Mathematics Commons^™

Additive Energies Of Subsets Of Discrete Cubes, Xuancheng Shao

Mathematics Faculty Publications

For a positive integer n ≥ 2, define tn to be the smallest number such that the additive energy E (A) of any subset A ⊂ {0, 1, · · · , n − 1}d and any d is at most |A|tn . Trivially, we have tn ≤ 3 and tn ≥ 3 − logn 3n3 2n3 + n by considering A = {0, 1, · · · , n − 1}d. In this note, we investigate the behaviour of tn for large n and obtain the following non-trivial bounds: 3 − (1 + on→∞(1)) logn 3√3 4 ≤ tn …

A Limit Order Book Model For High Frequency Trading With Rough Volatility, Yun S. Chen-Shue

Graduate Thesis and Dissertation 2023-2024

We introduce a financial model for limit order book with two main features: First, the limit orders and market orders for the given asset both appear and interact with each other. Second, the high frequency trading (HFT, for short) activities are allowed and described by the scaling limit of nearly-unstable multi-dimensional Hawkes processes with power law decay. The model eventually becomes a stochastic partial differential equation (SPDE, for short) with the diffusion coefficient determined by a Volterra integral equation governed by a Hawkes process, whose Hurst exponent is less than 1/2, which makes the volatility path of the stochastic PDE …

On The Transmuted Distributions; Properties And Application, Jacob D. Kretzer

Theses, Dissertations and Capstones

The transmuted distributions first appeared in (2007) after Shaw and Buckley constructed a quadratic rank transmutation map (QRTM), G(u) = (1 + λ)u − λu², as a transformation of a cumulative distribution function of a random variable X, to generate the transmuted-X distribution. In (2017), Jayakumar & Babu defined the T -transmuted-X family of distributions by incorporating a transmuted-X into a transformer-transformed class of distributions (Aljarrah et al., 2014). This thesis surveys the main properties of the transmuted-X distribution, such as density shapes, moments, and entropy. Detailed attention will be given …

Post Developmental Mathematics: Experiences In College Algebra For Stem Students, Maria Cruciani

Undergraduate Research (Journal)

Students majoring in a STEM discipline whose sequence of collegiate mathematics begins at the developmental level follow a unique progression towards degree completion. With an elongated sequence of mathematics courses, these students have already had exposure to collegiate mathematics when enrolling in a college algebra course. A structured multiple case study provided a context for understanding students’ perceptions about how their developmental mathematics experiences may have influenced their experiences in college algebra. Qualitative data was gathered through interviews with three students who are majoring in a STEM field of study. The selected students had similar quantitative literacy expectations for their …

Bounded Point Derivations On Roadrunner Sets, Evan Abshire

Theses, Dissertations and Capstones

This paper is concerned primarily with a type of subset of the complex plane known as a Roadrunner set, and its admittance of a bounded point derivation with respect to a given norm on the complex plane. The four norms we are concerned with are the Uniform norm, Lipschitz norm, L^p norm, and Campanato semi-norm. The purpose of this thesis is to provide researchers in approximation theory with more tools for them to accomplish their goals such as the proof of theorems regarding generalized derivatives. A connection has historically been established between the existence of bounded point derivations and …

The Dual Boundary Complex Of The Moduli Space Of Cyclic Compactifications, Toby Anderson

HMC Senior Theses

Moduli spaces provide a useful method for studying families of mathematical objects. We study certain moduli spaces of algebraic curves, which are generalizations of familiar lines and conics. This thesis focuses on, Δ^(r,n), the dual boundary complex of the moduli space of genus-zero cyclic curves. This complex is itself a moduli space of graphs and can be investigated with combinatorial methods. Remarkably, the combinatorics of this complex provides insight into the geometry and topology of the original moduli space. In this thesis, we investigate two topologically invariant properties of Δ^(r,n). We compute its Euler characteristic and …

Solving Robert Wilson’S 𝑡 ≠ 2 Conjecture On Graham Sequences, Krishna Rajesh

HMC Senior Theses

Ron Graham's sequence is a surprising bijection from the natural numbers to the non-prime integers, which is constructed by looking at sequences whose product is square. In this thesis we will resolve a 22-year-old conjecture about this bijection, by construction of explicit sequences in a modified number theoretic context. Additionally, we will discuss the history of this problem, and give computational techniques for computing this bijection, levering ideas from linear algebra over the finite field of two elements.

Simulation Of Optimal Control Of Vaccination For Svihr Covid-19 Epidemic Model, Kantanop Yimfan

Chulalongkorn University Theses and Dissertations (Chula ETD)

Infectious disease modeling plays a crucial role in understanding and managing epidemic outbreaks. Mathematical models, combined with control strategies, can help guide effective interventions and policy making. In this study, we examined an epidemic model, so-called SV IHR, which is an extended SIR model with additional states to incorporate vaccination and treatment. Our goal is to determine parameters known as controls; specifically, the vaccination proportion, through the framework of the optimal control problems. We defined an objective function and explored the control strategies that optimize it. Numerical simulations were then performed to illustrate the dynamics of the epidemic both with …

The Cohomology Of The Extended Morava Stabilizer Group, With Trivial Coefficients, At Large Primes, Mohammad Behzad Kang

Wayne State University Dissertations

The goal of this thesis is to calculate the cohomology of the extended height n Morava stabilizer group, with trivial coefficients, for all heights n and all primes p>>n. To do this, we construct a family of deformations – parametrized over an affine line and smooth away from a single point – of Ravenel's Lie algebra model for the Morava stabilizer group. The singular fiber of the resulting bundle of differential graded algebras is the Chevalley-Eilenberg DGA of Ravenel's Lie algebra model, while the smooth fibers have very understandable cohomology. From here, tools such as parallel transport, connections, and …

Ookami: An A64fx Computing Resource, A. C. Calder, E. Siegmann, C. Feldman, S. Chheda, Dennis C. Smolarski Sj, F. D. Swesty, A. Curtis, J. Dey, D. Carlson, B. Michalowicz, R. J. Harrison

Mathematics and Computer Science

We present a look at Ookami, a project providing community access to a testbed supercomputer with the ARM-based A64FX processors developed by a collaboration between RIKEN and Fujitsu and deployed in the Japanese supercomputer Fugaku. We provide an overview of the project and details of the hardware, and describe the user base and education/training program. We present highlights from previous performance studies of two astrophysical simulation codes and present a strong scaling study of a full 3D supernova simulation as an example of the the machine’s capability.

Preliminary Results Of Pythagorean N-Tuples, Cara Admiraal

Science University Research Symposium (SURS)

This presentation will introduce the idea of extending the Pythagorean Theorem in higher dimensions. First, I will highlight and recognize key patterns of primitive Pythagorean Triples by examining visual and algebraic representations. I will then present key findings and questions surrounding the idea of a Pythagorean quadruple, quintuple, and n-tuple. Lastly, I will propose different branches of exploration that will be researched in the coming months.

Examining Course Achievement In An Undergraduate Psychology Statistics Course Through The Lens Of Machine Learning Techniques, Sunny Nguyet Le

CGU Theses & Dissertations

The Introductory to Psychology Statistics course stands as a notable challenge for psychology majors, often acting as a gatekeeper course. This study embarks on two primary objectives using machine learning techniques: (1) to identify the determinants of overall course achievement, specifically course grade, and (2) to investigate the influence of statistics anxiety and statistics self-efficacy, when both are present, on overall course grade. Employing a machine-learning approach, both objectives were effectively addressed. The study involved the development of a self-reported questionnaire consisting of perceptions of statistics anxiety and statistics self-efficacy, along with other demographic and academic background variables. Conducted at …

Proof Of The Toponogov Conjecture On Complete Surfaces, Brendan Guilfoyle, Wilhelm Klingenberg

Department of Mathematics Publications

We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample. Corollaries of the main theorem include a Hawking-Penrose singularity-type theorem, as well as the proof of a conjecture of Milnor’s from 1965 in the convex case.

Every Feasibly Computable Reals-To-Reals Function Is Feasibly Uniformly Continuous, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that every computable function is continuous; moreover, it is computably continuous in the sense that for every ε > 0, we can compute δ > 0 such that δ-close inputs lead to ε-close outputs. It is also known that not all functions which are, in principle, computable, can actually be computed: indeed, the computation sometimes requires more time than the lifetime of the Universe. A natural question is thus: can the above known result about computable continuity of computable functions be extended to the case when we limit ourselves to feasible computations? In this paper, we prove that this …

From Normal Distribution To What? How To Best Describe Distributions With Known Skewness, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we only have partial information about the probability distribution -- e.g., all we know is its few moments. In such situations, it is desirable to select one of the possible probability distributions. A natural way to select a distribution from a given class of distributions is the maximum entropy approach. For the case when we know the first two moments, this approach selects the normal distribution. However, when we also know the third central moment -- corresponding to skewness -- a direct application of this approach does not work. Instead, practitioners use several heuristic techniques, techniques …

Solid Angle Measure Approximation Methods For Polyhedral Cones, Allison Fitisone

Theses and Dissertations--Mathematics

Polyhedral cones are of interest in many fields, like geometry and optimization. A simple, yet fundamental question we may ask about a cone is how large it is. As cones are unbounded, we consider their solid angle measure: the proportion of space that they occupy. Beyond dimension three, definitive formulas for this measure are unknown. Consequently, devising methods to estimate this quantity is imperative. In this dissertation, we endeavor to enhance our understanding of solid angle measures and provide valuable insights into the efficacy of various approximation techniques.

Ribando and Aomoto independently discovered a Taylor series formula for solid angle …

A Geometric Model For Syzygies Over 2-Calabi–Yau Tilted Algebras Ii, Ralf Schiffler, Khrystyna Serhiyenko

Mathematics Faculty Publications

In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra B, we construct a polygon S with a checkerboard pattern in its interior, which defines a category Diag(S). The indecomposable objects of Diag(S) are the 2-diagonals in S, and its morphisms are certain pivoting …

Steklov Eigenvalue Problems On Nearly Spherical And Annular Domains, Nathan Philip Schroeder

CGU Theses & Dissertations

We consider Steklov eigenvalues on nearly spherical and nearly annular domains in d dimensions where d is any given positive integer. By using the Green-Beltrami identity for spherical harmonic functions, the derivatives of Steklov eigenvalues with respect to the domain perturbation parameter can be determined by the eigenvalues of a matrix involving the integral of the product of three spherical harmonic functions. By using the addition theorem for spherical harmonic functions, we determine conditions when the trace of this matrix becomes zero. These conditions can then be used to determine when spherical and annular regions are critical points while we …

The Law Of The Iterated Logarithm For Lp-Norms Of Kernel Estimators Of Cumulative Distribution Functions, Fuxia Cheng

Faculty Publications – Mathematics

In this paper, we consider the strong convergence of Lp-norms (p ≥ 1) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the Lp-norms of empirical processes is extended to the kernel estimator of the CDF.

Point Modules And Line Modules Of Certain Quadratic Quantum Projective Spaces, Jose E. Lozano

Mathematics Dissertations - Archive

During the past 36 years, some research in noncommutative algebra has been driven by attempts to classify AS-regular algebras of global dimension four. Such algebras are often considered to be noncommutative analogues of polynomial rings. In the 1980s, Artin, Tate, and Van den Bergh introduced a projective scheme that parametrizes the point modules over a graded algebra generated by elements of degree one. In 2002, Shelton and Vancliff introduced the concept of line scheme, which is a projective scheme that parametrizes line modules.

This dissertation is in two parts. In the first part, we consider a 1-parameter family of quadratic …