Relevant bibliographies by topics / Random Polytope

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Random Polytope'

Author: Grafiati
Published: 4 June 2021
Last updated: 15 February 2022

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Journal articles on the topic "Random Polytope"

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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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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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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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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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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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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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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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, these expectations are O(1), O(n (d–1)/(d+4)), O(1) (for d = 2 only), and O(n (d–1)/(d+3)).
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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, these expectations are O(1), O(n(d–1)/(d+4)), O(1) (for d = 2 only), and O(n(d–1)/(d+3)).
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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 for the surface area and the number of facets ofXn.
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Dissertations / Theses on the topic "Random Polytope"

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Leydold, Josef, and Wolfgang Hörmann. "A Sweep-Plane Algorithm for Generating Random Tuples in Simple Polytopes." Department of Statistics and Mathematics, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, 1997. http://epub.wu.ac.at/476/1/document.pdf.

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A sweep-plane algorithm by Lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes. In our method an affine hyperplane is swept through the given polytope until a random fraction (sampled from a proper univariate distribution) of the volume of the polytope is covered. Then the intersection of the plane with the polytope is a simple polytope with smaller dimension. In the second part we apply this method to construct a black-box algorithm for log-concave and T-concave multivariate distributions by means of transformed density rejection. (author's abstract)<br>Series: Preprint Series / Department of Applied Statistics and Data Processing
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Wu, Lei. "Random inscribed polytopes." Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2006. http://wwwlib.umi.com/cr/ucsd/fullcit?p3210646.

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Thesis (Ph. D.)--University of California, San Diego, 2006.<br>Title from first page of PDF file (viewed June 7, 2006). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 60-65).
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Geffroy, Arthur. "Contribution a l'étude locale et globale de l'enveloppe convexe d'un échantillon aléatoire." Rouen, 1997. http://www.theses.fr/1997ROUES017.

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On définit le concept de plage d'appui d'un échantillon aléatoire, les plages d'appui étant certaines parties de l'enveloppe convexe pouvant coïncider avec l'enveloppe complète. On établit des formules générales pour l'espérance du nombre de sommets ou du nombre de cotes d'une plage d'appui quelconque, ainsi que l'espérance de sa longueur. Ces formules sont ensuite calculées dans le cas d'une loi normale dans le plan, d'une loi uniforme dans un polygone convexe, puis d'une loi uniforme dans une courbe convexe lisse. Des simulations informatiques permettent finalement d'étudier les vitesses de convergence vers les résultats asymptotiques précédemment trouvés (dans le début de cette thèse ainsi que dans d'autres travaux), et d'obtenir des estimations plus fines pour la moyenne, la variance et la loi du nombre de sommets de l'enveloppe convexe.
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Qi, Weinan. "On Resampling Schemes for Uniform Polytopes." Thesis, Université d'Ottawa / University of Ottawa, 2017. http://hdl.handle.net/10393/36057.

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The convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. For example, approximating the boundary of a finite set is used by many authors in environmental studies and medical research. To approximate the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately, the asymptotic results are mostly very complicated. To address this complication, we suggest a consistent bootstrapping scheme for certain cases. Our resampling technique is used for both semi-parametric and non-parametric cases. Let X1,X2,...,Xn be a sequence of i.i.d. random points uniformly distributed on an unknown convex set. Our bootstrapping scheme relies on resampling uniformly from the convex hull of X1,X2,...,Xn. In this thesis, we study the asymptotic consistency of certain functionals of convex hulls. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. We also provide a conjecture for the application of our bootstrapping scheme to Gaussian polytopes. Moreover, some other relevant consistency results for the regular bootstrap are developed.
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Calka, Pierre. "Mosaïques, enveloppes convexes et modèle Booléen : quelques propriétés et rapprochements." Habilitation à diriger des recherches, Université René Descartes - Paris V, 2009. http://tel.archives-ouvertes.fr/tel-00448249.

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Ce mémoire est consacré à trois modèles classiques de géométrie aléatoire : les mosaïques, les enveloppes convexes et le modèle booléen. Dans la première partie, on étudie les mosaïques poissonniennes d'hyperplans isotropes et plus particulièrement leur zéro-cellule qui est un polyèdre convexe aléatoire de l'espace euclidien. Deux cas particuliers de zéro-cellules sont la cellule typique de Poisson-Voronoi et la cellule de Crofton. On donne une formule explicite pour la loi du nombre de côtés d'une zéro-cellule en dimension deux. On s'intéresse au comportement asymptotique de cette loi et on fait le lien avec le problème de Sylvester des points en position convexe. On décrit ensuite la loi du rayon circonscrit ainsi que le comportement asymptotique du polyèdre à grand rayon inscrit au moyen de théorèmes limites. De cette manière et aussi par l'utilisation de la fréquence fondamentale, on apporte des précisions à l'énoncé de la conjecture de D. G. Kendall. La seconde partie a pour objet les enveloppes convexes de processus ponctuels de Poisson isotropes dans la boule-unité. On établit un résultat de type grandes déviations pour le nombre de sommets. On montre ensuite la convergence de la frontière de l'enveloppe après changement d'échelle et on en déduit des résultats de valeurs extrêmes, estimations de variance, théorèmes centraux limites et principes d'invariance pour certaines caractéristiques. Dans la troisième partie, on s'intéresse enfin aux modèles de recouvrement de type booléen de l'espace euclidien. Dans un premier travail, on applique une variante du modèle sans interpénétration des objets à la modélisation d'un phénomène de fissuration. On étudie ensuite la convergence de la composante connexe de l'origine d'un modèle booléen vers la cellule de Crofton en dimension deux. On s'intéresse enfin à la fonction de visibilité de cette composante connexe pour laquelle on obtient une estimée de la queue de distribution et des résultats de valeurs extrêmes.
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Turchi, Nicola [Verfasser], Christoph [Gutachter] Thäle, and Peter [Gutachter] Eichelsbacher. "High-dimensional asymptotics for random polytopes / Nicola Turchi ; Gutachter: Christoph Thäle, Peter Eichelsbacher ; Fakultät für Mathematik." Bochum : Ruhr-Universität Bochum, 2019. http://d-nb.info/1182682375/34.

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Temesvari, Daniel [Verfasser], Christoph [Gutachter] Thäle, and Monika [Gutachter] Ludwig. "Discrete stochastic geometry : beta-polytopes, random cones and empty simplices / Daniel Temesvari ; Gutachter: Christoph Thäle, Monika Ludwig ; Fakultät für Mathematik." Bochum : Ruhr-Universität Bochum, 2019. http://d-nb.info/1182682162/34.

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Beermann, Mareen. "Random Polytopes." Doctoral thesis, 2015. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2015062313276.

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Random polytopes can be constructed in many different ways. In this thesis two certain kinds are considered - random polytopes as the convex hull of random points and as the intersection of finitely many random half spaces. Concerning these two models different issues are treated.
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Grygierek, Jens Jan. "Random Geometric Structures." Doctoral thesis, 2020. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-202001302552.

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We construct and investigate random geometric structures that are based on a homogeneous Poisson point process. We investigate the random Vietoris-Rips complex constructed as the clique complex of the well known gilbert graph as an infinite random simplicial complex and prove that every realizable finite sub-complex will occur infinitely many times almost sure as isolated complex and also in the case of percolations connected to the unique giant component. Similar results are derived for the Cech complex. We derive limit theorems for the f-vector of the Vietoris-Rips complex on the unit cube centered at the origin and provide a central limit theorem and a Poisson limit theorem based on the model parameters. Finally we investigate random polytopes that are given as convex hulls of a Poisson point process in a smooth convex body. We establish a central limit theorem for certain linear combinations of intrinsic volumes. A multivariate limit theorem involving the sequence of intrinsic volumes and the number of i-dimensional faces is derived. We derive the asymptotic normality of the oracle estimator of minimal variance for estimation of the volume of a convex body.
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Pivovarov, Peter. "Volume distribution and the geometry of high-dimensional random polytopes." Phd thesis, 2010. http://hdl.handle.net/10048/1170.

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Thesis (Ph. D.)--University of Alberta, 2010.<br>Title from pdf file main screen (viewed on July 13, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta. Includes bibliographical references.
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Books on the topic "Random Polytope"

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Handelman, David. Positive polynomials, convex integral polytopes, and a random walk problem. Springer-Verlag, 1987.

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Handelman, David E. Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem. Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/bfb0078909.

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Positive Polynomials Convex Integral Polytopes And A Random Walk Problem. Springer, 1987.

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Book chapters on the topic "Random Polytope"

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Suck, Reinhard. "The Equivalence Relation Polytope and Random Classification and Clustering." In Studies in Classification, Data Analysis, and Knowledge Organization. Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/978-3-642-61159-9_31.

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Dalla, L., and D. Larman. "Volumes of a random polytope in a convex set." In DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, 1991. http://dx.doi.org/10.1090/dimacs/004/13.

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Hug, Daniel. "Random Polytopes." In Stochastic Geometry, Spatial Statistics and Random Fields. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-33305-7_7.

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Calka, Pierre. "Asymptotic Properties of Random Polytopes." In Lecture Notes in Computer Science. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-25040-3_22.

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Brazitikos, Silouanos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou. "Random polytopes in isotropic convex bodies." In Mathematical Surveys and Monographs. American Mathematical Society, 2014. http://dx.doi.org/10.1090/surv/196/11.

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Mendelson, Shahar. "On the Geometry of Random Polytopes." In Lecture Notes in Mathematics. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-46762-3_8.

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Kaibel, Volker. "Low-Dimensional Faces of Random 0/1-Polytopes." In Integer Programming and Combinatorial Optimization. Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-25960-2_30.

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Kaibel, Volker, and Anja Remshagen. "On the Graph-Density of Random 0/1-Polytopes." In Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques. Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/978-3-540-45198-3_27.

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Dwyer, Rex, and Ravi Kannan. "Convex hull of randomly chosen points from a polytope." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/3-540-18099-0_25.

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Dey, Santanu S., Yatharth Dubey, and Marco Molinaro. "Branch-and-Bound Solves Random Binary IPs in Polytime." In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA). Society for Industrial and Applied Mathematics, 2021. http://dx.doi.org/10.1137/1.9781611976465.35.

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Conference papers on the topic "Random Polytope"

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Emiris, Ioannis Z., and Vissarion Fisikopoulos. "Efficient Random-Walk Methods for Approximating Polytope Volume." In Annual Symposium. ACM Press, 2014. http://dx.doi.org/10.1145/2582112.2582133.

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Hosoe, Yohei, Yuji Nagira, and Tomomichi Hagiwara. "Random polytope representation of discrete-time uncertain stochastic switched systems and robust stabilization." In 2016 European Control Conference (ECC). IEEE, 2016. http://dx.doi.org/10.1109/ecc.2016.7810590.

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Chandrasekaran, Karthekeyan, and Santosh S. Vempala. "Integer feasibility of random polytopes." In ITCS'14: Innovations in Theoretical Computer Science. ACM, 2014. http://dx.doi.org/10.1145/2554797.2554838.

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Nagira, Yuji, Yohei Hosoe, and Tomomichi Hagiwara. "Gain-scheduled state feedback synthesis for systems characterized by random polytopes." In 2015 54th IEEE Conference on Decision and Control (CDC). IEEE, 2015. http://dx.doi.org/10.1109/cdc.2015.7402318.

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Hosoe, Yohei, and Tomomichi Hagiwara. "Extension of the concept of random polytopes and robust stabilization synthesis." In 2015 European Control Conference (ECC). IEEE, 2015. http://dx.doi.org/10.1109/ecc.2015.7330970.

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Kannan, Ravi, and Hariharan Narayanan. "Random walks on polytopes and an affine interior point method for linear programming." In the 41st annual ACM symposium. ACM Press, 2009. http://dx.doi.org/10.1145/1536414.1536491.

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Vershynin, Roman. "Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method." In 2006 47th Annual IEEE Conference on Foundations of Computer Science. IEEE, 2006. http://dx.doi.org/10.1109/focs.2006.19.

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Hosoe, Yohei, and Dimitri Peaucelle. "S-variable approach to robust stabilization state feedback synthesis for systems characterized by random polytopes." In 2016 European Control Conference (ECC). IEEE, 2016. http://dx.doi.org/10.1109/ecc.2016.7810589.

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Andrianova, Olga G., and Alexey A. Belov. "On Robust Performance Analysis of Linear Systems with Polytopic Uncertainties Affected by Random Disturbances." In 2019 20th International Carpathian Control Conference (ICCC). IEEE, 2019. http://dx.doi.org/10.1109/carpathiancc.2019.8766038.

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Belov, Alexey A. "Random Disturbance Attenuation in Discrete-time Polytopic Systems: Performance Analysis and State-Feedback Control*." In 2020 European Control Conference (ECC). IEEE, 2020. http://dx.doi.org/10.23919/ecc51009.2020.9143952.

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