Subseries and category

Some Theorems on Subseries

1. Absolutely convergent series. A simple calculation reveals that the arithmetic mean value of all subsums (including the void sum) of a given finite sum sn=ui+U2 + • • • +un is equal to sn/2. In this section we shall show (see Theorem 1 below) that an integral mean value can be found, consistent with the preceding, for the sums of all infinite subseries of a given absolutely convergent series...

A Note on Subseries Convergence

Summing Curious, Slowly Convergent, Harmonic Subseries

The harmonic series diverges. But if we delete from it all terms whose denominators contain any string of digits such as “9”, “42”, or “314159”, then the sum of the remaining terms converges. These series converge far too slowly to compute their sums directly. We describe an algorithm to compute these and related sums to high precision. For example, the sum of the series whose denominators cont...

Subseries of Lecture Notes in Computer Science

This work proposes two versions of an Artificial Immune System (AIS) a relatively recent computational intelligence paradigm – for predicting protein functions described in the Gene Ontology (GO). The GO has functional classes (GO terms) specified in the form of a directed acyclic graph, which leads to a very challenging multi-label hierarchical classification problem where a protein can be ass...

On the category of geometric spaces and the category of (geometric) hypergroups

‎In this paper first we define the morphism between geometric spaces in two different types‎. ‎We construct two categories of $uu$ and $l$ from geometric spaces then investigate some properties of the two categories‎, ‎for instance $uu$ is topological‎. ‎The relation between hypergroups and geometric spaces is studied‎. ‎By constructing the category $qh$ of $H_{v}$-groups we answer the question...