Pseudodifferential multi-product representation of the solution operator of a parabolic equation

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on $\R^n$ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.Comment: Comm. Partial Differential Equations to appear (2009) 28 page

Semiclassical theory for many-body Fermionic systems

We present a treatment of many-body Fermionic systems that facilitates an expression of the well-known quantities in a series expansion of the Planck's constant. The ensuing semiclassical result contains to a leading order of the response function the classical time correlation function of the observable followed by the Weyl-Wigner series, on top of these terms are the periodic-orbit correction terms. The treatment given here starts from linear response assumption of the many-body theory and in its connection with semiclassical theory, it makes no assumption of the integrability of classical dynamics underlying the one-body quantal system. Applications of the framework are also discussed.Comment: 18 pages, Te

A semi-classical trace formula for Schrödinger operators

Let S ℏ =−ℏΔ+ V , with V smooth. If 0< E 2 <lim inf V(x) , the spectrum of S ℏ near E 2 consists (for ℏ small) of finitely-many eigenvalues, λ j (ℏ). We study the asymptotic distribution of these eigenvalues about E 2 as ℏ→0; we obtain semi-classical asymptotics for with , in terms of the periodic classical trajectories on the energy surface . This in turn gives Weyl-type estimates for the counting function . We make a detailed analysis of the case when the flow on B E is periodic.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46475/1/220_2005_Article_BF02099074.pd

Frequency, mutual exclusivity and clinical associations of myositis autoantibodies in a combined European cohort of idiopathic inflammatory myopathy patients

Objectives: To determine prevalence and co-existence of myositis specific autoantibodies (MSAs) and myositis associated autoantibodies (MAAs) and associated clinical characteristics in a large cohort of idiopathic inflammatory myopathy (IIM) patients. Methods: Adult patients with confirmed IIM recruited to the EuroMyositis registry (n = 1637) from four centres were investigated for the presence of MSAs/MAAs by radiolabelled-immunoprecipitation, with confirmation of anti-MDA5 and anti-NXP2 by ELISA. Clinical associations for each autoantibody were calculated for 1483 patients with a single or no known autoantibody by global linear regression modelling. Results: MSAs/MAAs were found in 61.5% of patients, with 84.7% of autoantibody positive patients having a sole specificity, and only three cases (0.2%) having more than one MSA. The most frequently detected autoantibody was anti-Jo-1 (18.7%), with a further 21 specificities each found in 0.2–7.9% of patients. Autoantibodies to Mi-2, SAE, TIF1, NXP2, MDA5, PMScl and the non-Jo-1 tRNA-synthetases were strongly associated (p < 0.001) with cutaneous involvement. Anti-TIF1 and anti-Mi-2 positive patients had an increased risk of malignancy (OR 4.67 and 2.50 respectively), and anti-SRP patients had a greater likelihood of cardiac involvement (OR 4.15). Interstitial lung disease was strongly associated with the anti-tRNA synthetases, antiMDA5, and anti-U1RNP/Sm. Overlap disease was strongly associated with anti-PMScl, anti-Ku, anti-U1RNP/Sm and anti-Ro60. Absence of MSA/MAA was negatively associated with extra-muscular manifestations. Conclusions: Myositis autoantibodies are present in the majority of patients with IIM and identify distinct clinical subsets. Furthermore, MSAs are nearly always mutually exclusive endorsing their credentials as valuable disease biomarkers

Introduction to the theory of linear partial differential equations

Introduction to the Theory of Linear Partial Differential Equation