Grassmannian flows and applications to nonlinear partial differential equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.Comment: 26 pages, 2 figure

CO I Barcoding Reveals New Clades and Radiation Patterns of Indo-Pacific Sponges of the Family Irciniidae (Demospongiae: Dictyoceratida)

DNA barcoding is a promising tool to facilitate a rapid and unambiguous identification of sponge species. Demosponges of the order Dictyoceratida are particularly challenging to identify, but are of ecological as well as biochemical importance.Here we apply DNA barcoding with the standard CO1-barcoding marker on selected Indo-Pacific specimens of two genera, Ircinia and Psammocinia of the family Irciniidae. We show that the CO1 marker identifies several species new to science, reveals separate radiation patterns of deep-sea Ircinia sponges and indicates dispersal patterns of Psammocinia species. However, some species cannot be unambiguously barcoded by solely this marker due to low evolutionary rates.We support previous suggestions for a combination of the standard CO1 fragment with an additional fragment for sponge DNA barcoding

Phylogenetically and spatially close marine sponges harbour divergent bacterial communities

Recent studies have unravelled the diversity of sponge-associated bacteria that may play essential roles in sponge health and metabolism. Nevertheless, our understanding of this microbiota remains limited to a few host species found in restricted geographical localities, and the extent to which the sponge host determines the composition of its own microbiome remains a matter of debate. We address bacterial abundance and diversity of two temperate marine sponges belonging to the Irciniidae family - Sarcotragus spinosulus and Ircinia variabilis – in the Northeast Atlantic. Epifluorescence microscopy revealed that S. spinosulus hosted significantly more prokaryotic cells than I. variabilis and that prokaryotic abundance in both species was about 4 orders of magnitude higher than in seawater. Polymerase chain reaction-denaturing gradient gel electrophoresis (PCR-DGGE) profiles of S. spinosulus and I. variabilis differed markedly from each other – with higher number of ribotypes observed in S. spinosulus – and from those of seawater. Four PCR-DGGE bands, two specific to S. spinosulus, one specific to I. variabilis, and one present in both sponge species, affiliated with an uncultured sponge-specific phylogenetic cluster in the order Acidimicrobiales (Actinobacteria). Two PCR-DGGE bands present exclusively in S. spinosulus fingerprints affiliated with one sponge-specific phylogenetic cluster in the phylum Chloroflexi and with sponge-derived sequences in the order Chromatiales (Gammaproteobacteria), respectively. One Alphaproteobacteria band specific to S. spinosulus was placed in an uncultured sponge-specific phylogenetic cluster with a close relationship to the genus Rhodovulum. Our results confirm the hypothesized host-specific composition of bacterial communities between phylogenetically and spatially close sponge species in the Irciniidae family, with S. spinosulus displaying higher bacterial community diversity and distinctiveness than I. variabilis. These findings suggest a pivotal host-driven effect on the shape of the marine sponge microbiome, bearing implications to our current understanding of the distribution of microbial genetic resources in the marine realm.This work was financed by the Portuguese Foundation for Science and Technology (FCT - http://www.fct.pt) through the research project PTDC/MAR/101431/2008. CCPH has a PhD fellowship granted by FCT (Grant No. SFRH/BD/60873/2009). JRX’s research is funded by a FCT postdoctoral fellowship (grant no. SFRH/BPD/62946/2009). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Global Diversity of Sponges (Porifera)

With the completion of a single unified classification, the Systema Porifera (SP) and subsequent development of an online species database, the World Porifera Database (WPD), we are now equipped to provide a first comprehensive picture of the global biodiversity of the Porifera. An introductory overview of the four classes of the Porifera is followed by a description of the structure of our main source of data for this paper, the WPD. From this we extracted numbers of all ‘known’ sponges to date: the number of valid Recent sponges is established at 8,553, with the vast majority, 83%, belonging to the class Demospongiae. We also mapped for the first time the species richness of a comprehensive set of marine ecoregions of the world, data also extracted from the WPD. Perhaps not surprisingly, these distributions appear to show a strong bias towards collection and taxonomy efforts. Only when species richness is accumulated into large marine realms does a pattern emerge that is also recognized in many other marine animal groups: high numbers in tropical regions, lesser numbers in the colder parts of the world oceans. Preliminary similarity analysis of a matrix of species and marine ecoregions extracted from the WPD failed to yield a consistent hierarchical pattern of ecoregions into marine provinces. Global sponge diversity information is mostly generated in regional projects and resources: results obtained demonstrate that regional approaches to analytical biogeography are at present more likely to achieve insights into the biogeographic history of sponges than a global perspective, which appears currently too ambitious. We also review information on invasive sponges that might well have some influence on distribution patterns of the future

Integration von Produkten halbzahliger Besselfunktionen mit Potenzen von X

Bei der Anwendung des Ritz'schen oder Galerkin'schen Verfahrens für dreidimensionale eliptische Aufgaben benutzt man als Koordinatenfunktionen häufig die Funktionen $\phi_{klm} = c_{klm} \frac{1}{\sqrt{r}}J_{k+\frac{1}{2}} (\gamma_{k, l}r) P^{m}_{k}(cos \vartheta) ^{cos m \varphi}_{sin m \varphi}$ für r $\le$ 1, oder $\phi_{klm} = c_{klm} \frac{1}{\sqrt{r}}J_{k+\frac{1}{2}} (\frac{\gamma_{k,l}}{r}) P^{m}_{k} (cos \vartheta) ^{cos m \varphi}_{sin m \varphi}$ für r $\geq$ 1, wobei $\gamma_{k,l}$ die 1-te Nullstelle der Bessel-Funktion J$_{k+\frac{1}{2}}$ ist. Wenn wir noch die vorgegebene Gleichung als A u = f schreiben, tauchen in den oben erwähnten Verfahren die Skalarprodukte (A $\phi_{klm}$, $\phi_{\alpha\beta\gamma}$) auf. Dazu müssen wir oft (wenn der Operator A bezüglich r separabel ist) die Integrale der Form (1.1) $\int x^{n} \sqrt{x} J_{m+\frac{1}{2}}(\gamma x)dx$ und $\int x^{k} J_{m+\frac{1}{2}} (\gamma_{1}x) J_{n+\frac{1}{2}} (\gamma_{2}x)dx$ ausrechnen. Es handelt sich dabei um die Integration schnell oszillierenden Funktionen (bei größerem $\gamma$), die numerisch nur schwer erfaßbar ist und dadurch auch sehr viel Rechenzeit kostet. Die Genauigkeit ist (bei Gauß'scher Quadratur) im Abschnitt 5. gezeigt. Andererseits sind gerade die halbzahligen Bessel-Funktionen geschlossen darstellbar, wie man an der Formel (2.1) sieht, und die Integrale (1.1) kann man analytisch ausrechnen. Die Methode ist in den Abschnitten 2. und 3. erklärt. Im Abschnitt 4. sind alle Formeln für (1.1) zusannnengestellt; das sind die Formeln für die unbestimmten Integrale (1.1). Es kommt jetzt auf die Integrationsgrenzen an. Für die von Null verschiedenen Grenzen können wir die Formeln aus 4. direkt benutzen; aber im Nullpunkt erscheinen in einzelnen Gliedern die Potenz- und Logarithmus-Singularitäten,obwohl das Integral als Summe von diesen Gliedern im Nullpunkt regulär ist. Die Grenzwerte der Integrale für x gegen Null werden im Abschnitt 5. berechnet. Die Formeln aus 4. und 5. haben z.B. die Anwendung des Galerkinschen Verfahrens auf die Diffussionsgleichung mit Driftterm ermöglicht, d.h. auf Gleichungen vom Typ $-\Delta u + \bigtriangledown u \bigtriangledown (-\frac{c(\vartheta, \varphi)}{r^{3}})$ = 0 im Äußeren einer Kugel