Nonuniform estimates of the stability of the normal law under random perturbations of the scale parameter and some of their applications

Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 84–92.     

Boundary-value problem for the nonlinear Schrödinger equation

A mixed boundary-value problem for the nonlinear Schrödinger equation and its generalization is studied by the method used for the inverse scattering problem. A connection is established between conservation laws and boundary conditions in integrable boundary-value problems for higher nonlinear Schrödinger equations. It is shown that the generalized boundary-value problem requires a joint consideration of regular and singular solutions for the nonlinear Schrödinger equation with repulsion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 151–165, 1988.     

Reducing systems of linear differential equations to a passive form

In this paper, an application of the Riquer-Thomas-Janet theory is described for the problem of transforming a system of partial differential equations into a passive form, i.e., to a special form which contains explicitly both the equations of the initial system and all their nontrivial differential consequences. This special representation of a system markedly facilitates the subsequent integration of the corresponding differential equations. In this paper, the modern approach to the indicated problem is presented. This is the approach adopted in the Knuth-Bendix procedure [13] for critical-pair/completion and then Buchberger's algorithm for completion of polynomial ideal bases [13] (or, alternatively, for the construction of Groebner bases for ideals in a differential operator ring [14]). The algorithm of reduction to the passive form for linear system of partial differential equations and its implementation in the algorithmic language REFAL, as well as the capabilities of the corresponding program, are outlined. Examples illustrating the power and efficiency of the system are presented.     

Van der Waerden continuity theorem for semisimple lie groups

As was proved by van der Waerden in 1933, every finite-dimensional locally bounded representation of a semisimple compact Lie group is continuous. This is the famous “van der Waerden continuity theorem,” and it motivated a vast literature. In particular, relationships between the assertion of the theorem (and of the inverse, in a sense, to this theorem) and some properties of the Bohr compactifications of topological groups were established, which led to the introduction and the study of certain classes of the so-called van der Waerden groups and algebras. Until now, after more than 70 years have passed, the van der Waerden theorem appears in monographs and surveys in diverse forms; new proofs were found and then simplified in important special cases. In this paper, we prove that the statement of the van der Waerden theorem remains valid for all (not necessarily compact) real semisimple Lie groups, i.e., any given finite-dimensional representation of a real semisimple Lie group is continuous if and only if this representation is locally bounded. More subtle results are also obtained. The main theorem contains several conditions equivalent to the continuity condition for a finite-dimensional representation. In particular, it is proved that every finite-dimensional representation of a real semisimple Lie group is continuous if and only if the restriction of this representation to the “compact” part, to the Abelian part, or to the nilpotent part of the Iwasawa decomposition is locally bounded, and the original van der Waerden theorem is also somewhat refined. For instance, the following assertion holds: every finite-dimensional representation of a compact semisimple Lie group is continuous if and only if the restriction of this representation to some maximal torus is locally bounded. Dedicated to the memory of George Mackey (1916–2006) Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by the INTAS grant.     

Logarithmic L 2-Small Ball Asymptotics with Respect to a Self-Similar Measure for Some Gaussian Processes

We find logarithmic small ball asymptotics for the L₂-norm with respect to self-similar measures for a certain class of Gaussian processes including Brownian motion, Ornstein-Uhlenbeck process, and their integrated counterparts. Bibliography: 46 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 190–213.     

Scattering of surface plasmon polaritons by a nanoparticle with the inclusion of the magnetic dipole contribution

Using the method of the tensor Green’s function of the wave equation, the differential and total cross sections for the scattering of surface optical electromagnetic waves (surface plasmon polaritons) by a spherical nanoparticle into surface plasmon polaritons and light have been obtained with the inclusion of the magnetic dipole contribution. Using the example of noble metal systems, it has been shown that the magnetic dipole contribution may significantly affect the angular dependence of the differential cross sections and increase their anisotropy as the plasmon wavelength increases.