Digital Commons Network^™

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 …

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

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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

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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?

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.

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

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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

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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

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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

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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

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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.

The Awards Project: Promoting Good Practices In Award Selection, Betty Mayfield, Francis Su

All HMC Faculty Publications and Research

Every year the MAA honors many members of our community with a wide variety of prizes, awards, and certificates for excellence in teaching, writing, scholarship, and service (see maa.org/awards). The winners exemplify our ideals as an association; consequently, they are often viewed as role models and leaders. So it is important to ask: Do these awards, as a whole, reflect the outstanding contributions of the breadth of association membership?

Uniqueness Of Nonnegative Solutions For Semipositone Problems On Exterior Domains, Alfonso Castro, Lakshmi Sankar, Ratnasingham Shivaji

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We consider the problem

−Δu = λK(|x|)f(u), x∈Ω

u=0 if |x|=r₀

u→0 as |x|→∞,

where λ is a positive parameter, Δu = div(∇u)is the Laplacian of u, Ω = {x ∈ Rⁿ; n > 2,|x| > r₀}, K ∈ C¹([r₀,∞),(0,∞)) is such that lim r→∞ K(r) = 0 and f ∈ C¹([0,∞),R) is a concave function which is sublinear at ∞ and f(0) < 0. We establish the uniqueness of nonnegative radial solutions when λ is large.

3d Imaging And Mechanical Modeling Of Helical Buckling In Medicago Truncatula Plant Roots, Jesse L. Silverberg, Roslyn D. Noar, Michael S. Packer, Maria J. Harrison, Christopher L. Henley, Itai Cohen, Sharon J. Gerbode

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We study the primary root growth of wild-type Medicago truncatula plants in heterogeneous environments using 3D time-lapse imaging. The growth medium is a transparent hydrogel consisting of a stiff lower layer and a compliant upper layer. We find that the roots deform into a helical shape just above the gel layer interface before penetrating into the lower layer. This geometry is interpreted as a combination of growth-induced mechanical buckling modulated by the growth medium and a simultaneous twisting near the root tip. We study the helical morphology as the modulus of the upper gel layer is varied and demonstrate that …

Locust Dynamics: Behavioral Phase Change And Swarming, Chad M. Topaz, Maria R. D'Orsogna, Leah Edelstein-Keshet, Andrew J. Bernoff

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Locusts exhibit two interconvertible behavioral phases, solitarious and gregarious. While solitarious individuals are repelled from other locusts, gregarious insects are attracted to conspecifics and can form large aggregations such as marching hopper bands. Numerous biological experiments at the individual level have shown how crowding biases conversion towards the gregarious form. To understand the formation of marching locust hopper bands, we study phase change at the collective level, and in a quantitative framework. Specifically, we construct a partial integrodifferential equation model incorporating the interplay between phase change and spatial movement at the individual level in order to predict the dynamics of …

Black Hole Thermalization, D0 Brane Dynamics, And Emergent Spacetime, Paul L. Riggins '12, Vatche Sahakian

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When matter falls past the horizon of a large black hole, the expectation from string theory is that the configuration thermalizes and the information in the probe is rather quickly scrambled away. The traditional view of a classical unique spacetime near a black hole horizon conflicts with this picture. The question then arises as to what spacetime does the probe actually see as it crosses a horizon, and how does the background geometry imprint its signature onto the thermal properties of the probe. In this work, we explore these questions through an extensive series of numerical simulations of D0 branes. …

How The Cucumber Tendril Coils And Overwinds, Sharon J. Gerbode, Joshua R. Puzey, Andrew G. Mccormick, L. Mahadevan

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The helical coiling of plant tendrils has fascinated scientists for centuries, yet the underlying mechanism remains elusive. Moreover, despite Darwin’s widely accepted interpretation of coiled tendrils as soft springs, their mechanical behavior remains unknown. Our experiments on cucumber tendrils demonstrate that tendril coiling occurs via asymmetric contraction of an internal fiber ribbon of specialized cells. Under tension, both extracted fiber ribbons and old tendrils exhibit twistless overwinding rather than unwinding, with an initially soft response followed by strong strain-stiffening at large extensions. We explain this behavior using physical models of prestrained rubber strips, geometric arguments, and mathematical models of elastic …

Gromov-Witten Theory Of P^1 X P^1 X P^1, Dagan Karp, Dhruv Ranganathan

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We prove equivalences between the Gromov-Witten theories of toric blowups of P^1xP^1xP^1 and P^3. In particular, we prove that the all genus, virtual dimension zero Gromov-Witten theory of the blowup of P^3 at points precisely coincides with that of the blowup at points of P^1xP^1xP^1, for non-exceptional classes. It follows that the all-genus stationary Gromov-Witten theory of P^1xP^1xP^1 coincides with that of P^3 in low degree. We also prove there exists a toric symmetry of the Gromov-Witten theory of P^1xP^1xP^1 analogous to and intimately related to Cremona symmetry of P^3. Enumerative applications are given.

Resonant Solutions And Turning Points In An Elliptic Problem With Oscillatory Boundary Conditions, Alfonso Castro, Rosa Pardo

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We consider the elliptic equation -Δu + u = 0 with nonlinear boundary conditions ∂u/∂n = λu + g(λ,x,u), where the nonlinear term g is oscillatory and satisfies g(λ,x,s)/s→0 as |s|→0. We provide sufficient conditions on g for the existence of sequences of resonant solutions and turning points accumulating to zero.

Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su

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We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.

R₀ Analysis Of A Spatiotemporal Model For A Stream Population, H. W. Mckenzie, Y. Jin, Jon T. Jacobsen, M. A. Lewis

All HMC Faculty Publications and Research

Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem impacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R₀ for advection-diffusion-reaction equations and on related measures. We apply the measures to population persistence in rivers under various flow regimes. This work lays …

Adventures In Teaching: A Professor Goes To High School To Learn About Teaching Math, Darryl H. Yong

All HMC Faculty Publications and Research

During the 2009–2010 academic year I did something unusual for a university mathematician on sabbatical: I taught high school mathematics in a large urban school district. This might not be so strange except that my school does not have a teacher preparation program and only graduates a few students per year who intend to be teachers. Why did I do this? I, like many of you, am deeply concerned about mathematics education and I wanted to see what a typical high school in my city is like. Because I regularly work with high school mathematics teachers, I wanted to experience …

On The Hardness Of Counting And Sampling Center Strings, Christina Boucher, Mohamed Omar

All HMC Faculty Publications and Research

Given a set S of n strings, each of length ℓ, and a nonnegative value d, we define a center string as a string of length ` that has Hamming distance at most d from each string in S. The #CLOSEST STRING problem aims to determine the number of center strings for a given set of strings S and input parameters n, ℓ, and d. We show #CLOSEST STRING is impossible to solve exactly or even approximately in polynomial time, and that restricting #CLOSEST STRING so that any one of the parameters n, ℓ, or d is fixed leads to …

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

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I still recall my thrill and disappointment when I read Mathematical Carnival, by Martin Gardner. I was thrilled because, as my high school teacher had recommended, mathematics was presented in a playful way that I had never seen before. I was disappointed because it contained a formula that I thought I had "invented" a few years earlier. I have always had a passion for mental calculation, and the following formula appears in Gardner's chapter on "Lightning Calculators." It was used by the mathematician A. C. Aitken to mentally square large numbers.

Dynamic Server Allocation At Parallel Queues, Susan E. Martonosi

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We explore whether dynamically reassigning servers to parallel queues in response to queue imbalances can reduce average waiting time in those queues. We use approximate dynamic programming methods to determine when servers should be switched, and we compare the performance of such dynamic allocations to that of a pre-scheduled deterministic allocation. Testing our method on both synthetic data and data from airport security checkpoints at Boston Logan International Airport, we find that in situations where the uncertainty in customer arrival rates is significant, dynamically reallocating servers can substantially reduce waiting time. Moreover, we find that intuitive switching strategies that are …

Evolution Of Spur-Length Diversity In Aquilegia Petals Is Achieved Solely Through Cell-Shape Anisotropy, Joshua R. Puzey, Sharon J. Gerbode, Scott A. Hodges, Elena M. Kramer, L. Mahadevan

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The role of petal spurs and specialized pollinator interactions has been studied since Darwin. Aquilegia petal spurs exhibit striking size and shape diversity, correlated with specialized pollinators ranging from bees to hawkmoths in a textbook example of adaptive radiation. Despite the evolutionary significance of spur length, remarkably little is known about Aquilegia spur morphogenesis and its evolution. Using experimental measurements, both at tissue and cellular levels, combined with numerical modelling, we have investigated the relative roles of cell divisions and cell shape in determining the morphology of the Aquilegia petal spur. Contrary to decades-old hypotheses implicating a discrete meristematic zone …

Combinatorics Of Two-Toned Tilings, Arthur T. Benjamin, Phyllis Chinn, Jacob N. Scott '11, Greg Simay

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We introduce the function a(r, n) which counts tilings of length n + r that utilize white tiles (whose lengths can vary between 1 and n) and r identical red squares. These tilings are called two-toned tilings. We provide combinatorial proofs of several identities satisfied by a(r, n) and its generalizations, including one that produces kth order Fibonacci numbers. Applications to integer partitions are also provided.

Characteristics Of Optimal Solutions To The Sensor Location Problem, David R. Morrison '08, Susan E. Martonosi

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In Bianco et al. (2001), the authors present the Sensor Location Problem: that of locating the minimum number of traffic sensors at intersections of a road network such that the traffic ow on the entire network can be determined. They offer a necessary and sufficient condition on the set of monitored nodes in order for the ow everywhere to be determined. In this paper, we present a counterexample that demonstrates that the condition is not actually sufficient (though it is still necessary). We present a stronger necessary condition for ow calculability, and show that it is a sufficient condition in …

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

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We present mentally efficient algorithms for mentally squaring and cubing 2-digit and 3-digit numbers and for finding cube roots of numbers with 2-digit or 3-digit answers.