Mathematics Commons^™

Results From Electrostatic Calibrations For Measuring The Casimir Force In The Cylinder-Plane Geometry, Q. Wei, D. A. R. Dalvit, F. C. Lombardo, F. D. Mazzitelli, R. Onofrio

Dartmouth Scholarship

We report on measurements performed on an apparatus aimed to study the Casimir force in the cylinder-plane configuration. The electrostatic calibrations evidence anomalous behaviors in the dependence of the electrostatic force and the minimizing potential upon distance. We discuss analogies and differences of these anomalies with respect to those already observed in the sphere-plane configuration. At the smallest explored distances we observe frequency shifts of non-Coulombian nature preventing the measurement of the Casimir force in the same range. We also report on measurements performed in the parallel-plane configuration, showing that the dependence on distance of the minimizing potential, if present …

Noncommutative Topology And The World’S Simplest Index Theorem, Erik Van Erp

Dartmouth Scholarship

In this article we outline an approach to index theory on the basis of methods of noncommutative topology. We start with an explicit index theorem for second-order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how it is derived as a special case of an index theorem for hypoelliptic operators on contact manifolds. Finally, we discuss the noncommutative topology that is employed in the proof of this theorem. The article is intended to illustrate that noncommutative topology can be a powerful tool …

Explicit And Implicit Methods In Solving Differential Equations, Timothy Bui

Honors Scholar Theses

Differential equations are equations that involve an unknown function and derivatives. Euler's method are efficient methods to yield fairly accurate approximations of the actual solutions. By manipulating such methods, one can find ways to provide good approximations compared to the exact solution of parabolic partial differential equations and nonlinear parabolic differential equations.

A Predictive Model For Secondary Rna Structure Using Graph Theory And A Neural Network., Denise Renee Koessler

Electronic Theses and Dissertations

In this work we use a graph-theoretic representation of secondary RNA structure found in the database RAG: RNA-As-Graphs. We model the bonding of two RNA secondary structures to form a larger structure with a graph operation called merge. The resulting data from each tree merge operation is summarized and represented by a vector. We use these vectors as input values for a neural network and train the network to recognize a tree as RNA-like or not based on the merge data vector.

The network correctly assigned a high probability of RNA-likeness to trees identified as RNA-like in the RAG database, …

Total Domination Dot Critical And Dot Stable Graphs., Stephanie Anne Marie Mcmahon

Electronic Theses and Dissertations

Two vertices are said to be identifed if they are combined to form one vertex whose neighborhood is the union of their neighborhoods. A graph is total domination dot-critical if identifying any pair of adjacent vertices decreases the total domination number. On the other hand, a graph is total domination dot-stable if identifying any pair of adjacent vertices leaves the total domination number unchanged. Identifying any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most two. Among other results, we characterize total domination dot-critical trees with total …

Σary, Minnesota State University Moorhead, Mathematics Department

Math Department Newsletters

No abstract provided.

Discrete Fractional Calculus And Its Applications To Tumor Growth, Sevgi Sengul

Masters Theses & Specialist Projects

Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the "difference" of any function, from 1st order up to the n-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make n- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus …

An Algorithm To Generate Two-Dimensional Drawings Of Conway Algebraic Knots, Jen-Fu Tung

Masters Theses & Specialist Projects

The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(n) time where n is the number of crossings …

Development Of A Discrete Mathematics Textbook And Guide For High School Teachers, Rebecca A. Stokes

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

This project was to create the beginnings a textbook for teacher's to supplement instruction of a discrete mathematics course at the high school level. The development of the text was guided by past and current efforts to place discrete mathematics in high school curriculum. A review of the literature and experiences of instructors were viewed and analyzed to guide the construction of the textbook. The text was written with the goal to give teachers information about the topics of discrete mathematics, extra resources, lesson ideas , and optional worksheets for students. Several lessons were created and one was implemented in …

Classical Foundations For A Quantum Theory Of Time In A Two-Dimensional Spacetime, Nathan Thomas Carruth

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

We consider the set of all spacelike embeddings of the circle S1 into a spacetime R1 × S1 with a metric globally conformal to the Minkowski metric. We identify this set and the group of conformal isometries of this spacetime as quotients of semidirect products involving diffeomorphism groups and give a transitive action of the conformal group on the set of spacelike embeddings. We provide results showing that the group of conformal isometries is a topological group and that its action on the set of spacelike embeddings is continuous. Finally, we point out some directions for future research.

Dancing With Heretics: Essays On Orthodoxy, Questioning And Faith, Darren M. Edwards

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

While much has been written about the conflicts, supposed or actual, between logic and faith, science and religion, few accounts of the personal turmoil these conflicts can cause exist. Likewise, many of these nonfiction accounts are written from a distinctly polarized place leaning either to science or faith.

In this thesis, I mix research and history with memoir and a sense of poetry to explore my personal experience with this conflict. At its outset, I hoped for this project to capture my struggle as an orthodox member of The Church of Jesus Christ of Latter-day Saints (LDS) in dealing with …

Dynamics Groups Of Asynchronous Cellular Automata, Michael Macauley, Jon Mccammond, Henning S. Mortveit

Publications

We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is π-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible 2²³ = 256 cellular automaton rules are π-independent. In the article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information …

The Life Of Evariste Galois And His Theory Of Field Extension, Felicia N. Adams

Senior Honors Theses

Evariste Galois made many important mathematical discoveries in his short lifetime, yet perhaps the most important are his studies in the realm of field extensions. Through his discoveries in field extensions, Galois determined the solvability of polynomials. Namely, given a polynomial P with coefficients is in the field F and such that the equation P(x) = 0 has no solution, one can extend F into a field L with α in L, such that P(α) = 0. Whereas Galois Theory has numerous practical applications, this thesis will conclude with the examination and proof of the fact that it is impossible …

On The Numerical Range And Spectrum Of The Weighted Shift Operator In [Iota]², Gina-Louise Santamaria

Theses, Dissertations and Culminating Projects

In this paper, we examine the weighted shift operator in l² as described in Yoo & Rho [15], which is an example of what is known as a hyponormal weighted shift. Using the methods of Tam [13], in conjuction with properties of the weighted shift, we determine the numerical range of Yoo &; Rho’s unilateral weighted shift operator.

It is well-established that the spectrum of a bounded linear operator is always included in the closure of the numerical range. In particular, for a bounded linear operator, the point and compression spectra are contained within the numerical range itself [7]. …

Using Matrix Pencils To Solve Discrete Sturm-Liouville Problems With Nonlinear Boundary Conditions, Michael Kofi Wilson

Theses, Dissertations and Culminating Projects

This thesis deals with discrete second order Sturm-Liouville Boundary Value Problems (DSLBVP) where the parameter as part of the Sturm-Liouville difference equation appears nonlinearly in the boundary conditions. We focus on analyzing the case with cubic nonlinearity in the boundary condition. First, we describe the problem by a matrix equation with nonlinear variables such that solving the DSLBVP is equivalent to solving the matrix equation. Second, we formulate the problem as a nonlinear eigenvalue problem. We further reduce the problem to finding eigenvalues of a matrix pencil in the form A - X B . Under certain conditions, such a …

On Directionally Dependent Subdifferentials, Ivan Ginchev, Boris S. Mordukhovich

Mathematics Research Reports

In this paper directionally contextual concepts of variational analysis, based on dual-space constructions similar to those in [4, 5], are introduced and studied. As an illustration of their usefulness, necessary and also sufficient optimality conditions in terms of directioual subdifferentials are established, and it is shown that they can be effective in the situations where known optimality conditions in terms of nondirectional subdifferentials fail.

Analyzing Fractals, Kara Mesznik

Renée Crown University Honors Thesis Projects - All

For my capstone project, I analyzed fractals. A fractal is a picture that is composed of smaller images of the larger picture. Each smaller picture is self- similar, meaning that each of these smaller pictures is actually the larger image just contracted in size through the use of the Contraction Mapping Theorem and shifted using linear and affine transformations.

Fractals live in something called a metric space. A metric space, denoted (X, d), is a space along with a distance formula used to measure the distance between elements in the space. When producing fractals …

Time Series Models For Computing Activation In Fmri, Daniel W. Adrian, Ranjan Maitra, Daniel B. Rowe

Mathematics, Statistics and Computer Science Faculty Research and Publications

No abstract provided.

Option Pricing And Stable Trading Strategies In The Presence Of Information Asymmetry, Anirban Dutta

Dissertations

Pricing derivatives is one of the central issues in mathematical finance. The seminal work of Black, Scholes and Merton has been the cornerstone of option pricing since their introduction in 1973. Their work influenced the pricing theory of other derivatives as well.

This derivative pricing theory has two primary shortcomings. Firstly, the theoretical pricing in such theories are not accompanied by a stable trading strategy. Secondly, they often assume that the market agents use a uniform model for the underlying instrument and that the market prices of the derivatives reveal all the information about the underlying instrument.

Theoreticians like Grossman …

Combinatorics And Topology Of Curves And Knots, Bailey Ann Ross

Boise State University Theses and Dissertations

The genus of a graph is the minimal genus of a surface into which the graph can be embedded. Four regular graphs play an important role in low dimensional topology since they arise from curves and virtual knot diagrams. Curves and virtual knots can be encoded combinatorially by certain signed words, called Gauss codes and Gauss paragraphs. The purpose of this thesis is to investigate the genus problem for these combinatorial objects: Given a Gauss word or Gauss paragraph, what is the genus of the curve or virtual knot it represents?

Developmental Understanding Of The Equals Sign And Its Effects On Success In Algebra, Ryan W. Brown

Boise State University Theses and Dissertations

For some students, the equals symbol is viewed as a directive to carry out a procedure, instead of a symbol expressing mathematical equivalence. The purpose of this study was to develop and to pilot a questionnaire to measure students’ understandings of relational equivalence as implied by their interpretations and use of the equals symbol. The results of this questionnaire were compared with student testing data with the goal of determining a correlation between understanding of symbolic equivalence and success in a typical algebra course. It was found that students who demonstrated an ability to define and articulate an appropriate meaning …

Stably Free Modules Over The Klein Bottle, Andrew Misseldine

Boise State University Theses and Dissertations

This paper is concerned with constructing countably many, non-free stably free modules for the Klein bottle group. The work is based on the papers “Stably Free, Projective Right Ideals" by J.T. Stafford (1985) and “Projective, Nonfree Modules Over Group Rings of Solvable Groups" by V. A. Artamonov (1981). Stafford proves general results that guarantee the existence of non-free stably frees for the Klein bottle group but has not made the argument explicit. Artamonov allows us to construct infinitely many non-free stably free modules. This paper will also construct presentations and sets of generators for these modules. This paper concludes with …

A Mathematical Model Of The Biofluidmechanics Of The Non-Newtonian Mucus Layer Of The Tear Film In The Human Eye, Douglas M. Platt

Theses, Dissertations and Culminating Projects

The human eye is a complicated and delicate organ. The structure of the eye is such that it provides for clear vision of the world. The cornea and conjunctiva at the front of the eye are avascular structures that require nutrients and moisture to be provided by the tear film. The tear film also provides for the removal of debris from the surface of the eye. The tear film has several layers: a lipid layer, an aqueous solution, and mucus. The optical clarity and structural uniformity of the tear film is maintained by the blinking motion of the eyelid. During …

Assessing The Impact Of A Computer-Based College Algebra Course, Ningjun Ye

Dissertations

USM piloted the Math Zone in Spring 2007, a computer-based program in teaching MAT 101and MAT 099 in order to improve student performance. This research determined the effect of the re-design of MAT 101 on student achievements in comparison to a traditional approach to the same course. Meanwhile, the study investigated possible effects of the Math Zone program on students’ attitude toward studying mathematics.

This study shows that there was no statistically significant difference on MAT101 final exam scores between the Math Zone students and the Classroom students in Fall 2007, Spring 2008 and Fall 2008. At the same time, …

Schubert Polynomials And Classes Of Hessenberg Varieties, Dave Anderson, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a " Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, …

Certain Two-Parameter Representations Of The Lie Algebra Sl(2,C), Scott Sidoli

Mathematics and Statistics

Classical Lie algebras, like sl(2,C) can be represented using differential operators that act on polynomial space. These operators will take a different form when they are used on the space of polynomials of several variables and when the differentials are taken to be of higher order. We recall some known realizations and discuss possible deformations. In our two-parameter case we describe decomposition into indecomposable components.

Analysis Of Discrete Data Under Order Restrictions, Jeff Campbell

College of Graduate Studies: Theses & Dissertations

Strategies for the analysis of discrete data under order restrictions are discussed. We consider inference for sequences of binomial populations, and the corresponding risk difference, relative risk and odds ratios. These concepts are extended to deal with independent multinomial populations. Natural orderings such as stochastic ordering and cumulative ratio probability ordering are discussed. Methods are developed for the estimation and testing of differences between binomial as well as multinomial populations under order restrictions. In particular, inference for sequences of ordered binomial probabilities and cumulative probability ratios in multinomial populations are presented. Closed-form estimates of the multinomial parameters under order restrictions …

Neutrosophic Physics: More Problems, More Solutions, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

When considering the laws of theoretical physics, one of the physicists says that these laws – the actual expressions of the laws of mathematics and logics being applied to physical phenomena – should be limited according to the physical meaning we attribute to the phenomena. In other word, there is an opinion that a theoretical physicist should put some limitations onto mathematics, in order to “reduce” it to the observed reality. No doubt, we can do it. However, if following this way, we would arrive at only mathematical models of already known physical phenomena. Of course, this might be useful …

A Criterion For Identifying Stressors In Non-Linear Equations Using Gröbner Bases, Elisabeth Marie Palchak

Honors Theses

No abstract provided.

The Effect Of Explicit Timing On Math Performance Using Interspersal Assignments With Students With Mild/Moderate Disabilities, Fangjuan Hou

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Explicit timing and interspersal assignments have been validated as effective methods to facilitate students' math practice. However, no researchers have explored the combinative effect of these two methods. In Study 1, we extended the literature by comparing the effect of explicit timing with interspersal assignments, and interspersal assignments without timing. Generally, participants' rate of digits correct on easy and hard addition problems was higher during the explicit timing condition than during the untimed condition. However, the participants' rate of digits correct decreased after initial implementation of the explicit timing condition.

Motivation plays a crucial role in maintaining performance levels and …