The consistency principle and an axiomatization of the α-core

This paper examines the α-core of strategic games by means of the consistency principle. I provide a new definition of a reduced game for strategic games. And I define consistency (CONS) and two forms of converse consistency (COCONS and COCONS^*) under this definition of reduced games. Then I axiomatize the α-core for families of strategic games with bounded payoff functions by the axioms CONS, COCONS^*, weak Pareto optimality (WPO) and one person rationality (OPR). Furthermore, I show that these four axioms are logically independent. In proving this, I also axiomatize the α-individually rational solution by CONS, COCONS and OPR for the same families of games. Here the α-individually rational solution is a natural extension of the classical `maximin' solution. Received: June 1998/Final version: 6 July 2001     

The existence of TU α-core in normal form games

This paper provides a TU α-core existence result in a large class of normal form games. In the oligopoly markets of a homogeneous good, the TU α-core is non-empty if all profit functions are continuous and concave. In a general game, the existence of TU α-core follows from the weak separability, the compactness and convexity of choice sets, and the concavity and continuity of payoff functions. Received: November 1997/final version: August 1998     

On σ-Embedded and σ-n-Embedded Subgroups of Finite Groups

Siberian Mathematical Journal - Let G be a finite group, and let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and σ(G) = {σi | σi ∩...     

Wiener's criterion and Γ-convergence

Dirichlet problems with homogeneous boundary conditions in (possibly irregular) domains and stationary Schrödinger equations with (possibly singular) nonnegative potentials are considered as special cases of more general equations of the form –u + µu = 0, whereµ is an arbitrary given nonnegative Borel measure in ⁿ . The stability and compactness of weak solutions under suitable variational perturbations ofµ is investigated and stable pointwise estimates for the modulus of continuity and the energy of local solutions are obtained.     

Γ-convergence of free-discontinuity problems

We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u.We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.     

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.     

The Randić energy of generalized double sun

Czechoslovak Mathematical Journal - We show that the family of trees defined as generalized double sun of odd order satisfies the conjecture for the Randi? energy proposed by I. Gutman, B....     

Some examples and counterexamples of convergence of σ-algebras and filtrations

We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of σ-algebras to filtrations generated by different discretizations of a single process. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 295–306, July–September, 2000. Translated by V. Mackevičius     

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and -convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (P _h) of optimal control problems and a control problem (P), we determine some general conditions, involvingG-convergence and -convergence, under which the sequence of the optimal pairs of the problems (P _h) converges to the optimal pair of problem (P).The authors wish to thank Professor E. De Giorgi for many stimulating discussions.     

Refinements of the β-core and the strong equilibrium and the Aumann proposition

The Aumann proposition establishes that an outcome in the -core of the one-shot gameG is the same as the strong equilibrium utility allocations of the associated supergame. We refine the -core by requiring that the threat strategies used to deter deviations be credible and examine the consequences for the Aumann proposition in the case of the refinement. We show that a refinement of the strong-equilibrium notion is always in the refined -core. The converse is not necessarily true unless the utility allocations are linear.I am grateful to T. Ichiishi for several discussions that helped in the development of this paper. This is part of the author's Ph.D. dissertation submitted at the university of Iowa in December 1985.     

The interpolation ofℓ r sequences by hp functions

The sequence spaceH ^P (z)={{f (z_h)}:f H ^p} is defined for a fixed sequence Z={z_k} of different points of the open unit disk and the Hardy class HP of analytic functions in the disk. For an arbitrary p[1, ) is constructed a point sequence Z= {z_k} such that ¹h ^p(z), but ^r h^p (Z) for r > 1. It follows from a well-known result of L. Carleson that the inclusions ^r h (Z) for all r[1,] are equivalent.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 503–508, April, 1977.     

Atomic intersection of σ-fields and some of its consequences

Let ${(\Omega, \mathcal{F}, P)}$ be a probability space. For each ${\mathcal{G}\subset\mathcal{F}}$ , define ${\overline{\mathcal{G}}}$ as the σ-field generated by ${\mathcal{G}}$ and those sets ${F\in \mathcal{F}}$ satisfying ${P(F)\in\{0,1\}}$ . Conditions for P to be atomic on ${\cap_{i=1}^k\overline{\mathcal{A}_i}}$ , with ${\mathcal{A }_1,\ldots,\mathcal{A}_k\subset\mathcal{F}}$ sub-σ-fields, are given. Conditions for P to be 0-1-valued on ${\cap_{i=1}^k \overline{\mathcal{A}_i}}$ are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.     

A relation between Г-convergence of functionals and their associated gradient flows

Giorgi conjectured in 1979 that if a sequence of functionals converges in the sense of Г-convergence to a limiting functional, then the corresponding gradient flows will converge as well after changing timescale appropriately. It is shown that this conjecture holds true for a rather wide kind of functionals. Project supported by the National Natural Science Foundation of China (Grant No. 19701018).     

The Infinite—dimensional σ—widths of Besov Classes

Let (X(Rd),|.|X) be a normed space of real functions on Rd.Let >0 and Pα be the operator in X(Rd) defined by Pα f(x)=χα(x) f(x) (where χα(x) is thecharacteristic function of the cube Idα=［-α,α］d). Let L be asubspace of X(Rd). Set PαL=Pα f:f∈L. Suppose L islocally-finite dimensional, I.e., dim(Pα L,X)<+∞ for every >0.Then the following quantity is said to be the average dimension of L in X(in the sense of LED)(see ［1］).     

On the L1-convergence and behavior of coefficients of Fourier–Vilenkin series

In this paper, we prove the following statement that is true for both bounded and some type of unbounded Vilenkin systems: for any \( \varepsilon \in (0,1)\), there exists a measurable set \(E\subset [0,1)\) of measure bigger than \(1-\varepsilon \) such that for any function \(f \in L^{1}[0,1)\), it is possible to find a function \(g\in L^{1}[0,1)\) coinciding with f on E, Fourier series of g with respect to Vilenkin system are convergent in \(L^{1}\)-norm and the absolute values of non zero Fourier coefficients of g are monotonically decreasing.     

The construction of infinite families of any k-tight optimal and singular k-tight optimal directed double loop networks

The double loop network(DLN)is a circulant digraph with n nodes and outdegree 2.It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems.Given the number n of nodes,how to construct a DLN which has minimum diameter?This problem has attracted great attention.A related and longtime unsolved problem is:for any given non-negative integer k,is there an infinite family of k-tight optimal DLN?In this paper,two main results are obtained:(1)for any k≥0,the infinite families of k-tight optimal DLN can be constructed,where the number n(k,e,c)of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c.(2)for any k≥0, an infinite family of singular k-tight optimal DLN can be constructed.     

Compact convergence of σ-fields and relaxed conditional expectation

The collection of sub-σ-fields of a Borel measure space when endowed with the topology of strong convergence is in general not a compact space. The paper offers a completion of this space which makes it compact. The elements which are added to the space are called relaxed σ-fields. A notion of relaxed conditional expectation with respect to a relaxed σ-field is identified. The relaxed conditional expectation is a probability measure-valued map. It is shown that the conditional expectation operator is continuous on the completion of the space. Other properties of conditional expectation are lifted to and interpreted in the relaxed framework. Received: 22 February 1999 / Revised version: 23 October 2000 / Published online: 8 May 2001