappa$ are specified, we have the notion of a $\kappa$-frame. \item[4.] In the case that only finite joins are specified, we have the notion of a bounded distributive lattice. \end{itemize} \end{note}
The remainder of this section gives a lot of information about $\mathcal{H}_{\mathcal{S}}L$, the free frame over the $\calS$-frame{} $L$, as well as $\mathcal{C}_{\sels}L$, the frame of $\sels$-congruence{s} of $L$, and the relationship between the two. These results come from \cite{jfas_hcozforqm} on $\mathcal{H}_{\mathcal{S}}L$, \cite {jfascov} and \cite{jfasmadden} on $\mathcal{C}_{\sels}L$.
In the definition below, $L$ is an $\calS$-frame{}.
\begin{definition}\label{aaaa} \begin{enumerate}\item[(a)] A subset $J$ of an $L$ is an \textit{$\mathcal{S}$-ideal of $L$} if $J$ is a non-empty downset closed under designated joins (the latter meaning that if $X\subseteq J$, for $X$ a designated subset of $L$, then $\bigvee X\in J$). \item[(b)] The collection of all $\mathcal{S}$-ideal s of $L$ will be denoted $\mathcal{H}_{\mathcal{S}} L$, and called the \textit{$\mathcal{S}$-ideal{} frame of $L$}. It is in fact the free frame over $L$.
\item[(c)] For $I\in \mathcal{H}_{\mathcal{S}}L$, $t\in(\operatorname{\downarrow} x)\vee I\Longleftrightarrow t\leq x\vee s$, for some $s\in I$.
\item[(d)] We call $\theta\subseteq L\times L$ an \textit{$\sels$-congruence{} on }$L$ if it satisfies the following:\\ (C1) $\theta$ is an equivalence relation on $L$.\\ (C2) $(a,b), (c,d)\in\theta$ implies that $(a\wedge c,b\wedge d)\in\theta$.\\ (C3) If $\{(a_{\alpha},b_{\alpha}):\alpha\in\mathcal{A}\}\subseteq\theta$ and $\{a_{\alpha}:\alpha\in\mathcal{A}\}$ and $\{b_{\alpha}:\alpha\in\mathcal{A}\}$ are designated subsets of $L$, then $(\bjoinl{\alpha\in\mathcal{A}}a_{\alpha},\bjoinl{\alpha\in\mathcal{A}}b_{\alpha})\in\theta$. \item[(e)] The collection of all $\sels$-congruence{s} on $L$ is denoted by $\mathcal{C}_{\sels}L$; it is in fact a (full) frame with meet given by intersection. \item[(f)] \begin{enumerate}
\item[(i)] Let $A\subseteq L\times L$. We use the notation $\langle A\rangle$ to denote the smallest $\sels$-congruence{} containing $A$. This exists by completeness of $\mathcal{C}_{\sels}L$.
\item[(ii)] We define $\nabla\hskip-2pt_a=\{(x,y):x\vee a=y\vee a\}$ and $\Delta_a=\{(x,y):x\wedge a=y\wedge a\}$; these are $\sels$-congruence{s} on $L$.
\item[(iii)] It is easily seen that $\nabla\hskip-2pt_a=\bigcap\{\theta:\theta\in\mathcal{C}_{\sels}L\textrm{ and }(0,a)\in\theta\}=\langle(0,a)\rangle$ and that $\Delta_a=\bigcap\{\theta:\theta\in\mathcal{C}_{\sels}L\textrm{ and }(a,1)\in\theta\}=\langle(a,1)\rangle$.
\item[(iv)] For $a\leq b$, it follows that $\Delta_a\cap\nabla_b=\langle(a,b)\rangle$.
\item[(v)] The congruence $\nabla_1=L\times L$ is the top element and $\nabla_0=\{(x,x):x\in L\}$ (called the \textit{diagonal}) is the bottom element of $\mathcal{C}_{\sels}L$.
\end{enumerate}
\item[(g)] The following hold in $\mathcal{C}_{\sels}L$. \begin{enumerate} \item[(i)] For any $\theta\in\mathcal{C}_{\sels}L$, $\theta=\bigvee\{\nabla_b\wedge\Delta_a:(a,b)\in\theta,a\leq b\}$. \item[(ii)] $\nabla\hskip-2pt_a\vee\theta=\{(x,y):(x\vee a,y\vee a)\in\theta\}$. \item[(iii)] $\Delta_a\vee\theta=\{(x,y):(x\wedge a,y\wedge a)\in\theta\}$. \item[(iv)] For any $I\in\mathcal{H}_{\mathcal{S}}L$, $\bjoinl{x\in I}\nabla_x=\bcupl{x\in I}\nabla_x$.
\end{enumerate} \item[(h)] The function $\nabla:L\to\mathcal{C}_{\sels}L$ given by $\nabla(a)=\nabla_a$ is an $\calS$-frame{} embedding. It has the universal property that if $f:L\to M$ is an $\calS$-frame{} map into a frame $M$ with complemented image, then there exists a frame map $\bar{f}:\mathcal{C}_{\sels}L\to M$ such that $f=\bar{f}\circ\nabla$. \item[(i)] We also note that for frame maps $f$ and $g$ with domain $\mathcal{C}_{\sels}L$, if $f\circ\nabla=g\circ \nabla$ then $f=g$.
\item[(j)] A useful congruence for our purposes is the \textit{Madden} congruence, denoted $\pi_L$ below: \begin{enumerate} \item[(i)] For $x\in L$, set $P_x=\{t\in L:t\wedge x=0\}$. \item[(ii)] For $x\in L$, $P_x$ is an $\mathcal{S}$-ideal, and in $\mathcal{H}_{\mathcal{S}}L$, $P_x=(\operatorname{\downarrow} x)^*$, the pseudocomplement of $\operatorname{\downarrow} x$. \item[(iii)] Let $\pi_L=\{(x,y):P_x=P_y\}$; $\pi_L$ is an $\sels$-congruence{}. \item[(iv)] The quotient map induced by the Madden congruence, $p:L\to L/{\pi_L}$ is dense, onto and the universal such. We refer to this as the \textit{Madden quotient} of $L$. (See \cite{jfasmadden}.) \end{enumerate} \end{enumerate} \end{definition}
\begin{definition}\label{ab} For any $\calS$-frame{} $L$, define $e_L:\mathcal{H}_{\mathcal{S}}L\to\mathcal{C}_{\sels}L$ to be the unique frame map such that $e_L(\operatorname{\downarrow} a)=\nabla_a$ for all $a\in L$; that is, making the following diagram commute: \begin{center}
\begin{tikzpicture}
\node at (1.2,2.9){$L$};
\node at (4.3,2.9){$\mathcal{H}_{\mathcal{S}}L$};
\node at (1.2,1.3){$\mathcal{C}_{\sels}L $};
\draw[<-](1.2,1.6)--(1.2,2.6);
\draw[->](1.5,2.9)--(3.8,2.9);
\draw[<-](1.7,1.5)--(3.8,2.6);
\node at (1,2.1){$\nabla $};
\node at (2.6,3.2){$\downarrow\! $};
\node at (2.8,1.7){$e_L $};
\end{tikzpicture} \end{center} \vskip5pt That this map $e_L$ exists follows from the freeness of $\mathcal{H}_{\mathcal{S}}L$ as a frame over $L$. See \cite{jfas_hcozforqm}. \end{definition}
\begin{note}\label{ac} For any $\calS$-frame{} $L$, $\mathcal{H}_{\mathcal{S}}L$ is isomorphic to a subframe of $\mathcal{C}_{\sels}L$; that is, the free frame over $L$ is isomorphic to a subframe of the frame of $\sels$-congruence{s} of $L$. \end{note}
\section{Right and left adjoints}
We use the following standard terminology:
\begin{definition}\label{A} Let $h:L\to M$ be an $\calS$-frame{} map.\\ A function $r:M\to L$ is a \textit{right adjoint} of $h$ if $$h(x)\leq m\Longleftrightarrow x\leq r(m)\textrm{ for all }x\in L, m\in M.$$ A function $l:M\to L$ is a \textit{left adjoint} of $h$ if $$l(m)\leq x\Longleftrightarrow m\leq h(x) \textrm{ for all }x\in L, m\in M.$$ \end{definition}
We make no claim that all $\calS$-frame{} maps have right (or left) adjoints; this is false (see Example \ref{D}). However, clearly if an $\calS$-frame{} map has a right or left adjoint, such is unique.
\begin{lemma}\label{C} Let $h:L\to M$ be an $\calS$-frame{} map. \begin{enumerate} \item If $h$ has a right adjoint $r$, then for all $m\in M$, $$r(m)=\bigvee\{x\in L:h(x)\leq m\}.$$ \item If $h$ has a left adjoint $\ell$, then for all $m\in M$, $$l(m)=\bigwedge\{x\in L:m\leq h(x)\}.$$ \end{enumerate} \end{lemma}
We note that the existence of the above joins and meets has to be established since an $\calS$-frame{} need not be complete.
\begin{example}\label{D} This is an example of an $\calS$-frame{} map which has neither a right nor a left adjoint.
Let $L$ be the $\sigma$-frame consisting of all countable and cocountable subsets of $\mathbb{R}$, and $\mathbf{2}$ denote the $2$-element chain. Define $h:L\to \mathbf{2}$ by $h(C)=0$ if $C$ is countable and $h(D)=1$ if $D$ is cocountable. Then $h$ is a $\sigma$-frame map. However it has no right adjoint since there is no largest $A\in L$ with $h(A)=0$. Similarly it has no left adjoint. \end{example}
\begin{proposition}\label{E} Let $h:L\to M$ be an $\calS$-frame{} map. \begin{enumerate} \item Suppose that $h$ has a right adjoint, $r$. Then $h$ preserves all existing joins and $r$ preserves all existing meets. \item Suppose that $h$ has a left adjoint $L$. Then $h$ preserves all existing meets and $\ell$ preserves all existing joins. \end{enumerate} \end{proposition}
\section{Closed and open maps}
\begin{definition}\label{I} Let $h:L\to M$ be an $\calS$-frame{} map.
We call $h$ \emph{closed} if, for all $m \in M$, there exists $x\in L$ with $(h\times h)^{-1}(\nabla_m)=\nabla_x$.
We call $h$ \emph{open} if, for all $m\in M$, there exists $x\in L$ with $(h\times h)^{-1}(\Delta_m)=\Delta_x$. \end{definition}
We know that (see \cite{jfasmadden}) that $\mathcal{C}_{\sels}$ is a functor from \sframe{}s{} to frames which is natural in the sense that for any $\calS$-frame{} $h:L\to M$ we have a frame map $\mathcal{C}_{\sels} h:\mathcal{C}_{\sels}L\to \mathcal{C}_{\sels}M$ making the following diagram commute:
\begin{center}
\begin{tikzpicture}
\node at(1,1.2){$M$};
\node at(1,3){$L$};
\node at(4.5,3){$\mathcal{C}_{\sels}L$};
\node at(4.5,1.2){$\mathcal{C}_{\sels}M$};
\node at(0.7,2.1){$h$};
\node at(2.6,3.3){$\nabla_L $};
\node at(2.6,0.8){$ \nabla_M$};
\node at(5.2,2.1){$\mathcal{C}_{\sels} h$};
\draw[<-](1,1.5)--(1,2.8);
\draw[->](1.5,3)--(4,3) ;
\draw[->](1.5,1.2)--(4,1.2);
\draw[<-](4.7,1.5)--(4.7,2.8);
\end{tikzpicture} \end{center} \vskip5pt
Now $(h\times h)^{-1}$ is the right adjoint of $\mathcal{C}_{\sels} h$, because, for $\theta\in\mathcal{C}_{\sels}L$, $\mathcal{C}_{\sels} h(\theta)$ is the $\sels$-congruence{} of $M$ generated by $(h\times h)[\theta]$, so for all $\theta\in\mathcal{C}_{\sels}L,\phi\in\mathcal{C}_{\sels}M$, $$\mathcal{C}_{\sels} h(\theta)\subseteq\phi\Longleftrightarrow\theta\subseteq(h\times h)^{-1}(\phi).$$
\begin{theorem}\label{J} Let $h:L\to M$ be an $\calS$-frame{} map. \begin{enumerate}
\item[(a)] The map $h$ is closed iff $h$ has a right adjoint, $r$, and for all $x\in L, m\in M$,
$$r(h(x)\vee m)=x\vee r(m).$$
\item[(b)] The map $h$ is open iff $h$ has a left adjoint, $l$, and for all $x\in L,m\in M$,
$$l(h(x)\wedge m)=x\wedge l(m).$$ \end{enumerate} \end{theorem}
\begin{theorem}\label{K} Let $L$ be an $\calS$-frame{} and $\theta$ an $\sels$-congruence{} on $L$.\\ (a) The quotient map $q:L\to L/\theta$ is closed if and only if $\theta$ is a closed $\sels$-congruence; i.e. $\theta=\nabla_a$ for some $a\in L$.\\ (b) The quotient map $q:L\to L/\theta$ is open if and only if $\theta$ is an open $\sels$-congruence{}; that is, $\theta=\Delta_a$ for some $a\in L$. \end{theorem}
\begin{definition} Let $h:L\to M$ be an $\calS$-frame{} map. We say that $h$ is \emph{dense} (resp., \emph{codense}) if for all $a\in L, h(a)=0$ (resp., $h(a)=1$) implies that $a=0$ (resp., $a=1$).\end{definition} \begin{lemma}\label{L} Let $h:L\to M$ be an $\calS$-frame{} map. \begin{enumerate} \item[(a)] If $h$ is dense and closed, then $h$ is one-one. If $h$ is dense, closed and onto, then $h$ is an isomorphism. \item[(b)] If $h$ is codense and open, then $h$ is one-one. If $h$ is codense, open and onto, then $h$ is an isomorphism. \end{enumerate} \end{lemma}
\begin{lemma}\label{M} Suppose that $f:L\to M$ and $g:M\to N$ are $\calS$-frame{} maps. \begin{enumerate} \item[(a)] \begin{enumerate} \item[(i)] If $f$ and $g$ are both closed, then $g\circ f$ is closed. \item[(ii)] If $g\circ f$ is closed and $g$ is one-one, then $f$ is closed. \item[(iii)] If $g\circ f$ is closed and $f$ is onto, then $g$ is closed. \end{enumerate} \item[(b)] As above but replace ``closed'' by ``open''. \end{enumerate} \end{lemma}
\section{Boolean properties for partial frames}
The material in this section comes from \cite{jfasBoolVar}.
We begin by recalling how matters stand in the case of full frames. A Boolean frame is simply a frame that is also a Boolean algebra, that is, every element has a complement. However, Booleanness can also be characterized in a different way. For any frame $M$, let $M_{**}=\{x^{**}:x\in M\}$ where $x^*=\bigvee\{z\in M:z\wedge x=0\}$, the pseudocomplement of $x$. The frame map $p:M\to M_{**}$ given by $p(x)=x^{**}$ is called the Booleanization of $M$. It is the least dense quotient of $M$, but is also the unique dense Boolean quotient of $M$. A frame is then Boolean if and only if it is isomorphic to its Booleanization.
Following Madden's lead in \cite{madden}, in \cite{jfasmadden} we constructed a least dense quotient for partial frames. The codomain need not be Boolean, however, as Madden already noted in the case of $\kappa$-frames. We use his terminology, ``d-reduced'', to refer to those partial frames isomorphic to their least dense quotients. We refer the reader to Definition \ref{aaaa}(l) for our notation and terminology in this regard.
The next result characterizes those $\calS$-frame{s}, $L$, that are Boolean algebras, in several ways. These involve the free frame over $L$, the congruence frame of $L$ and the the relationship between these two entities. \begin{proposition}\label{ci} Let $L$ be an $\calS$-frame. The following are equivalent: \begin{enumerate} \item $L$ is Boolean; that is, every element of $L$ is complemented. \item All principal $\mathcal{S}$-ideal{s} in $\mathcal{H}_{\mathcal{S}}L$ are complemented. \item The embedding $e:\mathcal{H}_{\mathcal{S}}L\to\mathcal{C}_{\sels}L$ is an isomorphism. \item Every $\sels$-congruence{} $\theta$ of $L$ is an arbitrary join of $\sels$-congruence{s} of the form $\nabla_a$, for some $a\in L$. \end{enumerate} \end{proposition}
In our experience with partial frames, it has often proved useful to compare properties for a partial frame with the analogous properties for the corresponding free frame. We do this now for Booleanness.
We recall that, if $M$ is a frame and $x\in M$, we call $x$ a \textit{dense} element of $M$ if $x^*=0$.
\begin{proposition}\label{cj} Let $L$ be an $\calS$-frame. The following are equivalent: \begin{enumerate} \item The frame $\mathcal{H}_{\mathcal{S}}L$ is Boolean. \item $\operatorname{\downarrow} 1$ is the only dense element of $\mathcal{H}_{\mathcal{S}}L$. \item The $\calS$-frame{} embedding $\nabla: L\to\mathcal{C}_{\sels}L$ is an isomorphism. \item Every $\theta\in\mathcal{C}_{\sels}L$ has the form $\theta=\nabla_a$, for some $a\in L$. \end{enumerate} \end{proposition}
We now provide four provably distinct conditions akin to Booleanness for partial frames. In the setting of (full) frames they all amount to every element being complemented.
\begin{theorem}\label{da} Let $L$ be an $\calS$-frame. In the following list of conditions, each one implies the succeeding one, but not conversely. \begin{enumerate} \item $\mathcal{H}_{\mathcal{S}}L$ is a Boolean frame. \item $L$ is a Boolean frame. \item $L$ is a Boolean $\calS$-frame. \item $L$ is a d-reduced $\calS$-frame. \end{enumerate} \end{theorem} \bproof (a)$\Rightarrow$(b): ($\not\Leftarrow$) See Example \ref{db}.\\ (b)$\Rightarrow$(c): ($\not\Leftarrow$): See Example \ref{dc}.\\ (c)$\Rightarrow$(d): ($\not\Leftarrow$): See Example \ref{bh}. \end{pf} \begin{example}\label{bh} Let $\mathcal{S}$ designate countable subsets, and consider the $\sigma$-frame $L=\mathcal{P}_C(\mathbb{R})$, which consists of all countable subsets of $\mathbb{R}$ together with $\mathbb{R}$ as the top element. Countable join is union, binary meet is intersection.\\ Here $(X,Y)\in \sym_0$ if and only if, for any countable subset $U$, $U\cap X=\emptyset\Longleftrightarrow U\cap Y=\emptyset$, which makes $X=Y$. So $\sym_0=\Delta$, which makes $\mathcal{P}_C(\mathbb{R})$ d-reduced. However, $\mathcal{P}_C(\mathbb{R})$ is clearly not Boolean. \end{example} \begin{example}\label{db} Let $\mathcal{S}$ designate countable subsets, and let $\mathcal{L}$ consist of all subsets of $\mathbb{R}$. Clearly $\mathcal{L}$ is a Boolean frame. We show that $\mathcal{H}_{\mathcal{S}}\mathcal{L}$ is not Boolean, using condition (d) of Proposition \ref{cj}.\\ Let $\mathcal{I}=\{X\subseteq\mathbb{R}: X\cap(\mathbb{R}\backslash\mathbb{Q}) \textrm{ is countable}\}$. Then $\mathcal{I}$ is a $\sigma$-ideal of $\mathcal{L}$; that is, a downset closed under countable unions. By Definition \ref{aaaa}(g)(iv), $\bjoinl{X\in\mathcal{I}}\nabla_X=\bigcup\limits_{X\in\mathcal{I}}\nabla_X$ and this cannot have the form $\nabla_Z$ for any $Z\in\mathcal{L}$, since that would require $Z\supseteq X$ for all $X\in\mathcal{I}$, and hence $Z=\mathbb{R}$; a contradiction.\phantom{XXXX} \end{example}
\begin{example}\label{dc} Let $L$ consist of all countable and co-countable subsets of the real line, and let $\mathcal{S}$ designate countable subsets. Clearly $L$ is a Boolean $\sigma$-frame, but not a complete lattice, so not a frame. \end{example}
\section{Comparing congruences on a partial frame and its free frame}
The material in this section comes from \cite{jfasBoolVar}.
In this section, for a partial frame $L$, we compare $\mathcal{C}_{\sels}L$, the frame of $\sels$-congruence{s} of $L$, with $\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$, the frame of (frame) congruences on $\mathcal{H}_{\mathcal{S}}L$, the free frame over $L$. The universal property of the embedding $\nabla:L\to\mathcal{C}_{\sels}L$ provides a frame map $E_L:\mathcal{C}_{\sels}L\to\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$. We give an explicit description of this map, and show that it provides a natural transformation.
We then turn our attention to its right adjoint $D_L:\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)\to \mathcal{C}_{\sels}L$. Again, we provide an explicit description of this function, including an interesting and useful action on closed congruences (Lemma \ref{ec}).
\begin{definition}\label{dg} Let $L$ be an $\calS$-frame. Consider this diagram \begin{center}
\begin{tikzpicture}
\node at (0.4,1){$\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$};
\node at (0,3){$\mathcal{H}_{\mathcal{S}}L$};
\node at (0,5){$L$};
\node at (3.2,5){$\mathcal{C}_{\sels}L$};
\node at (-0.3,2){$\nabla$};
\node at (-0.3,4){$\operatorname{\downarrow}$};
\node at (1.6,5.3){$\nabla$};
\node at (2.2,3){$E_L$};
\draw[->](0.2,4.8)--(0.2,3.4);
\draw[->](0.2,2.8)--(0.2,1.4);
\draw[->](0.4,5.1)--(2.8,5.1);
\draw[->](3.1,4.8)--(0.6,1.4);
\end{tikzpicture} \end{center}
By the universal property of $\nabla:L\to\mathcal{C}_{\sels}L$ there exists a unique frame map $E_L:\mathcal{C}_{\sels}L\to\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$ such that $E_L\circ\nabla=\nabla\circ\operatorname{\downarrow}$; that is, for all $a\in L$ $$ E_L(\nabla_a)=\nabla_{\operatorname{\downarrow} a}.$$
\end{definition}
\begin{lemma}\label{dh} Let $L$ be an $\calS$-frame{}. \begin{enumerate} \item For $\theta\in\mathcal{C}_{\sels}L$, $E(\theta)$ is the frame congruence on $\mathcal{H}_{\mathcal{S}}L$ generated by $\{(\operatorname{\downarrow} x,\operatorname{\downarrow} y):(x,y)\in\theta\}$; this is denoted by $E(\theta)=\langle (\operatorname{\downarrow} x,\operatorname{\downarrow} y):(x,y)\in\theta\rangle$. \item The frame map $E:\mathcal{C}_{\sels}L\to\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$ is dense. \end{enumerate} \end{lemma}
\begin{corollary}\label{di} Let $L$ be an $\calS$-frame{} and $E:\mathcal{C}_{\sels}L\to\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$ given as in Definition \ref{dg}. For all $a\in L$: \begin{enumerate} \item $E(\nabla_a)=\nabla_{\operatorname{\downarrow} a}$ \item $E(\Delta_a)=\Delta_{\operatorname{\downarrow} a}$ \end{enumerate} \end{corollary}
\begin{remark}\label{dj} Let $L$ be an $\calS$-frame. the embedding $e:\mathcal{H}_{\mathcal{S}}L\to\mathcal{C}_{\sels}L$ of Definition \ref{ab} can be incorporated into the diagram of Definition \ref{dg} as follows: \begin{center}
\begin{tikzpicture}
\node at (0,0){$\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$};
\node at (0,2){$\mathcal{H}_{\mathcal{S}}L$};
\node at (0,4){$L$};
\draw[->](0.2,3.9)--(0.2,2.4);
\draw[->](0.2,1.7)--(0.2,0.4);
\node at (3.3,4){$\mathcal{C}_{\sels}L$};
\draw[->](0.4,4.1)--(2.9,4.1);
\draw[->](0.6,2.4)--(2.9,3.9);
\draw[->](3.3,3.7)--(0.9,0.3);
\node at (-0.2,1){$\nabla$};
\node at (-0.2,3){$\operatorname{\downarrow}$};
\node at (1.5,4.5){$\nabla$};
\node at (1.5,3.2){$e$};
\node at (2.3,2){$E$};
\end{tikzpicture} \end{center} Note that \begin{itemize}
\item the upper triangle commutes, since $e\circ \operatorname{\downarrow} =\nabla$.
\item the lower triangle commutes, since, for $I\in\mathcal{H}_{\mathcal{S}}L$,\\ $E\circ e(I)=E(\bjoinl{i\in I}\nabla_i)=\bjoinl{i\in I}E(\nabla_i)=\bjoinl{i\in I}\nabla_{\operatorname{\downarrow} i}=\nabla_I$.\\
Alternatively, this can be seen because the outer diagram commutes and every $\mathcal{S}$-ideal{} is generated by principal $\mathcal{S}$-ideal{s}. \end{itemize}
\end{remark} \begin{proposition}\label{eg} The function $E_L$ provides a natural transformation from the functor $\mathcal{C}_{\mathcal{S}}$ to the functor $\mathcal{C}\mathcal{H}_{\mathcal{S}}$. \end{proposition}
We now define, for any $\calS$-frame{} $L$, the function $D_L$. In a subsequent lemma, $D_L$ is seen to be the right adjoint of the frame map $E_L$.
\begin{definition}\label{ea} Let $L$ be an $\calS$-frame, and $\Phi$ a frame congruence on the frame $\mathcal{H}_{\mathcal{S}}L$. Define $$D_L(\Phi)=\{(x,y)\in L\times L: (\operatorname{\downarrow} x,\operatorname{\downarrow} y)\in\Phi\}.$$
\end{definition} \begin{lemma}\label{eb} Let $L$ be an $\calS$-frame. \begin{enumerate} \item For any frame congruence $\Phi$ on $\mathcal{H}_{\mathcal{S}}L$, $D_L(\Phi)$ is an $\sels$-congruence{} on $L$. \item The function $D_L:\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)\to\mathcal{C}_{\sels}L$ is the right adjoint of the frame map $E_L:\mathcal{C}_{\sels}L\to\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$ of Definition \ref{dg}. \item The function $D_L$ preserves bottom, top and arbitrary meets. \end{enumerate} \end{lemma}
We now provide further properties of $D$, including its action on certain special congruences. We note that the proof of Lemma \ref{ec}(a) uses the fact that, for $I$ an $\mathcal{S}$-ideal{} of an $\calS$-frame{} $L$, $\bjoinl{i\in I}\nabla_i=\bigcup\limits_{i\in I}\nabla_i$. This is not immediately obvious, but was proved in \cite{jfasheyt} Lemma 3.1.
\begin{lemma}\label{ec} Let $L$ be an $\calS$-frame, and $D$ as in Definition \ref{ea}. \begin{enumerate} \item For all $I\in\mathcal{H}_{\mathcal{S}}L$, $D(\nabla_I)=\bcupl{i\in I}\nabla_i.$ \item For all $a\in L$, \begin{enumerate} \item $D(\nabla_{\operatorname{\downarrow} a})=\nabla_a$ \item $D(\Delta_{\operatorname{\downarrow} a})=\Delta_a$ \end{enumerate} \item For $I\in\mathcal{H}_{\mathcal{S}}L$, $I$ is principal $\Longleftrightarrow D(\nabla_I)\vee D(\Delta_I)=\nabla$. \end{enumerate} \end{lemma}
\begin{definition} Let $M$ be a full frame. For any $a\in M$ we say $a$ is an \textit{$\sels$-Lindel\"{o}f{} element} of $M$ if the following condition holds:
If $a = \bigvee B$ for some $B\subseteq M$, then $a =\bigvee D$ for some designated subset $D$ of $M$ such that $D \subseteq B$. \end{definition} See \cite{jfas_hcozforqm} for details about this notion. In particular, Lemma 4.3 of that paper characterizes the $\sels$-Lindel\"{o}f{} elements of $\mathcal{H}_{\mathcal{S}}L$ as being the principal $\mathcal{S}$-ideal{s}.
The next result characterizes those rather special $\calS$-frame{s} $L$ for which $E_L$ is an isomorphism.
\begin{theorem}\label{ed} Let $L$ be an $\calS$-frame. The following are equivalent: \begin{enumerate} \item The embedding $\operatorname{\downarrow}:L\to\mathcal{H}_{\mathcal{S}}L$ is an isomorphism. \item Every $\mathcal{S}$-ideal{} of $L$ is principal. \item $L$ is a frame and every element of $L$ is $\sels$-Lindel\"{o}f. \item The frame map $E:\mathcal{C}_{\sels}L\to\mathcal{C}(\mathcal{H}_{\mathcal{S}}L)$ is an isomorphism. \end{enumerate} \end{theorem}
The equivalent conditions of Theorem \ref{ed} might seem rather strong. Here are some examples which show that these can obtain.
\begin{example}\label{ee} The conditions of Theorem \ref{ed} hold in the following examples: \begin{itemize}
\item $\mathcal{S}$ selects finite subsets and $L$ is a finite frame.
\item $\mathcal{S}$ selects countable subsets, and $L$ consists of the open subsets of the real line.
\item $\mathcal{S}$ selects finite subsets, or $\mathcal{S}$ selects countable subsets, and $L$ consists of the cofinite subsets of the real line, together with the empty set. \end{itemize} \end{example}
\section{Closed and open embeddings into the free frame and the congruence frame}
\begin{theorem}\label{U} Let $L$ be an $\calS$-frame{} and $\mbox{$\downarrow$}:L \to \mathcal{H}_{\mathcal{S}}L$ the embedding into its free frame. \begin{enumerate} \item[(a)] The map $\mbox{$\downarrow$}$ has a right adjoint iff $\mbox{$\downarrow$}$ is an isomorphism. \item[(b)] The map $\mbox{$\downarrow$}$ is closed iff $\mbox{$\downarrow$}$ is an isomorphism. \item[(c)] The map $\mbox{$\downarrow$}$ has a left adjoint iff $L$ is a complete lattice. \item[(d)] The map $\mbox{$\downarrow$}$ is open iff $L$ is a frame. \end{enumerate} \end{theorem}
\begin{theorem}\label{V} Let $L$ be an $\calS$-frame{} and $\nabla:L \to \mathcal{C}_{\sels}L$ the embedding into its congruence frame. \begin{enumerate} \item[(a)] The map $\nabla$ is closed iff $\nabla$ is an isomorphism. \item[(b)] The map $\nabla$ is open iff $L$ is a Boolean frame. \end{enumerate}\end{theorem}
\end{document} | arXiv |
\begin{document}
\begin{abstract}
Skew Dyck are a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ a south-west step $(-1,-1)$ is also allowed, provided that the path
does not intersect itself. Replacing the south-west step by a red south-east step, we end up with decorated Dyck paths. We analyze partial versions of them
where the path ends on a fixed level $j$, not necessarily at level 0. We exclusively use generating functions and derive them with the celebrated kernel method.
In the second part of the paper, a dual version is studied, where the paths are read from right to left. In this way, we have two types of up-steps, not two types of
down-steps, as before.
A last section deals with the variation that the negative territory (below the $x$-axis) is also allowed. Surprisingly, this is more involved in terms of computations. \end{abstract}
\maketitle
\section{Introduction}
Skew Dyck are a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ a south-west step $(-1,-1)$ is also allowed, provided that the path does not intersect itself. Otherwise, like for Dyck path, it must never go below the $x$-axis and end eventually (after $2n$ steps) on the $x$-axis. Here are a few references: \cite{Deutsch-italy, KimStanley, Baril-neu, Prodinger-hex}. The enumerating sequence is \begin{equation*}
1, 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, 171369, 751236, 3328218, 14878455,\dots, \end{equation*} which is A002212 in \cite{OEIS}.
Skew Dyck appeared very briefly in our recent paper \cite{Prodinger-hex}; here we want to give a more thorough analysis of them, using generating functions and the kernel method. Here is a list of the 10 skew paths consisting of 6 steps:
\begin{figure}
\caption{All 10 skew Dyck paths of length 6 (consisting of 6 steps).}
\end{figure}
We prefer to work with the equivalent model (resembling more traditional Dyck paths) where we replace each step $(-1,-1)$ by $(1,-1)$ but label it red. Here is the list of the 10 paths again (Figure 2):
\begin{figure}
\caption{The 10 paths redrawn, with red south-east edges instead of south-west edges.}
\end{figure}
The rules to generate such decorated Dyck paths are: each edge $(1,-1)$ may be black or red, but \begin{tikzpicture}[scale=0.3]\draw [thick](0,0)--(1,1); \draw [red,thick] (1,1)--(2,0);\end{tikzpicture} and \begin{tikzpicture}[scale=0.3] \draw [red,thick] (0,1)--(1,0);\draw [thick](1,0)--(2,1);\end{tikzpicture} are forbidden.
Our interest is in particular in \emph{partial} decorated Dyck paths, ending at level $j$, for fixed $j\ge0$; the instance $j=0$ is the classical case.
The analysis of partial skew Dyck paths was recently started in \cite{Baril-neu} (using the notion `prefix of a skew Dyck path') using Riordan arrays instead of our kernel method. The latter gives us \emph{bivariate} generating functions, from which it is easier to draw conclusions. Two variables, $z$ and $u$, are used, where $z$ marks the length of the path and $j$ marks the end-level. We briefly mention that one can, using a third variable $w$, also count the number of red edges.
Again, once all generating functions are explicitly known, many corollories can be derived in a standard fashion. We only do this in a few instances. But we would like to emphasize that the substitution \begin{equation*} x=\frac{v}{1+3v+v^2}, \end{equation*} which was used in \cite{HPW, Prodinger-hex} allows to write \emph{explicit enumerations}, using the notion of a (weighted) trinomial coefficient: \begin{equation*} \binom{n;1,3,1}{k}:=[t^k](1+3t+t^2)^n. \end{equation*}
The second part of the paper deals with a dual version, where the paths are read from right to left.
\section{Generating functions and the kernel method} \label{dunno}
We catch the essence of a decorated Dyck path using a state-diagram:
\begin{figure}
\caption{Three layers of states according to the type of steps leading to them (up, down-black, down-red).}
\end{figure} It has three types of states, with $j$ ranging from 0 to infinity; in the drawing, only $j=0..8$ is shown. The first layer of states refers to an up-step leading to a state, the second layer refers to a black down-step leading to a state and the third layer refers to a red down-step leading to a state. We will work out generating functions describing all paths leading to a particular state. We will use the notations $f_j,g_j,h_j$ for the three respective layers, from top to bottom. Note that the syntactic rules of forbidden patterns \begin{tikzpicture}[scale=0.3]\draw [thick](0,0)--(1,1); \draw [red,thick] (1,1)--(2,0);\end{tikzpicture} and \begin{tikzpicture}[scale=0.3] \draw [red,thick] (0,1)--(1,0);\draw [thick](1,0)--(2,1);\end{tikzpicture} can be clearly seen from the picture. The functions depend on the variable $z$ (marking the number of steps), but mostly we just write $f_j$ instead of $f_j(z)$, etc.
The following recursions can be read off immediately from the diagram: \begin{gather*} f_0=1,\quad f_{i+1}=zf_i+zg_i,\quad i\ge0,\\ g_i=zf_{i+1}+zg_{i+1}+zh_{i+1},\quad i\ge0,\\
h_i=zh_{i+1}+zg_{i+1},\quad i\ge0. \end{gather*} And now it is time to introduce the promised \emph{bivariate} generating functions: \begin{equation*} F(z,u)=\sum_{i\ge0}f_i(z)u^i,\quad G(z,u)=\sum_{i\ge0}g_i(z)u^i,\quad H(z,u)=\sum_{i\ge0}h_i(z)u^i. \end{equation*} Again, often we just write $F(u)$ instead of $F(z,u)$ and treat $z$ as a `silent' variable. Summing the recursions leads to \begin{align*} \sum_{i\ge0}u^if_{i+1}&=\sum_{i\ge0}u^izf_i+\sum_{i\ge0}u^izg_i,\\ \sum_{i\ge0}u^ig_i&=\sum_{i\ge0}u^izf_{i+1}+\sum_{i\ge0}u^izg_{i+1}+\sum_{i\ge0}u^izh_{i+1},\\ \sum_{i\ge0}u^ih_i&=\sum_{i\ge0}u^izh_{i+1}+\sum_{i\ge0}u^izg_{i+1}. \end{align*} This can be rewritten as \begin{align*}
\frac1u(F(u)-1)&=zF(u)+zG(u),\\*
G(u)&=\frac zu(F(u)-1)+\frac zu(G(u)-G(0))+\frac zu(H(u)-H(0)),\\*
H(u)&= \frac zu(G(u)-G(0))+\frac zu(H(u)-H(0)). \end{align*} This is a typical application of the kernel method. For a gentle example-driven introduction to the kernel method, see \cite{Prodinger-kernel}. First, \begin{align*} F(u)&=\frac{z^2uG(0)+z^2uH(0)+z^2u-u-z^3+2z}{-{z}^{3}-u+2z+z{u}^{2}-{z}^{2}u},\\ G(u)&=\frac{z(H(0)-uzH(0)+z^2+G(0)-zuG(0)-zu)}{-{z}^{3}-u+2z+z{u}^{2}-{z}^{2}u},\\ H(u)&=\frac{z(-uzH(0)-z^2-zuG(0)+G(0)-z^2H(0)+H(0)-z^2G(0))}{-{z}^{3}-u+2z+z{u}^{2}-{z}^{2}u}. \end{align*} The denominator factors as $z(u-r_1)(u-r_2)$, with \begin{equation*} r_1=\frac{1+z^2+\sqrt{1-6z^2+5z^4}}{2z},\quad r_2=\frac{1+z^2-\sqrt{1-6z^2+5z^4}}{2z}. \end{equation*} Note that $r_1r_2=2-z^2$.
Since the factor $u-r_2$ in the denominator is ``bad,'' it must also cancel in the numerators. From this we conclude as a first step \begin{equation*} G(0) = \frac{1-2z^2H(0)-3z^2-\sqrt{1-6z^2+5z^4}}{2z^2}, \end{equation*} and by further simplification \begin{equation*} H(0)=\frac{1-4z^2+z^4+(z^2-1)\sqrt{1-6z^2+5z^4}}{2-z^2}. \end{equation*} Thus (with $W=\sqrt{1-6z^2+5z^4}=\sqrt{(1-z^2)(1-5z^2)}$\,) \begin{align*} F(u)&=\frac{-1-z^2-W}{2z(u-r_1)}=\frac{1+z^2+W}{2zr_1(1-u/r_1)},\\ G(u)&=\frac{-1+z^2+W}{2z(u-r_1)}=\frac{1-z^2-W}{2zr_1(1-u/r_1)},\\ H(u)&=\frac{-1+3z^2+W}{2z(u-r_1)}=\frac{1-3z^2-W}{2zr_1(1-u/r_1)}. \end{align*} The total generating function is \begin{equation*} S(u)=F(u)+G(u)+H(u)=\frac{3-3z^2-W}{2zr_1(1-u/r_1)}. \end{equation*}
The coefficient of $u^jz^n$ in $S(u)$ counts the partial paths of length $n$, ending at level $j$.
We will write $s_j=[u^j]S(u)$. Furthermore \begin{align*} f_j=[u^j] F(u)&=[u^j]\frac{1+z^2+W}{2zr_1(1-u/r_1)},\\ g_j=[u^j] G(u)&=[u^j]\frac{1-z^2-W}{2zr_1(1-u/r_1)},\\ h_j=[u^j] H(u)&=[u^j]\frac{1-3z^2-W}{2zr_1(1-u/r_1)}. \end{align*} At this stage, we are only interested in \begin{equation*} s_j=f_j+g_j+h_j=[u^j]\frac{3-3z^2-W}{2zr_1(1-u/r_1)}=\frac{3-3z^2-W}{2zr_1^{j+1}}, \end{equation*} which is the generating function of all (partial) paths ending at level $j$. Parity considerations give us that only coefficients $[z^n]s_j$ are non-zero if $n\equiv j\bmod2$. To make this more transparent, we set \begin{equation*} P(z)=zr_1=\frac{1+z^2+\sqrt{1-6z^2+5z^4}}{2}, \end{equation*} and then \begin{equation*}
s_j=f_j+g_j+h_j=z^j\frac{3-3z^2-W}{2P^{j+1}}. \end{equation*} Now we read off coefficients. We do this using residues and contour integration. The path of integration, in both variables $x$ resp.\ $v$ is a small circle or an equivalent contour. \begin{align*}
[z^{2m+j}]s_j&=[z^{2m}]\frac{3-3z^2-W}{2P^{j+1}}=
[x^m]\frac{3-3x-\sqrt{1-6x+5x^2}}{2\Big(\frac{1+x-\sqrt{1-6x+5x^2}}{2}\Big)^{j+1}}\\
&=[x^m]\frac{3-3\frac v{1+3v+v^2}-\frac{1-v^2}{1+3v+v^2}}{2\big(\frac{v(v+2)}{1+3v+v^2}\big)^{j+1}}\\
&=[x^m]\frac{(1+v)(1+2v)}{v^{j+1}(v+2)^{j+1}}(1+3v+v^2)^j\\
&=\frac1{2\pi i}\oint\frac{dx}{x^{m+1}}\frac{(1+v)(1+2v)}{v^{j+1}(v+2)^{j+1}}(1+3v+v^2)^j\\
&=\frac1{2\pi i}\oint\frac{dv}{v^{m+1}}\frac{(1+v)(1+2v)(1-v^2)}{v^{j+1}(v+2)^{j+1}}(1+3v+v^2)^{m-1+j}\\
&=[v^{m+j+1}]\frac{(1+v)^2(1+2v)(1-v)}{(v+2)^{j+1}}(1+3v+v^2)^{m-1+j}. \end{align*} Note that \begin{equation*}(1+v)^2(1+2v)(1-v)= -9+27( v+2 ) -29( v+2 ) ^{2}+13( v+2) ^{3}-2( v+2 ) ^{4}; \end{equation*} consequently \begin{align*} [v^k]&\frac{(1+v)^2(1+2v)(1-v)}{(v+2)^{j+1}}\\ &=-9\frac1{2^{j+1+k}}\binom{-j-1}{k} +27\frac1{2^{j+k}}\binom{-j}{k} -29\frac1{2^{j-1+k}}\binom{-j+1}{k}\\& +13\frac1{2^{j-2+k}}\binom{-j+2}{k} -2\frac1{2^{j-3+k}}\binom{-j+3}{k}=:\lambda_{j;k}. \end{align*} With this abbreviation we find \begin{equation*}
[v^{m+j+1}]\frac{(1+v)^2(1+2v)(1-v)}{(v+2)^{j+1}}(1+3v+v^2)^{m-1+j}
=\sum_{k=0}^{m+j+1}\lambda_{j;k}\binom{m-1+j;1,3,1}{m+j+1-k}. \end{equation*} This is not extremely pretty but it is \emph{explicit} and as good as it gets. Here are the first few generating functions: \begin{align*} s_0&=1+z^2+3z^4+10z^6+36z^8+137z^{10}+543z^{12}+\cdots\\* s_1&=z+2z^3+6z^5+21z^7+79z^9+311z^{11}+1265z^{13}+\cdots\\ s_2&=z^2+3z^4+10z^6+37z^8+145z^{10}+589z^{12}+2455z^{14}+\cdots\\ s_3&=z^3+4z^5+15z^7+59z^9+241z^{11}+1010z^{13}+4314^{15}+\cdots\\ \end{align*} We could also give such lists for the functions $f_j$, $g_j$, $h_j$, if desired. We summarize the essential findings of this section: \begin{theorem} The generating function of decorated (partial) Dyck paths, consisting of $n$ steps, ending on level $j$, is given by
\begin{equation*} S(z,u)=\frac{3-3z^2-\sqrt{1-6z^2+5z^4}}{2zr_1(1-u/r_1)},
\end{equation*}
with
\begin{equation*} r_1=\frac{1+z^2+\sqrt{1-6z^2+5z^4}}{2z}.
\end{equation*}
Furthermore
\begin{equation*} [u^j]S(z,u)=\frac{3-3z^2-\sqrt{1-6z^2+5z^4}}{2zr_1^{j+1}}.
\end{equation*} \end{theorem}
\section{Open ended paths}
If we do not specify the end of the paths, in other words we sum over all $j\ge0$, then at the level of generating functions this is very easy, since we only have to set $u:=1$. We find \begin{align*} S(1)&=-\frac{(z+1)(z^2+3z-2)+(z+2)\sqrt{1-6z^2+5z^4}}{2z(z^2+2z-1)}\\ &=1+z+2z^2+3z^3+7z^4+11z^5+26z^6+43z^7+102z^8+175z^9+416z^{10}+\cdots. \end{align*}
\section{Counting red edges}
We can use an extra variable, $w$, to count additionally the red edges that occur in a path. We use the same letters for generating functions. Eventually, the coefficient $[z^nu^jw^k]S$ is the number of (partial) paths consisting of $n$ steps, leading to level $j$, and having passed $k$ red edges. The endpoint of the original skew path has then coordinates $(n-2k,j)$. The computations are very similar, and we only sketch the key steps.
\begin{equation*}
f_0=1,\quad f_{i+1}=zf_i+zg_i,\quad i\ge0, \end{equation*} \begin{equation*}
g_i=zf_{i+1}+zg_{i+1}+zh_{i+1},\quad i\ge0, \end{equation*} \begin{equation*}
h_i=wzh_{i+1}+wzg_{i+1},\quad i\ge0; \end{equation*} \begin{align*}
\frac1u(F(u)-1)&=zF(u)+zG(u),\\*
G(u)&=\frac zu(F(u)-1)+\frac zu(G(u)-G(0))+\frac zu(H(u)-H(0)),\\*
H(u)&= \frac {wz}u(G(u)-G(0))+\frac {wz}u(H(u)-G(0)); \end{align*} \begin{align*}
F(u)&=\frac{z^2uG(0)+z^2uH(0)+z^2u-u-wz^3+z+wz}{-w{z}^{3}-u+z+wz+z{u}^{2}-w{z}^{2}u},\\
G(u)&=\frac{z(H(0)-uzH(0)+wz^2+G(0)-zuG(0)-zu)}{-w{z}^{3}-u+z+wz+z{u}^{2}-w{z}^{2}u},\\
H(u)&=\frac{wz(-uzH(0)-z^2-zuG(0)+G(0)-z^2H(0)+H(0)-z^2G(0))}{-w{z}^{3}-u+z+wz+z{u}^{2}-w{z}^{2}u}. \end{align*} The denominator factors as $z(u-r_1)(u-r_2)$, with \begin{align*}
r_1&=\frac{1+wz^2+\sqrt{1-(4+2w)z^2+(4w+w^2)z^4}}{2z},\\*
r_2&=\frac{1+wz^2-\sqrt{1-(4+2w)z^2+(4w+w^2)z^4}}{2z}. \end{align*} Note the factorization $1-(4+2w)z^2+(4w+w^2)z^4=(1-z^2w)(1-(4+w)z^2)$. Since the factor $u-r_2$ in the denominator is ``bad,'' it must also cancel in the numerators. From this we eventually find, with the abbreviation $W=\sqrt{1-(4+2w)z^2+(4w+w^2)z^4}\,$) \begin{align*}
F(u)&=\frac{-1-wz^2-W}{2z(u-r_1)},\\
G(u)&=\frac{-1+wz^2+W}{2z(u-r_1)},\\
H(u)&=\frac{-1+(2+w)z^2+W}{2z(u-r_1)}. \end{align*} The total generating function is \begin{equation*}
S(u)=F(u)+G(u)+H(u)=\frac{-2-w+z^2(w+w^2)+ wW}{2z(u-r_1)}.
\end{equation*} The special case $u=0$ (return to the $x$-axis) is to be noted: \begin{equation*}
S(0)=\frac{-2-w+z^2(w+w^2)+ wW}{-2zr_1}=\frac{1-wz^2-W}{2z^2}. \end{equation*} Since there are only even powers of $z$ in this function, we replace $x=z^2$ and get \begin{align*}
S(0)&=\frac{1-wx-\sqrt{1-(4+2w)x+(4w+w^2)x^2}}{2x}\\
&=1+x+(w+2)x^2+(w^2+4w+5)x^3+(w^3+6w^2+15w+14)x^4+\cdots. \end{align*} Compare the factor $(w^2+4w+5)$ with the earlier drawing of the 10 paths.
There is again a substitution that allows for better results: \begin{equation*} z=\frac{v}{1+(2+w)v+v^2}, \quad\text{then}\quad S(0)=1+v. \end{equation*} Reading off coefficients can now be done using modified trinomial coefficients: \begin{equation*} \binom{n;1,2+w,1}{k}=[t^k]\bigl(1+(2+w)t+t^2\bigr)^n. \end{equation*} Again, we use contour integration to extract coefficients: \begin{align*} [x^n](1+v)&=\frac1{2\pi i}\oint \frac{dx}{x^{n+1}}(1+v)\\ &=\frac1{2\pi i}\oint \frac{dx}{v^{n+1}}\frac{1-v^2}{(1+(2+w)v+v^2)^2}(1+(2+w)v+v^2)^{n+1}(1+v)\\ &=[v^n](1-v)(1+v)^2(1+(2+w)v+v^2)^{n-1}\\ &=\binom{n-1;1,2+w,1}{n}+\binom{n-1;1,2+w,1}{n-1}\\* &\qquad-\binom{n-1;1,2+w,1}{n-2}-\binom{n-1;1,2+w,1}{n-3}. \end{align*}
Now we want to count the average number of red edges. For that, we differentiate $S(0)$ w.r.t.\ $w$, followed by $w:=1$. This leads to \begin{equation*} \frac{-1+6x-5x^2+(1+3x)\sqrt{1-6x+5x^2}}{2(1-x)(1-5x)}. \end{equation*}
A simple application of singularity analysis leads to \begin{equation*} \frac{\frac1{2\sqrt5}[x^n]\frac1{\sqrt{1-5x}}}{-\sqrt5[x^n]\sqrt{1-5x}}\sim \frac {n}{5}. \end{equation*} So, a random path consisting of $2n$ steps has about $n/5$ red steps, on average.
For readers who are not familiar with singularity analysis of generating functions \cite{FlOd90, FS}, we just mention that one determines the local expansion around the dominating singularity, which is at $z=\frac15$ in our instance. In the denominator, we just have the total number of skew Dyck paths, according to the sequence A002212 in \cite{OEIS}.
In the example of Figure~2, the exact average is $6/10$, which curiously is exactly the same as $3/5$.
We finish the discussion by considering fixed powers of $w$ in $S(0)$, counting skew Dyck paths consisting of zero, one, two, three, \dots red edges. We find \begin{align*} [w^0]S(0)&=\frac{1-\sqrt{1-4x}}{2x},\\ [w^1]S(0)&=\frac{1-2x-\sqrt{1-4x}}{2\sqrt{1-4x}},\\ [w^2]S(0)&=\frac{x^3}{(1-4x)^{3/2}},\\ [w^3]S(0)&=\frac{x^4(1-2x)}{(1-4x)^{5/2}},\\ [w^4]S(0)&=\frac{x^5(1-4x+5x^2)}{(1-4x)^{7/2}}, \quad\&\text{c}. \end{align*} The generating function $[w^0]S(0)$ is of course the generating function of Catalan numbers, since no red edges just means: ordinary Dyck paths. We can also conclude that the asymptotic behaviour is of the form $n^{k-3/2}4^n$, where the polynomial contribution gets higher, but the exponential growth stays the same: $4^n$. This is compared to the scenario of an \emph{arbitrary} number of red edges, when we get an exponential growth of the form $5^n$.
\section{Dual skew Dyck paths}
The mirrored version of skew Dyck paths with two types of up-steps, $(1,1)$ and $(-1,1)$ are also cited among the objects in A002212 in \cite{OEIS}. We call them dual skew paths and drop the `dual' when it isn't necessary. When the paths come back to the $x$-axis, no new enumeration is necessary, but this is no longer true for paths ending at level $j$.
Here is a list of the 10 skew paths consisting of 6 steps:
\begin{figure}
\caption{All 10 dual skew Dyck paths of length 6 (consisting of 6 steps).}
\end{figure}
We prefer to work with the equivalent model (resembling more traditional Dyck paths) where we replace each step $(-1,-1)$ by $(1,-1)$ but label it blue. Here is the list of the 10 paths again (Figure 2):
\begin{figure}
\caption{All 10 dual skew Dyck paths of length 6 (consisting of 6 steps).}
\end{figure}
The rules to generate such decorated Dyck paths are: each edge $(1,-1)$ may be black or blue, but \begin{tikzpicture}[scale=0.3]\draw [thick](0,1)--(1,0); \draw [cyan,thick] (1,0)--(2,1);\end{tikzpicture} and \begin{tikzpicture}[scale=0.3] \draw [cyan,thick] (0,0)--(1,1);\draw [thick](1,1)--(2,0);\end{tikzpicture} are forbidden.
Our interest is in particular in \emph{partial} decorated Dyck paths, ending at level $j$, for fixed $j\ge0$; the instance $j=0$ is the classical case.
The analysis of partial skew Dyck paths was recently started in \cite{Baril-neu} (using the notion `prefix of a skew Dyck path') using Riordan arrays instead of our kernel method. The latter gives us \emph{bivariate} generating functions, from which it is easier to draw conclusions. Two variables, $z$ and $u$, are used, where $z$ marks the length of the path and $j$ marks the end-level. We briefly mention that one can, using a third variable $w$, also count the number of blue edges.
The substitution \begin{equation*}
x=\frac{v}{1+3v+v^2}, \end{equation*} which was used in \cite{HPW, Prodinger-hex} is the key to the success and allows to write \emph{explicit enumerations}, using the notion of a (weighted) trinomial coefficient: \begin{equation*}
\binom{n;1,3,1}{k}:=[t^k](1+3t+t^2)^n. \end{equation*}
\section{Generating functions and the kernel method}
We catch the essence of a decorated (dual skew) Dyck path using a state-diagram:
\begin{figure}
\caption{Three layers of states according to the type of steps leading to them (down, up-black, up-blue).}
\end{figure} It has three types of states, with $j$ ranging from 0 to infinity; in the drawing, only $j=0..8$ is shown. The first layer of states refers to an up-step leading to a state, the second layer refers to a black down-step leading to a state and the third layer refers to a blue down-step leading to a state. We will work out generating functions describing all paths leading to a particular state. We will use the notations $c_j,a_j,b_j$ for the three respective layers, from top to bottom. Note that the syntactic rules of forbidden patterns \begin{tikzpicture}[scale=0.3]\draw [thick,cyan](0,0)--(1,1); \draw [thick] (1,1)--(2,0);\end{tikzpicture} and \begin{tikzpicture}[scale=0.3] \draw [thick] (0,1)--(1,0);\draw [thick,cyan](1,0)--(2,1);\end{tikzpicture} can be clearly seen from the picture. The functions depend on the variable $z$ (marking the number of steps), but mostly we just write $a_j$ instead of $a_j(z)$, etc.
The following recursions can be read off immediately from the diagram: \begin{gather*}
a_0=1,\quad a_{i+1}=za_i+zb_i+zc_i,\quad i\ge0,\\
b_i=za_{i+1}+zb_{i+1},\quad i\ge0,\\
c_{i+1}=za_{i}+zc_{i},\quad i\ge0. \end{gather*} And now it is time to introduce the promised \emph{bivariate} generating functions: \begin{equation*}
A(z,u)=\sum_{i\ge0}a_i(z)u^i,\quad
B(z,u)=\sum_{i\ge0}b_i(z)u^i,\quad
C(z,u)=\sum_{i\ge0}c_i(z)u^i. \end{equation*} Again, often we just write $A(u)$ instead of $A(z,u)$ and treat $z$ as a `silent' variable. Summing the recursions leads to \begin{align*}
\sum_{i\ge0}u^ia_i &=1+u\sum_{i\ge0}u^i(za_i+zb_i+zc_i)\\
&=1+uzA(u)+uzB(u)+uzC(u),\\
\sum_{i\ge0}u^ib_i &= \sum_{i\ge0}u^i(za_{i+1}+zb_{i+1})\\
&=\frac zu\sum_{i\ge1}u^ia_i+\frac zu\sum_{i\ge1}u^ib_i,\\
\sum_{i\ge1}u^ic_i &=uz\sum_{i\ge0}u^ia_i+uz\sum_{i\ge0}u^ic_i. \end{align*} This can be rewritten as \begin{align*}
A(u)&=1+uzA(u)+uzB(u)+uzC(u),\\
B(u)&=\frac zu(A(u)-a_0)+\frac zu(B(u)-b_0),\\
C(u)&=c_0+uzA(u)+uzC(u). \end{align*} Note that $a_0=1$, $c_0=0$. Simplification leads to \begin{equation*}
C(u)=\frac{uzA(u)}{1-uz} \end{equation*} and \begin{equation*}
B(u)=\frac{z(A(u)-1-B(0))}{u-z} \end{equation*} leaving us with just one equation \begin{equation*}
A(u)={\frac { \left( z-u+u{z}^{2}+u{z}^{2}B(0) \right) \left( uz-1
\right) }{{u}^{2}{z}^{3}+u{z}^{2}-2{u}^{2}z-z+u}}. \end{equation*} This is a typical application of the kernel method, \cite{Prodinger-kernel}. \begin{equation*}
{u}^{2}{z}^{3}+u{z}^{2}-2{u}^{2}z-z+u=z(z^2-2)(u-s_1)(u-s_2) \end{equation*} The denominator factors as $2z(z^2-2)(u-s_1)(u-s_2)$, with \begin{equation*}
s_1=\frac{1+z^2+\sqrt{1-6z^2+5z^4}}{2z(2-z^2)},\quad s_2=\frac{1+z^2-\sqrt{1-6z^2+5z^4}}{2z(2-z^2)}. \end{equation*} Note that $s_1s_2=\frac{1}{2-z^2}$. Since the factor $u-s_2$ in the denominator is ``bad,'' it must also cancel in the numerators. From this we conclude (again with the abbreviation $W=\sqrt{1-6z^2+5z^4}\,$) \begin{equation*}
B(0) = \frac{zs_2}{1-2zs_2}, \end{equation*} and further \begin{equation*}
A(u)
=\frac{(1-uz)(1+z^2+W)}{2z(z^2-2)(u-s_1)}, \end{equation*} \begin{equation*}
B(u)=\frac{1-2z^2-W}{z(2-z^2)(u-s_1)}, \end{equation*} \begin{equation*}
C(u)=\frac{1+z^2+W}{2(z^2-2)}\frac{u}{u-s_1}, \end{equation*} and for the function of main interest \begin{equation*}
G(u)=A(u)+B(u)+C(u)=\frac{3z^2-3+W}{2z(2-z^2)(u-s_1)}. \end{equation*} Note that \begin{align*}
\frac1{s_1}&=\frac{1+z^2-\sqrt{1-6z^2+5z^4}}{2z}=zS,\\
\frac1{s_2}&=\frac{1+z^2+\sqrt{1-6z^2+5z^4}}{2z}. \end{align*} Then \begin{align*}
[u^j]G(u)&=[u^j]\frac{3z^2-3+W}{2z(z^2-2)s_1(1-u/s_1)}\\
&=\frac{3z^2-3+W}{2z(z^2-2)s_1^{j+1}}
=\frac{3z^2-3+W}{2(z^2-2)}z^{j}S^{j+1}. \end{align*} So $[u^j]G(u)$ contains only powers of the form $z^{j+2N}$. Now we continue \begin{align*}
[z^{j+2N}u^j]G(u)&
=[z^{2N}]\frac{3z^2-3+W}{2(z^2-2)}S^{j+1}
\\&=[x^{N}]\frac{3x-3+\sqrt{1-6x+5x^2}}{2(x-2)}\bigg(\frac{1+x-\sqrt{1-6x+5x^2}}
{2x}\bigg)^{j+1}\\
&=[x^{N}](v+1)(v+2)^{j} \end{align*} which is the generating function of all (partial) paths ending at level $j$.
Now we read off coefficients. We do this using residues and contour integration. The path of integration, in both variables $x$ resp.\ $v$ is a small circle or an equivalent contour; \begin{align*}
[z^{j+2N}u^j]G(u)&=[x^{N}](v+1)(v+2)^{j}\\
&=\frac1{2\pi i}\oint \frac{dx}{x^{N+1}}(v+1)(v+2)^{j}\\
&=\frac1{2\pi i}\oint \frac{dv}{v^{N+1}}(1+3v+v^2)^{N+1}\frac{(1-v^2)}{(1+3v+v^2)^2}(v+1)(v+2)^{j}\\
&=[v^{N}](1+3v+v^2)^{N-1}(1-v)(1+v)^2(v+2)^{j}. \end{align*} Note that \begin{equation*}(1-v)(1+v)^2=
3-7( v+2 ) +5( v+2 ) ^{2}- ( v+2) ^{3}; \end{equation*} consequently \begin{align*}
[z^{j+2N}u^j]G(u)&=[v^{N}](1+3v+v^2)^{N-1}\Big[3-7( v+2 ) +5( v+2 ) ^{2}- ( v+2) ^{3}
\Big](v+2)^{j}. \end{align*} We abbreviate: \begin{align*}
\mu_{j;k}&=[v^{k}]\Big[3(v+2)^{j}-7(v+2)^{j+1} +5(v+2)^{j+2}- (v+2)^{j+3}\Big]\\
&=3\binom{j}{k}2^{j-k}-7\binom{j+1}{k}2^{j+1-k}+5\binom{j+2}{k}2^{j+2-k}-\binom{j+3}{k}2^{j+3-k}. \end{align*} With this notation we get \begin{equation*}
[z^{j+2N}u^j]G(u)
=\sum_{0\le k\le N-1}\mu_{j;k}\binom{N-1;1,3,1}{N-k}. \end{equation*} Here are the first few generating functions: \begin{align*}
G_0&=1+{z}^{2}+3{z}^{4}+10{z}^{6}+36{z}^{8}+137{z}^{10}+543{z}^{
12}+2219{z}^{14}
+\cdots\\*
G_1&=2z+3{z}^{3}+10{z}^{5}+36{z}^{7}+137{z}^{9}+543{z}^{11}+
2219{z}^{13}+9285{z}^{15}
+\cdots\\
G_2&=4{z}^{2}+8{z}^{4}+29{z}^{6}+111{z}^{8}+442{z}^{10}+1813{z
}^{12}+7609{z}^{14}+32521{z}^{16}
+\cdots\\
G_3&=8{z}^{3}+20{z}^{5}+78{z}^{7}+315{z}^{9}+1306{z}^{11}+5527
{z}^{13}+23779{z}^{15}+103699{z}^{17}
+\cdots\\ \end{align*} We could also give such lists for the functions $a_j$, $b_j$, $c_j$, if desired. We summarize the essential findings of this section: \begin{theorem} The generating function of decorated (partial) dual skew Dyck paths, consisting of $n$ steps, ending on level $j$, is given by
\begin{equation*}
G(z,u)=\frac{3z^2-3+\sqrt{1-6z^2+5z^4}}{2z(2-z^2)(u-s_1)},
\end{equation*}
with
\begin{equation*}
s_1=\frac{2z}{1+z^2-\sqrt{1-6z^2+5z^4}}.
\end{equation*}
Furthermore
\begin{equation*}
[u^j]G(z,u)=\frac{3z^2-3+\sqrt{1-6z^2+5z^4}}{2(z^2-2)}z^jS^{j+1},
\end{equation*}
with
\begin{equation*}
S=\frac{1+z^2-\sqrt{1-6z^2+5z^4}}{2z^2}.
\end{equation*} \end{theorem}
\section{Open ended paths}
If we do not specify the end of the paths, in other words we sum over all $j\ge0$, then at the level of generating functions this is very easy, since we only have to set $u:=1$. We find \begin{align*}
G(1)&=\frac{(1+z)(1-3z)}{2z(z^2+2z-1)-\sqrt{1-6z^2+5z^4}}\\
&=1+2z+5{z}^{2}+11{z}^{3}+27{z}^{4}+62{z}^{5}+151{z}^{6}+
354{z}^{7}+859{z}^{8}+2036{z}^{9}+\cdots. \end{align*}
\section{Counting blue edges}
We can use an extra variable, $w$, to count additionally the blue edges that occur in a path. We use the same letters for generating functions. Eventually, the coefficient $[z^nu^jw^k]S$ is the number of (partial) paths consisting of $n$ steps, leading to level $j$, and having passed $k$ blue edges. The endpoint of the original skew path has then coordinates $(n-2k,j)$. The computations are very similar, and we only sketch the key steps.
\begin{gather*}
a_0=1,\quad a_{i+1}=za_i+zb_i+zc_i,\quad i\ge0,\\
b_i=za_{i+1}+zb_{i+1},\quad i\ge0,\\
c_{i+1}=wza_{i}+wzc_{i},\quad i\ge0. \end{gather*} This leads to \begin{align*}
A(u)&=1+uzA(u)+uzB(u)+uzC(u),\\
B(u)&=\frac zu(A(u)-a_0)+\frac zu(B(u)-b_0),\\
C(u)&=c_0+wuzA(u)+wuzC(u). \end{align*} Solving, \begin{equation*}
S(u)=A(u)+B(u)+C(u)={\frac {u-wu{z}^{2}-zA(0)-zB(0)+uw{z}^{2}A(0)+uw{z}^{2}B(0)}{{u}^{2}{z}^{3}w+u-w{u}^{2}z-{u}^{2}z-z+wu{z}^{2}}}. \end{equation*} The denominator factors as $-z(1+w-z^2w)(u-s_1)(u-s_2)$, with \begin{align*}
s_1&={\frac {1+{z}^{2}w+\sqrt {1-2\,{z}^{2}w+{z}^{4}{w}^{2}-4\,{z}^{2
}+4{z}^{4}w}}{2z \left( 1+w-{z}^{2}w \right) }},\\*
s_2&= {\frac {1+{z}^{2}w-\sqrt {1-2\,{z}^{2}w+{z}^{4}{w}^{2}-4\,{z}^{2
}+4{z}^{4}w}}{2z \left( 1+w-{z}^{2}w \right) }} . \end{align*} Note the factorization $1-(4+2w)z^2+(4w+w^2)z^4=(1-z^2w)(1-(4+w)z^2)$. Since the factor $u-r_2$ in the denominator is ``bad,'' it must also cancel in the numerators. From this we eventually find, with the abbreviation $W=\sqrt{1-(4+2w)z^2+(4w+w^2)z^4}\,$) \begin{equation*}
G(0)={\frac {1-{z}^{2}w-W }{2{z}^{2}}}, \end{equation*} and further \begin{equation*}
G(u)=\frac {w-{z}^{2}{w}^{2}-wW+2-2{z}^{2}w}
{2z \left( -w -1+{z}^{2}w \right) (u-s_1)}. \end{equation*} The special case $u=0$ (return to the $x$-axis) is to be noted: \begin{equation*}
G(0)=1+{z}^{2}+ \left( w+2 \right) {z}^{4}+ \left( {w}^{2}+4w+5 \right)
{z}^{6}+ \left( w+2 \right) \left( {w}^{2}+4w+7 \right) {z}^{8}+\cdots. \end{equation*} Compare the factor $(w^2+4w+5)$ with the earlier drawing of the 10 paths. There is again a substitution that allows for better results: \begin{equation*}
z=\frac{v}{1+(2+w)v+v^2}, \quad\text{then}\quad G(0)=1+v. \end{equation*} Since $S(u)=G(u)$ with $S(u)$ from the first part of the paper, as it means the same objects, read from left to right resp.\ from right to left, no new analysis is required.
\section{Skew paths that can go into negative territory}
For Dyck paths and the standard random walk on the integers, the enumeration, if the negative territory is allowed, is easier. In our instance of paths equipped with an additional red down-step and the usual restrictions (up--red and red--up are forbidden) this is not so; it is rather more complicated. The paths may be described by another directed graph~Figure~\ref{arseneg}. \begin{figure}
\caption{Three layers of states according to the type of steps leading to them (up, down-black, down-red).}
\label{arseneg}
\end{figure}
We have the following recursions, \begin{gather*}
f_{i}=[i=0]+zf_{i-1}+zg_{i-1},\\
g_i=zf_{i+1}+zg_{i+1}+zh_{i+1},\\
h_i=zg_{i+1}+zh_{i+1}. \end{gather*} For negative indices we need to introduce separate sequences, \begin{equation*}
a_i=f_{-i}, \ b_i=g_{-i}, \ c_i=h_{-i}. \end{equation*} Then we find \begin{gather*}
f_{-i}=[-i=0]+zf_{-i-1}+zg_{-i-1},\\*
g_{-i}=zf_{-i+1}+zg_{-i+1}+zh_{-i+1},\\*
h_{-i}=zg_{-i+1}+zh_{-i+1} \end{gather*} and, rewriting, \begin{gather*}
a_{i}=[i=0]+za_{i+1}+zb_{i+1},\\
b_{i}=za_{i-1}+zb_{i-1}+zc_{i-1},\\
c_{i}=zb_{i-1}+zc_{i-1}. \end{gather*} Introducing \begin{equation*} F(u)=\sum_{i\ge0}f_iu^i,\ G(u)=\sum_{i\ge0}g_iu^i,\ H(u)=\sum_{i\ge0}h_iu^i \end{equation*} and \begin{equation*}
A(u)=\sum_{i\ge0}a_iu^i,\ B(u)=\sum_{i\ge0}b_iu^i,\ C(u)=\sum_{i\ge0}c_iu^i \end{equation*} we get the following 6 equations: \begin{align*}
F(u)-f_0&=zu(F(u)+G(u)),\\
G(u)&=\frac zu(F(u)+G(u)+H(u)-f_0-g_0-h_0),\\
H(u)&=\frac zu(G(u)+H(u)-g_0-h_0),\\
A(u)&=1+\frac zu(A(u)+B(u)-f_0-g_0),\\
B(u)-g_0&=zu(A(u)+B(u)+C(u)),\\
C(u)-h_0&=zu(B(u)+C(u)). \end{align*} Solving the system, \begin{align*}
F&={\frac {{z}^{2}ug_0+{z}^{2}uh_0+{z}^{2}u{f_0}-u{f_0}-{z}^{3}{f_0}+2 z{f_0}}{-{z}^{3}-u+2 z+z{u}^{2}-{z}^{2}u}},\\
G&=-{\frac {z \left( zu{h_0}-{z}^{2}{f_0}-{g_0}-{h_0}
+uz{f_0}+zu{g_0} \right) }{-{z}^{3}-u+2 z+z{u}^{2}-{z}^{2}u}},\\
H&=-{\frac {z \left( zu{h_0}+{z}^{2}{f_0}+zu{g_0}+{z}^{2}{
h_0}-{g_0}+{z}^{2}{g_0}-{h_0} \right) }{-{z}^{3}-u
+2 z+z{u}^{2}-{z}^{2}u}},\\
A&={\frac {2 {z}^{2}u{f_0}+{z}^{2}u{g_0}+{z}^{2}u{h_0}-2
z{u}^{2}-z{f_0}+u}{{z}^{2}u+{z}^{3}{u}^{2}+u-2 z{u}^{2}-z}},\\
B&={\frac {z{u}^{2}-{z}^{2}u{f_0}+u{g_0}-z{g_0}-{z}^{2}{u}
^{3}+{z}^{3}{u}^{2}{f_0}+{z}^{3}{u}^{2}{g_0}-z{u}^{2}{g_0}+z{u}^{2}{h_0}-{z}^{2}u{h_0}}{{z}^{2}u+{z}^{3}{u}^{2}+u
-2 z{u}^{2}-z}},\\
C&=-{\frac {-{z}^{2}{u}^{3}+{z}^{3}{u}^{2}{f_0}+{z}^{3}{u}^{2}{g_0}-u{h_0}-z{u}^{2}{g_0}+z{u}^{2}{h_0}+z{h_0}+{z
}^{2}u{g_0}}{{z}^{2}u+{z}^{3}{u}^{2}+u-2 z{u}^{2}-z}}, \end{align*} and \begin{equation*}
-{z}^{3}-u+2 z+z{u}^{2}-{z}^{2}u=z(u-r_1)(u-r_2) \end{equation*} with \begin{equation*}
r_{1,2}=\frac{1+z^2\pm\sqrt{1-6z^2+5z^4}}{2z}. \end{equation*} As usual, the factor $u-r_2$ must cancel out. The other denominators are \begin{equation*} {z}^{2}u+{z}^{3}{u}^{2}+u -2 z{u}^{2}-z=z(z^2-2)(u-s_1)(u-2_2) \end{equation*} and $s_1=1/r_2$, $s_2=1/r_1$. The factor $u-s_1$ must cancel out as well. This leads to
\begin{align*}
F(u)&=\frac {{z}^{2}g_0+{z}^{2}h_0+{z}^{2}f_0-f_0}{r_2z-1-{z}^{2}+zu},\\
G(u)&=-{\frac { \left(f_0+ g_0+h_0 \right) {z}^{2}}{r_2z-1-{z}^{2}+zu}},\\
H(u)&=-{\frac {{z}^{2} \left( g_0+h_0 \right) }{r_2z-1-{z}^{2}+zu}}, \end{align*} \scriptsize \begin{align*}
A(u)&=-{\frac {2 s_1z-2 {z}^{2}f_0-{z}^{2}g_0-{z}^{2}h_0-1+2 zu}{s_1{z}^{3}-2s_1z+1+{z}^{2}+u{z}^{3}-2 zu}},\\
B(u)&=-\frac{\mathcal{X}}{s_1 {z}^{3}
-2 s_1 z+1+{z}^{2}+u{z}^{3}-2 zu},\\
C(u)&={\frac {{s_1}^{2}{z}^{2}-s_1 {z}^{3}f_0-s_1 {z}^{3}g_0+s_1 zg_0-s_1 zh_0+s_1 {z}^{2}u+h-{z}^{2}g_0-u{z}^{3}f_0
-u{z}^ {3}g_0+zug_0-zuh_0+{z}^{2}{u}^{2}}{s_1 {z}^{3}-2 s_1 z+1+{z}^{2 }+u{z}^{3}-2 zu}} \end{align*} \normalsize with $\mathcal{X}= {s_1}^{2}{z}^{2}-s_1 z-s_1 {z}^{3}f_0-s_1
{z}^{3}g_0+s_1 zg_0-s_1 zh_0+s_1 {z}^{2}u+{z}^{2}f_0-g_0+{z}
^{2}h_0-zu-u{z}^{3}f_0-u{z}^{3}g_0+zug_0-zuh_0+{z}^{2}{u}^{2}$. The computation of $f_0$, $g_0$, $h_0$ requires some care. From the equations for $G$ and $H$ we conclude \begin{equation*}
g_0=\frac{z^2f_0+h_0}{1-z^2} \end{equation*} and from the expression for $H$, as just derived, we find \begin{equation*}
h_0=f_0{\frac {{z}^{4}}{r_2{z}^{3}-r_2z+1-2{z}^{2}}}. \end{equation*} So both, $g_0$ and $h_0$ are multiples of $f_0$. As any $f_0$ would solve the first 3 equations with the appropriate $g_0$, $h_0$, we need to resort to $A(u)$ since there we find that $f_0=1+\cdots$. By elimination, \begin{equation*}
A(u)={\frac {-2z{u}^{2}+u+g_0{z}^{2}u+{z}^{2}uh_0+2f_0{z}^{2}u-zf_)}{{z}^{2}u-z+{z}^{3}{u}^{2}-2z{u}^{2}+u}}. \end{equation*} Substitute $u=0$ and use $g_0$ and $h_0$ from before and then solve $A(0)=f_0$ leads to \begin{equation*}
f_0={\frac {r_2{z}^{3}-r_2z+1-2{z}^{2}}{-{z}^{6}+{z}^{4}+2r_2{z}^{3}-3{z}^{2}-r_2z+1}}. \end{equation*} It can be made explicit: \begin{equation*} f_0=\frac{1+z^2-\sqrt{1-6z^2+5z^4}}{2z^2(2-z^2)}. \end{equation*} Now everything is explicit: \begin{align*}
f_0&=1+z^2+2z^4+6z^6+21z^8+79z^{10}+311z^{12}+1265z^{14},\\
g_0&=z^2+3z^4+10z^6+37z^8+145z^{10}+589z^{12}+2455z^{14},\\
h_0&=z^4+5z^6+21z^8+87z^{10}+365z^{12}+1555z^{14}. \end{align*} The expressions for $g_0$ and $h_0$ are a bit long, but \begin{align*}
f_0+g_0+h_0&=\frac{1-3z^2+2z^4-\sqrt{1-6z^2+5z^4}}{2z^4(2-z^2)}\\
&=1+2z^2+6z^4+21z^6+79z^8+311z^{10}+1265z^{12}+5275z^{14}. \end{align*} The coefficients $1,2,6,21,\dots$ are sequence A033321 in \cite{OEIS}. In the comments to this sequence, the number of skew Dyck paths of semilength $n$ ending with a down step $(1,-1)$ is mentioned, something that follows from our results for $g_0$ in section~\ref{dunno}.
\end{document} | arXiv |
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| Wikipedia |
\begin{document}
\title{\sffamily\LARGE Best Approximation from the Kuhn-Tucker Set of\\ Composite Monotone Inclusions\footnote{Contact author: P. L. Combettes, \ttfamily{[email protected]}, phone: +33 1 4427 6319, fax: +33 1 4427 7200.}}
\author{ Abdullah Alotaibi,$\!^{\natural}$ ~Patrick L. Combettes,$\!^{\flat}$~ and ~Naseer Shahzad$\,^\natural$ \\[5mm] \small $^\natural$King Abdulaziz University\\ \small Department of Mathematics, P. O. Box 80203\\ \small Jeddah 21859, Saudi Arabia\\[5mm] \small $^\flat$Sorbonne Universit\'es -- UPMC Univ. Paris 06\\ \small UMR 7598, Laboratoire Jacques-Louis Lions\\ \small F-75005, Paris, France\\[4mm] }
\date{~}
\maketitle
\vskip 8mm
\begin{abstract} \noindent Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian composite monotone inclusion problem. Applications to systems of coupled monotone inclusions are presented. Our framework does not impose additional assumptions on the operators present in the formulation, and it does not require knowledge of the norm of the linear operators involved in the compositions or the inversion of linear operators. \end{abstract}
{\bfseries Keywords} best approximation, duality, Haugazeau, monotone operator, primal-dual algorithm, splitting algorithm, strong convergence
{\bfseries Mathematics Subject Classifications (2010)} Primary 47H05, 41A50; Secondary 65K05, 41A65, 90C25.
\maketitle
\section{Introduction}
Let $\ensuremath{{\mathcal H}}$ and $\ensuremath{{\mathcal G}}$ be real Hilbert spaces, let $L\colon\ensuremath{{\mathcal H}}\to\ensuremath{{\mathcal G}}$ be a bounded linear operator, and let $f\colon\ensuremath{{\mathcal H}}\to\ensuremath{\left]-\infty,+\infty\right]}$ and $g\colon\ensuremath{{\mathcal G}}\to\ensuremath{\left]-\infty,+\infty\right]}$ be proper lower semicontinuous convex functions. Classical Fenchel-Rockafellar duality \cite{Rock67} concerns the interplay between the optimization problem \begin{equation} \label{e:9F842h20a} \minimize{x\in\ensuremath{{\mathcal H}}}{f(x)+g(Lx)} \end{equation} and its dual \begin{equation} \label{e:9F842h20b} \minimize{v^*\in\ensuremath{{\mathcal G}}}{f^*(-L^*v^*)+g^*(v^*)}. \end{equation} An essential ingredient in the analysis of such dual problems is the associated Kuhn-Tucker set \cite{Rock74} \begin{equation} \label{e:9F842h20c} \boldsymbol{Z}=\menge{(x,v^*)\in\ensuremath{{\mathcal H}}\oplus\ensuremath{{\mathcal G}}} {-L^*v^*\in\partial f(x)\:\;\text{and}\;Lx\in\partial g^*(v^*)}, \end{equation} which involves the maximally monotone subdifferential operators $\partial f$ and $\partial g^*$. A fruitful generalization of \eqref{e:9F842h20a}--\eqref{e:9F842h20b} is obtained by pairing the inclusion $0\in Ax+L^*BLx$ on $\ensuremath{{\mathcal H}}$ with the dual inclusion $0\in -LA^{-1}(-L^*v^*)+B^{-1}v^*$ on $\ensuremath{{\mathcal G}}$, where $A$ and $B$ are maximally monotone operators acting on $\ensuremath{{\mathcal H}}$ and $\ensuremath{{\mathcal G}}$, respectively. Such operator duality has been studied in \cite{Ecks99,Penn00,Robi99,Robi01} and the first splitting algorithm for solving such composite inclusions was proposed in \cite{Siop11}. The strategy adopted in that paper was to use a standard 2-operator splitting method to construct a point in the Kuhn-Tucker set $\boldsymbol{Z}=\menge{(x,v^*)\in\ensuremath{{\mathcal H}}\oplus\ensuremath{{\mathcal G}}} {-L^*v^*\in Ax\:\;\text{and}\;Lx\in B^{-1}v^*}$ and hence obtain a primal-dual solution (see also \cite{Bot13a,Siop13,Svva12,Opti13,Bang13} for variants of this approach). In \cite{Genn13} we investigated a different strategy based on an idea first proposed in \cite{Svai08} for solving the inclusion $0\in Ax+Bx$. In this framework, at each iteration, one uses points in the graphs of $A$ and $B$ to construct a closed affine half-space of $\ensuremath{{\mathcal H}}\oplus\ensuremath{{\mathcal G}}$ containing $\boldsymbol{Z}$; the primal-dual update is then obtained as the projection of the current iterate onto it. The resulting Fej\'er-monotone algorithm provides only weak convergence to an unspecified Kuhn-Tucker point. In the present paper we propose a strongly convergent modification of these methods for solving the following best approximation problem.
\begin{problem} \label{prob:11} Let $\ensuremath{{\mathcal H}}$ and $\ensuremath{{\mathcal G}}$ be real Hilbert spaces, and set $\ensuremath{\boldsymbol{\mathcal K}}=\ensuremath{{\mathcal H}}\oplus\ensuremath{{\mathcal G}}$. Let $A\colon\ensuremath{{\mathcal H}}\to 2^{\ensuremath{{\mathcal H}}}$ and $B\colon\ensuremath{{\mathcal G}}\to 2^{\ensuremath{{\mathcal G}}}$ be maximally monotone operators, and let $L\colon\ensuremath{{\mathcal H}}\to\ensuremath{{\mathcal G}}$ be a bounded linear operator. Let $(x_0,v_0^*)\in\ensuremath{\boldsymbol{\mathcal K}}$, assume that the inclusion problem \begin{equation} \label{e:primal} \text{find}\;\;x\in\ensuremath{{\mathcal H}}\;\;\text{such that}\;\;0\in Ax+L^*BLx \end{equation} has at least one solution, and consider the dual problem \begin{equation} \label{e:dual} \text{find}\;\;v^*\in\ensuremath{{\mathcal G}}\;\;\text{such that}\;\; 0\in -LA^{-1}(-L^*v^*)+B^{-1}v^*. \end{equation} The problem is to find the best approximation $(\overline{x},\overline{v}^*)$ to $(x_0,v_0^*)$ from the associated Kuhn-Tucker set \begin{equation} \label{e:Aas7B09k21a} \boldsymbol{Z}=\menge{(x,v^*)\in\ensuremath{\boldsymbol{\mathcal K}}} {-L^*v^*\in Ax\:\;\text{and}\;Lx\in B^{-1}v^*}. \end{equation} \end{problem}
The principle of our algorithm goes back to the work of Yves Haugazeau \cite{Haug68} for finding the projection of a point onto the intersection of closed convex sets by means of projections onto the individual sets. Haugazeau's method was generalized in several directions and applied to a variety of problems in nonlinear analysis and optimization in \cite{Sico00}. In \cite{Moor01}, it was formulated as an abstract convergence principle for turning a class of weakly convergent methods into strongly convergent ones (see also \cite{Marq13} for recent related work). In the area of monotone inclusions, Haugazeau-like methods were used in \cite{Solo00} for solving $x\in A^{-1}0$ and in \cite{Moor01} for solving $x\in\bigcap_{i=1}^mA_i^{-1}0$. They were also used in splitting method for solving $0\in Ax+Bx$ as a modification of the forward-backward splitting algorithm in \cite{Hirs05} and \cite[Corollary~29.5]{Livre1}, and as a modification of the Douglas-Rachford algorithm in \cite{Joat06} and \cite{Zhan13}.
The paper is organized as follows. Section~\ref{sec:2} is devoted to a version of an abstract Haugazeau principle. The algorithms for solving Problem~\ref{prob:11} are presented in Section~\ref{sec:3}, where their strong convergence is established. In Section~\ref{sec:4}, we present an extension to systems of coupled monotone inclusions and consider applications to the relaxation of inconsistent common zero problems and to structured multivariate convex minimization problems.
\noindent {\bfseries Notation.} Our notation is standard and follows \cite{Livre1}, where the necessary background on monotone operators and convex analysis is available. The scalar product of a Hilbert space is denoted by
$\scal{\cdot}{\cdot}$ and the associated norm by $\|\cdot\|$. We denote respectively by $\ensuremath{\:\rightharpoonup\:}$ and $\to$ weak and strong convergence, and by $\ensuremath{\operatorname{Id}}\,$ the identity operator. Let $\ensuremath{{\mathcal H}}$ and $\ensuremath{{\mathcal G}}$ be real Hilbert space. The Hilbert direct sum of $\ensuremath{{\mathcal H}}$ and $\ensuremath{{\mathcal G}}$ is denoted by $\ensuremath{{\mathcal H}}\oplus\ensuremath{{\mathcal G}}$, and the power set of $\ensuremath{{\mathcal H}}$ by $2^{\ensuremath{{\mathcal H}}}$. Now let $A\colon\ensuremath{{\mathcal H}}\to 2^{\ensuremath{{\mathcal H}}}$. Then $\ensuremath{\text{\rm ran}\,} A$ is the range $A$, $\ensuremath{\text{\rm gra}\,} A$ the graph of $A$, $A^{-1}$ the inverse of $A$, and $J_A=(\ensuremath{\operatorname{Id}}\,+A)^{-1}$ the resolvent of $A$. The projection operator onto a nonempty closed convex subset $C$ of $\ensuremath{{\mathcal H}}$ is denoted by $P_C$ and $\Gamma_0(\ensuremath{{\mathcal H}})$ is the class of proper lower semicontinuous convex functions from $\ensuremath{{\mathcal H}}$ to $\ensuremath{\left]-\infty,+\infty\right]}$. Let $f\in\Gamma_0(\ensuremath{{\mathcal H}})$. The conjugate of $f$ is $\Gamma_0(\ensuremath{{\mathcal H}})\ni f^*\colon u^*\mapsto \sup_{x\in\ensuremath{{\mathcal H}}}(\scal{x}{u^*}-f(x))$ and the subdifferential of $f$ is $\partial f\colon\ensuremath{{\mathcal H}}\to 2^{\ensuremath{{\mathcal H}}}\colon x\mapsto\menge{u^*\in\ensuremath{{\mathcal H}}} {(\forall y\in\ensuremath{{\mathcal H}})\;\:\scal{y-x}{u^*}+f(x)\ensuremath{\leqslant} f(y)}$.
\section{An abstract Haugazeau algorithm} \label{sec:2}
In \cite[Th\'eor\`eme~3-2]{Haug68} Haugazeau proposed an ingenious method for projecting a point onto the intersection of closed convex sets in a Hilbert space using the projections onto the individual sets. Abstract versions of his method for projecting onto a closed convex set in a real Hilbert space were devised in \cite{Sico00} and \cite{Moor01}. In this section, we present a formulation of this abstract principle which is better suited for our purposes.
Let $\ensuremath{\boldsymbol{\mathcal H}}$ be a real Hilbert space. Given an ordered triplet $(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})\in\ensuremath{\boldsymbol{\mathcal H}}^3$, we define \begin{equation} H(\boldsymbol{x},\boldsymbol{y})=\menge{\boldsymbol{h}\in\ensuremath{\boldsymbol{\mathcal H}}} {\scal{\boldsymbol{h}-\boldsymbol{y}} {\boldsymbol{x}-\boldsymbol{y}}\ensuremath{\leqslant} 0}. \end{equation} Moreover, if $\boldsymbol{R}=H(\boldsymbol{x},\boldsymbol{y})\cap H(\boldsymbol{y},\boldsymbol{z})\neq\ensuremath{{\varnothing}}$, we denote by $Q(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})$ the projection of $\boldsymbol{x}$ onto $\boldsymbol{R}$. The principle of the algorithm to project a point $\boldsymbol{x}_0\in\ensuremath{\boldsymbol{\mathcal H}}$ onto a nonempty closed convex set $\boldsymbol{C}\subset\ensuremath{\boldsymbol{\mathcal H}}$ is to use at iteration $n$ the current iterate $\boldsymbol{x}_n$ to construct an outer approximation to $\boldsymbol{C}$ of the form $H(\boldsymbol{x}_0,\boldsymbol{x}_n)\cap H(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2})$; the update is then computed as the projection of $\boldsymbol{x}_0$ onto it, i.e., $\boldsymbol{x}_{n+1}=Q(\boldsymbol{x}_0,\boldsymbol{x}_n, \boldsymbol{x}_{n+1/2})$.
\begin{proposition} \label{p:j7yG9i-9jmL406} Let $\boldsymbol{C}$ be a nonempty closed convex subset of $\ensuremath{\boldsymbol{\mathcal H}}$ and let $\boldsymbol{x}_0\in\ensuremath{\boldsymbol{\mathcal H}}$. Iterate \begin{equation} \label{e:Aas7B09k20f} \begin{array}{l} \text{for}\;n=0,1,\ldots\\ \left\lfloor \begin{array}{l} \text{take}\;\boldsymbol{x}_{n+1/2}\in\ensuremath{\boldsymbol{\mathcal H}}\;\text{such that}\; \boldsymbol{C}\subset H(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2})\\ \boldsymbol{x}_{n+1}=Q\big(\boldsymbol{x}_0,\boldsymbol{x}_n, \boldsymbol{x}_{n+1/2}\big). \end{array} \right.\\ \end{array} \end{equation} Then the sequence $(\boldsymbol{x}_n)_{n\in\ensuremath{\mathbb N}}$ is well defined and the following hold: \begin{enumerate} \item \label{p:j7yG9i-9jmL406i} $(\forall n\in\ensuremath{\mathbb N})$ $\boldsymbol{C}\subset H(\boldsymbol{x}_0,\boldsymbol{x}_n)\cap H(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2})$. \item \label{p:j7yG9i-9jmL406ii}
$\sum_{n\in\ensuremath{\mathbb N}}\|\boldsymbol{x}_{n+1}-
\boldsymbol{x}_n\|^2<\ensuremath{{+\infty}}$. \item \label{p:j7yG9i-9jmL406iii}
$\sum_{n\in\ensuremath{\mathbb N}}\|\boldsymbol{x}_{n+1/2}-
\boldsymbol{x}_n\|^2<\ensuremath{{+\infty}}$. \item \label{p:j7yG9i-9jmL406iv} Suppose that, for every $\boldsymbol{x}\in\ensuremath{\boldsymbol{\mathcal H}}$ and every strictly increasing sequence $(k_n)_{n\in\ensuremath{\mathbb N}}$ in $\ensuremath{\mathbb N}$, $\boldsymbol{x}_{k_n}\ensuremath{\:\rightharpoonup\:}\boldsymbol{x}$ $\Rightarrow$ $\boldsymbol{x}\in\boldsymbol{C}$. Then $\boldsymbol{x}_n\to P_{\boldsymbol{C}}\boldsymbol{x}_0$. \end{enumerate} \end{proposition} \begin{proof} The proof is similar to those found in \cite[Section~3]{Moor01} and \cite[Section~3]{Sico00}. First, recall that the projector onto a nonempty closed convex subset $\boldsymbol{D}$ of $\ensuremath{\boldsymbol{\mathcal H}}$ is characterized by \cite[Theorem~3.14]{Livre1} \begin{equation} \label{e:our-old-mor2001} (\forall\boldsymbol{x}\in\ensuremath{{\mathcal H}})\quad P_{\boldsymbol{D}}\boldsymbol{x} \in\boldsymbol{D}\quad\text{and}\quad\boldsymbol{D}\subset H(\boldsymbol{x},P_{\boldsymbol{D}}\boldsymbol{x}). \end{equation}
\ref{p:j7yG9i-9jmL406i}: Let $n\in\ensuremath{\mathbb N}$ be such that $\boldsymbol{x}_n$ exists. Since by construction $\boldsymbol{C}\subset H(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2})$, it is enough to show that $\boldsymbol{C}\subset H(\boldsymbol{x}_0,\boldsymbol{x}_n)$. This inclusion is trivially true for $n=0$ since $H(\boldsymbol{x}_0,\boldsymbol{x}_0)=\ensuremath{\boldsymbol{\mathcal H}}$. Furthermore, it follows from \eqref{e:our-old-mor2001} and \eqref{e:Aas7B09k20f} that \begin{eqnarray} \label{e:xenon} \boldsymbol{C}\subset H(\boldsymbol{x}_0,\boldsymbol{x}_n) &\Rightarrow&\boldsymbol{C}\subset H(\boldsymbol{x}_0,\boldsymbol{x}_n)\cap H(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2})\nonumber\\ &\Rightarrow&\boldsymbol{C}\subset H\big(\boldsymbol{x}_0,Q(\boldsymbol{x}_0, \boldsymbol{x}_n,\boldsymbol{x}_{n+1/2})\big)\nonumber\\ &\Leftrightarrow&\boldsymbol{C}\subset H(\boldsymbol{x}_0,\boldsymbol{x}_{n+1}), \end{eqnarray} which establishes the assertion by induction. This also shows that $H(\boldsymbol{x}_0,\boldsymbol{x}_n)\cap H(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2})$ is a nonempty closed convex set and therefore that the projection $\boldsymbol{x}_{n+1}$ of $\boldsymbol{x}_0$ onto it is well defined.
\ref{p:j7yG9i-9jmL406ii}: Let $n\in\ensuremath{\mathbb N}$. By construction, $\boldsymbol{x}_{n+1}=Q(\boldsymbol{x}_0,\boldsymbol{x}_n, \boldsymbol{x}_{n+1/2})\in H(\boldsymbol{x}_0,\boldsymbol{x}_n)\cap H\big(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2}\big)$. Consequently, since $\boldsymbol{x}_n$ is the projection of $\boldsymbol{x}_0$ onto $H(\boldsymbol{x}_0,\boldsymbol{x}_n)$ and $\boldsymbol{x}_{n+1}\in H(\boldsymbol{x}_0,\boldsymbol{x}_n)$,
we have $\|\boldsymbol{x}_0-\boldsymbol{x}_n\|\ensuremath{\leqslant}
\|\boldsymbol{x}_0-\boldsymbol{x}_{n+1}\|$. On the other hand, since $P_{\boldsymbol{C}}\boldsymbol{x}_0\in \boldsymbol{C}\subset H(\boldsymbol{x}_0,\boldsymbol{x}_n)$,
we have $\|\boldsymbol{x}_0-\boldsymbol{x}_n\|\ensuremath{\leqslant}
\|\boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0\|$. It follows that $(\|\boldsymbol{x}_0-\boldsymbol{x}_k\|)_{k\in\ensuremath{\mathbb N}}$ converges and that \begin{equation} \label{e:KKn+=2-07g}
\lim\|\boldsymbol{x}_0-\boldsymbol{x}_k\|\ensuremath{\leqslant}
\|\boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0\|. \end{equation} On the other hand, since $\boldsymbol{x}_{n+1}\in H(\boldsymbol{x}_0,\boldsymbol{x}_n)$, we have \begin{equation}
\|\boldsymbol{x}_{n+1}-\boldsymbol{x}_n\|^2
\ensuremath{\leqslant}\|\boldsymbol{x}_{n+1}-\boldsymbol{x}_n\|^2 +2\scal{\boldsymbol{x}_{n+1}-\boldsymbol{x}_n} {\boldsymbol{x}_n-\boldsymbol{x}_0}
=\|\boldsymbol{x}_0-\boldsymbol{x}_{n+1}\|^2-
\|\boldsymbol{x}_0-\boldsymbol{x}_n\|^2. \end{equation}
Hence, $\sum_{k=1}^n\|\boldsymbol{x}_{k+1}-\boldsymbol{x}_k\|^2
\ensuremath{\leqslant}\|\boldsymbol{x}_0-\boldsymbol{x}_{n+1}\|^2\ensuremath{\leqslant}
\|\boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0\|^2$ and, in turn,
$\sum_{k\in\ensuremath{\mathbb N}}\|\boldsymbol{x}_{k+1}-\boldsymbol{x}_k\|^2<\ensuremath{{+\infty}}$.
\ref{p:j7yG9i-9jmL406iii}: For every $n\in\ensuremath{\mathbb N}$, we derive from the inclusion $\boldsymbol{x}_{n+1}\in H(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2})$ that \begin{align} \label{e:KKn+=2-07m}
\|\boldsymbol{x}_{n+1/2}-\boldsymbol{x}_n\|^2
&\ensuremath{\leqslant}\|\boldsymbol{x}_{n+1}-\boldsymbol{x}_{n+1/2}\|^2+
\|\boldsymbol{x}_n-\boldsymbol{x}_{n+1/2}\|^2\nonumber\\
&\ensuremath{\leqslant}\|\boldsymbol{x}_{n+1}-\boldsymbol{x}_{n+1/2}\|^2+ 2\scal{\boldsymbol{x}_{n+1}-\boldsymbol{x}_{n+1/2}} {\boldsymbol{x}_{n+1/2}-\boldsymbol{x}_n}
+\|\boldsymbol{x}_n-\boldsymbol{x}_{n+1/2}\|^2\nonumber\\
&=\|\boldsymbol{x}_{n+1}-\boldsymbol{x}_n\|^2. \end{align} Hence, it follows from \ref{p:j7yG9i-9jmL406ii} that
$\sum_{n\in\ensuremath{\mathbb N}}\|\boldsymbol{x}_{n+1/2}-\boldsymbol{x}_n\|^2<\ensuremath{{+\infty}}$.
\ref{p:j7yG9i-9jmL406iv}: Let us note that \eqref{e:KKn+=2-07g} implies that $(\boldsymbol{x}_n)_{n\in\ensuremath{\mathbb N}}$ is bounded. Now, let $\boldsymbol{x}$ be a weak sequential cluster point of $(\boldsymbol{x}_n)_{n\in\ensuremath{\mathbb N}}$, say $\boldsymbol{x}_{k_n}\ensuremath{\:\rightharpoonup\:} \boldsymbol{x}$. Then, by weak lower semicontinuity of
$\|\cdot\|$ \cite[Lemma~2.35]{Livre1} and \eqref{e:KKn+=2-07g}
$\|\boldsymbol{x}_0-\boldsymbol{x}\|\ensuremath{\leqslant}\varliminf
\|\boldsymbol{x}_0-\boldsymbol{x}_{k_n}\|\ensuremath{\leqslant}
\|\boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0\|= \inf_{\boldsymbol{y}\in\boldsymbol{C}}
\|\boldsymbol{x}_0-\boldsymbol{y}\|$. Hence, since $\boldsymbol{x}\in\boldsymbol{C}$, $\boldsymbol{x}=P_{\boldsymbol{C}}\boldsymbol{x}_0$ is the only weak sequential cluster point of the sequence $(\boldsymbol{x}_n)_{n\in\ensuremath{\mathbb N}}$ and it follows from \cite[Lemma~2.38]{Livre1} that $\boldsymbol{x}_n\ensuremath{\:\rightharpoonup\:} P_{\boldsymbol{C}}\boldsymbol{x}_0$. In turn \eqref{e:KKn+=2-07g} yields
$\|\boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0\|\ensuremath{\leqslant}
\varliminf\|\boldsymbol{x}_0-\boldsymbol{x}_n\|=
\lim\|\boldsymbol{x}_0-\boldsymbol{x}_n\|
\ensuremath{\leqslant}\|\boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0\|$. Thus, $\boldsymbol{x}_0-\boldsymbol{x}_n\ensuremath{\:\rightharpoonup\:} \boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0$ and
$\|\boldsymbol{x}_0-\boldsymbol{x}_n\|\to
\|\boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0\|$. We therefore derive from \cite[Lemma~2.41(i)]{Livre1} that $\boldsymbol{x}_0-\boldsymbol{x}_n\to \boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0$, i.e., $\boldsymbol{x}_n\to P_{\boldsymbol{C}}\boldsymbol{x}_0$. \end{proof}
\begin{remark} \label{r:9i-9jmL428} Suppose that, for some $n\in\ensuremath{\mathbb N}$, $\boldsymbol{x}_n\in \boldsymbol{C}$ in \eqref{e:Aas7B09k20f}. Then
$\|\boldsymbol{x}_0-P_{\boldsymbol{C}}\boldsymbol{x}_0\|\ensuremath{\leqslant}
\|\boldsymbol{x}_0-\boldsymbol{x}_n\|$ and, since we always have
$\|\boldsymbol{x}_0-\boldsymbol{x}_n\|\ensuremath{\leqslant}\|\boldsymbol{x}_0-
P_{\boldsymbol{C}}\boldsymbol{x}_0\|$, we conclude that $\boldsymbol{x}_n=P_{\boldsymbol{C}}\boldsymbol{x}_0$ and that the iterations can be stopped. \end{remark}
Algorithm \eqref{e:Aas7B09k20f} can easily be implemented thanks to the following lemma.
\begin{lemma} \label{l:haugazeauy} Let $(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})\in\ensuremath{\boldsymbol{\mathcal H}}^3$ and set $\boldsymbol{R}=H(\boldsymbol{x},\boldsymbol{y})\cap H(\boldsymbol{y},\boldsymbol{z})$. Moreover, set $\chi=\scal{\boldsymbol{x}-\boldsymbol{y}}{\boldsymbol{y}-
\boldsymbol{z}}$, $\mu=\|\boldsymbol{x}-\boldsymbol{y}\|^2$,
$\nu=\|\boldsymbol{y}-\boldsymbol{z}\|^2$, and $\rho=\mu\nu-\chi^2$. Then exactly one of the following holds: \begin{enumerate} \item \label{c:haugazeaui} $\rho=0$ and $\chi<0$, in which case $\boldsymbol{R}=\ensuremath{{\varnothing}}$. \item \label{c:haugazeauii} \emph{[}$\,\rho=0$ and $\chi\ensuremath{\geqslant} 0\,$\emph{]} or $\rho>0$, in which case $\boldsymbol{R}\neq\ensuremath{{\varnothing}}$ and \begin{equation} \label{e:j7yG9i-9jmL405} Q(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})= \begin{cases} \boldsymbol{z}, &\!\text{if}\;\rho=0\;\text{and}\; \chi\ensuremath{\geqslant} 0;\\[+0mm] \displaystyle \boldsymbol{x}+(1+\chi/\nu) (\boldsymbol{z}-\boldsymbol{y}), &\!\text{if}\;\rho>0\;\text{and}\; \chi\nu\ensuremath{\geqslant}\rho;\\ \displaystyle \boldsymbol{y}+(\nu/\rho) \big(\chi(\boldsymbol{x}-\boldsymbol{y}) +\mu(\boldsymbol{z}-\boldsymbol{y})\big), &\!\text{if}\;\rho>0\;\text{and}\;\chi\nu<\rho. \end{cases} \end{equation} \end{enumerate} \end{lemma} \begin{proof} See \cite[Th\'eor\`eme~3-1]{Haug68} for the original proof and \cite[Corollary~28.21]{Livre1} for an alternate derivation. \end{proof}
\section{Main result} \label{sec:3}
In this section, we devise a strongly convergent algorithm for solving Problem~\ref{prob:11} by coupling Proposition~\ref{p:j7yG9i-9jmL406} with the construction of \cite{Genn13} to determine the half-spaces $(H(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2}))_{n\in\ensuremath{\mathbb N}}$. First, we need a couple of facts.
\begin{proposition}{\rm \cite[Proposition~2.8]{Siop11}} \label{p:0kkhUj710-31z} In the setting of Problem~\ref{prob:11}, $\boldsymbol{Z}$ is a nonempty closed convex set and, if $(x,v^*)\in\boldsymbol{Z}$, then $x$ solves \eqref{e:primal} and $v^*$ solves \eqref{e:dual}. \end{proposition}
\begin{proposition}{\rm \cite[Proposition~2.5]{Genn13}} \label{p:genna3Hbl-915} In the setting of Problem~\ref{prob:11}, let $(a_n,a_n^*)_{n\in\ensuremath{\mathbb N}}$ be a sequence in $\ensuremath{\text{\rm gra}\,} A$, let $(b_n,b_n^*)_{n\in\ensuremath{\mathbb N}}$ be a sequence in $\ensuremath{\text{\rm gra}\,} B$, and let $(x,v^*)\in\ensuremath{\boldsymbol{\mathcal K}}$. Suppose that $a_n\ensuremath{\:\rightharpoonup\:}{x}$, $b^*_n\ensuremath{\:\rightharpoonup\:}{v}^*$, $a^*_n+L^*b^*_n\to 0$, and $La_n-b_n\to 0$. Then $\scal{a_n}{a_n^*}+\scal{b_n}{b_n^*}\to 0$ and $(x,v^*)\in\boldsymbol{Z}$. \end{proposition}
The next result features our general algorithm for solving Problem~\ref{prob:11}.
\begin{theorem} \label{t:9i-9jmL402} Consider the setting of Problem~\ref{prob:11}. Let $\varepsilon\in\ensuremath{\left]0,1\right[}$, let $\alpha\in\ensuremath{\left]0,+\infty\right[}$, and set, for every $(x,v^*)\in\ensuremath{\boldsymbol{\mathcal K}}$, \begin{multline} \label{e:9g45g2h29a} \boldsymbol{G}_\alpha(x,v^*)=
\Big\{(a,b,a^*,b^*)\in\ensuremath{\boldsymbol{\mathcal K}}\times\ensuremath{\boldsymbol{\mathcal K}}\;\big |\; (a,a^*)\in\ensuremath{\text{\rm gra}\,} A,\;(b,b^*)\in\ensuremath{\text{\rm gra}\,} B,\;\text{and}\\ \scal{x-a}{a^*+L^*v^*}+\scal{Lx-b}{b^*-v^*}
\ensuremath{\geqslant}\alpha\big(\|a^*+L^*b^*\|^2+\|La-b\|^2\big)\Big\}. \end{multline} Iterate \begin{equation} \label{e:j7yG9i-9jmL409h} \begin{array}{l} \text{for}\;n=0,1,\ldots\\ \left\lfloor \begin{array}{l} (a_n,b_n,a_n^*,b_n^*)\in\boldsymbol{G}_\alpha(x_n,v_n^*)\\ s^*_n=a^*_n+L^*b^*_n\\ t_n=b_n-La_n\\
\tau_n=\|s_n^*\|^2+\|t_n\|^2\\ \text{if}\;\tau_n=0\\ \left\lfloor \begin{array}{l} \theta_n=0\\ \end{array} \right.\\ \text{if}\;\tau_n>0\\ \left\lfloor \begin{array}{l} \lambda_n\in\left[\varepsilon,1\right]\\ \theta_n=\lambda_n\big(\scal{x_n}{s^*_n}+\scal{t_n}{v^*_n} -\scal{a_n}{a^*_n}-\scal{b_n}{b^*_n}\big)/\tau_n\\ \end{array} \right.\\ x_{n+1/2}=x_n-\theta_n s^*_n\\ v^*_{n+1/2}=v^*_n-\theta_n t_n\\ \chi_n=\scal{x_0-x_n}{x_n-x_{n+1/2}} +\scal{v_0^*-v_n^*}{v_n^*-v_{n+1/2}^*}\\
\mu_n=\|x_0-x_n\|^2+\|v_0^*-v_n^*\|^2\\
\nu_n=\|x_n-x_{n+1/2}\|^2+\|v_n^*-v_{n+1/2}^*\|^2\\ \rho_n=\mu_n\nu_n-\chi_n^2\\ \text{if}\;\rho_n=0\;\text{and}\;\chi_n\ensuremath{\geqslant} 0\\ \left\lfloor \begin{array}{l} x_{n+1}=x_{n+1/2}\\ v^*_{n+1}=v_{n+1/2}^* \end{array} \right.\\ \text{if}\;\rho_n>0\;\text{and}\;\chi_n\nu_n\ensuremath{\geqslant}\rho_n\\ \left\lfloor \begin{array}{l} x_{n+1}=x_0+(1+\chi_n/\nu_n)(x_{n+1/2}-x_n)\\ v^*_{n+1}=v_0^*+(1+\chi_n/\nu_n)(v_{n+1/2}^*-v_n^*) \end{array} \right.\\ \text{if}\;\rho_n>0\;\text{and}\;\chi_n\nu_n<\rho_n\\ \left\lfloor \begin{array}{l} x_{n+1}=x_n+(\nu_n/\rho_n)\big(\chi_n(x_0-x_n) +\mu_n(x_{n+1/2}-x_n)\big)\\ v^*_{n+1}=v_n^*+(\nu_n/\rho_n)\big(\chi_n(v_0^*-v_n^*) +\mu_n(v_{n+1/2}^*-v_n^*)\big). \end{array} \right.\\ \end{array} \right.\\ \end{array} \end{equation} Then \eqref{e:j7yG9i-9jmL409h} generates infinite sequences $(x_n)_{n\in\ensuremath{\mathbb N}}$ and $(v_n^*)_{n\in\ensuremath{\mathbb N}}$, and the following hold: \begin{enumerate} \item \label{t:9i-9jmL402i}
$\sum_{n\in\ensuremath{\mathbb N}}\|x_{n+1}-x_n\|^2<\ensuremath{{+\infty}}$ and
$\sum_{n\in\ensuremath{\mathbb N}}\|v^*_{n+1}-v^*_n\|^2<\ensuremath{{+\infty}}$. \item \label{t:9i-9jmL402ii}
$\sum_{n\in\ensuremath{\mathbb N}}\|s^*_n\|^2<\ensuremath{{+\infty}}$ and
$\sum_{n\in\ensuremath{\mathbb N}}\|t_n\|^2<\ensuremath{{+\infty}}$. \item \label{t:9i-9jmL402iii} Suppose that $x_n-a_n\ensuremath{\:\rightharpoonup\:} 0$ and $v_n^*-b_n^*\ensuremath{\:\rightharpoonup\:} 0$. Then $x_n\to\overline{x}$ and $v_n^*\to\overline{v}^*$. \end{enumerate} \end{theorem} \begin{proof} We are going to show that the claims follow from Proposition~\ref{p:j7yG9i-9jmL406} applied in $\ensuremath{\boldsymbol{\mathcal K}}$ to the set $\boldsymbol{Z}$ of \eqref{e:Aas7B09k21a}, which is nonempty, closed, and convex by Proposition~\ref{p:0kkhUj710-31z}. First, let us set \begin{equation} \label{e:j7yG9i-9jmL411b} (\forall n\in\ensuremath{\mathbb N})\quad \boldsymbol{x}_n=(x_n,v_n^*)\quad\text{and}\quad \boldsymbol{x}_{n+1/2}=(x_{n+1/2},v_{n+1/2}^*). \end{equation} We deduce from \eqref{e:j7yG9i-9jmL409h} that \begin{align} \label{e:j7yG9i-9jmL410x} &\hskip -4mm (\forall (x,v^*)\in\ensuremath{\boldsymbol{\mathcal K}})(\forall n\in\ensuremath{\mathbb N})\quad \scal{x}{s^*_n}+\scal{t_n}{v^*}-\scal{a_n}{a^*_n}- \scal{b_n}{b^*_n}\nonumber\\ &\hskip 44mm =\scal{x}{a^*_n+L^*b^*_n}+\scal{b_n-La_n}{v^*}- \scal{a_n}{a^*_n}-\scal{b_n}{b^*_n}\nonumber\\ &\hskip 44mm =\scal{x-a_n}{a_n^*+L^*v^*}+ \scal{Lx-b_n}{b_n^*-v^*}. \end{align} Next, let us show that \begin{equation} \label{e:j7yG9i-9jmL411u} (\forall n\in\ensuremath{\mathbb N})\quad\boldsymbol{Z}\subset H\big(\boldsymbol{x}_n,\boldsymbol{x}_{n+1/2}\big). \end{equation} To this end, let $\boldsymbol{z}=(x,v^*)\in\boldsymbol{Z}$ and let $n\in\ensuremath{\mathbb N}$. We must show that $\scal{\boldsymbol{z}-\boldsymbol{x}_{n+1/2}}{\boldsymbol{x}_{n}- \boldsymbol{x}_{n+1/2}}\ensuremath{\leqslant} 0$. If $\tau_n=0$, then $\boldsymbol{x}_{n+1/2}=\boldsymbol{x}_{n}$ and the inequality is trivially satisfied. Now suppose that $\tau_n>0$. Then \eqref{e:j7yG9i-9jmL410x} and \eqref{e:9g45g2h29a} yield \begin{align} \label{e:9i-9jmL417} \theta_n &=\lambda_n\frac{\scal{x_n}{s^*_n}+\scal{t_n}{v^*_n} -\scal{a_n}{a^*_n}-\scal{b_n}{b^*_n}}{\tau_n}\nonumber\\ &=\lambda_n\frac{\scal{x_n-a_n}{a_n^*+L^*v_n^*}+ \scal{Lx_n-b_n}{b_n^*-v_n^*}}{\tau_n}\nonumber\\ &\ensuremath{\geqslant}\varepsilon\alpha\nonumber\\ &>0. \end{align} On the other hand, it follows from \eqref{e:j7yG9i-9jmL409h} and \eqref{e:Aas7B09k21a} that $a_n^*\in Aa_n$ and $-L^*v^*\in Ax$. Hence, since $A$ is monotone, $\scal{x-a_n}{a_n^*+L^*v^*}\ensuremath{\leqslant} 0$. Similarly, since $v^*\in B(Lx)$ and $b_n^*\in Bb_n$, the monotonicity of $B$ implies that $\scal{Lx-b_n}{b_n^*-v^*}\ensuremath{\leqslant} 0$. Consequently, we derive from \eqref{e:j7yG9i-9jmL409h}, \eqref{e:j7yG9i-9jmL410x}, and \eqref{e:9g45g2h29a} that \begin{align} \label{e:9g45g2h30} &\hskip -6mm \scal{\boldsymbol{z}-\boldsymbol{x}_{n+1/2}}{\boldsymbol{x}_{n}- \boldsymbol{x}_{n+1/2}}/\theta_n\nonumber\\ &=\scal{\boldsymbol{z}}{\boldsymbol{x}_n-\boldsymbol{x}_{n+1/2}}/ \theta_n+\scal{\boldsymbol{x}_{n+1/2}} {\boldsymbol{x}_{n+1/2}-\boldsymbol{x}_n}/\theta_n\nonumber\\ &=\scal{x}{x_n-x_{n+1/2}}/\theta_n+ \scal{v^*}{v_n^*-v_{n+1/2}^*}/\theta_n\nonumber\\ &\quad\;+\scal{x_{n+1/2}}{x_{n+1/2}-x_n}/\theta_n+ \scal{v^*_{n+1/2}}{v^*_{n+1/2}-v^*_n}/\theta_n\nonumber\\ &=\scal{x}{s_n^*}+\scal{t_n}{v^*}-\scal{x_n}{s_n^*}-
\scal{t_n}{v^*_n}+\theta_n\big(\|s_n^*\|^2+\|t_n\|^2\big) \nonumber\\ &=\scal{x}{s_n^*}+\scal{t_n}{v^*} -\scal{x_n}{s_n^*}-\scal{t_n}{v^*_n}+ \lambda_n\big(\scal{x_n}{s^*_n}+\scal{t_n}{v^*_n} -\scal{a_n}{a^*_n}-\scal{b_n}{b^*_n}\big) \nonumber\\ &=\scal{x}{s_n^*}+\scal{t_n}{v^*} -\scal{a_n}{a_n^*}-\scal{b_n}{b_n^*}\nonumber\\ &\quad\;-(1-\lambda_n)\big(\scal{x_n}{s^*_n}+\scal{t_n}{v^*_n} -\scal{a_n}{a_n^*}-\scal{b_n}{b_n^*}\big)\nonumber\\ &=\scal{x-a_n}{a_n^*+L^*v^*}+\scal{Lx-b_n}{b_n^*-v^*}\nonumber\\ &\quad\;-(1-\lambda_n)\big(\scal{x_n-a_n}{a_n^*+L^*v_n^*}+ \scal{Lx_n-b_n}{b_n^*-v_n^*}\big)\nonumber\\ &\ensuremath{\leqslant}\scal{x-a_n}{a_n^*+L^*v^*}+\scal{Lx-b_n}{b_n^*-v^*}
-\alpha(1-\lambda_n)\big(\|a_n^*+L^*b_n^*\|^2+
\|La_n-b_n\|^2\big)\nonumber\\ &\ensuremath{\leqslant}\scal{x-a_n}{a_n^*+L^*v^*}+\scal{Lx-b_n}{b_n^*-v^*}\nonumber\\ &\ensuremath{\leqslant} 0. \end{align} This verifies \eqref{e:j7yG9i-9jmL411u}. It therefore follows from \eqref{e:j7yG9i-9jmL405} that \eqref{e:j7yG9i-9jmL409h} is an instance of \eqref{e:Aas7B09k20f}.
\ref{t:9i-9jmL402i}: It follows from \eqref{e:j7yG9i-9jmL411b} and Proposition~\ref{p:j7yG9i-9jmL406}\ref{p:j7yG9i-9jmL406ii}
that $\sum_{n\in\ensuremath{\mathbb N}}\|x_{n+1}-x_n\|^2+
\sum_{n\in\ensuremath{\mathbb N}}\|v^*_{n+1}-v^*_n\|^2=
\sum_{n\in\ensuremath{\mathbb N}}\|\boldsymbol{x}_{n+1}-
\boldsymbol{x}_n\|^2<\ensuremath{{+\infty}}$.
\ref{t:9i-9jmL402ii}: Let $n\in\ensuremath{\mathbb N}$. We consider two cases. \begin{itemize} \item $\tau_n=0$: Then \eqref{e:j7yG9i-9jmL409h} yields
$\|s^*_n\|^2+\|t_n\|^2=0=\|\boldsymbol{x}_{n+1/2}-
\boldsymbol{x}_n\|^2/(\alpha\varepsilon)^2$. \item $\tau_n>0$: Then it follows from \eqref{e:9g45g2h29a} and \eqref{e:j7yG9i-9jmL409h} that \begin{align} \label{e:j7yG9i-9jmL409s}
\|s^*_n\|^2+\|t_n\|^2 &=\tau_n\nonumber\\ &\ensuremath{\leqslant}\frac{\big(\scal{x_n-a_n}{a_n^*+L^*v_n^*}+ \scal{Lx_n-b_n}{b_n^*-v_n^*}\big)^2}{\alpha^2\tau_n}\nonumber\\ &=\frac{\big(\scal{x_n}{s^*_n}+\scal{t_n}{v^*_n}-\scal{a_n}{a^*_n} -\scal{b_n}{b^*_n}\big)^2}{\alpha^2\tau_n}\nonumber\\ &\ensuremath{\leqslant}\frac{\lambda_n^2 \big(\scal{x_n}{s^*_n}+\scal{t_n}{v^*_n}-\scal{a_n}{a^*_n}- \scal{b_n}{b^*_n}\big)^2}{\alpha^2\varepsilon^2\tau_n}\nonumber\\ &=\frac{\theta_n^2\tau_n}{\alpha^2\varepsilon^2}\nonumber\\
&=\frac{\|x_{n+1/2}-x_n\|^2+\|v^*_{n+1/2}-v^*_n\|^2} {\alpha^2\varepsilon^2}\nonumber\\
&=\frac{\|\boldsymbol{x}_{n+1/2}-\boldsymbol{x}_n\|^2} {\alpha^2\varepsilon^2}. \end{align} \end{itemize} Altogether, it follows from Proposition~\ref{p:j7yG9i-9jmL406}\ref{p:j7yG9i-9jmL406iii}
that $\sum_{n\in\ensuremath{\mathbb N}}\|s^*_n\|^2+\sum_{n\in\ensuremath{\mathbb N}}\|t_n\|^2<\ensuremath{{+\infty}}$.
\ref{t:9i-9jmL402iii}: Take $x\in\ensuremath{{\mathcal H}}$, $v^*\in\ensuremath{{\mathcal G}}$, and a strictly increasing sequence $(k_n)_{n\in\ensuremath{\mathbb N}}$ in $\ensuremath{\mathbb N}$, such that $x_{k_n}\ensuremath{\:\rightharpoonup\:} x$ and $v^*_{k_n}\ensuremath{\:\rightharpoonup\:} v^*$. We derive from \ref{t:9i-9jmL402ii} and \eqref{e:j7yG9i-9jmL409h} that $a^*_n+L^*b^*_n\to 0$ and $La_n-b_n\to 0$. Hence, the assumptions yield \begin{equation} \label{e:j7yG9i-9jmL411a} a_{k_n}\ensuremath{\:\rightharpoonup\:}{x},\quad b^*_{k_n}\ensuremath{\:\rightharpoonup\:}{v^*},\quad a^*_{k_n}+L^*b^*_{k_n}\to 0,\quad\text{and}\quad La_{k_n}-b_{k_n}\to 0. \end{equation} On the other hand, \eqref{e:9g45g2h29a} also asserts that $(\forall n\in\ensuremath{\mathbb N})$ $(a_n,a_n^*)\in\ensuremath{\text{\rm gra}\,} A$ and $(b_n,b_n^*)\in\ensuremath{\text{\rm gra}\,} B$. Altogether, Proposition~\ref{p:genna3Hbl-915} implies that $(x,v^*)\in\boldsymbol{Z}$. In view of Proposition~\ref{p:j7yG9i-9jmL406}\ref{p:j7yG9i-9jmL406iv}, the proof is complete. \end{proof}
\begin{remark} \label{r:9i-9jmL419} Here are a few observations pertaining to Theorem~\ref{t:9i-9jmL402}. \begin{enumerate} \item These results appear to provide the first algorithmic framework for composite inclusions problems that does not require additional assumptions on the constituents of the problem to achieve strong convergence. \item If the second half of \eqref{e:j7yG9i-9jmL409h} is by-passed, i.e., if we set $x_{n+1}=x_{n+1/2}$ and $v^*_{n+1}=v^*_{n+1/2}$, and if the relaxation parameter $\lambda_n$ is chosen in the range $[\varepsilon,2-\varepsilon]$, one recovers the algorithm of \cite[Corollary~3.3]{Genn13}. However, this algorithm provides only weak convergence to an unspecified Kuhn-Tucker point, whereas \eqref{e:j7yG9i-9jmL409h} guarantees strong convergence to the best Kuhn-Tucker approximation to $(x_0,v_0^*)$. This can be viewed as another manifestation of the weak-to-strong convergence principle investigated in \cite{Moor01} in a different setting ($\ensuremath{{\mathfrak{T}}}\,$-class operators). \end{enumerate} \end{remark}
The following proposition is an application of Theorem~\ref{t:9i-9jmL402} which describes a concrete implementation of \eqref{e:j7yG9i-9jmL409h} with a specific rule for selecting $(a_n,b_n,a_n^*,b_n^*)\in\boldsymbol{G}_\alpha(x_n,v_n^*)$.
\begin{proposition} \label{p:j7yG9i-9jmL407} Consider the setting of Problem~\ref{prob:11}. Let $\varepsilon\in\ensuremath{\left]0,1\right[}$ and iterate \begin{equation} \label{e:9i-9jmL403a} \begin{array}{l} \text{for}\;n=0,1,\ldots\\ \left\lfloor \begin{array}{l} (\gamma_n,\mu_n)\in [\varepsilon,1/\varepsilon]^2\\ a_n=J_{\gamma_n A}(x_n-\gamma_n L^*v_n^*)\\ l_n=Lx_n\\ b_n=J_{\mu_n B}(l_n+\mu_n v_n^*)\\ s^*_n=\gamma_n^{-1}(x_n-a_n)+\mu_n^{-1}L^*(l_n-b_n)\\ t_n=b_n-La_n\\
\tau_n=\|s_n^*\|^2+\|t_n\|^2\\ \text{if}\;\tau_n=0\\ \left\lfloor \begin{array}{l} \theta_n=0\\ \end{array} \right.\\ \text{if}\;\tau_n>0\\ \left\lfloor \begin{array}{l} \lambda_n\in\left[\varepsilon,1\right]\\
\theta_n=\lambda_n\big(\gamma_n^{-1}\|x_n-a_n\|^2+\mu_n^{-1}
\|l_n-b_n\|^2\big)/\tau_n\\ \end{array} \right.\\ x_{n+1/2}=x_n-\theta_n s^*_n\\ v^*_{n+1/2}=v^*_n-\theta_n t_n\\ \chi_n=\scal{x_0-x_n}{x_n-x_{n+1/2}} +\scal{v_0^*-v_n^*}{v_n^*-v_{n+1/2}^*}\\
\mu_n=\|x_0-x_n\|^2+\|v_0^*-v_n^*\|^2\\
\nu_n=\|x_n-x_{n+1/2}\|^2+\|v_n^*-v_{n+1/2}^*\|^2\\ \rho_n=\mu_n\nu_n-\chi_n^2\\ \text{if}\;\rho_n=0\;\text{and}\;\chi_n\ensuremath{\geqslant} 0\\ \left\lfloor \begin{array}{l} x_{n+1}=x_{n+1/2}\\ v^*_{n+1}=v_{n+1/2}^* \end{array} \right.\\ \text{if}\;\rho_n>0\;\text{and}\;\chi_n\nu_n\ensuremath{\geqslant}\rho_n\\ \left\lfloor \begin{array}{l} x_{n+1}=x_0+(1+\chi_n/\nu_n)(x_{n+1/2}-x_n)\\ v^*_{n+1}=v_0^*+(1+\chi_n/\nu_n)(v_{n+1/2}^*-v_n^*) \end{array} \right.\\ \text{if}\;\rho_n>0\;\text{and}\;\chi_n\nu_n<\rho_n\\ \left\lfloor \begin{array}{l} x_{n+1}=x_n+(\nu_n/\rho_n)\big(\chi_n(x_0-x_n) +\mu_n(x_{n+1/2}-x_n)\big)\\ v^*_{n+1}=v_n^*+(\nu_n/\rho_n)\big(\chi_n(v_0^*-v_n^*) +\mu_n(v_{n+1/2}^*-v_n^*)\big). \end{array} \right.\\ \end{array} \right.\\ \end{array} \end{equation} Then \eqref{e:9i-9jmL403a} generates infinite sequences $(x_n)_{n\in\ensuremath{\mathbb N}}$ and $(v_n^*)_{n\in\ensuremath{\mathbb N}}$, and the following hold: \begin{enumerate} \item \label{p:j7yG9i-9jmL407i}
$\sum_{n\in\ensuremath{\mathbb N}}\|x_{n+1}-x_n\|^2<\ensuremath{{+\infty}}$ and
$\sum_{n\in\ensuremath{\mathbb N}}\|v^*_{n+1}-v^*_n\|^2<\ensuremath{{+\infty}}$. \item \label{p:j7yG9i-9jmL407ii}
$\sum_{n\in\ensuremath{\mathbb N}}\|s^*_n\|^2<\ensuremath{{+\infty}}$ and
$\sum_{n\in\ensuremath{\mathbb N}}\|t_n\|^2<\ensuremath{{+\infty}}$. \item \label{p:j7yG9i-9jmL407iii}
$\sum_{n\in\ensuremath{\mathbb N}}\|x_n-a_n\|^2<\ensuremath{{+\infty}}$ and
$\sum_{n\in\ensuremath{\mathbb N}}\|Lx_n-b_n\|^2<\ensuremath{{+\infty}}$. \item \label{p:j7yG9i-9jmL407iv} $x_n\to\overline{x}$ and $v_n^*\to\overline{v}^*$. \end{enumerate} \end{proposition} \begin{proof} Let us define \begin{equation} \label{e:j7yG69gjn.-19o}
\alpha=\frac{\varepsilon}{1+\|L\|^2+2(1-\varepsilon^{2})
\text{max}\big\{1,\|L\|^2\big\}} \end{equation} and \begin{equation} \label{e:j7yG69gjn.-19r} (\forall n\in\ensuremath{\mathbb N})\quad a^*_n=\gamma_n^{-1}(x_n-a_n)-L^*v_n^*\quad\text{and}\quad b^*_n=\mu_n^{-1}(Lx_n-b_n)+v_n^*. \end{equation} Then it is shown in \cite[proof of Proposition~3.5]{Genn13} that \begin{equation} \label{e:9i-9jmL419e} (\forall n\in\ensuremath{\mathbb N})\quad (a_n,b_n,a_n^*,b_n^*)\in\boldsymbol{G}_\alpha(x_n,v_n^*) \end{equation} and \begin{equation} \label{e:9i-9jmL419f}
(\forall n\in\ensuremath{\mathbb N})\quad\|x_n-a_n\|^2\ensuremath{\leqslant} 2\varepsilon^{-2}
\big(\|s^*_n\|^2+\varepsilon^{-2}\|L\|^2\,\|t_n\|^2\big). \end{equation} We deduce from \eqref{e:j7yG69gjn.-19r} and \eqref{e:9i-9jmL419e} that \eqref{e:9i-9jmL403a} is a special case of \eqref{e:j7yG9i-9jmL409h}. Consequently, assertions \ref{p:j7yG9i-9jmL407i} and \ref{p:j7yG9i-9jmL407ii} follow from their counterparts in Theorem~\ref{t:9i-9jmL402}. To show \ref{p:j7yG9i-9jmL407iii} it suffices to note that \eqref{e:9i-9jmL419f} and \ref{p:j7yG9i-9jmL407ii} imply that \begin{equation} \label{e:293}
\sum_{n\in\ensuremath{\mathbb N}}\|x_n-a_n\|^2<\ensuremath{{+\infty}} \end{equation} and hence that
$\sum_{n\in\ensuremath{\mathbb N}}\|Lx_n-b_n\|^2<\ensuremath{{+\infty}}$ since\begin{equation} \label{e:69gjn.-08g} (\forall n\in\ensuremath{\mathbb N})\quad
\|Lx_n-b_n\|^2=\|L(x_n-a_n)+La_n-b_n\|^2\ensuremath{\leqslant}
2\big(\|L\|^2\,\|x_n-a_n\|^2+\|t_n\|^2\big). \end{equation} In turn, \eqref{e:j7yG69gjn.-19r} yields \begin{equation} \label{e:294}
\sum_{n\in\ensuremath{\mathbb N}}\|v_n^*-b_n^*\|^2
=\sum_{n\in\ensuremath{\mathbb N}}\mu_n^{-2}\|Lx_n-b_n\|^2
\ensuremath{\leqslant}\varepsilon^{-2}\sum_{n\in\ensuremath{\mathbb N}}\|Lx_n-b_n\|^2<\ensuremath{{+\infty}}. \end{equation} Altogether, \ref{p:j7yG9i-9jmL407iv} follows from \eqref{e:293}, \eqref{e:294}, and Theorem~\ref{t:9i-9jmL402}\ref{t:9i-9jmL402iii}. \end{proof}
\begin{remark} \label{r:9i-9jmL429} In \eqref{e:9i-9jmL403a}, the identity $\tau_n=0$ can be used as a stopping rule. Indeed, $\tau_n=0$ $\Leftrightarrow$ $(a_n^*+L^*b_n^*,b_n-La_n)=(0,0)$ $\Leftrightarrow$ $(-L^*b_n^*,La_n)=(a_n^*,b_n)\in Aa_n\times B^{-1}b_n^*$ $\Leftrightarrow$ $(a_n,b_n^*)\in\boldsymbol{Z}$. On the other hand, it follows from \eqref{e:9i-9jmL419f} and \eqref{e:69gjn.-08g} that $\tau_n=0$ $\Rightarrow$ $(x_n,v_n^*)=(a_n,b_n^*)$. Altogether, Remark~\ref{r:9i-9jmL428} yields $(x_n,v_n^*)=P_{\boldsymbol{Z}}(x_0,v_0^*)= (\overline{x},\overline{v}^*)$. \end{remark}
\begin{remark} An important feature of algorithm \eqref{e:9i-9jmL403a} which is inherited from that of \cite[Proposition~3.5]{Genn13} is
that it does not require the knowledge of $\|L\|$ or necessitate potentially hard to implement inversions of linear operators. \end{remark}
\section{Application to systems of monotone inclusions} \label{sec:4}
As discussed in \cite{Genn13,Sico10,Atto11,Bot13b,Bric13,Siop13,Juan12}, various problems in applied mathematics can be modeled by systems of coupled monotone inclusions. In this section, we consider the following setting.
\begin{problem} \label{prob:12} Let $m$ and $K$ be strictly positive integers, let $(\ensuremath{{\mathcal H}}_i)_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}$ and $(\ensuremath{{\mathcal G}}_k)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$ be real Hilbert spaces, and set $\ensuremath{\boldsymbol{\mathcal K}}=\ensuremath{{\mathcal H}}_1\oplus\cdots\ensuremath{{\mathcal H}}_m\oplus\ensuremath{{\mathcal G}}_1\oplus\cdots\oplus\ensuremath{{\mathcal G}}_K$. For every $i\in\{1,\ldots,m\}$ and every $k\in\{1,\ldots,K\}$, let $A_i\colon\ensuremath{{\mathcal H}}_i\to 2^{\ensuremath{{\mathcal H}}_i}$ and $B_k\colon\ensuremath{{\mathcal G}}_k\to 2^{\ensuremath{{\mathcal G}}_k}$ be maximally monotone, let $z_i\in\ensuremath{{\mathcal H}}_i$, let $r_k\in\ensuremath{{\mathcal G}}_k$, and let $L_{ki}\colon\ensuremath{{\mathcal H}}_i\to\ensuremath{{\mathcal G}}_k$ be linear and bounded. Let $(\boldsymbol{x}_0,\boldsymbol{v}_0^*)= (x_{1,0},\ldots,x_{m,0},v_{1,0}^*,\ldots,v_{K,0}^*)\in\ensuremath{\boldsymbol{\mathcal K}}$, assume that the coupled inclusions problem \begin{multline} \label{e:lk87b'kk-24p} \text{find}\;\;\overline{x}_1\in\ensuremath{{\mathcal H}}_1,\ldots,\overline{x}_m\in\ensuremath{{\mathcal H}}_m \;\;\text{such that}\\ (\forall i\in\{1,\ldots,m\})\quad z_i\in A_i\overline{x}_i+\ensuremath{\displaystyle\sum}_{k=1}^KL_{ki}^* \bigg(B_k\bigg(\ensuremath{\displaystyle\sum}_{j=1}^mL_{kj}\overline{x}_j-r_k\bigg)\bigg) \end{multline} has at least one solution, and consider the dual problem \begin{multline} \label{e:lk87b'kk-24d} \text{find}\;\;\overline{v}_1^*\in\ensuremath{{\mathcal G}}_1,\ldots,\overline{v}^*_K \in\ensuremath{{\mathcal G}}_K \;\;\text{such that}\\ (\forall k\in\{1,\ldots,K\})\quad -r_k\in-\ensuremath{\displaystyle\sum}_{i=1}^mL_{ki}\bigg(A_i^{-1} \bigg(z_i-\ensuremath{\displaystyle\sum}_{l=1}^KL_{li}^*\overline{v}^*_l\bigg)\bigg) +B_k^{-1}\overline{v}^*_k. \end{multline} The problem is to find the best approximation $(\overline{x}_1,\ldots,\overline{x}_m,\overline{v}_1^*,\ldots, \overline{v}_K^*)$ to $(\boldsymbol{x}_0,\boldsymbol{v}_0^*)$ from the associated Kuhn-Tucker set \begin{multline} \label{e:9g45g2h07k} \boldsymbol{Z}=\bigg\{(x_1,\ldots,x_m,v_1^*,\ldots,v^*_K)\in\ensuremath{\boldsymbol{\mathcal K}}
\;\bigg |\; (\forall i\in\{1,\ldots,m\})\;\;z_i-\sum_{k=1}^KL_{ki}^*v_k^*\in A_ix_i\:\;\text{and}\\ (\forall k\in\{1,\ldots,K\})\;\;\sum_{i=1}^mL_{ki}x_i-r_k\in B_k^{-1}v_k^*\bigg\}. \end{multline} \end{problem}
The next result presents a strongly convergent method for solving Problem~\ref{prob:12}. Let us note that existing methods require stringent additional conditions on the operators to achieve strong convergence, produce only unspecified points in the Kuhn-Tucker set, and necessitate the knowledge of the norms of the linear operators present in the model \cite{Sico10,Siop13}. These shortcomings are simultaneously circumvented in the proposed algorithm.
\begin{proposition} \label{p:9g45g2h07} Consider the setting of Problem~\ref{prob:12}. Let $\varepsilon\in\ensuremath{\left]0,1\right[}$ and iterate \begin{equation} \label{e:j7yG9i-9jmL407a} \begin{array}{l} \text{for}\;n=0,1,\ldots\\ \left\lfloor \begin{array}{l} (\gamma_n,\mu_n)\in [\varepsilon,1/\varepsilon]^2\\ \text{for}\;i=1,\ldots,m\\ \left\lfloor \begin{array}{l} a_{i,n}=J_{\gamma_n A_i}\big(x_{i,n}+\gamma_n \big(z_i-\sum_{k=1}^KL_{ki}^*v_{k,n}^*\big)\big)\\ \end{array} \right.\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} l_{k,n}=\sum_{i=1}^mL_{ki}x_{i,n}\\ b_{k,n}=r_k+J_{\mu_n B_k}\big(l_{k,n}+\mu_nv_{k,n}^*-r_k\big)\\ t_{k,n}=b_{k,n}-\sum_{i=1}^mL_{ki}a_{i,n}\\ \end{array} \right.\\ \text{for}\;i=1,\ldots,m\\ \left\lfloor \begin{array}{l} s^*_{i,n}=\gamma_n^{-1}(x_{i,n}-a_{i,n})+ \mu_n^{-1}\sum_{k=1}^KL_{ki}^*(l_{k,n}-b_{k,n})\\ \end{array} \right.\\
\tau_n=\sum_{i=1}^m\|s_{i,n}^*\|^2+\sum_{k=1}^K\|t_{k,n}\|^2\\ \text{if}\;\tau_n=0\\ \left\lfloor \begin{array}{l} \theta_n=0\\ \end{array} \right.\\ \text{if}\;\tau_n>0\\ \left\lfloor \begin{array}{l} \lambda_n\in\left[\varepsilon,1\right]\\ \theta_n=\lambda_n\big(\gamma_n^{-1}\sum_{i=1}^m
\|x_{i,n}-a_{i,n}\|^2+\mu_n^{-1}
\sum_{k=1}^K\|l_{k,n}-b_{k,n}\|^2\big)/\tau_n\\ \end{array} \right.\\ \text{for}\;i=1,\ldots,m\\ \left\lfloor \begin{array}{l} x_{i,n+1/2}=x_{i,n}-\theta_n s^*_{i,n}\\ \end{array} \right.\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} v^*_{k,n+1/2}=v^*_{k,n}-\theta_n t_{k,n} \end{array} \right.\\ \chi_n=\sum_{i=1}^m\scal{x_{i,0}-x_{i,n}}{x_{i,n}-x_{i,n+1/2}} +\sum_{k=1}^K\scal{v_{k,0}^*-v_{k,n}^*}{v_{k,n}^*-v_{k,n+1/2}^*}\\
\mu_n=\sum_{i=1}^m\|x_{i,0}-x_{i,n}\|^2+\sum_{k=1}^K
\|v_{k,0}^*-v_{k,n}^*\|^2\\
\nu_n=\sum_{i=1}^m\|x_{i,n}-x_{i,n+1/2}\|^2+
\sum_{k=1}^K\|v_{k,n}^*-v_{k,n+1/2}^*\|^2\\ \rho_n=\mu_n\nu_n-\chi_n^2\\ \text{if}\;\rho_n=0\;\text{and}\;\chi_n\ensuremath{\geqslant} 0\\ \left\lfloor \begin{array}{l} \text{for}\;i=1,\ldots,m\\ \left\lfloor \begin{array}{l} x_{i,n+1}=x_{i,n+1/2}\\ \end{array} \right.\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} v^*_{k,n+1}=v_{k,n+1/2}^*\\ \end{array} \right.\\ \end{array} \right.\\ \text{if}\;\rho_n>0\;\text{and}\;\chi_n\nu_n\ensuremath{\geqslant}\rho_n\\ \left\lfloor \begin{array}{l} \text{for}\;i=1,\ldots,m\\ \left\lfloor \begin{array}{l} x_{i,n+1}=x_{i,0}+(1+\chi_n/\nu_n)(x_{i,n+1/2}-x_{i,n})\\ \end{array} \right.\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} v^*_{k,n+1}=v_{k,0}^*+(1+\chi_n/\nu_n)(v_{k,n+1/2}^*-v_{k,n}^*) \end{array} \right.\\ \end{array} \right.\\ \text{if}\;\rho_n>0\;\text{and}\;\chi_n\nu_n<\rho_n\\ \left\lfloor \begin{array}{l} \text{for}\;i=1,\ldots,m\\ \left\lfloor \begin{array}{l} x_{i,n+1}=x_{i,n}+(\nu_n/\rho_n)\big(\chi_n(x_{i,0}-x_{i,n}) +\mu_n(x_{i,n+1/2}-x_{i,n})\big)\\ \end{array} \right.\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} v^*_{k,n+1}=v_{k,n}^*+(\nu_n/\rho_n) \big(\chi_n(v_{k,0}^*-v_{k,n}^*) +\mu_n(v_{k,n+1/2}^*-v_{k,n}^*)\big). \end{array} \right.\\ \end{array} \right.\\ \end{array} \right.\\ \end{array} \end{equation} Then \eqref{e:j7yG9i-9jmL407a} generates infinite sequences $(x_{1,n})_{n\in\ensuremath{\mathbb N}}$, \ldots, $(x_{m,n})_{n\in\ensuremath{\mathbb N}}$, $(v_{1,n}^*)_{n\in\ensuremath{\mathbb N}}$, \ldots, $(v_{K,n}^*)_{n\in\ensuremath{\mathbb N}}$, and the following hold: \begin{enumerate} \item \label{p:9g45g2h07i} Let $i\in\{1,\ldots,m\}$. Then
$\sum_{n\in\ensuremath{\mathbb N}}\!\|s^*_{i,n}\|^2\!<\!\ensuremath{{+\infty}}$,
$\sum_{n\in\ensuremath{\mathbb N}}\!\|x_{i,n+1}-x_{i,n}\|^2\!<\!\ensuremath{{+\infty}}$,
$\sum_{n\in\ensuremath{\mathbb N}}\!\|x_{i,n}-a_{i,n}\|^2\!<\!\ensuremath{{+\infty}}$, and $x_{i,n}\to\overline{x}_i$. \item \label{p:9g45g2h07ii} Let $k\in\{1,\ldots,K\}$. Then
$\sum_{n\in\ensuremath{\mathbb N}}\!\|t_{k,n}\|^2\!<\!\ensuremath{{+\infty}}$,
$\sum_{n\in\ensuremath{\mathbb N}}\!\|v^*_{k,n+1}-v^*_{k,n}\|^2\!<\!\ensuremath{{+\infty}}$,
$\sum_{n\in\ensuremath{\mathbb N}}\!\|\sum_{i=1}^mL_{ki}x_{i,n}-b_{k,n}\|^2\!<\!\ensuremath{{+\infty}}$, and $v^*_{k,n}\to\overline{v}_k^*$. \end{enumerate} \end{proposition} \begin{proof} Let us set $\ensuremath{{\mathcal H}}=\bigoplus_{i=1}^m\ensuremath{{\mathcal H}}_i$ and $\ensuremath{{\mathcal G}}=\bigoplus_{k=1}^K\ensuremath{{\mathcal G}}_k$, and let us introduce the operators \begin{equation} \label{e:9g45g2h10a} \begin{cases} A\colon\ensuremath{{\mathcal H}}\to 2^{\ensuremath{{\mathcal H}}}\colon (x_i)_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}\mapsto \cart_{\!i=1}^{\!m}(-z_i+A_ix_i)\\ B\colon\ensuremath{{\mathcal G}}\to 2^{\ensuremath{{\mathcal G}}}\colon (y_k)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}\mapsto\cart_{\!k=1}^{\!K}B_k(y_k-r_k)\\ L\colon\ensuremath{{\mathcal H}}\to\ensuremath{{\mathcal G}}\colon (x_i)_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}\mapsto \big(\sum_{i=1}^mL_{ki}x_i\big)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}. \end{cases} \end{equation} Then $L^*\colon\ensuremath{{\mathcal G}}\to\ensuremath{{\mathcal H}}\colon (y_k)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K} \mapsto(\sum_{k=1}^KL_{ki}^*y_k)_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}$ and, in this setting, Problem~\ref{prob:11} becomes Problem~\ref{prob:12}. Next, for every $n\in\ensuremath{\mathbb N}$, let us introduce the variables $a_n=(a_{i,n})_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}$, $s^*_n=(s^*_{i,n})_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}$, $x_n=(x_{i,n})_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}$, $x_{n+1/2}=(x_{i,n+1/2})_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}$, $b_n=(b_{k,n})_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$, $l_n=(l_{k,n})_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$, $t_n=(t_{k,n})_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$, $v_n^*=(v^*_{k,n})_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$, and $v_{n+1/2}^*=(v^*_{k,n+1/2})_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$. Since \cite[Propositions~23.15 and 23.16]{Livre1} assert that \begin{multline} (\forall n\in\ensuremath{\mathbb N})(\forall (x_i)_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}\in\ensuremath{{\mathcal H}}) (\forall (y_k)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}\in\ensuremath{{\mathcal G}})\quad J_{\gamma_n A}(x_i)_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}=\big(J_{\gamma_n A_i} (x_{i}+\gamma_n z_i)\big)_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}\\ \text{and}\quad J_{\mu_n B}(y_k)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}= \big(r_k+J_{\mu_n B_k}(y_{k}-r_k)\big)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}, \end{multline} \eqref{e:9i-9jmL403a} reduces in the present scenario to \eqref{e:j7yG9i-9jmL407a}. Thus, the results follow from Proposition~\ref{p:j7yG9i-9jmL407}. \end{proof}
\begin{example} \label{ex:9i-9jmL421} Let $A$, $(B_k)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$, and $(S_k)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$ be maximally monotone operators acting on a real Hilbert space $\ensuremath{{\mathcal H}}$. We revisit a problem discussed in \cite[Section~4]{Siop13}, namely the relaxation of the possibly inconsistent inclusion problem \begin{equation} \label{e:cras1995} \text{find}\;\;\overline{x}\in\ensuremath{{\mathcal H}}\;\;\text{such that}\;\; 0\in A\overline{x}\cap\bigcap_{k=1}^KB_k\overline{x} \end{equation} to \begin{equation} \label{e:2012-11-29p} \text{find}\;\;\overline{x}\in\ensuremath{{\mathcal H}}\;\;\text{such that}\;\; 0\in A\overline{x}+\sum_{k=1}^K(B_k\ensuremath{\mbox{\small$\,\square\,$}} S_k)\overline{x}, \quad\text{where}\quad B_k\ensuremath{\mbox{\small$\,\square\,$}} S_k= \big(B^{-1}_k+S^{-1}_k\big)^{-1}. \end{equation} We assume that \eqref{e:2012-11-29p} has at least one solution and that, for every $k\in\{1,\ldots,K\}$, $S_k^{-1}$ is at most single-valued and strictly monotone, with $S_k^{-1} 0=\{0\}$. Hence, \eqref{e:2012-11-29p} is a relaxation of \eqref{e:cras1995} in the sense that if the latter happens to have solutions, they coincide with those of the former \cite[Proposition~4.2]{Siop13}. As shown in \cite{Siop13}, this framework captures many relaxation schemes, and a point $\overline{x}_1\in\ensuremath{{\mathcal H}}$ solves \eqref{e:2012-11-29p} if and only if $(\overline{x}_1,\overline{x}_2,\ldots,\overline{x}_m)$ solves \eqref{e:lk87b'kk-24p}, where $m=K+1$, $\ensuremath{{\mathcal H}}_1=\ensuremath{{\mathcal H}}$, $A_1=A$, $z_1=0$, and, for every $k\in\{1,\ldots,K\}$, \begin{equation} \label{e:9i-9jmL423} \begin{cases} \ensuremath{{\mathcal H}}_{k+1}=\ensuremath{{\mathcal H}}\\ \ensuremath{{\mathcal G}}_{k}=\ensuremath{{\mathcal H}}\\ A_{k+1}=S_k\\ z_{k+1}=0\\ r_k=0 \end{cases} \qquad\text{and}\quad \begin{cases} L_{k1}=\ensuremath{\operatorname{Id}}\,\\ (\forall i\in\{2,\ldots,m\})\:\; L_{ki}= \begin{cases} -\ensuremath{\operatorname{Id}}\,,&\text{if}\;\;i=k+1;\\ 0,&\text{otherwise.} \end{cases} \end{cases} \end{equation} Thus \eqref{e:j7yG9i-9jmL407a} can be reduced to \begin{equation} \label{e:9i-9jmL422a} \begin{array}{l} \text{for}\;n=0,1,\ldots\\ \left\lfloor \begin{array}{l} (\gamma_n,\mu_n)\in [\varepsilon,1/\varepsilon]^2\\ a_{1,n}=J_{\gamma_n A}\big(x_{1,n}-\gamma_n \sum_{k=1}^Kv_{k,n}^*\big)\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} a_{k+1,n}=J_{\gamma_n S_k}\big(x_{k+1,n}+\gamma_n v_{k,n}^*\big)\\ l_{k,n}=x_{1,n}-x_{k+1,n}\\ b_{k,n}=J_{\mu_n B_k}\big(l_{k,n}+\mu_nv_{k,n}^*\big)\\ t_{k,n}=b_{k,n}+a_{k+1,n}-a_{1,n}\\ s^*_{k+1,n}=\gamma_n^{-1}(x_{k+1,n}-a_{k+1,n})+ \mu_n^{-1}(b_{k,n}-l_{k,n})\\ \end{array} \right.\\ s^*_{1,n}=\gamma_n^{-1}(x_{1,n}-a_{1,n})+ \mu_n^{-1}\sum_{k=1}^K(l_{k,n}-b_{k,n})\\
\tau_n=\sum_{k=1}^{K+1}\|s_{k,n}^*\|^2+\sum_{k=1}^K\|t_{k,n}\|^2\\ \text{if}\;\tau_n=0\\ \left\lfloor \begin{array}{l} \theta_n=0\\ \end{array} \right.\\ \text{if}\;\tau_n>0\\ \left\lfloor \begin{array}{l} \lambda_n\in\left[\varepsilon,1\right]\\ \theta_n=\lambda_n\big(\gamma_n^{-1}\sum_{k=1}^{K+1}
\|x_{k,n}-a_{k,n}\|^2+\mu_n^{-1}
\sum_{k=1}^K\|l_{k,n}-b_{k,n}\|^2\big)/\tau_n\\ \end{array} \right.\\ x_{1,n+1/2}=x_{1,n}-\theta_n s^*_{1,n}\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} x_{k+1,n+1/2}=x_{k+1,n}-\theta_n s^*_{k+1,n}\\ v^*_{k,n+1/2}=v^*_{k,n}-\theta_n t_{k,n} \end{array} \right.\\ \chi_n=\sum_{k=1}^{K+1}\scal{x_{k,0}-x_{k,n}}{x_{k,n}-x_{k,n+1/2}} +\sum_{k=1}^K\scal{v_{k,0}^*-v_{k,n}^*}{v_{k,n}^*-v_{k,n+1/2}^*}\\
\mu_n=\sum_{k=1}^{K+1}\|x_{k,0}-x_{k,n}\|^2+\sum_{k=1}^K
\|v_{k,0}^*-v_{k,n}^*\|^2\\
\nu_n=\sum_{k=1}^{K+1}\|x_{k,n}-x_{k,n+1/2}\|^2+
\sum_{k=1}^K\|v_{k,n}^*-v_{k,n+1/2}^*\|^2\\ \rho_n=\mu_n\nu_n-\chi_n^2\\ \text{if}\;\rho_n=0\;\text{and}\;\chi_n\ensuremath{\geqslant} 0\\ \left\lfloor \begin{array}{l} x_{1,n+1}=x_{1,n+1/2}\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} x_{k+1,n+1}=x_{k+1,n+1/2}\\ v^*_{k,n+1}=v_{k,n+1/2}^*\\ \end{array} \right.\\ \end{array} \right.\\ \text{if}\;\rho_n>0\;\text{and}\;\chi_n\nu_n\ensuremath{\geqslant}\rho_n\\ \left\lfloor \begin{array}{l} x_{1,n+1}=x_{1,0}+(1+\chi_n/\nu_n)(x_{1,n+1/2}-x_{1,n})\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} x_{k+1,n+1}=x_{k+1,0}+(1+\chi_n/\nu_n)(x_{k+1,n+1/2}-x_{k+1,n})\\ v^*_{k,n+1}=v_{k,0}^*+(1+\chi_n/\nu_n)(v_{k,n+1/2}^*-v_{k,n}^*) \end{array} \right.\\ \end{array} \right.\\ \text{if}\;\rho_n>0\;\text{and}\;\chi_n\nu_n<\rho_n\\ \left\lfloor \begin{array}{l} x_{1,n+1}=x_{1,n}+(\nu_n/\rho_n)\big(\chi_n(x_{1,0}-x_{1,n}) +\mu_n(x_{1,n+1/2}-x_{1,n})\big)\\ \text{for}\;k=1,\ldots,K\\ \left\lfloor \begin{array}{l} x_{k+1,n+1}=x_{k+1,n}+(\nu_n/\rho_n)\big(\chi_n(x_{k+1,0}-x_{k+1,n}) +\mu_n(x_{k+1,n+1/2}-x_{k+1,n})\big)\\ v^*_{k,n+1}=v_{k,n}^*+(\nu_n/\rho_n) \big(\chi_n(v_{k,0}^*-v_{k,n}^*) +\mu_n(v_{k,n+1/2}^*-v_{k,n}^*)\big), \end{array} \right.\\ \end{array} \right.\\ \end{array} \right.\\ \end{array} \end{equation} and it follows from Proposition~\ref{p:9g45g2h07} that $(x_{1,n})_{n\in\ensuremath{\mathbb N}}$ converges strongly to a solution $\overline{x}_1$ to the relaxed problem \eqref{e:2012-11-29p}. Let us note that the algorithm proposed in \cite[Proposition~4.2]{Siop13} to solve \eqref{e:2012-11-29p} requires that $A$ be uniformly monotone at $\overline{x}_1$ to guarantee strong convergence, whereas this assumption is not needed here. In addition, the scaling parameters used in the resolvents of the monotone operators in \cite[Proposition~4.2]{Siop13} must be identical at each iteration and bounded by a fixed constant: $(\forall n\in\ensuremath{\mathbb N})$ $\gamma_n=\mu_n\in[\varepsilon,(1-\varepsilon)/\sqrt{K+1}]$. By contrast, the parameters $\mu_n$ and $\gamma_n$ in \eqref{e:9i-9jmL422a} may differ and they can be arbitrarily large since $\varepsilon$ can be arbitrarily small, which could have some beneficial impact in terms of speed of convergence. \end{example}
As a second illustration of Proposition~\ref{p:9g45g2h07}, we consider the following multivariate minimization problem.
\begin{problem} \label{prob:13} Let $m$ and $K$ be strictly positive integers, let $(\ensuremath{{\mathcal H}}_i)_{1\ensuremath{\leqslant} i\ensuremath{\leqslant} m}$ and $(\ensuremath{{\mathcal G}}_k)_{1\ensuremath{\leqslant} k\ensuremath{\leqslant} K}$ be real Hilbert spaces, and set $\ensuremath{\boldsymbol{\mathcal K}}=\ensuremath{{\mathcal H}}_1\oplus\cdots\ensuremath{{\mathcal H}}_m\oplus\ensuremath{{\mathcal G}}_1\oplus\cdots\oplus\ensuremath{{\mathcal G}}_K$. For every $i\in\{1,\ldots,m\}$ and every $k\in\{1,\ldots,K\}$, let $f_i\in\Gamma_0(\ensuremath{{\mathcal H}}_i)$ and $g_k\in\Gamma_0(\ensuremath{{\mathcal G}}_k)$, let $z_i\in\ensuremath{{\mathcal H}}_i$, let $r_k\in\ensuremath{{\mathcal G}}_k$, and let $L_{ki}\colon\ensuremath{{\mathcal H}}_i\to\ensuremath{{\mathcal G}}_k$ be linear and bounded. Let $(\boldsymbol{x}_0,\boldsymbol{v}_0^*)= (x_{1,0},\ldots,x_{m,0},v_{1,0}^*,\ldots,v_{K,0}^*)\in\ensuremath{\boldsymbol{\mathcal K}}$ and assume that \begin{equation} \label{e:2012-10-21a} (\forall i\in\{1,\ldots,m\})\quad z_i\in\ensuremath{\text{\rm ran}\,}\bigg(\partial f_i+\sum_{k=1}^KL_{ki}^*\circ \partial g_k\circ\bigg(\sum_{j=1}^mL_{kj}\cdot-r_k\bigg)\bigg). \end{equation} Consider the primal problem \begin{equation} \label{e:9i-9jmL420p} \minimize{x_1\in\ensuremath{{\mathcal H}}_1,\ldots,\,x_m\in\ensuremath{{\mathcal H}}_m}{\sum_{i=1}^m \big(f_i(x_i)-\scal{x_i}{z_i}\big)+\sum_{k=1}^K g_k\bigg(\sum_{i=1}^mL_{ki}x_i-r_k\bigg)} \end{equation} and the dual problem \begin{equation} \label{e:9i-9jmL420d} \minimize{v^*_1\in\ensuremath{{\mathcal G}}_1,\ldots,\,v^*_K\in\ensuremath{{\mathcal G}}_K}{\sum_{i=1}^m f_i^*\bigg(z_i-\sum_{k=1}^KL_{ki}^*v^*_k\bigg) +\sum_{k=1}^K\big(g^*_k(v^*_k)+\scal{v^*_k}{r_k}\big)}. \end{equation} The objective is to find the best approximation $(\overline{x}_1,\ldots,\overline{x}_m,\overline{v}_1^*,\ldots, \overline{v}_K^*)$ to $(\boldsymbol{x}_0,\boldsymbol{v}_0^*)$ from the associated Kuhn-Tucker set \begin{multline} \label{e:9i-9jmL420k} \boldsymbol{Z}=\bigg\{(x_1,\ldots,x_m,v_1^*,\ldots,v^*_K)\in\ensuremath{\boldsymbol{\mathcal K}}
\;\bigg |\; (\forall i\in\{1,\ldots,m\})\;\;z_i-\sum_{k=1}^KL_{ki}^*v_k^*\in \partial f_i(x_i)\:\;\text{and}\\ (\forall k\in\{1,\ldots,K\})\;\;\sum_{i=1}^mL_{ki}x_i-r_k\in \partial g_k^*(v_k^*)\bigg\}. \end{multline} \end{problem}
The following corollary provides a strongly convergent method to solve Problem~\ref{prob:13}. Recall that the Moreau proximity operator \cite{Mor62b} of a function $\varphi\in\Gamma_0(\ensuremath{{\mathcal H}})$ is $\ensuremath{\text{\rm prox}}_\varphi=J_{\partial\varphi}$, i.e., the operator which maps every point $x\in\ensuremath{{\mathcal H}}$ to the unique minimizer of the function
$y\mapsto\varphi(y)+\|x-y\|^2/2$.
\begin{corollary} \label{c:9g45g2h13} Consider the setting of Problem~\ref{prob:13}. Let $\varepsilon\in\ensuremath{\left]0,1\right[}$ and execute \eqref{e:j7yG9i-9jmL407a}, where $J_{\gamma_n A_i}$ is replaced by $\ensuremath{\text{\rm prox}}_{\gamma_n f_i}$ and $J_{\mu_n B_k}$ is replaced by $\ensuremath{\text{\rm prox}}_{\mu_n g_k}$. Then the following hold: \begin{enumerate} \item $(\overline{x}_1,\ldots,\overline{x}_m)$ solves \eqref{e:9i-9jmL420p} and $(\overline{v}^*_1,\ldots,\overline{v}^*_m)$ solves \eqref{e:9i-9jmL420d}. \item For every $i\in\{1,\ldots,m\}$, $x_{i,n}\to\overline{x}_i$. \item For every $k\in\{1,\ldots,K\}$, $v^*_{k,n}\to\overline{v}^*_k$. \end{enumerate} \end{corollary} \begin{proof} Let us define $(\forall i\in\{1,\ldots,m\})$ $A_i=\partial f_i$ and $(\forall k\in\{1,\ldots,K\})$ $B_k=\partial g_k$. Then, as shown in the proof of \cite[Proposition~5.4]{Siop13}, \eqref{e:2012-10-21a} implies that Problem~\ref{prob:12} assumes the form of Problem~\ref{prob:13} and that Kuhn-Tucker points provide primal and dual solutions. Hence, applying Proposition~\ref{p:9g45g2h07} in this setting yields the claims. \end{proof}
\end{document} | arXiv |
Formula to calculate password cracking time in years, taking into account Moore's law and known adversary guessing power [closed]
We know that the biggest human rights violators in human history are capable of one trillion password guesses per second as of approximately January 2013.
Assume that the 1 trillion guesses per second is not a dictionary attack, but a brute force search of all possible permutations of the available characters in the password.
Assume this rate of password guessing is the same speed regardless of their computing equipment. Assume they have special equipment e.g. GPUs/ASICs capable of performing the industry standard Password Based Key Derivation Function (PBKDF2-SHA2 with large number of iterations) for each password guess and they can still guess 1 trillion password combinations per second. Therefore their actual hardware will not factor into the equation, just the 1 trillion guesses per second they can perform.
Discard the assumption of weak passwords and assume the password is very strong and made up of uniformly and randomly selected characters available on a standard US keyboard layout including special characters (95 possible characters total).
We also know from Moore's law that transistor count on an integrated circuit doubles every two years, which loosely translates to doubling of computing power every two years. So in January 2015 they will be able to guess 2 trillion passwords per second. In January 2017 they can guess 4 trillion per second and so on. Assume this trend will continue regardless of speculation that this law may come to an end and it needs to be factored into the formula.
To successfully guess a password, it often only requires 2^n-1 attempts. That needs to be factored into the equation. Please also factor this into the formula.
What I would like is a reusable formula which takes into account the known adversary power of 1 trillion guesses per second as of January 2013 and its future power with regards to Moore's law. I would like to dynamically enter in the total number of password characters and the current date. The formula will return a calculation of how many years this password will be secure from brute force search from that current date.
Apologies if this is the incorrect StackExchange forum, but I think it is in the right place as I am after a correct mathematical formula which I can then turn into a software function. Feel free to move it if that is more appropriate.
permutations computer-science computational-complexity combinations mathematical-physics
eichroyseichroys
closed as unclear what you're asking by RE60K, Najib Idrissi, kjetil b halvorsen, graydad, Narasimham Apr 11 '15 at 20:13
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
$\begingroup$ I'm thinking this is best for Security.SE... $\endgroup$ – Mario Carneiro Apr 11 '15 at 10:20
$\begingroup$ There's no mathematicians there! $\endgroup$ – eichroys Apr 11 '15 at 10:20
$\begingroup$ Not so sure about that, from the answers I've seen in the past. If you get a good answer, I think that this level of math will not pose a problem over there. For this to be a good fit for math.SE, I would recommend cutting out all the text and just giving the model; as it is there is way too much "word problem" stuff. $\endgroup$ – Mario Carneiro Apr 11 '15 at 10:21
$\begingroup$ I put the question into the context of a security question, but it reduces to a simple maths problem. I.e. how many years to try every combination of 95 chars on an x length string, based on x operations per second at this date, and take into account a doubling of operations per second every two years. $\endgroup$ – eichroys Apr 11 '15 at 10:25
$\begingroup$ I don't think the question is relevant. Most systems throw you out if you make too many incorrect guesses. Your reasoning only applies if the device has been physically taken (eg a stolen encrypted hard disk) and it doesn't have a limit on guesses. But no one expects it to be invulnerable in those situations anyway. $\endgroup$ – user117644 Apr 11 '15 at 10:41
Let $t_0$ be the current time in years from January 2013, and $n$ be the number of bits in the password. If $y$ is the number of attempts since the NSA started trying to hack your password, then we have the equation
$$\frac{dy}{dt}=10^{12}\cdot60\cdot60\cdot24\cdot365\cdot 2^{t/2}=:k\,2^{t/2}.$$
The big number $k$ is for converting the $10^{12}$ attacks-per-second figure into years, and the derivative is because this is measuring the accumulation of attacks done.
Now, we want to find the number of attacks that occur between now and some future time $t_0+t$, which we obtain by integrating this:
$$y(t_0+t)=\int_{t_0}^{t_0+t}k\,2^{x/2}\,dx=\left.\frac{2k}{\log 2}2^{x/2}\right|_{t_0}^{t_0+t}=\frac{2k}{\log 2}2^{t_0/2}(2^{t/2}-1).$$
Now, we are interested in finding when this number of attacks exceeds the maximum that our password can tolerate, which is roughly $2^n$ (I'm ignoring the $-1$ because the difference is negligible compared to other approximations of the model):
$$y(t_0+t)=2^n\implies2^{t/2}+1=\frac{2^n\log 2}{2k\,2^{t_0/2}}\implies t=2\log_2\left[\frac{2^n\log 2}{2k\,2^{t_0/2}}-1\right].$$
And there is your formula, given inputs $t_0$ and $n$, and the constant $k$. (Edit: I notice I have not factored in the information about keyboard layouts into this analysis, since the "$2^n-1$ attempts" part is already enough to answer the question. If the passwords are not bit strings but instead strings of characters from an alphabet of $95$ symbols, replace $2^n$ with ${95}^n$; nothing else is affected.)
Discussion: Looking at the form of the formula, we can get a feel for the implications of Moore's law in action. The denominator involves factors $k$ and $t_0$ that you can't do much about (we can't choose the era we live in), but $n$ is of course under our control, so it helps to isolate that part. Note that we can rewrite the equation as $t=2\log_2(2^{n+a}-1)$, where $a=\log_2\frac{\log 2}{2k\,2^{t_0/2}}$ is a constant; thus the overall speed of computing can be offset simply by adding a constant amount to your password length.
For large $n$, the $-1$ factor becomes negligible, and we get $t\sim 2\log_2(2^{n+a})=2(n+a)$. Thus the time to break the password goes up by about 2 years for every extra bit in the password. If we are working from a $95$-symbol alphabet, this changes to $t\sim 2\log_2(95^{n+a'})\approx13.1(n+a')$, so that each extra US keyboard character adds 13 years to the password strength.
Mario CarneiroMario Carneiro
While the mathematics of this problem are well covered by Mario Carneiro, the reality in the OP is rather distorted. Ignoring the problems with Moore's law as it is. The main issue you will run into very soon is the energy consumed.
To further simplify the problem we ignore the fact that energy costs money and just focus on the actual limits of how much energy we can produce. And then to make it easier on myself we crib Bruce Schneier's work from Applied Cryptography as reposted here so I don't have to do all the calculations myself.
One of the consequences of the second law of thermodynamics is that a certain amount of energy is necessary to represent information. To record a single bit by changing the state of a system requires an amount of energy no less than $kT$, where $T$ is the absolute temperature of the system and $k$ is the Boltzman constant. (Stick with me; the physics lesson is almost over.)
Given that $k =1.38 \cdot 10^{-16} \mathrm{erg}/{^\circ}\mathrm{Kelvin}$, and that the ambient temperature of the universe is $3.2{^\circ}\mathrm K$, an ideal computer running at $3.2{^\circ}\mathrm K$ would consume $4.4 \cdot 10^{-16}$ ergs every time it set or cleared a bit. To run a computer any colder than the cosmic background radiation would require extra energy to run a heat pump.
Now, the annual energy output of our sun is about $1.21 \cdot 10^{41}$ ergs. This is enough to power about $2.7 \cdot 10^{56}$ single bit changes on our ideal computer; enough state changes to put a 187-bit counter through all its values. If we built a Dyson sphere around the sun and captured all of its energy for 32 years, without any loss, we could power a computer to count up to $2^{192}$. Of course, it wouldn't have the energy left over to perform any useful calculations with this counter.
But that's just one star, and a measly one at that. A typical supernova releases something like $10^{51}$ ergs. (About a hundred times as much energy would be released in the form of neutrinos, but let them go for now.) If all of this energy could be channeled into a single orgy of computation, a 219-bit counter could be cycled through all of its states.
These numbers have nothing to do with the technology of the devices; they are the maximums that thermodynamics will allow. And they strongly imply that brute-force attacks against 256-bit keys will be infeasible until computers are built from something other than matter and occupy something other than space.
Now to look at a more realistic example (and do at least a little work) let us look at the output of the world as given by wikipedia for 2008 World energy supply. That gives us a base supply of 143,851 terawatt hours which is $5.178636 × 10^{27}$ ergs. Lets round up and call it $10^{28}$ ergs. Take the bit setting energy and round it down to $10^{-16}$ ergs per bit. That gives us $10^{44}$ bit flips. That's about enough to run a 144 bit counter through its paces. Notice that we are using ALL of the worlds energy for a year and making super unrealistic assumptions (like being able to run a computer at the ambient temperature of the universe).
Altogether, already below 128 bits of entropy it's orders of magnitudes easier and cheaper to bribe or beat the key or password out of someone than to try and crack it. More over this assessment will not change in the near future unless some world changing physics happen.
DRFDRF
$\begingroup$ GREAT! :D Amazing info $\endgroup$ – RE60K Apr 11 '15 at 13:33
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Model-based adaptive phase I trial design of post-transplant decitabine maintenance in myelodysplastic syndrome
Seunghoon Han1, 2,
Yoo-Jin Kim3,
Jongtae Lee1, 2,
Sangil Jeon1, 2,
Taegon Hong1, 2,
Gab-jin Park1, 2,
Jae-Ho Yoon3,
Seung-Ah Yahng3,
Seung-Hwan Shin3,
Sung-Eun Lee3,
Ki-Seong Eom3,
Hee-Je Kim3,
Chang-Ki Min3,
Seok Lee3 and
Dong-Seok Yim1, 2Email author
Journal of Hematology & Oncology20158:118
© Han et al. 2015
This report focuses on the adaptive phase I trial design aimed to find the clinically applicable dose for decitabine maintenance treatment after allogeneic hematopoietic stem cell transplantation in patients with higher-risk myelodysplastic syndrome and secondary acute myeloid leukemia.
The first cohort (three patients) was given the same initial daily dose of decitabine (5 mg/m2/day, five consecutive days with 4-week intervals). In all cohorts, the doses for Cycles 2 to 4 were individualized using pharmacokinetic-pharmacodynamic modeling and simulations. The goal of dose individualization was to determine the maximum dose for each patient at which the occurrence of grade 4 (CTC-AE) toxicities for both platelet and neutrophil counts could be avoided. The initial doses for the following cohorts were also estimated with the data from the previous cohorts in the same manner.
In all but one patient (14 out of 15), neutrophil count was the dose-limiting factor throughout the cycles. In cycles where doses were individualized, the median neutrophil nadir observed was 1100/mm3 (grade 2) and grade 4 toxicity occurred in 5.1 % of all cycles (while it occurred in 36.8 % where doses were not individualized). The initial doses estimated for cohorts 2 to 5 were 4, 5, 5.5, and 5 mg/m2/day, respectively. The median maintenance dose was 7 mg/m2/day.
We determined the acceptable starting dose and individualized the maintenance dose for each patient, while minimizing the toxicity using the adaptive approach. Currently, 5 mg/m2/day is considered to be the most appropriate starting dose for the regimen studied.
Clinicaltrials.gov NCT01277484
Model-based drug development
Adaptive design
Population pharmacokinetics-pharmacodynamics
Phase I clinical trial
DNA methylation is the best-known epigenetic marker for cancer development [1]. In some hematologic malignancies including myelodysplastic syndrome (MDS), DNA methylation results not only in increased cell proliferation but also in silencing of genes which regulate growth and differentiation [2]. Based upon those mechanisms, the use of a DNA hypomethylating agent (HMA) for hematologic malignancies has been expanded. Accordingly, clinical researches to optimize HMA therapy [3, 4] or to explore epigenetic mechanisms for new drug development have been widely performed [5, 6].
Decitabine (Dacogen®, 5-aza-2′-deoxycytidine) is a HMA that exerts its antitumor activity by inhibiting DNA methylation at low doses and by arresting DNA synthesis at high doses [7, 8]. For several decades, decitabine has been one of the most intensely studied anticancer agents in the field of hematology due to its sophisticated development history [9–11], as well as its impressive clinical outcomes against many hematologic diseases [7, 9, 12–17]. For MDS, the approved indication for decitabine, numerous efforts have been made to optimize the dosing regimen according to patient characteristics, including the regimen evaluated in this study (five consecutive days of dosing with 4-week interval) [12, 18–22].
Recently, HMA maintenance therapy after allogeneic hematopoietic stem cell transplantation (allo-HSCT) has been suggested as a potentially attractive approach to minimize relapse and to improve graft survival [23–25]. Several studies on azacitidine (Vidaza®, 5-azacytidine) reported low toxicity, along with its potential to increase the number of hematopoietic stem cells [11, 26–29]; thus, similar approaches using decitabine were initiated [22]. In this context, we designed and performed a phase I study that aimed to find a clinically applicable dosage regimen for decitabine maintenance treatment after allo-HSCT in patients with higher-risk MDS and secondary acute myeloid leukemia (AML).
Our study design incorporated two major considerations: (1) the purpose of the maintenance therapy was to maintain disease-free status in the patient while simultaneously preserving graft function, and (2) the dosage regimen should be determined using the smallest number of patients possible. Considering these aspects, without a confident estimation of the appropriate starting dose, traditional fixed-dose escalation schemes [30] were considered inappropriate for the following reasons: (1) fatal toxicity (e.g., graft failure) might occur in some subjects, (2) the study might need too many patients to find the optimal dose [31], and (3) dose differences between cohorts might be too large or small. Thus, we introduced an adaptive dose individualization design based upon pharmacokinetic (PK)-pharmacodynamic (PD) modeling for the neutropenia and thrombocytopenia caused by decitabine.
Dose individualization of anticancer drugs using PK-PD modeling has been theoretically proposed using simulated data [32, 33]; however, our report is the first to implement dose individualization using PK-PD modeling in patients in a phase I clinical trial. We endeavored to titrate the appropriate dose for each patient, with the goal of identifying the highest possible dose that did not result in severe hematologic toxicities. We also anticipated that this approach would more quickly accomplish the study's objectives and avoid having to test several cohorts for the dose escalation. This report focuses on the study design, the PK-PD model development for hematologic toxicities caused by decitabine, and the usefulness of our adaptive approach as it applies to subject safety.
Patient characteristics
Five patients with secondary AML evolving from MDS and 11 with MDS (9 males, 7 females) were enrolled (Table 1). All the patients received the myeloablative condition regimen and peripheral blood stem cells from the related (n = 6) or unrelated (n = 10) donors. The engraftment achievement of platelet and neutrophil counts was confirmed for all patients by an experienced hematologist upon enrollment. Graft-versus-host disease (GVHD) prophylaxis was calcineurin inhibitors (cyclosporine for related and tacrolimus for unrelated donors) plus short-course methotrexate. Antithymocyte globulin was given to all patients. Decitabine was administered at a median of 86 days (range, 56–90 days) after transplantation. At the time of decitabine treatment, acute (≤ overall grade 2) or chronic GVHD was observed in nine and one patients, respectively. The clinical features are given in Table 2.
Age (year)
41.0 ± 17.1
55.7 ± 5.8
Sex (male/female)
1.68 ± 0.05
Body surface area (m2)
1.6 ± 0.1
Patient characteristics and doses given in each subject, cohort, and cycle
Subject number
Sex/age
WHO diagnosis
GVHD gradea
Cycle (mg/m2/day for 5 days)
RAEB-2
5b,c
7.5b,c
GVHD graft-versus-host disease, RAEB refractory anemia with excess blast, MSD matched sibling donor, PMUD partially matched unrelated donor, MUD matched unrelated donor
aAssessed at the time of decitabine initiation
bIndividual dose titration (IDT) by the PK-PD model was not applied
cThe cycles where grade 4 toxicities occurred
Patient disposition and dataset
Patient dispositions are detailed in Fig. 1. In cohort 1, the third patient dropped out of the study without PD sampling; thus, we substituted with an additional patient, since PK-PD results from three patients were needed to obtain the initial dose for cohort 2. Fourteen patients completed all the study-related procedures until Cycle 4, and maintenance dose was determined for each patient at the end of Cycle 4 (Table 2).
Patient disposition
For each subject, PK sampling was performed according to the protocol, and the average number of PD observations used in individual dose titration (IDT) was 5.76/cycle for both neutrophils and platelets. Among 58 treatment cycles of 15 patients, the doses for Cycles 2 to 4 (a total of 39 cycles) were determined through PK-PD model-based adaptive dose individualization. Cycle 2 doses in four patients were clinically determined for the following reasons: no significant blood cell count decrease after cycle 1 (subjects 8 and 10) and not enough time for PK-PD modeling and IDT from sudden changes in visit schedules for Cycle 2 dosing (subjects 11 and 12). The actual dosing interval was 34.5 ± 8.7 days (mean ± SD).
Estimated doses and safety outcomes
In all but one patient (14 out of 15), the absolute neutrophil count (ANC) was the dose-limiting factor throughout all cycles. During the cycles in which IDT was performed, the median ANC nadir observed was 1100/mm3 (range, 300/mm3 to 2680/mm3). The maintenance dose determined with four cycle data was higher than the initial doses in 10 out of the 15 patients. The initial doses (Cycle 1 doses) estimated by cohort dose estimation (CDE) were 4, 5, 5.5, and 5 mg/m2/day for cohorts 2, 3, 4, and 5, respectively. The median individual maintenance dose of decitabine was 7 mg/m2/day (Table 2). Maintenance doses for the patients with Cycle 1 data inadequate for PK-PD modeling could be estimated using three cycle data (Cycles 2, 3, and 4) with acceptable model fits.
A total of nine dose-limiting toxicities (DLT, platelet count for one case and absolute neutrophil count for eight cases) were observed. Among these toxicities, seven cases occurred in non-IDT cycles (six in Cycle 1 and one in Cycle 2 with clinically determined doses). In the observed toxicities, 36.8 % of the non-IDT cycles (7 out of 19 cycles) showed dose-limiting toxicities, which was an approximately seven times higher occurrence rate than that observed in the IDT cycles (5.1 %, 2 out of 39 cycles).
Overall mixed-effect PK-PD analysis
A total of 95 PK observations and 622 PD observations (311 for ANC and 311 for platelet count, PC) were used in the overall mixed-effect PK-PD analysis. The one patient whose dose-limiting factor was PC was excluded from this analysis, whose disease entity was considered not to be similar to others, as she suffered from immune thrombocytopenia after transplantation and was managed with steroids. Among the data, 6.9 % (4 out of 58 cycles) was obtained from the cycles where IDT was not applied.
A two-compartment model was found to best describe the PK data. The between-subject variability (BSV) for CL (clearance from the central compartment) was the only random effect which could be estimated, except for the proportional residual error. The basic structure of the PD model was identical to that used for IDT and CDE for both PC and ANC (transit compartment model with feedback mechanism):
$$ \begin{array}{c}\frac{dA(1)}{dt}\kern0.5em =\kern0.5em {k}_{\mathrm{tr}}\cdot \kern0.5em A(1)\cdot \kern0.5em \left\{\left(1-\mathrm{SLOPE}\cdot C\right)\cdot {\left(\mathrm{BASE}/A(5)\right)}^{\mathrm{GAMMA}}-1\right\}\\ {}\frac{dA(2)}{dt}\kern0.5em =\kern0.5em {k}_{\mathrm{tr}}\cdot \kern0.5em \left(A(1)-A(2)\right)\\ {}\frac{dA(3)}{dt}\kern0.5em =\kern0.5em {k}_{\mathrm{tr}}\cdot \kern0.5em \left(A(2)-A(3)\right)\\ {}\frac{dA(4)}{dt}\kern0.5em =\kern0.5em {k}_{\mathrm{tr}}\cdot \kern0.5em \left(A(3)-A(4)\right)\\ {}\frac{dA(5)}{dt}\kern0.5em =\kern0.5em {k}_{\mathrm{tr}}\cdot \kern0.5em \left(A(4)-A(5)\right)\end{array} $$
where A(N) is the cell count in the Nth compartment and C is the plasma decitabine concentration. A detailed description for the parameters is presented in Table 3. BASE is a parameter indicating the level of cell count maintained at baseline or at the period without drug effect. For platelets, an asymptotic structure describing gradual cell count increase over cycles improved the model significantly, and thus the following structure substituted the simple BASE parameter:
Final parameter estimates and bootstrap outcomes
Population typical value
Between-subject variability
Bootstrap median (95 % CI)
Estimate (as CV%)
Pharmacokinetic parameters
L/h·m2
88.3 (72.2–108)
20.5 (13.0–26.8)
V c
Volume of central compartment
V p
Volume of peripheral compartment
Intercompartmental clearance
Pharmacodynamic parameters for platelet
k tr,P
h−1
Rate constant of inter-compartmental platelet movement
0.0246 (0.0236–0.0254)
SLOPE P
Drug effect on platelet count
20.8 (1.30–56.0)
BASE P
/mm3
Baseline platelet count
53,700 (36,100–95,800)
GAMMA P
Shape factor for platelet count fluctuation
0.299 (0.264–0.325)
Maximum degree of platelet count recovery expected
58700 (24200 – 98300)
72.9 (19.5 – 150)
Rate constant for asymptotic platelet count recovery
0.000513 (0.000213–0.000691)
Pharmacodynamic parameters for neutrophil
k tr,N
Rate constant of inter-compartmental neutrophil movement
SLOPE N
Drug effect on neutrophil count
BASE N
Baseline neutrophil count
GAMMA N
Shape factor for neutrophil count fluctuation
Residual error
σ PK 2
Variance of residual error (proportional) for PK
σ PD,P 2
Variance of residual error (additive) for platelet count
σ PD,N 2
Variance of residual error (additive) for neutrophil count
Proportion of successful convergence: 78.8 % for PK model, 78.0 % for PD model
NE not estimated
$$ {\mathrm{BASE}}_p+\mathrm{IMP}\kern0.5em \left(1\kern0.5em {\displaystyle {-\kern0.5em e}^{-\mathrm{I}\mathrm{M}\mathrm{K}*\mathrm{TIME}}}\right) $$
where IMP is the empirical value of the maximum PC recovery expected, IMK is the rate constant for asymptotic PC recovery, and TIME is the time from the initiation of decitabine treatment.
No meaningful covariate was found in either the patient demographic or clinical variables. The parameter descriptions and estimates are given in Table 3. Simulated time courses of ANC changes, under the maintenance dosage of 5 mg/m2/day for four treatment cycles, are presented in Fig. 2.
Prediction of neutrophil count change when 5 mg/m2 dose is given for five consecutive days with 4-week interval. (From 1000 simulations using the final PK-PD model)
Clinical course and non-hematological events
During four cycles of the dose-finding phase of this study, one patient (subject 3) died of pneumonia (protocol violation) while the other two (subject 1 and subject 11) also suffered from pneumonia but fully recovered. One of the three cases developed decitabine-induced neutropenia (subject 11, withdrawn). Aggravation of existing acute or chronic GVHD was not observed, while chronic GVHD was diagnosed in two patients (one in mild and the other in moderate form). Herpes zoster was a complication in three patients.
We succeeded in administering the maximum dose allowed for each patient, with minimized toxicity. The dose for each cycle was determined based upon the observed cell counts in the previous cycle(s) which are the ultimate outcome of patient characteristics and drug effect. Thus, the dose can be considered as a reflection of the vulnerability of the graft, the sensitivity to decitabine, and any possible drug interactions affecting cell counts. This method meant that using a large number of cohorts, as typically required in the traditional dose escalation scheme, could be avoided. Moreover, the doses of four patients were reduced from their initial doses because of their relatively vulnerable PD characteristics. The treatment of these patients might have been discontinued if a traditional, fixed-dose design had been used. Most importantly, our study design showed the significant advantage that all dose individualization steps were accomplished with a favorable toxicity profile, judging from the proportion of cycles that exhibited grade 4 toxicities. When IDT was applied, the proportion of cycles exhibiting grade 4 toxicities dropped to approximately one-seventh the level (36.8 versus 5.1 %) compared with the non-IDT cycles. Thus, model-based dose individualization can be a useful option in early-phase clinical trial designs, in particular when the initial dose cannot be set with sufficient confidence.
The PK properties of decitabine in Korean patients obtained here are similar to those in previous studies. Liu et al. [34] and Cashen et al. [35] reported that the PK properties of decitabine could be well described with a two-compartment model. The distributional characteristics from these two studies could be indirectly compared using the maximum concentration (C max) predicted upon the completion of decitabine infusion. From previous reports, the maximum concentration of decitabine was within the range of 60–70 ng/mL, which was obtained approximately 1 h after initiation of infusion, when decitabine was administered at a rate of 5 mg/m2/h (3-h infusion of a 15 mg/m2 dose) [35, 36]. This observation is consistent with our finding that the predicted C max after 1-h infusion of 5 mg/m2 was 66.0 ng/mL. In addition, the average terminal half-life was also similar (0.31 h in this study); thus, the decitabine concentration is predicted to drop below 5 % of C max within 1.5–2 h after the completion of infusion.
The baseline cell count increase over cycles was modeled for platelet level. This was a consistent finding to the results from previous reports regarding the contribution of decitabine to cell proliferation [37–41]. For neutrophil counts, doses estimated by neutrophil count nadirs were gradually escalated over cycles until reaching the maintenance dose in ten patients while baseline cell count increase was not meaningful. Gradual deflation in the width of the prediction interval for ANC, resulting from improved precision of the model along with increased data points obtained throughout the cycles, seems to be one possible explanation. Dose escalation from this prediction interval deflation lowers the predicted median of course while maintaining the lower 25 % prediction interval above 500/mm3 (grade 4 toxicity).
We also found it necessary to modify the interval between cycles that was initially planned as 4 weeks in this study. Although both PC and ANC were recovered to the baseline after decitabine dosing, our PK-PD model predicted that the time to nadir was 3.5 weeks and that the time to recovery from the influence of the last dose (ANC >1000/mm3) was approximately 5 weeks for the ANC. This prediction was consistent with the actual dosing interval practiced in this study (34.5 days on average). This finding implies that the 4-week interval may not be long enough to initiate the next cycle. Moreover, as illustrated in Fig. 2, the lowest value of ANC appears to be achieved in the second cycle (6–7 weeks after treatment initiation). Thus, the initial nadir of ANC within the first 4-week cycle should not be mistaken for the lowest ANC value throughout the cycles. This could also have been a reason for failure in dose determination if traditional fixed-dose escalation based on the first cycle nadir was recruited. To optimize the dosing regimen that may overcome this difficult property of decitabine, an initial loading dose may be considered before giving maintenance doses.
We exemplified the adaptive dose titration approach, based upon a quantitative exposure-toxicity model, in this study. This approach seemed most useful, since this method enabled rapid and precise dose individualization. The most appropriate initial dose was determined to be 5 mg/m2/day for five consecutive days. Throughout the course of data analysis, issues such as extending between-cycle intervals and the use of loading doses were also raised. Cohort 6 is ongoing for exploration of the adequacy of the recommended starting dose, and additional report will be provided after completion of 12 cycles of treatment of all participants.
Ethics, consent, and permission
This study was designed and conducted in accordance with the principles of the Declaration of Helsinki and the good clinical practice guidelines of Korea. The independent institutional review board of Seoul St. Mary's Hospital approved this study protocol before the initiation of any study-related procedure, and written informed consent was obtained from every subject. The registration number of this trial at "ClinicalTrials.gov" is NCT01277484.
Patient eligibility
Patients starting decitabine treatment on days 42–90 after allo-HSCT and meeting the following criteria were considered eligible: adult aged ≤65; recipient of allo-HSCT for higher-risk (intermediate 2 or high risk) MDS, as assessed by the International Prognostic Scoring System [42], and/or AML evolving from MDS; disease remission with appropriate recoveries of PC >30,000/mm3 and ANC >1000/mm3, both of which were maintained for more than 7 days without any transfusions or growth factors; absence of grade III/IV acute GVHD; Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2; and no evidence of renal or hepatic impairment.
Patients were assigned to cohorts according to their order of enrollment. A cohort consisted of three patients to whom the same initial daily dose of decitabine (according to body surface area) was given. The initial dose for cohort 1 was 5 mg/m2/day. The designated dose was infused intravenously over 60 min daily for five consecutive days in each cycle, and the cycle was repeated every 4 weeks up to Cycle 12. However, dosing was suspended if blood cell counts insufficiently recovered (PC <30,000/mm3, ANC <1000/mm3).
For cycles 2 to 4, the dose for each cycle was estimated using IDT according to PK-PD modeling and simulations based on blood cell count data accumulated until the time of dose estimation (just before administration). The maximum dose at which the occurrence of grade 4 hematologic toxicity (dose-limiting toxicity, PC <25,000/mm3 or ANC <500/mm3) could be avoided at the lower limit of the 50 % prediction interval (25th percentile), according to 500 simulations, was determined to be the dose for the next cycle. If the data from the previous cycle were not adequate for PK-PD modeling (e.g., no significant blood cell count decreases), the dose was determined based upon the hematologist's clinical decision [43]. Only the upper limit of the dose increment was pre-determined that the next cycle dose cannot exceed 150 % of the previous dose. The dose determined at Cycle 4 for each individual was maintained thereafter.
The fixed initial dose for each cohort was also estimated using PK-PD modeling and simulations and was based on the observations from the previous cohorts (CDE). For cohort 2, all of the data obtained before the initiation of treatment for the first patient in cohort 2 were used for the initial dose estimation; however, only Cycle 1 data from the previous cohorts were used for cohort 3, 4, and 5. A new cohort was not initiated before completion of the first cycle in the last patient of the previous cohort.
A schematic diagram of the overall study design is presented in Fig. 3.
Overall schema of the study design. Individual dose titration was performed for the next cycle based on the observations from the previous cycle (solid straight arrows). Cohort dose estimation was performed to determine initial doses (broken line arrows): (i) for cohort 2, using all data obtained from cohort 1 until the initiation of cohort 2; (ii) for cohorts 3–5, using only Cycle 1 data of previous subjects. The dose of Cycle 4 was maintained until the completion of decitabine treatment (Cycle 12) (dotted lines)
PK and PD samplings
To determine plasma concentration measurements, seven whole-blood samples (10 mL each) were collected using EDTA tubes before dosing and then at 20, 40, 60, 90, 120, and 180 min after initiation of the first dose infusion of Cycle 1. The samples were immediately cooled in an ice bath and then centrifuged (3000 rpm, 4 °C, for 10 min) within 1 h from the last sampling time. After centrifugation, 4 mL of plasma from each sample was aliquoted into four microtubes (1 mL each), and 10 μL of 10 mg/mL tetrahydrouridine (THU) solution was added to each microtube. Microtubes were stored at −70 °C until plasma concentration assays.
As PD (toxicity) markers, PCs and ANCs were monitored at scheduled follow-up visits (weekly until Cycle 4 and biweekly thereafter). The procedures for obtaining PCs and ANCs followed the routine clinical practices for automated complete blood cell counts at Seoul St. Mary's Hospital.
Plasma concentration measurements
Plasma samples were analyzed using liquid chromatography coupled with tandem mass spectrometry (API 4000, ABSciex, Canada) based upon a previously reported method [34]. The lower limit of quantification (LLOQ) was 0.5 ng/mL. The coefficients of correlation (r 2) were greater than 0.9975 in the range of 0.5–100 ng/mL decitabine, as determined by weighted linear regression (1/concentration). The precision (% coefficient of variation) and mean intra- and inter-day accuracies were below 11.57 % and 95.55–102 %, respectively.
PK-PD modeling and simulation
A mixed-effect analysis was performed using NONMEM (ver. 7.2, Icon Development Solution, Ellicott City, MD, USA). During the early phase of this study (e.g., IDT for cohort 1 and CDE for cohort 2), during which sufficient PD data to build a robust model were unavailable, we adopted the PD model proposed by Wallin et al. (2009) [33]. This model was used in conjunction with the one-compartment, first-order elimination PK model to build the initial PK-PD model. Therefore, only the values of the PK-PD parameters for each individual were estimated at this step. Then, as data accumulated, we performed additional modeling to find a better PK-PD model structure that optimally fits the data. Multi-compartment PK models, in addition to PD structures such as baseline cell count increase, were tested in the modeling process.
Random effects were also taken into consideration. The structure to describe the residual error, which refers to the deviation of each observation from the value predicted by the PK-PD model, was initially applied to both IDT and CDE procedures as follows:
$$ {\mathrm{DV}}_{ij}={\mathrm{IPRED}}_{ij}\cdot \left(1+{\varepsilon}_{\mathrm{prop},ij}\right)+{\varepsilon}_{\mathrm{add},ij} $$
where DV ij is the jth measured concentration or blood cell count in the ith individual, IPRED ij is the model-predicted value for the corresponding observation (DV ij ), and ε prop,ij and ε add,ij are the residual variabilities with means of 0 and variances of σ prop 2 and σ add 2, respectively. For the CDE step, BSV (η i ) of each PK and PD parameter was tested as follows:
$$ {P}_{ij}={TVP}_j\cdot \exp \left({\eta}_i\right) $$
where P ij is the jth model parameter in the ith individual and TVP j is the typical value of the jth model parameter. The BSV for each parameter was assumed to follow a normal distribution, with a mean of 0 and differing values of variance (described using the symbol ω i 2).
The first-order conditional estimation with interaction option (FOCE-I) method was used whenever applicable. Model adequacies were assessed based upon goodness-of-fit plots, likelihood ratio tests (LRT), and model stability measures (e.g., successful convergence, matrix singularity, and significant digits). Cutoff criteria incorporated a p value of 0.05 (e.g., 3.84 for one parameter addition, 5.99 for two) for LRT to determine statistically significant improvements in the model.
Covariate analysis was performed for potential covariates, including demographic variables (sex, age, baseline body weight, and surface area) and clinical variables (mainly results from laboratory tests). After covariate screening via visual correlation check-ups and generalized additive modeling (GAM) procedures, the variables selected from the screening were tested as fixed effects for a certain PK-PD parameter, using LRT and decreases in BSV for the corresponding parameter.
This research was supported by the Janssen Pharmaceutical Companies of Johnson & Johnson.
An immediate family member of the author Yoo-Jin Kim has been employed, has had a leadership role, and has owned stock of the Janssen Pharmaceutical Companies of Johnson & Johnson.
SH, Y-JK, JL, S-EL, C-KM, and D-SY wrote the manuscript. SH, Y-JK, J-HY, S-AY, S-EL, K-SE, H-JK, C-KM, SL, and D-SY designed the research. SH, Y-JK, JL, SJ, TH, G-JP, S-HS, S-EL, and D-SY performed the research. SH, Y-JK, JL, SJ, TH, and D-SY analyzed data. All authors read and approved the final manuscript.
Department of Pharmacology, College of Medicine, The Catholic University of Korea, 222 Banpo-Daero, Seochogu, Seoul, Republic of Korea
PIPET (Pharmacometrics Institute for Practical Education and Training), 222 Banpo-Daero, Seochogu, Seoul, Republic of Korea
Catholic Blood and Marrow Transplantation Center, Seoul St. Mary's Hospital, The Catholic University of Korea, 222 Banpo-Daero, Seochogu, Seoul, Republic of Korea
Esteller M. Epigenetics in cancer. N Engl J Med. 2008;358:1148–59.View ArticlePubMedGoogle Scholar
Issa JP, Baylin SB, Herman JG. DNA methylation changes in hematologic malignancies: biologic and clinical implications. Leukemia. 1997;11:S7–11.PubMedGoogle Scholar
van der Helm LH, Scheepers ER, Veeger NJ, Daenen SM, Mulder AB, van den Berg E, et al. Azacitidine might be beneficial in a subgroup of older AML patients compared to intensive chemotherapy: a single centre retrospective study of 227 consecutive patients. J Hematol Oncol. 2013;6:29.PubMed CentralView ArticlePubMedGoogle Scholar
Pleyer L, Stauder R, Burgstaller S, Schreder M, Tinchon C, Pfeilstocker M, et al. Azacitidine in patients with WHO-defined AML - results of 155 patients from the Austrian Azacitidine Registry of the AGMT-Study Group. J Hematol Oncol. 2013;6:32.PubMed CentralView ArticlePubMedGoogle Scholar
Bachegowda L, Gligich O, Mantzaris I, Schinke C, Wyville D, Carrillo T, et al. Signal transduction inhibitors in treatment of myelodysplastic syndromes. J Hematol Oncol. 2013;6:50.PubMed CentralView ArticlePubMedGoogle Scholar
Hájková H, Fritz MH, Haškovec C, Schwarz J, Šálek C, Marková J, et al. CBFB-MYH11 hypomethylation signature and PBX3 differential methylation revealed by targeted bisulfite sequencing in patients with acute myeloid leukemia. J Hematol Oncol. 2014;7:66.PubMed CentralView ArticlePubMedGoogle Scholar
Jabbour E, Issa JP, Garcia-Manero G, Kantarjian H. Evolution of decitabine development: accomplishments, ongoing investigations, and future strategies. Cancer. 2008;112:2341–51.View ArticlePubMedGoogle Scholar
Santini V, Kantarjian HM, Issa JP. Changes in DNA methylation in neoplasia: pathophysiology and therapeutic implications. Ann Intern Med. 2001;134:573–86.View ArticlePubMedGoogle Scholar
deVos D, van Overveld W. Decitabine: a historical review of the development of an epigenetic drug. Ann Hematol. 2001;84:S3–8.View ArticleGoogle Scholar
Lübbert M. DNA methylation inhibitors in the treatment of leukemias, myelodysplastic syndromes and hemoglobinopathies: clinical results and possible mechanisms of action. Curr Top Microbiol Immunol. 2000;249:135–64.PubMedGoogle Scholar
Estey EH. Epigenetics in clinical practice: the examples of azacitidine and decitabine in myelodysplasia and acute myeloid leukemia. Leukemia. 2013;27:1803–12.View ArticlePubMedGoogle Scholar
Kantarjian HM, Thomas XG, Dmoszynska A, Wierzbowska A, Mazur G, Mayer J, et al. Multicenter, randomized, open-label, phase III trial of decitabine versus patient choice, with physician advice, of either supportive care or low-dose cytarabine for the treatment of older patients with newly diagnosed acute myeloid leukemia. J ClinOncol. 2012;30:2670–7.View ArticleGoogle Scholar
Issa JP, Gharibyan V, Cortes J, Jelinek J, Morris G, Verstovsek S, et al. Phase II study of low-dose decitabine in patients with chronic myelogenous leukemia resistant to imatinibmesylate. J ClinOncol. 2005;23:3948–56.View ArticleGoogle Scholar
Oki Y, Kantarjian HM, Gharibyan V, Jones D, O'brien S, Verstovsek S, et al. Phase II study of low-dose decitabine in combination with imatinibmesylate in patients with accelerated or myeloid blastic phase of chronic myelogenous leukemia. Cancer. 2007;109:899–906.View ArticlePubMedGoogle Scholar
Saunthararajah Y, Molokie R, Saraf S, Sidhwani S, Gowhari M, Vara S, et al. Clinical effectiveness of decitabine in severe sickle cell disease. Br J Haematol. 2008;141:126–9.View ArticlePubMedGoogle Scholar
Kantarjian H, Issa JP, Rosenfeld CS, Bennett JM, Albitar M, DiPersio J, et al. Decitabine improves patient outcomes in myelodysplastic syndromes: results of a phase III randomized study. Cancer. 2006;106:1794–803.View ArticlePubMedGoogle Scholar
Santos FP, Kantarjian H, Garcia-Manero G, Issa JP, Ravandi F. Decitabine in the treatment of myelodysplastic syndromes. Expert Rev Anticancer Ther. 2010;10:9–22.View ArticlePubMedGoogle Scholar
Wijermans PW, Lübbert M, Verhoef G, Klimek V, Bosly A. An epigenetic approach to the treatment of advanced MDS; the experience with the DNA demethylating agent 5-aza-2'-deoxycytidine (decitabine) in 177 patients. Ann Hematol. 2005;84:S9–17.View ArticleGoogle Scholar
Lübbert M, Suciu S, Baila L, Rüter BH, Platzbecker U, Giagounidis A, et al. Low-dose decitabine versus best supportive care in elderly patients with intermediate- or high-risk myelodysplastic syndrome (MDS) ineligible for intensive chemotherapy: final results of the randomized phase III study of the European Organisation for Research and Treatment of Cancer Leukemia Group and the German MDS Study Group. J Clin Oncol. 2011;29:1987–96.View ArticlePubMedGoogle Scholar
Kantarjian HM, O'Brien S, Huang X, Garcia-Manero G, Ravandi F, Cortes J, et al. Survival advantage with decitabine versus intensive chemotherapy in patients with higher risk myelodysplastic syndrome: comparison with historical experience. Cancer. 2007;109:1133–7.PubMed CentralView ArticlePubMedGoogle Scholar
Steensma DP, Baer MR, Slack JL, Buckstein R, Godley LA, Garcia-Manero G, et al. Multicenter study of decitabine administered daily for 5 days every 4 weeks to adults with myelodysplastic syndromes: the alternative dosing for outpatient treatment (ADOPT) trial. J Clin Oncol. 2009;27:3842–8.View ArticlePubMedGoogle Scholar
Joeckel TE, Lübbert M. Clinical results with the DNA hypomethylating agent 5-aza-2'-deoxycytidine (decitabine) in patients with myelodysplastic syndromes: an update. Semin Hematol. 2012;49:330–41.View ArticlePubMedGoogle Scholar
Goodyear O, Agathanggelou A, Novitzky-Basso I, Siddique S, McSkeane T, Ryan G, et al. Induction of a CD8+ T-cell response to the MAGE cancer testis antigen by combined treatment with azacitidine and sodium valproate in patients with acute myeloid leukemia and myelodysplasia. Blood. 2010;116:1908–18.View ArticlePubMedGoogle Scholar
Choi J, Ritchey J, Prior JL, Holt M, Shannon WD, Deych E, et al. In vivo administration of hypomethylating agents mitigate graft-versus-host disease without sacrificing graft-versus-leukemia. Blood. 2010;116:129–39.PubMed CentralView ArticlePubMedGoogle Scholar
Schroeder T, Fröbel J, Cadeddu RP, Czibere A, Dienst A, Platzbecker U, et al. Salvage therapy with azacitidine increases regulatory T cells in peripheral blood of patients with AML or MDS and early relapse after allogeneic blood stem cell transplantation. Leukemia. 2013;27:1910–3.View ArticlePubMedGoogle Scholar
Jabbour E, Giralt S, Kantarjian H, Garcia-Manero G, Jagasia M, Kebriaei P, et al. Low‐dose azacitidine after allogeneic stem cell transplantation for acute leukemia. Cancer. 2009;115:1899–905.PubMed CentralView ArticlePubMedGoogle Scholar
de Lima M, Giralt S, Thall PF, de Padua SL, Jones RB, Komanduri K, et al. Maintenance therapy with low-dose azacitidine after allogeneic hematopoietic stem cell transplantation for recurrent acute myelogenous leukemia or myelodysplastic syndrome: a dose and schedule finding study. Cancer. 2010;116:5420–31.View ArticlePubMedGoogle Scholar
Schroeder T, Czibere A, Platzbecker U, Bug G, Uharek L, Luft T, et al. Azacitidine and donor lymphocyte infusions as first salvage therapy for relapse of AML or MDS after allogeneic stem cell transplantation. Leukemia. 2013;27:1229–35.View ArticlePubMedGoogle Scholar
Platzbecker U, Wermke M, Radke J, Oelschlaegel U, Seltmann F, Kiani A, et al. Azacitidine for treatment of imminent relapse in MDS or AML patients after allogeneic HSCT: results of the RELAZA trial. Leukemia. 2012;26:381–9.PubMed CentralView ArticlePubMedGoogle Scholar
Storer BE. Design and analysis of phase I clinical trials. Biometrics. 1989;45:925–37.View ArticlePubMedGoogle Scholar
Piantadosi S, Fisher JD, Grossman S. Practical implementation of a modified continual reassessment method for dose-finding trials. Cancer ChemotherPharmacol. 1998;41:429–36.Google Scholar
Wallin JE, Friberg LE, Karlsson MO. Model-based neutrophil-guided dose adaptation in chemotherapy: evaluation of predicted outcome with different types and amounts of information. Basic Clin Pharmacol Toxicol. 2010;106:234–42.View ArticlePubMedGoogle Scholar
Wallin JE, Friberg LE, Karlsson MO. A tool for neutrophil guided dose adaptation in chemotherapy. Comput Methods Programs Biomed. 2009;93:283–91.View ArticlePubMedGoogle Scholar
Liu Z, Marcucci G, Byrd JC, Grever M, Xiao J, Chan KK. Characterization of decomposition products and preclinical and low dose clinical pharmacokinetics of decitabine (5-aza-2'-deoxycytidine) by a new liquid chromatography/tandem mass spectrometry quantification method. Rapid Commun Mass Spectrom. 2006;20:1117–26.View ArticlePubMedGoogle Scholar
Cashen AF, Shah AK, Todt L, Fisher N, DiPersio J. Pharmacokinetics of decitabine administered as a 3-h infusion to patients with acute myeloid leukemia (AML) or myelodysplastic syndrome (MDS). Cancer Chemother Pharmacol. 2008;61:759–66.View ArticlePubMedGoogle Scholar
Karahoca M, Momparler RL. Pharmacokinetic and pharmacodynamic analysis of 5-aza-2'-deoxycytidine (decitabine) in the design of its dose-schedule for cancer therapy. Clin Epigenetics. 2013;5:3–18.PubMed CentralView ArticlePubMedGoogle Scholar
Milhem M, Mahmud N, Lavelle D, Araki H, DeSimone J, Saunthararajah Y, et al. Modification of hematopoietic stem cell fate by 5aza 2'deoxycytidine and trichostatin A. Blood. 2004;103:4102–10.View ArticlePubMedGoogle Scholar
Young JC, Wu S, Hansteen G, Du C, Sambucetti L, Remiszewski S, et al. Inhibitors of histone deacetylases promote hematopoietic stem cell self-renewal. Cytotherapy. 2004;6:328–36.View ArticlePubMedGoogle Scholar
Chung YS, Kim HJ, Kim TM, Hong SH, Kwon KR, An S, et al. Undifferentiated hematopoietic cells are characterized by a genome-wide undermethylation dip around the transcription start site and a hierarchical epigenetic plasticity. Blood. 2009;114:4968–78.View ArticlePubMedGoogle Scholar
Suzuki M, Harashima A, Okochi A, Yamamoto M, Nakamura S, Motoda R, et al. 5-Azacytidine supports the long-term repopulating activity of cord blood CD34(+) cells. Am J Hematol. 2004;77:313–5.View ArticlePubMedGoogle Scholar
Negrotto S, Ng KP, Jankowska AM, Bodo J, Gopalan B, Guinta K, et al. CpG methylation patterns and decitabine treatment response in acute myeloid leukemia cells and normal hematopoietic precursors. Leukemia. 2012;26:244–54.PubMed CentralView ArticlePubMedGoogle Scholar
Greenberg P, Cox C, LeBeau MM, Fenaux P, Morel P, Sanz G, et al. International scoring system for evaluating prognosis in myelodysplastic syndromes. Blood. 1997;89:2079–88.PubMedGoogle Scholar
Kantarjian H, Oki Y, Garcia-Manero G, Huang X, O'Brien S, Cortes J, et al. Results of a randomized of 3 schedules of low-dose decitabine in higher-risk myelodysplastic syndrome and chronic myelomonocytic leukemia. Blood. 2007;109:52–7.View ArticlePubMedGoogle Scholar | CommonCrawl |
Department of Algebra
| Seminars | History of the Department | Areas of Research | Main Results | Awards | International Relations | Publications |
Parshin Aleksei Nikolaevich
Doctor Phys.-Math. Sci., Academician of RAS, Head of Department
office: 525; tel.: +7 (499) 941 01 79, +7 (495) 984 81 41 * 39 33; e-mail: [email protected]
Principal fields of research: Аlgebraic number theory and Galois theory. Algebraic geometry and n-dimensional local fields and their applications to arithmetics, geometry of manifolds, integrable systems, and quatum field theory. History of mathematics.
Gorchinskiy Sergey Olegovich
Doctor Phys.-Math. Sci., Senior Scientific Researcher
office: 409; tel.: +7 (499) 941 01 79, +7 (495) 984 81 41 * 35 33; e-mail: [email protected]
Principal fields of research: Algebraic geometry, arithmetic geometry, higher-dimensional adeles, K-theory, algebraic cycles.
Kulikov Viktor Stepanovich
Doctor Phys.-Math. Sci., Professor, Leading Scientific Rsearcher
office: 524; tel.: +7 (499) 941 01 79, +7 (495) 984 81 41 * 36 70; e-mail: [email protected]
Principal fields of research: Algebraic geometry and topology of algebraical manifolds.
Nikulin Vyacheslav Valentinovich
Doctor Phys.-Math. Sci., Leading Scientific Researcher
e-mail: [email protected]
Principal fields of research: Algebraic Geometry. Integer-valued quadratic forms generated by reflections in hyperbolic spaces. Automorphic forms. Lorentzian Kac-Moody algebras.
Osipov Denis Vasil'evich
office: 540; tel.: +7 (499) 941 01 79, +7 (495) 984 81 41 * 39 32; e-mail: [email protected]
Principal fields of research: Algebraic geometry. Algebraical number theory. Integrable systems.
Popov Vladimir Leonidovich
Doctor Phys.-Math. Sci., Professor, Corresponding Member of RAS, Chief Scientific Researcher
office: 524; tel.: +7 (499) 941 01 79, +7 (495) 984 81 41 * 36 70; e-mail: [email protected]
Principal fields of research: Algebraic transformation groups. Invariant theory. Algebraic groups and their representation theory. Homogeneous spaces. Lie groups and Lie algebras. Algebro-geometric aspects of algebraic transformation group theory. Affine algebraic geometry. Automorphism groups of algebraic varieties. Discrete reflection groups.
Abrashkin Viktor Aleksandrovich
Doctor Phys.-Math. Sci., Out-Of-Staff Member
e-mail: [email protected]
Personal page: http://maths.dur.ac.uk/~dma0va/
Principal fields of research: Galois moduli of finite group schemes. $p$-Adic representations for the Galois group of local fields. The Iwasawa theory. Theory of $p$-extensions of local and global fields. Highest theory of branching.
Mikhailov Roman Valer'evich
e-mail: [email protected]
Principal fields of research: Group theory, topology, category theory, algebraic K-theory.
Trepalin Andrey Sergeevich
Scientific Researcher
e-mail: [email protected]
Tyurin Dmitry Nikolaevich
e-mail: [email protected]
Kostrikin Aleksei Ivanovich (12.02.1929 – 22.09.2000)
Doctor Phys.-Math. Sci., Corresponding Member of USSR Academy of Sciences
Shafarevich Igor Rostislavovich (03.06.1923 – 19.02.2017)
Doctor Phys.-Math. Sci., Academician of RAS
Personal page: http://www.mi-ras.ru/~shafarev
Principal fields of research: Algebraic number theory. Algebraic geometry. Theory of Lie groups and Lie algebras. Commutative and associative algebras.
Tyurin Andrei Nikolaevich (24.02.1940 – 27.10.2002)
Doctor Phys.-Math. Sci., Corresponding Member of RAS
Voronin Sergei Mikhailovich (11.03.1948 – 18.10.1997)
Doctor Phys.-Math. Sci.
Seminar of the Department of Algebra and of the Department of Algebraic Geometry (Shafarevich Seminar)
Seminar Chairmen: A. N. Parshin; D. O. Orlov
Steklov Mathematical Institute, room 540 (Gubkina 8)
Seminar on Arithmetic Algebraic Geometry
Seminar Chairman: A. N. Parshin
Steklov Mathematical Institute, Room 540 (8 Gubkina)
Arithmetic geometry seminar
Seminar Chairmen: S. O. Gorchinskiy; D. V. Osipov; S. Yu. Rybakov; V. A. Vologodsky
Seminar of the Department of Algebra
Seminar Chairmen: I. R. Shafarevich; A. N. Parshin
The Department of Algebra was created in the middle of 1930's. B. N. Delone was the first head. The list of people working at the department in the late 1930's and 1940's includes: O. Yu. Schmidt, S. A. Chunikhin, I. M. Gelfand, A. I. Malcev.
Starting from 1946 I. R. Shafarevich is working at the department, being its head from 1960 to 1995. Many people in the department are his pupils: A. I. Lapin (1950 and 1957–1969), A. I. Kostrikin (1956–2000; starting from 1977 he was also the head of the Chair of Algebra at the Moscow State University), S. P. Demushkin (1959–1975), A. B. Zhizhchenko (1959–1965), Yu. I. Manin (since 1960), A. N. Tyurin (1963–2002), V. A. Demyanenko (1967–1969), A. N. Parshin (since 1968, starting from 1995 he is the head of the department), S. Yu. Arakelov (PhD student from 1971 to 1974), V. V. Nikulin (1987–2002, out-of-staff member since 2002), V. A. Kolyvagin (1988–2004, out-of-staff member from 2004 to 2011), V. A. Abrashkin (1996–2002, out-of-staff member since 2002), Vik. S. Kulikov (PhD student from 1974 to 1977, then a member since 1997).
The following people have been working at the department: S. P. Novikov (1960–1975), F. A. Bogomolov (PhD student from 1970 to 1973, employee from 1973 to 1993, out-of-staff member until 2011), M. M. Kapranov (1986–1990), S. A. Stepanov (1987–2000), A. T. Fomenko (1998–2001).
Now the following people also work at the department: A. I. Bondal (since 1994), D. O. Orlov (since 1996), D. V. Osipov (since 1999), V. L. Popov (since 2002), D. B. Kaledin (since 2002), A. G. Kuznetsov (since 2002), R. V. Mikhailov (since 2004), V. V. Shokurov (since 2004), S. O. Gorchinskiy (since 2007), C. A. Shramov (since 2008), I. D. Shkredov (since 2010), A. I. Efimov (since 2010).
In 2009 the Department of Algebra was united with the Department of Number Theory. The following people have thus entered the department: G. I. Arkhipov, M. M. Grinenko (out-of-staff member since 2011), M. A. Korolev, V. V. Przyjalkowski, A. V. Pukhlikov, I. S. Rezvyakova.
In 2012 the Department of Algebraic Geometry was created on the base of the Department of Algebra and Number Theory. The following people are the members of the new department: D. O. Orlov (head of the department), A. I. Bondal, M. M. Grinenko, A. I. Efimov, D. B. Kaledin, A. G. Kuznetsov, V. V. Przyjalkowski, A. V. Pukhlikov, V. V. Shokurov, C. A. Shramov.
Algebraic number theory, Galois theory, Lie groups and Lie algebras, theory of algebraic groups, algebraic geometry (especially the category theory of coherent sheaves, birational geometry, invariant theory), arithmetic of algebraic varieties, algebraic and differential topology, mathematical physics, combinatorial group theory, homological algebra, representations of groups, mirror symmetry, theory of adeles.
1. Algebraic number theory and Galois theory
Construction of a general reciprocity law (I. R. Shafarevich, A. I. Lapin), solution of the inverse problem of the Galois theory for solvable groups (I. R. Shafarevich).
Description of p-extensions of local and global fields (I. R. Shafarevich, S. P. Demushkin, H. Koch), solution of the problem of class field tower (E. S. Golod and I. R. Shafarevich), the structure of the Galois group for local fields (V. A. Abrashkin).
Theory of Euler systems (V. A. Kolyvagin).
2. Lie groups and Lie algebras
Semi-simple subgroups of Lie groups, nilmanifolds (A. I. Malcev).
Theory of infinite-dimensional representations of classical Lie groups (I. M. Gelfand and M. A. Naimark).
Solution of the weak Burnside problem for an arbitrary prime exponent (A. I. Kostrikin).
Classification of simple Lie algebras in the positive characteristic (A. I. Kostrikin and I. R. Shafarevich).
Integral lattices and orthogonal decompositions of Lie algebras (A. I. Kostrikin).
Theory of Kac–Moody Lorentz algebras (V. A. Gritsenko, V. V. Nikulin).
3. Algebraic geometry
Geometry of algebraic varieties: pencils of elliptic curves (I. R. Shafarevich), the Gauss–Manin connection (Yu. I. Manin), finiteness theorems for families of curves (A. N. Parshin and S. Yu. Arakelov).
Theory of vector bundles: classification and Torelli type theorems for vector bundles over algebraic curves, the problem of a bundle of quadrics, vector bundles over an infinite-dimensional projective space (A. N. Tyurin).
Theory of algebraic K3 surfaces and manifolds with a trivial canonical class: Torelli theorem (I. I. Piatetski-Shapiro, I. R. Shafarevich), structure of K3 surfaces in positive characteristic (A. N. Rudakov, I. R. Shafarevich), group of automorphisms, topological classification (V. V. Nikulin), surjectivity of the period map (V. S. Kulikov), classification of complex manifolds with trivial canonical class (F. A. Bogomolov).
Solution of three-dimensional Lьroth problem (V. A. Iskovskikh, Yu. I. Manin).
Flat and projective structures on Riemann surfaces (A. N. Tyurin).
Theory of stable vector bundles on algebraic varieties (F. A. Bogomolov).
Arithmetic groups in hyperbolic spaces and integral lattices (V. V. Nikulin).
Smooth invariants of algebraic surfaces (V. Y. Pidstrigach, A. N. Tyurin).
Derived categories of coherent sheaves on algebraic varieties and equivalences between them for varieties with ample and anti-ample canonical class, as well as for abelian varieties (M. M. Kapranov, A. I. Bondal, D. O. Orlov).
Derived categories of coherent sheaves on a symplectic resolution of an arbitrary singularity (D. B. Kaledin).
Theorem on integral kernel for an equivalence between derived categories of coherent sheaves on possibly singular projective varieties (D. O. Orlov, V. A. Lunts).
Theory of homological projective duality (A. Kuznetsov).
Derived categories of coherent sheaves on isotropic Grassmannians (A. G. Kuznetsov, A. E. Polishchuk).
Prym varieties, their difference with Jacobians, and applications to the three-dimensional birational geometry (V. V. Shokurov).
Minimal model program and its applications to higher-dimensional geometry. Moduli of polarized log pairs and positivity of the module part in the adjunction formula (V. V. Shokurov).
Application of unramified Brauer group to the unirationality problem for algebraic varieties (F. A. Bogomolov).
Topology of algebraic surfaces: Chisini conjecture for generic projections of algebraic surfaces onto projective plane, counterexamples in deformation theory, description of components of the Hurwitz spaces of coverings of algebraic curves (Vik. S. Kulikov).
Birational geometry of Fano varieties: description of the structures of a rationally connected fibration on Fano double spaces of index 2 and dimension 5 and above, computation of the group of birational automorphisms and the proof of non-rationality (A. V. Pukhlikov).
Invariant theory: algebraic groups as groups of automorphisms of algebras, solution of the problem of rationality of the function field on a connected semisimple algebraic group over the subfield of central functions, description of Cayley groups (V. L. Popov).
Applications of algebraic geometry and Tannakian categories to the differential Galois theory, parametrized Picard–Vessiot extensions (S. O. Gorchinskiy, A. I. Ovchinnikov).
Deformation quantization of algebraic varieties over a field of positive characteristic. Noncommutative analogues of the Cartier morphism and the Frobenius map for cyclic homology (D. B. Kaledin).
4. Arithmetic of algebraic varieties
Diophantine equations of degree three (B. N. Delone and D. K. Faddeev).
Arithmetic of elliptic curves and abelian varieties: theory of principal homogeneous spaces (I. R. Shafarevich), unboundedness of rank over function fields (A. I. Lapin), boundedness of p-torsion of elliptic curves (Yu. I. Manin), estimates for the torsion of elliptic curves (V. A. Demyanenko), canonical heights of abelian varieties (A. N. Parshin), l-adic representations of Galois groups associated with abelian varieties, the group of points of finite order (F. A. Bogomolov), proof of the nonexistence of smooth abelian schemes over Z (V. A. Abrashkin).
Finiteness theorems in Diophantine geometry: proof of the Mordell conjecture on rational points over function fields (Yu. I. Manin), method of ramified coverings (A. N. Parshin), finiteness of the Tate–Shafarevich group for modular curves (V. A. Kolyvagin).
Arithmetic surfaces (Arakelov geometry).
Arithmetic of rational and cubic surfaces (Yu. I. Manin, V. A. Iskovskikh).
Theory of p-adic L-functions and modular forms (Yu. I. Manin).
Theory of n-dimensional local fields and its applications to class field theory, vector bundles and the theory of algebraic groups (A. N. Parshin).
Theory of adeles: measure theory and harmonic analysis on adelic spaces of two-dimensional schemes (D. V. Osipov, A. N. Parshin), symbols and reciprocity laws (D. V. Osipov), adelic resolutions for sheaves (S. O. Gorchinskiy, D. V. Osipov).
5. Algebraic and differential topology
Theory of cohomological operations. Description of complex cobordisms. Classification of simply connected smooth manifolds of dimension ≷ 4. Proof of topological invariance of Pontryagin classes. Theory of foliations on smooth manifolds. Foundations of Hermitian K-theory (S. P. Novikov).
Theory of derived functors for non-additive functors (R. V. Mikhailov, L. Breen).
Functorial methods in the unstable homotopy theory (R. V. Mikhailov).
6. Mathematical physics
Solution of the periodic problem for the KdV equation by methods of algebraic geometry (S. P. Novikov).
Classification of instantons (V. G. Drinfeld, Yu. I. Manin).
Models of classical field theory: supergeometry, the Yang–Mills theory, and string theory (Yu. I. Manin, M. M. Kapranov).
Description of instantons on noncommutative spaces and noncommutative twistor transform (A. Kapustin, A. G. Kuznetsov, D. O. Orlov).
Homological mirror symmetry and the category of D-branes for Landau–Ginzburg models (D. O. Orlov). Homological mirror symmetry for curves of genus greater than one (A. I. Efimov) and del Pezzo surfaces (D. O. Orlov).
7. Combinatorial group theory and applications
Theory of central series for groups (R. V. Mikhailov).
Description of homotopy groups of spheres in terms of group theory (R. V. Mikhailov, J. Wu).
8. Representation theory
Moduli space of representations of Lie algebras in positive characteristic (I. R. Shafarevich, A. N. Rudakov).
Classification and character theory for irreducible representations with finite weight of discrete Heisenberg groups (A. N. Parshin, S. A. Arnal).
Among the members of the department there are recipients of the Fields Medal (S. P. Novikov), the Lenin Prize (I. R. Shafarevich, Yu. I. Manin, S. P. Novikov), the State USSR Prize (A. I. Kostrikin, S. A. Stepanov), the Lomonosov Prize (A. I. Kostrikin), the Alexander von Humboldt Prize (A. N. Parshin), the European Mathematical Society Prize (A. G. Kuznetsov), Russian Federation President Prize in Science and Innovation for Young Scientists (A. G. Kuznetsov) and others.
Department members have been repeatedly invited to the International Congresses of Mathematicians as speakers:
I. R. Shafarevich (Stokholm, 1962; Nice, 1970),
A. I. Kostrikin (Stokholm, 1962; Nice, 1970),
S. P. Novikov (Stokholm, 1962; Moscow, 1966; Nice, 1970),
Yu. I. Manin (Stokholm, 1962; Nice, 1970; Helsinki, 1978; Berkeley, 1986),
A. N. Parshin (Nice, 1970),
S. Yu. Arakelov (Vancouver, 1974),
F. A. Bogomolov (Helsinki, 1978),
V. V. Nikulin (Berkeley, 1986),
V. L. Popov (Berkeley, 1986),
V. A. Kolyvagin (Kioto, 1990),
A. I. Bondal (Pekin, 2002),
D. O. Orlov (Pekin, 2002),
D. B. Kaledin (Hyderabad, 2010).
The Department of Algebra and Number Theory has many relations with Russian and foreign mathematicians. Department visitors include:
V. Alexeev, A. N. Andrianov, R. V. Bezrukavnikov, A. A. Beilinson, N. A. Vavilov, B. B. Venkov, A. M. Vershik, V. A. Voevodsky, V. E. Voskresensky, S. V. Vostokov, V. Ginzburg, V. A. Gritsenko, V. I. Guletskii, A. S. Dzhumadildaev, N. V. Durov, Yu. L. Ershov, Yu. G. Zarhin, M. M. Kapranov, A. A. Klyachko, M. L. Kontsevich, V. A. Lunts, S. A. Merkulov, I. A. Panin, A. A. Panchishkin, F. V. Petrov, V. P. Platonov, A. E. Polishchuk, Yu. G. Prokhorov, A. A. Rosly, A. N. Skorobogatov, A. L. Smirnov, S. G. Tankeev, A. S. Tikhomirov, N. A. Tyurin, L. D. Faddeev, V. M. Kharlamov, I. A. Cheltsov, V. I. Janchevsky, W. Baily, L. Bers, L. Breen, F. Campana, J. W. S. Cassels, A. Corti, P. Deligne, H. Esnault, G. van der Geer, D. Gieseker, Ph. Griffiths, M. Harris, M. Hazewinkel, F. Hirzebruch, R. Holzapfel, E. Kaehler, E. Kani, L. Katzarkov, B. Keller, H. Koch, S. Lang, R. P. Langlands, R. MacPherson, Y. Miyaoka, D. Mumford, M. S. Narasimhan, A. Neeman, H. Opolka, T. Pantev, I. B. Passi, G. Prasad, M. Raghunathan, M. Reid, N. Schappacher, T. Shioda, J.-P. Serre, C. S. Seshadri, J. Tate, A. Todorov, J.-L. Verdier, E. Vieweg, M. Wodzicki, G. Wuestholz, D. Zagier, E.-W. Zink, T. Zink, and many others.
The department actively cooperates with many institutes and universities, including:
Moscow State University, Novosibirsk State University, Independent University of Moscow, St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg State University, Yaroslavl State Pedagogical University, ETH (Switzerland), Harish-Chandra Research Institute (India), IAS (U.S.A.), IHES (France), IPMU (Japan), ICTP (Italy), London Imperial College (UK), MPIM (Germany), POSTECH (South Korea), Punjab University (India), RIMS (Japan), TIFR (India), University of Durham (UK), University of Edinburgh (UK), University of Liverpool (UK), University of Warwick (UK), University of Vienna (Austria).
1. V. V. Nikulin, "Classification of degenerations and Picard lattices of Kahlerian K3 surfaces with symplectic automorphism group D_6", Izv. RAN. Ser. Mat. (to appear)
2. V. V. Nikulin, Algebra, number theory, and algebraic geometry, Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich, Tr. Mat. Inst. Steklova, 307, Steklov Mathematical Institute of RAS, Moscow, 2019 (to appear)
3. A. B. Zheglov, D. V. Osipov, "Lax pairs for linear Hamiltonian systems", Siberian Mathematical Journal, 2019 (to appear) , arXiv: 1901.11130
4. Vik. S. Kulikov, Izv. RAN. Ser. Mat. (to appear)
5. Vik. S. Kulikov, "On germs of finite morphisms of smooth surfaces", Algebra, number theory, and algebraic geometry, Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich, Tr. Mat. Inst. Steklova, 307, Steklov Mathematical Institute of RAS, Moscow, 2019 (to appear)
6. V. L. Popov, "Three plots about the Cremona groups", Izv. RAN. Ser. Mat., 2019 (to appear) , arXiv: 1810.00824
7. Vladimir L. Popov, Variations on the theme of Zariski's Cancellation Problem, 2019 , 15 pp., arXiv: 1901.07030
8. Vladimir L. Popov, "On conjugacy of stabilizers of reductive group actions", Mathematical Notes, 105:4 (2019), 580–581 , arXiv: 1901.10858
9. V. L. Popov, "Orbit closures of the Witt group actions", Proc. Steklov Inst. Math., 307, 2019 (to appear)
10. V. L. Popov, "Rational differential forms on the variety of flexes of plane cubics", Uspekhi Mat. Nauk (to appear)
11. S. O. Ivanov, R. V. Mikhailov, V. A. Sosnilo, "Vysshie kopredely, proizvodnye funktory i kogomologii", Matem. sb., 210:9 (2019) (to appear)
12. V. V. Nikulin, "Classification of Picard lattices of K3 surfaces", Izv. Math., 82:4 (2018), 752–816 (cited: 1)
13. Valery Gritsenko, Viacheslav V. Nikulin, "Lorentzian Kac–Moody algebras with Weyl groups of 2-reflections", Proceedings of London Mathematical Society, 116:3 (2018), 485–533 (cited: 1) (cited: 1)
14. Viacheslav V. Nikulin, Classification of degenerations and Picard lattices of Kahlerian K3 surfaces with small finite symplectic automorphism groups, 2018 , 39 pp., arXiv: 1804.00991
15. D. V. Osipov, "Adelic quotient group for algebraic surfaces", St. Petersburg Mathematical Journal, 30 (2019), 111-122 , arXiv: 1706.09826
16. D. V. Osipov, "Arithmetic surfaces and adelic quotient groups", Izv. Math., 82:4 (2018), 817-836 , arXiv: 1801.02282 (cited: 1)
17. A. B. Zheglov, D. V. Osipov, "On first integrals of linear Hamiltonian systems", Dokl. Math., 98:3 (2018), 616–618
18. Vik. S. Kulikov, "On divisors of small canonical degree on Godeaux surfaces", Sb. Math., 209:8 (2018), 1155–1163 (cited: 1) (cited: 1)
19. Vik. S. Kulikov, On the variety of the inflection points of plane cubic curves, 2018 , 27 pp., arXiv: 1810.01705
20. Vik. S. Kulikov, On the almost generic covers of the projective plane, 2018 , 13 pp., arXiv: 1812.01313
21. Vladimir L. Popov, "The Jordan property for Lie groups and automorphism groups of complex spaces", Math. Notes, 103:5 (2018), 811–819
22. Vladimir L. Popov, Yuri G. Zarhin, Root systems in number fields, 2018 , 15 pp., arXiv: 1808.01136
23. Vladimir L. Popov, Three plots about the Cremona groups, 2018 , 27 pp., arXiv: 1810.00824
24. Victor G. Kac, Vladimir L. Popov, Editors, Lie Groups, Geometry, and Representation Theory. A Tribute to the Life and Work of Bertram Kostant, Series ISSN 0743-1643, ISBN 978-3-030-02191-7, Progress in Mathematics, 326, First Edition, Birkhäuser Basel (Copyright Holder: Springer Nature Switzerland AG), Basel, 2018 , X, 538 pp. www.springer.com/us/book/9783030021900
25. Vladimir L. Popov, Yuri G. Zarhin, Root symstems in number fields, Preprint MPIM 18-38, Max-Planck-Institut für Mathematik, Bonn, 2018 , 19 pp. www.mpim-bonn.mpg.de/preblob/5898
26. Vladimir L. Popov, "Modality of representations, and packets for $\theta$-groups", Lie Groups, Geometry, and Representation Theory. A Tribute to the Life and Work of Bertram Kostant, Prog. Math., 326, Birkhäuser Basel (Copyright Holder: Springer Nature Switzerland AG), Basel, 2018, 459–479 , arXiv: 1707.07720
27. V. L. Popov, "Compressible finite groups of birational automorphisms", Dokl. Math., 98:2 (2018), 413–415
28. V. L. Popov, Yu. G. Zarhin, "Types of root systems in number fields", Dokl. Math., 98:3 (2018), 600–602
29. A. N. Parshin, Vestnik RAN, 88:11 (2018), 982–984
30. S. Gorchinskiy, V. Guletskiǐ, "Positive model structures for abstract symmetric spectra", Appl. Categ. Struct., 26:1 (2018), 29–46 , arXiv: 1108.3509v3 (cited: 1) (cited: 2)
31. S. O. Gorchinskiy, D. M. Krekov, "An explicit formula for the norm in the theory of fields of norms", Russian Math. Surveys, 73:2 (2018), 369–371
32. S. O. Gorchinskiy, D. N. Tyurin, "Relative Milnor $K$-groups and differential forms of split nilpotent extensions", Izv. Math., 82:5 (2018), 880–913
33. S.Gorchinskiy, C.Shramov, Unramified Brauer group and its applications, Translations of Mathematical Monographs, 246, American Mathematical Society, Providence, 2018 , xvii+179 pp.
34. S.O.Gorchinskii, K.A.Shramov, Nerazvetvlennaya gruppa Brauera i ee prilozheniya, MTsNMO, Moskva, 2018 , 200 pp.
35. Andrey Trepalin, "Quotients of del Pezzo surfaces of high degree", Transactions of the American Mathematical Society, 370:9 (2018), 6097–6124 , arXiv: https://arxiv.org/abs/1312.6904
36. Andrey Trepalin, "Quotients of del Pezzo surfaces of degree $2$", Moscow Mathematical Journal, 18:3 (2018), 557–597 , arXiv: https://arxiv.org/abs/1709.02006
37. Valery Gritsenko, Viacheslav V. Nikulin, Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac–Moody algebras, 2017 , 15 pp., arXiv: 1702.07551
38. V. V. Nikulin, "Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. III", Izv. Math., 81:5 (2017), 985–1029 (cited: 1) (cited: 1)
39. Viacheslav V. Nikulin, Classification of Picard lattices of K3 surfaces, 2017 , 68 pp., arXiv: 1707.05677
40. V. A. Gritsenko, V. V. Nikulin, "Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac–Moody algebras", Trans. Moscow Math. Soc., 78 (2017), 75–83
41. Denis V. Osipov, "Second Chern numbers of vector bundles and higher adeles", Bull. Korean Math. Soc., 54:5 (2017), 1699–1718 , arXiv: 1706.07354 (cited: 2) (cited: 2)
42. Vik. S. Kulikov, E. I. Shustin, "On $G$-Rigid Surfaces", Proc. Steklov Inst. Math., 298 (2017), 133–151 (cited: 3) (cited: 2)
43. Vik. S. Kulikov, "The Hesse curve of a Lefschetz pencil of plane curves", Russian Math. Surveys, 72:3 (2017), 574–576
44. Victor Abrashkin, "Groups of automorphisms of local fields of period $p^M$ and nilpotent class $<p$", Ann. Inst. Fourier (Grenoble), 67:2 (2017), 605–635 (cited: 1) (cited: 1)
45. Victor Abrashkin, "Groups of automorphisms of local fields of period $p$ and nilpotent class $<p$, I", Int. J. Math., 28:6 (2017), 1750043 , 34 pp. (cited: 1) (cited: 1)
46. Victor Abrashkin, "Groups of automorphisms of local fields of period $p$ and nilpotent class $<p$, II", Int. J. Math., 28:10 (2017), 1750066 , 32 pp.
47. Vladimir L. Popov, "Do we create mathematics or do we gradually discover theories which exist somewhere independently of us?", Eur. Math. Soc. Newsl., 107 (2017), 37
48. V. L. Popov, "Borel subgroups of Cremona groups", Mathematical Notes, 102:1 (2017), 60-67 (cited: 3) (cited: 1)
49. Vladimir L. Popov, Algebraic groups whose orbit closures contain only finitely many orbits, 2017 , 12 pp., arXiv: 1707.06914v1
50. Vladimir L. Popov, "Bass' triangulability problem", Algebraic varieties and automorphism groups, Adv. Stud. Pure Math., 75, Math. Soc. Japan, Kinokuniya, Tokyo, 2017, 425–441 bookstore.ams.org/aspm-75/, arXiv: 1504.03867
51. Vladimir L. Popov, "Discrete groups generated by complex reflections", VI-th conference on algebraic geometry and complex analysis for young mathematicians of Russia (Northern (Arctic) Federal University named after M. V. Lomonosov, Koryazhma, Arkhangelsk region, Russia, August 25–30, 2017), Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 2017, 13–14 www.mathnet.ru/php/conference.phtml?confid=1006&option_lang=eng
52. V. L. Popov, "On modality of representations", Dokl. Math., 96:1 (2017), 312–314 (cited: 1)
53. Sergey Gorchinskiy, "Integral Chow motives of threefolds with $K$-motives of unit type", Bull. Korean Math. Soc., 54:5 (2017), 1827–1849 , arXiv: 1703.06977 (cited: 2) (cited: 2)
54. S. Yu. Rybakov, A. S. Trepalin, "Minimal cubic surfaces over finite fields", Sb. Math., 208:9 (2017), 1399–1419 (cited: 4) (cited: 2)
55. Andrey Trepalin, "Minimal del Pezzo surfaces of degree $2$ over finite fields", Bull. Korean Math. Soc., 54:5 (2017), 1779–1801 , arXiv: https://arxiv.org/abs/1611.02832
56. V. V. Nikulin, "Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. II", Izv. Math., 80:2 (2016), 359–402 (cited: 2) (cited: 1)
57. Valery Gritsenko, Viacheslav V. Nikulin, Lorentzian Kac–Moody algebras with Weyl groups of 2-reflection, 2016 , 73 pp., arXiv: 1602.08359
58. Viacheslav V. Nikulin, "Kählerian K3 surfaces and Niemeier lattices, II", Adv. Stud. Pure Math., 69, 2016, 421–471 (cited: 2)
59. Denis Osipov, Xinwen Zhu, "The two-dimensional Contou-Carrère symbol and reciprocity laws", J. Algebraic Geom., 25 (2016), 703–774 , arXiv: 1305.6032 (cited: 5) (cited: 5)
60. D. V. Osipov, A. N. Parshin, "Representations of the Discrete Heisenberg Group on Distribution Spaces of Two-Dimensional Local Fields", Proc. Steklov Inst. Math., 292 (2016), 185–201 , arXiv: 1510.02423 (cited: 1)
61. Sergey O. Gorchinskiy, Denis V. Osipov, "Continuous homomorphisms between algebras of iterated Laurent series over a ring", Proc. Steklov Inst. Math., 294 (2016), 47–66 (cited: 8) (cited: 3)
62. S. O. Gorchinskiy, D. V. Osipov, "Higher-dimensional Contou-Carrère symbol and continuous automorphisms", Funct. Anal. Appl., 50:4 (2016), 268–280 (cited: 10) (cited: 8)
63. Viktor S. Kulikov, Eugenii Shustin, "On rigid plane curves", Eur. J. Math., 2:1 (2016), 208–226 , arXiv: 1501.03777 (cited: 1) (cited: 2)
64. Vik. S. Kulikov, "Plane rational quartics and K3 surfaces", Proc. Steklov Inst. Math., 294 (2016), 95–128 (cited: 4) (cited: 1)
65. Vik. S. Kulikov, "K3 poverkhnosti s deistviyami gruppy $S_4$ i ratsionalnye kvartiki", Mezhdunarodnaya konferentsiya po algebraicheskoi geometrii, kompleksnomu analizu i kompyuternoi algebre (Filial S(A)FU im. M. V. Lomonosova, g. Koryazhma Arkhangelskoi oblasti, Rossiya, 3–9 avgusta 2016 g.), Matematicheskii institut im. V.A. Steklova Rossiiskoi akademii nauk, Moskva, 2016, 40–42 http://www.mathnet.ru/ConfLogos/805/thesis.pdf
66. Vik.S. Kulikov, "A remark on classical Pluecker's formulae", Ann. Fac. Sci. Toulouse. Math., 25:5 (2016), 959–967 , arXiv: 1101.5042
67. Vladimir L. Popov, "Birational splitting and algebraic group actions", Eur. J. Math., 2:1 (2016), 283–290 https://www.math.uni-bielefeld.de/LAG/man/552.pdf, arXiv: 1502.02167 (cited: 3) (cited: 2)
68. V. L. Popov, G. V. Sukhotskii, Analiticheskaya geometriya. Uchebnik i praktikum, Bakalavr. Akademicheskii kurs, 2-e izd., per. i dop., Yurait, Moskva, 2016 , 232 pp. http://urait.ru/catalog/388730
69. V. L. Popov, "Algebras of General Type: Rational Parametrization and Normal Forms", Proc. Steklov Inst. Math., 292:1 (2016), 202–215 (cited: 1) (cited: 1)
70. V. L. Popov, "Subgroups of the Cremona groups: Bass' problem", Dokl. Math., 93:3 (2016), 307–309
71. V. L. Popov, "Rationality of (co)adjoint orbits", International conference on algebraic geometry, complex analysis and computer algebra (Northern (Arctic) Federal University named after M. V. Lomonosov, Koryazhma, Arkhangelsk region, Russia, August 03–09, 2016), Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 2016, 84–85 http://www.mathnet.ru/ConfLogos/805/thesis.pdf
72. Roman Mikhailov, Inder Bir S. Passi, "Generalized dimension subgroups and derived functors", J. Pure Appl. Algebra, 220:6 (2016), 2143–2163 (cited: 2) (cited: 2)
73. Sergei O. Ivanov, Roman Mikhailov, "On a problem of Bousfield for metabelian groups", Adv. Math., 290 (2016), 552–589 (cited: 4) (cited: 3)
74. V. G. Bardakov, R. Mikhailov, V. V. Vershinin, J. Wu, "On the pure virtual braid group $PV_3$", Comm. Algebra, 44:3 (2016), 1350–1378 (cited: 6) (cited: 7)
75. Roman Mikhailov, "On transfinite nilpotence of the Vogel–Levine localization", Forum Math., 28:2 (2016), 333–338 (cited: 1) (cited: 1)
76. Roman Mikhailov, "A one-relator group with long lower central series", Forum Math., 28:2 (2016), 327–331 (cited: 1) (cited: 1)
77. Roman Mikhailov, Inder Bir S. Passi, "The subgroup determined by a certain ideal in a free group ring", J. Algebra, 449 (2016), 400–407 (cited: 1) (cited: 1)
78. Roman Mikhailov, Jie Wu, "On the metastable homotopy of mod 2 Moore spaces", Algebr. Geom. Topol., 16:3 (2016), 1773–1797
79. I. V. Beloshapka, S. O. Gorchinskiy, "Irreducible representations of finitely generated nilpotent groups", Sb. Math., 207:1 (2016), 41–64 (cited: 2)
80. S. Gorchinskiy, V. Guletskiǐ, "Symmetric powers in abstract homotopy categories", Adv. Math., 292 (2016), 707–754 (cited: 3) (cited: 4)
81. Andrey Trepalin, "Quotients of conic bundles", Transformation Groups, 21:1 (2016), 275–295 , arXiv: https://arxiv.org/abs/1312.6867 (cited: 3) (cited: 2)
82. Andrey Trepalin, "Quotients of cubic surfaces", European Journal of Mathematics, 2:1 (2016), 333–359 , arXiv: https://arxiv.org/abs/1506.05138 (cited: 1) (cited: 1)
83. V. V. Nikulin, Degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups, II, 2015 , 55 pp., arXiv: 1504.00326v4
84. V. V. Nikulin, "Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups", Izv. Math., 79:4 (2015), 740–794 (cited: 4) (cited: 1) (cited: 1) (cited: 1)
85. V. V. Nikulin, "Degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups.", Conference on K3 surfaces and related topics (KIAS, Seoul, Korea, 16–20 November), 2015 , 1 pp. http://home.kias.re.kr/MKG/h/K3surfaces/
86. S. O. Gorchinskiy, D. V. Osipov, "Explicit formula for the higher-dimensional Contou-Carrère symbol", Russian Math. Surveys, 70:1 (2015), 171–173 (cited: 8) (cited: 1) (cited: 1) (cited: 3)
87. S. O. Gorchinskiy, D. V. Osipov, "A higher-dimensional Contou-Carrère symbol: local theory", Sb. Math., 206:9 (2015), 1191–1259 , arXiv: 1505.03829 (cited: 12) (cited: 1) (cited: 1) (cited: 7)
88. D. V. Osipov, "The Discrete Heisenberg Group and Its Automorphism Group", Math. Notes, 98:1 (2015), 185–188 , arXiv: 1505.00348 (cited: 3) (cited: 1) (cited: 1) (cited: 1)
89. S. O. Gorchinskiy, D. V. Osipov, "Tangent Space to Milnor $K$-Groups of Rings", Proc. Steklov Inst. Math., 290 (2015), 26–34 , arXiv: 1505.03780 (cited: 5) (cited: 1) (cited: 1) (cited: 2)
90. F. A. Bogomolov, Vik. S. Kulikov, "The ambiguity index of an equipped finite group", Eur. J. Math., 1:2 (2015), 260–278 , arXiv: 1404.5763 (cited: 2)
91. Vik. S. Kulikov, "Dualizing coverings of the plane", Izv. Math., 79:5 (2015), 1013–1042 (cited: 4) (cited: 2)
92. Viktor S. Kulikov, Eugenii Shustin, "Duality of planar and spacial curves: new insight", Eur. J. Math., 1:3 (2015), 462–482 , arXiv: 1412.1944 (cited: 1)
93. Vik. S. Kulikov, "O klassicheskikh formulakh Plyukkera", V Shkola-konferentsiya po algebraicheskoi geometrii i kompleksnomu analizu dlya molodykh matematikov Rossii. Tezisy dokladov. (g. Koryazhma Arkhangelskoi oblasti, Filial S(A)FU im. M.V. Lomonosova, 17–22 avgusta 2015 g.), MIAN, M., 2015, 50–54 http://www.mathnet.ru/ConfLogos/604/thesis-Koryazhma.pdf
94. I. R. Shafarevich, Collected mathematical papers, Reprint of the 1989 edition, Springer Collect. Works Math., Springer, Heidelberg, 2015 , x+769 pp.
95. V. Abrashkin, "Ramification estimate for Fontaine–Laffaille Galois modules", J. Algebra, 427 (2015), 319–328
96. Vladimir L. Popov, "Around the Abhyankar–Sathaye conjecture", Documenta Mathematica, 2015, Extra Volume:Alexander S. Merkurjev's Sixtieth Birthday (The Book Series, Vol. 7), 513–528 https://www.math.uni-bielefeld.de/documenta/vol-merkurjev/popov.html, arXiv: 1409.6330 (ISSN 1431-0643 (INTERNET), 1431-0635 (PRINT))
97. V. L. Popov, "Finite subgroups of diffeomorphism groups", Proc. Steklov Inst. Math., 289 (2015), 221–226 , arXiv: 1310.6548v2 (cited: 9) (cited: 5)
98. V. L. Popov, "Problema Bassa o trianguliruemosti podgrupp grupp Kremony", V shkola-konferentsiya po algebraicheskoi geometrii i kompleksnomu analizu dlya molodykh matematikov Rossii (g. Koryazhma Arkhangelskoi oblasti, Filial Severnogo (Arkticheskogo) federalnogo universiteta im. M. V. Lomonosova, 17–22 avgusta 2015 g.), Matematicheskii institut im. V.A. Steklova Rossiiskoi akademii nauk, Moskva, 2015, 83–87 http://www.mathnet.ru/ConfLogos/604/thesis-Koryazhma.pdf
99. V. L. Popov, "Number of components of the nullcone", Proc. Steklov Inst. of Math., 290 (2015), 84–90 , arXiv: 1503.08303 (cited: 2) (cited: 2)
100. Vladimir L. Popov, "On the equations defining affine algebraic groups", Pacific J. Math., 279:1-2, Special issue. In memoriam: Robert Steinberg (2015), 423–446 http://msp.org/pjm/2015/279-1/p19.xhtml, arXiv: 1508.02860 (cited: 1)
101. S. O. Ivanov, R. Mikhailov, "A higher limit approach to homology theories", J. Pure Appl. Algebra, 219:6 (2015), 1915–1939 (cited: 1) (cited: 3) (cited: 1) (cited: 4)
102. R. Mikhailov, K. E. Orr, "Group localization and two problems of Levine", Math. Z., 280:1 (2015), 355–366
103. A. N. Parshin, "On the direct image conjecture in the relative Langlands programme", Russian Math. Surveys, 70:5 (2015), 961–963 (cited: 1)
104. Sergey Gorchinskiy, Alexei Rosly, "A polar complex for locally free sheaves", Int. Math. Res. Not. IMRN, 2015:10 (2015), 2784–2829
105. V. V. Nikulin, "Elliptic fibrations on K3 surfaces", Proc. Edinb. Math. Soc. (2), 57:1 (2014), 253–267 (cited: 1) (cited: 1)
106. V. V. Nikulin, Degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups, 2014 , 70 pp., arXiv: 1403.6061v3
107. V. V. Nikulin, "Kahlerian K3 surfaces and Niemeier lattices", Workshop: Automorphic forms, Lie algebras and String theory (Lille University II, March 3–6), Lille, France, 2014 , 28 pp. http://www.ihes.fr/~vanhove/Lille2014/index.html
108. V. V. Nikulin, "Degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups", Conference: Moduli spaces of real and complex varieties (Angers University, June 2–6), Angers, France, 2014 , 1 pp. http://www.math.univ-angers.fr/~mangolte/Angers-2014-abstracts.pdf
109. H. Kurke, D. Osipov, A. Zheglov, "Commuting differential operators and higher-dimensional algebraic varieties", Selecta Math. (N.S.), 20:4 (2014), 1159–1195 , arXiv: 1211.0976 (cited: 4) (cited: 1)
110. Vik. S. Kulikov, V. M. Kharlamov, "On numerically pluricanonical cyclic coverings", Izv. Math., 78:5 (2014), 986–1005
111. Vik.S. Kulikov, On the Galois groups of the dualizing coverings for plane curves, 2014 , 10 pp., arXiv: 1403.1426
112. V. S. Kulikov, "Chislenno kratnokanonicheskie tsiklicheskie nakrytiya", V Rossiisko-armyanskoe soveschanie po matematicheskoi fizike, kompleksnomu analizu i smezhnym voprosam (28 sentyabrya – 3 oktyabrya 2014 g., Erevan), Erevan, 2014, 32–33
113. V. Abrashkin, "Group schemes of period 2", Manuscripta Math., 143:3-4 (2014), 317–353
114. V. L. Popov, "Quotients by conjugation action, cross-sections, singularities,and representation rings", Representation Theory and Analysis of Reductive Groups: Spherical Spaces and Hecke Algebras (Mathematisches Forschungsinstitut Oberwolfach, 19 January – 25 January 2014), Oberwolfach Reports, 11, no. 1, European Mathematical Society, 2014, 156–159
115. V. L. Popov, "On infinite dimensional algebraic transformation groups", Transform. Groups, 19:2, special issue dedicated to E. B. Dynkin's 90th anniversary (2014), 549–568 https://www.math.uni-bielefeld.de/LAG/man/523.pdf, arXiv: 1401.0278 (cited: 7) (cited: 1) (cited: 7)
116. V. L. Popov, "Jordan groups and automorphism groups of algebraic varieties", Automorphisms in birational and affine geometry, Springer Proceedings in Mathematics & Statistics, 79, Springer, 2014, 185–213 https://www.math.uni-bielefeld.de/LAG/man/508.pdf, arXiv: 1307.5522 (cited: 20)
117. V. L. Popov, "Jordaness of the automorphism groups of varieties and manifolds", Modern Problems of Mathematics and Natural Sciences (Koryazhma, September 15–18, 2014), Northern (Arctic) Federal M. V. Lomonosov University, Koryazhma, 2014, 66–70
118. S. V. Ivanov, R. Mikhailov, "On zero-divisors in group rings of groups with torsion", Canad. Math. Bull., 57:2 (2014), 326–334 , arXiv: 1209.1443 (cited: 1) (cited: 1) (cited: 1)
119. G. Baumslag, R. Mikhailov, "Residual properties of groups defined by basic commutators", Groups Geom. Dyn., 8:3 (2014), 621–642
120. R. Mikhailov, "Homotopical and Combinatorial Aspects of the Theory of Normal Series in Groups", Proc. Steklov Inst. Math., 286, suppl. 1 (2014), S1–S135
121. A. N. Parshin, "A holomorphic version of the Tate-Iwasawa method for unramified $L$-functions. I", Sb. Math., 205:10 (2014), 1473–1491 (cited: 1) (cited: 1)
122. S. Gorchinskiy, A. Ovchinnikov, "Isomonodromic differential equations and differential categories", J. Math. Pures Appl. (9), 102:1 (2014), 48–78 , arXiv: 1202.0927 (cited: 7) (cited: 7) (cited: 11)
123. Andrey S. Trepalin, "Rationality of the quotient of $\mathbb{P}^2$ by finite group of automorphisms over arbitrary field of characteristic zero", Cent. Eur. J. Math., 12:2 (2014), 229–239 , arXiv: https://arxiv.org/abs/1110.6338
124. V. V. Nikulin, "Kählerian K3 surfaces and Niemeier lattices. I", Izv. Math., 77:5 (2013), 954–997 (cited: 10) (cited: 7)
125. V. V. Nikulin, "Kahlerian K3 surfaces and Niemeier lattices", The 6th MSJ-SI-Development of Moduli Theory, Conference dedicated to 60th birthday of Mukai (Kyoto, RIMS, 17–21 June 2013), Research Institute of Mathematical Sciences (RIMS), Kyoto University, Japan, 2013, 1
126. V. V. Nikulin, "Kahlerian K3 surfaces and Niemeier lattices", Project: Mock modular forms, Moonshine and String Theory, 2013 (New York State, USA, 25 September 2013), Simons Center for Geomery and Physics, Stony Brook University, 2013, 1–1
127. D. V. Osipov, "The unramified two-dimensional Langlands correspondence", Izv. Math., 77:4 (2013), 714–741 , arXiv: 1210.3780 (cited: 3) (cited: 2) (cited: 2) (cited: 3)
128. D. V. Osipov, "Noncommutative reciprocity laws on algebraic surfaces: the case of tame ramification", Sb. Math., 204:12 (2013), 1797–1810 , arXiv: 1307.1995
129. D. V. Osipov, "Dvumernoe sootvetstvie Lenglendsa", Algebra i teoriya chisel: sovremennye problemy i prilozheniya: Tezisy dokladov XI Mezhdunarodnoi konferentsii (Saratov, 9–14 sentyabrya 2013 g.), ISBN 978-5-292-04189-4, Izdatelstvo Saratovskogo universiteta, Saratov, 2013, 64–65 old.sgu.ru/files/nodes2013/67438/Tezis.pdf
130. D. V. Osipov, "Vvedenie v teoriyu vysshikh adelei", programma lektsii, Letnyaya shkola-konferentsiya po algebraicheskoi geometrii i kompleksnomu analizu dlya molodykh uchenykh Rossii (Yaroslavl, YaGPU, 20–25 maya 2013 g.), Matematicheskii institut im. V. A. Steklova RAN, Moskva, 2013, 8–9
131. D. V. Osipov, Kategornye metody v teorii vysshikh adelei i ikh primenenie, Diss. … dokt. fiz.-matem. nauk, Matematicheskii institut im. V. A. Steklova RAN, Moskva, 2013 , 195 pp. avtoreferat dissertatsii
132. Vik. S. Kulikov, "Factorizations in finite groups", Sb. Math., 204:2 (2013), 237–263 (cited: 3) (cited: 2)
133. Vik. S. Kulikov, V. M. Kharlamov, "Covering semigroups", Izv. Math., 77:3 (2013), 594–626 (cited: 5) (cited: 1) (cited: 1) (cited: 3)
134. F. Bogomolov, V. S. Kulikov, "On the irreducibility of Hilbert scheme of surfaces of minimal degree", Cent. Eur. J. Math., 11:2 (2013), 254–263 (cited: 1) (cited: 1) (cited: 1)
135. Vik. S. Kulikov, "Covering semigroups", Complex Algebraic Geometry (Mathematisches Forschungsinstitut Oberwolfach, 26 May – 1 June 2013), Oberwolfach Reports, Report No. 27/2013, 10, no. 2, European Mathematical Society, 2013, 1595–1598 http://www.mfo.de/occasion/1322/www_view
136. I. R. Shafarevich, "A problem about the tenth discriminant", St. Petersburg Math. J., 25:4 (2014), 699–711 (cited: 1) (cited: 1)
137. V. Abrashkin, "Projective varieties with bad semi-stable reduction at 3 only", Doc. Math., 18 (2013), 547–619 (cited: 1) (cited: 1) (cited: 1)
138. V. L. Popov, "Tori in the Cremona groups", Izv. Math., 77:4, special issue on the occasion of I. R. Shafarevich's 90th anniversary (2013), 742–771 https://www.math.uni-bielefeld.de/LAG/man/474.pdf (cited: 8) (cited: 2) (cited: 2) (cited: 3)
139. V. L. Popov, "Some subgroups of the Cremona groups", Affine algebraic geometry, Proceedings of the conference on the occasion of M. Miyanishi's 70th birthday (Osaka, Japan, 3–6 March 2011), World Scientific Publishing Co., Singapore, 2013, 213–242 https://www.math.uni-bielefeld.de/LAG/man/448.pdf (cited: 8)
140. V. L. Popov, "Algebraic groups and the Cremona group", Algebraic groups (Mathematisches Forschungsinstitut Oberwolfach, 7 April – 13 April 2013), Oberwolfach Reports, 10, no. 2, European Mathematical Society, 2013, 1053–1055
141. V. L. Popov, "Rationality and the FML invariant", Journal of the Ramanujan Mathematical Society, 28A (2013), 409–415 http://www.mathjournals.org/jrms/2013-028-000/2013-28A-SPL-017.html, https://www.math.uni-bielefeld.de/LAG/man/485.pdf (special Issue-2013 dedicated to C. S. Seshadri's 80th birthday) (cited: 2)
142. R. Mikhailov, J. Wu, "Combinatorial group theory and the homotopy groups of finite complexes", Geom. Topol., 17:1 (2013), 235–272 , arXiv: 1108.3055 (cited: 1) (cited: 1) (cited: 1)
143. J. Wu, R. V. Mikhailov, "Homotopy groups as centres of finitely presented groups", Izv. Math., 77:3 (2013), 581–593
144. A. N. Parshin, "Generalizations of the Langlands program", Lecture course, The Langlands program and adelic theory (June 1–8, 2013, SPb), Euler Mathematical Institute, 2013 , http://www.pdmi.ras.ru/EIMI/2013/Lpat/schedule.html
145. A. N. Parshin, "Base change and automorphic induction in relative dimension 1", The Langlands program and adelic theory (June 10–14, 2013, SPb), Euler Mathematical Institute, 2013 , http://www.pdmi.ras.ru/EIMI/2013/Lpat/abstracts.pdf
146. A. N. Parshin, "Teoriya predstavlenii i algebraicheskaya geometriya (novyi vzglyad na starye zadachi)", I. M. Gelfand i sovremennaya matematika (17–19 dekabrya 2013 g.), MGU, 2013 , http://gelfand100.mech.math.msu.su/cgi-bin/gelfand100.fcgi?lang=ru&page=abstracts
147. S. Gorchinskiy, V. Guletskii, "Non-trivial elements in the Abel–Jacobi kernels of higher-dimensional varieties", Adv. Math., 241 (2013), 162–191 , arXiv: 1009.1431 (cited: 1) (cited: 1)
148. H. Gillet, S. Gorchinskiy, A. Ovchinnikov, "Parameterized Picard–Vessiot extensions and Atiyah extensions", Adv. Math., 238 (2013), 322–411 , arXiv: 1110.3526 (cited: 22) (cited: 14) (cited: 26)
149. S. Gorchinskiy, D. Orlov, "Geometric phantom categories", Publ. Math. Inst. Hautes Études Sci., 117:1 (2013), 329–349 , arXiv: 1209.6183 (cited: 19) (cited: 7) (cited: 17)
150. S. O. Gorchinskiy, "Generation of modules and transcendence degree of zero-cycles", Izv. Math., 77:4 (2013), 696–699
151. R. Ya. Budylin, S. O. Gorchinskiy, "Intersections of adelic groups on a surface", Sb. Math., 204:12 (2013), 1701–1711 (cited: 2) (cited: 2)
152. Vik. S. Kulikov, "Factorization semigroups and irreducible components of the Hurwitz space. II", Izv. Math., 76:2 (2012), 356–364 (cited: 4) (cited: 4) (cited: 4) (cited: 2)
153. F. Bogomolov, V. S. Kulikov, "On the diffeomorphic type of the complement to a line arrangement in a projective plane", Cent. Eur. J. Math., 10:2 (2012), 521–529 (cited: 1)
154. Vik. S. Kulikov, "Dopolnenie k state: Yu. G. Zarkhin, «Mnogochleny ot odnoi peremennoi i rangi nekotorykh kasatelnykh otobrazhenii»", Matem. zametki, 91:4 (2012), 547–550
155. I. R. Shafarevich, A. Remizov, Linear algebra and geometry, Springer-Verlag, Berlin, 2012
156. V. L. Popov, "Problems for the problem session", International conference "Groups of Automorphisms in Birational and Affine Geometry" (Levico Terme (Trento), October 29th – November 3rd, 2012), 2012 , 2 pp. http://www.science.unitn.it/cirm/Trento_postersession.html
157. G. Baumslag, R. Mikhailov, K. E. Orr, "A new look at finitely generated metabelian groups", Combinatorial and computational group theory with cryptography, Contemp. Math., 582, Amer. Math. Soc., Providence, RI, 2012, 21–37 , arXiv: 1203.5431
158. V. Bardakov, R. Mikhailov, J. Wu, V. Vershinin, "Brunnian braids on surfaces", Alg. Geom. Top., 12 (2012), 1607–1648 , arXiv: 0909.3387 (cited: 9) (cited: 6) (cited: 10)
159. R. Mikhailov, On the splitting of polynomial functors, 2012 , arXiv: 1202.0586
160. A. N. Parshin, "Questions and remarks to the Langlands programme", Russian Math. Surveys, 67:3 (2012), 509–539 (cited: 3) (cited: 3) (cited: 3) (cited: 2)
161. S. A. Arnal', A. N. Parshin, "On irreducible representations of discrete Heisenberg groups", Math. Notes, 92:3 (2012), 295–301 (cited: 5) (cited: 5) (cited: 5) (cited: 4)
162. S. Gorchinskiy, V. Guletskii, "Motives and representability of algebraic cycles on threefolds over a field", J. Algebraic Geom., 21:2 (2012), 347–373 , arXiv: 0806.0173v2 (cited: 15) (cited: 8) (cited: 15)
163. S. Gorchinskiy, V. Guletskii, "Transcendence degree of zero-cycles and the structure of Chow motives", Cent. Eur. J. Math., 10:2 (2012), 559–568 , arXiv: 1009.1434 (cited: 1) (cited: 1) (cited: 1)
164. V. V. Nikulin, "The transition constant for arithmetic hyperbolic reflection groups", Izv. Math., 75:5 (2011), 971–1005 (cited: 4) (cited: 1) (cited: 1) (cited: 2)
165. Viacheslav V. Nikulin, "Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups", Proc. Steklov Inst. Math., 273 (2011), 229–237 (cited: 1)
166. D. V. Osipov, A. N. Parshin, "Harmonic analysis on local fields and adelic spaces. II", Izv. Math., 75:4 (2011), 749–814 , arXiv: 0912.1577 (cited: 11) (cited: 3) (cited: 3) (cited: 6)
167. Denis Osipov, Xinwen Zhu, "A categorical proof of the Parshin reciprocity laws on algebraic surfaces", Algebra Number Theory, 5:3 (2011), 289–337 , arXiv: 1002.4848 (cited: 9) (cited: 10)
168. D. V. Osipov, A. N. Parshin, "Harmonic analisys and the Riemann-Roch theorem", Dokl. Math., 84:3 (2011), 826–829 , arXiv: 1107.0408 (cited: 3) (cited: 3) (cited: 3) (cited: 2)
169. Vik. S. Kulikov, "Factorization semigroups and irreducible components of the Hurwitz space", Izv. Math., 75:4 (2011), 711–748 (cited: 6) (cited: 6) (cited: 6) (cited: 4)
170. J.-L. Colliot-Thélène, B. Kunyavskiĭ, V. L. Popov, Z. Reichstein, "Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?", Compos. Math., 147:2 (2011), 428–466 (cited: 8) (cited: 6)
171. V. L. Popov, "Cross-sections, quotients, and representation rings of semisimple algebraic groups", Transform. Groups, 16:3, special issue dedicated to Tonny Springer on the occasion of his 85th birthday (2011), 827–856 (cited: 4) (cited: 4) (cited: 4)
172. V. L. Popov, "On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties", Affine algebraic geometry: the Russell Festschrift, CRM Proceedings and Lecture Notes, 54, Amer. Math. Soc., 2011, 289–311 https://www.math.uni-bielefeld.de/LAG/man/375.pdf (cited: 21)
173. V. L. Popov, "Invariant rational functions on semisimple Lie algebras and the Gelfand–Kirillov conjecture", Algebra and Mathematical Logic, International conference commemorating $100$th birthday of professor V. V. Morozov (Kazan, September 25–30, 2011), Kazan Federal Univ., Kazan, 2011, 19
174. L. Breen, R. Mikhailov, "Derived functors of nonadditive functors and homotopy theory", Algebr. Geom. Topol., 11:1 (2011), 327–415 (cited: 6) (cited: 4) (cited: 7)
175. R. Mikhailov, I. B. S. Passi, J. Wu, "Symmetric ideals in group rings and simplicial homotopy", J. Pure Appl. Algebra, 215:5 (2011), 1085–1092 (cited: 5) (cited: 6) (cited: 5)
176. R. Mikhailov, J. Wu, A combinatorial description of homotopy groups of spheres, 2011 , 27 pp., arXiv: 1108.3055
177. A. N. Parshin, "Notes on the Poisson formula", St. Petersburg Math. J., 23:5 (2012), 781–818 (cited: 2) (cited: 3) (cited: 3) (cited: 2) | CommonCrawl |
\begin{document}
\begin{abstract} For $b\leq -2$ and $e \geq 2$, let $S_{e,b}:\mathbb{Z}\to\mathbb{Z}_{\geq 0}$ be the function taking an integer to the sum of the $e$-powers of the digits of its base $b$ expansion. An integer $a$ is a {\em $b$-happy number} if there exists $k\in\mathbb{Z}^+$ such that $S_{2,b}^k(a) = 1$. We prove that an integer is $-2$-happy if and only if it is congruent to 1 modulo 3 and that it is $-3$-happy if and only if it is odd. Defining a $d$-sequence to be an arithmetic sequence with constant difference $d$ and setting $d = \gcd(2,b - 1)$, we prove that if $b \leq -3$ odd or $b \in \{-4,-6,-8,-10\}$, there exist arbitrarily long finite sequences of $d$-consecutive $b$-happy numbers. \end{abstract}
\title{Sequences of Consecutive Happy Numbers in\ Negative Bases}
\section{Introduction}
As is standard, a positive integer $a$ can be uniquely expanded in the base $b\geq 2$ as $a=\sum_{i=0}^na_ib^i$, where $0\leq a_i\leq b-1$ and $a_n\neq 0$. This definition can be extended to negative bases $b\leq -2$ in an analogous manner, with $0 \leq a_i \leq |b| - 1$. The study of negative bases was introduced in the 1885 work of Vittorio Gr\"{u}nwald \cite{VG}. As with positive bases, positive integers have unique representation in negative bases, but the same holds for negative numbers in negative bases, with no need for a negative sign: Any number written in a negative base with an odd number of digits is necessarily positive, while any written with an even number of digits is necessarily negative. For example, converting between base 10 and base $-10$, we have $2017=(18197)_{(-10)}$ and $-2017=(2023)_{(-10)}$.
We begin by adapting the definition of generalized happy numbers and the corresponding function given in~\cite{GT01} to include the case of negative bases. It is natural, in this case, to extend the domain of the function to include all integers.
\begin{definition}
Let $b \leq -2$ and $e\geq 2$ be integers, and let $a\in \mathbb{Z}-\{0\}$ be given by $a=\sum_{i=0}^na_ib^i$ where $0\leq a_i\leq |b|-1$, for each $0\leq i\leq n$. Define the function $S_{e,b}:\mathbb{Z}\to\mathbb{Z}_{\geq 0}$ by $S_{e,b}(0) = 0$ and, for $a \neq 0$, \[S_{e,b}(a)=\sum_{i=0}^{n}a_i^e.\] \end{definition}
Note that if $b \leq -2$ and $k > 0$, for each $a\neq 0$, $S_{e,b}^k(a) > 0$. \begin{definition} An integer $a$ is an \emph{$e$-power $b$-happy number} if, for some $k \in \mathbb{Z}^+$, $S_{e,b}^k(a) = 1$. A \emph{$b$-happy number} is a $2$-power $b$-happy number. \end{definition}
The following definition and the others in the sequel are either directly from or adapted from~\cite{GT07}. \begin{definition} A {\em $d$-consecutive} sequence is defined to be an arithmetic sequence with constant difference $d$. \end{definition}
In Section~\ref{cyclesection}, we determine the fixed points and cycles of the functions $S_{2,b}$ for $-10 \leq b \leq -2$. In Section~\ref{sequencesection}, we generalize the work of El-Sedy and Siksek~\cite{siksek} and the work of Grundman and Teeple~\cite{GT07} on sequences of consecutive $b$-happy numbers. In particular, Grundman and Teeple showed that there exist arbitrarily long finite $d$-consecutive sequences of $b$-happy numbers, where $b\geq 2$ and $d=\gcd(2,b-1)$~\cite[Corollary 2]{GT07}. We prove that this result does not hold for $b = -2$, but does hold for all odd negative bases and for even negative bases $-10 \leq b \leq -4$.
\section{Cycles and Fixed Points} \label{cyclesection}
In this section, we first determine a bound, dependent on the given base $b \leq -2$, such that each fixed point and at least one point in every cycle is less than this bound. We then use this result to compute all cycles and fixed points of $S_{2,b}$ for $-10 \leq b \leq -2$.
\begin{theorem}\label{thm:GetSmaller}
Let $b\leq -2$. If $a > (|b|-1)(|b|^2 - |b| + 1)$, then $0 < S_{2,b}(a) < a$. \end{theorem}
\begin{proof}
Let $a$ and $b$ be as in the hypothesis. Then $a = \sum_{i=0}^na_ib^i$ with $n$ even, $0\leq a_i\leq |b| - 1$, $a_n\neq 0$. Observe that \begin{align} a - S_{2,b}(a) &=\sum_{i=0}^na_ib^i-\sum_{i=0}^na_i^2 =\sum_{i=0}^{n}a_i(b^i-a_i)\nonumber \\
&=\sum_{j=1}^{\frac{n}{2}}a_{2j}(|b|^{2j}-a_{2j})-\sum_{j=1}^{\frac{n}{2}}a_{2j-1}\left(|b|^{2j-1} + a_{2j-1}\right)
+ a_0(|b|^0 - a_0). \label{boundeq1} \end{align}
{\bf Case: $\mathbf{n \geq 4}$.}
Since, $a_n \geq 1$ and, for each $i$, $0\leq a_i\leq |b| - 1$, minimizing each term in~(\ref{boundeq1}) yields
\[a - S_{2,b}(a) \geq 1(b^n - 1) - \sum_{j=1}^{\frac{n}{2}}(|b| - 1)\left(|b|^{2j-1} + (|b| - 1)\right) + (|b| - 1)(1 - (|b| - 1)).\] Noting that
\[\sum_{j=1}^{\frac{n}{2}} |b|^{2j-1} = \frac{|b|}{|b|^2 - 1} \left(|b|^n - 1\right), \] we have \begin{align}
a - S_{2,b}(a) &\geq (b^n - 1) - (|b| - 1)\left(\frac{|b|}{|b|^2 - 1} \left(|b|^n - 1\right)+ \frac{n}{2}(|b| - 1)\right) - (|b| - 1)(|b| - 2) \nonumber \\
&= \frac{b^n - 1}{|b| + 1} - \frac{n}{2}(|b| - 1)^2 - (|b| - 1)(|b| - 2) \nonumber \\
&= \frac{1}{|b| + 1} \left(|b|^n - \frac{n}{2}(|b|^2 - 1)(|b| - 1) - (|b|^2 - 1)(|b| - 2) - 1\right) \label{b=2} \\
&> \frac{1}{|b| + 1} \left(|b|^n - \frac{n}{2}|b|^3 - |b|^3 \right) \label{b>2} \\
&> \frac{1}{|b| + 1} \left(|b|^{n-3} - \frac{n}{2} - 1 \right). \label{finalineq} \end{align}
Note that the function $f(x) = 2^{x-3} - x/2 - 1$ is an increasing function for $x \geq 5$ and that $f(5) > 0$. Thus, for $n \geq 5$, since $b \leq -2$,
\[|b|^{n-3} - n/2 - 1 \geq 2^{n-3} - n/2 - 1 > 0,\] and so, by inequality~(\ref{finalineq}), $a - S_{2,b}(a) > 0$. Now, for $n = 4$ and $b < -2$, using inequality~(\ref{b>2}),
$a - S_{2,b}(a) > \frac{1}{|b| + 1} \left(|b|^4 - 3|b|^3 \right) \geq 0$, and for $n = 4$ and $b = -2$, using inequality~(\ref{b=2}), $a - S_{2,b}(a) > 0$.
{\bf Case: $\mathbf{n < 4}$.}
In this case, $(|b| - 1)(|b|^2 - |b| + 1) < a \leq (|b|-1)(|b|^2+1)$. So $a = a_2b^2 + a_1 b + a_0$ with $a_2 = |b| - 1$, $0 \leq a_1 \leq |b| - 2$, and $0 \leq a_0 \leq |b| - 1$. Thus, \begin{align*} a - S_{2,b}(a) &=
a_{2}(|b|^{2}-a_{2}) - a_{1}\left(|b| + a_{1}\right) + a_0(1 - a_0) \\
&\geq (|b| - 1)(|b|^2 - (|b| - 1)) - (|b| - 2)(|b| + (|b| - 2)) + (|b| - 1)(1 - (|b| - 1)) \\
& = |b|^3 - 5|b|^2 + 11|b| - 7 > 0, \end{align*} since $b \leq -2$. \end{proof}
Note that when $b = -2$, the bound in Theorem~\ref{thm:GetSmaller} is 4. Since 3 is a fixed point of $S_{2,-2}$, the given bound is best possible.
The following corollary is immediate.
\begin{cor} \label{bound}
Let $b \leq -2$. Every fixed point of $S_{2,b}$ is less than or equal to $(|b| - 1)(|b|^2 - |b| + 1)$ and every cycle of $S_{2,b}$ contains a number that is less than or equal to $(|b| - 1)(|b|^2 - |b| + 1)$. \end{cor}
Using Corollary~\ref{bound} and a direct computer search, we determine all fixed points and cycles in the bases $-10 \leq b \leq -2$. The results are given in Table \ref{tab:1}.
\begin{table}[ht] \centering
\begin{tabular}{|c|p{1cm}|p{10.5cm}|p{1.5cm}|p{1.5cm}|}\hline Base& Fixed Points & Cycles& Smallest happy number $> 1$& Largest happy number $< -1$\\\hline\hline $-2$&1,2,3& None&4&-2\\\hline $-3$&1&(2,4,6)&3&-1\\\hline $-4$&1&(6,14)&16&-4\\\hline $-5$&1,10,11&(2,4,16,6,18,14,26), (9,33,29,17)&25&-5\\\hline $-6$&1&(2,4,16,33,11,51,29,30)&36&-6\\\hline $-7$&1, 41&(2,4,16,30,14,26,42), (5,25,33,35), (6,36)&49&-7\\\hline $-8$&1, 46&(11,59), (30,62,38,53)&64&-8\\\hline $-9$&1&(6,36,26,114,76,18,50,42,62,74), (9,65), (27,37)&5&-5\\\hline $-10$&1&(19,163,29,146,76,46,73), (35,75)&100&-10\\\hline \end{tabular}
\caption{Base 10 representation of fixed points and cycles of $S_{2,b}$ for $-10 \leq b \leq -2$.} \label{tab:1} \end{table} \begin{definition} \label{Udef} For $e \geq 2$ and $b \leq -2$, let
\[U_{e,b} = \{a \in \mathbb{Z}^+ | \mbox{ for some } m\in \mathbb{Z}^+,\ S_{e,b}^m(a) = a\}.\] \end{definition}
The following straightforward lemmas are used throughout this work. \begin{lemma}\label{Ulemma} Fix $b \leq -2$. For each $a\neq 0$, there exists some $k \in \mathbb{Z}^+$ such that $S_{2,b}^k(a) \in U_{2,b}$. \end{lemma}
\begin{lemma}\label{paritylemma} Fix $b \leq -2$, $a\in \mathbb{Z}$, and $k\in \mathbb{Z}^+$. If $b$ is odd, then \[S_{2,b}^k(a) \equiv a \pmod 2.\] \end{lemma}
\begin{proof} Fix $a$, $b$, and $k$ as in the lemma. Noting that the result is trivial if $a = 0$, let $a = \sum_{i=0}^{n}a_ib^i$. Then \[a = \sum_{i=0}^{n}a_ib^i \equiv \sum_{i=0}^{n}a_i \equiv \sum_{i=0}^{n}a_i^2 \equiv S_{2,b}(a) \pmod 2. \] A simple induction argument completes the proof. \end{proof}
\section{Consecutive $\mathbf b$-Happy Numbers} \label{sequencesection}
In this section, we consider sequences of consecutive $b$-happy numbers for negative $b$. Grundman and Teeple~\cite{GT07} proved, for each base $b \geq 2$, that, letting $d = \gcd(2,b-1)$, there exist arbitrarily long finite sequences of $d$-consecutive $b$-happy numbers. We prove the following theorem using ideas from both of~\cite{siksek,GT07}. Note that part (1) of the theorem demonstrates that the results in~\cite{GT07} do not generalize directly to negative bases.
\begin{theorem} \label{seqthm} Let $b \leq -2$. \begin{enumerate} \item There is an infinitely long sequence of $3$-consecutive $-2$-happy numbers. In particular, $a\in \mathbb{Z}^+$ is $-2$-happy if and only if $a \equiv 1 \pmod 3$. \item There is an infinitely long sequence of $2$-consecutive $-3$-happy numbers. In particular, $a\in \mathbb{Z}^+$ is $-3$-happy if and only if $a \equiv 1 \pmod 2$. \item For $b\in \{-4,-6,-8,-10\}$, there exist arbitrarily long finite sequences of consecutive $b$-happy numbers. \item For $b$ odd, there exist arbitrarily long finite sequences of $2$-consecutive $b$-happy numbers. \end{enumerate} \end{theorem}
The smaller even negative bases not covered by Theorem~\ref{seqthm} are addressed in the following conjecture.
\begin{conj} For $b\leq -12$ and even, there exist arbitrarily long finite sequences of consecutive $b$-happy numbers. \end{conj}
We begin by proving the first two cases of Theorem~\ref{seqthm}. The other two cases follow immediately from Corollary~\ref{finalcor}, stated and proved at the end of this section.
\begin{lemma} \label{-2-3} A positive integer $a$ is $-2$-happy if and only if $a \equiv 1 \pmod 3$ and is $-3$-happy if and only if $a$ is odd. \end{lemma}
\begin{proof}If $a=\sum_{i=0}^{n}a_i(-2)^i$ with $a_i\in\{0,1\}$ for all $0\leq i \leq n$, then \[S_{2,-2}(a) = \sum_{i=0}^{n}a_i^2 = \sum_{i=0}^{n}a_i \equiv \sum_{i=0}^{n}a_i(-2)^i \equiv a \pmod 3.\] Thus, if $a$ is $-2$-happy, $a \equiv 1 \pmod 3$. Now, suppose that $a \equiv 1 \pmod 3$. By Lemma~\ref{Ulemma}, there exists a $k\in \mathbb{Z}^+$ such that $S_{2,-2}^k(a)\in U_{2,-2} = \{1,2,3\}$. Since $S_{2,-2}^k(a)\equiv a \equiv 1 \pmod 3$, $S_{2,-2}^k(a) = 1$ and so $a$ is a $-2$-happy number.
By Lemma~\ref{paritylemma}, if $a$ is a $-3$-happy number, then $a$ is odd. Since $U_{2,-3} = \{1,2,4,6\}$, Lemmas~\ref{Ulemma} and~\ref{paritylemma} together imply that if $a \equiv 1 \pmod 2$, then $a$ is a $-3$-happy number. \end{proof}
The following definitions are from~\cite{GT07}.
\begin{definition} \label{good} Let $e \geq 2$ and $b \leq -2$. A finite set $T$ is {\em $(e,b)$-good} if, for each $u \in U_{e,b}$, there exist $n$, $k \in {\mathbb{Z}}^+$ such that for each $t \in T$, $S_{e,b}^k(t+n) = u$. \end{definition}
\begin{definition} Let $I: {\mathbb{Z}}^+ \rightarrow {\mathbb{Z}}^+$ be defined by $I(t) = t + 1$. \end{definition}
We will prove that for each odd $b \leq -5$ and for $b\in \{-4,-6,-8,-10\}$, a finite set $T$ of positive integers is $(2,b)$-good if and only if all of the elements of $T$ are congruent modulo $d = \gcd(2,b-1)$. Lemma~\ref{Flemma} and its proof are analogous to~\cite[Lemma 4 and proof]{GT07}.
\begin{lemma} \label{Flemma} Fix $e \geq 2$ and $b \leq -2$. Let $T \subseteq \mathbb{Z}^+$ be finite. Let $F:{\mathbb{Z}}^+ \rightarrow {\mathbb{Z}}^+$ be the composition of a finite sequence of the functions $S_{e,b}$ and $I$. If $F(T)$ is $(e,b)$-good, then $T$ is $(e,b)$-good. \end{lemma} \begin{proof} Fix $e \geq 2$, $b \leq -2$, and a finite set of positive integers, $T$. Clearly, if $I(T)$ is $(e,b)$-good, then $T$ is $(e,b)$-good. Using a simple induction argument, it suffices to show that if $S_{e,b}(T)$ is $(e,b)$-good, then $T$ is $(e,b)$-good.
Let $S_{e,b}(T)$ be $(e,b)$-good and $u \in U_{e,b}$. Then, by the definition of $(e,b)$-good, there exist $n^\prime$ and $k^\prime$ such that for each $s \in S_{e,b}(T)$, $S_{e,b}^{k^\prime}(s + n^\prime) = u$. Let $\ell$ be the number of base $b$ digits of the largest element of $T$ and let $\ell^\prime = \ell$ or $\ell + 1$ such that $n^\prime + \ell^\prime$ is odd. Let \[n = \sum_{i = \ell^\prime}^{n^\prime + \ell^\prime -1} b^i = \underbrace{11\ldots11}_{n^\prime}\underbrace{00\ldots00}_{\ell^\prime} \in \mathbb{Z}^+.\]
Then $S_{e,b}(n) = n^\prime$ and for each $t \in T$, $S_{e,b}(t+n) = S_{e,b}(t) + n^\prime$. Let $k = k^\prime +1$. Then for each $t\in T$, \[S_{e,b}^{k}(t+n) = S_{e,b}^{k^\prime}(S_{e,b}(t+n)) = S_{e,b}^{k^\prime}(S_{e,b}(t) + n^\prime) = u.\] So $T$ is $(e,b)$-good. \end{proof}
\begin{lemma}\label{Flemmas} Let $b\in \{-4,-6,-8,-10\}$ and let $0 < t_2 < t_1$ be integers. Then there exists a function $F$ of the type described in Lemma~\ref{Flemma} such that $F(t_1) = F(t_2)$. \end{lemma}
\begin{proof} Let $k\in \mathbb{Z}^+$ such that $S_{2,b}^{k}(t_1), S_{2,b}^{k}(t_2) \in U_{2,b}$, and let $F_1 = S_{2,b}^k$. If $F_1(t_1) = F_1(t_2)$, we are done, and so we assume otherwise. From Table~\ref{tab:1}, we have \begin{align*} U_{2,-4} &= \{1,6,14\},\\ U_{2,-6} &= \{1,2,4,11,16,29,30,33,51\},\\ U_{2,-8} &= \{1,11,30,38,46,53,59,62\},\\ U_{2,-10} &=\{1,19,29,35,46,73,75,76,146,163\}. \end{align*}
{\bf Case: $\mathbf{b = -4}$.} Let $F_2 = S_{2,-4}^2I$ and $F_3 = S_{2,-4}^5I^3$. Note that \begin{align*} F_2(6) &= S_{2,-4}^2(7) = S_{2,-4}^2(133_{(-4)}) = S_{2,-4}(19) = S_{2,-4}(103_{(-4)}) = 10 \text{ and} \\ F_2(1) &= S_{2,-4}^2(2) = S_{2,-4}(4) = S_{2,-4}(130_{(-4)}) = 10. \end{align*} Thus, if $\{F_1(t_1),F_1(t_2)\} = \{1,6\}$, then let $F = F_2F_1$, so that $F(t_1) = F(t_2)$. And if $\{F_1(t_1),F_1(t_2)\} = \{1,14\}$, then, noting that $S_{2,-4}(14) = 6$, let $F = F_2F_1S_{2,-4}$.
Finally, observe that \begin{align*} F_3(6) &= S_{2,-4}^5(9) = S_{2,-4}^5(121_{(-4)}) = S_{2,-4}^4(6) = S_{2,-4}^4(132_{(-4)}) = S_{2,-4}^3(14) = S_{2,-4}^2(6) = 6 \text{ and} \\ F_3(14) &= S_{2,-4}^5(17) = S_{2,-4}^5(101_{(-4)}) = S_{2,-4}^4(2) = S_{2,-4}^3(4) = S_{2,-4}^2(10) = S_{2,-4}(9) = 6. \end{align*} Hence, if $\{F_1(t_1),F_1(t_2)\} = \{6,14\}$, let $F = F_3F_1$.
{\bf Case: $\mathbf{b = -6}$.} Let $F_2 = S^\ell_{2,-6} I^{36-F_1(t_1)}$ where $\ell\in \mathbb{Z}^+$ such that $F_2F_1(t_2) \in U_{2,-6}$. Note that $F_2F_1(t_1) = S^\ell_{2,-6} (36) = 1$, regardless of the choice of $\ell$. If $F_2F_1(t_2) = 1$, we are done. If not, since $(2,4,16,33,11,51,29,30)$ is a cycle, we can modify our choice of $\ell$ (making it larger, if necessary) to guarantee that $F_2F_1(t_2) = 2$.
Now let $F_3 = S_{2,-6}^6I^{7}$. Noting that $F_3(1) = F_3(2)$, we set $F = F_3F_2F_1$.
{\bf Case: $\mathbf{b = -8}$.} Let $F_2 = S^\ell_{2,-8} I^{64-F_1(t_1)}$ where $\ell\in \mathbb{Z}^+$ such that $F_2F_1(t_2) \in U_{2,-8}$. Since $F_2F_1(t_1) = S^\ell_{2,-8} (64) = 1$, if $F_2F_1(t_2) = 1$, we are done. Otherwise, using Table~\ref{tab:1}, we can choose a possibly larger value of $\ell$ so that $F_2F_1(t_2) \in\{30,59,46\}$. If $F_2F_1(t_2) \in \{30,59\}$, let $F_3 = S_{2,-8}^8I^{2}$. Noting that $F_3(1) = F_3(30) = F_3(59)$, we set $F = F_3F_2F_1$. If instead $F_2F_1(t_2)= 46$, then let $F_4 = S_{2,-8}^9I^7$. Since $F_4(1) = F_4(46)$, setting $F = F_4F_2F_1$ completes this case.
{\bf Case: $\mathbf{b = -10}$.} Let $F_2 = S^\ell_{2,-10} I^{100-F_1(t_1)}$ where $\ell\in \mathbb{Z}^+$ such that $F_2F_1(t_2) \in U_{2,-10}$. Since $F_2F_1(t_1) = 1$, if $F_2F_1(t_2) =S^\ell_{2,-10}(100)= 1$, we are done. If not, we can choose $\ell$ so that $F_2F_1(t_2) \in \{19,35\}$. If $F_2F_1(t_2) = 19$, let $F_3 = S_{2,-10}^3I^{22}$ and set $F = F_3F_2F_1$. If instead $F_2F_1(t_2) =35$, let $F_4 = S_{2,-10}^{16}I$ and set $F = F_4F_2F_1$, completing the proof. \end{proof}
We now apply the methods in~\cite{GT07} to odd negative~bases, noting that the original proof does not carry over, since, for $b$ negative, $b - 1 \neq |b| - 1$.
\begin{lemma} \label{oddcongruencelemma}
Fix $b \leq -5$ odd, $v^\prime \in 2\mathbb{Z}^+$, and $r^\prime \in \mathbb{Z}^+$ such that $b^{2r^\prime} > v^\prime$. There exists $0 \leq c < |b| - 1$ such that \begin{equation} \label{congruence}
2c \equiv 4r^\prime - S_{2,b}\left((|b| - 1)\sum_{i=0}^{r^\prime - 1} b^{2i+1} + v^\prime - 1\right) - 1 \pmod{b - 1}. \end{equation} \end{lemma}
\begin{proof} Since $b$ is odd and $v^\prime$ is even, the input to $S_{2,b}$ in~(\ref{congruence}) is odd. Thus, by Lemma~\ref{paritylemma}, the output is also odd. Hence, we can choose
\[c \equiv 2r^\prime - \frac{1}{2}\left( S_{2,b}\left((|b| - 1)\sum_{i=0}^{r^\prime - 1} b^{2i+1} + v^\prime - 1\right) + 1\right) \; \left(\bmod\; {\frac{b - 1}{2}}\right),\]
with $0 \leq c < |\frac{b - 1}{2}| < |b| - 1$, since $b \leq -5$. \end{proof}
\begin{lemma} \label{Ftheoremodd} Fix $b \leq -5$ odd and let $t_1$, $t_2\in \mathbb{Z}^+$ be congruent modulo $2$ with $t_2 < t_1$. Then there exists a function $F$ of the type described in Lemma~\ref{Flemma} such that $F(t_1) = F(t_2)$. \end{lemma} \begin{proof} First note that if $t_1$ and $t_2$ have the same non-zero digits, then $S_{2,b}(t_1) = S_{2,b}(t_2)$, and so $F = S_{2,b}$ suffices.
Next, if $t_1 \equiv t_2 \pmod{b - 1}$, let $v\in \mathbb{Z}^+$ such that $t_2 - t_1 = (b - 1)v$. Choose $r \in \mathbb{Z}^+$ so that $b^{2r} > b^2v + t_1$, and let $m = b^{2r} + v - t_1 > 0$. Then \[I^m(t_1) = t_1 + b^{2r} + v - t_1 = b^{2r} + v\] and \[I^m(t_2)= t_2 + b^{2r} + v - t_1 = b^{2r} + v+(b-1)v=b^{2r}+bv.\] Since $b^{2r} > b^2v$, it follows that $I^m(t_1)$ and $I^m(t_2)$ have the same non-zero digits. Thus, it suffices to let $F = S_{2,b} I^{m}$.
Finally, if $t_1 \not\equiv t_2 \pmod{b - 1}$, let
$v^\prime = t_1 - t_2 \in 2\mathbb{Z}^+$. Choose $r^\prime \in \mathbb{Z}^+$ such that $b^{2r^\prime} > b^2t_1$. By Lemma~\ref{oddcongruencelemma}, since $b^2t_1 > v^\prime$, there exists $0 \leq c < |b| - 1$ such that congruence~(\ref{congruence}) holds.
Let \[{m^\prime} = cb^{2r^\prime } +
\sum_{i=0}^{r^\prime - 1} (|b| - 1)b^{2i} - t_2 \geq 0.\] Then \[S_{2,b}(t_2 + m^\prime) = S_{2,b}\left(cb^{r^\prime } +
\sum_{i=0}^{r^\prime - 1} (|b| - 1)b^{2i}\right) =
c^2 + r^\prime(|b| - 1)^2.\] And \begin{align*} S_{2,b}\left(t_1 + m^\prime\right) &= S_{2,b}\left(cb^{2r^\prime } +
\sum_{i=0}^{r^\prime - 1} (|b| - 1)b^{2i} + v^\prime\right)\\ &= S_{2,b}\left((c + 1)b^{2r^\prime } +
\sum_{i=0}^{r^\prime - 1} (|b| - 1)b^{2i+1} + v^\prime - 1\right)\\
&= (c + 1)^2 + S_{2,b}\left(\sum_{i=0}^{r^\prime - 1} (|b| - 1)b^{2i+1} + v^\prime - 1\right). \end{align*}
It follows that \[S_{2,b}(t_1 + m^\prime) - S_{2,b}(t_2 + m^\prime) =
2c + 1 + S_{2,b}\left(\sum_{i=0}^{r^\prime - 1} (|b| - 1)b^{2i+1} + v^\prime - 1\right) - r^\prime(b + 1)^2.\] Using congruence~(\ref{congruence}), this yields \[S_{2,b}(t_1 + m^\prime) - S_{2,b}(t_2 + m^\prime) \equiv 4r^\prime - r^\prime(b + 1)^2 \equiv 0 \pmod{b - 1}.\]
Therefore, $S_{2,b} (I^{m^\prime}(t_1)) \equiv S_{2,b}(I^{m^\prime}(t_2))\ \pmod{b-1}$. Applying the earlier argument to these two numbers, yielding an appropriate value of $m \in \mathbb{Z}^+$, we let $F = S_{2,b} I^{m} S_{2,b} I^{m^\prime}$. \end{proof}
\begin{theorem} \label{mainthm} Fix $b \leq -5$ odd or $b \in \{-4,-6,-8,-10\}$. Let $d = \gcd(2,b - 1)$. A finite set $T$ of positive integers is $(2,b)$-good if and only if all of the elements of $T$ are congruent modulo $d$. \end{theorem}
\begin{proof} Fix a finite set of positive integers, $T$. First, assume that $T$ is $(2,b)$-good. If $b$ is even, then $d = 1$, and the congruence result is trivial. If $b$ is odd, fix $u\in U_{2,b}$. Then there exists $n$, $k \in {\mathbb{Z}}^+$ such that for each $t \in T$, $S_{2,b}^k(t+n) = u$. It follows from Lemma~\ref{paritylemma} that, for each $t \in T$, $t + n \equiv u \pmod 2$. Hence, the elements of $T$ are congruent modulo $d = 2$.
For the converse, assume that the elements of $T$ are congruent modulo $d$. If $T$ is empty, then vacuously it is $(2,b)$-good. If $T = \{t\}$, then given $u \in U_{2,b}$, by definition, there exist $x\in \mathbb{Z}^+$ such that $S_{2,b}(x) = u$. Fix some $r \in 2\mathbb{Z}^+$ such that $t \leq b^rx$. Then, letting $n = b^rx-t$ and $k = 1$, since $S_{2,b}^k(t + n) = S_{2,b}(t+(b^rx-t)) = S_{2,b}(x) = u$, $T$ is $(2,b)$-good.
Now assume that $|T| = N > 1$ and assume, by induction, that any set of fewer than $N$ elements all of which are congruent modulo $d$ is $(2,b)$-good. Let $t_1 > t_2 \in T$. By Lemmas~\ref{Flemmas} and~\ref{Ftheoremodd}, there exists a function $F$ as in Lemma~\ref{Flemma} such that $F(t_1) = F(t_2)$. This implies that $F(T)$ has fewer than $N$ elements. Further, since the elements of $T$ are congruent modulo $d$, the same holds for $I(T)$ and, by Lemma~\ref{paritylemma}, for $S_{2,b}(T)$, implying that the same holds for $F(T)$. Thus, by the induction hypothesis, $F(T)$ is $(2,b)$-good and so, by Lemma~\ref{Flemma}, $T$ is $(2,b)$-good. \end{proof}
\begin{cor} \label{finalcor} For $b \leq -3$ odd or $b \in \{-4,-6,-8,-10\}$ and $d = \gcd(2,b - 1)$, there exist arbitrarily long finite sequences of $d$-consecutive $b$-happy numbers. \end{cor}
\begin{proof} By Lemma~\ref{-2-3}, every odd positive integer is $-3$-happy. So the corollary holds for $b = -3$.
For $b < -3$, given $N\in \mathbb{Z}^+$, let $T = \{1 + dt \;|\; 0\leq t \leq N-1\}$. By Theorem~\ref{mainthm}, $T$ is $(2,b)$-good.
By Definition~\ref{good}, there exist $n$, $k \in {\mathbb{Z}}^+$ such that for each $t \in T$, $S_{2,b}^k(t+n) = 1$. Thus, $\{1 + n + dt\; |\; 0\leq t \leq N-1\}$ is a sequence of $N$ $d$-consecutive $b$-happy numbers, as desired. \end{proof}
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\end{document} | arXiv |
Highly Damage-Resistant Thin Film Saturable Absorber Based on Mechanically Functionalized SWCNTs
Daewon Kang1,
Sourav Sarkar2,
Kyung-Soo Kim1 &
Soohyun Kim ORCID: orcid.org/0000-0002-2329-11171
Nanoscale Research Letters volume 17, Article number: 11 (2022) Cite this article
57 Accesses
Thin-film saturable absorbers (SAs) are extensively used in mode-locked fiber laser due to the robust and simple application methods that arise because SAs are alignment-free and self-standing. Single-walled carbon nanotubes (SWCNTs) are the most suitable low dimensional material uesd for SAs because of their high nonlinearity and the wavelength control of absorption based on tube diameters. The most challenging problem with the use of CNT-based thin film SAs is thermal damage caused during high power laser operation, which mainly occurs due to aggregation of CNTs. We have demonstrated improved thermal damage resistance and enhanced durability of a film-type SA based on functionalization of SWCNTs, which were subjected to a mechanical functionalization procedure to induce covalent structural modifications on the SWCNT surface. Increased intertube distance was shown by X-ray diffraction, and partial functionalization was shown by Raman spectroscopy. This physical change had a profound effect on integration with the host polymer and resolved aggregation problems. A free-standing SA was fabricated by the drop casting method, and improved uniformity was shown by scanning electron microscopy. The SA was analyzed using various structural and thermal evaluation techniques (Raman spectroscopy, thermogravimetric analysis, etc.). Damage tests at different optical powers were also performed. To the best of our knowledge, a comprehensive analysis of a film-type SA is reported here for the first time. The partially functionalized SWCNT (fSWCNT) SA shows significant structural integrity after intense damage tests and a modulation depth of 25.3%. In passively mode-locked laser operation, a pulse width of 152 fs is obtained with a repetition rate of 77.8 MHz and a signal-to-noise ratio of 75 dB. Stable operation of the femtosecond fiber laser over 200 h verifies the enhanced durability of the fSWCNT SA.
Passively mode-locked erbium-doped fiber lasers (EDFLs) have been popular in generating ultrashort optical pulses with widespread applications in industrial and scientific fields. Complex technical areas such as high-resolution microscopy, biophotonics, optical signal processing and optical metrology have extensively employed EDFLs due to their simple, reliable operation and consistent ability to operate in the picosecond and femtosecond regimes [1,2,3,4,5]. The working principle of a mode-locked fiber laser closely depends on the nonlinearity of an optical element called a saturable absorber (SA) present in the laser cavity whose intensity-dependent response is crucial for ultrashort pulse generation. Although the mode-locking technology based on semiconductor saturable absorber mirrors (SESAMs) is widely applied, the high cost and complexity in operation have had adverse effects on their wide-ranging application [6, 7].
Low-cost manufacturing combined with attractive features such as high nonlinearity has enabled the use of low-dimensional materials such as carbon nanotubes (CNTs) or graphene as alternatives to SESAMs. Film-type SAs with a free-standing capability can be applied to all fiber lasers in the simplest way. However, low-dimensional material-based SA films have also suffered significantly from low thermal stability and a high degradation rate owing to unreliable preparation processes, rendering them unsuitable for durable laser applications. Several studies using microfiber (for evanescent wave interactions) have been performed to overcome these limitations, but polarization sensitivity and high loss problems are caused by side-polished fibers [8, 9]. Additionally, tapered fibers with a few µm waist diameter are broken by minor impacts and exposure to environmental changes for a long time [10]. Recently, various studies have been performed to improve the damage resistance of SAs with tungsten disulfide, MoS\(_{2}\) and In\(_{2}\)Se\(_{3}\) [11,12,13].
Saturable absorption is a nonlinear phenomenon in which photonic materials undergo saturation of optical absorption under high-intensity illumination, which constitutes the basis of passive mode locking [14, 15]. Although all materials show a certain degree of saturable absorption, it occurs close to their optical damage threshold. Conversely, in CNTs, this absorption occurs at modest light intensities, while varying diameters offers better control over the operational wavelength [16]. Some of the key requirements of ideal SAs include superior control over saturable absorption and thermal durability to ensure stable, uninterrupted laser operation. CNT-polymer composite SAs with ease of producing large-area thin films with optical uniformity have the potential to satisfy some of these conditions [17]. One of the reasons behind the current capability limitations of SAs is the lack of uniformity in the distribution of CNTs in the polymer matrix, which greatly reduces the overall composite film performance. Conventional dispersion methods using surfactants such as sodium dodecyl sulfate (SDS) have some drawbacks (increased optical loss due to impurities, etc.). We present a solution to this problem by using fCNTs in a polymeric double layer film framework, which shows remarkable thermal stability and damage resistivity under stringent laser operation. Several reported works on high-performing composite polymers involved surface modification of graphene through ultrasonication techniques [18]. We hypothesized that mechanical modification or functionalization will be helpful in the formation of stable dispersions of CNTs in solvent media, which in turn will lead to perfect integration of nanofillers (CNTs) into the polymer matrix [19, 20]. Composite materials with uniformly dispersed nanofillers in the polymer matrix have displayed improved structural integrity combined with a higher thermal threshold in previously reported works [21].
As presented in this paper, SWCNTs were partially functionalized, resulting in an increase in the intertube distance. The improved uniformity of the the functionalized SWCNT (fSWCNT) SA is shown by scanning electron microscopy (SEM) images. Structural and chemical evaluation of SAs was also performed by applying X-ray diffraction (XRD), Raman and Fourier transform infrared (FTIR) techniques. The fSWCNT SA is proven to be thermally stable at high temperatures exceeding 650 \(^{\circ }{\mathrm {C}}\) according to thermogravimetric analysis (TGA) and to exhibit sustained damage resistivity in an EDFL operating at 1550 \(\text {nm}\) over a wide range of output powers. After exposing the SA to high-power continuous waves in the 1550 nm range, it was applied to an EDFL, and stable mode locking was confirmed. To our knowledge, this is the first comprehensive analysis of damage resistance involving one-dimensional material-reinforced polymer SA films. Our results demonstrate that fSWCNT SAs exhibit remarkable performance, including high damage resistance, due to the uniform distribution and femtosecond pulse generation in EDFLs. This new class of damage-resistant SAs is suitable for mass production at a low cost with good reproducibility, so it has the potential to usher in more advanced development of low-dimensional material-based polymeric SAs in the future.
CNTs are classified as single-walled CNTs (SWCNTs) and multiwalled CNTs (MWCNTs) according to the number of layers constituting them. SWCNTs are cylinders composed of a layer of a graphene strip, and the structural characteristics can be defined in terms of the chirality by a chiral vector (\(C_{\mathrm{h}}\)). The width of the strip is equal to the circumference of the SWCNT and can be expressed as
$$\begin{aligned} C_{\mathrm{h}}=n\vec {a_1}+m\vec {a_2}\equiv (n,m) \end{aligned}$$
where \(\vec {a_1}\) and \(\vec {a_2}\) are two linearly independent unit vectors of the hexagonal lattice and n and m are integers. The unique electronic properties of SWCNTs depend on the chiral vector and are divided into metallic (\(n-m=3k\)) and semiconducting (\(n-m\ne 3k\)), where k is an integer. Semiconducting SWCNTs are mainly used in optical applications and are selected according to the applied wavelength band. We purchased super-purified SWCNTs produced in the HiPco\(^{TM}\) (high-pressure carbon monoxide) process from NanoIntegris. The diameter of individual SWCNTs was between 0.8 and 1.2 nm (mean diameter: 1.0 nm), which is suitable for mode locking of EDFLs operating in the 1550 nm wavelength band.
Polyvinyl alcohol (PVA), poly methyl methacrylate (PMMA) and sodium carboxymethylcellulose (NaCMC) are commonly used for the host polymer of SA films but have several limitations. The low glass transition temperatures of PVA (85 \(^{\circ }\)C) and PMMA (105 \(^{\circ }\)C) limit the thermal endurance, and NaCMC is vulnerable to moisture. Polydimethylsiloxane (PDMS) was selected because of its high thermal stability below 350 \(^{\circ }\)C and moisture resistace. PDMS was purchased from Dow Corning, and chloroform (solvent of PDMS) was purchased from Sigma Aldrich.
CNT-Polymer SA Preparation
Preparation of a uniform and stable dispersion of CNTs is the first step in the fabrication of any CNT-based composite material due to their strongly aggregated state. Direct manual mixing of CNTs and a polymer matrix mostly results in poor distribution of CNTs. Agglomeration of CNTs generally causes severe problems in composite performance because it limits mechanical, thermal and optical properties by hindering the flow of energy through the interconnected network of the polymer. We selected the common solvent chloroform (\({\text {CHCl}_{3}}\)) to facilitate the integration of CNTs and PDMS. Even though several solvents are available for PDMS, chloroform was chosen because of the relative ease of evaporation at a later stage of the experiment. The limited chemical reactivity of functionalized CNTs (fCNTs) enables them to unbundle themselves during reaction and establish bonding interactions with the active chemical groups in the polymer matrix, thereby ensuring a proper and uniform dispersion of the nanofillers.
A carefully calibrated ultrasonication-assisted procedure induces partial surface modification (or functionalization) of the CNTs by transforming the sp\(^{2}\) hybridized atoms in the network into the sp\(^{3}\) configuration. The electron-rich sites of the sp\(^{3}\) network enable efficient unbundling, disruption of the \(\pi -\pi\) attachment and, finally, uniform attachment to the polymer matrix. This phenomenon eliminates the possibility of agglomeration by nanofillers, which is the most common problem observed in composite materials [22]. Figure 1 shows a schematic diagram of the preparation procedure. We ultrasonicated 5 mg of CNTs in chloroform to prepare a uniform and stable dispersion. We used 350 W probe ultrasonication (ULSSO HITECH, ULH700S) for approximately 3 h to introduce surface defects and minimize the degree of agglomeration of the CNTs. After the preparation of the CNT dispersion, we added it to 1 g of PDMS with constant stirring for another 60 min and then ultrasonicated the mixture for 30 min. We added a hardener to the CNT-PDMS mixture at an amout equaling 1/5th of that of the polymer and stirred again for 10 min. We dropped the mixture into a glass petri dish in a controlled manner and left it to settle for 30 min at 90 °C. Then, another layer of polymer mixture was added using a similar procedure, and the double-layered film was left to cure for 90 min at 90 °C.
Schematic representation of partial functionalization of SWCNTs and overall film-type SA preparation process
Fiber Laser Setup
We set up a femtosecond fiber laser to test the mode locking capability of the fSWCNT-PDMS SAs. In soliton mode locking, when the soliton pulse goes through the fiber cavity, no temporal or spectral changes in the pulses occur as a result of the combined effects of chromatic dispersion and nonlinearity (self-phase modulation). Even though soliton lasers have some limitations in practical applications due to spectral sideband generation, among others issues, they are relatively stable and easy to control.
The laser setup is shown in Fig. 2a. The optical fiber cavity is composed of a 980/1550 wavelength-division multiplexer (WDM), an isolator, a 50:50 coupler and an erbium-doped fiber (EDF). It was designed in a ring configuration, and all single-mode (SM) fibers were fusion spliced. To operate the fiber laser in the soliton mode-locked regime, negative net dispersion is required. Therefore, the lengths of the positively dispersive EDF and negatively dispersive SMF-28 and HI1060 were adjusted. The length of the EDF (0.033 ps\(^{2}\)/m at 1550 nm) was 93 cm, SMF-28 (\(-0.023\) ps\(^{2}\)/m at 1550 nm) was 159 cm and HI1060 (\(-0.007\) ps\(^{2}\)/m at 1550 nm) was 16 cm. The total intracavity group velocity dispersion (GVD) was \(-0.007\) ps\(^{2}\)/m, and the total length of the cavity was 268 cm.
Schematics of the a passively mode-locked EDFL setup and b nonlinear absorbance measuring system
Erbium (Er) belongs to the group of rare earth materials and was used in the form of the trivalent ion Er\(^{3+}\) as the laser-active dopant of the gain medium. The EDF was used for the lasing medium, and it was laser pumped by an external energy source. The WDM connects the pump laser diode (LD) to the laser cavity with the HI1060, which delivers both a 980 nm pump source and a 1550 nm pulsed laser. The SA film was inserted between the FC/APC connectors located between the WDM and coupler.
To evaluate the performance of SAs, two kinds of representative indicators, the modulation depth and nonsaturable loss, should be determined. The modulation depth is the maximum change in absorption (or reflection) of SAs that emerges due to incident light. This is the most important parameter: the laser cannot be mode locked in the case of a modulation depth lower than a sufficient value. In the soliton regime, a moderate modulation depth can be used, but in the case of other regimes (stretched pulse, etc.), a very high value is required for mode locking because of the high instability. The nonsaturable loss is typically an unwanted factor of an SA, and its value is the absorption rate when the SA is saturated at high intensity. In the case of a thin film SA, an excessively high concentration of nanomaterial and a high thickness of the film lead to a high value of the nonsaturable loss.
In general, SAs tend to exhibit increasd transmittance under high-intensity light. Therefore, to determine the saturation absorption characteristics, the power incident on the SA must be changed and how the absorption rate changes accordingly measured. A schematic diagram of the nonlinear absorbance measuring system is presented in Fig. 2b. This system was designed to measure optical properties by the power-scan (p-scan) method, which measures the transmittance of the SA according to the incident optical intensity. There are three different parts in the system: an oscillator, an amplifier, and an absorbance measuring device in the system. All the three parts are connected by SMF-28 and isolator is located between each to maintaining forward direction delivery. The entire system was developed in the form of an all-fiber system. The configuration of the fiber oscillator is the same as in Fig. 2a except for the length of each optical fiber. It generates femtosecond pulses in the wavelength band in which the SA will be operated. In the fiber amplifier, the EDF is forward pumped by an additional laser diode. The erbium ions are excited into metastable states and amplify the seed laser via stimulated emission. The low-power seed laser emitted from the oscillator passes thorough the fiber amplifier, and very high-intensity pulses enter the absorbance measuring device. The intensity of light incident on the SA is controlled through a variable attenuator (Thorlabs, VOA50), and light is divided in a ratio of 9:1 by an optical fiber coupler. Ninety percent of the light passes through the SA, and the power ofthe transmitted light is measured. The power of the remaining 10% of the light is measured directly by another power meter. Since the optical power emitted from the 90% terminal of the coupler can be inferred through the optical power measured at the 10% terminal, the light incident on the SA can be identified. The transmittance of the SA can be calculated using this value and the optical power of light transmitted through the SA. The two power meters are connected to a computer that enables real-time monitoring.
Structural Analysis of the CNTs and fCNTs
Structural analysis of the CNT and fCNT samples was carried out by applying XRD and Raman spectroscopy techniques. XRD was helpful in ascertaining the degree of the difference in the crystallinity of the samples. We analyzed the extent of damage or reorganization of structural bonding in the CNT walls as a result of intensive ultrasonication via Raman spectroscopy.
The typical crystalline nature of the CNT structure results in a strong peak at \(\sim\)26° in the XRD spectrum corresponding to the (002) plane, which generally indicates interlayer distance of approximately 0.34 nm (by applying Bragg's law) [23]. The (100) peak for CNTs reflect the hexagonal graphite structure [24]. The XRD result of CNTs is shown at the bottom of Fig. 3a. In the case of SWCNTs, the (002) peak is slightly shifted toward the larger interlayer distance than in the case of MWCNTs, and is broad and asymmetric. These results demonstrate that the (002) peak reflecting the intertube distance (outer-wall contacts) and the broadening is caused by curvature effects due to the cylindrical shape [25]. The (100) peak is shown in the inset to appear at \(\sim 43^{\circ }\), which corresponds to a spacing of \(\sim\)0.2 nm [26]. When the crystallinity is disturbed in graphitic materials, this results in the collapse of the C=C bonding integral to \(\text {sp}^{2}\) hybridization and creates \(\text {sp}^{3}\) bonding [27]. These structural disturbances can be initiated either by disturbing the bonds with outside passive forces or by inserting foreign functional groups [28]. In this case strong ultrasonication in the solvent resulted of structural modification in the CNT walls, which caused disappearance of the (002) peak [29]. The change of (002) peak according to the time of ultrasonication can be confirmed in Additional file 1. In the XRD pattern of 1-hour sonicated sample, (002) peak becomes broad and moves toward the direction of low angle. And in the XRD pattern of 2-hour sonicated sample, (002) peak becomes even more broader and moves further toward the direction of increasing intertube distance. Lastly, we confirmed that (002) peak eventually disappeared in the XRD pattern of 3-hour sonicated sample (Additional file 1: Figure S1).
a Comparison of XRD results for SWCNT and fSWCNT powders (inset: enlarged graph of SWCNT data in the 20–50° range). Comparison of b Raman analysis and c FTIR results of SWCNT and fSWCNT powders
We obtained further confirmation of this structural modification in the CNT samples by the Raman spectroscopic results in Fig. 3b. The Raman spectra of CNT and fCNT samples consist of two main peaks called the defect peak (D peak) and graphitic peak (G peak). While the G peak represents sp\(^{2}\) hybridized C=C bonding formation, the D peak typically denotes disordered graphitic structures or sp\(^{3}\) hybridized carbon atoms [30]. The CNTs showed D and G peaks at 1336 cm\(^{-1}\) and 1588 cm\(^{-1}\), respectively, while the peaks were at 1323 cm\(^{-1}\) and 1590 cm\(^{-1}\) for fCNTs in the same order. The ratio of the peak intensites (I\(_{G}\)/I\(_{D}\)) reveals the extent of the structural integrity of the samples [31]. Applying passive external forces such as rigorous ultrasonication causes stress on the C=C bonds that induces them to open up, creating electron-rich carbon centers and bringing considerable damage to the CNT wall [32]. We calculated the relative \(I_{G}/I_{D}\) ratios of the CNT and fCNT samples after ultrasonication, which showed increased relative D peak intensity in the latter sample. The overall \(I_{G}/I_{D}\) ratios were 1.23 and 1.06 for the CNTs and fCNTs, respectively, indicating structural modification because of mechanically induced functionalization. Since we wanted to largely preserve the innate characteristics of the CNT structure, we call this process a partial functionalization in which structural damage was initiated with the sole purpose of including a uniform bonding interaction between the nano-filler and the polymer matrix.
FTIR spectrosocpy was carried out to confirm that functional groups were attached to SWCNTs during ultrasonication in chloroform. The results measured through the KBr sample pellet are shown in Fig. 3c. Chloroform is mainly physiabsorbed into SWCNTs, but there is also the possibility of partially forming the SWCNT-\(\text {CCl}_{2}\) complex [33]. In previous research, SWCNT-\(\text {CCl}_\text {2}\) showed a peak around 2800 cm\(^{-1}\), but in this result, it can be confirmed that there is no related functional group because the corresponding peak is absent. The characteristic absorption band for the C=C bond stretching vibrations appearing at about 1600 cm\(^{-1}\). In general, SWCNTs show a weak IR spectrum because the difference in charge state between carbon atoms is small. In functionalized SWCNTs, generation of induced electric dipoles is enhanced and the size of the absorption peak increases as the symmetry of the nanotube is broken. It can be seen in Fig. 3c that the surface modification caused by a partial defect increased the peak located at 1633 cm\(^{-1}\). The broad absorption band around 3400 cm\(^{-1}\) can be caused by oxidation in the process of purification by the manufacturer, and is also greatly affected by moisture. The reason that fSWNCT shows increased absortpion at 3432 cm\(^{-1}\) is considered to be because the defect site absorbed moisture during drying for FTIR measurement. It is considered that the increase in the absorption band centered at 3432 cm is due to the absorption of moisture by the defective site during the drying process.
Analysis of SA Thin Films
Structural and Chemical Analysis
Generally, CNTs are supplied in powder form and most of them are bundled as shown in Fig. 4c. Without functionalization, CNTs remains aggregated, and the degree of dispersion is inferior, as shown in Fig. 4a. Figure 4b shows uniformly distributed CNTs without severe aggregation in the fSWCNT-PDMS composite film. EDS (energy dispersive spectroscopy) analysis shows 6.8 wt% of residual iron (Fe) catalyst in powdered CNTs. Peaks for the elements of PDMS, i.e., oxygen (O), silicon (Si) and carbon (c), are shown in the data for SWCNT-PDMS composites which is in good agreement with previously reported PDMS matrix data [34].
SEM images and EDS results for a SWCNT-PDMS, b fSWCNT-PDMS composite films and c SWCNT powder. d Comparison of XRD results for pure PDMS and SA thin films
We performed XRD studies to analyze the structural features of the polymeric SA films, and Fig. 4d shows the results. Pure PDMS shows a characteristic halo at 11.51°. The SWCNT-PDMS and fSWCNT-PDMS samples show sharp peaks at 11.98° and 11.49°. When mixed with fSWCNTs, the intensity of the halo peak from PDMS and SWCNT-PDMS decreases, but the peak remains at a similar position, possibly due to uniform mixing and superimposition with the fSWCNT peak [35].
We performed Raman spectroscopy on the samples consisting of pure PDMS SA films and composite SAs to ascertain the structural changes, if any, as a result of nanofiller addition. In Fig. 5a, the SA films show peaks typical of PDMS, but in the composite SA films, there are extra peaks due to the fSWCNT addition. The separate and distinct peaks at 491 cm\(^{-1}\), 708 cm\(^{-1}\), and 785 cm\(^{-1}\) indicate the presence of Si–O–Si symmetric stretching, Si–C symmetric stretching and Si–C asymmetric stretching, respectively. We identified \(\text {CH}_{3}\) symmetric rocking and asymmetric bending vibrations with clear peaks at 860 cm\(^{-1}\) and 1413 cm\(^{-1}\), respectively. The peaks at 2904 cm\(^{-1}\) and 2963 \(\text {cm}^{-1}\) represents \(\text {CH}_{3}\) symmetric stretching and \(\text {CH}_{3}\) asymmetric stretching vibrations, respectively [36]. The changes in the intensities and the broadening of some characteristic peaks along with the emergence of prominent D, G and G\(^\prime\) peaks at approximately 1317 \(\text {cm}^{-1}\), 1589 \(\text {cm}^{-1}\) and 2610 \(\text {cm}^{-1}\) confirm the presence of and attachment between the polymer and nanofiller.
Comparison of a Raman analysis and b FTIR results of pure PDMS and fSWCNT SA thin films
FTIR spectroscopy was carried out to analyze the presence of different functional groups in the pure PDMS and fSWCNT-PDMS SA films, and the results are shown in Fig. 5b. The different functional groups present in the PDMS and fSWCNTs are mainly instrumental for the bonding interactions between the nano-filler and the polymer matrix, thereby affecting the overall properties of the final SA films. The absence of any major or minor peaks at 3300 \(\text {cm}^{-1}\) and 1700 \(\text {cm}^{-1}\), which typically indicate the presence of hydroxyl and carboxylic groups, confirms the total evaporation of acetic acid residues from the composite films due to the curing process. The peaks at 2905 \(\text {cm}^{-1}\) and 2965 \(\text {cm}^{-1}\) are indicative of symmetric and asymmetric \(-\text {CH}_{3}\) stretching in \(\equiv \text {Si}-\text {CH}_{3}\) from PDMS. The major peak at 843 \(\text {cm}^{-1}\) denotes the rocking peak of stretching bands originating from Si–C bonds. The sharp peak at approximately 790 \(\text {cm}^{-1}\) corresponds to the rocking vibration of the Si–\((\text {CH}_{3})_{2}\) group, and the peak at 690 \(\text {cm}^{-1}\) is related to Si–C stretching. The peaks at approximatley 1414 \(\text {cm}^{-1}\) and 1260 \(\text {cm}^{-1}\) are related to asymmetric and symmetric stretching of C–H bond from Si–\((\text {CH}_{3})_{2}\). Multiple peaks at approximately 1011 \(\text {cm}^{-1}\) and 1055 \(\text {cm}^{-1}\) identify the stretching of Si–O–Si from the long chain structure of PDMS [37]. The peak intensity of the films containing fSWCNTs appears to slightly decrease with better dispersion of the SWCNTs in the films, possibly due to the formation of Si-C bonds between the PDMS structure and fSWCNTs. We did not find any significant difference between the pure PDMS film and the fSWCNT-PDMS film apart from some difference in intensity at approximately 912 \(\text {cm}^{-1}\), which is related to Si–H bending [38].
Figure 6 shows the TGA results of the SA samples. Since ultrashort pulse laser operation generates considerable electric power in the laser cavity, the SAs inevitably suffer thermal damage over time. Before placing the SAs inside the laser cavity, we tried to determine their thermal threshold by TGA. In this experiment, small samples of SA thin films were placed in nitrogen \((\text {N}_{2})\) environment and heated. The rate of temperature increase was fixed at 10 °C/min, and the temperature was recorded from 100 to 700 \(^{\circ }\)C. This experiment helped identify the thermal degradation threshold by measuring the total mass of the samples with respect to the increasing temperature as well as their end mass after the experiment was completed. The thermal stability is directly proportional to the extent of uniform attachment between the polymer and the nanofiller. PDMS starts to decompose into its organic and inorganic parts at approximately its boiling temperature of 420 °C. During this pyrolytic process, it gives SiOx (\(x = 1, 2\)) compounds and carbon ash while losing mass weight at a significant rate [39]. In this case, the CNT-PDMS and fCNT-PDMS thin films exhibit similar onset temperatures for the initiation of structural damage. While the fCNT-PDMS SA films retained almost 60% of the initial mass, the PDMS SA and CNT-PDMS films retained only 27% and 45% of the starting mass, indicating significant structural cohesion [40]. The covalent bonds between the fCNTs and PDMS matrix due to both the unique nano filler properties and the double layer fabrication process prevented total dissolution of the composite structure [41].
Comparison of TGA results of pure PDMS and film-type SAs
Damage Test
We put the composite SA samples through rigorous damage testing experiments. Since a continuous wave laser is more likely to cause severe thermal damage in thin film SAs, we employed it for this purpose. The damage testing conditions featured high-powered fiber laser operation at a significantly elevated capacity compared to that required in ordinary conditions for an extended amount of time. The rationale behind running the fiber laser under these stringent conditions was to ensure significant damage to the SA film samples. The experiment lasted continuously for 12 h. The samples showed sustained damage resistivity over a wide range from 4 to 13 mW before the degradation accelerated.
Figure 7a shows the variation in the optical power of the light transmitted through the SA samples for 12 h. There was almost no change in the absorption of the fSWCNT-PDMS SA for incident light power of 8 mW or less. When the SA is exposed to a power of 10 mW, the absorption is very slightly increased. This occurs due to graphitization of SWCNTs and increases the temperature of the SA. As a result, SWCNTs are burned out, and the absorption of the SA is decreased due to the lowered SWCNT concentration. At 12 mW incident power, graphitization and burning out occur simultaneously, and the change in absorbance is balanced. The amount of change is very small and stability is maintained under 12 mW power. The absorbance of the SA decreases noticeably at 13 mW incident power. This is caused by burning out of SWNTs in limited areas where dispersion is not perfectly achieved. The fCNT-PDMS SA shows significantly improved stability compared to previous research [42], in which the burning out stages started at 5 mW and the damage threshold was 8 mW.
a Variation in the output power during the damage test of a film-type SA (inset: output power of light after transmission through the SA). Comparison of damage test results of b SWCNT-PDMS and c fSWCNT-PDMS thin films
We analyzed the damage-tested samples by Raman spectroscopy and compared their extent of structural integrity with the Raman result. The results are presented in Fig. 7b, c. We identified 13 mW as the threshold for the samples and measured the ratio of the intensities of the graphitic peak and the defect peak in the Raman spectrum at this point to establish the overall structural integrity of the sample. We tested a different sample prepared following exactly the same process consisting of SWCNTs without functionalization at 13 mW compared its results with those of our original test samples. After analysis, the extent of structural damage in the fSWCNT-based SA sample became clear by comparing the Raman spectra [43]. As shown in Fig. 7b, the \(I_{G}/I_{D}\) ratio of the fSWCNT-PDMS SA film was 20.6 before commencing the damage test and remained at 16.5 after the damage test, retaining over 80% of the initial structural composition. In contrast, without functionalization, the SWCNT-based PDMS SA retained only approximately 51% of the initial structural composition after the 12 h damage test, as shown in Fig. 7c.
Laser Performance
A performance test of fSWCNT-PDMS SA films was conducted on the sample that had been subjected to the 12 h damage test at 12 mW incident power. An SA is an optical component whose absorption rate decreases as the intensity of light passing through it increases. The saturation of the absorption rate of the SA can be phenomenologically explained based on a two-level electronic model. The absorption rate is commonly modeled as
$$\begin{aligned} \alpha (I)=\alpha _0\left( 1+\frac{I}{I_{s}}\right) ^{-1} \end{aligned}$$
where \(\alpha (I)\) is the intensity-dependent absorption coefficient of the SA, \(\alpha _0\) is the linear absorption coefficient at low intensity, I is the light intensity and \(I_s\) is the saturation intensity of the SA [44]. The saturation absorption model in Eq. (2) is a model that calculates the absorption rate for an ideal SA, and when the incident light intensity is infinite, the absorption rate converges to zero. However, in reality, an intrinsic loss inevitably occurs due to elastic scattering, etc., which is related to the nonsaturable absorption coefficient (\(\alpha _{ns}\)). Therefore, the absorption model of an actual SA can be expressed as
$$\begin{aligned} \alpha (I)=\alpha _0\left( 1+\frac{I}{I_{s}}\right) ^{-1}+\alpha _{ns} \end{aligned}$$
The saturation absorption characteristics of the SA samples were measured with a nonlinear absorbance measurement system that delivered \(\sim\)300 fs pulses at a central wavelength of 1550 nm. The nonlinear transmittance increases from 11.6% to 22.0%. By fitting the measured data to Eq. (3), the saturation intensity (\(I_{s}\)) is 43 MW/\(\text {cm}^{2}\) (0.37 mW in average power) and the modulation depth (\(\Delta \alpha\)) is 25.3%, as shown in Fig. 8a. In the measured data, some spikes occurred when the measuring range of the optical power meter was changed.
a Normalized saturable absorption of the fSWCNT-PDMS SA. b Stability test of the passively mode-locked EDFL for 200 h (inset: enlarged graph for 1 h)
Figure 8b shows the long-term stability of the fSWCNT-PDMS SA, and it ensures durable operation of mode-locked fiber lasers. The inset shows the enlarged figure for 1 h. The small perturbations and periodic changes are thought to be caused by other factors, such as the temperature control of the laser diode.
The soliton fiber laser demonstrated self-starting single-pulse mode locking between 90 and 140 mW pump power. Figure 9 shows the optical spectrum, autocorrelation trace, pulse train and radio frequency (RF) signal spectrum from the EDFL setup. The average output power is 2.9 mW when it operates at a pump power of 140 mW. The existence of Kelly sidebands, a characteristic of soliton pulses, is shown in Fig. 9a. The optical spectrum exhibits a typical soliton spectrum shape (sech2) with a central wavelength of 1576 nm. The full-width at half-maximum (FWHM) bandwidth is 19.8 nm. Figure 9b shows the autocorrleation trace. The FWHM pulse duration is 238 fs, and assuming a sech2 pulse shape, the inferred pulse duration is 152 fs. The time-bandwidth product is 0.373 which means that pulses are close to transform-limited. The pulse train is shown in Fig. 9c, and the repetition rate is 77.82 MHz. Multiple pulses are obtained with equidistant spacing at high pump powers and quintuple pulsing is maintained up to 340 mw. The RF signal spectrum around the fundamental repetition rate is shown in Fig. 9d. From the signal-to-noise ratio (SNR) of \(\sim\)75 dB, we can verify the stability of pulsed laser operation with low amplitude fluctuation and timing jitter. Compared with the mode-locked fiber laser using the CNT-PDMS composite of the previous research, it can be confirmed that the spectral bandwidth, pulse duration, and SNR are improved based on the superior SA performance [45].
Pulse output properties of the laser: a optical spectrum, b autocorrelation trace, c pulse trains under different pump powers, d RF spectrum at the fundamental repetition rate with a 100 Hz resolution bandwidth (top), and RF spectrum with 1 GHz with 30 kHz resolution bandwidth
Partially functionalized SWCNTs were uniformly incorporated within a PDMS polymer matrix, and fSWCNT SAs showed high damage resistance with enhanced durability. The mechanical functionalization process solved the problem of CNT aggregation without using any surfactant or effecting chemical functional groups, both of which compromise the inherent nonlinear characteristics in conventional methods. The SAs demonstrate superior thermal stability and damage resistance during TGA and rigorous damage tests, so they ensure high-power ultrashort laser operation by virtue of their superior material characteristics and preparation procedure. Stable mode locking was acheived by placing the fSWCNTs into an EDFL. The ultrashort pulse trains exhibited a 77.82 MHz repetition rate, a 152 fs pulse duration and an SNR of 75 dB in single pulse generation. Stable femtosecond laser operation was also demonstrated for over 200 h, so the proposed fSWCNT SA provides a promising solution for laser applications that require ultrashort pulsed lasers with long-term durability. We demonstrated new approaches for developing damage-resistant polymer SA films and comprehensive analyses for self-standing SA films. Various related studies on SAs that use other low dimensional materials and photonic devices for pulsed fiber lasers with high output energy are expected based on this work.
Experimental Section
Fiber Components
A fiber Bragg grating-stabilized 980 nm pump laser diode (JDSU, S30) connected to a tabletop laser and a Peltier controller (OsTecch, ds11-t85) was used as the laser source. Three kinds of SM optical fibers, SMF-28, HI1060 (high-index) and ER80-4/125 (erbium-doped fiber for 980 nm pumping with emission at 1550 nm), were purchased from Thorlabs. All SMF-28 (transmission from 1260 to 1650 nm), HI-1060 (transmission from 980 to 1650 nm) and ER80-4/125 (80±8 dB/m core absorption peak at 1530 nm) fibers had a diameter of 125 µm. The WDM combines two wavelengths (980 and 1550 nm) into a single fiber output, the isolator (1550 nm center wavelength) prevents reverse transmission of light, and the \(1 \times 2\) coupler splits the light at a 50:50 ratio. Three fiber optic components, the WDM (WD9850BA), isolator (IO-H-1550) and coupler (TW1550R5A1), were purchased from Thorlabs. We used a cleaver (NorthLab Photonics, ProCleave SD/SFC) and a fusion splicer (UCL SWIFT, S3) for fiber processing.
Laser Characterization
The fiber laser characterization consisted of measuring the spectral bandwidth, pulse duration, repetition rate and SNR of the output pulses. An indium gallium arsenide (InGaAs) photodetector (EOT, ET-3010) was used for optical signal acquisition. The spectral bandwidth was measured with an optical spectrum analyzer (Anritsu, MS9710C), and the pulse duration was measured with an autocorrelator (APE, Mini TPA/PD). The repetition rate was measured through a pulse train from an oscilloscope (InfiniiVision, DS07104A), and the SNR was measured through an RF signal from a spectrum analyzer (Advantest, R3477). The optical power was measured with two kinds of power meters: an integrating sphere photodiode power sensor (PM100D) and a compact fiber photodiode power sensor (S155C), which were both manufactured by Thorlabs.
Polymer Composite Characterization
We analyzed the structural aspects of the CNT, fCNT and SA samples using XRD. Both high-resolution powder XRD (RIGAKU, SmartLab) and high-resolution thin film XRD (PANalytical, X'Pert-PRO MRD) were operated between 5° and 50° (2\(\theta\)). High-resolution Raman spectroscopic analysis (HORIBA, LabRAM HR Evolution Visible-NIR) was carried out with a 633 nm laser source. Chemical characterization typically involved FTIR spectroscopy using a Nicolet iS50 from Thermo Fisher Scientific Instrument. Thermal stability studies involved TGA experiments on a TG209 F1 Libra from Netzsch.
The data and materials in this article are fully available without restriction from the corresponding author on reasonable request.
Saturable absorber
CNT:
Carbon nanotube
SWCNT:
Single-walled carbon nanotube
fSWCNT:
Functionalized single-walled carbon nanotube
EDFL:
Erbium-doped fiber laser
SESAM:
Semiconductor saturable absorber mirror
SDS:
sodium dodecyl sulfate
HiPco:
High pressure carbon monoxide
Poly methyl methacrylate
NaCMC:
Sodium carboxymethylcellulose
EDF:
Erbium-doped fiber
SMF:
Single mode fiber
GVD:
Group velocity dispersion
Er:
LD:
P-scan:
Power-scan
G peak:
Graphitic peak
D peak:
Defect peak
FWHM:
Full-width half-maximum
SNR:
SM:
High-index
WDM:
Wavelength-division multiplexer
InGaAs:
Indium gallium arsenide
RF:
Set SY, Yaguchi H, Tanaka Y, Jablonski M (2004) Ultrafast fiber pulsed lasers incorporating carbon nanotubes. IEEE J Sel Top Quantum Electron 10:137–146
Pegoraro AF, Ridsdale A, Moffatt DJ, Pezacki JP, Thomas BK, Fu L, Dong L, Fermann ME, Stolow A (2009) All-fiber cars microscopy of live cells. Opt Express 17:20700–20706
Jones DJ, Diddams SA, Ranka JK, Stentz A, Windeler RS, Hall JL, Cundiff ST (2000) Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis. Science 288:635–639
Debnath PC, Yeom D-I (2021) Ultrafast fiber lasers with low-dimensional saturable absorbers: status and prospects. Sensors 21:3676
Chen Z, Wang Y, Lv R, Liu S, Wang J, Wang Y (2020) High-power erbium-doped fiber laser with a carbon nanotubes-doped sol-gel glass mode-locker. Opt Fiber Technol 58:102189
Keller U, Weingarten KJ, Kartner FX, Kopf D, Braun B, Jung ID, Fluck R, Honninger C, Matuschek N, Der Au JA (1996) Semiconductor saturable absorber mirrors (SESAM's) for femtosecond to nanosecond pulse generation in solid-state lasers. IEEE J Sel Top Quantum Electron 2:435–453
Okhotnikov O, Grudinin A, Pessa M (2004) Ultra-fast fibre laser systems based on SESAM technology: new horizons and applications. New J Phys 6:177
Mkrtchyan AA, Gladush YG, Galiakhmetova D, Yakovlev V, Ahtyamov VT, Nasibulin AG (2019) Dry-transfer technique for polymer-free single-walled carbon nanotube saturable absorber on a side polished fiber. Opt Mater Express 9:1551–1561
Sheng Q, Feng M, Xin W, Han T, Liu Y, Liu Z, Tian J (2013) Actively manipulation of operation states in passively pulsed fiber lasers by using graphene saturable absorber on microfiber. Opt Express 21:14859–14866
Lee H, Kwon WS, Kim JH, Kang D, Kim S (2015) Polarization insensitive graphene saturable absorbers using etched fiber for highly stable ultrafast fiber lasers. Opt Express 23:22116–22122
Chen H, Chen Y, Yin J, Zhang X, Guo T, Yan P (2016) High-damage-resistant tungsten disulfide saturable absorber mirror for passively q-switched fiber laser. Opt Express 24:16287–16296
Ma P, Lin W, Zhang H, Xu S, Yang Z (2019) High-power large-energy rectangular mode-locked Er-doped fiber laser based on high-damage-threshold mos2 saturable absorber. IEEE Photonics J 11:1–12
Han X, Zhang H, Jiang S, Zhang C, Li D, Guo Q, Gao J, Man B (2019) Improved laser damage threshold of In2Se3 saturable absorber by PVD for high-power mode-locked Er-doped fiber laser. Nanomaterials 9:1216
Ostojic G, Zaric S, Kono J, Strano M, Moore V, Hauge R, Smalley R (2004) Interband recombination dynamics in resonantly excited single-walled carbon nanotubes. Phys Rev Lett 92:117402
Martinez A, Sun Z (2013) Nanotube and graphene saturable absorbers for fibre lasers. Nat Photonics 7:842–845
Steinberg D, Rosa HG, de Souza EAT (2017) Influence of carbon nanotubes saturable absorbers diameter on mode-locking erbium-doped fiber laser performance. J Lightwave Technol 35:4804–4808
Schibli T, Minoshima K, Kataura H, Itoga E, Minami N, Kazaoui S, Miyashita K, Tokumoto M, Sakakibara Y (2005) Ultrashort pulse-generation by saturable absorber mirrors based on polymer-embedded carbon nanotubes. Opt Express 13:8025–8031
Shen B, Zhai W, Lu D, Wang J, Zheng W (2012) Ultrasonication-assisted direct functionalization of graphene with macromolecules. RSC Adv 2:4713–4719
Sun Y-P, Fu K, Lin Y, Huang W (2002) Functionalized carbon nanotubes: properties and applications. Acc Chem Res 35:1096–1104
Cheng Q, Debnath S, O'Neill L, Hedderman TG, Gregan E, Byrne HJ (2010) Systematic study of the dispersion of SWNTs in organic solvents. J Phys Chem C 114:4857–4863
Sahoo NG, Rana S, Cho JW, Li L, Chan SH (2010) Polymer nanocomposites based on functionalized carbon nanotubes. Prog Polym Sci 35:837–867
Liu C-X, Choi J-W (2012) Improved dispersion of carbon nanotubes in polymers at high concentrations. Nanomaterials 2:329–347
Soleimani H, Yahya N, Baig M, Khodapanah L, Sabet M, Burda M, Oechsner A, Awang M (2015) Synthesis of carbon nanotubes for oil-water interfacial tension reduction. Oil Gas Res 1:1000104
Atchudan R, Cha BG, Lone N, Kim J, Joo J (2019) Synthesis of high-quality carbon nanotubes by using monodisperse spherical mesoporous silica encapsulating iron oxide nanoparticles. Korean J Chem Eng 36:157–165
Futaba DN, Yamada T, Kobashi K, Yumura M, Hata K (2011) Macroscopic wall number analysis of single-walled, double-walled, and few-walled carbon nanotubes by X-ray diffraction. J Am Chem Soc 133:5716–5719
Aftab S, Hussain S, Siddique M, Nawaz H (2010) Comprehensive study of trends in the functionalization of CNTs using same oxidizing acids in different conditions. Der Pharm Chem 2:354–365
Qi G-Q, Cao J, Bao R-Y, Liu Z-Y, Yang W, Xie B-H, Yang M-B (2013) Tuning the structure of graphene oxide and the properties of poly (vinyl alcohol)/graphene oxide nanocomposites by ultrasonication. J Mater Chem A 1:3163–3170
Sankal S, Kaynak C (2013) Using various techniques to characterize oxidative functionalized and aminosilanized carbon nanotubes for polyamide matrix. J Reinf Plast Compos 32:75–86
Miyata Y, Yanagi K, Maniwa Y, Tanaka T, Kataura H (2008) Diameter analysis of rebundled single-wall carbon nanotubes using X-ray diffraction: verification of chirality assignment based on optical spectra. J Phys Chem C 112:15997–16001
Nakamiya T, Ueda T, Ikegami T, Ebihara K, Tsuda R (2008) Thermal analysis of carbon nanotube film irradiated by a pulsed laser. Curr Appl Phys 8:400–403
Costa S, Borowiak-Palen E, Kruszynska M, Bachmatiuk A, Kalenczuk R (2008) Characterization of carbon nanotubes by Raman spectroscopy. Mater Sci Pol 26:433–441
Yang D-Q, Rochette J-F, Sacher E (2005) Functionalization of multiwalled carbon nanotubes by mild aqueous sonication. J Phys Chem B 109:7788–7794
Girão EC, Liebold-Ribeiro Y, Batista JA, Barros EB, Fagan SB, Mendes Filho J, Dresselhaus MS, Souza Filho AG (2010) Functionalization of single-wall carbon nanotubes through chloroform adsorption: theory and experiment. Phys Chem Chem Phys 12:1518–1524
Stancu V, Galatanu A, Enculescu M, Onea M, Popescu B, Palade P, Aradoaie M, Ciobanu R, Pintilie L (2021) Influences of dispersions' shapes and processing in magnetic field on thermal conductibility of PDMS-FE3O4 composites. Materials 14:3696
Kim JH, Hwang J-Y, Hwang HR, Kim HS, Lee JH, Seo J-W, Shin US, Lee S-H (2018) Simple and cost-effective method of highly conductive and elastic carbon nanotube/polydimethylsiloxane composite for wearable electronics. Sci Rep 8:1–11
Shahzad M, Giorcelli M, Shahzad N, Guastella S, Castellino M, Jagdale P, Tagliaferro A (2013) Study of carbon nanotubes based polydimethylsiloxane composite films. J Phys Conf Ser 439:012010
Silva EAd, Windmöller D, Silva GG, Figueiredo KCdS (2017) Polydimethylsiloxane membranes containing multi-walled carbon nanotubes for gas separation. Mater Res 20:1454–1460
Javanmardi MS, Ameri E (2020) Pervaporation characteristics of PDMS/PMHS nanocomposite membranes inclusive multi-walled carbon nanotubes for improvement of acetic acid-methanol esterification reaction. Polym Bull 77:2591–2609
Ouyang M, Yuan C, Muisener R, Boulares A, Koberstein J (2000) Conversion of some siloxane polymers to silicon oxide by UV/ozone photochemical processes. Chem Mater 12:1591–1596
Wu H, Xia H, Zhang X, Zhang H, Liu H, Sun J (2020) Polydimethylsiloxane/multi-walled carbon nanotube nanocomposite film prepared by ultrasonic-assisted forced impregnation with a superior photoacoustic conversion efficiency of 9.98 × 10−4. J Nanophoton 14:046003
Shen X-J, Liu Y, Xiao H-M, Feng Q-P, Yu Z-Z, Fu S-Y (2012) The reinforcing effect of graphene nanosheets on the cryogenic mechanical properties of epoxy resins. Compos Sci Technol 72:1581–1587
Ryu SY, Kim K-S, Kim J, Kim S (2012) Degradation of optical properties of a film-type single-wall carbon nanotubes saturable absorber (SWNT-SA) with an Er-doped all-fiber laser. Opt Express 20:12966–12974
Hussain S, Shah KA, Islam S (2013) Investigation of effects produced by chemical functionalization in single-walled and multi-walled carbon nanotubes using raman spectroscopy. Mater Sci Pol 31:276–280
Jhon YI, Lee JH (2021) Saturable absorption dynamics of highly stacked 2d materials for ultrafast pulsed laser production. Appl Sci 11:2690
Hernandez-Romano I, Davila-Rodriguez J, Mandridis D, Sanchez-Mondragon JJ, May-Arrioja DA, Delfyett PJ (2011) Hybrid mode locked fiber laser using a PDMS/SWCNT composite operating at 4 GHz. J Lightwave Technol 29:3237–3242
This research was supported by the Global Center for Open Research with Enterprise (G-CORE) project funded by the Korean Advanced Institute of Science and Technology (KAIST) and K LAB Co., Ltd.
This work was supported by KAIST and K LAB Co., Ltd. (N11170176, N11180013, N11180209, N11190003).
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 34141, Daejeon, Republic of Korea
Daewon Kang, Kyung-Soo Kim & Soohyun Kim
K LAB Co., Ltd., 34014, Daejeon, Republic of Korea
Sourav Sarkar
Daewon Kang
Kyung-Soo Kim
Soohyun Kim
DK performed the experiment and prepared the manuscript. SS modified the manuscript and conducted the chemical experiment. SK guided and supervised the whole project. All authors read and approved the final manuscript.
Correspondence to Kyung-Soo Kim or Soohyun Kim.
Figure S1. Change of (002) peak in XRD patterns according to ultrasonication time. Figure S2. Normalized saturable absorption of the SWCNT-PDMS SA.
Kang, D., Sarkar, S., Kim, KS. et al. Highly Damage-Resistant Thin Film Saturable Absorber Based on Mechanically Functionalized SWCNTs. Nanoscale Res Lett 17, 11 (2022). https://doi.org/10.1186/s11671-021-03648-2
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Nano-Micro Letters
January 2020 , 12:23 | Cite as
Growth of Carbon Nanocoils by Porous α-Fe2O3/SnO2 Catalyst and Its Buckypaper for High Efficient Adsorption
Yongpeng Zhao
Jianzhen Wang
Hui Huang
Tianze Cong
Shuaitao Yang
Huan Chen
Jiaqi Qin
Muhammad Usman
Zeng Fan
Lujun Pan
High-purity (~ 99%) carbon nanocoils (CNCs) without the amorphous carbon layer were synthesized by using porous α-Fe2O3/SnO2 catalyst.
The highest yield of the CNCs can reach ~ 9098% after a 6 h growth, which is much higher than those mentioned in previous reports.
A CNC Buckypaper was successfully prepared and utilized as an efficient adsorbent for the removal of methylene blue dye with the adsorption efficiency of 90.9%.
High-purity (99%) carbon nanocoils (CNCs) have been synthesized by using porous α-Fe2O3/SnO2 catalyst. The yield of CNCs reaches 9,098% after a 6 h growth. This value is much higher than the previously reported data, indicating that this method is promising to synthesize high-purity CNCs on a large scale. It is considered that an appropriate proportion of Fe and Sn, proper particle size distribution, and a loose-porous aggregate structure of the catalyst are the key points to the high-purity growth of CNCs. Benefiting from the high-purity preparation, a CNC Buckypaper was successfully prepared and the electrical, mechanical, and electrochemical properties were investigated comprehensively. Furthermore, as one of the practical applications, the CNC Buckypaper was successfully utilized as an efficient adsorbent for the removal of methylene blue dye from wastewater with an adsorption efficiency of 90.9%. This study provides a facile and economical route for preparing high-purity CNCs, which is suitable for large-quantity production. Furthermore, the fabrication of macroscopic CNC Buckypaper provides promising alternative of adsorbent or other practical applications.
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Carbon nanocoils Porous α-Fe2O3/SnO2 Catalyst Buckypaper Methylene blue adsorption
The online version of this article ( https://doi.org/10.1007/s40820-019-0365-y) contains supplementary material, which is available to authorized users.
Carbon nanocoils (CNCs), one of the distinctive types of carbon nanomaterials, have attracted wide interests due to their unique helical morphology and attractive properties. Owing to their inherent properties, CNCs hold many potential applications in a wide range of technologies, such as micro-mechanical units [1, 2], strain sensors [3, 4], electromagnetic wave absorbers [5, 6, 7, 8, 9, 10], electromagnetic wave shielding [11], field-emission displays [12, 13], nanoactuators [14, 15], supercapacitors [16, 17, 18, 19, 20], anodes for lithium ion batteries [21], and nanocomposite photocatalyst [22]. To achieve these applications, large-scale, low-cost, and high-purity production methods are essential.
Catalytic chemical vapor deposition (CVD) method is widely used to synthesize CNCs because of its controllable reaction process, economical cost, and convenient for industrial large-scale production. In this method, selection of appropriate catalysts is crucial for synthesis of CNCs. Therefore, diversified types of catalysts, including Fe [23, 24], Co [25], Ni [26, 27], Cu [28], and multi-component alloys catalysts such as Fe/Sn [29, 30, 31], Fe/Sn/In [32, 33], K/Au [34], K/Ag [35], BaSrTiO3/Sn [36], Na/K [37], Ni/P [38], and TiC [39] have been investigated for growth of CNCs. Although some improvements were made in raising the purity and yield of CNCs using different systems of catalysts, the low CNC purity is still a challenging issue. The main problem is that the high-purity CNCs are mainly present on the surface of carbon deposits, and there is always an amorphous carbon layer with a thickness ranging from several to tens of microns between the CNC layer and substrate [40, 41, 42]. This amorphous carbon layer mixed in the products seriously reduces the purity of CNCs and introduces additional problems of purification. The main reason for this problem is considered to be that the proportion of catalyst particles suitable for the growth of CNCs is not high in the whole input catalysts. In addition, the density and morphology of the initial state catalysts on the substrate are also the key points for the growth of CNCs. In order to overcome this problem, some valuable work has been performed, Hirahara et al. successfully improved the growth efficiency of CNCs by introducing an extra SnO2 buffer layer between the catalyst layer and substrate, the thickness of by-product carbon layer was reduced by 50%, and the growth rate was improved 200% compared with the substrate without coating SnO2 [41]. Takehiro et al. reduced the thickness of by-product carbon layer to 1/3 by designing a patterned catalyst thin film based on the principle of suppressing catalyst collision [42]. However, the use of lithography or magnetron sputtering technology does not make it possible for large-scale industrial production of CNCs. In any case, a facile and low-cost approach to achieve high-purity CNCs is a crucial but unsettled issue. On the other hand, the production of macroscopical freestanding Buckypapers by using carbon nanomaterials, such as carbon nanotube [43], graphene [44, 45], and carbon nanofiber [46] as building blocks, becomes an important step toward their potential applications. Therefore, the successful preparation of CNC Buckypaper is a marker for CNCs to be synthesized in high purity with large quantity.
The porous α-Fe2O3/SnO2 catalyst shows excellent ability to synthesize CNCs with high efficiency, and it can be easily prepared by a one-pot solvothermal method with low-cost precursor. By using this catalyst in a CVD process, high-purity CNCs were synthesized, without the amorphous carbon layer and the yield of 9098% was achieved after a 6 h growth. Based on the experimental results, the growth mechanism of synthesizing high-purity CNC was investigated. Benefiting from the high-purity and efficient preparation, a CNC Buckypaper was prepared for the first time and the electrical, mechanical, and electrochemical properties were investigated. Finally, as one of the practical applications, the CNC Buckypaper was successfully utilized as an efficient adsorbent for the removal of methylene blue dye.
2 Experimental Methods
2.1 Preparation of Porous α-Fe2O3/SnO2 Catalyst
In a typical experiment, 0.05 mmol soluble Fe3+ salt was dissolved in N, N-dimethylformamide (DMF); then, a certain amount of soluble Sn4+ salt with a molar ratio of Fe3+ to Sn4+ from 1:0 to 3:1 were added in the solution correspondingly. After ultrasonication for 30 min, the mixture was transferred into a 100-mL Teflon-lined stainless autoclave and heated at 180 °C for 30 h. After reaction, the autoclave was cooled to room temperature naturally. The generated catalyst powder was collected by vacuum filtration using the cellulose membrane with pore size of 0.22 μm, washed with deionized (DI) water and absolute ethanol for three times, and finally dried at 60 °C for 3 h.
2.2 Synthesis of High-Purity CNCs
The catalyst powder (20 mg) was dispersed into 20 mL absolute ethanol. After ultrasonication for 30 min, 50 μL catalyst dispersions were spin-coated on a Si substrate (size: 15 × 15 mm2) with a rotation speed of 2000 rpm for 30 s and dried at 40 °C for 10 min. By repeating the spin-coating process, the catalyst films with different densities were obtained. CNCs were produced on these substrates using an atmospheric pressure CVD system at 710 °C for 30 min by introducing a mixture of 235 sccm Ar and 25 sccm C2H2 gases. During heating and cooling processes, the CVD system was flushed with 250 sccm Ar and the schematic of CVD apparatus with substrate position is shown in Fig. S1. The purity of CNCs is given by Eq. 1:
$${\text{Purity}}_{\text{CNC}} = \frac{{N_{\text{total}} - N_{\text{CNF}} }}{{N_{\text{total}} }} \times 100\%$$
where Ntotal is the number of all CNCs and carbon nanofibers (CNFs), and NCNF is the number of CNFs on the substrate. The number of the CNCs and CNFs was quantified by observing the SEM images of the top-view and cross-sectional SEM images. Furthermore, CNFs with spring-like, twist-like, or braided-like structure were defined as CNCs.
The yield of CNCs is calculated by Eq. 2:
$${\text{Yield}}_{\text{CNC}} = \frac{{M_{\text{total}} - M_{\text{Catalyst}} }}{{M_{\text{Catalyst}} }} \times 100\%$$
where Mtotal is the total mass of CVD product, and MCatalyst is the mass of the catalyst.
2.3 Fabrication of CNC Buckypaper
The as-grown CNCs (200 mg) were removed from the substrates and dispersed in 100 mL nitric acid (68 wt%) at 60 °C for 2 h. This was followed by washing the suspension several times with DI water. After that, 50 mg acid-treated CNCs were dispersed in DI water (100 mL) and treated by ultrasonication in a bath sonicator for 30 min. Then, the CNC dispersions were poured onto a cellulose membrane with pore size of 0.22 μm and filtrated by a vacuum filtration setup. After filtration, the filter paper was dried in an oven at 60 °C for 24 h, and then, a freestanding CNC Buckypaper was peeled off from the filter membrane. The schematic of fabrication process is shown in Fig. S2.
2.4 Characterization
The morphologies of products were characterized using a field-emission scanning electron microscope (FE-SEM, NOVA NanoSEM 450) and a transmission electron microscope (TEM, JEOL JEM-2100). Energy-dispersive X-ray spectroscopy (EDX), high-resolution transmission electron microscopy (HRTEM), and element mapping of the samples were also carried out. X-ray photoelectron spectroscopy (XPS, VG ESCALAB 250Xi), X-ray diffraction (XRD, PANalytical BV Empyrean), Raman spectroscopy (Renishaw in via plus, 532.8 nm laser excitation) were used to characterize the chemical compositions and structures of the samples. The Brunauer–Emmett–Teller (BET) surface area measurement was recorded at 77 K (QUADRASORB SI-KR/MP, Quantachrome, USA). The mechanical property of the CNC Buckypaper characterized by a tensile machine Yl-S370, and the electrical property was monitored using an Agilent Technologies B2902A. The electrochemical measurements of the CNC paper were carried out using a CHI660E electrochemical workstation. Adsorption characteristics of methylene blue on CNC Buckypaper and CNC powder were measured by using a UV–Vis spectrophotometer (PerkinElmer, Lambda 750 s).
3 Results and Discussion
3.1 Growth of High-Purity CNCs
3.1.1 Effects of Molar Ratios of Fe and Sn
In order to optimize the composition of Sn in catalyst, we prepared five kinds of nanoparticle catalysts with various molar ratios of Fe and Sn. Figure 1a–j shows a series of top-view and cross-sectional SEM images of carbon deposits on Si substrates using Fe/Sn catalysts with different molar ratios of 1:0, 60:1, 30:1, 10:1, and 3:1, respectively. It is found that with the change of Sn compositions in catalysts, the morphologies of carbon deposits are significantly different. As shown in Fig. 1a, b, the deposits are carbon nanoparticles when the catalyst does not contain Sn and no CNCs or carbon nanotubes are synthesized. With the increase in Sn content, CNCs with different morphologies are successfully synthesized as shown in Fig. 1c–h. Under the Fe/Sn molar ratio of 60:1 (as shown in Fig. 1c), the carbon deposits are spring- and twist-like CNCs with an average line diameter of approximately 160 nm. However, it cannot be ignored that the purity of CNCs is only about 50% and the by-product was identified clearly from Fig. 1d. CNCs with larger average line diameter and average coil diameter are successfully synthesized under the Fe/Sn molar ratios of 30:1 and 10:1 as shown in Fig. 1e and g, respectively. Nevertheless, a dense by-product layer between the base of the CNCs and substrate is observed in Fig. 1f. The enlarged image of the area indicated by the box in Fig. 1f shows the morphology of by-product layer, which is mainly composed of carbon-containing catalytic metal particles [42].
Top-view and cross-sectional SEM images of the carbon deposits prepared by the catalysts with different Fe/Sn molar ratios of a, b 1:0, c, d 60:1, e, f 30:1, g, h 10:1, and i, j 3:1. k Raman spectra and l the respective ID/IG and FWHW values for the carbon deposits; m the thickness of carbon layer and purity of CNCs synthesized at different Fe/Sn molar ratios
It is gratifying that under the Fe/Sn molar ratio of 10:1, as shown in Figs. 1h and S3, although some thin and irregular carbon nanowires are observed on the surface of substrate, the by-product layer has been eliminated completely and the CNCs with nearly 99% purity are obtained (Originated from ~ 211 CNCs and CNFs estimated by the top-view SEM images. Among them, there are 1 CNFs without spiral morphology, as shown in Fig. S3a. We also give the purity based on the section cross-sectional SEM image. As shown in Fig. S3b, a total number of 236 CNCs and 6 CNFs were identified). This purity is much higher than any of the reported values, suggesting that the catalyst having Fe/Sn molar ratio of 10:1 has high catalytic activity. In other words, the proportion of the "true" catalyst suitable for the growth of CNCs is greatly increased under this condition, and high-purity CNCs can be synthesized in large-scale by this kind of catalyst. When the Fe/Sn molar ratio of catalyst reaches 3:1 (Fig. 1i, j), the product becomes irregular and short CNFs. These results confirm that the content of Sn has important effects on the performance of catalyst, not only on the purity of CNCs, but also on the morphology of products. The carbon deposits prepared by catalysts with different molar ratios of Fe and Sn were studied by Raman spectroscopy at an excitation laser wavelength of 532 nm, as shown in Fig. 1k. There are two main peaks in the spectra: One is around 1322 cm−1, known as the D-band, which is originated from structural defects in carbon materials; the other one is around 1593 cm−1 named as G-band originated from graphite structure. The area ratio of the D-band and G-band is defined as ID/IG which is used to evaluate the degree of graphitization. As shown in Fig. 1l, with the increase in Sn content in the catalyst, the ID/IG ratio of the corresponding carbon deposit increases from 1.03 to 1.90, implying the increase in the amorphization of the carbon deposits. The full width at half maximum (FWHM) of the D-band also increases with the increase in Sn content, indicating that the unsaturated carbon atoms are more abundant for the carbon deposits prepared by catalysts with higher Sn/Fe ratio. The thickness of carbon layer and the purity of the CNCs prepared by catalysts with different molar ratios of Fe and Sn are presented in Fig. 1m. It is found that the purity of CNCs increases first and then decreases with the increase in Sn content in the catalyst, indicating that the appropriate ratio of Fe and Sn is needed for the high-efficiency growth of CNCs.
Figure 2a is the TEM image of a single spring-like CNC with line diameter of 220 nm, coil diameter of 430 nm, and pitch of 500 nm, and its HRTEM image is shown in Fig. 2b. It is found that the lattice is partially ordered, indicating that many graphite grains (sp2 structured) are embedded in an amorphous network (sp3 structured), and the circles show that the sp2 grains have an average size of approximately 5 nm. The HRTEM image certifies that the CNCs synthesized under the Fe/Sn molar ratios of 10:1 have a polycrystalline-amorphous structure [47]. Figure 2c, d is the representative TEM and HRTEM images of a single CNF (from the deposit prepared by the catalyst with Fe/Sn molar ratio of 3:1) with a line diameter of approximately 120 nm. Unlike the structure of CNC, Fig. 2d shows that the lattice of CNF is completely disordered and the structure is amorphous.
TEM and HRTEM images of a single CNC (a, b) and a single CNF (c, d)
3.1.2 Effects of Catalyst Densities
Our previous studies have shown that optimizing the film thickness or density of the catalyst significantly affects the morphology and purity of the synthesized carbon products [48, 49]. However, these are achieved by spin coating the catalyst precursor solution containing Fe and Sn or by adjusting the thicknesses of the Fe and Sn thin films in the magnetron sputtering process. Besides, the aggregation state of catalyst particles is also an important factor affecting the growth of CNCs. Therefore, we focus on the effect of changing the aggregation density of catalyst particles on the growth of CNCs. Figure S4 is a series of SEM images of catalyst aggregation prepared with different spin-coating times, and the samples are labeled as S1, S3, S5, S10, S15, and S30 corresponding to the coating times of 1, 3, 5, 10, 15, and 30, respectively. As shown in Fig. S4, the area density of the catalysts show a substantial increase from 7.1 × 108 to 1.91 × 1010 cm−2.
Figure 3 shows the cross-sectional SEM images of CNCs synthesized using catalyst films prepared with different spin-coating times. It is observed that the CNCs synthesized by the spin-coated catalysts with different film thicknesses of S1 to S30 are basically the same in morphology and line diameter. Meanwhile, the growth density of CNCs increases with spin-coating times of catalyst. This may be due to the fact that the catalysts prepared are in the form of aggregates rather than monodisperse ones. It is observed from Fig. 3 that the carbon deposits have a bi-layer structure, i.e., a short fibrous carbon layer (inner part, confirmed by the enlarged images in Fig. 3b, c) and CNC layer (upper part). It is gratifying to find that the dense amorphous carbon layer disappears in all the samples of S1 to S30, which is quite different form the tri-layered structure (shown in Fig. 1f). The low-magnification SEM images of S5, S10, and S30 are shown in Fig. S5. These images show that high-purity CNCs can be synthesized efficiently under different catalyst aggregations, and no by-product carbon layer is produced. The low-magnification cross-sectional SEM image of the CNCs synthesized with coating times of fifteen is shown in Fig. S6a. It is observed that the whole product consists of CNCs, the height of the dense CNC layer reaches 80 μm, and many CNCs are higher than 100 μm. The enlarged images of Fig. S5a at different positions are shown in Fig. S6b–d. In each image, the uniform production of high-purity CNCs is well identified. In addition, as shown in Fig. 3g, the density of the CNCs increases from 0.07 to 1.35 μm−2 with increase in the density of the catalyst dispersions. Meanwhile, the thickness of short fibrous layer shows a similar increase trend. Furthermore, the intensity ratio of D to G peaks is shown in Fig. 3h. As the density of the catalyst increases, a very slight change of ID/IG is observed. Therefore, we can confirm that the increase in catalyst density will not significantly affect the level of defects and disorder in CNCs. Hence, we believe that this facile strategy of preparing Fe/Sn catalyst particles to control the growth density of CNCs provides opportunities to boost their practical applications.
Cross-sectional SEM images of CNCs synthesized using Fe/Sn catalyst films with spin-coating times of a one, b three, c five, d ten, e fifteen, and f thirty times. g Effects of varying Fe/Sn catalyst density on the thickness of short fibrous carbon layers and density of CNCs. h Raman spectra of CNCs synthesized at different Fe/Sn catalyst films
3.1.3 Yield of High-Purity CNCs
Firstly, the dependence of yield and thickness of the CNCs on growth time was investigated carefully. As shown in Fig. 4a, with the increase in reaction time, the yield of CNCs increases apparently. It is noteworthy that the yield of the CNCs reaches 9,098% after a 6 h growth, which is much higher than those reported in the literature so far [23, 24, 29, 37]. This result suggests that the as-prepared catalyst has an excellent catalytic activity. Next, we measured the thicknesses of CNC layers at different growth times. Figure 4b–f shows cross-sectional SEM images of the CNCs grown for 10, 30, 60, 180, and 360 min, respectively. The relationship between the thicknesses of the CNC layers with growth time is plotted in Fig. 4a. It is found that the height of CNC layer continuously increased with growth time, and the maximum thickness of the CNC layer reaches 306 μm after the reaction for 6 h. This is the highest value compared with those reported recently [34, 35, 36, 40, 41, 42]. It is noted that both curves of the yield and thickness versus time are well matched, suggesting that the carbon deposits are almost CNCs. Furthermore, both of the curves rise with growth time without saturation, indicating that the catalyst remains high efficiency even after 6 h reaction.
Plots of yield and thickness of CNCs versus growth time; cross-sectional SEM images of CNCs grown for b 10, c 30, d 60, e 180, and f 360 min. Optical photographs of the substrates g before and h after the 'scale-up' CVD reaction. The scale bar for b–f is 300 μm. i, j Top and back sides SEM images of the carbon deposits prepared with different substrates of 1, 3, 5, 7, 10, and 12. The scale bar for i and j is 5 μm
Based on the results obtained, we performed a 'scale-up' experiment using 20 mg catalyst supported by 12 pieces of alumina substrates (size: 28 × 22 mm2, dip coating the catalyst on both sides of the substrate, labeled as 1 to 12, respectively.) in a quartz tube with inner diameter of 30 mm, as shown in Fig. 4g. After 1 h reaction, 729 mg carbon deposits were produced (as shown in Fig. 4h). The top and back sides of six substrates, labeled as 1, 3, 5, 7, 10, and 12, were examined by SEM carefully. Figure 4i, j shows a series of top and back sides SEM images of carbon deposits on substrates, and the results show that the CNCs with high purity are successfully synthesized in each position. This result suggests that nearly 150 cm2 area of high-purity CNCs can be obtained in a quartz tube with inner diameter of 30 mm. Since this process is simply operable and easily scalable, it is expected to be a promising method for large-scale commercial production of CNCs.
As listed in Table 1, we summarize various catalysts for the growth of CNCs reported in the literature. Among these reports, several catalysts achieved high-purity growth of CNCs, but their preparation processes are either complex/inefficient or use of chemical reagents containing noble metals, which are not suitable for mass synthesis. In addition, the high-purity CNCs/helical carbon nanotubes reported in Refs. [23] and [28] are short braided and do not have the morphologies of spring, which limits their applications in some fields. Furthermore, the yield of our CNCs in this work has reached a new record over the reported data. Therefore, it is clear that the as-prepared α-Fe2O3/SnO2 catalyst exhibits excellent performance with the characteristics of high catalytic efficiency, low-cost, and facile preparation.
Comparison of various catalysts assisted growth of CNCs reported in the literature
Puritya
Economyb
Thickness of carbon layer (μm)
Refs.
Ag/K
Thermal evaporation
Several microns
Au/K
BaSrTiO3/Sn
Mechanical mixing
Na/K
Solution method
Sol–Gel method
Fe/In/Sn
Fe/Sn
Tens of microns
Several of microns
Solvothermal method
aExcellent: The products are basically carbon nanocoils, and there is no by-product carbon layer; Good: The products are composed of carbon nanocoils and carbon layer
BGood: The equipment used is common, and the chemical reagents used are low cost. Fair: use of expensive equipment or chemical reagents containing noble metals
3.2 Growth Mechanism of High-Purity CNCs
3.2.1 Analyses of the Catalyst
In order to well understand the growth mechanism of high-purity CNCs from as-prepared catalyst, it is necessary to make clear the structure and composition of the catalyst particles. The catalysts prepared under the Fe/Sn molar ratio of 10:1 were analyzed in details. The SEM image of the catalyst film formed on Si substrate is shown in Fig. 5a. The catalysts are in the form of loose-porous nanoparticle aggregates and the nanoparticle with sizes distributed from 100 to 400 nm. It is observed from TEM image shown in Fig. 5b that the particle aggregates have an average size of approximately 200 nm, which consist of a large number of small and homogenous particles. HRTEM image of the designated area is shown in Fig. 5c. The lattice spacings of 0.176 and 0.270 nm correspond to the (211) plane of SnO2 (JCPDF No. 41-1445) and the (104) plane of α-Fe2O3 (JCPDF No. 33-0664), respectively. Furthermore, the mesoporous are also observed in the TEM and HRTEM images of the composite particles. Figure 5d shows the EDX spectrum of the catalyst particles. It is observed that the main components of the prepared catalysts are Fe, Sn, and O, with the molar ratio of Fe and Sn is 9.88:1 that is almost the same as the initial input molar ratio of Fe and Sn. The additional peak of silicon in the spectrum is derived from the supported Si substrate. XRD spectrum obtained from the catalysts is shown in Fig. 5e. All peaks in the spectrum can be well indexed to hematite (JCPDF No. 33-0664), indicating the formation of α-Fe2O3. No peak in the spectrum comes from SnO2 (JCPDF No. 41-1445), which is resulted from the small ratio of Sn in the catalysts. (XRD patterns and SEM images of the catalysts with different Fe/Sn molar ratios are given by Fig. S7.) The further evidence for the existence of Sn is the results of XPS shown in Fig. 5f–h. Figure 5f is the spectrum of Fe 2p, in which two peaks at 710.6 and 724.5 eV correspond to Fe 2p3/2 and Fe 2p1/2, respectively [50]. Figure 5g shows the spectrum of Sn 3d. There are two peaks at 486.2 and 494.6 eV corresponding to Sn 3d5/2 and Sn 3d3/2, respectively, which are originated from + 4 oxidation states of SnO2. Furthermore, the binding energy of 716.1 eV corresponding to Sn 3p3/2 is observed in Fig. 5f, which also supports the presence of SnO2 in the catalysts [51, 52, 53]. The O 1s spectrum is shown in Fig. 5h, with two peaks at 529.5 and 530.7 eV. The peak at 529.5 eV is derived from the defects and chemisorbed oxygen on the surface of Fe2O3, and the peak at 530.7 eV is attributed to the lattice oxygen in the α-Fe2O3/SnO2 composite [52]. A typical N2 adsorption–desorption isotherm of the catalysts is shown in Fig. 5i. The typical type IV isotherm in the relative pressure (P/P0) range of 0.45–0.90 indicates a well-developed meso-porosity in the catalyst nanoparticles, which is well in line with the SEM and HRTEM results. Furthermore, the BET surface area of the catalyst is measured to be 142.8 m2 g−1. Such a high value is due to the small particle size and porous structure of the catalyst. These porous aggregates have large contact area between supplied acetylene gas and particles, and thus improve the efficiency of CNC growth.
Structural and component analysis of the catalysts with Fe/Sn molar ratio of 10:1. a SEM, b TEM, and c HRTEM images, d EDX, and e XRD spectra, f Fe 2p and Sn 3p, g Sn 3d, and h O 1s XPS spectra, i N2 adsorption/desorption isotherm for the catalyst particles
Figure 6a shows a typical TEM image of the catalyst at the tip of an as-grown CNC synthesized by the α-Fe2O3/SnO2 catalyst under the molar ratio of 10:1. It is observed that the catalyst appears to have an irregular polyhedral shape and consists of two kinds of phases of with and without carbon layer. Figure 6b shows HRTEM image of the box area in the catalyst particle, which displays three typical lattice interlayer distances of 0.251, 0.298, and 0.34 nm and could be assigned to the (110) and (220) crystal planes of α-Fe2O3 and (110) planes of SnO2, respectively. Considering that Sn has a non-wetting interaction with graphite compared with that of the Fe [31], it is reasonable to believe that SnO2 attached to the surface of α-Fe2O3 most probably decreases the catalytic activity of this region. In order to solid our viewpoint, pure SnO2 was used as a catalyst for CVD reactions under the same conditions (710°, 235 sccm Ar, 25 sccm C2H2, 300 s), and the TEM and HRTEM results are given by Fig. S8. Only a small amount of amorphous carbon (Fig. S8b) were deposited on the surface of SnO2, suggesting that SnO2's ability to decompose C2H2 gas and deposit carbon is insufficient. Furthermore, the elemental mapping of catalyst particles at the tips of two CNCs synthesized by the catalysts with Fe/Sn molar ratios of 10:1 and 3:1 is shown in Fig. 6c, d, respectively. It is observed from Fig. 6c that the distribution area of Sn is obviously smaller than that of Fe when the molar ratio of Fe and Sn is 10:1.
TEM image of the tip of an as-grown CNC synthesized by the Fe/Sn catalyst with molar ratio of 10:1; b HRTEM image of the part indicated by a dashed box in a; c, d The elemental mapping of tip particles with Fe/Sn molar ratios of 10:1 and 3:1
However, when the molar ratio is 3:1 (shown in Fig. 6d), the distribution area of Sn is basically equal to that of Fe and the grown fiber is no longer helical, but a curved and short CNF. Therefore, under our experimental conditions, the role played by SnO2 is summarized as follows: (I) The presence of SnO2 reduces the local catalytic activity of the α-Fe2O3 and prevents the catalyst from covered by the carbon. (II) The non-uniform distribution of SnO2 leads to the heterogeneous deactivation of the Fe2O3 catalyst, which leads to the anisotropy of the catalyst and promotes the helical nanocarbon growth.
3.2.2 Growth Mechanism of CNCs
Although several CNC growth mechanisms for Fe/Sn-based catalytic systems have been proposed, the origins and function of by-products have not been well understood yet. It is found that with increase in the spin-coating times (as mentioned in Sect. 3.2), the thickness of short fibrous layer gradually increases to a steady state. To investigate in details the origination of the short fibrous carbon layer, the product at the beginning of the CVD process was examined. Figure 7 shows a series of SEM images of catalyst aggregates and deposits after feeding C2H2 (4 sccm) at 710 °C for 10, 30, 100, and 300 s.
SEM images of the catalyst aggregates and deposits on the substrates after feeding C2H2 (4 sccm) at 710 °C for a 10 s, b 30 s, c 100 s, d 300 s. e Schematic of growth pathway of high-purity CNCs
It is found that the morphology of catalyst aggregates changes with the C2H2 feeding time from 10 to 300 s. When the feeding time is increased from 10 to 30 s (Fig. 7a, b), a lot of fine particles are gradually formed on the surface of the catalyst aggregates. After feeding C2H2 for 100 s (Fig. 7c), some fibrous carbon and initial CNCs with a CNC–CNF hybrid structure have been synthesized. These results suggest that CNCs synthesized on the catalyst aggregates are likely to go through two stages: fibrous growth stage and spiral growth stage. It is accepted from mechanics point of view that the helical motion of a CNC generates a torsional moment on its base, which means that CNC itself requires a reaction force from the catalyst-carbon aggregate [54]. One reasonable explanation is that at the initial stage of CNC formation, the catalyst aggregate does not accumulate much carbon particles or fibrous carbon; therefore, it cannot provide enough solid base fixation for spiral growth. With the accumulation of carbon particles or fibrous carbon in the aggregate, the adhesion force between fiber and aggregate gradually increases. When the adhesion force can balance the torsional moment of its spiral growth, CNC begins to grow. It is also observed that the short fibrous layer is mainly formed at the root position of CNCs, which is considered to be derived from the catalyst particles not suitable for the growth of CNCs. With feeding C2H2 for 300 s, as shown in Fig. 7d, a large number of CNCs are grown from the surface of the catalyst aggregates, indicating that the catalyst particles in the form of aggregates are highly effective on the synthesis of CNCs. Thus, based on our experimental and analytic results, a growth pathway of CNCs is proposed, as shown by schematic diagrams in Fig. 7e. Herein, the classic vapor–liquid–solid model is used to explain the growth process of CNCs. The CVD growth process of CNCs is divided into three stages. At stage (i), the catalytically active phase of α-Fe2O3 particle assists the dissociation of C–H bonds and converts C2H2 into C atoms and H2, and then, these C atoms nucleate at precipitation phase and form carbon fiber, which is quite consistent with the experimental results observed in Figs. 6a, b and 7a. The presence of SnO2 reduces the local catalytic activity of the catalyst nanoparticle and prevents the catalyst covered by the carbon. Therefore, the amorphous carbon layer is greatly reduced and the catalyst efficiency is also significantly improved. It is worth noting that large specific surface area of the catalyst particles and the porous structure of the aggregates ensure their full contact with acetylene gas. Meanwhile, the porous structure of the catalyst aggregates provides necessary space for the growth of CNCs, which effectively improves the utilization of catalysts. At the next growth stage (ii), with increase in the amount of carbon deposition, a number of CNC, CNF, and CNC/CNF hybrid structures are grown from the catalyst aggregates, which are adhered or entangled with each other. It is reasonable to consider that the proper aggregation of catalyst particles is helpful for the root fixation during the growth of CNCs. Considering that the helical motion of a CNC during its growth generates a torsional moment on its base, therefore, the mutual adhesion and winding of CNC, CNF, and CNC/CNF hybrid provide the necessary rotary balancing moment for highly efficient growth of CNCs. At stage (iii), owning to the stable base fixation introduced by the adjacent CNCs and short fibrous carbon layer, as well as the non-uniform distribution of Sn on the tip catalyst particle induces the anisotropy of the catalyst, the CNC is grown with relatively uniform coil diameter and pitch.
3.3.1 Electrical, Mechanical, and Electrochemical properties of CNC Buckypaper
Due to the high-purity and large-scale preparation, a CNC Buckypaper has been successfully prepared. To the best of our knowledge, it is the first time that Buckypaper was prepared by CNCs. As shown in Fig. 8a–c, the diameter and thickness of the obtained CNC Buckypaper were about 35 mm and 80 μm, respectively. Due to the helical structure and long length of the synthesized CNCs, the CNC Buckypaper is flexible and has low density and rich porosity (bulk density: 0.075 g cm−3). To comprehensively understand the basic properties of CNC Buckypaper, the electrical, mechanical and electrochemical properties have been investigated. As shown in Fig. 8d, the conductivity and sheet resistance tested using four points probe were 5.7 S cm−1 and 47.1 Ω/□, respectively. As shown in Fig. 8e, the result of maximum strain range was 1.67%, which was little larger than that of carbon nanotube Buckypaper [55]. Meanwhile, the ultimate tensile strength reaches nearly 1 MPa. Electrochemical capacitive properties of the CNC Buckypaper were evaluated by cyclic voltammetry (CV) and galvanostatic charge/discharge (GCD) measurements. Figure S9a shows CV curves of the CNC paper at scan rates of 10, 20, 50, 100, and 200 mV s−1 with a potential window ranging from − 0.45 to 0.45 V in a 6 M KOH solution. The specific capacitances at various current densities are plotted in Fig. 8f and the highest area specific capacitance reaches 30.2 mF cm−2 (at the current density of 0.8 mA cm−2). These results suggested that CNC Buckypaper based capacitor shows a good capacitive behavior with the characteristic of double-layer capacitor.
CNC Buckypaper; b and c top-view and cross-sectional SEM images of CNC Buckypaper; d electrical, e mechanical, and f electrochemical properties of CNC Buckypaper. Insert of e galvanostatic charge/discharge measurement. g Photographs of a 10 ppm methylene blue solution (left) and the clear solution (right) obtained by soaking the CNC paper for 120 min. h UV–Vis spectra of pristine (10 ppm, black curve) and CNC paper-treated (red curve) methylene blue solution. i Photographs and j UV–Vis spectra of a 10 ppm methylene blue passing through the needle with pristine plug (left) or plug/CNC mix plug (right)
3.3.2 CNC Buckypaper as Adsorbent for Removal of Methylene Blue
Based on the above results, we believe that the CNC Buckypaper has potential applications in many fields. Considering the advantages of its low density and rich porosity, it is a reasonable choice to utilize CNC Buckypaper as an adsorbent for the removal of pollutants from waste water. Figure 8g shows photographs of a 10 ppm methylene blue (5 mL) solution before (left) and after (right) soaking the CNC paper (2.25 cm2, 10.1 mg) for 120 min. UV–Vis spectra of methylene blue dye is shown in Fig. 8h. An adsorption efficiency of 88.6% is obtained, suggesting that the CNC Buckypaper has a good adsorption performance for methylene blue. Furthermore, a continuous-flow filtering experiment was performed to remove methylene blue dye in the solution. As shown in Fig. 8i, 10 mg of CNCs were packed into the filtration system (confirmed by insert of Fig. 8j), an aqueous solution of methylene blue dye (10 ppm) was pressed to pass through the packed CNC film at 298 K. The color disappearance clearly suggests that most of the methylene blue dye is adsorbed by the CNC membrane, and UV–Vis spectra of methylene blue dye confirms that the adsorption efficiency is 90.9%. Meanwhile, the adsorption capacity of CNCs was also be evaluated by UV–Vis spectra of methylene blue after adsorption at different time. As shown in Fig. S10, the adsorption capacity of methylene blue onto CNCs is 57.3 mg g−1, which is nearly twice of that for carbon nanotubes [56]. It is reasonable that the good adsorption ability of CNC originates from their relatively large specific surface area (131.2 m2 g−1, as shown in Fig. S11) and rough surface (confirmed by insert of Fig. S10b).
CNCs with high purity of ~ 99% have been synthesized by using porous α-Fe2O3/SnO2 catalyst particles under Fe/Sn molar ratio of 10:1. Furthermore, the density of high-purity CNCs can be easily controlled by changing the density of the catalyst aggregates. The carbon deposit has little amorphous carbon layer, and the yield of the CNCs reaches 9098% in a 6 h reaction. Both the purity and yield of the CNCs are much higher than those reported in the literature. It is confirmed that the appropriate proportion of Fe and Sn, proper particle size distribution, and the loose-porous aggregates of the catalysts are the key points to the high-purity growth of the CNCs. Benefiting from the high-purity and efficient production, a CNC Buckypaper has been successfully prepared and the electrical, mechanical, and electrochemical properties were investigated comprehensively. Furthermore, the CNC Buckypaper was successfully utilized as an efficient adsorbent for the removal of methylene blue dye with an adsorption efficiency of 90.9%. We strongly believe that this work has a significant guiding importance in terms of efficient and large-quantity synthesis of high-purity CNCs at high yield. On the other hand, the fabrication of macroscopic CNC Buckypaper provides promising alternative for pollutant adsorption or other practical applications.
This work was financially supported by the National Natural Science Foundation of China (Nos. 51661145025, 51972039, and 51803018).
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X. Chen, S. Zhang, D.A. Dikin, W. Ding, R.S. Ruoff, L. Pan, Y. Nakayama, Mechanics of a carbon nanocoil. Nano Lett. 3(9), 1299–1304 (2003). https://doi.org/10.1021/nl034367o CrossRefGoogle Scholar
A. Volodin, D. Buntinx, M. Ahlskog, A. Fonseca, J.B. Nagy, C. Van Haesendonck, Coiled carbon nanotubes as self-sensing mechanical resonators. Nano Lett. 4(9), 1775–1779 (2004). https://doi.org/10.1021/nl0491576 CrossRefGoogle Scholar
C. Li, L. Pan, C. Deng, P. Wang, Y. Huang, H. Nasir, A flexible, ultra-sensitive strain sensor based on carbon nanocoil network fabricated by an electrophoretic method. Nanoscale 9(28), 9872–9878 (2017). https://doi.org/10.1039/c7nr01945a CrossRefGoogle Scholar
C. Deng, L. Pan, D. Zhang, C. Li, H. Nasir, A super stretchable and sensitive strain sensor based on a carbon nanocoil network fabricated by a simple peeling-off approach. Nanoscale 9(42), 16404–16411 (2017). https://doi.org/10.1039/c7nr05486f CrossRefGoogle Scholar
G. Wang, Z. Gao, S. Tang, C. Chen, F. Duan et al., Microwave absorption properties of carbon nanocoils coated with highly controlled magnetic materials by atomic layer deposition. ACS Nano 6(12), 11009–11017 (2012). https://doi.org/10.1021/nn304630h CrossRefGoogle Scholar
D.-L. Zhao, Z.-M. Shen, Preparation and microwave absorption properties of carbon nanocoils. Mater. Lett. 62(21–22), 3704–3706 (2008). https://doi.org/10.1016/j.matlet.2008.04.032 CrossRefGoogle Scholar
S. Zhao, Z. Gao, C. Chen, G. Wang, B. Zhang et al., Alternate nonmagnetic and magnetic multilayer nanofilms deposited on carbon nanocoils by atomic layer deposition to tune microwave absorption property. Carbon 98, 196–203 (2016). https://doi.org/10.1016/j.carbon.2015.10.101 CrossRefGoogle Scholar
L. Liu, P. He, K. Zhou, T. Chen, Microwave absorption properties of carbon fibers with carbon coils of different morphologies (double microcoils and single nanocoils) grown on them. J. Mater. Sci. 49(12), 4379–4386 (2014). https://doi.org/10.1007/s10853-014-8137-z CrossRefGoogle Scholar
R. Cui, L. Pan, D. Zhang, H. Nasir, Electromagnetic microwave absorption properties of carbon nanocoils/tissue. Diam. Relat. Mater. 77, 53–56 (2017). https://doi.org/10.1016/j.diamond.2017.05.014 CrossRefGoogle Scholar
W. Cao, C. Ma, S. Tan, M. Ma, P. Wan, F. Chen, Ultrathin and flexible CNTs/MXene/cellulose nanofibrils composite paper for electromagnetic interference shielding. Nano-Micro Lett. 11, 72 (2019). https://doi.org/10.1007/s40820-019-0304-y CrossRefGoogle Scholar
G.-H. Kang, S.-H. Kim, S. Park, Enhancement of shielding effectiveness for electromagnetic wave radiation using carbon nanocoil-carbon microcoil hybrid materials. Appl. Surf. Sci. 477, 264–270 (2019). https://doi.org/10.1016/j.apsusc.2017.10.007 CrossRefGoogle Scholar
L. Pan, T. Hayashida, M. Zhang, Y. Nakayama, Field emission properties of carbon tubule nanocoils. Jpn. J. Appl. Phys. 40(3B), L235 (2001). https://doi.org/10.1143/JJAP.40.L235 CrossRefGoogle Scholar
W.Y. Sung, J.G. Ok, W.J. Kim, S.M. Lee, S.C. Yeon, H.Y. Lee, Y.H. Kim, Synthesis and field emission characteristics of carbon nanocoils with a high aspect ratio supported by copper micro-tips. Nanotechnology 18(24), 245603 (2007). https://doi.org/10.1088/0957-4484/18/24/245603 CrossRefGoogle Scholar
P. Wang, L. Pan, C. Li, J. Zheng, Optically actuated carbon nanocoils. NANO 13(10), 1850112 (2018). https://doi.org/10.1142/s1793292018501126 CrossRefGoogle Scholar
H. Ma, X. Zhang, R. Cui, F. Liu, M. Wang et al., Photo-driven nanoactuators based on carbon nanocoils and vanadium dioxide bimorphs. Nanoscale 10(23), 11158–11164 (2018). https://doi.org/10.1039/c8nr03622e CrossRefGoogle Scholar
R.B. Rakhi, W. Chen, H.N. Alshareef, Conducting polymer/carbon nanocoil composite electrodes for efficient supercapacitors. J. Mater. Chem. 22(11), 5177–5183 (2012). https://doi.org/10.1039/c2jm15740c CrossRefGoogle Scholar
V. Barranco, V. Celorrio, M.J. Lazaro, J.M. Rojo, Carbon nanocoils as unusual electrode materials for supercapacitors. J. Electrochem. Soc. 159(4), A464–A469 (2012). https://doi.org/10.1149/2.083204jes CrossRefGoogle Scholar
D. Su, L. Pan, X. Fu, H. Ma, Facile synthesis of CNC–MnO2 hybrid as a supercapacitor electrode. Appl. Surf. Sci. 324, 349–354 (2015). https://doi.org/10.1016/j.apsusc.2014.10.141 CrossRefGoogle Scholar
A. Childress, K. Ferri, A.M. Rao, Enhanced supercapacitor performance with binder-free helically coiled carbon nanotube electrodes. Carbon 140, 377–384 (2018). https://doi.org/10.1016/j.carbon.2018.08.064 CrossRefGoogle Scholar
S. Shi, G. Wan, L. Wu, Z. He, K. Wang, Y. Tang, X. Xu, G. Wang, Ultrathin manganese oxide nanosheets uniformly coating on carbon nanocoils as high-performance asymmetric supercapacitor electrodes. J. Colloid. Interface Sci. 537, 142–150 (2019). https://doi.org/10.1016/j.jcis.2018.11.006 CrossRefGoogle Scholar
L. Wang, Z. Liu, Q. Guo, G. Wang, J. Yang, P. Li, X. Wang, L. Liu, Electrochemical properties of carbon nanocoils and hollow graphite fibers as anodes for rechargeable lithium ion batteries. Electrochim. Acta 199, 204–209 (2016). https://doi.org/10.1016/j.electacta.2016.03.160 CrossRefGoogle Scholar
X.-W. Guo, C.-H. Hao, C.-Y. Wang, S. Sarina, X.-N. Guo, X.-Y. Guo, Visible light-driven photocatalytic heck reaction over carbon nanocoil supported pd nanoparticles. Catal. Sci. Tech. 6(21), 7738–7743 (2016). https://doi.org/10.1039/c6cy01322h CrossRefGoogle Scholar
F. Meng, Y. Wang, Q. Wang, X. Xu, M. Jiang, X. Zhou, P. He, Z. Zhou, High-purity helical carbon nanotubes by trace-water-assisted chemical vapor deposition: large-scale synthesis and growth mechanism. Nano Res. 11(6), 3327–3339 (2018). https://doi.org/10.1007/s12274-017-1897-4 CrossRefGoogle Scholar
X. Qi, W. Zhong, Y. Deng, C. Au, Y. Du, Synthesis of helical carbon nanotubes, worm-like carbon nanotubes and nanocoils at 450°C and their magnetic properties. Carbon 48(2), 365–376 (2010). https://doi.org/10.1016/j.carbon.2009.09.038 CrossRefGoogle Scholar
X. Zhang, X. Zhang, D. Bernaerts, G. Van Tendeloo, S. Amelinckx et al., The texture of catalytically grown coil-shaped carbon nanotubules. Europhys. Lett. 27(2), 141 (1994). https://doi.org/10.1209/0295-5075/27/2/011 CrossRefGoogle Scholar
E.-X. Ding, H.-Z. Geng, W.-Y. Wang, L. Mao, Y. Wang et al., Growth of morphology-controllable carbon nanocoils from ni nanoparticle prepared by spray-coating method. Carbon 82, 604–607 (2015). https://doi.org/10.1016/j.carbon.2014.10.094 CrossRefGoogle Scholar
R. Baker, M. Barber, P. Harris, F. Feates, R. Waite, Nucleation and growth of carbon deposits from the nickel catalyzed decomposition of acetylene. J. Catal. 26(1), 51–62 (1972). https://doi.org/10.1016/0021-9517(72)90032-2 CrossRefGoogle Scholar
G. Wang, G. Ran, G. Wan, P. Yang, Z. Gao, S. Lin, C. Fu, Y. Qin, Size-selective catalytic growth of nearly 100% pure carbon nanocoils with copper nanoparticles produced by atomic layer deposition. ACS Nano 8(5), 5330–5338 (2014). https://doi.org/10.1021/nn501709h CrossRefGoogle Scholar
D. Li, L. Pan, J. Qian, D. Liu, Highly efficient synthesis of carbon nanocoils by catalyst particles prepared by a sol–gel method. Carbon 48(1), 170–175 (2010). https://doi.org/10.1016/j.carbon.2009.08.045 CrossRefGoogle Scholar
D.W. Li, L.J. Pan, J.J. Qian, H. Ma, High efficient synthesis of carbon nanocoils by catalysts produced by a fe and sn containing solution. Adv. Mater. Res. 60–61, 251–255 (2009). https://doi.org/10.4028/www.scientific.net/AMR.60-61.251 CrossRefGoogle Scholar
W. Wang, K. Yang, J. Gaillard, P.R. Bandaru, A.M. Rao, Rational synthesis of helically coiled carbon nanowires and nanotubes through the use of tin and indium catalysts. Adv. Mater. 20(1), 179–182 (2008). https://doi.org/10.1002/adma.200701143 CrossRefGoogle Scholar
M. Zhang, Y. Nakayama, L. Pan, Synthesis of carbon tubule nanocoils in high yield using iron-coated indium tin oxide as catalyst. Jpn. J. Appl. Phys. 39(12A), L1242 (2000). https://doi.org/10.1143/JJAP.39.L1242 CrossRefGoogle Scholar
K. Nishimura, L. Pan, Y. Nakayama, In situ study of Fe/ITO catalysts for carbon nanocoil growth by x-ray diffraction analysis. Jpn. J. Appl. Phys. 43(8A), 5665–5666 (2004). https://doi.org/10.1143/jjap.43.5665 CrossRefGoogle Scholar
T.Y. Tsou, C.Y. Lee, H.T. Chiu, K and au bicatalyst assisted growth of carbon nanocoils from acetylene: effect of deposition parameters on field emission properties. ACS Appl. Mater. Interfaces 4(12), 6505–6511 (2012). https://doi.org/10.1021/am3015372 CrossRefGoogle Scholar
W.-C. Liu, H.-K. Lin, Y.-L. Chen, C.-Y. Lee, H.-T. Chiu, Growth of carbon nanocoils from K and Ag cooperative bicatalyst assisted thermal decomposition of acetylene. ACS Nano 4(7), 4149–4157 (2010). https://doi.org/10.1021/nn901926r CrossRefGoogle Scholar
J. Sun, A.A. Koós, F. Dillon, K. Jurkschat, M.R. Castell, N. Grobert, Synthesis of carbon nanocoil forests on BaSrTiO3 substrates with the aid of a Sn catalyst. Carbon 60, 5–15 (2013). https://doi.org/10.1016/j.carbon.2013.03.027 CrossRefGoogle Scholar
X. Qi, W. Zhong, X. Yao, H. Zhang, Q. Ding et al., Controllable and large-scale synthesis of metal-free carbon nanofibers and carbon nanocoils over water-soluble naxky catalysts. Carbon 50(2), 646–658 (2012). https://doi.org/10.1016/j.carbon.2011.08.076 CrossRefGoogle Scholar
D. Ding, J. Wang, A. Dozier, Symmetry-related growth of carbon nanocoils from ni–p based alloy particles. J. Appl. Phys. 95(9), 5006–5009 (2004). https://doi.org/10.1063/1.1689734 CrossRefGoogle Scholar
S. Yang, X. Chen, N. Kikuchi, S. Motojima, Catalytic effects of various metal carbides and Ti compounds for the growth of carbon nanocoils (cncs). Mater. Lett. 62(10–11), 1462–1465 (2008). https://doi.org/10.1016/j.matlet.2007.08.086 CrossRefGoogle Scholar
X. Fu, L. Pan, Q. Wang, C. Liu, Y. Sun, M. Asif, J. Qin, Y. Huang, Controlled synthesis of carbon nanocoils on monolayered silica spheres. Carbon 99, 43–48 (2016). https://doi.org/10.1016/j.carbon.2015.11.028 CrossRefGoogle Scholar
K. Hirahara, Y. Nakayama, The effect of a tin oxide buffer layer for the high yield synthesis of carbon nanocoils. Carbon 56, 264–270 (2013). https://doi.org/10.1016/j.carbon.2013.01.007 CrossRefGoogle Scholar
T. Gohara, K. Takei, T. Arie, S. Akita, Reduction of carbon byproducts for high-purity carbon nanocoil growth by suppressing catalyst collision. Carbon 89, 225–231 (2015). https://doi.org/10.1016/j.carbon.2015.03.044 CrossRefGoogle Scholar
M. Endo, H. Muramatsu, T. Hayashi, Y.A. Kim, M. Terrones, M.S. Dresselhaus, Nanotechnology: 'Buckypaper' from coaxial nanotubes. Nature 433, 476 (2005). https://doi.org/10.1038/nature433476a CrossRefGoogle Scholar
D.A. Dikin, S. Stankovich, E.J. Zimney, R.D. Piner, G.H. Dommett, G. Evmenenko, R.S. Ruoff, Preparation and characterization of graphene oxide paper. Nature 448, 457–460 (2007). https://doi.org/10.1038/nature06016 CrossRefGoogle Scholar
B. Brown, B. Swain, J. Hiltwine, D.B. Brooks, Z. Zhou, Carbon nanosheet buckypaper: a graphene-carbon nanotube hybrid material for enhanced supercapacitor performance. J. Power Sour. 272, 979–986 (2014). https://doi.org/10.1016/j.jpowsour.2014.09.015 CrossRefGoogle Scholar
M.O. Memon, S. Haillot, K. Lafdi, Carbon nanofiber based buckypaper used as a thermal interface material. Carbon 49, 3820–3828 (2011). https://doi.org/10.1016/j.carbon.2011.05.015 CrossRefGoogle Scholar
Y. Sun, C. Wang, L. Pan, X. Fu, P. Yin, H. Zou, Electrical conductivity of single polycrystalline-amorphous carbon nanocoils. Carbon 98, 285–290 (2016). https://doi.org/10.1016/j.carbon.2015.11.025 CrossRefGoogle Scholar
X. Fu, L. Pan, D. Li, N. Zhou, Y. Sun, Controlled synthesis of carbon nanocoils with selective coil diameters and structures by optimizing the thickness of catalyst film. Carbon 93, 361–369 (2015). https://doi.org/10.1016/j.carbon.2015.05.059 CrossRefGoogle Scholar
D. Li, L. Pan, Growth of carbon nanocoils using Fe–Sn–O catalyst film prepared by a spin-coating method. J. Mater. Res. 26(16), 2024–2032 (2011). https://doi.org/10.1557/jmr.2011.227 CrossRefGoogle Scholar
H. Shan, C. Liu, L. Liu, J. Zhang, H. Li et al., Excellent toluene sensing properties of SnO2-Fe2O3 interconnected nanotubes. ACS Appl. Mater. Interfaces. 5(13), 6376–6380 (2013). https://doi.org/10.1021/am4015082 CrossRefGoogle Scholar
N. Jayababu, M. Poloju, M.V. Ramana Reddy, Facile synthesis of SnO2-Fe2O3 core-shell nanostructures and their 2-methoxyethanol gas sensing characteristics. J. Alloy. Compd. 780, 523–533 (2019). https://doi.org/10.1016/j.jallcom.2018.11.413 CrossRefGoogle Scholar
L. Liu, S. Li, L. Wang, C. Guo, Q. Dong, W. Li, Enhancement ethanol sensing properties of Nio-SnO2 nanofibers. J. Am. Ceram. Soc. 94(3), 771–775 (2011). https://doi.org/10.1111/j.1551-2916.2010.04137.x CrossRefGoogle Scholar
Y. Ding, B. Liu, J. Zou, H. Liu, T. Xin, L. Xia, Y. Wang, A-Fe2O3-SnO2 heterostructure composites: a high stability anode for lithium-ion battery. Mater. Res. Bull. 106, 7–13 (2018). https://doi.org/10.1016/j.materresbull.2018.05.014 CrossRefGoogle Scholar
D. Li, L. Pan, Necessity of base fixation for helical growth of carbon nanocoils. J. Mater. Res. 27(2), 431–439 (2012). https://doi.org/10.1557/jmr.2011.401 CrossRefGoogle Scholar
T.H. da Cunha, S. de Oliveira, I.L. Martins, V. Geraldo, D. Miquita et al., High-yield synthesis of bundles of double-and triple-walled carbon nanotubes on aluminum flakes. Carbon 133, 53–61 (2018). https://doi.org/10.1016/j.carbon.2018.03.014 CrossRefGoogle Scholar
Y. Yao, F. Xu, M. Chen, Z. Xu, Z. Zhu, Adsorption behavior of methylene blue on carbon nanotubes. Bioresource techn. 101(9), 3040–3046 (2010). https://doi.org/10.1016/j.biortech.2009.12.042 CrossRefGoogle Scholar
© The Author(s) 2020
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
1.School of PhysicsDalian University of TechnologyDalianPeople's Republic of China
2.School of MicroelectronicsDalian University of TechnologyDalianPeople's Republic of China
3.State Key Laboratory of Fine Chemicals, School of Chemical EngineeringDalian University of TechnologyDalianPeople's Republic of China
4.Department of PhysicsKhawaja Fareed University of Engineering and Information TechnologyRahim Yar KhanPakistan
Zhao, Y., Wang, J., Huang, H. et al. Nano-Micro Lett. (2020) 12: 23. https://doi.org/10.1007/s40820-019-0365-y
Received 21 October 2019
Accepted 04 December 2019
DOI https://doi.org/10.1007/s40820-019-0365-y
Publisher Name Springer Singapore | CommonCrawl |
Monoidal t-norm logic
In mathematical logic, monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;[1] it extends the logic of commutative bounded integral residuated lattices (known as Höhle's monoidal logic, Ono's FLew, or intuitionistic logic without contraction) by the axiom of prelinearity.
Motivation
In fuzzy logic, rather than regarding statements as being either true or false, we associate each statement with a numerical confidence in that statement. By convention the confidences range over the unit interval $[0,1]$, where the maximal confidence $1$ corresponds to the classical concept of true and the minimal confidence $0$ corresponds to the classical concept of false.
T-norms are binary functions on the real unit interval [0, 1], which in fuzzy logic are often used to represent a conjunction connective; if $a,b\in [0,1]$ are the confidences we ascribe to the statements $A$ and $B$ respectively, then one uses a t-norm $*$ to calculate the confidence $a*b$ ascribed to the compound statement ‘$A$ and $B$’. A t-norm $*$ has to satisfy the properties of
commutativity $a*b=b*a$,
associativity $(a*b)*c=a*(b*c)$,
monotonicity — if $a\leqslant b$ and $c\leqslant d$ then $a*c\leqslant b*d$,
and having 1 as identity element $1*a=a$.
Notably absent from this list is the property of idempotence $a*a=a$; the closest one gets is that $a*a\leqslant 1*a=a$. It may seem strange to be less confident in ‘$A$ and $A$’ than in just $A$, but we generally do want to allow for letting the confidence $a*b$ in a combined ‘$A$ and $B$’ be less than both the confidence $a$ in $A$ and the confidence $b$ in $B$, and then the ordering $a*b<a\leqslant b$ by monotonicity requires $a*a\leqslant a*b<a$. Another way of putting it is that the t-norm can only take into account the confidences as numbers, not the reasons that may be behind ascribing those confidences; thus it cannot treat ‘$A$ and $A$’ differently from ‘$A$ and $B$, where we are equally confident in both’.
Because the symbol $\wedge $ via its use in lattice theory is very closely associated with the idempotence property, it can be useful to switch to a different symbol for conjunction that is not necessarily idempotent. In the fuzzy logic tradition one sometimes uses $\&$ for this "strong" conjunction, but this article follows the substructural logic tradition of using $\otimes $ for the strong conjunction; thus $a*b$ is the confidence we ascribe to the statement $A\otimes B$ (still read ‘$A$ and $B$’, perhaps with ‘strong’ or ‘multiplicative’ as qualification of the ‘and’).
Having formalised conjunction $\otimes $, one wishes to continue with the other connectives. One approach to doing that is to introduce negation as an order-reversing map $[0,1]\longrightarrow [0,1]$, then defining remaining connectives using De Morgan's laws, material implication, and the like. A problem with doing so is that the resulting logics may have undesirable properties: they may be too close to classical logic, or if not conversely not support expected inference rules. An alternative that makes the consequences of different choices more predictable is to instead continue with implication $\to $ as the second connective: this is overall the most common connective in axiomatisations of logic, and it has closer ties to the deductive aspects of logic than most other connectives. A confidence counterpart $\Rightarrow $ of the implication connective may in fact be defined directly as the residuum of the t-norm.
The logical link between conjunction and implication is provided by something as fundamental as the inference rule modus ponens $A,A\to B\vdash B$: from $A$ and $A\to B$ follows $B$. In the fuzzy logic case that is more rigorously written as $A\otimes (A\to B)\vdash B$, because this makes explicit that our confidence for the premise(s) here is that in $A\otimes (A\to B)$, not those in $A$ and $A\to B$ separately. So if $a$ and $b$ are our confidences in $A$ and $B$ respectively, then $a\Rightarrow b$ is the sought confidence in $A\to B$, and $a*(a\Rightarrow b)$ is the combined confidence in $A\otimes (A\to B)$. We require that
$a*(a{\mathbin {\Rightarrow }}b)\leqslant b$
since our confidence $b$ for $B$ should not be less than our confidence $a*(a\Rightarrow b)$ in the statement $A\otimes (A\to B)$ from which $B$ logically follows. This bounds the sought confidence $a\Rightarrow b$, and one approach for turning $\Rightarrow $ into a binary operation like $*$ would be to make it as large as possible while respecting this bound:
$a{\mathbin {\Rightarrow }}b\equiv \sup \left\{x\in [0,1]\;{\big |}\;a*x\leqslant b\right\}$.
Taking $x=0$ gives $a*x=a*0\leqslant 1*0=0\leqslant b$, so the supremum is always of a nonempty bounded set and thus well-defined. For a general t-norm there remains the possibility that $f_{a}(x)=a*x$ has a jump discontinuity at $x=a{\mathbin {\Rightarrow }}b$, in which case $a*(a{\mathbin {\Rightarrow }}b)$ could come out strictly larger than $b$ even though $a{\mathbin {\Rightarrow }}b$ is defined as the least upper bound of $x$s satisfying $a*x\leqslant b$; to prevent that and have the construction work as expected, we require that the t-norm $*$ is left-continuous. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic.
More algebraically, we say that an operation $\Rightarrow $ is a residuum of a t-norm $*$ if for all $a$, $b$, and $c$ it satisfies
$a*b\leq c$ if and only if $a\leq (b{\mathbin {\Rightarrow }}c)$.
This equivalence of numerical comparisons mirrors the equivalence of entailments
$A\otimes B\vdash C$ if and only if $A\vdash B\to C$
that exists because any proof of $C$ from the premise $A\otimes B$ can be converted into a proof of $B\to C$ from the premise $A$ by doing an extra implication introduction step, and conversely any proof of $B\to C$ from the premise $A$ can be converted into a proof of $C$ from the premise $A\otimes B$ by doing an extra implication elimination step. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold.
Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation $\neg x=(x{\mathbin {\Rightarrow }}0).$ In this way, the left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives (see the section Standard semantics below) determine the truth values of complex propositional formulae in [0, 1]. Formulae that always evaluate to 1 are then called tautologies with respect to the given left-continuous t-norm $*,$ or $*{\mbox{-}}$tautologies. The set of all $*{\mbox{-}}$tautologies is called the logic of the t-norm $*,$ since these formulae represent the laws of fuzzy logic (determined by the t-norm) that hold (to degree 1) regardless of the truth degrees of atomic formulae. Some formulae are tautologies with respect to all left-continuous t-norms: they represent general laws of propositional fuzzy logic that are independent of the choice of a particular left-continuous t-norm. These formulae form the logic MTL, which can thus be characterized as the logic of left-continuous t-norms.[2]
Syntax
Language
The language of the propositional logic MTL consists of countably many propositional variables and the following primitive logical connectives:
• Implication $\rightarrow $ (binary)
• Strong conjunction $\otimes $ (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation $\otimes $ follows the tradition of substructural logics.
• Weak conjunction $\wedge $ (binary), also called lattice conjunction (as it is always realized by the lattice operation of meet in algebraic semantics). Unlike in BL and stronger fuzzy logics, weak conjunction is not definable in MTL and has to be included among the primitive connectives.
• Bottom $\bot $ (nullary — a propositional constant; $0$ or ${\overline {0}}$ are common alternative tokens and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL).
The following are the most common defined logical connectives:
• Negation $\neg $ (unary), defined as
$\neg A\equiv A\rightarrow \bot $
• Equivalence $\leftrightarrow $ (binary), defined as
$A\leftrightarrow B\equiv (A\rightarrow B)\wedge (B\rightarrow A)$
In MTL, the definition is equivalent to $(A\rightarrow B)\otimes (B\rightarrow A).$
• (Weak) disjunction $\vee $ (binary), also called lattice disjunction (as it is always realized by the lattice operation of join in algebraic semantics), defined as
$A\vee B\equiv ((A\rightarrow B)\rightarrow B)\wedge ((B\rightarrow A)\rightarrow A)$
• Top $\top $ (nullary), also called one and denoted by $1$ or ${\overline {1}}$ (as the constants top and zero of substructural logics coincide in MTL), defined as
$\top \equiv \bot \rightarrow \bot $
Well-formed formulae of MTL are defined as usual in propositional logics. In order to save parentheses, it is common to use the following order of precedence:
• Unary connectives (bind most closely)
• Binary connectives other than implication and equivalence
• Implication and equivalence (bind most loosely)
Axioms
A Hilbert-style deduction system for MTL has been introduced by Esteva and Godo (2001). Its single derivation rule is modus ponens:
from $A$ and $A\rightarrow B$ derive $B.$
The following are its axiom schemata:
${\begin{array}{ll}{\rm {(MTL1)}}\colon &(A\rightarrow B)\rightarrow ((B\rightarrow C)\rightarrow (A\rightarrow C))\\{\rm {(MTL2)}}\colon &A\otimes B\rightarrow A\\{\rm {(MTL3)}}\colon &A\otimes B\rightarrow B\otimes A\\{\rm {(MTL4a)}}\colon &A\wedge B\rightarrow A\\{\rm {(MTL4b)}}\colon &A\wedge B\rightarrow B\wedge A\\{\rm {(MTL4c)}}\colon &A\otimes (A\rightarrow B)\rightarrow A\wedge B\\{\rm {(MTL5a)}}\colon &(A\rightarrow (B\rightarrow C))\rightarrow (A\otimes B\rightarrow C)\\{\rm {(MTL5b)}}\colon &(A\otimes B\rightarrow C)\rightarrow (A\rightarrow (B\rightarrow C))\\{\rm {(MTL6)}}\colon &((A\rightarrow B)\rightarrow C)\rightarrow (((B\rightarrow A)\rightarrow C)\rightarrow C)\\{\rm {(MTL7)}}\colon &\bot \rightarrow A\end{array}}$
The traditional numbering of axioms, given in the left column, is derived from the numbering of axioms of Hájek's basic fuzzy logic BL.[3] The axioms (MTL4a)–(MTL4c) replace the axiom of divisibility (BL4) of BL. The axioms (MTL5a) and (MTL5b) express the law of residuation and the axiom (MTL6) corresponds to the condition of prelinearity. The axioms (MTL2) and (MTL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2012) and (Cintula, 2005). All the other axioms were shown to be independent (Chvalovský, 2012).
Semantics
Like in other propositional t-norm fuzzy logics, algebraic semantics is predominantly used for MTL, with three main classes of algebras with respect to which the logic is complete:
• General semantics, formed of all MTL-algebras — that is, all algebras for which the logic is sound
• Linear semantics, formed of all linear MTL-algebras — that is, all MTL-algebras whose lattice order is linear
• Standard semantics, formed of all standard MTL-algebras — that is, all MTL-algebras whose lattice reduct is the real unit interval [0, 1] with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any left-continuous t-norm
MTL-algebras
Algebras for which the logic MTL is sound are called MTL-algebras. They can be characterized as prelinear commutative bounded integral residuated lattices. In more detail, an algebraic structure $(L,\wedge ,\vee ,\ast ,\Rightarrow ,0,1)$ is an MTL-algebra if
• $(L,\wedge ,\vee ,0,1)$ is a bounded lattice with the top element 0 and bottom element 1
• $(L,\ast ,1)$ is a commutative monoid
• $\ast $ and $\Rightarrow $ form an adjoint pair, that is, $z*x\leq y$ if and only if $z\leq x\Rightarrow y,$ where $\leq $ is the lattice order of $(L,\wedge ,\vee ),$ for all x, y, and z in $L$, (the residuation condition)
• $(x\Rightarrow y)\vee (y\Rightarrow x)=1$ holds for all x and y in L (the prelinearity condition)
Important examples of MTL algebras are standard MTL-algebras on the real unit interval [0, 1]. Further examples include all Boolean algebras, all linear Heyting algebras (both with $\ast =\wedge $), all MV-algebras, all BL-algebras, etc. Since the residuation condition can equivalently be expressed by identities,[4] MTL-algebras form a variety.
Interpretation of the logic MTL in MTL-algebras
The connectives of MTL are interpreted in MTL-algebras as follows:
• Strong conjunction by the monoidal operation $\ast $
• Implication by the operation $\Rightarrow $ (which is called the residuum of $\ast $)
• Weak conjunction and weak disjunction by the lattice operations $\wedge $ and $\vee ,$ respectively (usually denoted by the same symbols as the connectives, if no confusion can arise)
• The truth constants zero (top) and one (bottom) by the constants 0 and 1
• The equivalence connective is interpreted by the operation $\Leftrightarrow $ defined as
$x\Leftrightarrow y\equiv (x\Rightarrow y)\wedge (y\Rightarrow x)$
Due to the prelinearity condition, this definition is equivalent to one that uses $\ast $ instead of $\wedge ,$ thus
$x\Leftrightarrow y\equiv (x\Rightarrow y)\ast (y\Rightarrow x)$
• Negation is interpreted by the definable operation $-x\equiv x\Rightarrow 0$
With this interpretation of connectives, any evaluation ev of propositional variables in L uniquely extends to an evaluation e of all well-formed formulae of MTL, by the following inductive definition (which generalizes Tarski's truth conditions), for any formulae A, B, and any propositional variable p:
${\begin{array}{rcl}e(p)&=&e_{\mathrm {v} }(p)\\e(\bot )&=&0\\e(\top )&=&1\\e(A\otimes B)&=&e(A)\ast e(B)\\e(A\rightarrow B)&=&e(A)\Rightarrow e(B)\\e(A\wedge B)&=&e(A)\wedge e(B)\\e(A\vee B)&=&e(A)\vee e(B)\\e(A\leftrightarrow B)&=&e(A)\Leftrightarrow e(B)\\e(\neg A)&=&e(A)\Rightarrow 0\end{array}}$
Informally, the truth value 1 represents full truth and the truth value 0 represents full falsity; intermediate truth values represent intermediate degrees of truth. Thus a formula is considered fully true under an evaluation e if e(A) = 1. A formula A is said to be valid in an MTL-algebra L if it is fully true under all evaluations in L, that is, if e(A) = 1 for all evaluations e in L. Some formulae (for instance, p → p) are valid in any MTL-algebra; these are called tautologies of MTL.
The notion of global entailment (or: global consequence) is defined for MTL as follows: a set of formulae Γ entails a formula A (or: A is a global consequence of Γ), in symbols $\Gamma \models A,$ if for any evaluation e in any MTL-algebra, whenever e(B) = 1 for all formulae B in Γ, then also e(A) = 1. Informally, the global consequence relation represents the transmission of full truth in any MTL-algebra of truth values.
General soundness and completeness theorems
The logic MTL is sound and complete with respect to the class of all MTL-algebras (Esteva & Godo, 2001):
A formula is provable in MTL if and only if it is valid in all MTL-algebras.
The notion of MTL-algebra is in fact so defined that MTL-algebras form the class of all algebras for which the logic MTL is sound. Furthermore, the strong completeness theorem holds:[5]
A formula A is a global consequence in MTL of a set of formulae Γ if and only if A is derivable from Γ in MTL.
Linear semantics
Like algebras for other fuzzy logics,[6] MTL-algebras enjoy the following linear subdirect decomposition property:
Every MTL-algebra is a subdirect product of linearly ordered MTL-algebras.
(A subdirect product is a subalgebra of the direct product such that all projection maps are surjective. An MTL-algebra is linearly ordered if its lattice order is linear.)
In consequence of the linear subdirect decomposition property of all MTL-algebras, the completeness theorem with respect to linear MTL-algebras (Esteva & Godo, 2001) holds:
• A formula is provable in MTL if and only if it is valid in all linear MTL-algebras.
• A formula A is derivable in MTL from a set of formulae Γ if and only if A is a global consequence in all linear MTL-algebras of Γ.
Standard semantics
Standard are called those MTL-algebras whose lattice reduct is the real unit interval [0, 1]. They are uniquely determined by the real-valued function that interprets strong conjunction, which can be any left-continuous t-norm $\ast $. The standard MTL-algebra determined by a left-continuous t-norm $\ast $ is usually denoted by $[0,1]_{\ast }.$ In $[0,1]_{\ast },$ implication is represented by the residuum of $\ast ,$ weak conjunction and disjunction respectively by the minimum and maximum, and the truth constants zero and one respectively by the real numbers 0 and 1.
The logic MTL is complete with respect to standard MTL-algebras; this fact is expressed by the standard completeness theorem (Jenei & Montagna, 2002):
A formula is provable in MTL if and only if it is valid in all standard MTL-algebras.
Since MTL is complete with respect to standard MTL-algebras, which are determined by left-continuous t-norms, MTL is often referred to as the logic of left-continuous t-norms (similarly as BL is the logic of continuous t-norms).
Bibliography
• Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
• Esteva F. & Godo L., 2001, "Monoidal t-norm based logic: Towards a logic of left-continuous t-norms". Fuzzy Sets and Systems 124: 271–288.
• Jenei S. & Montagna F., 2002, "A proof of standard completeness of Esteva and Godo's monoidal logic MTL". Studia Logica 70: 184–192.
• Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212.
• Cintula P., 2005, "Short note: On the redundancy of axiom (A3) in BL and MTL". Soft Computing 9: 942.
• Cintula P., 2006, "Weakly implicative (fuzzy) logics I: Basic properties". Archive for Mathematical Logic 45: 673–704.
• Chvalovský K., 2012, "On the Independence of Axioms in BL and MTL". Fuzzy Sets and Systems 197: 123–129, doi:10.1016/j.fss.2011.10.018.
References
1. Ono (2003).
2. Conjectured by Esteva and Godo who introduced the logic (2001), proved by Jenei and Montagna (2002).
3. Hájek (1998), Definition 2.2.4.
4. The proof of Lemma 2.3.10 in Hájek (1998) for BL-algebras can easily be adapted to work for MTL-algebras, too.
5. A general proof of the strong completeness with respect to all L-algebras for any weakly implicative logic L (which includes MTL) can be found in Cintula (2006).
6. Cintula (2006).
| Wikipedia |
Could a 21 meter space telescope detect the nearest exoplanets?
For reference, Hubble's mirror is 2.4 meters wide, the upcoming James Webb's 6.5 meters, and the proposed ATLAST 8 or 16 meters. Let's assume a mirror nearly ten times Hubble's size, 21 meters, is sent up and benefits from a coronagraph. Could it detect the planets of stars like Proxima Centauri, at 4.2 light years, and likewise Epsilon Eridani, at ~10.5 light years?
By detect, not so much a pretty picture but enough light and data to confirm orbits and overall characteristics. Only other assumption would be that this telescope observes UV-O-NIR spectra, say between 200 to 1000 nanometer wavelengths.
mission-design technology space-telescope exoplanet telescope
RedlioxRedliox
$\begingroup$ So in other words, you want to be able to resolve the planet? $\endgroup$ – Phiteros Dec 13 '17 at 0:49
$\begingroup$ Note that lens size might not be the most serious limitation. To observe a planet light-years away, you have to aim the telescope at exactly the right spot. That pretty much means you have to know the orbit beforehand. $\endgroup$ – Emilio M Bumachar Dec 16 '17 at 16:50
$\begingroup$ For those that didn't get the memo :) ATLAST (Advanced Technology Large Astronomical Space Telescope) has now morphed into LUVOIR (Large Ultra Violet/Optical/InfraRed Surveyor). $\endgroup$ – Vince 49 Jan 2 '18 at 22:23
As of 2017, 22 exoplanets have been imaged directly. The most distant of those is 1200 ly away.
This shows 4 of them orbiting HR 8799, which is 128 ly away:
These observations are good enough to determine planet orbits, and to do spectroscopy.
So we can already observe large exoplanets at much longer distances, if they orbit far enough away from their star. Smaller planets closer to their star are more difficult to see.
In the image above there's a black disk at the center. This is used to block out the star's light. It covers the star, plus a radius of about 5 AU, so any Earth-like planets in the habitable zone are masked in this image.
This is necessary because a telescope's optics and imaging system aren't perfect: the star's light is not confined to the star's diameter, it gets spread out a bit. This spread-out starlight is brighter than planets in close orbits, so those planets are lost in the noise.
One way to combat this is an occulter, i.e. a physical object in front of the telescope that blocks out the star's light. Most of these are small disks at the front end of the telescope.
NASA is working on a starshade: a much larger disk placed 50,000 km in front of a space telescope. This would be able to block out the star's light more accurately, allowing the telescope to see exoplanets in the habitable zone (so at distances of ~1 AU for smallish stars). The plan was to use the starshade with WFIRST (a 2.4 m telescope), but recent concerns about costs could see the starshade removed from this mission.
So the solution for direct imaging of exoplanets may not have to be a larger mirror.
HobbesHobbes
$\begingroup$ The planets of HR8799 are giants of 5 or even 7 times the mass of Jupiter and a distance of several tenths of astronomical units. The smallest directly observed planet is less than 2 times the mass of Jupiter. Direct observation of extrasolar planets with a mass and distance like our Earth will take some time. $\endgroup$ – Uwe Dec 13 '17 at 13:57
$\begingroup$ @Uwe -- I believe it is "several tens" of AUs, not several tenths. $\endgroup$ – antlersoft Dec 13 '17 at 20:48
$\begingroup$ If there were an award fro the best use of GIFs in SE, this one would win. Beautiful! $\endgroup$ – uhoh Dec 14 '17 at 5:15
The resolution of a telescope in radians is given (to a very good first-order approximation) by $\theta_{res}=1.22\frac{\lambda}{D}$ where $\lambda$ is the wavelength and $D$ is the diameter of the telescope aperture. Using the numbers you give, a 21 meter telescope at 200 nanometers would have a resolution of 0.002 arcseconds. Now, there are many other factors which will limit our resolution, but you can consider this, the diffraction-limited resolution, to be the highest possible resolution you can achieve. Anything with a smaller angular size than this will be indistinguishable.
So our next step is to figure out what angular size an exoplanet will subtend on the sky. Let's go with a Jupiter-size planet, to increase our chances. This means a planetary diameter of $~1.4*10^8$ meters. This object's angular size (in radians) will be given by $\theta_{size}=\frac{diameter}{distance}$. We can actually set this expression equal to our equation for resolution and solve for distance. This will give us an approximation for how far away a planet can be before we won't be able to resolve it. Solving for $distance$ yields:
$$ distance = \frac{diameter\cdot D}{1.22 \lambda} $$
Again, using our numbers, this gives us a distance of $1.2*10^{16}$ meters, or 1.3 light years.
So, ultimately the answer is no. A 21 meter primary aperture will not be enough to resolve an exoplanet. So let's see how big our mirror would have to be, by rearranging our equation:
$$ D = \frac{1.22\cdot \lambda \cdot distance}{diameter} $$ If we want to resolve a Jupiter size exoplanet at the distance of Epsilon Eridani, it would need to be 173 meters in diameter.
Now, this is all assuming that we don't have to worry about other things, like glare from the star, which presents its own set of problems. But we can get around this by doing things like optical interferometry, which allows us to increase the effective size of our telescope without having to build bigger mirrors.
PhiterosPhiteros
$\begingroup$ Your answer seems to be about resolving planet as a disk (as opposed to seeing it as a point). I think we don't need that much resolution to confirm orbital parameters. $\endgroup$ – Alexander Dec 13 '17 at 1:27
$\begingroup$ I was basing my answer more off the "overall characteristics" part of the question. If all you care about is separating the exoplanet from the star, you could calculate the pixel scale of the telescope. But I don't think there's enough information in the question to answer that. You would need the focal ratio of the telescope (which you could make up whatever number for it), but you would also need to know the point-spread function of the parent star (unless it's blocked), and the separation between the star and the planet. $\endgroup$ – Phiteros Dec 13 '17 at 1:33
$\begingroup$ And in this case, you would still see the planet as a point, but it would be a point separate from other objects, so you could say that you were looking only at the planet and not other stuff as well. $\endgroup$ – Phiteros Dec 13 '17 at 1:36
$\begingroup$ that's fine, but I don't see why the planet's diameter should be part of the equation while the orbit's diameter (semimajor axis) is not. $\endgroup$ – Alexander Dec 13 '17 at 5:43
$\begingroup$ @uhoh that presentation is really great. But no matter what secondary optics you add, you can't get better than the diffraction limited resolution. That's the absolute best resolution you can accomplish. $\endgroup$ – Phiteros Dec 13 '17 at 19:50
Yes. The ESO's VLT used the wobble method to detect Proxima Centauri b, a planet with a radius estimated at 0.8–1.5 R⊕ and a semi-major axis estimated at 0.0485 (+0.0041,−0.0051) AU, at a distance of 4.224 ly.
The Hubble Telescope can see the Proxima Centauri system's Sun.
The ground based VLT consists of four individual telescopes, each with a primary mirror 8.2 m across, they can achieve an angular resolution of about 0.001 arc-second. In single telescope mode of operation the angular resolution is about 0.05 arc-second.
Best case ground based conditions give a seeing disk diameter of ~0.4 arcseconds and are found at high-altitude observatories on small islands such as Mauna Kea or La Palma.
At the best high-altitude mountaintop observatories, the wind brings in stable air which has not previously been in contact with the ground, sometimes providing seeing as good as 0.4".
Under bad conditions a ground based telescope over 10 meters with poor seeing can limit the resolution to be about the same as given by a space-based 10–20 cm telescope.
Ground based telescopes must look through the atmosphere, which is opaque in many infrared bands (see figure of atmospheric transmission). Even where the atmosphere is transparent, many of the target chemical compounds, such as water, carbon dioxide, and methane, also exist in the Earth's atmosphere, vastly complicating analysis.
Existing space telescopes such as Hubble cannot study these bands since their mirrors are not cool enough (the Hubble mirror is maintained at about 15 degrees C) and hence the telescope itself radiates strongly in the IR bands.
The JWST telescope has an expected mass about half of Hubble Space Telescope's, but its primary mirror (a 6.5 meter diameter gold-coated beryllium reflector) will have a collecting area about five times as large (25 m^2 or 270 sq ft vs. 4.5 m^2 or 48 sq ft).
The JWST is oriented toward near-infrared astronomy, but can also see orange and red visible light, as well as the mid-infrared region, depending on the instrument.
JWST's primary mirror is a 6.5-meter-diameter gold-coated beryllium reflector with a collecting area of 25 m^2.
From the JWST FAQ:
At which wavelengths will Webb observe?
Webb will work from 0.6 to 28 micrometers, ranging from visible gold-colored light through the invisible mid-infrared. The short wavelength end is set by the gold coating on the primary mirror. The long wavelength cut-off is set by the sensitivity of the detectors in the Mid-Infrared Instrument.
How faint can Webb see?
Webb is designed to discover and study the first stars and galaxies that formed in the early Universe. To see these faint objects, it must be able to detect things that are ten billion times as faint as the faintest stars visible without a telescope. This is 10 to 100 times fainter than Hubble can see.
What are the main science goals of Webb?
Webb has four mission science goals:
Search for the first galaxies or luminous objects that formed after the Big Bang.
Determine how galaxies evolved from their formation until the present.
Observe the formation of stars from the first stages to the formation of planetary systems.
Measure the physical and chemical properties of planetary systems and investigate the potential for life in those systems.
How far will Webb look?
One of the main goals of Webb is to detect some of the very first star formation in the Universe. This is thought to happen somewhere between redshift 15 and 30 (redshift explained in question 45). At those redshifts, the Universe was only one or two percent of its current age. The Universe is now 13.7 billion years old, and these redshifts correspond to 100 to 250 million years after the Big Bang. The light from the first galaxies has traveled for about 13.5 billion years, over a distance of 13.5 billion light-years.
Will Webb see planets around other stars?
The Webb will be able to detect the presence of planetary systems around nearby stars from their infrared light (heat). It will be able to see directly the reflected light of large planets - the size of Jupiter - orbiting around nearby stars. It will also be possible to see very young planets in formation, while they are still hot. Webb will have coronagraphic capability, which blocks out the light of the parent star of the planets.
This is needed, as the parent star will be millions of times brighter than the planets orbiting it. Webb will not have the resolution to see any details on the planets; it will only be able to detect a faint light speckle next to the bright parent star.
Webb will also study planets that transit across their parent star. When the planet goes between the star and Webb, the total brightness will drop slightly. The amount that the brightness drops tells us the size of the planet. Webb can even see starlight that passes through the planet's atmosphere, measure its constituent gasses and determine whether the planet has liquid water on its surface. When the planet passes behind the star, the total brightness drops, and we can again determine more of the planet's characteristics.
Super short version: They're launching a slightly larger telescope than you have asked for that will reach ("detect", not provide close-up photos) virtually to the known edge of the universe.
$\begingroup$ How deep into the infrared spectrum can a space telescope study without needing either coolant or extensive sun shielding? $\endgroup$ – Redliox Dec 15 '17 at 19:10
$\begingroup$ Some of the sensors on the JWST can see from 0.6 microns to 5 microns using the heatshield only (it's 5 sheets of plastic film); without using the jwst.nasa.gov/cryocooler.html - it uses helium gas, and very little, so not a coolant in the usual sense, or not much of it. It's designed to be light weight. More info about the Sunshield: jwst.nasa.gov/sunshield.html . $\endgroup$ – Rob Dec 15 '17 at 19:39
At the present time, there is a plan to incorporate a planetary coronagraph on WFIRST based on the PISCES integral field spectrograph (IFS). The instrument will use a local occulter, rather than a starshade. The coronagraph will not resolve the planetary disk, but will be able to see a point image (PSF) separate from the host star. This will enable the IFS to extract a spectrum. The "smoking gun" for a habitable planet is a combination of water vapor, oxygen, and methane. [Methane can not endure over geologic time in the presence of free oxygen, so there must be a constant source (life?)].
Lessons learned from the coronagraph on WFIRST will be used to optimize the design of a revised planetary coronagraph on LUVOIR, or whatever the eventual name of the follow on to Hubble. A starshade is not only expensive (as Hobbs pointed out), but it also takes a long time to re-target. In an ideal world, and with sufficient funds, the local occulter would be used to get a first order spectrum of promising planets, then the most promising would be re-visited with a starshade to extract more detailed spectra.
Additionally, JPL is planning to propose the HabEx Mission which would incorporate a planetary coronagraph, also including an IFS. I believe it's important to point out that their site says: ... directly image planetary systems around Sun-like stars. This means they will be able to image luminous points (PSFs) that are distinguishable from the host star and adjacent planets--not planetary disks.
Of course, priorities change and funds come and go, so none of this is written in stone.
WFIRST/PISCES Integral Spectrograph: https://arxiv.org/abs/1707.07779
Proposed HabEx Mission: https://www.jpl.nasa.gov/habex/
Vince 49Vince 49
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Can we make super-massive Telescopes to image exoplanets? | CommonCrawl |
\begin{definition}[Definition:Total Preordering]
Let $S$ be a set.
Let $\precsim$ be a preordering on $S$.
Then $\precsim$ is a '''total preordering''' on $S$ {{iff}} $\precsim$ is connected.
That is, {{iff}} there is no pair of elements of $S$ which is non-comparable:
:$\forall x, y \in S: x \precsim y \lor y \precsim x$
\end{definition} | ProofWiki |
Design, simulation and testing of a cloud platform for sharing digital fabrication resources for education
Gianluca Cornetta1,
Javier Mateos1,
Abdellah Touhafi2 &
Gabriel-Miro Muntean3
Journal of Cloud Computing volume 8, Article number: 12 (2019) Cite this article
Cloud and IoT technologies have the potential to support applica- tions that are not strictly limited to technical fields. This paper shows how digital fabrication laboratories (Fab Labs) can leverage cloud technologies to enable resource sharing and provide remote access to distributed expensive fabrication resources over the internet. We call this new concept Fabrication as a Service (FaaS), since each resource is exposed to the internet as a web ser- vice through REST APIs. The cloud platform presented in this paper is part of the NEWTON Horizon 2020 technology-enhanced learning project. The NEWTON Fab Labs architecture is described in detail, from system concep- tion and simulation to system cloud deployment and testing in NEWTON project small and large-scale pilots for teaching and learning STEM subjects.
Most developed countries are experiencing a shortage of scientists; for example, the proportion of students graduating in STEM (Science, Technology, Engi- neering and Mathematics) subjects in Europe has reduced from 12% to 9% since 2000 [1]. There are strong evidences that young people disengagement from STEM subjects begins during secondary education [2] since students perceive scientific subjects as difficult and they consider science-related careers as less lucrative and more demanding compared to other disciplines. Govern- ments worldwide are putting great efforts in order to reverse this process and the European Union, in particular, has made a huge investment to fund large scale technology-enhanced-learning (TEL) projects like NEWTON in order to foster the passion for scientific disciplines among the younger generations. The goal of NEWTON project is avoiding early student dropout from the scientific stream, for this reason it is mainly targeted to primary and secondary school students. NEWTON aims at developing student-centered non-formal (i.e. out- side the education system) and informal (i.e. based on self-learning) teaching methodologies that leverage the latest innovative technologies to deliver more effectively learning contents and make STEM subjects more appealing. In such context, Fab Labs [3, 4] have been proven to be an innovative and effective teaching tool to attract students to STEM subjects. A Fab Lab is a small-scale workshop with a set of flexible computer-controlled tools and machines such as 3D printers, laser cutters, computer numerically-controlled (CNC) machines, printed circuit board millers and other basic fabrication tools which can allow the student to experiment and to prove theoretical concepts by prototyping. Thus, a Fab Lab is a place where the students can learn with a hands-on ap- proach based on experimentation and where they can materialize their ideas in engaging and stimulating ways and supervise the whole fabrication process. The Fab Lab concept is gaining worldwide interest and both governments and population are starting to recognize the importance of digital fabrication tech- nologies even as early as primary and secondary level education.Footnote 1 A direct consequence is that the number of Fab Labs is continuously increasing and to date there exists a worldwide network of more than 1100 Fab Labs located in more than 40 countries, which are coordinated by the Fab Lab Foundation.
The main factor that is actually limiting a wider diffusion of the Fab Lab concept is the lab set up cost.Footnote 2 Fabrication machines and materials are expen- sive and not all educational institutions, especially in primary and secondary education streams may afford the costs to start and especially maintain a Fab Lab. Surprisingly, all the research efforts put to date in the digital fabrication area have been aimed at demonstrating the effectiveness of Fab Labs in education [5] and at incorporating digital fabrication in the curricula [6,7,8]. However, to the best of authors knowledge no attempt has been made to address the challenges faced enhancing the Fab Lab functionality by providing support for pervasive and ubiquitous Internet access and resource sharing. That's when the concept of Fabrication as a Service (FaaS) comes into play. FaaS has been introduced in [9] and is an architecture designed to enable remote access to Fab Labs as a Cloud-based service. This approach is a necessary evolution of Fab Labs, allowing them to become available to a wider community over the Internet.
As described in [9], the NEWTON Fab Lab platform relies on a loosely- coupled set of microservices running either on cloud or on the Fab Lab premises. These microservices implement: (1) the communication layer to interconnect all the networked Fab Labs, (2) the Fab Lab software abstraction layer, and (3) the fabrication machines software abstraction layer. Each microservice ex- poses a set of REST (REpresentational State Transfer) APIs (Application Programming Interface) used for system integration and for communication with third-party services and applications. These APIs enable the development of application and protocols to implement remote access and resource sharing of the underlying digitally-controlled hardware (i.e. the fabrication machines). The cloud infrastructure acts as the Hub node of a spoke-hub architecture where the interconnected Fab Labs represent the spoke nodes. The Fab Lab infrastructure can be accessed though a Fab Lab gateway that implements the Fab Lab abstraction layer as well as security and API requests rate-limiting policies. Each machine in a Fab Lab is wrapped by a software abstraction layer that provides mechanisms to monitor the machine status as well as the status of the queued jobs. The Hub node keeps a registry of all the interconnected Fab Labs. The registry includes information on Fab Lab location, infrastructure, bill of materials and fabrication machines' load. The registry is updated in real-time using machine-to-machine communication protocols. The Cloud Hub acts also as a router that seamlessly relays the incoming fabrication requests to the Fab Lab that is geographically closer to requester's location, has availability of fabrication resources and matches the machine and material types specified in the fabrication request.
In this paper we dive deeper into the FaaS concept and the design and de- velopment of the NEWTON Fab Lab platform by analyzing in detail the soft- ware and hardware architecture as well as the design tradeoffs. The manuscript is organized as follows: Section 2 describes the system architecture and the service integration into Amazon AWS (Amazon Web Services) infrastructure. Each of the three tiers (i.e. cloud hub, Fab Lab gateway and machine wrap- per) is analyzed in depth and a comprehensive description of all the software modules is provided. Section 3 reports the results of the tests performed to stress the platform performance, the measured data has been used to build a simple simulation model on top of CloudSim simulatorFootnote 3 in order to per- form a rough estimation of the system performance and to find possible system bottlenecks under realistic operating scenarios. In Section 4 we analyze the de- ployment costs of the architecture described in this paper whereas, in Section 5, we evaluate the educational impact of the designed platform and present the data collected and the result obtained during NEWTON small- and large-scale pilots. Finally, in Section 6 we summarize our achievements, draw up some conclusions and analyze possible related research topics and future develop- ments.
Most of the digital fabrication machines used in a standard Fab Lab deployment are not open source, this means that hardware and software specifications are not available to developers and writing drivers and applications for that equipment entails a serious challenge to reverse-engineering the software in order to understand its behavior and write new open-source drivers and inter- faces. Another major design constraint to NEWTON Fab Lab is the lack of internet connectivity of the available fabrication machines. In order to over- come this limitation a hardware and software wrapper must be built on top of the fabrication equipment in order to provide the system with the capability to expose a Fab Lab to the internet as a web service. We call this hard- ware/software wrapper a Pi-wrapper since it is implemented on a Raspberry Pi embedded computing board. However, for security reasons, a machine is not directly exposed to the internet but lies behind a Fab Lab Gateway. The Fab Lab Gateway dynamically collects in real time the information from all the machine wrappers, builds a snapshot of all the services available in the Fab Lab and exposes them through a set of APIs that can be consumed by the Cloud Hub application.
The NEWTON Fab Lab architecture is a three-tier spoke-hub architecture in which the interconnected Fab Labs (i.e. the spokes) can communicate through a centralized hub located on cloud premises. The digital fabrication equipment of each Fab Lab is not directly exposed to the internet but can be accessed through a Fab Lab gateway that implement filtering and security policies. Finally, each digital fabrication machine has a software wrapper that exposes the underlying hardware though a set of REST APIs. Both the Fab Lab gateway and the machine wrapper are implemented using inexpensive off-the-shelf microcontroller boards. In our specific case, we use Raspberry PIs boards to implement the gateway and the machine wrappers; for this reason, we also refer to them as Pi-Gateway and Pi-Wrapper respectively. Fig. 1 depicts the simplified architecture of the NEWTON Fab Lab infrastructure. In order to allow inter-Fab Lab communication, each networked Fab Lab should have at least one public IP address Addr:ePort. The router/gateway maps the inbound traffic into a private address pAddr:pPort by means of a Network Address Table (NAT) and a Port Address Table (PAT). Similarly, the router performs the same task on the outbound traffic by forwarding it to the default gateway or by redirecting the requests for a private address to the private network. The message flow between the cloud application and the networked Fab Labs is managed by a cloud-deployed message broker that implements a publish/subscribe protocol. Spoke and hub nodes form a Virtual Private Net- work (VPN) in which the Fab lab gateway and the virtual machine instances on cloud premises communicate securely over the internet using private IP addresses though an IPSec (IP Secure) tunnel. IPsec is a suite of protocols for managing secure encrypted communications at the IP Packet Layer. The cloud and Fab Lab gateways are the tunnel endpoints deployed on local and cloud premises respectively.
NEWTON Fab Labs simplified system architecture
The cloud hub
The Cloud Hub is the centralized communication hub for all the networked NEWTON Fab Labs, tightly integrated into AWS (Amazon Web Services) web services infrastructure. More specifically, the cloud hub infrastructure requires the following AWS managed services:
Route 53 as the Domain Name Service (DNS).
S3 as the backend storage for the application cluster.
Internet Gateway to expose to the internet the underlying public infrastructure.
Figure 2 depicts the minimum infrastructure requirements for the cloud hub. The deployment requires five EC2 (Elastic Compute Cloud) instances. Two m3.medium instances are necessary to deploy the service networking infrastructure, whereas, three m4.large instances are necessary to deploy the cluster with the Platform as a Service Infrastructure (PaaS) to manage the Fab Lab cloud services. Digital fabrication services (i.e. the fabrication machines soft- ware wrappers and the underlying hardware) can be accessed through a set of REST APIs described in [10]. The cloud service networking infrastructure is formed by:
A VyOSFootnote 4 software-defined router to forward the incoming traffic from both the internet gateway and the IPSec tunnel to the service cluster in the private sub-network.
A reverse proxy to route the traffic forwarded by the VyOS router to the target service running on the service cluster.
Cloud Hub deployment on Amazon AWS infrastructure
The VyOS router is also used to manage the cloud end of the IPSec tunnel that connects the cloud hub to the Fab Labs network. Thus, the cloud hub and the interconnected Fab Labs form a unique VPN in which cloud and on- premise services communicate over an encrypted channel using private IPs.
The PaaS infrastructure is deployed on top of Flynn.Footnote 5 Flynn can be considered as a grid of Docker containers, rather than a traditional cluster. Each host will run containerized services and applications that can be deployed and scaled individually. Fig. 3 shows a simplified diagram depicting a Flynn grid deployment across a cluster of three hosts. Flynn architecture is split into two layers. Layer 0 provides basic services such as host management, service discovery and scheduling, whereas layer 1 implements the PaaS business logic (GitHub interface, Slug Builder, Slug Runner, etc.). Referring to Fig. 2, the layer 0 services are:
The Host Service (HS) that implements the interface between Flynn ser- vices and Docker. The Host Service is the only one that must run across all the Flynn hosts
The Scheduler (S). The scheduler distributes the containers among the instances given the current state of the grid and the resource allocation in each node.
Example of Flynn grid deployment across three hosts
The layer 1 services are:
The GitHub frontend (G). This module accepts Git connections through SSH and Git pushes; then deploys them in the Flynn grid.
The Controller (C) exposes APIs to control the whole infrastructure.
The Router (R) is a TCP/HTPP router/load balancer that distributes the incoming requests through the instances deployed in the Flynn grid. In order to implement a high-availability there must be several instances of this module across all the Flynn hosts.
The Slug Builder (SB) is a module that builds a slug starting from a Git push received by the Flynn Git front-end (G). A slug is a compressed and pre-packaged copy of an application optimized for distribution to the Flynn PaaS.
The Slug Runner (SR) is a module that allocates and instantiates several Docker containers (depending on the scaling parameters) to deploy and execute the code contained in a slug.
The Application (A) is a module that implements the application code (for example, the Cloud Hub and the Service Registry in our specific case).
The fab lab gateway
The Fab Lab gateway (i.e. the Pi-Gateway) is the entry point to the local network and to the digital fabrication infrastructure of a Fab Lab. Fig. 4 depicts the Pi-Gateway software architecture. The architecture is modular and distributed over four layers. The Communication Layer is a proxy server that implements the communication protocols between the cloud hub and the gateway (HTTP and HTTPS are both supported). The incoming requests are forwarded to the API Wrapper Layer that implements simple APIs to com- municate with the underlying Fab Lab infrastructures and a simple reactive websocket protocol to update in real-time the Fab Lab status in the cloud hub infrastructure. The proxy configuration is managed by a command line interface (CLI). Both the CLI and the API wrapper layer leverage the Middle- ware Layer functions to implement the business logic and the communication protocols. Middleware provides primitive functions to implement websocket communications, logging, process management (using programmatically the APIs provided by the PM2Footnote 6module), transactional e-mail (using an AWS Simple E-mail Service client) and persistence layer interfacing. Open API 2.0 (Swagger) support is also integrated in the middleware layer and APIs specifi- cations are described in [11]. Finally, the Data Layer (persistence layer) is used to store the proxy and the Fab Lab configurations. We use a NoSQL model and RedisFootnote 7 module) as the key-value store.
Fab Lab Gateway (Pi-Gateway) software architecture
The machine wrapper
The Machine Wrapper (i.e. the Pi-Wrapper) provides the connected machine with a software abstraction layer by exposing the machine functionalities through a set of APIs. Fig. 5 depicts the software architecture of the Pi- Wrapper. The software architecture is modular and distributed over five layers. The Communication Layer implements the HTTP server and the APIs interface to manage and schedule fabrication batches. The Presentation Layer implements the user interfaces to set up and manage a connected fabrication machine. An MVC (Model View Controller) programming paradigm is used at this stage; namely, a route in the browser triggers a controller function that dynamically generates and renders an HTML view using the data stored in the persistence layer (i.e. data base). The Application Layer implements the business logic. The business logic and the user interface rely on the middle- ware functions implemented in the Middleware Layer. More specifically, the middleware includes custom and third-party methods to manage security and authentication, machine to machine communications and interfacing, HTML views rendering, system logging, data base connection and access, and ADC (Analog to Digital Converter) drivers to sample data from the machine moni- toring circuit as described in [9]. Open API 2.0 (Swagger) support is integrated in the application middle- ware, this makes the Pi-Wrapper a very developer- friendly software since APIs and data models documentation is embedded into the application, in addition a developer can test the API using the Swagger User Interface that is also embedded in the Pi-Wrapper. Swagger Pi-Wrapper API specifications are described in [12]. Finally, the Data Layer is used to store session information as well as User and Machine data models. We use a NoSQL model and MongoDBFootnote 8 as the data store.
Machine wrapper (Pi-Wrapper) software architecture
Machine to machine communication
The communications between client applications and the remote NEWTON Fab Labs rely on a protocol stack which includes a simple publish/subscribe protocol. The fabrication equipment is accessed through the Fab Lab Gate- way that routes incoming commands to a given machine depending on both availability and the specific task to be carried on. The communication protocol relies on a server-to-server model in which some nodes act as message brokers collecting the incoming messages and re- laying them towards a destination node. A fabrication job is routed to a networked Fab Lab by the Cloud Hub message broker; however, the message broker on the cloud side has not direct visibility of the Fab Lab network infrastructure. Its main task is to connect a client to the Fab Lab infrastructure or to perform inter-Fab Labs message routing. The networked machines in a Fab Lab can be accessed through the Fab Lab Gateway only. The gateway main task is routing the outbound traffic to the networked equipment and managing intra-Fab Lab communications. Fig. 6 presents a simplified timing diagram that describes the communication between the cloud infrastructure and a networked Fab Lab. The message exchange has four stages:
link establishment;
topic subscription;
communication;
disconnection (not illustrated for the sake of simplicity).
Overview of the Inter- and Intra-Fab Lab Messaging Flow
Once the TCP links between the machine and the Fab Lab Gateway on one side, and the Fab Lab Gateway and the Cloud hub broker on the other side, have been established, both the Gateway and the Hub subscribe to topics they are interested in. The topic string is generated using the unique name and connection ID sent by the server that initiates the communications to the destination server during the link establishment. Both the link establishment and the subscription phases are terminated by an ACK message (Init ACK for the link establishment and Subscription ACK for the subscription phase). In other word, the Fab Lab Gateway and the Cloud Hub implement a double broker architecture: the former collects all the incoming messages from the Fab Lab machines whereas the latter collects all the incoming messages from the networked Fab Lab Gateways. The double broker architecture allows the implementation of Fab Lab access and security policies and of custom mes- sage filters mechanisms. Once the subscription phase has terminated, the end nodes start exchanging messages. Each published message can be acknowledged by an optional Publication ACK message. The use of a Publication ACK is mandatory in those cases when it is necessary to guarantee the delivery of a message and to implement retransmission mechanisms to increase the QoS of the protocol.
Test, modelling and simulation
The system infrastructure has been tested in real scenarios through small-scale pilots that have involved the participation of six schools and universities lo- cated in three European countries as part of the EU-funded NEWTON project. The test pilots have been used to stress the system infrastructure and evaluate the performance of the proposed algorithms for task scheduling and fabrica- tion resources allocation. In order to detect system peak performance, system infrastructure and APIs have been also load tested using Locust.Footnote 9 Locust allows to simulate user behavior using a Python script. We have designed a set of simple use cases that stresses all the Fab Lab APIs and provides a unified picture of the system performances.
The test scenario implements the use cases described in Table 1. These use cases have been translated into a Python script that is parsed by Locust in order to generate the requests for the infrastructure under test. Locust can be further configured so that the user behaviour described in that script can be associated to an arbitrary number of virtual users in order to stress the system response under different load conditions.
Table 1 Fab Lab modules test cases
Load tests
The Fab Lab infrastructure described in the previous sections has been load tested in the following emulated scenarios:
50 concurrent users with a hatch rate of 5 users per second.
100 concurrent users with a hatch rate of 5 users per second.
All the incoming requests are forwarded to the same fabrication machine, each test has a duration of 2 minutes and, as mentioned before, each simulated user performs the operations described in Table 1 which means that the following HTTP requests are sent to the Fab Lab APIs:
GET the available Fab Lab status.
POST a job to the available Fab Lab.
GET the status information of the submitted job.
DELETE the submitted job.
GET the information of the jobs running in the available Fab Lab.
The most time-consuming operation is the POST request to submit a fabrication job since it involves the following steps:
Uploading the image on the cloud hub.
Sending the image to the Fab Lab Gateway.
Sending the image to the target fabrication machine.
Update the jobs queue in the fabrication machine.
Fig 7 shows the load tests results for the three scenarios under test (i.e., the cases with 50, 100 and 150 concurrent users respectively). Fig. 7 a summarizes the overall results for all the request types, whereas Fig. 7 b depicts the results only for POST requests. Test results are excellent, considering the Fab Lab infrastructure has been deployed on inexpensive Raspberry Pi III boards. For example, the 90% of the incoming requests are served in maximum 680 ms for 50-user scenario, 1100 ms for the 100-user scenario, and 5100 ms for 150-user scenario. Of course, as outlined earlier in this section, the most time-consuming operations are the POST requests whose delay can be as high as 9141 ms in the case of 150 concurrent users. An overview of the measurements performed using Locust is summarized in Tables 2, 3 and 4. The tables report the median, minimum, maximum and average response time in milliseconds for each one of the API called by our simulated scenario for all the test cases studied (namely for the 50-, 100- and 150-user load respectively). The measured values confirm the excellent performance already outlined by Fig. 6. The total average response times for the 50-, 100- and 150-user test cases are 452 ms, 568 ms and 1680 ms respectively, whereas the maximum average response times are 801 ms, 1158 ms and 3883 ms respectively. An average response time of 3883 ms is acceptable and, according to Fig. 7 a allows, on the average, the completion of the 100% of the requests for the 50-user scenario, the 99% of the requests for the 100-user scenario and almost the 80% of the total requests for the 150-user scenario.
Percentage of Requests Completed in a Given Time Interval a Total Requests, and b POST Requests
Table 2 Summary of System Performance for 50 Users Load (values are in ms)
Table 3 Summary of system performance for 100 Users Load (values are in ms)
Platform modelling
The system stressed by the load tests described in Section 3.1 is a minimum deployment formed by the Cloud Hub located in the eu-central-1 AWS region (i.e., in the Amazon AWS data center in Frankfurt) and a single spoke node (i.e., the San Pablo-CEU Fab Lab located in Madrid). Thus, in order to estimate the performances of larger deployments across several AWS regions, a simulation model is necessary. The cloud infrastructure under test, depicted in Fig. 8, is very complex and requires up to six levels of AWS services (Route 53, Elastic Load Balancing, Autoscaling, EC2 instances, S3 storage and optionally, Cloudfront CDN services). This, in turn entails several challenges tied to infrastructure and application setup, administration, and behaviour predictability. On one hand, the promise of scalability, redundancy and on-demand service deployment makes a cloud implementation a very appealing solution. On the other hand, all these advantages come at the price of several issues that can make cloud application development and management a challenging task. More specifically, the issues with cloud deployment are related to the following impact factors:
Performance: Disk IO operations can be a serious issue and limit the performance of a cloud deployment. In a cloud infrastructure, the network and the underlying storage are shared among customers. If, for example, another customer sends large amounts of write requests to the cloud stor- age system, your application may experience slowdowns and its latency becomes unpredictable. Moreover, also the upstream network is shared among customer, so one can experience bottlenecks there too. Unluckily, cloud vendors use to offer to their customers large storage, but not fast storage.
Transparency: Transparency and simplicity are key factors when debug- ging either an application or an infrastructure. Unfortunately, cloud ser- vices are, in many cases very opaque and tend to hide underlying hardware and network problems. Cloud infrastructure is a shared service, and, for this reason, cloud users may experience issues that do not occur in dedi- cated infrastructure. More specifically, cloud infrastructure customers, share hardware resources such as CPU, RAM, disk and network, thus the workload of other users can saturate a computing node and heavily affect the performance of your application.
Complexity and scalability: Fig. 8 gives an idea of the complexity of the cloud architecture that has been deployed to ensure NEWTON Fab Labs connectivity. This entails the interaction of up to six different AWS service layers that require expertise for set-up and configuration. Moreover, Elastic Load Balancing and scalability are not straightforward in AWS and require the deployment and configuration of additional services (namely, CloudWatch and CloudFormation) that incur extra costs and complexity.
NEWTON Fab Labs global infrastructure deployment
Finally, as mentioned in Section 2.1, we have deployed a PaaS (Platform as a Service) infrastructure on top of the cloud infrastructure depicted in Fig 8. The PaaS simplifies application and service deployment in a cloud environment but adds other software layers and additional complexity to the underlying infrastructure, making the application behaviour even more unpredictable. In order to build a simulation model as close as possible to the real behaviour of the cloud infrastructure, we have followed the steps reported in the sequel:
We have instrumented the Cloud Hub server in order to measure the server latency to process an incoming request.
We have developed a fake client that performs fabrication requests at ran- dom times and have measured the elapsed times from request arrival to request dispatch to the selected Fab Lab. This time represents the server latency that is necessary to serve a request.
We have performed latency measurements for several server configurations, scaling the number of containers allocated to the database and to the Cloud Hub application.
We have used the measured data to build a simple regression model to predict the server latency as a function of the incoming requests and of the number of allocated containers.
We have deployed a test infrastructure across several AWS data centers in order to ensure the maximum geographic coverage as depicted in Fig. 8. The Fab Lab network implements a spoke-hub architecture in which each spoke relies on the Registry Server of the Cloud Hub for service detection and trac routing.
We have performed several measurements on the cloud infrastructure in order to determine latency and bandwidth across the networked Data Cen- ters.
We have used RIPE AtlasFootnote 10 data to build a latency and bandwidth model for the connections among a client and a Data Center and a Data Cen- ter and the target Fab Lab for each geographic region covered by AWS infrastructure.
We have used the experimental data gathered in Steps (6, 7) and the simple predictive model developed in Step (4) to build a delay model for the NEWTON Fab Lab infrastructure.
We have built an ad-hoc simulator on top of CloudSim [13] to simulate the behavior and the performance of the NEWTON Fab Labs network under different load conditions and using the delay model implemented at step (8).
Cloud hub delay estimation
The Cloud hub server has been instrumented in order to capture the the incom- ing POST requests and to measure the time elapsed from the request arrival and its subsequent forwarding to the selected Fab Lab. The measurements have been performed for several requesting users and server configurations. For each simulation set up the measurements have been performed 10 times at random intervals. We assume that the number n of requesting users is a power of 2 with 1 ≤ n ≤ 64 and that the number c of Docker containers allocated to the Cloud hub server is also a power of 2 with 1 ≤ c ≤ 8. For each configuration under test we compute the mean, the median, the standard deviation and the geometric mean of the measured latencies. The measurements are reported in Tables 5, 6, 7 and 8. Tables 5, 6, 7 and 8 summarize the statistical distributions of the measured delays for several application deployments. As also observed in [9], the measured values exhibit a high standard deviation. Moreover, observing the minimum, the median and the maximum values, one can infer that the measured latencies have a tail distribution (either lognormal or Pareto). This tail behaviour, as reported in [14], is typical for networked and internet appli- cations. More specifically, we have found that the distribution of the measured values, whose statistical behaviour is summarized in Tables 5, 6, 7 and 8, matches a Pareto type I distribution.Footnote 11 Due to the high dispersion of the measured data, the mean values are not meaningful and may lead to wrong conclusions since the arithmetic mean is heavily affected by the outliers. A more objec- tive analysis must rely on the minimum and median values of the latency as well as on its geometric mean since, unlike arithmetic mean, it is less sensitive to the effect of the outliers. Analyzing Tables 5, 6, 7 and 8 as a whole, one could easily observe that the minimum, the median and the geometric mean of the measured delays decrease as expected (with some outliers) as the number of allocated containers scales up. However, this is not the case for the maximum delays. As mentioned before, Downey [14] showed that this high variability is very typical in internet applications. In our specific case, the high dispersion of the measured values is due to the unpredictable latency introduced by the cloud infrastructure. As pinpointed in Section 3.2, a cloud deployment has some drawbacks that arise from the fact that several customers are sharing the same virtualized hardware and network infrastructure. Consequently, the performance of a cloud application is heavily affected by the other customers' application that are loading the underlying infrastructure at the same time. We have deliberately performed our measurements at random times to trigger this variable behavior and the effect of the other AWS customers application load on the performance of our platform. To this latency, we should also add the latency introduced by the virtual networking routing infrastructure de- ployed by Flynn. However, recall that the impact of the maximum delay on the overall performance is minimum; since, in a tail distribution the probability of a high delay is very low.
Table 5 Cloud Hub latency (in ms) with one container allocated to the appli- cation
Table 6 Cloud Hub latency (in ms) with two containers allocated to the application
Table 7 Cloud Hub latency (in ms) with four containers allocated to the application
Table 8 Cloud Hub latency (in ms) with eight containers allocated to the application
Communication latency and bandwidth
In order to build a realistic simulation model, we need to estimate communica- tion latency and bandwidth among the nodes that form the Fab Lab network as well as the maximum concurrency level that each node can support. This goal is accomplished through the following steps:
We estimate the network latency Lcj from client to Data center j and Lfj Fab Lab to Data Center j in the same AWS region. To do this, we use the real measurements provided by RIPE Atlas network. RIPE Atlas is a public network located in the last mile and formed by more than 16.000 measurement probes capable of measuring connectivity between internet endpoints on demand.
We estimate the network uplink and downlink bandwidth between client and Data Center j (Buplink,cj and Bdownlink,cj respectively) and Fab Lab and Data Center j (Buplink,fj and Bdownlink,fj respectively) in the same AWS region. To do this, we use the Clouharmony speed test service.Footnote 12 How- ever, this service allows measuring the desired parameters only between the client browser and the target Data Center. This means, that we are able to track performance only within Europe and must make the simplifying assumption that the network performances within the same AWS region are approximately the same using the measurements performed in Europe as the reference values.
We measure the Data center i to Data center j network latency Lij using ping and traceroute. Traceroute is even better than ping since it allows testing the response time of each network segment along the path. There- fore, this tool can not only measure but also locate the latency across the routers that form the packets path.
We measure the Data center i to Data center j uplink and downlink band- with (Buplink,ij and Bdownlink,ij respectively) using iPerf3 tool.
The delay D of the system response after a fabrication (POST) request has been issued is computed as follows:
$$ {\displaystyle \begin{array}{l}D\kern0.5em =\kern0.5em {L}_{cj}+\kern0.5em {t}_{uplink, cj\kern0.5em }+\kern0.5em {L}_{jk\kern0.5em }+\kern0.5em {t}_{uplink, jk\kern0.5em }+\kern0.5em {L}_{kj\kern0.5em }+\kern0.5em {t}_{uplink, kj}\\ {}\kern4.5em +\kern0.5em {L}_{jf\kern0.5em }+\kern0.5em {t}_{uplink, cj\kern0.5em }+\kern0.5em {L}_{fj\kern0.5em }+{t}_{uplink, fj\kern0.5em }+\kern0.5em {L}_{jc\kern0.5em }+\kern0.5em {t}_{uplink, jc}\end{array}} $$
where j denotes a Datacenter located in a spoke node, whereas k denotes the Data center located in the hub node. The delay D of a response is hence the packet round-trip time necessary to follow the path the goes from the client to the spoke node, from the spoke to the hub node and then to the spoke again, from the spoke to the selected fab lab and then to the spoke again, and finally to the client. Observe that the data transfer time tij between nodes i and j in Equation 1 is computed as:
$$ {t}_{ij\kern0.5em }=\kern0.5em \frac{S_{ij}}{B_{ij}} $$
where Sij and Bij represent, respectively, the number of bytes transmitted and the measured bandwidth between nodes i and j. Table 9 summarizes the average latencies measured from client to Data center from different world regions.
Table 9 Summary of the latencies (in ms) of client-to-Data center connection
Table 10 reports the uplink and downlink bandwidth between a client and a Data center located in the same AWS region. More specifically, these mea- surements refer to a client and a Data center located in Europe since, as we pointed out earlier, the Cloudharmony speed test service only allows perform- ing measurements from the client browser to the target Data center. We will use the values of Table 9 as reference for all the AWS supported region that form the NEWTON Fab Lab network architecture.
Table 10 AWS uplink and downlink bandwidths (Mb/s)
Table 11 reports the Data-center-to-Data-center latency. For each possible connection, we report minimum, average and maximum latency as well as the standard deviation with respect the average latency.
Table 11 Summary of the hub-to-spoke latencies (ms)
Finally, Table 12 summarizes the measured uplink and downlink band- widths for the Data center to Data center connections.
Table 12 Summary of the Inter-Data center uplink and downlink bandwidths (Mb/s)
Concurrency level
We use Apache BenchmarkFootnote 13 to estimate the maximum concurrency level that can be effectively borne by a node of the NEWTON cloud infrastructure. This allows us to estimate the maximum number of concurrent requests that can be served by the cloud infrastructure and to configure suitably the simulator that models NEWTON Fab Lab infrastructure. The Cloud Hub APIs provide a root (/) endpoint that supports both HTTP and HTTPS protocols and returns a response with a 200-status code and a body with an empty JSON (JavaScript Object Notation) object. We use this endpoint to ping the Cloud Hub sever; however, we can also use the same endpoint to perform simple load tests on our infrastructure. Nonetheless, you have to keep in mind that the result obtained in this way are optimistic since the authentication server and the underlying data base are not stressed. Although Apache Benchmark tool generates very detailed reports, we are only interested in detecting which is the maximum number of concurrent requests that breaks our server leading to a timeout error. In order to do this, we stress our server during a prolonged period with an increasing number of concurrent requests until it breaks. Table 13 summarizes the percentiles measured when a minimum cloud deployment (with only one container allocated to the cloud hub application) is stressed by 20.000 requests with concurrency levels 10, 50 and 100 respectively. Observing the percentiles of the measurements, we note that in all the scenarios under test the response delays exhibit a tail distribution. In addition, increasing the concurrency level of the incoming requests leads to larger tail delays, being 100 the maximum concurrency level that can be supported by the cloud configuration under test. However, the measurements carried out are qualitative and are only useful to set-up our simulation model with reasonable values. In fact, the measurements performed have been carried out just for a short period of time, thus they do not consider the delay variability of the cloud infrastructure as pointed out previously. Moreover, the measured times refer to the response latency for a simple API endpoint that returns a 200-OK response; consequently, they do not consider the extra latency to access to the underlying data base to retrieve the Fab Lab information. For all the aforementioned reasons, it seems reasonable to assume that, in a real deployment, the Fab Lab infrastructure can support without problems up to 50 concurrent accesses and manage approximately 1000 requests per second (by scaling up the number of containers allocated to the cloud application).
Table 13 Summary of the response times (in ms) for 20.000 incoming requests
Simulator implementation and simulation results
The measurements performed on the Cloud Hub infrastructure reported in Tables 5 to 8 show, as expected, non-normal distribution of the measured data that seems loosely correlated to the number of requests and the number of containers allocated to the application, which makes very difficult to make reliable predictions of the Cloud Hub application latency. Lognormal and Pareto distributions are those that better model server response time [14]. For this reason, the proposed prediction scheme does not predict the latency of the Cloud Hub application; this would make no sense, since, as stated before in a cloud environment several customers share the same network and infrastructure which makes very hard to predict the server behaviour in a given instant. What we do instead, is using the measured data to predict the shape of a type
I Pareto distribution that models the performance of our cloud infrastructure under different load conditions and number of allocated containers. We then use the prediction to generate, in our simulator, a random latency X(r, n) that is a function of the number r of incoming requests and of the number n of allocated containers, with that Pareto distribution starting from a uniform random variable U ∈ (0, 1) using the following equation:
$$ X\left(r,n\right)=\frac{{\hat{x}}_i\left(r,n\right)}{{\left(1-U\right)}^{1/\hat{\alpha}\left(r,n\right)}} $$
Where βˆ(r, n) = xˆi(r, n) is the prediction of the Pareto distribution scaleparameter and αˆ(r, n) is the prediction of the Pareto distribution shape parameter. Both βˆ and αˆ are computed using a simple regression model as a function of r and n. The simulation software has been built according to the following hypothesis:
The CPU load of each instance of the cluster must not exceed the 50%.
The requests are evenly distributed among the cluster instances.
The incoming requests are evenly distributed within a given instance among blocks of 8 Docker containers, being 64 the scaling threshold.Footnote 14
The cluster minimum configuration can manage up to 50 concurrent ac- cesses.
The following pseudo-code snippet describes the block allocation and latency estimation process implemented by our simulator:
The algorithm estimates the delay of the infrastructure response and follows the steps described next. First an array to hold the estimations of the response delay is initialized (line 1). Afterwards, the number of incoming requests is computed and the number of containers necessary to manage all the incoming requests is allocated in each of the virtual machines that form the cluster (lines 3 to 7). Then, the number of requests that must be forwarded to each allocated block of containers is computed (line 8). After that, for each allocated block, the shape of the Pareto distribution of the possible delays is computed (lines 9 to 13). Recall that, as stated before, the Pareto distribution shape and scale parameters are computed by performing a multivariate linear regression on the measured data whose statistics are summarized in Tables 5, 6, 7 and 8. Finally, the values of the shape and scale parameters for the given number of requests and allocated containers are used to estimate the system response latency using Equation 3.
Thus, our simulator relies on the measurements reported in Sections 3.3 to 3.5 to build a network bandwidth and latency model and on Equation 3 to estimate the delay of the spoke and hub nodes taking into account the variabil- ity introduced by the cloud shared infrastructure. The overall system delay, i.e. the packet round-trip time from a fabrication request issued by a client until the system acknowledge is computed using Equation 1. Experiments have been designed to analyze the behaviour of the NEWTON Fab Lab infrastructure with the following users' distribution: 250, 500, 1000, and 1500. Each user can issue from one to five requests; moreover, for each load configuration, the number of containers allocated to the application will scale as multiples of 8 from 8 to 128 (for 16 possible configurations). Finally, the simulated infrastructure must cover requests from four AWS availability zones (Europe, North and Central America, South America and Asia-Pacific) in order to ensure a globally optimal service to all the world regions. Table 14 summarizes the experiments configurations. The variable simulation parameters are the num- ber of users, the number of requests per user, and the number of containers allocated to each instance of the cluster. All the other parameters are fixed. This means that for each possible user configuration 80 experiments must be performed (i.e. the number of requests per user times the number of possible containers configurations). For the sake of simplicity, we also assume a uniform user distribution among different AWS regions.
Table 14 Experiments conguration
The scaling threshold is set to 1024 requests, i.e. the request count per tar- get of each Elastic Load Balancing (ELB) target group must be kept as close as 1024 for the Autoscaling group.Footnote 15 More specifically, assume that you have configured an Autoscaling group with a minimum of three instances (i.e. the minimum PaaS cluster configuration) and a maximum of six instances within an ELB group of a given AWS region. Setting a threshold of 1024 means that each instance of your cluster should receive approximately 1024 requests. If the overall number of incoming requests is larger, the number of instances should be scaled up to match the target threshold as close as possible. For example, if the cluster has three instances and the number of incoming requests is, say 3800, the system should scale up by one instance (i.e. from three to four), so that each instance handles 3800/4 = 950 requests. Finally, note that with the simulation set up depicted in Table 14, the maximum number of incoming requests from a given region do not exceed 5000; thus, with a threshold of 1024 it is not necessary to have more than five virtual machines in the Autoscaling group. Prior to running all the experiments, we have to make sure that the mathematical model we have developed for the cloud application behaves as expected. To do this we simply check that the simulated latency of the NEWTON cloud infrastructure matches a Pareto distribution. After running all the simulations whose set-up is detailed in Table 14 we obtain the Pareto-like distribution of the response latency depicted in Fig. 9. Recall that, as detailed in Table 14, our simulation scenario assumes fabrication requests with a 5 MB attachment (since this is the typical image le size of a design submitted for fabrication). In addition, we have also assumed that the users (and hence the service requests) are evenly distributed among all the Data centers that form the NEWTON Fab Lab cloud infrastructure.
Distribution of the response latency for NEWTON Fab Lab infrastructure
Table 15 represents the percentiles for the distribution of Fig. 9. Observe that 50% of the requests are served within 8000 ms and 99% of the requests within 38.000 ms, being 49.000 ms the worst-case simulated delay. These are indeed excellent results considering that:
As highlighted earlier in this paper, cloud infrastructure is shared among many customers, leading to very variable delays.
The simulated latency also includes the transmission time of the design le (assumed to be 5 MB) attached to a request (that must go from the client to the spoke or hub node of NEWTON infrastructure and finally to the target Fab Lab).
In the worst-case scenario the communication delay depends on the follow- ing path: client - spoke - hub - spoke - Fab Lab - spoke - client. Thus, the latency of a response can be very high due to the communication overhead introduced by each node in the communication path.
Table 15 Percentile table of the simulated NEWTON Fab Lab cloud infrastructure latency
After running the set of experiments described in in Table 14, for each Data center in the network, we obtain the performance estimations summarized from Table 16, 17, 18, 19, 20 and 21. For each simulation scenario and Data center, we report minimum, maximum, average, median and standard deviation of the simulated latency.
Table 16 Summary of the Data centers performance (250 users scenario)
Table 19 Summary of the Data centers performance (1000 users scenario)
Observing the simulation results we can easily infer that the cloud system infrastructure behaves as expected since:
The response delay increases with the number of requests.
The Europe Data center is the one that exhibits the longest delays because it is the hub of our infrastructure and must always process all the incoming requests.
The Data center latency exhibits a high variability, which reflects the performance fluctuations of the cloud infrastructure due to resource and network sharing with other customers as outlined previously.
The response latency exhibits a Pareto-like distribution, which is typical of internet networked systems.
Infrastructure costs
The NEWTON infrastructure must comprise four Data Centers to ensure max- imum coverage in all the AWS supported regions. The Data Centers implement a spoke-hub architecture being the Frankfurt node (eu-central-1 AWS region) the hub. Spokes must be located in United States (eu-east-1 AWS region), South America (sa-east-1 AWS region) and Singapore (ap-southeast-1 AWS region). The main infrastructure and application (i.e. the registry service, the Fab Lab monitoring service, the Fab Lab connection/routing service) is hosted on the hub, whereas the spokes only run a simple client to query the service registry and the router. With this approach we limit the more expensive vir- tual machines (i.e. the m4.large instances) to the network hub, whereas the spokes may rely on cheaper virtual machines (i.e. t2.micro instances).
In its minimum configuration, the NEWTON cloud infrastructure relies on the following Amazon AWS services:
Between five and eight Elastic Cloud Computing (EC2) instances.
Between five and eight EBS volumes allocated for each EC2 instance.
Route53 DNS service.
S3 storage to implement the blobstore for the PaaS infrastructure.
Optionally, the CloudFront content delivery network (CDN) service.
The EC2 instances that form the PaaS infrastructure are configured to be autoscaled, according to the platform load, between three and five instances. This, in turn, requires setting-up other two AWS services:
CloudWatch to monitor platform metrics and trigger the autoscaling.
CloudFormation, to dynamically build and deploy new instances of the PaaS platform.
CloudWatch has a free tier. Each month, AWS customers receive 10 met- rics (applicable to detailed monitoring for Amazon EC2 instances or custom metrics), 10 alarms, 5 GB of log size, 5 GB of archived log size, 3 dash- boards and 1 million API requests at no charge. This should be sufficient for NEWTON cloud infrastructure to operate safely without incurring extra costs. Conversely, CloudFormation is a free service.
Table 22 summarizes the running costs (VAT not included) of the hub node of the Fab Lab cloud infrastructure. Amazon AWS also offers to its cus- tomers dedicated instances and dedicated hosts. These solutions isolate your infrastructure from that of the other customers, leading to a more stable and controllable behaviour. Deploying a dedicated instance on AWS will incur an additional cost of $2 /h. This means that the monthly running costs of an EC2 instance will increase by $1440 if we want that instance to be dedicated. Conversely, the monthly cost of a dedicated host of m4 type in the eu-central-1 region (Frankfurt) is $2366,09. The spoke node infrastructure is very simple and is formed by one to three autoscaled t2.micro EC2 instances. This in- frastructure must be deployed in all the spoke nodes of the NEWTON Fab Lab network: eu-east-1 (N. Virginia), sa-east-1 (Sao Paulo) and ap-southeast-1 (Singapore).
Table 22 NEWTON Fab Labs hub node monthly running costs on AWS infrastructure
Tables 23, 24 and 25 report the running costs of the infrastructure for each one of the AWS regions in which the spoke nodes must be deployed.
Table 23 NEWTON Fab Labs eu-east-1 spoke node monthly running costs on AWS infrastructure
Table 24 NEWTON Fab Labs sa-east-1 spoke node monthly running costs on AWS infrastructure
Table 25 NEWTON Fab Labs ap-southeast-1 spoke node monthly running costs on AWS infrastructure
Finally, Table 26 summarizes the overall monthly costs necessary to run the whole NEWTON Fab Lab infrastructure. Thus, the infrastructure running costs of a minimum deployment may vary between $1386,33 and $1811,89 per month (VAT not included).
Table 26 NEWTON Fab Labs cloud infrastructure overall monthly running costs
Fab labs impact in education
The NEWTON project Fab Labs, as small workshops offering flexible remote digital fabrication, were tested in an educational context. The goal of these tests was to establish the degree of success of the proposed new learning paradigm learning by doing in terms of both student learning outcome, and most importantly their degree of satisfaction. Students from two schools: Saint Patricks Boys National School in Dublin, Ireland and CEU Monteprincipe School in Madrid, Spain were exposed to NEWTON Fab Labs as part of the NEWTON education initiative. The 39 students, aged between 10 and 13, were asked to model 3D ceramic vases using a third-party design software, prepare the digital files and send them over the Internet to the Fab Lab be printed. Following the usage of the NEWTON Fab Lab technology, the students were asked to fill a usability questionnaire. Fig 10 illustrates the average scores obtained after processing the results of the questionnaire. 87% of the participants from both schools reported that they had fun using the NEWTON Fab Lab technologies and indicated that they would recommend Fab Lab solutions to their friends. This is a great outcome and demonstrates how Fab Lab can have a highly positive impact on student increased satisfaction while learning. Future work will present in details the results of the deployment of Fab Lab in education.
Average scores for the Fab Lab usability questionnaire
FaaS Fab Lab deployment has been performed as part of the NEWTON plat- form. The platform is now in production phase and includes the cloud hub (deployed on an Amazon AWS EC2 cluster) and the on-premises interface in- frastructure (implemented with inexpensive Raspberry Pi III boards) that has been deployed and is presently under test at CEU Madrid, Spain. This deployment has helped gaining significant insights on several design and implementation aspects and trade-offs that include hardware design and interfacing, system monitoring and cloud deployment, data security as well as service deployment and orchestration in a multi-cloud environment. Several architectural aspects and implementations have been evaluated and tested so far, with particular emphasis on:
system replicability and scalability;
system costs and maintainability;
service availability and auto-discovery in multi-cloud environments;
API architecture and design;
functional and load tests design.
The next step is setting-up the system staging environment that involves networking and interfacing to the cloud hub the Fab Labs at CEU Madrid and Vrije University of Brussels, Belgium. This will enable testing the sys- tem in a distributed, yet still controlled environment. FaaS enhances existing Fab Lab capabilities by providing the digital fabrication equipment with the possibility to communicate over the Internet in order to remotely control fabrication activities. Using this approach, the fabrication facilities are exposed to the Internet as software services, which may be consumed by third-party applications. FaaS practical deployment strongly relies on IoT and Cloud architectural and software paradigms and requires design and development of specific hardware and software interfaces that allow pervasive connectivity. The hardware interface design was not difficult and has been accomplished by using standard and inexpensive off-the-shelf components. Conversely, firmware and software development were highly challenging and has involved solving several complex problems related to equipment monitoring and real time communications. The paper describes FaaS deployment in the context of NEWTON next generation Fab Labs; however, the proposed solution is general, hardware-independent and targets all those scenarios which involve collaborative fabrications. We foresee that this capability will have a huge impact not only on education, but also on industry helping to develop new business models in which fab-less companies may schedule medium or large-scale fabrication batches hiring third-party remote fabrication services.
The NEWTON Fab Lab modules are available with MIT license through the NEWTON Fab Lab project page on Git Hub at https://gcornetta.github.io/gwWrapper/. The experimental data is available at https://github.com/ gcornetta/data and it is licensed under Creative Common 4.0 BY-NC-SA.
"National Curriculum in England: Design and Technology Programmes Study", UK Department of Education, 2013, https://www.gov.uk/government/publications/national-curriculum-in-england-design-and-technology-programmes-of-study
The minimum deployment costs of a Fab Lab compliant with the Fab Foundation (https://www.fabfoundation.org/) specifications can be as high as 200.000 [$]
http://www.cloudbus.org/cloudsim/
https://vyos.io
https://flynn.io
https://keymetrics.io/pm2/
https://redis.io
https://www.mongodb.com/
Project Website: https://locust.io
https://atlas.ripe.net
The experimental data has been open-sourced and is available at https://github.com/gcornetta/data
https://cloudharmony.com/speedtest
https://httpd.apache.org/docs/2.4/programs/ab.html
This design choice is due to the fact that our simple prediction functions are defined for 1 ≤ r ≤ 64 requests and 1 ≤ n ≤ 8 containers. Also, consider that Flynn does not natively support the container autoscaling feature implemented by our simulator. In order to enable container autoscaling you should use other container orchestration platforms such as DC/OS or Rancher instead of Flynn, provided you may afford higher deployment costs.
Please note that in a real (i.e. not simulated) AWS deployment you need to enable the CloudWatch service to measure the metrics necessary to trigger autoscaling and the CloudFormation service to create and deploy an instance of the PaaS cluster node.
ADC:
Application Programming Interface
AWS:
CDN:
CLI:
CNC:
Computer Numerically-Controlled
EC2:
Elastic Compute Cloud
ELB:
Elastic Load Balancing
FaaS:
Fabrication as a Service
Host Service
HTTP Secure
IoT:
Internet of the Things
IPSec:
JavaScript Object Notation
JSON Web Token
MVC:
Model View Controller
Network Address Table
PaaS:
PAT:
Port Address Table
QoS:
REpresentational State Transfer
SB:
Slug Builder
SOA:
Slug Runner
Science, Technology, Engineering and Mathemathics
TCP:
Transmission Control Protocol
VPN:
Convert B (2005) Europe and the crisis in scientic vocations. Eur J Educ 40(4):361–366
Henriksen EK, Dillon J, Ryder J (eds) (2015) Understanding student participation and choice in science and technology education. Springer, Dordrecht, p 412
Gershenfeld N (2012) How to make almost anything: the digital fabrication revolution. Foreign Affairs, 91(6):43–57
Blikstein P (2013) Digital fabrication and making in education: the democratization of invention. In: Walter-Hermann J, Büching C (eds) Fab labs: of machine, makers and inventors. Bielefeld, Transcript Publishers, pp 203–222
Martin T, Brasiel S, Graham D, Smith S, Gurko K, Fields DA (2014) Fab lab professional development: changes in teacher and student STEM content knowledge. Digital Fabrication in Education Conference, FabLearn, Stanford
Gul LF, Simisic L (2014) Integration of digital fabrication in architectural curricula. Digital Fabrication in Education Conference, FabLearn Europe, Aarhus
Tesconi S, Arias L (2014) MAKING as a tool to competence-based school programming. Digital Fabrication in Education Conference, FabLearn Europe, Aarhus
Padeld N, Haldrup M, Hobye M (2014) Empowering academia through modern fabrication practices. Digital Fabrication in Education Conference, FabLearn Europe, Aarhus
Cornetta G, Touhafi A, Mateos FJ, Muntean G-M (2018) A cloud-based architecture for remote access to digital fabrication services for education. IEEE International Con- ference on Cloud Computing Technologies and Applications, Cloudtech, Brussels
G. Cornetta, and F. J. Mateos. "Fab lab modules: cloud hub." On- line documentation available at https://github.com/gcornetta/cloudhubAPI#documentation-and-developer-support. (2019)
Cornetta G, Mateos FJ (2019) Fab lab modules: fab lab wrapper (pi-gateway) APIs On-line documentation available at https://github.com/gcornetta/gwWrapper#fablab-apis
Cornetta G, Mateos FJ (2019) Fab lab modules: machine wrapper Online documentation available at https://github.com/gcornetta/piwrapper#machine-apis
Calheiros RN, Ranjan R, Beloglazov A, De Rose CAF, Buyya R (2010) CloudSim: a toolkit for modeling and simulation of cloud computing environments and evaluation of resource provisioning algorithms. Software Practice and Experience 41:23–50 Wiley Online Library. https://doi.org/10.1002/spe.995
Downey A (2005) Lognormal and Pareto distributions in the internet. Comput Commun 28:790–801. https://doi.org/10.1016/j.comcom.2004.11.001
The authors declare that they have no conflict of interest.
The work described in this paper is part of the NEWTON project, which has been funded by the European Union under the Horizon 2020 Research and Innovation Programme with Grant Agreement no. 688503.
Department of Information Engineering, San Pablo-CEU University, Campus de Montepŕıncipe, 28668, Boadilla del Monte, Madrid, Spain
Gianluca Cornetta
& Javier Mateos
Department Electronics and Informatics, Vrije Universiteit Brussels, Brussels, Belgium
Abdellah Touhafi
School of Electronic Engineering, Dublin City University, Glasnevin, Dublin, 9, Ireland
Gabriel-Miro Muntean
Search for Gianluca Cornetta in:
Search for Javier Mateos in:
Search for Abdellah Touhafi in:
Search for Gabriel-Miro Muntean in:
GC is the main author of this research paper, as well as the software architect and main programmer of the NEWTON Fab Lab platform. FJM has con- tributed to the software development of the NEWTON Fab Lab platform and has developed the cloud simulator to estimate the platform performance. AT and GMM supervised and reviewed the associated experiments, contributed to the literature review and general organization of the paper. All authors read and approved the final manuscript.
Correspondence to Gianluca Cornetta.
Cornetta, G., Mateos, J., Touhafi, A. et al. Design, simulation and testing of a cloud platform for sharing digital fabrication resources for education. J Cloud Comp 8, 12 (2019) doi:10.1186/s13677-019-0135-x
Fabrication as a service (FaaS)
Cloud architectures
Internet of the things (IoT)
Cloud information technologies in education | CommonCrawl |
Jessie Forbes Cameron
Jessie Forbes Cameron (1883 – 1968) was a British mathematician who in 1912 became the first woman to complete her doctorate in mathematics at the University of Marburg in Germany.[1]
Jessie Forbes Cameron
Born8 January 1883
Stanley, Scotland
Died27 March 1968 (1968-03-28) (aged 85)
Southwold, England
Other namesJessie Forbes Thompson
Alma materUniversity of Marburg
OccupationMathematician
SpouseEdward Vincent Thompson
Children3
Life and work
Jessie Cameron was born on 8 January 1883 in Stanley, Scotland, one of eight children whose parents were James Cameron, a school principal at a village school in Perthshire, and his wife Jessie Forbes.[2]
After attending the Perth Academy in Scotland, Jessie Cameron studied for four semesters at University of Edinburgh. From 1905 to 1908, she studied mathematics at Newnham College, which is part of University of Cambridge, in England, and earned a Magister degree (MA).[1][2] There, she was ranked the tenth best in her class (earning her the distinction "10th Wrangler"), passed the "Mathematical Tripos" and graduated with a Bachelor of Arts (BA).[2]
Postgraduate studies
Cameron moved to the University of Göttingen, in Germany, to take two more semesters of math, and finally, she enrolled at the University of Marburg, for three semesters. Under the supervision of distinguished mathematician Kurt Hensel,[3] Cameron wrote her dissertation On the decomposition of a prime number in a composed body.[1][2]
Before her degree was officially completed, however, there was one additional barrier for her to surmount. It seems she had completed the work and won the approval of her advisor, Dr. Hensel, without realizing a lesser-known caveat for graduation from a German university. According to Lorch-Göllner, she received the following letter from the university's Dean of Faculty of Philosophy on 10 November 1911.[1]
"Although your doctoral thesis was judged favorably by the representatives of mathematics, especially by Privy Councilor Hensel, your admission to the rigorous examination is unfortunately subject to legal difficulties, which have to be fixed before I make an appointment. According to the regulations of our faculty's doctoral regulations, admission to a doctorate is dependent on proof that at least six semesters were studied at a university in the German Reich or a foreign university set up in the German way. The universities of Great Britain are not among the latter.
"You only studied five semesters at German universities and your ten semesters, which you have spent in Scotland and England, cannot be credited easily ... [you will need] a special dispensation from the Minister of Spiritual and Educational Affairs. ...
"I wrote to the Minister for this purpose eight days ago. It is hoped that he will give his approval, and then I would presumably be able to schedule the day of rigorosum before Christmas as you requested it."[1]
She soon received the Minister's dispensation and passed her exams in "mathematics, physics and philosophy" on 20 December 1911, with the accolade magna cum laude. Thus, Cameron became the first female to earn a PhD in mathematics at that university and her dissertation was published in 1912.[1]
On 28 September 1912 Cameron married the lawyer Edward Vincent Thompson, and she returned to Newnham College for a year beginning in 1912 as an "Assistant Lecturer." With this appointment, according to Lorch-Göllner, she was "the only [female] of the first math students after her doctorate - if even temporarily - to work as a mathematics lecturer in a scientific institution."[1]
Later years
In 1913, the couple moved to London so Edward Thompson could pursue his career with a position at the British Treasury. Jessie Thompson gave birth to a daughter and two sons and continued her association with Newnham College until 1927. During the First World War, the Thompson family moved to Berkhamsted, England, where Jessie began working with the British National Council of Women.[2]
Jessie Forbes (Cameron) Thompson died on 27 March 1968 in Southwold, England, at the age of 85.[2]
Published work
• Cameron, Jessie Forbes. About the decomposition of a prime number in a composed body. University of Marburg, 1912.
Literature
• Cameron, Jessie Forbes. In: Newnham College Register, Vol. 1, 1905, pages 184-185.
• Francesca M. Wilson: Jessie Forbes Thompson (born Cameron), 1883-1968 (Newnham 1905-1909 and 1912-1913). In: Newnham College Roll Letter, Cambridge 1969, pages 63–64.
References
1. Lorch-Göllner, Silke (1 March 2018). "Die "mutvoll Trotzigen": Die ersten Mathematikstudentinnen der Königlich Preußischen Universität Marburg". Mathematische Semesterberichte (in German). 65 (1): 35–64. doi:10.1007/s00591-017-0211-6. ISSN 1432-1815. S2CID 126315878.
2. Habermann, Katharina (2016). "Jessie Cameron - Philipps-Universität Marburg - Fb. 12 - Mathematik und Informatik". www.uni-marburg.de (in German). Retrieved 27 June 2020.{{cite web}}: CS1 maint: url-status (link)
3. Hasse, Helmut (1948). "Kurt Hensel zum Gedächtnis" (PDF) (in German). Retrieved 28 June 2020.{{cite web}}: CS1 maint: url-status (link)
External links
• Renate Tobies (ed.): Despite all male culture Women in mathematics and natural sciences. Frankfurt am Main / New York 1997, page 137
Authority control
International
• ISNI
• VIAF
National
• Germany
• United States
Academics
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos
KRM Home
A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror
December 2016, 9(4): 687-714. doi: 10.3934/krm.2016012
A degenerate $p$-Laplacian Keller-Segel model
Wenting Cong 1, and Jian-Guo Liu 2,
School of Mathematics, Jilin University, Changchun 130012, China
Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708
Received February 2016 Revised April 2016 Published September 2016
This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak solution for the $p$-Laplacian Keller-Segel system with the supercritical diffusion exponent $1 < p < \frac{3d}{d+1}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(3-p)}{p}}$ norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.
Keywords: fast diffusion, monotone operator, global existence, Chemotaxis, critical space, non-Newtonian filtration..
Mathematics Subject Classification: Primary: 35K65, 35K92, 92C1.
Citation: Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012
T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598. Google Scholar
S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z. Google Scholar
S. Bian, J.-G. Liu and C. Zou, Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$, Kinet. Relat. Models, 7 (2014), 9-28. doi: 10.3934/krm.2014.7.9. Google Scholar
A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7. Google Scholar
F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874. Google Scholar
F. E. Browder, Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523. doi: 10.2307/1970660. Google Scholar
L. Chen, J.-G. Liu and J. Wang, Multidimensional degenerate Keller-Segel system with critical diffusion exponent $2n/(n+2)$, SIAM J. Math. Anal., 44 (2012), 1077-1102. doi: 10.1137/110839102. Google Scholar
L. Chen and J. Wang, Exact criterion for global existence and blow up to a degenerate Keller-Segel system, Doc. Math., 19 (2014), 103-120. Google Scholar
L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire}, 15 (1998), 493-516. doi: 10.1016/S0294-1449(98)80032-2. Google Scholar
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. Google Scholar
E. DiBenedetto and M. A. Herrero, Non-negative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1 < p < 2$, Arch. Rational Mech. Anal., 111 (1990), 225-290. doi: 10.1007/BF00400111. Google Scholar
P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310. doi: 10.1007/BF02392210. Google Scholar
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar
I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602. doi: 10.1137/110823584. Google Scholar
O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva and S. Smith, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968. Google Scholar
J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. Google Scholar
E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar
J.-G. Liu and J. Wang, A note on $L^{\infty}$-bound and uniqueness to a degenerate Keller-Segel model, Acta Appl. Math., 142 (2016), 173-188. doi: 10.1007/s10440-015-0022-5. Google Scholar
G. J. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1038-1041. doi: 10.1073/pnas.50.6.1038. Google Scholar
G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2. Google Scholar
B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564. doi: 10.1007/s10492-004-6431-9. Google Scholar
B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007. Google Scholar
Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180. Google Scholar
Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type, J. Differential Equations, 250 (2011), 3047-3087. doi: 10.1016/j.jde.2011.01.016. Google Scholar
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013. Google Scholar
J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, The Clarendon Press, Oxford University Press, Oxford, 2007. Google Scholar
Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791. Google Scholar
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Wenting Cong Jian-Guo Liu | CommonCrawl |
⌂ → L-functions → Rational → 2 → 2880
Rational L-function search results
$\zeta$ zeros
Lowest zero Conductor Analytic conductor Central character Analytic rank
Degree Bad \(p\) include exclude exactly subset Root analytic conductor Motivic weight Spectral label
Root number Primitive Origin Exclude origin
1 -1 yes no Dirichlet character Artin representation Elliptic curve/Q Elliptic curve/NF Genus 2 curve/Q Classical modular form Hilbert modular form Bianchi modular form GL3 Maass form GL4 Maass form GSp4 Maass form Dirichlet character Artin representation Elliptic curve/Q Elliptic curve/NF Genus 2 curve/Q Classical modular form Hilbert modular form Bianchi modular form GL3 Maass form GL4 Maass form GSp4 Maass form
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▲ root analytic conductor analytic conductor first zero conductor columns to display ✓ label ✓ root analytic conductor ✓ analytic conductor ✓ degree ✓ conductor ✓ central character ✓ mu ✓ nu ✓ motivic weight ✓ primitive ✓ root number ✓ order of vanishing ✓ first zero ✓ origin
Results (35 matches)
$\alpha$
$A$
$d$
$N$
$\chi$
$\mu$
$\nu$
$w$
$\epsilon$
$r$
First zero
2-2880-20.19-c0-0-1 $1.19$ $1.43$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 20.19 $$ $0.0$ $0$ ✓ $1$ $0$ $0.938223$ Artin representation 2.2880.4t3.a Artin representation 2.2880.4t3.a.a Modular form 2880.1.j.a Modular form 2880.1.j.a.1279.1
2-2880-1.1-c1-0-1 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.568764$ Elliptic curve 2880.a Modular form 2880.2.a.a Modular form 2880.2.a.a.1.1
2-2880-1.1-c1-0-10 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.862970$ Elliptic curve 2880.m Modular form 2880.2.a.m Modular form 2880.2.a.m.1.1
2-2880-1.1-c1-0-11 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.885365$ Elliptic curve 2880.bd Modular form 2880.2.a.bd Modular form 2880.2.a.bd.1.1
2-2880-1.1-c1-0-12 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.910745$ Elliptic curve 2880.bb Modular form 2880.2.a.bb Modular form 2880.2.a.bb.1.1
2-2880-1.1-c1-0-13 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.991628$ Elliptic curve 2880.bc Modular form 2880.2.a.bc Modular form 2880.2.a.bc.1.1
2-2880-1.1-c1-0-14 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $1.00571$ Elliptic curve 2880.o Modular form 2880.2.a.o Modular form 2880.2.a.o.1.1
2-2880-1.1-c1-0-15 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $1.00870$ Elliptic curve 2880.be Modular form 2880.2.a.be Modular form 2880.2.a.be.1.1
2-2880-1.1-c1-0-17 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $1.01518$ Elliptic curve 2880.r Modular form 2880.2.a.r Modular form 2880.2.a.r.1.1
2-2880-1.1-c1-0-18 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $1.01814$ Elliptic curve 2880.bg Modular form 2880.2.a.bg Modular form 2880.2.a.bg.1.1
2-2880-1.1-c1-0-19 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $1.12013$ Elliptic curve 2880.bh Modular form 2880.2.a.bh Modular form 2880.2.a.bh.1.1
2-2880-1.1-c1-0-2 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.589344$ Elliptic curve 2880.c Modular form 2880.2.a.c Modular form 2880.2.a.c.1.1
2-2880-1.1-c1-0-21 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $1.23621$ Elliptic curve 2880.bf Modular form 2880.2.a.bf Modular form 2880.2.a.bf.1.1
2-2880-1.1-c1-0-22 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.38232$ Elliptic curve 2880.b Modular form 2880.2.a.b Modular form 2880.2.a.b.1.1
2-2880-1.1-c1-0-23 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.41105$ Elliptic curve 2880.f Modular form 2880.2.a.f Modular form 2880.2.a.f.1.1
2-2880-1.1-c1-0-24 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.43825$ Elliptic curve 2880.g Modular form 2880.2.a.g Modular form 2880.2.a.g.1.1
2-2880-1.1-c1-0-25 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.44764$ Elliptic curve 2880.i Modular form 2880.2.a.i Modular form 2880.2.a.i.1.1
2-2880-1.1-c1-0-27 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.52785$ Elliptic curve 2880.k Modular form 2880.2.a.k Modular form 2880.2.a.k.1.1
2-2880-1.1-c1-0-28 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.52992$ Elliptic curve 2880.j Modular form 2880.2.a.j Modular form 2880.2.a.j.1.1
2-2880-1.1-c1-0-29 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.53717$ Elliptic curve 2880.u Modular form 2880.2.a.u Modular form 2880.2.a.u.1.1
2-2880-1.1-c1-0-3 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.643193$ Elliptic curve 2880.e Modular form 2880.2.a.e Modular form 2880.2.a.e.1.1
2-2880-1.1-c1-0-30 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.54848$ Elliptic curve 2880.t Modular form 2880.2.a.t Modular form 2880.2.a.t.1.1
2-2880-1.1-c1-0-31 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.55188$ Elliptic curve 2880.l Modular form 2880.2.a.l Modular form 2880.2.a.l.1.1
2-2880-1.1-c1-0-32 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.57588$ Elliptic curve 2880.w Modular form 2880.2.a.w Modular form 2880.2.a.w.1.1
2-2880-1.1-c1-0-33 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.58792$ Elliptic curve 2880.n Modular form 2880.2.a.n Modular form 2880.2.a.n.1.1
2-2880-1.1-c1-0-34 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.59198$ Elliptic curve 2880.x Modular form 2880.2.a.x Modular form 2880.2.a.x.1.1
2-2880-1.1-c1-0-35 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.63003$ Elliptic curve 2880.y Modular form 2880.2.a.y Modular form 2880.2.a.y.1.1
2-2880-1.1-c1-0-36 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.65306$ Elliptic curve 2880.z Modular form 2880.2.a.z Modular form 2880.2.a.z.1.1
2-2880-1.1-c1-0-37 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.66014$ Elliptic curve 2880.ba Modular form 2880.2.a.ba Modular form 2880.2.a.ba.1.1
2-2880-1.1-c1-0-38 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $-1$ $1$ $1.67076$ Elliptic curve 2880.q Modular form 2880.2.a.q Modular form 2880.2.a.q.1.1
2-2880-1.1-c1-0-4 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.662596$ Elliptic curve 2880.d Modular form 2880.2.a.d Modular form 2880.2.a.d.1.1
2-2880-1.1-c1-0-6 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.792978$ Elliptic curve 2880.s Modular form 2880.2.a.s Modular form 2880.2.a.s.1.1
2-2880-1.1-c1-0-7 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.794943$ Elliptic curve 2880.h Modular form 2880.2.a.h Modular form 2880.2.a.h.1.1
2-2880-1.1-c1-0-8 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.802224$ Elliptic curve 2880.p Modular form 2880.2.a.p Modular form 2880.2.a.p.1.1
2-2880-1.1-c1-0-9 $4.79$ $22.9$ $2$ $2^{6} \cdot 3^{2} \cdot 5$ 1.1 $$ $1.0$ $1$ ✓ $1$ $0$ $0.805470$ Elliptic curve 2880.v Modular form 2880.2.a.v Modular form 2880.2.a.v.1.1 | CommonCrawl |
Fresnel integral
The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:
$S(x)=\int _{0}^{x}\sin \left(t^{2}\right)\,dt,\quad C(x)=\int _{0}^{x}\cos \left(t^{2}\right)\,dt.$
The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid).
Definition
The Fresnel integrals admit the following power series expansions that converge for all x:
${\begin{aligned}S(x)&=\int _{0}^{x}\sin \left(t^{2}\right)\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+3}}{(2n+1)!(4n+3)}},\\C(x)&=\int _{0}^{x}\cos \left(t^{2}\right)\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+1}}{(2n)!(4n+1)}}.\end{aligned}}$
Some widely used tables[1][2] use π/2t2 instead of t2 for the argument of the integrals defining S(x) and C(x). This changes their limits at infinity from 1/2·√π/2 to 1/2[3] and the arc length for the first spiral turn from √2π to 2 (at t = 2). These alternative functions are usually known as normalized Fresnel integrals.
Euler spiral
Main article: Euler spiral
The Euler spiral, also known as Cornu spiral or clothoid, is the curve generated by a parametric plot of S(t) against C(t). The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering.
From the definitions of Fresnel integrals, the infinitesimals dx and dy are thus:
${\begin{aligned}dx&=C'(t)\,dt=\cos \left(t^{2}\right)\,dt,\\dy&=S'(t)\,dt=\sin \left(t^{2}\right)\,dt.\end{aligned}}$
Thus the length of the spiral measured from the origin can be expressed as
$L=\int _{0}^{t_{0}}{\sqrt {dx^{2}+dy^{2}}}=\int _{0}^{t_{0}}dt=t_{0}.$
That is, the parameter t is the curve length measured from the origin (0, 0), and the Euler spiral has infinite length. The vector (cos(t2), sin(t2)) also expresses the unit tangent vector along the spiral, giving θ = t2. Since t is the curve length, the curvature κ can be expressed as
$\kappa ={\frac {1}{R}}={\frac {d\theta }{dt}}=2t.$
Thus the rate of change of curvature with respect to the curve length is
${\frac {d\kappa }{dt}}={\frac {d^{2}\theta }{dt^{2}}}=2.$
An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering: if a vehicle follows the spiral at unit speed, the parameter t in the above derivatives also represents the time. Consequently, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.
Sections from Euler spirals are commonly incorporated into the shape of rollercoaster loops to make what are known as clothoid loops.
Properties
C(x) and S(x) are odd functions of x,
$C(-x)=-C(x),\quad S(-x)=-S(x).$
Asymptotics of the Fresnel integrals as x → ∞ are given by the formulas:
${\begin{aligned}S(x)&={\sqrt {{\tfrac {1}{8}}\pi }}\operatorname {sgn} x-\left[1+O\left(x^{-4}\right)\right]\left({\frac {\cos \left(x^{2}\right)}{2x}}+{\frac {\sin \left(x^{2}\right)}{4x^{3}}}\right),\\[6px]C(x)&={\sqrt {{\tfrac {1}{8}}\pi }}\operatorname {sgn} x+\left[1+O\left(x^{-4}\right)\right]\left({\frac {\sin \left(x^{2}\right)}{2x}}-{\frac {\cos \left(x^{2}\right)}{4x^{3}}}\right).\end{aligned}}$
Using the power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, where they become analytic functions of a complex variable.
C(z) and S(z) are entire functions of the complex variable z.
The Fresnel integrals can be expressed using the error function as follows:[4]
${\begin{aligned}S(z)&={\sqrt {\frac {\pi }{2}}}\cdot {\frac {1+i}{4}}\left[\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right)-i\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right)\right],\\[6px]C(z)&={\sqrt {\frac {\pi }{2}}}\cdot {\frac {1-i}{4}}\left[\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right)+i\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right)\right].\end{aligned}}$
or
${\begin{aligned}C(z)+iS(z)&={\sqrt {\frac {\pi }{2}}}\cdot {\frac {1+i}{2}}\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right),\\[6px]S(z)+iC(z)&={\sqrt {\frac {\pi }{2}}}\cdot {\frac {1+i}{2}}\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right).\end{aligned}}$
Limits as x approaches infinity
The integrals defining C(x) and S(x) cannot be evaluated in the closed form in terms of elementary functions, except in special cases. The limits of these functions as x goes to infinity are known:
$\int _{0}^{\infty }\cos \left(t^{2}\right)\,dt=\int _{0}^{\infty }\sin \left(t^{2}\right)\,dt={\frac {\sqrt {2\pi }}{4}}={\sqrt {\frac {\pi }{8}}}\approx 0.6267.$
This can be derived with any one of several methods. One of them[5] uses a contour integral of the function
$e^{-z^{2}}$
around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the bisector of the first quadrant y = x with x ≥ 0, and a circular arc of radius R centered at the origin.
As R goes to infinity, the integral along the circular arc γ2 tends to 0
$\left|\int _{\gamma _{2}}e^{-z^{2}}\,dz\right|=\left|\int _{0}^{\frac {\pi }{4}}e^{-R^{2}(\cos t+i\sin t)^{2}}\,Re^{it}dt\right|\leq R\int _{0}^{\frac {\pi }{4}}e^{-R^{2}\cos 2t}\,dt\leq R\int _{0}^{\frac {\pi }{4}}e^{-R^{2}\left(1-{\frac {4}{\pi }}t\right)}\,dt={\frac {\pi }{4R}}\left(1-e^{-R^{2}}\right),$
where polar coordinates z = Reit were used and Jordan's inequality was utilised for the second inequality. The integral along the real axis γ1 tends to the half Gaussian integral
$\int _{\gamma _{1}}e^{-z^{2}}\,dz=\int _{0}^{\infty }e^{-t^{2}}\,dt={\frac {\sqrt {\pi }}{2}}.$
Note too that because the integrand is an entire function on the complex plane, its integral along the whole contour is zero. Overall, we must have
$\int _{\gamma _{3}}e^{-z^{2}}\,dz=\int _{\gamma _{1}}e^{-z^{2}}\,dz=\int _{0}^{\infty }e^{-t^{2}}\,dt,$
where γ3 denotes the bisector of the first quadrant, as in the diagram. To evaluate the left hand side, parametrize the bisector as
$z=te^{i{\frac {\pi }{4}}}={\frac {\sqrt {2}}{2}}(1+i)t$
where t ranges from 0 to +∞. Note that the square of this expression is just +it2. Therefore, substitution gives the left hand side as
$\int _{0}^{\infty }e^{-it^{2}}{\frac {\sqrt {2}}{2}}(1+i)\,dt.$
Using Euler's formula to take real and imaginary parts of e−it2 gives this as
${\begin{aligned}&\int _{0}^{\infty }\left(\cos \left(t^{2}\right)-i\sin \left(t^{2}\right)\right){\frac {\sqrt {2}}{2}}(1+i)\,dt\\[6px]&\quad ={\frac {\sqrt {2}}{2}}\int _{0}^{\infty }\left[\cos \left(t^{2}\right)+\sin \left(t^{2}\right)+i\left(\cos \left(t^{2}\right)-\sin \left(t^{2}\right)\right)\right]\,dt\\[6px]&\quad ={\frac {\sqrt {\pi }}{2}}+0i,\end{aligned}}$
where we have written 0i to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting
$I_{C}=\int _{0}^{\infty }\cos \left(t^{2}\right)\,dt,\quad I_{S}=\int _{0}^{\infty }\sin \left(t^{2}\right)\,dt$
and then equating real and imaginary parts produces the following system of two equations in the two unknowns IC and IS:
${\begin{aligned}I_{C}+I_{S}&={\sqrt {\frac {\pi }{2}}},\\I_{C}-I_{S}&=0.\end{aligned}}$
Solving this for IC and IS gives the desired result.
Generalization
The integral
$\int x^{m}e^{ix^{n}}\,dx=\int \sum _{l=0}^{\infty }{\frac {i^{l}x^{m+nl}}{l!}}\,dx=\sum _{l=0}^{\infty }{\frac {i^{l}}{(m+nl+1)}}{\frac {x^{m+nl+1}}{l!}}$
is a confluent hypergeometric function and also an incomplete gamma function[6]
${\begin{aligned}\int x^{m}e^{ix^{n}}\,dx&={\frac {x^{m+1}}{m+1}}\,_{1}F_{1}\left({\begin{array}{c}{\frac {m+1}{n}}\\1+{\frac {m+1}{n}}\end{array}}\mid ix^{n}\right)\\[6px]&={\frac {1}{n}}i^{\frac {m+1}{n}}\gamma \left({\frac {m+1}{n}},-ix^{n}\right),\end{aligned}}$
which reduces to Fresnel integrals if real or imaginary parts are taken:
$\int x^{m}\sin(x^{n})\,dx={\frac {x^{m+n+1}}{m+n+1}}\,_{1}F_{2}\left({\begin{array}{c}{\frac {1}{2}}+{\frac {m+1}{2n}}\\{\frac {3}{2}}+{\frac {m+1}{2n}},{\frac {3}{2}}\end{array}}\mid -{\frac {x^{2n}}{4}}\right).$
The leading term in the asymptotic expansion is
$_{1}F_{1}\left({\begin{array}{c}{\frac {m+1}{n}}\\1+{\frac {m+1}{n}}\end{array}}\mid ix^{n}\right)\sim {\frac {m+1}{n}}\,\Gamma \left({\frac {m+1}{n}}\right)e^{i\pi {\frac {m+1}{2n}}}x^{-m-1},$
and therefore
$\int _{0}^{\infty }x^{m}e^{ix^{n}}\,dx={\frac {1}{n}}\,\Gamma \left({\frac {m+1}{n}}\right)e^{i\pi {\frac {m+1}{2n}}}.$
For m = 0, the imaginary part of this equation in particular is
$\int _{0}^{\infty }\sin \left(x^{a}\right)\,dx=\Gamma \left(1+{\frac {1}{a}}\right)\sin \left({\frac {\pi }{2a}}\right),$
with the left-hand side converging for a > 1 and the right-hand side being its analytical extension to the whole plane less where lie the poles of Γ(a−1).
The Kummer transformation of the confluent hypergeometric function is
$\int x^{m}e^{ix^{n}}\,dx=V_{n,m}(x)e^{ix^{n}},$
with
$V_{n,m}:={\frac {x^{m+1}}{m+1}}\,_{1}F_{1}\left({\begin{array}{c}1\\1+{\frac {m+1}{n}}\end{array}}\mid -ix^{n}\right).$
Numerical approximation
For computation to arbitrary precision, the power series is suitable for small argument. For large argument, asymptotic expansions converge faster.[7] Continued fraction methods may also be used.[8]
For computation to particular target precision, other approximations have been developed. Cody[9] developed a set of efficient approximations based on rational functions that give relative errors down to 2×10−19. A FORTRAN implementation of the Cody approximation that includes the values of the coefficients needed for implementation in other languages was published by van Snyder.[10] Boersma developed an approximation with error less than 1.6×10−9.[11]
Applications
The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects.[12] More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see track transition curve.[13] Other applications are rollercoasters[12] or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.
Gallery
• Plot of the Fresnel integral function S(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the Fresnel integral function C(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the Fresnel auxiliary function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the Fresnel auxiliary function F(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
See also
• Böhmer integral
• Fresnel zone
• Track transition curve
• Euler spiral
• Zone plate
• Dirichlet integral
Notes
1. Abramowitz & Stegun 1983, eqn 7.3.1–7.3.2.
2. Temme 2010.
3. Abramowitz & Stegun 1983, eqn 7.3.20.
4. functions.wolfram.com, Fresnel integral S: Representations through equivalent functions and Fresnel integral C: Representations through equivalent functions. Note: Wolfram uses the Abramowitz & Stegun convention, which differs from the one in this article by factors of √π⁄2.
5. Another method based on parametric integration is described for example in Zajta & Goel 1989.
6. Mathar 2012.
7. Temme 2010, §7.12(ii).
8. Press et al. 2007.
9. Cody 1968.
10. van Snyder 1993.
11. Boersma 1960.
12. Beatty 2013.
13. Stewart 2008, p. 383.
References
• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 7". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Alazah, Mohammad (2012). "Computing Fresnel integrals via modified trapezium rules". Numerische Mathematik. 128 (4): 635–661. arXiv:1209.3451. Bibcode:2012arXiv1209.3451A. doi:10.1007/s00211-014-0627-z. S2CID 13934493.
• Beatty, Thomas (2013). "How to evaluate Fresnel Integrals" (PDF). FGCU Math - Summer 2013. Retrieved 27 July 2013.
• Boersma, J. (1960). "Computation of Fresnel Integrals". Math. Comp. 14 (72): 380. doi:10.1090/S0025-5718-1960-0121973-3. MR 0121973.
• Bulirsch, Roland (1967). "Numerical calculation of the sine, cosine and Fresnel integrals". Numer. Math. 9 (5): 380–385. doi:10.1007/BF02162153. S2CID 121794086.
• Cody, William J. (1968). "Chebyshev approximations for the Fresnel integrals" (PDF). Math. Comp. 22 (102): 450–453. doi:10.1090/S0025-5718-68-99871-2.
• Hangelbroek, R. J. (1967). "Numerical approximation of Fresnel integrals by means of Chebyshev polynomials". J. Eng. Math. 1 (1): 37–50. Bibcode:1967JEnMa...1...37H. doi:10.1007/BF01793638. S2CID 122271446.
• Mathar, R. J. (2012). "Series Expansion of Generalized Fresnel Integrals". arXiv:1211.3963 [math.CA].
• Nave, R. (2002). "The Cornu spiral". (Uses π/2t2 instead of t2.)
• Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007). "Section 6.8.1. Fresnel Integrals". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
• van Snyder, W. (1993). "Algorithm 723: Fresnel integrals". ACM Trans. Math. Softw. 19 (4): 452–456. doi:10.1145/168173.168193. S2CID 12346795.
• Stewart, James (2008). Calculus Early Transcendentals. Cengage Learning EMEA. ISBN 978-0-495-38273-7.
• Temme, N. M. (2010), "Error Functions, Dawson's and Fresnel Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• van Wijngaarden, A.; Scheen, W. L. (1949). Table of Fresnel Integrals. Verhandl. Konink. Ned. Akad. Wetenschapen. Vol. 19.
• Zajta, Aurel J.; Goel, Sudhir K. (1989). "Parametric Integration Techniques". Mathematics Magazine. 62 (5): 318–322. doi:10.1080/0025570X.1989.11977462.
External links
• Cephes, free/open-source C++/C code to compute Fresnel integrals among other special functions. Used in SciPy and ALGLIB.
• Faddeeva Package, free/open-source C++/C code to compute complex error functions (from which the Fresnel integrals can be obtained), with wrappers for Matlab, Python, and other languages.
• "Fresnel integrals", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Roller Coaster Loop Shapes". Archived from the original on September 23, 2008. Retrieved 2008-08-13.
• Weisstein, Eric W. "Fresnel Integrals". MathWorld.
• Weisstein, Eric W. "Cornu Spiral". MathWorld.
Nonelementary integrals
• Elliptic integral
• Error function
• Exponential integral
• Fresnel integral
• Logarithmic integral function
• Trigonometric integral
| Wikipedia |
Calculus of constructions
In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants.
Some of its variants include the calculus of inductive constructions[1] (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity).
General traits
The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible within CoC to define functions from terms to terms, as well as terms to types, types to types, and types to terms.
The CoC is strongly normalizing, and hence consistent.[2]
Usage
The CoC has been developed alongside the Coq proof assistant. As features were added (or possible liabilities removed) to the theory, they became available in Coq.
Variants of the CoC are used in other proof assistants, such as Matita and Lean.
The basics of the calculus of constructions
The calculus of constructions can be considered an extension of the Curry–Howard isomorphism. The Curry–Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic. The calculus of constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will also call "propositions").
Terms
A term in the calculus of constructions is constructed using the following rules:
• $\mathbf {T} $ is a term (also called type);
• $\mathbf {P} $ is a term (also called prop, the type of all propositions);
• Variables ($x,y,\ldots $) are terms;
• If $A$ and $B$ are terms, then so is $(AB)$;
• If $A$ and $B$ are terms and $x$ is a variable, then the following are also terms:
• $(\lambda x:A.B)$,
• $(\forall x:A.B)$.
In other words, the term syntax, in BNF, is then:
$e::=\mathbf {T} \mid \mathbf {P} \mid x\mid e\,e\mid \lambda x{\mathbin {:}}e.e\mid \forall x{\mathbin {:}}e.e$
The calculus of constructions has five kinds of objects:
1. proofs, which are terms whose types are propositions;
2. propositions, which are also known as small types;
3. predicates, which are functions that return propositions;
4. large types, which are the types of predicates ($\mathbf {P} $ is an example of a large type);
5. $\mathbf {T} $ itself, which is the type of large types.
Judgments
The calculus of constructions allows proving typing judgments:
$x_{1}:A_{1},x_{2}:A_{2},\ldots \vdash t:B$
Which can be read as the implication
If variables $x_{1},x_{2},\ldots $ have, respectively, types $A_{1},A_{2},\ldots $, then term $t$ has type $B$.
The valid judgments for the calculus of constructions are derivable from a set of inference rules. In the following, we use $\Gamma $ to mean a sequence of type assignments $x_{1}:A_{1},x_{2}:A_{2},\ldots $; $A,B,C,D$ to mean terms; and $K,L$ to mean either $\mathbf {P} $ or $\mathbf {T} $. We shall write $B[x:=N]$ to mean the result of substituting the term $N$ for the free variable $x$ in the term $B$.
An inference rule is written in the form
${\frac {\Gamma \vdash A:B}{\Gamma '\vdash C:D}}$
which means
If $\Gamma \vdash A:B$ is a valid judgment, then so is $\Gamma '\vdash C:D$
Inference rules for the calculus of constructions
1. ${{} \over \Gamma \vdash \mathbf {P} :\mathbf {T} }$ :\mathbf {T} }}
2. ${{} \over {\Gamma ,x:A,\Gamma '\vdash x:A}}$
3. ${\Gamma \vdash A:K\qquad \qquad \Gamma ,x:A\vdash B:L \over {\Gamma \vdash (\forall x:A.B):L}}$
4. ${\Gamma \vdash A:K\qquad \qquad \Gamma ,x:A\vdash N:B \over {\Gamma \vdash (\lambda x:A.N):(\forall x:A.B)}}$
5. ${\Gamma \vdash M:(\forall x:A.B)\qquad \qquad \Gamma \vdash N:A \over {\Gamma \vdash MN:B[x:=N]}}$
6. ${\Gamma \vdash M:A\qquad \qquad A=_{\beta }B\qquad \qquad \Gamma \vdash B:K \over {\Gamma \vdash M:B}}$
Defining logical operators
The calculus of constructions has very few basic operators: the only logical operator for forming propositions is $\forall $. However, this one operator is sufficient to define all the other logical operators:
${\begin{array}{ccll}A\Rightarrow B&\equiv &\forall x:A.B&(x\notin B)\\A\wedge B&\equiv &\forall C:\mathbf {P} .(A\Rightarrow B\Rightarrow C)\Rightarrow C&\\A\vee B&\equiv &\forall C:\mathbf {P} .(A\Rightarrow C)\Rightarrow (B\Rightarrow C)\Rightarrow C&\\\neg A&\equiv &\forall C:\mathbf {P} .(A\Rightarrow C)&\\\exists x:A.B&\equiv &\forall C:\mathbf {P} .(\forall x:A.(B\Rightarrow C))\Rightarrow C&\end{array}}$
Defining data types
The basic data types used in computer science can be defined within the calculus of constructions:
Booleans
$\forall A:\mathbf {P} .A\Rightarrow A\Rightarrow A$
Naturals
$\forall A:\mathbf {P} .(A\Rightarrow A)\Rightarrow (A\Rightarrow A)$
Product $A\times B$
$A\wedge B$
Disjoint union $A+B$
$A\vee B$
Note that Booleans and Naturals are defined in the same way as in Church encoding. However, additional problems arise from propositional extensionality and proof irrelevance.[3]
See also
• Pure type system
• Lambda cube
• System F
• Dependent type
• Intuitionistic type theory
• Homotopy type theory
References
1. Calculus of Inductive Constructions, and basic standard libraries : Datatypes and Logic.
2. Coquand, Thierry; Gallier, Jean H. (July 1990). "A Proof of Strong Normalization for the Theory of Constructions Using a Kripke-Like Interpretation". Technical Reports (Cis): 14.
3. "Standard Library | The Coq Proof Assistant". coq.inria.fr. Retrieved 2020-08-08.
Sources
• Coquand, Thierry; Huet, Gérard (1988). "The Calculus of Constructions" (PDF). Information and Computation. 76 (2–3): 95–120. doi:10.1016/0890-5401(88)90005-3.
• Also available freely accessible online: Coquand, Thierry; Huet, Gérard (1986). The calculus of constructions (Technical report). INRIA, Centre de Rocquencourt. 530.
Note terminology is rather different. For instance, ($\forall x:A.B$) is written [x : A] B.
• Bunder, M. W.; Seldin, Jonathan P. (2004). "Variants of the Basic Calculus of Constructions". CiteSeerx: 10.1.1.88.9497.
• Frade, Maria João (2009). "Calculus of Inductive Constructions" (PDF). Archived from the original (talk) on 2014-05-29. Retrieved 2013-03-03.
• Huet, Gérard (1988). "Induction Principles Formalized in the Calculus of Constructions" (PDF). In Fuchi, K.; Nivat, M. (eds.). Programming of Future Generation Computers. North-Holland. pp. 205–216. ISBN 0444704108. Archived from the original (PDF) on 2015-07-01. — An application of the CoC
| Wikipedia |
The least common multiple of $1!+2!$, $2!+3!$, $3!+4!$, $4!+5!$, $5!+6!$, $6!+7!$, $7!+8!$, and $8!+9!$ can be expressed in the form $a\cdot b!$, where $a$ and $b$ are integers and $b$ is as large as possible. What is $a+b$?
Note that we can factor $n!+(n+1)!$ as $n!\cdot [1+(n+1)] = n!\cdot(n+2)$. Thus, we have \begin{align*}
1!+2! &= 1!\cdot 3 \\
2!+3! &= 2!\cdot 4 \\
3!+4! &= 3!\cdot 5 \\
4!+5! &= 4!\cdot 6 \\
5!+6! &= 5!\cdot 7 \\
6!+7! &= 6!\cdot 8 \\
7!+8! &= 7!\cdot 9 \\
8!+9! &= 8!\cdot 10
\end{align*}The last two numbers are $9\cdot 7!$ and $(8\cdot 10)\cdot 7!$, so their least common multiple is equal to $\mathop{\text{lcm}}[9,8\cdot 10]\cdot 7!$. Since $9$ and $8\cdot 10$ are relatively prime, we have $\mathop{\text{lcm}}[9,8\cdot 10] = 9\cdot 8\cdot 10$, and so $$\mathop{\text{lcm}}[7!+8!,8!+9!] = 9\cdot 8\cdot 10\cdot 7! = 10!.$$Finally, we note that all the other numbers in our list ($1!+2!,2!+3!,\ldots,6!+7!$) are clearly divisors of $10!$. So, the least common multiple of all the numbers in our list is $10!$. Writing this in the form specified in the problem, we get $1\cdot 10!$, so $a=1$ and $b=10$ and their sum is $\boxed{11}$. | Math Dataset |
\begin{definition}[Definition:Parametric Operator]
Let $A$ be an index set.
Let $\Omega$ be a indexed family of operators $\left({\Omega_{\alpha} }\right)_A = \left\{ {\Omega_\alpha: \alpha \in A}\right\}$ indexed by the index $\alpha$.
{{explain|Definition of "operator" in this context.}}
Then $\Omega$ is a '''parametric operator''' with '''parameter $\alpha$''' in the '''parameter set $A$'''.
Category:Definitions/Set Theory
\end{definition} | ProofWiki |
Quasicontraction semigroup
In mathematical analysis, a C0-semigroup Γ(t), t ≥ 0, is called a quasicontraction semigroup if there is a constant ω such that ||Γ(t)|| ≤ exp(ωt) for all t ≥ 0. Γ(t) is called a contraction semigroup if ||Γ(t)|| ≤ 1 for all t ≥ 0.
See also
• Contraction (operator theory)
• Hille–Yosida theorem
• Lumer–Phillips theorem
References
• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. xiv+434. ISBN 0-387-00444-0. MR2028503
| Wikipedia |
With subtle effects, we need a lot of data, so we want at least half a year (6 blocks) or better yet, a year (12 blocks); this requires 180 actives and 180 placebos. This is easily covered by $11 for Doctor's Best Best Lithium Orotate (5mg), 200-Count (more precisely, Lithium 5mg (from 125mg of lithium orotate)) and $14 for 1000x1g empty capsules (purchased February 2012). For convenience I settled on 168 lithium & 168 placebos (7 pill-machine batches, 14 batches total); I can use them in 24 paired blocks of 7-days/1-week each (48 total blocks/48 weeks). The lithium expiration date is October 2014, so that is not a problem
Regardless of your goal, there is a supplement that can help you along the way. Below, we've put together the definitive smart drugs list for peak mental performance. There are three major groups of smart pills and cognitive enhancers. We will cover each one in detail in our list of smart drugs. They are natural and herbal nootropics, prescription ADHD medications, and racetams and synthetic nootropics.
One item always of interest to me is sleep; a stimulant is no good if it damages my sleep (unless that's what it is supposed to do, like modafinil) - anecdotes and research suggest that it does. Over the past few days, my Zeo sleep scores continued to look normal. But that was while not taking nicotine much later than 5 PM. In lieu of a different ml measurer to test my theory that my syringe is misleading me, I decide to more directly test nicotine's effect on sleep by taking 2ml at 10:30 PM, and go to bed at 12:20; I get a decent ZQ of 94 and I fall asleep in 16 minutes, a bit below my weekly average of 19 minutes. The next day, I take 1ml directly before going to sleep at 12:20; the ZQ is 95 and time to sleep is 14 minutes.
The evidence? Found helpful in reducing bodily twitching in myoclonus epilepsy, a rare disorder, but otherwise little studied. Mixed evidence from a study published in 1991 suggests it may improve memory in subjects with cognitive impairment. A meta-analysis published in 2010 that reviewed studies of piracetam and other racetam drugs found that piracetam was somewhat helpful in improving cognition in people who had suffered a stroke or brain injury; the drugs' effectiveness in treating depression and reducing anxiety was more significant.
With this experiment, I broke from the previous methodology, taking the remaining and final half Nuvigil at midnight. I am behind on work and could use a full night to catch up. By 8 AM, I am as usual impressed by the Nuvigil - with Modalert or something, I generally start to feel down by mid-morning, but with Nuvigil, I feel pretty much as I did at 1 AM. Sleep: 9:51/9:15/8:27
Finally, it's not clear that caffeine results in performance gains after long-term use; homeostasis/tolerance is a concern for all stimulants, but especially for caffeine. It is plausible that all caffeine consumption does for the long-term chronic user is restore performance to baseline. (Imagine someone waking up and drinking coffee, and their performance improves - well, so would the performance of a non-addict who is also slowly waking up!) See for example, James & Rogers 2005, Sigmon et al 2009, and Rogers et al 2010. A cross-section of thousands of participants in the Cambridge brain-training study found caffeine intake showed negligible effect sizes for mean and component scores (participants were not told to use caffeine, but the training was recreational & difficult, so one expects some difference).
Since coffee drinking may lead to a worsening of calcium balance in humans, we studied the serial changes of serum calcium, PTH, 1,25-dihydroxyvitamin D (1,25(OH)2D) vitamin D and calcium balance in young and adult rats after daily administration of caffeine for 4 weeks. In the young rats, there was an increase in urinary calcium and endogenous fecal calcium excretion after four days of caffeine administration that persisted for the duration of the experiment. Serum calcium decreased on the fourth day of caffeine administration and then returned to control levels. In contrast, the serum PTH and 1,25(OH)2D remained unchanged initially, but increased after 2 weeks of caffeine administration…In the adult rat group, an increase in the urinary calcium and endogenous fecal calcium excretion and serum levels of PTH was found after caffeine administration. However, the serum 1,25(OH)2D levels and intestinal absorption coefficient of calcium remained the same as in the adult control group.
Organizations, and even entire countries, are struggling with "always working" cultures. Germany and France have adopted rules to stop employees from reading and responding to email after work hours. Several companies have explored banning after-hours email; when one Italian company banned all email for one week, stress levels dropped among employees. This is not a great surprise: A Gallup study found that among those who frequently check email after working hours, about half report having a lot of stress.
Another classic approach to the assessment of working memory is the span task, in which a series of items is presented to the subject for repetition, transcription, or recognition. The longest series that can be reproduced accurately is called the forward span and is a measure of working memory capacity. The ability to reproduce the series in reverse order is tested in backward span tasks and is a more stringent test of working memory capacity and perhaps other working memory functions as well. The digit span task from the Wechsler (1981) IQ test was used in four studies of stimulant effects on working memory. One study showed that d-AMP increased digit span (de Wit et al., 2002), and three found no effects of d-AMP or MPH (Oken, Kishiyama, & Salinsky, 1995; Schmedtje, Oman, Letz, & Baker, 1988; Silber, Croft, Papafotiou, & Stough, 2006). A spatial span task, in which subjects must retain and reproduce the order in which boxes in a scattered spatial arrangement change color, was used by Elliott et al. (1997) to assess the effects of MPH on working memory. For subjects in the group receiving placebo first, MPH increased spatial span. However, for the subjects who received MPH first, there was a nonsignificant opposite trend. The group difference in drug effect is not easily explained. The authors noted that the subjects in the first group performed at an overall lower level, and so, this may be another manifestation of the trend for a larger enhancement effect for less able subjects.
One idea I've been musing about is the connections between IQ, Conscientiousness, and testosterone. IQ and Conscientiousness do not correlate to a remarkable degree - even though one would expect IQ to at least somewhat enable a long-term perspective, self-discipline, metacognition, etc! There are indications in studies of gifted youth that they have lower testosterone levels. The studies I've read on testosterone indicate no improvements to raw ability. So, could there be a self-sabotaging aspect to human intelligence whereby greater intelligence depends on lack of testosterone, but this same lack also holds back Conscientiousness (despite one's expectation that intelligence would produce greater self-discipline and planning), undermining the utility of greater intelligence? Could cases of high IQ types who suddenly stop slacking and accomplish great things sometimes be due to changes in testosterone? Studies on the correlations between IQ, testosterone, Conscientiousness, and various measures of accomplishment are confusing and don't always support this theory, but it's an idea to keep in mind.
This would be a very time-consuming experiment. Any attempt to combine this with other experiments by ANOVA would probably push the end-date out by months, and one would start to be seriously concerned that changes caused by aging or environmental factors would contaminate the results. A 5-year experiment with 7-month intervals will probably eat up 5+ hours to prepare <12,000 pills (active & placebo); each switch and test of mental functioning will probably eat up another hour for 32 hours. (And what test maintains validity with no practice effects over 5 years? Dual n-back would be unusable because of improvements to WM over that period.) Add in an hour for analysis & writeup, that suggests >38 hours of work, and 38 \times 7.25 = 275.5. 12,000 pills is roughly $12.80 per thousand or $154; 120 potassium iodide pills is ~$9, so \frac{365.25}{120} \times 9 \times 5 = 137.
Oxiracetam is one of the 3 most popular -racetams; less popular than piracetam but seems to be more popular than aniracetam. Prices have come down substantially since the early 2000s, and stand at around 1.2g/$ or roughly 50 cents a dose, which was low enough to experiment with; key question, does it stack with piracetam or is it redundant for me? (Oxiracetam can't compete on price with my piracetam pile stockpile: the latter is now a sunk cost and hence free.)
An entirely different set of questions concerns cognitive enhancement in younger students, including elementary school and even preschool children. Some children can function adequately in school without stimulants but perform better with them; medicating such children could be considered a form of cognitive enhancement. How often does this occur? What are the roles and motives of parents, teachers, and pediatricians in these cases? These questions have been discussed elsewhere and deserve continued attention (Diller, 1996; Singh & Keller, 2010).
Table 5 lists the results of 16 tasks from 13 articles on the effects of d-AMP or MPH on cognitive control. One of the simplest tasks used to study cognitive control is the go/no-go task. Subjects are instructed to press a button as quickly as possible for one stimulus or class of stimuli (go) and to refrain from pressing for another stimulus or class of stimuli (no go). De Wit et al. (2002) used a version of this task to measure the effects of d-AMP on subjects' ability to inhibit a response and found enhancement in the form of decreased false alarms (responses to no-go stimuli) and increased speed of correct go responses. They also found that subjects who made the most errors on placebo experienced the greatest enhancement from the drug.
I've been actively benefitting from nootropics since 1997, when I was struggling with cognitive performance and ordered almost $1000 worth of smart drugs from Europe (the only place where you could get them at the time). I remember opening the unmarked brown package and wondering whether the pharmaceuticals and natural substances would really enhance my brain.
Going back to the 1960s, although it was a Romanian chemist who is credited with discovering nootropics, a substantial amount of research on racetams was conducted in the Soviet Union. This resulted in the birth of another category of substances entirely: adaptogens, which, in addition to benefiting cognitive function were thought to allow the body to better adapt to stress.
Another important epidemiological question about the use of prescription stimulants for cognitive enhancement concerns the risk of dependence. MPH and d-AMP both have high potential for abuse and addiction related to their effects on brain systems involved in motivation. On the basis of their reanalysis of NSDUH data sets from 2000 to 2002, Kroutil and colleagues (2006) estimated that almost one in 20 nonmedical users of prescription ADHD medications meets criteria for dependence or abuse. This sobering estimate is based on a survey of all nonmedical users. The immediate and long-term risks to individuals seeking cognitive enhancement remain unknown.
Rogers RD, Blackshaw AJ, Middleton HC, Matthews K, Hawtin K, Crowley C, Robbins TW. Tryptophan depletion impairs stimulus-reward learning while methylphenidate disrupts attentional control in healthy young adults: Implications for the monoaminergic basis of impulsive behaviour. Psychopharmacology. 1999;146:482–491. doi: 10.1007/PL00005494. [PubMed] [CrossRef]
At this point I began to get bored with it and the lack of apparent effects, so I began a pilot trial: I'd use the LED set for 10 minutes every few days before 2PM, record, and in a few months look for a correlation with my daily self-ratings of mood/productivity (for 2.5 years I've asked myself at the end of each day whether I did more, the usual, or less work done that day than average, so 2=below-average, 3=average, 4=above-average; it's ad hoc, but in some factor analyses I've been playing with, it seems to load on a lot of other variables I've measured, so I think it's meaningful).
Nor am I sure how important the results are - partway through, I haven't noticed anything bad, at least, from taking Noopept. And any effect is going to be subtle: people seem to think that 10mg is too small for an ingested rather than sublingual dose and I should be taking twice as much, and Noopept's claimed to be a chronic gradual sort of thing, with less of an acute effect. If the effect size is positive, regardless of statistical-significance, I'll probably think about doing a bigger real self-experiment (more days blocked into weeks or months & 20mg dose)
The use of cognitive enhancers by healthy individuals sparked debate about ethics and safety. Cognitive enhancement by pharmaceutical means was considered a form of illicit drug use in some places, even while other cognitive enhancers, such as caffeine and nicotine, were freely available. The conflict therein raised the possibility for further acceptance of smart drugs in the future. However, the long-term effects of smart drugs on otherwise healthy brains were unknown, delaying safety assessments.
Caffeine dose dependently decreased the 1,25(OH)(2)D(3) induced VDR expression and at concentrations of 1 and 10mM, VDR expression was decreased by about 50-70%, respectively. In addition, the 1,25(OH)(2)D(3) induced alkaline phosphatase activity was also reduced at similar doses thus affecting the osteoblastic function. The basal ALP activity was not affected with increasing doses of caffeine. Overall, our results suggest that caffeine affects 1,25(OH)(2)D(3) stimulated VDR protein expression and 1,25(OH)(2)D(3) mediated actions in human osteoblast cells.
Scientists found that the drug can disrupt the way memories are stored. This ability could be invaluable in treating trauma victims to prevent associated stress disorders. The research has also triggered suggestions that licensing these memory-blocking drugs may lead to healthy people using them to erase memories of awkward conversations, embarrassing blunders and any feelings for that devious ex-girlfriend.
Stimulants are the smart drugs most familiar to people, starting with widely-used psychostimulants caffeine and nicotine, and the more ill-reputed subclass of amphetamines. Stimulant drugs generally function as smart drugs in the sense that they promote general wakefulness and put the brain and body "on alert" in a ready-to-go state. Basically, any drug whose effects reduce drowsiness will increase the functional IQ, so long as the user isn't so over-stimulated they're shaking or driven to distraction.
This continued up to 1 AM, at which point I decided not to take a second armodafinil (why spend a second pill to gain what would likely be an unproductive set of 8 hours?) and finish up the experiment with some n-backing. My 5 rounds: 60/38/62/44/5023. This was surprising. Compare those scores with scores from several previous days: 39/42/44/40/20/28/36. I had estimated before the n-backing that my scores would be in the low-end of my usual performance (20-30%) since I had not slept for the past 41 hours, and instead, the lowest score was 38%. If one did not know the context, one might think I had discovered a good nootropic! Interesting evidence that armodafinil preserves at least one kind of mental performance.
Upon examining the photographs, I noticed no difference in eye color, but it seems that my move had changed the ambient lighting in the morning and so there was a clear difference between the two sets of photographs! The before photographs had brighter lighting than the after photographs. Regardless, I decided to run a small survey on QuickSurveys/Toluna to confirm my diagnosis of no-change; the survey was 11 forced-choice pairs of photographs (before-after), with the instructions as follows:
The effect? 3 or 4 weeks later, I'm not sure. When I began putting all of my nootropic powders into pill-form, I put half a lithium pill in each, and nevertheless ran out of lithium fairly quickly (3kg of piracetam makes for >4000 OO-size pills); those capsules were buried at the bottom of the bucket under lithium-less pills. So I suddenly went cold-turkey on lithium. Reflecting on the past 2 weeks, I seem to have been less optimistic and productive, with items now lingering on my To-Do list which I didn't expect to. An effect? Possibly.
As already mentioned, AMPs and MPH are classified by the U.S. Food and Drug Administration (FDA) as Schedule II substances, which means that buying or selling them is a felony offense. This raises the question of how the drugs are obtained by students for nonmedical use. Several studies addressed this question and yielded reasonably consistent answers.
Historically used to help people with epilepsy, piracetam is used in some cases of myoclonus, or muscle twitching. Its actual mechanism of action is unclear: It doesn't act exactly as a sedative or stimulant, but still influences cognitive function, and is believed to act on receptors for acetylcholine in the brain. Piracetam is used off-label as a 'smart drug' to help focus and concentration or sometimes as a way to allegedly boost your mood. Again, piracetam is a prescription-only drug - any supply to people without a prescription is illegal, and supplying it may result in a fine or prison sentence.
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Many people find that they experience increased "brain fog" as they age, some of which could be attributed to early degeneration of synapses and neural pathways. Some drugs have been found to be useful for providing cognitive improvements in these individuals. It's possible that these supplements could provide value by improving brain plasticity and supporting the regeneration of cells.10
Another common working memory task is the n-back task, which requires the subject to view a series of items (usually letters) and decide whether the current item is identical to the one presented n items back. This task taxes working memory because the previous items must be held in working memory to be compared with the current item. The easiest version of this is a 1-back task, which is also called a double continuous performance task (CPT) because the subject is continuously monitoring for a repeat or double. Three studies examined the effects of MPH on working memory ability as measured by the 1-back task, and all found enhancement of performance in the form of reduced errors of omission (Cooper et al., 2005; Klorman et al., 1984; Strauss et al., 1984). Fleming et al. (1995) tested the effects of d-AMP on a 5-min CPT and found a decrease in reaction time, but did not specify which version of the CPT was used.
AMP was first investigated as an asthma medication in the 1920s, but its psychological effects were soon noticed. These included increased feelings of energy, positive mood, and prolonged physical endurance and mental concentration. These effects have been exploited in a variety of medical and nonmedical applications in the years since they were discovered, including to treat depression, to enhance alertness in military personnel, and to provide a competitive edge in athletic competition (Rasmussen, 2008). Today, AMP remains a widely used and effective treatment for ADHD (Wilens, 2006).
Fish oil (Examine.com, buyer's guide) provides benefits relating to general mood (eg. inflammation & anxiety; see later on anxiety) and anti-schizophrenia; it is one of the better supplements one can take. (The known risks are a higher rate of prostate cancer and internal bleeding, but are outweighed by the cardiac benefits - assuming those benefits exist, anyway, which may not be true.) The benefits of omega acids are well-researched.
By which I mean that simple potassium is probably the most positively mind altering supplement I've ever tried…About 15 minutes after consumption, it manifests as a kind of pressure in the head or temples or eyes, a clearing up of brain fog, increased focus, and the kind of energy that is not jittery but the kind that makes you feel like exercising would be the reasonable and prudent thing to do. I have done no tests, but feel smarter from this in a way that seems much stronger than piracetam or any of the conventional weak nootropics. It is not just me – I have been introducing this around my inner social circle and I'm at 7/10 people felt immediately noticeable effects. The 3 that didn't notice much were vegetarians and less likely to have been deficient. Now that I'm not deficient, it is of course not noticeable as mind altering, but still serves to be energizing, particularly for sustained mental energy as the night goes on…Potassium chloride initially, but since bought some potassium gluconate pills… research indicates you don't want to consume large amounts of chloride (just moderate amounts).
Related to the famous -racetams but reportedly better (and much less bulky), Noopept is one of the many obscure Russian nootropics. (Further reading: Google Scholar, Examine.com, Reddit, Longecity, Bluelight.ru.) Its advantages seem to be that it's far more compact than piracetam and doesn't taste awful so it's easier to store and consume; doesn't have the cloud hanging over it that piracetam does due to the FDA letters, so it's easy to purchase through normal channels; is cheap on a per-dose basis; and it has fans claiming it is better than piracetam.
Clarke and Sokoloff (1998) remarked that although [a] common view equates concentrated mental effort with mental work…there appears to be no increased energy utilization by the brain during such processes (p. 664), and …the areas that participate in the processes of such reasoning represent too small a fraction of the brain for changes in their functional and metabolic activities to be reflected in the energy metabolism of the brain… (p. 675).
I have no particularly compelling story for why this might be a correlation and not causation. It could be placebo, but I wasn't expecting that. It could be selection effect (days on which I bothered to use the annoying LED set are better days) but then I'd expect the off-days to be below-average and compared to the 2 years of trendline before, there doesn't seem like much of a fall.
Smart drugs could lead to enhanced cognitive abilities in the military. Also known as nootropics, smart drugs can be viewed similarly to medical enhancements. What's important to remember though, is that smart drugs do not increase your intelligence; however, they may improve cognitive and executive functions leading to an increase in intelligence.
Between midnight and 1:36 AM, I do four rounds of n-back: 50/39/30/55%. I then take 1/4th of the pill and have some tea. At roughly 1:30 AM, AngryParsley linked a SF anthology/novel, Fine Structure, which sucked me in for the next 3-4 hours until I finally finished the whole thing. At 5:20 AM, circumstances forced me to go to bed, still having only taken 1/4th of the pill and that determines this particular experiment of sleep; I quickly do some n-back: 29/20/20/54/42. I fall asleep in 13 minutes and sleep for 2:48, for a ZQ of 28 (a full night being ~100). I did not notice anything from that possible modafinil+caffeine interaction. Subjectively upon awakening: I don't feel great, but I don't feel like 2-3 hours of sleep either. N-back at 10 AM after breakfast: 25/54/44/38/33. These are not very impressive, but seem normal despite taking the last armodafinil ~9 hours ago; perhaps the 3 hours were enough. Later that day, at 11:30 PM (just before bed): 26/56/47.
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What is three dimensional aromaticity?
I have recently come across a statement which states:
The relative stability of the 1,3-dehydro-5,7-adamantanediyl dication is ascribed to its three-dimensional aromaticity. [1]
My understanding of aromatic nature is, that the molecule must be cyclic and planar, every atom in the ring must be $\ce{sp^2}$-hybridized and the molecule must have [4n+2] $\pi$ electrons, where $n\in\mathbb{N}$ (Hückel's rule). The molecular orbitals also have to have the correct orientation and phase for aromaticity (2).
In my understanding 1,3-dehydro-5,7-adamantanediyl dication does not follow these rules for Y-aromaticity (3).
So my question is: What is three-dimensional aromaticity and how does it stabilise the 1,3-dehydro-5,7-adamantanediyl dication?
Firme, C. L.; Antunes, O. A. C.; Esteves, P. M. Electronic nature of the aromatic adamantanediyl ions and its analogues. J. Braz. Chem. Soc. 2008, 19 (1), 140–149. DOI: 10.1590/S0103-50532008000100020.
LibreTexts
What is Y-aromaticity? Is the trinitromethanide anion aromatic?
organic-chemistry aromaticity
Gaurang Tandon
Chakravarthy KalyanChakravarthy Kalyan
$\begingroup$ en.wikipedia.org/wiki/Adamantane#Adamantane_cations $\endgroup$
– Mithoron
$\begingroup$ Two more contributing structures, for a total of six. There is one with a bond on each edge of the "aromatic" tetrahedron. $\endgroup$
– Oscar Lanzi
$\begingroup$ This 3d-aromaticity has nothing to do with regular Hückel-aromaticity. The analogy is that thoses structures are mesomeric, that is they do not describe actual, discernible molecular structures. The inner bond of that three-ring, like the double bonds in benzene, does not exist as is, but is delocalised over all possible positions. $\endgroup$
Aromaticity in Organic chemistry textbooks covers the Huckel version, which abide by the rules you have correctly listed above. However, these rules are mainly a tool to identify an organic molecule which has a high delocalization of electrons across a ring structure (a valuable tool in identifying a molecule's stability/reactivity/etc...).
So, for 3-dimensional aromaticity, the molecule must reach a certain threshold of delocalization in the ring structure (which, in 3D, becomes more of a sphere). Rules such as trigonal planar hybridization in 3D are no longer useful, and thus different approaches are taken to measure the resonance state of a system, and thus its level of aromaticity.
Nozawa et al. (2016, cited below) discusses use of "[t]he bond length alternation (BLA) in cyclic compounds" as "a good indicator of aromaticity, as it allows an evaluation of the degree of effective $\pi$-electron delocalization." These values, using a "harmonic oscillator model of aromaticity (HOMA)," are close to 1 for aromatic molecules by definition, and Nozawa obtains these from a variety of experimental and computational techniques to compare aromaticity in normally anti-aromatic structures.
For your above structure, as some comments pointed out as well, they are mesomeric, or resonance structures of the molecule illustrating the ability for the electrons to delocalize to an aromatic state. Firme et al's article mentions "delocalization indexes among bridged atoms," what I assume to be yet another similar measure of aromaticity related to the BLA using HOMA quantification.
Hope this helps! Please let me know if there is any confusion, as this response is a bit packed with summary detail.
References: 1. Nozawa, R. et al. Stacked antiaromatic porphyrins. Nature Communications 7, 13620 (2016).
Len_spragueLen_sprague
$\begingroup$ By the way, are you a journalist or translator or so? The use of square brackets to enclose very minor text change in the citation seems rather confusing than useful here, in my opinion. $\endgroup$
– mykhal
$\begingroup$ @mykhal, nope. Chemist and avid writer. It's simply the appropriate form for in-text citations when slightly modifying the punctuation and/or word choice to fit the sentence flow. Note is appreciated, but I don't believe there is any loss of meaning to be had. $\endgroup$
– Len_sprague
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Greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted $\gcd(x,y)$. For example, the GCD of 8 and 12 is 4, that is, $\gcd(8,12)=4$.[1][2]
In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc.[3][4][5][6] Historically, other names for the same concept have included greatest common measure.[7]
This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see § In commutative rings below).
Overview
Definition
The greatest common divisor (GCD) of two nonzero integers a and b is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer. The GCD of a and b is generally denoted gcd(a, b).[8]
This definition also applies when one of a and b is zero. In this case, the GCD is the absolute value of the non zero integer: gcd(a, 0) = gcd(0, a) = |a|. This case is important as the terminating step of the Euclidean algorithm.
The above definition cannot be used for defining gcd(0, 0), since 0 × n = 0, and zero thus has no greatest divisor. However, zero is its own greatest divisor if greatest is understood in the context of the divisibility relation, so gcd(0, 0) is commonly defined as 0. This preserves the usual identities for GCD, and in particular Bézout's identity, namely that gcd(a, b) generates the same ideal as {a, b}.[9][10][11] This convention is followed by many computer algebra systems.[12] Nonetheless, some authors leave gcd(0, 0) undefined.[13]
The GCD of a and b is their greatest positive common divisor in the preorder relation of divisibility. This means that the common divisors of a and b are exactly the divisors of their GCD. This is commonly proved by using either Euclid's lemma, the fundamental theorem of arithmetic, or the Euclidean algorithm. This is the meaning of "greatest" that is used for the generalizations of the concept of GCD.
Example
The number 54 can be expressed as a product of two integers in several different ways:
$54\times 1=27\times 2=18\times 3=9\times 6.$
Thus the complete list of divisors of 54 is $1,2,3,6,9,18,27,54$. Similarly, the divisors of 24 are $1,2,3,4,6,8,12,24$. The numbers that these two lists have in common are the common divisors of 54 and 24, that is,
$1,2,3,6.$
Of these, the greatest is 6, so it is the greatest common divisor:
$\gcd(54,24)=6.$
Computing all divisors of the two numbers in this way is usually not efficient, especially for large numbers that have many divisors. Much more efficient methods are described in § Calculation.
Coprime numbers
Main article: Coprime integers
Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1.[14] For example, 9 and 28 are coprime.
A geometric view
For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can thus be divided into a grid of 12-by-12 squares, with two squares along one edge (24/12 = 2) and five squares along the other (60/12 = 5).
Applications
Reducing fractions
Further information: Irreducible fraction
The greatest common divisor is useful for reducing fractions to the lowest terms.[15] For example, gcd(42, 56) = 14, therefore,
${\frac {42}{56}}={\frac {3\cdot 14}{4\cdot 14}}={\frac {3}{4}}.$
Least common multiple
Further information: Least common multiple
The least common multiple of two integers that are not both zero can be computed from their greatest common divisor, by using the relation
$\operatorname {lcm} (a,b)={\frac {|a\cdot b|}{\operatorname {gcd} (a,b)}}.$
Calculation
Using prime factorizations
Greatest common divisors can be computed by determining the prime factorizations of the two numbers and comparing factors. For example, to compute gcd(48, 180), we find the prime factorizations 48 = 24 · 31 and 180 = 22 · 32 · 51; the GCD is then 2min(4,2) · 3min(1,2) · 5min(0,1) = 22 · 31 · 50 = 12 The corresponding LCM is then 2max(4,2) · 3max(1,2) · 5max(0,1) = 24 · 32 · 51 = 720.
In practice, this method is only feasible for small numbers, as computing prime factorizations takes too long.
Euclid's algorithm
Main article: Euclidean algorithm
The method introduced by Euclid for computing greatest common divisors is based on the fact that, given two positive integers a and b such that a > b, the common divisors of a and b are the same as the common divisors of a – b and b.
So, Euclid's method for computing the greatest common divisor of two positive integers consists of replacing the larger number by the difference of the numbers, and repeating this until the two numbers are equal: that is their greatest common divisor.
For example, to compute gcd(48,18), one proceeds as follows:
${\begin{aligned}\gcd(48,18)\quad &\to \quad \gcd(48-18,18)=\gcd(30,18)&&\to \quad \gcd(30-18,18)=\gcd(12,18)\\&\to \quad \gcd(12,18-12)=\gcd(12,6)&&\to \quad \gcd(12-6,6)=\gcd(6,6).\end{aligned}}$
So gcd(48, 18) = 6.
This method can be very slow if one number is much larger than the other. So, the variant that follows is generally preferred.
Euclidean algorithm
Main article: Euclidean algorithm
A more efficient method is the Euclidean algorithm, a variant in which the difference of the two numbers a and b is replaced by the remainder of the Euclidean division (also called division with remainder) of a by b.
Denoting this remainder as a mod b, the algorithm replaces (a, b) by (b, a mod b) repeatedly until the pair is (d, 0), where d is the greatest common divisor.
For example, to compute gcd(48,18), the computation is as follows:
${\begin{aligned}\gcd(48,18)\quad &\to \quad \gcd(18,48{\bmod {1}}8)=\gcd(18,12)\\&\to \quad \gcd(12,18{\bmod {1}}2)=\gcd(12,6)\\&\to \quad \gcd(6,12{\bmod {6}})=\gcd(6,0).\end{aligned}}$
This again gives gcd(48, 18) = 6.
Lehmer's GCD algorithm
Main article: Lehmer's GCD algorithm
Lehmer's algorithm is based on the observation that the initial quotients produced by Euclid's algorithm can be determined based on only the first few digits; this is useful for numbers that are larger than a computer word. In essence, one extracts initial digits, typically forming one or two computer words, and runs Euclid's algorithms on these smaller numbers, as long as it is guaranteed that the quotients are the same with those that would be obtained with the original numbers. The quotients are collected into a small 2-by-2 transformation matrix (a matrix of single-word integers) to reduce the original numbers. This process is repeated until numbers are small enough that the binary algorithm (see below) is more efficient.
This algorithm improves speed, because it reduces the number of operations on very large numbers, and can use hardware arithmetic for most operations. In fact, most of the quotients are very small, so a fair number of steps of the Euclidean algorithm can be collected in a 2-by-2 matrix of single-word integers. When Lehmer's algorithm encounters a quotient that is too large, it must fall back to one iteration of Euclidean algorithm, with a Euclidean division of large numbers.
Binary GCD algorithm
Main article: Binary GCD algorithm
The binary GCD algorithm uses only subtraction and division by 2. The method is as follows: Let a and b be the two non-negative integers. Let the integer d be 0. There are five possibilities:
• a = b.
As gcd(a, a) = a, the desired GCD is a × 2d (as a and b are changed in the other cases, and d records the number of times that a and b have been both divided by 2 in the next step, the GCD of the initial pair is the product of a and 2d).
• Both a and b are even.
Then 2 is a common divisor. Divide both a and b by 2, increment d by 1 to record the number of times 2 is a common divisor and continue.
• a is even and b is odd.
Then 2 is not a common divisor. Divide a by 2 and continue.
• a is odd and b is even.
Then 2 is not a common divisor. Divide b by 2 and continue.
• Both a and b are odd.
As gcd(a,b) = gcd(b,a), if a < b then exchange a and b. The number c = a − b is positive and smaller than a. Any number that divides a and b must also divide c so every common divisor of a and b is also a common divisor of b and c. Similarly, a = b + c and every common divisor of b and c is also a common divisor of a and b. So the two pairs (a, b) and (b, c) have the same common divisors, and thus gcd(a,b) = gcd(b,c). Moreover, as a and b are both odd, c is even, the process can be continued with the pair (a, b) replaced by the smaller numbers (c/2, b) without changing the GCD.
Each of the above steps reduces at least one of a and b while leaving them non-negative and so can only be repeated a finite number of times. Thus eventually the process results in a = b, the stopping case. Then the GCD is a × 2d.
Example: (a, b, d) = (48, 18, 0) → (24, 9, 1) → (12, 9, 1) → (6, 9, 1) → (3, 9, 1) → (3, 3, 1) ; the original GCD is thus the product 6 of 2d = 21 and a= b= 3.
The binary GCD algorithm is particularly easy to implement on binary computers. Its computational complexity is
$O((\log a+\log b)^{2})$
The computational complexity is usually given in terms of the length n of the input. Here, this length is $n=\log a+\log b,$ and the complexity is thus
$O(n^{2})$.
Other methods
If a and b are both nonzero, the greatest common divisor of a and b can be computed by using least common multiple (LCM) of a and b:
$\gcd(a,b)={\frac {|a\cdot b|}{\operatorname {lcm} (a,b)}}$,
but more commonly the LCM is computed from the GCD.
Using Thomae's function f,
$\gcd(a,b)=af\left({\frac {b}{a}}\right),$
which generalizes to a and b rational numbers or commensurable real numbers.
Keith Slavin has shown that for odd a ≥ 1:
$\gcd(a,b)=\log _{2}\prod _{k=0}^{a-1}(1+e^{-2i\pi kb/a})$
which is a function that can be evaluated for complex b.[16] Wolfgang Schramm has shown that
$\gcd(a,b)=\sum \limits _{k=1}^{a}\exp(2\pi ikb/a)\cdot \sum \limits _{d\left|a\right.}{\frac {c_{d}(k)}{d}}$
is an entire function in the variable b for all positive integers a where cd(k) is Ramanujan's sum.[17]
Complexity
The computational complexity of the computation of greatest common divisors has been widely studied.[18] If one uses the Euclidean algorithm and the elementary algorithms for multiplication and division, the computation of the greatest common divisor of two integers of at most n bits is $O(n^{2}).$ This means that the computation of greatest common divisor has, up to a constant factor, the same complexity as the multiplication.
However, if a fast multiplication algorithm is used, one may modify the Euclidean algorithm for improving the complexity, but the computation of a greatest common divisor becomes slower than the multiplication. More precisely, if the multiplication of two integers of n bits takes a time of T(n), then the fastest known algorithm for greatest common divisor has a complexity $O\left(T(n)\log n\right).$ This implies that the fastest known algorithm has a complexity of $O\left(n\,(\log n)^{2}\right).$
Previous complexities are valid for the usual models of computation, specifically multitape Turing machines and random-access machines.
The computation of the greatest common divisors belongs thus to the class of problems solvable in quasilinear time. A fortiori, the corresponding decision problem belongs to the class P of problems solvable in polynomial time. The GCD problem is not known to be in NC, and so there is no known way to parallelize it efficiently; nor is it known to be P-complete, which would imply that it is unlikely to be possible to efficiently parallelize GCD computation. Shallcross et al. showed that a related problem (EUGCD, determining the remainder sequence arising during the Euclidean algorithm) is NC-equivalent to the problem of integer linear programming with two variables; if either problem is in NC or is P-complete, the other is as well.[19] Since NC contains NL, it is also unknown whether a space-efficient algorithm for computing the GCD exists, even for nondeterministic Turing machines.
Although the problem is not known to be in NC, parallel algorithms asymptotically faster than the Euclidean algorithm exist; the fastest known deterministic algorithm is by Chor and Goldreich, which (in the CRCW-PRAM model) can solve the problem in O(n/log n) time with n1+ε processors.[20] Randomized algorithms can solve the problem in O((log n)2) time on $\exp \left(O\left({\sqrt {n\log n}}\right)\right)$ processors (this is superpolynomial).[21]
Properties
• For positive integers a, gcd(a, a) = a.
• Every common divisor of a and b is a divisor of gcd(a, b).
• gcd(a, b), where a and b are not both zero, may be defined alternatively and equivalently as the smallest positive integer d which can be written in the form d = a⋅p + b⋅q, where p and q are integers. This expression is called Bézout's identity. Numbers p and q like this can be computed with the extended Euclidean algorithm.
• gcd(a, 0) = |a|, for a ≠ 0, since any number is a divisor of 0, and the greatest divisor of a is |a|.[2][5] This is usually used as the base case in the Euclidean algorithm.
• If a divides the product b⋅c, and gcd(a, b) = d, then a/d divides c.
• If m is a positive integer, then gcd(m⋅a, m⋅b) = m⋅gcd(a, b).
• If m is any integer, then gcd(a + m⋅b, b) = gcd(a, b). Equivalently, gcd(a mod b,b) = gcd(a,b).
• If m is a positive common divisor of a and b, then gcd(a/m, b/m) = gcd(a, b)/m.
• The GCD is a commutative function: gcd(a, b) = gcd(b, a).
• The GCD is an associative function: gcd(a, gcd(b, c)) = gcd(gcd(a, b), c). Thus gcd(a, b, c, ...) can be used to denote the GCD of multiple arguments.
• The GCD is a multiplicative function in the following sense: if a1 and a2 are relatively prime, then gcd(a1⋅a2, b) = gcd(a1, b)⋅gcd(a2, b).
• gcd(a, b) is closely related to the least common multiple lcm(a, b): we have
gcd(a, b)⋅lcm(a, b) = |a⋅b|.
This formula is often used to compute least common multiples: one first computes the GCD with Euclid's algorithm and then divides the product of the given numbers by their GCD.
• The following versions of distributivity hold true:
gcd(a, lcm(b, c)) = lcm(gcd(a, b), gcd(a, c))
lcm(a, gcd(b, c)) = gcd(lcm(a, b), lcm(a, c)).
• If we have the unique prime factorizations of a = p1e1 p2e2 ⋅⋅⋅ pmem and b = p1f1 p2f2 ⋅⋅⋅ pmfm where ei ≥ 0 and fi ≥ 0, then the GCD of a and b is
gcd(a,b) = p1min(e1,f1) p2min(e2,f2) ⋅⋅⋅ pmmin(em,fm).
• It is sometimes useful to define gcd(0, 0) = 0 and lcm(0, 0) = 0 because then the natural numbers become a complete distributive lattice with GCD as meet and LCM as join operation.[22] This extension of the definition is also compatible with the generalization for commutative rings given below.
• In a Cartesian coordinate system, gcd(a, b) can be interpreted as the number of segments between points with integral coordinates on the straight line segment joining the points (0, 0) and (a, b).
• For non-negative integers a and b, where a and b are not both zero, provable by considering the Euclidean algorithm in base n:[23]
gcd(na − 1, nb − 1) = ngcd(a,b) − 1.
• An identity involving Euler's totient function:
$\gcd(a,b)=\sum _{k|a{\text{ and }}k|b}\varphi (k).$
• $\sum _{k=1}^{n}\gcd(k,n)=n\prod _{p|n}\left(1+\nu _{p}(n)\left(1-{\frac {1}{p}}\right)\right)$ where $\nu _{p}(n)$ is the p-adic valuation.
Probabilities and expected value
In 1972, James E. Nymann showed that k integers, chosen independently and uniformly from {1, ..., n}, are coprime with probability 1/ζ(k) as n goes to infinity, where ζ refers to the Riemann zeta function.[24] (See coprime for a derivation.) This result was extended in 1987 to show that the probability that k random integers have greatest common divisor d is d−k/ζ(k).[25]
Using this information, the expected value of the greatest common divisor function can be seen (informally) to not exist when k = 2. In this case the probability that the GCD equals d is d−2/ζ(2), and since ζ(2) = π2/6 we have
$\mathrm {E} (\mathrm {2} )=\sum _{d=1}^{\infty }d{\frac {6}{\pi ^{2}d^{2}}}={\frac {6}{\pi ^{2}}}\sum _{d=1}^{\infty }{\frac {1}{d}}.$
This last summation is the harmonic series, which diverges. However, when k ≥ 3, the expected value is well-defined, and by the above argument, it is
$\mathrm {E} (k)=\sum _{d=1}^{\infty }d^{1-k}\zeta (k)^{-1}={\frac {\zeta (k-1)}{\zeta (k)}}.$
For k = 3, this is approximately equal to 1.3684. For k = 4, it is approximately 1.1106.
In commutative rings
See also: divisor (ring theory)
The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.
If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b (that is, if there are elements x and y in R such that d·x = a and d·y = b). If d is a common divisor of a and b, and every common divisor of a and b divides d, then d is called a greatest common divisor of a and b.
With this definition, two elements a and b may very well have several greatest common divisors, or none at all. If R is an integral domain then any two GCD's of a and b must be associate elements, since by definition either one must divide the other; indeed if a GCD exists, any one of its associates is a GCD as well. Existence of a GCD is not assured in arbitrary integral domains. However, if R is a unique factorization domain, then any two elements have a GCD, and more generally this is true in GCD domains. If R is a Euclidean domain in which euclidean division is given algorithmically (as is the case for instance when R = F[X] where F is a field, or when R is the ring of Gaussian integers), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.
The following is an example of an integral domain with two elements that do not have a GCD:
$R=\mathbb {Z} \left[{\sqrt {-3}}\,\,\right],\quad a=4=2\cdot 2=\left(1+{\sqrt {-3}}\,\,\right)\left(1-{\sqrt {-3}}\,\,\right),\quad b=\left(1+{\sqrt {-3}}\,\,\right)\cdot 2.$
The elements 2 and 1 + √−3 are two maximal common divisors (that is, any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + √−3, but they are not associated, so there is no greatest common divisor of a and b.
Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form pa + qb, where p and q range over the ring. This is the ideal generated by a and b, and is denoted simply (a, b). In a ring all of whose ideals are principal (a principal ideal domain or PID), this ideal will be identical with the set of multiples of some ring element d; then this d is a greatest common divisor of a and b. But the ideal (a, b) can be useful even when there is no greatest common divisor of a and b. (Indeed, Ernst Kummer used this ideal as a replacement for a GCD in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.)
See also
• Bézout domain
• Lowest common denominator
• Unitary divisor
Notes
1. Long (1972, p. 33)
2. Pettofrezzo & Byrkit (1970, p. 34)
3. Kelley, W. Michael (2004), The Complete Idiot's Guide to Algebra, Penguin, p. 142, ISBN 978-1-59257-161-1.
4. Jones, Allyn (1999), Whole Numbers, Decimals, Percentages and Fractions Year 7, Pascal Press, p. 16, ISBN 978-1-86441-378-6.
5. Hardy & Wright (1979, p. 20)
6. Some authors treat greatest common denominator as synonymous with greatest common divisor. This contradicts the common meaning of the words that are used, as denominator refers to fractions, and two fractions do not have any greatest common denominator (if two fractions have the same denominator, one obtains a greater common denominator by multiplying all numerators and denominators by the same integer).
7. Barlow, Peter; Peacock, George; Lardner, Dionysius; Airy, Sir George Biddell; Hamilton, H. P.; Levy, A.; De Morgan, Augustus; Mosley, Henry (1847), Encyclopaedia of Pure Mathematics, R. Griffin and Co., p. 589.
8. Some authors use (a, b),[1][2][5] but this notation is often ambiguous. Andrews (1994, p. 16) explains this as: "Many authors write (a,b) for g.c.d.(a, b). We do not, because we shall often use (a,b) to represent a point in the Euclidean plane."
9. Thomas H. Cormen, et al., Introduction to Algorithms (2nd edition, 2001) ISBN 0262032937, p. 852
10. Bernard L. Johnston, Fred Richman, Numbers and Symmetry: An Introduction to Algebra ISBN 084930301X, p. 38
11. Martyn R. Dixon, et al., An Introduction to Essential Algebraic Structures ISBN 1118497759, p. 59
12. e.g., Wolfram Alpha calculation and Maxima
13. Jonathan Katz, Yehuda Lindell, Introduction to Modern Cryptography ISBN 1351133012, 2020, section 9.1.1, p. 45
14. Weisstein, Eric W. "Greatest Common Divisor". mathworld.wolfram.com. Retrieved 2020-08-30.
15. "Greatest Common Factor". www.mathsisfun.com. Retrieved 2020-08-30.
16. Slavin, Keith R. (2008). "Q-Binomials and the Greatest Common Divisor". INTEGERS: The Electronic Journal of Combinatorial Number Theory. University of West Georgia, Charles University in Prague. 8: A5. Retrieved 2008-05-26.
17. Schramm, Wolfgang (2008). "The Fourier transform of functions of the greatest common divisor". INTEGERS: The Electronic Journal of Combinatorial Number Theory. University of West Georgia, Charles University in Prague. 8: A50. Retrieved 2008-11-25.
18. Knuth, Donald E. (1997). The Art of Computer Programming. Vol. 2: Seminumerical Algorithms (3rd ed.). Addison-Wesley Professional. ISBN 0-201-89684-2.
19. Shallcross, D.; Pan, V.; Lin-Kriz, Y. (1993). "The NC equivalence of planar integer linear programming and Euclidean GCD" (PDF). 34th IEEE Symp. Foundations of Computer Science. pp. 557–564. Archived (PDF) from the original on 2006-09-05.
20. Chor, B.; Goldreich, O. (1990). "An improved parallel algorithm for integer GCD". Algorithmica. 5 (1–4): 1–10. doi:10.1007/BF01840374. S2CID 17699330.
21. Adleman, L. M.; Kompella, K. (1988). "Using smoothness to achieve parallelism". 20th Annual ACM Symposium on Theory of Computing. New York. pp. 528–538. doi:10.1145/62212.62264. ISBN 0-89791-264-0. S2CID 9118047.{{cite book}}: CS1 maint: location missing publisher (link)
22. Müller-Hoissen, Folkert; Walther, Hans-Otto (2012), "Dov Tamari (formerly Bernhard Teitler)", in Müller-Hoissen, Folkert; Pallo, Jean Marcel; Stasheff, Jim (eds.), Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift, Progress in Mathematics, vol. 299, Birkhäuser, pp. 1–40, ISBN 978-3-0348-0405-9. Footnote 27, p. 9: "For example, the natural numbers with gcd (greatest common divisor) as meet and lcm (least common multiple) as join operation determine a (complete distributive) lattice." Including these definitions for 0 is necessary for this result: if one instead omits 0 from the set of natural numbers, the resulting lattice is not complete.
23. Knuth, Donald E.; Graham, R. L.; Patashnik, O. (March 1994). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley. ISBN 0-201-55802-5.
24. Nymann, J. E. (1972). "On the probability that k positive integers are relatively prime". Journal of Number Theory. 4 (5): 469–473. Bibcode:1972JNT.....4..469N. doi:10.1016/0022-314X(72)90038-8.
25. Chidambaraswamy, J.; Sitarmachandrarao, R. (1987). "On the probability that the values of m polynomials have a given g.c.d." Journal of Number Theory. 26 (3): 237–245. doi:10.1016/0022-314X(87)90081-3.
References
• Andrews, George E. (1994) [1971], Number Theory, Dover, ISBN 978-0-486-68252-5
• Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950
• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 71081766
Further reading
• Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.5.2: The Greatest Common Divisor, pp. 333–356.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 31.2: Greatest common divisor, pp. 856–862.
• Saunders Mac Lane and Garrett Birkhoff. A Survey of Modern Algebra, Fourth Edition. MacMillan Publishing Co., 1977. ISBN 0-02-310070-2. 1–7: "The Euclidean Algorithm."
Number-theoretic algorithms
Primality tests
• AKS
• APR
• Baillie–PSW
• Elliptic curve
• Pocklington
• Fermat
• Lucas
• Lucas–Lehmer
• Lucas–Lehmer–Riesel
• Proth's theorem
• Pépin's
• Quadratic Frobenius
• Solovay–Strassen
• Miller–Rabin
Prime-generating
• Sieve of Atkin
• Sieve of Eratosthenes
• Sieve of Pritchard
• Sieve of Sundaram
• Wheel factorization
Integer factorization
• Continued fraction (CFRAC)
• Dixon's
• Lenstra elliptic curve (ECM)
• Euler's
• Pollard's rho
• p − 1
• p + 1
• Quadratic sieve (QS)
• General number field sieve (GNFS)
• Special number field sieve (SNFS)
• Rational sieve
• Fermat's
• Shanks's square forms
• Trial division
• Shor's
Multiplication
• Ancient Egyptian
• Long
• Karatsuba
• Toom–Cook
• Schönhage–Strassen
• Fürer's
Euclidean division
• Binary
• Chunking
• Fourier
• Goldschmidt
• Newton-Raphson
• Long
• Short
• SRT
Discrete logarithm
• Baby-step giant-step
• Pollard rho
• Pollard kangaroo
• Pohlig–Hellman
• Index calculus
• Function field sieve
Greatest common divisor
• Binary
• Euclidean
• Extended Euclidean
• Lehmer's
Modular square root
• Cipolla
• Pocklington's
• Tonelli–Shanks
• Berlekamp
• Kunerth
Other algorithms
• Chakravala
• Cornacchia
• Exponentiation by squaring
• Integer square root
• Integer relation (LLL; KZ)
• Modular exponentiation
• Montgomery reduction
• Schoof
• Trachtenberg system
• Italics indicate that algorithm is for numbers of special forms
| Wikipedia |
Klaus Hulek
Klaus Hulek (born 19 August 1952 in Hindelang) is a German mathematician, known for his work in algebraic geometry and in particular, his work on moduli spaces.
Life
Klaus Hulek studied Mathematics from 1971 at Ludwig Maximilian University of Munich graduating in 1976 with his Diplom. In 1974/75 he studied at Brasenose College of the University of Oxford, where he obtained a master's degree.
He obtained his doctorate under the supervision of Wolf Barth at the University of Erlangen–Nuremberg in 1979. His thesis was "Stable rank 2 vector bundles on $\mathbb {P} ^{2}$ with odd first Chern class".[1] In 1982/83 he held a Post-doctorate at Brown University and after that he returned to Erlangen as a research scientist, where he completed his habilitation in 1984, gaining the title Privatdozent.
From 1985, Hulek was a professor at the University of Bayreuth, and in 1990 he moved to Leibniz University Hannover, where he was also vice-president for research from 2005 to January 2015.
Hulek is an editor of the journal Mathematische Nachrichten. Since 2016 he has been editor in chief of zbMATH (formerly Zentralblatt für Mathematik). Hulek was vice president of the German Mathematical Society (DMV) from January 2019 to May 2020.
His former doctoral students include Andreas Gathmann and Matthias Schütt.
List of works
• Elementare algebraische Geometrie, Vieweg 2000, 2. Auflage 2012
• English translation Elementary algebraic geometry, American Mathematical Society 2003
• Projective Geometry of Elliptic Curves, Asterisque, Band 137, 1986
• with Constantin Kahn, Steven Weintraub Moduli spaces of Abelian surfaces: compactification, degenerations, and theta functions, de Gruyter 1993
• with Wolf Barth, Chris Peters, Antonius van de Ven Compact complex surfaces, Springer Verlag, 2. Auflage 2004 (Ergebnisse der Mathematik und ihrer Grenzgebiete)
• Edited with Fabrizio Catanese, Chris Peters, Miles Reid New trends in algebraic geometry, Cambridge University Press, London Mathematical Society Lecturenotes Series 264, 1999 (Conference Warwick 1996)
• Edited with Fabrizio Catanese, Hélène Esnault, Alan Huckleberry, Thomas Peternell Global Aspects of Complex Geometry, Springer Verlag 2006
• Edited with Wolf Barth, Herbert Lange Abelian Varieties, de Gruyter 1995 (Proc. Egloffstein Conference)
• Edited with Wolfgang Ebeling, Knut Smoczyk Complex and Differential Geometry, Springer Verlag 2011 (Conference Hannover 2009)
• Edited with Thomas Peternell, Michael Schneider, Frank-Olaf Schreyer Complex algebraic varieties, Lecture Notes in Mathematics 1507, Springer Verlag 1992 (Congference Bayreuth 1990)
• Elliptic curves, abelian surfaces and the icosahedron (German), Jahresbericht des DMV, Band 91, 1989, S. 126-147
• Geometry of the Horrocks-Mumford bundle, Proc. Symp. Pure Math., 46, Teil 2, 1987, S. 69-85
• Riemann Surfaces, in Francoise, Naber, Tsun (Eds.) Encyclopedia of Mathematical Physics, Elsevier 2006
• The Kodaira dimension of the moduli of K3 surfaces, Invent. Math. 169 (2007), 519-567 (with V. Gritsenko, G. K. Sankaran)
• The class of the locus of intermediate Jacobians of cubic threefolds, Invent. Math. 190 (2012), No. 1, 119-168 (with S. Grushevsky)
External links
Wikimedia Commons has media related to Klaus Hulek.
• Homepage
• Videos from Klaus Hulek (in German) in the AV-Portal of the German National Library of Science and Technology
References
1. Klaus Hulek at the Mathematics Genealogy Project
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| Wikipedia |
Hyperinteger
In nonstandard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence (1, 2, 3, ...) in the ultrapower construction of the hyperreals.
Discussion
The standard integer part function:
$\lfloor x\rfloor $
is defined for all real x and equals the greatest integer not exceeding x. By the transfer principle of nonstandard analysis, there exists a natural extension:
${}^{*}\!\lfloor \,\cdot \,\rfloor $
defined for all hyperreal x, and we say that x is a hyperinteger if $x={}^{*}\!\lfloor x\rfloor .$ Thus the hyperintegers are the image of the integer part function on the hyperreals.
Internal sets
The set $^{*}\mathbb {Z} $ of all hyperintegers is an internal subset of the hyperreal line $^{*}\mathbb {R} $. The set of all finite hyperintegers (i.e. $\mathbb {Z} $ itself) is not an internal subset. Elements of the complement $^{*}\mathbb {Z} \setminus \mathbb {Z} $ are called, depending on the author, nonstandard, unlimited, or infinite hyperintegers. The reciprocal of an infinite hyperinteger is always an infinitesimal.
Nonnegative hyperintegers are sometimes called hypernatural numbers. Similar remarks apply to the sets $\mathbb {N} $ and $^{*}\mathbb {N} $. Note that the latter gives a non-standard model of arithmetic in the sense of Skolem.
References
• Howard Jerome Keisler: Elementary Calculus: An Infinitesimal Approach. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
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| Wikipedia |
\begin{document}
\title{Quantum dynamics of PT-symmetrically kicked particle confined in a 1D box} \author{J. Yusupov$^a$, S. Rakhmanov$^b$, D. U. Matrasulov$^a$ and H. Susanto$^c$} \affiliation{
$^a$ Turin Polytechnic University in Tashkent, 17 Niyazov Str., 100095, Tashkent, Uzbekistan\\ $^b$National University of Uzbekistan, Vuzgorodok, Tashkent 100174,Uzbekistan\\ $^c$Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK}
\begin{abstract} We study quantum particle dynamics in a box and driven by PT-symmetric, delta-kicking complex potential. Such dynamical characteristics as the average kinetic energy as function of time and quasi-energy at different values of the kicking parameters. Breaking of the PT-symmetry at certain values of the non-Hermitian kicking parameter is shown. Experimental realization of the model is also discussed. \end{abstract}
\maketitle
\section{Introduction}
PT-symmetric quantum systems attracted much attention during past two decades after the discovery of the fact that non-Hermitian, but PT-symmetric system can have a set of eigenstates with real eigenvalues \ci{CMB1}. In other words, self-adjointness of the Hamiltonian is not necessary condition for being the eigenvalues real. Currently quantum physics of PT-symmetric such systems has become rapidly developing topic of contemporary physics and great progress is made in the study of different aspects of such systems(see, e.g., papers \ci{CMB2}-\ci{ptbox} for review of recent developments on the topic). These studies allowed to construct complete theory of PT-symmetric quantum systems, including PT-symmetric field theory \ci{CMB07,CMB11}. Experimental realization of such systems was also subject for extensive research. The latter has been done mainly in optics \ci{Makris,Nature,UFN,Konotop}. Some other PT-symmetric systems are discussed recently in the literature \ci{Chit,Longhi}. PT-symmetric relativistic system are also studied in \ci{PTSD1,PTSD2}. General condition for PT-symmetry in quantum systems has been derived in terms of so-called CPT-symmetric inner product \ci{CMB5,CMB9,CMB11}. Similarly to the case of Hermiticity, PT-symmetry in quantum systems can be introduced either through the complex potential, or by imposing proper boundary conditions, which provide such symmetry via the CPT-inner product \ci{CMB5,CMB11}. Different types of complex potentials providing PT-symmetry in Hamiltonian have been considered in \ci{CMB9,CMB11,Kottos,Longhi}. PT-symmetric particle-in-box system, where the box boundary conditions provide PT-symmetry of the system, have been studied in \ci{CMB13,Znojil,Znojil2,ptbox}. Certain progress is also done in nonlinear extension of PT-symmetric systems \ci{UFN,Konotop,Panos}.
In this paper we consider quantum particle confined in a 1D box and driven by a PT-symmetric, delta-kicking potential with the focus on the role of non-Hermitian parameter on such characteristics as everage kinetic, total energy and quasienergy. Here we mention that some time ago, both the classical and the quantum dynamics of systems interacting with a delta-kicking potential have been extensively studied in the context of nonlinear dynamics and quantum chaos theory \ci{Casati}-\ci{Casati1}. Kicked quantum particle dynamics in a box have been also considered in \ci{Roy,Well,Hu}. For kicked systems, the classical dynamics is characterized by diffusive growth of the average kinetic energy as a function of time, while for corresponding quantum systems such growth suppressed (except the special cases of so-called quantum resonances). The latter is called quantum localization of classical chaos \ci{Casati}-\ci{Casati1}. The dynamics of kicked nonrelativistic system is governed by single parameter, product of the kicking strength and kicking period.\\ We note that earlier, PT-symmetrically kicked systems have been considered in the Refs.\ci{Kottos,Longhi} in the context of quantum chaos theory. In \ci{Kottos} PT-symmetrically kicked rotor is studied by developing one-parameter scaling theory for non-Hermitian parameter and focusing on the gain, loss effects. In \ci{Longhi} PT-symmetrically kicked quantum rotor is studied by analyzing quasienergy spectrum and evolution of the momentum distribution at different values of the non-Hermitian parameter. Here we consider PT-symmetrically kicked confined system, by focusing on the role of confinement and non-Hermitian part of the kicking potential. Usual way for creating of kicked quantum system is confining of the system in a standing wave cavity. PT-symmetric analog of such system could be realized in a cavity with the losses. Another option, putting the system in a transverse beam propagation inside a passive optical resonator with combined phase and loss gratings, was discussed, e.g., \ci{Longhi}. An optical waveguide which is driven ny PT-symmetric optical field can be considered as another version of the model we are going to treat. This paper is organized as follows. In the next section we briefly recall Hermitian counterpart of our system, quantum particle confined in a 1D box and driven by delta-kicking potential. In section III we consider similar system with PT-symmetric delta-kicks. Section IV presents some concluding remarks. \begin{figure}
\caption{ Few quasienergy levels as a function of the wave number for different $K=\epsilon T$ , $K=0.1$ (a), $K=1$ (b),$K=1.5$ (c) and $L=10$ }
\label{qe1}
\end{figure}
\section{Kicked quantum particle dynamics in a box}
Hermitian counterpart of the system we are going to study, is a quantum particle confined in one-dimensional box of size $L$ and driven by external delta-kicking potential given by $$ U(x,t) =\epsilon \cos(\frac{2\pi x}{\mu}) \sum_l \delta(t-lT), $$ where $\mu,$ $\epsilon$ and $T$ are the wavelength, kicking strength and period, respectively. Such system was considered earlier in the context of quantum chaos theory e.g., in \ci{Roy,Well,Li} and described by the following time-dependent Schr\"{o}dinger equation: \begin{equation} i\frac{\partial}{\partial t} \Psi(x,t) =\left[-\frac{1}{2}\frac{d^2}{dx^2} +U(x,t)\right]\Psi(x,t), \label{kdb1} \end{equation} The wave function, $\Psi(x,t)$ fulfills the box boundary conditions given by \be \Psi(0,t) = \Psi(L,t). \lab{bc01}\ee Exact solution of Eq.\re{kdb1} can be obtained within the single kicking period \cite{Casati,Well} by expanding the wave function, $\Psi(x,t)$ in terms of the complete set of the eigenfunctions of the unperturbed system as
\begin{equation} \Psi(x,t) = \sum_n A_n(t) \psi_n(x) \label{1111} \end{equation} where $\psi_n(x)=\sqrt{2/L}\sin{(\pi n x/L)}$. Eqs.\re{kdb1} and \re{1111} lead to quantum mapping for the wave function amplitudes, $A_n(t)$ which is given by \begin{equation} A_n(t+T)=\sum_l A_l(t) U_{ln}e^{-iE_l T}, \label{evol} \end{equation} where $$ U_{ln} =\int_0^L \psi^*_n(x) e^{-i\epsilon \cos (2\pi x/\mu)}\psi_l (x)dx. $$ \begin{figure}
\caption{ (Color online) The average kinetic energy of kicked particle in a box as a function of kick number for different kicking strength for $L=3.3$, $T=0.01$ and $\mu=1.3$}
\label{ke1}
\end{figure}
and \be E_l=(\pi l/L)^2 \lab{en1}\ee The amplitudes fulfill the norm conservation given by \be N(t) = \sum_n |A_n(t)|^2 =1. \lab{norm1}\ee We use this condition for controlling of the accuracy of numerical computations. Thus the evolution of the wave function within the single kicking period can be written as $$ \Psi(x,t+T) = \hat U\Psi(x,t), \label{11111} $$ where the one-period evolution operator is given by \be \hat U =\exp(-i\frac{\partial^2}{2\partial x^2})\exp(-i\beta V(x)) \exp(-i\frac{\partial^2}{2\partial x^2}),\ee where $$\beta =\frac{\pi T}{\mu^2}.$$ For such operator, one can consider the eigenvalue problem given by \be \hat U\phi_n =\lambda_n\phi_n, \ee where the eigenvalues, $\lambda_n$ are called quasienergy levels of the kicked system. In Fig. \ref{qe1} few quasienergy levels are plotted as a function of the wave number, $k =2\pi/\mu$ at different values of the parameter $K=\epsilon T$. As $K$ is higher, as stronger the fluctuations of the quasienergy levels.
Having found amplitudes and wave function, one can compute the average kinetic energy, which is defined as $$ <E(t)_{kin}> =
-\frac{1}{2}<\Psi(x,t)|\frac{d^2}{dx^2}|\Psi(x,t)>$$ \be=\sum_n E_n|A_n(t)|^2, \lab{kinetic} \ee where $E_n$ are given by Eq.\re{en1}. Fig. \ref{ke1} presents plots of the average kinetic energy, $<E_k(t)>$ at different values of the kicking strength, $\varepsilon$ for fixed kicking period $T$. Unlike the kicked rotor $<E_k(t)>$ grows during some initial time and suppression with the subsequent decrease occurs for large enough number of kick ($N=t/T$). For very large number of kicks one can observe periodic or quasi-periodic time-dependence of $<E_k(t)>$. Such behavior in some kicked quantum systems have been discussed in \ci{Hogg}. Another feature of kicked quantum particle confined in a box is the absence of quantum resonance. It should be noted that the dynamics of kicked particle confined in a box depends on two factors, such as interaction with the kicking force and bouncing of particle from the box walls. Depending on the sign of of cosine in the kicking potential, the kicking force can be attractive and repulsive. When the kicking potential is repulsive particle gains the energy, while in case of attractive potential it losses its energy. Therefore depending on which area in the box, i.e. on the area where the kicking force is positive or negative, acceleration or deceleration of the particle may occur. Vey important factor is "synchronization" of the kicking force and bouncing of particle from the box wall. It also may cause acceleration and deceleration of the particle.
\begin{figure}
\caption{ (Color online) The norm as a function of kick number at different values of the $\gamma$ for $\epsilon=0.1$ , $L=3.3$ , $T=0.01$ and $\mu=1.3$}
\label{norm01}
\end{figure}
\section{$\mathcal{PT}-$symmetrically kicked quantum particle a one dimensional box} PT-symmetric analog of the above system can be constructed by adding into the kicking potential an imaginary part. Then PT-symmetric kicking potential can be written as \be V_{PT}(x,t) =f(t)\left[\epsilon \cos{(2\pi x/\mu)} + i \gamma \sin{(2\pi x/\mu)}\right], \lab{ptk} \ee
where $\epsilon$ and $T$ are the kicking strength and period, respectively, $\gamma\geq 0$ is the non-Hermitian parameter that measures the strength of the imaginary part of the potential and $ f(t)= \sum_l \delta(t-lT)$. The dynamics of the system is governed by the following time-dependent Schr\"{o}dinger equation: \begin{equation} i\frac{\partial}{\partial t} \Psi(x,t) =H_{PT}\Psi(x,t), \label{perturbed1} \end{equation} where $H_{PT}$ is the Schr\"{o}dinger operator containing potential $U_{PT}$. The same boundary conditions as those in Eq.\re{bc01}. Exact solution of Eq.(\ref{perturbed1}) can be obtained similarly to the case of Hermitian counterpart and one gets quantum mapping for the evolution of the amplitude, $A_n(t)$ within the one kicking period, $T$: \begin{equation} A_n(t+T)=\sum_l A_l(t) V_{ln}e^{-iE_l T}, \label{evol} \end{equation} where \be V_{ln} =\int \psi^*_n(x) e^{-i\epsilon \cos (2\pi x/\mu)}e^{\gamma \sin (2\pi x/\mu)}\psi_l (x)dx \lab{kick2}\ee and $E_l=(\pi l/L)^2$. The evolution operator corresponding to Eq.\re{evol} can be written as \be \hat U_{PT} =\exp(-i\frac{\partial^2}{2\partial x^2})\exp(-i\beta V(x)) \exp(-i\frac{\partial^2}{2\partial x^2}),\ee where $$\beta =\frac{\pi T}{\mu^2}.$$
\begin{figure}
\caption{ (Color online) The average kinetic energy of $\mathcal{PT}-$symmetrically kicked particle in a box as a function of kick number for different kicking strength for $\gamma=0.1$, $L=3.3$ , $T=0.01$ and $\mu=1.3$}
\label{ke2}
\end{figure}
\begin{figure}
\caption{ The real part (a),(c) and imaginary (b),(d) parts of few quasienergy levels as a function of the wave number, $k=2\pi/\mu$, for $\gamma=1$ for $\epsilon=0.1$ (a),(b) and $\epsilon=1$ (c),(d) for $L=10$ }
\label{qe2}
\end{figure}
\begin{figure}
\caption{ The real part (a),(c) and imaginary (b),(d) parts few quasienergy levels as a function of non-Hermitian parameter, $\gamma$ for $\epsilon=0.1$ (a),(b) and $\epsilon=1$ (c),(d) for $L=10$}
\label{qe3}
\end{figure}
\begin{figure}
\caption{The average total energy computed as the expectation value of the operator $H_{PT}$ for $\gamma =0.01$ (a) and $\gamma =0.1$
at $\epsilon =0.1$, $T=0.01$.}
\label{tot1}
\end{figure}
For a quantum systems with complex PT-symmetric potentials, the norm conservation is broken, i.e., the amplitudes, $A_n(t)$ do not fulfill Eq.\re{norm1}. Fig.\ref{norm01} presents plots of the norm as a function of time at different values of $\gamma$ for fixed $\epsilon$ and $T$. As higher is $\gamma$, as stronger is the breaking of the norm conservation. In Fig. \ref{ke2} the average kinetic energy is plotted as a function time. Although the profile of plot is almost similar to that of Hermitian counterpart, the values of $<E_k(t)>$ are much higher than that in Hermitian case.
Similarly to the Hermitian case, one can compute quasienergy levels for PT-symmetric system as the eigenvalues of the operator $U_{PT}$. Fig.\ref{qe2} presents few quasienergy levels as a function of the wave number, $k=2\pi/\mu$. In Fig. \ref{qe3} few quasi-energy levels defined as the eigenvalues of the operator $U_{PT}$ are plotted as a function of non-Hermitian parameter, $\gamma$. Important feature of PT-symmetric systems with complex potentials is the fact that is the expectation value of the Hamiltonian operator is always real. This holds true also in case of time-dependent operator. Fig. \ref{tot1} where the average total energy, $<E_{tot}(t)>$ i.e. the expectation value of the operator $H_{PT}$ is plotted as a function of time at different values of $\gamma$.
\section{Conclusions} We studied quantum dynamics of a particle confined in a 1d box and driven by PT-symmetric, delta-kicking potential. Different characteristics of the dynamics, such as the time-dependence of the average kinetic energy, quasienergy and the average total energy are analyzed using the exact solution of the time-dependent Schrodinger equation for single kicking period. It is found that no unbound acceleration in PT-symmetric quantum regime is possible, as the average kinetic energy is the periodic or quasi-periodic in time. However, in PT-symmetrically driven system the gain of energy and acceleration are more intensive than those for the Hermitian counterpart. The above model can be realized in different versions using optical systems where it is possible to create PT-symmetric kicking potential. Such kicking field could be realized e.g., in an optical cavity with losses and gains. Confining an optical pulse in such cavity would be a version for our model. Another option is considering a PT-symmetric periodic optical structure, e.g., array of optical waveguides driven by laser field. In the absence of external perturbation such system is described by the Helmholtz equation with periodic boundary condition, which is an analog of the box boundary condition. Therefore the driven waveguide array can be considered as an analog of the above model.
\end{document} | arXiv |
Isn't Domain of a variable nothing but a constraint?
In Constraint programming we have Variables and their Domains and then all the constraints, but if you at the concept of a domain of a variable it is nothing but another type of constraint, you are saying that this variable can take all these values. Is there any particular reason why domain is defined as a different concept than constraints?
type-theory typing constraint-programming curry-howard constraint-satisfaction
AnkurAnkur
$\begingroup$ May I ask what is the context of your question? Is it a general question about the logic of constraint programming and the design of constraint programming languages. Or is it a more mundane question on how to use domains and constraints in constraint programming? $\endgroup$ – babou May 28 '15 at 15:27
As you observe, restricting the domain of a variable has exactly the same effect as applying a unary constraint to it.
One situation where you might prefer to use unary constraints rather than restricted domains is when you want to control very tightly the relations that are allowed to be used in constraints. For example, if you want to investigate the computational complexity of CSP with a particular class of constraint languages. On the other hand, such investigations often assume that all unary relations are included in the constraint language, which is equivalent to fixing a global domain but allowing the domain of any variable to be any subset of that. (This is known as the "conservative" case because of certain algebraic properties of the constraint langauges.)
David RicherbyDavid Richerby
Even though a domain may be considered just another type of constraint, there do exist good reasons to keep them separated, and it may be easier to think of them from a pure mathematical standpoint. Domains should in a sense be seen as the definition of the variable in terms of Type - e.g. Integer or Real etcetera. The domains can also be seen as the Master bounds. Defining the domain type (Integer / Real) is also important to help the constraint solver determine what solving method to use for each specific constraint.
I would like to illustrate it with the following example;
Consider the following constraint problem for Pythagoras theorem.
Variables: a b c
Domains: a [1..10] Integer b [1..10] Integer c [1..10] Integer
Constraint: a x a + b x b = c x c
We actually have three unknowns but the finite sets of the domains will allow the constraint solver to find the following solutions for a,b,c;
a = 3,4,6,8 b = 3,4,6,8 c = 5,10
Now let us change the domains from Integer to Real, to the following;
a [1..10] Real b [1..10] Real c [1..10] Real
Simply by changing the domains, it will no longer be possible to find all solutions as there do exist an infinite number of solutions.
Therefore, the domains should in general be seen as the main definition of the total search space (in terms of type and bounds) while the constraints should be seen as the definition of the problem.
Jimmy RichardssonJimmy Richardsson
I suppose what is called domain for constraint programming corresponds to what would be called type in most contexts, and particularly in most programming languages.
The issue of types is an old one, and I am unfortunately not knowledgeable enough to give you a precise account. But this may hopefully serve as an introduction.
The idea of theorizing types is attributed to Bertrand Russell, to serve as an alternative to set theory and do away with some paradoxes of naive set theory.
What it says, intuitively, is that terms are always supposed to be typed and that operators or predicate make sense only when applied to the proper types. The idea is quite close to static type checking in programming languages. But I suggest you read at least the beginning of the wilipedia page on Type Theory, which seems rather clear.
Thus, as a first approximation, it seems quite possible to do as suggested in the question: consider the domains as simple constraints, on a universal sets of values. That is just returning to naive set theory and its paradoxes, which was however sufficient to do a lot of good mathematics for some time.
But it seems that a cleaner way to do things is to separate values into domains were operations and predicates have meaning. Otherwise, is the value true odd or even, and what is its square root? This is the way to ensure that the language you use (whether constraints or other) will have properly defined semantics, relying hopefully on a consistent logic.
But can we have both consistency and domains as constraints, and is there a cost?
The benefits of typing may come at the cost of Turing Completeness. For example, while (untyped) $\lambda$-calculus is Turing complete, simply typed $\lambda$-calculus is more restricted. Whether typing can be consistent with Turing Completeness is an issue I would not address at my competence level.
An important point of all this however is that typing systems have been shown to be isomorphic to logical predicates, through what is known as the Curry-Howard isomorphism. So this would vindicate from a theoretical perspective the view that domains can be seen as a constraint expressed by some predicate to be satisfied. From this point of view, there is no difference between a formula (to be satisfied) and a type, given appropriate formalization apparatus with a consistent logic.
However, it may be (I do not know) that distinguishing domains from other predicates may be a way of enforcing consistency of the underlying logical system (to be checked).
Now, if you wish for more details, it would be best to asked a full-fledged type theorist, as I am getting onto grounds for which I have no chart.
From a more mundane point of view, you may consider that expressing constraints makes sense only with respect to some domain, and there is in practice a hierarchical structures that enforces the existence of some sense (semantics) to what can be written, and also allows for more efficient implementations. That is pretty much, I believe, why static types were introduced in programming languages, as they had to refer to the encoding techniques used for the corresponding values (though I would not underestimate the influence of logicians on early programming language design, especially Lisp and Algol, that were both extremely influencial).
Of course, if the kind of domains you are willing to consider in your constraint programming language is very restricted, you are probably safe from paradoxes, and could, if you wish, freely see domains as unary constraints as already suggested by David Richerby's answer.
baboubabou
$\begingroup$ I don't think that most of this is really relevant to constraints. In particular, the question isn't asking for a type theory of constraints, or really about how one might design a constraint language. And, in the constraint world, it's not really an issue that there's no such thing as the square root of a Boolean value: "square root" is a relation between integers so there's no $y$ such that $\mathrm{sqrt}(\mathrm{true},y)$ holds, so you'll never satisfy a constraint about square roots by assigning a Boolean to something that's supposed to be an integer. $\endgroup$ – David Richerby May 28 '15 at 11:42
$\begingroup$ And the paradoxes of set theory will never come up because there's no mechanism for self-reference. $\endgroup$ – David Richerby May 28 '15 at 11:42
$\begingroup$ @DavidRicherby I realize that constraint programming does not necessarily get into such issues, though it does mix with other programming paradigms. However, I do believe it may be useful to take a larger view and see whether what works because it is "simple" (I do not mean easy, or trivial) can be inscribed into a more general view of things. In this sense, I do believe that what I said is relevant. I do realize that this can deal with "type errors" such as sqrt(true,x), and my example may not be pedagogical since the real issue is not a naive one. But I think what I said does stand. $\endgroup$ – babou May 28 '15 at 12:15
$\begingroup$ @DavidRicherby You are saying there are no problem in doing it because there is no mechanism for self reference. This justification was not in your answer, which would lead me to think that it said "yes, we can!", but not really why we can. So possibly my answer contributed as much. One purpose of my answer is to suggest that there are probably deep reasons why it makes sense to do it, and keep a consistent system, rather than, yes, we can hack it. Maybe such as having no mechanism for self-reference. $\endgroup$ – babou May 28 '15 at 12:22
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Elizabeth Stephansen
(Mary Ann) Elizabeth Stephansen (10 March 1872 – 23 February 1961) was a Norwegian mathematician and educator. She was one of the first Norwegian women to be awarded a doctorate degree.[1][2]
Biography
Stephansen was born in Bergen, Norway. She was the eldest daughter of Anton Stephan Stephansen (1845–1929) and Gerche Reimers Jahn (1848–1935). Her father was a merchant and owner of a textile shop. He later established the textile factory, Espelandfos Spinderi & Tricotagefabrik, in Arna. She was educated at the Bergen Cathedral School graduating in 1891. She was fluent in the German language and traveled to Switzerland to continue her education. She attended Eidgenössische Polytechnikum in Zurich and graduated in 1896. Her thesis Ueber partielle Differentialgleichungen vierter Ordnung die ein intermediäres Integral besitzen was published in 1902. She obtained her doctorate (Dr. Philos.) in absentia from the University of Zurich in the fall of 1902.[3][4]
In 1902–1903, she traveled to the University of Göttingen under a government grant to attend lectures by noted German mathematicians, Ernst Zermelo, David Hilbert and Felix Klein. She first served as a teacher of mathematics at Bergen Cathedral School and Bergen Technical School. Between 1905–1906, she completed mathematical research and wrote further papers on difference equations. From 1906 until her retirement in 1937 she worked at the Agricultural College of Norway at Ås in Akershus. She first taught physics and mathematics. In 1921, she was appointed Docent in mathematics.[5][6]
In retirement, she lived at the family farm at Espeland in Arna (Espeland, gnr. 289, gårdsbruk i Arna) which her father had first acquired during 1918. After the liberation of Norway in 1945, she was awarded the King's Medal of Merit (Kongens fortjenstmedalje) for the assistance she rendered to Norwegian prisoners held at the Nazi operated Espeland concentration camp (Espeland fangeleir). She died during 1961 at Espeland in the borough of Arna and was buried at Solheim Cemetery in the Årstad district of Bergen. [7][8][9]
References
1. "Elizabeth Stephansen". lokalhistoriewiki.no. Retrieved March 1, 2018.
2. O'Connor, J J; Robertson, E F (August 2005). "Mary Ann Elizabeth Stephansen". School of Mathematics and Statistics. University of St Andrews, Scotland. Retrieved March 1, 2018.
3. "Espelandfos Spinderi & Tricotagefabrik". Norsk Teknisk Museum. Retrieved March 1, 2018.
4. "Elizabeth Stephansen". Store norske leksikon. Retrieved December 15, 2016.
5. "Elizabeth Stephansen". Agnes Scott College – Atlanta, Georgia. Retrieved March 1, 2018.
6. Hag, Kari. "Elizabeth Stephansen". Norsk biografisk leksikon. Retrieved December 15, 2016.
7. "Espeland (Arna)". Bergen byleksikon. Retrieved March 1, 2018.
8. "Espeland fangeleir". Bergen byleksikon. Retrieved March 1, 2018.
9. "Kongens fortjenstmedalje". lokalhistoriewiki. Retrieved April 1, 2018.
Other sources
• Kari Hag and Peter Lindqvist (1997) Elizabeth Stephansen: A pioneer Skrifter det Kongelige Norske Videnskabers Selskab 2, 1-23.
• Catharine M. C. Haines (2001) "Stephansen, Mary Ann Elizabeth" in International Women in Science: A Bibliographical Dictionary to 1950 (Santa Barbara, CA: ABC-CLIO, Inc., pgs 295-296) ISBN 1-57607-090-5
Related reading
• Gila Hanna, ed. (2006) Towards Gender Equity in Mathematics Education (Volume 3 of New ICMI Study Series. Springer Science & Business Media) ISBN 9780306472053
External links
• Biographies of Women Mathematicians: Elizabeth Stephansen
Authority control
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• ISNI
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| Wikipedia |
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\newcommand{{\rm div}}{{\rm div}} \def\texttt{(a1)}{\texttt{(a1)}} \def\texttt{(a2)}{\texttt{(a2)}} \newcommand{{\mathcal{ A}}}{{\mathcal{ A}}} \newcommand{{\bar{\opA}}}{{\bar{{\mathcal{ A}}}}} \newcommand{\widetilde}{\widetilde} \newcommand{\varepsilon}{\varepsilon} \newcommand{\varphi}{\varphi} \newcommand{\vartheta}{\vartheta} \newcommand{\varrho}{\varrho} \newcommand{\partial}{\partial} \newcommand{{\mathcal{W}}}{{\mathcal{W}}} \newcommand{{\rm supp}}{{\rm supp}}
\def{\mathbb{R}}{{\mathbb{R}}} \def{\mathbb{N}}{{\mathbb{N}}} \def{[0,\infty)}{{[0,\infty)}} \def{\mathbb{R}}{{\mathbb{R}}} \def{\mathbb{N}}{{\mathbb{N}}} \def{\mathbf{l}}{{\mathbf{l}}} \def{\bar{u}}{{\bar{u}}} \def{\bar{g}}{{\bar{g}}} \def{\bar{G}}{{\bar{G}}} \def{\bar{a}}{{\bar{a}}} \def{\bar{v}}{{\bar{v}}} \def{\wt\gamma}{{\widetilde\gamma}} \def{\tens{w}}{{\tens{w}}} \def{\tens{\eta}}{{\tens{\eta}}} \def{\tens{\xi}}{{\tens{\xi}}} \def{\tens{u}}{{\tens{u}}} \def{\bar{\tens{u}}}{{\bar{\tens{u}}}} \def{\tens{\eta}}{{\tens{\eta}}} \def{\tens{\phi}}{{\tens{\phi}}} \def{\tens{v}}{{\tens{v}}} \def{\tens{\lambda}}{{\tens{\lambda}}} \def{\mathcal{V}}{{\mathcal{V}}} \def{\bar{\tens{{v}}}}{{\bar{\tens{{v}}}}} \def{\tens{\vp}}{{\tens{\varphi}}} \def{\tens{F}}{{\tens{F}}} \def{\tens{f}}{{\tens{f}}} \def{\tens{g}}{{\tens{g}}} \def{\tens{\ell_\vr}}{{\tens{\ell_\varrho}}} \def{\tens{\wt u}}{{\tens{\widetilde u}}} \def{{\mathfrak{A}}}{{{\mathfrak{A}}}}
\def{D\teu}{{D{\tens{u}}}} \def{D\tebu}{{D{\bar{\tens{u}}}}} \def{D\teet}{{D{\tens{\eta}}}} \def{D\teph}{{D{\tens{\phi}}}} \def{D\tev}{{D{\tens{v}}}} \def{D\tew}{{D{\tens{w}}}} \def{D\tebv}{{D{\bar{\tens{{v}}}}}} \def{D\tewtu}{{D{\tens{\wt u}}}} \defD{\tevp}{D{{\tens{\vp}}}} \def{D\teelvr}{{D{\tens{\ell_\vr}}}}
\def{\tens{\dv}}{{\tens{{\rm div}}}} \def{\tens{\mu}}{{\tens{\mu}}} \def\tens{a}{\tens{a}}
\def\text{I}{\text{I}} \def\text{II}{\text{II}} \def\text{III}{\text{III}} \def{\bar{\mu}}{{\bar{\mu}}} \def{\mathbb{R}^{n}}{{\mathbb{R}^{n}}} \def{\mathbb{R}^{m}}{{\mathbb{R}^{m}}} \def{\mathbb{R}^{n}}{{\mathbb{R}^{n}}} \def{\mathsf{Id}}{{\mathsf{Id}}} \def{\mathsf{P}}{{\mathsf{P}}} \def{\mathsf{P_j}}{{\mathsf{P_j}}} \def{\mathbb{R}^{n\times m}}{{\mathbb{R}^{n\times m}}} \def{\mathbb{R}^{N}}{{\mathbb{R}^{N}}} \def{\mathcal{M}(\Omega,\Rm)}{{\mathcal{M}(\Omega,{\mathbb{R}^{m}})}}
\newtheorem{coro}{\bf Corollary}[section] \newtheorem{theo}[coro]{\bf Theorem} \newtheorem{lem}[coro]{\bf Lemma} \newtheorem{rem}[coro]{\bf Remark} \newtheorem{defi}[coro]{\bf Definition} \newtheorem{ex}[coro]{\bf Example} \newtheorem{fact}[coro]{\bf Fact} \newtheorem{prop}[coro]{\bf Proposition}
\newcommand{\textit{\texttt{data}}}{\textit{\texttt{data}}}
\parindent 1em
\begin{abstract} We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for $p$-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth. \end{abstract}
\setcounter{tocdepth}{1} \tableofcontents
\section{Introduction}
A broad and profuse branch of the theory of nonlinear partial differential systems stems from seminal ideas by Ural'tseva~\cite{Ural} and Uhlenbeck~\cite{Uhl}. There is a solid stream of recent studies on regularity of solutions to general growth elliptic systems and related minimizers of vectorial functionals ~\cite{bemi,CiMa-ARMA2014,DiEt,DiMa,DF1,DFMin,DSV1,DSV2,GPT,lieb,Marc2006,MarcPapi,str}. Our aim is to contribute to the field by providing pointwise estimates for very weak solutions to measure data problems in the terms of potentials of relevant Orlicz growth. As a consequence, we infer sharp description of fine properties of solutions being the exact analogues of the ones available in the classical linear potential theory or in the case of $p$-Laplace systems~\cite{KuMi2018}.
Let us stress that {weak solutions do} not have to exist for arbitrary measure datum. Thus we employ a notion of very weak solutions obtained by an approximation studied since~\cite{BG}. Despite they
can be unbounded, they can be controlled by a certain potential. There are known deep classical results for scalar problems~\cite{KiMa92,KiMa94} settling the nonlinear potential theory for solutions to $-\Delta_p u=-{\rm div} (|Du|^{p-2}Du)=\mu$, $1<p<\infty$, followed by their variable exponent version~\cite{LuMaMa} and recently by the Orlicz one~\cite{CGZG-Wolff,Maly-Orlicz}, as well as a counterpart proven for systems involving $p$-Laplace operator~\cite{CiSch,KuMi2018}. We study the nonstandard growth version of pointwise estimates involving suitably generalized potential of the Wolff type and infer their regularity consequences.
In fact, we investigate very weak solutions ${{\tens{u}}}:\Omega\to{\mathbb{R}^{m}}$ to measure data elliptic systems involving nonlinear operators
\begin{equation}
\label{intro:eq:main} -{ {\tens{\dv}}}\left( {a(x)}\frac{g(|{D\teu}|)}{|{D\teu}|} {D\teu} \right)={\tens{\mu}}\quad\text{in}\quad \Omega, \end{equation} where $a\in C(\Omega)$ is bounded and separated from zero, $g=G'$, ${\tens{\mu}}\in\mathcal{M}(\Omega,{\mathbb{R}^{m}})$ is a bounded measure, whereas ${\tens{\dv}}$ stands for the ${\mathbb{R}^{m}}$-valued divergence operator. Here we admit $G$ to be a Young function satisfying both $\Delta_2$- and $\nabla_2$-conditions, which follows from the condition \begin{equation}\label{iG-sG} 2 \leq i_G=\inf_{t>0}\frac{tg(t)}{G(t)}\leq \sup_{t>0}\frac{tg(t)}{G(t)}=s_G<\infty.\end{equation}
See {\it Assumption }{\bf (A-vect)} in Section~\ref{sec:formulation-n-reg-cons} for all details. We stress that we impose the typical assumption of quasi-diagonal structure of ${\mathcal{ A}}$ naturally covering the case of (possibly weighted) $p$-Laplacian when $G_p(s)=s^{p},$ for every $p\geq 2$, together with operators governed by the Zygmund-type functions $G_{p,\alpha}(s)=s^p\log^\alpha(1+s)$, $p\geq 2,\,\alpha\in {\mathbb{R}}$, as well as their multiplications and compositions with various parameters. In turn, we generalize the corresponding results of~\cite{CiSch,KuMi2018} to embrace also $p$-Laplace systems with continuous coefficients, and -- on the other hand -- cover the natural scope of operators with similar structure and Orlicz growth. Let us stress that the operator we consider is {\em not} assumed to enjoy homogeneity of a~form ${\mathcal{ A}}(x,k\xi)=|k|^{p-2}k{\mathcal{ A}}(x,\xi)$. Consequently, our class of solutions is {\em not} invariant with respect to scalar multiplication.
Obtaining sharp regularity results for solutions to nonlinear systems is particularly challenging. In the scalar case one can infer continuity or H\"older continuity of the solution to $-{\rm div} {\mathcal{ A}}(x,Du)=\mu$ when the dependence of the operator on the spacial variable is merely bounded and measurable and the growth of ${\mathcal{ A}}$ with respect to the second variable is governed by an arbitrary doubling Young function (cf. \cite{CGZG-Wolff}). The same is not possible for systems even with far less complicated growth and null datum, cf.~\cite{DeG,JMVS,SY} and~\cite[Section~3]{Min-Dark}. To justify why we restrict our attention to operators having a specific form as in~\eqref{intro:eq:main}, let us point out that the typical assumption of a so-called Uhlenbeck structure is imposed in order to control energy of solutions. The continuity of the coefficients is a minimal assumption to get continuity of the solution in the view of counterexample of~\cite{DeG}.
The studies on the potential theory to measure data problems dates back to \cite{HaMa2,Ma}. We refer to~\cite{KuMi2014} for an overview of the nonlinear potential theory, to \cite{HedWol,KiMa92,KiMa94,LiMa} for cornerstones of the field, and~\cite{adams-hedberg,hekima} for well-present background of the $p$-growth case. In the scalar case, in their seminal works Kilpel\"ainen and Mal\'y in~\cite{KiMa92,KiMa94} provided optimal Wolff potential estimates for $p$-superharmonic functions $u$ generating a nonnegative measure $\mu$ from above and below \begin{equation}
\label{wolff-p-est} \tfrac 1c{\mathcal{W}}_p^{\mu}(x_0,R)\leq u(x_0)\leq c\Big(\inf_{B_R(x_0)}u+{\mathcal{W}}_p^{\mu}(x_0,R)\Big)\quad\text{for some }\ c=c(n,p) \end{equation}with {the} so-called Wolff potential \[ {\mathcal{W}}_p^{\mu}(x_0,R)=\int_0^R \left(r^{p-n}{\mu(B_r(x_0))}\right)^{\frac{1}{p-1}}\,\frac{dr}{r}= \int_0^R \left(\frac{\mu(B_r(x_0))}{r^{n-1}}\right)^{\frac{1}{p-1}}\,dr,\] see also~\cite{KoKu,tru-wa}. In the linear case ($p=2$) the estimates of~\eqref{wolff-p-est} become the classical Riesz potential bounds. The precise Orlicz counterpart of this result with nonnegative measure $\mu$ is proven with the nonstandard growth potential \begin{equation*}
{\mathcal{W}}^{\mu}_G(x_0,R)=\int_0^R g^{-1}\left(\frac{\mu(B_r(x_0))}{r^{n-1}}\right)\,dr, \end{equation*} see \cite{CGZG-Wolff,Maly-Orlicz}. To our best knowledge, the only reference one can find on the related results for systems are~\cite{CiSch,KuMi2018} that involves problems with the $p$-Laplace operator. The estimate related to~\eqref{wolff-p-est} provided therein establishes the upper bound only. Note however that no lower bound can be available in the vectorial case. Indeed, it origins in the lack of the possibility of proving maximum principle. Our method of proof relies on the ideas of \cite{KuMi2018}. We employ a properly adapted Orlicz version of ${\mathcal{ A}}$-harmonic approximation relevant for measure data problems (Theorem~\ref{theo:Ah-approx} in Section~\ref{sec:Ah-approx}) and careful estimates on concentric balls.
Let us also mention that a lot of attention is attracted by potential estimates on gradients of solutions, generalized harmonic approximation, and their application in the theory of partial regularity \cite{Baroni-Riesz,BaHa,BCDKS,Byun4,Byun5,Byun6,DiLeStVe,DSV3,DuGr,DuMi2010-2,KuMi2016p,KuMi2014} which is an open path from now on.
The paper is organized as follows. Our assumptions, main results and their regularity consequences are presented in Section~\ref{sec:formulation-n-reg-cons}. Section~\ref{sec:prelim} is devoted to notation and information on the setting. In particular see Section~\ref{ssec:sols} for the precise definition of the notion of very weak solutions we employ. Section~\ref{sec:Ah-approx} provides the most important tool of paper -- a measure data ${\mathcal{ A}}$-harmonic approximation. Section~\ref{sec:mainproof} contains the proofs of comparison estimates, the sufficient condition for ${\tens{u}}$ to be in VMO of Proposition~\ref{prop:vmo}, the potential estimates of Theorems~\ref{theo:pointwise}, the continuity criterion of Theorem~\ref{theo:continuity} and the H\"older continuity criterion of Theorem~\ref{theo:H-cont}.
\section{Main result and its consequences }\label{sec:formulation-n-reg-cons}
\subsection{The statement of the problem} Let us present an essential notation and details of the measure data problem we study.
\noindent{\underline{\it Essential notation}.} By {`$\cdot$'} we denote the {scalar product of two vectors, i.e. for ${{\tens{\xi}}}=(\xi_1,\dots,\xi_m)\in {\mathbb{R}^{m}}$ and
${\tens{\eta}}= (\eta_1,\dots,\eta_m)\in {\mathbb{R}^{m}}$ we have ${{\tens{\xi}}}\cdot{{\tens{\eta}}} = \sum_{i=1}^m \xi_i \eta_i$}; by {`$:$'} -- {the Frobenius product of the second-order tensors, i.e. for ${\xi}=[\xi_{j}^\alpha]_{j=1,\dots,n,\, \alpha=1,\dots,m}$ and $\eta=[\eta_{j}^\alpha]_{j=1,\dots,n,\, \alpha=1,\dots,m}$ we have
\[{\xi}: {\eta} =\sum_{\alpha=1}^m \sum_{j=1}^n \xi_{j}^\alpha \eta_{j}^\alpha.\]}
By `{$\otimes$}' we denote {the tensor product of two vectors, i.e for ${{{\tens{\xi}}}}=(\xi_1,\dots,\xi_k)\in {{\mathbb{R}}^k}$ and
${{{\tens{\eta}}}}= (\eta_1,\dots,\eta_\ell)\in {\mathbb{R}}^\ell$, we have ${\tens{\xi}}\otimes{\tens{\eta}}:=[\xi_i\eta_j]_{i=1,\dots,k,\,j=1,\dots,\ell},$ that is \[{{\tens{\xi}}}\otimes{{\tens{\eta}}}:=\begin{pmatrix}
\xi_{1}\eta_{1} & \xi_{1}\eta_{2} & \cdots & \xi_{1}\eta_{\ell} \\
\xi_{2}\eta_{1} & \xi_{2}\eta_{2} & \cdots & \xi_{2}\eta_{\ell} \\
\vdots & \vdots & & \vdots \\
\xi_{k}\eta_{1} & \xi_{k}\eta_{2} & \cdots & \xi_{k}\eta_{\ell}
\end{pmatrix}\in {\mathbb{R}}^{k\times \ell}.\]}
\noindent{\underline{\it Assumption {\bf (A-vect)}}.} Given a bounded, open, {Lipschitz} set $\Omega\subset{\mathbb{R}^{n}}$, $n\geq 2$, we investigate solutions ${{\tens{u}}}:\Omega\to{\mathbb{R}^{m}}$ to the problem \begin{equation}
\label{eq:mu}\begin{cases}-{\tens{\dv}} {\mathcal{ A}}(x,{D\teu})={\tens{\mu}}\quad\text{in }\ \Omega,\\
{{\tens{u}}}=0\quad\text{on }\ \partial\Omega\end{cases} \end{equation}
with a datum ${\tens{\mu}}$ being a vector-valued bounded Radon measure and a function ${\mathcal{ A}}:\Omega\times{\mathbb{R}^{n\times m}}\to{\mathbb{R}^{n\times m}}$ is a weighted operator of Orlicz growth expressed by the means of $g(t):=G'(t)$, where an $N$-function $G \in C^2((0,\infty))\cap C({[0,\infty)})$ satisfies $i_G\geq 2$ with $i_G$ given by~\eqref{iG-sG}. {Let $g\in\Delta_2\cap\nabla_2$.} Namely, ${\mathcal{ A}}$ is assumed to admit a~form\begin{equation}
\label{opA:def}{\mathcal{ A}}(x,\xi)=a(x)\frac{g(|\xi|)}{|\xi|}\,\xi,\end{equation} where $a:\Omega\to[c_a,C_a],$ $0<c_a<C_a$ is a continuous function with a modulus of continuity $\omega_a$.
We define a potential
\begin{equation} \label{Wolff-potential}
\mathcal{W}^{\mu}_G(x_0,R)=\int_0^R g^{-1}\left(\frac{|{\tens{\mu}}|(B_r(x_0))}{r^{n-1}}\right)\,dr. \end{equation} For the case of referring to the dependence of some quantities on the parameters of the problem, we collect them as \[\textit{\texttt{data}}=\textit{\texttt{data}}(i_G,s_G,c_a,C_a,\omega_a,n,m).\]Having~\eqref{opA:def}, one can infer the strong monotonicity of the vector field ${\mathcal{ A}}$ of a form given by Lemma~\ref{lem:DiEt-mon}.
Let us define the notion of very weak solutions we employ.
\noindent{\underline{\it Solutions Obtained as a Limit of Approximation}.} A map ${\tens{u}}\in W^{1,g}(\Omega,{\mathbb{R}^{m}})$ is called a SOLA to~\eqref{eq:mu} under the regime of {\rm Assumption {\bf (A-vect)}}, if there exists a sequence $({\tens{u}}_{h})\subset W^{1,G }(\Omega,{\mathbb{R}^{m}})$ of local energy solutions to the systems \[-{\tens{\dv}}{\mathcal{ A}}(x,D{\tens{u}}_{h})={{\tens{\mu}}_h}\] such that ${\tens{u}}_{h}\to {\tens{u}}$ locally in $W^{1,g}(\Omega,{\mathbb{R}^{m}})$ and $({\tens{\mu}}_h)\subset C^\infty (\Omega,{\mathbb{R}^{m}})$ is a sequence of smooth maps that converges to ${\tens{\mu}}$ weakly in the sense of measures and satisfies \begin{equation}
\label{conv-of-meas}
\limsup_h |{\tens{\mu}}_h |(B) \leq |{\tens{\mu}}|(B) \end{equation} for every ball $B\subset\Omega$.
Observe that the above approximation property immediately implies that a SOLA ${\tens{u}}$ is a distributional solution to~\eqref{eq:mu}, that is, \[\int_\Omega{\mathcal{ A}}(x,{D\teu}):D{\tevp}\,dx=\int{\tens{\vp}} \,d{\tens{\mu}}\qquad\text{for every }{\tens{\vp}}\in C^\infty (\Omega,{\mathbb{R}^{m}}).\]
\subsection{Main results}\label{ssec:main-results} Our main accomplishment reads as follows.
\begin{theo}[Pointwise Wolff potential estimates]\label{theo:pointwise}
Suppose ${\tens{u}}:\Omega\to{\mathbb{R}^{m}}$ is a SOLA to~\eqref{eq:mu} with ${\mathcal{ A}}$ satisfying {\rm Assumption {\bf (A-vect)}} and ${\tens{\mu}}\in{\mathcal{M}(\Omega,\Rm)}$. Let $B_r(x_0)\Subset\Omega$ with $r<R_0$ for some $R_0=R_0(\textit{\texttt{data}})$. If ${\mathcal{W}}^{\tens{\mu}}_G(x_0,r)$ is finite, then $x_0$ is a Lebesgue's point of ${\tens{u}}$ and\begin{equation}
\label{Wolff-osc-est}
|{\tens{u}}(x_0)-({\tens{u}})_{B_r(x_0)}|\leq C_{\mathcal{W}}\left({\mathcal{W}}^{\tens{\mu}}_G(x_0,r)+\barint_{B_r(x_0)}|{\tens{u}}-({\tens{u}})_{B_r(x_0)}|\,dx\right)
\end{equation} holds for $C_{\mathcal{W}}>0$ depending only on $\textit{\texttt{data}}$. In particular, we have the following pointwise estimate \begin{flalign}
\label{eq:u-est} |{\tens{u}}(x_0 )|\leq C_{\mathcal{W}}\left(\mathcal{W}^{{\tens{\mu}}}_G (x_0 ,r)+\barint_{B_r(x_0)} |{\tens{u}}(x)|dx\right). \end{flalign} \end{theo}
\subsection{Local behaviour of very weak solutions} \label{ssec:loc-beh} Potential estimates are known to be an efficient tool to bring precise information on the local behaviour of solutions. We refer to~\cite{KuMi2014} for clearly presented overview of consequences of estimates like~\eqref{eq:u-est} in studies on $p$-superharmonic functions together with a bunch of related references and to~\cite{CGZG-Wolff} for similar results for $\mathcal{A}$-harmonic functions where the operator $\mathcal{A}$ is exhibiting Orlicz type of growth. Notice however we referred to the scalar case. In the vectorial one, the only investigations on the potential estimates to solutions to measure data problem we are aware of are available in~\cite{CiSch,KuMi2018} for $p$-Laplace systems.
Let us present the regularity consequences of Wolff potential estimates to $p$-Laplace system with continuous coefficients and the natural scope of operators with similar structure and Orlicz growth.
We start with finding a density condition around a point $x_0$ implying that a solution has vanishing mean oscillations at $x_0$. The proposition below does not follow from Theorem \ref{theo:pointwise}, for the proof see Section~\ref{sec:mainproof}.
\begin{prop}[VMO criterion] \label{prop:vmo} Suppose ${\mathcal{ A}}$ satisfies {\rm Assumption {\bf (A-vect)}} and ${\tens{\mu}}\in{\mathcal{M}(\Omega,\Rm)}$. Let ${\tens{u}}$ be a SOLA to~\eqref{eq:mu} and let $B_r(x_0)\Subset\Omega.$ If\begin{equation}
\label{mu-shrinks}\lim_{\varrho\to 0}\varrho\, g^{-1}\left(\frac{|{\tens{\mu}}|(B_\varrho(x_0))}{\varrho^{n-1}}\right)=0,
\end{equation}
then ${\tens{u}}$ has vanishing mean oscillations at $x_0$, i.e.
\begin{equation}
\label{u-VMO-x0}\lim_{\varrho\to 0}\,\barint_{B_\varrho(x_0)}|{\tens{u}}-({\tens{u}})_{B_\varrho(x_0)}|\,dx=0.
\end{equation}
\end{prop}
An application of Theorem~\ref{theo:pointwise} is the following continuity criterion proven also in Section~\ref{sec:mainproof}.
\begin{theo}[Continuity criterion]
\label{theo:continuity} Suppose ${\tens{u}}$ be a SOLA to~\eqref{eq:mu} under the regime of {\rm Assumption {\bf (A-vect)}} and $B_r(x_0)\Subset\Omega.$ If\begin{equation}
\label{Wolff-shrinks}
\lim_{\varrho\to 0}\sup_{x\in B_r(x_0)}{\mathcal{W}}^{\tens{\mu}}_G(x,\varrho)=0,
\end{equation} then ${\tens{u}}$ is continuous in $B_r(x_0).$
\end{theo}
If ${\tens{\mu}}=0$, then ${\mathcal{W}}^{\tens{\mu}}_G(x,\varrho)=0$, thus trivially we have the following consequence. \begin{coro}\label{coro:ah-cont} Under {\rm Assumption {\bf (A-vect)}} if ${\tens{u}}$ is an ${\mathcal{ A}}$-harmonic map in $\Omega,$ then ${\tens{u}}$ is continuous in every $\Omega'\Subset\Omega.$ \end{coro}
Condition~\eqref{Wolff-shrinks} holds true provided the datum belongs to a Lorentz-type space. In order to define it we recall some definitions. We denote by $f^\star$ the decreasing rearrangement of a measurable function $f:\Omega\to{\mathbb{R}}$ by $$
f^\star(t) = \sup \{ s \geq 0 \colon |\{x\in {\mathbb{R}}^n:f(x)>s\}| > t \}, $$ the maximal rearrangement by $$ f^{\star \star}(t) =\frac 1t \int_0^t f^\star(s) \, ds\quad\text{and}\quad f^{\star \star}(0)= f^{\star}(0). $$ Following \cite{stein-w} by Lorentz space $L(\alpha, \beta)(\Omega)$ for $\alpha,\beta>0$ we mean the class of measurable functions such that $$
\int_0^\infty \left( t^{1/ \alpha} f^{\star \star}(t)\right)^\beta\,\frac{dt}{t}
<\infty. $$ {The following fact is proven in Appendix.}
\begin{lem}\label{lem:Wolff-est} Suppose ${\tens{\mu}}={\tens{F}}:{\mathbb{R}^{n}}\to{\mathbb{R}^{m}}$ is a locally integrable map vanishing outside $\Omega$, then there exists a constant $c=c(n,i_G,s_G)>0$, such that\[{\mathcal{W}}_G^{{\tens{F}}}(x,R)\leq c\int_0^{|B_R|} t^\frac{1}{n} g^{-1}\left(t^\frac{1}{n} |{\tens{F}}|^{\star\star}(x)\right)\,\frac{dt}{t}=: \mathcal{I}_R.\] \end{lem} Note that if $\mathcal{I}_R<\infty$, the datum ${\tens{F}}$ is in the dual space to $W^{1,G}(B_R)$ and, consequently, we deal with a weak solution, see~\cite{ACCZG}. Let us present a corollary of Theorem~\ref{theo:continuity} holding due to Lemma~\ref{lem:Wolff-est}.
\begin{coro}
\label{coro:cont-2} If ${\tens{u}}$ is a weak solution to $-{\tens{\dv}}{\mathcal{ A}}(x,{D\teu})={\tens{F}}$ with ${\mathcal{ A}}$ satisfying {\rm Assumption {\bf (A-vect)}} and ${\tens{F}}:\Omega\to{\mathbb{R}^{m}}$ such that \begin{equation}
\label{cond-Lor}
\mathcal{I}:=\int_0^{|\Omega|} t^\frac{1}{n} g^{-1}\left(t^\frac{1}{n} |{\tens{F}}|^{\star\star}(x)\right)\,\frac{dt}{t} <\infty\end{equation}
for $\Omega_0\Subset\Omega$, then ${\tens{u}}\in C(\Omega_0,{\mathbb{R}^{m}})$ {and $\|{\tens{u}}\|_{L^\infty(\Omega_0,{\mathbb{R}^{m}})}\leq c(\textit{\texttt{data}}) \mathcal{I}.$ This bound is optimal and attained by a radial solution on a ball, see \cite{ACCZG}. Moreover, as special cases we get that ${\tens{u}}$ is continuous under the regularity restrictions on $|{\tens{F}}|=f$ of \cite[Example~1 (A) and Example~2 (A)]{ACCZG}, still within our regime requiring $g\in\Delta_2\cap\nabla_2$ and $i_G\geq 2$.}\end{coro} The above corollary results in the following extension of \cite[Theorem 10.6]{KuMi2018} to the weighted case.
\begin{rem}\label{rem:p-Lorentz}\rm If ${\tens{u}}$ is a weak solution to $-{ {\tens{\dv}}}\left(a(x)|{D\teu}|^{p-2} {D\teu} \right)={\tens{F}}$ for $p\geq 2$ and $0<a\in C(\Omega)$ is separated from zero and $|{\tens{F}}|$ belongs locally to the Lorentz space $L(\tfrac{n}{p},\tfrac{1}{p-1})(\Omega)$ then ${\tens{u}}$ is continuous in $\Omega$. \end{rem} As another application of Corollary~\ref{coro:cont-2} let us present its consequences for the Zygmund case. \begin{rem} \rm Suppose that $2\leq p<n,$ {$\alpha\geq 0$}, $0<a\in C(\Omega)$ separated from zero is bounded, and ${\tens{u}}$ is a weak solution to \begin{equation}
\label{eq-log-Fu} -{ {\tens{\dv}}}\left(a(x)|{D\teu}|^{p-2}\log^\alpha({\rm e}+|{D\teu}|) \,{D\teu} \right)={\tens{F}}.\end{equation} Observe that in this case $g^{-1}(\lambda)\approx \lambda^\frac{1}{p-1}\log^{-\frac{\alpha}{p-1}}({\rm e}+\lambda)$. If ${\tens{F}}$ satisfies~\eqref{cond-Lor}, then ${\tens{u}}$ is continuous in $\Omega$. \end{rem}
Wolff potential estimates can be used to find a relevant conditions on a measure ${\tens{\mu}}$ to infer H\"older continuity of solutions. One of the natural ones is expressed in the Orlicz modification of the Morrey-type scale. \begin{theo}[H\"older continuity criterion] \label{theo:H-cont} Suppose ${\tens{u}}$ be a SOLA to~\eqref{eq:mu} under the regime of {\rm Assumption {\bf (A-vect)}}. Assume further that for ${\tens{\mu}}$ there exist positive constants $c=c(\textit{\texttt{data}})>0$ and $\theta\in (0,1)$ such that \begin{equation}
\label{mu-control}
|{\tens{\mu}}|(B_r(x))\leq c r^{n-1}g(r^{\theta-1})\approx r^{n-\theta}G(r^{\theta-1}) \end{equation} for each $B_r(x)\Subset\Omega$ with sufficiently small radius. Then ${\tens{u}}$ is locally H\"older continuous in $\Omega$.
\end{theo} For $p$-growth problems condition~\eqref{mu-control} reads as $|{\tens{\mu}}|(B_r(x))\leq c r^{n-p+\theta(p-1)}$ well known since~\cite{Car,KiMa92,kizo,RaZi}. Moreover, in the scalar case~\eqref{mu-control} is proven in~\cite{CGZG-Wolff} to be equivalent to H\"older continuity of solutions, while in~\cite{ChKa} to characterize removable sets for H\"older continuous solutions. In the vectorial case we cannot get equivalence, because by Theorem~\ref{theo:pointwise} we are equipped with one-sided estimate only. Specializing Theorem~\ref{theo:H-cont}, we have the following results.
\begin{rem} \rm Suppose $p\geq 2$, positive $a\in C(\Omega)$ is separated from zero, and ${\tens{u}}$ is a SOLA to \[ -{ {\tens{\dv}}}\left(a(x)|{D\teu}|^{p-2} {D\teu} \right)={\tens{\mu}}\] with $|{\tens{\mu}}|(B_r(x))\leq c r^{n-p+\theta(p-1)}$ for some $c>0,$ $\theta\in(0,1)$ and all sufficiently small $r>0.$ Then ${\tens{u}}$ is locally H\"older continuous in $\Omega$. \end{rem} \begin{rem} \rm Suppose $p\geq 2,$ $\alpha\in{\mathbb{R}}$, positive $a\in C(\Omega)$ is separated from zero, and ${\tens{u}}$ is a SOLA to \begin{equation}
\label{eq-log-mu}- {\tens{\dv}}\left(a(x)|{D\teu}|^{p-2}\log^\alpha({\rm e}+|{D\teu}|) {D\teu} \right)={\tens{\mu}}
\end{equation} with $|{\tens{\mu}}|(B_r(x))\leq c r^{n-p+\theta(p-1)}\log^\alpha({\rm e}+r^{\theta-1}) $ for some $c>0,$ $\theta\in(0,1)$ and all sufficiently small $r>0.$ Then ${\tens{u}}$ is locally H\"older continuous in $\Omega$. \end{rem}
The sufficient condition for~\eqref{mu-control} and, in turn, for the H\"older continuity of the solution is to assume that $|{\tens{\mu}}|=|{\tens{F}}|$ belongs to a~relevant Marcinkiewicz-type space. Following~\cite{oneil} for a continuous increasing function $\psi:(0,|\Omega|)\to (0,\infty)$ we say that $f\in L(\psi,\infty)(\Omega)$ if the maximal rearrangement $f^{\star\star}$ of $f$ satisfies
\[\sup_{s\in(0,|\Omega|)}\frac{f^{\star\star}(s)}{\psi^{-1}(1/s)}<\infty.\] There holds the following consequence of Theorem~\ref{theo:H-cont}.
\begin{coro}\label{coro:O-Marc-dens} Suppose ${\tens{u}}$ is a SOLA to $-{\tens{\dv}}{\mathcal{ A}}(x,{D\teu})={\tens{F}}$ under the regime of {\rm Assumption {\bf (A-vect)}} and $|{\tens{F}}|$ belongs locally to the Marcinkiewicz-type space $L(\psi,\infty)(\Omega)$ with $\psi^{-1}(1/\lambda)=\lambda^{-\frac{1}{n}}g\big(\lambda^{\frac{\theta-1}{n}}\big)$ for some $\theta\in(0,1)$,
then ${\tens{u}}$ is locally H\"older continuous. \end{coro} For justification that indeed under the assumptions of Corollary~\ref{coro:O-Marc-dens} the condition~\eqref{mu-control} is satisfied see the calculations provided for the scalar case \cite[Section~2]{CGZG-Wolff}. The above fact has the best possible consequence in the $p$-Laplace case.
\begin{rem}\rm \label{rem:p-Marc-dens} If $p\geq 2$, positive $a\in C(\Omega)$ is separated from zero, ${\tens{u}}$ is a SOLA to $-{ {\tens{\dv}}}\left(a(x)|{D\teu}|^{p-2} {D\teu} \right)={\tens{F}}$ and $|{\tens{F}}|$ belongs locally to the Marcinkiewicz space $L(\frac{n}{p+\theta(p-1)},\infty)(\Omega)$ for some $\theta\in(0,1)$, i.e. $\sup_{\lambda>0}\left(\lambda^\frac{n}{p+\theta(p-1)}\big|\{x\in\Omega_0:\,|{\tens{F}}(x)|>\lambda\}|\right)<\infty$ for any $\Omega_0\Subset\Omega$, then ${\tens{u}}$ is locally H\"older continuous in $\Omega$. \end{rem}
\begin{rem}\label{rem:G-Marc-dens} \rm When $G(t)\approx t^p\log^\alpha({\rm e}+t),$ $p\geq 2,$ $\alpha\in{\mathbb{R}}$, ${\tens{u}}$ is a SOLA to \eqref{eq-log-Fu} with ${\tens{F}}$ such that \[\sup_{\lambda>0}\left(\lambda^\frac{n}{p+\theta(p-1)}\log^{-\frac{\alpha(1-\theta)}{p+\theta(p-1)}}({\rm e}+\lambda^\frac{n}{1-\theta})\big|\{x\in\Omega_0:\,|{\tens{F}}(x)|>\lambda\}|\right)<\infty,\] for any $\Omega_0\Subset\Omega$, then ${\tens{u}}$ is locally H\"older continuous in $\Omega$. \end{rem}
\section{Preliminaries}\label{sec:prelim}
\subsection{Notation} We shall adopt the customary convention of denoting by $c$ a constant that may vary from line to line. To skip rewriting a constant, we use $\lesssim$. By $a\approx b$, we mean $a\lesssim b$ and $b\lesssim a$. To stress the dependence of the intrinsic constants on the parameters of the problem, we write $\lesssim_\textit{\texttt{data}}$ or $\approx_\textit{\texttt{data}}$. By $B_R$ we denote a ball skipping prescribing its center, when {it} is not important. By $cB_R=B_{cR}$ we mean a ball with the same center as $B_R$, but with rescaled radius $cR$. We make use of the truncation operator, $T_k:{\mathbb{R}^{m}}\to{\mathbb{R}^{m}}$, defined as follows
\begin{equation}\label{Tk}T_k({\tens{\xi}}):=\min\left\{1,\frac{k}{|{\tens{\xi}}|}\right\}{\tens{\xi}}. \end{equation} Then, of course, $D T_k:{\mathbb{R}^{m}}\to{\mathbb{R}}^{m\times m}$ is given by\begin{equation}
\label{DTk}DT_k({\tens{\xi}})=\begin{cases}
{\mathsf{Id}}&\text{if }\ |{\tens{\xi}}|\leq k,\\
\frac{k}{|{\tens{\xi}}|}\left({\mathsf{Id}}-\frac{{\tens{\xi}} \otimes {\tens{\xi}}}{|{\tens{\xi}}|^2}\right)&\text{if }\ |{\tens{\xi}}|> k.
\end{cases} \end{equation}
For a~measurable set $U\subset {\mathbb{R}^{n}}$ with finite and positive $n$-dimensional Lebesgue measure $|U|>0$ and ${\tens{f}}\colon U\to \mathbb{R}^{k}$, $k\ge 1$ being a measurable map, we define \begin{flalign*}
({\tens{f}})_U=\barint_{U}{\tens{f}}(x) \, dx =\frac{1}{|U|}\int_{U}{\tens{f}}(x) \,dx. \end{flalign*}
By $C^{0,\gamma}(U)$, $\gamma \in (0,1]$, we mean the family of H\"older continuous functions, i.e. measurable functions $f\colon U\to \mathbb{R}^k$ for which \begin{flalign*}
[{\tens{f}}]_{0,\gamma}:=\sup_{\substack{x,y\in U,\\x\not =y}}\frac{|{\tens{f}}(x)-{\tens{f}}(y)|}{|{x-y}|^{\gamma}}<\infty. \end{flalign*} We {describe the ellipticity of a vector field ${\mathcal{ A}}$} using a function ${\mathcal{V}}:{\mathbb{R}^{n\times m}}\to{\mathbb{R}^{n\times m}}$ \begin{equation}
\label{V-def}
{\mathcal{V}}({\xi})= \left(\frac{g(|{\xi}|)}{|{\xi}|}\right)^{1/2}\xi . \end{equation}
\subsection{Basic definitions} References for this section {are~\cite{rao-ren,KR}}.
We say that a function $G: [0, \infty) \to [0, \infty]$ {is a Young function} if it is convex, vanishes at $0$, and is neither identically equal to 0, nor to infinity. A Young function $G$ which is finite-valued, vanishes only at $0$ and satisfies the additional growth conditions \begin{equation*}\lim _{t \to 0}\frac{G (t)}{t}=0 \qquad \hbox{and} \qquad \lim _{t \to \infty }\frac{G (t)}{t}=\infty \end{equation*} is called an $N$-function. The complementary~function $\widetilde{G}$ (called also the Young conjugate, or the Legendre transform) to a nondecreasing function $G:{[0,\infty)}\to{[0,\infty)}$ is given by the following formula \[\widetilde{G}(s):=\sup_{t>0}(s\cdot t-G(t)).\] If $G$ is a Young function, so is $\widetilde{G}$. If $G$ is an $N$-function, so is $\widetilde{G}$.
Having Young functions $G,\widetilde G$, we are equipped with Young's inequality reading as follows\begin{equation} \label{in:Young} ts\leq G(t)+\widetilde{G}(s)\quad\text{for all }\ s,t\geq 0. \end{equation}
We say that a function $G:{[0,\infty)}\to{[0,\infty)}$ satisfies $\Delta_2$-condition if there exist $c_{\Delta_2},t_0>0$ such that $G(2t)\leq c_{\Delta_2}G(t)$ for $t>t_0.$
We say that $G$ satisfy $\nabla_2$-condition if $\widetilde{G}\in\Delta_2.$
Note that it is possible that $G$ satisfies only one of the conditions $\Delta_2/\nabla_2$. For instance, for $G(t) =( (1+|t|)\log(1+|t|)-|t|)\in\Delta_2$, its complementary function is $\widetilde{G}(s)= (\exp(|s|)-|s|-1 )\not\in\Delta_2$. See~\cite[Section~2.3, Theorem~3]{rao-ren} for equivalence of various definitions of these conditions and~\cite{CGZG,DFL} for illustrating the subtleties. In particular, $G\in\Delta_2\cap\nabla_2$ if and only if $1<i_G\leq s_G<\infty$, see~\eqref{iG-sG}. This assumption implies a comparison with power-type functions i.e.
$\frac{G(t)}{t^{i_G}}$
is non-decreasing
and
$\frac{G(t)}{t^{s_G}}$
is non-increasing, but it is stronger than being sandwiched between power functions.
\begin{lem}\label{lem:equivalences}
If {an $N$-function} $G\in\Delta_2\cap\nabla_2$, then
$g(t)t\approx G(t)$
and
$\widetilde G(g(t))\approx G(t)$
with the constants depending only on the growth indexes of $G$, that is $i_G$ and $s_G$. Moreover, $g^{-1}(2t)\leq c g^{-1}(t)$ with $c=c(i_G,s_G).$
\end{lem}
Due to Lemma~\ref{lem:equivalences} and~\cite[Lemmas~3 and~21]{DiEt}, we have the following relations. \begin{lem}\label{lem:DiEt-mon} If $G$ is an $N$-function of class $C^2((0,\infty))\cap C([0,\infty))$, $G{,g}\in\Delta_2\cap\nabla_2$, ${\mathcal{ A}}$ is given by~\eqref{opA:def}, then for every $\xi,\eta\in{\mathbb{R}^{n\times m}}$ it holds \begin{equation}
\label{opA:strict-monotonicity}
\left({\mathcal{ A}}(x,\xi)-{\mathcal{ A}}(x,\eta)\right):{(\xi-\eta)}\gtrsim_{\textit{\texttt{data}}}\, \frac{g(|\xi|+|\eta|)}{|\xi|+|\eta|}|\xi-\eta|^2\approx_{\textit{\texttt{data}}} \left| {\mathcal{V}}(\xi)-{\mathcal{V}}(\xi) \right|^2, \end{equation} and\begin{equation}
\label{relation:g-V} g(|\xi|+|\eta|)|\xi-\eta|\approx_{\textit{\texttt{data}}} G^\frac{1}{2}(|\xi|+|\eta|) |{\mathcal{V}}(\xi)-{\mathcal{V}}(\eta)|. \end{equation} \end{lem}
\subsection{Orlicz spaces}
Basic reference for this section is~\cite{adams-fournier}, where the theory of Orlicz spaces is presented for scalar functions. The proofs for functions with values in ${\mathbb{R}^{m}}$ can be obtained by obvious modifications.
We study the solutions to PDEs in the Orlicz-Sobolev spaces equipped with a~modular function $G\in C^1 {((0,\infty))}$ - a strictly increasing and convex function such that $G(0)=0$ and satisfying~\eqref{iG-sG}. Let us define a modular \begin{equation}
\label{modular}
\varrho_{G,U}({\tens{\xi}})=\int_U G(|{\tens{\xi}}|)\,dx. \end{equation}
For any bounded $\Omega\subset{\mathbb{R}^{n}}$, by Orlicz space ${L}^G(\Omega,{\mathbb{R}^{m}})$ we understand the space of measurable functions endowed with the Luxemburg norm
\[||{\tens{f}}||_{L^G(\Omega)}=\inf\left\{\lambda>0:\ \ \varrho_{G,\Omega}\left( \tfrac{1}{\lambda} |{\tens{f}}|\right)\leq 1\right\}.\]
We define the Orlicz-Sobolev space $W^{1,G}(\Omega)$ as follows \begin{equation*}
W^{1,G}(\Omega,{\mathbb{R}^{m}})=\big\{{\tens{f}}\in {W^{1,1}(\Omega,{\mathbb{R}^{m}})}:\ \ |{\tens{f}}|,|D{{\tens{f}}}|\in L^G(\Omega,{\mathbb{R}^{m}})\big\}, \end{equation*}where the gradient is understood in the distributional sense, endowed with the norm \[
\|{\tens{f}}\|_{W^{1,G}(\Omega,{\mathbb{R}^{m}})}=\inf\bigg\{\lambda>0 :\ \ \varrho_{G,\Omega}\left( \tfrac{1}{\lambda} |{\tens{f}}|\right)+ \varrho_{G,\Omega}\left( \tfrac{1}{\lambda} |D{{\tens{f}}}|\right)\leq 1\bigg\} \] and by $W_0^{1,G}(\Omega,{\mathbb{R}^{m}})$ we denote the closure of $C_c^\infty(\Omega,{\mathbb{R}^{m}})$ under the above norm.
Since condition~\eqref{iG-sG} imposed on $G$ implies $G,\widetilde{G}\in\Delta_2$, the Orlicz-Sobolev space $W^{1,G}(\Omega,{\mathbb{R}^{m}})$ we deal with is separable and reflexive. Moreover, one can apply arguments of \cite{Gossez} to infer density of smooth functions in $W^{1,G}(\Omega,{\mathbb{R}^{m}})$.
The counterpart of the H\"older inequality in this setting reads \begin{equation}
\label{in:Hold} \|{\tens{f}}{\tens{g}}\|_{L^1(\Omega,{\mathbb{R}^{m}})}\leq 2\|{\tens{f}}\|_{L^G(\Omega,{\mathbb{R}^{m}})}\|{\tens{g}}\|_{L^{\widetilde{G}}(\Omega,{\mathbb{R}^{m}})} \end{equation} for all ${\tens{f}}\in L^G(\Omega,{\mathbb{R}^{m}})$ and $ {\tens{g}}\in L^{\widetilde{G}}(\Omega,{\mathbb{R}^{m}})$.
\subsection {The operator} We notice that in such regime the operator $\mathsf{A}_{G}$ acting as \begin{flalign*} \langle\mathsf{A}_{G}u,{\tens{\phi}}\rangle:=\int_{\Omega}{\mathcal{ A}}(x,{D\teu}): {D\teph} \, dx\quad \text{for}\quad {\tens{\phi}}\in C^{\infty}_{0}(\Omega,{\mathbb{R}^{m}}) \end{flalign*} is well defined on a reflexive and separable Banach space $W^{1,G}(\Omega,{\mathbb{R}^{m}})$ and $\mathsf{A}_{G}(W^{1,G}(\Omega,{\mathbb{R}^{m}}))\subset (W^{1,G}(\Omega,{\mathbb{R}^{m}}))'$. Indeed, when ${\tens{u}}\in W^{1,G}(\Omega,{\mathbb{R}^{m}})$ and ${\tens{\phi}}\in C_c^\infty(\Omega,{\mathbb{R}^{m}})$, structure condition~\eqref{opA:def}, H\"older's inequality~\eqref{in:Hold}, and Lemma~\ref{lem:equivalences} justify that \begin{flalign*}
|{\langle \mathsf{A}_{G}u,{\tens{\phi}} \rangle}|\le &\,c\,\int_{\Omega}g(|{{D\teu}}|){|{D\teph}|} \, dx \le c\left \| g(|{{D\teu}}|) \right \|_{L^{\widetilde G(\cdot)} }\|{|{D\teph}|}\|_{L^{G}}\nonumber \\
\le &\, c\|{|{D\teu}|}\|_{L^{ G}}\|{|{D\teph}|}\|_{L^{G}}\le c\|{{\tens{\phi}}}\|_{W^{1,G}}. \end{flalign*}
\subsection{Definitions of solutions and comments on existence results} \label{ssec:sols}
We stress that the problems are considered under the regime of {\rm Assumption {\bf (A-vect)}}.
A function ${\tens{v}}\in W_{loc}^{1,G}(\Omega,{\mathbb{R}^{m}})$ is called an \underline{${\mathcal{ A}}$-harmonic map} in $\Omega\subset{\mathbb{R}^{n}}$ provided \begin{equation}\label{pre-hom-eq}
\int_\Omega {\mathcal{ A}}(x,{D\tev}):D{\tevp}\,dx=0\qquad\text{for all }\ {\tens{\vp}}\in C_c^\infty(\Omega,{\mathbb{R}^{m}}). \end{equation} As a consequence of our main result, in Corollary~\ref{coro:ah-cont}, we prove that ${\mathcal{ A}}$-harmonic maps are continuous. In fact, by Campanato's characterization \cite[Theorem~2.9]{giusti} one can infer from Proposition~\ref{prop:osc} H\"older continuity $C^{0,\gamma}(\Omega,{\mathbb{R}^{m}})$ of ${\mathcal{ A}}$-harmonic maps with any exponent $\gamma\in (0,1).$
A function ${\tens{u}}\in W^{1,G}_{loc}(\Omega,{\mathbb{R}^{m}})$ is called {a} {weak solution} to~\eqref{eq:mu}, if\begin{equation}
\label{eq:main-mu-weak}\int_\Omega {\mathcal{ A}}(x,{D\teu}): D{\tevp}\,dx=\int_\Omega{\tens{\vp}}\,d{\tens{\mu}}(x)\quad\text{for every }\ {\tens{\vp}}\in C^\infty_c(\Omega,{\mathbb{R}^{m}}). \end{equation} Recall that $W^{1,G}_{0}(\Omega,{\mathbb{R}^{m}})$ is separable and by its definition ${C_c^\infty}(\Omega,{\mathbb{R}^{m}})$ is dense there.
\begin{rem}[Existence and uniqueness of weak solutions] \rm For ${\tens{\mu}}\in (W^{1,G}_{0}(\Omega,{\mathbb{R}^{m}}))',$ due to the strict monotonicity of the operator, there exists a unique weak solution to~\eqref{eq:mu}, see~\cite[Section~3.1]{KiSt}. \label{rem:weak-sol} \end{rem}
Recall that the notion of SOLA is defined in Section~\ref{ssec:main-results}. The problem~\eqref{eq:mu} admits a solution of this type for arbitrary bounded measure. \begin{prop}\label{prop:exist} If a vector field ${\mathcal{ A}}$ satisfies {\rm Assumption {\bf (A-vect)}} and ${\tens{\mu}}\in{\mathcal{M}(\Omega,\Rm)}$, then there exists a SOLA $u\in W^{1,g}(\Omega)$ to~\eqref{eq:mu}. \end{prop} The idea to prove it is to
consider \[{\tens{f}}_k(x):= \int_{\mathbb{R}^{m}} \varrho_k(x-y)\, d{\tens{\mu}}(y),\]
where $\varrho_k$ stands for a standard mollifier i.e. for a nonnegative, smooth, and even function such that $\int_{\mathbb{R}^{m}}\varrho(s){\rm\,d}s=1$ we define $\varrho_k(s)=k^n\varrho(ks)$ for $k\in {\mathbb{N}}$. Of course \[{\tens{f}}_k\xrightharpoonup[]{*}{\tens{\mu}}\qquad\text{and}\qquad\sup_k\|{\tens{f}}_k\|_{L^1(\Omega)}\leq|{\tens{\mu}}|({\mathbb{R}^{n}})<\infty.\] By Remark~\ref{rem:weak-sol} one finds ${\tens{u}}_k\in W_0^{1,G}(\Omega,{\mathbb{R}^{m}})$ such that for every ${\tens{\vp}}\in W_0^{1,G}(\Omega,{\mathbb{R}^{m}})$ it holds that \[\int_\Omega {\mathcal{ A}}(x,{D\teu}_k): D{\tevp}\,dx=\int_\Omega{\tens{\vp}}\,{\tens{f}}_k dx.\] The existence of SOLA by passing to the limit can be justified by modification of arguments of \cite{DHM}.
\noindent See \cite{ACCZG,CGZG,IC-measure-data,CiMa} for related existence and regularity results in the scalar case and~\cite{DHM,DS,FR,L1,L2} for vectorial existence results in various regimes.
\subsection{Auxiliary results}
\begin{lem}\label{lem:excess} For $\tens{g}:B\to{\mathbb{R}}^k$, $k\geq 1$ and any ${\tens{\xi}}\in{\mathbb{R}}^k$ it holds that
\[\barint_B|\tens{g}-(\tens{g})_B|\,dx\leq
2\,\barint_B|\tens{g}-{\tens{\xi}}|\,dx.\] \end{lem}
We have the following corollary of the Cavalieri Principle. \begin{lem} \label{lem:cava}
If $\nu\in\mathcal{M}(\Omega)$ has a density $\omega$ (i.e. $d\nu=\omega(x)\,dx$ with $\omega\in L^1(\Omega)$) and $(1+|f|)^{-(\gamma+1)}\omega\in L^1({\mathbb{R}^{n}})$ for some $\gamma>0$, then\[\int_0^\infty \frac{\nu(\{|f|<t\})}{(1+t)^{2+\gamma}}\,dt=\frac{1}{1+\gamma}\int_{\mathbb{R}^{n}}\frac{d\nu}{(1+|f|)^{\gamma+1}}.\] \end{lem}
\begin{lem}[\cite{giusti}, Lemma~6.1] \label{lem:absorb1} Let $\phi:[R/2,3R/4]\to[0,\infty)$ be a function such that \[\phi(r_1)\leq \frac{1}{2}\phi(r_2)+\mathsf{A}+\frac{\mathsf{B}}{(r_2-r_1)^\beta}\qquad\text{for every}\qquad R/2\leq r_1<r_2\leq 3{R}/4\] with $\mathsf{A,B}\geq 0$ and $\beta>0$. Then there exists $c=c(\beta)$, such that\[\phi(R/2)\leq c \left(\mathsf{A}+\frac{\mathsf{B}}{R^\beta}\right).\] \end{lem}
\begin{lem}[\cite{HanLin}, Lemma~3.4] \label{lem:absorb2} Let $\phi(t)$ be a nonnegative and nondecreasing function on $[0,R]$. Suppose that \[\phi(\rho) \leq \mathsf{A} \left[ \left(\frac{\rho}{r}\right)^{\alpha} + \epsilon \right] \phi(r) + \mathsf{B} r^{\beta}\] for any $0 < \rho \leq r \leq R$, with $\mathsf{A},\mathsf{B},\alpha,\beta$ nonnegative constants and $\beta < \alpha$. Then for any $\gamma \in (\beta, \alpha)$, there exists a constant $\epsilon_{0} = \epsilon_{0}(\mathsf{A},\alpha, \beta, \gamma)$ such that if $\epsilon < \epsilon_0$, then for all $0 < \rho \leq r \leq R$ we have \[ \phi(\rho) \leq c \left \{ \left( \frac{\rho}{r} \right)^{\gamma} \phi(r) + \mathsf{B} \rho^{\beta} \right\}\] where $c$ is a positive constant depending on $\mathsf{A}, \alpha, \beta, \gamma$. \end{lem}
The next lemma is a self-improving property for the reverse H\"older inequalities.
\begin{lem}[\cite{hekima}, Lemma~3.38]\label{lem:self} Let $0<r<q<p<\infty$, $0< \rho < R \leq 1$ and $w \in L^{p}(B_1)$. If the following reverse H\"older inequality holds $$\bigg( \int_{B_{\sigma'}} w^p \, dx \bigg)^{1/p} \leq \frac{c_0}{(\sigma -\sigma')^{\varkappa}} \bigg( \int_{B_{\sigma}} w^{q} \, dx \bigg)^{1/q} + c_1$$ for some constants $c_0$ and $c_1$, whenever $\rho \leq \sigma' \leq \sigma \leq R$. Then there exists $c=c(c_0,\xi,p,q,r)$ such that $$\bigg( \int_{B_{\rho}} w^p \, dx \bigg)^{1/p} \leq \frac{c}{(R - \rho)^{\widetilde{\varkappa}}} \bigg( \int_{B_{R}} w^{r} \, dx \bigg)^{1/r} + c_1,$$ where $$\widetilde{\varkappa}=\frac{\varkappa r(p-q)}{q(p-r)}.$$ \end{lem}
The following modular version of the Sobolev-Poincar\'e inequality follows almost directly as in~\cite{Baroni-Riesz}, but we present the proof for vector-valued functions for the sake of completeness. \begin{prop} \label{prop:OrSob}
Suppose $\Omega$ is a bounded Lipschitz domain in ${\mathbb{R}^{n}}$, $m,n\geq 1$, and $G:{[0,\infty)} \to{[0,\infty)}$ is an $N$-function such that $G\in\Delta_2\cap\nabla_2$. Then there exist a constant $C=C(n,m,|\Omega|,G)>0$, such that for every ${\tens{u}}\in W_0^{1,G}(\Omega,{\mathbb{R}^{m}})$
\[\int_\Omega G^{n'}(|{\tens{u}}|)\,dx\leq C\left(\int_\Omega G (|{D\teu}|)\,dx\right)^{n'}.\]
\end{prop} \begin{proof} We provide the proof only in the case of continuously differentiable $G$. Otherwise every time one can find a sufficiently smooth function $G_\circ$ comparable to $G$, i.e.~such that there exists $c>0$, such that $G_\circ(t)/c\leq G(t)\leq cG_\circ(t).$ Moreover, we start with the proof for fixed ${\tens{u}}\in C_c^\infty(\Omega,{\mathbb{R}^{m}})$ and then conclude by the density argument. The classical Sobolev in $W^{1,1}$ inequality gives
\begin{equation}
\label{SobN'}
\left(\int_\Omega G^{n'}(|{\tens{u}}|)\,dx\right)^\frac{1}{n'}\leq c\,\int_\Omega |{D(G(| {\tens{u}}|))}|\,dx .
\end{equation} Since $G\in\Delta_2,$ it satisfies $g(t)\leq c\, G(t)/t$ and $G^*\left(\frac{G(t)}{t}\right)\leq {G(t)}.$ Thus by the Young inequality we arrive at
\begin{equation}
\label{grad-vp-est}\begin{split}
|{D(G(| {\tens{u}}|))}|&= g(|{\tens{u}}|)\big|{D|{\tens{u}}|}\big|\leq c \frac{G(|{\tens{u}}|)}{{\tens{u}}}|{D\teu}|\\
&\leq\varepsilon G^*\left(\frac{G(|{\tens{u}}|)}{|{\tens{u}}|}\right)+c \, G(|{D\teu}|)\leq \varepsilon G(|{\tens{u}}|)+c\, G(|{D\teu}|).\end{split}
\end{equation}
Summing up, we have
\[\left(\int_\Omega G^{n'}(|{\tens{u}}|)\,dx\right)^\frac{1}{n'}\leq C\varepsilon \int_\Omega G(| {\tens{u}}|) \,dx+Cc_\varepsilon\int_\Omega G(|{D\teu}|)\,dx,\] where according to the H\"older inequality we obtain
\[\left(\int_\Omega G^{n'}(|{\tens{u}}|)\,dx\right)^\frac{1}{n'} \leq \varepsilon C|\Omega|^\frac{1}{n}\left( \int_\Omega G^{n'} (| {\tens{u}}|) \,dx\right)^\frac{1}{n'}+Cc_\varepsilon\int_\Omega G(|{D\teu}|)\,dx.\]
Now we can choose $\varepsilon$ small enough to absorb it on the right-hand side and obtain the claim for ${\tens{u}}\in C_c^\infty(\Omega,{\mathbb{R}^{m}})$. Since smooth function are dense in $W_0^{1,G}(\Omega,{\mathbb{R}^{m}})$ by standard approximation argument we get the claim for all ${\tens{u}}\in W_0^{1,G}(\Omega,{\mathbb{R}^{m}})$. \end{proof}
\subsection{Properties of ${\mathcal{ A}}$-harmonic maps}
Let us establish some fundamental properties of ${\mathcal{ A}}$-harmonic maps. \begin{prop}[Caccioppoli estimate]\label{prop:Cacc} If ${\tens{v}} \in W_0^{1,G} (\Omega,{\mathbb{R}^{m}})$ is a nonnegative ${\mathcal{ A}}$-harmonic map, ${\tens{\lambda}}\in{\mathbb{R}^{m}}$, and $7/8\leq\sigma'<\sigma\leq 1,$ then there exists $c=c(\textit{\texttt{data}})>0$, such that \begin{equation} \label{inq:caccI}
\barint_{B_{\sigma' r}}G(|{D\tev}|)\,dx\leq\frac{c}{(\sigma'-\sigma)^{s_G}}\barint_{B_{\sigma r}}G\left(\frac{|{\tens{v}}-{\tens{\lambda}}|}{r}\right) \,dx. \end{equation} \end{prop}
\begin{proof} Let us pick a~cutoff function $\eta\in C_c^\infty({B_{\sigma r}},{\mathbb{R}^{m}})$ such that $\mathds{1}_{B_{\sigma' r}}\leq\eta\leq \mathds{1}_{B_{\sigma r}}$ and $|{D\teet}|\leq c_1/(\sigma'-\sigma)$. We use $\xi=\eta^q (v-{\tens{\lambda}})$ as a test function to get \[\int_\Omega {\mathcal{ A}}(x,{D\tev}): {D\tev}\,\eta^q \,dx =\int_\Omega {\mathcal{ A}}(x,{D\tev}):(-q\eta^{q-1}({\tens{v}}-{\tens{\lambda}})\otimes {D\teet})\, \,dx .\] Therefore, due to coercivity of ${\mathcal{ A}}$ and the Cauchy-Schwartz inequality we have
\begin{equation*}\int_{B_{\sigma r}}G(|{D\tev}|)\eta^q\,dx\leq c\,\int_{B_{\sigma r}}\frac{g(|{D\tev}|)}{|{D\tev}|}\eta^{q-1}|{D\tev}:(({\tens{v}}-{\tens{\lambda}})\otimes{D\teet})| \,dx=:\mathcal{K} \end{equation*} Noting that $q$ is large enough to satisfy $s_G'\geq q'$, we have in turn that $\widetilde{ G}(\eta^{q-1}t)\leq c \eta^q\widetilde{G}(t)$ and \[\widetilde{G}(\eta^{q-1}g(t))\leq c \eta^q \widetilde{G}(g(t))\leq c \eta^q G(t).\] Then, using Young inequality~\eqref{in:Young} applied to the integrand of $\mathcal{K}$ we get \begin{equation*}
\begin{split}\mathcal{K}&\leq \varepsilon \int_{B_{\sigma r}}\widetilde{G}(\eta^{q-1}|{D\tev}|)\, dx+ c_\varepsilon \int_{B_{\sigma r}}G\left(|{\tens{v}}-{\tens{\lambda}}|\,|{D\teet}| /c_1 \right)\, dx\\& \leq \varepsilon c\,\int_{B_{\sigma r}} \eta^{q }G(|{D\tev}| ) \,dx+ c_\varepsilon \int_{B_{\sigma r}} G\left(|{\tens{v}}-{\tens{\lambda}}|\,|{D\teet}| /c_1 \right)\, dx \end{split} \end{equation*} with arbitrary $\varepsilon<1$. Choosing $\varepsilon$ small enough to absorb the term, and finally by properties of $\eta$ we obtain~\eqref{inq:caccI}. \end{proof}
An ${\mathcal{ A}}$-harmonic function ${\tens{v}}$ is a minimizer of a functional
\[ {\tens{v}} \mapsto \int_{\Omega} G(|{D\tev}|) \, dx. \] Therefore, taking the operator independent of the spacial variable\begin{equation}
\label{opA-no-x}
{\mathcal{ A}}(x,\xi)=c_a\frac{g(|\xi|)}{|\xi|}\xi\end{equation} ${\mathcal{ A}}$-harmonic functions are Lipschitz regular by the following fact.
\begin{lem}[\cite{DSV1}, Lemma~5.8] Suppose $G$ is an $N$-function of class $C^2((0,\infty))\cap C([0,\infty))$, $G,g\in\Delta_2\cap\nabla_2$, ${\tens{w}}$ is ${\mathcal{ A}}$-harmonic in $\Omega$ for ${\mathcal{ A}}$ given by~\eqref{opA-no-x}. Let $B\subset 2B\Subset\Omega$. \label{lem:DSV1} Then there exists $c=c(\textit{\texttt{data}})>0$ such that \[\sup_{B} G(|D{\tens{w}}|) \leq c\, \barint_{2B} G(|D{\tens{w}}|) \, dx.\] \end{lem}
\begin{prop} If ${\tens{v}}\in W^{1,G}(\Omega,{\mathbb{R}^{m}})$ is ${\mathcal{ A}}$-harmonic, then for any $\varsigma \in (0,1)$ there exists $R_0=R_0(\textit{\texttt{data}},\varsigma) \in (0,1]$ such that for any $0< R \leq R_0$ and $B_R\subset \subset \Omega$ it holds that \label{prop:osc} \begin{equation}\label{osc-dec-est}
\barint_{B_{\delta R}} |{\tens{v}} - ({\tens{v}})_{B_{\delta R}}| \, dx \leq c_{\rm o} \delta^{1+(\varsigma - 1)/{s_{G}}} \barint_{B_{R}} |{\tens{v}} - ({\tens{v}})_{B_{R}}| \, dx \end{equation} whenever $\delta \in (0,1/4],$ where $c_{\rm o}\geq 1$ is a constant depending only on $\textit{\texttt{data}}$ and $\varsigma$. \end{prop} \begin{proof}
In this proof, we use a classical perturbation argument, see for instance \cite[Theorem 3.8]{HanLin}. It suffices to prove \eqref{osc-dec-est} for ${\tens{v}}$ solving \[-{\tens{\dv}}\left(a(x)\frac{g(|{D\tev}|)}{|{D\tev}|}{D\tev}\right)=0\quad\text{ in }\quad B_1.\] Indeed, the general case can be deduced then by considering $\widetilde{{\tens{v}}}(x)= {{\tens{v}}(x_0+Rx)}/{R}$ solving
$-{\tens{\dv}}{\bar{\opA}}(x,{D\widetilde{{\tens{v}}}})=0$ on $B_1(0)$ with \[{\bar{\opA}}(x,\xi)=\bar{a}(x)\frac{\bar g(|\xi|)}{|\xi|}\xi=\frac{1}{g(R)}{\mathcal{ A}}(x_0+Rx,R\xi)= {a(x_0+Rx)}\frac{g(R|\xi|)}{g(R)|\xi|}{\xi}.\] In this case, the modulus $\omega_{\bar a}$ of continuity of $\bar a$ satisfies $\omega_{\bar a}(r) = \omega_{a}(rR)$.
Note first Propositions~\ref{prop:OrSob} and~\ref{prop:Cacc} imply that for any $7/8 \leq \sigma' < \sigma \leq 1$
\[ \left( \barint_{B_{\sigma'}} G^{n'}\left( \frac{|{\tens{v}} - ({\tens{v}})_{B_{\sigma'}}|}{\sigma'} \right) \, dx \right)^{1/n'} \leq \frac{c}{(\sigma-\sigma')^{s_G}} \barint_{B_{\sigma}} G\left( \frac{|{\tens{v}}-({\tens{v}})_{B_{\sigma}}|}{\sigma} \right) \, dx.\] From the doubling property of $G$ and the upper and lower bound on $\sigma'$ and $\sigma$, we have
\[ \left( \barint_{B_{\sigma'}} G^{n'}\left( |{\tens{v}} - ({\tens{v}})_{B_{\sigma'}}| \right) \, dx \right)^{1/n'} \leq \frac{c}{(\sigma-\sigma')^{s_G}} \barint_{B_{\sigma}} G\left( |{\tens{v}}-({\tens{v}})_{B_{\sigma}}| \right) \, dx. \] Using the triangular inequality and Jensen's inequality \begin{align*}
& \left( \barint_{B_{\sigma'}} G^{n'}\left( |{\tens{v}} - ({\tens{v}})_{B_{1}}| \right) \, dx \right)^{1/n'} \\
&\qquad\qquad \leq c \left( \barint_{B_{\sigma'}} G^{n'}\left( |{\tens{v}} - ({\tens{v}})_{B_{\sigma'}}| \right) \, dx \right)^{1/n'} + c\,G \left( |({\tens{v}})_{B_{\sigma'}} - ({\tens{v}})_{B_{1}}| \right) \\
&\qquad\qquad \leq \frac{c}{(\sigma-\sigma')^{s_G}} \barint_{B_{\sigma}} G\left( |{\tens{v}}-({\tens{v}})_{B_{\sigma}}| \right) \, dx + c\,\barint_{B_{\sigma'}}G\left( |{\tens{v}} - ({\tens{v}})_{B_{1}}| \right) \, dx \\
&\qquad\qquad \leq c\left(\frac{1}{(\sigma-\sigma')^{s_G}} + 1 \right) \barint_{B_{\sigma}} G\left( |{\tens{v}}-({\tens{v}})_{B_{1}}| \right) \, dx + c\,G \left( |({\tens{v}})_{B_{1}} - ({\tens{v}})_{B_{\sigma}}| \right) \\
&\qquad\qquad \leq \frac{c}{(\sigma-\sigma')^{s_G}} \barint_{B_{\sigma}} G\left( |{\tens{v}}-({\tens{v}})_{B_{1}}| \right) \, dx, \end{align*} for $c=c(\textit{\texttt{data}})$. From Lemma~\ref{lem:self}, for any $t \in (0,1/s_{G})$, we have \begin{align} \label{self-imp-vexc}
\left( \barint_{B_{7/8}} G^{n'}\left(|{\tens{v}} - ({\tens{v}})_{B_{7/8}}|\right) \, dx \right)^{1/n'}
& \leq c \left( \barint_{B_{7/8}} G^{n'}\left(|{\tens{v}} - ({\tens{v}})_{B_{1}}|\right) \, dx \right)^{1/n'} \notag\\
& \leq c\left( \barint_{B_{1}} G^{t}\left(|{\tens{v}} - ({\tens{v}})_{B_{1}}|\right) \, dx \right)^{1/t} \notag \\
& \leq c \,G \left( \barint_{B_{1}} |{\tens{v}} - ({\tens{v}})_{B_{1}}| \, dx \right). \end{align} Here, in the last line, as $t \mapsto G^{t}(t)$ is a concave function for every $t \in (0, 1/s_{G})$ we have used Jensen's inequality.
Let ${\tens{w}} \in {\tens{v}} + W^{1,G}_{0}(B_{\sigma}, {\mathbb{R}}^m)$ for any $\sigma\in(1,1/2)$ be the weak solution to \begin{equation}\label{pre-ref-eq}
- {\tens{\dv}} \left(a(x_0) \frac{g(|{D\tew}|)}{|{D\tew}|}{D\tew}\right) = 0 \qquad \text{in } B_{\sigma}. \end{equation} Recall that the function ${\mathcal{V}}$ describing ellipticity of ${\mathcal{ A}}$ is defined in~\eqref{V-def} and $a$ is a function bounded below by ${c_a}>0$ and with a modulus of continuity $\omega_a$. Testing~\eqref{pre-ref-eq} and~\eqref{pre-hom-eq} against $({\tens{w}} - {\tens{v}})$, and applying Lemma~\ref{lem:DiEt-mon}, for any $\epsilon \in (0,1]$ we see \begin{align*}
& \int_{B_{\sigma}} c_a \left| {\mathcal{V}}({D\tew})-{\mathcal{V}}({D\tev}) \right|^2 \, dx \\
&\qquad\qquad\leq c \int_{B_{\sigma}} a(x_0) \left(\frac{g(|{D\tew}|)}{|{D\tew}|} {D\tew}- \frac{g(|{D\tev}|)}{|{D\tev}|} {D\tev}\right) : ({D\tew} - {D\tev}) \, dx \\
&\qquad\qquad = c \int_{B_{\sigma}} (a(x_0) - a(x)) \frac{g(|{D\tev}|)}{|{D\tev}|} {D\tev} : ({D\tew}-{D\tev}) \, dx \\
&\qquad\qquad \leq c \,\omega_{a}(\sigma)\int_{B_{\sigma}} g(|{D\tew}| + |{D\tev}|)\, |{D\tew}-{D\tev}| \, dx \\
&\qquad\qquad \leq c\, \epsilon \int_{B_{\sigma}} |{\mathcal{V}}({D\tew})-{\mathcal{V}}({D\tev})|^2 \, dx + c\,\frac{\omega_{a}(1/2)^2}{\epsilon} \int_{B_{\sigma}} G(|{D\tew}|+|{D\tev}|) \, dx, \end{align*} where in the last line we used Young's inequality. As $c=c(\textit{\texttt{data}})$ and $c_a>0$, we can take $\epsilon$ small enough to absorb the first term on the right-hand side. Then Jensen's inequality implies \begin{flalign}\nonumber
\int_{B_{\sigma}} \left| {\mathcal{V}}({D\tew})-{\mathcal{V}}({D\tev}) \right|^2 \, dx &\leq c\, \omega_{a}(1/2)^2 \int_{B_{\sigma}} [G(|{D\tev}|) + G(|{D\tew}|)] \, dx.\end{flalign} Since ${\tens{w}}$ is a minimizer of the integral functional
\[ {\tens{w}} \mapsto \int_{B_{\sigma}} G(|{D\tew}|) \, dx, \] we have \begin{flalign}\label{pre-excess-1}
\int_{B_{\sigma}} \left| {\mathcal{V}}({D\tew})-{\mathcal{V}}({D\tev}) \right|^2 \, dx \leq c\, \omega_{a}(1/2)^2 \int_{B_{\sigma}} G(|{D\tev}|) \, dx. \end{flalign}
Since we know the Lipschitz regularity of ${\tens{w}}$ provided by Lemma~\ref{lem:DSV1} \[\sup_{B_{\sigma/2}} G(|D{\tens{w}}|) \leq c \,\barint_{B_{\sigma}} G(|D{\tens{w}}|) \, dx,\] it follows from \eqref{pre-excess-1} that \begin{align*}
\int_{B_{\delta \sigma}} G(|{D\tev}|) \, dx &\leq c\,
\int_{B_{\delta \sigma}} |{\mathcal{V}}({D\tev})|^2 \, dx \\
& \leq c\, \int_{B_{\delta \sigma}} |{\mathcal{V}}({D\tev}) - {\mathcal{V}}({D\tew})|^2 \, dx + \int_{B_{\delta \sigma}} |{\mathcal{V}}({D\tew})|^2 \, dx \\
& \leq c\, \int_{B_{\sigma}} |{\mathcal{V}}({D\tev}) - {\mathcal{V}}({D\tew})|^2 \, dx + \delta^{n} \int_{B_{\sigma}} |{\mathcal{V}}({D\tew})|^2 \, dx \\
& \leq c \left( \delta^{n} + \omega_{a}(1/2)^2 \right) \int_{B_{\sigma}} G(|{D\tev}|) \, dx. \end{align*} We take $R_0=R_{0}(\textit{\texttt{data}},\varsigma)$ small enough to ensure that $\omega_{a}(R_0)^2<\epsilon_{0}$, where $\epsilon_0$ is a constant given in Lemma~\ref{lem:absorb2}. Then Lemma~\ref{lem:absorb2}, Proposition~\ref{prop:Cacc} and \eqref{self-imp-vexc} give \begin{equation}\label{pre-excess-2}
\barint_{B_{\delta}} G(|{D\tev}|) \, dx \leq c \delta^{\varsigma - 1} \barint_{B_{1/2}} G(|{D\tev}|) \, dx \leq c \delta^{\varsigma - 1} G \left( \int_{B_1} |{\tens{v}} - ({\tens{v}})_{B_1}| \, dx \right) \end{equation} where $c$ depends only on $\textit{\texttt{data}}$ and $\varsigma$.
Using the Sobolev-Poincar\'e inequality in $W^{1,1}$ and Jensen's inequality, we conclude that \begin{align*}
\barint_{B_{\delta}} |{\tens{v}} - ({\tens{v}})_{B_{\delta}}| \, dx
& \leq c \,\delta \, \barint_{B_{\delta}} |{D\tev}| \, dx \\
& \leq c \,\delta \, G^{-1} \left( \barint_{B_{\delta}} G(|{D\tev}|) \, dx \right) \\
& \leq c \, \delta^{1+(\varsigma - 1)/i_{G}} \int_{B_1} |{\tens{v}} - ({\tens{v}})_{B_1}| \, dx, \end{align*} what completes the proof. \end{proof}
\section{Measure data ${\mathcal{ A}}$-harmonic approximation}\label{sec:Ah-approx}
In this section we provide the tool of crucial meaning for our further reasoning -- the approximation of a $W^{1,G}$-function by an ${\mathcal{ A}}$-harmonic map for weighted operator ${\mathcal{ A}}$ of an Orlicz growth given by \eqref{opA:def}. Results in this spirit can be found in \cite{DuGr,DuMi2004,DSV3}, but most preeminently for the approximation relevant for application to measure data problems we refer to \cite[Theorem~4.1]{KuMi2018}.
We define an auxiliary function\begin{equation}\label{Hs-def}
H_s(t)=\int_{0}^{t} \frac{g(r)^{1-s} G(r)^{s}}{r} \, dr\qquad\text{for }\ s \in [0,1/2). \end{equation} It is readily checked that when $s>\max\{2-i_{G},0\}$, $H_s$ is a Young function satisfying \begin{equation}
\label{Hs-prop-1} H_s(t) \approx g(t)^{1-s} G(t)^{s} \end{equation} with intrinsic constants depending only on $i_G,s_G$ and $s$. In fact, $H_s\in\Delta_2\cap\nabla_2$ since \begin{equation}
\label{Hs-prop-2} 0 < s + i_G -2\leq \frac{t H_s''(t)}{H_s'(t)} = \frac{(1-s) t g'(t)}{g(t)} + \frac{s t g(t)}{G(t)} \leq s + s_G -2.\end{equation} Furthermore, there exist $\epsilon,c,t_0>0$, such that $H_s(t)\geq c t^{1+\epsilon}$ and $H_s(t)\geq c g^{1+\epsilon}(t)$ for all $t\geq t_0.$
\begin{theo}\label{theo:Ah-approx} Under Assumption {\bf (A-vect)} let $\varepsilon>0$, $\gamma\in (0,1/(s_Gn))$, and \begin{equation}
\label{s-range}
\max\{2-i_{G},0\} < s <s_{\rm m}:=\frac{i_G-\gamma s_G n}{i_G+ s_Gn}. \end{equation}Suppose that ${\tens{u}}\in W^{1,G}(B_r(x_0),{\mathbb{R}^{m}})$ satisfies\begin{equation}
\label{u-male}\barint_{B_r(x_0)}|{\tens{u}}|\,dx\leq Mr,\quad M\geq 1 \end{equation} then there exists $\delta=\delta(\textit{\texttt{data}},s,M,\varepsilon)\in(0,1]$ such that if ${\tens{u}}$ is almost ${\mathcal{ A}}$-harmonic in a sense that for every ${\tens{\vp}}\in W^{1,G}_0(B_r(x_0),{\mathbb{R}^{m}})\cap L^\infty(B_r(x_0),{\mathbb{R}^{m}})$ it holds \begin{equation}
\label{Du-male}\left|\barint_{B_r(x_0)}
{\mathcal{ A}}(x,{D\teu}):D{\tevp}\,dx\right|\leq\frac{\delta}{r}\|{\tens{\vp}}\|_{L^\infty(B_r(x_0),{\mathbb{R}^{m}})}, \end{equation} then there exists an ${\mathcal{ A}}$-harmonic map ${\tens{v}}\in W^{1,G}(B_{r/2}(x_0),{\mathbb{R}^{m}})$ satisfying
\begin{equation}\label{Du-blisko-Dv}
\barint_{B_{r/2}(x_0)}H_s(|{D\teu}-{D\tev}|)\,dx
\leq \varepsilon \end{equation} together with\begin{equation}\label{v-male}
\barint_{B_{r/2}(x_0)}|{\tens{v}}|\,dx\leq 2^nMr
\quad\text{and}\quad
\barint_{B_{r/2}(x_0)}H_s(|{D\tev}|)\,dx \leq cH_s(M), \end{equation} where $c=c(\textit{\texttt{data}})>0.$ \end{theo} \begin{rem} \label{rem:iG-male} The limitation that $G$ has to be superquadratic ($i_G\geq 2$) can be a~little bit relaxed in Theorem~\ref{theo:Ah-approx} and later on the restriction is not needed. The key property is to ensure that the range of admissible $s$ from \eqref{s-range} is {nonempty. We} need to assume that $i_G$ is either bigger or equal to $2$, or close to $2$ in a sense that \[2-i_G <\frac{i_G-1}{i_G+ s_Gn}.\] \end{rem} \begin{proof}The plan is to first establish suitable a priori estimates for the rescaled problem and then proceed with the proof via contradiction. The proof is presented in $6$ steps.
\textbf{Step 1. Scaling. } We fix arbitrary ${\tens{\vp}}\in W^{1,G}_0(B_r(x_0),{\mathbb{R}^{m}})\cap L^\infty(B_r(x_0),{\mathbb{R}^{m}})$ satisfying~\eqref{Du-male}. Let us change variables putting\begin{equation}
\label{bar}
{\bar{\tens{u}}}(x):=\frac{{\tens{u}}(x_0+rx)}{Mr},\ {\bar{\opA}}(x_0+rx,{\tens{\xi}})={\mathcal{ A}}(x_0+rx,M{\tens{\xi}}),\ \text{and}\ {\tens{\eta}}(x):=\frac{{\tens{\vp}}(x_0+rx)}{r}. \end{equation} Then ${\bar{\opA}}$ satisfies the same conditions as ${\mathcal{ A}}$ with the functions $ {\bar{g}}(t):=g(Mt)$ and ${\bar{G}}(t):=G(Mt)/M$ (with ${\bar{G}}'={\bar{g}}$), and the constants depending on $\textit{\texttt{data}}.$ Of course in such a case $i_G=i_{\bar{G}}$ and $s_G=s_{\bar{G}}.$
Having~\eqref{u-male} and~\eqref{Du-male}, by denoting the unit ball by $B_1$, we get further that \begin{flalign}
\label{bu-male}&\barint_{B_1}|{\bar{\tens{u}}}|\,dx\leq 1,\\
\label{Dbu-male} &\left|\barint_{B_1} {\bar{\opA}}(x,{D\tebu}):{D\teet}\,dx\right|\leq {\delta}\|{\tens{\eta}}\|_{L^\infty(B_1,{\mathbb{R}^{m}})}. \end{flalign}
\textbf{Step 2. A priori estimates. } We choose $q\geq s_G$, pick \begin{equation}
\label{eta}{\tens{\eta}}:=\phi^qT_k({\bar{\tens{u}}})\quad\text{with some}\ \phi\in C_c^\infty(B_1),\ 0\leq\phi\leq 1,\ k\geq 0, \end{equation}
and denote\[{\mathsf{P}}:=\frac{{\bar{\tens{u}}}\otimes{\bar{\tens{u}}}}{|{\bar{\tens{u}}}|^2}.\] Then \begin{flalign*}
{D\teet}=&\mathds{1}_{\{|{\bar{\tens{u}}}|\leq k\}}(\phi^q {D\tebu}+q\phi^{q-1}{\bar{\tens{u}}}\otimes D\phi)\\
&+\mathds{1}_{\{|{\bar{\tens{u}}}|> k\}}(\phi^q ({{\mathsf{Id}}}-{{\mathsf{P}}}) {D\tebu}+q\phi^{q-1}{\bar{\tens{u}}}\otimes D\phi). \end{flalign*} We use~\eqref{eta} in~\eqref{Dbu-male} to get\begin{flalign}
\nonumber& \left|\int_{B_1\cap \{|{\bar{\tens{u}}}|\leq k\}} {\bar{\opA}}(x,{D\tebu}):(\phi^q {D\tebu}+q\phi^{q-1}{\bar{\tens{u}}}\otimes D\phi)\,dx\right.\\
&\left.+\int_{B_1\cap \{|{\bar{\tens{u}}}|> k\}} {\bar{\opA}}(x,{D\tebu}):(\phi^q ({\mathsf{Id}}-{\mathsf{P}}) {D\tebu}+q\phi^{q-1}{\bar{\tens{u}}}\otimes D\phi)\,dx\right|\leq {\delta}|B_1|\|{\tens{\eta}}\|_{L^\infty(B_1,{\mathbb{R}^{m}})}.\label{est-1} \end{flalign} Since ${\bar{\opA}}$ has the quasi-diagonal structure resulting from~\eqref{opA:def} and
\[ {D\tebu}:\big(({\mathsf{Id}}-{\mathsf{P}}){D\tebu}\big)=|{D\tebu}|^2-\frac{D_j{\bar{\tens{u}}}^\alpha{\bar{\tens{u}}}^\alpha D_j{\bar{\tens{u}}}^\beta{\bar{\tens{u}}}^\beta}{|{\bar{\tens{u}}}|^2}=|{D\tebu}|^2-\frac{\sum_{j=1}^m\langle D_j {\bar{\tens{u}}} ,{\bar{\tens{u}}}\rangle ^2}{|{\bar{\tens{u}}}|^2}\geq 0, \] we infer that \begin{equation}\label{calc-eta}{\bar{\opA}}(x, {D\tebu}):\big(({\mathsf{Id}}-{\mathsf{P}}){D\tebu}\big)\geq 0. \end{equation}
By rearranging terms in~\eqref{est-1}, applying {Lemma~\ref{lem:equivalences},} noting that $\|{\tens{\eta}}\|_{L^\infty(B_1,{\mathbb{R}^{m}})}\leq k$, and dropping a~nonnegative term due to~\eqref{calc-eta}, we get for some $c=c(\textit{\texttt{data}},q)$\begin{flalign}
\nonumber \int_{B_1\cap \{|{\bar{\tens{u}}}|\leq k\}}{\bar{G}}(|{D\tebu}|) &\phi^q\,dx\leq c \int_{B_1\cap \{|{\bar{\tens{u}}}|\leq k\}} \frac{{\bar{g}}(|{D\tebu}|)}{|{D\tebu}|}\phi^{q-1}\big|{D\tebu}:({\bar{\tens{u}}}\otimes D\phi)\big|\,dx\\
+\,c&\int_{B_1\cap \{|{\bar{\tens{u}}}|> k\}} \frac{k}{|{\bar{\tens{u}}}|} \frac{{\bar{g}}(|{D\tebu}|)}{|{D\tebu}|}\phi^{q-1}\big|{D\tebu}:({\bar{\tens{u}}}\otimes D\phi)\big|\,dx+ c|B_1|\delta k.\nonumber \end{flalign} We estimate the first term on the right-hand side of the last display by the use of Young inequality with a parameter, use Lemma~\ref{lem:equivalences}, and we absorb one term. The second term can be estimated by Schwartz inequality. Altogether we obtain \begin{flalign}\nonumber
\int_{B_1\cap \{|{\bar{\tens{u}}}|< k\}}{\bar{G}}(|{D\tebu}|) \phi^q\,dx&\leq c \int_{B_1\cap \{|{\bar{\tens{u}}}|< k\}}{\bar{G}}(|{\bar{\tens{u}}}|\,|D\phi|) \,dx+ c|B_1|\delta k\\
&+\,ck \int_{B_1\cap \{|{\bar{\tens{u}}}|\geq k\}} {{\bar{g}}(|{D\tebu}|)}\phi^{q-1}| D\phi|\,dx \label{st2-apriori} \end{flalign} for some $c=c(\textit{\texttt{data}},q)$.
\textbf{Step 3. Summability of ${\bar{\tens{u}}}$ and $D{\bar{\tens{u}}}$. } We choose $k=t$ in~\eqref{st2-apriori}, then multiply this line by $(1+t)^{-(\gamma+2)},$ where $\gamma>0,$ integrate it from zero to infinity and apply Cavalieri's principle (Lemma~\ref{lem:cava}) twice (with $\nu_1={\bar{G}}(|{D\tebu}|) \phi^q$ and $\nu_2={\bar{G}}(|{\bar{\tens{u}}}|\,|D\phi|)$. Altogether we get
\begin{flalign}\nonumber
\frac{1}{1+\gamma} \int_{B_1 }\frac{{\bar{G}}(|{D\tebu}|) \phi^q}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\,dx&\leq
\frac{c}{1+\gamma} \int_{B_1 }\frac{{\bar{G}}(|{\bar{\tens{u}}}|\,|D\phi|) }{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\,dx+ \frac{c}{\gamma}\delta\\
&+\, c\,\int_0^\infty \frac{t}{(1+t)^{\gamma+2}}\int_{B_1\cap \{|{\bar{\tens{u}}}|> t\}} {{\bar{g}}(|{D\tebu}|)}\phi^{q-1}| D\phi|\,dx\,dt.\label{post-cava} \end{flalign} The right-most term in the last display can be estimated as follows
\begin{flalign}\nonumber c\,\int_0^\infty \frac{t}{(1+t)^{\gamma+2}}\int_{B_1\cap \{|{\bar{\tens{u}}}|> t\}}& {{\bar{g}}(|{D\tebu}|)}\phi^{q-1}| D\phi|\,dx\,dt\\\nonumber
&\leq c \int_0^\infty \frac{1}{(1+t)^{\gamma+1}}dt\,\int_{B_1 } {{\bar{g}}(|{D\tebu}|)}\phi^{q-1}|D\phi|\,dx \\
&\leq \frac{c}{\gamma} \int_{B_1 } {{\bar{g}}(|{D\tebu}|)}\phi^{q-1}| D\phi|\,dx .\label{do-younga} \end{flalign} To estimate it further we note that $q$ is large enough to satisfy $s_G'\geq q'$, {there exist $c_0,c_1>0$ depending on $i_G,s_G$, such that we have ${{\bar{G}}}^*(c_0\phi^{q-1}{\bar{g}}(t))\leq c_1 \phi^q {{\bar{G}}}^*({\bar{g}}(t))\leq \phi^q {\bar{G}}(t).$} Then, using Young inequality~\eqref{in:Young} applied to the integrand in \eqref{do-younga} and the above observation we get \begin{equation*}
\begin{split}\int_{B_1 } &{{\bar{g}}(|{D\tebu}|)}\phi^{q-1}|D\phi|\,dx\\
&\leq \frac{1}{2(1+\gamma)} \int_{B_{1}}\frac{{{\bar{G}}^*}\left( {c_0}{\phi^{q-1}{\bar{g}}(|{D\tebu}|) }\right)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} \, dx + \widetilde c \int_{B_{1}}\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)^{1+\gamma}|D\phi|\right)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\,dx \\
& \leq \frac{1}{2(1+\gamma)} \int_{B_{1}}\frac{{\bar{G}}\left(|{D\tebu}| \right)\phi^{q }}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} \,dx + \widetilde c \int_{B_{1}}\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)^{1+\gamma}|D\phi|\right)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\, dx \end{split} \end{equation*} with $\widetilde c=\widetilde c(\gamma,i_G,s_G)$. By applying this estimate in~\eqref{post-cava} and rearranging terms we obtain
\begin{flalign}
\int_{B_1 }\frac{{\bar{G}}(|{D\tebu}|) \phi^q}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\,dx&\leq
{c} \int_{B_{1}}\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)^{1+\gamma}|D\phi|\right)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\, dx + c\,\delta\frac{1+\gamma}{\gamma} .\label{post-cava-2} \end{flalign}
{Observe that $1/(s_G n)< i_G-1$, as otherwise the condition required by Remark~\ref{rem:iG-male} is violated}. Recall that since $\gamma<1/(s_G n)$, we have $ \gamma< i_G-1$. Let us set\begin{equation}
\vartheta(x):=\frac{{\bar{G}}((1+|{\bar{\tens{u}}}|)\phi^q)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} \label{vartheta} \end{equation}
and notice that since $|{D|{\bar{\tens{u}}}|}|\leq|{D\tebu}|$, by Jensen's inequality, and Lemma~\ref{lem:equivalences} we can estimate \begin{flalign}
\nonumber \left|{D\vartheta}\right|&=\frac{\left|{D( {\bar{G}}((1+|{\bar{\tens{u}}}|)\phi^q)}(1+|{\bar{\tens{u}}}|)^{1+\gamma}-{D\left((1+|{\bar{\tens{u}}}|)^{1+\gamma}\right)}G((1+|{\bar{\tens{u}}}|)\phi^q)\right|}{(1+|{\bar{\tens{u}}}|)^{2+2\gamma}}\\\nonumber
&\leq c \, \frac{{\bar{g}}((1+|{\bar{\tens{u}}}|)\phi^q)\left|{D((1+|{\bar{\tens{u}}}|)\phi^q)}\right|}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} + {(1+\gamma)} \frac{\left|D{{\bar{\tens{u}}}} \right|{\bar{G}}((1+|{\bar{\tens{u}}}|)\phi^q)} {(1+|{\bar{\tens{u}}}|)^{2+\gamma}}\\\nonumber
&\leq c \, \frac{{\bar{g}}((1+|{\bar{\tens{u}}}|)\phi^q)|{D{\bar{\tens{u}}}}|\phi^q}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\\ \nonumber
&\quad + c \, \frac{{\bar{g}}((1+|{\bar{\tens{u}}}|)\phi^q)(1+|{\bar{\tens{u}}}|)\phi^{q-1}\left|{D\phi}\right|}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} + {(1+\gamma)} \frac{ \left|{D {\bar{\tens{u}}}}\right|} {1+|{\bar{\tens{u}}}|}\vartheta\\
&{\leq c(\gamma)\frac{ \left|{D {\bar{\tens{u}}}}\right|} {1+|{\bar{\tens{u}}}|}\vartheta+c \, \frac{\left|{D\phi}\right|}{\phi}\vartheta .} \label{Dvt} \end{flalign}
For the later use in \textbf{Step 6}, we emphasize the dependence of constants on~$\gamma$ by denoting $c(\gamma)$. Note that every $c(\gamma)$ in \eqref{Dvt}-\eqref{int-vt} is an increasing function of~$\gamma$.
Since $q \geq s_G$, for any $\kappa \in [1,i_{G})$ we see \begin{align*}
\left|{D\left(\vartheta^{\frac{1}{\kappa}}\right)}\right|
\leq \frac{1}{\kappa} \vartheta^{\frac{1}{\kappa}-1} \left|{D\vartheta}\right|
\leq {c(\gamma)} \, \vartheta^{\frac{1}{\kappa}} \frac{\left|{D{\bar{\tens{u}}}}\right|}{1+|{\bar{\tens{u}}}|} + {c }\, \vartheta^{\frac{1}{\kappa}} \frac{\left|{D \phi} \right|}{\phi} \end{align*} and \begin{align}\label{high-vt}
\left|{D\left(\vartheta^{\frac{1}{\kappa}}\right)}\right|^{\kappa}
&\leq c(\gamma) \frac{G((1+|{\bar{\tens{u}}}|)\phi^{q})}{\left[(1+|{\bar{\tens{u}}}|) \phi^{q} \right]^{\kappa}} \frac{\left(|{D\tebu} | \phi^{q} \right)^{\kappa}}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}
+ c \frac{G(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} |{D \phi}|^{\kappa}. \end{align}
To proceed further, we define an auxiliary function $h_\kappa$ by setting $$h_{\kappa}^{-1}(t):= \int_{0}^{t} \frac{1}{\left[G^{-1}(\tau)\right]^{\kappa}} \, d\tau.$$ A straightforward calculation gives \begin{equation}\label{sob-aux-1} -1 < - \frac{\kappa}{i_G} \leq \frac{t [h_{\kappa}^{-1}]''(t)}{[h_{\kappa}^{-1}]'(t)}
= - \frac{\kappa t [G^{-1}]'(t)}{G^{-1}(t)} \leq - \frac{\kappa}{s_G} <0, \end{equation} which implies that $h_{\kappa}^{-1}$ is an increasing concave function on $[0, \infty)$. Note that \eqref{sob-aux-1} also gives $$h_{\kappa}\left( \frac{G(t)}{t^{\kappa}} \right) \approx G(t)$$ with intrinsic constants depending on $i_G,s_G$ and $\kappa$ only. Moreover, we have $${\bar{g}}(t)[h_{\kappa}^{-1}]'({\bar{G}}(t))=[h_{\kappa}^{-1}({\bar{G}}(t))]' = \frac{d}{dt} \left( \int_{0}^{G(t)} \frac{1}{[G^{-1}(\tau)]^{\kappa}} \, d \tau \right) = \frac{{\bar{g}}(t)}{t^{\kappa}},$$ and so \begin{align*} [h_{\kappa}^{*}]^{-1}(G(t)) \approx h_{\kappa}'(h_{\kappa}^{-1}({\bar{G}}(t)) = \frac{1}{[h_{\kappa}^{-1}]'(h_{\kappa}(h_{\kappa}^{-1}(G(t))))}= \frac{1}{[h_{\kappa}^{-1}]'(G(t))} = t^{\kappa}. \end{align*} Hence, $h_{\kappa}^{*}(t) \approx {\bar{G}}(t^{1/\kappa})$.
Applying Young's inequality~\eqref{in:Young} with the pair of Young functions $(h_{\kappa},h_{\kappa}^{*})$ to \eqref{high-vt}, for any $\epsilon_0 \in (0,1)$ we discover \begin{align*}
\left|{D\left(\vartheta^{\frac{1}{\kappa}}\right)}\right|^{\kappa}
& \leq \epsilon_{0} \frac{G((1+|{\bar{\tens{u}}}|)\phi^{q})}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} + c(\epsilon_0)c(\gamma) \frac{{\bar{G}}\left(|{D\tebu} | \phi^{q} \right)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}
+ c \frac{G(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} |{D \phi}|^{\kappa}. \end{align*} By the classical Sobolev inequality we get that\begin{equation}
\label{Sob}
\left( \int_{B_1} |\vartheta|^\frac{n}{n-\kappa} \,dx\right)^\frac{n-\kappa}{n}\leq c
\int_{B_1} \left|{D \left(\vartheta^{\frac{1}{\kappa}}\right)}\right|^{\kappa} \,dx. \end{equation} Merging \eqref{post-cava}, \eqref{post-cava-2}, \eqref{vartheta}, \eqref{Dvt} and \eqref{Sob} and taking $\epsilon_0$ small enough, we get \begin{equation}\label{int-vt}
\left(\int_{B_1}\left( \frac{{\bar{G}}((1+|{\bar{\tens{u}}}|)\phi^q)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^\frac{n}{n-\kappa}\,dx\right)^\frac{n-\kappa}{n}\leq
\bar c \left( \int_{B_{1}}\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)^{1+\gamma}|D\phi|\right)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\, dx + \frac{1}{\gamma}\right). \end{equation} It is worth to mention that $\bar c=\bar{c}(\textit{\texttt{data}},\gamma,\delta)>0$ depends on $\textit{\texttt{data}}, \gamma$ and $\delta$ and it is an increasing function of $\gamma$ and $\delta$. Since $\gamma\in (0,1/(s_G n))$ is fixed, we may choose $\alpha$ such that \begin{equation}
\label{alpha-range} \alpha \in \left(1,\frac{n}{n-\kappa}\right) \qquad \text{and} \qquad \frac{\alpha s_G\gamma}{\alpha-1} \leq 1. \end{equation} Then
\[\int_{B_{1}}(1+|{\bar{\tens{u}}}|)^{\frac{\alpha s_G\gamma}{\alpha-1}} \, dx\leq \int_{B_{1}} 1+|{\bar{\tens{u}}}|\,dx\leq 1+|B_1|\] and we can estimate
\begin{flalign}\label{int-vt-2} &\int_{B_{1}}\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}})|)^{1+\gamma}|D\phi|\right)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\, dx \leq \int_{B_{1}}\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)|D\phi|\right)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma-\gamma s_G}}\, dx \notag \\
&\qquad\qquad\qquad\leq\left(\int_{B_{1}}(1+|{\bar{\tens{u}}}|)^{\frac{\alpha s_G\gamma}{\alpha-1}} \, dx\right)^\frac{\alpha-1}{\alpha}\left(\int_{B_{1}} \left(\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)|D\phi|\right) }{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^{\alpha} \, dx\right)^\frac{1}{\alpha}\notag \\
&\qquad\qquad\qquad\leq c\left(\int_{B_{1}} \left(\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)|D\phi|\right) }{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^{\alpha} \, dx\right)^\frac{1}{\alpha}. \end{flalign}
Thus, from \eqref{int-vt} and \eqref{int-vt-2} we obtain \color{black}
\begin{equation}\label{do-int-2} \left(\int_{B_1}\left( \frac{{\bar{G}}((1+|{\bar{\tens{u}}}|)\phi^q)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^\frac{n}{n-\kappa}\,dx\right)^\frac{n-\kappa}{n}\leq c\left(\int_{B_{1}}\left(\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)|D\phi|\right) }{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^{\alpha}\, dx\right)^\frac{1}{\alpha} + c . \end{equation}
For $7/8 \leq r_1 < r_2\leq 1$, we take a cut-off function $\phi$ satisfying \[\phi\equiv 1\quad\text{on }\ B_{r_1}\qquad\text{and}\qquad |D\phi|\leq\frac{100}{r_2-r_1}.\] It then follows from the doubling property of ${\bar{G}}$ and Lemma~\ref{lem:self} that for any $\upsilon \in (0,1/s_G)$
\begin{flalign*} \left(\int_{B_{7/8}}\left( \frac{{\bar{G}}(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^\frac{n}{n-\kappa}\,dx\right)^\frac{n-\kappa}{n}
& \leq c\left(\int_{B_{1}}\left(\frac{{\bar{G}}\left(1+|{\bar{\tens{u}}}|\right) }{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^{\upsilon}\, dx\right)^\frac{1}{\upsilon} + c \\
& \leq c \left(\int_{B_{1}} ({\bar{G}}\left(1+|{\bar{\tens{u}}}|\right) )^{\upsilon}\, dx\right)^\frac{1}{\upsilon} + c\\
& \leq {\bar{G}} \left(\int_{B_{1}} (1+|{\bar{\tens{u}}}|)\, dx \right) + c. \end{flalign*} In the last line, we have used Jensen's inequality with the concave function $t \mapsto {\bar{G}}(t)^{\upsilon}$. Recalling \eqref{bu-male}, we obtain \begin{flalign}\label{do-int-3}
\left(\int_{B_{7/8}}\left( \frac{{\bar{G}}(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^\frac{n}{n-\kappa}\,dx\right)^\frac{n-\kappa}{n} \leq c \end{flalign} for some $c=c(\textit{\texttt{data}},\gamma)>0.$
To proceed further, we recall $H_s$ defined in~\eqref{Hs-def} with $s$ from~\eqref{s-range}. By~\eqref{Hs-prop-1} function $H_s$ satisfies also $H_s(Mt) \approx_{\textit{\texttt{data}}} {\bar{g}}(t)^{1-s} {\bar{G}}(t)^{s} $. We apply Young's inequality, \eqref{post-cava-2}, and \eqref{do-int-3} with suitable choice of $\phi$. In turn we see \begin{flalign} \label{H-s}
\int_{B_{\rho_1}} & H_s(M|{D\tebu}|) \,dx
\lesssim_{\textit{\texttt{data}}} \int_{B_{ \rho_1}} \frac{{\bar{g}}(|{D\tebu}|)^{1-s}{\bar{G}}(|{D\tebu}|)^{s}} {(1+|{\bar{\tens{u}}}|)^{1+\gamma}}(1+|{\bar{\tens{u}}}|)^{1+\gamma}\,dx \notag \\
& \leq \int_{B_{ \rho_1}} \frac{{\bar{G}}(|{D\tebu}|)} {(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\,dx
+ \int_{B_{ \rho_1}} \frac{ {\bar{g}}(|{D\tebu}|) (1+|{\bar{\tens{u}}}|)^{\frac{1+\gamma} {1-s}}}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} \,dx \notag \\
& \leq \int_{B_{ \rho_1}} \frac{{\bar{G}}(|{D\tebu}|)} {(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\,dx
+ \int_{B_{ \rho_1}} \frac{ {\bar{G}}(1+|{\bar{\tens{u}}}|) (1+|{\bar{\tens{u}}}|)^{\frac{(s + \gamma) s_G}{(1-s)}}}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}} \,dx \notag \\
&\notag \leq \int_{B_{ \rho_1}} \frac{{\bar{G}}(|{D\tebu}|)} {(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\,dx\\
&\quad+ \left(\int_{B_{ \rho_1}}\left( \frac{{\bar{G}}(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{1+\gamma}}\right)^\frac{n}{n-\kappa}\,dx\right)^\frac{n-\kappa}{n}
\left( \int_{B_{ \rho_1}} (1+|{\bar{\tens{u}}}|)^\frac{(s + \gamma) s_G n}{(1-s)\kappa}\,dx \right)^\frac{\kappa}{n} \end{flalign} In order to use \eqref{bu-male} we need to have \begin{equation}\label{s1} \frac{(s + \gamma) s_G n}{(1-s)\kappa} \leq 1. \end{equation} Observe that by Remark \ref{rem:iG-male} we have $2-i_G <({i_G-\gamma s_G n})/({i_G+ s_Gn}),$ and we can choose $\kappa \in [1, i_G)$ such that \begin{equation*}
\max\{2-i_{G},0\} < s \leq {\frac{\kappa-\gamma s_G n}{\kappa+ s_Gn}}<s_{\rm m} \end{equation*} and the bound \eqref{s1} follows. Then \eqref{H-s} combined with \eqref{post-cava-2}, \eqref{int-vt-2} and \eqref{do-int-3} implies for $s<s_{\rm m}$ that \begin{equation}
\label{H-apriori}\int_{B_{3/4}}H_s(M|{D\tebu}|)\,dx\leq c_{\rm ap}=c_{\rm ap}(\textit{\texttt{data}},\gamma). \end{equation}
\textbf{Step 4. Contradiction argument. } We state the counter--assumption.
Namely, we assume that there exists $\varepsilon$ and a sequences of balls $\{B_{r_j}(x_j)\}$ and almost ${\mathcal{ A}}$-harmonic maps $\{{\tens{u}}_j\}\subset W^{1,G}(B_{r_j}(x_j),{\mathbb{R}^{m}})$ such that \begin{equation}
\label{u-male-j}\barint_{B_{r_j}(x_j)}|{\tens{u}}_j|\,dx\leq Mr_j,\quad M\geq 1, \end{equation} \begin{equation}
\label{Du-male-j}\left|\barint_{B_{r_j}(x_j)}
{\mathcal{ A}}(x,{D\teu}_j):D{\tevp}\,dx\right|\leq\frac{2^{-j}}{r_j}\|{\tens{\vp}}\|_{L^\infty(B_{r_j}(x_j),{\mathbb{R}^{m}})} \end{equation} for all ${\tens{\vp}}\in W^{1,G}_0(B_{r_j}(x_j),{\mathbb{R}^{m}})\cap L^\infty(B_{r_j}(x_j),{\mathbb{R}^{m}}),$ but such that
\begin{equation}\label{Du-daleko-Dv}
\barint_{B_{r_j/2}(x_j)}H_s(|{D\teu}_j-{D\tev}|)\,dx
> \varepsilon \end{equation} whenever ${\tens{v}}\in W^{1,G}(B_{r/2}(x_0),{\mathbb{R}^{m}})$ is an ${\mathcal{ A}}$-harmonic map in $ B_{r/2}(x_0)$ satisfying \begin{equation}\label{v-male-j}
\barint_{B_{r_j/2}(x_0)}|{\tens{v}}|\,dx\leq 2^nMr_j
\quad\text{and}\quad
\left(\barint_{B_{r_j/2}(x_j)}H_s(M|{D\tev}|)\,dx\right)\leq c, \end{equation} where $c=c(\textit{\texttt{data}})>0.$
Let ${\bar{\tens{u}}}$ be scaled as in~\eqref{bar}, but with the use of $x_j$ and $r_j,$ that is we set\[{\bar{\tens{u}}}_j(x):=\frac{{\tens{u}}(x_j+r_jx)}{Mr_j}\quad \text{and}\quad {\bar{\opA}}(x_j+r_jx,{\tens{\xi}}):={\mathcal{ A}}(x_j+r_jx,M{\tens{\xi}}).\] In such a case by~\eqref{u-male-j} we get that\begin{equation}
\label{buj-male} \barint_{B_1}|{\bar{\tens{u}}}_j|\,dx\leq 1, \end{equation} so by~\eqref{Du-male-j} we infer that for all ${\tens{\eta}}={\tens{\vp}}(x_j+r_jx)/r_j\in W^{1,G}_0(B_1,{\mathbb{R}^{m}})\cap L^\infty(B_1,{\mathbb{R}^{m}}) $ it holds \begin{equation}
\label{Dbu-male-j} \left|\barint_{B_1}
{\bar{\opA}}(x,{D\tebu}_j):{D\teet}\,dx\right|\leq {2^{-j}}\|{\tens{\eta}}\|_{L^\infty(B_1,{\mathbb{R}^{m}})} \end{equation} and\begin{equation}
\label{HsM}
\barint_{B_1}H_s(M |{D\tebu}_j-{D\tebv}|)\,dx > \varepsilon \end{equation} whenever ${\bar{\tens{{v}}}}\in W^{1,G}(B_{1/2}(x_0),{\mathbb{R}^{m}})$ is an ${\bar{\opA}}$-harmonic map in $ B_{1/2}(x_0)$ satisfying \begin{equation}\label{bv-male}
\barint_{B_{1/2} }|{\bar{\tens{{v}}}}|\,dx\leq 2^n
\quad\text{and}\quad
\left(\barint_{B_{1/2} }H_s(M|{D\tebv}|)\,dx\right)\leq c, \end{equation} where $c=c(\textit{\texttt{data}})>0.$
Since ${\bar{\tens{u}}}$ satisfies~\eqref{buj-male}, we have~\eqref{H-apriori} for $s_{\rm o}<s_{\rm m}$ from~\eqref{s-range}. Therefore
\[\int_{B_{3/4}}H_{s_{\rm o}}(M|{D\tebu}_j|)\,dx\leq C\quad\text{for}\quad C=C(\textit{\texttt{data}},\gamma).\] We fix any $s<s_{\rm o}$ from the range~\eqref{s-range}. Then we pick $\epsilon>0$ for which there exist $ c,t_0>0$, such that $H_s(t)\geq c t^{1+\epsilon}$ and $H_s(t)\geq c g^{1+\epsilon}(t)$ for all $t\geq t_0.$ In turn, we conclude with the following estimates uniform in $j$ \begin{equation}
\label{unif-int-Duj}
\int_{B_{3/4}}g^{1+\epsilon}(M|{{D\tebu}_j}|)\,dx\leq c_1\quad\text{and}\quad
\int_{B_{3/4}}(M|{{D\tebu}_j}|)^{1+\epsilon}\,dx\leq c_2 \end{equation} with $c_1,c_2$ depending on $\textit{\texttt{data}}$ and $\gamma$ only. Further we infer that there exist \[\text{${\tens{\wt u}}\in W^{1,H_s}(B_{3/4},{\mathbb{R}^{m}}),\quad$ $ {{\mathfrak{A}}}\in L^{1+\epsilon}(B_{3/4},{\mathbb{R}^{n\times m}}),\quad$ and $\quad\mathfrak{h}\in L^{H_s}(B_{3/4})$}\] such that up to a subsequence \begin{equation}
\begin{split}\label{lots-of-conv}
&{{D\tebu}_j}- {{D\tewtu}}\rightharpoonup 0\qquad \text{in } \ L^{H_s}(B_{3/4},{\mathbb{R}^{n\times m}}),\\
&|{{D\tebu}_j}-{{D\tewtu}}| \rightharpoonup \mathfrak{h}\qquad \text{in } \ L^{H_s}(B_{3/4}),\\
&{\bar{\opA}}(x,{{D\tebu}_j}) \rightharpoonup {{\mathfrak{A}}}\qquad \text{in } \ L^{1+\epsilon}(B_{3/4},{\mathbb{R}^{n\times m}}),\\
&{\bar{\tens{u}}}_j\to {\tens{\wt u}}\quad \text{ strongly in $\ L^{H_s}(B_{3/4},{\mathbb{R}^{m}})\ $ and a.e. in $B_{3/4}$.}
\end{split} \end{equation}
By~\eqref{buj-male}, lower semicontinuity of a functional ${\tens{\vp}}\mapsto\barint_{B_{1/2}}H_s(M|D{\tevp}| )\,dx$, and~\eqref{H-apriori} we have \begin{equation}\label{wtu-male}
\barint_{B_{1/2} }|{\tens{\wt u}}|\,dx\leq 2^n
\quad\text{and}\quad
\barint_{B_{1/2} }H_s(M|{{D\tewtu}}|)\,dx\leq c, \end{equation}
\textbf{Step 5. Strong convergence of gradients. } Our aim is now to prove that \begin{equation}
\label{strong-conv-grad} {{D\tebu}_j}\to{{D\tewtu}}\qquad\text{in }\ L^{H_s}(B_{3/4},{\mathbb{R}^{n\times m}}). \end{equation} For this we need to show that $\mathfrak{h}\in L^{H_s}(B_{3/4})$ from~\eqref{lots-of-conv} satisfies \begin{equation}
\label{h=0} \mathfrak{h}=0 \end{equation} a.e. in $B_{3/4}$. This almost everywhere and weak convergence in $L^1$ implies strong $L^1$-convergence of ${{D\tebu}_j}\to{{D\tewtu}}$ in $B_{3/4}$. Using the monotonicity property of $H_s$ and~\eqref{bv-male}, for sufficiently small $\widetilde\epsilon \in (0, \frac{1}{s_{H_{s}}})$ we have \begin{align}\label{aux-esty1}
& \int_{B_{3/4}} H_s ( M |{{D\tebu}_j} - {{D\tewtu}}|) \,dx \\
&\quad \leq \int_{B_{3/4}} H_s( M |{{D\tebu}_j} - {{D\tewtu}}|)^{\widetilde\epsilon^2} H_s( M |{{D\tebu}_j}| + M |{{D\tewtu}}|)^{1-\widetilde\epsilon^2} \,dx \notag \\
&\quad \leq \bigg( \int_{B_{3/4}} H_s( M |{{D\tebu}_j} - {{D\tewtu}}|)^{\widetilde \epsilon} \,dx \bigg)^{\widetilde\epsilon} \bigg( \int_{B_{3/4}} H_s( M |{{D\tebu}_j}| + M |{{D\tewtu}}|)^{1 + \widetilde\epsilon} \,dx \bigg)^{1-\widetilde\epsilon} \notag \\
&\quad \leq c \bigg( \int_{B_{3/4}} H_s( M |{{D\tebu}_j} - {{D\tewtu}}|)^{\widetilde\epsilon} \,dx \bigg)^{\widetilde\epsilon}. \notag \end{align} Denoting $$\Psi (t) = \int_{0}^{t} \frac{H_{s}^{-1}(\tau^{1/\widetilde\epsilon})}{\tau} \, d \tau,$$ one can immediately check $$\frac{t \Psi''(t)}{\Psi'(t)}
= \frac{t^{1/\widetilde \epsilon}}{\widetilde \epsilon H_{s}' ( H_{s}^{-1}(t^{1/\widetilde \epsilon}) ) H_{s}^{-1}(t^{1/\widetilde \epsilon}) } - 1 \geq \frac{1}{\widetilde \epsilon s_{H_{s}}} -1 >0,$$ and so $\Psi$ is a Young function. We then apply Jensen's inequality to \eqref{aux-esty1} to obtain
$$\int_{B_{3/4}} H_s ( M |{{D\tebu}_j} - {{D\tewtu}}|) \,dx
\leq c \bigg[ H_{s} \bigg( M \int_{B_{3/4}} |{{D\tebu}_j} - {{D\tewtu}}| \,dx \bigg) \bigg]^{\widetilde\epsilon^2} \stackrel{j \to \infty}{\longrightarrow} 0.$$ Hence, it remains to show \eqref{h=0} to obtain \eqref{strong-conv-grad}.
We pick $\bar{x}$ being a Lebesgue's point simultaneously for ${{\tens{\wt u}}}, {{D\tewtu}}, \mathfrak{h},{{\mathfrak{A}}}$, that is \begin{flalign}
\label{Leb-point}
\lim_{\varrho\to 0}\barint_{B_\varrho(\bar{x})}&H_s(M|{{\tens{\wt u}}}-{{\tens{\wt u}}}(\bar{x})|)+H_s(M|{{D\tewtu}}-{{D\tewtu}}(\bar{x})|)\\
&+H_s(M|\mathfrak{h}-\mathfrak{h}(\bar{x})|)+|{{\mathfrak{A}}}-{{\mathfrak{A}}}(\bar{x})|^{1+\epsilon}\,dx=0\nonumber \end{flalign} and \begin{equation}
\label{fin-val-barx}
|{{\tens{\wt u}}}(\bar{x})|+|{{D\tewtu}}(\bar{x})|+|\mathfrak{h}(\bar{x})|+|{{\mathfrak{A}}}(\bar{x})|<\infty. \end{equation} Almost every point of $B_{3/4}$ satisfies this conditions. Thus it is enough to show that~\eqref{h=0} holds for $\bar{x}$.
We restrict our attention to $\varrho$ small enough for $B_\varrho(\bar x)\subset B_{3/4}$ and we set the linearization of ${{\tens{\wt u}}}$ at $\bar{x}$\begin{equation}
\label{l-vr} {\tens{\ell_\vr}}(x):=({{\tens{\wt u}}})_{B_{\varrho}(\bar x)} + {{D\tewtu}}(\bar{x}):(x-\bar{x}). \end{equation}Having the classical Poincar\'e inequality and~\eqref{Leb-point}, we obtain that \begin{equation}
\label{poinc-zero}
\lim_{\varrho\to 0}\barint_{B_\varrho(\bar x)}\left|\frac{{{\tens{\wt u}}}-{\tens{\ell_\vr}}}{\varrho}\right|^{1+\epsilon}\,dx\leq c \lim_{\varrho\to 0}\barint_{B_\varrho(\bar x)}\left|{{D\tewtu}}-{{D\tewtu}}(\bar{x})\right|^{1+\epsilon}\,dx=0. \end{equation}
Let us set\[\text{I}_{j,\varrho}^0:=\barint_{B_{\varrho/2}(\bar{x})}|{{D\tebu}_j}-{{D\tewtu}}|\,dx.\]
By~\eqref{lots-of-conv} we have the weak convergence of $|{{D\tebu}_j}-{{D\tewtu}}|\to\mathfrak{h}$ in $L^{H_s}(B_\varrho).$ Since $\bar{x}$ is a~Lebesgue's point of ${D\tewtu}$ we infer that\begin{equation}
\label{h-of-barx}\mathfrak{h}(\bar{x})=\lim_{\varrho\to 0}\lim_{j\to\infty}\text{I}_{j,\varrho}^0. \end{equation} In order to prove that $\mathfrak{h}(\bar{x})=0$, let us write\begin{flalign}
\nonumber \text{I}_{j,\varrho}^0=\barint_{B_{\varrho/2}(\bar{x})}|{{D\tebu}_j}-{{D\tewtu}}|\,dx&=\barint_{B_{\varrho/2}(\bar{x})}\mathds{1}_{\{|{\bar{\tens{u}}}_j-{\tens{\ell_\vr}}|\geq\varrho\}}|{{D\tebu}_j}-{{D\tewtu}}|\,dx\\
&\ \ + \barint_{B_{\varrho/2}(\bar{x})}\mathds{1}_{\{|{\bar{\tens{u}}}_j-{\tens{\ell_\vr}}|<\varrho\}}|{{D\tebu}_j}-{{D\tewtu}}|\,dx:=\text{I}^1_{j,\varrho}+\text{I}^2_{j,\varrho}\label{2ndline} \end{flalign} and prove the convergence of both terms first when $j\to \infty$ and then $\varrho\to 0$.
We start with $\text{I}^1_{j,\varrho}$. Let us observe that\begin{flalign*}
\text{I}^1_{j,\varrho}&\leq\barint_{B_{\varrho/2}(\bar{x})}\mathds{1}_{\{|{\bar{\tens{u}}}_j-{{\tens{\wt u}}}|\geq\varrho\}}|{{D\tebu}_j}-{{D\tewtu}}|\,dx\\
& \quad +\barint_{B_{\varrho/2}(\bar{x})}\mathds{1}_{\{|{\tens{\wt u}}-{\tens{\ell_\vr}}|\geq\varrho\}}|{{D\tebu}_j}-{{D\tewtu}}|\,dx=:\text{I}^{1,1}_{j,\varrho}+\text{I}^{1,2}_{j,\varrho}. \end{flalign*} Notice that $\text{I}^{1,1}_{j,\varrho}$ vanishes as $j\to\infty$. Indeed, $\text{I}^{1,1}_{j,\varrho}\geq 0$ and by H\"older inequality we have\begin{flalign*}
\text{I}^{1,1}_{j,\varrho}&\leq \frac{1}{|B_{\varrho/2}(\bar{x})|} \left(\int_{B_{\varrho/2}(\bar{x})}|{{D\tebu}_j}-{{D\tewtu}}|^{1+\epsilon}\,dx\right)^{\frac{1}{1+\epsilon}}\left(|\{x\in B_{3/4}:\ |{\bar{\tens{u}}}_j-{{\tens{\wt u}}}|\geq\varrho/2\}|\right)^\frac{\epsilon}{1+\epsilon}. \end{flalign*}
Since by~\eqref{lots-of-conv} one has that $|{\bar{\tens{u}}}_j-{{\tens{\wt u}}}|\to 0$ strongly in $L^1(B_{3/4})$, so \[\lim_{j\to\infty} |\{x\in B_{3/4}:\ |{\bar{\tens{u}}}_j-{{\tens{\wt u}}}|\geq\varrho/2\}|=0.\] The rest of the terms are bounded as $\epsilon$ is chosen such that~\eqref{unif-int-Duj} is true and $|{{D\tewtu}}|$ shares the same a priori estimates as $|{D\tebu}_j|$. Therefore, we infer that $\lim_{j\to\infty} \text{I}^{1,1}_{j,\varrho}=0$. On the other hand, $\text{I}^{1,2}_{j,\varrho}$ is convergent when $j\to\infty$, because of the weak convergence of $|{{D\tebu}_j}-{{D\tewtu}}|\to\mathfrak{h}$ in $L^{H_s}(B_\varrho)$. Hence, we get
\[\lim_{j\to\infty}\text{I}^{1,2}_{j,\varrho}=\barint_{B_{\varrho/2}(\bar{x})}\mathds{1}_{\{|{\tens{\wt u}}-{\tens{\ell_\vr}}|\geq\varrho\}}\mathfrak{h}\,dx=:\text{I}^{1,2}_\varrho.\] We can estimate further \begin{flalign*}
\text{I}^{1,2}_\varrho&\leq \left(\barint_{B_\varrho(\bar x)}\mathfrak{h}^{1+\epsilon}\,dx\right)^\frac{1}{1+\epsilon}\left(\barint_{B_\varrho(\bar x)}\mathds{1}_{\{|{\tens{\wt u}}-{\tens{\ell_\vr}}|\geq\varrho/2\}}\,dx\right)^\frac{\epsilon}{1+\epsilon}\\
&\leq c\left[\left(\barint_{B_\varrho(\bar x)}|\mathfrak{h}-\mathfrak{h}(\bar{x})|^{1+\epsilon}\,dx\right)^\frac{1}{1+\epsilon}+\mathfrak{h}(\bar{x})\right]\left(\barint_{B_\varrho(\bar x)}\left|\frac{{\tens{\wt u}}-{\tens{\ell_\vr}}}{\varrho}\right|^{1+\epsilon}\,dx\right)^\frac{\epsilon}{1+\epsilon} \end{flalign*} which tends to $0$ as $\varrho\to 0$ as $\bar{x}$ is a Lebesgue's point of $\mathfrak{h}$ as in~\eqref{Leb-point} and the last bracket converges to $0$ due to~\eqref{poinc-zero}. Altogether, we have that $\text{I}^{1}_{j,\varrho}$ vanishes in the limit, so we will now concentrate on $\text{I}^2_{j,\varrho}$ for which we have \begin{flalign*}
\text{I}^2_{j,\varrho}&\leq \barint_{B_{\varrho/2}(\bar{x})}\mathds{1}_{\{|{\bar{\tens{u}}}_j-{\tens{\ell_\vr}}|<\varrho\}}|{{D\tebu}_j}-D{{\tens{\ell_\vr}}}|\,dx+2^n\barint_{B_{\varrho}(\bar{x})}\mathds{1}_{\{|{\bar{\tens{u}}}_j-{\tens{\ell_\vr}}|<\varrho\}}|D{{\tens{\ell_\vr}}}-{{D\tewtu}}|\,dx\\
&=:\text{I}^{2,1}_{j,\varrho}+\text{I}^{2,2}_{j,\varrho}. \end{flalign*} By~\eqref{lots-of-conv},~\eqref{Leb-point}, and~\eqref{l-vr} we have that $\lim_{\varrho\to 0}\limsup_{j\to\infty}\text{I}^{2,2}_{j,\varrho}=0.$ Proving the convergence \begin{equation}
\label{I22-conv}\limsup_{\varrho\to 0}\limsup_{j\to\infty}\text{I}^{2,1}_{j,\varrho}=0 \end{equation}
requires more arguments. We take \[\text{$\phi\in C_c^\infty(B_\varrho(\bar x))\quad$ with $0\leq \phi\leq 1$, $\ \phi \equiv 1$ on $B_{\varrho/2}(\bar x)\ $ and $\ |D\phi|\leq 4/\varrho.$}\] Let \[{\tens{\eta}}=\phi T_\varrho ({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}),\] where the truncation is defined in~\eqref{Tk}. Let us denote \begin{equation}
\label{PjP}{\mathsf{P_j}}:=\frac{({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})\otimes ({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|^2}\qquad\text{and}\qquad {\mathsf{P}}:=\frac{({{\tens{\wt u}}}-{\tens{\ell_\vr}})\otimes ({{\tens{\wt u}}}-{\tens{\ell_\vr}})}{|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|^2} \end{equation}
when $|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|\neq 0$ and $|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|\neq 0,$ respectively. Within this notation we have that\begin{flalign*}
&\left({\bar{\opA}}(x,{{D\tebu}_j})-{\bar{\opA}}(x,D{{\tens{\ell_\vr}}})\right):{D\teet}\\
&\qquad=\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|<\varrho\}}\left[\left({\bar{\opA}}(x,{{D\tebu}_j})-{\bar{\opA}}(x,D{{\tens{\ell_\vr}}})\right):{D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}\right]\phi\\
&\qquad\quad+\frac{\varrho\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|\geq\varrho\}}}{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|}\left[\left({\bar{\opA}}(x,{{D\tebu}_j})-{\bar{\opA}}(x,D{{\tens{\ell_\vr}}})\right):({\mathsf{Id}}-{\mathsf{P_j}}){D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}\right]\phi\\
&\qquad\quad+\left({\bar{\opA}}(x,{{D\tebu}_j})-{\bar{\opA}}(x,D{{\tens{\ell_\vr}}})\right):\left[T_\varrho({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})\otimes D\phi\right] \\
&\qquad=: G_{j,\varrho}^{1}(x)+G_{j,\varrho}^2(x)+G_{j,\varrho}^3(x) . \end{flalign*} Moreover, we recall that since ${\tens{\ell_\vr}}$ is affine, whenever $B\subset B_1$ it holds that\[\int_B{\bar{\opA}}(x,D{{\tens{\ell_\vr}}}):D{\tevp}\,dx=0\qquad\text{for every }\ {\tens{\vp}}\in W^{1,1}_0(B,{\mathbb{R}^{m}}).\] Therefore, by~\eqref{Dbu-male-j} it is justified to write that\begin{equation}
\label{4.46}
0\leq\barint_{B_\varrho(\bar x)}G^1_{j,\varrho}(x)\,dx\leq 2^{-j}\varrho^{1-n}-\barint_{B_\varrho(\bar x)}G^2_{j,\varrho}(x)\,dx-\barint_{B_\varrho(\bar x)}G^3_{j,\varrho}(x)\,dx. \end{equation}The first term in the above display is nonnegative because of the monotonicity of~${\bar{\opA}}.$ Instrumental for proving that $\text{I}_{j,\varrho}^{2,1}\to 0$ is to establish that \begin{equation}
\label{G1-conv} \limsup_{\varrho\to 0}\limsup_{j\to\infty}\barint_{B_\varrho(\bar x)}G^1_{j,\varrho}(x)\,dx=0, \end{equation} which will be proven provided one justifies that the last two terms of~\eqref{4.46} vanish in the limit. We will show first that\begin{equation}\label{G2-conv} \limsup_{\varrho\to 0} \limsup_{j\to\infty} \left(-\,\barint_{B_\varrho(\bar x)} G^2_{j,\varrho}\,dx\right)\leq 0. \end{equation} The quasi-diagonal structure of ${\bar{\opA}}$ ensures that ${\bar{\opA}}(x,{{D\tebu}_j}):[({\mathsf{Id}}-{\mathsf{P_j}}){{D\tebu}_j}]\geq 0$, see~\eqref{calc-eta}. Therefore,\begin{flalign}
\nonumber &\big({\bar{\opA}}(x,{{D\tebu}_j})-{\bar{\opA}}(x,D{{\tens{\ell_\vr}}})\big):({\mathsf{Id}}-{\mathsf{P_j}}){D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}\\
&\qquad\geq -{\bar{\opA}}(x,{{D\tebu}_j}):({\mathsf{Id}}-{\mathsf{P_j}})D{{\tens{\ell_\vr}}}-{\bar{\opA}}(x,D{{\tens{\ell_\vr}}}):({\mathsf{Id}}-{\mathsf{P_j}}){D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}.\label{monot-ell-vr} \end{flalign}
Recall that ${\mathsf{P_j}}$ and ${\mathsf{P}}$, defined in~\eqref{PjP}, are bounded. Notice that for $j\to\infty$ we have $\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|\geq\varrho\}}{\mathsf{P_j}}\to \mathds{1}_{\{|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|\geq\varrho\}}{\mathsf{P}} $ almost everywhere and thus, by the Lebesgue's dominated convergence theorem, also strongly in $L^t(B_{3/4})$ for every $t\geq 1.$ Moreover, $\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|\geq\varrho\}}|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|^{-1}\to \mathds{1}_{\{|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|\geq\varrho\}}|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|^{-1}$ almost everywhere and, as a uniformly bounded sequence of functions, it converges also strongly in $L^t(B_{3/4})$ for every $t\geq 1.$ Having this,~\eqref{monot-ell-vr}, and~\eqref{lots-of-conv}, we obtain\begin{flalign}
\nonumber \limsup_{j\to\infty} &\left(-\barint_{B_\varrho(\bar x)} G^2_{j,\varrho}\,dx\right)\leq \barint_{B_\varrho(\bar x)}{{\mathfrak{A}}}:({\mathsf{Id}}-{\mathsf{P}})D{{\tens{\ell_\vr}}}\frac{\varrho\mathds{1}_{\{|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|\geq\varrho\}}}{|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|}\,dx\\
&\qquad\qquad+\barint_{B_\varrho(\bar x)}{\bar{\opA}}(x,D{{\tens{\ell_\vr}}}):({\mathsf{Id}}-{\mathsf{P}}){D({{\tens{\wt u}}}-{\tens{\ell_\vr}})}\frac{\varrho\mathds{1}_{\{|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|\geq\varrho\}}}{|{{\tens{\wt u}}}-{\tens{\ell_\vr}}|}\,dx=:\text{II}^1_\varrho+\text{II}^2_\varrho. \end{flalign} We can estimate \begin{flalign*}
|\text{II}^1_{\varrho}|&\leq c\,\barint_{B_\varrho(\bar{x})}|{{\mathfrak{A}}}|\left|\frac{{{\tens{\wt u}}}-{\tens{\ell_\vr}}}{\varrho}\right|^{\epsilon}\,dx\leq c\left(\barint_{B_\varrho(\bar{x})}|{{\mathfrak{A}}}|^{1+\epsilon}\,dx\right)^{\frac{1}{1+\epsilon}}\left(\barint_{B_\varrho(\bar{x})}\left|\frac{{{\tens{\wt u}}}-{\tens{\ell_\vr}}}{\varrho}\right|^{1+\epsilon}\,dx\right)^{\frac{\epsilon}{1+\epsilon}} \end{flalign*} where the first term is bounded and the second convergent to zero by~\eqref{poinc-zero}. On the other hand, by~\eqref{l-vr} and \eqref{fin-val-barx} we may estimate \begin{flalign*}
|\text{II}^2_{\varrho}|&\leq c\,\barint_{B_\varrho(\bar{x})}|{D({{\tens{\wt u}}}-{\tens{\ell_\vr}})}|\left|\frac{{{\tens{\wt u}}}-{\tens{\ell_\vr}}}{\varrho}\right|^\epsilon\,dx\\
&\leq c\left(\barint_{B_\varrho(\bar{x})}|{{D\tewtu}}-{{D\tewtu}}(\bar{x})|^{1+\epsilon}\,dx\right)^\frac{1}{1+\epsilon}\left(\barint_{B_\varrho(\bar{x})}\left|\frac{{{\tens{\wt u}}}-{\tens{\ell_\vr}}}{\varrho}\right|^{1+\epsilon}\,dx\right)^\frac{\epsilon}{1+\epsilon} \end{flalign*} where, again, the first term is bounded and the second convergent to zero by~\eqref{poinc-zero}. Summing up the information from the last three displays we get~\eqref{G2-conv}.
Now we concentrate on justifying that\begin{equation}
\label{G3-conv} \lim_{\varrho\to 0}\lim_{j\to\infty}\left| \barint_{B_\varrho(\bar x)}G^3_{j,\varrho}(x)\,dx\right|=0. \end{equation} Let us observe that because~\eqref{lots-of-conv} provides weak convergence of ${\bar{\opA}}(x,{{D\tebu}_j})$ to ${{\mathfrak{A}}}$ in $L^{1+\epsilon}$ and strong convergence ${{\bar{\tens{u}}}_j}$ to ${{\tens{\wt u}}}$ in $L^{1+\epsilon}$, we obtain that \[\lim_{j\to\infty} \barint_{B_\varrho(\bar x)}G^3_{j,\varrho}(x)\,dx=\barint_{B_\varrho(\bar x)}\big({{\mathfrak{A}}}-{\bar{\opA}}(x,D{{\tens{\ell_\vr}}})\big):[T_\varrho({{\tens{\wt u}}}-{\tens{\ell_\vr}})\otimes D\phi]\,dx.\] By H\"older inequality and the choice of $\phi$, we estimate further \begin{flalign*}
\lim_{j\to\infty} \barint_{B_\varrho(\bar x)}G^3_{j,\varrho}(x)\,dx &\leq \left(\barint_{B_\varrho(\bar x)}|{{\mathfrak{A}}}-{{\mathfrak{A}}}(\bar{x})|^{1+\epsilon}+|{{\mathfrak{A}}}(\bar{x})|^{1+\epsilon} + g(|{{D\tewtu}}(\bar{x})|)^{1+\epsilon} \, dx \right)^\frac{1}{1+\epsilon}\\
&\quad \cdot \left(\barint_{B_\varrho(\bar x)}\left(\frac{\min\{\varrho,|{{\tens{\wt u}}} -{\tens{\ell_\vr}}|\}}{\varrho}\right)^{\frac{1+\epsilon}{\epsilon}}\,dx\right)^{\frac{\epsilon}{1+\epsilon}}, \end{flalign*}
where the first integral on the right-hand side is finite and the second term converges to zero. Indeed, since $0 \leq \min\{ \rho, |{{\tens{\wt u}}} - l_{\rho}|\} \leq \rho$, we have
\begin{flalign*} \barint_{B_\varrho(\bar x)}\left(\frac{\min\{\varrho,|{{\tens{\wt u}}} -{\tens{\ell_\vr}}|\}}{\varrho}\right)^{\frac{1+\epsilon}{\epsilon}}\,dx & \leq \barint_{B_\varrho(\bar x)}\left(\frac{\min\{\varrho,|{{\tens{\wt u}}} -{\tens{\ell_\vr}}|\}}{\varrho}\right)^{1+\epsilon}\,dx \\
&\leq \barint_{B_\varrho(\bar x)}\left(\frac{|{{\tens{\wt u}}} -{\tens{\ell_\vr}}|}{\varrho}\right)^{1+\epsilon}\,dx \xrightarrow[\varrho\to 0]{}0,\end{flalign*} where the last convergence results from~\eqref{poinc-zero}. Therefore, we get~\eqref{G3-conv}.
We have shown~\eqref{G2-conv} and \eqref{G3-conv}, so because of~\eqref{4.46} the limit \eqref{G1-conv} follows. Hence, we are in the position to prove~\eqref{I22-conv}. In the view of~\eqref{opA:strict-monotonicity}, \eqref{G1-conv} implies that \begin{equation}\label{pre-grad-conv}
\limsup_{\varrho\to\infty}\limsup_{j\to\infty}\barint_{B_\varrho(\bar x)}\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|<\varrho\}} \frac{g(|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|)}{|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|}|{D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}|^2\,dx=0. \end{equation} At this stage, we calculate similarly to \eqref{aux-esty1} in order to show \eqref{I22-conv}. For any $\hat\epsilon<\frac{2}{s_{G}}$, it is readily checked that $t \mapsto t g(t)^{-\hat\epsilon/(2- \hat\epsilon)}$ is a monotone increasing function. Then \begin{flalign}
& \nonumber \barint_{B_\varrho(\bar x)}\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|<\varrho\}}|{D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}|\,dx\\
& \nonumber \leq \barint_{B_\varrho(\bar x)}\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|<\varrho\}} \left(|{D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}|^2\frac{g(|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|)}{|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|}\right)^{\frac{\hat\epsilon}{2}} \frac{(|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|)^{1-\frac{\hat\epsilon}{2}}}{{g}(|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|)^{\frac{\hat\epsilon}{2}}}\,dx\\
& \nonumber \leq \left(\barint_{B_\varrho(\bar x)}\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|<\varrho\}} |{D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}|^2\frac{g(|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|)}{|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|} \,dx \right)^{\frac{\hat\epsilon}{2}}\\
& \nonumber \quad \cdot \left( \barint_{B_\varrho(\bar x)}\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|<\varrho\}} \frac{|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|}{g(|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|)^{\frac{\hat\epsilon}{2-\hat\epsilon}}}\,dx \right)^{1-\frac{\hat\epsilon}{2}}\\
&\nonumber \leq c \left(\barint_{B_\varrho(\bar x)}\mathds{1}_{\{|{{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}}|<\varrho\}} |{D({{\bar{\tens{u}}}_j}-{\tens{\ell_\vr}})}|^2 \frac{g(|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|)}{|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}|} \,dx\right)^\frac{\hat\epsilon}{2}\\
&\nonumber \quad\cdot
\left(\barint_{B_\varrho(\bar x)} (|{{D\tebu}_j}|+|D{{\tens{\ell_\vr}}}| + 1) \,dx\right)^{1-\frac{\hat\epsilon}{2}}, \end{flalign} where $c=c(g)>0$. Noting that the very last term in the above display is bounded, by~\eqref{pre-grad-conv} we infer that~\eqref{I22-conv} holds.
Summing up all the convergences of this step, we get in~\eqref{h-of-barx} that $\mathfrak{h}(\bar{x})=0$ and, consequently,~\eqref{h=0} holds almost everywhere in $B_{3/4}.$ As explained in the beginning of this step, this suffices to get the final aim of \textbf{Step~5}, that is strong convergence of gradients~\eqref{strong-conv-grad}.
\textbf{Step 6. ${\bar{\opA}}$-harmonicity of the limit map and conclusion by contradiction.} Having~\eqref{strong-conv-grad}, we can pass to the limit in~\eqref{Dbu-male-j} with $j\to\infty$ getting that\begin{equation}
\label{wt-u-harm} \barint_{B_{1/2}} {\bar{\opA}}(x,{{D\tewtu}}):{D\teet}\,dx=0\qquad\text{ for $\ \ {\tens{\eta}}\in C_c^\infty(B_{1/2},{\mathbb{R}^{m}})$.}
\end{equation} Therefore, if ${{\tens{\wt u}}}\in W^{1,G}(B_{1/2},{\mathbb{R}^{m}})$, then it will be proven to be ${\bar{\opA}}$-harmonic. Indeed, since we know~\eqref{wtu-male}, it is allowed to take ${\bar{\tens{{v}}}}={{\tens{\wt u}}}$ in \eqref{HsM}. Note that in such a case~\eqref{wtu-male} is precisely the restriction on the test function from~\eqref{bv-male}. Then, in the view of~\eqref{strong-conv-grad}, taking $j$ large enough, we will get the desired contradiction. Hence, it remains to prove that $|{{D\tewtu}}|\in L^G(B_{1/2})$.
We have~\eqref{do-int-3} for each ${{\bar{\tens{u}}}_j}$ with the constant independent of $j$, so by the lower semicontinuity we can write that \begin{flalign*}
\left(\int_{B_{7/8}}\left( \frac{{\bar{G}}(1+|{{\tens{\wt u}}}|)}{(1+|{{\tens{\wt u}}}|)^{1+{\wt\gamma}}}\right)^\frac{n}{n-\kappa}\,dx\right)^\frac{n-\kappa}{n} &\leq\liminf_{j\to\infty}\left(\int_{B_{7/8}}\left( \frac{{\bar{G}}(1+|{{\bar{\tens{u}}}_j}|)}{(1+|{{\bar{\tens{u}}}_j}|)^{1+{\wt\gamma}}}\right)^\frac{n}{n-\kappa}\,dx\right)^\frac{n-\kappa}{n}\\ &\leq c \end{flalign*} for some $c=c(\textit{\texttt{data}},{\wt\gamma})>0.$ Analogically, by~\eqref{st2-apriori} for ${{\bar{\tens{u}}}_j}$ and with $\delta=2^{-j}$, by letting $j\to\infty$, Fatou's lemma on the left-hand side of the resultant inequality and \eqref{strong-conv-grad} on its right-hand side, we get \begin{flalign}\nonumber
\int_{B_{3/4}\cap \{|{{\tens{\wt u}}}|< t\}}{\bar{G}}(|{{D\tewtu}}|) \phi^q\,dx\leq &\,c_* \int_{B_{3/4}\cap \{|{{\tens{\wt u}}}|< t\}}{\bar{G}}(|{{\tens{\wt u}}}|\,|D\phi|) \,dx\\
&+\,c_*\,t \int_{B_{3/4}\cap \{|{{\tens{\wt u}}}|\geq t\}} {{\bar{g}}(|{{D\tewtu}}|)}\phi^{q-1}| D\phi|\,dx\nonumber \end{flalign}
for every $t>0$ and $\phi\in C_c^\infty(B_{3/4})$ with $\phi\geq 0$, and $c_*=c_*(\textit{\texttt{data}},q).$ We proceed as in the beginning of {\bf Step 3}. We multiply the above display by $(1+t)^{-({\wt\gamma}+1)},$ ${\wt\gamma}>0$ to be chosen sufficiently small in a few lines, integrate it from zero to infinity and apply Cavalieri's principle (Lemma~\ref{lem:cava}) twice (with $\nu_1={\bar{G}}(|{D\tewtu}|) \phi^q$ and $\nu_2={\bar{G}}(|{\tens{\wt u}}|\,|D\phi|)$. Altogether we get
\begin{flalign}\nonumber
\frac{1}{{\wt\gamma}} \int_{B_{3/4}}&\frac{{\bar{G}}(|{{D\tewtu}}|) \phi^q}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\,dx\leq
\frac{c_*}{{\wt\gamma}} \int_{B_{3/4}}\frac{{\bar{G}}(|{{\tens{\wt u}}}|\,|D\phi|) }{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\,dx\\
&+ c_* \int_0^\infty \frac{1}{(1+t)^{{\wt\gamma}}}\int_{B_{3/4}\cap\{|{{\tens{\wt u}}}|\geq t\} } {{\bar{g}}(|{{D\tewtu}}|)}\phi^{q-1}| D\phi|\,dx \,dt=\text{III}_1+\text{III}_2.\nonumber \end{flalign} To estimate further the very last term we note that $q$ is large enough to satisfy $s_G'\geq q'$, and so Lemma~\ref{lem:equivalences} implies that ${{\bar{G}}}^*(\phi^{q-1}{\bar{g}}(t))\leq c_G \phi^q {\bar{G}}(t).$ Then, using Young inequality~\eqref{in:Young} and by taking ${\wt\gamma} \in (0,1/(2c_*c_G+1)],$ get \begin{equation*}
\begin{split}\text{III}_2&\leq \frac{c_*}{1-{\wt\gamma}}\int_{B_{3/4} } {{\bar{g}}(|{{D\tewtu}}|)}{(1+|{{\tens{\wt u}}}|)^{1-{\wt\gamma}}}\phi^{q-1}| D\phi|\,dx\\
&\leq \frac{1}{2{\wt\gamma} c_G} \int_{B_{3/4}}\frac{{{\bar{G}}^*}\left( {\phi^{q-1}{\bar{g}}(|{{D\tewtu}}|) }\right)}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}} \, dx+ \frac{1}{2{\wt\gamma}}\int_{B_{3/4}}\frac{{\bar{G}}\left(\frac{2{\wt\gamma} c_* c_G}{1-{\wt\gamma}}(1+|{{\tens{\wt u}}}|)|D\phi|\right)}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\,dx \\
& \leq \frac{1}{2{\wt\gamma}} \int_{B_{3/4}}\frac{{\bar{G}}\left(|{{D\tewtu}}| \right)\phi^{q }}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}} \,dx + \frac{c_* c_G}{1- {\wt\gamma}} \int_{B_{3/4}}\frac{{\bar{G}}\left((1+|{{\tens{\wt u}}}|) |D\phi|\right)}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\, dx \end{split} \end{equation*} where in the last line we used that $\frac{2{\wt\gamma} c_* c_G} {1-{\wt\gamma}}<1$ can be taken out of the integrand by Jensen's inequality. Summing up we obtain
\begin{flalign}
\int_{B_{3/4} }\frac{{\bar{G}}(|{{D\tewtu}}|) \phi^q}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\,dx&\leq
\,{C} \int_{B_{3/4}}\frac{{\bar{G}}\left((1+|{{\tens{\wt u}}}|)|D\phi|\right)}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\, dx \label{post-cava-final-step} \end{flalign} where $C=C(\textit{\texttt{data}}) >0$. Then similar calculations to \eqref{Dvt}-\eqref{int-vt} yield that \begin{equation*}
\left(\int_{B_{3/4}}\left( \frac{{\bar{G}}((1+|{\bar{\tens{u}}}|)\phi^q)}{(1+|{\bar{\tens{u}}}|)^{{\wt\gamma}}}\right)^\frac{n}{n-1}\,dx\right)^\frac{n-1}{n}\leq
c \int_{B_{3/4}}\frac{{\bar{G}}\left((1+|{\bar{\tens{u}}}|)|D\phi|\right)}{(1+|{\bar{\tens{u}}}|)^{{\wt\gamma}}}\, dx. \end{equation*} holds with $c=c(\textit{\texttt{data}})>0$. Indeed, in \textbf{Step 4}, we have checked that the above $c=c(\textit{\texttt{data}},{\wt\gamma})$ is an increasing function of ${\wt\gamma}$. As we consider small ${\wt\gamma}$, $c$ in fact depends only on $\textit{\texttt{data}}$. For $5/8 \leq r_{1} < r_{2} \leq 3/4$ we take $\phi \in C_{c}^{\infty}(B_{r_2})$ to satisfy
$$\phi \equiv 1 \quad \text{on} \quad B_{r_1} \qquad \text{and} \qquad |{D \phi}| \leq \frac{100}{r_2 - r_1}.$$ Then the doubling property of ${\bar{G}}$ and Lemma \ref{lem:self} gives \begin{equation}\label{to-est}
\left(\int_{B_{5/8}}\left( \frac{{\bar{G}}(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{{\wt\gamma}}}\right)^\frac{n}{n-1}\,dx\right)^\frac{n-1}{n}\leq c \left( \int_{B_{3/4}} \left(\frac{{\bar{G}}(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{{\wt\gamma}}} \right)^{\frac{1}{2s_G}}\, dx \right)^{2s_G}. \end{equation} We now restrict ourselves to ${\wt\gamma} \in\big(0,\min\{1/(2c_*c_G+1), \frac{i_G}{2}\}\big)$ and define $$\Psi_{{\wt\gamma}}(t) = \int_{0}^{t} \frac{1}{\tau} \left( \frac{{\bar{G}}(\tau)}{\tau^{{\wt\gamma}}} \right)^{\frac{1}{2s_G}} \, d \tau,$$ which is an increasing concave function on $[0,\infty)$ satisfying $$\frac{t \Psi_{{\wt\gamma}}''(t)}{\Psi_{{\wt\gamma}}'(t)}=\frac{t g(t)}{2s_G {\bar{G}}(t)} - 1 - \frac{{\wt\gamma}}{2s_G} \in (-1,-1/2) \quad \text{and} \quad \Psi_{{\wt\gamma}}(t) \approx \left(\frac{{\bar{G}}(t)}{t^{{\wt\gamma}}}\right)^{\frac{1}{2s_G}}.$$ Then Jensen's inequality gives \begin{flalign}
\int_{B_{3/4}} &\left(\frac{{\bar{G}}(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{{\wt\gamma}}} \right)^{\frac{1}{2s_G}}\, dx
\leq c \int_{B_{3/4}} \Psi_{{\wt\gamma}}(1+|{\bar{\tens{u}}}|)\, dx \nonumber \leq c\, \Psi_{{\wt\gamma}} \left( \int_{B_{3/4}} (1+|{\bar{\tens{u}}}|) \, dx \right)\\
&\qquad\quad \leq c\, \Psi_{{\wt\gamma}} \left( \int_{B_{3/4}} (1+|{\bar{\tens{u}}}|) \, dx + 1\right)
\leq c\, {\bar{G}}^\frac{1}{2s_G} \left( \int_{B_{3/4}} (1+|{\bar{\tens{u}}}|) \, dx + 1\right)
\leq c, \label{LG-final-step} \end{flalign} with $c=c(\textit{\texttt{data}})>0.$ We used that if $t>1$ is arbitrarily fixed there exists $c>0$ independent of ${\wt\gamma}$ such that for all $t>1$ and all ${\wt\gamma}$, we have $\Psi_{\wt\gamma}(t)\leq c \left({{\bar{G}}(t)} \right)^{\frac{1}{2s_G}}$. By H\"older inequality,~\eqref{to-est} and \eqref{LG-final-step} we get that \begin{equation}\label{to-est-2}
\int_{B_{5/8}}\frac{{\bar{G}}(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{{\wt\gamma}}}\,dx\leq \left(\int_{B_{5/8}}\left( \frac{{\bar{G}}(1+|{\bar{\tens{u}}}|)}{(1+|{\bar{\tens{u}}}|)^{{\wt\gamma}}}\right)^\frac{n}{n-1}\,dx\right)^\frac{n-1}{n}\leq c \end{equation} with $c=c(\textit{\texttt{data}})>0.$
We now consider \eqref{post-cava-final-step} with a cutoff function $\phi \in C_{c}^{\infty}(B_{5/8})$ satisfying
$$\phi \equiv 1 \quad \text{in} \quad B_{1/2} \qquad \text{and} \qquad |{D \phi}| \leq 100$$ and combine it with \eqref{to-est-2}, to obtain for some $c=c(\textit{\texttt{data}})>0$ that \begin{flalign*}
\int_{B_{1/2} }\frac{{\bar{G}}(|{{D\tewtu}}|)}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\,dx&\leq
{c} \int_{B_{3/4}}\frac{{\bar{G}}(1+|{{\tens{\wt u}}}|)}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\, dx \leq c. \end{flalign*} Therefore, using Fatou's lemma we justify that \begin{flalign*}
\int_{B_{1/2} }{{\bar{G}}(|{{D\tewtu}}|)}\,dx&\leq \limsup_{{\wt\gamma}\to 0}
\int_{B_{1/2} }\frac{{\bar{G}}(|{{D\tewtu}}|)}{(1+|{{\tens{\wt u}}}|)^{{\wt\gamma}}}\,dx\leq c. \end{flalign*}
Consequently, we conclude that $|{D {{\tens{\wt u}}}}| \in L^{G}(B_{1/2})$. This completes the proof of Theorem \ref{theo:Ah-approx}. \end{proof}
\section{Proof of Wolff potential estimates} \label{sec:mainproof} \subsection{Comparison estimate} We need one more auxiliary estimate yielding comparison between energy of a weak solution and an ${\mathcal{ A}}$-harmonic function. \begin{lem}\label{lem:A-h-appr} Under Assumption {\bf (A-vect)} suppose $u\in W^{1,G}(B_r,{\mathbb{R}^{m}})$ is a weak solution to~\eqref{eq:mu} in $B_r=B_r(x_0),$ $r<1$ and let $\varepsilon\in(0,1)$. Then there exists a positive constant $c_{\rm s}=c_{\rm s}(\textit{\texttt{data}},\varepsilon)$ and a map ${\tens{v}}$ being ${\mathcal{ A}}$-harmonic in $B_{r/2}$ and such that\begin{equation}
\label{comp-est}\barint_{B_{r/2}}|{D\teu}-{D\tev}|\,dx\leq \frac{\varepsilon}{r}\barint_{B_r}|{\tens{u}}-({\tens{u}})_{B_r}|\,dx+c_{\rm s}g^{-1}\left(\frac{|{\tens{\mu}}|(B_r)}{r^{n-1}}\right). \end{equation} \end{lem} \begin{proof}Let us fix
\[\lambda:=\frac{1}{r}\barint_{B_r}|{\tens{u}}-({\tens{u}})_{B_r}|\,dx+g^{-1}\left(\delta \frac{|{\tens{\mu}}|(B_r)}{r^{n-1}}\right)\] with $\delta=\delta(\textit{\texttt{data}},\varepsilon)$ from Theorem~\ref{theo:Ah-approx} with $M=1$. If $\lambda=0,$ then ${\tens{u}}$ is constant and ${\tens{v}}={\tens{u}}$. Otherwise $\lambda>0$ and we can argue by scaling\[{\bar{\tens{u}}}:=\frac{{\tens{u}}-({\tens{u}})_{B_r}}{\lambda},\qquad\bar{{\tens{\mu}}}:=\frac{{\tens{\mu}}}{g(\lambda)},\qquad{\bar{\opA}}(x,{\tens{\xi}})=\frac{{\mathcal{ A}}(x,\lambda{\tens{\xi}})}{g(\lambda)}.\] Then
\[\left|\barint_{B_r} {\bar{\opA}}(x,{D\tebu}):D{\tevp}\,dx\right|\leq \frac{\|{\tens{\vp}}\|_{L^\infty(B_r)}|{\tens{\mu}}|(B_r)}{g(\lambda) r^{n-1}}\leq\frac{\delta}{r}\|{\tens{\vp}}\|_{L^\infty(B_r)}.\] By definition of ${\bar{\tens{u}}}$ and $\lambda$ we notice that
\[\barint_{B_r}|{\bar{\tens{u}}}|\,dx\leq r.\] Therefore, by Theorem~\ref{theo:Ah-approx} applied to ${\bar{\tens{u}}}$ we get that there exists ${\bar{\tens{{v}}}}$ being ${\bar{\opA}}$-harmonic in $B_{r/2}$ and such that
\[\barint_{B_{r/2}}|{D\tebu}-{D\tebv}|\,dx\leq \varepsilon.\] Then~\eqref{comp-est} follows by rescaling back with ${\tens{v}}=\lambda{\bar{\tens{{v}}}}$ which is ${\mathcal{ A}}$-harmonic. \end{proof}
\subsection{Estimates on concentric balls} This subsection is devoted to prove some properties of weak solutions to~\eqref{eq:mu} with ${\tens{\mu}}\in C^\infty(\Omega,{\mathbb{R}^{m}})$ holding over a family of concentric balls $\{B^j\}$. Before we pass to this, let us fix some notation and parameters. Recall that we have chosen $R_0 = R_0(\textit{\texttt{data}}, \varsigma)$ in Proposition~\ref{prop:osc}. We take an arbitrary constant $\alpha_V \in (0,1)$ and take $\varsigma=\varsigma(s_G,\alpha_V)$ to satisfy $\alpha_{D}:=\frac{\alpha_V+1}{2} \leq 1+(\varsigma-1)/s_G$. To prove Theorem~\ref{theo:pointwise}, it is enough to take $\alpha_V=\frac{1}{2}$, but for the later use in the proof of Theorem~\ref{theo:H-cont}, we have taken $\alpha_V$ arbitrarily. We now choose\begin{equation}
\label{sigma}\sigma_0:= \min\left\{ \left(\frac{1}{2^{n+6} c_{\rm o}}\right)^{\frac{1-\alpha_V}{2}},\frac{1}{4}\right\}. \end{equation}
If $r\in(0,R_0)$ is given, for every $j\in {\mathbb{N}}\cup\{0\}$ let us fix \[r_j:=\sigma^{j+1}r,\qquad\qquad B^j:=\overline{B_{r_j}(x_0)},\] so that $r_{-1}=r.$ We denote\begin{equation}
\label{AjEj}
E_j:=\barint_{B^j}|{\tens{u}}-({\tens{u}})_{B^j}|\,dx. \end{equation}
\begin{lem}
\label{lem:8.1}Suppose Assumption {\bf (A-vect)} is satisfied. If ${\tens{u}}\in W^{1,G}(\Omega,{\mathbb{R}^{m}})$ is a~weak solution to~\eqref{eq:mu} with ${\tens{\mu}}\in C^\infty(\Omega,{\mathbb{R}^{m}})$, $j\in{\mathbb{N}}$ is fixed, $E_j$ is given by~\eqref{AjEj}, while $0<\sigma\leq \sigma_0$ is arbitrary, then we have that\begin{equation}
\label{8.12} E_{j+1}\leq c_{\rm D}\sigma^{\alpha_D}E_j+c_{\rm E}r_jg^{-1}\left(\frac{|{\tens{\mu}}|(B^j)}{r_j^{n-1}}\right)
\end{equation} for $c_{\rm D}=c_{\rm D}(\textit{\texttt{data}},\alpha_V) =2^{n+4}c_{\rm o}$ and $c_{\rm E}=c_{\rm E}(\textit{\texttt{data}},\alpha_V)={2^{n+2}c_{\rm s}c_{\rm P}c_{\rm o}}\sigma^{-n},$ where $c_{\rm P}$ is the constant from Poincar\'e inequality in $W^{1,1}(\Omega,{\mathbb{R}^{m}})$. \end{lem} \begin{proof} We may apply Lemma~\ref{lem:A-h-appr} in $B_r=B_{r_j}(x_0)$ to get that there exists an ${\mathcal{ A}}$-harmonic map ${\tens{v}}_j\in W^{1,G}(B_{r_{j/2}},{\mathbb{R}^{m}})$ in $B_{r_{j/2}}$ such that\begin{equation}
\label{8.13} \barint_{\frac{1}{2}B^j}|{D\teu}-{D\tev}_j|\,dx\leq\frac{\varepsilon}{r_j}{E_j}+c_{\rm s} g^{-1}\left(\frac{|{\tens{\mu}}|(B^j)}{r_j^{n-1}}\right).
\end{equation} By Poincar\'e inequality in $W^{1,1}(\Omega,{\mathbb{R}^{m}})$ for ${\tens{w}}_j={\tens{u}}-{\tens{v}}_j$ we have
\[\barint_{\frac{1}{2}B^j}|{\tens{w}}_j-({\tens{w}}_j)_{\frac{1}{2}B^j}|\,dx\leq c_{\rm P} r_j \,\barint_{\frac{1}{2}B^j}|{D\teu}-{D\tev}_j|\,dx.\] Then \begin{equation}
\label{est-diff-w_j}
\barint_{\frac{1}{2}B^j}|{\tens{w}}_j-({\tens{w}}_j)_{\frac{1}{2}B^j}|\,dx\leq {\varepsilon}c_{\rm P}\,E_j+c_{\rm s} c_{\rm P}\,r_j g^{-1}\left(\frac{|{\tens{\mu}}|(B_{r_j})}{r_j^{n-1}}\right). \end{equation} Thus by Lemma~\ref{lem:excess}, the triangle inequality,~\eqref{est-diff-w_j}, and Proposition~\ref{prop:osc}, we estimate \begin{flalign*} E_{j+1}&=
\barint_{B^{j+1}}|{\tens{u}}-({\tens{u}})_{B^{j+1}}|\,dx\\
&\leq \,\barint_{B^{j+1}}|{\tens{w}}_{j}-({\tens{w}}_{j})_{B^{j+1}}|\,dx+\, \barint_{B^{j+1}}|{\tens{v}}_{j}-({\tens{v}}_{j})_{B^{j+1}}|\,dx\\
&\leq \,2\,\barint_{B^{j+1}}|{\tens{w}}_{j}-({\tens{w}}_{j})_{\frac 12 B^{j}}|\,dx+\, 2 c_{\rm o}\sigma^{1+(\varsigma - 1)/s_{G}}\,\barint_{\frac{1}{2}B^{j}}|{\tens{v}}_{j}-({\tens{v}}_{j})_{\frac{1}{2}B^{j}}|\,dx\\
&\leq \left(\frac{2^{n+1}}{\sigma^{n}} + 2c_{\rm o}\sigma^{\alpha_V} \right) \, \barint_{\frac{1}{2}B^{j}}|{\tens{w}}_j-({\tens{w}}_j)_{\frac{1}{2}B^j}|\,dx+\, 2 c_{\rm o} \sigma^{\alpha_V} \barint_{\frac{1}{2}B^{j}}|{\tens{u}}-({\tens{u}})_{\frac{1}{2}B^{j}}|\,dx\\
&\leq 2^{n+2} c_{\rm o}\sigma^{\alpha_V} E_{j} + \left(\frac{2^{n+1}}{\sigma^{n}} + 2 c_{\rm o}\sigma^{\alpha_V} \right) \,\barint_{\frac{1}{2}B^{j}}|{\tens{w}}_j-({\tens{w}}_j)_{\frac{1}{2}B^j}|\,dx\\
&\leq \left(2^{n+2} c_{\rm o}\sigma^{\alpha_V} + 2 \varepsilon c_{\rm o} c_{\rm P} \sigma^{\alpha_V} + \varepsilon c_{\rm P}\frac{2^{n+1}}{\sigma^{n}} \right)\,E_j\\& \quad +c_{\rm s} c_{\rm P} \left(\frac{2^{n+1}}{\sigma^{n}} + 2 c_0\sigma^{\alpha_V} \right) \,r_j g^{-1}\left(\frac{|{\tens{\mu}}|(B_{r_j})}{r_j^{n-1}}\right). \end{flalign*} By choosing $\varepsilon=\frac{\sigma^{n+\alpha_V}}{c_{\rm P}}$ we complete the proof. \end{proof}
\begin{prop}\label{prop:comp-exc} Suppose Assumption {\bf (A-vect)} is satisfied. If ${\tens{u}}\in W^{1,G}(\Omega,{\mathbb{R}^{m}})$ is a weak solution to~\eqref{eq:mu} with ${\tens{\mu}}\in C^\infty(\Omega,{\mathbb{R}^{m}})$, then there exists a constant $c_V=c_V(\textit{\texttt{data}},\alpha_V)\geq 1$ such that for every $\tau\in(0,1]$ we have\begin{flalign*}
\barint_{B_{\tau r}(x_0)}&|{\tens{u}}-({\tens{u}})_{B_{\tau r(x_0)}}|\,dx\\
&\leq c_V\tau^{\alpha_V}\barint_{B_r(x_0)}|{\tens{u}}-({\tens{u}})_{B_r(x_0)}|\,dx+c_V\sup_{0<\varrho<r}\varrho\, g^{-1}\left(\frac{|{\tens{\mu}}|(B_\varrho(x_0))}{\varrho^{n-1}}\right). \end{flalign*} \end{prop}\begin{proof} Lemma~\ref{lem:8.1} implies that for $j\in{\mathbb{N}}\cup\{0\}$ it holds that
\[E_{j+1}\leq c_{\rm D}\sigma^{\alpha_D}E_j+c_{\rm E}r_jg^{-1}\left(\frac{|{\tens{\mu}}|(B^j)}{r_j^{n-1}}\right).\] Iterating this estimate we get that for any $k \in{\mathbb{N}}\cup\{0\}$
\[E_{k+1}\leq (c_{D}\sigma^{\alpha_D})^{k+1} E_0+c_{\rm E} \sum_{j=0}^k (c_{\rm D} \sigma^{\alpha_{\rm D}})^{j}r_j\,g^{-1}\left(\frac{|{\tens{\mu}}|(B^j)}{r_j^{n-1}}\right),\] where $c=c(\textit{\texttt{data}})$. Recalling $\alpha_{D}=\frac{\alpha_V+1}{2}$, $c_{\rm D}=2^{n+4}c_{\rm o}$ and \eqref{sigma}, we see \[ c_{\rm D} \sigma^{\alpha_{\rm D}} \leq \frac{\sigma^{\alpha_V}}{4}. \] By Lemma~\ref{lem:excess} and direct computation we have for any $k\in{\mathbb{N}}\cup\{0\}$ that\begin{equation}\label{Ek-est}
E_k\leq \sigma^{k \alpha_V}\barint_{B_r(x_0)}|{\tens{u}}-({\tens{u}})_{B_r(x_0)}|\,dx + 2 c_{\rm E}\sup_{0<\varrho<r}\varrho\,g^{-1}\left(\frac{|{\tens{\mu}}|(B_{\varrho}(x_0))}{\varrho^{n-1}}\right). \end{equation} We take $\tau\in(0,\sigma)$ and $k\geq 1$ such that $\sigma^{k+1}<\tau\leq\sigma^k.$ Then by Lemma~\ref{lem:excess} and \eqref{Ek-est} we obtain \begin{flalign*}
& \barint_{B_{\tau r}(x_0)}|{\tens{u}}-({\tens{u}})_{B_{\tau r}(x_0)}|\,dx \\
&\leq\frac{2\sigma^{kn}}{\tau^n} \barint_{B_{r\sigma^k}(x_0)}|{\tens{u}}-({\tens{u}})_{B^{k-1}}|\,dx\leq \frac{2E_{k-1}}{\sigma^n}\\
&\leq \frac{\sigma^{(k+1)\alpha_V}}{\sigma^{n+2\alpha_V}} \barint_{B_r(x_0)}|{\tens{u}}-({\tens{u}})_{B_r(x_0)}|\,dx + \frac{2 c_{\rm E}}{\sigma^n}\sup_{0<\varrho<r}\varrho\,g^{-1}\left(\frac{|{\tens{\mu}}|(B_{\varrho}(x_0))}{\varrho^{n-1}}\right)\\
&\leq \frac{\tau^{\alpha_V}}{\sigma^{n+2 \alpha_V}} \barint_{B_r(x_0)}|{\tens{u}}-({\tens{u}})_{B_r(x_0)}|\,dx +\frac{2 c_{\rm E}}{\sigma^n}\sup_{0<\varrho<r}\varrho\,g^{-1}\left(\frac{|{\tens{\mu}}|(B_{\varrho}(x_0))}{\varrho^{n-1}}\right). \end{flalign*} By taking $c_V=c_V(\textit{\texttt{data}},\alpha_V)=2c_{\rm E}\sigma^{-n-2\alpha_V}$ we conclude the claim for $\tau\in(0,\sigma).$ For completing the range of $\tau\in[\sigma,1]$ it suffices to note that
\[\barint_{B_{\tau r}(x_0)}|{\tens{u}}-({\tens{u}})_{B_{\tau r}(x_0)}|\,dx\leq \frac{2}{\sigma^n}
\barint_{B_{r}(x_0)}|{\tens{u}}-({\tens{u}})_{B_{r}(x_0)}|\,dx.\] \end{proof}
\subsection{SOLA ${\tens{u}}$ belongs to VMO} \begin{proof}[Proof of Proposition~\ref{prop:vmo}] Suppose ${\tens{\mu}}\in C^\infty(\Omega,{\mathbb{R}^{m}})$ and ${\tens{u}}\in W^{1,G}(\Omega,{\mathbb{R}^{m}})$ is a weak solution to~\eqref{eq:mu}. By Proposition~\ref{prop:comp-exc} we can find constants $c_V=c_V(\textit{\texttt{data}},\alpha_V)$ we have\begin{equation}\label{est-weak}
\barint_{B_{\tau r}}|{\tens{u}}-({\tens{u}})_{B_{\tau r}}|\,dx\leq c_V\tau^{\alpha_V}\barint_{B_r}|{\tens{u}}-({\tens{u}})_{B_r}|\,dx+c_V\sup_{0<\varrho<r}\varrho\, g^{-1}\left(\frac{|{\tens{\mu}}|(B_\varrho)}{\varrho^{n-1}}\right). \end{equation} Let us consider a SOLA ${\tens{u}}\in W^{1,g}(\Omega,{\mathbb{R}^{m}}) $ existing due to Proposition~\ref{prop:exist}. Suppose $({\tens{u}}_h)$ and $({\tens{\mu}}_h)$ are approximating sequences from definition of SOLA, see Section~\ref{ssec:main-results}. Inequality~\eqref{est-weak} hold for each ${\tens{u}}_h$ and ${\tens{\mu}}_h$. We have to motivate passing to the limit with $h\to\infty$. Since~\eqref{conv-of-meas} holds, we can write~\eqref{est-weak} for the original SOLA too. From now on this kind of solution is considered.
Our aim now is to show that ${\tens{u}}$ is VMO at $x_0$ provided \eqref{mu-shrinks} is assumed. Let $\delta\in(0,1)$. By~\eqref{mu-shrinks} we find a positive radius $r_{1,\delta}<r$ such that \[c_V\sup_{0<\varrho<r_{1,\delta}}\varrho\, g^{-1}\left(\frac{|{\tens{\mu}}|(B_\varrho(x_0))}{\varrho^{n-1}}\right)\leq \frac{\delta}{2}\] and then $\tau_\delta$ so small that
\[c_V\tau_\delta^{\alpha_V}\barint_{B_{r_{1,\delta}}}|{\tens{u}}-({\tens{u}})_{B_{r_{1,\delta}}(x_0)}|\,dx\leq\frac{\delta}{2}.\] For $r_\delta:=\tau_\delta r_{1,\delta}$ from estimate~\eqref{est-weak} (applied with $r=r_{1,\delta}$) it follows that
\[\sup_{0<\varrho<r_\delta}\barint_{B_{\varrho}(x_0)}|{\tens{u}}-({\tens{u}})_{B_{\varrho(x_0)}}|dx\leq\delta,\] that is that ${\tens{u}}$ has vanishing mean oscillation at $x_0$. \end{proof}
\subsection{Proofs of Theorems~\ref{theo:pointwise},~\ref{theo:continuity} and~\ref{theo:H-cont}} We start with the proof of pointwise Wolff potential estimate, then pass to continuity and H\"older continuity criteria. \begin{proof}[Proof of Theorem~\ref{theo:pointwise}] We notice that having $E_j$ defined in~\eqref{AjEj} with $r=r^j$ we can fix $\sigma$ in Lemma~\ref{lem:8.1} to get that\begin{equation}
\label{est-2-st-1-theo-est}
E_{j+1}\leq \frac{1}{2}E_j+cr^jg^{-1}\left(\frac{|{\tens{\mu}}|(B^j)}{r_j^{n-1}}\right)\qquad\text{for every }\ j\in{\mathbb{N}}\cup\{0\}. \end{equation} We sum up inequalities from~\eqref{est-2-st-1-theo-est} to obtain
\[\sum_{j=1}^{k+1} E_j\leq\frac{1}{2}\sum_{j=0}^k E_j+c\sum_{j=0}^k r_jg^{-1}\left(\frac{|{\tens{\mu}}|(B^j)}{r_j^{n-1}}\right),\qquad k\in{\mathbb{N}}\cup\{0\}.\] By rearranging terms we have
\[\sum_{j=1}^{k+1} E_j\leq 2 E_0+c\sum_{j=0}^k r_jg^{-1}\left(\frac{|{\tens{\mu}}|(B^j)}{r_j^{n-1}}\right).\] We notice that for some $c=c(\textit{\texttt{data}})$ we can estimate
\[\sum_{j=0}^k r_jg^{-1}\left(\frac{|{\tens{\mu}}|(B^j)}{r_j^{n-1}}\right)\leq c\,\int_0^r g^{-1}\left(\frac{|{\tens{\mu}}|(B_\varrho)}{\varrho^{n-1}}\right)\,d\varrho=c\,{\mathcal{W}}^{\tens{\mu}}_G(x_0,r).\] Last two displays imply that \[\sum_{j=1}^{k+1} E_j\leq 2 E_0+c{\mathcal{W}}^{\tens{\mu}}_G(x_0,r).\]
For every $m,k\in{\mathbb{N}}$ such that $m<k$ we have\begin{flalign*}
|({\tens{u}})_{B^k}-({\tens{u}})_{B^m}|&\leq \sum_{j=m}^{k-1} |({\tens{u}})_{B^{j+1}}-({\tens{u}})_{B^j}|\leq \sigma^{-n} \sum_{j=m}^{k+1}E_j \leq \sigma^{-n} \sum_{j=0}^{k+1}E_j\\ &\leq 2\sigma^{-n} E_0+c\sigma^{-n}{\mathcal{W}}^{\tens{\mu}}_G(x_0,r)\\
&\leq 2\sigma^{-n}\barint_{B_r(x_0)}|{\tens{u}}-({\tens{u}})_{B_r(x_0)}|\,dx+c\sigma^{-n}{\mathcal{W}}^{\tens{\mu}}_G(x_0,r), \end{flalign*} where $\sigma=\sigma(\textit{\texttt{data}})$ and $c=c(\textit{\texttt{data}}).$ For $j\to\infty$, $(({\tens{u}})_{B^j})_j$ is a Cauchy sequence that converges to ${\tens{u}}(x_0)$, that is \[\lim_{\varrho\to 0}({\tens{u}})_{B_\varrho(x_0)}={\tens{u}}(x_0)\] and $x_0$ is a Lebesgue's point of ${\tens{u}}$. This completes the proof of \eqref{Wolff-osc-est}, while~\eqref{eq:u-est} follows as a direct corollary.\end{proof}
Let us concentrate on the continuity criterion. \begin{proof}[Proof of Theorem~\ref{theo:continuity}] Our aim is to infer continuity of ${\tens{u}}$ in $B_r(x_0)$ knowing that~\eqref{Wolff-shrinks} holds. We will show that for every $\delta>0$ and $x_1\in B_r(x_0)$ we can find $r_\delta\in (0,{\rm dist}\,(B_r(x_0),\partial\Omega))$ such that\begin{equation}
\label{osc-u}{\rm osc}_{B_{r_\delta}(x_1)}\,{\tens{u}}<\delta. \end{equation} Without loss of generality we assume that ${\tens{\mu}}$ is defined on whole ${\mathbb{R}^{n}}$, as we can extend it by zero outside $\Omega$. By~\eqref{Wolff-shrinks} we can take $\varrho_1$ small enough for \begin{equation}
\label{small-wolff} \sup_{x\in B_r(x_0)}{\mathcal{W}}^{\tens{\mu}}_G(x,\varrho_1)\leq \frac{\delta}{16}. \end{equation}
Let $r_\delta>0$ to be chosen in a moment. We take an arbitrary point $x_2\in B_{r_\delta}(x_1)$ and estimate \begin{flalign}\nonumber |{\tens{u}}(x_1)-{\tens{u}}(x_2)| &\leq |{\tens{u}}(x_1)-({\tens{u}})_{B_{2r_\delta}(x_1)}|+|({\tens{u}})_{B_{2r_\delta}(x_1)}-({\tens{u}})_{B_{r_\delta}(x_2)}|\\
&\quad+|({\tens{u}})_{B_{r_\delta}(x_2)}-{\tens{u}}(x_2)|=:A_1+A_2+A_3.\label{telescope} \end{flalign}
We start with estimating $A_2$ by noting that \[A_2=|({\tens{u}})_{B_{r_\delta}(x_2)}-({\tens{u}})_{B_{2r_\delta}(x_1)}| \leq \barint_{B_{r_\delta}(x_2)}|{\tens{u}}-({\tens{u}})_{B_{2r_\delta}(x_1)}|\,dx.\] Since~\eqref{Wolff-shrinks} implies~\eqref{mu-shrinks}, Proposition~\ref{prop:vmo} implies that ${\tens{u}}$ has vanishing mean oscillations at~$x_1$. Therefore there exists $\varrho_2\in(0,\min\{\varrho_1,{\rm dist}(x_1,\partial B_r(x_0))/4\})$ such that for every $\varrho\leq \varrho_2$ it holds\[\barint_{B_{\varrho}(x_1)}|{\tens{u}}-({\tens{u}})_{B_{\varrho}(x_1)}|\,dx\leq\frac{\delta}{2^{n+4}}.\]
We choose $r_\delta=\varrho_2/2$ and observe that~\eqref{small-wolff} imply that\begin{flalign*}\barint_{B_{r_\delta}(x_2)}|{\tens{u}}-({\tens{u}})_{B_{2 r_\delta}(x_1)}|\,dx
&\leq 2^{n}\,\barint_{B_{2r_\delta}(x_1)}|{\tens{u}}-({\tens{u}})_{B_{2 r_\delta}(x_1)}|\,dx\leq\frac{\delta}{16}. \end{flalign*}
In turn $ A_2 \leq\frac{\delta}{16}.$ By Theorem~\ref{theo:pointwise} and~\eqref{small-wolff} we get that $x_1$ and $x_2$ are Lebesgue's points and \[A_1+A_3=|{\tens{u}}(x_1)-({\tens{u}})_{B_{2r_\delta}(x_1)}|+|({\tens{u}})_{B_{r_\delta}(x_2)}-{\tens{u}}(x_2)|\leq \frac{\delta}4 .\]
Applying these observation we get from~\eqref{telescope} that\begin{flalign*} |{\tens{u}}(x_1)-{\tens{u}}(x_2)|&\leq \frac{\delta}{2}. \end{flalign*} Since $x_2$ was an arbitrary point of $B_{r_\delta}(x_1)$, we have \eqref{osc-u} justified, which completes the proof. \end{proof}
We are in the position to prove the H\"older continuity criterion. \begin{proof}[Proof of Theorem~\ref{theo:H-cont}.] Notice that assumption~\eqref{mu-control} implies that there exists $c=c(\textit{\texttt{data}})>0$, such that for all sufficiently small $r$ we have\begin{equation*}
{\mathcal{W}}^{\tens{\mu}}_G(x,r)\leq c r^\theta. \end{equation*} Applying assumption~\eqref{mu-control} to Proposition~\ref{prop:comp-exc} with $\alpha_V=\frac{\theta+1}{2}$, we have \begin{flalign*}
\barint_{B_{\rho}} |{\tens{u}}-({\tens{u}})_{B_{\rho}}|\,dx\leq c \left( \frac{\rho}{r} \right)^{\frac{\theta+1}{2}} \barint_{B_r}|{\tens{u}}-({\tens{u}})_{B_r}|\,dx + c r^{\theta} \end{flalign*} for any $0 < \rho < r \leq R_0$. Now we apply Lemma~\ref{lem:absorb2} to see
\[ \barint_{B_{\rho}} |{\tens{u}}-({\tens{u}})_{B_{\rho}}|\,dx\leq c \left( \frac{\rho}{r} \right)^{\theta} \barint_{B_r}|{\tens{u}}-({\tens{u}})_{B_r}|\,dx + c \rho^{\theta} \] By Campanato's characterization \cite[Theorem~2.9]{giusti}, we complete the proof. \end{proof}
\section*{Appendix}
\begin{proof}[Proof of Lemma~\ref{lem:Wolff-est}]Set $x\in\Omega_0,$ $R_k=2^{1-k}R$ and $B_k=B_{R_k}(x)$ for $k=0,1,\dots\,$. As ${\tens{F}}$ is taken in a place of a measure with a slight abuse of notation we write $|{\tens{F}}|(B_{R_k}(x))=\int_{B_k}|{\tens{F}}(y)|\,dy.$ We notice that we have
\[{\mathcal{W}}_G^{|{\tens{F}}|}(x,R)=\sum_{k=1}^\infty \int_{R_{k+1}}^{R_k} g^{-1}\left(\frac{|{\tens{F}}|(B_r(x))}{r^{n-1}}\right)\,dr\lesssim \sum_{k=1}^\infty R_{k} g^{-1}\left(\frac{|{\tens{F}}|(B_{R_k}(x))}{R_k^{n-1}}\right)\,.\]
To estimate the series we employ the decreasing rearrangement $|{\tens{F}}|^\star$ of $|{\tens{F}}|$ and its maximal rearrangement $|{\tens{F}}|^{\star\star}$. When $w_n$ is the volume of the unit ball, we have that \begin{flalign*}
\frac{|{\tens{F}}|(B_{R_k}(x))}{R_k^{n-1}}&=\frac{1}{R_k^{n-1}}\int_{B_{R_k}(x)}|{\tens{F}}(y)|\,dy\\&\leq {w_n R_k}\, \barint_0^{w_nR_k^n}|{\tens{F}}|^\star(t)\,dt={w_n R_k}\, |{\tens{F}}|^{\star\star}({w_nR_k^n})\,. \end{flalign*}
Then we have \begin{flalign}\label{1na8} R_kg^{-1}\left(\frac{|{\tens{F}}|(B_{R_k}(x))}{R_k^{n-1}}\right)&\lesssim R_kg^{-1}\left({w_n R_k}\, |{\tens{F}}|^{\star\star}({w_nR_k^n})\right)\\ &\lesssim \int_{w_n R_k^{n}}^{w_n R_{k-1}^n}\rho^\frac{1}{n}g^{-1}\left(\rho^\frac{1}{n} \, |{\tens{F}}|^{\star\star}(\rho)\right)\,\frac{d\rho}{\rho}\nonumber \end{flalign} with implicit constants independent of $k$. Therefore
\begin{flalign*}\sup_{x\in\Omega_0}{\mathcal{W}}_G^{|{\tens{F}}|}(x,R)&\lesssim \sum_{k=1}^\infty \int_{w_n R_k^{n}}^{w_n R_{k-1}^n} \rho^{\frac{1}{n}-1} g^{-1}\left(\rho^\frac{1}{n}\, |{\tens{F}}|^{\star\star}(\rho)\right)\,\frac{d\rho}{\rho}\\
&= \int_{0}^{w_n R^n} \rho^\frac{1}{n} g^{-1}\left(\rho^\frac{1}{n}\, |{\tens{F}}|^{\star\star}(\rho)\right)\,\frac{d\rho}{\rho}\,. \end{flalign*} \end{proof}
\noindent{\bf Acknowledgements} I. Chlebicka is supported by NCN grant no. 2019/34/E/ST1/00120. A. Zatorska-Goldstein is supported by NCN grant no. 2019/33/B/ST1/00535. Y. Youn is supported by NRF grant no. 2020R1C1C1A01009760.
\end{document} | arXiv |
\begin{document}
\title{Normal Functions and the Geometry of Moduli Spaces of Curves}
\author[Richard Hain]{Richard Hain} \address{Department of Mathematics\\ Duke University\\ Durham, NC 27708-0320}
\email{[email protected]}
\thanks{Supported in part by grants DMS-0103667 and DMS-1005675 from the National Science Foundation.}
\date{\today}
\maketitle
\tableofcontents
\section{Introduction}
A normal function on a complex manifold $T$ is a special kind of holomorphic section of a bundle ${\mathcal J}({\mathbb V}) \to T$ of compact complex tori constructed from a weight $-1$ variation of Hodge structure ${\mathbb V}$ over $T$. Normal functions in their modern formulation arose in the work \cite{griffiths} of Griffiths as a tool for understanding algebraic cycles in a complex projective manifold.
In this paper we give two examples to illustrate how normal functions might be a useful, if unconventional, tool for understanding the geometry of moduli spaces of curves. The first is to give a partial answer to a question of Eliashberg, which arose in symplectic field theory \cite{sft}. Namely, we compute the class in rational cohomology of the pullback of the $0$-section of the universal jacobian ${\mathcal J}_{g,n}^c\to {\mathcal M}_{g,n}^c$ over the moduli space of $n$-pointed, stable projective curves of compact type of genus $g$ along the section defined by $$ F_{\mathbf{d}} : [C;x_1,\dots,x_n] \mapsto \sum_{j=1}^n d_j [x_j] \in \Jac C, $$ where ${\mathbf{d}}=(d_1,\dots,d_n)\in {\mathbb Z}^n$ satisfies $\sum_{j=1}^n d_j = 0$. This section is a normal function.
The second application is to give an alternative and complementary approach to the slope inequalities of the type discovered by Moriwaki \cite{moriwaki,moriwaki:new} such as his result that the divisor class $$ M := (8g+4)\lambda_1 - g\delta_0 - 4\sum_{h=1}^{\lfloor g/2\rfloor} h(g-h)\delta_h $$ on ${\overline{\M}}_g$ has non-negative degree on all complete curves in ${\overline{\M}}_g$ that do not lie in the boundary divisor ${\Delta} := {\overline{\M}}_g-{\mathcal M}_g$.\footnote{A weaker version of this inequality had been proved previously by Cornalba and Harris in \cite{cornalba-harris}.} (Here the ${\delta}_h$ ($0\le h \le g/2$) denote the classes of the components of ${\Delta}$.) This alternative approach leads to actual and conjectural strengthenings of his inequalities: we show that the Moriwaki divisor $M$ has non-negative degree on all complete curves in ${\mathcal M}_g^c$ and conjecture that $M$ has non-negative degree on all complete curves in ${\overline{\M}}_g$ that do not lie in the boundary divisor ${\Delta}_0:= {\overline{\M}}_g - {\mathcal M}_g^c$.
Each normal function section $\nu$ of $J({\mathbb V}) \to T$ determines a class $c(\nu)$ in $H^1(T,{\mathbb V})$. When $H^0(T,{\mathbb V}_{\mathbb Q})$ vanishes, the normal function $\nu$ is determined, mod torsion, by its characteristic class $c(\nu)$. In such cases, there is a rigid relationship between normal functions and cohomology. When $T = {\mathcal M}_{g,n}^c$, $g\ge 3$, and ${\mathbb V}$ is a variation of Hodge structure corresponding to a non-trivial rational representation of $\Sp_g$ that does not contain the trivial representation, the group of normal function sections of $J({\mathbb V})$ is finitely generated and is known modulo torsion, \cite[\S8]{hain:normal}. This result is recalled in Appendix~\ref{sec:vmhs}. When ${\mathbb V}$ corresponds to the fundamental representation of $\Sp_g$, $J({\mathbb V})$ is the universal jacobian ${\mathcal J}_{g,n}^c$ over ${\mathcal M}_{g,n}^c$ and the class of the pullback $F_{\mathbf{d}}^\ast\eta_g$ of the zero section of ${\mathcal J}_{g,n}^c$ can be expressed in terms of the classes of certain basic normal functions defined on ${\mathcal M}_{g,n}^c$.
All variations of Hodge structure ${\mathbb V}$ of geometric origin have a {\em polarization}; that is, an invariant inner product $S : {\mathbb V}^{\otimes 2} \to {\mathbb Q}$ that satisfies the Riemann-Hodge bilinear relations on each fiber. When $V$ has odd weight, the polarization is skew symmetric. Each invariant, skew-symmetric inner product $S : {\mathbb V}^{\otimes 2} \to {\mathbb Q}$ gives rise to a class $S\circ c(\nu)^2$ in $H^2(T,{\mathbb Q})$. It is the image of $c(\nu)^{\otimes 2}$ under the composition of the cup product with the map induced by $S$: $$ \xymatrix{ H^1(T,{\mathbb V})^{\otimes 2} \ar[r]^{\smile} & H^2(T,{\mathbb V}^{\otimes 2}) \ar[r]^{S_\ast} & H^2(T,{\mathbb Q}). } $$ The class $S\circ c(\nu)^2$ has a natural de~Rham representative which is a non-negative $(1,1)$-form when $S$ is a polarization. Moriwaki's inequality for complete curves in ${\mathcal M}_g^c$ is an immediate consequence of this semi-positivity and the fact that the class of the Moriwaki divisor equals the square $S\circ c(\nu)^2$ of the class of the most fundamental normal function over ${\mathcal M}_g$ --- viz., the normal function associated to the cycle $C-C^-$ in $\Jac C$ that was first studied by Ceresa \cite{ceresa}.
When ${\mathbb V}$ is a weight $-1$ polarized variation of Hodge structure over $T$, there is a naturally metrized line bundle ${\mathcal B}$ over $J({\mathbb V})$, which is called the {\em biextension line bundle}. The curvature of its pullback along a normal function $\nu : T \to J({\mathbb V})$ is the natural de~Rham representative of $S\circ c(\nu)^2$. This line bundle extends naturally to any compactification $\overline{T}$ of $T$, even if ${\mathbb V}$ (and hence $\nu$) does not extend to $\overline{T}$. This extension is characterized by the property that the metric extends across the codimension 1 strata of $\overline{T}-T$. Surprisingly, this extension is {\em not} natural under pullback to a smooth variety as the metric on the extended line bundle may be singular on strata of codimension $\ge 2$ of $\overline{T}-T$. This curious phenomenon is called {\em height jumping}. Height jumping and its relevance to refined slope inequalities is discussed in Section~\ref{sec:moriwaki}.
The classes $c(\nu)$ form part of a larger structure. When $T={\mathcal M}_{g,n}$ they should be regarded as twisted tautological cohomology classes. To explain this, we need to introduce a certain graded commutative algebra associated to ${\mathcal M}_{g,n}$. Denote the coordinate ring of the symplectic group $\Sp_g$ by ${\mathcal O}(\Sp_g)$. Left and right multiplication induce commuting left and right actions of the mapping class group $\pi_1({\mathcal M}_{g,n},\ast)$ on it via the standard representation $\pi_1({\mathcal M}_{g,n},x_o) \to \Sp_g({\mathbb Q})$. Using the right action, one obtains a local system ${\pmb{\O}}$ of ${\mathbb Q}$-algebras over ${\mathcal M}_{g,n}$. Since ${\pmb{\O}}$ is a local system of commutative ${\mathbb Q}$-algebras, its cohomology $$ A_{g,n}^{\bullet} := H^{\bullet}({\mathcal M}_{g,n},{\pmb{\O}}) $$ is a graded commutative ${\mathbb Q}$-algebra. The left $\Sp_g$-action on ${\pmb{\O}}$ gives it the structure of a graded commutative algebra in the category of $\Sp_g$-modules.
The algebraic analogue of the Peter-Weyl Theorem implies that there is an $\Sp_g\times\Sp_g$-equivariant isomorphism $$ {\mathcal O}(\Sp_g) \cong \bigoplus_\lambda \End(V_\lambda)^\ast \cong \bigoplus_\lambda V_\lambda\boxtimes V_\lambda^\ast, $$ where $\{V_\lambda\}$ is a set of representatives of the isomorphism classes of irreducible $\Sp_g$-modules. There is thus an isomorphism $$ A_{g,n}^{\bullet} \cong \bigoplus_\lambda H^{\bullet}({\mathcal M}_{g,n},{\mathbb V}_\lambda)\otimes V_\lambda^\ast $$ where ${\mathbb V}_\lambda$ denotes the local system over ${\mathcal M}_{g,n}$ that corresponds to $V_\lambda$.
The classes $c(\nu)$ are more fundamental than their squares $S\circ c(\nu)^2$, which are known to be tautological classes. For this reason, we define the tautological subalgebra $T_{g,n}^{\bullet}$ of $A_{g,n}^{\bullet}$ to be the graded subalgebra generated by the classes $c(\nu)\otimes V_\lambda^\ast$ of the normal function sections $\nu$ of the $J({\mathbb V}_\lambda)$.\footnote{Each ${\mathbb V}_\lambda$ is the local system that underlies a polarized variation of Hodge structure over ${\mathcal M}_{g,n}$. It is unique up to Tate twist. The only ${\mathbb V}_\lambda$ that admit non-torsion normal functions have weight $-1$.} It is finitely generated. The classification of normal functions \cite{hain:normal} over ${\mathcal M}_{g,n}$, and the work of Kawazumi and Morita \cite{kawa-morita} imply that the ring of $\Sp_g$-invariants $(T_{g,n}^{\bullet})^{\Sp_g}$ is Faber's tautological ring $R_{g,n}$ (in cohomology) \cite{faber} of ${\mathcal M}_{g,n}$. The computations of Morita \cite{morita:taut}, Kawazumi and Morita \cite{kawa-morita}, and those of this paper, may be regarded as computations in $T_{g,n}^{\bullet}$. For this reason, we propose that the ring $T_{g,n}^{\bullet}$ is more fundamental than its $\Sp_g$-invariant part $R_{g,n}$. It would be interesting to define and study a Chow analogue of $T_{g,n}^{\bullet}$. The significance of this algebra and its relation to normal functions is discussed in Appendix~\ref{sec:big_picture}.
\noindent{\it Advice to the reader:} Although normal functions have long been a part of algebraic geometry (examples were first considered by Poincar\'e), they are not currently part of the standard repertoire of modern algebraic geometry. Their modern definition (Definition~\ref{def:normal}), in terms of extensions of variations of Hodge structure, requires an understanding of variations of (mixed) Hodge structure. However, if the reader is prepared to believe that the local systems associated to locally topologically trivial families of algebraic varieties are motivic, and so are variations of mixed Hodge structure, then the definition should be natural.
We assume the reader is familiar with the basic definitions and constructions of Hodge theory. In particular, the reader should know the definition of Hodge structures, mixed Hodge structures, and variations of Hodge structure. The book \cite{peters-steenbrink} by Peters and Steenbrink is a good source of basic material on these topics. The paper \cite{hain:cdm} contains a brief exposition of Schmid's work \cite{schmid} on the asymptotic properties of variations of Hodge structure that emphasizes the case of degenerations of curves. Finally, the recent survey of normal functions \cite{kerr-pearlstein} by Kerr and Pearlstein should be a useful supplement, although its emphasis is quite different from that of this article.
\noindent{\it Acknowledgments:} I am grateful to Yasha Eliashberg for posing his question to me in 2001 and to Gavril Farkas for his interest, which resulted in this paper seeing the light of day. Theorem~\ref{thm:eliashberg} was proved during visits to the University of Sydney and the Universit\'e de Nice during the author's sabbatical in 2002--03. Many thanks to both institutions for their support, and to my respective hosts, Gus Lehrer and Arnaud Beauville, for their hospitality. I am especially grateful to Samuel Grushevsky, Robin de Jong and Dmitry Zharkov for their interest in this work and for their constructive comments and corrections. Sam and Dmitry pointed out an error in the coefficient of ${\delta}_h^P$ in Theorem~\ref{thm:eliashberg}; Sam and Robin isolated the error, which was a missing term in the formula Theorem~\ref{phi2}.
I would also like to thank Gregory Pearlstein for his numerous constructive comments on the manuscript, and also Renzo Cavalieri for communicating his related results \cite{cavalieri-marcus} with Steffen Marcus. The section on the genus 1 case of Eliashberg's problem was written as a result of correspondence with him. Finally, I would like to thank the referee for helpful comments.
\section{Notation and Conventions}
All varieties (and stacks) will be defined over the complex numbers. Denote the moduli space of stable $n$-pointed curves of genus $g$ by ${\overline{\M}}_{g,n}$. This is defined when $2g-2+n>0$ and will be viewed as a stack or as a complex analytic orbifold. As such, it is smooth. Denote the Zariski open subset corresponding to the set of $n$-pointed smooth curves by ${\mathcal M}_{g,n}$ and by ${\mathcal M}_{g,n}^c$ the Zariski open subset consisting of $n$-pointed curves of compact type. Note that ${\mathcal M}_{g,n}^c = {\overline{\M}}_{g,n}-{\Delta}_0$, where ${\Delta}_0$ denotes the boundary divisor of ${\overline{\M}}_{g,n}$ whose generic point is an irreducible, geometrically connected curve with one node.
The moduli stack of principally polarized abelian varieties of dimension $g$ will be denoted by ${\mathcal A}_g$. The universal curve of compact type will be denoted by ${\mathcal C}_g^c \to {\mathcal M}_g^c$ and its restriction to ${\mathcal M}_g$ by ${\mathcal C}_g$. All of these moduli spaces are globally the quotient of a smooth variety by a finite group, \cite{looijenga,boggi-pikaart}.
Vector bundles, variations of (mixed) Hodge structure, etc on a stack $T$ that is the quotient of a smooth variety $S$ by a finite group $G$, are $G$-invariant bundles, variations of (mixed) Hodge structure, etc, over $S$. Since all stacks that occur in this paper are of this form, we will not distinguish between stacks and varieties, as working on one of these stacks is working equivariantly on a finite cover that is a variety.
The category of ${\mathbb Z}$-mixed Hodge structures by ${\sf{MHS}}$. For $d\in{\mathbb Z}$, the Hodge structure of type $(-d,-d)$ whose underlying lattice is isomorphic to ${\mathbb Z}$ will be denoted by ${\mathbb Z}(d)$. We shall denote by ${\Gamma} V$ the set $\Hom_{\sf{MHS}}({\mathbb Z}(0),V)$ of Hodge classes of type $(0,0)$ of the mixed Hodge structure $V$ . The category of admissible variations of mixed Hodge structure over a smooth variety $X$ will be denoted by ${\sf{MHS}}(X)$.
All cohomology groups will be with ${\mathbb Q}$ coefficients unless otherwise stated. Similarly, the Chow group of codimension $d$ cycles on a stack $X$, {\em tensored with ${\mathbb Q}$}, will be denoted by $CH^d(X)$.
\section{Eliashberg's Problem} \label{sec:eliashberg}
To motivate the discussion of normal functions and related topics in subsequent sections, we begin with a brief discussion of the universal jacobian and Eliashberg's problem. Some readers may prefer to begin with Sections~\ref{tori} and \ref{sec:normal}. Recall that a stable curve $C$ of genus $g$ is of {\em compact type} if its dual graph is a tree. This condition is equivalent to the condition that its jacobian $\Jac C := \Pic^0 C$ be an abelian variety.
\subsection{The universal jacobian} We begin with a review of the transcendental construction of the jacobian of the universal curve over ${\mathcal M}_g$. This is a special case of Griffiths' construction of families of intermediate jacobians and normal functions, which are reviewed in Section~\ref{sec:normal}.
First recall the transcendental construction of the jacobian of a smooth projective curve $C$, which we recast in the language of Hodge structures. It will be generalized in Section~\ref{sec:normal} where Griffiths intermediate jacobians are introduced. The Hodge Theorem implies that \begin{equation} \label{eqn:decomp} H^1(C,{\mathbb C}) \cong H^{1,0}(C) \oplus H^{0,1}(C) \end{equation} where $H^{1,0}(C)$ denotes the space $H^0(C,\Omega_C^1)$ of holomorphic 1-forms on $C$ and $H^{0,1}(C)$ its complex conjugate, the space of anti-holomorphic 1-forms. The first integral cohomology group $H^1(C,{\mathbb Z})$ endowed with the decomposition (\ref{eqn:decomp}) is the prototypical Hodge structure of weight 1. Its dual $$ H_1(C,{\mathbb C}) = H^{-1,0}(C) \oplus H^{0,-1}(C) $$ is a Hodge structure of weight $-1$, where $H^{-p,-q}(C)$ is defined to be the dual of $H^{p,q}(C)$. The Hodge filtration $$ H_1(C,{\mathbb C}) = F^{-1}H_1(C) \supset F^0H_1(C) \supset F^1H_1(C) = 0. $$ of $H_1(C)$ is defined by $$ F^p H_1(C) = \bigoplus_{\substack{s\ge p\cr s+t=-1}} H^{s,t}(C). $$ The projection onto $H^{1,0}(C)$ induces an isomorphism $$ H_1(C,{\mathbb C})/F^0 \cong H^{-1,0}(C) \cong H^0(C,\Omega^1_C)^\ast := \Hom_{\mathbb C}(H^0(C,\Omega^1_C),{\mathbb C}). $$ The composite $$ H_1(C,{\mathbb Z}) \hookrightarrow H_1(C,{\mathbb C}) \to H_1(C,{\mathbb C})/F^0 \cong H^0(C,\Omega^1_C)^\ast $$ is the map that takes the homology class of the 1-cycle $\gamma$ to the functional $$ \int_\gamma := \Big\{\omega \mapsto \int_\gamma \omega\Big\} \in H^0(C,\Omega_C^1)^\ast. $$ It is injective and its image is a lattice. The jacobian of $C$ is the quotient $$ \Jac C := H^0(C,\Omega^1_C)^\ast/H_1(C,{\mathbb Z}) \cong H_1(C,{\mathbb C})/(H_1(C,{\mathbb Z}) + F^0H_1(C)). $$ Every divisor $D$ of degree $0$ on $C$ can be written as the boundary $D=\partial \gamma$ of a real $1$-chain $\gamma$. The Abel-Jacobi mapping $$ \{\text{divisors of degree 0 on $C$}\}/\text{rational equivalence} \to \Jac C $$ is defined by taking the divisor class of the boundary of the 1-chain $\gamma$ to the functional $\int_\gamma$. Abel's Theorem implies that it is a group isomorphism. This construction works equally well when $C$ is a curve of compact type. In this case, the Hodge structure on $H_1(C)$ is the direct sum of the Hodge structures of its irreducible components. This construction can also be carried out for families of complete curves, where each fiber is either smooth or of compact type. Below we carry this out for the universal curve of compact type. The family of Hodge structures associated to such a family is an example of a {\em variation of Hodge structure}.
To the universal curve $\pi : {\mathcal C}_g^c \to {\mathcal M}_g^c$ of compact type we associate the variation of Hodge structure $$ {\mathbb H} := R^1 \pi_\ast {\mathbb Z} $$ and the corresponding holomorphic vector bundle ${\mathcal H} := {\mathbb H} \otimes_{\mathbb Z} {\mathcal O}_{{\mathcal M}_g^c}$. The fiber of ${\mathcal H}$ over the moduli point of $C$ is $H^1(C,{\mathbb C})$. This bundle has a flat holomorphic connection $\nabla$. The local monodromy transformations about the divisor $\Delta_0:={\overline{\M}}_{g,n}-{\mathcal M}_{g,n}^c$ are given by the Picard-Lefschetz formula and are therefore unipotent. Consequently, ${\mathcal H}$ has a canonical extension (in the sense of Deligne \cite{deligne:ode}) to a vector bundle ${\overline{{\cH}}}$ over ${\overline{\M}}_g$.\footnote{See \cite{hain:cdm} for a concise exposition.} It is characterized by the property that the connection is regular and that its residue at each smooth point of $\Delta_0$ is nilpotent. Since the monodromy of ${\mathbb H}$ about ${\Delta}_0$ is non-trivial, the local system ${\mathbb H}$ does not extend across ${\Delta}_0$. Consequently, ${\mathcal M}_{g,n}^c$ is the maximal Zariski open subset of ${\overline{\M}}_{g,n}$ to which ${\mathbb H}$ extends.
The {\em Hodge bundle} ${\mathcal F} := {\mathcal F}^1{\mathcal H}$ is the sub-bundle of ${\mathcal H}$ whose fiber over the moduli point of $C$ is $H^{1,0}(C)$. It is holomorphic and extends, by a result of Deligne \cite{deligne:ode}, to a holomorphic sub-bundle of $\overline{\F}^1$ of ${\overline{{\cH}}}$.\footnote{The fiber ${\mathcal F}_C$ of the Hodge bundle over the stable curve $C$ can be described as follows. Denote the normalization of $C$ by $\nu : {\widetilde{C}} \to C$. Let $D\subset {\widetilde{C}}$ be the inverse image of the double points of $C$. Then ${\mathcal H}_C$ is the subset of $H^0({\widetilde{C}},\Omega^1_{\widetilde{C}}(D))$ consisting of those $w$ such that $\Res_P w + \Res_Q w = 0$ whenever $P\neq Q$ and $\nu(P) = \nu(Q)$. It is naturally isomorphic to $F^1H^1(C_{\vec{v}})$ where $H^1(C_{\vec{v}})$ denotes the limit MHS on the first order smoothing $C_{\vec{v}}$ of $C$ associated to a tangent vector ${\vec{v}}$ of ${\overline{\M}}_g$ at $[C]$ which is not tangent to the boundary divisor.} (See \cite[\S4]{hain:cdm} for an exposition.) There is a natural projection $\check{\mathcal F} \to {\mathcal J}_g$ from the dual of the Hodge bundle to the universal jacobian, which is a covering map on each fiber. The kernel of the projection $$ \check{\mathcal F}_C \to {\mathcal J}_{g,C} $$ at the moduli point of the stable curve $C$ is the image of $H_1(C-C^{\mathrm{sing}},{\mathbb Z})$ under the integration map $H_1(C-C^{\mathrm{sing}},{\mathbb Z})\rightarrow \check{\mathcal F}_C$. The restriction of ${\mathcal J}_g$ to ${\overline{\M}}_g - {\Delta}_0^{\mathrm{sing}}$ is a Hausdorff complex analytic orbifold, a fact which follows, for example, from \cite[Prop.~2.9]{zucker}. It is the analytic orbifold associated to the restriction of the universal $\Pic^0$ stack over ${\overline{\M}}_g$ to ${\overline{\M}}_g - {\Delta}_0^{\mathrm{sing}}$, which is constructed in \cite{caporaso}.
Observe that the fiber of ${\mathcal J}_g \to {\overline{\M}}_g$ over the moduli point of a stable curve $C$ is an abelian variety if and only if $C$ is of compact type. From the construction, it is clear that the normal bundle of the zero section of ${\mathcal J}_g$ is the dual $\check{\mathcal F}$ of the Hodge bundle.
\subsection{Eliashberg's question}
Suppose that $2g-2+n>0$. Given an integer vector ${\mathbf{d}}=(d_1,\dots, d_n)$ with $\sum_j d_j=0$, we have the {\em rational} section $F_{\mathbf{d}}$ $$ \xymatrix{
& {\mathcal J}_g \ar[d] \cr {\overline{\M}}_{g,n}\ar[r]\ar@{.>}[ur]^{F_{\mathbf{d}}} & {\overline{\M}}_g } $$ of the universal jacobian defined by $$ F_{\mathbf{d}} : [C;x_1,\dots,x_d] \mapsto \bigg[\sum_{j=1}^n d_j x_j\bigg] \in \Jac C $$ when $C$ is smooth. It is holomorphic over ${\overline{\M}}_{g,n}-\Delta_0^{\mathrm{sing}}$, the complement of the singular locus of ${\Delta}_0$.
Denote the class of the zero section of ${\mathcal J}_g$ in $H^{2g}({\mathcal J}_g,{\mathbb Q})$ by $\eta_g$.
\begin{problem}[Eliashberg] Compute the class in $H^{2g}({\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}})$ of the pullback $F_{\mathbf{d}}^\ast \eta_g$ of the zero section of ${\mathcal J}_g$. \end{problem}
Denote the $j$th Chern class of the Hodge bundle by $\lambda_j$. When all $d_j$ are zero, the section $F_{\mathbf{d}}$ is defined on all of ${\overline{\M}}_{g,n}$. Since the normal bundle of the zero section is $\check{\mathcal F}$, the dual of the Hodge bundle, we have:
\begin{proposition} \label{prop:res_lambda} If ${\mathbf{d}} = 0$, then $F_{\mathbf{d}}^\ast\eta_g = (-1)^g \lambda_g \in H^{2g}({\overline{\M}}_{g,n},{\mathbb Q})$. \end{proposition}
\begin{remark} This result also holds in the Chow ring. \end{remark}
\section{Families of Compact Tori} \label{tori}
The restriction of the universal jacobian to ${\mathcal M}_{g,n}^c$ is a family of compact tori. This section is a discussion of some general properties of families of compact tori.
\begin{definition} A family of compact (real) $r$-dimensional tori is a smooth fiber bundle $f: T \to B$, each of whose fibers is a compact, connected abelian Lie group. This bundle is locally (but typically not globally) trivial as a bundle of Lie groups. \end{definition}
We shall assume throughout that $B$ is connected. The identity section will be denoted by $s : B \to T$. Denote the fiber of $T$ over $b\in B$ by $T_b$.
For a coefficient ring $R$, denote the local system over $B$ whose fiber over $b$ is $H_1(T_b,R)$ by ${\mathbb H}_R$. The following assertion is easily proved.
\begin{proposition} If $f : T \to B$ is a family of compact tori, then there is a natural bijection $$ T \to {\mathbb H}_{\mathbb R}/{\mathbb H}_{\mathbb Z} $$ which commutes with the projections to $B$ and is a group homomorphism on each fiber. \qed \end{proposition}
The flat structure on ${\mathbb H}_{\mathbb R}$ descends to a flat structure on $T={\mathbb H}_{\mathbb R}/{\mathbb H}_{\mathbb Z}$.
\begin{corollary} Every bundle of compact tori has a natural flat structure in which the torsion multi-sections are leaves. Equivalently, a bundle of compact real tori has a natural trivialization over each contractible subset of $B$ in which the torsion sections are constant. \qed \end{corollary}
\subsection{The class of a section} \label{sec:sect_class}
Each section $s$ of a family $T\to B$ of compact tori determines a class $c(s) \in H^1(B,{\mathbb H}_{\mathbb Z})$. We review three standard constructions of this class.
The first is sheaf theoretic. Denote the sheaf of $C^\infty$ real-valued functions on $B$ by ${\mathcal E}_B$. The flat vector bundle associated to ${\mathbb H}_{\mathbb R}$ has sheaf of sections ${\mathcal H} := {\mathbb H}_{\mathbb R} \otimes_{\mathbb R} {\mathcal E}_B$. Denote the sheaf of smooth sections of $T\to B$ by ${\mathcal T}$. Then one has a short exact sequence $$ 0 \to {\mathbb H}_{\mathbb Z} \to {\mathcal H} \to {\mathcal T} \to 0 $$ of sheaves. Taking cohomology yields the exact sequence $$ 0 \to H^0(B,{\mathbb H}_{\mathbb Z}) \to H^0(B,{\mathcal H}) \to H^0(B,{\mathcal T}) \overset{c}{\to} H^1(B,{\mathbb H}_{\mathbb Z}) \to 0. $$ The connecting homomorphism is well defined up to a sign. With the appropriate choice it takes a section $s$ to its characteristic class. Note that the vanishing of $c(s)$ implies that $s$ is homotopic to the zero section. Thus $H^1(B,{\mathbb H}_{\mathbb Z})$ can be identified with the group of homotopy classes of smooth sections of $T \to B$.
The second description is obtained by regarding $H^1(B,{\mathbb H}_{\mathbb Z})$ as congruence classes of extensions $$ 0 \to {\mathbb H}_{\mathbb Z} \to {\mathbb E} \to {\mathbb Z}_B \to 0 $$ of local systems over $B$. Given a section $s$ of $T\to B$, we can construct such an extension ${\mathbb E}$ as the local system whose fiber over $b\in B$ is $H_1({\widehat{T}}_b,{\mathbb Z})$, where $$ {\widehat{T}}_b := T_b\cup_h [0,1] $$ where $h(0)= 0$ and $h(1)=s(b)$. There is a short exact sequence $$ 0 \to H_1(T_b) \to H_1({\widehat{T}}_b) \to {\mathbb Z} \to 0 $$ in which $H_1([0,1],\{0,1\})$ is identified with ${\mathbb Z}$ by taking the generator to be the class of a path from $1$ to $0$.
When $s(b)\neq 0$, $H_1({\widehat{T}}_b) \cong H_1(T_b,\{0,s(b)\}) \cong H^{r-1}(T_b-\{0,s(b)\})$. The first description of $c$ is determined only up to a sign. This description fixes the sign.
The third description uses de Rham cohomology. Each $b\in B$ has an open neighbourhood $U$ where $s$ lifts to a section $\tilde{s} : U \to {\mathcal H}$ of the flat vector bundle ${\mathcal H}$. When $U$ is connected, such a lift is unique up to translation by a local section of ${\mathbb H}_{\mathbb Z}$. The 1-form $d\tilde{s}$ on $U$ with values in ${\mathbb H}_{\mathbb R}$ is therefore independent of the choice of the lift $\tilde{s}$. The de Rham representative of $c(s)$ is the class that is locally represented by $d\tilde{s}$.
\subsubsection{Equivalence of these constructions} Here is a quick sketch of the equivalence of these definitions. Choose an open covering ${\mathcal U} = \{U_\alpha\}$ of $T$ such that $U_{\alpha_0} \cap \dots \cap U_{\alpha_q}$ is contractible for all multi-indices $(\alpha_0,\dots,\alpha_q)$. Such an open covering can be constructed by taking each $U_\alpha$ to be a geodesically convex ball with respect to some riemannian metric on $T$. The complex $C^{\bullet}({\mathcal U},{\mathcal F})$ of {C}ech cochains with coefficients in ${\mathcal F}$ computes $H^{\bullet}(T,{\mathcal F})$ when ${\mathcal F}$ is ${\mathbb H}_{\mathbb Z}$, ${\mathbb H}_{\mathbb R}$, ${\mathcal H}$ and ${\mathcal T}$. Suppose that $s$ is a smooth section of $T \to B$. Its restriction to $U_\alpha$ can be lifted to a section $s_\alpha$ of ${\mathcal H}$. The difference $c_{\alpha\beta} := s_\beta - s_\alpha$ is a section of ${\mathbb H}_{\mathbb Z}$ over $U_\alpha\cap U_\beta$. The class of $c(s)$ is represented by the cocycle $(c_{\alpha\beta}) \in C^1({\mathcal U},{\mathbb H}_{\mathbb Z})$.
The class of an extension $$ 0 \to {\mathbb H}_{\mathbb Z} \to {\mathbb E} \to {\mathbb Z}_T \to 0. $$ is computed by choosing sections $e_\alpha$ of ${\mathbb E} \to {\mathbb Z}_T$ over each $U_\alpha$. The class of the extension is represented by the cocycle $(e_\beta-e_\alpha)_{\alpha\beta} \in C^1({\mathcal U},{\mathbb H}_{\mathbb Z})$. In the case above, one can take the local section $e_\alpha$ to be the class in $H_1(T_b,\{0,s(b)\})$ of the image of the path in the universal covering ${\mathbb H}_{{\mathbb R},b}$ of $T_b$ that goes from $0$ to $s_\alpha(b)$. Then $e_\beta - e_\alpha = c_{\alpha\beta}$, as required.
Denote the de~Rham sheaf of smooth ${\mathbb R}$-valued forms on $T$ by ${\mathcal E}^{\bullet}_T$. Standard arguments imply that the inclusions $$ C^{\bullet}({\mathcal U},{\mathbb H}_{\mathbb R}) \hookrightarrow C^{\bullet}({\mathcal U},{\mathcal E}^{\bullet}_T\otimes {\mathbb H}_{\mathbb R}) \hookleftarrow {\mathcal E}^{\bullet}_T \otimes {\mathbb H}_{\mathbb R} $$ induce isomorphisms on homology. The standard zig-zag argument implies that the class of the cocycle $(c_{\alpha\beta})$ is represented by the element of $$ E^{\bullet}(T,{\mathbb H}_{\mathbb R}) := H^0(T,{\mathcal E}^{\bullet}_T \otimes {\mathbb H}_{\mathbb R}) $$ whose restriction to $U_\alpha$ is $ds_\alpha$.
\subsection{Invariant cohomology classes}
The flat structure of a family of compact tori $T\to B$ can be used to construct natural de Rham representatives of cohomology classes on $T$. We will say that a differential form $w$ on a manifold $M$ with a foliation ${\mathcal L}$ is {\it parallel with respect to} ${\mathcal L}$ if the Lie derivative of $w$ with respect to each vector field tangent to ${\mathcal L}$ vanishes. A family of tori is foliated as it is a flat family of tori.
The following lemma is proved in \cite[Lemma~5.1]{hain-reed:arakelov}.
\begin{lemma} \label{lem:gen_lifts} If $f : T \to B$ is a family of compact tori, there is a natural mapping $$ \sigma : H^0(B, R^k f_\ast {\mathbb R}) \to H^k(T,{\mathbb R}) $$ whose composition with the projection $$ H^k(T,{\mathbb R}) \to H^0(B, R^k f_\ast {\mathbb R}) $$ is the identity. Moreover, for each $u\in H^0(B,R^k f_\ast {\mathbb R})$, the extended class $\sigma(u)$ has a natural differential form representative $w_u$ whose restriction to each fiber is a closed, translation-invariant differential form, and which is parallel with respect to the flat structure. This class has the property that its restriction to every leaf (such as the zero section and every torsion multi-section) is zero. \qed \end{lemma}
\subsection{The Poincar\'e dual of the zero section}
Let $r$ be the real dimension of the fiber of $f : T \to B$. If $B$ and $T$ are oriented and $B$ is connected, then $$ H^0(B,R^r f_\ast {\mathbb R}) \cong {\mathbb R}. $$ Let $u$ be the element of this group whose value on one (and hence all) fibers is $1$. Denote the class $\sigma(u) \in H^r(T,{\mathbb R})$ by $\psi$.
\begin{proposition} \label{cpact_tori} If the base $B$ is a compact manifold (possibly with boundary), then the Poincar\'e dual of the zero section is $\psi$. \end{proposition}
\begin{proof} Set $d = \dim_{\mathbb R} B$. For $e\in {\mathbb Z}$ define $[e] : T \to T$ to be the map whose restriction to each fiber is multiplication by $e$. Since $[e]$ induces multiplication by $e^k$ on $R^k f_\ast {\mathbb Q}$, it follows that the Leray spectral sequence degenerates at $E_2$. It also follows that the eigenvalues of the induced mapping $[e]^\ast$ on $H^k(T)$ lie in $\{1,e,\dots,e^k\}$. Since none of these eigenvalues is zero when $e\neq 0$, $[e]^\ast$ is invertible on rational cohomology when $e\neq 0$. Similar assertions hold for the homology spectral sequence $$ H_s(B,\partial B;{\mathbb H}_t) \implies H_{s+t}(T,\partial T), $$ where ${\mathbb H}_t$ denotes the local system whose fiber over $b\in B$ is $H_t(T_b)$. Since $$ H_{r+d}(T,\partial T) = H_d(B,\partial B; {\mathbb H}_r) $$ it follows that $[e]_\ast [T] = e^r[T]$, where $[T]$ denotes the fundamental class of $(T,\partial T)$.
Now assume that $e>1$. Then the collapsing of the Leray spectral sequence implies that the dimension of the $e^r$-eigenspace of $H^r(T)$ is one. Since the form $w_\psi$ that naturally represents $\psi$ has the property that $$ [e]^\ast w_\psi = e^r w_\psi, $$ and since $\psi$ is non-trivial (as it has non trivial integral over a fiber), it follows that $\psi$ spans this eigenspace.
Note that the class $[Z] \in H_d(T,\partial T)$ of the zero-section is an eigenvector of $[e]_\ast$ with eigenvalue $1$. Denote the Poincar\'e dual of the zero section by $\eta_Z$. It is characterized by the property that $$ [T] \cap \eta_Z = [Z] \in H_d(T,\partial T), $$ where $\cap$ denotes the cap product \cite[p.~254]{spanier} $$ \cap : H_{d+r}(T,\partial T) \otimes H^r(T) \to H_d(T,\partial T). $$ Since $e_\ast[Z]=[Z]$, standard properties of the cap product \cite[Assertion 16, p.~254]{spanier}, we have $$ [e]_\ast\big([T] \cap [e]^\ast \eta_Z\big) = \big([e]_\ast [T]\big) \cap \eta_Z = e^r [T] \cap \eta_Z = [e]_\ast \big([T] \cap e^r \eta_Z \big). $$ Since $[e]_\ast$ and capping with $[T]$ are both isomorphisms, it follows that $[e]^\ast \eta_Z = e^r \eta_Z$. Since $\eta_Z$ and $\psi$ both lie in the $e^r$-eigenspace and agree on each fiber, they are equal. \end{proof}
Since the normal bundle of the zero section $Z$ is the flat bundle ${\mathbb H}_{\mathbb R}$, the Euler class of the normal bundle of $Z$ vanishes in rational cohomology.
\begin{corollary} \label{vanishing} The restriction of the Poincar\'e dual of the zero section $Z$ of a family of compact tori to $Z$ vanishes in rational cohomology. \qed \end{corollary}
Combined with Proposition~\ref{prop:res_lambda}, this implies the well-known property of the top Chern class of the Hodge bundle.
\begin{corollary} The restriction of $\lambda_g$ to ${\mathcal M}_g^c$ vanishes in rational cohomology. \end{corollary}
\subsection{The class $S\circ c(s)^2$}
Suppose that $f:T\to B$ is a flat family of tori and that ${\mathbb H}$ is the corresponding local system such that $T={\mathbb H}_{\mathbb R}/{\mathbb H}_{\mathbb Z}$. A flat, skew-symmetric inner product $S : {\mathbb H}_{\mathbb R} \otimes {\mathbb H}_{\mathbb R} \to {\mathbb R}$ gives an element of $H^0(T,R^2f_\ast{\mathbb R}_T)$. Lemma~\ref{lem:gen_lifts} implies that $S$ determines a closed $2$-form $\phi_S$ on $T$. It is characterized by the properties: \begin{enumerate}
\item its restriction to the fiber $T_b$ of $T$ is the translation invariant $2$-form on $T_b$ that corresponds to $S$,
\item it is parallel with respect to the flat structure on $T$,
\item its restriction to the zero-section of $T$ is zero.
\end{enumerate}
The following result is easily proved using the de~Rham description of $c(s)$.
\begin{proposition} If $s$ is a holomorphic section of $T\to B$, then the form $s^\ast \phi_S$ represents the cohomology class $S\circ c(s)^2 \in H^2(T,{\mathbb R})$. \qed \end{proposition}
\section{Normal Functions} \label{sec:normal}
Normal functions are our primary tool. They are holomorphic sections of families of intermediate jacobians that satisfy certain infinitesimal and asymptotic conditions. In this section, we recall Griffiths construction of intermediate jacobians and of the normal function associated to a family of homologically trivial algebraic cycles in a family of smooth projective varieties.
\subsection{Intermediate Jacobians}
Suppose that $Y$ is a compact K\"ahler manifold and that $Z$ is an algebraic $d$-cycle in $Y$ where $0\le d < \dim Y$. One has the exact sequence $$
0 \to H_{2d+1}(Y) \to H_{2d+1}(Y,Z) \to H_{2d}(|Z|) \to H_{2d}(Y) \to \cdots $$
of integral homology groups associated to the pair $(Y,|Z|)$, where $|Z|$ denotes the support of $Z$. It is an exact sequence of mixed Hodge structure. The class of the cycle $Z$ defines a morphism of mixed Hodge structures $$
c_Z : {\mathbb Z}(d) \to H_{2d}(|Z|), $$ where ${\mathbb Z}(d)$ denotes the Hodge structure of type $(-d,-d)$ whose underlying lattice is isomorphic to ${\mathbb Z}$. If $Z$ is null homologous, we can pull back the above sequence along $c_Z$ to obtain an extension $$ 0 \to H_{2d+1}(Y) \to E_Z \to {\mathbb Z}(d) \to 0 $$ in ${\sf{MHS}}$, the category of mixed Hodge structures. Tensoring with ${\mathbb Z}(-d)$ gives an extension $$ 0 \to H_{2d+1}(Y,{\mathbb Z}(-d)) \to E_Z(-d) \to {\mathbb Z}(0) \to 0 $$ and thus a class $e_Z$ in $$ \Ext^1_{\sf{MHS}}({\mathbb Z}(0),H_{2d+1}(Y,{\mathbb Z}(-d))). $$ Note that, since $H_{2d+1}(Y)$ has weight $-(2d+1)$, $H_{2d+1}(Y,{\mathbb Z}(-d))$ has weight $-1$.
Suppose that $V$ is a Hodge structure of weight $-1$ whose underlying lattice $V_{\mathbb Z}$ is torsion free. The associated jacobian $$ J(V) := V_{\mathbb C}/(V_{\mathbb Z} + F^0 V_{\mathbb C}) $$ is a compact complex torus. In general, $J(V)$ is not an abelian variety. When $V$ is the weight $-1$ Hodge structure $H_{2d+1}(Y,{\mathbb Z}(-d))$ (mod its torsion), $J(V)$ is the $d$th Griffiths intermediate jacobian of $Y$.
There is a natural isomorphism (see \cite{carlson} or \cite[\S 3.5]{peters-steenbrink}, for example) $$ \Ext^1_{\sf{MHS}}({\mathbb Z}(0),V) \cong J(V). $$ The class $e_Z$ of a homologically trivial $d$-cycle $Z$ in $Y$ can thus be viewed as a class $$ e_Z \in J(H_{2d+1}(Y,{\mathbb Z}(d))) $$ in the $d$th Griffiths intermediate jacobian. This class can be described explicitly by Griffiths' generalization \cite{griffiths} of the Abel-Jacobi construction , which we now recall. First observe that the standard pairing between $H_{2d+1}(Y)$ and $H^{2d+1}(Y)$ induces an isomorphism $$ H_{2d+1}(Y,{\mathbb Z}(-d))/F^0 \cong \Hom_{\mathbb C}(F^{d+1}H^{2d+1}(Y),{\mathbb C}). $$ The natural mapping $H_{2d+1}(Y,{\mathbb Z}(-d)) \to H_{2d+1}(Y,{\mathbb Z}(-d))/F^0$ corresponds to the integration mapping $$ H_{2d+1}(Y,{\mathbb Z}) \to \Hom_{\mathbb C}(F^{d+1}H^{2d+1}(Y),{\mathbb C}) $$ that takes the homology class $z$ to $\xi \mapsto \int_z \xi$. A homologically trivial $d$-cycle $Z$ in $Y$ can be written as the boundary $\partial {\Gamma}$ of a (topological) $(2d+1)$-chain ${\Gamma}$. Note that ${\Gamma}$ is well defined up to the addition of an integral $(2d+1)$-cycle. Classical Hodge theory implies that each element $u$ of $F^{d+1}H^{2d+1}(Y)$ can be represented by a closed $C^\infty$ form $\xi_u$ in the $(d+1)$st level of the Hodge filtration on the de~Rham complex of $Y$ and that any two such forms differ by the exterior derivative of a form in the same level $F^{d+1}$ of the de~Rham complex. The point $e_Z$ is represented by the element $$ \int_{\Gamma} : u \mapsto \int_{\Gamma} \xi_u. $$ of $$ \Hom_{\mathbb C}(F^{d+1}H^{2d+1}(Y),{\mathbb C})/H_{2d+1}(Y,{\mathbb Z}) \cong J(H_{2d+1}(Y,{\mathbb Z}(-d))). $$ Stokes' Theorem implies that the image of this functional in the intermediate jacobian depends only on $Z$ and not on the choice of ${\Gamma}$ or $\xi_u$.
This construction generalizes the classical construction for $0$-cycles on curves that was sketched in Section~\ref{sec:eliashberg}. More generally, it generalizes the classical construction for $0$-cycles, where $J(H_1(Y)) \cong \Alb Y$, and for divisors, where $J(H_{2d-1}(Y))$ $\cong \Pic^0 Y$ and $d=\dim Y$.
\subsection{Normal Functions}
Suppose that ${\overline{X}}$ is a complex projective manifold and that $X = {\overline{X}} - D$ where $D$ is a normal crossings divisor in ${\overline{X}}$. Suppose that ${\mathbb V}$ is a variation of Hodge structure over $X$ of weight $-1$. Denote by $J({\mathbb V})\to X$ the corresponding bundle of intermediate jacobians; the fiber over $x\in X$ is $J(V_x)$, where $V_x$ is the fiber of ${\mathbb V}$ over $x$. It is a family of compact tori.
The discussion of Section~\ref{sec:sect_class} implies that a holomorphic section $\nu : X \to J({\mathbb V})$ determines a cohomology class $c(\nu) \in H^1(X,{\mathbb V})$ and a local system ${\mathbb E} \to X$ which is an extension $$ 0 \to {\mathbb V} \to {\mathbb E} \to {\mathbb Z}_X \to 0. $$ The point $\nu(x) \in J(V_x)\cong \Ext^1_{\sf{MHS}}({\mathbb Z}(0),V_x)$ determines a mixed Hodge structure on the fiber $E_x$ of ${\mathbb E}$ over $x\in X$ so that $$ 0 \to V_x \to E_x \to {\mathbb Z}(0) \to 0 $$ is an extension in ${\sf{MHS}}$. That $\nu$ is holomorphic implies that this family of MHSs varies holomorphically with $x\in X$.
\begin{example} \label{ex:cycles} Families of homologically trivial algebraic cycles give rise to such extensions. Suppose that $\overline{Y}$ is a complex projective manifold and that $f:\overline{Y} \to {\overline{X}}$ is a morphism whose restriction to $X$ is a family $Y \to X$ of projective manifolds. Suppose that that $Z$ is an algebraic $d$-cycle in $Y$ such that the restriction $Z_x$ of $Z$ to the fiber over each $x\in X$ is a homologically trivial $d$-cycle in $Y_x$. Applying the construction of the previous section fiber-by-fiber, one obtains an extension ${\mathbb E}_Z$ of ${\mathbb Z}_X(0)$ by the variation of Hodge structure ${\mathbb V}$ of weight $-1$ whose fiber over $x \in X$ is $V_x = H_{2d+1}(Y_x,{\mathbb Z}(-d))$. The family $\{E_x\}_{x\in X}$ of extensions of MHS corresponds to a holomorphic section $\nu$ of the bundle $J({\mathbb V}) \to X$ of intermediate jacobians. \end{example}
The section $\nu$ is not an arbitrary holomorphic section. It satisfies the {\em Griffiths infinitesimal period relation}\footnote{This states that if $\tilde{\nu}$ is a local holomorphic lift of a normal function $\nu$ to a section of ${\mathbb E}\otimes{\mathcal O}_X$, then its derivative $\nabla \tilde{\nu}$ is a section of ${\mathcal F}^{-1}({\mathbb E}\otimes{\mathcal O}_X)\otimes \Omega^1_X$.} at each $x\in X$ and also satisfies strong (and technical) conditions as $Y_x$ degenerates. A succinct way to state these conditions is to say that the corresponding family ${\mathbb E}$ of MHS is an {\em admissible variation of MHS} in the sense Steenbrink-Zucker \cite{steenbrink-zucker} and Kashiwara \cite{kashiwara}. (The definition and an exposition of admissible variations of MHS can be found in \cite[\S 14.4.1]{peters-steenbrink}.)
All ``naturally defined local systems'' over a smooth variety $X$ that arise from families of varieties over $X$, such as the one constructed in Example~\ref{ex:cycles}, are admissible variations of MHS.\footnote{These results are due to Steenbrink-Zucker \cite{steenbrink-zucker}, Navarro et al \cite{guillen} and Saito \cite {saito:mhm}. Precise statements can be found in \cite[Thm.~14.51]{peters-steenbrink}.} The admissibility conditions axiomatize the infinitesimal and asymptotic properties satisfied by such geometrically defined local systems.
\begin{definition}[{Hain \cite[\S6]{hain:normal}, Saito \cite{saito}}] \label{def:normal} A section $\nu$ of a family $J({\mathbb V})\to X$ of intermediate jacobians is a {\it normal function} when the corresponding family of MHS ${\mathbb E}$ is an admissible variation of MHS. \end{definition}
The preceding discussion implies that the section $\nu$ of $J({\mathbb V}) \to X$ associated to a family of null homologous cycles is a normal function. Several concrete examples of normal functions over moduli spaces of curves will be given in Section~\ref{sec:normal_geom}. A detailed discussion of normal functions can be found in \cite[\S 2.11]{kerr-pearlstein}, where they are called {\em admissible} normal functions.
Denote the category of admissible variations of mixed Hodge structure over a smooth variety $X$ by ${\sf{MHS}}(X)$. It is abelian. The definition of normal functions implies that normal function sections of $J({\mathbb V}) \to X$ correspond to elements of $\Ext^1_{{\sf{MHS}}(X)}({\mathbb Z}_X(0),{\mathbb V})$. In the appendix to \cite{hain:rat_pts} it is proved that one has an exact sequence \begin{equation} \label{eqn:seqce} 0 \to \Ext^1_{\sf{MHS}}({\mathbb Z}(0),H^0(X,{\mathbb V}_{\mathbb Z})) \overset{j}{\to} \Ext^1_{{\sf{MHS}}(X)}({\mathbb Z}_X(0),{\mathbb V}_{\mathbb Z}) \overset{\delta}{\to} H^1(X,{\mathbb V}_{\mathbb Z}) \end{equation} where $\delta$ takes a normal function $\nu$ to its class $c(\nu)$. An immediate consequence is the following rigidity property of normal functions.
\begin{proposition} \label{prop:class} If $H^0(X,{\mathbb V}_{\mathbb Q})=0$, then the group $\Ext^1_{{\sf{MHS}}(X)}({\mathbb Z}(0)_X,{\mathbb V}_{\mathbb Z})$ is finitely generated and each normal function section $\nu$ of $J({\mathbb V})$ is determined, up to a torsion section, by its class $c(\nu) \in H^1(X,{\mathbb V})$. \end{proposition}
\subsection{Extending normal functions}
The following result guarantees that the normal functions that we will define over ${\mathcal M}_{g,n}$ extend to ${\mathcal M}_{g,n}^c$. A proof can be found, for example, in \cite[Thm.~7.1]{hain:normal}.
\begin{proposition} \label{prop:extends} Suppose that ${\mathbb V}$ is a variation of Hodge structure of weight $-1$ over a smooth variety $X$. If $\nu$ is a normal function section of $J({\mathbb V})$ that is defined on the complement $X-Y$ of a closed subvariety $Y$ of $X$, where $Y\neq X$, then $\nu$ extends across $Y$ to a normal function section of $J({\mathbb V})$ over $X$. \end{proposition}
Note that, in this result, the variation ${\mathbb V}$ is defined over $X$, not just over $X-Y$. The proposition asserts that normal functions defined generically on $X$ extend to $X$. It does not assert that if ${\mathbb V}$ is defined only over $X-Y$, then a normal function section of $J({\mathbb V})$ over $X-Y$ will extend to $X$. Before discussing this problem, one first has to construct an extension of $J({\mathbb V})$ to $X$. The existence of such extensions is discussed in \cite{zucker} and \cite{kerr-pearlstein}, for example.
\subsection{Some variations of Hodge structure} \label{sec:vhs}
Denote the moduli stack of principally polarized abelian varieties of dimension $g$ by ${\mathcal A}_g$. Here we introduce three variations of Hodge structure over ${\mathcal A}_g$ whose pullbacks to ${\mathcal M}_{g,n}^c$ have a special geometric significance. Throughout we suppose that $g\ge 2$.
Denote the universal abelian variety over ${\mathcal A}_g$ by $f:{\mathcal X} \to {\mathcal A}_g$. The local system $$ {\mathbb H} := R^1 f_\ast {\mathbb Z}_{\mathcal X}(1) $$ is a variation of Hodge structure over ${\mathcal A}_g$ of weight $-1$. Its pullback to ${\mathcal M}_{g,n}^c$ along the period mapping ${\mathcal M}_{g,n}^c \to {\mathcal A}_g$ will also be denoted by ${\mathbb H}$. The corresponding family of intermediate jacobians $J({\mathbb H})$ over ${\mathcal A}_g$ is naturally isomorphic to ${\mathcal X}$, the universal principally polarized abelian variety.
The construction of the universal jacobian ${\mathcal J}_g$ in Section~\ref{sec:eliashberg} implies that its restriction to ${\mathcal M}_g^c$ is a bundle of intermediate jacobians.
\begin{proposition} \label{prop:jac} If $g\ge 2$, then $J({\mathbb H}) \to {\mathcal A}_g$ is naturally isomorphic to the universal abelian variety and its pullback to ${\mathcal M}_g^c$ is isomorphic to the universal jacobian ${\mathcal J}_g^c$. \end{proposition}
The canonical polarization defines a morphism $S_H : {\mathbb H}\otimes{\mathbb H} \to {\mathbb Z}(1)$ into the constant variation of Hodge structure ${\mathbb Z}(1)$. It can be regarded as a section of $(\Lambda^2{\mathbb H})(-1)$, which we shall also denote by $S_H$.
Denote by ${\mathbb L}$ the variation of Hodge structure $(\Lambda^3{\mathbb H})(-1)$. It has weight $-1$. Wedging with $S_H$ defines an inclusion $$ {\mathbb H} \hookrightarrow {\mathbb L},\qquad x\mapsto x\wedge S_H $$ of variations of Hodge structure. Set ${\mathbb V} = {\mathbb L}/{\mathbb H}$. Note that when $g=2$, ${\mathbb L}\cong {\mathbb H}$ and ${\mathbb V}=0$. Denote the fibers of ${\mathbb H}$, ${\mathbb L}$ and ${\mathbb V}$ over the moduli point of $\Jac C$ by $H_C$, $L_C$ and $V_C$, respectively.
\begin{remark} The variation ${\mathbb L}$ over ${\mathcal M}_g^c$ is isomorphic to the variation $R^3 \pi_\ast {\mathbb Z}(2)$ where $\pi : {\mathcal J}_g^c \to {\mathcal M}_g$ denotes the projection. Its fiber over $[C]$ is $H^3(\Jac C,{\mathbb Z}(2))$. The twist ${\mathbb Z}(2)$ lowers the weight from $3$ to $-1$. Its quotient ${\mathbb V}$ is an integral form of the ${\mathbb Q}$-variation of Hodge structure whose fiber over $[C]$ is the primitive part of $H^3(\Jac C,{\mathbb Q}(2))$. \end{remark}
\subsection{Normal Functions over ${\mathcal M}^c_g$ and ${\mathcal M}_{g,n}^c$} \label{sec:normal_geom}
Suppose that $g\ge 1$. An irreducible representation $V$ of $\Sp_g$ determines a variation of Hodge structure ${\mathbb V}$ over ${\mathcal M}_{g,n}$ that is unique up to Tate twist.\footnote{This is very well-known. A proof can be found, for example, in \cite[\S9]{hain:normal}.} Since every such variation of Hodge structure ${\mathbb V}$ extends to ${\mathcal M}_{g,n}^c$, Proposition~\ref{prop:extends} implies that every normal function section of $J({\mathbb V})$ defined over ${\mathcal M}_{g,n}$ extends to a normal function on ${\mathcal M}_{g,n}^c$.
By Proposition~\ref{prop:jac}, the bundle of intermediate jacobians that corresponds to the fundamental representation of $\Sp_g$ is the pullback of the universal jacobian $J({\mathbb H}) = {\mathcal J}_g$ to ${\mathcal M}_{g,n}^c$. Its group of sections $\Ext^1_{{\sf{MHS}}({\mathcal M}_{g,1})}({\mathbb Z}(0),{\mathbb H})$ is free of rank 1. Its generator ${\mathcal K}$ is the normal function that takes the moduli point $[C,x]$ of the pointed curve $(C,x)$ to $$ {\mathcal K}([C,x]) := (2g-2)[x] - K_C \in \Jac(C) $$ where $K_C$ denotes the canonical divisor class of $C$.
When $n\ge 1$, we can pull back ${\mathcal K}$ along the $j$th projection ${\mathcal M}_{g,n} \to {\mathcal M}_{g,1}$ to obtain the normal function $$ {\mathcal K}_j : {\mathcal M}_{g,n} \to J({\mathbb H})\qquad j = 1,\dots, n. $$ Explicitly $$ {\mathcal K}_j([C,x_1,\dots,x_n]) = (2g-2)[x_j] - K_C \in \Jac(C). $$
When $n\ge 2$ we also have the normal functions $$ {\mathcal D}_{j,k} : {\mathcal M}_{g,n} \to J({\mathbb H})\qquad 1 \le j<k \le n $$ defined by $$ {\mathcal D}_{j,k}([C,x_1,\dots,x_n]) = [x_j] - [x_k] \in \Jac(C). $$
\begin{proposition}[{cf.\ \cite[Thm.~12.3]{hain:normal}}] If $g\ge 3$ and $n\ge 2$, then the group of normal function sections (indeed, all sections) of $J({\mathbb H}) \to {\mathcal M}_{g,n}$ is torsion free and is generated by ${\mathcal K}_1,\dots,{\mathcal K}_n$ and the ${\mathcal D}_{j,k}$ where $1\le j < k \le n$. \end{proposition}
The most interesting normal function over ${\mathcal M}_{g,n}$ is constructed from the Ceresa cycle in the universal jacobian. Suppose that $C$ is a smooth projective curve of genus $g\ge 3$ and that $x\in C$. Then one has the imbedding $$ \mu_x : C \to \Jac C $$ that takes $y$ to $[y]-[x]$. Its image is an algebraic $1$-cycle in $\Jac C$ that we denote by $C_x$. Let $i$ be the involution $u\mapsto -u$ of $\Jac C$. Set $C_x^-=i_\ast C_x$. Since $i^\ast : H^1(\Jac C) \to H^1(\Jac C)$ is multiplication by $-1$, it follows that $i^\ast : H^k(\Jac C) \to H^k(\Jac C)$ is multiplication by $(-1)^k$. This implies that the $1$-cycle $C_x - C_x^-$, called the {\em Ceresa cycle}, is homologically trivial. It therefore determines a class $$ \nu_x(C) \in J(H_3(\Jac C,{\mathbb Z}(-1))) $$ and a normal function: $$ \xymatrix{
J({\mathbb L}) \ar[r] & {\mathcal M}_{g,1}\ar@/_1pc/[l]|\tilde{\nu} } $$ whose value at $[C,x]$ is $\nu_x(C)$. The inclusion ${\mathbb H} \hookrightarrow {\mathbb L}$ induces an inclusion $$ j: {\mathcal J}_{g,1} = J({\mathbb H}) \hookrightarrow J({\mathbb L}). $$ It is proved in \cite{pulte} that if $x,y\in C$, then $$ \tilde{\nu}_x(C) - \tilde{\nu}_y(C) = 2j([x]-[y]) \in J(L_C) $$ so that the image $\nu(C)$ of $\nu_x(C)$ in $J(V_C)$ does not depend on the choice of $x\in C$. This implies that $\tilde{\nu}$ is pulled back from a normal function $$ \xymatrix{
J({\mathbb V}) \ar[r] & {\mathcal M}_{g}\ar@/_1pc/[l]|\nu. } $$ We will abuse notation and also denote its pullback to ${\mathcal M}_{g,n}$ by $\nu$.
\begin{proposition}[{\cite[Thm.~8.3]{hain:normal}}] If $g\ge 3$ and $n\ge 0$, then the group of normal function sections of $J({\mathbb V}) \to {\mathcal M}_{g,n}$ is freely generated by $\nu$. \end{proposition}
Proposition~\ref{prop:extends} implies that the normal functions $\nu, {\mathcal K}_j, \delta_{j,k}$ extend canonically to normal functions over ${\mathcal M}_{g,1}^c$.
\section{Biextension Line Bundles} \label{sec:biextensions}
This section is a brief review of facts about biextension line bundles from \cite{hain:biext}, \cite{lear}, and \cite{hain-reed:arakelov}. It is needed to prove that the square of the class of a normal function extends naturally to a class on ${\overline{\M}}_{g,n}$ even though the normal function itself does not extend.
Suppose that ${\mathbb U}$ is a variation of Hodge structure over an algebraic manifold $X$ of weight $-1$ endowed with a flat inner product $S$ that satisfies the condition $$ S\in \Hom_{\sf{MHS}}(U_x^{\otimes 2},{\mathbb Z}(1))\text{ for all }x\in X. $$ Equivalently, $$ S(U_x^{p,q},\overline{U_x^{r,s}}) = 0 \text{ unless $p=r$ and $q=s$}. $$ Set $\check{\U} := \Hom_{\mathbb Z}({\mathbb U},{\mathbb Z}_X(1))$. This also a variation of Hodge structure of weight $-1$. There is a natural isomorphism $$ \Ext^1_{{\sf{MHS}}(X)}({\mathbb Z}_X(0),\check{\U}) \cong \Ext^1_{{\sf{MHS}}(X)}({\mathbb U},{\mathbb Z}_X(1)). $$
The {\em biextension line bundle ${\mathcal B}$} is a line bundle over $J({\mathbb U})\times_X J(\check{\U})$. Denote the associated ${\mathbb{G}_m}$-bundle by ${\mathcal B}^\ast$. The fiber of the projection \begin{equation} \label{eqn:biext} {\mathcal B}^\ast \to J({\mathbb U})\times_X J(\check{\U}) \to X \end{equation} over $x\in X$ is the set of all mixed Hodge structures whose weight graded quotients are ${\mathbb Z}(0)$, $V_x$ and ${\mathbb Z}(1)$ via a fixed isomorphism. These are called {\em biextensions}. The projection (\ref{eqn:biext}) takes the biextension $B$ to the pair of extensions $B/{\mathbb Z}(1)$ and $W_{-1}B$. A detailed exposition of the construction of ${\mathcal B}$ is given in \cite[\S7]{hain-reed:arakelov}.
A (Hodge) biextension is a section $\beta$ of (\ref{eqn:biext}) that corresponds to an admissible variation of MHS ${\mathbb B}$ over $X$ with weight graded quotients ${\mathbb Z}_X(0)$, ${\mathbb U}$ and ${\mathbb Z}_X(1)$. Its fiber over $x\in X$ is the biextension $\beta(x)$. The composite of a biextension $\beta$ with the projection ${\mathcal B}^\ast \to J({\mathbb U})\times_X J(\check{\U})$ is a pair of normal functions that determines the extension ${\mathbb B}/{\mathbb Z}(1)$ of ${\mathbb Z}_X(0)$ by ${\mathbb U}$ and the extension $W_{-1}{\mathbb B}$ of ${\mathbb U}$
by ${\mathbb Z}_X(1)$. The biextension line bundle has a canonical metric $|\phantom{x}|_{\mathcal B}$. A biextension $\beta$ thus determines the real-valued function $\log|\beta|_{\mathcal B} : X \to {\mathbb R}$.
The pairing $S$ also defines a morphism ${\mathbb U} \to \check{\U}$ of variations of Hodge structure over $X$, and therefore a map $i_S : J({\mathbb U})\to J(\check{\U})$. Pulling back the line bundle ${\mathcal B}$ along the map $$ (\id,i_S) : {\mathcal J}({\mathbb U}) \to J({\mathbb U})\times_X J(\check{\U}) $$ we obtain a metrized line bundle $\widehat{\B} \to J({\mathbb U})$. By \cite[Prop.~7.3]{hain-reed:arakelov}, the curvature of $\widehat{\B}$ is the translation-invariant, parallel 2-form $2{\omega}_S$ on $J({\mathbb U})$ that corresponds to the bilinear form $2S$. Points of the associated ${\mathbb C}^\ast$-bundle $\widehat{\B}^\ast$ correspond to ``symmetric biextensions''.
Denote by $\phi_S \in H^2(J({\mathbb U}))$ the class of ${\omega}_S$. Since $2{\omega}_S$ represents $c_1(\widehat{\B})$, the class $2\phi_S$ is integral.
Suppose now that $X = {\overline{X}} - Y$, where ${\overline{X}}$ is smooth and $Y$ is a subvariety. Each normal function section $\nu$ of $J({\mathbb U})$ thus determines a metrized holomorphic line bundle $\nu^\ast \widehat{\B}$ over $X$. One can ask whether it extends as a metrized line bundle to ${\overline{X}}$. Lear's thesis \cite{lear} implies that a power of it extends to a {\em continuously} metrized holomorphic line bundle over $X - Y^{\mathrm{sing}}$.
\begin{theorem}[Lear \cite{lear}] \label{thm:lear} If $\dim X = 1$ and $\nu$ is a normal function section of $J({\mathbb U}) \to X$, then there exists an integer $N\ge 1$ such that the metrized holomorphic line bundle $\nu^\ast\widehat{\B}^{\otimes N}$ over $X$ extends to a holomorphic line bundle over ${\overline{X}}$ with a continuous metric. Moreover, if $\beta$ is a biextension section defined over $X$ that projects to $\nu$, and if ${\mathbb D}$ is a disk in ${\overline{X}}$ with holomorphic coordinate $t$ that is centered at a point of ${\overline{X}}-X$, then there is a rational number $p/q$, which depends only on the monodromy about the origin of ${\mathbb D}$ of the VMHS over ${\mathbb D}^\ast$ corresponding to $\beta$, such that \begin{equation} \label{eqn:asymptotics}
\big|\log |\beta(t)|_{\mathcal B} - \frac{p}{q}\log|t|\big| \end{equation} is bounded in a neighbourhood of $t=0$. \end{theorem}
Note that the continuity of the metric ensures that the extension is uniquely determined. Since ${\overline{X}}$ is smooth, every line bundle over ${\overline{X}}-Y^{\mathrm{sing}}$ extends uniquely to a line bundle over ${\overline{X}}$. Lear's Theorem thus implies the following result in the case $\dim X \ge 1$.
\begin{corollary} If $\nu$ is a normal function section of $J({\mathbb U}) \to X$, then there exists an integer $N\ge 1$ such that the metrized holomorphic line bundle $\nu^\ast\widehat{\B}^{\otimes N}$ over $X$ extends to a holomorphic line bundle $\overline{\B}_{N,\nu}$ over ${\overline{X}}$ whose metric extends continuously over ${\overline{X}}-Y^{\mathrm{sing}}$. \end{corollary}
This result implies that the class $\nu^\ast\phi_S \in H^2(X)$ of $\nu^\ast{\omega}_S$ has a natural extension to a class in $H^2({\overline{X}})$; namely $c_1(\overline{\B}_{N,\nu})/2N$.
\begin{corollary} \label{cor:extension} If $\nu$ is a normal function section of $J({\mathbb U}) \to X$, then the class $\nu^\ast \phi_S$ has a natural extension to a class $\widehat{\nu^\ast \phi_S} \in H^2({\overline{X}})$. \end{corollary}
Lear's Theorem implies that the multiplicity of each boundary divisor in $\widehat{\nu^\ast\phi_S}$ is determined by the asymptotics (\ref{eqn:asymptotics}) of the restriction of the biextension to a disk transverse to the boundary divisor.
The previous result suggests that $\nu^\ast{\omega}_S$, regarded as a current on ${\overline{X}}$, is a natural representative of $\widehat{\nu^\ast\phi_S}$.
\begin{conjecture} \label{conj:integrable} If $X$ is a curve, then the $2$-form $\nu^\ast {\omega}_S$ is integrable on $X$ and $$ \int_X \nu^\ast {\omega}_S = \frac{1}{2N}\int_{{\overline{X}}} c_1(\overline{\B}_{N,\nu}). $$ \end{conjecture}
It is known that, in general, the metric does not extend continuously over $Y^{\mathrm{sing}}$ due to the phenomenon of ``height jumping'' which we shall discuss in Section~\ref{sec:moriwaki} and which has been explained by Brosnan and Pearlstein in \cite{brosnan-pearlstein:heights}.
\section{Polarizations}
Polarizations play an important and subtle (if sometimes neglected) role in Hodge theory due to their positivity properties.
\subsection{Polarizations}
A {\em polarization} on a Hodge structure $H$ of weight $k$ is a $(-1)^k$ symmetric bilinear form $S$ on $H_{\mathbb Q}$ satisfying the Riemann-Hodge bilinear relations: \begin{enumerate} \item $S(H^{p,q},\overline{H^{r,s}}) = 0$ unless $p=r$ and $q=s$; \item $i^{p-q}S(v,\overline{v}) > 0$ when $v\in H^{p,q}$ and $v\neq 0$. \end{enumerate} A bilinear form $S$ is a {\em weak polarization} on $H$ if it satisfies the first condition and the weaker version $i^{p-q}S(v,\overline{v}) \ge 0$ for all $v\in H^{p,q}$ of the second.
Suppose that $Y$ is a smooth projective variety of dimension $n$. Denote the hyperplane class by $w$. For $k\le n$, define a bilinear form $S$ on $H^k(Y)$ by \begin{equation} \label{eqn:polarization} S(u,v) = \int_Y u\wedge v \wedge w^{n-k}. \end{equation} This is a non-degenerate, $(-1)^k$ symmetric bilinear form. However, it is not a polarization in general. The Riemann-Hodge bilinear relations imply that the restriction of $(-1)^{k(k-1)/2} S$ to $PH^k(Y)$, the primitive part of $H^k(Y)$, is a polarization. These provide the principal examples of polarized Hodge structures.
A variation of Hodge structure ${\mathbb V}$ over a base $X$ is {\it polarized} by $S$ if $S$ is a flat bilinear form on the variation which restricts to a polarization on each fiber.
\subsection{Some polarized variations of Hodge structure over ${\mathcal A}_g$}
The variations ${\mathbb H}$, ${\mathbb L}$ and ${\mathbb V}$ defined in Section~\ref{sec:vhs} have natural polarizations. The Riemann bilinear relations imply that the variation ${\mathbb H}$ over ${\mathcal A}_g$ is polarized by the inner product $S_H$ introduced in Section~\ref{sec:vhs}. The corresponding polarization is easily described on the pullback of ${\mathbb H}$ to ${\mathcal M}_g$. In this case, the fiber $H_C$ of ${\mathbb H}$ over the moduli point $[C]$ of a smooth projective curve $C$ is $H^1(C,{\mathbb Z}(1))$. Under this isomorphism, $S_H$ corresponds to the inner product $$ S(u,v) = \int_C u \wedge v. $$ on $H^1(C)$.
The intersection form $S_H$ extends to the skew symmetric bilinear form $$ S_L : {\mathbb L} \otimes {\mathbb L} \to {\mathbb Z}(1) $$ defined by $$ S_L(x_1\wedge x_2 \wedge x_3,y_1\wedge y_2 \wedge y_3) = \det(S_H(x_i,y_j)). $$ Note that this is {\em not} dual to the inner product $S$ on $H^3(\Jac C)(1)$ defined in equation (\ref{eqn:polarization}) above as is easily seen by a direct computation.
Denote the fiber $\Lambda^3 H_C$ of ${\mathbb L}$ over $[C]$ by $L_C$ and the fiber of ${\mathbb V}$ over $[C]$ by $V_C$. Define $c: L_C \to H_C$ by \begin{equation} \label{eqn:contraction} c(x\wedge y \wedge z) = S_H(y,z)x + S_H(z,x)y + S_H(x,y)z. \end{equation} Regard $S_H$ as an element of $\Lambda^2 H_L$. The projection $p: L_C \to V_C$ has a canonical $\Sp_g$-invariant splitting $j$. It is defined by $$ j(p(x\wedge y \wedge z)) = x\wedge y \wedge z - S_H \wedge c(x\wedge y \wedge z)/(g-1). $$ A skew symmetric bilinear form on ${\mathbb V}$ can be defined by $$ S_{V}(u,v) = (g-1) S_L(j(u),j(v)). $$ This form is integral and primitive.
\begin{proposition} \label{polar} The variations $({\mathbb H},S_H)$, $({\mathbb L},S_L)$ and $({\mathbb V},S_{V})$ are polarized variations of Hodge structure over ${\mathcal A}_g$, as are their pullbacks to ${\mathcal M}_{g,n}^c$. \end{proposition}
\begin{proof} We have already seen that $({\mathbb H},S_H)$ is a polarized variation of Hodge structure. For the rest, it suffices to show that $\Lambda^3 H_1(C)$ is polarized by $S_L$. To do this, choose a basis $u_1,\dots,u_g$ of $H^{-1,0}$ that is orthonormal under the positive definite hermitian inner product $$ (u,v) = i^{-1}S_H(u,\overline{v}). $$ Then, for example, $$ i^{-3-0}S_L (u_1\wedge u_2 \wedge u_3, \overline{u}_1\wedge \overline{u}_2,\overline{u}_3) = \det(i^{-1}S_H(u_j,\overline{u}_k)) = 1 > 0. $$ and $$ i^{-2-(-1)}S_L (u_1\wedge u_2 \wedge \overline{u}_3, \overline{u}_1\wedge \overline{u}_2,u_3) = - i^{-1}i^3\det(i^{-1}S_H(u_j,\overline{u}_k)) = -i^2 = 1 >0 . $$ The remaining computations follow by taking complex conjugates. \end{proof}
The contraction (\ref{eqn:contraction}) induces a projection $c : {\mathbb L} \to {\mathbb H}$ of variations of Hodge structure. The canonical quotient mapping $p : {\mathbb L} \to {\mathbb V}$ is also a morphism of variation of Hodge structure. The polarizations $S_H$ of ${\mathbb H}$ and $S_V$ of ${\mathbb V}$ can be pulled back along these projections to obtain the invariant inner product $c^\ast S_H + p^\ast S_V$ on ${\mathbb L}$. For later use, we record the following fact:
\begin{lemma}[{\cite[Prop.~18]{hain-reed:chern}}] \label{lem:reln} If $g\ge 2$, then $c^\ast S_H + p^\ast S_V = (g-1)S_L$. \qed \end{lemma}
\section{Cohomology Classes}
By Lemma~\ref{lem:gen_lifts} each invariant inner product on a variation of MHS ${\mathbb U}$ over a smooth variety $T$ gives rise to a parallel, translation invariant $2$-form ${\omega}$ on the associated bundle of intermediate jacobians $J({\mathbb U})$ and its cohomology class $\phi \in H^2(J({\mathbb U}))$.
The polarizations $S_H$, $S_L$ and $S_V$ of the variations of Hodge structure ${\mathbb H}$, ${\mathbb L}$ and ${\mathbb V}$ over ${\mathcal A}_g$ therefore give rise to cohomology classes $$ \phi_H \in H^2(J({\mathbb H})),\quad \phi_L \in H^2(J({\mathbb L})) \text{ and } \phi_V \in H^2(J({\mathbb V})). $$ Denote their parallel, canonical translation invariant representatives by ${\omega}_H$, ${\omega}_L$ and ${\omega}_V$.
Recall that $J({\mathbb H}) = {\mathcal J}_g^c$, the universal jacobian over ${\mathcal M}_g^c$ and that $\eta_g \in H^{2g}({\mathcal J}_g^c)$ denotes the Poincar\'e dual of the $0$-section $Z_g$ of ${\mathcal J}_g^c$. A standard and elementary computation shows that if $C$ is a smooth projective curve of genus $g$, then $$ \int_{\Jac C} {\omega}_H^g = g! $$ Combining this with Proposition~\ref{cpact_tori} yields the following result, which can also be deduced from \cite[Cor.~2.2]{voisin}.
\begin{proposition} \label{prop:res_phi} The class $\eta_g$ of the zero section of ${\mathcal J}_g^c$ in $H^{2g}({\mathcal J}_g^c)$ is $\phi_H^g/g!$ \qed \end{proposition}
Denote the restriction of ${\mathcal J}_g$ to ${\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}}$ by ${\mathcal J}_g'$. Zucker's Theorem \cite{zucker} implies that every normal function section $\mu$ of ${\mathcal J}_g$ defined over ${\mathcal M}_{g,n}^c$ extends to a section (also denoted $\mu$) of ${\mathcal J}_g'$ defined over ${\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}}$.
\begin{proposition} \label{prop:phi_extends} The class $\phi_H \in H^2({\mathcal J}_g^c)$ extends naturally to a class $\hat{\phi}_H \in H^2({\mathcal J}_g')$. It is characterized by the property $$ [e]^\ast \hat{\phi}_H = e^2 \hat{\phi}_H\text{ for all } e\in{\mathbb Z} $$ and has the property that $\widehat{\mu^\ast \phi_H} = \mu^\ast \hat{\phi}_H$ for all normal function sections $\mu$ of ${\mathcal J}_g^c$ defined over ${\mathcal M}_{g,n}^c$. \end{proposition}
\begin{proof}[Sketch of Proof] The first rational cohomology of the smooth finite orbi-covering ${\overline{\M}}_{g-1,2}$ of ${\Delta}_0$ vanishes. The Gysin sequence thus gives an exact sequence $$ {\mathbb Q}{\delta}_0 \to H^2({\mathcal J}_g') \to H^2({\mathcal J}_g^c) \to 0 $$ of rational cohomology. For each integer $e>1$, the endomorphism $[e]$ of ${\mathcal J}_g'$ induces an action on this sequence. It acts trivially on the left-hand term. Since $[e]^\ast \phi_H = e^2 \phi_H$ in $H^2({\mathcal J}_g^c)$, it follows that $\phi_H$ has a unique lift $\hat{\phi}_H$ to $H^2({\mathcal J}_g')$ with the property that $[e]^\ast\hat{\phi}_H = e^2\hat{\phi}_H$. The restriction of $\hat{\phi}_H$ to the zero section ${\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}}$ of ${\mathcal J}_g'$ vanishes as $[e]$ preserves the zero section and acts trivially on $H^2({\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}})$, and since $[e]^\ast\hat{\phi}_H = e^2\hat{\phi}_H$.
To complete the proof, we need several facts about biextension line bundles. Suppose that $f : Y \to X$ is a morphism of smooth varieties and that ${\mathbb U}$ is a VHS over $X$ polarized by $S$. Then the constructions of \cite{hain:biext} imply that one has a commutative diagram $$ \xymatrix{ \widehat{\B}_Y \ar[r]^{f_{\mathcal B}}\ar[d] & \widehat{\B}_X\ar[d] \cr J(f^\ast{\mathbb U}) \ar[r]^{J(f)}\ar[d] & J({\mathbb U}) \ar[d] \cr Y \ar[r]^f & X } $$ of biextension line bundles, where $f_{\mathcal B}$ is a morphism of metrized line bundles. This implies that $J(f)^\ast\phi_S = \phi_{f^\ast S}$. The next fact, which follows from Lear's Theorem, is that if $X'$ and $Y'$ are smooth varieties in which $X$ and $Y$ are Zariski dense, and where $X'-X$ is a smooth divisor in $X'$, and if ${\delta}$ is a normal function section of $J({\mathbb U}) \to X$, then \begin{equation} \label{eqn:nat} \widehat{(f^\ast{\delta})^\ast\phi_{f^\ast S}} = f^\ast(\widehat{{\delta}^\ast \phi_S}) \in H^2(Y'). \end{equation}
We now apply this with $X={\mathcal J}_g^c$, $X'={\mathcal J}_g'$, $Y = {\mathcal M}_{g,n}^c$ and $Y'={\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}}$. The variation ${\mathbb U}$ is the standard variation ${\mathbb H}$, so that ${\mathcal J}({\mathbb U}) = {\mathcal J}_g^c\times_{{\mathcal M}_g^c}{\mathcal J}_g^c$. Important here is the fact that ${\mathcal J}_g^c$ is an algebraic variety. The normal function ${\delta}$ will be the diagonal section of ${\mathcal J}_g^c \times_{{\mathcal M}_g^c}{\mathcal J}_g^c \to {\mathcal J}_g^c$. Finally, $f : Y \to X$ will be a normal function $\mu : {\mathcal M}_{g,n}^c \to {\mathcal J}_g$, which is a morphism as ${\mathcal J}_g^c \to {\mathcal M}_g^c$ is a family of (semi)-abelian varieties: $$ \xymatrix{ {\mathcal J}_{g,n}' \ar[d] \ar[r]^(.43){({\Delta},\mu)} & {\mathcal J}_g'\times_{{\overline{\M}}_g}{\mathcal J}_g' \ar[d]_{\pi_1} \cr {\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}} \ar[r]_(.55)\mu \ar@/^1pc/[u]^\mu & {\mathcal J}_g'\ar@/_1pc/[u]_{\Delta} } $$ The class $\phi_S \in H^2({\mathcal J}_g'\times_{{\overline{\M}}_g}{\mathcal J}_g')$ is $\pi_2^\ast \phi_H$, where $\pi_1$ and $\pi_2$ are the two projections ${\mathcal J}_g'\times_{{\overline{\M}}_g}{\mathcal J}_g'\to {\mathcal J}_g'$. It is now a tautology that $ \widehat{\Delta^\ast\phi_S} = \hat{\phi}_H \in H^2({\mathcal J}_g'). $ This and the naturality statement (\ref{eqn:nat}) now imply that $\widehat{\mu^\ast\phi_H} = \mu^\ast(\widehat{{\Delta}^\ast\phi_S}) = \mu^\ast\hat{\phi}_H$. \end{proof}
It is important to note that Proposition~\ref{prop:res_phi} does not hold over ${\overline{\M}}_g$. This is because the restriction of $\hat{\phi}_H$ to the zero section vanishes, whereas the conormal bundle of the zero section is the Hodge bundle, whose top Chern class is non-trivial.
We shall also need the invariant inner product $\Delta$ on ${\mathbb H}\oplus{\mathbb H}$ that is defined by $$ S_\Delta\big((u_1,v_1),(u_2,v_2)\big) = S_H(u_1,v_2)-S_H(u_2,v_1) $$ Even though it preserves the Hodge filtration, it is not a weak polarization as can be seen by restricting it to the diagonal of $H\oplus H$ (where it is positive) and anti-diagonal (where it is negative). Denote the associated cohomology class in $H^2({\mathcal J}({\mathbb H}\oplus{\mathbb H}))$ by $\phi_\Delta$. The class $\phi_{\Delta}$ extends naturally to a class $\hat{\phi}_{\Delta} \in H^2({\mathcal J}_g'\times_{{\overline{\M}}_g}{\mathcal J}_g')$. The proof is similar to that of Proposition~\ref{prop:phi_extends} and is left to the reader.
\subsection{The classes ${\mathcal K}^\ast\phi_H$, $\nu^\ast\phi_V$, $\tilde{\nu}^\ast\phi_L$ and $({\mathcal K}\times{\mathcal K})^\ast\phi_{\Delta}$}
Pulling back the classes $\phi_H$, $\phi_V$, $\phi_L$ and $\phi_{\Delta}$ along the normal functions ${\mathcal K}$, $\nu$, $\tilde{\nu}$ and the normal function section ${\mathcal K}\times{\mathcal K}$ of $$ J({\mathbb H}\oplus{\mathbb H}) = J({\mathbb H})\times_{{\mathcal M}_{g,2}}J({\mathbb H}) \to {\mathcal M}_{g,2} $$ defined by $$ {\mathcal K}\times {\mathcal K} : [C;x_1,x_2] \mapsto ({\mathcal K}(x_1),{\mathcal K}(x_2)) \in \Jac(C)\times \Jac(C). $$ we obtain rational cohomology classes ${\mathcal K}^\ast\phi_H \in H^2({\mathcal M}_{g,1}^c)$, $\nu^\ast\phi_V \in H^2({\mathcal M}_g^c)$, $\tilde{\nu}^\ast \phi_L \in H^2({\mathcal M}_{g,1}^c)$, and $({\mathcal K}\times{\mathcal K})^\ast \phi_{\Delta} \in H^2({\mathcal M}_{g,2}^c)$. Lear's Theorem implies (via Cor.~\ref{cor:extension}) that they extend naturally to classes $\widehat{\nu^\ast\phi_V} \in H^2({\overline{\M}}_g)$, $\widehat{\tilde{\nu}^\ast \phi_L} \in H^2({\overline{\M}}_{g,1})$, and $\widehat{({\mathcal K}\times{\mathcal K})^\ast \phi_{\Delta}} \in H^2({\overline{\M}}_{g,1})$.
\begin{proposition} \label{prop:reln} If $g\ge 2$, then $(g-1)\widehat{\tilde{\nu}^\ast \phi_L} = \widehat{\nu^\ast\phi_V} + \widehat{{\mathcal K}^\ast \phi_H} \in H^2({\overline{\M}}_{g,1})$. \end{proposition}
\begin{proof} This follows immediately from Lemma~\ref{lem:reln} and the fact \cite{pulte} that $c\circ \tilde{\nu} = {\mathcal K}$ and $p \circ \tilde{\nu} = \nu$. \end{proof}
Our next task is to compute each of these classes. First we need to fix notation for the natural classes in $H^2({\overline{\M}}_{g,n})$.
\section{Divisor Classes}
Denote the set $\{x_1,x_2,\dots,x_n\}$ of marked points by $I$. Denote the relative dualizing sheaf of the universal curve $\pi :{\mathcal C} \to {\overline{\M}}_{g,n}$ over ${\overline{\M}}_{g,n}$ by $w$. Its pushforward $\pi_\ast{\omega}$ is locally free of rank $g$. Recall that $\lambda_1$ denotes the first Chern class of the Hodge bundle $\pi_\ast w$. The classes $\psi_j,\ x_j\in I$ are defined by $$ \psi_j := x_j^\ast c_1(w). $$ When $n=1$, we will denote $\psi_1$ by $\psi$. Note that this definition is different from the standard definition. With this definition the $\psi$ classes are natural with respect to the forgetful maps ${\overline{\M}}_{g,n+1} \to {\overline{\M}}_{g,n}$.
Each component of the boundary divisor of ${\mathcal M}_{g,n}$ in ${\overline{\M}}_{g,n}$ has as its generic point a stable $n$-pointed curve of genus $g$ with exactly one node. These components are: \begin{itemize}
\item $\Delta_0$: the generic point is an irreducible, geometrically connected $n$-pointed curve with one node;
\item $\Delta_0^P$, where $P$ is a subset of $I$ with $|P|\ge 2$: the generic point is a reducible curve with two geometrically connected components joined at a single node, one of which has genus 0 --- the points in $P$ lie on the genus 0 component minus its node, the remaining points $P^c := I - P$ lie on the other (genus $g$) component minus the node;
\item $\Delta_{h}^{P}$, where $0 < h < g$ and $P\subseteq I$ (possibly empty): the generic point is a reducible curve with exactly one node and two geometrically connected irreducible components, one of genus $h$, the other of genus $g-h$; the points in $P$ lie on the genus $h$ component minus the node, and the rest $P^c$ lie on the other component minus the node. Note that $\Delta_{g-h}^{P^c} = \Delta_h^P$.
\end{itemize}
Denote the classes of the divisors $$
\Delta_0,\ \Delta_0^P\ (P \subseteq I,\ |P|\ge 2),\ \Delta_h^P\ (0 < h < g,\ P\subseteq I) $$ by $$
{\delta}_0,\ {\delta}_0^P\ (P \subseteq I,\ |P|\ge 2),\ {\delta}_h^P\ (0 < h < g,\ P\subseteq I), $$ respectively. It is well-known that the classes $$
\lambda_1,\ \psi_j\ (x_j\in I),\ {\delta}_0,\ {\delta}_0^P\ (P \subseteq I,\ |P|\ge 2),\ {\delta}_h^P\ (0 < h \le g/2,\ P\subseteq I) $$ comprise a basis of $H^2({\overline{\M}}_{g,n})$.
One also has the Miller-Morita-Mumford classes $$ {\kappa}_j := \pi_\ast c_1(w)^{j+1} \in H^{2j}({\overline{\M}}_g) $$ which are defined for $j\ge 1$. It follows from Grothendieck-Riemann-Roch (cf.\ \cite{mumford}) that $\kappa_1 = 12 \lambda_1 - {\delta}$, where $$ {\delta} = \sum_{j=0}^{\lfloor g/2\rfloor} {\delta}_j \in H^2({\overline{\M}}_g). $$ Define ${\kappa}_j, {\delta} \in H^{\bullet}({\overline{\M}}_{g,n})$ to be the pullbacks of ${\kappa}_j,{\delta}\in H^{\bullet}({\overline{\M}}_g)$ under the natural morphisms ${\overline{\M}}_{g,n}\to{\overline{\M}}_g$. We thus have the alternate basis $$
{\kappa}_1,\ \psi_j\ (x_j\in I),\ {\delta}_0,\ {\delta}_0^P\ (P \subseteq I,\ |P|\ge 2),\ {\delta}_h^P\ (1\le h \le g/2,\ P\subseteq I) $$ of $H^2({\overline{\M}}_{g,n})$.
\begin{remark} All of these divisor classes can be regarded as classes in $H^2({\overline{\M}}_{g,n})$ or in $CH^1({\overline{\M}}_{g,n})$. Note that these two groups are isomorphic, so that any relation between divisor classes that holds in cohomology also holds in the Chow ring. \end{remark}
\section{Formulas for $\nu^\ast \phi_V$ and ${\mathcal K}^\ast\phi_H$} \label{sec:formulas}
The computation of $F_{\mathbf{d}}^\ast \eta_g$ will be reduced to the computation of the pullbacks of the classes $\phi_H$ and $\phi_{\Delta}$ along the normal functions ${\mathcal K}_j$ and $\delta_{i,j}$. In this section we compute these basic classes. The formulas reflect the structure of Torelli groups.
The Moriwaki divisor is the class $\widehat{\nu^\ast\phi_V}$:
\begin{theorem}[{\cite[Thm.~1.3]{hain-reed:arakelov}}] \label{thm:nu} If $g\ge 2$, then $$ 2\widehat{\nu^\ast\phi_V} = (8g+4)\lambda_1-g\delta_0-4\sum_{h=1}^{[g/2]}h(g-h){\delta}_h \in H^2({\overline{\M}}_g,{\mathbb Z}). $$ \end{theorem}
The case $g\ge 3$ is \cite[Thm.~1.3]{hain-reed:arakelov}. When $g=2$, $\nu=0$ and the result follows from Mumford's computation \cite{mumford} that $10\lambda_1 = \delta_0 + 2\delta_1$. Suitably interpreted, it holds in genus 1 as $12\lambda_1=\delta_0$ in $H^2({\overline{\M}}_{1,1},{\mathbb Z})$.
Proposition~\ref{prop:phi_extends} implies that $\widehat{{\mathcal K}^\ast\phi_H}= {\mathcal K}^\ast \hat{\phi}_H$ and $\widehat{({\mathcal K}\times{\mathcal K})^\ast \phi_{\Delta}} = ({\mathcal K}\times{\mathcal K})^\ast \hat{\phi}_{\Delta}$.
\begin{theorem} \label{thm:phi} If $g\ge 2$, then $$ 2{\mathcal K}^\ast \hat{\phi}_H = 4g(g-1)\psi - {\kappa}_1 - \sum_{h=1}^{g-1} (2h-1)^2\, {\delta}_{g-h}^{\{x\}} \in H^2({\overline{\M}}_{g,1},{\mathbb Z}). $$ \end{theorem}
This result holds trivially in genus $1$ as ${\mathcal K}\equiv 0$ and because ${\kappa}_1 = 12\lambda_1 - {\delta}_0 = 0$.
\begin{proof}[Sketch of Proof] Since ${\kappa}_1 = 12\lambda_1 - {\delta}$, $$ {\kappa}_1 + \sum_{h=1}^{g-1} (2h-1)^2 {\delta}_{g-h}^{\{x\}} = 12\lambda_1 - {\delta}_0 + \sum_{h=1}^{g-1} 4h(h-1){\delta}_{g-h}^{\{x\}}. $$ So it suffices to prove that $$ 2{\mathcal K}^\ast \hat{\phi}_H = 4g(g-1)\psi - 12\lambda_1 + {\delta}_0 - \sum_{h=1}^{g-1} 4h(h-1){\delta}_{g-h}^{\{x\}}. $$ This formula, modulo boundary terms, was proved by Morita \cite[(1.7)]{morita:jacobians}. Another proof is given in \cite[Thm.~1]{hain-reed:chern}.
The coefficient of ${\delta}_h^{\{x\}}$ is computed using the method of \cite[\S11]{hain-reed:arakelov}. The Torelli group $T_g$ is replaced by the Torelli group $T_{g,1}$ associated to a 1-pointed surface. Instead of taking $V = \Lambda^3 H /(\theta\wedge H)$, we take it to be $H$. The quadratic form $q$ (which is denoted $S_V$ in this paper) is replaced by $c^\ast S_H$, where $c:\Lambda^3 H \to H$ is the contraction (\ref{eqn:contraction}). We sketch the monodromy computation using the notation of \cite[\S11]{hain-reed:arakelov}.
The coefficient of ${\delta}_h^{\{x\}}$ is $-\hat{\tau}(\sigma_h)$, where $\sigma_h$ is a Dehn twist about a separating simple closed curve that divides a pointed, genus $g$ reference surface into a surface of genus $h$ (that does not contain the point) and a surface of genus $g-h$, and where $\hat{\tau}$ is a representation of $T_{g,1}$ into the Heisenberg group associated to $(H,S_H)$.
There is a symplectic basis $a_1,b_1,\dots, a_g,b_g$ of $H_{\mathbb Z}$ such that $a_1,b_1,\dots,a_h,b_h$ is a basis of $H'$, the first homology of the genus $h$ subsurface. Set ${\omega}' = a_1\wedge b_1 + \dots + a_h\wedge b_h$, the symplectic form of $H'$. If $u\in H$, then $c(u\wedge {\omega}') = (h-1)u$. Thus \begin{multline*} \hat{\tau}_h(\sigma_h) = \frac{8}{2h-2}\sum_{j=1}^h S_H\big(c(a_j\wedge {\omega}'),c(b_j\wedge{\omega}')\big) \cr = \frac{8(h-1)^2}{2h-2} \sum_{j=1}^h S_H(a_j,b_j) = 4h(h-1). \end{multline*}
It remains to compute the coefficient of ${\delta}_0$. The most direct way to compute it is by restricting to a curve in the hyperelliptic locus. First note that if $C$ is a hyperelliptic curve and $x \in C$ is a Weierstrass point, then ${\mathcal K}(C,x) = 0$. Call such a pair $(C,x)$ a {\em hyperelliptic pointed curve}. Suppose that $T$ is a smooth, complete curve and that $f : T \to {\overline{\M}}_{g,1}$ is a morphism where $f(t)$ is the moduli point of an irreducible hyperelliptic pointed curve for each $t\in T$.\footnote{For example, we can take $T = {\mathbb P}^1$ and $f$ the morphism associated to the family $$ v^2 = (u-t)u\prod_{j=1}^{2g}(u-a_j), $$ where $t\in{\mathbb C}$ and the $a_j$ are distinct non-zero complex numbers. A section of Weierstrass points is given by $x=(0,0)$.} The normal function $f^\ast{\mathcal K}$ vanishes identically on $T$, which implies that $f^\ast{\mathcal K}^\ast\widehat{\B}$ is trivial as a metrized line bundle over $T-f^{-1}{\Delta}_0$. Its extension as a metrized line bundle to $T$ is therefore trivial. This implies the vanishing of $$ f^\ast {\mathcal K}^\ast\hat{\phi}_H \in H^2(T). $$ On the other hand, standard techniques can be used to show that $$ f^\ast(8\lambda_1 + 4g\psi - {\delta}_0) = 0. $$ The Cornalba-Harris relation \cite[Prop.~4.7]{cornalba-harris} implies that $$ f^\ast\big((8g+4)\lambda_1 - g{\delta}_0 \big) = 0 \in \Pic T. $$ It follows that $$ f^\ast(4g(g-1)\psi - 12\lambda_1 + {\delta}_0) = (g-1)f^\ast(8\lambda_1 + 4g\psi - {\delta}_0) -f^\ast\big((8g+4)\lambda_1 - g{\delta}_0 \big) = 0. $$ These two facts together imply that the coefficient of ${\delta}_0$ in ${\mathcal K}^\ast\hat{\phi}_H$ is $1$. \end{proof}
Since $\kappa_1 = 12\lambda_1 - \delta$, Theorems~\ref{thm:nu} and \ref{thm:phi} and Proposition~\ref{prop:reln} imply the following result when $g\ge 2$. The case $g=1$ follows from the fact that $\psi = \lambda_1$ and the well-known relation ${\delta}_0=12\lambda_1$ in $\Pic{\overline{\M}}_{1,1}$.
An immediate consequence of Lemma~\ref{lem:reln} and the two previous results is the following formula for $\widehat{\tilde{\nu}^\ast\phi_L}$.
\begin{corollary} For all $g\ge 1$, $$ 2\widehat{\tilde{\nu}^\ast \phi_L} = 8\lambda_1 + 4g\psi - {\delta}_0 - 4\sum_{h=1}^{g-1} h\delta_{g-h}^{\{x\}} \in H^2({\overline{\M}}_{g,1},{\mathbb Z}). $$ \end{corollary}
The next result is needed in the solution of Eliashberg's problem.
\begin{theorem} \label{phi2} If $g \ge 2$, then in $H^2({\overline{\M}}_{g,2},{\mathbb Z})$ we have \begin{multline*} ({\mathcal K}\times{\mathcal K})^\ast \hat{\phi}_\Delta = (2g-2)(\psi_1 + \psi_2) - {\kappa}_1 - (2g-2)^2{\delta}_0^{\{x_1,x_2\}} \cr - \sum_{h=1}^{g-1} (2h-1)^2\, {\delta}_{g-h}^{\{x_1,x_2\}} + (2h-1)(2(g-h)-1)\big({\delta}_h^{\{x_1\}}+{\delta}_{g-h}^{\{x_2\}}\big)/2. \end{multline*} \end{theorem}
Note that in this and subsequent formulas, we will often sum from $h=1$ to $h=g-1$ and over all subsets $P$ of $I$. Because of this, some terms will appear twice as ${\delta}_h^P = {\delta}_{g-h}^P$. We do this to emphasize the symmetry of the formulas and to facilitate later computations.
\begin{proof} Modulo the coefficients of the ${\delta}_h^{\{x_1\}}$, this formula can be computed \begin{enumerate}
\item by restricting $({\mathcal K}\times{\mathcal K})^\ast\phi_{\Delta}$ to any fiber $C\times C$ and
\item from Theorem~\ref{thm:phi} by restricting to the diagonal ${\overline{\M}}_{g,1} \to {\overline{\M}}_{g,2}$, noting that the restriction of $\hat{\phi}_\Delta$ to the diagonal ${\mathcal J}_g$ of ${\mathcal J}_g\times_{{\mathcal M}_{g,2}^c}{\mathcal J}_g$ is $2\hat{\phi}_H$.
\end{enumerate} These computations are straightforward, once one notes that the divisor ${\Delta}_0^{\{x_1,x_2\}}$ is the ``diagonal'' ${\mathcal M}_{g,1}^c \to {\mathcal M}_{g,2}^c$ in ${\mathcal M}_{g,2}^c$ and that the Chern class of its normal bundle is $\psi$. If we restrict to a single curve $C$, then in $H^2(C\times C)$ \begin{align*} ({\mathcal K}\times{\mathcal K})^\ast \hat{\phi}_\Delta &= (2g-2)^2 \sum_{j=1}^g \big(a_j^{(1)}\wedge b_j^{(2)}- b_j^{(1)}\wedge a_j^{(2)}\big) \cr &= (2g-2)^2 \big([\text{point}]^{(1)} + [\text{point}]^{(2)} - [\text{diagonal}]\big) \cr &= (2g-2)(\psi_1+\psi_2) - (2g-2)^2{\delta}_0^{\{x_1,x_2\}}. \end{align*} Here $a_1,\dots, b_g$ is a symplectic basis of $H_1(C)$ and, for $x\in H^1(C)$, $x^{(k)}$ denotes the pullback of $x$ under the $k$th projection $p_k : C^2 \to C$.
It remains to compute the coefficient of ${\delta}_h^{x_1}$ when $0<h<g$. This we do using a test curve suggested by Sam Grushevsky. Since $\hat{\phi}_{\Delta}$ is invariant when the two factors of ${\mathcal J}_g^c\times_{{\mathcal M}_{g,2}^c}{\mathcal J}_g^c$ are swapped, it follows that the formula for $({\mathcal K}\times{\mathcal K})^\ast\hat{\phi}_{\Delta}$. is symmetric in $x_1$ and $x_2$. Since ${\delta}_h^{\{x_1\}} = {\delta}_{g-h}^{\{x_2\}}$, the formula is also invariant when $h$ is replaced by $g-h$. We therefore conclude that \begin{multline} \label{eqn:expression} ({\mathcal K}\times{\mathcal K})^\ast \hat{\phi}_\Delta = (2g-2)(\psi_1 + \psi_2) - {\kappa}_1 - (2g-2)^2{\delta}_0^{\{x_1,x_2\}} \cr - \sum_{h=1}^{g-1} (2h-1)^2\, {\delta}_{g-h}^{\{x_1,x_2\}} + \sum_{h=1}^{g-1} c_h{\delta}_h^{\{x_1\}} \end{multline} where $c_h = c_{g-h}$.
Suppose that $0<h<g$. Fix pointed smooth projective curves $(C',P')$ and $(C'',P'')$ with $g(C')=h$ and $g(C'') = g-h$. Let $C$ be the nodal genus $g$ curve with three components $C'$, $C''$ and ${\mathbb P}^1$, where $C'$ is attached to ${\mathbb P}^1$ by identifying $P'\in C'$ with $0\in {\mathbb P}^1$ and $P''\in C''$ with $\infty \in {\mathbb P}^1$. For $t\in {\mathbb P}^1-\{0,1,\infty\}$ let $C_t$ be the stable $2$-pointed curve $(C;1,t)$. The closure $T$ of the curve $$ {\mathbb P}^1 - \{0,1,\infty\} \to {\mathcal M}_{g,2}^c,\quad t \mapsto [C_t] $$ \begin{figure}
\caption{The $2$-pointed curve $C_t$}
\label{fig:test_curve}
\end{figure}
is a copy of ${\overline{\M}}_{0,4} \cong {\mathbb P}^1$ imbedded in ${\mathcal M}_{g,2}^c$.
The restriction of ${\mathcal K}\times{\mathcal K}$ to $T$ takes the constant value $$ \big((2h-2)P'-K_{C'},(2(g-h)-2)P''-K_{C''}\big) \in \Jac C'\times \Jac C'' \cong \Jac C, $$ which implies that $({\mathcal K}\times{\mathcal K})^\ast \phi_{\Delta} = 0$. The coefficient $c_h$ is computed by evaluating the right hand side (RHS) of (\ref{eqn:expression}) on $T$.
The curve $T$ is contained in ${\Delta}_h^{\{x_1,x_2\}} \cap {\Delta}_{g-h}^{\{x_1,x_2\}}$ and intersects the three boundary divisors ${\Delta}_0^{\{x_1,x_2\}}$, ${\Delta}_h^{\{x_1\}}$ and ${\Delta}_h^{\{x_2\}}$ transversely in three distinct points. These are the three boundary points of ${\overline{\M}}_{0,4} \cong T$. It does not intersect any other boundary divisors. Consequently, $$ \int_T {\delta}_0^{\{x_1,x_2\}} = \int_T {\delta}_h^{\{x_1\}} = \int_T {\delta}_h^{\{x_2\}} = 1. $$
The projection formula can be used to evaluate the other terms of the RHS of (\ref{eqn:expression}) on $T$. Let $$ q: {\overline{\M}}_{g,2} \to {\overline{\M}}_g, \quad p_j : {\overline{\M}}_{g,2} \to {\overline{\M}}_{g,1}\quad j =1,2 $$ denote the natural projections, where $p_j([C;x_1,x_2]) = [C,x_j]$. Note that $q$ and the $p_j$ collapse $T$ to a point. Since $\psi_j = p_j^\ast \psi$ and $\kappa_1 = q^\ast \kappa_1$, the projection formula implies that $$ \int_T \kappa_1 = \int_T q^\ast \kappa_1 = \int_{q_\ast T} \kappa_1 = 0 \text{ and } \int_T \psi_j = \int_T p_j^\ast \psi = \int_{p_j\ast T} \psi = 0. $$ Since $p_1^\ast {\delta}_h^{\{x\}} = {\delta}_h^{\{x_1,x_2\}} + {\delta}_h^{\{x_1\}}$, the projection formula implies that $$ \int_T {\delta}_h^{\{x_1,x_2\}} = - \int_T {\delta}_h^{\{x_1\}} = -1. $$ Similarly, $\int_T {\delta}_{g-h}^{\{x_1,x_2\}} = -1$.
Evaluating the expression (\ref{eqn:expression}) on $T$ we obtain $$ 0 = 0 + 0 - (2g-2)^2 + (2h-1)^2 + \big(2(g-h)-1\big)^2 + c_h + c_{g-h}. $$ Since $c_h = c_{g-h}$, this implies that $c_h = (2h-1)\big(2(g-h)-1\big)$. \end{proof}
\section{Solution of Eliashberg's Problem over ${\mathcal M}_{g,n}^c$ when $g>1$} \label{sec:cpt_type}
In this section, we solve Eliashberg's problem over ${\mathcal M}_{g,n}^c$ when $g > 2$. A complete solution in the genus 1 case is given in the following section. The solution in genus $>1$ is a direct consequence of Proposition~\ref{prop:res_phi} and Theorem~\ref{pullback_phi} below. Related work on Eliashberg's problem has been obtained independently by Cavalieri and Marcus \cite{cavalieri-marcus} via Gromov-Witten theory.
Fix an integral vector ${\mathbf{d}}=(d_1,\dots, d_n)$ with $\sum_j d_j=0$. As in the introduction, we have the section $$ F_{\mathbf{d}} : {\mathcal M}_{g,n}^c \to {\mathcal J}_g $$ of the universal jacobian defined by $$ F_{\mathbf{d}} : (x_1,\dots,x_d) \mapsto \bigg[\sum_{j=1}^n d_j x_j\bigg] \in \Jac C. $$ For each subset $P$ of $\{x_1,\dots,x_n\}$, set $d_P = \sum_{j\in P}d_j$. Since $d_P + d_{P^c} =0$, $d_P^2{\delta}_h^P = d_{P^c}^2{\delta}_{g-h}^{P^c}$.
\begin{theorem} \label{thm:eliashberg} If $g\ge 2$, then in $H^{2g}({\mathcal M}_{g,n}^c)$ we have $$ F_{\mathbf{d}}^\ast \eta_g = \frac{1}{g!}\bigg(\sum_{j=1}^n d_j^2\, \psi_j/2 - \sum_{P\subseteq I} \sum_{\{x_j,x_k\}\subseteq P} d_j d_k\, {\delta}_0^P - \frac{1}{4} \sum_{P\subseteq I}\sum_{h=1}^{g-1} d_P^2\, {\delta}_h^P\bigg)^g. $$ \end{theorem}
Recall that our definition of the $\psi_j$ differs from the commonly used one. Here $\psi_j := x_j^\ast c_1({\omega})$, where ${\omega}$ is the relative dualizing sheaf of the universal curve. Note, too, that since ${\delta}_h^P={\delta}_h^{P^c}$ and since we are summing from $h=1$ to $h=g-1$ in this and other results in this section, some boundary divisors occur twice in this expression.
I do not know if this formula holds in the Chow ring. Although this formula makes sense in $H^{2g}({\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}})$, it does not hold there. For example, when ${\mathbf{d}}=0$ the left hand side is the non-trivial class $(-1)^g\lambda_g$, whereas the right-hand side vanishes. A result of Ekedahl and van der Geer \cite{ekd-vdg} implies that, in $CH^g({\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}})$, $\lambda_g$ is $(-1)^g \zeta(1-2g)$ times a natural class, where $\zeta(s)$ denotes the Riemann zeta function. This suggests that the class $$ F_{\mathbf{d}}^\ast\eta_g - \frac{1}{g!}(F_{\mathbf{d}}^\ast \hat{\phi}_H)^g \in CH^g({\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}}) $$ should be interesting. In particular, does Proposition~\ref{prop:res_phi} hold in $CH^g({\mathcal J}_g^c)$? This makes sense, as $\phi_H = \theta - \lambda_1/2$ in $\Pic {\mathcal J}_g^c$. (Cf.\ the proof of Theorem~\ref{thm:variant} below.)
\subsection{The approach and reduction}
Denote the pullback of ${\mathcal J}_g$ to ${\overline{\M}}_{g,n}$ by ${\mathcal J}_{g,n}$ and its restriction to ${\mathcal M}_{g,n}^c$ by ${\mathcal J}_{g,n}^c$. We will consider a more general situation. Namely, we'll assume that ${\mathbf{d}}\in{\mathbb Z}^n$ and $m\in{\mathbb Z}$ satisfy $$ \sum_{j=1}^n d_j = (2g-2)m. $$ Then one has the section $$ F_{\mathbf{d}} : [C;x_1,\dots,x_n] \mapsto \sum_{j=1}^n d_j x_j - m K_C \in \Jac C $$ of ${\mathcal J}_{g,n}^c$ over ${\mathcal M}_{g,n}^c$. Our goal is to compute $F_{\mathbf{d}}^\ast \phi_H$.
Set $$ {\mathcal J}_{g,n}^m := \underbrace{{\mathcal J}_{g,n}\times_{{\overline{\M}}_{g,n}} \times {\mathcal J}_{g,n}\times_{{\overline{\M}}_{g,n}} \cdots \times_{{\overline{\M}}_{g,n}} {\mathcal J}_{g,n}}_m. $$ Denote its restriction to ${\mathcal M}_{g,n}^c$ by ${\mathcal J}_{g,n}^{c,m}$. Let $$ {\mathcal K}_n : {\overline{\M}}_{g,n} - {\Delta}_0^{\mathrm{sing}} \to {\mathcal J}_{g,n}^n $$ be the $n$th power of ${\mathcal K}$ --- that is, the section of ${\mathcal J}_{g,n}^{c,n} \to {\mathcal M}^c_{g,n}$ defined by $$ {\mathcal K}_n : (C;x_1,\dots,x_n) \mapsto \big({\mathcal K}(x_1),{\mathcal K}(x_2),\dots,{\mathcal K}(x_n)\big) \in \big(\Jac C\big)^n. $$ Note that ${\mathcal K}_2={\mathcal K}\times {\mathcal K}$.
\begin{proposition} Define ${\mathbf{d}} : {\mathcal J}_{g,n}^n \to {\mathcal J}_{g,n}^n$ by $$ {\mathbf{d}} : (u_1,\dots,u_n) \mapsto (d_1 u_1,\dots,d_n u_n) $$ and $\trace_n : {\mathcal J}_{g,n}^n \to {\mathcal J}_{g,n}$ by $$ \trace_n : (u_1,\dots,u_n) \mapsto u_1 + \dots + u_n. $$ Then the mapping $$ \xymatrix{ {\overline{\M}}_{g,n} -{\Delta}_0^{\mathrm{sing}} \ar[r]^(0.6){{\mathcal K}_n} & {\mathcal J}_{g,n}^n \ar[r]^{{\mathbf{d}}} & {\mathcal J}_{g,n}^n \ar[r]^{\trace_n} & {\mathcal J}_{g,n} } $$ equals $(2g-2) F_{\mathbf{d}}$. \qed \end{proposition}
This formula allows the reduction of the computation of $F_{\mathbf{d}}^\ast \phi_H$ to more basic computations. Denote the pullback of $\phi_H$ under the $j$th projection $$ p_j : {\mathcal J}_{g,n}^{c,n} \to {\mathcal J}_{g,n}^c $$ by $\phi_{j,j}$. For $j\neq k$, denote the pullback of $\phi_\Delta$ under the $(j,k)$th projection $$ p_{j,k} : {\mathcal J}_{g,n}^{c,n} \to {\mathcal J}_{g,n}^{c,2} \quad (u_1,\dots,u_n) \mapsto (u_j,u_k) $$ by $\phi_{j,k}$.
\begin{lemma} \label{sum} With notation as above, $$ {\mathbf{d}}^\ast \phi_{j,k} = d_j d_k\, \phi_{j,k} \text{ and } \trace_n^\ast \phi_H = \sum_{j \le k} \phi_{j,k}. $$ \end{lemma}
\begin{proof} Since all of the classes $\phi_{j,k}$ are represented by parallel, translation invariant forms, to prove the result, it suffices to prove the result in the cohomology of the jacobian $J := \Jac C$ of a single smooth projective curve $C$.
Set $J = \Jac C$. Note that the ring homomorphism $$ H^{\bullet}(J) \to H^{\bullet}(J)^{\otimes n} $$ induced by the addition map $J^n \to J$ is, in degree 1, given by $$ x \mapsto \sum_{\substack{a+b=n-1\cr a,b\ge 0}} 1^{\otimes a} \otimes x \otimes 1^{\otimes b}. $$ The formula for $\trace_n^\ast$ follows using the fact that this is a ring homomorphism.
The formula for ${\mathbf{d}}^\ast$ follows as the map $[e] : J \to J$ is multiplication by $e$ on $H^1(J)$, and therefore multiplication by $e^k$ on $H^k(J)$. \end{proof}
Recall that $\phi_H \in H^2({\mathcal J}_g^c)$ and $\phi_{\Delta} \in H^2({\mathcal J}_g^c\times_{{\mathcal M}_g^c}{\mathcal J}_g)$ extend naturally to classes $$ \hat{\phi}_H \in H^2({\mathcal J}_g')\text{ and } \hat{\phi}_{\Delta} \in H^2({\mathcal J}_g'\times_{{\overline{\M}}_g}{\mathcal J}_g'), $$ where ${\mathcal J}_g'$ denotes the universal jacobian over ${\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}}$. Using the scaling by the action of $(e_1,\dots,e_n) \in {\mathbb Z}^n$ on the cohomology of the $n$th power of ${\mathcal J}_g' \to {\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}}$ one can show that Lemma~\ref{sum} holds when $\phi_H$ and $\phi_{\Delta}$ are replaced by the extended classes $\hat{\phi}_H$ and $\hat{\phi}_{\Delta}$. We therefore have:
\begin{corollary} \label{cor:pullback} If $g\ge 2$, then $$ (2g-2)^2\,F_{\mathbf{d}}^\ast \hat{\phi}_H = \sum_{j=1}^n d_j^2\,\pi_j^\ast {\mathcal K}^\ast \hat{\phi}_H + \sum_{1\le j<k \le n} d_j d_k\, \pi_{j,k}^\ast({\mathcal K}\times{\mathcal K})^\ast \hat{\phi}_\Delta \in H^2({\mathcal M}_{g,n}^c), $$ where $\pi_j : {\overline{\M}}_{g,n} \to {\overline{\M}}_{g,1}$ and $ \pi_{j,k} : {\overline{\M}}_{g,n} \to {\overline{\M}}_{g,2}$ denote the natural projections. \end{corollary}
The following result is the main computation of this section.
\begin{theorem} \label{pullback_phi} If $g \ge 2$ and ${\mathbf{d}}\in {\mathbb Z}^n$ and $m\in{\mathbb Z}$ satisfy $\sum d_j = (2g-2)m$, then in $H^2({\overline{\M}}_{g,n})$ we have \begin{multline*} F_{\mathbf{d}}^\ast\hat{\phi}_H = -{m^2}\kappa_1/2 + \sum_{j=1}^n (d_jm+d_j^2/2)\,\psi_j - \sum_{P\subseteq I} \sum_{\{x_j,x_k\}\subseteq P} d_j d_k\, {\delta}_0^P \cr -\frac{1}{4} \sum_{P\subseteq I}\sum_{h=1}^{g-1} \big(d_P - (2h-1)m\big)^2\, {\delta}_h^P. \end{multline*} \end{theorem}
Recall that ${\delta}_h^P={\delta}_{g-h}^{P^c}$. Note that, in this expression, the coefficients of ${\delta}_h^P$ and ${\delta}_{g-h}^{P^c}$ are equal as $d_P + d_{P^c} = (2g-2)m$. Note, too, that one recovers Theorem~\ref{thm:phi} when $n=m=1$ and $d_1=2g-2$.
Theorem~\ref{thm:eliashberg} follows directly from Proposition~\ref{prop:res_phi} and the case $m=0$. The corresponding genus 1 statement is proved in the following section. The proof below fails in genus 1 as we cannot divide by $2g-2$.
\begin{proof} This follows from Theorems~\ref{thm:phi} and \ref{phi2} and Corollary~\ref{cor:pullback}. First observe that the coefficient of ${\kappa}_1$ in $(2g-2)^2F_{\mathbf{d}}^\ast\hat{\phi}_H$ is $$ -\sum_j d_j^2/2 - \sum_{j<k} d_j d_k = -(d_1 + \cdots + d_n)^2/2 = -(2g-2)^2m^2/2. $$ Next, the coefficient of $\psi_j$ in $(2g-2)^2\,F_{\mathbf{d}}^\ast \hat{\phi}_H$ is \begin{align*} (2g-2)\sum_{k\neq j} d_jd_k + \frac{1}{2}\, 4g(g-1)d_j^2 &= (2g-2)\big[(d_1 + \cdots + d_n - d_j)d_j + g\,d_j^2\big] \cr &= (2g-2)^2 d_j m + (2g-2)(g-1)d_j^2 \cr &= (2g-2)^2(d_j m +d_j^2/2). \end{align*}
Since $$ \pi_{j,k}^\ast {\delta}_0^{\{x_1,x_2\}} = \sum_{\{x_j,x_k\}\subseteq P} {\delta}_0^P, \quad j \neq k $$ the coefficient of ${\delta}_0^P$ in the expression for $(2g-2)^2 F_{\mathbf{d}}^\ast \hat{\phi}_H$ is $$ -(2g-2)^2 \sum_{\{x_j,x_k\}\subseteq P} d_jd_k\, {\delta}_0^P. $$
When $1\le h \le g-1$ and $j\neq k$ $$ \pi_j^\ast {\delta}_h^{\{x\}} = \sum_{x_j \in P} {\delta}_h^P,\quad \pi_{j,k}^\ast {\delta}_h^{\{x_1,x_2\}} = \sum_{\{x_j,x_k\} \subseteq P} {\delta}_h^P, \text{ and } \pi_{j,k}^\ast {\delta}_h^{x_1} = \sum_{\substack{x_j \in P\cr x_k \in P^c}} {\delta}_h^P. $$ Then, computing formally, we see that the coefficient of ${\delta}_h^P$ in $(2g-2)^2F_{\mathbf{d}}^\ast\hat{\phi}_H$ is the coefficient of ${\delta}_h^P$ in \begin{multline*} -\big(2(g-h)-1\big)^2\sum_j d_j^2\, \pi_j^\ast\, {\delta}_h^{\{x\}}/2 -\big(2(g-h)-1\big)^2 \sum_{j<k} d_jd_k\,\pi_{j,k}^\ast\, {\delta}_h^{\{x_1,x_2\}} \cr + (2h-1)\big(2(g-h)-1\big)\sum_{j < k} d_j d_k \pi_{j,k}^\ast \big({\delta}_h^{x_1}+ {\delta}_{g-h}^{x_2}\big)/2, \end{multline*} which is \begin{align*} &-\frac{\big(2(g-h)-1\big)^2}{2} \bigg(\sum_{x_j \in P} d_j^2 + \sum_{\substack{x_j,x_k\in P\cr x_j \neq x_k}} d_j d_k\bigg) + \frac{(2h-1)\big(2(g-h)-1\big)}{2} \sum_{\substack{x_j\in P\cr x_k \in P^c}} d_j d_k \cr &= -\frac{1}{2}\Big(\big(2(g-h)-1\big)^2 d_P^2 -\big(2(g-h)-1\big)(2h-1)d_Pd_{P^c}\Big). \end{align*} Since ${\delta}_h^P = {\delta}_{g-h}^{P^c}$, the coefficient of ${\delta}_h^P$ can be chosen (and will be chosen) to be the average of the formally computed coefficients of ${\delta}_h^P$ and ${\delta}_{g-h}^{P^c}$, which is \begin{align*} &-\frac{1}{4}\Big(\big(2(g-h)-1\big)^2 d_P^2 -\big(2(g-h)-1\big)(2h-1)d_Pd_{P^c}+(2h-1)^2 d_{P^c}^2\Big) \cr &= -\frac{1}{4}\Big(\big(2(g-h)-1\big)d_P -(2h-1)d_{P^c}\Big)^2 \cr &= -\frac{1}{4}\Big(\big(2(g-h)-1\big)d_P +(2h-1)\big(d_P -(2g-2)m\big)\Big)^2 \cr &= -\frac{(2g-2)^2}{4}\big(d_P -(2h-1)m\big)^2 \end{align*} as $d_P + d_{P^c}=(2g-2)m$. \end{proof}
\subsection{Variants}
Theorem~\ref{pullback_phi} can be adapted to establish more general results. Suppose that $r$ divides $2g-2$ and suppose that ${\mathbf{d}} = (d_1,\dots,d_n)\in {\mathbb Z}^n$ and $m\in {\mathbb Z}$ satisfy $$ \sum_{j=1}^n d_j = m(2g-2)/r. $$ Let $f : X \to {\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}}$ be a morphism over which there is a globally defined $r$th root $\alpha$ of the relative dualizing sheaf. Then one has the section $E_{\mathbf{d}}$ of ${\mathcal J}_{g,n}^c$ over $X$ defined by $$ E_{\mathbf{d}} : x \mapsto - m\alpha + \sum_{j=1}^n d_j x_j \in \Jac C. $$ where $f(x) = [C;x_1,\dots,x_n]$. As above, $d_P = \sum_{x_j \in P} d_j$.
\begin{theorem} \label{thm:variant} The class $E_{\mathbf{d}}^\ast \hat{\phi}_H \in H^2(X)$ is the pullback along $f$ of the class \begin{multline*} -(m/r)^2\kappa_1/2 + \sum_{j=1}^n (d_j(m/r)+d_j^2/2)\,\psi_j - \sum_{P\subseteq I} \sum_{\{x_j,x_k\}\subseteq P} d_j d_k\, {\delta}_0^P \cr -\frac{1}{4} \sum_{P\subseteq I}\sum_{h=1}^{g-1} \big(d_P - (2h-1)(m/r)\big)^2\, {\delta}_h^P \in H^2({\overline{\M}}_{g,n}). \end{multline*} \end{theorem}
\begin{proof} Since the diagram $$ \xymatrix{ & {\mathcal J}_{g,n}\ar[d] \cr X \ar[r]_(.3)f\ar[ur]^{rE_{\mathbf{d}}} & {\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}} \ar@/_1pc/[u]_{F_{r{\mathbf{d}}}} } $$ commutes (i.e., $F_{r{\mathbf{d}}}\circ f = r E_{\mathbf{d}}$), and since the extended class $\hat{\phi}_H$ satisfies $[e]^\ast\hat{\phi}_H = e^2\hat{\phi}_H$, we have $$ E_{\mathbf{d}}^\ast \hat{\phi}_H = r^{-2} E_{r{\mathbf{d}}}\hat{\phi}_H = r^{-2} f^\ast F_{r{\mathbf{d}}}\hat{\phi}_H. $$ The result now follows from Theorem~\ref{pullback_phi}. \end{proof}
This result can be used to give a partial solution to a problem posed to me by Joe Harris. Suppose that ${\mathbf{d}} = (d_1,\dots,d_n) \in {\mathbb Z}^n$ satisfies $\sum_{j=1}^n d_j = g-1$. Then one has the section $$ G_{\mathbf{d}} : [C;x_1,\dots,x_n] \mapsto \sum_j d_j x_j \in \Pic^{g-1} C $$ of the relative Picard bundle $$ {\mathcal P}_{g,n} := \Pic^{g-1}_{{\mathcal C}_{g,n}/{\mathcal M}_{g,n}} $$ over ${\mathcal M}_{g,n}$. This contains the divisor $W$ of effective divisor classes of degree $g-1$. The pullback $G_{\mathbf{d}}^\ast W$ is the divisor in ${\mathcal M}_{g,n}$ consisting of those $[C;x_1,\dots,x_n]$ where $h^0(C,\sum d_j x_j)>0$. Denote its closure in ${\overline{\M}}_{g,n}$ by $W_{\mathbf{d}}$. Harris' problem is to compute the class of $W_{\mathbf{d}}$ in terms of standard classes. This is a subtle problem as ${\mathcal P}_{g,n}$ is not separated over ${\mathcal M}_{g,n}^c$.
Although we cannot solve this problem, we can solve the following closely related problem. Let $X \to {\overline{\M}}_{g,n}-{\Delta}_0^{\mathrm{sing}}$ be any dominant morphism on which there is a globally defined theta characteristic $\alpha$. Denote the inverse image of ${\mathcal M}_{g,n}^c$ in $X$ by $X^c$ and the inverse image in $X$ of ${\mathcal M}_{g,n}$ by $X^o$. Denote the universal jacobian over $X$ by ${\mathcal J}_X$ and its restrictions to $X^c$ and $X^o$ by ${\mathcal J}_X^c$ and ${\mathcal J}_X^o$, respectively. Denote the pullback of ${\mathcal P}_{g,n}$ to $X^o$ by ${\mathcal P}_X$. Then $\alpha$ defines an isomorphism of ${\mathcal P}_X$ with ${\mathcal J}_X^o$. Under this isomorphism $G_{\mathbf{d}}$ corresponds to the section $$ F_{\mathbf{d}} : [C;x_1,\dots,x_n] \mapsto -\alpha + \sum_{j=1}^n d_j x_j \in \Jac C $$ and the pullback of the divisor $W$ to ${\mathcal P}_X$ corresponds to the divisor $\Theta_\alpha$ which is defined locally by the theta function $\vartheta_\alpha(z,\Omega)$ that corresponds to $\alpha$.
The section $F_{\mathbf{d}}$ extends to a section of ${\mathcal J}_X$ over $X$ and the divisor $\Theta_\alpha$ extends to ${\mathcal J}_X$. One therefore has the class $F_{\mathbf{d}}^\ast\Theta_\alpha$ in $H^2(X)$. Its restriction to $H^2(X^o)$ is the pullback of the class of $W_{\mathbf{d}}$.
\begin{theorem} \label{thm:harris} The class of $F_{\mathbf{d}}^\ast \Theta_\alpha$ in $H^2(X)$ is \begin{multline*} {\delta}_0/8 -\lambda_1 + \sum_{j=1}^n (d_j+d_j^2)\,\psi_j/2 - \sum_{P\subseteq I} \sum_{\{x_j,x_k\}\subseteq P} d_j d_k\, {\delta}_0^P \cr -\frac{1}{4} \sum_{P\subseteq I}\sum_{h=1}^{g-1} \big(d_P^2 - (2h-1)d_P + h^2 - h\big)\, {\delta}_h^P. \end{multline*} \end{theorem}
The class $-{\delta}_0/8+F_{\mathbf{d}}^\ast\Theta_\alpha$ is the pullback of an integral class from ${\overline{\M}}_{g,n}$ as ${\delta}_h^P = {\delta}_{g-h}^{P^c}$ and because each of $d_j + d_j^2$, $d_P^2 - (2h-1)d_P$ and $h^2 - h$ is even.
\begin{proof} The first step is to show that the class $\theta_\alpha$ of $\Theta_\alpha$ satisfies \begin{equation} \label{eqn:theta} \theta_\alpha = \hat{\phi}_H + \lambda_1/2 \in H^2({\mathcal J}_g'), \end{equation} where ${\mathcal J}_g'$ denotes the universal jacobian over ${\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}}$. Granted this, the result follows from Theorem~\ref{thm:variant} as $$ F_{\mathbf{d}}^\ast(\theta_\alpha) = F_{\mathbf{d}}^\ast(\hat{\phi}_H + \lambda_1/2) = \lambda_1/2 + F_{\mathbf{d}}^\ast(\hat{\phi}_H) $$ and $$ {\kappa}_1 = 12\lambda_1 - {\delta} = 12\lambda_1 - {\delta}_o -\frac{1}{2} \sum_{j=1}^h \sum_{P\subseteq I} {\delta}_h^P. $$
To prove (\ref{eqn:theta}), first note that the relation $\theta_\alpha = \phi_H + \lambda_1/2$ holds in $H^2({\mathcal J}_g^c)$. This is because the restrictions of $\theta_\alpha$ and $\phi_H$ to each fiber of ${\mathcal J}_g^c \to {\mathcal M}_g^c$ are equal and because the restriction of $\phi_H$ to the zero section vanishes, while the restriction of $\theta_\alpha$ to the zero section has class $\lambda_1/2$ as the corresponding theta null $\vartheta_\alpha(0,\Omega)$ is a modular form of weight $1/2$ for some finite index subgroup of $\Sp_g({\mathbb Z})$. Since $\hat{\phi}_H$, $\theta_\alpha$ and $\lambda_1$ are all classes of line bundles, and since ${\mathcal J}_g - {\mathcal J}_g^c$ is the restriction of ${\mathcal J}_g'$ to $\Delta_0$, it follows that $$ \theta_\alpha = \hat{\phi}_H + \lambda_1/2 + c{\delta}_0 \in H^2({\mathcal J}_g') $$ Restricting both sides to the zero section implies that $$ \lambda_1/2 = 0 + \lambda_1/2 + c{\delta}_0 \in H^2({\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}}) \cong H^2({\overline{\M}}_g), $$ which implies that $c=0$, as required. \end{proof}
\section{Eliashberg's Problem in Genus $1$}
The solution of Eliashberg's problem over ${\mathcal M}_{g,n}^c$ in genus $\ge 2$ given in the previous section fails when $g=1$ as we cannot divide by $2g-2$. However, a variant of the methods of the previous section gives a complete solution in genus $1$.
When $g=1$, the class of $F_{\mathbf{d}}^\ast\eta_1$ naturally lives in $H^2({\overline{\M}}_{1,n+1})$ as the locus of indeterminacy ${\Delta}_0^{\mathrm{sing}}$ of $F_{\mathbf{d}}$ has codimension $\ge 2$, whereas $F_{\mathbf{d}} \eta_1$ is the class of a divisor, and thus extends uniquely from ${\overline{\M}}_{1,n+1}-{\Delta}_0^{\mathrm{sing}}$ to ${\overline{\M}}_{1,n+1}$.
\begin{theorem} \label{thm:elliptic} If ${\mathbf{d}}=(d_0,\dots,d_n)\in {\mathbb Z}^{n+1}$ satisfies $\sum_j d_j = 0$, then $$ F_{\mathbf{d}}^\ast \eta_1 = \big(-1+(d_0^2 + \dots + d_n^2)/2\big)\lambda_1
- \sum_{P\subseteq I} \sum_{\{x_j,x_k\}\subseteq P} d_j d_k\, {\delta}_0^P \in H^2({\overline{\M}}_{1,n+1}). $$ \end{theorem}
The restriction of this class to ${\mathcal M}_{1,n+1}$ has been computed independently by Cavalieri and Marcus \cite{cavalieri-marcus} using different methods.
Denote the universal elliptic curve ${\mathcal J}_1 \to {\overline{\M}}_{1,1}$ by ${\mathcal E}$. Note that ${\mathcal E} = {\overline{\M}}_{1,2}-{\Delta}_0^{\mathrm{sing}}$. Its restriction to ${\mathcal M}_{1,1}$ is the universal elliptic curve ${\mathcal E}^c$ of ``compact type''. Since ${\mathcal E}$ and ${\overline{\M}}_{1,2}$ differ in codimension $2$, their second cohomology and Picard groups are isomorphic: $$ H^2({\overline{\M}}_{1,2}) \cong (\Pic {\overline{\M}}_{1,2})\otimes{\mathbb Q} \cong (\Pic {\mathcal E})\otimes{\mathbb Q} \cong H^2({\mathcal E}). $$ These groups are 2-dimensional with basis ${\delta}:={\delta}_0^{\{x_1,x_2\}}$, the class of the zero section $D$ of ${\mathcal E} \to {\overline{\M}}_{1,1}$, and ${\delta}_0$, the class of the fiber over the cusp of ${\overline{\M}}_{1,1}$. The class $\lambda_1$ of the Hodge bundle is ${\delta}_0/12$.
We will deduce Theorem~\ref{thm:elliptic} from the corresponding result for the $n$th power $$ {\mathcal E}^n = \underbrace{{\mathcal E}\times_{{\overline{\M}}_{1,1}} \cdots \times_{{\overline{\M}}_{1,1}} {\mathcal E}}_n. $$ of the universal elliptic curve over ${\overline{\M}}_{1,1}$. A point in ${\mathcal E}^n$ corresponds to the isomorphism class of an $n$-pointed elliptic curve $(E;x_0,x_1,\dots,x_n)$ where $x_0 = 0$, the identity.\footnote{Note that an $n$-pointed elliptic curves is an $(n+1)$-pointed genus $1$ curve.} Let $p : {\mathcal E}^n \to {\overline{\M}}_{1,1}$ be the canonical projection. For each ${\mathbf{d}} = (d_0,\dots,d_n)$ as above, there is a section of the pullback $p^\ast {\mathcal E} \to {\mathcal E}^n$ of the universal elliptic curve defined by $$ F_{\mathbf{d}}(E;x_0,\dots,x_n) = \sum_{j=0}^n d_j x_j = \sum_{j=1}^n d_j x_j \in E. $$
For $0\le j < k \le n$ let $D_{j,k}$ be the divisor in ${\mathcal E}^n$ where $x_j = x_k$. Denote its class in $H^2({\mathcal E}^n)$ by ${\delta}_{j,k}$. Let ${\Delta}_0$ be the inverse image of the moduli point of the nodal cubic under the projection ${\mathcal E}^n \to {\overline{\M}}_{1,1}$. Then $$ H^2({\mathcal E}^n) \cong (\Pic {\mathcal E}^n) \otimes {\mathbb Q} = {\mathbb Q}{\delta}_0 \oplus \bigoplus_{0\le j < k \le n} {\mathbb Q} {\delta}_{j,k}. $$ The pullback of ${\delta}_{j,k}$ under the morphism $\pi : {\overline{\M}}_{1,n+1}-{\Delta}_0^{\mathrm{sing}} \to {\mathcal E}^n$ is $$ \pi^\ast {\delta}_{jk} = \sum_{\{x_j,x_k\}\subseteq P} {\delta}_0^P \in \Pic {\overline{\M}}_{1,n+1}. $$ Since $d_0 + \dots + d_n = 0$, $d_0^2 + \dots + d_n^2 = -2\sum_{0\le j < k \le n } d_jd_k$. Since $F_{\mathbf{d}}^\ast \lambda_1 = \lambda_1$, to prove Theorem~\ref{thm:elliptic} it suffices to prove that \begin{equation} \label{eqn:reduction} F_{\mathbf{d}}^\ast({\delta} + \lambda_1) = - \sum_{0\le j<k\le n} d_jd_k ({\delta}_{j,k}+\lambda_1) \in H^2({\mathcal E}^n). \end{equation}
The key step in the proof of this statement is to show that $\delta + \lambda_1$ is a parallel, translation invariant class. This statement is made precise in the following lemma. For a positive integer $e$, let $[e] : {\mathcal E} \to {\mathcal E}$ be multiplication by $e$.
\begin{lemma} \label{lem:phihat} The class $\phi_H \in H^2({\mathcal E}^c)$ extends uniquely to a class $\hat{\phi}_H$ in $H^2({\mathcal E})$ that vanishes on the zero-section $D$. It is given by $$ \hat{\phi}_H = {\delta} + \lambda_1 \in H^2({\mathcal E}). $$ and is characterized by the two properties $$ \int_{{\Delta}_0} \hat{\phi}_H = 1 \text{ and } [e]^\ast \hat{\phi}_H = e^2 \hat{\phi}_H \text{ for all integers $e>1$.} $$ \end{lemma}
\begin{proof} The exact sequence $$ 0 \to {\mathbb Q}{\delta}_0 \to H^2({\mathcal E}) \to H^2({\mathcal E}^c) \to 0. $$ is invariant under $[e]^\ast$, which acts trivially on the kernel and by multiplication by $e^2$ on the quotient. Since $\phi_H$ spans the right-hand group, it follows that it has a unique lift $\hat{\phi}_H$ to $H^2({\mathcal E})$ with the property that $[e]^\ast\hat{\phi}_H = e^2\hat{\phi}_H$.
Since $[e]_\ast [D] = [D]$, we have $$ (e^2 - 1)\int_D\hat{\phi}_H = \int_D [e]^\ast\hat{\phi}_H - \int_D\hat{\phi}_H = \int_{([e]_\ast - 1)D} \hat{\phi}_H = 0. $$ Since $e > 1$, this implies the vanishing $\int_D \hat{\phi}_H = 0$ of $\hat{\phi}_H$ on $D$. Since ${\Delta}_0$ is the class of the fiber of ${\mathcal E} \to {\overline{\M}}_{1,1}$, $\int_{{\Delta}_0}\hat{\phi}_H = 1$. Since ${\delta}$ and ${\delta}_0$ span $H^2({\mathcal E})$ and since its intersection pairing is non-singular, these two properties characterize $\hat{\phi}_H$.
To prove that $\hat{\phi}_H = \delta + \lambda_1$, it suffices to show that $({\delta}+\lambda_1)\cdot {\delta} = 0$ and that $({\delta}+\lambda_1)\cdot {\delta}_0 = 1$. Since the Chen class of the normal bundle of $D$ in ${\mathcal E}$ is $-\lambda_1$, we have $$ \int_D{\delta} = D^2 = -\int_D \lambda_1, $$ which implies that $({\delta}+\lambda_1)\cdot{\delta} = 0$. Since ${\delta}_0$ is the class of a fiber of ${\mathcal E} \to {\overline{\M}}_{1,1}$, ${\delta}_0 \cdot \lambda_1 = 0$, so that ${\delta}_0\cdot({\delta}+\lambda_1) = {\delta}_0\cdot {\delta} = 1$. \end{proof}
Each unordered pair $\{j,k\}$ of integers in $[0,n]$ determines a parallel class $\hat{\phi}_{j,k} \in H^2({\mathcal E}^n)$ as follows. Let $p_{j,k} : {\mathcal E}^n \to {\mathcal E}$ be the projection that takes $[E:x_0,\dots,x_n]$ to $[E;x_j,x_k]=[E;0,x_k-x_j]$. Observe that $D_{j,k} = p_{j,k}^\ast D$. Set $\hat{\phi}_{j,k} = p_{j,k}^\ast \hat{\phi}_H$. Lemma~\ref{lem:phihat} implies that $\hat{\phi}_{j,k} = {\delta}_{j,k} + \lambda_1$. Combining this with (\ref{eqn:reduction}), we deduce that, to prove Theorem~\ref{thm:elliptic}, it suffices to show that \begin{equation} \label{eqn:reduction2} F_{\mathbf{d}}^\ast \hat{\phi}_H = - \sum_{0\le j<k\le n} d_jd_k \hat{\phi}_{j,k}. \end{equation}
\begin{lemma} The class $\phi_{\Delta} \in H^2({\mathcal E}^c\times_{{\mathcal M}_{1,1}}{\mathcal E}^c)$ extends to a class $\hat{\phi}_{\Delta}$ in $H^2({\mathcal E}^2)$ with the property that $[e]^\ast \hat{\phi}_{\Delta} = e^2\hat{\phi}_{\Delta}$. Specifically, \begin{equation} \label{eqn:phiD} \hat{\phi}_{\Delta} = p_{0,1}^\ast\hat{\phi}_H + p_{0,2}^\ast\hat{\phi}_H - p_{1,2}^\ast\hat{\phi}_H \in H^2({\mathcal E}^2). \end{equation} \end{lemma}
\begin{proof} The class given by (\ref{eqn:phiD}) is clearly an eigenvector of $[e]$ with eigenvalue $e^2$. To prove the result, it suffices to prove that its restriction to $({\mathcal E}^c)^2 := {\mathcal E}^c\times_{{\mathcal M}_{1,1}}{\mathcal E}^c$ is $\phi_{\Delta}$. Since the restriction to $({\mathcal E}^c)^2$ of all classes in the formula are represented by parallel, translation invariant forms, it suffices to check the formula for a single smooth elliptic curve $E$. Let $a,b$ be a symplectic basis of $H^1(E)$. Identify $H^{\bullet}(E^2)$ with $H^{\bullet}(E)\otimes H^{\bullet}(E)$. Then a routine computation shows that the restriction of $p_{1,1}^\ast\phi_H$ to $E^2$ is $$ (a\wedge b)\otimes 1 + 1\otimes(a\wedge b) - (a\otimes b - b\otimes a)
= \big(p_{0,1}^\ast \phi_H + p_{0,2}^\ast \phi_H - \phi_{\Delta}\big)|_{E^2}, $$ as required. \end{proof}
For each $0\le j < k \le n$ define $\hat{\phi}^{j,k}_{\Delta} = p_{j,k}^\ast \hat{\phi}_{\Delta}$.
\begin{corollary} If $0 < j < k\le n$, then $\hat{\phi}^{j,k}_{\Delta} = \hat{\phi}_{0,j} + \hat{\phi}_{0,k} - \hat{\phi}_{j,k}$. \end{corollary}
To prove (\ref{eqn:reduction2}), factor $F_{\mathbf{d}} : {\mathcal E}^n \to {\mathcal E}$ as follows: $$ \xymatrix{ {\mathcal E}^n \ar[r]_(0.45){(d_1,\dots,d_n)} \ar@/^1pc/[rr]^{F_{\mathbf{d}}} & {\mathcal E}^n \ar[r]_{\trace_n} & {\mathcal E} } $$ where $(d_1,\dots,d_n) : [E;0,x_1,\dots,x_n] \mapsto [E:0,d_1x_1,\dots,d_nx_n]$. Note that the formula $$ \trace_n^\ast \hat{\phi}_H = \sum_{j=1}^n \hat{\phi}_{0,j} + \sum_{1\le j < k\le n} \hat{\phi}_{\Delta}^{j,k} $$ holds in $H^2({\mathcal E}^n)$ as it holds when restricted to $({\mathcal E}^c)^n$, because both sides are eigenvectors of $[e]$ with eigenvalue $e^2$, and because both sides vanish on the divisor $x_1 + \dots + x_n = 0$ by an argument similar to the one used in the proof of Lemma~\ref{lem:phihat}.
Adapting the arguments of the previous section to this case, we see that \begin{align*} F_{\mathbf{d}}^\ast \hat{\phi}_H &= \sum_{j=1}^n d_j^2 \hat{\phi}_{0,j} + \sum_{1\le j < k\le n} d_jd_k \hat{\phi}_{\Delta}^{j,k} \cr & = \sum_{j=1}^n d_j^2 \hat{\phi}_{0,j} + \sum_{1\le j < k\le n} d_jd_k (\hat{\phi}_{0,j} + \hat{\phi}_{0,k} - \hat{\phi}_{j,k})\cr & = \sum_{j=1}^n \big(d_j^2 + (d_1 + \dots + d_n - d_j)d_j\big){\delta}_{0,j} - \sum_{1\le j < k\le n} d_jd_k \hat{\phi}_{j,k} \cr & = \sum_{j=1}^n -d_0d_j {\delta}_{0,j} -
\sum_{1\le j < k\le n} d_jd_k \hat{\phi}_{j,k} \cr
& = -\sum_{0\le j < k\le n} d_jd_k \hat{\phi}_{j,k}. \end{align*}
\section{Normal Functions and Positivity}
Suppose that ${\mathbb V}$ is a polarized variation of Hodge structure over $X$ of weight $-1$ endowed with a weak polarization $S$.
\begin{theorem} \label{positivity} If $\nu : X \to J({\mathbb V})$ is a normal function, then $\nu^\ast{\omega}_S$ is a non-negative $(1,1)$-form on $X$. \end{theorem}
\begin{proof} As previously remarked, $J({\mathbb V})$ is isomorphic, as a bundle of tori, to ${\mathbb V}_{\mathbb R}/{\mathbb V}_{\mathbb Z}$. The normal function $\nu$ thus corresponds to a section $s : X \to {\mathbb V}_{\mathbb R}/{\mathbb V}_{\mathbb Z}$. Locally this lifts to a section (also denoted by s) of ${\mathbb V}_{\mathbb R}$. We can view this as as section of $$ {\mathcal V} := {\mathbb V}\otimes {\mathcal O}_X $$ This is a flat holomorphic vector bundle. Denote its Hodge filtration by $$ {\mathcal V} \supseteq \cdots \supseteq {\mathcal F}^p \supseteq {\mathcal F}^{p+1} \supseteq \cdots $$ Each ${\mathcal F}^p$ is a holomorphic sub-bundle of ${\mathcal V}$. Since ${\mathbb V}$ has weight $-1$, it splits as the sum $$ {\mathcal V} = {\mathcal F}^0 \oplus \overline{\F}^0, $$ where $\overline{\F}^p$ denotes the complex conjugate of ${\mathcal F}^p$ in ${\mathcal V}$. Note that ${\mathcal F}^0$ is a holomorphic sub-bundle, while $\overline{\F}^0$ is not, in general, holomorphic. Decompose $s$ as $$ s = p + n $$ where $p$ is a smooth section of ${\mathcal F}^0$ and $n$ is a smooth section of $\overline{\F}^0$. Since $s$ is real, $p$ and $n$ are complex conjugates of each other.
We can compute the differentials of $s$, $n$ and $p$ with respect to the flat structure on ${\mathbb V}_{\mathbb R}$. Since ${\omega}_S$ is parallel, we have $$ \nu^\ast {\omega}_S = s^\ast{\omega}_S = S(ds,ds) = S(\partial s + \overline{\del} s,\partial s + \overline{\del} s) = 2 S(\partial s,\overline{\del} s). $$ This is clearly of type $(1,1)$.
Next we prove that $\nu^\ast {\omega}_S$ is non-negative. Since a 2-form is positive if and only if it is positive on every holomorphic arc in $X$, we may assume that $X$ is the unit disk. Let $t$ be a holomorphic coordinate in $X$.
The Griffiths infinitesimal period relation for normal functions implies that $\partial f/\partial t \in {\mathcal F}^{-1}$ for any smooth local lift $f : X \to {\mathcal V}$ of the normal function. Here, and in what follows, the partial derivatives are taken with respect to the natural flat connection on ${\mathcal V}$. Since $n$ and $p+n$ are both smooth local lifts of $\nu$, \begin{equation} \label{griff_inf} \frac{\partial p}{\partial t}\in {\mathcal F}^{-1} \text{ and } \frac{\partial n}{\partial t} \in {\mathcal F}^{-1}. \end{equation}
Since ${\mathcal F}^0$ is a holomorphic sub-bundle of ${\mathcal V}$, $\partial p/\partial \overline{t} \in {\mathcal F}^0$. Since $n$ is the conjugate of $p$, we have $\partial n/{\partial t} \in \overline{\F}^0$. Combining this with Griffiths infinitesimal period relation (\ref{griff_inf}), we conclude that ${\partial n}/{\partial t} \in {\mathcal F}^{-1}\cap \overline{\F}^0$ and, by taking complex conjugates, that ${\partial p}/{\partial \overline{t}} \in {\mathcal F}^0\cap \overline{\F}^{-1}$.
Next compute the pullback of ${\omega}_S$: as above, \begin{align*} \nu^\ast {\omega}_S &= 2 S(\partial s,\overline{\del} s) \cr &= 2 S(\partial n + \partial p, \overline{\del} n + \overline{\del} p) \cr &= 2 S\left(\frac{\partial n}{\partial t} + \frac{\partial p}{\partial t}, \frac{\partial n}{\partial \overline{t}} + \frac{\partial p}{\partial \overline{t}}\right) dt \wedge d\overline{t} \cr &= 2S(v(t),\overline{v(t)}) dt \wedge d\overline{t} \end{align*} where $v(t) := {\partial n}/{\partial t} + {\partial p}/{\partial t}$. Set $t = x + i y$. Since $v(t) \in H^{-1,0}_t$ for all $t$, it follows that $\nu^\ast {\omega}_S = 2i^{-1-0}S(v,\overline{v}) dx\wedge dy$, which is non-negative. \end{proof}
\begin{corollary} \label{cor:positivity} Suppose that ${\mathbb V}$ is a variation of Hodge structure of weight $-1$ over a smooth complex algebraic variety $X$. If $S$ is a weak polarization of ${\mathbb V}$ and $\nu$ is a normal function section of $J({\mathbb V}) \to X$, then for all complete curves $T$ in $X$ $$ \int_T \nu^\ast {\omega}_S \ge 0 $$ with equality if and only if the infinitesimal invariant of the normal function vanishes on $T$. \end{corollary}
Suppose that ${\overline{X}}$ is a smooth completion of $X$. If Conjecture~\ref{conj:integrable} holds, then we can conclude that the natural extension of $\nu^\ast\phi_S$ to a class in $H^2({\overline{X}})$ has non-negative degree on all complete curves $T$ in ${\overline{X}}$ that do not lie in ${\overline{X}}-X$.
\section{Slope Inequalities} \label{sec:moriwaki}
As an immediate consequence of Corollary~\ref{cor:positivity} with the computations in Section~\ref{sec:formulas}, we obtain the following versions Moriwaki's inequalities \cite{moriwaki,moriwaki:new}. The second assertion below was obtained independently by the author in the late 1990s (cf.\ \cite[p.~195]{moriwaki:new}).
\begin{theorem} \label{thm:moriwaki_weak} If $g\ge 2$, then \begin{enumerate} \item the divisor $$ (8g+4)\lambda_1 - 4\sum_{h=1}^{[g/2]} h(g-h){\delta}_h $$ has non-negative degree on each complete curve in ${\mathcal M}^c_g$,
\item the divisors \begin{align*} 4g(g-1)\psi - \kappa_1 - \sum_{h=1}^{g-1} (2h-1)^2\delta_{g-h}^{\{x\}} \cr 8\lambda_1 + 4g\psi - {\delta}_0 - \sum_{h=1}^{g-1} h\delta_{g-h}^{\{x\}} \in H^2({\mathcal M}_{g,1}^c) \end{align*} have non-negative degree on each complete curve in ${\mathcal M}_{g,1}^c$.
\end{enumerate} \end{theorem}
The semi-positivity of the $2$-forms representing these classes implies that their powers are also semi-positive.
\begin{corollary} If $g\ge 2$ and $k\ge 1$, then \begin{enumerate}
\item the cohomology class $$ \Big((8g+4)\lambda_1 - 4\sum_{h=1}^{[g/2]} h(g-h){\delta}_h\Big)^k \in H^{2k}({\mathcal M}^c_g) $$ has non-negative degree on each complete $k$-dimensional subvariety of ${\mathcal M}^c_g$.
\item the cohomology classes $$ \Big(4g(g-1)\psi - \kappa_1 - \sum_{h=1}^{g-1} (2h-1)^2\delta_h^{\{x\}}\Big)^k $$ and $$ \Big(8\lambda_1 + 4g\psi - {\delta}_0 - \sum_{h=1}^{g-1} h\delta_h^{\{x\}}\Big)^k $$ in $H^{2k}({\mathcal M}^c_{g,1})$ have non-negative degree on each complete $k$-dimensional subvariety of ${\mathcal M}^c_{g,1}$. \end{enumerate} \end{corollary}
This result also follows from Kleiman's criterion \cite[Thm.~1]{kleiman}, as J\'anos Koll\'ar pointed out to me.
The statements in Theorem~\ref{thm:moriwaki_weak} are weaker than Moriwaki's in the sense that they apply only to complete curves in ${\mathcal M}_{g,n}^c$ where $n=0,1$, but stronger than Moriwaki's as his versions apply only to complete curves in ${\overline{\M}}_{g,n}$ that do not lie in the boundary divisor ${\Delta}$. These two versions suggest the following stronger version of Moriwaki's inequalities, which would follow from Conjecture~\ref{conj:integrable} if it were true.
\begin{conjecture} \label{conj:moriwaki} For all $g\ge 2$: \begin{enumerate} \item the divisor $$ M := (8g+4)\lambda_1 - 4\sum_{h=1}^{[g/2]} h(g-h){\delta}_h $$ has non-negative degree on each complete curve in ${\overline{\M}}_g$ that does not lie in ${\Delta}_0$;
\item the divisors \begin{align*} W_H := 4g(g-1)\psi - \kappa_1 - \sum_{h=1}^{g-1} (2h-1)^2\delta_{g-h}^{\{x\}}\cr W_L := 8\lambda_1 + 4g\psi - {\delta}_0 - \sum_{h=1}^{g-1} h\delta_{g-h}^{\{x\}} \in H^2({\mathcal M}_{g,1}^c) \end{align*} have non-negative degree on each complete curve in ${\overline{\M}}_{g,1}$ that does not lie in ${\Delta}_0$;
\item and for all ${\mathbf{d}} = (d_1,\dots,d_n) \in {\mathbb Z}^d$ with $\sum d_j = m$, the class \begin{multline*} -{m^2}\kappa_1/2 + \sum_{j=1}^n (d_jm+d_j^2/2)\,\psi_j - \sum_{P\subseteq I} \sum_{\{x_j,x_k\}\subseteq P} d_j d_k\, {\delta}_0^P \cr -\frac{1}{4} \sum_{P\subseteq I}\sum_{h=1}^{g-1} \big(d_P - (2h-1)m\big)^2\, {\delta}_h^P \end{multline*} has non-negative degree on all complete curves in ${\overline{\M}}_{g,n}$ that do not lie in ${\Delta}_0$.
\end{enumerate} \end{conjecture}
Other slope inequalities of this type may be proved more directly, rather than deducing them from the fundamental slope inequalities above. For example, since $F_{\mathbf{d}}$ is a normal function, the class $F_{\mathbf{d}}^\ast \phi_H$ is represented by a non-negative $(1,1)$-form on ${\mathcal M}_{g,n}^c$. Theorem~\ref{pullback_phi} thus implies the following positivity statement:
\begin{proposition} If $g\ge 2$ and $(d_1,\dots,d_n)\in {\mathbb Z}^n$ satisfies $\sum_{j=1}^n d_j = (2g-2)m$, then for each $k\ge 1$, \begin{multline*} \bigg(-{m^2}\kappa_1/2 + \sum_{j=1}^n (d_jm+d_j^2/2)\,\psi_j - \sum_{P\subseteq I} \sum_{\{x_j,x_k\}\subseteq P} d_j d_k\, {\delta}_0^P \cr -\frac{1}{4} \sum_{P\subseteq I}\sum_{h=1}^{g-1} \big(d_P - (2h-1)m\big)^2\, {\delta}_h^P\bigg)^k \end{multline*} has non-negative degree on all complete $k$-dimensional subvarieties of ${\mathcal M}_{g,n}^c$. \end{proposition}
When $k=g$ and $m=g$, this implies that the pullback $F_{\mathbf{d}}^\ast \eta_g$ of the zero section has non-negative degree on all complete, $g$-dimensional subvarieties of ${\mathcal M}_{g,n}^c$.
\subsection{The jumping divisor} \label{sec:jump_div}
To better understand the behaviour of the degree of the Moriwaki divisors on curves in ${\overline{\M}}_{g,n}$ that pass through ${\Delta}_0^{\mathrm{sing}}$ but are not contained in ${\Delta}_0$, we need to consider the phenomenon of ``height jumping'' and the associated notion of a {\em jumping divisor}.
Suppose that $X={\overline{X}}-D$ where ${\overline{X}}$ is a smooth projective variety and $D$ is a normal crossings divisor in ${\overline{X}}$. Suppose that ${\mathbb U}$ is a weight $-1$ variation of Hodge structure over $X$ that is polarized by $S$. Suppose that $\nu$ is a normal function section of $J({\mathbb U})$. Then one has the symmetric biextension line bundle $\nu^\ast\widehat{\B}$ over $X$. Lear's Theorem implies that a positive power $\widehat{\B}_{N,\nu}$ of $\nu^\ast\widehat{\B}$ extends naturally to a line bundle over ${\overline{X}}$. It is characterized by the property that the metric on $\nu^\ast\widehat{\B}$ extends to a continuous metric on this line bundle over ${\overline{X}}-D^{\mathrm{sing}}$. For clarity of exposition, we suppose that the power is $1$, so that $\nu^\ast\widehat{\B}$ itself extends.\footnote{Otherwise, replace it by the power that does extend. In all known examples the power is $1$.} Denote the extended line bundle $\widehat{\B}_{1,\nu}$ by ${\mathcal B}_{\overline{X}}$.
Now suppose that $f : T \to {\overline{X}}$ is a morphism from a smooth projective curve to ${\overline{X}}$ whose image is not contained in $D$. Set $T'=T-f^{-1}(D)$. Applying Lear's Theorem to the normal function section $f^\ast\nu$ of $J(f^\ast {\mathbb U}) \to T'$, we obtain the Lear extension ${\mathcal B}_T$ of $(\nu\circ f)^\ast\widehat{\B}$ to $T$. If the image of $T$ avoids $D^{\mathrm{sing}}$, then $f^\ast {\mathcal B}_{\overline{X}} \cong {\mathcal B}_T$. This is because ${\mathcal B}_{\overline{X}}$ is metrized over ${\overline{X}} - D^{\mathrm{sing}}$, so that its pullback to $T$ is the unique metrized extension to $T$ of $(\nu\circ f)^\ast\widehat{\B}$. In general, there is a $0$-cycle $J$ on $T$, supported on $T-T'$, such that $$ f^\ast{\mathcal B}_{\overline{X}} \cong {\mathcal B}_T(J). $$ If $f(T)\cap D^{\mathrm{sing}}$ is empty, then $J=0$. We will call $J$ the {\em jumping divisor of $\nu$ on $T$}. It is not always trivial, as we shall explain below. The jumping divisor encodes ``height jumping''.
\subsection{Height jumping} Set $d=\dim X$. Assume that ${\mathbb D}^d$ is a polydisk in ${\overline{X}}$ with coordinates $(t_1,\dots,t_d)$, and that its intersection with $D$ is the divisor $t_1 \dots t_m = 0$. Assume that the monodromy of ${\mathbb U}$ about the branch $t_j = 0$ of $D$ is unipotent for each $j\in \{1,\dots,m\}$.\footnote{This condition is satisfied by the variations ${\mathbb H}$, ${\mathbb L}$ and ${\mathbb V}$ over ${\mathcal M}_{g,n}$.} Suppose that $$ \beta \in H^0\big(({\mathbb D}^\ast)^m\times {\mathbb D}^{d-m},\nu^\ast\widehat{\B}\big) $$ is a biextension section of $\nu^\ast\widehat{\B}$ defined over the punctured polydisk. The associated {\em height function} $({\mathbb D}^\ast)^m\times {\mathbb D}^{d-m} \to {\mathbb R}^+$ is the function $$
(t_1,\dots,t_d)\mapsto \log|\beta(t_1,\dots,t_d)|. $$
Suppose that $P\in T$ and that $f(P)$ is the origin of ${\mathbb D}^d$. Suppose that ${\mathbb D}$ is a disk in $T$ with coordinate $t$ with $t(P)=0$. The restriction of $f$ to ${\mathbb D}$ is a holomorphic arc $f : {\mathbb D} \to {\mathbb D}^d$. Set $$ r_j := \ord_{t=0} f^\ast t_j,\quad j = 1,\dots, m. $$ There is a rational number $q(r_1,\dots,r_m)$, which depends only on the exponents $r_j$, such that $$
\log|\beta(f(t))|_{\mathcal B} \sim q(r_1,\dots,r_m) \log|t|. $$ One might expect that $q(r_1,\dots,r_m)$ is linear. Surprisingly, this is not the case.
To better understand this, write $$ q(r_1,\dots,r_m) = q_0(r_1,\dots,r_m) + j(r_1,\dots,r_m) $$ where $q_0$ is the linear function $$ q_0(r_1,\dots,r_m) = \sum_{j=1}^m r_j q({\mathbf{e}}_j) $$ and ${\mathbf{e}}_1,\dots,{\mathbf{e}}_m$ is the standard basis of ${\mathbb Z}^m$. We shall call $j$ the {\em jump function} of $\beta$ at $P$. When $j$ vanishes, the height behaves as expected. Surprisingly, the height can jump. I first observed this when trying to understand Moriwaki's inequality, as explained in the following example. Although I was aware of height jumping through this example, I had no explanation for it. Recently Brosnan and Pearlstein \cite{brosnan-pearlstein:heights} have given a complete explanation of this phenomenon.
For the curve $f :T \to {\overline{X}}$ and $P\in T$, define $j_P= j(r_1,\dots,r_m)$. The jumping function determines the jumping divisor.
\begin{proposition} The jumping divisor $J$ of $\nu$ on $T$ associated to $f:T\to {\overline{X}}$ is the $0$-cycle $$ J = \sum_{P\in T} j_P P $$ on $T$. \end{proposition}
This is proved, using techniques similar to those described in \cite[\S8]{hain-reed:arakelov}, by considering the asymptotics as one approaches $P$ of the length $$
|s|_{\overline{X}}/|s|_T $$ of a section $s$ of $f^\ast {\mathcal B}_{\overline{X}}\otimes {\mathcal B}_T^{-1}$ that trivializes it over
$T'$. Here $|\phantom{x}|_{\overline{X}}$ denote the natural metric of $(\nu\circ f)^\ast
\widehat{\B}$ over $T'$ and $|\phantom{x}|_T$ denotes the natural metric on ${\mathcal B}_T$.
\subsection{An example of height jumping}
In this example, $g\ge 3$, $X = {\mathcal M}_g^c$, ${\mathbb U}$ is the variation ${\mathbb V}$ and $\nu$ is the normal function associated to the Ceresa cycle. Denote the Lear extension of $\nu^\ast\widehat{\B}$ to ${\overline{\M}}_g$ by ${\mathcal B}_{\overline{\M}}$. The main result of \cite{hain-reed:arakelov} implies that $c_1({\mathcal B}_{\overline{\M}})$ is the Moriwaki divisor $$ M:= (8g+4)\lambda_1 - g{\delta}_0 - \sum_{h=1}^{\lfloor g/2\rfloor} 4h(g-h){\delta}_h $$
Let ${\overline{{\cH}}}_g :=\{[C] \in {\overline{\M}}_{g} : C \text{ is hyperelliptic}\}$ be the locus of hyperelliptic curves in ${\overline{\M}}_g$. Set ${\mathcal H}^c_g = {\overline{{\cH}}}_g \cap {\mathcal M}_g^c$. The normal function $\nu$ vanishes identically on ${\mathcal H}_g^c$. This implies that the line bundle $\nu^\ast\widehat{\B}$ over ${\mathcal H}^c_g$ is trivial as metrized holomorphic line bundle. Consequently, its Lear extension to ${\overline{{\cH}}}_g$, which we denote by ${\mathcal B}_{\overline{{\cH}}}$, is a trivial as a metrized holomorphic line bundle over ${\overline{{\cH}}}_g$.
Recall that the boundary ${\overline{{\cH}}}_g - {\mathcal H}_g$ of ${\overline{{\cH}}}_g$ is a union of divisors $$ {\Delta}_h,\quad 1\le h \le g/2 \text{ and } \Xi_h,\quad 0\le h \le (g-1)/2, $$ where ${\Delta}_h$ is the restriction of the boundary divisor ${\Delta}_h$ of ${\overline{\M}}_g$ to ${\overline{{\cH}}}_g$; where $\Xi_0$ is the divisor whose generic point is an irreducible, geometrically connected hyperelliptic curve with one node; and where $\Xi_h$, ($h\neq 0$) is the locus whose generic point is a hyperelliptic curve with two nodes that are conjugate under the hyperelliptic involution and whose normalization is the union of two smooth, geometrically connected hyperelliptic curves, one of genus $h$ and the other of genus $g-h-1$. Denote the class of $\Xi_h$ in $H^2({\overline{{\cH}}}_g)$ by $\xi_h$.
Since ${\mathcal B}_{\overline{X}}$ is metrized over ${\overline{\M}}_g-{\Delta}_0^{\mathrm{sing}}$, the restrictions of ${\mathcal B}_{\overline{\M}}$ and ${\mathcal B}_{\overline{{\cH}}}$ to $$ {\overline{{\cH}}}_g - \bigcup_{h>0} \Xi_h $$ are isomorphic as metrized line bundles. This implies that $$ j^\ast {\mathcal B}_{\overline{\M}} \otimes {\mathcal B}_{\overline{{\cH}}}^{-1} \cong {\mathcal O}(J) $$ where $j$ denotes the inclusion ${\overline{{\cH}}}_g \hookrightarrow {\overline{\M}}_g$ and where $J$ is a linear combination of the $\Xi_h$, $h>0$. Note that, since ${\mathcal B}_{\overline{{\cH}}}$ is trivial, $$
{\mathcal O}(J) \cong j^\ast {\mathcal B}_{\overline{\M}} \cong {\mathcal O}(M)|_{{\overline{{\cH}}}_g} \in \Pic {\overline{{\cH}}}_g. $$ The restriction of the Moriwaki divisor to ${\overline{{\cH}}}_g$ is easily seen to be $$ j^\ast M = (8g+4)\lambda_1 - g\xi_0 - \sum_{h=1}^{\lfloor(g-1)/2 \rfloor} 2g \xi_h - 4\sum_{h=1}^{\lfloor g/2\rfloor} h(g-h){\delta}_h. $$ On the other hand, Cornalba and Harris \cite{cornalba-harris} have shown that $$ (8g+4)\lambda_1 - g\xi_0 - \sum_{h=1}^{\lfloor(g-1)/2 \rfloor} 2(h+1)(g-h)\xi_h - 4\sum_{h=1}^{\lfloor g/2\rfloor} h(g-h){\delta}_h = 0 $$ in $\Pic {\overline{{\cH}}}_g$. Together these imply that $$ J = \sum_{h=1}^{\lfloor(g-1)/2 \rfloor} 2h(g-h-1)\Xi_h. $$
It is now easy to construct an example of height jumping. Suppose that $f : T \to {\overline{\M}}_g$ is a curve whose image lies in the hyperelliptic locus and is not contained in ${\Delta}_0$. If $h>0$ and if the image of $f$ intersects $\Xi_h$ transversely at a smooth point $f(P)$, then the computations above imply that $$ j_P = 2(h+1)(g-h)-2g = 2h(g-h-1) > 0. $$ This implies that $$ \deg_T M = \deg_T J > 0. $$
Moriwaki's inequality and positivity in Hodge theory suggest that the jumping divisor associated to any curve in ${\overline{\M}}_g$ should be effective. Denote by $J_T$ the jumping divisor associated to a morphism $f:T \to {\overline{\M}}_g$ whose image is not contained in ${\Delta}_0$.
\begin{conjecture}[weak form] \label{conj:jump_weak} For all projective curves $f: T\to {\overline{\M}}$ whose image is not contained in ${\Delta}_0$, the jumping divisor $J_T$ is effective. \end{conjecture}
This and Conjecture~\ref{conj:integrable}, if true, imply a stronger version of Moriwaki's inequalities as, for example, $$ \deg_T M = \deg_T {\mathcal B}_T + \deg_T J_T \ge \deg_T J_T \ge 0. $$ This would imply that the degree of Moriwaki's divisor on most curves not contained in ${\Delta}_0$ that pass through ${\Delta}_0^{\mathrm{sing}}$ would be strictly positive as $\deg_T M$ would be bounded below by the degree of its jumping divisor.
Similarly, one can conjecture that for all projective curves $f: T \to {\overline{\M}}_{g,1}$ whose image is not contained in ${\Delta}_0$, the jumping divisor associated to the biextension line bundle associated to the normal function section ${\mathcal K}$ of $J({\mathbb H})$ is always effective.
In general, one might hope that in the situation described in Paragraph~\ref{sec:jump_div}, the jumping divisor $J$ associated to a curve $f: T \to {\overline{X}}$ whose image is not contained in $D$, is effective.
\appendix
\section{Normal Functions over ${\mathcal M}_{g,n}$} \label{sec:vmhs}
For completeness, we state the classification (mod torsion) of normal functions over ${\mathcal M}_{g,n}$ associated to variations of Hodge structure whose monodromy representation factors through a rational representation of $\Sp_g$. It follows quite directly from results proved in \cite[\S8]{hain:normal}.\footnote{Note that there are two typos on page~121. Line~4 should begin $\dim \Gamma H^1({\Gamma}_{g,r}^n,V(\lambda))$, and there is a $2$ missing from the right-hand side of line $-7$.}
The isomorphism classes of irreducible rational representations of the ${\mathbb Q}$-group $\Sp_g$ are indexed by partitions $\lambda$ $$ n = \lambda_1 + \dots + \lambda_h,\quad \lambda_1\ge \lambda_2 \ge \dots\ge \lambda_h, \quad h\le g $$ of integers $n$ into at most $g$ parts. Denote the local system over ${\mathcal A}_g$ that corresponds to the partition $\lambda$ by ${\mathbb V}_\lambda$. It underlies a
${\mathbb Q}$-variation of Hodge structure of weight $-|\lambda|$, where $$
|\lambda| := \lambda_1 + \dots + \lambda_h. $$ These can be pulled back to variations of Hodge structure over ${\mathcal M}_{g,n}$ along the period map. Note that ${\mathbb H} = {\mathbb V}_{[1]}$ and that ${\mathbb V}={\mathbb V}_{[1^3]}(-1)$.
Recall that ${\Gamma} A$ denotes the set of rational $(0,0)$ classes of a ${\mathbb Q}$-Hodge structure $A$. If $A$ is polarized, then $A = {\Gamma} A \oplus ({\Gamma} A)^\perp$ in the category of ${\mathbb Q}$-Hodge structures.
\begin{theorem} \label{thm:vmhs} Suppose that $2g-2+n>0$. If ${\mathbb U}$ is a polarized variation of Hodge structure over ${\mathcal M}_{g,n}$ whose monodromy representation factors through a rational representation of $\Sp_g$, then there is an isomorphism of variations of ${\mathbb Q}$-Hodge structure $$ {\mathbb U} \cong \bigoplus_\lambda A_\lambda \otimes_{\mathbb Q} {\mathbb V}_\lambda, $$ where $A_\lambda$ is the Hodge structure $H^0({\mathcal M}_{g,n},\Hom({\mathbb V}_\lambda,{\mathbb U}))$. Moreover, if $A$ is a polarized Hodge structure, then \begin{enumerate}
\item $\Ext^1_{{\sf{MHS}}({\mathcal M}_{g,n})}({\mathbb Q}(0),A \otimes {\mathbb V}_\lambda)$ vanishes unless $\lambda=[1]$ or $\lambda=[1^3]$,
\item $\Ext^1_{{\sf{MHS}}({\mathcal M}_{g,n})}({\mathbb Q}(0),A \otimes {\mathbb H})$ vanishes when ${\Gamma} A=0$.
\item $\Ext^1_{{\sf{MHS}}({\mathcal M}_{g,n})}({\mathbb Q}(0),{\mathbb H})$ has basis ${\mathcal K}_1,\dots,{\mathcal K}_n$,
\item $\Ext^1_{{\sf{MHS}}({\mathcal M}_{g,n})}({\mathbb Q}(0),A \otimes {\mathbb V})$ vanishes when ${\Gamma} A=0$,
\item $\Ext^1_{{\sf{MHS}}({\mathcal M}_{g,n})}({\mathbb Q}(0), {\mathbb V})$ is one-dimensional, spanned by $\nu$.
\end{enumerate} \end{theorem}
Crudely stated, this result says that all normal functions over ${\mathcal M}_{g,n}$ associated to variations of Hodge structure that are representations of $\Sp_g$ can be expressed, modulo torsion, as rational linear combinations of the basic normal functions $\nu$ and ${\mathcal K}_1,\dots,{\mathcal K}_n$. For example, $(2g-2)\delta_{j,k} = {\mathcal K}_j - {\mathcal K}_k$.
\section{The Big Picture} \label{sec:big_picture}
The philosophy behind this work is that a significant amount of the geometry of ${\overline{\M}}_{g,n}$ is encoded in the category ${\sf{MHS}}_{g,n}$ of those admissible variations of ${\mathbb Z}$-MHS over ${\overline{\M}}_{g,n}$ whose weight graded quotients are subquotients of Tate twists ${\mathbb H}^{\otimes m}(d)$ of tensor powers of the fundamental local system $$ {\mathbb H} := R^1\pi_\ast {\mathbb Z}(1) $$ associated to the universal curve $\pi : {\mathcal C} \to {\mathcal M}_{g,n}$. The normal functions discussed in this paper are objects of ${\sf{MHS}}_{g,n}$.
The pure objects of ${\sf{MHS}}_{g,n}$ are the variations of Hodge structure that correspond to the irreducible representations of the group $\GSp_g$ of symplectic similitudes.\footnote{This is well known. An explanation can be found in \cite[\S8.1]{hain:rat_pts}.} These are all pulled back from variations of Hodge structure over ${\mathcal A}_g$ along the period mapping ${\mathcal M}_{g,n} \to {\mathcal A}_g$. Theorem~\ref{thm:vmhs} implies that the only non-trivial extensions between these (mod torsion) are of the form \begin{equation} \label{eqn:extension} 0 \to {\mathbb V}_b \to {\mathbb E} \to {\mathbb V}_a \to 0 \end{equation} where the weights $w_a$ and $w_b$ of the pure variations ${\mathbb V}_a$ and ${\mathbb V}_b$ satisfy $w_a = 1 + w_a$. This is because the extension (\ref{eqn:extension}) is determined by the extension $$ 0 \to \Hom_{\mathbb Z}({\mathbb V}_a,{\mathbb V}_b) \to {\mathbb E}' \to {\mathbb Z}_{{\mathcal M}_{g,n}}(0) \to 0 $$ obtained by applying $\Hom_{\mathbb Z}({\mathbb V}_a,\phantom{x})$ to (\ref{eqn:extension}) and then pulling back along the identity ${\mathbb Z}(0) \to \Hom_{\mathbb Z}({\mathbb V}_a,{\mathbb V}_a)$. That is, the $1$-extensions in ${\sf{MHS}}_{g,n}$ correspond to normal functions (mod torsion) over ${\mathcal M}_{g,n}$.
The question arises as to how one can understand ${\sf{MHS}}_{g,n}$. A first approximation is to understand the monodromy representations of objects of ${\sf{MHS}}_{g,n}$. The (orbifold) fundamental group of ${\mathcal M}_{g,n}$ is the mapping class group: $$ \pi_1({\mathcal M}_{g,n},[C;P]) \cong {\Gamma}_{C,P} := \pi_0\Diff^+(C,P), $$ where $P=\{x_1,\dots,x_n\}$ and $\pi_0\Diff^+(C,P)$ denotes the group of connected components of the group of orientation preserving diffeomorphisms of $C$ that fix $P$ pointwise.\footnote{This group depends only on $g$ and $n$ and is often denoted by ${\Gamma}_{g,n}$.} The action of ${\Gamma}_{C,P}$ on $H_1(C,{\mathbb Z})$ induces a homomorphism $$ \rho : {\Gamma}_{C,P} \to \Aut\big(H_1(C,{\mathbb Z}),\text{intersection form}\big) =: \Sp(H_1(C,{\mathbb Z})) \cong \Sp_g({\mathbb Z}) $$ which is well known to be surjective. Its kernel is the {\em Torelli group}, $T_{C,P}$.
Because each object ${\mathbb V}$ of ${\sf{MHS}}_{g,n}$ is filtered by its weight filtration $$ \cdots \subseteq W_{j-1}{\mathbb V} \subseteq W_j {\mathbb V} \subseteq W_{j+1}{\mathbb V} \subseteq \cdots $$ where each $\Gr^W_j {\mathbb V} := W_j{\mathbb V}/W_{j-1}{\mathbb V}$ is a pure variation of Hodge structure, the Zariski closure (over ${\mathbb Q}$) of its monodromy representation $$ \rho_V : {\Gamma}_{C,P} \to \Aut(V_{C,P}) $$ is an extension $$ 1 \to U_V \to G_V \to \Sp(H_1(C)) \to 1 $$ of algebraic ${\mathbb Q}$-groups where $U_V$ is unipotent. It is thus natural to consider all Zariski dense representations ${\Gamma}_{C,P} \to G({\mathbb Q})$ where $G$ is a ${\mathbb Q}$ algebraic group that is an extension of $\Sp(H_1(C))$ by a unipotent group and where ${\Gamma}_{C,P} \to G \to \Sp(H_1(C))$ is the standard representation $\rho$. These form an inverse system. Their inverse limit is known as the {\em completion of ${\Gamma}_{C,P}$ relative to $\rho$}; it is studied in \cite{hain:torelli}.
The completion of ${\Gamma}_{C,P}$ relative to $\rho$ is an extension $$ 1 \to {\mathcal U}_{C,P} \to {\mathcal G}_{C,P} \to \Sp(H_1(C)) \to 1 $$ of affine ${\mathbb Q}$-groups, where ${\mathcal U}_{C,P}$ is prounipotent. There is a canonical homomorphism ${\Gamma}_{C,P} \to {\mathcal G}_{C,P}({\mathbb Q})$. Denote the corresponding sequence of Lie algebras by \begin{equation} \label{eqn:ses} 0 \to {\mathfrak u}_{C,P} \to {\mathfrak g}_{C,P} \to {\mathfrak{sp}}(H_1(C)) \to 0. \end{equation} It is proved in \cite{hain:malcev} and \cite{hain:torelli} that for each choice of a base point $[C,P]$ of ${\mathcal M}_{g,n}$, the sequence (\ref{eqn:ses}) is a short exact sequence of Lie algebras in ${\sf{MHS}}$.
Define a {\em Hodge representation} of ${\Gamma}_{C,P}$ to be a MHS $V$ and a pair of representations $\phi_{\mathbb Z} : {\Gamma}_{C,P} \to \Aut_{\mathbb Z} V$ and $\phi : {\mathcal G}_{C,P} \to \Aut V$ such that the diagram $$ \xymatrix{ {\Gamma}_{C,P} \ar[r]^{\phi_{\mathbb Z}}\ar[d] & \Aut_{\mathbb Z} V \ar[d] \cr {\mathcal G}_{C,P}({\mathbb Q}) \ar[r]^\phi & \Aut_{\mathbb Q} V } $$ commutes and the induced homomorphism $d\phi : {\mathfrak g}_{C,P} \to \End V$ is a morphism of MHS. The main result of \cite{vmhs} implies that ${\sf{MHS}}_{g,n}$ is equivalent to the category of Hodge representations of ${\mathfrak g}_{C,P}$, a statement informally conjectured by Deligne. Because of this, extensions in ${\sf{MHS}}_{g,n}$ are determined up to isogeny by Lie algebra cohomology via the following isomorphisms (cf.\ \cite[Cor.~3.7]{hain:nab}): $$ \Ext^{\bullet}_{{\sf{MHS}}_{g,n}}({\mathbb Q}(0),{\mathbb V}) \cong H^{\bullet}({\mathcal G}_{C,P},V_{C,P}) \cong H^{\bullet}({\mathfrak u}_{C,P},V_{C,P})^{{\mathfrak{sp}}(H_1(C))}. $$ Although Theorem~\ref{thm:vmhs} was originally proved by a direct argument, it is most natural to regard it as a consequence of this general result and known facts about the cohomology of mapping class and Torelli groups.
The relevance of the algebra $A_{g,n}^{\bullet}$ defined in the introduction is that there is a natural $\Sp_g$-equivariant algebra homomorphism $$ H^{\bullet}({\mathfrak u}_{C,P}) \to A_{g,n}^{\bullet} $$ which is an isomorphism in degrees $0$ and $1$ and injective in degree $2$. The construction is described in \cite{hain:malcev} in a more general context. The algebra $T_{g,n}^{\bullet}$ is simply the subalgebra of $A_{g,n}^{\bullet}$ generated by the image of $H^1({\mathfrak u})$ --- that is, by normal functions.
The coefficients of the boundary component ${\delta}_h^P$, ($h>0$) in the formulas in Section~\ref{sec:formulas} are determined by the image in the second weight graded quotient of ${\mathfrak u}_{C,P}$ of the Dehn twist corresponding to a loop about ${\Delta}_h^P$. The coefficients of ${\delta}_0^P$ can be determined similarly, although one has to work with the appropriate relative weight filtration.
One remaining question is whether more information can be obtained by considering other natural categories of MHS over ${\mathcal M}_{g,n}$ (possibly with a level structure), such as the category of variations of MHS obtained from the Prym construction. At present not enough is know about the topology of the Prym construction to understand this problem, although recent progress has been made by Putman \cite{putman}.
\end{document} | arXiv |
\begin{document}
\title{ROM-based multiobjective optimization of elliptic PDEs via numerical continuation}
\author[1]{Stefan Banholzer}
\author[2]{Bennet Gebken}
\author[2]{Michael Dellnitz}
\author[2]{Sebastian Peitz}
\author[1]{Stefan Volkwein}
\affil[1]{\normalsize Department of Mathematics and Statistics, University of Konstanz, Germany}
\affil[2]{\normalsize Department of Mathematics, Paderborn University, Germany}
\maketitle
\begin{abstract}
Multiobjective optimization plays an increasingly important role in modern applications, where several objectives are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the {set of optimal compromises} (the \emph{Pareto set}) between the conflicting objectives.
Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging which is particularly problematic when the objectives are costly to evaluate as is the case for models governed by partial differential equations (PDEs). To decrease the numerical effort to an affordable amount, surrogate models can be used to replace the expensive PDE evaluations.
Existing multiobjective optimization methods using model reduction are limited either to low parameter dimensions or to few (ideally two) objectives. In this article, we present a combination of the reduced basis model reduction method with a continuation approach using inexact gradients. The resulting approach can handle an arbitrary number of objectives while yielding a significant reduction in computing time.
\end{abstract}
\section{Introduction}
The dilemma of deciding between multiple, equally important goals is present in almost all areas of engineering and economy. A prominent example comes from production, where we want to produce a product at minimal cost while simultaneously preserving a high quality. In the same manner, multiple goals are present in most technical applications, maximizing the velocity while minimizing the energy consumption of electric vehicles \cite{PSOB17} being only one of many examples. These conflicting goals result in \emph{multiobjective optimization problems} (MOPs) \cite{Ehr05}, where we want to optimize all objectives simultaneously. Since the objectives are in general contradictory, there exists an infinite number of \emph{optimal compromises}. The set of these compromise solutions is called the \emph{Pareto set}, and the goal in multiobjective optimization is to approximate this set in an efficient manner, which is significantly more expensive than solving a single objective problem. Due to this, the development of efficient numerical approximation methods is an active area of research, and methods range from scalarization \cite{Ehr05,H2001} over set-oriented approaches \cite{DSH2005} and continuation \cite{H2001} to evolutionary algorithms \cite{CLV07}. Recent advances have paved the way to new challenging application areas for multiobjective optimization such as feedback control or problems constrained by partial differential equations (PDEs); cf.~\cite{PD18b} for a survey.
In the presence of PDE constraints, the computational effort can quickly become infeasible such that special means have to be taken in order to accelerate the computation.
To this end, surrogate models form a promising approach for significantly reducing the computational effort. A widely used approach is to directly construct a mapping from the parameter to the objective space using as few function evaluations of the expensive model as possible, cf.~\cite{THH+15,CSHM17} for extensive reviews. In the case of PDE constraints, an alternative approach is via dimension reduction techniques such as Proper Orthogonal Decomposition (POD) \cite{Sir87,KV01} or the reduced basis (RB) method \cite{GP05}. In these methods, a small number of high-fidelity solutions is used to construct a low-dimensional surrogate model for the PDE which can be evaluated significantly faster while guaranteeing convergence using error estimates. In recent years, several methods have been proposed where model reduction is used in multiobjective optimization and optimal control. In \cite{IUV17} and \cite{ITV16}, scalarization using the so-called weighted sum method was combined with RB and POD, respectively. In \cite{BBV16,BBV17}, convex problems were solved using reference point scalarization and POD, and set-oriented approaches were used in \cite{BDPV18,BDPV17}. A comparison of both was performed in \cite{POBD19} for the Navier--Stokes equations.
In this article we combine an extension of the continuation methods presented in \cite{H2001,SDD2005} to inexact gradients (Section \ref{p5sec:ContinuationMethodWithInexactness}) with a reduced basis approach for elliptic PDEs (Section~\ref{p5sec:MultiobjectiveParameterOptimizationWithRB}). To deal with the error introduced by the RB approach, we combine the KKT conditions for MOPs with error estimates for the RB method to obtain a tight superset of the Pareto set. For the example considered here, the proposed method yields a speed-up factor of approximately 63 compared to the direct solution of the expensive problem (Section~\ref{p5sec:NumericalResults}). Additionally, our approach allows us to control the quality of the result by controlling the errors for each objective function individually.
\section{A continuation method for MOPs with inexact objective gradients} \label{p5sec:ContinuationMethodWithInexactness}
In this section, we will begin by briefly introducing the basic concepts of multiobjective optimization upon which we will build in this article (see \cite{Ehr05,H2001} for detailed introductions). Afterwards, we will discuss the continuation method for MOPs and present two modifications of it that can deal with inexact gradient information.
\subsection{Multiobjective optimization} \label{p5sec:MultiobjectiveOptimization}
The goal of multiobjective optimization is to minimize several conflicting criteria at the same time. In other words, we want to minimize an objective $J = (J_1,...,J_k) : \mathbb{R}^n \rightarrow \mathbb{R}^k$ that is vector valued. It maps the \emph{variable space} $\mathbb{R}^n$ to the \emph{image space} $\mathbb{R}^k$. In contrast to single-objective optimization (i.e., $k = 1$), there exists no natural total order of the image space $\mathbb{R}^k$ for $k > 1$. As a result, the classical concept of \emph{optimality} has to be generalized:
\begin{definition} \label{p5def:Pareto_optimal}
\begin{itemize}
\item[(a)] $\bar{u} \in \mathbb{R}^n$ is called \emph{(globally) Pareto optimal} if there is no other point $u \in \mathbb{R}^n$ such that $J_i(u) \leq J_i(\bar{u})$ for all $i \in \{1,...,k\}$ and $J_j(u) < J_j(\bar{u})$ for some $j \in \{1,...,k\}$.
\item[(b)] The set $P$ of all Pareto optimal points is called the \emph{Pareto set}. Its image under $J$ is the \emph{Pareto front}.
\end{itemize}
\end{definition}
The Pareto set is the solution of the \emph{multiobjective optimization problem (MOP)}
\begin{equation} \label{p5eq:MOP}
\min_{u \in \mathbb{R}^n} J(u). \tag{MOP}
\end{equation}
Constrained MOPs can be formulated analogously by restricting $u$ in Definition \ref{p5def:Pareto_optimal} to a subset $U \subseteq \mathbb{R}^n$. Similar to the scalar-valued case, if $J$ is differentiable, we can use the derivative of $J$ to obtain necessary conditions for Pareto optimality, the \emph{Karush-Kuhn-Tucker (KKT) conditions} \cite{H2001}:
\begin{theorem}
Let $\bar{u}$ be a Pareto optimal point of \eqref{p5eq:MOP}. Then there exist multipliers
\begin{equation*}
\alpha \in \Delta_k := \left\{ \alpha \in (\mathbb{R}^{\geq 0})^k : \sum_{i = 1}^k \alpha_i = 1 \right\}
\end{equation*}
such that
\begin{equation} \label{p5eq:KKT} \tag{KKT}
DJ(\bar{u})^\top \alpha = \sum_{i = 1}^k \alpha_i \nabla J_i(\bar{u}) = 0.
\end{equation}
\end{theorem}
For $k = 1$, this reduces to the well-known optimality condition $\nabla J(\bar{u}) = 0$. If $J$ is non-convex, then the points satisfying \eqref{p5eq:KKT} form a proper superset of the Pareto set $P$:
\begin{definition}
If $\bar{u} \in \mathbb{R}^n$ and $\bar{\alpha} \in \Delta_k$ satisfy \eqref{p5eq:KKT}, then $\bar{u}$ is called \emph{Pareto critical} with corresponding \emph{KKT vector} $\bar{\alpha}$, containing the \emph{KKT multipliers} $\bar{\alpha}_i$, $i \in \{1,...,k\}$. The set $P_c$ of all Pareto critical points is called the \emph{Pareto critical set}.
\end{definition}
When solving an MOP, an initial step can be to compute the Pareto critical set. This set possesses additional structure which can be exploited in numerical schemes. Introducing the function
\begin{equation*}
F : \mathbb{R}^n \times (\mathbb{R}^{>0})^k \rightarrow \mathbb{R}^{n+1},
(u,\alpha) \mapsto
\begin{pmatrix}
\sum_{i = 1}^k \alpha_i \nabla J_i(u) \\
1 - \sum_{i = 1}^k \alpha_i
\end{pmatrix},
\end{equation*}
we see that Pareto critical points and their corresponding KKT vectors can be described as the zero level set of $F$. As shown by Hillermeier \cite{H2001}, this has the following implication:
\begin{theorem} \label{p5thm:hillermeier}
Let $J$ be twice continuously differentiable.
\begin{itemize}
\item[(a)] Let $\mathcal{M} := \{ (u,\alpha) \in \mathbb{R}^n \times (\mathbb{R}^{>0})^k : F(u,\alpha) = 0 \}$. If the Jacobian of $F$ has full rank everywhere, i.e.,
\begin{equation} \label{p5eq:rank_DF}
rk(DF(u,\alpha)) = n + 1 \quad \forall (u,\alpha) \in \mathcal{M},
\end{equation}
then $\mathcal{M}$ is a $(k-1)$-dimensional differentiable submanifold of $\mathbb{R}^{n+k}$. The tangent space of $\mathcal{M}$ at $(u,\alpha)$ is given by
\begin{equation*}
T_{(u,\alpha)} \mathcal{M} = ker(DF(u,\alpha)).
\end{equation*}
\item[(b)] Let $(u,\alpha) \in \mathcal{M}$ such that \eqref{p5eq:rank_DF} holds in $(u,\alpha)$. Then there is an open set $U \subseteq \mathbb{R}^n \times \mathbb{R}^k$ with $(u,\alpha) \in U$ such that $\mathcal{M} \cap U$ is a manifold as in \emph{(a)}. In other words, $\mathcal{M}$ locally possesses a manifold structure in all points satisfying \eqref{p5eq:rank_DF}.
\end{itemize}
\end{theorem}
Theorem \ref{p5thm:hillermeier} forms the basis for the continuation method we use in this article.
\subsection{Continuation method with exact gradients}
\label{p5subsec:ExactContinuation}
We only give a brief description of the method here and refer to \cite{SDD2005} and \cite{H2001} for details. By Theorem \ref{p5thm:hillermeier}, the Pareto critical set is -- except for the boundary -- the projection of the differentiable manifold $\mathcal{M} \subseteq \mathbb{R}^n \times \mathbb{R}^k$ onto its first $n$ components. In \cite{GPD2019} it has been shown that generically, this also holds for the first-order approximations, i.e., the projection of the tangent space of $\mathcal{M}$ yields the tangent cone of $P_c$. Given a Pareto critical point $\bar{u} \in P_c$, this means that we can find first-order candidates for new Pareto critical points in the vicinity of $\bar{u}$ by moving in the projected tangent space of $\mathcal{M}$. The idea of the continuation method is to do this iteratively to explore the entire Pareto critical set.
Instead of approximating $P_c$ by a set of points, we use a set-oriented numerical approach; cf.~\cite{SDD2005} for details. This has the key advantage that it is easy to check whether a certain part of the set has already been computed, which is difficult when working with points. Additionally, a covering of $P_c$ by boxes makes it easy to obtain (and exploit) its topological properties.
In the approach, we evenly divide the variable space $\mathbb{R}^n$ into hypercubes or \emph{boxes} $B$ with radius $r > 0$:
\begin{align} \label{p5eq:def_boxes}
\mathcal{B}(r) &:= \{ [-r,r]^n + (2 i_1 r,...,2 i_n r)^\top : (i_1,...,i_n) \in \mathbb{Z}^n \}.
\end{align}
\begin{remark}
For ease of notation and readability, we will only consider the case where points $u \in \mathbb{R}^n$ are contained in single boxes. In other words, we only consider the case where $u$ is in the interior of a box and not in the intersection of multiple boxes. Since this is the generic case, this has no impact on the numerical methods we will propose later.
$\blacksquare$
\end{remark}
For $u \in \mathbb{R}^n$ let $B(u,r)$ be the box containing $u$. We want to compute the subset of $\mathcal{B}(r)$ covering the Pareto critical set for a given radius $r$, i.e.,
\[
\mathcal{B}_c(r) := \{ B \in \mathcal{B}(r) : P_c \cap B \neq \emptyset \}.
\]
Since we are interested in a covering via boxes instead of an approximation via points, when moving in a tangent direction of the critical set, we will search for \emph{tangent boxes} instead of single points. For $u \in \mathbb{R}^n$ let
\begin{equation*}
N(u,r) := \{ B \in \mathcal{B}(r) : B(u,r) \cap B \neq \emptyset \}
\end{equation*}
be the set of neighboring boxes of $B(u,r)$. Starting from a box $B(\bar{u},r)$ containing a critical point $\bar{u}$ with KKT vector $\bar{\alpha}$, we want to explore the neighboring boxes covering the projected tangent space at $\bar{u}$, i.e.,
\begin{equation} \label{p5eq:exact_tangent_boxes}
\mathcal{B}'(\bar{u},r) = \{ B' \in \mathcal{B}(r) : B' \in N(\bar{u},r), \ B' \cap \bar{u} + pr_u(T_{(\bar{u},\bar{\alpha})} \mathcal{M}) \neq \emptyset \}.
\end{equation}
Here, $pr_u : \mathbb{R}^{n+k} \rightarrow \mathbb{R}^n$ is the projection of the tangent space onto the first $n$ components, i.e., the variable space. The typical situation is visualized in Figure \ref{p5fig:exact_cont}.
\begin{figure}
\caption{Tangent boxes (black) of the initial box (grey) containing $\bar{u}$, which is contained in the Pareto critical set $P_c$ (dashed). The red line indicates the projection of the tangent space of $\mathcal{M}$ onto the variable space}
\label{p5fig:exact_cont}
\end{figure}
As the tangent space of the Pareto critical set is only a linear approximation, a corrector step is required to verify that a given tangent box actually contains part of the Pareto critical set. This means that there has to be at least one $u\in B$ satisfying \eqref{p5eq:KKT}. To this end, for a box $B$, we consider the problem
\begin{equation}\label{p5eq:KKT_box_test}\tag{PC-Box}
\min_{u \in B, \alpha \in \Delta_k} \| DJ(u)^\top \alpha \|_2^2
\end{equation}
Let $\theta(B)$ be the optimal value of this problem. Then obviously
\begin{equation*}
\theta(B) = 0 \Leftrightarrow B \cap P_c \neq \emptyset.
\end{equation*}
In particular, if $\theta(B) = 0$ and $(\bar{u},\bar{\alpha})$ is the solution of \eqref{p5eq:KKT_box_test}, then $\bar{u}$ is Pareto critical with corresponding KKT vector $\bar{\alpha}$. After solving \eqref{p5eq:KKT_box_test} in each tangent box, all boxes with $\theta(B) = 0$ are added to a queue and a new iteration of the method is started with the first element in the queue. The method stops when the queue is empty, i.e., when there is no neighboring box of the current set of boxes that contains part of the Pareto critical set. For the remainder of this article, we will refer to this method as the \emph{exact continuation method}.
\subsection{Continuation method with inexact gradients} \label{p5subsec:ContinuationMethodInexact}
Using ROM to solve the state equation of an MOP of an elliptic PDE will introduce an error in the objective functions and the corresponding gradients, which has to be taken into account in order to ensure Pareto criticality of the solution. We here present a method that calculates a tight superset of the Pareto critical set via numerical continuation, using upper bounds for the errors in the approximated gradients. Formally, we now assume that for each gradient $\nabla J_i$, we only have an approximation $\nabla J_i^r$ such that
\begin{equation} \label{p5eq:error_bounds}
\sup_{u \in \mathbb{R}^n} \left\lVert \nabla J_i(u) - \nabla J^r_i(u) \right\rVert_2 \leq \epsilon_i, \ i \in \{1,...,k\},
\end{equation}
with upper bounds $\epsilon = (\epsilon_1, ..., \epsilon_k)^\top \in \mathbb{R}^k$. Let $P_c$ and $P^r_c$ be the Pareto critical sets corresponding to $(\nabla J_i)_i$ and $(\nabla J_i^r)_i$, respectively. The following lemma shows how these error bounds translate to error bounds for the KKT conditions:
\begin{lemma} \label{p5lem:KKT_error}
Let $\bar{u} \in \mathbb{R}^n$ be Pareto critical for $J$ with KKT vector $\bar{\alpha} \in \Delta_k$. Then
\begin{equation*}
\| DJ^r(\bar{u})^\top \bar{\alpha} \|_2 \leq \sum_{i = 1}^k \bar{\alpha}_i \epsilon_i \leq \| \epsilon \|_\infty.
\end{equation*}
\end{lemma}
\begin{proof}
From the estimate
\begin{align*}
\left\lVert DJ^r(\bar{u})^\top \bar{\alpha} \right\rVert_2
&= \left\lVert DJ^r(\bar{u})^\top \bar{\alpha} - DJ(\bar{u})^\top \bar{\alpha} \right\rVert_2
= \left\lVert \sum_{i = 1}^k (\nabla J_i^r(\bar{u}) - \nabla J_i(\bar{u}))^\top \bar{\alpha}_i \right\rVert_2 \\
&\leq \sum_{i = 1}^k \left\lVert \nabla J_i^r(\bar{u}) - \nabla J_i(\bar{u}) \right\rVert_2 \bar{\alpha}_i
\leq \sum_{i = 1}^k \bar{\alpha}_i \epsilon_i
\leq \left\lVert \epsilon \right\rVert_\infty
\end{align*}
we derive the claim.
\end{proof}
\begin{remark}
Lemma \ref{p5lem:KKT_error} can be generalized to equality and inequality constrained MOPs using the constrained version of the optimality conditions from \cite{H2001}. In this case, in the norm on the left-hand side of the inequality in Lemma \ref{p5lem:KKT_error}, one additionally has to add a linear combination of the gradients of the equality and inequality constraints.
$\blacksquare$
\end{remark}
Lemma \ref{p5lem:KKT_error} shows that we have to weaken the conditions for Pareto criticality of the reduced objective function to obtain a superset of the actual Pareto critical set $P_c$. Formally, let
\begin{align*}
P_1^r &:= \left\{ u \in \mathbb{R}^n : \min_{\alpha \in \Delta_k} \| DJ^r(u)^\top \alpha \|_2^2 \leq \| \epsilon \|_\infty^2 \right\}, \\
P_2^r &:= \left\{ u \in \mathbb{R}^n : \min_{\alpha \in \Delta_k} \left( \| DJ^r(u)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2 \right) \leq 0 \right\}.
\end{align*}
$P_1^r$ was also considered in \cite{PD2017} in the context of descent directions, where the solution of $\min_{\alpha \in \Delta_k} \| DJ^r(u)^\top \alpha \|_2^2$ is the squared length of the steepest descent direction in $u$. The condition for a point being in $P_1^r$ only depends on the maximal error $\| \epsilon \|_\infty$ and can be seen as a relaxed version of the KKT conditions for the inexact objective function. In contrast to this, the condition in $P_2^r$ actually considers the individual error bounds. By Lemma \ref{p5lem:KKT_error},
\begin{equation*}
P_c \subseteq P_2^r \subseteq P_1^r \text{ and } P^r_c \subseteq P_2^r \subseteq P_1^r,
\end{equation*}
i.e., both $P_1^r$ and $P_2^r$ are supersets of $P_c$ and $P_c^r$ (the points $\bar{u}$ for which the inexact gradients satisfy \eqref{p5eq:KKT}). In fact, $P_2^r$ is a tight superset of $P_c$ in the following sense:
\begin{lemma} \label{p5lem:P2_tight}
Let $\tilde{u} \in P_2^r$. Then there is some continuously differentiable $\tilde{J} : \mathbb{R}^n \rightarrow \mathbb{R}^k$ with
\begin{equation*}
\sup_{u \in \mathbb{R}^n} \| \nabla \tilde{J}_i(u) - \nabla J^r_i(u) \|_2 \leq \epsilon_i \ \forall i \in \{1,...,k\}
\end{equation*}
such that $\tilde{u}$ is Pareto critical for $\tilde{J}$.
\end{lemma}
\begin{proof}
Let
\begin{align*}
\tilde{\alpha} &\in \text{argmin}_{\alpha \in \Delta_k} \left( \| DJ^r(u)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2 \right), \\
\nu &:= DJ^r(\tilde{u})^\top \tilde{\alpha}, \\
g(u) &:= - \left( \frac{1}{\tilde{\alpha}^\top \epsilon} \sum_{i = 1}^n \nu_i u_i \right) \epsilon, \\
\tilde{J}(u) &:= J^r(u) + g(u).
\end{align*}
Since $\tilde{u} \in P_2^r$ by assumption, we have $\| \nu \|_2 \leq \tilde{\alpha}^\top \epsilon$. Thus
\begin{equation*}
\| \nabla \tilde{J}_i(u) - \nabla J^r_i(u) \|_2 = \| \nabla g_i(u) \|_2 = \frac{\epsilon_i}{\tilde{\alpha}^\top \epsilon} \| \nu \|_2 \leq \epsilon_i \quad \forall u \in \mathbb{R}^n \text{ and } \forall i \in \{1,...,k\},
\end{equation*}
and
\begin{align*}
D \tilde{J}(\tilde{u})^\top \tilde{\alpha} = \nu + \sum_{i = 1}^k \tilde{\alpha}_i \nabla g_i(\tilde{u}) = \nu - \sum_{i = 1}^k \tilde{\alpha}_i \frac{\epsilon_i}{\tilde{\alpha}^\top \epsilon} \nu = 0,
\end{align*}
which proves the lemma.
\end{proof}
Lemma \ref{p5lem:P2_tight} shows that for each point $\tilde{u}$ in $P_2^r$, there is an objective function satisfying the error bounds \eqref{p5eq:error_bounds} for which $\tilde{u}$ is Pareto critical. As a result, $P_2^r$ is the tightest superset of $P_c$ we can hope for if we only have the estimates in \eqref{p5eq:error_bounds}. The following example shows both supersets for a simple MOP (cf.~\cite{PD2017}).
\begin{example} \label{p5example:simple}
Let
\begin{equation*} \label{p5eq:MOP_simple_example}
J^r : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \ u \mapsto
\begin{pmatrix}
(u_1 - 1)^2 + (u_2 - 1)^4 \\
(u_1 + 1)^2 + (u_2 + 1)^2
\end{pmatrix}.
\end{equation*}
We consider the two error bounds $\epsilon^1 = (0.2, 0.05)^\top$ and $\epsilon^2 = (0, 0.2)^\top$. The corresponding supersets $P_1^r$ and $P_2^r$ are shown in Figure \ref{p5fig:simple_example}.
\begin{figure}
\caption{$P_1^r$ and $P_2^r$ for different error bounds $\epsilon$}
\label{p5fig:simple_example}
\end{figure}
As $\| \epsilon_1 \|_\infty = \| \epsilon_2 \|_\infty = 0.2$, $P_1^r$ is identical for both error bounds. Considering each component of $J^r$ individually, the critical points of $J_1^r$ and $J_2^r$ are located at $u^1 = (1,1)^\top$ and $u^2 = (-1,-1)^\top$, respectively. For $P_2^r$, we see that the difference between $P_c^r$ and $P_2^r$ becomes smaller the closer we get to the critical point of the objective function with the smaller error bound. This can be expected, as the influence (or weight) of $\nabla J^r_i(u)$ in the KKT conditions \eqref{p5eq:KKT} becomes larger the closer $u$ is to $u^i$. In particular, in Figure \ref{p5fig:simple_example}(b), the difference between $P_2^r$ and $P_c^r$ at $(1,1)^\top$ becomes zero, as $\epsilon^2_1 = 0$.
$\Diamond$
\end{example}
If we set $\epsilon_i = \| \epsilon \|_\infty$ for all $i \in \{1,...,k\}$, then $P_1^r = P_2^r$. Thus, we will from now on only consider $P_2^r$. As shown in the previous example, the ``dimension'' of $P_2^r$ is higher than the ``dimension'' of $P_c^r$. More precisely, $P_2^r$ contains the closure of an open subset of $\mathbb{R}^n$, which is shown in the following lemma:
\begin{lemma} \label{p5lem:dim_P2}
Let $\nabla J^r_i$ be continuous for all $i \in \{1,...,k\}$. Let
\begin{equation*}
A := \left\{ u \in \mathbb{R}^n : \min_{\alpha \in \Delta_k} \left( \| DJ^r(u)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2 \right) < 0 \right\}.
\end{equation*}
Then
\begin{enumerate}
\item[(a)] $P_2^r$ is closed. In particular, $\overline{A} \subseteq P_2^r$.
\item[(b)] $A$ is open.
\end{enumerate}
\end{lemma}
\begin{proof}
(a) The case $P_2^r = \emptyset$ is trivial, so we assume that $P_2^r \neq \emptyset$. Let $\bar{u} \in \overline{P_2^r}$. Then there is a sequence $(u^i)_i \in P_2^r$ with $\lim_{i \rightarrow \infty} u^i = \bar{u}$. Consider the sequence $(\alpha^i)_i \in \Delta_k$ with
\begin{equation*}
\alpha^i \in \text{argmin}_{\alpha \in \Delta_k} \left( \| DJ^r(u^i)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2 \right).
\end{equation*}
By compactness of $\Delta_k$, we can assume w.l.o.g.~that there is some $\bar{\alpha} \in \Delta_k$ with $\lim_{i \rightarrow \infty} \alpha^i = \bar{\alpha}$. Let
\begin{equation*}
\Psi : \mathbb{R}^n \times \Delta_k \rightarrow \mathbb{R}, \quad (u,\alpha) \mapsto \| DJ^r(u)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2.
\end{equation*}
By our assumption, $\Psi$ is continuous. From $\Psi(u^i,\alpha^i) < 0$ for all $i \in \mathbb{N}$ it follows that $\Psi(\bar{u},\bar{\alpha}) \leq 0$, which yields $\bar{u} \in P_2^r$. \\
(b) The case $A = \emptyset$ is again trivial such that we assume $A \neq \emptyset$. Let $\bar{u} \in A$ with
\begin{equation*}
\bar{\alpha} \in \text{argmin}_{\alpha \in \Delta_k} \left( \| DJ^r(\bar{u})^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2 \right).
\end{equation*}
Let $\psi : \mathbb{R}^n \rightarrow \mathbb{R}$, $u \mapsto \| DJ^r(u)^\top \bar{\alpha} \|_2^2 - (\bar{\alpha}^\top \epsilon)^2$. Then $\psi(\bar{u}) < 0$ and by our assumption, $\psi$ is continuous. Therefore, there is some open set $U \subseteq \mathbb{R}^n$ with $\bar{u} \in U$ such that $\psi(u) < 0$ for all $u \in U$. Since
\begin{equation*}
\min_{\alpha \in \Delta_k} \left( \| DJ^r(u)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2 \right) \leq \psi(u) < 0 \quad \forall u \in U
\end{equation*}
we have $U \subseteq A$ such that $A$ is open.
\end{proof}
We will now present two strategies for the numerical computation of $P^r_2$. Analogously to the case with exact gradients, we will approximate $P_2^r$ via the box covering
\begin{equation*}
\mathcal{B}_c^r(r) := \{ B \in \mathcal{B}(r) : B \cap P_2^r \neq \emptyset \}.
\end{equation*}
\subsubsection{Strategy 1}
The idea of our first method is to mimic the exact continuation method to calculate $\mathcal{B}_c^r$. For this, there are mainly two modifications we have to make:
\begin{enumerate}
\item By Lemma \ref{p5lem:dim_P2}, $P_2^r$ is not a lower-dimensional object in $\mathbb{R}^n$, so it makes no sense to use tangent information to find first-order candidates as in \eqref{p5eq:exact_tangent_boxes}. Instead, we have to consider all neighboring boxes.
\item The problem \eqref{p5eq:KKT_box_test} has to be replaced by a problem that checks the defining inequality of $P_2^r$.
\end{enumerate}
As a replacement for \eqref{p5eq:KKT_box_test}, we consider the following problem:
\begin{align} \label{p5eq:KKTred_box_test}
\min_{u \in B, \alpha \in \Delta_k} & \| DJ(u)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2. \tag{$\epsilon$PC-Box}
\end{align}
Let $\theta_\epsilon(B)$ be the optimal value of this problem. Note that $\theta_\epsilon(B) < 0$ is sufficient to verify that a box $B$ contains part of $P_2^r$. As a result, we do not need to solve \eqref{p5eq:KKTred_box_test} exactly. For example, when using an iterative method for the solution of \eqref{p5eq:KKTred_box_test}, we can stop when the function value is negative. The above mentioned changes yield Algorithm \ref{p5algo:boxcon_eps}.
\begin{algorithm}
\caption{Strategy 1: Box-Continuation Algorithm with Inexact Gradients}
\label{p5algo:boxcon_eps}
\begin{algorithmic}[1]
\item[] \hspace{-\algorithmicindent} Given: Radius $r > 0$ of boxes.
\State Choose an initial point $u_0 \in P_2^r$ and initialize $\mathcal{B} = \{ B(u_0,r) \}$ and a queue $Q = \{ u_0 \}$.
\While{$Q \neq \emptyset$}
\State Remove the first element $\bar{u}$ from $Q$.
\For{$B' \in N(\bar{u},r) \setminus \mathcal{B}$}
\State Solve \eqref{p5eq:KKTred_box_test} for $B'$. Let $\theta_\epsilon(B')$ be the optimal value and $(u',\alpha')$ be
\item[] \hspace{\algorithmicindent}\hspace{\algorithmicindent}the solution.
\If{$\theta_\epsilon(B') \leq 0$}
\State Add $u'$ to $Q$ and $B'$ to $\mathcal{B}$.
\EndIf
\EndFor
\EndWhile
\end{algorithmic}
\end{algorithm}
Due to the loss of low-dimensionality of $P_2^r$, the formulation of the continuation method becomes much simpler. As a consequence, it is straightforward to show that Algorithm \ref{p5algo:boxcon_eps} yields the desired covering $\mathcal{B}_c^r(r)$.
When executing the exact continuation method directly using inexact gradients (i.e., forgetting about the inexactness) and comparing it to Algorithm \ref{p5algo:boxcon_eps} (with the same box radius), the former will generally be much faster than the latter. A suitable way to evaluate the run time is to compare the number of times Problems \eqref{p5eq:KKT_box_test} and \eqref{p5eq:KKTred_box_test} need to be solved, respectively, as they require the majority of the computing time and are equally difficult to solve. (Here, we assume that both problems are solved with equal precision.) For each box added to the collection $\mathcal{B}$ in either algorithm, one of these problems has to be solved. Consequently, the longer run time of Algorithm \ref{p5algo:boxcon_eps} is partly due to the fact that $P_2^r$ is a superset of $P_c^r$, which means that more boxes are required to cover $P_2^r$ than $P_c^r$. However, even if the error bounds $\epsilon$ are small such that $P_2^r$ and $P_c^r$ are almost equal, Algorithm \ref{p5algo:boxcon_eps} will be slower. This is due to the fact that instead of only the tangent boxes, all neighboring boxes have to be tested with \eqref{p5eq:KKTred_box_test} in each loop of Algorithm \ref{p5algo:boxcon_eps}. While this does not matter in the interior of $P_2^r$ (as all neighboring boxes are in fact in $P_2^r$ in that case), it is very inefficient at the boundary of $P_2^r$. This is the motivation for the second strategy.
\subsubsection{Strategy 2}
By Lemma \ref{p5lem:dim_P2}, $P_2^r$ has the same dimension as the space of variables $\mathbb{R}^n$. This means that it can be described much more efficiently by its topological boundary $\partial P_2^r$. To be more precise, $\mathbb{R}^n \setminus \partial P_2^r$ consists of different connected components that lie either completely inside or completely outside $P_2^r$. So if we know $\partial P_2^r$, we merely have to test one point of each connected component if it is contained in $P_2^r$ or not to completely determine $P_2^r$. Therefore, the idea of our second strategy is to only compute $\partial P_2^r$.
Let
\begin{equation} \label{p5eq:def_varphi}
\varphi : \mathbb{R}^n \rightarrow \mathbb{R}, \quad u \mapsto \min_{\alpha \in \Delta_k} \left( \| DJ^r(u)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2 \right).
\end{equation}
This map is well-defined since $\Delta_k$ is compact, i.e., the minimum always exists. By Lemma \ref{p5lem:dim_P2}, we have $\partial P_2^r \subseteq \varphi^{-1}(0)$. Our goal is to compute $\varphi^{-1}(0)$ via a continuation approach. To this end, we first have to show that $\varphi$ is differentiable. We will do this by investigating the properties of the optimization problem in \eqref{p5eq:def_varphi}, i.e., of the problem
\begin{align} \label{p5eq:P2_problem}
\min_{\alpha \in \mathbb{R}^k} \ & \omega(\alpha), \nonumber \\
s.t. \quad & \sum_{i = 1}^k \alpha_i = 1, \\
\quad & \alpha_i \geq 0 \quad \forall i \in \{1,...,k\}, \nonumber
\end{align}
for
\begin{equation*}
\omega(\alpha) := \| DJ^r(u)^\top \alpha \|_2^2 - (\alpha^\top \epsilon)^2 = \alpha^\top (DJ^r(u) DJ^r(u)^\top - \epsilon \epsilon^\top) \alpha.
\end{equation*}
This leads to the following result.
\begin{theorem} \label{p5thm:varphi_differentiable}
Let $\bar{u} \in \varphi^{-1}(0)$ such that \eqref{p5eq:P2_problem} has a unique solution $\bar{\alpha} \in \Delta_k$ with $\bar{\alpha}_i > 0$ for all $i \in \{1,...,k\}$. Let \eqref{p5eq:P2_problem} be uniquely solvable in a neighborhood of $\bar{u}$. Then there is an open set $U \subseteq \mathbb{R}^n$ with $\bar{u} \in U$ such that $\varphi|_U$ is continuously differentiable.
\end{theorem}
\begin{proof}
See Appendix~\ref{p5app:ProofPhiDiff}.
\end{proof}
For a standard continuation approach, we also have to show that $\varphi^{-1}(0)$ is a manifold. By the Level Set Theorem (cf.~\cite{L2012}, Corollary 5.14), to properly show that $\varphi^{-1}(0)$ is a manifold in a neighborhood of some $\bar{u} \in \varphi^{-1}(0)$, we would have to show that $D\varphi|_U(\bar{u}) \neq 0$ (cf.~\eqref{p5eq:D_phi_expl}). From the theoretical point of view, this poses a problem as there is no obvious way to achieve this. In practice however, we can test this by checking if the norm of $D\varphi|_U(\bar{u})$ is below a certain threshold. If this is the case, and if $\varphi^{-1}(0)$ is indeed not a manifold, we again have to consider all neighboring boxes as tangent boxes as in strategy 1. Otherwise, if $D\varphi|_U(\bar{u}) \neq 0$, we can compute the tangent space $T_{\bar{u}}$ of $\varphi^{-1}(0)$ at $\bar{u}$ via
\begin{equation*}
T_{\bar{u}} = ker(D \varphi(\bar{u})).
\end{equation*}
Finally, in analogy to \eqref{p5eq:KKT_box_test} and \eqref{p5eq:KKTred_box_test}, we will use the following problem to test if a box $B$ contains part of $\partial P_2^r$:
\begin{equation} \label{p5eq:bdryP2r_test}
\min_{u \in B} \varphi(u)^2. \tag{$\partial \epsilon$PC-Box}
\end{equation}
The resulting continuation method is presented in Algorithm \ref{p5algo:boxcon_boundary}.
\begin{remark} \label{p5rem:S2_subproblem}
\begin{enumerate}
\item Since for every evaluation of $\varphi$ the solution of the quadratic problem \eqref{p5eq:P2_problem} has to be computed, \eqref{p5eq:bdryP2r_test} is significantly more difficult to solve than \eqref{p5eq:KKTred_box_test}. Additionally, we are looking for the points $u$ where $\varphi(u) = 0$, i.e., where the problem \eqref{p5eq:P2_problem} is not positive definite. This increased difficulty of Strategy 2 is compensated by the fact that far fewer boxes have to be checked with \eqref{p5eq:bdryP2r_test} than with \eqref{p5eq:KKTred_box_test} in Strategy 1.
\item When all $\epsilon_i = \bar{\epsilon}$ are equal, $\varphi(u) = -\bar{\epsilon}^2$ for all $u \in P_c^r$, i.e., $\varphi$ is constant on the Pareto critical set $P_c^r$. This means that local solvers may fail to find a minimum of $\varphi$ when the box $B$ in \eqref{p5eq:bdryP2r_test} has a nonempty intersection with $P^r$. An obvious but expensive way to circumvent this problem is to start the local solver multiple times with different initial points. Alternatively, one can use sufficient conditions for a box $B$ containing part of $\varphi^{-1}(0)$ before actually solving \eqref{p5eq:bdryP2r_test}. For example, by the intermediate value theorem, if there are two points in $B$ where $\varphi$ has different sign, we immediately know that $\varphi(u) = 0$ for some $u \in B$. (But note that for this method, we still need to find a point in $\varphi^{-1}(0) \cap B$ to be able to calculate the tangent space of $\varphi^{-1}(0)$).
\item In practice, error bounds which are zero can cause problems for the stability of Strategy 2. For example, in Figure \ref{p5fig:simple_example}(b), the width of $P_2^r$ becomes arbitrarily small near $(1,1)^\top$. As a result, Strategy 2 may jump between different parts of the boundary and thus miss certain parts. Additionally, since the boundary of $P_2^r$ typically intersects the Pareto critical set $P_c$ in this case, \eqref{p5eq:bdryP2r_test} may be difficult to solve (as in 2.). Thus, in practice, one should use error bounds that are slightly larger than zero, even if the corresponding gradients are exact.
\end{enumerate}
$\blacksquare$
\end{remark}
\begin{algorithm}
\caption{Strategy 2: Boundary-Continuation Algorithm for Inexact Gradients}
\label{p5algo:boxcon_boundary}
\begin{algorithmic}[1]
\item[] \hspace{-\algorithmicindent} Given: Radius $r > 0$ of boxes.
\State Choose an initial point $u_0 \in \partial P_2^r$ and initialize $\mathcal{B} = \{ B(u_0,r) \}$ and a queue $Q = \{ u_0 \}$.
\While{$Q \neq \emptyset$}
\State Remove the first element $\bar{u}$ from $Q$.
\State If $\left\lVert D \varphi(\bar{u}) \right\rVert_2$ is small set $T = \mathbb{R}^n$. Otherwise, compute the tangent space
\item[] \hspace{\algorithmicindent}$T = ker(D \varphi(\bar{u}))$.
\item[]
\item[] \hskip\algorithmicindent \textbf{Predictor:}
\State Find all neighboring boxes of $B(\bar{u},r)$ that have a nonempty intersection
\item[] \hskip\algorithmicindent with $\bar{u} + T$ and have not been considered before, i.e.,
\begin{equation*}
\mathcal{B}'(\bar{u},r) = \{ B' \in \mathcal{B}(r) : B' \cap B(\bar{u},r) \neq \emptyset, \ B' \cap \bar{u} + T \neq \emptyset \} \setminus \mathcal{B}.
\end{equation*}
\item[] \hskip\algorithmicindent \textbf{Corrector:}
\For{$B' \in \mathcal{B}'(\bar{u},r)$}
\State Solve \eqref{p5eq:bdryP2r_test} for $B'$. Let $\theta(B')$ be the optimal value and $u'$ be the
\item[]\hspace{\algorithmicindent}\hspace{\algorithmicindent}solution.
\If{$\theta(B') = 0$}
\State Add $u'$ to $Q$ and $B'$ to $\mathcal{B}$.
\EndIf
\EndFor
\EndWhile
\end{algorithmic}
\end{algorithm}
\subsection{Globalization approach} \label{p5sec:globalization}
Note that all algorithms presented in this section so far approximate either $P_c$, $P_2^r$ or $\partial P_2^r$ by starting in an initial point $u_0$ and then locally exploring in all (tangent) directions. Thus, if the set we want to approximate is disconnected, we can only compute the connected component that contains $u_0$. In the following, we will describe how we can solve this problem, i.e., how our methods can be globalized.
As mentioned earlier, an advantage of using boxes in the continuation method instead of points is the fact that it is easy to detect whether a region has already been explored. In particular, this allows us to start the continuation in multiple initial points at the same time, by simply adding all of them to the queue $Q$ in step 1 of Algorithms \ref{p5algo:boxcon_eps} or \ref{p5algo:boxcon_boundary} (and initializing the covering $\mathcal{B}$ with the corresponding boxes). As a result, to globalize our methods, we merely have to find an initial set $U_0$ of points such that the intersection of $U_0$ with each connected component is nonempty.
For obtaining an initial set, we make use of the optimization problems that verify if a box contains part of the set we want to approximate, i.e., the problems \eqref{p5eq:KKT_box_test}, \eqref{p5eq:KKTred_box_test} and \eqref{p5eq:bdryP2r_test}. The idea is to consider a box covering as in \eqref{p5eq:def_boxes} with large radius $R$ and then simply test each box for relevant points using these problems. Let $B_0$ be a compact superset of the set that we want to approximate (i.e., of $P_c$, $P_2^r$ or $\partial P_2^r$), e.g., a large outer box. For ease of notation, we assume that $B_0$ is a union of boxes in $\mathcal{B}(R)$. For the case of the Pareto critical set $P_c$, i.e., the globalization of the exact continuation method, the resulting method is presented in Algorithm \ref{p5algo:globalization}. The corresponding globalization methods for Algorithm \ref{p5algo:boxcon_eps} and \ref{p5algo:boxcon_boundary} are obtained by replacing \eqref{p5eq:KKT_box_test} in step 3 by \eqref{p5eq:KKTred_box_test} and \eqref{p5eq:bdryP2r_test}, respectively.
\begin{algorithm}
\caption{Global Initialization}
\label{p5algo:globalization}
\begin{algorithmic}[1]
\item[] \hspace{-\algorithmicindent} Given: Outer box $B_0$, Radius $R > 0$ of boxes.
\State Initialize $U_0 = \emptyset$.
\For{$B \in \mathcal{B}(R)$ with $B \cap B_0 \neq \emptyset$}
\State Solve \eqref{p5eq:KKT_box_test} for $B$. Let $\theta(B)$ be the optimal value and $\bar{u}$ be the solution.
\If{$\theta(B) = 0$}
\State Add $\bar{u}$ to $U_0$.
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
The radius $R$ has to be chosen such that for each connected component, there is at least one box in our covering that only has an intersection with the desired component. In theory, $R$ can obviously become very small if two different connected components are very close to each other. In this case, Algorithm \ref{p5algo:globalization} becomes infeasible to use, as the number of boxes that have to be tested becomes too large. In practice however, the components are often sufficiently far apart such that a large radius is sufficient and only few boxes have to be considered.
For the globalization of the exact continuation method and Algorithm \ref{p5algo:boxcon_eps}, we only have to take the non-connectivity of $P_c$ and $P_2^r$ into account. For Algorithm \ref{p5algo:boxcon_boundary}, an additional problem may arise since the boundary $\partial P_2^r$ does not necessarily need to be smooth. Non-smoothness of $\partial P_2^r$ is caused by points in which $\varphi$ is not differentiable. (By Theorem \ref{p5thm:varphi_differentiable}, these are points where the solution of \eqref{p5eq:bdryP2r_test} is not unique.) In these points, $\partial P_2^r$ does not posses a tangent space, and our method will be unable to continue. As a result, we have to ensure in the initialization of Algorithm \ref{p5algo:boxcon_boundary} that we choose an initial point in $U_0$ on each smooth component of $\partial P_2^r$. Visually, these can be thought of as the faces of $P_2^r$.
We conclude this section with some remarks on the practical use of Algorithm \ref{p5algo:globalization}.
\begin{remark}
\begin{enumerate}
\item For MOPs with a high-dimensional variable space, Algorithm \ref{p5algo:globalization} quickly becomes infeasible due to the exponential growth of the number of boxes in $\mathcal{B}(R)$. For these cases, an initialization based on points instead of boxes should be used, for example by applying methods from global optimization to modified versions of \eqref{p5eq:KKT_box_test}, \eqref{p5eq:KKTred_box_test} and \eqref{p5eq:bdryP2r_test}, where $u$ is not constrained to a box $B$.
\item Instead of directly looping over all boxes in step 2 of Algorithm \ref{p5algo:globalization}, in some cases it might be more beneficial to first execute a few steps of the subdivision algorithm (cf.~\cite{DSH2005}) to quickly discard boxes that are far away from the Pareto critical set.
\end{enumerate}
$\blacksquare$
\end{remark}
\section{Multiobjective optimization of an elliptic PDE using the RB method}
\label{p5sec:MultiobjectiveParameterOptimizationWithRB}
In this section we will present a multiobjective (parameter) optimization problem of an elliptic advection-diffusion-reaction equation and show how the reduced basis method can be applied in view of the continuation method for inexact gradients from Section \ref{p5subsec:ContinuationMethodInexact} (see Algorithms \ref{p5algo:boxcon_eps} and \ref{p5algo:boxcon_boundary}).
\subsection{Multiobjective optimization of an elliptic PDE}
\label{p5subsec:MultiobjectiveParameterOptimization}
Given a domain $\Omega \subset \mathbb{R}^d$, $d \in \{2,3\}$, we consider the problem
\begin{align}
\min_{y,u} \; \mathcal{J}(y,u) := \left( \begin{array}{c} \frac{1}{2} \left\Vert y - y^1_d \right\Vert_{L^2(\Omega)}^2 \\
\vdots \\
\frac{1}{2} \left\Vert y - y^{k-1}_d \right\Vert_{L^2(\Omega)}^2 \\
\frac{1}{2} \left\Vert u \right\Vert_{\mathbb{R}^{m}}^2
\end{array} \right) \tag{MPOP} \label{p5eq:MPOP}
\end{align}
s.t.
\begin{align}
\begin{array}{r l l}
- \sum_{i=1}^{m'} \kappa_i \chi_{\Omega_i}(x) \Delta y(x) + c \, b(x) \cdot \nabla y(x) + r \, y(x) & = f(x) &\text{for } x \in \Omega, \\
\frac{\partial y}{\partial \eta} (x) & = 0 &\text{for } x \in \partial\Omega,
\end{array} \tag{EPDE} \label{p5eq:EllipticPDE}
\end{align}
and the bilateral box constraints
\begin{align}
u_a \leq u \leq u_b, \tag{BC} \label{p5eq:BoxConstraints}
\end{align}
where $u = (u_1,\ldots,u_{m}) = (\kappa_1,\ldots,\kappa_{m'},c,r) \in \mathbb{R}^m$ is the parameter of dimension $m := m' + 2$, $U_{\textsl{ad}} := \{ u \in \mathbb{R}^m \mid u_a \leq u \leq u_b \}$ is the admissible parameter set, and $y \in L^2(\Omega) =: H$ is the state variable. \\
The domain $\Omega$ is divided into $m'$ pairwise disjoint subdomains $\Omega = \Omega_1 \dot\cup \ldots \dot\cup \, \Omega_{m'}$, such that $\kappa_i$ is the diffusion coefficient on $\Omega_i$. The vector field $b \in L^{\infty}(\Omega,\mathbb{R}^d)$ is the given advection, whose strength and orientation can be controlled by the parameter $c \in \mathbb{R}$. Moreover, the reaction coefficient is given by the parameter $r > 0$, and $f \in H$ is the inhomogeneity on the right-hand side of the equation. On the boundary we impose homogeneous Neumann boundary conditions. \\
The cost functions $\mathcal{J}_1,\ldots,\mathcal{J}_{k-1} : H \times U_{\textsl{ad}} \to \mathbb{R}^k$ are of tracking type with respect to the desired states $y_d^1,\ldots,y_d^{k-1} \in H$, and the cost function $\mathcal{J}_k : H \times U_{\textsl{ad}} \to \mathbb{R}^k$ measures the parameter cost. \\
Setting $V := H^1(\Omega)$ and using the parameter-dependent bilinear form $a(u;\cdot,\cdot): V \times V \to \mathbb{R}$ defined by
\begin{align*}
a(u,\varphi,\psi) := & \sum_{i=1}^m u_i a_i(\varphi,\psi) \\
:= & \sum_{i=1}^{m'} \kappa_i \int_{\Omega_i} \nabla \varphi(x) \cdot \nabla \psi(x) \, dx + c \int_\Omega b(x) \cdot \nabla \varphi(x) \psi(x) \, dx \\
& + r \int_{\Omega} \varphi(x) \psi(x) \, dx,
\end{align*}
for all $u \in U_{\textsl{ad}}$ and $\varphi,\psi \in V$, and the linear functional $F: V \to \mathbb{R}$ given by $F(\varphi) := \langle f , \varphi \rangle_H$ for all $\varphi \in V$,
we can write \eqref{p5eq:EllipticPDE} in its weak formulation as: Find $y \in V$ such that
\begin{align}
a(u;y,\varphi) = F(\varphi) \quad \text{for all } \varphi \in V \label{p5eq:WeakFormulationPDE}
\end{align}
is satisfied.
It is possible to show the unique solvability of \eqref{p5eq:WeakFormulationPDE} under some conditions on the parameter $u$.
\begin{theorem}
\label{p5thm:UniqueSolvabilityPDE}
There are $\kappa_{\textsl{min}} \in (0,\infty)^{m'}$, $c_{\textsl{min}}, c_{\textsl{max}} \in \mathbb{R}$ with $c_{\textsl{min}} < c_{\textsl{max}}$ and $r_{\textsl{min}} \in (0,\infty)$ such that \eqref{p5eq:WeakFormulationPDE} has a unique solution $y(u) \in V$ for every parameter $u = \left(\kappa,c,r\right) \in \mathbb{R}^m$ with $\kappa > \kappa_{\textsl{min}}$, $c_{\textsl{min}} < c < c_{\textsl{max}}$ and $r > r_{\textsl{min}}$.
\end{theorem}
\begin{proof}
It is straightforward to show that for all parameters $u \in \mathbb{R}^m$ the bilinear form $a(u;\cdot,\cdot)$ and the linear functional $F$ are continuous, and that there are $\kappa_{\textsl{min}} \in (0,\infty)^{m'}$, $c_{\textsl{min}}, c_{\textsl{max}} \in \mathbb{R}$ with $c_{\textsl{min}} < c_{\textsl{max}}$ and $r_{\textsl{min}} \in (0,\infty)$ such that $a(u;\cdot,\cdot)$ is coercive for all $u = \left(\kappa,c,r\right) \in \mathbb{R}^m$ with $\kappa > \kappa_{\textsl{min}}$, $c_{\textsl{min}} < c < c_{\textsl{max}}$ and $r > r_{\textsl{min}}$. Now the Lax-Milgram Theorem can be applied to show the unique solvability of \eqref{p5eq:WeakFormulationPDE}.
\end{proof}
With Theorem \ref{p5thm:UniqueSolvabilityPDE} in mind we can introduce the solution operator of the elliptic PDE.
\begin{definition}
\label{p5def:SolutionOperatorStateEquation}
Define the set $U_{\textsl{eq}} := (\kappa_{\textsl{min}},\infty) \times (c_{\textsl{min}}, c_{\textsl{max}}) \times (r_{\textsl{min}},\infty)$ with the constants from Theorem \ref{p5thm:UniqueSolvabilityPDE}. Let $\mathcal{S}: U_{\textsl{eq}} \to V \hookrightarrow H$ be defined as the solution operator of \eqref{p5eq:WeakFormulationPDE}, i.e., the function $y := \mathcal{S}(u)$ solves the weak formulation \eqref{p5eq:WeakFormulationPDE} for any parameter $u \in U_{\textsl{eq}}$.
\end{definition}
\begin{remark}
In the following we suppose that it holds $U_{\textsl{ad}} \subset U_{\textsl{eq}}$.
$\blacksquare$
\end{remark}
Using the explicit dependence of the state $y$ on the parameter $u$ for all $u \in U_{\textsl{ad}}$, the essential cost functions $J_1,\ldots,J_k: U_{\textsl{ad}} \to \mathbb{R}$ can be defined.
\begin{definition}
\label{p5def:ReducedCostFunctions}
For any $i \in \{1,\ldots,k\}$ let the essential cost function $J_i:U_{\textsl{ad}} \to \mathbb{R}$ be given by $J_i(u) := \mathcal{J}_i(\mathcal{S}(u),u)$ for all $u \in U_{\textsl{ad}}$.
\end{definition}
For applying the continuation method from Section \ref{p5sec:ContinuationMethodWithInexactness}, which is based on Theorem \ref{p5thm:hillermeier}, to solve this multiobjective parameter optimization problem, the cost functions $J_1,\ldots,J_k$ need to be twice continuously differentiable. This is the statement of the next lemma.
\begin{lemma}
\label{p5lem:CostFunctionsTwiceContDiff}
The cost functions $J_1,\ldots,J_k$ are twice continuously differentiable.
\end{lemma}
\begin{proof}
It is clear that the cost function $J_k$ is twice continuously differentiable. Furthermore, it is possible to show that the solution operator $\mathcal{S}$ of \eqref{p5eq:WeakFormulationPDE} is twice continuously differentiable (this can be shown by rewriting \eqref{p5eq:WeakFormulationPDE} in the form $e(y,u) = 0$ and then using the implicit function theorem, cf.~\cite[Section 1.6]{Hinze2009}). From this it immediately follows that the cost functions $J_1,\ldots,J_{k-1}$ are twice continuously differentiable as well.
\end{proof}
For later use, we need an explicit formula for the gradients $\nabla J_1,\ldots,\nabla J_k$. Therefore, we introduce the so-called adjoint equation for all $i \in \{1,\ldots,k-1\}$: Find $p \in V$ such that it holds
\begin{align}
a(u;\varphi,p) = \langle y_d^i - \mathcal{S}(u), \varphi \rangle_H \quad \text{for all } \varphi \in V. \label{p5eq:AdjointEquation}
\end{align}
With the same arguments as in Theorem \ref{p5thm:UniqueSolvabilityPDE} it is possible to show that \eqref{p5eq:AdjointEquation} has a unique solution for all $u \in U_{\textsl{eq}}$.
\begin{definition}
\label{p5def:SolutionOperatorAdjointEquation}
Denote by $\mathcal{A}_i: U_{\textsl{eq}} \to V \hookrightarrow H$ the solution operator of the adjoint equation \eqref{p5eq:AdjointEquation} for all $i \in \{1,\ldots,k-1\}$.
\end{definition}
Now a small computation shows that
\begin{align*}
J_i'(u) h = \langle \mathcal{S}(u) - y_d^i , \mathcal{S}'(u)h \rangle_H = \partial_u a(u;\mathcal{S}(u),\mathcal{A}_i(u))h,
\end{align*}
which yields
\begin{align}
\nabla J_i(u) = \left( \begin{array}{c}
\partial_{u} a(u;\mathcal{S}(u),\mathcal{A}_i(u)) e_1 \\
\vdots \\
\partial_{u} a(u;\mathcal{S}(u),\mathcal{A}_i(u)) e_m
\end{array} \right) = \left( \begin{array}{c}
a_1(\mathcal{S}(u),\mathcal{A}_i(u)) \\
\vdots \\
a_m(\mathcal{S}(u),\mathcal{A}_i(u))
\end{array} \right) \label{p5eq:RepresentationGradient}
\end{align} for all $i \in \{1,\ldots,k-1\}$. Lastly, it is obvious that $\nabla J_k(u) = u$.
\subsection{The reduced basis method}
\label{p5subsec:RB}
For computing the Pareto critical set of the problem \eqref{p5eq:MPOP} by the exact continuation method introduced in Section \ref{p5subsec:ExactContinuation}, the problem \eqref{p5eq:KKT_box_test} has to be solved numerous times. However, already one gradient evaluation of all cost functions $\nabla J_1(u),\ldots,\nabla J_k(u)$ involves the solution of one state and $k-1$ adjoint equations. Thus, using a finite element discretization for the weak formulations \eqref{p5eq:WeakFormulationPDE} and \eqref{p5eq:AdjointEquation}, which leads to large linear equation systems, is numerically very costly and time consuming. Therefore, the use of \textit{reduced-order modelling} (ROM) is a common tool to lower the computational costs. \\ The idea of ROM is to use a low-dimensional subspace $V^r \subset V$ as a surrogate for the infinite-dimensional space $V$ in the weak formulations \eqref{p5eq:WeakFormulationPDE} and \eqref{p5eq:AdjointEquation}. Given a finite-dimensional reduced-order space $V^r \subset V$, the reduced-order state equation reads: Find $y^r \in V^r$ such that
\begin{align}
a(u;y^r,\varphi) = F(\varphi) \quad \text{for all } \varphi \in V^r \label{p5eq:WeakFormulationRBStateEquation}
\end{align}
is satisfied. \\
With the same arguments as in Theorem \ref{p5thm:UniqueSolvabilityPDE} it can be shown that \eqref{p5eq:WeakFormulationRBStateEquation} has a unique solution for all $u \in U_{\textsl{eq}}$. Therefore, we can follow the procedure of Section \ref{p5subsec:MultiobjectiveParameterOptimization} and introduce the solution operator $\mathcal{S}^r: U_{\textsl{eq}} \to V^r \subset V \hookrightarrow H$ of the ROM state equation \eqref{p5eq:WeakFormulationRBStateEquation} and consequently the ROM essential cost functions $J_1^r,\ldots,J_k^r$, which are defined by $J_i^r(u) := \mathcal{J}_i(\mathcal{S}^r(u),u)$ for all $u \in U_{\textsl{ad}}$ and all $i \in \{1,\ldots,k\}$. Again, it can be shown that the functions $J_1^r,\ldots,J_k^r$ are twice continuously differentiable so that they fit into the framework of Theorem \ref{p5thm:hillermeier}. The gradient of the cost functions can also be displayed by the reduced-order adjoint equations
\begin{align}
a(u;\varphi,p^r) = \langle y_d^i - \mathcal{S}^r(u), \varphi \rangle_H \quad \text{for all } \varphi \in V^r, \label{p5eq:RBAdjointEquation}
\end{align}
for all $i \in \{1,\ldots,k-1\}$, whose solution operator we denote by $\mathcal{A}_i^r : U_{\textsl{eq}} \to V^r \subset V \hookrightarrow H$. With this definition it holds
\begin{align}
\nabla J_i^r(u) = \left( \begin{array}{c}
\partial_{u} a(u;\mathcal{S}^r(u),\mathcal{A}^r_i(u)) e_1 \\
\vdots \\
\partial_{u} a(u;\mathcal{S}^r(u),\mathcal{A}^r_i(u)) e_m
\end{array} \right) = \left( \begin{array}{c}
a_1(\mathcal{S}^r(u),\mathcal{A}^r_i(u)) \\
\vdots \\
a_m(\mathcal{S}^r(u),\mathcal{A}^r_i(u))
\end{array} \right) \label{p5eq:RepresentationGradientRB}
\end{align}
for all $i \in \{1,\ldots,k-1\}$. Moreover, we have $\nabla J_k^r(u) = u = \nabla J_k(u)$.
In this paper we use a particular model-order reduction technique, namely the \textit{reduced basis} (RB) method (see e.g. \cite{Rozza2007,Hesthaven2016,Quarteroni2016}). In the RB method the snapshot space $V^r$ is spanned by solutions of the state equation and the adjoint equations to different parameter values $u \in U_{\textsl{ad}}$. The reduced basis is then given by an orthonormal basis $(\Phi_1,\ldots,\Phi_N)$ of the space $V^r$. \\ By using the RB method we introduce an error in the state equation, which transfers to the cost functions, its gradients and eventually to the Pareto critical set, which we want to compute. In Section \ref{p5subsec:ContinuationMethodInexact} two strategies were presented to deal with the inflicted inexactness in the gradients of the multiobjective optimization problem. Both are based on the estimates \eqref{p5eq:error_bounds} for the errors in the gradients of the cost functions. Thus, when applying the RB method we need to ensure these estimates. This is done by using the well-known greedy algorithm (cf.~\cite{Buffa2012}). Given a sufficiently fine finite parameter training set $\mathcal{P} \subset U_{\textsl{ad}}$ new solution snapshots are computed until the error in the gradients of all cost functions is smaller than the predefined error tolerance for all parameters in $\mathcal{P}$. The parameter for the new snapshots is thereby chosen as the one for which the error in the gradient is the largest. The procedure is summarized in Algorithm \ref{p5algo:GreedyAdjoint}.
\begin{algorithm}
\caption{Greedy Algorithm}
\label{p5algo:GreedyAdjoint}
\begin{algorithmic}[1]
\item[] \hspace{-\algorithmicindent} Given: Parameter set $\mathcal{P} \subset U_{\textsl{ad}}$, greedy tolerances $\varepsilon_1,\ldots,\varepsilon_k > 0$.
\State Choose $u \in \mathcal{P}$, compute $\mathcal{S}(u)$,$\mathcal{A}_1(u),\ldots,\mathcal{A}_{k-1}(u)$.
\State Set $V^r = \textsl{span} \{ \mathcal{S}(u), \mathcal{A}_1(u),\ldots,\mathcal{A}_{k-1}(u) \}$ and compute the reduced basis by orthonormalization.
\While{$\max_{u \in \mathcal{P}} \max_{i \in \{1,\ldots,k-1\}} \left\lVert \nabla J_i(u) - \nabla J_i^r(u) \right\rVert_2 > \epsilon_i$}
\State Choose $(\bar{u},i) = \arg\max_{u \in \mathcal{P}, \, i \in \{1,\ldots,k-1\}} \left\lVert \nabla J_i(u) - \nabla J_i^r(u) \right\rVert_2$.
\State Compute $\mathcal{S}(\bar{u})$ and $\mathcal{A}_i(\bar{u})$.
\State Set $V^r = \textsl{span} \left\{ V^r \cup \{ \mathcal{S}(\bar{u}), \mathcal{A}_i(\bar{u}) \} \right\}$ and compute the reduced basis by
\item[] \hskip\algorithmicindent orthonormalization.
\EndWhile
\end{algorithmic}
\end{algorithm}
\subsection{Error estimation for the gradients}
In the greedy procedure in Algorithm \ref{p5algo:GreedyAdjoint}, the error between the full-order and the reduced-order gradients has to be evaluated. There are two strategies to do so.
\begin{enumerate}
\item The full-order gradients are computed and stored at the beginning of the greedy procedure. Therefore, in each greedy iteration, only the reduced-order gradients have to be computed and the error can be easily evaluated. Of course, this implies large computational costs at the beginning of the greedy procedure. This method is called strong greedy algorithm (cf.~\cite{Haasdonk2013,Buffa2012}).
\item An a-posteriori error estimator for the errors in the gradient is used, which can be efficiently evaluated. This results in computational costs for the greedy algorithm, which only depend on the reduced-order dimension $N$.
\end{enumerate}
To be able to follow the second strategy we introduce a rigorous a-posteriori error estimator for the error in the gradient of the cost functions. \\
Using the gradient representations \eqref{p5eq:RepresentationGradient} and \eqref{p5eq:RepresentationGradientRB} we can write for $i \in \{1,\ldots,k-1\}$
\begin{align*}
\left\lVert \nabla J_i(u) - \nabla J^r_i(u) \right\rVert_2^2 = \sum_{j=1}^{m} \left| a_j(\mathcal{S}(u),\mathcal{A}_i(u)) - a_j(\mathcal{S}^r(u),\mathcal{A}^r_i(u)) \right|^2. \end{align*}
Due to the bilinearity and the continuity of $a_1,\ldots,a_m$ and the triangle inequality, we can further write
\begin{align}
& \left| a_j(\mathcal{S}(u),\mathcal{A}_i(u)) - a_j(\mathcal{S}^r(u),\mathcal{A}^r_i(u)) \right| \nonumber \\
\leq & \left| a_j(\mathcal{S}(u) - \mathcal{S}^r(u),\mathcal{A}^r_i(u)) \right| + \left| a_j(\mathcal{S}(u) - \mathcal{S}^r(u),\mathcal{A}_i(u)-\mathcal{A}^r_i(u)) \right| \nonumber \\
& + \left| a_j(\mathcal{S}^r(u),\mathcal{A}_i(u)-\mathcal{A}^r_i(u)) \right| \label{p5eq:GradientEstimateTriangleInequality} \\
\leq & C_j \left( \left\Vert \mathcal{S}(u) - \mathcal{S}^r(u) \right\Vert_V \left\Vert \mathcal{A}^r_i(u) \right\Vert_V + \left\Vert \mathcal{S}(u) - \mathcal{S}^r(u) \right\Vert_V \left\Vert \mathcal{A}_i(u) - \mathcal{A}^r_i(u) \right\Vert_V \right. \nonumber \\
& \left. \qquad + \left\Vert \mathcal{S}^r(u) \right\Vert_V \left\Vert \mathcal{A}_i(u) - \mathcal{A}^r_i(u) \right\Vert_V \right) \label{p5eq:GradientEstimateFinal}
\end{align}
for all $j \in \{1,\ldots,m\}$. \\
Therefore, we need a-posteriori error estimators for the state and the adjoint equations in order to be able to estimate the approximation error induced in the gradients. To this end, we use the following well-known estimators (cf. \cite{Rozza2007}).
\begin{align*}
\left\Vert \mathcal{S}(u) - \mathcal{S}^r(u) \right\Vert_V & \leq \frac{\left\Vert r_\mathcal{S}(u) \right\Vert_{V'}}{\alpha(u)} =: \Delta_{\mathcal{S}}(u), \\
\left\Vert \mathcal{A}_i(u) - \mathcal{A}^r_i(u) \right\Vert_V & \leq \frac{\left\Vert r_{\mathcal{A}_i}(u) \right\Vert_{V'}}{\alpha(u)} + \Delta_{\mathcal{S}}(u) =: \Delta_{\mathcal{A}_i}(u),
\end{align*}
where the residuals $r_\mathcal{S}(u)$ and $r_{\mathcal{A}_i}(u)$ are given by
\begin{align*}
\langle r_\mathcal{S}(u) , \varphi \rangle_{V',V} & := F(\varphi) - a(u;\mathcal{S}^r(u),\varphi) \quad & \text{for all } \varphi \in V, \\
\langle r_{\mathcal{A}_i}(u) , \varphi \rangle_{V',V} & := \langle y_d^i - \mathcal{S}^r(u) , \varphi \rangle_H - a(u;\varphi,\mathcal{A}_i^r(u)) \quad & \text{for all } \varphi \in V. \\
\end{align*}
For methods on how to estimate $\alpha(u)$ and to evaluate the terms $\left\Vert r_\mathcal{S}(u) \right\Vert_{V'}$ and $\left\Vert r_{\mathcal{A}_i}(u) \right\Vert_{V'}$ efficiently, we refer for example to \cite{Rozza2007}.
\begin{remark}
Since $J_k = J_k^r$, the gradients of the two functions also coincide, so that the $\nabla J_k$ is approximated exactly by $\nabla J_k^r$.
$\blacksquare$
\end{remark}
\section{Numerical results} \label{p5sec:NumericalResults}
In this section we will numerically investigate the application of the continuation method presented in Section \ref{p5sec:ContinuationMethodWithInexactness} to the PDE-constrained multiobjective optimization problem using the reduced basis method in Section \ref{p5sec:MultiobjectiveParameterOptimizationWithRB}. \\
For the discretization of the state and adjoint equations we used linear finite elements with 714 degrees of freedom.
\subsection{Generation of the reduced basis}
For investigating the generation of the reduced basis by the greedy algorithm in Algorithm \ref{p5algo:GreedyAdjoint}, we consider the MPOP
\begin{align}
\left( \begin{array}{c}
J_1(u) \\
J_2(u) \end{array} \right) = \left( \begin{array}{c} \frac{1}{2} \left\Vert \mathcal{S}(u) - y^1_d \right\Vert_H^2 \\
\frac{1}{2} \left\Vert u \right\Vert_{\mathbb{R}^{4}}^2
\end{array} \right) \label{p5eq:MPOPExampleRBGeneration}
\end{align}
with $u = (\kappa_1,\kappa_2,c,r)$, $\Omega_1 = (0,1) \times (0,0.5)$, $\Omega_2 = (0,1) \times (0.5,1)$, and the admissible parameter set
\[
U_{ad} = \{ u = (\kappa_1,\kappa_2,c,r) \in \mathbb{R}^4 \mid 0.2 \leq \kappa_i \leq 5 \, (i = 1,2), \; c = 0, \; r = 0.5 \}.
\]
The reason for setting $c = 0$ in this example is that the coercivity constant $\alpha(u)$ of the bilinear form $a(u;\cdot,\cdot)$ is explicitly given by $\alpha(u) = \min \{ \kappa_1,\kappa_2,r \}$ for all $u \in U_{ad}$, so that we expect a good efficiency of the error estimator of both the state and adjoint equations.
\begin{figure}
\caption{Overestimations for $1000$ randomly selected parameter values}
\label{p5fig:RBGeneration_Overestimation}
\end{figure}
This is verified by the results shown in Figure \ref{p5fig:RBGeneration_Overestimation} (a), where the efficiency of the error estimator for both equations is shown for a given reduced basis for 1000 randomly chosen parameter values. However, the resulting efficiency of the error estimator for the error in the gradient is between $10^3$ and $10^6$ (see Figure \ref{p5fig:RBGeneration_Overestimation} (b)) and thus not well suited for a greedy procedure, which depends on a good error estimation. The huge overestimation of the error estimator is mainly due to the use of the triangle inequality \eqref{p5eq:GradientEstimateTriangleInequality} and the continuity estimates \eqref{p5eq:GradientEstimateFinal}, as can be seen in Figure \ref{p5fig:RBGeneration_Overestimation} (b). \\
\begin{table}[h!]
\caption{Number of basis functions for different error bounds} \centering
\begin{tabular}{|l || c | c|} \hline Error bound & Strong Greedy & Error Estimate \\
\hline $\varepsilon = 1e-6$ & 24 & 56 \\ $\varepsilon = 1e-5$ & 20 & 50 \\ $\varepsilon = 1e-4$ & 16 & 40 \\ $\varepsilon = 1e-3$ & 12 & 32 \\ $\varepsilon = 1e-2$ & 12 & 26 \\ $\varepsilon = 1e-1$ & 10 & 20 \\ \hline \end{tabular}
\label{p5tab:RBGenerationComparison} \end{table}
Compared to the strong greedy algorithm, we can see in Table \ref{p5tab:RBGenerationComparison} that this overestimation results in far more basis elements than actually needed to reach the given error bound. Since we want to investigate the influence of the error bounds in the estimate \eqref{p5eq:error_bounds} on the problem, we want that the estimate \eqref{p5eq:error_bounds} is satisfied sharply by the RB. Therefore, we will not use the error estimator to generate the basis, but instead use the strong greedy algorithm.
\subsection{Application of the continuation methods to an MPOP}
For the numerical investigation of the continuation method applied to a PDE-constrained multiobjective parameter optimization problem together with the use of the reduced basis method, we consider the MPOP
\begin{align}
\left( \begin{array}{c}
J_1(u) \\
J_2(u) \\
J_3(u) \\
J_4(u) \end{array} \right) = \left( \begin{array}{c} \frac{1}{2} \left\Vert \mathcal{S}(u) - \mathcal{S}((0.7,0.8,0.5)) \right\Vert_H^2 \\
\frac{1}{2} \left\Vert \mathcal{S}(u) - \mathcal{S}((2,0.5,0.5)) \right\Vert_H^2 \\
\frac{1}{2} \left\Vert \mathcal{S}(u) - \mathcal{S}((3,-0.5,0.5)) \right\Vert_H^2 \\
\frac{1}{2} \left\Vert u \right\Vert_{\mathbb{R}^{3}}^2
\end{array} \right) \label{p5eq:MPOPExampleNumerical}
\end{align} with $u = (\kappa,c,r)$ and \[ U_{ad} = \{ u = (\kappa,c,r) \in \mathbb{R}^3 \mid 0.5 \leq \kappa \leq 3, \; -1 \leq c \leq 1, \; r = 0.5 \}, \] i.e., the reaction parameter $r$ is a constant so that we only optimize the diffusivity in the whole domain $\Omega$ and the strength and orientation of the advection field $b$. Thus, this can be seen as a problem with two parameters.
As described before, the reduced basis is generated by the strong greedy Algorithm \ref{p5algo:GreedyAdjoint}, where the error bounds $\epsilon_1,\ldots,\epsilon_4$ are chosen in accordance with the estimate \eqref{p5eq:error_bounds}. As a reference, the exact solution of \eqref{p5eq:MPOPExampleNumerical} (via exact continuation and FEM discretization of the weak formulations) is shown in Figure \ref{p5fig:example_FEM_solution}.
\begin{figure}
\caption{The Pareto critical set of \eqref{p5eq:MPOPExampleNumerical}}
\label{p5fig:example_FEM_solution}
\end{figure}
\begin{remark}
Since \eqref{p5eq:MPOPExampleNumerical} is constrained to a box, we have to use a constrained version of the exact continuation method (cf.~\cite{H2001}) to calculate Pareto critical points that lie on the boundaries of \eqref{p5eq:MPOPExampleNumerical}. But note that for this example, all Pareto critical points on the boundary are also Pareto critical if we ignore the constraints. In other words, for each Pareto critical point $\bar{u}$ on the boundary, there is a sequence of Pareto critical point in the interior that converges to $\bar{u}$. By continuity of $DJ$, the gradients of the (active) inequality constraints in the KKT conditions can be ignored. As a result, we can treat \eqref{p5eq:MPOPExampleNumerical} as an unconstrained problem that we only solve in a certain area.
$\blacksquare$
\end{remark}
As a first test, we will compare the time needed to compute the exact solution of \eqref{p5eq:MPOPExampleNumerical} with the time needed for Strategy 1 and 2. For the error bounds we choose $\epsilon = (0.03, 0.03, 0.01, 0.01)$ and for the box radius we choose $r = \frac{3 - 0.5}{2^9} \approx 0.0049$. The results are shown in Figure \ref{p5fig:runtime_compare}.
\begin{figure}
\caption{Results of Strategy 1 (left) and 2 (right) for the MPOP \eqref{p5eq:MPOPExampleNumerical} with $\epsilon = (0.03,0.03,0.01,0.01)$}
\label{p5fig:runtime_compare}
\end{figure}
All three methods were implemented in Matlab. For the solution of the subproblems \eqref{p5eq:KKT_box_test}, \eqref{p5eq:KKTred_box_test} and \eqref{p5eq:bdryP2r_test}, the SQP-Algorithm of \verb+fmincon+ was used. (For increased stability during the continuation, each subproblem where the SQP-Algorithm found an optimal value larger than zero was restarted using the Interior-Point-Method and the Active-Set-Method of \verb+fmincon+). The runtime, number of boxes and number of subproblems needed are shown in Table \ref{p5tab:comparison}.
\begin{table}[h]
\caption{Comparison of the performance of the exact continuation method, Strategy 1 and Strategy 2 for Example \eqref{p5eq:MPOPExampleNumerical}. The number of subproblems is split up in subproblems for the continuation and initialization (cf.~Section \ref{p5sec:globalization})}
\begin{tabular}{ | l | l | l | l |}
\hline
Algorithm & \# Boxes & \# Subproblems & Runtime (in seconds) \\ \hline
Exact cont. & $15916$ & $18721 + 25$ & $17501 s$ \\ \hline
Strategy 1 & $21750$ & $24490 + 25$ & $1426 s$ \\ \hline
Strategy 2 & $899$ & $1027 + 225$ & $276 s$ \\
\hline
\end{tabular}
\label{p5tab:comparison}
\end{table}
When comparing Strategy 1 and Strategy 2, we see that Strategy 2 needs about $20$ times fewer boxes and solutions of subproblems than Strategy 1. This is to be expected, since Strategy 2 only computes a covering of the boundary of $P_2^r$, i.e., of a lower dimensional set. When comparing the actual runtime, Strategy 2 is about $5$ times faster than Strategy 1, since the subproblems in Strategy 2 are more expensive to solve than the ones in Strategy 1 (cf.~Remark \ref{p5rem:S2_subproblem}). Finally, Strategy 2 is about $63$ times faster than the exact continuation method with FEM discretization, illustrating the large increase in efficiency we gain from our approach.
Although it is a lot quicker to use inexact gradients from ROM instead of the exact gradients via FEM, it is important to keep in mind that our methods are computing a superset of the actual Pareto critical set. For example, in Figure \ref{p5fig:runtime_compare}, the right side of the lower connected component is only approximated poorly by $P_2^r$. Therefore, we will now investigate the influence of the error bounds $\epsilon = (\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4)$ on $P_2^r$, by applying Strategy 2 with reduced bases for different values of $\epsilon$. Note that in all our tests we set $\epsilon_4 = 0.01$, although the error in the gradient of the fourth cost function is zero for all parameters. This is done to make the solution of \eqref{p5eq:bdryP2r_test} in line 7 of Algorithm \ref{p5algo:boxcon_boundary} numerically stable (cf.~Remark \ref{p5rem:S2_subproblem}).\\
The results of our experiment can be seen in Figure \ref{p5fig:Strategy2DifferentEpsilon}. Generally, as expected, the boundary $\partial P_2^r$ encloses the Pareto critical set $P_c$ sharper and sharper for decreasing $\epsilon$. Moreover, we observe that it is crucial to choose an $\epsilon$ which is not too large: For the value $\epsilon = (0.1,0.1,0.1,0.01)$ the shape of the boundary $\partial P_2^r$ implies that the set $P_2^r$ is connected, i.e., we lose the topological information that the Pareto critical set actually consists of two connected components. Decreasing $\epsilon$ to $\epsilon = (0.0885,0.0885,0.0885,0.01)$ we are in the limit case in which the boundary $\partial P_2^r$ touches the box constraints at around $(2.3,1)$, so that this is the approximate $\epsilon$ for which we regain the basic topological information of a disconnected Pareto critical set. \\ If we compare the results for $\epsilon = (0.03,0.03,0.03,0.01)$, $\epsilon = (0.03,0.03,0.01,0.01)$ and $\epsilon = (0.03,0.01,0.01,0.01)$, the influence of changing one component of $\epsilon$ becomes obvious. For $\epsilon = (0.03,0.03,0.03,0.01)$ the set $\partial P_2^r$ encloses the set $P_c$ quite sharply at the upper connected component and at the left part of the lower connected component, where the second and third component of the corresponding KKT-multipliers $\alpha$ are small. On the other hand, in the right part of the lower connected component of $P_c$, where the second and third component of the corresponding KKT-multipliers are relatively large, the deviation of $\partial P_2^r$ to $P_c$ is still large. Consequently, first reducing $\epsilon_3$ and then also $\epsilon_2$ from 0.03 to 0.01 leads to a clearly visible sharper enclosing of this part of $P_c$.
\begin{figure}
\caption{Results of Strategy 2 for different values of $\epsilon$}
\label{p5fig:Strategy2DifferentEpsilon}
\end{figure}
\section{Conclusion and outlook}
In this article, we present a way to efficiently solve multiobjective parameter optimization problems of elliptic PDEs by combining the reduced basis method from PDE-constrained optimization with the continuation approach from multiobjective optimization, which computes a box covering of the Pareto critical set. Using the RB method in this setting introduces an error in the objective functions and their gradients that has to be considered when solving the MPOP. To this end, we require that the reduced basis guarantees error bounds for the gradients of the objective functions. These error bounds are then incorporated into the KKT optimality conditions for MOPs to derive a tight superset $P_2^r$ of the actual Pareto (critical) set. This superset can be computed using a straightforward modification of the continuation method for MOPs (Strategy 1). Since $P_2^r$ has the same dimensions as the variable space of the MOP, we afterwards present a second method that only computes the boundary $\partial P_2^r$ of $P_2^r$ (Strategy 2). We do this by showing that $\partial P_2^r$ can be written as the level set of a differentiable mapping, which again enables the use of a continuation approach to compute it. For constructing the reduced basis, we use a greedy procedure which incorporates, and thus ensures, the error bounds for the gradients of the objective functions. \\ Our numerical tests show that the presented a-posteriori error estimator for the error in the gradients is not well-suited for the application in a greedy procedure due to its bad efficiency. Therefore, a strong greedy algorithm is used to build the reduced basis. Concerning the solution of the MPOP we investigate two aspects: First, the runtimes of our methods are compared. In our case, Strategy 1 is about 13 times and Strategy 2 about $63$ times faster than the exact solution of the MPOP (via the classical continuation method with FEM discretization). Second, the influence of the error bound for the gradients of the objective functions is investigated. As expected, a smaller error bound leads to a tighter covering of the Pareto critical set. Moreover, we observe that single components of the error bound strongly influence the tightness of the covering in areas, in which the corresponding components of the KKT-multipliers are large. Thus, by individually adapting the single components of the error bound, we can nicely control the tightness of the covering.
For future work, there are some theoretical and practical aspects that should be investigated further:
\begin{itemize}
\item As mentioned in Remark \ref{p5rem:S2_subproblem}, in certain situations there can be difficulties when solving the problem \eqref{p5eq:bdryP2r_test}. In these situations, specialized methods that take these difficulties into account should be developed and used instead of standard methods for constrained optimization.
\item If the number of objectives of the MPOP is larger than the number of variables, it may be possible to combine our approaches in this article with the hierarchical decomposition of the Pareto critical set presented in \cite{GPD2019}.
\item The development of a more efficient a-posteriori error estimator for the error in the gradients of the objective functions would allow to use it in the greedy procedure. In that way, the expensive strong greedy procedure would be avoided in the offline phase. One way to do so might be the application of localized RB methods, see e.g. \cite{Ohlberger2017}.
\item As explained in the globalization approach in Section \ref{p5sec:globalization}, we have to use multiple initial points to ensure that we find all connected components of $P_2^r$ (and faces of $\partial P_2^r$). Due to the local nature of the continuation method, this approach can potentially be parallelized, increasing the efficiency of our methods even more.
\item If a decision maker is present with a certain preference, it may be worth to steer our continuation method in a direction that results from that preference instead of approximating the complete Pareto set. For the case with exact gradients, this was done in \cite{SCM2019}.
\end{itemize}
\subsection*{Acknowledgment} This research was funded by the DFG Priority Programme 1962 ``Non-smooth and Complementarity-based Distributed Parameter Systems''.
\appendix
\section{Proof of Theorem \ref{p5thm:varphi_differentiable}}\label{p5app:ProofPhiDiff}
To prove Theorem \ref{p5thm:varphi_differentiable}, we first have to investigate some of the properties of the optimization problem \eqref{p5eq:P2_problem}. This problem is quadratic with linear equality and inequality constraints. We will first investigate the uniqueness of the solution in the following lemma.
\begin{lemma} \label{p5lem:alpha_unique_1}
Let $u \in \varphi^{-1}(0)$ and let $\alpha^1$ and $\alpha^2$ be two solutions of \eqref{p5eq:P2_problem} with $\alpha^1 \neq \alpha^2$. Then $\omega(\alpha) = 0$ for all $\alpha \in span( \{ \alpha^1, \alpha^2 \} )$ and
\begin{equation} \label{p5eq:unique_1}
span( \{ \alpha^1, \alpha^2 \} ) \cap ker(DJ(u)^\top) \neq \emptyset.
\end{equation} \end{lemma} \begin{proof}
For $c_1, c_2 \in \mathbb{R} \setminus \{ 0 \}$ we have
\begin{align*}
&\omega(c_1 \alpha^1 + c_2 \alpha^2) \\
&= (c_1 \alpha^1 + c_2 \alpha^2)^\top (DJ^r(u) DJ^r(u)^\top - \epsilon \epsilon^\top) (c_1 \alpha^1 + c_2 \alpha^2) \\
&= c_1^2 \omega(\alpha^1) + 2 ({c_1 \alpha^1}^\top DJ^r(u) DJ^r(u)^\top c_2 \alpha^2 - {c_1 \alpha^1}^\top \epsilon \epsilon^\top c_2 \alpha^2) + c_2^2 \omega(\alpha^2) \\
&= 2 ({c_1 \alpha^1}^\top DJ^r(u) DJ^r(u)^\top c_2 \alpha^2 - {c_1 \alpha^1}^\top \epsilon \epsilon^\top c_2 \alpha^2) \\
&= 2 c_1 c_2 ( (DJ^r(u)^\top \alpha^1)^\top (DJ^r(u)^\top \alpha^2) - (\epsilon^\top \alpha^1) (\epsilon^\top \alpha^2) ).
\end{align*}
From $\omega(\alpha^1) = \omega(\alpha^2) = 0$ it follows that $\epsilon^\top \alpha^1 = \| DJ^r(u)^\top \alpha^1 \|$ and $\epsilon^\top \alpha^2 = \| DJ^r(u)^\top \alpha^2 \|$. Let $\sphericalangle$ be the angle between $DJ^r(u)^\top \alpha^1$ and $DJ^r(u)^\top \alpha^2$. Then
\begin{align} \label{p5eq:omega_add}
&\omega(c_1 \alpha^1 + c_2 \alpha^2) \nonumber \\
&= 2 c_1 c_2 (cos(\sphericalangle) \| DJ^r(u)^\top \alpha^1 \| \| DJ^r(u)^\top \alpha^2 \| - \| DJ^r(u)^\top \alpha^1 \| \| DJ^r(u)^\top \alpha^2 \|) \nonumber \\
&= 2 c_1 c_2 (cos(\sphericalangle) - 1) \| DJ^r(u)^\top \alpha^1 \| \| DJ^r(u)^\top \alpha^2 \|.
\end{align}
Assume $cos(\sphericalangle) \neq 1$ (i.e., $cos(\sphericalangle) - 1 < 0$), $\| DJ^r(u)^\top \alpha^1 \| \neq 0$ and $\| DJ^r(u)^\top \alpha^2 \| \neq 0$. If we choose $c_1 = t$ and $c_2 = 1 - t$ for $t \in (0,1)$, then $t \alpha^1 + (1-t) \alpha^2 \in \Delta_k$ and $\omega(t \alpha^1 + (1-t) \alpha^2) < 0$, which contradicts $u \in \varphi^{-1}(0)$. If $\| DJ^r(u)^\top \alpha^1 \| = 0$ or $\| DJ^r(u)^\top \alpha^2 \| = 0$ then \eqref{p5eq:unique_1} holds for $\bar{\alpha} = \alpha^1$ or $\bar{\alpha} = \alpha^2$, respectively. If $cos(\sphericalangle) - 1 = 0$ then $DJ^r(u)^\top \alpha^1$ and $DJ^r(u)^\top \alpha^2$ are linearly dependent, so there are $\bar{c}_1$, $\bar{c}_2 \in \mathbb{R} \setminus \{ 0 \}$ such that $DJ^r(u)^\top \bar{\alpha} = 0$ for $\bar{\alpha} = \bar{c}_1 \alpha^1 + \bar{c}_2 \alpha^2$. In particular, in any case we must have $\omega(\alpha) = 0$ for all $\alpha \in span(\{ \alpha^1, \alpha^2 \})$. \end{proof}
The previous lemma implies that for $k = 2$, the solution of \eqref{p5eq:P2_problem} for $u \in \varphi^{-1}(0)$ is non-unique iff $DJ^r(u) DJ^r(u)^\top - \epsilon \epsilon^\top = 0$. For $k > 2$, we can only have non-uniqueness if \eqref{p5eq:unique_1} holds. If we consider the dimensions of the spaces in \eqref{p5eq:unique_1}, we see that in the generic case, it can only hold if \begin{align*} & dim(span( \{ \alpha^1, \alpha^2 \} ) \cap ker(DJ(u)^\top)) \geq 1 \\ \Leftrightarrow \ & 2 + k - rk(DJ(u)^\top) - k \geq 1 \\ \Leftrightarrow \ & rk(DJ(u)^\top) \leq 1, \end{align*} i.e., if all gradients of the objectives are linearly dependent in $u$. This motivates us to assume that in general, the solution of \eqref{p5eq:P2_problem} is unique for almost all $u \in \varphi^{-1}(0)$.
We will now investigate the differentiability of $\varphi$. Our strategy is to apply the implicit function theorem to the KKT conditions of \eqref{p5eq:P2_problem} to obtain a differentiable function $\phi$ that maps a point $u \in \mathbb{R}^n$ onto the solution of \eqref{p5eq:P2_problem} in $u$. This would imply the differentiability of $\varphi$ via concatenation with $\omega$. An obvious problem here is the fact that \eqref{p5eq:P2_problem} has inequality constraints which, when activated or deactivated under variation of $u$, lead to non-differentiabilities in $\phi$. Note that an inequality constraint being active means that one component of $\alpha$ is zero, i.e., one of the objective functions has no impact on the current problem. Thus, for our theoretical purposes, if there is an active inequality constraint in \eqref{p5eq:P2_problem} we will just ignore the corresponding objective function. This approach is strongly related to the hierarchical decomposition of the Pareto critical set (cf.~\cite{GPD2019}).
For the reasons mentioned above, we will now consider the case where the solution of \eqref{p5eq:P2_problem} is strictly positive in each component. The following lemma shows a technical result that will be used in a later proof. \begin{lemma} \label{p5lem:alpha_unique_2}
Let $u \in \varphi^{-1}(0)$ and let $\bar{\alpha} \in \Delta_k$ be a solution of \eqref{p5eq:P2_problem} with $\alpha_i > 0$ $\forall i \in \{1,...,k\}$. Then $\bar{\alpha}$ is unique if and only if there is no $\beta \in \mathbb{R}^k \setminus \{ 0 \}$ with $\omega(\beta) = 0$ and $\sum_{i = 1}^k \beta_i = 0$. \end{lemma} \begin{proof}
We will show that $\alpha$ is non-unique if and only if there is some $\beta \in \mathbb{R}^k$ with $\omega(\beta) = 0$ and $\sum_{i = 1}^k \beta_i = 0$. \\
$\Rightarrow$: Let $\tilde{\alpha}$ be another solution of \eqref{p5eq:P2_problem}. Then, as in the proof of Lemma \ref{p5lem:alpha_unique_1}, we must have $\omega(c_1 \bar{\alpha} + c_2 \tilde{\alpha}) = 0$ for all $c_1, c_2 \in \mathbb{R}$. This means we can choose $\beta = \bar{\alpha} - \tilde{\alpha}$. \\
$\Leftarrow$: Let $\beta \in \mathbb{R}^k$ with $\omega(\beta) = 0$ and $\sum_{i = 1}^k \beta_i = 0$. Let $s > 0$ be small enough such that $\bar{\alpha} + s \beta \in \Delta_k$. Then, as in \eqref{p5eq:omega_add}, we have
\begin{align*}
\omega(\bar{\alpha} + s \beta) = 2 s (cos(\sphericalangle) - 1) \| DJ^r(u)^\top \bar{\alpha} \| \| DJ^r(u)^\top \beta \| \leq 0.
\end{align*}
Since by assumption $\varphi(u) = 0$ we must have $\omega(\bar{\alpha} + s \beta) = 0$, so $\bar{\alpha} + s \beta$ is another solution of \eqref{p5eq:P2_problem}. \end{proof}
To be able to use the KKT conditions of \eqref{p5eq:P2_problem} to obtain its solution, we have to make sure that these conditions are sufficient. Since \eqref{p5eq:P2_problem} is a quadratic problem, this means we have to show that the matrix in the objective $\omega$ is positive semidefinite. \begin{lemma} \label{p5lem:P2_pos_semidef}
Let $u \in \varphi^{-1}(0)$ and let $\bar{\alpha} \in \Delta_k$ be the unique solution of \eqref{p5eq:P2_problem} with $\bar{\alpha}_i > 0$ $\forall i \in \{1,...,k\}$. Then $\omega(\beta) \geq 0$ for all $\beta \in \mathbb{R}^k$. In particular, $DJ(u) DJ(u)^\top - \epsilon \epsilon^\top$ is positive semidefinite. \end{lemma} \begin{proof}
Assume there is some $\beta \in \mathbb{R}^k$ with $\omega(\beta) < 0$, i.e., $\epsilon^\top \beta > \| DJ^r(u)^\top \beta \|$. We distinguish between two cases: \\
Case 1: $\sum_{i = 1}^k \beta_i = 0$: Similar to the proof of Lemma \ref{p5lem:alpha_unique_1} we get
\begin{align*}
\omega(\bar{\alpha} + s \beta) &< 2 s ( (DJ^r(u)^\top \bar{\alpha})^\top (DJ^r(u)^\top \beta) - (\epsilon^\top \bar{\alpha}) (\epsilon^\top \beta) ) \\
&< 2 s (cos(\sphericalangle) - 1) \| DJ^r(u)^\top \bar{\alpha} \| \| DJ^r(u)^\top \beta \| \leq 0
\end{align*}
for all $s > 0$. In particular, since $\bar{\alpha}$ is positive, there is some $\bar{s} > 0$ such that $\bar{\alpha} + \bar{s} \beta \in \Delta_k$ with $\omega(\bar{\alpha} + \bar{s} \beta) < 0$, which is a contradiction. \\
Case 2: $\sum_{i = 1}^k \beta_i \neq 0$. W.l.o.g. assume that $\sum_{i = 1}^k \beta_i = 1$. Consider
\begin{equation*}
\bar{\omega} : \mathbb{R} \rightarrow \mathbb{R}, \quad s \mapsto \omega(\bar{\alpha} + s (\beta - \bar{\alpha})).
\end{equation*}
Then $\bar{\omega}(0) = 0$ and $\bar{\omega}(1) < 0$. By assumption we must have $\bar{\omega}(s) > 0$ for all $s$ such that $\bar{\alpha} + s (\beta - \bar{\alpha}) \in \Delta_k$. By continuity of $\bar{\omega}$ there must be some $s^*$ with $\bar{\omega}(s^*) = 0$. Let $\bar{\beta} := \bar{\alpha} + s^* (\beta - \bar{\alpha})$. Using \eqref{p5eq:omega_add} we get
\begin{align*}
\omega(\bar{\alpha} + t s^* (\beta - \bar{\alpha})) &= \omega((1 - t) \bar{\alpha} + t \bar{ \beta}) \\
&= 2 t (1-t) (cos(\sphericalangle) - 1) \| DJ^r(u)^\top \bar{\alpha} \| \| DJ^r(u)^\top \bar{\beta} \| \leq 0
\end{align*}
for all $t \in (0,1)$, which is a contradiction. \end{proof}
The previous results now allow us to prove Theorem \ref{p5thm:varphi_differentiable}. \paragraph{Theorem \ref{p5thm:varphi_differentiable}}
\textit{
Let $\bar{u} \in \varphi^{-1}(0)$ such that \eqref{p5eq:P2_problem} has a unique solution $\bar{\alpha} \in \Delta_k$ with $\bar{\alpha}_i > 0$ for all $i \in \{1,...,k\}$. Let \eqref{p5eq:P2_problem} be uniquely solvable in a neighborhood of $\bar{u}$. Then there is an open set $U \subseteq \mathbb{R}^n$ with $\bar{u} \in U$ such that $\varphi|_U$ is continuously differentiable.
} \begin{proof}
The KKT conditions for \eqref{p5eq:P2_problem} are
\begin{align} \label{p5eq:P2_KKT}
(DJ(u) DJ(u)^\top - \epsilon \epsilon^\top) \alpha -
\begin{pmatrix}
\lambda + \mu_1 \\
\vdots \\
\lambda + \mu_k
\end{pmatrix}
&= 0, \nonumber \\
\sum_{i = 1}^k \alpha_i - 1 &= 0, \nonumber \\
\alpha_i &\geq 0 \ \forall i \in \{1,...,k\}, \\
\mu_i &\geq 0 \ \forall i \in \{1,...,k\}, \nonumber \\
\mu_i \alpha_i &= 0 \ \forall i \in \{1,...,k\}. \nonumber
\end{align}
for $\lambda \in \mathbb{R}$ and $\mu \in \mathbb{R}^k$. By Lemma \ref{p5lem:P2_pos_semidef} these conditions are sufficient for optimality. By our assumption there is an open set $U'$ with $\bar{u} \in U'$ such that the solution of \eqref{p5eq:P2_problem} is unique and positive. Thus, on $U'$, \eqref{p5eq:P2_KKT} is equivalent to
\begin{align*}
(DJ(u) DJ(u)^\top - \epsilon \epsilon^\top) \alpha -
\begin{pmatrix}
\lambda \\
\vdots \\
\lambda
\end{pmatrix}
&= 0, \\
\sum_{i = 1}^k \alpha_i - 1 &= 0.
\end{align*}
for some $\lambda \in \mathbb{R}$. This system can be rewritten as $G(u,(\alpha,\lambda)) = 0$ for
\begin{equation*}
G : \mathbb{R}^n \times \mathbb{R}^{k+1} \rightarrow \mathbb{R}^{k+1}, \quad (u,(\alpha,\lambda)) \mapsto
\begin{pmatrix}
(DJ(u) DJ(u)^\top - \epsilon \epsilon^\top) \alpha - (\lambda,...,\lambda)^\top \\
\sum_{i = 1}^k \alpha_i - 1
\end{pmatrix}.
\end{equation*}
Derivating $G$ with respect to $(\alpha,\lambda)$ yields
\begin{equation*}
D_{(\alpha,\lambda)} G(u,(\alpha,\lambda)) =
\begin{pmatrix}
(DJ(u) DJ(u)^\top - \epsilon \epsilon^\top) & (-1,...,-1)^\top \\
(1,...,1) & 0
\end{pmatrix}
\in \mathbb{R}^{(k+1) \times (k+1)}.
\end{equation*}
Let $\bar{\lambda} \in \mathbb{R}$ such that $G(\bar{u},(\bar{\alpha},\bar{\lambda})) = 0$. (Note that uniqueness of $\bar{\alpha}$ implies uniqueness of $\bar{\lambda}$ here.) For $D_{(\alpha,\lambda)} G(\bar{u},(\bar{\alpha},\bar{\lambda}))$ to be singular, there would have to be some $v = (v^1,v^2) \in \mathbb{R}^{k+1}$ with
\begin{align*}
0 = D_{(\alpha,\lambda)} G(\bar{u},(\bar{\alpha},\bar{\lambda})) v =
\begin{pmatrix}
(DJ(\bar{u}) DJ(\bar{u})^\top - \epsilon \epsilon^\top) v^1 - (v^2,...,v^2)^\top \\
\sum_{i = 1}^k v_i^1
\end{pmatrix}
\end{align*}
and thus
\begin{align*}
0 &= {v^1}^\top (DJ(\bar{u}) DJ(\bar{u})^\top - \epsilon \epsilon^\top) v^1 - {v^1}^\top (v^2,...,v^2)^\top \\
&= {v^1}^\top (DJ(\bar{u}) DJ(\bar{u})^\top - \epsilon \epsilon^\top) v^1 - v_2 \sum_{i = 1}^k v_i^1 \\
&= w(v^1).
\end{align*}
By Lemma \ref{p5lem:alpha_unique_2}, this is a contradiction to the assumption that $\bar{\alpha}$ is a unique solution of \eqref{p5eq:P2_problem}. So $D_{(\alpha,\lambda)} G(\bar{u},(\bar{\alpha},\bar{\lambda}))$ has to be regular. This means we can apply the implicit function theorem to obtain open sets $U \subseteq U' \subseteq \mathbb{R}^n$, $V \subseteq \mathbb{R}^{k+1}$ with $\bar{u} \in U$, $(\bar{\alpha},\bar{\lambda}) \in V$ and a continuously differentiable function $\phi = (\phi_\alpha, \phi_\lambda): U \rightarrow V$ with
\begin{equation*}
G(u,(\alpha,\lambda)) = 0 \ \Leftrightarrow \ (\alpha,\lambda) = \phi(u) \quad \forall u \in U,(\alpha,\lambda) \in V.
\end{equation*}
In particular,
\begin{equation} \label{p5eq:varphi_phi}
\varphi|_U(u) = \min_{\alpha \in \Delta_k} \left( \| DJ(u)^\top \alpha \|^2 - (\alpha^\top \epsilon)^2 \right) = \| DJ(u)^\top \phi_\alpha(u) \|^2 - (\phi_\alpha(u)^\top \epsilon)^2,
\end{equation}
so $\varphi|_U$ is continuously differentiable. \end{proof}
\begin{remark}
From the proof of Theorem \ref{p5thm:varphi_differentiable} we can even derive an explicit formula for the derivative of $\varphi|_U$ in $\bar{u}$: First of all, the derivative of the implicit function $\phi$ is given by
\begin{align*}
&D\phi(\bar{u}) \\
&= - G_{(\alpha,\lambda)}(\bar{u},(\bar{\alpha},\bar{\lambda}))^{-1} G_u(\bar{u},(\bar{\alpha},\bar{\lambda})) \\
&= -
\begin{pmatrix}
DJ(\bar{u}) DJ(\bar{u})^\top - \epsilon \epsilon^\top & -1_{k \times 1} \\
1_{1 \times k} & 0
\end{pmatrix}^{-1} \cdot \\
& \hspace{60pt} \left(
\begin{pmatrix}
\bar{\alpha}^\top DJ(\bar{u}) \nabla^2 J_1(\bar{u}) \\
\vdots \\
\bar{\alpha}^\top DJ(\bar{u}) \nabla^2 J_k(\bar{u}) \\
0_{1 \times n}
\end{pmatrix}
+
\begin{pmatrix}
DJ(\bar{u}) \sum_{i = 1}^k \bar{\alpha}_i \nabla^2 J_i(\bar{u}) \\
0_{1 \times n}
\end{pmatrix}
\right).
\end{align*}
By applying the chain rule to \eqref{p5eq:varphi_phi} we obtain
\begin{align} \label{p5eq:D_phi_expl}
&D\varphi|_U(\bar{u}) \nonumber \\
&= 2 (DJ(\bar{u})^\top \bar{\alpha})^\top \sum_{i = 1}^k \bar{\alpha}_i \nabla^2 J_i(\bar{u}) + \left( 2 (DJ(\bar{u})^\top \bar{\alpha})^\top DJ(\bar{u})^\top - 2 (\bar{\alpha}^\top \epsilon) \epsilon^\top \right) D\phi_\alpha(\bar{u}) \nonumber \\
&= 2 (DJ(\bar{u})^\top \bar{\alpha})^\top \sum_{i = 1}^k \bar{\alpha}_i \nabla^2 J_i(\bar{u}) + 2 \bar{\alpha}^\top \left( DJ(\bar{u}) DJ(\bar{u})^\top - \epsilon \epsilon^\top \right) D\phi_\alpha(\bar{u}) \nonumber \\
&= 2 (DJ(\bar{u})^\top \bar{\alpha})^\top \sum_{i = 1}^k \bar{\alpha}_i \nabla^2 J_i(\bar{u}) + 2 (\bar{\lambda},...,\bar{\lambda}) D\phi_\alpha(\bar{u}).
\end{align}
$\blacksquare$ \end{remark}
\end{document} | arXiv |
Uniformization (set theory)
In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if $R$ is a subset of $X\times Y$, where $X$ and $Y$ are Polish spaces, then there is a subset $f$ of $R$ that is a partial function from $X$ to $Y$, and whose domain (the set of all $x$ such that $f(x)$ exists) equals
$\{x\in X\mid \exists y\in Y:(x,y)\in R\}\,$
Such a function is called a uniformizing function for $R$, or a uniformization of $R$.
To see the relationship with the axiom of choice, observe that $R$ can be thought of as associating, to each element of $X$, a subset of $Y$. A uniformization of $R$ then picks exactly one element from each such subset, whenever the subset is non-empty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to the axiom of choice.
A pointclass ${\boldsymbol {\Gamma }}$ is said to have the uniformization property if every relation $R$ in ${\boldsymbol {\Gamma }}$ can be uniformized by a partial function in ${\boldsymbol {\Gamma }}$. The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.
It follows from ZFC alone that ${\boldsymbol {\Pi }}_{1}^{1}$ and ${\boldsymbol {\Sigma }}_{2}^{1}$ have the uniformization property. It follows from the existence of sufficient large cardinals that
• ${\boldsymbol {\Pi }}_{2n+1}^{1}$ and ${\boldsymbol {\Sigma }}_{2n+2}^{1}$ have the uniformization property for every natural number $n$.
• Therefore, the collection of projective sets has the uniformization property.
• Every relation in L(R) can be uniformized, but not necessarily by a function in L(R). In fact, L(R) does not have the uniformization property (equivalently, L(R) does not satisfy the axiom of uniformization).
• (Note: it's trivial that every relation in L(R) can be uniformized in V, assuming V satisfies the axiom of choice. The point is that every such relation can be uniformized in some transitive inner model of V in which the axiom of determinacy holds.)
References
• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.
| Wikipedia |
The mutation rate of rpoB gene showed an upward trend with the increase of MIRU10, MIRU39 and QUB4156 repetitive number
Fan Su1 na1,
Lei Cao1 na1,
Xia Ren1,
Jian Hu1,
Grace Tavengana1,
Huan Wu2,
Yumei Zhou1,
Yuhan Fu1,
Mingfei Jiang3 &
Yufeng Wen ORCID: orcid.org/0000-0001-9338-918X1
BMC Genomics volume 24, Article number: 26 (2023) Cite this article
Mycobacterial interspersed repetitive unit-variable number tandem repeat (MIRU-VNTR) is a frequently used typing method for identifying the Beijing genotype of Mycobacterium tuberculosis (Mtb), which is easily transformed into rifampicin (RIF) resistance. The RIF resistance of Mtb is considered to be highly related with the mutation of rpoB gene. Therefore, this study aimed to analyze the relationship between the repetitive number of MIRU loci and the mutation of rpoB gene.
An open-source whole-genome sequencing data of Mtb was used to detect the mutation of rpoB gene and the repetitive number of MIRU loci by bioinformatics methods. Cochran-Armitage analysis was performed to analyze the trend of the rpoB gene mutation rate and the repetitive number of MIRU loci.
Among 357 rifampicin-resistant tuberculosis (RR-TB), 304 strains with mutated rpoB genes were detected, and 6 of 67 rifampicin susceptible strains were detected mutations. The rpoB gene mutational rate showed an upward trend with the increase of MIRU10, MIRU39, QUB4156 and MIRU16 repetitive number, but only the repetitive number of MIRU10, MRIU39 and QUB4156 were risk factors for rpoB gene mutation. The Hunter-Gaston discriminatory index (HGDI) of MIRU10 (0.65) and QUB4156 (0.62) was high in the overall sample, while MIRU39 (0.39) and MIRU16 (0.43) showed a moderate discriminatory Power.
The mutation rate of rpoB gene increases with the addition of repetitive numbers of MIRU10, QUB4156 and MIRU39 loci.
In recent years, the global tuberculosis (TB) epidemic continues to be serious. Drug-resistant tuberculosis, especially those resistant to rifampicin (RR-TB), has become one of the major obstacles to achieve the goal of TB elimination [1]. According to the Global Tuberculosis Report 2020 of World Health Organization (WHO), there were an estimated 465,000 (range, 400,000–535,000) incident cases of RR-TB, and China accounts for 14% of them [2].
More than 95% of rifampicin resistance is associated with mutations in the rpoB gene of Mycobacterium tuberculosis (Mtb), with 97% of mutations occurring within the 81 bp rifampicin-resistant determining region (RRDR) of this gene [3]. Besides, it has been proved that sequence mutation out of RRDR may be involved in the formation of rifampicin cross resistance [4]. In China, Beijing genotype tuberculosis occupies a dominant position of Mtb, Uddin MKM et al. [5] had proved the mutation of rpoB gene was a risk factor of rifampicin resistance for Beijing genotype TB.
In recent years, mycobacterial interspersed repetitive units-variable number tandem repeats (MIRU-VNTR) had been widely used in the typing of TB. Combined with Spoligotyping, MIRU-VNTR typing can distinguish Beijing family genotype with other genotype strains by cluster analysis [6]. Besides, different MIRU loci showed different discriminatory power for Beijing and non-Beijing genotype strains and significant differences were found in mutation of the rpoB gene between two genotype [7, 8]. On this basis, we hypothesized that there may be a correlation between the mutation of rpoB gene and the repetitive number of MIRU loci.
With the popularization of whole-genome sequencing technology, the mutation of known drug resistance genes and MIRU-VNTR information can be obtained based on the analysis of Mtb Illumina, Pacific Biosciences or Oxford Nanopore sequencing data [9, 10]. Therefore, we conducted this study to explore the relationship of rpoB gene mutation and MIRU loci with sequencing data.
Sample information was acquired from one study of the Chinese Center for Disease Control and Prevention (Chinese CDC) [11], including the phenotypic drug resistance of each strain, type of patient from which the strain originated, etc. Whole-genome sequencing raw data were deposited at NCBI Sequence Read Archive (SRP134826) and Genome Sequence Archive (CRA000786) (https://ngdc.cncb.ac.cn/search/?dbId=gsa&q = CRA000786)
RpoB gene mutation determination
In the first step, the sequencing data was submitted to remove linker and low-quality base treatment (filtering the bases with Phred < 20) using Fastp (https://github.com/OpenGene/fastp) software. Secondly, BWA (http://bio-bwa.sourceforge.net/bwa.shtml) software was used to compare the above sequence data with the genome template sequence of Mycobacterium tuberculosis standard strain (H37Rv) (obtained from the gene sequence database GenBank access: NC 000962.3 maintained by the National Institutes of Health). In the third step, according to the comparison results, sequencing data samples were screened that the sequencing depth is more than 10× and the genome coverage is more than 95%. Finally, SNPs of each strain compared with H37Rv were identified using Samtools (https://github.com/samtools/samtools/issues), and the lowest value of comparison quality was set to 30. Then VarScan 2 (http://varscan.sourceforge.net) software was used to further identify and screen SNP fixed mutations with a frequency of more than 75% and supported by at least 10 sequences. The whole genome SNPs detected in this study were compared with known rpoB gene mutations (obtained from GenBank gene database) to obtain the mutation information of rpoB gene of each strain [12], and only the non-synonymous mutations were recorded.
MIRU loci repetitive number determination
The sequencing data outputted from Fastp were assessed by FastQC (http://www.bioinformatics.babraham.ac.uk/projects/fastqc) to guarantee good reads quality. Spades (https://github.com/ablab/spades) was carried to assemble second generation sequencing data to long sequence, the finally assembled data were assessed by QUAST (http://bioinf.spbau.ru/quast) and BUSCO (https://busco.ezlab.org). MIRUReader (https://github.com/phglab/MIRUReader) was used to get the repetitive number of 24 MIRU loci (MIRU02, MTUB04, ETRC, MIRU04, MIRU40, MIRU10, MIRU16, MTUB21, MIRU20, QUB11B, ETRA, Mtub29, Mtub30, ETRB, MIRU23, MIRU24, MIRU26, MIRU27, Mtub34, MIRU31, Mtub39, QUB26, QUB4156, MIRU39) directly from long sequence reads [9].
HGDI calculation
$$HGDI=1-\frac{1}{N\left(N-1\right)}{\sum}_{j=1}^s nj\left(j-1\right)$$
N stands for the total number of strains, nj is the number of strains with the jth genotype, and s is the number of different genotypes at the MIRU-VNTR loci.
IBM SPSS 18.0 and GraphPad 7 were implemented for statistical analysis. Chi-square test or t-test was conducted to compare the differences in variables of general characteristics between TB groups with mutational and non-mutational rpoB gene. All variables with a P-value < 0.10 on Chi-Square test and t-test were included in a multivariate conditional logistic regression model to investigate the relationship of the mutation of rpoB gene and the repetitive number of the MIRU loci. Besides, Cochran-Armitage analysis was conducted to determine the trend of rpoB gene mutation rate and the repetitive number of MIRU loci.
The rpoB mutation results of the study samples
There were 424 TB samples included in our study, 357 (84.2%) strains extracted from them were RR-TB, and 67 (15.8%) were rifampicin sensitive strains. Among RR-TB, rpoB genes of 304 strains were detected mutations, and 6 of 67 susceptible strains were detected mutations. RpoB gene mutational rate between strains of retreated cases (83.41%) and new cases (62.32%) showed a significant difference (χ2 = 24.0 P < 0.05).
Relation between rpoB gene and 24-loci MIRU-VNTR
The mutation rate of rpoB gene showed an upward trend with the increase of MIRU10, MIRU39, QUB4156 and MIRU16 repetitive number after the Cochran-Armitage analysis (Fig. 1). However, only the repetitive number of MIRU10, MRIU39 and QUB4156 were risk factors for rpoB gene mutation after adjusted by category (retreated or new cases) and MIRU23 (Table 1).
Cochran-Armitage analysis of rpoB gene mutation rate and MIRU repeated numbers. The abscissa of the black dot on the broken line represents the repetitive number of different MIRU loci, and the ordinate represents the corresponding rpoB gene mutation rate. The broken line trend reflects whether the rpoB mutation rate increases or decreases with the increase of MIRU loci repetitive number. When P < 0.05, there was a significant overall trend between them
Table 1 Logistic regression analysis of the rpoB gene mutation and the repetitive number of 24 MIRU loci
Allelic diversity of the MIRU loci
As shown in the Table 2, two loci (MIRU10, QUB4156) were highly discriminative (Hunter-Gaston discriminatory index, HGDI> 0.6), two loci (MIRU39, MIRU16) were moderately discriminative (HGDI> 0.3) among all 24 loci studied. The allelic diversity of the 4 loci were different between the rpoB gene mutational strains and non-mutational strains. It was worth noting that MIRU39 showed a moderately discriminablility in rpoB gene mutational strains, while a low discriminablility in rpoB gene non-mutational strains.
Table 2 Allelic diversity of four loci in rpoB gene mutation and non-mutation isolates
In this study, we assessed the associated risk factors for rpoB gene mutation in data sourced areas. The rpoB gene mutation rate of retreated TB patients (83.41%) was higher than that of new cases (62.32%), it has been proved that the RIF resistance rate of retreated tuberculosis is higher than that of new cases in previous studies [13, 14], the higher rate may since that patients with retreated pulmonary tuberculosis often fail in the initial treatment due to unreasonable or irregular anti-tuberculosis treatment, resulting in the dominant growth of drug-resistant tuberculosis bacteria, and it's drug resistance mechanism is related to the mutation of rpoB gene which coding RNA polymerase β-subunit [3]. Notably, we found no rpoB gene mutation in partial RR-TB strains, but mutations in sensitive strains, the inconsistency between gene resistance and phenotype resistance may be caused by heterogeneity of Mtb. The presence of low-frequency RR-TB and the predominance of sensitive Mtb in the specimen may result in ineffective extraction of drug-resistant DNA if the specimen is not handled properly, while the proportional method of drug sensitivity suggested that it was RR-TB [15]. Patients may have been treated with multiple anti-tuberculosis drugs before sputum specimens were sent for testing, resulting in multiple Mtb states in sputum specimens, which can also lead to this result [16]. And mutations in the rpoB gene leading to low levels of rifampicin resistance may be the reason that these strains with mutations in the rpoB gene were detected as sensitive [17].
Different VNTR loci always has different discrimination ability between Beijing and non-Beijing genotype Mtb [18]. In our study, MIRU10, MIRU39, QUB4156 and MIRU16 all showed a difference in allellic diversity between the Beijing and non-Beijing genotype strains, but only MIRU39 showed remarkable difference (△HGDI > 0.2).
VNTR is a highly polymorphic and highly repetitive DNA fragment, which is characterized by variety and wide distribution. The distribution of VNTR in Mtb showed high individual specificity [19]. In recent years, MIRU-VNTR had been widely used in the typing of tuberculosis, some loci, such as MIRU10, MIRU39 and QUB4156 could genotype Mtb with high discriminatory power [20,21,22]. In this study, we found that strains with high MIRU10, MIRU39 QUB4156 or MIRU16 repetitive numbers may often have a high rpoB gene mutation rate, but only the repetitive number of MIRU10, MRIU39 and QUB4156 were risk factors for rpoB gene mutation after adjusting by category (retreated or new cases) and MIRU23.
MIRU loci are located in the spacer of DNA coding genes, and their specific functions are not clear. Some scholars [23,24,25] believed that the difference in the copy number of MIRU sites upstream of the coding gene will lead to the difference in the number of ribosomal binding sites (RBS), thus affecting the transcription and expression level of the gene. The coding product encoded by the fadB gene downstream of MIRU10 is an oxidoreductase that binds to flavin adenine dinucleotide (FAD), the oxidative stress response induced by this gene may be one of the mechanisms of anti-tuberculosis drugs killing bacteria [26]. With the increase of MIRU10 loci repetitive number, it may increase the inhibition of fadB gene expression [27], finally resulting in RIF resistance. EccCa1 gene which downstream of MIRU39 is part of the ESX-1 specialized secretion system, which delivers several virulence factors to host cells during infection, including the key virulence factors ESAT-6 and CFP-10 [28, 29]. The increase of MIRU39 repetitive number may target up-regulation of eccCa1 gene expression, resulting in increased bacterial virulence. The coding product encoded by the murT gene downstream of QUB4156 is involved in the pathway peptidoglycan biosynthesis, which is part of cell wall biogenesis [30]. The increase of QUB4156 repetitive number may enhance the virulence of Mtb by promoting the synthesis of cell wall.
The mutation rate of rpoB gene increased with the addition of MIRU10, MRIU39 and QUB4156 repetitive numbers, we speculated that these MIRU loci caused the RIF resistance in Mtb respectively through different ways. Repetitive numbers of MIRU loci are relatively easy to detect in the laboratory [31]. We hope that this upward trend can deepen the understanding of the function of MIRU10, QUB4156 and MIRU39 loci and the mechanism of RIF resistance. However, this experiment is limited to the research characteristics of molecular epidemiology, which need to be further verified by experimental research.
The mutation rate of rpoB gene increased with the addition of the number of repeats at MIRU10, QUB4156 and MIRU39 loci.
The sequencing datasets used and/or analyzed during the current study are available from NCBI Sequence Read Archive (SRP134826) and Genome Sequence Archive (CRA000786) (https://ngdc.cncb.ac.cn/search/?dbId=gsa&q=CRA000786).
Caminero Luna JA, Pérez Mendoza G, Rodríguez de Castro F. Multi-drug resistant tuberculosis, ten years later. Tuberculosis multirresistentediez años después. Med Clin (Barc). 2021;156(8):393–401. https://doi.org/10.1016/j.medcli.2020.08.018.
WHO, Global Tuberculosis Report 2020. (https://apps.who.int/iris/bitstream/handle/10665/336069/9789240013131-eng.pdf),20220603.
Zaw MT, Emran NA, Lin Z. Mutations inside rifampicin-resistance determining region of rpoB gene associated with rifampicin-resistance in mycobacterium tuberculosis. J Infect Public Health. 2018;11(5):605–10. https://doi.org/10.1016/j.jiph.2018.04.005.
Portelli S, Myung Y, Furnham N, Vedithi SC, Pires DEV, Ascher DB. Prediction of rifampicin resistance beyond the RRDR using structure-based machine learning approaches. Sci Rep. 2020;10(1):18120. https://doi.org/10.1038/s41598-020-74648-y.
Uddin MKM, Rahman A, Ather MF, et al. Distribution and frequency of rpoB mutations detected by Xpert MTB/RIF assay among Beijing and non-Beijing rifampicin resistant mycobacterium tuberculosis isolates in Bangladesh. Infect Drug Resist. 2020;10(13):789–97. https://doi.org/10.2147/IDR.S240408.
Chee CBE, Lim LKY, Ong RTH, et al. Whole genome sequencing analysis of multidrug-resistant tuberculosis in Singapore, 2006-2018. Eur J Clin Microbiol Infect Dis. 2021;40(5):1079–83. https://doi.org/10.1007/s10096-020-04100-6.
Shi J, Zheng D, Zhu Y, et al. Role of MIRU-VNTR and spoligotyping in assessing the genetic diversity of mycobacterium tuberculosis in Henan Province, China. BMC Infect Dis. 2018;18(1):447. https://doi.org/10.1186/s12879-018-3351-y.
Gupta A, Sinha P, Nema V, et al. Detection of Beijing strains of MDR M. tuberculosis and their association with drug resistance mutations in katG, rpoB, and embB genes. BMC Infect Dis. 2020;20(1):752. https://doi.org/10.1186/s12879-020-05479-5.
Tang CY, Ong RT. MIRUReader: MIRU-VNTR typing directly from long sequencing reads. Bioinformatics. 2020;36(5):1625–6. https://doi.org/10.1093/bioinformatics/btz771.
Maladan Y, Krismawati H, Wahyuni T, et al. The whole-genome sequencing in predicting mycobacterium tuberculosis drug susceptibility and resistance in Papua, Indonesia. BMC Genomics. 2021;22(1):844. https://doi.org/10.1186/s12864-021-08139-3.
Huang H, Ding N, Yang T, et al. Cross-sectional whole-genome sequencing and epidemiological study of multidrug-resistant mycobacterium tuberculosis in China. Clin Infect Dis. 2019;69(3):405–13. https://doi.org/10.1093/cid/ciy883.
Walker TM, Kohl TA, Omar SV, et al. Whole-genome sequencing for prediction of mycobacterium tuberculosis drug susceptibility and resistance: a retrospective cohort study. Lancet Infect Dis. 2015;15(10):1193–202. https://doi.org/10.1016/S1473-3099(15)00062-6.
Liu L, Zhao X, Wu X, et al. Prevalence and molecular characteristics of drug-resistant mycobacterium tuberculosis in Hainan, China: from 2014 to 2019. BMC Microbiol. 2021;21(1):185. https://doi.org/10.1186/s12866-021-02246-7.
Li D, Song Y, Yang P, Li X, Zhang AM, Xia X. Genetic diversity and drug resistance of mycobacterium tuberculosis in Yunnan, China. J Clin Lab Anal. 2019;33(5):e22884. https://doi.org/10.1002/jcla.22884.
Zheng C, Li S, Luo Z, et al. Mixed infections and rifampin Heteroresistance among mycobacterium tuberculosis clinical isolates. J Clin Microbiol. 2015;53(7):2138–47. https://doi.org/10.1128/JCM.03507-14.
Sun G, Luo T, Yang C, Dong X, Li J, Zhu Y, et al. Dynamic population changes in mycobacterium tuberculosis during acquisition and fixation of drug resistance in patients. J Infect Dis. 2012;206(11):1724–33. https://doi.org/10.1093/infdis/jis601.
Gopie FA, Commiesie E, Baldi S, et al. Should treatment of low-level rifampicin mono-resistant tuberculosis be different? J Clin Tuberc Other Mycobact Dis. 2021;23:100222. https://doi.org/10.1016/j.jctube.2021.100222.
Li Y, Pang Y, Zhang T, Xian X, Yang J, Wang R, et al. Genotypes of mycobacterium tuberculosis isolates circulating in Shaanxi Province, China. PLoS One. 2020;15(12):e0242971. https://doi.org/10.1371/journal.pone.0242971.
Al-Mahrouqi S, Ahmed R, Al-Azri S, Al-Hamidhi S, Balkhair AA, Al-Jardani A, et al. Dynamics of mycobacterium tuberculosis lineages in Oman, 2009 to 2018. Pathogens. 2022;11(5):541. https://doi.org/10.3390/pathogens11050541.
Elsayed MSAE. Applicability of using 15 MIRU-VNTR loci for genotyping of Mycobacterium avium subsp. paratuberculosis from two cattle farms in Egypt. Mol Biol Rep. 2019;46(6):6253–62. https://doi.org/10.1007/s11033-019-05065-2.
Antonenko PB, Kresyun VI, Antonenko KO. Clusters of mycobacterium tuberculosis genotypes in Odesa region. Mikrobiol Z. 2016;78(2):103–10. Russian 30141603.
Ghavidel M, Tadayon K, Mosavari N, et al. Introducing the best six loci in mycobacterial interspersed repetitive unit-variable-number tandem repeat (MIRU-VNTR) typing for mycobacterium tuberculosis genotyping. Rep Biochem Mol Biol. 2019;8(3):335–46. https://doi.org/10.1186/s12879-020-05479-5.
Refaya AK, Sivakumar S, Sundararaman B, Narayanan S. Polymorphism in the RD1 locus and its effect on downstream genes among south Indian clinical isolates of mycobacterium tuberculosis. J Med Microbiol. 2012;61(Pt 10):1352–9. https://doi.org/10.1099/jmm.0.044453-0.
Yu-Feng W, Chao J, Xian-Feng C. Drug-resistant tuberculosis can be predicted by mycobacterial interspersed repetitive unit locus. Front Microbiol. 2015;6:147. https://doi.org/10.3389/fmicb.2015.00147.
Cheng XF, Jiang C, Zhang M, et al. Mycobacterial interspersed repetitive unit can predict drug resistance of mycobacterium tuberculosis in China. Front Microbiol. 2016;7:378. https://doi.org/10.3389/fmicb.2016.00378.
Cox JAG, Taylor RC, Brown AK, Attoe S, Besra GS, Fütterer K. Crystal structure of mycobacterium tuberculosis FadB2 implicated in mycobacterial β-oxidation. Acta Crystallogr D Struct Biol. 2019;75(Pt 1):101–8. https://doi.org/10.1107/S2059798318017242.
Pérez-Lago L, Navarro Y, Herranz M, Bouza E, García-de-Viedma D. Differences in gene expression between clonal variants of mycobacterium tuberculosis emerging as a result of microevolution. Int J Med Microbiol. 2013;303(8):674–7. https://doi.org/10.1016/j.ijmm.2013.09.010.
Clemmensen HS, Knudsen NPH, Rasmussen EM, et al. An attenuated mycobacterium tuberculosis clinical strain with a defect in ESX-1 secretion induces minimal host immune responses and pathology. Sci Rep. 2017;7:46666. https://doi.org/10.1038/srep46666.
Soler-Arnedo P, Sala C, Zhang M, Cole ST, Piton J. Polarly localized EccE1 is required for ESX-1 function and stabilization of ESX-1 membrane proteins in mycobacterium tuberculosis. J Bacteriol. 2020;202(5):e00662–19. https://doi.org/10.1128/JB.00662-19.
Maitra A, Nukala S, Dickman R, et al. Characterization of the MurT/GatD complex in mycobacterium tuberculosis towards validating a novel anti-tubercular drug target. JAC Antimicrob Resist. 2021;3(1):dlab028. https://doi.org/10.1093/jacamr/dlab028.
Shafipour M, Shirzad-Aski H, Ghaemi EA, et al. Mycobacterium tuberculosis typing using allele-specific oligonucleotide multiplex PCR (ASO-PCR) method. Curr Microbiol. 2021;78(12):4009–13. https://doi.org/10.1007/s00284-021-02659-7.
Thanks are due to China CDC for sharing their sequencing data in Genome Sequence Archive.
This work was supported by Teaching Reform and Quality Improvement Plan of Education Department of Anhui Province-Famous Teacher (2019jxms066), and Public Health Collaborative Innovation Project of Provincial Undergraduate Medical Colleges (GXXT-2020-22).
Fan Su and Lei Cao contributed equally to this work.
School of Public Health, Wannan Medical College, Wuhu, Anhui Province, China
Fan Su, Lei Cao, Xia Ren, Jian Hu, Grace Tavengana, Yumei Zhou, Yuhan Fu & Yufeng Wen
School of Laboratory Medicine, Wannan Medical College, Wuhu, Anhui Province, China
Huan Wu
School of Clinical Medicine, Wannan Medical College, Wuhu, Anhui Province, China
Mingfei Jiang
Fan Su
Lei Cao
Xia Ren
Jian Hu
Grace Tavengana
Yumei Zhou
Yuhan Fu
Yufeng Wen
Yufeng Wen designed the research. Fan Su and Lei Cao participated in data analysis and drafted the manuscript. Yuhan Fu, Yumei Zhou, Grace Tavengana, Mingfei Jiang, Huan Wu, Jian Hu and Xia Ren helped analyze data and manuscript development. Yufeng Wen provided research funding and software support. All authors read and approved the final manuscript.
Correspondence to Yufeng Wen.
Su, F., Cao, L., Ren, X. et al. The mutation rate of rpoB gene showed an upward trend with the increase of MIRU10, MIRU39 and QUB4156 repetitive number. BMC Genomics 24, 26 (2023). https://doi.org/10.1186/s12864-023-09120-y
RpoB gene
MIRU10
QUB4156
Repetitive number | CommonCrawl |
Boundary parallel
In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a boundary component of M.
An example
Consider the annulus $I\times S^{1}$. Let π denote the projection map
$\pi \colon I\times S^{1}\rightarrow S^{1},\quad (x,z)\mapsto z.$
If a circle S is embedded into the annulus so that π restricted to S is a bijection, then S is boundary parallel. (The converse is not true.)
If, on the other hand, a circle S is embedded into the annulus so that π restricted to S is not surjective, then S is not boundary parallel. (Again, the converse is not true.)
• An example wherein π is not bijective on S, but S is ∂-parallel anyway.
• An example wherein π is bijective on S.
• An example wherein π is not surjective on S.
| Wikipedia |
\begin{document}
\begin{center} \LARGE{\textbf{Bicomplex Riesz-Fischer Theorem}} \end{center}
\begin{center} \bf{K. S. Charak$^{1 }$,\quad R. Kumar$^{2}$,\quad D. Rochon$^{3}$ } \end{center}
\begin{center} {$^{1}$ Department of Mathematics, University of Jammu,\\ Jammu-180 006, INDIA.\\ E-mail: [email protected] } \end{center}
\begin{center}{$^{2}$ Department of Mathematics, University of Jammu,\\ Jammu-180 006, INDIA.\\ E-mail: [email protected] } \end{center}
\begin{center} {$^{3}$ D\'epartement de math\'ematiques et d'informatique,\\ Universit\'e du Qu\'ebec \`a Trois-Rivi\`eres, C.P. 500, Trois-Rivi\`eres, Qu\'ebec, Canada G9A 5H7. \\ E-mail: [email protected],\\ Web: www.3dfractals.com} \end{center}
\begin{abstract} \noindent This paper continues the study of infinite dimensional bicomplex Hilbert spaces introduced in previous articles on the topic. Besides obtaining a Best Approximation Theorem, the main purpose of this paper is to obtain a bicomplex analogue of the Riesz-Fischer Theorem. There are many statements of the Riesz-Fischer (R-F) Theorem in the literature, some are equivalent, some are consequences of the original versions. The one referred to in this paper is the R-F Theorem which establishes that the spaces $l^2$ is the canonical model space. \end{abstract}
\noindent \textbf{Keywords: }Bicomplex numbers, Bicomplex algebra, Generalized Hilbert spaces, Riesz-Fischer Theorem.\\ \noindent \textbf{AMS [2010]: }Primary 16D10; Secondary 30G35, 46C05, 46C50.
\section{Introduction} Hilbert spaces over the field of complex numbers are indispensable for mathematical structure of quantum mechanics \cite{JVN} which in turn play a great role in molecular, atomic and subatomic phenomena. The work towards the generalization of quantum mechanics to bicomplex number system have been recently a topic in different quantum mechanical models \cite{CV, CVM, BK, Rochon2, Rochon3}. More specifically, in \cite{GMR2, GMR} the authors made an in depth study of bicomplex Hilbert spaces and operators acting on them. After obtaining reasonable results responsible for investigations on finite and infinite dimensional bicomplex Hilbert spaces and applications to quantum mechanics \cite{GMR3, GMR4, GeRo, MMR}, they in \cite{GMR} asked for extension of Riesz-Fischer Theorem and Spectral Theorem on infinite dimensional Hilbert spaces. Recently, the bicomplex analogue of the Spectral Decomposition Theorem was proven using bicomplex eigenvalues \cite{RKC2}. In this paper, we obtain a bicomplex analogue of the Riesz-Fischer Theorem \cite{Hansen, Horvath} on infinite dimensional Hilbert spaces. Our proof of R-F Theorem is essentially different from its complex Hilbert space analogue in the sense that we do not make use of the so called Parseval's identity as done in general Hilbert spaces over $\ensuremath{\mathbb{R}}$ or $\ensuremath{\mathbb{C}}$. To support our results, we prove A Best Approximation Theorem and we show that the bicomplex analogue of $l^2$, the space of all (real, complex or bicomplex) sequences $\{ w_l \}$ such that
$\sum_{l=1}^{\infty}|w_l|^{2}<\infty$, is a bicomplex Hilbert space. As for the standard quantum mechanics, this specific result is fundamental to understand the space where live the wave functions of the bicomplex Quantum Harmonic Oscillator \cite{GMR3, Rochon2, Rochon3}.
\section{Preliminaries}
This section first summarizes a number of known results on the algebra of bicomplex numbers, which will be needed in this paper. Much more details as well as proofs can be found in~\cite{Price, Rochon1, Rochon2, Rochon3}. Basic definitions related to bicomplex modules and scalar products are also formulated as in~\cite{GMR2, Rochon3}, but here we make no restrictions to finite dimensions following definitions of \cite{GMR}.
\subsection{Bicomplex Numbers}\label{Bicomplex Numbers}
\subsubsection{Definition}\label{Definition of bicomplex numbers} The set $\mathbb{M}(2)$ of \emph{bicomplex numbers} is defined as \begin{align}
\mathbb{M}(2):=\{ w=z_1+z_2\mathbf{i_2}~|~z_1,z_2\in\mathbb{C}(\mathbf{i_1}) \}, \label{2.1} \end{align} where $\mathbf{i_1}$ and $\mathbf{i_2}$ are independent imaginary units such that ${\rm \bf{i}}_{\bf{1}}^2=-1={\rm \bf{i}}_{\bf{2}}^2$. The product of $\mathbf{i_1}$ and $\mathbf{i_2}$ defines a hyperbolic unit $\mathbf{j}$ such that $\mathbf{j}^2=1$. The product of all units is commutative and satisfies \begin{equation*} \mathbf{i_1}\mathbf{i_2}=\mathbf{j}, \qquad \mathbf{i_1}\mathbf{j}=-\mathbf{i_2}, \qquad \mathbf{i_2}\mathbf{j}=-\mathbf{i_1}. \label{2.2} \end{equation*} With the addition and multiplication of two bicomplex numbers defined in the obvious way, the set $\mathbb{M}(2)$ makes up a commutative ring. They are a particular case of the so-called \textit{Multicomplex Numbers} (denoted $\mathbb{M}(n)$) \cite{Price, GR} and \cite{Vaijac}. In fact, bicomplex numbers $$\mathbb{M}(2)\cong {\rm Cl}_{\Bbb{C}}(1,0) \cong {\rm Cl}_{\Bbb{C}}(0,1)$$ are unique among the complex Clifford algebras (see \cite{BDS,DSS} and \cite{Ryan}) in the sense that this set form a commutative, but not division algebra.
Three important subsets of $\mathbb{M}(2)$ can be specified as
\begin{align*}
\mathbb{C}(\ik{k}) &:= \{ x+y\ik{k}~|~x,y\in\mathbb{R} \}, \qquad k=1,2 ;\label{2.3}\\
\mathbb{D} &:= \{ x+y\mathbf{j}~|~x,y\in\mathbb{R} \} . \end{align*}
Each of the sets $\mathbb{C}(\ik{k})$ is isomorphic to the field of complex numbers, while $\mathbb{D}$ is the set of so-called \emph{hyperbolic numbers}, also called duplex numbers (see, e.g. \cite {Sob}, \cite {Rochon1}).
\subsubsection{Conjugation and Moduli}\label{Bicomplex conjugation}
Three kinds of conjugation can be defined on bicomplex numbers. With $w$ specified as in~\eqref{2.1} and the bar ($\,\bar{\mbox{}}\,$) denoting complex conjugation in $\mathbb{C}(\mathbf{i_1})$, we define \begin{equation*} w^{\dag_1}:=\bar{z}_1+\bar{z}_2\mathbf{i_2},\label{2.5} \qquad w^{\dag_2}:=z_1-z_2\mathbf{i_2}, \qquad w^{\dag_3}:=\bar{z}_1-\bar{z}_2\mathbf{i_2} . \end{equation*} It is easy to check that each conjugation has the following properties: \begin{equation*} (s+t)^{\dag_k}=s^{\dag_k}+t^{\dag_k}, \qquad \left(s^{\dag_k} \right)^{\dag_k}=s, \qquad (s\cdot t)^{\dag_k}=s^{\dag_k}\cdot t^{\dag_k} . \label{2.6} \end{equation*} Here $s,t\in\mathbb{M}(2)$ and $k=1,2,3$.
With each kind of conjugation, one can define a specific bicomplex modulus as
\begin{align*}
|w|_\mathbf{i_1}^2&:=w\cdot w^{\dag_2}=z_1^2+z_2^2~\in\mathbb{C}(\mathbf{i_1}),\label{2.7a}\\
|w|_\mathbf{i_2}^2&:=w\cdot w^{\dag_1}=\left(|z_1|^2-|z_2|^2\right) + 2 \, \textrm{Re}(z_1\bar{z}_2)\mathbf{i_2}~\in\mathbb{C}(\mathbf{i_2}),\\
|w|_\mathbf{j}^2&:=w\cdot w^{\dag_3}=\left(|z_1|^2+|z_2|^2\right) - 2 \, \textrm{Im}(z_1\bar{z}_2)\mathbf{j}~\in\mathbb{D}. \end{align*}
It can be shown that $|s\cdot t|_k^2=|s|_k^2\cdot|t|_k^2$, where $k=\mathbf{i_1},\mathbf{i_2}$ or $\mathbf{j}$.
In this paper we will often use the Euclidean $\mathbb{R}^4$-norm defined as \begin{equation*}
|w|:=\sqrt{|z_1|^2+|z_2|^2}=\sqrt{\textrm{Re}(|w|_\mathbf{j}^2)} \; . \label{2.8} \end{equation*}
Clearly, this norm maps $\mathbb{M}(2)$ into $\mathbb{R}$. We have $|w|\geq0$, and $|w|=0$ if and only if $w=0$. Moreover~\cite{Rochon1}, for all $s,t\in\mathbb{M}(2)$, \begin{equation*}
|s+t|\leq|s|+|t|, \qquad |s\cdot t|\leq \sqrt{2} \, |s|\cdot|t|. \label{2.9} \end{equation*}
\subsubsection{Idempotent Basis}\label{Idempotant basis}
The operations of the bicomplex algebra is considerably simplified by the introduction of two bicomplex numbers $\mathbf{e_1}$ and $\mathbf{e_2}$ defined as \begin{equation*} \mathbf{e_1}:=\frac{1+\mathbf{j}}{2},\qquad\mathbf{e_2}:=\frac{1-\mathbf{j}}{2}.\label{2.10} \end{equation*}
In fact $\mathbf{e_1}$ and $\mathbf{e_2}$ are hyperbolic numbers. They make up the so-called \emph{idempotent basis} of the bicomplex numbers. One easily checks that ($k=1,2$)
\begin{equation} \mathbf{e}_{\mathbf{1}}^2=\mathbf{e_1}, \quad \mathbf{e}_{\mathbf{2}}^2=\mathbf{e_2}, \quad \mathbf{e_1}+\mathbf{e_2}=1, \quad \mathbf{e}_{\mathbf{k}}^{\dag_3}=\e{k} , \quad \mathbf{e_1}\mathbf{e_2}=0 . \label{2.11} \end{equation}
Any bicomplex number $w$ can be written uniquely as
\begin{equation} w = z_1+z_2\mathbf{i_2} = z_{\widehat{1}} \mathbf{e_1} + z_{\widehat{2}} \mathbf{e_2} , \label{2.12} \end{equation} where \begin{equation*} z_{\widehat{1}}= z_1-z_2\mathbf{i_1} \quad \mbox{and} \quad z_{\widehat{2}}= z_1+z_2\mathbf{i_1} \label{2.12a} \end{equation*} both belong to $\mathbb{C}(\mathbf{i_1})$. Note that \begin{equation*}
|w| = \frac{1}{\sqrt{2}}
\sqrt{|z_{\widehat{1}} |^2 + |z_{\widehat{2}} |^2} \, . \label{norm7} \end{equation*} The caret notation (${\widehat{1}}$ and ${\widehat{2}}$) will be used systematically in connection with idempotent decompositions, with the purpose of easily distinguishing different types of indices. As a consequence of~\eqref{2.11} and~\eqref{2.12}, one can check that if $\sqrt[n]{z_{\widehat{1}}}$ is an $n$th root of $z_{\widehat{1}}$ and $\sqrt[n]{z_{\widehat{2}}}$ is an $n$th root of $z_{\widehat{2}}$, then $\sqrt[n]{z_{\widehat{1}}} \, \mathbf{e_1} + \sqrt[n]{z_{\widehat{2}}} \, \mathbf{e_2}$ is an $n$th root of $w$.
The uniqueness of the idempotent decomposition allows the introduction of two projection operators as \begin{align*} P_1: w \in\mathbb{M}(2)&\mapsto z_{\widehat{1}} \in\mathbb{C}(\mathbf{i_1}),\label{2.14}\\ P_2: w \in\mathbb{M}(2)&\mapsto z_{\widehat{2}} \in\mathbb{C}(\mathbf{i_1}). \end{align*} The $P_k$ ($k = 1, 2$) satisfies \begin{equation*} [P_k]^2=P_k, \qquad P_1\mathbf{e_1}+P_2\mathbf{e_2}=\mathbf{Id}, \label{2.16} \end{equation*} and, for $s,t\in\mathbb{M}(2)$, \begin{equation*} P_k(s+t)=P_k(s)+P_k(t), \qquad P_k(s\cdot t)=P_k(s)\cdot P_k(t) .\label{2.17} \end{equation*}
The product of two bicomplex numbers $w$ and $w'$ can be written in the idempotent basis as
\begin{align*} w \cdot w' = (z_{\widehat{1}} \mathbf{e_1} + z_{\widehat{2}} \mathbf{e_2}) \cdot (z'_{\widehat{1}} \mathbf{e_1} + z'_{\widehat{2}} \mathbf{e_2}) = z_{\widehat{1}} z'_{\widehat{1}} \mathbf{e_1} + z_{\widehat{2}} z'_{\widehat{2}} \mathbf{e_2} .\label{2.20} \end{align*}
Since 1 is uniquely decomposed as $\mathbf{e_1} + \mathbf{e_2}$, we can see that $w \cdot w' = 1$ if and only if $z_{\widehat{1}} z'_{\widehat{1}} = 1 = z_{\widehat{2}} z'_{\widehat{2}}$. Thus $w$ has an inverse if and only if $z_{\widehat{1}} \neq 0 \neq z_{\widehat{2}}$, and the inverse $w^{-1}$ is then equal to $(z_{\widehat{1}})^{-1} \mathbf{e_1} + (z_{\widehat{2}})^{-1} \mathbf{e_2}$. A nonzero $w$ that does not have an inverse has the property that either $z_{\widehat{1}} = 0$ or $z_{\widehat{2}} = 0$, and such a $w$ is a divisor of zero. Zero divisors make up the so-called \emph{null cone} $\mathcal{NC}$. That terminology comes from the fact that when $w$ is written as in~\eqref{2.1}, zero divisors are such that $z_1^2 + z_2^2 = 0$.
Any hyperbolic number can be written in the idempotent basis as $x_{\widehat{1}} \mathbf{e_1} + x_{\widehat{2}} \mathbf{e_2}$, with $x_{\widehat{1}}$ and $x_{\widehat{2}}$ in~$\mathbb{R}$. We define the set~$\mathbb{D}_+$ of positive hyperbolic numbers as
\begin{equation*}
\mathbb{D}_+:= \{ x_{\widehat{1}} \mathbf{e_1} + x_{\widehat{2}} \mathbf{e_2} ~|~ x_{\widehat{1}}, x_{\widehat{2}} \geq 0 \}. \label{2.21} \end{equation*}
Since $w^{\dag_3} = \bar{z}_{\widehat{1}} \mathbf{e_1} + \bar{z}_{\widehat{2}} \mathbf{e_2}$, it is clear that $w \cdot w^{\dag_3} \in \mathbb{D}_+$ for any $w$ in $\mathbb{M}(2)$.
\subsection{$\mathbb{M}(2)$-Module and Scalar Product}\label{Module}
The set of bicomplex numbers is a commutative ring. Just like vector spaces are defined over fields, modules are defined over rings. A module~$M$ defined over the ring of bicomplex numbers is called an $\mathbb{M}(2)$-\emph{module}~\cite{Rochon3, GMR2, GMR}.
Let $M$ be an $\mathbb{M}(2)$-module. For $k=1, 2$, we define $V_k$ as the set of all elements of the form $\e{k} \ket{\psi}$, with $\ket{\psi} \in M$. Succinctly, $V_1:=\eo M$ and $V_2:=\et M$. In fact, $V_k$ is a vector space over $\mathbb{C}(\mathbf{i_1})$ and any element $\ket{v_k} \in V_k$ satisfies $\ket{v_k} = \e{k} \ket{v_k}$ for $k=1,2$. For arbitrary $\mathbb{M}(2)$-modules, vector spaces $V_1$ and $V_2$ bear no structural similarities. For more specific modules, however, they may share structure. It was shown in~\cite{GMR2} that if~$M$ is a finite-dimensional free $\mathbb{M}(2)$-module, then $V_1$ and~$V_2$ have the same dimension.
For any $\ket{\psi}\in M$, there exist a unique decomposition
\begin{align} \ket{\psi} = \ket{v_1} + \ket{v_2}, \label{2.31} \end{align}
where $v_k\in V_k$, $k=1,2$.
It will be useful to rewrite \eqref{2.31} as
\begin{align*} \ket{\psi} = \ket{\psi_\mathbf{1}} + \ket{\psi_\mathbf{2}} , \end{align*}
where
\begin{align*} \ket{\psi_\mathbf{1}} := \eo\ket{\psi} && \text{and} && \ket{\psi_\mathbf{2}} := \et\ket{\psi} . \end{align*}
In fact, the $\mathbb{M}(2)$-module $M$ can be viewed as a vector space $M'$ over $\ensuremath{\mathbb{C}}(\bo)$, and $M'=V_1\oplus V_2.$ From a set-theoretical point of view, $M$ and $M'$ are identical. In this sense we can say, perhaps improperly, that the \textbf{module} $M$ can be decomposed into the direct sum of two vector spaces over $\ensuremath{\mathbb{C}}(\bo)$, i.e.\ $M=V_1\oplus V_2.$
\subsubsection{Bicomplex Scalar Product}\label{bicomplex sc}
A \emph{bicomplex scalar product} maps two arbitrary kets $\ket{\psi}$ and $\ket{\phi}$ into a bicomplex number $(\ket{\psi}, \ket{\phi})$, so that the following always holds ($s \in \mathbb{M}(2)$):
\begin{enumerate} \item $(\ket{\psi}, \ket{\phi} + \ket{\chi}) =(\ket{\psi}, \ket{\phi}) + (\ket{\psi}, \ket{\chi})$; \item $(\ket{\psi}, s \ket{\phi}) = s (\ket{\psi},\ket{\phi})$; \item $(\ket{\psi}, \ket{\phi}) = (\ket{\phi}, \ket{\psi})^{\dagger_3}$; \item $(\ket{\psi}, \ket{\psi}) =0~\Leftrightarrow~\ket{\psi}=0$. \end{enumerate}
The bicomplex scalar product was defined in~\cite{Rochon3} where, as in this paper, the physicists' convention is used for the order of elements in the product.
Property $3$ implies that $(\ket{\psi}, \ket{\psi})\in\mathbb{D}$, while properties 2 and 3 together imply that $(s \ket{\psi}, \ket{\phi}) = s^{\dagger_3} (\ket{\psi},\ket{\phi})$. However, in this work we will also require the bicomplex scalar product $\(\cdot,\cdot\)$ to be \textit{hyperbolic positive}, i.e.
\begin{align*} (\ket{\psi},\ket{\psi})\in\mathbb{D}_{+},\mbox{ }\forall\ket{\psi}\in M. \end{align*}
This is a necessary condition if we want to recover the standard quantum mechanics from the bicomplex one (see \cite{GMR3}).
\begin{definition} Let $M$ be a $\mathbb{T}$-module and let $(\cdot,\cdot)$ be a bicomplex scalar product defined on $M$. The space $\{M, (\cdot,\cdot)\}$ is called a $\mathbb{M}(2)$-inner product space, or bicomplex pre-Hilbert space. When no confusion arises, $\{M, (\cdot,\cdot)\}$ will simply be denoted by~$M$. \end{definition}
In this work, we will sometimes use the Dirac notation
\begin{align*} (\ket{\psi},\ket{\phi})=\braket{\psi}{\phi} \end{align*}
for the scalar product. The one-to-one correspondence between \emph{bra} $\bra{\cdot}$ and \emph{ket} $\ket{\cdot}$ can be established from the Bicomplex Riesz Representation Theorem \cite[Th. 3.7]{GMR}. As in \cite{GeRo}, subindices will be used inside the ket notation. In fact, this is simply a convenient way to deal with the Dirac notation in $V_1$ and $V_2$. Note that the following projection of a bicomplex scalar product: \begin{equation*} (\cdot,\cdot)_{\widehat{k}}:=P_k((\cdot,\cdot)):M\times M\longrightarrow \ensuremath{\mathbb{C}}(\bo) \end{equation*} is a \textbf{standard scalar product} on $V_k$, for $k=1,2$. One easily show \cite{GMR} that
\begin{align} (\ket{\psi}, \ket{\phi}) &= \mathbf{e_1}\P{1}{(\ket{\psi_\mathbf{1}}, \ket{\phi_\mathbf{1}})} + \mathbf{e_2}\P{2}{(\ket{\psi_\mathbf{2}}, \ket{\phi_\mathbf{2}})} \notag\\ &=\mathbf{e_1}\scalarmath{\ket{\psi_\mathbf{1}}}{\ket{\phi_\mathbf{1}}}_{\widehat{1}}+\mathbf{e_2}\scalarmath{\ket{\psi_\mathbf{2}}}{\ket{\phi_\mathbf{2}}}_{\widehat{2}}. \notag\\ &=\mathbf{e_1}\braket{\psi_\mathbf{1}}{\phi_\mathbf{1}}_{\widehat{1}}+\mathbf{e_2}\braket{\psi_\mathbf{2}}{\phi_\mathbf{2}}_{\widehat{2}}\label{2.36}. \end{align}
We point out that a bicomplex scalar product is \textbf{completely characterized} by the two standard scalar products $\scalarmath{\cdot}{\cdot}_{\widehat{k}}$ on $V_k$. In fact, if $\scalarmath{\cdot}{\cdot}_{\widehat{k}}$ is an arbitrary scalar product on $V_k$, for $k=1,2$, then $\scalarmath{\cdot}{\cdot}$ defined as in \eqref{2.36} is a bicomplex scalar product on $M$.
From this scalar product, we can define a \textbf{norm} on the vector space $M'$: \begin{align}
\big{|}\big{|}\ket{\phi}\big{|}\big{|} &:= \frac{1}{\sqrt{2}} \sqrt{\scalarmath{\ket{\phi_\mathbf{1}}} {\ket{\phi_\mathbf{1}}}_{\widehat{1}} + \scalarmath{\ket{\phi_\mathbf{2}}}{\ket{\phi_\mathbf{2}}}_{\widehat{2}}} \notag\\
&=\frac{1}{\sqrt{2}} \sqrt{ \big{|}\ket{\phi_\mathbf{1}}\big{|}^{2}_{1}
+ \big{|}\ket{\phi_\mathbf{2}}\big{|}^{2}_{2}} \, . \label{T-norm} \end{align} Here we wrote \begin{equation*}
\big{|}\ket{\phi_\mathbf{k}} \big{|}_{k} = \sqrt{\scalarmath{\ket{\phi_\mathbf{k}}} {\ket{\phi_\mathbf{k}}}_{\widehat{k}}} \, ,
\end{equation*}
where $|\cdot|_k$ is the natural scalar-product-induced norm on~$V_k$. Moreover, \begin{equation}
\big{|}\big{|}\ket{\phi}\big{|}\big{|} = \frac{1}{\sqrt{2}} \sqrt{\scalarmath{\ket{\phi_\mathbf{1}}} {\ket{\phi_\mathbf{1}}}_{\widehat{1}} + \scalarmath{\ket{\phi_\mathbf{2}}}{\ket{\phi_\mathbf{2}}}_{\widehat{2}}} \notag
= \big{|} \sqrt{\scalarmath{\ket{\phi}}{\ket{\phi}}} \big{|}. \label{T-norma} \end{equation}
\begin{definition}
Let $M$ be an $\mathbb{M}(2)$-module and let $M'$ be the associated vector space. We say that $\|\cdot\|:M\longrightarrow \mathbb{R}$ is a \textbf{$\mathbb{M}(2)$-norm} on $M$ if the following holds:
\noindent 1. $\|\cdot\|:M'\longrightarrow \mathbb{R}$ is a norm;\\
2. $\big{\|}w\cdot \ket{\psi}\big{\|}\leq \sqrt{2}
\big{|}w\big{|}\cdot\big{\|}\ket{\psi}\big{\|}$, $\forall w\in\mathbb{T}$, $\forall \ket{\psi}\in M$. \label{norm} \end{definition}
\noindent A $\mathbb{M}(2)$-module with a \textbf{$\mathbb{M}(2)$-norm} is called a
\textbf{normed $\mathbb{M}(2)$-module}. It is easy to check that $\|\cdot\|$ in \eqref{T-norm} is a \textbf{$\mathbb{M}(2)$-norm} on $M$ and that the $\mathbb{M}(2)$-module $M$ is \textbf{complete} with respect to the following metric on $M$: \begin{equation*}
d(\ket{\phi},\ket{\psi})=\big{|}\big{|}\ket{\phi}-\ket{\psi}\big{|}\big{|} \end{equation*} if and only if $V_1$ and $V_2$ are complete (see \cite{GMR}). \begin{definition} A bicomplex Hilbert space is a $\mathbb{M}(2)$-inner product space $M$ which is complete with respect to the induced $\mathbb{M}(2)$-norm \eqref{T-norm}. \label{Hilbert} \end{definition}
\section{Main results}
Throughout the text, by a \textbf{bicomplex Hilbert space} we shall mean an infinite dimensional bicomplex Hilbert space. A normed $\mathbb{M}(2)$-module with a Schauder $\mathbb{M}(2)$-basis is called a \textbf{countable $\mathbb{M}(2)$-module}.
\begin{definition} A bicomplex Hilbert space $M$ is said to be \textit{separable by a basis} if it has a Schauder $\mathbb{M}(2)$-basis. \end{definition}
We note that by Theorem 3.10 in \cite{GMR}, any Schauder $\mathbb{M}(2)$-basis of $M$ can be orthonormalized.
\begin{remark} A topological space $S$ is called \textit{separable} if it admits a countable dense subset $W$. \end{remark}
\begin{proposition}
Let $\braket{\cdot}{\cdot}$ be a bicomplex inner product in the bicomplex Hilbert space $M$ and let $||\cdot||$ be the induced norm. If the sequences $\{\ket{\psi_n}\}$ and $\{\ket{\phi_n}\}$ in $M$ converge to $\{\ket{\psi}\}$ and $\{\ket{\phi}\}$ respectively, then the sequence of inner products $\{ \braket{{\psi_n}}{{\phi_n}}\}$ converges to $\braket{{\psi}}{{\phi}}$. \label{SCALIM} \end{proposition} \begin{proof} First observe that: $\braket{{\psi_n}}{{\phi_n}}-\braket{{\psi}}{{\phi}}$ \begin{eqnarray*} &=& \braket{{\psi_n}}{{\phi_n}}-\braket{{\psi}}{{\phi_n}}+\braket{{\psi}}{{\phi_n}}-\braket{{\psi}}{{\phi}}\\ &=& \braket{\psi_n - \psi}{\phi_n} + \braket{\psi}{\phi_n - \phi}\\ &=& \braket{{\psi_n}-{\psi}}{{\phi_n}-{\phi}} +\braket{{\psi_n}-{\psi}}{{\phi}} + \braket{{\psi}}{{\phi_n}-{\phi}}. \end{eqnarray*} From this we get by the \textbf{bicomplex Schwarz inequality} (\cite{GMR}, Theorem 3.8):
$\big{|}\braket{{\psi_n}}{{\phi_n}}-\braket{{\psi}}{{\phi}}\big{|}$ \begin{eqnarray*}
&=& \big{|}\braket{{\psi_n}-{\psi}}{{\phi_n}-{\phi}} +\braket{{\psi_n}-{\psi}}{{\phi}}
+ \braket{{\psi}}{{\phi_n}-{\phi}}\big{|}\\
&\leq& \big{|}\braket{{\psi_n}-{\psi}}{{\phi_n}-{\phi}}\big{|}
+\big{|}\braket{{\psi_n}-{\psi}}{{\phi}}\big{|}
+\big{|}\braket{{\psi}}{{\phi_n}-{\phi}}\big{|}\\
&\leq& \big{[}\sqrt{2}\big{|}\big{|}\ket{\psi_n}-\ket{\psi}\big{|}\big{|}\cdot\big{|}\big{|}\ket{\phi_n}-\ket{\phi}\big{|}\big{|}
+\sqrt{2}\big{|}\big{|}\ket{\psi_n}-\ket{\psi}\big{|}\big{|}\cdot\big{|}\big{|}\ket{\phi}\big{|}\big{|}\\
& &+\sqrt{2}\big{|}\big{|}\ket{\psi}\big{|}\big{|}\cdot\big{|}\big{|}\ket{\phi_n}-\ket{\phi}\big{|}\big{|}\big{]}. \end{eqnarray*} The proposition now follows easily. \end{proof}
\begin{theorem}[Best Approximation Theorem] Let $\{ \ket{\psi_n}\}$ be an arbitrary orthonormal sequence in the bicomplex Hilbert space $M=H_1\oplus H_2$, and let $\alpha_1,\ldots,\alpha_n$ be a set of bicomplex numbers. Then for all $\ket{\psi}\in M$,
$$\big{|}\big{|}\ket{\psi}-\sum_{l=0}^{n}\alpha_l \ket{\psi_l}\big{|}\big{|}\geq \big{|}\big{|}\ket{\psi}-\sum_{l=0}^{n}\braket{{\psi_l}}{{\psi}} \ket{\psi_l}\big{|}\big{|}.$$ \label{BEST} \end{theorem} \begin{proof} By definition of the bicomplex inner product, the set $\{\ket{\psi_{n\mathbf{k}}}\}$ is also an arbitrary orthonormal sequence in the Hilbert space $H_k$ for $k=1,2$. Therefore, using the classical Best Approximation Theorem (see \cite{Hansen}, P.61) on the Hilbert spaces $H_1$ and $H_2$, we obtain for $k=1,2$: \begin{equation*}
\big{|}\ket{\psi_\mathbf{k}}-\sum_{l=0}^{n}P_k(\alpha_l) \ket{\psi_{l\mathbf{k}}}\big{|}_{k}\geq
\big{|}\ket{\psi_\mathbf{k}}-\sum_{l=0}^{n}\braket{{\psi_{l\mathbf{k}}}}{{\psi_\mathbf{k}}}_{\widehat{k}} \ket{\psi_{l\mathbf{k}}}\big{|}_{k}. \end{equation*} Hence, by definition of the $\mathbb{M}(2)$-norm, we have that \begin{eqnarray*}
\big{|}\big{|}\ket{\psi}-\sum_{l=0}^{n}\alpha_l \ket{\psi_l}\big{|}\big{|} &=& \frac{1}{\sqrt{2}} \sqrt{\sum_{k=1}^{2} \big{|}\ket{\psi_\mathbf{k}}-\sum_{l=0}^{n}P_k(\alpha_l) \ket{\psi_{l\mathbf{k}}}\big{|}_{k}^{2}}\\
&\geq& \frac{1}{\sqrt{2}} \sqrt{\sum_{k=1}^{2} \big{|}\ket{\psi_\mathbf{k}}-\sum_{l=0}^{n}
\braket{{\psi_{l\mathbf{k}}}}{{\psi_\mathbf{k}}}_{\widehat{k}} \ket{\psi_{l\mathbf{k}}}\big{|}_{k}^{2}}\\
&=& \big{|}\big{|}\ket{\psi}-\sum_{l=0}^{n}\braket{{\psi_l}}{{\psi}} \ket{\psi_l}\big{|}\big{|}. \end{eqnarray*} \end{proof}
An important consequence of the Best Approximation Theorem is that an orthonormal basis for a dense subspace of a bicomplex Hilbert space is actually an orthonormal basis in the full bicomplex Hilbert space. This is very useful result for the construction of specific orthonormal basis in separable Hilbert spaces. The precise result is as follows.
\begin{theorem} Let $N$ be a dense subspace of the bicomplex Hilbert space $M$, and assume that $\{ \ket{m_l} \}$ is an orthonormal Schauder $\mathbb{M}(2)$-basis for $N$. Then $\{ \ket{m_l} \}$ is also an orthonormal Schauder $\mathbb{M}(2)$-basis for $M$. \label{DENSE} \end{theorem} \begin{proof} Since $\{ \ket{m_l} \}$ is a Schauder $\mathbb{M}(2)$-basis for $N$, any $\ket{\psi}\in N$ admits a unique expansion as an infinite series $\ket{\psi}=\sum_{l=1}^{\infty} \alpha_l \ket{m_l}$. In fact, $$\ket{\psi}=\sum_{l=1}^{\infty} \braket{{m_l}}{{\psi}} \ket{m_l}.$$ This follows by Proposition \ref{SCALIM} and the short computation $$\braket{{m_l}}{{\psi}}=\braket{{m_l}}{\lim_{n\rightarrow \infty}\sum_{k=1}^{n}\alpha_k {m_k}} =\lim_{n\rightarrow \infty}\braket{{m_l}}{\sum_{k=1}^{n}\alpha_k {m_k}} =\alpha_l,$$ valid for all $l\in\mathbb{N}$. Now, to complete the proof, let us prove that any ket $\ket{\phi}\in M$ admits the same expansion form: \begin{equation} \ket{\phi}=\sum_{l=1}^{\infty} \braket{{m_l}}{{\phi}} \ket{m_l}. \label{Unique} \end{equation}
To prove this assertion, let an arbitrary $\epsilon>0$ be given. Since, $N$ is dense in $M$, we can choose $\ket{\psi}\in N$, such that $\big{|}\big{|}\ket{\phi}-\ket{\psi} \big{|}\big{|}<\frac{\epsilon}{2}$. Now write $\ket{\psi}=\sum_{l=1}^{\infty} \braket{{m_l}}{{\psi}} \ket{m_l}$, and choose $n_0\in\mathbb{N}$ such that
$$n\geq n_0 \Rightarrow \big{|}\big{|} \ket{\psi}-\sum_{l=1}^{n} \braket{{m_l}}{{\psi}} \ket{m_l} \big{|}\big{|}<\frac{\epsilon}{2}.$$ By the Best Approximation Theorem, we then get for all $n\geq n_0$, \begin{eqnarray*}
\big{|}\big{|} \ket{\phi}-\sum_{l=1}^{n} \braket{{m_l}}{{\phi}} \ket{m_l} \big{|}\big{|} &\leq&
\big{|}\big{|} \ket{\phi}-\sum_{l=1}^{n} \braket{{m_l}}{{\psi}} \ket{m_l} \big{|}\big{|} \\
&\leq& \big{|}\big{|}\ket{\phi}-\ket{\psi} \big{|}\big{|}+\big{|}\big{|}\ket{\psi}- \sum_{l=1}^{n} \braket{{m_l}}{{\psi}} \ket{m_l}\big{|}\big{|}\\ &\leq& \frac{\epsilon}{2}+\frac{\epsilon}{2}. \end{eqnarray*} Hence, \begin{equation*} \ket{\phi}=\lim_{n\rightarrow\infty} \sum_{l=1}^{n} \braket{{m_l}}{{\phi}} \ket{m_l}=\sum_{l=1}^{\infty} \braket{{m_l}}{{\phi}} \ket{m_l}. \end{equation*} This prove that $\{ \ket{m_l} \}$ is an orthonormal Schauder $\mathbb{M}(2)$-basis for $M$. \end{proof}
The next result shows that all separable bicomplex Hilbert spaces are separable by a basis.
\begin{lemma} Every separable bicomplex Hilbert space $M$ has an orthonormal Schauder $\mathbb{M}(2)$-basis. \label{SHS03} \end{lemma} \begin{proof} By the definition of separability, $M$ contains a countable, dense subset $W$ of kets in $M$. Consider the linear subspace $U$ in $M$ consisting of all finite bicomplex linear combinations of kets in $W$ - the \textit{bicomplex linear span} of $W$. Clearly, $U$ is a dense sub-$\mathbb{M}(2)$-module in $M$. By the construction of $U$ we can eliminate kets from the countable set $W$ one after the other to get a (bicomplex) linearly independent set $\{ \ket{\phi_n} \}$ (finite, or countable) of kets in $U$ that spans $U$. However, a sub-$\mathbb{M}(2)$-module $U$ in $M$ of finite dimension is a complete space, thus a closed set in $M$, and then $U=\bar{U}=M$ a contradiction with our hypothesis. Therefore, the set $\{ \ket{\phi_n} \}$ is a countable (bicomplex) linearly independent set of kets in $U$. Now, since no $\ket{\phi_n}$ (and thus no $\braket{{\phi_n}}{{\phi_n}}$) can belongs to the null cone, the classical Gram-Schmidt process can be applied (see \cite{GMR2}, P.14). Hence, we can turn the sequence $\{ \ket{\phi_n} \}$ into an orthonormal sequence $\{ \ket{\psi_n} \}$ with the property that for all $n\in\ensuremath{\mathbb{N}}$, $$\mbox{span} \{ \ket{\phi_n} \}_{l=1}^{n}=\mbox{span} \{ \ket{\psi_l} \}_{l=1}^{n}$$ Since $\{ \ket{\psi_l} \}$ is orthonormal, we can use $\{ \ket{\psi_l} \}$ as a Schauder $\mathbb{M}(2)$-basis to generate a linear subspace $N$ in $M$ (for the unicity, see the proof of Theorem \ref{DENSE}). Then $N$ is a dense sub-$\mathbb{M}(2)$-module in $M$, since $U$ is a dense sub-$\mathbb{M}(2)$-module in $N$. The latter follows since any ket $\ket{\psi}\in N$ can be expanded into a series $\ket{\psi}=\sum_{l=1}^{\infty}\alpha_l \ket{\psi_l}$, showing that $\ket{\psi}=\lim_{n\rightarrow \infty}\sum_{l=1}^{n}\alpha_l \ket{\psi_l}$, and hence that $\ket{\psi}$ is the limit of a sequence of kets in $U$.
By construction, $\{ \ket{\psi_l} \}$ is an orthonormal Schauder $\mathbb{M}(2)$-basis for $N$ and hence by Theorem \ref{DENSE} also for $M$. \end{proof}
\begin{theorem} If $M$ is a separable bicomplex Hilbert space, then $H_k$ ($k=1,2$) is an infinite dimensional separable complex Hilbert space. \label{SHS} \end{theorem} \begin{proof} From Lemma \ref{SHS03}, $M=H_1\oplus H_2$ has an orthonormal Schauder $\mathbb{M}(2)$-basis $\{ \ket{\psi_l} \}$. It is easy to see that $\{ \ket{\psi_{l\mathbf{k}}} \}$ is also an orthonormal Schauder basis for $H_k$ ($k=1,2$). Hence, $H_k$ ($k=1,2$) is separable by a basis. Now, from Theorem 3.3.6. in \cite{Hansen}, $H_k$ ($k=1,2$) is an infinite dimensional separable complex Hilbert space. \end{proof}
\begin{definition} Denote by $l^2_2$, the space of all (real, complex or bicomplex) sequences $\{ w_l \}$ such that $$
\sum_{l=1}^{\infty}|w_l|^{2}<\infty. $$ \end{definition}
The bicomplex $l^2_2$ space is clearly an $\mathbb{M}(2)$-module. The norm of the associated vector space $({l^2_2})'$ over $\mathbb{C}(\ik{1})$ is defined by \begin{equation*}
||\{ w_l \}||_2=\Big( \sum_{l=1}^{\infty}|w_l|^{2} \Big)^{\frac{1}{2}}. \end{equation*}
\begin{theorem} $l^2_2$ is a bicompex Hilbert space. \end{theorem} \begin{proof} Let us prove that $({l^2_2})'=(\eo l^2) \oplus (\et l^2)$. This comes automatically from the fact that any bicomplex sequence $\{ w_l \}$ can be decomposed as the following sum of two sequences in $\mathbb{C}(\ik{1})$: $$\{ w_l \}=\eo\{ z_{1l}-z_{2l}\ik{1} \}+\et\{z_{1l}+z_{2l}\ik{1} \}.$$
To complete the proof, we need to verify that the norm $||\cdot||_2$
coincides with the induced $\mathbb{M}(2)$-norm of the bicomplex Hilbert space $(\eo l^2) \oplus (\et l^2)$. Let $||\cdot||$ be the induced $\mathbb{M}(2)$-norm of the bicomplex Hilbert space $(\eo l^2) \oplus (\et l^2)$. Thus
$$\big{|}\big{|}\{ w_l \}\big{|}\big{|}=
\frac{1}{\sqrt{2}} \sqrt{ \big{|}\{ z_{1l}-z_{2l}\ik{1} \}\big{|}^{2}_{1}
+ \big{|}\{z_{1l}+z_{2l}\ik{1} \}\big{|}^{2}_{2}}$$
where $\big{|}\cdot\big{|}_{1}=\big{|}\cdot\big{|}_{2}$ is the classical norm on $l^2$. Hence, \begin{align*}
\big{|}\big{|}\{ w_l \}\big{|}\big{|}
&= \frac{1}{\sqrt{2}} \sqrt{ \big{|}\{ z_{1l}-z_{2l}\ik{1} \}\big{|}^{2}_{1}
+ \big{|}\{z_{1l}+z_{2l}\ik{1} \}\big{|}^{2}_{1}}\\
&= \frac{1}{\sqrt{2}} \sqrt{ \sum_{l=1}^{\infty}|z_{1l}-z_{2l}\ik{1}|^{2}
+\sum_{l=1}^{\infty}|z_{1l}+z_{2l}\ik{1}|^{2}}\\
&=\sqrt{\sum_{l=1}^{\infty}\frac{[|z_{1l}-z_{2l}\ik{1}|^{2}
+|z_{1l}+z_{2l}\ik{1}|^{2}]}{2}}\\
&=||\{ w_l \}||_2. \end{align*}
\end{proof}
We are now ready for the proof of the main result on the structure of infinite dimensional, separable bicomplex Hilbert space. We show that the space of square summable bicomplex sequences $l^2_2$ Define the projection $T_\mathbf{k}:M\longrightarrow V_k$ as \begin{equation*} T_\mathbf{k} \ket{\phi} := \e{k}T(\ket{\phi}), \; \forall \ket{\phi} \in M, \; k=1,2. \notag \end{equation*}
\noindent With this definition we have the following Lemma.
\begin{lemma} Let $M_1, M_2$ be two $\mathbb{M}(2)$-modules and $T:M_1\rightarrow M_2$ be a bicomplex linear function. Then $\forall\ket{\phi}\in M_1$ we have \begin{equation*} T_{\mathbf{k}}(\ket{\phi})=T(\ket{\phi_{\mathbf{k}}}), \ \ (k=1,2). \end{equation*} \label{ISO} \end{lemma} \begin{proof} \begin{align*} T_{\mathbf{k}}(\ket{\phi}) &= \ek (T(\ket{\phi}))\\ &= \ek (T(\ket{\phi_{\mathbf{1}}}+\ket{\phi_{\mathbf{2}}}))\\ &= T((\ket{\phi_{\mathbf{k}}})). \end{align*} \end{proof}
\begin{theorem}[Riesz-Fischer] Every separable bicomplex Hilbert space $M$ is isometrically isomorphic to the bicomplex Hilbert space $l^2_2$. \end{theorem} \begin{proof} From Lemma \ref{SHS03}, since $M=H_1\oplus H_2$ is a separable bicomplex Hilbert space, it has an orthonormal Schauder $\mathbb{M}(2)$-basis: $$\{\ket{m_1}, \dots ,\ket{m_l}, \dots\}.$$ Then each $\ket{\psi}\in M$ admits a unique decomposition as $$\ket{\psi}=\sum_{l=1}^{\infty}w_l\ket{m_l}, \ \ w_l\in\mathbb{M}(2).$$
Since the infinite series above converges, by Theorem 3.11 in
\cite{GMR}, the series $\sum_{l=1}^{\infty}\left|w_l\right|^{2}$ converges in $\ensuremath{\mathbb{R}}$ and thus $\{w_l\}\in l^2_2$. Now, define a map $T:M\rightarrow l^2_2$ as $$T(\ket{\phi})=\{w_l\}_{l=1}^{\infty} \ \ \forall \ket{\phi}\in M.$$ $T$ is a well defined map: Let $\ket{\phi}, \ \ket{\psi} \in M$ be such that $\ket{\phi}=\ket{\psi}$. Hence, $\sum_{l=1}^{\infty}w_l\ket{m_l} =\sum_{l=1}^{\infty}{w_l}{\prime}\ket{m_l}$ and then by the uniqueness of the representation we find that $w_l ={w_l}^{\prime}$ for each $l\in \ensuremath{\mathbb{N}}$, which further implies that $T(\ket{\phi})=T(\ket{\psi})$. Next, we show that $T$ is \textbf{bicomplex} linear. Let $\ket{\phi}, \ \ket{\psi} \in M$ and $\alpha, \ \beta \in \mathbb {T}$. Then, \begin{eqnarray*} T(\alpha \ket{\phi}+\beta \ket{\psi}) &=& T(\alpha\sum_{l=1}^{\infty}w_l\ket{m_l} +\beta \sum_{l=1}^{\infty}{w_l}{\prime}\ket{m_l})\\ &=& T(\sum_{l=1}^{\infty}(\alpha w_l)\ket{m_l} + \sum_{l=1}^{\infty}(\beta {w_l}{\prime})\ket{m_l})\\ &=& T(\sum_{l=1}^{\infty}(\alpha w_l + \beta {w_l}{\prime})\ket{m_l})\\ &=& \{\alpha w_l +\beta {w_l}^{\prime} \}\\ &=& \alpha \{ w_l\}+\beta \{{w_l}^{\prime} \}\\ &=& \alpha T(\ket{\phi})+\beta T(\ket{\psi}). \end{eqnarray*} Now, since $\{ \ket{w_l} \}$ is an orthonormal basis in $M$, by Equation \eqref{Unique} in Theorem \ref{DENSE}, every ket $\ket{\phi}\in M$ admits the unique expansion $$\ket{\phi}=\sum_{l=1}^{\infty} \braket{{m_l}}{{\phi}} \ket{m_l}.$$ Hence, $T$ is \textbf{injective}, since $T(\ket{\phi})=\{ \braket{{m_l}}{{\phi}} \}=0$ implies $\braket{{m_l}}{{\phi}}=0$ for all $l\in\mathbb{N}$, and thus $\ket{\phi}=0$. Moreover, $T$ is \textbf{surjective}, since for any element $\{ \alpha_l \}\in l^2_2$, the series $\ket{\xi}=\sum_{l=1}^{\infty}\alpha_l\ket{m_l}$ is convergent (Theorem 3.11 in \cite{GMR}). Finally we shall show that $T$ is an isometry. By Lemma \ref{ISO}, we have that \begin{eqnarray}
\big{\|}T(\ket{\phi})\big{\|}
&=& \big{|}\sqrt{\scalarmath{T(\ket{\phi})}{T(\ket{\phi})}}\big{|}\notag\\
&=& \big{|}\sqrt {\eo \scalarmath{T(\ket{\phi_{\mathbf{1}}})}{T(\ket{\phi_{\mathbf{1}}})}_{\widehat{1}} +
\et \scalarmath{T(\ket{\phi_{\mathbf{2}}})}{T(\ket{\phi_{\mathbf{2}}})}_{\widehat{2}}}\big{|}\notag\\ \label{iso9001} \end{eqnarray}
By Theorem \ref{SHS}, the classical Riesz-Fischer Theorem can be applied to $H_k$ where $T:H_k\rightarrow \ek l^2$ for $k=1,2$. Then we find that \begin{eqnarray*} \scalarmath{T(\ket{\phi_{\mathbf{k}}})}{T(\ket{\phi_{\mathbf{k}}})}_{\widehat{k}} &=&
{\big{|}T(\ket{\phi_{\mathbf{k}}})\big{|}^{2}_k} \\
&=& { \big{|}\ket{\phi_{\mathbf{k}}}\big{|}^{2}_k} \\ &=& \braket{{\phi_{\mathbf{k}}}}{{\phi_{\mathbf{k}}}}_{\widehat{k}} \end{eqnarray*}
for $k=1,2$, where $\big{|}\cdot\big{|}_{1}=\big{|}\cdot\big{|}_{2}$ is the classical norm on $l^2$. Thus, from Equation \eqref{iso9001}, we get that \begin{equation*}
\big{\|}T(\ket{\phi})\big{\|}=\big{\|}\ket{\phi}\big{\|}. \end{equation*} This proves that $T$ is an isometry. Hence $M$ is isometrically isomorphic to the bicomplex Hilbert space $l^2_2$. \end{proof}
\section*{Acknowledgment} DR is grateful to the Natural Sciences and Engineering Research Council of Canada for financial support.
\end{document} | arXiv |
\begin{definition}[Definition:Peano Curve]
A '''Peano curve''' is a fractal curve which can be written as a Lindenmayer system.
:800px
{{stub}}
{{NamedforDef|Giuseppe Peano|cat = Peano}}
\end{definition} | ProofWiki |
Research | Open | Published: 30 July 2015
Face scanning and spontaneous emotion preference in Cornelia de Lange syndrome and Rubinstein-Taybi syndrome
Hayley Crawford1,2,
Joanna Moss2,3,
Joseph P. McCleery4,
Giles M. Anderson5 &
Chris Oliver2
Journal of Neurodevelopmental Disordersvolume 7, Article number: 22 (2015) | Download Citation
Existing literature suggests differences in face scanning in individuals with different socio-behavioural characteristics. Cornelia de Lange syndrome (CdLS) and Rubinstein-Taybi syndrome (RTS) are two genetically defined neurodevelopmental disorders with unique profiles of social behaviour.
Here, we examine eye gaze to the eye and mouth regions of neutrally expressive faces, as well as the spontaneous visual preference for happy and disgusted facial expressions compared to neutral faces, in individuals with CdLS versus RTS.
Results indicate that the amount of time spent looking at the eye and mouth regions of faces was similar in 15 individuals with CdLS and 17 individuals with RTS. Both participant groups also showed a similar pattern of spontaneous visual preference for emotions.
These results provide insight into two rare, genetically defined neurodevelopmental disorders that have been reported to exhibit contrasting socio-behavioural characteristics and suggest that differences in social behaviour may not be sufficient to predict attention to the eye region of faces. These results also suggest that differences in the social behaviours of these two groups may be cognitively mediated rather than subcortically mediated.
The processing of social information is crucial for understanding the social world in which we live. In order to identify people during social interactions, we must process their facial features and characteristics. Furthermore, information gained from the face, such as expressions of emotion, can inform whether it is necessary to alter our interaction style. Exploring the face to spontaneously distinguish emotional expressions is part of successful social interaction. The eye region, in particular, has been proposed to be highly important for social interaction, due to the role it plays in conveying emotional states and communicative intent [1, 2].
Different face scanning has been reported in the literature for individuals who exhibit impairments in social interaction [3–5]. However, the majority of these studies have focussed primarily on individuals with autism spectrum disorder (ASD), although more recent studies have investigated visual exploration of social stimuli in Williams syndrome (WS). These two neurodevelopmental disorders are each associated with impairments and atypicalities in social interactions, but the presentation of these impairments is dramatically different. For example, individuals with ASD have often been reported to exhibit social withdrawal and reduced eye contact, whereas individuals with WS have been reported to exhibit hyper-sociability and heightened eye gaze [3–5].
In addition to distinctive patterns of eye looking in individuals with neurodevelopmental disorders associated with divergent socio-behavioural characteristics, reduced eye looking has been well documented in individuals with amygdala damage. For example, failure to spontaneously fixate to the eye region of static faces has been reported in a patient with bilateral amygdala damage [6]. Reduced eye contact during real social interactions has also been shown in a patient with amygdala damage [7], and a positive relationship between amygdala activation and looking to the eye region of faces has been documented in ASD [8]. Although evidence for amygdala dysfunction in ASD exists, it is somewhat inconsistent [9, 10]. In addition, reduced eye looking in ASD has been proven to be more inconsistent than once thought. Indeed, many studies have reported no difference in the amount of time individuals spend looking at the eye region compared to typically developing controls [11–13]. Rather, reduced eye looking in ASD has recently become most commonly associated with dynamic stimuli as opposed to static stimuli [12]. This suggests that reduced eye looking in ASD may be mediated by higher order cognitive mechanisms as opposed to biologically mediated amygdala dysfunction, with which reduced eye looking to static faces is a more consistent finding than in ASD.
It has been suggested that impaired facial emotion recognition may be due to reduced looking at the eye region, which has been argued to be important for communicating emotional expressions [2, 14, 15]. In support of this, eye contact has been reported to predict performance on a facial emotion recognition task in individuals with ASD [14], and increased emotion recognition performance has also been reported in those looking longer to the eye region [15]. Furthermore, neuropsychological patients with damage to the amygdala have been found to exhibit both reduced looking to the eye region of static faces and reduced ability to discriminate facial expressions of emotion [6, 16]. Both of these findings are in line with the hypothesis that looking to the eye region is important for successful emotion recognition, which in turn has been suggested to be important for successful social interaction [17]. However, more recent studies have also suggested that the eye region may not be as crucial as once thought, with a recent study showing reduced eye contact in a group of participants with ASD who also displayed intact emotion recognition skills [18]. Furthermore, a number of studies have reported relatively intact emotional face processing in ASD when comparison groups are well matched [19], thus highlighting the mixed nature of the findings regarding emotion recognition and looking to the eye region in ASD.
The aforementioned studies have revealed a putative pathway from eye gaze behaviour during the viewing of social stimuli and social characteristics in ASD versus WS [3–5]. Whilst these studies report clear findings that reflect the characteristic social behaviours of the groups studied, the two disorders are also associated with socio-behavioural profiles argued to be at polar ends of a spectrum [20]. Whether or not similar group level or individual associations are replicable with different neurodevelopmental disorders associated with contrasting socio-behavioural profiles has not yet been investigated. Cornelia de Lange syndrome (CdLS) and Rubinstein-Taybi syndrome (RTS) show a divergent pattern of social abilities. Whether or not the social behaviours exhibited by individuals with CdLS and RTS, namely social withdrawal/anxiety and social interest, can be linked to visual exploration of social information in the same was as previously reported in ASD and WS is of interest to the present study.
CdLS is a genetic disorder affecting approximately 1 in 40,000 live births [21] and is associated with intellectual disability, specific physical characteristics such as distinctive facial features and limb abnormalities and increased rates of ASD symptomatology [22–24]. CdLS is primarily caused by a deletion in the NIPBL gene located on chromosome 5 [25–27], whilst fewer cases have been reported that are caused by mutations on the SMC3 gene on chromosome 10 [28], the SMC1 gene [29], the HDAC8 gene [30], and the RAD21 gene [31].
Although CdLS is associated with an increased prevalence of ASD, the social impairments are subtly different. Most notably, individuals with CdLS have been reported to exhibit extreme social anxiety alongside selective mutism, whereas those with ASD typically withdraw from social interaction, but this withdrawal is not commonly or primarily attributed to extreme anxiety [24]. In addition, individuals with CdLS have been reported to exhibit reduced eye contact during situations which require initiation of speech [32] but less impaired eye contact than individuals with ASD [24]. One possible explanation for this pattern of eye looking in CdLS is that decreased eye contact is associated with anxiety and social withdrawal tendencies during situations with high social demand, whereas less impaired eye contact during a range of situations may be reflective of relatively higher social motivation in this population [24]. Interestingly, increased eye contact is also reported in typically developing individuals experiencing social anxiety [33]. Furthermore, a role for amygdala dysfunction in social anxiety has been postulated [34, 35].
The social impairments documented in CdLS are not dissimilar to those seen in fragile X syndrome (FXS), with both groups generally being reported or suggested to exhibit heightened social anxiety alongside heightened social motivation [36]. Eye-tracking methodology has previously been used to investigate face scanning in individuals with FXS, with the majority of these studies reporting reduced looking to the eye region in comparison to typically developing individuals [37–40] and in comparison to children with ASD [41]. Interestingly, FXS is associated with amygdala dysfunction [42], which, with consistent reports of reduced looking to the eye region of faces, may indicate that the social impairments observed in this group are somewhat subcortically mediated by amygdala dysfunction. Furthermore, direct comparisons of the brain structure in individuals with FXS and ASD have revealed smaller amygdala size in those with FXS compared to those with ASD [10]. This provides further support for the notion that eye looking in ASD may be mediated by higher order cognitive mechanisms as opposed to biologically mediated by amygdala dysfunction, as is more likely to be the case in FXS.
Whilst eye contact has been investigated in real-life social situations in people with CdLS, the use of eye-tracking methodology to investigate face scanning has not previously been attempted. Investigating the visual exploration of social information such as faces and emotional expressions using robust methodological techniques is important due to the relationship this may have to the striking social impairments present in this group. Furthermore, brain-imaging studies have not been conducted in CdLS. Therefore, studying eye looking and emotion processing, which has been associated with amygdala damage, may further our understanding of amygdala function in an under-researched population.
RTS is also a genetic syndrome associated with intellectual disability, affecting approximately one in 100,000–125,000 live births [43]. Mutations in the CREB-binding protein gene (CBP) account for approximately 40 % of cases, whereas mutations in the EP300 gene account for a limited number of cases [44–48]. Whilst studies investigating the social characteristics of RTS are limited, those that have been conducted to date suggest that social skills are largely intact in this group relative to their level of intellectual functioning [49]. Individuals with RTS have been reported to initiate and maintain social contact despite cognitive impairments [47]. One study, for example, described three children with RTS as being friendly and as making good social contacts [50]. Families of children with RTS have also described them as friendly and loving [50–52]. Increased social interest in children with RTS particularly relating to eye contact, initiating play, and use of facial expressions, compared to a group of children matched for age, gender, and developmental ability [53] has also been reported. However, this report of increased social interest given the level of intellectual ability may be age-specific, as an increase in anxiety and depression in adults with RTS compared to children with RTS has recently been reported [54]. However, as the majority of research indicates typical levels of social interest in this group, if not increased, it would be interesting to investigate the visual exploration of social information in this syndrome group. Similarly to CdLS, eye-tracking methodology has not yet been used to investigate looking patterns to social stimuli in this group.
The current study uses eye-tracking methodology to investigate spontaneous emotion preference for happy versus neutral and disgust versus neutral facial expressions and face scanning in individuals with CdLS versus RTS. Happiness and disgust were the expressions used in the present study due to their contrast in emotional valence. Many negative emotional expressions, such as sadness, fear, and anger, can often be experienced cognitively with no distinctive facial expression. For example, one may not always display a frown when experiencing sadness. Disgust was chosen as the negative emotional expression of interest for the current study as it is depicted facially. Patterns of eye gaze across the eye, mouth, and other regions of the face were also measured during "standard" trials, which presented pairs of faces posed in neutral expressions, in order to examine and compare gaze to the eye region across participant groups.
The aim of this study is to determine whether or not previous findings in individuals with contrasting profiles of social behaviour, namely ASD (reduced) and WS (enhanced), replicate in the visual exploration of social information of two syndrome groups that exhibit similarly contrasting socio-behavioural characteristics. As impaired eye looking in static faces is associated with amygdala dysfunction, this study aims to further the understanding of whether the documented differences in social behaviour of CdLS and RTS are subcortically or cognitively mediated. Based on previous literature indicating differences in face processing for groups with divergent profiles of social behaviour, we hypothesised that those with CdLS and RTS would show contrasting patterns of looking to the eye region. Specifically, based on reports of heightened social anxiety in people with CdLS [24, 32] and reports of social interest being relatively intact in individuals with RTS [53], we predicted that individuals with CdLS would exhibit less looking to the eye region than individuals with RTS.
The methods used here are identical to those used and described in a previous study conducted by the authors to investigate face processing in Fragile X Syndrome and Autism Spectrum Disorder [41].
Fifteen individuals with CdLS (seven female, Mage = 18.42, SD = 9.78) and 17 individuals with RTS (10 female, Mage = 17.33, SD = 10.14) were included in the analyses. An additional one participant with RTS was tested but did not provide reliable data due to providing over 40 % invalid trials in one condition. A trial during which the participant did not look at either face was considered invalid. Table 1 presents the characteristics of the final study populations. As Table 1 shows, participants with CdLS and RTS were matched on chronological age, gender, severity of autistic impairments, as measured by the social communication questionnaire (SCQ; [55]), and global and communicative adaptive behaviour abilities, as measured by the Vineland adaptive behaviour scale (VABS; [56]). SCQ data was not returned for one participant with RTS. Participants were recruited through the Cerebra Centre for Neurodevelopmental Disorders, University of Birmingham (UoB) participant database, through the Cornelia de Lange Foundation UK and Ireland, and through the Rubinstein-Taybi syndrome UK support group. All participants had a confirmed diagnosis from a professional (paediatrician, general practitioner, or clinical geneticist). Two participants with CdLS were tested at the UoB. All remaining participants were tested at syndrome support group family meetings. This study was reviewed and approved by the School of Psychology Ethics Committee at the UoB. Written consent was obtained from participants aged 16 years and over and parents of children under 16 years of age before their participation in the study.
Table 1 Participant characteristics and alpha level for comparison between CdLS and RTS participants
The experimental procedures described here are the same as those used by the authors in a previous study, which reported a difference in looking times to the eye region of facial stimuli in individuals with FXS and ASD [41]. The stimuli were generated by the Experiment Builder programme (SR Research, Ontario, Canada) and presented on a 19-in. CRT screen at a screen resolution of 1024 × 768. Participants placed their head on a chin rest 0.6 m from the screen, in a dimly lit room with windows blacked-out to avoid luminance changes. Chin rest and desk heights were adjusted so that eye gaze was central to the display screen. Eye movements were recorded using an Eyelink 1000 Tower Mount system, which runs with a spatial accuracy of 0.5–1 visual angle (°), a spatial resolution of 0.01°, and a temporal resolution of 2 (500 Hz). A five-point calibration was performed prior to each experimental block, as well as mid-block if necessary. A single-point drift correction to the calibration was made prior to every fifth trial. The eye-tracking camera was linked to a separate host PC to the one displaying the search stimuli. EyeLink software (SR research, Ontario, Canada) was used to control the camera and collect data and was synchronised via an Ethernet cable with display PC.
During the eye-tracking task, an animated dolphin measuring 0.96 × 1.43° of visual angle was used for calibration, as well as for drift correction and fixation "cross" prior to each trial. The 38 static colour photographs of male and female adult human faces were taken from the MacBrain Face Stimulus SetFootnote 1 [57]. During each trial, two faces were presented side-by-side. On the majority of trials, both faces displayed a neutral facial expression. For the remainder of trials, one of the two faces displayed a happy or disgusted expression. The faces displayed a straight-ahead gaze and an open mouth. Only the face, hair, and neck were visible. Faces subtended an average of 14.30 × 18.59° of visual angle were displayed on a white background. They were positioned side-by-side, separated by a gap of 7.179° of visual angle.
In addition to participants completing the eye-tracking task, the participant's primary caregiver completed a demographic questionnaire providing information about the participants' gender, date of birth, verbal ability (more/less than 30 signs/words), and mobility (ability to walk unaided). Information about the participant's diagnosis was also collected from caregivers including the specific diagnosis given, who gave the diagnosis, and when. Participants' primary caregivers also completed the SCQ [55], to assess behaviours associated with ASD such as social functioning and communication skills. A score of 15 or above is suggested by the authors of the SCQ to indicate the presence of an ASD. The Communication Skills, Daily Living Skills, and Socialisation Skills of the Vineland Adaptive Behaviour Scale—Second Edition, Survey Interview Form [56] was administered to primary caregivers to assess participants' adaptive behaviour abilities. The interview yields an adaptive behaviour composite (ABC) from the three domains. Standard scores, which are based on a sample of 3000 children, can be calculated for each domain and the ABC and reflect performance relative to participant chronological age. The standard scores for the communication domain and the ABC were used in the present study to ensure participants were matched on communicative and global adaptive behaviour abilities.
Eye-tracking task
All participants were instructed to remain still during testing. At the start of the eye-tracking task, the eye-tracker was calibrated using a five-point calibration. During calibration, participants fixated on an animated blue dolphin as it moved positions from the centre of the screen to various locations around the edges of the display area. The calibration was repeated until all participants achieved a full five-point calibration. In between each trial, the animated dolphin reappeared at the centre of the screen to act as a point of fixation. Every five trials, this individually presented dolphin served as a one-point drift correct to adjust calibration of the eye-tracker accounting for small head movements. If necessary, re-calibration was undertaken at this point and the trials resumed once calibration was successful.
Participants were presented with 80 trials, during which two faces were presented side-by-side for 1500 ms. The animated dolphin was displayed for 1000 ms in between trials, except for trials when a drift correct was performed. This was a passive viewing task. Therefore, participants were instructed to look wherever they wished whilst the faces were presented on the screen but to look at the dolphin that appeared between trials. Participants completed one of two experimental blocks, each with trials in a different pseudo-random trial presentation order. As a result of randomization, in one experimental block, 10 of 80 trials were "emotion" trials in which one emotionally expressive face was presented alongside one neutrally expressive face; in the other experimental block, 11 of 80 trials were "emotion" trials. The experimental block assigned to participants was counterbalanced within and across participant groups. The remaining trials were "standard" trials, in which two neutrally expressive faces were presented in order to habituate participants to the category of neutral facial expressions. To ensure participant's habituation to neutrally expressive faces, the beginning of the testing block commenced with at least seven "standard" trials prior to the presentation of any "emotion" trials. Throughout the remainder of the experiment, "emotion" trials were separated by a minimum of four "standard" trials. During "emotion" trials, the emotionally expressive face displayed either happiness or disgust and was equally likely to appear on the left or right side of the screen. Happy faces were presented during approximately half of the emotion trials in both experimental blocks. Disgust was presented during the remainder of the emotion trials. The eye-tracking task generally lasted less than 10 min but total experiment time varied slightly across participants due to differences in the amount of time it took to obtain successful calibration and whether participants accepted the option to take additional breaks during the drift-correct trials.
Participants completed the eye-tracking task, and parents of participants completed the SCQ and the VABS. The eye-tracking task was completed first. Parents completed the SCQ either whilst their child performed the eye-tracking task or at home and returned it to the researchers. The VABS was either administered face to face following the eye-tracking task or over the telephone following the testing session.
Fixations were assessed as occurring when eye movement did not exceed a velocity threshold of 30°/s, an acceleration threshold of 8000°/s2, or a motion threshold of 0.1°, and the pupil was not missing for three or more samples in a sequence. A fixation was assigned to a particular area of the face when the fixation coordinates were within a rectangular area (termed the "region(s) of interest" or ROI) assigned to the area in question. Face ROI was a rectangular shape positioned automatically to cover the face, hair, and neck of the models presented on the left and right side of the screen, whilst ROI for the left eye, right eye, and mouth for each individual face were identified manually using coordinates (see Fig. 1). The ROI for all stimuli were identical to those previously reported in a study using the same paradigm to investigate eye looking and emotion preference in FXS and ASD [41]. All data were subjected to the Shapiro-Wilk test for normality.Footnote 2 The mean number of trials with missing data (where participants did not look at either facial stimulus) was 4.133 for participants with CdLS and 4.82 for participants with RTS. Except where mentioned, the alpha level for significance was 0.05.
Example of face ROI, left and right eyes ROI, and mouth ROI; face ROI was a rectangular shape positioned automatically to cover the face, hair, and neck of models, whilst fixation coordinates within the rectangular areas were assigned to eyes and mouth ROI for each model, respectively
Participants with CdLS spent, on average, 37.8 % of trial time looking at the facial stimuli and 25.4 % of trial time looking at other areas on the screen. This was similar for participants with RTS as they spent 38.5 % of trial time looking at the facial stimuli and 25.9 % of trial time looking at other areas on the screen. On average, data from 35.8 and 35.6 % of trial time from participants with CdLS and RTS, respectively, were lost due to saccades, blinks, and inattention. Previously published data from typically developing children and adults [41], who completed the same paradigm, yielded similar percentages of lost data (41.4 and 34.3 %, respectively). Spontaneous emotion preference data are presented as proportion of trial spent looking, in seconds, at faces posed in happy, disgust, and neutral facial expressions. Eyes and mouth looking data were only analysed during standard trials, on which both faces presented neutral expressions. Eye looking data are presented as a ratio of the time spent looking at the eyes to the time spent looking at the face:
$$ \frac{\mathrm{Mean}\ \mathrm{time}\ \left(\mathrm{in}\ \mathrm{ms}\right)\ \mathrm{spent}\ \mathrm{looking}\ \mathrm{at}\ \mathrm{the}\ \mathrm{left}\ \mathrm{eye} + \mathrm{mean}\ \mathrm{time}\ \left(\mathrm{in}\ \mathrm{ms}\right)\ \mathrm{spent}\ \mathrm{looking}\ \mathrm{at}\ \mathrm{right}\ \mathrm{eye}}{\mathrm{Mean}\ \mathrm{time}\ \left(\mathrm{in}\ \mathrm{ms}\right)\ \mathrm{spent}\ \mathrm{looking}\ \mathrm{at}\ \mathrm{neutral}\ \mathrm{faces}} $$
Mouth looking data are presented as a ratio of the time spent looking at the mouth to the time spent looking at the face:
$$ \frac{\mathrm{Mean}\ \mathrm{time}\ \left(\mathrm{in}\ \mathrm{ms}\right)\ \mathrm{spent}\ \mathrm{looking}\ \mathrm{at}\ \mathrm{the}\ \mathrm{mouth}\ \mathrm{region}}{\mathrm{Mean}\ \mathrm{time}\ \Big(\mathrm{in}\ \mathrm{ms}\ \mathrm{spent}\ \mathrm{looking}\ \mathrm{at}\ \mathrm{neutral}\ \mathrm{faces}} $$
There were no between-group differences in the amount of time spent looking at the screen (t(30) = −0.639, p = 0.528) or in the amount of time participants spent looking at faces relative to the background of the screen (t(30) = 0.538, p = 0.594).
Eyes/mouth looking time
Data reflect the amount of time in milliseconds that was spent looking at the left eye ROI, the right eye ROI, and the mouth ROI. In order to account for different looking time on faces, the average time each participant spent looking at the eyes and mouth of the neutral faces presented during standard trials was divided by the average amount of time that participant spent looking at both neutral faces. Emotional face (i.e. oddball) trials were not included in these analyses due to the low percentages of trials that they represented, as well as the fact that participant looking time was split between neutral and emotional faces on these trials.
To ensure that participants did not demonstrate a looking bias to the left or right eye in faces, t tests were conducted for each group to compare looking time to the left and right eyes relative to the amount of time spent looking at the face which revealed no significant differences (CdLS: t(14) = 0.557, p = 0.586; RTS: t(16) = −1.759, p = 0.098). Therefore, the time spent looking to the left and right eye, relative to the amount of time spent looking at faces, was summed for further analyses in order to investigate overall looking patterns to the eyes. Figure 2 depicts the ratio of time each group spent looking at the eye region of faces.
The amount of time spent looking at the eye region of neutral faces; the amount of time, in milliseconds, spent looking within the eyes ROI divided by the amount of time, in milliseconds, spent looking at the entire face ROI of neutral faces. Error bars represent standard error
To compare looking time to the eye region of the faces, an independent sample t test was conducted. The analysis revealed no significant between-group difference in the ratio of time spent looking at the eyes to the time spent looking at faces (t(30) = −0.158, p = 0.875).
In order to compare looking time to the mouth region of the faces relative to the rest of the face, an independent sample t test was conducted. The analysis revealed no significant between-group difference in the ratio of time spent looking at the mouth (t(18) = −1.537, p = 0.142). Figure 3 depicts the ratio of time each group spent looking at the mouth to the rest of the face. Due to the wide range of ages and abilities of participants included in this study, an analysis of covariance (ANCOVA) was conducted, which revealed no effect of syndrome group on the amount of time spent looking at the eyes or mouth relative to the amount of time spent looking at the face, when chronological age was controlled for (eye looking: F (1, 29) = 0.038, p = .846; mouth looking: F (1, 29) = 2.505, p = 0.124) and when global adaptive behaviour ability was controlled for (eye looking: F (1, 29) = 0.017, p = 0.896; mouth looking: F (1, 29) = 2.613, p = 0.117). Figure 4 presents the heat maps for each participant group to depict the distribution and duration of looking to neutral faces.
The amount of time spent looking at the mouth region of neutral faces; the amount of time, in milliseconds, spent looking within the mouth ROI divided by the amount of time, in milliseconds, spent looking at the entire face ROI of neutral faces. Error bars represent standard error
Heat maps depicting the distribution of looking on all neutral trials. The heat map is based on the duration of fixations across the display for all participants. The eyes and mouth were not exactly lined up across all trials due to natural variation in the position of features across the different facial stimuli
Spontaneous emotion preference
The proportion of the trial spent looking at faces displaying a happy expression was calculated for happy faces and neutral faces presented side-by-side with happy faces. This process was repeated for dwell time percentage on faces displaying a disgusted expression and for neutral faces presented alongside disgusted faces. Paired sample t tests were conducted for each group to investigate whether participants spent a significantly higher proportion of the trial looking at happy relative to neutral faces during happy-neutral trials and disgust relative to neutral faces during disgust-neutral trials. These t tests revealed that both participant groups spent a higher proportion of the trial looking at disgust compared to neutral faces (CdLS: t(14) = 2.761, p = 0.015; RTS: t(16) = 5.997, p < 0.001) but not happy compared to neutral faces (CdLS: t(14) = 0.617, p = 0.547; RTS: t(16) = 0.799, p = 0.436).
The analysis conducted thus far indicated that both participant groups look more at disgust faces than neutral faces but not happy faces compared to neutral faces. However, this analysis does not allow for a between-group comparison. Therefore, a looking preference for happy faces was calculated by subtracting the proportion of the trial spent looking at neutral faces during happy-neutral trials from the proportion of the trial spent looking at happy faces. This was repeated to calculate the disgust preference. Happy and disgust preferences were compared between groups using an independent samples t test. This test indicated no between-group difference of happy preference (t(30) = −0.115, p = 0.909) or disgust preference (t(30) = 1.414, p = 0.168). Figure 5 depicts the proportion of extra time spent looking at happy and disgust faces compared to neutral faces during oddball trials. In summary, all participants spent a higher proportion of time looking at disgust versus neutral faces but not happy versus neutral faces. Due to the wide range of ages and abilities of participants included in this study, an ANCOVA was conducted, which revealed no effect of syndrome group on happy or disgust preference, when chronological age was controlled for (happy preference: F (1, 29) = 0.009, p = 0.924; disgust preference: F (1, 29) = 1.941, p = 0.174) or when global adaptive behaviour was controlled for (happy preference: F (1, 29) = 0.028, p = 0.868; disgust preference: F (1, 29) = 1.923, p = 0.176.
Looking preference for happy and disgust faces, compared to neutral faces; the proportion of trial time that participants spent looking at happy faces divided by neutral faces during happy-neutral trials (happy preference), and the proportion of trial time that participants spent looking at disgusted faces divided by neutral faces during disgust-neutral trails (disgust preference). Error bars represent standard error
In the present study, we investigated looking patterns to the eyes and mouth, as well as spontaneous emotion preference, in individuals with CdLS and RTS. In line with previous literature that provides evidence for different patterns of visual exploration of social stimuli in groups displaying divergent social behaviours, it was hypothesised that individuals with CdLS and RTS would also demonstrate different patterns of face scanning due to their contrasting socio-behavioural profiles. Specifically, we predicted that individuals with CdLS would display lower levels of looking to the eye region than those with RTS, due to the reports of social anxiety and withdrawal reported in CdLS [32] and due to the heightened social interest reported in RTS [53]. The results demonstrate that individuals with CdLS and RTS displayed similar looking patterns to the eye region of the face. These findings do not support the hypothesis of a difference between groups with contrasting profiles of social behaviour exhibiting different face processing techniques. Furthermore, as existing literature points to a role for amygdala dysfunction in reduced looking to the eye region of static faces [6], the results from the present study indicate that the documented differences in CdLS and RTS are unlikely to be subcortically mediated.
Spontaneous looking patterns were assessed by examining and comparing the ratio of time spent looking at the eyes and mouth during the standard trials (neutral face pairs). The results indicated that participants with CdLS and RTS looked at the eye region of the faces a similar amount. These findings are unlikely to be driven by chronological age, autistic impairments, and global and adaptive behaviour ability levels as these variables were matched across participants. Whilst the expected group differences between CdLS and RTS did not emerge, it is unlikely that the lack of group differences in the present study is a result of the paradigm used and its sensitivity to highlight group differences. Using the same paradigm, the authors previously reported that participants with FXS exhibit reduced looking to the eye region of the faces, in comparison to those with ASD, as used here [41]. Although previously published data from participants with ASD and TD children and adults [41] were not presented here, those data were compared to data presented for participants with CdLS and RTS in the current study. Interestingly, no differences were found between any groups suggesting typical eye gaze in both those with CdLS and RTS. Reduced eye looking in FXS compared to ASD using the same measure lend support to the notion that the social impairments in FXS are somewhat subcortically mediated by amygdala dysfunction, which has been reported in this population [42]. However, amygdala dysfunction is a less consistent finding in ASD [9] and may go some way toward explaining inconsistent results regarding looking to the eye region of faces [12]. In the present study, no differences in looking to the eye region between individuals with CdLS, associated with social withdrawal and anxiety, and individuals with RTS, associated with social interest, were found. These results indicate that firstly, the documented differences in social behaviour in CdLS and RTS may not be subcortically mediated. Consequently, this suggests that the social anxiety reported in CdLS may be cognitively mediated, rather than associated with amygdala dysfunction, which has implications for both basic science and clinical intervention in relation to individuals with CdLS.
Previous literature comparing visual exploration of social stimuli in ASD and WS has consistently reported less eye looking in ASD, associated with social withdrawal, and increased eye looking in WS associated with hyper-sociability [3–5]. However, the present study reports similar eye gaze patterns in two different neurodevelopmental disorders also associated with clearly contrasting profiles of social behaviour. From previous reports of social behaviour in CdLS and RTS appearing to differ and from previous studies of face scanning in disorders with contrasting socio-behavioural characteristics, differing levels of eye gaze were predicted in these groups. However, such differences were not observed in the present study, and both groups showed typical levels of eye gaze. One possible explanation for these results concerns the nature of the differences in social skills between those with CdLS and those with RTS. Whilst there are documented differences in the social behaviours of the two disorders studied here, the differences are perhaps not as extreme as those described in ASD and WS, arguably at polar ends of a sociability spectrum. The lack of a clear distinction of visual exploration of social stimuli in the current groups, whose associated socio-behavioural characteristics are contrasting, suggests that studying social cognition across individuals with different genetic syndromes and neurodevelopmental disorders is often more complicated than the impaired or enhanced profile of results that emerge from those with ASD and WS. The results from this study suggest that clear differences in socio-behavioural characteristics are not sufficient to predict attention to social information. Furthermore, whilst eye contact has been reported to be a good indicator of social functioning in ASD and WS, this may not be the case for all neurodevelopmental disorders. Existing evidence exists to support this interpretation. Specifically, some studies have reported a developmental shift in the relationship between reduced eye looking and social disability. For example, reduced eye looking has been associated with higher levels of social disability in toddlers [58, 59], but there appears to be no such relationship in school-age children [60] or adults with ASD [61]. The mean age of participants in the present study was 17–18 years. Therefore, it may be the case that visual attention to social information may be more predictive of socio-behavioural impairments in the early years of life as opposed to throughout early adulthood.
It is important to consider the interpretation of these findings in light of the limitations of the study. Firstly, behavioural data on these two participant groups were not collected alongside the eye-tracking data. Although it is common in the existing literature for data to be presented on either behavioural or cognitive measures, previous studies documenting looking patterns to social stimuli have previously used participant groups with well-defined socio-behavioural characteristics, such as ASD and Williams syndrome. As the social behaviour of CdLS and RTS is comparatively under-researched, it would have been beneficial to collect such data on the individuals participating in the present study and this is a focus for future research. Instead, for the present study it was necessary to utilise previous literature to document the socio-behavioural characteristics of CdLS and RTS, and interpret the current results in light of existing literature. Whilst the sample size is comparable to existing studies investigating visual attention to social stimuli in genetic syndromes versus typical development, there may be limited power in the current study to detect smaller differences between two participant groups where different socio-behavioural features are documented yet require more extensive exploration. In addition, the conclusions stated here should be considered alongside the potential limitation that this study documents eye looking during a laboratory-based task of passive viewing of facial stimuli, which, whilst providing robust and novel findings, does not mirror real-world experiences of social interactions. Due to the laboratory-based setting, the facial stimuli presented may be less anxiety provoking than real faces, which could impact the way in which they are processed. Future research should consider the differential effects of laboratory and real-world experiences in visual exploration of faces in children and adults with neurodevelopmental disorders. Finally, whilst IQ measures were not administered for the present study, the VABS adaptive behaviour composite and communication standard score provide standard and reliable measures of adaptive behaviour abilities and verbal abilities, respectively, that are comparable across the CdLS and RTS groups. Adaptive behaviour was assessed over IQ measures due to the difficulty associated with selecting an IQ test that can be administered to individuals with a range of chronological ages and abilities. In addition, due to the level of intellectual ability of participants in this study, it was deemed more appropriate to use parental report of adaptive behaviour abilities, which focus on typical performance of everyday skills, as opposed to IQ measures, which focus on optimal performance of tasks that are associated with performance or cognitive demands. Furthermore, general IQ has been reported to correlate with the communication subscale of the VABS [62, 63], which did not differ between the CdLS and RTS participant groups in the current study. It should be noted that, to our knowledge, this is the first study documenting the use of eye-tracking technology in individuals with CdLS and RTS. Due to the level of intellectual disability associated with these genetic syndromes, the use of a passive viewing task was deemed most appropriate. Importantly, the overall levels of task engagement reflect those demonstrated by TD children and adults on the same paradigm. Therefore, these levels of task engagement, which may be considered relatively low, most likely reflect the nature of the task used. As passive viewing tasks do not require a response, there is no cost to the participant to look away from the screen.
In addition to the findings on face scanning, the results from the current study showed that implicit emotion preference did not differ in either individuals with CdLS or individuals with RTS. In the current study, spontaneous emotion preference was assessed using a novel oddball paradigm in conjunction with a preferential looking measure. Participants were presented with pairs of neutral faces (standard trials), with neutral-disgust, and neutral-happy pairs (oddball trials) presented infrequently. Participants in both groups looked longer at faces posed in disgusted expressions compared with neutral faces during the target trials, whereas no participant group looked longer at the faces posed in happy expressions compared to neutral faces. This pattern of results mirrors those previously reported for TD children and adults [41].
As described above, participants in both groups exhibited significant preferential looking to disgust relative to neutral expressions but did not exhibit a preference for looking to happy relative to neutral expressions. Two potential explanations for these findings are proposed. Firstly, it is possible that disgusted faced gain an attentional advantage over happy faces due to the relative novelty of disgusted faces (see [64] for a review). Disgusted faces are not seen in everyday life as often as happy faces. Therefore, the novelty of the disgusted faces may have captured the attention of participants to a greater extent than the happy faces. Alternatively, the negativity bias, whereby individuals attend more to negative information than to positive information due to its increased informational value in detecting threatening stimuli [65, 66], may also contribute to the results reported in the current study. Disgusted expressions may be perceived as a cue to threat, due to its association with negative affect, thus capturing an individual's attention more so than non-threatening, positive facial expressions.
The results of this study show similar face scanning in two neurodevelopmental disorders with contrasting profiles of social behaviour. Individuals with CdLS and RTS looked at the eye region of faces a similar amount. Spontaneous emotion preference was also observed to be similar in those with CdLS and RTS in the current study and mirror that previously reported in TD individuals [41]. These findings suggest that such coarse measures as attention to the eyes may not be sensitive to differences in socio-behavioural characteristics unless the differences are as extreme as those seen in ASD and WS. These findings also suggest that documented differences in the socio-behavioural characteristics of individuals with CdLS and RTS may be cognitively rather than subcortically mediated, due to the well-documented association between impaired eye looking and amygdala dysfunction. Future experimental eye-tracking and other research should focus on other aspects of social cognitive functioning and social behaviour in individuals with CdLS, RTS, and other genetic syndromes, in an effort to elucidate pathways from genetic disorders to behaviour in atypical and impaired social functioning.
Development of the MacBrain Face Stimulus Set was overseen by Nim Tottenham and supported by the John D. and Catherine T. MacArthur Foundation Research Network on Early Experience and Brain Development. Please contact Nim Tottenham at [email protected] for more information concerning the stimulus set.
Where results from non-parametric tests, used when data were not normally distributed, did not differ from results from the equivalent parametric tests, the results from the parametric tests are reported.
ABC:
adaptive behaviour composite
CdLS:
FXS:
RTS:
TD:
typically developing
UoB:
VABS:
Vineland adaptive behaviour scale
WS:
Baron-Cohen S, Joliffe T, Mortimore C, Robertson M. Another advanced test of theory of mind: evidence from very high functioning adults with autism or asperger syndrome. J Child Psychol Psychiatry. 1997;38:813–22.
Baron-Cohen S, Wheelwright S, Hill J, Raste Y, Plumb I. The "reading the mind in the eyes" test revised version: a study with normal adults, and adults with asperger syndrome or high-functioning autism. J Child Psychol Psychiatry. 2001;42:241–51.
Riby DM, Hancock PJB. Viewing it differently: social scene perception in Williams syndrome and autism. Neuropsychologia. 2008;46:2855–60.
Riby DM, Doherty-Sneddon G, Bruce V. Exploring face perception in disorders of development: evidence from Williams syndrome and autism. J Neuropsychol. 2008;2:47–64.
Riby DM, Hancock PJB. Do faces capture the attention of individuals with Williams syndrome or autism? evidence from tracking eye movements. J Autism Dev Disord. 2009;39:421–31.
Kennedy DP, Adolphs R. Impaired fixation to eyes following amygdala damage arises from abnormal bottom-up attention. Neuropsychologia. 2010;48:3392–8.
Spezio ML, Huang PY, Castelli F, Adolphs R. Amygdala damage impairs eye contact during conversations with real people. J Neurosci. 2007;27:3994–7.
Dalton KM, Nacewicz BM, Johnstone T, Schaefer HS, Gernsbacher MA, Goldsmith HH, et al. Gaze fixation and the neural circuitry of face processing in autism. Nat Neurosci. 2005;8:519–26.
Sweeten TL, Posey DJ, Shekhar A, McDougle CJ. The amygdala and related structures in the pathophysiology of autism. Pharmacol Biochem Behav. 2002;71:449–55.
Cody Hazlett H, Pelphrey KA, Piven J. Structural and functional magnetic resonance imaging of autism. Int J Dev Neurosci. 2002;20:421–38.
Kirchner JC, Hatri A, Heekeren HR, Dziobeck I. Autistic symptomatology, face processing abilities, and eye fixation patterns. J Autism Dev Disord. 2011;41:158–67.
Speer LL, Cook AE, McMahon WM, Clark E. Face processing in children with autism effects of stimulus contents and type. Autism. 2007;11:265–77.
Guillon Q, Hadjikhani N, Badual S, Rogé B. Visual social attention in autism spectrum disorder: insights from eye-tracking studies. Neurosci Biobehav Rev. 2014;42:279–97.
Kliemann D, Dziobeck I, Hatri A, Steimke R, Heekeren HR. Atypical reflexive gaze patterns on emotional faces in autism spectrum disorders. J Neurosci. 2010;30:12281–7.
Bal E, Harden E, Lamb D, Van Hecke AV, Denver JW, Porges SW. Emotion recognition in children with autism spectrum disorders: relations to eye gaze and autonomic state. J Autism Dev Disord. 2010;40:358–70.
Adolphs R, Tranel D, Damasio H, Damasio A. Impaired recognition of emotion in facial expressions following bilateral damage to the human amygdala. Nature. 1994;372:669–72.
Izard C, Fine S, Schultz D, Mostow A, Ackerman B, Youngstrom E. Emotion knowledge as a predictor of social behaviour and academic competence in children at risk. Psychol Sci. 2001;12:18–23.
Sawyer AA, Williamson P, Young RL. Can gaze avoidance explain why individuals with Asperger's syndrome can't recognise emotions from facial expressions? J Autism Dev Disord. 2012;42:606–18.
Harms MB, Martin A, Wallace GL. Facial emotion recognition in autism spectrum disorders: a review of behavioral and neuroimaging studies. Neuropsychol Rev. 2010;20:290–322.
Tager-Flusberg H, Skwerer DP, Joseph RM. Model syndromes for investigating social cognitive and affective neuroscience: a comparison of autism and Williams syndrome. Soc Cogn Affect Neurosci. 2006;1:175–82.
Beck B. Epidemiology of Cornelia de Lange's syndrome. Acta Paediatr. 1976;65:631–8.
Berney TP, Ireland M, Burn J. Behavioural phenotype of Cornelia de Lange syndrome. Arch Dis Child. 1999;81:333–6.
Moss J, Oliver C, Berg K, Kaur G, Jephcott L, Cornish K. Prevalence of autism spectrum phenomenology in Cornelia de Lange and Cri du Chat syndromes. Am J Ment Retard. 2008;113:278–91.
Moss J, Howlin P, Magiati I, Oliver C. Characteristics of autism spectrum disorder in Cornelia de Lange syndrome. J Child Psychol Psychiatry. 2012;53:883–91.
Gillis LA, McCallum J, Kaur M, DeScipio C, Yaeger D, Mariani A, et al. NIPBL mutational analysis in 120 individuals with Cornelia de Lange syndrome and evaluation of genotype-phenotype correlations. Am J Hum Genet. 2004;75:610–23.
Krantz ID, McCallum J, DeScipio C, Kaur M, Gillis LA, Yaeger D, et al. Cornelia de Lange syndrome is caused by mutations in NIPBL, the human homolog of Drosophila melanogaster Nipped-B. Nat Gen. 2004;36:631–5.
Miyake N, Visser R, Kinoshita A, Yoshiura K-I, Niikawa N, Kondoh T, et al. Four novel NIPBL mutations in Japanese patients with Cornelia de Lange syndrome. Am J Med Genet A. 2005;135:103–5.
Deardorff MA, Kaur M, Yaeger D, Rampuria A, Korolev S, Pie J, et al. Mutations in cohesin complex members SMC3 and SMC1A cause a mild variant of Cornelia de Lange syndrome with predominant mental retardation. Am J Hum Genet. 2007;80:485–94.
Musio A, Selicorni A, Focarelli ML, Gervasini C, Milani D, Russo S, et al. X-linked Cornelia de Lange syndrome owing to SMC1L1 mutations. Nat Gen. 2006;38:528–30.
Deardorff MA, Bando M, Nakato R, Watrin E, Itoh T, Minamino M, et al. HDAC8 mutations in Cornelia de Lange syndrome affect the cohesin acetylation cycle. Nature. 2012;489:313–7.
Minor A, Shinawi M, Hogue JS, Vineyard M, Hamlin DR, Tan C, et al. Two novel RAD21 mutations in patients with mild Cornelia de Lange syndrome-like presentation and report of the first familial case. Genet Med. 2014;537:279–84.
Richards C, Moss J, O'Farrell L, Kaur G, Oliver C. Social anxiety in Cornelia de Lange syndrome. J Autism Dev Disord. 2009;39:1155–62.
Wieser M, Pauli P, Alpers GW, Muhlberger A. Is eye to eye contact really threatening and avoided in social anxiety? An eye-tracking and psychophysiology study. J Anxiety Disord. 2009;23.
Evans KC, Wright CI, Wedig MM, Gold AL, Pollack MH, Rauch SL. A functional MRI study of amygdala response to angry schematic faces in social anxiety disorder. Depress Anxiety. 2008;25:495–505.
Amarel DG. The primate amygdala and the neurobiology of social behaviour: implications for understanding social anxiety. Biol Psychiatry. 2002;51:11–7.
Cornish K, Turk J, Levitas A. Fragile X syndrome and autism: common developmental pathways? Curr Pediatr Rev. 2007;3:61–8.
Dalton KM, Holsen L, Abbeduto L, Davidson RJ. Brain function and gaze fixation during facial-emotion processing in fragile X and autism. Autism Res. 2008;1:231–9.
Holsen LM, Dalton KM, Johnstone T, Davidson RJ. Prefrontal social cognition network dysfunction underlying face encoding and social anxiety in fragile X syndrome. Neuroimage. 2008;43:592–604.
Farzin F, Rivera SM, Hessl D. Brief report: visual processing of faces in individuals with fragile X syndrome: an eye tracking study. J Autism Dev Disord. 2009;39:946–52.
Farzin F, Scaggs F, Hervey C, Berry-Kravis E, Hessl D. Reliability of eye tracking and pupillometry measures in individuals with fragile X syndrome. J Autism Dev Disord. 2011;41:1515–22.
Crawford H, Moss J, Anderson GM, Oliver C, McCleery JP. Implicit discrimination of basic facial expressions of positive/negative emotion in fragile X syndrome and autism spectrum disorder, American journal on intellectual and developmental disabilities. 2015.
Hessl D, Rivera S, Koldewyn K, Cordeiro L, Adamns J, Tassone F, et al. Amygdala dysfunction in men with the fragile x premutation. Brain. 2007;130:404–16.
Hennekam RC, Van Den Boogaard M-J, Sibbles BJ, Van Spijker HG. Rubinstein-taybi syndrome in the Netherlands. Am J Med Genet. 1990;37:17–29.
Petrif F, Giles RH, Dauwerse HG, Saris JJ, Hennekam RC, Masuno M, et al. Rubinstein-Taybi syndrome caused by mutations in the transcriptional co-activator CBP. Nature. 1995;376:348–51.
Coupry I, Roudaut C, Stef M, Delrue MA, Marche M, Burgelin I, et al. Molecular analysis of the CBP gene in 60 patients with Rubinstein-Taybi syndrome. J Med Genet. 2002;39:415–21.
Kalkhoven E, Roelfsema JH, Teunissen H, den Boer A, Ariyurek Y, Zantema A, et al. Loss of CBP acetyltransferase activity by PHD finger mutations in Rubinstein-Taybi syndrome. Hum Mol Genet. 2003;12:441–50.
Hennekam RC. Rubinstein-Taybi syndrome. Eur J Hum Genet. 2006;14:981–5.
Roelfsema JH, White SJ, Ariyürek Y, Bartholdi D, Niedrist D, Papadia F, et al. Genetic heterogeneity in Rubinstein-Taybi syndrome: mutations in both the CBP and EP300 genes cause disease. Am J Hum Genet. 2005;76:572–80.
Hennekam RC, Baselier AC, Beyaert E, Bos A. Psychological and speech studies in Rubinstein-Taybi syndrome. Am J Ment Retard. 1992;96:645–60.
Gotts EE, Liemohn WP. Behavioural characteristics of three children with the broad-thumb-hallux (Rubinstein-Taybi) syndrome. Biol Psychiatry. 1977;12:413–23.
Baxter G, Beer J. Rubinstein-Taybi syndrome. Phys Rep. 1992;70:451–6.
Stevens CA, Carey JC, Blackburn BL. Rubinstein-Taybi syndrome: a natural history study. Am J Med Genet. 1990;37:30–7.
Galéra C, Taupiac E, Fraisse S, Naudion S, Toussaint E, Rooryck-Thambo C, et al. Socio-behavioral characteristics of children with Rubinstein-Taybi syndrome. J Autism Dev Disord. 2009;39:1252–60.
Yagihashi T, Kenjiro K, Nobuhiko O, Mizuno S, Kenji K, Takao T, et al. Age-dependent change in behavioural feature in Rubinstein-Taybi syndrome. Congenit Anom. 2012;52:82–6.
Rutter M, Bailey A, Lord C. The social communication questionnaire. Los Angeles, CA: Western Psychological Services; 2003.
Sparrow SS, Cicchetti DV, Balla DA. Vineland-II adaptive behavior scales: survey forms manual. Circle Pines, MN: AGS Publishing; 2005.
Tottenham N, Tanaka JW, Leon AC, McCarry T, Nurse M, Hare TA, et al. The NimStim set of facial expressions: judgments from untrained research participants. Psychiatry Res. 2009;168:242–9.
Jones W, Carr K, Klin A. Absence of preferential looking to the eyes of approaching adults predicts level of social disability in 2-year-olds with autism. Arch Gen Psychiatry. 2008;65.
Campbell DJ, Shic F, Macari S, Chawarska K. Gaze response to dyadic bids at 2 years related to outcomes at 3 years in autism spectrum disorders: a subtyping analysis. J Autism Dev Disord. 2014;44:431–42.
Rice K, Moriuchi JM, Jones W, Klin A. Parsing heterogeneity in autism spectrum disorders: visual scanning of dynamic social scenes in school-aged children. J Am Acad Child Adolesc Psychiatry. 2012;51:238–43.
Klin A, Jones W, Schultz R, Volkmar F, Cohen D. Visual fixation patterns during viewing of naturalistic social situations as predictors of social competence in individuals with autism. Arch Gen Psychiatry. 2002;59:809–16.
Perry A, Flanagan HE, HGeiser JD, Freeman NL. Brief report: the Vineland adaptive behavior scales in young children with autism spectrum disorders at different cognitive levels. J Autism Dev Disord. 2009;39:1066–78.
Bölte S, Poustka F. The relation between general cognitive level and adaptive behaviour domains in individuals with autism with and without comorbid mental retardation. Child Psychiatry Hum Dev. 2002;33:165–72.
Desimone R, Duncan J. Neural mechanisms of selective visual attention. Annu Rev Neurosci. 1995;18:193–222.
Peeters G, Czapinski J. Positive-negative asymmetry in evaluations: the distinction between affective and informational negativity effects. Eur Rev Soc Psychol. 1990;1:33–60.
Öhman A, Mineka S. Fears, phobias, and preparedness: toward an evolved module of fear and fear learning. Psychol Rev. 2001;108:483.
We would like to thank all participants and their families for taking part in this study. We would also like to thank the CdLS Foundation UK and Ireland and the Rubinstein-Taybi syndrome support group UK for the assistance in the recruitment of the participants. The research reported here was supported by a grant from the Economic and Social Research Council (Grant Number: ES/I901825/1) awarded to HC and by Cerebra.
Centre for Research in Psychology, Behaviour and Achievement, Coventry University, Coventry, CV1 5FB, UK
Hayley Crawford
Cerebra Centre for Neurodevelopmental Disorders, School of Psychology, University of Birmingham, Edgbaston, B15 2TT, UK
, Joanna Moss
& Chris Oliver
Institute of Cognitive Neuroscience, University College London, 17 Queen Square, London, WC1N 3AR, UK
Joanna Moss
Center for Autism Research, Children's Hospital of Philadelphia, 3535 Market Street, Philadelphia, PA, 19104, USA
Joseph P. McCleery
School of Psychology, Oxford Brookes University, Headington Campus, Oxford, OX3 0BP, UK
Giles M. Anderson
Search for Hayley Crawford in:
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Correspondence to Hayley Crawford.
All authors contributed to the planning of the study, the planning of analyses, and the writing of the paper. GMA and JPM were involved in developing and programming the eye-tracking task. HC, JM, JPM, and CO were involved in recruitment, data collection, and interpretation of the data. HC conducted the analysis and wrote the first draft of the paper. All authors read and approved the final manuscript.
Eye-tracking
Emotion preference
Eye gaze | CommonCrawl |
I am not alone in thinking of the potential benefits of smart drugs in the military. In their popular novel Ghost Fleet: A Novel of the Next World War, P.W. Singer and August Cole tell the story of a future war using drug-like nootropic implants and pills, such as Modafinil. DARPA is also experimenting with neurological technology and enhancements such as the smart drugs discussed here. As demonstrated in the following brain initiatives: Targeted Neuroplasticity Training (TNT), Augmented Cognition, and High-quality Interface Systems such as their Next-Generational Nonsurgical Neurotechnology (N3).
Our 2nd choice for a Brain and Memory supplement is Clari-T by Life Seasons. We were pleased to see that their formula included 3 of the 5 necessary ingredients Huperzine A, Phosphatidylserine and Bacopin. In addition, we liked that their product came in a vegetable capsule. The product contains silica and rice bran, though, which we are not sure is necessary.
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As already mentioned, AMPs and MPH are classified by the U.S. Food and Drug Administration (FDA) as Schedule II substances, which means that buying or selling them is a felony offense. This raises the question of how the drugs are obtained by students for nonmedical use. Several studies addressed this question and yielded reasonably consistent answers.
Cost-wise, the gum itself (~$5) is an irrelevant sunk cost and the DNB something I ought to be doing anyway. If the results are negative (which I'll define as d<0.2), I may well drop nicotine entirely since I have no reason to expect other forms (patches) or higher doses (2mg+) to create new benefits. This would save me an annual expense of ~$40 with a net present value of <820 ($); even if we count the time-value of the 20 minutes for the 5 DNB rounds over 48 days (0.2 \times 48 \times 7.25 = 70), it's still a clear profit to run a convincing experiment.
A large review published in 2011 found that the drug aids with the type of memory that allows us to explicitly remember past events (called long-term conscious memory), as opposed to the type that helps us remember how to do things like riding a bicycle without thinking about it (known as procedural or implicit memory.) The evidence is mixed on its effect on other types of executive function, such as planning or ability on fluency tests, which measure a person's ability to generate sets of data—for example, words that begin with the same letter.
We hope you find our website to be a reliable and valuable resource in your search for the most effective brain enhancing supplements. In addition to product reviews, you will find information about how nootropics work to stimulate memory, focus, and increase concentration, as well as tips and techniques to help you experience the greatest benefit for your efforts.
I take my piracetam in the form of capped pills consisting (in descending order) of piracetam, choline bitartrate, anhydrous caffeine, and l-tyrosine. On 8 December 2012, I happened to run out of them and couldn't fetch more from my stock until 27 December. This forms a sort of (non-randomized, non-blind) short natural experiment: did my daily 1-5 mood/productivity ratings fall during 8-27 December compared to November 2012 & January 2013? The graphed data28 suggests to me a decline:
Many people find it difficult to think clearly when they are stressed out. Ongoing stress leads to progressive mental fatigue and an eventual breakdown. Luckily, there are several ways that nootropics can help relieve stress. One is through the natural promotion of feelings of relaxation and the other is by replenishing the brain chemicals drained by stress.
Similarly, Mehta et al 2000 noted that the positive effects of methylphenidate (40 mg) on spatial working memory performance were greatest in those volunteers with lower baseline working memory capacity. In a study of the effects of ginkgo biloba in healthy young adults, Stough et al 2001 found improved performance in the Trail-Making Test A only in the half with the lower verbal IQ.
I have no particularly compelling story for why this might be a correlation and not causation. It could be placebo, but I wasn't expecting that. It could be selection effect (days on which I bothered to use the annoying LED set are better days) but then I'd expect the off-days to be below-average and compared to the 2 years of trendline before, there doesn't seem like much of a fall.
There is evidence to suggest that modafinil, methylphenidate, and amphetamine enhance cognitive processes such as learning and working memory...at least on certain laboratory tasks. One study found that modafinil improved cognitive task performance in sleep-deprived doctors. Even in non-sleep deprived healthy volunteers, modafinil improved planning and accuracy on certain cognitive tasks. Similarly, methylphenidate and amphetamine also enhanced performance of healthy subjects in certain cognitive tasks.
Many over the counter and prescription smart drugs fall under the category of stimulants. These substances contribute to an overall feeling of enhanced alertness and attention, which can improve concentration, focus, and learning. While these substances are often considered safe in moderation, taking too much can cause side effects such as decreased cognition, irregular heartbeat, and cardiovascular problems.
Bacopa Monnieri is probably one of the safest and most effective memory and mood enhancer nootropic available today with the least side-effects. In some humans, a majorly extended use of Bacopa Monnieri can result in nausea. One of the primary products of AlternaScript is Optimind, a nootropic supplement which mostly constitutes of Bacopa Monnieri as one of the main ingredients.
ATTENTION CANADIAN CUSTOMERS: Due to delays caused by it's union's ongoing rotating strikes, Canada Post has suspended its delivery standard guarantees for parcel services. This may cause a delay in the delivery of your shipment unless you select DHL Express or UPS Express as your shipping service. For more information or further assistance, please visit the Canada Post website. Thank you.
"Cavin's enthusiasm and drive to help those who need it is unparalleled! He delivers the information in an easy to read manner, no PhD required from the reader. 🙂 Having lived through such trauma himself he has real empathy for other survivors and it shows in the writing. This is a great read for anyone who wants to increase the health of their brain, injury or otherwise! Read it!!!"
Today piracetam is a favourite with students and young professionals looking for a way to boost their performance, though decades after Giurgea's discovery, there still isn't much evidence that it can improve the mental abilities of healthy people. It's a prescription drug in the UK, though it's not approved for medical use by the US Food and Drug Administration and can't be sold as a dietary supplement either.
One last note on tolerance; after the first few days of using smart drugs, just like with other drugs, you may not get the same effects as before. You've just experienced the honeymoon period. This is where you feel a large effect the first few times, but after that, you can't replicate it. Be careful not to exceed recommended doses, and try cycling to get the desired effects again.
It's basic economics: the price of a good must be greater than cost of producing said good, but only under perfect competition will price = cost. Otherwise, the price is simply whatever maximizes profit for the seller. (Bottled water doesn't really cost $2 to produce.) This can lead to apparently counter-intuitive consequences involving price discrimination & market segmentation - such as damaged goods which are the premium product which has been deliberately degraded and sold for less (some Intel CPUs, some headphones etc.). The most famous examples were railroads; one notable passage by French engineer-economist Jules Dupuit describes the motivation for the conditions in 1849:
But perhaps the biggest difference between Modafinil and other nootropics like Piracetam, according to Patel, is that Modafinil studies show more efficacy in young, healthy people, not just the elderly or those with cognitive deficits. That's why it's great for (and often prescribed to) military members who are on an intense tour, or for those who can't get enough sleep for physiological reasons. One study, by researchers at Imperial College London, and published in Annals of Surgery, even showed that Modafinil helped sleep-deprived surgeons become better at planning, redirecting their attention, and being less impulsive when making decisions.
The majority of nonmedical users reported obtaining prescription stimulants from a peer with a prescription (Barrett et al., 2005; Carroll et al., 2006; DeSantis et al., 2008, 2009; DuPont et al., 2008; McCabe & Boyd, 2005; Novak et al., 2007; Rabiner et al., 2009; White et al., 2006). Consistent with nonmedical user reports, McCabe, Teter, and Boyd (2006) found 54% of prescribed college students had been approached to divert (sell, exchange, or give) their medication. Studies of secondary school students supported a similar conclusion (McCabe et al., 2004; Poulin, 2001, 2007). In Poulin's (2007) sample, 26% of students with prescribed stimulants reported giving or selling some of their medication to other students in the past month. She also found that the number of students in a class with medically prescribed stimulants was predictive of the prevalence of nonmedical stimulant use in the class (Poulin, 2001). In McCabe et al.'s (2004) middle and high school sample, 23% of students with prescriptions reported being asked to sell or trade or give away their pills over their lifetime.
"We stumbled upon fasting as a way to optimize cognition and make yourself into a more efficient human being," says Manuel Lam, an internal medicine physician who advises Nootrobox on clinical issues. He and members of the company's executive team have implanted glucose monitors in their arms — not because they fear diabetes but because they wish to track the real-time effect of the foods they eat.
Ethical issues also arise with the use of drugs to boost brain power. Their use as cognitive enhancers isn't currently regulated. But should it be, just as the use of certain performance-enhancing drugs is regulated for professional athletes? Should universities consider dope testing to check that students aren't gaining an unfair advantage through drug use?
An unusual intervention is infrared/near-infrared light of particular wavelengths (LLLT), theorized to assist mitochondrial respiration and yielding a variety of therapeutic benefits. Some have suggested it may have cognitive benefits. LLLT sounds strange but it's simple, easy, cheap, and just plausible enough it might work. I tried out LLLT treatment on a sporadic basis 2013-2014, and statistically, usage correlated strongly & statistically-significantly with increases in my daily self-ratings, and not with any sleep disturbances. Excited by that result, I did a randomized self-experiment 2014-2015 with the same procedure, only to find that the causal effect was weak or non-existent. I have stopped using LLLT as likely not worth the inconvenience.
as scientific papers become much more accessible online due to Open Access, digitization by publishers, and cheap hosting for pirates, the available knowledge about nootropics increases drastically. This reduces the perceived risk by users, and enables them to educate themselves and make much more sophisticated estimates of risk and side-effects and benefits. (Take my modafinil page: in 1997, how could an average person get their hands on any of the papers available up to that point? Or get detailed info like the FDA's prescribing guide? Even assuming they had a computer & Internet?)
Remembering what Wedrifid told me, I decided to start with a quarter of a piece (~1mg). The gum was pretty tasteless, which ought to make blinding easier. The effects were noticeable around 10 minutes - greater energy verging on jitteriness, much faster typing, and apparent general quickening of thought. Like a more pleasant caffeine. While testing my typing speed in Amphetype, my speed seemed to go up >=5 WPM, even after the time penalties for correcting the increased mistakes; I also did twice the usual number without feeling especially tired. A second dose was similar, and the third dose was at 10 PM before playing Ninja Gaiden II seemed to stop the usual exhaustion I feel after playing through a level or so. (It's a tough game, which I have yet to master like Ninja Gaiden Black.) Returning to the previous concern about sleep problems, though I went to bed at 11:45 PM, it still took 28 minutes to fall sleep (compared to my more usual 10-20 minute range); the next day I use 2mg from 7-8PM while driving, going to bed at midnight, where my sleep latency is a more reasonable 14 minutes. I then skipped for 3 days to see whether any cravings would pop up (they didn't). I subsequently used 1mg every few days for driving or Ninja Gaiden II, and while there were no cravings or other side-effects, the stimulation definitely seemed to get weaker - benefits seemed to still exist, but I could no longer describe any considerable energy or jitteriness.
A synthetic derivative of Piracetam, aniracetam is believed to be the second most widely used nootropic in the Racetam family, popular for its stimulatory effects because it enters the bloodstream quickly. Initially developed for memory and learning, many anecdotal reports also claim that it increases creativity. However, clinical studies show no effect on the cognitive functioning of healthy adult mice.
Ngo has experimented with piracetam himself ("The first time I tried it, I thought, 'Wow, this is pretty strong for a supplement.' I had a little bit of reflux, heartburn, but in general it was a cognitive enhancer. . . . I found it helpful") and the neurotransmitter DMEA ("You have an idea, it helps you finish the thought. It's for when people have difficulty finishing that last connection in the brain").
Powders are good for experimenting with (easy to vary doses and mix), but not so good for regular taking. I use OO gel capsules with a Capsule Machine: it's hard to beat $20, it works, it's not that messy after practice, and it's not too bad to do 100 pills. However, I once did 3kg of piracetam + my other powders, and doing that nearly burned me out on ever using capsules again. If you're going to do that much, something more automated is a serious question! (What actually wound up infuriating me the most was when capsules would stick in either the bottom or top try - requiring you to very gingerly pull and twist them out, lest the two halves slip and spill powder - or when the two halves wouldn't lock and you had to join them by hand. In contrast: loading the gel caps could be done automatically without looking, after some experience.)
Unfortunately, cognitive enhancement falls between the stools of research funding, which makes it unlikely that such research programs will be carried out. Disease-oriented funders will, by definition, not support research on normal healthy individuals. The topic intersects with drug abuse research only in the assessment of risk, leaving out the study of potential benefits, as well as the comparative benefits of other enhancement methods. As a fundamentally applied research question, it will not qualify for support by funders of basic science. The pharmaceutical industry would be expected to support such research only if cognitive enhancement were to be considered a legitimate indication by the FDA, which we hope would happen only after considerably more research has illuminated its risks, benefits, and societal impact. Even then, industry would have little incentive to delve into all of the issues raised here, including the comparison of drug effects to nonpharmaceutical means of enhancing cognition.
As it happens, these are areas I am distinctly lacking in. When I first began reading about testosterone I had no particular reason to think it might be an issue for me, but it increasingly sounded plausible, an aunt independently suggested I might be deficient, a biological uncle turned out to be severely deficient with levels around 90 ng/dl (where the normal range for 20-49yo males is 249-839), and finally my blood test in August 2013 revealed that my actual level was 305 ng/dl; inasmuch as I was 25 and not 49, this is a tad low.
"Love this book! Still reading and can't wait to see what else I learn…and I am not brain injured! Cavin has already helped me to take steps to address my food sensitivity…seems to be helping and I am only on day 5! He has also helped me to help a family member who has suffered a stroke. Thank you Cavin, for sharing all your knowledge and hard work with us! This book is for anyone that wants to understand and implement good nutrition with all the latest research to back it up. Highly recommend!"
What if you could simply take a pill that would instantly make you more intelligent? One that would enhance your cognitive capabilities including attention, memory, focus, motivation and other higher executive functions? If you have ever seen the movie Limitless, you have an idea of what this would look like—albeit the exaggerated Hollywood version. The movie may be fictional but the reality may not be too far behind.
Most diehard nootropic users have considered using racetams for enhancing brain function. Racetams are synthetic nootropic substances first developed in Russia. These smart drugs vary in potency, but they are not stimulants. They are unlike traditional ADHD medications (Adderall, Ritalin, Vyvanse, etc.). Instead, racetams boost cognition by enhancing the cholinergic system.
Feeling behind, I resolved to take some armodafinil the next morning, which I did - but in my hurry I failed to recall that 200mg armodafinil was probably too much to take during the day, with its long half life. As a result, I felt irritated and not that great during the day (possibly aggravated by some caffeine - I wish some studies would be done on the possible interaction of modafinil and caffeine so I knew if I was imagining it or not). Certainly not what I had been hoping for. I went to bed after midnight (half an hour later than usual), and suffered severe insomnia. The time wasn't entirely wasted as I wrote a short story and figured out how to make nicotine gum placebos during the hours in the dark, but I could have done without the experience. All metrics omitted because it was a day usage.
Though their product includes several vitamins including Bacopa, it seems to be missing the remaining four of the essential ingredients: DHA Omega 3, Huperzine A, Phosphatidylserine and N-Acetyl L-Tyrosine. It missed too many of our key criteria and so we could not endorse this product of theirs. Simply, if you don't mind an insufficient amount of essential ingredients for improved brain and memory function and an inclusion of unwanted ingredients – then this could be a good fit for you.
The surveys just reviewed indicate that many healthy, normal students use prescription stimulants to enhance their cognitive performance, based in part on the belief that stimulants enhance cognitive abilities such as attention and memorization. Of course, it is possible that these users are mistaken. One possibility is that the perceived cognitive benefits are placebo effects. Another is that the drugs alter students' perceptions of the amount or quality of work accomplished, rather than affecting the work itself (Hurst, Weidner, & Radlow, 1967). A third possibility is that stimulants enhance energy, wakefulness, or motivation, which improves the quality and quantity of work that students can produce with a given, unchanged, level of cognitive ability. To determine whether these drugs enhance cognition in normal individuals, their effects on cognitive task performance must be assessed in relation to placebo in a masked study design.
…Phenethylamine is intrinsically a stimulant, although it doesn't last long enough to express this property. In other words, it is rapidly and completely destroyed in the human body. It is only when a number of substituent groups are placed here or there on the molecule that this metabolic fate is avoided and pharmacological activity becomes apparent.
"Cavin, you are phemomenal! An incredulous journey of a near death accident scripted by an incredible man who chose to share his knowledge of healing his own broken brain. I requested our public library purchase your book because everyone, those with and without brain injuries, should have access to YOUR brain and this book. Thank you for your legacy to mankind!"
"In the hospital and ICU struggles, this book and Cavin's experience are golden, and if we'd have had this book's special attention to feeding tube nutrition, my son would be alive today sitting right here along with me saying it was the cod liver oil, the fish oil, and other nutrients able to be fed to him instead of the junk in the pharmacy tubes, that got him past the liver-test results, past the internal bleeding, past the brain difficulties controlling so many response-obstacles back then. Back then, the 'experts' in rural hospitals were unwilling to listen, ignored my son's unexpected turnaround when we used codliver oil transdermally on his sore skin, threatened instead to throw me out, but Cavin has his own proof and his accumulated experience in others' journeys. Cavin's boxed areas of notes throughout the book on applying the brain nutrient concepts in feeding tubes are powerful stuff, details to grab onto and run with… hammer them!
More than once I have seen results indicating that high-IQ types benefit the least from random nootropics; nutritional deficits are the premier example, because high-IQ types almost by definition suffer from no major deficiencies like iodine. But a stimulant modafinil may be another such nootropic (see Cognitive effects of modafinil in student volunteers may depend on IQ, Randall et al 2005), which mentions:
Modafinil is not addictive, but there may be chances of drug abuse and memory impairment. This can manifest in people who consume it to stay up for way too long; as a result, this would probably make them ill. Long-term use of Modafinil may reduce plasticity and may harm the memory of some individuals. Hence, it is sold only on prescription by a qualified physician.
^ Sattler, Sebastian; Mehlkop, Guido; Graeff, Peter; Sauer, Carsten (February 1, 2014). "Evaluating the drivers of and obstacles to the willingness to use cognitive enhancement drugs: the influence of drug characteristics, social environment, and personal characteristics". Substance Abuse Treatment, Prevention, and Policy. 9 (1): 8. doi:10.1186/1747-597X-9-8. ISSN 1747-597X. PMC 3928621. PMID 24484640. | CommonCrawl |
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1. Global concern over drinking water: Code for analysis
perceptions, attitudes, and beliefs, drinking water, water insecurity, and harm
Sera L. Young and Joshua D. Miller
Sera Young
Software or Program Code
2. Motivators and barriers to cerebral palsy research participation
This data was a part of a REDCap survey opened from 5/6/2020 to 7/7/2020. Potential participants with cerebral palsy and their families voluntarily responded to various questions about the motivators, barriers, and goals associated with the decision to participate in cerebral palsy research. Included in this work are the dataset...
Sukal-Moulton, Theresa, Ingo, Carson, Zvolanek, Kristina, Hill, Nayo, Joshi, Divya, Roth, Heidi, Goyal, Vatsala, and Hruby, Alex
Vatsala Goyal
3. The Rule of Law in the United States: An Unfinished Project of Black Liberation
civil rights, abolition, critical race theory, rule of law, and constitutional law
Paul Gowder
Paul Anthony Gowder
4. Characteristics of early-stage research into human genes
Throughout the last two decades several scholars observed that present-day research into human genes rarely turns toward genes that had not already been extensively investigated in the past. Guided by hypotheses derived from studies of science and innovation, we present here a literature-wide data-driven meta-analysis to identify the specific scientific...
Stoeger, Thomas
Thomas Stoeger
5. Height and Depth in Black Literature
This thesis discussed how motifs of height and depth in Black literature highlight Black characters' inability to form positive racial identities. The thesis uses three canonical pieces of Black literature, Passing, Native Son, and Plum Bun, to demonstrate that Black characters who try to reimagine White space are forced into...
Stovall, Jessica Lee
Chris Diaz
6. Worlds Collide
Major American Author and James Baldwin
Lydia E. Wuorinen
Lydia Elizabeth Wuorinen
7. A Texan Summer: Part One
Care Home, Memory Loss, and Fiction
8. Evaluating How Students Comprehend Title IX Policies at the Undergraduate Level
Inter-Rater Reliablity, Statistical Analyses, Cohen's Kappa Coefficient, Cognitive Dissonance, Policy, Title IX, Sexual Assault, and Clery Act
9. Data-driven playback of natural tactile texture via broadband friction modulation
Roman Grigorii
Roman Vasilievich Grigorii
10. Building Material Selection and Use An Environmental Guide [2nd Edition]
building materials, disaster reconstruction, climate change, environment, disaster management, and sustainability
Stephen H. Carr, PhD, P.E. Professor Emeritus of Materials Science and Engineering and Chemical and Biological Engineering, Northwestern University, Missaka Hettiarachchi PhD, CEng Senior Fellow, World Wildlife Fund, William M. Miller, PhD Professor of Chemical & Biological Engineering, Northwestern University, Mike M. McMahon Strategic Partnerships Administrator, Institute for Sustainability and Energy at Northwestern University., Vidushi Dwivedi, MS Research Assistant, Institute for Sustainability and Energy at Northwestern University, Jennifer B. Dunn, PhD Associate Professor of Chemical & Biological Engineering, Northwestern University , and Anita Van Breda Senior Director, Environment and Disaster Management World Wildlife Fund
Jennifer Pepson-Elwood , Megan McConnell, Elham Ramyar, and Vasantha Wakkumbura
Mike McMahon
Alternate Identifier:
DOI 10.21985/n2-139p-gw08 and ISBN 978-0-578-31332-0
11. Child and Family Policy in the 21st Century: A Focus on Early Childhood Education and Parental Work
This dissertation consists of three studies related to parental employment, maternity leave policy, and early care and education. In study 1, my coauthor and I evaluate the impact of parenthood on men and women's job performance and career advancement using detailed data from the U.S. Marines. For parents who remain...
Public policy, Labor economics, and Education
Healy, Olivia
Scholarly Digital Publishing
http://dissertations.umi.com/northwestern:15746 and etdadmin_upload_844713
12. The Children of Timelessness: Contemplative Poetry of the Soviet Stagnation
This dissertation explores the poetry and culture of the late-Soviet era of Stagnation (1964-1985) through a broadly conceived cultural metaphor of stagnation. The five Russian poets and one American poet in this study- Viktor Krivulin, Alexei Parshchikov, Aleksandr Eremenko, Ivan Zhdanov, Elena Shvarts, and Lyn Hejinian- each engage with a...
Russian Intellectual History , End of History , Russian Poetry, Soviet Stagnation, Russian Postmodernism , and Soviet Underground
Modern literature, Slavic literature, and Slavic studies
Topoleski, Anthony
13. A Sociable Silence: Silence and Sympathy in the Victorian Novel
This dissertation argues that silence played a fundamental role in the Victorian novel and in Victorian novel writing, operating as a productive force in service of sympathetic exchange and creative labor. It examines Charles Lamb's and Thomas Carlyle's foundational roles in detaching silence from its traditional Romantic associations with solitude,...
Space, Victorian, Literature, Sympathy, Silence, and Environment
History and English literature
Mason, Sarah Michelle
14. Poly-pathway effects of dopamine covered iron oxide core - titanium dioxide shell nanoparticles
This thesis proposes a robust multi-pronged approach to study the effect of nanoparticles on cells. In the first place, this work is focused on investigation of the protein corona that accumulates on the surface of nanoparticles internalized by the cells and their poly-pathway effects on protein availability and messenger RNA...
SR-FTIR, Nanoparticles, PARP, XFM, and BIRC5
Biology, Nanotechnology, and Biochemistry
Lastra, Ruben Omar
etdadmin_upload_837951 and http://dissertations.umi.com/northwestern:15671
15. Misregulation of Mitochondria-Lysosome Contact Sites in GBA-linked Parkinson's Disease
Mitochondria-lysosome contacts are recently identified sites for mediating crosstalk between both organelles, but their role in normal and diseased human neurons remains unknown. We used super-resolution and live-cell microscopy in human iPSC-derived neurons to demonstrate that mitochondria-lysosome contacts can dynamically form in the soma, axons, and dendrites of human neurons,...
Lysosome, Mitochondria, iPSC, Interorganelle contacts, GBA, and Parkinson's disease
Neurosciences, Molecular biology, and Cellular biology
Kim, Soojin
16. A Mixed Methods Evaluation of Social Support and Homophily in Weight Loss
The prevalence of obesity within the US continues to rise, and many individuals elect to involve supportive others in their weight loss. Social support is generally helpful in weight management, but its mechanisms are less understood. One construct that deserves further attention is homophily, or the notion that "birds of...
Weight Loss, Homophily, Obesity, Social Support, Behavior Change, and Intervention
Clinical psychology, Behavioral psychology, and Behavioral sciences
Hoffman Marchese, Sara
17. Effect of Phylogenetic and Functional Diversity on Invasion Resistance in Restored Tallgrass Prairies
The question of how native species diversity affects a community's ability to resist invasive species has inspired decades of research. One of the oldest invasion biology hypotheses is that more species rich ecosystems are less invaded. While there has been strong support for this hypothesis, there is also strong evidence...
invasive species, community ecology, phylogenetic diversity, ecological restoration, and functional diversity
Plant sciences, Ecology, and Conservation biology
Ernst, Adrienne
18. Psychosocial Determinants Related to Medication Taking Behaviors in Patients with Type II Diabetes
Recent estimates indicate that 21 million US adults live with Type II Diabetes Mellitus (T2DM). The management of the condition often requires patients to take multiple prescription medications to prevent disease progression; yet prescribed regimens themselves can become burdensome. Studies have shown that for patients with T2DM, the average regimen...
Russell, Andrea M
19. Methods for Synthesizing and Translating Statistical Evidence in Education
This dissertation is a collection of three papers on synthesizing and translating statistical evidence in education research. Chapter 1 serves as an introduction and executive summary, and Chapters 2 - 4 contain the three substantive papers respectively. Chapter 2 presents methods for pooling sample variances across studies to improve properties...
meta-analysis, clearinghouse data, translation science, data visualization, statistical cognition, and evidence synthesis
Statistics and Education
Fitzgerald, Kaitlyn Grace
20. Haunting as Historical Thinking: Learning to Construct Whiteness in History Classrooms
How we remember, narrate and teach the past is an inherently political and ethical act. This is especially true when teaching about race and racism within the context of United States history. In this dissertation, I ask: how do young people narrate the durability of racial inequality in the United...
History and Education
Berry, Allena Ginneh
21. Excited State Dynamics of Non-Fullerene Acceptors Towards Solar Energy Conversion Applications: Multi-Chromophore Rylenediimide Arrays and Beyond
Recent developments in research concerning organic photovoltaics (OPVs) have overseen massive increases in device performance and the ascension of electron acceptor materials that outclass the preeminent acceptor compounds, buckminsterfullerene (C60) derivatives. New design strategies in the molecular structure of perylenediimides (PDIs) and fused-ring electron acceptors (FREAs) have increased single-junction photovoltaic...
Alzola, Joaquin Miguel
22. Mind Perception and Emotions in Evaluative Contexts
The prospect of being evaluated by others is oftentimes psychologically crippling. At the core of feeling evaluated is perceiving other minds capable of possessing (judgmental) thoughts and feelings. While research on mind perception in evaluative situations oftentimes examines positive aspects (e.g., increased prosocial behavior), this dissertation looks into how mind...
Social psychology and Management
Schweitzer, Shane
23. Sub-diffusion and Compartmentalization: How the Nuclear Pore Complex and Stochastic Movement are Utilized to Dynamically Organize the Eukaryotic Genome
In nearly all Eukaryotes, the membrane-enclosed nucleus contains the vast majority of the cellular genome. Within this sub-cellular compartment, the nuclear architecture facilitates genomic chromatin organization. Controlling chromosomal loci's spatial positioning relative to subnuclear structures and each other can have local and global effects on gene expression. Moreover, chromatin organization...
Statistics, Epigenetics, Genome, Yeast, Nucleus, and Diffusion
Molecular biology, Genetics, and Biophysics
Sumner, Michael Chas
24. Gold Nanostars: Importance of Nanoparticle Shape at The Single-Particle Level
Noble metal nanoparticles (NPs) have shown promise as imaging agents, drug delivery platforms, and plasmonic sensors. Anisotropic gold NPs, such as gold nanostars, have particularly received attention due to their shape-dependent optical and spectral properties. With their 3D anisotropic structure with branches protruding into different directions and high surface areas,...
Nanostar, DIC Microscopy, Nanoparticle, and Electron Tomography
Choo, Priscilla
25. Probing the Hydrogen-Bonding Network at Soft Matter and Metal:Metal:Oxide:Water Interfaces
The ubiquitous role of water in biochemical, electrochemical, and geochemical systems has driven scientific interest in studying the fundamental hydrogen-bonding interactions that water molecules exhibit in the presence of different materials.Specifically, we focus on the interactions characterizing water at the interface between two bulk media, as these are essential to...
Dalchand, Naomi
26. Development of Bioresponsive Small Molecule Probes for Molecular Magnetic Resonance Imaging
In an era of personalized medicine, the clinical community has become increasingly focused on understanding diseases at the cellular and molecular level. Magnetic resonance imaging (MRI) is a powerful imaging modality for acquiring anatomical and functional information. However, it has limited applications in field of molecular imaging due to low...
Magnetic resonance imaging, Bioresponsive probe, Contrast agent, Lanthanide Probe, and Molecular imaging
Li, Hao
27. Protein Stabilization with Metal–Organic Frameworks for Drug Delivery and Catalysis
Proteins are known to have diverse biomedical functions and excellent catalytic performance; however, they are also fragile outside living cells, challenging their use in industrial applications. Metal-organic frameworks (MOFs) are highly porous crystalline materials that consist of metal cluster nodes and organic linkers. With their rigid structures, MOFs can effectively...
Protein Stabilization, Drug Delivery, and Biocatalysis
Chen, Yijing
28. Development and Characterization of PEG-b-PPS Nanocarriers for Magnetic Resonance Imaging and Drug Delivery
Like many diseases, atherosclerotic cardiovascular disease is driven by the activity of inflammatory cells. Using molecular imaging to target and analyze populations of inflammatory cells is one promising strategy to non-invasively assess atherosclerosis progression. However, current molecular imaging contrast agents are not suited for such targeted imaging applications. Nanomaterial-based strategies...
Nanomaterials, Targeted delivery, Nanoparticles, Drug delivery, and Magnetic resonance imaging
Nanoscience, Biomedical engineering, and Nanotechnology
Modak, Mallika
29. Precision Medicine for Human Complex Diseases through Integrative Analysis of Novel Epigenetic Modifications
Human complex diseases such as common cancers and diabetes are characterized by high molecular heterogeneity contributed by both genetic and non- genetic factors. This molecular heterogeneity can not only complicate diagnosis, risk stratification and patient care, but also lead to differential therapeutic response and treatment efficiency. Therefore, understanding the molecular...
Precision medicine , Cancer , Epigenetics, Complex disease , and Diabetes
Epidemiology and Bioinformatics
Zeng, Chang
30. Methods for Nonlinear and Noisy Optimization
In this thesis, we aim to develop efficient algorithms with theoretical guarantees for noisy nonlinear optimization problems, with and without constraints, under various different assumptions. Apart from Chapter 1 which provides relevant backgrounds, the remaining of thesis is divided into four chapters. In Chapter 2, we establish the theoretical convergence...
LBFGS, noisy optimization, quasi-Newton, derivative-free optimization, nonlinear optimization, and BFGS
Computer science and Industrial engineering
Xie, Yuchen
31. Modeling the dynamical neural systems on different timescales
With neurons as its primary computational components, the brain operates at multiple timescales. In this thesis, we focus on two timescales: on a relatively slow timescale on the order of hours to days, the brain adapts to the environment it is exposed to and learns its circuitry by altering the...
Synchronization, Binocular matching, Primary visual cortex , Neuronal heterogeneity, Computational neuroscience, and Gamma rhythm
Xu, Xize
32. Investigating Nanoparticle-Cell Interactions at Single-Nanoconstruct Level
Nanoparticles (NPs) are emerging as attractive drug carriers in therapeutic and diagnostic applications. The physiochemical properties of NPs, such as particle size, shape, and surface chemistry, play important roles in the functions of engineered nanoconstructs−NP cores with surface ligands. Recent work has screened these properties by monitoring cellular uptake and/or...
Liu, Tingting
33. The Pocket: A Theory of Beats as Domains
This dissertation proposes a theory that views beats as probabilistic domains that I term "pockets," taking a vernacular term commonly used by jazz, funk, and popular music performers to describe the state of being in a good groove and making it concrete through empirical methods. Pockets have three key properties:...
rhythm, microtiming, music information retrieval, and corpus studies
Hosken, Fred
34. Spatial Thinking and the Learning of Mathematics in the Game of Go
Although there has been profound evidence showing the positive correlation between spatial abilities and math performances, we still know very little about how and why spatial thinking facilitates the learning of mathematics. This dissertation unpacks several aspects of mathematics that are embedded in learning and playing an ancient and rich...
Mathematics, Spatial thinking, Game , and Learning
Mathematics education, Education, and Educational psychology
Yu, Yanning
35. Noise-Protected Superconducting Quantum Circuits
Superconducting circuits are electrical circuits fabricated from superconducting materials. Due to the engineered strong interactions and suppressed sensitivity to environmental noise, such systems hold substantial promise as quantum bits, in which quantum information can be stored and manipulated with high fidelity. Over the decades, great efforts have been devoted to...
Two-level systems, Quantum computation, Superconducting qubits, Noise-protected qubits, Quantum information, and Circuit quantization
You, Xinyuan
36. Bayesian-robust Algorithms Analysis with Applications in Mechanism Design
This thesis studies Bayesian-robustness of algorithm design. The main perspective requires for a single fixed algorithm that its performance is an approximation of the optimal performance when its inputs are independent and identical draws (i.i.d.) from every unknown distribution which is an element of a known, large class of distributions....
mechanism design, prior independent algorithms, benchmark design, and lower bounds
Johnsen, Aleck
37. Uncovering Molecular Mechanisms of C9orf72- and NEK1-related ALS Pathogenesis Using iPSC-Derived Motor Neurons
Amyotrophic lateral sclerosis (ALS) is a devastating neurodegenerative disease characterized by motor neuron (MN) degeneration and resulting in progressive paralysis and death. ALS is genetically heterogeneous, disease pathophysiology is not completely understood, and there are no effective drug therapies. To develop broadly applicable therapeutics, we examine disease mechanisms in the...
nucleocytoplasmic transport, NEK1, microtubules, iPSCs, C9orf72, and ALS
Daley, Elizabeth
38. Immunomodulatory Receptors in Herpes Simplex Virus Pathogenesis
Herpes simplex virus (HSV) is a ubiquitous human pathogen capable of causing debilitating diseases such as herpes stromal keratitis (HSK) and herpes simplex encephalitis (HSE). Although HSV infection initiates disease pathogenesis, the resulting clinical manifestations are attributed to immunopathological events that occur following viral clearance. To elucidate the molecular mechanisms...
herpes simplex virus, herpes stromal keratitis, TAM receptors, herpes simplex encephalitis, and herpesvirus entry mediator
Virology and Immunology
Park, Seo Jin
39. Some Examples of Unique Equilibrium States and Measures of Maximal Entropy
We prove the uniqueness of equilibrium states for certain potentials satisfying the Bowen property for two flows related to geodesic flows on surfaces with sufficient hyperbolicity. Our first result is the uniqueness of equilibrium states for Hölder continuous potentials and the geometric potential for products of geodesic flows of rank...
geodesic flow, conjugate points, hyberbolic dynamics, measure of maximal entropy, thermodynamic formalism, and equilibrium state
McEnroe, Rachel Maxine
40. The Batalin-Vilkovisky Laplacian from Homological Perturbation Theory
The BV Laplacian Δ, first introduced by Batalin and Vilkovisky, is a second-order differential operator that appears in the quantum master equation for quantizing gauge theories. The geometric framework for the BV formalism was later recognized by Schwarz as the setting of odd symplectic geometry and Khudaverdian showed that Δ...
Physics and Mathematics
Kumar, Nilay
41. Multi-Scale Imaging of Nanomaterial-Tissue Interactions
Nanomaterials are increasingly incorporated in modern day life, from the biogenic viruses that cause pandemics and the mineral crystallites embedded alongside collagen in our bones, to the anthropogenic nanomaterials that are small but powerful components of sunscreen and paint, swimming pool algaecides and wound dressings, cancer treatments, bicycle frames, and...
Nanoscience and Materials Science
DiCorato, Allessandra Elizabeth
42. Topics in Meta-analysis with Few Studies
This dissertation consists of three papers on methods for meta-analysis with few studies. These papers are concerned with proper inference from meta-analysis models that combine data from a small number of studies using fixed and random-effects models. Chapter 1 provides an introduction to meta-analysis, the motivation for this work and...
Zejnullahi, Rrita
43. Numerical Simulation of Irradiation Induced Defects in Polycrystalline Solids
A framework is developed that models point defect diffusion and interaction with pre-existing microstructures during irradiation, including defect-defect interactions and defect sinks. This framework uses a modified diffusion potential that includes not only defect concentration, but also intrinsic stresses from the pre-existing microstructure. Various microstructures are studied in {Fe} by...
Point defects, Radiation, Triple junctions, Voids, Crack tip, and Grain boundaries
Nuclear engineering, Mechanical engineering, and Materials Science
Zarnas, Patrick
44. Inhibitory Effects of the MNK1/2 Inhibitor Tomivosertib in Acute Myeloid Leukemia
The treatment of AML remains to be a challenge due to the high rates of resistance and relapse experienced by patients after initial therapy. The MAPK-interacting kinases 1 and 2 (MNK1/2) have generated increasing interest as therapeutic targets for AML due to their critical role in malignant hematopoietic transformation via...
Biology, Molecular biology, and Biochemistry
Suarez Palacios, Milagros Melanie
45. The Ties That Bind Us to Earth: Neighborhoods and Interpersonal Relationships of Black Southern Marylanders, 1850-1910
This dissertation explores lasting familial relationships and friendships among southern African Americans from the antebellum years to the turn of the twentieth century. Focusing on southern Maryland, the dissertation shows how free and enslaved African Americans cultivated familial and non-familial relationships in towns and rural neighborhoods. Over the course of...
African American Studies and Black Studies
History, Black history, and American history
Rosado, Ana
46. Measurement of the W Boson Branching Fractions in Proton-Proton Collisions at 13 TeV Center-of-mass Energy with the CMS Experiment
The leptonic and inclusive hadronic decay branching fractions of the W boson are studied using 35.9 $fb^{-1}$ of proton-proton collision data collected at $\sqrt{s}=13~TeV$ during the 2016 run of the CMS experiment. Events characterized by the production of pairs of W bosons from \ttbar and \tW processes are selected. Multiple...
lepton flavor universality, W boson branching fraction, and CMS
Particle physics and Physics
Chen, Ziheng
47. Using automatic differentiation for coherent diffractive imaging applications
Coherent diffractive imaging (CDI) methods are techniques that image a sample by illuminating it with a coherent beam and recording the intensity diffraction pattern produced by the wavefront outgoing from the sample. These are lensless methods that are not limited in resolution by the physical characteristics of an objective lens,...
Applied physics and Physics
Kandel, Saugat
48. Tailoring Modular Spherical Nucleic Acids for DNA and RNA Delivery
Nucleic acids such as DNA or RNA of various lengths and structures have a wide scope of functions as therapeutic entities compared to conventional drugs. For instance, native and modified forms of nucleic acids can be used for gene silencing, genome editing, gene replacement, immune system modulation, and theranostics. While...
lipids, RNA, drug delivery, nanoparticles, and DNA
Sinegra, Andrew
49. The Greener Inhumanity of Renaissance Pastoral: A Posthumanist Reading of the Bucolic Literature of Early Modern England and Italy
In a new analysis of Renaissance pastoral that draws on ecocriticism, queer theory, and a historicist approach, this dissertation finds a green and inhuman world that opposes the modern view that humans differ significantly from, and enjoy a right of dominion over, nonhuman species and the environment. Through readings of...
posthumanism, Machiavelli, pastoral, nature, Shakespeare, and ecocriticism
Comparative literature, Italian literature, and English literature
Pederson, Alicia Sands
50. A Normative Lay Theory of Risk-taking
How do people make meaning of risk-taking? The present dissertation proposes a normative lay theory of risk-taking. The proposed model promotes the following core ideas: (a) Risk-taking is generally an ambiguous construct and requires the illumination of at least some dimensional parameters to disambiguate the risk behavior and risk-taker; (b)...
attitudes, responsible, lay theory, perceptions, reckless, and risk-taking
Cognitive psychology, Social psychology, and Psychology
Wages, James Ellis
51. Electrodynamics of particles in bulk fluid and on an interface
The focus on this thesis is on the dynamics of colloidal particles in an applied electric field in a uniform bulk fluid and on a fluid-fluid interface. In a bulk fluid, the dynamics of an isolated particle, one pair, and a cluster of particles under an applied nonuniform electric field...
collective motion, electrorotation, and colloidal particle
Fluid mechanics and Applied mathematics
Hu, Yi
52. Decentralized Persistent Shape Formation in Large-Scale Homogeneous Robotic Swarms
This research looks at the robotic shape formation problem, which is one of the fundamental problems in robotic swarm systems. Here, the task is to move a group of robots to form a user-specified shape. In this dissertation, the task of shape formation is divided to four problems: (i) using...
distributed systems, scheduling and planning, algorithms, sensor networks, and swarm systems
Computer science, Computer engineering, and Robotics
wang, hanlin
53. Investigating the Structure of Molybdenum Oxides and Sulfides for Catalytic Dehydrogenation
Molybdenum oxides and sulfides are earth-abundant materials known to catalyze a wide array of reactions, including dehydrogenation, hydrotreating, and higher alcohols synthesis. In particular, alkane and alcohol dehydrogenation are of interest given recent shifts in the energy landscape away from traditional petroleum feedstocks and towards natural gas and renewable energy...
heterogeneous catalysis, molybdenum, metal sulfides, metal oxides, and alkane dehydrogenation
Cheng, Emily
54. Associations between Empathy Development and Collective Music Making with Free Improvisation and Music Notation for Adolescent Musicians
The purpose of this study was (1) to examine the impact that small ensemble free improvisation experiences had on dispositional empathy development when compared with other forms of collective music making; and (2) to examine the relationship between co-performing musicians' empathy levels and their performance achievement in small ensembles using...
Free Improvisation, Collective Music Making, Performance Achievement, and Empathy Development
Schmidt, Casey
55. Optical Stark shifts of hybrid light-matter states in two-dimensional semiconductors
Light provides a high-speed and coherent medium for controlling quantum states. In semiconductors, coherent optical effects have been used extensively to lift spin degeneracy on ultrafast timescales and demonstrate high-fidelity control of quantum spin states. Extending this approach to novel pseudospins in monolayer transition metal dichalcogenides (TMDs), in this thesis...
Polariton, Stark effect, ultrafast optics, valley, two-dimensional materials, and exciton
Condensed matter physics, Applied physics, and Optics
LaMountain, Trevor
56. Changes in Motor Unit Firing Patterns as a Function of Age, Muscle, and Following a Unilateral Brain Injury: Ionotropic and Metabotropic Effects
Coordinated movement relies on the precise and controlled activation of populations of motor units, which convert the commands of the nervous system into muscle forces. Motor unit firing patterns are often nonlinear and generated through the response to a combination of ionotropic excitatory and inhibitory commands, as well as metabotropic...
Aging, Delta-F, Stroke, Persistent Inward Currents, Motor Unit, and Motoneuron
Neurosciences and Biomedical engineering
Hassan, Altamash Shamsul
57. Anxious to Play: Social and Emotional Forces that Restrict Women's Video Game Skill Development
Competitive gaming, or esports, is a high-skill endeavor embedded in a highly gendered social context. Using multiple methodological approaches, this dissertation argues that gender-gaming inequality is a result of changeable stereotypes that impact women throughout their lives. Specifically, gender-gaming stereotypes limit women's initial access to gaming, discourage their continued interest...
esports, expertise, learning, performance, stereotypes, and video games
Social psychology and Psychology
Nolla, Kyle
58. Surface Chemistry of Organic Molecules in Atmospheric and Indoor Environments Investigated by Sum Frequency Generation Spectroscopy
This work examines important heterogeneous processes of organic molecules on surfaces, in the contexts of atmospheric and indoor environments. In large forest ecosystems, biogenic secondary organic aerosols (SOAs) constitute a dominant fraction of organic particulate matter in the atmosphere. The formation of SOAs starts from the emission of volatile organic...
Physical chemistry, Environmental science, and Chemistry
Liu, Yangdongling
59. Physics-Informed Data-Driven Prediction and Design in Advanced Manufacturing Processes
Manufacturing processes are known for their intricacies in changing material shapes and properties. New generations of manufacturing technologies, known as flexible manufacturing, are moving toward design freedom, which allows producing parts with optimized geometries and high customizations at an affordable cost even for low-volume productions. Two prominent flexible manufacturing processes...
Reinforcement Learning, Neural Network, Additive Manufacturing, Manufacturing, and Artificial Intelligence
Computer science, Mechanical engineering, and Computational physics
Mozaffar, Mojtaba
60. The Gamma Conjecture for Local Mirror Symmetry: a Tropical Approach via the Gross-Siebert Program
In this thesis, we study the homological mirror symmetry for the Gross-Siebert program of local mirror symmetry. We construct a pair of mirror objects by lifting a tropical curve in the integral tropical manifold of the Gross-Siebert program. Furthermore, we evaluate the central charges of the mirror objects and show...
Wang, Junxiao
61. Engineering Low-Dimensional Layered Structures
The highly flexible nature of 2D materials has led to them becoming fundamental building blocks for achieving novel device physics and potential breakthroughs in practical technologies. 2D layers can be interfaced in a wide array of methods with themselves, other 2D layered materials, or materials of entirely different type or...
Monolayer, Exciton, Low-Dimensional, Heterojunction, Two-Dimensional, and Heterostructure
Condensed matter physics, Optics, and Materials Science
Stanev, Teodor Kosev
62. Museum Practitioners' Beliefs and Assumptions about Racial, Ethnic, and Cultural Diversity; Minoritized Learners' Sensemaking; and Their Institutional Context
This dissertation aims to: 1) characterize the range of beliefs museum practitioners have about racial, ethnic, and cultural diversity; 2) their understanding of the role of race, culture, and ethnicity in minoritized learners' sensemaking; and 3) the areas of tension and symmetry between practitioners' values and their perception of their...
Museum, Positioning Theory, Informal Learning, Science, Diversity, and Equity
Education, Teacher education, and Museum studies
Villanosa, Krystal
63. The Muscle-Powered Empire Organic Transport in Japan and Its Colonies, 1850 – 1930
This dissertation investigates the muscle-powered transport technologies that pervaded the Japanese empire. It examines the production, adoption, evolution, and decline of draft animals, rickshaws, human-powered railways, and push-car railways in Japan and colonial Taiwan, 1850-1930. Invented in Tokyo in 1870, rickshaws proliferated across Asia and became a symbol of modern...
History, Asian history, and Asian studies
Li, Youjia
64. Genetic Mechanisms of Benzimidazole Resistance in Caenorhabditis elegans
Parasitic nematode infections are common in both humans and livestock populations around the globe. In humans, these infections cause illness which can be debilitating. In livestock, parasitic nematode infections result in poor animal health and wellbeing as well as decreases in the yield of these animals. The decrease in yield...
Parasites, C. elegans, Anthelmintics, and beta-tubulin
Dilks, Clayton Matthew
65. Leading Teams in the Digital Age: Team Technology Adaptation in Human-Agent Teams
This dissertation imagines the near future of teamwork, when AI agents will join teams,interacting, collaborating, and completing tasks as a team member. Broadly, I seek to answer the questions: how do humans integrate a new AI teammate onto their team, and how does the AI teammate's function influence this integration...
team, team cognition, AI teammate, and team technology adaptation
Communication and Organizational behavior
Larson, Lindsay Elizabeth
66. A Victorian Disposition: Emotional Susceptibility in the Nineteenth-Century Novel
This dissertation argues that the nineteenth-century construction of "emotional susceptibility" turned a much-derided quirk of psychology—the long retention of one's earliest affective impressions—into a basis for radical interventions into thinking about attachment, ethics, and the Victorian novel. I focus in particular on the work of Henry Mackenzie, Charlotte Brontë, George...
Ethics, Impressibility, Affect, and Susceptibility
Cogswell, Clay
67. Multiscale in vivo Muscle Architecture in the Upper Limb: A Validation and Implementation of Novel Imaging Techniques for Quantification of Muscle Function and Plasticity
A major distinction among different skeletal muscles in the human body is the number, size, and arrangement of its cells, referred to as a muscle's architecture. Muscle architecture is indicative of a muscle's ability to contract and produce force and, like muscle function, is plastic. While neuromuscular plasticity is the...
Sarcomere, In vivo, Imaging, Muscle, and Wrist
Bioengineering and Biomechanics
Adkins, Amy
68. Geometric limits of Julia sets of unicritical polynomials under degree growth
In this thesis we study the geometric limits under degree growth of Julia sets and filled Julia sets for complex polynomials with a unique critical point at $z = 0$. Specifically, for $c \in \mathbb{S}^1$, we are interested in the limit of the associated sequence of Julia sets $J(f_{n,c})$ in...
Jensen, Signe Emalia
69. Synthesis and Characterization of Low-Dimensional Materials for Dynamic Reconfigurability in Mixed-Dimensional Heterostructures
This thesis describes the synthesis and photophysical characterization of low-dimensionalmaterials—including thin-film semiconductors, colloidal quantum dots, and molecules—with the broader motivation of integrating them into mixed-dimensional heterostructures with novel responses to external stimuli. Due to their high surface area to volume ratio and incomplete dielectric screening, mixed-dimensional heterostructures have high sensitivity...
Chemistry, Physics, and Materials Science
Olding, Jack Nicklaus
70. Self-Assembled Nanodielectrics and Combustion Processed Amorphous Metal Oxides as Unconventional Materials for Thin-Film Transistors
The demand for low cost, unconventional electronics requires new materials with unique characteristics that the traditionally used silicon-based technologies cannot provide. Metal oxide semiconductors, such has amorphous indium gallium zinc oxide (a-IGZO), have made impressive strides as alternatives to amorphous silicon for electronics applications. However, to achieve the full potential...
Combustion processing, Thin-film transistor, Metal Oxides, and SANDs
Chemistry, Inorganic chemistry, and Materials Science
Stallings, Katie Lynn
71. Complex Exposures to Social Determinants of Health through Young Adulthood and Associations with Mid-life Cardiovascular Health and Events: The Coronary Artery Risk Development in Young Adults (CARDIA) Study
In the U.S., approximately 840,000 Americans die from cardiovascular disease (CVD) each year, and it is the leading cause of morbidity and mortality worldwide. The prevalence of CVD is on the rise and widespread disparities in CVD exist across economic, racial, and ethnic groups. In order to address the rising...
informatics, cardiovascular health , social determinants of health, machine learning, and data science
Public health and Health sciences
Zimmerman, Lindsay
72. Three Essays on Religious Identity and the Cultural Authority of Science
Broadly speaking, this dissertation project seeks to address the following question: how do religious people think about the cultural authority of science, and to what extent does this vary across different contexts? Despite the predictions of classical modernization theorists, religious institutions continue to significantly shape public discourse—and rule-making—in the vast...
Religious Identity, Comparative Sociology, Science and Technology, and Public Opinion
Lee, John J.
73. Programming Assembly Pathways of Proteins Using DNA
The building blocks of life are proteins. These incredible nanostructures are responsible for forming the diverse infrastructure of living systems and for performing countless biological functions. In Nature, these materials and systems achieve structural complexity and function through highly regulated and controlled assembly of protein building blocks, driven by specific...
Supramolecular , Proteins, Self-assembly, and Hierarchical
Chemistry and Nanotechnology
Hayes, Oliver George
74. Calling on Courage: The Use of a Courage Intervention to Increase Engagement in Exposure for Specific Fears
Despite its demonstrated effectiveness, exposure therapy – repeatedly approaching a fear/anxiety trigger – is not widely used in the treatment of anxiety disorders. This may be due to its image as an aversive (and even harmful) approach to treatment and its reduced rates of compliance among patients. However, if exposure...
exposure, courage, therapy, fear, and anxiety
Eix, Amanda Kramer
75. Dependence as Independence, Instability as Immaturity: The Organizational Contradictions of Young Adult Homeless Centers Serving LGBTQ+ Clients
Although research has shown LGBTQ+ youth are overrepresented in counts of homeless youth, scholars have yet to investigate whether this trend exists among adults experiencing homelessness. This dissertation uses an organizational analysis of four Chicago homeless centers that cater to young adults to argue that most LGBTQ+ youth are not...
poverty, organizations, governance, sexuality, young adults, and homelessness
Sociology, Public policy, and LGBTQ studies
Lovell, Erik Stephen
76. Mapping Intimacies: Black Queer Women in Chicago's Urban-Digital Sphere
Theoretical and empirical inquiries into queer geographies have focused primarily on how white gay subjects navigate urban landscapes. Consequently, there has been little empirical work that examines (1) queer placemaking within Black and brown urban spaces; (2) placemaking among queer women of color; and (3) the relationship and interplay between...
black feminism, queer studies, digital ethnography, black queer studies, sexuality, and urban ethnography
Sociology and Gender studies
Adams-Santos, Dominique
77. Adsorption in Metal-Organic Frameworks for Energy Applications
Metal-organic frameworks are crystalline, nanoporous materials formed by metal nodes connected by organic ligands. MOFs represent an exciting approach to materials design where a material with desired properties can be made by choosing the compatible nodes, linkers and topologies independently. MOFs are highly porous and have high surface areas...
zeolitic imidazolate framework, n-alkanes, metal-organic frameworks, molecular simulations, and adsorption
Gopalan, Arun
78. Forms Follows Action: Performance in/against the City in New York and Los Angeles (1970-1985)
This critical/theoretical history of performance art investigates the relationship between the body of the artist and the infrastructure of the city in Los Angeles and New York City between 1970 and 1985, with specific attention to how performance art resists, renegotiates, and responds to architectural functionalism. Using performance studies as...
Movement, Los Angeles, Kinesthetic, New York City, Built environment, and Performance Art
Performing arts, Art history, and Architecture
Morelli, Didier
79. Role of the Type I IFN Response in Age-dependent Pathogenesis of HSV Encephalitis
Newborns are particularly susceptible to severe forms of herpes simplex virus type I (HSV-1) infection including encephalitis and multisystemic disseminated disease. The underlying age-dependent differences in the immune response that explain this increased susceptibility relative to the adult population remain largely understudied. Evidence from animal studies and genetic studies in...
HSV, Interferon Response, Neonatal Immunity, and Encephalitis
Giraldo Perez, Daniel
80. Homogeneous Organocalcium and Organolanthanide Catalysis for the Formation of C–N Bonds and Selective C–O Bond Cleavage
Among the most valuable applications of organometallic chemistry is its implementation in the field of catalysis. Many industrial processes rely heavily on catalysis, employing organometallic complexes in the production of commodity chemicals, fine chemicals, materials, and even in the discovery and development of pharmaceuticals. Through decades of intense study, homogeneous...
organolanthanide, homogeneous, catalysis, organic synthesis, rare earth, and organocalcium
Dicken, Rachel Dowrey
81. 'The Indians Say': Settler Colonialism and the Scientific Study of North America, 1722 to 1848
"'The Indians Say': Settler Colonialism and the Scientific Study of North America, 1722 to 1848" examines the issue of evidence and credibility within natural history by following the circulation of Indigenous testimony through Anglophone networks of scientific knowledge production. By merging the history of science with Native American and Indigenous...
natural history, history of science , settler colonialism , and testimony
Native American studies and American history
Jones, Emma Bennett
82. Transient Absorption Microscopy Study of Excited-State Dynamics and the Structural Origins in Metal Halide Perovskites
Metal halide perovskites have recently emerged as one of the most promising active layers in solar cells for their high power conversion efficiency (>25%) and ease of synthesis and deposition. Spatial heterogeneity is inevitable with current fabrication methods for both monocrystalline and polycrystalline perovskite thin films and crystals. The morphology-dependent...
Trap state, Metal halide perovskites, Grain Boundary, Carrier cooling, Transient absorption microscopy, and Global analysis
Jiang, Xinyi
83. Control of Light-Induced Electronic Behavior at Interfaces of Hybrid Nanomaterials
This dissertation explores ways to utilize physical parameters at the nanoscale interface to control the properties of mixed-dimensional heterojunctions (MDHJs). MDHJs combine the desirable properties of different classes of low-dimensional nanomaterials (materials that are quantum confined in at least one dimension). While MDHJs have achieved superlative performance for a variety...
nanomaterials, photophysics, time-resolved spectroscopy, mixed-dimensional, and low-dimensional
Physical chemistry, Nanoscience, and Materials Science
Padgaonkar, Suyog
84. The effects of hybrid histidine kinase BinK on <i>Vibrio fischeri</i> biofilm formation and host colonization
Symbiotic relationships involve a life-long interaction between host and bacteria, and there is much we do not understand about how these interactions are developed and maintained. During the horizontal recruitment of beneficial bacteria by hosts, a complex set of molecular signals and communication ensures specificity. On the bacterial side, these...
two-component signaling systems, Vibrio fischeri, symbiosis, and biofilm
Ludvik, Denise Alberta
85. Novel Functions of Mitochondria-Lysosome Contact Sites in Health and Neurological Disease
Inter-organelle contact sites have become increasingly appreciated as important regulators of cellular homeostasis, and disruption of inter-organelle contact site dynamics and function has been observed in various pathologies. Recently, inter-organelle contact sites between mitochondria and lysosomes were discovered, offering a new mechanism by which these two organelles may directly interact,...
Lysosome, Contact Sites, Imaging, Mitochondria, Neurological Disease, and Calcium
Biology, Neurosciences, and Medicine
Peng, Wesley J
86. Causal Heterogeneity in Social Essentialism: Shared Experiences and Shared Genes
We structure our lives around social groups – belonging to them and thinking about them. In this dissertation, I develop a new stereotype content measure to assess the attributes associated with groups in America today, propose and support a theory of sociocultural essentialism, and explore the strategic activation of sociocultural...
Social Groups, Stereotype Content, Entitativity, Common Fate, Marginalization, and Essentialism
Gallagher, Natalie McDaniel
87. Quantum-Chemical Screening of Redox-Active Metal–Organic Frameworks
Metal–organic frameworks (MOFs) are a class of crystalline materials composed of metal nodes connected by organic linkers. Due to their high degree of synthetic tunability, MOFs have been considered for a wide range of applications, including many that rely on a change in oxidation state. While most MOFs are generally...
Computational screening, Density functional theory, Metal-organic framework, and Machine learning
Rosen, Andrew Scott
88. Building a Rational Understanding of Novel 2D Hybrid Halide Perovskites in Bulk and in Films
Two-dimensional (2D) hybrid halide perovskites have been the response to their exciting but woefully unstable 3D counterparts. These 2D perovskites have been shown to have respectable stabilities as photovoltaic absorbers, yet they lag behind the 3D perovskites in terms of efficiency. With the need to catch up to the efficiencies...
films, mechanisms, GIWAXS, materials, perovskites, and solar cells
Hoffman, Justin Michael
89. Signatures of the Invisible: New Considerations for Xenon Dark Matter Searches
The nature of dark matter (DM) remains one of the largest open questions in modern physics despite the many searches for particle dark matter around the world. The search for dark matter particles with masses above 5 GeV/$c^2$ is led by the dual-phase xenon time-projection chamber (LXe-TPC) detector technology which...
dark matter, backgrounds, xenon, and instrumentation
Temples, Dylan Jason
90. Estimation of persistent inward currents in the human ankle flexor and extensor muscles
Movement is achieved by combining synaptic inputs from various sources and activating motor unit populations. Motor units are the quantal elements of motor control which act as a neuromechanical transducer that converts sensory inputs into motor output. Because of the tight neuromuscular junctions between motoneuron axon terminals and a large...
Motor Units, Delta-F, EMG, PICs, and Human Motor Control
Kim, Edward
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TWI514767B - A data-driven charge-pump transmitter for differential signaling - Google Patents
A data-driven charge-pump transmitter for differential signaling Download PDF
TWI514767B
TWI514767B TW101136887A TW101136887A TWI514767B TW I514767 B TWI514767 B TW I514767B TW 101136887 A TW101136887 A TW 101136887A TW 101136887 A TW101136887 A TW 101136887A TW I514767 B TWI514767 B TW I514767B
TW101136887A
TW201332290A (en
John W Poulton
Thomas Hastings Greer
William J Dally
2012-10-05 Application filed by Nvidia Corp filed Critical Nvidia Corp
2013-08-01 Publication of TW201332290A publication Critical patent/TW201332290A/en
2015-12-21 Publication of TWI514767B publication Critical patent/TWI514767B/en
230000011664 signaling Effects 0 title 1
H04L25/00—Baseband systems
H04L25/02—Details ; Arrangements for supplying electrical power along data transmission lines
H04L25/0264—Arrangements for coupling to transmission lines
H04L25/0272—Arrangements for coupling to multiple lines, e.g. for differential transmission
H01L2224/00—Indexing scheme for arrangements for connecting or disconnecting semiconductor or solid-state bodies and methods related thereto as covered by H01L24/00
H01L2224/01—Means for bonding being attached to, or being formed on, the surface to be connected, e.g. chip-to-package, die-attach, "first-level" interconnects; Manufacturing methods related thereto
H01L2224/02—Bonding areas; Manufacturing methods related thereto
H01L2224/04—Structure, shape, material or disposition of the bonding areas prior to the connecting process
H01L2224/05—Structure, shape, material or disposition of the bonding areas prior to the connecting process of an individual bonding area
H01L2224/0554—External layer
H01L2224/0555—Shape
H01L2224/05552—Shape in top view
H01L2224/05554—Shape in top view being square
H01L2224/42—Wire connectors; Manufacturing methods related thereto
H01L2224/47—Structure, shape, material or disposition of the wire connectors after the connecting process
H01L2224/48—Structure, shape, material or disposition of the wire connectors after the connecting process of an individual wire connector
H01L2224/481—Disposition
H01L2224/48135—Connecting between different semiconductor or solid-state bodies, i.e. chip-to-chip
H01L2224/48137—Connecting between different semiconductor or solid-state bodies, i.e. chip-to-chip the bodies being arranged next to each other, e.g. on a common substrate
H01L2224/49—Structure, shape, material or disposition of the wire connectors after the connecting process of a plurality of wire connectors
H01L2224/4912—Layout
H01L2224/49175—Parallel arrangements
H01L2924/00—Indexing scheme for arrangements or methods for connecting or disconnecting semiconductor or solid-state bodies as covered by H01L24/00
H01L2924/10—Details of semiconductor or other solid state devices to be connected
H01L2924/11—Device type
H01L2924/13—Discrete devices, e.g. 3 terminal devices
H01L2924/1304—Transistor
H01L2924/1306—Field-effect transistor [FET]
H01L2924/13091—Metal-Oxide-Semiconductor Field-Effect Transistor [MOSFET]
H01L2924/30—Technical effects
H01L2924/301—Electrical effects
H01L2924/30107—Inductance
H01L2924/3011—Impedance
H01L2924/30111—Impedance matching
Data driven charge pump transmitter for differential signaling
SUMMARY OF THE INVENTION The present invention is generally directed to transmitting signals between discrete integrated circuit devices, and more particularly to a differential signaling technique using a data driven switched capacitor transmitter or a bridged charge pump transmitter.
A single-ended signaling system uses a single signal conductor per bit stream to transfer it from one integrated circuit component (wafer) to another. In contrast, differential signaling systems explicitly require two signal conductors, so single-ended signaling is typically preset to have advantages when the number of external pins and signal conductors is limited by the limitations of the package.
However, a single-ended signaling system actually requires more than one signal conductor per channel and requires more circuitry. The current flowing from the transmitter to the receiver must be returned to the transmitter to form a complete electronic circuit, and in a single-ended signaling system, the returning circuit will flow over a common set of conductors, essentially Wait for the power supply terminal. In order to keep the return current flowing substantially close to the signal conductor, the common return terminals are typically physically planar in the package, such as a chip package or printed circuit board, while allowing the signal conductors to form a line or miniature band. Therefore, a single-ended signaling system requires more than N pins and conductors to carry N bit streams between the wafers, and the burden is substantially about 10-50%.
A single-ended signaling system requires a reference voltage at the receiver such that the receiver can distinguish between (substantially) representing two signal levels of "0" and "1". Conversely, the differential signaling system does not require a reference voltage: the receiver only needs to compare the voltages on the two symmetric conductors of the differential signaling system to distinguish the data values. There are many ways to establish a reference voltage for a single-ended signaling system. However, it is substantially difficult to ensure compliance with the value of the reference voltage between the transmitter and the receiver, which requires compliance to ensure consistent interpretation of the signals transmitted by the transmitter to the receiver.
A single-ended signaling system consumes more power for a given signal-to-noise ratio than an equivalent differential signaling system. In the case of a resistive terminal transmission line, a single-ended system must drive a current of +V/R 0 for a "1" to be transmitted to establish a voltage V above the reference voltage at the receiver, for a "0" The current -V/R0 must be pulled out to establish a voltage V below the reference voltage at the receiver, where R 0 is the terminating resistor. Therefore, the system consumes 2V/R 0 of current to establish the desired signal at the receiver. In comparison, when differential signaling is used, the transmitter requires only ±V/2R 0 of drive current to establish the same voltage (V) across the receiver terminals due to the symmetrical signal conductor pairing. A differential signaling system only needs to draw V/R 0 current from the power supply. Thus, even if a reference voltage at the receiver is perfectly matched to the transmitter, the single-ended signaling system is essentially only half the power efficiency of the differential signaling system.
Finally, single-ended systems are more susceptible to externally coupled noise sources than differential systems. For example, if the noise is electromagnetically coupled to a signal conductor of a single-ended system, the voltage generated by the coupling will arrive at the receiver as unremovable noise. Therefore, the noise budget of the signaling system must be responsible for all such noise sources. Unfortunately, this kind of noise coupling usually comes from an adjacent line in a single-ended signal, called cross-talk, and the noise source is proportional to the signal voltage level, so it cannot be borrowed. This is overcome by increasing the signal level. In differential signaling, the two symmetrical signal conductors can actually operate in close proximity to one another between a transmitter and a receiver, so that noise is symmetrically coupled into the two conductors. Therefore, many external noise sources affect the two lines approximately the same, and this common mode noise can be excluded at a receiver having a higher differential gain than the common mode gain.
Therefore, there is a need in the art for a technique for providing single-ended signaling that reduces the problem of establishing a reference voltage, reduces the common impedance of the return path of the signal, and crosstalk interference caused by the return path of the signal, and reduces the Power consumption of a single-ended signaling system.
The first A diagram shows an exemplary prior art single-ended signaling system 100 to illustrate the reference voltage problem, sometimes referred to as a "Pseudo-open-drain" (PODL) system. The single-ended signaling system 100 includes a transmitting device 101 and a receiving device 102. The transmitting device 101 operates by drawing a current Is from the power supply when transmitting a "0", and does not draw current when transmitting a "1" (allowing the termination resistors R0 and R1 to pull the signal high to Vdd) ). In order to develop a signal of size |V| at the receiving device 102, the signal swing must be 2V, so the current is 2V/(R0/2) when driving a "0", otherwise the current is zero. For an average of an equal number of "1"s and "0"s, the system consumes 2V/R0 from the power supply.
The signal at the input of the receiving device 102 is swung down from Vdd ("1") to Vdd - 2V ("0"). In order to distinguish the received data, the receiving device 102 requires a reference voltage of Vref = Vdd - V. There are three ways to generate the reference voltage, as shown in the first A, B, and C diagrams.
As shown in FIG. A, an external reference voltage Vref is generated by a resistor network located adjacent to the receiving device. The external reference voltage is transmitted to the receiving device via a dedicated pin 103 and distributed to some of the receivers sharing the external reference voltage. The first problem with the external reference voltage technique shown in FIG. A is that the external reference voltage is developed across the power supply terminals Vdd and GND across the external resistors R2a and R2b, and the resulting external The reference voltages cannot be matched to the voltage developed by the current sources in the transmitting device 101 because the current sources are completely free of the external resistors R2a and R2b.
A second problem is that the power supply voltage at the receiving device 102 can be different from the power supply voltage at the transmitting device 101 because the supply networks provided to the two communication chips have different impedances, and the two The chips draw different and variable currents. The third problem is that the noise injected into any of the transmitting lines 105 of the coupling transmitting device 101 to the receiving device 102 is not incident into the reference voltage, so the transmitting system must be directed to the hair The worst case of the noise voltage of the signal line 105 is budgeted. The fourth issue is the external power supply. The voltage level between the terminals Vdd and GND is different from the internal power supply network within the receiving device 102, again due to the supply impedance. Additionally, the configuration of the single-ended signaling system 100 is such that the currents in the shared supply terminals are related to the data. Accordingly, any data-related noise introduced into the inputs of the internal receiver amplifiers in the receiving device 102 is different from the external supply of the shared external reference voltages that are also input to the internal receiver amplifiers. News.
The first B diagram shows an exemplary prior art single-ended signaling system 120 that uses an internal reference voltage. The internal reference voltage Vref can improve these noise problems compared to a single-ended signaling system 100 that uses an external reference voltage. The single-ended signaling system 120 tracks the reference voltage level of the transmitting device 121 more closely than the single-ended signaling system 100. A proportional transmitter is included in the receiver circuit of the receiving device 122, which can generate an internal Vref associated with the terminating resistor and the transmitter current Is. Because the internal reference voltage is generated relative to the internal power supply network, the internal reference voltage is not subject to such supply-noise problems as the external voltage reference as shown in FIG. However, there is still a problem of noise coupling in conjunction with the external reference voltage method described in FIG. Additionally, because the current source (Is/2) used to generate the internal reference voltage is in a different wafer than the current sources in the transmitter device 122, the current source may not be tracked in the receiving device 122. These current sources are in 121.
The first C-picture shows an exemplary prior art single-ended signaling system 130 using an accompanying reference voltage Vref. Since the accompanying reference voltage is generated in the transmitting device 131 using a proportional transmitter, and the accompanying reference voltage is coupled to the same internal supply network as the data transmitters in the transmitting device 131, the accompanying reference is provided. Tracking between voltage and signal voltage can be improved. Therefore, the accompanying reference voltage voltage can be used to reasonably track the program-voltage-temperature variation of the transmitter device 131. The accompanying reference voltage is transmitted from the transmitting device 131 to the receiving device 132 in a line parallel to the transmitting line 135 carrying the data and as identical as possible to the transmitting line 135 transmitting the data.
External noise that can be coupled into the system, including some components of the power supply noise, as the external noise appears as a common mode noise between the accompanying reference voltage and any given signal of the transmit line 135. Can be eliminated. However, the elimination of the common mode noise is not perfectly effective because the accompanying reference voltage has a terminal impedance different from a data signal at the receiving device 132; since the accompanying reference voltage must be fanned out to a large number of receivers, The capacitance on the pin receiving Vref is always greater than the capacitance at a typical signal pin, so the noise is low through relative to a data signal.
The second A diagram shows current flow in a prior art single-ended signaling system 200 in which the ground plane is to be the return conductor for the common signal. The single-ended signaling system 200 illustrates the aforementioned return impedance problem. As shown in FIG. 2A, the single-ended signaling system 200 transmits a "0" by discharging current at the transmitting device 201. Half of this current flows out of the signal conductor (signal current flow 204), while the other half, the transmitter current flows 203, flows through the terminator of the conveyor 201. In this example, the return current is to flow on the ground (GND) plane, and if the signal conductors are to be referenced to the ground plane and there is no other supply, the electromagnetic coupling between the signal and the ground plane will cause the image Current flows on the ground plane directly below the signal conductor. In order to achieve a 50/50 current distribution at the transmitter, a path to the local current of the transmitter is provided by an internal bypass capacitor 205 in the transmitting device 201, i.e., the transmitter current flow 203 is returned to the terminal through the terminating resistor 206. Ground. Signal current 204 is returned to receiving device 202 via the ground plane and flows through its bypass capacitor 207 and internal termination resistor 208 into receiving device 202.
The second B diagram shows current flow in the prior art single-ended signaling system 200, where the ground plane acts as the common signal return conductor when a "1" is transmitted by the transmitting device 201 to the receiving device 202. The current source is turned off (and not shown) in the transfer device 201, and the termination resistor 206 in the transfer device 201 pulls the line HI up. Again, the return current is flowing on the ground plane, so the capacitor 215 needs to be bypassed in the transfer device 201 to carry the signal current flow. Move 214. The current path in the receiving device 202 is the same as that used to transmit a "0", as shown in the second A diagram, except for the direction of current flow that is transmitted.
There are several problems with the schemes shown in Figures 2A and 2B. First, if the impedance of the bypass capacitors 205 and 215 is not sufficiently low, some of the signal current will flow in the Vdd network. Any redirected current flowing through the Vdd network will in some way have to be recombined with the image current in the grounded network and reach the ground plane, the redirected current will have to flow through the external bypass Capacitors and other power supplies provide shunt impedance.
Second, the return current flows through the impedance of the common grounding network at the transmitting device 201 and the common grounding impedance 212 at the receiving device 202. Because the ground is a common return path, the signal current produces a voltage across the ground network impedances 211 and 212. At receiving device 202, this voltage across ground network impedances 211 and 212 produces noise in adjacent signal paths, providing a direct source of crosstalk interference.
Third, if the common ground return pin and the ground pin provide a distance from the signal pin of the return path, an inductance is associated with the formed current loop, increasing the effective ground impedance, and adding the common This crosstalk between the signals of the ground pin. In addition, the inductor is connected in series to the terminator and will cause reflections in the transmit channels to become another source of noise.
Finally, the current flow shown in Figures 2A and 2B is the transient flow that occurs when a data edge is transmitted. The steady state flow is quite different in both cases because the current must flow through the power supply over the Vdd and ground networks simultaneously. Because the steady state current path is different from the transient current path, there is a transition between the two conditions in which the transient current flows simultaneously in Vdd and the ground network to reduce the voltage across the supply impedance and generate more The noise.
It can be assumed that it is preferred to use the Vdd network to carry the return currents because the termination resistors are connected to the network. However, this choice will not solve the basic problem. It is still necessary to bypass capacitors 205 and 215 to direct the transient signal current, and the transient is still different from the steady state condition, so there is still a common supply resistance. Anti-crosstalk interference, as well as voltage noise from data conversion. The basic problem has two aspects: The first is that the shared supply impedance is the source of crosstalk interference and supply noise. Second, the separation of the signal currents between the two supplies makes it difficult to maintain the return current substantially adjacent to the signal current through the channel, which causes poor termination and reflection.
In single-ended signaling system 200, the current drawn from the power supply is related to the data. When a "0" is transmitted, the transmitter draws a current Is. Half of this current flows from the power supply via the terminator back to the receiving device 202 on the signal line, then to the ground via the current source in the transmitting device 201, and from there back to the power supply. The other half of the current flows from the power supply via the terminator of the transfer device 201 to the Vdd network of the transfer device 201, then through the current source, and back via the ground network.
When a "1" is transmitted, the steady state is such that no current flows through the power supply network. Therefore, when the data is thixotropic between "1" and "0", the peak-to-peak current in the Vdd of the transmitting device 201 and the grounding network is twice that of the transmitting current, and in the receiving device 202 Doubling; the varying current produces voltage noise on the internal power supplies of each of the transmitting device 201 and the receiving device 202 by falling across the supply impedances. When all of the data pins sharing the common combination of a Vdd/ground terminal are switched, the noise in the common impedances is added, and the size of the noise is directly from the signaling noise budget. It is difficult and expensive to combat this noise: reducing these supply impedances typically requires providing more power and grounding pins and/or adding more metal resources to the wafer to reduce impedance. The cost of improving the bypass on the wafer is area, such as a large thin oxide capacitor.
One solution to solve all three of this reference voltage, the return impedance, and the power supply noise problem is to utilize differential signaling. This reference problem does not exist when using a differential signaling. Since the symmetrical second transmit line carries all of the return current without the return impedance problem. The power supply current is nearly solid No, no information about the material being transmitted. However, differential signaling requires twice the signal pin for single-ended signaling and some power/ground pins.
Therefore, there is a need in the art for a technique for providing single-ended signaling that reduces the problem of establishing a reference voltage, reducing the impedance of the return path of the signal, and reducing the noise of the power supply. In addition, differential signaling is required, so common mode noise can be reduced.
One embodiment of the present invention provides a technique for transmitting signals using differential transmit line pairing. A differential transmitter combines a DC (to DC) converter (which includes a capacitor) with a 2:1 multiplexer to drive a pair of single-ended transmit lines. One of the transmit lines in each differential pair is driven to HI when another transmit line of the differential pair is pulled low via the ground plane, thereby minimizing between different differential transmit pairs A series of noise sources of noise are generated.
Various embodiments of the present invention include a transmitter circuit including a sub-circuit pre-charged with a capacitor and a discharge and multiplexer sub-circuit. The sub-circuit pre-charged by the capacitor includes a first capacitor disposed to be precharged to a supply voltage during a positive phase of a clock, and a second capacitor disposed to a negative phase of the clock The period is precharged to the supply voltage. The discharge and multiplexer sub-circuit is configured to couple the first capacitor to the first one of the first transmit line and the second transmit line during the negative phase of the clock to the first Transmitting a line to drive the first transmit line and configured to couple the second capacitor to the first transmit line of the differential transmit pair during the positive phase of the clock to drive the first transmit line.
The benefit of the mechanism disclosed herein is that differential signaling can reduce any common mode noise sources.
In the following description, numerous specific details are set forth to provide a more complete understanding of the invention. However, it will be understood by those skilled in the art that the present invention may be practiced without one or more of these specific details. In other instances, well-known features are not described in order to avoid obscuring the invention.
A single-ended signaling system can be constructed to use one of the power supply networks to simultaneously serve as the common signal return conductor and the common reference voltage. When a power supply or even a network under a voltage level that is not supplied as a power supply can be configured as the common signal return conductor and the common reference voltage, preferably the ground terminal, as here Further explanation. Thus, although the same reference is made to the ground reference signaling as illustrated in the following paragraphs, the same techniques can be applied to a signaling system called the positive supply network (Vdd). Or some kind of newly introduced common terminal.
The third A diagram illustrates a single-ended signaling system 300 in accordance with a ground reference of the prior art. In order for the ground reference single-ended signaling system 300 to function properly, the transmitting device 301 must drive a pair of voltages ±Vline that are symmetric to ground, which are shown as symmetric voltage pairs 305. It uses a switched capacitor DC-DC converter to generate the two signal voltages +/- Vs, and then a conventional transmitter drives ±Vline into the transmit line 307. ±Vline is generated by the "real" power supply to the device, which is assumed to have a much higher voltage. For example, the supply voltage to the device is approximately 1 volt, however the +/- VS voltage is approximately +/- 200 mV.
In the ground referenced signaling system 300, the ground (GND) plane 304 is this and is the only reference plane to which the transmit conductor refers. At the receiving device 302, the terminating resistor Rt is returned to a common connection to the GND plane, so the transmit current cannot flow back to the transmitting device 301 on any other conductor, thereby avoiding the previous cooperation of the second A and the second B. The problem of current separation as illustrated in the figure.
Please refer back to the third A picture, the line voltage size |Vline| is usually smaller than the transmission supply voltage |Vs|. For example, if the transmitting device 301 uses a self-source terminated voltage mode transmitter, the transmitter (conceptually) includes a serial connection to a terminal A pair of data driven switches of the resistor. When the transmitter is impedance matched to the transmit line 307, then |Vline| = 0.5 | Vs|.
The signal reference voltage is also defined by the GND plane 304 and the network (not shown). Preferably, the GND network is typically the lowest impedance and most robust network in a system, particularly with multiple power supplies. Therefore, the difference in voltage between different points in the GND network is as small as possible under the cost limit. Therefore, the reference noise is reduced to the most likely small amplitude. Since the reference voltage is a supply terminal (GND) and is not generated internally or externally, there is no matching problem between the signal and the reference voltage to be solved. In short, selecting GND 304 as the reference voltage avoids most of the problems outlined in Figures 2A, 2B, and 2C.
The GND network is also a good choice for the common signal return conductor (or network) and the common reference voltage because it avoids the power supply that occurs when the two communication devices are powered by different positive power supplies. Supply serialization issues.
Conceptually, the reference voltage and signal return path problems can be solved by citing two new power supplies that can generate the ±Vs voltages required for symmetric signaling on the transmit line 307 (at the transmitting device) 301)). Therefore, the main engineering challenge is how to use the power supply voltage to efficiently generate these ±Vs voltages.
It is assumed that an input offset voltage of the receiving device 302 can be eliminated, and the thermal noise level of the input origin is at a 1 mV rms, so that the input of the receiving device 302 needs to develop a signal of about 50 mV to overcome the uncompensated Offset, crosstalk, limited gain, other bounded noise sources, and thermal noise (unbounded noise sources). Assuming that the transmit line 307 will be equalized at the transmitter (as further explained herein), it is necessary to develop between the two transmit power supplies (+Vs and -Vs), assuming self-source termination. ±Vs level = ±200 mV.
The adjustment of the CMOS power supply voltage is relatively slow, so for the next generation of technology, the Vdd supply of the core device is expected to range from 0.75 to 1 volt. Therefore, the symmetrical Vs voltage is a small fraction of the power supply voltage. Want to turn The most efficient regulator to switch from a relatively high voltage to a low voltage is a switched capacitor DC-DC converter.
The third B diagram illustrates a switched capacitor DC-DC converter 310 that is arranged to generate +Vs in accordance with this prior art. The switched capacitor DC-DC converter 310 is arranged to generate +Vs from a Vdd supply voltage that is well above it. The converter 310 that produces the positive power supply +Vs operates in two phases. During φ1, the "flying" capacitor Cf is discharged, and both terminals of Cf are driven to GND. A flying capacitor has two terminals, none of which are directly coupled to a power supply, such as Vdd or GND. During φ2, Cf is charged to Vdd-Vs; the charging current flows through Cf and then enters the load Rload. The bypass/filter capacitor Cb having a capacitance value much larger than Cf stores the supply voltage +Vs, and stores the supply voltage +Vs during the φ1 interval, and supplies current to the Rload.
The third C diagram illustrates a switched capacitor DC-DC converter 315 configured to generate -Vs in accordance with an embodiment of the present invention. Similar to the switched capacitor DC-DC converter 310, the switched capacitor DC-DC converter 315 is arranged to generate -Vs from a Vdd well above it. The switched capacitor DC-DC converter 315 that produces the negative supply -Vs is identical in topology to the switched capacitor DC-DC converter 310, but the charge switches are reconfigured. During φ1, Cf is charged to Vdd. During φ2, Cf is discharged into the load Rload. Since the left hand terminal of the Cf is more positive than the right hand terminal, the voltage on the Rload is driven to be more negative on each operational cycle.
The current supplied to the load by the switched capacitor DC-DC converter 310 or 315 is proportional to the capacitance value Cf, the frequency of the φ1/φ2 clock, and the difference between Vdd and Vs. When the two supplies +Vs and -Vs are generated using the switched capacitor DC-DC converters 310 and 315 for the single-ended ground reference signaling system 300, from the switched capacitor DC-DC converters 310 and 315 The current drawn by each is based on the data to be transmitted. When the data=1, the current is supplied from the +Vs to the transmission line 307 via the transmission device 301, and the -Vs The supply was uninstalled. When the data = 0, the current is supplied by the -Vs supply, and the +Vs supply is unloaded.
The single-ended ground reference transmission system 300 has at least two significant features. The first is that if the two converters have the same efficiency, the current drawn from the Vdd supply has no data value, and this feature avoids the simultaneous switching problem that is implicit in most single-ended signaling methods. In particular, the simultaneous switching problem described in conjunction with the second A and B diagrams can be avoided. Second, the two switched capacitor DC-DC converters 310 and 315 require a control system to individually hold the output voltages +Vs and -Vs fixed in the face of varying loads. It is not practical to change the values of the flying capacitors Cf and Vdd and Vs, so the values of Cf, Vdd and Vs are usually fixed. However, the frequency of the switching clock can be used as a control variable.
The third D diagram individually illustrates a pair of schematic control loops for two switched capacitor DC-DC converters 310 and 315 of the third and third C diagrams in accordance with an embodiment of the present invention. Control loop 322 compares +Vs with a power supply reference voltage Vr, and if V(Vs) < V(Vr), the frequencies of φ1 and φ2 are increased to sink more current into the load. Control loop 324 operates by comparing an intermediate voltage between +Vs and -Vs with GND, thereby attempting to maintain the two supply voltages +Vs and -Vs symmetric about GND.
When the control loop and converter 320 system can be constructed as shown in the third D diagram, the two control loops 322 and 324 will likely be quite complex, as the control loops 322 and 324 may need to be processed in the load Rload. 100% change. Controlling the chopping on +Vs and -Vs requires operating the switched capacitor Cf at a high frequency and utilizing a large storage capacitor Cb. In fact, the clocks φ1, φ2, ψ0 and ψ1 will likely need to output multiple phases and drive each of a plurality of switching capacitors operating on different phases. All in all, this solution requires significant complexity and consumes a large area in a circuit on a die.
The voltage level of the single-ended signal can be generated by combining a transmitter and the switched capacitor DC-DC power supply into a single entity, which is also Includes a 2:1 clock-controlled data multiplexer to avoid the complexity and large area of the tuned switched capacitor converter. In addition to operating a switched capacitor power supply at a frequency controlled by a control loop, the switched capacitor converter is driven at the data clock rate. The data is driven onto the line by controlling the charging/discharging of the flying capacitor based on the data value to be transmitted.
Figure 4A illustrates a data driven charge pump transmitter 400 in accordance with an embodiment of the present invention. The structure of the data driven charge pump transmitter 400 incorporates a timed 2:1 data multiplexer with a charge pump DC-DC converter. The data driven charge pump transmitter 400 multiplexes the half rate dual bit stream dat{1,0} into a single full rate bit stream by transmitting dat1 when clk=HI, when clk=LO When, transfer dat0. The relationship between the clkP and the data signal is displayed in the clock and data signal 407.
The upper half of the structure, including sub-circuits 401 and 402, is the dat1 half of the multiplexer, where dat1P = HI and dat1N = LO when dat1 = HI. When clk=LO (clkP=LO and clkN=HI), Cf1p is discharged to the ground supply voltage, and both terminals of Cf1p are restored to GND. When Cf1p is discharged, Cf1n is charged to the power supply voltage. In other words, during a negative phase of the clock (when clkN = HI), each capacitor Cf1p and Cf1n is precharged to a supply voltage by pre-charging and flying capacitor circuits in sub-circuits 401 and 402. Cf1p and Cf1n are individually discharged and charged to the power supply voltage.
During a positive phase of the clk, when clk becomes HI (clkP = HI and clkN = LO), one of the two capacitors Cf1p or Cf1n drops the charge into the transmission line 405 according to the value of dat1. For example, when dat1 = HI, Cf1p is charged to Vdd-Vline, and the charging current drives the transmission line 405. The voltage level to which the transmit line 405 is driven is based on at least the values of Cf1p, Cf0p, Cf1n, and Cf0n, the Vdd value, the impedance R0, and the frequency of the clock. In one embodiment of the invention, the values of Cf1p, Cf0p, Cf1n, Cf0n, Vdd, R0 and the clock frequency are fixed at design time to drive the transmit line 405 to a voltage level of 100 mV. Cf1n does not change and remains charged. because Thus, Cf1n does not consume current on the next clk=LO phase. On the other hand, if dat1 = LO, Cf1p maintains discharge, and Cf1n discharges to the transmission line 405, which drives the transmission line 405 to a voltage as low as -Vline. During the positive phase of the clk, a 2:1 multiplexer operation is performed by the multiplexer and discharge circuitry in sub-circuits 401 and 402 to select one of the two capacitors to drive transmit line 405 to transmit dat1.
The lower half of the data-driven charge pump transmitter 400, including sub-circuits 403 and 404, performs the same action, but on the opposite phase of clk, and is controlled by dat0.
Since there is no charge storage capacitor (which may include an electrostatic discharge protection device in addition to the parasitic capacitance associated with the output), there may be significant chopping at the voltage level of the transmit line 405. Importantly, the chopping in the voltage level will be the bit rate at which the data is driven onto the transmit line 405. If there is significant symbol rate attenuation in the channel between the transmitting device and the receiving device, wherein the channel primarily includes the transmit line 405 and the ground plane associated with the transmit line 405, substantially including the package and printed circuit The plate conductor, the chopping in this voltage level will be strongly attenuated. However, even if the chopping in the voltage level is not attenuated, the chopping mainly affects when the data value is changing on the transmission line 405 (in time) away from the best detectable The amplitude of the signal at that point in time when the data value. The data-dependent attenuation of the transmission line can be corrected using an equalizer, as described in connection with the fourth C-picture.
There are some redundant components in the data driven charge pump transmitter 400. FIG. 4B illustrates a data driven charge pump transmitter 410 implemented using a CMOS transistor in accordance with an embodiment of the present invention, which is simplified compared to the data driven charge pump transmitter 400 illustrated in FIG. The series transistors implemented as dat‧clk can be replaced by a CMOS gate, which can pre-calculate the logical AND of dat1N and clkN, dat1N and clkP, dat0N and clkP, and dat0N and clkN, and drive a single transistor. .
Sub-circuits 413 and 414 pre-charged with flying capacitors pre-charge the capacitors Cf1p and Cf1n during the negative phase of the clock, while the positive phase of the clock The capacitors Cf0p and Cf0n are precharged during the bit period. The multiplexer and discharge circuitry are formed by the transistors in sub-circuits 413 and 414 that are not pre-charged with flying capacitors, which drive the transmit line 415 based on dat1 and dat0.
Because the output signal is driven to a voltage below the minimum supply voltage (ground), some of the devices in the data driven charge pump transmitter 410 operate under abnormal conditions. For example, when the transmit line 415 is driven to -Vs, the output source/drain terminals of the multiplex transistors 416 and 417 are driven below ground, so their associated N+/P junctions are forward biased. Pressure. If the signal swing is limited to hundreds of mV, this situation will not cause too much difficulty. If the forward biased junction becomes a problem, the negative drive NMOS transistor required by the data driven charge pump transmitter 410 can be implemented as an isolated p-type substrate in a deep N-well, as long as The structure can be used in this target manufacturing process technology. The isolated P-type substrate can be biased to a voltage below ground using another charge pump, but does not require a large current to be supplied thereto. A negatively biased P-type substrate avoids forward conduction in the device source/drain junction.
An additional problem arises with a negative conversion signal. It is assumed that the transmit line 415 is being driven to -Vs during clkP = 1 when the multiplex transistor 416 is enabled. The gates of both pre-charged transistor 418 and multiplex transistor 417 are driven to 0V (ground). However, one of the source/drain terminals of each of pre-charged transistor 418 and multiplex transistor 417 is now a negative voltage and thus becomes the source terminals of the individual devices. Because the gate to source voltages are now positive, the precharge transistor 418 and the multiplex transistor 417 are turned on and will clamp the negative output by conducting current to ground and limiting the negative oscillations that would be present. signal. However, this conduction does not become apparent until the negatively directed voltage is close to the threshold voltage of the precharge transistor 418 and the multiplex transistor 417, and in fact only as long as -Vs falls below 100 mV or is approximately below ground. This clamping current will be small. In a program that provides multiple threshold voltages, it is preferred to use a high threshold voltage transistor to implement the components of any data driven charge pump transmitter 410 that can be driven to a voltage below ground.
The fourth C diagram illustrates a data driven charge pump transmitter 420 implemented using CMOS gates and transistors in accordance with an embodiment of the present invention. The data driven charge pump transmitter 420 may not require a series arrangement in the data multiplex section of the charge pump transmitter. Pre-calculation! clkN‧! dat1P and! clkP‧! These NOR gates of dat0P are actually replaced by NAND and inverter because the series PFETs are slower in most current processes. The magnitude of the change in the size of the transistors can be somewhat balanced by the additional delay of the inverter.
The capacitors Cf1p and Cf1n are precharged during the negative phase of the clock, while the capacitors Cf0p and Cf0n are precharged during the positive phase of the clock. A 2:1 multiplexer and electronic discharge circuit drives the transmit line 425 based on dat0 during the negative phase of the clock during the positive phase of the clock during the positive phase of the clock.
If there is a strong frequency dependent attenuation on the channel, the data driven charge pump transmitters 410 and 420 shown in the fourth and fourth C diagrams, respectively, would require an equalizer.
The fourth D diagram illustrates a data driven charge pump transmitter 430 including an equalizer in accordance with an embodiment of the present invention. Sub-circuits 433 and 434 pre-charged with flying capacitors pre-charge the capacitors Cf1p and Cf1n during the negative phase of the clock, respectively, and pre-charge capacitors Cf0p and Cf0n during the positive phase of the clock. The transistors in the sub-circuits 413 and 414 or the equalizer 435 that are not pre-charged by the flying capacitor form a multiplexer and a discharge circuit that are based on dat1 and at that time during the positive phase of the clock. The transmit phase 432 is driven based on dat0 during the negative phase of the pulse. The circuits for precharging the flight capacitors and the multiplexed dual bit data onto the transmit line actually operate the same as the transmitter 410 of the fourth B diagram.
A capacitively coupled pulse mode transmitter equalizer 435 is connected in parallel to the data driven charge pump transmitter 430. When the output data changes value, the equalizer 435 pushes additional current to or from the transmit line 432 to increase the voltage of the transmit line 432 during the transition. The equalization constant can be changed by changing the ratio of Ceq to Cf. The equalizer 435 can be divided into a group of segments, each of which A section will have an "enable". By turning on certain portions of the segments, Ceq can be effectively changed to change the equalization constant. The data driven charge pump transmitter 430 circuit can be arranged in an array of identical segments and added to enable each segment to allow the voltage of the transmit line 432 to be adjusted according to operational requirements.
Please note that there is a problem with the equalization shown in the fourth D diagram in that the equalizer 435 has an additional gate delay (the inverter) after the multiplex function, so the voltage generated by the equalizer 435 The boost will be slightly delayed relative to the transition driven by the data driven charge pump transmitter 430. If desired, the clocks that drive the data-driven charge pump transmitter 430 can be delayed by an inverter delay, or the equalizer 435 can be implemented in the same manner as the data-driven charge pump transmitter 430. A fourth E diagram illustrates an equalizer 440 for applying pressure to the signal line voltage without additional gate delay in accordance with an embodiment of the present invention. The fourth F diagram illustrates yet another equalizer that uses the flying capacitors manipulated by the data bits dat{1, 0} instead of the clock to drive additional current to the signaling during data conversion. on-line.
The data driven charge pump transmitters 410, 420 and 430 shown in the fourth, fourth, and fourth D diagrams do not address the problem of return current flow in the transmitter. It is therefore necessary to avoid the return current separation that causes problems in conventional single-ended signaling systems. However, when the transmit line is driven positively at the data driven charge pump transmitters 410, 420 and 430, current flows from the power supply to the transmit line, so the return current must flow through the transmitter. power supply.
The current flowing from the power supply to the transmit line can be substantially effectively reduced by providing a bypass capacitor between the supply and the signal ground, so that most of the transmit current is drawn from the bypass capacitor And allowing the return current to flow locally to the transmitter. The addition of a small series resistance between the power supply of the transmitter and the positive supply terminal will further enhance the effect of forcing the return current to flow locally to the transmitter. The small series resistor, in conjunction with the bypass capacitor, isolates the supply from the chopping current required to charge the flying capacitors and also isolates the high frequency portion of the signal return current.
Figure 5A illustrates a switched capacitor transmitter 500 that directs return current into the grounded network in accordance with an embodiment of the present invention. The positive current converter sections of the switched capacitor transmitter 500, i.e., the positive converters 512 and 514, are modified from the data driven charge pump transmitters 410, 420, and 430, and therefore flow from the positive converters 512 and 514. This current to the transmit line 502 flows only in the grounded network. Cf1p and Cf0p are first precharged to the power supply voltage and then discharged to the transmit line 502. The negative converter portions of the switched capacitor transmitter 500 remain the same as the data driven charge pump transmitters 410, 420 and 430 because the output currents of the negative converters only flow in the GND network. . Switched capacitor transmitter 500 includes an equalizer 510 that can be implemented using the circuit of equalizer 435 or 440.
By adding some more switching transistors to the switched capacitor transmitter 500, it is possible to divide the grounding network into two parts: an internal network for precharging the flying capacitors, and as The external network that is part of the signaling system. Figure 5B illustrates a switched capacitor transmitter that acts on a separate ground return path for pre-charging and signal current. The circuit components of the negatively driven flying capacitors Cf0n and Cf1n are maintained in the same manner as in the fifth A diagram. Positive converters 522 and 523 have been modified with the addition of two transistors each. In positive converter 522, an NMOSfet driven by clkN carries a precharge current to the internal power supply grounding network, indicated by symbol 524. When the data is driven from the converter at dat1P=1, a second NMOSfet driven by dat1P carries the signal return current to the signal current return network, indicated by symbol 525. This configuration prevents the pre-charge currents in converters 522 and 523 from injecting noise into the signal current return network.
A practical problem that must be addressed is that capacitors are typically fabricated on a wafer using thin oxides. In other words, the capacitors are MOS transistors, typically varactors (NMOS capacitors). These capacitor structures have parasitic capacitances from their terminals to the substrate and to the surrounding conductors, often with some degree of asymmetry. For example, an NMOS varactor has a gate with the NMOS The most advantageous overlap capacitance of the source and drain terminals of the varactor, but the NWell body ohmically coupled to the source and drain terminals has a capacitance to the P-type substrate. In the case of a flying capacitor, this parasitic capacitance (preferably placed on the transmit line side of the capacitor) will have to be charged and discharged at each cycle. This current that is charging the parasitic capacitance cannot be used to drive the transmit line. This parasitic capacitance, along with switching losses, reduces the efficiency of the switched capacitor transmitter 500.
The Ceqa capacitors in the equalizer 510 of Figure 5B are all grounded, so the converter circuit using the Ceqa capacitors is likely to be more efficient than the converters using the flying capacitor Ceqb. The better efficiency of these Ceqa capacitors relative to the Ceqb capacitors can compensate for the different sizes of Ceqa and Ceqb.
FIG. 5C illustrates a "bridge" charge pump transmitter 540 in accordance with an embodiment of the present invention, wherein the output current flows only in the signal current return network 547, and the precharge current is only at the internal power supply Flow in the grounding network. When the internal power supply grounding network and the signal current return network 547 are separate networks for the purpose of noise isolation, they are typically maintained at the same potential because the signal current is returned to the network in the channel. This external portion is shared with the power supply ground. In the "bridge" charge pump transmitter 540, a single flight capacitor Cf1 and Cf0 is used for each phase of the clock (clk, where clkN is the inverted clkP). Regarding the switched capacitor converter 542, the flying capacitor Cf1 in the sub-circuit 541 charged with the flying capacitor is precharged to the power supply voltage when clk=LO. When clk=HI, the flying capacitor Cf1 throws the charge into the transmission line 545, and if the dat1=HI, the transmission line 545 is pulled to HI, and if dat1=LO, the transmission line 545 is pulled to the LO. .
The switched capacitor converter 544 performs the same operation at the opposite phase of the clock and is controlled by dat0. In particular, the flying capacitor Cf0 within the sub-circuit 543 pre-charged with the flying capacitor is pre-charged to the power supply voltage when clk = HI. When clk=LO, the flying capacitor Cf0 will be powered The load is dropped into the transmission line 545. If dat1 = HI, the transmission line 545 is pulled to HI, and if dat1 = LO, the transmission line 545 is pulled to LO. The transistors within the switched capacitor converters 542 and 544 that are not in the sub-circuits 541 and 543 pre-charged with the flying capacitor form a 2:1 multiplexer and discharge circuit.
In the "bridge" charge pump transmitter 540, the precharge current can be separated from the current in the signal current return network 547. For example, when the signal current return network 547 is isolated from other ground supplies, the precharge current does not flow into the signal current return network 547 and therefore does not occur in the network coupled to the signal current return network 547. Noise. Note that the four timing NFETs in each of the "bridged" connections can be logically broken down into two devices. However, when the flying capacitor Cf1 (or Cf0) is being pre-charged, the associated data bit is thixotropic, and there is a good chance that the datP- and datN-driven NFET can be turned on simultaneously during the rectification. , the current is drawn from the pre-charge. Therefore, in fact, the four timing NFETs must not be decomposed.
In order to avoid having to increase the four NFETs connected in series in each of the signal current paths, the device in the "bridge" charge pump transmitter 540 can use a pre-calculated gate. The fifth D diagram illustrates a bridge transmitter having a pre-calculated gate 550 in accordance with an embodiment of the present invention. The delay in the clock signals is balanced by gating the clock signals clkN and clkP to the data signals dat1N, dat1P, dat0P and dat0N. The gate control allows the signal delay within the transmitter 550 to closely match the delay within the equalizer 553 since both involve the same number of logic platforms. The delay through the first inverter in the multiplexer and equalizer 553 can be approximately matched to the NAND gates and the inverters associated with the NAND gates in the pre-calculated gates that generate d1NclkP Delay, and so on. Similarly, the delay of the inverter by directly driving the equalization capacitor Ceq can be matched to the delay of driving the output current to the two sets of four "bridge" transistors in the transmit line 557.
Depending on the process details, the bridge transmitter with pre-calculated gate 550 provides lower overall power than the "bridge" charge pump transmitter 540 and switched capacitor transmitter 500, albeit at the expense of causing chattering. Some extra power supply noise.
The circuit of the bridge transmitter having the pre-calculated gate 550 can be arranged in a significantly smaller area than the switched capacitor transmitter 500 because the flying capacitors in the sub-circuit 555 pre-charged with the flying capacitor are in each cycle Both are used, so compared to the switched capacitor transmitter 500 of FIG. 5A and the data-driven charge pump transmitters 400, 410, 420 and 430 of the fourth A, four B, four C and four D diagrams, respectively, There are half of the number of flying capacitors.
Each of the bridge transmitter, bridge charge pump transmitter 540 and switched capacitor transmitter 500 having a pre-calculated gate 550 includes a terminating resistor R0 on the transmit line. If a transmitter is return terminated, the terminating resistor must be larger than the characteristic impedance of the transmitting line because the charging pumps are not ideal current sources. In some cases, return termination is not necessary, and if necessary, the charge pump only needs to supply 1/2 of the current compared to a signal line that is terminated. Excluding return termination is an opportunity to save significant power.
A decision can be made as to whether such required flight capacitors are actually implemented in a CMOS fabrication technique. Assume that it is necessary to pass ±100mV to a 50Ω transmission line that is terminated. Each of these charge pumps must provide a current source of 100 mV / 25 Ω = 4 mA. Using I = CdV / dt, where dV = V (Vdd) - V (line) and dt = 1 UI, the required capacitance can be calculated once the supply voltage and the bit rate of the data are known. Assuming V(Vdd) = 0.9V and 1UI = 50psec, then C = 250fF. A 250fF capacitor can be implemented simply in a CMOS process. In a typical 28 micron CMOS process, the capacitance of an NMOS varactor is approximately 50 fF/μ2, so these flying capacitors will occupy an area of several μ2. In summary, these flying capacitors will have to be larger than the required values for the required calculated values due to switching losses and parasitic capacitance.
The transmitters, that is, the bridge transmitter having the pre-calculated gate 550 and the "bridge" charge pump transmitter 540, and the switched capacitor transmitter 500 drive the voltage of the signal line to a fixed portion of the power supply voltage. Where the portion is based on the operating frequency (the bit rate of the data) and the size of the flying capacitors. The power supply voltage is typically specified to vary by ±10%, so a bridge transmitter having a pre-calculated gate 550, a "bridge" charge pump transmitter 540, and a switched capacitor transmitter 500 will cause the signal voltage to vary in a similar manner.
The transmitter (and the equalizer) can be enclosed in a control loop if the voltage swing of the transmit line must be maintained at a tighter tolerance than the power supply change. Fifth E illustrates a regulator loop 560 for controlling the voltage on the transmit line 565 in accordance with an embodiment of the present invention. In regulator loop 560, switched capacitor transmitter 570 and an equalizer (not shown) are not operated by Vdd, but are operated by a regulated voltage Vreg, which is typically set lower than the wafer power supply Vdd. The lowest expected voltage. The primary regulating element is a pass transistor Ppass that charges a large filter capacitor Cfilt. The via transistor is driven by a comparator that compares a reference voltage Vref set to the desired line voltage with a Vrep of the output of a replica switched capacitor converter 572.
The replica switched capacitor converter 572 is a replica of one of the data driven switched capacitor converters in the transmitter 570, which may have been adjusted. The replica switched capacitor converter 572 cycles through each clk (any polarity of clk is used), drives its resistance equal to the impedance of the transmit line 562 (when the transmitter 570 does not have a return termination) or sends A resistor 574 (possibly adjusted) of one half of the impedance of the signal line 565 (when the transmitter 570 has a return termination). The load of the replica switched capacitor converter 572 includes a large capacitor Crep to remove chopping from the Vrep output. A regulator, such as regulator circuit 560, is typically designed such that the output filter (Cfilt) establishes the main pole in the closed loop conversion function. Additional components (not shown) may be included in the circuit to stabilize the loop.
The fifth F diagram illustrates a method for precharging a flying capacitor sub-circuit and driving the transmitting line on different phases of the clock in accordance with an exemplary embodiment of the present invention. Although the method steps are in conjunction with the fourth A, four B, four C and four D map data driven charge pump transmitters 400, 410, 420 and 430, the fifth A diagram of the switched capacitor transmitter 500, the fifth C The illustrated bridged charge pump transmitter 540 and the bridge transmitter having the pre-calculated gate 550 of FIG. 5D are illustrated, and those skilled in the art will appreciate that any system configured to perform the method steps in any order is known. It is within the scope of the invention.
The two different clock phases include a positive phase when clkP is HI and a negative phase when clkN is HI. The data is divided into two signals dat0 and dat1, where dat0 is valid when clkN is HI, and dat0 is valid when clkP is HI. At step 585, a first flying capacitor Cf is precharged by a sub-circuit pre-charged with a flying capacitor during the positive phase of the clock. At step 587, during the positive phase of the clock, a second flying capacitor Cf is discharged and the transmit line is driven by a multiplexer electronic circuit (HI or LO). At step 590, the second flying capacitor Cf is pre-charged by the sub-circuit pre-charged with the flying capacitor during the negative phase of the clock. At step 592, during the positive phase of the clock, the first flying capacitor Cf is discharged and the transmit line is driven by the multiplexer electronic circuit (HI or LO).
Single-ended transmit receiver circuit
Returning to the ground reference single-ended signaling system 300, the data-driven charging pump transmitters 400, 410, 420 and 430, the switched capacitor transmitter 500, the "bridge" charging pump transmitter 540 and the pre-calculated gate 550 are both received and received. The devices are paired, which can efficiently receive signals oscillating symmetrically to GND (0V). The receiver amplifies and level shifts the signals to CMOS levels (a logic level that is approximately thixotropic between Vdd and GND).
A conventional PMOS differential amplifier can be used as a receiver, but a lower power and simpler alternative is a ground gate (or common gate) amplifier. Figure 6A illustrates a grounding gate in accordance with an embodiment of the present invention Amplifier 600. If p0/p1 and n0/n1 are the same size, then p0/n0, a shortened inverter forming bias generator 602, can generate the bias voltage Vbias at the critical point of the inverter switching. The input amplifier 605 including p1/n1 has no signal on the transmission line 607, and its bit is at the same operating point, specifically, the inverter switches the critical point, so the output of the input amplifier 605 also stays. At V (Vbias). Input amplifier 605 provides current Iamp into transmit line 607. Therefore, the voltage on the transmission line 607 does not swing around 0V (GND), but the bias voltage Voff = Iamp‧R0, in which case only the receiver is terminated. When both ends of the transmission line 607 are terminated, Voff = 0.5 Iamp‧R0, and the bias resistor Rbias attached to the source of n0 is set at R0/2. In fact, the offset voltage is smaller than the signal swing Vs. Please note that the bias circuit p0/n0 and the bias resistor of the bias generator 602 can be reduced to save power.
In a conventional process technique, if the MOSFET of the grounded gate amplifier 600 is implemented using a low threshold device, the gain of the input amplifier 605 is approximately 5, so when the amplitude of the transmit line 607 is 100 mV, the input amplifier 605 An output voltage level can be generated that is nearly full swing to directly drive a CMOS sampler. A second stage amplifier 610 can be added if a higher gain is required. The output of the input amplifier 605 and the output of a continuous amplifier (eg, the second stage amplifier 610) are approximately symmetrically surrounded by V (Vbias), the inverter switching critical point, and approximately halfway between Vdd and GND. swinging.
The previous explanation ignores an effect implicit in the grounded gate amplifier. Because a grounded gate amplifier drives current into the input, such as the transmit line, the grounded gate amplifier has a finite input impedance that is approximately 1/gm of the input transistor n1. The impedance appears parallel to the terminating resistor R0, so the input will be terminated low unless the terminating resistor is adjusted upwards. In fact, the input amplifier 605 can be pulled out small enough to make the effect relatively small.
The description of the previous ground gate amplifier 600 implicitly assumes that p0 and p1 are matched, and n0 and n1 are matched. Inevitably, process variation and input source resistance The difference between Rbias and R0 will cause Vbias to move away from the actual switching threshold of p1/n1, thus causing an offset of this voltage at this output of input amplifier 605. As shown in Figure 6A, the output of input amplifier 605 does not oscillate symmetrically around Vbias and will be biased above or below the ideal swing.
To remove the introduced offset, an offset trimming mechanism can be used and a procedure to adjust the mechanism can be utilized. One of several possible ways to implement this offset trimming mechanism is to modify the bias generator. FIG. 6B illustrates an adjustable bias generator 620 in accordance with an embodiment of the present invention.
The adjustable bias generator 620 can replace the bias generator 602 in the grounded gate amplifier 600. The single source resistor Rbias in the bias generator 602 within the grounded gate amplifier 600 is replaced by an adjustable resistor that can be digitally trimmed by changing the value of the Radj[n] bus of the signal. A fixed resistor Rfixed sets the highest resistance of the adjustable resistor, and a set of binary weighted resistors Ra, 2Ra...(n-1)Ra can be selectively paralleled to Rfixed to reduce the effective resistance. These resistors are basically selected such that when Radj[n] is in the middle range, the overall resistance can match the terminating resistor R0. Additionally, in addition to adjusting Rbias, one or both of the transistors p0/n0 can be adjusted in a similar manner by providing a complex bias transistor in series with the control transistor to allow the effective width of the bias transistors. It can be changed digitally.
By establishing a digital trimming mechanism, a program that adjusts the mechanism is needed to remove the resulting offset. The method described herein does not require additional hardware beyond what is required by the grounded gate amplifier 600 to perform such receiving operations, except for a finite state machine to analyze the data from the receiver sampler and set the Trim the value above the adjustment bus Radj[n].
The sixth C diagram illustrates a method 640 for adjusting an offset trimming mechanism in accordance with an exemplary embodiment of the present invention. Although the method steps are described in conjunction with the system of Figure B, those skilled in the art will appreciate that any system configured to perform the method steps in any order is within the scope of the present invention.
Method 640 is performed to adjust adjustable bias generator 620. At step 645, the transmitting device that drives the transmitting line is turned off, so the transmitting line is set to its midpoint voltage Voff. At step 650, the Radj code that changes the resistor Rbias is set. At step 655, the finite state machine controlling Radj records a number of consecutive samples from the receiver sampler or samplers attached to the input amplifier with receiver clock thixotropy. Next at step 660, the values are filtered to determine if the average is greater than or less than 0.5. At step 665, the Radj code is changed to drive the offset toward the point where half of the sampler values are "1" and half is "0". Until this point is reached, the process returns to step 650 and the Radj code is re-adjusted. When it is determined that the average from the samplers is approximately 0.5, the receiver is assumed to be trimmed and the program exits at step 670.
At step 650, the finite state machine begins with the Radj code at a limit value and approaches the code on Radj toward another limit value. At this starting point, all of these samplers must output "1" or "0" depending on the details of the receiver. Since the code on Radj changes towards another limit value, there will be a point at which the digits from the samplers begin to thixify to the opposite digit value.
At step 660, the finite state machine filters the values from the samplers by averaging over a number of clock cycles. When one-half of the values from the four samplers is "1" and half is "0", averaging over a certain number of sampling clock cycles, the receiver can be assumed to be trimmed. The finite state machine can be implemented as a hardware, software, or combination of hardware and software that operates on a control processor.
Variations in this manner may include equal weighting of the Ra resistors and thermometer coding of the Rajj bus. Such an implementation would be helpful when a finishing operation had to be performed while using some of the programs other than the above to receive dynamic data. Another possible variation is that if the plurality of samplers in the receiver require individual offset trimming, four copies of the input amplifier can be provided (although It shares a common termination resistor) and includes four trimmable Vbias generators, each with its own Radj bus. The trimming procedure is quite similar to the above described procedure except that a plurality of finite state machines or a time multiplexed program will be required to perform the adjustment.
7A is a block diagram illustrating a processor/wafer 740 in accordance with one or more aspects of the present invention, including a ground reference single-ended transmit transmitter, such as a data driven charge pump transmitter from FIG. 410, 420 from the fourth C diagram, 430 from the fourth D diagram, the switched capacitor transmitter 500 from the fifth diagram, the bridge charge pump transmitter 540 from the fifth diagram C, or from the fifth D The bridge transmitter of the figure having a pre-calculated gate 550. Receiver circuit 765 can include a receiver configured to receive a single-ended input signal from other devices in a system, such as ground gate amplifier 600 from Figure 6A. Single-ended output signal 755 is generated by transmitter circuit 775. Receiver circuit 765 provides an input to core circuit 770. Core circuit 770 can be configured to process the input signals and produce an output. These outputs of core circuit 770 are received by transmitter circuit 775 and used to generate a single-ended output signal 755.
Figure 7B illustrates a block diagram of a computer system 700 that is configured to implement one or more aspects of the present invention. Computer system 700 includes a central processing unit (CPU) 702 and a system memory 704 that communicate via an interconnect path including a memory bridge 705. The memory bridge 705 can be, for example, a north bridge wafer that is connected to an I/O (input/output) bridge 707 via a bus or other communication path 706 (e.g., a HyperTransport link). The I/O bridge 707 can be, for example, a south bridge chip that receives user input from one or more user input devices 708 (eg, a keyboard, mouse) and forwards the via 706 and memory bridge 705 via the communication path 706 Input to the CPU 702. A parallel processing subsystem subsystem 712 is coupled to the memory bridge 705 via a bus or second communication path 713 (eg, PCI (Peripheral Component Interconnect) Express, accelerated graphics communication, or HyperTransport link); In an embodiment, parallel processing subsystem 712 A graphics subsystem that passes pixels to a display 710 (eg, a conventional cathode ray tube or liquid crystal monitor). A system disk 714 is also coupled to I/O bridge 707. A switch 716 provides a connection between the I/O bridge 707 and other components such as the network adapter 718 and the various embedded cards 720, 721. Other components (not explicitly shown) include a universal serial bus (USB, "Universal serial bus") or other 埠 connection, CD (CD) drive, digital video disc (DVD) drive, film recorder and the like. It can also be connected to an I/O bridge 707. The communication paths of the various components (including the specific name communication paths 706 and 713) shown in FIG. 7B can be implemented using any suitable protocol, such as PCI (Peripheral Component Interconnect), PCI Express. (PCI Express, PCI-E), AGP (Accelerated Graphics Port), HyperTransport, or any other bus or point-to-point protocol, and connections between different devices, can be used Different agreements known in the technology.
One or more of the devices shown in Figure 7B can use a single-ended signaling to receive or transmit signals. In particular, the transmitting device can be arranged to include a ground reference single-ended transmitter, such as data driven charge pump transmitter 410 from Figure 4B, 420 from Figure 4C, 430 from the fourth D-picture, The switched capacitor transmitter 500 from FIG. 5A, the bridged charge pump transmitter 540 from FIG. 5C, or the bridge transmitter having the pre-calculated gate 550 from the fifth D diagram. The receiving device can be arranged to include a grounded gate amplifier 600 from Figure 6A.
In one embodiment, parallel processing subsystem 712 incorporates circuitry that can be optimized for graphics and video processing, including, for example, video output circuitry, and constitutes a graphics processing unit (GPU). In another embodiment, parallel processing subsystem 712 incorporates circuitry that can be optimized for general purpose processing while retaining the underlying operational architecture, as will be described in more detail herein. In yet another embodiment, the parallel processing subsystem 712 can be integrated into one or more other system components in a single subsystem, such as in conjunction with the memory bridge 705, the CPU 702, And I/O bridge 707 forms a system on chip (SoC, "System on chip").
It will be appreciated that the systems shown herein are merely illustrative and that many variations and modifications are possible. The connection topology, including the number and configuration of bridges, the number of CPUs 702, and the number of parallel processing subsystems 712, can all be modified as needed. For example, in some embodiments, system memory 704 is directly coupled to CPU 702 and not coupled through a bridge, while other devices communicate with system memory 704 via memory bridge 705 and CPU 702. In other alternative topologies, parallel processing subsystem 712 is coupled to I/O bridge 707 or directly to CPU 702 rather than to memory bridge 705. In still other embodiments, I/O bridge 707 and memory bridge 705 can be integrated into a single wafer in addition to being one or more discrete devices. Large specific embodiments may include two or more CPUs 702, and two or more parallel processing subsystems 712. The particular components shown herein are all optional; for example, they can support any number of embedded cards or peripheral devices. In some embodiments, switch 716 is omitted and network switch 718 and embedded cards 720, 721 are directly connected to I/O bridge 707.
In summary, the dual-trigger low energy flip-flop circuit 300 or 350 is fully static because all nodes are driven high or low during all steady state periods of the circuits. Because the internal nodes are only thixotropic when the data changes and the load of the clock is only three transistor gates, the flip-flop circuit is low energy. Since the data inputs d and dN can change a gate delay after the rising edge of the clock, the hold time is relatively short. Moreover, the dual trigger low energy flip flop circuit 300 or 350 does not rely on the dimensional relationship between the different transistors to function properly. Therefore, the flip-flop circuit operation can remain stable even when the characteristics of the transistors vary due to the manufacturing process.
Differential letter
Differential signaling avoids most of the problems associated with single-ended messaging. The data-driven charging pump described in connection with the fourth A, four B, four C and four D drawings can also be used for Implement a low swing and send a letter. In order to implement differential signaling, two replicas of the switched capacitor transmitter (along with the associated auxiliary equalizer transmitter) may be used, wherein a first switched capacitor transmitter consists of a positive polarity of a data bit. The version is driven, and a second switched capacitor transmitter is driven by a negative version of the data bit. Each of the first switched capacitor transmitter and the second switched capacitor transmitter drives one of the two conductors of the differential transmission line. These first and second switched capacitor converters are capable of operating at lower power because the receiver detects the differential voltage between the two lines. In addition, one or more of the devices shown in FIG. B may use differential signaling to receive or transmit signals. The receiving device can be arranged to include an amplifier arranged to receive the differential signal.
Figure 8A illustrates a differential version of a data driven switched capacitor transmitter 800 in accordance with an embodiment of the present invention. In the data driven switched capacitor transmitter 800, the differential signal is transmitted on lineP 805 and lineN 810, where lineN 810 is complementary to lineP 805. To transmit the signal, one of line {P, N} is driven high by the positive data transmitter 812 or the negative data transmitter 814, while the other line is pulled to LO by the individual termination resistor R0. Therefore the common mode voltage is 1/2 Vs instead of 0. In transmitter 800, the voltages on each of lineP 805 and lineN 810 are thixotropic between a certain positive voltage Vs and ground. Additionally, the common mode voltage will have a chop equal to one-half of the chopping above any of the line terminals 801 and 803. The equalizer 830 can include the same circuitry as the equalizer 530 shown in FIG.
Sub-circuits 802 and 804 pre-charged with capacitors precharge the capacitor C1p to the power supply voltage during the negative phase of the clock, and pre-charge capacitor C0p to the power supply voltage during the positive phase of the clock. Sub-circuits 802 and 804 pre-charged with capacitors or the transistors in equalizer 830 form a multiplexer and discharge circuit that is based on dat1 and the clock during the positive phase of the clock. The differential phase is driven based on dat0 to drive the differential transmission lines, namely lineP 805 and lineN 810. Data driven switched capacitor transmission The current converters of the transmitter 800, the positive data transmitter 812 and the negative data transmitter 814 are arranged such that the positive data transmitter 812 or the negative data transmitter 814 flows to the two differential transmission lines. The current in one flows only in the grounded network.
FIG. 8B illustrates a switched capacitor differential transmitter 820 in accordance with an embodiment of the present invention. The switched capacitor differential transmitter 820 is taken from the bridged charge pump transmitter 540 of FIG. The switched capacitor differential transmitter 820 is of a fully balanced configuration and therefore does not have a net current from the power supply into the differential signal lines lineP 825 and lineN 830, except at the terminals of the flying capacitors Cf1 and Cf0. Current caused by unbalanced parasitic capacitance. These parasitic capacitances on the GND network will need to balance these parasitic capacitances as close as possible.
In accordance with an embodiment of the invention, the output current flows only in the signal current return network 847, and the precharge current flows only in the internal power supply ground network. When the internal power supply grounding network and the signal current return network 847 are separate networks for the purpose of noise isolation, they are typically maintained at the same potential because the signal current is returned to the network in the channel. This external portion is shared with the power supply ground.
A single flying capacitor Cf1 and Cf0 is used for each phase of the clock (clk, where clkN is the inverted clkP). Regarding the switched capacitor converter 854, the flying capacitor Cf0 in the sub-circuit 843 charged with the flying capacitor is precharged to the power supply voltage when clk=LO. When clk=HI, the flying capacitor Cf0 drops the electric charge into one of the differential transmission lines lineP 845 or lineN 850, and if the dat0=HI, the differential transmission line lineP 845 is HI, And if dat0=LO, the differential transmission line lineN 850 is pulled up to HI. When clk=HI, when dat0=HI, lineN 850 is pulled to LO via R0, and when dat0=LO, lineP 845 is pulled to LO via R0.
The switched capacitor converter 852 performs the same operation at the opposite phase of the clock and is controlled by dat1. In particular, preloading with flying capacitors The flying capacitor Cf1 within the electrical sub-circuit 852 is precharged to the power supply voltage when clk = HI. When clk=LO, the flying capacitor Cf1 drops the electric charge into one of the differential transmission lines lineP 845 or lineN 850, and if the dat1=HI, the differential transmission line lineP 845 is HI, And if dat1=LO, the differential transmission line lineN 850 is pulled up to HI. Forming a 2:1 multiplexer and discharge circuit individually for each of the transistors within the switched capacitor converters 852 and 854 that are not in the sub-circuits 841 and 843 pre-charged with flying capacitors .
Unlike the differential transmitter 800 of Figure 8A, the transmitter 840 of Figure 8B drives lineP and lineN (845, 850, respectively) between positive and negative voltages of approximately equal magnitude, namely +Vs and -Vs . Therefore, the common mode voltage on lineP, lineN is zero.
The eighth C diagram illustrates a method for precharging a flying capacitor sub-circuit and driving a differential transmitting line on different phases of the clock in accordance with an exemplary embodiment of the present invention. Although the method steps are described in conjunction with the data driven switched capacitor transmitter 800 of FIG. 8A and the bridged charge pump transmitter 840 of FIG. B, those skilled in the art will appreciate that the method is configured to perform in any order. Any system of the method steps is within the scope of the invention.
The two different clock phases include a positive phase when clkP is HI and a negative phase when clkN is HI. The data is divided into two signals dat0 and dat1, where dat0 is valid when clkN is HI, and dat0 is valid when clkP is HI. At step 885, a first flying capacitor Cf1 is pre-charged during a positive phase of the clock by a sub-circuit pre-charged with a flying capacitor. When the data driven switched capacitor transmitter 800 is used, the capacitor C0p is precharged to the power supply voltage during the positive phase of the clock. At step 887, during the positive phase of the clock, a second flying capacitor Cf is discharged, and one of the differential signaling lines is driven to HI by a multiplexer electronic circuit. The transmission line that is not driven high by the multiplexer electronic circuit is pulled to LO by the termination resistor R0.
At step 890, the second flying capacitor Cf is pre-charged by the sub-circuit pre-charged with the flying capacitor during the negative phase of the clock. When the data driven switched capacitor transmitter 800 is used, the capacitor C1p is precharged to the power supply voltage during the negative phase of the clock. At step 892, during the positive phase of the clock, the first flying capacitor Cf is discharged, and one of the differential signaling lines is driven by the multiplexer electronic circuit to HI. The transmission line that is not driven high by the multiplexer electronic circuit is pulled to LO by the termination resistor R0.
Figure 9A illustrates a differential version of a data driven charge pump data transfer system 900 in accordance with an embodiment of the present invention. The data driven charge pump data transfer system 900 includes a transmitter 901 including a sub-circuit 910 pre-charged with a capacitor and a multiplexer and discharge circuit 912. Transmitter 901 is a data driven charge pump that drives a differential signal to differential transmitter line 903 (lineP and lineN) to receiver 906. Transmitter 901 includes a pre-calculated gate that is configured to incorporate positive and negative versions of a data signal, and positive and negative versions of the clock to produce an output coupled to discharge and multiplexer sub-circuit 912. Receiver 906 includes a receiver input amplifier 907 that is configured to receive the differential signal and regenerate the original data stream to produce rxdat.
In the data driven charge pump data transfer system 900, only the positive charge pump is utilized in the sub-circuit 910 pre-charged with the capacitor, and the signals are at 0 volts (ground) on the differential transmit line 903 with some desired The line voltage Vline (assumed to be a fraction of the usable power supply voltage on the wafer containing the transmitter 901) is nominally thixotropic. Thus, the data driven charge pump data transfer system 900 can avoid problems associated with driving a signal below ground potential, such as forward biased source/drain junctions and unwanted transistor turn-on. As in the single-ended transmitter, that is, the fourth A, four B, four C and four D map data-driven charge pump transmitters 400, 410, 420 and 430, the fifth A-type switched capacitor transmitter 500, The bridged charge pump transmitter 540 of the fifth C diagram, the bridge transmitter 550 having the pre-calculated gate, determines the current driven into the lineP and the lineN during the clkN=1 period, and the dat1 determines to be driven to during the clkP=1 period. lineP and Current in lineN. For example, during clkN=1, if dat0P=1 (dat0N=0), on the first half of the cycle when clkN=0, the capacitor Cp0 is charged to the power supply voltage, and in the subsequent half cycle The portion of sub-circuit 910 including capacitor pre-charging including Cp0 delivers a current pulse to lineP. No current pulses are injected into the lineN during this time period, and the termination resistor 902 at the transmitters 901 and 904 at the receiver 906 supplies a pull-down current to pull the lineN toward 0 volts (ground). Note that during clkN=0, Cn0 is also charged to the power supply voltage, but Cn0 remains charged during the next clkN=1.
Typical voltage waveforms that can be observed on lineP and lineN during operation of data driven charge pump data transfer system 900 are shown in thumbnail 905. Note that if the consecutive bits to be transmitted have the same HI value, voltage ripple will be caused on lineP. As previously mentioned, this chopping rate is the bit rate, so this voltage chopping does not actually affect the ability of the receiver 906 to distinguish the correct data value at the center of each cell segment of the data transmission. If the consecutive bits have an LO value, the voltage on lineP is pulled to zero volts (ground) by termination resistors 902 and 904, and no chopping is observed. Although these individual line voltages have different waveforms for the HI and LO data values on lineP and lineN, the differential voltage V(lineP, lineN) will be completely symmetrical for the two data values, and its amplitude is approximately twice that of Vline. .
Those skilled in the art will immediately recognize that the transmitter 901 of Figure 9A can be simplified and reduced in area, as shown in Figure IXB. Figure IXB illustrates another differential version of a data driven charge pump transmitter 915 in accordance with an embodiment of the present invention. Similar to the transmitter 901 of FIG. A, the data driven charge pump transmitter 915 also includes a pre-calculated gate that is configured to combine the positive and negative versions of a data signal, and the positive and negative versions of the clock to generate The output of the discharge and multiplexer sub-circuit portion coupled to the data driven charge pump transmitter 915.
The capacitors Cp0 and Cn0 of the sub-circuit 910 pre-charged by the capacitor have been incorporated into a single device C0 of the sub-circuit 920 pre-charged by the capacitor in the data-driven charge pump transmitter 915, and C0 is commonly used to drive the differential The transmission line 913, that is, lineP and lineN. Similarly, capacitors Cp1 and Cn1 of sub-circuit 910 pre-charged with capacitors have been incorporated into a single device C1 of sub-circuit 921 pre-charged with capacitors in data-driven charge pump transmitter 915. C0 of sub-circuit 920 pre-charged with capacitor is pre-charged during clkN=0. When clkN = 1, {, N P } driven one of those two NFETs opened by clkN * dat0, and the driving current to lineP (dat0P = 1) or lineN (dat0N = 1). C1 of the sub-circuit 921 pre-charged with the capacitor is precharged on the other phase of the clock by the control of dat1{P, N}. When clkP = 1, one of the two NFETs driven by clkP * dat1 {P, N} is turned on and drives current into lineP (dat1P = 1) or lineN (dat1N = 1).
The data driven charge pump transmitter 915 in the transmitter 901 and the ninth diagram can be modeled as a current source. Therefore, the output currents of the transmitter 901 and the data-driven charge pump transmitter 915 can be added and subtracted, so any equalization or other signal adjustment technique that can be implemented by adding the differential current can use the transmitter 901 and the data. The charge pump transmitter 915 is driven to implement.
The ninth C diagram illustrates an equalized data driven charge pump transmitter 930 that is equalized using a 2-tap pre-emphasis FIR filter in accordance with an embodiment of the present invention. The equalized data-driven charge pump transmitter 930 has two portions, a master data transmitter 932 that outputs a differential current in the same manner as the data-driven charge pump transmitter 915 of FIG. That is, the equalizer 935. The internal capacitors of the equalizer 935 are drawn (and fabricated) to be smaller than those of the transmitter 932, so the equalizer 935 provides a smaller current per cycle of operation. In particular, C0e<C0, C1e<C1, but of course C0e=C1e and C0=C1. The internal structure of the equalizer 935 is the same as the transmitter 932 except that the equalizer 935 receives data that is delayed by half of the clock cycle. So, for example, equalizer 935 outputs a current during clkN=1, which represents the value of dat1 on the previous half of the clock. Please note that the output connections of the equalizer 935 are compared The output connections to transmitter 932 are perceived to be opposite, so the current from equalizer 935 is "opposite" to the current of transmitter 932.
The ninth D diagram illustrates the differential transmit lines of the data driven charge pump transmitter 930, i.e., the typical voltage waveforms 940 of lineP and lineN, in accordance with an embodiment of the present invention. When the previous bit is different from the current bit to be transmitted, the equalizer 935 and the transmitter 932 drive the current in the same direction, so that the line voltage on one of lineP or lineN rises to a full Vline (or always drop to 0 volts). On the other hand, if consecutive bits have the same value, the voltages on the differential lines are driven to intermediate values. For example, suppose dat0P = 0 and the previous dat1P = 0; in this example, transmitter 932 drives lineP to 0 volts, but equalizer 935 drives a small current into lineP to make its voltage 0 volts and Vline/2 between. Similarly, if dat0P=1 and the previous dat1P=1, the transmitter 932 drives lineP to HI, but the equalizer 935 does not drive current into the lineP. Therefore, lineP only rises to a voltage between Vline/2 and Vline. Therefore, the differential output voltage V(lineP)-V(lineN) can take one of four values: +2 * Vline, -2 * Vline, +2 * A * Vline and -2 * A * Vline, where A is a constant between 0 and 1. The value of this constant A sets the strength of the 2-tap filter used to equalize a channel with frequency dependent attenuation.
In summary, a practical transmitter would require a method for adjusting the value of the constant A defined in the previous paragraph. This can be achieved during design by appropriately adjusting the magnitude of the capacitances C{0, 1} and C{0, 1}e. In other embodiments, both the transmitter 932 and the equalizer 935 can be decomposed into equal or weighted segments, and the segments are turned on or off to set the relative strength of the transmitter 932 to the equalizer 935, Thereby the constant A is changed.
In other embodiments, the capacitively coupled or "linear" equalizer 435 shown in the fourth diagram is implemented in a differential version and connected in parallel with a transmitter, i.e., in place of the ninth C diagram. The izer 935 forms an equalized differential transmitter. In still other specific embodiments, as a unipolar charge pump, Figure IX The sub-circuit 910 shown in the capacitor pre-charging is replaced by a bipolar charge pump, such as sub-circuits 433 and 434 of the fourth D diagram pre-charging with the flying capacitor, the pre-charging circuit shown in FIG. Sub-circuits 541 and 543 pre-charged with flying capacitors as shown in FIG. 5C, and sub-circuit 555 pre-charged with flying capacitors. The bipolar charge pump can be configured to generate a differential signal pair having a common mode voltage intermediate the wafer supply voltage (Vdd) and ground. Since the differential signaling system does not require a fixed reference voltage and plane, since the reference is implied by the average of the two signal voltages on line {P, N}, the common mode voltage can be set to any convenient Value.
An embodiment of the invention can be implemented as a program product for use by a computer system. The program of the program product defines the functions of the specific embodiments (including the methods described herein) and can be included on a variety of computer readable storage media. Exemplary computer readable storage media include, but are not limited to: (i) non-writable storage media (eg, a read-only memory device in a computer, such as a CD-ROM disc that can be read by a CD-ROM, flashing Memory, ROM chip, or any other kind of solid non-volatile semiconductor memory) on which information can be stored permanently; and (ii) writeable to a storage medium (eg, a floppy disk in a disk drive, or A hard disk drive, or any kind of solid state random access semiconductor memory, on which information that can be changed can be stored.
The invention has been described above with reference to specific embodiments. It will be apparent to those skilled in the art that various modifications and changes can be made without departing from the spirit and scope of the invention. Accordingly, the foregoing description and drawings are to be regarded as illustrative
100‧‧‧Single-end signaling system
101‧‧‧Transfer device
102‧‧‧ receiving device
103‧‧‧ feet
105‧‧‧Send line
135‧‧‧ transmission line
203‧‧‧Transmitter current flow
204‧‧‧Signal current flow
205‧‧‧bypass capacitor
206‧‧‧Terminal resistor
211‧‧‧Common ground impedance
300‧‧‧ Ground reference transmission system
301‧‧‧Transportation device
304‧‧‧ Grounding
305‧‧‧Symmetric voltage pairing
310‧‧‧ converter
320‧‧‧Control loops and converters
322,324‧‧‧Control loop
400‧‧‧Data Driven Charge Pump Transmitter
401, 402, 403, 404‧‧ ‧ subcircuit
407‧‧‧clock and data signal
413,414‧‧‧Subcircuits pre-charged with flying capacitors
416‧‧‧Multiple Electric Crystals
418,419‧‧‧Precharged transistor
430‧‧‧Data Driven Charge Pump Transmission Feeder
435‧‧‧ equalizer
450‧‧‧ Equalizer Transmitter
500‧‧‧Switching capacitor transmitter
512,514‧‧‧ positive converter
522, 523‧‧ positive converter
540‧‧‧Bridged Charge Pump Transmitter
542,544‧‧‧Switched Capacitor Converter
547‧‧‧Signal current return network
547‧‧‧Signal grounding
550‧‧‧Bridge transmitter with pre-calculated gate
555‧‧‧Subcircuits pre-charged with flying capacitors
560‧‧‧Regulator loop
570‧‧‧transmitter
572‧‧‧Copy Switched Capacitor Converter
600‧‧‧ Grounding Gate Amplifier
602‧‧‧ bias generator
605‧‧‧Input amplifier
610‧‧‧Second stage amplifier
620‧‧‧Adjustable bias generator
700‧‧‧ computer system
702‧‧‧Central Processing Unit
704‧‧‧System Memory
705‧‧‧Memory Bridge
706,713‧‧‧Communication path
707‧‧‧Input/Output Bridge
708‧‧‧ Input device
710‧‧‧ display device
712‧‧‧Parallel Processing Subsystem
714‧‧‧System Disc
716‧‧‧Switch
718‧‧‧Network Adapter
720,721‧‧‧ embedded card
740‧‧‧Processor/wafer
751‧‧‧ single-ended input signal
755‧‧‧ single-ended output signal
765‧‧‧ Receiver Circuit
770‧‧‧ core circuit
775‧‧‧transmitter circuit
800‧‧‧Data Driven Switched Capacitor Transmitter
801,803‧‧‧ line terminal
802,804‧‧‧Subcircuits pre-charged with capacitors
812‧‧‧正 data transmitter
814‧‧‧negative data transmitter
840‧‧‧Bridge Charge Pump Transmitter
900‧‧‧Data Driven Charge Pump Data Transmission System
902,904‧‧‧ terminating resistor
903‧‧‧Differential transmission line
905‧‧‧Typical voltage waveform
906‧‧‧ Receiver
907‧‧‧ Receiver Input Amplifier
910‧‧‧Subcircuits pre-charged with capacitors
912‧‧‧Multiplexer and electronic circuit
920, 921‧‧‧Subcircuits pre-charged with capacitors
930‧‧‧ Equalized data driven charge pump transmitter
Therefore, a more detailed description of one of the embodiments of the present invention can be understood by reference to the specific embodiments, which are illustrated in the accompanying drawings. But it should be noted that such ancillary The drawings are merely illustrative of typical embodiments of the invention, and are not intended to
1A shows an exemplary single-ended signaling system illustrating a reference voltage problem in accordance with the prior art; FIG. 1B shows an exemplary single-ended signaling system using an internal reference voltage in accordance with the prior art; Figure C shows an exemplary single-ended signaling system using an accompanying reference voltage in accordance with the prior art; Figure 2A shows current flow in a single-ended signaling system according to the prior art, wherein the ground plane is required As the common signal return conductor; the second B diagram shows the current in the single-ended transmission as shown in FIG. 2A when a "1" is to be transmitted by the transmitting device to the receiving device according to the prior art. Flow; FIG. 3A illustrates a single-ended signaling system in accordance with a ground reference of the prior art; and FIG. 3B illustrates a switched capacitor DC-DC converter that generates +Vs according to the prior art setting; The figure illustrates a switched capacitor DC-DC converter that generates -Vs according to the prior art setting; the third D diagram illustrates two switched capacitor DC-DC converters according to the third and third C diagrams of the prior art. Pair of summary controls a circuit; a fourth embodiment illustrates a data driven charge pump transmitter in accordance with an embodiment of the present invention; and a fourth block diagram illustrates a data driven charge pump transfer using a CMOS transistor in accordance with an embodiment of the present invention. The device is simplified compared to the data driven charge pump of FIG. A; FIG. 4C illustrates a data driven charge pump transmitter implemented by using a CMOS gate and a transistor according to an embodiment of the invention. ; FIG. 4D illustrates a data driven charge pump transmitter including an equalizer in accordance with an embodiment of the present invention; and FIG. 4E illustrates pressurizing the signal line voltage according to an embodiment of the present invention without An equalizer having an additional gate delay; FIG. 5A illustrates a switched capacitor transmitter for directing return current into the ground network in accordance with an embodiment of the present invention; FIG. 5B illustrates a first embodiment of the present invention In a specific embodiment, the precharge current is maintained from an averaging circuit of the GND network; and FIG. 5C illustrates a "bridge" charge pump transmitter according to an embodiment of the invention, wherein the output current is only Flowing in the GND network; FIG. 5D illustrates a bridge transmitter having a pre-calculated gate in accordance with an embodiment of the present invention; and FIG. 5E illustrates a control for controlling in accordance with an embodiment of the present invention a regulator circuit for the voltage on the transmission line; a fifth F diagram illustrating a method for precharging a flying capacitor sub-circuit and driving on different phases of the clock in accordance with an exemplary embodiment of the present invention Method of transmitting a line; FIG. 6A illustrates a grounding gate amplifier according to an embodiment of the present invention; and FIG. 6B illustrates an adjustable biasing generator according to an embodiment of the present invention; A method for adjusting an offset trimming mechanism according to an exemplary embodiment of the present invention is illustrated; FIG. 7A is a diagram illustrating a single-ended signaling for a ground reference according to one or more aspects of the present invention and including the A block diagram of a processor/wafer of the transmitter and receiver circuits; and a seventh block diagram of a computer system configured to implement one or more aspects of the present invention. FIG. 8A illustrates a data-driven switched capacitor transmitter for differential pairing transmission according to an embodiment of the present invention; FIG. 8B illustrates a differential pairing method according to an embodiment of the present invention a bridged charge pump transmitter of the letter; FIG. 8C illustrates a method for precharging a capacitor subcircuit and driving a differential transmit line on different phases of the clock, in accordance with an exemplary embodiment of the present invention; 9A illustrates a differential version of a data driven charge pump data transfer system in accordance with an embodiment of the present invention; and FIG. 9B illustrates another differential of a data driven charge pump transmitter in accordance with an embodiment of the present invention. Version IX C illustrates an equalization transmitter using a 2-tap pre-emphasis FIR filter in accordance with an embodiment of the present invention; and a ninth D diagram illustrating such an embodiment in accordance with an embodiment of the present invention The data drives the differential transmit lines of the charge pump transmitter, ie the typical voltage waveforms of lineP and lineN.
A transmitter circuit comprising: a sub-circuit pre-charged with a capacitor, comprising a first capacitor configured to be precharged to a substantially constant Vdd supply voltage during a positive phase of a clock, and a first a second capacitor disposed to be precharged to the substantially constant Vdd supply voltage during a negative phase of the clock, wherein the first capacitor is configured as a first flying capacitor and the second capacitor is configured as a second flying capacitor And a discharge and multiplexer sub-circuit configured to couple the first capacitor to the differential transmit pair including a first transmit line and a second transmit line during the negative phase of the clock The first transmit line drives the first transmit line and is configured to couple the second capacitor to the first transmit line of the differential transmit pair during the positive phase of the clock to drive the a first transmit line, wherein the second transmit line is a negative transmit line, and the discharge and multiplexer subcircuit is further configured to transfer charge from the first flight capacitor and generate a current to the second transmit In the letter line When a data signal is driven low during the negative phase of the clock of the second transmission line is high.
The transmitter circuit of claim 1, wherein the first transmission line is a positive transmission line, and the discharge and multiplexer sub-circuit is further configured to transfer charge from the first flight capacitor and generate a A current is supplied to the first transmit line to drive the first transmit line high during a negative phase of the clock when a data signal is high.
A transmitter circuit comprising: a sub-circuit pre-charged with a capacitor, comprising a first capacitor configured to be precharged to a substantially constant Vdd supply voltage during a positive phase of a clock, and a first a second capacitor disposed to be precharged to the substantially constant Vdd supply voltage during a negative phase of the clock, wherein the first capacitor is configured as a first flying capacitor and the second capacitor is configured as a second flying capacitor ;and a discharge and multiplexer sub-circuit configured to couple the first capacitor to the differential transmit pair including a first transmit line and a second transmit line during the negative phase of the clock a first transmit line to drive the first transmit line and configured to couple the second capacitor to the first transmit line of the differential transmit pair during the positive phase of the clock to drive the first a transmission line, wherein the second transmission line is a negative transmission line, and the discharge and multiplexer subcircuit is further configured to transfer charge from the second flight capacitor and generate a current to the second transmission line The second transmit line is driven high during a positive phase of the clock when a data signal is low.
A transmitter circuit comprising: a sub-circuit pre-charged with a capacitor, comprising a first capacitor configured to be precharged to a substantially constant Vdd supply voltage during a positive phase of a clock, and a first a second capacitor disposed to be precharged to the substantially constant Vdd supply voltage during a negative phase of the clock, wherein the first capacitor is configured as a first flying capacitor and the second capacitor is configured as a second flying capacitor And a discharge and multiplexer sub-circuit configured to couple the first capacitor to the differential transmit pair including a first transmit line and a second transmit line during the negative phase of the clock The first transmit line drives the first transmit line and is configured to couple the second capacitor to the first transmit line of the differential transmit pair during the positive phase of the clock to drive the a first transmit line, wherein the sub-circuit pre-charging with a capacitor includes: the first flying capacitor having a first terminal coupled to the substantially constant Vdd supply voltage via a first clock-enabled transistor Which is activated during the phase of the clock timing, and when the enable pulse having a second via The transistor is coupled to a second terminal of a ground supply voltage that is also activated during a positive phase of the clock; and the second flying capacitor has a transistor coupled via a third clock enable to the ground supply a first terminal of the voltage that is activated during a negative phase of the clock and has a second terminal coupled to the substantially constant Vdd supply voltage via a fourth clock-enabled transistor, The negative phase period of the clock is also activated.
The transmitter circuit of claim 1, further comprising an equalizer coupled to the first transmit line and the second transmit line of the differential transmit pair, respectively, and configured to An additional current is boosted to one of the first transmit line and the second transmit line when a profile signal transmitted on the differential transmit pair transitions from low to high or from high to low.
The transmitter circuit of claim 1, further comprising a pre-calculated gate configured to combine the positive and negative versions of a data signal and the positive and negative versions of the clock to generate coupling to the discharge and The output of the sub-circuit of the tool.
A method for differential signaling, the method comprising: precharging a first capacitor to a substantially constant Vdd supply voltage during a positive phase of a clock; and applying a second capacitor during the positive phase of the clock And coupled to a first transmit line of a differential transmit pair, the differential transmit pair includes a first transmit line or a second transmit line, wherein the first capacitor is configured as a first flight capacitor And the second capacitor is configured as a second flying capacitor; precharging the second capacitor to the substantially constant Vdd supply voltage during a negative phase of the clock; and during the negative phase of the clock a capacitor coupled to the first transmit line of the differential transmit pair, wherein the second transmit line is a negative transmit line, and the discharge and multiplexer subcircuit is further configured to The first flying capacitor transmits a charge and generates a current to the second transmission The second transmit line is driven high during a negative phase of the clock when the data signal is low.
TW101136887A 2012-01-30 2012-10-05 A data-driven charge-pump transmitter for differential signaling TWI514767B (en)
US13/361,843 US9338036B2 (en) 2012-01-30 2012-01-30 Data-driven charge-pump transmitter for differential signaling
TW201332290A TW201332290A (en) 2013-08-01
TWI514767B true TWI514767B (en) 2015-12-21
TW101136887A TWI514767B (en) 2012-01-30 2012-10-05 A data-driven charge-pump transmitter for differential signaling
US8611437B2 (en) 2012-01-26 2013-12-17 Nvidia Corporation Ground referenced single-ended signaling
US9171607B2 (en) 2013-03-15 2015-10-27 Nvidia Corporation Ground-referenced single-ended system-on-package
US9076551B2 (en) 2013-03-15 2015-07-07 Nvidia Corporation Multi-phase ground-referenced single-ended signaling
US9153314B2 (en) 2013-03-15 2015-10-06 Nvidia Corporation Ground-referenced single-ended memory interconnect
US9170980B2 (en) 2013-03-15 2015-10-27 Nvidia Corporation Ground-referenced single-ended signaling connected graphics processing unit multi-chip module
EP2884643A1 (en) * 2013-12-11 2015-06-17 Nxp B.V. DC-DC voltage converter and conversion method
EP2884642B1 (en) * 2013-12-11 2016-10-19 Nxp B.V. DC-DC voltage converter and conversion method
EP2911281B1 (en) * 2014-02-24 2019-04-10 Nxp B.V. Electronic device
US9525441B2 (en) * 2014-12-11 2016-12-20 Intel Corporation Common mode noise introduction to reduce radio frequency interference
KR20160105101A (en) * 2015-02-27 2016-09-06 에스케이하이닉스 주식회사 Stack Package and Semiconductor Integrated Circuit Device Using Variable voltage
US10122392B2 (en) * 2016-08-18 2018-11-06 Advanced Micro Devices, Inc. Active equalizing negative resistance amplifier for bi-directional bandwidth extension
CN109213708A (en) * 2017-06-29 2019-01-15 华为技术有限公司 A kind of driver for the link transmitters that serially unstring
US20060232311A1 (en) * 2005-04-15 2006-10-19 Elpida Memory, Inc. Duty detection circuit and method for controlling the same
US20090002066A1 (en) * 2007-06-29 2009-01-01 Meng-Chang Lee Multi-Tap Direct Sub-sampling Mixing System for Wireless Receivers
US20110199122A1 (en) * 2009-08-31 2011-08-18 Panasonic Corporation Direct sampling circuit and receiver
US6125047A (en) 1993-12-14 2000-09-26 Seagate Technology, Inc. Regulated inverting power supply
US5696031A (en) 1996-11-20 1997-12-09 Micron Technology, Inc. Device and method for stacking wire-bonded integrated circuit dice on flip-chip bonded integrated circuit dice
US6188877B1 (en) 1997-07-03 2001-02-13 Ericsson Inc. Dual-band, dual-mode power amplifier with reduced power loss
US6754838B2 (en) 2001-01-26 2004-06-22 Hewlett-Packard Development Company, L.P. Method for reducing tuning etch in a clock-forwarded interface
US7702293B2 (en) 2001-11-02 2010-04-20 Nokia Corporation Multi-mode I/O circuitry supporting low interference signaling schemes for high speed digital interfaces
EP1333380A1 (en) 2002-01-30 2003-08-06 Stmicroelectronics, Ltd. DMA access generator
US7010637B2 (en) 2002-05-02 2006-03-07 Intel Corporation Single-ended memory interface system
JP4060696B2 (en) * 2002-12-13 2008-03-12 日本オプネクスト株式会社 Optical transmission module
US7173547B2 (en) 2004-03-22 2007-02-06 International Business Machines Incorporated Offset compensation in local-probe data storage devices
US7385607B2 (en) 2004-04-12 2008-06-10 Nvidia Corporation Scalable shader architecture
US7633505B1 (en) 2004-11-17 2009-12-15 Nvidia Corporation Apparatus, system, and method for joint processing in graphics processing units
US7375992B2 (en) 2005-01-24 2008-05-20 The Hong Kong University Of Science And Technology Switched-capacitor regulators
KR101119066B1 (en) 2005-08-12 2012-03-15 삼성전자주식회사 Multi-chip package
US7652922B2 (en) 2005-09-30 2010-01-26 Mosaid Technologies Incorporated Multiple independent serial link memory
US7728841B1 (en) 2005-12-19 2010-06-01 Nvidia Corporation Coherent shader output for multiple targets
TWI269462B (en) 2006-01-11 2006-12-21 Advanced Semiconductor Eng Multi-chip build-up package of an optoelectronic chip and method for fabricating the same
US7423458B2 (en) 2006-03-08 2008-09-09 Analog Devices, Inc. Multiple sampling sample and hold architectures
US20070211711A1 (en) 2006-03-08 2007-09-13 Clayton James E Thin multichip flex-module
US20080189457A1 (en) 2006-12-06 2008-08-07 International Business Machines Corporation Multimodal memory controllers
JP2009022093A (en) 2007-07-11 2009-01-29 Ricoh Co Ltd Multi-output power supply unit
US8892942B2 (en) 2007-07-27 2014-11-18 Hewlett-Packard Development Company, L.P. Rank sparing system and method
US7760010B2 (en) 2007-10-30 2010-07-20 International Business Machines Corporation Switched-capacitor charge pumps
TWI347097B (en) 2007-12-31 2011-08-11 Ind Tech Res Inst Circuit with programmable signal bandwidth and method thereof
US7583133B2 (en) 2008-01-25 2009-09-01 Texas Instruments Incorporated Self-oscillating regulated low-ripple charge pump and method
US7701763B2 (en) 2008-04-23 2010-04-20 Micron Technology, Inc. Leakage compensation during program and read operations
US7848153B2 (en) 2008-08-19 2010-12-07 Qimonda Ag High speed memory architecture
US8065540B2 (en) 2008-10-31 2011-11-22 Dell Products, Lp Power control for information handling system having shared resources
US20110063305A1 (en) 2009-09-16 2011-03-17 Nvidia Corporation Co-processing techniques on heterogeneous graphics processing units
JP2011114622A (en) 2009-11-27 2011-06-09 Panasonic Corp Filter automatic adjustment circuit and method, and radio communication equipment
JP5032623B2 (en) 2010-03-26 2012-09-26 株式会社東芝 Semiconductor memory device
KR101710658B1 (en) 2010-06-18 2017-02-27 삼성전자 주식회사 Three dimensional stacked structure semiconductor device having through silicon via and method for signaling thereof
JP2012059991A (en) 2010-09-10 2012-03-22 Toshiba Corp Semiconductor integrated circuit
US8395416B2 (en) 2010-09-21 2013-03-12 Intel Corporation Incorporating an independent logic block in a system-on-a-chip
US8352676B2 (en) 2010-10-26 2013-01-08 Hitachi, Ltd. Apparatus and method to store a plurality of data having a common pattern and guarantee codes associated therewith in a single page
JP6030298B2 (en) 2010-12-28 2016-11-24 株式会社半導体エネルギー研究所 Buffer storage device and signal processing circuit
US8621131B2 (en) 2011-08-30 2013-12-31 Advanced Micro Devices, Inc. Uniform multi-chip identification and routing system
US8624626B2 (en) 2011-11-14 2014-01-07 Taiwan Semiconductor Manufacturing Co., Ltd. 3D IC structure and method
TWI478490B (en) 2011-12-14 2015-03-21 Ind Tech Res Inst Charge domain filter and method therof
DE112012005383T5 (en) 2011-12-22 2014-09-11 International Business Machines Corp. Storage unit access system
US8538558B1 (en) 2012-03-01 2013-09-17 Texas Instruments Incorporated Systems and methods for control with a multi-chip module with multiple dies
US9111601B2 (en) 2012-06-08 2015-08-18 Qualcomm Incorporated Negative voltage generators
US8953384B2 (en) 2012-07-31 2015-02-10 Winbond Electronics Corporation Sense amplifier for flash memory
US8854123B1 (en) 2013-03-15 2014-10-07 Nvidia Corporation On-package multiprocessor ground-referenced single-ended interconnect
2012-09-28 DE DE201210217836 patent/DE102012217836A1/en active Pending
2012-10-05 TW TW101136887A patent/TWI514767B/en active
DE102012217836A1 (en) 2013-08-01
Kim et al. 2002 Adaptive supply serial links with sub-1-V operation and per-pin clock recovery
US6812767B2 (en) 2004-11-02 High speed source synchronous signaling for interfacing VLSI CMOS circuits to transmission lines
US6377638B1 (en) 2002-04-23 Signal transmission system for transmitting signals between lsi chips, receiver circuit for use in the signal transmission system, and semiconductor memory device applying the signal transmission system
US5959472A (en) 1999-09-28 Driver circuit device
US7199728B2 (en) 2007-04-03 Communication system with low power, DC-balanced serial link
US5646558A (en) 1997-07-08 Plurality of distinct multiplexers that operate as a single multiplexer
Poulton et al. 2007 A 14-mW 6.25-Gb/s transceiver in 90-nm CMOS
DE10101066B4 (en) 2015-03-12 Driver circuit, receiving circuit and signal transmission bus system
AU759089B2 (en) 2003-04-03 High speed signaling for interfacing VLSI CMOS circuits
JP6170663B2 (en) 2017-07-26 Flexible receiver architecture
DE60106541T2 (en) 2005-10-13 LVDS circuits connected in series for power
US7071756B1 (en) 2006-07-04 Clock multiplexing system
JP3866625B2 (en) 2007-01-10 Segmented mixed signal circuit device and switch circuit device
US5811992A (en) 1998-09-22 Dynamic clocked inverter latch with reduced charged leakage and reduced body effect
US6166971A (en) 2000-12-26 Method of and apparatus for correctly transmitting signals at high speed without waveform distortion
US7161378B2 (en) 2007-01-09 Semiconductor memory device with on die termination circuit
US6560290B2 (en) 2003-05-06 CMOS driver and on-chip termination for gigabaud speed data communication
US20060214695A1 (en) 2006-09-28 Keeper circuits having dynamic leakage compensation
US20140347108A1 (en) 2014-11-27 Method and apparatus for source-synchronous signaling
US6803790B2 (en) 2004-10-12 Bidirectional port with clock channel used for synchronization
US7777522B2 (en) 2010-08-17 Clocked single power supply level shifter
Wong et al. 2004 A 27-mW 3.6-gb/s I/O transceiver
US20060071691A1 (en) 2006-04-06 Input/output cells with localized clock routing
US6426656B1 (en) 2002-07-30 High speed, low-power inter-chip transmission system
O'Mahony et al. 2010 A 47$\,\times\, $10 Gb/s 1.4 mW/Gb/s Parallel Interface in 45 nm CMOS | CommonCrawl |
\begin{document}
\title{Isoparametric hypersurfaces in product spaces}
\author{João Batista Marques dos Santos}
\address{João Batista Marques dos Santos - Departamento de Matem\'atica, Universidade de Bras\'ilia, 70910-900, Bras\'ilia-DF, Brazil}
\email{[email protected]}
\thanks{The first author was supported by Capes and CNPq. }
\author{Jo\~ao Paulo dos Santos}
\address{Jo\~ao Paulo dos Santos - Departamento de Matem\'atica, Universidade de Bras\'ilia, 70910-900, Bras\'ilia-DF, Brazil}
\email{[email protected]} \thanks{The second author was supported by CNPq grant number 315614/2021-8.}
\subjclass[2020]{53C40, 53C42}
\keywords{isoparametric hypersurfaces, product spaces, parallel hypersurfaces}
\begin{abstract} In this paper, we characterize and classify the isoparametric hypersurfaces with constant principal curvatures in the product spaces $ \mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$, where $\mathbb{Q}^{2}_{c_{i}}$ is a space form with constant sectional curvature $c_{i}$, for $c_1 \neq c_2$. \end{abstract}
\maketitle
\section{Introduction}
A hypersurface $M^n$ of a Riemannian manifold $\widetilde{M}^{n+1}$ is said to be isoparametric if it has constant mean curvature as well as its nearby equidistant hypersurfaces (i.e., the correspondent mean curvatures depend only on the distance to $M$). Equivalently, we say that $M$ is isoparametric if it is the level set of some isoparametric function defined on $\widetilde{M}$. Following M. Domínguez-Vázquez \cite{notas-miguel}, the first notion of isoparametric function appeared in 1919 in the work of the Italian mathematician C. Somigliana \cite{somigliana1918sulle}, which deals with of the relations between the Huygens principle and geometric optics. This study represented the beginning of an important research line in Differential Geometry, namely the isoparametric hypersurfaces studied by renowned mathematicians such as Beniamino Segre, Élie Cartan, and Tullio Levi-Civita.
When the ambient space is a space form, i.e., a simply connected complete Riemannian manifold with constant sectional curvature, the previous definition of isoparametric hypersurface is equivalent to saying that the hypersurface has constant principal curvatures (see \cite{isoCartan} and \cite{notas-miguel}). However, in arbitrary ambient spaces of nonconstant curvature, the equivalence between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures may no longer be true. For instance, Q. M. Wang, in \cite{WangExIso}, found examples of isoparametric hypersurfaces in complex projective spaces that do not have constant principal curvatures. For more examples, we refer \cite{diaz2010inhomogeneous}, \cite{diaz2013isoparametric} and \cite{ge2015filtration}. Recently, A. Rodríguez-Vázquez, in \cite{RVazquezExIso}, found an example of a non-isoparametric hypersurface with constant principal curvatures. Another example was given by the authors, in a joint work with F. Guimarães \cite{isoparametric-mcf}.
In this work, we consider the Riemannian products of 2-dimensional space forms $ \mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$, with constant sectional curvatures $c_{1}$ and $c_2$, respectively, with $c_1 \neq c_2$, where $c_i=1,\,0$ or $-1$, $i=1,\,2$. The particular case where $c_{1} = 1$ and $c_{2} = 0$, that is, when the ambient space is $\mathbb{S}^2 \times \mathbb{R}^2$, was considered by J. Julio-Batalla in \cite{s2r2Batalla} where he obtained a complete classification of isoparametric hypersurfaces with constant principal curvatures. Using some ideas developed by F. Urbano in \cite{s2s2Urbano}, where isoparametric hypersurfaces of $\mathbb{S}^2 \times \mathbb{S}^2$ were classified, J. Julio-Batalla showed that if $\Sigma$ is an isoparametric hypersurface in $\mathbb{S}^2 \times \mathbb{R}^2$, with constant principal curvatures and unit normal $N = N_{1} + N_{2}$, then $\lvert N_{1} \rvert$ and $\lvert N_{2} \rvert$ are constant. The classification continues by showing that $\lvert N_{1} \rvert = 1$ and $\lvert N_{2} \rvert = 0$ or $\lvert N_{1} \rvert = 0$ and $\lvert N_{2} \rvert = 1$. Thus, the hypersurface families obtained are $\mathbb{S}^2 \times \mathbb{R}$, $\mathbb{S}^2 \times \mathbb{S}^1(r)$ (for $r \in \mathbb{R}^{+}$), or $\mathbb{S}^1(t) \times \mathbb{R}^2$ (for $t \in (0,1]$).
In this paper, we extend and improve the results of \cite{s2r2Batalla} in the following sense. Considering the ambient space $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$ with $c_{1} \neq c_{2}$, we prove
\begin{theorem}\label{theo1} Let $\Sigma$ be an isoparametric hypersurface in $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$, $c_{1} \neq c_{2}$, and unit normal $N = N_{1} + N_{2}$. Then the principal curvatures of $\Sigma$ are constant if and only if $\lvert N_{1} \rvert$ and $\lvert N_{2} \rvert$ are constant. \end{theorem}
In addition to the converse of a result obtained by J. Julio-Batalla, which states that if $\lvert N_{1} \rvert$ and $\lvert N_{2} \rvert$ are constant, then $\Sigma$ has constant principal curvatures, Theorem \ref{theo1} also provides the equivalence for the entire class of ambient spaces $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$, with $c_1 \neq c_2$. To get this Theorem, we use the theory of Jacobi fields, based on the ideas developed by M. Domínguez-Vázquez and J. M. Manzano in \cite{dominguez-manzano}, to analyze the extrinsic geometry of hypersurfaces parallel to $\Sigma$. It is interesting to note that Jacobi field theory allows us to obtain an alternative proof of J. Julio-Batalla's result. Moreover, we obtain the following general classification of isoparametric hypersurfaces with constant principal curvatures in $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$, $c_{1} \neq c_{2}$, which includes the classification for $\mathbb{S}^2 \times \mathbb{R}^2$ given in \cite{s2r2Batalla}:
\begin{theorem}\label{theo2} Let $\Sigma$ be an isoparametric hypersurface in $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$, $c_{1} \neq c_{2}$, with constant principal curvatures. Then, up to rigid motions, $\Sigma$ is an open subset of one of the following hypersurfaces: \begin{enumerate}[a)]
\item $\mathcal{C}^{1}(\kappa_{j})\times \mathbb{Q}^{2}_{c_{2}}$ or $\mathbb{Q}^{2}_{c_{1}} \times \mathcal{C}^{1}(\kappa_{j})$, where $\mathcal{C}^{1}(\kappa_{j})$ is a complete curve with constant geodesic curvature $\kappa_j$ in $\mathbb{Q}^{2}_{c_{j}}$. \item $\Psi(\mathbb{R}^3) \subset \mathbb{H}^2 \times \mathbb{R}^2$, where $\Psi : \mathbb{R}^3 \rightarrow \mathbb{H}^2 \times \mathbb{R}^2$ is an immersion given by \begin{equation} \begin{split} \Psi(t,u,v) &= e^{-b\, t}(\alpha(u),\vec{0})+ \Big(\cosh(-b\, t), 0, \sinh(-b\, t), V_0t \Big) \\ & \quad + \Big(\vec{0}, p_0 + W_0 v \Big), \end{split}\label{eq:parametrization-Psi} \end{equation}
where $\mathbb{H}^2 \subset \mathbb{L}^3$ is given as the standard model of the hyperbolic space in the Lorentz 3-space $\mathbb{L}^3$, the curve $\alpha$ is given by $\alpha(u)=\left( \dfrac{u^2}{2},\,u,\,-\dfrac{u^2}{2} \right)$, $V_0$ and $W_0$ are constant orthogonal vectors in $\mathbb{R}^2$ such that $||W_0||=1$ and $b=\sqrt{1-||V_0||^2}$. \end{enumerate} \end{theorem}
Recall that, besides of the geodesics, the complete curves $\mathcal{C}^1(\kappa_j) \subset \mathbb{Q}^{2}_{c_{j}}$ with constant geodesic curvature are given by: $\mathbb{S}^1(t) \subset \mathbb{S}^2$ for $t \in (0,1)$; circles, horocycles or hypercycles in $\mathbb{H}^2$; and $\mathbb{S}^1(r) \subset \mathbb{R}^2$ for $r \in \mathbb{R}^{+}$. Regarding the hypersurfaces given in Theorem \ref{theo2}.b), geometrically, $\Psi(\mathbb{R}^3$) provides a hypersurface given as a family of geodesically parallel surfaces given by the products $\mathcal{C}^1(1) \times \mathbb{R}$, where $\mathcal{C}^1(1) \subset \mathbb{H}^2$ is a horocycle (see Remark \ref{rmk:geometry-psi}). \\
The paper is organized as follows. In Section \ref{sec2}, we provide some preliminary concepts and notations that will be used throughout the work. Section \ref{sec3} is devoted to the proof of Theorems~\ref{theo1} and \ref{theo2}. Using Jacobi field theory, we start by proving Theorem~\ref{theo1}, which characterizes isoparametric hypersurfaces with constant principal curvatures in $ \mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$. Then, using Theorem~\ref{theo1}, we classify these hypersurfaces by proving Theorem~\ref{theo2}.
\section{Preliminary notions and results}\label{sec2}
Before proving our main results, let us present some background content on complex and product structures, the Jacobi field theory and isoparametric functions.
Let $\mathbb{Q}^{2}_{c_{1}}$ and $\mathbb{Q}^{2}_{c_{2}}$ be two 2-dimensional space forms with distinct constant sectional curvatures $c_{1}$ and $c_{2}$, respectively. For $i=1,2$, we denote by $L_{i}$ the standard complex structure in $\mathbb{Q}^{2}_{c_{i}}.$ If $\mathbb{Q}^{2}_{c_{i}}$ is the 2-dimensional sphere $\mathbb{S}^2$ of curvature $c_{i} = 1$, $L_{i}$ is given by \begin{align*} \begin{split} L_{i} : \ &T\mathbb{S}^2 \longrightarrow T\mathbb{S}^2 \\ & v \longrightarrow L_{i}(v) = p \times v, \end{split} \end{align*} for $p \in \mathbb{S}^2$, $ v \in T_{p}\mathbb{S}^2$, see \cite{dillen2012constant}. When $\mathbb{Q}^{2}_{c_{i}}$ is the hyperbolic space $\mathbb{H}^2$ of curvature $c_{i} = -1$, we will consider its standard Lorentzian model, i.e., $$\mathbb{H}^2 = \{(x_{1},x_{2},x_{3}) \in \mathbb{L}^3 \mid -x_{1}^2 + x_{2}^2 + x_{3}^2 = -1 \, \text{and} \, x_{1} > 0\},$$ where $\mathbb{L}^3$ is the 3-dimensional Minkowski space endowed with the Lorentzian cross product $\boxtimes$, defined by $$ (a_{1},a_{2},a_{3})\boxtimes(b_{1},b_{2},b_{3}) = (a_{3}b_{2} - a_{2}b_{3},a_{3}b_{1} - a_{1}b_{3},a_{1}b_{2} - a_{2}b_{1}).$$ In this model, $L_{i}$ is given by \begin{align*} \begin{split} L_{i} : \ &T\mathbb{H}^2 \longrightarrow T\mathbb{H}^2 \\ & v \longrightarrow L_{i}(v) = p \boxtimes v, \end{split} \end{align*} for $p \in \mathbb{H}^2$, $ v \in T_{p}\mathbb{H}^2$, see \cite{dillen2012constant} and \cite{gao2021lagrangian}. Finally, if $\mathbb{Q}^{2}_{c_{i}}$ is the space form $\mathbb{R}^2$ of curvature $c_i=0$, $L_{i}$ is defined by \begin{align*} \begin{split} L_{i} : \ &\mathbb{R}^2 \longrightarrow \mathbb{R}^2 \\ & v \longrightarrow L_{i}(q_{1},q_{2}) = (-q_{2},q_{1}), \end{split} \end{align*} see \cite{s2r2Batalla}.
We endow $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$ with the standard product metric, denoted by $\langle , \rangle$. Moreover, given $Y \in T(\mathbb{Q}^{2}_{c_{1}}~\times~\mathbb{Q}^{2}_{c_{2}})$, we write $Y=Y^{\mathbb{Q}^{2}_{c_{1}}} + Y^{\mathbb{Q}^{2}_{c_{2}}}$, where the components $Y^{\mathbb{Q}^{2}_{c_{1}}}$ and $Y^{\mathbb{Q}^{2}_{c_{2}}}$ of $Y$ are given as its tangent parts to $\mathbb{Q}^{2}_{c_{1}}$ and $\mathbb{Q}^{2}_{c_{2}}$, respectively. We define on $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$ the complex strutures \begin{equation*} J_{1} = L_{1} + L_{2}, \quad J_{2} = L_{1} - L_{2}, \end{equation*} and we denote by $\widetilde{\nabla}$ and $\widetilde{R}$ its Levi-Civita connection and curvature tensor, respectively.
Now, let us consider the product structure $P$ in $\mathbb{Q}^{2}_{c_{1}}~\times~\mathbb{Q}^{2}_{c_{2}}$ defined by $$P\Big(Y^{\mathbb{Q}^{2}_{c_{1}}} + Y^{\mathbb{Q}^{2}_{c_{2}}}\Big) = Y^{\mathbb{Q}^{2}_{c_{1}}} - Y^{\mathbb{Q}^{2}_{c_{2}}},$$ for any vector $Y \in T(\mathbb{Q}^{2}_{c_{1}}~\times~\mathbb{Q}^{2}_{c_{2}})$. Note that $P$ satisfies \begin{equation*} P = -J_{1}J_{2} = -J_{2}J_{1}. \end{equation*} Moreover, $P$ has the following properties: \begin{align*} &P^{2} = I \ (P \neq I), \quad \langle PY,Z \rangle = \langle Y,PZ \rangle, \quad \mbox{and} \quad (\widetilde{\nabla}_{Y}P)(Z) = 0, \end{align*} for any vector field $Y, Z \in T(\mathbb{Q}^{2}_{c_{1}}~\times~\mathbb{Q}^{2}_{c_{2}})$. Using the product structure $P$, $ \widetilde{R} $ is given by \begin{align*} \widetilde{R}(V,W,Z,Y) &= \frac{c_{1}}{4}\biggr\{\langle V,PY + Y \rangle\langle PW + W,Z \rangle - \langle W,PY + Y \rangle\langle PV + V,Z \rangle\biggr\} \nonumber \\ & \quad + \frac{c_{2}}{4}\biggr\{\langle V,PY - Y \rangle\langle PW - W,Z \rangle - \langle W,PY - Y \rangle\langle PV - V,Z \rangle\biggr\}, \end{align*} where $V, W, Z, Y \in T(\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}})$, see \cite{dillen2012constant}.
Let $\Sigma^3 \subset \mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$ be an oriented hypersurface with unit normal vector $N = N_{1} + N_{2}$ and Levi-Civita connection $ \nabla $. We define in $\Sigma^3$ a smooth function $C$ and a tangent vector field $X$ by \begin{equation}\label{funC} C = \langle PN,N \rangle \quad \mbox{and} \quad X = PN - CN. \end{equation}
Observe that $X$ is the tangential component of $PN$ and $\lvert X \rvert^2 = 1 - C^2$, which implies $-1 \leq C \leq 1 $.
Using the curvature tensor of $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$ and the vector field $X$ defined above, the Codazzi equation of $\Sigma$ is given by \begin{align*}
\nabla S(V,W,Z) - \nabla S(W,V,Z) &= \widetilde{R}(V,W,Z,N), \end{align*} where \begin{align*} \widetilde{R}(V,W,Z,N) &= \frac{c_{1}}{4}\biggr\{\langle V, PN + N \rangle \langle PW + W , Z \rangle - \langle W, PN + N \rangle \langle PV + V , Z \rangle \biggr\}\\
& \quad + \frac{c_{2}}{4}\biggr\{\langle V, PN - N \rangle \langle PW - W , Z \rangle - \langle W, PN - N \rangle \langle PV - V , Z \rangle \biggr\}\\
&= \frac{c_{1}}{4}\biggr\{\langle V, X \rangle \langle PW + W , Z \rangle - \langle W, X \rangle \langle PV + V , Z \rangle \biggr\}\\
& \quad + \frac{c_{2}}{4}\biggr\{\langle V, X \rangle \langle PW - W , Z \rangle - \langle W, X \rangle \langle PV - V , Z \rangle\biggr\}, \end{align*} with $V, W, Z \in T\Sigma$.
In this work, we will use the Jacobi field theory to analyze the extrinsic geometry of hypersurfaces equidistant to the hypersurface $\Sigma$. In what follows, we will give a brief description of this theory. For more details, we refer to \cite{bookOlmosCia, notas-miguel}.
Given a hypersurface $\Sigma^n$ of a Riemannian manifold $\widetilde{M}^{n+1}$ with unit normal vector field $N$, let $\varepsilon$ be a positive real number and, for $r \in (-\varepsilon, \varepsilon)$, consider the application \begin{equation}\label{paralellhypersurfaces} \begin{array}{rcl}
\Phi_r: \Sigma^n & \rightarrow & \widetilde{M}^{n+1}, \\
p &\mapsto& \exp_{p}(rN_{p}), \end{array} \end{equation} where $\exp_{p}: T_{p}\widetilde{M} \rightarrow \widetilde{M}$ denotes the exponential map of $\widetilde{M}^{n+1}$ at $p \in \Sigma$. For $\varepsilon>0$ small enough, the map $\Phi_r$ is smooth and it parametrizes the parallel displacement of $\Sigma$ at an oriented distance $r$ in the direction $N$. The parallel hypersurface $\Phi_r(\Sigma)$ will be denoted by $\Sigma_r$.
Let $\gamma_p: I \rightarrow \widetilde{M}$ be the geodesic parametrized by arc length with $0 \in I \subset \mathbb{R}$, $\gamma_p(0)=p \in \Sigma$ and $\dot{\gamma_p}(0) = N_{p}$. Let $\zeta_Y$ be the Jacobi field along $\gamma_p$ with initial conditions given by $$ \zeta_Y(0)=Y, \, \textnormal{ and } \, \zeta_Y'(0) = - AY, $$ where $A$ is the shape operator of $\Sigma$ associated with $N$. Then, a unit normal vector to $\Sigma_r$ at $\gamma_p(r)$ is given by $\dot{\gamma_p}(r)$ and its correspondent shape operator satisfies $$ A_r \zeta_Y(r) = - \zeta_Y'(r). $$ If we write $\zeta_Y(r)=D(r)\widetilde{P}_Y(r)$, where $D(r)$ is an endomorphism acting on $T_{\gamma_p(r)}\Sigma_r$ and $\widetilde{P}_Y(r)$ is the parallel transport of $Y$ along $\gamma_p$, then we have \begin{equation}\label{A-paralell} A_r = -(D'\circ D^{-1})(r). \end{equation} Consequently, by the Jacobi formula, the mean curvature of the hypersurface $\Sigma_r$ is given by \begin{equation} h(r) = - \dfrac{(\det D)'}{n \det D}(r). \label{H-parallel} \end{equation}
Finally, we introduce the notion of isoparametric function. A non-constant smooth function $f : \widetilde{M}^{n+1} \longrightarrow \mathbb{R} $ is called isoparametric if the gradient and the Laplacian of $f$ satisfy $$\lvert \nabla f \lvert^{2} = a(f) \quad \mbox{and} \quad \Delta f = b(f),$$ where $a,b: I \subset \mathbb{R} \longrightarrow \mathbb{R}$ are smooth functions. The smooth hypersurfaces $\Sigma_{r} = f^{-1}(r)$ for $r$ regular value of $f$ are called isoparametric hypersurfaces. In this case, the unit normal vector field is given by $N = \frac{\nabla f}{\lvert \nabla f \rvert}$. We observe that, by the conditions under the gradient and the Laplacian given in the definition of an isoparametric function, $\Sigma_r$ has constant mean curvature for each $r$ (i.e., depending only on $r$) and $N$ is a geodesic field, see \cite{notas-miguel}.
\section{Proof of the main results}\label{sec3}
To prove Theorems \ref{theo1} and \ref{theo2}, we combine the techniques developed by F. Urbano \cite{s2s2Urbano}, J. Julio-Batalla \cite{s2r2Batalla}, and Domínguez-Vázquez and Manzano \cite{dominguez-manzano}.
\begin{proof}[Proof of Theorem~\ref{theo1}] Let $\Sigma$ be an isoparametric hypersurface in $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$ with $c_{1}\neq c_{2}$ and unit normal $N = N_{1} + N_{2}$. In order to prove Theorem~\ref{theo1}, it is enough to show that the principal curvatures of $\Sigma$ are constant if and only if the function $C$, given in \eqref{funC}, is constant. In fact, as $\lvert N_{1} \rvert^2 = \frac{1+C}{2}$ and $\lvert N_{2} \rvert^2 = \frac{1-C}{2}$, it follows that $\lvert N_{1} \rvert$ and $\lvert N_{2} \rvert$ are constant if and only if $C$ is constant.
Recall that the family of hypersurfaces parallel to $\Sigma$ in the direction of $N$ is given by \eqref{paralellhypersurfaces} and the parallel hypersurface at an oriented distance $r$ is denoted by $\Sigma_r$. We first observe that, since $\Sigma$ is isoparametric and the product structure $P$ is parallel, the function $C$, defined on the family of parallel hypersurfaces, does not depend on the displacement parameter $r$, once $N(C) = 0$. In fact, since $C = \langle PN, \, N \rangle$ and $\nabla_{N}N=0$, we have $$N(C)= \langle \nabla_{N}N, PN \rangle + \langle N, P\nabla_{N}N \rangle = 0.$$
Now we prove that $C$ is constant along $\Sigma$. Let us recall that $|C| \leq 1$. Consider the open set $$ U = \big\{ p \in \Sigma \mid C^2(p) < 1 \big\}. $$ We can assume that $U \neq \varnothing$, otherwise $C^2=1$ on $\Sigma$. In this case, let us take in $U$ the following orthonormal frame \begin{equation*} B = \biggr\{ B_{1} = \frac{X}{\sqrt{1 - C^2}}, B_{2} = \frac{J_{1}N + J_{2}N}{\sqrt{2(1+C)}}, B_{3} = \frac{J_{1}N - J_{2}N}{\sqrt{2(1-C)}} \biggr\}, \end{equation*} where $X = PN - CN.$
Given $p \in \Sigma$, let $\gamma_{p}$ be a geodesic of $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$ with $\gamma_p(0) = p$ and $\dot{\gamma}_p(0) = N_p$. By the definition of $\Sigma_r$ we have that $\dot{\gamma}_{q}(r)$ is a normal vector to $\Sigma_{r}$ at $\gamma_q(r)$. Thus, we can extend the unit normal $N$ to $U\times(-\epsilon,\epsilon)$ by $N_{\gamma_{q}(r)} = \dot{\gamma}_{q}(r)$, $q \in U$. Consequently, we also can extend the fields $B_{i}$.
Recall that a Jacobi field along $\gamma_{p}$ is a vector field $\xi$ satisfying the Jacobi equation $\xi'' + R(\xi,\dot{\gamma}_{p})\dot{\gamma}_{p} = 0$. For each $j \in \{1,2,3\}$, take the Jacobi field $\xi_{j}$ along $\gamma_{p}$ with the initial conditions \begin{equation}\label{jacobi-initialcondicions} \xi_{j}(0) = B_{j} \quad \mbox{and} \quad \xi_{j}'(0) = -AB_{j}, \end{equation} where $A$ is the shape operator of $\Sigma$ associated with $N$.
Since these initial conditions are orthogonal to $\dot{\gamma}_{p}(0)$, each Jacobi field $\xi_{j}$ is also orthogonal to $N_{\gamma_{p}(r)} = \dot{\gamma}_{p}(r)$ and, hence, it can be written as $$\xi_{j} = b_{1j}B_{1} + b_{2j}B_{2} + b_{3j}B_{3},$$ for certain smooth functions $b_{ij}$ on $(-\epsilon,\epsilon)$.
Let us observe that $\nabla_{N}B_{i}=0$, for all $i = 1,2,3$. In fact, since $N(C)=0$ and $P$ is parallel, we have $\nabla_N X=0$, which implies $\nabla_N B_1=0$. Furthermore, since $J_i$ is also parallel, for $i=1,\,2$, we conclude that $\nabla_N B_j=0$, $j=2,\,3.$ Thus, we have, on the one hand, \begin{equation}\label{jacobi-second-derivative} \xi_{j}'' = b_{1j}''B_{1} + b_{2j}''B_{2} + b_{3j}''B_{3}. \end{equation} On the other hand, if we denote by $R^{c_i}$ the curvature tensor of $\mathbb{Q}^{2}_{c_{i}}$, we get \begin{align*} \widetilde{R}(B_{1},N)N &= R^{c_1}(B_{1}^{c_{1}},N_{1})N_{1} + R^{c_2}(B_{1}^{c_{2}},N_{2})N_{2} \\ &= \frac{1}{8\sqrt{1 - C^2}}\biggr(R^{c_{1}}(X + PX,N + PN)(N + PN) \\ & \quad + R^{c_{2}}(X - PX,N - PN)(N - PN) \biggr) \\ &= 0, \end{align*} since $X + PX = (1 - C)(N + PN)$ and $X - PX = -(1 + C)(N - PN)$. Now, using the curvature tensor formula of a manifold of constant sectional curvature, we get \begin{align*} \widetilde{R}(B_{2},N)N &= \frac{c_{1}\lvert N + PN \rvert^{2}}{4}B_{2}, \\ \widetilde{R}(B_{3},N)N &= \frac{c_{2}\lvert N - PN \rvert^{2}}{4}B_{3}. \end{align*} Therefore, \begin{align}\label{curvaturetensorRN} \begin{split} \widetilde{R}(\xi_{j},\dot{\gamma}_{p})\dot{\gamma}_{p} &= \widetilde{R}(\xi_{j},N)N \\ &= b_{1j}R(B_{1},N)N + b_{2j}R(B_{2},N)N + b_{3j}R(B_{3},N)N \\ &= b_{2j}\frac{c_{1}\lvert N + PN \rvert^{2}}{4}B_{2} + b_{3j}\frac{c_{2}\lvert N - PN \rvert^{2}}{4}B_{3} \\ &= b_{2j}\frac{c_{1}(1 + C)}{2}B_{2} + b_{3j}\frac{c_{2}(1 - C)}{2}B_{3}. \end{split} \end{align} Since $\xi_{j}$ is a Jacobi field, we have from \eqref{jacobi-second-derivative} and \eqref{curvaturetensorRN} the following homogeneous linear system of ordinary differential equations \begin{equation}\label{systemequations} b_{1j}'' = 0, \quad b_{2j}'' + \delta_{1}b_{2j} = 0, \quad b_{3j}'' + \delta_{2}b_{3j} = 0, \end{equation} where $\delta_{1} = \frac{c_{1}(1 + C)}{2}$ and $\delta_{2} = \frac{c_{2}(1 - C)}{2}$.
In the sequence, we describe the initial conditions of the system \eqref{systemequations}. Firstly, as $\xi_{j}(0) = B_{j}$, we get \begin{align}\label{initialconditions-bij} \begin{array}{llll} &b_{11}(0) = 1, &b_{12}(0) = 0, &b_{13}(0) = 0,\\ &b_{21}(0) = 0, &b_{22}(0) = 1, &b_{23}(0) = 0,\\ &b_{31}(0) = 0, &b_{32}(0) = 0, &b_{33}(0) = 1. \end{array} \end{align}
Secondly, let the shape operator of $\Sigma$ be determined by the relations $A B_{i} = \sigma_{i1}B_{1} + \sigma_{i2}B_{2} + \sigma_{i3}B_{3}$, for certain smooth functions $\sigma_{ij}$. Since $A$ is symmetric, we have $\sigma_{12} = \sigma_{21}$, $\sigma_{13} = \sigma_{31}$ and $\sigma_{32} = \sigma_{23}$. Furthermore, taking into account that $\xi_{j}' = \widetilde{\nabla}_{N}\xi_{j} = -A \xi_{j}$, we obtain \begin{align}\label{initialconditions-b'ij} \begin{array}{llll} &b_{11}'(0) = -\sigma_{11}, &b_{12}'(0) = -\sigma_{21}, &b_{13}'(0) = -\sigma_{31},\\ &b_{21}'(0) = -\sigma_{12}, &b_{22}'(0) = -\sigma_{22}, &b_{23}'(0) = -\sigma_{23},\\ &b_{31}'(0) = -\sigma_{13}, &b_{32}'(0) = -\sigma_{23}, &b_{33}'(0) = -\sigma_{33}. \end{array} \end{align} With the initial conditions \eqref{initialconditions-bij} and \eqref{initialconditions-b'ij}, the solution of system \eqref{systemequations} is given by \begin{align}\label{solution-system} \begin{split} b_{11}(r) &= -\sigma_{11}r + 1, \\ b_{12}(r) &= -\sigma_{12}r, \\ b_{13}(r) &= -\sigma_{13}r, \\ b_{21}(r) &= -\sigma_{12}S_{\delta_{1}}(r), \\ b_{22}(r) &= -\sigma_{22}S_{\delta_{1}}(r) + C_{\delta_{1}}(r), \\ b_{23}(r) &= -\sigma_{32}S_{\delta_{1}}(r), \\ b_{31}(r) &= -\sigma_{13}S_{\delta_{2}}(r), \\ b_{32}(r) &= -\sigma_{32}S_{\delta_{2}}(r), \\ b_{33}(r) &= -\sigma_{33}S_{\delta_{2}}(r) + C_{\delta_{2}}(r), \end{split} \end{align}
where we consider the auxiliary functions \begin{equation*} S_{\delta_{i}}(r)=\begin{cases}\frac{1}{\sqrt{-\delta_{i}}}\sinh(r\sqrt{-\delta_{i}})&\text{if } \delta_{i} < 0,\\ \frac{1}{\sqrt{\delta_{i}}}\sin(r\sqrt{\delta_{i}})&\text{if } \delta_{i} > 0, \end{cases}\qquad C_{\delta_{i}}(r)=\begin{cases}\cosh(r\sqrt{-\delta_{i}})&\text{if } \delta_{i} < 0,\\ \cos(r\sqrt{\delta_{i}})&\text{if } \delta_{i} > 0. \end{cases} \end{equation*} for $i \in \{1,2\}$.
For every $r$, the shape operator $A_{r}$ of $\Sigma_{r}$ with respect to the normal $\gamma_{p}'(r)$ is given by \eqref{A-paralell}, where $D(r)$ is linear endomorphism of $T_{\gamma_{p}(r)}\Sigma_{r}$, determined by the relations \begin{equation*} D(r)B_{j}(\gamma_{p}(r)) = \xi_{j}(r), \quad D'(r)B_{j}(\gamma_{p}(r)) = \xi_{j}'(r). \end{equation*}
Considering the orthonormal basis $\{B_{1}(\gamma_{p}(r)), B_{2}(\gamma_{p}(r)), B_{3}(\gamma_{p}(r))\}$ of $T_{\gamma_{p}(r)}\Sigma_{r}$, the matrix form of the operator $D(r)$ is given by \begin{align}\label{matrix-operatorD} D(r) = \left( \begin{array}{ccc} b_{11}(r) & b_{12}(r) & b_{13}(r) \\ b_{21}(r) & b_{22}(r) & b_{23}(r) \\ b_{31}(r) & b_{32}(r) & b_{33}(r) \end{array} \right), \end{align}
From now on, our strategy is given as follows. Firstly, we are going to get explicitly the formulas of $\det D(r)$ and $\frac{d}{dr}(\det D(r))$ in terms of the functions $b_{ij}$ and its derivatives. Secondly, will apply such formulas to construct \begin{equation*} f(r) = \frac{d}{dr}(\det D(r)) + 3h(r) \det D(r), \end{equation*} which vanishes identically on $(-\epsilon, \epsilon)$, by equation \eqref{H-parallel}. Finally, we will use the fact that $f \equiv 0$ as well as its derivatives to obtain some algebraic relations between the components of $A$ on the basis $\left\{ B_i \right\}_{i=1}^3$ and the function $C$.
From \eqref{solution-system} and \eqref{matrix-operatorD}, we have that \begin{align*} \det D(r) &= A_{1}rS_{\delta_{1}}(r)S_{\delta_{2}}(r) + A_{2}rS_{\delta_{1}}(r)C_{\delta_{2}}(r) + A_{3}rS_{\delta_{2}}(r)C_{\delta_{1}}(r) \\ & \quad + A_{4}S_{\delta_{1}}(r)S_{\delta_{2}}(r) - \sigma_{11}rC_{\delta_{1}}(r)C_{\delta_{2}}(r) - \sigma_{22}S_{\delta_{1}}(r)C_{\delta_{2}}(r) \\ & \quad - \sigma_{33}S_{\delta_{2}}(r)C_{\delta_{1}}(r) + C_{\delta_{1}}(r)C_{\delta_{2}}(r), \end{align*} where \begin{align}\label{eq:Ai} \begin{array}{lll} & A_{1} = -\det A, & A_{2} = \sigma_{11}\sigma_{22} - \sigma^2_{12}, \\ & A_{3} = \sigma_{11}\sigma_{33} - \sigma^2_{13}, & A_{4} = \sigma_{22}\sigma_{33} - \sigma^2_{23}. \end{array} \end{align}
Now, taking into account that $S'_{\delta_{i}}(r) = C_{\delta_{i}}(r)$ and $C'_{\delta_{i}}(r) = -\delta_{i}S_{\delta_{i}}(r)$, we obtain \begin{align*} \frac{d}{dr}(\det D(r)) &= A_{1}\left(S_{\delta_{1}}(r)S_{\delta_{2}}(r) + rC_{\delta_{1}}(r)S_{\delta_{2}}(r) + rS_{\delta_{1}}(r)C_{\delta_{2}}(r)\right) \\ & \quad + A_{2}\left(S_{\delta_{1}}(r)C_{\delta_{2}}(r) + rC_{\delta_{1}}(r)C_{\delta_{2}}(r) - r\delta_{2}S_{\delta_{1}}(r)S_{\delta_{2}}(r)\right) \\ & \quad + A_{3}\left(S_{\delta_{2}}(r)C_{\delta_{1}}(r) + rC_{\delta_{2}}(r)C_{\delta_{1}}(r) - r\delta_{1}S_{\delta_{2}}(r)S_{\delta_{1}}(r)\right) \\ & \quad + A_{4}\left(C_{\delta_{1}}(r)S_{\delta_{2}}(r) + S_{\delta_{1}}(r)C_{\delta_{2}}(r)\right) \\ & \quad - \sigma_{11}\left(C_{\delta_{1}}(r)C_{\delta_{2}}(r) - r\delta_{1}S_{\delta_{1}}(r)C_{\delta_{2}}(r) - r\delta_{2}C_{\delta_{1}}(r)S_{\delta_{2}}(r)\right) \\ & \quad - \sigma_{22}\left(C_{\delta_{1}}(r)C_{\delta_{2}}(r) - \delta_{2}S_{\delta_{1}}(r)S_{\delta_{2}}(r)\right) \\ & \quad - \sigma_{33}\left(C_{\delta_{2}}(r)C_{\delta_{1}}(r) - \delta_{1}S_{\delta_{2}}(r)S_{\delta_{1}}(r)\right) \\ & \quad - \delta_{1}S_{\delta_{1}}(r)C_{\delta_{2}}(r) - \delta_{2}C_{\delta_{1}}(r)S_{\delta_{2}}(r). \end{align*} Thus, the function $f$ is given explicitly as \begin{align}\label{function-f} \begin{split} f(r) &= A_{1}\big(S_{\delta_{1}}(r)S_{\delta_{2}}(r) + rC_{\delta_{1}}(r)S_{\delta_{2}}(r) + rS_{\delta_{1}}(r)C_{\delta_{2}}(r) \\ & \quad + 3rh(r)S_{\delta_{1}}(r)S_{\delta_{2}}(r)\big) \\ & \quad + A_{2}\big(S_{\delta_{1}}(r)C_{\delta_{2}}(r) + rC_{\delta_{1}}(r)C_{\delta_{2}}(r) - r\delta_{2}S_{\delta_{1}}(r)S_{\delta_{2}}(r) \\ & \quad + 3rh(r)S_{\delta_{1}}(r)C_{\delta_{2}}(r)\big) \\ & \quad + A_{3}\big(S_{\delta_{2}}(r)C_{\delta_{1}}(r) + rC_{\delta_{2}}(r)C_{\delta_{1}}(r) - r\delta_{1}S_{\delta_{2}}(r)S_{\delta_{1}}(r) \\ & \quad + 3rh(r)S_{\delta_{2}}(r)C_{\delta_{1}}(r)\big) \\ & \quad + A_{4}\big(C_{\delta_{1}}(r)S_{\delta_{2}}(r) + S_{\delta_{1}}(r)C_{\delta_{2}}(r) + 3h(r)S_{\delta_{1}}(r)S_{\delta_{2}}(r)\big) \\ & \quad - \sigma_{11}\big(C_{\delta_{1}}(r)C_{\delta_{2}}(r) - r\delta_{1}S_{\delta_{1}}(r)C_{\delta_{2}}(r) - r\delta_{2}C_{\delta_{1}}(r)S_{\delta_{2}}(r) \\ & \quad + 3rh(r)C_{\delta_{1}}(r)C_{\delta_{2}}(r)\big) \\ & \quad - \sigma_{22}\big(C_{\delta_{1}}(r)C_{\delta_{2}}(r) - \delta_{2}S_{\delta_{1}}(r)S_{\delta_{2}}(r) + 3h(r)S_{\delta_{1}}(r)C_{\delta_{2}}(r)\big) \\ & \quad - \sigma_{33}\big(C_{\delta_{2}}(r)C_{\delta_{1}}(r) - \delta_{1}S_{\delta_{2}}(r)S_{\delta_{1}}(r) + 3h(r)S_{\delta_{2}}(r)C_{\delta_{1}}(r)\big) \\ & \quad - \delta_{1}S_{\delta_{1}}(r)C_{\delta_{2}}(r) - \delta_{2}C_{\delta_{1}}(r)S_{\delta_{2}}(r) + 3h(r)C_{\delta_{1}}(r)C_{\delta_{2}}(r). \end{split} \end{align}
As $f \equiv 0$, so is its derivative. Then, taking the derivative in \eqref{function-f} and applying at $r=0$, we obtain the following relation: \begin{align} 0 = f'(0) &= 2(A_{2} + A_{3} + A_{4}) - 9h^{2}(0) + 3h'(0) - (\delta_{1} + \delta_{2}), \label{eq:f'(0)} \end{align} where $h(0)$ is the mean curvature of $\Sigma$.
Note that $A_{i}$, $\delta_{i}$, $h (0)$ and $h'(0)$, depend only, in principle, of the base point $p \in \Sigma$. However, by assumption, $\Sigma$ is isoparametric and hence, $h (0)$ and $h'(0)$ are constants throughout $\Sigma$, that is, it is independent of the chosen base point $p \in \Sigma$ of normal geodesic $\gamma_{p}$.
Furthermore, observe that \begin{equation*} 9h^{2}(0) = \sigma^{2}_{11} + \sigma^{2}_{22} + \sigma^{2}_{33} +2(\sigma_{11}\sigma_{22} + \sigma_{11}\sigma_{33} + \sigma_{22}\sigma_{33}), \end{equation*} and \begin{equation*} tr (A^2) = \sigma^{2}_{11} + \sigma^{2}_{22} + \sigma^{2}_{33} + 2(\sigma^{2}_{12} + \sigma^{2}_{13} + \sigma^{2}_{23}). \end{equation*} Thus, by the definitions of the $A_i's$ in \eqref{eq:Ai}, we have $2(A_{2} + A_{3} + A_{4}) - 9h^{2}(0) = -tr (A^2)$. Substituting in \eqref{eq:f'(0)}, we get \begin{equation*} tr (A^2) = 3h'(0) - (\delta_{1} + \delta_{2}), \end{equation*} where $\delta_{1} + \delta_{2} = \frac{1}{2}(C(c_{1} - c_{2}) + c_{1} + c_{2})$.
Therefore, if $\Sigma$ has constant principal curvatures $\mu_{1}$, $\mu_{2}$, $\mu_{3}$, then $tr(A^2) = \mu_{1}^2 + \mu_{2}^2 + \mu_{3}^2$ is constant and hence, $C$ is constant, since $c_1 \neq c_2$.
Conversely, suppose $C$ is constant. Since the gradient of the function $C$ is given by $\nabla C = -2A(X)$ (see [Lemma $1$, \cite{s2s2Urbano}]), then $ A(X) = 0$. Therefore, $ \sigma_{1j} = \sigma_{j1} = 0$, for all $j = 1,2,3$. Thus, we have $A_{1} = A_{2} = A_{3} = 0$ and we can rewrite \eqref{eq:f'(0)} as \begin{align*} 0 = 2A_{4} - 9h^{2}(0) + 3h'(0) - (\delta_{1} + \delta_{2}), \label{rewrited-eq:f'(0)} \end{align*} and, as a consequence, we have that $A_{4}$ is constant.
Moreover, as $ \sigma_{1j} = \sigma_{j1} = 0$, the characteristic polynomial $Q_{A}$ of $A$ is given by $$Q_{A}(\lambda) = -\lambda^{3} + 3h(0)\lambda^{2} - A_{4}\lambda.$$ Therefore, since $A_{4}$ is constant, it follows that the principal curvatures of $\Sigma$ are constant. \end{proof}
\begin{proof}[Proof of Theorem~\ref{theo2}] Let $\Sigma$ be an isoparametric hypersurface in $\mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$ with constant principal curvatures. By Theorem~\ref{theo1}, we have that $C$ is constant. If $C=1$ we have $PN = N$, and thus, $N = (N_{1},0)$. If $C=-1$ we have $PN = -N$, and then, $N = (0, N_{2})$. In such cases, $\Sigma$ is an open subset of $\mathcal{C}^{1}(\kappa_{j})\times \mathbb{Q}^{2}_{c_{2}}$ or $\mathbb{Q}^{2}_{c_{1}} \times \mathcal{C}^{1}(\kappa_{j})$, respectively, where $\mathcal{C}^{1}(\kappa_{j})$ is a complete curve in $\mathbb{Q}^{2}_{c_{j}}$ of constant geodesic curvature $\kappa_{j}$. In fact, let us suppose that $N=(N_1,0)$, then $\Sigma$ is an open subset of $\mathcal{C}^{1}\times \mathbb{Q}^{2}_{c_{2}}$, where $\mathcal{C}^{1}$ is a regular curve in $\mathbb{Q}^{2}_{c_1}$. Let $\psi$ be a parametrization by arc length of $\mathcal{C}^{1}$, with unit normal vector $n_{\psi} = \pm N_1$. Let $\{e_1,e_2,e_3\}$ a orthonormal frame in $\mathcal{C}^{1}\times \mathbb{Q}^{2}_{c_{2}}$, with $e_1 = \psi'$ and $\{e_2,e_3\}$ an orthonormal basis in $\mathbb{Q}^{2}_{c_{2}}$. If we denote the shape operator of $\Sigma$ by $A$, considering without loss of generality that $N_{1}= n_{\psi}$, we have \begin{align*} Ae_1 &= -\widetilde{\nabla}_{e_1}N_1 = -\widetilde{\nabla}^{\mathbb{Q}^{2}_{c_{1}}}_{\psi'}n_{\psi} = \kappa_{j}\psi' = \kappa_{j}e_1, \\ Ae_2 &= -\widetilde{\nabla}_{e_2}N_1 = 0, \\ Ae_3 &= -\widetilde{\nabla}_{e_3}N_1 = 0. \end{align*} Therefore, the curvature $\kappa_j$ of $\mathcal{C}^1$ is a principal curvature of $\Sigma$, which implies that $\kappa_j$ is constant. The case where $N=(0,N_2)$ is analogous.
In the sequence, we are going to prove that, if $|C|<1$, the only remaining possibility is the case when one $c_i$ is negative. Therefore, in what follows, let us assume that $ C \in (-1,1)$. In this case, as in the proof of Theorem \ref{theo1}, let us consider the frame \begin{equation*} B = \left\{ B_{1} = \frac{X}{\sqrt{1 - C^2}}, B_{2} = \frac{J_{1}N + J_{2}N}{\sqrt{2(1+C)}}, B_{3} = \frac{J_{1}N - J_{2}N}{\sqrt{2(1-C)}} \right\}, \end{equation*} and the function $f$ given in \eqref{function-f}. Again, taking derivatives in \eqref{function-f} and applying them at $r=0$, we obtain the following relations: \begin{align}
0 = f'(0) &= 2(A_{2} + A_{3} + A_{4}) - 9h^{2}(0) + 3h'(0) - (\delta_{1} + \delta_{2}), \label{eq:f'(0)1}\\ 0 = f''(0) &= 6A_{1} + 6h(0)(A_{2} + A_{3} + A_{4}) - 18h'(0)h(0) + 2\sigma_{11}(\delta_{1} + \delta_{2}) \label{eq:f''(0)1} \\ & \quad + 2\sigma_{22}\delta_{2} + 2\sigma_{33}\delta_{1} + 3h''(0), \nonumber
\end{align} where the functions $A_i$, $i=1,\dots,4$, are given in \eqref{eq:Ai}.
Let us recall that as $C$ is constant we have $\sigma_{1i}=\sigma_{i1}=0$, which imply that $A_{1} = A_{2} = A_{3} = 0$. Moreover, since $h(0)$ is the mean curvature of $\Sigma$, we also conclude that \begin{equation}\label{eq:h(0)} 3h(0) = \sigma_{22} + \sigma_{33}. \end{equation} Thus, we can rewrite \eqref{eq:f'(0)1} and \eqref{eq:f''(0)1} as follows: \begin{align} 0 &= 2(\sigma_{22}\sigma_{33} - \sigma^{2}_{23}) - 9h^{2}(0) + 3h'(0) - (\delta_{1} + \delta_{2}), \label{rewrited-eq:f'(0)2}\\ 0 &= 6h(0)(\sigma_{22}\sigma_{33} - \sigma^{2}_{23}) - 18h'(0)h(0) + 2\sigma_{22}\delta_{2} + 2\sigma_{33}\delta_{1} + 3h''(0). \label{rewrited-eq:f''(0)2} \end{align} Combining \eqref{eq:h(0)}, \eqref{rewrited-eq:f'(0)2} and \eqref{rewrited-eq:f''(0)2}, we have that \begin{equation*} 2\sigma_{33}(\delta_{1} - \delta_{2}) + 3h(0)(\delta_{1} + \delta_{2}) + 6h(0)\delta_{2} + 27h^{3}(0) - 27h'(0)h(0) + 3h''(0) = 0. \end{equation*} Note that $(\delta_{1} - \delta_{2}) = \frac{1}{2}(c_1 - c_2 + C(c_1 + c_2)) \neq 0$, since $C \in (-1,1)$ and $c_1 \neq c_2$. Therefore $\sigma_{33}$ is constant and hence, from \eqref{eq:h(0)} and \eqref{rewrited-eq:f'(0)2}, we have that $\sigma_{22}$ and $\sigma_{23}$ are also constant.
On the other hand, we are going to use Codazzi equation to compute $X(\sigma_{22})$, $X(\sigma_{23})$ and $X(\sigma_{33})$. As each $J_{i}$ is parallel and $A(X) = 0$ (since $\nabla C = -2A(X)$), we have $\nabla_{X}B_{j} = 0$ for all $j = 1,2,3$. In this way, since \begin{equation*} X(\sigma_{ij}) = X(A(B_{i},B_{j})) = \nabla A(X,B_{i},B_{j}) \end{equation*} it follows from the Codazzi equation that \begin{align*} X(\sigma_{22}) &= \nabla A(B_{2},X,B_{2}) + \frac{c_{1}}{4}\{\langle X,X \rangle \langle PB_{2} + B_{2},B_{2}\rangle -\langle B_{2},X \rangle \langle PX + X,B_{2}\rangle\} \\ & \quad + \frac{c_{2}}{4}\{\langle X,X \rangle \langle PB_{2} - B_{2},B_{2}\rangle -\langle B_{2},X \rangle \langle PX - X,B_{2}\rangle\} \\ &= -C\langle AB_{2},AB_{2}\rangle + \langle PAB_{2},AB_{2}\rangle + \frac{c_{1}\lvert X \rvert^2}{2} \\ &= \frac{c_{1}(1 - C^2)}{2} + (1 - C)\sigma_{22}^2 - (1 + C)\sigma_{23}^2, \end{align*} \begin{align*} X(\sigma_{23}) &= \nabla A(B_{2},X,B_{3}) + \frac{c_{1}}{4}\{\langle X,X \rangle \langle PB_{2} + B_{2},B_{3}\rangle -\langle B_{2},X \rangle \langle PX + X,B_{3}\rangle\} \\ & \quad + \frac{c_{2}}{4}\{\langle X,X \rangle \langle PB_{2} - B_{2},B_{3}\rangle -\langle B_{2},X \rangle \langle PX - X,B_{3}\rangle\} \\ &= -C\langle AB_{2},AB_{3}\rangle + \langle PAB_{2},AB_{3}\rangle \\ &= (1 - C)\sigma_{22}\sigma_{23} - (1 + C)\sigma_{23}\sigma_{33}, \end{align*} \begin{align*} X(\sigma_{33}) &= \nabla A(B_{3},X,B_{3}) + \frac{c_{1}}{4}\{\langle X,X \rangle \langle PB_{3} + B_{3},B_{3}\rangle -\langle B_{3},X \rangle \langle PX + X,B_{3}\rangle\} \\ & \quad + \frac{c_{2}}{4}\{\langle X,X \rangle \langle PB_{3} - B_{3},B_{3}\rangle -\langle B_{3},X \rangle \langle PX - X,B_{3}\rangle\} \\ &= -C\langle AB_{3},AB_{3}\rangle + \langle PAB_{3},AB_{3}\rangle - \frac{c_{2}\lvert X \rvert^2}{2} \\ &= \frac{c_{2}(C^2 - 1)}{2} + (1 - C)\sigma_{23}^2 - (1 + C)\sigma_{33}^2. \end{align*} Therefore, \begin{align} \frac{c_{1}(1 - C^2)}{2} + (1 - C)\sigma_{22}^2 - (1 + C)\sigma_{23}^2 &= 0,\label{eq:X(sigma_{22})=0} \\ \frac{c_{2}(C^2 - 1)}{2} + (1 - C)\sigma_{23}^2 - (1 + C)\sigma_{33}^2 &= 0,\label{eq:X(sigma_{33})=0} \\ (1 - C)\sigma_{22}\sigma_{23} - (1 + C)\sigma_{23}\sigma_{33} &= 0.\label{eq:X(sigma_{23})=0} \end{align}
Let us show that $\sigma_{23}=0$. Suppose by contradiction that $\sigma_{23} \neq 0$. From \eqref{eq:X(sigma_{23})=0}, we have \begin{equation}\label{eq3.20} (1 - C)^2\sigma_{22}^2 - (1 + C)^2\sigma_{33}^2 = 0. \end{equation} Now, multiplying \eqref{eq:X(sigma_{22})=0} by $1 - C$ and \eqref{eq:X(sigma_{33})=0} by $1 + C$, we have \begin{align} \frac{c_{1}(1 - C)(1 - C^2)}{2} + (1 - C)^2\sigma_{22}^2 - (1 - C^2)\sigma_{23}^2 &= 0, \label{eq3.21} \\ \frac{c_{2}(1 + C)(C^2 - 1)}{2} + (1 - C^2)\sigma_{23}^2 - (1 + C)^2\sigma_{33}^2 &= 0. \label{eq3.22} \end{align} Adding \eqref{eq3.21} to \eqref{eq3.22} and using \eqref{eq3.20}, we get \begin{equation} c_{1}(1 - C) = c_{2}(1 + C), \end{equation} Since $C \in (-1,1)$ and $c_{1} \neq c_{2}$, we have a contradiction. Therefore $\sigma_{23} = 0$.
If $\sigma_{23} = 0$, the system given by equations \eqref{eq:X(sigma_{22})=0}, \eqref{eq:X(sigma_{33})=0} and \eqref{eq:X(sigma_{23})=0} is reduced to \begin{equation} \sigma_{22}^2 = -\frac{c_{1}(1 + C)}{2}, \quad \sigma_{33}^2 = -\frac{c_{2}(1 - C)}{2}. \label{eq:reduced-system} \end{equation}
Observe that the only possibility of solving \eqref{eq:reduced-system} is to consider that one $c_{i}$ is negative and the other is zero. Then, without loss of generality, let us assume from now on that $c_{1} = -1$ and $c_{2} = 0$. Thus, the previous computation shows us that $\sigma_{ij}=0$, for $i \neq j$ and $\sigma_{11}=\sigma_{33}=0.$ Therefore, we conclude that $\{B_{1},B_{2},B_{3}\}$ must be a frame of principal directions of $\Sigma$, with principal curvatures \begin{align*} \mu_{1} = 0, \quad \mu_{2} = \pm \sqrt{\frac{1 + C}{2}}, \quad \mu_{3} = 0. \end{align*}
In what follows, we consider the case when $\mu_{2} = \sqrt{\frac{1 + C}{2}}$. The shape operator $A$ and the tangential component of the product structure $P^T$ are given, with respect to the frame $B$, respectively by \begin{align*} A = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & \sigma_{22} & 0 \\ 0 & 0 & 0 \end{array} \right),\quad P^T = \left( \begin{array}{ccc} -C & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array} \right). \end{align*}
Since $P$ and $J_i$ are parallel, we have that the Levi-Civita connection $\widetilde{\nabla}$ of $\mathbb{H}^2\times\mathbb{R}^2$ is given by \begin{align*} \begin{array}{llll} & \widetilde{\nabla}_{B_1}B_i = 0, & \widetilde{\nabla}_{B_2}B_3 = 0, & \widetilde{\nabla}_{B_3}B_2 = 0, \\ & \widetilde{\nabla}_{B_2}B_1 = - \sqrt{\frac{1 - C}{2}}B_{2}, & \widetilde{\nabla}_{B_2}B_2 = \frac{PN + N}{\sqrt{2(1 + C)}}, & \widetilde{\nabla}_{B_3}B_1 = 0, \\ &\widetilde{\nabla}_{B_3}B_3 = 0. \end{array} \end{align*} Note that $[B_1,B_3] = [B_2,B_3] = 0$. Now, let $\lambda$ a function such that \begin{equation*} B_1(\lambda) = -\lambda\sqrt{\frac{1 - C}{2}}, \quad B_2(\lambda) = 0 \quad \text{and} \quad B_3(\lambda) = 0. \end{equation*} In this way, we have \begin{align}\label{eq3.24} [B_1,\lambda B_2] = \biggr(B_{1}(\lambda) + \lambda\sqrt{\frac{1 - C}{2}}\biggr)B_{2} = 0 \end{align}
and $[\lambda B_2, B_3 ] = 0$. Therefore, there is a parametrization $\Psi: \Omega \subset \mathbb{R}^3 \longrightarrow \Sigma$, where $\Omega$ is an open subset of $\mathbb{R}^3$ with coordinates $(t,\,u,\,v)$, such that \begin{align*} \Psi_{t} = B_{1}, \quad \Psi_{u} = \lambda B_{2} \quad \text{and} \quad \Psi_{v} = B_{3}. \end{align*} Now, we are going to construct the parametrization $\Psi$. Since $\Psi_{v} = B_{3}$, $B_{3}$ has no component in $\mathbb{H}^2$, and $\widetilde{\nabla}_{B_3}B_3 = 0$, i.e., $B_3$ is a geodesic field of $\mathbb{H}^2 \times \mathbb{R}^2$, when we integrate it with respect to $v$, we have \begin{equation*} \Psi = \biggr(\Psi^{\mathbb{H}^2}(t,u), \beta(t,u) + B_{3}v\biggr), \end{equation*} where $\Psi^{\mathbb{H}^2}$ is the component of $\Psi$ in $\mathbb{H}^2$.
Before integrating with respect to the variable $u$, we first observe that $B_2$ has no component in $\mathbb{R}^2$ and $$ \widetilde{\nabla}_{B_2}B_2 = \nabla^{\mathbb{H}^2}_{B_2^{\mathbb{H}^2}}B_2^{\mathbb{H}^2} = \frac{PN + N}{\sqrt{2(1 + C)}}. $$ Therefore, \begin{align*} \langle \nabla^{\mathbb{H}^2}_{B_2^{\mathbb{H}^2}}B_2^{\mathbb{H}^2}, \nabla^{\mathbb{H}^2}_{B_2^{\mathbb{H}^2}}B_2^{\mathbb{H}^2} \rangle &= \frac{1}{2(1 + C)}\biggr(2\langle PN,N \rangle + \langle PN,PN \rangle + \langle N,N \rangle\biggr)\\ &=\frac{1}{2(1 + C)}(2C +2) = 1, \end{align*} that is, if $\varphi$ is a curve parametrized by arc length, with $\varphi' = B_2^{\mathbb{H}^2}$, then the geodesic curvature $k_g$ of $\varphi$ is $k_g = 1$, and hence $\varphi$ is a horocycle. Up to rigid motions, $\varphi$ is given by \begin{align*} \varphi(u) = \biggr( 1 + \frac{u^2}{2}, u, -\frac{u^2}{2}\biggr). \end{align*}
As $\Psi_{u} = \biggr(\Psi_{u}^{\mathbb{H}^2}, \beta_{u}\biggr)= \lambda B_{2} $, it follows that $\beta$ does not depend on $u$. Thus, $\Psi_{u}^{\mathbb{H}^2} = \lambda B_{2}= \lambda(t) \left( u,\,1,\,-u,\, 0,\,0 \right)$, once $B_2(\lambda)=B_3(\lambda)=0.$ When we integrate $\Psi_{u}^{\mathbb{H}^2}$ with respect to $u$, we have \begin{equation*} \Psi^{\mathbb{H}^2}(t,u) = \lambda(t)\biggr(\frac{u^2}{2},u,-\frac{u^2}{2}\biggr) + \Lambda(t), \end{equation*} where $\Lambda(t)$ is a smooth curve in $\mathbb{H}^2$. Hence, \begin{equation}\label{parametrization} \Psi(t,u,v) = \biggr(\lambda(t)\alpha(u) + \Lambda(t), \beta(t) + B_{3}v\biggr), \end{equation} with $\alpha(u) = \big(\frac{u^2}{2},u,-\frac{u^2}{2}\big)$.
Finally, we integrate $B_1 = \Psi_t = \biggr(\lambda'(t)\alpha(u) + \Lambda'(t), \beta'(t)\biggr)$. Since $\widetilde{\nabla}_{B_1}B_1 = 0$, $B_1$ is also a geodesic field of $\mathbb{H}^2 \times \mathbb{R}^2$. Therefore, $\beta(t) = p_{0} + V_0t$. Considering $\gamma(t) = \lambda(t)\alpha(u) + \Lambda(t)$, we have $\Psi_{t} = \biggr(\gamma'(t), V_0\biggr) = B_{1}$, with $V_0=B_{1}^{\mathbb{R}^2}$. It follows by the definition of $B_1$ that $\lVert B_{1}^{\mathbb{R}^2} \rVert = \sqrt{\frac{1 + C}{2}}$. As $\lVert \gamma' \rVert^2 + \lVert B_{1}^{\mathbb{R}^2} \rVert^2 = 1 $, we get $\lVert \gamma' \rVert = \lVert B_{1}^{\mathbb{H}^2} \rVert = \sqrt{\frac{1 - C}{2}} $.
Note that \begin{align*} \frac{D\gamma'}{dt} &= \frac{d\gamma'}{dt} - \lVert B_{1}^{\mathbb{H}^2} \rVert^2\gamma \\ &=\alpha(u)\biggr(\lambda''(t) - \lVert B_{1}^{\mathbb{H}^2} \rVert^2\lambda(t)\biggr) + \Lambda''(t) - \lVert B_{1}^{\mathbb{H}^2} \rVert^2\Lambda(t). \end{align*} Since $\gamma$ is a geodesic in $\mathbb{H}^2$, we have that \begin{align*} \lambda''(t) - \lVert B_{1}^{\mathbb{H}^2} \rVert^2\lambda(t) = 0 \quad \text{and} \quad \Lambda''(t) - \lVert B_{1}^{\mathbb{H}^2} \rVert^2\Lambda(t) = 0, \end{align*} and hence $\lambda(t)$ and $\Lambda(t)$ are given by \begin{align*} \lambda(t) &= b_{1}\cosh(r t) + b_{2}\sinh(r t), \\ \Lambda(t) &= V_{1}\cosh(r t) + V_{2}\sinh(r t), \end{align*} where $r = \pm \lVert B_{1}^{\mathbb{H}^2} \rVert$, $b_{i}$ are real constants and $V_{i}$ orthonormal vectors. If $\Lambda = (\Lambda_{1}, \Lambda_{2}, \Lambda_{3})$, using $\langle \gamma,\gamma\rangle = -1$, we obtain the following polynomial equation in $u$: $$ (\lambda - (\Lambda_{1} + \Lambda_{3}))u^2 + 2\Lambda_{2}u = 0, $$ that is, $$\lambda - (\Lambda_{1} + \Lambda_{3}) = 0 \quad \text{and} \quad \Lambda_{2}=0.$$ Therefore, if $V_{1} = (v_{11},v_{12},v_{13})$ and $V_{2} = (v_{21},v_{22},v_{23})$, we have $v_{12}=v_{22} = 0$, $b_{1} = v_{11} + v_{13}$ and $b_{1} = v_{21} + v_{23}$. Now, writing $V_{1} = (\cosh(a_{1}),0,\sinh(a_{1}))$ and $V_{2} = (\sinh(a_{1}),0,\cosh(a_{1}))$, we get $b_{1}=b_{2}=e^{a_1}$. Thus, we conclude that \begin{align*} \begin{split} \lambda(t) &= e^{r t}, \\ \Lambda(t) &= \biggr(\cosh(r t),0,\sinh(r t)\biggr). \end{split} \end{align*}
From \eqref{eq3.24}, it follows that $$r + \sqrt{\dfrac{1 - C}{2}} = 0.$$ Thus, we obtain that $r = -\lVert B_{1}^{\mathbb{H}^2} \rVert$, and therefore \begin{align} \begin{split} \lambda(t) &= e^{-\lVert B_{1}^{\mathbb{H}^2} \rVert t}, \\ \Lambda(t) &= \biggr(\cosh(-\lVert B_{1}^{\mathbb{H}^2} \rVert t),0,\sinh(-\lVert B_{1}^{\mathbb{H}^2} \rVert t)\biggr). \end{split}\label{eq3.26} \end{align}
Writing $b = ||B^{\mathbb{H}^2}_{1}||=\sqrt{1-||B^{\mathbb{R}^2}_{1}||^2} = \sqrt{1-||V_0||^2}$ and $W_0 = B_{3}$, when we replace \eqref{eq3.26} in \eqref{parametrization}, we obtain the parametrization \eqref{eq:parametrization-Psi}.
For the converse, suppose that $\Sigma$ is parametrized by \eqref{eq:parametrization-Psi}. Since \begin{align*} \Psi_{t} &= -b \biggr( e^{-b t}(\alpha(u),\vec{0})+ \Big(\sinh(-b t), 0, \cosh(-b t), -\frac{V_0}{b} \Big)\biggr),\\ \Psi_{u} &= e^{-b t}(\alpha'(u),\vec{0}), \\ \Psi_{v} &= \Big(\vec{0},W_0 \Big), \end{align*} we conclude that a unit normal vector field $N$ to $\Sigma$ is given by \begin{equation*}
N = -\lVert V_0\rVert \biggr( e^{-b t}(\alpha(u),\vec{0})+ \Big(\sinh(-b t), 0, \cosh(-b t)\Big), \frac{b}{\lVert V_0\rVert^2}V_0 \biggr).
\end{equation*}
Denoting by $\Tilde{D}$ the covariant derivative in $\mathbb{L}^3$, we obtain \begin{align*} \Tilde{D}_{\Psi_{t}}N &= b\lVert V_0\rVert \biggr( e^{-b t}\alpha(u) + \Big(\cosh(-b t), 0, \sinh(-b t) \Big), \vec{0} \biggr)\\ &= b\lVert V_0\rVert \Psi^{\mathbb{H}^2},\\ \Tilde{D}_{\Psi_{u}}N &= -\lVert V_0\rVert e^{-b t}(\alpha'(u),\vec{0})\\ &=-\lVert V_0\rVert \Psi_{u},\\ \Tilde{D}_{\Psi_{v}}N &= 0. \end{align*} It follows immediately from the derivatives above and the parametrization $\Psi$ that $$\langle \Tilde{D}_{\Psi_{u}}N,\Psi^{\mathbb{H}^2} \rangle = \langle \Tilde{D}_{\Psi_{v}}N,\Psi^{\mathbb{H}^2} \rangle = 0 \quad \text{and}\quad \langle \Tilde{D}_{\Psi_{t}}N,\Psi^{\mathbb{H}^2} \rangle = -b\lVert V_0\rVert.$$ Therefore, since $\widetilde{\nabla}_{V}W = \Tilde{D}_{V}W + \langle \Tilde{D}_{V}W, \Psi^{\mathbb{H}^2}\rangle \Psi^{\mathbb{H}^2},$ we get \begin{align*} \widetilde{\nabla}_{\Psi_{t}}N &= 0,\\ \widetilde{\nabla}_{\Psi_{u}}N &= -\lVert V_0\rVert \Psi_{u},\\ \widetilde{\nabla}_{\Psi_{v}}N &= 0, \end{align*} that is, $\Sigma$ has principal curvatures $\mu_{1} = 0, \, \mu_{2} = \lVert V_0\rVert$ and $\mu_{3} = 0.$ Finally, since \begin{equation*}
PN = -\lVert V_0\rVert \biggr( e^{-b t}(\alpha(u),\vec{0})+ \Big(\sinh(-b t), 0, \cosh(-b t)\Big), -\frac{b}{\lVert V_0\rVert^2}V_0 \biggr)
\end{equation*}
and $b=\sqrt{1-||V_0||^2}$, it follows that \begin{align*} C &= \langle PN,N \rangle \\ &= \lVert V_0\rVert^2\biggr( e^{-2b t}u^2 + e^{-b t}\Big(-\dfrac{u^2}{2}\sinh(-bt) - \dfrac{u^2}{2}\cosh(-bt)\Big) \\ & \quad \quad \quad \quad -\sinh^2(-bt) + \cosh^2(-bt) - \dfrac{b^2}{\lVert V_0\rVert^2}\biggr) \\ &= \lVert V_0\rVert^2\biggr( 1 - \dfrac{b^2}{\lVert V_0\rVert^2}\biggr)\\ &=2\lVert V_0\rVert^2 - 1, \end{align*} that is, $\lVert V_0\rVert = \sqrt{\dfrac{1 + C}{2}}$. Thus, we conclude the proof of the theorem. \end{proof}
\begin{remark} \label{rmk:geometry-psi} Following the notation established in the proof of Theorem \ref{theo2}, let us provide a geometric description of the hypersurface given by the parametrization $\Psi$. Note that a unit normal vector to the horocycle \begin{align*} \varphi(u) = \biggr( 1 + \frac{u^2}{2}, u, -\frac{u^2}{2}\biggr), \end{align*} is given by \begin{align*} n(u) = \biggr( \frac{u^2}{2}, u, 1 -\frac{u^2}{2}\biggr). \end{align*}
Fixing $u, \, v \in \mathbb{R}$, let us consider in $\mathbb{H}^2~\times~\mathbb{R}^2$ the following geodesic parametrized by arc length \begin{align*} \gamma(t) = \Big(\cosh(\omega t)\varphi(u) + \sinh(\omega t)n(u), g(v) + V_{0}t\Big), \end{align*} where $g(v)=p_{0} + W_{0}v$ is a geodesic in $\mathbb{R}^2$ with normal vector $V_{0}$. Since \begin{align*} \gamma'(t) = \Big(\omega\sinh(\omega t)\varphi(u) + \omega\cosh(\omega t)n(u), V_{0}\Big), \end{align*} it follows that \begin{equation*}
1 = ||\gamma'(t)||^2 = \omega^2 + ||V_{0}||^2, \end{equation*}
which implies $\omega = \pm \sqrt{1 - ||V_{0}||^2} = \pm b$. Considering $\omega = -b$, we get \begin{align*} \gamma(t) &= e^{-b\, t}(\alpha(u),\vec{0})+ \Big(\cosh(-b\, t), 0, \sinh(-b\, t), V_0t \Big) \\ & \quad + \Big(\vec{0}, p_0 + W_0 v \Big). \end{align*}
Varying the parameters $(t,u,v) \in \mathbb{R}^3$, the construction above provides exactly the parametrization $\Psi$. Therefore, the hypersurface $\Psi(\mathbb{R}^3$) is a family of geodesically parallel surfaces of $\mathbb{H}^2~\times~\mathbb{R}^2$, given by products of horocycles in $\mathbb{H}^2$ and straight lines in $\mathbb{R}^2$.
\end{remark}
\end{document} | arXiv |
\begin{document}
\title{Relative Property (T) and Linear Groups}
\begin{abstract}
Relative property (T) has recently been used to construct a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group $\Gamma$ admits a $\mathbb{R}$-special linear representation with non-amenable $\mathbb{R}$-Zariski closure if and only if it acts on an Abelian group $A$ (of finite nonzero $\mathbb{Q}$-rank) so that the corresponding group pair $(\Gamma \ltimes A, A)$ has relative property (T).
The proof is constructive. The main ingredients are Furstenberg's celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.
\end{abstract}
\section{Introduction}
Recall that if $\Gamma$ is a group and $A \leqslant \Gamma$ is a closed subgroup then the group pair $(\Gamma, A)$ is said to have relative property (T) if every unitary representation of $\Gamma$ with almost invariant vectors has $A$-invariant vectors. And $\Gamma$ is said to have property (T) if $(\Gamma, \Gamma)$ has relative property (T)
\footnote{We will assume throughout this paper that groups are locally compact and second countable, Hilbert spaces are separable, unitary representations are strongly continuous (in the usual sense), fields are of characteristic 0, and local fields are not discrete. Furthermore, all countable groups will be given the discrete topology, unless otherwise specified. }.
In 1967 D. Kazhdan used the relative property (T) of the group pair $(\sdp{\SL{2}{\mathbb{K}}}{\mathbb{K}^2}, \mathbb{K}^2)$ to show that $\SL{3}{\mathbb{K}}$ has property (T), for any local field $\mathbb{K}$ \cite[Lemmas 2 \cdot 3]{Kazhdan}. Later in 1973 G. A. Margulis used the relative property (T) of $(\sdp{\SL{2}{\mathbb{Z}}}{\mathbb{Z}^2}, \mathbb{Z}^2)$ \cite[Lemma 3.18]{Margulis} in order to construct the first explicit examples of families of expander graphs. It was he who later coined the term.
Recently relative property (T) has been used to construct a variety of new phenomena. Most notable is the recent work of S.Popa. He has shown that every countable subgroup of $\mathbb{R}_+^*$ is the fundamental group of some II$_1$-factor \cite{Popa3}, and constructed examples of II$_1$ factors with rigid Cartan subalgebra inclusion \cite{Popa1}. Also D. Gaboriau with S. Popa constructed uncountably many non-orbit equivalent (free and ergodic measure-preserving) actions of the free group $F_n$ (for $n \geqslant 2$) on the standard probability space. See \cite{PopaGab} and \cite{Popa2} and the references contained therein.
In a completely different direction, A. Navas, extending his previous work with property (T) groups, showed that relative property (T) group pairs acting on the circle by $C^2$ diffeomorphisms are trivial, in a suitable sense \cite{Navas}. Also, M. Kassabov and N. Nikolov \cite[Theorem 3]{KasNik} used relative property (T) to show that $\SL{n}{\mathbb{Z}[x_1, \dots, x_k]}$ has property ($\tau$) for $n \geqslant 3$.
We also refer to A. Valette's paper \cite{Valette} for more applications concerning, for example, the Baum-Connes conjecture.
Unfortunately, until recently the examples of group pairs with relative property (T) available in the literature have been scarce:
\begin{itemize}
\item If $n \geqslant 2$ then $(\sdp{\SL{n}{\mathbb{R}}}{\mathbb{R}^n}, \mathbb{R}^n)$ and $(\sdp{\SL{n}{\mathbb{Z}}}{\mathbb{Z}^n}, \mathbb{Z}^n)$ have relative property (T). \cite[10-Proposition]{HarpeValette}
\item If $\Gamma \leqslant \SL{2}{\mathbb{Z}}$ is not virtually cyclic then $(\sdp{\Gamma}{\mathbb{Z}^2}, \mathbb{Z}^2)$ has relative property (T). \cite[Example 2 Section 5]{Burger}
\item And, now, in a recent preprint of A. Valette \cite{Valette}: If $\Gamma$ is an arithmetic lattice in an absolutely simple Lie group then there exists a homomorphism $\Gamma \to \SL{N}{\mathbb{Z}}$ such that the corresponding pair $(\sdp{\Gamma}{\mathbb{Z}^N}, \mathbb{Z}^N)$ has relative property (T).
\end{itemize}
We remark that $\sdp{\SL{n}{\mathbb{R}}}{\mathbb{R}^n}$ actually has property (T) for $n \geqslant 3$ \cite{Wang} and so $(\sdp{\SL{n}{\mathbb{R}}}{\mathbb{R}^n}, A)$ has relative property (T) for any closed $A \leqslant \sdp{\SL{n}{\mathbb{R}}}{\mathbb{R}^n}$. Indeed, if $A \leqslant G \leqslant H$ are groups, and $G$ has property (T) then $(H,A)$ has relative property (T).
On the other extreme, if $S$ is an amenable group then $(S,A)$ has relative property (T) if and only if $A$ is compact. (See Lemma 8.3 in Section 8.) So, if one wants to find new examples of group pairs with relative property (T), they should not rely on the property (T) on one of the groups in question and they should be of the form $(\Gamma, A)$ where $\Gamma$ is non-amenable and $A$ is amenable but not compact.
Using these examples as a guide, one may ask to what extent can group pairs with relative property (T) be constructed? We offer the following as an answer to this question:
\begin{theorem} Let $\Gamma$ be a finitely generated group. The following are equivalent: \begin{enumerate}
\item There exists a homomorphism $\varphi : \Gamma \to \SL{n}{\mathbb{R}}$ such that the $\mathbb{R}$-Zariski-closure $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is non-amenable.
\item There exists an Abelian group $A$ of nonzero finite $\mathbb{Q}$-rank and a homomorphism $\varphi' \cdot \Gamma \to \text{Aut}(A)$ such that the corresponding group pair $(\sdp{\Gamma}{_{\varphi'}A}, A)$ has relative property (T). \end{enumerate} \end{theorem}
\textbf{Remark:} In the direction of $(1) \implies (2)$, more information can be given. Namely, we will specifically find that $A = \mathbb{Z}[S^{-1}]^N$ where $S$ is some finite set of rational primes, as is pointed out below. Also in the direction of $(2) \implies (1)$ we will find that $A$ can be taken to be of the form $\mathbb{Z}[S^{-1}]^N$.
\break \subsection{Outline of the proof of Theorem 1 in the direction $(1) \implies (2)$}
\subsubsection{From Transcendental to Arithmetic}
This step is a matter of showing that from an arbitrary representation $\varphi \cdot \Gamma\to \SL{n}{\mathbb{R}}$, such that the $\mathbb{R}$-Zariski closure $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is non-amenable, we may find an arithmetic representation $\psi \cdot \Gamma \to \SL{m}{\mathbb{Q}}$ such that the $\mathbb{R}$-Zariski closure $\cdot{\psi(\Gamma)}^Z(\mathbb{R})$ is non-amenable.
\subsubsection{Relative Property (T) for $\mathbb{R}^N$.}
We establish the existence of a subgroup $\Gamma_0 \trianglelefteqslant \Gamma$ of finite index and a ``nice'' representation $\cdot \cdot \Gamma_0 \to \SL{N}{\mathbb{Q}}$ such that $(\sdp{\Gamma_0}{_{\cdot} \mathbb{R}^N}, \mathbb{R}^N)$ has relative property (T). The representation $\cdot$ is a factor of $\psi|_{\Gamma_0}$.
\subsubsection{Fixing the Primes}
We show that, after conjugating the representation $\cdot$ by an element in $\GL{N}{\mathbb{Q}}$ if necessary, we may assume that $\cdot \cdot \Gamma_0 \to \SL{N}{\mathbb{Z}[S^{-1}]}$ and that $\cdot(\Gamma_0)$ is not $\mathbb{Q}_p$-precompact for each $p \in S$. The representation $\cdot$ is so nice that this allows us to conclude that $(\sdp{\Gamma_0}{_{\cdot} \mathbb{Q}_p^N}, \mathbb{Q}_p^N)$ has relative property (T) for each $p \in S$.
\subsubsection{Products and Induction}
The set $S$ of primes in Step 3 is finite, and we show that the relative property (T) passes to finite products. Namely, if $(\sdp{\Gamma_0}{_{\cdot} \mathbb{Q}_p^N}, \mathbb{Q}_p^N)$ has relative property (T) for each $p \in S\cup \cdot\infty\cdot$ then setting $V = \Prod{ p \in S\cup \cdot\infty\cdot }\mathbb{Q}_p^N$ we have that $\cdot \sdp{\Gamma_0}{V }, V \cdot$ has relative property (T).
Let $A = \mathbb{Z}[S^{-1}]^N$ and recall that the diagonal embedding $A \subset V$ is a lattice embedding. Since $\cdot(\Gamma_0) \leqslant \SL{N}{\mathbb{Z}[S^{-1}]}$ we have that $\Gamma_0$ acts on $A$ by automorphisms. Since $\sdp{\Gamma_0}{A}$ is a lattice in $\sdp{\Gamma_0}{V}$ we have that $(\sdp{\Gamma_0}{A}, A)$ has relative property (T).
\subsubsection{Extending up from a finite index subgroup}
We show that if $k = [\Gamma : \Gamma_0]$ then there is a homomorphism $\cdot' \cdot \Gamma \to \SL{kN}{\mathbb{Z}[S^{-1}]}$ such that $( \sdp{\Gamma}{A^k}, A^k)$ has relative property (T).
\subsection{Outline of the proof of Theorem 1 in the direction $(2) \implies (1)$}
\subsubsection{Managing $A$} We choose $A$ to be of minimal (non-zero) $\mathbb{Q}$-rank among all Abelian groups satisfying condition (2). Under the hypothesis, we show that we may assume that $A$ is torsion free and hence a subgroup of $\mathbb{Q}^n$ where $n$ is the $\mathbb{Q}$-rank of $A$. This yields that there are finite sets of primes $S_i$ such that, up to isomorphism, $A = \Oplus{i = 1}{n} \mathbb{Z}[S_i^{-1}]$.
\subsubsection{An Invariant subgroup of $A$}
We choose $m \in \{1, \dots, n\cdot$ such that $|S_m| \geqslant |S_i|$ for each $i \in \{1, \dots, n\cdot$. Letting $I_m = \set{i}{S_i = S_m}$ we get that $A_m = \Oplus{i \in I_m}{} \mathbb{Z}[S_m^{-1}]$ is $\Gamma$-invariant. By minimality of $A$ it follows that $A = A_m \cong \mathbb{Z}[S_m^{-1}]^n$. Set $S = S_m$.
\subsubsection{A is a lattice} Let $V = \mathbb{R}^n \times \Prod{p \in S}\mathbb{Q}_p^n$. Since $A \subset V$ is a co-compact lattice it follows that $(\sdp{\Gamma}{V}, V)$ has relative property (T).
\subsubsection{The $\mathbb{R}$-component} Since $\Prod{p \in S}\mathbb{Q}_p^n \subset V$ is $\Gamma$-invariant we have that $(\sdp{\Gamma}{\mathbb{R}^n},\mathbb{R}^n)$ has relative property (T).
\subsubsection{The Image of $\Gamma$} If $\varphi \cdot \Gamma \to \GL{n}{\mathbb{Q}}$ is the corresponding homomorphism, then $\ker(\varphi) \trianglelefteqslant \sdp{\Gamma}{\mathbb{R}^n}$ so that $(\sdp{\varphi(\Gamma)}{\mathbb{R}^n},\mathbb{R}^n)$ has relative property (T).
\subsubsection{The Zariski Closure} If $(\sdp{\varphi(\Gamma)}{\mathbb{R}^n},\mathbb{R}^n)$ has relative property (T) then $(\sdp{\cdot{\varphi(\Gamma)}^Z(\mathbb{R})}{\mathbb{R}^n},\mathbb{R}^n)$ has relative property (T). It is shown that this implies that $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is not amenable.
\subsection{Organization of the Paper}
We present the paper in the following order:
\subsubsection{Section 2} In Section 2 we discuss some algebraic preliminaries in order to make the rest of the exposition consistent and coherent.
\subsubsection{Section 3} In Section 3 we state and discuss the main theorems (Theorem 2 and Theorem 3) that will be used in the proof of Theorem 1 in the direction of $(1) \implies (2)$. Their roles are: \begin{itemize}
\item [] \begin{itemize}
\item [\textit{ Thm. 2}] To give a criterion on a group $\Gamma$ (we will call it Property (F$_p$)) for which we may construct group pairs $(\sdp{\Gamma}{\mathbb{Q}_p^n}, \mathbb{Q}_p^n)$ having relative property (T).
\item [\textit{ Thm. 3}] To give a criterion on a group $\Gamma$ for which there is a finite set of primes $S$ such that we may construct group pairs $(\sdp{\Gamma}{\mathbb{Z}[S^{-1}]^n}, \mathbb{Z}[S^{-1}]^n)$ having relative property (T). \end{itemize} \end{itemize}
\subsubsection{Section 4} In Section 4 we prove Theorem 2.
\subsubsection{Section 5} In Section 5 we prove Theorem 3 using Theorem 2.
\subsubsection{Section 6} In Section 6 we prove an algebro-geometric specialization proposition (Proposition 4). It exactly yields step 1 in the proof of Theorem 1 for the direction $(1) \implies (2)$.
\subsubsection{Section 7}
In Section 7 we prove Theorem 1 in the direction of $(1) \implies (2)$ essentially as a consequence of Proposition 6.1 and Theorem 3.
\subsubsection{Section 8} In Section 8, we prove Theorem 1 in the direction of $(2) \implies (1)$. The proof is simple, and is pretty much self contained.
\subsection{Acknowledgments:} I'd like to thank Alex Furman for being a truly excellent advisor. In particular he deserves a great deal of thanks for his many detailed readings of this paper and his instructive comments and suggestions. He also proposed the original idea behind this work. I'd also like to thank Alain Valette for sending me a preprint of his paper \cite{Valette}. It came at an opportune time as it allowed for the generalization of the work I had in progress. I'd also like to thank him for his comments on this work.
This work is a part of my doctoral thesis.
\section{Algebraic Preliminaries}
\subsection{A word about Zariski Closures}\cite[Section AG.13]{Borel}, \cite[Section3.1]{Zimmer}
Let $k$ be a field and $\~K$ an algebraically closed field containing $k$. Recall that to every subset $V \subset \~K^n$ there corresponds an ideal $I_{\~K}(V) \subset \~K[x_1, \dots, x_n]$ such that $p \in I_{\~K}(V)$ if and only if $p|_V \equiv 0$. The set $V$ is said to be Zariski closed if $V = \set{a \in \~K^n}{p(a) = 0 \text{ for every } p \in I_{\~K}(V)}$, that is, if it is exactly the zero-set of its ideal.
Furthermore, $V$ is said to be defined over $k$ if there exists an ideal $I_k(V) \subset k[x_1, \dots, x_n]$ such that $I_k(V) \cdot \~K[x_1, \dots, x_n] = I_{\~K}(V)$. In such a case we write \begin{equation} V(k) := \set{a \in k^n}{p(a) = 0 \text{ for every }p \in I_k(V)} \nonumber \end{equation} to denote the $k$-points of $V$. Observe that it could happen that $V(k) = \phi$ despite the fact that $I_k(V) \neq k[x_1, \dots, x_n]$. (Take for example $\~K= \mathbb{C}$ and $k = \mathbb{R}$ and $V = \cdot i, -i\cdot \subset \~K^1$. Then $I_\mathbb{R}(V) = (x^2 +1)$ is defined over $\mathbb{Q}$ and $V(\mathbb{R}) = \phi$. This is why we need to work with algebraically closed fields to begin with!) Fortunately, the situation for groups is significantly better.
Recall that $\GL{n}{\~K}$ is an algebraic (i.e. Zariski closed) group defined over $\mathbb{Q}$.
\begin{prop}\cite[Proposition 3.1.8]{Zimmer} Suppose that $G(\~K) \leqslant \GL{n}{\~K}$ is an algebraic group such that $G(k) := \GL{n}{k}\cap G(\~K)$ is Zariski dense in $G(\~K)$. Then $G(\~K)$ is defined over $k$. \end{prop}
\begin{prop}[Chevalley]\cite[Theorem 3.1.9]{Zimmer} If $G(\~K)$ is an algebraic group defined over $k$ then $G(k)$ is Zariski dense in $G(\~K)$. \end{prop}
Note that this means in particular, that if $G(\~K) \leqslant \GL{n}{\~K}$ is Zariski closed, nontrivial, and defined over $k$ then $G(k)$ is nontrivial as well!
Now, if $\Gamma \leqslant \GL{n}{k}$ is any subgroup, then the $\~K$-Zariski closure is denoted by $\cdot{\Gamma}^Z(\~K)$. We say $\~K$-Zariski closure since this depends on the algebraically closed field $\~K$. Indeed, if $\~K'$ is another algebraically closed field containing $k$, then by the above propositions, $\Gamma$ is also Zariski dense in $\cdot{\Gamma}^Z(\~K')$.
Observe that this notion is well defined even if the field is not algebraically closed. Namely, let $F$ be a field containing $k$ and let $\~F$ be its algebraic closure. We define the $F$-Zariski closure of $\Gamma$ to be $\cdot{\Gamma}^Z(F) := \cdot{\Gamma}^Z(\~F) \cap \GL{n}{F}$. In general we make use of this when it has additional topological content. For example if $k= \mathbb{Q}$ and $F = \mathbb{Q}_p$ for some prime $p$. Then the group $\cdot{\Gamma}^Z(F)$ is a $p$-adic group and has a lot of nice additional structure.
\subsection{Restriction of Scalars:}
Let $K$ be a finite separable extension of a field $k$ (of any characteristic) and $\cdot := \cdot \sigma \cdot K \to \~k\cdot$ be the set of $k$-linear embeddings of $K$ into $\~k$ a fixed separable closure of $k$. There is a functor called the \emph{restriction of scalars} functor which maps the category of linear algebraic $K$-groups and $K$-morphisms into the category of linear algebraic $k$-groups and $k$-morphisms. Namely, let $H$ be an algebraic $K$-group defined by the ideal $I \subset K[X]$. Then, for each $\sigma \in \cdot$ the algebraic group $^\sigma H$ is defined by $\sigma(I) \subset \sigma(K)[X]$, the ideal obtained by applying $\sigma$ to the coefficients of the polynomials in $I$. The restriction of scalars of $H$ is $\mathcal{R}_{K/k}H \cong \Prod{\sigma \in G}^\sigma H$. It has the following properties \cite[Section 6.17]{BorelTits} \cite[ Proposition 6.1.3] {Zimmer} \cite[Section 12.4 ]{Springer}: \begin{enumerate}
\item There is a $K$-morphism $\cdot \cdot \mathcal{R}_{K/k}H \to H$ such that the pair $(\mathcal{R}_{K/k}H , \cdot)$ is unique up to $k$-isomorphism.
\item If $H'$ is a $k$-group and $\cdot \cdot H' \to H$ is a $K$-morphism then there exists a unique $k$-morphism $\cdot' \cdot H' \to \mathcal{R}_{K/k}H $ such that $\cdot = \cdot\circ \cdot'$.
\item If $K'$ is any field containing $K$ then $\mathcal{R}_{K/k}H (K') \cong \Prod{\sigma \in G}^\sigma H(K')$.
\item The algebraic type of the group is respected. Namely, if $H$ has the property of being reductive (respectively semi-simple, parabolic, or Cartan) then $\mathcal{R}_{K/k}H$ is reductive (respectively semi-simple, parabolic, or Cartan).
\item The algebraic type of subgroups is respected. Namely, if $P \leqslant H$ is a $K$-Cartan subgroup (respectively $K$-maximal torus, $K$-parabolic subgroup) then $\mathcal{R}_{K/k}P \leqslant \mathcal{R}_{K/k}H$ is a $k$-Cartan subgroup (respectively $k$-maximal torus, $k$-parabolic subgroup).
\item There is a correspondence of rational points: Consider the diagonal embedding $\Delta \cdot H(K) \to \Prod{\sigma \in \cdot} ^\sigma H(K)$ defined pointwise by $h \mapsto \Prod{\sigma \in \cdot} \sigma(h)$. Then we have the correspondence $\mathcal{R}_{K/k}H (k) \cong \Delta(H(K))$.
\end{enumerate}
\noindent \textbf{ Disclaimer:} In the sequel we consider the isomorphism $\mathcal{R}_{K/k}H \cong \Prod{\sigma \in \cdot}^\sigma H$ as equality.
\section{The Main Theorems 2 and 3}
Note that if $\Gamma$ is a finitely generated group and $\varphi \cdot \Gamma \to \SL{n}{\cdot\mathbb{Q}}$ is an algebraic representation, then there is a field $K_\varphi$ which is a normal finite extension of $\mathbb{Q}$ such that $\varphi(\Gamma) \leqslant \SL{n}{K_\varphi}$. (Take for example, the normal field generated by the entries of some finite generating set for $\varphi(\Gamma)$.)
With this notation in place, we give the following definition, which will be used to find group pairs with relative property (T).
\begin{definition} Let $\Gamma$ be a finitely generated non-amenable group and $p \in \{2, 3, 5, \dots, \infty\cdot$ a rational prime. Then $\Gamma$ is said to satisfy property (F$_p$) (after Furstenberg) if there exists an algebraic homomorphism $\varphi \cdot \Gamma \to \SL{n}{\cdot\mathbb{Q}}$ satisfying the following conditions:
\begin{enumerate}
\item The $\cdot\mathbb{Q}$-Zariski closure $H = \cdot{\varphi(\Gamma)}^Z(\cdot\mathbb{Q})$ is $\cdot\mathbb{Q}$-simple.
\item There are no $\varphi(\Gamma)$-fixed vectors.
\item The natural diagonal embedding $\Delta: \varphi(\Gamma) \to \mathcal{R}_{K_\varphi/\mathbb{Q}}H(\mathbb{Q})$ is not pre-compact in the $p$-adic topology. \end{enumerate}
In such a case, we say that the representation $\varphi$ realizes property (F$_p$) for $\Gamma$
\end{definition}
Recall that the archimedean valuation on $\mathbb{Q}$ is called the prime at infinity. So, according to convenience, we use both notations $\mathbb{R}$ and $\mathbb{Q}_\infty$ to denote the completion of $\mathbb{Q}$ with respect to the archimedean valuation.
\begin{theorem} Let $\Gamma$ be a group satisfying property (F$_p$). Then, there exists a rational representation $\varphi' \cdot \Gamma \to \SL{N}{\mathbb{Q}}$ such that $(\sdp{\Gamma}{_{\varphi'} \mathbb{Q}_p^N}, \mathbb{Q}_p^N)$ has relative property (T). \end{theorem}
\begin{theorem} Suppose that $\Gamma$ is a group with property (F$_\infty$). Then there exists a finite set of primes $S \subset \mathbb{Z}$ and a representation $\rho \cdot \Gamma \to \SL{N}{\mathbb{Z}[S^{-1}]}$ such that, if $A = \mathbb{Z}[S^{-1}]^N$ then $(\sdp{\Gamma}{_{\rho}A}, A)$ has relative property (T). \end{theorem}
\textbf{Remark:} Conditions (1) and (2) of property (F$_p$) can be seen as an irreducibility requirement. With this in mind, we see that Theorems 2 and 3 say that irreducibility and unboundedness are sufficient ingredients to cook up a relative property (T) group pair.
\section{Theorem 2}
\subsection{How to Find Relative Property (T)}
Our first task is to establish a sufficient condition for the presence of relative property (T); one that lends itself to the present context. The following is due to M. Burger [Propositions 2 and 7]\cite{Burger}. In what follows $\mathbb{K}$ is a local field and $\cdot\mathbb{K} \cong \textnormal{Hom}(\mathbb{K}, S^1)$ is the unitary dual. Recall that $\cdot\mathbb{K}$ is topologically isomorphic to $\mathbb{K}$ \cite[Theorem 7-1-10 ]{Goldstein}. As such we will often not distinguish between $\GL{n}{\mathbb{K}}$ and $\GL{n}{\cdot\mathbb{K}}$.
\begin{prop}[Burger's Criterion for Relative Property (T)] Suppose that $\varphi \cdot \Gamma \to \GL{N}{\mathbb{K}}$ is such that there is no $\Gamma$-invariant probability measures on $\Ps{\cdot\mathbb{K}^N}$. Then, $(\sdp{\Gamma}{_\varphi \mathbb{K}^N}, \mathbb{K}^N)$ has relative property (T). \end{prop}
\begin{proof}
Let $\rho \cdot \sdp{\Gamma}{\mathbb{K}^N} \to \mbox{$\mathcal{U}$}(\cdot)$ be a unitary representation with $\Gamma$-almost invariant vectors and $\cdot \cdot \mathcal{B}(\cdot\mathbb{K}^N) \to \textrm{Proj}(\cdot)$ the projection valued measure associated to $\rho|_{\mathbb{K}^N}$, where $\mathcal{B}(\cdot\mathbb{K}^N)$ denotes the Borel $\sigma$-algebra of $\cdot\mathbb{K}^N$. Recall that $\cdot$ has the following properties:
\begin{enumerate} \item $\cdot(\cdot\mathbb{K}^N) = \mathrm{Id}$
\item For every $v \in \cdot$ the measure $B \mapsto \left\langle\cdot(B)v, v\right\rangle$ is a positive Borel measure on $\cdot\mathbb{K}^N$ with total mass $\|v\cdot^2$.
\item For every $\gamma \in \Gamma$ we have that \begin{center} $\rho(\gamma^{-1}) \cdot(B) \rho(\gamma) = \cdot(\gamma^*B)$. \end{center}
\item The projection onto the subspace of $\mathbb{K}^N$-invariant vectors is $\cdot(\{0\cdot)$. \end{enumerate}
Let $v_n \in \cdot$ be a sequence of $(\epsilon_n, F_\Gamma)$-almost invariant unit vectors where $\epsilon_n \to 0$ and $F_\Gamma$ is a finite generating set for $\Gamma$. Define the probability measures $\mu_n(B):= \left\langle\cdot(B)v_n, v_n\right\rangle$.
\begin{clam}
The sequence of measures $\cdot\mu_n\cdot$ is almost $\Gamma$-invariant. Namely $\cdot \gamma_*\mu_n - \mu_n\cdot \leqslant 2\epsilon_n$ for each $\gamma \in F_\Gamma$. \end{clam}
\begin{proof}
Let $B \subseteq \cdot\mathbb{K}^N$ be a Borel set and $\gamma \in F_\Gamma$. Then
\begin{equation} \begin{split}
|\mu_n(\gamma^* B) - \mu_n(B)| & = |\left\langle \pi(\gamma^{-1})\cdot(B)\pi(\gamma)v_n, v_n \right\rangle - \left\langle\cdot(B)v_n, v_n \right\rangle| \cdot
& \leqslant | \left\langle \pi(\gamma^{-1}) \cdot(B) \pi(\gamma) v_n, v_n \right\rangle - \left\langle \pi(\gamma^{-1}) \cdot(B) v_n, v_n \right\rangle | \cdot
& \qquad \qquad \qquad \qquad \qquad \qquad \quad + | \left\langle \pi(\gamma^{-1}) \cdot(B) v_n, v_n \right\rangle - \left\langle \cdot(B) v_n, v_n \right\rangle| \cdot
&= |\left\langle\pi(\gamma^{-1}) \cdot(B) (\pi(\gamma)v_n - v_n), v_n\right\rangle| + |\left\langle\cdot(B)v_n, (\pi(\gamma)v_n - v_n)\right\rangle| \cdot
&\leqslant \cdot \pi(\gamma^{-1})\cdot(B)\cdot \cdot \cdot \pi(\gamma)v_n - v_n \cdot + \cdot\cdot(B)\cdot \cdot \cdot \pi(\gamma)v_n - v_n \cdot \leqslant 2\epsilon_n. \end{split} \nonumber \end{equation}
Thus the sequence of probability measures $\cdot\mu_n\cdot$ is almost $\Gamma$-invariant. \end{proof}
Suppose by contradiction that the group pair $(\sdp{\Gamma}{\mathbb{K}^N},\mathbb{K}^N)$ fails to have relative property (T). Then for each $n$, $\mu_n(\{0\cdot) = 0$. This allows us to pass to the associated projective space.
Namely let $p \cdot \cdot\mathbb{K}^N \backslash \{0\cdot \to \Ps{\cdot\mathbb{K}^N}$ be the natural projection. Define the probability measures $\nu_n := p_*\mu_n$. It is clear that they also satisfy the following inequality for any $\gamma \in F_\Gamma$:
\begin{equation}
\cdot \gamma_*\nu_n - \nu_n\cdot \leqslant 2\epsilon_n \nonumber \end{equation}
Exploiting the compactness of $\Ps{\cdot\mathbb{K}^N}$, we get that a weak-$*$ limit point of $\cdot\nu_n\cdot$ will necessarily be $\Gamma$-invariant, a contradiction of the hypothesis that there are no $\Gamma$-invariant probability measures on $\Ps{\cdot\mathbb{K}^N}$. \end{proof}
This is a powerful criterion when taken together with the following:
\begin{lemma}[Furstenberg's Lemma] \cite[Lemma 3.2.1, Corollary 3.2.2]{Zimmer} Let $\mu$ be a Borel probability measure on $\Ps{\mathbb{K}^N}$. Suppose that $\Gamma \leqslant \PGL{N}{\mathbb{K}}$ leaves $\mu$ invariant. If $\Gamma$ is not precompact then there exists a nonzero subspace $V \subsetneq \mathbb{K}^N$ which is invariant under a finite index subgroup of $\Gamma$ and such that $\mu[V] >0$. \end{lemma}
These two statements will be used to show the presence of relative property (T) once we have a nice representation to work with. The representation will be provided by the following considerations.
\subsection{The Tensor Representation:}
Let $K$ be a finite normal extension of $\mathbb{Q}$ with Galois group $G$. Consider the vector space $W(K) =\underset{\sigma \in G}{\otimes} K^n$ and the representation of $\mathcal{R}_{K/\mathbb{Q}}\SL{n}{K} \cong \Prod{ \sigma \in G} ^\sigma \SL{n}{K}$ on $W(K)$, defined by $\cdot : \Prod{\sigma \in G} g_\sigma \mapsto \Otimes{\sigma \in G} g_\sigma$. This induces a representation $\Delta_\cdot \cdot \SL{n}{K} \to \SL{}{W(K)}$ defined by $\Delta_\cdot = \cdot \circ \Delta$.
There are two reasons which make this an excellent representation to work with. The first is due to Y. Benoist and is taken from \cite[Lemma 1]{Valette}.
\begin{lemma} The faithful representation $\Delta_\cdot \cdot \SL{n}{K} \to \SL{}{W(K)}$ is defined over $\mathbb{Q}$ and there is a $\mathbb{Q}$-subspace $W(\mathbb{Q})$ of $W(K)$ such that the map $K \otimes W(\mathbb{Q}) \to W(K)$ is an $\SL{n}{K}$-equivariant isomorphism. \end{lemma}
The second reason is observed in \cite[Item 1, page 9]{Valette}:
\begin{lemma}
If $H(K) \leqslant \SL{n}{K}$ is a group without fixed vectors in $K^n$ then for each $\sigma_0 \in G$ the restricted representation $\tau_0 = \tau \cdot |_ {{^{\sigma_0}} {H(K)}} \cdot. \cdot ^{\sigma_0} H(K) \to \SL{}{W(K)}$ also has no invariant vectors. \end{lemma}
\begin{proof}
Although we are thinking of $^{\sigma_0}H(K)$ as being a subgroup of $\SL{n}{K}$, for the sake of clarity it is necessary to denote by $\rho_0 \cdot {^{\sigma_0}H(K)} \to \SL{n}{K}$ the identity representation, so that $\rho_0(^{\sigma_0}H(K)) = {^{\sigma_0}H(K)}$.
With this notation, it is clear that $\tau_0 \cdot ^{\sigma_0}H(K) \to \SL{}{W(K)}$ is given by $\tau_0 = \rho_0 \Otimes{\sigma \neq \sigma_0} \mathds{1}$, where $\mathds{1}$ denotes the trivial representation. Namely, $^{\sigma_0}H(K)$ acts trivially on each tensor-factor except the one corresponding to $\sigma_0$, where it acts via $\rho_0$.
Also recall the fact that
\begin{equation} \Otimes{\sigma \in G} K^n \cong \cdot\Otimes{\sigma \neq \sigma_0} K^n\cdot \otimes K^n \cong \textnormal{Hom} \cdot \cdot \Otimes{\sigma \neq \sigma_0} K^n \cdot^*, K^n \cdot. \nonumber \end{equation}
Under this isomorphism, a vector which is $^{\sigma_0}H(K)$-invariant corresponds to a $K$-linear map which intertwines $( \Otimes{\sigma \neq \sigma_0} \mathds{1}^*, ( \Otimes{\sigma \neq \sigma_0} K^n )^* )$ with $(\tau_0, K^n)$. Since the dual of a trivial representation is trivial, it follows that the image of such a map consists of $\rho_0( ^{\sigma_0}H)$-invariant vectors.
We then have that $(\tau_0, \Otimes{\sigma \in G} K^n)$ contains a \emph{non-zero} $^{\sigma_0}H(K)$-invariant vector if and only if $(\rho_0, K^n)$ contains the trivial representation; that is, if and only if $(\rho_0, K^n)$ contains a $^{\sigma_0}H(K)$-invariant vector. And, since $H(K)$ does not have invariant vectors in $K^n$ neither does $^{\sigma_0}H(K)$. \end{proof}
Before the proof of Theorem 2, we establish a little more notation: Let $F$ be a field containing $\mathbb{Q}$. Then we write $W(F) = W(\mathbb{Q}) \otimes F$. If $F$ contains $K$ then naturally $W(F) \cong \underset{\sigma \in G}{\otimes} F^n$.
\subsection{The Proof of Theorem 2:}
We retain the notation established above. Recall that if $\Gamma$ is a group satisfying property (F$_p$) then there is a field $K$ which is a finite normal extension of $\mathbb{Q}$ and a representation $\varphi \cdot \Gamma \to \SL{n}{K}$ such that
\begin{enumerate}
\item The Zariski-closure $H = \cdot{\varphi(\Gamma)}^Z$ is $\cdot\mathbb{Q}$-simple.
\item There are no $\varphi(\Gamma)$-fixed vectors.
\item The natural diagonal embedding $ \Delta \cdot \varphi(\Gamma) \to \mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q})$ is not pre-compact in the $p$-adic topology. \end{enumerate}
\begin{proof} Consider the representation of $\varphi' \cdot \Gamma \to \SL{}{W(\mathbb{Q})}$ which is defined as $\varphi' = \tau \circ \Delta \circ \varphi$. We claim that $(\sdp{\Gamma}{_{\varphi'} W(\mathbb{Q}_p)}, W(\mathbb{Q}_p))$ has relative property (T).
If not then by Burger's Criterion (Proposition 4.1) there exists a $\Gamma$-invariant probability measure $\mu$ on $\Ps{W(\cdot{\mathbb{Q}_p})}$. Since $\varphi'$ factors through the diagonal embedding in item (3) above, it follows that $\varphi'(\Gamma) \leqslant \SL{}{W(\mathbb{Q}_p)}$ is not pre-compact, and hence the corresponding projective image in $\PGL{}{W(\cdot{\mathbb{Q}_p})}$ is also not pre-compact (since $\SL{}{W(\cdot{\mathbb{Q}_p})}$ has finite center). By Furstenberg's Lemma, there exists a non-trivial subspace $V \subsetneq W(\cdot{\mathbb{Q}_p})$ such that
\begin{enumerate}
\item There is a subgroup of finite index in $\Gamma$ which preserves $V$.
\item The mass $\mu[V]>0$.
\item $V$ is of minimal dimension among all subspaces satisfying (1) and (2). \end{enumerate}
We aim to show that this is impossible:
Observe that $V$ is actually $\mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}_p)$-invariant. Indeed, since preserving a subspace is a Zariski-closed condition (consider the corresponding parabolic subgroup), if $\Gamma$ has a finite index subgroup which preserves $V$ then so must the Zariski-closure $\mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}_p)$. Since $H$ is $\cdot\mathbb{Q}$-simple, it is Zariski-connected and therefore so is $\mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}_p)$. It follows that all of $\mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}_p)$, and in particular $\Gamma$, preserves $V$.
We claim that the map $\mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}_p) \to \SL{}{V}$ is a faithful continuous homomorphism. Continuity is automatic because the representation is linear. (Observe that the semisimplicity of $\mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}_p)$ guarantees that the image is in $\SL{}{V}$ versus $\GL{}{V}$.)
Since $\varphi'(\Gamma) \leqslant \SL{}{W(\mathbb{Q})}$ it follows that the subspace $V$ is defined over an algebraic field $F \subset \cdot\mathbb{Q}$, and we may as well assume that $K \subset F$. Let $V(F)$ be the $F$-span of an $F$-basis of $V$. Then, we have the representation $\mathcal{R}_{K/\mathbb{Q}}H(F) \to \SL{}{V(F)}$.
Recall that property (3) of the restriction of scalars says that $\mathcal{R}_{K/\mathbb{Q}}H(F) \cong \Prod{\sigma \in G} ^\sigma H(F)$, where $G$ is the Galois group of $K/\mathbb{Q}$. Now observe that since each $^\sigma H$ is $\cdot\mathbb{Q}$-simple, the kernel is either trivial, or contains $^{\sigma_0} H(F)$ for some $\sigma_0 \in G$. Assume that the kernel is not trivial. This means that $^{\sigma_0}H(F)$ acts trivially on $V(F)$, i.e. that each vector in $V(F)$ is fixed by $^{\sigma_0}H(F)$. We claim that this is impossible:
Indeed, by Lemma 4.3, there are no $^{\sigma_0}H(K)$-invariant vectors in $W(K)$. This means that $W(F)$ cannot have $^{\sigma_0}H(F)$-invariant vectors. If $v \in W(F)$ is $^{\sigma_0}H(F)$-invariant then it is $^{\sigma_0}H(K)$-invariant which means that $v \in W(K)$ (since the equations for $v$ are linear with coefficients in $K$), a contradiction.
Thus, the representation $\mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}_p) \to \SL{}{V}$ is faithful and continuous. Since $\Delta \circ \varphi (\Gamma) \leqslant \mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}_p) $ is not precompact, it follows that the corresponding representation $\Gamma \to \SL{}{V}$ is also not precompact.
Now, consider the induced measure:
\begin{equation} \mu_0(B) = \mu(B \cap [V]) / \mu[V]. \nonumber \end{equation}
It is clearly $\Gamma$-invariant. Furthermore, since $V$ was chosen to be of minimal dimension by Furstenberg's lemma, it follows that the image of $\Gamma$ in $\PGL{}{V}$ is pre-compact, which is a contradiction.
Thus, there are no $\Gamma$-invariant probability measures on $\Ps{W(\cdot{\mathbb{Q}_p})}$ and so by Burger's Criterion, the group pair $(\sdp{\Gamma}{W(\mathbb{Q}_p)}, {W(\mathbb{Q}_p)})$ has relative property (T).
\end{proof}
\section{Theorem 3}
Recall that if $\Gamma$ has property (F$_\infty$) then there exists a representation $\varphi \cdot \Gamma \to \SL{n}{K}$ (with $d = [K:\mathbb{Q}] <\infty$) such that
\begin{enumerate}
\item The Zariski-closure $H = \cdot{\varphi(\Gamma)}^Z$ is $\cdot\mathbb{Q}$-simple.
\item The representation $\varphi$ does not contain the trivial representation, that is, there are no $\varphi(\Gamma)$-fixed vectors.
\item[*(3)] The natural diagonal embedding $\Delta: \varphi(\Gamma) \to \mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q})$ is not pre-compact in the $\infty$-adic (that is the $\mathbb{R}$) topology. \end{enumerate} \break We now turn to the proof of Theorem 3:
\begin{proof}
Let $N = n^d$. We retain the notation from the proof of Theorem 2 and set $\mathbb{Q}^N \cong W(\mathbb{Q})$. Recall that this gives rise to:
\begin{equation} \varphi' \cdot \Gamma \overset{\varphi}{\to} H(K) \overset{\Delta}{\hookrightarrow} \mathcal{R}_{K/\mathbb{Q}}H(\mathbb{Q}) \overset{\tau}{\hookrightarrow} \SL{N}{\mathbb{Q}} \nonumber \end{equation}
and $(\sdp{\Gamma}{_{\varphi'}\mathbb{R}^N}, \mathbb{R}^N)$ has relative property (T) by Theorem 2.
Note that proof of Theorem 2 also shows that if there exists a prime $p$ such that condition *(3) holds at $p$ (that is if $\Delta \circ\varphi(\Gamma)$ is also not precompact in the $p$-adic topology) then $(\sdp{\Gamma}{_{\varphi'}\mathbb{Q}_p^N}, \mathbb{Q}_p^N)$ has relative property (T). (For the same $\varphi'$!) Let $S \subset \mathbb{Z}$ be the set of primes such that if $p \in S$ then condition *(3) holds at $p$.
Next, let $S_0 \subset \mathbb{Z}$ be the set of primes such that if $p \in S_0$ then $p$ appears as a denominator in some entry of $\varphi'(\Gamma)$. Since $\Gamma$ is finitely generated, $S_0$ is finite and by definition $\varphi'(\Gamma) \leqslant \SL{N}{\mathbb{Z}[S_0^{-1}]}$.
Recall that going to infinity in the $p$-adic topology amounts to being \Quote{increasingly divided by $p$}. By observing that $\tau$ is faithful, we see that $S \subset S_0$ and so $S$ is also finite. Consider the following:
\begin{lemma} Let $S$ and $S_0 = S \cup \{p\cdot$ be two distinct sets of primes. If $\Gamma \leqslant \SL{N}{\mathbb{Z}[S_0^{-1}] }$ is such that the natural embedding $\Gamma \leqslant \SL{n}{\mathbb{Q}_p}$ is precompact, then there exists an element $g\in \GL{n}{\mathbb{Z}[p^{-1}]}$ such that $g\Gamma g^{-1} \leqslant \SL{n}{\mathbb{Z}[S^{-1}]}$. \end{lemma}
\begin{proof}
Recall that all maximal compact subgroups of $\GL{n}{\mathbb{Q}_p}$ are conjugate and that $\GL{n}{\mathbb{Z}_p} \leqslant \GL{n}{\mathbb{Q}_p}$ is one such subgroup. The fact that it is both compact and open means that $\mathcal{B}_v := \GL{n}{\mathbb{Q}_p}/\GL{n}{\mathbb{Z}_p}$ is discrete. (The notation $\mathcal{B}_v$ is intended to remind the reader familiar with the Bruhat-Tits building for $\GL{n}{\mathbb{Q}_p}$ that $\mathcal{B}_v$ is the vertex set of the building, though we will not need to make use of that here.)
Also recall that the subgroup $\GL{n}{\mathbb{Z}[p^{-1}]} \leqslant \GL{n}{\mathbb{Q}_p}$ is dense, and since $\mathcal{B}_v$ is discrete, it follows that $\mathcal{B}_v = \GL{n}{\mathbb{Z}[p^{-1}]}/\GL{n}{\mathbb{Z}}$. (Observe that $\GL{n}{\mathbb{Z}} = \GL{n}{\mathbb{Z}[p^{-1}]} \cap \GL{n}{\mathbb{Z}_p}$.)
Now since the maximal compact subgroups of $\GL{n}{\mathbb{Q}_p}$ are in one to one correspondence with $\mathcal{B}_v$, we see that if $K \leqslant \GL{n}{\mathbb{Q}_p}$ is a maximal compact subgroup, then there exists an element $g \in \GL{n}{\mathbb{Z}[p^{-1}]}$ such that $K = g^{-1}\GL{n}{\mathbb{Z}_p}g$.
So, if $\Gamma \leqslant \SL{n}{\mathbb{Z}[S_0^{-1}]} \leqslant \GL{n}{\mathbb{Q}_p}$ is precompact then $\Gamma \leqslant K$ for some maximal compact subgroup $K$ of $\GL{n}{\mathbb{Q}_p}$ and by the above argument, there exists an element $g \in \GL{n}{\mathbb{Z}[p^{-1}]}$ such that \begin{equation}
g\Gamma g^{-1} \leqslant \GL{n}{\mathbb{Z}_p} \cap \SL{n}{\mathbb{Z}[S_0^{-1}]} = \SL{n}{\mathbb{Z}[S^{-1}]}. \nonumber \end{equation} \end{proof}
Now note that conjugation, as in Lemma 5.1, amounts to a change of basis. It is clear that if $(\sdp{\Gamma}{_{\varphi'} \mathbb{Q}_p^N}, \mathbb{Q}_p^N)$ has relative property (T) then so does $(\sdp{\Gamma}{_{\varphi''} \mathbb{Q}_p^N}, \mathbb{Q}_p^N)$ where $\varphi''$ is a conjugate representation of $\varphi'$. So, by Lemma 5.1, after conjugating if necessary, we may assume that $\varphi'(\Gamma) \leqslant \SL{N}{\mathbb{Z}[S^{-1}]}$ and that $(\sdp{\Gamma}{_{\varphi'} \mathbb{Q}_p^N}, \mathbb{Q}_p^N)$ has relative property (T) for each $p \in S \cup \cdot\infty\cdot$.
By Lemma 5.2 (below), we have that the following group pair has relative property (T): \begin{equation} \cdot\sdp{\Gamma}{(\Prod{ p \in S \cup \cdot\infty\cdot} \mathbb{Q}_p^N )}, \Prod{ p \in S \cup \cdot\infty\cdot} \mathbb{Q}_p^N \cdot \nonumber \end{equation}.
Finally, recall that the diagonal embedding $\mathbb{Z}[S^{-1}]^N \subset \Prod{ p \in S \cup \cdot\infty\cdot} \mathbb{Q}_p^N$ is a co-compact lattice embedding. And, since $\varphi'(\Gamma) \leqslant \SL{N}{\mathbb{Z}[S^{-1}]}$ it follows that this lattice is preserved by $\Gamma$. Therefore, $\sdp{\Gamma}{\mathbb{Z}[S^{-1}]^N}$ is a lattice in $\sdp{\Gamma}{(\Prod{p \in S \cup \cdot\infty\cdot} \mathbb{Q}_p^N)}$. Since lattices of this type inherit relative property (T) \cite[Proposition 3.1]{Jolissaint} this means that $(\sdp{\Gamma}{_{\varphi'}\mathbb{Z}[S^{-1}]^N}, \mathbb{Z}[S^{-1}]^N)$ has relative property (T). \end{proof}
In the above proof, we made use of the following handy lemma:
\begin{lemma} Suppose that $\Gamma$ is a group acting by automorphisms on two groups $V_1$ and $V_2$. If $(\sdp{\Gamma}{V_1}, V_1)$ and $(\sdp{\Gamma}{V_2}, V_2)$ both have relative property (T) then $(\sdp{\Gamma}{(V_1 \times V_2)}, V_1 \times V_2)$ also have relative property (T). \end{lemma}
This is a corollary to the following general fact. The reader may notice the similarity between it and an analogous well known result about groups with property (T) and exact sequences.
\begin{lemma} Suppose that $0 \to A_0 \to A \to A_1 \to 0$ is an exact sequence and that $\Gamma$ acts by automorphisms on $A$ and leaves $A_0$-invariant. If $(\sdp{\Gamma}{A_0}, A_0)$ and $(\sdp{\Gamma}{A_1}, A_1)$ have relative property (T) then so does $(\sdp{\Gamma}{A}, A)$. \end{lemma}
\begin{proof} Let $\pi \cdot \sdp{\Gamma}{A} \to \mbox{$\mathcal{U}$}(\cdot)$ be a unitary representation with almost invariant unit vectors $\{v_n\cdot \subset \cdot$. Then the space of $A_0$-invariant vectors $\cdot_0$ is non-trivial. Let $P \cdot \cdot \to \cdot_0$ and $P^\bot \cdot \cdot \to \cdot_0^\bot$ be the corresponding orthogonal projections. Observe that, since $A_0 \trianglelefteqslant \sdp{\Gamma}{A}$, the subspaces $\cdot_0$ and $\cdot_0^\bot$ are $\sdp{\Gamma}{A}$-invariant and the corresponding projections commute with $\pi(\sdp{\Gamma}{A})$.
We claim that for $n$ sufficiently large $\|P(v_n)\cdot^2 \geqslant 1/2$. Otherwise, there is a subsequence $n_j$ such that $\|P^\bot(v_{n_j})\cdot^2 = 1 - \|P(v_{n_j})\cdot^2 > 1/2$. Then
\begin{eqnarray}
\cdot \pi(\gamma)P^\bot(v_{n_j}) - P^\bot(v_{n_j}) \cdot^2 & = &\cdot P^\bot(\pi(\gamma)v_{n_j} - v_{n_j}) \cdot^2 \nonumber\cdot
\leqslant \cdot \pi(\gamma)v_{n_j} - v_{n_j} \cdot^2 & < & 2 \cdot \pi(\gamma)v_{n_j} - v_{n_j} \cdot^2 \cdot \|P^\bot(v_{n_j})\cdot^2 \nonumber \end{eqnarray}
This of course means that if $v_{n_j}$ is $(K, \epsilon)$ invariant then $P^\bot(v_{n_j})$ is $(K, \sqrt{2}\epsilon)$-invariant. So, $\{P^\bot(v_{n_j})\cdot \in \cdot_0^\bot$ is a sequence of almost-invariant vectors, which is of course a contradiction: Indeed, $\cdot_0^\bot$ does not contain $A_0$-invariant vectors, so it can not contain $\sdp{\Gamma}{A_0}$-almost invariant vectors.
Therefore, for $n$ sufficiently large, $\|P(v_n)\cdot^2 \geqslant 1/2$. The same argument above shows that the restricted homomorphism $\pi_0 \cdot \sdp{\Gamma}{A} \to \mbox{$\mathcal{U}$}(\cdot_0)$ has almost invariant vectors $\{P(v_n)\cdot$. And since this homomorphism factors through $\sdp{\Gamma}{A_1}$ we obtain the existence of a nonzero $A_1$-invariant vector. \end{proof}
\textbf{Remark:} Lemma 5.1 can be obtained in two other ways. One is a similar argument appealing to the CAT(0) structure of the Bruhat-Tits building for $\GL{n}{\mathbb{Q}_p}$ via a center of mass construction. Another is to observe that two maximal compact-open subgroups of $\GL{n}{\mathbb{Q}_p}$ are commensurable in the sense that their common intersection is a finite index subgroup in each. So, we may assume the result after passing to a finite index subgroup of $\Gamma$.
\section{Algebro-Geometric Specialization}
In order to prove Theorem 1, in the direction of $(1) \implies (2)$, we need two basic ingredients. The first is to use the hypothesis (i.e. finite generation and the existence of a linear representation whose image has a non-amenable $\mathbb{R}$-Zariski closure) in order to cook up a rational (or algebraic) representation to which we can apply Theorem 3, which is of course the second ingredient. This section is devoted to finding such a specialization, which is provided by the following:
\begin{prop} Let $\Gamma$ be a finitely generated group. If there exists a linear representation $\varphi \cdot \Gamma\to \SL{n}{\mathbb{R}}$ such that the Zariski closure $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is non-amenable then there exists a representation $\psi \cdot \Gamma \to \SL{m}{\mathbb{Q}}$ (possibly in a higher dimension) so that the Zarski closure $\cdot{\psi(\Gamma)}^Z(\mathbb{R})$ is semisimple and not compact. \end{prop}
Recall that a semisimple $\mathbb{R}$-algebraic group is amenable if and only if it is compact. This follows from Whitney's theorem \cite[Theorem 3]{Whitney} (which says that a $\mathbb{R}$-algebraic group has finitely many components as a $\mathbb{R}$-Lie group) and from \cite[Corollary 4.1.9]{Zimmer} which states that a connected semisimple $\mathbb{R}$-Lie group is amenable if and only if it is compact. So, the proposition guarantees that we may find, from an arbitrary $\mathbb{R}$-representation, a $\mathbb{Q}$-representation which preserves the property of having non-amenable $\mathbb{R}$-Zariski closure. The techniques used in the proof of this proposition are standard: the restriction of scalars functor and specializations of purely transcendental rings over $\mathbb{Q}$.
However, we will also need a criterion which can distinguish when the image of a representation has non-amenable $\mathbb{R}$-Zariski closure. This is provided by the following:
\begin{prop} Let $\Gamma$ be a finitely generated group. Then, there exists a normal finite index subgroup $\Gamma_n \trianglelefteqslant \Gamma$ so that for any homomorphism $\varphi \cdot \Gamma \to \GL{n}{\mathbb{R}}$ the following are equivalent:
\break \begin{enumerate}
\item The $\mathbb{R}$-Zariski closure $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is amenable.
\item The traces of the commutator subgroup $\varphi([\Gamma_n, \Gamma_n])$ are uniformly bounded; that is \begin{equation}
|\textsl{tr}(\varphi([\Gamma_n, \Gamma_n]))| \leqslant n. \nonumber \end{equation}
\end{enumerate} \end{prop}
\textbf{Remark:} It is a fact (see Subsection 6.3, Lemma 6.6), that if a subgroup of $\GL{n}{\mathbb{R}}$ has bounded traces, then it's $\mathbb{R}$-Zariski closure is amenable (actually it is a compact extension of a unipotent group). Therefore, in the direction of (2) implies (1), there is nothing special about $[\Gamma_n, \Gamma_n]$. Namely, any co-amenable normal subgroup of $\Gamma$ would do. The more subtle direction is that of (1) implies (2). It is in this direction that we must work to find a suitable $\Gamma_n$. Under the added assumption that $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is Zariski-connected the result follows from classical structure theory of Zariski-connected $\mathbb{R}$-algebraic groups with $\Gamma_n = \Gamma$.
However, we must address the fact that the image of a general representation $\varphi \cdot \Gamma \to \GL{n}{\mathbb{R}}$, need not have Zariski-connected Zariski-closure. It turns out that for an arbitrary (reductive) $\mathbb{R}$-algebraic group, there is a finite index subgroup (with uniformly bounded index) which ``behaves as if'' it were connected (see Subsection 6.2, Lemma 6.3). Namely, it has most of the nice structure properties of Zariski-connected groups (see Subsection 6.1, Lemma 6.2). It turns out that the uniform bound on the index of this subgroup, together with its ``pseudo-connectedness'' properties are exactly what we need to find a suitable $\Gamma_n$ which is done in Subsection 6.4. We then prove Proposition 6.2 in Subsection 6.5 and Proposition 6.1 in Subsection 6.6.
\subsection{Some Algebraic Facts}
Throughout this section, we will be dealing exclusively with $\mathbb{R}$-Zariski closures. As such we will write $G$ instead of $G(\mathbb{R})$, when speaking of $\mathbb{R}$-Zariski closed groups, and we will just say Zariski-closed or algebraic. Also, when we say connected, we mean Zariski-connected. We now develop the necessary lemmas to prove Proposition 6.2.
\begin{definition} An algebraic group $G$ is said to be reductive if any closed unipotent normal subgroup is trivial. \end{definition}
Observe that it is common to require in the definition of a reductive group that either $G$ be Zariski-connected or that any closed \emph{connected} normal unipotent subgroup of $G$ be trivial.
However, in characteristic zero, the two notions are the same since algebraic unipotent groups are always Zariski-connected. This follows by \begin{itemize}
\item Chevalley's Theorem: \cite[Theorem 11.2]{Humphreys} If $H \leqslant G$ are two algebraic groups, then there exists a rational representation $G \to \GL{N}{\mathbb{R}}$ and a vector $v \in \mathbb{R}^n$ such that $H = \mathrm{stab}_G(\mathbb{R}\cdot v)$.
\item The image of a unipotent element under a rational homomorphism is unipotent.
\item Unipotent elements have infinite order in characteristic zero. \end{itemize}
\break To be complete, we also give the following definition:
\begin{definition} An algebraic group $G$ is said to be semisimple if any closed solvable normal subgroup is finite. \end{definition}
And now onto the lemmas; the first of which shows that we may restrict our attention to reductive groups, since doing so does not affect the hypotheses and conclusions of Proposition 6.2.
\begin{lemma} Suppose that $L \leqslant \GL{n}{\mathbb{C}}$ is a $\mathbb{C}$-closed group and $U \trianglelefteqslant L$ is the maximal unipotent normal subgroup. There is a representation $\pi \cdot L \to \GL{n}{\mathbb{C}}$ such that $\ker(\pi) = U$ and $\textsl{tr}(g) = \textsl{tr}(\pi(g))$ for every $g \in L$. \end{lemma}
\begin{proof} Choose a Jordan-H\"older series for $\mathbb{C}^n$ as an $L$-module: \begin{equation} 0 = V_0 \subset V_1\subset \cdots \subset V_k = \mathbb{C}^n \nonumber \end{equation} so that the corresponding representation $\rho_i \cdot L \to \GL{}{V_i/V_{i-1}}$ is irreducible. Then for each $i$, the image $\rho_i(U)$ is again unipotent, and by the Lie-Kolchin Theorem there is a vector $v_i \in V_i/V_{i-1}$ which is fixed by $\rho_i(U)$. But since $\rho_i(U)\trianglelefteqslant \rho_i(L)$ and $\rho_i(L)$ acts irreducibly on $V_i/V_{i-1}$ it follows that $U \leqslant \ker(\rho_i)$.
Choosing a basis, for $\mathbb{C}^n$ which respects this Jordan-H\"older series, we see that \begin{equation} L \leqslant \cdot \begin{array}{cccc}
\rho_1(L) & * & \cdots & * \cdot
0 & \rho_2(L) & \cdots & * \cdot
\vdots & \vdots & \ddots & \vdots \cdot
0 & 0 & \cdots & \rho_k(L) \cdot \end{array} \cdot \nonumber \end{equation}
and
\begin{equation} U \leqslant \cdot \begin{array}{cccc}
I_{n_1} & * & \cdots & * \cdot
0 & I_{n_2} & \cdots & * \cdot
\vdots & \vdots & \ddots & \vdots \cdot
0 & 0 & \cdots & I_{n_k} \cdot \end{array} \cdot, \nonumber \end{equation}
where $n_i$ is the dimension of $V_i/V_{i-1}$ and $I_{n_i}$ is the $n_i \times n_i$ identity matrix.
Let $V = \underset{i = 1}{\overset{k}{\bigoplus}} V_i/V_{i-i}$ be the corresponding $n$-dimensional vector space. Consider the homomorphism $ \pi \cdot L \to \underset{i = 1}{\overset{k}{\bigoplus}} \rho_i (L) \leqslant \GL{}{V}$. By construction, $\textsl{tr}(\pi(g)) = \Sum{i = 1}{k} \textsl{tr}(\rho_i(g)) = \textsl{tr}(g)$ for each $g \in L$. Furthermore, it is clear that $\ker(\pi)$ is unipotent, and contains $U$. Since $U$ is maximal it follows that $\ker(\pi) = U$. \end{proof}
The following is a corollary to the proof above:
\begin{cor} Let $G$ be a $\mathbb{R}$-algebraic reductive group. Then every $\mathbb{C}$-representation of $G$ is the direct sum of $G$-irreducible sub-representations. \end{cor}
This next lemma is classical. These are exactly the ``nice'' properties of connected (and reductive) groups that were alluded to above.
\begin{lemma} Let $G_0$ be a connected reductive group. Then the following hold: \begin{enumerate}
\item The radical $R(G_0) = Z(G_0)^o$, where $Z(G_0)^o$ is the identity component of the center of $G_0$.
\item The intersection $[G_0, G_0] \cap Z(G_0)$ is finite.
\item $G_0 = [G_0, G_0] \cdot Z(G_0)$.
\item The commutator subgroup $[G_0, G_0]$ is semisimple. \end{enumerate} \end{lemma}
\begin{proof} For assertions (1) and (2) we cite \cite[Lemma 19.5]{Humphreys}.
Assertion (3) follows from (2) by noting that $G_0/[G_0, G_0] \cdot Z(G_0)$ is a connected Abelian semisimple group, and therefore trivial.
Assertion (4) follows from (3) and (2): Let $R \trianglelefteqslant [G_0, G_0]$ be a closed solvable normal subgroup. Since $G_0$ is reductive, $G_0/R(G_0)$ is semisimple. Then, $R/R\cap R(G_0)$ is closed and solvable and hence finite. Since $[G_0, G_0] \cap R(G_0)$ is finite, it follows that $R$ is finite. \end{proof}
This next lemma yields the want-to-be connected group that was alluded to above.
\begin{lemma} Let $G_0$ be a connected reductive group of finite index in $G \leqslant \GL{n}{\mathbb{R}}$. Then there exists a subgroup $G_1 \trianglelefteqslant G$ such that \begin{enumerate}
\item $G_0 \trianglelefteqslant G_1$.
\item The index $[G:G_1] \leqslant n!$.
\item The commutator subgroup $[G_1, G_1]$ contains $[G_0, G_0]$ as a finite index normal subgroup. (And hence $[G_1, G_1]$ is semisimple.) \end{enumerate} \end{lemma}
We first prove the following special case:
\begin{lemma} Let $G_0$ be a connected reductive group of finite index in $G$. Suppose that $G \leqslant \GL{n}{\mathbb{C}}$ is an irreducible representation. Then, there exists a subgroup $G_1 \trianglelefteqslant G$ with the following properties:
\begin{enumerate}
\item $G_0 \trianglelefteqslant G_1$
\item The index $[G : G_1] \leqslant n!$.
\item If $Z(G_0)$ and $Z(G_1)$ are the centers of $G_0$ and $G_1$ respectively then $Z(G_0) \leqslant Z(G_1)$. \end{enumerate} \end{lemma}
\begin{proof}
Since $G_0$ is reductive, the representation on $\mathbb{C}^n$ decomposes as a direct sum of irreducible sub-representations. Let $V \subset \mathbb{C}^n$ be one such.
Now, since $G_0 \trianglelefteqslant G$ it follows that for each $g \in G$ the subspace $gV$ is also an irreducible $G_0$-sub-representation. Hence if $gV \cap V \neq \{0\cdot$ then $gV = V$.
\begin{clam} There exists $\{g_1, \dots, g_l\cdot \subset G$ such that $\mathbb{C}^n = \Oplus{j = 1}{l} g_jV$. \end{clam}
\begin{proof} Let $g_1 = 1$. Then either $V= \mathbb{C}^n$, or there exists a $g_2 \in G$ such that $V \cap g_2 V = \{0\cdot$. In this latter case we have that $V\oplus g_2 V \subset \mathbb{C}^n$.
Inductively, suppose that we have found $\{g_1, \dots, g_k\cdot \subset G$ such that the corresponding $g_jV$ are linearly independent. Namely so that $\Oplus{j=1}{k} g_jV \subset \mathbb{C}^n$ is a direct sum of $G_0$-irreducible sub-representations.
Observe that $\Oplus{j=1}{k} g_jV$ is $G_0$-invariant. And since the $G$-translates of $V$ are $G_0$-irreducible sub-representations we get the following dichotomy:
\begin{enumerate}
\item There exists a $g_{k+1} \in G$ such that $g_{k+1}V \cap \Oplus{j=1}{k} g_jV = 0$, or
\item $gV \subset \Oplus{j=1}{k} g_jV$ for each $g \in G$. \end{enumerate}
In case (1) we may conclude that $\Oplus{j=1}{k+1} g_jV \subset \mathbb{C}^n$ is a direct sum of $G_0$-irreducible sub-representations.
In case (2) we must have that $gg_iV \subset \Oplus{j=1}{k} g_jV$ for each $i = 1, \dots, k$, and $g \in G$. This means that $\Oplus{j=1}{k} g_jV$ is $G$-invariant, and hence $\Oplus{j=1}{k} g_jV = \mathbb{C}^n$.
\end{proof}
This induces a homomorphism $\sigma \cdot G \to \textrm{Sym}(l)$ where the $\textrm{Sym}(l)$ denotes the symmetric group on $l$-symbols. Let $G_1 = \Cap{j = 1}{l} \mathrm{stab}_G(g_jV)$. Then clearly, $G_1 = \ker(\sigma)$, so that $G_1$ satisfies properties (1) and (2) as promised above. (Note that $l \leqslant n$.)
Furthermore, all of the $G_0$-irreducible subspaces are $G_1$-invariant and hence these are also $G_1$-irreducible subspaces. By Schur's Lemma, the centers of $G_0$ and $G_1$ are block-scalar matrices of the same type, and therefore, $G_1$ also satisfies property (3) as it was promised to do. \end{proof}
In order to pass from Lemma 6.4 to Lemma 6.3 we will need the following:
\begin{lemma} If $G_0 \trianglelefteqslant G$ is a finite index subgroup then $[G, G_0]$ is a normal finite index subgroup of $[G, G]$. \end{lemma}
\begin{proof} Since $G_0 \trianglelefteqslant G$ it follows that $[G, G_0] \trianglelefteqslant G$ (and in particular $[G,G_0] \trianglelefteqslant [G,G]$). Hence, to show that the index of $[G, G_0]$ in $[G,G]$ is finite, it is sufficient to show that if $[G, G_0] =1$ then $[G,G]$ is finite. (Just take the quotient of $G$ by $[G, G_0]$ if necessary, and use the general fact that for any homomorphism $h \cdot G \to H$ and any subgroups $A , B \leqslant G$ the following equality holds: $h([A,B]) = [h(A),h(B)]$.)
If $[G, G_0] =1$, it follows that $G$ centralizes $G_0$. That is, $G_0 \leqslant Z(G)$. Then,
\begin{equation} [G:Z(G)] \leqslant [G:G_0] <\infty \nonumber \end{equation}
This implies that $[G,G] <\infty$ (see \cite[Lemma 17.1.A]{Humphreys}). \end{proof}
\subsection{Proof of Lemma 6.3}
\begin{proof} The assumptions are that $G_0 \trianglelefteqslant G \leqslant \GL{n}{\mathbb{R}}$ where $G_0$ is connected, reductive and of finite index in $G$. This means that $G$ is also reductive and so by Corollary 6.1 we have that the representation of $G$ on $\mathbb{C}^n = \Oplus{i = 1}{k} \mathbb{C}^{n_i}$ is the direct sum of $G$-irreducible subrepresentations. By considering each irreducible piece and applying Lemma 6.4, we see that there exists a subgroup $G_1 \trianglelefteqslant G$ of index at most $\overset{k}{\Prod{i = 1}} (n_i)! \leqslant n!$ such that $G_0 \trianglelefteqslant G_1$ and $Z(G_0) \leqslant Z(G_1)$.
We claim that $[G_1, G_0] = [G_0, G_0]$:
Let $ x \in G_1, y \in [G_0, G_0]$, and $z \in Z(G_0) \leqslant Z(G_1)$. Recall that $[G_0, G_0] \trianglelefteqslant G_1$ so that
\begin{equation} [x,yz] = [x,y] = (xyx^{-1})y^{-1} \in [G_0, G_0] \nonumber \end{equation}
Since $G_0 = [G_0, G_0] \cdot Z(G_0)$ it follows that $[G_1, G_0]$ is generated by elements in $[G_0, G_0]$ and therefore, $[G_1, G_0] \leqslant [G_0, G_0]$. On the other hand, $[G_0, G_0] \leqslant [G_1, G_0]$ so $[G_1, G_0] = [G_0, G_0]$.
Now, since $G_0$ has finite index in $G_1$ by Lemma 6.5, we see that $[G_0, G_0] = [G_1, G_0]$ is a finite index normal subgroup of $[G_1, G_1]$ and we are done. \end{proof}
\subsection{The Trace Connection}
So, far, we have addressed only the structure of the algebraic groups in question, and have ignored the role of the trace. We now discuss how the trace ties in to the picture.
Recall that if $\Gamma \leqslant \GL{n}{\mathbb{R}}$ is a precompact group then all of its eigen values have norm 1 and hence its traces are uniformly bounded by $n$. Also recall that the Zariski closure of a precompact group is compact and therefore amenable. The following shows that the converse also holds. Namely:
\begin{lemma} Let $\Gamma \leqslant \GL{n}{\mathbb{R}}$ be a group. If the set of traces $\textsl{tr}(\Gamma) := \set{\textsl{tr}(\gamma)}{\gamma \in \Gamma}$ is bounded then the Zariski-closure $\cdot{\Gamma}^Z(\mathbb{R})$ is amenable. \end{lemma}
\subsubsection{Some Useful Facts}
We will need the following:
\begin{fact} \cite[Corollary 1.3(c)]{Bass} Let $\Gamma \leqslant \GL{}{V}$ be a group acting irreducibly on the complex vector space $V$. If the traces of $\Gamma$ are bounded then $\Gamma$ is precompact (in the $\mathbb{C}$-topology). \end{fact}
\begin{claim}
Let $\Gamma \leqslant \GL{n}{\mathbb{C}}$ be a subgroup such that $B = \Sup{\gamma \in \Gamma}|\textsl{tr}(\gamma)| < \infty$. Then all $\Gamma$-eigen values have norm 1 and $B = n$. \end{claim}
\begin{proof}
By contradiction suppose that there is some $\gamma \in \Gamma$ with an eigen value of norm not equal to 1. Then upon passing to $\gamma^{-1}$ if necessary, we may assume that $\gamma$ has an eigen value of norm strictly greater than 1.
Order the eigen values so that
\begin{equation}
|\lambda_1| = \cdots =|\lambda_m| > |\lambda_{m+1}| \geqslant \cdots |\lambda_n|. \nonumber \end{equation}
Since the traces of $\Gamma$ are bounded we get that for each $k \in \mathbb{N}$
\begin{equation}
\cdot|\textsl{tr}(\gamma^k)\cdot| = \cdot|\Sum{j=1}{n} \lambda_j^k \cdot| \leqslant B. \nonumber \end{equation}
\noindent The triangle inequality gives us that
\begin{equation}
\cdot|\Sum{j=1}{m} \frac{\lambda_j^k}{\lambda_1^k} \cdot| \leqslant \frac{ B}{|\lambda_1^k|} +\Sum{j=m+1}{n}\cdot| \frac{\lambda_j^k}{\lambda_1^k} \cdot| \to 0. \nonumber \end{equation}
\noindent By Claim 6.2 (see below) we get that $\Sum{j=1}{m} \frac{\lambda_j^k}{\lambda_1^k} = m$ and so
\begin{equation}
1 \leqslant m = \cdot|\Sum{j=1}{m} \frac{\lambda_j^k}{\lambda_1^k} \cdot| \to 0 \nonumber \end{equation} a contradiction.
Therefore, all eigen values of $\Gamma$ have norm 1 and the supremum $B = \Sup {\gamma \in \Gamma}|\textsl{tr}(\gamma)|$ is attained at the identity.
\end{proof}
\begin{claim} If $\Sum{j =1}{n} e^{ik\theta_j}$ converges as $k \to \infty$ then $\Sum{j =1}{n} e^{ik\theta_j} \equiv n$ for all $k$.
\end{claim}
\begin{proof} Consider the action of $\mathbb{Z}$ by the rotation on the $n$-torus $\mbox{$\mathbb{T}$}^n$ corresponding to $(e^{i\theta_1}, \dots, e^{i\theta_n})$. Let us definte the closed subgroup \begin{equation} S = \cdot{\left\langle(e^{ik\theta_1}, \dots, e^{ik\theta_n}) \cdot k \in \mathbb{Z}\right\rangle}. \nonumber \end{equation}
Now, if $S$ is discrete then it is finite, which means that the identity $(1, \dots, 1) \in \mbox{$\mathbb{T}$}^n$ is a periodic point. On the other hand, if $S$ is not discrete then the identity is an accumulation point of the sequence $\cdot(e^{ik\theta_1}, \dots, e^{ik\theta_n}) \cdot k \in \mathbb{Z}\cdot$. Either way, there is a subsequence $k_l \to +\infty$ such that \begin{equation} \Lim{l \to \infty}(e^{i{k_l}\theta_1}, \dots, e^{i{k_l}\theta_n}) = (1, \dots, 1). \nonumber \end{equation}
This shows that if $\Sum{j =1}{n} e^{ik\theta_j}$ converges then \begin{equation} \Lim{k \to \infty} \Sum{j =1}{n} e^{ik\theta_j} = \Lim{l \to \infty} \Sum{j =1}{n} e^{i{k_l}\theta_j} = n. \nonumber \end{equation}
Since the sequence is convergent, any subsequence converges to the same limit. Therefore the same argument shows that if $(e^{i\psi_1}, \dots, e^{i\psi_n}) \in S$ then
\begin{equation} \Sum{j = 1}{n}e^{i\psi_j} = n. \nonumber \end{equation}
In particular, this holds for $(e^{i\psi_1}, \dots, e^{i\psi_n}) = (e^{i\theta_1}, \dots, e^{i\theta_n})$. Since 1 is an extreme point of the unit disk we conclude that $e^{i\theta_j} = 1$ for each $j =1, \dots, n$.
\end{proof}
\subsubsection{The Proof of Lemma 6.6}
\begin{proof} By Lemma 6.1 we may assume that $G = \cdot{\Gamma}^Z(\mathbb{R}) \leqslant \GL{n}{\mathbb{R}}$ is reductive. Using Corollary 6.1, we decompose $\mathbb{C}^n = \Oplus{i \in I}{}V_i$ into a direct sum of $G$-irreducible sub-representations. Since $\Gamma$ is Zariski-dense in $G$, this of course means that each $V_i$ is also a $\Gamma$-irreducible sub-representation. We aim to show that $G$ is compact. To this end, it is sufficient to show that $\Gamma$ is pre-compact in $\GL{n}{V_i}$ for each $i \in I$ since the homomorphism $G \to \Prod{i \in I} \GL{n}{V_i}$ is rational and injective.
By Claim 6.1, $\Gamma$ has bounded traces in each $\GL{n}{V_i}$. And by Fact 6.1, $\Gamma$ is precompact in each $\GL{n}{V_i}$ since it acts irreducibly on $V_i$. \end{proof}
\subsection{Choosing $\Gamma_n$ for Proposition 6.2}
Recall that condition (2) of Lemma 6.3 guarantees a uniform bound on the index of the groups in question. We now show how we will make use of that fact to find our $\Gamma_n$:
\begin{lemma} Let $\Gamma$ be a finitely generated group and let \begin{equation}
H_N \cdot \cdot\pi \cdot \Gamma \to F| F \text{ is a group of order at most }N\cdot. \nonumber \end{equation}
Then $\Gamma(N) = \Cap{\pi \in H_N}{} \ker(\pi)$ is a finite index (normal) subgroup of $\Gamma$. \end{lemma}
\begin{proof} This is a straightforward consequence of two facts:
\begin{enumerate}
\item[Fact 1:] There are finitely many groups of order at most $N$.
\item[Fact 2:] There are finitely many homomorphisms from a finitely generated group to a fixed finite group. \end{enumerate} \end{proof}
\subsection{The proof of Proposition 5}
\begin{proof} Let $\Gamma_n = \Gamma(n!)$ as in Lemma 6.7. Then, $\Gamma_n$ is a finite index normal subgroup of $\Gamma$. Let $\varphi \cdot \Gamma \to \GL{n}{\mathbb{R}}$ be any homomorphism.
$(2) \implies (1)$: If the set of traces $\textsl{tr}(\varphi([\Gamma_n, \Gamma_n]))$ is uniformly bounded, then by Lemma 6.6 the Zariski closure $\cdot{\varphi([\Gamma_n,\Gamma_n])}^Z(\mathbb{R})$ is amenable. Therefore, $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is amenable as it is a virtually Abelian extension of $\cdot{\varphi([\Gamma_n,\Gamma_n])}^Z(\mathbb{R})$.
$(1) \implies (2)$: Suppose that $G := \cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is amenable. As was mentioned several times, by Lemma 6.1 it does no harm to assume that $G$ is reductive. Let $G_0$ be the Zariski connected component of 1 and let $G_1$ be as in Lemma 6.3. Then, $[G_0, G_0]$ being Zariski-connected, semisimple, and amenable, it is compact. Since $[G_1, G_1]$ contains $[G_0, G_0]$ as a finite index subgroup, it follows that $[G_1, G_1]$ is compact.
Thus, if $\varphi(\Gamma_n) \leqslant G_1$ then we are done. But, this follows by construction: Recall that $\Gamma_n \leqslant \ker(\pi)$ for every homomorphism $\pi \cdot \Gamma \to F$ where $F$ is a finite group of order at most $n!$. Since $G_1 \trianglelefteqslant G$ and the index $[G:G_1] \leqslant n!$ we must have that $\varphi(\Gamma_n) \leqslant G_1$.
\end{proof}
\subsection{Finally: The Proof of Proposition 6.1}
\begin{proof}
To conserve notation, we assume that $\Gamma \leqslant \SL{n}{\mathbb{R}}$. Let $K$ be the field generated by the entries of some finite generating set for $\Gamma$ so that $\Gamma \leqslant \SL{n}{K}$. Then, since $K$ is finitely
generated, it is a finite and hence separable extension of $\mathbb{Q}(t_1, \dots, t_s) \subset \mathbb{R}$, where $t_1, \dots, t_s \in K$ are algebraically independent transcendentals. So, after applying the restriction of scalars if necessary, we may assume that $\Gamma \leqslant \SL{n}{\mathbb{Q}(t_1, \dots, t_s)}$. (We note that property (3) of the restriction of scalars, guarantees that the hypothesis is preserved.)
The proof is by induction on the transcendence degree of $\mathbb{Q}(t_1, \dots, t_s)/\mathbb{Q}$.
\textbf{Base Case:} Suppose $s=0$.
Let $G = \cdot{\Gamma}^Z$ be the Zariski-closure. Since $\Gamma \leqslant \SL{n}{\mathbb{Q}}$ it follows that $G$ and its radical $R(G)$ are defined over $\mathbb{Q}$. Fixing a representation of $G/R(G) (\mathbb{Q}) \leqslant \SL{n}{\mathbb{Q}}$ we have the desired result.
\textbf{Induction Hypothesis:} Assume it is true for $s-1$.
Since $\Gamma$ is finitely generated, it follows that there exist irreducible polynomials $\cdot_1, \dots, \cdot_l \in \mathbb{Q}[t_1, \dots, t_s]$ such that if we set $\mathcal{R} = \mathbb{Q}[t_1, \dots, t_s, \cdot_1^{-1}, \dots, \cdot_l^{-1}]$ then $\Gamma \leqslant \SL{n}{\mathcal{R}}$.
Observe that by Proposition 6.2, $[\Gamma_n, \Gamma_n] \leqslant \SL{n}{\mathbb{R}}$ has unbounded traces since $\cdot{\Gamma}^Z(\mathbb{R})$ is non-amenable. So, up to a relabeling of the transcendentals there are two cases to consider:
\begin{enumerate}
\item [Case 1:] The unbounded traces of $[\Gamma_n,\Gamma_n]$ are independent of $t_s$, that is
\begin{equation}
\set{\textsl{tr}(\gamma)}{\gamma \in [\Gamma_n,\Gamma_n] \text{ and } |\textsl{tr}(\gamma)| \geqslant n+2} \cap \mathcal{R} \subset \mathbb{Q}(t_1, \dots, t_{s-1}). \nonumber \end{equation}
\item [Case 2:] There is an element in $[\Gamma_n,\Gamma_n]$ with large trace which is non-constant as a rational function in $t_s$. Namely, there is a $\gamma \in [\Gamma_n,\Gamma_n]$ such that $|\textsl{tr}(\gamma)| \geqslant n+2$ and $\textsl{tr}(\gamma) \in \mathcal{R} \backslash \mathbb{Q}(t_1, \dots, t_{s-1})$.
\end{enumerate}
We now need to say how we will specialize the transcendental $t_s$. First consider the denominators $\cdot_i$ as polynomials in $t_s$. Since there are finitely many, the bad set \begin{equation} B = \set{ a \in \mathbb{R}}{\cdot_i(t_1, \dots, t_{s-1}, a) = 0 \text{ for some } i = 1, \dots l} \nonumber \end{equation} is finite. (Recall that the kernel of a ring homomorphism cannot contain any invertible elements.) Now, we choose the specialization in each case:
\break \begin{enumerate}
\item [Case 1:] Choose $a \in \mathbb{Q} \backslash B$.
\item [Case 2:] Let $\gamma \in [\Gamma_n,\Gamma_n]$ be such that
\begin{equation} r(t_s)=\textsl{tr}(\gamma) \nonumber \end{equation}
is a nonconstant rational function in $t_s$ and such that $|r(t_s)|\geqslant n+2$. Then, $r(x)$ is a continuous function in some neighborhood of $t_s \in \mathbb{R} \backslash B$ and so there is an $a \in \mathbb{Q} \backslash B$ such that $|r(a)| \geqslant n+1$. \end{enumerate}
Now, fix an embedding $\mathbb{Q}(t_1, \dots, t_{s-1}) \subset \mathbb{R}$ and let \begin{equation}
\psi \cdot \SL{n}{\mathbb{Q}(t_1, \dots, t_s)} \to \SL{n}{\mathbb{Q}(t_1, \dots, t_{s-1})} \nonumber \end{equation} be the homomorphism induced from the ring homomorphism $t_s \mapsto a$. Observe that this is well defined since we are dealing with unimodular matrices.
To apply the induction hypothesis, we must show that the Zariski-closure $\cdot{\psi(\Gamma)}^Z(\mathbb{R})$ is again non-amenable. This is immediate by Proposition 6.2 since by construction, there is a $\gamma \in [\Gamma_n,\Gamma_n]$ such that $|\textsl{tr}(\psi(\gamma))| \geqslant n+1$. Since the traces of a subgroup of $\SL{n}{\mathbb{R}}$ are either uniformly bounded by $n$ or unbounded, we see that $\psi([\Gamma_n,\Gamma_n])$ has unbounded traces and the proposition is proved. \end{proof}
\section{Proof of Theorem 1 in the direction $(1) \implies (2)$}
We instead prove the following:
\begin{theorem} Suppose that $\Gamma$ is a finitely generated group which admits a linear representation $\varphi \cdot \Gamma \to \SL{n}{\mathbb{R}}$ such that the $\mathbb{R}$-Zariski closure $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is non-amenable. Then there exists a finite set of primes $S \subset \mathbb{Z}$ and a homomorphism $\cdot \cdot \Gamma \to \SL{N}{\mathbb{Z}[S^{-1}]}$ such that, if $A=\mathbb{Z}[S^{-1}]^N$ then $(\sdp{\Gamma}{_\cdot A}, A)$ has relative property (T). \end{theorem}
The proof is in two basic steps: \begin{enumerate}
\item [\textbf{Step A:}] Show that under the hypothesis of Theorem 7.1 there is a finite index subgroup $\Gamma_0 \trianglelefteqslant \Gamma$ satisfying property (F$_\infty$).
\item [\textbf{Step B:}] Show that if $\Gamma_0 \trianglelefteqslant \Gamma$ is a finite index subgroup such that $(\Gamma_0 \ltimes A, A)$ has relative property (T) then there is an action of $\Gamma$ on $A^k$ (with $A$ as in Theorem 7.1 and $k = [\Gamma:\Gamma_0]$) such that $(\Gamma \ltimes A^k, A^k)$ has relative property (T).
\end{enumerate}
It is clear that Steps A and B prove Theorem 7.1 by Theorem 3.
\subsubsection{Proof of Step A}
\begin{proof}
By Proposition 6.1 there exists a rational representation $\psi \cdot \Gamma \to \SL{m}{\mathbb{Q}}$ such that $\cdot{\psi(\Gamma)}^Z (\mathbb{R})$ is semisimple and not compact.
Let $\Gamma_0 \trianglelefteqslant \Gamma$ be the normal subgroup of finite index such that $\cdot{\psi(\Gamma_0)}^Z(\mathbb{R})$ is the Zariski-connected component of the identity of $\cdot{\psi(\Gamma)}^Z(\mathbb{R})$. Then, $G(\mathbb{R}):=\cdot{\psi(\Gamma_0)}^Z(\mathbb{R})$ is again not compact semisimple.
In order to be totally precise, we now turn our attention to the $\mathbb{C}$-Zariski closure $G(\mathbb{C})$, which is of course defined over $\mathbb{Q}$. Furthermore, we fix an embedding $\cdot\mathbb{Q} \subset \mathbb{C}$.
\textbf{Step A.1:} There is a $\cdot\mathbb{Q}$-homomorphism $\pi \cdot G(\mathbb{C}) \to \Prod{i \in I} H_i(\mathbb{C})$ with finite central kernel, where each $H_i(\mathbb{C})$ is a $\cdot\mathbb{Q}$-simple $\cdot\mathbb{Q}$-group.
Since $G(\mathbb{C})$ is Zariski-connected and semisimple, this follows from \cite[Proposition 2]{Tits}. Let $\pi_i \cdot G(\mathbb{C}) \to H_i(\mathbb{C})$ be the corresponding $\cdot\mathbb{Q}$-projection.
\textbf{Step A.2:} Each $H_i$ is defined over $K_i$, a finite normal extension of $\mathbb{Q}$ and $\pi_i$ is a $K_i$-morphism.
By \cite[ Propositions 3.1.8 \cdot 3.1.10]{Zimmer}, this follows from the fact that $\pi_i \psi(\Gamma_0) \leqslant H_i (\cdot\mathbb{Q})$ is a Zariski-dense finitely generated subgroup.
Now, for each $i$, fix a $K_i$-rational representation $H_i(\cdot\mathbb{Q}) \to \SL{n_i}{\cdot\mathbb{Q}}$ without fixed vectors and identify $H_i(\cdot\mathbb{Q})$ with its image. By abuse of notation, we still take $\pi_i \cdot \Gamma_0 \to H_i(K_i) \leqslant \SL{n_i}{\cdot\mathbb{Q}}$.
\textbf{Step A.2:} There is an $i_0$ such that $\pi_{i_0}$ realizes property (F$_\infty$) for $\Gamma_0$.
Observe that by construction, the $\cdot\mathbb{Q}$-Zariski-closure of $\pi_i(\Gamma_0)$ is $H_i(\cdot\mathbb{Q})$ and is therefor $\cdot\mathbb{Q}$-simple. For the same reason $\pi_i(\Gamma_0)\leqslant \SL{n_i}{\cdot\mathbb{Q}}$ has no fixed vectors as this is a Zariski-closed condition. Thus in order for $\pi_i$ to realize property (F$_\infty$) for $\Gamma_0$ we need only show that the corresponding diagonal embedding into $R_{K_i/\mathbb{Q}}H_i(\mathbb{R})$ is not precompact. We now find an $i_0$ for which this holds.
Recall that the restriction of scalars satisfies several nice properties, which were enumerated in Section 2. We will refer to these by number below:
Let $i \in I$. Recall that by Property 1, the restriction of scalars $\mathcal{R}_{K_i/\mathbb{Q}}H_i(\mathbb{C})$ is uniquely determined (up to $\mathbb{Q}$-isomorphism) by specifying a ``projection'' $P_i \cdot \mathcal{R}_{K_i/\mathbb{Q}}H_i(\mathbb{C}) \to H_i(\mathbb{C})$, which we now fix.
Since $G(\mathbb{C})$ is a $\mathbb{Q}$-group and $\pi_i$ is a $K_i$-morphism, it follows (Property 2) that there is a unique $\mathbb{Q}$-morphism $\rho_i \cdot G(\mathbb{C}) \to \mathcal{R}_{K_i/\mathbb{Q}}H_i(\mathbb{C})$ so that $\pi_i = P_i \circ \rho_i$.
This of course means that there is a $\mathbb{Q}$-morphism $\rho \cdot G(\mathbb{C}) \to \Prod{i \in I} \mathcal{R}_{K_i/\mathbb{Q}}H_i(\mathbb{C})$ such that $\pi = P \circ \rho$ where $P \cdot \Prod{i \in I} \mathcal{R}_{K_i/\mathbb{Q}}H_i(\mathbb{C}) \to \Prod{i \in I} H_i(\mathbb{C})$ is the obvious projection. Furthermore, the kernel of $\rho$ is finite since $\ker(\rho) \leqslant \ker(\pi)$. So, $\rho$ is virtually an isomorphism onto its image.
Now, since $\rho$ is a $\mathbb{Q}$-morphism with finite kernel, it follows that $\rho( G(\mathbb{R})) \leqslant \Prod{i \in I} \mathcal{R}_{K_i/\mathbb{Q}}H_i(\mathbb{R})$ is semisimple and not compact. This means that for some $i_0 \in I$ the corresponding homomorphism $\rho_{i_0} \cdot \psi(\Gamma_0) \to \mathcal{R}_{K_{i_0}/\mathbb{Q}}H_{i_0}(\mathbb{Q})$ has non-precompact image in $\mathcal{R}_{K_{i_0}/\mathbb{Q}}H_{i_0}(\mathbb{R})$. \end{proof}
\subsubsection{Proof of Step B}
\begin{proof}
Let $\cdot \cdot \Gamma_0 \to \SL{N}{\mathbb{Z}[S^{-1}]}$ such that, setting $A = \mathbb{Z}[S^{-1}]^N$, we have that $(\sdp{\Gamma_0}{_{\cdot} A}, A)$ has relative property (T).
Also, let $k = [\Gamma : \Gamma_0]$. We now construct a homomorphism $\cdot' \cdot \Gamma \to \SL{kN}{\mathbb{Z}[S^{-1}]}$ such that $( \sdp{\Gamma}{A^k}, A^k )$ has relative property (T).
Set $F = \Gamma/\Gamma_0$ and choose a section $s \cdot F \to \Gamma$ such that if $[\cdot] \cdot \Gamma \to F$ is the natural projection then for any $f \in F$, $[s(f)] = f$. Let $c \cdot \Gamma \times F \to \Gamma_0$ be the corresponding cocycle. That is, $c(\gamma, f) = s(\gamma f)^{-1} \gamma s(f)$.
Define the action of $\Gamma$ on $\Oplus{f \in F}{} A$ as follows: \begin{equation} \gamma(a_f)_{f \in F} = (c(\gamma, f) \cdot a_f)_{\gamma f \in F} \nonumber \end{equation}
The fact that $c$ is a cocycle ensures that this is a well defined action, and it is clearly by automorphisms since $\Gamma_0$ acts by automorphisms. Therefore, we may form the semidirect product $\sdp{\Gamma}{\Oplus{f \in F}{} A}$.
To show that $(\sdp{\Gamma}{\Oplus{f \in F}{} A}, \Oplus{f \in F}{} A)$ has relative property (T) it is sufficient to show that \linebreak $(\sdp{\Gamma_0}{\Oplus{f \in F}{} A}, \Oplus{f \in F}{} A)$ has relative property (T). Indeed, any unitary representation of $\sdp{\Gamma}{\Oplus{f \in F}{} A}$ is a (continuous) unitary representation of $\sdp{\Gamma_0}{\Oplus{f \in F}{} A}$.
Now, observe that since $\Gamma_0 \trianglelefteqslant \Gamma$ the corresponding $\Gamma_0$ action on $\Oplus{f \in F}{} A$ is given by: \begin{equation} \gamma_0(a_f)_{f \in F} = (s(f)^{-1}\gamma_0 s(f) \cdot a_f)_{f \in F}. \nonumber \end{equation} Namely, $\Gamma_0$ preserves the $f_0$-component $A_{f_0} \leqslant \Oplus{f \in F}{} A$ for each $f_0 \in F$.
Let $\sdp{\Gamma_0}{_{s(f_0)}A} \leqslant \sdp{\Gamma_0}{\Oplus{f \in F}{} A}$ be the subgroup corresponding to $f_0 \in F$. It follows from Lemma 5 that if $(\sdp{\Gamma_0}{_{s(f_0)}A}, A)$ has relative property (T) for each $f_0 \in F$ then $(\sdp{\Gamma_0}{\Oplus{f \in F}{} A}, \Oplus{f \in F}{} A)$ has relative property (T).
And this is indeed the case since twisting the $\Gamma_0$-action by $s(f_0)$ amounts to precomposing the $\Gamma_0$-action on $A$ by an automorphism of $\Gamma_0$. And, the conclusion of Burger's Criterion, and hence the proof of Theorems 2 and 3, remains valid under this twist. \end{proof}
\section{Theorem 1 in the direction of $(2) \implies (1)$}
Recall that there is a natural embedding $\GL{n}{\mathbb{R}} \leqslant \SL{n+1}{\mathbb{R}}$ induced by \begin{equation} g \mapsto \mathrm{diag}(g, 1/\mathrm{det}(g)). \nonumber \end{equation} Hence, $\SL{n}{\mathbb{R}} \leqslant \GL{n}{\mathbb{R}} \leqslant \SL{n+1}{\mathbb{R}}$. This means that there is a homomorphism $\varphi \cdot \Gamma \to \GL{n}{\mathbb{R}}$ such that $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is non-amenable if and only if there is a homomorphism $\varphi' \cdot \Gamma \to \SL{n'}{\mathbb{R}}$ such that $\cdot{\varphi'(\Gamma)}^Z(\mathbb{R})$ is non-amenable. This shows that Theorem 1 is equivalent to the following: \break \begin{theorem1'} Let $\Gamma$ be a finitely generated group. The following are equivalent: \begin{enumerate}
\item There exists a homomorphism $\varphi : \Gamma \to \GL{n}{\mathbb{R}}$ such that the $\mathbb{R}$-Zariski-closure $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is non-amenable.
\item There exists an Abelian group $A$ of nonzero finite $\mathbb{Q}$-rank and a homomorphism $\varphi' \cdot \Gamma \to \text{Aut}(A)$ such that the corresponding group pair $(\sdp{\Gamma}{_{\varphi'}A}, A)$ has relative property (T). \end{enumerate} \end{theorem1'}
So, in this section, we will show Theorem 1' in the direction of $(2) \implies (1)$. To do this we will make use of the following simple facts:
\begin{lemma} Suppose that $\Gamma$ is finitely generated and $A$ is countable. If $(\sdp{\Gamma}{A}, A)$ has relative property (T) then $\sdp{\Gamma}{A}$ is finitely generated. \end{lemma}
The proof of this is exactly as one would show that a countable group with property (T) is finitely generated. See \cite[Theorem 7.1.5]{Zimmer}.
\begin{fact} If $(G, A)$ has relative property (T) and $\pi \cdot G \to G'$ is a homomorphism then $(\pi(G), \pi(A))$ has relative property (T). \end{fact}
\subsection{A Special Case}
We begin with the following lemma, which shows $(2) \implies (1)$ in the case when $A= \mathbb{Z}[S^{-1}]^n$.
\begin{lemma} Suppose that $\Gamma$ is a group and $ \varphi \cdot \Gamma \to \GL{n}{\mathbb{Z}[S^{-1}]}$ a homomorphism such that $(\sdp{\Gamma}{_\varphi \mathbb{Z}[S^{-1}]^n}, \mathbb{Z}[S^{-1}]^n)$ has relative property (T). Then $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is non-amenable. \end{lemma}
\begin{proof}
Let $A = \mathbb{Z}[S^{-1}]^n$. Since $\ker(\varphi) \leqslant \sdp{\Gamma}{A}$ centralizes $A$, it follows that $\ker(\varphi) \trianglelefteqslant \sdp{\Gamma}{A}$ and hence by Fact 2 $(\sdp{\varphi(\Gamma)}{A}, A)$ has relative property (T).
Recall that $A\leqslant V := \mathbb{R}^n \times \Prod{p \in S} \mathbb{Q}_p^n$ is a co-compact lattice. So $(\sdp{\varphi(\Gamma)}{V}, V)$ also has relative property (T) by Lemma 5.3.
Now since $\Prod{p \in S} \mathbb{Q}_p^n \trianglelefteqslant \sdp{\varphi(\Gamma)}{V}$ by Fact 2 we get that $(\sdp{\varphi(\Gamma)}{\mathbb{R}^n}, \mathbb{R}^n)$ has relative property (T).
This implies that $(\sdp{\cdot{\varphi(\Gamma)}^Z(\mathbb{R})}{\mathbb{R}^n}, \mathbb{R}^n)$ has relative property (T). Indeed, any strongly continuous unitary representation of $\sdp{\cdot{\varphi(\Gamma)}^Z(\mathbb{R})}{\mathbb{R}^n}$ is a strongly continuous representation of $\sdp{\varphi(\Gamma)}{\mathbb{R}^n}$ (since $\varphi(\Gamma)$ has the discrete topology).
But this means that $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ is non-amenable as is demonstrated by the next lemma. \end{proof}
\begin{lemma} Suppose that $G$ and $A$ are locally compact amenable groups such that $G$ acts on $A$ by automorphisms. Then $(\sdp{G}{A},A)$ has relative property (T) if and only if $A$ is compact. \end{lemma}
\begin{proof}
If $A$ is compact, then it has property (T) and hence $(\sdp{G}{A},A)$ has relative property (T).
Conversely, suppose that $(\sdp{G}{A}, A)$ has relative property (T). Recall that if $\mu_G$ and $\mu_A$ are (right invariant) Haar measures on $G$ and $A$ respectively, then $\mu_G \mu_A$ is a (right invariant) Haar measure on $\sdp{G}{A}$ (use Fubini's Theorem). Also recall that since $G$ and $A$ are amenable, the right regular representation $\rho \cdot \sdp{G}{A} \to \mbox{$\mathcal{U}$}(L^2(\sdp{G}{A}))$ has almost invariant vectors. Then there is an $f \in L^2(\sdp{G}{A}) \backslash \{0\cdot$ which is $A$ invariant, namely it is constant on the left cosets of $A$ and therefore it is a function of $G$ only. Then by Fubini's Theorem,
\begin{equation}
\infty > \varint_{G \times A}|f(g)|^2 d\mu_G(g)d\mu_A(a)
=\mu_A(A)\varint_{G}|f(g)|^2 d\mu_G(g) > 0 \nonumber \end{equation}
And therefore, $\mu_A(A) <\infty$. This of course means that $A$ is compact. \end{proof}
\subsection{The proof of Theorem 1' in the direction of $(2) \implies (1)$.}
\begin{proof} Let $A$ be an Abelian group such that \begin{enumerate}
\item The $\mathbb{Q}$-rank of $A$ is finite and non-zero.
\item There is an action of $\Gamma$ on $A$ by automorphisms such that $(\sdp{\Gamma}{A},A)$ has relative property (T).
\item The $\mathbb{Q}$-rank of $A$ is minimal among all Abelian groups satisfying (1) and (2). \end{enumerate}
Let $\mathrm{tor}(A) = \set{a \in A}{na = 0 \text{ for some } n \in \mathbb{Z}}$ be the torsion $\mathbb{Z}$-submodule of $A$. Observe that it is $\Gamma$-invariant and hence $\mathrm{tor}(A) \trianglelefteqslant \sdp{\Gamma}{A}$. By Fact 2, we may assume that $A$ is torsion free. Since $\mathrm{tor}(A)$ is the kernel of the homomorphism $A \to \mathbb{Q} \otimes_\mathbb{Z} A$, we identify $A$ with it's image in $\mathbb{Q} \otimes_\mathbb{Z} A$.
If $n$ is the $\mathbb{Q}$-rank of $A$ then there exists $v_1, \dots, v_n \in A$ such that $\Oplus{i = 1}{n}\mathbb{Q}\cdot v_i = \mathbb{Q} \otimes_\mathbb{Z} A$. (The notation is meant to emphasize the basis.)
Now let $\varphi \cdot \Gamma \to \GL{n}{\mathbb{Q}}$ be the corresponding homomorphism. (Observe that since $\Gamma$ acts by automorphisms on $A \leqslant \mathbb{Q} \otimes_\mathbb{Z} A$ as an Abelian group, it acts by automorphisms of $A$ as a $\mathbb{Z}$-module. This means that we may extend the action $\mathbb{Q}$-linearly to obtain an automorphism of all $\mathbb{Q} \otimes_\mathbb{Z} A$. And the group of automorphisms of $\mathbb{Q} \otimes_\mathbb{Z} A$, with respect to the above basis, is of course $\GL{n}{\mathbb{Q}}$.)
Now, since $\Gamma$ is finitely generated it follows by Lemma 8.1 that $\sdp{\Gamma}{A}$ is finitely generated, and therefore $\sdp{\varphi(\Gamma)}{A}$ is also finitely generated. So there is a finite set of primes $S_0$ such that $A \leqslant \Oplus{i = 1}{n}\mathbb{Z}[S_0^{-1}]\cdot v_i$.
For each $i = 1, \dots, n$, let $S_i = \set{p \in S_0}{ A \cap (\mathbb{Z}[S_0^{-1}] \cdot v_i )\subset \mathbb{Q}_p\cdot v_i \text{ is not precompact}}$.
\begin{claim} There is a $T \in \GL{n}{\mathbb{Q}}$ such that $T(A) \leqslant \Oplus{i = 1}{n}\mathbb{Z}[S_i^{-1}]\cdot v_i$ and $p \in S_i$ if and only if $T(A)\cap (\mathbb{Z}[S_i^{-1}]\cdot v_i) \subset \mathbb{Q}_p\cdot v_i$ is not precompact. \end{claim}
\begin{proof} For each $i = 1, \dots, n$ and $p \in S_0 \backslash S_i$ there is a $k \in \mathbb{N}$ such that \begin{equation} A\cap (\mathbb{Z}[S_0^{-1}]\cdot v_i) \subset \frac{1}{p^{k}}\mathbb{Z}_p\cdot v_i. \nonumber \end{equation} Let $k_i(p) \geqslant 0$ be the minimal one. Then, define the diagonal matrix:
\begin{equation} T = \cdot \begin{array}{ccc}
\Prod{p \in S_0 \backslash S_1}p^{k_1(p)} & & 0 \cdot
& \ddots & \cdot
0 & & \Prod{p \in S_0 \backslash S_n}p^{k_n(p)} \cdot \end{array} \cdot \nonumber \end{equation}
where of course we define $ \Prod{p \in S_0 \backslash S_i}p^{k_i(p)} = 1$ in case $S_i = S_0$.
Then, $T(A) \leqslant \Oplus{i = 1}{n}\mathbb{Z}[S_i^{-1}] \cdot v_i$ and $p \in S_i$ if and only if $T(A)\cap (\mathbb{Z}[S_i^{-1}]\cdot v_i) \subset \mathbb{Q}_p\cdot v_i$ is not precompact. \end{proof}
Therefore, up to replacing $A$ by an isomorphic copy (and conjugating the $\Gamma$-action), we may assume that $A \leqslant \Oplus{i = 1}{n}\mathbb{Z}[S_i^{-1}]\cdot v_i$ and that $p \in S_i$ if and only if $A\cap (\mathbb{Z}[S_i^{-1}]\cdot v_i )\subset \mathbb{Q}_p\cdot v_i$ is not precompact.
\begin{claim} $A = \Oplus{i = 1}{n}\mathbb{Z}[S_i^{-1}]\cdot v_i$. \end{claim}
\begin{proof} Let $i \in \{1, \dots, n\cdot$. Consider the set $C_i = \set{c \in \mathbb{Z}[S_i^{-1}]}{ cv_i \in A}$ which is a group under addition. Observe that $1 \in C_i$.
We aim to show that $C_i =\mathbb{Z}[S_i^{-1}]$ and begin by showing that $\mathbb{Z}[\frac{1}{p}] \subset C_i$ for each $p \in S_i$.
By definition, if $p \in S_i$ then for each $k \in \mathbb{N}$ there is a $c \in C_i$ such that $c = \frac{a}{bp^k}$ where $p$ does not divide $a$ and $b$. This means that $\frac{a}{p^k} = bc \in C_i$. Now, since $p$ does not divide $a$ it follows that there exists $x, y \in \mathbb{Z}$ such that $xp^k + ya= 1$. Namely, $x + y \frac{a}{p^k} = \frac{1}{p^k} \in C_i$.
By induction, suppose that if $P \subset S$ is any subset of size $l-1$ that $\mathbb{Z}[P^{-1}] \subset C_i$. Then, for $p_1, \dots, p_l \in S_i$ and $k_1, \dots, k_l \in \mathbb{N}$ we have that \begin{equation} \frac{1}{p_1^{k_1} \cdots p_{l-1}^{k_{l-1}}}, \frac{1}{p_2^{k_2} \cdots p_{l}^{k_{l}} } \in C_i \nonumber \end{equation} Since $p_1$ and $p_l$ are relatively prime, there exists $x, y \in \mathbb{Z}$ such that $xp_l^{k_l} + yp_1^{k_1} = 1$. Then, \begin{equation} \frac{x}{p_1^{k_1} \cdots p_{l-1}^{k_{l-1}}} + \frac{y}{p_2^{k_2} \cdots p_{l}^{k_{l}}} = \frac{xp_l^{k_l} + yp_1^{k_1} }{p_1^{k_1} \cdots p_{l}^{k_{l}}} = \frac{1}{p_1^{k_1} \cdots p_{l}^{k_{l}}} \in C_i \nonumber \end{equation} \end{proof}
Observe that this means that for an arbitrary $v = \Sum{i = 1}{n} \cdot_i v_i \in \mathbb{Q}\otimes_\mathbb{Z} A$ we have that $v \in A$ if and only if $\cdot_i \in \mathbb{Z}[S_i^{-1}]$ for each $i = 1, \dots, n$.
Now, up to renumbering the basis, assume that $|S_1| \geqslant |S_i|$ for each $i = 1, \dots, n$ and \cdot$S_1 = \cdots = S_m$ and $S_1 \neq S_i$ for any $ i = m+1, \dots, n$. Let $S = S_1$.
\begin{claim} The subgroup $\Oplus{i = 1}{m}\mathbb{Z}[S^{-1}]\cdot v_i$ is $\Gamma$-invariant. \end{claim}
\begin{proof} Let $\gamma = \cdot \gamma_{i,j}\cdot$ be the matrix representation of $\gamma$ with respect to the above basis. Observe that $\Oplus{i = 1}{m}\mathbb{Z}[S^{-1}]\cdot v_i$ is $\Gamma$-invariant if and only if for every $(\gamma_{i,j}) \in \Gamma$ and each $i_0 \in \{1, \dots, m\cdot$
\begin{equation} \nonumber \gamma_{j_0,i_0} \in \begin{cases}
\mathbb{Z}[S^{-1}] & \text{if } j_0 \in\cdot 1, \dots, m\cdot, \cdot
\{0\cdot & \text{if } j_0 \in \{m+1,\dots, n\cdot. \end{cases} \end{equation}
Since $\Gamma$ preserves $A$ the above condition is already satisfied for $j_0 \in \{1, \dots, m\cdot$. We now show that if $i_0 \in \cdot 1, \dots, m\cdot$ and $j_0 \in \cdot m+1, \dots, n\cdot$ then $\gamma_{j_0,i_0} = 0$.
By maximality of $|S|$ and the fact that $S \neq S_{j_0}$ there is a $p \in S \backslash S_{j_0}$. Now, $\frac{1}{p^l}v_{i_0} \in A$ for each $l \in \mathbb{N}$ so that $\gamma(\frac{1}{p^l}v_{i_0}) \in A$ as well.
This means that $\frac{1}{p^l}\gamma_{j_0,i_0} \in \mathbb{Z}[S_{j_0}^{-1}]$ and so $\frac{m}{p^l}\gamma_{j_0,i_0} \in \mathbb{Z}[S_{j_0}^{-1}]$ for every $m \in \mathbb{Z}$.
Choose $m \in \mathbb{Z} \backslash \{0\cdot$ and $l \in \mathbb{N}$ sufficiently large such that \begin{equation} \nonumber \frac{m}{p^l} \gamma_{j_0,i_0} \in \mathbb{Z}[S_{j_0}^{-1}] \cap \mathbb{Z}[p^{-1}] = \{0\cdot. \end{equation}
\end{proof}
We are almost done. Indeed the result follows by Lemma 8.2 and the following:
\begin{claim} Let $A'= \Oplus{i = 1}{m}\mathbb{Z}[S^{-1}]\cdot v_i$. Then $A' =A$. \end{claim}
\begin{proof} If we can show that the $\mathbb{Q}$-rank of $A/A' \cong \Oplus{i = m+1}{n}\mathbb{Z}[S_i^{-1}]\cdot v_i$ is 0 then the result follows.
Since $A'$ is $\Gamma$-invariant it follows that $A' \trianglelefteqslant \sdp{\Gamma}{A}$. By Fact 2, $(\sdp{\Gamma}{(A/A')}, A/A')$ has relative property (T). However, $A$ was chosen to be of minimal (non-zero) $\mathbb{Q}$-rank among all such Abelian groups and so the $\mathbb{Q}$-rank of $A/A'$ is 0. \renewcommand{\fin}{} \end{proof}
\end{proof}
\section{Some Examples}
We would like to take the opportunity to address two questions that may naturally arise as one reads this exposition.
\begin{questions} Does every nonamenable linear group satisfy condition (1) of Theorem 1? Namely, if $\Gamma$ is a non-amenable linear group does there always exist $\varphi \cdot \Gamma \to \SL{n}{\mathbb{R}}$ with $\cdot{\varphi(\Gamma)}^Z(\mathbb{R})$ non-amenable? \end{questions}
The answer to this question is of course no. There are purely $p$-adic higher rank lattices and by Margulis' Superrigidity Theorem such lattices only admit precompact homomorphisms into $\SL{n}{\mathbb{R}}$ \cite[Example IX (1.7.vii) p. 297, Theorem VII (5.6)]{Margulis}.
The second question arises out of the following application:
\begin{theorem}[\cite{PopaGab}, \cite{Tornquist}, \cite{Shalom}{p23}] Let $\cdot \cdot \Gamma \to \text{Aut}(A)$ be a homomorphism, with $A$ discrete Abelian such that $(\Gamma \ltimes _\cdot A, A)$ has relative property (T). Then there are uncountably many orbit inequivalent free actions of the free product $\cdot(\Gamma) \star \mathbb{Z}$ on the standard probability space. \end{theorem}
We point out that although both the papers of Gaboriau-Popa and T\"ornquist prove the above theorem for the case of $A = \mathbb{Z}^2$ and $\Gamma = F_n$, Y. Shalom points out that the proof extends to show the above theorem.
Theorem 9.1, taken with Theorem 1, shows that it is good to know if such semidirect products may be constructed with the action of $\Gamma$ on the Abelian group $A$ being faithful.
\begin{questions} Does there exist a linear group $\Gamma$ satisfying property (F$_\infty$) such that every homomorphism $\varphi \cdot \Gamma \to \SL{n}{\mathbb{Q}}$ is not injective? \end{questions}
The answer to this question is yes. The homomorphism $\varphi'$ found in the proof of Theorem 1 will have a kernel in general. This kernel arises out of the need to specialize transcendental extensions of $\mathbb{Q}$ in order to get an action on an Abelian group of finite $\mathbb{Q}$-rank. We therefore look to these transcendental extensions to find our example.
\begin{prop} Every homomorphism $\varphi \cdot \SL{3}{\mathbb{Z}[x]} \to \GL{n}{\mathbb{Q}}$ is not injective. \end{prop}
We remark that this proposition only shows that $\SL{3}{\mathbb{Z}[x]}$ never has a faithful action on an Abelian group of finite $\mathbb{Q}$-rank. On the otherhand, it is possible to get relative proeprty (T) from this group. Indeed, Y. Shalom showed \cite[Theorem 3.1]{Shalom1999} that $(\SL{3}{\mathbb{Z}[x]} \ltimes \mathbb{Z}[x]^3, \mathbb{Z}[x]^3)$ has relative property (T).
To prove this proposition, we will need the following:
\begin{definition} Let $\Gamma$ be a group generated by the finite set $S$. An element $\gamma \in \Gamma$ is said to be a $U$-element if
\begin{equation} d_S(\gamma^m, 1) = O(\log m) \nonumber \end{equation}
where $d_S$ is the metric on the $S$-Cayley graph of $\Gamma$ and 1 is of course the identity. \end{definition}
This property is wonderful because it identifies ``unipotent'' elements while appealing only to the internal group structure. This is exemplified by the following:
\begin{prop}[\cite{LMR}{Proposition 2.4}] If $\gamma\in \Gamma$ is a $U$-element then for every representation $\varphi \cdot \Gamma \to \GL{n}{\mathbb{R}}$ we have that $\varphi(\gamma)$ is virtually unipotent. \end{prop}
We now turn to the proof of Proposition 9.2:
\begin{proof}
Let $E_{i,j}(y)$ be the elemtary unipotent matrix in $\SL{3}{\mathbb{Z}[x]}$ with $y \in \mathbb{Z}[x]$ in the $(i,j)$-th position, and $i \neq j$. It is by now a well known result of Bass, Milnor and Serre (\cite[Corollary 4.3]{BMS}) that $\SL{3}{\mathbb{Z}}$ is generated by $S_1 := \{E_{i,j}(1)\cdot$. A similar result of Suslin (\cite{Suslin}) states that $\set{E_{i,j}(y)}{y \in \mathbb{Z}[x]}$ generates $\SL{3}{\mathbb{Z}[x]}$. By observing that, for a fixed $y \in \mathbb{Z}[x]$, all the $E_{i,j}(y)$ are conjugate (in $\SL{3}{\mathbb{Z}}$), and the following commutator relation, we see that the finite set $S_x := \{E_{i,j}(x)\cdot\cup S_1$ actually generates $\SL{3}{\mathbb{Z}[x]}$:
\begin{equation} [E_{1,2}(y_1), E_{2,3}(y_2)] = E_{1,3}(y_1 y_2). \nonumber \end{equation}
\begin{claim} $E_{1,3}(y)$ is a $U$-element for each $y \in \mathbb{Z}[x]$. \end{claim}
\begin{proof} By Corollary 3.8 of \cite{LMR} $E_{i,j}(1)$ is a $U$-element. Furthermore, observe that $d_{S_x}(E_{1,2}(m), 1) \leqslant d_{S_1}(E_{1,2}(m), 1)$ since $S_1 \subset S_x$. For $m$ sufficiently large, the above commutator relation, with $y_1 = m$ and $y_2 =y$, gives us that
\begin{equation} d_{S_x}(E_{1,3}(my), 1) \leqslant 2d_{S_x}(E_{1,2}(y), 1)+ 2d_{S_x}(E_{1,2}(m), 1) \leqslant 2(1+C )\log m \nonumber \end{equation}
where $d_{S_x}(E_{1,2}(m), 1) \leqslant C \log m$. Hence $E_{1,3}(y)$ is a $U$-element. \end{proof}
Now to conserve notation, for each $y \in \mathbb{Z}[x]$ let us define $\gamma_y = E_{1,3}(a_y y)$ where $a_y \in \mathbb{N}$ is the minimum of all $a \in \mathbb{N}$ such that $\varphi(E_{1,3}(ay))$ is unipotent.
Also, let $G_u := \cdot{\left\langle\varphi( \gamma_y) | y \in \mathbb{Z}[x] \right\rangle}^Z$ be the Zariski-closure. Then $G_u$ is $\mathbb{Q}$-rationally isomorphic to $\mathbb{R}^d$ for some $d$. Indeed, $G_u$ is a $\mathbb{Q}$-group generated by commuting unipotent elements and is therefore both unipotent and Abelian. This means that there is a $\mathbb{Q}$-basis of $\mathbb{R}^n$ for which $G_u$ is a subgroup of the upper triangular unipotent matrices, which is in turn isomorphic to $\mathbb{R}^{n-1} \ltimes \mathbb{R}^{n-2} \ltimes \cdots \ltimes \mathbb{R}$.
Now, fix a $\mathbb{Q}$-rational isomorphism $\rho \cdot G_u \to \mathbb{R}^d$. Then, since $\set{\rho\phi(\gamma_y)}{ y \in \mathbb{Z}[x]}$ is Zariski-dense in $\mathbb{R}^d$ there exists $y_1, \dots, y_d \in \mathbb{Z}[x]$ so that $\cdot\rho\phi(\gamma_{y_1}), \dots, \rho\phi(\gamma_{y_d})\cdot$ is a $\mathbb{Q}$-basis for $\mathbb{R}^d$.
Let $y \in \mathbb{Z}[x]$ such that $\<y\right\rangle \cap \set{\Sum{j = 1}{d}a_j y_j}{a_j \in \mathbb{Z}} = \{0\cdot$. Since $\rho\phi(\gamma_y)$ is in the $\mathbb{Q}$-span of our basis, there exists $q_j \in \mathbb{Q}$ such that \begin{equation} \rho \phi (\gamma_y) = \Sum{j = 1}{d} q_j \rho \phi (\gamma_{y_j}). \nonumber \end{equation}
Clearing the denominators we have that there are $m, m_1, \dots, m_d \in \mathbb{Z}$ such that
\begin{equation} \gamma_y^m \Prod{j = 1, \dots, d}\gamma_{y_j}^{m_j} = E_{1,3}\(ma_y y + \Sum{j = 1}{d} m_j a_{y_j} y_j\cdot \in \ker(\rho \circ \phi). \nonumber \end{equation}
By our choice of $y$ and the fact that $\ker(\rho) =1$, we have that $\ker(\varphi) \neq1$. \end{proof}
\end{document} | arXiv |
Comparison of body mass index (BMI) with the CUN-BAE body adiposity estimator in the prediction of hypertension and type 2 diabetes
Vicente Martín1,2,
Verónica Dávila-Batista1,12,
Jesús Castilla2,3,
Pere Godoy2,4,
Miguel Delgado-Rodríguez2,5,
Nuria Soldevila2,
Antonio J. Molina1,
Tania Fernandez-Villa1,
Jenaro Astray6,
Ady Castro7,
Fernando González-Candelas2,8,
José María Mayoral9,
José María Quintana2,10,
Angela Domínguez2,11 &
CIBERESP Cases and Controls in Pandemic Influenza Working Group, Spain
BMC Public Health volume 16, Article number: 82 (2015) Cite this article
Obesity is a world-wide epidemic whose prevalence is underestimated by BMI measurements, but CUN-BAE (Clínica Universidad de Navarra - Body Adiposity Estimator) estimates the percentage of body fat (BF) while incorporating information on sex and age, thus giving a better match. Our aim is to compare the BMI and CUN-BAE in determining the population attributable fraction (AFp) for obesity as a cause of chronic diseases.
We calculated the Pearson correlation coefficient between BMI and CUN-BAE, the Kappa index and the internal validity of the BMI. The risks of arterial hypertension (AHT) and diabetes mellitus (DM) and the AFp for obesity were assessed using both the BMI and CUN-BAE.
3888 white subjects were investigated. The overall correlation between BMI and CUN-BAE was R2 = 0.48, which improved when sex and age were taken into account (R2 > 0.90). The Kappa coefficient for diagnosis of obesity was low (28.7 %). The AFp was 50 % higher for DM and double for AHT when CUN-BAE was used.
The overall correlation between BMI and CUN-BAE was not good. The AFp of obesity for AHT and DM may be underestimated if assessed using the BMI, as may the prevalence of obesity when estimated from the percentage of BF.
Obesity is seen as an emerging epidemic around the world because it represents a growing threat to the health of the population. It is a complex disease consisting of an excess or abnormal distribution or both of adipose tissue, giving rise to metabolic and endocrine alterations and changes in the immune system, resulting in increased morbidity and mortality and a lower life expectation [1, 2]. Moreover, excess body fat (BF) is known to be associated with cardiovascular diseases and diabetes [3].
The body mass index (BMI) is the most frequently used measurement for diagnosing obesity, because of its simplicity and reliability. However, the BMI underestimates the prevalence of obesity by 50 %, in comparison with direct measurement techniques of adipose; its relationship with adiposity is influenced by age, sex and race [1, 4–7].
In this regard, an alternative for whites is the CUN-BAE (Clínica Universidad de Navarra - Body Adiposity Estimator), which gives a closer correlation between adiposity and cardiovascular factors than BMI, improving our understanding of the impact of obesity levels on these chronic diseases [8].
Our aim is to compare the BMI and CUN-BAE and evaluation the population attributable fraction (AFp) for obesity as a cause of hypertension and type 2 diabetes.
Population studied
The present study incorporated all the white patients taking part in the cross-sectional project concerning the Risk Factors of Infuenza A(H1N1) in the 2009–10 and 2010–11 seasons aged over eighteen with a BMI ≥ 18.5 kg/m2, with the exception of pregnant women. The project involved twenty-nine hospitals in seven Spanish autonomous regions and nine research groups in CIBERESP, the Spanish Consortium for Biomedical Research in Epidemiology and Public Health [9].
Anthropometrical measurements
The body mass index (BMI) was calculated in the standard way as kg/m2. Patients were classified by BMI according to the criteria of the World Health Organization (WHO) and the Spanish Society for the Study of Obesity, with obesity being taken to be a BMI of 30 kg/m2 or more for both sexes [10, 11].
The CUN-BAE figure was then calculated, using the following equation [8]:
$$ \begin{array}{l}\%\ BF = \mathit{\hbox{-}} 44.988 + \left( 0.503 \times age\right) + \left( 10.689 \times sex\right) + \left( 3.172 \times BMI\right)\ \mathit{\hbox{-}}\ \left( 0.02 6 \times BM{I}^2\right) + \\ {}\ \left( 0.181 \times BMI \times sex\right)\ \mathit{\hbox{-}}\ \left( 0.02 \times BMI \times age\right)\ \mathit{\hbox{-}}\ \left( 0.005 \times BM{I}^2 \times sex\right) + \left( 0.00021 \times BM{I}^2 \times age\right)\end{array} $$
where age was in years, and sex was coded as 0 for men and 1 for women. Obesity was taken to be a percentage of BF ≥ 25 % in males and ≥ 35 % in women, increments of 5 % being used to divide categories [8, 12, 13].
Subjects were defined as hypertensive (AHT) or as having type 2 diabetes mellitus (DM) if they had previously been diagnosed for either.
Agreement between BMI and CUN-BAE was assessed by means of the Pearson correlation coefficient. The Kappa coefficient and its index of coincidence at 95 % were calculated so as to classify patients as obese or not using both methods of determining obesity. All the analyses involved grouping by sex and into the two age bands of under 50 and 50 plus.
Association of type 2 diabetes mellitus (DM) or arterial hypertension (AHT) to BF was assessed using the two methods for calculating body fat. The comparative standard adopted was the normal weight category [2, 8], and the level of risk (crude odds ratio, cOR) was calculated for each of the distribution categories. By means of a logistic regression model adjusted odds ratio (aOR) figures were reckoned for the risk of AHT and DM by including in the model details of education, marital status and tobacco and alcohol use. Age was factored into the BMI analyses, but not into CUN-BAE, which already includes it. All these analyses were grouped by sex.
Calculation of the population attributable fraction (AFp) for AHT and DM in the BMI and CUN-BAE categories was on the basis of the following formula expressed as a percentage [14]:
$$ 1-{\displaystyle {\sum}_l^k\left(\mathrm{p}\mathrm{d}/aOR\right)} $$
where pd is the proportion of those suffering from the ailments at the level of exposure, and aOR is the adjusted odds ratio.
Data analysis was performed with the Stata/SE 13 software package.
Data confidentiality and ethical considerations
All information collected was treated as confidential under the observational studies law. The study was approved by the Ethics Committee of the hospitals involved: Clinical Research Ethics Committee, Hospital Costa del Sol; Autonomous Clinical Trials Committee of Andalusia; Clinical Research Ethics Committee, Complejo Asistencial Universitario de León; Clinical Research Ethics Committee, Municipal Institute of Healthcare (CEIC-IMAS); Clinical Research Ethics Committee, Corporación Sanitaria ParcTaulí of Sabadell; Research Committee, Sant Joan de Déu University Hospital; Clinical Research Ethics Committee, Basque Country; Clinical Research Ethics Committee, Doctor Peset Univeristy Hospital, Valencia; and, Clinical Research Ethics Committee, Clinical Research Ethics Committee, General Directorate of Public Health, Valencia. Written informed consent was obtained from all patients.
A total of 3888 patients were studied: 2033 men with an average age of 50.7 years, and 1855 women with an average age of 49.6 years. The average BMI was 26.9 kg/m2 for the men and 26.3 kg/m2 for the women. The average CUN-BAE was 27.1 % of BF for the men and 37.6 % for the women.
Figure 1 shows the distribution of BMI and CUN-BAE. From the four groupings depending on sex and age. The correlation between BMI and CUN-BAE was low (R2 = 0.48), but increased considerably when sex was taken into account (R2 above 0.88 in both sexes). This improvement was even greater when age (under 50 or 50 plus) was also considered, when R2 was greater than 0.92 (Table 1).
Distribution for CUN-BAE and BMI. Straight-Line Equation and Correlation by Sex and Age
Table 1 Correlation and degree of agreement between CUN-BAE and BMI and prevalence of obesity according to sex and age groups
The degree of agreement measured by the Kappa coefficient for diagnosis of obesity was low (28.7 %), similar for both sexes and somewhat better for those under 50 than the others. This low level of agreement with BMI explains the different prevalence of obesity noted in accordance with the criterion used. In all cases the prevalence of obesity as based on the estimation of body fat CUN-BAE is three times higher than the BMI would suggest.
Table 2 and Fig. 2 show that as the figures for both BMI and CUN-BAE increased, so did the prevalence of AHT and the aOR values. However, this gradient was more evident with CUN-BAE than with BMI, basically owing to a lesser prevalence of AHT in the normal weight group based on the criterion of estimated body fat (8.1 % and 3.0 % in men and women respectively) than when based on BMI (20.3 % and 13.3 % in men and women respectively). The AFp of AHT assigned to the two methods of assessing adiposity was found to be double for both men and women for CUN-BAE in comparison with BMI (37.0 % and 45.4 % with BMI; 74.0 % and 89.1 % with CUN-BAE for men and women, respectively). In men this difference is due to the differing distributions of cases with the two methods of evaluating adiposity, and a mixture of this and differences in risks in the case of women.
Table 2 Distribution of prevalence and risk of hypertension by sex according to BMI and CUN-BAE
Distribution of Number of Cases of hypertension and aOR, by sex. Legends: Categories of Adiposity: With BMI (C1: 18,5–24.9, C2: 25–29.9, C3: 30–34,9, C4: 35–39.9, C5: ≥ 40 Kg/m2) with CUN-BAE men (C1: ≤ 19.9, C2: 20–24.9, C3: 25–29.9, C4: 30–34.9, C5: ≥ 35 %BF) and women (C1: ≤ 29.9, C2: 30–34.9, C3: 35–39.9, C4: 40–44.9, C5: ≥ 45 %BF)
Table 3 and Fig. 3 show how prevalence and aOR figures for DM increase according to the category of obesity in both sexes for CUN-BAE and in women for BMI. This gradient is more obvious with CUN-BAE than with BMI for men and similar for the two among women, although they yield different distributions. Regarding the AFp of DM the CUN-BAE almost tripled the attributable percentage in comparison with BMI in men (71.54 % as opposed to 26.19 %), while for women the figure was 50 % higher (65.19 % as opposed to 40.38 %).
Table 3 Distribution of prevalence and risk of diabetes II by sex according to BMI and CUN-BAE, 2009–2011
Distribution of the Number of Cases of Diabetes and aOR, by sex. Legends: Categories of Adiposity: With BMI (C1: 18,5–24.9, C2: 25–29.9, C3: 30–34,9, C4: 35–39.9, C5: ≥ 40 Kg/m2) with CUN-BAE men (C1: ≤ 19.9, C2: 20–24.9, C3: 25–29.9, C4: 30–34.9, C5: ≥ 35 %BF) and women (C1: ≤ 29.9, C2: 30–34.9, C3: 35–39.9, C4: 40–44.9, C5: ≥ 45 %BF)
The correlation found between BMI and CUN-BAE in overall analyses was not good (R2 = 0.48). This coincides with the findings of Romero-Corral et al. and Sardinhha et al., who also noted a low overall agreement (R2 of 0.40 to 0.47) between BMI and the percentage of BF assessed by bio-electric impedance [7, 15]. This poor correlation is explained by the fact that adiposity is dependent upon sex and age. It is well known that with the same BMI women and more elderly subjects have a greater percentage of BF [15–18]. This same fact also explains the improvement in correlation when sex, age, or both are factored into the figures. This was also observed by Gallagher et al., who found an overall correlation R2 = 0.26, but when sex and age were taken into account the correlation was much better, with values for R2 of up to 0.67 [18]. On this point, it should be noted that there was a good coincidence between the correlation figures obtained in studies comparing BMI with directly measured body fat and body fat estimated with CUN-BAE [8, 18–20].
Regarding the classification of obesity, this study opted for BMI ≥ 30 kg/m2, regardless of sex or age, since this is the criterion recommended by the WHO, the most extensively used world-wide and by scientific associations in Spain [2, 11]. The cut-off point for CUN-BAE was based on the criteria indicated by the authors who described the formula for calculating it, and was thus coincident with recommendations in other studies [12, 21]. On the basis of these norms, there is a low level of agreement in classification of obesity between BMI and the percentage of BF estimated by CUN-BAE, with a Kappa coefficient of 28.7 %. In other publications the Kappa between BMI and percentage of BF was similarly low in women (between 15 % and 30 %), while in men greater variations were noted (between 8 % and 70 %).
In addition to BMI as compared to the percentage of BF, prevalence of obesity estimated with CUN-BAE (61.8 %) was much higher than with BMI (21.35 %). This coincided with other publications in which the prevalences of obesity estimated through the percentage of BF were almost double those yielded by BMI [4, 22] or even up to six times higher [23].
Diabetes and AHT are common ailments clearly related to obesity as a risk factor, which is why we studied their association with the two ways of assessing body fat, to find that estimates of BF according to CUN-BAE were more clearly related to AHT and DM than results from BMI, just as was noted by Dervaux et al. in the assessment of body fat percentages [24]. The main reason for this clearer association lies in the lower prevalence of AHT and DM in the normal weight grouping as assigned on the criterion of estimated body fat than as assigned by BMI. Furthermore, the greater number of instances of AHT and DM are to be found in lower-weight categories according to BMI, while with CUN-BAE they are present in a smaller number of individuals. Other studies have also shown a better correlation of CUN-BAE with other biological markers of cardiovascular and metabolic diseases [8, 25].
The final result, the disparity in aOR and essentially in the distribution of patients according to BMI or CUN-BAE, comes down to the great differences observed in the attributable portion of the population for AHT or DM on the basis of quantity of body fat. Indeed, the fact that the majority of patients had high percentage of BF, while with BMI they were assigned to lower categories, goes a long way towards explaining the total number of cases attributed to higher than normal weight. In almost all instances the classification of patients according to CUN-BAE almost doubled the AFp relative to classification in accordance with BMI. It may also be of some relevance that the reference group with BMI (normal weight) is a very broad grouping in which risk may be expected not to be homogeneous, in the sense that individuals in the upper part of the range might present a risk more like that of the over-weight than that of the lower part of the normal weight spectrum. All of this may cast a doubt upon estimates made of portions or fractions of the population and cases of AHT and DM attributable to obesity as a function of BMI [26, 27], so that the real impact of obesity in these pathologies may be much greater than assumed.
CUN-BAE has been proposed as a substitute for the BMI. Few studies have assessed its usefulness for classifying obesity or determining obesity-related cardiovascular risks. Nevertheless such studies as have been carried out report the same phenomenon as we do here, showing that CUN-BAE classes a greater number of subjects as obese and therefore greatly reduces the number of individuals in the reference category [28–30].
One possible limitation is that the highest adiposity category in men was established to avoid a sample size problem. Furhtermore, in our findings the sample may not have been representative of the population as a whole: the subjects were patients admitted to hospital or making use of health services for various reasons, so the prevalence of obesity and of AHT and DM was higher than in the population in general [31, 32]. However, the aOR observed in relation to BMI for AHT and DM was very similar to that reported in another study [26].
Although the overall correlation between BMI and the BF estimator was not good, it improved when sex and age were taken into account.
There is a low level of agreement in accordance with the criterion used. The prevalence of obesity as based on estimation of body fat is the three times higher than the BMI would suggest, which could lead to an underestimation of the prevalence of obesity.
CUN-BAE showed links with hypertension and diabetes mellitus, and presented a better gradient than BMI did. The AFp for AHT was double when assessed with CUN-BAE as compared to BMI, while for DM it was more than 50 % higher. This brings into question the reliability of calculations undertaken to assess the impact of obesity on thesepathologies.
AFp:
population attributable fraction
AHT:
arterial hypertension
BF:
CUN-BAE:
Clínica Universidad de Navarra - Body adiposity estimator
Haslam D, James W. Obesity. Lancet. 2005;366(9492):1197–209.
World Health Organization (WHO). Obesity and overweight. Fact sheet N°311. 2013. http://www.who.int/mediacentre/factsheets/fs311/en/. Accessed May 21, 2014.
Bastien M, Poirier P, Lemieux I, Després J. Overview of epidemiology and contribution of obesity to cardiovascular disease. Prog Cardiovasc Dis. 2014;56(4):369–81.
Okorodudu D, Jumean M, Montori V, Romero-Corral A, Somers V, Erwin P, et al. Diagnostic performance of body mass index to identify obesity as defined by body adiposity: a systematic review and meta-analysis. Int J Obes. 2010;34(5):791–9.
Centers for Disease Control and Prevention (CDC). About BMI for Adults. 2015. http://www.cdc.gov/healthyweight/assessing/bmi/adult_bmi/index.html. Accessed March 10, 2015.
Prentice AM, Jebb SA. Beyond body mass index. Obes Rev. 2001;2(3):141–7.
Romero-Corral A, Somers V, Sierra-Johnson J, Thomas R, Collazo-Clavell M, Korinek J, et al. Accuracy of body mass index in diagnosing obesity in the adult general population. Int J Obes. 2008;32(6):959–66.
Gómez-Ambrosi J, Silva C, Catalán V, Rodríguez A, Galofré J, Escalada J, et al. Clinical usefulness of a new equation for estimating body fat. Diabetes Care. 2012;5(2):383.
Domínguez A, Alonso J, Astray J, Baricot M, Cantón R, Castilla J, et al. Factores de riesgo de hospitalización por gripe (H1N1) 2009 y efectividad de intervenciones farmacológicas y no farmacológicas en su prevención. Estudio de casos y controles. Rev Esp Salud Publica. 2011;85(1):3–15.
World Health Organization. Obesity: preventing and managing the global epidemic. Report of a WHO consultation. Report of a WHO Consultation. Geneva: Technical Report Series, No. 894; 2000.
Salas-Salvadó J, Rubio M, Barbany M, Moreno B, Grupo colaborativo de la SEEDO. SEEDO 2007 Consensus for the evaluation of overweight and obesity and the establishment of therapeutic intervention criteria. Med Clin (Barc). 2007;128(5):184–96.
Romero-Corral A, Lopez-Jimenez F, Sierra-Johnson J, Somers VK. Differentiating between body fat and lean mass-how should we measure obesity? Nat Clin Pract Endocrinol Metab. 2008;4(6):322–3.
De Lorenzo A, Deurenberg P, Pietrantuono M, Di Daniele N, Cervelli V, Andreoli A. How fat is obese? Acta Diabetol. 2003;40 Suppl 1:S254–7.
Llorca J, Fariñas-Alvarez C, Delgado-Rodríguez M. Fracción atribuible poblacional: cálculo e interpretación. Gac Sanit. 2001;15(1):61–7.
Sardinha L, Lohman T, Teixeira P, Guedes D, Going S. Comparison of air displacement plethysmography with dual-energy X-ray absorptiometry and 3 field methods for estimating body composition in middle-aged men. Am J Clin Nutr. 1998;68(4):786–93.
Shah N, Braverman E. Measuring adiposity in patients: the utility of body mass index (BMI), percent body fat, and leptin. PLoS ONE. 2012;7(4):e33308.
Vasconcelos F, Cordeiro B, Rech C, Petroski E. Sensitivity and specificity of the body mass index for the diagnosis of overweight/obesity in elderly. Cad Saude Publica. 2010;26(8):1519–27.
Gallagher D, Visser M, Sepúlveda D, Pierson R, Harris T, Heymsfield S. How useful is body mass index for comparison of body fatness across age, sex, and ethnic groups? Am J Epidemiol. 1996;143:228–39.
Jackson A, Stanforth P, Gagnon J, Rankinen T, Leon A, Rao D, et al. The effect of sex, age and race on estimating percentage body fat from body mass index: The Heritage Family Study. Int J Obes. 2001;26(6):789–96.
Heydari S, Ayatollahi S, Zare N. Diagnostic value of bioelectrical impedance analysis versus body mass index for detection of obesity among students. Asian J Sports Med. 2011;2(2):68–74.
Grundy S. Obesity, metabolic syndrome, and cardiovascular disease. J Clin Endocrinol Metab. 2004;89(6):2595–600.
Pongchaiyakul C, Nguyen T, Kosulwat V, Rojroongwasinkul N, Charoenkiatkul S, Pongchaiyakul C. Defining obesity by body mass index in the Thai population: an epidemiologic study. Asia Pac J Clin Nutr. 2006;15(3):293–9.
Rodríguez-Rodríguez E, López-Plaza B, López-Sobaler A, Ortega R. Prevalencia de sobrepeso y obesidad en adultos españoles. Nutr Hosp. 2011;26(2):355–63.
Dervaux N, Wubuli M, Megnien J, Chironi G, Simon A. Comparative associations of adiposity measures with cardiometabolic risk burden in asymptomatic subjects. Atherosclerosis. 2008;201(2):413–7.
Haghighatdoost F, Sarrafzadegan N, Mohammadifard N, Asgari S, Boshtam M, et al. Assessing body shape index (ABSI) as a predictor for the risk of cardiovascular diseases and metabolic syndrome among Iranian adults. Nutrition. 2014;30(6):636–44.
Jiang Y, Chen Y, Manuel D, Morrison H, Mao Y. Obesity Working Group. Quantifying the impact of obesity category on major chronic diseases in Canada. Sci World J. 2007;7:1211–21.
Wilson P, D'Agostino R, Sullivan L, Parise H, Kannel W. Overweight and obesity as determinants of cardiovascular risk: the Framingham experience. Arch Intern Med. 2002;162(16):1867–72.
Gómez-Ambrosi J, Silva C, Galofré J, Escalada J, Santos S, Millán D, et al. Body mass index classification misses subjects with increased cardiometabolic risk factors related to elevated adiposity. Int J Obes. 2012;36(2):286–94.
Zubiaga Toro L, Ruiz-Tovar Polo J, Díez-Tabernilla M, Giner Bernal L, Arroyo Sebastián A, Calpena RR. CUN-BAE formula and biochemical factors as predictive markers of obesity and cardiovascular disease in patients before and after sleeve gastrectomy. Nutr Hosp. 2014;30(2):281–6.
Fuster-Parra P, Bennasar-Veny M, Tauler P, Yañez A, López-González A, Aguiló A. A comparison between multiple regression models and CUN-BAE equation to predict body fat in adults. PLoS ONE. 2015;10(3):e0122291.
Instituto Nacional de Estadistica. Encuesta Nacional de Salud 2011 – 2012. Nota de Prensa. Ministerio de Sanidad, Servicios Sociales e Igualdad; 2013. Available from: http://www.ine.es/prensa/np770.pdf
Van Kerkhove M, Vandemaele K, Shinde V, Jaramillo-Gutierrez G, Koukounari A, Donnelly C, et al. Risk factors for severe outcomes following 2009 influenza A (H1N1) infection: a global pooled analysis. PLoS Med. 2011;8(7):e1001053.
We thank the physicians of the Sentinel Network of the participating Spanish regions and the study interviewers for their help and collaboration.
Funding: This study was supported by the Ministry of Science and Innovation, Carlos III Institute of Health, Programme of Research on Influenza A/H1N1 (Grant GR09/0030), and the Catalan Agency for the Management of Grants for University Research (AGAUR Grant number 2009/ SGR 42).
Verónica Dávila-Batista: Predoctoral contract financed by the Ministry of Education of the Junta de Castilla y Leon and the European Social Fund.
The other members of the CIBERESP Cases and Controls in Pandemic Influenza Working Group are: Andalusia: Ernestina Azor, Jerónimo Carrillo, Rosa Moyano, Juan Antonio Navarro, Manuel Vázquez, Francisco Zafra (Médico Centinela); Mª Fe Bautista Martín, Jose Mª Navarro, Irene Pedrosa Corral, Mercedes Pérez Ruiz (Laboratorio de Referencia de Gripe); Virtudes Gallardo, Esteban Pérez (Servicio de Epidemiología), José Ramón Maldonado (Hospital de Torrecárdenas), Áurea Morillo (Hospital Virgen del Rocío), Mª Carmen Ubago (Hospital Virgen de las Nieves). Castile and Leon: Demetrio Carriedo, Florentino Díez, Isabel Fernández-Natal, Silvia Fernández (Compl. Asist. Universitario, León); Javier Castrodeza, Carolina Rodríguez, Sonia Tamames (Consejería de Sanidad de la Junta de Castilla y León); Pilar Sanz (Universidad de León); Raul Ortiz de Lejarazu (Centro Nacional de Gripe de la Universidad de Valladolid); Alberto Pérez (Servicio de Vigilancia Epidemiológica); Pedro Redondo (Servicio Territorial de Sanidad y Bienestar Social); Ana Pueyo, José Luis Viejo (Complejo Asistencial de Burgos). Catalonia: Jordi Alonso (IMIM-Hospital del Mar); Ferrán Barbé (Hospital Arnau de Vilanova); Lluis Blanch, Gemma Navarro (Hospital de Sabadell); Xavier Bonfill, Joaquin López-Contreras, Virginia Pomar, María Teresa Puig (Hospital de Sant Pau); Eva Borràs, Ana Martínez, Núria Torner (Dirección General de Salud Pública); Francesc Calafell (Universitat Pompeu Fabra); Joan Cayla, Cecilia Tortajada (Agencia de Salud Publica de Barcelona); I Garcia, Juan Ruiz (Hospital Germans Trias i Pujol); Juan Jose Garcia (Hospital Sant Joan de Deu); Joaquim Gea, Juan Pablo Horcajada (Universitat Pompeu Fabra _CIBER Enfermedades Respiratorias); Ned Hayes (Hospital Clínic_CRESIB); Fernando Moraga (Hospital Vall d'Hebrón); Tomas Pumarola (Laboratorio de Referencia de Gripe); Jordi Dorca (Hospital de Bellvitge); Marc Sáez (Universidad de Girona); A Agustí, Antoni Trilla, Ana Vilella (Hospital Clínic de Barcelona), Maretva Baricot, Olatz Garín (CIBERESP). Madrid Community: Ricard Génova, Margarita García Barquero, Elisa Gil, Susana Jiménez, Fernando Martín, María Luisa Martínez, Silvia Sánchez (Subdirección de Promoción de la Salud y Prevención); Rafael Cantón, Ana Robustillo (Hospital Ramón y Cajal); Carlos Álvarez, Ana Hernandez Voth, Francisco Pozo (Hospital 12 de octubre), José Ramón Paño (Hospital La Paz). Navarre: Antonia Martínez, Leyre Martínez (Inst. de Salud Pública), María Ruiz, Patricia Fanlo, Francisco Gil, Victor Martínez-Artola (Compl. Hosp. Navarra), María Eugenia Ursua, Maite Sota, María Teresa Virto, Juana Gamboa, Felipe Pérez-Afonso (Médico Centinelas). The Basque Country: Urko Aguirre, Alberto Caspelastegui, Pedro Pablo España, Susana García (Hospital Galdakao); Javier Arístegui, Amaia Bilbao, Antonio Escobar (Hospital Basurto); Itziar Astigarraga, José María Antoñana (Hospital de Cruces); Gustavo Cilla, Javier Korta, Emilio Pérez Trallero (Hospital Donostia), José Luis Lobo (Hospital Txagorritxu), Francisco J. Troya (Hospital de Santiago). Valencia Community: María Morales (Hospital General Universitario).
We would also like to acknowledge Dr Javier Llorca, Javier Gómez-Ambrosi and Dr Gema Frühbeck, for their comments and help.
Grupo de Investigación Interacciones Gen-Ambiente y Salud - Universidad de León (Gigas), León, Spain
Vicente Martín, Verónica Dávila-Batista, Antonio J. Molina & Tania Fernandez-Villa
CIBER Epidemiología y Salud Pública, Madrid, Spain
Vicente Martín, Jesús Castilla, Pere Godoy, Miguel Delgado-Rodríguez, Nuria Soldevila, Fernando González-Candelas, José María Quintana & Angela Domínguez
Instituto de Salud Pública de Navarra, Pamplona, Spain
Jesús Castilla
Departament de Salut, Generalitat de Catalunya, Barcelona, Spain
Pere Godoy
División de Medicina Preventiva y Salud Pública, Universidad de Jaén, Jaén, Spain
Miguel Delgado-Rodríguez
Subdirección de Vigilancia. Comunidad de Madrid, Madrid, Spain
Jenaro Astray
CIBER Enfermedades Respiratorias, Madrid, Spain
Ady Castro
Unidad Mixta Genómica y Salud CSISP (FISABIO)-Universitat de València, Valencia, Spain
Fernando González-Candelas
Servicio de Vigilancia de Andalucía, Sevilla, Spain
José María Mayoral
Fundación Vasca de Innovación e Investigación Sanitarias, Sondika, Spain
José María Quintana
Departament de Salut Pública, Universitat de Barcelona, Barcelona, Spain
Angela Domínguez
Facultad de Ciencias de la Salud. Campus de Vegazana. Universidad de León, 24071, León, Spain
Verónica Dávila-Batista
Vicente Martín
Nuria Soldevila
Antonio J. Molina
Tania Fernandez-Villa
Correspondence to Verónica Dávila-Batista.
The authors declare that they have no competing interests
VM directed the study; VM and VDB analysed the data and wrote the manuscript, AD JC PG NS MDR AJM TFV JA AC FGC JMM JMQ and the CIBERESP Working Group conceived, performed and designed the study; and AD JC PG NS MDR AJM TFV JA AC FGC JMM JMQ contributed to the supervision of the study and critical analysis of the article. All authors read and approved the final manuscript.
Martín, V., Dávila-Batista, V., Castilla, J. et al. Comparison of body mass index (BMI) with the CUN-BAE body adiposity estimator in the prediction of hypertension and type 2 diabetes. BMC Public Health 16, 82 (2015). https://doi.org/10.1186/s12889-016-2728-3
CUN-BAE
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