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Articles 1 - 26 of 26
Full-Text Articles in Physical Sciences and Mathematics
Voting, The Symmetric Group, And Representation Theory, Zajj Daugherty '05, Alexander K. Eustis '06, Gregory Minton '08, Michael E. Orrison
Voting, The Symmetric Group, And Representation Theory, Zajj Daugherty '05, Alexander K. Eustis '06, Gregory Minton '08, Michael E. Orrison
All HMC Faculty Publications and Research
We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied QSn-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur's Lemma), this allows us to recast and extend some well-known results in the field of voting theory.
On Similarity Classes Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
On Similarity Classes Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
CMC Faculty Publications and Research
A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of well-rounded sublattices of Z2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has the structure of a non-commutative infinitely generated monoid. We discuss the structure of a given similarity class, and define a zeta function corresponding to each similarity class. We relate it to Dedekind zeta of Z[i], and investigate the growth of some related Dirichlet series, which reflect on …
A Preliminary Mathematical Model Of Skin Dendritic Cell Trafficking And Induction Of T Cell Immunity, Amy H. Lin Erickson, Alison Wise, Stephen Fleming, Margaret Baird, Zabeen Lateef, Annette Molinaro, Miranda Teboh-Ewungkem, Lisette G. De Pillis
A Preliminary Mathematical Model Of Skin Dendritic Cell Trafficking And Induction Of T Cell Immunity, Amy H. Lin Erickson, Alison Wise, Stephen Fleming, Margaret Baird, Zabeen Lateef, Annette Molinaro, Miranda Teboh-Ewungkem, Lisette G. De Pillis
All HMC Faculty Publications and Research
Chronic inflammation is a process where dendritic cells (DCs) are constantly sampling antigen in the skin and migrating to lymph nodes where they induce the activation and proliferation of T cells. The T cells then travel back to the skin where they release cytokines that induce/maintain the inflammatory condition. This process is cyclic and ongoing. We created a differential equations model to reflect the initial stages of the inflammatory process. In particular, we modeled antigen stimulation of DCs in the skin, movement of DCs from the skin to a lymph node, and the subsequent activation of T cells in the …
Compression Theorems For Periodic Tilings And Consequences, Arthur T. Benjamin, Alex K. Eustis '06, Mark A. Shattuck
Compression Theorems For Periodic Tilings And Consequences, Arthur T. Benjamin, Alex K. Eustis '06, Mark A. Shattuck
All HMC Faculty Publications and Research
We consider a weighted square-and-domino tiling model obtained by assigning real number weights to the cells and boundaries of an n-board. An important special case apparently arises when these weights form periodic sequences. When the weights of an nm-tiling form sequences having period m, it is shown that such a tiling may be regarded as a meta-tiling of length n whose weights have period 1 except for the first cell (i.e., are constant). We term such a contraction of the period in going from the longer to the shorter tiling as "period compression". It turns out that …
Asymptotic Dynamics Of Attractive-Repulsive Swarms, Andrew J. Leverentz '08, Chad M. Topaz, Andrew J. Bernoff
Asymptotic Dynamics Of Attractive-Repulsive Swarms, Andrew J. Leverentz '08, Chad M. Topaz, Andrew J. Bernoff
All HMC Faculty Publications and Research
We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractive-repulsive social interactions. The kernel’s first moment and its limiting behavior at the origin determine whether the population asymptotically spreads, contracts, or reaches steady state. For the spreading case, the dynamics approach those of the porous medium equation. The widening, compactly supported population has edges that behave like traveling waves whose speed, density, and slope we calculate. …
A Semilinear Wave Equation With Smooth Data And No Resonance Having No Continuous Solution, Jose F. Caicedo, Alfonso Castro
A Semilinear Wave Equation With Smooth Data And No Resonance Having No Continuous Solution, Jose F. Caicedo, Alfonso Castro
All HMC Faculty Publications and Research
We prove that a boundary value problem for a semilinear wave equation with smooth nonlinearity, smooth forcing, and no resonance cannot have continuous solutions. Our proof shows that this is due to the non-monotonicity of the nonlinearity.
Topics In Compressed Sensing, Deanna Needell
Topics In Compressed Sensing, Deanna Needell
CMC Faculty Publications and Research
Compressed sensing has a wide range of applications that include error correction, imaging, radar and many more. Given a sparse signal in a high dimensional space, one wishes to reconstruct that signal accurately and efficiently from a number of linear measurements much less than its actual dimension. Although in theory it is clear that this is possible, the difficulty lies in the construction of algorithms that perform the recovery efficiently, as well as determining which kind of linear measurements allow for the reconstruction. There have been two distinct major approaches to sparse recovery that each present different benefits and shortcomings. …
Counting On Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
Counting On Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
All HMC Faculty Publications and Research
Chebyshev polynomials have several elegant combinatorial interpretations. Specificially, the Chebyshev polynomials of the first kind are defined by T0(x) = 1, T1(x) = x, and Tn(x) = 2x Tn-1(x) - Tn-2(x). Chebyshev polynomials of the second kind Un(x) are defined the same way, except U1(x) = 2x. Tn and Un are shown to count tilings of length n strips with squares and dominoes, where the tiles are given weights and sometimes color. Using these interpretations, many identities satisfied by Chebyshev polynomials can be given …
Review: A Class Of Solutions To The Quantum Colored Yang-Baxter Equation, Gizem Karaali
Review: A Class Of Solutions To The Quantum Colored Yang-Baxter Equation, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Mathematical Model Creation For Cancer Chemo-Immunotherapy, Lisette G. De Pillis, K Renee Fister, Weiqing Gu, Craig Collins, Michael Daub, David Gross '08, James Moore '07, Benjamin Preskill '09
Mathematical Model Creation For Cancer Chemo-Immunotherapy, Lisette G. De Pillis, K Renee Fister, Weiqing Gu, Craig Collins, Michael Daub, David Gross '08, James Moore '07, Benjamin Preskill '09
All HMC Faculty Publications and Research
One of the most challenging tasks in constructing a mathematical model of cancer treatment is the calculation of biological parameters from empirical data. This task becomes increasingly difficult if a model involves several cell populations and treatment modalities. A sophisticated model constructed by de Pillis et al., Mixed immunotherapy and chemotherapy of tumours: Modelling, applications and biological interpretations, J. Theor. Biol. 238 (2006), pp. 841–862; involves tumour cells, specific and non-specific immune cells (natural killer (NK) cells, CD8 T cells and other lymphocytes) and employs chemotherapy and two types of immunotherapy (IL-2 supplementation and CD8 T-cell infusion) as treatment modalities. …
Why Did Lagrange "Prove" The Parallel Postulate?, Judith V. Grabiner
Why Did Lagrange "Prove" The Parallel Postulate?, Judith V. Grabiner
Pitzer Faculty Publications and Research
In 1806, Joseph-Louis Lagrange read a memoir "proving" Euclid's parallel postulate to the Institut de France in Paris. The memoir still exists in manuscript, and we’ll look at what it says. We ask why he tried to prove the postulate, and why he attacked the problem in the way that he did. We also look at how the ideas in this manuscript are related to such things as Lagrange’s philosophy of mathematics, artists’ ideas about space, Newtonian mechanics, and Leibniz's Principle of Sufficient Reason. Finally, we reflect on how this episode changes our views about eighteenth-century attitudes toward geometry, space, …
Integral Orthogonal Bases Of Small Height For Real Polynomial Spaces, Lenny Fukshansky
Integral Orthogonal Bases Of Small Height For Real Polynomial Spaces, Lenny Fukshansky
CMC Faculty Publications and Research
Let PN(R) be the space of all real polynomials in N variables with the usual inner product < , > on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form representing this inner product on the space of coefficient vectors of all polynomials in PN(R) of degree ≤ M. We exhibit two applications of this formula. First, given a finite dimensional subspace V of PN(R) defined over Q, we prove the existence of an orthogonal basis for (V, < , >), consisting of polynomials of small height …
Search Bounds For Zeros Of Polynomials Over The Algebraic Closure Of Q, Lenny Fukshansky
Search Bounds For Zeros Of Polynomials Over The Algebraic Closure Of Q, Lenny Fukshansky
CMC Faculty Publications and Research
We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of unions of subspaces. All bounds on the height are explicit.
A Different Pencil Too Good To Be Ignored? A First Look At Wolfram|Alpha, Gizem Karaali, Bruce Yoshiwara
A Different Pencil Too Good To Be Ignored? A First Look At Wolfram|Alpha, Gizem Karaali, Bruce Yoshiwara
Pomona Faculty Publications and Research
So you came across several news pieces about Wolfram|Alpha and you’re wondering what all the fuss is about. Let’s start with the basics.
Predicting The Drug Release Kinetics Of Matrix Tablets, Boris Baeumer, Lipika Chatterjee, Peter Hinow, Thomas Rades, Ami E. Radunskaya, Ian Tucker
Predicting The Drug Release Kinetics Of Matrix Tablets, Boris Baeumer, Lipika Chatterjee, Peter Hinow, Thomas Rades, Ami E. Radunskaya, Ian Tucker
Pomona Faculty Publications and Research
In this paper we develop two mathematical models to predict the release kinetics of a water soluble drug from a polymer/excipient matrix tablet. The first of our models consists of a random walk on a weighted graph, where the vertices of the graph represent particles of a drug, excipient and polymer, respectively. The graph itself is the contact graph of a multidisperse random sphere packing. The second model describes the dissolution and the subsequent diffusion of the active drug out of a porous matrix using a system of a partial differential equations. The predictions of both models show good qualitative …
Review: Spectra Of Toeplitz Operators And Compositions Of Muckenhoupt Weights With Blaschke Products, Stephan Ramon Garcia
Review: Spectra Of Toeplitz Operators And Compositions Of Muckenhoupt Weights With Blaschke Products, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Complex Symmetric Partial Isometries, Stephan Ramon Garcia, Warren R. Wogen
Complex Symmetric Partial Isometries, Stephan Ramon Garcia, Warren R. Wogen
Pomona Faculty Publications and Research
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 provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension $\leq 4$ is complex symmetric.
Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, And Similarity, Joseph A. Cima, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, And Similarity, Joseph A. Cima, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Pomona Faculty Publications and Research
A truncated Toeplitz operator A φ : K Θ → K Θ is the compression of a Toeplitz operator T φ : H 2 → H 2 to a model space K Θ ≔ H 2 ⊖ Θ H 2 . For Θ inner, let T Θ denote the set of all bounded truncated Toeplitz operators on K Θ . Our main result is a necessary and sufficient condition on inner functions Θ 1 and Θ 2 which guarantees that T Θ 1 and T Θ 2 are spatially isomorphic (i.e., U T Θ 1 = T Θ 2 U …
Every Graph Has An Embedding In S³ Containing No Non-Hyperbolic Knot, Erica Flapan, Hugh Howards
Every Graph Has An Embedding In S³ Containing No Non-Hyperbolic Knot, Erica Flapan, Hugh Howards
Pomona Faculty Publications and Research
In contrast with knots, whose properties depend only on their extrinsic topology in S³ , there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in S³ . For example, it was shown by Conway and Gordon that every embedding of the complete graph K_7 in S³ contains a non-trivial knot. Later it was shown that for every m ∈ N there is a complete graph K_n such that every embedding of K_n in S³ contains a knot Q whose minimal crossing number is at least m. …
Review: On Quantum Yang-Baxter Coherent Algebra Sheaves, Gizem Karaali
Review: On Quantum Yang-Baxter Coherent Algebra Sheaves, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Graded Structure And Hopf Structures In Parabosonic Algebra. An Alternative Approach To Bosonisation, Gizem Karaali
Review: Graded Structure And Hopf Structures In Parabosonic Algebra. An Alternative Approach To Bosonisation, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Dynamical Yang-Baxter Maps With An Invariance Condition, Gizem Karaali
Review: Dynamical Yang-Baxter Maps With An Invariance Condition, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Solutions For The Constant Quantum Yang-Baxter Equation From Lie (Super)Algebras, Gizem Karaali
Review: Solutions For The Constant Quantum Yang-Baxter Equation From Lie (Super)Algebras, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Set-Theoretic Solutions Of The Yang-Baxter Equation, Graphs And Computations, Gizem Karaali
Review: Set-Theoretic Solutions Of The Yang-Baxter Equation, Graphs And Computations, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Hankel And Toeplitz Transforms On H¹: Continuity, Compactness And Fredholm Properties, Stephan Ramon Garcia
Review: Hankel And Toeplitz Transforms On H¹: Continuity, Compactness And Fredholm Properties, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
The Norm And Modulus Of A Foguel Operator, Stephan Ramon Garcia
The Norm And Modulus Of A Foguel Operator, Stephan Ramon Garcia
Pomona Faculty Publications and Research
We develop a method for calculating the norm and the spectrum of the modulus of a Foguel operator. In many cases, the norm can be computed exactly. In others, sharp upper bounds are obtained. In particular, we observe several connections between Foguel operators and the Golden Ratio.