Solution to a problem of A. D. Sands

Abstract

Let G be a finite additive abelian group, and suppose that A and B are subsets of G. We say that G = A(&B if every element ge G can be uniquely written in the form g = a + b, where aeA, beB. The study of such decompositions (usually called factorizations in the literature) was initiated by G. Hajos [3] in connection with his solution to a problem of Minkowski in the geometry of numbers.

Probing the binding sites of exchanged chlorophyll a in LH2 by Raman and site‐selection fluorescence spectroscopies

“…Photosynthetic membranes were then isolated by rupturing the whole cells in a French press. Finally, the membranes were solubilised using LDAO and LH2 proteins puri¢ed as previously described [13,17].…”

“…The pigment exchange protocol has been extensively described elsewhere [13,17]. In short, all of the Bchla-B800 molecules were released from their binding sites by incubating a LH2 sample in bu¡er containing Triton BG-10 at a pH of 4.75 at 30³C for 1 h. B850 complexes lacking Bchla-B800 were then puri¢ed by ion exchange chromatography using phosphocellulose as the absorbent.…”

“…Recently, it has been shown that Bchla-B800 molecules within the LH2 protein from Rps. acidophila can be selectively released and replaced with native Bchla (esteri¢ed with phytol) with near 100% e¤ciency [13]. By using Raman spectroscopy in resonance with the Soret transitions of the Bchla molecules, it was shown that these molecules are correctly bound within the reconstituted complex [14].…”

“…Using these modi¢ed complexes, the B800CB850 energy transfer mechanism was investigated to evaluate the Fo « rster theory [18]. Although suggested by circular dichroism measurements [13], it has still not been formally proved that the reconstituted LH2s containing non-native (bacterio)chlorin molecules in the B800 sites are correctly incorporated into the protein. This is especially true for those pigments, such as Chla, that are not naturally found in purple bacteria.…”

ELEMENTARY p-GROUPS WITH THE RÉDEI PROPERTY

“…Fraser and Gordon [4] proved that if p is a prime and p ≥ 5, then an elementary p-group of order p (p+1) does not possess the Rédei property. Dinitz [2] improved upon this result showing that if p is a prime p ≥ 5, then an elementary p-group of order p 4 does not have the Rédei property.…”