39,970 research outputs found
Invariant hypersurfaces for derivations in positive characteristic
Let be an integral -algebra of finite type over an algebraically
closed field of characteristic . Given a collection of
-derivations on , that we interpret as algebraic vector fields on
, we study the group spanned by the hypersurfaces of
invariant for modulo the rational first integrals of .
We prove that this group is always a finite -vector space, and we
give an estimate for its dimension. This is to be related to the results of
Jouanolou and others on the number of hypersurfaces invariant for a foliation
of codimension 1. As an application, given a -algebra between and
, we show that the kernel of the pull-back morphism is a finite -vector space. In particular, if is a
UFD, then the Picard group of is finite.Comment: 16 page
A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems
In this paper, we prove a Pontryagin Maximum Principle for constrained
optimal control problems in the Wasserstein space of probability measures. The
dynamics, is described by a transport equation with non-local velocities and is
subject to end-point and running state constraints. Building on our previous
work, we combine the classical method of needle-variations from geometric
control theory and the metric differential structure of the Wasserstein spaces
to obtain a maximum principle stated in the so-called Gamkrelidze form.Comment: 35 page
Cohomology of regular differential forms for affine curves
Let be a complex affine reduced curve, and denote by its first
truncated cohomology group, i.e. the quotient of all regular differential
1-forms by exact 1-forms. First we introduce a nonnegative invariant
that measures the complexity of the singularity of at the point
. Then, if denotes the first singular homology group of with
complex coefficients, we establish the following formula: Second we consider a family of curves given
by the fibres of a dominant morphism , where is an
irreducible complex affine surface. We analyze the behaviour of the function
. More precisely, we show that it is constant on a
Zariski open set, and that it is lower semi-continuous in general.Comment: 16 page
Discussion of "Second order topological sensitivity analysis" by J. Rocha de Faria et al
The article by J. Rocha de Faria et al. under discussion is concerned with
the evaluation of the perturbation undergone by the potential energy of a
domain (in a 2-D, scalar Laplace equation setting) when a disk
of small radius centered at a given location
\hat{\boldsymbol{x}\in\Omega is removed from , assuming either
Neumann or Dirichlet conditions on the boundary of the small `hole' thus
created. In each case, the potential energy of the
punctured domain \Omega_{\epsilon}=\Omega\setminus\B_{\epsilon} is expanded
about so that the first two terms of the perturbation are given.
The first (leading) term is the well-documented topological derivative of
. The article under discussion places, logically, its main focus on the
next term of the expansion. However, it contains incorrrect results, as shown
in this discussion. In what follows, equations referenced with Arabic numbers
refer to those of the article under discussion.Comment: International Journal of Solids and Structures (2007) to appea
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