6,033 research outputs found
Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes III. Quasinormal Pulsations of Schwarzschild and Kerr Black Holes
In recent papers, we and colleagues have introduced a way to visualize the
full vacuum Riemann curvature tensor using frame-drag vortex lines and their
vorticities, and tidal tendex lines and their tendicities. We have also
introduced the concepts of horizon vortexes and tendexes and 3-D vortexes and
tendexes (regions where vorticities or tendicities are large). Using these
concepts, we discover a number of previously unknown features of quasinormal
modes of Schwarzschild and Kerr black holes. These modes can be classified by
mode indexes (n,l,m), and parity, which can be electric [(-1)^l] or magnetic
[(-1)^(l+1)]. Among our discoveries are these: (i) There is a near duality
between modes of the same (n,l,m): a duality in which the tendex and vortex
structures of electric-parity modes are interchanged with the vortex and tendex
structures (respectively) of magnetic-parity modes. (ii) This near duality is
perfect for the modes' complex eigenfrequencies (which are well known to be
identical) and perfect on the horizon; it is slightly broken in the equatorial
plane of a non-spinning hole, and the breaking becomes greater out of the
equatorial plane, and greater as the hole is spun up; but even out of the plane
for fast-spinning holes, the duality is surprisingly good. (iii)
Electric-parity modes can be regarded as generated by 3-D tendexes that stick
radially out of the horizon. As these "longitudinal," near-zone tendexes rotate
or oscillate, they generate longitudinal-transverse near-zone vortexes and
tendexes, and outgoing and ingoing gravitational waves. The ingoing waves act
back on the longitudinal tendexes, driving them to slide off the horizon, which
results in decay of the mode's strength. (iv) By duality, magnetic-parity modes
are driven in this same manner by longitudinal, near-zone vortexes that stick
out of the horizon. [Abstract abridged.]Comment: 53 pages with an overview of major results in the first 11 pages, 26
figures. v2: Very minor changes to reflect published version. v3: Fixed Ref
Optimal classification in sparse Gaussian graphic model
Consider a two-class classification problem where the number of features is
much larger than the sample size. The features are masked by Gaussian noise
with mean zero and covariance matrix , where the precision matrix
is unknown but is presumably sparse. The useful features,
also unknown, are sparse and each contributes weakly (i.e., rare and weak) to
the classification decision. By obtaining a reasonably good estimate of
, we formulate the setting as a linear regression model. We propose a
two-stage classification method where we first select features by the method of
Innovated Thresholding (IT), and then use the retained features and Fisher's
LDA for classification. In this approach, a crucial problem is how to set the
threshold of IT. We approach this problem by adapting the recent innovation of
Higher Criticism Thresholding (HCT). We find that when useful features are rare
and weak, the limiting behavior of HCT is essentially just as good as the
limiting behavior of ideal threshold, the threshold one would choose if the
underlying distribution of the signals is known (if only). Somewhat
surprisingly, when is sufficiently sparse, its off-diagonal
coordinates usually do not have a major influence over the classification
decision. Compared to recent work in the case where is the identity
matrix [Proc. Natl. Acad. Sci. USA 105 (2008) 14790-14795; Philos. Trans. R.
Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009) 4449-4470], the current
setting is much more general, which needs a new approach and much more
sophisticated analysis. One key component of the analysis is the intimate
relationship between HCT and Fisher's separation. Another key component is the
tight large-deviation bounds for empirical processes for data with
unconventional correlation structures, where graph theory on vertex coloring
plays an important role.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1163 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Elation generalised quadrangles of order (s,p), where p is prime
We show that an elation generalised quadrangle which has p+1 lines on each
point, for some prime p, is classical or arises from a flock of a quadratic
cone (i.e., is a flock quadrangle).Comment: 14 page
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