Double Aztec rectangles

We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies et al. (1992) [2], [3]). Moreover, we consider the connection between the generating function and MacMahon's q-enumeration of plane partitions fitting in a given box     

Lozenge tilings of hexagons with arbitrary dents

Eisenkölbl gave a formula for the number of lozenge tilings of a hexagon on the triangular lattice with three unit triangles removed from along alternating sides. In earlier work, the first author extended this to the situation when an arbitrary set of unit triangles is removed from along alternating sides of the hexagon. In this paper we address the general case when an arbitrary set of unit triangles is removed from along the boundary of the hexagon.     

A periodicity theorem for the octahedron recurrence

The octahedron recurrence lives on a 3-dimensional lattice and is given by . In this paper, we investigate a variant of this recurrence which lives in a lattice contained in . Following Speyer, we give an explicit non-recursive formula for the values of this recurrence and use it to prove that it is periodic of period n+m. We then proceed to show various other hidden symmetries satisfied by this bounded octahedron recurrence. An earlier version of this work has circulated under the name “A coboundary category defined using the octahedron recurrence.”     

A four-color theorem for periodic tilings

There exist exactly 4044 topological types of 4-colorable tile-4-transitive tilings of the plane. These can be obtained by systematic application of two geometric algorithms, edge-contraction and vertex-truncation, to all tile-3-transitive tilings of the plane.     

A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on (−∞,∞) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.     

A min-max theorem for plane bipartite graphs

We consider a partitioning problem, defined for bipartite and 2-connected plane graphs, where each node should be covered exactly once by either an edge or by a cycle surrounding a face. The objective is to maximize the number of face boundaries in the partition. This problem arises in mathematical chemistry in the computation of the Clar number of hexagonal systems. In this paper we establish that a certain minimum weight covering problem of faces by cuts is a strong dual of the partitioning problem. Our proof relies on network flow and linear programming duality arguments, and settles a conjecture formulated by Hansen and Zheng in the context of hexagonal systems [P. Hansen, M. Zheng, Upper Bounds for the Clar Number of Benzenoid Hydrocarbons, Journal of the Chemical Society, Faraday Transactions 88 (1992) 1621-1625].     

A duality theorem for crossed products of hopf algebras

Abstract

We study the automorphism and collineation groups of plane curves, i.e., in ?², that are not necessarily smooth, and obtain bounds for these curves in terms of, their degree and the number of singularities. We also introduce the notion of bad curves and good curves, and show that all bad curves are rational.     

A Sturmian separation theorem for symplectic difference systems

We establish a Sturmian separation theorem for conjoined bases of 2n-dimensional symplectic difference systems. In particular, we show that the existence of a conjoined basis without focal points in a discrete interval (0,N+1] implies that any other conjoined basis has at most n focal points (counting multiplicities) in this interval.     

A local Paley-Wiener theorem for compact symmetric spaces

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M=U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.     

A convexity theorem for three tangled Hamiltonian torus actions,and super-integrable systems

A completely integrable system on a symplectic manifold is called super-integrable when the number of independent integrals of motion is more than half the dimension of the manifold. Several important completely integrable systems are super-integrable: the harmonic oscillators, the Kepler system, the non-periodic Toda lattice, etc. Motivated by an additional property of the super-integrable system of the Toda lattice (Agrotis et al., 2006) [2], we will give a generalization of the Atiyah and Guillemin–Sternberg?s convexity theorem.     

A Vizing-type theorem for matching forests

A well-known Theorem of Vizing states that one can colour the edges of a graph by Δ+ colours, such that edges of the same colour form a matching. Here, Δ denotes the maximum degree of a vertex, and the maximum multiplicity of an edge in the graph. An analogue of this Theorem for directed graphs was proved by Frank. It states that one can colour the arcs of a digraph by Δ+ colours, such that arcs of the same colour form a branching. For a digraph, Δ denotes the maximum indegree of a vertex, and the maximum multiplicity of an arc. We prove a common generalization of the above two theorems concerning the colouring of mixed graphs (these are graphs having both directed and undirected edges) in such a way that edges of the same colour form a matching forest.     

A structural theorem for planar graphs with some applications

In this note, we prove a structural theorem for planar graphs, namely that every planar graph has one of four possible configurations: (1) a vertex of degree 1, (2) intersecting triangles, (3) an edge xy with d(x)+d(y)≤9, (4) a 2-alternating cycle. Applying this theorem, new moderate results on edge choosability, total choosability, edge-partitions and linear arboricity of planar graphs are obtained.     

A Hecke correspondence theorem for automorphic integrals with symmetric rational period functions on the Hecke groups

Marvin Knopp showed that entire automorphic integrals with rational period functions satisfy a Hecke correspondence theorem, provided the rational period functions have poles only at 0 or ∞. For other automorphic integrals the corresponding Dirichlet series has a functional equation with a remainder term that arises from the nonzero poles of the rational period function. In this paper we prove a Hecke correspondence theorem for a class of automorphic integrals with rational period functions on the Hecke groups. We restrict our attention to automorphic integrals of weight that is twice an odd integer and to rational period functions that satisfy a symmetry property we call “Hecke-symmetry.” Each remainder term satisfies two relations (the second of which is new in this paper) corresponding to the two relations for the rational period function.