Mathematics Commons^™

New Examples Of Self-Dual Near-Extremal Ternary Codes Of Length 48 Derived From 2-(47,23,11) Designs, Sanja Rukavina, Vladimir Tonchev

Michigan Tech Publications, Part 2

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number A12 of codewords of minimum weight 12, and raised the question about the existence of codes for other values of A12. In this note, we use symmetric 2-(47,23,11) designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of A12.

Approval Gap Of Weighted K-Majority Tournaments, Jeremy Coste, Breeann Flesch, Joshua D. Laison, Erin Mcnicholas, Dane Miyata

Theory and Applications of Graphs

A $k$-majority tournament $T$ on a finite set of vertices $V$ is defined by a set of $2k-1$ linear orders on $V$, with an edge $u \to v$ in $T$ if $u>v$ in a majority of the linear orders. We think of the linear orders as voter preferences and the vertices of $T$ as candidates, with an edge $u \to v$ in $T$ if a majority of voters prefer candidate $u$ to candidate $v$. In this paper we introduce weighted $k$-majority tournaments, with each edge $u \to v$ weighted by the number of voters preferring $u$.

We define the …

Bernstein Polynomials Method For Solving Multi-Order Fractional Neutral Pantograph Equations With Error And Stability Analysis, M. H. T. Alshbool

All Works

In this investigation, we present a new method for addressing fractional neutral pantograph problems, utilizing the Bernstein polynomials method. We obtain solutions for the fractional pantograph equations by employing operational matrices of differentiation, derived from fractional derivatives in the Caputo sense applied to Bernstein polynomials. Error analysis, along with Chebyshev algorithms and interpolation nodes, is employed for solution characterization. Both theoretical and practical stability analyses of the method are provided. Demonstrative examples indicate that our proposed techniques occasionally yield exact solutions. We compare the algorithms using several established analytical methods. Our results reveal that our algorithm, based on Bernstein series …

A Comprehensive Uncertainty Quantification Methodology For Metrology Calibration And Method Comparison Problems Via Numeric Solutions To Maximum Likelihood Estimation And Parametric Bootstrapping, Aloka B. S. N. Dayarathne

All Graduate Theses and Dissertations, Fall 2023 to Present

In metrology, the science of measurements, straight line calibration models are frequently employed. These models help understand the instrumental response to an analyte, whose chemical constituents are unknown, and predict the analyte’s concentration in a sample. Techniques such as ordinary least squares and generalized least squares are commonly used to fit these calibration curves. However, these methods may yield biased estimates of slope and intercept when the calibrant, substance used to calibrate an analytical procedure with known chemical constituents (x-values), carries uncertainty. To address this, Ripley and Thompson (1987) proposed functional relationship estimation by maximum likelihood (FREML), which considers uncertainties …

On The Existence Of Periodic Traveling-Wave Solutions To Certain Systems Of Nonlinear, Dispersive Wave Equations, Jacob Daniels

All Graduate Theses and Dissertations, Fall 2023 to Present

A variety of physical phenomena can be modeled by systems of nonlinear, dispersive wave equations. Such examples include the propagation of a wave through a canal, deep ocean waves with small amplitude and long wavelength, and even the propagation of long-crested waves on the surface of lakes. An important task in the study of water wave equations is to determine whether a solution exists. This thesis aims to determine whether there exists solutions that both travel at a constant speed and are periodic for several systems of water wave equations. The work done in this thesis contributes to the subfields …

A Central Limit Theorem For The Number Of Excursion Set Components Of Gaussian Fields, Dmitry Beliaev, Michael Mcauley, Stephen Muirhead

Articles

For a smooth stationary Gaussian field f on R^d and level ℓ ∈ R, we consider the number of connected components of the excursion set {f ≥ ℓ} (or level set {f = ℓ}) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for ‘additive’ geometric functionals of the excursion/level sets (e.g. the volume or …

On Cheeger Constants Of Knots, Robert Lattimer

Electronic Theses, Projects, and Dissertations

In this thesis, we will look at finding bounds for the Cheeger constant of links. We will do this by analyzing an infinite family of links call two-bridge fully augmented links. In order to find a bound on the Cheeger constant, we will look for the Cheeger constant of the link’s crushtacean. We will use that Cheeger constant to give us insight on a good cut for the link itself, and use that cut to obtain a bound. This method gives us a constructive way to find an upper bound on the Cheeger constant of a two-bridge fully augmented link. …

Ramanujan Type Congruences For Quotients Of Klein Forms, Timothy Huber, Nathaniel Mayes, Jeffery Opoku, Dongxi Ye

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients …

Vectors And Vector Borne Disease: Models For The Spread Of Curly Top Disease And Culex Mosquito Abundance, Rachel M. (Frantz) Georges

All Graduate Theses and Dissertations, Fall 2023 to Present

Mathematical models are useful tools in managing infectious disease. When designed appropriately, these models can provide insight into disease incidence patterns and transmission rates. In this work, we present several models that provide information that is useful in monitoring diseases spread by insects.

In the first part of this dissertation, we present two models that predict disease incidence patterns for Curly Top disease (CT) in tomato crops. CT affects a wide variety of plants and is spread through the bite of the Beet Leafhopper. This disease is particularly devastating to tomato crops. When infected, tomato plants present with stunted growth …

Information Based Approach For Detecting Change Points In Inverse Gaussian Model With Applications, Alexis Anne Wallace

Electronic Theses, Projects, and Dissertations

Change point analysis is a method used to estimate the time point at which a change in the mean or variance of data occurs. It is widely used as changes appear in various datasets such as the stock market, temperature, and quality control, allowing statisticians to take appropriate measures to mitigate financial losses, operational disruptions, or other adverse impacts. In this thesis, we develop a change point detection procedure in the Inverse Gaussian (IG) model using the Modified Information Criterion (MIC). The IG distribution, originating as the distribution of the first passage time of Brownian motion with positive drift, offers …

Tasks For Learning Trigonometry, Sydnee Andreasen

All Graduate Reports and Creative Projects, Fall 2023 to Present

Many studies have been done using task-based learning within different mathematics courses. Within the field of trigonometry, task-based learning is lacking. The following research aimed to create engaging, mathematically rich tasks that meet the standards for the current trigonometry course at Utah State University and align with the State of Utah Core Standards for 7th through 12th grades. Four lessons were selected and developed based on the alignment of standards, the relevance to the remainder of the trigonometry course, and the relevance to courses beyond trigonometry. The four lessons that were chosen and developed were related to trigonometric ratios, graphing …

Art And Math Via Cubic Polynomials, Polynomiography And Modulus Visualization, Bahman Kalantari

LASER Journal

Throughout history, both quadratic and cubic polynomials have been rich sources for the discovery and development of deep mathematical properties, concepts, and algorithms. In this article, we explore both classical and modern findings concerning three key attributes of polynomials: roots, fixed points, and modulus. Not only do these concepts lead to fertile ground for exploring sophisticated mathematics and engaging educational tools, but they also serve as artistic activities. By utilizing innovative practices like polynomiography—visualizations associated with polynomial root finding methods—as well as visualizations based on polynomial modulus properties, we argue that individuals can unlock their creative potential. From crafting captivating …

Euler Archive Spotlight: Multiple Search Options, Christopher Goff

Euleriana

The Euler Archive houses PDF versions of almost all of Euler's original publications. While most visitors search the archive via a work's Eneström number, the Archive can be searched via source publication name, date written, or decade of publication. The Archive also provides context for Euler's publications through short pieces of historical information.

Euler And A Proof Of The Functional Equation For The Riemann Zeta-Function He Could Have Given, Alexander Aycock

Euleriana

We explain how Euler could have proved a functional equation, which is equivalent to the one for the Riemann zeta-function, that he conjectured in his paper {\it ``Remarques sur un beau rapport entre les series des puissances tant directes que reciproques"} \cite{E352} (E352: ``Remarks on the beautiful relation between the series of the direct and reciprocal powers").

Euler And The Gaussian Summation Formula For The Hypergeometric Series, Alexander Aycock

Euleriana

We show that in his paper {\it ``Plenior expositio serierum illarum memorabilium, quae ex unciis potestatum binomii formantur"} \cite{E663} (E663: ``A more thorough exposition of those memorable series that are formed from the binomial coefficients") Euler could have found the Gaussian summation formula for the hypergeometric series from his own formulas in that same paper, if he actually set the task for himself.

Euler And Homogeneous Difference Equations With Linear Coefficients, Alexander Aycock

Euleriana

We present a method outlined by Euler in his paper{\it ``De fractionibus continuis observationes"} \cite{E123} (E123: ``Observations on continued fractions") that can be used to solve homogeneous difference equations with linear coefficients. We will illustrate his ideas by applying it to two familiar examples and explain how it can be understood from a more modern point of view.

On The Cases In Which The Formula X^4+Kxxyy+Y^4 Can Be Reduced To A Square, Georg Ehlers

Euleriana

Euler’s key idea for equating the Quartic in the title to a square is to set k=P+surd(Q). From this he derives P=f·x^2 and Q=4f·y^2+4 and solves the Pell equation for y. He then discusses various extensions to rational numbers that leave k an integer. Euler provides incomplete tables for integers k with |k|square.

Research On Arithmetic, Erik R. Tou

Euleriana

In this English translation, some of Joseph-Louis Lagrange's early number theory is presented. Here, he laid out a theory of binary quadratic forms with special attention to the representation problem: determining those integers which may be represented by a given form, and cataloguing the possible forms of their divisors.

Number Theory And More, Christopher Goff, Erik Tou

Euleriana

An introduction to the contents in Issue 1, Volume 4 of Euleriana.

Caterpillar, Lobster, X Graphs, Gerald Melin, Landon Seward, Will Mahowald, Xavier Jones

Celebrating Scholarship and Creativity Day (2018-)

We studied a combinatorial game played between two players ("Alpha", who goes first, and "Beta", who goes second). The idea is that there are a lot of lightbulbs in a large warehouse, and they take turns turning a light bulb on. When a light bulb is turned on, it illuminates the area directly by it as well as the areas immediately surrounding it. The player who is the one to make all of the warehouse illuminated is the winner. This can be modeled on a graph. The two players take turns (1) selecting a vertex that has not yet been …