A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the rate of convergence to equilibrium is studied within the formalism of differential calculus on Riemannian manifolds. Under explicit geometric assumptions on the...

In 2010, Jain et al. [JJUW10] provided the first proof that QIP=PSPACE. Jain et al. used a constructive argument which applied the matrix multiplicative weight update method to a semidefinite programming problem formulation of a QMAM protocol to achieve this proof. The proof that QIP= PSPACE was alternately characterized by Xiaodi Wu [Wu10] that same year; Xiaodi Wu’s method, however, was simpl...

We provide an analytical study on the stability of equilibria of rigid rodlike nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with the Maier-Saupe intermolecular potential. We simplify the expression of the free energy of an orientational distribution function of rodlike LCP molecules by properly selecting a coordinate system and then investigate its stability ...

Abstract − The asymptotic behavior of a class discrete-time Hopfield neural network is studied in this paper. Some properties for this class discrete-time neural network, such as the boundedness of motion trajectory, the uniqueness and the absolute stability of equilibrium point etc, are obtained. In this paper, the sufficient conditions related to the existence of unique equilibrium point and ...

In a Shapley-Shubik assignment problem with a supermodular output matrix, we consider games in which each firm makes a take-it-orleave-it salary offer to one applicant, and a match is made only when the offer is accepted by her. We consider both one-shot and multistage games. In either game, we show that there can be many equilibrium salary vectors which are higher or lower than the minimum com...

We present a simple primal-dual algorithm for computing approximate Nash equilibria in two-person zero-sum sequential games with incomplete information and perfect recall (like Texas Hold’em poker). Our algorithm only performs basic iterations (i.e matvec multiplications, clipping, etc., and no calls to external first-order oracles, no matrix inversions, etc.) and is applicable to a broad class...

A large class of sequential decision making problems under uncertainty with multiple competing decision makers can be modeled as stochastic games. It can be considered that the stochastic games are multiplayer extensions of Markov decision processes (MDPs). In this paper, we develop a reinforcement learning algorithm to obtain average reward equilibrium for irreducible stochastic games. In our ...

The equilibrium state probabilities for queues with batch Markovian arrival processes are determined in form of matrix expressions, in which the central item to be computed is the so called fundamental-period-matrix G . G appears as the solution of the non-linear matrix equation G = AνG ν Σ , or as the infinite sum over matrices Gν , which in turn are functions of the matrices Aν as has been sh...

McLennan and Tourky (2010) showed that “imitation games” provide a new view of the computation of Nash equilibria of bimatrix games with the Lemke–Howson algorithm. In an imitation game, the payoff matrix of one of the players is the identity matrix. We study the more general “unit vector games”, which are already known, where the payoff matrix of one player is composed of unit vectors. Our mai...

In the study of Markovian processes, one of the principal achievements is the equivalence between the Φ-Sobolev inequalities and an exponential decrease of the Φ-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix Φ-entropies. This result also specializes to spectral gap inequalit...