Equal signs additive sequences in Banach spaces

A sequence (e_n) spanning a Banach space E is called ESA or equal signs additive if the norm of a linear combination of the e_i's does not change when adjacent coefficients of equal sign are combined. Call the sequence (e_n) regular if neither E nor its dual contain an isomorphic copy of c₀. It is shown that a regular ESA sequence is a boundedly complete and 1-shrinking basis for its span which is thus quasi-reflexive. It is further possible to replace a regular ESA norm by an equivalent ESA norm rendering E isometrically isomorphic to its second dual. A sequence (e_n) is called IS or invariant under spreading if the norm of a linear combination of the e_i's does not change when the mutual distances of the terms in the sequence (but not their relative positions) change. We give a simple construction of an unconditional norm for an IS sequence, hence, in particular, for an ESA sequence. Also, an inverse construction is obtained: We prove that each unconditional IS basis gives rise to an ESA basis by means of an inversion formula; to nonequivalent IS unconditional bases correspond nonequivalent ESA bases. It follows that nonisomorphic ESA bases are plentiful.     

Computing finite commutative semigroups

Precedence results are used to improve existing algorithms for the enumeration of finite commutative semigroups. As an application 11,545,843 distinct commutative semigroups of order 9 were found.     

Subdirect decompositions of finite commutative semigroups

The Ponizovskîi decomposition of a finite commutative semigroup yields all shortest elementary subdirect decompositions of that semigroup.     

Photolytic rearrangement of germanium azides. Evidence for transient germa-imines

Photolysis of germanium azides gives rise to transient germa-imines which have been quantitatively trapped by pinacol. Evidence for the formation of diphenylgermylene in the photolysis of hexaphenylcyclodigermazane is also reported.