Physical Sciences and Mathematics Commons^™

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 …

Asymptotically-Optimal Topological Nearest-Neighbor Filtering, Read Sandström, Jory Denny, Nancy M. Amato

Department of Math & Statistics Faculty Publications

Nearest-neighbor finding is a major bottleneck for sampling-based motion planning algorithms. The cost of finding nearest neighbors grows with the size of the roadmap, leading to a significant computational bottleneck for problems which require many configurations to find a solution. In this work, we develop a method of mapping configurations of a jointed robot to neighborhoods in the workspace that supports fast search for configurations in nearby neighborhoods. This expedites nearest-neighbor search by locating a small set of the most likely candidates for connecting to the query with a local plan. We show that this filtering technique can preserve asymptotically-optimal …

Topology-Guided Roadmap Construction With Dynamic Region Sampling, Read Sandström, Diane Uwacu, Jory Denny, Nancy M. Amato

Department of Math & Statistics Faculty Publications

Many types of planning problems require discovery of multiple pathways through the environment, such as multi-robot coordination or protein ligand binding. The Probabilistic Roadmap (PRM) algorithm is a powerful tool for this case, but often cannot efficiently connect the roadmap in the presence of narrow passages. In this letter, we present a guidance mechanism that encourages the rapid construction of well-connected roadmaps with PRM methods. We leverage a topological skeleton of the workspace to track the algorithm's progress in both covering and connecting distinct neighborhoods, and employ this information to focus computation on the uncovered and unconnected regions. We demonstrate …

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.

Score Following With Hidden Tempo Using A Switching State-Space Model, Yucong Jiang, Chris Raphael

Department of Math & Statistics Faculty Publications

A score-following program traces the notes in a musical score during a performance. This capability is essential to many meaningful applications that synchronize audio with a score in an on-line fashion. Existing algorithms often stumble on certain difficult cases, one of which is piano music. This paper presents a new method to tackle such cases. The method treats tempo as a variable rather than a constant (with constraints), allowing the program to adapt to live performance variations. This is first expressed by a Kalman filter model at the note level, and then by an almost equivalent switching state-space model at …

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 …

The Price Is Right: Analyzing Bidding Behavior On Contestants’ Row, Paul Kvam

Department of Math & Statistics Faculty Publications

The TV game show “The Price is Right” features a bidding auction called Contestant’s Row that rewards the player (out of four) who bids closest to an item’s value without overbidding. By exploring 903 game outcomes from the 2000–2001 season, we show how player strategies are significantly inefficient, and compare the empirical results to probability outcomes for optimal bid strategies found in a recent study. Findings show that the last bidder would do better using the naïve strategy of bidding a dollar more than the highest of the three bids. We apply the EM algorithm in a novel way to …

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.

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 …

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 …

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.

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.

Range Spaces Of Co-Analytic Toeplitz Operators, Emmanuel Fricain, Andreas Hartmann, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges-Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern-Clark, we …

Equivalence Of Edge Bicolored Graphs On Surfaces, Oliver T. Dasbach, Heather M. Russell

Department of Math & Statistics Faculty Publications

Consider the collection of edge bicolorings of a graph that are cellularly embedded on an orientable surface. In this work, we count the number of equivalence classes of such colorings under two relations: reversing colors around a face and reversing colors around a vertex. In the case of the plane, this is well studied, but for other surfaces, the computation is more subtle. While this question can be stated purely graph theoretically, it has interesting applications in knot theory.

Developing A Minimally Structured Mathematical Mode Of Cancer Treatment With Oncolytic Viruses And Dendritic Cell Injections, Jane L. Gevertz, Joanna R. Wares

Department of Math & Statistics Faculty Publications

Mathematical models of biological systems must strike a balance between being sufficiently complex to capture important biological features, while being simple enough that they remain tractable through analysis or simulation. In this work, we rigorously explore how to balance these competing interests when modeling murine melanoma treatment with oncolytic viruses and dendritic cell injections. Previously, we developed a system of six ordinary differential equations containing fourteen parameters that well describes experimental data on the efficacy of these treatments. Here, we explore whether this previously developed model is the minimal model needed to accurately describe the data. Using a variety of …

A Tidy Data Model For Natural Language Processing Using Cleannlp, Taylor B. Arnold

Department of Math & Statistics Faculty Publications

Recent advances in natural language processing have produced libraries that extract low level features from a collection of raw texts. These features, known as annotations, are usually stored internally in hierarchical, tree-based data structures. This paper proposes a data model to represent annotations as a collection of normalized relational data tables optimized for exploratory data analysis and predictive modeling. The R package cleanNLP, which calls one of two state of the art NLP libraries (CoreNLP or spaCy), is presented as an implementation of this data model. It takes raw text as an input and returns a list of normalized tables. …

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 sub-optimal 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 …