Some theorems on hopficity

Some Theorems on Coverings

Some Theorems on Continua

PROOF. Let {Gn} and {Hn} denote monotonie decreasing sequences of open sets which close down upon .Pi and F2 respectively. We may suppose that d and Hi are so chosen that G i # i = 0. Now, by Mullikin's theorem, there is a connected subset of C— C(Gw+27n) which has a limit point in Gn and a limit point in Hn. The closure of such a connected set is a subcontinuum of C which "extends" from Gn to ...

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...

Some Theorems on Incremental Compression

The ability to induce short descriptions of, i.e. compressing, a wide class of data is essential for any system exhibiting general intelligence. In all generality, it is proven that incremental compression – extracting features of data strings and continuing to compress the residual data variance – leads to a time complexity superior to universal search if the strings are incrementally compress...

Note on Some Gap Theorems