Remarks on extremal solutions of differential equations with advanced argument

We show the existence of absolutely continuous extremal solutions to the problemx′(t)=f(t, x)h(t)))+g(t)),x(0)=x ₀, whereh is an arbitrary continuous deviated argument. Conditions for the uniqueness of solutions are given. Research partialy supported by grant UG BW 5100 - 5 - 0143 - 4     

On positivity and stability of linear time-varying Volterra equations

Linear time-varying Volterra integro-differential equations of non-convolution type are considered. Positivity is characterized and a sufficient condition for exponential asymptotic stability is presented.

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The second author thanks the Alexander von Humboldt Foundation for their support.     

Unbounded solutions of differential equations of second order

In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator

On certain non-linear differential equations in complex domains

We describe meromorphic solutions of certain non-linear (ordinary and partial) differential equations in complex domains. Received: 13 November 2007     

On two-point boundary value problems for systems of higher-order ordinary differential equations with singularities

The sufficient conditions of solvability and unique solvability of the two-point boundary value problems of Vallèe-Poussin and Cauchy-Niccoletti have been found for a system of ordinary differential equations of the form

Kneser-type oscillation criteria for self-adjoint two-term differential equations

It is proved that the even-order equationy ⁽²ⁿ⁾ +p(t)y=0 is (n,n) oscillatory at if

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder     

Existence theorems for nonlinear functional differential equations of neutral type

Conditions are found upon satisfaction of which the differential equation

On general algebraic mechanisms for producing centers in polynomial differential systems

We classify nondegenerate centers of systems of the form

On the algebraic points of a definable set

This paper studies diophantine properties of sets definable in an o-minimal structure over the real field. The main theorem of the author’s recent paper with A. J. Wilkie is refined, and used to deduce a strong result on the density of algebraic points of such sets.      

Skorohod stochastic differential equations of diffusion type

Leta, b beC ²(R ¹)-functions with bounded derivatives of first and second order. We study stochastic differential equations

On the existence of periodic solutions of Rayleigh equations

In this paper, we study the existence of periodic solutions of Rayleigh equation

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem

Decay properties for the solutions of a partial differential equation with memory

Decay properties in energy norm for solutions of a class of partial differential equations with memory are studied by means of frequency domain methods. Our results are optimal for this class, as we are able to characterize polynomial as well as exponential decay rates. The results apply to models for viscoelastic materials. An extension to a semilinearly perturbed problem is also included. Received: 9 July 2008, Revised: 16 September 2008     

Oscillatory solutions of nonhomogeneous linear differential equations

The complex oscillation of nonhomogeneous linear differential equations with transcendental coefficients is discussed. Results concerning the equation f ^(k)+a _k−1 f ^(k−1)+...+a ₀ f=F where a ₀,...,a _k−i and Fare entire functions, possessing an oscillatory solution subspace in which all solutions (with at most one exception) have infinite exponent of convergence of zeros are obtained. All solutions of the equation are also characterized when the coefficients a ₀,a ₁,...,a _k−1 are polynomials and F=h exp (p ₀), where p ₀ is a polynomial and h is an entire function. Author supported by Max-Planck-Gesellschaft and by NSFC.     

Asymptotically almost periodic solutions of abstract retarded functional differential equations of second order

We establish existence of asymptotically almost periodic mild solutions for a class of semi-linear second-order abstract retarded functional differential equations with infinite delay. Research supported in part by FONDECYT, grant 1050314.     

Discrete monodromy, pentagrams, and the method of condensation

This paper studies the pentagram map, a projectively natural iteration on the space of polygons. Inspired by a method from the theory of ordinary differential equations, the paper constructs roughly n algebraically independent invariants for the map, when it is defined on the space of n-gons. These invariants strongly suggest that the pentagram map is a discrete completely integrable system. The paper also relates the pentagram map to Dodgson’s method of condensation for computing determinants, also known as the octahedral recurrence. I dedicate this paper to Professor V. I. Arnold on the occasion of his 70th birthday     

Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

Amalgamation,interpolation and epimorphisms in algebraic logic

In his landmark paper on amalgamation published in Algebra Universalis in 1971, Don Pigozzi posed some open questions in connection with amalgamation of subclasses of cylindric algebras. Some of these questions were originally raised by Comer, Daigneault, Johnson, McKenzie and others. In this paper we give answers to all these as well as a number of other related questions. Most of the solutions were found by the authors of this paper. However, a few were contributed by others who will of course be given due credit at the appropriate points. This paper is dedicated to Don Pigozzi on his retirement. Presented by R. W. Quackenbush. The first author’s research was supported by the Hungarian National Foundation for Scientific Research grants no T30314 and T23234. The second author’s research was supported by the Hungarian National Foundation for Scientific Reserach OKTA grant no T30314. All the new results in this article were announced by Judit Madarász in the Workshop on Abstract Algebraic Logic, held in Centre de Recerca Matematica Bellaterra, Spain July 1–5, 1997. Received December 15, 2002; accepted in final form January 3, 2006.     

On left conjugacy closed loops with a nucleus of index two

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and middle nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity, but that is associated with the level of commutativity of G (amongst others, the centre of G has to be nontrivial). On the other hand, for each m ≥ 2 one can construct an LCC loop Q of order 2m in such a way that its left nucleus is trivial, and the right nucleus if of order m. If Q is involutorial, then it is a Bol loop. Work supported by institutional grant MSM 113200007 and by Grant Agency of Czech Republic, Grant 201/02/0594. The paper was written while the author was visiting Universitaet Hamburg in January 2004.