The Dirac equation without spinors

In the first part of the paper we give a tensor version of the Dirac equation. In the second part we formulate and analyse a simple model equation which for weak external fields appears to have properties similar to those of the 2--dimensional Dirac equation.Comment: 20 pages. Submitted for publication in the proceedings of the conference `Functional analysis, partial differential equations and applications', Rostock (Germany) 31 August--4 September 199

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

Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in Rd\R^d

We consider a class of second order linear nonautonomous parabolic equations in R^d with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator G(t,s) be compact in C_b(R^d) for t>s, and describe the asymptotic behavior of G(t,s)f as t-s goes to infinity in terms of a family of measures mu_s, s in R, solution of the associated Fokker-Planck equation

Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $\Omega$ in the Heisenberg group $\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $\Omega$ is the half-space given by $\langle \xi, \nu\rangle > d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $\Omega$ the inequality is sharp and takes the form \begin{equation} \int_\Omega |\nabla_{\mathbb{H}^n}u|^2 \, d\xi \geq \frac{1}{4}\int_{\Omega} \sum_{i=1}^n\frac{\langle X_i(\xi), \nu\rangle^2+\langle Y_i(\xi), \nu\rangle^2}{\textrm{dist}(\xi, \partial \Omega)^2}|u|^2\, d\xi, \end{equation} where $\textrm{dist}(\, \cdot\,, \partial \Omega)$ denotes the Euclidean distance from $\partial \Omega$.Comment: 14 page

D{\cal D}-deformed harmonic oscillators

We analyze systematically several deformations arising from two-dimensional harmonic oscillators which can be described in terms of $\cal{D}$-pseudo bosons. They all give rise to exactly solvable models, described by non self-adjoint hamiltonians whose eigenvalues and eigenvectors can be found adopting the quite general framework of the so-called $\cal{D}$-pseudo bosons. In particular, we show that several models previously introduced in the literature perfectly fit into this scheme.Comment: in press in International Journal of Theoretical Physic

An Adiabatic Theorem without a Gap Condition

The basic adiabatic theorems of classical and quantum mechanics are over-viewed and an adiabatic theorem in quantum mechanics without a gap condition is described.Comment: Talk at QMath 7, Prague, 1998. 10 pages, 7 figure

Fidelity of delivery and contextual factors influencing children’s level of engagement: Process evaluation of the Online Remote Behavioural Intervention for Tics (ORBIT) Trial (Preprint)

BACKGROUND: The Online Remote Behavioural Intervention for Tics (ORBIT) study was a multicentre randomized controlled trial of a complex intervention that consisted of an online behavioural intervention for children and young people (CYP) with tic disorders. In this first part of a two-stage process evaluation, we conducted a mixed-methods study exploring reach, dose, and fidelity of the intervention and contextual factors influencing engagement with the intervention. OBJECTIVE: This study aims to explore the fidelity of delivery and the contextual factors underpinning the ORBIT intervention. METHODS: Baseline study data and intervention usage metrics from participants in the intervention arm were used as quantitative implementation data (n=112). The experiences of being in the intervention were explored by semi-structured interviews with children (n=20) and parent (n=20) participants, therapists (n=4), and referring clinicians (n=6). A principal components analysis was used to create a comprehensive, composite measure of CYP’s engagement with the intervention. Engagement factor scores reflected relative uptake as assessed by a range of usage indices including chapters accessed, number of pages visited and number of logins. RESULTS: The intervention was implemented with high fidelity, and participants deemed the intervention acceptable and satisfactory. Engagement and adherence were high with child participants completing an average of 7.5/10 chapters and 100/112 (89.3%) participants completed a minimum of 4 chapters: the pre-defined threshold for effective dose. Compared to the total population of children with tic disorders, the sample tended to have more educated parents and live in more economically advantaged areas but socioeconomic factors were not related to engagement factor scores. Factors associated with higher engagement factor scores included participants enrolled at the London site vs. the Nottingham site (P=.011), self-referred vs. clinic-referred (P=.041), higher parental engagement as evidenced by number of parental chapters completed (ρ=0.73, n=111, P<.001) and more therapist time for parent (ρ=0.46, n=111, P<.001). A multiple linear regression indicated that parents’ chapter completion (β=.69, t110=10.18, P<.001) and therapist time for parent (β=.19, t110=2.95, P=.004) were the only significant independent predictors of engagement factor scores. CONCLUSIONS: Overall, the intervention had high fidelity of delivery and was evaluated positively by participants, although reach may have been constrained by the nature of the randomized controlled trial. Parental engagement and therapist time for parent were strong predictors of intervention implementation which has important implications for the design and implementation of digital therapeutic interventions into Child and Adolescent Mental Health Services. Clinical Trial: International Standard Randomized Controlled Trial Number (ISRCTN) 70758207; https://doi.org/10.1186/ISRCTN70758207 and ClinicalTrials.gov NCT03483493; https://clinicaltrials.gov/ct2/show/NCT0348349

Factors influencing the efficacy of an online behavioural intervention for children and young people with tics: Process evaluation of a randomised controlled trial

The Online Remote Behavioural Intervention for Tics (ORBIT) trial found that an internet-delivered, therapist-supported, and parent-assisted Exposure and Response Prevention (ERP) intervention reduced tic severity and improved clinical outcomes. This process evaluation aimed to explore mechanisms of impact and factors influencing efficacy. Participants were 112 children with a tic disorder and their parents randomised to the active intervention arm of the ORBIT trial. Child engagement was assessed by usage metrics and parent engagement by chapter completion. Experiences of the digital intervention were explored by semi-structured interviews. Outcomes (3-months post randomisation) were change in tic severity and overall clinical improvement. Tic severity reduced from baseline to 3-month follow-up and 36% were rated as much improved clinically. Greater tic severity at baseline predicted reduction in tic severity. Parental engagement was the only independent predictor of clinical improvement. There were no statistically significant mediators or moderators of the relationship between level of child engagement and outcome. From the qualitative findings, child participants appreciated working together with parents on the intervention and participants found the intervention engaging. ORBIT may be an effective and acceptable intervention for children and young people with tic disorders, with parental engagement being a key factor in successful outcomes

Measuring processes and the Heisenberg picture

In this paper, we attempt to establish quantum measurement theory in the Heisenberg picture. First, we review foundations of quantum measurement theory, that is usually based on the Schr\"{o}dinger picture. The concept of instrument is introduced there. Next, we define the concept of system of measurement correlations and that of measuring process. The former is the exact counterpart of instrument in the (generalized) Heisenberg picture. In quantum mechanical systems, we then show a one-to-one correspondence between systems of measurement correlations and measuring processes up to complete equivalence. This is nothing but a unitary dilation theorem of systems of measurement correlations. Furthermore, from the viewpoint of the statistical approach to quantum measurement theory, we focus on the extendability of instruments to systems of measurement correlations. It is shown that all completely positive (CP) instruments are extended into systems of measurement correlations. Lastly, we study the approximate realizability of CP instruments by measuring processes within arbitrarily given error limits.Comment: v

Evaluating semi-supervision methods for medical image segmentation: applications in cardiac magnetic resonance imaging

PURPOSE: Purpose Neural networks have potential to automate medical image segmentation but require expensive labeling efforts. While methods have been proposed to reduce the labeling burden, most have not been thoroughly evaluated on large, clinical datasets or clinical tasks. We propose a method to train segmentation networks with limited labeled data and focus on thorough network evaluation. APPROACH: We propose a semi-supervised method that leverages data augmentation, consistency regularization, and pseudolabeling and train four cardiac magnetic resonance (MR) segmentation networks. We evaluate the models on multiinstitutional, multiscanner, multidisease cardiac MR datasets using five cardiac functional biomarkers, which are compared to an expert’s measurements using Lin’s concordance correlation coefficient (CCC), the within-subject coefficient of variation (CV), and the Dice coefficient. RESULTS: The semi-supervised networks achieve strong agreement using Lin’s CCC (>0.8), CV similar to an expert, and strong generalization performance. We compare the error modes of the semi-supervised networks against fully supervised networks. We evaluate semi-supervised model performance as a function of labeled training data and with different types of model supervision, showing that a model trained with 100 labeled image slices can achieve a Dice coefficient within 1.10% of a network trained with 16,000+ labeled image slices. CONCLUSION: We evaluate semi-supervision for medical image segmentation using heterogeneous datasets and clinical metrics. As methods for training models with little labeled data become more common, knowledge about how they perform on clinical tasks, how they fail, and how they perform with different amounts of labeled data is useful to model developers and users