Application of graph combinatorics to rational identities of type A

To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).Comment: This is the complete version of the submitted fpsac paper (2009

On ergodic states, spontaneous symmetry breaking and the Bogoliubov quasi-averages

It is shown that Bogoliubov quasi-averages select the pure or ergodic states in the ergodic decomposition of the thermal (Gibbs) state. Our examples include quantum spin systems and many-body boson systems. As a consequence, we elucidate the problem of equivalence between Bose-Einstein condensation and the quasi-average spontaneous symmetry breaking (SSB) discussed for continuous boson systems. The multi-mode extended van den Berg-Lewis-Pul\'{e} condensation of type III demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages

On Uniformly finitely extensible Banach spaces

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno-Plichko, \emph{On automorphic Banach spaces}, Israel J. Math. 169 (2009) 29--45 and Castillo-Plichko, \emph{Banach spaces in various positions.} J. Funct. Anal. 259 (2010) 2098-2138. We show that they have the Uniform Approximation Property of Pe\l czy\'nski and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal --do there exist automorphic spaces other than $c_0(I)$ and $\ell_2(I)$?-- showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI --among them, the super-reflexive HI space constructed by Ferenczi-- and asymptotically $\ell_2$ spaces in the literature cannot be automorphic.Comment: This paper is to appear in the Journal of Mathematical Analysis and Application

Hamiltonian structure of thermodynamics with gauge

The state of a thermodynamic system being characterized by its set of extensive variables $q^{i}(i=1,...,n) ,$ we write the associated intensive variables $\gamma_{i},$ the partial derivatives of the entropy $S(q^{1},...,q^{n}) \equiv q_{0},$ in the form $\gamma_{i}=-p_{i}/p_{0}$ where $p_{0}$ behaves as a gauge factor. When regarded as independent, the variables $q^{i},p_{i}(i=0,...,n)$ define a space $\mathbb{T}$ having a canonical symplectic structure where they appear as conjugate. A thermodynamic system is represented by a $n+1$-dimensional gauge-invariant Lagrangian submanifold $\mathbb{M}$ of $\mathbb{T}.$ Any thermodynamic process, even dissipative, taking place on $\mathbb{M}$ is represented by a Hamiltonian trajectory in $\mathbb{T},$ governed by a Hamiltonian function which is zero on $\mathbb{M}.$ A mapping between the equations of state of different systems is likewise represented by a canonical transformation in $\mathbb{T}.$ Moreover a natural Riemannian metric exists for any physical system, with the $q^{i}$'s as contravariant variables, the $p_{i}$'s as covariant ones. Illustrative examples are given.Comment: Proofs corrections latex vali.tex, 1 file, 28 pages [SPhT-T00/099], submitted to Eur. Phys. J.