Mathematics Commons^™

The Quantum Gromov-Hausdorff Propinquity, Frédéric Latrémolière

Mathematics Preprint Series

We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the GromovHausdorff distance to noncommutative geometry and strengthens Rieffel’s quantum Gromov-Hausdorff distance and Rieffel’s proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*- algebras.

Computing A Logarithm Of A Unitary Matrix With General Spectrum [Dataset], Terry A. Loring

Math and Statistics Datasets

We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary matrix, and also skew-Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near -1, lead to very non-Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force an Hermitian output creates accuracy issues which are avoided by the considered algorithm. A modification is introduced to deal properly with the J-skew symmetric unitary matrices. Applications …

The Kronecker-Weber Theorem: An Exposition, Amber Verser

Lawrence University Honors Projects

This paper is an investigation of the mathematics necessary to understand the Kronecker-Weber Theorem. Following an article by Greenberg, published in The American Mathematical Monthly in 1974, the presented proof does not use class field theory, as the most traditional treatments of the theorem do, but rather returns to more basic mathematics, like the original proofs of the theorem. This paper seeks to present the necessary mathematical background to understand the proof for a reader with a solid undergraduate background in abstract algebra. Its goal is to make what is usually an advanced topic in the study of algebraic number …

Dynkin Diagrams And Fusion Algebras, Christopher D. Goff

College of the Pacific Faculty Presentations

No abstract provided.

Leslie Matrices For Logistic Population Modeling, Bruce Kessler

Mathematics Faculty Publications

Leslie matrices are taught as a method of modeling populations in a discrete-time fashion with more detail in the tracking of age groups within the population. Leslie matrices have limited use in the actual modeling of populations, since when the age groups are summed, it is basically equivalent to discrete-time modeling assuming exponential population growth. The logistic model of population growth is more realistic, since it takes into account a carrying capacity for the environment of the population. This talk will describe an adjustment to the Leslie matrix approach for population modeling that is both takes into account the carrying …

Left Baire-1 Compositors And Continuous Functions, Jonald P. Fenecios, Emmanuel A. Cabral

Mathematics Faculty Publications

We showed that the class of left Baire-1 compositors is precisely the class of continuous functions. This answers a characterization problem posed in the work by Zhao (2007, page 550) and settles in the affirmative the conjecture in the work by Fenecios and Cabral (2012, page 43). Moreover, based on the above result we provide a new proof that the class of functions BS,T where S is the class of all positive continuous functions and T is the class of all positive constants as defined in the work by Bakowska and Pawlak (2010) is exactly the class of all continuous …

Quotients Of Gaussian Primes, Stephan Ramon Garcia

Pomona Faculty Publications and Research

It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"

An Interface-Capturing Regularization Method For Solving The Equations For Two-Fluid Mixtures, Jian Du, Robert D. Guy, Aaron L. Fogelson, Grady B. Wright, James P. Keener

Mathematics Faculty Publications and Presentations

Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid solvent (water). The two-fluid model is a widely used approach to described gel mechanics, in which both network and solvent coexist at each point of space and their relative abundance is described by their volume fractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel and regions of pure solvent, resulting in a degenerate network momentum equation where the network volume fraction vanishes. To overcome this difficulty, …

Correlation Properties Of Collective Motion In Bacterial Suspensions, Shawn D. Ryan, Andrey Sokolov, Leonid Berlyand, Igor S. Aranson

Mathematics and Statistics Faculty Publications

The study of collective motion in bacterial suspensions has been of significant recent interest. To better understand the non-trivial spatio-temporal correlations emerging in the course of collective swimming in suspensions of motile bacteria, a simple model is employed: a bacterium is represented as a force dipole with size, through the use of a short-range repelling potential, and shape. The model emphasizes two fundamental mechanisms: dipolar hydrodynamic interactions and short-range bacterial collisions. Using direct particle simulations validated by a dedicated experiment, we show that changing the swimming speed or concentration alters the time scale of sustained collective motion, consistent with experiment. …

The Behavior Of Normality When Iteratively Finding The Normal To A Line In An Lp Geometry, David L. Farnsworth, Joshua M. Fitzhugh

Articles

The normal direction to the normal direction to a line in Minkowski geometries generally does not give the original line. We show that in lp geometries with p > 1 repeatedly finding the normal line through the origin gives sequences of lines that monotonically approach specific lines of symmetry of the unit circle. Which lines of symmetry that are approached depends upon the value of p and the slope of the initial line.

A Subelliptic Analogue Of Aronson-Serrin's Harnack Inequality, Luca Capogna, Giovanna Citti, Garrett Rea

Mathematics Sciences: Faculty Publications

We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE,(Formula presented.) in cylinders Ω × (0,T) where Ω ⊂ M is an open subset of a manifold M endowed with control metric d corresponding to a system of Lipschitz continuous vector fields X=(X_1,..., X_m) and a measure dσ. We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincaré inequality in the metric measure space (M,d,dσ). We also show that such hypothesis hold for a class of Riemannian metrics gε collapsing to a sub-Riemannian metric limε → …

Supplementary Balance Laws For Cattaneo Heat Propagation, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this work we determine for the Cattaneo heat propagation system all the supplementary balance laws (conservation laws ) of the same order (zero) as the system itself and extract the constitutive relations (expression for the internal energy) dictated by the Entropy Principle. The space of all supplementary balance laws (having the functional dimension 8) contains four original balance laws and their deformations depending on 4 functions of temperature (λ⁰(ϑ),K^A (ϑ), A = 1, 2, 3). The requirements of the II law of thermodynamics leads to the exclusion of three functional degrees (K^A= 0, A …

Ct Angiography In The Detection Of Carotid Body Enlargement In Patients With Hypertension And Heart Failure, Sreejit Nair, Ajay Gupta, Marat Fudim, Christopher Robinson, Vinay Ravi, Sandra M. Hurtado Rua, Zoar Engelman, Kyungmouk S. Lee, C. Douglas Philips, Akhilesh K. Sista

Mathematics and Statistics Faculty Publications

Introduction: The carotid body (CB) has previously been found to be enlarged and hyperactive in various disease states such as heart failure (HF), hypertension (HTN), and respiratory disease. Evaluation of CB size in these disease states using imaging has not been performed. The purpose of this case–control study was to compare CB sizes in patients with HF and HTN with those of controls using CT angiography. Methods: A retrospective review was performed on 323 consecutive patients who had neck computed tomography angiography (CTA) exams in 2011. Following extensive review, 17 HF and HTN patients and 14 controls were identified. Two …

Relationships Between Body Roundness With Body Fat And Visceral Adipose Tissue Emerging From A New Geometrical Model, Diana M. Thomas, Carl Bredlau, Anja Bosy-Westphal, Manfred Mueller, Wei Shen, Dympna Gallagher, Yuna Maeda, Andrew Mcdougall, Courtney M. Peterson, Eric Ravussin, Steven B. Heymsfield

Department of Mathematics Faculty Scholarship and Creative Works

Objective To develop a new geometrical index that combines height, waist circumference (WC), and hip circumference (HC) and relate this index to total and visceral body fat. Design and Methods Subject data were pooled from three databases that contained demographic, anthropometric, dual energy X-ray absorptiometry (DXA) measured fat mass, and magnetic resonance imaging measured visceral adipose tissue (VAT) volume. Two elliptical models of the human body were developed. Body roundness was calculated from the model using a well-established constant arising from the theory. Regression models based on eccentricity and other variables were used to predict %body fat and %VAT. Results …

Progressive Reliability Method And Its Application To Offshore Mooring Systems, Mir Emad Mousavi, Paolo Gardoni, Mehdi Maadooliat

Mathematics, Statistics and Computer Science Faculty Research and Publications

Assessing the reliability of complex systems (e.g. structures) is essential for a reliability-based optimal design that balances safety and costs of such systems. This paper proposes the Progressive Reliability Method (PRM) for the quantification of the reliability of complex systems. The proposed method is a closed-form solution for calculating the probability of failure. The new method is flexible to the definition of “failure” (i.e., can consider serviceability and ultimate-strength failures) and uses the rules of probability theory to estimate the failure probability of the system or its components. The method is first discussed in general and then illustrated in two …

Assessing Protein Conformational Sampling Methods Based On Bivariate Lag-Distributions Of Backbone Angles, Mehdi Maadooliat, Xin Gao, Jianhua Z. Huang

Mathematics, Statistics and Computer Science Faculty Research and Publications

Despite considerable progress in the past decades, protein structure prediction remains one of the major unsolved problems in computational biology. Angular-sampling-based methods have been extensively studied recently due to their ability to capture the continuous conformational space of protein structures. The literature has focused on using a variety of parametric models of the sequential dependencies between angle pairs along the protein chains. In this article, we present a thorough review of angular-sampling-based methods by assessing three main questions: What is the best distribution type to model the protein angles? What is a reasonable number of components in a mixture model …

On Closed Subsets Of Non-Commutative Association Schemes Of Rank 6, Jose Vera

Theses and Dissertations - UTB/UTPA

The notion of an association scheme is a generalization of the concept of a group. In fact, the so-called thin association schemes correspond in a well-understood way to groups. In this thesis, we look at the structure of non-commutative association schemes of rank 6. We will show that a non-normal closed subset of a noncommutative association scheme of rank 6, must have rank 2. The so-called Coxeter schemes of rank 6 which we present in Section 4 provide examples of association schemes of rank 6 with non-normal closed subsets of rank 2. It is shown that normal closed subsets of …

Mathematical Engineering And Control With Applications, Mamdouh M. El Kady, Martin Bohner, J. Liang, Mouffak Benchohra

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

A Topics Analysis Model For Health Insurance Claims, Jared Anthony Webb

Theses and Dissertations

Mathematical probability has a rich theory and powerful applications. Of particular note is the Markov chain Monte Carlo (MCMC) method for sampling from high dimensional distributions that may not admit a naive analysis. We develop the theory of the MCMC method from first principles and prove its relevance. We also define a Bayesian hierarchical model for generating data. By understanding how data are generated we may infer hidden structure about these models. We use a specific MCMC method called a Gibbs' sampler to discover topic distributions in a hierarchical Bayesian model called Topics Over Time. We propose an innovative use …

The Kalman Filter For Linear Systems On Time Scales, Martin Bohner, Nick Wintz

Mathematics and Statistics Faculty Research & Creative Works

We introduce the Kalman filter for linear systems on time scales, which includes the discrete and continuous versions as special cases. When the system is also stochastic, we show that the Kalman filter is an observer that estimates the system when the state is corrupted by noisy measurements. Finally, we show that the duality of the Kalman filter and the Linear Quadratic Regulator (LQR) is preserved in their unification on time scales. A numerical example is provided. © 2013 Elsevier Ltd.

The Kalman Filter For Linear Systems On Time Scales, Martin Bohner, Nick Wintz

Faculty Scholarship

We introduce the Kalman filter for linear systems on time scales, which includes the discrete and continuous versions as special cases. When the system is also stochastic, we show that the Kalman filter is an observer that estimates the system when the state is corrupted by noisy measurements. Finally, we show that the duality of the Kalman filter and the Linear Quadratic Regulator (LQR) is preserved in their unification on time scales. A numerical example is provided.

Minkowski And Beckenbach-Dresher Inequalities And Functionals On Time Scales, Rabia Bibi, Martin Bohner, Josip Pečarić, Sanja Varǒsanec

Mathematics and Statistics Faculty Research & Creative Works

We obtain integral forms of the Minkowski inequality and Beckenbach-Dresher inequality on time scales. Also, we investigate a converse of Minkowski's inequality and several functionals arising from the Minkowski inequality and the Beckenbach-Dresher inequality. © ELEMENT, Zagreb.

Orca, Kourtney Peters

Journal of Undergraduate Research

Throughout my research, the students were rarely given a verbal introduction to the tasks they faced. Instead, they had to rely on their intuition, pre-conceptions, and any previous knowledge to find solutions to the tasks at hand. They were given time and resources to explore their already existing understanding. This balance of time to think and distance from the teacher’s guidance is key, I think, to encouraging students to use their intuition and past experience to solve problems. According to Duckworth (Bransford, 2000), “Accomplished teachers ‘give learners reason’ by respecting and understanding learners’ prior experiences and understandings, assuming that these …

Probable Frames: Using A Probability Task To Explore Fraction Multiplication, Joseph G. Curtis, Dr. Robert Speiser

Journal of Undergraduate Research

When you toss a coin at the start of a football game and call “Heads!” the probability the coin will land “heads-up” is one-half. If the referee does not hear your prediction, he must toss again. The probability the second toss will land heads is again one-half. But what is the probability that both tosses were heads? In the double coin toss, the probability of two heads in a row is one-half times one-half: one-fourth. Although the rule (multiply the probabilities of the two independent events in order to calculate the probability of both events) is simple to carry out, …

Function And Distribution Of Apolipoprotein A1 In The Artery Wall Are Markedly Distinct From Those In Plasma, Joseph A. Didonato, Ying Huang, Kulwant S. Aulak, Orli Even-Or, Gary Gerstenecker, Valentin Gogonea, Yuping Wu, Paul L. Fox, W.H. Wilson Tang, Edward F. Plow, Jonathan D. Smith, Edward A. Fisher, Stanley L. Hazen

Mathematics and Statistics Faculty Publications

Background—Prior studies show that apolipoprotein A1 (apoA1) recovered from human atherosclerotic lesions is highly oxidized. Ex vivo oxidation of apoA1 or high-density lipoprotein (HDL) cross-links apoA1 and impairs lipid binding, cholesterol efflux, and lecithin-cholesterol acyltransferase activities of the lipoprotein. Remarkably, no studies to date directly quantify either the function or HDL particle distribution of apoA1 recovered from the human artery wall. Methods and Results—A monoclonal antibody (10G1.5) was developed that equally recognizes lipid-free and HDL-associated apoA1 in both native and oxidized forms. Examination of homogenates of atherosclerotic plaque–laden aorta showed >100-fold enrichment of apoA1 compared with normal aorta (P<0.001). Surprisingly, buoyant density fractionation revealed that only a minority (<3% of total) of apoA1 recovered from either lesions or normal aorta resides within an HDL-like particle (1.063≤d≤1.21). In contrast, the majority (>90%) …

Rainich-Type Conditions For Null Electrovacuum Spacetimes Ii, Charles G. Torre

Research Vignettes

In this second of two worksheets I continue describing local Rainich-type conditions which are necessary and sufficient for the metric to define a null electrovacuum. In other words, these conditions, which I will call the null electrovacuum conditions, guarantee the existence of a null electromagnetic field such that the metric and electromagnetic field satisfy the Einstein-Maxwell equations. When it exists, the electromagnetic field is easily constructed from the metric. In this worksheet I consider the null electrovacuum conditions which apply when a certain null geodesic congruence determined by the metric is twisting. I shall illustrate the these conditions using a …

Stability Analysis Of 4-Species Aβ Aggregation Model: A Novel Approach To Obtaining Physically Meaningful Rate Constants, G. Ghag, P. Ghosh, A. Mauro, V. Rangachari, Ashuwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

Protein misfolding and concomitant aggregation towards amyloid formation is the underlying biochemical commonality among a wide range of human pathologies. Amyloid formation involves the conversion of proteins from their native monomeric states (intrinsically disordered or globular) to well-organized, fibrillar aggregates in a nucleation-dependent manner. Understanding the mechanism of aggregation is important not only to gain better insight into amyloid pathology but also to simulate and predict molecular pathways. One of the main impediments in doing so is the stochastic nature of interactions that impedes thorough experimental characterization and the development of meaningful insights. In this study, we have utilized a …

Bytes Of Π, Fall 2013, Department Of Mathematics, Bridgewater State University

Department of Mathematics Newsletter

No abstract provided.

From Urysohn's Universal Metric Space To A Universal Space-Time, A. G. Aksoy, Z. Glassman, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A known Urysohn's result shows that there exists a universal} metric space, i.e., a metric space into every other (separable) metric space can be isomorphically embedded. Moreover, this universal metric space can be selected to be ultra-homogeneous -- every isomorphism of its two finite subsets can be extended to the isomorphism of the whole space.

Starting with Einstein's theories of Special and General relativity, space-times are described by a different type of structure -- a set (of events) equipped with the proper time t(a,b) between points a and b; such spaces are known as space-times with kinematic metric, or k-space-times. …

A Primer For Mathematical Modeling, Marla A. Sole

Publications and Research

With the implementation of the National Council of Teachers of Mathematics recommendations and the adoption of the Common Core State Standards for Mathematics, modeling has moved to the forefront of K-12 education. Modeling activities not only reinforce purposeful problem-solving skills, they also connect the mathematics students learn in school with the mathematics they will use outside of school. Instructors have found mathematical modeling difficult to teach. To successfully incorporate modeling activities I believe that curricular changes should be accompanied by professional development for curriculum developers, classroom teachers, and higher education professionals. This article serves as an introduction to modeling by …