Ergodic theorems along sequences and Hardy fields.

Let a(x) be a real function with a regular growth as x --> infinity. [The precise technical assumption is that a(x) belongs to a Hardy field.] We establish sufficient growth conditions on a(x) so that the sequence ([a(n)])(infinity)(n=1) is a good averaging sequence in L2 for the pointwise ergodic theorem. A sequence (an) of positive integers is a good averaging sequence in L2 for the pointwise ergodic theorem if in any dynamical system (Omega, Sigma, m, T) for f [symbol, see text] in L2(Omega) the averages [equation, see text] converge for almost every omicron in. Our result implies that sequences like ([ndelta]), where delta > 1 and not an integer, ([n log n]), and ([n2/log n]) are good averaging sequences for L2. In fact, all the sequences we examine will turn out to be good averaging for Lp, p > 1; and even for L log L. We will also establish necessary and sufficient growth conditions on a(x) so that the sequence ([a(n)]) is good averaging for mean convergence. Note that for some a(x) (e.g., a(x) = log2 x), ([a(n)]) may be good for mean convergence without being good for pointwise convergence.

Binding site size limit of the 2:1 pyrrole-imidazole polyamide-DNA motif.

Polyamides containing N-methylimidazole (Im) and N-methylpyrrole (Py) amino acids can be combined in antiparallel side-by-side dimeric complexes for sequence-specific recognition in the minor groove of DNA. Six polyamides containing three to eight rings bind DNA sites 5-10 bp in length, respectively. Quantitative DNase I footprint titration experiments demonstrate that affinity maximizes and is similar at ring sizes of five, six, and seven. Sequence specificity decreases as the length of the polyamides increases beyond five rings. These results provide useful guidelines for the design of new polyamides that bind longer DNA sites with enhanced affinity and specificity.

Coordination of aproximations in secondary school students’ understanding of limit concept

The aim of this research is to characterize the coordination of the processes of approximation related to the understanding of the limit of a function. We analyze the answers of 64 post-secondary school students to 7 problems considering the dynamic and metric conception of limit of a function. Results indicate that the metric understanding of the limit in terms of inequality supports that the student is capable of coordinating the approximations in the domain and in the range when lateral approximations coincide. However, the student is not capable of this coordination when lateral approximations do not coincide. This indicates that the metric understanding of the limit begins with the previous construction of the dynamic conception in case of coincidence of the lateral approximations in the range.

Of the invention of theorems : manuscript, [17--]

Mathematical notes, equations, theorems, and definitions. Probablynot by William Winthrop (Harvard AB 1770) as it is different handwriting and language habits from other of his manuscript.

Cameron's proposals to limit EU citizens' access to the UK: lawful or not, under EU rules? CEPS Commentary, 29 November 2013

This Commentary briefly examines each of Prime Minister Cameron's headline proposals to limit EU citizens’ access to social benefits. Are these proposals in breach of the UK’s obligations under EU law? The paper also makes some general comments about the end of the transitional arrangements for Bulgarian and Romanian work-seekers, and ‘benefits tourism’ in general.