Some recent developments in quantization of fractal measures

We give an overview on the quantization problem for fractal measures, including some related results and methods which have been developed in the last decades. Based on the work of Graf and Luschgy, we propose a three-step procedure to estimate the quantization errors. We survey some recent progress, which makes use of this procedure, including the quantization for self-affine measures, Markov-type measures on graph-directed fractals, and product measures on multiscale Moran sets. Several open problems are mentioned.Comment: 13 page

Matrix Elements of Four-quark Operators Relevant to Lifetime Difference \Delta \Gamma_{B_s} from QCD Sum Rules

We extract the matrix elements of four quark operators O_{L,S} relevant to the B_s and \bar B_s life time difference from QCD sum rules. We find the vacuum saturation approximation works reasonably well, i.e., within 10%. We discuss the implications of our results and compare them with the recent lattice QCD determination.Comment: 11 pages, 4 eps figures, Latex. Some reference renewe

Effect of unitary impurities in non-STM-types of tunneling in high-T_c superconductors

Based on an extended Hubbard model, we present calculations of both the local (i.e., single-site) and spatially-averaged differential tunneling conductance in d-wave superconductors containing nonmagnetic impurities in the unitary limit. Our results show that a random distribution of unitary impurities of any concentration can at most give rise to a finite zero-bias conductance (with no peak there) in spatially-averaged non-STM type of tunneling, in spite of the fact that local tunneling in the immediate vicinity of an isolated impurity does show a conductance peak at zero bias, whereas to give rise to even a small zero-bias conductance peak in the former type of tunneling the impurities must form dimers, trimers, etc. along the [110] directions. In addition, we find that the most-recently-observed novel pattern of the tunneling conductance around a single impurity by Pan et al. [Nature (London) 403,746 (2000)] can be explained in terms of a realistic model of the tunneling configuration which gives rise to the experimental results reported there. The key feature in this model is the blocking effect of the BiO and SrO layers which exist between the tunneling tip and the CuO_2 layer being probed.Comment: 9 pages, 7 ps-figures, to appear in Phys. Rev. B (Sep. 1, 2000); typos corrected, references added, figure 6 changed to expand the explanation on recent experimental measurements by S.H. Pan et al. [Nature (London) 403, 746 (2000)