Related papers: Rotor-Routing Induces the Only Consistent Sandpile…
Simple cycles on a digraph form a trace monoid under the rule that two such cycles commute if and only if they are vertex disjoint. This rule describes the spatial configuration of simple cycles on the digraph. Cartier and Foata have showed…
D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a…
Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all…
We develop a Hodge theory for relative simple normal crossing varieties over an Artinian base scheme. We introduce the notion of a mixed Hodge structure over an Artin ring, which axiomatizes the structure that is found on the cohomology of…
Most computational accounts of cognitive maps assume that stability is achieved primarily through sensory anchoring, with self-motion contributing to incremental positional updates only. However, biological spatial representations often…
This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to…
We study rotation-robust learning for image inputs using Convolutional Model Trees (CMTs) [1], whose split and leaf coefficients can be structured on the image grid and transformed geometrically at deployment time. In a controlled MNIST…
In the author's earlier work there appeared a new way to specify any smooth closed 4-manifold by a surface diagram, which consists of an orientable surface decorated with simple closed curves. These curves are cyclically indexed, and each…
We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.
We study surjective endomorphisms of projective bundles over toric varieties, achieving three main results. First, we provide a structural theorem describing endomorphisms of projectivized split bundles over arbitrary base varieties, which…
Several sequences of free cumulants that count binary plane trees correspond to sequences of classical cumulants that count the decreasing versions of the same trees. Using two new operations on colored binary plane trees that we call…
We bound the Castelnuovo-Mumford regularity and syzygies of the ideal of the singular set of a plane curve, and more generally of the conductor scheme of certain projectively Gorenstein varieties.
Sand pile models are dynamical systems describing the evolution from $N$ stacked grains to a stable configuration. It uses local rules to depict grain moves and iterate it until reaching a fixed configuration from which no rule can be…
In rotor walk on a finite directed graph, the exits from each vertex follow a prescribed periodic sequence. Here we consider the case of rotor walk where a particle starts from a designated source vertex and continues until it hits a…
We study the convergence of stochastic fixed point iterations in the consistent case (in the sense of Butnariu and Fl{\aa}m (1995)) in several different settings, under decreasingly restrictive regularity assumptions of the fixed point…
We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie…
The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we…
Chevyrev and Galbraith recently devised an algorithm which inputs a maximal order of the quaternion algebra ramified at one prime and infinity and constructs a supersingular elliptic curve whose endomorphism ring is precisely this maximal…
A well-known conjecture of Stanley is that the h-vector of a matroid is a pure O-sequence. There have been numerous papers with partial progress on this conjecture, but it is still wide open. In particular, for graphic matroids coming from…
We introduce a persistence-type invariant for finite weighted graphs based on combinatorial multivector dynamics. For each threshold parameter, a relation matrix determines a graph multivector field, whose induced directed dynamics admits a…