Related papers: On the coincidence of Pascal lines
Wilson lines, being comparators that render non-local operator products gauge invariant, are extensively used in QCD calculations, especially in small-$x$ calculations, calculations concerning validation of factorisation schemes and in…
The new concept of a system of hex equations is introduced as an overdetermined system of six five-point face-centered quad equations defined on six vertices of a hexagon. For a consistent system of hex equations, two variables on…
A configuration of points and lines is cyclic if it has an automorphism which permutes its points in a full cycle. A closed formula is derived for the number of non-isomorphic connected cyclic configurations of type (v_3), i.e., which have…
The generic monic polynomial of sixth degree features 6 a priori arbitrary coefficients. We show that if these 6 coefficients are appropriately defined in two different ways|in terms of 5 arbitrary parameters, then the 6 roots of the…
Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials, via the definition of convexity, its first order characterization, and its second order…
A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line, or block, is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points…
For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$…
An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in…
We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings and the number of perfect matchings. Most importantly, for bipartite graphs the polynomial encodes the number of…
A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if,…
Every polyhedral cone can be described either by its facets or by its extreme rays. Computation of one description from the other is a problem that can be very complex, i.e. one encounter the combinatorial explosion. We present here several…
Let $S$ be a smooth irreducible curve defined over $\overline{\mathbb{Q}}$, let $\mathcal{A}$ be an abelian scheme over $S$ and $\mathcal{C}$ a curve inside $\mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In this paper we prove…
We prove that over a Poncelet triangle family interscribed between two nested ellipses $\mathcal{E},\mathcal{E}_c$, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a $90^o$-rotated copy of…
Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in…
We classify SIC-POVMs of rank one in CP^2, or equivalently sets of nine equally-spaced points in CP^2, without the assumption of group covariance. If two points are fixed, the remaining seven must lie on a pinched torus that a standard…
We consider two families of Pascal-like triangles that have all ones on the left side and ones separated by $m-1$ zeros on the right side. The $m=1$ cases are Pascal's triangle and the two families also coincide when $m=2$. Members of the…
We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ with smooth components on which there exist families of finite sets (of unbounded size) with quadratically many…
Given a collection of points in the plane, classifying which subsets are collinear is a natural problem and is related to classical geometric constructions. We consider collections of points in a projective plane over a finite field such…
We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painlev\'e VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of B\"acklund…
Let $k$ be a positive integer and let $G$ be a graph with $n$ vertices. A connected $k$-subpartition of $G$ is a collection of $k$ pairwise disjoint sets (a.k.a. classes) of vertices in $G$ such that each set induces a connected subgraph.…