Mathematics Commons^™

Equations Over Hyperbolic Groups, Alexander Kai-Chi Taam

Dissertations, Theses, and Capstone Projects

We show that the Diophantine problem, for quadratic equations over a non-elementary torsion-free hyperbolic group, is NP-complete. Furthermore, given a finite system of equations over a torsion-free hyperbolic group $\Gamma$, there is an algorithm which constructs a canonical $Hom$-diagram and a complete set of induced $\Gamma$-NTQ systems, for $\Gamma_{R(S)}$. Finally, the class of $\Gamma$-limit groups is the same as that of iterated generalized doubles over $\Gamma$.

On String Topology Operations And Algebraic Structures On Hochschild Complexes, Manuel Rivera

Dissertations, Theses, and Capstone Projects

The field of string topology is concerned with the algebraic structure of spaces of paths and loops on a manifold. It was born with Chas and Sullivan's observation of the fact that the intersection product on the homology of a smooth manifold $M$ can be combined with the concatenation product on the homology of the based loop space on $M$ to obtain a new product on the homology of $LM$, the space of free loops on $M$. Since then, a vast family of operations on the homology of $LM$ have been discovered.

In this thesis we focus our attention on …

The Geometry And Combinatorics Of Closed Geodesics On Hyperbolic Surfaces, Chris Arettines

Dissertations, Theses, and Capstone Projects

In this thesis, we obtain combinatorial algorithms that determine the minimal number of self-intersections necessary for a free homotopy class $[\gamma]$ on an orientable surface, using algebraic input. Using this same input, we describe another algorithm which determines whether or not a minimally intersecting curve in $[\gamma]$ is \textit{filling}, that is, whether or not the complement is a disjoint union of disks or punctured disks. Next, we use these algorithms as inspiration for proving the existence of filling curves which self-intersect $2g-1$ times, which is the minimal number of intersections possible. The combinatorial viewpoint that is developed can then be …

Counting Restricted Integer Partitions, David Dakota Blair

Dissertations, Theses, and Capstone Projects

The first chapter examines $p_b(n)$, the number of partitions of $n$ into powers of $b$, along with a family of identities which can be deduced by iterating a recurrence satisfied by $p_b(n)$ in a suitable way. These identities can then be used to calculate $p_b(n)$ for large values of $n$.

The second chapter restricts these types of partitions even further, limiting the multiplicity of each part. Its object of study is $p_{b,d}(n)$, that is, the number of partitions of $n$ into powers of $b$ repeating each power at most $d$ times. The methods of the first chapter are applied, and …

Algorithmic Properties Of Poly-Z Groups And Secret Sharing Using Non-Commutative Groups, Bren Cavallo

Dissertations, Theses, and Capstone Projects

Computational aspects of polycyclic groups have been used to study cryptography since 2004 when Eick and Kahrobaei proposed polycyclic groups as a platform for conjugacy based cryptographic protocols.

In the first chapter we study the conjugacy problem in polycyclic groups and construct a family of torsion-free polycyclic groups where the uniform conjugacy problem over the entire family is at least as hard as the subset sum problem. We further show that the conjugacy problem in these groups is in NP, implying that the uniform conjugacy problem is NP-complete over these groups. This is joint work with Delaram Kahrobaei. We also …

The Moduli Space Of Rational Maps, Lloyd William West

Dissertations, Theses, and Capstone Projects

We construct the moduli space, M_d, of degree d rational maps on ℙ¹ in terms of invariants of binary forms. We apply this construction to give explicit invariants and equations for M₃.

Using this construction, we give a method for solving the following problems: (1) explicitly construct, from a moduli point P ∈ M_d(k), a rational map ∅ with the given moduli; (2) find the field of definition of a rational map ∅. We work out the method in detail for the case d = 3.

Some Bernstein Type Results Of Graphical Self-Shrinkers With High Codimension In Euclidean Space, Hengyu Zhou

Dissertations, Theses, and Capstone Projects

A self-shrinker characterizes the type I singularity of the mean curvature flow. In this thesis we concern about some Bernstein type results of graphical self-shrinkers with high codimension in Euclidean space.

There are two main tools in our work. The first one is structure equations of graphical self-shriners in terms of parallel forms (Theorem 2.3.6). This is motivated by M.T.Wang's work ([Wan02]) on graphical mean curvature flows with arbitrary codimension in product manifolds. The second one is an integration technique (Lemma 2.4.5) based on the fact that every graphical graphical self-shrinker has the polynomial volume growth (Corollary 2.4.4). Because of …

Foliations With Geometric Structures: An Approach Through H-Principle., Sauvik Mukherjee Dr.

Doctoral Theses

A foliation on a manifold M can be informally thought of as a partition of M into injectively immersed submanifolds, called leaves. In this thesis we study foliations whose leaves carry some specific geometric structures.The thesis consists of two parts. In the first part we classify foliations on open manifolds whose leaves are either locally conformal symplectic or contact manifolds. These foliations can be described by some higher geometric structures - namely the Poisson and the Jacobi structures. In the second part of the thesis, we consider foliations on open contact manifolds whose leaves are contact submanifolds of the ambient …

Life As An Nfl Statistician, Dennis Lock

Mathematics Colloquium Series

Over the last few years, the fields of statistics and mathematics have become more prevalent and popular in professional sports (with the help of mainstream books and movies like Moneyball). The use of advanced (and non-advanced) statistical methods is growing across the sporting landscape from the front office to the media, and even into business and ticket sales. This talk will discuss Lock’s experiences building an analytics department with the Miami Dolphins as well as the general role of statistics in sports today. It will also including the recent analytics boom in the front office framework, the coinciding need for …

A Nonlinear Filter For Markov Chains And Its Effect On Diffusion Maps, Stefan Steinerberger

Yale Day of Data

Diffusion maps are a modern mathematical tool that helps to find structure in large data sets - we present a new filtering technique that is based on the assumption that errors in the data are intrinsically random to isolate and filter errors and thus boost the efficiency of diffusion maps. Applications include data sets from medicine (the Cleveland Heart Disease Data set and the Wisconsin Breast Cancer Data set) and engineering (the Ionosphere data set).

On The Analysis Of Some Recursive Equations In Probability., Arunangshu Biswas Dr.

Doctoral Theses

This thesis deals with recursive systems used in theoretical and applied probability. Recursive systems are stochastic processes {Xn}n≥1 where the Xn depends on the earlier Xn−1 and also on some increment process which is uncorrelated with the process Xn. The simplest example of a recursive system is the Random Walk, whose properties have been extensively studied. Mathematically a recursive system takes the form Xn = f(Xn−1, n), is the increment/ innovation procedure and f(·, ·) is a function on the product space of xn and n. We first consider a recursive system called Self-Normalized sums (SNS) corresponding to a sequence …

Convergence Rates And Hölder Estimates In Almost-Periodic Homogenization Of Elliptic Systems, Zhongwei Shen

Mathematics Faculty Publications

For a family of second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the coefficients. The results are used to investigate the problem of convergence rates. We also establish uniform Hölder estimates for the Dirichlet problem in a bounded C^1,α domain.

Radial Versus Othogonal And Minimal Projections Onto Hyperplanes In L_4^3, Richard Alan Warner

USF Tampa Graduate Theses and Dissertations

In this thesis, we study the relationship between radial projections, and orthogonal and minimal projections in l_4^3. Specifically, we calculate the norm of the maximum radial projection and we prove that the hyperplane constant, with respect to the radial projection, is not achieved by a minimal projection in this space. We will also show our numerical results, obtained using computer software, and use them to approximate the norms of the radial, orthogonal, and minimal projections in l_4^3. Specifically, we show, numerically, that the maximum minimal projection is attained for ker{1,1,1} as well as compute the norms for the maximum radial …

Computing Invariant Dynamics For Differential Equations: Spectral Methods, Errors, And Computer Assisted Proof, J. D. Mireles James

Mathematics Colloquium Series

The qualitative theory of dynamical systems is concerned with studying the long time behavior discrete and continuous time models such as nonlinear differential equations. The long time behavior of such models is organized by landmarks called invariant sets. For complicated nonlinear equations these invariant sets are difficult to study via pen and paper analysis, and we typically employ numerical simulations to gain insights into the dynamics. If we now think of these computer assisted insights as mathematical conjectures, then it is natural to ask how we might obtain proofs. Since the conjectures themselves originate with the computer it is not …

Art, Math, And Physics; All About For, Chris Brownell, Steve Pauls

The Transdisciplinary STEAM+ Journal

Anish Kapoor’s public sculpture “Cloud Gate” and Frame of Reference.

Ambiguity In Speaking Chemistry And Other Stem Content: Educational Implications, Mick D. Isaacson, Michelle Michaels

Journal of Science Education for Students with Disabilities

Ambiguity in speech is a possible barrier to the acquisition of knowledge for students who have print disabilities (such as blindness, visual impairments, and some specific learning disabilities) and rely on auditory input for learning. Chemistry appears to have considerable potential for being spoken ambiguously and may be a barrier to accessing knowledge and to learning. Educators in chemistry may be unaware of, or have limited awareness of, potential ambiguity in speaking chemistry and may speak chemistry ambiguously to their students. One purpose of this paper is to increase awareness of potential ambiguity in speaking chemistry and other STEM fields …

Conditional Dimension In Metric Spaces: A Natural Metric-Space Counterpart Of Kolmogorov-Complexity-Based Mutual Dimension, Vladik Kreinovich, Luc Longpre, Olga Kosheleva

Departmental Technical Reports (CS)

It is known that dimension of a set in a metric space can be characterized in information-related terms -- in particular, in terms of Kolmogorov complexity of different points from this set. The notion of Kolmogorov complexity K(x) -- the shortest length of a program that generates a sequence x -- can be naturally generalized to conditionalKolmogorov complexity K(x:y) -- the shortest length of a program that generates x by using y as an input. It is therefore reasonable to use conditional Kolmogorov complexity to formulate a conditional analogue of dimension. Such a generalization has indeed been proposed, under …

Constructive Mathematics Is Seemingly Simple But There Are Still Open Problems: Kreisel's Observation Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In his correspondence with Grigory Mints, the famous logician Georg Kreisel noticed that many results of constructive mathematics seem easier-to-prove than the corresponding classical (non-constructive) results -- although he noted that these results are still far from being simple and the corresponding open problems are challenging. In this paper, we provide a possible explanation for this empirical observation.

Occam's Razor Explains Matthew Effect, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Sociologists of science noticed that the results of many collaborative projects and discoveries are often attributed only to their most famous collaborators, even when the contributions of these famous collaborators were minimal. This phenomenon is known as the Matthew effect, after a famous citation from the Gospel of Matthew. In this article, we show that Occam's razor provides a possible explanation for the Matthew effect.

Combining Interval And Probabilistic Uncertainty: What Is Computable?, Vladik Kreinovich, Andrzej Pownuk, Olga Kosheleva

Departmental Technical Reports (CS)

In many practical problems, we need to process measurement results. For example, we need such data processing to predict future values of physical quantities. In these computations, it is important to take into account that measurement results are never absolutely exact, that there is always measurement uncertainty, because of which the measurement results are, in general, somewhat different from the actual (unknown) values of the corresponding quantities. In some cases, all we know about measurement uncertainty is an upper bound; in this case, we have an interval uncertainty, meaning that all we know about the actual value is that is …

Candy Crush Combinatorics, Dana Rowland

Mathematics Faculty Publications

In the popular game Candy Crush, differently colored candies are arranged in a grid and a player swaps adjacent candies in order to crush them by lining up three or more of the same color. At the beginning of each game, the grid cannot have three consecutive candies of the same color in a row or column, but it must be possible to swap two adjacent candies in order to get at least three consecutive candies of the same color. How many starting configurations are there? We derive recurrence relations to answer this question for a single line of candy, …

Problems In The Theory Of Extremal Graphs And Hypergraphs, Cory T. Palmer

University Grant Program Reports

Objective: Investigate the tree packing conjecture from graph theory and extremal numbers for hypergraphs.

Paradox Of Choice: A Possible Explanation, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

At first glance, we would expect that the more choices we have, the happier we will be. Experiments show, however, then when the number of choices increases, customers become less happy. In this paper, we provide a possible explanation for this paradox.

Properties Of Catlin’S Reduced Graphs And Supereulerian Graphs, Wei-Guo Chen, Zhi-Hong Chen, Mei Lu

Scholarship and Professional Work - LAS

A graph G is called collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is the reduction of G if it is obtained from G by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs G of order n with d(u) + d(v) ≥ 2(n/p − …

Host Switching Vs. Host Sharing In Overlapping Sylvatic Trypanosoma Cruzi Transmission Cycles, Christopher Kribs, Christopher Mitchell

Mathematics Faculty Publications - Archive

The principle of competitive exclusion is well established for multiple populations competing for the same resource, and simple models for multistrain infection exhibit it as well when cross-immunity precludes coinfections. However, multiple hosts provide niches for different pathogens to occupy simultaneously. This is the case for the vector-borne parasite Trypanosoma cruzi in overlapping sylvatic transmission cycles in the Americas, where it is enzootic. This study uses cycles in the USA involving two different hosts but the same vector species as a context for the study of the mechanisms behind the communication between the two cycles. Vectors dispersing in search of …

Towards An Automated Screening Tool For Pediatric Speech Delay, Roozbeh Sadeghian, Stephen A. Zahorian

Faculty Works

Speech delay is a childhood language problem that sometimes is resolved on its own but sometimes may cause more serious language difficulties later. This leads therapists to screen children for detection at early ages in order to eliminate future problems. Using the Goldman-Fristoe Test of Articulation (GFTA) method, therapists listen to a child's pronunciation of certain phonemes and phoneme pairs in specified words and judge the child's stage of speech development. The goal of this paper is to develop an Automatic Speech Recognition (ASR) tool and related speech processing methods which emulate the knowledge of speech therapists. In this paper …

Some Quantum Dynamical Semi-Groups With Quantum Stochastic Dilation, Lingaraj Sahu, Preetinder Singh

Communications on Stochastic Analysis

No abstract provided.

Stochastic Dynamics Of Determinantal Processes By Integration By Parts, Laurent Decreusefond, Ian Flint, Nicolas Privault, Giovanni Luca Torrisi

Communications on Stochastic Analysis

No abstract provided.

On The Regularity Of One Parameter Transformation Groups In Barreled Locally Convex Topological Vector Spaces, Maximilian Bock

Communications on Stochastic Analysis

No abstract provided.

Hidden Markov Change Point Estimation, Robert J Elliott, Sebastian Elliott

Communications on Stochastic Analysis

No abstract provided.