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Journal articles on the topic 'Random Polytope'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Dwyer, Rex A. "On the convex hull of random points in a polytope." Journal of Applied Probability 25, no. 4 (1988): 688–99. http://dx.doi.org/10.2307/3214289.

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The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(logd–1n) for any polytope, the expected number of vertices is Ω(logd–1n) for any simple polytope, and the expected number of facets is O(logd–1n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.

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2

Dwyer, Rex A. "On the convex hull of random points in a polytope." Journal of Applied Probability 25, no. 04 (1988): 688–99. http://dx.doi.org/10.1017/s0021900200041474.

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The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(log d–1 n) for any polytope, the expected number of vertices is Ω(log d–1 n) for any simple polytope, and the expected number of facets is O(log d–1 n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.

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3

Shiffman, Bernard, and Steve Zelditch. "Random polynomials with prescribed Newton polytope." Journal of the American Mathematical Society 17, no. 1 (2003): 49–108. http://dx.doi.org/10.1090/s0894-0347-03-00437-5.

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4

Paramasamy, S. "On the moments of random variables uniformly distributed over a polytope." International Journal of Mathematics and Mathematical Sciences 20, no. 1 (1997): 197–200. http://dx.doi.org/10.1155/s0161171297000240.

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SupposeX=(X1,X2,…,Xn)is a random vector uniformly distributed over a polytope. In this note, the author derives a formula forE(XirXjs…), (the expected value ofXirXjs…), in terms of the extreme points of the polytope.

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5

Böröczky, Károly J., and Rolf Schneider. "The Mean Width of Circumscribed Random Polytopes." Canadian Mathematical Bulletin 53, no. 4 (2010): 614–28. http://dx.doi.org/10.4153/cmb-2010-067-5.

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AbstractFor a given convex body K in ℝd, a random polytope K(n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of K(n) and K as n tends to infinity. For a simplicial polytope P, a precise asymptotic formula for the difference of the mean widths of P(n) and P is obtained.

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6

B�r�ny, Imre, and Christian Buchta. "Random polytopes in a convex polytope, independence of shape, and concentration of vertices." Mathematische Annalen 297, no. 1 (1993): 467–97. http://dx.doi.org/10.1007/bf01459511.

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7

Bárány, Imre, and Leoni Dalla. "Few points to generate a random polytope." Mathematika 44, no. 2 (1997): 325–31. http://dx.doi.org/10.1112/s0025579300012638.

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8

Affentranger, Fernando, and Rex A. Dwyer. "The convex hull of random balls." Advances in Applied Probability 25, no. 02 (1993): 373–94. http://dx.doi.org/10.1017/s0001867800025404.

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While the convex hull of n d-dimensional balls is not a polytope, it does have an underlying combinatorial structure similar to that of a polytope. In the worst case, its combinatorial complexity can be of order Ω(n [d/2]). The thrust of this work is to show that its complexity is typically much smaller, and that it can therefore be constructed more quickly on average than in the worst case. To this end, four models of the random d-ball are developed, and the expected combinatorial complexity of the convex hull of n independent random d-balls is investigated for each. As n grows without bound,

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9

Affentranger, Fernando, and Rex A. Dwyer. "The convex hull of random balls." Advances in Applied Probability 25, no. 2 (1993): 373–94. http://dx.doi.org/10.2307/1427658.

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While the convex hull of n d-dimensional balls is not a polytope, it does have an underlying combinatorial structure similar to that of a polytope. In the worst case, its combinatorial complexity can be of order Ω(n[d/2]). The thrust of this work is to show that its complexity is typically much smaller, and that it can therefore be constructed more quickly on average than in the worst case. To this end, four models of the random d-ball are developed, and the expected combinatorial complexity of the convex hull of n independent random d-balls is investigated for each. As n grows without bound,

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10

Küfer, K. H. "On the approximation of a ball by random polytopes." Advances in Applied Probability 26, no. 04 (1994): 876–92. http://dx.doi.org/10.1017/s0001867800026665.

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Letbe a sequence of independent and identically distributed random vectors drawn from thed-dimensional unit ballBdand letXnbe the random polytope generated as the convex hull ofa1,· ··,an.Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributedaiand2 we prove that the limiting distribution of Δ(Xn)/Ε(Δ(Xn)) forn→ ∞ (satisfies a 0–1 law. In particular, we show that Varforn→ ∞. We provide analogous results for spherically symmetric distributions inBdwith regularly varying tail. In addition, we indicate similar results

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11

Schneider, Rolf. "Interaction of Poisson hyperplane processes and convex bodies." Journal of Applied Probability 56, no. 4 (2019): 1020–32. http://dx.doi.org/10.1017/jpr.2019.65.

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AbstractGiven a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

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12

Küfer, K. H. "On the approximation of a ball by random polytopes." Advances in Applied Probability 26, no. 4 (1994): 876–92. http://dx.doi.org/10.2307/1427895.

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Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a1,· ··, an. Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn)/Ε (Δ (Xn)) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we ind

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13

Magazinov, A. "A Uniform Asymptotical Upper Bound for the Variance of a Random Polytope in a Simple Polytope." Modeling and Analysis of Information Systems 19, no. 6 (2015): 148–51. http://dx.doi.org/10.18255/1818-1015-2012-6-148-151.

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The present paper contains a sketch of the proof of an upper bound for the variance of the number of hyperfaces of a random polytope when the mother body is a simple polytope. Thus we verify a weaker version of the result in [1] stated without a proof. The article is published in the author’s wording.

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14

Hug, Daniel, and Matthias Reitzner. "Gaussian polytopes: variances and limit theorems." Advances in Applied Probability 37, no. 02 (2005): 297–320. http://dx.doi.org/10.1017/s0001867800000197.

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The convex hull ofnindependent random points in ℝd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number ofi-faces and for the variance of theith intrinsic volume of a Gaussian polytope in ℝd,d∈ℕ, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number ofi-faces and for theith intrinsic volume of a Gaussian polytope asn→∞.

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15

Hug, Daniel, and Matthias Reitzner. "Gaussian polytopes: variances and limit theorems." Advances in Applied Probability 37, no. 2 (2005): 297–320. http://dx.doi.org/10.1239/aap/1118858627.

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The convex hull of n independent random points in ℝd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number of i-faces and for the variance of the ith intrinsic volume of a Gaussian polytope in ℝd, d∈ℕ, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number of i-faces and for the ith intrinsic volume of a Gaussian polytope as n→∞.

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16

Hartzoulaki, M., and G. Paouris. "Quermassintegrals of a random polytope in a convex body." Archiv der Mathematik 80, no. 4 (2003): 430–38. http://dx.doi.org/10.1007/s00013-003-4593-4.

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17

Yılmaz, Şerife, Taner Büyükköroğlu, and Vakif Dzhafarov. "Random search of stable member in a matrix polytope." Journal of Computational and Applied Mathematics 308 (December 2016): 59–68. http://dx.doi.org/10.1016/j.cam.2016.05.020.

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18

Guédon, O., A. E. Litvak, and K. Tatarko. "Random polytopes obtained by matrices with heavy-tailed entries." Communications in Contemporary Mathematics 22, no. 04 (2019): 1950027. http://dx.doi.org/10.1142/s0219199719500275.

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Let [Formula: see text] be an [Formula: see text] random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope [Formula: see text] in [Formula: see text] (the absolute convex hull of rows of [Formula: see text]). In particular, we show that [Formula: see text] where [Formula: see text] depends only on parameters in small ball inequality. This extends results of [A. E. Litvak, A. Pajor, M

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19

Andrews, Lawrence C., and Herbert J. Bernstein. "The geometry of Niggli reduction:BGAOL–embedding Niggli reduction and analysis of boundaries." Journal of Applied Crystallography 47, no. 1 (2014): 346–59. http://dx.doi.org/10.1107/s1600576713031002.

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Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a higher-dimensional space to facilitate calculation of distances between cells is described. This distance metric is used to create a program,BGAOL, for Bravais lattice determination. Results fromBGAOLare compared with results from other metric based Bravais lattice determination algorithms. This embedding depends on understanding the boundary polytopes of the Niggli-reduced coneNin the six-dimensional spaceG6. This a

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20

Affentranger, Fernando. "The expected volume of a random polytope in a ball*." Journal of Microscopy 151, no. 3 (1988): 277–87. http://dx.doi.org/10.1111/j.1365-2818.1988.tb04688.x.

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21

DiVincenzo, D. P., and M. H. Brodsky. "Polytope-like order in random network models of amorphous semiconductors." Journal of Non-Crystalline Solids 77-78 (December 1985): 241–44. http://dx.doi.org/10.1016/0022-3093(85)90648-9.

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22

Dafnis, N., A. Giannopoulos, and A. Tsolomitis. "Asymptotic shape of a random polytope in a convex body." Journal of Functional Analysis 257, no. 9 (2009): 2820–39. http://dx.doi.org/10.1016/j.jfa.2009.06.027.

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23

Turchi, N., and F. Wespi. "Limit theorems for random polytopes with vertices on convex surfaces." Advances in Applied Probability 50, no. 4 (2018): 1227–45. http://dx.doi.org/10.1017/apr.2018.58.

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Abstract We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes of Kn. A normal approximation bound from Stein's method and estimates for surface bodies are among the tools involved.

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24

Buchta, Christian. "A note on the volume of a random polytope in a tetrahedron." Illinois Journal of Mathematics 30, no. 4 (1986): 653–59. http://dx.doi.org/10.1215/ijm/1256064237.

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25

Bäsel, Uwe. "The expected mean width of a random polytope associated with a ball." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 53, no. 2 (2012): 571–77. http://dx.doi.org/10.1007/s13366-012-0103-9.

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26

Affentranger, Fernando, and John A. Wieacker. "On the convex hull of uniform random points in a simpled-polytope." Discrete & Computational Geometry 6, no. 3 (1991): 291–305. http://dx.doi.org/10.1007/bf02574691.

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27

Parmentier, Axel, Victor Cohen, Vincent Leclère, Guillaume Obozinski, and Joseph Salmon. "Integer Programming on the Junction Tree Polytope for Influence Diagrams." INFORMS Journal on Optimization 2, no. 3 (2020): 209–28. http://dx.doi.org/10.1287/ijoo.2019.0036.

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Influence diagrams (ID) and limited memory influence diagrams (LIMID) are flexible tools to represent discrete stochastic optimization problems, with the Markov decision process (MDP) and partially observable MDP as standard examples. More precisely, given random variables considered as vertices of an acyclic digraph, a probabilistic graphical model defines a joint distribution via the conditional distributions of vertices given their parents. In an ID, the random variables are represented by a probabilistic graphical model whose vertices are partitioned into three types: chance, decision, and

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28

Enríquez, Marco, Francisco Delgado, and Karol Życzkowski. "Entanglement of Three-Qubit Random Pure States." Entropy 20, no. 10 (2018): 745. http://dx.doi.org/10.3390/e20100745.

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We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with

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29

BARVINOK, ALEXANDER. "What Does a Random Contingency Table Look Like?" Combinatorics, Probability and Computing 19, no. 4 (2010): 517–39. http://dx.doi.org/10.1017/s0963548310000039.

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Let R = (r1, . . ., rm) and C = (c1, . . ., cn) be positive integer vectors such that r1 + ⋯ + rm = c1 + ⋯ + cn. We consider the set Σ(R, C) of non-negative m × n integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D ∈ Σ(R, C) is close with high probability to a particular matrix (‘typical table’) Z defined as follows. We let g(x) = (x + 1)ln(x + 1) − x ln x for x ≥ 0 and let g(X) = ∑i,jg(xij) for a non-negative matrix X = (xij). Then g(X) is strictly concave and attains its maximum on the

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30

Cação, Rafael, Lucas Cortez, Ismael de Farias, Ernee Kozyreff, Jalil Khatibi Moqadam, and Renato Portugal. "Quantum Walk on the Generalized Birkhoff Polytope Graph." Entropy 23, no. 10 (2021): 1239. http://dx.doi.org/10.3390/e23101239.

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We study discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs), which arise in the solution-set to certain transportation linear programming problems (TLPs). It is known that quantum walks mix at most quadratically faster than random walks on cycles, two-dimensional lattices, hypercubes, and bounded-degree graphs. In contrast, our numerical results show that it is possible to achieve a greater than quadratic quantum speedup for the mixing time on a subclass of GBPG (TLP with two consumers and m suppliers). We analyze two types of initial states. If the walker starts on a

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31

LOVETT, SHACHAR, and SASHA SODIN. "ALMOST EUCLIDEAN SECTIONS OF THE N-DIMENSIONAL CROSS-POLYTOPE USING O(N) RANDOM BITS." Communications in Contemporary Mathematics 10, no. 04 (2008): 477–89. http://dx.doi.org/10.1142/s0219199708002879.

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It is well known that ℝN has subspaces of dimension proportional to N on which the ℓ1 norm is equivalent to the ℓ2 norm; however, no explicit constructions are known. Extending an earlier work by Artstein-Avidan and Milman, we prove that such a subspace can be generated using O(N) random bits.

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32

Smith, David N., and John F. Ferguson. "Constrained inversion of seismic refraction data using the controlled random search." GEOPHYSICS 65, no. 5 (2000): 1622–30. http://dx.doi.org/10.1190/1.1444850.

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When designing models for 2-D seismic refraction inversion, it is desirable to frame the models as polygonal units with well‐defined geophysical characteristics. Such inversion is strongly nonlinear. Random search methods for nonlinear inversion, such as the genetic algorithm and simulated annealing, have received much attention in recent years. Such methods are well suited when the model space is large and contains multiple global minima to the misfit function. Constraints are necessary in the automatic construction of polygonal models. Controlled random search (CRS) is a member of this class

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33

Kabluchko, Zakhar. "EXPECTED f ‐VECTOR OF THE POISSON ZERO POLYTOPE AND RANDOM CONVEX HULLS IN THE HALF‐SPHERE." Mathematika 66, no. 4 (2020): 1028–53. http://dx.doi.org/10.1112/mtk.12056.

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34

Onn, Shmuel, and Ishay Weissman. "Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes." Annals of Operations Research 189, no. 1 (2009): 331–42. http://dx.doi.org/10.1007/s10479-009-0567-7.

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35

Lee, Woo-Jong, and T. C. Woo. "Tolerances: Their Analysis and Synthesis." Journal of Engineering for Industry 112, no. 2 (1990): 113–21. http://dx.doi.org/10.1115/1.2899553.

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Tolerance, representing a permissible variation of a dimension in an engineering drawing, is synthesized by considering assembly stack-up conditions based on manufacturing cost minimization. A random variable and its standard deviation are associated with a dimension and its tolerance. This probabilistic approach makes it possible to perform trade-off between performance and tolerance rather than the worst case analysis as it is commonly practiced. Tolerance (stack-up) analysis, as an inner loop in the overall algorithm for tolerance synthesis, is performed by approximating the volume under th

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36

Leineweber, A., C. Wolf, P. Kalanke, et al. "From random stacking faults to polytypes: A 12-layer NiSn4 polytype." Journal of Alloys and Compounds 774 (February 2019): 265–73. http://dx.doi.org/10.1016/j.jallcom.2018.09.341.

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37

Chaturvedi, Anubhav, Máté Farkas, and Victoria J. Wright. "Characterising and bounding the set of quantum behaviours in contextuality scenarios." Quantum 5 (June 29, 2021): 484. http://dx.doi.org/10.22331/q-2021-06-29-484.

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The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in th

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38

Richardson, Ross M., Van H. Vu, and Lei Wu. "Random inscribing polytopes." European Journal of Combinatorics 28, no. 8 (2007): 2057–71. http://dx.doi.org/10.1016/j.ejc.2007.04.001.

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39

Braun, Gábor, and Sebastian Pokutta. "Random half-integral polytopes." Operations Research Letters 39, no. 3 (2011): 204–7. http://dx.doi.org/10.1016/j.orl.2011.03.003.

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40

Gruber, Peter M. "Expectation of random polytopes." Manuscripta Mathematica 91, no. 1 (1996): 393–419. http://dx.doi.org/10.1007/bf02567963.

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41

Meckes, M. W. "Volumes of symmetric random polytopes." Archiv der Mathematik 82, no. 1 (2004): 85–96. http://dx.doi.org/10.1007/s00013-003-4808-8.

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42

Chasapis, Giorgos, and Nikos Skarmogiannis. "Affine quermassintegrals of random polytopes." Journal of Mathematical Analysis and Applications 479, no. 1 (2019): 546–68. http://dx.doi.org/10.1016/j.jmaa.2019.06.037.

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43

Buchta, C., and R. F. Tichy. "Random polytopes on the torus." Proceedings of the American Mathematical Society 93, no. 2 (1985): 312. http://dx.doi.org/10.1090/s0002-9939-1985-0770543-1.

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44

B�r�czky, Jr., K., and M. Henk. "Random projections of regular polytopes." Archiv der Mathematik 73, no. 6 (1999): 465–73. http://dx.doi.org/10.1007/s000130050424.

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45

Vu, V. H. "Sharp concentration of random polytopes." GAFA Geometric And Functional Analysis 15, no. 6 (2005): 1284–318. http://dx.doi.org/10.1007/s00039-005-0541-8.

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46

Mendelson, S., A. Pajor, and M. Rudelson. "The Geometry of Random {-1,1}-Polytopes." Discrete & Computational Geometry 34, no. 3 (2005): 365–79. http://dx.doi.org/10.1007/s00454-005-1186-y.

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47

Richardson, Ross M., Van H. Vu, and Lei Wu. "An Inscribing Model for Random Polytopes." Discrete & Computational Geometry 39, no. 1-3 (2007): 469–99. http://dx.doi.org/10.1007/s00454-007-9012-3.

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48

Mankiewicz, Piotr, and Nicole Tomczak-Jaegermann. "Stability Properties of Neighbourly Random Polytopes." Discrete & Computational Geometry 41, no. 2 (2008): 257–72. http://dx.doi.org/10.1007/s00454-008-9092-8.

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49

Schütt, Carsten. "Random Polytopes and Affine Surface Area." Mathematische Nachrichten 170, no. 1 (2006): 227–49. http://dx.doi.org/10.1002/mana.19941700117.

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50

Bárány, Imre. "Random polytopes in smooth convex bodies." Mathematika 39, no. 1 (1992): 81–92. http://dx.doi.org/10.1112/s0025579300006872.

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