Thermodynamics of the model of equal spin-spin interactions

We consider the thermodynamics of the model of equal spin-spin interactions. We obtain exact expressions for the correlation functions and heat capacity of finite clusters applicable in the entire range of temperature and external fields. We analyze the obtained thermodynamic characteristics depending on the interaction parameters, the external magnetic field, and the number of particles in the cluster. We find an anomalous behavior of the heat capacity and other thermodynamic quantities due to elementary spin gap excitations occurring in the spectrum. The absence of a long-range order in the system is ensured by the presence of topological excitations (solitons). We study the effect of an anisotropic interaction parameter on the soliton structure.     

Optimal method for truncating a chain of equations for two-time Green’s functions

We use an example of a chain of equations describing a system of Bose particles with pairwise interaction to develop a method for decoupling the chain at its second element. We obtain an approximation of the interacting-modes type, which results in a system of nonlinear equations for one-, two-, and three-particle functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 503–510, June, 2006.     

Longitudinal Shear of Layered Anisotropic Media with Band-Type Inhomogeneities

Using the method of jump functions, we solve the antiplane problem of elasticity theory for a stack of anisotropic strips containing plane band-type inhomogeneities. We model these inclusions by jumps of the stress vector and the derivative of the displacement vector at the middle surfaces. Applying the Fourier integral transformation, we obtain the dependence of the components of the stress tensor and displacement vector on the external load and unknown jump functions. Taking into account the conditions of the interaction between a thin inclusion and an anisotropic medium, we reduce the problem to a system of singular integral equations for the jump functions. A specific example is considered as well.     

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.     

Bitopologies on Totally Ordered Sets

We study the category of ray bispaces, that is, the category whose objects are totally ordered sets with two topologies, each having a subbase of rays and so that the resulting bitopological space is pairwise weakly symmetric, and whose morphisms are the pairwise continuous functions. In contrast with the purely topological results of [5], we show that, (1) such spaces are utterly normal and hence monotonically normal (in the sense of [6]), and (2) (Intermediate Value Theorem) the pairwise continuous image of a pairwise connected bitopological space in a selective ray bispace is an interval. We also obtain conditions for the equality of the de Groot dual (see [4]) and the ray dual (see [5]) of a ray topology and show that a selective ray topology is compact if and only if it is skew compact.     

Correlation Function of the Two-Dimensional Ising Model on a Finite Lattice: II

We calculate the pair correlation function and the magnetic susceptibility in the anisotropic Ising model on the lattice with one infinite and one finite dimension with periodic boundary conditions imposed along the second dimension. Using the exact expressions for lattice form factors, we propose formulas for arbitrary spin matrix elements, thus providing a possibility to calculate all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. We analyze passing to the scaling limit.     

Correlation functions in the anomalous-dimension approximation for a multicomponent liquid

We analyze the features of solutions for pair correlation functions in the case of a multicomponent liquid. We obtain these solutions based on the Ornstein-Zernike equation. In the anomalous-dimension approximation, we find expressions for pair correlation functions in the case of a spatially unbounded multicomponent liquid. We show that all pair correlation functions for a system in the close vicinity of the critical state are described by a general expression similar to the expression for a pair correlation function in the case of a one-component liquid. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 124–129, October, 2007.     

Integral representation and embedding theorems for <Emphasis Type="Italic">n</Emphasis>-dimensional multianisotropic spaces with one anisotropic vertex

We prove embedding theorems for the multianisotropic Sobolev spaces generated by the completely regular Newton polyhedron. Under study is the case of the polyhedron with one anisotropic vertex. We obtain a special integral representation of functions in terms of the tuple of multi-indices of the Newton polyhedron.     

Meson correlation functions for a model with nonlocal four-fermion interaction

We study meson correlation functions for several models with a four-fermion quark interaction. We show that although the system average energy and quark condensate, as previously noted, are singular, the meson observables are finite, completely recognizable, and comparable to the experimental data on the energy scale. This permits using a wide set of Hamiltonians to model nonequilibrium states of quark and hadronic systems, which is a relevant problem in studying the physics of ultrarelativistic heavy-ion collisions. We obtain analytic expressions for the meson correlation functions in the Keldysh model.     

The Gel'fand widths of anisotropic Sobolev classes of smooth functions

In this paper, we obtain some Gel'fand widths of anisotropic Sobolev periodic classes of smooth functions, and average Gel'fand widths of anisotropic Sobolev classes of smooth functions.     

A diophantine problem concerning cubes

This paper deals with the problem of finding n integers such that their pairwise sums are cubes. We obtain eight integers, expressed in parametric terms, such that all the six pairwise sums of four of these integers are cubes, 9 of the 10 pairwise sums of five of these integers are cubes, 12 pairwise sums of six of these integers are cubes, 15 pairwise sums of seven of these integers are cubes and 18 pairwise sums of all the eight integers are cubes. This leads to infinitely many examples of four positive integers such that all of their six pairwise sums are cubes. Further, for any arbitrary positive integer n, we obtain a set of 2(n+1) integers, in parametric terms, such that 5n+1 of the pairwise sums of these integers are cubes. With a choice of parameters, we can obtain examples with 5n+2 of the pairwise sums being cubes.     

Asymptotic behavior of an eigenvalue of the two-particle discrete Schr?dinger operator

We consider a two-particle discrete Schr?dinger operator corresponding to a system of two identical particles on a lattice interacting via an attractive pairwise zero-range potential. We show that there is a unique eigenvalue below the bottom of the essential spectrum for all values of the coupling constant and two-particle quasimomentum. We obtain a convergent expansion for the eigenvalue.     

高速扩展平面应力裂纹尖端的各向异性塑性场

在裂纹尖端的应力分量都只是θ的函数的条件下,利用定常运动方程,Hill各向异性屈服条件及应力应变关系,我们得到高速扩展平面应力裂纹尖端的各向异性塑性场的一般解.将这个一般解用于四种各向异性特殊情形,我们就导出这四种特殊情形的一般解.最后,本文给出X=Y=Z情形的高速扩展平面应力Ⅰ型裂纹尖端的各向异性塑性场.     

Riesz rising sun lemma for several variables and the John-Nirenberg inequality

We obtain a multidimensional analog of the well-known Riesz rising sun lemma. We prove a more precise version of this lemma for space dimension d = 2 . We use these lemmas to establish an anisotropic analog of the John-Nirenberg inequality for functions of bounded mean oscillation with an exact constant in the exponent. Earlier, this exact constant was only known in the one-dimensional case.     

The three-dimensional problem for an anisotropic plate of arbitrary thickness

By expanding the components of the displacement vector in a certain system of functions of the transverse coordinate, we reduce the solution of the three-dimensional problem of the theory of elasticity of an anisotropic body to a series of two-dimensional problems. To determine the displacements we obtain a system of differential equations of infinite order with two independent variables. We show how to pass from the infinite system to a series of finite systems depending on the form of the external forces. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 11–19.     

Modeling Multiparticle Coherences in Solid-State Nuclear Spin Systems Using Infinite-Range Interaction

Using the exactly solvable model known in statistical physics as the infinite-range interaction model (the van der Waals model), we obtain expressions for the hierarchy of multispin multiquantum time correlation functions for a solid-state paramagnetic nuclear spin system with magnetic dipole–dipole interaction between nuclei. These functions are observed experimentally using contemporary multiquantum NMR spectroscopy methods. We write the complete set of orthogonal multispin operators explicitly and find the time dependence for amplitudes of multiparticle time correlation functions. The proposed model describes the behavior of multiparticle correlation functions well, at least for not very large numbers n of expansion harmonics. The theoretical results agree well with the experimental data obtained for hexamethylbenzol.     

Transmission dynamics of a two-strain pairwise model with infection age

We present and study a two-strain pairwise epidemic model with non-Markovian recovery process in which the recovery rate depends on infection age. The novel model is a hyperbolic system which can be transformed into a system of integro-differential equations by using the method of characteristics. For the two-strain pairwise model, the reproduction number with arbitrary recovery time distributions is obtained. We carry out rigorous analysis and obtain upper and lower estimates for the final epidemic size. Finally, the effects of three commonly used recovery time distributions on the reproduction number are compared by numerical simulations. We further present how the mean length of the infectious period affects the final epidemic size, the peak time and the duration of an epidemic.     

两两PQD序列的完全收敛性和强大数定律

给出两两PQD序列部分和的几个不等式,研究了一类广泛的随机变量序列两两PQD列的收敛性质,得到了与PA序列相类似的完全收敛性和强大数定律的结果.     

Continuity of Some Classes of Functions Related to Degenerate Parabolic Equations and Its Applications

We study some classes of functions satisfying the assumptions similar to but weaker than those for the classical B2 function classes used in the research of quasi-linear parabolic equations as well as the ones used in the research of degenerate parabolic equations including porous medium equations. Consequently, we prove that a function in such a class is continuous. As an application, we obtain the estimate for the continuous modulus of the solutions of a few degenerate parabolic equations in divergence form, including the anisotropic porous equations.     

On comparability of integral forms

We prove comparability of certain homogeneous anisotropic integral forms. As a consequence we obtain a Hardy type inequality generalising that for the fractional Laplacian. We give an application to anisotropic censored stable processes in Lipschitz domains.