On a conjecture of Bennewitz, and the behaviour of the Titchmarsh-Weyl matrix near a pole

For any real limit-$n$ $2n$th-order selfadjoint linear differential expression on $[0,\infty)$, Titchmarsh- Weyl matrices $M(\lambda)$ can be defined. Two matrices of particu lar interest are the matrices $M_D(\lambda)$ and $M_N(\lambda)$ assoc iated respectively with Dirichlet and Neumann boundary conditions at $x=0$. These satisfy $M_D(\lambda) = -M_{N}(\lambda)^{-1}$. It is known that when these matrices have poles (which can only lie on the real axis) the existence of valid HELP inequalities depends on their behaviour in the neighbourhood of these poles. We prove a conjecture of Bennewitz and use it, together with a new algorithm for computing the Laurent expansion of a Titchmarsh-Weyl matrix in the neighbourhood of a pole, to investigate the existence of HELP inequalities for a number of differential equations which have so far proved awkward to analys

Shooting methods for a PT-symmetric periodic eigenvalue problem

We present a rigorous analysis of the performance of some one-step discretization schemes for a class of PT-symmetric singular boundary eigenvalue problem which encompasses a number of different problems whose investigation has been inspired by the 2003 article of Benilov et al. (J Fluid Mech 497:201-224, 2003). These discretization schemes are analyzed as initial value problems rather than as discrete boundary problems, since this is the setting which ties in most naturally with the formulation of the problem which one is forced to adopt due to the presence of an interior singularity. We also devise and analyze a variable step scheme for dealing with the singular points. Numerical results show better agreement between our results and those obtained from small-ε asymptotics than has been shown in results presented hitherto

Stability of the inverse resonance problem on the line

In the absence of a half-bound state, a compactly supported potential of a Schr\"odinger operator on the line is determined up to a translation by the zeros and poles of the meropmorphically continued left (or right) reflection coefficient. The poles are the eigenvalues and resonances, while the zeros also are physically relevant. We prove that all compactly supported potentials (without half-bound states) that have reflection coefficients whose zeros and poles are \eps-close in some disk centered at the origin are also close (in a suitable sense). In addition, we prove stability of small perturbations of the zero potential (which has a half-bound state) from only the eigenvalues and resonances of the perturbation.Comment: 21 page

Maximal operators and Hilbert transforms along variable non-flat homogeneous curves

We prove that the maximal operator associated with variable homogeneous planar curves $(t, u t^{\alpha})_{t\in \mathbb{R}}$, $\alpha\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\mathbb{R}^2 \to \mathbb{R}$ is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with $(t, ut^{\alpha})_{t\in \mathbb{R}}$, $\alpha\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\mathbb{R}^2\to \mathbb{R}$ is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, $TT^*$ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.Comment: 38 page

Impact of the treatment of periodontitis on systemic health and quality of life: A systematic review

AIM: To investigate the effect of treatment of periodontitis on systemic health outcomes, pregnancy complications, and associated quality of life. MATERIALS AND METHODS: Systematic electronic searches were conducted to identify randomized controlled trials with minimum 6-month follow-up and reporting on the outcomes of interest. Qualitative and quantitative analyses were performed as deemed suitable. RESULTS: Meta-analyses confirmed reductions of high-sensitivity C-reactive protein (hs-CRP) [0.56 mg/L, 95% confidence interval (CI) (−0.88, −0.25), p < .001]; interleukin (IL)-6 [0.48 pg/ml, 95% CI (−0.88, −0.08), p = .020], and plasma glucose [1.33 mmol/l, 95% CI (−2.41, −0.24), p = .016], and increase of flow-mediated dilation (FMD) [0.31%, 95% CI (0.07, 0.55), p = .012] and diastolic blood pressure [0.29 mmHg, 95% CI (0.10, 0.49), p = .003] 6 months after the treatment of periodontitis. A significant effect on preterm deliveries (<37 weeks) was observed [0.77 risk ratio, 95% CI (0.60, 0.98), p = .036]. Limited evidence was reported on quality-of-life (QoL) outcomes in the included studies. CONCLUSIONS: Treatment of periodontitis results in systemic health improvements including improvement in cardiometabolic risk, reduction in systemic inflammation and the occurrence of preterm deliveries. Further research is however warranted to confirm whether these changes are sustained over time. Further, appropriate QoL outcomes should be included in the study designs of future clinical trials