Physical Sciences and Mathematics Commons^™

Counting Coffee Combinations: An Example Of The Fundamental Principle Of Counting, David R. Duncan, Bonnie H. Litwiller

Faculty Publications

Teachers in courses which involve probability and statistics are always looking for situations in which the Fundamental Principle of Counting (FPC) can be exemplified. We shall present an example involving a coffee vending machine at a highway rest stop.

Kernels Of Directed Graph Laplacians, John S. Caughman Iv, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Let G denote a directed graph with adjacency matrix Q and in- degree matrix D. We consider the Kirchhoff matrix L = D − Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper …

The Convergence Of V-Cycle Multigrid Algorithms For Axisymmetric Laplace And Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity …

Implementing Standards-Based Mathematics Curricula Into The Teaching Of Mathematics And Education Courses For Prospective Teachers, Shannon Driskell, J. Herrelko

Shannon O.S. Driskell

Driskell, S. (Co-PI), & Herrelko, J. (Co-PI), University of Dayton Learning Teaching Center Innovation Grants, $7,500, Year: 2006 - 2007.

The Tablet Pc For Faculty: A Pilot Project, Rob Weitz, Bert Wachsmuth, Danielle Mirliss

Bert Wachsmuth

This paper describes a pilot project with the purpose of evaluating the usefulness of tablet PCs for university professors. The focus is on the value of tablets primarily with respect to teaching and learning (and not for research or administrative work). Sixty-four professors, distributed across the various schools of a university, were provided with tablet PCs and were trained in their use.

Multivariate Expansion Associated With Sheffer-Type Polynomials And Operators, Tian-Xiao He, Leetsch Hsu, Peter Shiue

Tian-Xiao He

With the aid of multivariate Sheffer-type polynomials and differential operators, this paper provides two kinds of general expansion formulas, called respectively the first expansion formula and the second expansion formula, that yield a constructive solution to the problem of the expansion of A(ˆt)f([g(t)) (a composition of any given formal power series) and the expansion of the multivariate entire functions in terms of multivariate Sheffer-type polynomials, which may be considered an application of the first expansion formula and the Sheffer-type operators. The results are applicable to combinatorics and special function theory.

Construction Of Rational Points On Elliptic Curves Over Finite Fields, Andrew Shallue, Christiaan E. Van De Woestijne

Andrew Shallue

A New Type Of Orthogonality In Banach Spaces, Abeer Hasan

Abeer Hasan

On The Convergence Of The Summation Formulas Constructed By Using A Symbolic Operator Approach, Tian-Xiao He, Leetsch C. Hsu, Peter J.-S. Shiue

Tian-Xiao He

This paper deals with the convergence of the summation of power series of the form Σ_{a ≤ k ≤ b}f(k)x^k, where 0 ≤ a ≤ b < ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) a differentiable function defined on [a, b). Here, the summation is found by using the symbolic operator approach shown in [1]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Several examples such as the generalized Euler's transformation series will also be given. In addition, we will compare the convergence of the given series transforms.

Absolute Equal Distribution Of The Eigenvalues Of Discrete Sturm-Liouville Problems, William F. Trench

William F. Trench

No abstract provided.

Numerical Approximation To Ζ(2n+1), Tian-Xiao He, Michael J. Dancs

Tian-Xiao He

In this short paper, we establish a family of rapidly converging series expansions ζ(2n +1) by discretizing an integral representation given by Cvijovic and Klinowski [3] in Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math. 142 (2002) 435–439. The proofs are elementary, using basic properties of the Bernoulli polynomials.

Functional Perturbations Of Nonoscillatory Second Order Difference Equations, William F. Trench

William F. Trench

No abstract provided.

On The Generalized Möbius Inversion Formulas, Tian-Xiao He, Peter J. S. Shiue3, Leetsch C. Hsu

Tian-Xiao He

We provide a wide class of M¨obius inversion formulas in terms of the generalized M¨obius functions and its application to the setting of the Selberg multiplicative functions.

An Euler-Type Formula For Ζ(2k +1), Tian-Xiao He, Michael J. Dancs

Tian-Xiao He

In this short paper, we give several new formulas for ζ(n) when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler’s classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.

Universal Series By Trigonometric System In Weighted Spaces, Sergo Armenak Episkoposian (Yepiskoposyan)

Sergo Armenak Episkoposian (Yepiskoposyan)

No abstract provided.

An Euler-Type Formula For Zeta (2k+1), Tian-Xiao He

Tian-Xiao He

No abstract provided.

Combinatorial Stochastic Processes , Jim Pitman

Jim Pitman

This is a set of lecture notes for a course given at the St. Flour summer school in July 2002. The theme of the course is the study of various combinatorial models of random partitions and random trees, and the asymptotics of these models related to continuous parameter stochastic processes. Following is a list of the main topics treated: models for random combinatorial structures, such as trees, forests, permutations, mappings, and partitions; probabilistic interpretations of various combinatorial notions e.g. Bell polynomials, Stirling numbers, polynomials of binomial type, Lagrange inversion; Kingman's theory of exchangeable random partitions and random discrete distributions; connections …

Almost Split Morphisms, Preprojective Algebras And Multiplication Maps Of Maximal Rank, Steven P. Diaz, Mark Kleiner

Mathematics - All Scholarship

With a grading previously introduced by the second-named author, the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the multiplication maps in the commutative polynomial ring. The result follows from a more general theorem about the maximal rank property of a minimal almost split morphism, which also yields a quadratic inequality for the dimensions of indecomposable modules involved.

Finite-Dimensional Algebras With Smallest Resolutions Of Simple Modules, Shashidhar Jagadeeshan, Mark Kleiner

Mathematics - All Scholarship

Let lamda be an associative ring with identity and with the Jacobson radical r, let mod lamda be the category of finitely generated left lamda-modules, and let lamda^op be the opposite ring of lamda. All modules are left unital modules, and if X is a module then pd X is the projective dimension of X. If lamda is left artinian and M in mod labmda, we denote by P(M) a projective cover of M.

Coordinate Systems And Bounded Isomorphisms, David R. Pitts, David R. Pitts

Department of Mathematics: Faculty Publications

For a Banach D-bimoduleMover an abelian unital C*-algebraD, we define E1(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual M#. Equip E1(M) with the weak*-topology. We develop general properties of E1(M). It is properly viewed as a coordinate system for M when M C, where C is a unital C*-algebra containing D as a regular MASA with the extension property; moreover, E1(C) coincides with Kumjian’s twist in the context of C*-diagonals. We identify the C*-envelope of a subalgebra A of a C*-diagonal when D A C. For triangular subalgebras, each containing …

Foundations Of Generalized Cwatsets, Jesse Beder

Mathematical Sciences Technical Reports (MSTR)

We present a new, abstract definition for a generalized cwatset that produces notions of subcwatset and quotient cwatset that behave naturally. We use small cancellation theory to prove a result analogous to the statement that every group is isomorphic to some permutation group.

Change Bell Ringing And Mathematics: A High School Student Excursion Into Graph Theory And Group Theory, Kristi Carlson

Honors Theses

This unit was created as a way to introduce higher level mathematics concepts to advanced high school students. All five of the National Council of Teachers of Mathematics Process Standards are found in this unit. For most of the unit, students work within small groups

Existence And Uniqueness Of V-Asymptotic Expansions And Colombeau’S Generalized Numbers, Todor D. Todorov

Mathematics

We define a type of generalized asymptotic series called v-asymptotic. We show that every function with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with moderate growth has a unique v-asymptotic expansion. As an application to random matrix theory we calculate the coefficients and establish the uniqueness of the v-asymptotic expansion of an integral with a large parameter. As another application (with significance in the non-linear theory of generalized functions) we show that every Colombeau’s generalized number has a v-asymptotic expansion. A similar …

Smooth And Analytic Normal And Canonical Forms For Strict Feedforward Systems, Issa Amadou Tall, Witold Respondek

Miscellaneous (presentations, translations, interviews, etc)

Recently we proved that any smooth (resp. analytic) strict feedforward system can be brought into its normal form via a smooth (resp. analytic) feedback transformation. This will allow us to identify a subclass of strict feedforward systems, called systems in special strict feedforward form, shortly (SSFF), possessing a canonical form which is an analytic counterpart of the formal canonical form. For (SSFF)-systems, the step-by-step normalization procedure of Kang and Krener leads to smooth (resp. convergent analytic) normalizing feedback transformations. We illustrate the class of (SSFF)-systems by a model of an inverted pendulum on a cart.

A Mathematical Journey Through The Land Of The Maya, Philip Scalisi, Paul Fairbanks

Bridgewater Review

No abstract provided.

On The Existence Of Multiple Periodic Solutions For The Vector P-Laplacian Via Critical Point Theory, Haishen Lu, Donal O'Regan, Ravi P. Agarwal

Mathematics and System Engineering Faculty Publications

We study the vector p-Laplacian (∗){−(|u′|p−2u′)′=∇F(t,u)a.e.t∈[0,T],u(0)=u(T),u′(0)=u′(T),1

The Relationship Between The Number Of Shots And The Quality Of Gamma Knife Radiosurgeries, D Cheek, Allen G. Holder, M Fuss, B Salter

Mathematics Faculty Research

Radiosurgery is a non-invasive alternative to brain surgery that uses a single focused application of high radiation to destroy intracerebral target tissues. A Gamma Knife delivers such treatments by using 201 cylindrically collimated cobalt-60 sources that are arranged in a hemispherical pattern and aimed to a common focal point. The accumulation of radiation at the focal point, called a \shot" due to the spherical nature of the dose distribution, is used to ablate (or destroy) target tissue in the brain. If the target is small and spherical, it is easily treated by choosing one of four available collimators (4, 8, …

Five Years At The Magazine, Frank A. Farris

Mathematics and Computer Science

What do the editors of MAA journals do? What is so different about editing expository mathematics? After my five years as editor of Mathematics Magazine, I have strong opinions about these matters. The goal of most mathematics journals is to print the very latest results from the forefront of research. The goal of Mathematics Magazine is to remind us all why we loved mathematics in the first place, with stimulating articles and notes accessible to advanced undergraduates.

Disc Separation Of The Schur Complement Of Diagonally Dominant Matrices And Determinantal Bounds, Jianzhou Liu, Fuzhen Zhang

Mathematics Faculty Articles

We consider the Gersgorin disc separation from the origin for (doubly) diagonally dominant matrices and their Schur complements, showing that the separation of the Schur complement of a (doubly) diagonally dominant matrix is greater than that of the original grand matrix. As application we discuss the localization of eigenvalues and present some upper and lower bounds for the determinant of diagonally dominant matrices.

Multivalued Logic, Neutrosophy And Schrodinger Equation, Florentin Smarandache, Victor Christianto

Branch Mathematics and Statistics Faculty and Staff Publications

This book was intended to discuss some paradoxes in Quantum Mechanics from the viewpoint of Multi-Valued-logic pioneered by Lukasiewicz, and a recent concept Neutrosophic Logic. Essentially, this new concept offers new insights on the idea of ‘identity’, which too often it has been accepted as given. Neutrosophy itself was developed in attempt to generalize Fuzzy-Logic introduced by L. Zadeh. While some aspects of theoretical foundations of logic are discussed, this book is not intended solely for pure mathematicians, but instead for physicists in the hope that some of ideas presented herein will be found useful. The book is motivated by …