Intégalités de Sobolev et Temps d'Atteinte

We investigate the links between Sobolev and Nash inequalities, capacity and hitting times estimates and ultracontractive semigroups, in a non-symmetric setting.     

Estimates for the Rapid Decay of Concentration Functions of n-Fold Convolutions

We estimate the concentration functions of n-fold convolutions of one-dimensional probability measures. The Kolmogorov–Rogozin inequality implies that for nondegenerate distributions these functions decrease at least as O(n ^–1/2). On the other hand, Esseen⁽³⁾ has shown that this rate is o(n ^–1/2) iff the distribution has an infinite second moment. This statement was sharpened by Morozova.⁽⁹⁾ Theorem 1 of this paper provides an improvement of Morozova's result. Moreover, we present more general estimates which imply the rates o(n ^–1/2).     

Numerical integration of 2‐D integrals based on local bivariate C 1 quasi‐interpolating splines

In this paper cubature formulas based on bivariate C ¹ local polynomial splines with a four directional mesh [4] are generated and studied. Some numerical results with comparison with other methods are given. Moreover the method proposed is applied to the numerical evaluation of 2‐D singular integrals defined in the Hadamard finite part sense. Computational features, convergence properties and error bounds are proved. This revised version was published online in August 2006 with corrections to the Cover Date.     

New computational method for atoms in an intense electromagnetic field

A new representation is found for the interaction of intense circularly polarized light with an atom. A stationary centrosymmetric part, which depends on the field parameter a ₀=F/ω², is separated out from the atom-field interaction. The time-dependent part of the interaction is represented in the form of a multipole expansion with a ₀ taken into account. The application of this representation for calculating the nonlinear dynamic polarizability of a complicated atom in the random-phase approximation with exchange is studied. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 3, 189–193 (10 August 1998)     

Nonhomogeneous waring equations

It is proved that for an arbitrary positive integer k the equation n=x²+y²+z³+u³+v⁴+w¹⁴+t^4k+1 has a positive integer solution for all sufficiently large n. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 65–68.     

Contour-solid properties of finely holomorphic functions

We prove contour-solid theorems for holomorphic functions defined in finely open sets of the complex plane. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 712–723, May, 1998.     

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Book reviews

     

Records in the Fα-scheme. II. Limit laws

Limit theorems for the number of records in a sequence of independent nonidentically distributed random variables are obtained. A generalization of the so-called F^α-scheme is given. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 42–51. Translated by A. Sudakov.