648 research outputs found
Post-critical set and non existence of preserved meromorphic two-forms
We present a family of birational transformations in depending on
two, or three, parameters which does not, generically, preserve meromorphic
two-forms. With the introduction of the orbit of the critical set (vanishing
condition of the Jacobian), also called ``post-critical set'', we get some new
structures, some "non-analytic" two-form which reduce to meromorphic two-forms
for particular subvarieties in the parameter space. On these subvarieties, the
iterates of the critical set have a polynomial growth in the \emph{degrees of
the parameters}, while one has an exponential growth out of these subspaces.
The analysis of our birational transformation in is first carried out
using Diller-Favre criterion in order to find the complexity reduction of the
mapping. The integrable cases are found. The identification between the
complexity growth and the topological entropy is, one more time, verified. We
perform plots of the post-critical set, as well as calculations of Lyapunov
exponents for many orbits, confirming that generically no meromorphic two-form
can be preserved for this mapping. These birational transformations in ,
which, generically, do not preserve any meromorphic two-form, are extremely
similar to other birational transformations we previously studied, which do
preserve meromorphic two-forms. We note that these two sets of birational
transformations exhibit totally similar results as far as topological
complexity is concerned, but drastically different results as far as a more
``probabilistic'' approach of dynamical systems is concerned (Lyapunov
exponents). With these examples we see that the existence of a preserved
meromorphic two-form explains most of the (numerical) discrepancy between the
topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure
Formalized proof, computation, and the construction problem in algebraic geometry
An informal discussion of how the construction problem in algebraic geometry
motivates the search for formal proof methods. Also includes a brief discussion
of my own progress up to now, which concerns the formalization of category
theory within a ZFC-like environment
The Stokes Phenomenon and Some Applications
Multisummation provides a transparent description of Stokes matrices which is
reviewed here together with some applications. Examples of moduli spaces for
Stokes matrices are computed and discussed. A moduli space for a third
Painlev\'e equation is made explicit. It is shown that the monodromy identity,
relating the topological monodromy and Stokes matrices, is useful for some
quantum differential equations and for confluent generalized hypergeometric
equations
SecDec: A general program for sector decomposition
We present a program for the numerical evaluation of multi-dimensional
polynomial parameter integrals. Singularities regulated by dimensional
regularisation are extracted using iterated sector decomposition. The program
evaluates the coefficients of a Laurent series in the regularisation parameter.
It can be applied to multi-loop integrals in Euclidean space as well as other
parametric integrals, e.g. phase space integrals.Comment: 42 pages, 12 figures. Replaced by published version. Cuba library
included in the progra
Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials
Differential systems with a Fuchsian linear part are studied in regions
including all the singularities in the complex plane of these equations. Such
systems are not necessarily analytically equivalent to their linear part (they
are not linearizable) and obstructions are found as a unique nonlinear
correction after which the system becomes formally linearizable.
More generally, normal forms are found.
The corrections and the normal forms are found constructively. Expansions in
multiple orthogonal polynomials and their generalization to matrix-valued
polynomials are instrumental to these constructions.Comment: 24 page
Invariants of pseudogroup actions: Homological methods and Finiteness theorem
We study the equivalence problem of submanifolds with respect to a transitive
pseudogroup action. The corresponding differential invariants are determined
via formal theory and lead to the notions of k-variants and k-covariants, even
in the case of non-integrable pseudogroup. Their calculation is based on the
cohomological machinery: We introduce a complex for covariants, define their
cohomology and prove the finiteness theorem. This implies the well-known
Lie-Tresse theorem about differential invariants. We also generalize this
theorem to the case of pseudogroup action on differential equations.Comment: v2: some remarks and references addee
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