Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-positive type

The two-dimensional Hamiltonian system (*)  y'(x)=zJH(x)y(x),  x∈(a,b), where the Hamiltonian H takes non-negative 2x2-matrices as values, and $J:= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, has attracted a lot of interest over the past decades.  Special emphasis has been put on operator models and direct and inverse spectral theorems.  Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (*) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients. In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (*) was proposed, where the 'general Hamiltonian' is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class <N∞ of all generalized Nevanlinna functions, were established. In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval.  Our results can comprehensively be formulated as follows. — We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form. —  We determine the asymptotic growth of the fundamental solution when approaching the singularity. —  We show that each solution of the equation has 'polynomially regularized' boundary values at the singularity. Besides the intrinsic interest and depth of the presented results, our motivation is drawn from forthcoming applications: the present theorems form the core for our study of Sturm–Liouville equations with two singular endpoints and our further study of the structure theory of general Hamiltonians (both to be presented elsewhere)

Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints

Part I of this paper deals with two-dimensional canonical systems $y'(x)=yJH(x)y(x)$, $x\in(a,b)$, whose Hamiltonian $H$ is non-negative and locally integrable, and where Weyl's limit point case takes place at both endpoints $a$ and $b$. We investigate a class of such systems defined by growth restrictions on H towards a. For example, Hamiltonians on $(0,\infty)$ of the form $H(x):=\begin{pmatrix}x^{-\alpha}&0\\ 0&1\end{pmatrix}$ where $\alpha<2$ are included in this class. We develop a direct and inverse spectral theory parallel to the theory of Weyl and de Branges for systems in the limit circle case at $a$. Our approach proceeds via - and is bound to - Pontryagin space theory. It relies on spectral theory and operator models in such spaces, and on the theory of de Branges Pontryagin spaces. The main results concerning the direct problem are: (1) showing existence of regularized boundary values at $a$; (2) construction of a singular Weyl coefficient and a scalar spectral measure; (3) construction of a Fourier transform and computation of its action and the action of its inverse as integral transforms. The main results for the inverse problem are: (4) characterization of the class of measures occurring above (positive Borel measures with power growth at $\pm\infty$); (5) a global uniqueness theorem (if Weyl functions or spectral measures coincide, Hamiltonians essentially coincide); (6) a local uniqueness theorem. In Part II of the paper the results of Part I are applied to Sturm--Liouville equations with singular coefficients. We investigate classes of equations without potential (in particular, equations in impedance form) and Schr\"odinger equations, where coefficients are assumed to be singular but subject to growth restrictions. We obtain corresponding direct and inverse spectral theorems

Karamata's theorem for regularised Cauchy transforms

We prove Abelian and Tauberian theorems for regularised Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the distribution functions to the asymptotics of the regularised Cauchy transform

Estimates for the Weyl coefficient of a two-dimensional canonical system

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e.\ positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\Im q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is independent of the off-diagonal entries of $H$ and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for "most" Hamiltonians also $\Im q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $\mu_H$ of $H$ by means of Abelian--Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$ w.r.t.\ a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations

A very large deployable space antenna structure based on pantograph tensioned membranes

This paper provides investigation results on the membrane technology development of a high accuracy (low deformation and high eigenfrequency) membrane antenna structure and its demonstrator for space application

Ultrafine Particles - Air Quality and Climate: European Federation of Clean Air and Environmental Protection Associations (EFCA) International Symposium, Brussels, Belgium, July 5 and 6, 2022 - Proceedings

Ultrafine particles (UFP), the nano fraction of airborne particulate matter, are considered to be causing serious health problems and environmental effects. Combustion is a major source, also by producing volatile organic pollutants which are converted in the atmosphere through photochemical reactions. Increasing applications of man-made nanomaterials add to the problem, e.g. after incineration at the end of their lifetime. A further interest in UFP’s results from their specific role in atmospheric processes such as cloud formation and precipitation and, in fact, in climate. The relation between UFP and human health and that of UFP and climate are both areas of active research and cross-links between these fields are found nowadays. The subtitle of the conference series: “air quality and climate” reflects this development. Present policies to decrease exposure to particulate matter make use of the mass-based metrics PM10/PM2.5, which do not properly represent all risks for human health. EFCA is, therefore, in favour of the development of a fraction-by-fraction approach on particulate matter, both with respect to size and chemical composition. It already recommended European policymakers the introduction of Black Carbon Particles as additional metric in the Air Quality Directive. EFCA‘s 8th Ultrafine Particles Symposium 2022 featured the most recent scientific progress in the field and so contribute to policy-relevant developments which improve the dialogue with policymakers in Europe. The Symposium has gained visibility by permanently moving to Brussels and attracts an effective mix of EU representatives and scientists. EFCA and KIT, together with GUS and CEEES are pleased to organize this event again

Impaired Cardiac Baroreflex Sensitivity Predicts Response to Renal Sympathetic Denervation in Patients With Resistant Hypertension

ObjectivesThis study sought to evaluate cardiac baroreflex sensitivity (BRS) as a predictor of response to renal sympathetic denervation (RDN).BackgroundCatheter-based RDN is a novel treatment option for patients with resistant arterial hypertension. It is assumed that RDN reduces efferent renal and central sympathetic activity.MethodsFifty patients (age 60.3 ± 13.8 years [mean ± SD mean systolic blood pressure (BP) on ambulatory blood pressure monitoring (ABPM) 157 ± 22 mm Hg, despite medication with 5.4 ± 1.4 antihypertensive drugs) underwent RDN. Prior to RDN, a 30-min recording of continuous arterial BP (Finapres; TNO-TPD Biomedical Instrumentation, Amsterdam, the Netherlands) and high-resolution electrocardiography (1.6 kHz in orthogonal XYZ leads) was performed in all patients under standardized conditions. Cardiac BRS was assessed by phase-rectified signal averaging (BRSPRSA) according to previously published technologies. Response to RDN was defined as a reduction of mean systolic BP on ABPM by 10 mm Hg or more at 6 months after RDN.ResultsSix months after RDN, mean systolic BP on ABPM was significantly reduced from 157 ± 22 mm Hg to 149 ± 20 mm Hg (p = 0.003). Twenty-six of the 50 patients (52%) were classified as responders. BRSPRSA was significantly lower in responders than nonresponders (0.16 ± 0.75 ms/mm Hg vs. 1.54 ± 1.73 ms/mm Hg; p < 0.001). Receiver-operator characteristics analysis revealed an area under the curve for prediction of response to RDN by BRSPRSA of 81.2% (95% confidence interval: 70.0% to 90.1%; p < 0.001). On multivariable logistic regression analysis, reduced BRSPRSA was the strongest predictor of response to RDN, which was independent of all other variables tested.ConclusionsImpaired cardiac BRS identifies patients with resistant hypertension who respond to RDN

Deletion of the Ca2+-activated potassium (BK) alpha-subunit but not the BK-beta-1-subunit leads to progressive hearing loss

The large conductance voltage- and Ca2+-activated potassium (BK) channel has been suggested to play an important role in the signal transduction process of cochlear inner hair cells. BK channels have been shown to be composed of the pore-forming alpha-subunit coexpressed with the auxiliary beta-1-subunit. Analyzing the hearing function and cochlear phenotype of BK channel alpha-(BKalpha–/–) and beta-1-subunit (BKbeta-1–/–) knockout mice, we demonstrate normal hearing function and cochlear structure of BKbeta-1–/– mice. During the first 4 postnatal weeks also, BKalpha–/– mice most surprisingly did not show any obvious hearing deficits. High-frequency hearing loss developed in BKalpha–/– mice only from ca. 8 weeks postnatally onward and was accompanied by a lack of distortion product otoacoustic emissions, suggesting outer hair cell (OHC) dysfunction. Hearing loss was linked to a loss of the KCNQ4 potassium channel in membranes of OHCs in the basal and midbasal cochlear turn, preceding hair cell degeneration and leading to a similar phenotype as elicited by pharmacologic blockade of KCNQ4 channels. Although the actual link between BK gene deletion, loss of KCNQ4 in OHCs, and OHC degeneration requires further investigation, data already suggest human BK-coding slo1 gene mutation as a susceptibility factor for progressive deafness, similar to KCNQ4 potassium channel mutations. © 2004, The National Academy of Sciences. Freely available online through the PNAS open access option