Mathematics Commons^™

Factorization Of Tropical Polynomials, Nathan Grigg, Dr. Tyler Jarvis

Journal of Undergraduate Research

Tropical algebra, also called “min-plus” or “max-plus” algebra, is a relatively new topic in mathematics that has recently caught the interest of algebraic geometers, computer scientists, combinatorists, and other mathematicians. According to Andreas Gathmann , tropical algebra was pioneered by mathematician and computer scientist Imre Simon in the 1980s but did not receive widespread attention until a few years ago. Interestingly, the title “tropical algebra” is nothing more than a reference to Simon’s home country—Brazil.

Non-Linear Ray Tracing Visualizing The Impossible, Jared Duke, Dr. Matthew Anderson

Journal of Undergraduate Research

Ray tracing is a common technique for computational, photo-realistic simulation of the “real world.” However, most current approaches assume light propagation is linear: that is, light always travels in a straight line. Our aim was to create a ray tracing program that would handle the “bending” of light waves at relativistic speeds or by gravitational forces. This could in turn be used to assist scientists in understanding the numerical results from their research and to help others understand and visualize such complex natural phenomenon.

Assessing Cognitive Construction In Learning Statistics As Applied Through Mathematics Education, Aubrey Baxter, Dr. Hope Gerson

Journal of Undergraduate Research

Paul Fields in the statistics department was my first mentor. Together we tried to develop a new pedagogy for teaching statistics. This interested me because I may teach advanced placement statistics after I graduate from the Mathematics Education program. A position then came available to me as a research assistant for Dr. Hope Gerson. It was a better opportunity because it is in my field of study and it allowed me work with statistics and apply calculus. Together with Dr. Walter, Dr. Gerson taught an Honors Calculus 1 class for two semesters and an Honors Calculus 2 class for one …

Calculus Excel At Byu: Increasing Student Potential For Success, Karla Jeanese Hendricks, Dr. Michael Dorff

Journal of Undergraduate Research

Success in calculus can have significant implications both within an intended field of study and throughout academic pursuits. Of the 34 science, technology, engineering, and mathematics (STEM) majors at BYU, 25 of them require calculus for graduation and 20 of them require it as a prerequisite for other courses. A study at Bryant University in Rhode Island examined mathematics SAT scores, verbal SAT scores, percentile rank in high school graduating class, percentage score on a college mathematics placement exam, and calculus grades of actuarial students; it was found that the calculus grades were the “single best predictor of a student’s …

Why Rozenzweig-Style Midrashic Approach Makes Rational Sense: A Logical (Spinoza-Like) Explanation Of A Seemingly Non-Logical Approach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A 20 century German Jewish philosopher Franz Rosenzweig promoted a new approach to knowledge, an approach in which in addition to logical reasoning, coming up with stories with imagined additional details is also important. This approach is known as midrashic since it is similar to the use of similar stories -- known as midrashes -- in Judaism. While stories can make the material interesting, traditionally, such stories are not viewed as a serious part of scientific discovery. In this paper, we show that this seemingly non-logical approach can actually be explained in logical terms and thus, makes perfect rational sense.

Why In Mayan Mathematics, Zero And Infinity Are The Same: A Possible Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In Mayan mathematics, zero is supposed to be, in some sense, equal to infinity. At first glance, while this statement may have a deep philosophical meaning, it does not seem to make much mathematical sense. In this paper, we show, that this statement may be made mathematically reasonable. Specifically, on a real line, it is often useful to consider both −∞ and +∞ as a single infinity. When we deal with very small and very large numbers, it makes sense to use floating point representation, i.e., in effect, consider logarithms of the original values. In terms of logarithms, the original …

A Construction That Produces Wallis-Type Formulas, Joshua M. Fitzhugh, David L. Farnsworth

Articles

Generalizations of the geometric construction that repeatedly attaches rectangles to a square, originally given by Myer- son, are presented. The initial square is replaced with a rectangle, and also the dimensionality of the construction is increased. By selecting values for the various parameters, such as the lengths of the sides of the original rectangle or rectangular box in dimensions more than two and their relationships to the size of the attached rectangles or rectangular boxes, some interesting formulas are found. Examples are Wallis-type infinite-product formulas for the areas of p-circles with p > 1.

Book Review: Encyclopedia Of Mathematics And Society, Gizem Karaali

Pomona Faculty Publications and Research

The Encyclopedia of Mathematics and Society is an impressive achievement of collective effort and serious thought, an amazing collection of delightful, unexpected essays, a sourcebook for students and teachers alike. The experience of reading the EMS was, for this encyclopedia enthusiast, enlightening and enjoyable.

If any librarians out there are still wondering, let me be clear: I strongly recommend this encyclopedia. For individuals, the expense might well be an initial turn-off; keeping in mind that the hard copy books come with online access to the same content might alleviate some of the pain.

As for me? I expect to enjoy …

Can The Differentiated Math Classroom Be A Reality?, Carl Jeff Morris

Mathematics Graduate Theses

Imagine a math classroom that looked more like a Google workplace where collaboration, teamwork, and innovation are the centerpieces. Instead of just students sitting in rows passively learning about the nature of mathematics, picture a classroom of students in small circle discussing mathematics. This could range from groups working together on a problem solving strategy for a newly learned concept, students helping students using technology as an aid, another small group receiving intervention from the teacher, or individual students making or viewing instructional math videos. All of this according to research focused on classroom differentiation; specifically in middle school and …

The Generalized Sub-Fractional Brownian Motion, Aissa Sghir

Communications on Stochastic Analysis

No abstract provided.

An Exploratory Study Of Pre-Service Middle School Teachers’ Knowledge Of Algebraic Thinking, Marta Magiera, Leigh A. Van Den Kieboom, John C. Moyer

Mathematics, Statistics and Computer Science Faculty Research and Publications

Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers’ ability to use the features of the algebraic thinking (AT) habit of mind “Building Rules to Represent Functions” and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers’ own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed …

The Semigroups B2 And B0 Are Inherently Nonfinitely Based, As Restriction Semigroups, Peter R. Jones

Mathematics, Statistics and Computer Science Faculty Research and Publications

The five-element Brandt semigroup B₂ and its four-element subsemigroup B₀, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B₂ carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, ^-1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x^-1 and retaining the induced operations x ↦ x^{+ …}

An Investigation Of Teachers' Intentions And Reflections About Using Standards-Based And Traditional Textbooks In The Classroom, Bikai Nie, Tony Freedman, Stephen Hwang, Ning Wang, John Moyer, Jinfa Cai

Mathematics, Statistics and Computer Science Faculty Research and Publications

This study analyzed teachers’ intentions for and reflections on their use of Standards-based [Connected Mathematics Program (CMP)] textbooks and traditional (non-CMP) mathematics textbooks to guide instruction. In this investigation of the interplay between textbooks and instruction, we focused on learning goals, instructional tasks, teachers’ anticipation of students’ difficulties, and their perceptions of students’ achievement of learning goals. All of these are aspects of teachers’ intentions and reflections that have proved fruitful in comparing the roles of the CMP and non-CMP mathematics textbooks in our Longitudinal Investigation of the Effect of Curriculum on Algebra Learning project. Whereas the cognitive level …

Sub-Independence: An Expository Perspective, Gholamhossein Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

Limit theorems as well as other well-known results in probability and statistics are often based on the distribution of the sums of independent random variables. The concept of sub-independence, which is much weaker than that of independence, is shown to be sufficient to yield the conclusions of these theorems and results. It also provides a measure of dissociation between two random variables which is much stronger than uncorrelatedness.

Bayesian Decision Theoretic Approach To Directional Multiple Hypotheses Problems, Naveen K. Bansal, Klaus J. Miescke

Mathematics, Statistics and Computer Science Faculty Research and Publications

A multiple hypothesis problem with directional alternatives is considered in a decision theoretic framework. Skewness in the alternatives is considered, and it is shown that this skewness permits the Bayes rules to possess certain advantages when one direction of the alternatives is more important or more probable than the other direction. Bayes rules subject to constraints on certain directional false discovery rates are obtained, and their performances are compared with a traditional FDR rule through simulation. We also analyzed a gene expression data using our methodology, and compare the results to that of a FDR method.

A Bayesian Model For Cluster Detection, Jonathan Wakefield, Albert Y. Kim

Statistical and Data Sciences: Faculty Publications

The detection of areas in which the risk of a particular disease is significantly elevated, leading to an excess of cases, is an important enterprise in spatial epidemiology. Various frequentist approaches have been suggested for the detection of “clusters” within a hypothesis testing framework. Unfortunately, these suffer from a number of drawbacks including the difficulty in specifying a p-value threshold at which to call significance, the inherent multiplicity problem, and the possibility of multiple clusters. In this paper, we suggest a Bayesian approach to detecting “areas of clustering” in which the study region is partitioned into, possibly multiple, “zones” …

Stein's Method For Brownian Approximations, L Coutin, L Decreusefond

Communications on Stochastic Analysis

No abstract provided.

A Clark-Ocone Type Formula Under Change Of Measure For Lévy Processes With L^2-Lévy Measure, Ryoichi Suzuki

Communications on Stochastic Analysis

No abstract provided.

Identities And Inequalities For Cdo Tranche Sensitivities, Claas Becker, Ambar N Sengupta

Communications on Stochastic Analysis

No abstract provided.

Positive Harris Recurrence Of The Cir Process And Its Applications, Peng Jin, Vidyadhar Mandrekar, Barbara Rüdiger, Chiraz Trabelsi

Communications on Stochastic Analysis

No abstract provided.