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Articles 1171 - 1200 of 1985

Full-Text Articles in Mathematics

Compressed Sensing With Coherent And Redundant Dictionaries, Emmanuel J. Candès, Yonina C. Eldar, Deanna Needell, Paige Randall Oct 2010

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 …

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Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky Oct 2010

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 …

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"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 Oct 2010

"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 …

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Existence Of Solutions For A Semilinear Wave Equation With Non-Monotone Nonlinearity, Alfonso Castro, Benjamin Preskill '09 Oct 2010

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.

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Combinatorial Trigonometry With Chebyshev Polynomials, Arthur T. Benjamin, Larry Ericksen, Pallavi Jayawant, Mark Shattuck Aug 2010

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.

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Combinatorially Composing Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07 Aug 2010

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).

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Group Frames And Partially Ranked Data, Kwang B. Ketcham May 2010

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.

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Combinatorial Proofs Using Complex Weights, Bo Chen May 2010

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.

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Optimizing Restaurant Reservation Scheduling, Jacob Feldman May 2010

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 …

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Arithmetic On Specializable Continued Fractions, Ross C. Merriam May 2010

Arithmetic On Specializable Continued Fractions, Ross C. Merriam

HMC Senior Theses

No abstract provided.

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Understanding Voting For Committees Using Wreath Products, Stephen C. Lee May 2010

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.

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Computational Feasibility Of Increasing The Visibility Of Vertices In Covert Networks, Yaniv J. Ovadia May 2010

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 …

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A Nonlinear Ode Model Of Tumor Growth And Effect Of Immunotherapy And Chemotherapy Treatment In Colorectal Cancer, Hannah P. Savage May 2010

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 …

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Minimal Circuits For Very Incompletely Specified Boolean Functions, Richard Strong Bowen May 2010

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.

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A Multistage Incidence Estimation Model For Diseases With Differential Mortality, Alyssa W. Dray May 2010

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 …

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Signal Recovery From Inaccurate And Incomplete Measurements Via Regularized Orthogonal Matching Pursuit, Deanna Needell, Roman Vershynin Apr 2010

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.

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Randomized Kaczmarz Solver For Noisy Linear Systems, Deanna Needell Mar 2010

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, …

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Review: Nontangential Limits In Pt(Μ)-Spaces And The Index Of Invariant Subgroups, Stephan Ramon Garcia Jan 2010

Review: Nontangential Limits In Pt(Μ)-Spaces And The Index Of Invariant Subgroups, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

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Review: Common Cyclic Vectors For Unitary Operators, Stephan Ramon Garcia Jan 2010

Review: Common Cyclic Vectors For Unitary Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

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Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar Jan 2010

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.

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Mathematical Biology At An Undergraduate Liberal Arts College, Stephen C. Adolph, Lisette G. De Pillis Jan 2010

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.

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Stability And Dynamics Of Self-Similarity In Evolution Equations, Andrew J. Bernoff, Thomas P. Witelski Jan 2010

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 …

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Two-Player Envy-Free Multi-Cake Division, John Cloutier '03, Kathryn L. Nyman, Francis E. Su Jan 2010

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 …

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Review: The Spectrum Of Some Compressions Of Unilateral Shifts, Stephan Ramon Garcia Jan 2010

Review: The Spectrum Of Some Compressions Of Unilateral Shifts, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

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Unitary Equivalence To A Complex Symmetric Matrix: Geometric Criteria, Levon Balayan '09, Stephan Ramon Garcia Jan 2010

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.

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Review: Classification Of Quasi-Trigonometric Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali Jan 2010

Review: Classification Of Quasi-Trigonometric Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

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Review: Quantization Of Hamiltonian-Type Lie Algebras, Gizem Karaali Jan 2010

Review: Quantization Of Hamiltonian-Type Lie Algebras, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

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The Norm Of A Truncated Toeplitz Operator, Stephan Ramon Garcia, William T. Ross Jan 2010

The Norm Of A Truncated Toeplitz Operator, Stephan Ramon Garcia, William T. Ross

Pomona Faculty Publications and Research

We prove several lower bounds for the norm of a truncated Toeplitz operator and obtain a curious relationship between the H2 and H∞ norms of functions in model spaces.

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Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen Jan 2010

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 $(\dim \ker T, \dim \ker T^*)$.

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Review: Intertwining Symmetry Algebras Of Quantum Superintegrable Systems, Gizem Karaali Jan 2010

Review: Intertwining Symmetry Algebras Of Quantum Superintegrable Systems, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

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