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Toeplitz matrices and convergence

100%
Mildenberger H.
Fundamenta Mathematicae
|
2000
|
tom 165
|
nr 2
175-189
EN
We investigate $||χ_\mathbb A,2||$, the minimum cardinality of a subset of $2^ω$ that cannot be made convergent by multiplication with a single matrix taken from $\mathbb A$, for different sets $\mathbb A$ of Toeplitz matrices, and show that for some sets $\mathbb A$ it coincides with the splitting number. We show that there is no Galois-Tukey connection from the chaos relation on the diagonal matrices to the chaos relation on the Toeplitz matrices with the identity on $2^ω$ as first component. With Suslin c.c.c. forcing we show that $||χ_\mathbb M,2||$ < $\gb ∙ \gs$ is consistent relative to ZFC.
2
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The splitting number can be smaller than the matrix chaos number

63%
Mildenberger H., Shelah S.
Fundamenta Mathematicae
|
2002
|
tom 171
|
nr 2
167-176
EN
Let χ be the minimum cardinality of a subset of $^ω 2$ that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that 𝔰 < χ is consistent. We thus answer a question by Vojtáš. We give two kinds of models for the strict inequality. The first is the combination of an ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of $MA_{<κ}$ for some δ ∈ [ℵ₁,κ). It was a conjecture of Blass that 𝔰 = ℵ₁ < χ = κ holds in such a model. For the analysis of the second model we again use the creature forcing from the first model.
3
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Proper translation

63%
Mildenberger H., Shelah S.
Fundamenta Mathematicae
|
2011
|
tom 215
|
nr 1
1-38
EN
We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that $2^{ω} = ℵ₂$ together with the weak diamond for covering by thin trees, the weak diamond for covering by meagre sets, the weak diamond for covering by null sets, and "all Aronszajn trees are special" is consistent relative to ZFC. We iterate alternately forcings specialising Aronszajn trees without adding reals (the NNR forcing from ["Proper and Improper Forcing", Ch. V]) and < ω₁-proper $^{ω} ω$-bounding forcings adding reals. We show that over a tower of elementary submodels there is a sort of a reduction ("proper translation") of our iteration to the countable support iteration of simpler iterands. If we use only Sacks iterands and NNR iterands, this allows us to guess the values of Borel functions into small trees and thus derive the above mentioned weak diamonds.
4
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On absolutely divergent series

51%
Fuchino S., Mildenberger H., Shelah S., Vojtáš P.
Fundamenta Mathematicae
|
1999
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tom 160
|
nr 3
255-268
EN
We show that in the $ℵ_2$-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš' result that under $cf(\gc) = \gp$ the two algebras are isomorphic [15].
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