An Optimal Alternative Theorem and Applications to Mathematical Programming

Given a closed convex cone P with nonempty interior in a locally convex vector space, and a set , we provide various equivalences to the implication

Graph-like spaces: An introduction

Thomassen and Vella (Graph-like continua, augmenting arcs, and Menger’s Theorem, Combinatorica, doi:10.1007/s00493-008-2342-9) have recently introduced the notion of a graph-like space, simultaneously generalizing infinite graphs and many of the compact spaces recently used by Diestel or Richter (and their coauthors) to study cycle spaces and related problems in infinite graphs. This work is a survey to introduce graph-like spaces and shows how many of these works on compact spaces can be generalized to compact graph-like spaces.     

Multiple solutions for Neumann and periodic problems with singular ?-Laplacian

We use the critical point theory for convex, lower semicontinuous perturbations of C¹-functionals to establish existence of multiple radial solutions for some one parameter Neumann problems involving the operator . Similar results for periodic problems are also provided.     

Chen’s Theorem with Small Primes

Let N be a sufficiently large even integer. Let p denote a prime and P2 denote an almost prime with at most two prime factors. In this paper, it is proved that the equation N - p+ P2 (p ≤ AT0.945) is solvable.     

A new development for the Tikhonov Theorem in nonlinear singular perturbation systems

This paper deals with the exponential stability of nonlinear perturbation systems under a new condition. A novel criterion of exponential stability of nonlinear systems is firstly given in a general form. In this criterion, a new kind of characteristic value is introduced, which makes the exponential stability measurable. Based on this criterion, a new development for the Tikhonov Theorem in nonlinear singular perturbation systems is presented.     

Truth‐table Schnorr randomness and truth‐table reducible randomness

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth‐table Schnorr randomness (defined in 6 too only by martingales) and truth‐table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth‐table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth‐table Schnorr randomness. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

罗尔定理的三种推广形式

通过减弱罗尔定理成立的条件,给出并严格证明罗尔定理的三种推广形式,最后简单说明其应用．     

Cropper's question and Cruse's theorem about partial Latin squares

In 1974 Cruse gave necessary and sufficient conditions for an r × s partial latin square P on symbols σ₁,σ₂,…,σ_t, which may have some unfilled cells, to be completable to an n × n latin square on symbols σ₁,σ₂,…,σ_n, subject to the condition that the unfilled cells of P must be filled with symbols chosen from {σ_{t + 1},σ_{t + 2},…,σ_n}. These conditions consisted of r + s + t + 1 inequalities. Hall's condition applied to partial latin squares is a necessary condition for their completion, and is a generalization of, and in the spirit of Hall's Condition for a system of distinct representatives. Cropper asked whether Hall's Condition might also be sufficient for the completion of partial latin squares, but we give here a counterexample to Cropper's speculation. We also show that the r + s + t + 1 inequalities of Cruse's Theorem may be replaced by just four inequalities, two of which are Hall inequalities for P (i.e. two of the inequalities which constitute Hall's Condition for P), and the other two are Hall inequalities for the conjugates of P. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:268‐279, 2011     

General model of subtraction of stochastic variables. Attractor and stability analysis

We introduce a general process designed to model stochastic systems in which the dependence of random variables is not through addition only but combined addition and subtraction with bounded ranges, and whose probabilistic factors have compact support. We show that, still retaining much of the general essence of the Central Limit Theorem, this process presents a functional attractor which is neither Gaussian nor Lévy like, and is precisely akin numerically to a probability density function shown in previous works to have ubiquitous character, namely the two-parameter beta distribution.