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Revisiting the hexagonal lattice: on optimal lattice circle packing  [PDF]
Lenny Fukshansky
Mathematics , 2009,
Abstract: In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the important class of well-rounded lattices, on which the density function takes a particularly simple form. Our proof emphasizes the role of well-rounded lattices for discrete optimization problems.
Counting Hexagonal Lattice Animals  [PDF]
Mohamud Mohammed
Mathematics , 2002,
Abstract: We describe Maple packages for the automatic generation of generating functions(and series expansions) for counting lattice animals(fixed polyominoes), in the two-dimensional hexagonal lattice, of bounded but arbitrary width. Our Maple packages(complete with source code) are easy-to-use and available from my website.
Coincidences of a shifted hexagonal lattice and the hexagonal packing  [PDF]
Jeanine Concepcion H. Arias,Evelyn D. Gabinete,Manuel Joseph C. Loquias
Mathematics , 2013, DOI: 10.12693/APhysPolA.126.516
Abstract: A geometric study of twin and grain boundaries in crystals and quasicrystals is achieved via coincidence site lattices (CSLs) and coincidence site modules (CSMs), respectively. Recently, coincidences of shifted lattices and multilattices (i.e. finite unions of shifted copies of a lattice) have been investigated. Here, we solve the coincidence problem for a shifted hexagonal lattice. This result allows us to analyze the coincidence isometries of the hexagonal packing by viewing the hexagonal packing as a multilattice.
Hexagonal Lattice Points on Circles  [PDF]
Oscar Marmon
Mathematics , 2005,
Abstract: We study the hexagonal lattice $\mathbb{Z}[\omega]$, where $\omega^6=1$. More specifically, we study the angular distribution of hexagonal lattice points on circles with a fixed radius. We prove that the angles are equidistributed on average, and suggest the possibility of constructing a consistent discrete velocity model (DVM) for the Boltzmann equation, using a hexagonal lattice. Equidistribution on average is expressed in terms of cancellation in exponential sums. We introduce Hecke L-functions and investigate their analytic properties in order to derive estimates on sums of Hecke characters. Using a version of the Halberstam-Richert inequality, these estimates then yield the desired results for the exponential sums. As a further measure of equidistribution, we give a bound for the discrepancy.
Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions  [PDF]
Marc Timme,Frank van Bussel,Denny Fliegner,Sebastian Stolzenberg
Computer Science , 2009, DOI: 10.1088/1367-2630/11/2/023001
Abstract: Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.
The hexagonal versus the square lattice  [PDF]
Pieter Moree,Herman J. J. te Riele
Mathematics , 2002,
Abstract: We establish Schmutz Schaller's conjecture that the hexagonal lattice is `better' than the square lattice. Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8 the best known lattice sphere packings have `maximal lengths' and goes on to write: "In dimension 2 the conjecture means in particular that the hexagonal lattice is `better' than the square lattice. More precisely, let 0
Emanuele Munarini
Applicable Analysis and Discrete Mathematics , 2009, DOI: 10.2298/aadm0901157m
Abstract: An antiregular graph is a simple graph with the maximum number of vertices with different degrees. In this paper we study the characteristic polynomial, the admittance (or Laplacian) polynomial and the matching polynomial of a connected antiregular graph. For these polynomials we obtain recurrences and explicit formulas. We also obtain some spectral properties. In particular, we prove an interlacing property for the eigenvalues and we give some bounds for the energy.
Connections between the matching and chromatic polynomials  [cached]
E. J. Farrell,Earl Glen Whitehead
International Journal of Mathematics and Mathematical Sciences , 1992, DOI: 10.1155/s016117129200098x
Abstract: The main results established are (i) a connection between the matching and chromatic polynomials and (ii) a formula for the matching polynomial of a general complement of a subgraph of a graph. Some deductions on matching and chromatic equivalence and uniqueness are made.
Rational polynomials of simple type  [PDF]
Walter D. Neumann,Paul Norbury
Mathematics , 2000,
Abstract: We classify two-variable polynomials which are rational of simple type. These are precisely the two-variable polynomials with trivial homological monodromy.
Supermodes of Hexagonal Lattice Waveguide Arrays  [PDF]
J. Scott Brownless,Felix J. Lawrence,Sahand Mahmoodian,Kokou B. Dossou,Lindsay C. Botten,C. Martijn de Sterke
Physics , 2012, DOI: 10.1364/JOSAB.29.001338
Abstract: We present a semi-analytical formulation for calculating the supermodes and corresponding Bloch factors of light in hexagonal lattice photonic crystal waveguide arrays. We then use this formulation to easily calculate dispersion curves and predict propagation in systems too large to calculate using standard numerical methods.
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