Minimum-weight cycle covers and their approximability

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L⊆N.We investigate how well L-cycle covers of minimum weight can be approximated. For undirected graphs, we devise non-constructive polynomial-time approximation algorithms that achieve constant approximation ratios for all sets L. On the other hand, we prove that the problem cannot be approximated with a factor of 2−ε for certain sets L.For directed graphs, we devise non-constructive polynomial-time approximation algorithms that achieve approximation ratios of O(n), where n is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated with a factor of o(n) for certain sets L.To contrast the results for cycle covers of minimum weight, we show that the problem of computing L-cycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.     

Zeta functions of a class of elliptic curves over a rational function field of characteristic two

We show how to calculate the zeta functions and the orders of Tate-Shafarevich groups of the elliptic curves with equation over the rational function field , where is a power of 2. In the range , , odd of degree , the largest values obtained for are (one case), (one case) and (three cases). We observe and discuss a remarkable pattern for the distributions of signs in the functional equation and of fudge factors at places of bad reduction. These imply strong restrictions on the precise form of the Langlands correspondence for GL over local or global fields of characteristic two.

     

Chemistry and Biochemistry of Microbial α-Glucosidase Inhibitors

α-Glucosidases are among the most important carbohydrate-splitting enzymes. They catalyze the hydrolysis of α-glucosidic linkages. Their substrates are—depending on their specificity—oligo- and polysaccharides. Microbial inhibitors of α-amylases and other mammalian intestinal carbohydrate-splitting enzymes studied during the last few years have aroused medical interest in the treatment of metabolic diseases such as diabetes. Moreover, they extend the spectrum of microbial secondary metabolites which comprises an enormous variety of structures. They also contribute considerably to a better understanding of the mechanism of action of α-glucosidases. These inhibitors belong to different classes of substances. Those studied most thoroughly are microbial α-glucosidase inhibitors which are members of a homologous series of pseudooligosaccharides of the general formula (4). They all have a core in common which is essential for their inhibitory action, a pseudodisaccharide residue consisting of an unsaturated cyclitol unit, and a 4-amino-4,6-dideoxy- glucose unit. The—in many respects—most interesting representative of this homologous series is acarbose (5), a pseudotetrasaccharide exhibiting a very pronounced inhibitory effect on intestinal α-glucosidases such as sucrase, maltase and glucoamylase. The present paper will review this new field of microbial α-glucosidase inhibitors which has been studied with particular intensity during the past ten years.     

Synthesis and reactions of 6‐methylsulfanyl‐2,4‐dithioxo‐1,2,3,4‐tetrahydropyrimidin‐5‐carbonitrile

The synthesis of 6‐methylsulfanyl‐2,4‐dithioxo‐1,2,3,4‐tetrahydropyrimidine‐5‐carbonitrile 4 is described. Compound 4 was reacted with various alkylants. The reaction with chloroacetic acid derivatives results in the formation of thieno[2,3‐d]pyrimidines 8 . When methyl iodide was used 2,4,6‐tris(methylsul‐fanyl)pyrimidine‐5‐carbonitrile 5 was obtained. The substitution of the methylsulfanyl groups in compound 5 by several N‐nuclophiles leads to amino substituted pyrimidines.     

Synthése par deux voies différentes d'un systéme dihétérocyclique apparenté aux peltogynoides

5-Oxo-5H,7H-[2]benzopyrano[4,3-b][1]benzopyran ( 2 ) has been synthesized from 3-(o-hydroxybenzylidene)isochroman-1,4-dione ( 3 ) by reductive cyclization, and from 5H,7H[2]benzo-pyrano[4,3-b][1]benzopyran-5,7-dione ( 4 ) by selective hydro-genation. This second method affords the dihydro compound 6 or 6a,12a-dihydro-5H,7H-[2]benzopyrano[4,3-b] [1 ]benzopyran-5-one as reaction time increases.     

On the Importance of Intermediate Internal Charge Repulsion for the Synthesis of Multifunctional Pores

Intermediate internal charge repulsion (ICR) is required to create synthetic pores with large, stable, transmembrane, and variably functionalized space. This conclusion is drawn from maximal transport and, in one case, catalytic activity of p‐octiphenyl β‐barrel pores with internal lysine, aspartate, and histidine residues around pH 7, 6, and 4.5, respectively. pK_a Simulations corroborate the experimental correlation of intermediate ICR with activity and suggest that insufficient ICR causes pore ‘implosion' and excess ICR pore ‘explosion'. Esterolysis experiments support the view that the formation of stable space within multifunctional p‐octiphenyl β‐barrels requires more ICR in bilayer membranes than in H₂O. Multivalency effects are thought to account for p‐octiphenyl β‐barrel expansion with increasing number of β‐sheets, and proximity effects for unchanged pH profiles with increasing β‐sheet length. Q‐TOF‐nano‐ESI‐MS barrel‐denaturation experiments indicate that contributions from internal counterion effects are not negligible. The overall characteristics of p‐octiphenyl β‐barrel pores with internal lysine, aspartate, and histidine residues, unlike de novo ‘α‐barrels' and similarly to certain biological channels, underscore the usefulness of rigid‐rod molecules to preorganize complex multifunctional supramolecular architecture.     

4.3 Degradation and modification of cellulose acetates by biological systems

A survey is given on recent findings in the enzymology of cellulose acetate degradation. Acetyl esterases have been identified as the principal enzymes, initiating cellulose acetate degradation as a prerequisite for endoglucanase-catalyzed cellulose acetate depolymerisation. Acetyl esterases are provided by nature to deacetylate naturally occurring partly acetylated polysaccharides, i.e. xylan and chitin. Accordingly they are not designed to attack high DS cellulose acetate. Under these circumstances acetyl esterases require a pretreatment of cellulose acetate, leading to some reduction in DS, in case highly substituted material should be degraded. One of these treatments is composting under the conditions of which a partial deacetylation may occur under the action of heat and high pH, facilitating the accessibility for acetyl esterases. However from the present knowledge it cannot be excluded that certain microbial specialists exist, being capable to degrade high DS cellulose acetate.     

A New Approach in the Analysis of the Substituent Distribution of Carboxymethyl Celluloses

An approach is presented for the quantitative analysis of the substituent distribution in carboxymethyl cellulose (CMC). Standard substances have been isolated by preparative HPLC and characterised by ¹³C-NMR spectroscopy. The availability of these reference substances enabled a quantitative HPLC determination of glucose and its seven carboxymethyl derivatives by anion exchange chromatography and pulsed amperometric detection. A comparison of sulphuric acid and perchloric acid hydrolysis for conventional CMC samples gave higher yields for perchloric acid. The yield of the eight CMC building units decreased with increasing DS for both hydrolysis methods. Results were different, when samples from new synthesis concepts were investigated. While CMCs from induced phase separation showed increasing hydrolysis yields up to a DS of 1.9 regioselectively substituted 2,3-O-CMC gave lower yields with increasing DS. For 2,3-O-CMC samples sulphuric acid proved to be the superior hydrolysis medium.     

The virtual Compton amplitude in the generalized Bjorken region: Twist-2 contributions

A systematic derivation is presented of the twist-2 anomalous dimensions of the general quark and gluon light-ray operators in the generalized Bjorken region in leading order both for unpolarized and polarized scattering. Various representations of the anomalous dimensions are derived in the non-local and local light cone expansion and their properties are discussed in detail. Evolution equations for these operators are derived using different representations. General two- and single-variable evolution equations are presented for the expectation values of these operators for non-forward scattering. The Compton amplitude is calculated in terms of these distribution amplitudes. In the limit of forward scattering a new derivation of the integral relations between the twist-2 contributions to the structure functions is given. Special limiting cases which are derived from the general relations are discussed, as the forward case, near-forward scattering, and vacuum-meson transition. Solutions of the two-variable evolution equations for non-forward scattering are presented.