Related papers: Toric Ideals of Flow Polytopes
We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model…
The Chan-Robbins-Yuen polytope ($CRY_n$) of order $n$ is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph $K_{n+1}$ with netflow $(1,0,0, \ldots , 0, -1)$. The volume…
In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in $\mathbb{R}^3$, we prove that the curve evolves to leading order by…
Border bases arise as a canonical generalization of Gr\"obner bases. We provide a polyhedral characterization of all order ideals (and hence border bases) that are supported by a zero-dimensional ideal: order ideals that support a border…
We study a family of dissections of flow polytopes arising from the subdivision algebra. To each dissection of a flow polytope, we associate a polynomial, called the left-degree polynomial, which we show is invariant of the dissection…
We construct a Gr\"obner Basis of the relation ideal of a polynomial, give an interpolation formula for the basis elements and explain the connection of the interpolation formula to the Buchberger--M\"oller algorithm. We present a situation…
We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\ldots,v_n \in \mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines…
We study the following problem in computer vision from the perspective of algebraic geometry: Using $m$ pinhole cameras we take $m$ pictures of a line in $\mathbb P^3$. This produces $m$ lines in $\mathbb P^2$ and the question is which…
In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite…
Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient…
The theory of flows was used as a crucial tool in the recent proof by Margolis, Rhodes and Schilling that Krohn-Rhodes complexity is decidable. In this paper we begin a systematic study of aperiodic flows. We give the foundations of the…
In this paper we investigate the superconvergence properties of the discontinuous Galerkin method based on the upwind-biased flux for linear time-dependent hyperbolic equations. We prove that for even-degree polynomials, the method is…
We derive a linearly causal and stable third-order relativistic fluid-dynamical theory from the Boltzmann equation using the method of moments. For this purpose, we demonstrate that such theory must include novel degrees of freedom,…
We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the "longest increasing subsequence" have Ehrhart quasi-polynomials which are honest polynomials, even though they are just…
In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.
Following Lortz, we construct a family of smooth steady states of the ideal, incompressible Euler equation in three dimensions that possess no continuous Euclidean symmetry. As in Lortz, they do possess a planar reflection symmetry and, as…
In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the…
Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional…
This is part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected…
We generalize Tutte's integer flows and the $d$-dimensional Euclidean flows of Mattiolo, Mazzuoccolo, Rajn\'{i}k, and Tabarelli to \emph{$d$-dimensional $p$-normed nowhere-zero flows} and define the corresponding flow index $\phi_{d,p}(G)$…