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Articles 1 - 30 of 36
Full-Text Articles in Physical Sciences and Mathematics
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.
Compressed Sensing With Coherent And Redundant Dictionaries, Emmanuel J. Candès, Yonina C. Eldar, Deanna Needell, Paige Randall
Compressed Sensing With Coherent And Redundant Dictionaries, Emmanuel J. Candès, Yonina C. Eldar, Deanna Needell, Paige Randall
CMC Faculty Publications and Research
This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not only that compressed sensing is viable in this context, but also that accurate recovery is possible via an ℓ1-analysis optimization problem. We introduce a condition on the measurement/sensing matrix, which is a natural generalization of the now well-known restricted isometry property, and which guarantees accurate recovery of signals that are …
"Toward Integration: From Quantitative Biology To Mathbio-Biomath?", Pat Marsteller, Lisette G. De Pillis, Ann Findley, Karl Joplin, John Pelesko, Karen Nelson, Katerina Thompson, David Usher, Joseph Watkins
"Toward Integration: From Quantitative Biology To Mathbio-Biomath?", Pat Marsteller, Lisette G. De Pillis, Ann Findley, Karl Joplin, John Pelesko, Karen Nelson, Katerina Thompson, David Usher, Joseph Watkins
All HMC Faculty Publications and Research
In response to the call of BIO2010 for integrating quantitative skills into undergraduate biology education, 30 Howard Hughes Medical Institute (HHMI) Program Directors at the 2006 HHMI Program Directors Meeting established a consortium to investigate, implement, develop, and disseminate best practices resulting from the integration of math and biology. With the assistance of an HHMI-funded mini-grant, led by Karl Joplin of East Tennessee State University, and support in institutional HHMI grants at Emory and University of Delaware, these institutions held a series of summer institutes and workshops to document progress toward and address the challenges of implementing a more quantitative …
Existence Of Solutions For A Semilinear Wave Equation With Non-Monotone Nonlinearity, Alfonso Castro, Benjamin Preskill '09
Existence Of Solutions For A Semilinear Wave Equation With Non-Monotone Nonlinearity, Alfonso Castro, Benjamin Preskill '09
All HMC Faculty Publications and Research
For double-periodic and Dirichlet-periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with asymptotically linear nonlinearity, no resonance, and non-monotone nonlinearity when the forcing term is not flat on characteristics. The solutions are in L∞ when the forcing term is in L∞ and continous when the forcing term is continuous. This is in contrast with the results in [4], where the non-enxistence of continuous solutions is established even when forcing term is of class C∞ but is flat on a characteristic.
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN where N≥ 2. Let ZK be a union of varieties defined over K such that V ⊈ ZK. We prove the existence of a point of small height in V \ ZK, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of hypersurface containing ZK, where dependence on …
Combinatorial Trigonometry With Chebyshev Polynomials, Arthur T. Benjamin, Larry Ericksen, Pallavi Jayawant, Mark Shattuck
Combinatorial Trigonometry With Chebyshev Polynomials, Arthur T. Benjamin, Larry Ericksen, Pallavi Jayawant, Mark Shattuck
All HMC Faculty Publications and Research
We provide a combinatorial proof of the trigonometric identity cos(nθ) = Tncos(θ),
where Tn is the Chebyshev polynomial of the first kind. We also provide combinatorial proofs of other trigonometric identities, including those involving Chebyshev polynomials of the second kind.
Combinatorially Composing Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
Combinatorially Composing Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
All HMC Faculty Publications and Research
We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m, n ≥ 0, Tm(Tn(x)) = Tmn(x) and Um-1 (Tn(x))Un-1(x) = Umn-1(x).
Arithmetic On Specializable Continued Fractions, Ross C. Merriam
Arithmetic On Specializable Continued Fractions, Ross C. Merriam
HMC Senior Theses
No abstract provided.
Minimal Circuits For Very Incompletely Specified Boolean Functions, Richard Strong Bowen
Minimal Circuits For Very Incompletely Specified Boolean Functions, Richard Strong Bowen
HMC Senior Theses
In this report, asymptotic upper and lower bounds are given for the minimum number of gates required to compute a function which is only partially specified and for which we allow a certain amount of error. The upper and lower bounds match. Hence, the behavior of these minimum circuit sizes is completely (asymptotically) determined.
Combinatorial Proofs Using Complex Weights, Bo Chen
Combinatorial Proofs Using Complex Weights, Bo Chen
HMC Senior Theses
In 1961, Kasteleyn, Fisher, and Temperley gave a result for the number of possible tilings of a 2m 2n checkerboard with dominoes. Their proof involves the evaluation of a complicated Pfaffian. In this thesis we investigate combinatorial strategies to evaluate the sum of evenly spaced binomial coefficients, and present steps towards a purely combinatorial proof of the 1961 result.
Computational Feasibility Of Increasing The Visibility Of Vertices In Covert Networks, Yaniv J. Ovadia
Computational Feasibility Of Increasing The Visibility Of Vertices In Covert Networks, Yaniv J. Ovadia
HMC Senior Theses
Disrupting terrorist and other covert networks requires identifying and capturing key leaders. Previous research by Martonosi et al. (2009) defines a load metric on vertices of a covert network representing the amount of communication in which a vertex is expected to participate. They suggest that the visibility of a target vertex can be increased by removing other, more accessible members of the network. This report evaluates the feasibility of efficiently calculating the optimal subset of vertices to remove. We begin by proving that the general problem of identifying the optimally load maximizing vertex set removal is NP-complete. We then consider …
Group Frames And Partially Ranked Data, Kwang B. Ketcham
Group Frames And Partially Ranked Data, Kwang B. Ketcham
HMC Senior Theses
We give an overview of finite group frames and their applications to calculating summary statistics from partially ranked data, drawing upon the work of Rachel Cranfill (2009). We also provide a summary of the representation theory of compact Lie groups. We introduce both of these concepts as possible avenues beyond finite group representations, and also to suggest exploration into calculating summary statistics on Hilbert spaces using representations of Lie groups acting upon those spaces.
A Nonlinear Ode Model Of Tumor Growth And Effect Of Immunotherapy And Chemotherapy Treatment In Colorectal Cancer, Hannah P. Savage
A Nonlinear Ode Model Of Tumor Growth And Effect Of Immunotherapy And Chemotherapy Treatment In Colorectal Cancer, Hannah P. Savage
HMC Senior Theses
Colorectal cancer will kill approximately 50,000 people in the United States this year. Current treatment options, including surgery, chemotherapy, and radiation, are often able to force the cancer into remission, but better treatments are needed to help those who don't respond to current treatments. A new and promising treatment option, monoclonal-antibody therapy, has the potential to help reduce the deaths caused by colorectal cancer, but most monoclonal-antibody drugs are currently still in trial phases, and the variations in the dosing schedule of those currently approved for use have not been heavily explored. We have modified a nonlinear ODE tumor/treatment model …
A Multistage Incidence Estimation Model For Diseases With Differential Mortality, Alyssa W. Dray
A Multistage Incidence Estimation Model For Diseases With Differential Mortality, Alyssa W. Dray
HMC Senior Theses
According to theWorld Health Organization, surgically removable cataract remains the leading cause of blindness worldwide. In sub-Saharan Africa, cataract surgical rate targets should ideally be set based on cataract incidence (the number of new cataracts developed each year). Unfortunately, the longitudinal studies necessary to measure incidence have not yet been feasible in these areas. Our research instead proposes a method for estimating incidence based on available cataract prevalence data. We extend a method proposed by Podgor and Leske (1986) to estimate age-specific incidence from age-specific prevalence in single diseases with differential mortality. A two-stage disease extension is created in order …
Optimizing Restaurant Reservation Scheduling, Jacob Feldman
Optimizing Restaurant Reservation Scheduling, Jacob Feldman
HMC Senior Theses
We consider a yield-management approach to determine whether a restaurant should accept or reject a pending reservation request. This approach was examined by Bossert (2009), where the decision for each request is evaluated by an approximate dynamic program (ADP) that bases its decision on a realization of future demand. This model only considers assigning requests to their desired time slot. We expand Bossert's ADP model to incorporate an element of flexibility that allows requests to be assigned to a time slot that differs from the customer's initially requested time. To estimate the future seat utilization given a particular decision, a …
Understanding Voting For Committees Using Wreath Products, Stephen C. Lee
Understanding Voting For Committees Using Wreath Products, Stephen C. Lee
HMC Senior Theses
In this thesis, we construct an algebraic framework for analyzing committee elections. In this framework, module homomorphisms are used to model positional voting procedures. Using the action of the wreath product group S2[Sn] on these modules, we obtain module decompositions which help us to gain an understanding of the module homomorphism. We use these decompositions to construct some interesting voting paradoxes.
Signal Recovery From Inaccurate And Incomplete Measurements Via Regularized Orthogonal Matching Pursuit, Deanna Needell, Roman Vershynin
Signal Recovery From Inaccurate And Incomplete Measurements Via Regularized Orthogonal Matching Pursuit, Deanna Needell, Roman Vershynin
CMC Faculty Publications and Research
We demonstrate a simple greedy algorithm that can reliably recover a vector v ?? ??d from incomplete and inaccurate measurements x = ??v + e. Here, ?? is a N x d measurement matrix with Nv with O(n) nonzeros from its inaccurate measurements x in at most n iterations, where each iteration amounts to solving a least squares problem. The noise level of the recovery is proportional to ??{logn} ||e||2. In particular, if the error term e vanishes the reconstruction is exact.
Randomized Kaczmarz Solver For Noisy Linear Systems, Deanna Needell
Randomized Kaczmarz Solver For Noisy Linear Systems, Deanna Needell
CMC Faculty Publications and Research
The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b. Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It was proved that for overdetermined systems, the randomized Kaczmarz method converges with expected exponential rate, independent of the number of equations in the system. Here we analyze the case where the system Ax=b is corrupted by noise, so we consider the system where Ax is approximately b + r where r is an arbitrary error vector. We prove that in this noisy version, …
Review: The Semi-Dynamical Reflection Equation: Solutions And Structure Matrices, Gizem Karaali
Review: The Semi-Dynamical Reflection Equation: Solutions And Structure Matrices, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Voting In Agreeable Societies, Deborah E. Berg '06, Serguei Norine, Francis E. Su, Robin Thomas, Paul Wollan
Voting In Agreeable Societies, Deborah E. Berg '06, Serguei Norine, Francis E. Su, Robin Thomas, Paul Wollan
All HMC Faculty Publications and Research
No abstract provided in this article.
Two-Player Envy-Free Multi-Cake Division, John Cloutier '03, Kathryn L. Nyman, Francis E. Su
Two-Player Envy-Free Multi-Cake Division, John Cloutier '03, Kathryn L. Nyman, Francis E. Su
All HMC Faculty Publications and Research
We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may not exist for three cakes cut into three pieces each. However, there do exist such divisions for two cakes cut into three pieces each, and for three cakes cut into four pieces each. The …
Stability And Dynamics Of Self-Similarity In Evolution Equations, Andrew J. Bernoff, Thomas P. Witelski
Stability And Dynamics Of Self-Similarity In Evolution Equations, Andrew J. Bernoff, Thomas P. Witelski
All HMC Faculty Publications and Research
A methodology for studying the linear stability of self-similar solutions is discussed. These fundamental ideas are illustrated on three prototype problems: a simple ODE with finite-time blow-up, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, the use of dimensional analysis to derive scaling invariant similarity variables is discussed, as well as the role of symmetries in the context of stability of self-similar dynamics. The …
Local Versus Global Search In Channel Graphs, A.H. Hunter, Nicholas Pippenger
Local Versus Global Search In Channel Graphs, A.H. Hunter, Nicholas Pippenger
All HMC Faculty Publications and Research
Previous studies of search in channel graphs has assumed that the search is global; that is, that the status of any link can be probed by the search algorithm at any time. We consider for the first time local search, for which only links to which an idle path from the source has already been established may be probed. We show that some well known channel graphs may require exponentially more probes, on the average, when search must be local than when it may be global.
Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar
Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar
All HMC Faculty Publications and Research
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.
Mathematical Biology At An Undergraduate Liberal Arts College, Stephen C. Adolph, Lisette G. De Pillis
Mathematical Biology At An Undergraduate Liberal Arts College, Stephen C. Adolph, Lisette G. De Pillis
All HMC Faculty Publications and Research
Since 2002 we have offered an undergraduate major in Mathematical Biology at Harvey Mudd College. The major was developed and is administered jointly by the mathematics and biology faculty. In this paper we describe the major, courses, and faculty and student research and discuss some of the challenges and opportunities we have experienced.
Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen
Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen
Pomona Faculty Publications and Research
We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ so that $T = CT^*C$. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data $(\dim \ker T, \dim \ker T^*)$.
Review: Common Cyclic Vectors For Unitary Operators, Stephan Ramon Garcia
Review: Common Cyclic Vectors For Unitary Operators, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Review: The Spectrum Of Some Compressions Of Unilateral Shifts, Stephan Ramon Garcia
Review: The Spectrum Of Some Compressions Of Unilateral Shifts, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Unitary Equivalence To A Complex Symmetric Matrix: Geometric Criteria, Levon Balayan '09, Stephan Ramon Garcia
Unitary Equivalence To A Complex Symmetric Matrix: Geometric Criteria, Levon Balayan '09, Stephan Ramon Garcia
Pomona Faculty Publications and Research
We develop several methods, based on the geometric relationship between the eigenspaces of a matrix and its adjoint, for determining whether a square matrix having distinct eigenvalues is unitarily equivalent to a complex symmetric matrix. Equivalently, we characterize those matrices having distinct eigenvalues which lie in the unitary orbit of the complex symmetric matrices.