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Enumeration of index 3 and 4 subgroups of hyperbolic triangle symmetry groups Ma. Louise Antonette N. De Las Peñas*, I, Rene P. FelixII and Ma. Carlota B. DecenaI I Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines II Institute of Mathematics, University of the Philippines, Diliman, Quezon City, Philippines Received April 13, 2008; accepted August 12, 2008 Hyperbolic symmetry groups / Index 3 and 4 subgroups of triangle groups / Color symmetry / Perfect colorings Abstract. This paper explores the area of crystallogra

Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation Yong Chena,b and Xiaorui Hub a Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China b Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo 315211, China Reprint requests to Y. C.; E-mail: Z. Naturforsch. 64a, 8 – 14 (2009); received April 7, 2008 / revised June 3, 2008 The classical symmetry method and the modified Clarkson and Kruskal (C-K) method are used to obtain the Lie symmetry group of a

Symmetrie-Gruppe eines geschlossenen Weltalls F. Vollendorf (Z. Naturforsch. 30 a, 1 5 1 0 - 1 5 1 5 [1975] ; eingegangen am 19. August 1975) Symmetry Group of a closed Universe The fourteen degrees of freedom of a free falling and rotating body are described by means of a non-compact simple 14-parameter Lie group G 1 4 . It is shown that this group can be interpreted as the symmetry group of a spherically closed space. Finally an outlook is made to the combination of an internal and an external symmetry group in a nontrivial way. 1. Einleitung „Jeder

Solutions Generated from the Symmetry Group of the (2 + 1)-Dimensional Sine-Gordon System Hong-Cai Maa and Sen Yue Loua,b a Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, P. R. China b Department of Physics, Ningbo University, Ningbo, 315211, P. R. China Reprint requests to Prof. S. Y. L.; E-mail: Z. Naturforsch. 60a, 229 – 236 (2005); received October 25, 2004 Applying a symmetry group theorem on a two-straight-line soliton, some types of new localized multiply curved line excitations including the plateau-basin type

The Quaternion Group as a Symmetry Group Vi Hart and Henry Segerman Introduction A symmetry of an object is a geometric transformation which leaves the object unchanged. So, for example, an object with 3- fold rotational symmetry has three symmetries: rotation by 120°, rotation by 240°, and the trivial symmetry, where we do nothing. The symmetries of an object naturally form a group under composition. Care must be taken to differentiate between the symmetry group of an object, consisting of geometric transformations that leave the object unchanged, and

5 The symmetry group of chemical elements 5.1 Description of the system of elements Dmitri Mendeleev was the first to suggest that properties of the atoms depend on their positions in the whole integral system they form. If atoms are arranged in the order of increasing atomic weights, their chemical properties change periodically and homol- ogous series of elements with similar properties can be identified. Thus, the first clas- sification of particles originated long before the beginning of atomic physics when the very concepts of the atom and the molecule

Zeilschrift für Kristallographie 158, 1 - 2 6 (1982) © by Akademische Verlagsgesellschaft 1982 Normalizer groups and automorphism groups of symmetry groups Martin Gubler Institute of Crystallography and Petrography, Swiss Federal Institute of Technology (ΕΤΗ), CH-8092 Zürich, Switzerland Received: September 9, 1980; in revised form: January 23, 1981 Symmetry groups / Normalizer groups j Automorphism groups Abstract. Normalizer groups and automorphism groups are used to discuss questions of equivalence arising in the theory of colour groups and

DEMONSTRATIO MATHEMATICA Vol. X No 3 - 4 1977 Jerzy Pionka REPRESENTATIONS OF THE FINITE GROUPS BY SYMMETRY GROUPS OF OPERATIONS Let j§Q be the group of a l l permutations of the set K n = L e t ( X j f ( x . | , . . . , x n ) ) be a universal a l g e - bra . We denote by S ( f ) the symmetry group of the operation f i . e . the group of a l l permutations f of the set K s a - tysfying in CL the ident i ty f ( x 1 l . . . f x Q ) = f ( x ^ 1 ^ , . . . ) (see [ 2 ] ) . In t h i s paper we show that i f P i s a subgroup of § n then there e x i s t s a f i

Symmetry groups of prime knots up to 10 crossings Kouzi Kodama and Makoto Sakuma Dedicated to Professor Fujitsugu Hosokawa on his 60th birthday Let Κ be a smooth oriented knot in S3. The symmetry group Sym(S3, K) of Κ is defined as the mapping class group of the pair (£3 , K)\ that is, Sym(S3, K) = 7ToDiff(S3,K). This group contains information on chirality and invertibility of K , and essentially controls the rigid symmetries of Κ (cf. [BiZ2, Theorem 2.1]). The purpose of this paper is to study the symmetry groups of prime knots up to 10 crossings

Chapter 15 Nonrigid molecular systems with continuous axial symmetry groups Nonrigid molecular systems with continuous axial groups appearing in a descrip- tion of the geometric symmetry of their internal dynamics are numerous enough. The HCN / HNC system in the ground electronic state is a well-known example. The iso- meric forms HCN and HNC corresponding to the local minima of the effective nu- clear interaction potential have a linear equilibrium configuration [52]. The symmetry of motions in the local minima is characterized by groups coinciding with the