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Статті в журналах з теми "Orthogonal representation"

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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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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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11

Vasco, D. W. "Seismic source representation in orthogonal functions." Geophysical Journal International 102, no. 3 (September 1990): 531–35. http://dx.doi.org/10.1111/j.1365-246x.1990.tb04579.x.

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12

Li, Hong, Jin Ping Zhang, Fen Xia Wu, and Cong E. Tan. "Image Fusion with Sparse Representation." Advanced Materials Research 798-799 (September 2013): 737–40. http://dx.doi.org/10.4028/www.scientific.net/amr.798-799.737.

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Sparse representation is a new image representation theory. It can accurately represent the image information. In this paper, a novel fusion scheme using sparse representation is proposed. The sparse representation is conducted on overlapping patches. Each source image is divided into patches, and all the patches are transformed into vectors. Decompose the vectors into theirs sparse representations using orthogonal matching pursuit. Sparse coefficients are fused with the maximum absolute. The simulation results show that the proposed method can provide high-quality images.

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13

Asmuth, Charles. "Some Supercuspidal Representations of Sp4(k)." Canadian Journal of Mathematics 39, no. 1 (February 1, 1987): 1–7. http://dx.doi.org/10.4153/cjm-1987-001-9.

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The purpose of this paper is to produce explicit realizations of supercuspidal representations of Sp4(k) where k is a p-adic field with odd residual characteristic. These representations will be constructed using the Weil representation of Sp4(k) associated with a certain 4-dimensional compact orthogonal group OQ over k. The main problem addressed in this paper is the analysis of this representation; we need to find how the supercuspidal summands decompose into irreducible pieces.The problem of decomposing Weil representations has been studied quite a bit already. The Weil representations of SL2(k) associated to 2-dimensional orthogonal groups were used by Casselman [4] and Shalika [9] to produce all supercuspidals of SL2(k). The explicit formulas for these representations were used by Sally and Shalika ([10]) to compute the characters and finally to write down a Plancherel formula for that group.

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14

Ruiz-Medina, M. D., and M. J. Valderrama. "Orthogonal representations of random fields and an application to geophysics data." Journal of Applied Probability 34, no. 2 (June 1997): 458–76. http://dx.doi.org/10.2307/3215385.

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We present a brief summary of some results related to deriving orthogonal representations of second-order random fields and its application in solving linear prediction problems. In the homogeneous and/or isotropic case, the spectral theory provides an orthogonal expansion in terms of spherical harmonics, called spectral decomposition (Yadrenko 1983). A prediction formula based on this orthogonal representation is shown. Finally, an application of this formula in solving a real-data problem related to prospective geophysics techniques is presented.

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15

Ruiz-Medina, M. D., and M. J. Valderrama. "Orthogonal representations of random fields and an application to geophysics data." Journal of Applied Probability 34, no. 02 (June 1997): 458–76. http://dx.doi.org/10.1017/s0021900200101093.

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Анотація:

We present a brief summary of some results related to deriving orthogonal representations of second-order random fields and its application in solving linear prediction problems. In the homogeneous and/or isotropic case, the spectral theory provides an orthogonal expansion in terms of spherical harmonics, called spectral decomposition (Yadrenko 1983). A prediction formula based on this orthogonal representation is shown. Finally, an application of this formula in solving a real-data problem related to prospective geophysics techniques is presented.

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16

Cui, Minshan, and Saurabh Prasad. "Sparse representation-based classification: Orthogonal least squares or orthogonal matching pursuit?" Pattern Recognition Letters 84 (December 2016): 120–26. http://dx.doi.org/10.1016/j.patrec.2016.08.017.

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17

Proctor, Robert A. "A Schensted Algorithm Which Models Tensor Representations of the Orthogonal Group." Canadian Journal of Mathematics 42, no. 1 (February 1, 1990): 28–49. http://dx.doi.org/10.4153/cjm-1990-002-1.

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This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.

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18

Shepard, Ron, Scott R. Brozell, and Gergely Gidofalvi. "The Representation and Parametrization of Orthogonal Matrices." Journal of Physical Chemistry A 119, no. 28 (June 2, 2015): 7924–39. http://dx.doi.org/10.1021/acs.jpca.5b02015.

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19

Glynn, David G. "A geometrical representation theory for orthogonal arrays." Bulletin of the Australian Mathematical Society 49, no. 2 (April 1994): 311–24. http://dx.doi.org/10.1017/s0004972700016373.

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Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well-defined collection of ranks r. Having rank r means that it can be constructed as a cone cut by qs hyperplanes in projective space of dimension r over a field of order q.

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20

Hosny, Khalid M. "Image representation using accurate orthogonal Gegenbauer moments." Pattern Recognition Letters 32, no. 6 (April 2011): 795–804. http://dx.doi.org/10.1016/j.patrec.2011.01.006.

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21

Zhang, Jian, Jian Yang, Jianjun Qian, and Jiawei Xu. "Nearest orthogonal matrix representation for face recognition." Neurocomputing 151 (March 2015): 471–80. http://dx.doi.org/10.1016/j.neucom.2014.09.019.

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22

Zhao, Wei, Zheng Liu, Ziyu Guan, Binbin Lin, and Deng Cai. "Orthogonal Projective Sparse Coding for image representation." Neurocomputing 173 (January 2016): 270–77. http://dx.doi.org/10.1016/j.neucom.2014.10.106.

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23

Dobigeon, Nicolas, and Jean-Yves Tourneret. "Bayesian Orthogonal Component Analysis for Sparse Representation." IEEE Transactions on Signal Processing 58, no. 5 (May 2010): 2675–85. http://dx.doi.org/10.1109/tsp.2010.2041594.

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24

Scheunders, P. "An orthogonal wavelet representation of multivalued images." IEEE Transactions on Image Processing 12, no. 6 (June 2003): 718–25. http://dx.doi.org/10.1109/tip.2003.811502.

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25

Sun, Xiaobai, and Christian Bischof. "A Basis-Kernel Representation of Orthogonal Matrices." SIAM Journal on Matrix Analysis and Applications 16, no. 4 (October 1995): 1184–96. http://dx.doi.org/10.1137/s0895479894276369.

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26

Farouk, R. M., and Qamar A. A. Awad. "Image representation based on fractional order Legendre and Laguerre orthogonal moments." International Journal of ADVANCED AND APPLIED SCIENCES 8, no. 2 (February 2021): 54–59. http://dx.doi.org/10.21833/ijaas.2021.02.007.

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In this paper, we have introduced new sets of fractional order orthogonal basis moments based on Fractional order Legendre orthogonal Functions (FLeFs) and Fractional order Laguerre orthogonal Functions (FLaFs) for image representation. We have generated a novel set of Fractional order Legendre orthogonal Moments (FLeMs) from fractional order Legendre orthogonal functions and a new set of Fractional order Laguerre orthogonal Moments (FLaMs) from the fractional order Laguerre orthogonal functions. The new presented sets of (FLeMs) and (FLaMs) are tested with the recently introduced Fractional order Chebyshev orthogonal Moments (FCMs). This edge detection filter can be used successfully in the gray level image and color images. The new sets of fractional moments are used to reconstruct the gray level image. The numerical results show FLeMs and FLaMs are promised techniques for image representation. The computational time of the proposed techniques is compared with the computational time of Chebyshev orthogonal Moments techniques and gives better results. Also, the fractional parameters give the flexibility of studying global features of the image at different positions of moments.

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27

Chekanin, Alexander V., and Vladislav A. Chekanin. "Improved Packing Representation Model for the Orthogonal Packing Problem." Applied Mechanics and Materials 390 (August 2013): 591–95. http://dx.doi.org/10.4028/www.scientific.net/amm.390.591.

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The multidimensional NP-hard orthogonal bin packing problem is considered in the article. Usually the problem is solved using heuristic algorithms of discrete optimization which optimize a selection sequence of objects to be packed in containers. The quality and speed of getting the resulting packing for a given sequence of placing objects is determined by the used packing representation model. In the article presented a new packing representation model for constructing the orthogonal packing. The proposed model of potential containers describes all residual free spaces of containers in packing. The developed model is investigated on well-known standard benchmarks of three-dimensional orthogonal bin packing problem. The model can be used in development of applied software for the optimal allocation of orthogonal resources.

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28

Batioua, Imad, Rachid Benouini, Khalid Zenkouar, Said Najah, Hakim El Fadili, and Hassan Qjidaa. "3D Image Representation Using Separable Discrete Orthogonal Moments." Procedia Computer Science 148 (2019): 389–98. http://dx.doi.org/10.1016/j.procs.2019.01.047.

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29

Van gucht, Patrick, and Adhemar Bultheel. "State space representation for arbitrary orthogonal rational functions." Systems & Control Letters 49, no. 2 (June 2003): 91–98. http://dx.doi.org/10.1016/s0167-6911(02)00303-1.

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30

Angulo, J. M., and M. D. Ruiz-Medina. "On the orthogonal representation of generalized random fields." Statistics & Probability Letters 31, no. 3 (January 1997): 145–53. http://dx.doi.org/10.1016/s0167-7152(96)00026-0.

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31

Narita, Keiko, Noboru Endou, and Yasunari Shidama. "The Orthogonal Projection and the Riesz Representation Theorem." Formalized Mathematics 23, no. 3 (September 1, 2015): 243–52. http://dx.doi.org/10.1515/forma-2015-0020.

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Abstract In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces. Referring to the article [15], we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions [11], we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to [36], [9], [24] and [3] in the formalization.

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32

Ouknine, Youssef, and Mohamed Erraoui. "Noncanonical representation with an infinite-dimensional orthogonal complement." Statistics & Probability Letters 78, no. 10 (August 2008): 1200–1205. http://dx.doi.org/10.1016/j.spl.2007.11.015.

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33

Alhaidari, A. D. "Orthogonal polynomials derived from the tridiagonal representation approach." Journal of Mathematical Physics 59, no. 1 (January 2018): 013503. http://dx.doi.org/10.1063/1.5001168.

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34

Daniel, Derek J. "Orthogonal Representation of Weber's Function Using Hermite Polynomials." Journal of Approximation Theory 113, no. 1 (November 2001): 156–63. http://dx.doi.org/10.1006/jath.2001.3620.

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35

Aristidi, E. "Representation of Signals as Series of Orthogonal Functions." EAS Publications Series 78-79 (2016): 99–126. http://dx.doi.org/10.1051/eas/1678006.

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36

Charmchi, H., and J. A. Salehi. "Outer-Product Matrix Representation of Optical Orthogonal Codes." IEEE Transactions on Communications 54, no. 6 (June 2006): 983–89. http://dx.doi.org/10.1109/tcomm.2006.876839.

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37

Dragnev, Peter, and Erwin Miña-Díaz. "On a series representation for Carleman orthogonal polynomials." Proceedings of the American Mathematical Society 138, no. 12 (December 1, 2010): 4271. http://dx.doi.org/10.1090/s0002-9939-2010-10583-x.

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38

ZHONG, YIN, and HONG-GANG LUO. "ORTHOGONAL DIRAC SEMIMETAL ON HONEYCOMB LATTICE." International Journal of Modern Physics B 27, no. 07 (March 10, 2013): 1361002. http://dx.doi.org/10.1142/s021797921361002x.

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Recently, a concept of orthogonal metal has been introduced to reinterpret the disordered state of slave-spin representation in the Hubbard model as an exotic gapped metallic state. We have extended this concept to study the slave-spin representation of Hubbard model on the honeycomb lattice. It is found that a novel gapped metallic state coined orthogonal Dirac semimetal is identified. Such state corresponds to the disordered phase of slave-spin and has the same thermal-dynamical and transport properties as Dirac semimetal but its singe-particle excitation is gapped.

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39

Sezer, Ali Devin, and Ferruh Özbudak. "Approximation of bounds on mixed-level orthogonal arrays." Advances in Applied Probability 43, no. 02 (June 2011): 399–421. http://dx.doi.org/10.1017/s0001867800004912.

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Анотація:

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

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40

Sezer, Ali Devin, and Ferruh Özbudak. "Approximation of bounds on mixed-level orthogonal arrays." Advances in Applied Probability 43, no. 2 (June 2011): 399–421. http://dx.doi.org/10.1239/aap/1308662485.

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Анотація:

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

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41

Morris, Peter R. "Generalized Spherical Harmonics for Cubic-Triclinic Symmetry." Textures and Microstructures 24, no. 4 (January 1, 1995): 221–24. http://dx.doi.org/10.1155/tsm.24.221.

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An explicit representation is suggested for orthogonal generalized spherical harmonics with cubic-crystal and triclinic-sample symmetries. The representation employs sums and differences of orthogonal generalized spherical harmonics with cubic-crystal symmetry, previously described by Bunge for orthorhombic (or higher) sample symmetry, and is illustrated, for T:.ιμν, ι=4, 9, μ=1, ν=1 to 5. This representation facilitates crystallite orientation distribution (COD)analysis (aka ODF analysis) for these symmetries, using the Bunge formalism.

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42

Morris, Peter R. "Generalized Spherical Harmonics for Cubic-Triclinic Symmetry." Textures and Microstructures 29, no. 3-4 (January 1, 1997): 235–39. http://dx.doi.org/10.1155/tsm.29.235.

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An explicit representation is suggested for orthogonal generalized spherical harmonics with cubic-crystal and triclinic-sample symmetries. The representation employs sums and differences of orthogonal generalized spherical harmonics with cubic-crystal symmetry, previously described by Bunge for orthorhombic (or higher) sample symmetry, and is illustrated, for T∴iμυ, i = 4, 9, μ = 1, υ = 1 to 5. This representation facilitates crystallite orientation distribution (COD) analysis (aka ODF analysis) for these symmetries, using the Bunge formalism.

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43

MORIWAKI, MASAYASU. "MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP." International Journal of Mathematics 19, no. 10 (November 2008): 1187–201. http://dx.doi.org/10.1142/s0129167x08005084.

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Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

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44

Jiang, Dihua, Baiying Liu, and Bin Xu. "A reciprocal branching problem for automorphic representations and global Vogan packets." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 765 (August 1, 2020): 249–77. http://dx.doi.org/10.1515/crelle-2019-0016.

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AbstractLet G be a group and let H be a subgroup of G. The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G, determine the occurrence of an irreducible representation σ of H in the restriction of π to H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation π of G such that σ occurs in the restriction of π to H. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied to other classical groups as well.

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45

Víglaský, Viktor. "Hidden Information Revealed Using the Orthogonal System of Nucleic Acids." International Journal of Molecular Sciences 23, no. 3 (February 4, 2022): 1804. http://dx.doi.org/10.3390/ijms23031804.

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In this study, the organization of genetic information in nucleic acids is defined using a novel orthogonal representation. Clearly defined base pairing in DNA allows the linear base chain and sequence to be mathematically transformed into an orthogonal representation where the G–C and A–T pairs are displayed in different planes that are perpendicular to each other. This form of base allocation enables the evaluation of any nucleic acid and predicts the likelihood of a particular region to form non-canonical motifs. The G4Hunter algorithm is currently a popular method of identifying G-quadruplex forming sequences in nucleic acids, and offers promising scores despite its lack of a substantial rational basis. The orthogonal representation described here is an effort to address this incongruity. In addition, the orthogonal display facilitates the search for other sequences that are capable of adopting non-canonical motifs, such as direct and palindromic repeats. The technique can also be used for various RNAs, including any aptamers. This powerful tool based on an orthogonal system offers considerable potential for a wide range of applications.

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46

Huang, Tian Li, Wei Xin Ren, and Meng Lin Lou. "A New Spectral Representation of Earthquake Recordings Using an Improved Hilbert-Huang Transform." Advanced Materials Research 255-260 (May 2011): 1671–75. http://dx.doi.org/10.4028/www.scientific.net/amr.255-260.1671.

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A new spectral representation method of earthquake recordings using an improved Hilbert-Huang transform (HHT) is proposed in the paper. Firstly, the problem that the intrinsic mode functions (IMFs) decomposed by the empirical mode decomposition (EMD) in HHT is not exactly orthogonal is pointed out and improved through the Gram-Schmidt orthogonalization method which is referred as the orthogonal empirical mode decomposition (OEMD). Combined the OEMD and the Hilbert transform (HT) which is referred as the improved Hilbert-Huang transform (IHHT), the orthogonal intrinsic mode functions (OIMFs) and the orthogonal Hilbert spectrum (OHS) and the orthogonal Hilbert marginal spectrum (OHMS) are obtained. Then, the IHHT has been applied for the analysis of the El Centro earthquake recording. The obtained spectral representation result shows that the OHS gives more detailed and accurate information in a time–frequency–energy presentation than the Hilbert spectrum (HS) and the OHMS gives more faithful low-frequency energy presentation than the Fourier spectrum (FS) and the Hilbert marginal spectrum (HMS).

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47

Miranian, L. "Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory." Canadian Mathematical Bulletin 52, no. 1 (March 1, 2009): 95–104. http://dx.doi.org/10.4153/cmb-2009-012-3.

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AbstractIn the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel–Darboux formulas are presented for the first time.

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48

Peng, Siyuan, Wee Ser, Badong Chen, and Zhiping Lin. "Robust orthogonal nonnegative matrix tri-factorization for data representation." Knowledge-Based Systems 201-202 (August 2020): 106054. http://dx.doi.org/10.1016/j.knosys.2020.106054.

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Schneider, B. I., and Nicolai Nygaard. "Orthogonal Functions, Discrete Variable Representation, and Generalized Gauss Quadratures†." Journal of Physical Chemistry A 106, no. 45 (November 2002): 10773–76. http://dx.doi.org/10.1021/jp025552d.

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Fu, Xiaomei, Li Chen, and Jingyu Yang. "Non-orthogonal frequency division multiplexing based on sparse representation." IET Communications 12, no. 16 (October 9, 2018): 2005–9. http://dx.doi.org/10.1049/iet-com.2017.1260.

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