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Symmetry, Volume 3, Issue 1 (March 2011), Pages 1-125

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Research

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Open AccessArticle Symmetry in Complex Networks
Symmetry 2011, 3(1), 1-15; doi:10.3390/sym3010001
Received: 16 November 2010 / Revised: 4 January 2011 / Accepted: 7 January 2011 / Published: 10 January 2011
Cited by 6 | PDF Full-text (224 KB)
Abstract
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
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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. Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
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Open AccessArticle The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art
Symmetry 2011, 3(1), 54-71; doi:10.3390/sym3010054
Received: 9 February 2011 / Revised: 7 March 2011 / Accepted: 16 March 2011 / Published: 21 March 2011
Cited by 11 | PDF Full-text (258 KB)
Abstract
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
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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. Full article
Open AccessArticle Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian
Symmetry 2011, 3(1), 72-83; doi:10.3390/sym3010072
Received: 16 February 2011 / Accepted: 7 March 2011 / Published: 21 March 2011
Cited by 4 | PDF Full-text (232 KB)
Abstract 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. Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)

Review

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Open AccessReview Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications
Symmetry 2011, 3(1), 16-36; doi:10.3390/sym3010016
Received: 6 January 2011 / Revised: 7 February 2011 / Accepted: 11 February 2011 / Published: 14 February 2011
Cited by 6 | PDF Full-text (251 KB)
Abstract
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
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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. Full article
(This article belongs to the Special Issue Quantum Symmetry)
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Open AccessReview The First Appearance of Symmetry in the Human Lineage: Where Perception Meets Art
Symmetry 2011, 3(1), 37-53; doi:10.3390/sym3010037
Received: 17 January 2011 / Revised: 23 February 2011 / Accepted: 23 February 2011 / Published: 1 March 2011
Cited by 11 | PDF Full-text (596 KB)
Abstract
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
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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. Full article
(This article belongs to the Special Issue Symmetry Processing in Perception and Art)
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Open AccessReview Asymmetric Organocatalytic Reactions of α,β-Unsaturated Cyclic Ketones
Symmetry 2011, 3(1), 84-125; doi:10.3390/sym3010084
Received: 18 February 2011 / Revised: 8 March 2011 / Accepted: 9 March 2011 / Published: 22 March 2011
Cited by 20 | PDF Full-text (457 KB)
Abstract
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
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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. Full article
(This article belongs to the Special Issue Asymmetric Organocatalysis)

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