135 research outputs found
About the maximal rank of 3-tensors over the real and the complex number field
High dimensional array data, tensor data, is becoming important in recent
days. Then maximal rank of tensors is important in theory and applications. In
this paper we consider the maximal rank of 3 tensors. It can be attacked from
various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and
Atkinson-Lloyd(1980). They treated the problem in the complex field, and we
will present various bounds over the real field by proving several lemmas and
propositions, which is real counterparts of their results.Comment: 13 pages, no figure v2: correction and improvemen
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
The Volume Inside a Black Hole
The horizon (the surface) of a black hole is a null surface, defined by those
hypothetical "outgoing" light rays that just hover under the influence of the
strong gravity at the surface. Because the light rays are orthogonal to the
spatial 2-dimensional surface at one instant of time, the surface of the black
hole is the same for all observers (i.e. the same for all coordinate
definitions of "instant of time"). This value is 4*(pi)* (2Gm/c^2)^2 for
nonspinning black holes, with G= Newton's constant, c= speed of light, and m=
mass of the black hole.
The 3-dimensional spatial volume inside a black hole, in contrast, depends
explicitly on the definition of time, and can even be time dependent, or zero.
We give examples of the volume found inside a standard, nonspinning spherical
black hole, for several different standard time-coordinate definitions.
Elucidating these results for the volume provides a new pedagogical resource of
facts already known in principle to the relativity community, but rarely worked
out.Comment: 17 pages, 5 figure
Gluing construction of initial data with Kerr-de Sitter ends
We construct initial data sets which satisfy the vacuum constraint equa-
tions of General Relativity with positive cosmologigal constant. More pre-
silely, we deform initial data with ends asymptotic to Schwarzschild-de Sitter
to obtain non-trivial initial data with exactly Kerr-de Sitter ends. The method
is inspired from Corvino's gluing method. We obtain here a extension of a
previous result for the time-symmetric case by Chru\'sciel and Pollack.Comment: 27 pages, 3 figure
Cosmological Surrealism: More than ``Eternal Reality" is Needed
Inflationary Cosmology makes the universe ``eternal" and provides for
recurrent universe creation, ad infinitum -- making it also plausible to assume
that ``our" Big Bang was also preceeded by others, etc.. However, GR tells us
that in the ``parent" universe's reference frame, the newborn universe's
expansion will never start. Our picture of ``reality" in spacetime has to be
enlarged.Comment: 7 pages, TAUP N23
The physical meaning of the "boost-rotation symmetric" solutions within the general interpretation of Einstein's theory of gravitation
The answer to the question, what physical meaning should be attributed to the
so-called boost-rotation symmetric exact solutions to the field equations of
general relativity, is provided within the general interpretation scheme for
the ``theories of relativity'', based on group theoretical arguments, and set
forth by Erich Kretschmann already in the year 1917.Comment: 9 pages, 1 figure; text to appear in General Relativity and
Gravitatio
Scattering of scalar particles by a black hole
The absorption cross section for scalar particle impact on a Schwarzschild
black hole is found. The process is dominated by two physical phenomena. One of
them is the well-known greybody factor that arises from the energy-dependent
potential barrier outside the horizon that filters the incoming and outgoing
waves. The other is related to the reflection of particles on the horizon
(Kuchiev 2003). This latter effect strongly diminishes the cross section for
low energies, forcing it to vanish in the infrared limit. It is argued that
this is a general property, the absorption cross section vanishes in the
infrared limit for scattering of particles of arbitrary spin.Comment: 7 pages, revtex, 1 figur
Nonlinear Modulation of Multi-Dimensional Lattice Waves
The equations governing weakly nonlinear modulations of -dimensional
lattices are considered using a quasi-discrete multiple-scale approach. It is
found that the evolution of a short wave packet for a lattice system with cubic
and quartic interatomic potentials is governed by generalized Davey-Stewartson
(GDS) equations, which include mean motion induced by the oscillatory wave
packet through cubic interatomic interaction. The GDS equations derived here
are more general than those known in the theory of water waves because of the
anisotropy inherent in lattices. Generalized Kadomtsev-Petviashvili equations
describing the evolution of long wavelength acoustic modes in two and three
dimensional lattices are also presented. Then the modulational instability of a
-dimensional Stokes lattice wave is discussed based on the -dimensional
GDS equations obtained. Finally, the one- and two-soliton solutions of
two-dimensional GDS equations are provided by means of Hirota's bilinear
transformation method.Comment: Submitted to PR
Reflection, radiation and interference for black holes
Black holes are capable of reflection: there is a finite probability for any
particle that approaches the event horizon to bounce back. The albedo of the
black hole depends on its temperature and the energy of the incoming particle.
The reflection shares its physical origins with the Hawking process of
radiation, both of them arise as consequences of the mixing of the incoming and
outgoing waves that takes place on the event horizon.Comment: 10 pages, 1 figure, Revte
Sheared Flow As A Stabilizing Mechanism In Astrophysical Jets
It has been hypothesized that the sustained narrowness observed in the
asymptotic cylindrical region of bipolar outflows from Young Stellar Objects
(YSO) indicates that these jets are magnetically collimated. The j cross B
force observed in z-pinch plasmas is a possible explanation for these
observations. However, z-pinch plasmas are subject to current driven
instabilities (CDI). The interest in using z-pinches for controlled nuclear
fusion has lead to an extensive theory of the stability of magnetically
confined plasmas. Analytical, numerical, and experimental evidence from this
field suggest that sheared flow in magnetized plasmas can reduce the growth
rates of the sausage and kink instabilities. Here we propose the hypothesis
that sheared helical flow can exert a similar stabilizing influence on CDI in
YSO jets.Comment: 13 pages, 2 figure
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