An Equivalent Condition Between Poincaré Inequality and <Emphasis Type="Italic">T</Emphasis><Subscript>2</Subscript>-Transportation Cost Inequality

In this paper, We give an equivalent condition between Poincaré inequality and T ₂-transportation inequality, and by this result we find a series of measures to enhance the claim that Log-Sobolev inequality is stronger than T ₂-transportation cost inequality.     

Uniqueness of Gibbs states for quantum lattice systems

Summary. We prove uniqueness of Euclidean Gibbs states for certain quantum lattice systems with unbounded spins. We use Dobrushin’s uniqueness criterion. The necessary estimates for the Vasershtein distance between the corresponding one-point conditional distributions with boundary conditions differing only at one side, are obtained by proving a Log-Sobolev inequality on the infinite dimensional single spin (= loop) spaces. Some important classes of concrete examples to which all this applies are discussed. Received: 28 February 1996 / In revised form: 9 September 1996     

Uniqueness of Gibbs states on loop lattices

We prove the uniqueness of Gibbs states for certain quantum lattice systems with unbounded spins, based on Dobrushin's fundamental uniqueness criterion. The necessary estimates for the Vasershtein distance between one-point conditional distributions, with boundary conditions differing only at one site, are obtained from the Log-Sobolev inequality which holds on the infinite dimensional single spin (= loop) spaces. Detailed proofs are contained in [2] and [3]. The latter deals with the more general case of nonharmonic pair potentials with possibly infinite radius of interaction.     

Estimations and asymptotic behaviors of coherent entropic risk measure for sums of random variables

In this article, we provide an estimation and several asymptotic behaviors for the coherent entropic risk measure of compound Poisson process. We also establish an estimation for the coherent entropic risk measure of sum of i.i.d. random variables in virtue of Log-Sobolev inequality. As an application, we provide two deviation estimations of the tail probability for compound Poisson process. Finally, several simulation results are given to support our results.     

Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincaré inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on R₊ with the reference example of the Marcenko-Pastur distribution.     

Characterization of Talagrand's like transportation-cost inequalities on the real line

In this paper, we give necessary and sufficient conditions for Talagrand's like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimension-free concentration of measure property. Another byproduct is the characterization of modified Log-Sobolev inequalities for log-concave probability measures on .     

Strong Central Limit Theorem for isotropic random walks in {\mathbb{R}^d}

We prove an optimal Gaussian upper bound for the densities of isotropic random walks on ${\mathbb{R}^d}$ in spherical case (d ?? 2) and ball case (d ?? 1). We deduce the strongest possible version of the Central Limit Theorem for the isotropic random walks: if ${\tilde S_n}$ denotes the normalized random walk and Y the limiting Gaussian vector, then ${\mathbb{E} f(\tilde S_{n}) \rightarrow \mathbb{E} f(Y)}$ for all functions f integrable with respect to the law of Y. We call such result a ??Strong CLT??. We apply our results to get strong hypercontractivity inequalities and strong Log-Sobolev inequalities.     

A very simple proof of the LSI for high temperature spin systems

We present a very simple proof that the $O(n)$ model satisfies a uniform logarithmic Sobolev inequality (LSI) if the difference between the largest and the smallest eigenvalue of the coupling matrix is less than n. This condition applies in particular to the SK spin glass model at inverse temperature $\beta <1/4$. It is the first result of rapid relaxation for the SK model and requires significant cancellations between the ferromagnetic and anti-ferromagnetic spin couplings that cannot be obtained by existing methods to prove Log-Sobolev inequalities. The proof also applies to more general bounded and unbounded spin systems. It uses a single step of zero range renormalisation and Bakry–Emery theory for the renormalised measure.     

Elliptic operators with unbounded drift coefficients and Neumann boundary condition

We study the realization A_N of the operator in L2(Ω,μ) with Neumann boundary condition, where Ω is a possibly unbounded convex open set in , U is a convex unbounded function, DU(x) is the element with minimal norm in the subdifferential of U at x, and is a probability measure, infinitesimally invariant for . We show that A_N is a dissipative self-adjoint operator in L2(Ω,μ). Log-Sobolev and Poincaré inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by A_N.     

A trivalent graph with 58 vertices and girth 9

The concept of a partial three-space is due to Laskar and Dunbar, and is a three-dimensional analogue of a partial geometry. Here we determine all partial three-spaces S for which the S-planes are planes of PG(n, q), for which the S-lines are all the lines contained in the S-planes, for which the S-points are all the points in the S-planes, and for which the incidence relation is that of PG(n, q). More generally, we determine all partial three-spaces S for which the S-lines are lines of PG(n,q), for which the S-points are all the points on these lines, and for which the incidence relation is that of PG(n, q).     

W-Gleitgleitkinematik

The following gives some starting elements of a theory, which the author calls glide-glide-kinematics and which goes far beyond the kinematics of helical motion in ℝ ⁿ , which was studied up to now. We are mainly concerned here with “angle-preservering glide-glide-kinematics”, generalising (and giving as well some idea of) the distance-preservering glide-glide-kinematics, which we develop in a forthcoming book ([5]) in detail.      

Espaces de Banach representables

We study the Banach spaces which are isomorphic to a subspace ofl ^∞(N) which is analytic inR ^N. We prove structure theorems which show that some pathological situations cannot take place in this class. We show that a non-metrizable separable compact of Rosenthal has a continuous image which is not a compact of Rosenthal.      

Structure of approximate solutions for discrete-time control systems arising in economic dynamics

In this work we study the structure of approximate solutions of an autonomous discrete-time control system with a compact metric space of states X which is a subset of a finite-dimensional Euclidean space. This control system is described by a nonempty closed set Ω⊂X×X which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous function v:Ω→R¹ which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. Usually, in economic dynamics, the turnpike properties have been studied for systems such that all their good programs converge to a turnpike which was an interior point of Ω. In this paper we establish turnpike results for a large class of control systems for which the turnpike is not necessarily an interior point of Ω.     

On cubical graphs

It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of n-dimensional cubes. This has found applications, for example, in coding theory, data transmission, and linguistics. In this note, we study the structure of those graphs G, called cubical graphs (not to be confused with cubic graphs, those graphs for which all vertices have degree 3), which can be embedded into an n-dimensional cube. A basic technique used is the investigation of graphs which are critically nonembeddable, i.e., which can not be embedded but all of whose subgraphs can be embedded.     

Functions not first-return integrable

Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [0,1] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable. Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C^∞ on (0,1] yet fails to be first-return integrable.     

Some results concerning maximum Rényi entropy distributions

We consider the Student-t and Student-r distributions, which maximise Rényi entropy under a covariance condition. We show that they have information-theoretic properties which mirror those of the Gaussian distributions, which maximise Shannon entropy under the same condition. We introduce a convolution which preserves the Rényi maximising family, and show that the Rényi maximisers are the case of equality in a version of the Entropy Power Inequality. Further, we show that the Rényi maximisers satisfy a version of the heat equation, motivating the definition of a generalised Fisher information.     

Thurston?s relative inequalities and Reeb components

We construct a family of codimension 1 foliations in a 3-manifold for which Thurston?s relative inequality holds, but for which the absolute one is violated. For this, we introduce a variant of these inequalities, which we call the relative^(±) inequality. Also we determine the class of foliations for which the relative^(±) inequality holds.     

Construction of General Linear Methods with Runge–Kutta Stability Properties

We describe the construction of explicit general linear methods of order p and stage order q=p with s=p+1 stages which achieve good balance between accuracy and stability properties. The conditions are imposed on the coefficients of these methods which ensure that the resulting stability matrix has only one nonzero eigenvalue. This eigenvalue depends on one real parameter which is related to the error constant of the method. Examples of methods are derived which illustrate the application of the approach presented in this paper.     

Extensions of rooted trees and their applications

Informally, an almost-tree is a rooted graph which has the form of a tree in which some leaves are bent back to earlier nodes of the branch on which they sit. The paper provides a construction for unfurling any finite rooted graph to obtain a strong homomorphic image with desirable properties. As an application, it is shown that for each flow-diagram G there can be constructed an unfurling G¹ which is an almost-tree with semantic properties equivalent to those of G in a very strong sense.