823 research outputs found
Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities
We examine in detail the relative equilibria in the four-vortex problem where
two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and
\Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is
that for m > 0, the convex configurations all contain a line of symmetry,
forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for
all m but the isosceles trapezoid case exists only when m is positive. In fact,
there exist asymmetric convex configurations when m < 0. In contrast to the
Newtonian four-body problem with two equal pairs of masses, where the symmetry
of all convex central configurations is unproven, the equations in the vortex
case are easier to handle, allowing for a complete classification of all
solutions. Precise counts on the number and type of solutions (equivalence
classes) for different values of m, as well as a description of some of the
bifurcations that occur, are provided. Our techniques involve a combination of
analysis and modern and computational algebraic geometry
Seven-body central configurations: a family of central configurations in the spatial seven-body problem
The main result of this paper is the existence of a new family of central
configurations in the Newtonian spatial seven-body problem. This family is
unusual in that it is a simplex stacked central configuration, i.e the bodies
are arranged as concentric three and two dimensional simplexes.Comment: 15 pages 5 figure
Symmetry, bifurcation and stacking of the central configurations of the planar 1+4 body problem
In this work we are interested in the central configurations of the planar
1+4 body problem where the satellites have different infinitesimal masses and
two of them are diametrically opposite in a circle. We can think this problem
as a stacked central configuration too. We show that the configuration are
necessarily symmetric and the other sattelites has the same mass. Moreover we
proved that the number of central configuration in this case is in general one,
two or three and in the special case where the satellites diametrically
opposite have the same mass we proved that the number of central configuration
is one or two saying the exact value of the ratio of the masses that provides
this bifurcation.Comment: 9 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1103.627
The Lie-Poisson structure of the reduced n-body problem
The classical n-body problem in d-dimensional space is invariant under the
Galilean symmetry group. We reduce by this symmetry group using the method of
polynomial invariants. As a result we obtain a reduced system with a
Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The
reduction preserves the natural form of the Hamiltonian as a sum of kinetic
energy that depends on velocities only and a potential that depends on
positions only. Hence we proceed to construct a Poisson integrator for the
reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure
Convex central configurations of the 4-body problem with two pairs of equal masses
Agraïments: The first and third authors are partially supported by FAPEMIG grant APQ-001082/14. The third author is partially supported by CNPq grant 472321/2013-7 and by FAPEMIG grant PPM-00516-15. The second and third autors are supported by CAPES CSF-PVE grant 88881.030454/2013-01.MacMillan and Bartky in 1932 proved that there is a unique isosceles trapezoid central configuration of the 4--body problem when two pairs of equal masses are located at adjacent vertices. After this result the following conjecture was well known between people working on central configurations: The isosceles trapezoid is the unique convex central configuration of the planar 4--body problem when two pairs of equal masses are located at adjacent vertices. We prove this conjecture
Arguments géophysiques et géologiques en faveur de manifestations volcaniques récentes dans la région du lac Faguibine (Mali)
Carbon Nanotubes Synthesized in Channels of Alpo4-5 Single Crystals : First X-Ray Scattering Investigations
Following the synthesis of aligned single-wall carbon nanotubes in the
channels of AlPO4-5 zeolite single crystals, we present the first X-ray
diffraction and diffuse scattering results. They can be analysed in terms of a
partial filling of the zeolite channels by nanotubes with diameter around 4A.
The possible selection of only one type of nanotube during the synthesis, due
to the constraints imposed by the zeolite host, is discussed.Comment: to appear in Solid State Com
Bases for qudits from a nonstandard approach to SU(2)
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for
quantum information and quantum computation are constructed from angular
momentum theory and su(2) Lie algebraic methods. We report on a formula for
deriving in one step the (1+p)p qupits (i.e., qudits with d = p a prime
integer) of a complete set of 1+p mutually unbiased bases in C^p. Repeated
application of the formula can be used for generating mutually unbiased bases
in C^d with d = p^e (e > or = 2) a power of a prime integer. A connection
between mutually unbiased bases and the unitary group SU(d) is briefly
discussed in the case d = p^e.Comment: From a talk presented at the 13th International Conference on
Symmetry Methods in Physics (Dubna, Russia, 6-9 July 2009) organized in
memory of Prof. Yurii Fedorovich Smirnov by the Bogoliubov Laboratory of
Theoretical Physics of the JINR and the ICAS at Yerevan State University
Qudits of composite dimension, mutually unbiased bases and projective ring geometry
The Pauli operators attached to a composite qudit in dimension may
be mapped to the vectors of the symplectic module
( the modular ring). As a result, perpendicular vectors
correspond to commuting operators, a free cyclic submodule to a maximal
commuting set, and disjoint such sets to mutually unbiased bases. For
dimensions , and 18, the fine structure and the incidence
between maximal commuting sets is found to reproduce the projective line over
the rings , , ,
and ,
respectively.Comment: 10 pages (Fast Track communication). Journal of Physics A
Mathematical and Theoretical (2008) accepte
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