Mathematics Commons^™

On The Noisy Gradient Descent That Generalizes As Sgd, Jingfeng Wu, Wenqing Hu, Haoyi Xiong, Jun Huan, Vladimir Braverman, Zhanxing Zhu

Mathematics and Statistics Faculty Research & Creative Works

The gradient noise of SGD is considered to play a central role in the observed strong generalization abilities of deep learning. While past studies confirm that the magnitude and covariance structure of gradient noise are critical for regularization, it remains unclear whether or not the class of noise distributions is important. In this work we provide negative results by showing that noises in classes different from the SGD noise can also effectively regularize gradient descent. Our finding is based on a novel observation on the structure of the SGD noise: it is the multiplication of the gradient matrix and a …

Small Gaps Between Almost Primes, The Parity Problem, And Some Conjectures Of Erdős On Consecutive Integers Ii, Daniel A. Goldston, Sidney W. Graham, Apoorva Panidapu, Janos Pintz, Jordan Schettler, Cem Y. Yıldırım

Faculty Publications

We show that for any positive integer n, there is some fixed A such that d(x) = d(x +n) = A infinitely often where d(x) denotes the number of divisors of x. In fact, we establish the stronger result that both x and x +n have the same fixed exponent pattern for infinitely many x. Here the exponent pattern of an integer x > 1is the multiset of nonzero exponents which appear in the prime factorization of x.

A Single-Center Comparison Using Exoskeleton Rehabilitation For Cerebrovascular Accidents And Traumatic Brain Injury In A Cohort Of Hispanic Patients, Lisa R. Trevino, Kristina Vatcheva, Michael E. Auer, Angela Morales, Lama M. Abdurrahman, Sarajova Viswamitra, Annelyn Torres-Reveron

School of Mathematical & Statistical Sciences Faculty Publications

Background Traumatic brain injury (TBI) is one of the leading causes of disability in the United States. The EKSO GT Bionics® (EKSO®) is a robotic exoskeleton approved by the Federal Drug Administration (FDA) for rehabilitation following a cerebrovascular accident (CVA or stroke) and recently received approval for use in patients with TBI. The aim of the study was to examine if the use of exoskeleton rehabilitation in patients with TBI will produce beneficial outcomes.

Methods This retrospective chart-review reports the use of the (EKSO®) robotic device in the rehabilitation of patients with TBI compared to patients with CVA. We utilized …

Arithmetic Differential Operators On Z_P^R, Marshall Brandenburg

Mathematics & Statistics ETDs

Given a prime, p, we prove that if a function can be represented as a restricted power series, then the function is analytic.

Methods Of Uncertainty Quantification For Physical Parameters, Kellin Rumsey

Mathematics & Statistics ETDs

Uncertainty Quantification (UQ) is an umbrella term referring to a broad class of methods which typically involve the combination of computational modeling, experimental data and expert knowledge to study a physical system. A parameter, in the usual statistical sense, is said to be physical if it has a meaningful interpretation with respect to the physical system. Physical parameters can be viewed as inherent properties of a physical process and have a corresponding true value. Statistical inference for physical parameters is a challenging problem in UQ due to the inadequacy of the computer model. In this thesis, we provide a comprehensive …

Spacetime Groups, Ian M. Anderson, Charles G. Torre

All Physics Faculty Publications

A spacetime group is a connected 4-dimensional Lie group G endowed with a left invariant Lorentz metric h and such that the connected component of the isometry group of h is G itself. The Newman-Penrose formalism is used to give an algebraic classification of spacetime groups, that is, we determine a complete list of inequivalent spacetime Lie algebras, which are pairs, (g, n), with g being a 4-dimensional Lie algebra and n being a Lorentzian inner product on g. A full analysis of the equivalence problem for spacetime Lie algebras is given which leads to a completely …

Assessing The Validity Of Sentiment Analysis Measures Through Polychoric Correlation, Kelli N. Kasper

Mathematics & Statistics ETDs

Sentiment analysis methods extract the attitude of a text via systematic algorithms. To evaluate the validity of common sentiment analysis methods, we use polychoric correlation to compare computer-mediated methods and human-rated analogues. Our main topics of interest are the internal consistency of the raters' scores, the level of consensus among raters, and how well raters' scores correlate with those given by sentiment analysis methods for randomly collected Twitter data.

Our analysis found that there is good validity for methods that measure negative and positive sentiments in short texts, both in terms of inter-rater consistency and when comparing raters to computer-mediated …

Some Results On Seymour’S Second-Neighborhood Conjecture And On Decompositions Of Graphs, Farid Bouya

LSU Doctoral Dissertations

This dissertation consists of two parts. In the first part, I examine Seymour’s Second-Neighborhood Conjecture, which states that every orientation of every simple graph has at least one vertex v such that the number of vertices of out-distance 2 from v is at least as large as the number of vertices of out-distance 1 from it. I present alternative statements of this conjecture using the language of linear algebra, the last one being completely in terms of the inverse of some matrix. In the second part of this dissertation, comprising of Chapters 2 and 3, I examine two conjectures on …

An Improved Method For Spectroscopic Quality Classification, Elizabeth G. Mayer

Mathematics & Statistics ETDs

Spectral quality classification is a vital step in data cleaning before the

analysis of magnetic resonance spectroscopy (MRS) data can be done. This

analysis compares five methods of quality classification; three of these are

legacy methods, Maudsley et al. (2006), Zhang et al. (2018), and

Bustillo et al. (2020), and two newly created methods that used a random forests

classifier (RFC) to inform their classifications. We found that the random forest

classifier was the most accurate at predicting spectra quality (balanced

accuracy for RF of 88% vs legacy of 70%, 72%, or 72%). A

Random-Forests-Informed Filtering method (RFIFM) for quality …

Connecting Ancient Philosophers’ Math Theory To Modern Fractal Mathematics, Colin Mccormack

Parnassus: Classical Journal

No abstract provided.

Shake Slice And Shake Concordant Links, Anthony Bosman

Faculty Publications

© 2020 World Scientific Publishing Company. We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shaker-concordance for links and versions with stricter conditions on the embedded spheres that we …

A Study Of The Efficacy Of Machine Learning For Diagnosing Obstructive Coronary Artery Disease In Non-Diabetic Patients, Demond Larae Handley

Theses and Dissertations

According to the Centers for Disease Control and Prevention, about 18.2 million adults age 20 and older have Coronary Artery Disease in the United States. Early diagnosis is therefore of crucial importance to help prevent debilitating consequences, and principally death for many patients. In this study we use data containing gene expression values from peripheral blood samples in 198 non-diabetic patients, with the goal of developing an age and sex gene expression model for diagnosis of Coronary Artery Disease. We employ machine learning methods to obtain a classification based on genetic information, age and sex. Our implementation uses feed forward …

Laplacian Spectral Properties Of Signed Circular Caterpillars, Maurizio Brunetti

Theory & Applications of Graphs

A circular caterpillar of girth n is a graph such that the removal of all pendant vertices yields a cycle C_n of order n. A signed graph is a pair Γ =(G,σ), where G is a simple graph and σ: E(G) →{+1, -1} is the sign function defined on the set E(G) of edges of G. The signed graph Γ is said to be balanced if the number of negatively signed edges in each cycle is even, and it is said to be unbalanced otherwise. We determine some bounds for the first n Laplacian eigenvalues …

Influence Of The Nature Of The Medium On The Conformational States Of Macromolecules Of Anionic Polyelectrolytes And Their Flocculating Effect, Mapruza Dzhumamuratova, Amanbay Pirniyazov, Oleg Dormeshkin, Akhmed Reymov, Tleumuratov Kuanish, Saparbay Kalbaev

Karakalpak Scientific Journal

The properties of surfactants of anionic polyelectrolytes are investigated depending on the nature of the medium, conformational states neutralized by organic bases, differing in the length of the hydrocarbon radical. The concept of the role of the state of macromolecules in solution in the indicated processes has been developed. The anionic carboxyl-containing polyelectrolytes synthesized on the basis of methacrylic acid differed not only in the length of the hydrocarbon radical of the organic base - counterion, but also in their branching. The relationship between the state of polyelectrolyte macromolecules in solution and their ability to regulate the properties of natural …

Cross-Cultural Comparisons: The Art Of Computing The Greatest Common Divisor, Mary K. Flagg

Number Theory

No abstract provided.

A Classification Of Hull Operators In Archimedean Lattice-Ordered Groups With Unit, Ricardo Enrique Carrera, Anthony W. Hager

Mathematics Faculty Articles

The category, or class of algebras, in the title is denoted by W. A hull operator (ho) in W is a reflection in the category consisting of W objects with only essential embeddings as morphisms. The proper class of all of these is hoW. The bounded monocoreflection in W is denoted B. We classify the ho’s by their interaction with B as follows. A “word” is a function w : hoW → W^W obtained as a finite composition of B and x a variable ranging in hoW. The set of these,“Word”, is in a natural …

The Kinetic Study Of Dpt Using Mathematica As An Efficient Optimization Tool, Marwa Eldewaik, Mamdouh A. Gadalla Prof., M A. Radwan, M.A Sadek, Hany A. Elazab

Basic Science Engineering

Mathematica is an efficient and optimized tool for computing numeric and algebraic calculations as well as graphing two and three dimensional curves and surfaces. It is used in several aspects of science including physics, engineering, chemistry and even biology due to the impact of mathematics with almost the fields of science nowadays. Synthesis of Cyclotetramethylene Tetramine through the action of nitrating mixture formed of ammonium nitrate and fuming nitric acid on hexamine in presence of acetic acid, acetic anhydride and p-formaldehyde has been proven. DPT was prepared at different temperatures. The variation of some factors like: temperature and time has …

Italian Sociologists: A Community Of Disconnected Groups, Aliakbar Akbaritabar, Vincent Traag, Alberto Caimo, Flaminio Squazzoni

Articles

Examining coauthorship networks is key to study scientific collaboration patterns and structural characteristics of scientific communities. Here, we studied coauthorship networks of sociologists in Italy, using temporal and multi-level quantitative analysis. By looking at publications indexed in Scopus, we detected research communities among Italian sociologists. We found that Italian sociologists are fractured in many disconnected groups. The giant connected component of the Italian sociology could be split into five main groups with a mixture of three main disciplinary topics: sociology of culture and communication (present in two groups), economic sociology (present in three groups) and general sociology (present in three …

The Classification Of Countable Models Of Set Theory, John Clemens, Samuel Coskey, Samuel Dworetzky

Mathematics Faculty Publications and Presentations

We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well‐founded models of ZFC.

Structure Aware Runge–Kutta Time Stepping For Spacetime Tents, Jay Gopalakrishnan, Joachim Schöberl, Christoph Wintersteiger

Mathematics and Statistics Faculty Publications and Presentations

We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.