Parallel Tensor Research Articles

Tensor deconvolution: A method to locate equivalent sources from full tensor gravity data

We present a method dedicated to the interpretation of full tensor (gravity) gradiometry (FTG) data called tensor deconvolution. It is especially designed to benefit from the simultaneous use of all the FTG components and of the gravity field. In particular, it uses tensor scalar invariants as a basis for source location. The invariant expressions involve all of the independent components of the tensor. This method is a tensor analog of Euler deconvolution, but has the following advantages compared to the conventional Euler deconvolution method: (1) It provides a solution at every observation point, without the use of a sliding window. (2) It determines the structural index automatically; as a consequence, the structural index follows the variations of the field morphology. (3) It uses all components of the measured full gradient tensor and gravity field, thus reducing errors caused by random noise. It is based on scalar invariants that are by nature insensitive to the orientation of the measuring device. We tested our method on both noise-free and noise-contaminated data. These tests show that tensor solutions cluster in the vicinity of the center of causative bodies, whereas Euler solutions better outline their edges. Hence, these methods should be combined for improved contouring and depth estimation. In addition, we use a clustering method to improve the selection of solutions, which proves advantageous when data are noisy or when signals from close causative bodies interfere.

A classification of immersed hypersurfaces in spheres with parallel Blaschke tensor

In this paper, we give a complete classification of all immersed hypersurfaces in the unit sphere with parallel Blaschke tensors. For this classification, two kinds of new examples are constructed.

Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds

We determine the local structure of all pseudo-Riemannian manifolds $(M,g)$ in dimensions $n\ge4$ whose Weyl conformal tensor $W$ is parallel and has rank 1 when treated as an operator acting on exterior 2-forms at each point. If one fixes three discrete parameters: the dimension $n\ge4$, the metric signature $--...++$, and a sign factor $\epsilon=\pm1$ accounting for semidefiniteness of $W$, then the local-isometry types of our metrics $g$ correspond bijectively to equivalence classes of surfaces $\varSigma$ with equiaffine projectively flat torsionfree connections; the latter equivalence relation is provided by unimodular affine local diffeomorphisms. The surface $\varSigma$ arises, locally, as the leaf space of a codimension-two parallel distribution on $M$, naturally associated with $g$. We exhibit examples in which the leaves of the distribution form a fibration with the total space $M$ and base $\varSigma$, for a closed surface $\varSigma$ of any prescribed diffeomorphic type. Our result also completes a local classification of pseudo-Riemannian metrics with parallel Weyl tensor that are neither conformally flat nor locally symmetric: for those among such metrics which are not Ricci-recurrent, rank $W$ = 1, and so they belong to the class mentioned above; on the other hand, the Ricci-recurrent ones have already been classified by the second author.

Magnetization-recovery experiments for static and MAS-NMR of I = 3/2 nuclei

Multifrequency pulsed NMR experiments on quadrupole-perturbed I = 3/2 spins in single crystals are shown to be useful for measuring spin–lattice relaxation parameters even for a mixture of quadrupolar plus magnetic relaxation mechanisms. Such measurements can then be related to other MAS-NMR experiments on powders. This strategy is demonstrated by studies of 71Ga and 69Ga (both I = 3/2) spin–lattice relaxation behavior in a single-crystal (film) sample of gallium nitride, GaN, at various orientations of the axially symmetric nuclear quadrupole coupling tensor. Observation of apparent single-exponential relaxation behavior in I = 3/2 saturation-recovery experiments can be misleading when individual contributing rate processes are neglected in the interpretation. The quadrupolar mechanism (dominant in this study) has both a single-quantum process ( T 1Q1) and a double-quantum process ( T 1Q2), whose time constants are not necessarily equal. Magnetic relaxation (in this study most likely arising from hyperfine couplings to unpaired delocalized electron spins in the conduction band) also contributes to a single-quantum process ( T 1M). A strategy of multifrequency irradiation with observation of satellite and/or central transitions, incorporating different initial conditions for the level populations, provides a means of obtaining these three relaxation time constants from single-crystal 71Ga data alone. The 69Ga results provide a further check of internal consistency, since magnetic and quadrupolar contributions to its relaxation scale in opposite directions compared to 71Ga. For both perpendicular and parallel quadrupole coupling tensor symmetry axis orientations small but significant differences between T 1Q1 and T 1Q2 were measured, whereas for a tensor symmetry axis oriented at the magic-angle (54.74°) the values were essentially equal. Magic-angle spinning introduces a number of complications into the measurement and interpretation of the spin–lattice relaxation. Comparison of 71Ga and 69Ga MAS-NMR saturation-recovery curves with both central and satellite transitions completely saturated by a train of 90° pulses incommensurate with the rotor period provides the simplest means of assessing the contribution from magnetic relaxation, and yields results for the quadrupolar mechanism contribution that are consistent with those obtained from the film sample.

Real hypersurfaces of a nonflat complex space form in terms of the Ricci tensor

We know the fact that there are no real hypersurfaces with parallel Ricci tensors in a nonflat complex space form (cf. [5]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the Ricci tensor $S$ which are weaker than $\nabla S=0$. We characterize Hopf hypersurfaces of a nonflat complex space form.

An Atomic Charge−Charge Flux−Dipole Flux Atom-in-Molecule Decomposition for Molecular Dipole-Moment Derivatives and Infrared Fundamental Intensities

The molecular dipole moment and its derivatives are determined from atomic charges, atomic dipoles, and their fluxes obtained from AIM formalism and calculated at the MP2(FC)/6-311++G(3d,3p) level for 16 molecules: 6 diatomic hydrides, CO, HCN, OCS, CO2, CS2, C2H2, C2N2, H2O, H2CO, and CH4. Root-mean-square (rms) errors of 0.052 D and 0.019 e are found for the dipole moments and their derivatives calculated using AIM parameters when compared with those obtained directly from the MP2(FC)/6-311++G(3d,3p) calculations and 0.097 D and 0.049 e when compared to the experimental values. The major deviations occur for the NaH, HF, and H2O molecules. Parallel polar tensor elements for the diatomic and linear polyatomic molecules, except H2, HF, LiH, and NaH, have values resulting from cancellations of substantial contributions from atomic charge fluxes and atomic dipole fluxes. These fluxes have a large negative correlation coefficient, -0.97. IR fundamental intensity sums for CO, HCN, OCS, CO2, CS2, C2H2, C2N2, H2CO, and CH4 calculated using AIM charges, charge fluxes, and atomic dipole fluxes have rms errors of 14.9 km mol(-1) when compared with sums calculated directly from the molecular wave function and 36.2 km mol(-1) relative to experimental values. The classical model proposed here to calculate dipole-moment derivatives is compared with the charge-charge flux-overlap model long used by spectroscopists for interpreting IR vibrational intensities. The utility of the AIM atomic charges and dipoles was illustrated by calculating the forces exerted on molecules by a charged particle. AIM quantities were able to reproduce forces due to a +0.1 e particle over a 3-8-A separation range for the CO and HF molecules in collinear and perpendicular arrangements. These results show that IR intensities do contain information relevant to the study of intermolecular interactions.

ON NONABELIAN TENSOR ANALOGUES OF 2-ENGEL CONDITIONS

Tensor analogues of right 2-Engel elements in groups were introduced by D. P. Biddle and L.-C. Kappe. We investigate the properties of right 2-Engel tensor elements and introduce the concept of -groups.

Parallel imaging and diffusion tensor imaging for diffusion-weighted MRI of the liver: preliminary experience in healthy volunteers.

Our aim was to determine whether parallel imaging and diffusion tensor imaging affect the measurement of apparent diffusion coefficient (ADC) during diffusion-weighted MRI of the liver in healthy volunteers. We performed breath-hold single-shot echo-planar diffusion-weighted MRI of the liver in 10 healthy volunteers using conventional diffusion, conventional diffusion with parallel imaging, and diffusion tensor with parallel imaging sequences. TE values for the three sequences were 83, 74, and 63, respectively. Liver signal intensity was measured on all sequences and normalized to the SD of the measurement. Hepatic ADC was calculated by acquiring all sequences with b values of 0 and 500 sec/mm(2). The normalized liver signal intensity was higher on diffusion tensor with parallel imaging and conventional diffusion with parallel imaging than on conventional diffusion without parallel imaging for a b value of 500 sec/mm(2) (13.0 and 10.1 vs 9.1, respectively; p < 0.03) and for a b value of 0 sec/mm(2) (9.0 and 7.6 vs 6.9, respectively; without reaching a significant difference, p = 0.12). Hepatic ADC was not significantly different between sequences (p = 0.16). Higher signal intensity can be obtained when using parallel imaging and diffusion tensor imaging during diffusion-weighted MRI of the liver without compromising hepatic ADC measurement.

The nuclear magnetic resonance line shapes of Xe in the cages of clathrate hydrates.

We report, for the first time, a prediction of the line shapes that would be observed in the (129)Xe nuclear magnetic resonance (NMR) spectrum of xenon in the cages of clathrate hydrates. We use the dimer tensor model to represent pairwise contributions to the intermolecular magnetic shielding tensor for Xe at a specific location in a clathrate cage. The individual tensor components from quantum mechanical calculations in clathrate hydrate structure I are represented by contributions from parallel and perpendicular tensor components of Xe-O and Xe-H dimers. Subsequently these dimer tensor components are used to reconstruct the full magnetic shielding tensor for Xe at an arbitrary location in a clathrate cage. The reconstructed tensors are employed in canonical Monte Carlo simulations to find the Xe shielding tensor component along a particular magnetic field direction. The shielding tensor component weighted according to the probability of finding a crystal fragment oriented along this direction in a polycrystalline sample leads to a predicted line shape. Using the same set of Xe-O and Xe-H shielding functions and the same Xe-O and Xe-H potential functions we calculate the Xe NMR spectra of Xe atom in 12 distinct cage types in clathrate hydrates structures I, II, H, and bromine hydrate. Agreement with experimental spectra in terms of the number of unique tensor components and their relative magnitudes is excellent. Agreement with absolute magnitudes of chemical shifts relative to free Xe atom is very good. We predict the Xe line shapes in two cages in which Xe has not yet been observed.

REAL HYPERSUREAACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH PARALLEL SHAPE OPERATOR II

In this paper we consider the notion of ξ-invariant or (equation omitted)-invariant real hypersurfaces in a complex two-plane Grassmannian <TEX>$G_2$</TEX>( <TEX>$C^{m+2}$</TEX>) and prove that there do not exist such kinds of real hypersurfaces in <TEX>$G_2$</TEX>( <TEX>$C^{m+2}$</TEX>) with parallel second fundamental tensor on a distribution ζ defined by ζ = ξ U(equation omitted), where(equation omitted) = Span ｛ξ<TEX>$_1$</TEX>, ξ<TEX>$_2$</TEX>, ξ<TEX>$_3$</TEX>｝.X>｝.

Old and new results for superenergy tensors from dimensionally dependent tensor identities

It is known that some results for spinors, and in particular for superenergy spinors, are much less transparent and require a lot more effort to establish, when considered from the tensor viewpoint. In this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4-dimensional identities by straightforward tensor methods in a signature independent manner. In particular, we consider the quadratic identity for the Bel–Robinson tensor TabcxTabcy=δxy TabcdTabcd/4 and also the new conservation law for the Chevreton tensor, both of which have been obtained by spinor means; both of these results are rederived by tensor means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and, moreover, we are able to conclude that there are no direct higher dimensional analogs. In addition we demonstrate a simple way to show the nonexistense of such identities via counter examples; in particular we show that there is no nontrivial Bel tensor analog of this simple Bel–Robinson tensor quadratic identity. On the other hand, as a sample of the power of generalizing dimensionally dependent tensor identities from four to higher dimensions, we show that the symmetry structure, trace-free and divergence-free nature of the 4-dimensional Bel–Robinson tensor does have an analog for a class of tensors in higher dimensions.

ON SOME PSEUDOSYMMETRY TYPE HYPERSURFACES OF SEMI-EUCLIDEAN SPACES

A semi-Riemannian manifold (M;g), n  3, is called semisymmetricifR R = 0 (1)holds on M: It is well known that the class of semisymmetric mani-folds includes the set of locally symmetric manifolds (rR = 0) as aproper subset. Semisymmetric Riemannian manifolds were –rst stud-ied by E. Cartan, A. Lichnerowicz, R. S. Couty and N. S. Sinjukov. In[34] K. Nomizu asked the question if there exist complete, irreducibleand simply connected, Riemannian manifolds of dimension n  3 sat-isfying (1) and not locally symmetric. The –rst positive example wasconstructed by H. Takagi ([39]). A fundamental study on Riemann-ian semisymmetric manifolds was made by Z. I. Szabo ([36], [37], and[38]) and O. Kowalski (see [6] and the references therein). Semisym-metric semi-Riemannian manifolds, among others, were studied by A.Derdzinski and W. Roter. In particular, they have investigated semi-symmetric manifolds having parallel Weyl tensor (rC = 0) as well assemisymmetric manifolds with recurrent Weyl tensor (rC = C ). A

ON SUBGROUPS RELATED TO THE TENSOR CENTER

The tensor center of a group G is the set of elements a in G such that a ⊗ g = 1⊗ for all g in G.I t is ac haracteristic subgroup of G contained in its center. We introduce tensor analogues of various other subgroups of a group such as centralizers and 2-Engel elements and investigate their embedding in the group as well as interrelationships between those subgroups.

Optimizing parallel performance of unstructured volume rendering for the Earth Simulator

A scalable and high-performance parallel visualization subsystem has been developed in GeoFEM for the Earth Simulator (ES). As part of the ES project in Japan, the proposed subsystem is effective for the visualization of huge-scale unstructured datasets, and can be concurrent with computation on the same high-performance parallel computer. Moreover, some parallel visualization modules have obtained a good parallel performance, covering scalar, vector and tensor fields. This paper will take volume rendering method as an example to describe a number of efficient parallel performance optimization strategies we adopted for large-scale unstructured data visualization on SMP cluster machines, including suitable design of visualization method, the three-level hybrid parallelization which means message passing for inter-SMP node communication, loop directives for intra-SMP node parallelization, and vectorization for each processing element, plus dynamic load balancing. Good visualization images and high parallel performance have been achieved on the ES, thus demonstrating the feasibility and effectiveness of the proposed method.

Isospectral deformations of negatively curved Riemannian manifolds with boundary which are not locally isometric

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We present a method for constructing isospectral manifolds with different local geometry, generalizing an earlier technique. Examples include continuous families of isospectral negatively curved manifolds with boundary as well as various pairs of isospectral manifolds. The latter illustrate that the spectrum does not determine whether a manifold with boundary has negative curvature, whether it has constant Ricci curvature, and whether it has parallel curvature tensor, and the spectrum does not determine whether a closed manifold has constant scalar curvature.

Localization of scalar fluctuations in a dilatonic brane-world scenario

We derive and solve the full set of scalar perturbation equations for a class of Z 2-symmetric five-dimensional geometries generated by a bulk cosmological constant and by a 3-brane non-minimally coupled to a bulk dilaton field. The massless scalar modes, like their tensor analogues, are localized on the brane, and provide long-range four-dimensional dilatonic interactions, which are generically present even when matter on the brane carries no dilatonic charge. The shorter-range corrections induced by the continuum of massive scalar modes are always present: they persist even in the case of a trivial dilaton background (the standard Randall–Sundrum configuration) and vanishing dilatonic charges.

Variant of the bimetric theory of gravitation. II. Energy-momentum tensor of the gravitational field

Starting from the invariance of action relative to coordinate transformations, differential conservation laws are established in the bimetric theory of gravitation with a Lagrangian that is quadratic with respect to “intensities.” Explicit expressions are found for the canonical and metric energymomentum tensors of the gravitational field, as well as for the tensor analog of the Landau-Lifshitz pseudotensor.

Generalization of the Clausius-Clapeyron equation in a coupled thermomechanical model

Several modifications of the Clausius-Clapeyron equation for deformable media, including solid-phase transformations which depend on the change of additional parameters, are proposed. A model of the medium with tensor concentrations of the components for which the unique Clausius-Clapeyron equation is also valid is proposed. The tensor analog of the transition heat is introduced, and an expression for the total transition heat related to the energies of chemical bonds in the crystal lattice is obtained. At least for slow processes, the fundamental possibility of determining the self transition heat in the experiment is shown analytically.

Hypersurfaces with parallel affine curvature tensor R*

Hypersurfaces with parallel affine curvature tensor R*

Courbure sectiounelle et intégrales premières symétriques du flot géodésique: généralisation d'un théorème de S. Bochner

Let (M, g) be a Riemannian manifold. We prove that the space of symmetric tensors invariant under the geodesic flow, is a Lie algebra which contains, as a subalgebra, the Lie algebra of Killing vector fields, and which also contains the space of parallel symmetric tensors as an Abelian subalgebra. Morever, we give a Weitzenböck decomposition of some Laplace—Beltrami operator on symmetric tensors and prove a vanishing theorem which generalizes a theorem due to S. Bochner [2].