Mathematics Commons^™

Supplementary Files For: "Structure Identification For High-Dimensional Data In The Vicinity Of Bear Lake", Ben Shaw, Haley Burger, Brennan Bean, Kevin Moon

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This report focuses on seven water quality measurements taken at 43 different depths on the Bear Lake for the months of June - November in the years 2018 - 2023. These measurements create a high-dimensional dataset on which we apply state-of-the-art machine learning (ML) techniques to look for low-dimensional structure in the data. A similar effort was made for weather measurements taken near the lake. Our analysis revealed that water quality measurements tend to cluster (i.e., group together) by year, while weather measurements tend to cluster by time of the year. This suggests that the structure observed in the water …

Massera’S Theorem On Arbitrary Discrete Time Domains, Martin Bohner, Jaqueline G. Mesquita, Sabrina Streipert

Mathematics and Statistics Faculty Research & Creative Works

We present a general version of Massera's theorems for arbitrary discrete domains, based on a newly introduced definition for both linear and nonlinear equations. For scalar nonlinear equations, we identify sufficient conditions that ensure each µ-bounded solution approaches a periodic solution asymptotically. In the case of linear systems, we prove that the presence of a µ-bounded solution necessarily leads to a periodic solution. We also provide some examples to show the practical implications of our findings.

Long-Time Asymptotics For The Kadomtsev–Petviashvili I Equation With Small Initial Data, Samir Donmazov

Theses and Dissertations--Mathematics

We study the initial value problem for the Kadomtsev--Petviashvili I (KP I) equation (u_t + 6uu_x + u_xxx)_x = 3u_yy with small initial data belonging to a subspace of the energy space for the KP I equation. We establish the long-time asymptotics for solutions of the KP I equation using the inverse scattering transform formalism developed by Zhou. Within this framework, the inverse problem for the KP I equation is formulated as a nonlocal Riemann--Hilbert problem (RHP) in two spatial dimensions. As part of the asymptotic analysis, we determine the long-time behavior of the …

An Algorithm And Computation To Verify Legendre's Conjecture Up 7 · 1013, Jonathan Sorenson, Jonathan Webster

Computer Science and Software Engineering

We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integer n, there exists a prime between n² and (n + 1)². Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between n2 and n(n + 1) and also between n(n + 1) and (n + 1)². Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all n ≤ N in time O(N log N log …

On The Hölder Continuity Of The Brascamp-Lieb Constant, Ori Friesen

Mathematics, Statistics, and Computer Science Honors Projects

The Brascamp-Lieb inequality is a generalization of many well-known multilinear functional inequalities. The Brascamp-Lieb constant is the best constant that works for the Brascamp-Lieb inequality for a given tuple of input linear maps and powers. If we keep the powers constant while varying the input linear maps, the Brascamp-Lieb constant becomes a function of the linear maps. In this thesis, we explore the Hölder continuity of the Brascamp-Lieb constant. Specifically,we prove that the general 4-linear case of the Brascamp-Lieb inequality is locally Lipschitz continuous. Additionally, we provide an improvement of a previous result on the local Hölder continuity of the …

All We (And Llms) Need Is Fuzzy: An Argument, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Large Language Models (LLMs) like ChatGPT have spectacular successes -- but they also have surprising failures that an average person with common sense could easily avoid. It is therefore desirable to incorporate the imprecise ("fuzzy") common sense into LLMs. A natural question is: to what extent will this help? This way, we may avoid a few simple mistakes, but will it significantly improve the LLMs' performance? What portion of the gap between current LLMs and ideal perfect AI-based agents can be, in principle, covered by using fuzzy techniques? Judging by the fact that few researchers working on LLMs (and on …

How To Share A Success, How To Share A Crisis, And How All This Is Related To Fuzzy, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, a group of people needs to share a success. What is the fair way to share this success? Nobelist John Nash showed that under reasonable conditions, the group should select the alternative for which the product of utility gains is the largest possible. This solution makes perfect sense from the fuzzy-formalized commonsense viewpoint: it maximizes the degree of confidence that all participants are happy. A natural question is: can we extend this result to a different class of situations, when a group of people needs to share sacrifices caused by a crisis? In this paper, we …

Multi-Valued Variational Inequalities For Variable Exponent Double Phase Problems: Comparison And Extremality Results, Siegfried Carl, Vy Khoi Le, Patrick Winkert

Mathematics and Statistics Faculty Research & Creative Works

We prove existence and comparison results for multi-valued variational inequalities in a bounded domain Ω of the form (Formula presented.) where A:W1,H(Ω)→W1,H(Ω)∗ given by (Formula presented.) for u∈W1,H(Ω), is the double phase operator with variable exponents and W1,H(Ω) is the associated Musielak–Orlicz Sobolev space. First, an existence result is proved under some weak coercivity condition. Our main focus aims at the treatment of the problem under consideration when coercivity fails. To this end we establish the method of sub–super-solution for the multi-valued variational inequality in the space W1, H(Ω) based on appropriately defined sub- and super-solutions, which yields the existence …

Galoistheory And The Arithmetic-Geometric Series, Daniel Vargas

HMC Senior Theses

Motivated by classical works of Gauss and Euler on the AGM, Ono and his

collaborators Griffin et al. (2023); McSpirit and Ono (2023) have investigated

the union of AGM sequences over finite fields 𝔽𝑞, where 𝑞 ≡3 mod 4. A

recent preprint Kayath et al. (2024) extends some of their results to all finite

fields with odd characteristic. We refine these works when 𝑞≡5 mod 8. In

particular, we explicitly determine the components of these graphs and their

total population. We also use Galois-theoretic results to make progress in

the search for cycles over finite fields with odd characteristic.

Existence Results For A Discrete Fractional Boundary Value Problem, David Barilla, Martin Bohner, Giuseppe Caristi, Shapour Heidarkhani, Shahin Moradi

Mathematics and Statistics Faculty Research & Creative Works

In this study, we investigate the existence of at least one solution and the existence of an infinite number of solutions for a discrete fractional boundary value problem. Requiring an algebraic condition on the nonlinear term for small values of the parameter and requiring an additional asymptotical behavior of the potential at zero, we investigate the existence of at least one nontrivial solution for the problem. Moreover, under suitable assumptions on the oscillatory behavior of the nonlinearity at infinity, for exact collections of the parameter, we discuss the existence of a sequence of solutions for the problem. We also present …

On The Gumbel-Weibull{Cauchy} Distribution, Jennifer D. Pippin

Theses, Dissertations and Capstones

Developing new statistical distributions and seeking higher flexibility in modeling different shapes of data remain a strong emphasis in research. The T-R{Y } framework, introduced in [3], utilizes three statistical distributions in order to generate a new distribution. Many research papers appeared in literature to develop distributions based on the T-R{Y } framework. In this thesis, a member of the T-R{Y } framework, namely the Gumbel-Weibull{Cauchy} (GWC), is introduced. Statistical properties of the GWC are studied, such as the quantile function, the hazard function, transformations, Shannon entropy, the …

Floquet Theory For First-Order Delay Equations And An Application To Height Stabilization Of A Drone’S Flight, Martin Bohner, Alexander Domoshnitsky, Oleg Kupervasser, Alex Sitkin

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we proposed a version of the Floquet theory for delay differential equations. We demonstrated that very natural assumptions for control in technical applications can lead us to a one-dimensional fundamental system. This approach allowed researchers to work with classical methods used in the case of ordinary differential equations. On this basis, new original unexpected results on the exponential stability were proposed. For example, in the equation x' (4)+a(t)x(t—-T(7)) = 0, t € [0, co), we avoided the assumption on the smallness of the product sup,j9,.) 41 SUP;< {9,00) TD) < 3/2 for asymptotic stability. We obtained that in the case of w-periodic coefficient and delay, the fact that the period w was situated in a corresponding interval can lead to exponential stability. We then applied our new tests of stability to the stabilization of a drone's flight, where smallness of the noted above product could not be achieved from a technical point of view. For an equation with periodic coefficient and delay, we got a formula of the solution's representation on the semiaxis.

Gompertz Distribution On Time Scales, Wasiu Sule

Theses, Dissertations and Capstones

We shall investigate Gompertz dynamic equations within the context of time scales calculus, by exploring the mathematical foundations and applications of the Gompertz model, which is commonly used to describe growth phenomena in various fields such as biology and economics. This research seeks to analyze the Gompertz cumulative distribution functions (CDF) and probability density functions (PDF) across different time scales, including the real numbers R and integer multiples hN. Probability techniques will be used to derive the CDF and PDF associated with the Gompertz dynamic equations, and we will examine how varying the time scale impacts the characteristics …

A Numerical Method For Coefficient Reconstruction Of A Periodic Inverse Source Problem, Robert Ireri

Theses, Dissertations and Capstones

This thesis investigates a numerical method for solving the periodic inverse source problem governed by the Helmholtz equation. The problem involves reconstructing an unknown periodic source term from boundary measurements, which is inherently ill-posed. To address this challenge, we employ a quasi-reversibility method (QRM) combined with a basis function expansion to stabilize the inverse reconstruction. The forward problem is solved using the Lippmann-Schwinger equation, discretized via the trapezoidal rule, and the inverse problem is formulated as a constrained least-squares minimization. The discretized system is efficiently solved using sparse matrix techniques and regularization strategies. Numerical experiments demonstrate the robustness of the …

Discrete Fractional Gompertz Models, Rebecca Oduro

Theses, Dissertations and Capstones

This thesis explores the theory and application of discrete fractional Gompertz models—systems that integrate fractional difference operators into the classical Gompertz growth paradigm. By doing so, these models capture both discrete time steps and the long-range memory effects characteristic of fractional calculus. After outlining the fundamental notions of discrete calculus, discrete fractional sums and differences, and related special functions such as the discrete Mittag–Leffler function, we derive various fractional Gompertz-type equations. We prove the existence and uniqueness of solutions to these fractional difference equations, often employing discrete analogues of standard solution methods like variation of constants. We also investigate the …

A Unified Concept Of Periodicity On Any Time Scale And Applications, Martin Bohner, Jaqueline G. Mesquita, Sabrina H. Streipert

Mathematics and Statistics Faculty Research & Creative Works

We introduce a novel definition of periodicity on arbitrary time scales, dependent on a strictly increasing and differentiable function. This removes the commonly used and restrictive assumption of a periodic time scale to define periodic functions. Our new definition furthermore allows for a wider class of functions to be studied using the theory of periodic systems. After providing crucial properties of these periodic functions, such as the translation invariance of integrals of periodic functions, we apply the concept of this new periodicity to linear dynamic equations. We provide necessary and sufficient conditions for a linear dynamic equation to have such …

The Discrete Generalized Proportional Fractional Derivative, Martin Bohner, Rajrani Gupta

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we have introduced a discrete generalized proportional fractional derivative and generated Riemann-Liouville and Caputo discrete generalized proportional fractional derivatives. The Laplace transforms of the discrete generalized proportional fractional derivatives and integrals are also calculated.

A High-Order Eulerian–Lagrangian Runge–Kutta Finite Volume (El–Rk–Fv) Method For Scalar Nonlinear Conservation Laws, J. Chen, Joseph Nakao, J.-M. Qiu, Y. Yang

Mathematics & Statistics Faculty Works

We present a class of high-order Eulerian–Lagrangian Runge–Kutta finite volume methods that can numerically solve Burgers’ equation with shock formations, which could be extended to general scalar conservation laws. Eulerian–Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine–Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for …

Line Graphs Of Directed Graphs I, Vaidy Sivaraman, Daniel Slilaty

Mathematics and Statistics Faculty Publications

We determine the forbidden induced subgraphs for the intersection of the classes of chordal bipartite graphs and line graphs of acyclic directed graphs. This is a first step towards finding the forbidden induced subgraphs for the class of line graphs of directed graphs.

Localized Kernel Methods For Signal Processing, Sippanon Kitimoon

CGU Theses & Dissertations

This dissertation presents two signal processing methods using specially designed localized kernels for parameter recovery under noisy condition. The first method addresses the estimation of frequencies and amplitudes in multidimensional exponential models. It utilizes localized trigonometric polynomial kernels to detect the multivariate frequencies, followed by a more detailed parameter estimation. We compare our method with MUSIC and ESPRIT, which are classical subspace-based algorithms widely used for estimating the parameters of exponential signals. In the univariate case, the method outperforms MUSIC and ESPRIT under low signal-to-noise ratios. For the multivariate case, we develop a coordinate-wise projection and registration approach that achieves …