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

Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients

This is the post-print version of the Article. The official publised version can be accessed from the links below. Copyright @ 2013 Springer BaselEmploying the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.This research was supported by the grant EP/H020497/1: "Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems" from the EPSRC, UK

A smooth introduction to the wavefront set

The wavefront set provides a precise description of the singularities of a distribution. Because of its ability to control the product of distributions, the wavefront set was a key element of recent progress in renormalized quantum field theory in curved spacetime, quantum gravity, the discussion of time machines or quantum energy inequalitites. However, the wavefront set is a somewhat subtle concept whose standard definition is not easy to grasp. This paper is a step by step introduction to the wavefront set, with examples and motivation. Many different definitions and new interpretations of the wavefront set are presented. Some of them involve a Radon transform.Comment: 29 pages, 7 figure

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

Scattering frequencies and Gervey 3 singularities

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46619/1/222_2005_Article_BF01389032.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, anti-MDA5, 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

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