117 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Notes on the Localization of Generalized Hexagonal Cellular Networks

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    The act of accessing the exact location, or position, of a node in a network is known as the localization of a network. In this methodology, the precise location of each node within a network can be made in the terms of certain chosen nodes in a subset. This subset is known as the locating set and its minimum cardinality is called the locating number of a network. The generalized hexagonal cellular network is a novel structure for the planning and analysis of a network. In this work, we considered conducting the localization of a generalized hexagonal cellular network. Moreover, we determined and proved the exact locating number for this network. Furthermore, in this technique, each node of a generalized hexagonal cellular network can be accessed uniquely. Lastly, we also discussed the generalized version of the locating set and locating number

    High-order renormalization of scalar quantum fields

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    Thema dieser Dissertation ist die Renormierung von perturbativer skalarer Quantenfeldtheorie bei großer Schleifenzahl. Der Hauptteil der Arbeit ist dem Einfluss von Renormierungsbedingungen auf renormierte Greenfunktionen gewidmet. Zunächst studieren wir Dyson-Schwinger-Gleichungen und die Renormierungsgruppe, inklusive der Gegenterme in dimensionaler Regularisierung. Anhand zahlreicher Beispiele illustrieren wir die verschiedenen Größen. Alsdann diskutieren wir, welche Freiheitsgrade ein Renormierungsschema hat und wie diese mit den Gegentermen und den renormierten Greenfunktionen zusammenhängen. Für ungekoppelte Dyson-Schwinger-Gleichungen stellen wir fest, dass alle Renormierungsschemata bis auf eine Verschiebung des Renormierungspunktes äquivalent sind. Die Verschiebung zwischen kinematischer Renormierung und Minimaler Subtraktion ist eine Funktion der Kopplung und des Regularisierungsparameters. Wir leiten eine neuartige Formel für den Fall einer linearen Dyson-Schwinger Gleichung vom Propagatortyp her, um die Verschiebung direkt aus der Mellintransformation des Integrationskerns zu berechnen. Schließlich berechnen wir obige Verschiebung störungstheoretisch für drei beispielhafte nichtlineare Dyson-Schwinger-Gleichungen und untersuchen das asymptotische Verhalten der Reihenkoeffizienten. Ein zweites Thema der vorliegenden Arbeit sind Diffeomorphismen der Feldvariable in einer Quantenfeldtheorie. Wir präsentieren eine Störungstheorie des Diffeomorphismusfeldes im Impulsraum und verifizieren, dass der Diffeomorphismus keinen Einfluss auf messbare Größen hat. Weiterhin untersuchen wir die Divergenzen des Diffeomorphismusfeldes und stellen fest, dass die Divergenzen Wardidentitäten erfüllen, die die Abwesenheit dieser Terme von der S-Matrix ausdrücken. Trotz der Wardidentitäten bleiben unendlich viele Divergenzen unbestimmt. Den Abschluss bildet ein Kommentar über die numerische Quadratur von Periodenintegralen.This thesis concerns the renormalization of perturbative quantum field theory. More precisely, we examine scalar quantum fields at high loop order. The bulk of the thesis is devoted to the influence of renormalization conditions on the renormalized Green functions. Firstly, we perform a detailed review of Dyson-Schwinger equations and the renormalization group, including the counterterms in dimensional regularization. Using numerous examples, we illustrate how the various quantities are computable in a concrete case and which relations they satisfy. Secondly, we discuss which degrees of freedom are present in a renormalization scheme, and how they are related to counterterms and renormalized Green functions. We establish that, in the case of an un-coupled Dyson-Schwinger equation, all renormalization schemes are equivalent up to a shift in the renormalization point. The shift between kinematic renormalization and Minimal Subtraction is a function of the coupling and the regularization parameter. We derive a novel formula for the case of a linear propagator-type Dyson-Schwinger equation to compute the shift directly from the Mellin transform of the kernel. Thirdly, we compute the shift perturbatively for three examples of non-linear Dyson-Schwinger equations and examine the asymptotic growth of series coefficients. A second, smaller topic of the present thesis are diffeomorphisms of the field variable in a quantum field theory. We present the perturbation theory of the diffeomorphism field in momentum space and find that the diffeomorphism has no influence on measurable quantities. Moreover, we study the divergences in the diffeomorphism field and establish that they satisfy Ward identities, which ensure their absence from the S-matrix. Nevertheless, the Ward identities leave infinitely many divergences unspecified and the diffeomorphism theory is perturbatively unrenormalizable. Finally, we remark on a third topic, the numerical quadrature of Feynman periods

    Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition

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    A graph is well-covered if all its maximal independent sets have the same cardinality. This well studied concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph GG, a real-valued vertex weight function ww is said to be a well-covered weighting of GG if all its maximal independent sets are of the same weight. The set of all well-covered weightings of a graph GG forms a vector space over the field of real numbers, called the well-covered vector space of GG. Since the problem of recognizing well-covered graphs is co\mathsf{co}-NP\mathsf{NP}-complete, the problem of computing the well-covered vector space of a given graph is co\mathsf{co}-NP\mathsf{NP}-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graph. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies that well-covered fork-free graphs can be recognized in polynomial time and also generalizes some known results on cographs.Comment: 25 page

    Kalai's 3d3^{d}-conjecture for unconditional and locally anti-blocking polytopes

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    Kalai's 3d3^d-conjecture states that every centrally symmetric dd-polytope has at least 3d3^d faces. We give short proofs for two special cases: if PP is unconditional (that is, invariant w.r.t. reflection in any coordinate hyperplane), and more generally, if PP is locally anti-blocking (that is, looks like an unconditional polytope in every orthant). In both cases we show that the minimum is attained exactly for the Hanner polytopes

    Geometric Graphs with Unbounded Flip-Width

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    We consider the flip-width of geometric graphs, a notion of graph width recently introduced by Toru\'nczyk. We prove that many different types of geometric graphs have unbounded flip-width. These include interval graphs, permutation graphs, circle graphs, intersection graphs of axis-aligned line segments or axis-aligned unit squares, unit distance graphs, unit disk graphs, visibility graphs of simple polygons, β\beta-skeletons, 4-polytopes, rectangle of influence graphs, and 3d Delaunay triangulations.Comment: 10 pages, 7 figures. To appear at CCCG 202

    Comparing invariants of toric ideals of bipartite graphs

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    Let GG be a finite simple graph and let IGI_G denote its associated toric ideal in the polynomial ring RR. For each integer n2n\geq 2, we completely determine all the possible values for the tuple (reg(R/IG),deg(hR/IG(t)),pdim(R/IG),depth(R/IG),dim(R/IG))({\rm reg}(R/I_G), {\rm deg}(h_{R/I_G}(t)),{\rm pdim}(R/I_G), {\rm depth}(R/I_G),\dim(R/I_G)) when GG is a connected bipartite graph on nn vertices.Comment: 13 pages, comments welcom

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    Intersection Cographs and Aesthetics

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    Cographs are complete graphs with colored lines (edges); in an intersection cograph, the points (vertices) and lines (edges) are labeled by sets, and the line between each pair of points is (or represents) their intersection. This article first presents the elementary theory of intersection cographs: 15 are possible on 4 points; constraints on the triangles and quadrilaterals; some forbidden configurations; and how, under suitable constraints, to generate the points from the lines alone. The mathematical theory is then applied to aesthetics, using set cographs to describe the experience of a person enjoying a picture (Mu Qi), poem (Dickinson), play (Shakespeare), or piece of music (Anna Magdalena Bach)

    Injective split systems

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    A split system S\mathcal S on a finite set XX, X3|X|\ge3, is a set of bipartitions or splits of XX which contains all splits of the form {x,X{x}}\{x,X-\{x\}\}, xXx \in X. To any such split system S\mathcal S we can associate the Buneman graph B(S)\mathcal B(\mathcal S) which is essentially a median graph with leaf-set XX that displays the splits in S\mathcal S. In this paper, we consider properties of injective split systems, that is, split systems S\mathcal S with the property that medB(S)(Y)medB(S)(Y)\mathrm{med}_{\mathcal B(\mathcal S)}(Y) \neq \mathrm{med}_{\mathrm B(\mathcal S)}(Y') for any 3-subsets Y,YY,Y' in XX, where medB(S)(Y)\mathrm {med}_{\mathcal B(\mathcal S)}(Y) denotes the median in B(S)\mathcal B(\mathcal S) of the three elements in YY considered as leaves in B(S)\mathcal B(\mathcal S). In particular, we show that for any set XX there always exists an injective split system on XX, and we also give a characterization for when a split system is injective. We also consider how complex the Buneman graph B(S)\mathcal B(\mathcal S) needs to become in order for a split system S\mathcal S on XX to be injective. We do this by introducing a quantity for X|X| which we call the injective dimension for X|X|, as well as two related quantities, called the injective 2-split and the rooted-injective dimension. We derive some upper and lower bounds for all three of these dimensions and also prove that some of these bounds are tight. An underlying motivation for studying injective split systems is that they can be used to obtain a natural generalization of symbolic tree maps. An important consequence of our results is that any three-way symbolic map on XX can be represented using Buneman graphs.Comment: 22 pages, 3 figure
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