Narrow Results

This paper is concerned with an optimization problem related to the pseudo p-Laplacian eigenproblem, with Robin boundary conditions. The principal eigenvalue is minimized over a rearrangement class generated by a fixed positive function. Existence and optimality condition are proved. The popular case where the generator is a characteristic function is also considered. In this case the method of domain derivative is used to capture qualitative features of the optimal solutions.

A time-optimal procedure to transpose in situ a matrix stored over a distributed memory 2-dimensional mesh-connected parallel computer is shown. The matrix need not be square. Only nearest-neighbor blocking communications is used, and a small bounded buffer space is required.

Let Ω be an open domain in ℝⁿ, let k∈ℕ, . Using a natural extension of the L(p, q) spaces and a new form of the Pólya–Szegö symmetrization principle, we extend the sharp version of the Sobolev embedding theorem: to the limiting value . This result extends a recent result in [3] for the case k=1. More generally, if Y is a r.i. space satisfying some mild conditions, it is shown that . Moreover Y_n(∞,k) is not larger (and in many cases essentially smaller) than any r.i. space X(Ω) such that . This result extends, complements, simplifies and sharpens recent results in [10].

We show that the symmetric radial decreasing rearrangement can increase the fractional Gagliardo semi-norm in domains.

In this paper, by applying the well-known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.

The conversion of glycerol to acrolein is an undesirable event in whisky production, caused by infection of the broth with Klebsiella pneumoniae. This organism uses glycerol dehydratase to transform glycerol into 3-hydroxypropanal, which affords acrolein on distillation. The enzyme requires adenosylcobalamin (coenzyme B₁₂) as cofactor and a monovalent cation (e.g. K⁺). Diol dehydratase is a similar enzyme that converts 1,2-diols (C₂-C₄) including glycerol into an aldehyde and water. The subtle stereochemical features of these enzymes are exemplified by propane-1,2-diol: both enantiomers are substrates but different hydrogen and oxygen atoms are abstracted. The mechanism of action of the dehydratases has been elucidated by protein crystallography and ab initio molecular orbital calculations, aided by stereochemical and model studies. The 5'-deoxyadenosyl (adenosyl) radical from homolysis of the coenzyme's Co-C σ-bond abstracts a specific hydrogen atom from C-1 of diol substrate giving a substrate radical that rearranges to a product radical by 1,2-shift of hydroxyl from C-2 to C-1. The rearrangement mechanism involves an acid-base 'push-pull' in which migration of OH is facilitated by partial protonation by Hisα143, synergistically assisted by partial deprotonation of the non-migrating (C-1) OH by the carboxylate of Gluα170. The active site K⁺ ion holds the two hydroxyl groups in the correct conformation, whilst not significantly contributing to catalysis. Recently, diol dehydratases not dependent on coenzyme B₁₂ have been discovered. These enzymes utilize the same kind of diol radical chemistry as the coenzyme B₁₂-dependent enzymes and they also use the adenosyl radical as initiator, but this is generated from S-adenosylmethionine.