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## Metric Tensor (metric + tensor)
## Selected Abstracts## Metric spaces in NMR crystallography CONCEPTS IN MAGNETIC RESONANCE, Issue 4 2009David M. GrantAbstract The anisotropic character of the chemical shift can be measured by experiments that provide shift tensor values and comparing these experimental components, obtained from microcrystalline powders, with 3D nuclear shielding tensor components, calculated with quantum chemistry, yields structural models of the analyzed molecules. The use of a metric tensor for evaluating the mean squared deviations, d2, between two or more tensors provides a statistical way to verify the molecular structure governing the theoretical shielding components. The sensitivity of the method is comparable with diffraction methods for the heavier organic atoms (i.e., C, O, N, etc.) but considerably better for the positions of H atoms. Thus, the method is especially powerful for H-bond structure, the position of water molecules in biomolecular species, and other proton important structural features, etc. Unfortunately, the traditional Cartesian tensor components appear as reducible metric representations and lack the orthogonality of irreducible icosahedral and irreducible spherical tensors, both of which are also easy to normalize. Metrics give weighting factors that carry important statistical significance in a structure determination. Details of the mathematical analysis are presented and examples given to illustrate the reason nuclear magnetic resonance are rapidly assuming an important synergistic relationship with diffraction methods (X-ray, neutron scattering, and high energy synchrotron irradiation). © 2009 Wiley Periodicals, Inc.Concepts Magn Reson Part A 34A: 217,237, 2009. [source] ## Extensions of the 3-Dimensional Plasma Transport Code E3D CONTRIBUTIONS TO PLASMA PHYSICS, Issue 1-3 2004A. RunovAbstract One important aspect of modern fusion research is plasma edge physics. Fluid transport codes extending beyond the standard 2-D code packages like B2-Eirene or UEDGE are under development. A 3-dimensional plasma fluid code, E3D, based upon the Multiple Coordinate System Approach and a Monte Carlo integration procedure has been developed for general magnetic configurations including ergodic regions. These local magnetic coordinates lead to a full metric tensor which accurately accounts for all transport terms in the equations. Here, we discuss new computational aspects of the realization of the algorithm. The main limitation to the Monte Carlo code efficiency comes from the restriction on the parallel jump of advancing test particles which must be small compared to the gradient length of the diffusion coefficient. In our problems, the parallel diffusion coefficient depends on both plasma and magnetic field parameters. Usually, the second dependence is much more critical. In order to allow long parallel jumps, this dependence can be eliminated in two steps: first, the longitudinal coordinate x3 of local magnetic coordinates is modified in such a way that in the new coordinate system the metric determinant and contra-variant components of the magnetic field scale along the magnetic field with powers of the magnetic field module (like in Boozer flux coordinates). Second, specific weights of the test particles are introduced. As a result of increased parallel jump length, the efficiency of the code is about two orders of magnitude better. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Coherent state path integral and super-symmetry for condensates composed of bosonic and fermionic atoms FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 9-10 2007B. MieckAbstract A super-symmetric coherent state path integral on the Keldysh time contour is considered for bosonic and fermionic atoms which interact among each other with a common short-ranged two-body potential. We investigate the symmetries of Bose-Einstein condensation for the equivalent bosonic and fermionic constituents with the same interaction potential so that a super-symmetry results between the bosonic and fermionic components of super-fields. Apart from the super-unitary invariance U(L | S) of the density terms, we specialize on the examination of super-symmetries for pair condensate terms. Effective equations are derived for anomalous terms which are related to the molecular- and BCS- condensate pairs. A Hubbard-Stratonovich transformation from ,Nambu'-doubled super-fields leads to a generating function with super-matrices for the self-energy whose manifold is given by the orthosympletic super-group Osp(S,S | 2L). A nonlinear sigma model follows from the spontaneous breaking of the ortho-symplectic super-group Osp(S,S | 2L) to the coset decomposition Osp(S,S | 2L) \ U(L | S), U(L | S). The invariant subgroup U(L | S) for the vacuum or background fields is represented by the density terms in the self-energy whereas the super-matrices on the coset space Osp(S,S | 2L) \ U(L | S) describe the anomalous molecular and BCS- pair condensate terms. A change of integration measure is performed for the coset decomposition Osp(S,S | 2L) \ U(L | S) , U(L | S), including a separation of density and anomalous parts of the self-energy with a gradient expansion for the Goldstone modes. The independent anomalous fields in the actions can be transformed by the inverse square root of the metric tensor of Osp(S,S | 2L) \ U(L | S) so that the non-Euclidean integration measure with super-Jacobi-determinant can be removed from the coherent state path integral and Gaussian-like integrations remain. The variations of the independent coset fields in the effective actions result in classical field equations for a nonlinear sigma model with the anomalous terms. The dynamics of the eigenvalues of the coset matrices is determined by Sine-Gordon equations which have a similar meaning for the dynamics of the molecular- and BCS-pair condensates as the Gross-Pitaevskii equation for the coherent wave function in BEC phenomena. [source] ## Superalgebras of Dirac operators on manifolds with special Killing-Yano tensors FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 12 2006I.I. CotAbstract We present the properties of new Dirac-type operators generated by real or complex-valued special Killing-Yano tensors that are covariantly constant and represent roots of the metric tensor. In the real case these are just the so called complex or hyper-complex structures of the Kählerian manifolds. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. In this way the Dirac operators are related among themselves through continuous transformations associated with specific discrete ones. We show that the group of these continuous transformations can be only U(1) or SU(2). It is pointed out that the Dirac and Dirac-type operators can form ,, = 4 superalgebras whose automorphisms combine isometries with the SU(2) transformation generated by the Killing-Yano tensors. As an example we study the automorphisms of the superalgebras of Dirac operators on Minkowski spacetime. [source] ## Quadratic metric-affine gravity ANNALEN DER PHYSIK, Issue 4 2005D. VassilievAbstract We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler,Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp-wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non-Riemannian solution as a mathematical model for the neutrino. [source] ## Zones and sublattices of integral lattices ACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2004A. JannerMethods are presented for an analysis of zones and sublattices of integral lattices, whose relevance is revealed by sharp peaks in the frequency distribution of hexagonal and tetragonal lattices, as a function of the axial ratio . Starting from a few examples, zone symmetries, lattice,sublattice relations and integral scaling transformations are derived for hexagonal lattices with axial ratios , , and 1 (the isometric case) and for the related and tetragonal lattices. Sublattices and zones connected by linear rational transformations lead to rational equivalence classes of integral lattices. For properties like the axial ratio and the point-group symmetry (lattice holohedry), rational equivalence can be extended so that also metric tensors differing by an integral factor become equivalent. These two types of equivalence classes are determined for the lattices mentioned above. [source] |