The classification of closed surfaces using colored graphs

A short graph theoretic proof of the classification of closed surfaces is given. The new proof has the feature that the symmetric canonical graphs for the surfaces do not correspond to the canonical polygons in the usual proof of the classification.     

Separation index of a graph

The concepts of separation index of a graph and of a surface are introduced. We prove that the separation index of the sphere is 3. Also the separation index of any graph faithfully embedded in a surface of genus g is bounded by a funtion of g. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 53–61, 2002     

Co-Orthogonal Codes

We define, construct and sketch possible applications of a new class of non-linear codes: co-orthogonal codes, with possible applications in cryptography and parallel processing. We also describe a fast and general method for generating (non-linear) codes with prescribed dot-products with the help of multi-linear polynomials.     

Balanced extensions of graphs and hypergraphs

For a hypergraphG withv vertices ande _i edges of sizei, the average vertex degree isd(G)= ∑ie ₁/v. Callbalanced ifd(H)≦d(G) for all subhypergraphsH ofG. Let $$m(G) = \mathop {\max }\limits_{H \subseteqq G} d(H).$$ A hypergraphF is said to be abalanced extension ofG ifG?F, F is balanced andd(F)=m(G), i.e.F is balanced and does not increase the maximum average degree. It is shown that for every hypergraphG there exists a balanced extensionF ofG. Moreover everyr-uniform hypergraph has anr-uniform balanced extension. For a graphG let ext (G) denote the minimum number of vertices in any graph that is a balanced extension ofG. IfG hasn vertices, then an upper bound of the form ext(G) ₁ n ² is proved. This is best possible in the sense that ext(G)>c ₂ n ² for an infinite family of graphs. However for sufficiently dense graphs an improved upper bound ext(G) ₃ n can be obtained, confirming a conjecture of P. Erdõs.     

Studies related to the head-maturation pathway of bacteriophages T4 And T2:II. nuclear disruption, protein synthesis and particle formation with the mutant 43-.30-.46-

We describe the aberrant phage multiplication of the triple conditional lethal mutant 43-(polymerase).30-(ligase).46-(exonuclease) of bacteriophage T4D in which phage DNA replication is arrested but some late protein synthesis occurs (33). The nuclear disruption is indistinguishable from wild type. Forty-five empty small and empty large particles are assembled per cell when the multiplicity of infection (m.o.i.) is 100. This number corresponds closely to the 38 phage equivalents of cleaved major head protein determined biochemically. By reducing the m.o.i. the number of observable particles decreases, reaching 1-5 per cell at an m.o.i. of 5(+5). The total synthesis of phage related proteins is not significantly dependant on the m.o.i. The synthesis of late proteins is about 10% of that of wild type at high m.o.i. and decreases with the m.o.i. The different early and late proteins do not show the same relative proportions as in wild type and respond differently to an increased m.o.i. These and other results are discussed with respect to the role of phage DNA in prehead assembly and head maturation.     

Palladium-Catalyzed Allylic Coupling of 1,2,3-Triazolo[4,5-d]pyrimidines (8-Azapurines)

The palladium-catalyzed coupling of the sodium salt of 7-amino-1,2,3-triazolo[4,5-d]pyrimidine (8-azaadenine, 1) with allylic phosphates or carbonates resulted in mixtures of 2- and 3-substituted 1,2,3-triazolopyrimidines, which were separated by chromatography. 1-Substituted triazolopyrimidines were not isolated from these reactions. Regioselectivity (and stereoselectivity) was also observed for substitution of the allylic moiety when more than one isomer is possible from the reaction. The use of 5-amino-1,2,3-triazolo[4,5-d]pyrimidin-7-ones (8-azaguanine, 2), instead of 8-azaadenine, also resulted in mixtures. Alternate syntheses of the 3-allyl-1,2,3-triazolo[4,5-d]pyrimidines confirmed the structures of these compounds.     

Mechanism of the oxidation of L-ascorbic acid by the bis(pyridine-2,6-dicarboxylate)cobaltate(III) ion in aqueous solution

A detailed investigation of the oxidation of L-ascorbic acid (H₂A) by the title complex has been carried out using conventional spectrophotometry at 510 nm, over the ranges: 0.010 [ascorbate] _T 0.045 mol dm^–3, 3.62 pH 5.34, and 12.0 30.0 °C, 0.50 I 1.00 mol dm^–3, and at ionic strength 0.60 mol dm^–3 (NaClO₄). The main reaction products are the bis(pyridine-2,6-dicarboxylate)cobaltate(II) ion and l-dehydroascorbic acid. The reaction rate is dependent on pH and the total ascorbate concentration in a complex manner, i.e., k _obs = (k ₁ K ₁)[ascorbate] _T /(K ₁ + [H⁺]). The second order rate constant, k ₁ [rate constant for the reaction of the cobalt(III) complex and HA^–] at 25.0 °C is 2.31 ± 0.13 mol^–1 dm³ s^–1. H = 30 ± 4 kJ mol^–1 and S = –138 ± 13 J mol^–1 K^–1. K ₁, the dissociation constant for H₂A, was determined as 1.58 × 10^–4 mol dm^–3 at an ionic strength of 0.60 mol dm^–3, while the self exchange rate constant, k ₁₁ for the title complex, was determined as 1.28 × 10^–5 dm³ mol^–1 s^–1. An outer-sphere electron transfer mechanism has been proposed.     

Synthesis and near infrared luminescence of a tetrametallic Zn2Yb2 architecture from a trinuclear Zn3L2 Schiff base complex

Near infrared luminescence is observed in tetrametallic [Zn2Yb2L2(mu-OH)2Cl4].2MeCN which is obtained from the Zn3 Schiff-base complex [Zn3L2(NO3)2].MeOH, (H2L =N,N'-bis(5-bromo-3-methoxysalicylidene)propylene-1,3-diamine).     

Synthesis of two glycolipid antigens of the causative agent of Lyme disease

As a prelude to development of a human vaccine against Lyme disease, the first chemical synthesis of glycolipid antigens of Borrelia burkholderi is reported. First, cholesteryl β-d-galactopyranoside was synthesized and was converted to partially protected congeners having the HO-6 group of the galactose moiety unprotected. Treatment of these intermediates with palmitic and oleic acid, respectively, under dehydrative conditions followed by removal of the protecting groups afforded cholesteryl 6-O-palmitoyl/oleoyl-β-d-galactopyranosides that were identical to the glycolipids isolated from B. burkholderi.