Physical Sciences and Mathematics Commons^™

Fearless Friday: Kirsten Crear, Kirsten Crear

SURGE

Even in her last semester here at Gettysburg, Kirsten Crear ’14 is fearlessly working to make changes for the future of the campus community. This semester, Kirsten introduced a STEMinists club on campus that will give female students who are STEM (an acronym for Science-Technology-Engineering-Mathematics) majors the opportunity to come together and create a community, share and discuss the difficulties they face as women in their fields of study, and support and mentor each other as they prepare to enter their fields.

Kirsten is passionate, driven, and determined, taking the initiative to bring this group of women together on campus …

What Makes A ‘Good’ Definition: Analyzing Students’ Conceptions Of Mathematical Definitions And The Pedagogy Of Definitions In Math Education, Jesse Aston

Seaver College Research And Scholarly Achievement Symposium

What are the key differences between how students view mathematical definitions and how their professors think they do? In this study we follow students during a guided reconstruction exercise in which they create a definition for, in this case, the convergence of a sequence and we record the process on film. Several theoretical frameworks including RME are then used to analyze student responses and behavior throughout the process and determine the difference in perceptions between student and teacher; these differences inform the best method to present definitions to students.

For Each Mathematical Statement, Only Finitely Many Of Its Generalizations Are Useful: A Formal Proof Of E. Bishop's Idea, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Generalization is one of the main mathematical activities. Some generalizations turn out to be useful for working mathematics, while many other generalizations have so far been not very useful. E. Bishop believed that most fruitless-so-far generalizations are hopeless, that every mathematical statement has only a few useful generalizations. In this paper, we show that, under a natural definition of the notion of useful generalization, Bishop's belief can be proven -- moreover, it turns out that for each mathematical statement, only finitely many of its generalizations are useful.

Conditional Tests On Basins Of Attraction With Finite Fields, Ian H. Dinwoodie

Mathematics and Statistics Faculty Publications and Presentations

An iterative method is given for computing the polynomials that vanish on the basin of attraction of a steady state in discrete polynomial dynamics with finite field coefficients. The algorithm is applied to dynamics of a T cell survival network where it is used to compare transition maps conditional on a basin of attraction.

The Efficacy Of Mathematics Education, Eric Geimer

The STEAM Journal

Evidence supports the notion that mathematics education in the United States is inadequate. There is also evidence that mathematics education deficiencies extend internationally. The worldwide mathematics education deficit appears large enough that improving student performance in this educational problem area could yield great economic benefit. To improve the efficacy of mathematics education, education’s root problems must first be understood. Often supposed educational root problems are considered and contrasted against potential deficiencies of mathematics methodologies and curricula that are based on mainstream educational philosophies. The educational philosophies utilized to form early-grade mathematics methodologies and related curricula are judged to be the …

An Introduction To Fourier Analysis With Applications To Music, Nathan Lenssen, Deanna Needell

Journal of Humanistic Mathematics

In our modern world, we are often faced with problems in which a traditionally analog signal is discretized to enable computer analysis. A fundamental tool used by mathematicians, engineers, and scientists in this context is the discrete Fourier transform (DFT), which allows us to analyze individual frequency components of digital signals. In this paper we develop the discrete Fourier transform from basic calculus, providing the reader with the setup to understand how the DFT can be used to analyze a musical signal for chord structure. By investigating the DFT alongside an application in music processing, we gain an appreciation for …

Mathematics In Rural Mexico: Motivations And Barriers For Adult Women Learners, Janel Wilson Williams, Dr. Jacqualine Taylor Voyles

Journal of Undergraduate Research

Math educators have long realized that women in general have shied away from involvement in the mathematical world. Current research in the mathematics education field studies the motivations and barriers women have to studying math. My research extends this topic to a small village in Mexico: La Estancia. By studying factors that encourage or discourage adult Mexican women from studying math in rural settings, important influences can be identified. The identification of these elements can lead to the development of important intervention programs targeted at minimizing negative influences while capitalizing on and strengthening the existing positive influences.

Statistical Inference For Everyone (Sie), Brian S. Blais

Science and Technology Department Faculty Book Publications

In the field of statistical inference, there are two primary schools of thought. Each has its proponents, but it is generally accepted that on all problems covered in an introductory course, that both approaches are valid and lead to the same numerical values when applied to actual problems. Only one of these approaches is covered in a traditional course, which denies the students access to an entire field of statistical inference. The traditional approach, also called the frequentist or orthodox perspective, leads almost directly to problem above. The other approach, also called Probability Theory as Logic, derives all statistical inference …

Students Ahead Of The Curve In Regional Mathematics Competition, Tia Patsavas

News and Events

No abstract provided.

Cracking The Code On Stem: A People Strategy For Nevada's Economy, Jessica A. Lee, Mark Muro, Jonathan Rothwell, Scott Andes, Siddharth Kulkarni

Brookings Mountain West Publications

Nevada has in place a plausible economic diversification strategy—and it’s beginning to work. Now, the state and its regions need to craft a people strategy. Specifically, the state needs to boost the number of Nevadans who possess at least some postsecondary training in the fields of science, technology, engineering, or math—the so-called “STEM” disciplines (to which some leaders add arts and design to make it “STEAM”).

The moment is urgent—and only heightened by the projected worker needs of Tesla Motors’ planned “gigafactory” for lithium-ion batteries in Storey County.

Even before the recent Tesla commitment, a number of the more high-tech …

Investigating Engineering Practice Is Valuable For Mathematics Learning, Eileen Goold Dr

Conference Papers

While engineering mathematics curricula often prescribe a fixed body of mathematical knowledge, this study takes a different approach; second-year engineering students are additionally required to investigate and document an aspect of mathematics used in engineering practice. A qualitative approach is used to evaluate the impact that students’ investigations have on their mathematics learning and whether this approach creates greater value for students compared to curriculum mathematics learning. This article contains an account of students’ engagement with and their emotional responses to their investigations of professional engineers’mathematics usage.The study illustrates the positive changes in students’ mathematics attitudes arising from their insights …

Inversion Of The Broken Ray Transform, Roman Krylov

Electronic Theses and Dissertations

The broken ray transform (BRT) is an integral of a function along a union of two rays with a common vertex. Consider an X-ray beam scanning an object of interest. The ray undergoes attenuation and scatters in all directions inside the object. This phenomena may happen repeatedly until the photons either exit the object or are completely absorbed. In our work we assume the single scattering approximation when the intensity of the rays scattered more than once is negligibly small. Among all paths that the scattered rays travel inside the object we pick the one that is a union of …

Cracking The Code On Stem: A People Strategy For Nevada's Economy Executive Summary, Jessica A. Lee, Mark Muro, Jonathan Rothwell, Scott Andes, Siddharth Kulkarni

Brookings Mountain West Publications

Nevada has in place a plausible economic diversification strategy—and it’s beginning to work. Now, the state and its regions need to craft a people strategy. Specifically, the state needs to boost the number of Nevadans who possess at least some postsecondary training in the fields of science, technology, engineering, or math—the so-called “STEM” disciplines (to which some leaders add arts and design to make it “STEAM”).

The moment is urgent—and only heightened by the projected worker needs of Tesla Motors’ planned “gigafactory” for lithium-ion batteries in Storey County.

Even before the recent Tesla commitment, a number of the more high-tech …

Hankel Vector Moment Sequences And The Non-Tangential Regularity At Infinity Of Two Variable Pick Functions, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

A Pick function of variables is a holomorphic map from to , where is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series with real numbers that gives an asymptotic expansion on non-tangential approach regions to infinity. In 1921 H. Hamburger characterized which sequences can occur. We give an extension of Hamburger's results to Pick functions of two variables.

Lack Of Quantitative Training Among Early-Career Ecologists: A Survey Of The Problem And Potential Solutions, F. Barraquand, T. G. Ezard, P. Søgaard Jørgensen, Naupaka B. Zimmerman, S. Chamberlain, R. Salguero-Gómez, T. J. Curran, T. Poisot

Biology Faculty Publications

Proficiency in mathematics and statistics is essential to modern ecological science, yet few studies have assessed the level of quantitative training received by ecologists. To do so, we conducted an online survey. The 937 respondents were mostly early-career scientists who studied biology as undergraduates. We found a clear self-perceived lack of quantitative training: 75% were not satisfied with their understanding of mathematical models; 75% felt that the level of mathematics was “too low” in their ecology classes; 90% wanted more mathematics classes for ecologists; and 95% more statistics classes. Respondents thought that 30% of classes in ecology-related degrees should be …

Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño

All HMC Faculty Publications and Research

We prove the existence of a positive solution for a class of nonlin- ear elliptic systems with Neumann boundary conditions. The proof combines extensive use of a priori estimates for elliptic problems with Neumann boundary condition and Krasnoselskii's compression-expansion theorem

How We Got From There To Here: A Story Of Real Analysis, Eugene Boman, Robert Rogers

Milne Open Textbooks

The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.

This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) …

The Mathematics Of Spot It, Marcus Heemstra

The Journal of Undergraduate Research

Spot It™ is a fun, fast paced game for people of all ages. The game appears to have a rich mathematical structure, which we investigated, using combinatorial analysis. We constructed some similar Spot It type games with a different number of cards, and displayed that some games are impossible to complete. In particular, the original Spot It game has eight objects on each card; however, it is impossible to construct a complete game (a game in which every object appears with every other object) in which every card has seven objects. We have investigated the mathematics behind the game.

Re-Analysis Of The World Data On The Emc Effect And Extrapolation To Nuclear Matter, Stacy E. Karthas

Honors Theses and Capstones

The EMC effect has been investigated by physicists over the past 30 years since it was discovered at CERN. This effect shows that the internal nucleon structure varies when in the nuclear medium. Data from SLAC E139 and JLab E03 103 were studied with the Coulomb correction and directly compared. The Coulomb distortion had a greater impact on the JLab data due to slightly lower energies and different kinematics. The EMC ratio was found to have a dependence on a kinematic variable, which was removed when Coulomb corrections were applied. The methodology was developed to apply Coulomb corrections, which had …

The Analytic Hierarchy Process: A Mathematical Model For Decision Making Problems, Giang Huong Nguyen

Senior Independent Study Theses

The ability to make the right decision is an asset in many areas and lines of profession including social work, business, national economics, and international security. However, decision makers often have difficulty choosing the best option since they might not have a full understanding of their preferences, or lack a systematic approach to solve the decision making problems at hand. The Analytic Hierarchy Process (AHP) provides a mathematical model that helps the decision makers arrive at the most logical choice, based on their preferences. We investigate the theory of positive, reciprocal matrices, which provides the theoretical justification of the method …

Markov Chain Theory With Applications To Baseball, Cal D. Thomay

Senior Independent Study Theses

The applications of Markov chains span a wide range of fields to which models have been designed and implemented to simulate random processes. Markov chains are stochastic processes that are characterized by their memoryless property, where the probability of the process being in the next state of the system depends only on the current state and not on any of the previous states. This property is known as the Markov property. This thesis paper will first introduce the theory of Markov chains, along with explaining two types of Markov chains that will be beneficial in creating a model for analyzing …

Keeping Your Options Open: An Introduction To Pricing Options, Ryan F. Snyder

Senior Independent Study Theses

An option is a contract which gives the holder of the option the right, but not the obligation, to buy or sell a given security at a given price, which is called the strike price. For example, suppose Yahoo stock is currently trading at $10 per share. A person could buy an option that gives him or her the ability to purchase shares of Yahoo stock for $12 in one year. If the price of Yahoo stock is greater than $12 in one year, the holder of the option will make money. However, he or she will not use the …

Bad Boundary Behavior In Star Invariant Subspaces I, William T. Ross, Andreas Hartmann

Department of Math & Statistics Faculty Publications

We discuss the boundary behavior of functions in star invariant subspaces (BH²)¹, where B is a Blaschke product. Extending some results of Ahern and Clark, we are particularly interested in the growth rates of functions at points of the spectrum of B where B does not admit a derivative in the sense of Carathéodory.

Education And Women Mathematicians In The Middle East, Syeda Tooba Zahra

Legacy Theses & Dissertations (2009 - 2024)

Middle Eastern women have made important contributions in the field of Mathematics. The purpose of this thesis is to understand the educational background for women in the Middle East and look at the life and contribution of a sample of women mathematicians in representative Middle Eastern countries such as Turkey, Lebanon, Iran, and Pakistan.

Design Considerations For Visually-Aided Discussion Prompts: Emphasizing Mathematical Reasoning In Teacher Education, Anne Marie S. Marshall, Kadian M. Callahan

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

The availability and familiarity of online discussion tools create new instructional options that teacher educators can use to foster prospective teachers’ understanding of mathematics. In particular, online discussion blogs provide an avenue through which teacher educators can press prospective teachers to explore mathematical concepts and share their mathematical reasoning with peers. Furthermore, by incorporating visual stimulations as a design component of these discussion blogs, prospective teachers can make sense of and respond to others’ ideas about mathematical concepts with greater clarity. This paper shares preliminary findings of a research study that examined the extent to which the design of a …

A Twisted Dimer Model For Knots, Heather M. Russell, Moshe Cohen, Oliver Dasbach

Department of Math & Statistics Faculty Publications

We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group.

A Reduced Set Of Moves On One-Vertex Ribbon Graphs Coming From Links, Heather M. Russell, Susan Abernathy, Cody Armond, Moshe Cohen, Oliver T. Dasbach, Hannah Manuel, Chris Penn, Neal W. Stoltzfus

Department of Math & Statistics Faculty Publications

Every link in R³ can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.

A Bayesian Model Of Fertility Decisions In Relationship To Female Labor Force Participation, Rebecca C. Wardrop

Senior Independent Study Theses

Due to the increasing number of women in the labor force, opportunity costs associated with labor force participation are becoming an important factor in fertility decisions. Further, these decisions are assumed to be dynamic as the opportunity costs change as a woman progresses through her career. A Bayesian statistical model , which allows the distribution of the likelihood of having children to be updated as information is gathered, lends itself to the dynamicity of the decision-making process. A generalized model for fertility decisions in terms of labor force participation is created. I also discuss potentials for implementation and furthering the …

Generalist And Specialist Pollination Syndromes: When Are They Favoured? A Theoretical Approach To Predict The Conditions Under Which A Generalist Or Specialist Pollination Syndrome Is Favoured., Tyler L. Poppenwimer

Senior Independent Study Theses

No abstract provided.

Collected Papers, Vol. V, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.