Synthesis of basic elements of multidimensional multirate systems Part. 1. Nonseparable decimation matrices

A synthesis method is developed for multidimensional decimation matrices of given dimension for a specified number of channels which are used in multidimensional multirate systems. The method is based on application of eigenvalues and Gröbner bases techniques. A full parametrization of decimation matrices is effected for two-and three-dimensional cases. Peculiarities of a four-dimensional case are considered.     

Squeezed states and their applications to quantum evolution

In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant (E,H)-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.     

A complete classification of the two-point extensions of a multidimensional bernoulli shift

A theorem is proven which gives five characterizations of a multidimensional Bernoulli shift. The two-point extensions of a multidimensional Bernoulli shift are classified completely. If such an extension is weakly mixing then it must be Bernoulli; otherwise, it is isomorphic to one of 2 ⁿ specific trivial extensions. This result is extended to multidimensional Bernoulli flows and Bernoulli shifts of infinite entropy. This work supported in part by N.S.F. Grant DMS-85-04701 and by the University of Maryland Department of Mathematics.     

Multidimensional Inverse Scattering for First-Order Systems

A method for solving the inverse problem for a class of multidimensional first-order systems is given. The analysis yields equations which the scattering data must satisfy; these equations are natural candidates for characterizing admissible scattering data. The results are used to solve the multidimensional N-wave resonant interaction equations.     

Multivariate conditional versions of Spearman's rho and related measures of tail dependence

A new family of conditional-dependence measures based on Spearman's rho is introduced. The corresponding multidimensional versions are established. Asymptotic distributional results are derived for related estimators which are based on the empirical copula. Particular emphasis is placed on a new type of multidimensional tail-dependence measure and its relationship to other measures of tail dependence is shown. Multivariate tail dependence describes the limiting amount of dependence in the vertices of the copula's domain.     

Asymptotic value of a multidimensional normal probability integral

Laplace's method of asymptotic estimation of integrals is extended to deal with multidimensional normal probability integrals. A multidimensional normal distribution is integrated over a region which does not include the origin, and an asymptotic value is obtained for small values of the variance. The method is illustrated by an error probability calculation in a communication system using M-level differential phase shift keying.     

Uniqueness of the solution of the Tricomi-type problem for multidimensional hyperbolic-parabolic equation

A mixed domain is described in which the solution of a Tricomi-type homogeneous problem for a multidimensional hyperbolic-parabolic equation is trivial.     

高维有噪频谱有限函数的外推

本文给出的定理既适合于一维与高维,又适合于有噪与无噪的频变有限函数的外推,是对Sanz-Huang两个猜想的改进,同时讨论了外推数与真函数的逼近关系。     

Application of thermal potentials to solving heat conduction problems in a layer

We solve the boundary-value problem of nonstationary heat conduction in a layer between parallel planes by joint application of Fourier-Laplace integral transforms and thermal potentials. A relationship is established between the original multidimensional problem and some special one-dimensional problem of nonstationary heat conduction, which is solved numerically. As a result, we obtain a solution of the original multidimensional problem of heat conduction which can be conveniently used in practical calculations.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 54–60, 1988.     

Cooley-Tukey FFT在高维的算法

A new fast algorithm is presented for multidimensional DFT in this paper. This algorithm is derived based on an interesting coding technique for multidimensional integral point, named the technique vector coding. And called the algorithm VCFFT (vector coding fast Fourier transform). Since the VC-FFT is the extension of Cooley-Tukey algorithm from one-dimensional to multidimensional, its structure of program is simple as Cooley-Tukey FFT, and significantly reduces multiplications and recursive stages.     

Recent trends in random number and random vector generation

A survey of recent work in the areas of uniform pseudorandom number and uniform pseudorandom vector generation is presented. The emphasis is on methods for which a detailed theory is available. A progress report on the construction of quasirandom points for efficient multidimensional numerical integration is also given.     

A statistical model for global optimization by means of select and clone

A statistical model tor giobai optimization is constructed generalizing some properties ofthe Wiener process to the multidimensional case. A new approach, which is similar to the Branch and Bound approach, is proposed to the construction of algorithms based on statistical models. A two dimensional version of the algorithm is implemented, and test results are presented     

Simplicial differential geometric theory for language cortical dynamics

This paper starts by reformulating synthetic differential geometry on abstract simplicial complexes. A new, multidimensional simplicial differential geometry is built by the authors: multidimensional vector fields, multidimensional distributions of vector fields, multidimensional Lie brackets, etc. New concepts of attractors, invariant distributions, and accessibility are formulated in this framework. Using these, control systems are for the first time introduced, in a global manner, on simplicial complexes in order to describe neurodynamics. Invariant distributions, attractors, etc. for cortical controlled systems form a rich picture of the neurodynamics of aphasia (deblocking or facilitation).     

Multiple outlier detection in multivariate data using self-organizing maps title

Summary  The problem of detection of multidimensional outliers is a fundamental and important problem in applied statistics. The unreliability of multivariate outlier detection techniques such as Mahalanobis distance and hat matrix leverage has led to development of techniques which have been known in the statistical community for well over a decade. The literature on this subject is vast and growing. In this paper, we propose to use the artificial intelligence technique ofself-organizing map (SOM) for detecting multiple outliers in multidimensional datasets. SOM, which produces a topology-preserving mapping of the multidimensional data cloud onto lower dimensional visualizable plane, provides an easy way of detection of multidimensional outliers in the data, at respective levels of leverage. The proposed SOM based method for outlier detection not only identifies the multidimensional outliers, it actually provides information about the entire outlier neighbourhood. Being an artificial intelligence technique, SOM based outlier detection technique is non-parametric and can be used to detect outliers from very large multidimensional datasets. The method is applied to detect outliers from varied types of simulated multivariate datasets, a benchmark dataset and also to real life cheque processing dataset. The results show that SOM can effectively be used as a useful technique for multidimensional outlier detection.     

Method of “transition into space of derivatives”: 40 years of evolution

In 1975, the “method of transition into space of derivatives” was proposed. It is an efficiently verifiable frequency criterion for the existence of a nontrivial periodic solution in multidimensional models of automatic control systems with one differentiable nonlinear term. The method used the classical torus principle and refrained from any constructions in the phase space of the system under study. Moreover, the method allowed researchers to broaden the class of systems to which it could be applied. In this work, we give a survey of the results presenting generalization and expansion of the method. We also show the connection between the method of transition into space of derivatives, the well-known generalized Poincaré–Bendixson principle proposed by R. A. Smith, and the results of contemporary authors who are active in the theory of oscillations in multidimensional systems. In the recent years, the author obtained frequency criteria for the existence of orbitally stable cycles in multiinput multioutput (MIMO) control systems and methods for the construction of multidimensional systems having a unique equilibrium and an arbitrarily prescribed number of orbitally stable cycles, which are described in the paper. The extension of the generalized Poincaré–Bendixson principle to multidimensional systems with angular coordinate is presented. We show the application of described methods of investigation of oscillation processes in multidimensional dynamical systems to solving S. Smale’s problem in the chemical kinetics theory of biological cells and also to finding hidden attractors of the generalized Chua system and the minimal global attractor of a system with a polynomial nonlinear term. The publication is illustrated by numerous examples.     

Comparative performance evaluation under uncertainty

In this article CCPR, a multidimensional framework for comparative performance evaluation is proposed, which is elaborated and illustrated through a real-life case. A particular feature of the approach is that it takes account of and corrects for the influence of risks, which are beyond the control of the decision maker. Here risk is seen as a multidimensional measure; it is expressed by means of sensitivities to unexpected changes of a multitude of risk factors. Furthermore, the approach corrects for differences in characteristics between the firms that are being compared. Some characteristics are fixed and thus uncontrollable for the decision maker, whereas others may be changed and controlled by the decision maker. The approach also answers whether the changes in firm characteristics made by management have been appropriate or not.     

A scalable pairwise class interaction framework for multidimensional classification

We present a general framework for multidimensional classification that captures the pairwise interactions between class variables. The pairwise class interactions are encoded using a collection of base classifiers (Phase 1), for which the class predictions are combined in a Markov random field that is subsequently used for multidimensional inference (Phase 2); thus, the framework can be positioned between multilabel Bayesian classifiers and label transformation-based approaches. Our proposal leads to a general framework supporting a wide range of base classifiers in the first phase as well as different inference methods in the second phase. We describe the basic framework and its main properties, as well as strategies for ensuring the scalability of the framework. We include a detailed experimental evaluation based on a range of publicly available databases. Here we analyze the overall performance of the framework and we test the behavior of the different scalability strategies proposed. A comparison with other state-of-the-art multidimensional classifiers show that the proposed framework either outperforms or is competitive with the tested straw-men methods.     

关于有噪频谱有限函数的外推

有噪情况下频谱有限函数的外推是函数逼近理论及信号分析中一个重要的病态问题。本文在Ｔｉｋｈｏｎｏｖ正则化理论研究的基础上，研究这一个推问题，给出了不同于［１］中的方法，它在计算上改进了［１］中的方法，在范围上不仅适用于一维而且适用于高维的情况。     

Multidimensional greatest common divisor and Lehmer algorithms

A class of multidimensional greatest common divisor algorithms is studied. Their connection with the Jacobi algorithm is established and used to obtain theoretical properties such as the existence of digit frequencies. A technique of D. H. Lehmer's for Euclid's algorithm is generalized for efficient computation of the multidimensional algorithms. For triples of integers, two algorithms of interest are studied empirically.This work was partially supported by NSF grant #DCR75-07070.     

多元系统马氏田口方法的诊断与分析研究

马氏田口方法是研究多元系统的一种新方法,主要用于建立和优化多元系统的测量表,以便采用优化后的测量表进行诊断/预测。本文基于马氏田口方法四大基本步骤和MYT正交分解对马氏田口方法进行了拓展研究。首先,介绍了马氏田口方法的四大基本步骤;接着介绍了MYT正交分解法,并将MYT正交分解法应用于多元系统的马氏田口方法异常值潜在原因分析;最后,利用拓展后的马氏田口方法对某医院血粘度诊断系统进行了优化,同时对利用优化后的测量表进行诊断时发现的异常值进行了潜在原因分析,取得了良好的效果。