3,868 research outputs found
Isotropy, shear, symmetry and exact solutions for relativistic fluid spheres
The symmetry method is used to derive solutions of Einstein's equations for
fluid spheres using an isotropic metric and a velocity four vector that is
non-comoving. Initially the Lie, classical approach is used to review and
provide a connecting framework for many comoving and so shear free solutions.
This provides the basis for the derivation of the classical point symmetries
for the more general and mathematicaly less tractable description of Einstein's
equations in the non-comoving frame. Although the range of symmetries is
restrictive, existing and new symmetry solutions with non-zero shear are
derived. The range is then extended using the non-classical direct symmetry
approach of Clarkson and Kruskal and so additional new solutions with non-zero
shear are also presented. The kinematics and pressure, energy density, mass
function of these solutions are determined.Comment: To appear in Classical and Quantum Gravit
Analytical perturbative approach to periodic orbits in the homogeneous quartic oscillator potential
We present an analytical calculation of periodic orbits in the homogeneous
quartic oscillator potential. Exploiting the properties of the periodic
Lam{\'e} functions that describe the orbits bifurcated from the fundamental
linear orbit in the vicinity of the bifurcation points, we use perturbation
theory to obtain their evolution away from the bifurcation points. As an
application, we derive an analytical semiclassical trace formula for the
density of states in the separable case, using a uniform approximation for the
pitchfork bifurcations occurring there, which allows for full semiclassical
quantization. For the non-integrable situations, we show that the uniform
contribution of the bifurcating period-one orbits to the coarse-grained density
of states competes with that of the shortest isolated orbits, but decreases
with increasing chaoticity parameter .Comment: 15 pages, LaTeX, 7 figures; revised and extended version, to appear
in J. Phys. A final version 3; error in eq. (33) corrected and note added in
prin
Occurrence of periodic Lam\'e functions at bifurcations in chaotic Hamiltonian systems
We investigate cascades of isochronous pitchfork bifurcations of
straight-line librating orbits in some two-dimensional Hamiltonian systems with
mixed phase space. We show that the new bifurcated orbits, which are
responsible for the onset of chaos, are given analytically by the periodic
solutions of the Lam\'e equation as classified in 1940 by Ince. In Hamiltonians
with C_ symmetry, they occur alternatingly as Lam\'e functions of period
2K and 4K, respectively, where 4K is the period of the Jacobi elliptic function
appearing in the Lam\'e equation. We also show that the two pairs of orbits
created at period-doubling bifurcations of touch-and-go type are given by two
different linear combinations of algebraic Lam\'e functions with period 8K.Comment: LaTeX2e, 22 pages, 14 figures. Version 3: final form of paper,
accepted by J. Phys. A. Changes in Table 2; new reference [25]; name of
bifurcations "touch-and-go" replaced by "island-chain
Solutions for certain classes of Riccati differential equation
We derive some analytic closed-form solutions for a class of Riccati equation
y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are
C^{\infty}-functions. We show that if \delta_n=\lambda_n
s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}=
\lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and
s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has
a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the
generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also
investigated.Comment: 10 page
Concurrent acute pancreatitis and pericardial effusion
While pleural effusion and ascites secondary to acute pancreatitis are common, clinically relevant pericardial effusion and cardiac tamponade are observed rarely. In a study by Pezzilli et al., pleural effusion was noted in 7 of the 21 patients with acute pancreatitis whereas the authors detected pericardial effusion development in only three. The authors asserted that pleural effusion was associated with severe acute pancreatitis, while pericardial effusion and the severity of acute pancreatitis were not significantly related
Green's function for a Schroedinger operator and some related summation formulas
Summation formulas are obtained for products of associated Lagurre
polynomials by means of the Green's function K for the Hamiltonian H =
-{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of
a Mercer type theorem that arises in connection with integral equations. The
new approach introduced in this paper may be useful for the construction of
wider classes of generating function.Comment: 14 page
Rational Solutions of the Painleve' VI Equation
In this paper, we classify all values of the parameters , ,
and of the Painlev\'e VI equation such that there are
rational solutions. We give a formula for them up to the birational canonical
transformations and the symmetries of the Painlev\'e VI equation.Comment: 13 pages, 1 Postscript figure Typos fixe
Factorization and Lie point symmetries of general Lienard-type equation in the complex plane
We present a variational approach to a general Lienard-type equation in order
to linearize it and, as an example, the Van der Pol oscillator is discussed.
The new equation which is almost linear is factorized. The point symmetries of
the deformed equation are also discussed and the two-dimensional Lie algebraic
generators are obtained
On the Linearization of the First and Second Painleve' Equations
We found Fuchs--Garnier pairs in 3X3 matrices for the first and second
Painleve' equations which are linear in the spectral parameter. As an
application of our pairs for the second Painleve' equation we use the
generalized Laplace transform to derive an invertible integral transformation
relating two its Fuchs--Garnier pairs in 2X2 matrices with different
singularity structures, namely, the pair due to Jimbo and Miwa and the one
found by Harnad, Tracy, and Widom. Together with the certain other
transformations it allows us to relate all known 2X2 matrix Fuchs--Garnier
pairs for the second Painleve' equation with the original Garnier pair.Comment: 17 pages, 2 figure
Nonlocal aspects of -symmetries and ODEs reduction
A reduction method of ODEs not possessing Lie point symmetries makes use of
the so called -symmetries (C. Muriel and J. L. Romero, \emph{IMA J.
Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE
is used here to recover -symmetries of as
nonlocal symmetries. In this framework, by embedding into a
suitable system determined by the function ,
any -symmetry of can be recovered by a local symmetry of
. As a consequence, the reduction method of Muriel and
Romero follows from the standard method of reduction by differential invariants
applied to .Comment: 13 page
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