The Monte-Carlo algorithm for the solving of systems of linear algebraic equations by the Seidel method

The iteration algorithm is used to solve systems of linear algebraic equations by the Monte-Carlo method. Each next iteration is simulated as a random vector such that its expectation coincides with the Seidel approximation of the iteration process. We deduce a system of linear equations such that mutual correlations of components of the limit vector and correlations of two iterations satisfy them. We prove that limit dispersions of the random vector of solutions of the system exist and are finite.     

Relative efficiencies of estimates using patterned covariances or correlations in the multivariate normal estimation problem

Summary The relative efficiency of maximum likelihood estimates is studied when taking advantage of underlying linear patterns in the covariances or correlations when estimating covariance matrices. We compare the variances of estimates of the covariance matrix obtained under two nested patterns with the assumption that the more restricted pattern is the true state. Formulas for the asymptotic variances are given which are exact for linear covariance patterns when explicit maximum likelihood estimates exist. Several specific examples are given using complete symmetry, circular symmetry and general covariance patterns as well as an example involving a covariance matrix with a linear pattern in the correlations.     

Theory of the conductivity of ordered alloys

A theory of the electrical conductivity of alloys going beyond the coherent potential approximation and taking into account statistical correlations in the scattering of the electrons by the atoms is developed. The two-particle Green's functions for the electrons is calculated with allowance for scattering by pairs of atoms (the zeroth approximation corresponds to the coherent potential approximation), the correlations being described by means of the parameters of the short-and long-range order. It is shown that the change in the electron spectrum on ordering leads to a significant change of the alloy conductivity.Institute of Metal Physics, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 2, pp. 279–293, May 1992.     

On the parameterization of the CreditRisk model for estimating credit portfolio risk

The CreditRisk⁺ model is one of the industry standards for estimating the credit default risk for a portfolio of credit loans. The natural parameterization of this model requires the default probability to be apportioned using a number of (non-negative) factor loadings. However, in practice only default correlations are often available but not the factor loadings. In this paper we investigate how to deduce the factor loadings from a given set of default correlations. This is a novel approach and it requires the non-negative factorization of a positive semi-definite matrix which is by no means trivial. We also present a numerical optimization algorithm to achieve this.     

Many-particle densities

In classical systems with an arbitrary interaction determined by a set of many-particle potentials, the equations for many-particle correlations are reduced to the variational problem for the thermodynamic potential regarded as a functional depending on these correlations. The functional in the variational problem contains not only the total correlations but also the generalized direct correlations determined by the sum of chains composed of total correlations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 136–147 July, 2000.     

Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials

The aim of this paper is to prove a stretched-exponential bound for the decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations. A corollary is the Central Limit Theorem for the Teichmüller flow on the moduli space of abelian differentials with prescribed singularities.

     

Cantorian Fractal Spacetime and Information in Neural Network of the Human Brain

The neural networks of the human brain act as very efficient parallel processing computers co-ordinating memory related responses to a multitude of input signals from sensory organs. Information storage, update and appropriate retrieval are controlled at the molecular level by the neuronal cytoskeleton which serves as the internal communication network within neurons. Information flow in the highly ordered parallel networks of the filamentous protein polymers which make up the cytoskeleton may be compared to atmospheric flows which exhibit long-range spatiotemporal correlations, i.e. long-term memory. Such long-range spatiotemporal correlations are ubiquitous to real world dynamical systems and is recently identified as signature of self-organized criticality or chaos. The signatures of self-organized criticality i.e. long-range temporal correlations have recently been identified in the electrical activity of the brain. The physics of self-organized criticality or chaos is not yet identified. A recently developed non-deterministic cell dynamical system model for atmospheric flows predicts the observed long-range spatiotemporal correlations as intrinsic to quantum-like mechanics governing flow dynamics. The model visualises large scale circulations to form as the result of spatial integration of enclosed small scale perturbations with intrinsic two-way ordered energy flow between the scales. Such a concept maybe applied for the collection and integration of a multitude of signals at the cytoskeletal level and manifested in activation of neurons in the macroscale. The cytoskeleton networks inside neurons may be the elementary units of a unified dynamic memory circulation network with intrinsic global response to local stimuli. A cell dynamical system model for human memory circulation network analogous to atmospheric circulations network is presented in this paper. The model like the analysis of Koruga et al. make use of certain connections to the concept of Cantorian-Fractal spacetime.     

Memory Effects and Nonequilibrium Correlations in the Dynamics of Open Quantum Systems

We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.     

复杂动力巨系统中子系统行为间相关性研究的一种新方法

复杂开放巨系统内部关系错综复杂并具有动态特征,提出了非线性相互预测算法,分析一类复杂巨系统的内部多个子系统之间行为的相互关系及表征其非线性依赖关系的耦合强度.该方法在相空间重构的基础上,仅通过小数据和微信号可得到表征子系统间依赖关系的判别值,进而通过对判别值的分析得到了各子系统的行为特征及其相互作用关系.同时文中得到的子系统间的相互作用机制,可以为金融危机的理论分析提供非线性相互预测度量.     

Correlations whose squares are perspectivities

This article concerns itself with correlations of finite projective planes which enjoy the property that their squares are non-trivial perspectivities of the plane. Existence results are obtained and actual constructions are carried out, along the lines established in the study of polarities.It is shown that planes of non-square order do not admit correlations whose squares are elations. Also, that planes of odd non-square order do not admit correlations whose squares are homologies of odd order, while planes of even non-square order do not admit any correlations whose squares are homologies.In Desarguesian planes, the connection between the correlations that we are interested in and unitary polarities is made apparent. In the process, the first infinite family of correlations with exactly one absolute point is obtained.Finally, in a non-Desarguesian plane, an example is displayed of a correlation whose square is an elation.This research was supported in part by New York Institute of Technology.     

The correlation matrix problem in insurance and banking

In insurance, the analyst is often faced with a large number of inter-related variables for which correlations need to be estimated. Clearly, all correlations lie in the interval [?1,?1], but the numbers cannot be assigned independently. Here, the choices left to the analyst are considered from both a geometric and a probabilistic viewpoint. In practice, the underwriter or risk manager may fix some of the correlations and this paper considers, from both these viewpoints, what effect this has on the analyst's choice of correlations between the remaining variables.     

Effect of lunar atmospheric tide on the meson intensity at Kodaikanal

The lunar semidiurnal variation in the meson intensity at Kodaikanal has been determined by the Chapman-Miller method of analysis from bihourly records of the cosmic ray meson intensity registered by three-fold coincidences for two consecutive groups of data, each extending over a period of 18 months. The former period gives a result about two to three times the probable error and can be taken to be significant. While the correlations between the solar diurnal variation of meson intensity and barometric pressure are high and negative, the corresponding correlations between the lunar semidiurnal variations of meson intensity and barometric pressure are positive. The results obtained at Kodaikanal agree with those of Duperier obtained at London. But the latter period does not give any significant result outside the probable error. Hence further investigation has to be continued.     

Boundedness and stability criteria for linear ordinary differential equations of the second order

We establish some correlations for solutions of ordinary differential equations and the imaginary part of the complex solution of the corresponding Riccati equation. On the basis of these correlations and the I. M. Sobol’ theorem we prove some new stability and boundedness criteria for linear equations of the second order.     

The correlations of finite Desarguesian planes, Part I: Generalities

The polarities of Desarguesian planes have long been known. This author has undertaken to classify the correlations of finite Desarguesian planes in general. In [6] we have presented all the correlations with identity companion automorphism which are not polarities, of these planes. In this sequence of papers, we classify the correlations of planes of order $ p^{2^{i}(2n+1)}, n \neq 0 $, with companion automorphism ( $p^{2^{i}t}$ ), p an odd prime, $ t \neq 0 $. This represents a complete classification of the correlations of planes of odd nonsquare order (i = 0). Some of the correlations of planes of odd square order ($ t \neq 0 $ ) are also covered by the present analysis.When the companion automorphism is not trivial, the problem, naturally, becomes more involved, and a great deal begins to hinge upon the order of the plane being odd or even, and also a square or a nonsquare.The correlations of planes of order $ 2^{2^{i}(2n+1)}, n \neq 0 $, with companion automorphism $ 2^{2^{i}t}, t \neq 0 $, and especially those of planes of order $ p^{2^{i}(2n+1)}, i \neq 0 $, with companion automorphism $ p^{2^{j}(2r+1)}, j > i $ require a substantially different treatment, and will be the object of separate efforts.     

The distribution of the maximum likelihood-ratio statistic used for testing a normal sample for three outliers

We study the exact distribution of the likelihood-ratio statistic used for testing a normal sample for three upper (lower) outliers. We obtain recursive correlations for the integral distribution function of this statistic. We apply the obtained correlations for calculating critical values of the likelihood-ratio statistic which appear to be close to critical values of this statistic simulated by the Monte Carlo method. We give an example of the joint use of the likelihood-ratio statistic for testing a sample for more than one outlier.     

Analytic approach to the (an)harmonic crystal chains with self-consistent stochastic reservoirs

We consider the harmonic and anharmonic chains of oscillators with self-consistent stochastic reservoirs and derive an integral representation (à la Feynman-Kac) for the correlations, in particular, for the heat flow. For the harmonic chain, we give a new proof that its thermal conductivity is finite in the steady state. Based on this integral representation for the correlations and a perturbative analysis, the approach is quite general and can be extended to more intricate systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 138–146, July, 2008.     

The problem of identification of parameters by the distribution of the maximum random variable

Suppose that X₁, X₂,…, X_n are independently distributed according to certain distributions. Does the distribution of the maximum of {X₁, X₂,…, X_n} uniquely determine their distributions? In the univariate case, a general theorem covering the case of Cauchy random variables is given here. Also given is an affirmative answer to the above question for general bivariate normal random variables with non-zero correlations. Bivariate normal random variables with nonnegative correlations were considered earlier in this context by T. W. Anderson and S. G. Ghurye.     

相互粒子系统中的负相关(英文)

主要研究相互粒子系统中概率测度的负相关.我们得到判定概率测度是负相关的一个充分必要条件.最后证明了具有负相关的概率测度的线性组合及乘积测度仍是负相关的.     

On the theory of high-temperature superconductivity

The construction of a theory of the superconductivity of oxide metals based on the polar model is discussed. The matrix electron Green's function is calculated with allowance for both the electron—phonon interaction and strong Coulomb correlations. The possible raising ofT _c due to the lattice instability associated with strong anharmonicity of the oxygen ion vibrations in perovskitelike compounds is considered. Strong electron correlations are taken into account in a one-bandt—J model. Because of a local restriction that precludes doubly occupied states in a system with strong correlation, onlyd-wave pairing is possible.N. N. Bogolyubov (deceased)The present paper was presented at the Fifth International Symposium on Selected Problems of Statistical Mechanics (August 22–24, 1989, Dubna) and is published here with small additions made without the participation of N. N. Bogolyubov.Joint Institute for Nuclear Research, 141980, Dubna, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 371–383, December, 1992.     

On Pseudorandomness in Families of Sequences Derived from the Legendre Symbol

We construct a family of finite binary sequences which has a remarkable uniformity with respect to specification of several terms and which also has the property that every sequence in the family has small measures of normality, well distribution in arithmetical progressions and multiple correlations. We also construct a pseudorandom bit generator whose output consists of members of the family.