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

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

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

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

Quadricuspid aortic valve not discovered by transthoracic echocardiography

BACKGROUND: Quadricuspid aortic valve is a rare congenital heart defect. Several different anatomical variations of a quadricuspid aortic valve has been described and aortic regurgitation is the predominant valvular dysfunction associated with quadricuspid aortic valve. CASE PRESENTATION: A 68-year-old woman presented with almost a years history of increasing dyspnoea on exertion. The patient have had two previous transthoracic echocardiographic exams in the last six years and they had only documented moderate aortic regurgitation. Transoesophageal echocardiography displayed a rare case of quadricuspid aortic valve with three cusps of equal size and one larger cusp. The malformation was associated with severe aortic regurgitation. CONCLUSION: Liberal use of transoesophageal echocardiography is often warranted if optimal display of valvular morphology is desired

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 &lt; 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

Bone mineral accrual and gain in skeletal width in pre-pubertal school children is independent of the mode of school transportation – one-year data from the prospective observational pediatric osteoporosis prevention (POP) study

Background: Walking and cycling to school could be an important regular source of physical activity in growing children. The aim of this 12 months prospective observational study was thus to evaluate the effect of self-transportation to school on bone mineral accrual and gain in bone width in pre-pubertal children, both traits independently contributing to bone strength. Methods: Ninety-seven girls and 133 boys aged 7-9 years were recruited as a part of the Malmo Pediatric Osteoporosis Prevention (POP) Study in order to evaluate the influence of self-selected school transportation for the accrual of bone mineral and bone width. Children who walked or cycled to school were compared with children who went by bus or car. Bone mineral content (BMC) was measured by dual energy X-ray absorptiometry (DXA) in the lumbar spine (L2-L4), third lumbar vertebra (L3) and hip, and bone width was calculated at L3 and femoral neck (FN). Changes during the first 12 months were compared between the groups. Subjective duration of physical activity was estimated by a questionnaire and objective level of everyday physical activity at follow-up by accelerometers worn for four consecutive days. All children remained in Tanner stage 1 throughout the study. Comparisons were made by independent student's t-tests between means, ANCOVA and Fisher's exact tests. Results: There were no differences in baseline or annual changes in BMC or bone width when the transportation groups were compared. No differences were detected in objectively measured daily level of physical activity by accelerometer. All children reached above 60 minutes of moderate to intense daily physical activity per day, the international recommended level of daily physical activity according to the United Kingdom Expert Consensus Group. Conclusion: The everyday physical activity in these pre-pubertal children seems to be so high that the school transportation contributes little to their total level of physical activity. As a result, the choice of school transportation seems not to influence the accrual of bone mineral or gain in bone size during a I2-month follow-up period