Annales Mathematiques du Quebec最新文献

We generalise Pollack’s construction of plus/minus L-functions to certain cuspidal automorphic representations of (mathrm {GL_{2n}}) using the p-adic L-functions constructed in work of Barrera Salazar et al. (On p-adic l-functions for (text {GL}_{2n}) in finite slope shalika families, 2021). We use these to prove that the complex L-functions of such representations vanish at at most finitely many twists by characters of p-power conductor.

We first recall various formulations and approximations for the motion of an incompressible fluid, in the well-known setting of the Euler equations. Then, we address incompressible motions in porous media, through the Muskat system, which is a friction dominated first order analog of the Euler equations for inhomogeneous incompressible fluids subject to an external potential.

We construct examples of p-adic L-functions over universal deformation spaces for ({{,mathrm{GL},}}_2). We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions and Euler systems are not the usual eigenvarieties (parametrising nearly-ordinary families of automorphic representations), but other, larger spaces depending on a choice of a parabolic subgroup, which we call ‘big parabolic eigenvarieties’.

Let (G=A *B) be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.

We consider the case of scattering by several obstacles in ({mathbb {R}}^d), (d ge 2) for the Laplace operator (Delta ) with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators (Delta _1) and (Delta _2) obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator (g(Delta ) - g(Delta _1) - g(Delta _2) + g(Delta _0)) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function (xi _mathrm {rel}(lambda ) = -frac{1}{pi } {text {Im}}(Xi (lambda ))), where (Xi (lambda )) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of (xi _mathrm {rel}). In particular we prove that ({hat{xi }}_mathrm {rel}) is real-analytic near zero and we relate the decay of (Xi (lambda )) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function (Xi (lambda )) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.

We consider an automorphism of an arbitrary CAT(0) cube complex. We study its combinatorial displacement and we show that either the automorphism has a fixed point or it preserves some combinatorial axis. It follows that when a f.g. group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls. As an application Baumslag-Solitar groups and Heisenberg groups provide examples of groups having a proper action on measured spaces with walls, but no proper action on a discrete space with wall.

Let (ell ) be a rational prime. Previously, abelian (ell )-towers of multigraphs were introduced which are analogous to ({mathbb {Z}}_{ell })-extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the (ell )-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for ({mathbb {Z}}_{ell })-extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian (ell )-towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in ({mathbb {Z}}_{ell }) and then study the special value at (u=1) of the Artin-Ihara L-function (ell )-adically.

This paper is devoted to the analysis of Steklov eigenvalues and Steklov eigenfunctions on a class of warped product Riemannian manifolds (M, g) whose boundary (partial M) consists in two distinct connected components (Gamma _0) and (Gamma _1). First, we show that the Steklov eigenvalues can be divided into two families ((lambda _m^pm )_{m ge 0}) which satisfy accurate asymptotics as (m rightarrow infty ). Second, we consider the associated Steklov eigenfunctions which are the harmonic extensions of the boundary Dirichlet to Neumann eigenfunctions. In the case of symmetric warped product, we prove that the Steklov eigenfunctions are exponentially localized on the whole boundary (partial M) as (m rightarrow infty ). When we add an asymmetric perturbation of the metric to a symmetric warped product, we observe in almost all cases a flea on the elephant effect. Roughly speaking, we prove that “half” the Steklov eigenfunctions are exponentially localized on one connected component of the boundary, say (Gamma _0), and the other half on the other connected component (Gamma _1) as (m rightarrow infty ).

Our goal is the discussion of the problem of mathematical interpretation of basic postulates (or “principles”) of Quantum Mechanics: transitions to quantum stationary orbits, the wave-particle duality, and the probabilistic interpretation, in the context of semiclassical self-consistent Maxwell–Schrödinger equations. We discuss possible dynamical interpretation of these postulates relying on a new general mathematical conjecture on global attractors of G-invariant nonlinear Hamiltonian partial differential equations with a Lie symmetry group G. This conjecture is inspired by the results on global attractors of nonlinear Hamiltonian PDEs obtained by the author together with his collaborators since 1990 for a list of model equations with three basic symmetry groups: the trivial group, the group of translations, and the unitary group (mathbf {U}(1)). We sketch these results.