On the stable set polytope of a series-parallel graph

We give a short proof of Chvátal's conjecture that the nontrivial facets of the stable set polytope of a series-parallel graph all come from edges and odd holes.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by CP Rail.     

A test of the core solution in finite strategy non-sidepayment games

This paper reports a test of the core solution in cooperative non-sidepayment games where players have finite strategy sets. Two laboratory experiments were conducted with three-person and four-person games; in both experiments, the core solution was tested competitively against the von Neumann-Morgenstern stable set and the imputation set. Predictions from these solution concepts were computed under parameters of α-effectiveness and strict preference. Results show that the frequency of outcomes falling in core is substantially higher than that observed in previous experiments (most of which involve sidepayment games). In addition, goodness-of-fit tests show that the core solution predicts the observed outcomes more accurately than do the stable set or the imputation set.     

The stable set problem and the thinness of a graph

We introduce a poly-time algorithm for the maximum weighted stable set problem, when a certain representation is given for a graph. The algorithm generalizes the algorithm for interval graphs and that for graphs with bounded pathwidth. By a suitable application to the frequency assignment problem, we improved several solutions to relevant benchmark instances.     

A note of fuzzy programming and the pareto optimal set

We indicate some errors made in Buckley's paper on fuzzy programming and the Pareto optimal set [1].     

On the sensitivity of von Neumann and Morgenstern abstract stable sets: The stable and the individual stable bargaining set

The purpose of this paper is twofold: First, to study the properties of the notions of the stable and individual stable bargaining sets (SBS and ISBS). Second, to point out the sensitivity of the von Neumann and Morgenstern (vN&M) abstract stable set to the dominance relation that is being employed: Insisting that each member of the coalition be made better off yields the SBS, while requiring that at least one member of the coalition is better off and all others are not worse off yields the ISBS. Rather surprisingly, the SBS and the ISBS may have an empty intersection.We fully characterize both the SBS and the ISBS in 3-person games with transferable utilities, and we also show that in ordinally convex games these two sets coincide with the core. As a by-product we thus derive a new proof that such games have a nonempty core. The paper concludes with an open question.     

On the bargaining set, kernel and core of superadditive games

We prove that for superadditive games a necessary and sufficient condition for the bargaining set to coincide with the core is that the monotonic cover of the excess game induced by a payoff be balanced for each imputation in the bargaining set. We present some new results obtained by verifying this condition for specific classes of games. For N-zero-monotonic games we show that the same condition required at each kernel element is also necessary and sufficient for the kernel to be contained in the core. We also give examples showing that to maintain these characterizations, the respective assumptions on the games cannot be lifted. Received: March 1998/Revised version: December 1998     

Stable subsets of modules and the existence of a unit in associative rings

We obtain criteria for the existence of a (left) unit in rings (arbitrary, Artinian, Noetherian, prime, and so on) that are based on the systematic study of properties of stable subsets of modules and their stabilizers that generalize the technique of idempotents. We study a class of quasiunitary rings that is a natural extension of classes of rings with unit and of von Neumann (weakly) regular rings, which inherits may properties of these classes. Some quasiunitary radicals of arbitrary rings are constructed, and the size of these radicals “measures the probability” of the existence of a unit. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 596–611, April, 1997. Translated by A. I. Shtern     

关于向量组的秩与最大无关组的教学改革研究

对向量的秩与最大无关组的教学过程进行了分析研究，从教学内容组织与教学手段等方面提出了改革的方案．对有的教材的不足提出了改进办法.     

On the stable rank and reducibility in algebras of real symmetric functions

Let A_?(??) denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that A_?(??) has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains with holes, which is a generalization of the main theorem in [18]. A sufficient topological condition on the symmetric open set ?? is given for the corresponding real algebra A_?(??) to have Bass stable rank equal to 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On biaccessible points in the Julia set of a Cremer quadratic polynomial

We prove that the only possible biaccessible points in the Julia set of a Cremer quadratic polynomial are the Cremer fixed point and its preimages. This gives a partial answer to a question posed by C. McMullen on whether such a Julia set can contain any biaccessible point at all.

     

On the existence of a stable limit cycle to a piecewise linear system in R2

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type “focus-focus” and “focus-center.” Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case “focus-center,” we use a pure geometric approach. In the case “focus-focus,” we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two-dimensional system describing the qualitative behavior of a two-dimensional excitable membrane model.     

On the maximal multiplicity of block sizes in a random set partition

We study the asymptotic behavior of the maximal multiplicity M_n = M_n(σ) of the block sizes in a set partition σ of [n] = {1,2,…,n}, assuming that σ is chosen uniformly at random from the set of all such partitions. It is known that, for large n, the blocks of a random set partition are typically of size W = W(n), with We^W = n. We show that, over subsequences {n_k}_{k ≥ 1} of the sequence of the natural numbers, , appropriately normalized, converges weakly, as k→∞, to , where Z₁ and Z₂ are independent copies of a standard normal random variable. The subsequences {n_k}_{k ≥ 1}, where the weak convergence is observed, and the quantity u depend on the fractional part f_n of the function W(n). In particular, we establish that . The behavior of the largest multiplicity M_n is in a striking contrast to the similar statistic of integer partitions of n. A heuristic explanation of this phenomenon is also given.     

On the core,the Weber set and convexity in games with a priori unions

This paper deals with the concepts of core and Weber set with a priori unions à la Owen. As far as we know, the Owen approach to games with a priori unions has never been studied from the coalitional stability point of view. Thus we introduce the coalitional core and coalitional Weber set and characterize the class of convex games with a priori unions by means of the relationships between both solution concepts.     

On properties of the set of invariant lines of a Brouwer homeomorphism

Abstract

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points.     

On the set of Lorenz-maximal imputations in the core of a balanced game

This paper considers the set of Lorenz-maximal imputations in the core of a balanced cooperative game as a solution concept. It is shown that the Lorenz-solution concept satisfies a number of suitable properties such as desirability, continuity and the reduced game property. Moreover, the paper consideres alternative characterizations where it is shown that Lorenz-fairness is tantamount to the existence of an additive, strictly increasing and concave social welfare function. Finally the paper also provides axiomatic characterizations as well as two examples of application. Received: February 1999/Final version: June 2001     

On the value of a stopped set function process

For certain types of stochastic processes {X_n | n ∈ $\text{N}$}, which are integrable and adapted to a nondecreasing sequence of σ-algebras $\text{F}$_n on a probability space (Ω, $\text{F}$, P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {$\text{F}$_n | n ∈ $\text{N}$}, when is$\text{sup}{}_{\mathrm{s}}\text{∫}{}_{\mathrm{\Omega }}\text{| X}{}_{\mathrm{\sigma }}\text{| dP}$finite, and when is there a stopping time $\text{σ}$ for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even σ-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research.     

On the area and perimeter of a random convex hull in a bounded convex set

Suppose K is a compact convex set in ℝ² and X _i , 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X _i we study the asymptotic properties of certain statistics of the convex hull of the sample. Received: 24 July 1996/Revised version: 24 February 1998     

On the structure of the solution set of a third kind boundary value problem

We show that given any closed subset C of a real Banach space E, there is a continuous function f(t, x) which is Lipschitz continuous in its second variable such that the solution set of the corresponding third kind boundary value problem is homeomorphic to C (Theorem 1.1). In the special problem we give the infimum of Lipschitz constants L_f of such functions f(t, x) (Theorem 1.3).     

On the stability of inner and outer approximations of a convex compact set by a ball

The finite-dimensional problems of outer and inner estimation of a convex compact set by a ball of some norm (circumscribed and inscribed ball problems) are considered. The stability of the solution with respect to the error in the specification of the estimated compact set is generally characterized. A new solution criterion for the outer estimation problem is obtained that relates the latter to the inner estimation problem for the lower Lebesgue set of the distance function to the most distant point of the estimated compact set. A quantitative estimate for the stability of the center of an inscribed ball is given under the additional assumption that the compact set is strongly convex. Assuming that the used norm is strongly quasi-convex, a quantitative stability estimate is obtained for the center of a circumscribed ball.