Farey Series and the Riemann Hypothesis III

In this paper we shall make a further study on the the equivalence problem, i.e., equivalent conditions to the Riemann hypothesis in terms of Farey series by developing a rather analytical (-arithmetical) method to establish unexpected short interval results, namely results, therewith simultaneously clarifying the underlying reasons for results obtained in Part I.     

Farey Series and the Riemann Hypothesis. IV

This is a direct continuation of Part II of this series of papers and we shall not only improve (numerically) all our former results in Part II, but also prove new theorems, Theorem 4 on the function f(t) = log Γ(t+λ) and Theorem 5 for k-th power moments, k = 2,...,7, by elaborating our previous arguments. This revised version was published online in June 2006 with corrections to the Cover Date.     

Even subgraphs of bridgeless graphs and 2-factors of line graphs

By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible.     

The Effect of Intramolecular Hydrogen Bonding on the Intensity and Frequency of OH Stretching Band in Alicyclic Substances

The change in intensity and frequency of O-H stretching band upon intramolecular hydrogen bonding was sufficiently examined for various proton acceptors of many alicyclic substances. The order of proton acceptors to bring about the intensity increase of constant degree is: Some discussions are given on these.     

Visible-light photocatalysis of ZnO deposited on nanoporous Au

ZnO deposited on nanoporous Au showed photocatalytic decomposition toward methyl orange under visible light, unlike ZnO sputtered on flat Au without a nanoporous structure. First-principles calculations suggested that the surface lattice disorder in nanoporous Au induced a band gap narrowing and a large built-in electric field in the adjacent ZnO, resulting in the visible-light photocatalytic response.     

First synthesis of 1,3-oxaselenepanes

The first synthesis of 1,3-oxaselenepane derivatives by the reaction of aryl isoselenocyanates with 4-bromobutanol in the presence of sodium hydride in THF as a one-pot reaction is described. The Z/E isomerism for the exocyclic carbon-nitrogen double bond in the selenium heterocycles was observed for the first time.