On the elementary divisors of the Sylvester and Lyapunov maps

The paper addresses the problem of computing the elementary divisors of the tensor product of linear transformations using the analysis of the tensor products of polynomial models, as developed in Fuhrmann and Helmke [5]. We use this to study the elementary divisors of the Lyapunov and complementary Lyapunov maps.     

Stability of two player game structures

We have extended a two player game-theoretical model proposed by V. Gurvich [To theory of multi-step games, USSR Comput. Math and Math. Phys. 13 (1973)] and H. Moulin [The Strategy of Social Choice, North Holland, Amsterdam, 1983]: All the considered game situations are framed by the same game structure. The structure determines the families of potential decisions of the two players, as well as the subsets of possible outcomes allowed by pairs of such choices. To be a solution of a game, a pair of decisions has to determine a (pure) functional equilibrium of the situational pair of payoff mappings which transforms the realized outcome into real-valued rewards of the players. Accordingly we understand that a structure is stable, if it admits functional equilibria for all possible game situations; and that it is complete, if every situation that only partitions the potential outcomes, is dominated by one of the players. We have generalized and strengthened a theorem by V. Gurvich [Equilibrium in pure strategies, Soviet Math. Dokl. 38 (1989)], proving that a proper structure is stable iff it is complete. Additional results provide game-theoretical insight that focuses the inquiry on the complexity of the stability decision problem; in particular, for coherent structures.These results also have combinatorial importance because every structure is characterized by a pair of hypergraphs [C. Berge, Graphes et Hypergraphes, Dunod, 1970] over a common ground set. The structure is dual (complete/coherent) iff the clutter of one hypergraph equals (includes/is included in) the blocker of the other one. So, for non-void coherent structures, the stability decision problem is equivalent to the much studied subexponential [M.L. Fredman, L. Khachiyan, On the complexity of dualization of monotone disjunctive normal forms, J. Algorithms 21 (1996)] hypergraph duality decision problem.     

The partial inverse minimum spanning tree problem when weight increase is forbidden

In a partial inverse optimization problem there is an underlying optimization problem with a partially given solution. The objective is to find a minimal perturbation of some of the problem’s parameter values, in such a way that the partial solution becomes a part of the optimal solution.     

On realizations of point determining graphs, and obstructions to full homomorphisms

A graph is point determining if distinct vertices have distinct neighbourhoods. A realization of a point determining graph H is a point determining graph G such that each vertex-removed subgraph G-x which is point determining, is isomorphic to H. We study the fine structure of point determining graphs, and conclude that every point determining graph has at most two realizations.A full homomorphism of a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have uv an edge of G if and only if f(u)f(v) is an edge of H. For a fixed graph H, a full H-colouring of G is a full homomorphism of G to H. A minimal H-obstruction is a graph G which does not admit a full H-colouring, such that each proper induced subgraph of G admits a full H-colouring. We analyse minimal H-obstructions using our results on point determining graphs. We connect the two problems by proving that if H has k vertices, then a graph with k+1 vertices is a minimal H-obstruction if and only if it is a realization of H. We conclude that every minimal H-obstruction has at most k+1 vertices, and there are at most two minimal H-obstructions with k+1 vertices.We also consider full homomorphisms to graphs H in which loops are allowed. If H has ? loops and k vertices without loops, then every minimal H-obstruction has at most (k+1)(?+1) vertices, and, when both k and ? are positive, there is at most one minimal H-obstruction with (k+1)(?+1) vertices.In particular, this yields a finite forbidden subgraph characterization of full H-colourability, for any graph H with loops allowed.     

Solution of the least squares method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima. This research was supported in part by the Hungarian Scientific Research Fund, Grant No. OTKA K 60480.     

Some properties of invariant polynomials with matrix arguments and their applications in econometrics

Summary Further properties are derived for a class of invariant polynomials with several matrix arguments which extend the zonal polynomials. Generalized Laguerre polynomials are defined, and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix. The latter includes thek-class estimator in econometrics.     

Inequalities for formal power series and entire functions

We present several integral and exponential inequalities for formal power series and for both arbitrary entire functions of exponential type and generalized Borel transforms. They are obtained through certain limit procedures which involve the multiparameter binomial inequalities, integral inequalities for continuous functions, and weighted norm inequalities for analytic functions. Some applications to the confluent hypergeometric functions, Bessel functions, Laguerre polynomials, and trigonometric functions are discussed. Also some generalizations are given.     

Norm Bounds for Ehrhart Polynomial Roots

M. Beck et al. found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1+(d+1)!. We provide an improved bound which is quadratic in d and applies to a larger family of polynomials.     

Exploiting equalities in polynomial programming

We propose a novel approach for solving polynomial programs over compact domains with equality constraints. By means of a generic transformation, we show that existing solution schemes for the, typically simpler, problem without equalities can be used to address the problem with equalities.