Heun equations coming from geometry

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pull-backs of hypergeometric functions by rational Belyi functions with restricted ramification data give rise to Heun equations.     

Automorphisms of spine spaces

Automorphisms of non-trivial spine spaces are either, type-preserving, or type-exchanging, that is, they map stars onto stars or exchange them with tops. In the first case they are given by semi-linear bijections, in the latter, by sesqui-linear forms.     

Automorphisms in generalized spaces

A survey of results on the theory of automorphisms in classical spaces with connections, Finsler spaces, and spaces of support elements are presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 10, pp. 147–191, 1978.     

On the concept of generalized solution of operator equations in banach spaces

We introduce a new concept of generalized solution of operator equations with closed linear operator in a Banach space as an element of the completion of the space in certain locally convex topology. We prove a theorem on the existence and uniqueness of a generalized solution and give examples of finding the generalized solution for infinite systems of the linear algebraic equations. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9. pp. 1286–1290, September, 1996.     

Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations'' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.     

Kolmogorov diameters in solution spaces of systems of partial differential equations

Let E_p(0) denote the solutions (on 0<R ^N) of a system P(D) of partial differential equations with constant coefficients in a localizable analytically uniform space E (defined on 0). The relative Kolmogorov diameters of the neighbourhoods of 0 in E_p(0) are estimated from above and below, using the fundamental principle of Ehrenpreis. The diametral dimension, of E_p(0) is calculated and it is proved, that E_p(R ^N) and E_p(0) are nonisomorphic for (partially) bounded 0, in special cases.Dedicated to Heinz-Günter Tillmann on the occasion of his 60th birthday     

Asymptotically optimal approximation in fractional Sobolev spaces and the numerical solution of differential equations

We study two extremal problems in fractional Sobolev spaces which were motivated by the consideration of optimal finite difference methods for approximating the solution of a well posed Cauchy problem for linear partial differential equations. The limiting forms as the mesh increment tends to zero of the solutions of the extremal problem are shown to be generalized spline functions. Computations illustrating the effectiveness of these generalized spline functions are included.This research was partially supported by ONR Contract N 0014-69-C-0023.     

Existence of the periodic solution to the nematic liquid crystal equations in weighted Sobolev spaces

This paper is concerned with the time-periodic solution to the simplified incompressible nematic liquid crystal equation. We prove the existence of the time-periodic solution of this equation with small external forces g₁ and g₂, satisfying the T-periodic conditions g_j(t)=g_j(t+T) for j=1,2 in weighted Sobolev spaces.     

Group algebras acting on the space of solutions of a special double confluent Heun equation

Theoretical and Mathematical Physics - We study properties of the space $$boldsymbol{Omega}$$ of solutions of a special double confluent Heun equation closely related to the model of a overdamped...     

Approximate solution of singular integro- differential equations in generalized Hölder spaces

We have elaborated the numerical schemes of collocation methods and mechanical quadrature methods for approximate solution of singular integro- differential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fejér points. Theoretical background of collocation methods and mechanical quadrature methods has been obtained in Generalized Hölder spaces.     

Regularity criterion of weak solution for the 3D Magneto-micropolar fluid equations in Besov spaces

In this paper, we consider the 3D Magneto-micropolar fluid equations in Besov spaces. We improve and extend some known regularity criterion of weak solution for the 3D Magneto-micropolar fluid equations by means of the Fourier localization technique and Bony’s para-product decomposition.