A proof of W. T. Gowers' theorem

W. T. Gowers' theorem asserts that for every Lipschitz function and 0$">, there exists an infinite-dimensional subspace of such that the oscillation of on is at most . The proof of this theorem has been reduced by W. T. Gowers to the proof of a new Ramsey type theorem. Our aim is to present a proof of the last result.

     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Optimal Error Estimates in Operator-Norm Approximations of Semigroups

We demonstrate that an idea related to the Central Limit Theorem and approximations by accompanying laws in probability theory is useful to get optimal convergence rates in some approximation formulas for operators. As examples we provide a bound for Euler approximations of bounded holomorphic semigroups; a bound for error in approximation of a power of operators by accompanying exponents, which is a useful tool in analysis of the Trotter–Kato formula, and can be considered as an extended version of Chernoff's lemma.     

Ergodic-theoretic properties of certain Bernoulli convolutions

The aim of this paper is to prove some fixed point theorems which generalize well known basic fixed point principles of nonlinear functional analysis. Moreover, we investigate the class of mappings f: X→ X, where X is a Banach space, for which one of the main conditions in the metric fixed point theory, namely the condition (1), is satisfied. We obtain essential applications of this fact. All our results are illustrated by suitable examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonlinear Boundary Stabilization of Nonuniform Timoshenko Beam

In this paper, the stabilization problem of nonuniform Timoshenko beam by some nonlinear boundary feedback controls is considered. By virtue of nonlinear semigroup theory, energy-perturbed approach and exponential multiplier method, it is shown that the vibration of the beam under the proposed control actiondecays exponentially or in negative power of time t as t→∞.     

Lévy–Khintchine Formula and Dual Convolution Semigroups Associated with Laguerre and Bessel Functions

In this work we consider a system of partial differential operators D ₁,D ₂ on K=[0,+[×R, whose eigenfunctions are the functions (x,t), (x,t)K, =((R0)×N)(0×[0,+[), which are related to the Laguerre functions for ((R 0)×N)(0,0) and which are the Bessel functions for (0×[0,+[). We provide K and with a convolution structure. We prove a Lévy–Khintchine formula on K, which permits us to characterize dual convolution semigroups on .     

Generalized Bicyclic Semigroups and Jones Semigroups

In this paper, we consider the generalized bicyclic semigroups B_n = a, b | aⁿb = 1 and the Jones semigroups A_n = a, b | aⁿ⁺¹b = a. They are the generalizations of the bicyclic semigroup B = a, b | ab = 1 and its analogous semigroup A = a, b | a²b = a discovered by P.R., Jones in 1987. The word problem for these kinds of semigroups is solved. It is proved that, for n 2, B_n are bisimple right inverse but not inverse semigroups and that the semigroup C = a, b | a²b = a, ab² = b is the smallest idempotent-free homomorphic image of A_n. Moreover, we also prove that A_n and A_m are mutually embeddable but not isomorphic with each other if n m. As a consequence, different kind of -nontrivial [0-]simple semigroups without idempotents are discussed.AMS 1991 Subject Classification: primary 20M10 secondary 20M05.Supported by NNSF of China (19671063) and KSRF of Sichuan Education Committee ([1999]127).     

On a Class of Elliptic-Parabolic Equations on Unbounded Intervals

We study a class of degenerate elliptic second order differential operators acting on some polynomial weighted function spaces on [0,+[. We show that these operators are the generators of C ₀-semigroups of positive operators which, in turn, are the transition semigroups associated with right-continuous normal Markov processes with state space [0,+]. Approximation and qualitative properties of both the semigroups and the Markov processes are investigated as well. Most of the results of the paper depend on a representation of the semigroups we give in terms of powers of particular positive operators of discrete type we introduced and studied in a previous paper.     

Rank Properties in Finite Semigroups II: The Small Rank and the Large Rank

This paper is concerned with the application of two possible definitions of rank to certain well-known semigroups.AMS Subject Classification (2000), 20M10     

Analyticity and Dynamic Behavior of a Damped Three-Layer Sandwich Beam

Using the Riesz basis approach, we study a sandwich beam that is composed of an outer stiff layer and a compliant middle layer. The dynamic behavior and analyticity of the system are obtained based on a detailed spectral analysis and Riesz basis generation. As a consequence, the analyticity of the solution and the exponential stability of the system are concluded. This work was supported by the National Natural Science Foundation of China, Program for New Century Excellent Talents in the Universities of China and by the National Research Foundation of South Africa.