### Canonical heights on the Jacobians of curves of genus 2 and the infinite descent

E. V. Flynn, N. P. Smart (1997)

Acta Arithmetica

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E. V. Flynn, N. P. Smart (1997)

Acta Arithmetica

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Joan Hart, Kenneth Kunen (1999)

Fundamenta Mathematicae

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We prove the following theorem: Given a⊆ω and $1\le \alpha <{\omega}_{1}^{CK}$, if for some $\eta <{\aleph}_{1}$ and all u ∈ WO of length η, a is ${\Sigma}_{\alpha}^{0}\left(u\right)$, then a is ${\Sigma}_{\alpha}^{0}$.We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: ${\Sigma}_{1}^{1}$-Turing-determinacy implies the existence of ${0}^{}$.

Ilijas Farah (1996)

Fundamenta Mathematicae

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We investigate some natural questions about the class of posets which can be embedded into ⟨ω,≤*⟩. Our main tool is a simple ccc forcing notion ${H}_{E}$ which generically embeds a given poset E into ⟨ω,≤*⟩ and does this in a “minimal” way (see Theorems 9.1, 10.1, 6.1 and 9.2).

Maciej Wojtkowski (1998)

Fundamenta Mathematicae

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We consider a class of Hamiltonian systems with linear potential, elastic constraints and arbitrary number of degrees of freedom. We establish sufficient conditions for complete hyperbolicity of the system.

Feliks Przytycki, Steffen Rohde (1998)

Fundamenta Mathematicae

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We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.

J.-L. Colliot-Thélène, A. N. Skorobogatov, Sir Peter Swinnerton-Dyer (1997)

Acta Arithmetica

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O. Alas, I. Protasov, M. Tkačenko, V. Tkachuk, R. Wilson, I. Yaschenko (1998)

Fundamenta Mathematicae

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We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.