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Let A be a nonlinear differential operator on an open set X in R^n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A (u) = 0 in the complement of S of class F satisfies this equation weakly in all of X. For the most extensively studied classes F we show conditions on S which guarantee that S is removable for F relative to A.
We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.
Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.
We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.
Inflammation in Cachexia
(2015)
Extract: Topics in psycholinguistics and the neurocognition of language rarely attract the attention of journalists or the general public. One topic that has done so, however, is the potential benefits of bilingualism for general cognitive functioning and development, and as a precaution against cognitive decline in old age. Sensational claims have been made in the public domain, mostly by journalists and politicians. Recently (September 4, 2014) The Guardian reported that “learning a foreign language can increase the size of your brain”, and Michael Gove, the UK's previous Education Secretary, noted in an interview with The Guardian (September 30, 2011) that “learning languages makes you smarter”. The present issue of BLC addresses these topics by providing a state-of-the-art overview of theoretical and experimental research on the role of bilingualism for cognition in children and adults.