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The Isoperimetric Inequality: Proofs By Convex And Differential Geometry, Penelope Gehring 2020 Potsdam University

The Isoperimetric Inequality: Proofs By Convex And Differential Geometry, Penelope Gehring

Rose-Hulman Undergraduate Mathematics Journal

The Isoperimetric Inequality has many different proofs using methods from diverse mathematical fields. In the paper, two methods to prove this inequality will be shown and compared. First the 2-dimensional case will be proven by tools of elementary differential geometry and Fourier analysis. Afterwards the theory of convex geometry will briefly be introduced and will be used to prove the Brunn--Minkowski-Inequality. Using this inequality, the Isoperimetric Inquality in n dimensions will be shown.


Quantitative Analysis Of A Stochastic Seitr Epidemic Model With Multiple Stages Of Infection And Treatment, Olusegun M. Otunuga, Mobolaji O. Ogunsolu 2020 Marshall University

Quantitative Analysis Of A Stochastic Seitr Epidemic Model With Multiple Stages Of Infection And Treatment, Olusegun M. Otunuga, Mobolaji O. Ogunsolu

Mathematics Faculty Research

We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered (SEITR) model with multiple stages of infection and treatment and explore the effects of treatments and external fluctuations in the transmission, treatment and recovery rates. We assume external fluctuations are caused by variability in the number of contacts between infected and susceptible individuals. It is shown that the expected number of secondary infections produced (in the absence of noise) reduces as treatment is introduced into the population. By defining RT,n and ℛT,n as the basic deterministic and stochastic reproduction numbers, respectively ...


Elementary Hyperreal Analysis, Logan Cebula 2020 The University of Akron

Elementary Hyperreal Analysis, Logan Cebula

Williams Honors College, Honors Research Projects

This text explores elementary analysis through the lens of non-standard analysis. The hyperreals will be proven to be implied by the existence of the reals via the axiom of choice. The notion of a hyperextension will be defined, and the so-called Transfer Principle will be proved. This principle establishes equivalence between results in real and hyperreal analysis. Sequences, subsequences, and limit suprema/infima will then be explored. Finally, integration will be considered.


Evaluating An Ordinal Output Using Data Modeling, Algorithmic Modeling, And Numerical Analysis, Martin Keagan Wynne Brown 2020 Murray State University

Evaluating An Ordinal Output Using Data Modeling, Algorithmic Modeling, And Numerical Analysis, Martin Keagan Wynne Brown

Murray State Theses and Dissertations

Data and algorithmic modeling are two different approaches used in predictive analytics. The models discussed from these two approaches include the proportional odds logit model (POLR), the vector generalized linear model (VGLM), the classification and regression tree model (CART), and the random forests model (RF). Patterns in the data were analyzed using trigonometric polynomial approximations and Fast Fourier Transforms. Predictive modeling is used frequently in statistics and data science to find the relationship between the explanatory (input) variables and a response (output) variable. Both approaches prove advantageous in different cases depending on the data set. In our case, the data ...


Eigenvalue Statistics And Localization For Random Band Matrices With Fixed Width And Wegner Orbital Model, Benjamin Brodie 2020 University of Kentucky

Eigenvalue Statistics And Localization For Random Band Matrices With Fixed Width And Wegner Orbital Model, Benjamin Brodie

Theses and Dissertations--Mathematics

We discuss two models from the study of disordered quantum systems. The first is the Random Band Matrix with a fixed band width and Gaussian or more general disorder. The second is the Wegner $n$-orbital model. We establish that the point process constructed from the eigenvalues of finite size matrices converge to a Poisson Point Process in the limit as the matrix size goes to infinity.

The proof is based on the method of Minami for the Anderson tight-binding model. As a first step, we expand upon the localization results by Schenker and Peled-Schenker-Shamis-Sodin to account for complex energies ...


Theory Of Lexicographic Differentiation In The Banach Space Setting, Jaeho Choi 2019 University of Maine

Theory Of Lexicographic Differentiation In The Banach Space Setting, Jaeho Choi

Electronic Theses and Dissertations

Derivative information is useful for many problems found in science and engineering that require equation solving or optimization. Driven by its utility and mathematical curiosity, researchers over the years have developed a variety of generalized derivatives. In this thesis, we will first take a look at Clarke’s generalized derivative for locally Lipschitz continuous functions between Euclidean spaces, which roughly is the smallest convex set containing all nearby derivatives of a domain point of interest. Clarke’s generalized derivative in this setting possesses a strong theoretical and numerical toolkit, which is analogous to that of the classical derivative. It includes ...


A Generalization Of Schroter's Formula To George Andrews, On His 80th Birthday, James McLaughlin 2019 West Chester University of Pennsylvania

A Generalization Of Schroter's Formula To George Andrews, On His 80th Birthday, James Mclaughlin

Mathematics Faculty Publications

We prove a generalization of Schroter's formula to a product of an arbitrary number of Jacobi triple products. It is then shown that many of the well-known identities involving Jacobi triple products (for example the Quintuple Product Identity, the Septuple Product Identity, and Winquist's Identity) all then follow as special cases of this general identity. Various other general identities, for example certain expansions of (q; q)(infinity) and (q; q)(infinity)(k), k >= 3, as combinations of Jacobi triple products, are also proved.


Stochastic Process And Its Role In The Development Of The Financial Market: Celebrating Professor Chow's Long And Successful Career, Xisuo L. Liu 2019 18 Thoroughbred Drive, Sherborn, MA 01770, USA

Stochastic Process And Its Role In The Development Of The Financial Market: Celebrating Professor Chow's Long And Successful Career, Xisuo L. Liu

Communications on Stochastic Analysis

No abstract provided.


Stochastic Partial Differential Equation Sis Epidemic Models: Modeling And Analysis, Nhu N. Nguyen, George Yin 2019 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

Stochastic Partial Differential Equation Sis Epidemic Models: Modeling And Analysis, Nhu N. Nguyen, George Yin

Communications on Stochastic Analysis

No abstract provided.


Action Functionals For Stochastic Differential Equations With Lévy Noise, Shenglan Yuan, Jinqiao Duan 2019 Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan, 430074, China

Action Functionals For Stochastic Differential Equations With Lévy Noise, Shenglan Yuan, Jinqiao Duan

Communications on Stochastic Analysis

No abstract provided.


Anticipating Exponential Processes And Stochastic Differential Equations, Chii Ruey Hwang, Hui-Hsiung Kuo, Kimiaki Saitô 2019 Institute of Mathematics, Academia Sinica, 6F Astronomy-Math Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan

Anticipating Exponential Processes And Stochastic Differential Equations, Chii Ruey Hwang, Hui-Hsiung Kuo, Kimiaki Saitô

Communications on Stochastic Analysis

No abstract provided.


Preface, Hui-Hsiung Kuo, George Yin 2019 Louisiana State University, Baton Rouge, LA 70803 USA

Preface, Hui-Hsiung Kuo, George Yin

Communications on Stochastic Analysis

No abstract provided.


Euler-Maruyama Method For Regime Switching Stochastic Differential Equations With Hölder Coefficients, Dung T. Nguyen, Son L. Nguyen 2019 Department of Applied Mathematics, Faculty of Applied Science, HCMC University of Technology, Vietnam

Euler-Maruyama Method For Regime Switching Stochastic Differential Equations With Hölder Coefficients, Dung T. Nguyen, Son L. Nguyen

Communications on Stochastic Analysis

No abstract provided.


Subdifferentials Of Value Functions In Nonconvex Dynamic Programming For Nonstationary Stochastic Processes, Boris S. Mordukhovich, Nobusumi Sagara 2019 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

Subdifferentials Of Value Functions In Nonconvex Dynamic Programming For Nonstationary Stochastic Processes, Boris S. Mordukhovich, Nobusumi Sagara

Communications on Stochastic Analysis

No abstract provided.


Hybrid Models And Switching Control With Constraints, Jose L. Menaldi, Maurice Robin 2019 Wayne State University

Hybrid Models And Switching Control With Constraints, Jose L. Menaldi, Maurice Robin

Communications on Stochastic Analysis

No abstract provided.


Non-Nested Monte Carlo Dual Bounds For Multi-Exercisable Options, Xiang Cheng, Zhuo Jin 2019 Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia

Non-Nested Monte Carlo Dual Bounds For Multi-Exercisable Options, Xiang Cheng, Zhuo Jin

Communications on Stochastic Analysis

No abstract provided.


Totalitarian Random Tug-Of-War Games In Graphs, Marcos Antón, Fernando Charro, Peiyong Wang 2019 Universitat Politècnica de Catalunya, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain

Totalitarian Random Tug-Of-War Games In Graphs, Marcos Antón, Fernando Charro, Peiyong Wang

Communications on Stochastic Analysis

No abstract provided.


Analysis Of Feast Spectral Approximations Using The Dpg Discretization, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall, Benjamin Q. Parker 2019 Portland State University

Analysis Of Feast Spectral Approximations Using The Dpg Discretization, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall, Benjamin Q. Parker

Jeffrey S. Ovall

A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as “FEAST”, has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of ...


Polynomial And Rational Convexity Of Submanifolds Of Euclidean Complex Space, Octavian Mitrea 2019 The University of Western Ontario

Polynomial And Rational Convexity Of Submanifolds Of Euclidean Complex Space, Octavian Mitrea

Electronic Thesis and Dissertation Repository

The goal of this dissertation is to prove two results which are essentially independent, but which do connect to each other via their direct applications to approximation theory, symplectic geometry, topology and Banach algebras. First we show that every smooth totally real compact surface in complex Euclidean space of dimension 2 with finitely many isolated singular points of the open Whitney umbrella type is locally polynomially convex. The second result is a characterization of the rational convexity of a general class of totally real compact immersions in complex Euclidean space of dimension n..


Analysis Of Feast Spectral Approximations Using The Dpg Discretization, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall, Benjamin Q. Parker 2019 Portland State University

Analysis Of Feast Spectral Approximations Using The Dpg Discretization, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall, Benjamin Q. Parker

Jay Gopalakrishnan

A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as “FEAST”, has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of ...


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