Estimates from above of certain double trigonometric sums

In this paper we consider double trigonometric sums. Expressions of this type appear in some problems of quantum chaos and number theory. We are interested in rotation numbers of bounded type. We prove a uniform linear bound on double trigonometric sums along the subsequence of denominators of the continued fraction. The proof uses elementary techniques and the analysis of cancellations in sums of certain oscillatory functions over rotations. We also include a proof of a result on discrepancy for rotations of bounded type and in the Appendix we give an elementary proof of a result by Hardy and Littlewood.     

Better <Emphasis Type="Italic">s</Emphasis>–<Emphasis Type="Italic">t</Emphasis>-tours by Gao trees

We consider the s–t-path TSP: given a finite metric space with two elements s and t, we look for a path from s to t that contains all the elements and has minimum total distance. We improve the approximation ratio for this problem from 1.599 to 1.566. Like previous algorithms, we solve the natural LP relaxation and represent an optimum solution \(x^*\) as a convex combination of spanning trees. Gao showed that there exists a spanning tree in the support of \(x^*\) that has only one edge in each narrow cut [i.e., each cut C with \(x^*(C)<2\)]. Our main theorem says that the spanning trees in the convex combination can be chosen such that many of them are such “Gao trees” simultaneously at all sufficiently narrow cuts.     

Consistency results on Newmark methods for dynamical contact problems

The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the classical Newmark method, the improved energy dissipative version due to Kane et al., and the contact-stabilized Newmark method recently suggested by Deuflhard et al. In the absence of contact, any such variant is equivalent to the Störmer–Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper.     

Total synthesis of marinomycins A-C and of their monomeric counterparts monomarinomycin A and iso-monomarinomycin A

Marinomycins A-C (1-3), and their monomeric analogues monomarinomycin A (m-1) and iso-monomarinomycin A (m-2), were synthesized by a convergent strategy from key building blocks ketophosphonate 5, aldehyde 6, and dienyl bromide carboxylic acid 7. The first attempt to construct marinomycin A [1, convertible to marinomycins B (2) and C (3) by light] by direct Suzuki-type dimerization/cyclization of boronic acid dienyl bromide 4 led to premature ring closure to afford, after global desilylation, monomarinomycin A (m-1) and iso-monomarinomycin A (m-2) in good yield and only small amounts (< or =2%) of the desired product. A subsequent stepwise approach based on Suzuki-type couplings improved considerably the overall yield of marinomycin A (1), and hence of marinomycins B (2) and C (3). Alternative direct dimerization approaches based on the Stille and Heck coupling reactions also led to monomarinomycins A (m-1 and m-2), but failed to deliver useful amounts of marinomycin A (1).     

1,4,7,10-tetraazacyclododecane metal complexes as potent promoters of phosphodiester hydrolysis under physiological conditions

Previously reported mono- and dinuclear Zn(II), Cu(II), and Ni(II) complexes of 1,4,7,10-tetrazacyclododecane ([12]aneN4 or cyclen) with different heterocyclic spacers (triazine, pyridine) of various lengths (bi- and tripyridine) or an azacrown-pendant have been tested for the hydrolysis of bis(4-nitrophenyl)phosphate (BNPP) under physiological conditions (pH 7-9, 25 degrees C). All Zn(II) complexes promote the hydrolysis of BNPP under physiological conditions, while those of Cu(II) and Ni(II) do not have a significant effect on the hydrolysis reaction. The hydrolysis kinetics in buffered solutions (0.05 M Bis/Tris, TRIS, HEPES, or CHES, I=0.1 M, NaCl) at 25 degrees C were determined by the initial slope method (product conversion<5%). Comparison of the second-order pH-independent rate constants (kBNPP, M(-1) s(-1)) for the mononuclear complexes ZnL1, ZnL3, and ZnL6, which are 6.1x10 (-5), 5.1x10(-5), and 5.7x10(-5), respectively, indicate that the heterocyclic moiety improves the rate of hydrolysis up to six times over the parent Zn([12]aneN4) complex (kBNPP=1.1x10(-5) M(-1) s(-1)). The reactive species is the Zn(II)-OH- complex, in which the Zn(II)-bound OH- acts as a nucleophile. For dinuclear complexes Zn2L2, Zn2L4, and Zn2L5, the rate of reaction is defined by the degree of cooperation between the metal centers, which is determined by the spacer length. Zn2L2 and Zn2L4 possessing shorter spacers are able to hydrolyze BNPP 1 to 2 orders of magnitudes faster than Zn2L5. The second-order rate constants k of Zn2L4 and Zn2L2 at pH 7, 8, and 9 are significantly higher than those of previously reported related complexes. The high BNPP hydrolytic activity may be related to pi-stacking and hydrophobic interactions between the aromatic spacer moieties and the substrate. Complexes Zn2L4 and Zn2L2 show hydrolytic activity at pH 7 and 8, which allows for the hydrolysis of activated phosphate esters under physiological conditions.     

Information theory for evaluating environmental classification systems

Environmental pollution data are often ranked in rule-based classification systems. These environmental data are separated in predetermined classes of a classification system for a better and smarter characterization of the state of pollution. Often the measured values are transformed, e.g. in pseudocolor maps, and can then be presented in maps. For some environmental compartments different classification systems for evaluating environmental loadings are used. Because of the dissimilarity of the various classification systems direct visual comparison is difficult. However, by means of information theory an objective comparison of these various classification systems based on their information content enables a decision to be made about which system is the most informative for objective assessment of the state of pollution. By means of the new measure multiple medium information content (multiple entropy) objective and simultaneous comparison of all channels (in an environmental classification system: pollutants) of each classification system is now possible. Furthermore the development of the state of pollution over the whole investigation period can be detected by means of information theory. On the basis of the conditions of the established rule-based systems the use of information theory enables definition of new ranges of classes in order to reach the optimum of information during conversion into the environmental classification system.