Physical Sciences and Mathematics Commons^™

Counting Structures In The Möbius Ladder, John P. Mcsorley

Articles and Preprints

The Möbius ladder, M_n, is a simple cubic graph on 2n vertices. We present a technique which enables us to count exactly many different structures of M_n, and somewhat unifies counting in M_n. We also provide new combinatorial interpretations of some sequences, and ask some questions concerning extremal properties of cubic graphs.

Ua66/10/2 Alumni Newsletter, Wku Mathematics

WKU Administration Documents

Newsletter created by and about the WKU Mathematics department.

Mathematical Models Of Tumors And Their Remote Metastases, Carryn Bellomo

Mathematics & Statistics Theses & Dissertations

Clinical observations and indications in the literature have led us to investigate several models of tumors. For example, it has been shown that a tumor has the ability to send out anti-growth factors, or inhibitors, to keep its remote metastases from growing. Thus, we model the depleting effect of such a growth inhibitor after the removal of the primary tumor (thus removing the source) as a function of time t and distance from the original tumor r.

It has also been shown clinically that oxygen and glucose are nutrients critical to the survival and growth of tumors. Thus, we model …

Reaction-Diffusion Models Of Cancer Dispersion, Kim Yvette Ward

Mathematics & Statistics Theses & Dissertations

The phenomenological modeling of the spatial distribution and temporal evolution of one-dimensional models of cancer dispersion are studied. The models discussed pertain primarily to the transition of a tumor from an initial neoplasm to the dormant avascular state, i.e. just prior to the vascular state, whenever that may occur. Initiating the study is the mathematical analysis of a reaction-diffusion model describing the interaction between cancer cells, normal cells and growth inhibitor. The model leads to several predictions, some of which are supported by experimental data and clinical observations $\lbrack25\rbrack$. We will examine the effects of additional terms on these characteristics. …

Sparse Equation-Eigen Solvers For Symmetric/Unsymmetric Positive-Negative-Indefinite Matrices With Finite Element And Linear Programming Applications, Hakakizumwami Birali Runesha

Civil & Environmental Engineering Theses & Dissertations

Vectorized sparse solvers for direct solutions of positive-negative-indefinite symmetric systems of linear equations and eigen-equations are developed. Sparse storage schemes, re-ordering, symbolic factorization and numerical factorization algorithms are discussed. Loop unrolling techniques are also incorporated in the coding to enhance the vector speed. In the indefinite solver, which employs various pivoting strategies, a simple rotation matrix is introduced to simplify the computer implementation. Efficient usage of the incore memory is accomplished by the proposed "restart memory management" schemes. A sparse version of the Interior Point Method, IPM, has also been implemented that incorporates the developed indefinite sparse solver for linear …

Neuro Fuzzy Reasoning For Pattern Classification And Object Recognition., Jayati Ghosh Dr.

Doctoral Theses

In real world, pattern classification and object recognition problems are faced with fuzzi- ness that is connected with diverse facets of cognitive activity of the human being. An origin of sources of fuzziness is related to labels expressed in feature space as well as to labels of classes taken into account in classification and /or recognition procedures. Though a lot of scientific efforts have already been dedicated to pattern recognition problems, especially to classification procedures, still pattern recognition is confronted with a continuous challenge coming from a human being who can perform lot of ex- tremely complex classification tasks by …

The Stable Manifold Theorem For Sde's (Probability Seminar, University Of California, Irvine), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

In this talk, we formulate a local stable manifold theorem for stochastic differential equations in Euclidean space, driven by multi-dimensional Brownian motion. We introduce the concept of hyperbolicity for stationary trajectories of a SDE. This is done using the Oseledec muliplicative ergodic theorem on the linearized SDE along the stationary solution. Using methods of (non-linear ergodic theory), we construct a stationary family of stable and unstable manifolds in a stationary neighborhood around the hyperbolic stationary trajectory of the non-linear SDE. The stable/unstable manifolds are dynamically characterized using anticipative stochastic calculus.

Computer Simplification Of Formulas In Linear Systems Theory, J. W. Helton, Mark Stankus, John J. Wavrik

Mathematics

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA. These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Gröbner Basis Algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations.

On the other hand, most of the computation involved in linear control theory is performed on matrices, and these do not commute. A typical issue of IEEE TRANSACTIONS ON AUTOMATIC CONTROL is full of linear systems and computations with their coefficient …

Games To Teach Mathematical Modelling, James A. Powell, Jim S. Cangelosi, Ann M. Harris

Mathematics and Statistics Faculty Publications

We discuss the use of in-class games to create realistic situations for mathematical modelling. Two games are presented which are appropriate for use in post-calculus settings. The first game reproduces predator-prey oscillations and the second game simulates disease propagation in a mixing population. When used creatively these games encourage students to model realistic data and apply mathematical concepts to understanding the data.

Yang's Gravitational Theory, Brendan Guilfoyle, Brien C. Nolan

Publications

Yang's pure space equations generalize Einstein's gravitational equations, while coming from gauge theory. We study these equations from a number of vantage points: summarizing the work done previously, comparing them with the Einstein equations and investigating their properties. In particular, the initial value problem is discussed and a number of results are presented for these equations with common energy-momentum tensors.

Wiener Tauberian Theorems On Semisimple Lie Group., Rudra Pada Sarkar Dr.

Doctoral Theses

In their celebrated study of Harmonic analysis on semi-simple Lie groups Ehrenpreis and Mautner [E-M] noticed that the analogue of the claasical Wiener Tauberian theorem resting on the unitary dual does not hold for semisimple Lle groups. A simple proof of this fact due to M. Duflo appears in (H). Ehrenpreis and Mautner went on in (E-M] to formulate the problem on the commutative Banach algebra of the SO2(R)-biinvariant functions in L1(SL2(R))1, and obtained two different versions of the theorem involving, this time, the dual of the Banach algebra which includes, beside the unitary dual of G, a part of …

Static Interconnection Networks And Parallel Algorithms For Efficient Problem Solving., T. Krishnan Dr.

Doctoral Theses

There is nothing more difficult to take in hand, more perilous to conduct, or more uncertain in its success, than to take the lead in the introduction of a new order of things.

On The Geometrisability Of Some Strongly Regular Graphs Related To Polar Spaces., Pratima Panigraphi Dr.

Doctoral Theses

GraphsA graph G = (VE) consists of a finite set V and a subset E of ). (Here () denotes the set of all 2-subsets of V.) Elements of Vare called the vertices and the elements of E are called the edges of the graph. So Vis the vertex set and E is the edge set of the graph G. Two vertices a, y are said to be adjacent if the pair {a, y} is an edge; otherwise they are non-adjacent. If two vertices are adjacent then each is called a neighbour of the other vertex.Sometimes the edges of a …

Proper And Unit Bitolerance Orders And Graphs, Kenneth P. Bogart, Garth Isaak

Dartmouth Scholarship

We say any order ≺ is a tolerance order on a set of vertices if we may assign to each vertex x an interval I_x of real numbers and a real number t_x called a tolerance in such a way that x ≺ y if and only if the overlap of I_x and I_y is less than the minimum of t_x and t_y and the center of I_x is less than the center of I_y. An order is a bitolerance order if and only if there are intervals I_x and …

The Set Of Hemispheres Containing A Closed Curve On The Sphere, Mary Kate Boggiano, Mark Desantis

Department of Math & Statistics Technical Report Series

Suppose you get in your car and take a drive on the sphere of radius R, so that when you return to your starting point the odometer indicates you've traveled less than 2πR. Does your path, γ, have to lie in some hemisphere?

This question was presented to us by Dr. Robert Foote of Wabash College. Previous authors chose two points, A and B, on γ such that these points divided γ into two arcs of equal length. Then they took the midpoint of the great circle arc joining A and B to be the North Pole and showed that …

Recognizing Constant Curvature Discrete Groups In Dimension 3, J. W. Cannon, E. L. Swenson

Faculty Publications

We characterize those discrete groups Gwhich can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space H^3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the 2-sphere.

Constructions Of Random Distributions Via Sequential Barycenters, Theodore P. Hill, Michael Monticino

Research Scholars in Residence

This article introduces and develops a constructive method for generating random probability measures with a prescribed mean or distribution of the means. The method involves sequentially generating an array of barycenters which uniquely defines a probability measure. Basic properties of the generated measures are presented, including conditions under which almost all the generated measures are continuous or almost all are purely discrete or almost all have finite support. Applications are given to models for average-optimal control problems and to experimental approximation of universal constants.

Positive Knots And Robinson's Attractor, Michael C. Sullivan

Articles and Preprints

We study knotted periodic orbits which are realized in an attractor of a certain ODE. These knots can be presented so as to have all positive crossings, but may not be restricted to positive braids.

Murnaghan-Nakayama Rules For Characters Of Iwahori-Hecke Algebras Of The Complex Reflection Groups G(R,P,N), Thomas Halverson, A. Ram

Thomas M. Halverson

No abstract provided.

The Approximate Functional Formula For The Theta Function And Diophantine Gauss Sums, Evangelos A. Coutsias, N.D. Kazarinoff

Branch Mathematics and Statistics Faculty and Staff Publications

By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small w, to Cornu spirals (C-spirals), we prove the precise renormalization formula. This formula, which sharpens Hardy and Littlewood's approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and …

New Families Of Semi-Regular Relative Difference Sets, James A. Davis, Jonathan Jedwab, Miranda Mowbray

Department of Math & Statistics Faculty Publications

We give two constructions for semi-regular relative difference sets (RDSs) in groups whose order is not a prime power, where the order u of the forbidden subgroup is greater than 2. No such RDSs were previously known. We use examples from the first construction to produce semi-regular RDSs in groups whose order can contain more than two distinct prime factors. For u greater than 2 these are the first such RDSs, and for u = 2 we obtain new examples.

Some Harmonic N-Slit Mappings, Michael Dorff

Faculty Publications

The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, ∆, such that f = h+g where h(z) = (see PDF), g(z) = (see PDF) . SOH will denote the subclass with b1 = 0. We present a collection of n-slit mappings (n ≥ 2) and prove that the 2-slit mappings are in SH while for n ≥ 3 the mappings are in SOH. Finally we show that these mappings establish the sharpness of a previous theorem by Clunie and Sheil-Small while disproving a conjecture about the inner mapping radius.

Kempe Revisited, Joan Hutchinson, Stan Wagon

Stan Wagon, Retired

No abstract provided.

On Lipschitzian, And Connected Matrices In: Linear Complementarity Problem., Sriparna Bandyopadhyay Dr.

Doctoral Theses

This dissertation deals with a number of questions related to the linear complementarity problem (LCP). Given A ∈ Rn*n and q ∈ Rnthe LCP is to find a vector z ∈ R" such that Az+q ≥0,≥ and 2'(Az + 9) = 0. There is a vast literature on LCP developed during the last four decades. LCP plays a crucial role in the study of Mathematical Progranming from the point of view of algorithms as well as applications. The questions on existence and multiplicity of solutions in LCP has led researchers to introduce and study a variety of matrix classes. Most …

An Embedding Of Schwartz Distributions In The Algebra Of Asymptotic Functions, Michael Oberguggenberger, Todor D. Todorov

Mathematics

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper "asymptotic function," similar to but different from J. F Colombeau's algebras of new generalized functions.

Spatial Estimates For Stochastic Flows In Euclidean Space, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow

Articles and Preprints

We study the behavior for large |x| of Kunita-type stochastic flows φ(t, ω, x) on R^d, driven by continuous spatial semimartingales. For this class of flows we prove new spatial estimates for large |x|, under very mild regularity conditions on the driving semimartingale random field. It is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering, stochastic functional and partial differential equations. Some examples and counterexamples are given.

Lee Weights Of Z/4z-Codes From Elliptic Curves, José Felipe Voloch, Judy L. Walker

Department of Mathematics: Faculty Publications

In [15: J. L. Walker, Algebraic geometric codes over rings], the second author defined algebraic geometric codes over rings. This definition was motivated by two recent trends in coding theory: the study of algebraic geometric codes over finite fields, and the study of codes over rings. In that paper, many of the basic parameters of these new codes were computed. However, the Lee weight, which is very important for codes over the ring Z/4Z, was not considered. In [14: J.-F. Voloch and J. L. Walker, Euclidean weights of codes from elliptic curves over rings], this …

Relationships Among The First Variation, The Convolution Product, And The Fourier-Feynman Transform, Chull Park, David Skough, David Storvick

Department of Mathematics: Faculty Publications

In this paper we examine the various relationships that exist among the first variation, the Fourier- Feynman transform, and the convolution product for functionals on Wiener space that belong to a Banach algebra S.

Error Estimation Of The Padé Approximation Of Transfer Functions Via The Lanczos Process, Zhaojun Bai, Qiang Ye

Mathematics Faculty Publications

Krylov subspace based moment matching algorithms, such as PVL (Padé approximation Via the Lanczos process), have emerged as popular tools for efficient analyses of the impulse response in a large linear circuit. In this work, a new derivation of the PVL algorithm is presented from the matrix point of view. This approach simplifies the mathematical theory and derivation of the algorithm. Moreover, an explicit formulation of the approximation error of the PVL algorithm is given. With this error expression, one may implement the PVL algorithm that adaptively determines the number of Lanczos steps required to satisfy a prescribed error tolerance. …

On Subwavelet Sets., Eugen J. Ionascu

Faculty Bibliography

Abstract. In this note we give a characterization of subwavelet sets and show that any point x ∈ R\0 has a neighborhood which is contained in a regularized wavelet set.