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Symmetry
Volume 5
Issue 1
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Symmetry, Volume 5, Issue 1 (March 2013) – 5 articles , Pages 1-118

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1066 KiB  
Article
Frame Transformation Relations and Symmetry Analysis of Fluxional Symmetric Rotor Dimers
by Horace T. Crogman, Bumgyunmiga Choi, Harrison B. Chen and William G. Harter
Symmetry 2013, 5(1), 86-118; https://doi.org/10.3390/sym5010086 - 5 Feb 2013
Cited by 2 | Viewed by 6082
Abstract
The theory of Frame transformation relations between the states of Born Oppenheimer and the weak coupling approximations is developed for polyatomic molecules. The symmetry relations are a generalization of the frame transformation relations derived by Harter and Crogman for coupled rotor molecules. A [...] Read more.
The theory of Frame transformation relations between the states of Born Oppenheimer and the weak coupling approximations is developed for polyatomic molecules. The symmetry relations are a generalization of the frame transformation relations derived by Harter and Crogman for coupled rotor molecules. A key internal symmetry label (named “soul”) is defined so that it remains a constant label for frame transformation relations, and is conserved during vibronic transitions, ionization, and even dissociation provided the nuclear spin-rotation interaction is relatively small. Simplified procedures are given for obtaining selection rules, statistical weights, and matrix elements of multipole operators for common molecules having various point symmetries. Full article
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Short Note
Perlman and Wellner’s Circular and Transformed Circular Copulas are Particular Beta and t Copulas
by M. C. Jones
Symmetry 2013, 5(1), 81-85; https://doi.org/10.3390/sym5010081 - 31 Jan 2013
Cited by 1 | Viewed by 4265
Abstract
All but one of the copulas in a recent paper in Symmetry by Perlman and Wellner can be identified as particular members of either the beta or t families of elliptical copulas. Full article
2516 KiB  
Article
Non-Crystallographic Symmetry in Packing Spaces
by Valery G. Rau, Leonty A. Lomtev and Tamara F. Rau
Symmetry 2013, 5(1), 54-80; https://doi.org/10.3390/sym5010054 - 9 Jan 2013
Cited by 4 | Viewed by 7485
Abstract
In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings [...] Read more.
In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings by policubes (polyominoes) and forms a structure. Such an approach establishes the computer design of abstract groups of symmetry. Every finite discrete model of the real structure is an element of symmetry groups, including non-crystallographic ones. The set packing spaces of the same order N characterizes discrete deformation transformations of the structure. Full article
(This article belongs to the Special Issue Polyhedra)
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Short Note
A Note on Lower Bounds for Colourful Simplicial Depth
by Antoine Deza, Tamon Stephen and Feng Xie
Symmetry 2013, 5(1), 47-53; https://doi.org/10.3390/sym5010047 - 7 Jan 2013
Cited by 2 | Viewed by 4761
Abstract
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal [...] Read more.
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14. Full article
(This article belongs to the Special Issue Polyhedra)
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4194 KiB  
Article
Taylor–Socolar Hexagonal Tilings as Model Sets
by Jeong-Yup Lee and Robert V. Moody
Symmetry 2013, 5(1), 1-46; https://doi.org/10.3390/sym5010001 - 28 Dec 2012
Cited by 7 | Viewed by 8876
Abstract
The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model [...] Read more.
The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice Q in the plane. If XQ denotes the set of all Taylor–Socolar tilings with centers on Q, then XQ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of Q.The Q-adic completion Q of Q is a natural factor of XQ and the natural mapping XQ → Q is bijective except at a dense set of points of measure 0 in /Q. We show that XQ consists of three LI classes under translation. Two of these LI classes are very small, namely countable Q-orbits in XQ. The other is a minimal dynamical system, which maps surjectively to /Q and which is variously 2 : 1, 6 : 1, and 12 : 1 at the singular points. We further develop the formula of what determines the parity of the tiles of a tiling in terms of the coordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull XQ; more precisely the two hulls are mutually locally derivable. Full article
(This article belongs to the Special Issue Polyhedra)
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