Stable sets and mean Li–Yorke chaos in positive entropy systems

It is shown that in a topological dynamical system with positive entropy, there is a measure-theoretically “rather big” set such that a multivariant version of mean Li–Yorke chaos happens on the closure of the stable or unstable set of any point from the set. It is also proved that the intersections of the sets of asymptotic tuples and mean Li–Yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively.     

Group classification and conservation laws of nonlinear filtration equation with a small parameter

Group classification of the perturbed nonlinear filtration equation is performed assuming that the perturbation is an arbitrary function of the dependent variable. The nonlinear self-adjointness of the equation under consideration is investigated. Using these results, the approximate conservation laws are constructed.     

Non-Classical Behavior of Atoms in an Interferometer

Using the time-dependent wave function we have studied the properties of the atomic transverse motion in an interferometer, and the cause of the non-classical behavior of atoms reported by Kurtsiefer, Pfau, and Mlynek [Nature 386, 150 (1997)]. The transverse wave function is derived from the solution of the two-dimensional Schrödinger's equation, written in the form of the Fresnel–Kirchhoff diffraction integral. It is assumed that the longitudinal motion is classical. Comparing data of the space distribution and of the transverse momentum distribution in interferometers with one and two open slits, it follows that the atomic motion is influenced by the atomic matter wave and violates the laws of classical mechanics. However, the negative values of Wigner's function should not be taken as evidence that the atoms in an interferometer violate the classical statistical law of the addition of positive probabilities. This inference follows from the comparison of properties of Wigner's function and of the de Broglian probability density in phase space.     

半格与Domain的表示

引入一种半格———Ｄ 半格，建立了Ｄ－半格的表示理论，利用它得到了Ｌ ｄｏｍａｉｎ（即局部代数格）的表示理论；证明了Ｌ ｄｏｍａｉｎ与稳定映射范畴对偶等价于稳定Ｄ 半格与Ｄ－半格同态范畴．     

WHEN CAN THE STABLE ALGEBRA DETERMINE THE STRUCTURE OF A C^＊-ALGEBRA ？

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Convergence of weighted sums of tight random elements

Convergence of weighted sums of tight random elements {V_n} (in a separable Banach space) which have zero expected values and uniformly bounded rth moments (r > 1) is obtained. In particular, if {a_nk} is a Toeplitz sequence of real numbers, then | Σ_k=1^∞a_nkf(V_k)| → 0 in probability for each continuous linear functional f if and only if 6Σ_k=1^∞a_nkV_k 6→ 0 in probability. When the random elements are independent and max_{1≤k≤n | ank | = $\text{O}$(n^?8)} for some $\text{0 <}\text{1}\text{s}\text{< r ? 1}$, then |Σ_k=1^∞a_nkV_k 6→ 0 with probability 1. These results yield laws of large numbers without assuming geometric conditions on the Banach space. Finally, these results can be extended to random elements in certain Fréchet spaces.     

Fractional and integral colourings

LetG=(V, E) be an undirected graph andc any vector in ℤ ^V(G) ₊. Denote byχ(G _c) (resp.η(G _c)) the chromatic number (resp. fractional chromatic number) ofG with respect toc. We study graphs for whichχ(G _c)−[η(G _c)]⩽1. We show that for the class of graphs satisfyingχ(G _c)=[η(G _c)] (a class generalizing perfect graphs), an analogue of the Duplication Lemma does not hold. We also describe a 2-vertex cut decomposition procedure related to the integer decomposition property. We use this procedure to show thatχ(G _c)=[η(G _c)] for series-parallel graphs andχ(G _c)⩽[η(G _c)]+1 for graphs that do not have the 4-wheel as a minor. The work of this author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERCC) under grant A9126.     

Measure-Valued Solutions to Initial-Boundary Value Problems for Certain Systems of Conservation Laws: Existence and Dynamics

A framework for studying initial-boundary value problems for systems of conservation laws, in what concerns to the existence of measure-valued solutions and their asymptotic behavior, is developed here with the helpful introduction of a class of flux maps which allow a rather complete treatment of these questions including systems of practical importance as those arising in multiphase flow in porous media. The systems of this class may, in general, admit umbilic points, submanifolds where genuine nonlinearity fails, as well as elliptic regions. We prove the existence of measure-valued solutions by using the vanishing viscosity method and, also, finite difference schemes. The main result about the dynamics of the measure-valued solutions is that for certain special boundary values, given by constant states, the time-averages of these m-v solutions converge weakly to the Dirac measure concentrated at those states, for a.e. space variable. The rate of convergence of the time-averages of the expected values can be estimated by properties of the flux maps only.

     

Approximations of stable actions on $ \Bbb {R} $-trees

This article shows how to approximate a stable action of a finitely presented group on an -tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an -tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on -trees and on the dynamical study of exotic components by D. Gaboriau. Received: 22 October, 1997     

On the Number of Elementary Submodels of an Unsuperstable Homogeneous Structure

We show that if M is a stable unsuperstable homogeneous structure, then for most κ ? |M|, the number of elementary submodels of M of power κ is 2^κ.