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The Surface Diffusion And The Willmore Flow For Uniformly Regular Hypersurfaces, Jeremy Lecrone, Yuanzhen Shao, Gieri Simonett

Department of Math & Statistics Faculty Publications

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are C^1+α–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are C^1+α–close to a sphere, and we prove …

A Template For Success: Celebrating The Work Of Judith Grabiner, Della Dumbaugh, Adrian Rice

Department of Math & Statistics Faculty Publications

Judith Grabiner is a mathematician who specializes in the history of mathematics. She is currently the Flora Sanborn Pitzer Professor Emerita of Mathematics at Pitzer College, one of the Claremont Colleges in Claremont, California. She has authored more than forty articles, as well as three books: The Origins of Cauchy’s Rigorous Calculus (1981), The Calculus as Algebra: J.-L. Lagrange, 1736–1813 (1990), and A Historian Looks Back: The Calculus as Algebra and Selected Writings (2010), which won the Beckenbach Prize from the Mathematical Association of America in 2014. She deliv- ered an invited address titled “The Centrality of Mathemat- ics in …

On Quasilinear Parabolic Equations And Continuous Maximal Regularity, Jeremy Lecrone, Gieri Simonett

Department of Math & Statistics Faculty Publications

We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.

Identifying Important Parameters In The Inflammatory Process With A Mathematical Model Of Immune Cell Influx And Macrophage Polarization, Marcella Torres, Jing Wang, Paul J. Yannie, Shobha Ghosh, Rebecca A. Segal, Angela M. Reynolds

Department of Math & Statistics Faculty Publications

In an inflammatory setting, macrophages can be polarized to an inflammatory M1 phenotype or to an anti-inflammatory M2 phenotype, as well as existing on a spectrum between these two extremes. Dysfunction of this phenotypic switch can result in a population imbalance that leads to chronic wounds or disease due to unresolved inflammation. Therapeutic interventions that target macrophages have therefore been proposed and implemented in diseases that feature chronic inflammation such as diabetes mellitus and atherosclerosis. We have developed a model for the sequential influx of immune cells in the peritoneal cavity in response to a bacterial stimulus that includes macrophage …

Critical Fault-Detecting Time Evaluation In Software With Discrete Compound Poisson Models, Min-Hsiung Hsieh, Shuen-Lin Jeng, Paul Kvam

Department of Math & Statistics Faculty Publications

Software developers predict their product’s failure rate using reliability growth models that are typically based on nonhomogeneous Poisson (NHP) processes. In this article, we extend that practice to a nonhomogeneous discrete-compound Poisson process that allows for multiple faults of a system at the same time point. Along with traditional reliability metrics such as average number of failures in a time interval, we propose an alternative reliability index called critical fault-detecting time in order to provide more information for software managers making software quality evaluation and critical market policy decisions. We illustrate the significant potential for improved analysis using wireless failure …

Mean Value Theorems For Riemannian Manifolds Via The Obstacle Problem, Brian Benson, Ivan Blank, Jeremy Lecrone

Department of Math & Statistics Faculty Publications

We develop some of the basic theory for the obstacle problem on Riemannian manifolds, and we use it to establish a mean value theorem. Our mean value theorem works for a very wide class of Riemannian manifolds and has no weights at all within the integral.

Perturbed Obstacle Problems In Lipschitz Domains: Linear Stability And Nondegeneracy In Measure, Ivan Blank, Jeremy Lecrone

Department of Math & Statistics Faculty Publications

We consider the classical obstacle problem on bounded, connected Lipschitz domains D⊂Rⁿ. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right-hand side and the boundary data for obstacle problems. In particular, we show that the Lebesgue measure of the symmetric difference between two contact sets is linearly comparable to the L¹-norm of perturbations in the data.

Finite Blaschke Products: A Survey, Stephan Ramon Garcia, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

A finite Blaschke product is a product of finitely many automorphisms of the unit disk. This brief survey covers some of the main topics in the area, including characterizations of Blaschke products, approximation theorems, derivatives and residues of Blaschke products, geometric localization of zeros, and selected other topics.

Multipliers Between Model Spaces, Emmanuel Fricain, Andreas Hartmann, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we examine the multipliers from one model space to another.

The Range And Valence Of A Real Smirnov Function, Timothy Ferguson, William T. Ross

Department of Math & Statistics Faculty Publications

We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees …

A Probability Model For Strategic Bidding On The Price Is Right, Paul H. Kvam

Department of Math & Statistics Faculty Publications

The TV game show “The Price is Right” features a bidding auction called “Contestants’ Row” that rewards the player (out of 4) who bids closest to an item’s value, without overbidding. This paper considers ways in which players can maximize a winning probability based on the player's bidding order. We consider marginal strategies in which players assume opponents are bidding individually perceived values of the merchandise. Based on preceding bids of others, players have information available to create strategies. We consider conditional strategies in which players adjust bids knowing other players are using strategies. The last bidder has a large …

Optimal Weak Parallelogram Constants For L-P Spaces, Raymond Cheng, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

Inspired by Clarkson's inequalities for L-p and continuing work from [5], this paper computes the optimal constant C in the weak parallelogram laws parallel to f + g parallel to(r )+ C parallel to f - g parallel to(r )= 2(r-1 )(parallel to f parallel to(r) + parallel to g parallel to(r)) for the L-p spaces, 1 < p < infinity.

A Unified Inter-Host And In-Host Model Of Antibiotic Resistance And Infection Spread In A Hospital Ward, Lester Caudill, Barry Lawson

Department of Math & Statistics Faculty Publications

As the battle continues against hospital-acquired infections and the concurrent rise in antibiotic resistance among many of the major causative pathogens, there is a dire need to conduct controlled experiments, in order to compare proposed control strategies. However, cost, time, and ethical considerations make this evaluation strategy either impractical or impossible to implement with living patients. This paper presents a multi-scale model that offers promise as the basis for a tool to simulate these (and other) controlled experiments. This is a “unified” model in two important ways: (i) It combines inter-host and in-host dynamics into a single model, and (ii) …

A Comprehensive Analysis Of Team Streakiness In Major League Baseball: 1962-2016, Paul H. Kvam, Zezhong Chen

Department of Math & Statistics Faculty Publications

A baseball team would be considered “streaky” if its record exhibits an unusually high number of consecutive wins or losses, compared to what might be expected if the team’s performance does not really depend on whether or not they won their previous game. If an average team in Major League Baseball (i.e., with a record of 81-81) is not streaky, we assume its win probability would be stable at around 50% for most games, outside of peculiar details of day-to-day outcomes, such as whether the game is home or away, who is the starting pitcher, and so on.

In this …

Approaching Cauchy’S Theorem, Stephan Ramon Garcia, William T. Ross

Department of Math & Statistics Faculty Publications

We hope to initiate a discussion about various methods for introducing Cauchy’s Theorem. Although Cauchy’s Theorem is the fundamental theorem upon which complex analysis is based, there is no “standard approach.” The appropriate choice depends upon the prerequisites for the course and the level of rigor intended. Common methods include Green’s Theorem, Goursat’s Lemma, Leibniz’ Rule, and homotopy theory, each of which has its positives and negatives.

Multipliers Of Sequence Spaces, Raymond Cheng, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

This paper is selective survey on the space l_A^p and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality.

Birkhoff–James Orthogonality And The Zeros Of An Analytic Function, Raymond Cheng, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

Bounds are obtained for the zeros of an analytic function on a disk in terms of the Taylor coefficients of the function. These results are derived using the notion of Birkhoff–James orthogonality in the sequence space ℓ^p with p ∈ (1,∞), along with an associated Pythagorean theorem. It is shown that these methods are able to reproduce, and in some cases sharpen, some classical bounds for the roots of a polynomial.

Evaluating Infection Prevention Strategies In Out-Patient Dialysis Units Using Agent-Based Modeling, Joanna R. Wares, Barry Lawson, Douglas Shemin, Erika D'Agata

Department of Math & Statistics Faculty Publications

Patients receiving chronic hemodialysis (CHD) are among the most vulnerable to infections caused by multidrug-resistant organisms (MDRO), which are associated with high rates of morbidity and mortality. Current guidelines to reduce transmission of MDRO in the out-patient dialysis unit are targeted at patients considered to be high-risk for transmitting these organisms: those with infected skin wounds not contained by a dressing, or those with fecal incontinence or uncontrolled diarrhea. Here, we hypothesize that targeting patients receiving antimicrobial treatment would more effectively reduce transmission and acquisition of MDRO. We also hypothesize that environmental contamination plays a role in the dissemination of …

Evaluating Infection Prevention Strategies In Out-Patient Dialysis Units Using Agent-Based Modeling, Joanna R. Wares, Barry Lawson, Douglas Shemin, Erika M. C. D'Agata

Department of Math & Statistics Faculty Publications

Patients receiving chronic hemodialysis (CHD) are among the most vulnerable to infections caused by multidrug-resistant organisms (MDRO), which are associated with high rates of morbidity and mortality. Current guidelines to reduce transmission of MDRO in the out-patient dialysis unit are targeted at patients considered to be high-risk for transmitting these organisms: those with infected skin wounds not contained by a dressing, or those with fecal incontinence or uncontrolled diarrhea. Here, we hypothesize that targeting patients receiving antimicrobial treatment would more effectively reduce transmission and acquisition of MDRO. We also hypothesize that environmental contamination plays a role in the dissemination of …

The Role Of Mathematical Modeling In Designing And Evaluating Antimicrobial Stewardship Programs, Lester Caudill, Joanna R. Wares

Department of Math & Statistics Faculty Publications

Antimicrobial agent effectiveness continues to be threatened by the rise and spread of pathogen strains that exhibit drug resistance. This challenge is most acute in healthcare facilities where the well-established connection between resistance and suboptimal antimicrobial use has prompted the creation of antimicrobial stewardship programs (ASPs). Mathematical models offer tremendous potential for serving as an alternative to controlled human experimentation for assessing the effectiveness of ASPs. Models can simulate controlled randomized experiments between groups of virtual patients, some treated with the ASP measure under investigation, and some without. By removing the limitations inherent in human experimentation, including health risks, study …

Classifying Coloring Graphs, Julie Beier, Janet Fierson, Ruth Haas, Heather M. Russell, Kara Shavo

Department of Math & Statistics Faculty Publications

Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k-colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs.

Concrete Examples Of H(B) Spaces, Emmanuel Fricain, Andreas Hartmann, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we give an explicit description of de Branges-Rovnyak spaces H(b) when b is of the form q^r, where q is a rational outer function in the closed unit ball of H^∞ and r is a positive number.

An Inner-Outer Factorization In ℓp With Applications To Arma Processes, Raymond Cheng, William T. Ross

Department of Math & Statistics Faculty Publications

The following inner-outer type factorization is obtained for the sequence space ℓ^p: if the complex sequence F = (F₀, F₁,F₂,...) decays geometrically, then for an p sufficiently close to 2 there exists J and G in ℓ^p such that F = J * G; J is orthogonal in the Birkhoff-James sense to all of its forward shifts SJ, S²J, S³J, ...; J and F generate the same S-invariant subspace of ℓ^p; and G is a cyclic vector for S on ℓ …

Real Complex Functions, Stephan Ramon Garcia, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to operator theory.

Partial Orders On Partial Isometries, William T. Ross, Stephan Ramon Garcia, Robert T. W. Martin

Department of Math & Statistics Faculty Publications

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBranges-Rovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate.

Weak Parallelogram Laws On Banach Spaces And Applications To Prediction, William T. Ross, R. Cheng

Department of Math & Statistics Faculty Publications

This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.

Chebyshev Polynomials And The Frohman-Gelca Formula, Heather M. Russell, Hoel Queffelec

Department of Math & Statistics Faculty Publications

Using Chebyshev polynomials, C. Frohman and R. Gelca introduced a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones–Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which emphasizes and demystifies the role played by Chebyshev polynomials.

The Robinson-Schensted Correspondence And A2-Web Bases, Heather M. Russell, Matthew Housley, Julianna Tymoczko

Department of Math & Statistics Faculty Publications

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n; n; n]: the reduced web basis associated to Kuperberg's combinatorial description of the spider category; and the left cell basis for the left cell construction of Kazhdan and Lusztig. In the case of [n; n], the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the image of these bases under classical maps: the Robinson-Schensted algorithm between permutations and Young tableaux and Khovanov-Kuperberg's bijection between …

A Survey On Reverse Carleson Measures, Emmanuel Fricain, Andreas Hartmann, William T. Ross

Department of Math & Statistics Faculty Publications

This is a survey on reverse Carleson measures for various Hilbert spaces of analytic functions. These spaces include the Hardy, Bergman, certain harmonically weighted Dirichlet, Paley-Wiener, Fock, model (backward shift invariant), and de Branges-Rovnyak spaces. The reverse Carleson measure for backward shift invariant subspaces in the non-Hilbert situation is new.

Weak Parallelogram Laws On Banach Spaces And Applications To Prediction, R. Cheng, William T. Ross

Department of Math & Statistics Faculty Publications

This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.