Associated primes of graded components of local cohomology modules

The -th local cohomology module of a finitely generated graded module over a standard positively graded commutative Noetherian ring , with respect to the irrelevant ideal , is itself graded; all its graded components are finitely generated modules over , the component of of degree . It is known that the -th component of this local cohomology module is zero for all > 0$">. This paper is concerned with the asymptotic behaviour of as .

The smallest for which such study is interesting is the finiteness dimension of relative to , defined as the least integer for which is not finitely generated. Brodmann and Hellus have shown that is constant for all (that is, in their terminology, is asymptotically stable for ). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that is a homomorphic image of a regular ring): our answer is precisely the set of contractions to of certain relevant primes of whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology.

Brodmann and Hellus raised various questions about such asymptotic behaviour when f$">. They noted that Singh's study of a particular example (in which ) shows that need not be asymptotically stable for . The second main aim of this paper is to determine, for Singh's example, quite precisely for every integer , and, thereby, answer one of the questions raised by Brodmann and Hellus.

     

Field applied (FMD) computer simulation of the frequency doubled optical Stark effect

It is shown by field applied molecular dynamics computer simulation (FMD) of liquid water that the frequency doubled optical Stark effect is accompanied by novel second order rise transients which have no known counterpart in orientational theory, based on Langevin/Kielich functions. In the statistically stationary, post transient, steady state, correlation function have been computed which are intricately dependent on the frequency of the pump laser. These transients can be obtained experimentally by modifications of optical Stark effect apparatus for measurement on a femtosecond time scale.     

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H⁴(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H⁴(BG, ℤ) to H³(G, ℤ). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.The authors acknowledge the support of the Australian Research Council. ALC thanks MPI für Mathematik in Bonn and ESI in Vienna and BLW thanks CMA of Australian National University for their hospitality during part of the writing of this paper.     

Active phase stabilization in Fourier-transform two-dimensional infrared spectroscopy

A scheme for active phase stabilization and absolute positioning in Fourier-transform two-dimensional (2D) IR spectroscopy is presented. The position accuracy is better than 20 nm(rms) corresponding to lambda/250 at 5 microm, which significantly improves the quality of Fourier-transform 2D IR spectra. Phase stabilization is added to a standard photon echo setup in a way that does not impair the flexibility of the experiment and the control over IR pulse parameters.     

A family of macrocyclic antibiotics with a mixed peptide-peptoid beta-hairpin backbone conformation

Macrocyclic peptidomimetics having a mixed peptide-peptoid backbone have been synthesized and shown to possess antibiotic activity against gram-positive and -negative bacteria with a low hemolytic activity against human erythrocytes; one is shown to adopt a regular beta-hairpin conformation by NMR in aqueous solution.