Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Abelian vectors and self-dual tensors in six-dimensional supergravity

In this note we describe the most general coupling of abelian vector and tensor multiplets to six-dimensional (1,0) supergravity. As was recently pointed out, it is of interest to consider more general Chern-Simons couplings to abelian vectors of the type H(r) = dB(r) - 1/2 c(rab)AadAb, with c(r) matrices that may not be simultaneously diagonalized. We show that these couplings can be related to Green-Schwarz terms of the form B(r)c(r)/abFaFb, and how the complete local Lagrangian, that embodies factorized gauge and supersymmetry anomalies (to be disposed of by fermion loops) is uniquely determined by Wess-Zumino consistency conditions, aside from an arbitrary quartic coupling for the gauginos. (C) 2000 Elsevier Science B.V.

Local Stereology of Tensors of Convex Bodies

In this paper, we present local stereological estimators of Minkowski tensors defined on convex bodies in ℝ d . Special cases cover a number of well-known local stereological estimators of volume and surface area in ℝ3, but the general set-up also provides new local stereological estimators of various types of centres of gravity and tensors of rank two. Rank two tensors can be represented as ellipsoids and contain information about shape and orientation. The performance of some of the estimators of centres of gravity and volume tensors of rank two is investigated by simulation.

Estimating Particle Shape and Orientation Using Volume Tensors

We develop statistical procedures for estimating shape and orientation of arbitrary three-dimensional particles. We focus on the case where particles cannot be observed directly, but only via sections. Volume tensors are used for describing particle shape and orientation, and we derive stereological estimators of the tensors. These estimators are combined to provide consistent estimators of the moments of the so-called particle cover density. The covariance structure associated with the particle cover density depends on the orientation and shape of the particles. For instance, if the distribution of the typical particle is invariant under rotations, then the covariance matrix is proportional to the identity matrix. We develop a non-parametric test for such isotropy. A flexible Lévy-based particle model is proposed, which may be analysed using a generalized method of moments in which the volume tensors enter. The developed methods are used to study the cell organization in the human brain cortex.

Stereological estimation of particle shape and orientation from volume tensors

In the present paper, we describe new robust methods of estimating cell shape and orientation in 3D from sections. The descriptors of 3D cell shape and orientation are based on volume tensors which are used to construct an ellipsoid, the Miles ellipsoid, approximating the average cell shape and orientation in 3D. The estimators of volume tensors are based on observations in several optical planes through sampled cells. This type of geometric sampling design is known as the optical rotator. The statistical behaviour of the estimator of the Miles ellipsoid is studied under a flexible model for 3D cell shape and orientation. In a simulation study, the lengths of the axes of the Miles ellipsoid can be estimated with CVs of about 2% if 100 cells are sampled. Finally, we illustrate the use of the developed methods in an example, involving neurons in the medial prefrontal cortex of rat.

Modelling word meaning using efficient tensor representations

Models of word meaning, built from a corpus of text, have demonstrated success in emulating human performance on a number of cognitive tasks. Many of these models use geometric representations of words to store semantic associations between words. Often word order information is not captured in these models. The lack of structural information used by these models has been raised as a weakness when performing cognitive tasks. This paper presents an efficient tensor based approach to modelling word meaning that builds on recent attempts to encode word order information, while providing flexible methods for extracting task specific semantic information.

Tensor query expansion : a cognitively motivated relevance model

In information retrieval, a user's query is often not a complete representation of their real information need. The user's information need is a cognitive construction, however the use of cognitive models to perform query expansion have had little study. In this paper, we present a cognitively motivated query expansion technique that uses semantic features for use in ad hoc retrieval. This model is evaluated against a state-of-the-art query expansion technique. The results show our approach provides significant improvements in retrieval effectiveness for the TREC data sets tested.

Individual user behaviour modelling for effective web recommendation

With the growth of the Web, E-commerce activities are also becoming popular. Product recommendation is an effective way of marketing a product to potential customers. Based on a user’s previous searches, most recommendation methods employ two dimensional models to find relevant items. Such items are then recommended to a user. Further too many irrelevant recommendations worsen the information overload problem for a user. This happens because such models based on vectors and matrices are unable to find the latent relationships that exist between users and searches. Identifying user behaviour is a complex process, and usually involves comparing searches made by him. In most of the cases traditional vector and matrix based methods are used to find prominent features as searched by a user. In this research we employ tensors to find relevant features as searched by users. Such relevant features are then used for making recommendations. Evaluation on real datasets show the effectiveness of such recommendations over vector and matrix based methods.