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APPLICATION OF THE CONTOUR INTEGRAL METHOD TO THE SOLUTION OF MULTIDIMENSIONAL MIXED PROBLEMS FOR PARABOLIC SYSTEMS OF SECOND ORDER

M. L. Rasulov
1970 SovietRxiv: ru-197001.38370

The paper applies the contour integral method to multidimensional mixed problems for second order parabolic systems in a three-dimensional domain, with boundary conditions that include a time derivative. Under assumptions on the spectrum of the coefficient matrix, regularity and boundedness of the c…

ON THE SOLVABILITY OF QUASILINEAR PARABOLIC SYSTEMS

S. I. Pokhozhaev
1970 SovietRxiv: ru-197001.38299

The paper studies boundary value problems for quasilinear parabolic systems in the Douglis, Nirenberg, and Solonnikov sense, with linear boundary conditions and zero initial data in anisotropic Sobolev spaces. Using an abstract solvability theorem for nonlinear equations in Banach spaces together wi…

SURFACES OF NEGATIVE EXTERNAL CURVATURE IN A RIEMANNIAN SPACE WITH NONPOSITIVE RIEMANNIAN CURVATURE

I. S. Brandt
1970 SovietRxiv: ru-197001.38153

The paper extends Efimov-type results on complete surfaces of negative curvature from Euclidean three-space to three-dimensional Riemannian spaces with nonpositive sectional curvature. For a complete surface with negative extrinsic curvature, it derives generalized fundamental equations in asymptoti…

FROM THE EXISTENCE OF A NONMEASURABLE SET OF TYPE \(A_2\) THERE FOLLOWS THE EXISTENCE OF AN UNCOUNTABLE SET CONTAINING NO PERFECT SUBSET OF TYPE \(CA\)

V. A. Lyubetskii
1970 SovietRxiv: ru-197001.38011

The paper addresses two classical descriptive set theory questions in their formal set theoretic formulation: the existence of an uncountable coanalytic set containing no perfect subset, and the existence of a non-Lebesgue-measurable set of projective type. It proves in Zermelo-Fraenkel set theory t…

ON FREDHOLM THEORY FOR NONLINEAR OPERATOR EQUATIONS

Unknown
1970 SovietRxiv: ru-197001.37583

This note develops partial Fredholm-type solvability results for nonlinear operator equations in Banach spaces with completely continuous operators that possess an asymptotic derivative. Using a finite-dimensional algebraic lemma and the Schauder-Tikhonov fixed point principle, it proves necessary a…

ISOMETRIC EMBEDDINGS AND IMMERSIONS

M. L. Gromov
1970 SovietRxiv: ru-197001.36592

This note formulates general theorems on the space of isometric immersions and embeddings of pseudo-Riemannian manifolds, extending results of Nash, Kuiper, Janet, Cartan, Burstin, and Smale-Hirsch type immersion theory. Using bundle-theoretic conditions on isometric morphisms, monomorphisms, and se…

ON THE DENSITIES OF MEASURES CORRESPONDING TO SOLUTIONS OF CERTAIN DIFFERENTIAL EQUATIONS WITH RANDOM FUNCTIONS

Unknown
1970 SovietRxiv: ru-197001.36511

The paper studies absolute continuity of measures generated by solutions of differential equations with random Gaussian forcing in finite-dimensional Euclidean space. It compares a linear equation with a corresponding nonlinear equation containing an additional drift term, formulates conditions invo…

ILL-POSED PROBLEMS AND GEOMETRIES OF BANACH SPACES

V. P. Tanana
1970 SovietRxiv: ru-197001.36399

This paper studies stability of the residual method for ill-posed linear operator equations of the first kind in Banach spaces, where only a convex closed neighborhood filter of the exact right-hand side is given. The method selects the minimum-norm element among points whose image lies in the presc…

ON THE STABILITY OF PERIODIC MOTIONS IN THE TRANSCENDENTAL CASE OF TWO PURELY IMAGINARY CHARACTERISTIC EXPONENTS

Yu. N. Bibikov
1970 SovietRxiv: ru-197001.35840

This paper studies Lyapunov stability of the zero solution of a periodic analytic system of differential equations whose linear approximation has two purely imaginary characteristic exponents and all remaining exponents with negative real parts. Focusing on Lyapunov’s generally transcendental case, …

A GENERAL FIRST MAIN THEOREM OF VALUE DISTRIBUTION THEORY

I. M. Degtyarev
1970 SovietRxiv: ru-197001.34528

This note extends first main theorems in value distribution theory for holomorphic mappings to compact complex target manifolds without imposing Kähler, homogeneous, or projective structure. The main construction proves that on any compact complex manifold with a normalized volume form there is a co…

ON THE SUBDIFFERENTIAL OF A COMPOSITE FUNCTIONAL

V. L. Levin
1970 SovietRxiv: ru-197001.34221

This note studies subdifferentials of composite convex functionals on Banach spaces, where a continuous convex mapping into a Banach lattice is composed with a monotone convex functional. It introduces a subdifferential for Banach lattice valued mappings and establishes conditions under which this s…

ON FINDING A MINIMAX ON A CONSTRAINED SET

V. F. Dem'yanov
1970 SovietRxiv: ru-197001.34077

The paper studies the minimization over a convex compact constrained set of a nonsmooth function defined as the maximum of finitely many continuously differentiable functions. It formulates stationarity in terms of the intersection of the convex hull of active gradients with the conjugate cone of fe…

APPROXIMATION OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS OF MANY VARIABLES BY GENERALIZED POLYNOMIALS (FINITE LINEAR SUPERPOSITIONS OF FUNCTIONS OF A SMALLER NUMBER OF VARIABLES)

Unknown
1970 SovietRxiv: ru-197001.33795

The paper studies approximation of continuous and differentiable functions on the unit cube by generalized polynomials, defined as finite linear superpositions of functions of fewer variables with respect to prescribed one variable systems. It establishes existence and characterization results for b…

THE SPECTRUM OF UNIFORM HOMOLOGIES

E. S. Tikhomirova
1970 SovietRxiv: ru-197001.33737

The paper develops a refinement of uniform homology groups for metric spaces, motivated by the inability of earlier groups to distinguish manifolds whose bounding chains grow with different diameter orders. For each admissible order of growth, it defines subsets determined by cycles requiring chains…

ISOMETRIC IMMERSIONS OF CLASS $C^1$

S. Z. Shefel'
1970 SovietRxiv: ru-197001.33462

This note studies isometric immersions of two-dimensional Riemannian metrics into Euclidean three-space by surfaces of class C1 under an added condition of regularity with respect to affine transformations. It introduces affine regularity and proves structural results for immersions of metrics of bo…

PROPERTIES OF ABSOLUTE CONVERGENCE OF MULTIDIMENSIONAL FOURIER SERIES FROM THE POINT OF VIEW OF THE GEOMETRY OF WEAK DISCONTINUITIES OF THE EXPANDED FUNCTIONS

Unknown
1970 SovietRxiv: ru-197001.33045

This paper studies how the geometry of weak discontinuities affects absolute and quasi-absolute convergence of multidimensional Fourier series and eigenfunction expansions of elliptic operators. For functions on the two-dimensional torus weakly discontinuous along a smooth curve, it introduces quasi…

On One Variant of Averaging in Integro-Differential Equations

A. N. Filatov, L. I. Talipova
1970 SovietRxiv: ru-197001.32607

This note studies an averaging procedure for nonlinear integro-differential equations with a small parameter, in which the memory term is replaced by an integral over an infinite interval and the resulting vector field is time averaged. Under continuity, Lipschitz, boundedness, and growth assumption…

THE SCHRÖDINGER EQUATION IN HILBERT SPACE

E. A. Begovatov
1970 SovietRxiv: ru-197001.32408

The paper studies a second order Schrödinger type differential operator in a Hilbert space of Bochner integrable functions, with an operator valued potential depending on the real variable. Under assumptions on sectorial boundedness, Hölder continuity, growth of the operator family, and compactness …

ON A METHOD FOR SOLVING PROBLEMS OF MATHEMATICAL PHYSICS

Ya. N. Fel'd
1970 SovietRxiv: ru-197001.32000

The paper proposes a general method for solving uniquely posed problems of mathematical physics by omitting different conditions, representing the resulting families of partial solutions as functional sets, and finding the original solution as their common element. It formulates both exact and far-z…

ON THE ZEROS OF PERIODIC DIRICHLET FUNCTIONS

A. F. Lavrik
1970 SovietRxiv: ru-197001.31932

The paper studies the location of zeros of two simple classes of periodic Dirichlet functions defined by Dirichlet series with real coefficients and functional equations relating values at s and 1 minus s. Using shortened functional equations, it derives explicit classifications of possible zero con…

ON THE \(S\)-NUMBERS OF POSITIVE OPERATORS

P. E. Sobolevskii
1970 SovietRxiv: ru-197001.31277

This paper studies estimates for the S-numbers of functions of positive linear operators in Banach spaces, including fractional powers, logarithms, iterated logarithms, and analytic semigroups. Using integral representations of fractional powers and logarithms together with standard inequalities for…

A PROBLEM WITH TWO FREE BOUNDARIES FOR AN ELLIPTIC EQUATION AND OPTIMAL STOPPING OF A MARKOV PROCESS

N. V. Krylov
1970 SovietRxiv: ru-197001.30948

This paper studies an elliptic boundary value problem with two free boundaries, formulated as finding a Sobolev function constrained between two obstacles and satisfying complementary sign conditions for an elliptic operator. The analysis connects this problem with a two-sided optimal stopping game …

On Oscillations of Systems Close to Essentially Nonlinear Ones

NGUYEN VAN DAO
1970 SovietRxiv: ru-197001.30617

This paper studies periodic oscillations of systems that are close to essentially nonlinear differential systems with a small parameter, focusing on the special case in which the first solvability conditions for the principal amplitudes are identically satisfied for part of the arbitrary constants. …

ON THE CLASSIFICATION OF TRISYMMETRIC SPACES

L. V. Sabinin
1970 SovietRxiv: ru-197001.30246

The note classifies nontrivial trisymmetric homogeneous Riemannian spaces with simple compact Lie groups of motions. It reformulates trisymmetry in terms of invoproducts and corresponding involutive sums of Lie algebras, proves auxiliary restrictions on the mirrors and isotropy subgroups, and reduce…

ON A SEQUENCE OF LINEAR POSITIVE OPERATORS

I. I. Ibragimov, A. D. Gadzhiev
1970 SovietRxiv: ru-197001.30214

This paper constructs a broad sequence of linear positive operators defined through analytic kernels and parameter sequences, encompassing classical Bernstein, Bernstein Chlodovsky, Mirakyan, and Baskakov operators as special cases. Using Korovkin type arguments, it establishes uniform convergence o…

ON THE UNIQUENESS OF THE SOLUTION OF THE CAUCHY PROBLEM FOR LINEAR EVOLUTION SYSTEMS WITH VARIABLE COEFFICIENTS

V. A. Kondratiev, S. D. Èidel'man
1970 SovietRxiv: ru-197001.29827

The paper studies uniqueness for the Cauchy problem for linear evolution systems with variable coefficients, including systems of arbitrary type solved with respect to time and systems whose spatial part is elliptic or parabolic in the sense of Petrovsky. It develops a weighted L1 estimate in a thin…

Differential pursuit games with constrained phase coordinates

A. B. Kurzhanskii
1970 SovietRxiv: ru-197001.29264

This paper studies differential games of approach for linear controlled systems with convex control constraints, constraints on selected phase coordinates, and terminal payoffs measuring distance to a target set. It formulates a programmed approach problem and an adjoint optimization problem in dist…

ON THE APPROXIMATION OF POLYNOMIALS OF TWO VARIABLES BY FUNCTIONS OF THE FORM \(\varphi(x)+\psi(y)\)

M.-B. A. Babaev
1970 SovietRxiv: ru-197001.29212

The paper studies uniform approximation on a rectangle of continuous functions of two variables by sums of one-variable functions, \(\varphi(x)+\psi(y)\). It identifies a class of supermodular-type functions for which the best approximation error is given explicitly by one quarter of the alternating…

The Cauchy Problem for Weakly Hyperbolic Equations

V. N. Shuhman
1970 SovietRxiv: ru-197001.29026

This paper studies the Cauchy problem for nonstrictly hyperbolic linear partial differential equations with variable coefficients in Gevrey-type classes. It formulates conditions on the real characteristic roots and uses pseudodifferential factorization, Gårding estimates, and induction on the order…

UNSTABLE EXTREMAL PROBLEMS

V. F. Sholohovich
1970 SovietRxiv: ru-197001.28327

This paper studies regularization of solvable but unstable extremal problems over closed convex subsets of real Banach spaces, for quasiconvex lower semicontinuous functionals. The proposed method replaces the original minimization by the search for the least norm element in approximate level sets, …

JUSTIFICATION OF THE AVERAGING METHOD

I. B. Simonenko
1970 SovietRxiv: ru-197001.28147

The paper justifies Bogolyubov’s averaging method for abstract parabolic equations in a Banach space, considering evolution equations with a rapidly varying nonlinear term and an unbounded linear operator generating an analytic semigroup. Under assumptions on fractional power spaces, compactness, di…

CONSTRUCTION OF SEPARATRIX SURFACES AND REGIONS OF STABILITY IN THE PHASE SPACES OF NONLINEAR SYSTEMS

B. N. Petrov, R. A. Nelepin, B. M. Shamrikov
1970 SovietRxiv: ru-197001.27622

The paper develops a method for constructing separatrix surfaces and stability regions in the phase spaces of nonlinear systems of the form consisting of a linear part plus a scalar nonlinearity. By transforming the system to diagonal coordinates and introducing two-dimensional revealing sections in…

On bicompact spaces with noncoinciding dimensions

Unknown
1970 SovietRxiv: ru-197001.27600

The paper develops quotient and decomposition constructions for bicompact spaces designed to separate covering dimension, small inductive dimension, and large inductive dimension. After formulating a general scheme, it constructs families of bicompacta built from ordered compacta, Cantor sets, and m…

ON LINEAR ESTIMATES OF REGRESSION

A. A. Tempel'man
1970 SovietRxiv: ru-197001.26483

This note studies estimation of unknown finite-dimensional regression functions in the mean of a random function with known correlation kernel. Using reproducing-kernel Hilbert spaces and extended kernels, it characterizes canonical isometries associated with the random function and derives uniforml…