A reduction technique for weighted grouping problems

Weighted grouping problems are shown to have an equivalent reduced form, which is often considerably smaller than the original problem. Although the reduction may be small for randomly generated problems, real-life problems often contain non-random properties that greatly increase the effect of reduction. We give an efficient algorithm to build the reduced problem instance, and analyse the expected amount of reduction for certain statistical distributions and real-life data. In addition, we briefly discuss the effect of reduction on traditional solving methods of the grouping problem. The results show clearly the usefulness of problem reduction: it is computationally cheap to apply and may make the reduced problem solvable in a practical time whilst the original one is not. The method is readily applicable to the job grouping problem of printed circuit board (PCB) printing industry.     

Outcome standards and core curricula: a new orientation for mathematics teachers in Germany

One of the pivotal innovations in Germany after publication and discussion of the results of international large scale assessment studies is the introduction of national educational standards for major subjects. The 16 German federal states have committed themselves to implement these standards by developing core curricula for the use in schools and by regularly testing the achievement of students. This development is expected to have considerable impact on the outcome of education. In this article we give, a picture of the current development in the German education system by describing the design of the core curriculum in one of the federal states (NRW) and discuss its function in bridging the gap between national educational standards and teaching practice     

A note on perfect Lee-codes over small alphabets

Lee-codes, correcting e-errors, over an alphabet with q = 2^m letters are considered. It is proved that a code can be perfect only if $\text{q |(}\text{n ? 1}\text{k}\text{)}$, where n is the block length and $\text{k = [}\text{2e}\text{q}\text{]}$.     

Stacking interactions and the twist of DNA

The importance of stacking interactions for the Twist and stability of DNA is investigated using the fully ab initio van der Waals density functional (vdW-DF). Our results highlight the role that binary interactions between adjacent sets of base pairs play in defining the sequence-dependent Twists observed in high-resolution experiments. Furthermore, they demonstrate that additional stability gained by the presence of thymine is due to methyl interactions with neighboring bases, thus adding to our understanding of the mechanisms that contribute to the relative stability of DNA and RNA. Our mapping of the energy required to twist each of the 10 unique base pair steps should provide valuable information for future studies of nucleic acid stability and dynamics. The method introduced will enable the nonempirical theoretical study of significantly larger pieces of DNA or DNA/amino acid complexes than previously possible.     

Organometallic chemistry of Ga(+): formation of an unusual gallium dimer in the coordination sphere of ruthenium

New insights into the distinct organometallic chemistry of the Ga(+) ion are presented. Ga(+) reacts as a strong electrophile with the electron rich ligand trismethylene-methane (C(CH(2))(3) (2-)) attached at Ru by insertion into a Ru--C bond. The resulting "gallamethylallyl" ligand behaves like strong nucleophile similar to known monovalent GaR species. This donor property leads to the dimeric structure of the product [{Ru(GaCp*)(3)[eta(3)-(CH(2))(2)C{CH(2)(mu-Ga)}]}(2)][(BAr(F))(2)] (4) (Cp*=C(5)Me(5), [BAr(F)]=[B{C(6)H(3)(CF(3))(2)}(4)]). Very unexpectedly, the two gallium ligands in this dimer are found in close vicinity to each other with a distance in the range of Ga--Ga bonds. Indeed, AIM calculations confirm a weak attractive closed shell Ga--Ga interaction. Finally, a novel example of a complex with substituent-free Ga(+) as a ligand was found in the compound [Ru(PCy(3))(2)(GaCp*)(2)(Ga)][BAr(F)] (6) (Cy=C(6)H(11), cyclohexyl), the very short Ru--Ga bond length confirming the assumption that Ga(+) represents a pure sigma/pi-accepting ligand in this case.     

Molecular dynamics simulations of the interactions between platinum clusters and carbon platelets

Molecular dynamics simulations have been performed with two reactive force fields to investigate the structure of a Pt100 cluster adsorbed on the three distinct sides of a carbon platelet. A revised Reax force field for the carbon-platinum system is presented. In the simulations, carbon platelet edges both with and without hydrogen termination have been studied. It is found that the initial mismatch between the atomic structure of the platelet egde and the adsorbed face of the Pt100 cluster leads to a desorption of a few platinum atoms from the cluster and the subsequent restructuring of the cluster. Consequently, the average Pt-Pt bond length is enlarged in agreement with experimental results. This change in the bond length is supposed to play an important role in the enhancement of the catalytic activity, which is demonstrated by studying the changes in the bond order of the platinum atoms. We found an overall shift to lower values as well as a loss of the well-defined peak structure in the bond-order distribution.