On the model companion fore-foldp-adically valued fields

We describe a model companion for the first-order theory ofe-foldp-adically valued fields. We obtain several characterizations of this model companion by means of geometro-algebraic, Galois-theoretic and measure-theoretic properties. This note corresponds to a chapter in the author's Ph.D. thesis, carried out in Tel-Aviv University under the supervision of Prof. Moshe Jarden. The work was partially supported by a grant from the G.I.F., the German-Israeli foundation for scientific research and development.     

On model complete differential fields

We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE . We show that this sub-family is usually definable (in particular if lives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.

     

The uniform companion for large differential fields of characteristic 0

We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic , which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields. As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudo-finite fields.

     

On non-linear partial differential equations with an infinite-dimensional conditional symmetry

The invariance of non-linear partial differential equations under a certain infinite-dimensional Lie algebra AN(z) in N spatial dimensions is studied. The special case A₁(2) was introduced in [J. Stat. Phys. 75 (1994) 1023] and contains the Schrödinger Lie algebra sch₁ as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of AN(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of AN(z) such that the well-known Monge-Ampère equation is the required additional condition. New exact solutions are found for some representatives of this class.     

On the dimensions of automorphism groups of 8-dimensional ternary fields I

Let be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed subgroup of its automorphism group which is taken with the compact-open topology. It is proved that is either isomorphic to the compact exceptional Lie group G₂, or the (covering) dimension of is at most 11. This bound can be decreased to 10, if the ternary fixed fieldF of is connected.     

On the dimensions of automorphism groups of eight-dimensional ternary fields,II

LetT be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed fieldF of is connected, then is either isomorphic to one of the compact Lie groupsG ₂ or SU₃, or the (covering) dimension of is at most 7.     

On the classification of the extremal self-dual codes over small fields with 2-transitive automorphism groups

There are seven binary extremal self-dual doubly-even codes which are known to have a 2-transitive automorphism group. Using representation theoretical methods we show that there are no other such codes, except possibly for length n = 1024. We also classify all extremal ternary self-dual and quaternary Hermitian self-dual codes.     

On an explicit representation of the solution of linear stochastic partial differential equations with delays

Based on the analysis of a certain class of linear operators on a Banach space, we provide a closed form expression for the solutions of certain linear partial differential equations with non-autonomous input, time delays and stochastic terms, which takes the form of an infinite series expansion.     

On stochastic partial differential equations with Unbounded coefficients

Second-order stochastic partial differential equations of parabolic type are considered. Generalizations of known theorems on existence, uniqueness and on approximations are presented. Thus, in particular the case of unbounded coefficients in investigated. Some examples illustrating the usefulness of the results are also given.     

On some partial differential inequalities with gradient terms

Using a modification of the nonlinear capacity method, we obtain necessary conditions for the solvability of some nonlinear partial differential equations and inequalities containing the polyharmonic operator and terms that depend on the norm of the gradient of the solution, both in the entire space and in bounded domains; in the latter case the coefficients of the inequality are allowed to have singularities.     

On the convergence of partial differential equations of parabolic type with rapidly oscillating coefficients to stochastic partial differential equations

In this paper we consider the convergence of the solutions to a sequence of partial differential equations of parabolic type with rapidly oscillating coefficients to the solutions of a stochastic partial differential equation. We use the martingale method and the characteristic functional to prove that the martingale problem has a unique solution. Our emphasis is in treating strongly mixing noises.This research was partially supported by Stifting Volkswagenwerk through Forschungszentrum BiBoS Universität Bielefeld and Grant-in-Aid for General Scientific Research (C) 61540162, the Ministry of Education, Science and Culture (Japan). Presented at the International Workshop on Diffusion Approximations and Related Topics, IISA, Laxenburg, Austria, 29 June to 3 July 1987.     

On partial stability of differential equations with time delay

Summary We investigate stability, uniform stability and equi-asymptotic stability with respect to the x-components and y-components of a differential equation with time delay. We also obtain necessary and sufficient conditions for the generalized asymptotic stability of the exponential type with respect to the components which generalizes the work of Corduneanu[3]. We make use of Lyapunov functionals and differential inequalities in our study. Entrata in Redazione il 7 luglio 1978.     

On the centralizer and the commutator subgroup of an automorphism

Let φ be an automorphism of a group G. In this paper, we study the influence of its centralizer \({C_G(\varphi)}\) on its commutator subgroup \({[G,\varphi]}\) when G is polycyclic or metabelian. For instance, when G is metabelian and φ fixed-point-free of prime order p, we prove that \({[G,\varphi]}\) is nilpotent of class ≤ p. Also, when G is polycyclic and φ of order 2, we show that if \({C_G(\varphi)}\) is finite, then so are \({G/[G,\varphi]}\) and \({[G,\varphi]'}\) .     

The model theory of differential fields revisited

The intent of this article is to provide a general and elementary account of the model theory of differential fields, collecting together various results (many without proof) and offering a few algebraic details for the logician reader. The first model-theoretic look at differential fields was taken by Abraham Robinson in the context of model completeness, while later developments have served to illustrate concepts developed by Morley and Shelah. Preparation of this paper supported in part by N.S.F. Grant MPS 75-08241.     

On the solution of partial differential equations with specified boundary conditions

The decomposition method is applied to solution of partial differential equations in two and three dimensions with specified boundary conditions.     

On a classification of the automorphism groups of polarized abelian surfaces over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of polarized abelian surfaces over finite fields.