Mathematics Commons^™

The Complexity Of Linear Algebra, Leann Kay Christensen

Theses Digitization Project

This study examines the complexity of linear algebra. Complexity means how much work, or the number of calculations or time it takes to perform a task. As linear algebra is used more and more in different fields, it becomes useful to study ways of reducing the amount of work required to complete basic procedures.

A Study Of Finite Symmetrical Groups, May Majid

Theses Digitization Project

This study investigated finite homomorphic images of several progenitors, including 2*⁵ : S₅, 2*⁶ : A₆, and 3*⁵ : C₅ The technique of manual of double coset enumeration is used to construct several groups by hand and computer-based proofs are given for the isomorphism types of the groups that are not constructed.

Enumeration And Symmetric Presentations Of Groups, With Music Theory Applications, Jesse Graham Train

Theses Digitization Project

The purpose of this project is to construct groups as finite homomorphic images of infinite semi-direct products. In particular, we will construct certain classical groups and subgroups of sporadic groups, as well groups with applications to the field of music theory.

Plasma Confinement: Mathematical Modeling Of A Fusion Reactor, James Scott Jones

Theses Digitization Project

This study will discuss currently used power sources and their drawbacks, leading to covering fusion as an energy source and its potential. Fusion has three significant important advantages: Fuel reserves, safety, and environment. A significant amount of fuel reserves comes from the natural occurrence in ocean water of deuterium at a 1 to 6700 ratio, accounting for the energy supply being on the order of 2 billion years. Fusion does not produce any greenhouse gases and its only 'exhaust' is that of harmless inert helium.

Hyperbolicity Equations For Knot Complements, Christopher Martin Jacinto

Theses Digitization Project

This study analyzes Carlo Petronio's paper, An Algorithm Producing Hyperbolicity Equations for a Link Complement in S³. Using the figure eight knot as an example, we will explain how Petronio's algorithm was able to decompose the knot complement of an alternating knot into tetrahedra. Then, using the vertex invariants of these tetrahedra, we will explain how Petronio was able to create hyperbolicity equations.

A Study Of Different Modeling Choices For Simulating Platelets Within The Immersed Boundary Method, Varun Shankar, Grady B. Wright, Aaron L. Fogelson, Robert M. Kirby

Mathematics Faculty Publications and Presentations

The Immersed Boundary (IB) method is a widely-used numerical methodology for the simulation of fluid–structure interaction problems. The IB method utilizes an Eulerian discretization for the fluid equations of motion while maintaining a Lagrangian representation of structural objects. Operators are defined for transmitting information (forces and velocities) between these two representations. Most IB simulations represent their structures with piecewise linear approximations and utilize Hookean spring models to approximate structural forces. Our specific motivation is the modeling of platelets in hemodynamic flows. In this paper, we study two alternative representations – radial basis functions (RBFs) and Fourier-based (trigonometric polynomials and spherical …

Generalized Least-Squares Regressions Ii: Theory And Classification, Nataniel Greene

Publications and Research

In the first paper of this series, a variety of known and new symmetric and weighted least-squares regression methods were presented with efficient derivations. This paper continues and generalizes the previous work with a theory for deriving, analyzing, and classifying all symmetric and weighted least-squares regression methods.

Bekenstein Bound Of Information Number N And Its Relation To Cosmological Parameters In A Universe With And Without Cosmological Constant, Ioannis Haranas, Ioannis Gkigkitzis

Physics and Computer Science Faculty Publications

Bekenstein has obtained is an upper limit on the entropy S, and from that, an information number bound N is deduced. In other words, this is the information contained within a given finite region of space that includes a finite amount of energy. Similarly, this can be thought as the maximum amount of information required to perfectly describe a given physical system down to its quantum level. If the energy and the region of space are finite then the number of information N required in describing the physical system is also finite. In this short letter two information number …

The Number Of Information Bits Related To The Minimum Quantum And Gravitational Masses In A Vacuum Dominated Universe, Ioannis Haranas, Ioannis Gkigkitzis

Physics and Computer Science Faculty Publications

Wesson obtained a limit on quantum and gravitational mass in the universe by combining the cosmological constant Λ, Planck’s constant h , the speed of light c, and also the gravitational constant G. The corresponding masses are 2.0 x 10^-62 and 2.3 x 10⁵⁴ kg respectively, and in general can be obtained with the help of a generic dimensional analysis, or from an analysis where the cosmological constant appears in a four dimensional space-time and as a result of a higher dimensional reduction. In this paper our goal is to establish a relation for both quantum …

Number Of Information And Its Relation To The Cosmological Constant Resulting From Landauer’S Principle, Ioannis Gkigkitzis, Ioannis Haranas, Samantha Kirk

Physics and Computer Science Faculty Publications

Using a recent published formula for the number of information N that results from Landauer’s principle we obtain an expression for the cosmological constant Λ . Next, assuming the universe as a system of mass M satisfying Landauer’s principle and eliminating its mass M from the given expression for the number of information, we obtain a new expression that agrees with the one derived by Lloyd. Furthermore, we modify the generalized entropy relation and three equivalent entropy expressions are obtained. Finally, in two different universes the time rate of change of the entropy is calculated. In a flat universe the …

Pointless Hyperelliptic Curves, Ryan P. Becker, Darren B. Glass

Math Faculty Publications

In this paper we consider the question of whether there exists a hyperelliptic curve of genus g which is defined over but has no rational points over for various pairs . As an example of such a result, we show that if p is a prime such that is also prime then there will be pointless hyperelliptic curves over of every genus.

Solitary Wave And Shock Wave Solutions Of The Variants Of Boussinesq Equations, Houria Triki, Abhinandan Chowdhury, Anjan Biswas

Math Faculty Publications

This paper obtains the solitary wave as well as the shock wave solutions of the variants of the Boussinesq equations in both (1+1) and (1+2) dimensions. The domain restrictions are also identified in the process.

A State-Dependent Delay Equation With Negative Feedback And ‘Mildly Unstable’ Rapidly Oscillating Periodic Solutions, Benjamin B. Kennedy

Math Faculty Publications

We consider state-dependent delay equations of the form x′(t)=f(x(t−d(x(t)))) where d is smooth and f is smooth, bounded, nonincreasing, and satisfies the negative feedback condition xf(x)x≠0. We identify a special family of such equations each of which has a ``rapidly oscillating" periodic solution p. The initial segment p0 of p is the fixed point of a return map R that is differentiable in an appropriate setting.

We show that, although all the periodic solutions p we consider are unstable, the stability can be made arbitrarily mild in the sense that, given ϵ>0, we can choose f and d such …

Symmetric Random Walks On Homeo+(R), B. Deroin, V. Klepstyn, A. Navas, Kamlesh Parwani

Faculty Research and Creative Activity

We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of "global stability at a finite distance": roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the …

A Gentle Introduction To Pythontex, Andrew Mertz, William Slough

Faculty Research and Creative Activity

No abstract provided.

School Choice As A One-Sided Matching Problem: Cardinal Utilities And Optimization, Sinan Aksoy, Alexander Adam Azzam, Chaya Coppersmith, Julie Glass, Gizem Karaali, Xueying Zhao, Xinjing Zhu

Pomona Faculty Publications and Research

The school choice problem concerns the design and implementation of matching mechanisms that produce school assignments for students within a given public school district. Previously considered criteria for evaluating proposed mechanisms such as stability, strategyproofness and Pareto efficiency do not always translate into desirable student assignments. In this note, we explore a class of one-sided, cardinal utility maximizing matching mechanisms focused exclusively on student preferences. We adapt a well-known combinatorial optimization technique (the Hungarian algorithm) as the kernel of this class of matching mechanisms. We find that, while such mechanisms can be adapted to meet desirable criteria not met by …

On The Norm Closure Problem For Complex Symmetric Operators, Stephan Ramon Garcia, Daniel E. Poore '11

Pomona Faculty Publications and Research

We prove that the set of all complex symmetric operators on a separable, infinite-dimensional Hilbert space is not norm closed.

On The Maximal Operators Of Vilenkin-Fejéer Means, George Tephnadze

Turkish Journal of Mathematics

The main aim of this paper is to prove that the maximal operator \overset{\sim}{\sigma}^{\ast}f:=\underset{n \in P} \sup\frac{ \sigma _nf }{\log^2 (n+1)} is bounded from the Hardy space H_{1/2} to the space L_{1/2}, where \sigma _nf are Fejér means of bounded Vilenkin-Fourier series.

Contact Hypersurfaces Of A Bochner-Kaehler Manifold, Amalendu Ghosh, Ramesh Sharma

Mathematics Faculty Publications

We have studied contact metric hypersurfaces of a Bochner-Kaehler manifold and obtained the following two results: (1) A contact metric constant mean curvature (C M C) hypersurface of a Bochner-Kaehler manifold is a (k, µ)-contact manifold, and (2) If M is a compact contact metric C M C hypersurface of a Bochner-Kaehler manifold with a conformal vector field V that is neither tangential nor normal anywhere, then it is totally umbilical and Sasakian, and under certain conditions on V , is isometric to a unit sphere.

Solutions Of Dynamic Equations On Time Scales With Jumps, Kayode Daniel Olumoyin

Theses, Dissertations and Capstones

To obtain the solution of first order dynamic equations on time scales with jumps, a good question to ask is, how many initial conditions will be needed? We shall show that you only need the initial condition that gives you either the initial position or the initial velocity. The solution at each left scattered point in the time scale can be obtained analytically. With this approach we shall write the general form of the solution of a first order dynamic equations on time scales with jumps. To do this we shall use the Hilger derivative, anti-derivatives, the Hilger Complex plane, …

Chromatic Bounds On Orbital Chromatic Roots, Dae Hyun Kim, Alexander H. Mun, Mohamed Omar

All HMC Faculty Publications and Research

Given a group G of automorphisms of a graph Γ, the orbital chromatic polynomial OPΓ,G(x) is the polynomial whose value at a positive integer k is the number of orbits of G on proper k-colorings of Γ. In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in R, extending Thomassen's famous result (see \cite{Thomassen}) that chromatic roots are dense in [32/27,∞). Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root …

Social Aggregation In Pea Aphids: Experiment And Random Walk Modeling, Christa Nilsen, John Paige, Olivia Warner, Benjamin Mayhew, Ryan Sutley, Matthew Lam '15, Andrew J. Bernoff, Chad M. Topaz

All HMC Faculty Publications and Research

From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid, Acyrthosiphon pisum. Specifically, we conduct experiments to track the motion of aphids walking in a featureless circular arena in order to deduce individual-level rules. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. …

Selma Karaali And Artemis Karaali, Gizem Karaali

Pomona Faculty Publications and Research

A blog about grandmothers in the fields of science, technology, engineering and mathematics.

A Method For Generating Realistic Correlation Matrices, Johanna S. Hardin, Stephan Ramon Garcia, David Golan

Pomona Faculty Publications and Research

Simulating sample correlation matrices is important in many areas of statistics. Approaches such as generating Gaussian data and finding their sample correlation matrix or generating random uniform $[-1,1]$ deviates as pairwise correlations both have drawbacks. We develop an algorithm for adding noise, in a highly controlled manner, to general correlation matrices. In many instances, our method yields results which are superior to those obtained by simply simulating Gaussian data. Moreover, we demonstrate how our general algorithm can be tailored to a number of different correlation models. Using our results with a few different applications, we show that simulating correlation matrices …

The Graphic Nature Of The Symmetric Group, J.L. Brumbaugh '13, Madeleine Bulkow '14, Luis Alberto Garcia '14, Stephan Ramon Garcia, Matt Michal '15, Andrew P. Turner '14

Pomona Faculty Publications and Research

We investigate a remarkable class of exponential sums which are derived from the symmetric groups and which display a diverse array of visually appealing features. Our interest in these expressions stems not only from their astounding visual properties, but also from the fact that they represent a novel and intriguing class of supercharacters.

The Brave New World Of Open Access & Creative Commons: A Humanistic Experiment In Mathematical Publishing, Gizem Karaali

Pomona Faculty Publications and Research

In January 2011 the Journal of Humanistic Mathematics (JHM) published its first issue. JHM (http://scholarship.claremont.edu/jhm) is an online-only, peer-reviewed, open-access journal which has passed the all-important ten-thousand-download barrier in its first anniversary. In order to remain faithful to the fundamental principles of open access, JHM uses Creative Commons licensing, where authors retain copyright of their work, but others are free to reuse them (with proper attribution). In this note I share and reflect upon our experience with open access and Creative Commons.

Existence And Qualitative Properties Of Solutions For Nonlinear Dirichlet Problems, Alfonso Castro, Jorge Cossio, Carlos Vélez

All HMC Faculty Publications and Research

Sign-changing solutions to semilinear elliptic problems in connection with their Morse indices. To this end, we first establish a priori bounds for one-sign solutions. Secondly, using abstract saddle point principles we find large augmented Morse index solutions. In this part, extensive use is made of critical groups, Morse index arguments, Lyapunov-Schmidt reduction, and Leray-Schauder degree. Finally, we provide conditions under which these solutions necessarily change sign and we comment about further qualitative properties.

Generalized Least-Squares Regressions I: Efficient Derivations, Nataniel Greene

Publications and Research

Ordinary least-squares regression suffers from a fundamental lack of symmetry: the regression line of y given x and the regression line of x given y are not inverses of each other. Alternative symmetric regression methods have been developed to address this concern, notably: orthogonal regression and geometric mean regression. This paper presents in detail a variety of least squares regression methods which may not have been known or fully explicated. The derivation of each method is made efficient through the use of Ehrenberg's formula for the ordinary least-squares error and through the extraction of a weight function g(b) which characterizes …

Exploring Diagonals In The Calkin-Wilf Tree, Matthew Gagne

Undergraduate Review

For centuries, people have been interested in patterns. Even in that which appears random, humans have been trying to understand the underlying order of things. Mathematicians throughout time have studied many phenomena, including infinite sequences of numbers and have been able, at times, to see structure. Many have found the satisfaction, even joy, of discovering patterns in sequences. A typical way to describe this is by a recursive formula. A recursive definition defines a term in the sequence using the previous terms in the sequence. Even more satisfying than a recursive formula is a closed formula. With this, one can …

The Fibonacci Sequence And Hosoya's Triangle, Jeffrey Lee Smith

Theses Digitization Project

The purpose of this thesis is to study the Fibonacci sequence in a context many are unfamiliar with. A triangular array of numbers, similar looking to Pascal's triangle, was constructed a few decades ago and is called Hosoya's triangle. Each element within the triangle is created using Fibonacci numbers.