Mathematics Commons^™

A Multi-Frequency Inverse Source Problem For The Helmholtz Equation, Sebastian Ignacio Acosta

Theses and Dissertations

The inverse source problem for the Helmholtz equation is studied. An unknown source is to be identified from the knowledge of its radiated wave. The focus is placed on the effect that multi-frequency data has on establishing uniqueness. In particular, we prove that data obtained from finitely many frequencies is not sufficient. On the other hand, if the frequency varies within an open interval of the positive real line, then the source is determined uniquely. An algorithm is based on an incomplete Fourier transform of the measured data and we establish an error estimate under certain regularity assumptions on the …

Solvability Of A Quadratic Hammerstein Integral Equation In The Class Of Functions Having Limits At Infinity, Ravi P. Agarwal, Józef Banas̈, Kamil Banas̈, Donal O'Regan

Mathematics and System Engineering Faculty Publications

The goal of the paper is to prove that a quadratic Hammerstein integral equation has solutions in the class of real functions denned, bounded, continuous on the real half-axis and having limits at infinity. The main tools used in our investigations are the technique of measures of noncompactness and the Darbo fixed point theorem. We provide an example illustrating our theory. © 2011 Rocky Mountain Mathematics Consortium.

Three-Dimensional Galois Representations And A Conjecture Of Ash, Doud, And Pollack, Vinh Xuan Dang

Theses and Dissertations

In the 1970s and 1980s, Jean-Pierre Serre formulated a conjecture connecting two-dimensional Galois representations and modular forms. The conjecture came to be known as Serre's modularity conjecture. It was recently proved by Khare and Wintenberger in 2008. Serre's conjecture has various important consequences in number theory. Most notably, it played a key role in the proof of Fermat's last theorem. A natural question is, what is the analogue of Serre's conjecture for higher dimensional Galois representations? In 2002, Ash, Doud and Pollack formulated a precise statement for a higher dimensional analogue of Serre's conjecture. They also provided numerous computational examples …

Asymptotic Distributions Of The Overshoot And Undershoots For The Lévy Insurance Risk Process In The Cramér And Convolution Equivalent Cases, Philip S. Griffin, Ross A. Maller, Kees Van Schaik

Mathematics - All Scholarship

Recent models of the insurance risk process use a Levy process to generalise the traditional Cramer-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Levy process which drifts to -infinity and satis es a Cramer or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramer case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding …

Non-Commutative Desingularization Of Determinantal Varieties, Ii, Ragar-Olaf Buchweitz, Graham J. Leuschke, Michel Van Den Bergh

Mathematics - All Scholarship

In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the existence of non-commutative resolutions in the general case in a manner which is still characteristic free. The explicit description of the resolution by generators and relations is deferred to a later paper. As an application of our results we prove that there is a fully faithful …

Common Edge-Unzippings For Tetrahedra, Joseph O'Rourke

Computer Science: Faculty Publications

It is shown that there are examples of distinct polyhedra, each with a Hamiltonian path of edges, which when cut, unfolds the surfaces to a common net. In particular, it is established for infinite classes of triples of tetrahedra.

A New Integrable Two-Component System With Cubic Nonlinearity, Junfeng Song, Changzheng Qu, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, a new integrable two-component system, mt=[m(uxvx−uv+uvx−uxv)]x,nt=[n(uxvx−uv+uvx−uxv)]x, where m=u−uxx and n=v−vxx, is proposed. Our system is a generalized version of the integrable system mt=[m(u2x−u2)]x, which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained.

Lesson’S Learned: A Journey In Computational Science, Ryan Botts, Lori Carter

ACMS Conference Proceedings 2011

Inspired by work on building a computational science program and student questions about modeling, we aim to discuss some of our experiences with computational science. We will first clarify what computational science is, why it is a legitimate science, why it is worth our students’ time and what makes it a challenging field. We will also discuss how computer scientists, mathematicians and laboratory scientists each have something different to contribute to the field.

A Bayesian Secondary Analysis In An Asthma Study, Samuel P. Wilcock, Vernon M. Chinchilli, Stephen P. Peters

ACMS Conference Proceedings 2011

A recent study published in the New England Journal of Medicine by the Asthma Clinical Research Network (ACRN) compared three different treatments for their effectiveness in treating adults with uncontrolled asthma. This paper will describe the study design and its results, then detail the beginnings of a secondary analysis using Bayesian methods to estimate the parameters of interest. The methods will be explained, and the preliminary estimates given and contextualized. The paper will conclude with a discussion of the next steps and the goals for further analysis of the data in this study.

What We Can Learn From Process Theology: Integrating Faith And Mathematics, Josh Wilkerson

ACMS Conference Proceedings 2011

In the inaugural issue of The Journal of the Association of Christians in the Mathematical Sciences, James Bradley, the founding editor, suggests fourteen areas that need to be addressed by Christian mathematicians who are series about integrating their faith and their work. One of those areas is the topic of this paper. Bradly frames the question: “Some thinkers (perhaps influenced by process theology) have asserted the idea that God’s creation is not a finished work but that he creates new mathematical objects through mathematicians. Is this idea theologically sound? Is it helpful for our understanding of mathematics?” I copy …

Google And The Mathematics Of Web Search, Michael Rempe

ACMS Conference Proceedings 2011

This article examines the algorithms used by Google to rank search engine results, called PageRank.

Calculus Communication Circle, Judith Palagallo

ACMS Conference Proceedings 2011

Calculus Communication Circle is a network for the professional development of Advanced Placement Calculus teachers. In Northeast Ohio the Circle provides a forum where teachers meet to share ideas about mathematics and the teaching of calculus. This article describes the creation of the Circle and the progress it has made over its three year existence.

Using Original Historical Mathematics Texts In The Classroom, Maria Zack

ACMS Conference Proceedings 2011

Incorporating information from the history of mathematics into undergraduate mathematics courses is an effective way to help students make connections between the mathematics they are learning and their general education courses. It also helps students to see mathematics as living subject that has changed over time. One way to introduce historical topics into a variety of classes is through the use of original mathematical texts. This paper describes how non-experts can make use of original texts and provides sources for identifying candidate texts.

The Mathematics Of Cubic Sudoku, Nicholas Zoller

ACMS Conference Proceedings 2011

In the last decade the Sudoku puzzle has fixed itself in America’s puzzle consciousness. Sudoku puzzles share space with crossword puzzles and word finds in newspaper puzzle sections, and several books have been written for the Sudoku playing community. Mathematicians are among the most dedicated Sudoku players. Although some are content with simply solving puzzle after puzzle, others have used tools from combinatorics and algebra to study its important properties.

We investigate a variant of Sudoku called Cubic Sudoku, as well as Cubic Sudoku’s simpler relative, Cubic Shidoku. We successfully count the number of Cubic Shidoku puzzles in two different …

Math History Study Abroad Program: Learning Math History In A Cultural Context, Donna Pierce

ACMS Conference Proceedings 2011

In January 2011 fifteen Whitworth University mathematics students and I, their professor, traveled through Europe to study the history of mathematics. The goal was to gain an understanding of how mathematical ideas have developed over time; how social, cultural and historical factors have influenced the development of mathematics and conversely, how mathematics contributed to society and human culture. Over a course of three weeks we traveled to three countries and over a half dozen cities, viewing the tools, papers and workbooks or these mathematicians, seeing their engineering and artistic creations, and learning from local experts as they guided us through …

Thinking Philosophically About Mathematics, Robert L. Brabenec

ACMS Conference Proceedings 2011

In my early years as a teacher of mathematics, the history of mathematics was seldom mentioned in the classroom. It was viewed as an unworthy topic that would detract from the presentation of mathematics itself. This opinion has dramatically changed over the years, and the history of mathematics is now embraced and used by many mathematicians in their teaching and even research. We might choose to ask a related question. How much philosophy is necessary or helpful for a mathematics teacher to know, and to use in his or her teaching? We see a growing interest in the philosophy of …

The Need For A Graphics Programming Course, Nathan Gossett

ACMS Conference Proceedings 2011

A discussion of the benefits of offering a course on programming Computer Graphics in an undergraduate Computer Science curriculum. A sample course outline is provided, as well as a discussion of ways to conduct lectures, labs and a list of suggested assignments. A discussion of “dos and don’t s will also be presented, including a list of required prerequisite courses and skills that students would need in order for the course to be a success.

Bringing Undergraduate Research Into The Classroom, Stephen Lovett

ACMS Conference Proceedings 2011

Mathematics graduate programs and companies that employ math majors often want to ascertain an applicant’s potential for research. However, in many undergraduate courses, assessments consist only of regular exercise sets, quizzes, and in-class tests. Without doing a senior research thesis or landing an official REUs, students do not regularly gain experience in or an appreciation for research. Courses in the humanities regularly require students to write in the discipline, progressively preparing them methodologically for “writing in the field.” This begs the question: could math departments do a little more to prepare our students to use mathematics beyond college?

In this …

Real Simulations And Simulated Reality, Wayne Iba

ACMS Conference Proceedings 2011

Movies such as The Matrix have stimulated popular interest in “brain in a vat” scenarios. Amidst the traditional questions of the mind, we tend to overlook an integral enabling component—the world simulation—which merits consideration in its own right. When facing the simulations in these imagined scenarios, we struggle with conceptual muddles regarding what is real and not. In this paper, I argue that simulated worlds are every bit as real as the one we inhabit. This turns out to be important when considering the possibility, as suggested by Nick Bostrom, that the world we experience as “real” is actually a …

Pk Mathematics, Jeremy Case

ACMS Conference Proceedings 2011

An examination of the historical development of mathematics and how mathematical history has changed by looking at mathematicians who were also PKs (Preacher’s Kids).