Directional independent component analysis with tensor representation

Author name used in this publication: David ZhangRefereed conference paper2007-2008 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Asymptotic structure of radiation in higher dimensions

We characterize a general gravitational field near conformal infinity (null, spacelike, or timelike) in spacetimes of any dimension. This is based on an explicit evaluation of the dependence of the radiative component of the Weyl tensor on the null direction from which infinity is approached. The behaviour similar to peeling property is recovered, and it is shown that the directional structure of radiation has a universal character that is determined by the algebraic type of the spacetime. This is a natural generalization of analogous results obtained previously in the four-dimensional case.Comment: 14 pages, no figures (two references added

N-Dimensional Principal Component Analysis

In this paper, we first briefly introduce the multidimensional Principal Component Analysis (PCA) techniques, and then amend our previous N-dimensional PCA (ND-PCA) scheme by introducing multidirectional decomposition into ND-PCA implementation. For the case of high dimensionality, PCA technique is usually extended to an arbitrary n-dimensional space by the Higher-Order Singular Value Decomposition (HO-SVD) technique. Due to the size of tensor, HO-SVD implementation usually leads to a huge matrix along some direction of tensor, which is always beyond the capacity of an ordinary PC. The novelty of this paper is to amend our previous ND-PCA scheme to deal with this challenge and further prove that the revised ND-PCA scheme can provide a near optimal linear solution under the given error bound. To evaluate the numerical property of the revised ND-PCA scheme, experiments are performed on a set of 3D volume datasets

On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell's Equations

Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the so-called Convection Theorem. Both approaches are thought to be necessary for cross-verification of the standard differential form of continuity equation. Any revision of these fundamental results might have important implications for all classical field theories. Rigorous reconsideration of time derivatives in Eulerian description shows that it evokes Minkowski metric for any flow field domain without any previous postulation. Mathematical approach is developed within the framework of congruences for general 4-dimensional differentiable manifold and the final result is formulated in form of a theorem. A modified version of the Convection Theorem provides a necessary cross-verification for a reconsidered differential form of continuity equation. Although the approach is developed for one-component (scalar) flow field, it can be easily generalized to any tensor field. Some possible implications for classical electrodynamics are also explored.Comment: no figure

Testing Isotropy of Cosmic Microwave Background Radiation

We introduce new symmetry-based methods to test for isotropy in cosmic microwave background radiation. Each angular multipole is factored into unique products of power eigenvectors, related multipoles and singular values that provide 2 new rotationally invariant measures mode by mode. The power entropy and directional entropy are new tests of randomness that are independent of the usual CMB power. Simulated galactic plane contamination is readily identified, and the new procedures mesh perfectly with linear transformations employed for windowed-sky analysis. The ILC -WMAP data maps show 7 axes well aligned with one another and the direction Virgo. Parameter free statistics find 12 independent cases of extraordinary axial alignment, low power entropy, or both having 5% probability or lower in an isotropic distribution. Isotropy of the ILC maps is ruled out to confidence levels of better than 99.9%, whether or not coincidences with other puzzles coming from the Virgo axis are included. Our work shows that anisotropy is not confined to the low l region, but extends over a much larger l range.Comment: 40 pages 15 figure