Recent Papers

Proof of an analogue of Mordell's conjecture for algebraic curves over function fields

This note announces a proof of a function field analogue of Mordell’s conjecture for algebraic curves of genus at least two over a finitely generated extension of a relatively algebraically closed field of constants of characteristic zero. The argument develops additive differential algebraic functi…

On the continuation of the solution of the Cauchy problem for parabolic systems

This paper studies the Cauchy problem for nonlinear parabolic equations and systems in the sense of Petrovskii, with emphasis on local solvability and continuation of solutions. Using uniqueness results for linear systems, interior estimates, and refined estimates for approximating linear problems, …

ON THE LEBESGUE FUNCTION OF ONE LINEAR APPROXIMATION PROCESS

The paper studies Lebesgue functions for a class of linear approximation processes applied to continuous biperiodic functions of two variables, using trigonometric polynomials whose coefficients are obtained from values at equally spaced interpolation nodes. Under monotonicity and sign conditions on…

On the Linear Verigin Problem

The paper studies the linear Verigin free-boundary problem for a one-dimensional parabolic equation with variable coefficients, motivated by fluid displacement in porous media. It formulates the problem with first, second, or third kind boundary conditions and general conjugation conditions across a…

Some Extremal Problems in the Class of Polynomials and Rational Functions

This paper studies extremal inequalities for polynomials and rational functions whose zeros are excluded from prescribed disks, refining classical bounds for maximum moduli and related results of Ankeny, Rivlin, Ibragimov, Mamedov, and Walsh. It establishes sharpened pointwise and maximum modulus es…

On a modified problem $\mathrm{T}_1$ for the equation $u_{xx}+\operatorname{sign}yn u_{yy}=0$

The paper studies a modified Tricomi-type boundary value problem for the Lavrent’ev-Bitsadze equation \(u_{xx}+\operatorname{sign} y\,u_{yy}=0\) in a mixed elliptic-hyperbolic domain with prescribed data on selected boundary arcs and characteristics, allowing certain constants to be determined as pa…

Metrizability and the $\Sigma$-product of bicompacta

This paper studies cardinal invariants and metrizability questions for bicompact spaces arising as continuous images of generalized Sigma-products and products of bicompacts. It proves bounds on the weight of such images in terms of the support cardinal, the weights of the factors, and the supremum …

On the measure of the set of $S$-numbers in a $p$-adic field

This note studies metric properties of Mahler’s classification of transcendental numbers in the field of Hensel p-adic numbers, focusing on the exponent governing lower bounds for values of integral polynomials at a p-adic number. Building on earlier work of Lock, Kasch, and Volkmann, it argues that…

Some Theorems on the Stabilization of the Solution of the Cauchy Problem for Parabolic Equations

This note proves stabilization results for solutions of the Cauchy problem for Petrovskii parabolic systems with time-dependent coefficients and bounded measurable initial data. The argument relates the existence of a limiting value as time tends to infinity to limiting means of the initial function…

Ergodic properties of geodesic flows on closed Riemannian manifolds of negative curvature

The paper formulates a general class of uniformly hyperbolic dynamical systems, including geodesic flows on closed Riemannian manifolds of negative curvature, through invariant splittings of tangent spaces with exponential contraction and expansion. It develops the associated stable and unstable fol…

On the Index and Normal Solvability of a Class of Functional Equations

This paper studies a class of functional equations on a Lyapunov contour involving a Carleman shift, Cauchy singular integrals, and Fredholm kernels. It proposes reducing the equation to a system of singular integral equations with Cauchy kernels, which yields necessary and sufficient conditions for…

On Permutational Unitary Operators in the Space $\Pi_\kappa$

The paper studies invariant subspaces for families of operators in a Pontryagin space with an indefinite scalar product of finite positive index. Its main result states that any family of pairwise commuting unitary operators in such a space has a common nonnegative invariant subspace whose dimension…

The S. L. Sobolev and S. M. Nikol'skii Problem for the Limiting Exponent

This note addresses a problem posed by S. L. Sobolev and S. M. Nikolskii concerning Sobolev embeddings at the limiting exponent into mixed-norm spaces. It proves that functions in the Sobolev space W_p^{(l)} on possibly unbounded domains belong to the corresponding mixed Lebesgue class under the sta…

On the Solvability Condition for the Homogeneous Riemann Problem on Closed Riemann Surfaces

This paper studies the homogeneous Riemann boundary value problem on a closed Riemann surface for the case in which the index is nonnegative but smaller than the genus, a range not covered by general solvability results. Using a Cauchy type integral, Abelian differentials, and Jacobi inversion, it d…

On the Theory of Generalized Groups

This paper studies generalized groups, or inverse semigroups, through several alternative axiomatic conditions on semigroups involving generalized inverses, idempotents, involution, and order. It proves chains and equivalences among eight conditions, showing in particular that they become equivalent…

On the Justification of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics

This paper studies the ergodic hypothesis for dynamical systems from statistical mechanics, especially systems of hard balls with elastic collisions, in response to earlier heuristic arguments based on exponential divergence of trajectories. It develops a sufficient condition for positive entropy of…

Propagation of Smoothness of Solutions to Differential Equations of Principal Type

This paper studies propagation of smoothness for solutions of constant coefficient partial differential equations of principal type, under real principal symbol and nonvanishing gradient on the characteristic cone. For bounded domains star-shaped with respect to a point, it gives a necessary and suf…

ON WEAK COMPLETE CONTINUITY OF A LINEAR MAPPING AND OF ITS ADJOINT

This note extends Gantmacher’s theorem on weak complete continuity of a linear mapping between Banach spaces and its adjoint to broader classes of separable locally convex spaces. It introduces three classes of spaces characterized by when weak complete continuity passes from a mapping to its adjoin…

On infinite-dimensional unitary representations of the group of second-order matrices with entries in a locally compact field

The paper studies irreducible infinite-dimensional unitary representations of the group of nonsingular two by two matrices over a locally compact nondiscrete field, with emphasis on the disconnected case. Using Mackey’s description of representations of the affine subgroup, Fourier analysis on the m…

Extremum Problems with Constraints

Dubovitskii and Milyutin propose a general variational framework for extremum problems with equality and inequality constraints in a complete normed space, including nonclassical problems such as optimal control. The method describes forbidden and admissible first variations as cones and reduces the…

Solvable Algebraic Groups

This paper studies solvable algebraic linear groups over fields of characteristic zero, with particular attention to the disconnected case where geometric methods are less effective. Using properties of unipotent algebraic groups, invariant unipotent subgroups, generalized tori, and conjugacy argume…

$(L_p,L_q)$-multipliers of Fourier integrals

The paper gives sufficient conditions for a function to be an \( (L_p,L_q) \)-multiplier of Fourier integrals in Euclidean space, extending earlier multiplier criteria of Marcinkiewicz and Mikhlin. The main theorem shows that if a multiplier candidate and its mixed first derivatives satisfy uniform …

Continuum as the Complete Limit Set of a Convergent Sequence of Analytic Functions

The paper studies boundary limit behavior of sequences of analytic functions in the unit disk, asking which plane continua can occur as complete limit sets. It proves that, for an arbitrary continuum satisfying the stated separation condition relative to a point w0, there exists a uniformly locally …

On a new criterion for the speciality of $\pi d$-groups with a prescribed number of classes of unattainable isoorder $\pi d$-subgroups

This paper studies finite pi d-groups in relation to the number r of classes of unattainable isoordinal pi d-subgroups and the number t of distinct prime pi-divisors of the group order. It gives simplified and strengthened proofs that such groups are solvable when r equals t minus 1, and also when r…

In our paper (V. I. Arnold and Ya. G. Sinai, “On small perturbations of automorphisms of the torus”), published in *DAN*, vol. 144, no. 4, 1962, the smoothness of the curves considered in Theorem 1 is insufficient for applying Theorem 2 in the proof of Theorem 3. Theorem 3 itself, however, is valid, as follows from D. V. Anosov’s work “Roughness of geodesic flows on compact manifolds of negative curvature” (*DAN*, vol. 145, no. 4, 1962).

On the computation of the rotation of completely continuous vector fields associated with the problem of periodic solutions of differential equations

The paper studies the computation of the rotation of completely continuous vector fields arising from the search for periodic solutions of ordinary differential equations with periodic right-hand side. It relates the rotation of integral-operator fields in the space of continuous functions to the fi…

Extremal Problems for Surfaces of Bounded Gaussian Curvature

This paper studies extremal problems for closed convex hypersurfaces in Euclidean space under one-sided bounds on Gaussian curvature. Using Minkowski’s theorems for convex polyhedra, approximation by surface area measures, and Aleksandrov’s uniqueness theorem, it proves volume comparison results for…

On a Problem in Value Distribution Theory

This paper addresses a problem in Nevanlinna value distribution theory concerning lower estimates for the ratio of the integrated counting functions of zeros and poles to the characteristic of a meromorphic function. Building on results of Nevanlinna and of Edrei and Fuchs, it proves that the lower …

On refinement methods using higher-order differences and $h^2$-extrapolation

The paper develops and justifies higher-order difference refinement methods for numerical solutions of Poisson boundary-value problems in rectangles, including Dirichlet and mixed nonhomogeneous boundary conditions. It decomposes the solution into explicitly constructed singular and polynomial or ha…

On Conditions for Representing a Function of Several Variables as a Sum of a Finite Number of Plane Waves in Given Directions

This paper studies when a sufficiently smooth function of several real variables can be represented as a finite sum of one variable functions evaluated on prescribed linear forms, equivalently as a sum of plane waves in fixed noncollinear directions. Using homogeneous polynomials vanishing on the co…

On Estimates of the Rate of Convergence of Solutions of Difference Equations to Solutions of Elliptic Equations with Discontinuous Coefficients and on a Numerical Method for Solving the Dirichlet Problem

The paper studies convergence estimates for approximating the Dirichlet problem for an elliptic equation in an arbitrary plane domain by embedding the domain in a rectangle and solving an auxiliary elliptic problem with discontinuous coefficients. It derives bounds for the convergence of the auxilia…

On the sum of differences of eigenvalues of two self-adjoint operators

The paper derives trace formulas for sums of differences between eigenvalues of two self-adjoint operators with common domain, motivated by singular Sturm-Liouville problems under finite perturbations. It establishes inequalities bounding partial sums of eigenvalue differences by diagonal sums of th…

The Picard theorem for entire bi-analytic functions

This paper studies value distribution for entire bianalytic functions of the form B(z) = phi(z) + conjugate(z) psi(z), contrasting them with the classical Picard theorem for entire analytic functions. Using normal and quasinormal family arguments together with generalized argument principle and Rouc…

On orthogonal series not summable by linear methods

This paper addresses a question of P. L. Ul'yanov concerning whether a nondecreasing majorant of the Lebesgue functions for a linear summability method can serve as a Weyl multiplier for almost everywhere summability of orthogonal series. It proves negative results for Toeplitz-regular linear method…

On the Decay of Harmonic Functions in a Cylinder

This note proves a Phragmen-Lindelof type uniqueness theorem for harmonic functions of three variables in a half-cylinder with arbitrary planar cross-section. Using a Laplace transform in the axial variable, Fourier integral representations, and estimates for the associated kernel, the authors show …

On a Constructive Duality Principle in the Class of Continuous Functions Decreasing Monotonically to Zero and Convex on the Half-Line

This note studies best uniform approximation on the half-line of bounded monotone convex functions by functions satisfying a prescribed modulus of continuity condition. For a convex unbounded modulus of continuity, it proves a duality theorem characterizing exactly which functions of the parameter M…

A Generalization of a Theorem of A. F. Timan

This paper generalizes Timan’s refinement of Jackson’s theorem for approximation of functions on finite intervals by algebraic polynomials. It establishes pointwise estimates for bounded functions in terms of arbitrary moduli of smoothness, first on a single interval and then on a finite union of in…

On the Spectrum of a Non-Self-Adjoint Differential Operator of Even Order

The paper studies the spectrum of a non-self-adjoint differential operator of even order on the whole real line with a generally complex exponentially decaying potential. Using the resolvent kernel of the unperturbed operator, Fredholm determinant techniques, analytic continuation, and estimates bas…

On canonical transformations in the second quantization representation

This paper studies linear canonical transformations in second quantization for fermion and Bose commutation relations, with particular attention to transformations that are not implemented by unitary operators in the original Hilbert space. It shows that every linear canonical transformation can be …

DIRICHLET SERIES WITH A SEQUENCE OF COMPLEX EXPONENTS HAVING ANGULAR DENSITY

This paper studies Dirichlet series with complex exponents whose sequence has angular density, extending earlier results on series with positive exponents or exponents asymptotically directed along the real axis. Using a method for determining convergence domains and a generalized Cramér Pólya type …

On Systems of Differential Equations with Constant Coefficients

This note extends the global theory of linear differential equations with constant coefficients from single equations and square systems to arbitrary underdetermined and overdetermined systems in spaces of functions and generalized functions allowing exponential growth. It formulates a solvability c…

Functors in Categories of Banach Spaces

This note develops a theory of functors on categories of Banach spaces, emphasizing duality, reflexivity, and operator rings. After defining bounded linear functors and natural mappings between them, it introduces dual functors through morphisms into tensor product functors and proves isometric iden…

On the limiting amplitude principle

This note studies the large time behavior of the solution to a forced Cauchy problem for a homogeneous positive definite elliptic operator with constant coefficients, focusing on the existence of a limiting amplitude. Using the Laplace transform and analytic continuation of the resolvent kernel, it …

Extension of an Equimorphism to Infinity

The paper studies whether equimorphisms of hyperbolic spaces extend naturally to mappings of the sphere of infinitely distant points, contrasting this behavior with the failure of such a statement in Euclidean spaces of dimension at least two. It proves that the image of any geodesic ray under an eq…

ON CONFORMAL MAPPINGS OF DOMAINS WITH CORNERS AND ON THE DIFFERENTIAL PROPERTIES OF SOLUTIONS OF THE POISSON EQUATION IN DOMAINS WITH CORNERS

This paper studies conformal mappings and boundary value problems in planar domains whose boundaries have a corner. Extending Kellogg’s regularity theorem, it establishes conditions under which a conformal map between domains with the same corner angle preserves Hölder smoothness up to the boundary …

On the Relationship Between Small Changes in a System of Differential Equations and the Corresponding Point Mapping

The paper studies whether arbitrary small perturbations of the Poincare point mapping near a fixed point of a closed trajectory can be realized by comparably small perturbations of the right hand sides of the generating system of differential equations. Using coordinates adapted to the periodic orbi…

A Theorem on Homeomorphisms and Local Improvement of Smoothness up to the Boundary for Solutions of Elliptic Equations

This note proves homeomorphism theorems for elliptic operators with general homogeneous boundary conditions in Hilbert scales, including spaces of negative norm. Using an interpolation theorem and coercive estimates, it establishes a complete set of mappings for properly elliptic boundary value prob…

On the Analytical Description of Certain Geometric Objects

The paper introduces algebraic R-operations on real-valued functions to encode logical relations between inequalities and thereby construct analytic descriptions of geometric objects. It establishes operations corresponding to negation, conjunction, and disjunction of sign conditions, showing that t…

Conditions for Preserving the Sign of the Green's Function of a Sturm–Liouville Problem

This paper studies when the Green’s function for a second order Sturm-Liouville boundary value problem with general separated boundary conditions exists and preserves a fixed sign. It derives necessary and sufficient criteria in terms of auxiliary twice differentiable functions satisfying endpoint a…

On the optimality of the principal lattice of Voronoi’s first type among lattices of the first type in any dimension

This paper studies lattice coverings of Euclidean space by balls centered at lattice points, focusing on lattices of the first type in Voronoi’s classification. It proves that Voronoi’s principal lattice of the first type, constructed from a regular Selling frame, gives the least covering density am…