On The Density Of The Odd Values Of The Partition Function, 2018 Michigan Technological University

#### 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 ...

Commutators, Little Bmo And Weak Factorization, 2018 Washington University in St. Louis

#### Commutators, Little Bmo And Weak Factorization, Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang

*Mathematics Faculty Publications*

In this paper, we provide a direct and constructive proof of weak factorization of h1 (ℝ×ℝ) (the predual of little BMO space bmo(ℝ×ℝ) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f Є h1 (ℝ×ℝ) there exist sequences {αkj} Є l and functions gjk, hkj Є L2 (ℝ2 ) such that [Equation Unavailable] in the sense of h1 (ℝ×ℝ), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm ║fh1║(ℝ×ℝ) is given in terms of ║gjk║ L2(ℝ2) and ║hkj║ L2(ℝ2). By duality, this ...

Onsager Reciprocal Relations: Microscopic (Onsager) Or Macroscopic (Sliepcevich), 2018 University of Arkansas - Main Campus

#### Onsager Reciprocal Relations: Microscopic (Onsager) Or Macroscopic (Sliepcevich), R. E. "Buddy" Babcock

*Chemical Engineering Faculty Publications and Presentations*

This paper is a combination of the discussion of two nineteenth century theoretical giantsLars Onsager and C. M. Sliepcevich, their works in general, and specifically the famousreciprocal relations of Onsager with respect to irreversible thermodynamics. Emphasis isplaced on their penetrating depth and breadth of analysis so inherently necessary in theirproblem-solving endeavors. The landscape of their work will be laid out for the readerby a comparison of Onsager’s microscopic statistical mechanics derivation of the famousreciprocal relationships and a macroscopic thermodynamic derivation published by C. M.Sliepsevich that led to considerable discussion in the literature in the 1960’s. Somelabelled this ...

Bounded Point Derivations On Certain Function Spaces, 2018 University of Kentucky

#### Bounded Point Derivations On Certain Function Spaces, Stephen Deterding

*Theses and Dissertations--Mathematics*

Let 𝑋 be a compact subset of the complex plane and denote by 𝑅^{𝑝}(𝑋) the closure of rational functions with poles off 𝑋 in the 𝐿^{𝑝}(𝑋) norm. We show that if a point 𝑥_{0} admits a bounded point derivation on 𝑅^{𝑝}(𝑋) for 𝑝 > 2, then there is an approximate derivative at 𝑥_{0}. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for 𝑅(𝑋), the uniform closure of rational functions with poles off 𝑋. In addition, we show that if ...

Positive Symmetric Solutions Of A Boundary Value Problem With Dirichlet Boundary Conditions, 2018 Eastern Kentucky University

#### Positive Symmetric Solutions Of A Boundary Value Problem With Dirichlet Boundary Conditions, Tek Nath Dhakal

*Online Theses and Dissertations*

We apply a recent extension of a compression-expansion fixed point theorem of function type to a second order boundary value problem with Dirichlet boundary conditions. We show the existence of positive symmetric solutions of this boundary value problem.

W1,P Regularity Of Eigenfunctions For The Mixed Problem With Nonhomogeneous Neumann Data, 2018 Murray State University

#### W1,P Regularity Of Eigenfunctions For The Mixed Problem With Nonhomogeneous Neumann Data, Kohei Miyazaki

*Murray State Theses and Dissertations*

We consider an eigenvalue problem with a mixed boundary condition, where a second-order differential operator is given in divergence form and satisfies a uniform ellipticity condition. We show that if a function u in the Sobolev space W^{1,p}_{D} is a weak solution to the eigenvalue problem, then u also belongs to W^{1,p}_{D} for some p>2. To do so, we show a reverse Hölder inequality for the gradient of u. The decomposition of the boundary is assumed to be such that we get both Poincaré and Sobolev-type inequalities up to the boundary.

Homogenization In Perforated Domains And With Soft Inclusions, 2018 University of Kentucky

#### Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell

*Theses and Dissertations--Mathematics*

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative *H*^{1}-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally ...

On Spectral Theorem, 2018 Colby College

#### On Spectral Theorem, Muyuan Zhang

*Honors Theses*

There are many instances where the theory of eigenvalues and eigenvectors has its applications. However, Matrix theory, which usually deals with vector spaces with finite dimensions, also has its constraints. Spectral theory, on the other hand, generalizes the ideas of eigenvalues and eigenvectors and applies them to vector spaces with arbitrary dimensions. In the following chapters, we will learn the basics of spectral theory and in particular, we will focus on one of the most important theorems in spectral theory, namely the spectral theorem. There are many different formulations of the spectral theorem and they convey the "same" idea. In ...

Applications Of Analysis To The Determination Of The Minimum Number Of Distinct Eigenvalues Of A Graph, 2018 Iowa State University

#### Applications Of Analysis To The Determination Of The Minimum Number Of Distinct Eigenvalues Of A Graph, Beth Bjorkman, Leslie Hogben, Scarlitte Ponce, Carolyn Reinhart, Theodore Tranel

*Mathematics Publications*

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues of several families of graphs and small graphs.

I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, 2018 University of Central Florida

#### I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, Jason L. Greuling

*Honors Undergraduate Theses*

Research has shown that a frame for an n-dimensional real Hilbert space oﬀers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the Han-Larson-Naimark Dilation Theorem, which gives us the necessary and suﬃcient conditions required to dilate a frame for an n-dimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an n-dimensional real Hilbert space does not ensure that its dilation will oﬀer phase retrieval. In this thesis, we will explore and provide what necessary and suﬃcient ...

Numerical Simulation For A Rising Bubble Interacting With A Solid Wall: Impact, Bounce, And Thin Film Dynamics, 2018 Old Dominion University

#### Numerical Simulation For A Rising Bubble Interacting With A Solid Wall: Impact, Bounce, And Thin Film Dynamics, Changjuan Zhang, Jie Li, Li-Shi Luo, Tiezheng Qian

*Mathematics & Statistics Faculty Publications*

Using an arbitrary Lagrangian-Eulerian method on an adaptive moving unstructured mesh, we carry out numerical simulations for a rising bubble interacting with a solid wall. Driven by the buoyancy force, the axisymmetric bubble rises in a viscous liquid toward a horizontal wall, with impact on and possible bounce from the wall. First, our simulation is quantitatively validated through a detailed comparison between numerical results and experimental data. We then investigate the bubble dynamics which exhibits four different behaviors depending on the competition among the inertial, viscous, gravitational, and capillary forces. A phase diagram for bubble dynamics has been produced using ...

Mod Rectangular Natural Neutrosophic Numbers, 2018 University of New Mexico

#### Mod Rectangular Natural Neutrosophic Numbers, Florentin Smarandache, K. Ilanthenral, W.B. Vasantha Kandasamy

*Mathematics and Statistics Faculty and Staff Publications*

In this book authors introduce the new notion of MOD rectangular planes. The functions on them behave very differently when compared to MOD planes (square). These are different from the usual MOD planes. Algebraic structures on these MOD rectangular planes are defined and developed. However we have built only MOD interval natural neutrosophic products

The Boundedness Of The Hardy-Littlewood Maximal Function And The Strong Maximal Function On The Space Bmo, 2018 Claremont Colleges

#### The Boundedness Of The Hardy-Littlewood Maximal Function And The Strong Maximal Function On The Space Bmo, Wenhao Zhang

*CMC Senior Theses*

In this thesis, we present the space BMO, the one-parameter Hardy-Littlewood maximal function, and the two-parameter strong maximal function. We use the John-Nirenberg inequality, the relation between Muckenhoupt weights and BMO, and the Coifman-Rochberg proposition on constructing A_{1} weights with the Hardy- Littlewood maximal function to show the boundedness of the Hardy-Littlewood maximal function on BMO. The analogous statement for the strong maximal function is not yet understood. We begin our exploration of this problem by discussing an equivalence between the boundedness of the strong maximal function on rectangular BMO and the fact that the strong maximal function maps ...

Survey Of Results On The Schrodinger Operator With Inverse Square Potential, 2018 Georgia Southern University

#### Survey Of Results On The Schrodinger Operator With Inverse Square Potential, Richardson Saint Bonheur

*Electronic Theses and Dissertations*

In this paper we present a survey of results on the Schrodinger operator with Inverse ¨ Square potential, L_{a}= −∆ + a/|x|^2 , a ≥ −( d−2/2 )^2. We briefly discuss the long-time behavior of solutions to the inter-critical focusing NLS with an inverse square potential(proof not provided). Later we present spectral multiplier theorems for the operator. For the case when a ≥ 0, we present the multiplier theorem from Hebisch [12]. The case when 0 > a ≥ −( d−2/2 )^2 was explored in [1], and their proof will be presented for completeness. No improvements on the sharpness of their proof ...

Old English Character Recognition Using Neural Networks, 2018 Georgia Southern University

#### Old English Character Recognition Using Neural Networks, Sattajit Sutradhar

*Electronic Theses and Dissertations*

Character recognition has been capturing the interest of researchers since the beginning of the twentieth century. While the Optical Character Recognition for printed material is very robust and widespread nowadays, the recognition of handwritten materials lags behind. In our digital era more and more historical, handwritten documents are digitized and made available to the general public. However, these digital copies of handwritten materials lack the automatic content recognition feature of their printed materials counterparts. We are proposing a practical, accurate, and computationally efficient method for Old English character recognition from manuscript images. Our method relies on a modern machine learning ...

Neutrosophic Logic: The Revolutionary Logic In Science And Philosophy -- Proceedings Of The National Symposium, 2018 University of New Mexico

#### Neutrosophic Logic: The Revolutionary Logic In Science And Philosophy -- Proceedings Of The National Symposium, Florentin Smarandache, Huda E. Khalid, Ahmed K. Essa

*Mathematics and Statistics Faculty and Staff Publications*

The first part of this book is an introduction to the activities of the National Symposium, as well as a presentation of Neutrosophic Scientific International Association (NSIA), based in New Mexico, USA, also explaining the role and scope of NSIA - Iraqi branch. The NSIA Iraqi branch presents a suggestion for the international instructions in attempting to organize NSIA's work. In the second chapter, the pivots of the Symposium are presented, including a history of neutrosophic theory and its applications, the most important books and papers in the advancement of neutrosophics, a biographical note of Prof. Florentin Smarandache in Arabic ...

Weighted Inequalities For Dyadic Operators Over Spaces Of Homogeneous Type, 2017 University of New Mexico

#### Weighted Inequalities For Dyadic Operators Over Spaces Of Homogeneous Type, David Edward Weirich

*Mathematics & Statistics ETDs*

A so-called space of homogeneous type is a set equipped with a quasi-metric and a doubling measure. We give a survey of results spanning the last few decades concerning the geometric properties of such spaces, culminating in the description of a system of dyadic cubes in this setting whose properties mirror the more familiar dyadic lattices in R^n . We then use these cubes to prove a result pertaining to weighted inequality theory over such spaces. We develop a general method for extending Bellman function type arguments from the real line to spaces of homogeneous type. Finally, we uses this ...

Statistical Analysis Of Momentum In Basketball, 2017 Bowling Green State University

#### Statistical Analysis Of Momentum In Basketball, Mackenzi Stump

*Honors Projects*

The “hot hand” in sports has been debated for as long as sports have been around. The debate involves whether streaks and slumps in sports are true phenomena or just simply perceptions in the mind of the human viewer. This statistical analysis of momentum in basketball analyzes the distribution of time between scoring events for the BGSU Women’s Basketball team from 2011-2017. We discuss how the distribution of time between scoring events changes with normal game factors such as location of the game, game outcome, and several other factors. If scoring events during a game were always randomly distributed ...

Infinite-Dimensional Measure Spaces And Frame Analysis, 2017 The University of Iowa

#### Infinite-Dimensional Measure Spaces And Frame Analysis, Palle Jorgensen, Myung-Sin Song

*SIUE Faculty Research, Scholarship, and Creative Activity*

We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space ℋ, we study three cases of measures. We first show that, for ℋ infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain ℋ. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.

Electromagnetic Resonant Scattering In Layered Media With Fabrication Errors, 2017 Louisiana State University and Agricultural and Mechanical College

#### Electromagnetic Resonant Scattering In Layered Media With Fabrication Errors, Emily Anne Mchenry

*LSU Doctoral Dissertations*

In certain layered electromagnetic media, one can construct a waveguide that supports a harmonic electromagnetic field at a frequency that is embedded in the continuous spectrum. When the structure is perturbed, this embedded eigenvalue moves into the complex plane and becomes a “complex resonance” frequency. The real and imaginary parts of this complex frequency have physical meaning. They lie behind anomalous scattering behaviors known collectively as “Fano resonance”, and people are interested in tuning them to specific values in optical devices. The mathematics involves spectral theory and analytic perturbation theory and is well understood [16], at least on a theoretical ...