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INFINITE IRREDUCIBLE SYSTEMS OF GROUP IDENTITIES

Unknown
1970 SovietRxiv: ru-197001.01873

This paper addresses the independence of infinite systems of group identities and gives an effective construction for a known problem on varieties of groups. For every odd integer n at least 4381, it proves that the identities of the form (x^{rn}y^{rn}x^{-rn}y^{-rn})^n = 1, with r ranging over prime…

ON NECESSARY AND SUFFICIENT CONDITIONS FOR POINTS OF THE MARKOV SPECTRUM TO BELONG TO THE LAGRANGE SPECTRUM

A. A. Bershtein
1970 SovietRxiv: ru-197001.01196

This paper studies when a value of the Markov spectrum in a specified interval also belongs to, or lies in the closure of, the Lagrange spectrum. Building on methods of Freiman, it formulates conditions in terms of two-sided sequences of continued-fraction partial quotients, convergence of such sequ…

SOME PROPERTIES OF PLANE AND SPATIAL MAPPINGS

I. S. Ovchinnikov
1970 SovietRxiv: ru-197001.00108

This note states several estimates for families of homeomorphic, quasiconformal, and Sobolev-type mappings of plane and spatial domains under integral bounds on derivatives or inverse derivatives. Using a previously established inequality, it derives lower bounds for the distance of image continua f…

ON A CLASS OF NONLINEAR OPERATOR EQUATIONS

A. G. Zarubin
1969 SovietRxiv: ru-196901.99727

The paper proves existence theorems for nonlinear operator equations in scales of Banach spaces under energy type inequalities and multiplicative norm estimates. Using the Leray Schauder principle, it gives conditions under which equations of the form Au + Ku = h have solutions, including versions b…

NONSTANDARD ANALYSIS AND HOMEOMORPHISM OF \(B\)-SPACES

D. V. Chudnovskii
1969 SovietRxiv: ru-196901.99487

The paper applies model theory and nonstandard analysis to questions of topological equivalence for Banach spaces, revisiting the theorem that all infinite-dimensional separable Banach spaces are homeomorphic. It constructs nonstandard Banach spaces through internal function spaces and ultrapowers, …

Generalized Steenrod Homology Groups

Unknown
1969 SovietRxiv: ru-196901.99448

The paper introduces a generalized construction of Steenrod homology groups for compact metric spaces. It starts with a category of locally finite simplicial complexes and closed simplicial embeddings satisfying union and gluing axioms, together with specified chain classes over a coefficient group …

BASES OF EIGENVECTORS, THE CHARACTERISTIC FUNCTION, AND INTERPOLATION PROBLEMS IN THE HARDY SPACE \(H^2\)

N. K. Nikol'skii, B. S. Pavlov
1969 SovietRxiv: ru-196901.99324

The paper studies when the eigenvectors of model contractions, realized as restrictions of the adjoint shift on invariant subspaces of vector-valued Hardy space, form a Riesz basis. Using the Sz.-Nagy and Foias characteristic function, it gives criteria for the basis property in terms of inner opera…

EXPLICIT REPRESENTATION AND ASYMPTOTICS AS \(t \to \infty\) OF THE SOLUTION OF THE CAUCHY PROBLEM FOR THE LINEARIZED SYSTEM OF A ROTATING COMPRESSIBLE FLUID

V. N. Maslennikova
1969 SovietRxiv: ru-196901.99092

This paper studies the Cauchy problem for the linearized hyperbolic system describing a rotating compressible fluid in three dimensions, with smooth rapidly decreasing initial data and, in the general representation, smooth forcing terms. It derives explicit integral formulas for the velocity and pr…

THE LOCALIZATION PRINCIPLE FOR ALMOST THE ENTIRE SEQUENCE OF PARTIAL SUMS OF A FOURIER SERIES WITH RESPECT TO A FUNDAMENTAL SYSTEM OF FUNCTIONS

V. A. Il'in
1969 SovietRxiv: ru-196901.98969

The paper studies a modified localization principle for Fourier series with respect to fundamental systems of Laplace operator eigenfunctions in an arbitrary N-dimensional domain. Instead of requiring convergence at a fixed point where the expanded function vanishes locally, it considers almost ever…

MULTILEVEL DYNAMIC GAMES

Yu. A. Flerov
1969 SovietRxiv: ru-196901.98522

The paper introduces multilevel dynamic games as a formal framework for complex hierarchical systems composed of relatively autonomous subsystems. It defines a finite tree of positions, local games at each position, probabilistic transitions between states across levels, and several important subcla…

PROPERTIES OF GREEN’S FUNCTIONS AND GENERALIZED SOLUTIONS OF ELLIPTIC BOUNDARY-VALUE PROBLEMS

Yu. P. Krasovskii
1969 SovietRxiv: ru-196901.98398

The paper studies elliptic boundary value problems for properly elliptic operators with boundary operators that cover the equation, without assuming that the boundary operator system is normal. It constructs a Green vector function for problems with finite dimensional solvability conditions, proves …

ON PERTURBATION OF THE CONTINUOUS SPECTRUM

V. È. Lyantse
1969 SovietRxiv: ru-196901.97902

The paper studies perturbations of the continuous spectrum for the multiplication operator in \(L_2(-\infty,\infty)\) under a class of infinite-dimensional completely regular perturbations represented as \(V=A^*B\). Using an operator-valued function \(K(\zeta)=1+BS_\zeta A^*\), it characterizes nonr…

TOPOLOGICAL EMBEDDINGS OF MANIFOLDS

Unknown
1969 SovietRxiv: ru-196901.97755

This note studies spaces of topological embeddings of manifolds, especially locally flat and locally simply connected embeddings in codimension other than two. It proves a local isotopy theorem implying local p-connectedness of the space of locally flat embeddings, using reductions to local cube emb…

SOME RESULTS ON SEQUENTIAL ESTIMATION OF A SHIFT PARAMETER

A. I. Shalyt
1969 SovietRxiv: ru-196901.97676

This paper studies estimation of a shift parameter for distributions supported on a bounded interval, with emphasis on equivariant Pitman estimates and their sequential analogues. It proves an upper bound for the mean squared error of the fixed-sample Pitman estimate in terms of the sample size and …

On a Boundary Value Problem in an Infinite Layer for Linear Differential-Difference Systems

1°.** Consider the system of linear differential-difference equations
1969 SovietRxiv: ru-196901.97442

This paper studies a boundary value problem in an infinite layer for systems of linear differential difference equations with constant coefficients, where some components are prescribed at the initial hyperplane and the remaining components at the terminal hyperplane. Using a determinant associated …

ON A GENERALIZATION OF KAKUTANI’S FIXED POINT THEOREM

N. I. Glebov
1969 SovietRxiv: ru-196901.97306

This note extends Glicksberg’s generalization of Kakutani’s fixed point theorem in locally convex Hausdorff linear topological spaces to a broader class of set-valued mappings. After recalling closed multifunctions, it introduces a weaker notion of partial closedness, defined by a limiting condition…

ON THE KERNELS OF SPECTRA OF FAMILIES OF OPERATORS

V. S. Ryaben'kii
1969 SovietRxiv: ru-196901.95917

This paper studies spectra associated with uniformly bounded families of finite-dimensional operators acting in normed spaces, introducing several notions of spectral kernels defined through asymptotic approximate eigenvalues. It establishes basic topological and inclusion properties of these kernel…

SEMI-INTEGRAL AND QUASI-NUCLEAR MAPPINGS

Unknown
1969 SovietRxiv: ru-196901.95759

This note studies relations among absolutely summing, semi-integral, quasi-nuclear, and generalized p- and q-quasi-nuclear mappings between locally convex and Banach spaces. Using tensor product formulations and nuclearity criteria, it characterizes absolutely summing operators via continuity of ind…

GAMES WITH INERTIA

I. V. Emelin, M. A. Krasnosel'skii, A. V. Pokrovskii
1969 SovietRxiv: ru-196901.95680

This paper introduces and studies multistage matrix games with inertia, in which each player’s admissible mixed strategies at a given step are constrained by the pure strategies realized in several preceding steps. The authors formalize inertia of finite depth, define limiting expected average payof…

ON A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS FOR WHICH QUESTIONS OF STABILITY IN THE LARGE AND OF INSTABILITY CAN BE SOLVED EFFECTIVELY

V. A. Yakubovich
1969 SovietRxiv: ru-196901.95314

The paper studies a class of nonlinear differential systems formed by a finite-dimensional linear part in feedback with a possibly history-dependent nonlinearity, under controllability, spectral, and quadratic inequality assumptions. Using a frequency-domain negativity condition for an associated He…

BASES OF EIGENVECTORS OF COMPLETELY NONUNITARY CONTRACTIONS

N. K. Nikol'skii, B. S. Pavlov
1969 SovietRxiv: ru-196901.95298

This note studies eigenvector systems of completely nonunitary contractions modeled as restrictions of the adjoint shift to invariant subspaces determined by inner operator functions. Under assumptions of isolated eigenvalues, complete eigenfunctions, and uniform minimality, it gives equivalent crit…

CONES WITH A TRANSITIVE GROUP

A. D. Aleksandrov
1969 SovietRxiv: ru-196901.94239

This paper studies cones in affine space whose associated family of parallel translated cones is invariant under a group acting transitively on nonvertex points, with attention to when such symmetry forces the cone to be elliptic. It proves several classification theorems: under closedness, sharpnes…

ONE BINARY ADDITIVE PROBLEM

Unknown
1969 SovietRxiv: ru-196901.94089

The paper studies a binary additive problem for representations of an integer N as N = n1 + n2, where all prime divisors of n1 are small and all prime divisors of n2 are large, relative to powers of N. It defines the counting function L(N, N1/alpha, N1/beta) and proves an asymptotic estimate for all…

ON SUBELLIPTIC PSEUDODIFFERENTIAL OPERATORS

Yu. V. Egorov
1969 SovietRxiv: ru-196901.93630

This note gives necessary and sufficient conditions for subelliptic estimates for pseudodifferential operators in a domain, expressed through the principal symbol and the leading symbols of iterated commutators of the operator with its formal adjoint. The argument introduces an integer invariant mea…

ON THE QUESTION OF THE DEPENDENCE OF HOLOMORPHIC DIFFERENTIALS ON THE MODULI OF RIEMANN SURFACES

S. L. Krushkal'
1969 SovietRxiv: ru-196901.93410

This note studies how holomorphic quadratic differentials on closed Riemann surfaces of genus greater than one depend on the moduli of the surface. Using the Bers-Teichmüller realization of moduli by normalized quasiconformal mappings and a variational formula for solutions of the Beltrami equation,…

TWO-LEVEL ITERATIVE SCHEMES

A. A. Samarskii
1969 SovietRxiv: ru-196901.93144

This note studies two-level iterative schemes for solving linear operator equations with a self-adjoint positive operator in a Hilbert space. Implicit schemes are reduced to equivalent explicit schemes, allowing convergence estimates to be expressed through norms of resolving or transition operators…

ELLIPTIC BOUNDARY-VALUE PROBLEMS

Ya. A. Sovin
1969 SovietRxiv: ru-196901.92729

This note formulates existence results for elliptic boundary value problems in bounded plane domains with corners and with internal curves of coefficient or boundary condition discontinuity reaching the boundary. Using weighted Sobolev spaces, the argument reduces local model problems near corners a…

CONSISTENCY OF A CERTAIN FRAGMENT OF QUINE’S SYSTEM NF

V. N. Grishin
1969 SovietRxiv: ru-196901.91992

This paper proves the consistency of the fragment NF3 of Quine’s New Foundations, consisting of those axioms whose stratification uses at most three type levels. Using Specker’s criterion, the argument reduces consistency to the construction of a model of three-type theory admitting an epsilon-isomo…

ON A SINGULAR NICOLETTI PROBLEM

I. T. Kiguradze
1969 SovietRxiv: ru-196901.91641

This note studies a singular Nicoletti boundary value problem for a system of first order ordinary differential equations with conditions imposed at prescribed points, allowing the right hand sides to have nonintegrable singularities at those points. Using Carathéodory type assumptions, comparison f…

ON THE PROBLEM OF COEFFICIENTS

I. A. Aleksandrov, V. Ya. Gutlyanskii
1969 SovietRxiv: ru-196901.90970

This paper studies non-uniqueness phenomena in the coefficient problem for normalized univalent functions in the unit disk, focusing on solutions of the differential-functional equations that give necessary extremality conditions for the coefficient functional. Using subclasses with p-fold rotationa…

MORSE THEORY OF NONMULTIPLE CLOSED CURVES

A. M. Vinogradov
1969 SovietRxiv: ru-196901.90596

Vinogradov develops a variant of Morse theory for incomplete spaces of the form M setminus K, where M is complete and K is a closed subset, by introducing responsible and noncritical points and proving a fundamental homotopy invariance lemma under admissibility conditions. The method is applied to t…

IMPLICIT FUNCTIONS AND THE AVERAGING PRINCIPLE OF N. N. BOGOLYUBOV—N. M. KRYLOV

P. P. Zabreiko, Yu. S. Kolesov, M. A. Krasnosel'skii
1969 SovietRxiv: ru-196901.90391

The paper develops a generalized implicit function theorem for operators in Banach spaces under weakened differentiability assumptions, requiring only control of the deviation from the derivative at the origin and uniform boundedness of inverse linearizations. This result is applied to bounded solut…