Physical Sciences and Mathematics Commons^™

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

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?"

Kaczmarz Algorithm With Soft Constraints For User Interface Layout, Noreen Jamil, Deanna Needell, Johannes Muller, Christof Lutteroth, Gerald Weber

CMC Faculty Publications and Research

The Kaczmarz method is an iterative method for solving large systems of equations that projects iterates orthogonally onto the solution space of each equation. In contrast to direct methods such as Gaussian elimination or QR-factorization, this algorithm is efficient for problems with sparse matrices, as they appear in constraint-based user interface (UI) layout specifications. However, the Kaczmarz method as described in the literature has its limitations: it considers only equality constraints and does not support soft constraints, which makes it inapplicable to the UI layout problem.

In this paper we extend the Kaczmarz method for solving specifications containing soft constraints, …

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

Review: Unitary Equivalence To Truncated Toeplitz Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Book Review: Encyclopedia Of Mathematics And Society, Gizem Karaali

Pomona Faculty Publications and Research

The Encyclopedia of Mathematics and Society is an impressive achievement of collective effort and serious thought, an amazing collection of delightful, unexpected essays, a sourcebook for students and teachers alike. The experience of reading the EMS was, for this encyclopedia enthusiast, enlightening and enjoyable.

If any librarians out there are still wondering, let me be clear: I strongly recommend this encyclopedia. For individuals, the expense might well be an initial turn-off; keeping in mind that the hard copy books come with online access to the same content might alleviate some of the pain.

As for me? I expect to enjoy …

Signal Space Cosamp For Sparse Recovery With Redundant Dictionaries, Mark A. Davenport, Deanna Needell, Michael B. Wakin

CMC Faculty Publications and Research

Compressive sensing (CS) has recently emerged as a powerful framework for acquiring sparse signals. The bulk of the CS literature has focused on the case where the acquired signal has a sparse or compressible representation in an orthonormal basis. In practice, however, there are many signals that cannot be sparsely represented or approximated using an orthonormal basis, but that do have sparse representations in a redundant dictionary. Standard results in CS can sometimes be extended to handle this case provided that the dictionary is sufficiently incoherent or well conditioned, but these approaches fail to address the case of a truly …

Lattice Point Counting And Height Bounds Over Number Fields And Quaternion Algebras, Lenny Fukshansky, Glenn Henshaw

CMC Faculty Publications and Research

An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a collection of height comparison inequalities for heights over a number …

Super-Resolution Via Superset Selection And Pruning, Laurent Demanet, Deanna Needell, Nam Nguyen

CMC Faculty Publications and Research

We present a pursuit-like algorithm that we call the "superset method" for recovery of sparse vectors from consecutive Fourier measurements in the super-resolution regime. The algorithm has a subspace identification step that hinges on the translation invariance of the Fourier transform, followed by a removal step to estimate the solution's support. The superset method is always successful in the noiseless regime (unlike L1-minimization) and generalizes to higher dimensions (unlike the matrix pencil method). Relative robustness to noise is demonstrated numerically.

Using Correlated Subset Structure For Compressive Sensing Recovery, Atul Divekar, Deanna Needell

CMC Faculty Publications and Research

Compressive sensing is a methodology for the reconstruction of sparse or compressible signals using far fewer samples than required by the Nyquist criterion. However, many of the results in compressive sensing concern random sampling matrices such as Gaussian and Bernoulli matrices. In common physically feasible signal acquisition and reconstruction scenarios such as super-resolution of images, the sensing matrix has a non-random structure with highly correlated columns. Here we present a compressive sensing recovery algorithm that exploits this correlation structure. We provide algorithmic justification as well as empirical comparisons.

Entropy Driven Crystal Formation On Highly Strained Substrates, John R. Savage, Stefan F. Hopp, Rajesh Ganapathy, Sharon J. Gerbode, Andreas Heuer, Itai Cohen

All HMC Faculty Publications and Research

In heteroepitaxy, lattice mismatch between the deposited material and the underlying surface strongly affects nucleation and growth processes. The effect of mismatch is well studied in atoms with growth kinetics typically dominated by bond formation with interaction lengths on the order of one lattice spacing. In contrast, less is understood about how mismatch affects crystallization of larger particles, such as globular proteins and nanoparticles, where interparticle interaction energies are often comparable to thermal fluctuations and are short ranged, extending only a fraction of the particle size. Here, using colloidal experiments and simulations, we find particles with short-range attractive interactions form …

Mathematical Modeling Of The Regulatory T Cell Effects On Renal Cell Carcinoma Treatment, Lisette G. De Pillis, Trevor Caldwell '12, Elizabeth Sarapata '13, Heather Williams '12

All HMC Faculty Publications and Research

We present a mathematical model to study the effects of the regulatory T cells (Treg) on Renal Cell Carcinoma (RCC) treatment with sunitinib. The drug sunitinib inhibits the natural self-regulation of the immune system, allowing the effector components of the immune system to function for longer periods of time. This mathematical model builds upon our non-linear ODE model by de Pillis et al. (2009) [13] to incorporate sunitinib treatment, regulatory T cell dynamics, and RCC-specific parameters. The model also elucidates the roles of certain RCC-specific parameters in determining key differences between in silico patients whose immune profiles allowed them to …

B Cell Chronic Lymphocytic Leukemia - A Model With Immune Response, Seema Nanda, Lisette G. De Pillis, Ami E. Radunskaya

All HMC Faculty Publications and Research

B cell chronic lymphocytic leukemia (B-CLL) is known to have substantial clinical heterogeneity. There is no cure, but treatments allow for disease management. However, the wide range of clinical courses experienced by B-CLL patients makes prognosis and hence treatment a significant challenge. In an attempt to study disease progression across different patients via a unified yet flexible approach, we present a mathematical model of B-CLL with immune response, that can capture both rapid and slow disease progression. This model includes four different cell populations in the peripheral blood of humans: B-CLL cells, NK cells, cytotoxic T cells and helper T …

Near-Optimal Compressed Sensing Guarantees For Total Variation Minimization, Deanna Needell, R. Ward

CMC Faculty Publications and Research

Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. This paper extends recent reconstruction guarantees for two-dimensional images x ∈ ℂN2 to signals x ∈ ℂNd of arbitrary dimension d ≥ 2 and to isotropic total variation problems. In this …

A Borsuk-Ulam Equivalent That Directly Implies Sperner's Lemma, Kathryn L. Nyman, Francis Su

All HMC Faculty Publications and Research

We show that Fan’s 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem. Moreover, unlike other Borsuk–Ulam equivalents, we show that this lemma directly implies Sperner’s Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk–Ulam theorem implies the Brouwer fixed-point theorem, or that the Lusternik–Schnirelmann–Borsuk theorem implies the KKM lemma.

Quantitative Approaches To Sustainability Seminars, Rachel Levy

All HMC Faculty Publications and Research

How can mathematicians contribute to education of about sustainability? Mathematicians study climate change, energy-related technologies, models of energy availability, production and consumption, and even the political and social aspects of sustainable legislation and practices. However, at this point, few courses on sustainability can be found in math department offerings. When we consider problems that our current and future students will face, energy sustainability certainly seems important. But how many of these ideas reach our classrooms?

Stable Image Reconstruction Using Total Variation Minimization, Deanna Needell, Rachel Ward

CMC Faculty Publications and Research

This article presents near-optimal guarantees for stable and robust image recovery from undersampled noisy measurements using total variation minimization. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be reconstructed to within the best s-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of suitably incoherent matrices.

A Model Of Dendritic Cell Therapy For Melanoma, Lisette G. De Pillis, Angela Gallegos, Ami E. Radunskaya

All HMC Faculty Publications and Research

Dendritic cells are a promising immunotherapy tool for boosting an individual’s antigen-specific immune response to cancer. We develop a mathematical model using differential and delay-differential equations to describe the interactions between dendritic cells, effector-immune cells, and tumor cells. We account for the trafficking of immune cells between lymph, blood, and tumor compartments. Our model reflects experimental results both for dendritic cell trafficking and for immune suppression of tumor growth in mice. In addition, in silico experiments suggest more effective immunotherapy treatment protocols can be achieved by modifying dose location and schedule. A sensitivity analysis of the model reveals which patient-specific …

The Lesson Of Grace In Teaching, Francis Su

All HMC Faculty Publications and Research

I want to talk about the biggest life lesson that I have learned, and that I continue to learn over and over again. It is deep and profound. It has changed the way I relate with people. It has reshaped my academic life. And it continually renovates the way I approach my students.

The Brave New World Of Open Access & Creative Commons: A Humanistic Experiment In Mathematical Publishing, Gizem Karaali

Pomona Faculty Publications and Research

In January 2011 the Journal of Humanistic Mathematics (JHM) published its first issue. JHM (http://scholarship.claremont.edu/jhm) is an online-only, peer-reviewed, open-access journal which has passed the all important ten-thousand-download barrier in its first anniversary. In order to remain faithful to the fundamental principles of open access, JHM uses Creative Commons licensing, where authors retain copyright of their work, but others are free to reuse them (with proper attribution). In this note I share and reflect upon our experience with open access and Creative Commons.

On The Matrix Equation Xa + Ax_T = 0, Ii, Alice Zhuo-Yu Chan '14, Luis Alberto Garcia '14, Stephan Ramon Garcia, Amy L. Shoemaker '14

Pomona Faculty Publications and Research

The matrix equation $XA + AX^T = 0$ was recently introduced by De Ter\'an and Dopico to study the dimension of congruence orbits. They reduced the study of this equation to a number of special cases, several of which have not been explicitly solved. In this note we obtain an explicit, closed-form solution in the difficult Type 0-I interaction case.

Uncle Sam’S Badge: Identity And Representation In The Usda Forest Service, 1905–2013, Char Miller

Pomona Faculty Publications and Research

Howard Abbey could recall the exact moment when he learned that he had passed the forest ranger’s examination for the newly established USDA Forest Service (USFS). In the early morning of Aug. 1, 1905, while he was managing a team of horses pulling a mowing machine on the McIntosh Ranch in the northern Sierra Nevada Mountains, Allen Ray Powers, a Forest Assistant on the Plumas Forest Reserve, rode up and “informed me that I was wanted at the Forest Supervisor’s office in Quincy.” Abbey handed over the reins to his boss and walked the 2 miles to town where he …

Virtual Machine Workloads: The Case For New Nas Benchmarks, Vasily Tarasov, Dean Hildebrand, Geoffrey H. Kuenning, Erez Zadok

All HMC Faculty Publications and Research

Network Attached Storage (NAS) and Virtual Machines (VMs) are widely used in data centers thanks to their manageability, scalability, and ability to consolidate resources. But the shift from physical to virtual clients drastically changes the I/O workloads to seen on NAS servers, due to guest file system encapsulation in virtual disk images and the multiplexing of request streams from different VMs. Unfortunately, current NAS workload generators and benchmarks produce workloads typical to physical machines.

This paper makes two contributions. First, we studied the extent to which virtualization is changing existing NAS workloads. We observed significant changes, including the disappearance of …

Social Aggregation In Pea Aphids: Experiment And Random Walk Modeling, Christa Nilsen, John Paige, Olivia Warner, Benjamin Mayhew, Ryan Sutley, Matthew Lam '15, Andrew J. Bernoff, Chad M. Topaz

All HMC Faculty Publications and Research

From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid, Acyrthosiphon pisum. Specifically, we conduct experiments to track the motion of aphids walking in a featureless circular arena in order to deduce individual-level rules. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. …

Barred Preferential Arrangements, Connor Thomas Ahlbach '13, Jeremy Usatine '14, Nicholas Pippenger

All HMC Faculty Publications and Research

A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with respect to one or more bars. We present various combinatorial identities for r_m‚_ℓ, the number of barred preferential arrangements of ℓ elements with m bars, using both algebraic and combinatorial arguments. Our main result is an expression for r_m,_ℓ as a linear combination of the r_k (= r_0,_ …

Nonlocal Aggregation Models: A Primer Of Swarm Equilibria, Andrew J. Bernoff, Chad M. Topaz

All HMC Faculty Publications and Research

Biological aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms have characteristic morphologies governed by the group members' intrinsic social interactions with each other and by their interactions with the external environment. Starting from a simple discrete model treating individual organisms as point particles, we derive a nonlocal partial differential equation describing the evolving population density of a continuum aggregation. To study equilibria and their stability, we use tools from the calculus of variations. In one spatial dimension, and for several choices of social forces, external forces, and domains, we find exact analytical expressions for the equilibria. …

Chromatic Bounds On Orbital Chromatic Roots, Dae Hyun Kim, Alexander H. Mun, Mohamed Omar

All HMC Faculty Publications and Research

Given a group G of automorphisms of a graph Γ, the orbital chromatic polynomial OPΓ,G(x) is the polynomial whose value at a positive integer k is the number of orbits of G on proper k-colorings of Γ. In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in R, extending Thomassen's famous result (see \cite{Thomassen}) that chromatic roots are dense in [32/27,∞). Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root …

Exploring The Baccalaureate Origin Of Domestic Ph.D. Students In Computing Fields, Susanne Hambrusch, Ran Libeskind-Hadas, Fen Zhao, David Rabson, Amy Csizmar Dalal, Ed Fox, Charles Isbell, Valerie Taylor

All HMC Faculty Publications and Research

Increasing the number of US students entering graduate school and receiving a Ph.D. in computer science is a goal as well as a challenge for many US Ph.D. granting institutions. Although the total computer science Ph.D. production in the U.S. has doubled between 2000 and 2010 (Figure 1), the fraction of domestic students receiving a Ph.D. from U.S. graduate programs has been below 50% since 2003 (Figure 2).

The goal of the Pipeline Project of CRA-E (PiPE) is to better understand the pipeline of US citizens and Permanent Residents (henceforth termed domestic students ) who apply, matriculate, and graduate from …

Existence And Qualitative Properties Of Solutions For Nonlinear Dirichlet Problems, Alfonso Castro, Jorge Cossio, Carlos Vélez

All HMC Faculty Publications and Research

Sign-changing solutions to semilinear elliptic problems in connection with their Morse indices. To this end, we first establish a priori bounds for one-sign solutions. Secondly, using abstract saddle point principles we find large augmented Morse index solutions. In this part, extensive use is made of critical groups, Morse index arguments, Lyapunov-Schmidt reduction, and Leray-Schauder degree. Finally, we provide conditions under which these solutions necessarily change sign and we comment about further qualitative properties.

Near-Optimal Compressed Sensing Guarantees For Anisotropic And Isotropic Total Variation Minimization, Deanna Needell, Rachel Ward

CMC Faculty Publications and Research

Consider the problem of reconstructing a multidimensional signal from partial information, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. Recently, guarantees for two-dimensional images were established. This paper extends these theoretical results to signals of arbitrary dimension and to both the anisotropic and isotropic total variation problems. To be precise, we show that …