Mathematics Commons^™

Ultraconnected And Critical Graphs, Jason Nicholas Grout

Theses and Dissertations

We investigate the ultraconnectivity condition on graphs, and provide further connections between critical and ultraconnected graphs in the positive definite partial matrix completion problem. We completely characterize when the join of graphs is ultraconnected, and prove that ultraconnectivity is preserved by Cartesian products. We completely characterize when adding a vertex to an ultraconnected graph preserves ultraconnectivity. We also derive bounds on the number of vertices which guarantee ultraconnectivity of certain classes of regular graphs. We give results from our exhaustive enumeration of ultraconnected graphs up to 11 vertices. Using techniques involving the Lovász theta parameter for graphs, we prove certain …

Sub-Supersolution Method For Quasilinear Parabolic Variational Inequalities, Siegfried Carl, Vy Khoi Le

Mathematics and Statistics Faculty Research & Creative Works

This paper is about a systematic attempt to apply the sub-supersolution method to parabolic variational inequalities. We define appropriate concepts of sub-supersolutions and derive existence, comparison, and extremity results for such inequalities.

Optimal Control Of Delay Systems With Differential And Algebraic Dynamic Constraints, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between "slow" and "fast" variables. We pursue a two-hold goal: to study variational stability for this class of control systems with respect to discrete …

Σary, Minnesota State University Moorhead, Mathematics Department

Math Department Newsletters

No abstract provided.

Four-Dimensional Non-Reductive Homogeneous Manifolds With Neutral Metrics, Andrew Renner

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A method due to É. Cartan was used to algebraically classify the possible four-dimensional manifolds that allow a (2, 2)-signature metric with a transitive group action which acts by isometries. These manifolds are classified according to the Lie algebra of the group action. There are six possibilities: four non-parameterized Lie algebras, one discretely parameterized family, and one family parameterized by R.

Statistical Pronouncements Iii, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

No abstract provided.

A Random Boolean Network Model And Deterministic Chaos, Mihaela Teodora Matache, Jack Heidel

Mathematics Faculty Publications

This paper considers a simple Boolean network with N nodes, each node’s state at time t being determined by a certain number of parent nodes, which may vary from one node to another. This is an extension of a model studied by Andrecut and Ali ( [5]) who consider the same number of parents for all nodes. We make use of the same Boolean rule as the authors of [5], provide a generalization of the formula for the probability of finding a node in state 1 at a time t and use simulation methods to generate consecutive states of the …

The Robustness Of Factor Analyses When The Data Does Not Conform To Standard Parametric Requirements, Haisong Peng

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

Objective: To access the robustness of factor analyses when the data does not conform to standard parametric requirements.

Methods: Data were simulated in package R. Maximum likelihood was used to fit and assess the factor models. Chi-square statistics were obtained to test hypotheses about the correct number of factors in simulated settings where the true number of factors was known. The number of true factors varied between 1 and 3; the number of observed variables was either 6 (for 1 factor) or 3 per factor for 2 or more factors.

Results: With standard normal factor populations, and normal errors added …

Semilinear Elliptic Equations In Unbounded Domains, Francois A. Van Heerden

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

We studied some semilinear elliptic equations on the entire space R^N. Our approach was variational, and the major obstacle was the breakdown in compactness due to the unboundedness of the domain. First, we considered an asymptotically linear Scltrodinger equation under the presence of a steep potential well. Using Lusternik-Schnirelmann theory, we obtained multiple solutions depending on the interplay between the linear, and nonlinear parts. We also exploited the nodal structure of the solutions. For periodic potentials, we constructed infinitely many homoclinic-type multibump solutions. This recovers the analogues result for the superlinear case. Finally, we introduced weights on the linear and …

Lorentz Homogeneous Spaces And The Petrov Classification, Adam Bowers

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A. Z. Petrov gave a complete list of all local group actions on a four-dimensional space-time that admit an invariant Lorentz metric, up to an equivalence relation. His list was compiled by directly constructing all possible Lie algebras of infinitesimal generators of group actions that preserve a Lorentz metric. The goal of this paper was to verify that classification by algebraically constructing a list of all possible three-dimensional homogeneous spaces and calculating which among them have a non-degenerate invariant metric.

Factorization Of Polynomials And Real Analytic Function, Radoslaw L. Stefanski

Honors Theses

In this project, we address the question: When can a polynomial p(x, y) of two variables be factored as p(x, y) = f(x)g(y), where f and g are polynomials of one variable. We answer this question, using linear algebra, and create a Mathematica program which carries out this factorization. For example,

3+3x-5x^3+y+xy-5/3x^3y+y^2+xy^2-5/3x^3y^2 = (1+x-5/3x^3)(3+y+y^2)

We then generalize this concept and ask: When can p(x,y) can be written as

p(x,y) = f1(x)g2(y)+f2(x)g2(y)+...+fr(x)gr(y)

where fj,gj are polynomials. This can certainly be done (for large enough r). Which is the minimum such r? Again, we have a Mathematica program which carries out this …

Discovering The Potential For Advancements In Intrusion Detection Systems, Kenneth J. Buonforte

Honors Theses

An intrusion detection system (IDS) is a collection of monitors strategically placed on a network or individual host in order to detect anomalous behavior. Since James Anderson introduced one of the first frameworks for an intrusion detection system in [1]. researchers have extended the capabilities of these systems. IDSs take many forms, from systems as basic as command line audit logs to those that emulate the defense mechanisms of the human immune system. However, as intrusion detection has evolved, scientists and administrators alike are beginning to question the need for this technology due to its lack of quantifiable performance and …

Two Theorems About Maximal Cohen--Macaulay Modules, Craig Huneke, Graham J. Leuschke

Mathematics - All Scholarship

This paper contains two theorems concerning the theory of maximal Cohen-Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen-Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen-Macaulay modules exist having ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen-Macaulay local ring of finite Cohen-Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning …

Securing Distributed Computations : In Search Of Reliable Large-Scale Compute Power And Refreshed Redundancy, Edward P. Kenney

Honors Theses

The Internet may be the single largest technological advance or significant societal change in the last century. Not only does it allow access to more information than any human could ever hope to digest, but it produces the potential of having millions of computers combining their computational forces for the betterment of a single cause .. Tl is is the fundamental goal of distributed computing. A distributed system is defined to be a network of machines with some degree of centralized direction. In a distributed computational system each machine will accept computational tasks from a supervisor in a master-slave relationship. …

Optimal Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

In this paper we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints. In contrast to parabolic dynamics, such systems have not been sufficiently studied in the literature. The reason is the lack of regularity in the case of hyperbolic dynamics. We present necessary optimality conditions for both Neumann and Dirichlet boundary control problems and discuss differences and relationships between them.

Asymptotic Stability Of Linear Difference Equations Of Advanced Type, Fozi M. Dannan, Saber Elaydi

Mathematics Faculty Research

Necessary and sufficient conditions are obtained for the asymptotic stability of difference equations of advanced type n of the form x(n) - ax(n+1) + bx(n+k) = 0, n = 0, 1, .. where a and b are arbitrary real numbers and k > 1. For a = 1, we establish an analogue of a result by Levin and May.

Skew-Product Dynamical Systems: Applications To Difference Equations, Saber Elaydi, Robert J. Sacker

Mathematics Faculty Research

No abstract provided.

Mathematical And Empirical Modeling Of Chemical Reactions In A Microreactor, Jing Hu

Doctoral Dissertations

This dissertation is concerned with mathematical and empirical modeling to simulate three important chemical reactions (cyclohexene hydrogenation and dehydrogenation, preferential oxidation of carbon monoxide, and the Fischer-Tropsch (F-T) synthesis in a microreaction system.

Empirical modeling and optimization techniques based on experimental design (Central Composite Design (CCD)) and response surface methodology were applied to these three chemical reactions. Regression models were built, and the operating conditions (such as temperature, the ratio of the reactants, and total flow rate) which maximize reactant conversion and product selectivity were determined for each reaction.

A probability model for predicting the probability that a certain species …

Editorial, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Understanding Polygons And Polyhedrons Using Flexagons, Aaron Tekulve

The Mathematics Enthusiast

The goal of this paper is to help students understand simple polygons and simple polyhedrons. First the project within this paper involves having students look at polygons. Though much of this information should have been learned in the fourth grade it is still important to review this material. Having the students define certain shapes illustrates their true understanding of the subject. The second part to the project within this paper is to use the student’s knowledge of polygons and build polyhedrons. In this paper the students will only have to concern themselves with squares and equilateral triangles. Here they will …

Teaching Symmetry In The Elementary Curriculum, Christy Knuchel

The Mathematics Enthusiast

Symmetry is a fundamental part of geometry, nature, and shapes. It creates patterns that help us organize our world conceptually. We see symmetry every day but often don’t realize it. People use concepts of symmetry, including translations, rotations, reflections, and tessellations as part of their careers. Examples of careers that incorporate these ideas are artists, craftspeople, musicians, choreographers, and not to mention, mathematicians.

It is important for students to grasp the concepts of geometry and symmetry while at the elementary level as a means of exposing them to things they see everyday that aren’t obviously related to mathematics but have …

Author Information

The Mathematics Enthusiast

No abstract provided.

Radius, Diameter, Circumference, Pi, Geometer's Sketchpad, And You!, Scott Edge

The Mathematics Enthusiast

I truly believe learning mathematics can be a fun experience for children of all ages. It is up to us, the teachers, to present math as an interesting application. The addition of computers into our ever-changing world has given us an important tool, which can assist us on our journey to teach math in new fun and interesting ways. The Program Geometer’s Sketchpad© is one of many mathematic programs we as teachers can use to better help kids understand different geometric concepts. I would like to use Geometer’s Sketchpad© in my classroom to help teach my students about circles and …

Tme Volume 1, Number 1

The Mathematics Enthusiast

No abstract provided.

Products Of Characters And Finite P-Groups Ii, Edith Adan-Bante

Faculty Publications

Let p be a prime number. Let G be a finite p-group and χ ∈ Irr(G). Denote by the ^-χ ∈ Irr(G) complex conjugate of χ . Assume that χ(1) = pⁿ. We show that the number of distinct irreducible constituents of the product of χ^-χ is at least 2n(p−1)+1.

Exploring Perimeter And Area With 4th Graders, Amber Lieberg Winkler

The Mathematics Enthusiast

I am going to teach basic introductory geometry skills to 4th graders using Geometry’s Sketchpad. At this age, children are only beginning to learn about geometry in their math classes, and I would like for the students to understand these basic concepts using technology. This problem is so important for the students to learn early, and learn correctly. These skills will involve finding the area and perimeter of regular polygons, basic skills using Geometry’s Sketchpad, and activities that will apply these introductory concepts; directly correlating within the national geometry standards of mathematics. Children use geometry everyday, even when they don’t …

Fermat's Last Theorem For Rational Exponents, Curtis D. Bennett, A. M. W. Glass, Gábor J. Székely

Mathematics, Statistics and Data Science Faculty Works

No abstract provided.

Two New Criteria For Comparison In The Bruhat Order, Brian Drake, Sean Gerrish, Mark Skandera

Dartmouth Scholarship

We give two new criteria by which pairs of permutations may be compared in defining the Bruhat order (of type $A$). One criterion utilizes totally nonnegative polynomials and the other utilizes Schur functions.

Transcendence Measures And Algebraic Growth Of Entire Functions, Dan Coman, Evgeny A. Poletsky

Mathematics - All Scholarship

In this paper we obtain estimates for certain transcendence measures of an entire function f. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z,w) in C2 along the graph of f. These inequalities provide, in turn, estimates for the number of zeros of the function P(z, f(z)) in the disk of radius r, in terms of the degree of P and of r. Our estimates hold for arbitrary entire functions f of finite order, and for a subsequence {nj} of degrees of polynomials. But for special classes of functions, including the Riemann zeta-function, they hold …

A Forbidden Subgraph Characterization Problem And A Minimal-Element Subset Of Universal Graph Classes, Michael D. Barrus

Theses and Dissertations

The direct sum of a finite number of graph classes H_1, ..., H_k is defined as the set of all graphs formed by taking the union of graphs from each of the H_i. The join of these graph classes is similarly defined as the set of all graphs formed by taking the join of graphs from each of the H_i. In this paper we show that if each H_i has a forbidden subgraph characterization then the direct sum and join of these H_i also have forbidden subgraph characterizations. We provide various results which in many cases allow us to exactly …