2,045 research outputs found
Continuity of the radius of convergence of differential equations on -adic analytic curves
This paper deals with connections on -adic analytic curves, in the sense
of Berkovich. The curves must be compact but the connections are allowed to
have a finite number of meromorphic singularities on them. For any choice of a
semistable formal model of the curve, we define an intrinsic notion of
normalized radius of convergence as a function on the curve, with values in
. For a sufficiently refined choice of the semistable model, we prove
continuity and logarithmic concavity of that function. We characterize
\emph{Robba connections}, that is connections whose sheaf of solutions is
constant on any open disk contained in the curve.Comment: 51 pages, 4 figure
Algebraic Solutions of the Lam\'e Equation, Revisited
A minor error in the necessary conditions for the algebraic form of the
Lam\'e equation to have a finite projective monodromy group, and hence for it
to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On
algebraic solutions of Lam\'e's differential equation", J. Differential
Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral
group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6
from an integer; this possibility was missed. The omission affects a recent
result on the monodromy of the Weierstrass form of the Lam\'e equation. [See R.
C. Churchill, "Two-generator subgroups of SL(2,C) and the hypergeometric,
Riemann, and Lam\'e equations", J. Symbolic Computation 28 (1999), 521-545.]
The Weierstrass form, which is a differential equation on an elliptic curve,
may have, after all, an octahedral projective monodromy group.Comment: 20 pages, elsart document class, no figure
-adic formulas and unit root -subcrystals of the hypergeometric system
We define the notion of {\it Dwork family of logarithmic -crystals}, a
typical example of which is the family of Gauss hypergeometricdifferential
systems, viewed as parametrized by their exponents of algebraic monodromy. The
-adic analytic dependence of the Frobenius operation upon those exponents,
is Dwork's "Boyarsky Principle". We discuss, in favorable cases, the -adic
analytic continuation of the unit root -subcrystal in the open tube of a
singularity, uniformly w.r.t. the exponents. We obtain a conceptual proof of
the Koblitz-Diamond formula -adically analog to Gauss' evaluation of
.Comment: 20 pages, Plain Te
On Dwork cohomology and algebraic D-modules
After works by Katz, Monsky, and Adolphson-Sperber, a comparison theorem
between relative de Rham cohomology and Dwork cohomology is established in a
paper by Dimca-Maaref-Sabbah-Saito in the framework of algebraic D-modules. We
propose here an alternative proof of this result. The use of Fourier transform
techniques makes our approach more functorial.Comment: latex, 8 page
Power laws statistics of cliff failures, scaling and percolation
The size of large cliff failures may be described in several ways, for
instance considering the horizontal eroded area at the cliff top and the
maximum local retreat of the coastline. Field studies suggest that, for large
failures, the frequencies of these two quantities decrease as power laws of the
respective magnitudes, defining two different decay exponents. Moreover, the
horizontal area increases as a power law of the maximum local retreat,
identifying a third exponent. Such observation suggests that the geometry of
cliff failures are statistically similar for different magnitudes. Power laws
are familiar in the physics of critical systems. The corresponding exponents
satisfy precise relations and are proven to be universal features, common to
very different systems. Following the approach typical of statistical physics,
we propose a "scaling hypothesis" resulting in a relation between the three
above exponents: there is a precise, mathematical relation between the
distributions of magnitudes of erosion events and their geometry. Beyond its
theoretical value, such relation could be useful for the validation of field
catalogs analysis. Pushing the statistical physics approach further, we develop
a numerical model of marine erosion that reproduces the observed failure
statistics. Despite the minimality of the model, the exponents resulting from
extensive numerical simulations fairly agree with those measured on the field.
These results suggest that the mathematical theory of percolation, which lies
behind our simple model, can possibly be used as a guide to decipher the
physics of rocky coast erosion and could provide precise predictions to the
statistics of cliff collapses.Comment: 20 pages, 13 figures, 1 table. To appear in Earth Surface Processes
and Lanforms (Rocky Coast special issue
Violation of the Einstein relation in Granular Fluids: the role of correlations
We study the linear response in different models of driven granular gases. In
some situations, even if the the velocity statistics can be strongly
non-Gaussian, we do not observe appreciable violations of the Einstein formula
for diffusion versus mobility. The situation changes when strong correlations
between velocities and density are present: in this case, although a form of
fluctuation-dissipation relation holds, the differential velocity response of a
particle and its velocity self-correlation are no more proportional. This
happens at high densities and strong inelasticities, but still in the
fluid-like (and ergodic) regime.Comment: 18 pages, 6 figures, submitted for publicatio
Influence of correlations on the velocity statistics of scalar granular gases
The free evolution of inelastic particles in one dimension is studied by
means of Molecular Dynamics (MD), of an inelastic pseudo-Maxwell model and of a
lattice model, with emphasis on the role of spatial correlations. We present an
exact solution of the 1d granular pseudo-Maxwell model for the scaling
distribution of velocities and discuss how this model fails to describe
correctly the homogeneous cooling stage of the 1d granular gas. Embedding the
pseudo-Maxwell gas on a lattice (hence allowing for the onset of spatial
correlations), we find a much better agreement with the MD simulations even in
the inhomogeneous regime. This is seen by comparing the velocity distributions,
the velocity profiles and the structure factors of the velocity field.Comment: Latex file: 6 pages, 5 figures (.eps). See also
http://axtnt3.phys.uniroma1.it/Maxwel
Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces
We consider a vector bundle with integrable connection (\cE,\na) on an
analytic domain U in the generic fiber \cX_{\eta} of a smooth formal p-adic
scheme \cX, in the sense of Berkovich. We define the \emph{diameter}
\delta_{\cX}(\xi,U) of U at \xi\in U, the \emph{radius} \rho_{\cX}(\xi) of the
point \xi\in\cX_{\eta}, the \emph{radius of convergence} of solutions of
(\cE,\na) at \xi, R(\xi) = R_{\cX}(\xi, U,(\cE, \na)). We discuss (semi-)
continuity of these functions with respect to the Berkovich topology. In
particular, under we prove under certain assumptions that \delta_{\cX}(\xi,U),
\rho_{\cX}(\xi) and R_{\xi}(U,\cE,\na) are upper semicontinuous functions of
\xi; for Laurent domains in the affine space, \delta_{\cX}(-,U) is continuous.
In the classical case of an affinoid domain U of the analytic affine line, R is
a continuous function.Comment: 19 pages. We have simplified and improved the expositio
- …