Feynman path integrals for polynomially growing potentials

A general class of infinite dimensional oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proved, as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. These results are applied to provide a rigorous Feynman path integral representation for the solution of the time-dependent Schrödinger equation with a quartic anharmonic potential. The Borel summability of the asymptotic expansion of the solution in power series of the coupling constant is also proved.     

The unified Taylor-Ito expansion

We consider the problem of the Taylor-Ito expansion for Ito processes in a neighborhood of a fixed time moment. The Taylor-Ito expansion known in literature is unified by a canonical system of repeated stochastic Ito integrals with polynomial weight functions. The unified expansion has some computational advantages, such as recurrent relations between the expansion coefficients, ordering of the expansion with respect to smallness of its terms, and a smaller number of applied repeated stochastic integrals of different types. The unified expansion is more convenient in constructing algorithms of numerical solution for stochastic Ito differential equations. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 186–204. Translated by S. Yu. Pilyugin.     

Asymptotic expansions for ornstein-uhlenbeck semigroups perturbed by potentials over banach spaces

An asymptotic expansion of Schilder-type integrals with general phase function on abstract Wiener spaces is given and good control on remainders is obtained. For Ornstein –Uhlenbeck semigroups perturbed by potentials on Banach spaces the asymptotic expansion is given in terms of explicitly discussed “classical orbits”, in the case of finitely many non-degenerate maxima of the phase function. A representation of the leading term by a solution of an infinite dimensional Sturm-Liouville problem is also provided     

New Representations of the Taylor–Stratonovich Expansion

The problem of the Taylor–Stratonovich expansion of the Itô random processes in a neighborhood of a point is considered. The usual form of the Taylor–Stratonovich expansion is transformed to a new representation, which includes the minimal quantity of different types of multiple Stratonovich stochastic integrals. Therefore, these representations are more convenient for constructing algorithms of numerical solution of stochastic differential Itô equations. Bibliography: 14 titles.     

On Taylor coefficients of Cauchy-type integrals of finite complex measures

Boundary values of Cauchy-type integrals of finite complex measures given on a unit circle, generally speaking, are not Lebesgue integrable, and therefore at expansion of Cauchy-type integrals in Taylor series, the expansion coefficients cannot be expressed by boundary values using the Lebesgue integral. In this paper, using the notion of A-integration and N-integration, we get a formula for calculating the Taylor expansion coefficients of Cauchy-type integrals of finite complex measures.     

F-粗积分与(F)-粗积分的关系

在F-粗积分、■-粗积分定义的基础上研究了F-粗积分与■-粗积分之间的关系及粗变化度与粗萎缩度和粗扩充度、粗变化率与粗萎缩率和粗扩充率的关系。     

Double asymptotic expansion in the problem of radiation heat exchange of annular fins of trapezoidal shape

We propose a method of double asymptotic expansion of an approximate analytic solution of the problem on thermal irradiation of annular radiator fins with a trapezoidal cross-section, which is reduced to a nonlinear differential equation of the second order with variable coefficients. The method is based on using the Poincaré small parameter method and method of phase integrals.     

The Magnus Expansion,Trees and Knuth’s Rotation Correspondence

W. Magnus introduced a particular differential equation characterizing the logarithm of the solution of linear initial value problems for linear operators. The recursive solution of this differential equation leads to a peculiar Lie series, which is known as Magnus expansion, and involves Bernoulli numbers, iterated Lie brackets and integrals. This paper aims at obtaining further insights into the fine structure of the Magnus expansion. By using basic combinatorics on planar rooted trees we prove a closed formula for the Magnus expansion in the context of free dendriform algebra. From this, by using a well-known dendriform algebra structure on the vector space generated by the disjoint union of the symmetric groups, we derive the Mielnik–Plebański–Strichartz formula for the continuous Baker–Campbell–Hausdorff series.     

F粗积分与它的度量(英文)

F-rough integrals is defined on the basis of the dual of function one direction S-rough sets,which has dynamic characteristics.Using F-rough integrals,the concepts of expansion measurement-expansion degree and expansion ratio are given.By expansion degree and expansion ratio the changing extent can be expressed with numbers,and the recognition principle of attribute effect on function equivalences is got.     

Asymptotic expansion of stochastic flows

Summary We study the asymptotic expansion in small time of the solution of a stochastic differential equation. We obtain a universal and explicit formula in terms of Lie brackets and iterated stochastic Stratonovich integrals. This formula contains the results of Doss [6], Sussmann [15], Fliess and Normand-Cyrot [7], Krener and Lobry [10], Yamato [17] and Kunita [11] in the nilpotent case, and extends to general diffusions the representation given by Ben Arous [3] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by Strichartz [14].     

The Displacements Produced in an Elastic Half-space by a Suddenly Applied Surface Force

The problem considered here is that of the sudden applicationof a constant force to the surface of a homogeneous, isotropicelastic half-space. The case of a normally applied force isconsidered under axially-symmetric conditions. This force isassumed to be either concentrated at a point or distributedover a circular disc. The method of solution is based on theuse of integral transforms and differs in some respects frommethods previously used for this type of problem. The displacementsat all points of the solid are calculated in terms of finiteintegrals. These integrals are evaluated numerically for selectedinterior points of the half-space for point loading, and forpoints on the axis of symmetry in the case of distributed loading.An expansion of the integrals which is valid for large valuesof the time is also obtained.     

A theorem concerning a product of a general class of polynomials and theH-function of several complex variables

A theorem concerning a product of a general class of polynomials and theH-function of several complex variables is given. Using this theorem certain integrals and expansion formula have been obtained. This general theorem is capable of giving a number of new, interesting and useful integrals, expansion formulae as its special cases.     

A theorem concerning a product of two general classes of polynomials and the multivariableH-function

A theorem concerning a product of two general classes of polynomials and the multivariableH-function is established. Certain integrals and expansion formulae have also been derived by the application of this theorem. This general theorem yields a number of new, interesting and useful theorems, integrals and expansion formulae as its particular cases.     

Waves from a Deep Source of Longitudinal Waves in an Elastic Half Space with a Free Boundary

The nonstationary propagation of waves on the surface of an elastic half space from a deep expansion source (model of an explosion in a half space) is examined. Exact solutions are obtained in the form of integrals with finite limits and the general solution is calculated. Algebraic expressions are obtained for the Rayleigh wave. The transition of Rayleigh waves at the surface of the half space is studied. Calculations of Rayleigh waves from discontinuous pulsed sources are presented.     

The Borel Transform and Its Use in the Summation of Asymptotic Expansions

The solution of connection problems on the real line (the x axis) often give asymptotic expansions which are either even or odd. This gives rise to 'identically zero' expansions, that is, an asymptotic expansion in which all terms are identically zero at the origin. We show that the Borel transform of these problems have solutions that provide integral representations of the solution. The evaluation of these integrals, as x →0, allows us to compute the exponentially small term that these 'identically zero' expansions represent.     

On the Method of Neumann Series for Highly Oscillatory Equations

The main purpose of this paper is to describe and analyse techniques for the numerical solution of highily oscillatory ordinary differential equations by exployting a Neumann expansion. Once the variables in the differential system are changed with respect to a rapidly rotating frame of reference, the Neumann method becomes very effective indeed. However, this effectiveness rests upon suitable quadrature of highly oscillatory multivariate integrals, and we devote part of this paper to describe how to accomplish this to high accuracy with a modest computational effort.     

Multiple stochastic integrals constructed by special expansions of products of the integrating stochastic processes

The paper deals with problems of constructing multiple stochastic integrals in the case when the product of increments of the integrating stochastic process admits an expansion as a finite sum of series with random coefficients. This expansion was obtained for a sufficiently wide class including centered Gaussian processes. In the paper, some necessary and sufficient conditions are obtained for the existence of multiple stochastic integrals defined by an expansion of the product of Wiener processes. It was obtained a recurrent representation for the Wiener stochastic integral as an analog of the Hu–Meyer formula.     

The Borel Transform and Its Use in the Summation of Asymptotic Expansions

The solution of connection problems on the real line (the x axis) often give asymptotic expansions which are either even or odd. This gives rise to "identically zero" expansions, that is, an asymptotic expansion in which all terms are identically zero at the origin. We show that the Borel transform of these problems have solutions that provide integral representations of the solution. The evaluation of these integrals, as x →0, allows us to compute the exponentially small term that these "identically zero" expansions represent.     

Optimal Planar Interception with Fixed End Conditions: Approximate Solutions

In a previous paper (Ref. 1), an exact solution of the optimal planar interception with fixed end conditions was derived in closed form. The optimal control was expressed as an explicit function of the state variables and two fixed parameters, obtained by solving a set of nonlinear algebraic equations involving elliptic integrals. In order to facilitate the optimal control implementation, the present paper derives a highly accurate simplified solution assuming that the ratio of the pursuer turning radius to the initial range is small. An asymptotic expansion further reduces the computational workload. Construction of a near-optimal open-loop control, based on the approximations, completes the present paper.     

Noncommutative Batalin–Vilkovisky geometry and matrix integrals

I study the new type of supersymmetric matrix models associated with any solution to the quantum master equation of the noncommutative Batalin–Vilkovisky geometry. The asymptotic expansion of the matrix integrals gives homology classes in the Kontsevich compactification of the moduli spaces, which I associated with the solutions to the quantum master equation in my previous paper. I associate with the Bernstein–Leites matrix superalgebra equipped with an odd differentiation, whose square is nonzero, the family of cohomology classes of the compactification. This family is the generating function for the products of the tautological classes. The simplest example of my matrix integrals in the case of dimension zero is a supersymmetric extension of the Kontsevich model of 2-dimensional gravity.