The modern Inspectorate : Her Majesty's Inspectorate of Schools in England and Wales, 1944-1991.

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Analysis of unbounded operators and random motion

We study infinite weighted graphs with view to \textquotedblleft limits at infinity,\textquotedblright or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means \textquotedblleft very\textquotedblright large) networks of resistors, or in statistical mechanics models for classical or quantum systems. But more generally our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If $X$ is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on $X$ evaluated on pairs of points in $X$. And the Hilbert norm-squared in $\mathcal{H}(X)$ will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian, or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space $\mathcal{H}(X)$ which measure quantitative notions of limits at infinity in $X$, one generalizes finite-energy harmonic functions in $\mathcal{H}(X)$, and the other a deficiency index of a natural operator in $\mathcal{H}(X)$ associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of \textquotedblleft boundaries\textquotedblright in more standard random walk models. Comparisons are made.Comment: 38 pages, 4 tables, 3 figure

Trends in sugar content of non-alcoholic beverages in Australia between 2015 and 2019 during the operation of a voluntary industry pledge to reduce sugar content

Objectives: To investigate changes in mean sugar content of non-alcoholic beverages (overall and sugar-sweetened beverages) available for purchase in Australia and to compare signatories versus non-signatories of the Australian Beverages Council voluntary pledge from 2018 Design: Retrospective observational study Setting: Australia Participants: About 1,500 non-alcoholic beverages per year included in the FoodSwitch Monitoring Datasets for 2015-2019 Results: Overall, mean sugar content fell by 1.3g/100mL (17.1%) from 7.5g/100mL in 2015 to 6.2g/100mL in 2019. SSBs have accounted for about 56% of all beverages available for purchase since 2015. Between 2015 and 2019, the sugar content of SSBs dropped by about 10% (0.8g/100mL). Soft drinks and milk-based drinks were the categories with the largest decrease in sugar content. The greater reduction in sugar observed for beverages overall than SSBs suggests at least some of the overall decrease in sugar content is due to the appearance of new products with low or no-sugar rather than reformulation. Over the same period, beverages with added non-nutritive sweeteners increased from 41% to 44%. The decrease in sugar content for all beverages and SSBs was, in general, larger for non-signatories than signatories of the voluntary industry pledge. Conclusions: Between 2015 and 2019, the small reduction in sugar content of non-alcoholic beverages in Australia resulted from the combined effects of introducing low or no-sugar products and reformulation of some categories of SSBs. Further policy and regulatory measures are required to reap the most benefit that sugar reduction among non-alcoholic beverages can bring to population health

International collaborative project to compare and track the nutritional composition of fast foods

BackgroundChronic diseases are the leading cause of premature death and disability in the world with over-nutrition a primary cause of diet-related ill health. Excess quantities of energy, saturated fat, sugar and salt derived from fast foods contribute importantly to this disease burden. Our objective is to collate and compare nutrient composition data for fast foods as a means of supporting improvements in product formulation.Methods/designSurveys of fast foods will be done in each participating country each year. Information on the nutrient composition for each product will be sought either through direct chemical analysis, from fast food companies, in-store materials or from company websites. Foods will be categorized into major groups for the primary analyses which will compare mean levels of saturated fat, sugar, sodium, energy and serving size at baseline and over time. Countries currently involved include Australia, New Zealand, France, UK, USA, India, Spain, China and Canada, with more anticipated to follow.DiscussionThis collaborative approach to the collation and sharing of data will enable low-cost tracking of fast food composition around the world. This project represents a significant step forward in the objective and transparent monitoring of industry and government commitments to improve the quality of fast foods.<br /

On approximate solutions of semilinear evolution equations

A general framework is presented to discuss the approximate solutions of an evolution equation in a Banach space, with a linear part generating a semigroup and a sufficiently smooth nonlinear part. A theorem is presented, allowing to infer from an approximate solution the existence of an exact solution. According to this theorem, the interval of existence of the exact solution and the distance of the latter from the approximate solution can be evaluated solving a one-dimensional "control" integral equation, where the unknown gives a bound on the previous distance as a function of time. For example, the control equation can be applied to the approximation methods based on the reduction of the evolution equation to finite-dimensional manifolds: among them, the Galerkin method is discussed in detail. To illustrate this framework, the nonlinear heat equation is considered. In this case the control equation is used to evaluate the error of the Galerkin approximation; depending on the initial datum, this approach either grants global existence of the solution or gives fairly accurate bounds on the blow up time.Comment: 33 pages, 10 figures. To appear in Rev. Math. Phys. (Shortened version; the proof of Prop. 3.4. has been simplified

Bohl-Perron type stability theorems for linear difference equations with infinite delay

Relation between two properties of linear difference equations with infinite delay is investigated: (i) exponential stability, (ii) \l^p-input \l^q-state stability (sometimes is called Perron's property). The latter means that solutions of the non-homogeneous equation with zero initial data belong to \l^q when non-homogeneous terms are in \l^p. It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted \l^r-space with an exponentially fading weight (the phase space). Our main result states that (i) $\Leftrightarrow$ (ii) whenever $(p,q) \neq (1,\infty)$ and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and \l^p-input \l^q-state stabilities does not depend on the choice of a phase space and parameters $p$ and $q$, respectively. \l^1-input \l^\infty-state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.Comment: To be published in Journal of Difference Equations and Application

Improved Lieb-Oxford exchange-correlation inequality with gradient correction

We prove a Lieb-Oxford-type inequality on the indirect part of the Coulomb energy of a general many-particle quantum state, with a lower constant than the original statement but involving an additional gradient correction. The result is similar to a recent inequality of Benguria, Bley and Loss, except that the correction term is purely local, which is more usual in density functional theory. In an appendix, we discuss the connection between the indirect energy and the classical Jellium energy for constant densities. We show that they differ by an explicit shift due to the long range of the Coulomb potential.Comment: Final version to appear in Physical Review A. Compared to the very first version, this one contains an appendix discussing the link with the Jellium proble

Semispectral measures as convolutions and their moment operators

The moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied, with paying attention to the natural domains of these unbounded operators. The results are then applied to conveniently determine the moment operators of the Cartesian margins of the phase space observables.Comment: 7 page