26,073 research outputs found
Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations
There is a fascinating interplay and overlap between recursion theory and
descriptive set theory. A particularly beautiful source of such interaction has
been Martin's conjecture on Turing invariant functions. This longstanding open
problem in recursion theory has connected to many problems in descriptive set
theory, particularly in the theory of countable Borel equivalence relations.
In this paper, we shall give an overview of some work that has been done on
Martin's conjecture, and applications that it has had in descriptive set
theory. We will present a long unpublished result of Slaman and Steel that
arithmetic equivalence is a universal countable Borel equivalence relation.
This theorem has interesting corollaries for the theory of universal countable
Borel equivalence relations in general. We end with some open problems, and
directions for future research.Comment: Corrected typo
Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries
We state the intrinsic form of the Hamiltonian equations of first-order
Classical Field theories in three equivalent geometrical ways: using
multivector fields, jet fields and connections. Thus, these equations are given
in a form similar to that in which the Hamiltonian equations of mechanics are
usually given. Then, using multivector fields, we study several aspects of
these equations, such as the existence and non-uniqueness of solutions, and the
integrability problem. In particular, these problems are analyzed for the case
of Hamiltonian systems defined in a submanifold of the multimomentum bundle.
Furthermore, the existence of first integrals of these Hamiltonian equations is
considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl
general symmetries} of the system is discussed. Noether's theorem is also
stated in this context, both the ``classical'' version and its generalization
to include higher-order Cartan-Noether symmetries. Finally, the equivalence
between the Lagrangian and Hamiltonian formalisms is also discussed.Comment: Some minor mistakes are corrected. Bibliography is updated. To be
published in J. Phys. A: Mathematical and Genera
Quasi-analyticity in Carleman ultraholomorphic classes
We give a characterization for two different concepts of quasi-analyticity in
Carleman ultraholomorphic classes of functions of several variables in
polysectors. Also, working with strongly regular sequences, we establish
generalizations of Watson's Lemma under an additional condition related to the
growth index of the sequence.Comment: To appear in Ann. Inst. Fourier, Grenobl
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