A Centre-Stable Manifold for the Focussing Cubic NLS in R1+3R^{1+3}

Consider the focussing cubic nonlinear Schr\"odinger equation in $R^3$: $$i\psi_t+\Delta\psi = -|\psi|^2 \psi.$$ It admits special solutions of the form $e^{it\alpha}\phi$, where $\phi$ is a Schwartz function and a positive ($\phi>0$) solution of $$-\Delta \phi + \alpha\phi = \phi^3.$$ The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form $e^{i(v \cdot + \Gamma)} \phi(\cdot - y, \alpha)$. We prove that any solution starting sufficiently close to a standing wave in the $\Sigma = W^{1, 2}(R^3) \cap |x|^{-1}L^2(R^3)$ norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that \mc N is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones. The proof is based on the modulation method introduced by Soffer and Weinstein for the $L^2$-subcritical case and adapted by Schlag to the $L^2$-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in $R^3$ for the nonselfadjoint Schr\"odinger operator obtained by linearizing around a standing wave solution.Comment: 56 page

Primordialists and Constructionists: a typology of theories of religion

This article adopts categories from nationalism theory to classify theories of religion. Primordialist explanations are grounded in evolutionary psychology and emphasize the innate human demand for religion. Primordialists predict that religion does not decline in the modern era but will endure in perpetuity. Constructionist theories argue that religious demand is a human construct. Modernity initially energizes religion, but subsequently undermines it. Unpacking these ideal types is necessary in order to describe actual theorists of religion. Three distinctions within primordialism and constructionism are relevant. Namely those distinguishing: a) materialist from symbolist forms of constructionism; b) theories of origins from those pertaining to the reproduction of religion; and c) within reproduction, between theories of religious persistence and secularization. This typology helps to make sense of theories of religion by classifying them on the basis of their causal mechanisms, chronology and effects. In so doing, it opens up new sightlines for theory and research

Genome-Wide Association Study in BRCA1 Mutation Carriers Identifies Novel Loci Associated with Breast and Ovarian Cancer Risk

BRCA1-associated breast and ovarian cancer risks can be modified by common genetic variants. To identify further cancer risk-modifying loci, we performed a multi-stage GWAS of 11,705 BRCA1 carriers (of whom 5,920 were diagnosed with breast and 1,839 were diagnosed with ovarian cancer), with a further replication in an additional sample of 2,646 BRCA1 carriers. We identified a novel breast cancer risk modifier locus at 1q32 for BRCA1 carriers (rs2290854, P = 2.7×10-8, HR = 1.14, 95% CI: 1.09-1.20). In addition, we identified two novel ovarian cancer risk modifier loci: 17q21.31 (rs17631303, P = 1.4×10-8, HR = 1.27, 95% CI: 1.17-1.38) and 4q32.3 (rs4691139, P = 3.4×10-8, HR = 1.20, 95% CI: 1.17-1.38). The 4q32.3 locus was not associated with ovarian cancer risk in the general population or BRCA2 carriers, suggesting a BRCA1-specific associat

Galaxy bulges and their massive black holes: a review

With references to both key and oft-forgotten pioneering works, this article starts by presenting a review into how we came to believe in the existence of massive black holes at the centres of galaxies. It then presents the historical development of the near-linear (black hole)-(host spheroid) mass relation, before explaining why this has recently been dramatically revised. Past disagreement over the slope of the (black hole)-(velocity dispersion) relation is also explained, and the discovery of sub-structure within the (black hole)-(velocity dispersion) diagram is discussed. As the search for the fundamental connection between massive black holes and their host galaxies continues, the competing array of additional black hole mass scaling relations for samples of predominantly inactive galaxies are presented.Comment: Invited (15 Feb. 2014) review article (submitted 16 Nov. 2014). 590 references, 9 figures, 25 pages in emulateApJ format. To appear in "Galactic Bulges", E. Laurikainen, R.F. Peletier, and D.A. Gadotti (eds.), Springer Publishin

Habitat preferences of a corallivorous reef fish: predation risk versus food quality

Many animals preferentially select a habitat from a range of those potentially available. However, the consequences of these preferences for distribution and abundance, and the underlying basis of habitat preferences are often unknown. The present study, conducted at Great Keppel Island, Australia, examined how distribution and abundance of an obligate corallivorous filefish, Oxymonacanthus longirostris, relates to coral architecture and diversity. The main drivers of the distribution and abundance of O. longirostris among reefs were coral species richness and availability of branching coral. Feeding territories had a higher percentage of Acropora coral than surrounding habitat. In addition, feeding territories had a higher percentage of the structurally important branching coral, Acropora nobilis, and a primary prey species, Acropora millepora. A series of pair-wise choice experiments in which both structural complexity and coral tissue quality were independently manipulated showed that habitat choice was primarily based on structural complexity and shelter characteristics. In addition, the choice for the preferred coral (A. nobilis) was stronger in the presence of a piscivorous fish. These results indicate that species-diverse coral habitats, which provide sufficient structural complexity along with nutritionally important prey, are essential for population persistence of this small, corallivorous reef fish