Integral Representations in Linearly Convex Polyhedra

We introduce the notion of mixed Levian (Levi determinant) of a system of k Hermitian and k linear forms. In terms of these Levians we obtain an integral formula for holomorphic functions in linearly convex polyhedra.Original Russian Text Copyright © 2005 Krivokolesko V. P. and Tsikh A. K.The first author was supported by a grant of the President of the Russian Federation and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1212.2003.1); the second author was supported by the Ministry for Education of the Russian Federation (St. Petersburg, Grant 02-1-138).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 579–593, May–June, 2005.     

On nonlinear integral equations and Λ-bounded variation

Summary We discuss the existence or the existence and uniqueness of global and local -bounded variation (BV) solutions as well as continuous BV-solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations formulated in terms of the Lebesgue integral. Since the space of functions of bounded variation in the sense of Jordan is a proper subspace of functions of -bounded variation and for some class of functions , the space of functions of bounded -variation in the sense of Young is also a proper subspace of the space under consideration, our results extend known results in the literature.     

Integrability of Generalized (Matrix) Ernst Equations in String Theory

We elucidate the integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric (d×d)-matrix Ernst potentials. These equations arise in string theory as the equations of motion for the truncated bosonic parts of the low-energy effective action for the respective dilaton and d×d matrix of moduli fields or for a string gravity model with a scalar (dilaton) field, a U(1) gauge vector field, and an antisymmetric 3-form field, all depending on only two space-time coordinates. We construct the corresponding spectral problems based on the overdetermined 2d×2d linear systems with a spectral parameter and the universal (i.e., solution-independent) structures of the canonical Jordan forms of their matrix coefficients. The additionally imposed existence conditions for each of these systems of two matrix integrals with appropriate symmetries provide specific (coset) structures of the related matrix variables. We prove that these spectral problems are equivalent to the original field equations, and we envisage an approach for constructing multiparametric families of their solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 214–225, August, 2005.     

Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

It is shown that, depending upon the orientation of the end tangents t₀,t₁ relative to the end point displacement vector p=p₁–p₀, the problem of G¹ Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points p₀,...,p_n in R³, compatible with a G¹ piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples.     

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump‐diffusion processes (JDP). In this contribution, we develop further our previous fast Monte Carlo method in the case of multivariate (and correlated) JDP. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises. Copyright © 2008 John Wiley & Sons, Ltd.     

On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p$p$ and a parameter λ>0$\lambda >0$. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂Rⁿ$B\subset {\mathbb{R}}^n$. Regularity of solutions strictly depends on the power p$p$ and the parameter λ$\lambda$. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.     

A NUMERICAL EMBEDDING METHOD FOR SOLVING THE NONLINEAR COMPLEMENTARITY PROBLEM(Ⅱ)-ALGORITHM AND ITS CONVERGENCE

In this paper,based on the results presented in part Ⅰ of this paper[18],we present a numerical embedding algorithm for solving the nonlinear complementarityproblem, and prove its convergence carefully. Numerical experiments show that thealgorithm is successful.     

C~n中单位球上的面积积分与BMO

设B是Cn中的单位球,S是Cn中的单位球面,Sα(f)是B上的面积积分.设f∈BMOA, 若存在正测度集E(?)S,使Sα(f)<∞在E上成立,则Sα(f)<∞在S上几乎处处成立,同时 Sα(f)∈ BMO(S)且存在常数C,使得||Sαf||*≤C||f||*.