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 Comptes Rendus : MathematiqueJournal Prestige (SJR): 0.757 Citation Impact (citeScore): 1Number of Followers: 0     Open Access journal ISSN (Print) 1631-073X - ISSN (Online) 1778-3569 Published by Académie des sciences  [7 journals]
• Volker Kempf - Brezzi–Douglas–Marini interpolation on
anisotropic simplices and prisms

• Abstract: L’erreur d’interpolation de Brezzi–Douglas–Marini sur les éléments anisotropes a été analysée dans deux publications récentes, la première se concentrant sur les simplices avec des estimations dans L 2 , l’autre considérant les parallelotopes avec des estimations en termes de normes L p . Notre contribution fournit des estimations généralisées pour les simplexes anisotropes pour le cas L p , 1≤p≤∞, et montre de nouvelles estimations pour les prismes anisotropes à base triangulaire.
PubDate: Thu, 12 Jan 2023 10:55:40 +000

• Dragan Stankov - The number of nonunimodular roots of a reciprocal
polynomial

• Abstract: Nous introduisons une suite P d de polynômes réciproques unitaires à coefficients entiers ayant les coefficients centraux fixes ainsi que les coefficients périphériques. Nous prouvons que le rapport du nombre de racines non unimodulaires de P d sur son degré d a une limite L lorsque d tend vers l’infini. Nous montrons que si les coefficients d’un polynôme peuvent être arbitrairement grands en module alors L peut être arbitrairement proche de 0. Il semble raisonnable de croire que si les coefficients sont bornés, alors l’analogue de la conjecture de Lehmer est vrai : soit L=0, soit il existe un écart tel que L ne puisse pas être arbitrairement proche de 0. Nous présentons un algorithme pour le calcul du rapport limite et une méthode numérique pour son approximation. Nous avons estimé le rapport limite pour une famille de polynômes déduits des puissances d’un nombre de Salem donné. Nous avons calculé le rapport limite des polynômes corrélés à de nombreux polynômes bivariés ayant une petite mesure de Mahler introduits par Boyd et Mossinghoff.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Priyankur Chaudhuri - An inductive approach to generalized abundance using
nef reduction

• Abstract: We use the canonical bundle formula for parabolic fibrations to give an inductive approach to the generalized abundance conjecture using nef reduction. In particular, we observe that generalized abundance holds for a klt pair (X,B) if the nef dimension n(K X +B+L)=2 and K X +B≥0 or n(K X +B+L)=3 and κ(K X +B)>0.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Alessandro Carderi - Asymptotic invariants of lattices in locally compact
groups

• Abstract: The aim of this work is to understand some of the asymptotic properties of sequences of lattices in a fixed locally compact group. In particular we will study the asymptotic growth of the Betti numbers of the lattices renormalized by the covolume and the rank gradient, the minimal number of generators also renormalized by the covolume. For doing so we will consider the ultraproduct of the sequence of actions of the locally compact group on the coset spaces and we will show how the properties of one of its cross sections are related to the asymptotic properties of the lattices.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Dawei Chen - Complete curves in the strata of differentials

• Abstract: Gendron proved that the strata of holomorphic differentials with prescribed orders of zeros do not contain complete algebraic curves by applying the maximum modulus principle to saddle connections. Here we provide an alternative proof for this result by using positivity of divisor classes on moduli spaces of curves.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Chenyan Wang; Huoxiong Wu - A note on singular oscillatory integrals with
certain rational phases

• Abstract: Let Ω be homogeneous of degree zero with mean value zero, P and Q real polynomials on ℝ n with Q(0)=0 and Ω∈B q 0,0 (S n-1 ) for some q>1. This note extends and improves a classical result of Stein and Wainger (Ann. Math. Stud. 112, pp. 307-355, (1986)) to the following general formp.v.∫ ℝ n e i(P(x)+1/Q(x)) Ω(x/ x ) x n dx≤B,where B depend only on ∥Ω∥ B q 0,0 (S n-1 ) , n and the degrees of P and Q, but not on their coefficients.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Ludovic Godard-Cadillac - Hölder estimate for the 3 point-vortex
problem with alpha-models

• Abstract: In this article we study quasi-geostrophic point-vortex systems in a general setting called alpha point-vortex. We study a particular case of vortex collapses called mono-scale collapses and this study gives the Hölder regularity for the 3-vortex problem. This result implies in particular that the trajectories of the vortices are convergent even in the case of a collapse.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• J. Marshall Ash; Stefan Catoiu et al. - A new proof of the GGR
conjecture

• Abstract: For each positive integer n, function f, and point x, the 1998 conjecture by Ginchev, Guerragio, and Rocca states that the existence of the nth Peano derivative f (n) (x) is equivalent to the existence of all n(n+1)/2 generalized Riemann derivatives,D k,-j f(x)=lim h→0 1 h n ∑ i=0 k (-1) i k if(x+(k-i-j)h),for j,k with 0≤j<k≤n. A version of it for n≥2 replaces all -j with j and eliminates all j=k-1. Both the GGR conjecture and its version were recently proved by the authors using non-inductive proofs based on highly non-trivial combinatorial algorithms. This article provides a second, inductive, algebraic proof to each of these theorems, based on a reduction to (Laurent) polynomials.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Xiaobiao Jia; Xuemei Li - Asymptotic behavior of solutions of
Monge–Ampère equations with general perturbations of boundary values

• Abstract: In this paper, we consider the asymptotic behavior of solutions of Monge–Ampère equations with general boundary value conditions in half spaces, which reveals the accurate effect of boundary value condition on asymptotic behavior and improves the result in [13].
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Alexei Tsygvintsev - On the existence of generalised syzygies in the
planar three-body problem

• Abstract: Nous considérons le problème plan des trois corps. On définit une syzygie généralisée comme une configuration où les trois corps ou leurs vitesses deviennent colinéaires. En supposant que le mouvement est borné et sans collision, nous fournissons une condition suffisante pour l’existence de telles configurations. Nos principaux outils sont élémentaires et basés sur la théorie de Sturm–Liouville.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Narad Rampersad - Prefixes of the Fibonacci word that end with a cube

• Abstract: The Fibonacci word f=010010100100101⋯ is one of the most well-studied words in the area of combinatorics on words. It is not periodic, but nevertheless contains many highly periodic factors (contiguous subwords). For example, it contains many cubes (i.e., non-empty words of the form xxx). We study the prefixes of the Fibonacci word that end with a cube. Using the computer prover Walnut, we obtain an exact description of the positions of the Fibonacci word at which a cube ends. This gives a certain measure of how close the Fibonacci word is to being periodic.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Pascal Fong; Sokratis Zikas - Connected algebraic subgroups of groups of
birational transformations not contained in a maximal one

• Abstract: We prove that for each n≥2, there exist a ruled variety X of dimension n and a connected algebraic subgroup of Bir(X) which is not contained in a maximal one.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Camille Laurent; Matthieu Léautaud - On uniform controllability of 1D
transport equations in the vanishing viscosity limit

• Abstract: We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schrödinger operator. They improve, in this particular situation, the results obtained in the companion paper [38].The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [40], together with estimates on the spectral gap [33, 1]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Heayong Shin; Young Wook Kim et al. - Addendum to the paper: Compact
embedded minimal surfaces in the Berger sphere

• Abstract: Nous construisons une famille à deux paramètres discrets de surfaces minimales compactes plongées dans la sphère de Berger qui peut être considérée comme l’analogue de l’hélicoïde de Karcher-Scherk.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Marc Girondot - Nouvelle méthodologie d’analyse bayésienne du report
des voix et de l’abstention lors du deuxième tour des élections

• Abstract: Le mécanisme de report des voix lors des élections à deux tours est une grande inconnue : l’électeur suit-il une consigne de vote, lorsqu’il y en a une, ou bien développe-t-il une stratégie personnelle de vote ' L’information issue des dépouillements des premiers et seconds tours est utilisée pour modéliser le comportement des électeurs comme un problème inverse résolu par une approche bayésienne. Il est alors possible d’estimer les flux de votes entre le premier et le second tour avec une très bonne précision afin de fournir les bases objectives qui serviront à une analyse politique ou sociologique objective.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• k -nearest+neighbors+rule+for+functional+data&rft.title=Comptes+Rendus+:+Mathematique&rft.issn=1631-073X&rft.date=2023&rft.volume=">Ahmad Younso - Consistency of the k -nearest neighbors rule for
functional data

• Abstract: Le problème de la classification non paramétrique par la règle des k- plus proches voisins dans un espace métrique général sera considéré. La consistance et la forte consistance du classifieur seront établies sous des conditions légères.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Zhong-Xuan Mao; Jing-Feng Tian - Monotonicity and complete monotonicity of

• Abstract: In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function -T ν,α,β (s) is completely monotonic in s and absolutely monotonic in ν if and only if β≥1, where T ν,α,β (s)=K ν 2 (s)-βK ν-α (s)K ν+α (s) defined on s>0 and K ν (s) is the modified Bessel function of the second kind of order ν. Finally, we determine the necessary and sufficient conditions for the functions s↦T μ,α,1 (s)/T ν,α,1 (s), s↦(T μ,α,1 (s)+T ν,α,1 (s))/(2T (μ+ν)/2,α,1 (s)), and s↦d n 1 dν n 1 T ν,α,1 (s)/d n 2 dν n 2 T ν,α,1 (s) to be monotonic in s∈(0,∞) by employing the monotonicity rules.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

hyperbolic-elliptic chemotaxis system with logistic source

• Abstract: This paper is concerned with the blow-up of solutions to the following hyperbolic-elliptic chemotaxis system:u t =-∇·(χu∇v)+g(u),x∈Ω,t>0,0=Δv-v+u,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a bounded domain Ω⊂ℝ n ,n≥1, with smooth boundary and the function g is assumed to generalize the logistic source:g(s)≤as-bs γ fors>0with 1<γ≤2. For b<χ and some suitable conditions on parameters of problem, we prove that the solutions of this problem blow up in finite time. This result extend the obtained results for this problem.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Chiun-Chang Lee; Masashi Mizuno et al. - On the uniqueness of linear
convection–diffusion equations with integral boundary conditions

• Abstract: We investigate a class of convection–diffusion equations in an expanding domain involving a parameter, where we consider integral boundary conditions that depend non-locally on unknown solutions. Generally, the uniqueness result of this type of equation is unclear. In this work, we obtain a uniqueness result when the domain is sufficiently large or small. This approach has the advantage of transforming the integral boundary conditions into new Dirichlet boundary conditions so that we can obtain refined estimates, and the comparison theorem can be applied to the equations. Furthermore, we show a domain such that under different boundary data, the equation in this domain can have infinitely numerous solutions or no solution. This work may contribute to the first understanding of the domain size’s effect on the existence and uniqueness of the linear convection–diffusion equation with integral-type boundary conditions.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

• Karine Beauchard; Jérémy Le Borgne et al. - On expansions for nonlinear
systems Error estimates and convergence issues

• Abstract: Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied.First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann’s infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation.Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input.Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann’s infinite product expansion.Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.
PubDate: Thu, 12 Jan 2023 10:55:39 +000

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