Related papers: Factorization structures, cones, and polytopes
We survey various notions of symmetry for toric varieties. These notions range from algebraic geometric, complex geometric, representation theoretic, combinatorial, convex geometric, to geometric stability. The main theorem gives the…
We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of…
We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is…
This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces do. Our method is a detailed analysis of a general…
We show factorization formulas for a class of partition functions of rational six vertex model. First we show factorization formulas for partition functions under triangular boundary. Further, by combining the factorization formulas with…
Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…
We discuss interrelations between: Cohn localizations of full square matrices; a Leavitt localization of a row; and the Jacobson quasi-inverses of quasi-regular elements. The latter Jacobson localizations appear naturally and easily in…
The construction of parametric curve and surface plays important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with…
We study the matrix factorizations defined by the generic plane projections of a curve singularity of $\mathbb{C}^3$. On the other hand, given a plane curve singularity $Y\subset \mathbb{C}^2$ we study the family of matrix factorizations…
The article presents simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is an ellipse, then the locus of vertexes of the cones is a hyperbola. The hyperbola…
Orbits of automorphism groups of partially ordered sets are not necessarily congruence classes, i.e. images of an order homomorphism. Based on so-called orbit categories a framework of factorisations and unfoldings is developed that…
This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal…
We classify projective toric manifolds whose dual variety is not a hypersurface in the dual projective space. Under the standard dictionary between toric geometry and convex geometry, they correspond to certain convex Delzant integer…
It is well known that the Lorenz system has $Z_2$-symmetry. Using introducted in math.DS/0105147 topological covering-coloring a new representation for the Lorenz system is obtained. Deleting coloring leads to the factorized Lorenz system…
We study the properties of groups that have presentations in which the square of each generator gives the identity and all relations are of even length. We consider the parabolic subgroups of such a group and show that every element has a…
We argue for Brackets Consistency to be a `Pillar of Geometry', i.e. a foundational approach, other Pillars being 1) Euclid's constructive approach, 2) the algebraic approach, 3) the projective approach, and 4) the geometrical automorphism…
Geometric properties of schemes obtained by gluing algebras of monoids, including separation and finiteness properties, irreducibility, normality, catenarity, dimension, and Serre's properties (S_k) and (R_k), are investigated. This is used…
Braid matrices $\hat{R}(\theta)$, corresponding to vector representations, are spectrally decomposed obtaining a ratio $f_{i}(\theta)/f_{i}(-\theta)$ for the coefficient of each projector $P_{i}$ appearing in the decomposition. This…
We discuss the $F$-matrices associated to the $R$-matrix of a general $N$-state vertex model whose statistical configurations encode $N-1$ U(1) symmetries. The factorization condition is shown for arbitrary weights being based only on the…
We consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case…