Physical Sciences and Mathematics Commons^™

Poincaré Duality, Christopher Michael Duran

Theses Digitization Project

This project is an expository study of the Poincaré duality theorem. Homology, cohomology groups of manifolds and other aglebraic and topological preliminaires are discussed.

Quantum Phases For A Generalized Harmonic Oscillator, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

An effective Hamiltonian for the generalized harmonic oscillator is determined by using squeezed state wavefunctions. The equations of motion over an extended phase space are determined and then solved perturbatively for a specific choice of the oscillator parameters. These results are used to calculate the dynamic and geometric phases for the generalized oscillator with this choice of parameters.

An Action For A Classical String, The Equation Of Motion And Group Invariant Classical Solutions, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

A string action which is essentially a Willmore functional is presented and studied. This action determines the physics of a surface in Euclidean three space which can be used to model classical string configurations. By varying this action an equation of motion for the mean curvature of the surface is obtained which is shown to govern certain classical string configurations. Several classes of classical solutions for this equation are discussed from the symmetry group point of view and an application is presented.

A Fragment On Euler's Constant In Ramanujan's Lost Notebook, Bruce C. Berndt, Timothy Huber

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

A formula for Euler’s constant found in Ramanujan’s lost notebook and also in a problem he submitted to the Journal of the Indian Mathematical Society is proved and discussed.

Adaptive Hp-Fem For Elliptic Problems In 3d On Irregular Meshes, David Andrs

Open Access Theses & Dissertations

This work deals with adaptive hp-FEM on irregular meshes. It shows the advantage of completely irregular meshes, using the arbitrary level hanging nodes. The above is demonstrated on a numerical example with distribution of electrical potencial.

Mathematics Of Voting, Darryl H. Yong

All HMC Faculty Publications and Research

Voting theory is a fascinating area of research involving mathematics, political scientists, and economists. The American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics chose mathematics and voting as the theme for Mathematics Awareness Month 2008. There is more information on mathematics and voting at www.mathaware.org/mam/08/. It is a mathematical topic that is rich yet accessible to students, pertinent to their lives, especially during this election year, and has the potential to draw students who may not have a strong affinity for mathematics to become interested in mathematics.

A Study Of Boundary Layer Fflow With No-Slip And Slip Boundary Conditions, Sandra B. Spillane

Doctoral

This thesis involves solving the two-dimensional boundary layer equations for axially symmetric fluid flow along a circular cylinder using the no-slip and slip boundary condition, and along a flat plate with the slip boundary condition. Initially, historical results in both areas are summarised. The research section of this thesis is concerned with extending these historical results. In the first research chapters, a Pade approximation and an Euler transformation are used to greatly extend the region of validity of the historical results. Following that is an investigation of the relaxation of the traditional no-slip boundary condition, which usually occurs when there …

Integrability Of A Singularly Perturbed Model Describing Gravity Water Waves On A Surface Of Finite Depth, Steven Little

Electronic Theses and Dissertations

Our work is closely connected with the problem of splitting of separatrices (breaking of homoclinic orbits) in a singularly perturbed model describing gravity water waves on a surface of finite depth. The singularly perturbed model is a family of singularly perturbed fourth-order nonlinear ordinary differential equations, parametrized by an external parameter (in addition to the small parameter of the perturbations). It is known that in general separatrices will not survive a singular perturbation. However, it was proven by Tovbis and Pelinovsky that there is a discrete set of exceptional values of the external parameter for which separatrices do survive the …

Padé Spline Functions, Tian-Xiao He

Scholarship

We present here the definition of Pad´e spline functions, their expressions, and the estimate of the remainders of pad´e spline expansions. Some algorithms are also given.

Super Fuzzy Matrices And Super Fuzzy Models For Social Scientists, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Amal

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of supermatrix for social scientists was first introduced by Paul Horst. The main purpose of his book was to introduce this concept to social scientists, students, teachers and research workers who lacked mathematical training. He wanted them to be equipped in a branch of mathematics that was increasingly valuable for the analysis of scientific data. This book introduces the concept of fuzzy super matrices and operations on them. The author has provided only those operations on fuzzy supermatrices that are essential for developing super fuzzy multi expert models. We do not indulge in labourious use of suffixes or …

N- Linear Algebra Of Type I And Its Applications, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

With the advent of computers one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure namely n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of nvector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several …

Methods In Industrial Biotechnology For Chemical Engineers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Industrial Biotechnology is an interdisciplinary topic to which tools of modern biotechnology are applied for finding proper proportion of raw mix of chemicals, determination of set points, finding the flow rates etc., This study is significant as it results in better economy, quality product and control of pollution. The authors in this book have given only methods of industrial biotechnology mainly to help researchers, students and chemical engineers. Since biotechnology concerns practical and diverse applications including production of new drugs, clearing up pollution etc. we have in this book given methods to control pollution in chemical industries as it has …

Chinese Neutrosophy And Taoist Natural Philosophy, Florentin Smarandache, Jiang Zhengjie

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Cultural Advantages In China: Tale Of Six Cities, Florentin Smarandache, Fu Yuhua, Victor Christianto

Branch Mathematics and Statistics Faculty and Staff Publications

Nowadays, plenty of factories from Europe and other developed countries have been relocated to this country, considering its tremendous economic scale and rapid growth rate during the past three decades. But most of what happens inside the China nowadays is deeply hidden from the outside world (“the foreigners” as China people would call). This fact is partly because most reports on China were written by the so‐called fly‐high experts who are busy completing their reports despite a busy schedule. Very few books or reports were written by people inside, or at least “foreigners” who spent a few years in China. …

The Art Of Teaching Mathematics, Garikai Campbell, Jon T. Jacobsen, Aimee S A Johnson, Michael E. Orrison Jr.

All HMC Faculty Publications and Research

On June 10–12, 2007, Harvey Mudd College hosted A Conference on the Art of Teaching Mathematics. The conference brought together approximately thirty mathematicians from the Claremont Colleges, Denison, DePauw, Furman, Middlebury, Penn State, Swarthmore, and Vassar to explore the topic of teaching as an art. Assuming there is an element of artistic creativity in teaching mathematics, in what ways does it surface and what should we be doing to develop this creativity?

Distribution Of The Number Of Encryptions In Revocation Schemes For Stateless Receivers, Christopher Eagle, Zhicheng Gao, Mohamed Omar, Daniel Panario, Bruce Richmond

All HMC Faculty Publications and Research

We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subset-difference scheme (SD). These are well-known tree based broadcast encryption schemes. Park and Blake in: Journal of Discrete Algorithms, vol. 4, 2006, pp. 215--238, give the mean number of encryptions for these schemes. We continue their analysis and show that the limiting distribution of the number of encryptions for these schemes is normal. This implies that the mean numbers of Park and Blake are good estimates for the number of necessary encryptions used by these schemes.

A Model For Rolling Swarms Of Locusts, Chad M. Topaz, Andrew J. Bernoff, Sheldon Logan '06, Wyatt Toolson '07

All HMC Faculty Publications and Research

We construct an individual-based kinematic model of rolling migratory locust swarms. The model incorporates social interactions, gravity, wind, and the effect of the impenetrable boundary formed by the ground. We study the model using numerical simulations and tools from statistical mechanics, namely the notion of H-stability. For a free-space swarm (no wind and gravity), as the number of locusts increases, the group approaches a crystalline lattice of fixed density if it is H-stable, and in contrast becomes ever denser if it is catastrophic. Numerical simulations suggest that whether or not a swarm rolls depends on the statistical mechanical properties of …

Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky

CMC Faculty Publications and Research

Let N >= 2 be an integer, F a quadratic form in N variables over (Q) over bar, and Z subset of (Q) over bar (N) an L-dimensional subspace, 1 <= L <= N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F). This provides an analogue over (Q) over bar of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over (Q) over bar. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over (Q) over bar. This extends previous results of the author over number fields. All bounds on height are explicit.

Students' Perceptions Of Sense Of Community In Abstract Algebra: Contributing Factors And Benefits, Hortensia Soto-Johnson, Nissa Yestness, Casey Dalton

Mathematical Sciences Faculty Publications

In this phenomenological study, we explore how multiple assessments contribute to creating a sense of community (SOC) in an undergraduate abstract algebra course. Strike (2004) describes community as a process rather than a feeling and outlines four characteristics of community: coherence, cohesion, care, and contact. In this report, we describe contributing factors to and perceived benefits of SOC that students provided in an open-ended interview. Our findings indicate students viewed the teacher and the classroom environment as the primary sources of creating a SOC. Our findings also suggest students believed the SOC of the classroom increased classroom interaction and opened …

Algebraic And Combinatorial Properties Of Certain Toric Ideals In Theory And Applications, Sonja Petrovic

University of Kentucky Doctoral Dissertations

This work focuses on commutative algebra, its combinatorial and computational aspects, and its interactions with statistics. The main objects of interest are projective varieties in Pⁿ, algebraic properties of their coordinate rings, and the combinatorial invariants, such as Hilbert series and Gröbner fans, of their defining ideals. Specifically, the ideals in this work are all toric ideals, and they come in three flavors: they are defining ideals of a family of classical varieties called rational normal scrolls, cut ideals that can be associated to a graph, and phylogenetic ideals arising in a new and increasingly popular area of …

The H-Vectors Of Matroids And The Arithmetic Degree Of Squarefree Strongly Stable Ideals, Erik Stokes

University of Kentucky Doctoral Dissertations

Making use of algebraic and combinatorial techniques, we study two topics: the arithmetic degree of squarefree strongly stable ideals and the h-vectors of matroid complexes.

For a squarefree monomial ideal, I, the arithmetic degree of I is the number of facets of the simplicial complex which has I as its Stanley-Reisner ideal. We consider the case when I is squarefree strongly stable, in which case we give an exact formula for the arithmetic degree in terms of the minimal generators of I as well as a lower bound resembling that from the Multiplicity Conjecture. Using this, we can …

Operations In Hilbert Space, Dennis Michael Gumaer

Theses Digitization Project

This thesis reviews some of the major topics in elementary Hilbert space theory. The theory of operators is developed by providing details regarding several types of operators, in particular compact operators. This study of compact operators is the start of the refinement of bounded linear operators to those which are also members of the Schatten p-class operators.

Studies In Free Module And It's Basis, Hsu-Chia Chen

Theses Digitization Project

The purpose of this project was to study some basic properties of free modules over a ring. A module with a basis is called a free module and a free module over a division ring (or field) is called a vector space. We show every vector has a basis and any two bases of a vector space have same cardinality. However, a free module over an arbitrary ring (with identity) does not have this property.

Symmetric Representation Of The Elements Of Finite Groups, Barbara Hope Gwinn-Edwards

Theses Digitization Project

The main purpose of this thesis is to construct finite groups as homomorphic images of infinite semi-direct products.

Valuations For Spike Train Prediction, Vladimir Itskov, Carina Curto, Kenneth D. Harris

Department of Mathematics: Faculty Publications

The ultimate product of an electrophysiology experiment is often a decision on which biological hypothesis or model best explains the observed data. We outline a paradigm designed for comparison of different models, which we refer to as spike train prediction. A key ingredient of this paradigm is a prediction quality valuation that estimates how close a predicted conditional intensity function is to an actual observed spike train. Although a valuation based on log likelihood (L) is most natural, it has various complications in this context. We propose that a quadratic valuation (Q) can be used as an alternative to L. …

Matrix Model Superpotentials And Ade Singularities, Carina Curto

Department of Mathematics: Faculty Publications

We use F. Ferrari’s methods relating matrix models to Calabi–Yau spaces in order to explain much of Intriligator and Wecht’s ADE classification of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. We find that ADE superpotentials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi–Yau with corresponding ADE singularities. Moreover, in the additional Ô, Â, Dˆ and Ê cases we find new singular geometries. These “hat” geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for …

Ldpc Codes From Voltage Graphs, Christine A. Kelley, Judy L. Walker

Department of Mathematics: Faculty Publications

Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of abelianinevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained.

C^1 Actions Of The Mapping Class Group On The Circle, Kamlesh Parwani

Faculty Research and Creative Activity

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.

The Two Variable Substitution Problem For Free Products Of Groups, Leo P. Comerford, Charles C. Edmunds

Faculty Research and Creative Activity

We consider equations of the form W(x,y) = U with U an element of a free product G of groups. We show that with suitable algorithmic conditions on the free factors of G, one can effectively determine whether or not the equations have solutions in G. We also show that under certain hypotheses on the free factors of G and the equation itself, the equation W(x,y) = U has only finitely many solutions, up to the action of the stabilizer of W(x,y) in Aut().

Inequalities And Exponential Approximations For Residual Life Reliability Functions, Broderick O. Oluyede, Marvis Pararai

Department of Mathematical Sciences Faculty Publications

Given that a unit is of age t, the remaining life after time t is random. The expected value of this random residual life is called the mean residual life at time t. Specifically, if T is the life of a component with distribution function F, then δF (t) = E(T −t|T > t) is called the mean residual life function (MRLF). It is well known that the class of distributions with decreasing mean residual life (DMR) contains the class of distributions with increasing hazard rate (IHR). In this note, …