Mathematics Commons^™

Singular Cr Structures Of Constant Webster Curvature And Applications, Chiara Guidi, Ali Maalaoui, Vittorio Martino

Mathematics

We consider the sphere (Formula presented.) equipped with its standard contact form. In this paper, we construct explicit contact forms on (Formula presented.), which are conformal to the standard one and whose related Webster metrics have constant Webster curvature; in particular, it is positive if (Formula presented.). As main applications, we provide two perturbative results. In the first one, we prove the existence of infinitely many contact forms on (Formula presented.) conformal to the standard one and having constant Webster curvature, where (Formula presented.) is a small perturbation of (Formula presented.). In the second application, we show that there exist …

Energy Extraction, Or Lack Thereof, Nishanth Gudapati

Mathematics

The problem of stability of rotating black holes is the subject of a long standing research program since the 1960s and remains an unresolved problem in general relativity. A major obstacle in the black hole stability problem is that the energy of waves propagating through rotating black holes spacetimes is not necessarily positive-definite, due to the so called ergo-region. This is a serious complication that limits the efficacy of most mathematical techniques. In this expository article, we report that, despite the ergo-region, there exists a positive-definite total energy for axisymmetric Maxwell, gravitational and electrovacuum perturbations of Kerr and Kerr–Newman black …

Prescribing The Q¯ ′ -Curvature On Pseudo-Einstein Cr 3-Manifolds, Ali Maalaoui

Mathematics

In this paper we study the problem of prescribing the Q ¯ ′ -curvature on embeddable pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive (resp. negative) CR pluriharmonic function, if ∫ M Q ′ d v θ > 0 (resp. ∫ M Q ′ d v θ < 0 ). In the second stage, we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality and non-normal ones that have a biharmonic leading term.

The available download on this page is the author manuscript accepted for publication. This version has undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading …

Finsler Pp-Waves And The Penrose Limit, Amir Babak Aazami, Miguel Ángel Javaloyes, Marcus C. Werner

Mathematics

We extend the notion of a Lorentzian pp-wave to that of Finsler spacetimes by providing a coordinate-independent definition of a Finsler pp-wave with respect to the Chern connection; our definition also includes the special case of a plane wave. This treatment introduces suitable lightlike coordinates, in analogy with the Lorentzian case, and utilizes the anisotropic calculus recently developed by one of the authors. We then extend Penrose’s “plane wave limit” to the setting of Finsler spacetimes. New examples of such Finsler pp-waves are also presented. © 2023, The Author(s).

Exact Parallel Waves In General Relativity, Cian Roche, Amir Babak Aazami, Carla Cederbaum

Mathematics

We conduct a review of the basic definitions and the principal results in the study of wavelike spacetimes, that is spacetimes whose metric models massless radiation moving at the speed of light, focusing in particular on those geometries with parallel rays. In particular, we motivate and connect their various definitions, outline their coordinate descriptions and present some classical results in their study in a language more accessible to modern readers, including the existence of “null coordinates” and the construction of Penrose limits. We also present a thorough summary of recent work on causality in pp-waves, and describe progress in addressing …

Function Spaces Via Fractional Poisson Kernel On Carnot Groups And Applications, Ali Maalaoui, Andrea Pinamonti, Gareth Speight

Mathematics

We provide a new characterization of homogeneous Besov and Sobolev spaces in Carnot groups using the fractional heat kernel and Poisson kernel. We apply our results to study commutators involving fractional powers of the sub-Laplacian. © 2022, The Hebrew University of Jerusalem.

Formal Oscillatory Distributions, Alexander Karabegov

Mathematics

The formal asymptotic expansion of an oscillatory in- tegral whose phase function has one nondegenerate critical point is a formal distribution supported at the critical point which is applied to the amplitude. This formal distribution is called a formal oscillatory integral (FOI). We introduce the notion of a formal oscillatory distribution supported at a point. We prove that a formal distribution is given by some FOI if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We also prove that a star product ? on a Poisson manifold M is natural in the sense of …

Analysis Of A Time-Delayed Hiv/Aids Epidemic Model With Education Campaigns, Sedar Ngoma, Dawit Denu, Rachidi B. Salako

Mathematics

We consider a time-delayed HIV/AIDS epidemic model with education dissemination and study the asymptotic dynamics of solutions as well as the asymptotic behavior of the endemic equilibrium with respect to the amount of information disseminated about the disease. Under appropriate assumptions on the infection rates, we show that if the basic reproduction number is less than or equal to one, then the disease will be eradicated in the long run and any solution to the Cauchy problem converges to the unique disease-free equilibrium of the model. On the other hand, when the basic reproduction number is greater than one, we …

Space-Efficient Prime Knot 7-Mosaics, Aaron Heap, Natalie Lacourt

Mathematics

The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles in 2018, where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic number 7. Specifically, we find the possible values for the number of non-blank tiles used in a space-efficient 7   7 mosaic of a prime knot are 27, 29, 31, 32, 34, 36, 37, 39, and 41. We also provide the possible layouts for the mosaics that lead to these …

Identifying Group Contributions In Nba Lineups With Spectral Analysis, Stephen Devlin, David Uminsky

Mathematics

We address the question of how to quantify the contributions of groups of players to team success. Our approach is based on spectral analysis, a technique from algebraic signal processing, which has several appealing features. First, our analysis decomposes the team success signal into components that are naturally understood as the contributions of player groups of a given size: individuals, pairs, triples, fours, and full five-player lineups. Secondly, the decomposition is orthogonal so that contributions of a player group can be thought of as pure: Contributions attributed to a group of three, for example, have been separated from the lower-order …

Hypersurfaces With Nonnegative Ricci Curvature In Hyperbolic Space, Vincent Bonini, Shiguang Ma, Jie Qing

Mathematics

Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier (Proc Symp Pure Math 54(3):37–44, 1993).

On Nonnegatively Curved Hypersurfaces In Hyperbolic Space, Vincent Bonini, Shiguang Ma, Jie Qing

Mathematics

In this paper we prove a conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.

A Network Diffusion Ranking Family That Includes The Methods Of Markov, Massey, And Colley, Stephen Devlin, Thomas Treloar

Mathematics

We present a one parameter family of ratings and rankings that includes the Markov method, as well as the methods of Colley and Massey as particular cases. The rankings are based on a natural network diffusion process that unites the methodologies above in a common framework and brings strong intuition to how and why they differ. We also explore the behavior of the ranking family using both real and simulated data.

Engaging Students In The Practice Of Statistics Through Undergraduate Research, Debra L. Hydorn

Mathematics

As statisticians, we engage in a variety of activities, some of which are regularly integrated into our undergraduate courses. However, the individual courses that comprise a mathematics or statistics degree program might not provide students with experiences in the broader range of activities that define the practice of statistics. To remedy this situation, faculty can consider developing and mentoring undergraduate research projects. This article briefly discusses the skills that comprise statistical practice along with some course and program options for helping students to develop these skills. Then, types of undergraduate research projects in statistics are described to help faculty generate …

Weakly Horospherically Convex Hypersurfaces In Hyperbolic Space, Vincent Bonini, Jie Qing, Jingyong Zhu

Mathematics

In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces ϕ:Mn→Hn+1 and a class of conformal metrics on domains of the round sphere Sn . Some of the key aspects of the correspondence and its consequences have dimensional restrictions n≥3 due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of Sn . In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the …

Discussion Of "Bridging The Gap Between Theory And Practice In Basic Statistical Process Monitoring", L. A. Jones-Farmer, Nathaniel Stevens

Mathematics

No abstract provided.

Topic Supervised Non-Negative Matrix Factorization, K. Macmillan, James D. Wilson

Mathematics

Topic models have been extensively used to organize and interpret the contents of large, unstructured corpora of text documents. Although topic models often perform well on traditional training vs. test set evaluations, it is often the case that the results of a topic model do not align with human interpretation. This interpretability fallacy is largely due to the unsupervised nature of topic models, which prohibits any user guidance on the results of a model. In this paper, we introduce a semi-supervised method called topic supervised non-negative matrix factorization (TS-NMF) that enables the user to provide labeled example documents to promote …

Statistical Modeling Of The Default Mode Brain Network Reveals A Segregated Highway Structure, P. E. Stillman, James D. Wilson, M. J. Denny, B. A. Desmarais, Shankar Bhamidi, S. J. Cranmer, Zhong-Lin Lu

Mathematics

We investigate the functional organization of the Default Mode Network (DMN) – an important subnetwork within the brain associated with a wide range of higher-order cognitive functions. While past work has shown the whole-brain network of functional connectivity follows small-world organizational principles, subnetwork structure is less well understood. Current statistical tools, however, are not suited to quantifying the operating characteristics of functional networks as they often require threshold censoring of information and do not allow for inferential testing of the role that local processes play in determining network structure. Here, we develop the correlation Generalized Exponential Random Graph Model (cGERGM) …

Quantifying Similarity In Reliability Surfaces Using The Probability Of Agreement, Nathaniel Stevens, C. M. Anderson-Cook

Mathematics

When separate populations exhibit similar reliability as a function of multiple explanatory variables, combining them into a single population is tempting. This can simplify future predictions and reduce uncertainty associated with estimation. However, combining these populations may introduce bias if the underlying relationships are in fact different. The probability of agreement formally and intuitively quantifies the similarity of estimated reliability surfaces across a two-factor input space. An example from the reliability literature demonstrates the utility of the approach when deciding whether to combine two populations or to keep them as distinct. New graphical summaries provide strategies for visualizing the results.

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 …

Mathematics Anxiety And Pre-Service Teachers, Margaret O'Toole

Mathematics

Mathematics anxiety has been a topic of interest since the 1950s. It has been shown that apprehensive elementary mathematics teachers unintentionally transfer anxiety to students in their classroom. In this study we assessed the change in 34 pre-service teacher's anxiety, self-efficacy and perception of ability during and after a content-specific mathematics course. This study used a mixed method approach for analyze data. The results suggests that anxiety decreased over the semester and perception of ability and self-efficacy increased. The levels of confidence with the material were recorded before and after three exams in the course. The pre-service teacher's change in …

Geometry Of Cubic Polynomials, Xavier Boesken

Mathematics

No abstract provided.

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 …

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 …

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 …

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 …