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Автор: Grafiati
Опубліковано: 30 травня 2022
Оновлено: 31 травня 2022

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Статті в журналах з теми "Orthogonal representation"

1

Rico, J. M., and J. Duffy. "A Representation of the Euclidean Group by Spin Groups, and Spatial Kinematics Mappings." Journal of Mechanical Design 112, no. 1 (March 1, 1990): 42–49. http://dx.doi.org/10.1115/1.2912577.

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A new derivation of the spin and biquaternion representation of the Euclidean group is presented. The derivation is based upon the even Clifford algebra representation of the orientation preserving orthogonal automorphisms of nondegenerate orthogonal spaces, also called spin representation. Embedding the degenerate orthogonal space IR1,0,3 into the nondegenerate orthogonal space IR1,4, and imposing certain conditions on the orthogonal automorphisms of IR1,4, one obtains a subgroup of the spin group. The action of this subgroup, on a subspace of IR1,4, is isomorphic to IR1,0,3, is precisely a Euclidean motion. The conditions imposed on the orthogonal automorphisms of IR1,4 lead to the biquaternion representation. Furthermore, the invariants of the representations are easily obtained. The derivation also allows the spin representation to be related to the action of the representation over an element of a three-dimensional vector space proposed by Porteous, and used by Selig. As a byproduct, the derivation provides an insightful interpretation of the dual unit used in both the spin representation and the biquaternion representation.
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2

Valverde, Cesar. "On Induced Representations Distinguished by Orthogonal Groups." Canadian Mathematical Bulletin 56, no. 3 (September 1, 2013): 647–58. http://dx.doi.org/10.4153/cmb-2012-008-0.

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Abstract.LetFbe a local non-archimedean field of characteristic zero. We prove that a representation ofGL(n,F) obtained from irreducible parabolic induction of supercuspidal representations is distinguished by an orthogonal group only if the inducing data is distinguished by appropriate orthogonal groups. As a corollary, we get that an irreducible representation induced from supercuspidals that is distinguished by an orthogonal group is metic.
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3

Yefremov, A. P. "Orthogonal representation of complex numbers." Gravitation and Cosmology 16, no. 2 (April 2010): 137–39. http://dx.doi.org/10.1134/s0202289310020064.

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4

Bellaïche, Joël, and Gaëtan Chenevier. "The sign of Galois representations attached to automorphic forms for unitary groups." Compositio Mathematica 147, no. 5 (July 27, 2011): 1337–52. http://dx.doi.org/10.1112/s0010437x11005264.

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AbstractLet K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.
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5

Ismail, Mourad E. H., and Dennis Stanton. "q-Integral and Moment Representations for q-Orthogonal Polynomials." Canadian Journal of Mathematics 54, no. 4 (August 1, 2002): 709–35. http://dx.doi.org/10.4153/cjm-2002-027-2.

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AbstractWe develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for q-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the q-exponential function εq.
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6

Shindo, Yuji, Akihisa Kameari, and Tetsuji Matsuo. "Efficient circuit representation of eddy-current fields." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 36, no. 5 (September 4, 2017): 1457–73. http://dx.doi.org/10.1108/compel-02-2017-0084.

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Purpose This paper aims to discuss the relationship between the continued fraction form of the analytical solution in the frequency domain, the orthogonal function expansion and their circuit realization to derive an efficient representation of the eddy-current field in the conducting sheet and wire/cylinder. Effective frequency ranges of representations are analytically derived. Design/methodology/approach The Cauer circuit representation is derived from the continued fraction form of analytical solution and from the orthogonal polynomial expansion. Simple circuit calculations give the upper frequency bounds where the truncated circuit and orthogonal expansion are applicable. Findings The Cauer circuit representation and the orthogonal polynomial expansions for the magnetic sheet in the E-mode and for the wire in the axial H-mode are derived. The upper frequency bound for the Cauer circuit is roughly proportional to N4 with N inductive elements, whereas the frequency bound for the finite element eddy-current analysis with uniform N elements is roughly proportional to N2. Practical implications The Cauer circuit representation is expected to provide an efficient homogenization method because it requires only several elements to describe the eddy-current field over a wide frequency range. Originality/value The applicable frequency ranges are analytically derived depending on the conductor geometry and on the truncation types.
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7

Chan, Chuan-Tsung, A. Mironov, A. Morozov, and A. Sleptsov. "Orthogonal Polynomials in Mathematical Physics." Reviews in Mathematical Physics 30, no. 06 (July 2018): 1840005. http://dx.doi.org/10.1142/s0129055x18400056.

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This is a review of ([Formula: see text]-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials, and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of ([Formula: see text]-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra. To the memory of Ludwig Dmitrievich Faddeev
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8

Robinson, G. M., and A. J. Keane. "Concise Orthogonal Representation of Supercritical Airfoils." Journal of Aircraft 38, no. 3 (May 2001): 580–83. http://dx.doi.org/10.2514/2.2803.

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9

Lee, Chung-Nim, Timothy Poston, and Azriel Rosenfeld. "Representation of orthogonal regions by vertices." CVGIP: Graphical Models and Image Processing 53, no. 2 (March 1991): 149–56. http://dx.doi.org/10.1016/1049-9652(91)90058-r.

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10

Yi, Seong-Baek. "An Orthogonal Representation of Estimable Functions." Communications for Statistical Applications and Methods 15, no. 6 (November 30, 2008): 837–42. http://dx.doi.org/10.5351/ckss.2008.15.6.837.

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Дисертації з теми "Orthogonal representation"

1

WANG, KWANG SHANG. "FINITE GROUPS FOR WHICH EVERY COMPLEX REPRESENTATION IS REALIZABLE." Diss., The University of Arizona, 1985. http://hdl.handle.net/10150/188019.

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In Chapter 2 we develop the concept of total orthogonality. A number of necessary conditions are derived. Necessary and sufficient conditions for total orthogonality are obtained for 2-groups and for split extensions of elementary abelian 2-groups. A complete description is given for totally orthogonal groups whose character degrees are bounded by 2. Brauer's problem is reduced for Frobenius groups to the corresponding problems for Frobenius kernels and complements. In Chapter 3 classes of examples are presented illustrating the concepts and results of Chapter 2. It is shown, in particular, that 2-Sylow subgroups of finite reflection groups, and of alternating groups, are totally orthogonal.
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2

Segura, Bermudez Jairo Alonso [Verfasser], and Christian [Akademischer Betreuer] Franzke. "On the Empirical Orthogonal Functions representation of the ocean circulation / Jairo Alonso Segura Bermudez ; Betreuer: Christian Franzke." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2020. http://d-nb.info/1214370209/34.

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3

Segura, Bermudez Jairo Alonso Verfasser], and Christian [Akademischer Betreuer] [Franzke. "On the Empirical Orthogonal Functions representation of the ocean circulation / Jairo Alonso Segura Bermudez ; Betreuer: Christian Franzke." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2020. http://nbn-resolving.de/urn:nbn:de:gbv:18-105495.

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4

Dinckal, Cigdem. "Decomposition Of Elastic Constant Tensor Into Orthogonal Parts." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612226/index.pdf.

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All procedures in the literature for decomposing symmetric second rank (stress) tensor and symmetric fourth rank (elastic constant) tensor are elaborated and compared which have many engineering and scientific applications for anisotropic materials. The decomposition methods for symmetric second rank tensors are orthonormal tensor basis method, complex variable representation and spectral method. For symmetric fourth rank (elastic constant) tensor, there are four mainly decomposition methods namely as, orthonormal tensor basis, irreducible, harmonic decomposition and spectral. Those are applied to anisotropic materials possessing various symmetry classes which are isotropic, cubic, transversely isotropic, tetragonal, trigonal and orthorhombic. For isotropic materials, an expression for the elastic constant tensor different than the traditionally known form is given. Some misprints found in the literature are corrected. For comparison purposes, numerical examples of each decomposition process are presented for the materials possessing different symmetry classes. Some applications of these decomposition methods are given. Besides, norm and norm ratio concepts are introduced to measure and compare the anisotropy degree for various materials with the same or di¤
erent symmetries. For these materials,norm and norm ratios are calculated. It is suggested that the norm of a tensor may be used as a criterion for comparing the overall e¤
ect of the properties of anisotropic materials and the norm ratios may be used as a criterion to represent the anisotropy degree of the properties of materials. Finally, comparison of all methods are done in order to determine similarities and differences between them. As a result of this comparison process, it is proposed that the spectral method is a non-linear decomposition method which yields non-linear orthogonal decomposed parts. For symmetric second rank and fourth rank tensors, this case is a significant innovation in decomposition procedures in the literature.
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5

Hagemann, Willem [Verfasser], and Christoph [Akademischer Betreuer] Weidenbach. "Symbolic orthogonal projections : a new polyhedral representation for reachability analysis of hybrid systems / Willem Hagemann. Betreuer: Christoph Weidenbach." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2015. http://d-nb.info/107952388X/34.

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6

Smaïli, Nasser-Eddine. "Les polynômes e-semi-classiques de classe zéro." Paris 6, 1987. http://www.theses.fr/1987PA066081.

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7

Boutahar, Jaouad. "Méthodes de réduction et de propagation d'incertitudes : application à un mùodèle de Chimie-Transport pour la modélisation et la stimulation des impacts." Marne-la-vallée, ENPC, 2004. https://pastel.archives-ouvertes.fr/tel-00007557.

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Dans une modélisation intégrée des impacts, l'objectif est de tester plusieurs scénarios d'entrées de modèle et/ ou d'identifier l'effet de l'incertitude des entrées sur les sorties de modèle. Dans les deux cas, un grand nombre de simulations de modèle sont nécessaires. Cela reste bien évidemment infaisable avec un modèle de Chimie-Transport à cause du temps CPU demandé. Pour surmonter cette difficulté, deux approches ont été étudiées dans cette thèse. La première consiste à construire un modèle réduit. Deux techniques ont été utilisées : la première est la méthode POD (Proper Orthogonal Decomposition) liée au comportement statistique du système. La seconde méthode est une méthode efficace de prétabulation fondée sur la troncature d'un développement multivariables de la relation Entrées/ sorties associé au modèle. La seconde est relative à la réduction du nombre de simulations demandé par la méthode Monte-Carlo classique de propagation d'incertitude. La technique utilisée ici est basée sur une représentation d'une sortie de modèle incertaine comme un développement de polynômes orthonormaux de variables d'entrées. Un autre point clé dans la modélisation intégrée d'impacts est de développer des stratégies de réduction des émissions en calculant des matrices de transfert sur plusieurs années de simulation. Une méthode efficace de calcul de ces matrices a été ainsi développée, notamment en définissant des scénarios "chimiquement" représentatifs. L'ensemble de ces méthodes a été appliqué au modèle POLAIR3D, modèle de Chimie-Transport développé dans le cadre de cette thèse
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8

Martínez, Bayona Jonàs. "Skeletal representations of orthogonal shapes." Doctoral thesis, Universitat Politècnica de Catalunya, 2013. http://hdl.handle.net/10803/134699.

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Skeletal representations are important shape descriptors which encode topological and geometrical properties of shapes and reduce their dimension. Skeletons are used in several fields of science and attract the attention of many researchers. In the biocad field, the analysis of structural properties such as porosity of biomaterials requires the previous computation of a skeleton. As the size of three-dimensional images become larger, efficient and robust algorithms that extract simple skeletal structures are required. The most popular and prominent skeletal representation is the medial axis, defined as the shape points which have at least two closest points on the shape boundary. Unfortunately, the medial axis is highly sensitive to noise and perturbations of the shape boundary. That is, a small change of the shape boundary may involve a considerable change of its medial axis. Moreover, the exact computation of the medial axis is only possible for a few classes of shapes. For example, the medial axis of polyhedra is composed of non planar surfaces, and its accurate and robust computation is difficult. These problems led to the emergence of approximate medial axis representations. There exists two main approximation methods: the shape is approximated with another shape class or the Euclidean metric is approximated with another metric. The main contribution of this thesis is the combination of a specific shape and metric simplification. The input shape is approximated with an orthogonal shape, which are polygons or polyhedra enclosed by axis-aligned edges or faces, respectively. In the same vein, the Euclidean metric is replaced by the L infinity or Chebyshev metric. Despite the simpler structure of orthogonal shapes, there are few works on skeletal representations applied to orthogonal shapes. Much of the efforts have been devoted to binary images and volumes, which are a subset of orthogonal shapes. Two new skeletal representations based on this paradigm are introduced: the cube skeleton and the scale cube skeleton. The cube skeleton is shown to be composed of straight line segments or planar faces and to be homotopical equivalent to the input shape. The scale cube skeleton is based upon the cube skeleton, and introduces a family of skeletons that are more stable to shape noise and perturbations. In addition, the necessary algorithms to compute the cube skeleton of polygons and polyhedra and the scale cube skeleton of polygons are presented. Several experimental results confirm the efficiency, robustness and practical use of all the presented methods.
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9

Braun, Oliver Verfasser], Gabriele [Akademischer Betreuer] [Nebe, and Gerhard [Akademischer Betreuer] Hiß. "Orthogonal representations of finite groups / Oliver Braun ; Gabriele Nebe, Gerhard Hiß." Aachen : Universitätsbibliothek der RWTH Aachen, 2016. http://d-nb.info/1130352382/34.

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10

Braun, Oliver [Verfasser], Gabriele [Akademischer Betreuer] Nebe, and Gerhard [Akademischer Betreuer] Hiß. "Orthogonal representations of finite groups / Oliver Braun ; Gabriele Nebe, Gerhard Hiß." Aachen : Universitätsbibliothek der RWTH Aachen, 2016. http://d-nb.info/1130352382/34.

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Книги з теми "Orthogonal representation"

1

Arithmetical investigations: Representation theory, orthogonal polynomials, and quantum interpolations. Berlin: Springer, 2008.

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2

1977-, Mano Gen, ed. The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p, q). Providence, R.I: American Mathematical Society, 2011.

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3

Kobayashi, Toshiyuki. The Schrodinger model for the minimal representation of the indefinite orthogonal group O(p,q). Kyoto, Japan: Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2007.

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4

The geometric and arithmetic volume of Shimura varieties of orthogonal type. Providence, Rhode Island, USA: American Mathematical Society, 2014.

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5

The endoscopic classification of representations orthogonal and symplectic groups. Providence, Rhode Island: American Mathematical Society, 2013.

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6

Kobayashi, Toshiyuki. Symmetry breaking for representations of rank one orthogonal groups. Providence, Rhode Island: American Mathematical Society, 2015.

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7

Riehm, C. R. Introduction to orthogonal, symplectic, and unitary representations of finite groups. Providence, R.I: American Mathematical Society, 2011.

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8

Kobayashi, Toshiyuki, and Birgit Speh. Symmetry Breaking for Representations of Rank One Orthogonal Groups II. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-2901-2.

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9

Jantzen, Chris. Degenerate principal series for symplectic and odd-orthogonal groups. Providence, RI: American Mathematical Society, 1996.

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10

Kalnins, E. G. Tensor products of special unitary and oscillator algebras. Hamilton, N.Z: University of Waikato, 1992.

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Частини книг з теми "Orthogonal representation"

1

Wybrow, Michael, Kim Marriott, and Peter J. Stuckey. "Orthogonal Hyperedge Routing." In Diagrammatic Representation and Inference, 51–64. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-31223-6_10.

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2

Bournez, Olivier, Oded Maler, and Amir Pnueli. "Orthogonal Polyhedra: Representation and Computation." In Hybrid Systems: Computation and Control, 46–60. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/3-540-48983-5_8.

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3

Marriott, Kim, Peter J. Stuckey, and Michael Wybrow. "Seeing Around Corners: Fast Orthogonal Connector Routing." In Diagrammatic Representation and Inference, 31–37. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-662-44043-8_4.

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4

Rallis, Stephen. "Special Eisenstein series on orthogonal groups." In L-Functions and the Oscillator Representation, 10–24. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/bfb0077896.

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5

Mohamed, Ashraf, and Christos Davatzikos. "Shape Representation via Best Orthogonal Basis Selection." In Medical Image Computing and Computer-Assisted Intervention – MICCAI 2004, 225–33. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-30135-6_28.

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6

Helmke, Stefan, Bernhard Goetze, Robert Scheffler, and Gregor Wrobel. "Interactive, Orthogonal Hyperedge Routing in Schematic Diagrams Assisted by Layout Automatisms." In Diagrammatic Representation and Inference, 20–27. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-86062-2_2.

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AbstractSchematic diagrams are used in graph-based engineering systems. They focus mainly on the structure of the design object. Graph-based engineering systems help to solve a concrete design task. This is primarily realized by the application of domain-specific languages. The layout of schematic diagrams is of particular importance, and a neat representation is desirable. But automatically generated layouts cannot always fully match the intention of a modeler. To improve automatic layouts and enable a user-specific representation, an algorithm that allows interactive changes of the orthogonal hyperedge geometry was implemented. In this paper, we present this algorithm and give an overview of such interactions. Additionally, several reductions of the hyperedge geometry are shown. Furthermore, a local, automatic routing considering interactions on the hyperedge geometry is presented. The consideration of domain-specific semantics and the possibility of interactive changes is a new approach. All algorithms were implemented in a self-developed software framework.
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7

Serre, Jean-Pierre. "On the mod p reduction of orthogonal representations." In Lie Groups, Geometry, and Representation Theory, 527–40. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-02191-7_18.

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8

Yosida, Kôsaku. "The Orthogonal Projection and F. Riesz’ Representation Theorem." In Functional Analysis, 81–101. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/978-3-642-61859-8_4.

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9

Chinta, Gautam, and Omer Offen. "Orthogonal Period of a GL 3(ℤ) Eisenstein Series." In Representation Theory, Complex Analysis, and Integral Geometry, 41–59. Boston: Birkhäuser Boston, 2011. http://dx.doi.org/10.1007/978-0-8176-4817-6_3.

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10

Vilenkin, N. Ja, and A. U. Klimyk. "Quantum Groups, q-Orthogonal Polynomials and Basic Hypergeometric Functions." In Representation of Lie Groups and Special Functions, 1–136. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-017-2881-2_1.

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Тези доповідей конференцій з теми "Orthogonal representation"

1

Zhijing Yang, Chunmei Qing, Bingo Wing-Kuen Ling, Wai Lok Woo, and Saeid Sanei. "Orthogonal orthogonal overcomplete kernel design for sparse representation." In 2012 8th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP 2012). IEEE, 2012. http://dx.doi.org/10.1109/csndsp.2012.6292723.

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2

Wei, Chen, Cai Zhanchuan, and Huang Jing. "Orthogonal GF Moments for Image Representation." In 2013 Seventh International Conference on Image and Graphics (ICIG). IEEE, 2013. http://dx.doi.org/10.1109/icig.2013.59.

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3

Chatterjee, Ayan, and Peter W. T. Yuen. "Rapid Estimation of Orthogonal Matching Pursuit Representation." In IGARSS 2020 - 2020 IEEE International Geoscience and Remote Sensing Symposium. IEEE, 2020. http://dx.doi.org/10.1109/igarss39084.2020.9323532.

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4

Robert, Arnaud, and Dirk Van Hertem. "Reduced Grid Representation through Proper Orthogonal Decomposition." In 2021 IEEE Madrid PowerTech. IEEE, 2021. http://dx.doi.org/10.1109/powertech46648.2021.9494759.

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5

Chen, Yunpeng, Xiaojie Jin, Jiashi Feng, and Shuicheng Yan. "Training Group Orthogonal Neural Networks with Privileged Information." In Twenty-Sixth International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/212.

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Learning rich and diverse representations is critical for the performance of deep convolutional neural networks (CNNs). In this paper, we consider how to use privileged information to promote inherent diversity of a single CNN model such that the model can learn better representations and offer stronger generalization ability. To this end, we propose a novel group orthogonal convolutional neural network (GoCNN) that learns untangled representations within each layer by exploiting provided privileged information and enhances representation diversity effectively. We take image classification as an example where image segmentation annotations are used as privileged information during the training process. Experiments on two benchmark datasets – ImageNet and PASCAL VOC – clearly demonstrate the strong generalization ability of our proposed GoCNN model. On the ImageNet dataset, GoCNN improves the performance of state-of-the-art ResNet-152 model by absolute value of 1.2% while only uses privileged information of 10% of the training images, confirming effectiveness of GoCNN on utilizing available privileged knowledge to train better CNNs.
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Li, Leida, Shushang Li, Guihua Wang, and Ajith Abraham. "An evaluation on circularly orthogonal moments for image representation." In 2011 International Conference on Information Science and Technology (ICIST). IEEE, 2011. http://dx.doi.org/10.1109/icist.2011.5765275.

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7

Balu, Aditya, Sambit Ghadai, Soumik Sarkar, and Adarsh Krishnamurthy. "Orthogonal Distance Fields Representation for Machine-Learning Based Manufacturability Analysis." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22487.

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Abstract Computer-aided Design for Manufacturing (DFM) systems play an essential role in reducing the time taken for product development by providing manufacturability feedback to the designer before the manufacturing phase. Traditionally, DFM rules are hand-crafted and used to accelerate the engineering product design process by integrating manufacturability analysis during design. Recently, the feasibility of using a machine learning-based DFM tool in intelligently applying the DFM rules have been studied. These tools use a voxelized representation of the design and then use a 3D-Convolutional Neural Network (3D-CNN), to provide manufacturability feedback. Although these frameworks work effectively, there are some limitations to the voxelized representation of the design. In this paper, we introduce a new representation of the computer-aided design (CAD) model using orthogonal distance fields (ODF). We provide a GPU-accelerated algorithm to convert standard boundary representation (B-rep) CAD models into ODF representation. Using the ODF representation, we build a machine learning framework, similar to earlier approaches, to create a machine learning-based DFM system to provide manufacturability feedback. As proof of concept, we apply this framework to assess the manufacturability of drilled holes. The framework has an accuracy of more than 84% correctly classifying the manufacturable and non-manufacturable models using the new representation.
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Godovitsyn, Maxim, Julia Zhivchikova, Nickolay Starostin, and Anton Shtanyuk. "Algorithm for Implementing Logical Operations on Sets of Orthogonal Polygons." In 31th International Conference on Computer Graphics and Vision. Keldysh Institute of Applied Mathematics, 2021. http://dx.doi.org/10.20948/graphicon-2021-3027-1088-1097.

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As part of the development CAD for design rule checks (DRC), it is necessary to use logical operations on orthogonal polygons that form the layout of an integrated circuit. Such operations as union, intersection, subtraction are performed over layers that contain orthogonal polygons. These operations are subject to stringent execution time requirements. The traditional representation of polygons in the form of bitmaps does not provide a quasi-linear dependence of time on the processed data size and requires development of new algorithms and polygon representation approaches. This paper contains a description of a modified sweeping line obscuring algorithm that achieves O(N log N) time. The algorithm uses three properties of the polygon: the separation of inner region from outer region by the edge, the belonging of edges to the set of either vertical or horizontal edges, and dissection of the layer plane into rectangular fragments which belong to either inner or outer region of the polygon. Procedures of input polygon contour representations that are dissected into sets of vertical and horizontal edges are described. As a result of performing logical operations, polygon edges of the resulting layer are formed. These edges, in turn, are converted into contour representations. The results of a computational experiment confirming the nature of the time dependences determined theoretically are presented. We propose the structure of a software system for DRC, built with the use of programming languages C++ and Lua.
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Mutelo, R. M., W. L. Woo, and S. S. Dlay. "Two Dimensional Orthogonal Wavelet Features for Image Representation and Recognition." In 2007 15th International Conference on Digital Signal Processing. IEEE, 2007. http://dx.doi.org/10.1109/icdsp.2007.4288567.

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Sieber, Moritz, Alexander Kuhn, Hans-Christian Hege, C. Oliver Paschereit, and Kilian Oberleithner. "Poster: A graphical representation of the spectral proper orthogonal decomposition." In 68th Annual Meeting of the APS Division of Fluid Dynamics. American Physical Society, 2015. http://dx.doi.org/10.1103/aps.dfd.2015.gfm.p0007.

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