208 research outputs found
Conformal compactification and cycle-preserving symmetries of spacetimes
The cycle-preserving symmetries for the nine two-dimensional real spaces of
constant curvature are collectively obtained within a Cayley-Klein framework.
This approach affords a unified and global study of the conformal structure of
the three classical Riemannian spaces as well as of the six relativistic and
non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both
Newton-Hooke and Galilean), and gives rise to general expressions holding
simultaneously for all of them. Their metric structure and cycles (lines with
constant geodesic curvature that include geodesics and circles) are explicitly
characterized. The corresponding cyclic (Mobius-like) Lie groups together with
the differential realizations of their algebras are then deduced; this
derivation is new and much simpler than the usual ones and applies to any
homogeneous space in the Cayley-Klein family, whether flat or curved and with
any signature. Laplace and wave-type differential equations with conformal
algebra symmetry are constructed. Furthermore, the conformal groups are
realized as matrix groups acting as globally defined linear transformations in
a four-dimensional "conformal ambient space", which in turn leads to an
explicit description of the "conformal completion" or compactification of the
nine spaces.Comment: 43 pages, LaTe
Random walks pertaining to a class of deterministic weighted graphs
In this note, we try to analyze and clarify the intriguing interplay between
some counting problems related to specific thermalized weighted graphs and
random walks consistent with such graphs
Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry
A new method to obtain trigonometry for the real spaces of constant curvature
and metric of any (even degenerate) signature is presented. The method
encapsulates trigonometry for all these spaces into a single basic
trigonometric group equation. This brings to its logical end the idea of an
absolute trigonometry, and provides equations which hold true for the nine
two-dimensional spaces of constant curvature and any signature. This family of
spaces includes both relativistic and non-relativistic homogeneous spacetimes;
therefore a complete discussion of trigonometry in the six de Sitter,
minkowskian, Newton--Hooke and galilean spacetimes follow as particular
instances of the general approach. Any equation previously known for the three
classical riemannian spaces also has a version for the remaining six
spacetimes; in most cases these equations are new. Distinctive traits of the
method are universality and self-duality: every equation is meaningful for the
nine spaces at once, and displays explicitly invariance under a duality
transformation relating the nine spaces. The derivation of the single basic
trigonometric equation at group level, its translation to a set of equations
(cosine, sine and dual cosine laws) and the natural apparition of angular and
lateral excesses, area and coarea are explicitly discussed in detail. The
exposition also aims to introduce the main ideas of this direct group
theoretical way to trigonometry, and may well provide a path to systematically
study trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe
Geometries for Possible Kinematics
The algebras for all possible Lorentzian and Euclidean kinematics with
isotropy except static ones are re-classified. The geometries
for algebras are presented by contraction approach. The relations among the
geometries are revealed. Almost all geometries fall into pairs. There exists correspondence in each pair. In the viewpoint of
differential geometry, there are only 9 geometries, which have right signature
and geometrical spatial isotropy. They are 3 relativistic geometries, 3
absolute-time geometries, and 3 absolute-space geometries.Comment: 40 pages, 7 figure
On the bicrossproduct structures for the family of algebras
It is shown that the family of deformed algebras has a different bicrossproduct
structure for each in analogy to the undeformed case.Comment: Latex2e file. 14 page
Chaotic Cascades with Kolmogorov 1941 Scaling
We define a (chaotic) deterministic variant of random multiplicative cascade
models of turbulence. It preserves the hierarchical tree structure, thanks to
the addition of infinitesimal noise. The zero-noise limit can be handled by
Perron-Frobenius theory, just as the zero-diffusivity limit for the fast dynamo
problem. Random multiplicative models do not possess Kolmogorov 1941 (K41)
scaling because of a large-deviations effect. Our numerical studies indicate
that deterministic multiplicative models can be chaotic and still have exact
K41 scaling. A mechanism is suggested for avoiding large deviations, which is
present in maps with a neutrally unstable fixed point.Comment: 14 pages, plain LaTex, 6 figures available upon request as hard copy
(no local report #
Big Entropy Fluctuations in Statistical Equilibrium: The Macroscopic Kinetics
Large entropy fluctuations in an equilibrium steady state of classical
mechanics were studied in extensive numerical experiments on a simple
2--freedom strongly chaotic Hamiltonian model described by the modified Arnold
cat map. The rise and fall of a large separated fluctuation was shown to be
described by the (regular and stable) "macroscopic" kinetics both fast
(ballistic) and slow (diffusive). We abandoned a vague problem of "appropriate"
initial conditions by observing (in a long run)spontaneous birth and death of
arbitrarily big fluctuations for any initial state of our dynamical model.
Statistics of the infinite chain of fluctuations, reminiscent to the Poincar\'e
recurrences, was shown to be Poissonian. A simple empirical relation for the
mean period between the fluctuations (Poincar\'e "cycle") has been found and
confirmed in numerical experiments. A new representation of the entropy via the
variance of only a few trajectories ("particles") is proposed which greatly
facilitates the computation, being at the same time fairly accurate for big
fluctuations. The relation of our results to a long standing debates over
statistical "irreversibility" and the "time arrow" is briefly discussed too.Comment: Latex 2.09, 26 pages, 6 figure
Strong Universality in Forced and Decaying Turbulence
The weak version of universality in turbulence refers to the independence of
the scaling exponents of the th order strcuture functions from the
statistics of the forcing. The strong version includes universality of the
coefficients of the structure functions in the isotropic sector, once
normalized by the mean energy flux. We demonstrate that shell models of
turbulence exhibit strong universality for both forced and decaying turbulence.
The exponents {\em and} the normalized coefficients are time independent in
decaying turbulence, forcing independent in forced turbulence, and equal for
decaying and forced turbulence. We conjecture that this is also the case for
Navier-Stokes turbulence.Comment: RevTex 4, 10 pages, 5 Figures (included), 1 Table; PRE, submitte
Integrable potentials on spaces with curvature from quantum groups
A family of classical integrable systems defined on a deformation of the
two-dimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed
through Hamiltonians defined on the non-standard quantum deformation of a sl(2)
Poisson coalgebra. All these spaces have a non-constant curvature that depends
on the deformation parameter z. As particular cases, the analogues of the
harmonic oscillator and Kepler--Coulomb potentials on such spaces are proposed.
Another deformed Hamiltonian is also shown to provide superintegrable systems
on the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constant
curvature that exactly coincides with z. According to each specific space, the
resulting potential is interpreted as the superposition of a central harmonic
oscillator with either two more oscillators or centrifugal barriers. The
non-deformed limit z=0 of all these Hamiltonians can then be regarded as the
zero-curvature limit (contraction) which leads to the corresponding
(super)integrable systems on the flat Euclidean and Minkowskian spaces.Comment: 19 pages, 1 figure. Two references adde
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