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Distributive pairs of elements in a structure

Unknown
1965 SovietRxiv: ru-196501.84065

The paper studies pairs of elements in a lattice satisfying additive or multiplicative distributive identities, emphasizing when one property implies the other and when strict examples can occur. It gives criteria for an additively distributive pair to be multiplicatively distributive within a princ…

Asymptotics of the solution of a boundary value problem for a second-order elliptic equation in domains in which one dimension is sufficiently small compared with the others

In the present paper we investigate the asymptotics of the solution of the following boundary-value problem.
1965 SovietRxiv: ru-196501.83757

This paper studies the asymptotic behavior of solutions to a boundary value problem for a second order uniformly elliptic equation in a cylindrical domain whose height is small compared with its other dimensions. After rescaling the thin coordinate, the solution is constructed as a sum of an interio…

NORMALLY SOLVABLE ELLIPTIC BOUNDARY-VALUE PROBLEMS

A. V. Bitsadze
1965 SovietRxiv: ru-196501.83446

This paper discusses normal solvability, Noetherianity, Fredholmness, and index for elliptic boundary value problems, emphasizing the need for criteria that determine when orthogonality conditions to adjoint homogeneous solutions are also sufficient for solvability. It analyzes the Dirichlet problem…

ON ESTIMATING THE NUMBER OF SOLUTIONS OF CERTAIN EQUATIONS

A. A. Karatsuba
1965 SovietRxiv: ru-196501.82990

The paper studies upper bounds for the number of integer solutions to equations of the form \(x_1^n+\cdots+x_k^n=y_1^n+\cdots+y_k^n\) with variables bounded by \(P\), a problem connected with estimates used in Waring’s problem. Using a main theorem on the number of solutions of related systems of co…

On Ill-Posed Problems in Linear Algebra and a Stable Method for Solving Them

1.** Consider the system of linear algebraic equations
1965 SovietRxiv: ru-196501.82900

The paper addresses the instability of determining normal solutions of linear algebraic systems, especially singular or nearly singular systems, where small perturbations in the matrix or right hand side can produce widely scattered solutions. It defines the normal solution as the minimum norm solut…

On the Application of the Parameter Variation Method to the Construction of Iterative Formulas of Increased Accuracy for Determining Numerical Solutions of Nonlinear Integral Equations

Unknown
1965 SovietRxiv: ru-196501.82768

The paper applies the parameter variation method to construct higher accuracy iterative formulas for numerical solutions of nonlinear integral equations. After replacing the integral equation by a finite system using an arbitrary quadrature formula, the residual is treated as a vector parameter, and…

Construction of the Complete Set of Solutions to the Homogeneous Dirichlet Problem for the Vibrating String Equation

R. A. Aleksandryan
1965 SovietRxiv: ru-196501.82031

This note studies generalized eigenfunctions of the homogeneous Dirichlet problem for the string vibration equation in a domain with piecewise smooth boundary, viewed through the self-adjoint operator associated with the problem. Using boundary automorphisms generated by characteristic directions an…

Some Boundary Value Problems for Systems of Second-Order Elliptic-Type Equations Not Satisfying the Ya. B. Lopatinskii Condition

1.** In the present paper the following problem is considered:
1965 SovietRxiv: ru-196501.81944

This paper studies boundary value problems for two-component second order elliptic systems with constant real matrix coefficients in a bounded simply connected plane domain, under a first order linear boundary condition that may fail the Lopatinskii condition. Using representations of solutions by a…

On Reflexive Spaces Generalizing Reflexive Orlicz Spaces

Unknown
1965 SovietRxiv: ru-196501.81738

This paper studies reflexivity in Banach spaces constructed from partially ordered KB-spaces by means of N-functions, extending classical results on reflexive Orlicz spaces. It first establishes characterizations of N-functions whose complementary functions also satisfy the Delta 2 condition, showin…

THE STRUCTURE OF PERIODIC LINEAR GROUPS AND ALGEBRAIC GROUPS

The following principal results are obtained in the paper:
1965 SovietRxiv: ru-196501.81192

This paper studies periodic linear groups over arbitrary fields and their connections with algebraic and Lie groups. It proves conjugacy of Sylow subgroups and a Schur Zassenhaus type theorem for periodic linear groups, gives examples of infinite simple locally finite linear groups, and completes th…

On Commutative Algebras of Operators in the Space $\Pi_k$

Unknown
1965 SovietRxiv: ru-196501.81174

This paper studies commutative symmetric algebras of bounded operators in a Pontryagin space of finite negative index and seeks their classification up to equivalence. It analyzes eigenfunctionals and root lineals, separates the contribution of nonreal eigenfunctionals, and constructs invariant prin…

Undecidability in the constructive predicate calculus of the class of formulas of type $\daleth\daleth\forall\exists$

Unknown
1965 SovietRxiv: ru-196501.80677

The paper proves undecidability results for restricted classes of formulas in constructive and minimal predicate calculi, focusing on formulas of the form double negation followed by universal and existential quantifier blocks with a quantifier-free matrix. It reduces the compatibility problem for s…

ON THE PROBLEM OF \(M\)-CONVEXITY

Unknown
1965 SovietRxiv: ru-196501.79913

The paper reformulates the problem of M-convexity for systems of linear partial differential equations with constant coefficients in terms of finite modules over the polynomial ring in differential operators. It relates the cohomology of sheaves of local homogeneous solutions to Ext groups via a Dol…

Some Questions in Approximation Theory

V. M. Tikhomirov
1965 SovietRxiv: ru-196501.79308

The note surveys several problems in approximation theory in Banach spaces: best approximation of individual elements by finite-dimensional subspaces, approximation of convex sets, Kolmogorov widths, and related codimensional widths. It gives general criteria for best approximation in spaces of cont…

On the Approximation of Periodic Functions in the Mean

Unknown
1965 SovietRxiv: ru-196501.78412

This paper studies analogues in \(L_p\) metrics of Favard-type inequalities relating the approximation of a periodic function by Fourier partial sums or trigonometric polynomials to the corresponding approximation of its derivative. It proves an \(L_p\) estimate for the deviation of a function from …

ON THE THEORY OF OPERATORS OF THE FORM $\dfrac{d}{dt}-A$

A. A. Dezin
1965 SovietRxiv: ru-196501.78357

This paper studies boundary value problems for operators of the form du/dt minus A in a Hilbert space, where A is an unbounded normal operator, with emphasis on constant coefficient differential operators on the torus. It analyzes scalar boundary conditions relating the values at t = 0 and t = a, es…

On the Summability of Fourier Series in Eigenfunctions of the Laplace Operator by Cesaro, Riesz, and Poisson–Abel Means

Unknown
1965 SovietRxiv: ru-196501.78327

This paper studies almost everywhere summability of Fourier series in eigenfunctions of the Laplace operator on an arbitrary multidimensional domain under boundary conditions yielding a complete orthonormal eigenfunction system. It proves that for every function in \(L_2(g)\), the eigenfunction expa…

On the Behavior of Solutions to the Equations of Hydrodynamics as the Reynolds Number Passes Through a Critical Value

Unknown
1965 SovietRxiv: ru-196501.78021

This paper studies the behavior of solutions to the incompressible Navier-Stokes equations in a bounded domain as viscosity, equivalently the Reynolds number, passes through a critical value. The problem is reduced by expansion in eigenfunctions of the Stokes operator to an infinite system of ordina…

Properties of Equimorphisms at Infinity

Unknown
1965 SovietRxiv: ru-196501.77910

This note studies how equi-morphisms of hyperbolic spaces act on the sphere at infinity, especially with respect to tangency in the spherical Poincaré model. It proves that tangency of a curve in hyperbolic space to the sphere at infinity is invariant under equi-morphisms, using a characterization b…

Local Structure of Darboux Surfaces

I. Kh. Sabitov
1965 SovietRxiv: ru-196501.77021

This note studies the local extrinsic geometry of the Darboux surfaces associated with an infinitesimal bending of a positively curved surface of class \(C^{n,\alpha}\). Using local coordinates, properties of the bending and rotation fields, and Stoilow’s theory of interior mappings, it analyzes the…

Decomposition of a Representation of a Reductive Lie Algebra into Representations of Its Regular Reductive Subalgebras of Maximal Rank

V. B. Mandelzweig
1965 SovietRxiv: ru-196501.76997

This paper studies the branching of an irreducible finite-dimensional representation of a complex reductive Lie algebra when restricted to a regular reductive subalgebra of the same rank. Using character identities, Kostant’s weight multiplicity formula, and finite difference operators applied to pa…

On the Convergence of Positive Functionals and Operators

M. A. KRASNOSEL′SKII, V. S. KLIMOV, E. A. LIFSHITS
1965 SovietRxiv: ru-196501.76190

This paper studies convergence criteria for sequences of positive linear functionals and operators in Banach spaces ordered by a cone, extending Korovkin-type results beyond spaces of continuous functions. It introduces smoothness points of a cone and saturated, uniformly saturated, and completely s…

On Deformations of Fibered Spaces

Unknown
1965 SovietRxiv: ru-196501.75720

This paper constructs a complete holomorphic family of deformations of a holomorphic principal fiber bundle with compact base and complex structure group. Using differential forms with values in the adjoint bundle, the deformation problem is expressed through the integrability equation for a form of…