Symmetry
http://www.mdpi.com/journal/symmetry
Latest open access articles published in Symmetry at http://www.mdpi.com/journal/symmetry<![CDATA[Symmetry, Vol. 6, Pages 896-908: Dynamical Response of Particles in Asymmetric Ratchet Potential]]>
http://www.mdpi.com/2073-8994/6/4/896
The dynamics and response of particles in an asymmetric potential of the ratchet type is considered. An approximate asymmetric ratchet potential, which allows us to apply an analytical approach to investigate the systems’ response, is proposed based on the power series interpolation scheme. The method of multiple scales (MMS) is then employed to obtain an approximate solution at the principal resonance. Comparison between results obtained from the approximated and exact potentials show excellent agreement. The frequency response curves (FRCs) for different forcing are presented. The numerically computed FRCs were compared with the theoretical result to show the extent of agreement. Furthermore, by using perturbation methods, the stable and unstable domains of the solution were obtained.Symmetry2014-10-2764Article10.3390/sym60408968969082073-89942014-10-27doi: 10.3390/sym6040896Usman MarteUchechukwu VincentAbdulahi NjahBiodun Badmus<![CDATA[Symmetry, Vol. 6, Pages 880-895: Supramolecular Chirality in Dynamic Coordination Chemistry]]>
http://www.mdpi.com/2073-8994/6/4/880
Labile metal complexes have a useful coordination bond; which is weaker than a covalent C–C bond and is reversibly and dynamically formed and dissociated. Such labile metal complexes also can be used to construct chiral shapes and offer dynamic conversion of chiral molecular shapes in response to external stimuli. This review provides recent examples of chirality induction and describes the dynamic conversion systems produced by chiral metal complexes including labile metal centers, most of which respond to external stimuli by exhibiting sophisticated conversion phenomena.Symmetry2014-10-2464Review10.3390/sym60408808808952073-89942014-10-24doi: 10.3390/sym6040880Hiroyuki Miyake<![CDATA[Symmetry, Vol. 6, Pages 851-879: Recent Advances in Microscopic Approaches to Nuclear Matter and Symmetry Energy]]>
http://www.mdpi.com/2073-8994/6/4/851
Nuclear matter is a convenient theoretical laboratory to test many-body theories. When neutron and proton densities are different, the isospin dependence of the nuclear force gives rise to the symmetry energy term in the equation of state. This quantity is a crucial mechanism in the formation of the neutron skin in nuclei, as well as in other systems and phenomena involved in the dynamics of neutrons and protons in neutron-rich systems, such as isospin-asymmetric heavy-ion collisions. In this article, we will review phenomenological facts about the symmetry energy and recent experimental efforts to constrain its density dependence and related quantities. We will then review our microscopic approach to the equation of state of symmetric and asymmetric nuclear matter and present a corresponding set of predictions. Our calculations utilize the Dirac–Brueckner–Hartree–Fock method and realistic meson-theoretic nucleon-nucleon potentials. Chiral perturbation theory is an alternative approach, based on a well-defined scheme, which allows one to develop nuclear forces at each order of the chiral expansion. We will present and discuss predictions based on chiral perturbation theory, where we employ consistent two- and three-body chiral interactions. Throughout the article, one of the focal points is the importance of pursuing ab initio methods towards a deeper understanding of the many-body system.Symmetry2014-10-2064Review10.3390/sym60408518518792073-89942014-10-20doi: 10.3390/sym6040851Francesca Sammarruca<![CDATA[Symmetry, Vol. 6, Pages 844-850: Polar Vector Property of the Stationary State of Condensed Molecular Matter]]>
http://www.mdpi.com/2073-8994/6/4/844
Crystalline phases undergoing 180\(^{\circ}\) orientational disorder of dipolar entities in the seed or at growing (hkl) faces will show a polar vector property described by \(\infty\) /mm symmetry. Seeds and crystals develop a bi-polar state (\(\infty\)/mm), where domains related by a mirror plane m allow for a \(\infty\) m symmetry in each domain. The polarity of domains is due to energetic favorable interactions at the object-to-nutrient interface. Such interactions are well reproduced by an Ising Hamiltonian. Two-dimensional Monte Carlo simulations performed for real molecules with full long-range interactions allow us to calculate the spatial distribution of the electrical polarization Pel. The investigation has been extended to liquid droplets made of dipolar entities by molecular dynamics simulations. We demonstrate the development of an m\(\bar{\infty}\) quasi bi-polar state leading to a charged surface.Symmetry2014-10-1364Article10.3390/sym60408448448502073-89942014-10-13doi: 10.3390/sym6040844Jürg HulligerLuigi CannavacciuoloMathias Rech<![CDATA[Symmetry, Vol. 6, Pages 781-843: Chlorophylls, Symmetry, Chirality, and Photosynthesis]]>
http://www.mdpi.com/2073-8994/6/3/781
Chlorophylls are a fundamental class of tetrapyrroles and function as the central reaction center, accessory and photoprotective pigments in photosynthesis. Their unique individual photochemical properties are a consequence of the tetrapyrrole macrocycle, the structural chemistry and coordination behavior of the phytochlorin system, and specific substituent pattern. They achieve their full potential in solar energy conversion by working in concert in highly complex, supramolecular structures such as the reaction centers and light-harvesting complexes of photobiology. The biochemical function of these structures depends on the controlled interplay of structural and functional principles of the apoprotein and pigment cofactors. Chlorophylls and bacteriochlorophylls are optically active molecules with several chiral centers, which are necessary for their natural biological function and the assembly of their supramolecular complexes. However, in many cases the exact role of chromophore stereochemistry in the biological context is unknown. This review gives an overview of chlorophyll research in terms of basic function, biosynthesis and their functional and structural role in photosynthesis. It highlights aspects of chirality and symmetry of chlorophylls to elicit further interest in their role in nature.Symmetry2014-09-1063Review10.3390/sym60307817818432073-89942014-09-10doi: 10.3390/sym6030781Mathias SengeAoife RyanKristie LetchfordStuart MacGowanTamara Mielke<![CDATA[Symmetry, Vol. 6, Pages 758-780: Twinning of Polymer Crystals Suppressed by Entropy]]>
http://www.mdpi.com/2073-8994/6/3/758
We propose an entropic argument as partial explanation of the observed scarcity of twinned structures in crystalline samples of synthetic organic polymeric materials. Polymeric molecules possess a much larger number of conformational degrees of freedom than low molecular weight substances. The preferred conformations of polymer chains in the bulk of a single crystal are often incompatible with the conformations imposed by the symmetry of a growth twin, both at the composition surfaces and in the twin axis. We calculate the differences in conformational entropy between chains in single crystals and chains in twinned crystals, and find that the reduction in chain conformational entropy in the twin is sufficient to make the single crystal the stable thermodynamic phase. The formation of cyclic twins in molecular dynamics simulations of chains of hard spheres must thus be attributed to kinetic factors. In more realistic polymers this entropic contribution to the free energy can be canceled or dominated by nonbonded and torsional energetics.Symmetry2014-09-0463Article10.3390/sym60307587587802073-89942014-09-04doi: 10.3390/sym6030758Nikos KarayiannisKaterina FoteinopoulouManuel Laso<![CDATA[Symmetry, Vol. 6, Pages 722-757: Recognition of Symmetric 3D Bodies]]>
http://www.mdpi.com/2073-8994/6/3/722
The paper deals with the recognition of symmetric three-dimensional (3D) bodies that can be rotated and translated. We provide a complete list of all existing combinations of rotation and reflection symmetries in 3D. We define 3D complex moments by means of spherical harmonics, and the influence of individual symmetry groups on complex moment values is studied. Each particular symmetry pre-defines certain moment values. These moments can no longer differentiate between two objects of the same symmetry, which decreases the recognition power of the feature set. They should not be included when constructing the invariants. Translation and rotation invariants up to the fourth order are presented and their performance is studied on both artificial and real data.Symmetry2014-09-0163Article10.3390/sym60307227227572073-89942014-09-01doi: 10.3390/sym6030722Tomáš SukJan Flusser<![CDATA[Symmetry, Vol. 6, Pages 704-721: Adhesive/Repulsive Codes in Vertebrate Forebrain Morphogenesis]]>
http://www.mdpi.com/2073-8994/6/3/704
The last fifteen years have seen the identification of some of the mechanisms involved in anterior neural plate specification, patterning, and morphogenesis, which constitute the first stages in the formation of the forebrain. These studies have provided us with a glimpse into the molecular mechanisms that drive the development of an embryonic structure, and have resulted in the realization that cell segregation in the anterior neural plate is essential for the accurate progression of forebrain morphogenesis. This review summarizes the latest advances in our understanding of mechanisms of cell segregation during forebrain development, with and emphasis on the impact of this process on the morphogenesis of one of the anterior neural plate derivatives, the eyes.Symmetry2014-08-1463Review10.3390/sym60307047047212073-89942014-08-14doi: 10.3390/sym6030704Florencia Cavodeassi<![CDATA[Symmetry, Vol. 6, Pages 677-703: Supramolecular Chirality: Solvent Chirality Transfer in Molecular Chemistry and Polymer Chemistry]]>
http://www.mdpi.com/2073-8994/6/3/677
Controlled mirror symmetry breaking arising from chemical and physical origin is currently one of the hottest issues in the field of supramolecular chirality. The dynamic twisting abilities of solvent molecules are often ignored and unknown, although the targeted molecules and polymers in a fluid solution are surrounded by solvent molecules. We should pay more attention to the facts that mostly all of the chemical and physical properties of these molecules and polymers in the ground and photoexcited states are significantly influenced by the surrounding solvent molecules with much conformational freedom through non-covalent supramolecular interactions between these substances and solvent molecules. This review highlights a series of studies that include: (i) historical background, covering chiral NaClO3 crystallization in the presence of d-sugars in the late 19th century; (ii) early solvent chirality effects for optically inactive chromophores/fluorophores in the 1960s–1980s; and (iii) the recent development of mirror symmetry breaking from the corresponding achiral or optically inactive molecules and polymers with the help of molecular chirality as the solvent use quantity.Symmetry2014-08-1363Review10.3390/sym60306776777032073-89942014-08-13doi: 10.3390/sym6030677Michiya Fujiki<![CDATA[Symmetry, Vol. 6, Pages 655-676: Lubricating and Waxy Esters. VI. Effect of Symmetry about Ester on Crystallization of Linear Monoester Isomers]]>
http://www.mdpi.com/2073-8994/6/3/655
The crystal structure development of jojoba-like esters incorporating either 1-decenoic acid and/or 1-decenol, namely octadec-9-enyl dec-9-enoate (JLE-281), and its isomer dec-9-enyl oleate (JLE-282) was investigated to reveal the effect of symmetry about the ester group on crystallization of aliphatic fatty monoesters. The phase transformation path was investigated with temperature-time resolved X-ray diffraction during stepped isothermal crystallization, and while cooling from the melt at a fixed rate. Startling differences in phase behavior were uncovered between the isomers. When stepped isothermals were used, selective extinctions occurred at a transition temperature for JLE-281 but not for JLE-282. The extinctions, which are due to dramatic changes in the electronic density of certain families of planes, indicate a phase transition attributed to a brusque rearrangement of the oxygen atoms in the crystal subcell. The phase transition did not occur when the JLEs were cooled continuously. The crucial role played by the position of the alkyl chain and its orientation relative to the easy rotation site of the C–O bond in the phase trajectories of the JLEs was particularly highlighted.Symmetry2014-08-0763Article10.3390/sym60306556556762073-89942014-08-07doi: 10.3390/sym6030655Laziz BouzidiShaojun LiSuresh Narine<![CDATA[Symmetry, Vol. 6, Pages 622-654: Symmetry Breaking in NMR Spectroscopy: The Elucidation of Hidden Molecular Rearrangement Processes]]>
http://www.mdpi.com/2073-8994/6/3/622
Variable-temperature NMR spectroscopy is probably the most convenient and sensitive technique to monitor changes in molecular structure in solution. Rearrangements that are rapid on the NMR time-scale exhibit simplified spectra, whereby non-equivalent nuclear environments yield time-averaged resonances. At lower temperatures, when the rate of exchange is sufficiently reduced, these degeneracies are split and the underlying “static” molecular symmetry, as seen by X-ray crystallography, becomes apparent. Frequently, however, such rearrangement processes are hidden, even when they become slow on the NMR time-scale, because the molecular point group remains unchanged. Judicious symmetry breaking, such as by substitution of a molecular fragment by a similar, but not identical moiety, or by the incorporation of potentially diastereotopic (chemically non-equivalent) nuclei, allows the elucidation of the kinetics and energetics of such processes. Examples are chosen that include a wide range of rotations, migrations and other rearrangements in organic, inorganic and organometallic chemistry.Symmetry2014-08-0463Review10.3390/sym60306226226542073-89942014-08-04doi: 10.3390/sym6030622Michael McGlinchey<![CDATA[Symmetry, Vol. 6, Pages 589-621: Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures]]>
http://www.mdpi.com/2073-8994/6/3/589
Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting restrictions can be expressed in terms of linear relations of the a2, b2 and a · b scalar products of the lattice basis vectors with rational coefficients. To characterize OD families and to check their validity, these lattice restrictions are expressed in the bases of different layers and combined. For a more familiar notation, they can be expressed in terms of the lattice parameters a, b and . Alternatively, the description of the lattice restrictions may be simplified by using centered lattices. The representation of the lattice restrictions in terms of scalar products is dependent on the chosen basis. A basis-independent classification of the lattice restrictions is outlined.Symmetry2014-07-2163Article10.3390/sym60305895896212073-89942014-07-21doi: 10.3390/sym6030589Berthold Stöger<![CDATA[Symmetry, Vol. 6, Pages 578-588: Coset Group Construction of Multidimensional Number Systems]]>
http://www.mdpi.com/2073-8994/6/3/578
Extensions of real numbers in more than two dimensions, in particular quaternions and octonions, are finding applications in physics due to the fact that they naturally capture symmetries of physical systems. However, in the conventional mathematical construction of complex and multicomplex numbers multiplication rules are postulated instead of being derived from a general principle. A more transparent and systematic approach is proposed here based on the concept of coset product from group theory. It is shown that extensions of real numbers in two or more dimensions follow naturally from the closure property of finite coset groups adding insight into the utility of multidimensional number systems in describing symmetries in nature.Symmetry2014-07-1163Article10.3390/sym60305785785882073-89942014-07-11doi: 10.3390/sym6030578Horia Petrache<![CDATA[Symmetry, Vol. 6, Pages 566-577: Gestalt Algebra—A Proposal for the Formalization of Gestalt Perception and Rendering]]>
http://www.mdpi.com/2073-8994/6/3/566
Gestalt Algebra gives a formal structure suitable for describing complex patterns in the image plain. This can be useful for recognizing hidden structure in images. The work at hand refers to the laws of perceptual psychology. A manifold called the Gestalt Domain is defined. Next to the position in 2D it also contains an orientation and a scale component. Algebraic operations on it are given for mirror symmetry as well as organization into rows. Additionally the Gestalt Domain contains an assessment component, and all the meaning of the operations implementing the Gestalt-laws is realized in the functions giving this component. The operation for mirror symmetry is binary, combining two parts into one aggregate as usual in standard algebra. The operation for organization into rows, however, combines n parts into an aggregate, where n may well be more than two. This is algebra in its more general sense. For recognition, primitives are extracted from digital raster images by Lowe’s Scale Invariant Feature Transform (SIFT). Lowe’s key-point descriptors can also be utilized. Experiments are reported with a set of images put forth for the Computer Vision and Pattern Recognition Workshops (CVPR) 2013 symmetry contest.Symmetry2014-07-0763Article10.3390/sym60305665665772073-89942014-07-07doi: 10.3390/sym6030566Eckart Michaelsen<![CDATA[Symmetry, Vol. 6, Pages 551-565: Autosolvation: Architecture and Selection of Chiral Conformers in Alkylcobalt Carbonyl Molecular Clocks]]>
http://www.mdpi.com/2073-8994/6/3/551
Autosolvation is an important factor in stabilizing the architecture of medium complicated molecules. It is a kind of “supramolecular force” acting in intramolecular manner, consisting of orbital-orbital interactions between polar groups, separated by more than one covalent bonds within the same molecule. This effect facilitates also the development of chiral conformations. Two typical alkylcobalt carbonyl type molecules are discussed here as examples of autosolvating intramolecular interactions, leading to dramatic selection of chiral conformers and indicating also to the limits of the effect. The conformers stabilized by autosolvation and their interconversion are excellent examples of a “molecular clockwork”. Operation mode of these molecular clockworks gives some insight into the intramolecular transfer of chiral information.Symmetry2014-07-0463Review10.3390/sym60305515515652073-89942014-07-04doi: 10.3390/sym6030551Róbert KurdiAttila TáborosiClaudia ZucchiGyula Pályi<![CDATA[Symmetry, Vol. 6, Pages 516-550: Symmetry Adapted Assur Decompositions]]>
http://www.mdpi.com/2073-8994/6/3/516
Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools.Symmetry2014-06-2763Article10.3390/sym60305165165502073-89942014-06-27doi: 10.3390/sym6030516Anthony NixonBernd SchulzeAdnan SljokaWalter Whiteley<![CDATA[Symmetry, Vol. 6, Pages 473-515: Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere]]>
http://www.mdpi.com/2073-8994/6/3/473
The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group Sp(2). It is shown that this oscillator system contains the essential features of Wigner’s little groups dictating the internal space-time symmetries of particles in the Lorentz-covariant world. The little groups are the subgroups of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. It is shown that the damping modes of the oscillator correspond to the little groups for massive and imaginary-mass particles respectively. When the system makes the transition from the oscillation to damping mode, it corresponds to the little group for massless particles. Rotations around the momentum leave the four-momentum invariant. This degree of freedom extends the Sp(2) symmetry to that of SL(2, c) corresponding to the Lorentz group applicable to the four-dimensional Minkowski space. The Poincaré sphere contains the SL(2, c) symmetry. In addition, it has a non-Lorentzian parameter allowing us to reduce the mass continuously to zero. It is thus possible to construct the little group for massless particles from that of the massive particle by reducing its mass to zero. Spin-1/2 particles and spin-1 particles are discussed in detail.Symmetry2014-06-2563Article10.3390/sym60304734735152073-89942014-06-25doi: 10.3390/sym6030473Sibel BaşkalYoung KimMarilyn Noz<![CDATA[Symmetry, Vol. 6, Pages 444-472: Chiral Liquid Crystals: Structures, Phases, Effects]]>
http://www.mdpi.com/2073-8994/6/2/444
The introduction of chirality, i.e., the lack of mirror symmetry, has a profound effect on liquid crystals, not only on the molecular scale but also on the supermolecular scale and phase. I review these effects, which are related to the formation of supermolecular helicity, the occurrence of novel thermodynamic phases, as well as electro-optic effects which can only be observed in chiral liquid crystalline materials. In particular, I will discuss the formation of helical superstructures in cholesteric, Twist Grain Boundary and ferroelectric phases. As examples for the occurrence of novel phases the Blue Phases and Twist Grain Boundary phases are introduced. Chirality related effects are demonstrated through the occurrence of ferroelectricity in both thermotropic as well as lyotropic liquid crystals. Lack of mirror symmetry is also discussed briefly for some biopolymers such as cellulose and DNA, together with its influence on liquid crystalline behavior.Symmetry2014-06-1662Review10.3390/sym60204444444722073-89942014-06-16doi: 10.3390/sym6020444Ingo Dierking<![CDATA[Symmetry, Vol. 6, Pages 427-443: Symmetry Detection in Visual Impairment: Behavioral Evidence and Neural Correlates]]>
http://www.mdpi.com/2073-8994/6/2/427
Bilateral symmetry is an extremely salient feature for the human visual system. An interesting issue is whether the perceptual salience of symmetry is rooted in normal visual development. In this review, we discuss empirical work on visual and tactile symmetry detection in normally sighted and visually impaired individuals. On the one hand, available evidence suggests that efficient visual symmetry detection may need normal binocular vision development. On the other hand, converging evidence suggests that symmetry can develop as a principle of haptic perceptual organization in individuals lacking visual experience. Certain features of visual symmetry detection, however, such as the higher salience of the patterns containing a vertical axis of symmetry, do not systematically apply to the haptic modality. The neural correlates (revealed with neuroimaging) associated with visual and haptic symmetry detection are also discussed.Symmetry2014-05-2662Review10.3390/sym60204274274432073-89942014-05-26doi: 10.3390/sym6020427Zaira CattaneoSilvia BonaCorinna BauerJuha SilvantoAndrew HerbertTomaso VecchiLotfi Merabet<![CDATA[Symmetry, Vol. 6, Pages 409-426: The Symmetric-Partitioning and Incremental-Relearning Classification and Back-Propagation-Network Tree Approach for Cycle Time Estimation in Wafer Fabrication]]>
http://www.mdpi.com/2073-8994/6/2/409
An innovative classification and back-propagation-network tree (CABPN tree) approach is proposed in this study to estimate the cycle time of a job in a wafer fabrication factory, which is one of the most important tasks in controlling the wafer fabrication factory. The CABPN tree approach is an extension from the traditional classification and regression tree (CART) approach. In CART, the cycle times of jobs of the same branch are estimated with the same value, which is far from accurate. To tackle this problem, the CABPN tree approach replaces the constant estimate with variant estimates. To this end, the cycle times of jobs of the same branch are estimated with a BPN, and may be different. In this way, the estimation accuracy can be improved. In addition, to determine the optimal location of the splitting point on a node, the symmetric partition with incremental re-learning (SP-IR) algorithm is proposed and illustrated with an example. The applicability of the CABPN tree approach is shown with a real case. The experimental results supported its effectiveness over several existing methods.Symmetry2014-05-2362Case Report10.3390/sym60204094094262073-89942014-05-23doi: 10.3390/sym6020409Toly Chen<![CDATA[Symmetry, Vol. 6, Pages 396-408: Invisibility and PT Symmetry: A Simple Geometrical Viewpoint]]>
http://www.mdpi.com/2073-8994/6/2/396
We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Möbius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.Symmetry2014-05-2262Article10.3390/sym60203963964082073-89942014-05-22doi: 10.3390/sym6020396Luis Sánchez-SotoJuan Monzón<![CDATA[Symmetry, Vol. 6, Pages 383-395: Hydrodynamic Helical Orientations of Nanofibers in a Vortex]]>
http://www.mdpi.com/2073-8994/6/2/383
In this review article, I report our recent studies on spectroscopic visualizations of macroscopic helical alignments of nanofibers in vortex flows. Our designed supramolecular nanofibers, formed through self-assemblies of dye molecules, helically align in torsional flows of a vortex generated by mechanical rotary stirring of the sample solutions. The nanofiber, formed through bundling of linear supramolecular polymers, aligns equally in right- and left-handed vortex flows. However, in contrast, a one-handedly twisted nanofiber, formed through helical bundling of the supramolecular polymers, shows unequal helical alignments in these torsional flows. When the helical handedness of the nanofiber matches that of the vortex flow, the nanofiber aligns more efficiently in the flowing fluid. Such phenomena are observed not only with the artificial helical supramolecular nanofibers but also with biological nanofibers such as double-stranded DNA.Symmetry2014-05-2062Review10.3390/sym60203833833952073-89942014-05-20doi: 10.3390/sym6020383Akihiko Tsuda<![CDATA[Symmetry, Vol. 6, Pages 368-382: Symmetry Perspectives on Some Auxetic Body-Bar Frameworks]]>
http://www.mdpi.com/2073-8994/6/2/368
Scalar mobility counting rules and their symmetry extensions are reviewed for finite frameworks and also for infinite periodic frameworks of the bar-and-joint, body-joint and body-bar types. A recently published symmetry criterion for the existence of equiauxetic character of an infinite framework is applied to two long known but apparently little studied hinged-hexagon frameworks, and is shown to detect auxetic behaviour in both. In contrast, for double-link frameworks based on triangular and square tessellations, other affine deformations can mix with the isotropic expansion mode.Symmetry2014-05-1562Article10.3390/sym60203683683822073-89942014-05-15doi: 10.3390/sym6020368Patrick FowlerSimon GuestTibor Tarnai<![CDATA[Symmetry, Vol. 6, Pages 345-367: Dynamic Processes in Prochiral Solvating Agents (pro-CSAs) Studied by NMR Spectroscopy]]>
http://www.mdpi.com/2073-8994/6/2/345
Several dynamic processes, including tautomerism and macrocyclic inversion, in 1H-NMR prochiral solvating agents (pro-CSAs) are investigated. Various features of pro-CSA, including modes of interaction for complex formation, stoichiometry, binding strength and temperature effects were compared for three representative pro-CSA molecules. Structural effects of conjugated tetrapyrrole pro-CSA on the mechanism of enantiomeric excess determination are also discussed. Detailed analysis of species (complexes) and dynamic processes occurring in solution and their 1H-NMR spectral manifestations at various temperatures is presented.Symmetry2014-05-1462Article10.3390/sym60203453453672073-89942014-05-14doi: 10.3390/sym6020345Jan LabutaShinsuke IshiharaKatsuhiko ArigaJonathan Hill<![CDATA[Symmetry, Vol. 6, Pages 329-344: Closed-Form Expressions for the Matrix Exponential]]>
http://www.mdpi.com/2073-8994/6/2/329
We discuss a method to obtain closed-form expressions of f(A), where f is an analytic function and A a square, diagonalizable matrix. The method exploits the Cayley–Hamilton theorem and has been previously reported using tools that are perhaps not sufficiently appealing to physicists. Here, we derive the results on which the method is based by using tools most commonly employed by physicists. We show the advantages of the method in comparison with standard approaches, especially when dealing with the exponential of low-dimensional matrices. In contrast to other approaches that require, e.g., solving differential equations, the present method only requires the construction of the inverse of the Vandermonde matrix. We show the advantages of the method by applying it to different cases, mostly restricting the calculational effort to the handling of two-by-two matrices.Symmetry2014-04-2962Article10.3390/sym60203293293442073-89942014-04-29doi: 10.3390/sym6020329F. De Zela<![CDATA[Symmetry, Vol. 6, Pages 308-328: The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks]]>
http://www.mdpi.com/2073-8994/6/2/308
A crystallographic bar-joint framework, C in Rd, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω (C), is a singleton. Moreover, the almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, ΦC(z), on the d-torus, Td, determined by a full rank translation symmetry group and an associated motif of joints and bars.Symmetry2014-04-2462Article10.3390/sym60203083083282073-89942014-04-24doi: 10.3390/sym6020308Ghada BadriDerek KitsonStephen Power<![CDATA[Symmetry, Vol. 6, Pages 295-307: Dynamical Relation between Quantum Squeezing and Entanglement in Coupled Harmonic Oscillator System]]>
http://www.mdpi.com/2073-8994/6/2/295
In this paper, we investigate into the numerical and analytical relationship between the dynamically generated quadrature squeezing and entanglement within a coupled harmonic oscillator system. The dynamical relation between these two quantum features is observed to vary monotically, such that an enhancement in entanglement is attained at a fixed squeezing for a larger coupling constant. Surprisingly, the maximum attainable values of these two quantum entities are found to consistently equal to the squeezing and entanglement of the system ground state. In addition, we demonstrate that the inclusion of a small anharmonic perturbation has the effect of modifying the squeezing versus entanglement relation into a nonunique form and also extending the maximum squeezing to a value beyond the system ground state.Symmetry2014-04-2362Article10.3390/sym60202952953072073-89942014-04-23doi: 10.3390/sym6020295Lock ChewNing Chung<![CDATA[Symmetry, Vol. 6, Pages 256-294: Supramolecular Chirality in Porphyrin Chemistry]]>
http://www.mdpi.com/2073-8994/6/2/256
Supramolecular chirality, being an intelligent combination of supramolecular chemistry and chiral science, plays a decisive role in the functioning of various natural assemblies and has attracted much attention from the scientific community, due to different applications in modern technologies, medicine, pharmacology, catalysis and biomimetic research. Porphyrin molecules are of particular interest to study this phenomenon owing to their unique spectral, physico-chemical and synthetic properties. This review highlights the most important types of chiral porphyrin structures by using the best-suited representative examples, which are frequently used in the area of supramolecular chirality.Symmetry2014-04-2362Review10.3390/sym60202562562942073-89942014-04-23doi: 10.3390/sym6020256Victor Borovkov<![CDATA[Symmetry, Vol. 6, Pages 234-255: Quantum Local Symmetry of the D-Dimensional Non-Linear Sigma Model: A Functional Approach]]>
http://www.mdpi.com/2073-8994/6/2/234
We summarize recent progress on the symmetric subtraction of the Non-Linear Sigma Model in D dimensions, based on the validity of a certain Local Functional Equation (LFE) encoding the invariance of the SU(2) Haar measure under local left transformations. The deformation of the classical non-linearly realized symmetry at the quantum level is analyzed by cohomological tools. It is shown that all the divergences of the one-particle irreducible (1-PI) amplitudes (both on-shell and off-shell) can be classified according to the solutions of the LFE. Applications to the non-linearly realized Yang-Mills theory and to the electroweak theory, which is directly relevant to the model-independent analysis of LHC data, are briefly addressed.Symmetry2014-04-1762Review10.3390/sym60202342342552073-89942014-04-17doi: 10.3390/sym6020234Andrea Quadri<![CDATA[Symmetry, Vol. 6, Pages 222-233: Domain Specificity in Human Symmetry Preferences: Symmetry is Most Pleasant When Looking at Human Faces]]>
http://www.mdpi.com/2073-8994/6/2/222
Visual symmetry has been found to be preferred to asymmetry in a variety of domains and across species. A number of theories propose to explain why symmetry is preferred. In this article, I compare a perceptual bias view, in which symmetry is preferred due to factors inherit to the visual system, and an evolutionary advantage view, in which symmetry is preferred due to selection pressures on partner preference. Preferences for symmetry in three stimulus types were determined by having symmetric and asymmetric versions of the same images rated for pleasantness: human female faces, macaque monkey faces, and abstract art. It was found that preferences for symmetry were strongest for human female faces and weakest for art. This finding builds on previous research suggesting that symmetry preferences for human faces are different from symmetry preferences in other domains and that simple perceptual bias explanations do not wholly explain human visual face symmetry preferences. While consistent with an evolutionary advantage view, these data are also potentially explainable via a perceptual bias view which accounts for experience of stimuli. The interplay between these two views is discussed in the context of the current study.Symmetry2014-04-1762Article10.3390/sym60202222222332073-89942014-04-17doi: 10.3390/sym6020222Anthony Little<![CDATA[Symmetry, Vol. 6, Pages 210-221: Metalloporphyrin Symmetry in Chiral Recognition and Enantioselective Catalysis]]>
http://www.mdpi.com/2073-8994/6/2/210
Symmetry plays a fundamental role in chiral recognition and enantioselective catalysis. Porphyrins possess a number of structural features that make them attractive for the stereocontrol of chiral recognition and metal-catalyzed asymmetric reactions. This article is a brief account of our studies on chiral recognition and enantioselective catalysis by optically active metalloporphyrins. Some of the studies on chiral recognition and asymmetric catalysis by metalloporphyrins performed by others have also been included when useful.Symmetry2014-04-1062Review10.3390/sym60202102102212073-89942014-04-10doi: 10.3390/sym6020210Gérard SimonneauxHassan SrourPaul MauxSoizic ChevanceDaniel Carrie<![CDATA[Symmetry, Vol. 6, Pages 189-209: Topological Symmetry Groups of Small Complete Graphs]]>
http://www.mdpi.com/2073-8994/6/2/189
Topological symmetry groups were originally introduced to study the symmetries of non-rigid molecules, but have since been used to study the symmetries of any graph embedded in R3. In this paper, we determine for each complete graph Kn with n ≤ 6, what groups can occur as topological symmetry groups or orientation preserving topological symmetry groups of some embedding of the graph in R3.Symmetry2014-04-0862Article10.3390/sym60201891892092073-89942014-04-08doi: 10.3390/sym6020189Dwayne ChambersErica Flapan<![CDATA[Symmetry, Vol. 6, Pages 171-188: Development of Symmetry Concepts for Aperiodic Crystals]]>
http://www.mdpi.com/2073-8994/6/2/171
An overview is given of the use of symmetry considerations for aperiodic crystals. Superspace groups were introduced in the seventies for the description of incommensurate modulated phases with one modulation vector. Later, these groups were also used for quasi-periodic crystals of arbitrary rank. Further extensions use time reversal and time translation operations on magnetic and electrodynamic systems. An alternative description of magnetic structures to that with symmetry groups, the Shubnikov groups, is using representations of space groups. The same can be done for aperiodic crystals. A discussion of the relation between the two approaches is given. Representations of space groups and superspace groups play a role in the study of physical properties. These, and generalizations of them, are discussed for aperiodic crystals. They are used, in particular, for the characterization of phase transitions between aperiodic crystal phases.Symmetry2014-03-3162Article10.3390/sym60201711711882073-89942014-03-31doi: 10.3390/sym6020171Ted Janssen<![CDATA[Symmetry, Vol. 6, Pages 164-170: Spacetime Metrics from Gauge Potentials]]>
http://www.mdpi.com/2073-8994/6/2/164
I present an approach to gravity in which the spacetime metric is constructed from a non-Abelian gauge potential with values in the Lie algebra of the group U(2) (or the Lie algebra of quaternions). If the curvature of this potential vanishes, the metric reduces to a canonical curved background form reminiscent of the Friedmann S3 cosmological metric.Symmetry2014-03-2762Article10.3390/sym60201641641702073-89942014-03-27doi: 10.3390/sym6020164Ettore Minguzzi<![CDATA[Symmetry, Vol. 6, Pages 148-163: Symmetry Aspects of Dislocation-Effected Crystal Properties: Material Strength Levels and X-ray Topographic Imaging]]>
http://www.mdpi.com/2073-8994/6/1/148
Several materials science type research topics are described in which advantageous use of crystal symmetry considerations has been helpful in ferreting the essential elements of dislocation behavior in determining material properties or for characterizing crystal/polycrystalline structural relationships; for example: (1) the mechanical strengthening produced by a symmetrical bicrystal grain boundary; (2) cleavage crack formation at the intersection within a crystal of symmetrical dislocation pile-ups; (3) symmetry aspects of anisotropic crystal indentation hardness measurements; (4) X-ray diffraction topography imaging of dislocation strains and subgrain boundary misorientations; and (5) point and space group aspects of twinning. Several applications are described in relation to the strengthening of grain boundaries in nanopolycrystals and of multiply-oriented crystal grains in polysilicon photovoltaic solar cell materials. A number of crystallographic aspects of the different topics are illustrated with a stereographic method of presentation.Symmetry2014-03-2061Review10.3390/sym60101481481632073-89942014-03-20doi: 10.3390/sym6010148Ronald Armstrong<![CDATA[Symmetry, Vol. 6, Pages 111-147: The Hunt for Supersymmetry at the Tevatron]]>
http://www.mdpi.com/2073-8994/6/1/111
During the Tevatron data-taking period from April 2001 to September 2011 (Run-II), several searches for supersymmetric particles were performed. The results from searches by the CDF and DØ collaborations are concisely reviewed. This includes results up to the summer conferences of 2013. Model-independent and model-dependent limits on new particle production are set, and interpretations in supersymmetric models are given. Several limits from the Large Electron Positron (LEP) era have been extended. Specific results are placed into the context of the Tevatron performance expectations and a few of the current results from searches at the Large Hadron Collider (LHC).Symmetry2014-03-1961Review10.3390/sym60101111111472073-89942014-03-19doi: 10.3390/sym6010111André Sopczak<![CDATA[Symmetry, Vol. 6, Pages 103-110: Pseudo Hermitian Interactions in the Dirac Equation]]>
http://www.mdpi.com/2073-8994/6/1/103
We consider a (2 + 1)-dimensional massless Dirac equation in the presence of complex vector potentials. It is shown that such vector potentials (leading to complex magnetic fields) can produce bound states, and the Dirac Hamiltonians are η-pseudo Hermitian. Some examples have been explicitly worked out.Symmetry2014-03-1761Article10.3390/sym60101031031102073-89942014-03-17doi: 10.3390/sym6010103Orlando PanellaPinaki Roy<![CDATA[Symmetry, Vol. 6, Pages 90-102: Using Symmetries (Beyond Geometric Symmetries) in Chemical Computations: Computing Parameters of Multiple Binding Sites]]>
http://www.mdpi.com/2073-8994/6/1/90
We show how transformation group ideas can be naturally used to generate efficient algorithms for scientific computations. The general approach is illustrated on the example of determining, from the experimental data, the dissociation constants related to multiple binding sites. We also explain how the general transformation group approach is related to the standard (backpropagation) neural networks; this relation justifies the potential universal applicability of the group-related approach.Symmetry2014-02-2561Article10.3390/sym6010090901022073-89942014-02-25doi: 10.3390/sym6010090Andres OrtizVladik Kreinovich<![CDATA[Symmetry, Vol. 6, Pages 89: Acknowledgement to Reviewers of Symmetry in 2013]]>
http://www.mdpi.com/2073-8994/6/1/89
The editors of Symmetry would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2013. [...]Symmetry2014-02-2461Editorial10.3390/sym601008989892073-89942014-02-24doi: 10.3390/sym6010089 Symmetry Editorial Office<![CDATA[Symmetry, Vol. 6, Pages 67-88: On General Off-Shell Representations of World Line (1D) Supersymmetry]]>
http://www.mdpi.com/2073-8994/6/1/67
Every finite-dimensional unitary representation of the N-extended world line supersymmetry without central charges may be obtained by a sequence of differential transformations from a direct sum of minimal Adinkras, simple supermultiplets that are identifiable with representations of the Clifford algebra. The data specifying this procedure is a sequence of subspaces of the direct sum of Adinkras, which then opens an avenue for the classification of the continuum of the so-constructed off-shell supermultiplets.Symmetry2014-02-0361Article10.3390/sym601006767882073-89942014-02-03doi: 10.3390/sym6010067Charles DoranTristan HübschKevin IgaGregory Landweber<![CDATA[Symmetry, Vol. 6, Pages 23-66: Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator]]>
http://www.mdpi.com/2073-8994/6/1/23
The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with Ζ2 x Ζ2 symmetry. They used symmetric bifurcation theory to predict the existence of four primary gaits and seven secondary gaits. We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. Rate models incorporate parameters corresponding to the strengths of connections in the CPG: positive for excitatory connections and negative for inhibitory ones. The three-dimensional space of connection strengths is partitioned into regions that correspond to the first local bifurcation from a fully symmetric equilibrium. The partition is polyhedral, and its symmetry group is that of a tetrahedron. It comprises two concentric tetrahedra, subdivided by various symmetry planes. The tetrahedral symmetry arises from the structure of the eigenvalues of the connection matrix, which is involved in, but not equal to, the Jacobian of the rate model at bifurcation points. Some of the results apply to rate equations on more general networks.Symmetry2014-01-0361Article10.3390/sym601002323662073-89942014-01-03doi: 10.3390/sym6010023Ian Stewart<![CDATA[Symmetry, Vol. 6, Pages 1-22: Peripheral Contour Grouping and Saccade Targeting: The Role of Mirror Symmetry]]>
http://www.mdpi.com/2073-8994/6/1/1
Integrating shape contours in the visual periphery is vital to our ability to locate objects and thus make targeted saccadic eye movements to efficiently explore our surroundings. We tested whether global shape symmetry facilitates peripheral contour integration and saccade targeting in three experiments, in which observers responded to a successful peripheral contour detection by making a saccade towards the target shape. The target contours were horizontally (Experiment 1) or vertically (Experiments 2 and 3) mirror symmetric. Observers responded by making a horizontal (Experiments 1 and 2) or vertical (Experiment 3) eye movement. Based on an analysis of the saccadic latency and accuracy, we conclude that the figure-ground cue of global mirror symmetry in the periphery has little effect on contour integration or on the speed and precision with which saccades are targeted towards objects. The role of mirror symmetry may be more apparent under natural viewing conditions with multiple objects competing for attention, where symmetric regions in the visual field can pre-attentively signal the presence of objects, and thus attract eye movements.Symmetry2014-01-0261Article10.3390/sym60100011222073-89942014-01-02doi: 10.3390/sym6010001Michaël SassiMaarten DemeyerJohan Wagemans<![CDATA[Symmetry, Vol. 5, Pages 344-354: Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points]]>
http://www.mdpi.com/2073-8994/5/4/344
Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the Brillouin zone. In this paper, we will illustrate this property by considering two-dimensional (2D)-hexagonal lattices characterized by a possible two-fold degenerate band at the K points with a linear dispersion (Dirac points). By combining scanning tunneling spectroscopy and angle-resolved photoemission, we study the electronic properties of a similar system: the Ag/Cu(111) interface reconstruction characterized by a hexagonal superlattice, and we show that the gap opening at the K points of the Brillouin zone of the reconstructed cell is due to the symmetry breaking of the wave vector group.Symmetry2013-12-0954Article10.3390/sym50403443443542073-89942013-12-09doi: 10.3390/sym5040344Guillaume VasseurYannick Fagot-RevuratBertrand KierrenMuriel SicotDaniel Malterre<![CDATA[Symmetry, Vol. 5, Pages 313-343: Interplay between Point-Group Symmetries and the Choice of the Bloch Basis in Multiband Models]]>
http://www.mdpi.com/2073-8994/5/4/313
We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis. We first give two analytically accessible examples. Then, we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches, such as the functional renormalization group.Symmetry2013-11-1154Article10.3390/sym50403133133432073-89942013-11-11doi: 10.3390/sym5040313Stefan MaierCarsten HonerkampQiang-Hua Wang<![CDATA[Symmetry, Vol. 5, Pages 287-312: Multiple Solutions to Implicit Symmetric Boundary Value Problems for Second Order Ordinary Differential Equations (ODEs): Equivariant Degree Approach]]>
http://www.mdpi.com/2073-8994/5/4/287
In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree theory, establishes the existence of multiple solutions together with a complete description of their symmetric properties. The abstract result is supported by a concrete example of an implicit system respecting D4-symmetries.Symmetry2013-11-0754Article10.3390/sym50402872873122073-89942013-11-07doi: 10.3390/sym5040287Zalman BalanovWieslaw KrawcewiczZhichao LiMylinh Nguyen<![CDATA[Symmetry, Vol. 5, Pages 271-286: A New Route to the Majorana Equation]]>
http://www.mdpi.com/2073-8994/5/4/271
In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby, use is made of the complex conjugation operator and Pauli spin matrices only. The eigenfunctions of the two-component complex Majorana equation are also calculated. The associated quantum fields are found to describe particles and antiparticles, which have opposite mean helicities and are not their own antiparticles, but correspond to two independent degrees of freedom. The four-component real Dirac equation in its Majorana representation is shown to be the natural outcome of the two-component complex Majorana equation. Both types of equations come in two forms, which correspond to the irreducible left- and right-chiral representations of the Lorentz group.Symmetry2013-09-2654Article10.3390/sym50402712712862073-89942013-09-26doi: 10.3390/sym5040271Eckart Marsch<![CDATA[Symmetry, Vol. 5, Pages 253-270: Supersymmetric Version of the Euler System and Its Invariant Solutions]]>
http://www.mdpi.com/2073-8994/5/3/253
In this paper, we formulate a supersymmetric extension of the Euler system of equations. We compute a superalgebra of Lie symmetries of the supersymmetric system. Next, we classify the one-dimensional subalgebras of this superalgebra into 49 equivalence conjugation classes. For some of the subalgebras, the invariants have a non-standard structure. For nine selected subalgebras, we use the symmetry reduction method to find invariants, orbits and reduced systems. Through the solutions of these reduced systems, we obtain solutions of the supersymmetric Euler system. The obtained solutions include bumps, kinks, multiple wave solutions and solutions expressed in terms of arbitrary functions.Symmetry2013-07-1253Article10.3390/sym50302532532702073-89942013-07-12doi: 10.3390/sym5030253A. GrundlandAlexander Hariton<![CDATA[Symmetry, Vol. 5, Pages 233-252: Symmetries Shared by the Poincaré Group and the Poincaré Sphere]]>
http://www.mdpi.com/2073-8994/5/3/233
Henri Poincaré formulated the mathematics of Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré sphere for polarization optics. It is shown that these two mathematical instruments can be derived from the two-by-two representations of the Lorentz group. Wigner’s little groups for internal space-time symmetries are studied in detail. While the particle mass is a Lorentz-invariant quantity, it is shown to be possible to address its variations in terms of the decoherence mechanism in polarization optics.Symmetry2013-06-2753Article10.3390/sym50302332332522073-89942013-06-27doi: 10.3390/sym5030233Young KimMarilyn Noz<![CDATA[Symmetry, Vol. 5, Pages 215-232: Fermi Surface Reconstruction due to Hidden Rotating Antiferromagnetism in N and P-Type High-TC Cuprates]]>
http://www.mdpi.com/2073-8994/5/2/215
The Fermi surface calculated within the rotating antiferromagnetism theory undergoes a topological change when doping changes from p-type to n-type, in qualitative agreement with experimental data for n-type cuprate Nd2−xCexCuO4 and p-type La2−xSrxCuO4. Also, the reconstruction of the Fermi surface, observed experimentally close to optimal doping in p-type cuprates, and slightly higher than optimal doping in the overdoped regime for this n-type high-TC cuprate, is well accounted for in this theory. This reconstruction is a consequence of the quantum criticality caused by the disappearance of rotating antiferromagnetism. The present results are in qualitative agreement with recently observed quantum oscillations in some high-TC cuprates. This paper presents new results about the application of the rotating antiferromagnetism theory to the study of the electronic structure for n-type materials.Symmetry2013-05-0752Article10.3390/sym50202152152322073-89942013-05-07doi: 10.3390/sym5020215Mohamed Azzouz<![CDATA[Symmetry, Vol. 5, Pages 119-214: Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry]]>
http://www.mdpi.com/2073-8994/5/2/119
Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “already known” topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking, but most typically manifest a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e., UOSp(N|2) and UOSp(N|4) SVBS states, whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with a superconducting property that interpolates various VBS states, depending on the value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate, regardless of the parity of bulk (super)spins. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states.Symmetry2013-04-2652Review10.3390/sym50201191192142073-89942013-04-26doi: 10.3390/sym5020119Kazuki HasebeKeisuke Totsuka<![CDATA[Symmetry, Vol. 5, Pages 86-118: Frame Transformation Relations and Symmetry Analysis of Fluxional Symmetric Rotor Dimers]]>
http://www.mdpi.com/2073-8994/5/1/86
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.Symmetry2013-02-0551Article10.3390/sym5010086861182073-89942013-02-05doi: 10.3390/sym5010086Horace CrogmanBumgyunmiga ChoiHarrison ChenWilliam Harter<![CDATA[Symmetry, Vol. 5, Pages 81-85: Perlman and Wellner’s Circular and Transformed Circular Copulas are Particular Beta and t Copulas]]>
http://www.mdpi.com/2073-8994/5/1/81
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.Symmetry2013-01-3151Short Note10.3390/sym501008181852073-89942013-01-31doi: 10.3390/sym5010081M. C. Jones<![CDATA[Symmetry, Vol. 5, Pages 54-80: Non-Crystallographic Symmetry in Packing Spaces]]>
http://www.mdpi.com/2073-8994/5/1/54
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.Symmetry2013-01-0951Article10.3390/sym501005454802073-89942013-01-09doi: 10.3390/sym5010054Valery RauLeonty LomtevTamara Rau<![CDATA[Symmetry, Vol. 5, Pages 47-53: A Note on Lower Bounds for Colourful Simplicial Depth]]>
http://www.mdpi.com/2073-8994/5/1/47
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.Symmetry2013-01-0751Short Note10.3390/sym501004747532073-89942013-01-07doi: 10.3390/sym5010047Antoine DezaTamon StephenFeng Xie<![CDATA[Symmetry, Vol. 5, Pages 1-46: Taylor–Socolar Hexagonal Tilings as Model Sets]]>
http://www.mdpi.com/2073-8994/5/1/1
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.Symmetry2012-12-2851Article10.3390/sym50100011462073-89942012-12-28doi: 10.3390/sym5010001Jeong-Yup LeeRobert Moody<![CDATA[Symmetry, Vol. 4, Pages 667-685: Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries]]>
http://www.mdpi.com/2073-8994/4/4/667
The eigenfunction approach used for discrete symmetries is deduced from the concept of quantum numbers. We show that the irreducible representations (irreps) associated with the eigenfunctions are indeed a shorthand notation for the set of eigenvalues of the class operators (character table). The need of a canonical chain of groups to establish a complete set of commuting operators is emphasized. This analysis allows us to establish in natural form the connection between the quantum numbers and the eigenfunction method proposed by J.Q. Chen to obtain symmetry adapted functions. We then proceed to present a friendly version of the eigenfunction method to project functions.Symmetry2012-11-3044Article10.3390/sym40406676676852073-89942012-11-30doi: 10.3390/sym4040667Renato Lemus<![CDATA[Symmetry, Vol. 4, Pages 644-666: A Peculiarly Cerebroid Convex Zygo-Dodecahedron is an Axiomatically Balanced “House of Blues”: The Circle of Fifths to the Circle of Willis to Cadherin Cadenzas]]>
http://www.mdpi.com/2073-8994/4/4/644
A bilaterally symmetrical convex dodecahedron consisting of twelve quadrilateral faces is derived from the icosahedron via a process akin to Fuller’s Jitterbug Transformation. The unusual zygomorphic dodecahedron so obtained is shown to harbor a bilaterally symmetrical jazz/blues harmonic code on its twelve faces that is related to such fundamental music theoretical constructs as the Circle of Fifths and Euler’s tonnetz. Curiously, the patterning within the aforementioned zygo-dodecahedron is discernibly similar to that observed in a ventral view of the human brain. Moreover, this same pattern is arguably evident during development of the embryonic pharynx. A possible role for the featured zygo-dodecahedron in cephalogenesis is considered. Recent studies concerning type II cadherins, an important class of proteins that promote cell adhesion, have generated data that is demonstrated to conform to this zygo-dodecahedral brain model in a substantially congruous manner.Symmetry2012-11-1544Article10.3390/sym40406446446662073-89942012-11-15doi: 10.3390/sym4040644David Becker<![CDATA[Symmetry, Vol. 4, Pages 626-643: Dirac Matrices and Feynman’s Rest of the Universe]]>
http://www.mdpi.com/2073-8994/4/4/626
There are two sets of four-by-four matrices introduced by Dirac. The first set consists of fifteen Majorana matrices derivable from his four γ matrices. These fifteen matrices can also serve as the generators of the group SL(4, r). The second set consists of ten generators of the Sp(4) group which Dirac derived from two coupled harmonic oscillators. It is shown possible to extend the symmetry of Sp(4) to that of SL(4, r) if the area of the phase space of one of the oscillators is allowed to become smaller without a lower limit. While there are no restrictions on the size of phase space in classical mechanics, Feynman’s rest of the universe makes this Sp(4)-to-SL(4, r) transition possible. The ten generators are for the world where quantum mechanics is valid. The remaining five generators belong to the rest of the universe. It is noted that the groups SL(4, r) and Sp(4) are locally isomorphic to the Lorentz groups O(3, 3) and O(3, 2) respectively. This allows us to interpret Feynman’s rest of the universe in terms of space-time symmetry.Symmetry2012-10-3044Article10.3390/sym40406266266432073-89942012-10-30doi: 10.3390/sym4040626Young S. KimMarilyn E. Noz<![CDATA[Symmetry, Vol. 4, Pages 603-625: N = (4,4) Supersymmetry and T-Duality]]>
http://www.mdpi.com/2073-8994/4/4/603
A sigma model with four-dimensional target space parametrized by chiral and twisted chiral N =(2,2) superfields can be extended to N =(4,4) supersymmetry off-shell, but this is not true for a model of semichiral fields, where the N = (4,4) supersymmetry can only be realized on-shell. The two models can be related to each other by T-duality. In this paper we perform a duality transformation from a chiral and twisted chiral model with off-shell N = (4,4) supersymmetry to a semichiral model. We find that additional non-linear terms must be added to the original transformations to obtain a semichiral model with N =(4,4) supersymmetry, and that the algebra closes on-shell as a direct consequence of the T-duality.Symmetry2012-10-2444Article10.3390/sym40406036036252073-89942012-10-24doi: 10.3390/sym4040603Malin Göteman<![CDATA[Symmetry, Vol. 4, Pages 581-602: Hexagonal Inflation Tilings and Planar Monotiles]]>
http://www.mdpi.com/2073-8994/4/4/581
Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces.Symmetry2012-10-2244Article10.3390/sym40405815816022073-89942012-10-22doi: 10.3390/sym4040581Michael BaakeFranz GählerUwe Grimm<![CDATA[Symmetry, Vol. 4, Pages 566-580: On the Notions of Symmetry and Aperiodicity for Delone Sets]]>
http://www.mdpi.com/2073-8994/4/4/566
Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.Symmetry2012-10-1044Article10.3390/sym40405665665802073-89942012-10-10doi: 10.3390/sym4040566Michael BaakeUwe Grimm<![CDATA[Symmetry, Vol. 4, Pages 545-565: Barrel Pseudotilings]]>
http://www.mdpi.com/2073-8994/4/3/545
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.Symmetry2012-08-3043Article10.3390/sym40305455455652073-89942012-08-30doi: 10.3390/sym4030545Undine Leopold<![CDATA[Symmetry, Vol. 4, Pages 537-544: A Higher Dimensional Description of the Structure of β-Mn]]>
http://www.mdpi.com/2073-8994/4/3/537
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.Symmetry2012-08-2743Article10.3390/sym40305375375442073-89942012-08-27doi: 10.3390/sym4030537Sven LidinDaniel Fredrickson<![CDATA[Symmetry, Vol. 4, Pages 517-536: Supersymmetric Extensions of Non-Relativistic Scaling Algebras]]>
http://www.mdpi.com/2073-8994/4/3/517
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.Symmetry2012-08-2443Review10.3390/sym40305175175362073-89942012-08-24doi: 10.3390/sym4030517Makoto SakaguchiKentaroh Yoshida<![CDATA[Symmetry, Vol. 4, Pages 507-516: Flexibility of Hydrogen Bond and Lowering of Symmetry in Proton Conductor]]>
http://www.mdpi.com/2073-8994/4/3/507
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.Symmetry2012-08-2343Article10.3390/sym40305075075162073-89942012-08-23doi: 10.3390/sym4030507Yukihiko YoshidaJunko HatoriHinako KawakamiYasumitsu MatsuoSeiichiro Ikehata<![CDATA[Symmetry, Vol. 4, Pages 474-506: Supersymmetric Sigma Model Geometry]]>
http://www.mdpi.com/2073-8994/4/3/474
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.Symmetry2012-08-2343Article10.3390/sym40304744745062073-89942012-08-23doi: 10.3390/sym4030474Ulf Lindström<![CDATA[Symmetry, Vol. 4, Pages 452-473: Supersymmetric Quantum Mechanics and Solvable Models]]>
http://www.mdpi.com/2073-8994/4/3/452
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.Symmetry2012-08-1643Review10.3390/sym40304524524732073-89942012-08-16doi: 10.3390/sym4030452Jonathan BougieAsim GangopadhyayaJeffry MallowConstantin Rasinariu<![CDATA[Symmetry, Vol. 4, Pages 441-451: Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations]]>
http://www.mdpi.com/2073-8994/4/3/441
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.Symmetry2012-08-0843Article10.3390/sym40304414414512073-89942012-08-08doi: 10.3390/sym4030441Laurent DelisleVéronique Hussin<![CDATA[Symmetry, Vol. 4, Pages 427-440: Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection]]>
http://www.mdpi.com/2073-8994/4/3/427
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.Symmetry2012-08-0743Article10.3390/sym40304274274402073-89942012-08-07doi: 10.3390/sym4030427Andrzej Okniński<![CDATA[Symmetry, Vol. 4, Pages 379-426: Symmetry-Adapted Fourier Series for the Wallpaper Groups]]>
http://www.mdpi.com/2073-8994/4/3/379
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.Symmetry2012-07-1743Article10.3390/sym40303793794262073-89942012-07-17doi: 10.3390/sym4030379Bart Verberck<![CDATA[Symmetry, Vol. 4, Pages 344-378: Particle-Dependent Deformations of Lorentz Symmetry]]>
http://www.mdpi.com/2073-8994/4/3/344
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.Symmetry2012-07-0343Article10.3390/sym40303443443782073-89942012-07-03doi: 10.3390/sym4030344Giovanni Amelino-Camelia<![CDATA[Symmetry, Vol. 4, Pages 336-343: Superspheres: Intermediate Shapes between Spheres and Polyhedra]]>
http://www.mdpi.com/2073-8994/4/3/336
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.Symmetry2012-07-0343Article10.3390/sym40303363363432073-89942012-07-03doi: 10.3390/sym4030336Susumu Onaka<![CDATA[Symmetry, Vol. 4, Pages 329-335: Topological Invariance under Line Graph Transformations]]>
http://www.mdpi.com/2073-8994/4/2/329
It is shown that the line graph transformation G ↦ L(G) of a graph G preserves an isomorphic copy of G as the nerve of a finite simplicial complex K which is naturally associated with the Krausz decomposition of L(G). As a consequence, the homology of K is isomorphic to that of G. This homology invariance algebraically confirms several well known graph theoretic properties of line graphs and formally establishes the Euler characteristic of G as a line graph transformation invariant.Symmetry2012-06-0842Article10.3390/sym40203293293352073-89942012-06-08doi: 10.3390/sym4020329Allen D. Parks<![CDATA[Symmetry, Vol. 4, Pages 302-328: Knots in Art]]>
http://www.mdpi.com/2073-8994/4/2/302
We analyze applications of knots and links in the Ancient art, beginning from Babylonian, Egyptian, Greek, Byzantine and Celtic art. Construction methods used in art are analyzed on the examples of Celtic art and ethnical art of Tchokwe people from Angola or Tamil art, where knots are constructed as mirror-curves. We propose different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture and sculpture.Symmetry2012-06-0542Article10.3390/sym40203023023282073-89942012-06-05doi: 10.3390/sym4020302Slavik JablanLjiljana RadovićRadmila SazdanovićAna Zeković<![CDATA[Symmetry, Vol. 4, Pages 285-301: Diagrammatics in Art and Mathematics]]>
http://www.mdpi.com/2073-8994/4/2/285
This paper explores two-way relations between visualizations in mathematics and mathematical art, as well as art in general. A collection of vignettes illustrates connection points, including visualizing higher dimensions, tessellations, knots and links, plotting zeros of polynomials, and new and rapidly developing mathematical discipline, diagrammatic categorification.Symmetry2012-05-2242Article10.3390/sym40202852853012073-89942012-05-22doi: 10.3390/sym4020285Radmila Sazdanovic<![CDATA[Symmetry, Vol. 4, Pages 276-284: Following Knots down Their Energy Gradients]]>
http://www.mdpi.com/2073-8994/4/2/276
This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot.Symmetry2012-04-2742Article10.3390/sym40202762762842073-89942012-04-27doi: 10.3390/sym4020276Louis H. Kauffman<![CDATA[Symmetry, Vol. 4, Pages 265-275: Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators]]>
http://www.mdpi.com/2073-8994/4/2/265
This paper introduces the idea of a maniplex, a common generalization of map and of polytope. The paper then discusses operators, orientability, symmetry and the action of the symmetry group.Symmetry2012-04-1642Article10.3390/sym40202652652752073-89942012-04-16doi: 10.3390/sym4020265Steve Wilson<![CDATA[Symmetry, Vol. 4, Pages 251-264: One-Sign Order Parameter in Iron Based Superconductor]]>
http://www.mdpi.com/2073-8994/4/1/251
The onset of superconductivity at the transition temperature is marked by the onset of order, which is characterized by an energy gap. Most models of the iron-based superconductors find a sign-changing (s±) order parameter [1–6], with the physical implication that pairing is driven by spin fluctuations. Recent work, however, has indicated that LiFeAs has a simple isotropic order parameter [7–9] and spin fluctuations are not necessary [7,10], contrary to the models [1–6]. The strength of the spin fluctuations has been controversial [11,12], meaning that the mechanism of superconductivity cannot as yet be determined. We report the momentum dependence of the superconducting energy gap, where we find an anisotropy that rules out coupling through spin fluctuations and the sign change. The results instead suggest that orbital fluctuations assisted by phonons [13,14] are the best explanation for superconductivity.Symmetry2012-03-2141Article10.3390/sym40102512512642073-89942012-03-21doi: 10.3390/sym4010251Sergey V. BorisenkoVolodymyr B. ZabolotnyyAlexnader A. KordyukDanil V. EvtushinskyTimur K. KimIgor V. MorozovRolf FollathBernd Büchner<![CDATA[Symmetry, Vol. 4, Pages 225-250: Classical Knot Theory]]>
http://www.mdpi.com/2073-8994/4/1/225
This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.Symmetry2012-03-0741Article10.3390/sym40102252252502073-89942012-03-07doi: 10.3390/sym4010225J. Scott Carter<![CDATA[Symmetry, Vol. 4, Pages 219-224: Hidden Symmetries in Simple Graphs]]>
http://www.mdpi.com/2073-8994/4/1/219
It is shown that each element s in the normalizer of the automorphism group Aut(G) of a simple graph G with labeled vertex set V is an Aut(G) invariant isomorphism between G and the graph obtained from G by the s permutation of V—i.e., s is a hidden permutation symmetry of G. A simple example illustrates the theory and the applied notion of system robustness for reconfiguration under symmetry constraint (RUSC) is introduced.Symmetry2012-03-0541Article10.3390/sym40102192192242073-89942012-03-05doi: 10.3390/sym4010219Allen D. Parks<![CDATA[Symmetry, Vol. 4, Pages 208-218: Self-Dual, Self-Petrie Covers of Regular Polyhedra]]>
http://www.mdpi.com/2073-8994/4/1/208
The well-known duality and Petrie duality operations on maps have natural analogs for abstract polyhedra. Regular polyhedra that are invariant under both operations have a high degree of both “external” and “internal” symmetry. The mixing operation provides a natural way to build the minimal common cover of two polyhedra, and by mixing a regular polyhedron with its five other images under the duality operations, we are able to construct the minimal self-dual, self-Petrie cover of a regular polyhedron. Determining the full structure of these covers is challenging and generally requires that we use some of the standard algorithms in combinatorial group theory. However, we are able to develop criteria that sometimes yield the full structure without explicit calculations. Using these criteria and other interesting methods, we then calculate the size of the self-dual, self-Petrie covers of several polyhedra, including the regular convex polyhedra.Symmetry2012-02-2741Article10.3390/sym40102082082182073-89942012-02-27doi: 10.3390/sym4010208Gabe Cunningham<![CDATA[Symmetry, Vol. 4, Pages 143-207: Intrinsic Symmetry Groups of Links with 8 and Fewer Crossings]]>
http://www.mdpi.com/2073-8994/4/1/143
We present an elementary derivation of the “intrinsic” symmetry groups for links of 8 or fewer crossings. We show that standard invariants are enough to rule out all potential symmetries outside the symmetry group of the group of the link for all but one of these links and present explicit isotopies generating the symmetry group for every link.Symmetry2012-02-2041Article10.3390/sym40101431432072073-89942012-02-20doi: 10.3390/sym4010143Michael BerglundJason CantarellaMeredith Perrie CaseyEleanor DannenbergWhitney GeorgeAja JohnsonAmelia KelleyAl LaPointeMatt MastinJason ParsleyJacob RooneyRachel Whitaker<![CDATA[Symmetry, Vol. 4, Pages 129-142: The 27 Possible Intrinsic Symmetry Groups of Two-Component Links]]>
http://www.mdpi.com/2073-8994/4/1/129
We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L′ obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ2 up to conjugacy. We are able to provide prime, nonsplit examples for 21 of these groups; some are classically known, some are new. We catalog the frequency at which each group appears among all 77,036 of the hyperbolic two-component links of 14 or fewer crossings in Thistlethwaite’s table. We also provide some new information about symmetry groups of the 293 non-hyperbolic two-component links of 14 or fewer crossings in the table.Symmetry2012-02-1741Article10.3390/sym40101291291422073-89942012-02-17doi: 10.3390/sym4010129Jason CantarellaJames CornishMatt MastinJason Parsley<![CDATA[Symmetry, Vol. 4, Pages 116-128: Defining the Symmetry of the Universal Semi-Regular Autonomous Asynchronous Systems]]>
http://www.mdpi.com/2073-8994/4/1/116
The regular autonomous asynchronous systems are the non-deterministic Boolean dynamical systems and universality means the greatest in the sense of the inclusion. The paper gives four definitions of symmetry of these systems in a slightly more general framework, called semi-regularity, and also many examples.Symmetry2012-02-1541Article10.3390/sym40101161161282073-89942012-02-15doi: 10.3390/sym4010116Serban E. Vlad<![CDATA[Symmetry, Vol. 4, Pages 39-115: Knots on a Torus: A Model of the Elementary Particles]]>
http://www.mdpi.com/2073-8994/4/1/39
Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, mass, charge, CPT invariance and more. There are no quarks to provide fractional charge, no gluons to sequester them within nucleons and no “colors” to avoid violating Pauli’s principle. Nor do we require the importation of an enigmatic Higgs boson to confer mass upon the particles of our world. All the requisite attributes emerge simply (and relativistically invariant) as a result of particle conformation and occupation in and of spacetime itself, a spacetime endowed with the imprimature of general relativity. Also emerging are some novel tools for systemizing the particle taxonomy as governed by the gauge group SU(2) and the details of particle degeneracy as well as connections to Hopf algebra, Dirac theory, string theory, topological quantum field theory and dark matter. One exception: it is found necessary to invoke the munificent geometry of the icosahedron in order to provide, as per the group “flavor” SU(3), a scaffold upon which to organize the well-known three generations—no more, no less—of the particle family tree.Symmetry2012-02-0941Article10.3390/sym4010039391152073-89942012-02-09doi: 10.3390/sym4010039Jack S. Avrin<![CDATA[Symmetry, Vol. 4, Pages 26-38: Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori]]>
http://www.mdpi.com/2073-8994/4/1/26
A symmetry group of a spatial graph Γ in S3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ.Symmetry2012-01-2041Article10.3390/sym401002626382073-89942012-01-20doi: 10.3390/sym4010026Toru Ikeda<![CDATA[Symmetry, Vol. 4, Pages 15-25: Towards Symmetry-Based Explanation of (Approximate) Shapes of Alpha-Helices and Beta-Sheets (and Beta-Barrels) in Protein Structure]]>
http://www.mdpi.com/2073-8994/4/1/15
Protein structure is invariably connected to protein function. There are two important secondary structure elements: alpha-helices and beta-sheets (which sometimes come in a shape of beta-barrels). The actual shapes of these structures can be complicated, but in the first approximation, they are usually approximated by, correspondingly, cylindrical spirals and planes (and cylinders, for beta-barrels). In this paper, following the ideas pioneered by a renowned mathematician M. Gromov, we use natural symmetries to show that, under reasonable assumptions, these geometric shapes are indeed the best approximating families for secondary structures.Symmetry2012-01-1941Article10.3390/sym401001515252073-89942012-01-19doi: 10.3390/sym4010015Jaime NavaVladik Kreinovich<![CDATA[Symmetry, Vol. 4, Pages 1-14: Convex-Faced Combinatorially Regular Polyhedra of Small Genus]]>
http://www.mdpi.com/2073-8994/4/1/1
Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3.Symmetry2011-12-2841Article10.3390/sym40100011142073-89942011-12-28doi: 10.3390/sym4010001Egon SchulteJörg M. Wills<![CDATA[Symmetry, Vol. 3, Pages 828-851: Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry]]>
http://www.mdpi.com/2073-8994/3/4/828
We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with p3m1, p4m, or p6m symmetry groups that have polyominoes or polyiamonds as fundamental domains. We display the algorithms’ output and give enumeration tables for small values of n. This expands earlier works [1,2] and is a companion to [3].Symmetry2011-12-1234Article10.3390/sym30408288288512073-89942011-12-12doi: 10.3390/sym3040828Hiroshi FukudaChiaki KanomataNobuaki MutohGisaku NakamuraDoris Schattschneider<![CDATA[Symmetry, Vol. 3, Pages 780-827: An Application of the Extended Global SO(3) × SO(3) × U(1) Symmetry of the Hubbard Model on a Square Lattice: The Spinon, η-Spinon, and c Fermion Description]]>
http://www.mdpi.com/2073-8994/3/4/780
In this paper we review recent results on the preliminary applications of the new-found extended global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on a bipartite lattice. Our results refer to the particular case of the bipartite square lattice. Specifically, we review a general description for such a model with nearest-neighbor transfer integral t and on-site repulsion U on a square lattice with N2a 1 sites consistent with its extended global symmetry. It refers to three types of elementary objects whose occupancy configurations generate the state representations of the model extended global symmetry. Such objects emerge from a suitable electron-rotated-electron unitary transformation. An application to the spin spectrum of the parent compound La2CuO4 is shortly reviewed.Symmetry2011-12-1234Article10.3390/sym30407807808272073-89942011-12-12doi: 10.3390/sym3040780Jose M. P. CarmeloMaria J. Sampaio<![CDATA[Symmetry, Vol. 3, Pages 767-779: Information Theory of Networks]]>
http://www.mdpi.com/2073-8994/3/4/767
The paper puts the emphasis on surveying information-theoretic network measures for analyzing the structure of networks. In order to apply the quantities interdisciplinarily, we also discuss some of their properties such as their structural interpretation and uniqueness.Symmetry2011-11-2934Article10.3390/sym30407677677792073-89942011-11-29doi: 10.3390/sym3040767Matthias Dehmer<![CDATA[Symmetry, Vol. 3, Pages 750-766: Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons]]>
http://www.mdpi.com/2073-8994/3/4/750
The language of gene expression displays topological symmetry. An important step during gene expression is the binding of transcriptional proteins to DNA promoters adjacent to a gene. Some proteins bind to many promoters in a genome, defining a regulon of genes wherein each promoter might vary in DNA sequence relative to the average consensus. Here we examine the linguistic organization of gene promoter networks, wherein each node in the network represents a promoter and links between nodes represent the extent of base pair-sharing. Prior work revealed a fractal nucleus in several σ-factor regulons from Escherichia coli. We extend these findings to show fractal nuclei in gene promoter networks from three bacterial species, E. coli, Bacillus subtilis, and Pseudomonas aeruginosa. We surveyed several non-σ transcription factors from these species and found that many contain a nucleus that is both visually and numerically fractal. Promoter footprint size scaled as a negative power-law with both information entropy and fractal dimension, while the latter two parameters scaled positively and linearly. The fractal dimension of the diffuse networks (dB = ~1.7) was close to that expected of a diffusion limited aggregation process, confirming prior predictions as to a possible mechanism for development of this structure.Symmetry2011-11-1834Article10.3390/sym30407507507662073-89942011-11-18doi: 10.3390/sym3040750Preston R. AldrichRobert K. HorsleyStefan M. Turcic<![CDATA[Symmetry, Vol. 3, Pages 699-749: d-Wave Superconductivity and s-Wave Charge Density Waves: Coexistence between Order Parameters of Different Origin and Symmetry]]>
http://www.mdpi.com/2073-8994/3/4/699
A review of the theory describing the coexistence between d-wave superconductivity and s-wave charge-density-waves (CDWs) is presented. The CDW gapping is identified with pseudogapping observed in high-Tc oxides. According to the cuprate specificity, the analysis is carried out for the two-dimensional geometry of the Fermi surface (FS). Phase diagrams on the σ0 − α plane—here, σ0 is the ratio between the energy gaps in the parent pure CDW and superconducting states, and the quantity 2α is connected with the degree of dielectric (CDW) FS gapping—were obtained for various possible configurations of the order parameters in the momentum space. Relevant tunnel and photoemission experimental data for high-Tc oxides are compared with theoretical predictions. A brief review of the results obtained earlier for the coexistence between s-wave superconductivity and CDWs is also given.Symmetry2011-10-2034Article10.3390/sym30406996997492073-89942011-10-20doi: 10.3390/sym3040699Toshikazu EkinoAlexander M. GabovichMai Suan LiMarek PękałaHenryk SzymczakAlexander I. Voitenko<![CDATA[Symmetry, Vol. 3, Pages 680-698: Symmetry and Evidential Support]]>
http://www.mdpi.com/2073-8994/3/3/680
This article proves that formal theories of evidential favoring must fail because they are inevitably language dependent. I begin by describing Carnap’s early confirmation theories to show how language dependence problems (like Goodman’s grue problem) arise. I then generalize to showthat any formal favoring theory satisfying minimal plausible conditions will yield different judgments about the same evidence and hypothesis when they are expressed in alternate languages. This does not just indict formal theories of favoring; it also shows that something beyond our evidence must be invoked to substantively favor one hypothesis over another.Symmetry2011-09-1633Article10.3390/sym30306806806982073-89942011-09-16doi: 10.3390/sym3030680Michael G. Titelbaum<![CDATA[Symmetry, Vol. 3, Pages 653-679: Lattices of Graphical Gaussian Models with Symmetries]]>
http://www.mdpi.com/2073-8994/3/3/653
In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, [1] introduced model classes which place equality restrictions on concentrations or partial correlations. The models can be represented by vertex and edge coloured graphs. The need for model selection methods makes it imperative to understand the structure of model classes. We identify four model classes that form complete lattices of models with respect to model inclusion, which qualifies them for an Edwards–Havránek model selection procedure [2]. Two classes turn out most suitable for a corresponding model search. We obtain an explicit search algorithm for one of them and provide a model search example for the other.Symmetry2011-09-0733Article10.3390/sym30306536536792073-89942011-09-07doi: 10.3390/sym3030653Helene Gehrmann<![CDATA[Symmetry, Vol. 3, Pages 636-652: Symmetry and the Brown-Freiling Refutation of the Continuum Hypothesis]]>
http://www.mdpi.com/2073-8994/3/3/636
Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for two reasons. First, the failure is unusual in that the symmetry intuitions are demonstrably coherent, even though other constraints make it impossible to find a probability model for the scenario. Second, the best probability models have properties analogous to non-conglomerability, motivating a proposed extension of that concept (and corresponding limits on Bayesian conditionalization).Symmetry2011-09-0633Article10.3390/sym30306366366522073-89942011-09-06doi: 10.3390/sym3030636Paul Bartha<![CDATA[Symmetry, Vol. 3, Pages 611-635: Symmetry, Invariance and Ontology in Physics and Statistics]]>
http://www.mdpi.com/2073-8994/3/3/611
This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. objective interpretations that are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion.Symmetry2011-09-0133Article10.3390/sym30306116116352073-89942011-09-01doi: 10.3390/sym3030611Julio Michael Stern<![CDATA[Symmetry, Vol. 3, Pages 600-610: High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument]]>
http://www.mdpi.com/2073-8994/3/3/600
Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. In particular, we provide a new approach for proving the following result of D’Aristotile, Diaconis, and Newman: Let the random matrix Hn be uniformly distributed according to Haar measure on the group of n × n orthogonal matrices, and let An be a non-random n × n real matrix such that tr (A'nAn) = 1. Then, as n→∞, √n tr AnHn converges in distribution to the standard normal distribution.Symmetry2011-08-2633Article10.3390/sym30306006006102073-89942011-08-26doi: 10.3390/sym3030600Donald St. P. Richards<![CDATA[Symmetry, Vol. 3, Pages 574-599: Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas]]>
http://www.mdpi.com/2073-8994/3/3/574
Do there exist circular and spherical copulas in ℝd? That is, do there exist circularly symmetric distributions on the unit disk in ℝ2 and spherically symmetric distributions on the unit ball in ℝd, d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in ℝ2 by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in ℝd are also described, and determined explicitly for d = 2.Symmetry2011-08-2333Article10.3390/sym30305745745992073-89942011-08-23doi: 10.3390/sym3030574Michael D. PerlmanJon A. Wellner<![CDATA[Symmetry, Vol. 3, Pages 564-573: Green’s Symmetries in Finite Digraphs]]>
http://www.mdpi.com/2073-8994/3/3/564
The semigroup DV of digraphs on a set V of n labeled vertices is defined. It is shown that DV is faithfully represented by the semigroup Bn of n ´ n Boolean matrices and that the Green’s L, R, H, and D equivalence classifications of digraphs in DV follow directly from the Green’s classifications already established for Bn. The new results found from this are: (i) L, R, and H equivalent digraphs contain sets of vertices with identical neighborhoods which remain invariant under certain one-sided semigroup multiplications that transform one digraph into another within the same equivalence class, i.e., these digraphs exhibit Green’s isoneighborhood symmetries; and (ii) D equivalent digraphs are characterized by isomorphic inclusion lattices that are generated by their out-neighborhoods and which are preserved under certain two-sided semigroup multiplications that transform digraphs within the same D equivalence class, i.e., these digraphs are characterized by Green’s isolattice symmetries. As a simple illustrative example, the Green’s classification of all digraphs on two vertices is presented and the associated Green’s symmetries are identified.Symmetry2011-08-1533Article10.3390/sym30305645645732073-89942011-08-15doi: 10.3390/sym3030564Allen D. Parks