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Topic: Discrete symmetry

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spontaneous symmetry breaking

symmetry of second derivatives

bilateral symmetry

CP symmetry

radial symmetry

P symmetry

T symmetry

CPT symmetry

mirror symmetry

rotational symmetry

gauge symmetries

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	Symmetry group - Wikipedia, the free encyclopedia
		The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations) under which the figure is invariant.
		Discrete symmetry groups come in two types: finite point groups, which include only rotations and reflections - they are in fact just the finite subgroups of O(n), and infinite lattice groups, which also include translations and possibly glide reflections.
		The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups.
	en.wikipedia.org /wiki/Symmetry_group (1574 words)

	Symmetry and Symmetry Breaking
		The extension of the concept of continuous symmetry from “global” symmetries (such as the Galilean group of spacetime transformations) to “local” symmetries is one of the important developments in the concept of symmetry in physics that took place in the twentieth century.
		The discrete symmetries C, P and T are connected by the so-called CPT theorem, demonstrated by Lüders in 1952, which states that the combination of C, P, and T is a general symmetry of physical laws.
		Symmetries may be used to explain (i) the form of the laws, and (ii) the occurrence (or non-occurrence) of certain events (this latter in a manner analogous to the way in which the laws explain why certain events occur and not others).
	plato.stanford.edu /entries/symmetry-breaking (9818 words)

	Chapter 5
		The non-polarity of a certain symmetry element is conditioned by the existence of adequate reflections (inversions) commuting with it, while bipolarity is caused by the presence of adequate central reflections or their equivalents.
		Because visually presentable continuous symmetry groups and maximal discrete symmetry groups generated by reflections or their equivalents are, also, the simplest ones in a constructional and in a visual sense, constructional simplicity is directly connected to the degree of symmetry.
		Since symmetry is one of the basic structure-organization laws in nature, the existence of natural models was one of the origins of ornaments and an inexhaustible source of ideas during all the history of ornamental art.
	www.emis.de /monographs/jablan/chap5.htm (5552 words)

	CONTROL and DYNAMICAL SYSTEMS - California Institute of Technology
		Discrete exterior calculus is modeled on a primal simplicial complex, and a dual circumcentric cell complex.
		Discrete connections on principal bundles are obtained by introducing the discrete Atiyah sequence, and considering splittings of the sequence.
		Discrete connections provide an intrinsic coordinatization of the reduced discrete space, and the necessary discrete geometry to develop more general discrete symmetry reduction techniques.
	www.cds.caltech.edu /research/seminars/seminar.php?id=9 (353 words)

	Summary_Chapter_4_JJ
		Symmetry in classical physics is here displayed in the case we have managed to have one of the configuration space coordinates to be the symmetry, as given in (4.1.1).
		Apart from symmetries related to a continuous parameter (such as rotations, translations etc..), there can also be a set of discrete symmetries, which belong to a finite, or a countable set of operations.
		The candidate symmetries S here are NOT symmetries of the physical situation, in the sense that the physical situation Mx and the physical situation Sx are physically distinct situations (which, if S turns out to be a symmetry, have similar behaviour under the dynamics).
	perso.wanadoo.fr /patrick.vanesch/nrqmJJ/Summary_Chapter_4_JJ.html (3226 words)

	Chapter 2.6
		In the mathematical theory of symmetry, the first complete list of the discrete symmetry groups of ornaments was given by E.S. Fedorov (1891b), although this problem was, even before that, the subject of study of many important mathematicians.
		The symmetry group p6 is the subgroup of the index 2 of the symmetry group p6m, and the symmetry group p3 is the subgroup of the index 2 of the symmetry group p6.
		The visual stationariness of an ornament is conditioned by the non-polarity of its axes and rotations, by the presence of reflections, especially perpendicular ones, by the absence of glide reflections suggesting alternating motions and by the absence of the enantiomorphism.
	www.emis.de /monographs/jablan/chap26.htm (12081 words)

	Chapter 2.4 (Site not responding. Last check: 2007-10-04)
		According to the principle of maximal symmetry, a survey of group-subgroup relations may, in a way, serve as an indicator of the frequency of occurrence of the particular symmetry groups of friezes in ornamental art, and also for recognition and evidence of symmetry substructures.
		Besides the objective elements of symmetry, the visual impression is influenced by the subjective elements referring to the physiological-psychological properties of the visual perception (e.g., perception of the "right" diagonal as "ascending" and the "left" as "descending", etc.), so that this dependence may be very complex.
		Through knowledge of the geometric-algebraic properties of the symmetry groups of friezes, the visual qualities of the corresponding friezes may be anticipated directly from the presentations and structures of their symmetry groups.
	www-irma.u-strasbg.fr /EMIS/monographs/jablan/chap24.htm (3852 words)

	Peter Hydon - Symmetries
		Such symmetries are important in many applications, but (generally speaking) it is not possible to calculate them all directly from the symmetry condition.
		Each discrete symmetry maps the set of Lie symmetries to itself; moreover the mapping is linear with constant coefficients.
		The discrete part of the variational complex is analogous to the de Rham complex.
	www.maths.surrey.ac.uk /personal/st/P.Hydon/sym.htm (927 words)

	[No title] (Site not responding. Last check: 2007-10-04)
		The program computes the invariants J and K in reduced form, determines the dimension of the symmetry group, and, in the case of a finite symmetry group, applies the Maple command solve to solve the two polynomial symmetry equations (3,4) to find explicit form of symmetries.
		The output of symm consists of the projective index of the form and the explicit formulae for its discrete projective symmetries.
		The program also notifies the user if the symmetry group is not discrete, or is in the maximal discrete symmetry class.
	www4.ncsu.edu /~iakogan/Maple/SympolCode.txt (229 words)

	Discrete gauge symmetries, baryon number and large extra dimensions
		Krauss and Wilczek have shown that an unbroken discrete gauge symmetry is respected by gravitationally mediated processes.
		This has led to a search for such a symmetry compatible with the standard model or MSSM that would protect protons from gravitationally mediated decay in a universe with a low scale for quantum gravity (large extra dimensions).
		The fact that the discrete symmetry must remain unbroken and have a gauge origin puts important restrictions on the space of possible discrete symmetries.
	stacks.iop.org /1126-6708/2005/i=03/a=034 (292 words)

	The Design of 2-Colour Wallpaper Patterns Using Methods Based on Chaotic Dynamics and Symmetry
		There are restrictions imposed by symmetry on the symmetries of attractors.
		Translational symmetries preserve colour, while glide reflection symmetries (along either of the glide lines a or b) reverse colour.
		Roughly speaking, our two-colourings depend on symmetry and dynamics in the sense that colours measure the probability of lying in one of the subpatterns as well as the statistics of the iteration on the subpattern.
	arpam.free.fr /mfield.html (5289 words)

	John Preskill's Publications
		Neutrino Masses and Family Symmetry (with B. Grinstein and M. Wise), Phys.
		Magnetic Wormholes and Topological Symmetry (with A. Gupta, J. Hughes, and M. Wise), Nucl.
		Cosmology and Broken Discrete Symmetry (with S.P. Trivedi, F. Wilczek and M. Wise), Nucl.
	www.theory.caltech.edu /people/preskill/pub.html (785 words)

	Diamond Theory: Symmetry in Binary Spaces
		Symmetry is often described as invariance under a group of transformations.
		An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.
		In the 4x4 case, D is a four-diamond figure (left, below) and G is a group of 322,560 permutations generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2x2 quadrants.
	m759.freeservers.com (1917 words)

	Discrete symmetry - Wikipedia, the free encyclopedia
		In theoretical physics, a discrete symmetry is a symmetry under the transformations of a discrete group - i.e.
		a topological group with a discrete topology whose elements form a finite or a countable set.
		This page was last modified 05:28, 19 February 2005.
	en.wikipedia.org /wiki/Discrete_symmetry (81 words)

	Heteroclinic Cycles and Modulated Travelling Waves in a System with D4 Symmetry -- from Mathematica Information Center
		We study a system of two complex ordinary differential equations with D4 symmetry describing the effects of the symmetry breaking O(2) to D4 on an interaction between Fourier modes with wavenumbers in the ratio 1 : 2.
		This change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag.
		The method of averaging is used to show that some of the quasiperiodic solutions of the O(2) system persist under the symmetry breaking, while others break into sets of periodic orbits.
	library.wolfram.com /infocenter/Articles/2817 (203 words)

	A Fearful Symmetry \| Musings
		Longtime readers of this blog will know that I think that discrete symmetries (particularly, those that suppress proton decay and flavour-changing neutral currents) are a serious challenge for ideas about the Landscape and, more generally, applications of the Anthropic Principle to String Theory Vacua (see, for instance, these three and posts).
		To suppress these bad operators (note that the observed proton lifetime is at least 20 orders of magnitude longer than the anthropic bound), one generally invokes a discrete symmetry.
		Such discrete symmetries typically only occur on subspaces of the moduli space of very high codimension.
	golem.ph.utexas.edu /~distler/blog/archives/000467.html (1177 words)

	The Math Forum - Math Library - Symmetry/Tesselltns (Site not responding. Last check: 2007-10-04)
		Topics include an introduction to symmetry; symmetry in the alphabet, in flags, in quilt blocks; reflectional and rotational symmetry; strip patterns; regular and irregular polygons; paper-folding and cutting; origami; tessellation in nature; kaleidoscopes; regular, semiregular, and demiregular tessellations; islamic tessellations; M. Escher; jigsaw puzzles, tessellating art; and explorations with TesselMania.
		Symmetry in nature: butterfly and moth images from a variety of sources.
		Materials about symmetry and classification of repeating patterns for students in grades 7-10 using wallpaper patterns, to be used as either an introduction or a review: a classroom-ready source of information.
	mathforum.org /library/topics/sym_tess (2543 words)

	Research Interests: Joanne D Cohn (Site not responding. Last check: 2007-10-04)
		However, instead of putting in a continuous symmetry in order to get inflation and then breaking the symmetry to get inflation to start and end, we use an unbroken discrete symmetry to induce the lowest order continuous symmetry and thus inflation.
		Also, if the potential is to have special points where the absolute values of the fields are different, for instance in order to produce a hybrid exit in a supergravity model, a discrete nonabelian symmetry is useful.
		The discreteness of the symmetries allows inflation to end via a hybrid or mutated hybrid mechanism, as neighboring points in field space can have different potentials because the exact symmetry is discrete.
	cfa-www.harvard.edu /~jcohn/res_ints.html (404 words)

	The Divisions of mathematics
		Algebra is principally concerned with symmetry, patterns, discrete sets, and the rules for manipulating arithmetic operations; one might think of this as the outgrowth of arithmetic and algebra classes in primary and secondary school.
		Geometry is concerned with shapes and sets, and the properties of them which are preserved under various kinds of motions.
		It's false to assume that mathematics consists of discrete subfields, it's false to assume that there is an objective way to gather those subfields into main divisions, and it's false to assume that there is an accurate two-dimensional positioning of the parts.
	www.math.niu.edu /~rusin/known-math/index/tour_div.html (1215 words)

	The Discrete Symmetry Transform in Computer Vision - Di Gesu, Valenti (ResearchIndex)
		A new Symmetry Operator for the analysis of sequences of images - Di Gesù, Valenti
		Di Ges`u V. and Valenti C., The Discrete Symmetry Transform in computer vision.
		23 the detection of the Axes of Symmetry of Symmetric and Almos..
	citeseer.ist.psu.edu /367846.html (575 words)

	Soliton solutions for a classical field theory with Z(3) symmetry (Site not responding. Last check: 2007-10-04)
		The study of interacting classical fields in 1+1 dimensions with a discrete internal symmetry presents some relevant features and applications [1].
		In this work we consider the case of the Z(3) symmetry and we search for the most simple soliton solutions; we exhibit these and their physical properties in a general, relativistically covariant notation.
		Other areas of application of cyclic, symmetric field theories could arise, for instance, in solid state physics, where the symmetry follows from the lattice structure of the crystals, and in elementary particle physics, where the color theory of quarks and gluons imposes some cyclic transformation properties of the quantum variables.
	www.sif.it /cimento/toca/110.04/05/05.html (2968 words)

	Nodal solitons and the nonlinear breaking of discrete symmetry
		These solitons have their origin in a novel mechanism of breaking of discrete symmetry by the presence of nonlinearities.
		These so-called nodal solitons are characterized by nodal lines determined by the discrete symmetry of the system.
		A. Ferrando, M. Zacarés, P. Andreés, P. Fernández de Córdoba, and J. Monsoriu, "Nodal solitons and the nonlinear breaking of discrete symmetry," Opt.
	www.opticsexpress.org /abstract.cfm?URI=OPEX-13-4-1072 (340 words)

	Discrete symmetry and GUT breaking
		We study the supersymmetric GUT models in which the supersymmetry and GUT gauge symmetry can be broken by a discrete symmetry.
		First, with the ansatz that there exist discrete symmetries in the branes' neighborhoods, we discuss the general reflection Z
		In those models, the extra dimensions can be large and the KK states can be set arbitrarily heavy.
	www.edpsciences.org /abstract/epjC/v24/p595 (158 words)

	Discrete Symmetry Noether theorem? (Site not responding. Last check: 2007-10-04)
		A variant of Noether's Theorem implies that exact conservation of discrete angular momentum must enforce asymptotic continuous angular isotropy as one looks at processes some level above the scale of the cellular array.
		Similarly, absolute microscopic conservation of discrete units of momentum must enforce asymptotic continuous translational symmetry.
		Conservation of discrete units of energy does the same for asymptotic continuous time symmetry.
	www.lns.cornell.edu /spr/2003-05/msg0051290.html (180 words)

	Abstract mv_1998a (Site not responding. Last check: 2007-10-04)
		Nathan C. Carter, Richard L. Eagles, Stephen M. Grimes, Andrew C. Hahn, and Clifford A. Reiter, Chaotic Attractors with Discrete Planar Symmetry, Chaos, Solitons and Fractals, 9 12 (1998) 2031-2054; errata10 7 (1999) 1261-1264.
		Chaotic behavior is known to be compatible with symmetry and illustrations are constructed using functions equivariant with respect to the desired symmetries.
		Earlier investigations determined families of equivariant functions for a few of the discrete symmetry groups in the plane; those results are extended to all the discrete symmetry groups of the plane.
	ww2.lafayette.edu /~reiterc/abstracts/mv_1998a.html (99 words)

	Physics Society, Directory (Site not responding. Last check: 2007-10-04)
		Identity and Individuality in Quantum Theory Assesses the metaphysical implications of quantum theory by considering the impact of the theory on our understanding of objects as individuals with well defined identity conditions.
		Digital Philosophy -- Discrete Physics Assuming that all quantities, including space and time, are finite and discrete.
		Symmetry in Physics Group of motions determines the conservation laws - the fundamental principles of physics.
	www.flashunion.org /ZmxzXzIwNDgxOA==.aspx (470 words)

	Chapter 4.1 (Site not responding. Last check: 2007-10-04)
		The idea of conformal symmetry was given in the monograph Colored Symmetry, its Generalizations and Applications by A.M. Zamorzaev, E.I. Galyarski and A.F. Palistrant (1978) as a generalization of similarity symmetry.
		According to the isomorphism existing between the types of the discrete symmetry groups of non-polar rods G
		Owing to the existence of a singular point, any figure the symmetry group of which is a conformal symmetry group is called a conformal symmetry rosette.
	www.maths.tcd.ie /EMIS/monographs/jablan/chap41.htm (699 words)

	3rd ENOC Proceedings: Bushes of normal modes for nonlinear mechanical systems with discrete symmetry (Site not responding. Last check: 2007-10-04)
		Bushes of normal modes for nonlinear mechanical systems with discrete symmetry
		The normal modes in a linear mechanical system with discrete symmetry are independent of each other.
		As an example, we list the one, two, and three-dimensional bushes for all possible free molecules with crystallographic point-group symmetry.
	www2.imm.dtu.dk /~mps/ENOC/proceedings/Chechin (193 words)

	Find in a Library
		Symmetry of discrete mathematical structures and their symmetry groups : a collection of essays
		To find a library, type in a postal code, state, province, or country.
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	worldcatlibraries.org /wcpa/ow/4c947dbef97813fda19afeb4da09e526.html (43 words)

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