Mathematics Commons^™

Finite Integration Method Using Chebyshev Polynomials For Solving Time-Dependent Linear Partial Differential Equations And Linear Fractional Order Differential Equations, Arnont Saengsiritongchai

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this thesis, we devise numerical algorithms base on the finite integration method (FIM) using Chebyshev polynomial to find numerical solutions of time-dependent linear partial differential equations and linear fractional order differential equations. The results show that for time-dependent linear partial differential equations, our algorithm gives a lot better accuracy than the traditional FIMs. Several examples illustrate that our algorithm for linear fractional order differential equations gives good approximate solution as well.

Generalized Symplectic Graphs And Generalized Orthogonal Graphs Over Finite Commutative Rings, Siripong Sirisuk

Chulalongkorn University Theses and Dissertations (Chula ETD)

Let R be a finite commutative ring with identity, n ∈ N and β a bilinear form on Rⁿ. In this dissertation, we count the numbers of free submodules and totally isotropic free submodules of Rⁿ of rank s by using the lifting idea. We define the generalized bilinear form graph whose vertex set is the set of totally isotropic free submodules of Rⁿ of rank s and the adjacency condition is given by some rank conditions. We study this graph when (Rⁿ, β) is a symplectic space and an orthogonal space. We can determine the degree of each vertex …

On The Density Of The Odd Values Of The Partition Function, Samuel Judge

Dissertations, Master's Theses and Master's Reports

The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities is that, under suitable …

An Efficient Method For Online Identification Of Steady State For Multivariate System, Honglun None Xu

Open Access Theses & Dissertations

Most of the existing steady state detection approaches are designed for univariate signals. For multivariate signals, the univariate approach is often applied to each process variable and the system is claimed to be steady once all signals are steady, which is computationally inefficient and also not accurate. The article proposes an efficient online method for multivariate steady state detection. It estimates the covariance matrices using two different approaches, namely, the mean-squared-deviation and mean-squared-successive-difference. To avoid the usage of a moving window, the process means and the two covariance matrices are calculated recursively through exponentially weighted moving average. A likelihood ratio …

A New Proof Of The Existence Of Embedded Surfaces With Anosov Geodesic Flow, Victor J. Donnay

Mathematics Faculty Research and Scholarship

We give a new proof of the existence of compact surfaces embedded in ℝ³ with Anosov geodesic flows. This proof starts with a noncompact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone condition. Using a sequence of explicit maps based on the standard torus embedding, we produce compact embedded surfaces that can be seen as small perturbations of the Anosov model system and hence are themselves Anosov.

In Quest Of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, And Summation Identities For Entire Functions, Rajitha Puwakgolle Gedara

Electronic Theses and Dissertations

The title of the dissertation gives an indication of the material involved with the connecting thread throughout being the classical Bernstein inequality (and its variants), which provides an estimate to the size of the derivative of a given polynomial on a prescribed set in the complex plane, relative to the size of the polynomial itself on the same set. Chapters 1 and 2 lay the foundation for the dissertation. In Chapter 1, we introduce the notations and terminology that will be used throughout. Also a brief historical recount is given on the origin of the Bernstein inequality, which dated back …

A Two-Dimensional Finite Element Model Of The Grain Boundary Based On Thermo-Mechanical Strain Gradient Plasticity, Yooseob Song, George Z. Voyiadjis

Civil Engineering Faculty Publications

In this work, a two-dimensional finite element model for the grain boundary flow rule is developed based on the thermo-mechanical gradient-enhanced plasticity theory. The proposed model is temperature-dependent. A special attention is given to physical and micromechanical nature of dislocation interactions in combination with thermal activation on stored and dissipated energy. Thermodynamic conjugate microforces are decomposed into energetic and dissipative components. Correspondingly, two different grain boundary material length scales are present in the proposed model. Finally, numerical examples are solved in order to explore characteristics of the proposed grain boundary flow rule.

Interlace Polynomials Of Friendship Graphs, Christina Eubanks-Turner, Aihua Li

Department of Mathematics Faculty Scholarship and Creative Works

In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdös, Rényi and Sos. Explicit formulas, special values, and behaviour of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph.

Inverse Function: Pre-Service Teachers’ Techniques And Meanings, Teo Paoletti, Irma E. Stevens, Natalie L.F. Hobson, Kevin C. Moore, Kevin R. Laforest

Department of Mathematics Faculty Scholarship and Creative Works

Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers’ and students’ inverse function meanings. Responding to this call, we characterize 25 pre-service teachers’ inverse function meanings as inferred from our analysis of clinical interviews. After summarizing relevant research, we describe the methodology and theoretical framework we used to interpret the pre-service teachers’ activities. We then present data highlighting the techniques the pre-service teachers used when responding to tasks that involved analytical and graphical representations of functions and inverse functions in both decontextualized and contextualized situations and discuss …

A Covariational Understanding Of Function: Putting A Horse Before The Cart, Teo Paoletti, Kevin C. Moore

Department of Mathematics Faculty Scholarship and Creative Works

Supporting students developing understandings of function has been a notoriously elusive task in mathematics education. We present Thompson and Carlson’s (2017) description of a covariational meaning of function and provide an example of a student who maintains meanings compatible with this description. We use this student’s activity to illustrate nuances in a covariational meaning of function and to highlight how such meanings can be powerful for students. In doing so, we argue that a student who has develop meanings compatible with the covariational meaning of function presented by Thompson and Carlson has the foundational meanings needed to understand a formal …

On The Mixtures Of Weibull And Pareto (Iv) Distribution: An Alternative To Pareto Distribution, I. Ghosh, Gholamhossein G. Hamedani, Naveen K. Bansal, Mehdi Maadooliat

Mathematics, Statistics and Computer Science Faculty Research and Publications

Finite mixture models have provided a reasonable tool to model various types of observed phenomena, specially those which are random in nature. In this article, a finite mixture of Weibull and Pareto (IV) distribution is considered and studied. Some structural properties of the resulting model are discussed including estimation of the model parameters via expectation maximization (EM) algorithm. A real-life data application exhibits the fact that in certain situations, this mixture model might be a better alternative than the rival popular models.

On Characterizations Of Mcllog, Ellogw, Pthl And K-Ge Distributions, Gholamhossein G. Hamedani, Nadeem Shafique Butt

Mathematics, Statistics and Computer Science Faculty Research and Publications

Huang S. and Oluyede (2016), Oluyede et al. (2016), Krishnarani (2016) and Rather and Rather (2017) consider the "McDonald Log-Logistic", the "Exponentiated Log-Logistic Weibull", the "Power Transformation Half-Logistic" and "k-Generalized Exponential" distributions, respectively, and study certain properties and applications of these distributions. The present short note is intended to complete, in some way, the above mentioned works via establishing certain characterizations of these distributions in different directions.

Characterizations And Infinite Divisibility Of Certain Recently Introduced Distributions Iii, Gholamhossein G. Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

Certain characterizations of recently proposed univariate continuous distributions are presented in different directions. This work may be a source of preventing reinventing and duplicating the existing distributions and calling them newly proposed distributions.

A Bayesian Variable Selection Approach Yields Improved Detection Of Brain Activation From Complex-Valued Fmri, Cheng-Han Yu, Raquel Prado, Hernando Ombao, Daniel B. Rowe

Mathematics, Statistics and Computer Science Faculty Research and Publications

Voxel functional magnetic resonance imaging (fMRI) time courses are complex-valued signals giving rise to magnitude and phase data. Nevertheless, most studies use only the magnitude signals and thus discard half of the data that could potentially contain important information. Methods that make use of complex-valued fMRI (CV-fMRI) data have been shown to lead to superior power in detecting active voxels when compared to magnitude-only methods, particularly for small signal-to-noise ratios (SNRs). We present a new Bayesian variable selection approach for detecting brain activation at the voxel level from CV-fMRI data. We develop models with complex-valued spike-and-slab priors on the activation …

Analysis Of High Performance Scientific Programming Workflows, Withana Kankanamalage Umayanganie Klaassen

Open Access Theses & Dissertations

Substantial time is spent on building, optimizing and maintaining large-scale software that is run on supercomputers. However, little has been done to utilize overall resources efficiently when it comes to including expensive human resources. The community is beginning to acknowledge that optimizing the hardware performance such as speed and memory bottlenecks contributes less to the overall productivity than does the development lifecycle of high-performance scientific applications. Researchers are beginning to look at overall scientific workflows for high performance computing. Scientific programming productivity is measured by time and effort required to develop, configure, and maintain a simulation experiment and its constituent …

Mathematical Modeling Of Tumor Growth For Free-Boundary Problem By Enhanced Finite Volume Method, Mashriq Ahmed Saleh

Open Access Theses & Dissertations

Modeling tumor growth due to infiltration of immune cells presents several challenges in numerical computations. First, it involves multiple cell species whose total number should be a constant, due to the incompressibility assumption; second, by mapping the Eulerian coordinate of the free-boundary problem onto a fixed logical domain, geometric source terms appear and they need to be addressed properly in numerical methods. In this work, we use a simplified model that contains two species and prescribed infiltration velocity and to show that the conventional finite volume methods fail to preserve the trivial (constant) solutions. To this end, we introduce the …

The P-Laplacian Problem Via An Euler Equation And The Basis Properties Of Its Eigenfunctions, Luis Suarez Salas

Open Access Theses & Dissertations

The Laplacian operator is used in many fields of science, such as fluidodynamics, mechanics and elasticity. Mathematically, much research has been devoted to develop a theory with which it and other variations can be understood. In this work, we present the p-Laplacian problem via an Euler equation. We then study the properties of its eigenfunctions which generalize the trigonometric functions sine and cosine. In connection with a Fourier series, we then show the generalized trigonometric functions possess basis properties for L^r((0,1)^d), d=1,2,3. Finally, we introduce the spaces of variable exponent and the analogue p(x)-Laplacian problem which has immense applications such …

On Numerical Stochastic Optimal Control Via Bellman's Dynamic Programming Principle, Prince Osei Aboagye

Open Access Theses & Dissertations

In this work, we present an application of Stochastic Control Theory to the Merton's portfolio optimization problem. Then, the dynamic programming methodology is applied to reduce the whole problem to solving the well-known HJB (Hamilton-Jacobi-Bellman) equation that arises from the Merton's portfolio optimization problem subject to the power utility function. Finally, a numerical method is proposed to solve the HJB equation and the optimal strategy. The numerical solutions are compared with the explicit solutions for optimal consumption and investment control policies.

A Mixed Finite Element Method For The Coupling Of Linear Elasticity And Stokes Flow, Maranda Bean

Open Access Theses & Dissertations

The complex interaction between fluids and structures require the coupling the laws concerning structure mechanics and fluid dynamics and are of vital importance to many scientific and engineering fields. We propose a method for modeling the coupling of a linearly elastic solid and slow fluid flow modeled by Stokes equations. The model equations are expressed in terms of displacement, velocity and stress. With these primary variables, we use a single mixed finite element space based on the Hellinger-Reissner variational principle for linear elasticity to discretize the resulting system spatially. This results in more accurate approximations for stress than those obtained …

The Rsa Cryptosystem, Rodrigo Iglesias

Williams Honors College, Honors Research Projects

This paper intends to present an overview of the RSA cryptosystem. Cryptosystems are mathematical algorithms that disguise information so that only the people for whom the information is intended can read it. The invention of the RSA cryptosystem in 1977 was a significant event in the history of cryptosystems. We will describe in detail how the RSA cryptosystem works and then illustrate the process with a realistic example using fictional characters. In addition, we will discuss how cryptosystems worked prior to the invention of RSA and the advantage of using RSA over any of the previous cryptosystems. This will help …

Meeting Real World Demands Of The Global Economy: An Employer's Perspective, Doreen Mcgunagle, Laura Zizka

Journal of Aviation/Aerospace Education & Research

Educational programs prepare students theoretically for the workplace, but many programs are still lacking in the real-world skills that the workplace requires. This is especially evident in Science, Technology, Engineering, and Math (STEM) education where today’s graduates hold a fundamental role in advancing science, medicine, sustainability, national security, and the economy, yet the programs to prepare them are falling short of employer expectations. At present, there is a lack of information on the necessary skills for workplace success that is specific to Airline, Aerospace, Defense (A&D) and related Industries’ STEM graduates. This paper attempts to fill this gap by offering …

Generalized Characteristics Of A Generic Polytope, Tommy Naugle

Electronic Theses and Dissertations

For a smooth hypersurface S ⊂ R 2n given by the level set of a Hamiltonian function H, a symplectic form ω on R2n induces a vector field XH which flows tangent to S. By the nondegeneracy of ω, there exists a distinguished line bundle LS whose characteristics are the integral curves of XH. When S is the boundary of a smooth convex domain K˜ ⊂ R 2n, then the least action among closed characteristics of LS is equal to the Ekeland-Hofer-Zehnder capacity, a symplectic invariant. From a result due to Artstein-Avidan and Ostrover, there exists a continuous extension of …

Mathematically Predicting The Aleut Tribe Population Using Archaeological Data, Jack Conrad Kiefer Ii, Paden Allen Thompson

Research on Capitol Hill

Sanak Island, located off the southern Alaska Peninsula, was home to the native Aleut peoples for thousands of years. Their hunter-gatherer society depended heavily on the arctic and marine ecosystem for food resources.

In 2015, a team of archaeologists from Idaho State and Utah State universities went to the island and collected data about the Aleut population size and their diet.

This study constructed a dynamical model to mathematically predict the Aleut population over time in order to gain insights into how food resources affected the Aleut people’s ability to survive.

Connecting The Arcs Motivational Model To Game Design For Mathematics Learning, Mario J. Toussaint, Victoria Brown

Transformations

Students’ performance in mathematics at all levels of education have not markedly improved for the past few decades. Stakeholders at tertiary institutions have been attempting to remedy this situation by implementing various types of intervention programs designed to increase students’ success in mathematics courses. Technology has been the primary method adopted by many educators to foster student engagement in mathematics. This paper describes an intervention program that uses the ARCS model to develop a serious game intended to increase students’ motivation and engagement in a mathematics course.

Labs For Calculus: Learning Through Collaborative Discovery, Ryan Mckinney

Summer Community of Scholars Posters (RCEU and HCR Combined Programs)

No abstract provided.

A Supply Chain Profile Of A School-Based Feeding Program Using The Centralized Kitchen Model, Eden Delight Miro, J. Lemuel Martin, Leslie Lopez, Joselito Secson, Carmela Oracion, Jhoel Loanzon

Mathematics Faculty Publications

There has recently been renewed interest and a growing demand for school feeding programs. In the Philippines, the government, through the Department of Education (DepEd), Department of Social Welfare and Development (DSWD), and non-government organizations such as the Ateneo Center for Educational Development (ACED), has initiated such programs to address the prevalence of malnutrition among Filipino school-age children. In 2011, ACED introduced the ACED Blueplate Centralized Kitchen (ABCK) model for large-scale school feeding. This study aims to provide a supply chain profile of the first and largest city-wide implementation of the ABCK model in the Philippines to date, which is …

String C-Groups Of Order 1024, Yasushi Gomi, Mark L. Loyola, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

This paper determines the nondegenerate string C-groups of order 1024. For groups of rank 3, we use the technique of central extension of string C-groups of order 512. For groups of rank at least 4, we compute for quotients of universal string C-groups.

Deep Linear Networks With Arbitrary Loss: All Local Minima Are Global, Thomas Laurent

Mathematics, Statistics and Data Science Faculty Works

We consider deep linear networks with arbitrary convex differentiable loss. We provide a short and elementary proof of the fact that all local minima are global minima if the hidden layers are either 1) at least as wide as the input layer, or 2) at least as wide as the output layer. This result is the strongest possible in the following sense: If the loss is convex and Lipschitz but not differentiable then deep linear networks can have sub-optimal local minima.

The Multilinear Structure Of Relu Networks, Thomas Laurent

Mathematics, Statistics and Data Science Faculty Works

We study the loss surface of neural networks equipped with a hinge loss criterion and ReLU or leaky ReLU nonlinearities. Any such network defines a piecewise multilinear form in parameter space. By appealing to harmonic analysis we show that all local minima of such network are non-differentiable, except for those minima that occur in a region of parameter space where the loss surface is perfectly flat. Non-differentiable minima are therefore not technicalities or pathologies; they are heart of the problem when investigating the loss of ReLU networks. As a consequence, we must employ techniques from nonsmooth analysis to study these …

Stochastic Maximum Principle For Partial Information Optimal Investment And Dividend Problem Of An Insurer, Yanping Ma

Mathematics, Statistics and Data Science Faculty Works

We study an optimal investment and dividend problem of an insurer, where the aggregate insurance claims process is modeled by a pure jump Lévy process. We allow the management of the dividend payment policy and the investment of surplus in a continuous-time financial market, which is composed of a risk free asset and a risky asset. The information available to the insurer is partial information. We generalize this problem as a partial information regular-singular stochastic control problem, where the control variable consists of regular control and singular control. Then maximum principles are established to give sufficient and necessary optimality conditions …