Equilibrium States of Random Economies with Locally Interacting Agents and Solutions to Stochastic Variational Inequalities in 〈L1,L∞〉

We study a stochastic model of an economy with locally interacting agents. The basis of the study is a deterministic model of dynamic economic equilibrium proposed by Polterovich. We generalize Polterovich's theory, in particular, in two respects. We introduce stochastics and consider a version of the model with local interactions between the agents. The structure of the interactions is described in terms of random fields on a directed graph. Equilibrium states of the system are solutions to certain variational inequalities in spaces of random vectors. By analyzing these inequalities, we establish an existence theorem for equilibrium, which generalizes and refines a number of previous results.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

Pointwise Hardy inequalities

If is an open set with the sufficiently regular boundary, then the Hardy inequality holds for and , where . The main result of the paper is a pointwise inequality , where on the right hand side there is a kind of maximal function. The pointwise inequality combined with the Hardy-Littlewood maximal theorem implies the Hardy inequality. This generalizes some recent results of Lewis and Wannebo.

     

On a Primal-Dual Analytic Center Cutting Plane Method for Variational Inequalities

We present an algorithm for variational inequalities VI( , Y) that uses a primal-dual version of the Analytic Center Cutting Plane Method. The point-to-set mapping is assumed to be monotone, or pseudomonotone. Each computation of a new analytic center requires at most four Newton iterations, in theory, and in practice one or sometimes two. Linear equalities that may be included in the definition of the set Y are taken explicitly into account.We report numerical experiments on several well—known variational inequality problems as well as on one where the functional results from the solution of large subproblems. The method is robust and competitive with algorithms which use the same information as this one.     

解线性不等式的神经网络

本文提出两个解线性不等式的Hopfiedl-Tank型的神经网络。第一个网络模拟同时松弛投影方法，第二个网络是二次规划方法。当线性不等式的解集非空时，这两个方法都给出该线性不等式的解。同时我们还给出了这两个网络的数值模拟。     

The Bell inequalities and the joint measurement of incompatible observables

A quantum-mechanical theory of joint nonideal measurement of incompatible polarization observables is applied to an EPR-like experiment. It is demonstrated that this experiment yields both information satisfying and information violating the Bell inequalities. The measurement is also discussed in the context of a local hidden-variables theory. It is argued that the violation of an additional assumption of reproducibility of the hidden variable rather than violation of locality may be responsible for the violation of the Bell inequalities.     

Quantum theory of rotation angles: The problem of angle sum and angle difference

We reconsider the problem of the sum and difference of two angle variables in quantum mechanics. The spectra of the sum and difference operators have widths of , but angles differing by are indistinguishable. This means that the angle sum and difference probability distributions must be cast into a range. We obtain probability distributions for the angle sum and difference and relate this problem to the representation of nonbijective canonical transformations. Received: 6 December 1997 / Revised: 15 April 1998 / Accepted: 7 May 1998     

On a class of martingale inequalities

Let Z = {Z₀, Z₁, Z₂,…} be a martingale, with difference sequence X₀ = Z₀, X_i = Z_i ? Z_{i ? 1}, i ≥ 1. The principal purpose of this paper is to prove that the best constant in the inequality λP(sup_i |X_i| ≥ λ) ≤ C sup_iE |Z_i|, for λ > 0, is C = (log 2)^?1. If Z is finite of length n, it is proved that the best constant is $\text{C}{}_{\mathrm{n}}\text{= [n(2}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{? 1)]}{}^{\mathrm{?1}}$. The analogous best constant C_n(z) when Z₀ ≡ z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity G_n(z, E) = sup_zP(max_i=1,…,n |X_i| ≥ 1), the supremum being taken over all martingales Z with Z₀ ≡ z and E|Z_n| = E. The expression for G_n(z,E) is derived by induction, using methods from the theory of moments.     

Coexistence of phases in Ising ferromagnets

We derive a new inequality for ferromagnetic Ising spin systems and then use it to obtain information about the number of phases which can coexist in such systems. We show in particular that for even interactions only two phases (up and down magnetization) can coexist below the critical temperature at zero magnetic field (h=0) whenever the energy is a continuous function of the temperature. We also prove that the derivatives with respect toh ath=0 of the odd correlation functions (triplet,...) diverge like the susceptibility in the vicinity of the critical temperature (at least for pair interactions). Our results also apply to higher order Ising spins (not just spin 1/2).Research supported in part by NSF Grant #MPS 75-20638 and USAFOR Grant #73-2430D.John Simon Guggenheim Fellow.