6,939 research outputs found
Canonical decomposition of linear differential operators with selected differential Galois groups
We revisit an order-six linear differential operator having a solution which
is a diagonal of a rational function of three variables. Its exterior square
has a rational solution, indicating that it has a selected differential Galois
group, and is actually homomorphic to its adjoint. We obtain the two
corresponding intertwiners giving this homomorphism to the adjoint. We show
that these intertwiners are also homomorphic to their adjoint and have a simple
decomposition, already underlined in a previous paper, in terms of order-two
self-adjoint operators. From these results, we deduce a new form of
decomposition of operators for this selected order-six linear differential
operator in terms of three order-two self-adjoint operators. We then generalize
the previous decomposition to decompositions in terms of an arbitrary number of
self-adjoint operators of the same parity order. This yields an infinite family
of linear differential operators homomorphic to their adjoint, and, thus, with
a selected differential Galois group. We show that the equivalence of such
operators is compatible with these canonical decompositions. The rational
solutions of the symmetric, or exterior, squares of these selected operators
are, noticeably, seen to depend only on the rightmost self-adjoint operator in
the decomposition. These results, and tools, are applied on operators of large
orders. For instance, it is seen that a large set of (quite massive) operators,
associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained
recently by P. Lairez, correspond to a particular form of the decomposition
detailed in this paper.Comment: 40 page
Globally nilpotent differential operators and the square Ising model
We recall various multiple integrals related to the isotropic square Ising
model, and corresponding, respectively, to the n-particle contributions of the
magnetic susceptibility, to the (lattice) form factors, to the two-point
correlation functions and to their lambda-extensions. These integrals are
holonomic and even G-functions: they satisfy Fuchsian linear differential
equations with polynomial coefficients and have some arithmetic properties. We
recall the explicit forms, found in previous work, of these Fuchsian equations.
These differential operators are very selected Fuchsian linear differential
operators, and their remarkable properties have a deep geometrical origin: they
are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing
on the factorised parts of all these operators, we find out that the global
nilpotence of the factors corresponds to a set of selected structures of
algebraic geometry: elliptic curves, modular curves, and even a remarkable
weight-1 modular form emerging in the three-particle contribution
of the magnetic susceptibility of the square Ising model. In the case where we
do not have G-functions, but Hamburger functions (one irregular singularity at
0 or ) that correspond to the confluence of singularities in the
scaling limit, the p-curvature is also found to verify new structures
associated with simple deformations of the nilpotent property.Comment: 55 page
Lateral-directional control of the x-15 airplane
Lateral directional control and stability characteristics of X-15 aircraf
Renormalization, isogenies and rational symmetries of differential equations
We give an example of infinite order rational transformation that leaves a
linear differential equation covariant. This example can be seen as a
non-trivial but still simple illustration of an exact representation of the
renormalization group.Comment: 36 page
Solution of ordinary differential equations by means of Lie series
Solution of ordinary differential equations by Lie series - Laplace transformation, Weber parabolic-cylinder functions, Helmholtz equations, and applications in physic
Painleve versus Fuchs
The sigma form of the Painlev{\'e} VI equation contains four arbitrary
parameters and generically the solutions can be said to be genuinely
``nonlinear'' because they do not satisfy linear differential equations of
finite order. However, when there are certain restrictions on the four
parameters there exist one parameter families of solutions which do satisfy
(Fuchsian) differential equations of finite order. We here study this phenomena
of Fuchsian solutions to the Painlev{\'e} equation with a focus on the
particular PVI equation which is satisfied by the diagonal correlation function
C(N,N) of the Ising model. We obtain Fuchsian equations of order for
C(N,N) and show that the equation for C(N,N) is equivalent to the
symmetric power of the equation for the elliptic integral .
We show that these Fuchsian equations correspond to rational algebraic curves
with an additional Riccati structure and we show that the Malmquist Hamiltonian
variables are rational functions in complete elliptic integrals. Fuchsian
equations for off diagonal correlations are given which extend our
considerations to discrete generalizations of Painlev{\'e}.Comment: 18 pages, Dedicated to the centenary of the publication of the
Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de
Paris by Richard Fuchs in 190
The solution of the Eulerian gyroscope equations by means of Lie series making use of recurrence formulas
Euler gyroscope equations solved by means of Lie series making use of recurrence formula
Magnetic domain walls displacement : automotion vs. spin-transfer torque
The magnetization dynamics equation predicts that a domain wall that changes
structure should undergo a displacement by itself - automotion - due to the
relaxation of the linear momentum that is associated with the wall structure.
We experimentally demonstrate this effect in soft nanostrips,transforming under
spin transfer torque a metastable asymmetric transverse wall into a vortex
wall. Displacements more than three times as large as under spin transfer
torque only are measured for 1~ns pulses. The results are explained by
analytical and numerical micromagnetics. Their relevance to domain wall motion
under spin transfer torque is emphasized
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