Related papers: The perimeter generating function for nondirected …
This paper presents an additional class of regular polyhedra--envelope polyhedra--made of regular polygons, where the arrangement of polygons (creating a single surface) around each vertex is identical; but dihedral angles between faces…
The evolution of many kinetic processes in 1+1 (space-time) dimensions results in 2d directed percolative landscapes. The active phases of these models possess numerous hidden geometric orders characterized by various types of large-scale…
Let $D_\lambda^{d,k}$ denote the family of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_\lambda^{d,k}:X_1^d+X_2^d=\lambda dX_1^kx_2^{d-k}, \end{align*} where $d\geq2$, $1\leq k\leq d-1$, and…
The linear orbit of a degree d hypersurface in $\mathbb{P}^n$ is its orbit under the natural action of PGL(n+1), in the projective space of dimension $N =\binom{n+d}{d} - 1$ parameterizing such hypersurfaces. This action restricted to a…
We show that the set of complex points in the moduli space of polynomials of degree d corresponding to post-critically finite polynomials is a set of algebraic points of bounded height. It follows that for any B, the set of conjugacy…
The Lasserre's reconstruction algorithm is extended to the D-polytopes with the construction of their shape space. Thus, the areas of d-skeletons $(1\leq d\leq D)$ can be expressed as functions of the areas and normal bi-vectors of the…
In [math-ph/0107005] we have proven that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions. This…
There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have…
Behling, Bello-Cruz, Lara-Urdaneta, Oviedo, and Santos showed that the circumcentric direction $d$ of a finitely generated polyhedral cone $\KK\subset\RR^n$ admits an inscribed Euclidean ball of radius $\norm{d}^2$ inside the polar cone…
In this article we study, given a pair of integers (d,g), the problem of existence of a smooth, irreducible, non-degenerate curve in the projective n-domensional space of degree d and genus g (the Halphen-Castelnuovo Problem). We define two…
We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids recently introduced by the second author to construct counter-examples to the Hirsch conjecture. We show that the `strong $d$-step Theorem'…
We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral…
We first extend the results of Chatterjee,Kumar,Shi,Volk(Computational Complexity 2022) by showing that the degree $d$ elementary symmetric polynomials in $n$ variables have formula lower bounds of $\Omega(d(n-d))$ over fields of positive…
This paper, about a fluid-like system of spatially confined particles, reveals the analytic structure for both, the canonical and grand canonical partition functions. The studied system is inhomogeneously distributed in a region whose…
A tight map is a map with some of its vertices marked, such that every vertex of degree $1$ is marked. We give an explicit formula for the number $N_{0,n}(d_1,\ldots,d_n)$ of planar tight maps with $n$ labeled faces of prescribed degrees…
Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function…
Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \emph{$m$-convex} polygons and are characterised by…
We study perimeters of connecting cycles for concentric circles. More precisely, we are interested in characterization of those connecting cycles which are critical points of perimeter considered as a function on the product of given…
Projectivizations of pointed polyhedral cones $C$ are positive geometries in the sense of Arkani-Hamed, Bai, and Lam. Their canonical forms look like $$ \Omega_C(x)=\frac{A(x)}{B(x)} dx, $$ with $A,B$ polynomials. The denominator $B(x)$ is…
An important goal of self-assembly is to achieve a preprogrammed structure with high fidelity. Here, we control the valence of DNA-functionalized emulsions to make linear and branched model polymers, or `colloidomers'. The distribution of…