1,361 research outputs found
Poincaré on the Foundation of Geometry in the Understanding
This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them
Angular harmonics of the excitonic polarization conversions effect
We suggest a phenomenological theory of the polarization conversions effect,
an excitonic analog of the first-order spatial dispersion phenomena which is,
however, observed in the photoluminescence rather than in the passing light.
The optical polarization response of a model system of electrically neutral
quantum dots subject to the magnetic field along the growth axis was calculated
by means of the pseudospin method. All possible forms of the polarization
response are determined by nine different field-dependent coefficients which
represent, therefore, a natural basis for classification of a variety of
conversions. Existing experimental data can be well inscribed in this
classification scheme. Predictions were made regarding two effects which have
not been addressed experimentally.Comment: 14 pages, 1 figure, 1 tabl
Restricted three-body problem in effective-field-theory models of gravity
One of the outstanding problems of classical celestial mechanics was the
restricted 3-body prob- lem, in which a planetoid of small mass is subject to
the Newtonian attraction of two celestial bodies of large mass, as it occurs,
for example, in the sun-earth-moon system. On the other hand, over the last
decades, a systematic investigation of quantum corrections to the Newtonian
potential has been carried out in the literature on quantum gravity. The
present paper studies the effect of these tiny quantum corrections on the
evaluation of equilibrium points. It is shown that, despite the extreme
smallness of the corrections, there exists no choice of sign of these
corrections for which all qualitative features of the restricted 3-body problem
in Newtonian theory remain unaffected. Moreover, first-order stability of
equilibrium points is characterized by solving a pair of algebraic equations of
fifth degree, where some coefficients depend on the Planck length. The
coordinates of stable equilibrium points are slightly changed with respect to
Newtonian theory, because the planetoid is no longer at equal distance from the
two bodies of large mass. The effect is conceptually interesting but too small
to be observed, at least for the restricted 3-body problems available in the
solar system.Comment: 20 pages, latex, 8 figure
A Renormalization Proof of the KAM Theorem for Non-Analytic Perturbations
We shall use a Renormalization Group (RG) scheme in order to prove the
classical KAM result in the case of a non-analytic perturbation (the latter
will be assumed to have continuous derivatives up to a sufficiently large
order). We shall proceed by solving a sequence of problems in which the
perturbations are analytic approximations of the original one. We shall finally
show that the sequence of the approximate solutions will converge to a
differentiable solution of the original problem.Comment: 33 pages, no figure
Classical small systems coupled to finite baths
We have studied the properties of a classical -body system coupled to a
bath containing -body harmonic oscillators, employing an model
which is different from most of the existing models with . We have
performed simulations for -oscillator systems, solving
first-order differential equations with and , in order to calculate the time-dependent energy exchange between the
system and the bath. The calculated energy in the system rapidly changes while
its envelope has a much slower time dependence. Detailed calculations of the
stationary energy distribution of the system (: an energy per
particle in the system) have shown that its properties are mainly determined by
but weakly depend on . The calculated is analyzed with the
use of the and - distributions: the latter is derived with
the superstatistical approach (SSA) and microcanonical approach (MCA) to the
nonextensive statistics, where stands for the entropic index. Based on
analyses of our simulation results, a critical comparison is made between the
SSA and MCA. Simulations have been performed also for the -body ideal-gas
system. The effect of the coupling between oscillators in the bath has been
examined by additional () models which include baths consisting of
coupled linear chains with periodic and fixed-end boundary conditions.Comment: 30 pages, 16 figures; the final version accepted in Phys. Rev.
Dirac monopole with Feynman brackets
We introduce the magnetic angular momentum as a consequence of the structure
of the sO(3) Lie algebra defined by the Feynman brackets. The Poincare momentum
and Dirac magnetic monopole appears as a direct result of this framework.Comment: 10 page
Binary systems: implications for outflows & periodicities relevant to masers
Bipolar molecular outflows have been observed and studied extensively in the
past, but some recent observations of periodic variations in maser intensity
pose new challenges. Even quasi-periodic maser flares have been observed and
reported in the literature. Motivated by these data, we have tried to study
situations in binary systems with specific attention to the two observed
features, i.e., the bipolar flows and the variabilities in the maser intensity.
We have studied the evolution of spherically symmetric wind from one of the
bodies in the binary system, in the plane of the binary. Our approach includes
the analytical study of rotating flows with numerical computation of
streamlines of fluid particles using PLUTO code. We present the results of our
findings assuming simple configurations, and discuss the implications.Comment: 5 pages, 3 figures, Proceedings IAU Symposium No. 287, 2012, Cosmic
masers - from OH to H
Electric charge in the field of a magnetic event in three-dimensional spacetime
We analyze the motion of an electric charge in the field of a magnetically
charged event in three-dimensional spacetime. We start by exhibiting a first
integral of the equations of motion in terms of the three conserved components
of the spacetime angular momentum, and then proceed numerically. After crossing
the light cone of the event, an electric charge initially at rest starts
rotating and slowing down. There are two lengths appearing in the problem: (i)
the characteristic length , where and are the
electric charge and mass of the particle, and is the magnetic charge of the
event; and (ii) the spacetime impact parameter . For , after a time of order , the particle makes sharply a quarter of a
turn and comes to rest at the same spatial position at which the event happened
in the past. This jump is the main signature of the presence of the magnetic
event as felt by an electric charge. A derivation of the expression for the
angular momentum that uses Noether's theorem in the magnetic representation is
given in the Appendix.Comment: Version to appear in Phys. Rev.
Poincar\'{e}'s Observation and the Origin of Tsallis Generalized Canonical Distributions
In this paper, we present some geometric properties of the maximum entropy
(MaxEnt) Tsallis- distributions under energy constraint. In the case q > 1,
these distributions are proved to be marginals of uniform distributions on the
sphere; in the case q < 1, they can be constructed as conditional distribu-
tions of a Cauchy law built from the same uniform distribution on the sphere
using a gnomonic projection. As such, these distributions reveal the relevance
of using Tsallis distributions in the microcanonical setup: an example of such
application is given in the case of the ideal gas.Comment: 2 figure
Revealing the state space of turbulent pipe flow by symmetry reduction
Symmetry reduction by the method of slices is applied to pipe flow in order
to quotient the stream-wise translation and azimuthal rotation symmetries of
turbulent flow states. Within the symmetry-reduced state space, all travelling
wave solutions reduce to equilibria, and all relative periodic orbits reduce to
periodic orbits. Projections of these solutions and their unstable manifolds
from their -dimensional symmetry-reduced state space onto suitably
chosen 2- or 3-dimensional subspaces reveal their interrelations and the role
they play in organising turbulence in wall-bounded shear flows. Visualisations
of the flow within the slice and its linearisation at equilibria enable us to
trace out the unstable manifolds, determine close recurrences, identify
connections between different travelling wave solutions, and find, for the
first time for pipe flows, relative periodic orbits that are embedded within
the chaotic attractor, which capture turbulent dynamics at transitional
Reynolds numbers.Comment: 24 pages, 12 figure
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