Mathematics Commons^™

The Maximum Rectilinear Crossing Number Of The Petersen Graph, Elie Feder, Heiko Harborth, Steven Herzberg, Sheldon Klein

Publications and Research

We prove that the maximum rectilinear crossing number of the Petersen graph is 49. First, we illustrate a picture of the Petersen graph with 49 crossings to prove the lower bound. We then prove that this bound is sharp by carefully analyzing the ten Cs's which occur in the Petersen graph and their properties.

Weighted Inverse Weibull And Beta-Inverse Weibull Distribution, Jing Xiong Kersey

College of Graduate Studies: Theses & Dissertations

The weighted inverse Weibull distribution and the beta-inverse Weibull distribution are considered. Theoretical properties of the inverse Weibull model, weighted inverse Weibull distribution including the hazard function, reverse hazard function, moments, moment generating function, coefficient of variation, coefficient of skewness, coefficient of kurtosis, Fisher information and Shanon entropy are studied. The estimation for the parameters of the length-biased inverse Weibull distribution via maximum likelihood estimation and method of moment estimation techniques are presented, as well as a test for the detection of length-biasedness in the inverse Weibull model. Furthermore, the beta-inverse Weibull distribution which is a weighted distribution is presented, …

Impulsive Discontinuous Hyperbolic Partial Differential Equations Of Fractional Order On Banach Algebras, Said Abbas, Ravi P. Agarwal, Mouffak Benchohra

Mathematics and System Engineering Faculty Publications

This article studies the existence of solutions and extremal solutions to partial hyperbolic differential equations of fractional order with impulses in Banach algebras under Lipschitz and Carathéodory conditions and certain monotonicity conditions.

Multiplicity Results For P-Sublinear P-Laplacian Problems Involving Indefinite Eigenvalue Problems Via Morse Theory, Kanishka Perera, Ravi P. Agarwal, Donal O'Regan

Mathematics and System Engineering Faculty Publications

We establish some multiplicity results for a class of p-sublinear p- Laplacian problems involving indefinite eigenvalue problems using Morse theory.

Curvature Measures For Nonlinear Regression Models Using Continuous Designs With Applications To Optimal Design, Timothy O'Brien, Somsri Jamroenpinyo, Chinnaphong Bumrungsup

Mathematics and Statistics: Faculty Publications and Other Works

We present and illustrate the methodology to calculate curvature measures for continuous designs, and extend design criteria to incorporate continuous designs. These design algorithms include quadratic design procedures, a subset design criterion, a second-order mean-square error design criterion, and a marginal curvature design methodology. A discussion of confidence intervals is also provided for continuous designs.

Statistical Analysis Of Wastewater Remediation And Bio-Fuels Production Of Algae, Jay D. Jones

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The Logan city wastewater treatment system consists of a series of seven large aerated ponds (460 acres) that biologically treats 15 million gallons per day of wastewater from Logan city and six other communities. Tighter regulations of allowed phosphorus levels in the effluent have recently been implemented due to environmental concerns of a downstream reservoir. The Biological Engineering program at Utah State University, the Bio-fuels Center, the Utah Water Research Laboratory (UWRL) and the city of Logan are working together to remediate the wastewater treatment system using microalgae. Algal growth requires the uptake of phosphorus. Thus, phosphorus in the effluent …

Evolving Models: A Density-Based Approach To Modeling Sexual Dimorphism And Adaptive Speciation, Audrey Smith

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

In this paper, we begin by extending existing deterministic and individual-based ecological models for sexual dimorphism and adaptive speciation into density-based mathematical models describing the density or number of individuals with various trait values, or phenotypes. These density-based models describe the dynamics of a population of males and females using both clonal and sexual reproduction. Each generation, the populations are subject to mating, mutation, and ecological dynamics including infraspecific competition and carrying capacity of the environment. By avoiding individual-based models, we are able to avoid simulations and instead achieve repeatable results.

Implementing these models numerically, we are able to show …

Improving Accuracy Of Large-Scale Prediction Of Forest Disease Incidence Through Bayesian Data Reconciliation, Ephraim M. Hanks

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Increasing the accuracy of predictions made from ecological data typically involves replacing or replicating the data, but the cost of updating large-scale data sets can be prohibitive. Focusing resources on a small sample of locations from a large, less accurate data set can result in more reliable observations, though on a smaller scale. We present an approach for increasing the accuracy of predictions made from a large-scale eco logical data set through reconciliation with a small, highly accurate data set within a Bayesian hierarchical modeling framework. This approach is illustrated through a study of incidence of eastern spruce dwarf mistletoe …

The Maximal Pure Spectrum Of An Abelian Group, Brendan Goldsmith, Ruediger Goebel

Articles

This paper introduces the notion of the maximal pure spectrum of an Abelian group - this is the set of isomorphism classes of maximal proper pure subgroups - and focuses on the situation in which this spectrum is small. The converse situation is also examined i.e. given a collection of isomorphism classes of groups, can one find an Abelian group having precisely this collection as its maximal pure spectrum. Finally, it is shown that in some familiar situations, the answers to these questions may be undecidable.

On Coalgebras Over Algebras, Adriana Balan, Alexander Kurz

Engineering Faculty Articles and Research

We extend Barr’s well-known characterization of the final coalgebra of a Set-endofunctor as the completion of its initial algebra to the Eilenberg-Moore category of algebras for a Set-monad M for functors arising as liftings. As an application we introduce the notion of commuting pair of endofunctors with respect to the monad M and show that under reasonable assumptions, the final coalgebra of one of the endofunctors involved can be obtained as the free algebra generated by the initial algebra of the other endofunctor.

Families Of Symmetries As Efficient Models Of Resource Binding, Vincenzo Ciancia, Alexander Kurz, Ugo Montanari

Engineering Faculty Articles and Research

Calculi that feature resource-allocating constructs (e.g. the pi-calculus or the fusion calculus) require special kinds of models. The best-known ones are presheaves and nominal sets. But named sets have the advantage of being finite in a wide range of cases where the other two are infinite. The three models are equivalent. Finiteness of named sets is strictly related to the notion of finite support in nominal sets and the corresponding presheaves. We show that named sets are generalisd by the categorical model of families, that is, free coproduct completions, indexed by symmetries, and explain how locality of interfaces gives good …

Algebraic Theories Over Nominal Sets, Alexander Kurz, Daniela Petrişan, Jiří Velebil

Engineering Faculty Articles and Research

We investigate the foundations of a theory of algebraic data types with variable binding inside classical universal algebra. In the first part, a category-theoretic study of monads over the nominal sets of Gabbay and Pitts leads us to introduce new notions of finitary based monads and uniform monads. In a second part we spell out these notions in the language of universal algebra, show how to recover the logics of Gabbay-Mathijssen and Clouston-Pitts, and apply classical results from universal algebra.

Incidence Functions, Yiyu Liao

Open Access Theses & Dissertations

In the mid 1960's, the incidence algebra was introduced in the seminal paper of Gian-Carlo Rota. He addressed the importance of the Mobius function in combinatorics. In particular, the incidence algebra of a locally finite poset plays an essentially unifying role in the theory of the Mobius function. One of the significant generalizations is the incidence algebra of a Mobius category introduced by Pierre Leroux. With the help from Mobius category, it was exciting to be able to extend the combinatorial results more broadly than just on posets. Before attempting to study this generalization of the Mobius function, we have …

On The Socles Of Characteristic Subgroups Of Abelian P-Groups, Brendan Goldsmith, Peter Danchev

Articles

Fully invariant subgroups of an Abelian p-group have been the object of a good deal of study, while characteristic subgroups have received somewhat less attention. Recently the socles of fully invariant subgroups have been studied and this led to the notion of a socle-regular group. The present work replaces the fully invariant subgroups with characteristic ones and leads in a natural way to the notion of a strongly socle-regular group. A surprising relationship, mirroring that between transitive and fully transitive groups, is obtained.

From Permutahedron To Aassociahedron, Colum Watt, Thomas Brady

Articles

For each finite real reflection group $W$, we identify a copy of the type-$W$ simplicial generalised associahedron inside the corresponding simplicial permutahedron. This defines a bijection between the facets of the generalised associahedron and the elements of the type $W$ non-crossing partition lattice which is more tractable than previous such bijections. We show that the simplicial fan determined by this associahedron coincides with the Cambrian fan for $W$.

The Similarity Problem For Indefinite Sturm-Liouville Operators With Periodic Coefficients, Aleksey Kostenko

Articles

We investigate the problem of similarity to a self-adjoint operator for $J$-positive Sturm-Liouville operators $L=\frac{1}{\omega}(-\frac{d^2}{dx^2}+q)$ with $2\pi$-periodic coefficients $q$ and $\omega$. It is shown that if 0 is a critical point of the operator $L$, then it is a singular critical point. This gives us a new class of $J$-positive differential operators with the singular critical point 0. Also, we extend the Beals and Parfenov regularity conditions for the critical point $\infty$ to the case of operators with periodic coefficients.

A Time Series Analysis Of The New Jersey Meadowlands Weather And Air Quality Data, Steven Spero

Theses, Dissertations and Culminating Projects

This research applies time series methods to determine relationships among a set of weather variables which are continually monitored in the Hackensack Meadowlands region of northern New Jersey. Weather data includes chemical and atmospheric factors. Chemical factors are Nitrogen Oxide, atmospheric Ozone, Carbon Monoxide, and Carbon Dioxide. Weather factors are wind speed, barometric pressure, air temperature, humidity, and solar radiation. Additionally, traffic density and time of week are brought in as categorical factors. This research attempts to (a) introduce the reader to various time series methodologies, (b) find a significant and efficient model for forecasting Nitrogen Oxide levels, and (c) …

Meaningful Distributed Instruction— Conceptual Previews For Symbolic Procedures, Edward C. Rathmell

Faculty Work

Understanding a symbolic procedure means far more than “getting the right answer.” A mathematical symbolic procedure or written skill involves step-bystep thinking that leads from a computational problem to a solution. Memorizing this step-by-step procedure may enable a student to answer the problem, even answer it correctly. Yes, that is important, but understanding means much more.

A Categorical Semantics For Fuzzy Predicate Logic, Lawrence Stout

Scholarship

The object of this study is to look at categorical approaches to many valued logic, both propositional and predicate, to see how different logical properties result from different parts of the situation. In particular, the relationship between the categorical fabric I introduced at Linz in 2004 and the Fuzzy Logics studied by Hajek (2003) [5], Esteva et al. (2003) [1], and Hajek (1998) [4], comes from restricting the kind of structures used for truth values. We see how the structure of the various kinds of algebras shows up in the categorical logic, giving a variant on natural deduction for these …

Categorical Approaches To Non-Commutative Fuzzy Logic, Lawrence Stout

Scholarship

In this paper we consider what it means for a logic to be non-commutative, how to generate examples of structures with a non-commutative operation * which have enough nice properties to serve as the truth values for a logic. Inference in the propositional logic is gotten from the categorical properties (products, coproducts, monoidal and closed structures, adjoint functors) of the categories of truth values. We then show how to extend this view of propositional logic to a predicate logic using categories of propositions about a type A with functors giving change of type and adjoints giving quantifiers. In the case …