Physical Sciences and Mathematics Commons^™

Calculus Students’ Difficulties In Using Variables As Changing Quantities, Susan S. Gray, Barbara J. Loud, Carole Sokolowski

Mathematics Faculty Publications

The study of calculus requires an ability to understand algebraic variables as generalized numbers and as functionally-related quantities. These more advanced uses of variables are indicative of algebraic thinking as opposed to arithmetic thinking. This study reports on entering Calculus I students’ responses to a selection of test questions that required the use of variables in these advanced ways. On average, students’ success rates on these questions were less than 50%. An analysis of errors revealed students’ tendencies toward arithmetic thinking when they attempted to answer questions that required an ability to think of variables as changing quantities, a characteristic …

A Combinatorial Solution To Intertwined Recurrences, Arthur T. Benjamin, Michael D. Hirschhorn

All HMC Faculty Publications and Research

We provide combinatorial derivations of solutions to intertwined second order linear recurrences (such as a_n = pb_n-1 + qa_n-2, b_n = ra_n-1 + sb_n-2) by counting tilings of length n strips with squares and dominoes of various colors and shades. A similar approach can be applied to intertwined third order recurrences with coefficients equal to one. Here we find that all solutions can be expressed in terms of tribonacci numbers. The method can also be easily extended to solve and combinatorially comprehend kth order Fibonacci recurrences.

Fibonacci Deteminants - A Combinatorial Approach, Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn

All HMC Faculty Publications and Research

In this paper, we provide combinatorial interpretations for some determinantal identities involving Fibonacci numbers. We use the method due to Lindström-Gessel-Viennot in which we count nonintersecting n-routes in carefully chosen digraphs in order to gain insight into the nature of some well-known determinantal identities while allowing room to generalize and discover new ones.

The Lsb Theorem Implies The Kkm Lemma, Gwen Spencer '05, Francis E. Su

All HMC Faculty Publications and Research

No abstract provided in this article.

Today's Mathematics Students, Carmen M. Latterell

The Mathematics Enthusiast

A common mistake that undergraduate mathematics professors make when teaching is to assume that students are younger versions of themselves. Since many mathematics professors are above average in intelligence and were quite good students, the assumption that students are just like themselves can cause pedagogical difficulties (Krantz, 1993). To teach effectively, it is important to understand students. Yet, understanding today's students is literally like bridging a generation gap (Hawk, 2005).

Lagrange: A Well-Behaved Function, Benjamin Harris

The Mathematics Enthusiast

This paper outlines the biography and achievements of Joseph Louis Lagrange (1736–1813) and includes a detailed explanation, with examples, of the Lagrange Multiplier method for optimizing multivariate functions subject to constraint. The Lagrange Multiplier is widely used in chemistry, physics, and economics, in particular. The paper considers the origin of economics’ use of the multiplier and provides a concrete example of how it is used in microeconomic theory. While the focus is on the multiplier’s application to microeconomics, the intended audience includes all teachers and students who encounter any of Lagrange’s contributions. Since Lagrange’s contributions to mathematics are numerous, so …

Can Our Learners Model In Mathematics?, Vimolan Mudaly

The Mathematics Enthusiast

Mathematical modeling of real world conditions should be part ofl mathematics classroom activities. In this paper I argue that when real world problems are taught at schools learners are not able to cope on their own, without the assistance of their educator. There is very little or no emphasis placed on this aspect of mathemtics at schools, although it is just beginning to make an appearance in our new Outcomes Based Curriculum. I also discuss an experiment conducted with Grade 10 learners (15 year old) and their responses to real world problems and the conditions that need to be considered. …

Separated Lie Models And The Homotopy Lie Algebra, Peter G. Bubenik

Mathematics and Statistics Faculty Publications

Abstract. A simply connected topological space X has homotopy Lie algebra π∗(ΩX) ⊗ Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property we call separated. The homology of a separated dgL has a particular form which lends itself to calculations. 1.

Numerical Methods With Ms Excel, M. El-Gebeily, B. Yushau

The Mathematics Enthusiast

In this note we show how MS Excel can be used to to perform numerical Integration, specifically Trapezoidal Rule and Simson’s rule. Futhermore, we illustrate how to generate Lagranges Interpolation polynomial.

Tme Volume 4, Number 1

The Mathematics Enthusiast

No abstract provided.

Adult Students' Reasoning In Geometry: Teaching Mathematics Through Collaborative Problem Solving In Teacher Education, Raymond Bjuland

The Mathematics Enthusiast

This article reports research that is concerned with pre-service teachers2 working collaboratively in a problem-solving context without teacher involvement. The aim is to focus on the students’ heuristic strategies employed in the solution process while working on two problems in geometry. Two episodes from the dialogues in one group of students with limited mathematical backgrounds have been chosen to illustrate some mathematical movement throughout the group meetings, from working with the first problem to working with the second one. The findings reveal that three categories of strategies, visualising, monitoring, and questioning, play an important role in order to make progress …

Erratum: An Improvement On The Article Taxi Cab Geometry: History And Applications, Tmme, Vol2, No.1, P. 38 - 64, Benjamin Urland

The Mathematics Enthusiast

As a high school student from Germany I did a Mathematics research paper entitled Taxicab Geometry: Fundamentals and Applications. During my research I found the article Taxicab Geometry: History and Applications by Chip Reinhardt which was published in the edition Vol.2, no. 1 (p. 38 – 64) of this journal. The article was very helpful for my coursework and I’d like to compliment the author and everyone who is involved in the journal on their work. In my coursework I created an application example similar to the one Chip Reinhardt used in his article. I solved my example in the …

The Interplay Of Processing Efficiency And Working Memory With The Development Of Metacognitive Performance In Mathematics, Areti Panaoura

The Mathematics Enthusiast

The present study outlines a specific three level hierarchy of the cognitive system and especially the relations of specific cognitive and metacognitive processes in mathematics. The emphasis is on the impact of the development of processing efficiency and working memory ability on the development of metacognitive abilities and mathematical performance. We had used instruments measuring pupils´ metacognitive ability, mathematical performance, working memory and processing efficiency. We administered them to 126 pupils (8-11 years old) three times, with breaks of 3-4 months between them. Results indicated that the development of each of the abilities was affected by the state of the …

Students' Conceptions Of Limits: High Achievers Versus Low Achievers, Kristina Juter

The Mathematics Enthusiast

Learning an advanced mathematical concept, limits of functions in this case, is not a linear development equal for all learners. Intentions and abilities influence students’ learning paths and results. Students’ learning developments of limits were studied in terms of concept images (Tall & Vinner, 1981) in the sense that their actions, such as problem solving and reasoning, were considered traces of their mental representations of concepts. High achievers’ developments were compared to low achievers’ developments to for the duration of a semester to reveal differences and similarities.

The Need For An Inclusive Framework For Students' Thinking In School Geometry, Jaguthsing Dindyal

The Mathematics Enthusiast

This study is the outcome of a research that investigated how students who were assigned varying levels of geometric thinking attempted problems requiring some amount of algebraic thinking in geometry. The study reports that students’ thinking in geometry also requires facility with algebra and as such there is a need for a framework that provides a more inclusive view of what constitutes geometric thinking in school mathematics.

Learning Mathematics With Understanding: A Critical Consideration Of The Learning Principle In The Principles And Standards For School Mathematics, Andreas J. Stylianides, Gabriel J. Stylianides

The Mathematics Enthusiast

Learning with understanding has increasingly received attention from educators and psychologists, and has progressively been elevated to one of the most important goals for all students in all subjects. However, the realization of this goal has been problematic, especially in the domain of mathematics. To this might have contributed the fact that, although the vision of students learning mathematics with understanding has often appeared in curriculum frameworks, this vision has tended to be poorly described, thereby offering limited support to curriculum development and policy. The Learning Principle in the Principles and Standards for School Mathematics, an influential mathematics curriculum framework …

Hartman-Grobman Theorems Along Hyperbolic Stationary Trajectories, Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, Paulo Régis C. Ruffino

Articles and Preprints

We extend the Hartman-Grobman theorems on discrete random dynamical systems (RDS), proved in [7], in two directions: For continuous RDS and for hyperbolic stationary trajectories. In this last case there exists a conjugacy between traveling neighbourhoods of trajectories and neighbourhoods of the origin in the corresponding tangent bundle. We present applications to deterministic dynamical systems.

Modeling Projection Neuron And Neuromodulatory Effects On A Rhythmic Neuronal Network, Nicholas Kintos

Dissertations

Projection neurons shape the activity of many neural networks. In particular, neuromodulatory substances, which are often released by projection neurons, alter the cellular and/or synaptic properties within a target network. However, neural networks in turn influence projection neuron input via synaptic feedback. This dissertation uses mathematical and biophysically-realistic modeling to investigate these issues in the gastric mill (chewing) motor network of the crab, Cancer borealis. The projection neuron MCN1 elicits a gastric mill rhythm in which the LG neuron and INTl burst in anti-phase due to their reciprocal inhibition. However, bath application of the neuromodulator PK elicits a similar gastric …

Some Geometrical Aspects Of The Cone Linear Complementarity Problem., Madhur Malik Dr.

Doctoral Theses

Cone Linear Complementarity ProblemLet V be a finite dimensional real inner product space and K be a closed convex cone in V. Given a linear transformation L : V → V and a vector q ∈ V the cone linear complementarity problem or linear complementarity problem over K, denoted as LCP(K, L, q), is to find a vector x ∈ K such thatL(x) + q ∈ K+ and hx, L(x) + qi = 0,where h., .i denotes an inner product on V and K is the dual cone of K defined as:K∗ := {y ∈ V : hx, yi ≥ …

Reflection In A Translation Invariant Surface, Brendan Guilfoyle, Wilhelm Klingenberg

Preprints

We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by constructing explicitly the focal set of the reflected line congruence (2-parameter family of oriented lines in R 3) with the aid of the natural complex structure on the space of all oriented affine lines.

A Proposal For Robust Temperature Compensation Of Circadian Rhythms, Christian I. Hong, Emery D. Conrad, John J. Tyson

Dartmouth Scholarship

The internal circadian rhythms of cells and organisms coordinate their physiological properties to the prevailing 24-h cycle of light and dark on earth. The mechanisms generating circadian rhythms have four defining characteristics: they oscillate endogenously with period close to 24 h, entrain to external signals, suffer phase shifts by aberrant pulses of light or temperature, and compensate for changes in temperature over a range of 10°C or more. Most theoretical descriptions of circadian rhythms propose that the underlying mechanism generates a stable limit cycle oscillation (in constant darkness or dim light), because limit cycles quite naturally possess the first three …

Integral And Nonnegativity Preserving Approximations Of Functions, Jiu Ding, Larry Eifler, N. H. Rhee

Faculty Publications

In this paper we consider the problem of approximating a function by continuous piecewise linear functions that preserve the integral and nonnegativity of the original function. (c) 2006 Elsevier Inc. All rights reserved.

Modular Invariants For Lattice Polarized K3 Surfaces, Adrian Clingher, Charles F. Doran

Adrian Clingher

No abstract provided.

The Q-Exponential Generating Function For Permutations By Consecutive Patterns And Inversions, Don Rawlings

Mathematics

The inverse of Fedou's insertion-shift bijection is used to deduce a general form for the q-exponential generating function for permutations by consecutive patterns (overlaps allowed) and inversion number from a result due to Jackson and Goulden for enumerating words by distinguished factors. Explicit q-exponential generating functions are then derived for permutations by the consecutive patterns 12…m, 12…(m−2)m(m−1), 1m(m−1)…2, and by the pair of consecutive patterns (123,132).

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

Mathematics

Using the theory of geodesics on surfaces of revolution, we show 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 does not hold in the wider class of closed surfaces with …

A Centre-Stable Manifold For The Focussing Cubic Nls In R 1+3, Marius Beceanu

Mathematics and Statistics Faculty Scholarship

Consider the focussing cubic nonlinear Schr\"odinger equation in R 3 : iψ t +Δψ=−|ψ| 2 ψ.

It admits special solutions of the form e itα ϕ , whereϕ is a Schwartz function and a positive (ϕ>0 ) solution of −Δϕ+αϕ=ϕ 3 .

The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form e i(v⋅+Γ) ϕ(⋅−y,α) . We prove that any solution starting sufficiently close to a standing wave in the Σ=W 1,2 (R 3 …

Extreme Values For The Area Of Rectangles With Vertices On Concentrical Circles, Eugen J. Ionascu

Faculty Bibliography

No abstract provided.

Heron Triangles With Two Fixed Sides, Eugen J. Ionascu

Faculty Bibliography

In this paper, we study the function H(a, b), which associates to every pair of positive integers a and b the number of positive integers c such that the triangle of sides a, b and c is Heron, i.e., has integral area. In particular, we prove that H(p, q) ≤ 5 if p and q are primes, and that H(a, b) = 0 for a random choice of positive integers a and b.

A Parametrization Of Equilateral Triangles Having Integer Coordinates., Eugen J. Ionascu

Faculty Bibliography

We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form p 2(m2 − mn + n2) for some integers m, n. We also show a similar characterization for the sides of a regular tetrahedron in Z 3 : such a tetrahedron exists if and only if the sides are of the form k √ 2, for some k ∈ N. The classification of all the equilateral triangles in Z 3 contained in a given …

A Proof Of Two Conjectures Related To Erdos-Debrunner Inequality, Eugen J. Ionascu

Faculty Bibliography

In this paper we prove some results which imply two conjectures proposed by Janous on an extension to the p-th power-mean of the Erdös–Debrunner inequality relating the areas of the four sub-triangles formed by connecting three arbitrary points on the sides of a given triangle.