Listing minimal edge-covers of intersecting families with applications to connectivity problems

Let G=(V,E) be a directed/undirected graph, let s,t∈V, and let F be an intersecting family on V (that is, X∩Y,X∪Y∈F for any intersecting X,Y∈F) so that s∈X and t∉X for every X∈F. An edge set I⊆E is an edge-cover of F if for every X∈F there is an edge in I from X to V−X. We show that minimal edge-covers of F can be listed with polynomial delay, provided that, for any I⊆E the minimal member of the residual family F_I of the sets in F not covered by I can be computed in polynomial time. As an application, we show that minimal undirected Steiner networks, and minimal k-connected and k-outconnected spanning subgraphs of a given directed/undirected graph, can be listed in incremental polynomial time.     

Some vector sequence transformations with applications to systems of equations

First, recursive algorithms for implementing some vector sequence transformations are given. In a particular case, these transformations are generalizations of Shanks transformation and the G-transformation. When the sequence of vectors under transformation is generated by linear fixed point iterations, Lanczos' method and the CGS are recovered respectively. In the case of a sequence generated by nonlinear fixed point iterations, a quadratically convergent method based on the -algorithm is recovered and a nonlinear analog of the CGS method is obtained.     

A result on sets,with applications to vector optimization

In this paper there is stated a result on sets in ordered linear spaces which can be used to show that some properties of the sets are inherited by their convex hulls under suitable conditions. As applications one gives a characterization of weakly efficient points and a duality result for nonconvex vector optimization problems.     

Conjugate duality in vector optimization and some applications to the vector variational inequality

The aim of this paper is to extend the so-called perturbation approach in order to deal with conjugate duality for constrained vector optimization problems. To this end we use two conjugacy notions introduced in the past in the literature in the framework of set-valued optimization. As a particular case we consider a vector variational inequality which we rewrite in the form of a vector optimization problem. The conjugate vector duals introduced in the first part allow us to introduce new gap functions for the vector variational inequality. The properties in the definition of the gap functions are verified by using the weak and strong duality theorems.     

A common fixed point theorem with applications to vector equilibrium problems

In this paper, using the Brouwer fixed point theorem, we establish a common fixed point theorem for a family of set-valued mappings. As applications of this result we obtain existence theorems for the solutions of two types of vector equilibrium problems, a Ky Fan-type minimax inequality and a generalization of a known result due to Iohvidov.     

Extensions of the generalized moment theorem and applications to distributed-parameter control systems

In this paper, we present extensions to the generalized moment theorem and apply it to optimal control problems for a certain class of distributed-parameter systems. We also apply it to the time-optimal control problem and extend the results of Ref. 1 pertaining to the largest controllable set, so that we can discuss the problem of recoverability for some distributed-parameter systems.The author wishes to express his gratitude to Professor P. K. C. Wang for his guidance and suggestions.     

Orthogonal matrix polynomials with respect to linear matrix moment functionals: Theory and applications

In this paper orthogonal matrix polynomials with respect to a right matrix moment functional are introduced. Basic results, important examples and applications to the approximation of matrix integrals are studied. Error bounds for the proposed matrix quadrature rules are given.     

Fast game dynamics coupled to slow population dynamics: A single population with hawk-dove strategies

We study a model of a population subdivided into two subpopulations corresponding to hawk and dove tactics. It is assumed that the hawk and dove individuals compete for a resource every Day, I.e., at a fast time scale. This fast part of the model is coupled to a slow part which describes the growth of the subpopulations and the long term effects of the encounters between the individuals which must fight to have an access to the resource. We aggregate the model into a single equation for the total population. It is shown that in the case of a constant game matrix, the total population grows according to a logistic curve whose τ and K parameters are related to the coefficients of the hawk-dove game matrix. Our result shows that high equilibrium density populations are mainly doves, whereas low equilibrium density populations are mainly hawks. We also study the case of a density dependent game matrix for which the gain is linearly decreasing with the total density.     

An approach to solutions of coupled semilinear partial differential equations with applications

In this work, an approach for finding the solution of coupled semi‐linear diffusion equations for initial value problems is presented. The formal exact solution is found and the Picard iteration is constructed. It is shown that the constructed sequence of solutions converges uniformly for some classes of initial value problems. The problem of dispersion of an oxygen demanding pollutant released into a uniform flow is studied. Copyright © 2000 John Wiley & Sons, Ltd.     

KKM type theorems with applications to generalized vector equilibrium problems in FC-spaces

A new coincidence theorem and some KKM type theorems for better admissible set-valued mappings are established in noncompact FC-space. As applications, some new existence theorems of solutions for several classes of generalized vector equilibrium problems are established in noncompact FC-spaces. The results represented in this paper unify and extend some corresponding known results in the literature.     

A generic approach to approximate efficiency and applications to vector optimization with set-valued maps

In this paper we focus on approximate minimal points of a set in Hausdorff locally convex spaces. Our aim is to develop a general framework from which it is possible to deduce important properties of these points by applying simple results. For this purpose we introduce a new concept of ε-efficient point based on set-valued mappings and we obtain existence results and properties on the behavior of these approximate efficient points when ε is fixed and by considering that ε tends to zero. Finally, the obtained results are applied to vector optimization problems with set-valued mappings.     

Perturbed iterative solution of coupled nonlinear systems with application to fluid dynamics

The perturbed iterative scheme developed in [3] is extended in this work to solve coupled systems of nonlinear equations. The algorithm consists of computing distinct perturbation parameters for each system at each iteration and adding these to corresponding nonlinear Gauss-Seidel iterates. It has been found computationally that such an algorithm significantly improves the convergence properties of Gauss-Seidel iterations. The method has been successfully applied to several coupled nonlinear systems of equations some of which are discussed in this work.     

Perturbed iterative solution of nonlinear equations with applications to fluid dynamics

In this work a technique has been developed to solve a set of nonlinear equations with the assumption that a solution exists. The algorithm involves nonlinear Gauss-Seidel iteractions and at each iteration the value of the iterate is added to a predetermined perturbation parameter which is computed in terms of quantities already known. This perturbation parameter has two properties: (i) it determines the mode of convergence, that means it shows how many more computations are required so that convergence may be achieved, and (ii) it accelerates the rate of convergence. The algorithm is computationally simple. Several nonlinear equations have been studied. The results seem to be encouraging.     

Strongly coupled elliptic systems and applications to Lotka–Volterra models with cross-diffusion

The aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations. The existence problem is for nonquasimonotone functions arising in the system, while the monotone iterations require some mixed monotone property of these functions. Applications are given to three Lotka–Volterra model problems with cross-diffusion and self-diffusion which are some extensions of the classical competition, prey–predator, and cooperating ecological systems. The monotone iterative schemes lead to some true positive solutions of the competition system, and to quasisolutions of the prey–predator and cooperating systems. Also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems.     

Reducing triangular systems of ODEs with rational coefficients,with applications to coupled Regge-Wheeler equations

We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This method is then used to solve the problem of conclusively deciding when certain rational ODE systems in upper triangular form can or cannot be reduced to diagonal form by differential operators with rational coefficients. As specific examples, we consider systems of coupled Regge-Wheeler equations, which have naturally appeared in previous work on vector and tensor perturbations on the Schwarzschild black hole spacetime. Our systematic approach reproduces and complements identities that have been previously found by trial and error methods.     

Continuity of operators in spaces of vector functions,with applications to the theory of bases

Banach spaces with unconditional martingale differences are investigated. In Sec. 1 a survey of their fundamental properties and connections with vector-valued harmonic analysis is given. In Sec. 2 new results are obtained regarding bases in the spaces E(X), where E is a symmetric space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 5–22, 1987.     

Generalized convex relations with applications to optimization and models of economic dynamics

We examine a notion of generalized convex set-valued mapping, extending the notions of a convex relation and a convex process. Under general conditions, we establish duality results for composite set-valued mappings and for convex programming problems involving convex set-valued mappings. We also present applications to the study of economic dynamical systems, by obtaining the characteristics of optimal paths generated by convex processes, and to optimization problems of a certain class of positively homogeneous increasing functions.