On Cubic Graphs Admitting an Edge-Transitive Solvable Group

Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip₃ or the complete graph K ₄, by a sequence of elementary-abelian covers. Another consequence of the main structural result is that the action of an arc-transitive solvable group on a connected cubic simple graph is at most 3-arc-transitive. As an application, a new infinite family of semisymmetric cubic graphs, arising as regular elementary abelian covering projections of K _3,3, is constructed.     

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

For 1/2<<1 fixed, letE (T) denote the error term in the asymptotic formula for . We obtain some new bounds forE (T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem.     

An Extension of Lomonosov's Techniques to non-compact Operators

The aim of this work is to generalize Lomonosov's techniques in order to apply them to a wider class of not necessarily compact operators. We start by establishing a connection between the existence of invariant subspaces and density of what we define as the associated Lomonosov space in a certain function space. On a Hilbert space, approximation with Lomonosov functions results in an extended version of Burnside's Theorem. An application of this theorem to the algebra generated by an essentially self-adjoint operator yields the existence of vector states on the space of all polynomials restricted to the essential spectrum of . Finally, the invariant subspace problem for compact perturbations of self-adjoint operators acting on a real Hilbert space is translated into an extreme problem and the solution is obtained upon differentiating certain real-valued functions at their extreme.

     

DFT studies on tautomerism of C5-substituted 1,2,4-triazoles

DFT (B3LYP/6-311++G**, B3PW91/6-311++G**) Gibbs free energy and single point CCSD(T)/6-311++G**//DFT total energy calculations were performed to investigate stability and tautomerism of C5-substituted 1,2,4-triazoles. Three different tautomers are possible for the substituted 1,2,4-triazoles: N1–H, N2–H, and N4–H. Unlike for the 1,2,3-triazoles, where the most stable is the N2–H tautomer regardless of substituent applied, for the 1,2,4-triazoles, the electron donating substituents (–OH, –F, –CN, –NH₂, and –Cl) and the C5-cation stablize the N2–H tautomer, whereas the electron withdrawing substituents (–CONH₂, –COOH, –CHO, –BH₂, and –CFO) and the C5-anion stablize the N1–H tautomer. Except for the C5-anion and C5-cation, the N4–H form is the least stable tautomer. The relative stability of the C5-substituted 1,2,4-triazole tautomers is strongly influenced by attractive and/or repulsive intramolecular interactions between substituent and electron donor or electron acceptor centres of the triazole ring.     

Stabilization of a subvalent oxidation state of bismuth in N,N-dimethylthioformamide solution: an EXAFS, UV-Vis, IR, and cyclic voltammetry study

At the dissolution of anhydrous bismuth(III) trifluoromethanesulfonate in N,N-dimethylthioformamide (DMTF) a deep red-orange complex, lambda(max) = 457 nm, is formed. Bismuth(III) is reduced by the solvent to a low-valent oxidation state stabilized by the sulfur-coordinating solvent DMTF. The obtained complex is weakly solvated seen by a low EXAFS amplitude and a slightly higher absorption energy of the L(III) edge than of the DMTF-solvated bismuth(III) ion. The EXAFS data reveal a dimeric bismuth complex solvated by a single DMTF molecule, which sulfur atom bridges the bismuth atoms. The Bi-S bond distance is 2.543(2) A, and the Bi...Bi distance is 3.929(7) A giving a Bi-S-Bi angle of 101.2(4) degrees. The very low number of coordinated solvent molecule shows that the lone electron pairs of the reduced bismuth ions are stereochemically active. Cyclic voltammetry investigations provide evidence that at least one bismuth atom in the dimer exists in an oxidation state lower than +III, seen by two peaks at approximately -0.36 and -0.57 V in the reduction half-cycle. The absence of EPR signals excludes the presence of bismuth(II) radicals.     

Factor analysis of deuterium isotope effects on 13C NMR chemical shifts in Schiff bases

We have analyzed deuterium isotope effects on (13)C chemical shifts in a series of o-hydroxy Schiff bases by applying factor analysis. Two orthogonal factors were obtained that explain about 80 and 10 % of the variance of the data. The numerical values of these factors can be related to 1H NMR chemical shifts of the proton involved in the intramolecular bonds delta(XH) (X = O or N). Such a relation allows one to identify clusters of compounds with different tautomeric forms of hydrogen bonding. Application of a similar approach to solution 13C NMR chemical shifts produces three important factors, which have a different structure to factors describing isotope effects. This illustrates well the different nature of chemical shifts and isotope effects. The three factors explain about 54, 15, and 13 % of variance. They can be rationalized and are strongly related to the electronic properties and location of substituents.     

New organogold(I) complexes: Synthesis, structure, and dynamic behavior of the polynuclear organogold diphenylmethane and diphenylethane derivatives

Two organogold derivatives of diphenylmethane and diphenylethane, Ph₃PAu(o-C₆H₄)CH₂(C₆H₄-o)AuPPh₃ (1) and Ph₃PAu(o-C₆H₄)(CH₂)₂(C₆H₄-o)AuPPh₃ (2), have been synthesized by the reaction of ClAuPPh₃ with Li(o-C₆H₄)CH₂(C₆H₄-o)Li and Li(o-C₆H₄)(CH₂)₂(C₆H₄-o)Li respectively. The interaction of 1 with dppe results in the replacement of the two PPh₃ groups to give a macrocyclic compound (3) that includes an Au Au bond. Compounds 1 and 2 react with one or two equivalents of [Ph₃PAu]BF₄ to form new types of cationic complex [CH₂(C₆H₄-o)₂(AuPPh₃)₃]BF₄ (4), [CH₂(C₆H₄-o)₂(AuPPh₃)₄](BF₄)₂ (5), and [(CH₂)₂(C₆H₄-o)₂(AuPPh₃)₄](BF₄)₂ (6). Complexes 1–6 have been characterized by X-ray diffraction studies, FAB MS, and IR as well as by ¹H and ³¹P NMR spectroscopy. A complicated system of Au H-C agostic interactions, involving the bridging alkyl groups (—CH₂— and CH₂-CH₂—) of diphenylmethane and diphenylethane ligands, has been found to occur in complexes 1–3 and 6.     

Cores,cutsets and the fixed point property

The purpose of this paper is the analysis and application of the concepts of a core (a pair of chains) and cutset in the fixed point theory for posets. The main results are:

1. (Theorem 3) If P is chain-complete and (*), it contains a cutset S such that every nonempty subset of S has a join or a meet in P, then P has the fixed point property (FPP),
2. (Theorem 5) If P or Q is chain-complete, Q satisfies (*) and both P and Q have the FPP, then P x Q has the FPP.
3. (Theorem 6) Let P or Q be chain-complete and there exist p∈P and a finite sequence f ₁, f ₂, ..., f _n of order-preserving mappings of P into P such that $$\left( {\forall x\varepsilon P} \right)x \leqslant f_1 \left( x \right) \geqslant f_2 \left( x \right) \leqslant \cdots \geqslant f_n \left( x \right) \leqslant p$$ If P and Q have the FPP then P x Q has the FPP.
4. (Theorem 7) If T is an ordered set with the FPP and {P _t :t∈T} is a disjoint family of ordered sets with the FPP then its ordered sum ∪{P _t :t∈T} has the FPP.