6 research outputs found
Spacelike Singularities and Hidden Symmetries of Gravity
We review the intimate connection between (super-)gravity close to a
spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody
algebras. We show that in this limit the gravitational theory can be
reformulated in terms of billiard motion in a region of hyperbolic space,
revealing that the dynamics is completely determined by a (possibly infinite)
sequence of reflections, which are elements of a Lorentzian Coxeter group. Such
Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras,
suggesting that these algebras yield symmetries of gravitational theories. Our
presentation is aimed to be a self-contained and comprehensive treatment of the
subject, with all the relevant mathematical background material introduced and
explained in detail. We also review attempts at making the infinite-dimensional
symmetries manifest, through the construction of a geodesic sigma model based
on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case
of the hyperbolic algebra E10, which is conjectured to be an underlying
symmetry of M-theory. Illustrations of this conjecture are also discussed in
the context of cosmological solutions to eleven-dimensional supergravity.Comment: 228 pages. Typos corrected. References added. Subject index added.
Published versio
Robust kernel distance multivariate control chart using support vector principles
It is important to monitor manufacturing processes in order to improve product
quality and reduce production cost. Statistical Process Control (SPC) is the
most commonly used method for process monitoring, in particular making
distinctions between variations attributed to normal process variability to
those caused by ‘special causes’. Most SPC and multivariate SPC (MSPC) methods
are parametric in that they make assumptions about the distributional properties
and autocorrelation structure of in-control process parameters, and, if
satisfied, are effective in managing false alarms/-positives and false-
negatives. However, when processes do not satisfy these assumptions, the
effectiveness of SPC methods is compromised. Several non-parametric control
charts based on sequential ranks of data depth measures have been proposed in
the literature, but their development and implementation have been rather slow
in industrial process control. Several non-parametric control charts based on
machine learning principles have also been proposed in the literature to
overcome some of these limitations. However, unlike conventional SPC methods,
these non-parametric methods require event data from each out-of-control process
state for effective model building. The paper presents a new non-parametric
multivariate control chart based on kernel distance that overcomes these
limitations by employing the notion of one-class classification based on support
vector principles. The chart is non-parametric in that it makes no assumptions
regarding the data probability density and only requires ‘normal’ or in-control
data for effective representation of an in-control process. It does, however,
make an explicit provision to incorporate any available data from out-of-control
process states. Experimental evaluation on a variety of benchmarking datasets
suggests that the proposed chart is effective for process mo