4,012 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Map enumeration from a dynamical perspective
This contribution summarizes recent work of the authors that combines methods
from dynamical systems theory (discrete Painlev\'e equations) and asymptotic
analysis of orthogonal polynomial recurrences, to address long-standing
questions in map enumeration. Given a genus , we present a framework that
provides the generating function for the number of maps that can be realized on
a surface of that genus. In the case of 4-valent maps, our methodology leads to
explicit expressions for map counts. For general even or mixed valence, the
number of vertices of the map specifies the relevant order of the derivatives
of the generating function that needs to be considered. Beyond summarizing our
own results, we provide context for the program highlighted in this article
through a brief review of the literature describing advances in map
enumeration. In addition, we discuss open problems and challenges related to
this fascinating area of research that stands at the intersection of
statistical physics, random matrices, orthogonal polynomials, and discrete
dynamical systems theory
An upper bound on geodesic length in 2D critical first-passage percolation
We consider i.i.d. first-passage percolation (FPP) on the two-dimensional
square lattice, in the critical case where edge-weights take the value zero
with probability . Critical FPP is unique in that the Euclidean lengths of
geodesics are superlinear, rather than linear, in the distance between their
endpoints. This fact was speculated by Kesten in 1986 but not confirmed until
2019 by Damron and Tang, who showed a lower bound on geodesic length that is
polynomial with degree strictly greater than . In this paper we establish
the first non-trivial upper bound. Namely, we prove that for a large class of
critical edge-weight distributions, the shortest geodesic from the origin to a
box of radius uses at most edges with high
probability, for any . Here is the polychromatic 3-arm
probability from classical Bernoulli percolation; upon inserting its
conjectural asymptotic, our bound converts to . In any
case, it is known that for some ,
and so our bound gives an exponent strictly less than . In the special case
of Bernoulli() edge-weights, we replace the additional factor of
with a constant and give an expectation bound.Comment: 62 pages, 14 figure
A computational approach for finding 6-List-critical graphs on the Torus
La coloració de grafs dibuixats a superfÃcies és un à rea antiga i molt estudiada de la teoria de grafs. Thomassen va demostrar que hi ha un nombre finit de grafs 6-crÃtics a qualsevol superfÃcie fixa i va proporcionar el conjunt explÃcit dels grafs 6-crÃtics al torus. Després, Postle va demostrar que hi ha un nombre finit de grafs 6-llista-crÃtics a qualsevol superfÃcie fixa. Amb l'objectiu de trobar el conjunt de grafs 6-llista-crÃtics al torus, desenvolupem i implementem tècniques algorÃtmiques per la cerca per ordinador de grafs crÃtics en diferents situacions de coloració per llistes.La coloración de grafos dibujados en superficies es un área antigua y muy estudiada de la teorÃa de grafos. Thomassen demostró que hay un número finito de grafos 6-crÃticos en cualquier superficie fija y proporcionó el conjunto explÃcito de los grafos 6-crÃticos en el toro. Después, Postle demostró que hay un número finito de grafos 6-lista-crÃticos en cualquier superficie fija. Con el objetivo de encontrar el conjunto de grafos 6-lista-crÃticos en el toro, desarrollamos e implementamos técnicas algorÃtmicas para la búsqueda por ordenador de grafos crÃticos en diferentes situaciones de coloración por listas.Coloring graphs embedded on surfaces is an old and well-studied area of graph theory. Thomassen proved that there are finitely many 6-critical graphs on any fixed surface and provided the explicit set of 6-critical graphs on the torus. Later, Postle proved that there are finitely many 6-list-critical graphs on any fixed surface. With the goal of finding the set of 6-list-critical graphs on the torus, we develop and implement algorithmic techniques for computer search of critical graphs in different list-coloring settings.Outgoin
Map enumeration on surfaces: from bijective techniques to asymptotic counting
L'objectiu principal del treball és l'enumeració de mapes a superfÃcies orientables amb l'ús majoritari de tècniques bijectives. Deduirem la fórmula exacta per mapes planars amb n arestes amb 3 mètodes diferents. Un d'ells explica en detall la relació que hi ha entre mapes i arbres planars, a més d'oferir una base per l'enumeració de mapes en gèneres majors. En aquestes superfÃcies, enumerarem asimptòticament els mapes a partir d'una intel·ligent bijecció entre mapes i mapes amb una sola cara, anomenats g-trees.El objetivo principal del trabajo es la enumeración de mapas en superficies orientables con el uso mayoritario de técnicas biyectivas. Deduciremos la fórmula exacta para mapas planos con n aristas con tres métodos diferentes. Uno de ellos explica en detalle la relación que hay entre mapas y árboles planos, a más de dar una base para la enumeración de mapas en superficies de género mayor. En dichas superficies, enumeraremos asintóticamente los mapas a partir de una biyección entre mapas y mapas con una sola cara, llamados g-trees.The main objective of this work is the enumeration of maps on orientable surfaces with the use of mostly bijective techniques. We are going to deduce the exact formula for planar maps with n edges by three different methods. One of them explains in detail the relation that planar maps and trees have, in addition to giving the foundations for the enumeration of maps in higher genus surfaces. In said surfaces, we are going to asymptotically enumerate maps via a clever bijection between maps and maps with one face, which are called g-trees
Cut vertices in random planar maps
The main goal of this paper is to determine the asymptotic behavior of the number X n of cut-vertices in random planar maps with n edges. It is shown that X n / n ¿ c in probability (for some explicit c > 0 ). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.Peer ReviewedPostprint (published version
Drawings of Complete Multipartite Graphs up to Triangle Flips
For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph Kn with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n16). The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph Km, n minus two edges and Km, n plus one edge for any m, n ≥ 4, as well as Kn minus a 4-cycle for any n ≥ 5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when removing or adding very few edges
Scanning Probe Microscopy Studies of Petroleum Chemistry: Substrate-Dependent Catalytic Properties of MoS2 and Automating Scanning Probe Microscopy with Machine Learning
With the growth of the population, society’s energy demands are mostly reliant on petroleum products that come from the refining of crude oil. Most of these refining reactions have been developed through averaging spectroscopic techniques, but scientists do not know exactly what is happening in these processes at the nano and atomic levels. This information is crucial when designing an efficient refining process that produces petroleum products that emit fewer harmful gases when combusting. Scanning probe microscopy techniques have become a powerful tool to look into the chemical structures found in petroleum products, to understand catalytic reactions in refining processes, and to find new non-combustible uses for these products. In this dissertation, I show how scanning probe microscopy (SPM) techniques, especially non-contact atomic force microscopy (NC-AFM) can provide an atomic-level understanding of the chemical structures and active catalytic sites that play a role in these refining processes. First, I studied hydrodesulfurization reactions that use molybdenum disulfide as a main catalyst to explore the effect of layer thickness, strain, and underlying substrates on its electronic and catalytic properties. Here, I present the first NC-AFM experiments investigating the active catalytic sites of molybdenum disulfide on industrially relevant substrates. Through these experiments, I found how NC-AFM techniques on insulators need to be improved to achieve high-resolution images that are comparable to those collected on metal substrates. Second, I created Auto-HR-AFM, a machine-learning script that collects optimal high-resolution NC-AFM images. Auto-HR-AFM is a modular and open-source script that provides an initial framework for a fully automated SPM. Expanding on this framework will widen the use of scanning probe microscopy techniques to non-experts and the automation will increase the time the system is kept running to collect large optimal datasets. Ultimately, these studies will broaden the use of high-resolution SPM techniques and help create more efficient catalysts and refining processes to produce cleaner and more efficient petroleum products
Northeastern Illinois University, Academic Catalog 2023-2024
https://neiudc.neiu.edu/catalogs/1064/thumbnail.jp
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