Fast rates for empirical vector quantization

We consider the rate of convergence of the expected loss of empirically optimal vector quantizers. Earlier results show that the mean-squared expected distortion for any fixed distribution supported on a bounded set and satisfying some regularity conditions decreases at the rate O(log n/n). We prove that this rate is actually O(1/n). Although these conditions are hard to check, we show that well-polarized distributions with continuous densities supported on a bounded set are included in the scope of this result.Comment: 18 page

When are the invariant submanifolds of symplectic dynamics Lagrangian?

Let L be a D-dimensional submanifold of a 2D-dimensional exact symplectic manifold (M, w) and let f be a symplectic diffeomorphism onf M. In this article, we deal with the link between the dynamics of f restricted to L and the geometry of L (is L Lagrangian, is it smooth, is it a graph...?). We prove different kinds of results. - for D=3, we prove that if a torus that carries some characteristic loop, then either L is Lagrangian or the restricted dynamics g of f to L can not be minimal (i.e. all the orbits are dense) with (g^k) equilipschitz; - for a Tonelli Hamiltonian of the cotangent bundle M of the 3-dimenional torus, we give an example of an invariant submanifold L with no conjugate points that is not Lagrangian and such that for every symplectic diffeomorphism f of M, if $f(L)=L$, then $L$ is not minimal; - with some hypothesis for the restricted dynamics, we prove that some invariant Lipschitz D-dimensional submanifolds of Tonelli Hamiltonian flows are in fact Lagrangian, C^1 and graphs; -we give similar results for C^1 submanifolds with weaker dynamical assumptions.Comment: 17 page

Large deviation functional of the weakly asymmetric exclusion process

We obtain the large deviation functional of a density profile for the asymmetric exclusion process of L sites with open boundary conditions when the asymmetry scales like 1/L. We recover as limiting cases the expressions derived recently for the symmetric (SSEP) and the asymmetric (ASEP) cases. In the ASEP limit, the non linear differential equation one needs to solve can be analysed by a method which resembles the WKB method

On hyperbolic analogues of some classical theorems in spherical geometry

We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise

Non-relativistic conformal symmetries and Newton-Cartan structures

This article provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational "dynamical exponent", $z$. The Schr\"odinger-Virasoro algebra of Henkel et al. corresponds to $z=2$. Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schr\"odinger Lie algebra, for which z=2. For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) and Lukierski, Stichel and Zakrzewski [alias "\alt" of Henkel], with $z=1$. Physical systems realizing these symmetries include, e.g., classical systems of massive, and massless non-relativistic particles, and also hydrodynamics, as well as Galilean electromagnetism.Comment: LaTeX, 47 pages. Bibliographical improvements. To appear in J. Phys.

Tame Class Field Theory for Global Function Fields

We give a function field specific, algebraic proof of the main results of class field theory for abelian extensions of degree coprime to the characteristic. By adapting some methods known for number fields and combining them in a new way, we obtain a different and much simplified proof, which builds directly on a standard basic knowledge of the theory of function fields. Our methods are explicit and constructive and thus relevant for algorithmic applications. We use generalized forms of the Tate-Lichtenbaum and Ate pairings, which are well-known in cryptography, as an important tool.Comment: 25 pages, to appear in Journal of Number Theor

Demazure roots and spherical varieties: the example of horizontal SL(2)-actions

Let $G$ be a connected reductive group, and let $X$ be an affine $G$-spherical variety. We show that the classification of $\mathbb{G}_{a}$-actions on $X$ normalized by $G$ can be reduced to the description of quasi-affine homogeneous spaces under the action of a semi-direct product $\mathbb{G}_{a}\rtimes G$ with the following property. The induced $G$-action is spherical and the complement of the open orbit is either empty or a $G$-orbit of codimension one. These homogeneous spaces are parametrized by a subset ${\rm Rt}(X)$ of the character lattice $\mathbb{X}(G)$ of $G$, which we call the set of Demazure roots of $X$. We give a complete description of the set ${\rm Rt}(X)$ when $G$ is a semi-direct product of ${\rm SL}_{2}$ and an algebraic torus; we show particularly that ${\rm Rt}(X)$ can be obtained explicitly as the intersection of a finite union of polyhedra in $\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{X}(G)$ and a sublattice of $\mathbb{X}(G)$. We conjecture that ${\rm Rt}(X)$ can be described in a similar combinatorial way for an arbitrary affine spherical variety $X$.Comment: Added Section 4; modified main result, Theorem 5.18 now; other change

Generalised Mertens and Brauer-Siegel Theorems

In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to make it tend to infinity and thus to point out the link between it and the famous Brauer-Siegel theorem. Using this we deduce an explicit version of the generalised Brauer-Siegel theorem under GRH, and a unified proof of this theorem for asymptotically exact families of almost normal number fields

Stable set of self map

The attracting set and the inverse limit set are important objects associated to a self-map on a set. We call \emph{stable set} of the self-map the projection of the inverse limit set. It is included in the attracting set, but is not equal in the general case. Here we determine whether or not the equality holds in several particular cases, among which are the case of a dense range continuous function on an Hilbert space, and the case of a substitution over left infinite words

Homotopy theory of higher categories

This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of $M$-precategories. The procedure can then be iterated, leading to model categories of $(\infty, n)$-categories