New primitive whose degree is a Mersenne exponent

All primitive trinomials over with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are and its reciprocal. Also two examples of primitive pentanomials over with degree 86243 (which is the 28th Mersenne exponent) are presented. The sieve used is briefly described.

     

Discovering network behind infectious disease outbreak

Stochasticity and spatial heterogeneity are of great interest recently in studying the spread of an infectious disease. The presented method solves an inverse problem to discover the effectively decisive topology of a heterogeneous network and reveal the transmission parameters which govern the stochastic spreads over the network from a dataset on an infectious disease outbreak in the early growth phase. Populations in a combination of epidemiological compartment models and a meta-population network model are described by stochastic differential equations. Probability density functions are derived from the equations and used for the maximal likelihood estimation of the topology and parameters. The method is tested with computationally synthesized datasets and the WHO dataset on the SARS outbreak.     

The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers

A set A of non‐negative integers is called a Sidon set if all the sums , with and a₁, , are distinct. A well‐known problem on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of . Results of Chowla, Erd?s, Singer and Turán from the 1940s give that . We study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ by a sparse, random subset R of , and ask how large a subset can be, if we require that S should be a Sidon set. Let be a random subset of of cardinality , with all the subsets of equiprobable. We investigate the random variable , where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m, as illustrated by the following abridged version of our results. Let be a fixed constant and suppose . We show that there is a constant such that, almost surely, we have . As it turns out, the function is a continuous, piecewise linear function of a that is non‐differentiable at two ‘critical’ points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function is constant. Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erd?s (On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 1–25, 2015     

Universality for bounded degree spanning trees in randomly perturbed graphs

We solve a problem of Krivelevich, Kwan and Sudakov concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph G_α on n vertices with δ(G_α) ≥ αn for α > 0 and we add to it the binomial random graph G(n,C/n), then with high probability the graph G_α∪G(n,C/n) contains copies of all spanning trees with maximum degree at most Δ simultaneously, where C depends only on α and Δ.     

Enzymatic Syntheses of N-Acetyllactosamine and N-Acetylallolactosamine by the Use of β-D-Galactosidases

Abstract

A β-D-galactosidase from Bacillus circulans induced β-D-galactopyranosyl transfer from lactose predominantly to a secondary (OH-4) rather than the primary hydroxyl group (OH-6) of 2-acetamido-2-deoxy-D-glucopyranose. 4-O-β-D-Galacto-pyranosyl-2-acetamido-2-deoxy-D-glucopyranose (N-acetyl-lactosamine) was thus readily synthesized on a gram scale and conveniently isolated by chromatography on a column of charcoal-Celite. On the other hand, the glycosyl transfer to the 6-position predominantly was efficiently induced to give 6-O-β-D-galactopyranosyl-2-acetamido-2-deoxy-D-glucopyranose (N-acetyl-allolactosamine) by consecutive use of β-D-galactosidases from Kluyveromyces lactis and B. circulans. These enzyme reactions were efficient enough to allow the one-pot preparation of the desired disaccharides.     

Partial Protection of Carbohydrate Derivatives. Part 13.1 Chemical Conversion of Kanamycin A Into 2′-Deoxykanamycin A, 2′-Epi-Kanamycin A,and 2′-Epi-Kanamycin B

Abstract

Hydrazinolyses of hexa-0-benzoyl-tetra-N-benzyloxycarbonyl-and N-ethoxycarbonylkanamycin A were performed and found to be sufficiently regioselective to give the corresponding 2′-hydroxyl derivatives in good yields under controlled conditions. The products were converted into the corresponding 2′-triflates, which were then subjected to nucleophilic substitution reactions with sodium benzenethioxide, sodium benzoate, and sodium azide to give the corresponding d-mannopyranosyl derivatives in good yields. Deprotection of the phenylthio (10) and azido (12) derivatives, and hydrogenolysis, gave 2′-deoxykankmycin A and 2′-epi-kanamycin B, respectively. Moreover, deprotection of the benzoyl compound 11 gave 2′-epi-kanamycin A.     

Experimental and computational study of intermolecular migration of N,N-dimethylcarbamoyl group from N(7) to N(1) on a 7-azaindoline derivative

N,N-Dimethylcarbamoylation of the anilinic nitrogen atom N(1) on the spiro 7-azaindoline consists of two steps. The first step is N,N-dimethylcarbamoylation of the pyridyl nitrogen atom N(7), leading to the formation of an isolable intermediate. The second step is intermolecular migration of the N,N-dimethylcarbamoyl group from the pyridyl nitrogen atom N(7) to the anilinic nitrogen atom N(1). We accomplished optimization of the reaction conditions based on the revealed reaction mechanism and a large scale synthesis of compound 3 in quantitative yield.     

Synthesis of 1,4-diaryl[60]fullerenes by bis-hydroarylation of C60 and their use in solution-processable, thin-film organic photovoltaic cells

An AlCl₃-mediated Friedel-Crafts reaction of arenes with C₆₀ affords two-fold hydroarylated compounds, C₆₀Ar₂H₂, which upon deprotonation with ^tBuOK and oxidation with CuBr·SMe₂ yield 1,4-diaryl[60]fullerenes, C₆₀Ar₂ (Ar = Ph, 4-Me-C₆H₄, 3,4-Me₂-C₆H₃, and 4-Ph-C₆H₄) in good yield. A solution-processed, thin-film organic photovoltaic device using C₆₀(4-PhC₆H₄)₂ as electron acceptor material showed a 2.3% power conversion efficiency.     

(S)-(+)-1-Methyl-2-(6,7,-dimethoxy-2,3-naphthalimido)ethyl trifluoromethanesulfonate as a fluorescence chiral derivatizing reagent for carboxylic acid enantiomers in high-performance liquid chromatography

A new triflate-type fluorescence chiral derivatizing reagent, (S)-(+)-1-methyl-2-(6,7-dimethoxy-2,3-naphthalimido)ethyl trifluoromethanesulfonate, [S-(+)-MDNE-OTf], has been developed for the determination of the enantiomers of carboxylic acids. By introducing the two methoxy groups on the naphthalimido ring moiety, the red shift in the fluorescence spectrum and a high resolution in reversed-mode separation of the diastereomers of chiral carboxylic acids have been achieved. The detection limits (S/N=3) with ultraviolet and fluorescence detection are 8 fmol (λ_max=283 nm) and 4 fmol (λ_ex=283 nm, λ_em=467 nm), respectively.     

Structural effects on biodegradation of aliphatic polyesters

Various types of aliphatic polyesters were prepared by both biosynthetic and chemosynthetic methods, and their biodegradation tests were carried out under aerobic conditions in the river water. Biodegradabilities of polyester films were evaluated by monitoring the time-dependent changes in the biochemical oxygen demand (BOD), weight loss (erosion) of polyester film, and dissolved organic carbon concentration (DOC) of test solution. The microbial copolyesters were degraded in the river water at a rapid rate, and the weight-loss- and BOD-biodegradabilities of the majority of biosynthetic polyesters were 100 % and 80±5 % for 28 days, respectively. In contrast, the biodegradabilities of chemosynthetic polyesters were strongly dependent of the chemical structure of monomeric units.