Uniqueness of radial solutions for the fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian $(-\Delta)^s$ with $s \in (0,1)$ for any space dimensions $N \geq 1$. By extending a monotonicity formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear equation $(-\Delta)^s u+ Vu = 0$ in $\mathbb{R}^N$ has at most one radial and bounded solution vanishing at infinity, provided that the potential $V$ is a radial and non-decreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schr\"odinger operator $H=(-\Delta)^s + V$ are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space $\mathbb{R}^{N+1}_+$, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation $(-\Delta)^s Q + Q - |Q|^{\alpha} Q = 0$ in $\mathbb{R}^N$ for arbitrary space dimensions $N \geq 1$ and all admissible exponents $\alpha >0$. This generalizes the nondegeneracy and uniqueness result for dimension N=1 recently obtained by the first two authors in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.Comment: 38 pages; revised version; various typos corrected; proof of Lemma 8.1 corrected; discussion of case \kappa_* =1 in the proof of Theorem 2 corrected with new Lemma A.2; accepted for publication in Comm. Pure. Appl. Mat

Fast flowing populations are not well mixed

In evolutionary dynamics, well-mixed populations are almost always associated with all-to-all interactions; mathematical models are based on complete graphs. In most cases, these models do not predict fixation probabilities in groups of individuals mixed by flows. We propose an analytical description in the fast-flow limit. This approach is valid for processes with global and local selection, and accurately predicts the suppression of selection as competition becomes more local. It provides a modelling tool for biological or social systems with individuals in motion.Comment: 19 pages, 8 figure

Can arterial wave augmentation in young adults help account for variability of cardiovascular risk in different British ethnic groups?

Objective: Traditional cardiovascular risk factors do not fully account for ethnic differences in cardiovascular disease. We tested if arterial function indices, particularly augmentation index (AIx), and their determinants from childhood could underlie such ethnic variability among young British adults in the 'DASH' longitudinal study. Methods: Dash, at http://dash.sphsu.mrc.ac.uk/, includes representative samples of six main British ethnic groups. Pulse wave velocity (PWV) and AIx were recorded using the Arteriograph device at ages 21-23 years in a subsample (n = 666); psychosocial, anthropometric, and blood pressure (BP) measures were collected then and in two previous surveys at ages 11-13 years and 14-16 years. For n = 334, physical activity was measured over 5 days (ActivPal). Results: Unadjusted values and regression models for PWVs were similar or lower in ethnic minority than in White UK young adults, whereas AIx was higher - Caribbean (14.9, 95% confidence interval 12.3-17.0%), West African (15.3, 12.9-17.7%), Indian (15.1, 13.0-17.2%), and Pakistani/Bangladeshi (15.7, 13.7-17.7%), compared with White UK (11.9, 10.2-13.6%). In multivariate models, adjusted for sex, central SBP, height, and heart rate, Indian and Pakistani/Bangladeshi young adults had higher AIx (β = 3.35, 4.20, respectively, P < 0.01) than White UK with a similar trend for West Africans and Caribbeans but not statistically significant. Unlike PWV, physical activity, psychosocial or deprivation measures were not associated with AIx, with borderline associations from brachial BP but no other childhood variables. Conclusion: Early adult AIx, but not arterial stiffness, may be a useful tool for testing components of excess cardiovascular risk in some ethnic minority groups

Model for Glass Transition in a Binary fluid from a Mode Coupling approach

We consider the Mode Coupling Theory (MCT) of Glass transition for a Binary fluid. The Equations of Nonlinear Fluctuating Hydrodynamics are obtained with a proper choice of the slow variables corresponding to the conservation laws. The resulting model equations are solved in the long time limit to locate the dynamic transition. The transition point from our model is considerably higher than predicted in existing MCT models for binary systems. This is in agreement with what is seen in Computer Simulation of binary fluids. fluids.Comment: 9 Pages, 3 Figure

Polarizibilities as a test of localized approximations to the self-interaction correction

We present applications of the recently introduced ``Generalized SIC-Slater'' scheme which provides a simple Self-Interaction Correction approximation in the framework of the Optimized Effective Potential. We focus on the computation of static polarizabilities which are known to constitute stringent tests for Density Functional Theory. We apply the new method to model H chains, but also to more realistic systems such as C4 (organic) chains, and less symmetrical systems such as a Na5 (metallic) cluster. Comparison is made with other SIC schemes, especially with the standard SIC-Slater one.Comment: 17 pages, 4 figures, 49 reference

Global attractors for the coupled suspension bridge system with temperature

This paper deals with the longterm properties of the thermoelastic nonlinear string-beam system related to the well-known Lazer-McKenna suspension bridge model. In particular no mechanical dissipation occurs in the equations, since the loss of energy is entirely due to thermal effects. The existence of regular global attractors for the associated solution semigroup is proved for time-independent supplies and any axial load

Chemo- and Thermosensory Responsiveness of Grueneberg Ganglion Neurons Relies on Cyclic Guanosine Monophosphate Signaling Elements

Neurons of the Grueneberg ganglion (GG) in the anterior nasal region of mouse pups respond to cool temperatures and to a small set of odorants. While the thermosensory reactivity appears to be mediated by elements of a cyclic guanosine monophosphate (cGMP) cascade, the molecular mechanisms underlying the odor-induced responses are unclear. Since odor-responsive GG cells are endowed with elements of a cGMP pathway, specifically the transmembrane guanylyl cyclase subtype GC-G and the cyclic nucleotide-gated ion channel CNGA3, the possibility was explored whether these cGMP signaling elements may also be involved in chemosensory GG responses. Experiments with transgenic mice deficient for GC-G or CNGA3 revealed that GG responsiveness to given odorants was significantly diminished in these knockout animals. These findings suggest that a cGMP cascade may be important for both olfactory and thermosensory signaling in the GG. However, in contrast to the thermosensory reactivity, which did not decline over time, the chemosensory response underwent adaptation upon extended stimulation, suggesting that the two transduction processes only partially overlap. Copyright (C) 2011 S. Karger AG, Base

Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $\mu$ as the unique measure of maximal entropy. Then we show that the measure $\mu$ has a local product structure and that the currents $\mu^\pm$ have a laminar structure. This allows us to deduce information about periodic points and heteroclinic intersections. For example, we prove that the support of $\mu$ coincides with the closure of the set of saddle points. The methods used combine the pluripotential theory with the theory of non-uniformly hyperbolic dynamical systems

The dynamical evolution of molecular clouds near the Galactic Centre - II. Spatial structure and kinematics of simulated clouds

The evolution of molecular clouds in galactic centres is thought to differ from that in galactic discs due to a significant influence of the external gravitational potential. We present a set of numerical simulations of molecular clouds orbiting on the 100-pc stream of the Central Molecular Zone (the central $\sim500$ pc of the Galaxy) and characterise their morphological and kinematic evolution in response to the background potential and eccentric orbital motion. We find that the clouds are shaped by strong shear and torques, by tidal and geometric deformation, and by their passage through the orbital pericentre. Within our simulations, these mechanisms control cloud sizes, aspect ratios, position angles, filamentary structure, column densities, velocity dispersions, line-of-sight velocity gradients, spin angular momenta, and kinematic complexity. By comparing these predictions to observations of clouds on the Galactic Centre 'dust ridge', we find that our simulations naturally reproduce a broad range of key observed morphological and kinematic features, which can be explained in terms of well-understood physical mechanisms. We argue that the accretion of gas clouds onto the central regions of galaxies, where the rotation curve turns over and the tidal field is fully compressive, is accompanied by transformative dynamical changes to the clouds, leading to collapse and star formation. This can generate an evolutionary progression of cloud collapse with a common starting point, which either marks the time of accretion onto the tidally-compressive region or of the most recent pericentre passage. Together, these processes may naturally produce the synchronised starbursts observed in numerous (extra)galactic nuclei