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ON ELEMENTARY STRUCTURAL PROPERTIES OF SEMIGROUPS

L. N. Shevrin
1966 SovietRxiv: ru-196601.62210

The paper introduces elementary refinements of structural characterizability for semigroups in terms of first-order axiomatizability and elementary definability of their subsemigroup lattices. It proves general non-closure results for classes of such structures, including cases involving infinite cy…

ON FUNDAMENTAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH A BESSEL DIFFERENTIAL OPERATOR

I. A. Kipriyanov, V. I. Kononenko
1966 SovietRxiv: ru-196601.62049

This note constructs fundamental solutions for certain constant coefficient partial differential operators involving the singular Bessel operator in one variable. It introduces a basic test function space, generalized functions, convolution, translation, and the mixed Fourier-Bessel transform, then …

On the Problem of Identity in Finitely Presented Semigroups

G. S. Makanin
1966 SovietRxiv: ru-196601.61813

This note addresses decision problems for finitely presented semigroups, with emphasis on constructing presentations with very few defining relations whose identity problem is undecidable. It proves that a semigroup on four generators with six defining relations has an unsolvable identity problem fo…

ON HIGHER APPROXIMATIONS OF THE AVERAGING METHOD OF N. N. BOGOLYUBOV—N. M. KRYLOV

P. P. Zabreiko, I. B. Ledovskaya
1966 SovietRxiv: ru-196601.61735

This paper studies higher order approximations in the Bogolyubov, Krylov averaging method for ordinary differential equations with a small parameter, with the aim of weakening unnecessary smoothness and regularity assumptions used in earlier justifications. It introduces Hölder-type conditions for t…

ON THE STABILITY OF CLASSES OF WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR AN EQUATION WELL-POSED IN THE SENSE OF I. G. PETROVSKII

V. M. Borok, Ya. I. Zhitomirskii
1966 SovietRxiv: ru-196601.61600

This note studies well-posed solvability of the Cauchy problem for a linear evolution equation with variable coefficients on the real line, assuming that the highest coefficient is constant and the remaining coefficients converge at infinity to those of a constant-coefficient equation well-posed in …

CRYSTALLIZATION OF A MULTICOMPONENT CHAIN

A. A. Chernov
1966 SovietRxiv: ru-196601.61361

This paper develops a probabilistic model for crystallization of a semi-infinite multicomponent chain whose particles attach to and detach from the chain end with transition frequencies depending only on the terminal pair. It derives equations for the survival probabilities of chain configurations, …

On Effectively \(\Phi\)-Inseparable Sets

Unknown
1966 SovietRxiv: ru-196601.60790

The paper introduces effective Phi-inseparability for two disjoint subsets of a metric space, extending earlier notions of effective non-enumerability and effective distinguishability. It defines inseparability through a continuous selection function on a compact metric base and studies how the prop…

ON THE DENSITY OF FINITE FUNCTIONS IN \(l_p^{(l)}\)

S. L. Sobolev
1966 SovietRxiv: ru-196601.60619

This note gives a simplified proof, for the case \(lp>n\), of a theorem concerning the density of finitely supported functions in discrete Sobolev type spaces \(l_p^{(l)}\). Sobolev proves that if a function belongs to \(L_p^{(l)}(E_n)\), then its values on any regular lattice form an element of \(l…

ON SOME CONDITIONAL EXTREMAL PROBLEMS IN THE CLASS OF ENTIRE FUNCTIONS OF FINITE DEGREE

I. I. Ibragimov, B. A. Rymarenko
1966 SovietRxiv: ru-196601.60541

The paper studies conditional extremal problems for real entire functions of finite exponential degree that are nonnegative on the real axis and belong to an Lp class, subject to admissible linear constraints. Using a Nikolskii type inequality and comparison with extremal problems in L1 and uniform …

DISTRIBUTION OF EIGENVALUES FOR SINGULAR DIFFERENTIAL OPERATORS

A. G. Kostyuchenko
1966 SovietRxiv: ru-196601.60455

This paper studies asymptotic formulas for the eigenvalue counting function of singular strongly elliptic differential operators with matrix-valued coefficients on unbounded domains. The approach relates the counting function to the trace of the Green matrix for an associated parabolic Cauchy proble…

ON THE DECISION PROBLEM

Unknown
1966 SovietRxiv: ru-196601.60321

The paper proves an undecidability result for a restricted fragment of the pure narrow predicate calculus with prenex formulas of the form involving a universal, existential, and universal quantifier prefix followed by an existential block. The argument reduces recursively inseparable halting and cy…

ON THE QUESTION OF DESCRIPTION OF THE DUALITY FUNCTOR

L. M. Kisina, A. S. Schwarz
1966 SovietRxiv: ru-196601.59589

This paper studies the description of dual functors in categories of topological spaces with a distinguished point and in more general D-categories. For continuous functors on suitable categories of completely regular spaces, it proves that the dual functor is determined by its restriction to a smal…

ONE-SIDED ESTIMATES UNDER CONDITIONS OF ASYMPTOTIC STABILITY OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH UNBOUNDED OPERATORS

Ya. D. Mamedov
1966 SovietRxiv: ru-196601.59409

This paper studies Lyapunov asymptotic stability of the zero solution for nonlinear differential equations in a Banach space with generally unbounded time-dependent linear operators. Using the Gateaux differentiability of the norm, it introduces one-sided estimates through the duality mapping and de…

ON THE MOMENTS OF THE NUMBER OF CLASSES OF PURELY RADICAL QUADRATIC FORMS OF NEGATIVE DETERMINANT

M. B. Barban, G. Gordover
1966 SovietRxiv: ru-196601.59205

This note studies asymptotic formulas for fixed moments of the number of classes of purely radical quadratic forms of negative determinant. Using Gauss’s formula, the problem is reduced to estimating moments of Dirichlet L-series at 1, and then to bounds in the critical strip via the A. I. Vinogrado…

DUALITY IN MEASURE THEORY IN LINEAR SPACES

A. M. Vershik
1966 SovietRxiv: ru-196601.59086

The paper develops a dual formulation of measure theory on linear spaces by associating a linear space with measure to the ring of measurable functions together with the closed generating subspace of measurable linear functionals. It shows that every such space is linearly isomorphic, modulo null se…

On Mutually Adjoint Boundary Value Problems for a System of Three First-Order Composite-Type Equations

A. Dzhuraev
1966 SovietRxiv: ru-196601.58737

This paper studies mutually adjoint boundary value problems for a first order system of three partial differential equations of composite type, reducible to a canonical form combining a real characteristic component with a Beltrami type equation. For a simply connected domain with prescribed boundar…

BOUNDARY-VALUE PROBLEMS FOR ABSTRACT ELLIPTIC EQUATIONS

V. A. Trenogin
1966 SovietRxiv: ru-196601.58528

This paper studies boundary value problems for second order abstract elliptic equations in a Banach space, with time dependent closed operators and general linear boundary conditions involving endpoint values and derivatives. Using analytic semigroups generated by square roots of the operators, the …

ON THE DECOMPOSITION OF THE DIRECT PRODUCT OF REPRESENTATIONS OF THE DISCRETE UNITARY SERIES \(D_\lambda^+\) OF THE LORENTZ GROUP \(\mathscr L_3\)

S. S. Sannikov
1966 SovietRxiv: ru-196601.58320

The paper studies the decomposition of tensor products of representations in the positive discrete unitary series of the Lorentz group L3, with parameters allowed to be arbitrary positive real numbers. Using a basis method, it first derives the irreducible decomposition of the product of two such re…

ON MULTIFREQUENCY OSCILLATIONS AND ROTATIONS

L. D. Akulenko
1966 SovietRxiv: ru-196601.57524

The paper studies systems of ordinary differential equations with a small parameter whose unperturbed part has an isolated rotational and oscillatory solution, while the perturbing terms are almost periodic in time and in angular variables. Under smoothness, Lipschitz, and nonresonance-type isolatio…

EMBEDDING THEOREMS FOR A CLASS OF FUNCTIONS WITH MIXED NORM

A. P. Uninskij
1966 SovietRxiv: ru-196601.57408

This note studies differentiable functions on Euclidean space whose derivatives have finite mixed norm and whose higher derivatives satisfy a multiple Hölder condition in the same norm. It gives a necessary and sufficient characterization of the class by representation as finite sums of series of en…

ON THE THEORY OF GEODESIC MAPPINGS OF RIEMANNIAN SPACES

N. S. Sinyukov
1966 SovietRxiv: ru-196601.56992

The paper studies the local analytic problem of determining all Riemannian spaces that admit a nontrivial geodesic mapping onto a given space, without restrictions on metric signature. It reformulates the existence of such mappings as a first order differential system for an auxiliary symmetric tens…

CARTAN PSEUDOGROUPS AND \(p\)-LIE ALGEBRAS

A. I. Kostrikin, I. R. Shafarevich
1966 SovietRxiv: ru-196601.56682

This note studies analogues in characteristic p of Cartan’s simple transitive pseudogroup Lie algebras, realized as derivation algebras of truncated polynomial rings. It describes the resulting Cartan type Lie p-algebras, general, special, Hamiltonian, and contact, and formulates the conjecture that…

SOLUTION OF THE HARDY–LITTLEWOOD PROBLEM AND OF ITS INDEFINITE ANALOGUE IN SECTORS AND CONTOURS

A. A. Polyanskii
1966 SovietRxiv: ru-196601.56230

This paper studies sector-restricted versions of the Hardy-Littlewood problem for representations of a large integer as a prime plus two squares, and of its indefinite analogue for primes of the form a fixed integer plus two squares. Combining Linnik’s dispersion method with unimodular transformatio…

THE STRUCTURE OF FUNDAMENTAL MATRICES OF \(R\)-SYSTEMS WITH ALMOST PERIODIC COEFFICIENTS

V. M. Millionshchikov
1966 SovietRxiv: ru-196601.56124

This paper studies linear differential systems whose coefficient matrices are uniformly almost periodic and belong to the class of R-systems, aiming to extend Floquet-type structural results beyond the periodic and quasiperiodic cases. It develops auxiliary lemmas on limiting increments of functions…

A THEOREM ON THE METRIZABILITY OF THE PREIMAGE OF A METRIC SPACE UNDER AN OPEN-CLOSED FINITE-TO-ONE MAPPING, AN EXAMPLE, AND UNSOLVED PROBLEMS

A. V. Arkhangel'skii
1966 SovietRxiv: ru-196601.55849

The paper studies preservation of metrizability under finite-to-one mappings in general topology, focusing on spaces that map openly and closedly onto metric spaces. It proves that the inverse image of a metric space under an open-closed finite-to-one continuous mapping is metrizable, by constructin…

ON THE THEORY OF PRIME NUMBERS

D. N. Lenskoi
1966 SovietRxiv: ru-196601.55724

The paper studies upper bounds for the number of integer points in multidimensional boxes at which several parameter-dependent integral polynomials simultaneously take prime absolute values. Under irreducibility, non-association, coefficient growth, and local solubility assumptions, it combines alge…