Non-Euclidean visibility problems

Chamizo, Fernando

doi:10.1007/bf02829784

Cited by 9 publications

(6 citation statements)

References 11 publications

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“…Group SL 2 (Z) is important in the context of many fundamental problems, for example from hyperbolic geometry [25,9,12], dynamical systems [19], Lorenz/modular knots [15], braid groups [20], particle physics, high energy physics [24], M/string theories [11], ray tracing analysis, music theory [17] and it plays a central role for the development of efficient solutions of 2 × 2 matrix problems [21].…”

mentioning

confidence: 99%

The Identity Problem for Matrix Semigroups in SL₂(ℤ) is NP-complete

Bell¹,

Hirvensalo²,

Potapov³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper, we show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of 2 × 2 matrices from the modular group PSL 2 (Z) and thus the Special Linear group SL 2 (Z) is solvable in NP. From this fact, we can immediately derive that the fundamental problem of whether a given finite set of matrices from SL 2 (Z) or PSL 2 (Z) generates a group or free semigroup is also decidable in NP. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhumäki, was in EXPSPACE mainly due to the translation of matrices into exponentially long words over a binary alphabet {s, r} and further constructions with a large nondeterministic finite state automaton that is built on these words. Our algorithm is based on various new techniques that allow us to operate with compressed word representations of matrices without explicit expansions. When combined with the known NP-hard lower bound, this proves that the membership problem for the identity problem, the group problem and the freeness problem in SL 2 (Z) are NP-complete.

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mentioning

confidence: 99%

The Identity Problem for Matrix Semigroups in SL₂(ℤ) is NP-complete

Bell¹,

Hirvensalo²,

Potapov³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In this section we establish notation in use throughout the paper, and we reduce the pair correlation problem to angles in the first quadrant. A similar reduction can be found in [Chamizo 2006], in the context of visibility problems for the hyperbolic lattice centered at i.…”

Section: Reduction To the First Quadrantmentioning

confidence: 63%

Untitled

2014

Algebra Number Theory

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We construct the derived scheme of stable sheaves on a smooth projective variety via derived moduli of finite graded modules over a graded ring. We do this by dividing the derived scheme of actions of Ciocan-Fontanine and Kapranov by a suitable algebraic gauge group. We show that the natural notion of GIT stability for graded modules reproduces stability for sheaves. 1. The derived scheme of simple graded modules 788 1A. The differential graded Lie algebra L 788 1B. The moduli stack of L 791 1C. The derived space of equivalence classes of simple modules 796 1D. The tangent complex 798 2. Stability 801 3. Moduli of sheaves 803 3A. The adjoint of the truncation functor 803 3B. Open immersion 804 3C. Stable sheaves 806 3D. Moduli of sheaves 807 3E. An amplification 808 4. Derived moduli of sheaves 810 References 811 MSC2010: 14D20.

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“…, y d+1 ) ∈ R d . If one can prove that the distance in the hyperplane {x 1 = k} between the point (5) and an element of the set (4) is less than for some k ∈ 0, V , this would plainly imply that (3) holds.…”

Section: Consider Now Anymentioning

confidence: 99%

“…Allen [1] extended this result to the case when the disk has a non-integer radius and Kruskal [20] dealt with the situation where the trees are centred at non-zero points of any lattice. Chamizo [5] also studied an analogue of this problem in hyperbolic spaces and Cusick [9] considered the case when the trees have the shape of any given convex body (Cusick relates this case with the Lonely Runner Conjecture -see [6,7] for further developments). G. Pólya [24] also took an interest in the visibility in a random and periodic forest, a topic related to the distribution of free path lengths in the Lorentz gas which is still an active domain of research -see [21] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

How Far Can You See in a Forest?

Adiceam

2015

Int Math Res Notices

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We address a visibility problem posed by Solomon & Weiss [26]. More precisely, in any dimension n := d + 1 ≥ 2, we construct a forest F with finite density satisfying the following condition : if > 0 denotes the radius common to all the trees in F, then the visibility V therein satisfies the estimate V( ) = O η,d −2d−η for any η > 0, no matter where we stand and what direction we look in. The proof involves Fourier analysis and sharp estimates of exponential sums.

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Non-Euclidean visibility problems

Cited by 9 publications

References 11 publications

The Identity Problem for Matrix Semigroups in SL₂(ℤ) is NP-complete

The Identity Problem for Matrix Semigroups in SL₂(ℤ) is NP-complete

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How Far Can You See in a Forest?

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Non-Euclidean visibility problems

Cited by 9 publications

References 11 publications

The Identity Problem for Matrix Semigroups in SL2(ℤ) is NP-complete

The Identity Problem for Matrix Semigroups in SL2(ℤ) is NP-complete

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How Far Can You See in a Forest?

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The Identity Problem for Matrix Semigroups in SL₂(ℤ) is NP-complete

The Identity Problem for Matrix Semigroups in SL₂(ℤ) is NP-complete