Mathematics Commons^™

A Modeling Study Of The History-Dependence Of Conduction Delay In Unmyelinated Axons, Yang Zhang

Dissertations

Conduction delay in an axon is the time required for an action potential to propagate between two positions. It is a function of the axon’s passive membrane properties, voltage-gated ion channels and the Na+/K+ pump, and can be substantially affected by neuromodulators. The conduction delay of action potential, generated by the pyloric dilator (PD) neuron unmyel i nated motor axon in the stomatogastric nervous system, shows significant variability with ongoing bursting or Poisson stimulation. When the axon is stimulated, the mean value (Dmean) and coefficient variation of conduction delay (CV-D) slowly increase with time (slow timescale effect), and the relationship …

Particle Filtering For Frequency Estimation From Acoustic Time-Series In Dispersive Media, Nattapol Aunsri

Dissertations

Acoustic signals propagating in the ocean carry information about geometry and environmental parameters within the propagation medium. Accurately retrieving this information leads us to effectively estimate parameters that are of utmost importance in environmental studies, climate monitoring, and defense. This dissertation focuses on the development of sequential Bayesian filtering methods to obtain accurate estimates of instantaneous frequencies using Short Term Fourier Transforms within the acoustic field measured at an array of hydrophones, which can be used in a subsequent step for the estimation of propagation related parameters. We develop a particle filter to estimate these frequencies along with modal amplitudes, …

Mathematics In Rural Mexico: Motivations And Barriers For Adult Women Learners, Janel Wilson Williams, Dr. Jacqualine Taylor Voyles

Journal of Undergraduate Research

Math educators have long realized that women in general have shied away from involvement in the mathematical world. Current research in the mathematics education field studies the motivations and barriers women have to studying math. My research extends this topic to a small village in Mexico: La Estancia. By studying factors that encourage or discourage adult Mexican women from studying math in rural settings, important influences can be identified. The identification of these elements can lead to the development of important intervention programs targeted at minimizing negative influences while capitalizing on and strengthening the existing positive influences.

Pre-Service University Math Education Programs In Japan, Vonn R. Christenson, Dr. Blake Peterson

Journal of Undergraduate Research

For years the United States has lagged behind the international community in mathematical performance at the elementary and secondary education levels. In the Third International Mathematics and Science Study (TIMSS), differences between the teaching style in Japan and the United States were studied, identified and recorded. Through this study, it was discovered that students in the United States “encounter mathematics that is at a lower level, is somewhat superficial, and is not as fully or coherently developed” as the mathematics in Japan.ⁱ

Weighted Inverse Weibull Distribution: Statistical Properties And Applications, Valeriia Sherina, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this paper, the weighted inverse Weibull (WIW) class of distributions is proposed and studied. This class of distributions contains several models such as: length-biased, hazard and reverse hazard proportional inverse Weibull, proportional inverse Weibull, inverse Weibull, inverse exponential, inverse Rayleigh, and Fr´echet distributions as special cases. Properties of these distributions including the behavior of the hazard function, moments, coefficients of variation, skewness, and kurtosis, R´enyi entropy and Fisher information are presented. Estimates of the model parameters via method of maximum likelihood (ML) are presented. Extensive simulation study is conducted and numerical examples are given.

Solving Laplace’S Equation On A Personal Computer, Michael Higley, Dr. Peter Bates

Journal of Undergraduate Research

Laplace’s equation is a differential equations that describes many physical properties in steady state systems. It can be used to describe heat distributions, displacements in elastic media, or electrostatic fields. My interest is in solving an electrostatics problem where the domain—the area in which the equation has physical meaning—is irregular enough that the only way to solve the problem is to use some sort of numerical approximation.

Ratios Of Positive Definite Matrices, Dena Plant, Dr. Wayne Barrett

Journal of Undergraduate Research

As our research began we focused on proving the hypothesis that for a 4 x 4 Positive Definite matrix A, the ratio

Length Minimizing Network For N Points On The K-Torus, W. Lauritz Petersen, Dr. Denise Halverson

Journal of Undergraduate Research

Our project consisted of several components with the end goal being: given n points on the ktorus (a donut with k number of holes), what would the shortest path network be to connect all the n points (network meaning a network of paths connecting each of the n-points).

Optimal Energy Recovery In Finite Time For Linearly Dissipative Systems, Mark Meilstrup, Dr. Scott Glasgow

Journal of Undergraduate Research

Many physical processes involve the loss or dissipation of energy. For example, when pushing a box across the floor, some of the energy that is spent will be lost to frictional forces. Our research is concerned with the study of the energy which is lost, and also the energy which is recoverable, in the more complex setting of a linearly dissipative optical system. We consider some electromagnetic pulse, such as a laser beam, entering and interacting with an absorbing medium, such as glass or some gas. When the pulse exits the medium, some of the energy has been absorbed, and …

Orca Research Report, Jeanna Chilton, Dr. Gary Lawlor

Journal of Undergraduate Research

The work I was able to do for the research on the minimal surface area of the octahedron was time consuming, but unfortunately did not bring about any profound knowledge for anyone besides myself.

On Cubic Multisections Of Eisenstein Series, Andrew Alaniz, Timothy Huber

School of Mathematical & Statistical Sciences Faculty Publications

A systematic procedure for generating cubic multisections of Eisenstein series is given. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo three. We prove that the resulting series are rational functions of η(τ) and η(3τ), where η is the Dedekind eta function. A more general treatment of cubic dissection formulas is given by describing the dissection operators in terms of linear transformations. These operators exhibit properties that mirror those of similarly defined quintic operators.

Algebraic Detection Of Flexibility Of Polyhedral Structures With Applications To Robotics And Chemistry, Stephen Fox Fcrh '11, Robert Lewis

The Fordham Undergraduate Research Journal

Many problems arise in biochemistry, robotics, and other fields in which flexibility of a polygonal or polyhedral structure plays an important role. In biochemistry, the flexibility and folding of molecules is an important factor in drug design and is a subject of ongoing research (Erickson et al. 2004). In robotics, stable configurations of manipulators (e.g., a mechanical arm grasping) as well as mechanical joints for locomotion (e.g., walking) must be calculated for safe, smooth movement.

Paved With Good Intentions: Analysis Of A Randomized Block Kaczmarz Method, Deanna Needell, Joel A. Tropp

CMC Faculty Publications and Research

The block Kaczmarz method is an iterative scheme for solving overdetermined least-squares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving …

Multiscale Geometric Modeling Of Macromolecules I: Cartesian Representation, Kelin Xia, Xin Feng, Zhan Chen, Yiying Tong, Guo-Wei Wei

Mathematical Sciences: Faculty Publications

This paper focuses on the geometric modeling and computational algorithm development of biomolecular structures from two data sources: Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian) representation. Molecular surface (MS) contains non-smooth geometric singularities, such as cusps, tips and self-intersecting facets, which often lead to computational instabilities in molecular simulations, and violate the physical principle of surface free energy minimization. Variational multiscale surface definitions are proposed based on geometric flows and solvation analysis of biomolecular systems. Our approach leads to geometric and potential driven Laplace–Beltrami flows for biomolecular surface evolution and formation. The …

A Simple Proof For The Number Of Tilings Of Quartered Aztec Diamonds, Tri Lai

Department of Mathematics: Faculty Publications

We get four quartered Aztec diamonds by dividing an Aztec diamond region by two zigzag cuts passing its center. W. Jockusch and J. Propp (in an unpublished work) found that the number of tilings of quartered Aztec diamonds is given by simple product formulas. In this paper we present a simple proof for this result.

Using Prior Knowledge And Learning From Experience In Estimation Of Distribution Algorithms, Mark Walter Hauschild

Dissertations

Estimation of distribution algorithms (EDAs) are stochastic optimization techniques that explore the space of potential solutions by building and sampling explicit probabilistic models of promising candidate solutions. One of the primary advantages of EDAs over many other stochastic optimization techniques is that after each run they leave behind a sequence of probabilistic models describing useful decompositions of the problem. This sequence of models can be seen as a roadmap of how the EDA solves the problem. While this roadmap holds a great deal of information about the problem, until recently this information has largely been ignored. My thesis is that …

Quantitative K-Theory And Spin Chern Numbers [Dataset], Terry A. Loring

Math and Statistics Datasets

We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this is two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine what values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to …

Assessing The Optimal Virulence Of Malaria‐Targeting Mosquito Pathogens: A Mathematical Study Of Engineered Metarhizium Anisopliae, Bernhard Konrad, Michael R. Lindstrom, Anja Gumpinger, Jielin Zhu, Daniel Coombs

School of Mathematical & Statistical Sciences Faculty Publications

Background

Metarhizium anisopliae is a naturally occurring fungal pathogen of mosquitoes. Recently, Metarhizium has been engineered to act against malaria by directly killing the disease agent within mosquito vectors and also effectively blocking onward transmission. It has been proposed that efforts should be made to minimize the virulence of the fungal pathogen, in order to slow the development of resistant mosquitoes following an actual deployment.

Results

Two mathematical models were developed and analysed to examine the efficacy of the fungal pathogen. It was found that, in many plausible scenarios, the best effects are achieved with a reduced or minimal pathogen …

A Sampling Of Popular Books For Numeracy Readers, Michael T. Catalano

Numeracy

Popular books on quantitative themes are seemingly more available than ever. In this book review, we look at five such books from a wide range of authors. Although the books are written for diverse audiences, all provide examples and discussion of concepts that could be used in courses with quantitative literacy objectives. The books are Guesstimation and Guesstimation 2.0 by Lawrence Weinstein and John A. Adam, and Weinstein, respectively; Turning Numbers into Knowledge: Mastering the Art of Problem Solving, by Jonathan G. Koomey; How to Measure Anything: Finding The Value of “Intangibles” in Business, by Douglas W. Hubbard; and …

Variations Of The Feast Eigenvalue Algorithm, Stephanie Kajpust

Dissertations, Master's Theses and Master's Reports - Open

FEAST is a recently developed eigenvalue algorithm which computes selected interior eigenvalues of real symmetric matrices. It uses contour integral resolvent based projections. A weakness is that the existing algorithm relies on accurate reasoned estimates of the number of eigenvalues within the contour. Examining the singular values of the projections on moderately-sized, randomly-generated test problems motivates orthogonalization-based improvements to the algorithm. The singular value distributions provide experimentally robust estimates of the number of eigenvalues within the contour. The algorithm is modified to handle both Hermitian and general complex matrices. The original algorithm (based on circular contours and Gauss-Legendre quadrature) is …

Fractal Powers In Serrin's Swirling Vortex Solutions, Pavel Bělík, Douglas P. Dokken, Kurt Scholz, Mikhail M. Shvartsman

Faculty Authored Articles

We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by Serrin [Phil. Trans. Roy. Soc. London, Series A, Math & Phys. Sci. 271(1214) (1972), 325–360], where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events [Mon. Wea. Rev. 133(9) (2005), 2535–2551; Mon. Wea. Rev. 128(7) (2000), 2135–2164; Mon. Wea. Rev. 133(1) (2005), 97–119], indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest a different scaling of the velocity/radial distance dependence. Motivated …

On The Dynamics Of Laguerre’S Iteration Method For Finding The Nth Roots Of Unity, Pavel Bělík, Heechan Kang, Andrew Walsh, Emma Winegar

Faculty Authored Articles

Previous analyses of Laguerre’s iteration method have provided results on the behavior of this popular method when applied to the polynomials 𝑝𝑛(𝑧) = 𝑧𝑛 − 1, 𝑛 ∈ N. In this paper, we summarize known analytical results and provide new results. In particular, we study symmetry properties of the Laguerre iteration function and clarify the dynamics of the method. We show analytically and demonstrate computationally that for each 𝑛 ≥ 5 the basin of attraction to the roots is a subset of an annulus that contains the unit circle and whose Lebesgue measure shrinks to zero as 𝑛 → ∞. …

Gaussian Integer Straus-Erdos, Eugen J. Ionascu

Faculty Bibliography

No abstract provided.

Pythagorean Triples Challenge, Thomas Moore

Mathematics Faculty Publications

No abstract provided.

Ideal Projections And Forcing Projections, Sean Cox, Martin Zeman

Mathematics and Applied Mathematics Publications

It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman-Magidor-Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove: (1) If I is a normal ideal on ω2 which satisfies stationary antichain catching, then there is an inner model with a Woodin cardinal; (2) For any n ∈ ω, it is consistent relative to large cardinals that there is a normal ideal I on ωn which satisfies projective antichain catching, yet I is not saturated (or even strong). This provides a negative answer to Open Question number 13 from …

A Linear Iteration Algorithm For A Second-Order Energy Stable Scheme For A Thin Film Model Without Slope Selection, Wenbin Chen, Cheng Wang, Xiaoming Wang, Steven M. Wise

Mathematics and Statistics Faculty Research & Creative Works

We present a linear iteration algorithm to implement a second-order energy stable numerical scheme for a model of epitaxial thin film growth without slope selection. the PDE, which is a nonlinear, fourth-order parabolic equation, is the L 2 gradient flow of the energy d x. the energy stability is preserved by a careful choice of the second-order temporal approximation for the nonlinear term, as reported in recent work (Shen et al. in SIAM J Numer Anal 50:105-125, 2012). the resulting scheme is highly nonlinear, and its implementation is non-trivial. in this paper, we propose a linear iteration algorithm to solve …

Coloring Hypercomplete And Hyperpath Graphs, Yusuf Ci̇van, Demet Taylan

Turkish Journal of Mathematics

Given a graph G with an induced subgraph H and a family F of graphs, we introduce a (hyper)graph H_H(G;F)=(V_H, E_H), the hyper-H (hyper)graph of G with respect to F, whose vertices are induced copies of H in G, and \{H_1,H_2,\ldots,H_r\} \in E_H if and only if the induced subgraph of G by the set \cup_{i=1}^r H_i is isomorphic to a graph F in the family F, and the integer r is the least integer for F with this property. When H is a k-complete or a k-path of G, we abbreviate H_{K_k}(G;F) and H_{P_k}(G;F) to H_k(G;F) and HP_k(G;F), respectively. …

Euler-Seidel Matrices Over F_P, Nesri̇n Tutaş

Turkish Journal of Mathematics

A Euler--Seidel matrix is determined by an infinite sequence whose elements are given by recursion. The recurrence relations are investigated for numbers and polynomials such as hyperharmonics, Lucas numbers, and Euler and Genocchi polynomials. Linear recurring sequences in finite fields are employed, for instance, in coding theory and in several branches of electrical engineering. In this work, we define the period of a Euler--Seidel matrix over a field F_p with p elements, where p is a prime number. We give some results for the matrix whose initial sequence is \{s_r(n)\}_{n=0}^{\infty}, where s_r(n)=\sum_{k=0}^n {\binom{n}{k}}^r, n \geq 0, and r is a …

On Betti Series Of The Universal Modules Of Second Order Derivations Of \Frac{K[X_1,X_2,...,X_S]}{(F)}, Ali̇ Erdoğan, Hali̇se Meli̇s Teki̇n Akçi̇n

Turkish Journal of Mathematics

Let R be a coordinate ring of an affine irreducible curve represented by \frac{k[x_1,x_2,...,x_s]}{(f)} and m be a maximal ideal of R. In this article, the Betti series of \Omega_2(R_m) is studied. We proved that the Betti series of \Omega_2(R_m), where \Omega_2(R_m) denotes the universal module of second order derivations of R_m, is a rational function under some conditions.

Some Results On T-Noncosingular Modules, Rachid Tribak

Turkish Journal of Mathematics

The notion of T-noncosingularity of a module has been introduced and studied recently. In this article, a number of new results of this property are provided. It is shown that over a commutative semilocal ring R such that Jac(R) is a nil ideal, every T-noncosingular module is semisimple. We prove that for a perfect ring R, the class of T-noncosingular modules is closed under direct sums if and only if R is a primary decomposable ring. Finitely generated T-noncosingular modules over commutative rings are shown to be precisely those having zero Jacobson radical. We also show that for a simple …