Physical Sciences and Mathematics Commons^™

Construction Of Homomorphic Images, Stephanie Ann Hilber

Theses Digitization Project

This thesis constructs several finite homomorphic images of infinite semi-direct products of the form 2*n:N.

The Universal Coefficient Theorem For Cohomology, Michael Anthony Rosas

Theses Digitization Project

This project is an expository survey of the Universal Coefficient Theorem for Cohomology. Algebraic preliminaries, homology, and cohomology are discussed prior to the proof of the theorem.

Mathematical Model Creation For Cancer Chemo-Immunotherapy, Lisette G. De Pillis, K Renee Fister, Weiqing Gu, Craig Collins, Michael Daub, David Gross '08, James Moore '07, Benjamin Preskill '09

All HMC Faculty Publications and Research

One of the most challenging tasks in constructing a mathematical model of cancer treatment is the calculation of biological parameters from empirical data. This task becomes increasingly difficult if a model involves several cell populations and treatment modalities. A sophisticated model constructed by de Pillis et al., Mixed immunotherapy and chemotherapy of tumours: Modelling, applications and biological interpretations, J. Theor. Biol. 238 (2006), pp. 841–862; involves tumour cells, specific and non-specific immune cells (natural killer (NK) cells, CD8 T cells and other lymphocytes) and employs chemotherapy and two types of immunotherapy (IL-2 supplementation and CD8 T-cell infusion) as treatment modalities. …

Phase Transitions In Materials With Thermal Memory: The Case Of Unequal Conductivities, John Murrough Golden

Articles

A model for thermally induced phase transitions in materials with thermal memory was recently proposed, where the equations determining heatflow were assumed to be the same in both phases. In this work, the model is generalized to the case of phase dependent heatflow relations. The temperature (or coldness) gradient is decomposed into two parts, each zero on one phase and equal to the temperature (or coldness) gradient on the other. However, they vary smoothly over the transition zone. These are treated as separate independent quantities in the derivation of field equations from thermodynamics. Heat flux is given by an integral …

1-D Schrödinger Operators With Local Point Interactions On A Discrete Set, Aleksey Kostenko, Mark M. Malamud

Articles

Spectral properties of 1-D Schrödinger operators HX,α := − d2 dx2 + xn∈X αnδ(x − xn) with local point interactions on a discrete set X = {xn}∞ n=1 are well studied when d∗ := infn,k∈N |xn − xk| > 0. Our paper is devoted to the case d∗ = 0. We consider HX,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions.

On The Sensitivity To Noise Of A Boolean Function, Mihaela Teodora Matache, Valentin Matache

Mathematics Faculty Publications

In this paper we generate upper and lower bounds for the sensitivity to noise of a Boolean function using relaxed assumptions on input choices and noise. The robustness of a Boolean network to noisy inputs is related to the average sensitivity of that function. The average sensitivity measures how sensitive to changes in the inputs the output of the function is. The average sensitivity of Boolean functions can indicate whether a specific random Boolean network constructed from those functions is ordered, chaotic, or in critical phase. We give an exact formula relating the sensitivity to noise and the average sensitivity …

The Existence Of Triple Positive Solutions Of Nonlinear Four-Point Boundary Value Problem With P-Laplacian, Xiang-Feng Li, Pei-Hao Zhao

Turkish Journal of Mathematics

This paper deals with the multiplicity results of positive solutions of one-dimensional singular p-Laplace equation (\varphi_p(u'(t)))'+a(t)f(t,u(t),u'(t))=0, 0

A Globally Convergent Numerical Method For Coefficient Inverse Problems, Natee Pantong

Mathematics Dissertations

In our terminology "globally convergent numerical method" means a numerical method, whose convergence to a good approximation for the correct solution is independent of the initial approximation. A new numerical imaging algorithm of reconstruction of optical absorption coefficients from near infrared light data with a continuous-wave has been purposed to solves a coefficient inverse problem for an elliptic equation with the data generated by the source running along a straight line. A regularization process, so-called "exterior forward problem", for preprocessing data with noise on the boundary has also been purpose for the problem related to matching fluid in experiment. A …

Studies In Statistical Inference, Sampling Techniques And Demography, Florentin Smarandache, Rajesh Singh, Jayant Singh

Branch Mathematics and Statistics Faculty and Staff Publications

This volume is a collection of five papers. Two chapters deal with problems in statistical inference, two with inferences in finite population, and one deals with demographic problem. The ideas included here will be useful for researchers doing works in these fields. The following problems have been discussed in the book: Chapter 1. In this chapter optimum statistical test procedure is discussed. The test procedures are optimum in the sense that they minimize the sum of the two error probabilities as compared to any other test. Several examples are included to illustrate the theory. Chapter 2. In testing of hypothesis …

Groups As Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Through this book, for the first time we represent every finite group in the form of a graph. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group. This study is innovative because through this description one can immediately look at the graph and say the number of elements in the group G which are self-inversed. Also study of different properties like the subgroups of a group, normal subgroups of a group, p-sylow subgroups of a group and conjugate elements of a group …

Investigation Of A Neutrosophic Group, A. Elrawy, Florentin Smarandache, Ayat A. Temraz

Branch Mathematics and Statistics Faculty and Staff Publications

We use a neutrosophic set, instead of an intuitionistic fuzzy because the neutrosophic set is more general, and it allows for independent and partial independent components, while in an intuitionistic fuzzy set, all components are totally dependent. In this article, we present and demonstrate the concept of neutrosophic invariant subgroups. We delve into the exploration of this notion to establish and study the neutrosophic quotient group. Further, we give the concept of a neutrosophic normal subgroup as a novel concept.

Topology And Infinite Graphs, Nicholas Blackburn Lowery

Honors Papers

The main focus of this paper will be on two very different areas in which topology is relevant to the study of infinite graphs. The first is the mechanics of compactness proofs, which use a particular group of lemmas to extend results about finite subgraphs to apply to an entire infinite graph. We will explore these results by using them to prove a result of de Bruijn and Erdos, that an infinite graph is k-colorable if its finite subgraphs are k-colorable, in several different ways.

The second area is a relatively new area of study pioneered by Diestel which redefines …

Integral Orthogonal Bases Of Small Height For Real Polynomial Spaces, Lenny Fukshansky

CMC Faculty Publications and Research

Let P_N(R) be the space of all real polynomials in N variables with the usual inner product < , > on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form representing this inner product on the space of coefficient vectors of all polynomials in P_N(R) of degree ≤ M. We exhibit two applications of this formula. First, given a finite dimensional subspace V of P_N(R) defined over Q, we prove the existence of an orthogonal basis for (V, < , >), consisting of polynomials of small height …

Search Bounds For Zeros Of Polynomials Over The Algebraic Closure Of Q, Lenny Fukshansky

CMC Faculty Publications and Research

We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of unions of subspaces. All bounds on the height are explicit.

The Classification Of Simple Lie Algebras In Maple, D. Russell Sadler

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Lie algebras are invaluable tools in mathematics and physics as they enable us to study certain geometric objects such as Lie groups and differentiable manifolds. The computer algebra system Maple has several tools in its Lie Algebras package to work with Lie algebras and Lie groups. The purpose of this paper is to supplement the existing software with tools that are essential for the classification of simple Lie algebras over C.

In particular, we use a method to find a Cartan subalgebra of a Lie algebra in polynomial time. From the Cartan subalgebra we can compute the corresponding root system. …

The Generalized Burnside And Representation Rings, Eric B. Kahn

University of Kentucky Doctoral Dissertations

Making use of linear and homological algebra techniques we study the linearization map between the generalized Burnside and rational representation rings of a group G. For groups G and H, the generalized Burnside ring is the Grothendieck construction of the semiring of G × H-sets with a free H-action. The generalized representation ring is the Grothendieck construction of the semiring of rational G×H-modules that are free as rational H-modules. The canonical map between these two rings mapping the isomorphism class of a G-set X to the class of its permutation module …

Aspects Of The Geometry Of Metrical Connections, Matthew J. Wells

University of Kentucky Doctoral Dissertations

Differential geometry is about space (a manifold) and a geometric structure on that space. In Riemann’s lecture (see [17]), he stated that “Thus arises the problem, to discover the matters of fact from which the measure-relations of space may be determined...”. It is key then to understand how manifolds differ from one another geometrically. The results of this dissertation concern how the geometry of a manifold changes when we alter metrical connections. We investigate how diverse geodesics are in different metrical connections. From this, we investigate a new class of metrical connections which are dependent on the class of smooth …

Q-Algebroids And Their Cohomology, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

A Q-algebroid is a graded Lie algebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a Q-groupoid. We associate to every Q-algebroid a double complex. As a special case, we define the Becchi-Rouet-Stora-Tyutin (BRST) model of a Lie algebroid, which generalizes the BRST model for equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman isomorphism of the BRST and Weil models, and we suggest a definition of a basic subcomplex which, however, requires a choice of a connection. Other examples include Roytenberg's homological double of a Lie bialgebroid, Ginzburg's model of equivariant Lie algebroid …

There Are Only Four, Danielle Josette Mccoy

Theses Digitization Project

This paper is an investigation of finite-dimensional normed algebras over the reals from both an abstract and concrete point of view.

The Geometry Of The Space Of Oriented Geodesics Of Hyperbolic 3-Space, Nikos Georgiou

Theses

In this thesis we construct a Kähler structure (J, Ω, G) on the space L(H³) of oriented geodesics of hyperbolic 3-space H³ and investigate its properties. We prove that (L(H³),J) is biholomorphic to (see thesis pdf), and that the Kähler metric G is of neutral signature, conformally flat and scalar flat.

We establish that the identity component of the isometry group of the metric G on L(H³) is isomorphic to the identity component of the hyperbolic isometry group. We show that the geodesics of G correspond to ruled minimal surfaces in H3, which …

Connections Between Computation Trees And Graph Covers, Deanna Dreher, Judy L. Walker

Department of Mathematics: Faculty Publications

Connections between graph cover pseudocodewords and computation tree pseudocodewords are investigated with the aim of bridging the gap between the theoretically attractive analysis of graph covers and the more intractable analysis of iterative message-passing algorithms that are intuitively linked to graph covers. Both theoretical results and numerous examples are presented.

On The Dynamics Of Quasi-Self-Matings Of Generalized Starlike Complex Quadratics And The Structure Of The Mated Julia Sets, Ross Flek

Dissertations, Theses, and Capstone Projects

It has been shown that, in many cases, Julia sets of complex polynomials can be "glued" together to obtain a new Julia set homeomorphic to a Julia set of a rational map; the dynamics of the two polynomials are reflected in the dynamics of the mated rational map. Here, I investigate the Julia sets of self-matings of generalized starlike quadratic polynomials, which enjoy relatively simple combinatorics. The points in the Julia sets of the mated rational maps are completely classified according to their topology. The presence and location of buried points in these Julia sets are addressed. The interconnections between …

Standing Waves Of Spatially Discrete Fitzhugh-Nagumo Equations, Joseph Segal

Electronic Theses and Dissertations

We study a system of spatially discrete FitzHugh-Nagumo equations, which are nonlinear differential-difference equations on an infinite one-dimensional lattice. These equations are used as a model of impulse propagation in nerve cells. We employ McKean's caricature of the cubic as our nonlinearity, which allows us to reduce the nonlinear problem into a linear inhomogeneous problem. We find exact solutions for standing waves, which are steady states of the system. We derive formulas for all 1-pulse solutions. We determine the range of parameter values that allow for the existence of standing waves. We use numerical methods to demonstrate the stability of …

Positive Solutions For The (N, P) Boundary Value Problem, Bo Yang

Faculty Articles

We consider the (n, p) boundary value problem in this paper. Some new upper estimates to positive solutions for the problem are obtained. Existence and nonexistence results for positive solutions of the problem are obtained by using the Krasnosel'skii fixed point theorem. An example is included to illustrate the results.

Rees Products Of Posets And Inequalities, Tricia Muldoon Brown

University of Kentucky Doctoral Dissertations

In this dissertation we will look at properties of two different posets from different perspectives. The first poset is the Rees product of the face lattice of the n-cube with the chain. Specifically we study the Möbius function of this poset. Our proof techniques include straightforward enumeration and a bijection between a set of labeled augmented skew diagrams and barred signed permutations which label the maximal chains of this poset. Because the Rees product of this poset is Cohen-Macaulay, we find a basis for the top homology group and a representation of the top homology group over the symmetric …

Iterative Methods For Computing Eigenvalues And Exponentials Of Large Matrices, Ping Zhang

University of Kentucky Doctoral Dissertations

In this dissertation, we study iterative methods for computing eigenvalues and exponentials of large matrices. These types of computational problems arise in a large number of applications, including mathematical models in economics, physical and biological processes. Although numerical methods for computing eigenvalues and matrix exponentials have been well studied in the literature, there is a lack of analysis in inexact iterative methods for eigenvalue computation and certain variants of the Krylov subspace methods for approximating the matrix exponentials. In this work, we proposed an inexact inverse subspace iteration method that generalizes the inexact inverse iteration for computing multiple and clustered …

Direct Products And The Intersection Map Of Certain Classes Of Finite Groups, Julia Chifman

University of Kentucky Doctoral Dissertations

The main goal of this work is to examine classes of finite groups in which normality, permutability and Sylow-permutability are transitive relations. These classes of groups are called T , PT and PST , respectively. The main focus is on direct products of T , PT and PST groups and the behavior of a collection of cyclic normal, permutable and Sylow-permutable subgroups under the intersection map. In general, a direct product of finitely many groups from one of these classes does not belong to the same class, unless the orders of the direct factors are relatively prime. Examples suggest that …

Series That Probably Converge To One, Thomas J. Pfaff, Max Tran

Publications and Research

No abstract provided.

Semi-Automated Frame Transformations Using Fft Analysis On 2-D Images, Francisco Javier Osuna

Open Access Theses & Dissertations

Cassini entered Saturn's orbit on July 1, 2004 beginning a four-year exploration of Saturn. In 2008 the mission was extended, and Cassini continues to collect and transmit images and data collected during its mission. In order to accurately interpret images, it is necessary to know the location and orientation of the camera provided the field of view when the image was collected. While the mission managers provide initial estimates of this orientation, scientific analysis requires better estimates than the initial data provided. Navigation is a process for improving the estimation of the true camera pointing vector as determined by features …

Equational Coalgebraic Logic, Alexander Kurz, Raul Leal

Engineering Faculty Articles and Research

Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible one-step behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for T-coalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing one-step translations between them, extending the results in [21] by making systematic use of Stone duality. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatization of Moss’s logic. The three logics are …