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Search: MSC category 11A55 ( Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] )

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1. CJM 2008 (vol 60 pp. 975)

Boca, Florin P.
An AF Algebra Associated with the Farey Tessellation
We associate with the Farey tessellation of the upper half-plane an AF algebra $\AA$ encoding the ``cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of $\AA$. Using the path algebra model for AF algebras we construct, for each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in $\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.

Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20

2. CJM 2007 (vol 59 pp. 85)

Foster, J. H.; Serbinowska, Monika
On the Convergence of a Class of Nearly Alternating Series
Let $C$ be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If $\alpha$ is of the first type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $\sum c_k/k$ converges. An example of a quadratic irrational of the first type is $\sqrt{2}$, and an example of the second type is $\sqrt{3}$. The analysis of this problem relies heavily on the representation of $ \alpha$ as a simple continued fraction and on properties of the sequences of partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$ and double partial sums $T(n)=\sum_{k=1}^n S(k)$.

Keywords:Series, convergence, almost alternating, convex, continued fractions
Categories:40A05, 11A55, 11B83

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