Kriterien in der Theorie der Gleichverteilung

It is the aim of this paper to introduce two new notions of discrepancy. They are defined by the formulas $$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z e^2 \pi i\omega \left( n \right)} \right)} - f\left( 0 \right)} \right|, and \hfill \\ \delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z \omega \left( n \right)} \right)} \cdot z - \int\limits_0^z {f\left( \zeta \right)d\zeta } } \right|, \hfill \\ \end{gathered} $$ wheref is a holomorphic function defined in the unit disc withf ^(k) (0)≠0 for allk∈?,r<1 is a positive number, and ω is a sequence in [0, 1]. The first of these discrepancies can be generalized for multidimensional sequences. ω is uniform distributed if and only if lim _N→∞ Δ _N ^r (ω;f)=0 resp. lim _N→∞δ _N ^r (ω;f)=0. These results are proved in a quantitative way by estimating the classical discrepancyD _N (ω) by means ofΔ _N ^r (ω;f) and δ _N ^r (ω;f): $$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Phi \left( {\Delta _N^r \left( {\omega ;f} \right)} \right), \hfill \\ \delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Psi \left( {\delta _N^r \left( {\omega ;f} \right)} \right). \hfill \\ \end{gathered} $$ The functions Φ and Ψ only depend onf andr. These estimations are based on the inequalities ofKoksma-Hlawka andErdös-Turán.     

Sulfide ion transfer from a vicinal dithiolate. A new route for the synthesis of mesocyclic trithiaethers

Four mesocyclic and one macrocyclic trithiaethers have been synthesized using a sulfide ion transfer from a vicinal dithiolate. The mechanism of the sulfide ion transfer is discussed. An improved method for synthesizing 2-hydroxymethyl-1,4,8,11-tetrathiacyclotetradecane is also reported.     

Preparative production of 5-desoxy-L-arabinose, xylitol and D-ribose from "diacetone glucose"]

Simple, reproducible preparative methods are described for the production of the title compounds from diacetone glucose.     

Solution of sparse quasi-square rectangular systems by Gaussian elimination

We present a general method for the linear least-squares solutionof overdetermined and underdetermined systems. The method isparticularly efficient when the coefficient matrix is quasi-square,that is when the number of rows and number of columns is almostthe same. The numerical methods for linear least-squares problemsand minimum-norm solutions do not generally take account ofthis special characteristic. The proposed method is based onLU factorization of the original quasi-square matrix A, assumingthat A has full rank. In the overdetermined case, the LU factorsare used to compute a basis for the null space of A^T. The right-handside vector b is then projected onto this subspace and the least-squaressolution is obtained from the solution of this reduced problem.In the case of underdetermined systems, the desired solutionis again obtained through the solution of a reduced system.The use of this method may lead to important savings in computationaltime for both dense and sparse matrices. It is also shown inthe paper that, even in cases where the matrices are quite small,sparse solvers perform better than dense solvers. Some practicalexamples that illustrate the use of the method are included.     

Photochemical patterning of a self-assembled monolayer of 7-diazomethylcarbonyl-2,4,9-trithiaadmantane on gold films via Wolff rearrangement

Photolithographic attachment of functional organic molecules via ester or amide linkages to self-assembled monolayers (SAMs) on gold thin films was achieved by employing a novel photoreactive surface anchor, 7-diazomethylcarbonyl-2,4,9-trithiaadmantane. The photoreactive SAM was prepared by the spontaneous physical adsorption of the photoreactive surface anchor onto gold surfaces. The alpha-diazo ketone moiety of the SAM was found to display the classical Wolff rearrangement reactivity to produce a ketene intermediate on the exposed area. Organic molecules such as alcohols and amines can thus be attached to the gold surfaces selectively by the facile in situ formation of ester or amide linkages. The structure and reactivity of the photoreactive surface anchor were characterized by real-time FT-IR, fluorescence, and polarization modulation infrared reflectance absorption spectroscopy (PM-IRRAS). The Wolff rearrangement reactivity of the SAM suggested that a "surface-isolated" carbonylcarbene may be generated when the SAM was exposed to 255-nm irradiation.     

The swap of integral and limit in constructive mathematics

Integration within constructive, especially intuitionistic mathematics in the sense of L. E. J. Brouwer, slightly differs from formal integration theories: Some classical results, especially Lebesgue's dominated convergence theorem, have tobe substituted by appropriate alternatives. Although there exist sophisticated, but rather laborious proposals, e.g. by E. Bishop and D. S. Bridges (cf. [2]), the reference to partitions and the Riemann‐integral, also with regard to the results obtained by R. Henstock and J. Kurzweil (cf. [9], [12]), seems to give a better direction. Especially, convergence theorems can be proved by introducing the concept of “equi‐integrability”. The paper is strongly motivated by Brouwer's result that each function fully defined on a compact interval has necessarily to be uniformly continuous. Nevertheless, there are, with only one exception (a corollary of Theorem 4.2), no references to the fan‐theorem or to bar‐induction. Therefore, the whole paper can be read within the setting of Bishop's access to constructive mathematics. Nothing of genuine full‐fledged Brouwerian intuitionism is used for the main results in this note (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ein metrischer Satz derC-Gleichverteilung

A metric theorem onC-uniform distribution will be proved and some applications will be mentioned, e. g. theC-uniform distribution for (in a certain sense) almost all continuously differentiable functions.     

Gleichverteilte Doppelfolgen und eine Abschätzung der C-Diskrepanz

In this paper a criterion forC-uniformly distributed differentiable functions is given by using uniformly distributed double sequences. This criterion allows to find lower bounds of theC-discrepancy of differentiable functions.