Related papers: A Note on Flips in Diagonal Rectangulations
We show that 3-fold terminal flips and divisorial contractions may be factored into a sequence of flops, blow-downs to a smooth curve in a smooth 3-fold or divisorial contractions to points with minimal discrepancies.
This paper is a direct continuation of the paper arXiv:2401.00053. By this reason neither introductory part of the paper nor the list of references are not duplicated. However for the reader convenience, the formulas from the first paper…
We provide isomorphism results for Hopf algebras that are obtained as graded twistings of function algebras on finite groups by cocentral actions of cyclic groups. More generally , we also consider the isomorphism problem for…
We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves…
We characterize the absolute retracts in the category of reflexive oriented graphs, that is, antisymmetric reflexive graphs, where morphisms between objects preserve arcs (which may be sent to loops). Here we show, by correcting a much…
We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and…
A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost…
This article is a short review on the relationship between convergent matrix integrals, formal matrix integrals, and combinatorics of maps. We briefly summarize results developed over the last 30 years, as well as more recent discoveries.…
We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible…
We are interested in constructing zonotopes which are combinatorially nonequivalent but have the same face vector. In this paper we introduce a quadrilateral flip operation on graphs. We show that, if one graph is obtained from another…
Given two combinatorial triangulations, how many edge flips are necessary and sufficient to convert one into the other? This question has occupied researchers for over 75 years. We provide a comprehensive survey, including full proofs, of…
In this paper we study two operations, Pasting and Reversing, defined from a natural way to be applied over some rings such as the ring of polynomials and the ring of linear differential operators, which is a differential ring. We obtain…
In this paper we define an internal binary operation between functions called in the text \emph{fractal convolution}, that applies a pair of mappings into a fractal function. This is done by means of a suitable Iterated Function System. We…
Let $S$ be a planar point set in general position, and let $\mathcal{P}(S)$ be the set of all plane straight-line paths with vertex set $S$. A flip on a path $P \in \mathcal{P}(S)$ is the operation of replacing an edge $e$ of $P$ with…
We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold into another balanced triangulation. These moves form a natural analog of bistellar…
This article demonstrates that convolutional operation can be converted to matrix multiplication, which has the same calculation way with fully connected layer. The article is helpful for the beginners of the neural network to understand…
We describe how orbital graphs can be used to improve the practical performance of many algorithms for permutation groups, including intersection and stabilizer problems. First we explain how orbital graphs can be integrated in partition…
Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogues: the multidimensional quadrilateral lattices, i.e. lattices x: Z^N -> R^M, whose…
The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the $\delta$ type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the…
The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated…