Mathematics Commons^™

Recent Analytic Development Of The Dynamic Q-Tensor Theory For Nematic Liquid Crystals, Xiang Xu

Mathematics & Statistics Faculty Publications

Liquid crystals are a typical type of soft matter that are intermediate between conventional crystalline solids and isotropic fluids. The nematic phase is the simplest liquid crystal phase, and has been studied the most in the mathematical community. There are various continuum models to describe liquid crystals of nematic type, and Q-tensor theory is one among them. The aim of this paper is to give a brief review of recent PDE results regarding the Q-tensor theory in dynamic configurations.

Solutions For Fermi Questions, April 2022, Larry Weinstein

Physics Faculty Publications

[Introduction] Answer: When an asteroid hits the Earth, it transfers thermal and kinetic energy to the ground. The kinetic energy moves rock and makes a crater. We'll start by estimating the kinetic energy needed to make a crater.

On The Geometry Of The Multiplier Space Of ℓPA, Christopher Felder, Raymond Cheng

Mathematics & Statistics Faculty Publications

For p ∊ (1, ∞)\ {2}, some properties of the space M_p of multipliers on ℓ^p_A are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for M_p. It is also shown that extremal multipliers on the ℓ^p_A spaces are exactly the monomials, in stark contrast to the p = 2 case.

A Drop In The Bucket?: Solutions For Fermi Questions, December 2022, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Rock Paintings, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Wittgenstein On Miscalculation And The Foundations Of Mathematics, Samuel J. Wheeler

Philosophy Faculty Publications

In Remarks on the Foundations of Mathematics, Wittgenstein notes that he has 'not yet made the role of miscalculating clear' and that 'the role of the proposition: "I must have miscalculated"...is really the key to an understanding of the "foundations" of mathematics.' In this paper, I hope to get clear on how this is the case. First, I will explain Wittgenstein's understanding of a 'foundation' for mathematics. Then, by showing how the proposition 'I must have miscalculated' differentiates mathematics from the physical sciences, we will see how this proposition is the key to understanding the foundations of mathematics.

Solutions For Fermi Questions, October 2022 Cloud Ripple Pattern, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Robust Testing Of Paired Outcomes Incorporating Covariate Effects In Clustered Data With Informative Cluster Size, Sandipan Dutta

Mathematics & Statistics Faculty Publications

Paired outcomes are common in correlated clustered data where the main aim is to compare the distributions of the outcomes in a pair. In such clustered paired data, informative cluster sizes can occur when the number of pairs in a cluster (i.e., a cluster size) is correlated to the paired outcomes or the paired differences. There have been some attempts to develop robust rank-based tests for comparing paired outcomes in such complex clustered data. Most of these existing rank tests developed for paired outcomes in clustered data compare the marginal distributions in a pair and ignore any covariate effect on …

Rock Paintings: Solutions For Fermi Questions, September 2022, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Question 1: Ketchup Packets; Question 2: Dog Hair, Larry Weinstein

Physics Faculty Publications

Question 2: Dog hair How much dog hair is shed (and subsequently vacuumed) in the U.S. each year? How many ketchup packets are used in America every year?.

Solutions For Fermi Questions, May 2022, Larry Weinstein

Physics Faculty Publications

[Introduction] Question: If the James Webb Space Telescope could be turned to face the Earth, how small an object could it resolve? How does that compare to the Hubble Space Telescope?.

To answer this, we need to know the location of the JWST, its size, and the wavelengths it can image. (This is one of those Fermi questions where we're not really estimating, but we are calculating unusual order-of-magnitude answers.) The JWST is orbiting the Earth and Sun at the second Lagrange point, L2, which is located 1.5 × 10⁶ km beyond the Earth (so that the Sun, Earth and …

Solutions For Fermi Questions, March 2022, Larry Weinstein

Physics Faculty Publications

Answers the questions:

Question 1: Ketchup packets

How many ketchup packets are used in America every year?

Question 2: Dog hair

How much dog hair is shed (and subsequently vacuumed) in the U.S. each year?

Question 1: Dinosaur Killer Crater; Question 2: Graduation Speeches, Larry Weinstein

Physics Faculty Publications

The Fermi Questions: In 2003, the "Dinosaur Killer" asteroid crater was imaged for the first time (see National Geographic News, March 7, 2003). The crater is believed to be 100 km in radius and 1 km deep. What was the kinetic energy of the asteroid that made the crater? Express your answer in joules and in megatons (1 MT = 4 ·10 J).

Question 1: Space Telescope; Question 2: Jwst2, Larry Weinstein

Physics Faculty Publications

[Introduction] Space telescope

Question 1: If the James Webb Space Telescope could be turned to face Earth, how small an object could it resolve? How does that compare to the Hubble Space Telescope? ...

Question 2: How much heat from Earth is blocked by the Sun and Earth shade of the James Webb Space Telescope? ...

Question: While I was at a Norfolk Tides baseball game, a foul ball landed in the section above me and showered some of my friends with beer. What is the probability of a foul ball landing directly in a cup of beer during one …

Cloud Ripple Pattern, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Wiener-Fliess Composition Of Formal Power Series: Additive Static Feedback And Shuffle Rational Series, Subbarao Venkatesh Guggilam

Electrical & Computer Engineering Theses & Dissertations

The problem statement for this dissertation is two-fold. The first problem considered is when does a Chen-Fliess series in an additive static feedback connection with a formal static map yield a closed-loop system with a Chen-Fliess series expansion? This work proves that such a closed-loop system always has a Chen-Fliess series representation. Furthermore, an algorithm based on the Hopf algebras for the shuffle group and the dynamic output feedback group is designed to compute the generating series of the closed-loop system. It is proved that the additive static feedback connection preserves local convergence and relative degree, but a counterexample shows …

Finite Difference Schemes For Integral Equations With Minimal Regularity Requirements, Wesley Cameron Davis

Mathematics & Statistics Theses & Dissertations

Volterra integral equations arise in a variety of applications in modern physics and engineering, namely in interactions that contain a memory term. Classical formulations of these problems are largely inflexible when considering non-homogeneous media, which can be problematic when considering long term interactions of real-world applications. The use of fractional derivative and integral terms naturally relax these restrictions in a natural way to consider these problems in a more general setting. One major drawback to the use of fractional derivatives and integrals in modeling is the regularity requirement for functions, where we can no longer assume that functions are as …

A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, Kumudu Gamage, Yan Peng

College of Sciences Posters

Interface problems have many applications in fluid dynamics, molecular biology, electromagnetism, material science, heat distribution in engines, and hyperthermia treatment of cancer. Mathematically, interface problems commonly lead to partial differential equations (PDE) whose in- put data are discontinuous or singular across the interfaces in the solution domain. Many standard numerical methods designed for smooth solutions poorly work for interface problems as solutions of the interface problems are mostly non-smoothness or discontinuous. Moving interface problems depends on the accuracy of the gradient of the solution at the interface. Therefore, it became essential to derive a method for interface problems that gives …

Ranked List Fusion And Re-Ranking With Pre-Trained Transformers For Arqmath Lab, Shaurya Rohatgi, Jian Wu, C. Lee Giles

Computer Science Faculty Publications

This paper elaborates on our submission to the ARQMath track at CLEF 2021. For our submission this year we use a collection of methods to retrieve and re-rank the answers in Math Stack Exchange in addition to our two-stage model which was comparable to the best model last year in terms of NDCG’. We also provide a detailed analysis of what the transformers are learning and why is it hard to train a math language model using transformers. This year’s submission to Task-1 includes summarizing long question-answer pairs to augment and index documents, using byte-pair encoding to tokenize formula and …

Formal Power Series Approach To Nonlinear Systems With Static Output Feedback, G.S. Venkatesh, W. Steven Gray

Electrical & Computer Engineering Faculty Publications

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen-Fliess series and static output feedback is applied. The first step is to reconsider the so called Wiener-Fliess connection consisting of a Chen-Fliess series followed by a memoryless function. Of particular importance will be the contractive nature of this map, which is needed to show that the closed-loop system has a Chen-Fliess series representation. To explicitly compute the generating series, two Hopf algebras are needed, the existing output feedback Hopf algebra used to describe dynamic output …

Multivariate Distributions Of Correlated Binary Variables Generated By Pair-Copulas, Huihui Lin, N. Rao Chaganty

Mathematics & Statistics Faculty Publications

Correlated binary data are prevalent in a wide range of scientific disciplines, including healthcare and medicine. The generalized estimating equations (GEEs) and the multivariate probit (MP) model are two of the popular methods for analyzing such data. However, both methods have some significant drawbacks. The GEEs may not have an underlying likelihood and the MP model may fail to generate a multivariate binary distribution with specified marginals and bivariate correlations. In this paper, we study multivariate binary distributions that are based on D-vine pair-copula models as a superior alternative to these methods. We elucidate the construction of these binary distributions …

Numerical Simulations Of Capsule Deformation Using A Dual Time-Stepping Lattice Boltzmann Method, Charles Armstrong, Yan Peng

Mathematics & Statistics Faculty Publications

In this work a quasisteady, dual time-stepping lattice Boltzmann method is proposed for simulation of capsule deformation. At each time step the steady-state lattice Boltzmann equation is solved using the full approximation storage multigrid scheme for nonlinear equations. The capsule membrane is modeled as an infinitely thin shell suspended in an ambient fluid domain with the fluid structure interaction computed using the immersed boundary method. A finite element method is used to compute the elastic forces exerted by the capsule membrane. Results for a wide range of parameters and initial configurations are presented. The proposed method is found to reduce …

A Class Of Copula-Based Bivariate Poisson Time Series Models With Applications, Mohammed Alqawba, Dimuthu Fernando, Norou Diawara

Mathematics & Statistics Faculty Publications

A class of bivariate integer-valued time series models was constructed via copula theory. Each series follows a Markov chain with the serial dependence captured using copula-based transition probabilities from the Poisson and the zero-inflated Poisson (ZIP) margins. The copula theory was also used again to capture the dependence between the two series using either the bivariate Gaussian or “t-copula” functions. Such a method provides a flexible dependence structure that allows for positive and negative correlation, as well. In addition, the use of a copula permits applying different margins with a complicated structure such as the ZIP distribution. Likelihood-based inference was …

Em Estimation For Zero- And K-Inflated Poisson Regression Model, Monika Arora, N. Rao Chaganty

Mathematics & Statistics Faculty Publications

Count data with excessive zeros are ubiquitous in healthcare, medical, and scientific studies. There are numerous articles that show how to fit Poisson and other models which account for the excessive zeros. However, in many situations, besides zero, the frequency of another count k tends to be higher in the data. The zero- and k-inflated Poisson distribution model (ZkIP) is appropriate in such situations The ZkIP distribution essentially is a mixture distribution of Poisson and degenerate distributions at points zero and k. In this article, we study the fundamental properties of this mixture distribution. Using stochastic representation, we …

Representer Theorems In Banach Spaces: Minimum Norm Interpolation, Regularized Learning And Semi-Discrete Inverse Problems, Rui Wang, Yusheng Xu

Mathematics & Statistics Faculty Publications

Learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation (MNI) problem, a regularized learning problem or, in general, a semi discrete inverse problem (SDIP), in either Hilbert spaces or Banach spaces. The goal of this paper is to systematically study solutions of these problems in Banach spaces. We aim at obtaining explicit representer theorems for their solutions, on which convenient solution methods can then be developed. For the MNI problem, the explicit representer theorems enable us to express the infimum …

Stable And Convergent Difference Schemes For Weakly Singular Convolution Integrals, Wesley Davis, Richard D. Noren

Mathematics & Statistics Faculty Publications

We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in C^γ [0, T], where 0 < γ <= 5. A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown …

Joint Modeling Of Rnaseq And Radiomics Data For Glioma Molecular Characterization And Prediction, Zeina A. Shboul, Norou Diawara, Arastoo Vossough, James Y. Chen, Khan M. Iftekharuddin

Electrical & Computer Engineering Faculty Publications

RNA sequencing (RNAseq) is a recent technology that profiles gene expression by measuring the relative frequency of the RNAseq reads. RNAseq read counts data is increasingly used in oncologic care and while radiology features (radiomics) have also been gaining utility in radiology practice such as disease diagnosis, monitoring, and treatment planning. However, contemporary literature lacks appropriate RNA-radiomics (henceforth, radiogenomics) joint modeling where RNAseq distribution is adaptive and also preserves the nature of RNAseq read counts data for glioma grading and prediction. The Negative Binomial (NB) distribution may be useful to model RNAseq read counts data that addresses potential shortcomings. …

Predicting The Winning Percentage Of Limited-Overs Cricket Using The Pythagorean Formula, Hasika K. W. Senevirathne, Ananda B.W. Manage

Mathematics & Statistics Faculty Publications

The Pythagorean Win-Loss formula can be effectively used to estimate winning percentages for sporting events. This formula was initially developed by baseball statistician Bill James and later was extended by other researchers to sports such as football, basketball, and ice hockey. Although one can calculate actual winning percentages based on the outcomes of played games, that approach does not take into account the margin of victory. The key benefit of the Pythagorean formula is its utilization of actual average runs scored and actual average runs allowed. This article presents the application of the Pythagorean Win-Loss formula to two different types …

Morphology-Dependent Resonances In Two Concentric Spheres With Variable Refractive Index In The Outer Layer: Analytic Solutions, Umaporn Nuntaplook, John A. Adam

Mathematics & Statistics Faculty Publications

In many applications constant or piecewise constant refractive index profiles are used to study the scattering of plane electromagnetic waves by a spherical object. When the structured media has variable refractive indices, this is more of a challenge. In this paper, we investigate the morphology dependent resonances for the scattering of electromagnetic waves from two concentric spheres when the outer shell has a variable refractive index. The resonance analysis is applied to the general solutions of the radial Debye potential for both transverse magnetic and transverse electric modes. Finally, the analytic conditions to determine the resonance locations for this system …

Question 1: Chess Moves; Question 2: Cat Litter, Larry Weinstein

Physics Faculty Publications

How many chess moves are made by Americans each year?

How much cat litter did Americans scoop last year?