356 research outputs found
Investigation of dynamical systems using tools of the theory of invariants and projective geometry
The investigation of nonlinear dynamical systems of the type
by means of reduction to
some ordinary differential equations of the second order in the form
is done. The main
backbone of this investigation was provided by the theory of invariants
developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century
and the projective geometry of E. Cartan. In our work two, in some sense
supplementary, systems are considered: the Lorenz system and the R\"o\ss ler system
. The invarinats for the ordinary
differential equations, which correspond to the systems mentioned abouve, are
evaluated. The connection of values of the invariants with characteristics of
dynamical systems is established.Comment: 18 pages, Latex, to appear in J. of Applied Mathematics (ZAMP
Study of the risk-adjusted pricing methodology model with methods of Geometrical Analysis
Families of exact solutions are found to a nonlinear modification of the
Black-Scholes equation. This risk-adjusted pricing methodology model (RAPM)
incorporates both transaction costs and the risk from a volatile portfolio.
Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM
equation. It gives us the possibility to describe an optimal system of
subalgebras and correspondingly the set of invariant solutions to the model. In
this way we can describe the complete set of possible reductions of the
nonlinear RAPM model. Reductions are given in the form of different second
order ordinary differential equations. In all cases we provide solutions to
these equations in an exact or parametric form. We discuss the properties of
these reductions and the corresponding invariant solutions.Comment: larger version with exact solutions, corrected typos, 13 pages,
Symposium on Optimal Stopping in Abo/Turku 200
Casimir force under the influence of real conditions
The Casimir force is calculated analytically for configurations of two
parallel plates and a spherical lens (sphere) above a plate with account of
nonzero temperature, finite conductivity of the boundary metal and surface
roughness. The permittivity of the metal is described by the plasma model. It
is proved that in case of the plasma model the scattering formalism of quantum
field theory in Matsubara formulation underlying Lifshitz formula is well
defined and no modifications are needed concerning the zero-frequency
contribution. The temperature correction to the Casimir force is found
completely with respect to temperature and perturbatively (up to the second
order in the relative penetration depth of electromagnetic zero-point
oscillations into the metal) with respect to finite conductivity. The
asymptotics of low and high temperatures are presented and contributions of
longitudinal and perpendicular modes are determined separately. Serving as an
example, aluminium test bodies are considered showing good agreement between
the obtained analytical results and previously performed numerical
computations. The roughness correction is formally included and formulas are
given permitting to calculate the Casimir force under the influence of all
relevant factors
Dynamical Casimir Effect in a one-dimensional uniformly contracting cavity
We consider particle creation (the Dynamical Casimir effect) in a uniformly
contracting ideal one-dimensional cavity non-perturbatively. The exact
expression for the energy spectrum of created particles is obtained and its
dependence on parameters of the problem is discussed. Unexpectedly, the number
of created particles depends on the duration of the cavity contracting
non-monotonously. This is explained by quantum interference of the events of
particle creation which are taking place only at the moments of acceleration
and deceleration of a boundary, while stable particle states exist (and thus no
particles are created) at the time of contracting.Comment: 13 pages, 4 figure
Ground state energy in a wormhole space-time
The ground state energy of the massive scalar field with non-conformal
coupling on the short-throat flat-space wormhole background is calculated
by using zeta renormalization approach. We discuss the renormalization and
relevant heat kernel coefficients in detail. We show that the stable
configuration of wormholes can exist for . In particular case of
massive conformal scalar field with , the radius of throat of stable
wormhole . The self-consistent wormhole has radius of throat
and mass of scalar boson ( and
are the Planck length and mass, respectively).Comment: revtex, 18 pages, 3 eps figures. accepted in Phys.Rev.
Lateral projection as a possible explanation of the nontrivial boundary dependence of the Casimir force
We find the lateral projection of the Casimir force for a configuration of a
sphere above a corrugated plate. This force tends to change the sphere position
in the direction of a nearest corrugation maximum. The probability distribution
describing different positions of a sphere above a corrugated plate is
suggested which is fitted well with experimental data demonstrating the
nontrivial boundary dependence of the Casimir force.Comment: 5 pages, 1 figur
On the ground state energy for a penetrable sphere and for a dielectric ball
We analyse the ultraviolet divergencies in the ground state energy for a
penetrable sphere and a dielectric ball. We argue that for massless fields
subtraction of the ``empty space'' or the ``unbounded medium'' contribution is
not enough to make the ground state energy finite whenever the heat kernel
coefficient is not zero. It turns out that for a penetrable
sphere, a general dielectric background and the dielectric ball. To our
surprise, for more singular configurations, as in the presence of sharp
boundaries, the heat kernel coefficients behave to some extend better than in
the corresponding smooth cases, making, for instance, the dilute dielectric
ball a well defined problem.Comment: 18 pages, 1 figure, subm. to Phys. Rev.
Thermodynamical aspects of the Casimir force between real metals at nonzero temperature
We investigate the thermodynamical aspects of the Casimir effect in the case
of plane parallel plates made of real metals. The thermal corrections to the
Casimir force between real metals were recently computed by several authors
using different approaches based on the Lifshitz formula with diverse results.
Both the Drude and plasma models were used to describe a real metal. We
calculate the entropy density of photons between metallic plates as a function
of the surface separation and temperature. Some of these approaches are
demonstrated to lead to negative values of entropy and to nonzero entropy at
zero temperature depending on the parameters of the system. The conclusion is
that these approaches are in contradiction with the third law of thermodynamics
and must be rejected. It is shown that the plasma dielectric function in
combination with the unmodified Lifshitz formula is in perfect agreement with
the general principles of thermodynamics. As to the Drude dielectric function,
the modification of the zero-frequency term of the Lifshitz formula is outlined
that not to violate the laws of thermodynamics.Comment: 8pages, 4 figures; Phys. Rev. A, to appea
Boundary dynamics and multiple reflection expansion for Robin boundary conditions
In the presence of a boundary interaction, Neumann boundary conditions should
be modified to contain a function S of the boundary fields: (\nabla_N +S)\phi
=0. Information on quantum boundary dynamics is then encoded in the
-dependent part of the effective action. In the present paper we extend the
multiple reflection expansion method to the Robin boundary conditions mentioned
above, and calculate the heat kernel and the effective action (i) for constant
S, (ii) to the order S^2 with an arbitrary number of tangential derivatives.
Some applications to symmetry breaking effects, tachyon condensation and brane
world are briefly discussed.Comment: latex, 22 pages, no figure
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