On the Schrodinger equation involving a critical Sobolev exponent and magnetic field

We consider the semilinear Schrodinger equation -Delta(A)u + V(x)u = Q(x)vertical bar u vertical bar(2* -2) u. Assuming that V changes sign, we establish the existence of a solution u not equal 0 in the Sobolev space H-A,V(1) + (R-N). The solution is obtained by a min-max type argument based on a topological linking. We also establish certain regularity properties of solutions for a rather general class of equations involving the operator -Delta(A).

Sharp Sobolev inequality involving a critical nonlinearity on a boundary

We consider the solvability of the Neumann problem for the equation -Delta u + lambda u = 0, partial derivative u/partial derivative v = Q(x)vertical bar u vertical bar(q-2)u on partial derivative Omega, where Q is a positive and continuous coefficient on partial derivative Omega, lambda is a parameter and q = 2(N - 1)/(N - 2) is a critical Sobolev exponent for the trace embedding of H-1(Omega) into L-q(partial derivative Omega). We investigate the joint effect of the mean curvature of partial derivative Omega and the shape of the graph of Q on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter lambda interferes with the spectrum of -Delta with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

Linking superiority bias in the interpersonal and intergroup domains

The author conducted 2 studies to explore the link between superiority bias in the interpersonal and intergroup domains. Australian university students evaluated the extent to which various personality traits were more or less applicable to themselves than to other Australian university students in general. They then evaluated the extent to which the same traits were more or less applicable to Australians than to people from other countries in general. As expected, the more participants evaluated themselves as superior to other university students, the more they evaluated Australians as a whole as superior to people from other countries. This link between interpersonal and intergroup superiority biases explained 22.1% of variance in Study 1 and 33.6% of variance in Study 2. The author interprets the results of the 2 studies as support for fundamental principles of social identity theory: (a) that self-concept consists of not only one's personal self but also the social groups to which one belongs and (b) that people are motivated to view both levels of self in a relatively positive fashion.

Various topological excitations in the SO(4) gauge field in higher dimensions

Following the original analysis Of Zhang and Hu for the 4-dimensional generalization of Quantum Hall effect, there has been much work from different viewpoints on the higher dimensional condensed matter systems. In this paper, we discuss three kinds of topological excitations in the SO(4) gauge field of condensed matter systems in 4-dimension-the instantons and anti-instantons, the 't Hooft-Polyakov monopoles, and the 2-membranes. Using the phi-mapping topological theory, it is revealed that there are 4-, 3-, and 2-dimensional topological currents inhering in the SO (4) gauge field, and the above three kinds of excitations can be directly and explicitly derived from these three kinds of currents, respectively. Moreover, it is shown that the topological charges of these excitations are characterized by the Hopf indices and Brouwer degrees of phi-mapping. (c) 2005 Elsevier Inc. All rights reserved.

Topological side-chain classification of beta-turns: Ideal motifs for peptidomimetic development

beta-turns are important topological motifs for biological recognition of proteins and peptides. Organic molecules that sample the side chain positions of beta-turns have shown broad binding capacity to multiple different receptors, for example benzodiazepines. beta-turns have traditionally been classified into various types based on the backbone dihedral angles (phi 2, psi 2, phi 3 and psi 3). Indeed, 57-68% of beta-turns are currently classified into 8 different backbone families (Type I, Type II, Type I', Type II', Type VIII, Type VIa1, Type VIa2 and Type VIb and Type IV which represents unclassified beta-turns). Although this classification of beta-turns has been useful, the resulting beta-turn types are not ideal for the design of beta-turn mimetics as they do not reflect topological features of the recognition elements, the side chains. To overcome this, we have extracted beta-turns from a data set of non-homologous and high-resolution protein crystal structures. The side chain positions, as defined by C-alpha-C-beta vectors, of these turns have been clustered using the kth nearest neighbor clustering and filtered nearest centroid sorting algorithms. Nine clusters were obtained that cluster 90% of the data, and the average intra-cluster RMSD of the four C-alpha-C-beta vectors is 0.36. The nine clusters therefore represent the topology of the side chain scaffold architecture of the vast majority of beta-turns. The mean structures of the nine clusters are useful for the development of beta-turn mimetics and as biological descriptors for focusing combinatorial chemistry towards biologically relevant topological space.

Cyclic MrIA: A stable and potent cyclic conotoxin with a novel topological fold that targets the norepinephrine transporter

Conotoxins, disulfide-rich peptides from the venom of cone snails, have created much excitement over recent years due to their potency and specificity for ion channels and their therapeutic potential. One recently identified conotoxin, MrIA, a 13-residue member of the chi-conotoxin family, inhibits the human norepinephrine transporter (NET) and has potential applications in the treatment of pain. In the current study, we show that the, beta-hairpin structure of native MrIA is retained in a synthetic cyclic version, as is biological activity at the NET. Furthermore, the cyclic version has increased resistance to trypsin digestion relative to the native peptide, an intriguing result because the cleavage site for the trypsin is not close to the cyclization site. The use of peptides as drugs is generally hampered by susceptibility to proteolysis, and so, the increase in enzymatic stability against trypsin observed in the current study may be useful in improving the therapeutic potential of MrIA. Furthermore, the structure reported here for cyclic MrIA represents a new topology among a growing number of circular disulfide-rich peptides.

Linking physiological processes with mangrove forest structure: phosphorus deficiency limits canopy development, hydraulic conductivity and photosynthetic carbon gain in dwarf Rhizophora mangle

Spatial gradients in mangrove tree height in barrier islands of Belize are associated with nutrient deficiency and sustained flooding in the absence of a salinity gradient. While nutrient deficiency is likely to affect many parameters, here we show that addition of phosphorus (P) to dwarf mangroves stimulated increases in diameters of xylem vessels, area of conductive xylem tissue and leaf area index (LAI) of the canopy. These changes in structure were consistent with related changes in function, as addition of P also increased hydraulic conductivity (K-s), stomatal conductance and photosynthetic assimilation rates to the same levels measured in taller trees fringing the seaward margin of the mangrove. Increased xylem vessel size and corresponding enhancements in stern hydraulic conductivity in P fertilized dwarf trees came at the cost of enhanced midday loss of hydraulic conductivity and was associated with decreased assimilation rates in the afternoon. Analysis of trait plasticity identifies hydraulic properties of trees as more plastic than those of leaf structural and physiological characteristics, implying that hydraulic properties are key in controlling growth in mangroves. Alleviation of P deficiency, which released trees from hydraulic limitations, reduced the structural and functional distinctions between dwarf and taller fringing tree forms of Rhizophora mangle.

Modelling the structure of latexin-carboxypeptidase A complex based on chemical cross-linking and molecular docking

We have determined the three-dimensional structure of the protein complex between latexin and carboxypeptidase A using a combination of chemical cross-linking, mass spectrometry and molecular docking. The locations of three intermolecular cross-links were identified using mass spectrometry and these constraints were used in combination with a speed-optimised docking algorithm allowing us to evaluate more than 3 x 10(11) possible conformations. While cross-links represent only limited structural constraints, the combination of only three experimental cross-links with very basic molecular docking was sufficient to determine the complex structure. The crystal structure of the complex between latexin and carboxypeptidase A4 determined recently allowed us to assess the success of this structure determination approach. Our structure was shown to be within 4 angstrom r.m.s. deviation of C alpha atoms of the crystal structure. The study demonstrates that cross-linking in combination with mass spectrometry can lead to efficient and accurate structural modelling of protein complexes.

Summarizing level-two topological relations in large spatial datasets

Summarizing topological relations is fundamental to many spatial applications including spatial query optimization. In this article, we present several novel techniques to effectively construct cell density based spatial histograms for range (window) summarizations restricted to the four most important level-two topological relations: contains, contained, overlap, and disjoint. We first present a novel framework to construct a multiscale Euler histogram in 2D space with the guarantee of the exact summarization results for aligned windows in constant time. To minimize the storage space in such a multiscale Euler histogram, an approximate algorithm with the approximate ratio 19/12 is presented, while the problem is shown NP-hard generally. To conform to a limited storage space where a multiscale histogram may be allowed to have only k Euler histograms, an effective algorithm is presented to construct multiscale histograms to achieve high accuracy in approximately summarizing aligned windows. Then, we present a new approximate algorithm to query an Euler histogram that cannot guarantee the exact answers; it runs in constant time. We also investigate the problem of nonaligned windows and the problem of effectively partitioning the data space to support nonaligned window queries. Finally, we extend our techniques to 3D space. Our extensive experiments against both synthetic and real world datasets demonstrate that the approximate multiscale histogram techniques may improve the accuracy of the existing techniques by several orders of magnitude while retaining the cost efficiency, and the exact multiscale histogram technique requires only a storage space linearly proportional to the number of cells for many popular real datasets.

Linking riparian vegetation spatial structure in Australian tropical savannas to ecosystem health indicators: semi-variogram analysis of high spatial resolution satellite imagery

Government agencies responsible for riparian environments are assessing the combined utility of field survey and remote sensing for mapping and monitoring indicators of riparian zone health. The objective of this work was to determine if the structural attributes of savanna riparian zones in northern Australia can be detected from commercially available remotely sensed image data. Two QuickBird images and coincident field data covering sections of the Daly River and the South Alligator River - Barramundie Creek in the Northern Territory were used. Semi-variograms were calculated to determine the characteristic spatial scales of riparian zone features, both vegetative and landform. Interpretation of semi-variograms showed that structural dimensions of riparian environments could be detected and estimated from the QuickBird image data. The results also show that selecting the correct spatial resolution and spectral bands is essential to maximize the accuracy of mapping spatial characteristics of savanna riparian features. The distribution of foliage projective cover of riparian vegetation affected spectral reflectance variations in individual spectral bands differently. Pan-sharpened image data enabled small-scale information extraction (< 6 m) on riparian zone structural parameters. The semi-variogram analysis results provide the basis for an inversion approach using high spatial resolution satellite image data to map indicators of savanna riparian zone health.