Mathematics Commons^™

Numerical Simulations Of Dense Granular Systems With And Without Cohesive Effects, Lenka Kovalcinova

Dissertations

Granular materials are collections of objects ranging from sand grains that form sand piles or even sand castles to collections of large objects such as a group of meteors in outer space. The considered range of sizes of granular particles is such that the effect of thermal fluctuations is not relevant. However, the interaction between the particles may be very complex, involving inelasticity and friction, in addition to repulsive and possibly attractive interaction forces. These interactions that may be history dependent, make the systems that consist of a large number of particles complex to analyze and difficult to understand using …

Efficient High-Order Integral Equation Methods For The Heat Equation, Shaobo Wang

Dissertations

Efficient high-order integral equation methods have been developed for solving the boundary value problems of the heat equation with complex geometries in two and three dimensions. First of all, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials. Some advantages of the integral formulation as compared with standard finite difference and finite element methods include reduction of the dimension of the problem by one, high order accuracy, unconditional stability, insensitivity to different geometries, and elimination of truncating the computational domain and the need of artificial …

A Survey Of Graphs Of Minimum Order With Given Automorphism Group, Jessica Alyse Woodruff

Math Theses

We survey vertex minimal graphs with prescribed automorphism group. Whenever possible, we also investigate the construction of such minimal graphs, confirm minimality, and prove a given graph has the correct automorphism group.

Stabilizing The Lorenz Flows Using A Closed Loop Quotient Controller, James P. Braselton, Yan Wu

Mathematical Sciences: Faculty Publications

In this study, we introduce a closed loop quotient controller into the three-dimensional Lorenz system. We then compute the equilibrium points and analyze their local stability. We use several examples to illustrate how cross-sections of the basins of attraction for the equilibrium points look for various parameter values. We then provided numerical evidence that with the controller, the controlled Lorenz system cannot exhibit chaos if the equilibrium points are locally stable.

Batched Stochastic Gradient Descent With Weighted Sampling, Deanna Needell, Rachel Ward

CMC Faculty Publications and Research

We analyze a batched variant of Stochastic Gradient Descent (SGD) with weighted sampling distribution for smooth and non-smooth objective functions. We show that by distributing the batches computationally, a significant speedup in the convergence rate is provably possible compared to either batched sampling or weighted sampling alone. We propose several computationally efficient schemes to approximate the optimal weights, and compute proposed sampling distributions explicitly for the least squares and hinge loss problems. We show both analytically and experimentally that substantial gains can be obtained

Bayesian Peer Calibration With Application To Alcohol Use, Miles Q. Ott, Joseph W. Hogan, Krista J. Gile, Crystal Linkletter, Nancy P. Barnett

Statistical and Data Sciences: Faculty Publications

Peers are often able to provide important additional information to supplement self-reported behavioral measures. The study motivating this work collected data on alcohol in a social network formed by college students living in a freshman dormitory. By using two imperfect sources of information (self-reported and peer-reported alcohol consumption), rather than solely self-reports or peer-reports, we are able to gain insight into alcohol consumption on both the population and the individual level, as well as information on the discrepancy of individual peer-reports. We develop a novel Bayesian comparative calibration model for continuous, count and binary outcomes that uses covariate information to …

On The Expression Of Higher Mathematics In American Sign Language, John Tabak

Journal of Interpretation

Abstract

The grammar and vocabulary of higher mathematics are different from the grammar and vocabulary of conversational English and conversational American Sign Language (ASL). Consequently, mathematical language presents interpreters with a unique set of challenges. This article characterizes those aspects of mathematical grammar that are peculiar to the subject. (A discussion of mathematical vocabulary and its expression in ASL can be found elsewhere (Tabak, 2014).) An increased awareness of the grammar of mathematical language will prove useful to those interpreters for the deaf and deaf mathematics professionals seeking to express higher mathematics in ASL.

In this article one will, for …

Invertibility Of Submatrices Of The Pascal Matrix And Birkhoff Interpolation, Scott N. Kersey

Mathematical Sciences: Faculty Publications

The infinite upper triangular Pascal matrix is T = [( j )i] for 0 ≤ i, j. It is easy to see that any leading principle square submatrix is triangular with determinant 1, hence invertible. In this paper, we investigate the invertibility of arbitrary square submatrices Tr, c comprised of rows r = [r0, … , rm ] and columns c = c0 , … , cm[] of T. We show that Tr, c is invertible r ≤ c i.e., ri ≤ ci for i = 0, …, m(), or equivalently, iff all diagonal entries are nonzero. To prove this …

Fast Cycles Detecting In Non-Linear Discrete Systems, Dmitriy Dmitrishin, Elena Franzheva, Alexander M. Stokolos

Mathematical Sciences: Faculty Publications

In the paper below we consider a problem of stabilization of a priori unknown unstable periodic orbits in non-linear autonomous discrete dynamical systems. We suggest a generalization of a non-linear DFC scheme to improve the rate of detecting T-cycles. Some numerical simulations are presented.

Prediction And Optimal Scheduling Of Advertisements In Linear Television, Mark J. Panaggio, Pak-Wing Fok, Ghan S. Bhatt, Simon Burhoe, Michael Capps, Christina J. Edholm, Fadoua El Moustaid, Tegan Emerson, Star-Lena Estock, Nathan Gold, Ryan Halabi, Madelyn Houser, Peter R. Kramer, Hsuan-Wei Lee, Qingxia Li, Weiqiang Li, Dan Lu, Yuzhou Qian, Louis F. Rossi, Deborah Shutt, Vicky Chuqiao Yang, Yingxiang Zhou

Mathematical Sciences Faculty Research

Advertising is a crucial component of marketing and an important way for companies to raise awareness of goods and services in the marketplace. Advertising campaigns are designed to convey a marketing image or message to an audience of potential consumers and television commercials can be an effective way of transmitting these messages to a large audience. In order to meet the requirements for a typical advertising order, television content providers must provide advertisers with a predetermined number of "impressions" in the target demographic. However, because the number of impressions for a given program is not known a priori and because …

Totally Null Surfaces In Neutral K¨Ahler 4-Manifolds, Nikos Georgiou, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

We study the totally null surfaces of the neutral K¨ahler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (α-planes) or anti-self-dual (β-planes) and so we consider α-surfaces and β-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the α-planes are integrable and α-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The β-surfaces are less known and our interest is mainly in their …

Improved Compressive Sensing Of Natural Scenes Using Localized Random Sampling, Victor J. Barranca, G. Kovačič, D. Zhou, D. Cai

Mathematics & Statistics Faculty Works

Compressive sensing (CS) theory demonstrates that by using uniformly-random sampling, rather than uniformly-spaced sampling, higher quality image reconstructions are often achievable. Considering that the structure of sampling protocols has such a profound impact on the quality of image reconstructions, we formulate a new sampling scheme motivated by physiological receptive field structure, localized random sampling, which yields significantly improved CS image reconstructions. For each set of localized image measurements, our sampling method first randomly selects an image pixel and then measures its nearby pixels with probability depending on their distance from the initially selected pixel. We compare the uniformly-random and localized …

Expert Elicitation On The Uncertainties Associated With Chronic Wasting Disease, Michael G. Tyshenko, Tamer Oraby, Shalu Darshan, Margit Westphal, Maxine C. Croteau, Willy Aspinall, Susie Elsaadany, Daniel Krewski, Neil Cashman

School of Mathematical & Statistical Sciences Faculty Publications

A high degree of uncertainty exists for chronic wasting disease (CWD) transmission factors in farmed and wild cervids. Evaluating the factors is important as it helps to inform future risk management strategies. Expert opinion is often used to assist decision making in a number of health, science, and technology domains where data may be sparse or missing. Using the "Classical Model" of elicitation, a group of experts was asked to estimate the most likely values for several risk factors affecting CWD transmission. The formalized expert elicitation helped structure the issues and hence provide a rational basis for estimating some transmission …

Quantum Dynamics, Entropy And Quantum Versions Of Maxwell’S Demon, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Several subjects which reside in the overlap area of quantum mechanics, statistical physics and thermodynamics are investigated in depth. This collection of subjects shares a common domain which is referred to as Maxwell’s demon. The classical version of this idea is introduced, and then, the contribution made by Szilard to the subject is presented. Several demons are considered, and it is shown that to best understand this area, quantum mechanics and the role information plays in it must be appreciated deeply.

Determinantal Representations Of Semihyperbolic Polynomials, Greg Knese

Mathematics Faculty Research

We prove a generalization of the Hermitian version of the Helton–Vinnikov determinantal representation for hyperbolic polynomials to the class of semihyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.

On The Geodesic Centers Of Polygonal Domains, Haitao Wang

Computer Science Faculty and Staff Publications

In this paper, we study the problem of computing Euclidean geodesic centers of a polygonal domain P of n vertices. We give a necessary condition for a point being a geodesic center. We show that there is at most one geodesic center among all points of P that have topologically-equivalent shortest path maps. This implies that the total number of geodesic centers is bounded by the size of the shortest path map equivalence decomposition of P, which is known to be O(n^{10}). One key observation is a pi-range property on shortest path lengths when points are moving. With these observations, …

From Chaos To Order Through Mixing, Dmitriy Dmitrishin, I. M. Skrinnik, Alexander M. Stokolos

Mathematical Sciences: Faculty Publications

In this article we consider the possibility of controlling the dynamics of nonlinear discrete systems. A new method of control is by mixing states of the system (or the functions of these states) calculated on previous steps. This approach allows us to locally stabilize a priori unknown cycles of a given length. As a special case, we have a cycle stabilization using nonlinear feedback. Several examples are considered.

Tolerant Compressed Sensing With Partially Coherent Sensing Matrices, Tobias Birnbaum, Yonina C. Eldar, Deanna Needell

CMC Faculty Publications and Research

We consider compressed sensing (CS) using partially coherent sensing matrices. Most of CS theory to date is focused on incoherent sensing, that is, columns from the sensing matrix are highly uncorrelated. However, sensing systems with naturally occurring correlations arise in many applications, such as signal detection, motion detection and radar. Moreover, in these applications it is often not necessary to know the support of the signal exactly, but instead small errors in the support and signal are tolerable. In this paper, we focus on d-tolerant recovery, in which support set reconstructions are considered accurate when their locations match the true …

Gender Representation On Journal Editorial Boards In The Mathematical Sciences, Chad M. Topaz, Shilad Sen

Faculty Publications

No abstract provided.

Generalizing The Gsvd, Mark Gockenbach

Department of Mathematical Sciences Publications

The generalized singular value decomposition (GSVD) of a pair of matrices is the natural tool for certain problems defined on Euclidean space, such as certain weighted least-squares problems, the result of applying Tikhonov regularization to such problems (sometimes called regularization with seminorms), and equality-constrained least-squares problems. There is an extension of the GSVD to pairs of bounded linear operators defined on Hilbert space that turns out to be a natural representation for analyzing the same problems in the infinite-dimensional setting.