Related papers: A note on friezes of type $\Lambda_p$
We consider the family of polynomials $p_{n}\left( x;z\right) ,$ orthogonal with respect to the inner product \[ \left\langle f,g\right\rangle = \int_{-z}^{z} f\left( x\right) g\left( x\right) e^{-x^{2}} \,dx. \] We show some properties…
The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects…
A recent paper computed the induced $p$-norm of a special class of circulant matrices $A(n,a,b) \in \mathbb{R}^{n \times n}$, with the diagonal entries equal to $a \in \mathbb{R}$ and the off-diagonal entries equal to $b \ge 0$. We provide…
Ranking of intuitionsitic fuzzy number plays a vital role in decision making and other intuitionistic fuzzy applications. In this paper, we propose a new ranking method of intuitionistic fuzzy number based on distance measure. We first…
We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive…
In this article we study mutation of friezes of type $D$. We provide a combinatorial formula for the entries in a frieze after mutation. The two main ingredients in the proof include a certain transformation of a type $D$ frieze into a sub…
Let $d(n)$ denote the number of divisors of $n$. In this paper, we study the average value of $d(a(p))$, where $p$ is a prime and $a(p)$ is the $p$-th Fourier coefficient of a normalized Hecke eigenform of weight $k \ge 2$ for $\Gamma_0(N)$…
The aim of this article is to count the $n$-tuples of positive integers $(a_{1},\ldots,a_{n})$ solutions of the equation $\begin{pmatrix} a_{n} & -1 \\[4pt] 1 & 0 \end{pmatrix} \begin{pmatrix} a_{n-1} & -1 \\[4pt] 1 & 0 \end{pmatrix} \cdots…
Given a standard graded polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic zero and a graded $k$-subalgebra $A=k[f_1,...,f_m]\subset R$, one relates the module $\Omega_{A/k}$ of K\"ahler $k$-differentials of $A$ to the…
Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of…
In this paper, we prove that the set of triangulations of a polygon can be equipped with an order to become a lattice. First, we define this order. In [HN99], authors defined the flip operator and then prove some properties of the graph of…
We introduce two classes of algebras coming from partial triangulations of marked surfaces. The first one, called frozen algebra of a partial triangulation, is generally of infinite rank and contains frozen Jacobian algebras of…
For an integer $m\geq 1$, a combinatorial manifold $\widetilde{M}$ is defined to be a geometrical object $\widetilde{M}$ such that for $\forall p\in\widetilde{M}$, there is a local chart $(U_p,\phi_p)$ enable $\phi_p:U_p\to…
We give a short and elementary proof that every Dynkin diagram admits finitely many (positive integral) friezes. This was originally proven by Gunawan-Muller using the geometry of cluster algebras. The proof here provides an explicit…
In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes),…
Broline, Crowe and Isaacs have computed the determinant of a matrix associated to a Conway-Coxeter frieze pattern. We generalise their result to the corresponding frieze pattern of cluster variables arising from the Fomin-Zelevinsky cluster…
We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of…
In this paper we give several criteria for the edge polytope of a fundamental FHM-graph to possess a regular unimodular triangulation in terms of some simple data of the the graph. We further apply our criteria to several examples of graphs…
We show that the existence of a non-trivial solution of $x^n+y^n=p^n$, with $p$ a prime number, is equivalent to the existence of a solution of a certain (over-determined) system of $(n-1)$-recursion relations ("zipper" equations) in…
For the general linear group $GL_n(k)$ over an algebraically closed field $k$ of characteristic $p$, there are two types of "twisting" operations that arise naturally on partitions. These are of the form $\lambda \rightarrow p\lambda$ and…