On Levi's Duality Between Permutations and Convergent Series

On Levfs Duality between Permutations and Convergent Series

This paper concerns a duality between conditionally convergent real series and permutations of their indices. It is widely known that if one is given a series E ^ which is conditionally convergent but not absolutely convergent, then there is a permutation n such that £«< # £#*(*)• Various authors have studied extensions of this result of Riemann (for example, Smith [8], Steinitz [9], Threlfall ...

Large Partial Random Permutations of Conditionally Convergent Series

This paper studies the behavior of sums of series under random permutations. The first result considers the case of uniformly distributed permutations on the set N of all integers, which requires finitely additive probability. The next result considers the limit of the first portion p, 0 ≤ p ≤ 1, of a uniform distribution on the first n integers, as n gets large. The focal point is when 0 < p <...

Duality between Time Series and Networks

Studying the interaction between a system's components and the temporal evolution of the system are two common ways to uncover and characterize its internal workings. Recently, several maps from a time series to a network have been proposed with the intent of using network metrics to characterize time series. Although these maps demonstrate that different time series result in networks with dis...

On Absolutely Convergent Dirichlet Series

satisfying zZ\ \^A < °°, if and only if f is) is bounded away from zero in the half-plane 0-3:0. This discovery amounts to a determination of the spectrum of the Banach-algebra element associated with fis), and it thus makes available for the theory of ordinary Dirichlet series the well-known theorem of Gelfand on analytic functions of Banach-algebra elements (see §24D of [7]) and its generaliz...

On Cantor's Series with Convergent