Mathematics

Mathematics

Undergraduate Research

Most undergraduate research at Eckerd culminates in a published thesis. Here are some recent examples of theses in mathematics.

A Family of Multiple Harmonic Sums and Multiple Zeta Star Value Identities

Title: A Family of Multiple Harmonic Sums and Multiple Zeta Star Value Identities
Student: Erin Linebarger
Faculty Director: Jianqiang Zhao

Abstract: In this thesis I present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al. This identity relates the multiple star harmonic sum of a sequence to the sum of several multiple harmonic sums of sequences whose weight is equal to the original sequence's weight.  I then apply this result to obtain a family of identities of multiple zeta star values.

A Non-Linear Approach for the Solution of a Conditional Fracture Propagation Problem

Title: A Non-Linear Approach for the Solution of a Conditional Fracture Propagation Problem
Student: Ivelin Georgiev
Faculty Director: Gerald Junevicus

Abstract: Using numerical techniques, we find an approximation to the solution for the width and length of a fracture propagating inside a rock at different times under specified conditions. The physics of the problem is introduced first, resulting in a partial differential equation with appropriately specified boundary and initial conditions. The facet that makes this boundary-value problem non-standard is the variation of the right-hand boundary with time; the determination of this behavior is a part of the solution. We then develop a finite difference analog for the equations and discuss two methods of solving the resulting system: a linearization using a binomial approximation and a non-linear approach. A program based on the non-linear approach is implemented, and the resulting approximations are considered.

Properties of the Duality Map in an Abstract Banach Space

Title: Properties of the Duality Map in an Abstract Banach Space
Student: Denise Mason
Faculty Director: David Kerr

Abstract: In every Banach space, the Hahn-Banach theorem guarantees the existence of a multi-valued “duality mapping” between the space and its dual. In this thesis, various properties of this map are explored. It will be shown that, under certain convexity conditions, the duality map is single-valued and thus establishes a function between the Banach space and its dual. Furthermore, this thesis discusses the relationship between the duality map and the differentiability of the norm, while it also develops necessary conditions under which this function is uniformly continuous on bounded subsets of its domain.

Properties of Accretive and Monotone Operators in Banach Spaces

Title: Properties of Accretive and Monotone Operators in Banach Spaces
Student: Soupy Alexander
Faculty Director: David Kerr

Abstract: In a real Banach space, an accretive operator T is "m-accretive" if the operator (T + nI) is onto for all natural numbers n. Associated with these operators are other operators called “resolvents.” In this thesis, we develop a "resolvent identity" which relates one resolvent to another and go on to prove a number of theorems concerning these resolvents and their associated Yosida approximants. The final section will discuss concepts of topological degree theory and provide several proofs using this theory.

Resolvents of m-Accretive Operators in Banach Spaces

Title: Resolvents of m-Accretive Operators in Banach Spaces
Student: Eric Lehr
Faculty Director: David Kerr

Abstract: In this thesis, we develop a "resolvent identity" for the accretive operator T defined on a real Banach space and show that if one resolvent of T is compact, they all are compact. We then go on to prove a number of theorems concerning these resolvents and their associated "Yosida approximants." Finally, some results generalizing these concepts to the class of "w-accretive" operators are provided.

Solvability of Nonlinear Equations of m-Accretive Operators in Banach Spaces

Title: Solvability of Nonlinear Equations of m-Accretive Operators in Banach Spaces
Student: Mark Croxford
Faculty Director: David Kerr

Abstract: In this thesis, we study accretive, m-accretive, and w-accretive operators mapping a Banach space into itself. We associate a “resolvent” with each accretive operator and show that these resolvents satisfy a number of interesting properties. We then relate the resolvents to another operator called the “Yosida approximant.” The methods in this thesis involve standard applications of inverse function theory and nonlinear functional analysis. These results compliment the works of Browder, Kartsatos and Kerr.

Topological Dimensions and Weierstrass Functions

Title: Topological Dimensions and Weierstrass Functions
Student: Nazarre N. Merchant
Faculty Director: David Kerr

Abstract: This thesis will explore the developments of dimension from the small and large inductive dimension through the Hausdorff dimension. Properties of the two topological dimensions are investigated including various relationships between the small and large inductive dimensions. Finally, the Hausdorff dimension will be used to look at ‘pathological’ functions including Weierstrass functions to examine non-integer dimension properties.

Constituents of Chaos

Title: Constituents of Chaos
Student: Sean Bird
Faculty Director: David Kerr

Abstract: Chaos is a new and popular field of mathematics. This thesis will serve as an introduction and will provide a taste of the excitement which chaos has in store. The upper-level undergraduate student with some background in dynamical systems and topology will find the main body of the paper approachable. We will demonstrate various chaotic maps, including hyberbolic toral automorphisms.

An Exploration into the Concept of Topological Dimension

Title: An Exploration into the Concept of Topological Dimension
Student: Christopher Stimac
Faculty Director: David Kerr

Abstract: The various connotations of “exploration” truly encapsulate the purposes of this thesis, as this paper represents an inquisitive journey through the most significant and salient aspects of inductive dimensions are elucidated and illustrated with examples. The culmination of the findings presented herein is the Fundamental Theorem of Dimension Theory. Many of the concepts related to finite-dimensional spaces are further investigated within the context of an infinite-dimensional space and the question of countability is addressed. An interesting example of a zero-dimensional subset with cardinality of the continuum is also discussed.

Our Graduates

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