Solvability of subprincipal type operators

In this paper we consider the solvability of pseudodifferential operators in the case when the principal symbol vanishes of order $k \ge 2$ at a nonradial involutive manifold $\Sigma_2$. We shall assume that the operator is of subprincipal type, which means that the $k$:th inhomogeneous blowup at $\Sigma_2$ of the refined principal symbol is of principal type with Hamilton vector field parallel to the base $\Sigma_2$, but transversal to the symplectic leaves of $\Sigma_2$ at the characteristics. When $k = \infty$ this blowup reduces to the subprincipal symbol. We also assume that the blowup is essentially constant on the leaves of $\Sigma_2$, and does not satisfying the Nirenberg-Treves condition (${\Psi}$). We also have conditions on the vanishing of the normal gradient and the Hessian of the blowup at the characteristics. Under these conditions, we show that $P$ is not solvable.Comment: Changed the formulation of Theorem 2.15, added an assuption. Corrected errors and clarified the arguments. Added reference

Risks associated with obesity in pregnancy, for the mother and baby: a systematic review of reviews

Maternal obesity is linked with adverse outcomes for mothers and babies. To get an overview of risks related to obesity in pregnant women, a systematic review of reviews was conducted. For inclusion, reviews had to compare pregnant women of healthy weight with women with obesity, and measure a health outcome for mother and/or baby. Authors conducted full-text screening, quality assurance using the AMSTAR tool and data extraction steps in pairs. Narrative analysis of the 22 reviews included show gestational diabetes, pre-eclampsia, gestational hypertension, depression, instrumental and caesarean birth, and surgical site infection to be more likely to occur in pregnant women with obesity compared with women with a healthy weight. Maternal obesity is also linked to greater risk of preterm birth, large-for-gestational-age babies, foetal defects, congenital anomalies and perinatal death. Furthermore, breastfeeding initiation rates are lower and there is greater risk of early breastfeeding cessation in women with obesity compared with healthy weight women. These adverse outcomes may result in longer duration of hospital stay, with concomitant resource implications. It is crucial to reduce the burden of adverse maternal and foetal/child outcomes caused by maternal obesity. Women with obesity need support to lose weight before they conceive, and to minimize their weight gain in pregnancy

Nonlinear Instability in a Semiclassical Problem

We consider a nonlinear evolution problem with an asymptotic parameter and construct examples in which the linearized operator has spectrum uniformly bounded away from Re z >= 0 (that is, the problem is spectrally stable), yet the nonlinear evolution blows up in short times for arbitrarily small initial data. We interpret the results in terms of semiclassical pseudospectrum of the linearized operator: despite having the spectrum in Re z < -c < 0, the resolvent of the linearized operator grows very quickly in parts of the region Re z > 0. We also illustrate the results numerically

The Egorov theorem for transverse Dirac type operators on foliated manifolds

Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.Comment: 27 pages, to appear in Journal of Geometry and Physic

Probabilistic Weyl laws for quantized tori

For the Toeplitz quantization of complex-valued functions on a $2n$-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law. In numerical experiments the same Weyl law also holds for ``false'' eigenvalues created by pseudospectral effects.Comment: 33 pages, 3 figures, v2 corrected listed titl

Childbirth experience questionnaire: validating its use in the United Kingdom

BACKGROUND: The Childbirth Experience Questionnaire (CEQ) was developed in Sweden in 2010 and validated in 920 primiparous women. It has not been validated in the United Kingdom (UK). Measuring the impact of an intervention on a woman's childbirth experience is arguably as important as measuring its impact on outcomes such as caesarean delivery and perinatal morbidity or mortality and yet surprisingly it is rarely done. The lack of a robust validated tool for evaluating labour experience in the UK is a topical issue in the UK at present. Indeed NICE say 'A standardised method to measure and quantify women's psychological and emotional wellbeing and their birth experiences is urgently required to support any study investigating the effectiveness of interventions, techniques or strategies during birth.' METHODS: The Childbirth Experience Questionnaire and part of the Care Quality Commission Maternity Survey (2010) was sent to 350 women at one month postnatal. The CEQ was sent again two weeks later. The CEQ was tested for face validity among 25 postnatal mothers. Demographic data and delivery data was used to establish construct validity of the CEQ using the method of known-groups validation. The results of the scored CEQ sent out twice were used to measure test-retest reliability of the CEQ by calculating the quadratic weighted index of agreement between the two scores. Criterion validity was measured by calculating the Pearson correlation coefficient for the CEQ and Maternity Survey scores. RESULTS: Face validity of the CEQ in a UK population was demonstrated with all respondents stating it was easy to understand and complete. A statistically significantly higher CEQ score for subgroups of women known to report a better birth outcome demonstrated construct validity of the CEQ. A weighted kappa of 0.68 demonstrated test-retest reliability of the CEQ. A Pearson correlation co-efficient of 0.73 demonstrated a strong correlation between the results of the CEQ and the results of the 'gold standard' assessment of childbirth experience in the UK: the Maternity Survey and hence criterion validity of the CEQ. CONCLUSIONS: The Childbirth Experience Questionnaire is a valid and reliable measure of childbirth experience in the UK population

Cardiorespiratory fitness is associated with hard and light intensity physical activity but not time spent sedentary in 10–14 year old schoolchildren: the HAPPY study

Sedentary behaviour is a major risk factor for developing chronic diseases and is associated with low cardiorespiratory fitness in adults. It remains unclear how sedentary behaviour and different physical activity subcomponents are related to cardiorespiratory fitness in children. The purpose of this study was to assess how sedentary behaviour and different physical activity subcomponents are associated with 10–14 year-old schoolchildren's cardiorespiratory fitness

Characterization of Single-Mode Vertical Cavity Surface-Emitting Lasers

A high-quality single-mode beam is desirable for the efficient use of lasers as light sources for optical data communications and interconnects, however there is little data which characterizes operating ranges and near-field beam qualities of Vertical Cavity Surface Emitting Lasers (VCSELs), which has resulted in a lack of analysis of these devices. Measures of beam quality include beam-quality factor (M2 ), Side-Mode-Suppression-Ratio (SMSR) and RMS linewidth. M2 is a measurement of how closely the beam is to an ideal Gaussian. SMSR is the difference, in dB, between the amplitude of the primary peak and the amplitude of the next highest peak of the output spectrum, with single-mode operation defined by a SMSR \u3e 30 dB. RMS linewidth is a second moment calculation involving the power spectral density, where smaller RMS linewidth indicates higher beam quality. Utilizing a novel vertical M2 setup in which on-wafer VCSEL M2 can be measured, a study was conducted on the relation between M2 , SMSR and RMS linewidth, for various oxide-confined VCSELs of varying aperture sizes and Photonic Crystal (PhC) VCSELs of varying aperture sizes and photonic crystal configurations. First, the operating range of the VCSEL was determined utilizing a Semiconductor Parameter Analyzer to obtain the LIV characteristics. Along with this measurement, spectral data was collected using an Optical Spectrum Analyzer at several key operating points, which allowed the RMS linewidths and SMSRs of the devices to be calculated at these points. The novel beam-profiler setup was used to measure the device’s M2 . Initial results show a strong correlation between the measures of beam quality, with increasing SMSR, corresponding to M2 values closer to 1, and single-mode operation characterized by a M2 of less than 1.5. A strong correlation between RMS linewidth and M2 was also seen, with increasing RMS linewidths corresponding to an increase in M2

Characterization of Single-Mode Vertical Cavity Surface-Emitting Lasers

A high-quality single-mode beam is desirable for the efficient use of lasers as light sources for optical data communications and interconnects, however there is little data which characterizes operating ranges and near-field beam qualities of Vertical Cavity Surface Emitting Lasers (VCSELs), which has resulted in a lack of analysis of these devices. Measures of beam quality include beam-quality factor (M2 ), Side-Mode-Suppression-Ratio (SMSR) and RMS linewidth. M2 is a measurement of how closely the beam is to an ideal Gaussian. SMSR is the difference, in dB, between the amplitude of the primary peak and the amplitude of the next highest peak of the output spectrum, with single-mode operation defined by a SMSR \u3e 30 dB. RMS linewidth is a second moment calculation involving the power spectral density, where smaller RMS linewidth indicates higher beam quality. Utilizing a novel vertical M2 setup in which on-wafer VCSEL M2 can be measured, a study was conducted on the relation between M2 , SMSR and RMS linewidth, for various oxide-confined VCSELs of varying aperture sizes and Photonic Crystal (PhC) VCSELs of varying aperture sizes and photonic crystal configurations. First, the operating range of the VCSEL was determined utilizing a Semiconductor Parameter Analyzer to obtain the LIV characteristics. Along with this measurement, spectral data was collected using an Optical Spectrum Analyzer at several key operating points, which allowed the RMS linewidths and SMSRs of the devices to be calculated at these points. The novel beam-profiler setup was used to measure the device’s M2 . Initial results show a strong correlation between the measures of beam quality, with increasing SMSR, corresponding to M2 values closer to 1, and single-mode operation characterized by a M2 of less than 1.5. A strong correlation between RMS linewidth and M2 was also seen, with increasing RMS linewidths corresponding to an increase in M2

Soliton compression and supercontinuum spectra in nonlinear diamond photonics

We numerically explore synthetic crystal diamond for realizing novel light sources in ranges which are up to now difficult to achieve with other materials, such as sub-10-fs pulse durations and challenging spectral ranges. We assess the performance of on-chip diamond waveguides for controlling light generation by means of nonlinear soliton dynamics. Tailoring the cross-section of such diamond waveguides allows to design dispersion profiles with custom zero-dispersion points and anomalous dispersion ranges exceeding an octave. Various propagation dynamics, including supercontinuum generation by soliton fission, can be realized in diamond photonics. In stark contrast to usual silica-based optical fibers, where such processes occur on the scale of meters, in diamond millimeter-scale propagation distances are sufficient. Unperturbed soliton-dynamics prior to soliton fission allow to identify a pulse self-compression scenario that promises record-breaking compression factors on chip-size propagation lengths