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

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

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

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

Explicit formula for the generating series of diagonal 3D rook paths

Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \emph{discovery and proof} of the fact that the generating series $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire

A Bayesian approach to constrained single- and multi-objective optimization

This article addresses the problem of derivative-free (single- or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria---the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions---as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single- and multi-objective constrained optimization

The complexity of acyclic conjunctive queries revisited

In this paper, we consider first-order logic over unary functions and study the complexity of the evaluation problem for conjunctive queries described by such kind of formulas. A natural notion of query acyclicity for this language is introduced and we study the complexity of a large number of variants or generalizations of acyclic query problems in that context (Boolean or not Boolean, with or without inequalities, comparisons, etc...). Our main results show that all those problems are \textit{fixed-parameter linear} i.e. they can be evaluated in time $f(|Q|).|\textbf{db}|.|Q(\textbf{db})|$ where $|Q|$ is the size of the query $Q$, $|\textbf{db}|$ the database size, $|Q(\textbf{db})|$ is the size of the output and $f$ is some function whose value depends on the specific variant of the query problem (in some cases, $f$ is the identity function). Our results have two kinds of consequences. First, they can be easily translated in the relational (i.e., classical) setting. Previously known bounds for some query problems are improved and new tractable cases are then exhibited. Among others, as an immediate corollary, we improve a result of \~\cite{PapadimitriouY-99} by showing that any (relational) acyclic conjunctive query with inequalities can be evaluated in time $f(|Q|).|\textbf{db}|.|Q(\textbf{db})|$. A second consequence of our method is that it provides a very natural descriptive approach to the complexity of well-known algorithmic problems. A number of examples (such as acyclic subgraph problems, multidimensional matching, etc...) are considered for which new insights of their complexity are given.Comment: 30 page