417 research outputs found
One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula
In \cite{Mul} one-parameter planar motion was first introduced and the
relations between absolute, relative, sliding velocities (and accelerations) in
the Euclidean plane were obtained. Moreover, the relations
between the Complex velocities one-parameter motion in the Complex plane were
provided by \cite{Mul}. One-parameter planar homothetic motion was defined in
the Complex plane, \cite{Kur}. In this paper, analogous to homothetic motion in
the Complex plane given by \cite{Kur}, one-parameter planar homothetic motion
is defined in the Hyperbolic plane. Some characteristic properties about the
velocity vectors, the acceleration vectors and the pole curves are given.
Moreover, in the case of homothetic scale identically equal to 1, the
results given in \cite{Yuc} are obtained as a special case. In addition, three
hyperbolic planes, of which two are moving and the other one is fixed, are
taken into consideration and a canonical relative system for one-parameter
planar hyperbolic homothetic motion is defined. Euler-Savary formula, which
gives the relationship between the curvatures of trajectory curves, is obtained
with the help of this relative system
KPZ in one dimensional random geometry of multiplicative cascades
We prove a formula relating the Hausdorff dimension of a subset of the unit
interval and the Hausdorff dimension of the same set with respect to a random
path matric on the interval, which is generated using a multiplicative cascade.
When the random variables generating the cascade are exponentials of Gaussians,
the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum
gravity appears. This note was inspired by the recent work of Duplantier and
Sheffield proving a somewhat different version of the KPZ formula for Liouville
gravity. In contrast with the Liouville gravity setting, the one dimensional
multiplicative cascade framework facilitates the determination of the Hausdorff
dimension, rather than some expected box count dimension.Comment: 14 page
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
On the noise-induced passage through an unstable periodic orbit II: General case
Consider a dynamical system given by a planar differential equation, which
exhibits an unstable periodic orbit surrounding a stable periodic orbit. It is
known that under random perturbations, the distribution of locations where the
system's first exit from the interior of the unstable orbit occurs, typically
displays the phenomenon of cycling: The distribution of first-exit locations is
translated along the unstable periodic orbit proportionally to the logarithm of
the noise intensity as the noise intensity goes to zero. We show that for a
large class of such systems, the cycling profile is given, up to a
model-dependent change of coordinates, by a universal function given by a
periodicised Gumbel distribution. Our techniques combine action-functional or
large-deviation results with properties of random Poincar\'e maps described by
continuous-space discrete-time Markov chains.Comment: 44 pages, 4 figure
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
First passage behaviour of fractional Brownian motion in two-dimensional wedge domains
We study the survival probability and the corresponding first passage time
density of fractional Brownian motion confined to a two-dimensional open wedge
domain with absorbing boundaries. By analytical arguments and numerical
simulation we show that in the long time limit the first passage time density
scales as t**{-1+pi*(2H-2)/(2*Theta)} in terms of the Hurst exponent H and the
wedge angle Theta. We discuss this scaling behaviour in connection with the
reaction kinetics of FBM particles in a one-dimensional domain.Comment: 6 pages, 4 figure
Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
We construct a N-dimensional Gaussian landscape with multiscale, translation
invariant, logarithmic correlations and investigate the statistical mechanics
of a single particle in this environment. In the limit of high dimension N>>1
the free energy of the system and overlap function are calculated exactly using
the replica trick and Parisi's hierarchical ansatz. In the thermodynamic limit,
we recover the most general version of the Derrida's Generalized Random Energy
Model (GREM). The low-temperature behaviour depends essentially on the spectrum
of length scales involved in the construction of the landscape. If the latter
consists of K discrete values, the system is characterized by a K-step Replica
Symmetry Breaking solution. We argue that our construction is in fact valid in
any finite spatial dimensions . We discuss implications of our results
for the singularity spectrum describing multifractality of the associated
Boltzmann-Gibbs measure. Finally we discuss several generalisations and open
problems, the dynamics in such a landscape and the construction of a
Generalized Multifractal Random Walk.Comment: 25 pages, published version with a few misprints correcte
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
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
Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics
Before we dive into the accessibility stream of nowadays indicatory
applications of octonions to computer and other sciences and to quantum physics
let us focus for a while on the crucially relevant events for todays revival on
interest to nonassociativity. Our reflections keep wandering back to the
two square identity and then via the four
square identity up to the eight square identity.
These glimpses of history incline and invite us to retell the story on how
about one month after quaternions have been carved on the bridge
octonions were discovered by , jurist and
mathematician, a friend of . As for today we just
mention en passant quaternionic and octonionic quantum mechanics,
generalization of equations for octonions and triality
principle and group in spinor language in a descriptive way in order not
to daunt non specialists. Relation to finite geometries is recalled and the
links to the 7stones of seven sphere, seven imaginary octonions units in out of
the cave reality applications are appointed . This way we are welcomed
back to primary ideas of , and other distinguished
fathers of quantum mechanics and quantum gravity foundations.Comment: 26 pages, 7 figure
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