On the microlocal properties of the range of systems of principal type

The purpose of this paper is to study microlocal conditions for inclusion relations between the ranges of square systems of pseudodifferential operators which fail to be locally solvable. The work is an extension of earlier results for the scalar case in this direction, where analogues of results by L. H\"ormander about inclusion relations between the ranges of first order differential operators with coefficients in $C^\infty$ which fail to be locally solvable were obtained. We shall study the properties of the range of systems of principal type with constant characteristics for which condition (\Psi) is known to be equivalent to microlocal solvability.Comment: Added Theorem 4.7, Corollary 4.8 and Lemma A.4, corrected misprints. The paper has 40 page

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

Spectral projections and resolvent bounds for partially elliptic quadratic differential operators

We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers-Fokker-Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral projections onto the ground state.Comment: 60 pages, 3 figures. J. Pseudo-Differ. Oper. Appl., to appear. Revised according to referee report, including minor changes to Corollary 1.8. The final publication will be available at link.springer.co

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

Objectively measured physical activity and fat mass in a large cohort of children

Background Previous studies have been unable to characterise the association between physical activity and obesity, possibly because most relied on inaccurate measures of physical activity and obesity. Methods and Findings We carried out a cross sectional analysis on 5,500 12-year-old children enrolled in the Avon Longitudinal Study of Parents and Children. Total physical activity and minutes of moderate and vigorous physical activity (MVPA) were measured using the Actigraph accelerometer. Fat mass and obesity (defined as the top decile of fat mass) were measured using the Lunar Prodigy dual x-ray emission absorptiometry scanner. We found strong negative associations between MVPA and fat mass that were unaltered after adjustment for total physical activity. We found a strong negative dose-response association between MVPA and obesity. The odds ratio for obesity in adjusted models between top and the bottom quintiles of minutes of MVPA was 0.03 (95% confidence interval [CI] 0.01-0.13, p-value for trend < 0.0001) in boys and 0.36 (95% CI 0.17-0.74, p-value for trend = 0.006) in girls. Conclusions We demonstrated a strong graded inverse association between physical activity and obesity that was stronger in boys. Our data suggest that higher intensity physical activity may be more important than total activity

High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.Comment: 26 pages, latex2

Gupta–Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times

We give a complete framework for the Gupta–Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta–Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero modes. The only requirement, the absence of zero-resonance states, is shown to be stable under compact perturbations of topology and metric. Usual deformation arguments based on the time-slice axiom then lead to a construction of Gupta–Bleuler representations on a large class of globally hyperbolic space-times. As usual, the field algebra is represented on an indefinite inner product space, in which the physical states form a positive semi-definite subspace. Gauge transformations are incorporated in such a way that the field can be coupled perturbatively to a Dirac field. Our approach does not require any topological restrictions on the underlying space-time

Adopting a healthy lifestyle when pregnant and obese - an interview study three years after childbirth

Background: Obesity during pregnancy is increasing and is related to life-threatening and ill-health conditions in both mother and child. Initiating and maintaining a healthy lifestyle when pregnant with body mass index (BMI) ≥ 30 kg/m2 can improve health and decrease risks during pregnancy and of long-term illness for the mother and the child. To minimise gestational weight gain women with BMI ≥ 30 kg/m2 in early pregnancy were invited to a lifestyle intervention including advice and support on diet and physical activity in Gothenburg, Sweden. The aim of this study was to explore the experiences of women with BMI ≥ 30 kg/m2 regarding minimising their gestational weight gain, and to assess how health professionals' care approaches are reflected in the women's narratives. Methods: Semi-structured interviews were conducted with 17 women who had participated in a lifestyle intervention for women with BMI ≥ 30 kg/m2 during pregnancy 3 years earlier. The interviews were digitally recorded and transcribed in full. Thematic analysis was used. Results: The meaning of changing lifestyle for minimising weight gain and of the professional's care approaches is described in four themes: the child as the main motivation for making healthy changes; a need to be seen and supported on own terms to establish healthy routines; being able to manage healthy activities and own weight; and need for additional support to maintain a healthy lifestyle. Conclusions: To support women with BMI ≥ 30 kg/m2 to make healthy lifestyle changes and limit weight gain during pregnancy antenatal health care providers should 1) address women's weight in a non-judgmental way using BMI, and provide accurate and appropriate information about the benefits of limited gestational weight gain; 2) support the woman on her own terms in a collaborative relationship with the midwife; 3) work in partnership to give the woman the tools to self-manage healthy activities and 4) give continued personal support and monitoring to maintain healthy eating and regular physical activity habits after childbirth involving also the partner and family

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

Person-centred care in interventions to limit weight gain in pregnant women with obesity - a systematic review

Background Person-centred care, asserting that individuals are partners in their care, has been associated with care satisfaction but the value of using it to support women with obesity during pregnancy is unknown. Excessive gestational weight gain is associated with increased risks for both mother and baby and weight gain therefore is an important intervention target. The aims of this review was to 1) explore to what extent and in what manner interventions assessing weight in pregnant women with obesity use person-centred care and 2) assess if interventions including aspects of person-centred care are more effective at limiting weight gain than interventions not employing person-centred care. Methods Ten databases were systematically searched in January 2014. Studies had to report an intervention offered to pregnant women with obesity and measure gestational weight gain to be included. All included studies were independently double coded to identify to what extent they included three defined aspects of person-centred care: 1) “initiate a partnership” including identifying the person’s circumstances and motivation; 2) “working the partnership” through sharing the decision-making regarding the planned action and 3) “safeguarding the partnership through documentation” of care preferences. Information on gestational weight gain, study quality and characteristics were also extracted. Results Ten studies were included in the review, of which five were randomised controlled trials (RCT), and the remaining observational studies. Four interventions included aspects of person-centred care; two observational studies included both “initiating the partnership”, and “working the partnership”. One observational study included “initiating the partnership” and one RCT included “working the partnership”. No interventions included “safeguarding the partnership through documentation”. Whilst all studies with person-centred care aspects showed promising findings regarding limiting gestational weight gain, so did the interventions not including person-centred care aspects. Conclusions The use of an identified person-centred care approach is presently limited in interventions targeting gestational weight gain in pregnant women with obesity. Hence to what extent person-centred care may improve health outcomes and care satisfaction in this population is currently unknown and more research is needed. That said, our findings suggest that use of routines incorporating person-centredness are feasible to include within these interventions