Negative Größen bei Diophant?

In this paper we champion Diophantus of Alexandria and Isabella Basmakova against Norbert Schappacher. In two publications ([46] and [47]) he puts forward inter alia two propositions: Questioning Diophantus' originality he considers affirmatively the possibility, that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Basmakova's claim (in [5]), that Diophantus uses negative numbers, a "nonsense", reproaching her for her "thoughtlessness". First, we disprove Schappacher's Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus' keywords leipsis and mparxis. Next, we discuss Schappacher's epistemology of the history of mathematics and defend Basmakova's methods. Furthermore, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a - b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis (pleyra) of a square number (tetragonos), which is afterwards computed by the formula (a - b)^2 = a^2 + b^2 - 2ab. Finally, we report how the topic "Diophantus and the negative numbers" has been dealt with by translators and commentators from Maximus Planudes onwards.

The role of Ken&Barbie in JAK/STAT signalling during development of Drosophila melanogaster

Cell-cell interactions during embryonic development are crucial in the co-ordination of growth, differentiation and maintenance of many different cell types. To achieve this co-ordination each cell must properly translate signals received from neighbouring cells, into spatially and temporally appropriate developmental responses. A surprisingly limited number of signal pathways are responsible for the differentiation of enormous variety of cell types. As a result, pathways are frequently 'reused' during development. Thus, in mammals the JAK/STAT pathway is required during early embryogenesis, mammary gland formation, hematopoiesis and, finally, plays a pivotal role in immune response. In the canonical way, the JAK/STAT pathway is represented by a transmembrane receptor associated with a Janus kinase (JAK), which upon stimulation by an extra-cellular ligand, phosphorylates itself, the receptor and, finally, the signal transducer and activator of transcription (STAT) molecules. Phosphorylated STATs dimerise and translocate to the nucleus where they activate transcription of target genes. The JAK/STAT pathway has been conserved throughout evolution, and all known components are present in the genome of Drosophila melanogaster. Besides hematopoietic and immunity functions, the pathway is also required during development for processes including embryonic segmentation, tracheal morphogenesis, posterior spiracle formation etc. This study describes Drosophila Ken&Barbie (Ken) as a selective regulator of JAK/STAT signalling. ken mutations identified in a screen for modulators of an eye overgrowth phenotype, caused by over-expression of the pathway ligand unpaired, also interact genetically with the pathway receptor domeless (dome) and the transcription factor stat92E. Over-expression of Ken can phenocopy developmental defects known to be caused by the loss of JAK/STAT signalling. These genetic interactions suggest that Ken may function as a negative regulator of the pathway. Ken has C-terminal Zn-finger domain, presumably for DNA binding, and N-terminal BTB/POZ domain, often found in transcriptional repressors. Using EGFP-fused construct expressed in vivo revealed nuclear accumulation of Ken. Therefore, it is proposed that Ken may act as a suppresser of STAT92E target genes. An in vitro assay, termed SELEX, determined that Ken specifically binds to a DNA sequence, with the essential for DNA recognition core overlapping that of STAT92E. This interesting observation suggests that not all STAT92E sites may also allow Ken binding. Strikingly, when effects of ectopic Ken on the expression of putative JAK/STAT pathway target genes were examined, only a subset of the genes tested, namely vvl, trh and kni, were down-regulated by Ken, whereas some others, such as eve and fj, appeared to be unresponsive. Further analysis of vvl, one of the genes susceptible to ectopic Ken, was undertaken. In the developing hindgut, expression of vvl is JAK/STAT pathway dependent, but remains repressed in the posterior spiracles, despite the stimulation of STAT92E by Upd in their primordia. Importantly, ken is also expressed in the developing posterior spiracles. Strikingly, up-regulation of vvl is observed in these tissues in ken mutant embryos. These imply that while ectopic Ken is sufficient to repress the expression of vvl in the hindgut, endogenous Ken is also necessary to prevent its activation in the posterior spiracles. It is therefore conceivable that ectopic vvl expression in the posterior spiracles of the ken mutants may be the result of de-repression of endogenous STAT92E activity. Another consequence of these observations is a fine balance that must exist between STAT92E and Ken activities. Apparently, endogenous level of Ken is sufficient to repress vvl, but not other, as yet unidentified, JAK/STAT pathway targets, whose presumable activation by STAT92E is required for posterior spiracle development as the embryos mutant for dome, the receptor of the pathway, show severe spiracle defects. These defects are also observed in the embryos mis-expressing Ken. Though it is possible that the posterior spiracle phenotype caused by higher levels of Ken results from a JAK/STAT pathway independent activity, it seems to be more likely that Ken acts in a dosage dependent manner, and extra Ken is able to further antagonise JAK/STAT pathway target genes. While STAT92E binding sites required for target gene expression have been poorly characterised, the existence of genome data allows the prediction of candidate STAT92E sites present in target genes promoters to be attempted. When a 6kb region containing the putative regulatory domains flanking the vvl locus are examined, only a single potential STAT92E binding site located 825bp upstream of the translational start can be detected. Strikingly, this site also includes a perfect Ken binding sequence. Such an in silico observation, though consistent with both Ken DNA binding assay in vitro and regulation of STAT92E target genes in vivo, however, requires further analysis. The JAK/STAT pathway is implicated in a variety of processes during embryonic and larval development as well as in imago. In each case, stimulation of the same transcription factor results in different developmental outcomes. While many potential mechanisms have been proposed and demonstrated to explain such pleiotropy, the present study indicates that Ken may represent another mechanism, with which signal transduction pathways are controlled. Ken selectively down-regulates a subset of potential target genes and so modifies the transcriptional profile generated by activated STAT92E - a mechanism, which may be partially responsible for differences in the morphogenetic processes elicited by JAK/STAT signalling during development.

On Oseen Resolvent Estimates: A Negative Result

We consider the resolvent problem for the scalar Oseen equation in the whole space R^3. We show that for small values of the resolvent parameter it is impossible to obtain an L^2-estimate analogous to the one which is valid for the Stokes resolvent, even if the resolvent parameter has positive real part.

Unveiling the components of the signaling pathway between endosomes and the phagocytic uptake machinery in Dictyostelium discoideum

The soil amoebae Dictyostelium discoideum take up particles from their environment in order to obtain nutrition. The particle transits through the cell within a phagosome that fuses with organelles of different molecular compositions, undergoing a gradual degradation by different sets of hydrolytic enzymes. Griffiths’ concept of “phagosome individuality” predicts signaling from phagosomes into the cytoplasm, which might regulate many aspects of cell physiology. The finding that Dictyostelium cells depleted of the lysozyme AlyA or over-expressing the esterase Gp70 exhibit increased uptake of food particles, led to the postulation of a signaling cascade between endocytic compartments and the cytoskeletal uptake machinery at the plasma membrane. Assuming that Gp70 acts downstream of AlyA, gene-expression profiling of both mutants revealed different and overlapping sets of misregulated genes that might participate in this signaling cascade. Based on these results, we analyzed the effects of the artificial misregulation of six candidate genes by over-expression or negative genetic interference, in order to reconstruct at least part of the signaling pathway. SSB420 and SSL793 were chosen as candidates for the first signaling step, as they were up-regulated in AlyA-null cells and remained unaltered in the Gp70 over-expressing cells. The over-expression of SSB420 enhanced phagocytosis and raised the expression levels of Gp70, supporting its involvement in the signaling pathway between AlyA and Gp70 as a positive regulator of phagocytosis. However, this was not the case of cells over-expressing SSL793, as this mutation had no effects on phagocytosis. For the signaling downstream of Gp70, we studied four commonly misregulated genes in AlyA-depleted and Gp70 over-expressing cells. The expression levels of SLB350, SSB389 and TipD were lower in both mutants and therefore these were assumed as possible candidates for the negative regulation of phagocytosis. Cells depleted of SLB350 exhibited an increased phagocytic activity and no effect on Gp70 expression, proving its participation in the signaling pathway downstream of Gp70. Unlike SLB350, the disruption of the genes coding for SSB389 and TipD had no effects on particle uptake, excluding them from the pathway. The fourth candidate was Yipf1, the only gene that was commonly up-regulated in both mutants. Yet, the artificial over-expression of this protein had no effects on phagocytosis, so this candidate is also not included in the signaling pathway. Furthermore, localizing the products of the candidate genes within the cell helped unveiling several cellular organelles that receive signals from the phagosome and transduce them towards the uptake machinery.

Time Discretization of the SST-generalized Navier-Stokes Equations: Positive and Negative Results

In the theory of the Navier-Stokes equations, the proofs of some basic known results, like for example the uniqueness of solutions to the stationary Navier-Stokes equations under smallness assumptions on the data or the stability of certain time discretization schemes, actually only use a small range of properties and are therefore valid in a more general context. This observation leads us to introduce the concept of SST spaces, a generalization of the functional setting for the Navier-Stokes equations. It allows us to prove (by means of counterexamples) that several uniqueness and stability conjectures that are still open in the case of the Navier-Stokes equations have a negative answer in the larger class of SST spaces, thereby showing that proof strategies used for a number of classical results are not sufficient to affirmatively answer these open questions. More precisely, in the larger class of SST spaces, non-uniqueness phenomena can be observed for the implicit Euler scheme, for two nonlinear versions of the Crank-Nicolson scheme, for the fractional step theta scheme, and for the SST-generalized stationary Navier-Stokes equations. As far as stability is concerned, a linear version of the Euler scheme, a nonlinear version of the Crank-Nicolson scheme, and the fractional step theta scheme turn out to be non-stable in the class of SST spaces. The positive results established in this thesis include the generalization of classical uniqueness and stability results to SST spaces, the uniqueness of solutions (under smallness assumptions) to two nonlinear versions of the Euler scheme, two nonlinear versions of the Crank-Nicolson scheme, and the fractional step theta scheme for general SST spaces, the second order convergence of a version of the Crank-Nicolson scheme, and a new proof of the first order convergence of the implicit Euler scheme for the Navier-Stokes equations. For each convergence result, we provide conditions on the data that guarantee the existence of nonstationary solutions satisfying the regularity assumptions needed for the corresponding convergence theorem. In the case of the Crank-Nicolson scheme, this involves a compatibility condition at the corner of the space-time cylinder, which can be satisfied via a suitable prescription of the initial acceleration.