Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

Let u and solve the problem

On Asymptotic Properties of Solutions of Diffusion Equations

In this work the authors study the conditions for the existence of diffusion equations

Differential Equations in Hilbert Space with Dissipative Symbols

In this article equations of the form

Inverse Approximation Theorem on an Infinite Union of Segments

Let , where the and satisfy the following relations: and . Denote by B the class of all entire functions of exponential type bounded on the real axis. Under certain assumptions on the rate of approximation on E of a bounded function f by functions in B ( varies), we get some information about the smoothness of f. Bibliography: 4 titles.     

Conditions of Local Asymptotic Normality for Gaussian Stationary Processes

Let {\bold x}[] be a stationary Gaussian process with zero mean and spectral density f, let be the -algebra induced by the random variables {\bold x}[], D(R¹), and let _t, t > 0, be the -algebra induced by the random variables x[],supp [-t,t]. Denote by (f) the Gaussian measure on generated by {\bold x}. Let _t(f) be the restriction of (f) to _t. Let f and g be nonnegative functions such that the measures _t(f) and _t(g) are absolutely continuous. Put

Some properties of the moduli of families of curves

Let A={a₁,...,a_n} and B={b₁,...,b_m} be systems of distinct points in , let be a family of homotopic classes H_i,i=1,..., j+m, of closed Jordan curves on, where the classes H_j+l, l=1,...,m, consist of curves that are homotopic to a point curve in b. Let =₁,..., _j+m be a system of positive numbers and letU be the modulus of the extremal-metric problem for the family and the system . In this paper we investigate the dependence of the modulusU=U(,A,B) on the parameters _i and on the disposition of the points a_k and b. One shows thatU is a smooth function of the indicated arguments and one obtains expressions for the derivatives U, U, and U. One gives some applications of these results.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 72–82, 1985.     

Double Extensions of Dynamical Systems and Construction of Mixing Filtrations. II. Quasihyperbolic Toral Automorphisms

Let T be an automorphism (an invertible measure preserving transformation) of a probability space and let U be a unitary operator on defined by . Let and be the generators of symmetric Markov transition semigroups acting on L₂. It is assumed that A_s and A_u satisfy the relations U_-1 A_s U =^-1 A_sand U^-1 A_u U A_u for some > 1. A nonnegative self-adjoint operator A on L₂ with the properties UA=AU and A_u+A_s A is called a T-invariant minorant for (A_u, A_s). Assuming that A_u and A_s commute, certain assumptions on a function f L₂ in terms of such a minorant are proposed under which the sequence (f T^k, k Z) satisfies the functional form of the central limit theorem and the law of the iterated logarithm. A special case of these assumptions was considered in an earlier paper by the author. Quasihyperbolic toral automorphisms are considered as an application. Bibliography: 9 titles.     

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Projection in a Space of Solenoidal Vector Fields

Let R ³ be a bounded domain, 0$$ " align="middle" border="0"> , a family of extending subdomains, and =(x) a positive function in be a space of -solenoidal vector fields, 0$$ " align="middle" border="0"> , a family of subspaces, G orthogonal projectors in onto . A unitary transformation that diagonalizes the family of projectors {G} is constructed: it takes to the operator of multiplication by the independent variable. The isometry of this transformation is proved with the help of the operator Riccati equation for the NeumanntoDirichlet mapping. Bibliography: 8 titles.     

On Interpolation Sequences of One Class of Functions, Analytic in the Half-Plane

A new criterion of solvability of the interpolation problem f( _n )=b_n in the class of functions f, analytic in the right half-plane and such that there exists c ₁(0;+) such that |f(z)|c ₁exp((c₁|z|)) for all z , where is a positive increasing continuous differentiable function on [0;+), for which (t)+ as t+ and there exists c ₂(0;+) such that

Note on the effectiveness factor of a first order reaction

Upper and lower bounds for the effectiveness factor are derived, whereu is the solution of u=c ² u in,u=1 on .

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Zusammenfassung Es werden obere und untere Schranken hergeleitet für den Diffusionsfaktor , wobeiu die Lösung ist von u=c ² u in,u=1 auf .

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This research was done while the author was visiting Cornell University, Ithaca, New York.     

Asymptotic properties of the likelihood ratio of independent observations

For observations of independent random quantities in the series scheme we study the asymptotic behavior of the logarithm of the likelihood ratio. We find conditions for it to be asymptotically infinitely divisible. and in the parametric case we find for it the decomposition in which () is a random variable converging in distribution to an infinitely divisible law with zero mean, finite dispersion, and Kolmogorov function foru in some subset of the real axis, while _n (u; )0 in probability.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 76–83, 1986.     

On the Existence of Solutions for Some Nondegenerate Nonlinear Wave Equations of Kirchhoff Type

Let be a bounded domain in #x211D;ⁿ with a smooth boundary . In this work we study the existence of solutions for the following boundary value problem:

Class Numbers of Indefinite Binary Quadratic Forms and the Residual Indices of Integers Modulo p

Let h(d) be the class number of properly equivalent primitive binary quadratic forms ax²+bxy+cy² with discriminant d=b²-4ac. The behavior of h(5p²), where p runs over primes, is studied. It is easy to show that there are few discriminants of the form 5p² with large class numbers. In fact, one has the estimate

The local stability of positive solutions to the Hammerstein equation with a nonmonotonic nemytskii operator

LetE be a Banach lattice which consists of some functionsu: over a fixed domain . This article is concerned with the local stability of a non-zero positive solutionu ^*E to the Hammerstein equationu(x)= x, wherek0, andf:×[0, )[0, ) is not necessarily increasing in the second variable. It is assumed thatf(x, 0)=0 and forx,u>0. Under some mild additional hypotheses onE, k, andf it is proved that the spectral radius of the Fréchet derivative atu ^* of the Hammerstein operator is less than one. Also the impact of local stability on secondary bifurcations is investigated. The proof of the main result is based on the spectral theory for completely continuous and irreducible positive operators on Banach lattices.     

Imbedding theorems for the invariant subspaces of the backward shift operator

For subspaces K ^p of the form of the Hardy space H^p and for measures with support in the closed unit circleclos , one finds conditions that ensure the imbedding KL^p(). One considers measures with support inclos , satisfying the following condition: for some number >0 and for all circles with center on the circumference, intersecting the set , we have the inequality ()C(). Here C does not depend on , while () is the radius of the circle . For such measures one has the imbedding K ^p L^p(). From here one derives a criterion for the imbedding K _A ² L²(), found by B. Cohn for inner functions , such that the set is connected for some positive . In the paper one also proves that a condition on , necessary and sufficient for the imbedding of K ^p into L^p(), must depend on p.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 38–51, 1986.     

A 2-extension of the field ℚ of rational numbersof rational numbers

It is proved that the rational number field has one, and only one, normal 2-extension (_{2, t8)}/with group isomorphic to .If is the maximal subfield of a real-closed field, which does not contain ,then the algebraic closure of is isomorphic to the field .Bibliography: 7titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 192–196.     

A self-adjoint “simultaneous crossing of the axis”

By constructing the corresponding Green's function in a trapezoidal domain, we establish the existence of self-adjoint realizations of incorporating boundary conditions of the formu(s, 0)=u(s, T)=0. Such operators correspond to the historically important concept of a simultaneous crossing of the axis for vibrating strings.