Mathematics Commons^™

The Use And Influence Of Technology In Mathematics Education, Ma. Louise Antonette N. De Las Peñas, Wei-Chi Yang

Mathematics Faculty Publications

The use of various types of technologies in the classroom and examinations is growing rapidly and is strongly influencing teaching and learning practices. In this paper, we will look at particular situations on how various technologies such as numerically capable calculators, graphics calculators, and technological tools that are CAS enabled or have CAS with Dynamic Geometry, impact students' learning. We also discuss briefly the educational opportunities that are made available by the emergence of graphics calculators with capabilities of handling electronic learning activities, such as Casio’s Class Pad (see [1]) and Casio’s 9860 graphics calculator.

Commensurability Classes Of Twist Knots, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

In this paper we prove that if M_K is the complement of a non-fibered twist knot K in S³, then M_K is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

A Map On The Space Of Rational Functions, G. Boros, J. Little, V. Moll, Edward Mosteig, R. Stanley

Mathematics, Statistics and Data Science Faculty Works

We describe dynamical properties of a map defined on the space of rational functions. The fixed points of F are classified and the long time behavior of a subclass is described in terms of Eulerian polynomials.

Derived Categories And The Analytic Approach To General Reciprocity Laws. Part I, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

We reformulate Hecke's open problem of 1923, regarding the Fourier-analytic proof of higher reciprocity laws, as a theorem about morphisms involving stratified topological spaces. We achieve this by placing Kubota's formulations of n-Hilbert reciprocity in a new topological context, suited to the introduction of derived categories of sheaf complexes. Subsequently, we begin to investigate conditions on associated sheaves and a derived category of sheaf complexes specifically designed for an attack on Hecke's eighty-year-old challenge.

From Loop Groups To 2-Groups, John C. Baez, Danny Stevenson, Alissa S. Crans, Urs Schreiber

Mathematics, Statistics and Data Science Faculty Works

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g_k each having Lie(G) as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on G. There appears to be no Lie 2-group …

Reversals And Transpositions Over Finite Alphabets, A. J. Radcliffe, A. D. Scott, E. L. Wilmer

Department of Mathematics: Faculty Publications

Extending results of Christie and Irving, we examine the action of reversals and transpositions on finite strings over an alphabet of size k. We show that determining reversal, transposition, or signed reversal distance between two strings over a finite alphabet is NP-hard, while for “dense” instances we give a polynomial-time approximation scheme. We also give a number of extremal results, as well as investigating the distance between random strings and the problem of sorting a string over a finite alphabet.

Global Stability Of Periodic Orbits Of Non-Autonomous Difference Equations And Population Biology, Saber Elaydi, Robert J. Sacker

Mathematics Faculty Research

Elaydi and Yakubu showed that a globally asymptotically stable(GAS) periodic orbit in an autonomous difference equation must in fact be a fixed point whenever the phase space is connected. In this paper we extend this result to periodic nonautonomous difference equations via the concept of skew-product dynamical systems. We show that for a k-periodic difference equation, if a periodic orbit of period r is GAS, then r must be a divisor of k. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and …

The Asymptotic Optimal Partition And Extensions Of The Nonsubstitution Theorem, Julio R. Hasfura-Buenaga, Allen G. Holder, Jeffrey Stuart

Mathematics Faculty Research

The data describing an asymptotic linear program rely on a single parameter, usually referred to as time, and unlike parametric linear programming, asymptotic linear programming is concerned with the steady state behavior as time increases to infinity. The fundamental result of this work shows that the optimal partition for an asymptotic linear program attains a steady state for a large class of functions. Consequently, this allows us to define an asymptotic center solution. We show that this solution inherits the analytic properties of the functions used to describe the feasible region. Moreover, our results allow significant extensions of an economics …

On Rough Singular Integrals Along Surfaces On Product Domains, Ahmad Al-Salman, Ali A. Al-Jarrah

Turkish Journal of Mathematics

In this paper, we study a class of singular integrals along surfaces on product domains with kernels in L(log L)^2(S^{n-1} \times S^{m-1}). We formulate a general theorem concerning the L^p boundedness of these operators. As a consequence of this theorem we establish L^p estimates of several classes of operators whose L^p boundedness in the one parameter setting is known. The condition L(log L)^2(B^{n-1} \times S^{m-1}) is known to be an optimal size condition

Exterior Differential Systems With Symmetry, Ian M. Anderson, Mark E. Fels

Mathematics and Statistics Faculty Publications

We use the theory of reduction of exterior differential systems with symmetry to study the problem of using a symmetry group of a differential equation to find noninvariant solutions.

Coffee To Go! Modeling Thermoclines In Multivariable Calculus, Andrea Bruder, Brynja R. Kohler

Mathematics and Statistics Faculty Publications

While mathematical modeling is an integral process in applied mathematics, students rarely encounter genuine modeling opportunities in their calculus courses. Here we introduce a laboratory experience as a natural starting point for calculus students to investigate multivariable functions. A layered system of coffee and milk serves as a physical model for temperature gradients in lakes or the atmosphere, where temperature depends on both a temporal and spatial variable. Students create, observe, and collect temperature data of their own, graph the data, and develop mathematical models to fit the data. We require students to write a report about their findings. This …

On The Integrability Of A Class Of Nonlinear Dispersive Wave Equations, Rossen Ivanov

Articles

We investigate the integrability of a class of 1+1 dimensional models describing nonlinear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.

Gender Differences In Reading And Math Scores On The Pennsylvania System Of School Assessment Grade Five And Grade Eight Tests, Steven J. Rush

Seton Hall University Dissertations and Theses (ETDs)

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Weak Factorizations, Fractions And Homotopies, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.

Positive Solutions For A Fourth Order Boundary Value Problem, Bo Yang

Faculty Articles

We consider a boundary value problem for the beam equation, in which the boundary conditions mean that the beam is embedded at one end and free at the other end. Some new estimates to the positive solutions to the boundary value problem are obtained. Some sufficient conditions for the existence of at least one positive solution for the boundary value problem are established. An example is given at the end of the paper to illustrate the main results.

Fixed Point Theory For Mönch-Type Maps Defined On Closed Subsets Of Fréchet Spaces: The Projective Limit Approach, Ravi P. Agarwal, Donal O'Regan, Jewgeni H. Dshalalow

Mathematics and System Engineering Faculty Publications

New Leray-Schauder alternatives are presented for Mönch-type maps defined between Fréchet spaces. The proof relies on viewing a Fréchet space as the projective limit of a sequence of Banach spaces.

Julian Cecil Stanley Papers, Zach S. Henderson Library, Special Collections

Finding Aids

This collection consists of the research materials of Johns Hopkins professor of psychology, Julian C. Stanley. Materials range from 1950-2005 and include his published reprints, abstracts, reports, letters, and seminar papers. The collection also includes reviews of research materials and articles by prominent psychologists in similar fields.

Find this collection in the University Libraries' catalog.

Matrix Model Superpotentials And Calabi–Yau Spaces: An A-D-E Classification, Carina Curto

Department of Mathematics: Faculty Publications

We use F. Ferrari’s methods relating matrix models to Calabi-Yau spaces in order to explain Intriligator and Wecht’s ADE classification of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. The connection between matrix models and N = 1 gauge theories can be seen as evidence for the Dijkgraaf–Vafa conjecture. We find that ADE superpotentials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yau’s with corresponding ADE singularities. Moreover, in the additional Ô, Â, Dˆ and Ê cases we find new singular geometries. These ‘hat’ geometries are …

Pseudo-Codewords Of Cycle Codes Via Zeta Functions, Ralf Koetter, Wen-Cheng W. Li, Pascal O. Vontobel, Judy L. Walker

Department of Mathematics: Faculty Publications

Cycle codes are a special case of low- density parity-check (LDPC) codes and as such can be decoded using an iterative message-passing decod- ing algorithm on the associated Tanner graph. The existence of pseudo-codewords is known to cause the decoding algorithm to fail in certain instances. In this paper, we draw a connection between pseudo- codewords of cycle codes and the so-called edge zeta function of the associated normal graph and show how the Newton polyhedron of the zeta function equals the fundamental cone of the code, which plays a crucial role in characterizing the performance of iterative de- coding …

Weak Solutions To The Cauchy Problem Of A Semilinear Wave Equation With Damping And Source Terms, Petronela Radu

Department of Mathematics: Faculty Publications

In this paper we prove local existence of weak solutions for a semilinear wave equation with power-like source and dissipative terms on the entire space ℝⁿ. The main theorem gives an alternative proof of the local in time existence result due to J. Serrin, G. Todorova and E. Vitillaro, and also some extension to their work. In particular, our method shows that sources that are not locally Lipschitz in L² can be controlled without any damping at all. If the semilinearity involving the displacement has a “good” sign, we obtain global existence of solutions.

Splitting Of Vector Bundles On Punctured Spectrum Of Regular Local Rings, Mahdi Majidi-Zolbanin

Dissertations, Theses, and Capstone Projects

In this dissertation we study splitting of vector bundles of small rank on punctured spectrum of regular local rings. We give a splitting criterion for vector bundles of small rank in terms of vanishing of their intermediate cohomology modules Hⁱ(U, E)_{2_i_n−3}, where n is the dimension of the regular local ring. This is the local analog of a result by N. Mohan Kumar, C. Peterson, and A. Prabhakar Rao for splitting of vector bundles of small rank on projective spaces.

As an application we give a positive answer (in a special case) to a conjecture …

Quantum Quasi-Paradoxes And Quantum Sorites Paradoxes, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

There can be generated many paradoxes or quasi-paradoxes that may occur from the combination of quantum and non-quantum worlds in physics. Even the passage from the micro-cosmos to the macro-cosmos, and reciprocally, can generate unsolved questions or counter-intuitive ideas. We define a quasi-paradox as a statement which has a prima facie self-contradictory support or an explicit contradiction, but which is not completely proven as a paradox. We present herein four elementary quantum quasi-paradoxes and their corresponding quantum Sorites paradoxes, which form a class of quantum quasi-paradoxes.

Fuzzy And Neutrosophic Analysis Of Women With Hiv/Aids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Fuzzy theory is one of the best tools to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indeterminacy. The only tool that can involve itself with indeterminacy is the neutrosophic model. Neutrosophic models are used in the analysis of the socio-economic problems of HIV/AIDS infected women patients living in rural Tamil Nadu. Most of these women are uneducated and live in utter poverty. Till they became seriously ill they worked as daily wagers. When these women got admitted in the hospital …

Fuzzy And Neutrosophic Analysis Of Periyar’S Views On Untouchability, Florentin Smarandache, Vasantha Kandasamy, K. Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

“Day in and day out we take pride in claiming that India has a 5000 year old civilization. But the way Dalits and those suppressed are being treated by the people who wield power and authority speaks volumes for the degradations of our moral structure and civilized standards.” Ex-President of India, the late K. R. Narayanan The New Indian Express, Saturday, 12 Nov. 2005 K.R.Narayanan was a lauded hero and a distinguished victim of his Dalit background. Even in an international platform when he was on an official visit to Paris, the media headlines blazed, ‘An Untouchable at Elysee’. He …

N-Algebraic Structures And S-N-Algebraic Structures, Florentin Smarandache, Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, for the first time we introduce the notions of Ngroups, N-semigroups, N-loops and N-groupoids. We also define a mixed N-algebraic structure. We expect the reader to be well versed in group theory and have at least basic knowledge about Smarandache groupoids, Smarandache loops, Smarandache semigroups and bialgebraic structures and Smarandache bialgebraic structures. The book is organized into six chapters. The first chapter gives the basic notions of S-semigroups, S-groupoids and S-loops thereby making the book self-contained. Chapter two introduces N-groups and their Smarandache analogues. In chapter three, Nloops and Smarandache N-loops are introduced and analyzed. Chapter four …

Introduction To Linear Bialgebra, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications. With the recent introduction of bimatrices (2005) we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialgebra and also give the applications of these algebraic …

Applications Of Bimatrices To Some Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. …

Repeating Decimals And Cyclotomic Polynomials, David H. Chapman

Dissertations and Theses @ UNI

This thesis makes a contribution to the study of repeating decimals. The primary focus of earlier investigations of repeating decimals have been in base 10. The purpose of this thesis is to determine when solutions exist for the following problem: given integers n ≥1 and b ≥ 2, what is the smallest prime p such that 1/p repeats exactly n digits in base b. In addition, when a solution p exists, it is shown that the cycles in base b of the form x/p correspond to the cosets of a subgroup of the multiplicative group on non-zero congruence classes modulo …

Bootstrap Prediction Intervals For Multivariate Time Series, Florian Sebastian Rueck

Doctoral Dissertations

"The theory and methodology of obtaining bootstrap prediction intervals for univariate time series using the forward representation of the series is extended to vector autoregressive (VAR) models. Kim has shown that simultaneous prediction intervals based on the Bonferroni method and the backward representation of the time series achieve coverage close to nominal when the parameter estimates are corrected for small sample bias. To utilize his method, it is necessary to assume that the innovations are normally distributed to maintain independence of the innovations associated with the backward representation of the time series. This assumption is not necessary if the forward …

Rational Hyperholomorphic Functions In R4, Daniel Alpay, Michael Shapiro, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of realizations and the third in terms of backward-shift invariance. Also introduced and studied are the counterparts of the Arveson space and Blaschke factors.