Showing papers on "Symmetry (geometry) published in 1982"
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CERN1
TL;DR: In this paper, an extension of N = 8 supergravity is presented, in which the natural symmetry group SO(8) is gauged and shown to be consistent with the dynamically realized SU(8)-symmetric symmetry.
233 citations
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TL;DR: In this article, the irreducible band representations of a space group are induced from a set of inequivalent relevant symmetry centers in the Wigner-Seitz cell, and a connection is established between representations and band representations by using the Born-von K\'arm\'an boundary conditions.
Abstract: All the irreducible band representations of a space group are shown to be induced from a set of inequivalent relevant symmetry centers in the Wigner-Seitz cell. A connection is established between representations and band representations of space groups by using the Born---von K\'arm\'an boundary conditions. Continuity chords are used for proving the equivalency theorem which enables one to distinguish between equivalent and inequivalent band representations. As examples we consider a one-dimensional crystal and the ${D}_{6h}^{4}$ space group for a hexagonal close-packed structure.
72 citations
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TL;DR: Redundancy in the perception of bilateral symmetry in dynamic dot textures was investigated using two-alternative forced-choice techniques and it was found that the symmetry information fell within a strip approximately 1 deg wide about the central axis of symmetry, irrespective of the size of the texture at the retina.
Abstract: Redundancy in the perception of bilateral symmetry in dynamic dot textures was investigated using two-alternative forced-choice techniques. It was found that the symmetry information utilized by the visual system fell within a strip approximately 1 deg wide about the central axis of symmetry, irrespective of the size of the texture at the retina. Outside this strip, the symmetry information was found to be completely redundant.
55 citations
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TL;DR: In this article, the unbroken discrete subgroup of a Peccei-Quinn symmetry may be effectively embedded in the rest of the continuous symmetry group to avoid the formation of domain walls.
48 citations
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TL;DR: In this article, it was shown that the method of McClelland for graphical factorization of Huckel matrices of conjugated systems can also be applied if the system has twofold symmetry but no plane of symmetry perpendicular to the molecular plane.
Abstract: It is pointed out that the method of McClelland [3] for the graphical factorization of Huckel matrices of conjugated systems can also be applied if the system has twofold symmetry but no plane of symmetry perpendicular to the molecular plane.
41 citations
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TL;DR: In this article, the problem of classifying the bordered Klein surfaces with maximal symmetry was studied and the automorphism groups of these surfaces were classified into two classes: A/*-simple and A*-quotient.
Abstract: A compact bordered Klein surface of (algebraic) genus g > 2 is said to have maximal symmetry if its automorphism group is of order 12(g — 1), the largest possible. In this paper we study the bordered surfaces with maximal symmetry and their automorphism groups, the A/*-groups. We are concerned with the topological type, rather than just the genus, of these surfaces and its relation to the structure of the associated M*-group. We begin by classifying the bordered surfaces with maximal symmetry of low topological genus. We next show that a bordered surface with maximal symmetry is a full covering of another surface with primitive maximal symmetry. A surface has primitive maximal symmetry if its automorphism group is M*-simple, that is, if its automorphism group has no proper M*-quotient group. Our results yield an approach to the problem of classifying the bordered Klein surfaces with maxima] symmetry. Next we obtain several constructions of full covers of a bordered surface. These constructions give numerous infinite families of surfaces with maximal symmetry. We also prove that only two of the M*-simple groups are solvable, and we exhibit infinitely many nonsolvable ones. Finally we show that there is a correspondence between bordered Klein surfaces with maximal symmetry and regular triangulations of surfaces. 0. Introduction. In the fundamental paper [6] Hurwitz showed that a compact Riemann surface of genus g > 2 has at most 84(g — 1) automorphisms. Recent research [7,8,16] has studied the values of g for which this bound is attained and the structure of the automorphism group in these cases. A compact bordered Klein surface [1] of genus g>2 has at most 12(g— 1) automorphisms [10]. In this paper we study the surfaces for which this bound is attained, the bordered surfaces which have \"maximal symmetry.\" We also examine the automorphism groups of these surfaces, the M*-groups [11]. We are concerned with the topological type, rather than just the genus, of these surfaces and its relation to the structure of the associated Af*-group. §1 contains preliminary results and definitions, while §2 classifies the bordered surfaces with maximal symmetry of low topological genus. §3 is the central section of the paper. Surfaces with maximal symmetry are seen to be \"full\" coverings of those with \"primitive\" maximal symmetry. The latter surfaces are those whose automorphism groups are \"A/*-simple.\" The classification problem for surfaces with maximal symmetry then breaks into three parts, which are considered in the next three sections. Received by the editors February 3, 1981 and, in revised form, September 22, 1981. 1980 Mathematics Subject Classification. Primary 14H30; Secondary 30F10, 57M10.
38 citations
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TL;DR: In this paper, the complete reduction of a fourth-rank Cartesian tensor into parts wich are irreducible under the three-dimensional rotation group is presented, and results are given for the general case of a tensor without index permutational symmetry, and also for tensors with specific index symmetry properties.
Abstract: The complete reduction of a fourth-rank Cartesian tensor into parts wich are irreducible under the three-dimensional rotation group is presented. Results are given for the general case of a tensor without index permutational symmetry, and also for tensors with specific index symmetry properties. Applications in the realm of crystal physics are indicated with particular reference to four-photon absorption, and a table of irreducible tensor representations is provided for the seven crystal classes.
34 citations
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30 citations
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TL;DR: The T = 13,1 structure has been confirmed by computer model fitting to optically low-pass filtered images of virus particles prepared by the mica-sandwich freezeetching technique.
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TL;DR: In this article, an infinite-parameter Lie algebra for the complex self-dual SU(N) Yang-Mills fields is extended from SL(N,C)xC(lambda) to SL(S,C,C/sup -1/)direct-sumSL(S N, C).
Abstract: The infinite-parameter ''hidden symmetry'' algebra for the complex self-dual SU(N) Yang-Mills fields is extended from SL(N,C)xC(lambda) to SL(N,C)xC(lambda,lambda/sup -1/)direct-sumSL(N, C). Furthermore, we also find an infinite set of infinitesimal ''hidden symmetry'' transformations, indexed by all integers, which can maintain the reality of the self-dual Yang-Mills potentials. Their infinite-parameter Lie algebra is shown to be related to an infinite-dimensional symmetric space with isotropy group SU(N)xR(lambda).
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TL;DR: In this paper, a scheme for the numbering and labelling of the variants nucleated in an anisotropic medium has been proposed for the identification of the martensite morphologies.
Abstract: The symmetry properties are among the simplest deterministic principles which allow us to recognize equivalencies between different crystalline morphologies. The basic idea is essentially that the total symmetry of the resulting morphology reflects the symmetry of the originating process : for example, the different types of variants induced by a phase transition are related to the coset decomposition of the space groups of the involved structures onto their intersection group. A scheme has been proposed for the numbering and labelling of the variants nucleated in an anisotropic medium. More recently the importance of the Intersection Group has been pointed out as one of the basic ingredients for the study of precipitate morphology. The symmetry considerations are not only an easy way of reducing the configurational space of a certain physical property to a non redundant subspace, but they also allow for the determination of the \"symmetry dictated extrema\" of the considered physical property. This promising way will certainly induce a new development in the understanding of martensite morphologies. The role of symmetry in the study of phase transitions is well recognized from the determination of the order of the transition as well as the description of the microstructure of the transformed phase. The classical theory of Landau on phase transitions (1) stipulates that for a second order transition, the space group of the product is induced by an irreducible representation of the space group of the parent phase, both phases being group-subgroup related ( 2 ) . Whatever the order ofthe transition, a single crystal of the parent phase transforms to a collection of domains of the product phase called variants that are separated by interfaces. The variants correspond to all possible ways, strictly equivalent, of realizing the transformation. The properties of the transformed phase are closely related to this microstructure. Martensitic transformations being first order transformations, the parent phase may coexist with the product and two kinds of interfaces have to be considered: homophase interfaces between two variants of martensite (or parent phase) and heterophase interfaces between martensite and parent phases. Using symmetry arguments for studying the transformation, the first step consists in determining the number of generated variants and the nature of the interfaces between them. A second step concerns the possible relations between the morphology of the bicrystals of interest (parent phase one variant of the transformed phase and two variants of the transformed phase) and the interfacial symmetry which has to be determined. Then, a relation between this interfacial symmetry and a term of energy will be given. In a second part some information about interfaces, like the coincidence site lattice, and several points about the martensitic transformation, like the symmetry induced by displacement waves, will be treated. There are only a few papers on symmetry and martensite. Cahn in 1977 (3) used symmetry for the phenomenological description of the transformation and discussed Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982402 C4-18 JOURNAL DE PHYSIQUE the occurrence of sub and supergroup generation. The present authors, in 1979, used symmetry for finding relations between the parent phase and the product, ignoring the intermediate steps. These two papers will provide the basic material for the present one. Also the present psper is more a sort of \"pot pourri\" on symmetry and phase transitions and does not pretend to give any solution of the martensitic transformation. NUMBERING OF THE VARIANTS AND INTERFACE OPERATIONS (4) 1. Numbering of the variants generated by a phase transition Symmetry arguments distinguish cases in which two physical situations are equivalent. These arguments are only sufficient conditions. Certain physical properties may be accidental.1~ equivalent. Applied to phase transition, the basic idea is essentially that the total symmetry of the resulting morphology reflects the symmetry of the original process (Curie's Laws). In the present context, the symmetries involved through two basic groups can be described as : i) the symmetry group of the parent crystal G which is a space group 0' ii) the symmetry group of the external medium (g). This latter group reflects the anisotropy of the sollicitation applied to the crystal and leads to the transition when the critical value is reached. This group can be continuous. If the sollicitation is the temperature, its invariance group is the symmetry group of a sphere (Sog); if it is an applied stress field along a n direction, its invariance group is that of a cylinder (Dmh [n ] ) . . . Obviously, all thermodynamical properties of the transformation are invariant through any symmetry elements belonging simultaneously to both groups : neither through the application of such elements. The intersection group : Ho = G o A (g) may be considered mathematically as the kernel of the transitior. All its elements act as the identity operator with respect to the thermodynamic properties. In other words, assuming that a certain embryo of the transformed phase has to be formed somewhere, any other embryos deduced from the first one through the operations of Ho are strictly equivalent. Therefore, Ho is called the group of isoprobability of nucleation of the transformation product. Once the embryo has formed, further growth will lead to the new structure of space group GI. The existence of the heterophase interface between the two structures will be studied in terms of symmetry further below. Due to the growth, it may happen that certain embryos in contact would form a single crystal, whereas other embryos with space groups differing in orientationtranslation would form a bicrystal and would be separated by an homophase interface. In order to number all the different variants of the transformed phase, one has to select in Ho all the subset of elements which do not belong to GI. This is performed by considering the new intersection group N 01 ' N~~ = H ~ A G~ which is the set of symmetry elements common to both Ho and GI. Now, the number, n of different possible variants of the G1 product phase is given by the number oF1t)imes NO1 is contained into Ho. Mathematically, nol is the index of the subgroup NO1 into Ho : it is the number of NO1 cosets onto Ho : n = index (NO1/Ho) 01 The complete group-tree characterizing the transition from G to G is sketched as shown : 1 The inverse transformation is described similarly but the sollicitation group (g') may be different from the first one. There is an important distinction between domains and variants : each domain belongs to one kind of variant. All possible variants generated can be numbered but the number of domains will depend on the actual conditions of the transformation. The numbering of variants is for the case of a single crystal of the parent phase Go. For a polycrystal parent phase, the group-tree has to be considered for each grain : two grains being deduced by a symmetry-translation operation, the variants in the two grains will also be deduced by,this operation. Finally, it is assumed that the transformation has no boundary conditions and the parent crystal is a perfect one. If there are some boundary conditions, they must be included in the group of the sollicitation which is then reduced to the common symmetry of both the sollicitation and the boundary conditions. Moreover, if the parent phase is faulted, obviously at the level of a defect the symmetry will be lowered but defects will essentially play an important role as possible sites for the nucleation of embryos and not for the numbering of variants which is governed by the \"location\" of G with respect to H . 1 2. Space operations and space group notations The most general isometry in space is written (air) where a, is a point operation (identity, inversion, rotation, rotation-inversion)' and :, the associated translation (Seitz notation (5)). If two crystals are related by such an operation, that means the homologous points are deduced by : (el:)&, = arI + J EII with cI belonging to crystal I, fII belonging to crystal 11. If the space group of crystal I is GI, the space group of crystal I1 is then conjugated by (a]:) : GII = (al~) GI (a1~1-I That means GI only differs from GI by orientation-translation. The operation caracterises the homophase interface but a crystal is invariant through any operation of its space group : if one operates on crystal I first by GI and then by (a I r) one gets the same crystal 11. So, an interface operation is characterized by a coset : the right product of the operation (a]:) initially choosen by the space group of the crystal : (all) GI (see also appendix 1) 3. Some examples of phase transitions i) Order disorder transition Au3Cu-transformation from the disordered state (Go = Fm3m (a,b,c;o)) at high temperature to the ordered state (GI = Pm3m (a,b,c;o)) at low temperature is a simple case of group-subgroup transition. The sollicitation is the temperature and, assuming there is no gradient, its 3 invariance group is S0,QDR : H = G f\\ (g) = Fm3m (a,b,c;o) 0 0 NO1= Hor\\G1 = G = Pm3m (a,b,c;o) 1 The number of variants is : \"01 = index (Fm3m ($,b,c;o)/Pm3m ($,b,c;o)) = 4 There are 3 different interfaces : which are the three well known antiphase boundaries, obtained by the decomposition of H into cosets of N . At the critical temperature, the symmetry of the parent phasg is broken and t% interface operations between variants are precisely the symmetry operations of the parent phase which were lost. If, one now considers the reverse transformation by increasing the temperature, the new group of isoprobability of nucleation is : H~ = G n ( g ) = Pm3m (~,b,c;o) 1 and C4-20 JOURMAL DE PHYSIQUE N10 = H /\\ G = Pm3m ($,b,c;o) 1 0 these two groups being identical, the number of g
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TL;DR: In this paper, the Wigner operators for point groups were applied to generate the required basis for spin-free multishell electron configurations in molecules to general non-Abelian point groups using the basis spanning the irreducible representation of the unitary group U(n).
Abstract: Symmetry adaptation of spin–free multishell electron configurations in molecules to general non-Abelian point groups has been carried out. Using the basis spanning the irreducible representation [2N/2−S, 12S, 0n−N/2−S] of the unitary group U(n) as primitives, the Wigner operators for point groups were applied to generate the required basis. In the process it was found that the segments of the Weyl tableaux could be handled individually. Using this and the matric algebra of permutation group a viable procedure has been developed for point groups adaptation. A program based on the procedure has been generated and implemented.
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TL;DR: A series of experiments showed generally that the effects are attributable to the axes of bilateral symmetry rather than to less restrictively defined axes of topological symmetry, although those axes may produce other illusory effects.
Abstract: The lines of a surrounding figure can induce illusory distortion in the apparent orientation of an enclosed line. The axes of symmetry of the figure have been implicated in this distortion. A series of experiments showed generally that the effects are attributable to the axes of bilateral symmetry rather than to less restrictively defined axes of topological symmetry, although those axes may produce other illusory effects. Furthermore, the distortions affect the whole line rather than only parts of it. These results are consistent with interactions of the lines in the orientation rather than position domain. It is suggested that axes of symmetry may play an important role in pattern recognition.
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TL;DR: An SU(5) model is constructed in which the unbroken discrete subgroup of a Peccei-Quinn symmetry is a trivial Z/sub 1/.
Abstract: In general, the unbroken discrete subgroup of a Peccei-Quinn symmetry gives rise to a domain problem: the formation of unacceptable domain walls in the course of the evolution of the early universe. In this paper, we construct an SU(5) model in which the unbroken discrete subgroup of a Peccei-Quinn symmetry is a trivial Z/sub 1/. That means that there is no domain problem for this model. It is pointed out that in models with more than one global U(1) symmetry it is easy to achieve a solution to the domain problem.
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TL;DR: In this article, a closed expression for the commutators of chiral models was obtained, which summarizes the same set of commutation relations as recently uncovered for the hidden symmetry with a different set of off-shell transformations.
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TL;DR: In this paper, the magnitude functions of the frequency responses of two-dimensional analog and digital transfer functions were derived so that the magnitude function of these transfer functions possess fourfold rotational symmetry.
Abstract: The presence of various types of symmetry in the frequency responses of a two-dimensional filter function reflects as constraints on its coefficients. In this paper, such constraints are derived for two-dimensional analog and digital transfer functions so that the magnitude functions of the frequency responses of these transfer functions possess fourfold rotational symmetry. The relationship between fourfold rotational symmetry and quadrantal, diagonal, and octagonal symmetries is also discussed.
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01 Dec 1982
TL;DR: In this article, a transition to a structure with a Mixed Order-Disorder and Distorted Structure is considered, where the authors apply the Landau Theory to the NiAs-type to MnP-type phase transition.
Abstract: Space Lattice Symmetry- Translational Periodicity- Proper Rotational Symmetry- Symmetries of Plane Lattices- Space Lattices- The Bravais Lattices- Space Group Symmetry- Proper and Improper Rotations- Combination of Rotations and Translations- Seitz Operators- Screw Axes and Glide Planes- Combination of Symmetry Operations- Reciprocal Space and Irreducible Representations of Space Groups- Reciprocal Lattice- Reciprocal Space- Irreducible Representations of Space Groups- Second-Order Phase Transitions- Thermodynamics of Second-Order Phase Transitions- Landau Theory (without symmetry)- Landau Theory, with Consideration of Symmetry, Applied to the NiAs-type to MnP-type Phase Transition- General Development of the Landau Theory with Consideration of Symmetry- Landau's 4th Condition- Another Example of the Application of Landau Theory: $$ Fm3m - R\bar{3}m $$ Order-Disorder Transition in Sc1?xS- Application of Landau Theory to a Transition to a Structure with a Mixed Order-Disorder and Distorted Structure
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TL;DR: In this paper, it was shown that an incompressible fluid body rotating about an axis with a given angular velocity ω is in equilibrium relative to the potential energy of its own gravitational field and the surface energy due to surface tension.
Abstract: Consider an incompressible fluid body (in outer space) rotating about an axis with a given angular velocity ω, and which is in equilibrium relative to the potential energy of its own gravitational field and the surface energy due to surface tension. We show that such a body possesses a plane of symmetry perpendicular to the axis of rotation such that any line parallel to the axis and meeting the body cuts it in a line segment whose center lies on the plane of symmetry. This extends an earlier result of L. Lichtenstein [4].
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TL;DR: In this article, a method for enumerating and constructing isomerization reactions of molecules exhibiting large amplitude nonrigid motions is proposed, which not only enumerates the isomers of non-rigid molecules and the corresponding rigid molecules but also the symmetry species spanned by the equivalent structures whose representative is an isomer.
Abstract: A method is formulated for enumerating and constructing isomerization reactions of molecules exhibiting large amplitude nonrigid motions. This method not only enumerates the isomers of nonrigid molecules and the corresponding rigid molecules but also the symmetry species spanned by the equivalent structures whose representative is an isomer. Consequently, using the method of correlating the symmetry species of a group to the symmetry species of its subgroup the splitting patterns of isomers of nonrigid molecule to those of rigid molecule are obtained. This provides an elegant method for both enumerating and constructing reaction graphs. The method is illustrated with examples.
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TL;DR: In this article, the authors extend the work of a previous paper (1980) in which they showed that some of the phase choices in the 3jm factors of certain point group embeddings affect the orientation of the symmetry axes.
Abstract: The authors extend the work of a previous paper (1980) in which they showed that some of the phase choices in the 3jm factors of certain point group embeddings affect the orientation of the symmetry axes. The occurrence of such choices is unpredictable and their effect is sometimes subtle but once a set of 3jm factors has been calculated they can determine the orientations of the symmetry axes and mirror planes and investigate how these orientations vary with the phase choices. The properties of the reflection-rotation groups are obtained from the the isomorphic pure rotation groups but care is necessary in interpreting the effects of the corresponding operations of, for example, D3 and C3v.
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TL;DR: In this paper, the cyclic regions in a crystal are described by finite factor groups of the crystallographic space group which contain the isogonal group as a subgroup, and a new approach to labeling is introduced based on the concept of cyclic region.
Abstract: Earlier work on unique labelling schemes in Oh symmetry is extended to include 48-atom shells and the symmetry coordinates of shells subject to periodicity conditions. A new approach to labelling is introduced based on the concept of cyclic regions in a crystal. These are described by finite factor groups of the crystallographic space group which contain the isogonal group as a subgroup. Some examples are worked out in detail.
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TL;DR: The group factorisation and correlation theorem methods used previously to provide a unique labeling of atomic shell symmetry coordinates are shown to give also a simple means for deriving explicit expressions for these coordinates as discussed by the authors.
Abstract: The group factorisation and correlation theorem methods used previously to provide a unique labelling of atomic shell symmetry coordinates are shown to give also a simple means for deriving explicit expressions for these coordinates. Results are given for the 12-atom shell and the two types of 24-atom shells in Oh symmetry.
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TL;DR: In this article, a normal coordinate analysis for bis(2,2,6,6-tetramethylheptane-3,5-dionato)uranyl has been performed.
Abstract: A normal coordinate analysis has been performed for bis(2,2,6,6-tetramethylheptane-3,5-dionato)uranyl according to two simplified models: (i) a 35-atom model consisting of one tetramethylheptanedionate ligand attached to uranyl; symmetry C 2 V and (ii) a 31-particle model of the whole complex, where the methyl groups are taken as point masses; symmetry D 2h . The construction of independent symmetry coordinates by the “method of fragments” is described. Calculated vibrational frequencies are reported along with mean amplitudes ( l - values) and perpendicular amplitude coefficients ( K - values) for selected interatomic distances.
01 Apr 1982