494 research outputs found
New solutions of Heun general equation
We show that in four particular cases the derivative of the solution of Heun
general equation can be expressed in terms of a solution to another Heun
equation. Starting from this property, we use the Gauss hypergeometric
functions to construct series solutions to Heun equation for the mentioned
cases. Each of the hypergeometric functions involved has correct singular
behavior at only one of the singular points of the equation; the sum, however,
has correct behavior
A transient network of telechelic polymers and microspheres : structure and rheology
We study the structure and dynamics of a transient network composed of
droplets of microemulsion connected by telechelic polymers. The polymer induces
a bridging attraction between droplets without changing their shape. A
viscoelastic behaviour is induced in the initially liquid solution,
characterised in the linear regime by a stretched exponential stress
relaxation. We analyse this relaxation in the light of classical theories of
transient networks. The role of the elastic reorganisations in the deformed
network is emphasized. In the non linear regime, a fast relaxation dynamics is
followed by a second one having the same rate as in the linear regime. This
behaviour, under step strain experiments, should induce a non monotonic
behaviour in the elastic component of the stress under constant shear rate.
However, we obtain in this case a singularity in the flow curve very different
from the one observed in other systems, that we interpret in terms of fracture
behaviour.Comment: 9 pages, 4 figure
On two-dimensional Bessel functions
The general properties of two-dimensional generalized Bessel functions are
discussed. Various asymptotic approximations are derived and applied to analyze
the basic structure of the two-dimensional Bessel functions as well as their
nodal lines.Comment: 25 pages, 17 figure
Nuclear Effects on Heavy Boson Production at RHIC and LHC
We predict W and Z transverse momentum distributions from proton-proton and
nuclear collisions at RHIC and LHC. A resummation formalism with power
corrections to the renormalization group equations is used. The dependence of
the resummed QCD results on the non-perturbative input is very weak for the
systems considered. Shadowing effects are discussed and found to be unimportant
at RHIC, but important for LHC. We study the enhancement of power corrections
due to multiple scattering in nuclear collisions and numerically illustrate the
weak effects of the dependence on the nuclear mass.Comment: 21 pages, 11 figure
Projective dynamics and classical gravitation
Given a real vector space V of finite dimension, together with a particular
homogeneous field of bivectors that we call a "field of projective forces", we
define a law of dynamics such that the position of the particle is a "ray" i.e.
a half-line drawn from the origin of V. The impulsion is a bivector whose
support is a 2-plane containing the ray. Throwing the particle with a given
initial impulsion defines a projective trajectory. It is a curve in the space
of rays S(V), together with an impulsion attached to each ray. In the simplest
example where the force is identically zero, the curve is a straight line and
the impulsion a constant bivector. A striking feature of projective dynamics
appears: the trajectories are not parameterized.
Among the projective force fields corresponding to a central force, the one
defining the Kepler problem is simpler than those corresponding to other
homogeneities. Here the thrown ray describes a quadratic cone whose section by
a hyperplane corresponds to a Keplerian conic. An original point of view on the
hidden symmetries of the Kepler problem emerges, and clarifies some remarks due
to Halphen and Appell. We also get the unexpected conclusion that there exists
a notion of divergence-free field of projective forces if and only if dim V=4.
No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure
On Virtual Displacement and Virtual Work in Lagrangian Dynamics
The confusion and ambiguity encountered by students, in understanding virtual
displacement and virtual work, is discussed in this article. A definition of
virtual displacement is presented that allows one to express them explicitly
for holonomic (velocity independent), non-holonomic (velocity dependent),
scleronomous (time independent) and rheonomous (time dependent) constraints. It
is observed that for holonomic, scleronomous constraints, the virtual
displacements are the displacements allowed by the constraints. However, this
is not so for a general class of constraints. For simple physical systems, it
is shown that, the work done by the constraint forces on virtual displacements
is zero. This motivates Lagrange's extension of d'Alembert's principle to
system of particles in constrained motion. However a similar zero work
principle does not hold for the allowed displacements. It is also demonstrated
that d'Alembert's principle of zero virtual work is necessary for the
solvability of a constrained mechanical problem. We identify this special class
of constraints, physically realized and solvable, as {\it the ideal
constraints}. The concept of virtual displacement and the principle of zero
virtual work by constraint forces are central to both Lagrange's method of
undetermined multipliers, and Lagrange's equations in generalized coordinates.Comment: 12 pages, 10 figures. This article is based on an earlier article
physics/0410123. It includes new figures, equations and logical conten
Structure factor of polymers interacting via a short range repulsive potential: application to hairy wormlike micelles
We use the Random Phase Approximation (RPA) to compute the structure factor,
S(q), of a solution of chains interacting through a soft and short range
repulsive potential V. Above a threshold polymer concentration, whose magnitude
is essentially controlled by the range of the potential, S(q) exhibits a peak
whose position depends on the concentration. We take advantage of the close
analogy between polymers and wormlike micelles and apply our model, using a
Gaussian function for V, to quantitatively analyze experimental small angle
neutron scattering profiles of semi-dilute solutions of hairy wormlike
micelles. These samples, which consist in surfactant self-assembled flexible
cylinders decorated by amphiphilic copolymer, provide indeed an appropriate
experimental model system to study the structure of sterically interacting
polymer solutions
Unification, KK-thresholds and the top Yukawa coupling in F-theory GUTs
In a class of F-theory SU(5) GUTs the low energy chiral mass spectrum is
obtained from rank one fermion mass textures with a hierarchical structure
organised by U(1) symmetries embedded in the exceptional E_8 group. In these
theories chiral fields reside on matter `curves' and the tree level masses are
computed from integrals of overlapping wavefuctions of the particles at the
triple intersection points. This calculation requires knowledge of the exact
form of the wavefuctions. In this work we propose a way to obtain a reliable
estimate of the various quantities which determine the strength of the Yukawa
couplings. We use previous analysis of KK threshold effects to determine the
(ratios of) heavy mass scales of the theory which are involved in the
normalization of the wave functions. We consider similar effects from the
chiral spectrum of these models and discuss possible constraints on the
emerging matter content. In this approach, we find that the Yukawa couplings
can be determined solely from the U(1) charges of the states in the
`intersection' and the torsion which is a topological invariant quantity. We
apply the results to a viable SU(5) model with minimal spectrum which satisfies
all the constraints imposed by our analysis. We use renormalization group
analysis to estimate the top and bottom masses and find that they are in
agreement with the experimental values.Comment: 28 pages, 2 figure
Projective dynamics and first integrals
We present the theory of tensors with Young tableau symmetry as an efficient
computational tool in dealing with the polynomial first integrals of a natural
system in classical mechanics. We relate a special kind of such first
integrals, already studied by Lundmark, to Beltrami's theorem about
projectively flat Riemannian manifolds. We set the ground for a new and simple
theory of the integrable systems having only quadratic first integrals. This
theory begins with two centered quadrics related by central projection, each
quadric being a model of a space of constant curvature. Finally, we present an
extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure
Fuchs versus Painlev\'e
We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e
VI. We then show that the polynomiality of the expressions of the correlation
functions (and form factors) in terms of the complete elliptic integral of the
first and second kind,
and , is a straight consequence of the fact that the differential
operators corresponding to the entries of Toeplitz-like determinants, are
equivalent to the second order operator which has as solution (or,
for off-diagonal correlations to the direct sum of and ). We show
that this can be generalized, mutatis mutandis, to the anisotropic Ising model.
The singled-out second order linear differential operator being replaced
by an isomonodromic system of two third-order linear partial differential
operators associated with , the Jacobi's form of the complete elliptic
integral of the third kind (or equivalently two second order linear partial
differential operators associated with Appell functions, where one of these
operators can be seen as a deformation of ). We finally explore the
generalizations, to the anisotropic Ising models, of the links we made, in two
previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and
elliptic curves. In particular the elliptic representation of Painlev\'e VI has
to be generalized to an ``Appellian'' representation of Garnier systems.Comment: Dedicated to the : Special issue on Symmetries and Integrability of
Difference Equations, SIDE VII meeting held in Melbourne during July 200
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