On characterizations of recursively enumerable languages

Summary Geffert has shown that earch recursively enumerable languageL over can be expressed in the formL{h(x) ^–1 g(x)x in ⁺} ^* where is an alphabet andg, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations ofL. For instance,L =(L ₀) ^* whereL ₀ is a minimal linear language and is the Dyck reductiona, A.     

ε-Entropy and Fractal Dimension of a State for a Gaussian Measure

Kolmogorov introduced the concept of -entropy to analyze information in classical continuous systems. The fractal dimension of a geometric set was introduced by Mandelbrot as a new criterion to analyze the geometric complexity of the set. The -entropy and the fractal dimension of a state in a general quantum system were introduced by one of the present authors (MO) in order to characterize chaotic properties of general states.In this paper, we show that -entropy of a state includes Kolmogorov's -entropy, and that the fractal dimension of a state describes fractal structure of Gaussian measures.     

A Delay-Averaged Logistic Model

For the equation x(t) = x(t) (1-(1/) _t-- ^t- x(u)du), > 0, > 0, > 0, conditions for the stability of a nonzero stationary solution under small perturbations are determined.     

Comparative complexity of the representation of languages by probabilistic automata

We analyze the effect of the degree of isolation of a cut point on the number of states P(U, ) of a probabilistic automaton representing the language U. We give an example of a language Uⁿ consisting of words of length n such that there exist numbers < for which P(Uⁿ, )/P(Uⁿ, )0 as n.Translated from Kibernetika, No. 3, pp. 21–25, May–June, 1989.     

On uniform convergence rates for Eulerian and Lagrangian finite element approximations of convection-dominated diffusion problems

Abstract Standard error estimates in the literature for finite element approximations of nonstationary convection-diffusion problems depend either reciprocally on the diffusion parameter or on higher order norms of the solution. Therefore, the estimates generally become worthless in the convection-dominated case with 0 < 1. In this work we develop a rigorous -uniform convergence theory for finite element discretizations of convection-dominated diffusion problems in Eulerian and Lagrangian coordinates. Here, the constants that arise in the error estimates depend on norms of the data and not of the solution and remain bounded in the hyperbolic limit 0. In particular in the Lagrangian case this requires modifications to standard finite element error analyses. In the Eulerian formulation -uniform convergence of order one half is proven whereas in the Lagrangian framework -uniform convergence of optimal order is established. The estimates are based on -uniform a priori estimates for the solution of the continuous problems which are derived first.     

Convergence of MUSCL Relaxing Schemes to the Relaxed Schemes for Conservation Laws with Stiff Source Terms

We consider the convergence and stability property of MUSCL relaxing schemes applied to conservation laws with stiff source terms. The maximum principle for the numerical schemes will be established. It will be also shown that the MUSCL relaxing schemes are uniformly l ¹- and TV-stable in the sense that they are bounded by a constant independent of the relaxation parameter , the Lipschitz constant of the stiff source term and the time step t. The Lipschitz constant of the l ¹ continuity in time for the MUSCL relaxing schemes is shown to be independent of and t. The convergence of the relaxing schemes to the corresponding MUSCL relaxed schemes is then established.     

Decision procedures for elementary sublanguages of set theory: XIII. Model graphs,reflection and decidability

Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on , , =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of , , =, {...}, , where {...} is the tuple operator and is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of , , =, {...}, , obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving is allowed.     

Finding a closest visible vertex pair between two polygons

Given nonintersecting simple polygonsP andQ, two verticespP andq Q are said to be visible if does not properly intersectP orQ. We present a parallel algorithm for finding a closest pair among all visible pairs (p,q),pP andqQ. The algorithm runs in time O(logn) using O(n) processors on a CREW PRAM, wheren=¦P¦+¦Q¦. This algorithm can be implemented serially in (n) time, which gives a new optimal sequential solution for this problem.This paper appeared in preliminary form as [1]. This work was supported in part by an AT&T BellLaboratories Graduate Fellowship, the Joint Services Electronics Program (U.S. Army, U.S. Navy, U.S. Air Force) under Contract N00014-90-J-1270, and NSF Grant CCR-89-22008. This work was done while the author was with the Department of Computer Science at the University of Illinois at Urbana-Champaign.     

An Estimation Problem for Quasilinear Stochastic Partial Differential Equations

A problem of estimating a functional parameter (x) and functionals () based on observation of a solution u (t, x) of the stochastic partial differential equation is considered. The asymptotic problem setting, as the noise intensity 0, is investigated.     

On the Epsilon-entropy of One Class of Ellipsoids in a Hamming Space

The asymptotic behavior of the -entropy of ellipsoids in an n-dimensional Hamming space whose coefficients take only two different values is investigated as n . Explicit expressions for the main terms of the asymptotic expansion of -entropy of such ellipsoids are obtained under various relations between and parameters that define these ellipsoids.