Showing papers on "Symmetry (geometry) published in 1985"
TL;DR: In this paper, a concept of symmetry is defined for general nonlinear control systems, and it is shown that such systems admit local and/or global decompositions in terms of lower dimensional subsystems and feedback loops.
Abstract: A concept of symmetry is defined for general nonlinear control systems. It is shown, under various technical conditions, that nonlinear control systems with symmetries admit local and/or global decompositions in terms of lower dimensional subsystems and feedback loops. The structure of the individual subsystems is dependent on the structure of the symmetry group; for example, if the symmetry group is Abelian, one of the subsystems is a quadrature. An additional feature of the decomposition is that the state-space dimensions of the subsystems sum to the state-space dimension of the original system.
153 citations
TL;DR: In this article, power transformations for achieving distributional symmetry are discussed and the Box-Cox method or a robust adaptation of it is found to be the generally most suitable method.
Abstract: SUMMARY Power transformations for achieving distributional symmetry are discussed. Estimates of the transformation power are based on general measures of symmetry. They are shown to be consistent and asymptotically normal. Use of the skewness coefficient as a measure of symmetry is shown to be optimal in an important special case. The methods are compared to the likelihood methods of Box & Cox (1964) and alternative methods of Hinkley (1975, 1977). The Box-Cox method or a robust adaptation of it (Carroll, 1980; Bickel & Doksum, 1981) is found to be the generally most suitable method.
127 citations
TL;DR: In this paper, a generalized dual method for generating quasicrystal structures with arbitrary orientational symmetry in two and three dimensions was introduced, which can be used to generate arbitrary orientations.
Abstract: We introduce a ``generalized dual method'' for generating quasicrystal structures with arbitrary orientational symmetry in two and three dimensions.
102 citations
TL;DR: The results of five experiments reported are interpreted as supporting the assumptions of the symmetry theory with the provision that local as well as global axes of symmetry are important.
92 citations
01 Jan 1985
TL;DR: An expression is obtained for the maximum number of axial symmetries of a tree which can be simultaneously displayed in a single drawing, and an algorithm is presented for constructing such a maximally-symmetric drawing.
Abstract: The automatic construction of good drawings of abstract graphs is a problem of practical importance. Displaying symmetry appears as one of the main criteria for achieving goodness. An expression is obtained for the maximum number of axial symmetries of a tree which can be simultaneously displayed in a single drawing, and an algorithm is presented for constructing such a maximally-symmetric drawing. Similar results are also obtained for rotational symmetries in trees. The algorithms run in time which is linear in the size of the tree, and hence are optimal.
54 citations
43 citations
TL;DR: This chapter presents the indications and strategies for interpretation of protein maps at lower resolution or with less accurate phasing, which vary with crystal packing density, crystallization medium, and accuracy of phasing.
Abstract: Publisher Summary This chapter presents the indications and strategies for interpretation of protein maps at lower resolution or with less accurate phasing. The resolution level around 5–6 A is one of the traditional milestones in the solution of a protein structure: it is a low point in most radial distributions of diffraction intensity, it is sufficient for location of heavy-atom positions, and it provides information about some overall features of the structure. The simplest information commonly obtained at low resolution is the distinction between protein and solvent density, which shows the molecular shape and packing. Subunit arrangement and symmetry can be seen, as can domains within a subunit if they are well separated. The ease and certainty with which molecular boundaries can be drawn at low resolution vary with crystal packing density (tight packing produces ambiguous contacts), crystallization medium (contrast is better in alcohols or PEG than with concentrated salts), and accuracy of phasing (which can be improved, for instance, by averaging over noncrystallographic symmetry).
24 citations
20 citations
TL;DR: In this article, the meaning of the assignment of probabilities to events implied by the kind of model commonly used by Statisticians is investigated and traditional frequentist understandings are reviewed and reinterpreted.
Abstract: We consider the meaning of the assignment of probabilities to events implied by the kind of model regularly used by Statisticians. Traditional frequentist understandings are reviewed and re...
20 citations
TL;DR: In this paper, a method for obtaining a complete set of independent relations among integral derivatives is presented based on the invariance under translations and rotations of quantum chemical one-and two-electron integrals.
Abstract: Based upon the invariance under translations and rotations of quantum chemical one‐ and two‐electron integrals, a method for obtaining a complete set of independent relations among integral derivatives is presented. Due to the unitary form of the operators corresponding to finite translations and rotations, this analysis is generally applicable to all orders of integral derivatives. It is shown that the number of dependent integral derivatives is equal to the number of such independent relations. These dependent integral derivatives can thus be straightforwardly determined in terms of the remaining derivatives which must be explicitly calculated. For example, out of a total of 21, 45, and 78 second‐derivative integrals for the two‐ , three‐ , and four‐center cases, respectively, only 1, 6, and 21 such integral derivatives need be explicitly calculated. The set of such independent and dependent integral derivatives can be chosen in a manner which imposes no restrictions on the allowable geometries of the n...
17 citations
TL;DR: In this paper, the total classification of the possible symmetries of the extremal points of adiabatic potential surfaces of the Jahn-Teller systems for all molecular symmetry point groups was elaborated on the basis of group-theoretical analysis.
Abstract: The total classification of the possible symmetries of the extremal points of adiabatic potential surfaces of the Jahn—Teller systems for all molecular symmetry point groups was elaborated on the basis of group-theoretical analysis. The method proceeds from the consecutive split of degenerate irreducible representations of the corresponding electron terms as a consequence of the symmetry descent.
TL;DR: In this paper, a simple yet general technique is presented for the problem of assessing symmetry about an unknown point with linear rank statistics, which is similar to the one presented in this paper.
TL;DR: In this paper, the recognition of forbidden icosahedral symmetry by both construction and experiment is discussed, and a minor upheaval occurs in the field of crystallography with respect to the discovery of forbidden symmetry.
Abstract: Crystallography is in for a minor upheaval, with the recognition of forbidden icosahedral symmetry by both construction and experiment.
TL;DR: The renormalization-group (RG) method' in reciprocal space yields a set of first-order differential equations (recursion relations) that characterizes the critical behavior at the continuous transition for the associated physical system.
Abstract: The renormalization-group (RG) method' in reciprocal space yields a set of first-order differential equations (recursion relations). A stable fixed point of the Hamiitonian flow determined by these equations characterizes the critical behavior at the continuous transition for the associated physical system. There are as many types of initial effective Hamiltonians and RG recursion relations as there are types of quartic potentials. The Hamiltonian is a fourth-degree polynomial expansion in the order parameter and usually only includes isotropic gradient terms. (Some systems also allow anisotropic gradient terms. We restrict our considerations here to contributions from isotropic terms only. ) A natural generalization of the potential obtained in the Landau theory gives this initial Hamiltonian to which RG methods are to be applied. The Landau theory' assumes the existence of an order parameter $, which is an n-component vector in the carrier space E of an active physically irreducible representation (irrep) I D (I ) of the higher symmetry group I . The matrices D(y), representing y E I\", are orthogonal matrices in n dimensions which satisfy the Landau' and Lifshitz conditions (\"active\" irrep). The Landau potential is obtained by constructing invariant polynomials in the components of $. To fourth degree the potential can be written
TL;DR: In this paper, it was shown that a comprehensive symmetry description in polycrystalline bodies needs black-white point groups rather than the usual (one-colour) groups that are sufficient for single crystals.
Abstract: It is shown that a comprehensive symmetry description in polycrystalline bodies needs black-white point groups rather than the usual (one-colour) groups that are sufficient for single crystals.
TL;DR: A theoretical analysis for the computation of the coincidence site lattice symmetry is presented in this paper, where three types of symmetry elements can exist and each one can be found by properly using the CSL's rotation matrix of the smallest-angle description.
Abstract: A theoretical analysis for the computation of the coincidence site lattice (CSL) symmetry is presented. It is shown that three types of symmetry elements can exist and each one can be found by properly using the CSL's rotation matrix of the smallest-angle description. Thus, from the existence of the subgroup H1, the order of which is directly connected with the number of the different orientations that the sublattice Λ11 can have, a low-symmetry H1 group implies more possibilities for the formation of the corresponding CSL. From the existence of the symmetry elements of the second type, the smallest-angle rotation matrix can be a symmetry element but only of the fourth or sixth order. From the third type of elements a connection between CSLs of different Σvalues can exist. Since the analytical form of this smallest-angle rotation matrix can be deduced for every crystallographic system, the procedure described here is of general use. Thus a new classification of the different CSLs is possible according to their symmetry group. This allows the study of the CSL model from the symmetry point of view.
TL;DR: In this article, the rank dependence of unitary group 6j and 3jm symbols has been analyzed using the composite Young tableaux notation, which is a back-to-back notation.
Abstract: We give tables of algebraic formulas for some nontrivial 6j symbols and 3jm symbols of the unitary groups. The tables demonstrate that the building‐up method can be used successfully to obtain the rank dependence of unitary group j and jm symbols. To emphasize the rank‐dependent nature of this calculation, we have employed the composite Young tableaux notation (or back‐to‐back notation) to label the unitary group irreps. In using this notation, the transpose conjugate symmetry of the corresponding composite Young diagram leads to a new symmetry of the unitary group 6j and 3jm symbols. The transposition of the groups Um and Un gives rise to a further symmetry of the 3jm symbols of Umn⊃Um×Un.
TL;DR: In this paper, the authors classified the resonances and root vectors in the problem of scattering by tetrahedral and cubic clusters of zero-range potentials and proved that the series of root vectors are complete in some subspaces of L/sub 2/ (S/sup 2/ ), using for this some facts from the theory of entire functions of exponential type.
Abstract: This paper classifies the resonances and root vectors in the problem of scattering by tetrahedral and cubic clusters of zero-range potentials. It is shown that the symmetry of scatterer leads to the following properties of the S matrix: the resonances (roots of the S matrix) and root vectors can be classified in accordance with special subgroups of the symmetry groups of the scatterers. It is also proved that the series of root vectors are complete in some subspaces of L/sub 2/ (S/sup 2/), using for this some facts from the theory of entire functions of exponential type.
01 Jan 1985
TL;DR: In this article, it was shown that solid manganese-aluminum alloy may have 5-fold symmetry on the atomic scale, which is the same as Pauling's conjecture.
Abstract: Linus Pauling has produced an alternative explanation of the observation that solid manganese–aluminium alloy may have 5-fold symmetry on the atomic scale. How can the two views be reconciled?
TL;DR: In this article, the average density in such a partially ordered medium has the superordinate symmetry ℒ 1, while the pair correlation function has the subordinate symmetry � 2 in two and three dimensions.
Abstract: Hermann proposed that mesomorphic media should be classified by assigning certain “statistical symmetry groups” to each possible partially ordered array Two translational groups introduced were called superordinate and subordinate We find that the average density in such a partially ordered medium has the superordinate symmetry ℒ1, while the pair correlation function has the subordinate symmetry ℒ2 A complete listing is made of all compatible combinations of ℒ1 and ℒ2 in two and three dimensions This leads to more possible symmetries than Hermann obtained, eg, also to nonstoichiometric crystals The order parameter space for the systems is found to be the quotient space ℒ1/ℒ2 In most cases it is identical to the order parameter space of low-dimensionalXY spin systems The Landau free energy is expanded as functional of the two-particle correlation functionK; the translation group is found to be ℒ1×ℒ2 A Landau mean-field theory can then be carried out by expanding the system free energy into a series of invariants of the active irreducible representations ofK and mapping the free energy onto that for anXY planar spin system We predict novel critical behavior for transitions between mesomorphic phases and “go nogo” selection rules for continuous transitions We give the structure factors for X-ray scattering so changes in all such phase transitions are observable The statistical symmetry groups, which describe point and translational symmetries of the mesophases, are classified Proposals are made to include quasi-long-range or topological order in the classification scheme
TL;DR: All isotropy subgroups (and thus all quasicontinuous symmetry changes) corresponding to k\ensuremath{\rightarrow} points of symmetry have been obtained for the 80 diperiodic space groups.
Abstract: All isotropy subgroups (and thus all quasicontinuous symmetry changes) corresponding to k\ensuremath{\rightarrow} points of symmetry have been obtained for the 80 diperiodic space groups. The detailed information for such phase transitions is given here for the diperiodic space group P(4/m) (2/m) (2/m). Only two distinct images (sets of representation matrices) occur for this example yielding Landau-Ginzburg-Wilson (LGW) Hamiltonians corresponding to the Ising and XY models, respectively. Minimization of the LGW Hamiltonians yields those transitions which are continuous in the mean-field description.
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TL;DR: The algebraic theory of groups is a vast subject with applications in many fields as discussed by the authors, and group theory is used as a tool for using symmetry to predict properties of chemical systems.
Abstract: The algebraic theory of groups is a vast subject with applications in many fields. Our interest in group theory is as a tool for using symmetry to predict properties of chemical systems. In this chapter we give some of the elementary properties of groups that will be used in subsequent chapters. We show how the terminology of group theory is used to describe the symmetry of molecules and (briefly) crystals.
TL;DR: In this paper, a group augmentation procedure is presented for the analytic determination of tensional invariants of arbitrary order for all point groups, and the results are an ideal pratical adjunct to the generalized selection rule approach, which focuses predominantly on the concept of symmetry invariant operator products.
TL;DR: In this article, the authors obtained the complete Lie-Backlund symmetry for a modified Harry-Dym system and deduced the bi-Hamiltonian structure associated with it, and showed that these Liebacklund symmetries generate the recursion operator inherent in the theory and the conserved quantities can also be computed according to the Dorfman prescription.
Abstract: The authors have obtained the complete Lie-Backlund symmetry for a modified Harry-Dym system and hence deduced the bi-Hamiltonian structure associated with it It is shown that these Lie-Backlund symmetries generate the recursion operator inherent in the theory and the conserved quantities can also be computed according to the Dorfman prescription
TL;DR: In this paper, Singerman et al. show that a bordered Klein surface of algebraic genus g is an M*-group if and only if it is generated by 3 non-trivial elements.
Abstract: by DAVID SINGERMAN(Received 8 September, 1983)1. Let X be a bordered Klein surface, by which we mean a Klein surface withnon-empty boundary. X is characterized topologically by its orientability, the number k ofits boundary components and the genus p of the closed surface obtained by filling in allthe holes. Th algebraice genus g of X is defined byf2p + fc — 1 if X is orientableg =