Open Access. Powered by Scholars. Published by Universities.®
- Discipline
-
- Arts and Humanities (820)
- Education (451)
- Science and Mathematics Education (332)
- Applied Mathematics (134)
- Creative Writing (129)
-
- Discrete Mathematics and Combinatorics (123)
- Poetry (106)
- Algebra (96)
- Higher Education (91)
- Other Mathematics (91)
- Philosophy (87)
- Social and Behavioral Sciences (77)
- Curriculum and Instruction (76)
- Scholarship of Teaching and Learning (69)
- Teacher Education and Professional Development (67)
- Geometry and Topology (66)
- Logic and Foundations of Mathematics (60)
- Computer Sciences (49)
- History (49)
- Statistics and Probability (45)
- Number Theory (44)
- Ordinary Differential Equations and Applied Dynamics (39)
- Analysis (33)
- Logic and Foundations (31)
- Educational Methods (30)
- Junior High, Intermediate, Middle School Education and Teaching (30)
- Life Sciences (30)
- Algebraic Geometry (28)
- Keyword
-
- Mathematics (107)
- Poetry (49)
- Combinatorics (39)
- Mathematics education (30)
- Education (25)
-
- Teaching (22)
- Algebra (20)
- Statistics (20)
- Creativity (17)
- Differential equations (17)
- Math (17)
- Geometry (16)
- Heights (16)
- Mathematical modeling (15)
- Motherhood (15)
- Pedagogy (15)
- Dirichlet problem (14)
- Algebra rings (13)
- Calculus (13)
- Fibonacci numbers (13)
- Lattice theory (13)
- Complex symmetric operator (12)
- Ethics (12)
- Functions (11)
- Number theory (11)
- Poem (11)
- Polynomials (11)
- Probability (11)
- STEM (11)
- Social justice (11)
- Publication Year
- Publication
-
- Journal of Humanistic Mathematics (708)
- Humanistic Mathematics Network Journal (409)
- All HMC Faculty Publications and Research (352)
- Pomona Faculty Publications and Research (144)
- HMC Senior Theses (104)
-
- CMC Faculty Publications and Research (86)
- CODEE Journal (73)
- CMC Senior Theses (29)
- CGU Theses & Dissertations (28)
- Scripps Senior Theses (16)
- The Transdisciplinary STEAM+ Journal (13)
- Pitzer Faculty Publications and Research (12)
- Pitzer Senior Theses (6)
- Pomona Senior Theses (2)
- WM Keck Science Faculty Papers (2)
- KGI Theses and Dissertations (1)
- Publication Type
- File Type
Articles 1141 - 1170 of 1985
Full-Text Articles in Mathematics
Noise, Delays, And Resonance In A Neural Network, Austin Quan
Noise, Delays, And Resonance In A Neural Network, Austin Quan
HMC Senior Theses
A stochastic-delay differential equation (SDDE) model of a small neural network with recurrent inhibition is presented and analyzed. The model exhibits unexpected transient behavior: oscillations that occur at the boundary of the basins of attraction when the system is bistable. These are known as delay-induced transitory oscillations (DITOs). This behavior is analyzed in the context of stochastic resonance, an unintuitive, though widely researched phenomenon in physical bistable systems where noise can play in constructive role in strengthening an input signal. A method for modeling the dynamics using a probabilistic three-state model is proposed, and supported with numerical evidence. The potential …
A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger
A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger
All HMC Faculty Publications and Research
We consider regular tessellations of the plane as infinite graphs in which q edges and q faces meet at each vertex, and in which p edges and p vertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the Euclidean plane; for 1/p + 1/q < 1/2, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all p ≥ 3 and q ≥ 3 with 1/p + 1/q ≤ 1/2, we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation.
Uniqueness Conditions For Low-Rank Matrix Recovery, Yonina C. Eldar, Deanna Needell, Yaniv Plan
Uniqueness Conditions For Low-Rank Matrix Recovery, Yonina C. Eldar, Deanna Needell, Yaniv Plan
CMC Faculty Publications and Research
Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractible approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. For example, m >= Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high probability. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever --- tractible or not. We show that for a family of random measurement ensembles, m >= 4nr - 4r^2 …
Acceleration Of Randomized Kaczmarz Method Via The Johnson-Lindenstrauss Lemma, Yonina C. Eldar, Deanna Needell
Acceleration Of Randomized Kaczmarz Method Via The Johnson-Lindenstrauss Lemma, Yonina C. Eldar, Deanna Needell
CMC Faculty Publications and Research
The Kaczmarz method is an algorithm for finding the solution to an overdetermined consistent system of linear equations Ax=b by iteratively projecting onto the solution spaces. The randomized version put forth by Strohmer and Vershynin yields provably exponential convergence in expectation, which for highly overdetermined systems even outperforms the conjugate gradient method. In this article we present a modified version of the randomized Kaczmarz method which at each iteration selects the optimal projection from a randomly chosen set, which in most cases significantly improves the convergence rate. We utilize a Johnson-Lindenstrauss dimension reduction technique to keep the runtime on the …
Bounds For Solid Angles Of Lattices Of Rank Three, Lenny Fukshansky, Sinai Robins
Bounds For Solid Angles Of Lattices Of Rank Three, Lenny Fukshansky, Sinai Robins
CMC Faculty Publications and Research
We find sharp absolute constants C1 and C2 with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C1,C2]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we …
What Do We Mean By Mathematical Proof?, Todd Cadwalladerolsker
What Do We Mean By Mathematical Proof?, Todd Cadwalladerolsker
Journal of Humanistic Mathematics
Mathematical proof lies at the foundations of mathematics, but there are several notions of what mathematical proof is, or might be. In fact, the idea of mathematical proof continues to evolve. In this article, I review the body of literature that argues that there are at least two widely held meanings of proof, and that the standards of proof are negotiated and agreed upon by the members of mathematical communities. The formal view of proof is contrasted with the view of proofs as arguments intended to convince a reader. These views are examined in the context of the various roles …
Aesthetic Considerations In Mathematics, Nathalie Sinclair
Aesthetic Considerations In Mathematics, Nathalie Sinclair
Journal of Humanistic Mathematics
Drawing on some of the principles of humanistic mathematics first outlined by Alvin White, this paper seeks to examine the way in which value judgments are implicated in the growth of the mathematics discipline. After a short overview of some of the roles ascribed to the mathematical aesthetic historically, I turn to more contemporary positioning of the aesthetic in order to develop a framework that offers insight into the particular values, assumptions and desires that constrain what is done in mathematics, how it is done and why. My goal, at least in part, is to bring together under one umbrella …
Welcome To The Journal Of Humanistic Mathematics, Mark Huber, Gizem Karaali
Welcome To The Journal Of Humanistic Mathematics, Mark Huber, Gizem Karaali
Journal of Humanistic Mathematics
No abstract provided.
Topological Symmetry Groups Of K4r+3, Dwayne Chambers '12, Erica Flapan, John D. O'Brien
Topological Symmetry Groups Of K4r+3, Dwayne Chambers '12, Erica Flapan, John D. O'Brien
Pomona Faculty Publications and Research
We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of a complete graph of the form K4r+3^in S³.
A Space-Filling, Nonregular Tetrahedron, Margaret Cagle, Joyce Frost, Christine Latulippe, Darryl H. Yong
A Space-Filling, Nonregular Tetrahedron, Margaret Cagle, Joyce Frost, Christine Latulippe, Darryl H. Yong
All HMC Faculty Publications and Research
This activity is an investigation of a special nonregular tetrahedron that can be arranged to fill space without leaving any internal gaps in the same way that certain planar figures tessellate the plane. These tetrahedra can be connected together with hinges to make fun and interesting puzzles. More background information can be found in the paper "An Amazing, Space-Filling, Non-Regular Tetrahedron" by Joyce Frost and Peg Cagle, published by the IAS/Park City Mathematics Institute (available at mathforum.org/pcmi/hstp/resources/dodeca/).
Coalitions And Cliques In The School Choice Problem, Sinan Aksoy, Alexander Adam Azzam, Chaya Coppersmith, Julie Glass, Gizem Karaali, Xueying Zhao, Xinjing Zhu
Coalitions And Cliques In The School Choice Problem, Sinan Aksoy, Alexander Adam Azzam, Chaya Coppersmith, Julie Glass, Gizem Karaali, Xueying Zhao, Xinjing Zhu
Pomona Faculty Publications and Research
The school choice mechanism design problem focuses on assignment mechanisms matching students to public schools in a given school district. The well-known Gale Shapley Student Optimal Stable Matching Mechanism (SOSM) is the most efficient stable mechanism proposed so far as a solution to this problem. However its inefficiency is well-documented, and recently the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) was proposed as a remedy for this weakness. In this note we describe two related adjustments to SOSM with the intention to address the same inefficiency issue. In one we create possibly artificial coalitions among students where some students modify their …
Review: Nearly Invariant Subspaces For Backwards Shifts On Vector-Valued Hardy Spaces, Stephan Ramon Garcia
Review: Nearly Invariant Subspaces For Backwards Shifts On Vector-Valued Hardy Spaces, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Review: Massey Products On Cycles Of Projective Lines And Trigonometric Solutions Of The Yang-Baxter Equations, Gizem Karaali
Review: Massey Products On Cycles Of Projective Lines And Trigonometric Solutions Of The Yang-Baxter Equations, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Unitary Equivalence To A Complex Symmetric Matrix: A Modulus Criterion, Stephan Ramon Garcia, Daniel E. Poore '11, Madeline K. Wyse '11
Unitary Equivalence To A Complex Symmetric Matrix: A Modulus Criterion, Stephan Ramon Garcia, Daniel E. Poore '11, Madeline K. Wyse '11
Pomona Faculty Publications and Research
We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We compare our approach to several existing methods [1, 19, 20] and present a number of examples.
Geologic Map Of The Ganiki Planitia Quadrangle (V–14), Venus, Eric B. Grosfils, Sylvan M. Long, Elizabeth M. Venechuk, Debra M. Hurwitz, Joseph W. Richards, Brian Kastl, Dorothy E. Drury, Johanna S. Hardin
Geologic Map Of The Ganiki Planitia Quadrangle (V–14), Venus, Eric B. Grosfils, Sylvan M. Long, Elizabeth M. Venechuk, Debra M. Hurwitz, Joseph W. Richards, Brian Kastl, Dorothy E. Drury, Johanna S. Hardin
Pomona Faculty Publications and Research
Our current research focuses on addressing four specific questions. Has the dominant style of volcanic expression within the quadrangle varied in a systematic fashion over time? Does the tectonic deformation within the quadrangle record significant regional patterns that vary spatially or temporally, and if so what are the scales, orientations and sources of the stress fields driving this deformation? If mantle upwelling and downwelling have played a significant role in the formation of Atla Regio and Atalanta Planitia as has been proposed, does the geology of Ganiki Planitia record evidence of northwest-directed lateral mantle flow connecting the two sites? Finally, …
Classical Kloosterman Sums: Representation Theory, Magic Squares, And Ramanujan Multigraphs, Patrick S. Fleming, Stephan Ramon Garcia, Gizem Karaali
Classical Kloosterman Sums: Representation Theory, Magic Squares, And Ramanujan Multigraphs, Patrick S. Fleming, Stephan Ramon Garcia, Gizem Karaali
Pomona Faculty Publications and Research
We consider a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of “magical” combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.
Quotient Sets And Diophantine Equations, Stephan Ramon Garcia, Vincent Selhorst-Jones '09, Daniel E. Poore '11, Noah Simon '08
Quotient Sets And Diophantine Equations, Stephan Ramon Garcia, Vincent Selhorst-Jones '09, Daniel E. Poore '11, Noah Simon '08
Pomona Faculty Publications and Research
Quotient sets U / U = {u / u¹: u, u¹ ϵ U} have been considered several times before in the MONTHLY. We consider more general quotient sets U / V and we apply our results to certain simultaneous Diophantine equations with side constraints.
Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Pomona Faculty Publications and Research
We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.
A Topological Approach To Molecular Chirality, Erica Flapan
A Topological Approach To Molecular Chirality, Erica Flapan
Pomona Faculty Publications and Research
Topology is the study of deformations of geometric figures. Chemistry is the study of molecular structures. At first glance these fields seem to have nothing in common. But let’s take a closer look to see how these fields come together in the study of molecular symmetries.
Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, Erica Flapan, Blake Mellor, Ramin Naimi
Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, Erica Flapan, Blake Mellor, Ramin Naimi
Pomona Faculty Publications and Research
We determine for which m the complete graph Km has an embedding in S3 whose topological symmetry group is isomorphic to one of the polyhedral groupsA4, A5 or S4.
Discrete Variable Representation Of The Angular Variables In Quantum Three-Body Scattering, David Caballero
Discrete Variable Representation Of The Angular Variables In Quantum Three-Body Scattering, David Caballero
CGU Theses & Dissertations
There are many numerical methods to study the quantum mechanical three-body scattering system using the Schrodinger equation. Traditionally, a partial-wave decomposition of the total wave function is carried out first, allowing the scattering system to be solved one partial wave at a time. This is convenient when the interaction is central, causing the total angular momentum to be conserved during the collision process. This is not possible in the presence of a non-central interaction such as a laser field, where the total angular momentum is not conserved during the collision process. The Discrete Variable Representation is a new method for …
A Primer Of Swarm Equilibria, Andrew J. Bernoff, Chad M. Topaz
A Primer Of Swarm Equilibria, Andrew J. Bernoff, Chad M. Topaz
All HMC Faculty Publications and Research
We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding continuum population density. Equilibrium solutions are extrema of an energy functional and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying …
Evidence Of The Harmonic Faraday Instability In Ultrasonic Atomization Experiments With A Deep, Inviscid Fluid, Andrew P. Higginbotham '09, Aaron Guillen '11, Nathan C. Jones '10, Thomas D. Donnelly, Andrew J. Bernoff
Evidence Of The Harmonic Faraday Instability In Ultrasonic Atomization Experiments With A Deep, Inviscid Fluid, Andrew P. Higginbotham '09, Aaron Guillen '11, Nathan C. Jones '10, Thomas D. Donnelly, Andrew J. Bernoff
All HMC Faculty Publications and Research
A popular method for generating micron-sized aerosols is to submerge ultrasonic (ω~MHz) piezoelectric oscillators in a water bath. The submerged oscillator atomizes the fluid, creating droplets with radii proportional to the wavelength of the standing wave at the fluid surface. Classical theory for the Faraday instability predicts a parametric instability driving a capillary wave at the subharmonic (ω/2) frequency. For many applications it is desirable to reduce the size of the droplets; however, using higher frequency oscillators becomes impractical beyond a few MHz. Observations are presented that demonstrate that smaller droplets may also be created by …
Strong Nonnegativity And Sums Of Squares On Real Varieties, Mohamed Omar, Brian Osserman
Strong Nonnegativity And Sums Of Squares On Real Varieties, Mohamed Omar, Brian Osserman
All HMC Faculty Publications and Research
Motivated by scheme theory, we introduce strong nonnegativity on real varieties, which has the property that a sum of squares is strongly nonnegative. We show that this algebraic property is equivalent to nonnegativity for nonsingular real varieties. Moreover, for singular varieties, we reprove and generalize obstructions of Gouveia and Netzer to the convergence of the theta body hierarchy of convex bodies approximating the convex hull of a real variety.
Applications Of Convex And Algebraic Geometry To Graphs And Polytopes, Mohamed Omar
Applications Of Convex And Algebraic Geometry To Graphs And Polytopes, Mohamed Omar
All HMC Faculty Publications and Research
No abstract provided.
Women In Mathematics: An Historical Account Of Women's Experiences And Achievement, Kendra D. Huff
Women In Mathematics: An Historical Account Of Women's Experiences And Achievement, Kendra D. Huff
CMC Senior Theses
For a long time, women have struggled to gain complete acceptance in the mathematics field. The purpose of this paper is to explore the history of women in the field of mathematics, the impact and experiences of current female mathematicians, and the common trends for women in the mathematics field, through literature review and personal interviews. This paper looks at the lives of four famous female mathematicians, as well as female mathematicians in the Claremont Colleges who were interviewed for this paper. Specifically this paper examines the discrimination they faced and how they overcame this discrimination, as well as the …
Twisted Virtual Biracks, Jessica Ceniceros
Twisted Virtual Biracks, Jessica Ceniceros
CMC Senior Theses
This thesis will take a look at a branch of topology called knot theory. We will first look at what started the study of this field, classical knot theory. Knot invariants such as the Bracket polynomial and the Jones polynomial will be introduced and studied. We will then explore racks and biracks along with the axioms obtained from the Reidemeister moves. We will then move on to generalize classical knot theory to what is now known as virtual knot theory which was first introduced by Louis Kauffman. Finally, we take a look at a newer aspect of knot theory, twisted …
The Mathematical Landscape, Antonio Collazo
The Mathematical Landscape, Antonio Collazo
CMC Senior Theses
The intent of this paper is to present the reader will enough information to spark a curiosity in to the subject. By no means is the following a complete formulation of any of the topics covered. I want to give the reader a tour of the mathematical landscape. There are plenty of further details to explore in each section, I have just touched the tip the iceberg. The work is basically in four sections: Numbers, Geometry, Functions, Sets and Logic, which are the basic building blocks of Math. The first sections are a exposition into the mathematical objects and their …
Sums Of Evenly Spaced Binomial Coefficients, Arthur T. Benjamin, Bob Chen '10, Kimberly Kindred
Sums Of Evenly Spaced Binomial Coefficients, Arthur T. Benjamin, Bob Chen '10, Kimberly Kindred
All HMC Faculty Publications and Research
We provide a combinatorial proof of a formula for the sum of evenly spaced binomial coefficients. This identity, along with a generalization, are proved by counting weighted walks on a graph.
Teaching Research: Encouraging Discoveries, Francis E. Su
Teaching Research: Encouraging Discoveries, Francis E. Su
All HMC Faculty Publications and Research
What does it take to turn a learner into a discoverer? Or to turn a teacher into a co-adventurer? A handful of experiences—from teaching a middle-school math class to doing research with undergraduates—have changed the way that I would answer these questions. Some of the lessons I’ve learned have surprised me.