Entropy and the fourth moment phenomenon

We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijn?s formula of information theory. When applied to sequences of functionals of a general Gaussian field, our results can be combined with the Carbery–Wright inequality in order to yield multidimensional entropic rates of convergence that coincide, up to a logarithmic factor, with those achievable in smooth distances (such as the 1-Wasserstein distance). In particular, our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components, with explicit rates of convergence in total variation that are independent of the order of the associated Wiener chaoses. The results proved in the present paper are outside the scope of other existing techniques, such as for instance the multidimensional Stein?s method for normal approximations.     

Bergeʼs maximum theorem for noncompact image sets

This note generalizes Berge?s maximum theorem to noncompact image sets. It also clarifies the results from Feinberg, Kasyanov and Zadoianchuk (2013) [7] on the extension to noncompact image sets of another Berge?s theorem, that states semi-continuity of value functions. Here we explain that the notion of a K$\mathbb{K}$-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a KN$\mathbb{K}\mathbb{N}$-inf-compact function (N$\mathbb{N}$ stands for “nets” in K$\mathbb{K}$-inf-compactness), which coincides with K$\mathbb{K}$-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.     

L p − L q Decay Estimates for the Wave Equations with Exponentially Growing Speed of Propagation

We find L p m L q decay estimates for a model hyperbolic equation with coefficient depending on time and having exponential growth as t M X . The approach in the article is based on a representation of a fundamental solution by means of confluent hypergeometric functions.     

Analysis of an eddy current and micromagnetic model

We consider a coupled eddy current and micromagnetic model describing the behaviour of dynamic electromagnetic phenomena in applications such as disk write heads. We first prove the existence of a weak solution to this nonlinear problem. Then we outline a numerical time-stepping scheme. Since the numerical method requires a nonstandard mixed boundary value eddy current problem to be solved at each time step, we show the existence and uniqueness of a solution for the corresponding eddy current problem. This is accomplished using an image principle and the verification of a suitable Babu?ka–Brezzi condition.     

High school students’ perceptions of geometrical proofs proving and refuting geometrical claims of the ‘for every … ’ and ‘there exists’ type

Proving and refuting mathematical claims constitute a significant element in the development of deductive thinking. These issues are mainly studied during geometry lessons and very little (if at all) in lessons of other mathematical disciplines. This study deals with high school students’ perceptions of proofs in the geometry. The study explores whether students know when to use a deductive proof and when an example is sufficient for proving or refuting geometrical claims. The findings indicate that in cases of simple claims, the students corroborate them by using a deductive proof. However, when the claim is more complex, the students tend to present both a proof and an example. Moreover, they are unsure whether using an example can constitute a method for proving a mathematical claim, believing that in mathematics everything must be proven. They believe that examples are used merely for illustration purposes rather than as a means of convincing. The research conclusions support the need for deepening and developing the students’ distinction between cases where examples are insufficient and cases where an example is sufficient for proving a claim.     

An Andô-Douglas type theorem in Riesz spaces with a conditional expectation

In this paper we formulate and prove analogues of the Hahn-Jordan decomposition and an Andô-Douglas-Radon-Nikodým theorem in Dedekind complete Riesz spaces with a weak order unit, in the presence of a Riesz space conditional expectation operator. As a consequence we can characterize those subspaces of the Riesz space which are ranges of conditional expectation operators commuting with the given conditional expectation operators and which have a larger range space. This provides the first step towards a formulation of Markov processes on Riesz spaces.