Numerical solution of matrix interpolation problems

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Function spaces defined by means of oscillation, with application

In the mathematical literature, a plethora of different meanings and formal definitions have been associated to the word "oscillation". In this text we will explore some of the function spaces defined by means of oscillation, in many different senses of the word, fitting into two modes: spaces in which oscillation is bounded and spaces in which oscillation is vanishing, i.e. arbitrarily small when measured on a sufficiently small set. This general framework will be made precise in a diversity of ways. The outline of the thesis is the following. In the first chapter a very large space of functions introduced by Brezis, Bourgain and Mironescu in 2015 and often denoted as $B$ is introduced. In Chapter 2 we explore the space of Lipschitz functions, in the most general setting of an arbitrary compact metric space and together with other function spaces from the family of Holder spaces. The last section of the Chapter 2 is devoted to some results obtained in the field of Optimal Control Theory. In Chapter $3$ we introduce Orlicz spaces for any choice of a Young function $\Psi$ and the closure of $L^\infty$ in $L^\Psi$, also known as the Morse space $M^\Psi$. We individuate a large subfamily of Orlicz spaces for which $(L^\Psi,M^\Psi)$ fits into a mathematical framework by K.M. Perfekt, deriving many functional properties of the couple.\\ In particular, in the last section of this chapter, we discuss some applications to the regularity of minima of some functionals in the calculus of variations. Chapters 4 and 5 are dedicated to the spaces $BMO$ and $VMO$ and their respective subcones $BLO$ and $VLO$, concluding the text with discussion of what are arguably the most natural "children" of the very large space $B$ discussed at the beginning

Gender differences in mathematics performance. Analysis of attainment and attitudes in mathematics of girls and boys; detailed appraisal of theories and pressures that influence girls' underachievement and underparticipation in the subject.

Statistics show that boys perform better in mathematics tests than girls. In order to make a refined assessment of the magnitude of gender differences in mathematics performance, a study was made of one thousand 16+ mathematics scripts to find the precise topics on which girls and boys differ significantly in performance. These concepts were found to be concerned with scale or ratio, spatial problems, space-time relationships and probability questions. Differences were found in performance between girls and boys at each ten-percentile level through the ability range. A longitudinal study also revealed differences in mathematics 'performance through the years of secondary education. There is no convincing evidence that the discrepancy can be accounted for by innate or genetic reasons. Intervention programmes have been found to improve the performance of girls in the weak areas of spatial awareness, proportionality and problem solving. In addition, a study was made of gender attitudes towards mathematics. Ten secondary schools were surveyed and the results revealed a marked decrease in the attitudes of third and fourth form girls. During these difficult adolescent years girls and boys are susceptible to strong internal and external pressures. Corresponding differences were also found across the ability range. These social pressures are concerned with teacher influence, social interaction, type of grouping, sex stereotyping, choices, teaching materials and careers advice

Harmonic approximation and Sarason's-type theorem

Hansen W, Netuka I. Harmonic approximation and Sarason's-type theorem. Journal of Approximation Theory. 2003;120(1):183-190.In this paper uniform approximation of bounded harmonic functions on an arbitrary open set in Euclidean space by harmonic functions arising as solutions of the classical or generalized Dirichlet problem is studied. In particular, an analogue of Sarason's H-infinity + C theorem (known from the theory of algebras of analytic functions) is established for harmonic functions. (C) 2002 Elsevier Science (USA). All rights reserved