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Journal articles on the topic 'Lie transformation groups'

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Published: 10 March 2023

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1

Miao, Xu, and Rajesh P. N. Rao. "Learning the Lie Groups of Visual Invariance." Neural Computation 19, no. 10 (October 2007): 2665–93. http://dx.doi.org/10.1162/neco.2007.19.10.2665.

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A fundamental problem in biological and machine vision is visual invariance: How are objects perceived to be the same despite transformations such as translations, rotations, and scaling? In this letter, we describe a new, unsupervised approach to learning invariances based on Lie group theory. Unlike traditional approaches that sacrifice information about transformations to achieve invariance, the Lie group approach explicitly models the effects of transformations in images. As a result, estimates of transformations are available for other purposes, such as pose estimation and visuomotor control. Previous approaches based on first-order Taylor series expansions of images can be regarded as special cases of the Lie group approach, which utilizes a matrix-exponential-based generative model of images and can handle arbitrarily large transformations. We present an unsupervised expectation-maximization algorithm for learning Lie transformation operators directly from image data containing examples of transformations. Our experimental results show that the Lie operators learned by the algorithm from an artificial data set containing six types of affine transformations closely match the analytically predicted affine operators. We then demonstrate that the algorithm can also recover novel transformation operators from natural image sequences. We conclude by showing that the learned operators can be used to both generate and estimate transformations in images, thereby providing a basis for achieving visual invariance.

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2

Al-Shomrani, M. M. "Lie Groups Analysis and Contact Transformations for Ito System." Abstract and Applied Analysis 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/342680.

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Generalized Ito systems of four coupled nonlinear evaluation equations are proposed. New classes of exact invariant solutions by using Lie group analysis are obtained. Moreover, we investigate the existence of a one-parameter group of contact transformations for a generalized Ito system. Consequently, we study the relationship between one-parameter group of a contact transformation and a one-parameter Lie point transformation for a generalized Ito system.

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3

FIGUEROA-O’FARRILL, JOSÉ M., and SONIA STANCIU. "POISSON LIE GROUPS AND THE MIURA TRANSFORMATION." Modern Physics Letters A 10, no. 36 (November 30, 1995): 2767–73. http://dx.doi.org/10.1142/s0217732395002908.

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We point out that the recent proof of the Kupershmidt-Wilson theorem by Cheng and Mas-Ramos is underpinned by the Poisson Lie property of the second Gel’fand-Dickey bracket. The supersymmetric Kupershmidt-Wilson theorem is also proved along the same lines.

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4

Cariñena, Jose F., Mariano A. Del Olmo, and Mariano Santander. "Locally operating realizations of transformation Lie groups." Journal of Mathematical Physics 26, no. 9 (September 1985): 2096–106. http://dx.doi.org/10.1063/1.526974.

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5

Lin, Feng, Haohang Xu, Houqiang Li, Hongkai Xiong, and Guo-Jun Qi. "Auto-Encoding Transformations in Reparameterized Lie Groups for Unsupervised Learning." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 10 (May 18, 2021): 8610–17. http://dx.doi.org/10.1609/aaai.v35i10.17044.

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Unsupervised training of deep representations has demonstrated remarkable potentials in mitigating the prohibitive expenses on annotating labeled data recently. Among them is predicting transformations as a pretext task to self-train representations, which has shown great potentials for unsupervised learning. However, existing approaches in this category learn representations by either treating a discrete set of transformations as separate classes, or using the Euclidean distance as the metric to minimize the errors between transformations. None of them has been dedicated to revealing the vital role of the geometry of transformation groups in learning representations. Indeed, an image must continuously transform along the curved manifold of a transformation group rather than through a straight line in the forbidden ambient Euclidean space. This suggests the use of geodesic distance to minimize the errors between the estimated and groundtruth transformations. Particularly, we focus on homographies, a general group of planar transformations containing the Euclidean, similarity and affine transformations as its special cases. To avoid an explicit computing of intractable Riemannian logarithm, we project homographies onto an alternative group of rotation transformations SR(3) with a tractable form of geodesic distance. Experiments demonstrate the proposed approach to Auto-Encoding Transformations exhibits superior performances on a variety of recognition problems.

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6

García‐Prada, Oscar, Mariano A. del Olmo, and Mariano Santander. "Locally operating realizations of nonconnected transformation Lie groups." Journal of Mathematical Physics 29, no. 5 (May 1988): 1083–90. http://dx.doi.org/10.1063/1.527946.

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7

Nesterenko, Maryna O. "Transformation groups on real plane and their differential invariants." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–17. http://dx.doi.org/10.1155/ijmms/2006/17410.

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Complete sets of bases of differential invariants, operators of invariant differentiation, and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of finite-dimensional Lie algebras on the real plane are revisited.

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8

Nikonov, V. I. "The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 20, no. 3 (September 6, 2018): 295–303. http://dx.doi.org/10.15507/2079-6900.20.201802.295-303.

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The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

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9

Nikonov, Vladimir I. "The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 20, no. 3 (September 6, 2018): 295–303. http://dx.doi.org/10.15507/2079-6900.20.201803.295-303.

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The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

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10

Russell, Thomas. "Gorman demand systems and lie transformation groups: A reply." Economics Letters 51, no. 2 (May 1996): 201–4. http://dx.doi.org/10.1016/0165-1765(96)00806-3.

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11

Caelli, Terry M., and Mario Ferraro. "Lie transformation groups, integral transforms, and invariant pattern recognition." Spatial Vision 8, no. 1 (1994): 33–44. http://dx.doi.org/10.1163/156856894x00224.

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12

Mansouri, A. R., D. P. Mukherjee, and S. T. Acton. "Constraining Active Contour Evolution via Lie Groups of Transformation." IEEE Transactions on Image Processing 13, no. 6 (June 2004): 853–63. http://dx.doi.org/10.1109/tip.2004.826128.

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13

Linden, Martin, and Helmut Reckziegel. "On the differentiability of maps into Lie transformation groups." Manuscripta Mathematica 63, no. 3 (September 1989): 377–79. http://dx.doi.org/10.1007/bf01168378.

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14

ANIELLO, PAOLO. "PERTURBATIVE SOLUTIONS OF DIFFERENTIAL EQUATIONS IN LIE GROUPS." International Journal of Geometric Methods in Modern Physics 02, no. 01 (February 2005): 111–25. http://dx.doi.org/10.1142/s0219887805000478.

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We show that, given a matrix Lie group [Formula: see text] and its Lie algebra [Formula: see text], a 1-parameter subgroup of [Formula: see text] whose generator is the sum of an unperturbed matrix Â0 and an analytic perturbation Â♢(λ) can be mapped — under suitable conditions — by a similarity transformation depending analytically on the perturbative parameter λ, onto a 1-parameter subgroup of [Formula: see text] generated by a matrix [Formula: see text] belonging to the centralizer of Â0 in [Formula: see text]. Both the similarity transformation and the matrix [Formula: see text] can be determined perturbatively, hence allowing a very convenient perturbative expansion of the original 1-parameter subgroup.

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15

Emerson, Victor F., G. Keith Humphrey, and Peter C. Dodwell. "Colored aftereffects contingent on patterns generated by Lie transformation groups." Perception & Psychophysics 37, no. 2 (March 1985): 155–62. http://dx.doi.org/10.3758/bf03202851.

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16

Chernoff, Paul R. "Irreducible representations of infinite-dimensional transformation groups and Lie algebras." Bulletin of the American Mathematical Society 13, no. 1 (July 1, 1985): 46–49. http://dx.doi.org/10.1090/s0273-0979-1985-15359-5.

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17

Dhooghe, Paul F. "Multilocal invariants for the classical groups." International Journal of Mathematics and Mathematical Sciences 2003, no. 1 (2003): 27–63. http://dx.doi.org/10.1155/s016117120301233x.

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Multilocal higher-order invariants, which are higher-order invariants defined at distinct points of representation space, for the classical groups are derived in a systematic way. The basic invariants for the classical groups are the well-known polynomial or rational invariants as derived from the Capelli identities. Higher-order invariants are then constructed from the former ones by means of total derivatives. At each order, it appears that the invariants obtained in this way do not generate all invariants. The necessary additional invariants are constructed from the invariant polynomials on the Lie algebra of the Lie transformation groups.

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18

Daping WENG. "Donaldson-Thomas transformation of double bruhat cells in semisimple Lie groups." Annales scientifiques de l'École normale supérieure 53, no. 2 (2020): 353–436. http://dx.doi.org/10.24033/asens.2424.

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19

Straume, Eldar. "Compact connected Lie transformation groups on spheres with low cohomogeneity. I." Memoirs of the American Mathematical Society 119, no. 569 (1996): 0. http://dx.doi.org/10.1090/memo/0569.

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20

Straume, Eldar. "Compact connected Lie transformation groups on spheres with low cohomogeneity. II." Memoirs of the American Mathematical Society 125, no. 595 (1997): 0. http://dx.doi.org/10.1090/memo/0595.

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21

Chernoff, P. R. "Irreducible Representations of Infinite-Dimensional Transformation Groups and Lie Algebras, I." Journal of Functional Analysis 130, no. 2 (June 1995): 255–82. http://dx.doi.org/10.1006/jfan.1995.1069.

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22

Netzer, N., and H. Reitberger. "On the convergence of iterated Pilgerschritt transformation in nilpotent Lie groups." Publicationes Mathematicae Debrecen 29, no. 3-4 (July 1, 2022): 309–14. http://dx.doi.org/10.5486/pmd.1982.29.3-4.11.

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23

Khabirov, S. V. "The main tasks of group analysis of differential equations of mechanics." Multiphase Systems 17, no. 1-2 (2022): 51–62. http://dx.doi.org/10.21662/mfs2022.1.005.

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Group analysis of differential equations uses the Lie theory of correspondence between continuous transformation groups and Lie algebras of first-order differentiation operators. Differential equations of mechanics necessarily admit an extensive group of transformations. Lie theory studies the structure of the algebra of this group. The group analysis of the equations of mechanics uses the structure of the admissible algebra to produce submodels and exact solutions, to study the boundary value problems of submodels and the behavior of the mechanical medium for exact solutions. The main tasks of group analysis are formulated and simple examples of their solutions are given.

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24

Shvedov, Oleg Yu. "Symmetries of semiclassical gauge systems." International Journal of Geometric Methods in Modern Physics 12, no. 10 (October 25, 2015): 1550110. http://dx.doi.org/10.1142/s0219887815501108.

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Semiclassical systems being symmetric under Lie group are studied. A state of a semiclassical system may be viewed as a set (X, f) of a classical state X and a quantum state f in the external classical background X. Therefore, the set of all semiclassical states may be considered as a bundle (semiclassical bundle). Its base {X} is the set of all classical states, while a fiber is a Hilbert space ℱX of quantum states in the external background X. Symmetry transformation of a semiclassical system may be viewed as an automorphism of the semiclassical bundle. Automorphism groups can be investigated with the help of sections of the bundle: to any automorphism of the bundle one assigns a transformation of section of the bundle. Infinitesimal properties of transformations of sections are investigated; correspondence between Lie groups and Lie algebras is discussed. For gauge theories, some points of the semiclassical bundle are identified: a gauge group acts on the bundle. For this case, only gauge-invariant sections of the semiclassical bundle are taken into account.

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25

Lu, Huanhuan, and Yufeng Zhang. "Lie Symmetry Analysis, Exact Solutions, Conservation Laws and Bäcklund Transformations of the Gibbons-Tsarev Equation." Symmetry 12, no. 8 (August 18, 2020): 1378. http://dx.doi.org/10.3390/sym12081378.

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In this paper, we mainly put the Lie symmetry analysis method on the Gibbons-Tsarev equation (GTe) to obtain some new results, including some Lie symmetries, one-parameter transformation groups, explicit invariant solutions in the form of power series. Subsequently, the self-adjointness of the GTe is singled out. It follows that the conservation laws associated with symmetries of GTe are constructed with the aid of Ibragimov’ method. Finally, we present the Bäcklund transformations so that more abundant solutions can be worked out.

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26

Bor, Winny Chepngetich, Owino M. Oduor, and John K. Rotich. "A Lie Symmetry Solutions of Sawada-Kotera Equation." JOURNAL OF ADVANCES IN MATHEMATICS 17 (July 30, 2019): 1–11. http://dx.doi.org/10.24297/jam.v17i0.8364.

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In this article, the Lie Symmetry Analysis is applied in finding the symmetry solutions of the fifth order Sawada-Kotera equation. The technique is among the most powerful approaches currently used to achieveprecise solutions of the partial differential equations that are nonlinear. We systematically show the procedure to obtain the solution which is achieved by developing infinitesimal transformation, prolongations, infinitesimal generatorsand invariant transformations hence symmetry solutions of the fifth order Sawada-Kotera equation. Key Words- Lie symmetry analysis. Sawada-Kotera equation. Symmetry groups. Prolongations. Invariant solutions. Power series solutions. Symmetry solutions.

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27

Zhao, Weidong, Muhammad Mobeen Munir, Hajra Bashir, Daud Ahmad, and Muhammad Athar. "Lie symmetry analysis for generalized short pulse equation." Open Physics 20, no. 1 (January 1, 2022): 1185–93. http://dx.doi.org/10.1515/phys-2022-0212.

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Abstract Lie symmetry analysis (LSA) is one of the most common, effective, and estimation-free methods to find the symmetries and solutions of the differential equations (DEs) by following an algorithm. This analysis leads to reduce the order of partial differential equations (PDEs). Many physical problems are converted into non-linear DEs and these DEs or system of DEs are then solved with several methods such as similarity methods, Lie Bäcklund transformation, and Lie group of transformations. LSA is suitable for providing the conservation laws corresponding to Lie point symmetries or Lie Bäcklund symmetries. Short pulse equation (SPE) is a non-linear PDE, used in optical fibers, computer graphics, and physical systems and has been generalized in many directions. We will find the symmetries and a class of solutions depending on one-parameter (ε) obtained from Lie symmetry groups. Then we will construct the optimal system for the Lie algebra and invariant solutions (called similarity solutions) from Lie subalgebras of generalized SPE.

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28

Parkhomenko, S. E. "Poisson–Lie T-Duality and N=2 Superconformal WZNW Models on Compact Groups." Modern Physics Letters A 12, no. 39 (December 21, 1997): 3091–103. http://dx.doi.org/10.1142/s0217732397003216.

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The supersymmetric generalization of Poisson–Lie T-duality in the N=2 superconformal WZNW models on the compact groups is considered. It is shown that the role of Drinfeld's doubles play the complexifications of the corresponding compact groups. These complex doubles are used to define the natural actions of the isotropic subgroups forming the doubles on the group manifolds of the N=2 superconformal WZNW models. The Poisson–Lie T-duality in N=2 superconformal U(2)-WZNW model is considered in details. It is shown that this model admits Poisson–Lie symmetries with respect to the isotropic subgroups forming Drinfeld's double Gl (2,C). Poisson–Lie T-duality transformation maps this model into itself but acts nontrivially on the space of classical solutions. Supersymmetric generalization of Poisson–Lie T-duality in N=2 superconformal WZNW models on the compact groups of higher dimensions is proposed.

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29

Wang, XiaoMin, Sudao Bilige, and YueXing Bai. "Potential Symmetries, Lie Transformation Groups and Exact Solutions of Kdv-Burgers Equation." Interdisciplinary journal of Discontinuity, Nonlinearity and Complexity 6, no. 1 (March 2017): 1–9. http://dx.doi.org/10.5890/dnc.2017.03.001.

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30

Chepngetich, Winny. "The lie symmetry analysis of third order Korteweg-de Vries equation." Journal of Physical and Applied Sciences (JPAS) 1, no. 1 (November 1, 2022): 38–43. http://dx.doi.org/10.51317/jpas.v1i1.299.

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This study sought to analyse the Lie symmetry of third order Korteweg-de Vries equation. Solving nonlinear partial differential equations is of great importance in the world of dynamics. Korteweg-de Vries equations are partial differential equations arising from the theory of long waves, modelling of shallow water waves, fluid mechanics, plasma fluids and many other nonlinear physical systems, and their effects are relevant in real life. In this study, Lie symmetry analysis is demonstrated in finding the symmetry solutions of the third-order KdV equation of the form. The study systematically showed the formula to find the specific solution attained by developing prolongations, infinitesimal transformations and generators, adjoint symmetries, variation symmetries, invariant transformation and integrating factors to obtain all the lie groups presented by the equation. In conclusion, infinitesimal generators, group transformations and symmetry solutions of third-order KdV equation are acquired using a method of Lie symmetry analysis. This was achieved by generating infinitesimal generators which act on the KdV equation to form infinitesimal transformations. It can be seen from the solutions of this paper that the Lie symmetry analysis method is an effective and best mathematical technique for studying linear and nonlinear PDEs and ODEs.

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31

Bibi, Khudija, and Khalil Ahmad. "Exact Solutions of Newell-Whitehead-Segel Equations Using Symmetry Transformations." Journal of Function Spaces 2021 (January 25, 2021): 1–8. http://dx.doi.org/10.1155/2021/6658081.

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In this article, Lie and discrete symmetry transformation groups of linear and nonlinear Newell-Whitehead-Segel (NWS) equations are obtained. By using these symmetry transformation groups, several group invariant solutions of considered NWS equations have been constructed. Furthermore, some more group invariant solutions are generated by using discrete symmetry transformation group. Graphical representations of some obtained solutions are also presented.

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32

Guo, Huijuan, Yan Zhou, Huiying Liu, and Xiaoxiang Hu. "Improved Cubature Kalman Filtering on Matrix Lie Groups Based on Intrinsic Numerical Integration Error Calibration with Application to Attitude Estimation." Machines 10, no. 4 (April 7, 2022): 265. http://dx.doi.org/10.3390/machines10040265.

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This paper investigates the numerical integration error calibration problem in Lie group sigma point filters to obtain more accurate estimation results. On the basis of the theoretical framework of the Bayes–Sard quadrature transformation, we first established a Bayesian estimator on matrix Lie groups for system measurements in Euclidean spaces or Lie groups. The estimator was then employed to develop a generalized Bayes–Sard cubature Kalman filter on matrix Lie groups that considers additional uncertainties brought by integration errors and contains two variants. We also built on the maximum likelihood principle, and an adaptive version of the proposed filter was derived for better algorithm flexibility and more precise filtering results. The proposed filters were applied to the quaternion attitude estimation problem. Monte Carlo numerical simulations supported that the proposed filters achieved better estimation quality than that of other Lie group filters in the mentioned studies.

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33

Chen, Yong, and Xiaorui Hu. "Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation." Zeitschrift für Naturforschung A 64, no. 1-2 (February 1, 2009): 8–14. http://dx.doi.org/10.1515/zna-2009-1-202.

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The classical symmetry method and the modified Clarkson and Kruskal (C-K) method are used to obtain the Lie symmetry group of a nonisospectral Kadomtsev-Petviashvili (KP) equation. It is shown that the Lie symmetry group obtained via the traditional Lie approach is only a special case of the symmetry groups obtained by the modified C-K method. The discrete group analysis is given to show the relations between the discrete group and parameters in the ansatz. Furthermore, the expressions of the exact finite transformation of the Lie groups via the modified C-K method are much simpler than those obtained via the standard approach.

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34

GOLODETS, VALENTIN YA, and SERGEY D. SINEL'SHCHIKOV. "On the conjugacy and isomorphism problems for stabilizers of Lie group actions." Ergodic Theory and Dynamical Systems 19, no. 2 (April 1999): 391–411. http://dx.doi.org/10.1017/s014338579913013x.

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The spaces of subgroups and Lie subalgebras with the group actions by conjugations are considered for real Lie groups. Our approach allows one to apply the properties of algebraically regular transformation groups to finding the conditions when those actions turn out to be type I. It follows, in particular, that in this case the stability groups for all the ergodic actions of such groups are conjugate (for example when the stability groups are compact). The isomorphism of the stability groups for ergodic actions is also established under some assumptions. A number of examples of non-conjugate and non-isomorphic stability groups are presented.

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ALDAYA, V., M. CALIXTO, and E. SÁNCHEZ-SASTRE. "EXTENDING THE STUECKELBERG MODEL FOR SPACETIME SYMMETRIES: COSMOLOGICAL IMPLICATIONS." Modern Physics Letters A 21, no. 37 (December 7, 2006): 2813–25. http://dx.doi.org/10.1142/s0217732306021876.

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The so-called Stueckelberg transformation is introduced as a prescription of minimal coupling following the procedure of the Utiyama theory, for both internal and spacetime symmetry Lie groups. As a natural example, the theory is applied to the Weyl group and the corresponding gauge gravitational theory is developed. This context appears to be a natural source to account for some sort of dark matter intrinsically related to the gauge-group parameter associated with scale transformations.

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36

Man, Jia. "Lie point symmetry algebras and finite transformation groups of the general Broer–Kaup system." Chinese Physics 16, no. 6 (June 2007): 1534–44. http://dx.doi.org/10.1088/1009-1963/16/6/007.

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37

Wiltshire, R. J. "The use of Lie transformation groups in the solution of the coupled diffusion equation." Journal of Physics A: Mathematical and General 27, no. 23 (December 7, 1994): 7821–29. http://dx.doi.org/10.1088/0305-4470/27/23/025.

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38

Chen, An Mei, Haw-minn Lu, and Robert Hecht-Nielsen. "On the Geometry of Feedforward Neural Network Error Surfaces." Neural Computation 5, no. 6 (November 1993): 910–27. http://dx.doi.org/10.1162/neco.1993.5.6.910.

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Many feedforward neural network architectures have the property that their overall input-output function is unchanged by certain weight permutations and sign flips. In this paper, the geometric structure of these equioutput weight space transformations is explored for the case of multilayer perceptron networks with tanh activation functions (similar results hold for many other types of neural networks). It is shown that these transformations form an algebraic group isomorphic to a direct product of Weyl groups. Results concerning the root spaces of the Lie algebras associated with these Weyl groups are then used to derive sets of simple equations for minimal sufficient search sets in weight space. These sets, which take the geometric forms of a wedge and a cone, occupy only a minute fraction of the volume of weight space. A separate analysis shows that large numbers of copies of a network performance function optimum weight vector are created by the action of the equioutput transformation group and that these copies all lie on the same sphere. Some implications of these results for learning are discussed.

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39

Iserles, Arieh, and Antonella Zanna. "On the Dimension of Certain Graded Lie Algebras Arising in Geometric Integration of Differential Equations." LMS Journal of Computation and Mathematics 3 (2000): 44–75. http://dx.doi.org/10.1112/s1461157000000206.

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AbstractMany discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

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40

Torrisi, Mariano, and Rita Traciná. "Lie Symmetries and Solutions of Reaction Diffusion Systems Arising in Biomathematics." Symmetry 13, no. 8 (August 20, 2021): 1530. http://dx.doi.org/10.3390/sym13081530.

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In this paper, a special subclass of reaction diffusion systems with two arbitrary constitutive functions Γ(v) and H(u,v) is considered in the framework of transformation groups. These systems arise, quite often, as mathematical models, in several biological problems and in population dynamics. By using weak equivalence transformation the principal Lie algebra, LP, is written and the classifying equations obtained. Then the extensions of LP are derived and classified with respect to Γ(v) and H(u,v). Some wide special classes of special solutions are carried out.

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41

Nadjafikhah, Mehdi, and Parastoo Kabi-Nejad. "Approximate Symmetries of the Harry Dym Equation." ISRN Mathematical Physics 2013 (December 23, 2013): 1–7. http://dx.doi.org/10.1155/2013/109170.

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We derive the first-order approximate symmetries for the Harry Dym equation by the method of approximate transformation groups proposed by Baikov et al. (1989, 1996). Moreover, we investigate the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. We compute the one-dimensional optimal system of subalgebras as well as point out some approximately differential invariants with respect to the generators of Lie algebra and optimal system.

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42

Yuan, Qing-Qing, and Sen-Yue Lou. "Lie Point Symmetry Algebras and Finite Transformation Groups of Baroclinic Mode for Rotating Stratified Flows." Communications in Theoretical Physics 55, no. 5 (May 2011): 878–82. http://dx.doi.org/10.1088/0253-6102/55/5/26.

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LUSANNA, LUCA. "THE SHANMUGADHASAN CANONICAL TRANSFORMATION, FUNCTION GROUPS AND THE EXTENDED SECOND NOETHER THEOREM." International Journal of Modern Physics A 08, no. 24 (September 30, 1993): 4193–233. http://dx.doi.org/10.1142/s0217751x93001727.

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After the definition of a class of well-behaved singular Lagrangians, an analysis of all the consequences of the extended second Noether theorem in the second-order formalism is made. The phase-space reformulation contains arbitrary first- and second-class constraints. An answer to the problem of the Dirac conjecture is given for this class of singular Lagrangians. By using the concepts of function groups and of the associated Shanmugadhasan canonical transformations, an attempt is made to arrive at a global formulation of the theorem, in which the original invariance under an “infinite continuous group” of transformations is replaced by weak quasi-invariance under an “infinite continuous group [Formula: see text],” whose algebra is an involutive distribution of Lie-Bäcklund vector fields generating the Noether transformations. Its phase-space counterpart is the involutive distribution associated with a special function group Ḡpm, which contains a function subgroup Ḡp connected (when in canonical form) to the Shanmugadhasan canonical transformations. Also, the various possible first-order formalisms are analyzed.

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Sultanov, A. Ya, M. V. Glebova, and O. V. Bolotnikova. "Lie algebras of differentiations of linear algebras over a field." Differential Geometry of Manifolds of Figures, no. 52 (2021): 123–36. http://dx.doi.org/10.5922/0321-4796-2021-52-12.

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In this paper, we study a system of linear equations that define the Lie algebra of differentiations DerA of an arbitrary finite-dimensional linear algebra over a field. A system of equations is obtained, which is satisfied by the components of an arbitrary differentiation with respect to a fixed basis of algebra A. This system is a system of linear homogeneous equations. The law of transformation of the matrix of this system is proved. The invariance of the rank of the matrix of this system in the transition to a new basis in algebra is proved. Next, we consider the possibility of applying the obtained results in differential geometry when estimating the dimensions of groups of affine transformations from above. As an example, the method of I. P. Egorov is given for studying the dimensions of Lie algebras of affine vector fields on smooth manifolds equipped with linear connections having non-zero torsion tensor fields.

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Cieśliński, Jan L., and Dzianis Zhalukevich. "Spectral Parameter as a Group Parameter." Symmetry 14, no. 12 (December 6, 2022): 2577. http://dx.doi.org/10.3390/sym14122577.

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A large class of integrable non-linear partial differential equations is characterized by the existence of the associated linear problem (in the case of two independent variables, known as a Lax pair) containing the so-called spectral parameter. In this paper, we present and discuss the conjecture that the spectral parameter can be interpreted as the parameter of some one-parameter groups of transformation, provided that it cannot be removed by any gauge transformation. If a non-parametric linear problem for a non-linear system is known (e.g., the Gauss–Weingarten equations as a linear problem for the Gauss–Codazzi equations in the geometry of submanifolds), then, by comparing both symmetry groups, we can find or indicate the integrable cases. We consider both conventional Lie point symmetries and the so-called extended Lie point symmetries, which are necessary in some cases. This paper is intended to be a review, but some novel results are presented as well.

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Lenz, Reiner. "Lie methods for color robot vision." Robotica 26, no. 4 (July 2008): 453–64. http://dx.doi.org/10.1017/s0263574707003906.

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SUMMARYWe describe how Lie-theoretical methods can be used to analyze color related problems in machine vision. The basic observation is that the nonnegative nature of spectral color signals restricts these functions to be members of a limited, conical section of the larger Hilbert space of square-integrable functions. From this observation, we conclude that the space of color signals can be equipped with a coordinate system consisting of a half-axis and a unit ball with the Lorentz groups as natural transformation group. We introduce the theory of the Lorentz group SU(1, 1) as a natural tool for analyzing color image processing problems and derive some descriptions and algorithms that are useful in the investigation of dynamical color changes. We illustrate the usage of these results by describing how to compress, interpolate, extrapolate, and compensate image sequences generated by dynamical color changes.

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Wulfman, C. E., and H. Rabitz. "Global sensitivity analysis of nonlinear chemical kinetic equations using lie groups: II. Some chemical and mathematical properties of the transformation groups." Journal of Mathematical Chemistry 3, no. 3 (July 1989): 261–97. http://dx.doi.org/10.1007/bf01169596.

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Reihani, Kamran. "K-theory of Furstenberg Transformation Group C*-algebras." Canadian Journal of Mathematics 65, no. 6 (December 1, 2013): 1287–319. http://dx.doi.org/10.4153/cjm-2013-022-x.

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AbstractThis paper studies the K-theoretic invariants of the crossed product C*-algebras associated with an important family of homeomorphisms of the tori Tn called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given n, the K-groups of those crossed products whose corresponding n × n integer matrices are unipotent of maximal degree always have the same rank an. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these K-groups is false. Using the representation theory of the simple Lie algebra sl(2;C), we show that, remarkably, an has a combinatorial significance. For example, every a2n+1 is just the number of ways that 0 can be represented as a sum of integers between–n and n (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple explicit formula for the asymptotic behavior of the sequence {an} is given. Finally, we describe the order structure of the K0-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

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SASAKURA, NAOKI. "AN INVARIANT APPROACH TO DYNAMICAL FUZZY SPACES WITH A THREE-INDEX VARIABLE." Modern Physics Letters A 21, no. 13 (April 30, 2006): 1017–28. http://dx.doi.org/10.1142/s0217732306020329.

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A dynamical fuzzy space might be described by a three-index variable [Formula: see text], which determines the algebraic relations [Formula: see text] among the functions fa on the fuzzy space. A fuzzy analogue of the general coordinate transformation would be given by the general linear transformation on fa. We study equations for the three-index variable invariant under the general linear transformation and show that the solutions can be generally constructed from the invariant tensors of Lie groups. As specific examples, we study SO(3) symmetric solutions and discuss the construction of a scalar field theory on a fuzzy two-sphere within this framework.

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Kontogiorgis, Stavros, and Christodoulos Sophocleous. "On the simplification of the form of Lie transformation groups admitted by systems of evolution differential equations." Journal of Mathematical Analysis and Applications 449, no. 2 (May 2017): 1619–36. http://dx.doi.org/10.1016/j.jmaa.2016.12.084.

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