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Henderson,M. V. and Reimer, C.W. (2003): Bibliography on the fine structure of diatom frustules (Bacillariophyceae). II. (and deletions, addenda and corrigenda for bibliography I.) - In: Witkowski, A. (ed.): Diatom Monographs. Vol. 3, ARG Gantner Verlag KG, Ruggell, 372 pp. Krammer, K. (2003): Cymbopleura, Delicata, Navicymbula, Gomphocymbellopsis, Afrocymbella. - In: Lange-Bertalot, H. (ed.): Diatoms of Europe. Vol. 4: Diatoms of the European inland waters and comparable habitats elsewhere. ARG Gantner Verlag KG, Ruggell, 530 pp. Lange-Bertalot, H., Cavacini, P., Tagliaventi, N. and Alfinito, S. (2003): Diatoms of Sardinia. Rare and 76 new species in rock pools and other ephemeral waters. - In: Lange-Bertalot, H. (ed.): Iconographia Diatomologica. Vol. 12: Annotated diatom micrographs. ARG Gantner Verlag KG, Ruggell, 438 pp. Metzeltin, D. and Lange-Bertalot,H. (2002): Diatoms from the “Island Continent” Madagascar. - In: Lange-Bertalot, H. (ed.): Iconographia Diatomologica. Vol. 11: Annotated diatom micrographs. ARG Gantner Verlag KG, Ruggell, 286 pp. Van DeVijver, B., Frenot, Y. and Beyens, L. (2002): Freshwater diatoms from Ile de la Possession (Crozet Archipelago, Subantarctica). - Bibliotheca Diatomologica. Band 46.Gebrüder Borntraeger Verlagsbuchhandlung, Berlin, Stuttgart, 412 pp.

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Abstract  

If a finite abelian group is factored into a direct product of its cyclic subsets, then at least one of the factors is periodic. This is a famous result of G. Hajós. We propose to replace the cyclicity of the factors by an abstract property that still guarantees that one of the factors is periodic. Then we present applications of this approach.

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The statement, that in a tiling by translates of ann-dimensional cube there are two cubes having common (n-1)-dimensional faces, is known as Keller's conjecture. We shall prove that there is a counterexample for this conjecture if and only if the following graphsΓ n has a 2n size clique. The 4n vertices ofΓ n aren-tuples of integers 0, 1, 2, and 3. A pair of thesen-tuples are adjacent if there is a position at which the difference of the corresponding components is 2 modulo 4 and if there is a further position at which the corresponding components are different. We will give the size of the maximal cliques ofΓ n forn≤5.

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Rédei's theorem asserts that if a finite abelian group is expressed as a direct product of subsets of prime cardinality, then at least one of the factors must be periodic. (A periodic subset is a direct product of some subset and a nontrivial subgroup.) A. D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality that is a power of a prime, then at least one of the factors is periodic. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subsets of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.

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Reynolds, C. (2006): Ecology of phytoplankton. Ecology, biodiversity and conservation. - Cambridge University Press, Cambridge, 535 pp.

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RUCK, E. C. and KOCIOLEK, J. P. (2004): Preliminary phylogeny of the family Surirellaceae (Bacillariophyta). - In: LANGE-BERTALOT, H. and KOCIOLEK, P. (eds): Bibliotheca Diatomologica, Band 50. J. Cramer in der Gebrüder Borntrager Verlagsbuchhandlung, Berlin, Stuttgart, 236 pp.;VANDEVIJVER, B., BEYENS, L. and LANGE-BERTALOT, H. (2004): The genus Stauroneis in the Arctic and (Sub-)Antarctic Regions. - In: LANGE-BERTALOT, H. and KOCIOLEK, P. (eds): Bibliotheca Diatomologica, Band 51. J. Cramer in der Gebrüder Borntrager Verlagsbuchhandlung, Berlin, Stuttgart, 317 pp.; WILLIAMS, D. M. and REID, G. (2006): Amphorotia nov. gen., a new genus in the family Eunotiaceae (Bacillariophyceae), based on Eunotia clevei Grunow in Cleve et Grunow. - In: WITKOWSKI, A. (ed.): Diatom Monographs, Vol. 6. A. R. G. Gantner Verlag, K. G. Ruggell, 153 pp.; WYNNE,M. J. (2005):Achecklist of benthic marine algae of the tropical and subtropical western Atlantic: second revision. - Nova Hedwigia, Beiheft 129. J. Cramer in der Gebr. Borntraeger Verlagsbuchhandlung, Berlin, Stuttgart, 152 pp.

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