Mathematics Commons^™

Stationary Probability Distributions Of Stochastic Gradient Descent And The Success And Failure Of The Diffusion Approximation, William Joseph Mccann

Theses

In this thesis, Stochastic Gradient Descent (SGD), an optimization method originally popular due to its computational efficiency, is analyzed using Markov chain methods. We compute both numerically, and in some cases analytically, the stationary probability distributions (invariant measures) for the SGD Markov operator over all step sizes or learning rates. The stationary probability distributions provide insight into how the long-time behavior of SGD samples the objective function minimum.

A key focus of this thesis is to provide a systematic study in one dimension comparing the exact SGD stationary distributions to the Fokker-Planck diffusion approximation equations —which are commonly used in …

Dimensionless Analysis Of Trajectories Of Cylindrical Objects Dropped Into Water In Two Dimensions, Yi Zhen

LSU New Orleans Theses and Dissertations

Nondimensionalization is powerful technique and is widely applied in the study of fluid mechanics and engineering because it helps to reduce the number of free parameters, identify the relative size of effects of parameters, and gain a deeper insight of the essential nature of phenomena. The nondimensionalization of 2D theory has been completed by the author (Zhen et.al.,2020) and new dimensionless equations of motion were obtained. In this study, new dimensionless dynamic equations are extended by incorporating new parameters to cope with various environmental conditions. The new dimensionless analysis of dropped cylindrical objects is consisted of four parts.

Part 1, …

A Gate-Based Monte Carlo Simulation Of A Dual-Layer Pixelized Gadolinium Oxyorthosilicate (Gso) Detector Performance And Response For Micro Pet Scanner, Leonid L Nkuba, Innocent J Lugendo, Idrissa S Amour

Tanzania Journal of Science

The purpose of this study was to simulate the GSO detector of a micro PET using GATE simulation platform. The performance and responses of the simulated GSO detector assembly were evaluated by comparing the simulated data to the experimental and XCOM data to validate the simulation platform and procedure. Based on NEMA NU-4 2008 protocols, the performance of GSO detector in terms of sensitivity was simulated and compared to the experimental data. Similarly, the GSO detector response to photons interaction was simulated and compared against the XCOM data for absorbed intensity ratio in the GSO detector and survived intensity ratio …

Mathematical Analysis Of Harvested Predator-Prey System With Prey Refuge And Intraspecific Competition, Alanus Mapunda, Thadei Sagamiko

Tanzania Journal of Science

In this paper, a predator-prey relationship in the presence of prey refuge was studied. The analysis of the dependence of locally stable equilibrium points on the parameters of the problem was carried out. Bifurcation and limit cycles for the model were analyzed to show the dynamical behaviour of the system. The results showed that the system is stable at a constant prey refuge m = 0.3 and prey harvesting rate H = 0.3. However, increasing m and decreasing H or vice versa, the predator-prey system remains stable. It was further observed that for a constant prey refuge m ≥ 0.78, …

Count Time Series Models For Road Traffic Accidents In Tanzania Mainland, Amina S Msengwa, Florence D Ngari

Tanzania Journal of Science

A pairwise analysis was conducted to assess the trends and factors associated with road traffic accidents in Tanzania. The Poisson and Negative Binomial Autoregressive Models were used to extend log linear functions by accounting time-varying components. A total of 85,514 road traffic accidents in Tanzania mainland that occurred from 2012 to 2017 were extracted from Tanzania Police Office records. Eleven factors were grouped into a human, vehicle, physical/environmental and pedestrian-related factors. The Likelihood ratio test, Akaike Information Criterion, Bayesian Information Criterion and residual ACF plots were used to evaluate the performance of the models in Dar es Salaam and other …

Local Zeta Functions And Koba–Nielsen String Amplitudes, Miriam Bocardo-Gaspar, Hugo García-Compeán, Edgar Y. López, Wilson A. Zúñiga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action …

Compare And Contrast Maximum Likelihood Method And Inverse Probability Weighting Method In Missing Data Analysis, Scott Sun

Mathematical Sciences Technical Reports (MSTR)

Data can be lost for different reasons, but sometimes the missingness is a part of the data collection process. Unbiased and efficient estimation of the parameters governing the response mean model requires the missing data to be appropriately addressed. This paper compares and contrasts the Maximum Likelihood and Inverse Probability Weighting estimators in an Outcome-Dependendent Sampling design that deliberately generates incomplete observations. WE demonstrate the comparison through numerical simulations under varied conditions: different coefficient of determination, and whether or not the mean model is misspecified.

Optimal Partial Boundary Condition For Degenerate Parabolic Equations, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

For the stability of the non-Newtonian fluid equation∂u∂t−div(a(x)|∇u|p−2∇u)−N∑i=1bi(x)Diu+c(x,t)u=f(x,t),where a(x)|x∈Ω>0, a(x)|x∈∂Ω=0 and bi(x)∈C1(¯¯¯Ω), we know that the degeneracy of a(x) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the …

Anticipating Linear Stochastic Differential Equations With Adapted Coefficients, Hui-Hsiung Kuo, Pujan Shrestha, Sudip Sinha

Journal of Stochastic Analysis

No abstract provided.

On Properties Of Weil Sums Of Binomials, Liem P. Nguyen

LSU Doctoral Dissertations

This dissertation explores questions regarding the Weil sum of binomials, a finite field character sum originated from information theory. The Weil spectrum counts distinct values of the Weil sum through invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in information theory, coding theory, and cryptography to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we prove the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero values in the Weil spectrum and …

Ambient Obstruction Solitons And Homogeneous Gradient Bach Solitons, Erin Renee Griffin

Dissertations - ALL

Differential geometry is a diverse field which applies principles from calculus to a more general set of objects. Endowing a smooth manifold with a Riemannian metric allows us to measure length and angle in a way such that length is positive. This enables us to examine measures of curvature on a manifold. The study of manifolds with such metrics is called Riemannian geometry. Using geometric flows associated with tensors, we are able to analyze the relationship between metrics and curvature. Examining solitons, specifically gradient solitons, is one way we investigate this relationship.

This thesis focuses on the geometric flows associated …

N-Bakry Emery Ricci Curvature & N-Quasi Einstein Metrics, Alice Wu Lim

Dissertations - ALL

We begin the thesis by giving an intuitive introduction to calculus on mani- folds for the non-mathematician. We then give a semi-intuitive description on Ricci curvature for the non-geometer. We give a description of the N-Bakry- Émery Ricci curvature and the N-quasi Einstein metric. The main results in this thesis are related to the N-Bakry-Émery Ricci curvature and the N-quasi Einstein metric.

Our first set of main results are as follows. We generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative N-Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called …

Torsion Subgroups Of Rational Elliptic Curves Over Odd Degree Galois Fields, Caleb Mcwhorter

Dissertations - ALL

The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with …

Hochschild Cohomology Of Short Gorenstein Rings, Mkrtich Ohanyan

Dissertations - ALL

Let k be a field of characteristic 0. In this thesis, we show that the Hochschild cohomology of the family of short Gorenstein k-algebras

sGor(N) =k[X_0,...,X_N](X_iX_j, X_i^2−X_j^2 | i,j= 0,...,N, i different from j), N≥2,

exhibits exponential growth. The proof uses Gröbner-Shirshov basis theory and along the way we describe an explicit monomial basis for the Koszul dual of sGor(N) for N≥2.

Integrated Shift And Drift Control Of A Non-Linear Growth Process., Soham Chakraborty Dr.

Doctoral Theses

In the field of Quality Control (QC), the term 'control' is used to mean many different things. For example, it may refer to an isolated control action (e.g. a process adjustment), a particular type of control architecture (e.g. feedback vs. feed forward control) or even a control discipline (e.g. Deming's PDCA cycle). So will be the case here. But the exact meaning will be clear from the context.Since the development of control chart by Shewhart [1] and the method of acceptance sampling by Dodge and Romig [2] during 1920's - 1940's, the field of quality control has been greatly enriched …

Knowing What We Know: Leveraging Community Knowledge Through Automated Text-Mining, Justin Gardner, Jonathan Tory Toole, Hemant Kalia, Garry Spink Jr., Gordon Broderick

Advances in Clinical Medical Research and Healthcare Delivery

No abstract provided.

Applications Of Nonstandard Analysis In Probability And Measure Theory, Irfan Alam

LSU Doctoral Dissertations

This dissertation broadly deals with two areas of probability theory and investigates how methods from nonstandard analysis may provide new perspectives in these topics. In particular, we use nonstandard analysis to prove new results in the topics of limiting spherical integrals and of exchangeability.

In the former area, our methods allow us to represent finite dimensional Gaussian measures in terms of marginals of measures on hyperfinite-dimensional spheres in a certain strong sense, thus generalizing some previously known results on Gaussian Radon transforms as limits of spherical integrals. This first area has roots in the kinetic theory of gases, which is …

Hessenberg Varieties Of Parabolic Type, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type A in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-known Kostka numbers count the maximal-dimensional irreducible components of …

A New Method To Generate Superoscillating Functions And Supershifts, Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Tomer Shushi, Daniele C. Struppa, Jeff Tollaksen

Mathematics, Physics, and Computer Science Faculty Articles and Research

Superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics and in many fields of science and technology such as optics, signal processing and antenna theory. In this paper, we introduce a new method to generate superoscillatory functions that allows us to construct explicitly a very large class of superoscillatory functions.

2-Adic Valuations Of Quadratic Sequences, Will Boultinghouse, Jane H. Long, Olena Kozhushkina, Justin Trulen

Faculty Publications

We determine properties of the 2-adic valuation sequences for general quadratic polynomials with integer coefficients directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at terminating nodes. We completely describe the periodic sequences in the bounded case. Throughout, we frame results in terms of trees and sequences.

The Edwards Model For Fractional Brownian Loops And Starbursts, Wolfgang Bock, Torben Fattler, Ludwig Streit

Journal of Stochastic Analysis

No abstract provided.

Association Of Hypothermia With Increased Mortality Rate In Sars-Cov-2 Infection, Naaz Fatteh, Glen E. Sutherland, Radleigh Santos, Rita Zeidan, Alexandra P. Gastesi, Christopher D. Naranjo

Mathematics Faculty Articles

Objective

We observed patients to have variable temperatures. The objective of the study was to identify if hypothermia in a patient infected with SARS-CoV-2 is associated with a higher than expected mortality.

Methods

We reviewed 331 charts from patients hospitalized with SARS-CoV-2 between March 9 to April 20, 2020.

Results

The probability of death was 2.06 times higher for those with hypothermia than those without [95% C.I. (1.25, 3.38)]. In ventilated patients, there were 32 deaths. Of those, 75% had been hypothermic. In a prior review of 10,000 non-SARS-CoV-2 patients with sepsis, the mortality rate in patients with hypothermia was …

Weighted Inequalities On Spaces Of Homogeneous Type, Naga Manasa Vempati

Arts & Sciences Graduate Student Theses and Dissertations

Let $(X,d,\mu )$ be a space of homogeneous type in the sense of Coifman andWeiss, i.e. $d$ is a quasi metric on $X$ and $\mu $ is a nonzero measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Borel measures on $(X,d,\mu )$, the two weight inequality for the Calder\'on-Zygmund operator is of the form \begin{align*} & \|T(f\cdot u)\|_{L^2(v)}\lesssim \|f\|_{L^2(u)}. \end{align*} Subject to the pair of weights satisfying a side condition, we have given a characterization of the boundedness of a Calder\'{o}n-Zygmund operator $T$ from $L^{2}(u)$ to $L^{2}(v)$ in terms of the $A_{2}$ condition …

Spring 2021

Scientia

From the Dean: A Decade of Purpose and Progress; Lab Notes: Alumna Wins Gordon Bell Special Prize, New Scholarships, Vaccination Site Volunteers; Women in Science Lecture, National Institutes of Health Grants, "Unequal Cities" Research; All Hands on Deck: Inspired pandemic approaches showcase interdisciplinary acumen in action; Unlocking Potential: Christopher Beasley thinks psychology is key to academic transformation for the formerly incarcerated; Puzzle Master: Bridget Tenner goes to pieces solving problems in cutting-edge mathematics

Interpolating Matrices, Alberto Dayan

Arts & Sciences Graduate Student Theses and Dissertations

This Thesis work aims to extend the notion of interpolating sequences to objects having a non-trivial algebraic structure, such as a sequence of square matrices A (of eventually unbounded dimensions). First, we will consider the case in which the spectra of A lie in the unit disc, a necessary assumption if we want to apply to each matrix of the sequence a bounded holomorphic function on the unit disc. Since analytic functions preserve algebraic properties of square matrices, such as their eigenspaces, we can't extend the definition of interpolating sequences to matrices by just considering targets which are bounded in …

Alòs Type Decomposition Formula For Barndorff-Nielsen And Shephard Model, Takuji Arai

Journal of Stochastic Analysis

No abstract provided.

Интегрирование Модифицированного Уравнения Кортевега-Де Фриза С Нагруженным Членом И Источником В Виде Суммы, Т.Ж. Алланазарова Ассистент Преподаватель

Science and Society

This paper considers the method of the inverse spectral problem for integrating the modified Korteweg-de Vries equation (mKdV) with a loaded term and a source in the form of a sum in the class of infinite-gap periodic functions.

An Exploration Of Manipulatives In Math Education, Jade Monte

Honors Program Theses and Projects

Pre-existing literature has shown that the education system needs to re-evaluate mathematical teaching practices in a manner that can boost students’ confidence in mathematics. Thus, the research is to investigate the use of manipulatives in reducing students’ anxiety by increasing their learning experience and engagement in mathematics. Furthermore, the purpose of this thesis is to explain the interconnectedness of math manipulatives, student engagement, and problem-solving. An in-depth literature review is conducted, which contains definitions, important benefits and methodologies of manipulatives, as well as the teacher’s role regarding these three terms. When manipulatives, student engagement, and problem-solving are in harmony, students …

We’Re Here To Get You There: A Statistical Analysis Of Bridgewater State University’S Transit System, Abigail Adams

Honors Program Theses and Projects

Bridgewater State University first established its on-campus transportation service in January of 1984. While it began only running as an on-campus service for students throughout the day, the service grew to expand by offering an off-campus connection to the neighboring city of Brockton and absorbed the night service system from the campus safety team. As BSU Transit continues to grow, the organization is seeking ways to improve their overall service and better prepare their fleet and driver pool to accommodate this growth. The purpose of this research is to analyze trends among the data collected by BSU Transit and assist …

Mixed Generalized Fractional Brownian Motion, Shaykhah Alajmi, Ezzedine Mliki

Journal of Stochastic Analysis

No abstract provided.