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The paper is devoted to Koosis' approach to the Carleson theorem. We show that this approach also works for other related questions. The main emphasis in this paper is on the method. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

We introduce interpolation analogs of the Schur Q-functions — the multiparameter Schur Q-functions. We obtain for them several results: a combinatorial formula, generating functions for one-row and two-row functions, vanishing and characterization properties, a Pieri-type formula, a Nimmo-type formula (a quotient of two Pfaffians), a Giambelli-Schur-type Pfaffian formula, a determinantal formula for the transition coefficients between multiparameter Schur Q-functions with different parameters. We write an explicit Pfaffian expression for the dimension of a skew shifted Young diagram. This paper is a continuation of the author's paper math. CO/0303169 and a partial projective analog of the paper q-alg/9605042 by A. Okounkov and G. Olshanski and the paper math. CO/0110077 by G. Olshanski, A. Regev, and A. Vershik. Bibliography: 36 titles. [ABSTRACT FROM AUTHOR]

In this paper, we study continuous approximations to the Clarke subdifferential and the Demyanov­Rubinov quasidifferential. Different methods have been proposed and discussed for the construction of the continuous approximations. Numerical methods for minimization of the locally Lipschitzian functions which are based on the continuous approximations are described and their convergence is studied. To test the proposed methods, numerical experiments have been carried out and discussed in the paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of abstract parabolic problems u′( t) = Au( t), u(0) = u with hyperbolic generator A. We develop a general approach to establish a discrete dichotomy in a very general setting in the case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic equilibrium can be split in such a way that the original initial value problem is reduced to initial value problems with exponentially decaying solutions in opposite time directions. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for the finite element method as well as finite difference methods. [ABSTRACT FROM AUTHOR]

We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced to the quadrature of one dimensional integrals over the halfline. We conclude the paper providing numerical tests which show that these formulas give very accurate approximations and confirm the predicted order of convergence. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

Let $$ F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} $$ be an étale map of degree deg F = d. An étale map $$ G \in \mathbb{C}{\left[ {X,Y} \right]^2} $$ is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =( X + A( X, Y), Y + B( X, Y)) and G ◦ F =( X + C( X, Y), Y + D( X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every $$ {\mathbb{C}^2} $$-automorphism F of degree d has a d-inverse approximation, namely, F−1. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles. [ABSTRACT FROM AUTHOR]

Let be the Fejér kernel, C be the space of contiuous 2π-periodic functions f with the norm $ {\left\| f \right\|} = {\mathop {\max }\limits_{x \in \mathbb{R}} }{\left| {f{\left( x \right)}} \right|} $ , let be the Jackson polynomials of the function f, and let be the Fejér sums of f. The paper presents upper bounds for certain quantities like which are exact in order for every function f ∈ C. Special attention is paid to the constants occurring in the inequalities obtained. Bibliography: 14 titles. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate problems of the Navier-Stokes approximation to kinetic equations in terms of the so-called Chapman-Enskog projection. One considers properties of the Chapman-Enskog projection for the Cauchy problem for moment approximations of the kinetic equation and primarily the Chapman-Enskog projection for the Boltzmann-Peierls kinetic equation. The existence of the Chapman-Enskog projection for the Cauchy problem is proved for the phase space of conservative variables (phenomena of nonlinear diffusion) and for the phase space of physical variables (the second sound projection). [ABSTRACT FROM AUTHOR]