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Mathematics Commons™

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2007

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Articles 271 - 300 of 541

Full-Text Articles in Mathematics

Galois Number Fields With Small Root Discriminant, John W. Jones, David P. Roberts Feb 2007

Galois Number Fields With Small Root Discriminant, John W. Jones, David P. Roberts

Mathematics Publications

We pose the problem of identifying the set K(G,Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632. We definitively treat the cases G = A4. A5, A6, and S4, S5, S6, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois groups SL3(2), A7, S7, PGL2(7), SL2(8), ΣL2(8), PGL2(9), PSL2(11), and …

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Calculus Students’ Difficulties In Using Variables As Changing Quantities, Susan S. Gray, Barbara J. Loud, Carole Sokolowski Feb 2007

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 …

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Approximations Of Continuous Newton's Method: An Extension Of Cayley's Problem, Jon T. Jacobsen, Owen Lewis '05, Bradley Tennis '06 Feb 2007

Approximations Of Continuous Newton's Method: An Extension Of Cayley's Problem, Jon T. Jacobsen, Owen Lewis '05, Bradley Tennis '06

All HMC Faculty Publications and Research

Continuous Newton's Method refers to a certain dynamical system whose associated flow generically tends to the roots of a given polynomial. An Euler approximation of this system, with step size h=1, yields the discrete Newton's method algorithm for finding roots. In this note we contrast Euler approximations with several different approximations of the continuous ODE system and, using computer experiments, consider their impact on the associated fractal basin boundaries of the roots

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The Lsb Theorem Implies The Kkm Lemma, Gwen Spencer '05, Francis E. Su Feb 2007

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.

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Fibonacci Deteminants - A Combinatorial Approach, Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn Feb 2007

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.

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A Combinatorial Solution To Intertwined Recurrences, Arthur T. Benjamin, Michael D. Hirschhorn Feb 2007

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 an = pbn-1 + qan-2, bn = ran-1 + sbn-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.

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The Need For An Inclusive Framework For Students' Thinking In School Geometry, Jaguthsing Dindyal Feb 2007

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.

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Lagrange: A Well-Behaved Function, Benjamin Harris Feb 2007

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 …

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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 Feb 2007

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 …

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Today's Mathematics Students, Carmen M. Latterell Feb 2007

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).

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Hartman-Grobman Theorems Along Hyperbolic Stationary Trajectories, Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, Paulo Régis C. Ruffino Feb 2007

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.

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Variational Principles For Set-Valued Mappings With Applications To Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich Feb 2007

Variational Principles For Set-Valued Mappings With Applications To Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich

Mathematics Research Reports

This paper primarily concerns the study of general classes of constrained multiobjective optimization problems (including those described via set-valued and vector-valued cost mappings) from the viewpoint of modern variational analysis and generalized differentiation. To proceed, we first establish two variational principles for set-valued mappings, which~being certainly of independent interest are mainly motivated by applications to multiobjective optimization problems considered in this paper. The first variational principle is a set-valued counterpart of the seminal derivative-free Ekeland variational principle, while the second one is a set-valued extension of the subdifferential principle by Mordukhovich and Wang formulated via an appropriate subdifferential notion for …

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Tme Volume 4, Number 1 Feb 2007

Tme Volume 4, Number 1

The Mathematics Enthusiast

No abstract provided.

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Separated Lie Models And The Homotopy Lie Algebra, Peter G. Bubenik Feb 2007

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.

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Step-Up Simultaneous Tests For Identifying Active Effects In Orthogonal Saturated Designs, Samuel S. Wu, Weizhen Wang Feb 2007

Step-Up Simultaneous Tests For Identifying Active Effects In Orthogonal Saturated Designs, Samuel S. Wu, Weizhen Wang

Mathematics and Statistics Faculty Publications

A sequence of null hypotheses regarding the number of negligible effects (zero effects) in orthogonal saturated designs is formulated. Two step-up simultaneous testing procedures are proposed to identify active effects (nonzero effects) under the commonly used assumption of effect sparsity. It is shown that each procedure controls the experimentwise error rate at a given alpha level in the strong sense.

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Adult Students' Reasoning In Geometry: Teaching Mathematics Through Collaborative Problem Solving In Teacher Education, Raymond Bjuland Feb 2007

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 …

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The Interplay Of Processing Efficiency And Working Memory With The Development Of Metacognitive Performance In Mathematics, Areti Panaoura Feb 2007

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 …

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Students' Conceptions Of Limits: High Achievers Versus Low Achievers, Kristina Juter Feb 2007

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.

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Can Our Learners Model In Mathematics?, Vimolan Mudaly Feb 2007

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. …

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Numerical Methods With Ms Excel, M. El-Gebeily, B. Yushau Feb 2007

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.

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Erratum: An Improvement On The Article Taxi Cab Geometry: History And Applications, Tmme, Vol2, No.1, P. 38 - 64, Benjamin Urland Feb 2007

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 …

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Modeling Projection Neuron And Neuromodulatory Effects On A Rhythmic Neuronal Network, Nicholas Kintos Jan 2007

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 …

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Some Geometrical Aspects Of The Cone Linear Complementarity Problem., Madhur Malik Dr. Jan 2007

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 ≥ …

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A Proposal For Robust Temperature Compensation Of Circadian Rhythms, Christian I. Hong, Emery D. Conrad, John J. Tyson Jan 2007

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 …

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Integral And Nonnegativity Preserving Approximations Of Functions, Jiu Ding, Larry Eifler, N. H. Rhee Jan 2007

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.

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Extreme Values For The Area Of Rectangles With Vertices On Concentrical Circles, Eugen J. Ionascu Jan 2007

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

Faculty Bibliography

No abstract provided.

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Heron Triangles With Two Fixed Sides, Eugen J. Ionascu Jan 2007

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.

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A Parametrization Of Equilateral Triangles Having Integer Coordinates., Eugen J. Ionascu Jan 2007

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 …

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A Proof Of Two Conjectures Related To Erdos-Debrunner Inequality, Eugen J. Ionascu Jan 2007

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.

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Vorticity Dynamics And Sound Generation In Two-Dimensional Fluid Flow, Raymond J. Nagem, Guido Sandri, David Uminsky Jan 2007

Vorticity Dynamics And Sound Generation In Two-Dimensional Fluid Flow, Raymond J. Nagem, Guido Sandri, David Uminsky

Mathematics

An approximate solution to the two-dimensional incompressible fluid equations is constructed by expanding the vorticity field in a series of derivatives of a Gaussian vortex. The expansion is used to analyze the motion of a corotating Gaussian vortex pair, and the spatial rotation frequency of the vortex pair is derived directly from the fluid vorticity equation. The resulting rotation frequency includes the effects of finite vortex core size and viscosity and reduces, in the appropriate limit, to the rotation frequency of the Kirchhoff point vortex theory. The expansion is then used in the low Mach number Lighthill equation to derive …

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