Apportionment methods

Most democratic countries use apportionment methods to transform election results into whole numbers, which usually give the number of seats in a legislative body that the parties obtained. Which apportionment method does this best can be specified by measuring the error between the allocated result and the ideal proportion. We show how to find an apportionment method which is best suited to a given error function. We also discuss several properties of apportionment methods that have been labelled paradoxa. In particular we explain the highly publicised “Alabama” Paradox for the Hare/Hamilton method and show that other popular apportionment methods come with their very own paradoxa.     

Local 2-Geodesic Transitivity of Graphs

An s-geodesic in a graph Γ is a path connecting two vertices at distance s. Being locally transitive on s-geodesics is not a monotone property: if an automorphism group G of a graph Γ is locally transitive on s-geodesics, it does not follow that G is locally transitive on shorter geodesics. In this paper, we characterise all graphs that are locally transitive on 2-geodesics, but not locally transitive on 1-geodesics.     

Characterization of Commercial Amylases for the Removal of Filter Cake on Petroleum Wells

Drilling fluid has many functions, such as carry cuttings from the hole permitting their separation at the surface, cool and clean the bit, reduce friction between the drill pipe and wellbore, maintain the stability of the wellbore, and prevent the inflow of fluids from the wellbore and form a thin, low-permeable filter cake. Filter cake removal is an important step concerning both production and injection in wells, mainly concerning horizontal completion. The drilling fluids are typically comprised of starch, the most important component of the filter cake. A common approach to remove this filter cake is the use of acid solutions. However, these are non-specific reactants. A possible alternative is the use of enzymatic preparations, like amylases, that are able to hydrolyze starch. Wells usually operate in drastic conditions for enzymatic preparations, such as high temperature, high salt concentration, and high pressure. Thus, the main objective of this work was to characterize four enzymatic preparations for filter cake removal under open hole conditions. The results showed that high salt concentrations (204,000 ppm NaCl) in completion fluid decreased amylolytic activity. All enzymatic preparations were able to catalyze starch hydrolysis at all temperatures tested (30, 65, 80, and 95 °C). An increase of amylolytic activity was observed with the increase of pressure (100, 500 and 1,000 psi) for one commercial amylase.     

Molecular dynamics simulations and free energy calculations on the enzyme 4‐hydroxyphenylpyruvate dioxygenase

4‐Hydroxyphenylpyruvate dioxygenase is a relevant target in both pharmaceutical and agricultural research. We report on molecular dynamics simulations and free energy calculations on this enzyme, in complex with 12 inhibitors for which experimental affinities were determined. We applied the thermodynamic integration approach and the more efficient one‐step perturbation. Even though simulations seem well converged and both methods show excellent agreement between them, the correlation with the experimental values remains poor. We investigate the effect of slight modifications on the charge distribution of these highly conjugated systems and find that accurate models can be obtained when using improved force field parameters. This study gives insight into the applicability of free energy methods and current limitations in force field parameterization. © 2011 Wiley Periodicals, Inc. J Comput Chem 2011     

Dipolar Rotors Orderly Aligned in Mesoporous Fluorinated Organosilica Architectures

New mesoporous covalent frameworks, based on hybrid fluorinated organosilicas, were prepared to realize a periodic architecture of fast molecular rotors containing dynamic dipoles in their structure. The mobile elements, designed on the basis of fluorinated p‐divinylbenzene moieties, were integrated into the robust covalent structure through siloxane bonds, and showed not only the rapid dynamics of the aromatic rings (ca. 10⁸ Hz at 325 K), as detected by solid‐state NMR spectroscopy, but also a dielectric response typical of a fast dipole reorientation under the stimuli of an applied electric field. Furthermore, the mesochannels are open and accessible to diffusing in gas molecules, and rotor mobility could be individually regulated by I₂ vapors. The iodine enters the channels of the periodic structure and reacts with the pivotal double bonds of the divinyl‐fluoro‐phenylene rotors, affecting their motion and the dielectric properties.     

Linking Deltahedral Zintl Clusters with Conjugated Organic Building Blocks: Synthesis and Characterization of the Zintl Triad [R‐Ge9‐CHCH?CHCH‐Ge9‐R]4−

The accessibility of triads with deltahedral Zintl clusters in analogy to fullerene–linker–fullerene triads is another example for the close relationship between fullerenes and Zintl clusters. The compound {[K(2.2.2‐crypt)]₄[RGe₉‐CHCH CHCH‐Ge₉R]}(toluene)₂ (R=(2Z,4E)‐7‐amino‐5‐aza‐hepta‐2,4‐dien‐2‐yl), containing two deltahedral [Ge₉] clusters linked by a conjugated (1Z,3Z)‐buta‐1,3‐dien‐1,4‐diyl bridge, was synthesized through the reaction of 1,4‐bis(trimethylsilyl)butadiyne with K₄Ge₉ in ethylenediamine and crystallized after the addition of 2.2.2‐cryptand and toluene. The compound was characterized by single‐crystal structure analysis as well asNMR and IR spectroscopy.     

The crank moments weighted by the parity of cranks

In this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\). When \(k=0\), the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\), where \(M_e(n)\) (resp. \(M_o(n)\)) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\).     

Solid supported chemical syntheses of both components of the lantibiotic lacticin 3147

Lantibiotics are antimicrobial peptides produced by bacteria. Some are employed for food preservation, whereas others have therapeutic potential due to their activity against organisms resistant to current antibiotics. They are ribosomally synthesized and posttranslationally modified by dehydration of serine and threonine residues followed by attack of thiols of cysteines to form monosulfide lanthionine and methyllanthionine rings, respectively. Chemical synthesis of peptide analogues is a powerful method to verify stereochemistry and access structure-activity relationships. However, solid supported synthesis of lantibiotics has been difficult due to problems in generating lanthionines and methyllanthionines with orthogonal protection and good stereochemical control. We report the solid-phase syntheses of both peptides of a two-component lantibiotic, lacticin 3147. Both successive and interlocking ring systems were synthesized on-resin, thereby providing a general methodology for this family of natural products.