Ramanujan sums as supercharacters

The theory of supercharacters, recently developed by Diaconis-Isaacs and Andre, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.Comment: 32 pages. Comments welcom

A primer of swarm equilibria

We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding continuum population density. Equilibrium solutions are extrema of an energy functional, and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying discrete model.The exact solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework. The equilibria typically are compactly supported and may contain $\delta$-concentrations or jump discontinuities at the edge of the support. We apply our methods to a model of locust swarms, which are observed in nature to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; quasi-two-dimensionality of the model plays a critical role.Comment: 38 pages, submitted to SIAM J. Appl. Dyn. Sy

Bounds for solid angles of lattices of rank three

We find sharp absolute constants $C_1$ and $C_2$ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval $[C_1,C_2]$. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in $\mathbb R^N$ with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in $\mathbb R^3$. Such spherical configurations come up in connection with the kissing number problem.Comment: 12 pages; to appear in the Journal of Combinatorial Theory

On effective Witt decomposition and Cartan-Dieudonne theorem

Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical theorem of Witt states that the bilinear space $(Z,F)$ can be decomposed into an orthogonal sum of hyperbolic planes, singular, and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's Lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of Cartan-Dieudonn{\'e} theorem. Namely, we show that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can be represented as a product of reflections of small heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $\sigma$.Comment: 16 pages, revised and corrected version, to appear in Canadian Journal of Mathematic

Super Solutions of the Dynamical Yang-Baxter Equation

We classify super dynamical r-matrices with zero weight, thus extending earlier results of Etingof and Varchenko to the graded case

Discrepancy convergence for the drunkard's walk on the sphere

We analyze the drunkard's walk on the unit sphere with step size theta and show that the walk converges in order constant/sin^2(theta) steps in the discrepancy metric. This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.Comment: 20 pages; to appear in Electron. J. Probab.; related work at http://www.math.hmc.edu/~su/papers.htm

Recognizing Graph Theoretic Properties with Polynomial Ideals

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, Groebner bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure

Bounds on generalized Frobenius numbers

Let $N \geq 2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. The Frobenius number of this $N$-tuple is defined to be the largest positive integer that has no representation as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. More generally, the $s$-Frobenius number is defined to be the largest positive integer that has precisely $s$ distinct representations like this. We use techniques from the Geometry of Numbers to give upper and lower bounds on the $s$-Frobenius number for any nonnegative integer $s$.Comment: We include an appendix with an erratum and addendum to the published version of this paper: two inaccuracies in the statement of Theorem 2.2 are corrected and additional bounds on s-Frobenius numbers are derive

Predicting Knot or Catenane Type of Site-Specific Recombination Products

Site-specific recombination on supercoiled circular DNA yields a variety of knotted or catenated products. We develop a model of this process, and give extensive experimental evidence that the assumptions of our model are reasonable. We then characterize all possible knot or catenane products that arise from the most common substrates. We apply our model to tightly prescribe the knot or catenane type of previously uncharacterized data.Comment: 17 pages, 4 figures. Revised to include link to the companion paper, arXiv:0707.3896v1, that provides topological proofs underpinning the conclusions of the current paper. References update

Spectral Equivalence of Bosons and Fermions in One-Dimensional Harmonic Potentials

Recently, Schmidt and Schnack (cond-mat/9803151, cond-mat/9810036), following earlier references, reiterate that the specific heat of N non-interacting bosons in a one-dimensional harmonic well equals that of N fermions in the same potential. We show that this peculiar relationship between specific heats results from a more dramatic equivalence between bose and fermi systems. Namely, we prove that the excitation spectrums of such bose and fermi systems are spectrally equivalent. Two complementary proofs are provided, one based on an analysis of the dynamical symmetry group of the N-body system, the other using combinatoric analysis.Comment: Six Pages, No Figures, Submitted to Phys. Rev.