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Symmetry, Volume 4, Issue 3 (September 2012), Pages 336-565

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Research

Jump to: Review

Open AccessArticle Superspheres: Intermediate Shapes between Spheres and Polyhedra
Symmetry 2012, 4(3), 336-343; doi:10.3390/sym4030336
Received: 16 May 2012 / Revised: 20 June 2012 / Accepted: 25 June 2012 / Published: 3 July 2012
Cited by 4 | PDF Full-text (786 KB) | HTML Full-text | XML Full-text
Abstract
Using an x-y-z coordinate system, the equations of the superspheres have been extended to describe intermediate shapes between a sphere and various convex polyhedra. Near-polyhedral shapes composed of {100}, {111} and {110} surfaces with round edges are treated in the present study, [...] Read more.
Using an x-y-z coordinate system, the equations of the superspheres have been extended to describe intermediate shapes between a sphere and various convex polyhedra. Near-polyhedral shapes composed of {100}, {111} and {110} surfaces with round edges are treated in the present study, where {100}, {111} and {110} are the Miller indices of crystals with cubic structures. The three parameters p, a and b are included to describe the {100}-{111}-{110} near-polyhedral shapes, where p describes the degree to which the shape is a polyhedron and a and b determine the ratios of the {100}, {111} and {110} surfaces. Full article
(This article belongs to the Special Issue Polyhedra)
Open AccessArticle Particle-Dependent Deformations of Lorentz Symmetry
Symmetry 2012, 4(3), 344-378; doi:10.3390/sym4030344
Received: 25 May 2012 / Revised: 21 June 2012 / Accepted: 26 June 2012 / Published: 3 July 2012
Cited by 3 | PDF Full-text (447 KB)
Abstract
I report results suggesting that it is possible to introduce laws of relativistic kinematics endowing different types of particles with suitably different deformed-Lorentz-symmetry properties. I also consider some possible applications of these results, among which I highlight those relevant for addressing a [...] Read more.
I report results suggesting that it is possible to introduce laws of relativistic kinematics endowing different types of particles with suitably different deformed-Lorentz-symmetry properties. I also consider some possible applications of these results, among which I highlight those relevant for addressing a long-standing challenge in the description of composite particles, such as atoms, within quantum-gravity-inspired scenarios with Planck-scale deformations of Lorentz symmetry. Some of the new elements here introduced in the formulation of relativistic kinematics appear to also provide the starting point for the development of a correspondingly novel mathematical formulation of spacetime-symmetry algebras. Full article
Open AccessArticle Symmetry-Adapted Fourier Series for the Wallpaper Groups
Symmetry 2012, 4(3), 379-426; doi:10.3390/sym4030379
Received: 11 June 2012 / Revised: 27 June 2012 / Accepted: 5 July 2012 / Published: 17 July 2012
Cited by 1 | PDF Full-text (20907 KB)
Abstract
Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between [...] Read more.
Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between the Fourier coefficients. Visualising the equivalencies of Fourier coefficients by means of discrete 2D maps reveals how direct-space symmetry is transformed into coefficient-space symmetry. Explicit expressions are given for the Fourier series and Fourier coefficient maps of both real and complex functions, readily applicable to the description of the properties of 2D materials like graphene or boron-nitride. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
Figures

Open AccessArticle Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection
Symmetry 2012, 4(3), 427-440; doi:10.3390/sym4030427
Received: 18 June 2012 / Revised: 15 July 2012 / Accepted: 26 July 2012 / Published: 7 August 2012
Cited by 6 | PDF Full-text (267 KB) | HTML Full-text | XML Full-text
Abstract
In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, [...] Read more.
In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 x 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 x 3 subequations which are thus a supersymmetric link between fermionic and bosonicdegrees of freedom. Full article
(This article belongs to the Special Issue Supersymmetry)
Open AccessArticle Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations
Symmetry 2012, 4(3), 441-451; doi:10.3390/sym4030441
Received: 25 May 2012 / Accepted: 27 July 2012 / Published: 8 August 2012
Cited by 1 | PDF Full-text (11001 KB) | HTML Full-text | XML Full-text
Abstract
We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons [...] Read more.
We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called virtual solitons whose nonlinear interactions produce no phase shifts. Full article
(This article belongs to the Special Issue Supersymmetry)
Open AccessArticle Supersymmetric Sigma Model Geometry
Symmetry 2012, 4(3), 474-506; doi:10.3390/sym4030474
Received: 6 July 2012 / Revised: 23 July 2012 / Accepted: 2 August 2012 / Published: 23 August 2012
PDF Full-text (467 KB) | HTML Full-text | XML Full-text
Abstract
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitian symmetric spaces. [...] Read more.
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitian symmetric spaces. Full article
(This article belongs to the Special Issue Supersymmetry)
Open AccessArticle Flexibility of Hydrogen Bond and Lowering of Symmetry in Proton Conductor
Symmetry 2012, 4(3), 507-516; doi:10.3390/sym4030507
Received: 3 July 2012 / Revised: 31 July 2012 / Accepted: 10 August 2012 / Published: 23 August 2012
Cited by 1 | PDF Full-text (476 KB) | HTML Full-text | XML Full-text
Abstract
In order to investigate why crystal symmetry lowers with increasing temperature by phase transition of TII–III (=369 K) in Cs3H(SeO4)2, in spite of the fact that crystal symmetry in the high-temperature phase of many ionic [...] Read more.
In order to investigate why crystal symmetry lowers with increasing temperature by phase transition of TII–III (=369 K) in Cs3H(SeO4)2, in spite of the fact that crystal symmetry in the high-temperature phase of many ionic conductors becomes higher by the phase transition, we have studied the relation between the change in crystal symmetry and the appearance of proton motion. It was found from the analysis of domains based on crystal structure that the number of possible geometrical arrangement of hydrogen bond in phase II becomes two times larger than that in phase III, derived from the lowering of crystal symmetry with increasing temperature. These results indicate that the lowering of crystal symmetry in phase II appears by the increase of the number of geometrical arrangements and by the enhancement of the flexibility of hydrogen bond. Considering that the enhancement of the flexibility of hydrogen bond yields mobile proton in phase II, it is deduced that mobile proton in phase II appears in exchange for the lowering of crystal symmetry at II–III phase transition. Full article
Open AccessArticle A Higher Dimensional Description of the Structure of β-Mn
Symmetry 2012, 4(3), 537-544; doi:10.3390/sym4030537
Received: 3 July 2012 / Revised: 22 August 2012 / Accepted: 22 August 2012 / Published: 27 August 2012
Cited by 4 | PDF Full-text (306 KB) | HTML Full-text | XML Full-text
Abstract
The structure of β-Mn crystallizes in space group P4132. The pseudo 8-fold nature of the 41 axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually [...] Read more.
The structure of β-Mn crystallizes in space group P4132. The pseudo 8-fold nature of the 41 axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually perpendicular 8-fold axes cannot generate P4132 on projection to 3-d space and how this may instead be accomplished from a six-dimensional model. A procedure for generating the actual structure of β-Mn lifted to six-dimensional space is given. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
Open AccessArticle Barrel Pseudotilings
Symmetry 2012, 4(3), 545-565; doi:10.3390/sym4030545
Received: 29 June 2012 / Revised: 13 August 2012 / Accepted: 19 August 2012 / Published: 30 August 2012
PDF Full-text (355 KB) | HTML Full-text | XML Full-text
Abstract
This paper describes 4-valent tiling-like structures, called pseudotilings, composed of barrel tiles and apeirogonal pseudotiles in Euclidean 3-space. These (frequently face-to-face) pseudotilings naturally rise in columns above 3-valent plane tilings by convex polygons, such that each column is occupied by stacked congruent barrel [...] Read more.
This paper describes 4-valent tiling-like structures, called pseudotilings, composed of barrel tiles and apeirogonal pseudotiles in Euclidean 3-space. These (frequently face-to-face) pseudotilings naturally rise in columns above 3-valent plane tilings by convex polygons, such that each column is occupied by stacked congruent barrel tiles or congruent apeirogonal pseudotiles. No physical space is occupied by the apeirogonal pseudotiles. Many interesting pseudotilings arise from plane tilings with high symmetry. As combinatorial structures, these are abstract polytopes of rank 4 with both finite and infinite 2-faces and facets. Full article
(This article belongs to the Special Issue Polyhedra)

Review

Jump to: Research

Open AccessReview Supersymmetric Quantum Mechanics and Solvable Models
Symmetry 2012, 4(3), 452-473; doi:10.3390/sym4030452
Received: 29 June 2012 / Revised: 20 July 2012 / Accepted: 31 July 2012 / Published: 16 August 2012
Cited by 8 | PDF Full-text (507 KB) | HTML Full-text | XML Full-text
Abstract
We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is [...] Read more.
We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of ħ-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ħ. Full article
(This article belongs to the Special Issue Supersymmetry)
Open AccessReview Supersymmetric Extensions of Non-Relativistic Scaling Algebras
Symmetry 2012, 4(3), 517-536; doi:10.3390/sym4030517
Received: 3 July 2012 / Revised: 7 August 2012 / Accepted: 8 August 2012 / Published: 24 August 2012
Cited by 3 | PDF Full-text (549 KB) | HTML Full-text | XML Full-text
Abstract
An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general value of dynamical critical exponent z [...] Read more.
An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general value of dynamical critical exponent z plays an important role in constructing the gravity duals for non-relativistic field theories. Supersymmetric extensions of symmetry algebras including the anisotropic scaling are very helpful to consider holographic relations accurately. We give a short summary on the classification of superalgebras with the anisotropic scaling as subalgebras of the following Lie superalgebras, psu(2,2|4), osp(8|4) and osp (8*|4), which appear in the study of AdS/CFT in type IIB string and M theories. It contains supersymmetric extensions of Schrödinger algebra and Lifshitz algebra. Full article
(This article belongs to the Special Issue Supersymmetry)

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