Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space our results cover the full range of the exponent $s \in (0,1)$ of the fractional Laplacians. We answer in particular an open problem raised by Frank and Seiringer \cite{FS}.Comment: 42 page

Detection of Emission from the CN Radical in the Cloverleaf Quasar at z=2.56

We report the detection of CN(N=3-2) emission towards the Cloverleaf quasar (z=2.56) based on observations with the IRAM Plateau de Bure Interferometer. This is the first clear detection of emission from this radical at high redshift. CN emission is a tracer of dense molecular hydrogen gas (n(H2) > 10^4 cm^{-3}) within star-forming molecular clouds, in particular in regions where the clouds are affected by UV radiation. The HCN/CN intensity ratio can be used as a diagnostic for the relative importance of photodissociation regions (PDRs) in a source, and as a sensitive probe of optical depth, the radiation field, and photochemical processes. We derive a lensing-corrected CN(N=3-2) line luminosity of L'(CN(3-2) = (4.5 +/- 0.5) x 10^9 K km/s pc^2. The ratio between CN luminosity and far-infrared luminosity falls within the scatter of the same relationship found for low-z (ultra-) luminous infrared galaxies. Combining our new results with CO(J=3-2) and HCN(J=1-0) measurements from the literature and assuming thermal excitation for all transitions, we find a CO/CN luminosity ratio of 9.3 +/- 1.9 and a HCN/CN luminosity ratio of 0.95 +/- 0.15. However, we find that the CN(N=3-2) line is likely only subthermally excited, implying that those ratios may only provide upper limits for the intrinsic 1-0 line luminosity ratios. We conclude that, in combination with other molecular gas tracers like CO, HCN, and HCO+, CN is an important probe of the physical conditions and chemical composition of dense molecular environments at high redshift.Comment: 6 pages, 5 figures, 1 table, to appear in ApJ (accepted May 23, 2007

An Improved Upper Bound for the Ring Loading Problem

The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on $n$ nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let $L$ be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with $L^*$ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that $L \leq L^* + 1.5D$, where $D$ is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with $L = L^* + 1.01D$. Recently, Skutella [Sku16] improved these bounds by showing that $L \leq L^* + \frac{19}{14}D$, and there exists an instance with $L = L^* + 1.1D$. We contribute to this line of research by showing that $L \leq L^* + 1.3D$. We also take a first step towards lower and upper bounds for small instances

The BCS Functional for General Pair Interactions

The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.Comment: Revised Version. To appear in Commun. Math. Phys

Multifractal analysis of the metal-insulator transition in anisotropic systems

We study the Anderson model of localization with anisotropic hopping in three dimensions for weakly coupled chains and weakly coupled planes. The eigenstates of the Hamiltonian, as computed by Lanczos diagonalization for systems of sizes up to $48^3$, show multifractal behavior at the metal-insulator transition even for strong anisotropy. The critical disorder strength $W_c$ determined from the system size dependence of the singularity spectra is in a reasonable agreement with a recent study using transfer matrix methods. But the respective spectrum at $W_c$ deviates from the ``characteristic spectrum'' determined for the isotropic system. This indicates a quantitative difference of the multifractal properties of states of the anisotropic as compared to the isotropic system. Further, we calculate the Kubo conductivity for given anisotropies by exact diagonalization. Already for small system sizes of only $12^3$ sites we observe a rapidly decreasing conductivity in the directions with reduced hopping if the coupling becomes weaker.Comment: 25 RevTeX pages with 10 PS-figures include

Two-dimensional lattice-fluid model with water-like anomalies

We investigate a lattice-fluid model defined on a two-dimensional triangular lattice, with the aim of reproducing qualitatively some anomalous properties of water. Model molecules are of the "Mercedes Benz" type, i.e., they possess a D3 (equilateral triangle) symmetry, with three bonding arms. Bond formation depends both on orientation and local density. We work out phase diagrams, response functions, and stability limits for the liquid phase, making use of a generalized first order approximation on a triangle cluster, whose accuracy is verified, in some cases, by Monte Carlo simulations. The phase diagram displays one ordered (solid) phase which is less dense than the liquid one. At fixed pressure the liquid phase response functions show the typical anomalous behavior observed in liquid water, while, in the supercooled region, a reentrant spinodal is observed.Comment: 9 pages, 1 table, 7 figure

Energy-level statistics at the metal-insulator transition in anisotropic systems

We study the three-dimensional Anderson model of localization with anisotropic hopping, i.e. weakly coupled chains and weakly coupled planes. In our extensive numerical study we identify and characterize the metal-insulator transition using energy-level statistics. The values of the critical disorder $W_c$ are consistent with results of previous studies, including the transfer-matrix method and multifractal analysis of the wave functions. $W_c$ decreases from its isotropic value with a power law as a function of anisotropy. Using high accuracy data for large system sizes we estimate the critical exponent $\nu=1.45\pm0.2$. This is in agreement with its value in the isotropic case and in other models of the orthogonal universality class. The critical level statistics which is independent of the system size at the transition changes from its isotropic form towards the Poisson statistics with increasing anisotropy.Comment: 22 pages, including 8 figures, revtex few typos corrected, added journal referenc

The Chandra X-ray Survey of Planetary Nebulae (ChanPlaNS): Probing Binarity, Magnetic Fields, and Wind Collisions

We present an overview of the initial results from the Chandra Planetary Nebula Survey (ChanPlaNS), the first systematic (volume-limited) Chandra X-ray Observatory survey of planetary nebulae (PNe) in the solar neighborhood. The first phase of ChanPlaNS targeted 21 mostly high-excitation PNe within ~1.5 kpc of Earth, yielding 4 detections of diffuse X-ray emission and 9 detections of X-ray-luminous point sources at the central stars (CSPNe) of these objects. Combining these results with those obtained from Chandra archival data for all (14) other PNe within ~1.5 kpc that have been observed to date, we find an overall X-ray detection rate of ~70%. Roughly 50% of the PNe observed by Chandra harbor X-ray-luminous CSPNe, while soft, diffuse X-ray emission tracing shocks formed by energetic wind collisions is detected in ~30%; five objects display both diffuse and point-like emission components. The presence of X-ray sources appears correlated with PN density structure, in that molecule-poor, elliptical nebulae are more likely to display X-ray emission (either point-like or diffuse) than molecule-rich, bipolar or Ring-like nebulae. All but one of the X-ray point sources detected at CSPNe display X-ray spectra that are harder than expected from hot (~100 kK) central star photospheres, possibly indicating a high frequency of binary companions to CSPNe. Other potential explanations include self-shocking winds or PN mass fallback. Most PNe detected as diffuse X-ray sources are elliptical nebulae that display a nested shell/halo structure and bright ansae; the diffuse X-ray emission regions are confined within inner, sharp-rimmed shells. All sample PNe that display diffuse X-ray emission have inner shell dynamical ages <~5x10^3 yr, placing firm constraints on the timescale for strong shocks due to wind interactions in PNe.Comment: 41 pages, 6 figures; submitted to the Astronomical Journa

Prevalence of acute dizziness and vertigo in cortical stroke

BACKGROUND: In posterior circulation stroke, vertigo can be a presenting feature. However, whether isolated hemispheric strokes present with vertigo is less clear, despite a few single case-reports in the literature. Here we, a) explored the prevalence of vertigo/dizziness in acute stroke and, b) considered the cortical distribution of these lesions in relation to both the known vestibular cortex and evolution of these symptoms. METHODS: We conducted structured interviews in 173 consecutive unselected patients admitted to the hyperacute stroke unit at the University College London Hospitals. The interview was used to evaluate whether the patient was suffering from dizziness and/or vertigo before the onset of the stroke and at the time of the stroke (acute dizziness/vertigo), and the nature of these symptoms. RESULTS: 112 patients had subcortical lesions and 53 patients had cortical infarcts, of which 21 patients reported acute dizziness. Out of these 21, five patients reported rotational vertigo. 17 of the total 53 patients had lesions in known vestibular cortical areas distributed within the insular and parietal opercular cortices. CONCLUSIONS: The prevalence of vertigo in acute cortical strokes was 9%, with no single locus of lesion overlap. There is growing evidence supporting a lateralised vestibular cortex, with speculation that cortical strokes affecting the right hemisphere, are more likely to cause vestibular symptoms than left-hemispheric strokes. We observed a trend for this association, with the right hemisphere affected in four of five patients who reported spinning vertigo at the onset of the stroke