L_p-DICHOTOMY OF LINEAR DIFFERENTIAL EQUATIONS IN AN ARBITRARY BANACH SPACE

The notion of Lp-dichotomy for linear differential equations with possibly unbounded operator is introduced. By help of Banach fixed point theorem sufficient conditions for the existence of bounded solutions of nonlinear differential equations with an Lp-dichotomous linear part are obtained.     

On the number of permutatively inequivalent basic sequences in a Banach space

Let X be a Banach space with a Schauder basis (en)_n∈N. The relation E₀ is Borel reducible to permutative equivalence between normalized block-sequences of (en)_n∈N or X is c₀ or ?p saturated for some 1?p<+∞. If (en)_n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c₀ or ?p, 1<p<+∞, or the relation E₀ is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X^∗. If (en)_n∈N is unconditional, then either X is isomorphic to ?₂, or X contains ω² subspaces or ω² quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases.     

Structure of Nonunital Purely Infinite Simple Rings

In this note, we study the notion of purely infinite simple ring in the case of nonunital rings, and we obtain an analog to Zhang's Dichotomy for σ-unital purely infinite simple C*-algebras in the purely algebraic context.     

Asymptotic behavior of dynamic equations ontime scales

As a way to unify a discussion of many kinds of problems for equations in the contionous and discrete case(but also in order to reveal discrepancies between both cases), a theory of "time scales" was proposed and developed by Sulbach and Hilger. In our paper we investigate the asymptoic behaviour of so-called dynamic equations on time scales, and sych dynamic equations are differentialequations in the continous case and difference equations in the discrete case. We offer a perturbation result that leads to a time scales version of Levinson's Fundamental Lemma. Crucial are a dichotomy condition and a growth condition on the perturbation. Also, in the case that Levinson's result cannot be applied immediately, we suggest several preliminary transformations that might lead to a situation where Levinson's lemma is applicable. Such tranformations have been suggested by Harris and Lutz in the continuous case and by Benzaid and Lutz in the discrete case. Both those cases are covered by our theory, plus cases "in between". Examples for such cases will also be discussed in this paper.     

The complexity of the minimum cost homomorphism problem for semicomplete digraphs with possible loops

For digraphs D and H, a mapping f:V(D)→V(H) is a homomorphism of D to H if uv∈A(D) implies f(u)f(v)∈A(H). For a fixed digraph H, the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H).An optimization version of the homomorphism problem was motivated by a real-world problem in defence logistics and was introduced in Gutin, Rafiey, Yeo and Tso (2006) [13]. If each vertex u∈V(D) is associated with costs c_i(u),i∈V(H), then the cost of the homomorphism f is ∑_u∈V(D)c_f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem forH and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs c_i(u),u∈V(D),i∈V(H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost.Although a complete dichotomy classification of the complexity of MinHOM(H) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM(H) were proved when H is a semicomplete digraph Gutin, Rafiey and Yeo (2006) [10], and a semicomplete multipartite digraph Gutin, Rafiey and Yeo (2008) [12] and [11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in Gutin and Kim (2008) [9].     

Asymptotic solutions and error estimates for linear systems of difference and differential equations

Classical results concerning the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are asymptotically constant, i.e., k+o(1) as t tends to infinity. Here we are interested in more precise information about the o(1) terms, specifically how they depend precisely on certain perturbation terms in the equation. Results along these lines were given by Gel'fond and Kubenskaya for scalar difference equations and we will both extend and generalize one of them as well as provide some corresponding results for differential equations.     

干摩擦问题的概周期性

本文研究了干摩擦问题的概周期性，得到了系统具有概周期解的条件。     

Integrability of rotationally symmetric n-harmonic maps

A rotationally symmetric n-harmonic map is a rotationally symmetric p-harmonic map between two n-dimensional model spaces such that p=n. We show that rotationally symmetric n-harmonic maps can be integrated and are n-harmonic diffeomorphism, and apply such results to investigate the asymptotic behaviors of these maps. We also derive this integrability using Lie theory.