A note on the convergence of transfer sequences inn-person games

It was shown byStearns [1968], and more recently byKalai, Maschler, andOwen [1973], that a certain iterative technique converges to points in the bargaining sets. We prove in this paper that the conditions on the iterative technique can be relaxed to a certain extent. A counter-example shows that no further relaxation can be allowed.     

Attractivity properties of α-contractions

It is known that if T:X → X is completely continuous where X is a Banach space, then point dissipative and compact dissipative are equivalent, and imply the existence of a maximal compact invariant set which is uniformly asymptotically stable and attracts bounded sets uniformly. If T is an α-contraction, it is not known whether point dissipative and compact dissipative are equivalent. However, it is known that if T is an α-contraction and compact dissipative, then there exists a maximal compact invariant set which is uniformly asymptotically stable and attracts a neighborhood of any compact set uniformly. In this paper we show that for most practical examples which give rise to α-contraction, point dissipative and compact dissipative are equivalent. For example, we show this is true for stable neutral functional differential equations, retarded functional differential equations of infinite delay, and strongly damped nonlinear wave equations. We conjecture that this should be true for almost any practical application which gives rise to an α-contraction.     

On a dynamic theory for the kernel of ann-person game

In this paper, a dynamic theory for the kernel ofn-person games given byBillera is studied. In terms of the (bargaining) trajectories associated with a game (i.e. solutions to the differential equations defined by the theory), an equivalence relation is defined. The “consistency” of these equivalence classes is examined. Then, viewing the pre-kernel as the set of equilibrium points of this system of differential equations, some topological, geometric, symmetry and stability properties of the pre-kernel are given.     

Uniqueness of Limit Cycles in a Predator-Prey Model with Sigmoid Functional Response

In this paper, we prove that a predator-prey model with sigmoid functional response and logistic growth for the prey has a unique stable limit cycle, if the equilibrium point is locally unstable. This extends the results of the literature where it was proved that the equilibrium point is globally asymptotically stable, if it is locally stable. For the proof, we use a combination of three versions of Zhang Zhifen''s uniqueness theorem for limit cycles in Li$\acute{\rm e}$nard systems to cover all possible limit cycle configurations. This technique can be applied to a wide range of differential equations where at most one limit cycle occurs.     

Stability and fixed points of point-dissipative systems

It is known that if T: X → X is completely continuous or if there exists an n₀ > 0 such that Tⁿ⁰ is completely continuous, then T point dissipative implies that there is a maximal compact invariant set which is uniformly asymptotically stable, attracts bounded sets, and has a fixed point (see Billotti and LaSalle [Bull. Amer. Math. Soc.6 1971]). The result is used, for example, in studying retarded functional differential equations, or parabolic partial differential equations. This result has been extended by Hale and Lopes [J. Differential Equations13 1973]. They get the result that if T is an α-contraction and compact dissipative then there is a maximal compact invariant set which is uniformly asymptotically stable, attracts neighborhoods of compact sets, and has a fixed point. The above result requires the stronger assumption of compact dissipative. The principal result of this paper is to get similar results under the weaker assumption of point dissipative. To do this we must make additional assumptions. We will show these assumptions are naturally satisfied by stable neutral functional differential equations and retarded functional differential equations with infinite delay. The result has applications to many other dynamical systems, of course.     

Stability and traveling waves of an epidemic model with relapse and spatial diffusion

An epidemic model with relapse and spatial diffusion is studied. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. By using the linearized method, the local stability of each of feasible steady states to this model is investigated. It is proven that if the basic reproduction number is less than unity, the disease-free steady state is locally asymptotically stable; and if the basic reproduction number is greater than unity, the endemic steady state is locally asymptotically stable. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder's fixed point theorem, the existence of a traveling wave solution which connects the two steady states is established. Furthermore, numerical simulations are carried out to complement the main results.     

Independence of irrelevant expansions

Nash characterized the only bargaining solution to satisfy a well-known list of axioms. Independence of Irrelevant Alternatives states invariance of the solution outcome under certain contractions of the bargaining problem. A dual of this axiom is proposed here, stating invariance under certainexpansions of the bargaining problem andNash's solution is characerized by substituting this axiom for IIA in Nash's original list. After a transposition from the domain of bargaining solutions to the domain of choice rules, and a weakening of Invariance with respect to Positive Affine Transformations toTranslation Invariance, this new list of axioms is shown to characterizeUtilitarian rules.     

Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications

We use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi-Lipschitzian mappings. As a consequence, several convergence theorems for quasi-nonexpansive mappings and asymptotically κ-strict pseudo-contractions are deduced. We also establish strong convergence of iterative sequences for finding a common element of the set of fixed point, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain the corresponding results due to Tada-Takahashi and Nakajo-Shimoji-Takahashi.     

On a generalization of distance sets

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X.Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d−1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.     

Topologie locale des méthodes de Newton cubiques: plan dynamique

For a cubic Newton map N, we obtain the following theorems: 1) The boundary of the immediate basin of each fixed critical point is locally connected. 2) The Julia set J(N) is locally connected provided either N has no irrational indifferent periodic point or N has no Siegel disc and the orbit of the non-fixed critical point doesn 't accumulate on the boundary of the fixed immediate basins. In particular, in contrast with Julia sets of polynomials, J(N) can be locally connected even if N has a periodic Cremer point.The proofs rely on the construction of articulated rays which are very special simple arcs landing on J(N).     

Equilibrium selection in a bargaining problem with transaction costs

It is the purpose of the paper to analyse a bargeining situation with the help of the equilibrium selection theory of John C. Harsanyi and Reinhard Selten. This theory selects one equilibrium point in every finite non-cooperative game. The bargaining problem is the following one: the two bargainers — player 1 and player 2 — simultaneously and independently propose a payoffx of player 1 in the interval 〈0, 1〉. If agreement is reached player 2's payoffs is 1?x. Otherwise both receive zero. Each playeri has a further alternativeW _i , namely not to bargain at all (i=1, 2). Thereby he avoids transaction costsc andd of bargaining which arise whether an agreement is reached or not. One may think of an illegal deal where bargaining involves a risk of being punished — independently whether the deal is made or not. The model has the form of a (K+1)×(K+1)-bimatrix game. It is assumed that there is an indivisable smallest money unit. The game hasK+1 pure strategy equilibrium points.K of them correspond to an agreement and the last one is the strategy pair where both players refuse to bargain. Each of theK+1 equilibrium points can be the solution of the game. The aim of the Harsanyi-Selten-theory is to select in a unique way one of these equilibrium points by an iterative process of elimination (by payoff dominance and risk dominance relationships) and substitution. For each parameter combination (c, d) a sequence of candidate sets arises which becomes smaller and smaller until finally a candidate set with exactly one equilibrium point — the solution of the game — is found. For the sake of shortness the paper will report results without detailed proofs, which can be found elsewhere [Leopold-Wildburger].     

An edge-deletion problem for locally finite graphs

For each positive integer n, let T_n be the tree in which exactly one vertex has degree n and all the other vertices have degree n + 1. A graph G is called stable if its edge set is nonempty and if deleting an arbitrary edge of G there is always a component of the residue graph which is isomorphic to G. The question whether there are locally finite stable graphs that are not isomorphic to one of the graphs T_n is answered affirmatively by constructing an uncountable family of pairwise nonisomorphic, locally finite, stable graphs. Further, the following results are proved: (1) Among the locally finite trees containing no subdivision of T₂, the oneway infinite path T₁ is the only stable graph. (2) Among the locally finite graphs containing no two-way infinite path, T₁ is also the only stable graph.     

Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity

In this paper, some SEIRS epidemiological models with vaccination and temporary immunity are considered. First of all, previously published work is reviewed. In the next section, a general model with a constant contact rate and a density-dependent death rate is examined. The model is reformulated in terms of the proportions of susceptible, incubating, infectious, and immune individuals. Next the equilibrium and stability properties of this model are examined, assuming that the average duration of immunity exceeds the infectious period. There is a threshold parameter R_o and the disease can persist if and only if R_o exceeds one. The disease-free equilibrium always exists and is locally stable if R_o < 1 and unstable if R_o > 1. Conditions are derived for the global stability of the disease-free equilibrium. For R_o > 1, the endemic equilibrium is unique and locally asymptotically stable.For the full model dealing with numbers of individuals, there are two critical contact rates. These give conditions for the disease, respectively, to drive a population which would otherwise persist at a finite level or explode to extinction and to cause a population that would otherwise explode to be regulated at a finite level. If the contact rate β(N) is a monotone increasing function of the population size, then we find that there are now three threshold parameters which determine whether or not the disease can persist proportionally. Moreover, the endemic equilibrium need no longer be locally asymptotically stable. Instead stable limit cycles can arise by supercritical Hopf bifurcation from the endemic equilibrium as this equilibrium loses its stability. This is confirmed numerically.     

Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models

In this paper we are concerned with the fractional-order predator-prey model and the fractional-order rabies model. Existence and uniqueness of solutions are proved. The stability of equilibrium points are studied. Numerical solutions of these models are given. An example is given where the equilibrium point is a centre for the integer order system but locally asymptotically stable for its fractional-order counterpart.     

Group ramsey theory

A subset S of a group G is said to be a sum-free set if $\text{S ∩ (S + S) = ?}$. Such a set is maximal if for every sum-free set T ? G, we have |T| ? |S|. Here, we generalize this concept, defining a sum-free set S to be locally maximal if for every sum free set T such that S ? T ? G, we have S = T. Properties of locally maximal sum-free sets are studied and the sets are determined (up to isomorphism) for groups of small order.     

Stability of an age-structured epidemiological model for hepatitis C

This article introduces an age-structured epidemiological model for the disease transmission dynamics of hepatitis C. We first show that the infection-free steady state is locally and globally asymptotically stable if the basic reproductive number ? ₀ is below one, in this case, the disease always dies out, then we prove that at least one endemic steady state exists when the reproductive number ? ₀ is above one, the stability conditions for the endemic steady states are also given.     

Locally minimal topological groups 1

The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G,τ) is called locally minimal if there exists a neighborhood U of 0 in τ such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than τ. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudoconvex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. Unlike the minimal topologies, the locally minimal ones are always available on “large” groups. To support this line, we prove that a bounded abelian group G admits a non-discrete locally minimal and locally GTG group topology iff |G|?c.     

Nonsmooth optimization via quasi-Newton methods

We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f, not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.     

Stability of properly efficient points and isolated minimizers of constrained vector optimization problems

In this paper the constrained vector optimization problem mic _C f(x), g(x) ? ? K, is considered, where\(f:\mathbb{R}^n \to \mathbb{R}^m \) and\(g:\mathbb{R}^n \to \mathbb{R}^p \) are locally Lipschitz functions and\(C \subset \mathbb{R}^m \) and\(K \subset \mathbb{R}^p \) are closed convex cones. Several solution concepts are recalled, among them the concept of a properly efficient point (p-minimizer) and an isolated minimizer (i-minimizer). On the base of certain first-order optimalitty conditions it is shown that there is a close relation between the solutions of the constrained problem and some unconstrained problem. This consideration allows to “double” the solution concepts of the given constrained problem, calling sense II optimality concepts for the constrained problem the respective solutions of the related unconstrained problem, retaining the name of sense I concepts for the originally defined optimality solutions. The paper investigates the stability properties of thep-minimizers andi-minimizers. It is shown, that thep-minimizers are stable under perturbations of the cones, while thei-minimizers are stable under perturbations both of the cones and the functions in the data set. Further, it is shown, that sense I concepts are stable under perturbations of the objective data, while sense II concepts are stable under perturbations both of the objective and the constraints. Finally, the so called structural stability is discused.     

The reactive bargaining set: Structure, dynamics and extension to NTU games

The reactive bargaining set (Granot [1994]) is the set of outcomes for which no justified objection exists. Here, in a justified objection the objector first watches how the target tries to act (if he has such an option), and then reacts by making a profit and ruining the target's attempt to maintain his share. In this paper we explore properties of the reactive bargaining set, set up the system of inequalities that defines it, and construct a dynamic system in the sense of Stearns' transfer scheme that leads the players to this set. We also extend the definition of the reactive bargaining set toNTU games in a way that keeps it nonempty. To shed light on its nature and its relative ease of computation, we compute the reactive bargaining set for games that played important role in the game theory literature.