E-Mail Alert

Add your e-mail address to receive forthcoming issues of this journal:

Journal Browser

Journal Browser

Special Issue "Feature Papers: Symmetry Concepts and Applications"

Quicklinks

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 December 2009)

Special Issue Editor

Assistant Editor
Ms. Cathy Wang

MDPI Wuhan Office, No.6 Jingan Road, 5.5 Creative Industry Park, Floor 25th. 430064 Wuhan, Hubei Province, China
Fax: +86 27 8761 2588

Published Papers (42 papers)

View options order results:
result details:
Displaying articles 1-42
Export citation of selected articles as:

Research

Jump to: Review

Open AccessArticle Chemical Reasoning Based on an Invariance Property: Bond and Lone Pair Pictures in Quantum Structural Formulas
Symmetry 2010, 2(3), 1559-1590; doi:10.3390/sym2031559
Received: 21 December 2009 / Revised: 9 July 2010 / Accepted: 22 July 2010 / Published: 23 July 2010
PDF Full-text (2526 KB)
Abstract
Chemists use one set of orbitals when comparing to a structural formula, hybridized AOs or NBOs for example, and another for reasoning in terms of frontier orbitals, MOs usually. Chemical arguments can frequently be made in terms of energy and/or electron density [...] Read more.
Chemists use one set of orbitals when comparing to a structural formula, hybridized AOs or NBOs for example, and another for reasoning in terms of frontier orbitals, MOs usually. Chemical arguments can frequently be made in terms of energy and/or electron density without the consideration of orbitals at all. All orbital representations, orthogonal or not, within a given function space are related by linear transformation. Chemical arguments based on orbitals are really energy or electron density arguments; orbitals are linked to these observables through the use of operators. The Valency Interaction Formula, VIF, offers a system of chemical reasoning based on the invariance of observables from one orbital representation to another. VIF pictures have been defined as one-electron density and Hamiltonian operators. These pictures are classified in a chemically meaningful way by use of linear transformations applied to them in the form of two pictorial rules and the invariance of the number of doubly, singly, and unoccupied orbitals or bonding, nonbonding, and antibonding orbitals under these transformations. The compatibility of the VIF method with the bond pair – lone pair language of Lewis is demonstrated. Different electron lone pair representations are related by the pictorial rules and have stability understood in terms of Walsh’s rules. Symmetries of conjugated ring systems are related to their electronic state by simple mathematical formulas. Description of lone pairs in conjugated systems is based on the strength and sign of orbital interactions around the ring. Simple models for bonding in copper clusters are tested, and the bonding of O2 to Fe(II) in hemoglobin is described. Arguments made are supported by HF, B3LYP, and MP2 computations. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessArticle Orientational Sampling Schemes Based on Four Dimensional Polytopes
Symmetry 2010, 2(3), 1423-1449; doi:10.3390/sym2031423
Received: 4 February 2010 / Revised: 27 May 2010 / Accepted: 30 June 2010 / Published: 7 July 2010
Cited by 5 | PDF Full-text (654 KB)
Abstract
The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may [...] Read more.
The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may be used in the numerical computation of solid-state nuclear magnetic resonance spectra, and in spherical tensor analysis procedures. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Emergent Dynamics of Five-Colour QCD Due to Dimensional Frustration
Symmetry 2010, 2(3), 1322-1337; doi:10.3390/sym2031322
Received: 24 November 2009 / Revised: 20 May 2010 / Accepted: 25 June 2010 / Published: 1 July 2010
PDF Full-text (261 KB)
Abstract
The consequences for five-colour QCD of a novel symmetry-breaking mechanism, published in an earlier paper, are further explored. In addition to the emergence of QED and three-colour QCD, there is also a candidate for the Z0μ. The representation theory [...] Read more.
The consequences for five-colour QCD of a novel symmetry-breaking mechanism, published in an earlier paper, are further explored. In addition to the emergence of QED and three-colour QCD, there is also a candidate for the Z0μ. The representation theory of SU (N) is applied to the matter sector and yields the quark and electron charge ratios, and a mechanism for generating fermion particle masses. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Fluctuating Asymmetry in Flies, What Does It Mean?
Symmetry 2010, 2(2), 1099-1107; doi:10.3390/sym2021099
Received: 23 February 2010 / Revised: 11 May 2010 / Accepted: 27 May 2010 / Published: 4 June 2010
Cited by 1 | PDF Full-text (951 KB)
Abstract
The degree of departure from perfect symmetry in organisms, fluctuating asymmetry (FA), is seen in most populations of animals. It has particular impact on choice of mate which lies within the world of sexual selection. Here I consider a relatively little studied [...] Read more.
The degree of departure from perfect symmetry in organisms, fluctuating asymmetry (FA), is seen in most populations of animals. It has particular impact on choice of mate which lies within the world of sexual selection. Here I consider a relatively little studied aspect of sexual selection, i.e. the effect of FA on contests between males for mates, based not on display ornament but rather on agility seen in the mating systems of many insects. The model organism considered is the ubiquitous chironomid midge. In these flies, mating takes place in the air, so symmetry in the length of wings bears directly on a male’s aerobatic ability on which successful mating depends. The role of parasites and predators in creating and responding to FA in the host/prey midge is considered. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Magnetization Dynamics Symmetry in Spin Torque Induced Magnetization Switching
Symmetry 2010, 2(2), 999-1021; doi:10.3390/sym2020999
Received: 27 December 2009 / Revised: 28 April 2010 / Accepted: 30 April 2010 / Published: 7 May 2010
Cited by 1 | PDF Full-text (266 KB)
Abstract
Magnetization dynamics symmetry plays important roles in magnetization switching. Here we study magnetic field and spin torque induced magnetization switching. Spin moment transferring from polarized itinerant electrons to local magnetization provides a magnetization switching mechanism without using external magnetic field. Besides its [...] Read more.
Magnetization dynamics symmetry plays important roles in magnetization switching. Here we study magnetic field and spin torque induced magnetization switching. Spin moment transferring from polarized itinerant electrons to local magnetization provides a magnetization switching mechanism without using external magnetic field. Besides its importance in fundamental magnetization switching dynamics, spin torque magnetization switching has great application potential for future nanoscale magnetoelectronic devices. The paper explores magnetization dynamics symmetry effects on spin torque induced magnetization switching, and its interactions with random fluctuations. We will illustrate the consequences of magnetization dynamics symmetry on the critical switching current magnitude and the thermal stability energy of spin torque induced magnetization switching, which are the two most important design criteria for nanoscale spin torque magnetic devices. The concept of Logarithmic magnetization susceptibility is used to extract symmetry and damping information on spin torque induced nonlinear magnetization dynamic processes, and provides paths to control spin torque induced switching in a fluctuating environment. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation
Symmetry 2010, 2(2), 868-883; doi:10.3390/sym2020868
Received: 21 December 2009 / Revised: 30 March 2010 / Accepted: 15 April 2010 / Published: 19 April 2010
PDF Full-text (308 KB)
Abstract
Invariant numerical schemes possess properties that may overcome the numerical properties of most of classical schemes. When they are constructed with moving frames, invariant schemes can present more stability and accuracy. The cornerstone is to select relevant moving frames. We present a [...] Read more.
Invariant numerical schemes possess properties that may overcome the numerical properties of most of classical schemes. When they are constructed with moving frames, invariant schemes can present more stability and accuracy. The cornerstone is to select relevant moving frames. We present a new algorithmic process to do this. The construction of invariant schemes consists in parametrizing the scheme with constant coefficients. These coefficients are determined in order to satisfy a fixed order of accuracy and an equivariance condition. Numerical applications with the Burgers equation illustrate the high performances of the process. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling
Symmetry 2010, 2(2), 848-867; doi:10.3390/sym2020848
Received: 10 December 2009 / Revised: 18 March 2010 / Accepted: 12 April 2010 / Published: 16 April 2010
PDF Full-text (236 KB)
Abstract
In this work, the non-isothermal Navier–Stokes equations are studied from the group theory point of view. The symmetry group of the equations is presented and discussed. Some standard turbulence models are analyzed with the symmetries of the equations. A class of turbulence [...] Read more.
In this work, the non-isothermal Navier–Stokes equations are studied from the group theory point of view. The symmetry group of the equations is presented and discussed. Some standard turbulence models are analyzed with the symmetries of the equations. A class of turbulence models which preserve the physical properties contained in the symmetry group is built. The proposed turbulence models are applied to an illustrative example of natural convection in a differentially heated cavity, and the results are presented. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Replication and Abstraction: Symmetry in Automated Formal Verification
Symmetry 2010, 2(2), 799-847; doi:10.3390/sym2020799
Received: 27 December 2009 / Revised: 3 March 2010 / Accepted: 4 March 2010 / Published: 14 April 2010
Cited by 10 | PDF Full-text (678 KB)
Abstract
This article surveys fundamental and applied aspects of symmetry in system models, and of symmetry reduction methods used to counter state explosion in model checking, an automated formal verification technique. While covering the research field broadly, we particularly emphasize recent progress in [...] Read more.
This article surveys fundamental and applied aspects of symmetry in system models, and of symmetry reduction methods used to counter state explosion in model checking, an automated formal verification technique. While covering the research field broadly, we particularly emphasize recent progress in applying the technique to realistic systems, including tools that promise to elevate the scope of symmetry reduction to large-scale program verification. The article targets researchers and engineers interested in formal verification of concurrent systems. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Application of Symmetry Methods to Low-Dimensional Heisenberg Magnets
Symmetry 2010, 2(2), 722-766; doi:10.3390/sym2020722
Received: 24 December 2009 / Revised: 15 February 2010 / Accepted: 11 March 2010 / Published: 9 April 2010
PDF Full-text (754 KB)
Abstract
An account of symmetry is very fruitful in studies of quantum spin systems. In the present paper we demonstrate how to use the spin SU(2) and the point symmetries in optimization of the theoretical condensed matter tools: the exact diagonalization, the renormalization [...] Read more.
An account of symmetry is very fruitful in studies of quantum spin systems. In the present paper we demonstrate how to use the spin SU(2) and the point symmetries in optimization of the theoretical condensed matter tools: the exact diagonalization, the renormalization group approach, the cluster perturbation theory. We apply the methods for study of Bose-Einstein condensation in dimerized antiferromagnets, for investigations of magnetization processes and magnetocaloric effect in quantum ferrimagnetic chain. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Symmetry and Asymmetry Level Measures
Symmetry 2010, 2(2), 707-721; doi:10.3390/sym2020707
Received: 23 November 2009 / Revised: 30 March 2010 / Accepted: 6 April 2010 / Published: 8 April 2010
Cited by 2 | PDF Full-text (260 KB)
Abstract
Usually, Symmetry and Asymmetry are considered as two opposite sides of a coin: an object is either totally symmetric, or totally asymmetric, relative to pattern objects. Intermediate situations of partial symmetry or partial asymmetry are not considered. But this dichotomy on the [...] Read more.
Usually, Symmetry and Asymmetry are considered as two opposite sides of a coin: an object is either totally symmetric, or totally asymmetric, relative to pattern objects. Intermediate situations of partial symmetry or partial asymmetry are not considered. But this dichotomy on the classification lacks of a necessary and realistic gradation. For this reason, it is convenient to introduce "shade regions", modulating the degree of Symmetry (a fuzzy concept). Here, we will analyze the Asymmetry problem by successive attempts of description and by the introduction of the Asymmetry Level Function, as a new Normal Fuzzy Measure. Our results (both Theorems and Corollaries) suppose to be some new and original contributions to such very active and interesting field of research. Previously, we proceed to the analysis of the state of art. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Symmetry as an Intrinsically Dynamic Feature
Symmetry 2010, 2(2), 554-581; doi:10.3390/sym2020554
Received: 4 March 2010 / Revised: 23 March 2010 / Accepted: 29 March 2010 / Published: 1 April 2010
Cited by 3 | PDF Full-text (1977 KB)
Abstract
Symmetry is one of the most prominent spatial relations perceived by humans, and has a relevant role in attentive mechanisms regarding both visual and auditory systems. The aim of this paper is to establish symmetry, among the likes of motion, depth or [...] Read more.
Symmetry is one of the most prominent spatial relations perceived by humans, and has a relevant role in attentive mechanisms regarding both visual and auditory systems. The aim of this paper is to establish symmetry, among the likes of motion, depth or range, as a dynamic feature in artificial vision. This is achieved in the first instance by assessing symmetry estimation by means of algorithms, putting emphasis on erosion and multi-resolution approaches, and confronting two ensuing problems: the isolation of objects from the context, and the pertinence (or lack thereof) of some salient points, such as the centre of mass. Next a geometric model is illustrated and detailed, and the problem of measuring symmetry in a world where symmetry is not perfect nor the only attention trigger is tackled. Two algorithmic lines, based on the so-called symmetry kernel and its evolution with pattern warping, and by correlation of blocks with varying sizes and positions, are proposed and investigated. An extended illustration of the power of symmetry as a feature, based on face expression recognition, concludes the paper. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Inverse Supersymmetry Breaking in S1 × R3
Symmetry 2010, 2(1), 366-387; doi:10.3390/sym2010366
Received: 5 October 2009 / Revised: 27 December 2009 / Accepted: 21 January 2010 / Published: 19 March 2010
PDF Full-text (273 KB)
Abstract
In this paper, we study the influence of hard supersymmetry breaking terms in a N = 1, d = 4 supersymmetric model, in S1 × R3 spacetime topology. It is shown that when the radius of the compact dimension [...] Read more.
In this paper, we study the influence of hard supersymmetry breaking terms in a N = 1, d = 4 supersymmetric model, in S1 × R3 spacetime topology. It is shown that when the radius of the compact dimension is large supersymmetry is unbroken, and dynamically breaks as the radius decreases. We point out that this resembles the inverse symmetry breaking of continuous symmetries at finite temperature (however, in the case of supersymmetry, the role of the temperature is played by the compact dimension’s radius). Furthermore, we also find a universality in the dependence of the critical length Lc as a function of a coupling g3, after comparing all cases. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Doubly-Special Relativity: Facts, Myths and Some Key Open Issues
Symmetry 2010, 2(1), 230-271; doi:10.3390/sym2010230
Received: 26 January 2010 / Accepted: 3 March 2010 / Published: 8 March 2010
Cited by 72 | PDF Full-text (411 KB)
Abstract
I report, emphasizing some key open issues and some aspects that are particularly relevant for phenomenology, on the status of the development of “doubly-special” relativistic (“DSR”) theories with both an observer-independent high-velocity scale and an observer-independent small-length/large-momentum scale, possibly relevant for the [...] Read more.
I report, emphasizing some key open issues and some aspects that are particularly relevant for phenomenology, on the status of the development of “doubly-special” relativistic (“DSR”) theories with both an observer-independent high-velocity scale and an observer-independent small-length/large-momentum scale, possibly relevant for the Planck-scale/quantum-gravity realm. I also give a true/false characterization of the structure of these theories. In particular, I discuss a DSR scenario without modification of the energy-momentum dispersion relation and without the қ-Poincaré Hopf algebra, a scenario with deformed Poincaré symmetries which is not a DSR scenario, some scenarios with both an invariant length scale and an invariant velocity scale which are not DSR scenarios, and a DSR scenario in which it is easy to verify that some observable relativistic (but non-special-relativistic) features are insensitive to possible nonlinear redefinitions of symmetry generators. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Time-Symmetric Boundary Conditions and Quantum Foundations
Symmetry 2010, 2(1), 272-283; doi:10.3390/sym2010272
Received: 1 January 2010 / Accepted: 4 March 2010 / Published: 8 March 2010
PDF Full-text (110 KB)
Abstract
Despite the widely-held premise that initial boundary conditions (BCs) corresponding to measurements/interactions can fully specify a physical subsystem, a literal reading of Hamilton’s principle would imply that both initial and final BCs are required (or more generally, a BC on a closed [...] Read more.
Despite the widely-held premise that initial boundary conditions (BCs) corresponding to measurements/interactions can fully specify a physical subsystem, a literal reading of Hamilton’s principle would imply that both initial and final BCs are required (or more generally, a BC on a closed hypersurface in spacetime). Such a time-symmetric perspective of BCs, as applied to classical fields, leads to interesting parallels with quantum theory. This paper will map out some of the consequences of this counter-intuitive premise, as applied to covariant classical fields. The most notable result is the contextuality of fields constrained in this manner, naturally bypassing the usual arguments against so-called “realistic” interpretations of quantum phenomena. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Symmetry Analysis in Mechanistic Studies of Nucleophilic Substitution and β-Elimination Reactions
Symmetry 2010, 2(1), 201-212; doi:10.3390/sym2010201
Received: 31 January 2010 / Accepted: 20 February 2010 / Published: 25 February 2010
Cited by 2 | PDF Full-text (299 KB)
Abstract
A mechanistic study of the bimolecular nucleophilic substitution (SN2) reaction for halomethane CH3X (X = Cl, Br, or I) is approached by using symmetry principles and molecular orbital theory. The electrophilicity of the functionalized sp3–carbon is [...] Read more.
A mechanistic study of the bimolecular nucleophilic substitution (SN2) reaction for halomethane CH3X (X = Cl, Br, or I) is approached by using symmetry principles and molecular orbital theory. The electrophilicity of the functionalized sp3–carbon is attributable to a 2p-orbital-based antibonding MO along the C–X bond. This antibonding MO, upon accepting an electron pair from a nucleophile, gives rise to dissociation of the C–X bond and formation of a new Nuc–C bond. Correlations are made between the molecular orbitals of reactants (Nuc- and CH3X) and products (NucCH3 and X-). Similar symmetry analysis has been applied to mechanistic study of the bimolecular b-elimination (E2) reactions of haloalkanes. It well explains the necessity of an anti-coplanar arrangement of the Cα–X and Cβ–H bonds for an E2 reaction (anti-elimination). Having this structural arrangement, the bonding Cα–X (σC-X) and antibonding Cβ–H (σC-H*) orbitals become symmetry–match. They can partially overlap resulting in increase in electron density in σC-H*, which weakens and polarizes the Cβ–H bond making the β-H acidic. An E2 reaction can readily take place in the presence of a base. The applications of symmetry analysis to the SN2 and E2 reactions represent a new approach to studying organic mechanisms. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessArticle On the Physical Reasons for the Extension of Symmetry Groups in Molecular Spectroscopy
Symmetry 2010, 2(1), 213-229; doi:10.3390/sym2010213
Received: 27 November 2009 / Revised: 11 February 2010 / Accepted: 24 February 2010 / Published: 25 February 2010
PDF Full-text (171 KB)
Abstract
Several situations of general interest, in which the symmetry groups usually applied to spectroscopy problems need to be extended, are reviewed. It is emphasized that any symmetry group of geometrical operations to be used in Molecular Spectroscopy should be extended for completeness [...] Read more.
Several situations of general interest, in which the symmetry groups usually applied to spectroscopy problems need to be extended, are reviewed. It is emphasized that any symmetry group of geometrical operations to be used in Molecular Spectroscopy should be extended for completeness by considering the time reversal operator, as far as the Hamiltonian is invariant with respect to the inversion of the direction of motion. This can explain the degeneracy of pairs of vibrational and rotational states spanning the so-called separably degenerate irreducible representations, in symmetric tops of low symmetry, and Kramers degeneracy in odd electron molecules in the absence of magnetic fields. An extension with account of time reversal is also useful to determine relative phase conventions on vibration-rotation wavefunctions, which render all vibration-rotation matrix elements real. An extension of a molecular symmetry group may be required for molecules which can attain different geometries by large amplitude periodical motions, if such motions are hindered and are not completely free. Special cases involving the internal rotation are discussed in detail. It is observed that the symmetry classification of vibrational modes involving displacements normal to the internal rotation axis is not univocal, but can be done in several ways, which actually correspond to different conventions on the separation of vibration and internal rotation in the adopted basis functions. The symmetry species of the separate vibrational and torsional factors of these functions depend on the adopted convention. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Effective Supramolecular Chirogenesis in Ethane-Bridged Bis-Porphyrinoids
Symmetry 2010, 2(1), 184-200; doi:10.3390/sym2010184
Received: 19 November 2009 / Revised: 26 January 2010 / Accepted: 20 February 2010 / Published: 23 February 2010
Cited by 7 | PDF Full-text (312 KB)
Abstract
This feature article gives a general introduction to the phenomenon of supramolecular chirogenesis using the most representative examples of different chirogenic assemblies on the basis of ethane-bridged bis-porphyrinoids. Supramolecular chirogenesis is based upon a smart combination of supramolecular chemistry and chirality sciences [...] Read more.
This feature article gives a general introduction to the phenomenon of supramolecular chirogenesis using the most representative examples of different chirogenic assemblies on the basis of ethane-bridged bis-porphyrinoids. Supramolecular chirogenesis is based upon a smart combination of supramolecular chemistry and chirality sciences and deals with various aspects of asymmetry induction, transfer, amplification, and modulation. These chiral processes are governed by numerous noncovalent supramolecular forces thus allowing a judicious, mechanistic, and dynamic control by applying a variety of internal and external influencing factors. Currently, supramolecular chirogenesis is widely used in different fields of fundamental and applied branches of science and modern technology, touching on such important issues as origin of chirality on the Earth, asymmetry sensing, enantioselective catalysis, nonlinear optics, polymer and materials science, pharmacy and medicine, nanotechnology, molecular and supramolecular devices, chiral memory, absolute configuration determination, etc. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Symmetries and (Related) Recursion Operators of Linear Evolution Equations
Symmetry 2010, 2(1), 98-111; doi:10.3390/sym2010098
Received: 28 December 2009 / Revised: 26 January 2010 / Accepted: 29 January 2010 / Published: 5 February 2010
PDF Full-text (262 KB)
Abstract
Significant cases of time-evolution equations, the linear Schr¨odinger and the Fokker–Planck equation are considered. It is known that equations of this type can be transformed, in some cases, into a highly simplified form. The properties of these equations in their initial and [...] Read more.
Significant cases of time-evolution equations, the linear Schr¨odinger and the Fokker–Planck equation are considered. It is known that equations of this type can be transformed, in some cases, into a highly simplified form. The properties of these equations in their initial and their simplified form are compared, showing in particular that this transformation partially prevents a clear understanding and a full application of the (physically relevant) notion of the so-called step up/down operators. These operators are shown to be recursion operators, related to the Lie point symmetries of the equations, which are also carefully discussed. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Symmetry in Infancy: Analysis of Motor Development in Autism Spectrum Disorders
Symmetry 2009, 1(2), 215-225; doi:10.3390/sym1020215
Received: 27 October 2009 / Accepted: 5 December 2009 / Published: 10 December 2009
Cited by 6 | PDF Full-text (163 KB)
Abstract
Motor asymmetry, defined as the lack of symmetry in movements or postures, is often observed briefly in many typically developing children. However, if such asymmetry persists, it may be a sign of neurological disease. Recent studies have suggested that motor asymmetries may [...] Read more.
Motor asymmetry, defined as the lack of symmetry in movements or postures, is often observed briefly in many typically developing children. However, if such asymmetry persists, it may be a sign of neurological disease. Recent studies have suggested that motor asymmetries may be an early symptom of Autism Spectrum Disorders (ASD). ASD involve a range of social, cognitive, and behavioral problems, at different degrees of functioning, which are thought to be the final common pathway of multiple etiological mechanisms. Furthermore, early identification of ASD has been recognized as a critical aspect for treatment. Our study aims to analyze symmetry in the motor milestones of infants with ASD compared with typically developing infants (TD) or infants with other developmental delay (DD) during the first year of life. Our results highlight that there are different patterns of motor symmetry in the groups. In particular, infants with ASD scored significantly poorer (higher levels of asymmetry) then the TD and DD infants. We also identified two subgroups of infants with ASD, one with a typical level and the other with a lower level of motor functioning. Implications of the study for diagnosis and treatment are described. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities
Symmetry 2009, 1(2), 201-214; doi:10.3390/sym1020201
Received: 23 October 2009 / Accepted: 24 November 2009 / Published: 26 November 2009
Cited by 4 | PDF Full-text (250 KB)
Abstract
We consider a bilateral birth-death process having sigmoidal-type rates. A thorough discussion on its transient behaviour is given, which includes studying symmetry properties of the transition probabilities, finding conditions leading to their bimodality, determining mean and variance of the process, and analyzing [...] Read more.
We consider a bilateral birth-death process having sigmoidal-type rates. A thorough discussion on its transient behaviour is given, which includes studying symmetry properties of the transition probabilities, finding conditions leading to their bimodality, determining mean and variance of the process, and analyzing absorption problems in the presence of 1 or 2 boundaries. In particular, thanks to the symmetry properties we obtain the avoiding transition probabilities in the presence of a pair of absorbing boundaries, expressed as a series. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Tetraquark Spectroscopy: A Symmetry Analysis
Symmetry 2009, 1(2), 155-179; doi:10.3390/sym1020155
Received: 23 October 2009 / Accepted: 13 November 2009 / Published: 23 November 2009
Cited by 9 | PDF Full-text (271 KB)
Abstract We present a detailed analysis of the symmetry properties of a four-quark wave function and its solution by means of a variational approach for simple Hamiltonians. We discuss several examples in the light and heavy-light meson sector. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus
Symmetry 2009, 1(2), 145-152; doi:10.3390/sym1020145
Received: 17 July 2009 / Revised: 13 September 2009 / Accepted: 14 September 2009 / Published: 8 October 2009
PDF Full-text (168 KB)
Abstract
A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix d [...] Read more.
A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclidean graph of a carbon nanotorus. We prove that this group is a semidirect product of a dihedral group by a group of order 2. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle The Relationship Between Morphological Symmetry and Immune Response in Wild-Caught Adult Bush-Crickets
Symmetry 2009, 1(2), 106-114; doi:10.3390/sym1020106
Received: 11 August 2009 / Revised: 17 September 2009 / Accepted: 24 September 2009 / Published: 28 September 2009
Cited by 1 | PDF Full-text (98 KB)
Abstract
Despite interest in the relationship between fluctuating asymmetry (FA), immune response and ecological factors in insects, little data are available from wild populations. In this study we measured FA and immune response in 370 wild-caught male bush-crickets, Metrioptera roeseli, from 20 experimentally [...] Read more.
Despite interest in the relationship between fluctuating asymmetry (FA), immune response and ecological factors in insects, little data are available from wild populations. In this study we measured FA and immune response in 370 wild-caught male bush-crickets, Metrioptera roeseli, from 20 experimentally introduced populations in southern-central Sweden. Individuals with more-symmetric wings had a higher immune response as measured by the cellular encapsulation of a surgically-implanted nylon monofilament. However, we found no relationship between measures of FA in other organs (i.e. tibia and maxillary palp) and immune response, suggesting that this pattern may reflect differing selection pressures. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Nuclei, Primes and the Random Matrix Connection
Symmetry 2009, 1(1), 64-105; doi:10.3390/sym1010064
Received: 16 July 2009 / Revised: 3 September 2009 / Accepted: 18 September 2009 / Published: 20 September 2009
Cited by 13 | PDF Full-text (563 KB)
Abstract
In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The [...] Read more.
In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics – random matrix theory – provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle A Stochastic Poisson Structure
Symmetry 2009, 1(1), 55-63; doi:10.3390/sym1010055
Received: 2 July 2009 / Revised: 26 July 2009 / Accepted: 12 August 2009 / Published: 20 August 2009
PDF Full-text (149 KB)
Abstract We define a Poisson structure on the Nualart-Pardoux test algebra associated to the path space of a finite dimensional Lie algebra. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging
Symmetry 2009, 1(1), 10-20; doi:10.3390/sym1010010
Received: 10 July 2009 / Revised: 5 August 2009 / Accepted: 6 August 2009 / Published: 6 August 2009
Cited by 1 | PDF Full-text (190 KB)
Abstract
If the Hamiltonian in the time independent Schrödinger equation, HΨ = , is invariant under a group of symmetry transformations, the theory of group representations can help obtain the eigenvalues and eigenvectors of H. A finite group that is [...] Read more.
If the Hamiltonian in the time independent Schrödinger equation, HΨ = , is invariant under a group of symmetry transformations, the theory of group representations can help obtain the eigenvalues and eigenvectors of H. A finite group that is not a symmetry group of H is nevertheless a symmetry group of an operator Hsym projected from H by the process of symmetry averaging. In this case H = Hsym + HR where HR is the nonsymmetric remainder. Depending on the nature of the remainder, the solutions for the full operator may be obtained by perturbation theory. It is shown here that when H is represented as a matrix [H] over a basis symmetry adapted to the group, the reduced matrix elements of [Hsym] are simple averages of certain elements of [H], providing a substantial enhancement in computational efficiency. A series of examples are given for the smallest molecular graphs. The first is a two vertex graph corresponding to a heteronuclear diatomic molecule. The symmetrized component then corresponds to a homonuclear system. A three vertex system is symmetry averaged in the first case to Cs and in the second case to the nonabelian C3v. These examples illustrate key aspects of the symmetry-averaging process. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)

Review

Jump to: Research

Open AccessReview Behind the Looking-Glass: A Review on Human Symmetry Perception
Symmetry 2010, 2(3), 1510-1543; doi:10.3390/sym2031510
Received: 25 June 2010 / Revised: 13 July 2010 / Accepted: 19 July 2010 / Published: 22 July 2010
Cited by 38 | PDF Full-text (5565 KB)
Abstract
The human visual system is highly proficient in extracting bilateral symmetry from visual input. This paper reviews empirical and theoretical work on human symmetry perception with a focus on recent issues such as its neural underpinnings. Symmetry detection is shown to be [...] Read more.
The human visual system is highly proficient in extracting bilateral symmetry from visual input. This paper reviews empirical and theoretical work on human symmetry perception with a focus on recent issues such as its neural underpinnings. Symmetry detection is shown to be a versatile, ongoing visual process that interacts with other visual processes. Evidence seems to converge towards the idea that  symmetry detection is subserved by a preprocessing stage involving spatial filters followed by information integration across the visual field in higher-tier cortical areas. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessReview Symmetries of the Central Vestibular System: Forming Movements for Gravity and a Three-Dimensional World
Symmetry 2010, 2(3), 1544-1558; doi:10.3390/sym2031544
Received: 18 May 2010 / Accepted: 16 July 2010 / Published: 22 July 2010
Cited by 2 | PDF Full-text (403 KB)
Abstract
Intrinsic dynamics of the central vestibular system (CVS) appear to be at least partly determined by the symmetries of its connections. The CVS contributes to whole-body functions such as upright balance and maintenance of gaze direction. These functions coordinate disparate senses (visual, [...] Read more.
Intrinsic dynamics of the central vestibular system (CVS) appear to be at least partly determined by the symmetries of its connections. The CVS contributes to whole-body functions such as upright balance and maintenance of gaze direction. These functions coordinate disparate senses (visual, inertial, somatosensory, auditory) and body movements (leg, trunk, head/neck, eye). They are also unified by geometric conditions. Symmetry groups have been found to structure experimentally-recorded pathways of the central vestibular system. When related to geometric conditions in three-dimensional physical space, these symmetry groups make sense as a logical foundation for sensorimotor coordination. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessReview Symmetric Matrix Fields in the Finite Element Method
Symmetry 2010, 2(3), 1375-1389; doi:10.3390/sym2031375
Received: 1 April 2010 / Revised: 3 June 2010 / Accepted: 30 June 2010 / Published: 6 July 2010
Cited by 4 | PDF Full-text (148 KB)
Abstract
The theory of elasticity is used to predict the response of a material body subject to applied forces. In the linear theory, where the displacement is small, the stress tensor which measures the internal forces is the variable of primal importance. However [...] Read more.
The theory of elasticity is used to predict the response of a material body subject to applied forces. In the linear theory, where the displacement is small, the stress tensor which measures the internal forces is the variable of primal importance. However the symmetry of the stress tensor which expresses the conservation of angular momentum had been a challenge for finite element computations. We review in this paper approaches based on mixed finite element methods. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessReview Symmetry and Asymmetry in Bouncing Gaits
Symmetry 2010, 2(3), 1270-1321; doi:10.3390/sym2031270
Received: 16 December 2009 / Revised: 18 June 2010 / Accepted: 24 June 2010 / Published: 25 June 2010
Cited by 8 | PDF Full-text (1898 KB) | Supplementary Files
Abstract
In running, hopping and trotting gaits, the center of mass of the body oscillates each step below and above an equilibrium position where the vertical force on the ground equals body weight. In trotting and low speed human running, the average vertical [...] Read more.
In running, hopping and trotting gaits, the center of mass of the body oscillates each step below and above an equilibrium position where the vertical force on the ground equals body weight. In trotting and low speed human running, the average vertical acceleration of the center of mass during the lower part of the oscillation equals that of the upper part, the duration of the lower part equals that of the upper part and the step frequency equals the resonant frequency of the bouncing system: we define this as on-offground symmetric rebound. In hopping and high speed human running, the average vertical acceleration of the center of mass during the lower part of the oscillation exceeds that of the upper part, the duration of the upper part exceeds that of the lower part and the step frequency is lower than the resonant frequency of the bouncing system: we define this as on-off-ground asymmetric rebound. Here we examine the physical and physiological constraints resulting in this on-off-ground symmetry and asymmetry of the rebound. Furthermore, the average force exerted during the brake when the body decelerates downwards and forwards is greater than that exerted during the push when the body is reaccelerated upwards and forwards. This landing-takeoff asymmetry, which would be nil in the elastic rebound of the symmetric spring-mass model for running and hopping, suggests a less efficient elastic energy storage and recovery during the bouncing step. During hopping, running and trotting the landing-takeoff asymmetry and the mass-specific vertical stiffness are smaller in larger animals than in the smaller animals suggesting a more efficient rebound in larger animals. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessReview Minimum Phi-Divergence Estimators and Phi-Divergence Test Statistics in Contingency Tables with Symmetry Structure: An Overview
Symmetry 2010, 2(2), 1108-1120; doi:10.3390/sym2021108
Received: 12 March 2010 / Revised: 7 May 2010 / Accepted: 10 June 2010 / Published: 11 June 2010
PDF Full-text (144 KB)
Abstract
In the last years minimum phi-divergence estimators (MϕE) and phi-divergence test statistics (ϕTS) have been introduced as a very good alternative to classical likelihood ratio test and maximum likelihood estimator for different statistical problems. The main purpose of [...] Read more.
In the last years minimum phi-divergence estimators (MϕE) and phi-divergence test statistics (ϕTS) have been introduced as a very good alternative to classical likelihood ratio test and maximum likelihood estimator for different statistical problems. The main purpose of this paper is to present an overview of the main results presented until now in contingency tables with symmetry structure on the basis of (MϕE) and (ϕTS). Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessReview Symmetry in Boolean Satisfiability
Symmetry 2010, 2(2), 1121-1134; doi:10.3390/sym2021121
Received: 31 December 2009 / Revised: 14 May 2010 / Accepted: 9 June 2010 / Published: 11 June 2010
Cited by 2 | PDF Full-text (182 KB)
Abstract
This paper reviews recent approaches on how to accelerate Boolean Satisfiability (SAT) search by exploiting symmetries in the problem space. SAT search algorithms traverse an exponentially large search space looking for an assignment that satisfies a set of constraints. The presence of [...] Read more.
This paper reviews recent approaches on how to accelerate Boolean Satisfiability (SAT) search by exploiting symmetries in the problem space. SAT search algorithms traverse an exponentially large search space looking for an assignment that satisfies a set of constraints. The presence of symmetries in the search space induces equivalence classes on the set of truth assignments. The goal is to use symmetries to avoid traversing all assignments by constraining the search to visit a few representative assignments in each equivalence class. This can lead to a significant reduction in search runtime without affecting the completeness of the search. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessReview Symmetry OUT, Asymmetry IN
Symmetry 2010, 2(2), 1033-1054; doi:10.3390/sym2021033
Received: 15 March 2010 / Revised: 23 April 2010 / Accepted: 18 May 2010 / Published: 25 May 2010
Cited by 2 | PDF Full-text (2461 KB)
Abstract
The formation of a perfect vertebrate body plan poses many questions that thrill developmental biologists. Special attention has been given to the symmetric segmental patterning that allows the formation of the vertebrae and skeletal muscles. These segmented structures derive from bilaterally symmetric [...] Read more.
The formation of a perfect vertebrate body plan poses many questions that thrill developmental biologists. Special attention has been given to the symmetric segmental patterning that allows the formation of the vertebrae and skeletal muscles. These segmented structures derive from bilaterally symmetric units called somites, which are formed under the control of a segmentation clock. At the same time that these symmetric units are being formed, asymmetric signals are establishing laterality in nearby embryonic tissues, allowing the asymmetric placement of the internal organs. More recently, a “shield” that protects symmetric segmentation from the influence of laterality cues was uncovered. Here we review the mechanisms that control symmetric versus asymmetric development along the left-right axis among vertebrates. We also discuss the impact that these studies might have in the understanding of human congenital disorders characterized by congenital vertebral malformations and abnormal laterality phenotypes. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessReview Lie Symmetries of Differential Equations: Classical Results and Recent Contributions
Symmetry 2010, 2(2), 658-706; doi:10.3390/sym2020658
Received: 2 January 2010 / Accepted: 30 March 2010 / Published: 8 April 2010
Cited by 20 | PDF Full-text (393 KB)
Abstract
Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of [...] Read more.
Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conservation laws, to the construction of links between different differential equations that turn out to be equivalent. This paper reviews some well known results of Lie group analysis, as well as some recent contributions concerned with the transformation of differential equations to equivalent forms useful to investigate applied problems. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessReview Fluctuating Asymmetry and Steroid Hormones: A Review
Symmetry 2010, 2(2), 541-553; doi:10.3390/sym2020541
Received: 6 January 2010 / Revised: 12 March 2010 / Accepted: 24 March 2010 / Published: 1 April 2010
Cited by 1 | PDF Full-text (63 KB)
Abstract
Fluctuating asymmetry (FA) represents random, minor deviations from perfect symmetry in paired traits. Because the development of the left and right sides of a paired trait is presumably controlled by an identical set of genetic instructions, these small imperfections are considered to [...] Read more.
Fluctuating asymmetry (FA) represents random, minor deviations from perfect symmetry in paired traits. Because the development of the left and right sides of a paired trait is presumably controlled by an identical set of genetic instructions, these small imperfections are considered to reflect genetic and environmental perturbations experienced during ontogeny. The current paper aims to identify possible neuroendocrine mechanisms, namely the actions of steroid hormones that may impact the development of asymmetrical characters as a response to various stressors. In doing so, it provides a review of the published studies on the influences of glucocorticoids, androgens, and estrogens on FA and concomitant changes in other health and fitness indicators. It follows the premise that hormonal measures may provide direct, non-invasive indicators of how individuals cope with adverse life conditions, strengthening the associations between FA and health, fitness, and behavior. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessReview Diastereoselective Desymmetrization of Symmetric Dienes and its Synthetic Application
Symmetry 2010, 2(2), 437-454; doi:10.3390/sym2020437
Received: 30 January 2010 / Revised: 4 March 2010 / Accepted: 4 March 2010 / Published: 25 March 2010
Cited by 7 | PDF Full-text (180 KB)
Abstract
The desymmetrization of symmetric compounds is a useful approach to obtain chiral building blocks. Readily available precursors with a prochiral unit could be converted into complex molecules with multiple stereogenic centers in a single step. In this review, recent advances in the [...] Read more.
The desymmetrization of symmetric compounds is a useful approach to obtain chiral building blocks. Readily available precursors with a prochiral unit could be converted into complex molecules with multiple stereogenic centers in a single step. In this review, recent advances in the desymmetrization of symmetric dienes in the diastereotopic group differentiating reaction and its synthetic application are presented. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessReview Fluctuating Asymmetry: Methods, Theory, and Applications
Symmetry 2010, 2(2), 466-540; doi:10.3390/sym2020466
Received: 22 December 2009 / Revised: 27 February 2010 / Accepted: 15 March 2010 / Published: 25 March 2010
Cited by 82 | PDF Full-text (2500 KB)
Abstract
Fluctuating asymmetry consists of random deviations from perfect symmetry in populations of organisms. It is a measure of developmental noise, which reflects a population’s average state of adaptation and coadaptation. Moreover, it increases under both environmental and genetic stress, though responses are [...] Read more.
Fluctuating asymmetry consists of random deviations from perfect symmetry in populations of organisms. It is a measure of developmental noise, which reflects a population’s average state of adaptation and coadaptation. Moreover, it increases under both environmental and genetic stress, though responses are often inconsistent. Researchers base studies of fluctuating asymmetry upon deviations from bilateral, radial, rotational, dihedral, translational, helical, and fractal symmetries. Here, we review old and new methods of measuring fluctuating asymmetry, including measures of dispersion, landmark methods for shape asymmetry, and continuous symmetry measures. We also review the theory, developmental origins, and applications of fluctuating asymmetry, and attempt to explain conflicting results. In the process, we present examples from the literature, and from our own research at “Evolution Canyon” and elsewhere. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessReview Symmetry in Generating Functions
Symmetry 2010, 2(1), 346-365; doi:10.3390/sym2010345
Received: 23 February 2010 / Accepted: 16 March 2010 / Published: 19 March 2010
PDF Full-text (163 KB)
Abstract
Generating functions play important roles in theory of orthogonal polynomials. In particular, it is important to consider generating functions that have symmetry. This paper is a survey on generating functions that define unitary operators. First, classical generating functions that define unitary operators [...] Read more.
Generating functions play important roles in theory of orthogonal polynomials. In particular, it is important to consider generating functions that have symmetry. This paper is a survey on generating functions that define unitary operators. First, classical generating functions that define unitary operators are discussed. Next, group theoretical approach to generating functions that have unitarity are discussed. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessReview Nanotribology of Symmetric and Asymmetric Liquid Lubricants
Symmetry 2010, 2(1), 320-345; doi:10.3390/sym2010320
Received: 15 December 2009 / Revised: 8 February 2010 / Accepted: 20 February 2010 / Published: 15 March 2010
Cited by 3 | PDF Full-text (1113 KB)
Abstract
When liquid molecules are confined in a narrow gap between smooth surfaces, their dynamic properties are completely different from those of the bulk. The molecular motions are highly restricted and the system exhibits solid-like responses when sheared slowly. This solidification behavior is [...] Read more.
When liquid molecules are confined in a narrow gap between smooth surfaces, their dynamic properties are completely different from those of the bulk. The molecular motions are highly restricted and the system exhibits solid-like responses when sheared slowly. This solidification behavior is very dependent on the molecular geometry (shape) of liquids because the solidification is induced by the packing of molecules into ordered structures in confinement. This paper reviews the measurements of confined structures and friction of symmetric and asymmetric liquid lubricants using the surface forces apparatus. The results show subtle and complex friction mechanisms at the molecular scale. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessReview Symmetry, Optical Properties and Thermodynamics of Neptunium(V) Complexes
Symmetry 2010, 2(1), 1-14; doi:10.3390/sym2010001
Received: 25 November 2009 / Revised: 19 December 2009 / Accepted: 21 December 2009 / Published: 28 December 2009
Cited by 11 | PDF Full-text (2096 KB)
Abstract
Recent results on the optical absorption and symmetry of the Np(V) complexes with dicarboxylate and diamide ligands are reviewed. The importance of recognizing the “silent” feature of centrosymmetric Np(V) species in analyzing the absorption spectra and calculating the thermodynamic constants of Np(V) [...] Read more.
Recent results on the optical absorption and symmetry of the Np(V) complexes with dicarboxylate and diamide ligands are reviewed. The importance of recognizing the “silent” feature of centrosymmetric Np(V) species in analyzing the absorption spectra and calculating the thermodynamic constants of Np(V) complexes is emphasized. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Figures

Open AccessReview Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions
Symmetry 2009, 1(2), 115-144; doi:10.3390/sym1020115
Received: 3 September 2009 / Accepted: 2 October 2009 / Published: 6 October 2009
Cited by 2 | PDF Full-text (266 KB)
Abstract
We review recent results on how to extend the supersymmetry SUSY normalism in Quantum Mechanics to linear generalizations of the time-dependent Schrödinger equation in (1+1) dimensions. The class of equations we consider contains many known cases, such as the Schrödinger equation for [...] Read more.
We review recent results on how to extend the supersymmetry SUSY normalism in Quantum Mechanics to linear generalizations of the time-dependent Schrödinger equation in (1+1) dimensions. The class of equations we consider contains many known cases, such as the Schrödinger equation for position-dependent mass. By evaluating intertwining relations, we obtain explicit formulas for the interrelations between the supersymmetric partner potentials and their corresponding solutions. We review reality conditions for the partner potentials and show how our SUSY formalism can be extended to the Fokker-Planck and thenonhomogeneous Burgers equation. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessReview Symmetry-Break in Voronoi Tessellations
Symmetry 2009, 1(1), 21-54; doi:10.3390/sym1010021
Received: 4 July 2009 / Accepted: 6 August 2009 / Published: 20 August 2009
Cited by 6 | PDF Full-text (926 KB)
Abstract
We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)

Journal Contact

MDPI AG
Symmetry Editorial Office
St. Alban-Anlage 66, 4052 Basel, Switzerland
symmetry@mdpi.com
Tel. +41 61 683 77 34
Fax: +41 61 302 89 18
Editorial Board
Contact Details Submit to Symmetry
Back to Top