4,012 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Map enumeration from a dynamical perspective

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    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 gg, 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

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    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 1/21/2. 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 11. 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 RR uses at most R2+ϵπ3(R)R^{2+\epsilon}\pi_3(R) edges with high probability, for any ϵ>0\epsilon > 0. Here π3(R)\pi_3(R) is the polychromatic 3-arm probability from classical Bernoulli percolation; upon inserting its conjectural asymptotic, our bound converts to R4/3+ϵR^{4/3 + \epsilon}. In any case, it is known that π3(R)≲R−δ\pi_3(R) \lesssim R^{-\delta} for some δ>0\delta > 0, and so our bound gives an exponent strictly less than 22. In the special case of Bernoulli(1/21/2) edge-weights, we replace the additional factor of RϵR^\epsilon 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

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    La coloració de grafs dibuixats a superfícies és un àrea antiga i molt estudiada de la teoria de grafs. Thomassen va demostrar que hi ha un nombre finit de grafs 6-crítics a qualsevol superfície fixa i va proporcionar el conjunt explícit dels grafs 6-crítics al torus. Després, Postle va demostrar que hi ha un nombre finit de grafs 6-llista-crítics a qualsevol superfície fixa. Amb l'objectiu de trobar el conjunt de grafs 6-llista-crítics al torus, desenvolupem i implementem tècniques algorítmiques per la cerca per ordinador de grafs crítics en diferents situacions de coloració per llistes.La coloración de grafos dibujados en superficies es un área antigua y muy estudiada de la teoría de grafos. Thomassen demostró que hay un número finito de grafos 6-críticos en cualquier superficie fija y proporcionó el conjunto explícito de los grafos 6-críticos en el toro. Después, Postle demostró que hay un número finito de grafos 6-lista-críticos en cualquier superficie fija. Con el objetivo de encontrar el conjunto de grafos 6-lista-críticos en el toro, desarrollamos e implementamos técnicas algorítmicas para la búsqueda por ordenador de grafos críticos 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

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    L'objectiu principal del treball és l'enumeració de mapes a superfícies 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 superfícies, 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

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    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

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    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

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    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

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    https://neiudc.neiu.edu/catalogs/1064/thumbnail.jp

    Graduate Council Minutes - February 16, 2023

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