Related papers: Obtaining trees of tangles from tangle-tree dualit…
In this article we prove some theorems related to the triplets of triangles, homological two by two. These theorems will be used later to build triplets of triangles two by two tri-homological.
We carry out a proof theoretic analysis of the wellfoundedness of recursive path orders in an abstract setting. We outline a very general termination principle and extract from its wellfoundedness proof subrecursive bounds on the size of…
Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille…
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
We look for partition theorems for large subtrees for suitable uncountable trees and colourings. We concentrate on sub-trees of $^{\kappa \ge} 2$ expanded by a well ordering of each level. Unlike earlier works, we do not ask the embedding…
A brief review of the status of duality symmetries in string theory is presented. The evidence is accumulating rapidly that an enormous group of duality symmetries, including perturbative T dualities and non-perturbative S-dualities,…
An extension of order theory is presented that serves as a formalism for the study of dendroidal sets analogously to way the formalism of order theory is used in the study of simplicial sets.
In this paper we introduce a variation on the multidimensional segment tree, formed by unifying different interpretations of the dimensionalities of the data structure. We give some new definitions to previously well-defined concepts that…
We give an alternative, more geometric, proof of the well-known Joyal-Tierney Theorem in locale theory by utilizing Priestley duality for frames.
We discuss a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. We give several equivalent descriptions, and prove some algebraic and combinatorial properties. In addition, we characterize…
Separation systems are posets with additional structure that form an abstract setting in which tangle-like clusters in graphs, matroids and other combinatorial structures can be expressed and studied. This paper offers some basic theory…
We present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The…
Given a graph or a matroid, a tree of tangles is a tree decomposition that displays the structure of the connectivity: every edge of the decomposition tree induces a separation, that is, a way to divide the graph or matroid into two parts;…
We give a simple proof of the "tree-width duality theorem" of Seymour and Thomas that the tree-width of a finite graph is exactly one less than the largest order of its brambles.
Let $T$ be a tree, we show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, and use it in order to get formulas for independence number and…
We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some…
We combine the two fundamental fixed-order tangle theorems of Robertson and Seymour into a single theorem that implies both, in a best possible way. We show that, for every $k \in \mathbb{N}$, every tree-decomposition of a graph $G$ which…
The ability to compare complex systems can provide new insight into the fundamental nature of the processes captured in ways that are otherwise inaccessible to observation. Here, we introduce the $n$-tangle method to directly compare two…
This paper provides a fresh perspective on the representation of distributive bilattices and of related varieties. The techniques of naturalduality are employed to give, economically and in a uniform way, categories ofstructures dually…
This is an introduction to the theory of disconjugacy for a second order linear differential equation. We give new proofs of some of basic results and obtain new sufficient conditions for disconjugacy (in particular, on the whole real…