Mathematics Commons^™

Benchmark Dose Modeling With Covariates For Nanomaterials, Sarah E. Davidson

Mathematics

In the last decade, the use of engineered nanomaterials (ENMs) such as titanium dioxide (TiO2), carbon nanotubes (CNTs), carbon nanofibers (CNFs), as well as a variety of other materials have become increasingly popular in commerce because of their many beneficial properties (e.g. ability to manufacture products that are lighter, stronger, and/or more compact). However, according to the National Institute of Occupational Safety and Health, with the development of new nanotechnology it is prudent to ensure the health and safety of workers who are producing or using these materials at the forefront. For many ENMs, occupational exposure limits (OELs) are not …

The Optimization Behind Support Vector Machines And An Application In Handwriting Recognition, Caitlin Snyder

Mathematics

Support Vector Machines(SVMs) are a unique tool for classification of data and are used for solving real world problems such as image classification and text analysis. This project explores the underlying mathematics optimization problem behind the algorithm in a Support Vector Machine. The original minimization problem is described and an equivalent maximization formulation is derived. Various two and three dimensional examples are given to illustrate how the optimization gives a useable result with both linearly and nonlinearly separable data. Finally, this tool is applied in a handwriting character recognition problem. Popular SVM kernels are explored and their respective accuracy percentages …

Hypersurfaces In Hyperbolic Space With Support Function, Vincent Bonini, José M. Espinar, Jie Qint

Mathematics

Based on [previous publication*], we develop a global correspondence between immersed hypersurfaces ϕ:Mⁿ→Hⁿ⁺¹ϕ:Mn→Hn+1 satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e^2ρg_Sⁿe2ρgSn on domains Ω in the boundary SⁿSn at infinity of Hⁿ⁺¹Hn+1, where ρ is the horospherical support function, ∂_∞ϕ(Mⁿ)=∂Ω∂∞ϕ(Mn)=∂Ω, and Ω is the image of the Gauss map G:Mⁿ→SⁿG:Mn→Sn. To do so we first establish results on when the Gauss map G:Mⁿ→SⁿG:Mn→Sn is injective. We also …

An Analysis Of Strategic Treatment Interruptions During Imatinib Treatment Of Chronic Myelogenous Leukemia With Imatinib-Resistant Mutations, Dana Paquin, David Sacco, John Shamshoian

Mathematics

Chronic myelogenous leukemia (CML) is a cancer of the white blood cells that results from increased and uncontrolled growth of myeloid cells in the bone marrow and the accumulation of these cells in the blood. The most common form of treatment for CML is imatinib, a tyrosine kinase inhibitor. Although imatinib is an effective treatment for CML and most patients treated with imatinib do attain some form of remission, imatinib does not completely eradicate all leukemia cells, and if treatment is stopped, all patients eventually relapse (Cortes, 2005). In Kim (2008), the authors developed a mathematical model for the dynamics …

Being Smart About Parts, Nathaniel Stevens, S H. Steiner, R J. Mackay

Mathematics

When conducting a measurement system assessment study, practitioners want as precise an estimate of the true repeatability and reproducibility (R&R) as possible so they can correctly decide whether the measurement system is acceptable. They are, however, faced with cost and time constraints that restrict the number of parts and repeated measurements that can be used in the study. By incorporating freely available production measurements (baseline data), they can reduce the number of parts in the study to two or three and still obtain a better estimate of the R&R than they would have otherwise. To avoid bias, they must ensure …

Assessing Agreement Between Two Measurement Systems: An Alternative To The Limits Of Agreement Approach, Nathaniel Stevens, S H. Steiner, R J. Mackay

Mathematics

The comparison of two measurement systems is important in medical and other contexts. A common goal is to decide if a new measurement system agrees suitably with an existing one, and hence whether the two can be used interchangeably. Various methods for assessing interchangeability are available, the most popular being the limits of agreement approach due to Bland and Altman. In this article, we review the challenges of this technique and propose a model-based framework for comparing measurement systems that overcomes those challenges. The proposal is based on a simple metric, the probability of agreement, and a corresponding plot which …

Ring Patterns And Their Bifurcations In A Nonlocal Model Of Biological Swarms, Andrea L. Bertozzi, T Kolokolnikov, Hui Sun, David Uminsky, J Von Brecht

Mathematics

In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions. The swarm is represented by particles and the dynamics are driven by a gradient flow of a non-local interaction potential which has a local repulsion long range attraction structure. We leverage a co-dimension one formulation of the continuum gradient flow to characterize the stability of ring solutions for general interaction kernels. In the regime of long-wave instability we show that the resulting ground state is as a low mode bifurcation away from the ring and use weakly nonlinear analysis to provide …

A Testing Based Extraction Algorithm For Identifying Significant Communities In Networks, James D. Wilson, Simi Wang, Peter J. Mucha, Shankar Bhamidi, Andrew B. Nobel

Mathematics

A common and important problem arising in the study of networks is how to divide the vertices of a given network into one or more groups, called communities, in such a way that vertices of the same community are more interconnected than vertices belonging to different ones. We propose and investigate a testing based community detection procedure called Extraction of Statistically Significant Communities (ESSC). The ESSC procedure is based on p-values for the strength of connection between a single vertex and a set of vertices under a reference distribution derived from a conditional configuration network model. The procedure automatically selects …

The Stratified Structure Of Spaces Of Smooth Orbifold Mappings, Joseph E. Borzellino, Victor Brunsden

Mathematics

We consider four notions of maps between smooth C^∞ orbifolds , with compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between and with the C^r topology carries the structure of a smooth C^∞ Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of C^r maps between and with the C^r topology carries the …

Hilbert Space Theory And Applications In Basic Quantum Mechanics, Matthew Gagne

Mathematics

We explore the basic mathematical physics of quantum mechanics. Our primary focus will be on Hilbert space theory and applications as well as the theory of linear operators on Hilbert space. We show how Hermitian operators are used to represent quantum observables and investigate the spectrum of various linear operators. We discuss deviation and uncertainty and briefly suggest how symmetry and representations are involved in quantum theory.

Multiclass Total Variation Clustering, X Bresson, T Laurent, David Uminsky, J Von Brecht

Mathematics

Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bi-partitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a general framework for multiclass total variation clustering that does not rely on recursion. The results greatly outperform previous total variation algorithms and compare well with state-of-the-art NMF approaches.

Elementary Orbifold Differential Topology, Joseph E. Borzellino, Victor Brunsden

Mathematics

Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f:O→P between smooth orbifolds O and P. We show that Sardʼs theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence of a smooth orbifold map imposes on local isotropy groups. As an application, we prove a Borsuk no retraction theorem for compact orbifolds with boundary and some obstructions to the existence of real-valued orbifold maps from local model orbifold …

When Is A Trigonometric Polynomial Not A Trigonometric Polynomial?, Joseph E. Borzellino, Morgan Sherman

Mathematics

No abstract provided.

Temporal Topos And U-Singularities, Goro C. Kato

Mathematics

Several papers and books by C. Isham, C.Isham-A. Doering, F. Van Oystaeyen, A.Mallios-I. Raptis, C. Mulvey, and Guts and Grinkevich, have been published on the methods of categories and sheaves to study quantum gravity. Needless to say, there are well-written treatises on quantum gravity whose methods are non-categorical and non-sheaf theoretic. This paper may be one of the first papers explaining the methods of sheaves with minimally required background that retains experimental applications. Temporal topos (t-topos) is related to the topos approach to quantum gravity being developed by Prof. Chris Isham of the Oxford-Imperial research group (with its foundations inthe …

A Generalized Birkhoff-Rott Equation For Two-Dimensional Active Scalar Problems, Hui Sun, David Uminsky, Andrea L. Bertozzi

Mathematics

In this paper we derive evolution equations for the two-dimensional active scalar problem when the solution is supported on one-dimensional curves. These equations are a generalization of the Birkhoff–Rott equation when vorticity is the active scalar. The formulation is Lagrangian and it is valid for nonlocal kernels K that may include both a gradient and an incompressible term. We develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time. We verify the model by simulating classic active scalar problems such as the vortex sheet problem (in the case of inviscid, …

Network-Based Criterion For The Success Of Cooperation In An Evolutionary Prisoner's Dilemma, Stephen Devlin, T Treloar

Mathematics

We consider an evolutionary prisoner's dilemma on a random network. We introduce a simple quantitative network-based parameter and show that it effectively predicts the success of cooperation in simulations on the network. The criterion is shown to be accurate on a variety of networks with degree distributions ranging from regular to Poisson to scale free. The parameter allows for comparisons of random networks regardless of their underlying topology. Finally, we draw analogies between the criterion for the success of cooperation introduced here and existing criteria in other contexts.

Convergence And Energy Landscape For Cheeger Cut Clustering, X Bresson, T Laurent, David Uminsky, J Von Brecht

Mathematics

This paper provides both theoretical and algorithmic results for the l ₁-relaxation of the Cheeger cut problem. The l₂- relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The l₁-relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The l₁-relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The first challenge is understanding convergence of algorithms. This paper provides the first complete proof of convergence for …

Stability And Clustering Of Self-Similar Solutions Of Aggregation Equations, Hui Sun, David Uminsky, Andrea L. Bertozzi

Mathematics

In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρ _t = ∇ · (ρ∇K * ρ) in Rd , d ⩾ 2, where K(r) = r ^γ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math.70, 2582–2603 (Year: 2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the …

An Axiomatic Approach To The Non-Linear Theory Of Generalized Functions And Consistency Of Laplace Transforms, Todor D. Todorov

Mathematics

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of …

Completeness Of The Leibniz Field And Rigorousness Of Infinitesimal Calculus, James F. Hall, Todor D. Todorov

Mathematics

We present a characterization of the completeness of the field of real numbers in the form of a collection of ten equivalent statements borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. As an application we exploit one of our results to argue that the Leibniz infinitesimal calculus in the 18^th century was already a rigorous branch of mathematics – at least much more rigorous than most contemporary mathematicians prefer to believe. By advocating our particular historical point of view, we hope to …

Strategic Treatment Interruptions During Imatinib Treatment Of Chronic Myelogenous Leukemia, Dana C. Paquin, Peter S. Kim, Peter P. Lee, Doron Levy

Mathematics

Although imatinib is an effective treatment for chronic myelogenous leukemia (CML), and nearly all patients treated with imatinib attain some form of remission, imatinib does not completely eliminate leukemia. Moreover, if the imatinib treatment is stopped, most patients eventually relapse (Cortes et al. in Clin. Cancer Res. 11:3425–3432, 2005). In Kim et al. (PLoS Comput. Biol. 4(6):e1000095, 2008), the authors presented a mathematical model for the dynamics of CML under imatinib treatment that incorporates the anti-leukemia immune response. We use the mathematical model in Kim et al. (PLoS Comput. Biol. 4(6):e1000095, 2008) to study and numerically simulate strategic treatment interruptions …

On The Equivariant Cohomology Of Rotation Groups And Stiefel Manifolds, William Kronholm

Mathematics

In this paper, we compute the RO(Z/2)-graded equivariant cohomology of rotation groups and Stiefel manifolds with particular involutions.

Monitoring Radiation Use In Cardiac Fluoroscopy Imaging Procedures, Nathaniel Stevens, S H. Steiner, I R. Smith, R J. Mackay

Mathematics

Objective: Timely identification of systematic changes in radiation delivery of an imaging system can lead to a reduction in risk for the patients involved. However, existing quality assurance programs involving the routine testing of equipment performance using phantoms are limited in their ability to effectively carry out this task. To address this issue we propose the implementation of an ongoing monitoring process that utilizes procedural data to identify unexpected large or small radiation exposures for individual patients, as well as to detect persistent changes in the radiation output of imaging platforms.

Methods: Data used in this study were obtained from …

Planning And Analysis Of Measurement Reliability Studies, S H. Steiner, Nathaniel Stevens, R Browne, R J. Mackay

Mathematics

In the traditional plan for assessing the reliability of a measurement system, a number of raters each measure the same group of subjects. If the system has a large number of raters, we recommend a new set of plans that have two advantages over the traditional plan. First, the proposed plans provide greater precision for estimating the intraclass correlation coefficient with the same total number of measurements. Second, the plans are flexible and can be adapted to constraints on the number of times any subject can be assessed or the number of times any rater can make an assessment. We …

Another Proof Of The Existence A Dedekind Complete Totally Ordered Field, James F. Hall, Todor D. Todorov

Mathematics

We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis.

Completeness Of Ordered Fields, James Forsythe Hall

Mathematics

The main goal of this project is to prove the equivalency of several characterizations of completeness of Archimedean ordered fields; some of which appear in most modern literature as theorems following from the Dedekind completeness of the real numbers, while a couple are not as well known and have to do with other areas of mathematics, such as nonstandard analysis. Continuing, we study the completeness of non-Archimedean fields, and provide several examples of such fields with varying degrees of properties, using nonstandard analysis to produce some relatively "nice" (in particular, they are Cantor complete) final examples. As a small detour, …

Correspondences Of Hypersurfaces In Hyperbolic Poincaré Manifolds And Conformally Invariant Pdes, Vincent Bonini, José M. Espinar, Jie Qing

Mathematics

On a hyperbolic Poincaré manifold, we derive an explicit relationship between the eigenvalues of Weyl-Schouten tensor of a conformal representative of the conformal infinity and the principal curvatures of the level sets of the associated geodesic defining function. This considerably simplifies the arguments and generalizes the results of Gálvez, Mira and the second author. In particular, we obtain the equivalence between Christoffel-type problems for hypersurfaces in a hyperbolic Poincar´e manifold and scalar curvature problems on the conformal infinity.

Consecutive Patterns: From Permutations To Column-Convex Polyominoes And Back, Don Rawlings, Mark Tiefenbruck

Mathematics

We expose the ties between the consecutive pattern enumeration problems associated with permutations, compositions, column-convex polyominoes, and words. Our perspective allows powerful methods from the contexts of compositions, column-convex polyominoes, and of words to be applied directly to the enumeration of permutations by consecutive patterns. We deduce a host of new consecutive pattern results,including a solution to the (2m+1)-alternating pattern problem on permutations posed by Kitaev.

Ozsváth-Szabó And Rasmussen Invariants Of Cable Knots, Cornelia A. Van Cott

Mathematics

We study the behavior of the Ozsváth–Szabó and Rasmussen knot concordance invariants τ and s on Km,n, the (m,n)–cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on Km,n differ from their value on the torus knot Tm,n by fixed constants for all but finitely many n>0. Combining this result together with Hedden’s extensive work on the behavior of τ on (m,mr+1)–cables yields bounds on the value of τ on any (m,n)–cable of K. In addition, several …

Reply To "Comment On 'Cooperation In An Evolutionary Prisoner's Dilemma On Networks With Degree-Degree Correlations' ", Stephen Devlin, T Treloar

Mathematics

We respond to the comment of Zhu et al. [Phys. Rev. E 82, 038101 (2010)] and show that the results in question are not misleading.