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Applied Mathematical Analysis and Computations I

1st SGMC, Statesboro, USA, April 2–3, 2021 (Virtual)

Editors: Divine Wanduku, Shijun Zheng, Haomin Zhou, Zhan Chen, Andrew Sills, Ephraim Agyingi

Publisher: Springer Nature Switzerland

Book Series : Springer Proceedings in Mathematics & Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Table of Contents

About this book

This volume convenes selected, peer-reviewed research and survey articles that address the modern state-of-the-art in varied areas of applied mathematical analysis. They primarily include presentations as well as invited contributions for the 1st Southern Georgia Mathematics Conference (SGMC) that was virtually held on April 2—3, 2021 at the Georgia Southern University, Statesboro, USA. Papers in this volume incorporate both advanced theory and methods from mathematical analysis, and cover myriad topics like imaging and inverse problems, evolutionary PDEs, symbolic computation, dynamics and data analysis, data science, computational mathematics, and more. This first volume focuses on mathematical analysis theory and applications.

These studies and findings contained herein will be of interest to researchers and graduate students working in the fields of mathematical analysis, modeling, data analysis and computation, with applications in many interdisciplinary applied sciences, as in statistics, physics, biology, and medical imaging. They are particularly relevant to those at the forefront of applied mathematical and statistical analysis, as well as data science and other computational science disciplines.

In its first edition, the Southern Georgia Mathematics Conference brought together 74 speakers from 70 different institutions, from the USA, Canada, Austria, and Botswana. Attendees included faculty, researchers, experts, graduate and undergraduate students from all over the world.

Frontmatter

Characterization of Carleson Measures via Spectral Estimates on Compact Manifolds with Boundary

Abstract

Given a compact Riemannian manifold M with boundary of dimension \(m\geq 2\), we study the space of functions \(E_L\) of \(L^2(M)\) generated by eigenfunctions of eigenvalues less than \(L\geq 1\) associated to Dirichlet Laplacian and Neumann Laplacian on M. The asymptotics of the reproducing kernel of the space \(E_L\) and a Bernstein type inequality for \(f\in E_L\) are discussed. Furthermore under suitable convexity assumptions on the boundary, the mean value inequalities of subharmonic functions associated to \(E_L\) in the scale \(\frac {1}{\sqrt {L}}\) are achieved on the metric ball with possible nonempty intersection with the boundary, which generalizes the classical mean value inequality on the interior geodesic ball by Li, Schoen, and Yau. Applying the asymptotic estimates, Bernstein type inequality and mean value inequality on these spaces \(E_L\), we show a characterization of the \(L^2\)-Carleson measures associated to Neumann Laplacian with the interior rolling R-ball condition on the boundary, and give a counterexample to invalid the characterization of the \(L^2\)-Carleson measures associated to Dirichlet Laplacian.

Xiangjin Xu

Perturbed Fourier Transform Associated with Schrödinger Operators

Abstract

We give an exposition on the \(L^2\) theory of the perturbed Fourier transform associated with a Schrödinger operator \(H=-d^2/dx^2 +V\) on the real line, where V is a real-valued finite measure. In the case \(V\in L^1\cap L^2\), we explicitly define the perturbed Fourier transform \(\mathcal {F}\) for H and obtain an eigenfunction expansion theorem for square integrable functions. This provides a complete proof of the inversion formula for \({\mathcal F}\) that covers the class of short range potentials in \((1+|x|)^{-\frac 12-\epsilon } L^2 \). Such paradigm has applications in the study of scattering problems in connection with the spectral properties and asymptotic completeness of the wave operators.

Shijun Zheng

Explicit Lump and Line Rogue Wave Solutions to a Modified Hietarinta Equation

Abstract

Lump solutions are spatially rationally localized solutions which usually arise as solutions to higher dimensional nonlinear partial differential equations often possessing Hirota bilinear forms. Under some parameter constraint, these solutions may lead to rogue wave solutions. In this article, we study lump and rogue wave solutions of a new nonlinear non-evolutionary equation in 2+1 dimensions with the aid of a computer algebra system. We present illustrative examples and analyze the dynamical behavior of the solutions using graphical representations.

Solomon Manukure, Morgan McAnally, Yuan Zhou, Demetrius Rowland, Gina Pantano

Applying the Maximum Entropy Technique to the Gaussian Dispersion Plume Model

Abstract

The Maximum Entropy (MaxEnt) technique is applied to the derivation of the Gaussian Dispersion Plume Model as well as to more complex transport phenomena such as the one-dimensional advection equation, the one-dimensional diffusion equation, the one-dimensional advection-diffusion equation, and finally to the multi-dimensional advection-diffusion equation. Further applications are discussed.

J. A. Secrest, J. M. Conroy, H. G. Miller

Time Asymptotic Behavior of Solutions to a Chemotaxis Model with Logarithmic Singularity

Abstract

The paper is a continuation of the author’s work in [14]. We consider a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We consider Cauchy problems with Cauchy data not bounded away from the logarithmic singularity. The model can be converted into a \(2\times 2\) system of hyperbolic-parabolic balance laws by inverse Hopf-Cole transformation, with Cauchy data connecting two different end states. The converted form was studied in [14] when Cauchy data are near a diffusive contact wave. The current paper is to study the scenario under the original model to gain understanding of the evolution of physical quantities when the logarithmic singularity plays an intrinsic role. For all three cases, singularity at \(-\infty \), at \(+\infty \), and at \(\pm \infty \), we obtain a clear picture of time asymptotic behavior of solutions.

Yanni Zeng

Classical Solution to an Interfacial Flow with Kinetic Undercooling in a Time-Dependent Gap Hele-Shaw Cell

Abstract

Hele-Shaw cells where the top plate is moving uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. This paper focuses on an interfacial flow with kinetic undercooling regularization in a radial Hele-Shaw cell with a time dependent gap \(b(t)\). We obtain the local existence of classical solution of the moving boundary problem when the initial data is close to a circle. The methodology is to use complex analysis and reduce the free boundary problem to a Riemann-Hilbert problem and an abstract Cauchy-Kovalevskaya evolution problem, then apply Nash-Moser iteration.

Xuming Xie

Divisibility Arising from Addition

Abstract

The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve, representation difficulties of the associated sequences of modular functions, and difficulties regarding the piecewise \(\ell \)-adic convergence of elements of the associated space of modular functions. However, our knowledge of the subject has developed substantially and continues to develop. In this very brief survey, we will discuss the utility of modular functions in proving partition congruences, both theoretical and computational, and many of the problems in the subject that are yet to be overcome.

Nicolas Allen Smoot

Using Words to Construct and Enumerate Maximum Nonattacking Chessboard Arrangements

Abstract

Words are ubiquitous objects in combinatorics that can be studied in their own right or be used to represent sets of other combinatorial objects. Additionally, mathematical questions related to chess, its pieces, its board, and the many extensions and generalizations, have been posed for hundreds of years. In particular, chess pieces can have move sets beyond those in the established game of chess. Here we use words to describe nonattacking arrangements of chess pieces on a rectangular \(2\times 2n\) chessboard. We then extend the notion of words on a single line to a matrix of letters, focusing on pieces who can move with attacks from a king’s move set. Bijections between matrices of letters in small alphabets and nonattacking maximum arrangements of pieces on rectangular chessboards are used to enumerate such arrangements.

Tricia Muldoon Brown

A Refinement of the Multinomial Distribution with Application

Abstract

A refinement of the multinomial distribution is presented where the number of inversions in the sequence of outcomes is tallied. This refinement of the multinomial distribution is its joint distribution with the number of inversions in the accompanying experiment. An application of this additional information is described in which the number of inversions acts as a proxy measure of homogeneity (or lack thereof) in the sequence of outcomes.

Andrew V. Sills

The Generalized Laws of Total Variance and Total Covariance

Abstract

The law of total variance states that the unconditional variance of a random variable Y is the sum of (a) the variance of the conditional expectation of Y given X and (b) the expectation of the conditional variance of Y given the random variable X. We show that the total variance of Y can be partitioned by using the relationship between Y and one or more random variables \(X_{1},\ldots ,X_{k}\), where \(k\geq 1\). An application in multivariate analysis is given. Further, we generalize the total law of total covariance and show that the generalized law of total variance is a special case. Some examples are referenced.

Charles W. Champ, Andrew V. Sills

Ansatz in a Nutshell: A Comprehensive Step-by-Step Guide to Polynomial, C-finite, Holonomic, and C2-finite Sequences

Abstract

Given a sequence 1, 1, 5, 23, 135, 925, 7285, 64755, 641075, 6993545, 83339745,…, how can we guess a formula for it? This article will quickly walk you through the concept of ansatz for classes of polynomial, C-finite, holonomic, and the most recent addition C²-finite sequences. For each of these classes, we discuss in detail various aspects of the guess and check, generating functions, closure properties, and closed-form solutions. Every theorem is presented with an accessible proof, followed by several examples intended to motivate the development of the theories. Each example is accompanied by a Maple program with the purpose of demonstrating use of the program in solving problems in this area. While this work aims to give a comprehensive review of existing ansatzes, we also systematically fill a research gap in the literature by providing theoretical and numerical results for the C²-finite sequences.

Tipaluck Krityakierne, Thotsaporn Aek Thanatipanonda

The Exponentiated Half-Logistic-Weibull Topp-Leone-G Family of Distributions

Abstract

A new family of distributions called the Exponentiated Half-Logistic-Weibull Topp-Leone-G (EHL-WTL-G) distribution is proposed and studied. The novel EHL-WTL-G distribution is flexible and compatible with a variety of real data sets, most importantly is the ability of the model to capture heavy-tailed data sets. Some structural properties of the new family of distributions including moments, conditional moments, probability weighted moments, distribution of the order statistics and Rényi entropy are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study to examine the bias and mean square error of the maximum likelihood estimators is presented and applications to real dataset to illustrates the usefulness of the model are given.

Whatmore Sengweni, Broderick Oluyede

Statistical Analysis of a New Markov Chain Model for Rumor Dynamics in Heterogeneous Complex Social Networks

Abstract

A new Markov chain (MC) model for rumor epidemic dynamics in heterogeneous complex social networks (HCSN) is investigated. The epidemic model is also a compartmental SEIR (susceptible-exposed-infected-removed), with an expanded multidimensional state space for the MC by utilizing two discrete time measures for representing both the “disease” states and the age in each state of the nodes in the HCSN. Moreover, characterizations of the MC at both the mean-field and global levels of the HCSN are given. A statistical analysis of the MC is conducted to find a parameter for the basic reproduction number, \( \mathfrak {R}_{0}\), of the “disease” in the network; and to find an unbiased and precise estimator for \( \mathfrak {R}_{0}\). The goodness-of-fit of the estimator is investigated by employing the mean squared error.

Divine Wanduku

Dynamics of Amensalism, Mutualism, and Predation in a Three Species Complex Ecosystem

Abstract

Understanding the intricate interactions within ecosystems is vital for comprehending the delicate balance that sustains life on our planet. This paper is concerned with modeling an ecosystem characterized by the simultaneous occurrence of amensalism, mutualism, and predation, exploring the interplay between these ecological relationships and their impact on species diversity and community stability. The model formulated is shown to admit only positive solutions that are also bounded. We determine the equilibrium points, conduct a comprehensive analysis of their stability and numerical computations of the proposed model are provided. Further, numerical simulations that demonstrated the existence of a Hopf bifurcation about the interior equilibrium point for several parameter values are also provided.

Ramsey (Rayla) Phuc, Ephraim O. Agyingi

Backmatter

Title: Applied Mathematical Analysis and Computations I
Editors: Divine Wanduku
Shijun Zheng
Haomin Zhou
Zhan Chen
Andrew Sills
Ephraim Agyingi
Publisher: Springer Nature Switzerland
Electronic ISBN: 978-3-031-69706-7
Print ISBN: 978-3-031-69705-0
DOI: https://doi.org/10.1007/978-3-031-69706-7