Related papers: Infinite friezes
We construct frieze patterns of type D_N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram…
Categories of models of algebraic theories have good categorical properties except for gluing. Building upon insights and examples from Synthetic Differential Geometry, we introduce a generalisation of models of algebraic theories to…
We prove that there is an finite number of friezes in type D_n, and we provide a formula to count them. As a corollary, we obtain formulas to count the number of friezes in types B_n, C_n and G_2. We conjecture finiteness (and precise…
We characterise $t$-perfect plane triangulations by forbidden induced subgraphs. As a consequence, we obtain that a plane triangulation is $h$-perfect if and only if it is perfect.
We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and…
Some aspects of a mathematical theory of rigidity and flexibility are developed for general infinite frameworks and two main results are obtained. In the first sufficient conditions, of a uniform local nature, are obtained for the existence…
In this note, we show that the sequence of maximum values in frieze patterns of type $A_n$ is the sequence of Fibonacci numbers, and that of frieze patterns of type $C_n$ is the sequence of odd Fibonacci numbers.
We show that for any positive forward density subset N \subset Z, there exists an integer m>0, such that, for all n>m, N contains almost perfect n-scaled reproductions of any previously chosen finite set of integers.
Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of…
We give an elementary account of the notion of Y-frieze patterns, explain some of their properties, and reveal their connection with Coxeter's frieze patterns.
We introduce a growth process which samples sections of uniform infinite causal triangulations by elementary moves in which a single triangle is added. A relation to a random walk on the integer half line is shown. This relation is used to…
Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander--Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with…
We study non-zero integral friezes for Dynkin types $A_n$, $B_n$, $C_n$, $D_n$ and $G_2$. These differ from standard Coxeter-Conway (positive) friezes by allowing any non-zero integer to appear. In each case we show that there are either…
An infinite permutation is a linear ordering of the set of non-negative integers. Generally, the properties of infinite permutations analogous to those of infinite words show some resemblances and some differences between permutations and…
The existence of the weak limit as n --> infinity of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random…
This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the…
We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…
Matrices are very popular and widely used in mathematics and other fields of science. Every mathematician has known the properties of finite-sized matrices since the time of study. In this paper, we consider the basic theory of infnite…
Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined…
The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a…