Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

A sharp exponential inequality for Lorentz-Sobolev spaces on bounded domains

This paper generalizes an inequality of Moser from the case that is in the Lebesgue space to certain subspaces, namely the Lorentz spaces , where . The conclusion is that is integrable, where . This is a higher degree of integrability than in the Moser inequality when . A formula for is given and it is also shown that no larger value of works.

     

A new bound for the smallest

Let denote the number of primes and let denote the usual integral logarithm of . We prove that there are at least integer values of in the vicinity of with . This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of , where appears to exceed by more than . The plots strongly suggest, although upper bounds derived to date for are not sufficient for a proof, that exceeds for at least integers in the vicinity of . If it is possible to improve our bound for by finding a sign change before , our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of and find that as departs from the region in the vicinity of , the density is , and that it varies from this by no more than over the next integers. This should be compared to Rubinstein and Sarnak.

     

HOW DOES THE WAY IN WHICH INDIVIDUAL STUDENTS BEHAVE AFFECT THE SHARED CONSTRUCTION OF MEANING?

Audio taped discussions between three students have been examined to shed light on the way in which the behaviour of individual students may affect the shared construction of meaning with graphical calculators. These discussions revealed a complex pattern of interaction between the students. Each student was responsible for defining his or her own role within the discourse and these roles appeared to change as the discussion progressed. With reference to the framework offered by Winbourne and Watson (1998), it is proposed that local communities of practice have been established and that the individual student's positioning within the community of practice determines their success as a learner and contributes towards the creation of shared knowledge.     

Controlled modulation of mesoporosities in metal (M = Al, Fe) phosphate solids

Mesoporous solids which possess average pore diameters between 7 and 20 nm, depending on the composition, have been prepared. The solids have the general formula Al₁₀₀P_χM₂₀ where M = Al or Fe, and χ = 0, 4.5, 9, 18, 36, 72 or 144. The initial addition of phosphorus as phosphate transforms the originally crystalline oxide/oxides into amorphous solids. These amorphous materials possess a narrow pore size distribution: 80–90% of the pores lie within 1–2 nm of the average pore diameter. Subsequent incremental amounts of phosphorus transform the material into a crystalline solid whilst the pore size distribution becomes much wider and the maximum moves towards larger pore diameters. Substitution of 20% of the aluminium by iron results, at a low phosphorus content, in pores with smaller pore volumes and smaller surface areas.

The data in the dV_p/dD_p = > D_p) graphs, where V_p is the incremental pore volume and D_p is the average pore diameter, can be approximated using an admixture of Gaussian and Lorentzian curves. For low phosphorus contents the dV_p/dD_p = (D_p) curves have a mainly Gaussian profile but the gradual addition of phosphorus transforms them to Lorentzian-type curves. An attempt to approximate the histograms dV_p =(D_p) with the minimum number of distribution curves made up of the corresponding Gaussian and Lorentzian components indicates that each successive addition of phosphorus creates a dominant new pore component at a larger pore diameter. At the same time, the components at smaller pore diameters are diminished and eventually disappear as more phosphorus is added.     

Use of orthoesters in the synthesis of meso-substituted porphyrins

The use of two orthoesters, trimethyl orthoacetate and trimethyl orthobenzoate, in the synthesis of porphyrins from 5-phenyldipyrromethanes is described. Previously unreported 5,15-diphenyl-10,20-dimethyl porphyrins can be accessed conveniently by this route. A relationship between the steric bulk of the orthoester and the amount of scrambling of the porphyrin products has been found. Strong electron-withdrawing substituents on the phenyl ring of the dipyrromethane also enhance scrambling.     

Heliuim-3 mass spectrometry for low-level tritium analysis of environmental samples

Helium-3 (³He) mass spectrometry for the analysis of low-level tritium (³H) concentrations (0.5 to 5 Bq·l^–1) in environmental sample matrices was compared with conventional low-level -decay counting methods. The mass-spectrometry method compared favorably, equalling or surpassing conventional decay-counting methods with respect to most criteria. Additional research and method refinements may make³He mass spectrometry the method of choice for routine, low-level to very-low-level (L<0.5 Bq·l^–1)³H measurements in a wide variety of environmental samples in the future.