Related papers: Subpolygons in Conway-Coxeter frieze patterns

A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to…

The Frobenius manifold structure on the space of rational functions with multiple simple poles is constructed. In particular, the dependence of the Saito-flat coordinates on the flat coordinates of the intersection form is studied. While…

In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math.…

In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the "topograph," Conway revisited the reduction of BQFs and the solution of…

Let $ (W,S)$ be a Coxeter system. We investigate the equation $ w(\Phi_{x}) = \Phi_{y}$ where $ w,x,y\in W$ and $ \Phi_{x}$, $\Phi_{y}$ denote the left inversion sets of $ x$ and $ y$. We then define a commutative square diagram called a…

We introduce a new notion for geometric families called self-coverability and show that homothets of convex polygons are self-coverable. As a corollary, we obtain several results about coloring point sets such that any member of the family…

For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this…

Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected…

We study the 3-form flux $H_{\m\n\l}$ associated with the semi-classical geometry of $G/H$ gauged WZW models. We derive a simple, general expression for the flux in an orthonormal frame and use it to explicitly verify conformal invariance…

In this paper one extends the binomial and trinomial coefficients to the concept of 'k-nomial' coefficients, and one obtains some properties of these. As an application one generalizes Pascal's triangle.

For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge in terms of Hausdorff distance. We prove that the exterior angles of control polygons under subdivision converge to 0 at…

$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…

We consider closed manifolds, which occur as intersections of products of spheres of the same dimension with certain hyperplanes. Among those are the so called (big) polygon- and chain spaces. The equivariant cohomology with respect to…

Given a semi-simple algebra equipped with a coproduct, the Clebsch--Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

We show that the space of classical Coxeter's frieze patterns can be viewed as a discrete version of a coadjoint orbit of the Virasoro algebra. The canonical (cluster) (pre)symplectic form on the space of frieze patterns is a discretization…

The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic…

The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated…

This paper investigates the combinatorics that gives rise to the Boltzmann probability distribution. Despite being one of the most important distributions in physics and other fields of science, the mathematics of the underlying model of…

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and…