Related papers: The Pentagram Integrals on Inscribed Polygons

Let $G=(V,E)$ be a finite, simple graph. We consider for each oriented graph $G_{\cal O}$ associated to an orientation ${\cal O}$ of the edges of $G$, the toric ideal $P_{G_{\cal O}}$. In this paper we study those graphs with the property…

Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete…

We prove NP-completeness of Yin-Yang / Shiromaru-Kuromaru pencil-and-paper puzzles. Viewed as a graph partitioning problem, we prove NP-completeness of partitioning a rectangular grid graph into two induced trees (normal Yin-Yang), or into…

In this paper we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let $d$ denote a nonnegative integer. Let…

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $\gamma$ of aspect ratio $\tan(r\cdot \pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do…

We give an algorithm for finding conformal mappings onto the upper half-plane and conformal modules of some types of polygons. The polygons are obtained by stretching along the real axis polyominoes i.e., polygons which are connected unions…

In this paper, we study graphs whose matching polynomial have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs.. We show that…

Orthogonal polynomials are of fundamental importance in many fields of mathematics and science, therefore the study of a particular family is always relevant. In this manuscript, we present a survey of some general results of the Hermite…

One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there…

In this article, we construct a generating set of rational invariants for the action of the orthogonal group $\text{O}(n)$ on the space $\mathbb{R}[x_1,\dots,x_n]_{2d}$ of real homogeneous polynomials of even degree $2d$. This generalizes a…

An odd diagram class is a set of permutations with the same odd diagram. Brenti, Carnevale and Tenner showed that each odd diagram class is an interval in the Bruhat order. They conjectured that such intervals are rank-symmetric. In this…

Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\quad…

We introduce a new class of transitive permutation groups which properly contains the automorphism groups of vertex-transitive graphs and digraphs. We then give a sufficient condition for a quotient of this family to remain in the family,…

The Laplacian matrix $L$ of a signed graph $G$ may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of $L$. This is achieved by finding a combinatorial formula for the Moore-Penrose inverse of an…

We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic…

Let $K$ be a complex bi-quadratic field with ring of integers $\mathcal{O}_{K}$. For $K = \mathbb{Q}(\sqrt{-m}$, $\sqrt{n}$), where $ m \equiv 3 \pmod 4 $ and $ n \equiv 1 \pmod 4$, we prove that every algebraic integer can be written as…

In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Latt\`{e}s map. In particular, this provides some…

Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective. More precisely, we say that a polygon with 2n sides joining 2n vertices A1, A2,..., A2n is well inscribed in a configuration Ln of n lines if…

The independence polynomial $i(G,x)$ of a graph $G$ is the generating function of the numbers of independent sets of each size. A graph of order $n$ is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu…

We obtain a connection between admissible elements for quadruples and Herrmann's endomorphisms. Herrmann constructed perfect elements $s_n$, $t_n$, $p_{i,n}$ in $D^4$ by means of some endomorphisms and showed that these perfect elements…