Mathematics Commons^™

Stochastic Properties Of Spacings In A Single-Outlier Exponential Model, Baha-Eldin Khaledi, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

Let X₁,..., X_n be independent exponential random variables with possibly different scale parameters. Kochar and Korwar [J. Multivar. Anal. 57 (1996)] conjectured that, in this case, the successive normalized spacings are increasing according to hazard rate ordering. In this article, we prove this conjecture in the case of a single-outlier exponential model when all except one of the parameters are identical. We also prove that the spacings are more dispersed and larger in the sense of hazard rate ordering when the vector of scale parameters is more dispersed in the sense of majorization.

The Multiplicities Of A Dual-Thin Q-Polynomial Association Scheme, Bruce E. Sagen, John S. Caughman Iv

Mathematics and Statistics Faculty Publications and Presentations

Let Y=(X,{Ri}1≤i≤D) denote a symmetric association scheme, and assume that Y is Q-polynomial with respect to an ordering E₀,...,E_D of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities mi (0≤i≤D) of Yis unimodal. Talking to Terwilliger, Stanton made the related conjecture that mi≤mi+1 and mi≤mD−i for i<D/2. We prove that if Y is dual-thin in the sense of Terwilliger, then the …

Wholes And Parts In General Systems Methodology, Martin Zwick

Complex Systems Faculty Publications and Presentations

Reconstructability analysis (RA) decomposes wholes, namely data in the form either of set-theoretic relations or multivariate probability distributions, into parts, namely relations or distributions involving subsets of variables. Data is modeled and compressed by variablebased decomposition, by more general state-based decomposition, or by the use of latent variables. Models, which specify the interdependencies among the variables, are selected to minimize error and complexity.

Dependence Among Spacings, Baha-Eldin Khaledi, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we study the dependence properties of spacings. It is proved that if X₁,..., X_n are exchangeable random variables which are TP₂ in pairs and their joint density is log-convex in each argument, then the spacings are MTP₂ dependent. Next, we consider the case of independent but nonhomogeneous exponential random variables. It is shown that in this case, in general, the spacings are not MTP₂ dependent. However, in the case of a single outlier when all except one parameters are equal, the spacings are shown to be MTP₂ dependent and, hence, …

An Efficient Method For Band Structure Calculations In 3d Photonic Crystals, David C. Dobson, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

A method for computing band structures for three-dimensional photonic crystals is described. The method combines a mixed finite element discretization on a uniform grid with a fast Fourier transform preconditioner and a preconditioned subspace iteration algorithm. Numerical examples illustrating the behavior of the method are presented.

About Non-Spherically Symmetric Deformations Of An Incompressible Neo-Hookean Sphere, Marek Elźanowski

Mathematics and Statistics Faculty Publications and Presentations

A class of non-spherically symmetric deformations of a neo-Hookean incompressible elastic ball is considered. It is shown that the only possible solution, the cavitated radially symmetric solution and the deformation of radial inflation and polar stretching. These are the same solutions as found by Polignone-Warne and Warne [6] for a smaller class of deformations. This fact shows once again that the radial deformations are the only deformations, at least within the class considered, which may support a formation of a cavity in the center of an incompressible, isotropic, elastic sphere.

Estimation Of A Monotone Mean Residual Life, Subhash C. Kochar, Hari Mukerjee, Francisco J. Samaniego

Mathematics and Statistics Faculty Publications and Presentations

In survival analysis and in the analysis of life tables an important biometric function of interest is the life expectancy at age x,M(x), defined by M(x)=E[X?x|X>x], where X is a lifetime. M is called the mean residual life function. In many applications it is reasonable to assume that M is decreasing (DMRL) or increasing (IMRL); we write decreasing (increasing) for nonincreasing (non-decreasing). There is some literature on empirical estimators of M and their properties. Although tests for a monotone M are discussed in the literature, we are not aware of any estimators of M under these order restrictions. In …

Multigrid For The Mortar Finite Element Method, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

A multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed. These problems are posed on domains partitioned into subdomains, each of which is independently triangulated in a multilevel fashion. The multilevel mortar finite element spaces based on such triangulations (which need not align across subdomain interfaces) are in general not nested. Suitable grid transfer operators and smoothers are developed which lead to a variable Vcycle preconditioner resulting in a uniformly preconditioned algebraic system. Computational results illustrating the theory are also presented.

State-Based Reconstructability Modeling For Decision Analysis, Michael S. Johnson, Martin Zwick

Complex Systems Faculty Publications and Presentations

Reconstructability analysis (RA) is a method for detecting and analyzing the structure of multivariate categorical data. Jones and his colleagues extended the original variable-based formulation of RA to encompass models defined in terms of system states (Jones 1982; Jones 1985; Jones 1985; Jones 1986; Jones 1989). In this paper, we demonstrate that Jones’ previous work comprises two separable ideas: the “g to k” transformation and state-based modeling. We relate the concept of state-based modeling to established variable-based RA methods (Klir 1985; Krippendorff 1986), and demonstrate that statebased modeling, when applied to event and decision tree models, is a valuable adjunct …

Mortar Estimates Independent Of Number Of Subdomains, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

The stability and error estimates for the mortar finite element method are well established. This work examines the dependence of constants in these estimates on shape and number of subdomains. By means of a Poincar´e inequality and some scaling arguments, these estimates are found not to deteriorate with increase in number of subdomains.

Prestructuring Neural Networks Via Extended Dependency Analysis With Application To Pattern Classification, George G. Lendaris, Thaddeus T. Shannon, Martin Zwick

Complex Systems Faculty Publications and Presentations

We consider the problem of matching domain-specific statistical structure to neural-network (NN) architecture. In past work we have considered this problem in the function approximation context; here we consider the pattern classification context. General Systems Methodology tools for finding problem-domain structure suffer exponential scaling of computation with respect to the number of variables considered. Therefore we introduce the use of Extended Dependency Analysis (EDA), which scales only polynomially in the number of variables, for the desired analysis. Based on EDA, we demonstrate a number of NN pre-structuring techniques applicable for building neural classifiers. An example is provided in which EDA …

Geometrical Models For Grain Dynamics, Giovani L. Vasconcelos, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We study models for the gravity-driven, dissipative motion of a single grain on an inclined rough surface. Imposing some conditions on the momentum loss due to the collisions between the particle and the surface, we arrive at a class of models in which the grain dynamics is described by one-dimensional maps. The dynamics of these maps is studied in detail. We prove the existence of various dynamical phases and show that the presence of these phases is independent of the restitution law (within the class considered).

Complexity Reduction In State-Based Modeling, Martin Zwick

Complex Systems Faculty Publications and Presentations

For a system described by a relation among qualitative variables (or quantitative variables "binned" into symbolic states), expressed either set-theoretically or as a multivariate joint probability distribution, complexity reduction (compression of representation) is normally achieved by modeling the system with projections of the overall relation. To illustrate, if ABCD is a four variable relation, then models ABC:BCD or AB:BC:CD:DA, specified by two triadic or four dyadic relations, respectively, represent simplifications of the ABCD relation. Simplifications which are lossless are always preferred over the original full relation, while simplifications which lose constraint are still preferred if the reduction of complexity more …

Complexity And Decomposability Of Relations, Martin Zwick

Complex Systems Faculty Publications and Presentations

A discrete multivariate relation, defined set-theoretically, is a subset of a cartesian product of sets which specify the possible values of a number of variables. Where three or more variables are involved, the highest order relation, namely the relation between all the variables, may or may not be decomposable without loss into sets of lower order relations which involve subsets of the variables. In a completely parallel manner, the highest order relation defined information-theoretically, namely the joint probability distribution involving all the variables, may or may not be decomposed without loss into lower-order distributions involving subsets of the variables. Decomposability …

Resolution Of Local Inconsistency In Identification, Douglas Ray Anderson, Martin Zwick

Complex Systems Faculty Publications and Presentations

This paper reports an algorithm for the resolution of local inconsistency in information-theoretic identification. This problem was first pointed out by Klir as an important research area in reconstructability analysis. Local inconsistency commonly arises when an attempt is made to integrate multiple data sources, i.e., contingency tables, which have differing common margins. For example, if one ha)s an AB table and a BC table, the B margins obtained from the two tables may disagree. If the disagreement can be assigned to sampling error, then one can arrive at a compromise B margin, adjust the original AB and BC tables to …

Geometrical Modeling Of Material Aging, Alexander Chudnovsky, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

Material aging is understood as changes of material properties with time. The aging is usually observed as an improvement of some properties and a deterioration of others. For example an increase of rigidity and strength and reduction in toughness with time are commonly observed in engineering materials. In an attempt to model aging phenomena on a continuum (macroscopical) level one faces three major tasks. The first is to identify an adequate age parameter that represents, on a macroscopic scale, the micro and sub microscopical features, underlying the aging phenomena such as nucleation, growth and coalescence of microdefects, physico-chemical transformations etc. …

On Non-Holonomic Second-Order Connections With Applications To Continua With Microstructure, Marek Elźanowski, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

Motivated by the theory of uniform elastic structures we try to determine the conditions for the local flatness of locally integrable connections on non-holonomic frame bundles of order 2. Utilizing the results of Yuen as well as our results for the holonomic case, we show that the locally integrable non-holonomic 2-connection is locally flat if, and only if, its projection to the bundle of linear frames is symmetric and the so-called inhomogeneity tensor vanishes. In the last section of this short paper we show how these results can be interpreted in the framework of the theory of uniformity of simple …

On Matching Ann Structure To Problem Domain Structure, George G. Lendaris, Martin Zwick, Karl Mathia

Complex Systems Faculty Publications and Presentations

To achieve reduced training time and improved generalization with artificial neural networks (ANN, or NN), it is important to use a reduced complexity NN structure. A "problem" is defined by constraints among the variables describing it. If knowledge about these constraints could be obtained a priori, this could be used to reduce the complexity of the ANN before training it. Systems theory literature contains methods for determining and representing structural aspects of constrained data (these methods are herein called GSM, general systems method). The suggestion here is to use the GSM model of the given data as a pattern for …

Organizational Inducements And Social Motives: A Game Theoretic Analysis, Richard G. Davis

Dissertations and Theses

Game theory was used to analyze compensation systems based on individual and group incentives. Payoff formulas were developed for these incentives assuming different preferences for individual and social outcomes. Two levels of contributions were considered: (1) Defection. The minimum acceptable level of contributions, and (2) Cooperation. A level of discretionary contributions above the minimum. The discretionary contributions associated with cooperation were represented as a cost to the individual.

A classification scheme for uniform n-person games was developed using the approach of Rappaport and Guyer (1966) for 2 x 2 games. This classification scheme defines the natural outcome (cooperation or defection) …

The Decay And Formation Of One-Dimensional Nonconservative Shocks, Marek Elźanowski, Marcelo Epstein

Mathematics and Statistics Faculty Publications and Presentations

The method of singular surfaces is used to obtain explicit conditions under which a one-dimensional acceleration wave develops into a shock when some dissipation mechanism is present. The conditions which secure the initial growth of the strong shock wave propagating into an undeformed nonlinear and dissipative medium are also derived. The analysis is pre- sented for a single balance law, one-dimensional elasticity, and the non- linear Maxwellian continuum.

Derivation And Test Of Predictions Of A Discrete Latent State Model For Signed Number Addition Test Performance, Kentaro Yamamoto

Dissertations and Theses

This study is an investigation of the performance of a discrete latent state model devised by Paulson (1982) to account for signed-number arithmetic test data gathered by Birenbaum and Tatsuoka (1980). One hundred twenty nine students took a test which consists of sixteen item types with four parallel arithmetic items of each type. The present study utilizes the five addition item types of four items each; hence, there are four parallel subtests. Responses to the addition items can be analyzed in terms of two components: the sign component (is the sign correct?), and the absolute value component (is the size …

An Improved Approach For "Optimization" Of Multiple Policy Objectives, Perihan Soliman Tawfik

Dissertations and Theses

This thesis involved three categories of activity; development and testing of an expanded version of ELECTRE II, also the development of a computer software program for ELECTRE II.

The expanded version of ELECTRE II took the form of an input aiding questionnaire along with a tailored structure to suit a particular problem. The contents of the questionnaire were based on geueral problem solving concepts (techniques, strategies) gleaned from the systems science literature. This questionnaire assumed a programmed instruction format in contrast to that of an interactive computer software package, so that it would not be prohibitive in terms of expenses …

Distribution Of The Sum Of Independent Unity-Truncated Logarithmic Series Variables, Russell James Wayland

Dissertations and Theses

Let X₁, X₂, ..., X_n be n independent and identically distributed random variables having the unity-truncated logarithmic series distribution with probability function given by f(x;θ) = αθ^X/x x ε T where α = [-log(1-θ) - θ]^-1, 0 < θ < 1, and T = {2,3,...,∞}. Define their sum as Z = X₁ + X₂ + ... + X_n.

We derive here the distribution of Z, denoted by p(z;n,θ), using the inversion formula for characteristic functions, in an explicit form in terms of a linear combination of Stirling numbers of the first kind. A recurrence relation for the probability function p(z;n,θ) is obtained …