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Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, Funder: Einstein Visiting Fellow Program, The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

We study linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We demonstrate that regularization by projection and variational regularization can be formulated by using the training data only and without making use of the forward operator. We study convergence and stability of the regularized solutions in view of Seidman (1980 J. Optim. Theory Appl. 30 535), who showed that regularization by projection is not convergent in general, by giving some insight on the generality of Seidman’s nonconvergence example. Moreover, we show, analytically and numerically, that regularization by projection is indeed capable of learning linear operators, such as the Radon transform.

Funder: Cambridge Commonwealth, European and International Trust; doi: https://doi.org/10.13039/501100003343, Funder: Natural Sciences and Engineering Research Council of Canada (NSERC), We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e., sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system $XA$, corresponding to an ensemble of two quantum states, with $X$ being classical and $A$ being quantum. We study the resource theory for two different classes of free operations: $(i)$ ${\rm{CPTP}}_A$, which consists of quantum channels acting only on $A$, and $(ii)$ conditional doubly stochastic (CDS) maps acting on $XA$. We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.

Electronic data capture is fast becoming the preferred method of collecting patient-reported outcome (PRO) data in clinical trials. Data collection can be site-based (clinical study site), and typically collected on a tablet, or field-based (subject's typical environment such as home, school, or workplace), and most often accomplished with handheld devices, such as a smartphone. While site and study subject compliance with protocol-specific data collection procedures using these devices is critical to trial success, so is the robustness of the device hardware and the software these devices use to capture the trial data. Technology failures and/or site or subject resistance to the electronic data capture protocol may lead a subject to record data on paper, which can result in undesirable data challenges. As such, both site and subject compliance issues and technology-related factors must be anticipated to adhere to the ePRO data collection plan. The objective of this paper is to provide the technology industry's best practice recommendations for optimizing ePRO data collection in clinical trials by proposing the inclusion of a planned approach to data collection that includes viable electronic backup strategies so that defaulting to a paper-based backup becomes unnecessary.

In an attempt to understand the soliton resolution conjecture, we consider the Sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree $n$) there exists a unique linearly stable soliton, which we call the $n$-kink. We give numerical evidence that the $n$-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree $n$. When the radius of the wormhole throat $a$ is large enough, the convergence to the $n$-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the $1$-kink using the Soffer-Weinstein weakly nonlinear perturbation theory., Comment: 19 pages, 10 figures, final version

Strain engineering is used to obtain desirable materials properties in a range of modern technologies. Direct nanoscale measurement of the three-dimensional strain tensor field within these materials has however been limited by a lack of suitable experimental techniques and data analysis tools. Scanning electron diffraction has emerged as a powerful tool for obtaining two-dimensional maps of strain components perpendicular to the incident electron beam direction. Extension of this method to recover the full three-dimensional strain tensor field has been restricted though by the absence of a formal framework for tensor tomography using such data. Here, we show that it is possible to reconstruct the full non-symmetric strain tensor field as the solution to an ill-posed tensor tomography inverse problem. We then demonstrate the properties of this tomography problem both analytically and computationally, highlighting why incorporating precession to perform scanning precession electron diffraction may be important. We establish a general framework for non-symmetric tensor tomography and demonstrate computationally its applicability for achieving strain tomography with scanning precession electron diffraction data.

Funder: University of Cambridge Centre for Doctoral Training, We consider the nearest-neighbour tight binding model of the honeycomb lattice in magnetic fields and discover surprizing new analytical results that fully explain fractal spectra and experimentally observed asymmetries in the density of states of molecular graphene. We describe a fractal Cantor spectrum for irrational magnetic flux through a honeycomb, and establish the existence of zero energy Dirac cones for each rational flux with fully explicit estimates on the cone angle. Our results give a substantially more refined description of subtleties in the de Haas–van Alphen and quantum Hall effects, and provide the first quantitative bounds on transport coefficients for the tight-binding model under disorder.

In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation., Revised after referee comments

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, In this work we study the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of auto conversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein and Majda in [17] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [16] and Grabowski and Smolarkiewicz [12]. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In [14] we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao and Titi [6], who succeeded in proving the global solvability of the primitive equations.

This study assessed the presence, amount, and sensitivity of personally identifiable information and personal health information found in hospital recycling bins in Toronto, Ontario, Canada.

The software “Multireader sample size program for diagnostic studies,” written by Kevin Schartz and Stephen Hillis, performs sample size computations for diagnostic reader-performance studies. The program computes the sample size needed to detect a specified difference in a reader-performance measure between two imaging modalities when using the analysis methods initially proposed by Dorfman, Berbaum, and Metz, and Obuchowski and Rockette, and later unified and improved by Hillis and colleagues. A commonly used reader-performance measure is the area under the receiver-operating-characteristic curve. The program has an easy-to-use step-by-step intuitive interface that walks the user through the entry of the needed information. It can be used with several different study designs, inference procedures, hypotheses, and input and output formats. The program is functional in Windows, OS X, and Linux. The methodology underlying the software is discussed for the most common diagnostic study design, where each reader evaluates each case using each modality.

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.