Method for diagonalizing pair correlation functions for a multicomponent liquid system

We consider the method for finding pair correlation functions for a multicomponent liquid system based on diagonalizing the initial system of integral equations for the pair correlation functions, which allows solving this system. We obtain asymptotic solutions for the correlation functions and analyze the introduced approximations from the physical standpoint. We interpret the obtained results physically.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 569–576, March, 2005.     

Pairwise Correlations in a Multicomponent Anisotropic Liquid

We investigate a multicomponent anisotropic liquid system. The first spatial moment of the direct correlation function is taken into account to obtain asymptotic expressions for the pairwise correlation functions. In this approximation, we obtain the pairwise correlation functions that describe the system behavior not only in the critical-state neighborhood but also in the noncritical domain. We show that the critical parameters for the anisotropic system differ from those for the isotropic system.     

A Matrix Operator Formalism for Investigating Spatially Bounded Multicomponent Liquid Systems

We consider a multicomponent spatially bounded liquid system and propose a method that allows analyzing such systems formally. We illustrate the application of the method with an approximate calculation of pair correlators and an analysis of the shifts of critical parameters due to the spatial bounds of the system.     

Calculating the profile of a multicomponent liquid with the plane-layer geometry

We use the multicomponent liquid system with the spatial restriction in the form of a plane parallel layer. For this system, we calculate the profile of the distribution of densities of different system components in the general form in the approximation of a smooth inhomogeneity under the condition that the external field acts perpendicularly to the layer plane. We investigate the gravitational field case separately. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 149–154, April, 2007.     

ON THE BOX DIMENSION FOR A CLASS OF NONAFFINE FRACTAL INTERPOLATION FUNCTIONS

We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.     

Fermionic construction of the partition function for multimatrix models and the multicomponent Toda lattice hierarchy

We use p-component fermions, p = 2, 3,..., to represent (2p−2)N-fold integrals as a fermionic vacuum expectation. This yields a fermionic representation for various (2p−2)-matrix models. We discuss links with the p-component Kadomtsev-Petviashvili hierarchy and also with the p-component Toda lattice hierarchy. We show that the set of all but two flows of the p-component Toda lattice hierarchy changes standard matrix models to new ones. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 265–277, August, 2007.     

Correlation functions of the XX Heisenberg magnet and random walks of vicious walkers

We investigate a relation between random walks on a one-dimensional periodic lattice and correlation functions of the XX Heisenberg spin chain. Operator averages over the ferromagnetic state play the role of generating functions of the number of paths traveled by so-called vicious random walkers (vicious walkers annihilate each other if they arrive at the same lattice site). We show that the two-point correlation function of spins calculated over eigenstates of the XX magnet can be interpreted as the generating function of paths traveled by a single walker in a medium characterized by a variable number of vicious neighbors. We obtain answers for the number of paths traveled by the described walker from a fixed lattice site to a sufficiently remote site. We provide asymptotic estimates of the number of paths in the limit of a large number of steps. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 2, pp. 179–193, May, 2009.     

一个分形函数的分数阶微积分函数

Based on the combination of fractional calculus with fractal functions, a new type of is introduced； the definition, graph, property and dimension of this function are discussed.     

Pair correlations in a multicomponent fluid placed in a porous medium

We consider a multicomponent fluid placed in a porous medium. The Ornstein—Zernike approximation is used to calculate the pair correlation functions for density fluctuations in the mixture components. We show that light scattering in the neighborhood of the critical state of the system is determined (in the single-scattering approximation) by the commonly known Ornstein—Zernike formula. We investigate the shift in the critical parameters due to the porous medium. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 525–528, March, 2006.     

Neural networks for of continuous optimal approximation functions in R^d

Using some regular matrices we present a method to express any multivariate algebraic polynomial of total order n in a normal form. Consequently, we prove constructively that, to approximate continuous target functions defined on some compact set of ? ^d , neural networks are at least as good as algebraic polynomials.     

Continuity of Julia set for a family of entire functions

Based on the work of McMullen about the continuity of Julia set for rational functions, in this paper, we discuss the continuity of Julia set and its Hausdorff dimension for a family of entire functions which satisfy some conditions.     

Data-sparse approximation to a class of operator-valued functions

In earlier papers we developed a method for the data-sparse approximation of the solution operators for elliptic, parabolic, and hyperbolic PDEs based on the Dunford-Cauchy representation to the operator-valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents. In the present paper, we discuss how these techniques can be applied to approximate a hierarchy of the operator-valued functions generated by an elliptic operator .

     

Complexity of approximation of functions of few variables in high dimensions

In DeVore et al. (2011) [7] we considered smooth functions on [0,1]^N which depend on a much smaller number of variables ? or continuous functions which can be approximated by such functions. We were interested in approximating those functions when we can calculate point values at points of our choice. The number of points we needed for non-adaptive algorithms was higher than that in the adaptive case. In this paper we improve on DeVore et al. (2011) [7] and show that in the non-adaptive case one can use the same number of points (up to a multiplicative constant depending on ?) that we need in the adaptive case.     

Correlation functions of the XXZ Heisenberg chain for Δ = 1/2

We present exact results for the correlation functions of the XXZ Heisenberg chain in the thermodynamic limit for the anisotropy parameter Δ = 1/2. We show that the results are given by integers. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 304–310, February, 2007.     

An outer approximation method for minimizing the product of several convex functions on a convex set

This paper addresses the minimization of the product ofp convex functions on a convex set. It is shown that this nonconvex problem can be converted to a concave minimization problem withp variables, whose objective function value is determined by solving a convex minimization problem. An outer approximation method is proposed for obtaining a global minimum of the resulting problem. Computational experiments indicate that this algorithm is reasonable efficient whenp is less than 4.This research was partly supported by Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture, Grant No. (C)03832018 and (C)04832010.     

试函数法在临界载荷近似值计算中的应用

以试函数拟合压杆挠曲线，用加权残值法求得临界载荷近似值，结果表明，近似值的精度与试函数接近曲线的程度有关．     

Polynomial approximation of piecewise analytic functions on a compact subset of the real line

We discuss Totik’s extension of the classical Bernstein theorem on polynomial approximation of piecewise analytic functions on a closed interval. The error of the best uniform approximation of such functions on a compact subset of the real line is studied.     

土工参数随机模型中相关函数的选择

本文研究土性剖面的随机模型,给出了相关文献中使用的方差衰减函数与相关距离的数学模型,并证明了相关函数的类型在某种条件下必为指数函数,纠正了相关文献中的错误概念。     

Data-sparse approximation to the operator-valued functions of elliptic operator

In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent

In the present paper, we consider various operator functions, the operator exponential negative fractional powers , the cosine operator function and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.

     

Quadratic approximation of penalty functions for solving large-scale linear programs

Using the least squares, modified Lagrangian function, and some other methods as examples, the capabilities of the new optimization technique based on the quadratic approximation of penalty functions that has been recently proposed by O. Mangasarian for a special class of linear programming problems are demonstrated. The application of this technique makes it possible to use unified matrix operations and standard linear algebra packages (including parallel ones) for solving large-scale problems with sparse strongly structured constraint matrices. With this technique, the computational schemes of some well-known algorithms can take an unexpected form.