On the convergence of second order spectra and multiplicity

Let A be a self-adjoint operator acting on a Hilbert space. The notion of second order spectrum of A relative to a given finite-dimensional subspace L has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. We establish in this paper a general framework allowing us to determine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of A. Our theoretical findings are supported by various numerical experiments on the computation of inclusions for eigenvalues of benchmark differential operators via finite element bases.Comment: 22 pages, 2 figures, 4 tables, research paper

Sharp eigenvalue enclosures for the perturbed angular Kerr-Newman Dirac operator

A certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients is examined. The strategy relies on computing the second order spectrum relative to subspaces of continuous piecewise linear functions. For smooth perturbations of the angular Kerr-Newman Dirac operator, explicit rates of convergence due to regularity of the eigenfunctions are established. Existing benchmarks are validated and sharpened by several orders of magnitude in the unperturbed setting.Comment: 27 pages, 2 figures, 5 tables. Some errors fixe

An estimate for the Morse index of a Stokes wave

Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two dimensional flow under gravity without surface tension. They can be described in terms of solutions of the Euler-Lagrange equation of a certain functional. This allows one to define the Morse index of a Stokes wave. It is well known that if the Morse indices of the elements of a set of non-singular Stokes waves are bounded, then none of them is close to a singular one. The paper presents a quantitative variant of this result.Comment: This version contains an additional reference and some minor change

On Approximation of the Eigenvalues of Perturbed Periodic Schrodinger Operators

This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so called quadratic projection method, in order to achieve convergence free from spectral pollution. We describe the theoretical foundations of the method in detail, and illustrate its effectiveness by several examples.Comment: 17 pages, 2 tables and 2 figure

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

Effect of Vitamin D Supplementation on Markers of Vascular Function: A Systematic Review and Individual Participant Meta- Analysis

Background-—Low 25-hydroxyvitamin D levels are associated with an increased risk of cardiovascular events, but the effect of vitamin D supplementation on markers of vascular function associated with major adverse cardiovascular events is unclear. Methods and Results-—We conducted a systematic review and individual participant meta-analysis to examine the effect of vitamin D supplementation on flow-mediated dilatation of the brachial artery, pulse wave velocity, augmentation index, central blood pressure, microvascular function, and reactive hyperemia index. MEDLINE, CINAHL, EMBASE, Cochrane Central Register of Controlled Trials, and http://www.ClinicalTrials.gov were searched until the end of 2016 without language restrictions. Placebo-controlled randomized trials of at least4 weeks duration were included. Individual participant data were sought from investigators on included trials. Trial-level metaanalysis was performed using random-effects models; individual participant meta-analyses used a 2-stage analytic strategy, examining effects in prespecified subgroups. 31trials (2751 participants) were included; 29 trials (2641participants) contributed data to trial-level meta-analysis, and24trials (2051 participants) contributed to individual-participant analyses. VitaminD3daily dose equivalents ranged from 900 to 5000 IU; duration was 4 weeks to12 months. Trial-level meta-analysis showed no significant effect of supplementation on macrovascularmeasures(flow-mediateddilatation,0.37%[95%confidenceinterval, 0.23to0.97]; carotid-femoralpulsewavevelocity, 0.00 m/s [95% confidence interval, 0.36 to 0.37]); similar results were obtained from individual participant data. Microvascular function showed a modest improvement in trial-level data only. No consistent benefit was observed in subgroup analyses or between different vitamin D analogues. Conclusions-—Vitamin D supplementation had no significant effect on most markers of vascular function in this analysis

Autonomic neuropathy predisposes to rosiglitazone-induced vascular leakage in insulin-treated patients with type 2 diabetes: a randomised, controlled trial on thiazolidinedione-induced vascular leakage

Contains fulltext : 88447.pdf (publisher's version ) (Closed access)AIMS/HYPOTHESIS: The mechanism of fluid-related complications caused by thiazolidinedione derivatives is unclear. One potential mechanism is thiazolidinedione-induced arterial vasodilatation, which results in vascular leakage and a fall in blood pressure, normally counterbalanced by sympathetic activation and subsequent renal fluid retention. We hypothesised that thiazolidinedione-induced vascular leakage will be particularly prominent in patients with autonomic neuropathy. METHODS: We conducted a randomised, double-blind, placebo-controlled, parallel study in 40 patients with type 2 diabetes on insulin treatment recruited from a university medical centre. The randomisation was performed by a central office using a randomisation schedule. Both treatment groups, placebo (n = 21) and rosiglitazone (n = 19), were stratified for sex and level of autonomic neuropathy as assessed by Ewing score (or=2.5). We investigated the effects of 16 weeks of treatment with rosiglitazone 4 mg twice daily on vascular leakage (transcapillary escape rate of albumin, TERalb), body weight, extracellular volume and plasma volume. RESULTS: Thirty-nine patients were included in the analysis. In patients with high Ewing scores (n = 16), rosiglitazone increased TERalb significantly (DeltaTERalb: rosiglitazone +2.43 +/- 0.45%/h, placebo -0.11 +/- 0.15%/h, p = 0.002), while rosiglitazone had no effect in the patients with low Ewing scores (n = 23). Rosiglitazone-induced increases in TERalb and Ewing score at baseline were correlated (r = 0.65, p = 0.02). There was no correlation between Ewing score and rosiglitazone-induced changes in fluid variables. One subject was withdrawn from the study because of atrial fibrillation. CONCLUSIONS/INTERPRETATION: Rosiglitazone may increase vascular leakage in insulin-treated patients with type 2 diabetes with autonomic neuropathy. Autonomic neuropathy did not exaggerate rosiglitazone-induced fluid retention. Therefore, autonomic neuropathy should be considered as a risk factor for thiazolidinedione-induced oedema, not for thiazolidinedione-induced fluid retention. TRIAL REGISTRATION: ClinicalTrials.gov NCT00422955. FUNDING: GlaxoSmithKline.1 september 201

Rosiglitazone and glimeperide: review of clinical results supporting a fixed dose combination

Type 2 diabetes has become a major burden to the health care systems worldwide. Among the drugs approved for this indication, glimepiride and rosiglitazone have gained substantial importance in routine use. While glimepiride stimulates β-cell secretion and leads to reduction of blood glucose values, rosiglitazone activates PPARγ and improves insulin resistance, at the vascular and metabolically active cells. Therefore, the combination of the two drugs may be an interesting approach to improve glycemic control and lower cardiovascular risk. A fixed combination of both drugs has been approved for clinical use in the US and EU. The combination of glimepiride and rosiglitazone is generally well tolerated and the use of a fixed combination may lead to improved adherence of the patients to their therapy. The purpose of this review is to evaluate the clinical data that have been published on this combination, appearing to represent a convenient way to obtain therapeutic targets in patients with type 2 diabetes mellitus