Related papers: Rack shadows and their invariants
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks,…
The hashing trick is a machine learning technique used to encode categorical features into a numerical vector representation of pre-defined fixed length. It works by using the categorical hash values as vector indices, and updating the…
We revisit hierarchical bracketing encodings from a practical perspective in the context of dependency graph parsing. The approach encodes graphs as sequences, enabling linear-time parsing with $n$ tagging actions, and still representing…
A synchrony subspace of R^n is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices form a lattice. Applications of these…
Schreier graphs, which possess both a graph structure and a Schreier structure (an edge-labeling by the generators of a group), are objects of fundamental importance in group theory and geometry. We study the Schreier structures with which…
Shadows in transitional disks are generally interpreted as signs of a misaligned inner disk. This disk is usually beyond the reach of current day high contrast imaging facilities. However, the location and morphology of the shadow features…
Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a…
Trace diagrams are structured graphs with edges labeled by matrices. Each diagram has an interpretation as a particular multilinear function. We provide a rigorous combinatorial definition of these diagrams using a notion of signed graph…
The class of closed graphs by a linear ordering on their sets of vertices is investigated. A recent characterization of such a class of graphs is analyzed by using tools from the proper interval graph theory.
It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper we describe a more subtle invariant: a family of biquandle endomorphisms obtained…
A spatial surface is a compact surface embedded in the $3$-sphere. We assume that a spatial surface is oriented and that each connected component of a spatial surface is neither a disk nor without a boundary. A diagram of a spatial surface…
In this article, we provide a review of the current state of the research of the black hole shadow, focusing on analytical (as opposed to numerical and observational) studies. We start with particular attention to the definition of the…
A RAC-drawing of a graph is a straight-line drawing in which every crossing occurs at a right-angle. We show that deciding whether a graph has a RAC-drawing is as hard as the existential theory of the reals, even if we know that every edge…
The aim of this paper is to define a homology theory for racks with finite rank N and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in [15]. For this purpose, we prove…
In recent years, various shadow detection methods from a single image have been proposed and used in vision systems; however, most of them are not appropriate for the robotic applications due to the expensive time complexity. This paper…
For any operator $M$ acting on an $N$-dimensional Hilbert space $H_N$ we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of $M$. The shadow of $M$ at point $z$ is defined…
In a polydiagonal subspace of the Euclidean space, certain components of the vectors are equal (synchrony) or opposite (anti-synchrony). Polydiagonal subspaces invariant under a matrix have many applications in graph theory and dynamical…
For a graph $G$, we define its great shadow $S(G)$ as a construction that duplicates each vertex $v$ in $G$ and sets this duplicated vertex adjacent to $v$ and all neighbors of $v$. Great graph shadows arise naturally in the routing of…
We introduce a notion of a sheaf of vector spaces on a graph, and develop the foundations of homology theories for such sheaves. One sheaf invariant, its "maximum excess," has a number of remarkable properties. It has a simple definition,…
A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this…