Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,

This paper studies the transport of a mass $\mu$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range This paper studies the transport of a mass $\mu$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius, thus allowing for continuation of the solution past the blowup time. The monotone constraint on the data is consistent with the typical blowup profiles observed in recent numerical studies of these singularities. We prove monotonicity is preserved for all time, even after blowup, in contrast to the case $\alpha >2$ where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential ($\alpha=2-d$), under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. This enables us to prove preservation of monotonicity using the classical theory of conservation laws. In the case $2 -d < \alpha < 2$ and at the critical exponent $p$ we exhibit initial data in $L^p$ for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local ill-posedness of solutions at the critical exponent.Comment: 30 page

Staving off extinction – more than luck and fate

National Environmental Science Programme Threatened Species Recovery Hu

A doubling measure on R(d) can charge a rectifiable curve

"Vegeu el resum a l'inici del document del fitxer adjunt"

Lipschitz harmonic capacity and Bilipschitz images of cantor sets

For bilipschitz images of Cantor sets in Rd we estimate the Lipschitz harmonic capacity and show this capacity is invariant under bilipschitz homeomorphisms

Some variational problems from image processing

"Vegeu el resum a l'inici del document del fitxer adjunt"

Image Restoration Using One-Dimensional Sobolev Norm Profiles of Noise and Texture

This work is devoted to image restoration (denoising and deblurring) by variational models. As in our prior work [Inverse Probl. Imaging, 3 (2009), pp. 43-68], the image (f) over tilde to be restored is assumed to be the sum of a cartoon component u (a function of bounded variation) and a texture component v (an oscillatory function in a Sobolev space with negative degree of differentiability). In order to separate noise from texture in a blurred noisy textured image, we need to collect some information that helps distinguish noise, especially Gaussian noise, from texture. We know that homogeneous Sobolev spaces of negative differentiability help capture oscillations in images very well; however, these spaces do not directly provide clear distinction between texture and noise, which is also highly oscillatory, especially when the blurring effect is noticeable. Here, we propose a new method for distinguishing noise from texture by considering a family of Sobolev norms corresponding to noise and texture. It turns out that the two Sobolev norm profiles for texture and noise are different, and this enables us to better separate noise from texture during the deblurring process.open0

Detectors for the James Webb Space Telescope Near-Infrared Spectrograph I: Readout Mode, Noise Model, and Calibration Considerations

We describe how the James Webb Space Telescope (JWST) Near-Infrared Spectrograph's (NIRSpec's) detectors will be read out, and present a model of how noise scales with the number of multiple non-destructive reads sampling-up-the-ramp. We believe that this noise model, which is validated using real and simulated test data, is applicable to most astronomical near-infrared instruments. We describe some non-ideal behaviors that have been observed in engineering grade NIRSpec detectors, and demonstrate that they are unlikely to affect NIRSpec sensitivity, operations, or calibration. These include a HAWAII-2RG reset anomaly and random telegraph noise (RTN). Using real test data, we show that the reset anomaly is: (1) very nearly noiseless and (2) can be easily calibrated out. Likewise, we show that large-amplitude RTN affects only a small and fixed population of pixels. It can therefore be tracked using standard pixel operability maps.Comment: 55 pages, 10 figure

Single Doses up to 800 mg of E-52862 Do Not Prolong the QTc Interval--A Retrospective Validation by Pharmacokinetic-Pharmacodynamic Modelling of Electrocardiography Data Utilising the Effects of a Meal on QTc to Demonstrate ECG Assay Sensitivity.

BACKGROUND: E-52862 is a Sigma-1 receptor antagonist (S1RA) currently under investigation as a potential analgesic medicine. We successfully applied a concentration-effect model retrospectively to a four-way crossover Phase I single ascending dose study and utilized the QTc shortening effects of a meal to demonstrate assay sensitivity by establishing the time course effects from baseline in all four periods, independently from any potential drug effects. METHODS: Thirty two healthy male and female subjects were included in four treatment periods to receive single ascending doses of 500 mg, 600 mg or 800 mg of E-52862 or placebo. PK was linear over the dose range investigated and doses up to 600 mg were well tolerated. The baseline electrocardiography (ECG) measurements on Day-1 were time-matched with ECG and pharmacokinetic (PK) samples on Day 1 (dosing day). RESULTS: In this conventional mean change to time-matched placebo analysis, the largest time-matched difference to placebo QTcI was 1.44 ms (90% CI: -4.04, 6.93 ms) for 500 mg; -0.39 ms (90% CI: -3.91, 3.13 ms) for 600 mg and 1.32 ms (90% CI: -1.89, 4.53 ms) for 800 mg of E-52862, thereby showing the absence of any QTc prolonging effect at the doses tested. In addition concentration-effect models, one based on the placebo corrected change from baseline and one for the change of QTcI from average baseline with time as fixed effect were fitted to the data confirming the results of the time course analysis. CONCLUSION: The sensitivity of this study to detect small changes in the QTc interval was confirmed by demonstrating a shortening of QTcF of -8.1 (90% CI: -10.4, -5.9) one hour and -7.2 (90% CI: -9.4, -5.0) three hours after a standardised meal. TRIAL REGISTRATION: EU Clinical Trials Register EudraCT 2010 020343 13