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Journal articles on the topic 'Hyperplanes arrangements'

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Author: Grafiati

Published: 25 May 2024

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Arvola, William A. "Complexified real arrangements of hyperplanes." manuscripta mathematica 71, no. 1 (December 1991): 295–306. http://dx.doi.org/10.1007/bf02568407.

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12

Abe, Takuro. "Divisionally free arrangements of hyperplanes." Inventiones mathematicae 204, no. 1 (August 6, 2015): 317–46. http://dx.doi.org/10.1007/s00222-015-0615-7.

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13

Gao, Ruimei, Xiupeng Cui, and Zhe Li. "Supersolvable orders and inductively free arrangements." Open Mathematics 15, no. 1 (May 6, 2017): 587–94. http://dx.doi.org/10.1515/math-2017-0052.

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Abstract In this paper, we define the supersolvable order of hyperplanes in a supersolvable arrangement, and obtain a class of inductively free arrangements according to this order. Our main results improve the conclusion that every supersolvable arrangement is inductively free. In addition, we assert that the inductively free arrangement with the required induction table is supersolvable.

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14

Cuntz, M., and D. Geis. "Combinatorial simpliciality of arrangements of hyperplanes." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 56, no. 2 (January 30, 2014): 439–58. http://dx.doi.org/10.1007/s13366-014-0190-x.

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15

Terao, Hiroaki, and Masahiko Yoshinaga. "Recent topics of arrangements of hyperplanes." Sugaku Expositions 31, no. 1 (March 20, 2018): 43–67. http://dx.doi.org/10.1090/suga/428.

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16

Meiser, S. "Point Location in Arrangements of Hyperplanes." Information and Computation 106, no. 2 (October 1993): 286–303. http://dx.doi.org/10.1006/inco.1993.1057.

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17

Paris, Luis. "Arrangements of hyperplanes with property D." Geometriae Dedicata 45, no. 2 (February 1993): 171–76. http://dx.doi.org/10.1007/bf01264519.

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18

Frønsdal, Christian. "q-algebras and arrangements of hyperplanes." Journal of Algebra 278, no. 2 (August 2004): 433–55. http://dx.doi.org/10.1016/j.jalgebra.2004.03.024.

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19

HUBARD, ALFREDO, and ROMAN KARASEV. "Bisecting measures with hyperplane arrangements." Mathematical Proceedings of the Cambridge Philosophical Society 169, no. 3 (October 31, 2019): 639–47. http://dx.doi.org/10.1017/s0305004119000380.

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20

Lee, Ki-Suk, and Mi-Yeon Kwon. "ARRANGEMENTS OF HYPERPLANES IN ℝ3AND THEIR FREENESS." Honam Mathematical Journal 31, no. 1 (March 25, 2009): 25–29. http://dx.doi.org/10.5831/hmj.2009.31.1.025.

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21

LEÓN TRUJILLO, Francisco James. "$\mathcal{D}$-Modules and Arrangements of Hyperplanes." Tokyo Journal of Mathematics 29, no. 2 (December 2006): 429–44. http://dx.doi.org/10.3836/tjm/1170348177.

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22

Mulmuley, Ketan, and Sandeep Sen. "Dynamic point location in arrangements of hyperplanes." Discrete & Computational Geometry 8, no. 3 (September 1992): 335–60. http://dx.doi.org/10.1007/bf02293052.

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23

Abe, Takuro. "Erratum to: Divisionally free arrangements of hyperplanes." Inventiones mathematicae 207, no. 3 (December 19, 2016): 1377–78. http://dx.doi.org/10.1007/s00222-016-0714-0.

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24

Dolgachev, Igor V. "Logarithmic sheaves attached to arrangements of hyperplanes." Journal of Mathematics of Kyoto University 47, no. 1 (2007): 35–64. http://dx.doi.org/10.1215/kjm/1250281067.

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25

Schechtman, Vadim V., and Alexander N. Varchenko. "Arrangements of hyperplanes and Lie algebra homology." Inventiones Mathematicae 106, no. 1 (December 1991): 139–94. http://dx.doi.org/10.1007/bf01243909.

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26

CHO, KOJI, and MASAAKI YOSHIDA. "VERONESE ARRANGEMENTS OF HYPERPLANES IN REAL PROJECTIVE SPACES." International Journal of Mathematics 23, no. 05 (May 2012): 1250061. http://dx.doi.org/10.1142/s0129167x12500619.

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27

Amend, Nils, Pierre Deligne, and Gerhard Röhrle. "On the -problem for restrictions of complex reflection arrangements." Compositio Mathematica 156, no. 3 (January 20, 2020): 526–32. http://dx.doi.org/10.1112/s0010437x19007796.

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Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$, let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing $Y$. We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$. We prove it in case of the monomial groups $W=G(r,p,\ell )$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

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28

Terao, Hiroaki, and Sergey Yuzvinsky. "Logarithmic forms on affine arrangements." Nagoya Mathematical Journal 139 (September 1995): 129–49. http://dx.doi.org/10.1017/s002776300000533x.

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Let V be an affine of dimension l over some field K. An arrangement A is a finite collection of affine hyperplanes in V. We call A an l-arrangement when we want to emphasize the dimension of V. We use [6] as a general reference. Choose an arbitrary point of V and fix it throughout this paper. We will use it as the origin.

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29

Kanarek, Herbert. "Gauß-Manin Connection arising from arrangements of hyperplanes." Illinois Journal of Mathematics 44, no. 4 (December 2000): 741–66. http://dx.doi.org/10.1215/ijm/1255984690.

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30

Roudneff, Jean-Pierre. "Cells with many facets in arrangements of hyperplanes." Discrete Mathematics 98, no. 3 (December 1991): 185–91. http://dx.doi.org/10.1016/0012-365x(91)90375-c.

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31

Edelsbrunner, H., J. O’Rourke, and R. Seidel. "Constructing Arrangements of Lines and Hyperplanes with Applications." SIAM Journal on Computing 15, no. 2 (May 1986): 341–63. http://dx.doi.org/10.1137/0215024.

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32

BOSE, PROSENJIT, HAZEL EVERETT, and STEPHEN WISMATH. "PROPERTIES OF ARRANGEMENT GRAPHS." International Journal of Computational Geometry & Applications 13, no. 06 (December 2003): 447–62. http://dx.doi.org/10.1142/s0218195903001281.

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An arrangement graph G is the abstract graph obtained from an arrangement of lines L, in general position by associating vertices of G with the intersection points of L, and the edges of G with the line segments joining the intersection points of L. A simple polygon (respectively path) of n sides in general position, induces a set of n lines by extension of the line segments into lines. The main results of this paper are: • Given a graph G, it is NP-Hard to determine if G is the arrangement graph of some set of lines. • There are non-Hamiltonian arrangement graphs for arrangements of six lines and for odd values of n>6 lines. • All arrangements of n lines contain a subarrangement of size [Formula: see text] with an inducing polygon. • All arrangements on n lines contain an inducing path consisting of n line segments. A Java applet implementing the algorithm for determining such a path is also provided. • All arrangements on n hyperplanes in Rd contain a simple inducing polygonal cycle of size n.

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33

Athanasiadis, Christos A. "A Combinatorial Reciprocity Theorem for Hyperplane Arrangements." Canadian Mathematical Bulletin 53, no. 1 (March 1, 2010): 3–10. http://dx.doi.org/10.4153/cmb-2010-004-7.

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AbstractGiven a nonnegative integer m and a finite collection of linear forms on ℚd, the arrangement of affine hyperplanes in ℚd defined by the equations α(x) = k for α ∈ and integers k ∈ [–m,m] is denoted by . It is proved that the coefficients of the characteristic polynomial of are quasi-polynomials in m and that they satisfy a simple combinatorial reciprocity law.

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34

COHEN, DANIEL C., and ALEXANDER I. SUCIU. "Characteristic varieties of arrangements." Mathematical Proceedings of the Cambridge Philosophical Society 127, no. 1 (July 1999): 33–53. http://dx.doi.org/10.1017/s0305004199003576.

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The kth Fitting ideal of the Alexander invariant B of an arrangement [Ascr ] of n complex hyperplanes defines a characteristic subvariety, Vk([Ascr ]), of the algebraic torus ([Copf ]*)n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of Vk([Ascr ]). For any arrangement [Ascr ], we show that the tangent cone at the identity of this variety coincides with [Rscr ]1k(A), one of the cohomology support loci of the Orlik–Solomon algebra. Using work of Arapura [1], we conclude that all irreducible components of Vk([Ascr ]) which pass through the identity element of ([Copf ]*)n are combinatorially determined, and that [Rscr ]1k(A) is the union of a subspace arrangement in [Copf ]n, thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.

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35

Dolgachev, I., and M. Kapranov. "Arrangements of hyperplanes and vector bundles on $P^n$." Duke Mathematical Journal 71, no. 3 (September 1993): 633–64. http://dx.doi.org/10.1215/s0012-7094-93-07125-6.

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36

ATHANASIADIS, CHRISTOS A. "GENERALIZED CATALAN NUMBERS, WEYL GROUPS AND ARRANGEMENTS OF HYPERPLANES." Bulletin of the London Mathematical Society 36, no. 03 (April 28, 2004): 294–302. http://dx.doi.org/10.1112/s0024609303002856.

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37

Guibas, L. J., D. Halperin, J. Matoušek, and M. Sharir. "Vertical decomposition of arrangements of hyperplanes in four dimensions." Discrete & Computational Geometry 14, no. 2 (September 1995): 113–22. http://dx.doi.org/10.1007/bf02570698.

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38

Blanco, Víctor, Alberto Japón, and Justo Puerto. "Optimal arrangements of hyperplanes for SVM-based multiclass classification." Advances in Data Analysis and Classification 14, no. 1 (July 26, 2019): 175–99. http://dx.doi.org/10.1007/s11634-019-00367-6.

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39

Aronov, Boris, Daniel Q. Naiman, János Pach, and Micha Sharir. "An invariant property of balls in arrangements of hyperplanes." Discrete & Computational Geometry 10, no. 4 (December 1993): 421–25. http://dx.doi.org/10.1007/bf02573987.

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40

Bespamyatnikh, Sergei, and Michael Segal. "Selecting distances in arrangements of hyperplanes spanned by points." Journal of Discrete Algorithms 2, no. 3 (September 2004): 333–45. http://dx.doi.org/10.1016/j.jda.2003.12.001.

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41

Liu, Ding. "A note on point location in arrangements of hyperplanes." Information Processing Letters 90, no. 2 (April 2004): 93–95. http://dx.doi.org/10.1016/j.ipl.2004.01.010.

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42

Terao, Hiroaki. "Chambers of arrangements of hyperplanes and Arrow's impossibility theorem." Advances in Mathematics 214, no. 1 (September 2007): 366–78. http://dx.doi.org/10.1016/j.aim.2007.02.006.

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43

Varchenko, A. "Bethe Ansatz for Arrangements of Hyperplanes and the Gaudin Model." Moscow Mathematical Journal 6, no. 1 (2006): 195–210. http://dx.doi.org/10.17323/1609-4514-2006-6-1-195-210.

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44

Wiens, Jonathan, and Sergey Yuzvinsky. "De Rham cohomology of logarithmic forms on arrangements of hyperplanes." Transactions of the American Mathematical Society 349, no. 4 (1997): 1653–62. http://dx.doi.org/10.1090/s0002-9947-97-01894-1.

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45

Fukuda, Komei, Shigemasa Saito, Akihisa Tamura, and Takeshi Tokuyama. "Bounding the number of k-faces in arrangements of hyperplanes." Discrete Applied Mathematics 31, no. 2 (April 1991): 151–65. http://dx.doi.org/10.1016/0166-218x(91)90067-7.

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46

Prodan, Ionela, Florin Stoican, Sorin Olaru, and Silviu-Iulian Niculescu. "Enhancements on the Hyperplanes Arrangements in Mixed-Integer Programming Techniques." Journal of Optimization Theory and Applications 154, no. 2 (March 24, 2012): 549–72. http://dx.doi.org/10.1007/s10957-012-0022-9.

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47

Haggui, Fathi, and Abdessami Jalled. "Hyperbolicity of the complement of arrangements of non complex lines." Filomat 34, no. 9 (2020): 3109–18. http://dx.doi.org/10.2298/fil2009109h.

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The goal of this paper is twofold. We study holomorphic curves f:C ? C3 avoiding four complex hyperplanes and a real subspace of real dimension five in C3 where we study the cases where the projection of f into the complex projective space CP2 is constant. On the other hand, we investigate the kobayashi hyperbolicity of the complement of five perturbed lines in CP2.

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48

Prudhom, Andrew, and Alexander Varchenko. "Potentials of a Family of Arrangements of Hyperplanes and Elementary Subarrangements." Moscow Mathematical Journal 19, no. 1 (2019): 153–80. http://dx.doi.org/10.17323/1609-4514-2019-19-1-153-180.

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49

Ezra, Esther, Sariel Har-Peled, Haim Kaplan, and Micha Sharir. "Decomposing Arrangements of Hyperplanes: VC-Dimension, Combinatorial Dimension, and Point Location." Discrete & Computational Geometry 64, no. 1 (December 17, 2019): 109–73. http://dx.doi.org/10.1007/s00454-019-00141-7.

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50

Paris, Luis. "Universal Cover of Salvetti's Complex and Topology of Simplicial Arrangements of Hyperplanes." Transactions of the American Mathematical Society 340, no. 1 (November 1993): 149. http://dx.doi.org/10.2307/2154550.

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