Worldlines and body points associated with an extensive property

Material structure of bodies that is usually assumed a-priory in continuum mechanics is constructed on the basis of a balance of a given extensive property on spacetime. Body points are identified with worldlines—the integral lines of the flux of the property. The geometric setting assumes that spacetime has only the structure of a differentiable manifold and no particular frame is assumed to be given.     

Semi-Iterative Minimum Cross-Entropy Algorithms for Rare-Events,Counting, Combinatorial and Integer Programming

We present a new generic minimum cross-entropy method, called the semi-iterative MinxEnt, or simply SME, for rare-event probability estimation, counting, and approximation of the optimal solutions of a broad class of NP-hard linear integer and combinatorial optimization problems (COP’s). The main idea of our approach is to associate with each original problem an auxiliary single-constrained convex MinxEnt program of a special type, which has a closed-form solution. We prove that the optimal pdf obtained from the solution of such a specially designed MinxEnt program is a zero variance pdf, provided the “temperature” parameter is set to minus infinity. In addition we prove that the parametric pdf based on the product of marginals obtained from the optimal zero variance pdf coincides with the parametric pdf of the standard cross-entropy (CE) method. Thus, originally designed at the end of 1990s as a heuristics for estimation of rare-events and COP’s, CE has strong connection with MinxEnt, and thus, strong mathematical foundation. The crucial difference between the proposed SME method and the standard CE counterparts lies in their simulation-based versions: in the latter we always require to generate (via Monte Carlo) a sequence of tuples including the temperature parameter and the parameter vector in the optimal marginal pdf’s, while in the former we can fix in advance the temperature parameter (to be set to a large negative number) and then generate (via Monte Carlo) a sequence of parameter vectors of the optimal marginal pdf’s alone. In addition, in contrast to CE, neither the elite sample no the rarity parameter is needed in SME. As result, the proposed SME algorithm becomes simpler, faster and at least as accurate as the standard CE. Motivated by the SME method we introduce a new updating rule for the parameter vector in the parametric pdf of the CE program. We show that the CE algorithm based on the new updating rule, called the combined CE (CCE), is at least as fast and accurate as its standard CE and SME counterparts. We also found numerically that the variance minimization (VM)-based algorithms are the most robust ones. We, finally, demonstrate numerically that the proposed algorithms, and in particular the CCE one, allows accurate estimation of counting quantities up to the order of hundred of decision variables and hundreds of constraints. This research was supported by the Israel Science Foundation (grant No 191-565).     

The Cross-Entropy Method for Continuous Multi-Extremal Optimization

In recent years, the cross-entropy method has been successfully applied to a wide range of discrete optimization tasks. In this paper we consider the cross-entropy method in the context of continuous optimization. We demonstrate the effectiveness of the cross-entropy method for solving difficult continuous multi-extremal optimization problems, including those with non-linear constraints.      

On input-output stability of time-varying nonlinear systems

In this work the question of bounded input-output stability of systems is investigated. The main tool of analysis is the generalized impulse response representation. Necessary and sufficient conditions for the stability of the operator are given. The analysis is carried out first for the linear operator and then the nonlinear operator is pursued. As is expected, difficulties are encountered for the nonlinear operator which do not occur for the linear case.     

The reference wave solution for a two-frequency wave propagating in a random medium

In this work a new reference wave method for solving parabolic-type equations is proposed. The performance of the method is demonstrated by applying it to the equation governing the propagation of the two-frequency mutual coherence function in a random medium. An analytic solution is presented for arbitrary correlation properties of the medium. It is shown that when approximating the transverse structure function of the medium by a quadratic form, the solution reduces to the exact result derived previously. Extensions to more general types of media are considered.     

Are quantum processes cosmologically induced?

Why, among ～10 ¹⁶ possibilities, does a wave function of a fast ion, spread over several centimeters, collapse in a crystal within a region measuring a few angstroms across? The most direct answer to this question would be the existence in this region of a cosmic particle, called a dybbuk, which induces the collapse. If dybbuks exist, deviations from conventional theory might be expected to occur for experiments involving ultrasmall space-time intervals less than the characteristic space-time interval of a dybbuk. It is shown that a dybbuk theory having the above characteristics can be constructed and can be tested experimentally by the decay of a0 ^? (J ^P) meson into two vector mesons.