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1
Charles, Balthazar. "Combinatorics and computations : Cartan matrices of monoids & minimal elements of Shi arrangements." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPASG063.
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Cette thèse présente le résultat de recherches sur deux thèmes combinatoires distincts: le calcul effectif des matrices de Cartan en théorie des représentations des monoïdes et l'exploration des propriétés des éléments minimaux dans les arrangements de Shi des groupes de Coxeter. Bien que disparates, ces deux domaines de recherche partagent l'utilisation de méthodes combinatoires et d'exploration informatique, soit en tant que fin en soi pour le premier domaine, soit comme aide à la recherche pour le second. Dans la première partie de la thèse, nous développons des méthodes pour le calcul effectif des tables de caractères et des matrices de Cartan dans la théorie des représentations des monoïdes. À cette fin, nous présentons un algorithme basé sur nos résultats pour le calcul efficace des points fixes sous une action similaire à une conjugaison, dans le but de mettre en œuvre la formule de [Thiéry '12] pour la matrice de Cartan. Après une introduction largement auto-contenue aux notions nécessaires, nous présentons nos résultats sur le comptage des points fixes, ainsi qu'une nouvelle formule pour la table de caractères des monoïdes finis. Nous évaluons les performances des algorithmes résultants en termes de temps d'exécution et d'utilisation mémoire. Nous observons qu'ils sont plus efficaces par plusieurs ordres de grandeur que les algorithmes non spécialisés pour les monoïdes. Nous espérons que l'implémentation (publique) résultant de ces travaux contribuera à la communauté des représentations des monoïdes en permettant des calculs auparavant difficiles. La deuxième partie de la thèse se concentre sur les propriétés des éléments minimaux dans les arrangements de Shi. Les arrangements de Shi ont été introduits dans [Shi '87] et sont l'objet de la Conjecture 2 dans [Dyer, Hohlweg '14]. Initialement motivés par cette conjecture, après une introduction aux notions nécessaires, nous présentons deux résultats. Premièrement, une démonstration directe dans le cas des groupes de rang 3. Deuxièmement, dans le cas particulier des groupes de Weyl, nous donnons une description des éléments minimaux des régions de Shi en étendant une bijection issue de [Athanasiadis, Linusson '99] et [Armstrong, Reiner, Rhoades '15] entre les fonctions de parking et les régions de Shi permettant d'effectuer le calcul pratique des éléments minimaux. Comme application, à partir des propriétés de ce calcul, nous donnons une démonstration de la conjecture pour les groupes de Weyl indépendante de leur classification. Ces résultats révèlent une interaction intrigante entre les partitions non-croisées et non-embrassées dans le cas des groupes de Weyl classiquesThis thesis presents an investigation into two distinct combinatorial subjects: the effective computation of Cartan matrices in monoid representation theory and the exploration of properties of minimal elements in Shi arrangements of Coxeter groups. Although disparate, both of these research focuses share a commonality in the utilization of combinatorial methods and computer exploration either as an end in itself for the former or as a help to research for the latter. In the first part of the dissertation, we develop methods for the effective computation of character tables and Cartan matrices in monoid representation theory. To this end, we present an algorithm based on our results for the efficient computations of fixed points under a conjugacy-like action, with the goal to implement Thiéry's formula for the Cartan matrix from [Thiéry '12]. After a largely self-contained introduction to the necessary background, we present our results for fixed-point counting, as well as a new formula for the character table of finite monoids. We evaluate the performance of the resulting algorithms in terms of execution time and memory usage and find that they are more efficient than algorithms not specialized for monoids by orders of magnitude. We hope that the resulting (public) implementation will contribute to the monoid representation community by allowing previously impractical computations. The second part of the thesis focuses on the properties of minimal elements in Shi arrangements. The Shi arrangements were introduced in [Shi '87] and are the object of Conjecture 2 from [Dyer, Hohlweg '14]. Originally motivated by this conjecture, we present two results. Firstly, a direct proof in the case of rank 3 groups. Secondly, in the special case of Weyl groups, we give a description of the minimal elements of the Shi regions by extending a bijection from [Athanasiadis, Linusson '99] and [Armstrong, Reiner, Rhoades '15] between parking functions and Shi regions. This allows for the effective computation of the minimal elements. From the properties of this computation, we provide a type-free proof of the conjecture in Weyl groups as an application. These results reveal an intriguing interplay between the non-nesting and non-crossing worlds in the case of classical Weyl groups
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Johnston, David. "Quasi-invariants of hyperplane arrangements." Thesis, University of Glasgow, 2012. http://theses.gla.ac.uk/3169/.
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The ring of quasi-invariants $Q_m$ can be associated with the root system $R$ and multiplicity function $m$. It first appeared in the work of Chalykh and Veselov in the context of quantum Calogero-Moser systems. One can define an analogue $Q_{\mathcal{A}}$ of this ring for a collection $\mathcal{A}$ of vectors with multiplicities. We study the algebraic properties of these rings. For the class of arrangements on the plane with at most one multiplicity greater than one we show that the Gorenstein property for $Q_{\mathcal{A}}$ is equivalent to the existence of the Baker-Akhiezer function, thus suggesting a new perspective on systems of Calogero-Moser type. The rings of quasi-invariants $Q_m$ have a well known interpretation as modules for the spherical subalgebra of the rational Cherednik algebra with integer valued multiplicity function. We explicitly construct the anti-invariant quasi-invariant polynomials corresponding to the root system $A_n$ as certain representations of the spherical subalgebra of the Cherednik algebra $H_{1/m}(S_{mn})$. We also study the relation of the algebra $\Lambda_{n,1,k}$ introduced by Sergeev and Veselov to the ring of quasi-invariants for the deformed root system $\mathcal{A}_n(k)$. We find the Poincar\'e series for a `symmetric part' of $Q_$ for positive integer values of $k$.
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Ziegler, Günter M. (Günter Matthias). "Algebraic combinatorics of hyperplane arrangements." Thesis, Massachusetts Institute of Technology, 1987. http://hdl.handle.net/1721.1/14854.
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Moseley, Daniel, and Daniel Moseley. "Group Actions on Hyperplane Arrangements." Thesis, University of Oregon, 2012. http://hdl.handle.net/1794/12373.
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In this dissertation, we will look at two families of algebras with connections to hyperplane arrangements that admit actions of finite groups. One of the fundamental questions to ask is how these decompose into irreducible representations. For the first family of algebras, we will use equivariant cohomology techniques to reduce the computation to an easier one. For the second family, we will use two decompositions over the intersection lattice of the hyperplane arrangement to aid us in computation.
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Bibby, Christin. "Abelian Arrangements." Thesis, University of Oregon, 2015. http://hdl.handle.net/1794/19273.
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An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. We are interested in the topology of the complement of an arrangement. If the arrangement is unimodular, we provide a combinatorial presentation for a differential graded algebra (DGA) that is a model for the complement, in the sense of rational homotopy theory. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers. If the arrangement is chordal, then this model is a Koszul algebra. In this case, studying its quadratic dual gives a combinatorial description of the Q-nilpotent completion of the fundamental group and the minimal model of the complement of the arrangement.
This dissertation includes previously unpublished co-authored material.
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Sleumer, Nora Helena. "Hyperplane arrangements : construction, visualization and applications /." [S.l.] : [s.n.], 2000. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=13502.
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Agosti, Claudia. "Cohomology of hyperplane and toric arrangements." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/19510/.
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L'algebra di coomologia del complementare di un arrangiamento torico è più complicata di quella del complementare di un arrangiamento di iperpiani, in quanto il toro complesso ha già di per sè una coomologia non banale e perchè l'intersezione di due sottotori in generale non è connessa. Nel 2005, De Concini e Procesi si sono concentrati sullo studio dell'algebra di coomologia del complementare degli arrangiamenti torici nel quale le intersezioni di sottotori sono sempre connesse (arrangiamenti torici unimodulari) ottenendone una presentazione sullo stile di quella data da Orlik e Solomon per gli arrangiamenti di iperpiani. Nel 2018, Callegaro, D'Adderio, Delucchi, Migliorini e Pagaria hanno generalizzato il lavoro di De Concini e Procesi fornendo una presentazione, sempre sullo stile di quella data da Orlik e Solomon, dell'algebra di coomologia di un generico arrangiamento torico. In questa tesi descriviamo tali presentazioni dell'algebra di coomologia, soffermandoci in particolare su alcuni esempi.
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Mücksch, Paul [Verfasser]. "Combinatorics and freeness of hyperplane arrangements and reflection arrangements / Paul Mücksch." Hannover : Technische Informationsbibliothek (TIB), 2018. http://d-nb.info/1169961169/34.
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Biyikoglu, Türker, Wim Hordijk, Josef Leydold, Tomaz Pisanski, and Peter F. Stadler. "Graph Laplacians, Nodal Domains, and Hyperplane Arrangements." Department of Statistics and Mathematics, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, 2002. http://epub.wu.ac.at/1036/1/document.pdf.
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Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e., the connected components of the maximal induced subgraphs of G on which an eigenvector \psi does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of \psi in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processing
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Moss, Aaron. "Basis Enumeration of Hyperplane Arrangements up to Symmetries." Thesis, Fredericton: University of New Brunswick, 2012. http://hdl.handle.net/1882/44593.
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This thesis details a method of enumerating bases of hyperplane arrangements up to symmetries. I consider here automorphisms, geometric symmetries which leave the set of all points contained in the arrangement setwise invariant. The algorithm for basis enumeration described in this thesis is a backtracking search over the adjacency graph implied on the bases by minimum-ratio simplex pivots, pruning at bases symmetric to those already seen. This work extends Bremner, Sikiri c, and Sch urmann's method for basis enumeration of polyhedra up to symmetries, including a new pivoting rule for nding adjacent bases in arrangements, a method of computing automorphisms of arrangements which extends the method of Bremner et al. for computing automorphisms of polyhedra, and some associated changes to optimizations used in the previous work. I include results of tests on ACEnet clusters showing an order of magnitude speedup from the use of C++ in my implementation, an up to 3x speedup with a 6-core parallel variant of the algorithm, and positive results from other optimizations.
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Hager, Amanda C. "Freeness of hyperplane arrangement bundles and local homology of arrangement complements." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/678.
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A recent result of Salvetti and Settepanella gives, for a complexified real arrangement, an explicit description of a minimal CW decomposition as well as an explicit algebraic complex which computes local system homology. We apply their techniques to discriminantal arrangements in two dimensional complex space and calculate the boundary maps which will give local system homology groups given any choice of local system. This calculation generalizes several known results; examples are given related to Milnor fibrations, solutions of KZ equations, and the LKB representation of the braid group.
Another algebraic object associated to a hyperplane arrangement is the module of derivations. We analyze the behavior of the derivation module for an affine arrangement over an infinite field and relate its derivation module to that of its cone. In the case of an arrangement fibration, we analyze the relationship between the derivation module of the total space arrangement and those of the base and fiber arrangements. In particular, subject to certain restrictions, we establish freeness of the total space arrangement given freeness of the base and fiber arrangements.
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Williams, Kristopher John. "The Milnor fiber associated to an arrangement of hyperplanes." Diss., University of Iowa, 2011. https://ir.uiowa.edu/etd/1277.
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Let f be a non-constant, homogeneous, complex polynomial in n variables. We may associate to f a fibration with typical fiber F known as the Milnor fiber. For regular and isolated singular points of f at the origin, the topology of the Milnor fiber is well-understood. However, much less is known about the topology in the case of non-isolated singular points. In this thesis we analyze the Milnor fiber associated to a hyperplane arrangement, ie, f is a reduced, homogeneous polynomial with degree one irreducible components in n variables. If n > 2then the origin will be a non-isolated singular point. In particular, we use the fundamental group of the complement of the arrangement in order to construct a regular CW-complex that is homotopy equivalent to the Milnor fiber. Combining this construction with some local combinatorics of the arrangement, we generalize some known results on the upper bounds for the first betti number of the Milnor fiber. For several classes of arrangements we show that the first homology group of the Milnor fiber is torsion free. In the final section, we use methods that depend on the embedding of the arrangement in the complex projective plane (ie not necessarily combinatorial data) in order to analyze arrangements to which the known results on arrangements do not apply.
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Ardila, Federico 1977. "Enumerative and algebraic aspects of matroids and hyperplane arrangements." Thesis, Massachusetts Institute of Technology, 2003. http://hdl.handle.net/1721.1/29287.
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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003.Includes bibliographical references (p. 109-115).
This thesis consists of three projects on the enumerative and algebraic properties of matroids and hyperplane arrangements. In particular, a central object of study is the Tutte polynomial, which stores much of the enumerative information of these objects. The first project is the study of the Tutte polynomial of an arrangement and, more generally, of a semimatroid. It has two components: an enumerative one and a matroid-theoretic one. We start by considering purely enumerative questions about the Tutte polynomial of a hyperplane arrangement. We introduce a new method for computing it, which generalizes several known results. We apply our method to several specific arrangements, thus relating the computation of Tutte polynomials to problems in enumerative combinatorics. As a consequence, we obtain several new results about classical combinatorial objects such as labeled trees, Dyck paths, semiorders and alternating trees. We then address matroid-theoretic aspects of arrangements and their Tutte polynomials. We start by defining semimatroids, a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. After discussing these objects in detail, we define and investigate their Tutte polynomial. In particular, we prove that it is the universal Tutte-Grothendieck invariant for semimatroids, and we give a combinatorial interpretation for its non-negative coefficients. The second project is the beginning of an attempt to study the Tutte polynomial from an algebraic point of view.
(cont.) Given a matroid representable over a field of characteristic zero, we construct a graded algebra whose Hilbert-Poincar6 series is a simple evaluation of the Tutte polynomial of the matroid. This construction is joint work with Alex Postnikov. The third project involves a class of matroids with very rich enumerative properties. We show how the set of Dyck paths of length 2n naturally gives rise to a matroid, which we call the Catalan matroid Cn. We describe this matroid in detail; among several other results, we show that Cn is self-dual, it is representable over the rationals but not over finite fields Fq with q < n - 2, and it has a nice Tutte polynomial. We then introduce a more general family of matroids, which we call shifted matroids. They are precisely the matroids whose independence complex is a shifted simplicial complex.
by Federico Ardila.
Ph.D.
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Tohaneanu, Stefan Ovidiu. "Homological algebra and problems in combinatorics and geometry." Texas A&M University, 2003. http://hdl.handle.net/1969.1/5789.
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This dissertation uses methods from homological algebra and computational commutative
algebra to study four problems. We use Hilbert function computations and
classical homology theory and combinatorics to answer questions with a more applied
mathematics content: splines approximation, hyperplane arrangements, configuration
spaces and coding theory.
In Chapter II we study a problem in approximation theory. Alfeld and Schumaker
give a formula for the dimension of the space of piecewise polynomial functions
(splines) of degree d and smoothness r. Schenck and Stiller conjectured that this formula
holds for all d 2r + 1. In this chapter we show that there exists a simplicial
complex such that for any r, the dimension of the spline space in degree d = 2r is
not given by this formula.
Chapter III is dedicated to formal hyperplane arrangements. This notion was
introduced by Falk and Randell and generalized to formality by Brandt and Terao.
In this chapter we prove a criteria for formal arrangements, using a complex constructed
from vector spaces introduced by Brandt and Terao. As an application,
we give a simple description of formality of graphic arrangements in terms of the
homology of the flag complex of the graph.
Chapter IV approaches the problem of studying configuration of smooth rational
curves in P2. Since an irreducible conic in P2 is a P1 (so a line) it is natural to ask if classical results about line arrangements in P2, such as addition-deletion type
theorem, Yoshinaga criterion or Terao's conjecture verify for such configurations. In
this chapter we answer these questions. The addition-deletion theorem that we find
takes in consideration the fine local geometry of singularities. The results of this
chapter are joint work with H. Schenck.
In Chapter V we study a problem in algebraic coding theory. Gold, Little and
Schenck find a lower bound for the minimal distance of a complete intersection evaluation
codes. Since complete intersections are Gorenstein, we show a similar bound for
the minimal distance depending on the socle degree of the reduced zero-dimensional
Gorenstein scheme. The results of this chapter are a work in progress.
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Wakefield, Max. "On the derivation module and apolar algebra of an arrangement of hyperplanes /." view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1188874511&sid=1&Fmt=2&clientId=11238&RQT=309&VName=PQD.
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Thesis (Ph. D.)--University of Oregon, 2006.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 83-84). Also available for download via the World Wide Web; free to University of Oregon users.
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Kebede, Sebsibew. "On Bernstein-Sato ideals and Decomposition of D-modules over Hyperplane Arrangements." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-129493.
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Zhang, Yang. "Combinatorics of Milnor fibres of reflection arrangements." Thesis, University of Sydney, 2020. https://hdl.handle.net/2123/22985.
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Recently, Brady, Falk and Watt introduced a simplicial complex which has the homotopy type of the Milnor fibre F_Q of the reflection arrangement associated to a finite Coxeter group W. This thesis is devoted to developing a combinatorial approach to computing the integral homology groups of F_Q based on this simplicial complex. Our main result is a chain complex of free abelian groups whose integral homology is isomorphic to that of F_Q. Each chain group is isomorphic to a tensor product of the integral group ring ZW with the top reduced homology group of a rank-selected subposet of the noncrossing partition (NCP) lattice of W. Associated to the NCP lattice of W we define two isomorphic graded Z-algebras A and B, which have similarities to the Orlik-Solomon algebra and characterise the homology of the NCP lattice. The algebra A is defined in terms of generators and relations, while the algebra B is defined in a combinatorial manner which has to do with the Hurwitz actions. In particular, each element of B (or A) produces an explicit cycle of the top reduced homology group of the corresponding interval or rank-selected subposet of the NCP lattice. This permits us to calculate the homology of our chain complex computationally. The actions of both W and the monodromy may also be partly described by our chain complex. In particular, we prove that the homology of the subcomplex of W-invariant chain groups is isomorphic to the homology of F_P=F_Q/W, the Milnor fibre of the discriminant of W. This recovers the result of Brady, Falk and Watt.
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Paolini, Giovanni. "Topology and combinatorics of affine reflection arrangements." Doctoral thesis, Scuola Normale Superiore, 2019. http://hdl.handle.net/11384/85743.
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Bailet, Pauline. "Arrangements d'hyperplans." Phd thesis, Université Nice Sophia Antipolis, 2014. http://tel.archives-ouvertes.fr/tel-01059809.
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Cette thèse étudie la fibre de Milnor d'un arrangement d'hyperplans complexe central, et l'opérateur de monodromie sur ses groupes de cohomologie. On s'intéresse à la problématique suivante : peut-on déterminer l'opérateur de monodromie, ou au moins les nombres de Betti de la fibre de Milnor, à partir de l'information contenue dans le treillis d'intersection de l'arrangement? On donne deux théorèmes d'annulation des sous-espaces propres non triviaux de l'opérateur de monodromie. Le premier résultat s'applique à une large classe d'arrangements, le deuxième à des arrangements de droites projectives tels qu'il existe une droite contenant exactement un point de multiplicité supérieure ou égale à trois. Dans le dernier chapitre, on considère la structure de Hodge mixte des groupes de cohomologie de la fibre de Milnor d'un arrangement central et essentiel dans l'espace complexe de dimension quatre. On donne ensuite l'équivalence entre la trivialité de la monodromie, la nullité des coefficients non entiers du spectre de l'arrangement, et la nullité des nombres de Hodge mixtes des groupes de cohomologie de la fibre de Milnor.
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Möller, Tilman Hendrik [Verfasser], Gerhard [Gutachter] Röhrle, Christian [Gutachter] Stump, and Graham [Gutachter] Denham. "Combinatorial properties of hyperplane arrangements and reflection arrangements / Tilman Hendrik Möller ; Gutachter: Gerhard Röhrle, Christian Stump, Graham Denham ; Fakultät für Mathematik." Bochum : Ruhr-Universität Bochum, 2019. http://d-nb.info/1185171819/34.
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Möller, Tilman [Verfasser], Gerhard [Gutachter] Röhrle, Christian [Gutachter] Stump, and Graham [Gutachter] Denham. "Combinatorial properties of hyperplane arrangements and reflection arrangements / Tilman Hendrik Möller ; Gutachter: Gerhard Röhrle, Christian Stump, Graham Denham ; Fakultät für Mathematik." Bochum : Ruhr-Universität Bochum, 2019. http://d-nb.info/1185171819/34.
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Bartz, Jeremiah. "Multinets in P^2 and P^3." Thesis, University of Oregon, 2013. http://hdl.handle.net/1794/13252.
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In this dissertation, a method for producing multinets from a net in P^3 is presented. Multinets play an important role in the study of resonance varieties of the complement of a complex hyperplane arrangement and very few examples are known. Implementing this method, numerous new and interesting examples of multinets are identified. These examples provide additional evidence supporting the conjecture of Pereira and Yuzvinsky that all multinets are degenerations of nets. Also, a complete description is given of proper weak multinets, a generalization of multinets.
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Lund, Benjamin. "Some Results in Discrete Geometry." University of Cincinnati / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1342463167.
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Vo, Phi Khanh. "Contributions à l'étude des arrangements : équivalences combinatoires et perturbations." Université Joseph Fourier (Grenoble), 1994. http://tel.archives-ouvertes.fr/tel-00344973.
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Cette thèse est une contribution à l'étude des arrangements. L'idée est le calcul de la combinatoire d'un arrangement de courbes ou surfaces compte tenu du fait que les données et les opérations ne seront connues qu'à une précision près. Dans cette démarche, il se pose un problème qui est de savoir si la combinatoire d'un arrangement est stable lorsque les éléments constitutifs sont perturbés. Un préliminaire indispensable est alors d'établir une définition rigoureuse adaptée à nos besoin concernant l'équivalence des arrangements. Le travail consiste essentiellement en un développement des notions mathématiques nécessaires pour étudier l'équivalence, la construction, les perturbations d'arrangements. Quelques résultats en terme d'analyse de complexité sont également énoncés. Des résultats sont obtenus sur les perturbations d'arrangements d'hyperplans en dimension quelconque. Dans le plan est étudiée une méthode particulière de calcul des arrangements des courbes, avec un exemple détaillé sur les cercles. Utilisant des transformations classiques de dualité, des applications des propriétés d'équivalence des arrangements d'hyperplans aux configurations de points et aux diagrammes de Voronoï sont aussi données
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Rattan, Amarpreet. "Parking Functions and Related Combinatorial Structures." Thesis, University of Waterloo, 2001. http://hdl.handle.net/10012/1028.
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The central topic of this thesis is parking functions. We give a survey of some of the current literature concerning parking functions and focus on their interaction with other combinatorial objects; namely noncrossing partitions, hyperplane arrangements and tree inversions. In the final chapter, we discuss generalizations of both parking functions and the above structures.
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Venturelli, Federico. "The Alexander polynomial of certain classes of non-symmetric line arrangements." Doctoral thesis, Università degli studi di Padova, 2019. http://hdl.handle.net/11577/3422691.
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The Alexander polynomial of a projective hypersurface V ϲ Pᶰ is the characteristic polynomial of the monodromy operator acting on Hᶰ¯¹(F, C), where F is the Milnor fibre of V; unless V is smooth, the problem of its computation is open. The singular hypersurfaces that have drawn the most attention are projectivisations Ᾱ of central hyperplane arrangements A C Cᶰ⁺ ¹, as one can hope to take advantage of the combinatorial nature of such objects; one can assume without loss of generality that n=2.
In this Thesis we prove that the Alexander polynomials of line arrangements Ᾱ C P² belonging to some particular non-symmetric classes are trivial: this constitutes evidence in favour of the validity of a conjecture due to Papadima and Suciu.
The Thesis is organised as follows.
In Chapter 1 we gather some known results on which we will build upon: the discussion of mixed Hodge structures on cohomology groups of algebraic varieties and the comparison between the polar and Hodge filtration are of particular importance; the construction of cubical hyperresolutions and their use in the definition of algebraic de Rham cohomology for singular algebraic varieties will be very useful too.
Chapter 2 is divided in two parts. The first one is mainly devoted to defining the Alexander polynomial and presenting a formula by Libgober for its computation in case V is a curve. The second part is a survey of known results around the problem of determining the Alexander polynomial of a line arrangement, and closes with a discussion of some interesting examples; we try to highlight how the symmetry of the arrangement affects its Alexander polynomial.
In Chapter 3 we introduce some classes of non-symmetric line arrangements Ᾱ and prove that their Alexander polynomials are trivial. The methods we use are essentially two: one is the combination of Libgober's formula with an easy deformation theory argument, thanks to which we can restrict ourselves to considering a finite number of “representative arrangements”; the other relies on associating to Ᾱ a threefold T fibred in surfaces over P¹ and on studying the monodromy around a special fibre of the latter. A key step of the second method is the proof of the existence of a Gysin morphism that connects the cohomology of T to that of a hyperplane section S: this result is of independent interest, as T and S do not satisfy the hypotheses usually required in order to obtain Lefschetz-type results.
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Dupont, Clément. "Périodes des arrangements d'hyperplans et coproduit motivique." Thesis, Paris 6, 2014. http://www.theses.fr/2014PA066207.
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Dans cette thèse, on s'intéresse à des questions relatives aux arrangements d'hyperplans du point de vue des périodes motiviques. Suivant un programme initié par Beilinson et al., on étudie une famille de périodes appelée polylogarithmes d'Aomoto et leurs variantes motiviques, vues comme éléments de l'algèbre de Hopf fondamentale de la catégorie des structures de Hodge-Tate mixtes, ou de la catégorie des motifs de Tate mixtes sur un corps de nombres. On commence par calculer le coproduit motivique d'une famille de telles périodes, appelées polylogarithmes de dissection génériques, en montrant qu'il est régi par une formule combinatoire. Ce résultat généralise un théorème de Goncharov sur les intégrales itérées. Puis, on introduit les bi-arrangements d'hyperplans, objets géométriques et combinatoires qui généralisent les arrangements d'hyperplans classiques. Le calcul de groupes de cohomologie relative associés aux bi-arrangements d'hyperplans est une étape cruciale dans la compréhension du coproduit motivique des polylogarithmes d'Aomoto. On définit des outils cohomologiques et combinatoires pour calculer ces groupes de cohomologie, qui éclairent dans un cadre global des objets classiques tels que l'algèbre d'Orlik-SolomonIn this thesis, we deal with some questions about hyperplane arrangements from the viewpoint of motivic periods. Following a program initiated by Beilinson et al., we study a family of periods called Aomoto polylogarithms and their motivic variants, viewed as elements of the fundamental Hopf algebra of the category of mixed Hodge-Tate structures, or the category of mixed Tate motives over a number field. We start by computing the motivic coproduct of a family of such periods, called generic dissection polylogarithms, showing that it is governed by a combinatorial formula. This result generalizes a theorem of Goncharov on iterated integrals. Then, we introduce bi-arrangements of hyperplanes, which are geometric and combinatorial objects which generalize classical hyperplane arrangements. The computation of relative cohomology groups associated to bi-arrangements of hyperplanes is a crucial step in the understanding of the motivic coproduct of Aomoto polylogarithms. We define cohomological and combinatorial tools to compute these cohomology groups, which recast classical objects such as the Orlik-Solomon algebra in a global setting
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Le, Giang T. "The Action Dimension of Artin Groups." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1469011775.
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Rada, Miroslav. "Algoritmy pro vybrané geometrické problémy nad zonotopy a jejich aplikace v optimalizaci a v analýze dat." Doctoral thesis, Vysoká škola ekonomická v Praze, 2009. http://www.nusl.cz/ntk/nusl-199386.
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The thesis unifies the most important author's results in the field of algorithms concerning zonotopes and their applications in optimization and statistics. The computational-geometric results consist of a new compact output-sensitive algorithm for enumerating vertices of a zonotope, which outperforms the rival algorithm with the same complexity-theoretic properties both theoretically and empirically, and a polynomial algorithm for arbitrarily precise approximation of a zonotope with the Löwner-John ellipsoid. In the application area, the thesis presents a result, which connects linear regression model with interval outputs with the zonotope matters. The usage of presented geometric algorithms for solving a nonconvex optimisation problem is also discussed.
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Shokrieh, Farbod. "Divisors on graphs, binomial and monomial ideals, and cellular resolutions." Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/52176.
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We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs.
We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their Z-graded Betti tables coincide.
As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.
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Rousset, Mireille. "Sommes de Minkowski de triangles." Phd thesis, Université Joseph Fourier (Grenoble), 1996. http://tel.archives-ouvertes.fr/tel-00005017.
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La modélisation géométrique d'un problème de gestion de la fabrication des mélanges (faisabilité simultanée de deux mélanges) fait apparaître des polytopes nouveaux résultant de la somme de triangles particuliers qui dans ce contexte sont appelés convexes de 2-mélanges. De façon plus générale, la somme de triangles peut être considérée comme la généralisation des zonotopes (somme de segments). De ce point de vue, l'étude menée ici fait apparaître que la propriété de zone associée à un segment du zonotope se généralise à trois demi-zones associées à chaque triangle; et que la complexité combinatoire (nombre de faces du polytope), par rapport au nombre de sommandes, est du même ordre de grandeur que celle des zonotopes. On traite également le problème de la construction de tels polytopes, des algorithmes optimaux en temps sont proposés. Concernant le problème particulier des mélanges, le premier cas non trivial est celui de mélanges à trois composantes qui nous place en dimension 6. L'appartenance d'un point au convexe de 2-mélanges détermine la faisabilité simultanée des mélanges. Les facettes de ce polytope sont décrites, en détail, dans le cas de la dimension 6, dans le but d'obtenir des conditions de faisabilité des deux mélanges. Le problème de la décomposition de polytopes en somme de Minkowski de polytopes plus simples est exposé, ainsi que les principaux résultats existant.
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"Graph Laplacians, Nodal Domains, and Hyperplane Arrangements." Department of Statistics and Mathematics, Abt. f. Angewandte Statistik u. Datenverarbeitung, 2002. http://epub.wu-wien.ac.at/dyn/dl/wp/epub-wu-01_9f2.
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Geldon, Todd Wolman. "Computing the Tutte Polynomial of hyperplane arrangements." 2009. http://hdl.handle.net/2152/6660.
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We are studying the Tutte Polynomial of hyperplane arrangements. We discuss some previous work done to compute these polynomials. Then we explain our method to calculate the Tutte Polynomial of some arrangements more efficiently. We next discuss the details of the program used to do the calculation. We use this program and present the actual Tutte Polynomials calculated for the arrangements E6, E7, and E8.text
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Biyikoglu, Türker, Wim Hordijk, Josef Leydold, Tomaz Pisanski, and Peter F. Stadler. "Graph Laplacians, Nodal Domains, and Hyperplane Arrangements." 2004. https://ul.qucosa.de/id/qucosa%3A32145.
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Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e. the connected components of the maximal induced subgraphs of G on which an eigenvector ψ does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of ψ in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures.
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Arvola, William Arthur. "The fundamental group of the complement of an arrangement of complex hyperplanes." 1991. http://catalog.hathitrust.org/api/volumes/oclc/24492863.html.
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Thesis (Ph. D.)--University of Wisconsin--Madison, 1991.Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf 67).
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Thieu, Dinh Phong. "On graded ideals over the exterior algebra with applications to hyperplane arrangements." Doctoral thesis, 2013. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2013092311626.
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Graded ideals over the polynomial ring are studied deeply with a huge of methods and results. Over the exterior algebra, there are not much known about the structures of minimal graded resolutions, Gröbner fans of graded ideals or the Koszul property of algebras defined by graded ideals. We study componentwise linearity, linear resolutions of graded ideals as well as universally, initially and strongly Koszul properties of graded algebras defined by a graded ideals over the exterior algebra. After that, we apply our results to Orlik-Solomon ideals of hyperplane arrangements and show in which way the exterior algebra is useful in the study of related combinatorial objects.
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Brandt, Keith Allan. "A combinatorial study of the module of derivations of an arrangement of hyperplanes." 1992. http://catalog.hathitrust.org/api/volumes/oclc/28726852.html.
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Thesis (Ph. D.)--University of Wisconsin--Madison, 1992.Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 86-87).
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Le, Van Dinh. "The broken circuit complex and the Orlik - Terao algebra of a hyperplane arrangement." Doctoral thesis, 2016. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2016021714257.
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My thesis is mostly concerned with algebraic and combinatorial aspects of the
theory of hyperplane arrangements. More specifically, I study the Orlik-Terao algebra of a hyperplane arrangement and the broken circuit complex of a matroid. The Orlik-Terao algebra is a useful tool for studying hyperplane arrangements, especially for characterizing some non-combinatorial properties. The broken circuit complex, on the one hand, is closely related to the Orlik-Terao algebra, and on the other hand, plays a crucial role in the study of many combinatorial problem: the coefficients of the characteristic polynomial of a matroid are encoded in the f-vector of the broken circuit complex of the matroid. Among main results of the thesis are characterizations of the complete intersection and Gorenstein properties of the broken circuit complex and the Orlik-Terao algebra. I also study the h-vector of the broken circuit complex of a series-parallel network and relate certain entries of that vector to ear decompositions of the network. An application of the Orlik-Terao algebra in studying the relation space of a hyperplane arrangement is also included in the thesis.
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Vo, Phi Khanh. "Contributions à l'étude des arrangements: Equivalences combinatoires et perturbations." Phd thesis, 1994. http://tel.archives-ouvertes.fr/tel-00344973.
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Cette thèse est une contribution à l'étude des arrangements. L'idée est le calcul de la combinatoire d'un arrangement de courbes ou surfaces compte tenu du fait que les données et les opérations ne seront connues qu'à une précision près. Dans cette démarche, il se pose un problème qui est de savoir si la combinatoire d'un arrangement est stable lorsque les éléments constitutifs sont perturbés. Un préliminaire indispensable est alors d'établir une définition rigoureuse adaptée à nos besoin concernant l'équivalence des arrangements. Le travail consiste essentiellement en un développement des notions mathématiques nécessaires pour étudier l'équivalence, la construction, les perturbations d'arrangements. Quelques résultats en terme d'analyse de complexité sont également énoncés. Des résultats sont obtenus sur les perturbations d'arrangements d'hyperplans en dimension quelconque. Dans le plan est étudiée une méthode particulière de calcul des arrangements des courbes, avec un exemple détaillé sur les cercles. Utilisant des transformations classiques de dualité, des applications des propriétés d'équivalence des arrangements d'hyperplans aux configurations de points et aux diagrammes de Voronoï sont aussi données
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Narkawicz, Anthony Joseph. "Cohomology Jumping Loci and the Relative Malcev Completion." Diss., 2007. http://hdl.handle.net/10161/441.
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(10724076), Daniel L. Bath. "Bernstein--Sato Ideals and the Logarithmic Data of a Divisor." Thesis, 2021.
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We study a multivariate version of the Bernstein–Sato polynomial, the so-called Bernstein–Sato ideal, associated to an arbitrary factorization of an analytic germ f - f1···fr. We identify a large class of geometrically characterized germs so that the DX,x[s1,...,sr]-annihilator of fs11···fsrr admits the simplest possible description and, more-over, has a particularly nice associated graded object. As a consequence we are able to verify Budur’s Topological Multivariable Strong Monodromy Conjecture for arbitrary factorizations of tame hyperplane arrangements by showing the zero locus of the associated Bernstein–Sato ideal contains a special hyperplane. By developing ideas of Maisonobe and Narvaez-Macarro, we are able to find many more hyperplanes contained in the zero locus of this Bernstein–Sato ideal. As an example, for reduced, tame hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial contained in [−1,0) are combinatorially determined; for reduced, free hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial are all combinatorially determined. Finally, outside the hyperplane arrangement setting, we prove many results about a certain DX,x-map ∇A that is expected to characterize the roots of the Bernstein–Sato ideal.