arXiv subjects

Lukas Nabergall

Publications and source records attributed to Lukas Nabergall.

4 recordsLinked to original sources

arXiv2024

Tubings, chord diagrams, and Dyson--Schwinger equations

We give series solutions to single insertion place propagator-type systems of Dyson--Schwinger equations using binary tubings of rooted trees. These solutions are combinatorially transparent in the sense that each tubing has a straightforward contribution. The Dyson--Schwinger equations solved here are more general than those previously solved by chord diagram techniques, including systems and non-integer values of the insertion parameter $s$. We remark on interesting combinatorial connections and properties.

math.CO↗

arXiv2021

The combinatorics of a tree-like functional equation for connected chord diagrams

We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating functions of two different subsets of weighted connected chord diagrams: arbitrary diagrams and diagrams forbidding so-called top cycle subdiagrams. These equations generalize the classic specification for increasing ordered trees and their solution uses a novel decomposition, simplifying and generalizing previous results. The resulting tree perspective on chord diagrams leads to new enumerative insights through the study of novel diagram classes. We present a recursive bijection between connected top-cycle-free diagrams with $n$ chords and triangulations of a disk with $n+1$ vertices, thereby counting the former. This connects to combinatorial maps, Catalan intervals, and uniquely sorted permutations, leading to new conjectured bijective relationships between diagram classes defined by forbidding graphical subdiagrams and imposing connectedness properties and a variety of other rich combinatorial objects. We conclude by exhibiting and studying a direct bijection between diagrams of size $n$ with a single terminal chord and diagrams of size $n-1$.

math.CO↗

arXiv2020

An Antidote for Hawkmoths: On the prevalence of structural chaos in non-linear modeling

This paper deals with the question of whether uncertainty regarding model structure, especially in climate modeling, exhibits a kind of "chaos." Do small changes in model structure, in other words, lead to large variations in ensemble predictions? More specifically, does model error destroy forecast skill faster than the ordinary or "classical" chaos inherent in the real-world attractor? In some cases, the answer to this question seems to be "yes." But how common is this state of affairs? And are there precise mathematical results that can help us answer this question? We examine some efforts in the literature to answer this last question in the affirmative and find them to be unconvincing.

physics.hist-ph↗

arXiv2019

Insertions Yielding Equivalent Double Occurrence Words

A double occurrence word (DOW) is a word in which every symbol appears exactly twice; two DOWs are equivalent if one is a symbol-to-symbol image of the other. We consider the so called repeat pattern ($αα$) and the return pattern ($αα^R$), with gaps allowed between the $α$'s. These patterns generalize square and palindromic factors of DOWs, respectively. We introduce a notion of inserting repeat/return words into DOWs and study how two distinct insertions into the same word can produce equivalent DOWs. Given a DOW $w$, we characterize the structure of $w$ which allows two distinct insertions to yield equivalent DOWs. This characterization depends on the locations of the insertions and on the length of the inserted repeat/return words and implies that when one inserted word is a repeat word and the other is a return word, then both words must be trivial (i.e., have only one symbol). The characterization also introduces a method to generate families of words recursively.

math.CO↗