Upper bounds for growth in the Ryll-Nardzewski function of anω-categorical,ω-stable theory

We establish a sharp upper bound for growth in the sequences _k (T):=the number ofk-types consistent withT, forT ω-categorical andω-stable. This paper forms part of the author’s doctoral dissertation written at the University of Maryland under the direction of David W. Kueker.     

Constructive models of N1-categorical theories

An example of an N₁-categorical, but not N₀-categorical, theory, for which only a prime model is constructible, is constructed.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 885–888, June, 1978.     

Countable models of ℵ1-categorical theories

Countable models of ℵ₁-categorical theories are classified. It is shown that such a theory has only a countable number of nonisomorphic countable models. The author was partially supported by NSF grants GP-1621 and GP-4257 during the period those results were obtained.     

An auxiliary result in the theory of transcendental numbers

This is an improvement on some estimates of exponential polynomials proved by Gelfond, Mahler, and Baker. This type of estimate is useful in the theory of transcendental numbers.     

Basic formulas and languages Part I. The theory

A general mathematical framework to deal with (the decidability status of) properties of derivations in E0L systems (forms) is developed. It is based on the theory of well-quasi-orders. This paper (the first of two parts) deals with the mathematical theory of the proposed approach.     

A. O. Gel'fond's method in the theory of transcendental numbers

Obtained in this work, with the help of a method of A. O. Gel'fond, are several results on algebraic independence of the values of an exponential function at nonalgebraic points.Translated from Matematicheskie Zametki, Vol. 10, No. 4, pp. 415–426, October, 1971.     

The rank of the covariance matrix of an evanescent field

Evanescent random fields arise as a component of the 2D Wold decomposition of homogeneous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space-time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance matrix of a finite dimension sample from an evanescent random field. It is shown that the rank of this covariance matrix is completely determined by the evanescent field spectral support parameters, alone. Thus, the problem of estimating the rank lends itself to a solution that avoids the need to estimate the rank from the sample covariance matrix. We show that this result can be immediately applied to considerably simplify the estimation of the rank of the interference covariance matrix in the STAP problem.     

Model-completeness of a theory and evaluation of formulas

Translated from Algebra i Logika, Vol 29, No. 1, pp. 15–28, January–February, 1990.     

On symmetries in the theory of finite rank singular perturbations

For a nonnegative self-adjoint operator A₀ acting on a Hilbert space H singular perturbations of the form A₀+V, are studied under some additional requirements of symmetry imposed on the initial operator A₀ and the singular elements ψj. A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A₀+V. The results are applied for the investigation of singular perturbations of the Schrödinger operator in L₂(R³) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions.