From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results. On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput's ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter's $(t,s)$-sequences

Tractability of multivariate analytic problems

In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$ and $\varepsilon^{-1}$ function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with $\varepsilon^{-1}$ replaced by $1+\log \varepsilon^{-1}$. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of $s$ and $1+\log \varepsilon^{-1}$. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions

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Transcriptional regulation of prostate kallikrein-like genes by androgen.

Using gene-specific synthetic oligonucleotides the expression and regulation of kallikrein-like genes in the human prostatic cancer cell line LNCaP were studied. Prostate-specific antigen (PSA) and human glandular kallikrein (hGK-1) together constitute a subfamily of serine proteases exclusively produced in the human prostate. RNA analysis revealed that both genes are expressed in LNCaP cells with PSA basal levels being 2-fold higher than hGK-1 levels. Both mRNAs are induced over a period of 24 h in the presence of 3.3 nM of the synthetic androgen mibolerone. Stimulation of PSA RNA is about 5- fold,whereas hGK-1 stimulation is less pronounced. Nuclear run-on analysis revealed that androgen induction of kallikrein-like genes in LNCaP cells is a rapid event (c3 h) occurring at the level of transcription initiation. Treatment of cells with cycloheximide demonstrates that, while PSA/hGK-1 basal transcription strictly depends on continuous protein synthesis, transcriptional induction by androgen does not. This suggests the direct involvement of the androgen receptor in the induction process independent of additional labile protein factors necessary for kallikrein basal transcription. A binding motif is present in the PSA and hGK-1 promoters, closely resembling the consensus sequence for steroidresponsive elements. The androgen antagonist cyproterone acetate was also able to stimulate transcription of kallikrein-like genes in LNCaP cells. In contrast, androgen-dependent transcriptional suppression of the protooncogene c-myc was strongly counteracted by cyproterone acetate. Thus, antiandrogens act differentially on androgen-regulated prostate-specific (PSA, hGK-1) and growthrelated (c-myc) gene expression in LNCaP cells