A note on correlated equilibrium

The set of correlated equilibria for a bimatrix game is a closed, bounded, convex set containing the set of Nash equilibria. We show that every extreme point of a maximal Nash set is an extreme point of the above convex set. We also give an example to show that this result is not true in the payoff space, i.e. there are games where no Nash equilibrium payoff is an extreme point of the set of correlated equilibrium payoffs.     

A quantum generalization of equilibrium statistical thermodynamics: Effective macroparameters

We propose a generalization of statistical thermodynamics in which quantum effects are taken into account on the macrolevel without explicitly using the operator formalism while traditional relations between the macroparameters are preserved. In a generalized thermostat model, thermal equilibrium is characterized by an effective temperature bounded from below. We introduce fundamental theoretical macroparameters: the effective entropy and the effective action. Because the effective entropy is nonzero at low temperatures, we can write the third law of thermodynamics in the form postulated by Nernst. The effective action at any temperature coincides with the product of standard deviations of the coordinate and momentum in the Heisenberg uncertainty relation and is therefore bounded from below. We establish that the ratio of the effective action to the effective entropy in the low-temperature limit is determined by a holistic stochastic-action constant depending on the Planck and Boltzmann constants. We show that the same results can be obtained in the framework of a modified version of thermofield dynamics in which the quantum oscillator is described by a temperature-dependent complex macroscopic wave function. We study the discrepancy between the behavior of the action-to-entropy ratio in the low-temperature limit in our proposed theory and that in quantum equilibrium statistical mechanics, which can be verified experimentally. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 183–196, January, 2008.     

A new generalization of the Yannelis–Prabhakar equilibrium existence theorem for abstract economies

In this paper, we prove an intersection theorem for an infinite family of correspondences defined on non-compact spaces, and apply this result to give a new generalization of the Yannelis–Prabhakar existence theorem for equilibrium of abstract economies.     

A generalization of the Nash equilibrium theorem on bimatrix games

In this article, we consider a two-person game in which the first player picks a row representative matrixM from a nonempty set $A$ ofm ×n matrices and a probability distributionx on {1,2,...,m} while the second player picks a column representative matrixN from a nonempty set ? ofm ×n matrices and a probability distribution y on 1,2,...,n. This leads to the respective costs ofx ^t My andx ^t Ny for these players. We establish the existence of an ?-equilibrium for this game under the assumption that $A$ and ? are bounded. When the sets $A$ and ? are compact in ?^mxn, the result yields an equilibrium state at which stage no player can decrease his cost by unilaterally changing his row/column selection and probability distribution. The result, when further specialized to singleton sets, reduces to the famous theorem of Nash on bimatrix games.     

A new generalization of binomial coefficients

Let t be a fixed parameter and x some indeterminate. We give some properties of the generalized binomial coefficients $\genfrac{\langle }{\rangle}{0pt}{}{x}{k}$ inductively defined by $k/x \genfrac{\langle}{\rangle}{0pt}{}{x}{k}= t\genfrac{\langle}{\rangle}{0pt}{}{x-1}{k-1} +(1-t)\genfrac{\langle}{\rangle}{0pt}{}{x-2}{k-2}$ .     

A new generalization of the Lelong number

We will introduce a quantity which measures the singularity of a plurisubharmonic function φ relative to another plurisubharmonic function ψ, at a point a. We denote this quantity by ν _a,ψ (φ). It can be seen as a generalization of the classical Lelong number in a natural way: if ψ=(n?1)log|????a|, where n is the dimension of the set where φ is defined, then ν _a,ψ (φ) coincides with the classical Lelong number of φ at the point a. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form {z:ν _z,ψ (φ)≥c} where c>0, are in fact analytic sets, provided that the weight ψ satisfies some additional conditions.     

A new generalization of complex Stancu operators

In this paper, we investigate approximation properties of the complex form of an extension of the Bernstein polynomials, defined by Stancu by means of a probabilistic method. We obtain quantitative upper estimates for simultaneous approximation and the exact order of approximation by these operators attached to analytic functions in closed disks. Also, we prove that the new generalized complex Stancu operators preserve the univalence, starlikeness, convexity, and spirallikeness in the unit disk.     

A new generalization of Moulton affine planes

We consider a variation on W. A. Pierce's construction of Moulton planes. For any pseudo-ordered fieldF, the pairs of elements ofF are taken as points, and straight lines are given by the equationsx=c,y=mx+n withm≥0 andg(y=mf(x)+n withm < 0, wheref andg are mappings ofF into itself which have to satisfy a number of conditions.     

A new generalization of the KMOV cryptosystem

The KMOV scheme is a public key cryptosystem based on an RSA modulus \(n=pq\) where p and q are large prime numbers with \(p\equiv q\equiv 2\pmod 3\). It uses the points of an elliptic curve with equation \(y^2\equiv x^3+b\pmod n\). In this paper, we propose a generalization of the KMOV cryptosystem with a prime power modulus of the form \(n=p^{r}q^{s}\) and study its resistance to the known attacks.     

关于Wielandt-Hoffman不等式的简化证明及其推广

利用矩阵变换及控制论的知识给出了wielandt-Hoffman定理的新证法,从而简化了原证法,并推广了wielandt-Hoffman不等式.     

Oddness of the number of equilibrium points: A new proof

A new proof is offered for the theorem that, in “almost all” finite games, the number of equilibrium points isfinite andodd. The proof is based on constructing a one-parameter family of games with logarithmic payoff functions, and studying the topological properties of the graph of a certain algebraic function, related to the graph of the set of equilibrium points for the games belonging to this family. In the last section of the paper, it is shown that, in the space of all games of a given size, those “exceptional” games which fail to satisfy the theorem (by having an even number or an infinity of equilibrium points) is a closed set of measure zero.     

Selection of a correlated equilibrium in Markov stopping games

This paper deals with an extension of the concept of correlated strategies to Markov stopping games. The Nash equilibrium approach to solving nonzero-sum stopping games may give multiple solutions. An arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players’ decisions according to some optimality criterion. This is a form of equilibrium selection. Examples of correlated equilibria in nonzero-sum games related to the best choice problem are given. Several concepts of criteria for selecting a correlated equilibrium are used.     

R-andT-Groupoids: A generalization of groups

Aequationes mathematicae -     

A new method to detect periodically correlated structure

In this paper, we introduce a new method to test whether a discrete-time periodically correlated model explains an observed time series. The proposed method is based on the estimation of the support of spectral measure. Comparisons between our procedure and the methods which were proposed by Broszkiewicz-Suwaj et al. (Phys A 336:196–205, 2004) show that our testing procedure is more powerful. We investigate the performance of the proposed method by using real and simulated datasets.     

A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium

We study the Hamiltonian system (HS) where H ε C² ( ^2N, ) satisfies H (0) = 0, H′ (0) = 0 and the quadratic form ) is non-degenerate. We fix τ₀ > 0 and assume that ²N E F decomposes into linear subspaces E and F which are invariant under the flow associated to the linearized system (LHS) = JH″ (0) x and such that each solution of (LHS) in E is τ₀-periodic whereas no solution of (LHS) in F − 0 is τ₀-periodic. We write σ(τ₀) = σ_Q(τ₀) for the signature of the quadratic form Q restricted to E. If σ(τ₀) ≠ 0 then there exist periodic solutions of (HS) arbitrarily close to 0. More precisely we show, either there exists a sequence x_k → 0 of τ_k-periodic orbits on the energy level H⁻¹ (0) with τ_k → τ₀; or for each λ close to 0 with λσ(τ₀) > 0 the energy level H⁻¹ (λ) contains at least distinct periodic orbits of (HS) near 0 with periods near τ₀. This generalizes a result of Weinstein and Moser who assumed QE to be positive definite.