A Framework for Generalising the Newton Method and Other Iterative Methods from Euclidean Space to Manifolds

The Newton iteration is a popular method for minimising a cost function on Euclidean space. Various generalisations to cost functions defined on manifolds appear in the literature. In each case, the convergence rate of the generalised Newton iteration needed establishing from first principles. The present paper presents a framework for generalising iterative methods from Euclidean space to manifolds that ensures local convergence rates are preserved. It applies to any (memoryless) iterative method computing a coordinate independent property of a function (such as a zero or a local minimum). All possible Newton methods on manifolds are believed to come under this framework. Changes of coordinates, and not any Riemannian structure, are shown to play a natural role in lifting the Newton method to a manifold. The framework also gives new insight into the design of Newton methods in general.Comment: 36 page

Classicalization of Gravitons and Goldstones

We establish a close parallel between classicalization of gravitons and derivatively-coupled Nambu-Goldstone-type scalars. We show, that black hole formation in high energy scattering process represents classicalization with the classicalization radius given by Schwarzschild radius of center of mass energy, and with the precursor of black hole entropy being given by number of soft quanta composing this classical configuration. Such an entropy-equivalent is defined for scalar classicalons also and is responsible for exponential suppression of their decay into small number of final particles. This parallel works in both ways. For optimists that are willing to hypothesize that gravity may indeed self-unitarize at high energies via black hole formation, it illustrates that the Goldstones may not be much different in this respect, and they classicalize essentially by similar dynamics as gravitons. In the other direction, it may serve as an useful de-mystifier of via-black-hole-unitarization process and of the role of entropy in it, as it illustrates, that much more prosaic scalar theories essentially do the same. Finally, it illustrates that in both cases classicalization is the defining property for unitarization, and that it sets-in before one can talk about accompanying properties, such as entropy and thermality of static classicalons (black holes). These properties are by-products of classicalization, and their equivalents can be defined for non-gravitational cases of classicalization.Comment: 23 page

Predator Dispersal Determines the Effect of Connectivity on Prey Diversity

Linking local communities to a metacommunity can positively affect diversity by enabling immigration of dispersal-limited species and maintenance of sink populations. However, connectivity can also negatively affect diversity by allowing the spread of strong competitors or predators. In a microcosm experiment with five ciliate species as prey and a copepod as an efficient generalist predator, we analysed the effect of connectivity on prey species richness in metacommunities that were either unconnected, connected for the prey, or connected for both prey and predator. Presence and absence of predator dispersal was cross-classified with low and high connectivity. The effect of connectivity on local and regional richness strongly depended on whether corridors were open for the predator. Local richness was initially positively affected by connectivity through rescue of species from stochastic extinctions. With predator dispersal, however, this positive effect soon turned negative as the predator spread over the metacommunity. Regional richness was unaffected by connectivity when local communities were connected only for the prey, while predator dispersal resulted in a pronounced decrease of regional richness. The level of connectivity influenced the speed of richness decline, with regional species extinctions being delayed for one week in weakly connected metacommunities. While connectivity enabled rescue of prey species from stochastic extinctions, deterministic extinctions due to predation were not overcome through reimmigration from predator-free refuges. Prey reimmigrating into these sink habitats appeared to be directly converted into increased predator abundance. Connectivity thus had a positive effect on the predator, even when the predator was not dispersing itself. Our study illustrates that dispersal of a species with strong negative effects on other community members shapes the dispersal-diversity relationship. When connections enable the spread of a generalist predator, positive effects of connectivity on prey species richness are outweighed by regional extinctions through predation

Observation of discrete time-crystalline order in a disordered dipolar many-body system

Understanding quantum dynamics away from equilibrium is an outstanding challenge in the modern physical sciences. It is well known that out-of-equilibrium systems can display a rich array of phenomena, ranging from self-organized synchronization to dynamical phase transitions. More recently, advances in the controlled manipulation of isolated many-body systems have enabled detailed studies of non-equilibrium phases in strongly interacting quantum matter. As a particularly striking example, the interplay of periodic driving, disorder, and strong interactions has recently been predicted to result in exotic "time-crystalline" phases, which spontaneously break the discrete time-translation symmetry of the underlying drive. Here, we report the experimental observation of such discrete time-crystalline order in a driven, disordered ensemble of $\sim 10^6$ dipolar spin impurities in diamond at room-temperature. We observe long-lived temporal correlations at integer multiples of the fundamental driving period, experimentally identify the phase boundary and find that the temporal order is protected by strong interactions; this order is remarkably stable against perturbations, even in the presence of slow thermalization. Our work opens the door to exploring dynamical phases of matter and controlling interacting, disordered many-body systems.Comment: 6 + 3 pages, 4 figure

Phylogenetic Patterns of Colonization and Extinction in Experimentally Assembled Plant Communities

Evolutionary history has provided insights into the assembly and functioning of plant communities, yet patterns of phylogenetic community structure have largely been based on non-dynamic observations of natural communities. We examined phylogenetic patterns of natural colonization, extinction and biomass production in experimentally assembled communities.We used plant community phylogenetic patterns two years after experimental diversity treatments (1, 2, 4, 8 or 32 species) were discontinued. We constructed a 5-gene molecular phylogeny and statistically compared relatedness of species that colonized or went extinct to remaining community members and patterns of aboveground productivity. Phylogenetic relatedness converged as species-poor plots were colonized and speciose plots experienced extinctions, but plots maintained more differences in composition than in phylogenetic diversity. Successful colonists tended to either be closely or distantly related to community residents. Extinctions did not exhibit any strong relatedness patterns. Finally, plots that increased in phylogenetic diversity also increased in community productivity, though this effect was inseparable from legume colonization, since these colonists tended to be phylogenetically distantly related.We found that successful non-legume colonists were typically found where close relatives already existed in the sown community; in contrast, successful legume colonists (on their own long branch in the phylogeny) resulted in plots that were colonized by distant relatives. While extinctions exhibited no pattern with respect to relatedness to sown plotmates, extinction plus colonization resulted in communities that converged to similar phylogenetic diversity values, while maintaining differences in species composition

From resource to document: Scaffolding content and organising student learning in teachers’ documentation work on the teaching of series

We examine teachers’ use of resources as they prepare to teach the topic of numerical series of real numbers in order to identify how their personal relationship with mathematical content—and its teaching—interacts with their use of a commonly used textbook. We describe this interplay between textbook and personal relationship, a term coined in the Anthropological Theory of the Didactic (ATD, Chevallard, 2003), in the terms of documentation work (resources, aims, rules of action, operational invariants), a key construct from the documentational approach (DA, Gueudet & Trouche, 2009). We do so in the case of five post-secondary teachers who use the same textbook as a main resource to teach the topic. Documentational analysis of interviews with the teachers led to the identification of their aims and rules of action (the what and how of their resource use as they organise their teaching of the topic) as well as the operational invariants (the why for this organisation of their teaching). We describe the teachers’ documentation work in two sets of aims/rules of action: scaffolding mathematical content (series as a stepping stone to learning about Taylor polynomials and Maclaurin series) and organising student learning about series through drill exercises, visualisation, examples, and applications. Our bridging (networking) of theoretical constructs originating in one theoretical framework (personal relationship, ATD) with the constructs of a different, yet compatible, framework (documentation work, DA) aims to enrich the latter (teachers’ documentation work) with the individual agency (teachers’ personal relationship with the topic) provided by the former

Internet-based medical education: a realist review of what works, for whom and in what circumstances

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Frequencies of BCR-ABL1 fusion transcripts among Sudanese chronic myeloid leukaemia patients

The incidence of one or other rearrangement in chronic myeloid leukemia (CML) patients varies in different reported series. In this study we report the frequencies of BCR-ABL1 fusion transcript variants studied in 43 CML patients from Sudan. The study includes 46 Sudanese patients, three of which negative for the BCR-ABL1 fusion transcript. More than half of 43 positive patients showed b2a2 fusion transcript (53.5%), while (41.9%) showed b3a2 transcript and the remaining (4.6%) coexpression of b3a2/ b2a2 and b3a2/b2a2/e19a2. We detected neither coexpression of p210/p190 nor e1a2 alone. Male patients showed a tendency to express b2a2, while female tende to express b3a2 (p = 0.017). Moreover, a single nucleotide polymorphism was detected in BCR exon 13 in one out of four patients and this patient showed only b2a2 expression. In conclusion, we observed a significant correlation between sex and type of BCR-ABL1 transcript, an observation that deserves further investigation

Complex Odor from Plants under Attack: Herbivore's Enemies React to the Whole, Not Its Parts

Background: Insect herbivory induces plant odors that attract herbivores ’ natural enemies. Assuming this attraction emerges from individual compounds, genetic control over odor emission of crops may provide a rationale for manipulating the distribution of predators used for pest control. However, studies on odor perception in vertebrates and invertebrates suggest that olfactory information processing of mixtures results in odor percepts that are a synthetic whole and not a set of components that could function as recognizable individual attractants. Here, we ask if predators respond to herbivoreinduced attractants in odor mixtures or to odor mixture as a whole. Methodology/Principal Findings: We studied a system consisting of Lima bean, the herbivorous mite Tetranychus urticae and the predatory mite Phytoseiulus persimilis. We found that four herbivore-induced bean volatiles are not attractive in pure form while a fifth, methyl salicylate (MeSA), is. Several reduced mixtures deficient in one component compared to the full spider-mite induced blend were not attractive despite the presence of MeSA indicating that the predators cannot detect this component in these odor mixtures. A mixture of all five HIPV is most attractive, when offered together with the noninduced odor of Lima bean. Odors that elicit no response in their pure form were essential components of the attractive mixture. Conclusions/Significance: We conclude that the predatory mites perceive odors as a synthetic whole and that th