Mathematics Commons^™

Weakly $2$-Absorbing Submodules Of Modules, Sedigheh Moradi, Abdulrasool Azizi

Turkish Journal of Mathematics

Let $M$ be a module over a commutative ring $R.$ A proper submodule $N$ of $M$ is called weakly $2$-absorbing, if for $r,s\in R$ and $x\in M$ with $0\neq rsx\in N,$ either $rs\in (N:M)$ or $rx\in N$ or $sx\in N.$ We study the behavior of $(N:M)$ and $\sqrt{(N:M)},$ when $N$ is weakly $2$-absorbing. The weakly $2$-absorbing submodules when $R=R_1\oplus R_2$ are characterized. Moreover we characterize the faithful modules whose proper submodules are all weakly $2$-absorbing.

A Generalization Of Turaev’S Virtual String Cobracket And Self-Intersections Of Virtual Strings, Patricia Cahn

Mathematics Sciences: Faculty Publications

Previously we defined an operation µ that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding question for virtual strings, and conjecture that µ gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that µ gives a bound on the minimal self-intersection number of a virtual string which is …

A Coin Vibrational Motor Swimming At Low Reynolds Number, Alice C. Quillen, Hesam Askari, Douglas H. Kelley, Tamar Friedmann, Patrick W. Oakes

Mathematics Sciences: Faculty Publications

Low-cost coin vibrational motors, used in haptic feedback, exhibit rotational internal motion inside a rigid case. Because the motor case motion exhibits rotational symmetry, when placed into a fluid such as glycerin, the motor does not swim even though its oscillatory motions induce steady streaming in the fluid. However, a piece of rubber foam stuck to the curved case and giving the motor neutral buoyancy also breaks the rotational symmetry allowing it to swim. We measured a 1 cm diameter coin vibrational motor swimming in glycerin at a speed of a body length in 3 seconds or at 3 mm/s. …

Twenty-Seven Element Albertian Semifields, Thomas Joel Hughes

Open Access Theses & Dissertations

In 1958 Abraham Adrian Albert published his findings on the now famous class of semifields known as twisted fields or, as we refer to them in this Thesis, Albertian semifields. Since the publication of his findings, interest in the topic of semifields has grown and, naturally, some progress has been made in the subject, but there is still much to be studied. Moreover, the greater portion of modern efforts have been made to develop a theory of semifields, with less attention paid to more practical considerations. In this Thesis, we will narrow our investigations to a specific subclass of Albert's …

An A Posteriori Error Estimator For The C0 Interior Penalty Approximations Of Fourth Order Elliptic Boundary Value Problem On Quadrilateral Meshes, Mohammad Arifur Rahman

Open Access Theses & Dissertations

Numerical solutions of fourth order elliptic problems with finite element methods has been the topic of research in computational mechanics for over 50 years. Traditional approaches to solve these problems include using C1 conforming finite element methods which demand the C1 continuity of the underlying shape functions, which is computationally very expensive. In this work, we will present the C0 Interior Penalty Galerkin approximation of the fourth order elliptic problems which relies only on continuous i.e., C0 shape functions, which is much cheaper to implement. The spatial discretization is based on quadrilateral meshes and the underlying C0 shape functions are …

The Diameter Of A Rouquier Block, Andrew Mayer

Williams Honors College, Honors Research Projects

For my Honors Research Project, I will be researching special properties of Rouquier blocks that represent the partitions of integers. This problem is motivated by ongoing work in representation theory of the symmetric group. For each integer n and each prime p, there is an object called a Rouquier block; this block can be visualized as a collection of points in a plane, each corresponding to a partition. In this group of points, we say a pair of points is “connected” if certain conditions on the partitions are met. We compare each partition with each other partition, add edges when …

New Facets Of The Balanced Minimal Evolution Polytope, Logan Keefe

Williams Honors College, Honors Research Projects

The balanced minimal evolution (BME) polytope arises from the study of phylogenetic trees in biology. It is a geometric structure which has a variant for each natural number n. The main application of this polytope is that we can use linear programming with it in order to determine the most likely phylogenetic tree for a given genetic data set. In this paper, we explore the geometric and combinatorial structure of the BME polytope. Background information will be covered, highlighting some points from previous research, and a new result on the structure of the BME polytope will be given.

Price Signaling In A Two-Market Duopoly, Matthew Hughes

Williams Honors College, Honors Research Projects

Within any industry, firms typically produce related products over multiple subsequent periods in an attempt to build consumer loyalty and achieve continued sales. Apple releases new iPhones and car companies produce new models every year, relying on consumers believing each new product is of high quality. Firms rely on the spillover effects from previous markets, where firms are able to more easily demonstrate their product's quality to the consumers before purchase. The goal is to find a range of prices which allows the high quality firm to distinguish its type to consumers via the price pH and if spillover effects …

Cameron-Liebler Line Classes And Partial Difference Sets, Uthaipon Tantipongipat

Honors Theses

The work consists of three parts. The first is a study of Cameron-Liebler line classes which receive much attention recently. We studied a new construction of infinite family of Cameron-Liebler line classes presented in the paper by Tao Feng, Koji Momihara, and Qing Xiang (rst introduced in 2014), and summarized our attempts to generalize this construction to discover any new Cameron-Liebler line classes or partial difference sets (PDSs) resulting from the Cameron-Liebler line classes. The second is our approach to finding PDS in non-elementary abelian groups. Our attempt eventually led to the same general construction of PDS presented in John …

Partitioning Groups With Difference Sets, Rebecca Funke

Honors Theses

This thesis explores the use of difference sets to partition algebraic groups. Difference sets are a tool belonging to both group theory and combinatorics that provide symmetric properties that can be map into over mathematical fields such as design theory or coding theory. In my work, I will be taking algebraic groups and partitioning them into a subgroup and multiple McFarland difference sets. This partitioning can then be mapped to an association scheme. This bridge between difference sets and association schemes have important contributions to coding theory.

A Comparative Analysis Of An Interior-Point Method And A Sequential Quadratic Programming Method For The Markowitz Portfolio Management Problem, Zhifu Xiao

Honors Papers

In this paper, I give a brief introduction of the general optimization problem as well as the convex optimization problem. The portfolio selection problem, as a typical type of convex optimization problem, can be easily solved in polynomial time. However, when the number of available stocks in the portfolio becomes large, there might be a significant difference in the running time of different polynomial-time solving methods. In this paper, I perform a comparative analysis of two different solving methods and discuss the characteristics and differences.

Asymptotic And Optimal Liouville Properties For Wolff Type Integral Systems, John Villavert

School of Mathematical & Statistical Sciences Faculty Publications

This article examines the properties of positive solutions to fully nonlinear systems of integral equations involving Hardy and Wolff potentials. The first part of the paper establishes an optimal existence result and a Liouville type theorem for the integral systems. Then, the second part examines the decay rates of positive bound states at infinity. In particular, a complete characterization of the asymptotic properties of bounded and decaying solutions is given by showing that such solutions vanish at infinity with two principle rates: the slow decay rates and the fast decay rates. In fact, the two rates can be fully distinguished …

Stability Analysis And Hopf Bifurcation Of Density-Dependent Predator-Prey Systems With Beddington-Deangelis Functional Response, Xin Jiang, Zhikun She, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this article, we study a density-dependent predator-prey system with the Beddington-DeAngelis functional response for stability and Hopf bifurcation under certain parametric conditions. We start with the condition of the existence of the unique positive equilibrium, and provide two sufficient conditions for its local stability by the Lyapunov function method and the Routh-Hurwitz criterion, respectively. Then, we establish sufficient conditions for the global stability of the positive equilibrium by proving the non-existence of closed orbits in the first quadrant R²+. Afterwards, we analyze the Hopf bifurcation geometrically by exploring the monotonic property of the trace of the Jacobean matrix with …

Hepatitis C Virus In Mexican Americans: A Population-Based Study Reveals Relatively High Prevalence And Negative Association With Diabetes, Gordon P. Watt, Kristina Vatcheva, Laura Beretta, Jen-Jung Pan, Michael Fallon, Joseph B. Mccormick, Susan P. Fisher-Hoch

School of Mathematical & Statistical Sciences Faculty Publications

This study aimed to estimate the prevalence and risk factors for hepatitis C virus (HCV) infection in Mexican Americans living in South Texas. We tested plasma for the presence of HCV antibody from the Cameron County Hispanic Cohort (CCHC), a randomized, population-based cohort in an economically disadvantaged Mexican American community on the United States/Mexico border with high rates of chronic disease. A weighted prevalence of HCV antibody of 2·3% [n = 1131, 95% confidence interval (CI) 1·2-3·4] was found. Participants with diabetes had low rates of HCV antibody (0·4%, 95% CI 0·0-0·9) and logistic regression revealed a statistically significant negative …

A Polyhedral Model Of Partitions With Bounded Differences And A Bijective Proof Of A Theorem Of Andrews, Beck, And Robbins, Felix Breuer, Brandt Kronholm

School of Mathematical & Statistical Sciences Faculty Publications

The smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied through analytic methods by Andrews, Beck, and Robbins. Our approach is geometric: We model partitions with bounded differences as lattice points in an infinite union of polyhedral cones. Surprisingly, this infinite union tiles a single simplicial cone. This construction then leads to a bijection that can be interpreted on a purely combinatorial level.

Tabulating Pseudoprimes And Tabulating Liars, Andrew Shallue

Scholarship

This paper explores the asymptotic complexity of two problems related to the Miller-Rabin-Selfridge primality test. The first problem is to tabulate strong pseudoprimes to a single fixed base $a$. It is now proven that tabulating up to $x$ requires $O(x)$ arithmetic operations and $O(x\log{x})$ bits of space. The second problem is to find all strong liars and witnesses, given a fixed odd composite $n$. This appears to be unstudied, and a randomized algorithm is presented that requires an expected $O((\log{n})^2 + |S(n)|)$ operations (here $S(n)$ is the set of strong liars). Although interesting in their own right, a notable application …

The Neural Dynamics Of Somatosensory Processing And Adaptation Across Childhood: A High-Density Electrical Mapping Study, Neha Uppal, John J. Foxe, John Butler, Frantzy Acluche, Sophie Molholm

Articles

Young children are often hyperreactive to somatosensory inputs hardly noticed by adults, as exemplified by irritation to seams or labels in clothing. The neurodevelopmental mechanisms underlying changes in sensory reactivity are not well understood. Based on the idea that neurodevelopmental changes in somatosensory processing and/or changes in sensory adaptation might underlie developmental differences in somatosensory reactivity, high-density electroencephalography was used to examine how the nervous system responds and adapts to repeated vibrotactile stimulation over childhood. Participants aged 6–18 yr old were presented with 50-ms vibrotactile stimuli to the right wrist over the median nerve at 5 blocked interstimulus intervals (ranging …

Unique Characterization Of Materials With Memory, John Murrough Golden

Articles

In general, materials with linear memory constitutive relations are characterized by a relaxation function. This leads to a situation where the free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. An alternative procedure is proposed which characterizes a material by the kernel of the rate of dissipation functional. Using some recent results, we find that a unique free energy and relaxation function may then be deduced. An example is given for discrete spectrum materials. Also, the new results are used to show that …

A Topological Study Of Stochastic Dynamics On Cw Complexes, Michael Joseph Catanzaro

Wayne State University Dissertations

In this dissertation, we consider stochastic motion of subcomplexes of a CW complex, and explore the implications on the underlying space. The random process on the complex is motivated from Ito diffusions on smooth manifolds and Langevin processes in physics. We associate a Kolmogorov equation to this process, whose solutions can be interpretted in terms of generalizations of electrical, as well as stochastic, current to higher dimensions. These currents also serve a key function in relating the random process to the topology of the complex. We show the average current generated by such a process can be written in a …

Mathematical Writing Assignment For Deeper Understanding And Process Writing, Colton Magnant, Saeed Nasseh, Teresa Flateby

Mathematical Sciences: Faculty Publications

Brief Description: The broad goals of this writing assignment are two-fold: 1) To delve deeper into the inner workings of a chosen proof and explore fundamental motivation of the chosen result. 2) To enhance student learning in the area of academic writing in the discipline of mathematics.

By walking the students through a process of academic writing, we address the following DQP proficiencies: Specialized Knowledge, Applied and Collaborative Learning and Intellectual Skills - Use of Information Resources, Mathematics-Specific Intellectual and Practical Skills and Communicative Fluency.

Background and context: This assignment has been used in a Mathematical Structures (introduction-to-proofs) course and …

The Beta Lindley-Poisson Distribution With Applications, Broderick O. Oluyede, Gayan Warahena-Liyanage, Mavis Pararai

Mathematical Sciences: Faculty Publications

The beta Lindley-Poisson (BLP) distribution which is an extension of the Lindley-Poisson Distribution is introduced and its properties are explored. This new distribution represents a more flexible model for the lifetime data. Some statistical properties of the proposed distribution including the expansion of the density function, hazard rate function, moments and moment generating function, skewness and kurtosis are explored. Renyi entropy and the distribution of the order statistics are given. The maximum likelihood estimation technique is used to estimate the model parameters and finally applications of the model to real data sets are presented for the illustration of the usefulness …

A New Class Of Generalized Power Lindley Distribution With Applications To Lifetime Data, Broderick O. Oluyede, Tiantian Yang, Boikanyo Makubate

Mathematical Sciences: Faculty Publications

In this paper, a new class of generalized distribution called the Kumaraswamy Power Lindley (KPL) distribution is proposed and studied. This class of distributions contains the Kumaraswamy Lindley (KL), exponentiated power Lindley (EPL), power Lindley (PL), generalized or exponentiated Lindley (GL), and Lindley (L) distributions as special cases. Series expansion of the density is obtained. Statistical properties of this class of distributions, including hazard function, reverse hazard function, monotonicity property, shapes, moments, reliability, quantile function, mean deviations, Bonferroni and Lorenz curves, entropy and Fisher information are derived. Method of maximum likelihood is used to estimate the parameters of this new …

Multi-Type Display Calculus For Dynamic Epistemic Logic, Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano, Vlasta Sikimić

Engineering Faculty Articles and Research

In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The displayapproach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki’s dynamic epistemic logic, and enjoys Belnap-style cut …

Combinatorial Optimization Of Subsequence Patterns In Words, Matthew R. Just

College of Graduate Studies: Theses & Dissertations

Packing patterns in words concerns finding a word with the maximum number of a prescribed pattern. The majority of the work done thus far is on packing patterns into permutations. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently, the packing density for all but two (up to equivalence) permutation patterns up to length 4 can be obtained. In this thesis we consider the analogous question for colored patterns and permutations. By introducing the concept of colored blocks …

Restricted Lie Algebras With Maximal 0-Pim, Jorg Feldvoss, Salvatore Siciliano, Thomas Weigel

University Faculty and Staff Publications

In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one dimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p ^MT(L), where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced …

Characterizing Mathematics Graduate Student Teaching Assistants’ Opportunities To Learn From Teaching, Yvonne Lai, Wendy Smith, Nathan Wakefield, Erica R. Miller, Julia St. Goar, Corbin M. Groothuis, Kelsey M. Wells

Department of Mathematics: Faculty Publications

Exemplary models to inform novice instruction and the development of graduate teaching assistants (TAs) exist. What is missing from the literature is the process of how graduate students in model professional development programs make sense of and enact the experiences offered. A first step to understanding TAs’ learning to teach is to characterize how and whether they link observations of student work to hypotheses about student thinking and then connect those hypotheses to future teaching actions. A reason to be interested in these connections is that their strength and coherence determine how well TAs can learn from experiences. We found …

R0 Analysis Of A Benthic-Drift Model For A Stream Population, Qihua Huang, Yu Jin, Mark A. Lewis

Department of Mathematics: Faculty Publications

One key issue for theory in stream ecology is how much stream flow can be changed while still maintaining an intact stream ecology, instream flow needs (IFNs); the study of determining IFNs is challenging due to the complex and dynamic nature of the interaction between the stream environ- ment and the biological community. We develop a process-oriented benthic-drift model that links changes in the flow regime and habitat availability with population dynamics. In the model, the stream is divided into two zones, drift zone and benthic zone, and the population is divided into two interacting compartments, individuals residing in the …

Existence Of Periodic Orbits In Nonlinear Oscillators Of Emden-Fowler Form, S.C. Mancas, Haret C. Rosu

Publications

The nonlinear pseudo-oscillator recently tackled by Gadella and Lara is mapped to an Emden–Fowler (EF) equation that is written as an autonomous two-dimensional ODE system for which we provide the phase-space analysis and the parametric solution. Through an invariant transformation we find periodic solutions to a certain class of EF equations that pass an integrability condition. We show that this condition is necessary to have periodic solutions and via the ODE analysis we also find the sufficient condition for periodic orbits. EF equations that do not pass integrability conditions can be made integrable via an invariant transformation which also allows …

Nongauge Bright Soliton Of The Nonlinear Schrodinger (Nls) Equation And A Family Of Generalized Nls Equations, M. A. Reyes, D. Gutierrez-Ruiz, S. C. Mancas, H. C. Rosu

Publications

We present an approach to the bright soliton solution of the nonlinear Schrödinger (NLS) equation from the standpoint of introducing a constant potential term in the equation. We discuss a “nongauge” bright soliton for which both the envelope and the phase depend only on the traveling variable. We also construct a family of generalized NLS equations with solitonic sechpsechp solutions in the traveling variable and find an exact equivalence with other nonlinear equations, such as the Korteveg–de Vries (KdV) and Benjamin–Bona–Mahony (BBM) equations when p=2.

Mathematical Frameworks For Consciousness, Menas C. Kafatos, Ashok Narasimhan

Mathematics, Physics, and Computer Science Faculty Articles and Research

If Awareness is fundamental in the universe, mathematical frameworks are better suited to reveal its fundamental aspects than physical models. Awareness operates through three fundamental laws which apply at all levels of reality and is characterized by three universal powers. We explore and summarize in general terms mathematical formalisms that may take us as close as possible to conscious awareness, beginning with the primary relationships between the observer with the observed, using a Hilbert space approach. We also examine insights from category theory, and the calculus of indications or laws of forms. Mathematical frameworks as fundamental languages of our interaction …