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Symmetry, Volume 8, Issue 3 (March 2016) – 9 articles

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386 KiB  
Article
Analytical Solutions of Temporal Evolution of Populations in Optically-Pumped Atoms with Circularly Polarized Light
by Heung-Ryoul Noh
Symmetry 2016, 8(3), 17; https://doi.org/10.3390/sym8030017 - 19 Mar 2016
Cited by 3 | Viewed by 3725
Abstract
We present an analytical calculation of temporal evolution of populations for optically pumped atoms under the influence of weak, circularly polarized light. The differential equations for the populations of magnetic sublevels in the excited state, derived from rate equations, are expressed in the [...] Read more.
We present an analytical calculation of temporal evolution of populations for optically pumped atoms under the influence of weak, circularly polarized light. The differential equations for the populations of magnetic sublevels in the excited state, derived from rate equations, are expressed in the form of inhomogeneous second-order differential equations with constant coefficients. We present a general method of analytically solving these differential equations, and obtain explicit analytical forms of the populations of the ground state at the lowest order in the saturation parameter. The obtained populations can be used to calculate lineshapes in various laser spectroscopies, considering transit time relaxation. Full article
(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)
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228 KiB  
Article
Pseudospin Symmetry as a Bridge between Hadrons and Nuclei
by Joseph N. Ginocchio
Symmetry 2016, 8(3), 16; https://doi.org/10.3390/sym8030016 - 18 Mar 2016
Cited by 3 | Viewed by 3242
Abstract
Atomic nuclei exhibit approximate pseudospin symmetry. We review the arguments that this symmetry is a relativistic symmetry. The condition for this symmetry is that the sum of the vector and scalar potentials in the Dirac Hamiltonian is a constant. We give the generators [...] Read more.
Atomic nuclei exhibit approximate pseudospin symmetry. We review the arguments that this symmetry is a relativistic symmetry. The condition for this symmetry is that the sum of the vector and scalar potentials in the Dirac Hamiltonian is a constant. We give the generators of pseudospin symmetry. We review some of the predictions that follow from the insight that pseudospin symmetry has relativistic origins . We show that approximate pseudospin symmetry in nuclei predicts approximate spin symmetry in anti-nucleon scattering from nuclei. Since QCD sum rules predict that the sum of the scalar and vector potentials is small, we discuss the quark origins of pseudospin symmetry in nuclei and spin symmetry in hadrons. Full article
(This article belongs to the Special Issue Symmetry in Hadrons and Nuclei)
254 KiB  
Article
Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations
by Rutwig Campoamor-Stursberg
Symmetry 2016, 8(3), 15; https://doi.org/10.3390/sym8030015 - 17 Mar 2016
Cited by 5 | Viewed by 3694
Abstract
A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints [...] Read more.
A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems. Full article
(This article belongs to the Special Issue Symmetry and Integrability)
290 KiB  
Article
Status of X(1835) and pp¯ Interaction from Chiral Symmetry
by Yong-Feng Liu and Xian-Wei Kang
Symmetry 2016, 8(3), 14; https://doi.org/10.3390/sym8030014 - 17 Mar 2016
Cited by 5 | Viewed by 3920
Abstract
After the observation of the strong near-threshold enhancement of proton-antiproton mass spectrum in J / ψ γ p p ¯ decay, lots of theoretical investigations have been available such as new resonance, the final-state p p ¯ interaction (FSI), [...] Read more.
After the observation of the strong near-threshold enhancement of proton-antiproton mass spectrum in J / ψ γ p p ¯ decay, lots of theoretical investigations have been available such as new resonance, the final-state p p ¯ interaction (FSI), p p ¯ bound state (or baryonium), glueball, or other exotic (tetra-quark) states. Here, we provide a short review on the current status, especially on the pertinent discussions concerning its relation to p p ¯ interaction, for which the emphasis is put on the recently constructed chiral potential. Full article
(This article belongs to the Special Issue Symmetry in Hadrons and Nuclei)
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430 KiB  
Article
Dual Pairs of Holomorphic Representations of Lie Groups from a Vector-Coherent-State Perspective
by David J. Rowe and Joe Repka
Symmetry 2016, 8(3), 12; https://doi.org/10.3390/sym8030012 - 16 Mar 2016
Cited by 1 | Viewed by 3655
Abstract
It is shown that, for both compact and non-compact Lie groups, vector-coherent-state methods provide straightforward derivations of holomorphic representations on symmetric spaces. Complementary vector-coherent-state methods are introduced to derive pairs of holomorphic representations which are bi-orthogonal duals of each other with respect to [...] Read more.
It is shown that, for both compact and non-compact Lie groups, vector-coherent-state methods provide straightforward derivations of holomorphic representations on symmetric spaces. Complementary vector-coherent-state methods are introduced to derive pairs of holomorphic representations which are bi-orthogonal duals of each other with respect to a simple Bargmann inner product. It is then shown that the dual of a standard holomorphic representation has an integral expression for its inner product, with a Bargmann measure and a simply-defined kernel, which is not restricted to discrete-series representations. Dual pairs of holomorphic representations also provide practical ways to construct orthonormal bases for unitary irreps which bypass the need for evaluating the integral expressions for their inner products. This leads to practical algorithms for the application of holomorphic representations to model problems with dynamical symmetries in physics. Full article
(This article belongs to the Special Issue Symmetry and Duality)
7276 KiB  
Article
Bézier Triangles with G2 Continuity across Boundaries
by Chang-Ki Lee, Hae-Do Hwang and Seung-Hyun Yoon
Symmetry 2016, 8(3), 13; https://doi.org/10.3390/sym8030013 - 15 Mar 2016
Cited by 3 | Viewed by 6656
Abstract
PN (point-normal) triangles are cubic Bézier triangles which meet at their edges to surface a triangular mesh, but this only achieves G0 continuity. We define blending regions that span the edges shared by adjacent pairs of triangular domains and blend the corresponding [...] Read more.
PN (point-normal) triangles are cubic Bézier triangles which meet at their edges to surface a triangular mesh, but this only achieves G0 continuity. We define blending regions that span the edges shared by adjacent pairs of triangular domains and blend the corresponding Bézier triangles using a univariate blending function formulated in terms of barycentric coordinates. This produces G2 continuity across boundaries while preserving G1 continuity at vertices. The sharpness of the blends can be controlled locally by varying the extent of these blending regions. We demonstrate the effectiveness of our technique by showing several modeling examples. Full article
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333 KiB  
Article
A Combinatorial Approach to Time Asymmetry
by Martin Tamm
Symmetry 2016, 8(3), 11; https://doi.org/10.3390/sym8030011 - 15 Mar 2016
Cited by 6 | Viewed by 4523
Abstract
In this paper, simple models for the multiverse are analyzed. Each universe is viewed as a path in a graph, and by considering very general statistical assumptions, essentially originating from Boltzmann, we can make the set of all such paths into a finite [...] Read more.
In this paper, simple models for the multiverse are analyzed. Each universe is viewed as a path in a graph, and by considering very general statistical assumptions, essentially originating from Boltzmann, we can make the set of all such paths into a finite probability space. We can then also attempt to compute the probabilities for different kinds of behavior and in particular under certain conditions argue that an asymmetric behavior of the entropy should be much more probable than a symmetric one. This offers an explanation for the asymmetry of time as a broken symmetry in the multiverse. The focus here is on simple models which can be analyzed using methods from combinatorics. Although the computational difficulties rapidly become enormous when the size of the model grows, this still gives hints about how a full-scale model should behave. Full article
(This article belongs to the Special Issue Symmetry and Symmetry Breaking in Statistical Systems)
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359 KiB  
Article
Polarity Formation in Molecular Crystals as a Symmetry Breaking Effect
by Luigi Cannavacciuolo and Jürg Hulliger
Symmetry 2016, 8(3), 10; https://doi.org/10.3390/sym8030010 - 11 Mar 2016
Cited by 7 | Viewed by 3760
Abstract
The transition of molecular crystals into a polar state is modeled by a one-dimensional Ising Hamiltonian in multipole expansion and a suitable order parameter. Two symmetry breakings are necessary for the transition: the translational and the spin flip invariance—the former being broken by [...] Read more.
The transition of molecular crystals into a polar state is modeled by a one-dimensional Ising Hamiltonian in multipole expansion and a suitable order parameter. Two symmetry breakings are necessary for the transition: the translational and the spin flip invariance—the former being broken by geometric constraints, the latter by the interaction of the first non-zero multipole with the next order multipole. Two different behaviors of the thermal average of the order parameter as a function of position are found. The free energy per lattice site converges to a finite value in the thermodynamic limit showing the consistency of the model in a macroscopic representation. Full article
(This article belongs to the Special Issue Symmetry and Symmetry Breaking in Statistical Systems)
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1034 KiB  
Article
Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures
by Tyler Reese, Randy Paffenroth and Joseph D. Fehribach
Symmetry 2016, 8(3), 9; https://doi.org/10.3390/sym8030009 - 29 Feb 2016
Cited by 3 | Viewed by 5617
Abstract
We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing [...] Read more.
We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented by Fehribach, which can be defined independent of any matrix relations. In particular, we present simple cases in which vector graphs are guaranteed to be Kirchhoff or non-Kirchhoff. Next, we review Maxwell’s method of drawing reciprocal figures as he presented in 1864, using modern mathematical language. We then demonstrate cases in which R2-vector graphs defined from Maxwell reciprocals are “dual” Kirchhoff graphs. Given an example in which Maxwell’s theories are not sufficient to define vector graphs, we begin to explore other methods of developing dual Kirchhoff graphs. Full article
(This article belongs to the Special Issue Symmetry and Duality)
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