205,752 research outputs found
Response solutions for arbitrary quasi-periodic perturbations with Bryuno frequency vector
We study the problem of existence of response solutions for a real-analytic
one-dimensional system, consisting of a rotator subjected to a small
quasi-periodic forcing. We prove that at least one response solution always
exists, without any assumption on the forcing besides smallness and
analyticity. This strengthens the results available in the literature, where
generic non-degeneracy conditions are assumed. The proof is based on a
diagrammatic formalism and relies on renormalisation group techniques, which
exploit the formal analogy with problems of quantum field theory; a crucial
role is played by remarkable identities between classes of diagrams.Comment: 30 pages, 12 figure
Existence of periodic solutions for the periodically forced SIR model
We prove that the seasonally-forced SIR model with a T-periodic forcing has a
periodic solution with period T whenever the basic reproductive number R0>1.
The proof uses the Leray-Schauder degree theory. We also describe some
numerical results in which we compute the T-periodic solution, where in order
to obtain the T-periodic solution when the behavior of the system is
subharmonic or chaotic, we use a Galerkin scheme
Lattice initial segments of the hyperdegrees
We affirm a conjecture of Sacks [1972] by showing that every countable
distributive lattice is isomorphic to an initial segment of the hyperdegrees,
. In fact, we prove that every sublattice of any
hyperarithmetic lattice (and so, in particular, every countable locally finite
lattice) is isomorphic to an initial segment of . Corollaries
include the decidability of the two quantifier theory of
and the undecidability of its three quantifier theory. The key tool in the
proof is a new lattice representation theorem that provides a notion of forcing
for which we can prove a version of the fusion lemma in the hyperarithmetic
setting and so the preservation of . Somewhat surprisingly,
the set theoretic analog of this forcing does not preserve . On
the other hand, we construct countable lattices that are not isomorphic to an
initial segment of
Realizability algebras: a program to well order R
The theory of classical realizability is a framework in which we can develop
the proof-program correspondence. Using this framework, we show how to
transform into programs the proofs in classical analysis with dependent choice
and the existence of a well ordering of the real line. The principal tools are:
The notion of realizability algebra, which is a three-sorted variant of the
well known combinatory algebra of Curry. An adaptation of the method of forcing
used in set theory to prove consistency results. Here, it is used in another
way, to obtain programs associated with a well ordering of R and the existence
of a non trivial ultrafilter on N
Extracting Herbrand trees in classical realizability using forcing
International audienceKrivine presented in [Kri10] a methodology to combine Cohen's forcing with the theory of classical realizability and showed that the forcing condition can be seen as a reference that is not subject to backtracks. The underlying classical program transformation was then analyzed by Miquel [Miq11] in a fully typed setting in classical higher-order arithmetic (PAω⁺). As a case study of this methodology, we present a method to extract a Herbrand tree from a classical realizer of inconsistency, following the ideas underlying the compactness theorem and the proof of Herbrand's theorem. Unlike the traditional proof based on König's lemma (using a fixed enumeration of atomic formulas), our method is based on the introduction of a particular Cohen real. It is formalized as a proof in PAω⁺, making explicit the construction of generic sets in this framework in the particular case where the set of forcing conditions is arithmetical. We then analyze the algorithmic content of this proof
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