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Relevant bibliographies by topics / Regular polytopes

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Academic literature on the topic 'Regular polytopes'

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Published: 10 March 2023

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Journal articles on the topic "Regular polytopes"

1

Lalvani, Haresh. "Higher Dimensional Periodic Table Of Regular And Semi-Regular Polytopes." International Journal of Space Structures 11, no. 1-2 (April 1996): 155–71. http://dx.doi.org/10.1177/026635119601-222.

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This paper presents a higher-dimensional periodic table of regular and semi-regular n-dimensional polytopes. For regular n-dimensional polytopes, designated by their Schlafli symbol {p,q,r,…u,v,w}, the table is an (n-1)-dimensional hypercubic lattice in which each polytope occupies a different vertex of the lattice. The values of p,q,r,…u,v,w also establish the corresponding n-dimensional Cartesian co-ordinates (p,q,r,…u,v,w) of their respective positions in the hypercubic lattice. The table is exhaustive and includes all known regular polytopes in Euclidean, spherical and hyperbolic spaces, in addition to others candidate polytopes which do not appear in the literature. For n-dimensional semi-regular polytopes, each vertex of this hypercubic lattice branches into analogous n-dimensional cubes, where each n-cube encompasses a family with a distinct semi-regular polytope occupying each vertex of each n-cube. The semi-regular polytopes are obtained by varying the location of a vertex within the fundamental region of the polytope. Continuous transformations within each family are a natural fallout of this variable vertex location. Extensions of this method to less regular space structures and to derivation of architectural form are in progress and provide a way to develop an integrated index for space structures. Besides the economy in computational processing of space structures, integrated indices based on unified morphologies are essential for establishing a meta-structural knowledge base for architecture.

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Schulte, Egon, and Asia Ivić Weiss. "Free Extensions of Chiral Polytopes." Canadian Journal of Mathematics 47, no. 3 (June 1, 1995): 641–54. http://dx.doi.org/10.4153/cjm-1995-033-7.

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AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

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CONNOR, THOMAS, DIMITRI LEEMANS, and MARK MIXER. "ABSTRACT REGULAR POLYTOPES FOR THE O'NAN GROUP." International Journal of Algebra and Computation 24, no. 01 (February 2014): 59–68. http://dx.doi.org/10.1142/s0218196714500052.

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In this paper, we consider how the O'Nan sporadic simple group acts as the automorphism group of an abstract regular polytope. In particular, we prove that there is no regular polytope of rank at least five with automorphism group isomorphic to O′N. Moreover, we classify all rank four regular polytopes having O′N as their automorphism group.

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Comes, Jonathan. "Regular Polytopes." Mathematics Enthusiast 1, no. 2 (October 1, 2004): 30–37. http://dx.doi.org/10.54870/1551-3440.1007.

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Hou, Dong-Dong, Yan-Quan Feng, and Dimitri Leemans. "Existence of regular 3-polytopes of order 2𝑛." Journal of Group Theory 22, no. 4 (July 1, 2019): 579–616. http://dx.doi.org/10.1515/jgth-2018-0155.

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AbstractIn this paper, we prove that for any positive integers {n,s,t} such that {n\geq 10}, {s,t\geq 2} and {n-1\geq s+t}, there exists a regular polytope with Schläfli type {\{2^{s},2^{t}\}} and its automorphism group is of order {2^{n}}. Furthermore, we classify regular polytopes with automorphism groups of order {2^{n}} and Schläfli types {\{4,2^{n-3}\},\{4,2^{n-4}\}} and {\{4,2^{n-5}\}}, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Period. Math. Hungar. 53 2006, 1–2, 231–255].

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Boya, Luis J., and Cristian Rivera. "On Regular Polytopes." Reports on Mathematical Physics 71, no. 2 (April 2013): 149–61. http://dx.doi.org/10.1016/s0034-4877(13)60026-9.

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Cuypers, Hans. "Regular quaternionic polytopes." Linear Algebra and its Applications 226-228 (September 1995): 311–29. http://dx.doi.org/10.1016/0024-3795(95)00149-l.

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McMullen, Peter, and Egon Schulte. "Flat regular polytopes." Annals of Combinatorics 1, no. 1 (December 1997): 261–78. http://dx.doi.org/10.1007/bf02558480.

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Coxeter, H. S. M. "Regular and semi-regular polytopes. II." Mathematische Zeitschrift 188, no. 4 (December 1985): 559–91. http://dx.doi.org/10.1007/bf01161657.

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Coxeter, H. S. M. "Regular and semi-regular polytopes. III." Mathematische Zeitschrift 200, no. 1 (March 1988): 3–45. http://dx.doi.org/10.1007/bf01161745.

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Dissertations / Theses on the topic "Regular polytopes"

1

Duke, Helene. "A Study of the Rigidity of Regular Polytopes." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1366271197.

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Beteto, Marco Antonio Leite. "Less conservative conditions for the robust and Gain-Scheduled LQR-state derivative controllers design /." Ilha Solteira, 2019. http://hdl.handle.net/11449/180976.

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Orientador: Edvaldo Assunção
Resumo: Neste trabalho é proposta a resolução do problema do regulador linear quadrático (Linear Quadratic Regulator - LQR) via desigualdades matriciais lineares (Linear Matrix Inequalities - LMIs) para sistemas lineares e invariantes no tempo sujeitos a incertezas politópicas, bem como para sistemas lineares sujeitos a parâmetros variantes no tempo (Linear Parameter Varying - LPV). O projeto dos controladores é baseado na realimentação derivativa. A escolha da realimentação derivativa se dá devido à sua fácil implementação em certas aplicações como, por exemplo, no controle de vibrações. Os sinais usados na realimentação são aceleração e velocidade, sendo obtidos por meio de acelerômetros. Por meio do método proposto é possível obter condições LMIs para a síntese de controladores que garantam a estabilização do sistema em malha fechada, sendo que os controladores possuem desempenho otimizado. Para a formulação das condições LMIs, uma função de Lyapunov do tipo quadrática é utilizada. Exemplos teóricos e simulações são utilizados como forma de validação dos métodos propostos, além de mostrar que os novos resultados apresentam condições menos conservadoras. Além disso, ao final é apresentada uma implementação prática em um sistema de suspensão ativa, produzida pela Quanser®.
Abstract: The resolution of linear quadratic regulator (LQR) problem via linear matrix inequalities (LMIs) for linear time-invariant systems subject to polytopic uncertainties, as linear systems subjects to linear parameter varying (LPV), is proposed in this work. The controllers' designs are based on the state derivative feedback. The aim to the choice of the state derivative feedback is your easy implementation in a class of mechanical systems, such as in vibration control, for example. The signals used for feedback are acceleration and velocity, it is obtained by means of accelerometers. Through the proposed method it is possible to obtain LMIs conditions for the synthesis of controllers that guarantee the stabilisation of the closed-loop system, being that the controllers have optimised performance. For the LMIs conditions formulations, a Lyapunov function of type quadratic is used. As a form of validation, theoretical examples and simulations are performed, besides to show that the new results are less conservative. Furthermore, a practical implementation in an active suspension system, produced by Quanser®, is performed.
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Bruni, Matteo. "Incremental Learning of Stationary Representations." Doctoral thesis, 2021. http://hdl.handle.net/2158/1237986.

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Humans and animals, during their life, continuously acquire new knowledge over time while making new experiences. They learn new concepts without forgetting what already learned, they typically use a few training examples (i.e. a child could recognize a giraffe after seeing a single picture) and they are able to discern what is known from what is unknown (i.e. unknown faces). In contrast, current supervised learning systems, work under the assumption that all data is known and available during learning, training is performed offline and a test dataset is typically required. What is missing in current research is a way to bridge the human learning capabilities in an artificial learning system where learning is performed incrementally from a data stream of infinite length (i.e. lifelong learning). This is a challenging task that is not sufficiently studied in the literature. According to this, in this thesis, we investigated different aspects of Deep Neural Network models (DNNs) to obtain stationary representations. Similar to fixed representations these representations can remain compatible between learning steps and are therefore well suited for incremental learning. Specifically, in the first part of the thesis, we propose a memory-based approach that collects and preserves all the past visual information observed so far, building a comprehensive and cumulative representation. We exploit a pre-trained fixed representation for the task of learning the appearance of face identities from unconstrained video streams leveraging temporal-coherence as a form of self-supervision. In this task, the representation allows us to learn from a few images and to detect unknown subjects similar to how humans learn. As the proposed approach makes use of a pre-trained fixed representation, learning is somewhat limited. This is due to the fact that the features stored in the memory bank remain fixed (i.e. they are not undergoing learning) and only the memory bank is learned. To address this issue, in the second part of the thesis, we propose a representation learning approach that can be exploited to learn both the feature and the memory without considering their joint learning (computationally prohibitive). The intuition is that every time the internal feature representation changes the memory bank must be relearned from scratch. The proposed method can mitigate the need of feature relearning by keeping the compatibility of features between learning steps thanks to feature stationarity. We show that the stationarity of the internal representation can be achieved with a fixed classifier by setting the classifier weights according to values taken from the coordinate vertices of the regular polytopes in high dimensional space. In the last part of the thesis, we apply the previously stationary representation method in the task of class incremental learning. We show that the method is as effective as the standard approaches while exhibiting novel stationarity properties of the internal feature representation that are otherwise non-existent. The approach exploits future unseen classes as negative examples and learns features that do not change their geometric configuration as novel classes are incorporated in the learning model. We show that a large number of classes can be learned with no loss of accuracy allowing the method to meet the underlying assumption of incremental lifelong learning.

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Books on the topic "Regular polytopes"

1

Coxeter, H. S. M. Regular complex polytopes. 2nd ed. Cambridge [England]: Cambridge University Press, 1991.

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Chang, Peter Chung Yuen. Quantum field theory on regular polytopes. Manchester: University of Manchester, 1993.

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3

Mostly surfaces. Providence, R.I: American Mathematical Society, 2011.

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4

Coxeter, H. S. M. Regular Polytopes. Dover Publications, Incorporated, 2012.

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Coxeter, H. S. M. Regular Polytopes. Dover Publications, Incorporated, 2012.

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Coxeter, H. S. M. Regular Polytopes. Dover Publications, 2013.

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7

Doran, B., Egon Schulte, M. Ismail, Peter McMullen, and G. C. Rota. Abstract Regular Polytopes. Cambridge University Press, 2004.

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Mcmullen, Peter, and Egon Schulte. Abstract Regular Polytopes. Cambridge University Press, 2002.

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9

McMullen, Peter. Geometric Regular Polytopes. University of Cambridge ESOL Examinations, 2020.

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10

Schulte, Egon, and Peter McMullen. Abstract Regular Polytopes. Cambridge University Press, 2009.

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Book chapters on the topic "Regular polytopes"

1

Johnson, D. L. "Regular Polytopes." In Springer Undergraduate Mathematics Series, 155–66. London: Springer London, 2001. http://dx.doi.org/10.1007/978-1-4471-0243-4_12.

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2

McMullen, Peter. "Rigidity of Regular Polytopes." In Rigidity and Symmetry, 253–78. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-0781-6_13.

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McMullen, Peter. "Modern Developments in Regular Polytopes." In Polytopes: Abstract, Convex and Computational, 97–124. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-0924-6_5.

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4

Lee, C. "Regular triangulations of convex polytopes." In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 443–56. Providence, Rhode Island: American Mathematical Society, 1991. http://dx.doi.org/10.1090/dimacs/004/35.

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De Loera, Jesús A., Jörg Rambau, and Francisco Santos. "Regular Triangulations and Secondary Polytopes." In Triangulations, 209–74. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-12971-1_5.

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Schulte, Egon. "Classification of Locally Toroidal Regular Polytopes." In Polytopes: Abstract, Convex and Computational, 125–54. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-0924-6_6.

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McMullen, Peter. "New Regular Compounds of 4-Polytopes." In Bolyai Society Mathematical Studies, 307–20. Berlin, Heidelberg: Springer Berlin Heidelberg, 2018. http://dx.doi.org/10.1007/978-3-662-57413-3_12.

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Schulte, Egon. "Regular Incidence Complexes, Polytopes, and C-Groups." In Discrete Geometry and Symmetry, 311–33. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-78434-2_18.

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Downs, Martin, and Gareth A. Jones. "Möbius Inversion in Suzuki Groups and Enumeration of Regular Objects." In Symmetries in Graphs, Maps, and Polytopes, 97–127. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30451-9_5.

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Banchoff, Thomas F. "Torus Decompostions of Regular Polytopes in 4-space." In Shaping Space, 257–66. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-0-387-92714-5_20.

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Conference papers on the topic "Regular polytopes"

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Shahid, Salman, Sakti Pramanik, and Charles B. Owen. "Minimum bounding boxes for regular cross-polytopes." In the 27th Annual ACM Symposium. New York, New York, USA: ACM Press, 2012. http://dx.doi.org/10.1145/2245276.2245447.

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Bueno, Jose Nuno A. D., Kaio D. T. Rocha, Lucas B. Marcos, and Marco H. Terra. "Mode-Independent Regulator for Polytopic Markov Jump Linear Systems*." In 2022 30th Mediterranean Conference on Control and Automation (MED). IEEE, 2022. http://dx.doi.org/10.1109/med54222.2022.9837134.

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