1,706 research outputs found

    Template iterations with non-definable ccc forcing notions

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    We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if κ\kappa is a measurable cardinal and θ<κ<μ<λ\theta<\kappa<\mu<\lambda are uncountable regular cardinals, then there is a ccc poset forcing s=θ<b=μ<a=λ\mathfrak{s}=\theta<\mathfrak{b}=\mu<\mathfrak{a}=\lambda. Another application is to get models with large continuum where the groupwise-density number g\mathfrak{g} assumes an arbitrary regular value.Comment: To appear in the Annals of Pure and Applied Logic, 45 pages, 2 figure

    Restrictions and extensions of semibounded operators

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    We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure

    Middle divisors and σ\sigma-palindromic Dyck words

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    Given a real number λ>1\lambda > 1, we say that dnd|n is a λ\lambda-middle divisor of nn if nλ<dλn. \sqrt{\frac{n}{\lambda}} < d \leq \sqrt{\lambda n}. We will prove that there are integers having an arbitrarily large number of λ\lambda-middle divisors. Consider the word  ⁣n ⁣λ:=w1w2...wk{a,b}, \langle\! \langle n \rangle\! \rangle_{\lambda} := w_1 w_2 ... w_k \in \{a,b\}^{\ast}, given by wi:={aif uiDn\(λDn),bif ui(λDn)\Dn, w_i := \left\{ \begin{array}{c l} a & \textrm{if } u_i \in D_n \backslash \left(\lambda D_n\right), \\ b & \textrm{if } u_i \in \left(\lambda D_n\right)\backslash D_n, \end{array} \right. where DnD_n is the set of divisors of nn, λDn:={λd:dDn}\lambda D_n := \{\lambda d: \quad d \in D_n\} and u1,u2,...,uku_1, u_2, ..., u_k are the elements of the symmetric difference DnλDnD_n \triangle \lambda D_n written in increasing order. We will prove that the language Lλ:={ ⁣n ⁣λ:nZ1} \mathcal{L}_{\lambda} := \left\{\langle\! \langle n \rangle\! \rangle_{\lambda} : \quad n \in \mathbb{Z}_{\geq 1} \right\} contains Dyck words having an arbitrarily large number of centered tunnels. We will show a connection between both results

    Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

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    In this paper, we associate an invariant αx(L)\alpha_{x}(L) to an algebraic point xx on an algebraic variety XX with an ample line bundle LL. The invariant α\alpha measures how well xx can be approximated by rational points on XX, with respect to the height function associated to LL. We show that this invariant is closely related to the Seshadri constant ϵx(L)\epsilon_{x}(L) measuring local positivity of LL at xx, and in particular that Roth's theorem on P1\mathbf{P}^1 generalizes as an inequality between these two invariants valid for arbitrary projective varieties.Comment: 55 pages, published versio
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