Maretsugu Yamasaki

Classification of infinite networks and its application

In this paper we consider nonlinear, infinite networks of purely resistive type where the voltage across a branch of the network is proportional to a fixed power of the current flowing in the branch. It is known that the study of currents in such networks amounts to studying the space of the functions on a network which have finite Dirichlet sums of orderp. Such a study was carried out in [7], [9], and [11]–[14] under the assumption that every node is connected to only finitely many different nodes of the network. In this paper we drop this assumption and work with general countable networks. We prove that most results of the locally finite case, and especially the classification theory, hold true in a more general context. Moreover, we give necessary and sufficient conditions for a network to have only constant Dirichlet finitep-harmonic functions. The relationship with discrete Markov processes is also pointed out.

Discrete Dirichlet integral formula

Abstract The (discrete) Dirichlet integral is one of the most important quantities in the discrete potential theory and the network theory. In many situations, the dissipation formula which assures that the Dirichlet integral of a function u is expressed as the sum of - u ( x )[Δ u ( x )] seems to play an essential role, where Δ u ( x ) denotes the (discrete) Laplacian of u . This formula can be regarded as a special case of the discrete analogue of Greens Formula. In this paper, we aim to determine the class of functions which satisfy the dissipation formula.

Semi-Infinite Transportation Problems*

Abstract A semi-infinite transportation dual-program pair is specified which involves general pairings of linear spaces stemming from an infinite number of destination requirements, but where in the primal program least-cost flows of goods are sought from only a finite number of origins to these destinations. Building on the work of M. J. Todd [Solving the generalized market area problem, Management Sci. 24 (1978), 1549–1554] a finite-dimensional dual unconstrained concave program is developed for the primal semi-infinite program but without certain measure-theoretic restrictions on the cost functions themselves. Optimality conditions for the dual-program pair are specified involving generalized column number conditions which parellel but extend those of the classical finite-dimensional transportation problem.

Verification for existence of solutions of linear complementarity problems

Abstract Recently, G. Alefeld, X. Chen and F. Potra [Numer. Math. 83 (1999) 265–315] presented a verification method for solutions of linear complementarity problems (LCPs). This paper is an attempt to obtain more useful information from the output of this verification method. In particular, existing results can only claim the nonexistence of solutions in a given interval. We will use the Farkas lemma to check if the interval contains a negative certification which shows the nonexistence of solutions in the whole space R n . Moreover, we will study how to choose a good nonsingular matrix A in the interval operator for P 0 -matrix linear complementarity problems. We report numerical results to illustrate the efficiency of the proposed technique.

Quasiharmonic classification of infinite networks

Abstract As a discrete analog to the quasiharmonic classification of Riemannian manifolds due to Nakai and Sario, we give a characterization of an infinite network by the class of discrete quasiharmonic functions on it. Some potential-theoretic properties of the network will be discussed in this paper.

Maximization of input conductances of an infinite network

Abstract For a locally finite infinite network in which the conductance of certain branches is variable, we shall investigate two maximization problems of input conductances of the network under the constraint that the sum of the variable branch conductances has a fixed bound. We shall study the existence and some properties of an optimal solution of our problems.