A summary of Euler’s work on the pentagonal number theorem

In this article, we give a summary of Leonhard Euler’s work on the pentagonal number theorem. First we discuss related work of earlier authors and Euler himself. We then review Euler’s correspondence, papers and notebook entries about the pentagonal number theorem and its applications to divisor sums and integer partitions. In particular, we work out the details of an unpublished proof of the pentagonal number theorem from Euler’s notebooks. As we follow Euler’s discovery and proofs of the pentagonal number theorem, we pay attention to Euler’s ideas about when we can consider a mathematical statement to be true. Finally, we discuss related results in the theory of analytic functions.     

A duration-dependent negative potential in the cichlid electroretinogram.

A negative potential can be evoked in the local electroretinogram of the cichlid fish Cichlasoma octofasciatum by light stimuli of duration longer than 70 msec. This response superficially resembles the proximal negative response, but differs in some waveform components and dependence upon stimulus configuration and duration.     

ATP-dependent recognition of eukaryotic origins of DNA replication by a multiprotein complex.

A multiprotein complex that specifically recognizes cellular origins of DNA replication has been identified and purified from the yeast Saccharomyces cerevisiae. We observe a strong correlation between origin function and origin recognition by this activity. Interestingly, specific DNA binding by the origin recognition complex is dependent upon the addition of ATP. We propose that the origin recognition complex acts as the initiator protein for S. cerevisiae origins of DNA replication.     

Sea urchin elongation factor 1δ (EF1δ) and evidence for cell cycle-directed localization changes of a sub-fraction of the protein at M phase

Eukaryotic elongation factor 1 (eEF1) is a translational multimolecular complex reported in higher eukaryotes to be a target of CDK1/cyclin B, the universal regulator of M phase, but whose role in the cell cycle remains to be determined. A specific polyclonal antibody was produced and used to characterize the delta subunit of sea urchin elongation factor 1 (SgEF1) in early embryos, a powerful model for investigating cell cycle regulation. The SgEF1 protein was present in unfertilized eggs as two isoforms of 35 and 37 kDa, issued from two different mRNAs. The two canonical eEF1 partners, eEF1 and eEF1, were shown to co-immunoprecipitate with the SgEF1 isoforms. Both isoforms were associated in a macromolecular complex, which resolved upon gel filtration chromatography at a molecular weight > 400 kDa, suggesting association with other yet unidentified partners. After fertilization, the amount as well as the ratio of both SgEF1 isoforms remained constant during the first cell division as judged by Western blotting. Immunofluorescence analysis showed that a pool of the protein concentrated as a ring at the embryo nuclear location around the period of nuclear envelope breakdown and was visualized later as two large spheres around the mitotic spindle poles. Thus, the eEF1 protein shows cell cycle-specific localization changes in sea urchin embryos.Received 27 May 2003; received after revision 1 July 2003; accepted 4 July 2003     

Role for a Drosophila Myb-containing protein complex in site-specific DNA replication

There is considerable interest in the developmental, temporal and tissue-specific patterns of DNA replication in metazoans. Site-specific DNA replication at the chorion loci in Drosophila follicle cells leads to extensive gene amplification, and the organization of the cis-acting DNA elements that regulate this process may provide a model for how such regulation is achieved. Two elements important for amplification of the third chromosome chorion gene cluster, ACE3 and Ori-beta, are directly bound by Orc (origin recognition complex), and two-dimensional gel analysis has revealed that the primary origin used is Ori-beta (refs 7-9). Here we show that the Drosophila homologue of the Myb (Myeloblastosis) oncoprotein family is tightly associated with four additional proteins, and that the complex binds site-specifically to these regulatory DNA elements. Drosophila Myb is required in trans for gene amplification, showing that a Myb protein is directly involved in DNA replication. A Drosophila Myb binding site, as well as the binding site for another Myb complex member (p120), is necessary in cis for replication of reporter transgenes. Chromatin immunoprecipitation experiments localize both proteins to the chorion loci in vivo. These data provide evidence that specific protein complexes bound to replication enhancer elements work together with the general replication machinery for site-specific origin utilization during replication.     

The landing of the NEAR-Shoemaker spacecraft on asteroid 433 Eros

The NEAR-Shoemaker spacecraft was designed to provide a comprehensive characterization of the S-type asteroid 433 Eros (refs 1,2,3), an irregularly shaped body with approximate dimensions of 34 x 13 x 13 km. Following the completion of its year-long investigation, the mission was terminated with a controlled descent to its surface, in order to provide extremely high resolution images. Here we report the results of the descent on 12 February 2001, during which 70 images were obtained. The landing area is marked by a paucity of small craters and an abundance of 'ejecta blocks'. The properties and distribution of ejecta blocks are discussed in a companion paper. The last sequence of images reveals a transition from the blocky surface to a smooth area, which we interpret as a 'pond'. Properties of the 'ponds' are discussed in a second companion paper. The closest image, from an altitude of 129 m, shows the interior of a 100-m-diameter crater at 1-cm resolution.