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Symmetry, Volume 3, Issue 2 (June 2011), Pages 126-388

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Research

Jump to: Review

Open AccessArticle Monochrome Symmetric Subsets in Colorings of Finite Abelian Groups
Symmetry 2011, 3(2), 126-133; doi:10.3390/sym3020126
Received: 13 January 2011 / Revised: 17 March 2011 / Accepted: 21 March 2011 / Published: 24 March 2011
Cited by 1 | PDF Full-text (279 KB)
Abstract 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. Full article
Open AccessArticle Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?
Symmetry 2011, 3(2), 134-154; doi:10.3390/sym3020134
Received: 21 February 2011 / Revised: 21 March 2011 / Accepted: 28 March 2011 / Published: 31 March 2011
Cited by 3 | PDF Full-text (391 KB)
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Quantum Symmetry)
Open AccessArticle Primary Amino Acid Lithium Salt-Catalyzed Asymmetric Michael Addition of Carbon Nucleophiles to Enones
Symmetry 2011, 3(2), 155-164; doi:10.3390/sym3020155
Received: 18 February 2011 / Revised: 6 April 2011 / Accepted: 7 April 2011 / Published: 8 April 2011
Cited by 13 | PDF Full-text (134 KB)
Abstract 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. Full article
(This article belongs to the Special Issue Asymmetric Organocatalysis)
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Open AccessArticle Facile and Convenient One-Pot Process for the Synthesis of Spirooxindole Derivatives in High Optical Purity Using (−)-(S)-Brevicolline as an Organocatalyst
Symmetry 2011, 3(2), 165-170; doi:10.3390/sym3020165
Received: 10 March 2011 / Revised: 11 April 2011 / Accepted: 12 April 2011 / Published: 20 April 2011
Cited by 12 | PDF Full-text (179 KB)
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Asymmetric Organocatalysis)
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Open AccessArticle Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry
Symmetry 2011, 3(2), 171-206; doi:10.3390/sym3020171
Received: 9 March 2011 / Revised: 6 April 2011 / Accepted: 12 April 2011 / Published: 27 April 2011
Cited by 11 | PDF Full-text (658 KB)
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Quantum Symmetry)
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Open AccessArticle Visual Discrimination of the 17 Plane Symmetry Groups
Symmetry 2011, 3(2), 207-219; doi:10.3390/sym3020207
Received: 1 April 2011 / Revised: 25 April 2011 / Accepted: 4 May 2011 / Published: 11 May 2011
PDF Full-text (1097 KB)
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry Processing in Perception and Art)
Open AccessArticle Similar Symmetries: The Role of Wallpaper Groups in Perceptual Texture Similarity
Symmetry 2011, 3(2), 246-264; doi:10.3390/sym3020246
Received: 1 March 2011 / Revised: 15 April 2011 / Accepted: 19 May 2011 / Published: 25 May 2011
Cited by 6 | PDF Full-text (6643 KB)
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry Processing in Perception and Art)
Open AccessArticle Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between
Symmetry 2011, 3(2), 305-324; doi:10.3390/sym3020305
Received: 11 January 2011 / Revised: 24 May 2011 / Accepted: 27 May 2011 / Published: 7 June 2011
Cited by 5 | PDF Full-text (202 KB)
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)
Open AccessArticle Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D2
Symmetry 2011, 3(2), 325-364; doi:10.3390/sym3020325
Received: 15 March 2011 / Revised: 24 May 2011 / Accepted: 27 May 2011 / Published: 9 June 2011
PDF Full-text (1457 KB)
Abstract
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 [...] Read more.
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]. Full article
(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)
Open AccessArticle Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation
Symmetry 2011, 3(2), 365-388; doi:10.3390/sym3020365
Received: 10 February 2011 / Revised: 27 May 2011 / Accepted: 30 May 2011 / Published: 10 June 2011
Cited by 4 | PDF Full-text (473 KB) | Supplementary Files
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry Processing in Perception and Art)
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Review

Jump to: Research

Open AccessReview Organocatalytic Enantioselective Henry Reactions
Symmetry 2011, 3(2), 220-245; doi:10.3390/sym3020220
Received: 29 March 2011 / Revised: 10 May 2011 / Accepted: 11 May 2011 / Published: 23 May 2011
Cited by 41 | PDF Full-text (1519 KB)
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Asymmetric Organocatalysis)
Open AccessReview Prolinethioamides versus Prolinamides in Organocatalyzed Aldol Reactions—A Comparative Study
Symmetry 2011, 3(2), 265-282; doi:10.3390/sym3020265
Received: 7 February 2011 / Revised: 23 May 2011 / Accepted: 24 May 2011 / Published: 1 June 2011
Cited by 10 | PDF Full-text (330 KB)
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Asymmetric Organocatalysis)
Open AccessReview Enriching the Symmetry of Maxwell Equations through Unprecedented Magnetic Responses of Artificial Metamaterials and Their Revolutionary Applications
Symmetry 2011, 3(2), 283-304; doi:10.3390/sym3020283
Received: 17 May 2011 / Revised: 24 May 2011 / Accepted: 26 May 2011 / Published: 3 June 2011
PDF Full-text (2313 KB)
Abstract
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 [...] Read more.
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. Full article

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