Physical Sciences and Mathematics Commons^™

Problems To Discover And To Boost Mathematical Talent In Early Grades: A Challenging Situations Approach, Viktor Freiman

The Mathematics Enthusiast

Several studies of mathematical giftedness conducted in the past two decades reveal the importance of creation of learning and teaching environment favourable to the identification and nurturing mathematically talented students. Based on psychological, methodological and didactical models created by Krutetskii (1976), Shchedrovtiskii (1968), Brousseau (1997) and Sierpinska (1994), we have developed our challenging situation approach. During 7 years of field study in the elementary K-6 classroom, we collected sufficient amount of data that demonstrate how these challenging situations help to discover and to boost mathematical talent in very young children keeping and increasing their interest towards more advanced mathematics curriculum. …

Building Blocks Problem Related To Harmonic Series, Yutaka Nishiyama

The Mathematics Enthusiast

In this discussion I give an explanation of the divergence and convergence of infinite series through the building blocks problem and at the same time I touch on the fact that mathematics is not just about manipulating complicated numerical formulas but also a field in which logical ways of thought are learnt. I emphasize that in order to overcome the aversion of university students to mathematics, teachers must pour their energies into developing study materials taken from topics relevant to the students.

Modeling Interdisciplinary Activities Involving Mathematics And Philosophy, Steffen M. Iversen

The Mathematics Enthusiast

In this paper a didactical model is presented. The goal of the model is to work as a didactical tool, or conceptual frame, for developing, carrying through and evaluating interdisciplinary activities involving the subject of mathematics and philosophy in the high schools. Through the terms of Horizontal Intertwining, Vertical Structuring and Horizontal Propagation the model consists of three phases, each considering different aspects of the nature of interdisciplinary activities. The theoretical modelling is inspired by work which focuses on the students abilities to concept formation in expanded domains (Michelsen, 2001, 2005a, 2005b). Furthermore the theoretical description rest on a series …

A C-Symplectic Free S1-Manifold With Contractible Orbits And Cat = 1/2 Dim, Christopher Allday, John Oprea

Mathematics and Statistics Faculty Publications

An interesting question in symplectic geometry concerns whether or not a closed symplectic manifold can have a free symplectic circle action with orbits contractible in the manifold. Here we present a c-symplectic example, thus showing that the problem is truly geometric as opposed to topological. Furthermore, we see that our example is the only known example of a c-symplectic manifold having non-trivial fundamental group and Lusternik-Schnirelmann category precisely half its dimension.

On Hamilton’S Characteristic Functions For Reflection, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

We review the complex differential geometry of the space of oriented affine lines in R3 and give a description of Hamilton’s characteristic functions for reflection in an oriented C1 surface in terms of this geometry.

Mathematically Promising Students From The Space Age To The Information Age, Linda Jensen Sheffield

The Mathematics Enthusiast

On October 4, 1957, with the launch of Sputnik 1 by the Soviet Union, the world entered the Space Age and the United States became quite concerned that the Soviet Union had a head start in the space race. A year later, realizing that the support of gifted and talented mathematics and science students was critical to national security, the United States federal government passed the National Defense Education Act (NDEA), providing aid to education in the United States at all levels, primarily to stimulate the advancement of education in science, mathematics, and modern foreign languages. Also, during this time, …

Tme's Worldwide Circulation Statistics

The Mathematics Enthusiast

No abstract provided.

Not Out Of The Blue: Historical Roots Of Mathematics Education In Italy, Fulvia Furinghetti

The Mathematics Enthusiast

In this note I outline some elements of the history of mathematics education in Italy. Initially the chief characters were mathematicians who played a role in designing curricula and in editing textbooks. The development of the Italian community of mathematics educators towards the present day trend in research was fostered by participation in international activities after the Second World War. I also identify some elements of continuity with the past to stress the influence of some mathematicians in the development of present research.

Algorithmic Problems In Junior Contests In Latvia, Agnis Andžans, Inese Berzina, Dace Bonka

The Mathematics Enthusiast

Mathematical contests are of great importance for advanced education in Latvia today. Their content must be well-balanced and must correspond to the inner logic and recent trends of mathematics. A classification of algorithmic problems and characteristic examples are considered.

A Sequence Of Polynomials For Approximating Arctangent, Herbert A. Medina

Mathematics, Statistics and Data Science Faculty Works

No abstract provided.

Toric Modular Forms And Nonvanishing Of L-Functions, Lev A. Borisov, Paul E. Gunnells

Paul Gunnells

In a previous paper \cite{BorGunn}, we defined the space of toric forms $\TTT(l)$, and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f,1)≠0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

Is Hollywood Good For Mathematics? (A Brief Discussion)., Christopher D. Goff

College of the Pacific Faculty Presentations

Mathematics has played a role in many recent films, such as Good Will Hunting, A Beautiful Mind, and Proof. But how exactly is mathematics being portrayed? What ramifications does this portrayal have on our ”students,” broadly defined? In Good Will Hunting, mathematics is Will’s ticket out of his lower-class life and into a prestigious existence. In A Beautiful Mind, mathematical talent and schizophrenia are identified, at least in part. In Proof, mathematics not only links a daughter to her late father, but it also provides a backdrop for much of the story’s conflict. In this discussion, we will begin by …

Semidefinite Programming And Stability Of Dynamical System, Kazumi Niki Stovall

Mathematics Theses

In the first part of the thesis we present several interior point algorithms for solving certain positive definite programming problems. One of the algorithms is adapted for finding out whether there exists or not a positive definite matrix which is a real linear combination of some given symmetric matrices A1,A2, . . . ,Am. In the second part of the thesis we discuss stability of nonlinear dynamical systems. We search using algorithms described in the first part, for Lyapunov functions of a few forms. A suitable Lyapunov function implies the existence of a hyperellipsoidal attraction region for the dynamical system, …

On The Reliability Of An N-Component System, Don Rawlings, Lawrence Sze

Mathematics

Under assumptions compatible with the theory of Markov chains, we use a property of Vandermonde matrices to examine the reliability of an n-component system of production or service.

On The Existence Of Infinitely Many Closed Geodesics On Orbifolds Of Revolution, Joseph Borzellino, Christopher R. Jordan-Squire, Gregory C. Petrics, D. Mark Sullivan

Mathematics

Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing that any two-dimensional orbifold of revolution homeomorphic to S2 must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be regarded as a topological two-sphere with metric singularities, we will have extended Bangert’s theorem on the existence of infinitely many closed geodesics on any smooth Riemannian two-sphere. In addition, we give an example of a two-sphere cone-manifold of revolution which possesses a single closed geodesic, thus showing that Bangert’s result …

Writing Projective Representations Over Subfields, Stephen P. Glasby, C. R. Leedham-Green, E. A. O'Brien

All Faculty Scholarship for the College of the Sciences

Let G=〈X〉be an absolutely irreducible subgroup of GL(d, K), and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL(d, F).K×, where K× denotes the centre of GL(d, K). If the derived group of G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs O(|X|d³+d²log|F|) arithmetic operations in K. This work forms part of a recognition project based on Aschbacher’s classification of maximal subgroups of GL(d, K).

Spanning Eulerian Subgraphs In Claw-Free Graphs, Zhi-Hong Chen, Hong-Jian Lai, Weiqi Luo, Yehomg Shao

Scholarship and Professional Work - LAS

A graph is claw-free if it has no induced K 1,3, subgraph. A graph is essential 4-edge-connected if removing at most three edges, the resulting graph has at most one component having edges. In this note, we show that every essential 4-edge-connected claw free graph has a spanning Eulerian subgraph with maximum degree at most 4.

Decentralized Convex-Type Equilibrium In Nonconvex Models Of Welfare Economics Via Nonlinear Prices, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to applications of modern tools of variational analysis to equilibrium models of welfare economics involving nonconvex economies with infinite-dimensional commodity spaces. The main results relate to generalized/ extended second welfare theorems ensuring an equilibrium price support at Pareto optimal allocations. Based on advanced tools of generalized differentiation, we establish refined results of this type with the novel usage of nonlinear prices at the three types to optimal allocations: weak Pareto, Pareto, and strong Pareto. The usage of nonlinear (vs. standard linear) prices allow us to decentralized price equilibria in fully nonconvex models similarly to linear prices …

Simultaneous Transitional And Multiplicative Tiling And Wavelet Sets In R2, Eugen J. Ionascu

Faculty Bibliography

Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent years, particularly for arbitrary dilation matrices, has led to the study of multiplicative tilings by the powers of a matrix. In this paper we consider the following simultaneous tiling problem: Given a lattice in Z ∈ Rd and a matrix A ∈ GL (d, R), does there exist a measurable set T such that both {T + α : α ∈ L} and {AnT : n ∈ Z} are tilings of Rd? This problem …

Vector Theory Of Spontaneous Lorentz Violation, Shu Hong Fung

Undergraduate Research Symposium (UGRS)

Classical electromagnetism predicts two massless propagating modes, which are known as the two polarizations of the photon. On the other hand, if the Lorentz symmetry of classical electromagnetism is spontaneously broken, the new theory will still have two massless Nambu-Goldstone modes resembling the photon. If the Lorentz symmetry is broken by a bumblebee potential that allows for excitations out of the minimum, then massive modes arise. Furthermore, in curved spacetime, such massive modes will be created through a process other than the usual Higgs mechanism because of the dependence of the bumblebee potential on both the vector field and the …

The Integral Cohomology Of The Group Of Loops, Craig Jensen, Jon Mccammond, John Meier

Mathematics Faculty Publications

No abstract provided.

Spectral Analysis Of The Supreme Court, B Lawson, M Orrison, David Uminsky

Mathematics

Should you decide that spectral analysis is worth looking into (as we hope to convince you), then you will be happy to know that there are efficient algorithms for doing spectral analysis. Perhaps more interestingly, at least from a mathematical perspective, these algorithms involve an intriguing mixture of ideas and techniques from linear algebra, abstract algebra, numerical analysis, and graph theory.

The focus of this paper is the linear algebraic framework in which the spectral analysis of voting data like that above is carried out. As we will show, this framework can be used to pinpoint voting coalitions in small …

On Self-Adjoint And J-Self-Adjoint Dirac-Type Operators: A Case Study, Stephen L. Clark, Fritz Gesztesy

Mathematics and Statistics Faculty Research & Creative Works

We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schrödinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl-Titchmarsh half-line m-coefficients (and 2 × 2 matrix-valued M-matrices) in the non-self-adjoint context and derive some of their basic properties. We conclude with an illustrative example showing that crossing spectral arcs in the non-self-adjoint context imply the blowup of the norm of spectral projections in the limit where the crossing point …

Multiple Lebesgue Integration On Time Scales, Gusein Sh. Guseinov, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

We study the process of multiple Lebesgue integration on time scales. The relationship of the Riemann and the Lebesgue multiple integrals is investigated.

Singular Second-Order Multipoint Dynamic Boundary Value Problems With Mixed Derivatives, Hua Luo, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

We study a certain singular second-order m-point boundary value problem on a time scale and establish the existence of a solution. The proof of our main result is based upon the Leray-Schauder continuation theorem.

Boundedness In Functional Dynamic Equations On Time Scales, Elvan Akin, Youssef N. Raffoul

Mathematics and Statistics Faculty Research & Creative Works

Using nonnegative definite Lyapunov functionals, we prove general theorems for the boundedness of all solutions of a functional dynamic equation on time scales. We apply our obtained results to linear and nonlinear Volterra integro-dynamic equations on time scales by displaying suitable Lyapunov functionals.

Modeling Of Bioinspired Sensors For Flow Separation Detection For Micro Air Vehicles, Belinda A. Batten, John R. Singler, Benjamin T. Dickinson

Mathematics and Statistics Faculty Research & Creative Works

Autonomous micro air vehicle (MAV) flight faces inherent stability challenges. One challenge is controlling flow separation over the airfoil and an autonomous control system for MAV flight may be enhanced with closed loop separation control. In this work, we focus on modeling biologically inspired hair cell sensors for future flow control applications. We model the sensor output and present examples and numerical results.

Feedback Control Of Low Dimensional Models Of Transition To Turbulence, John A. Burns, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

The problem of controlling or delaying transition to turbulence in shear flows has been the subject of numerous papers over the past twenty years. This period has seen the development of several low dimensional models for parallel shear flows in an attempt to explain the failure of classical linear hydrodynamic stability theory to correctly predict transition. In recent years, ideas from robust control theory have been employed to attack this problem. In this paper we use these models to develop a scenario for transition that employs both classical bifurcation theory and robust control theory. In addition, we present numerical results …

The Poweratlas: A Power And Sample Size Atlas For Microarray Experimental Design And Research, Grier P. Page, Jode W. Edwards, Gary L. Gadbury, Prashanth Yelisetti, Jelai Wang, Prinal Trivedi, David B. Allison

Mathematics and Statistics Faculty Research & Creative Works

Microarrays permit biologists to simultaneously measure the mRNA abundance of thousands of genes. An important issue facing investigators planning microarray experiments is how to estimate the sample size required for good statistical power. What is the projected sample size or number of replicate chips needed to address the multiple hypotheses with acceptable accuracy? Statistical methods exist for calculating power based upon a single hypothesis, using estimates of the variability in data from pilot studies. There is, however, a need for methods to estimate power and/or required sample sizes in situations where multiple hypotheses are being tested, such as in microarray …

The Emergence Of Large-Scale Coherent Structure Under Small-Scale Random Bombardments, Andrew Majda, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We provide mathematical justification of the emergence of large-scale coherent structure in a two-dimensional fluid system under small-scale random bombardments with small forcing and appropriate scaling assumptions. the analysis shows that the large-scale structure emerging out of the small-scale random forcing is not the one predicted by equilibrium statistical mechanics. But the error is very small, which explains earlier successful prediction of the large-scale structure based on equilibrium statistical mechanics. © 2005 Wiley Periodicals, Inc.