Generalized and weighted Strichartz estimates

In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schr\"odinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension 2 and 3.Comment: Final version, to appear in the Communications on Pure and Applied Analysis. 33 pages. 2 more references adde

The Strauss conjecture on asymptotically flat space-times

By assuming a certain localized energy estimate, we prove the existence portion of the Strauss conjecture on asymptotically flat manifolds, possibly exterior to a compact domain, when the spatial dimension is 3 or 4. In particular, this result applies to the 3 and 4-dimensional Schwarzschild and Kerr (with small angular momentum) black hole backgrounds, long range asymptotically Euclidean spaces, and small time-dependent asymptotically flat perturbations of Minkowski space-time. We also permit lower order perturbations of the wave operator. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and the assumed localized energy estimate, which is used in the remaining compact set.Comment: Final version, to appear in SIAM Journal on Mathematical Analysis. 17 page

Self-similar Solutions of the Cubic Wave Equation

We prove that the focusing cubic wave equation in three spatial dimensions has a countable family of self-similar solutions which are smooth inside the past light cone of the singularity. These solutions are labeled by an integer index $n$ which counts the number of oscillations of the solution. The linearized operator around the $n$-th solution is shown to have $n+1$ negative eigenvalues (one of which corresponds to the gauge mode) which implies that all $n>0$ solutions are unstable. It is also shown that all $n>0$ solutions have a singularity outside the past light cone which casts doubt on whether these solutions may participate in the Cauchy evolution, even for non-generic initial data.Comment: 14 pages, 1 figur

Risk of cancer in patients with dermatomyositis or polymyositis, and follow-up implications: a Scottish population-based cohort study

We conducted a national, retrospective population-based cohort study of 705 patients hospitalized with a first diagnosis of dermatomyositis (DM) or polymyositis (PM) during 1982–1996 based on linkage of hospital discharge, cancer registration, and death records in Scotland. Risks of cancer were assessed by calculating standardized incidence ratios (SIR). A first malignancy was diagnosed concurrently or subsequently in 50 patients with DM (SIR 7.7, 95% CI 5.7–10.1), and 40 patients with PM (2.1, 1.5–2.9). Significantly elevated risks were observed for lung, cervix uteri, and ovarian cancer in patients with DM, and for Hodgkin’s disease in patients with PM. The excess risk of cancer was highest around the time of diagnosis, and for patients with DM remained high for at least 2 years. Risks were elevated for both sexes but only significantly so for females, and were highest in patients aged 45–74 years at the time of diagnosis for DM and 15–44 for PM. © 2001 Cancer Research Campaign http://www.bjcancer.co

Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcricital case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.Comment: 16 pages. Couple of typos corrected, to appear in Annales de l'Institut Henri Poincar

Matched sizes of activating and inhibitory receptor/ligand pairs are required for optimal signal integration by human Natural Killer cells

It has been suggested that receptor-ligand complexes segregate or co-localise within immune synapses according to their size, and this is important for receptor signaling. Here, we set out to test the importance of receptor-ligand complex dimensions for immune surveillance of target cells by human Natural Killer (NK) cells. NK cell activation is regulated by integrating signals from activating receptors, such as NKG2D, and inhibitory receptors, such as KIR2DL1. Elongating the NKG2D ligand MICA reduced its ability to trigger NK cell activation. Conversely, elongation of KIR2DL1 ligand HLA-C reduced its ability to inhibit NK cells. Whereas normal-sized HLA-C was most effective at inhibiting activation by normal-length MICA, only elongated HLA-C could inhibit activation by elongated MICA. Moreover, HLA-C and MICA that were matched in size co-localised, whereas HLA-C and MICA that were different in size were segregated. These results demonstrate that receptor-ligand dimensions are important in NK cell recognition, and suggest that optimal integration of activating and inhibitory receptor signals requires the receptor-ligand complexes to have similar dimensions

Should the poultry red mite Dermanyssus gallinae be of wider concern for veterinary and medical science?

The poultry red mite Dermanyssus gallinae is best known as a threat to the laying-hen industry; adversely affecting production and hen health and welfare throughout the globe, both directly and through its role as a disease vector. Nevertheless, D. gallinae is being increasingly implemented in dermatological complaints in non-avian hosts, suggesting that its significance may extend beyond poultry. The main objective of the current work was to review the potential of D. gallinae as a wider veterinary and medical threat. Results demonstrated that, as an avian mite, D. gallinae is unsurprisingly an occasional pest of pet birds. However, research also supports that these mites will feed from a range of other animals including: cats, dogs, rodents, rabbits, horses and man. We conclude that although reported cases of D. gallinae infesting mammals are relatively rare, when coupled with the reported genetic plasticity of this species and evidence of permanent infestations on non-avian hosts, potential for host-expansion may exist. The impact of, and mechanisms and risk factors for such expansion are discussed, and suggestions for further work made. Given the potential severity of any level of host-expansion in D. gallinae, we conclude that further research should be urgently conducted to confirm the full extent of the threat posed by D. gallinae to (non-avian) veterinary and medical sectors

Transmission-Blocking Vaccines: Focus on Anti-Vector Vaccines against Tick-Borne Diseases

Tick-borne diseases are a potential threat that account for significant morbidity and mortality in human population worldwide. Vaccines are not available to treat several of the tick-borne diseases. With the emergence and resurgence of several tick-borne diseases, emphasis on the development of transmission-blocking vaccines remains increasing. In this review, we provide a snap shot on some of the potential candidates for the development of anti-vector vaccines (a form of transmission-blocking vaccines) against wide range of hard and soft ticks that include Ixodes, Haemaphysalis, Dermacentor, Amblyomma, Rhipicephalus and Ornithodoros species