Mathematics Commons^™

Lagrange's Theory Of Analytical Functions And His Ideal Of Purity Of Method, Giovanni Ferraro, Marco Panza

MPP Published Research

We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and we concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially on the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of …

Robust 𝒍𝟏 And 𝒍∞ Solutions Of Linear Inequalities, Maziar Salahi

Applications and Applied Mathematics: An International Journal (AAM)

Infeasible linear inequalities appear in many disciplines. In this paper we investigate the 𝑙₁ and 𝑙_∞ solutions of such systems in the presence of uncertainties in the problem data. We give equivalent linear programming formulations for the robust problems. Finally, several illustrative numerical examples using the cvx software package are solved showing the importance of the robust model in the presence of uncertainties in the problem data.

Finitely Presented Modules Over The Steenrod Algebra In Sage, Michael J. Catanzaro

Wayne State University Theses

No abstract provided.

Analyzing Common Algebra-Related Misconceptions And Errors Of Middle School Students., Sarah B. Bush

Electronic Theses and Dissertations

The purpose of this study was to examine common algebra-related misconceptions and errors of middle school students. In recent years, success in Algebra I is often considered the mathematics gateway to graduation from high school and success beyond. Therefore, preparation for algebra in the middle grades is essential to student success in Algebra I and high school. This study examines the following research question: What common algebra-related misconceptions and errors exist among students in grades six and eight as identified on student responses on an annual statewide standardized assessment? In this study, qualitative document analysis of existing data was used …

Support Varieties And Representation Type Of Self-Injective Algebras, Jorg Feldvoss, Sarah Witherspoon

University Faculty and Staff Publications

We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg [7]: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed in [23]. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half- quantum groups, and Nichols (quantum symmetric) algebras.

Completeness Of Interacting Particles, Pavel Abramski

Doctoral

This thesis concerns the completeness of scattering states of n _-interacting particles in one dimension. Only the repulsive case is treated, where thereare no bound states and the spectrum is entirely absolutely continuous, so the scattering Hilbert space is the whole of L2(Rn). The thesis consists of 4 chapters: The first chapter describes the model, the scattering states as given by the Bethe Ansatz, and the main completeness problem. The second chapter contains the proof of the completeness relation in the case of two particles: n = 2. This case had in fact already been treated by B. Smit (1997), …

The Eigenvalues Of A Tridiagonal Matrix In Biogeography, Boris Igelnik, Daniel J. Simon

Electrical and Computer Engineering Faculty Publications

We derive the eigenvalues of a tridiagonal matrix with a special structure. A conjecture about the eigenvalues was presented in a previous paper, and here we prove the conjecture. The matrix structure that we consider has applications in biogeography theory.

The Square Discrete Exponentiation Map, A Wood

Mathematical Sciences Technical Reports (MSTR)

We will examine the square discrete exponentiation map and its properties. The square discrete exponentiation map is a variation on a commonly seen problem in cryptographic algorithms. This paper focuses on understanding the underlying structure of the functional graphs generated by this map. Specifically, this paper focuses on explaining the in-degree of graphs of safe primes, which are primes of the form p = 2q + 1, where q is also prime.

Base-Free Formulas In The Lattice-Theoretic Study Of Compacta, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

The languages of finitary and infinitary logic over the alphabet of bounded lattices have proven to be of considerable use in the study of compacta. Significant among the sentences of these languages are the ones that are base free, those whose truth is unchanged when we move among the lattice bases of a compactum. In this paper we define syntactically the expansive sentences, and show each of them to be base free. We also show that many well-known properties of compacta may be expressed using expansive sentences; and that any property so expressible is closed under inverse limits and …

Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre

Honors Theses

It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored uniquely into primes. However, if K is a finite extension of the rational numbers, and OK its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into prime ideals. This result allows us to extend many properties of the integers to these rings. If we a finite extension L of K and OL of OK , we find that …

Noncomputable Functions In The Blub-Shub-Smale Model, Wesley Calvert, Ken Kramer, Russell Miller

Publications and Research

Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d + 1. We add a number of further results on relative computability of these sets and their unions. Then we show that the halting problem for BSS-computation is not decidable below any countable oracle set, and give a more …

Algebraic Solutions To Overdefined Systems With Applications To Cryptanalysis, Eric Crockett

Mathematical Sciences Technical Reports (MSTR)

Cryptographic algorithms are based on a wide variety of difficult problems in mathematics. One of these problems is finding a solution to a system of multivariate quadratic equations (MQ). A generalization of this problem is to find a solution to a system of higher order non-linear equations. Both of these problems are NP-hard over any field. Many cryptosystems such as AES, Serpent, Toyocrypt, and others can be reduced to some form of the MQ problem. In this paper we analyze the relinearization and XL algorithms for solving overdetermined systems of non-linear equations, as well as two variations of the XL …

Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, Ann Johnston

HMC Senior Theses

Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool …

Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell

Department of Mathematics: Dissertations, Theses, and Student Research

The Hilbert-Samuel function measures the length of powers of a zero-dimensional ideal in a local ring. Samuel showed that over a local ring these lengths agree with a polynomial, called the Hilbert-Samuel polynomial, for sufficiently large powers of the ideal. We examine the coefficients of this polynomial in the case the ideal is generated by a system of parameters, focusing much of our attention on the second Hilbert coefficient. We also consider the Hilbert-Kunz function, which measures the length of Frobenius powers of an ideal in a ring of positive characteristic. In particular, we examine a conjecture of Watanabe and …

Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer

Department of Mathematics: Dissertations, Theses, and Student Research

This work is primarily concerned with the study of artinian modules over commutative noetherian rings.

We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.

In the last chapter we prove that if the Betti …

Groups And Semigroups Generated By Automata, David Mccune

Department of Mathematics: Dissertations, Theses, and Student Research

In this dissertation we classify the metabelian groups arising from a restricted class of invertible synchronous automata over a binary alphabet. We give faithful, self-similar actions of Heisenberg groups and upper triangular matrix groups. We introduce a new class of semigroups given by a restricted class of asynchronous automata. We call these semigroups ``expanding automaton semigroups''. We show that this class strictly contains the class of automaton semigroups, and we show that the class of asynchronous automaton semigroups strictly contains the class of expanding automaton semigroups. We demonstrate that undecidability arises in the actions of expanding automaton semigroups and semigroups …

On The Homology Of Automorphism Groups Of Free Groups., Jonathan Nathan Gray

Doctoral Dissertations

Following the work of Conant and Vogtmann on determining the homology of the group of outer automorphisms of a free group, a new nontrivial class in the rational homology of Outer space is established for the free group of rank eight. The methods started in [8] are heavily exploited and used to create a new graph complex called the space of good chord diagrams. This complex carries with it significant computational advantages in determining possible nontrivial homology classes.
Next, we create a basepointed version of the Lie operad and explore some of it proper- ties. In particular, we prove a …

A Survey Of Modern Mathematical Cryptology, Kenneth Jacobs

Chancellor’s Honors Program Projects

No abstract provided.

Characteristic Polynomial Of Arrangements And Multiarrangements, Mehdi Garrousian

Electronic Thesis and Dissertation Repository

This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck.

In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction.

In chapter 3, we use some …

Annihilators Of Local Cohomology Modules, Laura Lynch

Department of Mathematics: Dissertations, Theses, and Student Research

In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider the annihilators of local cohomology modules. We examine these annihilators at various cohomological degrees, in particular at the cohomological dimension and at the height or the grade of the defining ideal. We also investigate the dimension of these annihilators at various degrees and we refine our results by specializing to particular types of rings, for example, Cohen Macaulay rings, unique factorization domains, and rings of small dimension.

Adviser: Thomas Marley

Descent Systems, Eulerian Polynomials And Toric Varieties, Letitia Mihaela Golubitsky

Electronic Thesis and Dissertation Repository

It is well-known that the Eulerian polynomials, which count permutations in S_n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the Sn -orbit for a generic weight in the weight lattice of Sn . Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this thesis we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight fixed by only the simple reflections …

Epistemic Strategies For Solving Two-Dimensional Physics Problems, Mary Elyse Hing-Hickman

Physics Theses & Dissertations

An epistemic strategy is one in which a person takes a piece of knowledge and uses it to create new knowledge. Students in algebra and calculus based physics courses use epistemic strategies to solve physics problems. It is important to map how students use these epistemic strategies to solve physics problems in order to provide insight into the problem solving process.

In this thesis three questions were addressed: (1) What epistemic strategies do students use when solving two-dimensional physics problems that require vector algebra? (2) Do vector preconceptions in kinematics and Newtonian mechanics hinder a student's ability to apply the …

Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette

Department of Mathematics: Dissertations, Theses, and Student Research

In this work, I offer an alternative presentation theory for C*-algebras with applicability to various other normed structures. Specifically, the set of generators is equipped with a nonnegative-valued function which ensures existence of a C*-algebra for the presentation. This modification allows clear definitions of a "relation" for generators of a C*-algebra and utilization of classical algebraic tools, such as Tietze transformations.

Much More Than Symbolics: The Early History Of Algebra And Its Significance For Introductory Algebra Education, Calvin Jongsma

Faculty Work Comprehensive List

No abstract provided.

The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle

Mathematics Faculty Publications

We will give a positive answer for the unimodality of the Hilbert functions in the smallest open case, that of Artinian level monomial algebras of type three in three variables.

Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, Subhajit Kumar Ganguly

Subhajit Kumar Ganguly

The following paper deals with the distribution of prime numbers, the twin prime numbers and the Goldbach conjecture. Starting from the simple assertion that prime numbers are never even, a rule for the distribution of primes is arrived at. Following the same approach, the twin prime conjecture and the Goldbach conjecture are found to be true.

Predictors Of Student Outcomes In Developmental Math At A Public Community And Technical College, Linda Darlene Hunt

Theses, Dissertations and Capstones

With the wide range of abilities of community college students, proper course placement is crucial. Therefore, having better predictors of success can help improve placement of students for their achievement. This study analyzed student predictors, instructor predictors, and classroom predictors in relation to student final exam score and student final grade in Elementary Algebra and Intermediate Algebra classes. Student predictors included gender, ACT math score, SAT math score, community college enrollment, math pretest score, and ASC grade. Instructor predictors included gender, employment status, Mozart music use, and ALEKS software use. Classroom predictors included time of day, number of class meetings …

Inverse Limits With Set Valued Functions, Van C. Nall

Department of Math & Statistics Faculty Publications

We begin to answer the question of which continua can be homeomorphic to an inverse limit with a single upper semi-continuous bonding map from [O, 1) to 2^(O,l). Several continua including (0, 1) x (0, 1) and all compact manifolds with dimension greater than one cannot be homeomorphic to such an inverse limit. It is also shown that if the upper semi-continuous bonding maps have only zero dimensional point values, then the dimension of the inverse limit does not exceed the dimension of the factor spaces.

Ken Kunen: Algebraist, Michael Kinyon

Mathematics Preprint Series

Ken Kunen is justifiably best known for his work in set theory and topology. What I would guess many of his friends and students in those areas do not know is that Ken also did important work in algebra, especially in quasigroup and loop theory. In fact, I think it is not an exaggeration to say that his work in loop theory, both alone and in collaboration, revolutionized the field. In this paper, I would like to describe some of his accomplishments to nonspecialists. My point of view is personal, of course, and so I will give the most attention …

Classical Kloosterman Sums: Representation Theory, Magic Squares, And Ramanujan Multigraphs, Patrick S. Fleming, Stephan Ramon Garcia, Gizem Karaali

Pomona Faculty Publications and Research

We consider a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of “magical” combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.