http://www.mdpi.com/journal/symmetry Latest open access articles published in Symmetry at http://www.mdpi.com/journal/symmetry 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. Symmetry 2013-05-07 5 2 Article 10.3390/sym5020215 215 232 2073-8994 2013-05-07 doi: 10.3390/sym5020215 Mohamed Azzouz 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. Symmetry 2013-04-26 5 2 Review 10.3390/sym5020119 119 214 2073-8994 2013-04-26 doi: 10.3390/sym5020119 Kazuki Hasebe Keisuke Totsuka 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. Symmetry 2013-02-05 5 1 Article 10.3390/sym5010086 86 118 2073-8994 2013-02-05 doi: 10.3390/sym5010086 Horace Crogman Bumgyunmiga Choi Harrison Chen William Harter 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. Symmetry 2013-01-31 5 1 Short Note 10.3390/sym5010081 81 85 2073-8994 2013-01-31 doi: 10.3390/sym5010081 M. C. Jones 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. Symmetry 2013-01-09 5 1 Article 10.3390/sym5010054 54 80 2073-8994 2013-01-09 doi: 10.3390/sym5010054 Valery Rau Leonty Lomtev Tamara Rau 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. Symmetry 2013-01-07 5 1 Short Note 10.3390/sym5010047 47 53 2073-8994 2013-01-07 doi: 10.3390/sym5010047 Antoine Deza Tamon Stephen Feng Xie 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. Symmetry 2012-12-28 5 1 Article 10.3390/sym5010001 1 46 2073-8994 2012-12-28 doi: 10.3390/sym5010001 Jeong-Yup Lee Robert Moody 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. Symmetry 2012-11-30 4 4 Article 10.3390/sym4040667 667 685 2073-8994 2012-11-30 doi: 10.3390/sym4040667 Renato Lemus 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. Symmetry 2012-11-15 4 4 Article 10.3390/sym4040644 644 666 2073-8994 2012-11-15 doi: 10.3390/sym4040644 David Becker 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. Symmetry 2012-10-30 4 4 Article 10.3390/sym4040626 626 643 2073-8994 2012-10-30 doi: 10.3390/sym4040626 Young S. Kim Marilyn E. Noz 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. Symmetry 2012-10-24 4 4 Article 10.3390/sym4040603 603 625 2073-8994 2012-10-24 doi: 10.3390/sym4040603 Malin Göteman 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. Symmetry 2012-10-22 4 4 Article 10.3390/sym4040581 581 602 2073-8994 2012-10-22 doi: 10.3390/sym4040581 Michael Baake Franz Gähler Uwe Grimm 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. Symmetry 2012-10-10 4 4 Article 10.3390/sym4040566 566 580 2073-8994 2012-10-10 doi: 10.3390/sym4040566 Michael Baake Uwe Grimm 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. Symmetry 2012-08-30 4 3 Article 10.3390/sym4030545 545 565 2073-8994 2012-08-30 doi: 10.3390/sym4030545 Undine Leopold 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. Symmetry 2012-08-27 4 3 Article 10.3390/sym4030537 537 544 2073-8994 2012-08-27 doi: 10.3390/sym4030537 Sven Lidin Daniel Fredrickson 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. Symmetry 2012-08-24 4 3 Review 10.3390/sym4030517 517 536 2073-8994 2012-08-24 doi: 10.3390/sym4030517 Makoto Sakaguchi Kentaroh Yoshida 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. Symmetry 2012-08-23 4 3 Article 10.3390/sym4030507 507 516 2073-8994 2012-08-23 doi: 10.3390/sym4030507 Yukihiko Yoshida Junko Hatori Hinako Kawakami Yasumitsu Matsuo Seiichiro Ikehata 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. Symmetry 2012-08-23 4 3 Article 10.3390/sym4030474 474 506 2073-8994 2012-08-23 doi: 10.3390/sym4030474 Ulf Lindström 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. Symmetry 2012-08-16 4 3 Review 10.3390/sym4030452 452 473 2073-8994 2012-08-16 doi: 10.3390/sym4030452 Jonathan Bougie Asim Gangopadhyaya Jeffry Mallow Constantin Rasinariu 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. Symmetry 2012-08-08 4 3 Article 10.3390/sym4030441 441 451 2073-8994 2012-08-08 doi: 10.3390/sym4030441 Laurent Delisle Véronique Hussin 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. Symmetry 2012-08-07 4 3 Article 10.3390/sym4030427 427 440 2073-8994 2012-08-07 doi: 10.3390/sym4030427 Andrzej Okniński 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. Symmetry 2012-07-17 4 3 Article 10.3390/sym4030379 379 426 2073-8994 2012-07-17 doi: 10.3390/sym4030379 Bart Verberck 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. Symmetry 2012-07-03 4 3 Article 10.3390/sym4030344 344 378 2073-8994 2012-07-03 doi: 10.3390/sym4030344 Giovanni Amelino-Camelia 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. Symmetry 2012-07-03 4 3 Article 10.3390/sym4030336 336 343 2073-8994 2012-07-03 doi: 10.3390/sym4030336 Susumu Onaka 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. Symmetry 2012-06-08 4 2 Article 10.3390/sym4020329 329 335 2073-8994 2012-06-08 doi: 10.3390/sym4020329 Allen D. Parks 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. Symmetry 2012-06-05 4 2 Article 10.3390/sym4020302 302 328 2073-8994 2012-06-05 doi: 10.3390/sym4020302 Slavik Jablan Ljiljana Radović Radmila Sazdanović Ana Zeković 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. Symmetry 2012-05-22 4 2 Article 10.3390/sym4020285 285 301 2073-8994 2012-05-22 doi: 10.3390/sym4020285 Radmila Sazdanovic 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. Symmetry 2012-04-27 4 2 Article 10.3390/sym4020276 276 284 2073-8994 2012-04-27 doi: 10.3390/sym4020276 Louis H. Kauffman 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. Symmetry 2012-04-16 4 2 Article 10.3390/sym4020265 265 275 2073-8994 2012-04-16 doi: 10.3390/sym4020265 Steve Wilson 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. Symmetry 2012-03-21 4 1 Article 10.3390/sym4010251 251 264 2073-8994 2012-03-21 doi: 10.3390/sym4010251 Sergey V. Borisenko Volodymyr B. Zabolotnyy Alexnader A. Kordyuk Danil V. Evtushinsky Timur K. Kim Igor V. Morozov Rolf Follath Bernd Büchner 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. Symmetry 2012-03-07 4 1 Article 10.3390/sym4010225 225 250 2073-8994 2012-03-07 doi: 10.3390/sym4010225 J. Scott Carter 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. Symmetry 2012-03-05 4 1 Article 10.3390/sym4010219 219 224 2073-8994 2012-03-05 doi: 10.3390/sym4010219 Allen D. Parks 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. Symmetry 2012-02-27 4 1 Article 10.3390/sym4010208 208 218 2073-8994 2012-02-27 doi: 10.3390/sym4010208 Gabe Cunningham 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. Symmetry 2012-02-20 4 1 Article 10.3390/sym4010143 143 207 2073-8994 2012-02-20 doi: 10.3390/sym4010143 Michael Berglund Jason Cantarella Meredith Perrie Casey Eleanor Dannenberg Whitney George Aja Johnson Amelia Kelley Al LaPointe Matt Mastin Jason Parsley Jacob Rooney Rachel Whitaker 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. Symmetry 2012-02-17 4 1 Article 10.3390/sym4010129 129 142 2073-8994 2012-02-17 doi: 10.3390/sym4010129 Jason Cantarella James Cornish Matt Mastin Jason Parsley 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. Symmetry 2012-02-15 4 1 Article 10.3390/sym4010116 116 128 2073-8994 2012-02-15 doi: 10.3390/sym4010116 Serban E. Vlad 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. Symmetry 2012-02-09 4 1 Article 10.3390/sym4010039 39 115 2073-8994 2012-02-09 doi: 10.3390/sym4010039 Jack S. Avrin 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. Symmetry 2012-01-20 4 1 Article 10.3390/sym4010026 26 38 2073-8994 2012-01-20 doi: 10.3390/sym4010026 Toru Ikeda 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. Symmetry 2012-01-19 4 1 Article 10.3390/sym4010015 15 25 2073-8994 2012-01-19 doi: 10.3390/sym4010015 Jaime Nava Vladik Kreinovich 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. Symmetry 2011-12-28 4 1 Article 10.3390/sym4010001 1 14 2073-8994 2011-12-28 doi: 10.3390/sym4010001 Egon Schulte Jörg M. Wills 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]. Symmetry 2011-12-12 3 4 Article 10.3390/sym3040828 828 851 2073-8994 2011-12-12 doi: 10.3390/sym3040828 Hiroshi Fukuda Chiaki Kanomata Nobuaki Mutoh Gisaku Nakamura Doris Schattschneider 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. Symmetry 2011-12-12 3 4 Article 10.3390/sym3040780 780 827 2073-8994 2011-12-12 doi: 10.3390/sym3040780 Jose M. P. Carmelo Maria J. Sampaio 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. Symmetry 2011-11-29 3 4 Article 10.3390/sym3040767 767 779 2073-8994 2011-11-29 doi: 10.3390/sym3040767 Matthias Dehmer 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. Symmetry 2011-11-18 3 4 Article 10.3390/sym3040750 750 766 2073-8994 2011-11-18 doi: 10.3390/sym3040750 Preston R. Aldrich Robert K. Horsley Stefan M. Turcic 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. Symmetry 2011-10-20 3 4 Article 10.3390/sym3040699 699 749 2073-8994 2011-10-20 doi: 10.3390/sym3040699 Toshikazu Ekino Alexander M. Gabovich Mai Suan Li Marek Pękała Henryk Szymczak Alexander I. Voitenko 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. Symmetry 2011-09-16 3 3 Article 10.3390/sym3030680 680 698 2073-8994 2011-09-16 doi: 10.3390/sym3030680 Michael G. Titelbaum 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. Symmetry 2011-09-07 3 3 Article 10.3390/sym3030653 653 679 2073-8994 2011-09-07 doi: 10.3390/sym3030653 Helene Gehrmann 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). Symmetry 2011-09-06 3 3 Article 10.3390/sym3030636 636 652 2073-8994 2011-09-06 doi: 10.3390/sym3030636 Paul Bartha 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. Symmetry 2011-09-01 3 3 Article 10.3390/sym3030611 611 635 2073-8994 2011-09-01 doi: 10.3390/sym3030611 Julio Michael Stern 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. Symmetry 2011-08-26 3 3 Article 10.3390/sym3030600 600 610 2073-8994 2011-08-26 doi: 10.3390/sym3030600 Donald St. P. Richards 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. Symmetry 2011-08-23 3 3 Article 10.3390/sym3030574 574 599 2073-8994 2011-08-23 doi: 10.3390/sym3030574 Michael D. Perlman Jon A. Wellner 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. Symmetry 2011-08-15 3 3 Article 10.3390/sym3030564 564 573 2073-8994 2011-08-15 doi: 10.3390/sym3030564 Allen D. Parks http://www.mdpi.com/2073-8994/3/3/541 Cyclotron frequency of a crystal electron is, in general, not an easily accessible parameter. Nevertheless, its calculation can be simplified when the symmetry properties of the band structure and those of the motion equations in the magnetic field are simultaneously taken into account. In effect, a combined symmetry of the electron Hamiltonian and that of the Lorentz equation provide us with a non-linear oscillator problem of high symmetry. In the next step, the kinetic energy of the oscillator can be separated from the whole of electron energy and applied in a new kind of calculation of the cyclotron frequency which is much more simple than before. In consequence, a detailed approach to the electron circulation, also in more complex band structures, becomes a relatively easy task. For different crystal lattices of cubic symmetry taken as examples the cyclotron frequency of the present and a former method are compared numerically giving the same results. Symmetry 2011-08-10 3 3 Article 10.3390/sym3030541 541 563 2073-8994 2011-08-10 doi: 10.3390/sym3030541 Stanislaw Olszewski Tomasz Roliński http://www.mdpi.com/2073-8994/3/3/524 An analysis of the path integral approach to quantum theory motivates the hypothesis that two experiments with the same classical action should have dual ontological descriptions. If correct, this hypothesis would not only constrain realistic interpretations of quantum theory, but would also act as a constructive principle, allowing any realistic model of one experiment to generate a corresponding model for its action-dual. Two pairs of action-dual experiments are presented, including one experiment that violates the Bell inequality and yet is action-dual to a single particle. The implications generally support retrodictive and retrocausal interpretations. Symmetry 2011-07-27 3 3 Article 10.3390/sym3030524 524 540 2073-8994 2011-07-27 doi: 10.3390/sym3030524 Ken B. Wharton David J. Miller Huw Price http://www.mdpi.com/2073-8994/3/3/503 As a contribution to the mysteries of human symmetry perception, reaction time data were collected on the detection of symmetry or repetition violations, in the context of short term visual memory studies. The histograms for reaction time distributions are rather narrow in the case of symmetry judgments. Their analysis was performed in terms of a simple kinetic model of a mental process in two steps, a slow one for the construction of the representation of the images to be compared, and a fast one, in the 50 ms range, for the decision. There was no need for an additional ‘mental rotation’ step. Symmetry seems to facilitate the construction step. I also present here original stimuli showing a color equalization effect across a symmetry axis, and its counterpart in periodic patterns. According to a “folded sheet model”, when a shape is perceived, the brain automatically constructs a mirror-image representation of the shape. Based in part on the reaction time analysis, I present here an alternative “transparent sheet” model in which the brain constructs a single representation, which can be accessed from two sides, thus generating simultaneously a pattern and its mirror-symmetric partner. Filtering processes, implied by current models of symmetry perception could intervene at an early stage, by nucleating the propagation of similar perceptual groupings in the two symmetric images. Symmetry 2011-07-22 3 3 Article 10.3390/sym3030503 503 523 2073-8994 2011-07-22 doi: 10.3390/sym3030503 Jacques Ninio http://www.mdpi.com/2073-8994/3/3/487 Our paper analyzes some aspects of Uncertainty Measures. We need to obtain new ways to model adequate conditions or restrictions, constructed from vague pieces of information. The classical entropy measure originates from scientific fields; more specifically, from Statistical Physics and Thermodynamics. With time it was adapted by Claude Shannon, creating the current expanding Information Theory. However, the Hungarian mathematician, Alfred Rényi, proves that different and valid entropy measures exist in accordance with the purpose and/or need of application. Accordingly, it is essential to clarify the different types of measures and their mutual relationships. For these reasons, we attempt here to obtain an adequate revision of such fuzzy entropy measures from a mathematical point of view. Symmetry 2011-07-20 3 3 Article 10.3390/sym3030487 487 502 2073-8994 2011-07-20 doi: 10.3390/sym3030487 Angel Garrido http://www.mdpi.com/2073-8994/3/3/472 An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum matching. If sk is the number of independent sets of size k in G, then I(G; x) = s0 + s1x + s2x2 + ... + sαxα, α = α (G), is called the independence polynomial of G (Gutman and Harary, 1986). If sj = sαj for all 0 ≤ j ≤ [α/2], then I(G; x) is called symmetric (or palindromic). It is known that the graph G ° 2K1, obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanović, 1998). In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to show that for every graph G and for each non-negative integer k ≤ μ (G), one can build a graph H, such that: G is a subgraph of H, I (H; x) is symmetric, and I (G ° 2K1; x) = (1 + x)k · I (H; x). Symmetry 2011-07-15 3 3 Article 10.3390/sym3030472 472 486 2073-8994 2011-07-15 doi: 10.3390/sym3030472 Vadim E. Levit Eugen Mandrescu http://www.mdpi.com/2073-8994/3/3/457 Mirror symmetry is often thought to be particularly salient to human observers because it engages specialized mechanisms that evolved to sense symmetrical objects in nature. Although symmetry is indeed present in many of our artifacts and markings on wildlife, studies have shown that sensitivity to mirror symmetry does not serve an alerting function and sensitivity to symmetry decreases in a rather unremarkable way when it is presented away from the centre of the visual field. Here we show that symmetrical targets are vulnerable to the same interference as other stimuli when surrounded by non-target elements. These results provide further evidence that symmetry is not special to the early visual system, and reinforce the notion that our fascination with symmetry is more likely attributable to cognitive and aesthetic factors than to specialized, low level mechanisms in the visual system. Symmetry 2011-07-13 3 3 Article 10.3390/sym3030457 457 471 2073-8994 2011-07-13 doi: 10.3390/sym3030457 Gabrielle Roddy Rick Gurnsey http://www.mdpi.com/2073-8994/3/3/443 Symmetric patterns are more appealing to human observers than asymmetric ones. Here, we investigate the visual information processing mechanisms underlying this aesthetic preference. All stimuli were derived from phase scrambled versions of forty face or nature images. In addition to the scrambled images, there were four other types of test image: symmetric, in which one part of the image was a reflection of another around an axis; repetitive, in which one part of the image was a copy of the other; anti-symmetric, similar to symmetric but with the contrast of one side reversed; and interleaved patterns, in which half of the symmetric pattern was replaced by a scrambled image. The number of axes ranged from 1 to 16 for all image types. The task of our 20 observers was to give a preference rating to each image on a 6-point Lickert scale. The preference rating increased with the number of axes for all stimulus types. The observers showed a similar preference for symmetric and repetitive patterns and slightly less preference for anti-symmetric patterns. The preference for interleaved patterns was much less than for other types of stimuli. Preference for an image cannot be explained by either the ecological significance of its content or the slope of its amplitude spectrum. Instead, preference can be accounted for by the complexity of the image. Symmetry 2011-07-11 3 3 Article 10.3390/sym3030443 443 456 2073-8994 2011-07-11 doi: 10.3390/sym3030443 Chien-Chung Chen Jo-Hsuan Wu Chia-Ching Wu http://www.mdpi.com/2073-8994/3/3/402 The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version. Symmetry 2011-07-07 3 3 Article 10.3390/sym3030402 402 442 2073-8994 2011-07-07 doi: 10.3390/sym3030402 Mihai Prunescu http://www.mdpi.com/2073-8994/3/3/389 From the physics point of view, time is now best described through General Relativity as part of space-time, which is a dynamical object encoding gravity. Time possesses also some intrinsic irreversibility due to thermodynamics and quantum mechanical effects. This irreversibility can look puzzling since time-like loops (and hence time machines) can appear in General Relativity (for example in the Gödel universe, a solution of Einstein’s equations). We take this apparent discrepancy as a warning bell, pointing out that time as we understand it might not be fundamental and that whatever theory lying beyond General Relativity may not include time as we know it as a fundamental structure. We propose therefore, following the philosophy of analog models of gravity, that time and gravity might not be fundamental per se, but only emergent features. We illustrate our proposal using a toy-model where we show how the Lorentzian signature and Nordström gravity (a diffeomorphisms invariant scalar gravity theory) can emerge from a timeless non-dynamical space. This article received the fourth prize at the essay competition of the Foundational Questions Institute on the nature of time. Symmetry 2011-06-29 3 3 Article 10.3390/sym3030389 389 401 2073-8994 2011-06-29 doi: 10.3390/sym3030389 Florian Girelli Stefano Liberati Lorenzo Sindoni http://www.mdpi.com/2073-8994/3/2/365 Symmetry has been shown to be a very effective a priori constraint in solving a 3D shape recovery problem. Symmetry is useful in 3D recovery because it is a form of redundancy. There are, however, some fundamental limits to the effectiveness of symmetry. Specifically, given two arbitrary curves in a single 2D image, one can always find a 3D mirror-symmetric interpretation of these curves under quite general assumptions. The symmetric interpretation is unique under a perspective projection and there is a one parameter family of symmetric interpretations under an orthographic projection. We formally state and prove this observation for the case of one-to-one and many-to-many point correspondences. We conclude by discussing the role of degenerate views, higher-order features in determining the point correspondences, as well as the role of the planarity constraint. When the correspondence of features is known and/or curves can be assumed to be planar, 3D symmetry becomes non-accidental in the sense that a 2D image of a 3D asymmetric shape obtained from a random viewing direction will not allow for 3D symmetric interpretations. Symmetry 2011-06-10 3 2 Article 10.3390/sym3020365 365 388 2073-8994 2011-06-10 doi: 10.3390/sym3020365 Tadamasa Sawada Yunfeng Li Zygmunt Pizlo http://www.mdpi.com/2073-8994/3/2/325 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 pmm, pmg, pgg or cmm symmetry [1]. These symmetry groups are members of the crystal class D2 among the 17 two-dimensional symmetry groups [2]. We display the algorithms’ output and give enumeration tables for small values of n. This work is a continuation of our earlier works for the symmetry groups p3, p31m, p3m1, p4, p4g, p4m, p6, and p6m [3–5]. Symmetry 2011-06-09 3 2 Article 10.3390/sym3020325 325 364 2073-8994 2011-06-09 doi: 10.3390/sym3020325 Hiroshi Fukuda Chiaki Kanomata Nobuaki Mutoh Gisaku Nakamura Doris Schattschneider http://www.mdpi.com/2073-8994/3/2/305 Whereas quantum computing circuits follow the symmetries of the unitary Lie group, classical reversible computation circuits follow the symmetries of a finite group, i.e., the symmetric group. We confront the decomposition of an arbitrary classical reversible circuit with w bits and the decomposition of an arbitrary quantum circuit with w qubits. Both decompositions use the control gate as building block, i.e., a circuit transforming only one (qu)bit, the transformation being controlled by the other w−1 (qu)bits. We explain why the former circuit can be decomposed into 2w − 1 control gates, whereas the latter circuit needs 2w − 1 control gates. We investigate whether computer circuits, not based on the full unitary group but instead on a subgroup of the unitary group, may be decomposable either into 2w − 1 or into 2w − 1 control gates. Symmetry 2011-06-07 3 2 Article 10.3390/sym3020305 305 324 2073-8994 2011-06-07 doi: 10.3390/sym3020305 Alexis De Vos Stijn De Baerdemacker http://www.mdpi.com/2073-8994/3/2/283 The major issue regarding magnetic response in nature—“negative values for the permeability μ of material parameters, especially in terahertz or optical region” makes the electromagnetic properties of natural materials asymmetric. Recently, research in metamaterials has grown in significance because these artificial materials can demonstrate special and, indeed, extraordinary electromagnetic phenomena such as the inverse of Snell’s law and novel applications. A critical topic in metamaterials is the artificial negative magnetic response, which can be designed in the higher frequency regime (from microwave to optical range). Artificial magnetism illustrates new physics and new applications, which have been demonstrated over the past few years. In this review, we present recent developments in research on artificial magnetic metamaterials including split-ring resonator structures, sandwich structures, and high permittivity-based dielectric composites. Engineering applications such as invisibility cloaking, negative refractive index medium, and slowing light fall into this category. We also discuss the possibility that metamaterials can be suitable for realizing new and exotic electromagnetic properties. Symmetry 2011-06-03 3 2 Review 10.3390/sym3020283 283 304 2073-8994 2011-06-03 doi: 10.3390/sym3020283 Yueh-Chun Lai Cheng-Kuang Chen Tsung-Yu Huang Ieng-Wai Un Yu-Hang Yang Ta-Jen Yen http://www.mdpi.com/2073-8994/3/2/265 Various organocatalysts have been developed for the aldol reaction but particular attention has been paid to prolinamide derivatives. They are easy to prepare and their catalytic activity can be readily tuned through structural modification. In this review, the comparison of catalytic activities between prolinethioamides and their respective amides in direct asymmetric aldol reactions is presented. Symmetry 2011-06-01 3 2 Review 10.3390/sym3020265 265 282 2073-8994 2011-06-01 doi: 10.3390/sym3020265 Dorota Gryko Mikołaj Chromiński Dominika J. Pielacińska http://www.mdpi.com/2073-8994/3/2/246 Periodic patterns and symmetries are striking visual properties that have been used decoratively around the world throughout human history. Periodic patterns can be mathematically classified into one of 17 different Wallpaper groups, and while computational models have been developed which can extract an image's symmetry group, very little work has been done on how humans perceive these patterns. This study presents the results from a grouping experiment using stimuli from the different wallpaper groups. We find that while different images from the same wallpaper group are perceived as similar to one another, not all groups have the same degree of self-similarity. The similarity relationships between wallpaper groups appear to be dominated by rotations. Symmetry 2011-05-25 3 2 Article 10.3390/sym3020246 246 264 2073-8994 2011-05-25 doi: 10.3390/sym3020246 Alasdair D. F. Clarke Patrick R. Green Fraser Halley Mike J. Chantler http://www.mdpi.com/2073-8994/3/2/220 A large number of interesting organocatalytic enantioselective protocols have been explored and successfully applied in the last decade. Among them, the Henry (nitroaldol) reaction represents a powerful carbon-carbon bond-forming procedure for the preparation of valuable synthetic intermediates, such as enantioenriched nitro alcohols, which can be further transformed in a number of important nitrogen and oxygen-containing compounds. This area of research is still in expansion and a more complex version of this useful process has recently emerged, the domino Michael/Henry protocol, affording highly functionalized cycles with multiple stereogenic centers. Symmetry 2011-05-23 3 2 Review 10.3390/sym3020220 220 245 2073-8994 2011-05-23 doi: 10.3390/sym3020220 Yolanda Alvarez-Casao Eugenia Marques-Lopez Raquel P. Herrera http://www.mdpi.com/2073-8994/3/2/207 Within most of the 17 plane symmetry groups, individual symmetry operations act in multiple, nonequivalent ways. This, and the fact that many groups can be realized on the basis of different unit cells and generating regions, poses difficulties for visual discrimination and identification. Because of inherent confounds, only few of the groups can be studied by traditional experimental methodology. The use of an oddity paradigm and specific tiling patterns that camouflage groups in complex textures are recommended as partial remedy to this impasse. In order to prepare readers for an appreciation of the aforementioned issues and to provide a rationale for their investigation, the reporting of experiments and the discussion of methodological problems is preceded by a brief overview of the role which symmetry has played in the visual arts. Symmetry 2011-05-11 3 2 Article 10.3390/sym3020207 207 219 2073-8994 2011-05-11 doi: 10.3390/sym3020207 Klaus Landwehr http://www.mdpi.com/2073-8994/3/2/171 Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with probability theory? If so, does it extend or generalize probability theory? In this paper, we answer these questions, and precisely determine the relationship between quantum theory and probability theory, by explicitly deriving both theories from first principles. In both cases, the derivation depends upon identifying and harnessing the appropriate symmetries that are operative in each domain. We prove, for example, that quantum theory is compatible with probability theory by explicitly deriving quantum theory on the assumption that probability theory is generally valid. Symmetry 2011-04-27 3 2 Article 10.3390/sym3020171 171 206 2073-8994 2011-04-27 doi: 10.3390/sym3020171 Philip Goyal Kevin H. Knuth http://www.mdpi.com/2073-8994/3/2/165 The paper presents an application of the asymmetry approach to spirooxindoles via Brevicolline, Cinchonidine or Cinchonine catalyzed one-pot multicomponent synthesis. Brevicolline, in comparison with Cinchonidine or Cinchonine, catalyzes the reaction of isatins, acetylacetone/ethyl 3-oxobutanoate and malononitrile, with the formation of spiro[oxindole-3,4'-4'H-pirane] derivatives in an optically active form in very good to excellent yields. Symmetry 2011-04-20 3 2 Article 10.3390/sym3020165 165 170 2073-8994 2011-04-20 doi: 10.3390/sym3020165 Fliur Macaev Natalia Sucman Felix Shepeli Marina Zveaghintseva Vsevolod Pogrebnoi http://www.mdpi.com/2073-8994/3/2/155 Asymmetric Michael addition of carbon nucleophiles, nitroalkanes and a β-ketoester, to enones was investigated by using a primary amino acid lithium salt as a catalyst. Symmetry 2011-04-08 3 2 Article 10.3390/sym3020155 155 164 2073-8994 2011-04-08 doi: 10.3390/sym3020155 Masanori Yoshida Keisuke Hirama Mao Narita Shoji Hara http://www.mdpi.com/2073-8994/3/2/134 Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that “quantisation commutes with reduction” and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity. Symmetry 2011-03-31 3 2 Article 10.3390/sym3020134 134 154 2073-8994 2011-03-31 doi: 10.3390/sym3020134 Karim P. Y. Thébault http://www.mdpi.com/2073-8994/3/2/126 A subset S of a group G is symmetric if there is an element g є G such that gS-1g = S. We study some Ramsey type functions for symmetric subsets in finite Abelian groups. Symmetry 2011-03-24 3 2 Article 10.3390/sym3020126 126 133 2073-8994 2011-03-24 doi: 10.3390/sym3020126 Yuliya Zelenyuk http://www.mdpi.com/2073-8994/3/1/84 The 1,4-conjugate addition of nucleophiles to α,β-unsaturated carbonyl compounds represents one fundamental bond-forming reaction in organic synthesis. The development of effective organocatalysts for the enantioselective conjugate addition of malonate, nitroalkane and other carbon and heteroatom nucleophiles to cycloenones constitutes an important research field and has been explored in recent years. At the same time, asymmetric Diels-Alder reactions have been developed and often a mechanism has been demonstrated to be a double addition rather than synchronous. This review aims to cover literature up to the end of 2010, describing all the different organocatalytic asymmetric 1,4-conjugate additions even if they are listed as transfer hydrogenation, cycloadditions or desymmetrization of aromatic compounds. Symmetry 2011-03-22 3 1 Review 10.3390/sym3010084 84 125 2073-8994 2011-03-22 doi: 10.3390/sym3010084 Renato Dalpozzo Giuseppe Bartoli Giorgio Bencivenni http://www.mdpi.com/2073-8994/3/1/72 We show that in long-time the heat semi-group on a path group associated to a Bilaplacian on the group tends to the Haar distribution on a path group. Symmetry 2011-03-21 3 1 Article 10.3390/sym3010072 72 83 2073-8994 2011-03-21 doi: 10.3390/sym3010072 Remi Leandre http://www.mdpi.com/2073-8994/3/1/54 Much is already known about single mirror symmetry, but multiple mirror symmetry is still understood poorly. In particular, perceptually, multiple symmetry does not seem to behave as suggested by the number of symmetry axes alone. Here, theoretical ideas on single symmetry perception and their extensions to multiple symmetry are discussed alongside empirical findings on multiple symmetry perception. The evidence suggests that, apart from the number of axes, also their relative orientation is perceptually relevant. This, in turn, suggests that perception is responsible for the preponderance of 3-fold and 5-fold symmetries in flowers as well as for their absence in decorative art. Symmetry 2011-03-21 3 1 Article 10.3390/sym3010054 54 71 2073-8994 2011-03-21 doi: 10.3390/sym3010054 Peter A. Van der Helm http://www.mdpi.com/2073-8994/3/1/37 Although symmetry may be important for understanding the selection of form in art over the historical period, this preference may have originally stemmed from certain basic perceptual mechanism that initially arose during prehistory. The first signs of an awareness to symmetry can be found in the archaeological record with the arrival of Acheulean handaxes, especially those dating from 500,000 years ago onwards, which are typified by a prodigious bilateral symmetry. As handaxes represent the earliest material record of an interest in symmetry by the human lineage, they provide a privileged means of understanding why this kind of form came to be valued by later human groups, particularly in relation to “art”. Although still controversial, the preference for symmetry at such an early date has been linked to various aspects of perception relating to enduring evolutionary factors. In this regard, it will be demonstrated how the preference for symmetrical Acheulean tools arose out of long standing perceptual correlates relating to ecological factors that predated the arrival of hominins. Symmetry 2011-03-01 3 1 Review 10.3390/sym3010037 37 53 2073-8994 2011-03-01 doi: 10.3390/sym3010037 Derek Hodgson http://www.mdpi.com/2073-8994/3/1/16 Among the symmetries in physics, the rotation symmetry is most familiar to us. It is known that the spherical harmonics serve useful purposes when the world is rotated. Squeeze transformations are also becoming more prominent in physics, particularly in optical sciences and in high-energy physics. As can be seen from Dirac’s light-cone coordinate system, Lorentz boosts are squeeze transformations. Thus the squeeze transformation is one of the fundamental transformations in Einstein’s Lorentz-covariant world. It is possible to define a complete set of orthonormal functions defined for one Lorentz frame. It is shown that the same set can be used for other Lorentz frames. Transformation properties are discussed. Physical applications are discussed in both optics and high-energy physics. It is shown that the Lorentz harmonics provide the mathematical basis for squeezed states of light. It is shown also that the same set of harmonics can be used for understanding Lorentz-boosted hadrons in high-energy physics. It is thus possible to transmit physics from one branch of physics to the other branch using the mathematical basis common to them. Symmetry 2011-02-14 3 1 Review 10.3390/sym3010016 16 36 2073-8994 2011-02-14 doi: 10.3390/sym3010016 Young S. Kim Marilyn E. Noz http://www.mdpi.com/2073-8994/3/1/1 In this paper, we analyze a few interrelated concepts about graphs, such as their degree, entropy, or their symmetry/asymmetry levels. These concepts prove useful in the study of different types of Systems, and particularly, in the analysis of Complex Networks. A System can be defined as any set of components functioning together as a whole. A systemic point of view allows us to isolate a part of the world, and so, we can focus on those aspects that interact more closely than others. Network Science analyzes the interconnections among diverse networks from different domains: physics, engineering, biology, semantics, and so on. Current developments in the quantitative analysis of Complex Networks, based on graph theory, have been rapidly translated to studies of brain network organization. The brain's systems have complex network features—such as the small-world topology, highly connected hubs and modularity. These networks are not random. The topology of many different networks shows striking similarities, such as the scale-free structure, with the degree distribution following a Power Law. How can very different systems have the same underlying topological features? Modeling and characterizing these networks, looking for their governing laws, are the current lines of research. So, we will dedicate this Special Issue paper to show measures of symmetry in Complex Networks, and highlight their close relation with measures of information and entropy. Symmetry 2011-01-10 3 1 Article 10.3390/sym3010001 1 15 2073-8994 2011-01-10 doi: 10.3390/sym3010001 Angel Garrido http://www.mdpi.com/2073-8994/2/4/1945 We argue that quantum theory should proceed not from a spacetime background but from a Lie algebra, which is treated as a symmetry algebra. Then the fact that the cosmological constant is positive means not that the spacetime background is curved but that the de Sitter (dS) algebra as the symmetry algebra is more relevant than the Poincare or anti de Sitter ones. The physical interpretation of irreducible representations (IRs) of the dS algebra is considerably different from that for the other two algebras. One IR of the dS algebra splits into independent IRs for a particle and its antiparticle only when Poincare approximation works with a high accuracy. Only in this case additive quantum numbers such as electric, baryon and lepton charges are conserved, while at early stages of the Universe they could not be conserved. Another property of IRs of the dS algebra is that only fermions can be elementary and there can be no neutral elementary particles. The cosmological repulsion is a simple kinematical consequence of dS symmetry on quantum level when quasiclassical approximation is valid. Therefore the cosmological constant problem does not exist and there is no need to involve dark energy or other fields for explaining this phenomenon (in agreement with a similar conclusion by Bianchi and Rovelli). Symmetry 2010-11-19 2 4 Article 10.3390/sym2041945 1945 1980 2073-8994 2010-11-19 doi: 10.3390/sym2041945 Felix M. Lev http://www.mdpi.com/2073-8994/2/4/1925 We are currently investigating measurable esthetic characteristics in persons considered “attractive” by the media. Three-dimensional soft-tissue facial asymmetry was quantified in 380 attractive (148 males, 232 females) and 669 control (397 males, 272 females) healthy persons aged 4–30 years. The coordinates of 50 facial landmarks were collected by a computerized digitizer, and asymmetry computed. Soft-tissue facial asymmetries reduced as a function of age in all cases. Attractive children were more symmetric than control children, but the reverse was true for young adults. The effect of symmetry on attractiveness seems to change as a function of age. Symmetry 2010-11-09 2 4 Article 10.3390/sym2041925 1925 1944 2073-8994 2010-11-09 doi: 10.3390/sym2041925 Chiarella Sforza Alberto Laino Gaia Grandi Luca Pisoni Virgilio Ferruccio Ferrario http://www.mdpi.com/2073-8994/2/4/1846 The paper introduces general considerations on structural properties of aromatic, antiaromatic and non-aromatic conjugated systems in terms of potential energy along bond length alternation and distortion coordinates, taking as examples benzene, cyclobutadiene and cyclooctatetraene. Pentalene, formally derived from cyclooctatetraene by cross linking, is also considered as a typical antiaromatic system. The main interest is concerned with [n]annulenes and model [n]annulene molecular systems, n ranging from 10 to 18. The rich variety of conformational and  configurational isomers and of dynamical processes among them is described. Specific attention is devoted to bridged [10]- and [14]annulenes in the ground and lowest excited states as well as to s-indacene and biphenylene. Experimental data obtained from vibrational and electronic spectroscopies are discussed and compared with ab initio calculation results. Finally, porphyrin, tetraoxaporphyrin dication and diprotonated porphyrin are presented as annulene structures adopting planar/non-planar geometries depending on the steric hindrance in the inner macrocycle ring. Radiative and non-radiative relaxation processes from excited state levels have been observed by means of time-resolved fluorescence and femtosecond transient absorption spectroscopy. A short account is also given of porphycene, the structural isomer of porphyrin, and of porphycene properties. Symmetry 2010-11-05 2 4 Article 10.3390/sym2041846 1846 1924 2073-8994 2010-11-05 doi: 10.3390/sym2041846 Cristina Gellini Pier Remigio Salvi http://www.mdpi.com/2073-8994/2/4/1810 We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only in the approximation when de Sitter energies are much less than the characteristic of the field. As a consequence, the very notions of particles and antiparticles are only approximate and such additive quantum numbers as the electric, baryon and lepton charges are conserved only in this approximation. There can be no neutral elementary particles and the spin-statistics theorem can be treated simply as a requirement that standard quantum theory should be based on complex numbers. Symmetry 2010-11-01 2 4 Article 10.3390/sym2041810 1810 1845 2073-8994 2010-11-01 doi: 10.3390/sym2041810 Felix M. Lev http://www.mdpi.com/2073-8994/2/4/1776 A concise discussion of spin-1/2 field equations with a special focus on Majorana spinors is presented. The Majorana formalism which describes massive neutral fermions by the help of two-component or four-component spinors is of fundamental importance for the understanding of mathematical aspects of supersymmetric and other extensions of the Standard Model of particle physics, which may play an increasingly important role at the beginning of the LHC era. The interplay between the two-component and the four-component formalism is highlighted in an introductory way. Majorana particles are predicted both by grand unified theories, in which these particles are neutrinos, and by supersymmetric theories, in which they are photinos, gluinos and other states. Symmetry 2010-10-27 2 4 Article 10.3390/sym2041776 1776 1809 2073-8994 2010-10-27 doi: 10.3390/sym2041776 Andreas Aste http://www.mdpi.com/2073-8994/2/4/1763 The paper presents an application of symmetry approach to finance. This symmetry approach comes from the gauge field theory in Physics. We revise the pricing model of financial derivatives in a financial market in a gauge symmetry view, and rewrite  it as a partial differential equation on a fiber bundle in covariant differential form so as to have invariance in form. The paper shows the form of the pricing equation can keep invariant under all the local num´eraire transformations, this symmetry behind the pricing equation of derivatives is revealed. In addition a  corresponding relationship between the curvature of the fiber bundle and the arbitrage in finance arises. Symmetry 2010-10-21 2 4 Article 10.3390/sym2041763 1763 1775 2073-8994 2010-10-21 doi: 10.3390/sym2041763 Shipeng Zhou Liuqing Xiao http://www.mdpi.com/2073-8994/2/4/1745 The paper reviews the polyanionic hexagons of silicon and germanium, focusing on aromaticity. The chair-like structures of hexasila- and hexagermabenzene are similar to a nonaromatic cyclohexane (CH2)6 and dissimilar to aromatic D6h-symmetric benzene (CH)6, although silicon and germanium are in the same group of the periodic table as carbon. Recently, six-membered silicon and germanium rings with extra electrons instead of conventional substituents, such as alkyl, aryl, etc., were calculated by us to have D6h symmetry and to be aromatic. We summarize here our main findings and the background needed to reach them, and propose a synthetically accessible molecule. Symmetry 2010-09-30 2 4 Review 10.3390/sym2041745 1745 1762 2073-8994 2010-09-30 doi: 10.3390/sym2041745 Masae Takahashi http://www.mdpi.com/2073-8994/2/3/1710 We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in economic space determines particular symmetries of the network, while the observed evolution of reciprocity is consistent with a symmetry breaking taking place in production space. Our results show that networks can be strongly affected by symmetry-breaking phenomena occurring in embedding spaces, and that stochastic network symmetries can successfully suggest, or rule out, possible underlying mechanisms. Symmetry 2010-09-27 2 3 Article 10.3390/sym2031710 1710 1744 2073-8994 2010-09-27 doi: 10.3390/sym2031710 Franco Ruzzenenti Diego Garlaschelli Riccardo Basosi http://www.mdpi.com/2073-8994/2/3/1683 In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefore propose a definition of stochastic symmetry based on graph ensembles and use it to review the main results of network theory from an unusual perspective. The results discussed here and in a companion paper show that stochastic symmetry highlights the most informative topological properties of real networks, even in noisy situations unaccessible to exact techniques. Symmetry 2010-09-27 2 3 Review 10.3390/sym2031683 1683 1709 2073-8994 2010-09-27 doi: 10.3390/sym2031683 Diego Garlaschelli Franco Ruzzenenti Riccardo Basosi http://www.mdpi.com/2073-8994/2/3/1653 The degree of p-electron (de)localization and aromaticity of a series of polybenzenoid hydrocarbons (PBHs) has been analyzed through the π-contribution to the electron localization function (ELFπ), calculated at the B3LYP/6-311G(d,p) hybrid density functional theory level. The extent of p-electron delocalization in the various hexagons of a PBH was determined through analysis of the bifurcation values of the ELFp basins (BV(ELFp)), the spans in the bifurcation values in each hexagon (ΔBV(ELFπ)), and the ring-closure bifurcation values of the ELFπ (RCBV(ELFπ)). These computed results were compared to the qualitative description of local aromaticities of the different hexagons in terms of Clar structures with p-sextets. Benzene, [18]annulene, and thirty two PBHs were analyzed at their equilibrium geometries, and benzene and triphenylene were also analyzed at bond length distorted structures. In general, the description of PBHs in terms of Clar valence structures is supported by the ELFp properties, although there are exceptions. For PBHs at their equilibrium geometries there is a clear sigmoidal relationship between the CC bond lengths and the amount of p-electron (de)localization at these bonds, however, this relationship is lost for bond distorted geometries. In the latter cases, we specifically examined benzene in D3h symmetric “1,3,5-cyclohexatriene” structures and triphenylene in eight different structures. From the distorted benzenes and triphenylenes it becomes clear that there is a distinct tendency for the p-electron network to retain delocalization (aromaticity). The ELFp analysis thus reveals an antidistortive rather than a distortive behavior of the p-electrons in these investigated compounds. Symmetry 2010-08-20 2 3 Article 10.3390/sym2031653 1653 1682 2073-8994 2010-08-20 doi: 10.3390/sym2031653 Jun Zhu Christian Dahlstrand Joshua R. Smith Sébastien Villaume Henrik Ottosson http://www.mdpi.com/2073-8994/2/3/1625 From elemental particles to human beings, matter is dissymmetric with respect to mirror symmetry. In 1860, Pasteur conjectured that biomolecular handedness— homochirality—may originate from certain inherent dissymmetric forces existing in the universe. Kipping, a pioneer of organosilicon chemistry, was interested in the handedness of sodium chlorate during his early research life. Since Kipping first synthesized several Si-Si bonded oligomers bearing phenyl groups, Si-Si bonded high polymers carrying various organic groups—polysilanes—can be prepared by sodium-mediated condensation of the corresponding organodichlorosilanes. Among these polysilanes, optically active helical polysilanes with enantiomeric pairs of organic side groups may be used for testing the mirror symmetry-breaking hypothesis by weak neutral current (WNC) origin in the realm of chemistry and material science. Several theoretical studies have predicted that WNC-existing chiral molecules with stereogenic centers and/or stereogenic bonds allow for distinguishing between image and mirror image molecules. Based on several amplification mechanisms, theorists claimed that minute differences, though still very subtle, may be detectable by precise spectroscopic and physicochemical measurements if proper chiral molecular pairs were employed. The present paper reports comprehensively an inequality between six pairs of helical polysilane high polymers, presumably, detectable by (chir)optical and achiral 29Si-/13C- NMR spectra, and viscometric measurements. Symmetry 2010-08-13 2 3 Article 10.3390/sym2031625 1625 1652 2073-8994 2010-08-13 doi: 10.3390/sym2031625 Michiya Fujiki http://www.mdpi.com/2073-8994/2/3/1591 Asymmetry and symmetry coexist in natural and human processes.  The vital role of symmetry in art has been well demonstrated. This article highlights the complementary role of asymmetry. Further we show that the interaction of asymmetric action (recursion) and symmetric opposition (sinusoidal waves) are instrumental in generating creative features (relatively low entropy, temporal complexity, novelty (less recurrence in the data than in randomized copies and complex frequency composition). These features define Bios, a pattern found in musical compositions and in poetry, except for recurrence instead of novelty. Bios is a common pattern in many natural and human processes (quantum processes, the expansion of the universe, gravitational waves, cosmic microwave background radiation, DNA, physiological processes, animal and human populations, and economic time series). The reduction in entropy is significant, as it reveals creativity and contradicts the standard claim of unavoidable decay towards disorder. Artistic creations capture fundamental features of the world. Symmetry 2010-07-30 2 3 Article 10.3390/sym2031591 1591 1624 2073-8994 2010-07-30 doi: 10.3390/sym2031591 Hector Sabelli Atoor Lawandow Abbe R. Kopra http://www.mdpi.com/2073-8994/2/3/1559 Chemists use one set of orbitals when comparing to a structural formula, hybridized AOs or NBOs for example, and another for reasoning in terms of frontier orbitals, MOs usually. Chemical arguments can frequently be made in terms of energy and/or electron density without the consideration of orbitals at all. All orbital representations, orthogonal or not, within a given function space are related by linear transformation. Chemical arguments based on orbitals are really energy or electron density arguments; orbitals are linked to these observables through the use of operators. The Valency Interaction Formula, VIF, offers a system of chemical reasoning based on the invariance of observables from one orbital representation to another. VIF pictures have been defined as one-electron density and Hamiltonian operators. These pictures are classified in a chemically meaningful way by use of linear transformations applied to them in the form of two pictorial rules and the invariance of the number of doubly, singly, and unoccupied orbitals or bonding, nonbonding, and antibonding orbitals under these transformations. The compatibility of the VIF method with the bond pair – lone pair language of Lewis is demonstrated. Different electron lone pair representations are related by the pictorial rules and have stability understood in terms of Walsh’s rules. Symmetries of conjugated ring systems are related to their electronic state by simple mathematical formulas. Description of lone pairs in conjugated systems is based on the strength and sign of orbital interactions around the ring. Simple models for bonding in copper clusters are tested, and the bonding of O2 to Fe(II) in hemoglobin is described. Arguments made are supported by HF, B3LYP, and MP2 computations. Symmetry 2010-07-23 2 3 Article 10.3390/sym2031559 1559 1590 2073-8994 2010-07-23 doi: 10.3390/sym2031559 Joseph Alia http://www.mdpi.com/2073-8994/2/3/1544 Intrinsic dynamics of the central vestibular system (CVS) appear to be at least partly determined by the symmetries of its connections. The CVS contributes to whole-body functions such as upright balance and maintenance of gaze direction. These functions coordinate disparate senses (visual, inertial, somatosensory, auditory) and body movements (leg, trunk, head/neck, eye). They are also unified by geometric conditions. Symmetry groups have been found to structure experimentally-recorded pathways of the central vestibular system. When related to geometric conditions in three-dimensional physical space, these symmetry groups make sense as a logical foundation for sensorimotor coordination. Symmetry 2010-07-22 2 3 Review 10.3390/sym2031544 1544 1558 2073-8994 2010-07-22 doi: 10.3390/sym2031544 Gin McCollum Douglas A. Hanes http://www.mdpi.com/2073-8994/2/3/1510 The human visual system is highly proficient in extracting bilateral symmetry from visual input. This paper reviews empirical and theoretical work on human symmetry perception with a focus on recent issues such as its neural underpinnings. Symmetry detection is shown to be a versatile, ongoing visual process that interacts with other visual processes. Evidence seems to converge towards the idea that  symmetry detection is subserved by a preprocessing stage involving spatial filters followed by information integration across the visual field in higher-tier cortical areas. Symmetry 2010-07-22 2 3 Review 10.3390/sym2031510 1510 1543 2073-8994 2010-07-22 doi: 10.3390/sym2031510 Matthias Sebastian Treder http://www.mdpi.com/2073-8994/2/3/1485 The HOMA (Harmonic Oscillator Model of Aromaticity) index, reformulated in 1993, has been very often applied to describe π-electron delocalization for mono- and polycyclic π-electron systems. However, different measures of π-electron delocalization were employed for the CC, CX, and XY bonds, and this index seems to be inappropriate for compounds containing heteroatoms. In order to describe properly various resonance effects (σ-π hyperconjugation, n-π conjugation, π-π conjugation, and aromaticity) possible for heteroatomic π-electron systems, some modifications, based on the original HOMA idea, were proposed and tested for simple DFT structures containing C, N, and O atoms. An abbreviation HOMED was used for the modified index. Symmetry 2010-07-12 2 3 Article 10.3390/sym2031485 1485 1509 2073-8994 2010-07-12 doi: 10.3390/sym2031485 Raczyńska Hallman Kolczyńska Stępniewski http://www.mdpi.com/2073-8994/2/3/1461 We propose a group-theoretical approach to the generalized oscillator algebra Aκ recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case κ ≥ 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case κ < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for Aκ in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices. Symmetry 2010-07-12 2 3 Article 10.3390/sym2031461 1461 1484 2073-8994 2010-07-12 doi: 10.3390/sym2031461 Atakishiyev Kibler Wolf http://www.mdpi.com/2073-8994/2/3/1450 The origin of homochirality in life remains a mystery that some believe is essential for life, and which may result from chiral symmetry breaking interactions with galactic organic material. Symmetry 2010-07-09 2 3 Review 10.3390/sym2031450 1450 1460 2073-8994 2010-07-09 doi: 10.3390/sym2031450 Cline http://www.mdpi.com/2073-8994/2/3/1423 The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may be used in the numerical computation of solid-state nuclear magnetic resonance spectra, and in spherical tensor analysis procedures. Symmetry 2010-07-07 2 3 Article 10.3390/sym2031423 1423 1449 2073-8994 2010-07-07 doi: 10.3390/sym2031423 Mamone Pileio Levitt http://www.mdpi.com/2073-8994/2/3/1401 The ground state of a system with symmetry can be described by a group G. This symmetry group G can be discrete or continuous. Thus for a crystal G is a finite group while for the vacuum state of a grand unified theory G is a continuous Lie group. The ground state symmetry described by G can change spontaneously from G to one of its subgroups H as the external parameters of the system are modified. Such a macroscopic change of the ground state symmetry of a system from G to H correspond to a “phase transition”. Such phase transitions have been extensively studied within a framework due to Landau. A vast range of systems can be described using Landau’s approach, however there are also systems where the framework does not work. Recently there has been growing interest in looking at such non-Landau type of phase transitions. For instance there are several “quantum phase transitions” that are not of the Landau type. In this short review we first describe a refined version of Landau’s approach in which topological ideas are used together with group theory. The combined use of group theory and topological arguments allows us to determine selection rule which forbid transitions from G to certain of its subgroups. We end by making a few brief remarks about non-Landau type of phase transition. Symmetry 2010-07-07 2 3 Article 10.3390/sym2031401 1401 1422 2073-8994 2010-07-07 doi: 10.3390/sym2031401 Sen