Mathematics Commons^™

Some Results Concerning Permutation Polynomials Over Finite Fields, Stephen Lappano

USF Tampa Graduate Theses and Dissertations

Let p be a prime, p a power of p and 𝔽_q the finite field with q elements. Any function φ: 𝔽_q → 𝔽_q can be unqiuely represented by a polynomial, 𝔽_φ of degree < q. If the map x ↦ F_φ(x) induces a permutation on the underlying field we say F_φ is a permutation polynomial. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial f, when does the map x ↦ F( …

Longitudinal Success Of Calculus I Reform, Doug Bullock, Kathrine E. Johnson, Janet Callahan

Mathematics Faculty Publications and Presentations

This paper describes the second year of an ongoing project to transform calculus instruction at Boise State University. Over the past several years, Calculus I has undergone a complete overhaul that has involved a movement from a collection of independent, uncoordinated, personalized, lecture-based sections, into a single coherent multi-section course with an activelearning pedagogical approach. The overhaul also significantly impacted the course content and learning objectives. The project is now in its fifth semester and has reached a steady state where the reformed practices are normative within the subset of instructors who might be called upon to teach Calculus I. …

One-Bit Compressive Sensing Of Dictionary-Sparse Signals, Richard Baraniuk, Simon Foucart, Deanna Needell, Yaniv Plan, Mary Wootters

CMC Faculty Publications and Research

One-bit compressive sensing has extended the scope of sparse recovery by showing that sparse signals can be accurately reconstructed even when their linear measurements are subject to the extreme quantization scenario of binary samples—only the sign of each linear measurement is maintained. Existing results in one-bit compressive sensing rely on the assumption that the signals of interest are sparse in some fixed orthonormal basis. However, in most practical applications, signals are sparse with respect to an overcomplete dictionary, rather than a basis. There has already been a surge of activity to obtain recovery guarantees under such a generalized sparsity model …

The Total Acquisition Number Of Random Graphs, Deepak Bal, Patrick Bennett, Andrzej Dudek, Paweł Prałat

Department of Mathematics Faculty Scholarship and Creative Works

Let G be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex u to a neighbouring vertex v can be moved, provided that the weight on v is at least as large as the weight on u. The total acquisition number of G, denoted by at(G), is the minimum possible size of the set of vertices with positive weight at the end of the process. LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of p = p(n) such that at(G(n, p)) = 1 with high probability, where G(n, …

Hamiltonian Formulations And Symmetry Constraints Of Soliton Hierarchies Of (1+1)-Dimensional Nonlinear Evolution Equations, Solomon Manukure

USF Tampa Graduate Theses and Dissertations

We derive two hierarchies of 1+1 dimensional soliton-type integrable systems from two spectral problems associated with the Lie algebra of the special orthogonal Lie group SO(3,R). By using the trace identity, we formulate Hamiltonian structures for the resulting equations. Further, we show that each of these equations can be written in Hamiltonian form in two distinct ways, leading to the integrability of the equations in the sense of Liouville. We also present finite-dimensional Hamiltonian systems by means of symmetry constraints and discuss their integrability based on the existence of sufficiently many integrals of motion.

Tables Of The Existence Of Equiangular Tight Frames, Matthew C. Fickus, Dustin G. Mixon

Faculty Publications

A Grassmannian frame is a collection of unit vectors which are optimally incoherent. To date, the vast majority of explicit Grassmannian frames are equiangular tight frames (ETFs). This paper surveys every known construction of ETFs and tabulates existence for sufficiently small dimensions.

Compressive Sensing Reconstruction Of Feed-Forward Connectivity In Pulse-Coupled Nonlinear Networks, Victor J. Barranca, D. Zhou, D. Cai

Mathematics & Statistics Faculty Works

Utilizing the sparsity ubiquitous in real-world network connectivity, we develop a theoretical framework for efficiently reconstructing sparse feed-forward connections in a pulse-coupled nonlinear network through its output activities. Using only a small ensemble of random inputs, we solve this inverse problem through the compressive sensing theory based on a hidden linear structure intrinsic to the nonlinear network dynamics. The accuracy of the reconstruction is further verified by the fact that complex inputs can be well recovered using the reconstructed connectivity. We expect this Rapid Communication provides a new perspective for understanding the structure-function relationship as well as compressive sensing principle …

Putnam's Inequality And Analytic Content In The Bergman Space, Matthew Fleeman

USF Tampa Graduate Theses and Dissertations

In this dissertation we are interested in studying two extremal problems in the Bergman space. The topics are divided into three chapters.

In Chapter 2, we study Putnam’s inequality in the Bergman space setting. In [32], the authors showed that Putnam’s inequality for the norm of self-commutators can be improved by a factor of 1 for Toeplitz operators with analytic symbol φ acting on the Bergman space A2(Ω). This improved upper bound is sharp when φ(Ω) is a disk. We show that disks are the only domains for which the upper bound is attained.

In Chapter 3, we consider the …

Upper And Lower Solution Method For Boundary Value Problems At Resonance, Samerah Al Mosa, Paul W. Eloe

Mathematics Faculty Publications

We consider two simple boundary value problems at resonance for an ordinary differential equation. Employing a shift argument, a regular fixed point operator is constructed. We employ the monotone method coupled with a method of upper and lower solutions and obtain sufficient conditions for the existence of solutions of boundary value problems at resonance for nonlinear boundary value problems. Three applications are presented in which explicit upper solutions and lower solutions are exhibited for the first boundary value problem. Two applications are presented for the second boundary value problem. Of interest, the upper and lower solutions are easily and explicitly …

Optimizing Quantization For Lasso Recovery, Xiaoyi Gu, Shenyinying Tu, Hao-Jun Michael Shi, Mindy Case, Deanna Needell, Yaniv Plan

CMC Faculty Publications and Research

This letter is focused on quantized Compressed Sensing, assuming that Lasso is used for signal estimation. Leveraging recent work, we provide a framework to optimize the quantization function and show that the recovered signal converges to the actual signal at a quadratic rate as a function of the quantization level. We show that when the number of observations is high, this method of quantization gives a significantly better recovery rate than standard Lloyd-Max quantization. We support our theoretical analysis with numerical simulations.

Characterization Of Simplices Via The Bezout Inequality For Mixed Volumes, Christos Saroglou, Ivan Soprunov, Arten Zvavitch

Mathematics and Statistics Faculty Publications

We consider the following Bezout inequality for mixed volumes: V (K1, . . . ,Kr, Δ[n − r])Vn(Δ)r−1 ≤ r i=1 V (Ki, Δ[n − 1]) for 2 ≤ r ≤ n. It was shown previously that the inequality is true for any -dimensional simplex and any convex bodies in . It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies in . In this paper we prove that this is indeed the case if we assume that is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of …

Intestinal Microbiota-Generated Metabolite Trimethylamine-N-Oxide And 5-Year Mortality Risk In Stable Coronary Artery Disease: The Contributory Role Of Intestinal Microbiota In A Courage-Like Patient Cohort, Vichai Senthong, Zeneng Wang, Xinmin S. Li, Yiying Fan, Yuping Wu, W. H. Wilson Tang, Stanley L. Hazen

Mathematics and Statistics Faculty Publications

Background: Trimethylamine-N-oxide (TMAO), a metabolite derived from gut microbes and dietary phosphatidylcholine, is linked to both coronary artery disease pathogenesis and increased cardiovascular risks. The ability of plasma TMAO to predict 5-year mortality risk in patients with stable coronary artery disease has not been reported. This study examined the clinical prognostic value of TMAO in patients with stable coronary artery disease who met eligibility criteria for a patient cohort similar to that of the Clinical Outcomes Utilizing Revascularization and Aggressive Drug Evaluation (COURAGE) trial. Methods and Results: We examined the relationship between fasting plasma TMAO and all-cause mortality over 5-year …

Plasma Trimethylamine N-Oxide, A Gut Microbe–Generated Phosphatidylcholine Metabolite, Is Associated With Atherosclerotic Burden, Vichai Senthong, Xinmin S. Li, Timothy Hudec, John Coughlin, Yuping Wu, Bruce Levison, Zeneng Wang, Stanley L. Hazen, W.H. Wilson Tang

Mathematics and Statistics Faculty Publications

Background: Trimethylamine N-oxide (TMAO), a gut microbiota metabolite from dietary phosphatidylcholine, has mechanistic links to atherosclerotic coronary artery disease (CAD) pathogenesis and is associated with adverse outcomes. Objectives: This study sought to examine the relationship between plasma TMAO levels and the complexity and burden of CAD and degree of subclinical myonecrosis. Methods: We studied 353 consecutive stable patients with evidence of atherosclerotic CAD detected by elective coronary angiography between 2012 and 2014. Their high-sensitivity cardiac troponin T (hs-cTnT) levels were measured. SYNTAX (Synergy Between Percutaneous Coronary Intervention With Taxus and Cardiac Surgery) scores and lesion characteristics were used to quantify …

Critical Thinking Skills And Academic Maturity: Emerging Results From A Five-Year Quality Enhancement Plan (Qep) Study, Ian N. Toppin, Shadreck Chitsonga

Journal of Inquiry and Action in Education

The QEP that was implemented in this study focused on enhancing students’ critical thinking skills. A pretest/ posttest approach was used to assess students’ critical thinking progress in freshman level core English and Math courses. An intervention was performed involving intensive instruction and assignments relating to a set of reasoning strategies such as: analytical, analogical, inductive, deductive, and comparative reasoning, among others. When students performed well on assignments by applying the reasoning strategies, it was assumed that critical thinking occurred. However, pre/ posttest results in these classes were often disappointing, and seemed at times to suggest that freshmen are not …

Maximum Waring Ranks Of Monomials And Sums Of Coprime Monomials, Erik Holmes, Paul Plummer, Jeremy Siegert, Zach Teitler

Mathematics Faculty Publications and Presentations

We show that monomials and sums of pairwise coprime monomials in four or more variables have Waring rank less than the generic rank, with a short list of exceptions. We asymptotically compare their ranks with the generic rank.

Cohomology Of Certain Polyhedral Product Spaces, Elizabeth A. Vidaurre

Dissertations, Theses, and Capstone Projects

The study of torus actions led to the discovery of moment-angle complexes and their generalization, polyhedral product spaces. Polyhedral products are constructed from a simplicial complex. This thesis focuses on computing the cohomology of polyhedral products given by two different classes of simplicial complexes: polyhedral joins (composed simplicial complexes) and $n$-gons. A homological decomposition of a polyhedral product developed by Bahri, Bendersky, Cohen and Gitler is used to derive a formula for the case of polyhedral joins. Moreover, methods from and results by Cai will be used to give a full description of the non-trivial cup products in a real …

Cayley Graphs Of Semigroups And Applications To Hashing, Bianca Sosnovski

Dissertations, Theses, and Capstone Projects

In 1994, Tillich and Zemor proposed a scheme for a family of hash functions that uses products of matrices in groups of the form $SL_2(F_{2^n})$. In 2009, Grassl et al. developed an attack to obtain collisions for palindromic bit strings by exploring a connection between the Tillich-Zemor functions and maximal length chains in the Euclidean algorithm for polynomials over $F_2$.

In this work, we present a new proposal for hash functions based on Cayley graphs of semigroups. In our proposed hash function, the noncommutative semigroup of linear functions under composition is considered as platform for the scheme. We will also …

Stochastic Processes And Their Applications To Change Point Detection Problems, Heng Yang

Dissertations, Theses, and Capstone Projects

This dissertation addresses the change point detection problem when either the post-change distribution has uncertainty or the post-change distribution is time inhomogeneous. In the case of post-change distribution uncertainty, attention is drawn to the construction of a family of composite stopping times. It is shown that the proposed composite stopping time has third order optimality in the detection problem with Wiener observations and also provides information to distinguish the different values of post-change drift. In the case of post-change distribution uncertainty, a computationally efficient decision rule with low-complexity based on Cumulative Sum (CUSUM) algorithm is also introduced. In the time …

P-Adic L-Functions And The Geometry Of Hida Families, Joseph Kramer-Miller

Dissertations, Theses, and Capstone Projects

A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this talk we explain results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable $p$-adic $L$-functions varying over families can detect geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special $L$-values and then $p$-adically interpolating congruences using …

Quaternion Algebras And Hyperbolic 3-Manifolds, Joseph Quinn

Dissertations, Theses, and Capstone Projects

I use a classical idea of Macfarlane to obtain a complex quaternion model for hyperbolic 3-space and its group of orientation-preserving isometries, analogous to Hamilton’s famous result on Euclidean rotations. I generalize this to quaternion models over number fields for the action of Kleinian groups on hyperbolic 3-space, using arithmetic invariants of the corresponding hyperbolic 3-manifolds. The class of manifolds to which this technique applies includes all cusped arithmetic manifolds and infinitely many commensurability classes of cusped non-arithmetic, compact arithmetic, and compact non-arithmetic manifolds. I obtain analogous results for actions of Fuchsian groups on the hyperbolic plane. I develop new …