Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space R N . Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

Using dyadic observation to explore equitable learning opportunities in classroom instruction

Because of poverty, many children do not receive adequate prenatal care, nutrition, or early childhood education. These inequities combine to ensure that many students enter school with considerably less academic content knowledge and skills for learning than their peers. Teachers and schools did not create these gaps, but they must address them. The impact of schools in reducing gaps has been explored for decades only to yield inconsistent findings. One possible reason for these contradictory results is because these studies ignore classroom process. We argue for the inclusion of process in research on opportunity and achievement gaps to better articulate if schools provide inequitable learning opportunities. Further, we argue for dyadic (teacher to individual student) measurement of classroom process because commonly-used observation instruments only measure teachers’ interactions with the whole class. These instruments obscure differential teacher treatment that may exist in some classrooms. To improve policy and practice, we call for supplementing extant measures of teachers’ whole-class interactions (process) and student outcome (product) measures with those that measure dyadic interactions to learn how opportunities to learn in classrooms and schools are distributed among students to reduce, sustain, or enhance learning gaps.

Probability Spaces

This chapter defines probability measures and probability spaces in a general context, as a case of the concepts introduced in Chapter 3. The axioms of probability are explained, and the important concepts of conditional probability and independence are introduced and linked to the role of product spaces and product measures.

On symmetric intersecting families of vectors

A family of vectors in [k] n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k] n invariant under a transitive group of symmetries is o(k n ), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.

Integration Theory

The Lebesgue measure on ?n (presented in section 8.4) is a pivotal component of this chapter. The approach in the chapter is to extend the positive linear functional provided by the Riemann integral on the space of continuous, compactly supported functions on ?n (presented in section 8.1). An excursion on Radon measures is included at the end of section 8.4. The rest of the sections are largely independent of sections 8.1 and 8.4 and constitute a deep introduction to general measure and integration theories. Topics include measurable spaces and measurable functions, Carathéodory’s theorem, abstract integration and convergence theorems, complex measures and the Radon-Nikodym theorem, Lp spaces, product measures and Fubini’s theorem, and a good collection of approximation theorems. The closing section of the book provides a glimpse of Fourier analysis and gives a nice conclusion to the discussion of Fourier series and orthogonal polynomials started in section 4.10.

The Reliability and Validation of the Aquatic Movement Protocol as an Instrument for Assessing Aquatic Motor Competence in Primary Aged Children

There is a dearth of research in aquatic motor competency, a key requirement for primary physical education in order to become physically literate. This study proposes a new assessment protocol for aquatic motor competence and sets out to examine the validity of the Aquatic Movement Protocol (AMP) in children between 7 and 9 years of age. Testing of Gross Motor Development—second edition was implemented to assess general motor competence, including a composite of 10-m running sprint time and standing long jump distance. Aquatic motor competence was assessed by the AMP. Univariate analyses of covariance were used to examine whether assessment of general motor competence differed as opposed to aquatic motor competence. Process and product measures of dryland motor competence were analyzed using male and female subjects measuring three aquatic motor competences (low, medium, and high). Cronbach’s alpha and exploratory factor analyses were implemented to show both construct and concurrent validity of the AMP. Children classified as high for aquatic motor competence had significantly higher general motor competence (p = .001). Those who achieved a higher composite score for faster sprint speeds and longer jump distances had significantly higher aquatic motor competence (p = .001). Cronbach’s alpha of .908, showed internal consistency of the AMP. Results extracted one factor from analysis with an eigenvalue = 6.2; %variance = 62.1, with loadings higher than 0.5. This data suggests that the items on the AMP measure a single construct that we would call “Aquatic Motor Competence.” This study demonstrates that the AMP is a valid measure of aquatic motor competence in primary aged children.

Soliton Decomposition of the Box-Ball System

The box-ball system (BBS) was introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg-de Vries equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size k solitons in each k-slot. The dynamics of the components is linear: the kth component moves rigidly at speed k. Let $\zeta $ be a translation-invariant family of independent random vectors under a summability condition and $\eta $ be the ball configuration with components $\zeta $ . We show that the law of $\eta $ is translation invariant and invariant for the BBS. This recipe allows us to construct a large family of invariant measures, including product measures and stationary Markov chains with ball density less than $\frac {1}{2}$ . We also show that starting BBS with an ergodic measure, the position of a tagged k-soliton at time t, divided by t converges as $t\to \infty $ to an effective speed $v_k$ . The vector of speeds satisfies a system of linear equations related with the generalised Gibbs ensemble of conservative laws.

Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic

In order to study the spread of an epidemic over a region as a function of time, we introduce an entropy ratio U describing the uniformity of infections over various states and their districts, and an entropy concentration coefficient C=1−U. The latter is a multiplicative version of the Kullback-Leibler distance, with values between 0 and 1. For product measures and self-similar phenomena, it does not depend on the measurement level. Hence, C is an alternative to Gini’s concentration coefficient for measures with variation on different levels. Simple examples concern population density and gross domestic product. Application to time series patterns is indicated with a Markov chain. For the Covid-19 pandemic, entropy ratios indicate a homogeneous distribution of infections and the potential of local action when compared to measures for a whole region.