6,666 research outputs found
Generic Automorphisms and Green Fields
We show that the generic automorphism is axiomatisable in the green field of
Poizat (once Morleyised) as well as in the bad fields which are obtained by
collapsing this green field to finite Morley rank. As a corollary, we obtain
"bad pseudofinite fields" in characteristic 0. In both cases, we give geometric
axioms. In fact, a general framework is presented allowing this kind of
axiomatisation. We deduce from various constructibility results for algebraic
varieties in characteristic 0 that the green and bad fields fall into this
framework. Finally, we give similar results for other theories obtained by
Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories
having the definable multiplicity property. We also close a gap in the
construction of the bad field, showing that the codes may be chosen to be
families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap
in the construction of the bad fiel
Infinity in string cosmology: A review through open problems
We review recent developments in the field of string cosmology with
particular emphasis on open problems having to do mainly with geometric
asymptotics and singularities. We discuss outstanding issues in a variety of
currently popular themes, such as tree-level string cosmology asymptotics,
higher-order string correction effects, M-theory cosmology, braneworlds, and
finally ambient cosmology.Comment: 37 pages, to appear in the IJMPD, v2: matches published versio
Linear bilevel problems: Genericity results and an efficient method for computing local minima
The paper is concerned with linear bilevel problems. These nonconvex problems are known to be NP-complete. So, no efficient method for solving the global bilevel problem can be expected. In this paper we give a genericity analysis of linear bilevel problems and present a new algorithm for computing efficiently local minimizers. The method is based on the given structural analysis and combines ideas of the Simplex method with projected gradient steps
Isospectral potentials and conformally equivalent isospectral metrics on spheres, balls and Lie groups
We construct pairs of conformally equivalent isospectral Riemannian metrics
and on spheres and balls for certain
dimensions , the smallest of which is , and on certain compact simple
Lie groups. In the case of Lie groups, the metric is left-invariant. In the
case of spheres and balls, the metric is not the standard metric but may be
chosen arbitrarily close to the standard one. For the same manifolds we
also show that the functions and are isospectral potentials
for the Schr\"odinger operator . To our knowledge, these
are the first examples of isospectral potentials and of isospectral conformally
equivalent metrics on simply connected closed manifolds.Comment: 34 pages, AMS-TeX; revised subsection 5.
Supercritical Nonlinear Schr\"odinger equations: Quasi-Periodic Solutions
We construct time quasi-periodic solutions to the energy supercritical
nonlinear Schr\"odinger equations on the torus in arbitrary dimensions. This
introduces a new approach, which could have general applicability.Comment: 62 pages; Duke Math. J. (to appear
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