Mathematics Commons^™

Polynomial Generalizations Of Two-Variable Ramanujan Type Identities, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

We provide finite analogs of a pair of two-variable q-series identities from Ramanujan's lost notebook and a companion identity.

Lorentzian Sasaki-Einstein Metrics On Connected Sums Of S² X S³, Ralph R. Gomez

Mathematics & Statistics Faculty Works

We settle completely an open problem formulated by Boyer and Galicki in [5] which asks whether or not # kS² x S³ are negative Sasakian manifolds for all k. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki eta-Einstein and Lorentzian Sasaki-Einsteinmetrics on these five-manifolds for all k and moreover all of these can be realized as links of isolated hypersurface singularities defined by weighted homogenous polynomials. The key step is to construct infinitely many hypersurfaces in weighted projective space that contain branch divisors Delta = Sigma(i) (1 - 1/m(i)) D-i such that the D-i …

What Do We Mean By Mathematical Proof?, Todd Cadwalladerolsker

Journal of Humanistic Mathematics

Mathematical proof lies at the foundations of mathematics, but there are several notions of what mathematical proof is, or might be. In fact, the idea of mathematical proof continues to evolve. In this article, I review the body of literature that argues that there are at least two widely held meanings of proof, and that the standards of proof are negotiated and agreed upon by the members of mathematical communities. The formal view of proof is contrasted with the view of proofs as arguments intended to convince a reader. These views are examined in the context of the various roles …

Aesthetic Considerations In Mathematics, Nathalie Sinclair

Journal of Humanistic Mathematics

Drawing on some of the principles of humanistic mathematics first outlined by Alvin White, this paper seeks to examine the way in which value judgments are implicated in the growth of the mathematics discipline. After a short overview of some of the roles ascribed to the mathematical aesthetic historically, I turn to more contemporary positioning of the aesthetic in order to develop a framework that offers insight into the particular values, assumptions and desires that constrain what is done in mathematics, how it is done and why. My goal, at least in part, is to bring together under one umbrella …

Welcome To The Journal Of Humanistic Mathematics, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Pattern Formation In Oscillatory Systems, Hui Wu

Dissertations

Synchronization is a kind of ordinary phenomenon in nature, the study of it includes many mathematical branches. Phase space is one of the most powerful inventions of modern mathematical science. There are two variables, the position and velocity, that can describe the 2-dimensional phase space system. For example, the state of pendulum may be specified by its position and its velocity, so its phase space is 2-dimensional. The state of the system at a given time has a unique corresponding point in the phase space. In order to describe the motion of an oscillator, we can talk about its motion …

The (1,2)-Step Competition Graph Of A Tournament, Kim A. S. Factor, Sarah Merz

Mathematics, Statistics and Computer Science Faculty Research and Publications

The competition graph of a digraph, introduced by Cohen in 1968, has been extensively studied. More recently, in 2000, Cho, Kim, and Nam defined the m-step competition graph. In this paper, we offer another generalization of the competition graph. We define the (1,2)-step competition graph of a digraph D, denoted C_1,2(D), as the graph on V(D) where {x,y}∈E(C_1,2(D)) if and only if there exists a vertex z≠x,y, such that either d_D−y( …

Mathematical Modeling And Computation Of Channel Flow Over Discrete Structures, Rolando J. Olivares, Daniel N. Riahi

School of Mathematical & Statistical Sciences Faculty Publications

In this paper mathematical modeling and computation of channel flow over small discrete structures are carried out under some reasonable conditions. A mathematical model for such a flow problem, which is based on a relevant system of partial differential equations and Fourier analysis, is studied using perturbation and nonlinear stability methods, and the resulting flow solutions over two types of discrete structures are computed under both stable and unstable conditions. It was found, in particular, that for a subcritical domain with the Reynolds number R slightly less than its critical value Rc, which is defined as the value below which …

Fundamental Transversals On The Complexes Of Polyhedra, Joy D'Andrea

USF Tampa Graduate Theses and Dissertations

We present a formal description of `Face Fundamental Transversals' on the faces of the Complexes of polyhedra (meaning threedimensional polytopes). A Complex of a polyhedron is the collection of the vertex points of the polyhedron, line segment edges and polygonal faces of the polyhedron. We will prove that for the faces of any 3-dimensional complex of a polyhedron under face adjacency relations, that a `Face Fundamental Transversal' exists, and it is a union of the connected orbits of faces that are intersected exactly once. While exploring the problem of finding a face fundamental transversal, we have found a partial result …

B ̈Acklund Transformations For Darboux Integrable Equations, Ian M. Anderson

Mathematics and Statistics Faculty Presentations

Lie symmetry reduction is typically viewed as an integration method for differential systems of finite type, that is, systems of ordinary differential equations.

In this talk I shall present two new, recent applications of Lie symmetry reduction to the study of partial differential equations.

The first gives a remarkably simple method for constructing B ̈acklund transformations.

The second also gives a simple, very general method for constructing Darboux integrable equations.

The combination of these result in a new method for constructing B ̈acklund transformations for Darboux integrable equations.

The utility of this group theoretic approach will be illustrated by a …

On The Classification Of Warped Product Einstein Metrics, Chenxu He, Peter Petersen, William Wylie

Mathematics - All Scholarship

In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the m-Quasi Einstein equation, but we will also call it the (lambda,n+m)-Einstein equation. In this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim to allow the base to have non-empty boundary. This is a natural case to consider since a manifold without boundary often occurs as a warped product over a manifold with boundary, and in this case we get some …

Evolution Of Ligand Specificity In Vertebrate Corticosteroid Receptors, Adam S. Arterbery, Daniel J. Fergus, Elizabeth A. Fogarty, John Mayberry, David L. Deitcher, W. L. Kraus, Andrew H. Bass

College of the Pacific Faculty Articles

Background. Corticosteroid receptors include mineralocorticoid (MR) and glucocorticoid (GR) receptors. Teleost fishes have a single MR and duplicate GRs that show variable sensitivities to mineralocorticoids and glucocorticoids. How these receptors compare functionally to tetrapod MR and GR, and the evolutionary significance of maintaining two GRs, remains unclear. Results. We used up to seven steroids (including aldosterone, cortisol and 11-deoxycorticosterone [DOC]) to compare the ligand specificity of the ligand binding domains of corticosteroid receptors between a mammal (Mus musculus) and the midshipman fish (Porichthys notatus), a teleost model for steroid regulation of neural and behavioral plasticity. Variation in mineralocorticoid sensitivity was …

First Radar Observations In The Vicinity Of The Plasmapause Of Pulsed Ionospheric Flows Generated By Bursty Bulk Flows, N. A. Frissell, J. B. H. Baker, J. M. Ruohoniemi, L. B. N. Clausen, Z. C. Kale, I. J. Rae, L. Kepko, K. Oksavik, R. A. Greenwald, M. L. West

Department of Mathematics Faculty Scholarship and Creative Works

[1] Recent expansion of the SuperDARN network to mid-latitudes and the addition of a new high-time resolution mode provides new opportunities to observe mid-latitude ultra-low frequency waves and other ionospheric sub-auroral features at high temporal resolution. On 22 February 2008, the Blackstone SuperDARN radar and THEMIS ground magnetometers simultaneously observed substorm Pi2 pulsations. Similarities in measurements from the Blackstone radar and a magnetometer at Remus suggest a common generating mechanism. Cross-phase analysis of magnetometer data places these measurements at the ionospheric projection of the plasmapause, while fine spatial and temporal details of the radar data show evidence of field line …

Which Chessboards Have A Closed Knight's Tour Within The Rectangular Prism?, Joseph Demaio, Mathew Bindia

Faculty Articles

A closed knight's tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the m x n rectangular chessboards that admit a closed knight's tour. In honor of the upcoming twentieth anniversary of the publication of Schwenk's paper, this article extends his result by classifying the i x j x k rectangular prisms that admit a closed knight's tour.

Convex Polyhedra Realizing Given Face Areas, Joseph O'Rourke

Computer Science: Faculty Publications

Given n ≥ 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.

On The Erdos-Straus Conjecture, Eugen J. Ionascu

Faculty Bibliography

Paul Erdos conjectured that for every n ∈ N, n ≥ 2, there exist a, b, c natural numbers, not necessarily distinct, so that 4 n = 1 a + 1 b + 1 c (see [3]). In this paper we prove an extension of Mordell’s theorem and formulate a conjecture which is stronger than Erdos’ conjecture.

Regular Tetrahedra Whose Vertices Have Integer Coordinates, Eugen J. Ionascu

Faculty Bibliography

In this paper we introduce theoretical arguments for constructing a procedure that allows one to find the number of all regular tetrahedra that have coordinates in the set {0, 1, ..., n}. The terms of this sequence are twice the values of the sequence A103158 in the Online Encyclopedia of Integer Sequences [16]. These results lead to the consideration of an infinite graph having fractal nature which is tightly connected to the set of orthogonal 3-by-3 matrices with rational coefficients. The vertices of this graph are the primitive integer solutions of the Diophantine equation a 2 + b 2 + …

Making Algebra More Accessible: How Steep Can It Be For Teachers?, Diana Cheng, Polina Sabinin

Mathematics Faculty Publications

Teacher educators need to support middle grades teachers in developing mathematical knowledge for teaching algebraic concepts. In particular, teachers should become familiar with possible introductions and sequencing to the concept of slope, and common middle school students’ limited conceptions about measuring the steepness of an incline. Steepness can be expressed directly in terms of an angle or indirectly as a slope. Encouraging middle school students to find a measure of steepness using a ratio may help support students’ transition to multiplicative thinking. This mixed – methods study explores middle school students’ responses in solving a comparison problem involving the steepness …

Extending A Mips Simulator For Fpga Support, Pete Sanderson

Mathematics Faculty Scholarship

This project enriches computer organization education by providing another avenue for execution of MIPS code. The initial and existing avenue is the MARS simulator for MIPS code, a well-established open source product for educational environments, in use at universities and colleges worldwide (http://www.cs.missouristate.edu/MARS). The new avenue is a special adaptation of MIPS circuitry on a Field Programmable Gate Array (FPGA) platform. FPGAs such as the Altera DE2 are very attractive in an educational environment because of their flexibility, re-use, and low cost. The FPGA MIPS circuit includes VGA graphical output for real-time display of the MIPS machine state, including registers, …

Balanced Pod For Model Reduction Of Linear Pde Systems: Convergence Theory, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

We consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems. The algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations. The algorithm is related to the proper orthogonal decomposition, and it was first proposed for systems of ordinary differential equations by Rowley (Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3), 997-1013, 2005). For the convergence analysis, we consider the algorithm in terms of the Hankel operator of the system, rather than the product of the system Gramians as originally proposed by Rowley. For …