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Articles 991 - 1020 of 1985
Full-Text Articles in Mathematics
Improving Equity And Education: Why And How, Patricia Clark Kenschaft
Improving Equity And Education: Why And How, Patricia Clark Kenschaft
Journal of Humanistic Mathematics
Progress is possible. When I graduated from college, only 5\% of the new U.S. doctorates in mathematics went to women; now it is about 30\%. There is of course room (and need) for more progress. This paper begins with an account of my research about women and black mathematicians. The latter group claimed that racial equality can be achieved only when better elementary school mathematics education is available to all children in this country. That motivated me to lead a seven-year, grant-supported program to work with elementary school teachers and children in nine New Jersey districts, including Newark, Paterson, and …
Liberal Arts Inspired Mathematics: A Report Or How To Bring Cultural And Humanistic Aspects Of Mathematics To The Classroom As Effective Teaching And Learning Tools, Anders K H Bengtsson
Liberal Arts Inspired Mathematics: A Report Or How To Bring Cultural And Humanistic Aspects Of Mathematics To The Classroom As Effective Teaching And Learning Tools, Anders K H Bengtsson
Journal of Humanistic Mathematics
This is the report of a project on ways of teaching university-level mathematics in a humanistic way. The main part of the project recounted here involved a journey to the United States during the fall term of 2012 to visit several liberal arts colleges in order to study and discuss mathematics teaching. Several themes that came up during my conversations at these colleges are discussed in the text: the invisibility of mathematics in everyday life, the role of calculus in American mathematics curricula, the "is algebra necessary?'' discussion, teaching mathematics as a language, the transfer problem in learning, and the …
Galileo And Aristotle's Wheel, Olympia Nicodemi
Galileo And Aristotle's Wheel, Olympia Nicodemi
Journal of Humanistic Mathematics
At the beginning of his last major work, Galileo tackles an old paradox, Aristotle's Wheel, in order to produce a model of the continuum that explains (at least to him) how line segments of different length could be put into a one-to-one correspondence. His argument seems like a playful digression. However, it is precisely this type of a one-to-one correspondence that he needs to support his work on free fall. In this article, we investigate how Galileo's model for the wheel paradox informs his work on free fall. We also examine some of the reasons his results on free fall---results …
Paved With Good Intentions: Analysis Of A Randomized Block Kaczmarz Method, Deanna Needell, Joel A. Tropp
Paved With Good Intentions: Analysis Of A Randomized Block Kaczmarz Method, Deanna Needell, Joel A. Tropp
CMC Faculty Publications and Research
The block Kaczmarz method is an iterative scheme for solving overdetermined least-squares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving …
Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott
Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott
HMC Senior Theses
Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe …
A Mathematical Framework For Unmanned Aerial Vehicle Obstacle Avoidance, Sorathan Chaturapruek
A Mathematical Framework For Unmanned Aerial Vehicle Obstacle Avoidance, Sorathan Chaturapruek
HMC Senior Theses
The obstacle avoidance navigation problem for Unmanned Aerial Vehicles (UAVs) is a very challenging problem. It lies at the intersection of many fields such as probability, differential geometry, optimal control, and robotics. We build a mathematical framework to solve this problem for quadrotors using both a theoretical approach through a Hamiltonian system and a machine learning approach that learns from human sub-experts' multiple demonstrations in obstacle avoidance. Prior research on the machine learning approach uses an algorithm that does not incorporate geometry. We have developed tools to solve and test the obstacle avoidance problem through mathematics.
A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.
A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.
HMC Senior Theses
The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.
Cycle Lengths Of Θ-Biased Random Permutations, Tongjia Shi
Cycle Lengths Of Θ-Biased Random Permutations, Tongjia Shi
HMC Senior Theses
Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θK, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The mth moment of the rth longest cycle of a random permutation is Θ(nm), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or …
Characterizing Forced Communication In Networks, Samuel C. Gutekunst
Characterizing Forced Communication In Networks, Samuel C. Gutekunst
HMC Senior Theses
This thesis studies a problem that has been proposed as a novel way to disrupt communication networks: the load maximization problem. The load on a member of a network represents the amount of communication that the member is forced to be involved in. By maximizing the load on an important member of the network, we hope to increase that member's visibility and susceptibility to capture. In this thesis we characterize load as a combinatorial property of graphs and expose possible connections between load and spectral graph theory. We specifically describe the load and how it changes in several canonical classes …
Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09
Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09
Pomona Faculty Publications and Research
We characterize which automorphisms of an arbitrary complete bipartite graph Kn,m can be induced by a homeomorphism of some embedding of the graph in S3.
Knotted And Linked Products Of Recombination On T(2,N)#T(2,M) Substrates, Erica Flapan, Jeremy Grevet, Qi Li, Chen Daisy Sun, Helen Wong
Knotted And Linked Products Of Recombination On T(2,N)#T(2,M) Substrates, Erica Flapan, Jeremy Grevet, Qi Li, Chen Daisy Sun, Helen Wong
Pomona Faculty Publications and Research
We develop a topological model of site-specific recombination that applies to substrates which are the connected sum of two torus links of the form T(2,n)#T(2,m). Then we use our model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two families, which we illustrate.
An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin
An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin
Pomona Faculty Publications and Research
Suppose E is a subset of the unit circle T and Hinfinity C Linfinity is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to znHinfinity. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.
Mathematical And Physical Aspects Of Complex Symmetric Operators, Stephan Ramon Garcia, Emil Prodan, Mihai Putinar
Mathematical And Physical Aspects Of Complex Symmetric Operators, Stephan Ramon Garcia, Emil Prodan, Mihai Putinar
Pomona Faculty Publications and Research
Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables.
On Approximation Schemes And Compactness, Asuman Güven Aksoy, Jose M. Almira
On Approximation Schemes And Compactness, Asuman Güven Aksoy, Jose M. Almira
CMC Faculty Publications and Research
We present an overview of some results about characterization of compactness in which the concept of approximation scheme has had a role. In particular, we present several results that were proved by the second author, jointly with Luther, a decade ago, when these authors were working on a very general theory of approximation spaces. We then introduce and show the basic properties of a new concept of compactness, which was studied by the first author in the eighties, by using a generalized concept of approximation scheme and its associated Kolmogorov numbers, which generalizes the classical concept of compactness.
The Apple Doesn’T Fall Far From The (Metric) Tree: Equivalence Of Definitions, Asuman Güven Aksoy, Sixian Jin
The Apple Doesn’T Fall Far From The (Metric) Tree: Equivalence Of Definitions, Asuman Güven Aksoy, Sixian Jin
CMC Faculty Publications and Research
In this paper we prove the equivalence of definitions for metric trees and for δ-Hperbolic spaces. We point out how these equivalences can be used to understand the geometric and metric properties of δ-Hperbolic spaces and its relation to CAT(κ) spaces.
Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño
Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño
All HMC Faculty Publications and Research
We prove the existence of a positive solution for a class of nonlin- ear elliptic systems with Neumann boundary conditions. The proof combines extensive use of a priori estimates for elliptic problems with Neumann boundary condition and Krasnoselskii's compression-expansion theorem
Infinitely Many Rotationally Symmetric Solutions To A Class Of Semilinear Laplace-Beltrami Equations On The Unit Sphere, Emily M. Fischer
Infinitely Many Rotationally Symmetric Solutions To A Class Of Semilinear Laplace-Beltrami Equations On The Unit Sphere, Emily M. Fischer
HMC Senior Theses
I show that a class of semilinear Laplace-Beltrami equations has infinitely many solutions on the unit sphere which are symmetric with respect to rotations around some axis. This equation corresponds to a singular ordinary differential equation, which we solve using energy analysis. We obtain a Pohozaev-type identity to prove that the energy is continuously increasing with the initial condition and then use phase plane analysis to prove the existence of infinitely many solutions.
Topological Symmetry Groups Of Small Complete Graphs, Erica Flapan, Dwayne Chambers
Topological Symmetry Groups Of Small Complete Graphs, Erica Flapan, Dwayne Chambers
Pomona Faculty Publications and Research
Topological symmetry groups were originally introduced to study the symmetries of non-rigid molecules, but have since been used to study the symmetries of any graph embedded in R3. In this paper, we determine for each complete graph Kn with n ≤ 6, what groups can occur as topological symmetry groups or orientation preserving topological symmetry groups of some embedding of the graph in R3.
Well-Rounded Zeta-Function Of Planar Arithmetic Lattices, Lenny Fukshansky
Well-Rounded Zeta-Function Of Planar Arithmetic Lattices, Lenny Fukshansky
CMC Faculty Publications and Research
We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at s=1 with a real pole of order 2, improving upon a result of Stefan Kühnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less than or equal to N is O(N log N) as N → ∞. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence …
On The Geometry Of Cyclic Lattices, Lenny Fukshansky, Xun Sun
On The Geometry Of Cyclic Lattices, Lenny Fukshansky, Xun Sun
CMC Faculty Publications and Research
Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices. Our main result is a counting estimate for the number of well-rounded …
Small Zeros Of Quadratic Forms Outside A Union Of Varieties, Wai Kiu Chan, Lenny Fukshansky, Glenn R. Henshaw
Small Zeros Of Quadratic Forms Outside A Union Of Varieties, Wai Kiu Chan, Lenny Fukshansky, Glenn R. Henshaw
CMC Faculty Publications and Research
Let be a quadratic form in variables defined on a vector space over a global field , and be a finite union of varieties defined by families of homogeneous polynomials over . We show that if contains a nontrivial zero of , then there exists a linearly independent collection of small-height zeros of in , where the height bound does not depend on the height of , only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace of the quadratic space such that is not …
Arithmetical Graphs, Riemann-Roch Structure For Lattices, And The Frobenius Number Problem, Jeremy Usatine
Arithmetical Graphs, Riemann-Roch Structure For Lattices, And The Frobenius Number Problem, Jeremy Usatine
HMC Senior Theses
If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the …
Reed's Conjecture And Cycle-Power Graphs, Alexa Serrato
Reed's Conjecture And Cycle-Power Graphs, Alexa Serrato
HMC Senior Theses
Reed's conjecture is a proposed upper bound for the chromatic number of a graph. Reed's conjecture has already been proven for several families of graphs. In this paper, I show how one of those families of graphs can be extended to include additional graphs and also show that Reed's conjecture holds for a family of graphs known as cycle-power graphs, and also for their complements.
Experiments On Surfactants And Thin Fluid Films, Peter Megson
Experiments On Surfactants And Thin Fluid Films, Peter Megson
HMC Senior Theses
We investigate the spatiotemporal dynamics of a surfactant monolayer on a thin fluid film spreading inward into a region devoid of surfactant, a system motivated by the alveolus of the human lung. We perform experiments that simultaneously measure the fluid height profile and the fluorescence intensity due to our fluorescent surfactant, NBD-PC. We perform experiments on both a Newtonian layer of glycerol and a shear-thinning fluid layer consisting of xanthan gum mixed with glycerol. We can very successfully extract height profiles on the xanthan gum fluid, although the simultaneous measurement of fluorescent intensity profiles proved problematic, as the laser tended …
Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy
Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy
CMC Senior Theses
In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.
Using Topology To Explore Mathematics Education Reform, Carling Sugarman
Using Topology To Explore Mathematics Education Reform, Carling Sugarman
HMC Senior Theses
Mathematics education is a constant topic of conversation in the United States. Many attempts have been made historically to reform teaching methods and improve student results. Particularly, past ideas have emphasized problem-solving to make math feel more applicable and enjoyable. Many have additionally tackled the widespread problem of “math anxiety” by creating lessons that are more discussion-based than drill-based to shift focus from speed and accuracy. In my project, I explored past reform goals and some added goals concerning students' perceptions of mathematics. To do so, I created and tested a pilot workshop in topology, a creative and intuitive field, …
There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich
There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich
HMC Senior Theses
L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.
Extended Book Review: Mathematics In Popular Culture: Essays On Appearances In Film, Fiction, Games, Television And Other Media, Edited By Jessica K. Sklar And Elizabeth S. Sklar; Loving+Hating Mathematics: Challenging The Myths Of Mathematical Life, By Reuben Hersh And Vera John-Steiner; Mathematicians: An Outer View Of The Inner World, By Mariana Cook, Gizem Karaali
Pomona Faculty Publications and Research
I was delighted to have the opportunity to review three books on a topic near and dear to my heart. In recent years it has become a passion of mine to think of and speak about the place of mathematics in the real world, in the world of those who are not doing mathematics for a living. I care about the applications and the implications of mathematics, but more than that, I care about the feelings and the impressions attached to it. Often math anxiety or skepticism comes up; the latter may be due to how frequently others (mis)use statistics, …
Quotients Of Gaussian Primes, Stephan Ramon Garcia
Quotients Of Gaussian Primes, Stephan Ramon Garcia
Pomona Faculty Publications and Research
It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"
Two-Part Reconstruction In Compressed Sensing, Yanting Ma, Dror Baron, Deanna Needell
Two-Part Reconstruction In Compressed Sensing, Yanting Ma, Dror Baron, Deanna Needell
CMC Faculty Publications and Research
Two-part reconstruction is a framework for signal recovery in compressed sensing (CS), in which the advantages of two different algorithms are combined. Our framework allow s to accelerate the reconstruction procedure without compromising the reconstruction quality. To illustrate the efficacy of ou r two-part approach, we extend the author’s previous Sudocodes algorithm and make it robust to measurement noise. In a 1- bit CS setting, promising numerical results indicate that our algorithm offers both a reduction in run-time and improvement in reconstruction quality