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This paper gives a complete characterization of the reachable space for a system described by the 1D heat equation with L2 (with respect to time) Dirichlet boundary controls at both ends. More precisely, we prove that this space coincides with the sum of two spaces of analytic functions (of Bergman type). These results are then applied to give a complete description of the reachable space via inputs which are n-times differentiable functions of time. Moreover, we establish a connection between the norm in the obtained sum of Bergman spaces and the cost of null controllability in small time. Finally we show that our methods yield new complex analytic results on the sums of Bergman spaces in infinite sectors.

We investigate the analytic classification of two dimensional neighborhoods of an elliptic curve with torsion normal bundle. We provide the complete analytic classification for those neighborhoods in the simplest formal class and we indicate how to generalize this construction to general torsion case., Comment: Working version 2 (new pictures, enhanced presentation of sectorial normalization, sketch of the proof of classification in the general torsion case)

We reprove the fact, due to Backus, that the Poincar{\'e} operator in ellipsoids admits a pure point spectrum with polynomial eigenfunctions.We then show that the eigenvalues of the Poincar{\'e} operator restricted to polynomial vector fields of fixed degree admitsa limit repartition given by a probability measure that we construct explicitely. For that, we use Fourier integral operators and ideas comingfrom Alan Weinstein and the first author in the seventies. The starting observation is that the orthogonal polynomials in ellipsoids satisfy a PDE.

We are interested in the optimal growth in terms of $L^p$-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic function on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between the $\mathcal{U}$-frequent hypercyclicity and the hypercyclicity.

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular , if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to f guaranteeing that g is a polynomial in f. We apply this to hypergeometric functions and iterated integrals.

Let $A$ be a generator of an analytic semigroup having a H{\"o}rmander functional calculus on $X = L^p(\Omega ,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for H{\"o}rmander spectral multipliers decaying sufficiently fast at $\infty$, there holds a maximal estimate $\| \sup_{t \geq 0} |m(tA)f|\, \|_{L^p(\Omega ,Y)} \lesssim \|f\|_{L^p(\Omega ,Y)}$. We also show square function estimates $\left\| \left( \sum_k \sup _{t \geq 0} |m_k(tA)f_k|^2 \right)^{\frac12} \right\|_{L^p(\Omega ,Y)} \lesssim \left\| \left( \sum _k |f_k|^2 \right)^{\frac12} \right\|_{L^p(\Omega ,Y)}$ for suitable families of spectral multipliers $m_k$, which are even new for the euclidean Laplacian on scalar valued $L^p(\mathbb{R}^d)$. As corollaries, we obtain maximal estimates for wave propagators and Bochner--Riesz means. Finally, we illustrate the results by giving several examples of operators $A$ that admit a H{\"o}rmander functional calculus on some $L^p(\Omega ,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context., Comment: International Mathematics Research Notices, Oxford University Press (OUP), 2022

In this paper we consider rough differential equations with affine boundary conditions. Using rough paths and fixed-point techniques, we study properties of the solutions. We also prove a convergence theorem and apply our results to differential equations driven by Brownian motions and fractional Brownian motions.

We discuss the best methods available for computing the gamma function $\Gamma(z)$ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function $\log \Gamma(z)$, the digamma function $\psi(z)$, and derivatives $\Gamma^{(n)}(z)$ and $\psi^{(n)}(z)$. Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results.

Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus its formalization in a proof assistant is to be designed to fit different goals and projects. Once Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is the Tonelli theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. Therefore, we need to define and prove results on product spaces, hoping that they can easily derive from the existing ones on a single space. This article describes the formal definition and proof in Coq of product $\sigma$-algebras, product measures and their uniqueness, the construction of iterated integrals, up to the Tonelli theorem. We also advertise the \emph{Lebesgue induction principle} provided by an inductive type for {\nonnegative} measurable functions.; L'intégrale de Lebesgue est un outil mathématique bien connu, utilisé par exemple en théorie des probabilités, en analyse réelle et pour les mathématiques appliquées. Sa formalisation dans un assistant de preuve doit donc être conçue pour s'adapter des buts et des projets différents. Une fois que l'intégrale de Lebesgue est définie formellement et que les premiers lemmes sont prouvés, il se pose naturellement la question de la commodité d'usage de la formalisation. Pour la contrôler, le théorème de Tonelli est une extension utile. Ce dernier établi que l'intégrale (double) d'une fonction mesurable positive de deux variables peut être calculée par des intégrales itérées et que l'on peut intervertir l'ordre d'intégration. Nous devons donc définir et prouver des résultats sur les espaces produits, en espérant qu'ils peuvent facilement découler des résultats existants sur un espace simple. Cet article décrit la définition formelle et la preuve en Coq des tribus produits, des l'existence et l'unicité des mesures produits, de la construction des intégrales itérées, jusqu'au théorème de Tonelli. Nous annonçons également le \emph{principe d'induction de Lebesgue}, qui est obtenu à partir d'un type inductif pour les fonctions mesurables positives.

International audience Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus, its formalization in a proof assistant is to be designed to fit different goals and projects. Once the Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is Tonelli's theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. This article describes the formal definition and proof in Coq of product sigma-algebras, product measures and their uniqueness, the construction of iterated integrals, up to Tonelli's theorem. We also advertise the Lebesgue induction principle provided by an inductive type for nonnegative measurable functions.

This thesis is mainly concerned with the problem of realization of homogeneous Besov and Sobolev spaces on the whole space, and some half-spaces. This problem of realization of function spaces appears naturally when studying the global well-posedness in-time and the regularity problems of some parabolic equations in unbounded domains. The constructions proposed in this thesis extend those initiated by Bahouri, Chemin, Danchin, Hieber, Mucha and Tolksdorf in various articles and monographs.We mainly review the density, real and complex interpolation results, as well as some results for traces on the boundary. A major difficulty comes from the fact that some of the considered normed vector spaces cannot be complete, nor be completed at the risk of not being constituted of elements identifiable with distributions.The lack of completeness for some spaces requires then a new construction of the tools in order to be able to exploit the operator theory, in particular for the trace estimates in the case of global-in-time parabolic maximal regularity for Lebesgue spaces. A reconstruction of the theory of interpolation and homogeneous operators has been carried out by Danchin, Hieber, Mucha and Tolksdorf, in order to obtain global-in-time estimates for maximal regularity of the Da Prato-Grisvard type for parabolic equations arising from injective, non-invertible sectorial operators. We use this framework to establish a new type of global-in-time maximal regularity results, with adapted trace estimates, where we replace the Lebesgue space in time by a homogeneous Sobolev space.The revisited theory of interpolation and homogeneous operators in combination with our construction of homogeneous spaces and their properties are applied to the study of the Hodge Laplacian on the flat upper half-space in arbitrary dimension. From this analysis, we derive the Hodge/Helmholtz decomposition, for any degree of differential forms, of homogeneous Sobolev and Besov spaces which is essentially optimal from the regularity viewpoint. Moreover, it also allows us to deduce many parabolic maximal regularity results for various Stokes or Maxwell evolutionary systems subject to various boundary conditions with global-in-time estimates. Those results may be of interest in fluid mechanics and electromagnetism.Finally, we also focus on the construction and realization of homogeneous function spaces on the open sets given by epigraphs of uniformly Lipschitz real-valued functions. We also propose a construction of homogeneous function spaces on the boundary, as well as a trace theorem essentially optimal from the point of view of regularity with homogeneous estimates from the point of view of norms.; Cette thèse s'attèle principalement au problème de réalisation des espaces de Besov et Sobolev homogènes sur l'espace entier, et certains demi-espaces. Ce problème de réalisation des espaces de fonctions apparaît naturellement lors de l'étude du caractère bien posé global en temps et des problèmes de régularité de certaines équations paraboliques dans les domaines non-bornés. Les constructions proposées dans cette thèse étendent celles initiées par Bahouri, Chemin, Danchin, Hieber, Mucha et Tolksdorf au cours de différents articles et monographes.On passera en revue principalement les résultats de densité, d'interpolation réelle et complexe, ainsi que les résultats de trace sur le bord. Une difficulté majeure vient du fait que certains des espaces vectoriels normés considérés ne peuvent pas être complets, ni complétés au risque de ne plus être constitués d'éléments identifiables à des distributions.Le manque de complétude pour certains espaces requiert alors une nouvelle construction des outils afin de pouvoir exploiter la théorie des opérateurs et en particulier la régularité maximale parabolique globale en temps dans les espaces de Lebesgue. Une reconstruction de la théorie de l'interpolation et des opérateurs homogènes a été effectuée par Danchin, Hieber, Mucha et Tolksdorf, afin d'obtenir dans ce cadre des estimées globales en temps pour une régularité maximale du type Da Prato-Grisvard pour des équations paraboliques issues d'opérateurs sectoriels injectifs, non-inversibles. On se sert de ce cadre afin d'établir un nouveau type de régularité maximale globale en temps, avec une estimation de trace adaptée, où l'on remplace l'espace de Lebesgue en temps par un espace de Sobolev homogène.La théorie revisitée de l'interpolation et des opérateurs homogènes en combinaison avec notre construction des espaces homogènes et leurs propriétés sont appliqués à l'étude du Laplacien de Hodge sur le demi-espace plat en dimension arbitraire. On déduit de cette analyse la décomposition de Hodge/Helmholtz, pour tout degré de formes différentielles, des espaces de Sobolev et Besov homogènes qui se trouve être essentiellement optimale du point de vue de la régularité. En outre, cela nous permet de déduire de nombreux résultats de régularité maximale pour divers systèmes d'évolution de Stokes ou Maxwell assujettis à diverses conditions au bord. Ceux-ci peuvent être d'un intérêt certain en mécanique des fluides et en électromagnétisme.Enfin, on se concentrera sur la construction et la réalisation des espaces de fonctions homogènes sur les ouverts qui sont des épigraphes de fonctions uniformément lipschitziennes à valeurs réelles. On proposera également une construction des espaces homogènes sur le bord, ainsi qu'un théorème de trace essentiellement optimal du point de vue de la régularité avec des estimées homogènes du point de vue des normes.

This paper introduces a couple of new time-frequency transforms, designed to adapt their scale to specific features of the analyzed function. Such an adaptation is implemented via so-called focus functions, which control the window scale as a function of the time variable, or the frequency variable. In this respect, these transforms are non-linear, which makes the analysis more complex than usual.Under appropriate assumptions, some norm control can be obtained for both transforms in L^2(R) spaces, which extend the classical continuous frame norm control and guarantees well-definedness on L^2. Given the non-linearity of the transforms, the existence of inverse transforms is not guaranteed anymore, and is an open question. However, the results of this paper represent a first step towards a more general theory.Besides mathematical results, some elementary examples of time and frequency focus functions are provided, which can serve as staring point for concrete applications.

Let $G$ be a locally compact unimodular group, let $1\leq p

We are interested in the optimal growth in terms of $L^p$-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic function on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between the $\mathcal{U}$-frequent hypercyclicity and the hypercyclicity.

We introduce and study a new model for the progression of Alzheimer's disease incorporating the interactions of Aβ-monomers, oligomers, microglial cells and interleukins with neurons through different mechanisms such as protein polymerization, inflammation processes and neural stress reactions. In order to understand the complete interactions between these elements, we study a spatially-homogeneous simplified model that allows to determine the effect of key parameters such as degradation rates in the asymptotic behavior of the system and the stability of equilibriums. We observe that inflammation appears to be a crucial factor in the initiation and progression of Alzheimer's disease through a phenomenon of hysteresis, which means that there exists a critical threshold of initial concentration of interleukins that determines if the disease persists or not in the long term. These results give perspectives on possible anti-inflammatory treatments that could be applied to mitigate the progression of Alzheimer's disease. We also present numerical simulations that allow to observe the effect of initial inflammation and concentration of monomers in our model.

International audience; A study of the greatest possible ratio of the smallest absolute value of a higher derivative of some function, defined on a bounded interval, to the L p-norm of the function.

In this short paper, we consider a sliding punch problem under recently proposed model of wear which is based on the Riemann-Liouville fractional integral relation between pressure and worn volume, and incorporates another additional effect pertinent to relaxation. A particular case of oscillatory (time-harmonic) load is studied. The time-dependent stationary state is identified in terms of eigenfunctions of an auxiliary integral operator. Convergence to this stationary state is quantified. Moreover, numerical simulations have been conducted in order to illustrate the obtained results and study qualitative dependence on two main model parameters.

We review the central results concerning wavelet methods in multifractal analysis, which consists in analysis of the pointwise singularities of a signal, and we describe its recent extension to multivariate multifractal analysis, which deals with the joint analysis of several signals; we focus on the mathematical questions that this new techniques motivate. We illustrate these methods by an application to data recorded on marathon runners.; Nous passons en revue les résultats centraux concernant les méthodes d'ondelettes dans l'analyse multifractale, qui consiste à analyser les singularités ponctuelles d'un signal, et nous décrivons sa récente extension à l'analyse multifractale multivariée, qui traite de l'analyse conjointe de plusieurs signaux ; nous nous concentrons sur les questions mathématiques que ces nouvelles techniques motivent. Nous illustrons ces méthodes par une application aux données enregistrées sur des marathoniens.

International audience; We report on a formalization of the change of variables formula in integrals, in the mathlib library for Lean. Our version of this theorem is extremely general, and builds on developments in linear algebra, analysis, measure theory and descriptive set theory. The interplay between these domains is transparent thanks to the highly integrated development model of mathlib.

In this work we consider the Landau-de Gennes model for liquid crystals with an external electromagnetic field to model the occurrence of the saturn ring effect under the assumption of rotational equivariance. After a rescaling of the energy, a variational limit is derived. Our analysis relies on precise estimates around the singularities and the study of a radial auxiliary problem in regions, where a continuous director field exists. Studying the limit problem, we explain the transition between the dipole and saturn ring configuration and the occurrence of a hysteresis phenomenon, giving a rigorous explanation of what was conjectured previously by [H. Stark, Eur. Phys. J. B 10, 311-321 (1999)]., Comment: 42 pages, 6 figures

It is well known that rational approximation theory involves degenerate hypergeometric functions and, in particular, the Pad\'e approximation of the exponential function is closely related to Kummer hypergeometric functions. Recently, in the context of the study of the exponential stability of the trivial solution of delay-differential equations, a new link between the degenerate hypergeometric function and the zeros distribution of the characteristic function associated with linear delay-differential equations was emphasized. Such a link allowed the characterization of a property of time-delay systems known as multiplicity-induced-dominancy (MID), which opened a new direction in designing low-complexity controllers for time-delay systems by using a partial pole placement idea. Thanks to their relations to hypergeometric functions, we explore in this paper links between the spectrum of delay-differential equations and Pad\'e approximations of the exponential function. This note exploits and further comments recent results from [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Comptes Rendus. Math\'ematique, arXiv:2107.11363] and [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Bulletin des Sciences Math\'ematiques, arXiv:2106.03378].

We review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coefficient Helmholtz equation, Zakharov–Shabat system and Kubelka–Munk equations. We show that they all can be reduced to solving one simple antilinear ordinary differential equation u′x=fxux¯ or its nonhomogeneous version u′x=fxux¯+gx, x∈0,x0⊂R. We point out some of the advantages of the proposed reformulation and call for further investigation of the obtained ODE.

First we propose a new way of combining two polynomial projectors on spaces of functions of few variables to obtain a polynomial projector on a space of functions of many variables. We present various algebraic properties of our construction and study the main approximation properties of the new projectors. We prove in the second chapter that the Newton product of efficient polynomial projectors is still efficient. Various polynomial approximation theorems are established involving Newton product projectors on spaces of holomorphic functions on a neighborhood of a regular compact set, on spaces of entire functions of given growth and on spaces of differentiable functions. Efficient explicit new projectors are presented. Finally we study the following problem : given a valid set X of interpolation points for Lagrange interpolation of degree d in n variables we study how many subsets of X can be chosen in order to obtain a valid set of interpolation points of degree d — 1. This leads to an estimate of the number of Newton structures for X which, in turngives the number of different unisolvent sets that can be obtainend by the process of interwinning defined in the first chapter.; Dans le premier chapitre nous définissons un nouveau procédé pour amalgamer des projecteurs polynomiaux à moins de variables pour en obtenir un en plus de variables. Nous présentons ensuite de nombreuses propriétés algébriques de cette construction. En second lieu nous montrons que cette notion permet d'obtenir des résultats d'approximations polynomiales ( via convergences) pour certains de type de fonctions à plusieurs variables. Là encore certaines approches sont originales. Enfin nous étudions des situations combinatoires et géométriques mettant en lumière la notion de produit de Newton définie au premier chapitre.

In [David-Toro 15] and [David-Engelstein-Toro 19], (some of) the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli in [Alt-Caffarelli 81] and Alt, Caffarelli and Friedman in [Alt-Caffarelli-Friedman 84]. In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to [DT15, DET19. AC 81] and [ACF84] which deal only with the "Laplacian" setting). We prove Lipschitz regularity up to, and across, the free boundary, generalizing the results of [David-Toro 15] to the variable coefficient setting., Comment: 41 pages. Revised version has minor corrections and additions. To appear in Math Z

International audience; Deformation of morphisms along leaves of foliations define the tangential foliation on the corresponding space of morphisms. We prove that codimension one foliations having a tangential foliation with at least one non-algebraic leaf are transversely homogeneous with structure group determined by the codimension of the non-algebraic leaf in its Zariski closure. As an application, we provide a structure theorem for degree three foliations on $P^3$ .

In this manuscript, we give a new proof of strong minimality of certain automorphic functions, originally results of Freitag and Scanlon (2017), Casale, Freitag, and Nagloo (2020), Bl\'azquez-Sanz, Casale, Freitag, and Nagloo (2020). Our proof is shorter and conceptually different than those presently in the literature., Comment: 8 pages

International audience; We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weighted p−Laplace term. A well chosen diffeomorphism between R and (−1, 1) is used for the estimates of the approximated solution, and is involved in the above weight. We show that this approximation converges to a weak sense of the problem for general right-hand-side, and to the entropy solution in the case where the right-hand-side is in L 1 .

The book develops the fundamental ideas of the famous Kac-Rice formula for vectorvalued random fields. This formula allows to compute the expectation and moments of the measure, and integrals with respect to this measure, of the sets of levels of such fields. After a presentation of the historical context of the Kac-Rice formula, we give an elementary demonstration of the co-area formula. This formula replaces the change of variable formula in multiple integrals and a direct application of this formula gives the Kac-Rice formula for almost all levels. We emphasize the necessity of having the formula for all levels, because for some applications one needs, for example, the formula for level zero.

In this paper, we obtain estimates for the solutions to the classical B{\'e}zout equation that are analogous to Carleson's solution to the corona theorem for the bounded analytic functions on the open unit disk. As an application, we extend some results of Luo and obtain a corona theorem for the multipliers of a class of de Branges--Rovnyak spaces.

In this paper, we present two new representations of the alternating Zeta function. We show that for any s $\in$ C this function can be computed as a limit of a series of determinant. We then express these determinants as the expectation of a functional of a random vector with Dixon-Anderson density. The generalization of this representation to more general alternating series allows us to evaluate a Selberg-type integral with a generalized Vandermonde determinant.

The macroscopic Hausdorff dimension Dim H (E) of a set E $\subset$ R d was introduced by Barlow and Taylor to quantify a "fractal at large scales" behavior of unbounded, possibly discrete, sets E. We develop a method based on potential theory in order to estimate this dimension in R d. Then, we apply this method to obtain Marstrand-like projection theorems: given a set E $\subset$ R 2 , for almost every $\theta$ $\in$ [0, 2$\pi$], the projection of E on the straight line passing through 0 with angle $\theta$ has dimension equal to min(Dim H (E) , 1).

The classical Szego polynomial approximation theorem states that the polynomials are dense in the space $L^2(\rho)$, where $\rho$ is a measure on the unit circle, if and only if the logarithmic integral of the measure $\rho$ diverges. In this note we give a quantitative version of Szego's theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure $\rho$ on a sufficiently rare subset of the circle., Comment: To appear in Israel Journal of Mathematics

International audience; The periodically forced light-limited Droop model represents microalgae growth under co-limitation by light and a single substrate, accounting for periodic fluctuations of factors such as light and temperature. In this paper, we describe the global dynamics of this model, considering general monotone growth and uptake rate functions. Our main result gives necessary and sufficient conditions for the existence of a positive periodic solution i.e. a periodic solution characterized by the presence of microalgae) which is globally attractive. In our approach, we reduce the model to a cooperative planar periodic system. Using results on periodic Kolmogorov equations and on monotone sub-homogeneous dynamical systems, we describe the global dynamics of the reduced system. Then, using the theory of asymptotically periodic semiflows, we extend the results on the reduced system to the original model. To illustrate the applicability of the main result, we include an example considering a standard microalgae population model.

We prove the first positive results concerning boundary value problems in the upper half-space of second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so, we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In the way, we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. The major new challenge, compared to the earlier results by one of us under time and transversally independence of the coefficients, is to handle non-local half-order derivatives in time which are unavoidable in our situation., Comment: Upload of the published version

International audience; In the paper, the authors introduce a notion "multivariate exponential polynomials" which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Fàa di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multi-variate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

The Bochner integral is a generalization of the Lebesgue integral, for functions taking their values in a Banach space. Therefore, both its mathematical definition and its formalization in the Coq proof assistant are more challenging as we cannot rely on the properties of real numbers. Our contributions include an original formalization of simple functions, Bochner integrability defined by a dependent type, and the construction of the proof of the integrability of measurable functions under mild hypotheses (weak separability). Then, we define the Bochner integral and prove several theorems, including dominated convergence and the equivalence with an existing formalization of Lebesgue integral for nonnegative functions.

Stieltjes' constants $\gamma_n$ are the coefficients in the Laurent series for the zeta function $\zeta(s)$ at the pole $s=1$. We present new sequences of approximations for Stieltjes' constants obtained by generalizing the ``remainder Pad\'e approximants'' method introduced by the first named author in 1996. Here, we replace Pad\'e approximants (of which poles are connected to zeros of orthogonal polynomials) by Pad\'etype approximants introduced by Brezinski of which poles are chosen {\em a priori}. The particular case $\gamma_1$ is also treated separately using ordinary Pad\'e approximants. The last section of the paper deals with approximations of the Riemann zeta function in the complex plane.

We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.

We prove the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [7], extending it to the Lagrangian framework.

The Malgrange-Galois groupoid of Painlev\'e IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange-Galois groupoid of Painlev\'e VI family including all parameters as new dependent variables. We conclude it is the pseoudogroup of transformations preserving parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of Painlev\'e VI depending analytically on parameters does not satisfy any new partial differential equation (including derivatives w. r. t. parameters) which is not derived from Painlev\'e VI., Comment: 35 pages. Revised version

We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\to\infty$ as $n\to\infty$. Formulas obtained are essentially different for relatively small and large diagonal elements $b_{n}$. Our analysis is intimately linked with spectral theory of Jacobi operators $J$ with coefficients $a_{n}, b_{n}$ and a study of the corresponding second order difference equations. We introduce the Jost solutions $f_{n}(z)$, $n\geq -1$, of such equations by a condition for $n\to\infty$ and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schr\"odinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions $P_{n}(z)$ by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $P_{n}(z)$ as $n \to\infty$ in terms of the Wronskian of the solutions $ P_{n} (z) $ and $ f_{n} (z)$. The formulas obtained for $P_{n}(z)$ generalize the asymptotic formulas for the classical Hermite polynomials where $a_{n}=\sqrt{(n+1)/2}$ and $b_{n}=0$., Comment: 86 pages

International audience; We prove lower semicontinuity of the Galois groupoid of a vector field dependingon parameters. Apply to Painlevé equations, this result can be used to compute theirsGalois groupoids for general values of parameters.; Nous montrons un résultat de semi-continuité du groupoïde de Galois d'un champs de vecteur dépendant d'un paramètre. Appliqué aux équations de Painlevé, ce résultat nous permet de calculer le groupoïde de Galois de ces équations pour des valeurs générales des paramètres.

We study stability of the eigenvalues of the generator of a one dimensional reversible diffusion process satisfying some natural conditions. The proof is based on Stein's method. In particular, these results are applied to the Normal distribution (via the Ornstein-Uhlenbeck process), to Gamma distributions (via the Laguerre process) and to Beta distributions (via Jacobi process).

Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs involving the use of integration, directly or indirectly. By its capability to extend the (Riemann) integral to a wide class of irregular functions, and to functions defined on more general spaces than the real line, the Lebesgue integral is perfectly suited for use in mathematical fields such as probability theory, numerical mathematics, and real analysis. In this article, we present the Coq formalization of $$\sigma $$ -algebras, measures, simple functions, and integration of nonnegative measurable functions, up to the full formal proofs of the Beppo Levi (monotone convergence) theorem and Fatou’s lemma. More than a plain formalization of the known literature, we present several design choices made to balance the harmony between mathematical readability and usability of Coq theorems. These results are a first milestone toward the formalization of $$L^p$$ spaces such as Banach spaces.

International audience; In this paper, we revisit some classical and recent works on modelling sliding contact with wear and propose their generalisation. Namely, we upgrade the relation between the pressure and the wear rate by incorporating some non-local time-dependence. To this effect, we use a combination of fractional calculus and relaxation effects. Moreover, we consider a possibility when the load is not constant in time. The proposed model is analysed and solved. The results are illustrated numerically and comparison with similar models is discussed.

This work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index $\frac{1}{2} \leq \mu, Comment: arXiv admin note: text overlap with arXiv:1804.04895

A classical problem due to Abel is to determine if a differential equation $y'=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta$ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta$, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta$ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty\}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y'=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=\eta y$ with an algebraic solution when $x\eta(x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks., Comment: In this version we correct some important typos

We prove that we can identify three types of pointwise behaviour in the regularity of the (generalized) Rosenblatt process. This extends to a non Gaussian setting previous results known for the (fractional) Brownian motion. On this purpose, fine bounds on the increments of the Rosenblatt process are needed. Our analysis is essentially based on various wavelet methods., Comment: 53 pages

Given a family G of rectangles, to which one associates a tree [G], one defines a natural number $\lambda$ [G] called its analytic split and satisfying, for all 1 < p < $\infty$ log($\lambda$ [G]) p MG p p where MG is the Hardy-Littlewood type maximal operator associated to the family G. As an application, we completely characterize the boundeness of planar rarefied directional maximal operators on L p for 1 < p < $\infty$. Precisely, if $\Omega$ is an arbitrary set of angles in [0, $\pi$ 4), we prove that any rarefied basis B of the directional basis R $\Omega$ yields an operator MB that has the same L p-behavior than the directional maximal operator M $\Omega$ for 1 < p < $\infty$.

We propose an analysis of heart rate marathon runners implemented by computing a multifractal spectrum based on p-exponents. We draw physiological conclusions about their performance. Finally, we link this analysis with the disturbances of the heart rate autoregulation during the marathon, which had been put in evidence up to now only by scales of feeling of marathon runners during the race., Comment: in French language. GRETSI 2022, XXVIII{\`e}me Colloque Francophone de Traitement du Signal et des Images, Sep 2022, Nancy, France