Relevant bibliographies by topics / Hyperplanes arrangements

Academic literature on the topic 'Hyperplanes arrangements'

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Published: 25 May 2024

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Journal articles on the topic "Hyperplanes arrangements"

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Bergerová, Diana. "Symmetry of f-Vectors of Toric Arrangements in General Position and Some Applications." PUMP Journal of Undergraduate Research 7 (February 15, 2024): 96–123. http://dx.doi.org/10.46787/pump.v7i0.3921.

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A toric hyperplane is the preimage of a point in a circle of a continuous surjective group homomorphism from the n-torus to the circle. A toric hyperplane arrangement is a finite collection of such hyperplanes. In this paper, we study the combinatorial properties of toric hyperplane arrangements on n-tori which are spanning and in general position. Specifically, we describe the symmetry of f-vectors arising in such arrangements and a few applications of the result to count configurations of hyperplanes.
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Gao, Ruimei, Qun Dai, and Zhe Li. "On the freeness of hypersurface arrangements consisting of hyperplanes and spheres." Open Mathematics 16, no. 1 (April 23, 2018): 437–46. http://dx.doi.org/10.1515/math-2018-0041.

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AbstractLet V be a smooth variety. A hypersurface arrangement 𝓜 in V is a union of smooth hypersurfaces, which locally looks like a union of hyperplanes. We say 𝓜 is free if all these local models can be chosen to be free hyperplane arrangements. In this paper, we use Saito’s criterion to study the freeness of hypersurface arrangements consisting of hyperplanes and spheres, and construct the bases for the derivation modules explicitly.
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Pfeiffer, Götz, and Hery Randriamaro. "The Varchenko determinant of a Coxeter arrangement." Journal of Group Theory 21, no. 4 (July 1, 2018): 651–65. http://dx.doi.org/10.1515/jgth-2018-0009.

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AbstractThe Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula for this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their associated determinants.
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Faenzi, Daniele, Daniel Matei, and Jean Vallès. "Hyperplane arrangements of Torelli type." Compositio Mathematica 149, no. 2 (December 14, 2012): 309–32. http://dx.doi.org/10.1112/s0010437x12000577.

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AbstractWe give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.
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Orlik, Peter, and Hiroaki Terao. "Commutative algebras for arrangements." Nagoya Mathematical Journal 134 (June 1994): 65–73. http://dx.doi.org/10.1017/s0027763000004852.

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Let V be a vector space of dimension l over some field K. A hyperplane H is a vector subspace of codimension one. An arrangement is a finite collection of hyperplanes in V. We use [7] as a general reference.
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Jambu, Michel, and Luis Paris. "Factored arrangements of hyperplanes." Kodai Mathematical Journal 17, no. 3 (1994): 402–8. http://dx.doi.org/10.2996/kmj/1138040032.

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Linhart, J. "Arrangements of oriented hyperplanes." Discrete & Computational Geometry 10, no. 4 (December 1993): 435–46. http://dx.doi.org/10.1007/bf02573989.

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Zaslavsky, Thomas. "EXTREMAL ARRANGEMENTS OF HYPERPLANES." Annals of the New York Academy of Sciences 440, no. 1 Discrete Geom (May 1985): 69–87. http://dx.doi.org/10.1111/j.1749-6632.1985.tb14540.x.

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Gallet, Matteo, and Elia Saini. "The diffeomorphism type of small hyperplane arrangements is combinatorially determined." Advances in Geometry 19, no. 1 (January 28, 2019): 89–100. http://dx.doi.org/10.1515/advgeom-2018-0015.

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Abstract It is known that there exist hyperplane arrangements with the same underlying matroid that admit non-homotopy equivalent complement manifolds. Here we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and the same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, that is, they depend only on the underlying matroid. To prove this, we associate to every such matroid a topological space, that we call the reduced realization space; its connectedness, shown by means of symbolic computation, implies the desired result.
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Abe, Takuro, Hiroaki Terao, and Masahiko Yoshinaga. "Totally free arrangements of hyperplanes." Proceedings of the American Mathematical Society 137, no. 04 (November 5, 2008): 1405–10. http://dx.doi.org/10.1090/s0002-9939-08-09755-4.

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Dissertations / Theses on the topic "Hyperplanes arrangements"

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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 classiques
This 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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Books on the topic "Hyperplanes arrangements"

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Orlik, Peter, and Hiroaki Terao. Arrangements of Hyperplanes. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/978-3-662-02772-1.

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1951-, Terao Hiroaki, ed. Arrangements of hyperplanes. Berlin: Springer-Verlag, 1992.

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Dimca, Alexandru. Hyperplane Arrangements. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-56221-6.

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Yoshinaga, Masahiko. Hyperplane arrangements and Lefschetz's hyperplane section theorem. Kyoto, Japan: Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2005.

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Alexeev, Valery. Moduli of Weighted Hyperplane Arrangements. Edited by Gilberto Bini, Martí Lahoz, Emanuele Macrí, and Paolo Stellari. Basel: Springer Basel, 2015. http://dx.doi.org/10.1007/978-3-0348-0915-3.

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De Concini, Corrado, and Claudio Procesi. Topics in Hyperplane Arrangements, Polytopes and Box-Splines. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-0-387-78963-7.

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Claudio, Procesi, ed. Topics in hyperplane arrangements, polytopes and box-splines. New York: Springer, 2011.

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Barg, Alexander, and O. R. Musin. Discrete geometry and algebraic combinatorics. Providence, Rhode Island: American Mathematical Society, 2014.

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Orlik, Peter, and Hiroaki Terao. Arrangements of Hyperplanes. Springer London, Limited, 2013.

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Orlik, Peter, and Hiroaki Terao. Arrangements of Hyperplanes. Springer Berlin / Heidelberg, 2010.

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Book chapters on the topic "Hyperplanes arrangements"

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Grünbaum, Branko. "Arrangements of Hyperplanes." In Convex Polytopes, 432–54. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9_18.

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Ovchinnikov, Sergei. "Hyperplane Arrangements." In Universitext, 207–35. New York, NY: Springer New York, 2011. http://dx.doi.org/10.1007/978-1-4614-0797-3_7.

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De Concini, Corrado, and Claudio Procesi. "Hyperplane Arrangements." In Topics in Hyperplane Arrangements, Polytopes and Box-Splines, 25–68. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-0-387-78963-7_2.

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Kastner, Lars, and Marta Panizzut. "Hyperplane Arrangements in polymake." In Lecture Notes in Computer Science, 232–40. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-52200-1_23.

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Alexeev, Valery. "Weighted Stable Hyperplane Arrangements." In Advanced Courses in Mathematics - CRM Barcelona, 75–92. Basel: Springer Basel, 2015. http://dx.doi.org/10.1007/978-3-0348-0915-3_5.

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Denham, Graham. "Homological Aspects of Hyperplane Arrangements." In Arrangements, Local Systems and Singularities, 39–58. Basel: Birkhäuser Basel, 2009. http://dx.doi.org/10.1007/978-3-0346-0209-9_2.

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De Concini, Corrado, and Claudio Procesi. "Toric Arrangements." In Topics in Hyperplane Arrangements, Polytopes and Box-Splines, 241–67. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-0-387-78963-7_14.

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Dimca, Alexandru. "Hyperplane Arrangements and Their Combinatorics." In Universitext, 15–43. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-56221-6_2.

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Massey, David B. "Lê numbers and hyperplane arrangements." In Lê Cycles and Hypersurface Singularities, 61–67. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/bfb0094415.

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Stanley, Richard. "An introduction to hyperplane arrangements." In Geometric Combinatorics, 389–496. Providence, Rhode Island: American Mathematical Society, 2007. http://dx.doi.org/10.1090/pcms/013/08.

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Conference papers on the topic "Hyperplanes arrangements"

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Mulmuley, Ketan, and Sandeep Sen. "Dynamic point location in arrangements of hyperplanes." In the seventh annual symposium. New York, New York, USA: ACM Press, 1991. http://dx.doi.org/10.1145/109648.109663.

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Stoican, Florin, Ionela Prodan, and Sorin Olaru. "On the hyperplanes arrangements in mixed-integer techniques." In 2011 American Control Conference. IEEE, 2011. http://dx.doi.org/10.1109/acc.2011.5990908.

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Hagerup, Torben, H. Jung, and E. Welzl. "Efficient parallel computation of arrangements of hyperplanes in d dimensions." In the second annual ACM symposium. New York, New York, USA: ACM Press, 1990. http://dx.doi.org/10.1145/97444.97696.

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Jambu, Michel. "Arrangements of Hyperplanes, Lower Central Series, Chen Lie Algebras and Resonance Varieties." In The International Conference on Algebra 2010 - Advances in Algebraic Structures. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814366311_0022.

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"Cutting hyperplane arrangements." In the sixth annual symposium, edited by Jiří Matoušek. New York, New York, USA: ACM Press, 1990. http://dx.doi.org/10.1145/98524.98528.

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JAMBU, MICHEL. "KOSZUL ALGEBRAS AND HYPERPLANE ARRANGEMENTS." In Proceedings of the Second International Congress in Algebra and Combinatorics. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812790019_0011.

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JAMBU, MICHEL. "HYPERGEOMETRIC FUNCTIONS AND HYPERPLANE ARRANGEMENTS." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0005.

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Stoican, Florin, Ionela Prodan, and Sorin Olaru. "Enhancements on the hyperplane arrangements in mixed integer techniques." In 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 2011). IEEE, 2011. http://dx.doi.org/10.1109/cdc.2011.6161361.

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Ioan, Daniel, Sorin Olaru, Ionela Prodan, Florin Stoican, and Silviu-Iulian Niculescu. "Parametrized Hyperplane Arrangements for Control Design with Collision Avoidance Constraints." In 2019 IEEE 15th International Conference on Control and Automation (ICCA). IEEE, 2019. http://dx.doi.org/10.1109/icca.2019.8899977.

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Aronov, Boris, Jiří Matoušek, and Micha Sharir. "On the sum of squares of cell complexities in hyperplane arrangements." In the seventh annual symposium. New York, New York, USA: ACM Press, 1991. http://dx.doi.org/10.1145/109648.109682.

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Reports on the topic "Hyperplanes arrangements"

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Paul, Thomas J. Enumerative Geometry of Hyperplane Arrangements. Fort Belvoir, VA: Defense Technical Information Center, May 2012. http://dx.doi.org/10.21236/ada575879.

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