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A Completely Integrable System And Parametric Representation Of Solutions Of The Wadati-Konno-Ichikawa Hierarchy, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

A finite-dimensional involutive system is presented, and the Wadati-Konno- Ichikawa (WKI) hierarchy of nonlinear evolution equations and their commutator representations are discussed in this article. By this finite-dimensional involutive system, it is proven that under the so-called Bargmann constraint between the potentials and the eigenfunctions, the eigenvalue problem (called the WKI eigenvalue problem) studied by Wadati, Konno, and Ichikawa [J. Phys. Sot. Jpn. 47, 1698 (1979)] is nonlinearized as a completely integrable Hamiltonian system in the Liouville sense. Moreover, the parametric representation of the solution of each equation in the WKI hierarchy is obtained by making use of the solution …

Periodicity And Indecomposability, William Thomas Ingram

Mathematics and Statistics Faculty Research & Creative Works

In this paper we characterize the existence of periodic points of odd period greater than one for unimodal mappings of an interval onto itself. The interesting juxtaposition of this condition with the occurrence in inverse limits of the well-known Brouwer-Janiszewski-Knaster continuum is explored. Also obtained is a characterization of indecomposability of certain inverse limits using a single unimodal bonding map. © 1995 American Mathematical Society.

Introduction To Linear Algebra: Models, Methods, And Theory, Alan Tucker

Department of Applied Mathematics & Statistics Faculty Books

This book develops linear algebra around matrices. Vector spaces in the abstract are not considered, only vector spaces associated with matrices. This book puts problem solving and an intuitive treatment of theory first, with a proof-oriented approach intended to come in a second course, the same way that calculus is taught. The book's organization is straightforward: Chapter 1 has introductory linear models; Chapter 2 has the basics of matrix algebra; Chapter 3 develops different ways to solve a system of equations; Chapter 4 has applications, and Chapter 5 has vector-space theory associated with matrices and related topics such as pseudoinverses …

The Genus 22 Crossing Number Of K9, Adrian Riskin

Mathematics

Our main result is that a 1971 conjecture due to Paul Kainen is false. Kainen's conjecture implies that the genus 2 crossing number of K 9 is 3. We disprove the conjecture by showing that the actual value is 4. The method used is a new one in the study of crossing numbers, involving proof of the impossibility of certain genus 2 embeddings of Ks.

Computations For A Vibrating System Diagonalize The Variance, J. N. Boyd, P. N. Raychowdhury

Mathematics and Applied Mathematics Publications

The transformations to diagonalize potential energy matrices for coupled harmonic oscillators will also diagonalize the variance when written in matrix form. After a brief review of a geometrical interpretation of the variance, the transformations are described and an example is given.

Conjugacy Classes Of Triple Products In Finite Groups, Kevin Hutson, Emily Salvo

Mathematical Sciences Technical Reports (MSTR)

For an underlying finite group G, we establish estimates on the number of triples that bind a certain set to one conjugacy class, or else breaks it into two conjugacy classes.

Point-Valued Mappings Of Sets, Matt Insall

Mathematics and Statistics Faculty Research & Creative Works

Let X be a metric space and let CB(X) denote the closed bounded subsets of X with the Hausdorff metric. Given a complete subspace Y of CB(X), two fixed point theorems, analogues of results in [1], are proved, and examples are given to suggest their applicability in practice.

Brain State In A Convex Body, S. Hui, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

We study a generalization of the brain-state-in-a-box (BSB) model for a class of nonlinear discrete dynamical systems where we allow the states of the system to lie in an arbitrary convex body. The states of the classical BSB model are restricted to lie in a hypercube. Characterizations of equilibrium points of the system are given using the support function of a convex body. Also, sufficient conditions for a point to be a stable equilibrium point are investigated. Finally, we study the system in polytopes. The results in this special case are more precise and have simpler forms than the corresponding …

Asymptotic Analysis Of The Linearized Navier-Stokes Equations In A Channel, Roger Temam, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

In this article we study and derive explicit formulas for the boundary layers occurring in the linearized channel flows in the limit of small viscosity. Our study is based on classical boundary layer techniques combined with a new global treatment of the pressure term. © 1995, Khayyam Publishing.

Stability Properties And Integrability Of The Resolvent Of Linear Volterra Equations, Muhammad Islam, Paul W. Eloe

Mathematics Faculty Publications

Integrability of the resolvent and the stability properties of the zero solution of linear Volterra integrodifferential systems are studied. In particular, it is shown that, the zero solution is uniformly stable if and only if the resolvent is integrable in some sense. It is also shown that, the zero solution is uniformly asymptotically stable if and only if the resolvent is integrable and an additional condition in terms of the resolvent and the kernel is satisfied. Finally, the integrability of the resolvent is obtained under an explicit condition.

An Exploration Of Doubly Transitive Designs In Affine Space, Michelle Persons

Honors Theses, 1963-2015

Through a combination of linear algebra, geometry, and algebraic structures, one can prove that certain families of geometric designs have groups of automorphisms that are doubly transitive. These geometric designs can be defined as edge colorings of complete graphs. Automorphisms of these geometric designs are permutations of the vertices of the graphs, which are also permutations of the colors of the edges such that given two edges of the same color, their images are also the same color. All cases considered are on the vector space Fn for a field F and the group of automorphisms is a subgroup of …

Analysis Of Rule Sets Generated By The Cn2, Id3, And Multiple Convergence Symbolic Learning Methods, Elizabeth M. Boll, Daniel C. St. Clair

Mathematics and Statistics Faculty Research & Creative Works

The ability to learn has long been an area of interest to researchers in artificial intelligence. Symbolic inductive learning algorithms have evolved as a class of algorithms that can be used to learn concepts from training examples. The knowledge acquired is represented in the form of rules. Since symbolic learning methods develop distinctive sets of rules when given identical training data, questions arise as to the quality of the different rule sets produced. The results of this research provide techniques for comparing and analyzing rule sets. Numerous rule sets were generated using three well-known symbolic learning methods; Quinlan's ID3, Clark …

Isometries In Three Space And The Seven Solid Crystal Systems, Iris Magee

Theses Digitization Project

No abstract provided.

Analysis On A Hyperplane Of The Quaternions, Pamela Jean Whelchel

Theses Digitization Project

No abstract provided.

Smooth Densities For Degenerate Stochastic Delay Equations With Hereditary Drift, Denis R. Bell, Salah-Eldin A. Mohammed

Articles and Preprints

We establish the existence of smooth densities for solutions of R^d-valued stochastic hereditary differential systems of the form

dx(t) = H(t,x)dt + g(t, x(t - r))dW(t).

In the above equation, W is an n-dimensional Wiener process, r is a positive time delay, H is a nonanticipating functional defined on the space of paths in R^d and g is an n x d matrix-valued function defined on [0, ∞) x R^d, such that gg^* has …

Analysis Of A Recurrence Arising From A Construction For Nonblocking Networks, Nicholas Pippenger

All HMC Faculty Publications and Research

Define f on the integers n > 1 by the recurrence f(n) = min( n, min_m|n( 2f(m) + 3f(n/m) ). The function f has f(n) = n as its upper envelope, attained for all prime n.

The goal of this paper is to determine the corresponding lower envelope. It is shown that this has the form f(n) ~ C(log n)^{1 + 1/γ} for certain constants γ and C, in the sense that for any ε > 0, the inequality f(n) ≤ (C + ε)(log n)^{1 + 1/γ} holds for infinitely many n, while f(n) ≤ (C + ε)(log …

Analysis Of A Recurrence Arising From A Construction For Non-Blocking Networks, Nicholas Pippenger

All HMC Faculty Publications and Research

Define f on the integers $n > 1$ by the recurrence $f( n ) = \min \{ n,\min _{m|n} 2f( m ) + 3f( n/m ) \}$. The function f has $f( n ) = n$ as its upper envelope, attained for all prime n. The goal of this paper is to determine the corresponding lower envelope. It is shown that this has the form $f( n ) \sim C( \log n )^{1 + 1/\gamma } $ for certain constants $\gamma $ and C, in the sense that for any $\varepsilon > 0$, the inequality $f( n ) \leq ( …

Describing And Distinguishing Knots, Lisa A. Padgett

Theses Digitization Project

No abstract provided.

Trace And Eigenvalue Inequalities Of Ordinary And Hadamard Products For Positive Semidefinite Hermitian Matrices, Bo-Ying Wang, Fuzhen Zhang

Mathematics Faculty Articles

Let Aand Bbe $n \times n$ positive semidefinite Hermitian matrices, let $\alpha $ and $\beta $ be real numbers, let $ \circ $ denote the Hadamard product of matrices, and let $A_k $ denote any $k \times k$ principal submatrix of A. The following trace and eigenvalue inequalities are shown: \[ \operatorname{tr}(A \circ B)^\alpha \leq \operatorname{tr}(A^\alpha \circ B^\alpha ),\quad\alpha \leq 0\,{\text{ or }}\,\alpha \geq 1, \]\[ \operatorname{tr}(A \circ B)^\alpha \geq \operatorname{tr}(A^\alpha \circ B^\alpha ),\quad 0 \leq \alpha \leq 1, \]\[ \lambda^{1/ \alpha } (A^\alpha \circ B^\alpha ) \leq \lambda ^{1/\beta } (A^\beta \circ B^\beta ),\quad\alpha \leq \beta ,\alpha \beta \ne …

Mathematical And Theological Beliefs: A Cognitive Science Perspective, Ron Benbow

ACMS Journal 2004

In recent years, research studies have shown that control decisions and processes, beliefs about the nature of mathematics, attitudes, and other affective variables have enormous impact on the mathematical performance of students. This paper gives an overview of the research on mathematical beliefs and reviews some work done in Christian education relating to theological beliefs. It then compares the two.

Mathematics From The Viewpoint Of Science In Context, Johan Deklerk

ACMS Journal 2004

This is the first of a series of papers presented over several years by deKlerk exploring the notion of how mathematics might be taught from a Christian perspective. In this paper, he introduces the notion of context as the basis for such an approach. He then discusses seven contexts a teacher can employ.

New Mathematical Properties Of The Banzhaf Value, Irinel C. Dragan

Mathematics Technical Papers - Archive

In a paper by P. Dubey and L.S. Shapley an axiomatic definition of the Banzhaf value has been extracted from an axiomatic definition of the Banzhaf power index (see [6]). Briefly speaking, the Banzhaf value axioms can be obtained from the Shapley value axioms by replacing efficiency by a similar axiom. This fact suggest that other recently discovered properties of the Shapley value can lead to similar properties for the Banzhaf value and this is the motivation for the present work.

Optimal Klappenspiel, Arthur T. Benjamin, Derek Stanford '93

All HMC Faculty Publications and Research

The game Klappenspiel ("flipping game") is a traditional German game of flipping tiles according to dice rolls. In this paper, we derive the optimal strategy for this game by using dynamic programming. We show that the probability of winning using the optimal strategy is 0.30%.

Heron's Problem In The Minkowski Plane, Mostafa Ghandehari

Mathematics Technical Papers - Archive

Heron's problem in the Euclidean plane is that of minimizing the length of a path that joins two points on one side of a line and intersects the line. A generalization in the Minkowski plane is given and used to discuss reflection properties of conics.

A Semiclassical, Dynamical Model Of The Water Molecule, Donald Greenspan

Mathematics Technical Papers - Archive

A model of the ground state water molecule is formulated dynamically and studied by computer simulations. Excellent approximations of NMR determined bond angles and bond lengths are achieved through electron pairing and simulation of ice I.

A New Method For Approximating Quantum Dynamical Phenomena Over Short (And Sometimes Long) Time Periods, Donald Greenspan

Mathematics Technical Papers - Archive

Approximate electron and proton motions are simulated for short time periods for using a special case of the quantum mechanical Ehrenfest equations. Attention is directed only to ground state. A numerical method is employed which conserves the ground state energy at all times. Large proton motions are found, with electron trajectories oscillating between the protons, but concentrating around them.

Duality On Noncompact Manifolds And Complements Of Topological Knots, Gerard A. Venema

University Faculty Publications and Creative Works

Let be the image of a topological embedding of into. In this paper the homotopy groups of the complement. are studied. In contrast with the situation in the smooth and piecewise-linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through n - 2. If the first nonstandard homotopy group of the complement occurs above the middle dimension, then the end of must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of if both …

Computer Modelling Of A Dynamical Water Molecule, Donald Greenspan

Mathematics Technical Papers - Archive

A model of a ground state water molecule is formulated dynamically and studied by computer simulations. The enigmatic 104.5° bond angle is approximated by introducing the effects of neighboring molecules in a crystaline state. The numerical methodology used conserves the system energy at all times. Illustrative examples are described and discussed.

A Sturm Separation Theorem For A Linear 2nth Order Self-Adjoint Differential Equation, Charles T. Fulton, Wu Limin, Steven Pruess

Mathematics and System Engineering Faculty Publications

For the 2nth order equation, (−1)nv(2n)+qv=0, with q continuous, we obtain a Sturm Separation theorem, involving n+1 solutions of the equation, which is somewhat analogous to the classical result that the zeros of two linearly independent solutions of the second order equation separate each other.

Research Announcement: Recursive Construction For Families Of Difference Sets, James A. Davis, Jonathan Jedwab

Department of Math & Statistics Faculty Publications

A (v, k, λ) difference set is a k-element subset D of a group G of order v for which the multiset {d₁d₂^-1 : d1, d_2, ∈ D} contains each nonzero element of G exactly λ times; n = k-λ.