Symmetry

Symmetry, Vol. 4, Pages 39-115: Knots on a Torus: A Model of the Elementary Particles

Jack S. Avrin — Thu, 09 Feb 2012 00:00:00 CET

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, Vol. 4, Pages 26-38: Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori

Toru Ikeda — Fri, 20 Jan 2012 00:00:00 CET

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, Vol. 4, Pages 15-25: Towards Symmetry-Based Explanation of (Approximate) Shapes of Alpha-Helices and Beta-Sheets (and Beta-Barrels) in Protein Structure

Jaime Nava — Thu, 19 Jan 2012 00:00:00 CET

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, Vol. 4, Pages 1-14: Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Egon Schulte — Wed, 28 Dec 2011 00:00:00 CET

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, Vol. 3, Pages 828-851: Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

Hiroshi Fukuda — Mon, 12 Dec 2011 00:00:00 CET

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, Vol. 3, Pages 780-827: An Application of the Extended Global SO(3) × SO(3) × U(1) Symmetry of the Hubbard Model on a Square Lattice: The Spinon, η-Spinon, and c Fermion Description

Jose M. P. Carmelo — Mon, 12 Dec 2011 00:00:00 CET

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, Vol. 3, Pages 767-779: Information Theory of Networks

Matthias Dehmer — Tue, 29 Nov 2011 00:00:00 CET

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, Vol. 3, Pages 750-766: Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons

Preston R. Aldrich — Fri, 18 Nov 2011 00:00:00 CET

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, Vol. 3, Pages 699-749: d-Wave Superconductivity and s-Wave Charge Density Waves: Coexistence between Order Parameters of Different Origin and Symmetry

Toshikazu Ekino — Thu, 20 Oct 2011 00:00:00 CEST

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, Vol. 3, Pages 680-698: Symmetry and Evidential Support

Michael G. Titelbaum — Fri, 16 Sep 2011 00:00:00 CEST

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, Vol. 3, Pages 653-679: Lattices of Graphical Gaussian Models with Symmetries

Helene Gehrmann — Wed, 07 Sep 2011 00:00:00 CEST

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, Vol. 3, Pages 636-652: Symmetry and the Brown-Freiling Refutation of the Continuum Hypothesis

Paul Bartha — Tue, 06 Sep 2011 00:00:00 CEST

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, Vol. 3, Pages 611-635: Symmetry, Invariance and Ontology in Physics and Statistics

Julio Michael Stern — Thu, 01 Sep 2011 00:00:00 CEST

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, Vol. 3, Pages 600-610: High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument

Donald St. P. Richards — Fri, 26 Aug 2011 00:00:00 CEST

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, Vol. 3, Pages 574-599: Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas

Michael D. Perlman — Tue, 23 Aug 2011 00:00:00 CEST

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, Vol. 3, Pages 564-573: Green’s Symmetries in Finite Digraphs

Allen D. Parks — Mon, 15 Aug 2011 00:00:00 CEST

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, Vol. 3, Pages 541-563: Symmetry Aspects of the Band Structure and Motion Equations Applied in Calculating the Cyclotron Frequency of Electrons in Metals

Stanislaw Olszewski — Wed, 10 Aug 2011 00:00:00 CEST

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, Vol. 3, Pages 524-540: Action Duality: A Constructive Principle for Quantum Foundations

Ken B. Wharton — Wed, 27 Jul 2011 00:00:00 CEST

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, Vol. 3, Pages 503-523: Folded Sheet Versus Transparent Sheet Models for Human Symmetry Judgments

Jacques Ninio — Fri, 22 Jul 2011 00:00:00 CEST

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, Vol. 3, Pages 487-502: Classifying Entropy Measures

Angel Garrido — Wed, 20 Jul 2011 00:00:00 CEST

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, Vol. 3, Pages 472-486: On Symmetry of Independence Polynomials

Vadim E. Levit — Fri, 15 Jul 2011 00:00:00 CEST

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, Vol. 3, Pages 457-471: Mirror Symmetry Is Subject to Crowding

Gabrielle Roddy — Wed, 13 Jul 2011 00:00:00 CEST

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, Vol. 3, Pages 443-456: Reduction of Image Complexity Explains Aesthetic Preference for Symmetry

Chien-Chung Chen — Mon, 11 Jul 2011 00:00:00 CEST

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, Vol. 3, Pages 402-442: Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content

Mihai Prunescu — Thu, 07 Jul 2011 00:00:00 CEST

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, Vol. 3, Pages 389-401: Is the Notion of Time Really Fundamental?

Florian Girelli — Wed, 29 Jun 2011 00:00:00 CEST

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, Vol. 3, Pages 365-388: Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation

Tadamasa Sawada — Fri, 10 Jun 2011 00:00:00 CEST

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, Vol. 3, Pages 325-364: Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D2

Hiroshi Fukuda — Thu, 09 Jun 2011 00:00:00 CEST

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, Vol. 3, Pages 305-324: Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between

Alexis De Vos — Tue, 07 Jun 2011 00:00:00 CEST

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, Vol. 3, Pages 283-304: Enriching the Symmetry of Maxwell Equations through Unprecedented Magnetic Responses of Artificial Metamaterials and Their Revolutionary Applications

Yueh-Chun Lai — Fri, 03 Jun 2011 00:00:00 CEST

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, Vol. 3, Pages 265-282: Prolinethioamides versus Prolinamides in Organocatalyzed Aldol Reactions—A Comparative Study

Dorota Gryko — Wed, 01 Jun 2011 00:00:00 CEST

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, Vol. 3, Pages 246-264: Similar Symmetries: The Role of Wallpaper Groups in Perceptual Texture Similarity

Alasdair D. F. Clarke — Wed, 25 May 2011 00:00:00 CEST

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, Vol. 3, Pages 220-245: Organocatalytic Enantioselective Henry Reactions

Yolanda Alvarez-Casao — Mon, 23 May 2011 00:00:00 CEST

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, Vol. 3, Pages 207-219: Visual Discrimination of the 17 Plane Symmetry Groups

Klaus Landwehr — Wed, 11 May 2011 00:00:00 CEST

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, Vol. 3, Pages 171-206: Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry

Philip Goyal — Wed, 27 Apr 2011 00:00:00 CEST

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, Vol. 3, Pages 165-170: Facile and Convenient One-Pot Process for the Synthesis of Spirooxindole Derivatives in High Optical Purity Using (−)-(S)-Brevicolline as an Organocatalyst

Fliur Macaev — Wed, 20 Apr 2011 00:00:00 CEST

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, Vol. 3, Pages 155-164: Primary Amino Acid Lithium Salt-Catalyzed Asymmetric Michael Addition of Carbon Nucleophiles to Enones

Masanori Yoshida — Fri, 08 Apr 2011 00:00:00 CEST

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, Vol. 3, Pages 134-154: Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?

Karim P. Y. Thébault — Thu, 31 Mar 2011 00:00:00 CEST

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, Vol. 3, Pages 126-133: Monochrome Symmetric Subsets in Colorings of Finite Abelian Groups

Yuliya Zelenyuk — Thu, 24 Mar 2011 00:00:00 CET

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, Vol. 3, Pages 84-125: Asymmetric Organocatalytic Reactions of α,β-Unsaturated Cyclic Ketones

Renato Dalpozzo — Tue, 22 Mar 2011 00:00:00 CET

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, Vol. 3, Pages 72-83: Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian

Remi Leandre — Mon, 21 Mar 2011 00:00:00 CET

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, Vol. 3, Pages 54-71: The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art

Peter A. Van der Helm — Mon, 21 Mar 2011 00:00:00 CET

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, Vol. 3, Pages 37-53: The First Appearance of Symmetry in the Human Lineage: Where Perception Meets Art

Derek Hodgson — Tue, 01 Mar 2011 00:00:00 CET

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, Vol. 3, Pages 16-36: Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications

Young S. Kim — Mon, 14 Feb 2011 00:00:00 CET

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, Vol. 3, Pages 1-15: Symmetry in Complex Networks

Angel Garrido — Mon, 10 Jan 2011 00:00:00 CET

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, Vol. 2, Pages 1945-1980: Positive Cosmological Constant and Quantum Theory

Felix M. Lev — Fri, 19 Nov 2010 00:00:00 CET

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, Vol. 2, Pages 1925-1944: Three-Dimensional Facial Asymmetry in Attractive and Normal People from Childhood to Young Adulthood

Chiarella Sforza — Tue, 09 Nov 2010 00:00:00 CET

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, Vol. 2, Pages 1846-1924: Structures of Annulenes and Model Annulene Systems in the Ground and Lowest Excited States

Cristina Gellini — Fri, 05 Nov 2010 00:00:00 CET

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, Vol. 2, Pages 1810-1845: Introduction to a Quantum Theory over a Galois Field

Felix M. Lev — Mon, 01 Nov 2010 00:00:00 CET

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, Vol. 2, Pages 1776-1809: A Direct Road to Majorana Fields

Andreas Aste — Wed, 27 Oct 2010 00:00:00 CEST

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, Vol. 2, Pages 1763-1775: An Application of Symmetry Approach to Finance: Gauge Symmetry in Finance

Shipeng Zhou — Thu, 21 Oct 2010 00:00:00 CEST

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, Vol. 2, Pages 1745-1762: Polyanionic Hexagons: X6n– (X = Si, Ge)

Masae Takahashi — Thu, 30 Sep 2010 00:00:00 CEST

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, Vol. 2, Pages 1710-1744: Complex Networks and Symmetry II: Reciprocity and Evolution of World Trade

Franco Ruzzenenti — Mon, 27 Sep 2010 00:00:00 CEST

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, Vol. 2, Pages 1683-1709: Complex Networks and Symmetry I: A Review

Diego Garlaschelli — Mon, 27 Sep 2010 00:00:00 CEST

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, Vol. 2, Pages 1653-1682: On the Importance of Clar Structures of Polybenzenoid Hydrocarbons as Revealed by the π-Contribution to the Electron Localization Function

Jun Zhu — Fri, 20 Aug 2010 00:00:00 CEST

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, Vol. 2, Pages 1625-1652: Mirror Symmetry Breaking in Helical Polysilanes: Preference between Left and Right of Chemical and Physical Origin

Michiya Fujiki — Fri, 13 Aug 2010 00:00:00 CEST

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, Vol. 2, Pages 1591-1624: Asymmetry, Symmetry and Beauty

Hector Sabelli — Fri, 30 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1559-1590: Chemical Reasoning Based on an Invariance Property: Bond and Lone Pair Pictures in Quantum Structural Formulas

Joseph Alia — Fri, 23 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1544-1558: Symmetries of the Central Vestibular System: Forming Movements for Gravity and a Three-Dimensional World

Gin McCollum — Thu, 22 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1510-1543: Behind the Looking-Glass: A Review on Human Symmetry Perception

Matthias Sebastian Treder — Thu, 22 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1485-1509: On the Harmonic Oscillator Model of Electron Delocalization (HOMED) Index and its Application to Heteroatomic π-Electron Systems

Raczyńska — Mon, 12 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1461-1484: SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

Atakishiyev — Mon, 12 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1450-1460: Possible Physical Mechanisms in the Galaxy to Cause Homochiral Biomaterials for Life

Cline — Fri, 09 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1423-1449: Orientational Sampling Schemes Based on Four Dimensional Polytopes

Mamone — Wed, 07 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1401-1422: Symmetry, Symmetry Breaking and Topology

Sen — Wed, 07 Jul 2010 00:00:00 CEST

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, Vol. 2, Pages 1390-1400: How to Find the Fries Structures for Benzenoid Hydrocarbons

Ciesielski — Tue, 06 Jul 2010 00:00:00 CEST

An efficient algorithm leading to the Fries canonical structure is presented for benzenoid hydrocarbons. This is a purely topological approach, which is based on adjacency matrices and the Hadamard procedure of matrix multiplication. The idea is presented for naphthalene, as an example. The Fries canonical-structures are also derived for anthracene, coronene, triphenylene, phenanthrene, benz[a]pyrene, and one large benzenoid system. The Fries concept can be convenient for obtaining Clar structures with the maximum number of sextets, which in turn effectively represent π-electron (de)localization in benzenoid hydrocarbons.

Symmetry, Vol. 2, Pages 1375-1389: Symmetric Matrix Fields in the Finite Element Method

Awanou — Tue, 06 Jul 2010 00:00:00 CEST

The theory of elasticity is used to predict the response of a material body subject to applied forces. In the linear theory, where the displacement is small, the stress tensor which measures the internal forces is the variable of primal importance. However the symmetry of the stress tensor which expresses the conservation of angular momentum had been a challenge for finite element computations. We review in this paper approaches based on mixed finite element methods.

Symmetry, Vol. 2, Pages 1338-1374: Phase Diagram and Critical Properties within an Effective Model of QCD: The Nambu–Jona-Lasinio Model Coupled to the Polyakov Loop

Costa — Tue, 06 Jul 2010 00:00:00 CEST

We investigate the phase diagram of the so-called Polyakov–Nambu–Jona-Lasinio model at finite temperature and non-zero chemical potential with three quark flavors. Chiral and deconfinement phase transitions are discussed and the relevant order-like parameters are analyzed. The results are compared with simple thermodynamic expectations and lattice data. We present the phase diagram in the (T, μB) plane, paying special attention to the critical end point: as the strength of the flavor-mixing interaction becomes weaker, the critical end point moves to low temperatures and can even disappear.

Symmetry, Vol. 2, Pages 1322-1337: Emergent Dynamics of Five-Colour QCD Due to Dimensional Frustration

Walker — Thu, 01 Jul 2010 00:00:00 CEST

The consequences for five-colour QCD of a novel symmetry-breaking mechanism, published in an earlier paper, are further explored. In addition to the emergence of QED and three-colour QCD, there is also a candidate for the Z0μ. The representation theory of SU (N) is applied to the matter sector and yields the quark and electron charge ratios, and a mechanism for generating fermion particle masses.

Symmetry, Vol. 2, Pages 1270-1321: Symmetry and Asymmetry in Bouncing Gaits

Giovanni A. Cavagna — Fri, 25 Jun 2010 00:00:00 CEST

In running, hopping and trotting gaits, the center of mass of the body oscillates each step below and above an equilibrium position where the vertical force on the ground equals body weight. In trotting and low speed human running, the average vertical acceleration of the center of mass during the lower part of the oscillation equals that of the upper part, the duration of the lower part equals that of the upper part and the step frequency equals the resonant frequency of the bouncing system: we define this as on-offground symmetric rebound. In hopping and high speed human running, the average vertical acceleration of the center of mass during the lower part of the oscillation exceeds that of the upper part, the duration of the upper part exceeds that of the lower part and the step frequency is lower than the resonant frequency of the bouncing system: we define this as on-off-ground asymmetric rebound. Here we examine the physical and physiological constraints resulting in this on-off-ground symmetry and asymmetry of the rebound. Furthermore, the average force exerted during the brake when the body decelerates downwards and forwards is greater than that exerted during the push when the body is reaccelerated upwards and forwards. This landing-takeoff asymmetry, which would be nil in the elastic rebound of the symmetric spring-mass model for running and hopping, suggests a less efficient elastic energy storage and recovery during the bouncing step. During hopping, running and trotting the landing-takeoff asymmetry and the mass-specific vertical stiffness are smaller in larger animals than in the smaller animals suggesting a more efficient rebound in larger animals.

Symmetry, Vol. 2, Pages 1250-1269: Will Science and Consciousness Ever Meat? Complexity, Symmetry and Qualia

Roger Vergauwen — Fri, 25 Jun 2010 00:00:00 CEST

Within recent discussions in the Philosophy of Mind, the nature of conscious phenomenal states or qualia (also called ‘raw feels’ or the feel of ‘what it is like to be’) has been an important focus of interest. Proponents of Mind-Body Type-Identity theories have claimed that mental states can be reduced to neurophysiological states of the brain. Others have denied that such a reduction is possible; for them, there remains an explanatory gap. In this paper, functionalist, physicalist, epiphenomenalist, and biological models of the mind are discussed and compared. Donald Davidson’s Anomalous Monism is proposed as a unifying framework for a non-reductive theory of qualia and consciousness. Downward Causation, Emergence through Symmetry-breaking, and Dynamical Systems Theory are used to show how consciousness and qualia emerge from their neural substrate and can also be causally efficacious.

Symmetry, Vol. 2, Pages 1201-1249: Loss of Temporal Homogeneity and Symmetry in Statistical Systems: Deterministic Versus Stochastic Dynamics

Gujrati — Thu, 24 Jun 2010 00:00:00 CEST

A detailed analysis of deterministic (one-to-one) and stochastic (one-to-many) dynamics establishes that dS/dt > 0 is only consistent with the latter, which contains violation of temporal symmetry and homogeneity. We observe that the former only supports dS/dt = 0 and cannot give rise to Boltzmann’s molecular chaos assumption. The ensemble average is more meaningful than the temporal average, especially in non-equilibrium statistical mechanics of systems confined to disjoint phase space components, which commonly occurs at low temperatures. We propose that the stochasticity arises from extra degrees of freedom, which are not part of the system. We provide a simple resolution of the recurrence and irreversibility paradoxes.

Symmetry, Vol. 2, Pages 1180-1200: Experimental Test of L- and D-Amino Acid Binding to L- and D-Codons Suggests that Homochirality and Codon Directionality Emerged with the Genetic Code

Root-Bernstein — Wed, 23 Jun 2010 00:00:00 CEST

L-amino acids bind preferentially to their D-codons, but almost nothing is known about whether D-amino acids correspondingly prefer L-codons, or how codon directionality affects amino acid binding. To investigate these issues, two D-RNA-oligonucleotides having inverse base sequences (D-CGUA and D-AUGC) and their corresponding L-RNA-oligonucleotides (L-CGUA and L-AUGC) were synthesized and their affinity determined for Gly and eleven pairs of L- and D-amino acids. The data support the hypothesis (Root-Bernstein, Bioessays 2007; 29: 689–698) that homochirality and codon directionality emerged as a function of the origin of the genetic code itself. Further tests involving amplification methods are proposed.

Symmetry, Vol. 2, Pages 1156-1179: A Critical Assessment of the Performance of Magnetic and Electronic Indices of Aromaticity

Solà — Mon, 14 Jun 2010 00:00:00 CEST

The lack of reference aromatic systems in the realm of inorganic aromatic compounds makes the evaluation of aromaticity in all-metal and semimetal clusters a difficult task. To date, calculation of nucleus-independent chemical shifts (NICS) has been the most widely used method to discuss aromaticity in these systems. In the first part of this work, we briefly review our previous studies, showing some pitfalls of the NICS indicator of aromaticity in organic molecules. Then, we refer to our study on the performance of some aromaticity indices in a series of 15 aromaticity tests, which can be used to analyze the advantages and drawbacks of aromaticity descriptors. It is shown that indices based on the study of electron delocalization are the most accurate among those analyzed in the series of proposed tests, while NICS(1)zz and NICS(0)πzz present the best behavior among NICS indices. In the second part, we discuss the use of NICS and electronic multicenter indices (MCI) in inorganic clusters. In particular, we evaluate the aromaticity of two series of all-metal and semimetal clusters with predictable aromaticity trends by means of NICS and MCI. Results show that the expected trends are generally better reproduced by MCI than NICS. It is concluded that NICS(0)π and NICS(0)πzz are the kind of NICS that perform the best among the different NICS indices analyzed for the studied series of inorganic compounds.

Symmetry, Vol. 2, Pages 1135-1155: A Review of New Analytic Techniques for Quantifying Symmetry in Locomotion

Hsiao-Wecksler — Mon, 14 Jun 2010 00:00:00 CEST

We present a review of novel techniques developed by our research group to improve quantitative assessment of human movement, especially assessments related to symmetric and asymmetric gait patterns. These new methods use motion capture data of the lower limb joints (e.g., joint and body segment angular position and/or velocity, or joint center locations) and include: (1) Regions of Deviation (ROD) analysis, (2) complexity and variability of phase portraits, and (3) multivariate shape-alignment and decomposition. We provide example demonstrations of these techniques using data from infants, typical and atypically developing children, simulated injuries of a knee or ankle, and wheelchair propulsion.

Symmetry, Vol. 2, Pages 1121-1134: Symmetry in Boolean Satisfiability

Aloul — Fri, 11 Jun 2010 00:00:00 CEST

This paper reviews recent approaches on how to accelerate Boolean Satisfiability (SAT) search by exploiting symmetries in the problem space. SAT search algorithms traverse an exponentially large search space looking for an assignment that satisfies a set of constraints. The presence of symmetries in the search space induces equivalence classes on the set of truth assignments. The goal is to use symmetries to avoid traversing all assignments by constraining the search to visit a few representative assignments in each equivalence class. This can lead to a significant reduction in search runtime without affecting the completeness of the search.

Symmetry, Vol. 2, Pages 1108-1120: Minimum Phi-Divergence Estimators and Phi-Divergence Test Statistics in Contingency Tables with Symmetry Structure: An Overview

Pardo — Fri, 11 Jun 2010 00:00:00 CEST

In the last years minimum phi-divergence estimators (MϕE) and phi-divergence test statistics (ϕTS) have been introduced as a very good alternative to classical likelihood ratio test and maximum likelihood estimator for different statistical problems. The main purpose of this paper is to present an overview of the main results presented until now in contingency tables with symmetry structure on the basis of (MϕE) and (ϕTS).

Symmetry, Vol. 2, Pages 1099-1107: Fluctuating Asymmetry in Flies, What Does It Mean?

McLachlan — Fri, 04 Jun 2010 00:00:00 CEST

The degree of departure from perfect symmetry in organisms, fluctuating asymmetry (FA), is seen in most populations of animals. It has particular impact on choice of mate which lies within the world of sexual selection. Here I consider a relatively little studied aspect of sexual selection, i.e. the effect of FA on contests between males for mates, based not on display ornament but rather on agility seen in the mating systems of many insects. The model organism considered is the ubiquitous chironomid midge. In these flies, mating takes place in the air, so symmetry in the length of wings bears directly on a male’s aerobatic ability on which successful mating depends. The role of parasites and predators in creating and responding to FA in the host/prey midge is considered.

Symmetry, Vol. 2, Pages 1081-1098: The Relationships between Symmetry and Attractiveness and Mating Relevant Decisions and Behavior: A Review

Wade — Wed, 26 May 2010 00:00:00 CEST

Evolutionary theory based research shows that attractiveness is based on biological correlates that index appropriate estrogen and testosterone levels. Symmetry affects or plays a role in the perception of many of these correlates of attractiveness. Additionally, since attractiveness affects infidelity perception and reactions, sexual satisfaction, and personality perception, symmetry also affects these areas. This paper reviews the literature on symmetry showing how symmetry affects: the correlates of attractiveness, sexual satisfaction, personality, and infidelity perceptions and reactions.

Symmetry, Vol. 2, Pages 1055-1080: Photochirogenesis: Photochemical Models on the Origin of Biomolecular Homochirality

Meinert — Tue, 25 May 2010 00:00:00 CEST

Current research focuses on a better understanding of the origin of biomolecular asymmetry by the identification and detection of the possibly first chiral molecules that were involved in the appearance and evolution of life on Earth. We have reasons to assume that these molecules were specific chiral amino acids. Chiral amino acids have been identified in both chondritic meteorites and simulated interstellar ices. Present research reasons that circularly polarized electromagnetic radiation was identified in interstellar environments and an asymmetric interstellar photon-molecule interaction might have triggered biomolecular symmetry breaking. We review on the possible prebiotic interaction of ‘chiral photons’ in the form of circularly polarized light, with early chiral organic molecules. We will highlight recent studies on enantioselective photolysis of racemic amino acids by circularly polarized light and experiments on the asymmetric photochemical synthesis of amino acids from only one C and one N containing molecules by simulating interstellar environments. Both approaches are based on circular dichroic transitions of amino acids that will be presented as well.

Symmetry, Vol. 2, Pages 1033-1054: Symmetry OUT, Asymmetry IN

Lourenço — Tue, 25 May 2010 00:00:00 CEST

The formation of a perfect vertebrate body plan poses many questions that thrill developmental biologists. Special attention has been given to the symmetric segmental patterning that allows the formation of the vertebrae and skeletal muscles. These segmented structures derive from bilaterally symmetric units called somites, which are formed under the control of a segmentation clock. At the same time that these symmetric units are being formed, asymmetric signals are establishing laterality in nearby embryonic tissues, allowing the asymmetric placement of the internal organs. More recently, a “shield” that protects symmetric segmentation from the influence of laterality cues was uncovered. Here we review the mechanisms that control symmetric versus asymmetric development along the left-right axis among vertebrates. We also discuss the impact that these studies might have in the understanding of human congenital disorders characterized by congenital vertebral malformations and abnormal laterality phenotypes.

Symmetry, Vol. 2, Pages 1022-1032: Origin of Homochirality of Amino Acids in the Biosphere

Kojo — Thu, 13 May 2010 00:00:00 CEST

Discussions are made concerning realistic mechanisms for the origin of L-amino acids in the biosphere. As the most plausible mechanism, it is proposed that a mixture of racemic amino acids in the prebiotic sea caused spontaneous and effective optical resolution through self crystallization, even if asymmetric synthesis of a single amino acid has never occurred without the aid of an optically active molecule. This hypothesis is based on recrystallization of a mixture of D,L-amino acids in the presence of excess of D,L-asparagine (Asn). The enantiomeric excess (ee) of each amino acid in the resulting crystals indicates that crystallization of co-existing amino acids with the configuration same as that of Asn took place, although it was incidental whether the enrichment occurred in L- or D-amino acids. In addition, the resulting ee was sufficiently high (up to 100%) to account for the predominance of L-amino acids on the earth.

Symmetry, Vol. 2, Pages 999-1021: Magnetization Dynamics Symmetry in Spin Torque Induced Magnetization Switching

Wang — Fri, 07 May 2010 00:00:00 CEST

Magnetization dynamics symmetry plays important roles in magnetization switching. Here we study magnetic field and spin torque induced magnetization switching. Spin moment transferring from polarized itinerant electrons to local magnetization provides a magnetization switching mechanism without using external magnetic field. Besides its importance in fundamental magnetization switching dynamics, spin torque magnetization switching has great application potential for future nanoscale magnetoelectronic devices. The paper explores magnetization dynamics symmetry effects on spin torque induced magnetization switching, and its interactions with random fluctuations. We will illustrate the consequences of magnetization dynamics symmetry on the critical switching current magnitude and the thermal stability energy of spin torque induced magnetization switching, which are the two most important design criteria for nanoscale spin torque magnetic devices. The concept of Logarithmic magnetization susceptibility is used to extract symmetry and damping information on spin torque induced nonlinear magnetization dynamic processes, and provides paths to control spin torque induced switching in a fluctuating environment.

Symmetry, Vol. 2, Pages 970-998: Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

Fatibene — Thu, 29 Apr 2010 00:00:00 CEST

We review the Lagrangian formulation of (generalised) Noether symmetries in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called “Natural Theories” and “Gauge-Natural Theories” that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors, etc.). It is discussed how the use of Poincar´e–Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as “generators of Noether symmetries”, energy and reduced energy flow, Bianchi identities, weak and strong conservation laws, covariant conservation laws, Hamiltonian-like conservation laws (such as, e.g., so-calledADMlaws in General Relativity) with emphasis on the physical interpretation of the quantities calculated in specific cases (energy, angular momentum, entropy, etc.). A few substantially new and very recent applications/examples are presented to better show the power of the methods introduced: one in Classical Mechanics (definition of strong conservation laws in a frame-independent setting and a discussion on the way in which conserved quantities depend on the choice of an observer); one in Classical Field Theories (energy and entropy in General Relativity, in its standard formulation, in its spin-frame formulation, in its first order formulation “à la Palatini” and in its extensions to Non-Linear Gravity Theories); one in Quantum Field Theories (applications to conservation laws in Loop Quantum Gravity via spin connections and Barbero–Immirzi connections).

Symmetry, Vol. 2, Pages 950-969: Synthesis and Reactions of Dibenzo[a,e]pentalenes

Saito — Wed, 21 Apr 2010 00:00:00 CEST

Pentalene has recently received a considerable amount of attention as a ligand in sandwich-type transition metal complexes. In contrast, dibenzo[a,e]pentalene (hereafter denoted as dibenzopentalene), which is more π-extended than pentalene, has received less attention, despite its potential usefulness as a building block of ladder-type π-conjugated molecules, which have recently received growing interest. However, very recently, several novel efficient methods for the synthesis of dibenzopentalenes have been reported. This review surveys recent advances in the synthesis and reactions of dibenzopentalenes and describes the aromaticity of their ionic species.

Symmetry, Vol. 2, Pages 935-949: Chiroptical Properties of Amino Acids: A Density Functional Theory Study

Adrian-Scotto — Mon, 19 Apr 2010 00:00:00 CEST

Amino acids are involved in many scientific theories elucidating possible origins of life on Earth. One of the challenges when discussing the evolutionary origin of biopolymers such as proteins and oligonucleotides in living organisms is the phenomenon that these polymers implement monomers of exclusively one handedness, a feature called biomolecular homochirality. Many attempts have been made to understand this process of racemic symmetry breaking. Assuming an extraterrestrial origin of the molecular building blocks of living organisms, their susceptibility to asymmetric photolysis by the absorption of circularly polarized electromagnetic radiation in interstellar space was proposed. In order to predict whether the interaction of circularly polarized light with various racemic amino acids can induce an enantiomeric excess, we investigated the electronic and chiroptical properties of the amino acids valine and isovaline by a molecular modelling approach based on quantum chemistry (Density Functional Theory). The average spectra of both L-valine and L-isovaline have been produced on the basis of Boltzmann population analysis using computed spectra for the various conformations of each amino acid.

Symmetry, Vol. 2, Pages 916-934: Miscellania about Entropy, Energy, and Available Free Energy

Müller — Mon, 19 Apr 2010 00:00:00 CEST

While the main concepts of thermodynamics are universal, the application to specific systems is not. Thus, the universal concepts combined with specific constitutive relations permit the derivation of important results in such fields as diverse as physics, chemistry, physical chemistry, chemical engineering and rheology. In all of these fields equilibrium is characterized either by a maximum of entropy or by a minimum of available free energies, depending on boundary data. In the latter case there is a compromise between the entropic tendency to grow and the energetic tendency to decrease. After some historical considerations the situation is illustrated for several specific cases: planetary atmospheres, osmosis and elastic rubber molecules, pertaining to physics, chemistry and rheology respectively. Afterwards, in the later parts of the article, thermodynamics considerations are extrapolated to remote fields, to wit evolutionary genetics and sociology.

Symmetry, Vol. 2, Pages 907-915: Study of Dynamical Chiral Symmetry Breaking in (2 + 1) Dimensional Abelian Higgs Model

Li — Mon, 19 Apr 2010 00:00:00 CEST

In this paper, we study the dynamical mass generation in the Abelian Higgs model in 2 + 1 dimensions. Instead of adopting the approximations in [Jiang H et al., J. Phys. A 41 2008 255402.], we numerically solve the coupled Dyson–Schwinger Equations (DSEs) for the fermion and gauge boson propagators using a specific truncation for the fermion-photon vertex ansatz and compare our results with the corresponding ones in the above mentioned paper. It is found that the results quoted in the above paper remain qualitatively unaffected by refining the truncation scheme of the DSEs, although there exist large quantitative differences between the results presented in the above paper and ours. In addition, our numerical results show that the critical number of fermion flavor Nc decreases steeply with the the gauge boson mass ma (or the ratio of the Higgs mass mh to the gauge boson mass ma, r = mh/ma) increasing. It is thus easier to generate a finite fermion mass by the mechanism of DCSB for a small ratio r for a given ma.

Symmetry, Vol. 2, Pages 884-906: Symmetry versus Asymmetry in the Molecules of Life: Homomeric Protein Assemblies

Kojić-Prodić — Mon, 19 Apr 2010 00:00:00 CEST

The essay is dedicated to the relation of symmetry and asymmetry-chirality in Nature. The Introduction defines symmetry and its impact on basic definitions in science and human activities. The following section Chirality of molecules reveals breifly development of notion of chirality and its significance in living organisms and science. Homochirality is a characteristic hallmark of life and its significance is presented in the section Homochirality of Life. Proteins, important constituents of living cells performing versatile functions are chiral macromolecules composed of L-amino acids. In particular, the protein assemblies are of a great importance in functions of a cell. Therefore, they have attracted researches to examine them from different points of view. Among proteins of known three-dimensional structures about 50–80% of them exist as homomeric protein complexes. Protein monomers lack any intrinsic, underlying symmetry, i.e. enantiomorphic protein molecules involve left-handed amino acids but their asymmetry does not appear to extend to the level of quaternary structures (homomeric complexes) as observed by Chothia in 1991. In the section Homomeric assemblies we performed our analysis of very special cases of homomers revealing non-crystallographic symmetry in crystals. Homochiral proteins can crystallize only in enantiomorphic space groups. Among 230 existing space groups 65 are enantiomorphic containing limited symmetry elements that are rotation and screw-rotation axes. Any axis of rotation symmetry of a crystal lattice must be two-fold, three-fold, four-fold, or six-fold. Five-fold, seven-fold, and higher-fold rotation symmetry axes are incompatible with the symmetry under spatial displacement of the three-dimensional crystal lattice.

Symmetry, Vol. 2, Pages 868-883: Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation

Chhay — Mon, 19 Apr 2010 00:00:00 CEST

Invariant numerical schemes possess properties that may overcome the numerical properties of most of classical schemes. When they are constructed with moving frames, invariant schemes can present more stability and accuracy. The cornerstone is to select relevant moving frames. We present a new algorithmic process to do this. The construction of invariant schemes consists in parametrizing the scheme with constant coefficients. These coefficients are determined in order to satisfy a fixed order of accuracy and an equivariance condition. Numerical applications with the Burgers equation illustrate the high performances of the process.

Symmetry, Vol. 2, Pages 848-867: The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling

Al Sayed — Fri, 16 Apr 2010 00:00:00 CEST

In this work, the non-isothermal Navier–Stokes equations are studied from the group theory point of view. The symmetry group of the equations is presented and discussed. Some standard turbulence models are analyzed with the symmetries of the equations. A class of turbulence models which preserve the physical properties contained in the symmetry group is built. The proposed turbulence models are applied to an illustrative example of natural convection in a differentially heated cavity, and the results are presented.

Symmetry, Vol. 2, Pages 799-847: Replication and Abstraction: Symmetry in Automated Formal Verification

Wahl — Wed, 14 Apr 2010 00:00:00 CEST

This article surveys fundamental and applied aspects of symmetry in system models, and of symmetry reduction methods used to counter state explosion in model checking, an automated formal verification technique. While covering the research field broadly, we particularly emphasize recent progress in applying the technique to realistic systems, including tools that promise to elevate the scope of symmetry reduction to large-scale program verification. The article targets researchers and engineers interested in formal verification of concurrent systems.

Symmetry, Vol. 2, Pages 767-798: The Role of Stochastic Models in Interpreting the Origins of Biological Chirality

Lente — Mon, 12 Apr 2010 00:00:00 CEST

This review summarizes recent stochastic modeling efforts in the theoretical research aimed at interpreting the origins of biological chirality. Stochastic kinetic models, especially those based on the continuous time discrete state approach, have great potential in modeling absolute asymmetric reactions, experimental examples of which have been reported in the past decade. An overview of the relevant mathematical background is given and several examples are presented to show how the significant numerical problems characteristic of the use of stochastic models can be overcome by non-trivial, but elementary algebra. In these stochastic models, a particulate view of matter is used rather than the concentration-based view of traditional chemical kinetics using continuous functions to describe the properties system. This has the advantage of giving adequate description of single-molecule events, which were probably important in the origin of biological chirality. The presented models can interpret and predict the random distribution of enantiomeric excess among repetitive experiments, which is the most striking feature of absolute asymmetric reactions. It is argued that the use of the stochastic kinetic approach should be much more widespread in the relevant literature.

Symmetry, Vol. 2, Pages 722-766: Application of Symmetry Methods to Low-Dimensional Heisenberg Magnets

Bostrem — Fri, 09 Apr 2010 00:00:00 CEST

An account of symmetry is very fruitful in studies of quantum spin systems. In the present paper we demonstrate how to use the spin SU(2) and the point symmetries in optimization of the theoretical condensed matter tools: the exact diagonalization, the renormalization group approach, the cluster perturbation theory. We apply the methods for study of Bose-Einstein condensation in dimerized antiferromagnets, for investigations of magnetization processes and magnetocaloric effect in quantum ferrimagnetic chain.

Symmetry, Vol. 2, Pages 707-721: Symmetry and Asymmetry Level Measures

Garrido — Thu, 08 Apr 2010 00:00:00 CEST

Usually, Symmetry and Asymmetry are considered as two opposite sides of a coin: an object is either totally symmetric, or totally asymmetric, relative to pattern objects. Intermediate situations of partial symmetry or partial asymmetry are not considered. But this dichotomy on the classification lacks of a necessary and realistic gradation. For this reason, it is convenient to introduce "shade regions", modulating the degree of Symmetry (a fuzzy concept). Here, we will analyze the Asymmetry problem by successive attempts of description and by the introduction of the Asymmetry Level Function, as a new Normal Fuzzy Measure. Our results (both Theorems and Corollaries) suppose to be some new and original contributions to such very active and interesting field of research. Previously, we proceed to the analysis of the state of art.

Symmetry, Vol. 2, Pages 658-706: Lie Symmetries of Differential Equations: Classical Results and Recent Contributions

Oliveri — Thu, 08 Apr 2010 00:00:00 CEST

Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conservation laws, to the construction of links between different differential equations that turn out to be equivalent. This paper reviews some well known results of Lie group analysis, as well as some recent contributions concerned with the transformation of differential equations to equivalent forms useful to investigate applied problems.

Symmetry, Vol. 2, Pages 609-657: Spontaneous Symmetry Breaking and Nambu–Goldstone Bosons in Quantum Many-Body Systems

Brauner — Wed, 07 Apr 2010 00:00:00 CEST

Spontaneous symmetry breaking is a general principle that constitutes the underlying concept of a vast number of physical phenomena ranging from ferromagnetism and superconductivity in condensed matter physics to the Higgs mechanism in the standard model of elementary particles. I focus on manifestations of spontaneously broken symmetries in systems that are not Lorentz invariant, which include both nonrelativistic systems as well as relativistic systems at nonzero density, providing a self-contained review of the properties of spontaneously broken symmetries specific to such theories. Topics covered include: (i) Introduction to the mathematics of spontaneous symmetry breaking and the Goldstone theorem. (ii) Minimization of Higgs-type potentials for higher-dimensional representations. (iii) Counting rules for Nambu–Goldstone bosons and their dispersion relations. (iv) Construction of effective Lagrangians. Specific examples in both relativistic and nonrelativistic physics are worked out in detail.

Symmetry, Vol. 2, Pages 582-608: Broken Time Translation Symmetry as a Model for Quantum State Reduction

Van Wezel — Thu, 01 Apr 2010 00:00:00 CEST

The symmetries that govern the laws of nature can be spontaneously broken, enabling the occurrence of ordered states. Crystals arise from the breaking of translation symmetry, magnets from broken spin rotation symmetry and massive particles break a phase rotation symmetry. Time translation symmetry can be spontaneously broken in exactly the same way. The order associated with this form of spontaneous symmetry breaking is characterised by the emergence of quantum state reduction: systems which spontaneously break time translation symmetry act as ideal measurement machines. In this review the breaking of time translation symmetry is first compared to that of other symmetries such as spatial translations and rotations. It is then discussed how broken time translation symmetry gives rise to the process of quantum state reduction and how it generates a pointer basis, Born’s rule, etc. After a comparison between this model and alternative approaches to the problem of quantum state reduction, the experimental implications and possible tests of broken time translation symmetry in realistic experimental settings are discussed.

Symmetry, Vol. 2, Pages 554-581: Symmetry as an Intrinsically Dynamic Feature

Di Gesu — Thu, 01 Apr 2010 00:00:00 CEST

Symmetry is one of the most prominent spatial relations perceived by humans, and has a relevant role in attentive mechanisms regarding both visual and auditory systems. The aim of this paper is to establish symmetry, among the likes of motion, depth or range, as a dynamic feature in artificial vision. This is achieved in the first instance by assessing symmetry estimation by means of algorithms, putting emphasis on erosion and multi-resolution approaches, and confronting two ensuing problems: the isolation of objects from the context, and the pertinence (or lack thereof) of some salient points, such as the centre of mass. Next a geometric model is illustrated and detailed, and the problem of measuring symmetry in a world where symmetry is not perfect nor the only attention trigger is tackled. Two algorithmic lines, based on the so-called symmetry kernel and its evolution with pattern warping, and by correlation of blocks with varying sizes and positions, are proposed and investigated. An extended illustration of the power of symmetry as a feature, based on face expression recognition, concludes the paper.

Symmetry, Vol. 2, Pages 541-553: Fluctuating Asymmetry and Steroid Hormones: A Review

Benderlioglu — Thu, 01 Apr 2010 00:00:00 CEST

Fluctuating asymmetry (FA) represents random, minor deviations from perfect symmetry in paired traits. Because the development of the left and right sides of a paired trait is presumably controlled by an identical set of genetic instructions, these small imperfections are considered to reflect genetic and environmental perturbations experienced during ontogeny. The current paper aims to identify possible neuroendocrine mechanisms, namely the actions of steroid hormones that may impact the development of asymmetrical characters as a response to various stressors. In doing so, it provides a review of the published studies on the influences of glucocorticoids, androgens, and estrogens on FA and concomitant changes in other health and fitness indicators. It follows the premise that hormonal measures may provide direct, non-invasive indicators of how individuals cope with adverse life conditions, strengthening the associations between FA and health, fitness, and behavior.

Symmetry, Vol. 2, Pages 466-540: Fluctuating Asymmetry: Methods, Theory, and Applications

Graham — Thu, 25 Mar 2010 00:00:00 CET

Fluctuating asymmetry consists of random deviations from perfect symmetry in populations of organisms. It is a measure of developmental noise, which reflects a population’s average state of adaptation and coadaptation. Moreover, it increases under both environmental and genetic stress, though responses are often inconsistent. Researchers base studies of fluctuating asymmetry upon deviations from bilateral, radial, rotational, dihedral, translational, helical, and fractal symmetries. Here, we review old and new methods of measuring fluctuating asymmetry, including measures of dispersion, landmark methods for shape asymmetry, and continuous symmetry measures. We also review the theory, developmental origins, and applications of fluctuating asymmetry, and attempt to explain conflicting results. In the process, we present examples from the literature, and from our own research at “Evolution Canyon” and elsewhere.