We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.
{"title":"Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions","authors":"Carlos E. Arreche, T. Dreyfus, J. Roques","doi":"10.5802/JEP.143","DOIUrl":"https://doi.org/10.5802/JEP.143","url":null,"abstract":"We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"43 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121013143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrodinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($qgeq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $qgeq 1$, providing a complete characterization of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstrass's $wp$ function).
{"title":"Maximal determinants of Schrödinger operators on bounded intervals","authors":"C. Aldana, Jean-Baptiste Caillau, P. Freitas","doi":"10.5802/jep.128","DOIUrl":"https://doi.org/10.5802/jep.128","url":null,"abstract":"We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrodinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($qgeq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $qgeq 1$, providing a complete characterization of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstrass's $wp$ function).","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131690344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $gge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the H"older property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained H"older estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove H"older regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.
{"title":"Hölder regularity for the spectrum of translation flows","authors":"A. Bufetov, B. Solomyak","doi":"10.5802/JEP.146","DOIUrl":"https://doi.org/10.5802/JEP.146","url":null,"abstract":"The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $gge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the H\"older property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained H\"older estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove H\"older regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125647547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results. We consider constant coefficients heat-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one dimensional torus. We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as much controls as equations. When the control acts only on the transport (resp. parabolic) component, we prove an algebraic necessary and sufficient condition, on the coupling term, for the null controllability. The whole study relies on a careful spectral analysis, based on perturbation theory. The negative controllability result in small time is proved on solutions localized on high hyperbolic frequencies, that solve a pure transport equation up to a compact term. The proof of the positive result in large time relies on a spectral decomposition into low, and asymptotically parabolic or hyperbolic frequencies.
{"title":"Null-controllability of linear parabolic-transport systems","authors":"K. Beauchard, Armand Koenig, K. L. Balc'h","doi":"10.5802/jep.127","DOIUrl":"https://doi.org/10.5802/jep.127","url":null,"abstract":"Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results. \u0000We consider constant coefficients heat-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one dimensional torus. \u0000We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as much controls as equations. When the control acts only on the transport (resp. parabolic) component, we prove an algebraic necessary and sufficient condition, on the coupling term, for the null controllability. \u0000The whole study relies on a careful spectral analysis, based on perturbation theory. The negative controllability result in small time is proved on solutions localized on high hyperbolic frequencies, that solve a pure transport equation up to a compact term. The proof of the positive result in large time relies on a spectral decomposition into low, and asymptotically parabolic or hyperbolic frequencies.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"356 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133725225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.
{"title":"Quotients of groups of birational transformations of cubic del Pezzo fibrations","authors":"J. Blanc, E. Yasinsky","doi":"10.5802/jep.136","DOIUrl":"https://doi.org/10.5802/jep.136","url":null,"abstract":"We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123882000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present the classification of positive harmonic functions on the Heisenberg group in the case of the southwest measure.
在西南测度的情况下，给出了海森堡群上正调和函数的分类。
{"title":"Positive harmonic functions on the Heisenberg group II","authors":"Y. Benoist","doi":"10.5802/JEP.163","DOIUrl":"https://doi.org/10.5802/JEP.163","url":null,"abstract":"We present the classification of positive harmonic functions on the Heisenberg group in the case of the southwest measure.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134461174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a class of multi-species birth-and-death processes going almost surely to extinction and admitting a unique quasi-stationary distribution (qsd for short). When rescaled by $K$ and in the limit $Kto+infty$, the realizations of such processes get close, in any fixed finite-time window, to the trajectories of a dynamical system whose vector field is defined by the birth and death rates. Assuming that this dynamical has a unique attracting fixed point, we analyzed in a previous work what happens for large but finite $K$, especially the different time scales showing up. In the present work, we are mainly interested in the following question: Observing a realization of the process, can we determine the so-called engineering resilience? To answer this question, we establish two relations which intermingle the resilience, which is a macroscopic quantity defined for the dynamical system, and the fluctuations of the process, which are microscopic quantities. Analogous relations are well known in nonequilibrium statistical mechanics. To exploit these relations, we need to introduce several estimators which we control for times between $log K$ (time scale to converge to the qsd) and $exp(K)$ (time scale of mean time to extinction).
{"title":"Quasi-stationary distributions and resilience: what to get from a sample?","authors":"J. Chazottes, P. Collet, S. Martínez, S. Méléard","doi":"10.5802/jep.132","DOIUrl":"https://doi.org/10.5802/jep.132","url":null,"abstract":"We study a class of multi-species birth-and-death processes going almost surely to extinction and admitting a unique quasi-stationary distribution (qsd for short). When rescaled by $K$ and in the limit $Kto+infty$, the realizations of such processes get close, in any fixed finite-time window, to the trajectories of a dynamical system whose vector field is defined by the birth and death rates. Assuming that this dynamical has a unique attracting fixed point, we analyzed in a previous work what happens for large but finite $K$, especially the different time scales showing up. In the present work, we are mainly interested in the following question: Observing a realization of the process, can we determine the so-called engineering resilience? To answer this question, we establish two relations which intermingle the resilience, which is a macroscopic quantity defined for the dynamical system, and the fluctuations of the process, which are microscopic quantities. Analogous relations are well known in nonequilibrium statistical mechanics. To exploit these relations, we need to introduce several estimators which we control for times between $log K$ (time scale to converge to the qsd) and $exp(K)$ (time scale of mean time to extinction).","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123751829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields $b_epsilon(x)$, the solutions of which $u_epsilon(t,x)$ agree at $t=0$ with a bounded sequence of $L^p_{rm loc}(mathbb{R}^N)$ for some $pin(1,infty)$. Assuming that the sequence $b_epsiloncdotnabla w_epsilon^1$ is compact in $L^q_{rm loc}(mathbb{R}^N)$ ($q$ conjugate of $p$) for some gradient field $nabla w_epsilon^1$ bounded in $L^N_{rm loc}(mathbb{R}^N)^N$, and that there exists a uniformly bounded sequence $sigma_epsilon>0$ such that $sigma_epsilon,b_epsilon$ is divergence free if $N!=!2$ or is a cross product of $(N!-!1)$ bounded gradients in $L^N_{rm loc}(mathbb{R}^N)^N$ if $N!geq!3$, we prove that the sequence $sigma_epsilon,u_epsilon$ converges weakly to a solution to a linear transport equation. It turns out that the compactness of $b_epsiloncdotnabla w_epsilon^1$ is a substitute to the ergodic assumption of the classical two-dimensional periodic case, and allows us to deal with non-periodic vector fields in any dimension. The homogenization result is illustrated by various and general examples.
{"title":"Homogenization of linear transport equations. A new approach","authors":"M. Briane","doi":"10.5802/jep.122","DOIUrl":"https://doi.org/10.5802/jep.122","url":null,"abstract":"The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields $b_epsilon(x)$, the solutions of which $u_epsilon(t,x)$ agree at $t=0$ with a bounded sequence of $L^p_{rm loc}(mathbb{R}^N)$ for some $pin(1,infty)$. Assuming that the sequence $b_epsiloncdotnabla w_epsilon^1$ is compact in $L^q_{rm loc}(mathbb{R}^N)$ ($q$ conjugate of $p$) for some gradient field $nabla w_epsilon^1$ bounded in $L^N_{rm loc}(mathbb{R}^N)^N$, and that there exists a uniformly bounded sequence $sigma_epsilon>0$ such that $sigma_epsilon,b_epsilon$ is divergence free if $N!=!2$ or is a cross product of $(N!-!1)$ bounded gradients in $L^N_{rm loc}(mathbb{R}^N)^N$ if $N!geq!3$, we prove that the sequence $sigma_epsilon,u_epsilon$ converges weakly to a solution to a linear transport equation. It turns out that the compactness of $b_epsiloncdotnabla w_epsilon^1$ is a substitute to the ergodic assumption of the classical two-dimensional periodic case, and allows us to deal with non-periodic vector fields in any dimension. The homogenization result is illustrated by various and general examples.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115250412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Patrick Cheridito, P. Patie, A. Srapionyan, A. Vaidyanathan
In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operator. We show that these operators extend to the generator of an ergodic Markov semigroup with a unique invariant probability measure and study its spectral and convergence properties. In particular, we give a series expansion of the semigroup in terms of explicitly defined polynomials, which are counterparts of the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators/semigroups, with the local Jacobi operator/semigroup serving as a reference object for transferring properties to the non-local ones.
{"title":"On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity","authors":"Patrick Cheridito, P. Patie, A. Srapionyan, A. Vaidyanathan","doi":"10.5802/JEP.148","DOIUrl":"https://doi.org/10.5802/JEP.148","url":null,"abstract":"In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operator. We show that these operators extend to the generator of an ergodic Markov semigroup with a unique invariant probability measure and study its spectral and convergence properties. In particular, we give a series expansion of the semigroup in terms of explicitly defined polynomials, which are counterparts of the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators/semigroups, with the local Jacobi operator/semigroup serving as a reference object for transferring properties to the non-local ones.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134379366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Poincare torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of the group subschemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink's conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformization of the Poincare torsor, but other, more geometric, approaches are also discussed.
{"title":"Pink’s conjecture on unlikely intersections and families of semi-abelian varieties","authors":"D. Bertrand, B. Edixhoven","doi":"10.5802/jep.126","DOIUrl":"https://doi.org/10.5802/jep.126","url":null,"abstract":"The Poincare torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of the group subschemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink's conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformization of the Poincare torsor, but other, more geometric, approaches are also discussed.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"56 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124476685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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