Showing papers by "Institut Élie Cartan de Lorraine published in 2021"

Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale

Abstract: We associate to every irreducible representation of a reductive group over a local field of equal characteristics a local Langlands parameter up to semisimplification and prove the compatibility with the global parameterization constructed by the second author. Our method involves stacks of restricted shtukas (which are analogues of truncated Barsotti-Tate groups) and nearby cycles over arbitrary bases.

Universality of cell differentiation trajectories revealed by a reconstruction of transcriptional uncertainty landscapes from single-cell transcriptomic data

Abstract: We employed our previously-described single-cell gene expression analysis CALISTA (Clustering And Lineage Inference in Single-Cell Transcriptional Analysis) to evaluate transcriptional uncertainty at the single-cell level using a stochastic mechanistic model of gene expression. We reconstructed a transcriptional uncertainty landscape during cell differentiation by visualizing single-cell transcriptional uncertainty surface over a two dimensional representation of the single-cell gene expression data. The reconstruction of transcriptional uncertainty landscapes for ten publicly available single-cell gene expression datasets from cell differentiation processes with linear, single or multi-branching cell lineage, reveals universal features in the cell differentiation trajectory that include: (i) a peak in single-cell uncertainty during transition states, and in systems with bifurcating differentiation trajectories, each branching point represents a state of high transcriptional uncertainty; (ii) a positive correlation of transcriptional uncertainty with transcriptional burst size and frequency; (iii) an increase in RNA velocity preceeding the increase in the cell transcriptional uncertainty. Finally, we provided biological interpretations of the universal rise-then-fall profile of the transcriptional uncertainty landscape, including a link with the Waddington’s epigenetic landscape, that is generalizable to every cell differentiation system.

Existence of strong solutions for a system of interaction between a compressible viscous fluid and a wave equation

Abstract: In this article, we consider a fluid-structure interaction system where the fluid is viscous and compressible and where the structure is a part of the boundary of the fluid domain and is deformable. The fluid is governed by the barotropic compressible Navier-Stokes system whereas the structure displacement is described by a wave equation. We show that the corresponding coupled system admits a unique strong solution for an initial fluid density and an initial fluid velocity in $H^3$ and for an initial deformation and an initial deformation velocity in $H^4$ and $H^3$ respectively. The reference configuration for the fluid domain is a rectangular cuboid with the elastic structure being the top face. We use a modified Lagrangian change of variables to transform the moving fluid domain into the rectangular cuboid and then analyze the corresponding linear system coupling a transport equation (for the density), a heat-type equation, and a wave equation. The corresponding results for this linear system and estimations of the coefficients coming from the change of variables allow us to perform a fixed point argument and to prove the existence and uniqueness of strong solutions for the nonlinear system, locally in time.

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Abstract: The aim of this work is to study the exponential stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed internal feedback. We first consider the case where the weight of the feedback with delay is smaller than the weight of the feedback without delay and prove the local exponential stability result by two methods: the first one by a Lyapunov method (which holds for restrictive length of the domain but allow to have an estimation on the decay rate) and the second one by an observability inequality for any length (without estimation of the decay rate). We also prove a semiglobal stabilization result for any length. Secondly we study the case where the support of the feedback without delay is not included in the feedback with delay and give a local exponential stability result if the weight of the delayed feedback is small enough. Some numerical simulations are given to illustrate these results.

Lp theory for the interaction between the incompressible Navier-Stokes system and a damped plate

Abstract: We consider a viscous incompressible fluid governed by the Navier-Stokes system written in a domain where a part of the boundary is moving as a damped beam under the action of the fluid. We prove the existence and uniqueness of global strong solutions for the corresponding fluid-structure interaction system in an Lp-Lq setting. The main point in the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped beam. We show that this linear system possesses the maximal regularity property by proving the R-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.

Non Uniform Rational B-Splines and Lagrange approximations for time-harmonic acoustic scattering: accuracy and absorbing boundary conditions

Abstract: In this paper, the performance of the finite element method based on Lagrange basis functions and the Non Uniform Rational B-Splines (NURBS) based Iso-Geometric Analysis (IGA) are systematically studied for solving time-harmonic acoustic scattering problems. To assess their performance, the numerical examples are presented with truncated absorbing boundary conditions. In the first two examples , we eliminate the domain truncation error by applying second-order Bayliss-Gunzburger-Turkel (BGT-2) Absorbing Boundary Condition (ABC) and modifying the exact solution. Hence, the calculated error is an indicator of the numerical accuracy in the bounded computational domain with no artificial domain truncation error. Next, we apply a higher order local ABC based on the Karp's and Wilcox's far-field expansions for 2D and 3D problems, respectively. The performance of both methods in solving exterior problems is compared. The introduced auxiliary surface functions are also estimated using the corresponding basis functions. The influence of various parameters, viz., order of the approximating polynomial, number of degrees of freedom, wave number and the boundary conditions (BGT-2 and number of terms in the far-field expansions) on the accuracy and convergence rate is systematically studied. It is inferred that, irrespective of the order of the polynomial, IGA yields higher accuracy per degree of freedom when compared to the conventional finite element method with Lagrange basis.

Observability, Identifiability and Epidemiology A survey

Abstract: In this review, we recall the concepts of Identifiability and Observability of dynamical systems, and analyse them in the framework of Mathematical Epidemiology. We show that, even for simple and well known models of the literature, these properties are not always fulfilled. We then consider the problem of practical identifiability and observability, which are connected to sensitivity and numerical condition numbers. We also recall the concept of observers to reconstruct state variable of the model which are not observed, and show how it can used with epidemiological models.

Resonances of the Laplace operator on homogeneous vector bundles on symmetric spaces of real rank one

Abstract: We study the resonances of the Laplacian acting on the compactly supported sec- tions of a homogeneous vector bundle over a Riemannian symmetric space of the non- compact type. The symmetric space is assumed to have rank-one but the irreducible representation τ of the maximal compact K defining the vector bundle is arbitrary. We determine the resonances. Under the additional assumption that τ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We find their Langlands parameters, their wave front sets and determine which of them are unitarizable.

T-coercivity for the asymptotic analysis of scalar problems with sign-changing coefficients in thin periodic domains

Abstract: We study a scalar problem in thin periodic composite media formed by two materials, a classical one and a metamaterial (also known as negative material). By applying T-coercivity methods and homogenization techniques specific to the thin periodic domains under consideration, for two geometric settings, we derive the homogenized limit problems, which both exhibit dimension-reduction effects.

ODE-based Double-preconditioning for Solving Linear Systems $A^{\alpha}x = b$ and $f(A)x=b$

Abstract: This paper is devoted to the computation of the solution to fractional linear algebraic systems using a differential-based strategy to evaluate matrix-vector products $A^\alpha x$, with $\alpha \in \mathbb{R}^{*}_{+}$. More specifically, we propose ODE-based preconditioners for efficiently solving fractional linear systems in combination with traditional sparse linear system preconditioners. Different types of preconditioners are derived (Jacobi, Incomplete LU, Pade) and numerically compared. The extension to systems $f (A)x = b$ is finally considered.

Pollution sources reconstruction based on the topological derivatve method

Abstract: The topological derivative method is used to solve a pollution sources reconstruction problem governed by a steady-state convection-diffusion equation. To be more precise, we are dealing with a shape optimization problem which consists of reconstruction of a set of pollution sources in a fluid medium by measuring the concentration of the pollutants within some subregion of the reference domain. The shape functional measuring the misfit between the known data and solution of the state equation is minimized with respect to a set of ball-shaped anomalies representing the pollution sources. The necessary conditions for optimality are derived with help of the topological derivative method which consists in expanding the shape functional asymptotically and then truncated it up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. Two different cases are considered. Firstly, when the velocity of the leakages is given and we reconstruct the support of the unknown sources, including their locations and sizes. In the second case, we consider the size of the pollution sources to be known and find out the mean velocity of the leakages and their locations. Numerical examples are presented showing the capability of the proposed algorithm in reconstructing multiple pollution sources in both cases.

Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations

Abstract: Minimizing the so-called “Dirichlet energy” with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.

What matters to patients? A mixed method study of the importance and consideration of oncology patient demands.

Abstract: Background: A patient-centred approach is increasingly the mandate for healthcare delivery, especially with the growing emergence of chronic conditions. A relevant but often overlooked obstacle to delivering person-centred care is the identification and consideration of all demands based on individual experience, not only disease-based requirements. Mindful of this approach, there is a need to explore how patient demands are expressed and considered in healthcare delivery systems. This study aims to: (i)understand how different types of demands expressed by patients are taken into account in the current delivery systems operated by Health Care Organisations (HCOs) (ii)explore the often overlooked content of specific non-clinical demands (i.e. demands related to interactions between disease treatments and every daylife). Method: We adopted a mixed method in two cancer centres, representing exemplary cases of organisational transformation: - (i) circulation of a questionnaire to assess the importance that patients attach to every clinical(C) and non-clinical (NC) demand identified in an exploratory inquiry, and the extent to which each demand has been taken into account based on individual experiences; - (ii) a qualitative analysis based on semi-structured interviews exploring the content of specific NC demands. Results: Further to the way in which the questionnaires were answered (573 answers/680 questionnaires printed) and the semi-structured interviews (36) with cancer patients, results show that NC demands are deemed by patients to be almost as important as C demands (C=6.53/7VS.NC=6.13), but are perceived to be considered to less of an extent in terms of pathway management (NC=4.02VSC= 5.65), with a significant variation depending on the type of non-clinical demands expressed. Five types of NC demands can be identified: demands relating to daily life, alternative medicine, structure of the treatment pathway, administrative and logistic assistance and demands relating to new technologies. Conclusions This study shows that HCOs should be able to consider non-clinical demands in addition to those referring to clinical needs. These demands require revision of the healthcare professionals’mandate and transition from a supply-orientated system towards a demand-driven approach throughout the care pathway. Other sectors have developed hospitality management, mass customisation and personalisation to scale up approaches that could serve as inspiring examples.

The bogomolov-beauville-yau decomposition for klt projective varieties with trivial first chern class -without tears-

Abstract: We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following the same steps as in the general case, treated next.

Stochastic analysis of emergence of evolutionary cyclic behavior in population dynamics with transfer

Abstract: Horizontal gene transfer consists in exchanging genetic materials between microorganisms during their lives This is a major mechanism of bacterial evolution and is believed to be of main importance in antibiotics resistance We consider a stochastic model for the evolution of a discrete population structured by a trait taking finitely many values, with density-dependent competition Traits are vertically inherited unless a mutation occurs, and can also be horizontally transferred by unilateral conjugation with frequency dependent rate Our goal is to analyze the trade-off between natural evolution to higher birth rates and transfer, which drives the population towards lower birth rates Simulations show that evolutionary outcomes include evolutionary suicide or cyclic re-emergence of small populations with well-adapted traits We focus on a parameter scaling where individual mutations are rare but the global mutation rate tends to infinity This implies that negligible sub-populations may have a strong contribution to evolution Our main result quantifies the asymptotic dynamics of subpopulation sizes on a logarithmic scale We characterize the possible evolutionary outcomes with explicit criteria on the model parameters An important ingredient for the proofs lies in comparisons of the stochastic population process with linear or logistic birth-death processes with immigration For the latter processes, we derive several results of independent interest

Bijections between walks inside a triangular domain and Motzkin paths of bounded amplitude

Abstract: This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0°, 60°, 120°, 180°, 240°, 300°) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height. We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a 2^n-to-one function from generic triangular walks to triangular walks only using directions 0°, 120°, 240°. The second is based on an extension of Mortimer and Prellberg's results to triangular walks starting not only at a corner of the triangle, but at any point inside it. It has a linear-time complexity and is in fact adjustable: by changing some set of parameters called a scaffolding, we obtain a wide range of different bijections. Finally, we extend our results to higher dimensions. In particular, by adapting the previous proofs, we discover an unexpected bijection between three-dimensional walks in a pyramid and two-dimensional simple walks confined in a bounded domain shaped like a waffle.

On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals

Abstract: We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding light on the preferential position of individuals.

Feedback stabilization of parabolic systems with input delay

Abstract: This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the \begin{document}$ N $\end{document} -dimensional linear reaction-convection-diffusion equation with \begin{document}$ N\geq 1 $\end{document} and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.

Fredholm transformation on Laplacian and rapid stabilization for the heat equations

Abstract: We revisit the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with the Fredholm transformation upon Laplace operators in a sharp functional setting, which is the major objective of this work, from the Riesz basis properties and the operator equality to the stabilizing spaces. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.

Convexity in a masure II

Abstract: Masures are generalizations of Bruhat-Tits buildings. They were introduced by Gaussent and Rousseau in order to study Kac-Moody groups over valued fields. We prove that the intersection of two apartments of a masure is convex. Using this, we simplify the axiomatic definition of masures given by Rousseau.

Propagation for KPP bulk-surface systems in a general cylindrical domain

Abstract: In this paper, we investigate propagation phenomena for KPP bulk-surface systems in a cylindrical domain with general section and heterogeneous coefficients. As for the scalar KPP equation, we show that the asymptotic spreading speed of solutions can be computed in terms of the principal eigenvalues of a family of self-adjoint elliptic operators. Using this characterization, we analyze the dependence of the spreading speed on various parameters, including diffusion rates and the size and shape of the section of the domain. In particular, we provide new theoretical results on several asymptotic regimes like small and high diffusion rates and sections with small and large sizes. These results generalize earlier ones which were available in the radial homogeneous case. Finally, we numerically investigate the issue of shape optimization of the spreading speed. By computing its shape derivative, we observe, in the case of homogeneous coefficients, that a disk either maximizes or minimizes the speed, depending on the parameters of the problem, both with or without constraints. We also show the results of numerical shape optimization with non homogeneous coefficients, when the disk is no longer an optimizer.

Existence of contacts for the motion of a rigid body into a viscous incompressible fluid with the Tresca boundary conditions

Abstract: We consider a fluid-structure interaction system composed by a rigid ball immersed into a viscous in-compressible fluid. The motion of the structure satisfies the Newton laws and the fluid equations are the standard Navier-Stokes system. At the boundary of the fluid domain, we use the Tresca boundary conditions, that permit the fluid to slip tangentially on the boundary under some conditions on the stress tensor. More precisely, there is a threshold determining if the fluid can slip or not and there is a friction force acting on the part where the fluid can slip. Our main result is the existence of contact in finite time between the ball and the exterior boundary of the fluid for this system in the bidimensional case and in presence of gravity.

Strong approximation of particular one-dimensional diffusions

Abstract: This paper develops a new technique for the path approximation of one-dimensional stochastic processes, more precisely the Brownian motion and families of stochastic differential equations sharply linked to the Brownian motion (usually known as L and G-classes). We are interested here in the e-strong approximation. We propose an explicit and easy to implement procedure that constructs jointly, the sequences of exit times and corresponding exit positions of some well chosen domains. The main results control the number of steps to cover a fixed time interval and the convergence theorems for our scheme. We combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also described in order to complete the theoretical results.

A coupling between integral equations and on-surface radiation conditions for diffraction problems by non convex scatterers

Abstract: The aim of this paper is to introduce an orignal coupling procedure between surface integral equation formulations and on-surface radiation condition (OSRC) methods for solving two-dimensional scattering problems for non convex structures. The key point is that the use of the OSRC introduces a sparse block in the surface operator representation of the wave field while the integral part leads to an improved accuracy of the OSRC method in the non convex part of the scattering structure. The procedure is given for both the Dirichlet and Neumann scattering problems. Some numerical simulations show the improvement induced by the coupling method.

A simple master Theorem for discrete divide and conquer recurrences

Abstract: The aim of this note is to provide a Master Theorem for some discrete divide and conquer recurrences: $$X_{n}=a_n+\sum_{j=1}^m b_j X_{\lfloor{\frac{n}{m_j}}\rfloor},$$ where the $m_i$'s are integers with $m_i\ge 2$. The main novelty of this work is there is no assumption of regularity or monotonicity for $(a_n)$. Then, this result can be applied to various sequences of random variables $(a_n)_{n\ge 0}$, for example such that $\sup_{n\ge 1}\mathbb{E}(|a_n|)<+\infty$.

Algebraic differential equations of periods integrals

Abstract: We explain that in the study of the asymptotic expansion at the origin of a period integral like γz ω/df or of a hermitian period like f =s ρ.ω/df ∧ ω /df the computation of the Bernstein polynomial of the "fresco" (filtered differential equation) associated to the pair of germs (f, ω) gives a better control than the computation of the Bernstein polynomial of the full Brieskorn module of the germ of f at the origin. Moreover, it is easier to compute as it has a better functoriality and smaller degree. We illustrate this in the case where f ∈ C[x 0 ,. .. , x n ] has n + 2 monomials and is not quasi-homogeneous, by giving an explicite simple algorithm to produce a multiple of the Bernstein polynomial when ω is a monomial holomorphic volume form. Several concrete examples are given.