3-bounded property in a triangle-free distance-regular graph

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D, b, \alpha, \beta)$ and $D\geq 3$. Assume the intersection numbers $a_1=0$ and $a_2\not=0$. We show $\Gamma$ is 3-bounded in the sense of the article [D-bounded distance-regular graphs, European Journal of Combinatorics(1997)18, 211-229].Comment: 13 page

Discrete complex analysis on planar quad-graphs

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.Comment: 49 pages, 8 figure

Translation surfaces and the curve graph in genus two

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ of the curve graph of $S$. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic, or by a concatenation of two parallel saddle connections (satisfying some additional properties) on $X$. The subgraph $\hat{\mathcal{C}}_{\rm cyl}$ is by definition $\mathrm{GL}^+(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm\"uller disk in the curve graph. We will show that $\hat{\mathcal{C}}_{\rm cyl}$ is always connected and has infinite diameter. The group ${\rm Aff}^+(X,\omega)$ of affine automorphisms of $(X,\omega)$ preserves naturally $\hat{\mathcal{C}}_{\rm cyl}$, we show that ${\rm Aff}^+(X,\omega)$ is precisely the stabilizer of $\hat{\mathcal{C}}_{\rm cyl}$ in ${\rm Mod}(S)$. We also prove that $\hat{\mathcal{C}}_{\rm cyl}$ is Gromov-hyperbolic if $(X,\omega)$ is completely periodic in the sense of Calta. It turns out that the quotient of $\hat{\mathcal{C}}_{\rm cyl}$ by ${\rm Aff}^+(X,\omega)$ is closely related to McMullen's prototypes in the case $(X,\omega)$ is a Veech surface in $\mathcal{H}(2)$. We finally show that this quotient graph has finitely many vertices if and only if $(X,\omega)$ is a Veech surface for $(X,\omega)$ in both strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$.Comment: 47 pages, 17 figures. Minor changes, some proofs improved. Comments welcome

Developing a Mathematical Model for Bobbin Lace

Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. An overview of how bobbin lace is made is presented and illustrated with a simple, traditional bookmark design. Research on the topology of textiles and braid theory form a base for the current work and is briefly summarized. We define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search. Results of this new approach are presented and, by comparison to existing bobbin lace patterns, it is demonstrated that this model reveals new patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as Computer Scientists and Mathematician

Disparity map generation based on trapezoidal camera architecture for multiview video

Visual content acquisition is a strategic functional block of any visual system. Despite its wide possibilities, the arrangement of cameras for the acquisition of good quality visual content for use in multi-view video remains a huge challenge. This paper presents the mathematical description of trapezoidal camera architecture and relationships which facilitate the determination of camera position for visual content acquisition in multi-view video, and depth map generation. The strong point of Trapezoidal Camera Architecture is that it allows for adaptive camera topology by which points within the scene, especially the occluded ones can be optically and geometrically viewed from several different viewpoints either on the edge of the trapezoid or inside it. The concept of maximum independent set, trapezoid characteristics, and the fact that the positions of cameras (with the exception of few) differ in their vertical coordinate description could very well be used to address the issue of occlusion which continues to be a major problem in computer vision with regards to the generation of depth map

Periodic boundary conditions on the pseudosphere

We provide a framework to build periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary conditions in the Euclidean plane, we introduce all the needed mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane. We stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems and we illustrate how to implement such periodic boundary conditions in two examples, the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices.Comment: 30 pages, minor corrections, accepted to J. Phys.

Happy endings for flip graphs

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio