Remarks on unit indices of imaginary abelian number fields II

Satz 41 in Hasse's monograph [4] is modified because of the incorrectness of Satz 29. We investigate unit indices of imaginary abelian number fields of type (2,2) and (2,2, 2), some of which give counterexamples to Satz 41. We moreover give a relation between unit indices of certain imaginary fields of type (2,2²) and those of their real cyclic subfields.     

Covering vertices of a graph by k disjoint cycles

Let d, k and n be three integers with k3, d4k−1 and n3k. We show that if d(x)+d(y)d for each pair of nonadjacent vertices x and y of a graph G of order n, then G contains k vertex-disjoint cycles converting at least min{d,n} vertices of G.     

Buchsbaum homogeneous algebras with minimal multiplicity

In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous k-algebras A in terms of the dimension d, the codimension c, the initial degree q, and the length of the local cohomology modules of A. Next, we introduce the notion of Buchsbaum k-algebras with minimal multiplicity of degree q, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.     

Some Recent Progress and Applications in Graph Minor Theory

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough' structure of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed. Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, Grant number 16740044, by Sumitomo Foundation, by C & C Foundation and by Inoue Research Award for Young Scientists Supported in part by the Research Grant P1–0297 and by the CRC program On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia     

Circularly polarized light-induced electrogyration in the Au nanoparticles on the ITO

We have found that the gold nanoparticles on the ITO substrates might be considered promising materials for circularly polarized light-induced linear electrogyration (EG). The maximal achieved value of the EG susceptibility described by third-order axial tensor caused by probe circularly polarized light at a wavelength of 1060 nm was equal to about 13 deg/mm at pulsed electric field strength 30.0 V/cm with a duration of about 1 ms. We have revealed that the maximal EG coefficient is achieved for the samples possessing maximal resistivity. The investigated composites possess long-lived EG grating which decreases by not more than 12% after 120 min of laser treatment. Applying a non-circular pump light leads to the diminishing of the observed EG.     

Lithographic design of photosensitive polyarylates based on reaction development patterning

The factors affecting pattern‐forming properties in reaction development patterning were examined with polyarylates with various bisphenol moieties. The developability of the photosensitive polyarylates was dependent on the properties of the subtituent (R) in the bisphenol moieties. The development time decreased in the following order: R?C(CH₃)₂ > fluorenyl unit ? phenolphthalein unit > C(CF₃)₂ > SO₂. This order agreed with that of the reactivity between the polyarylates and ethanolamine, and these orders can be explained by pK_a of the bisphenol used to prepare the polyarylates. The development with NH₂? R′? OH resulted in successful positive‐tone pattern formation. However, pattern formation with the developers containing NH₂? R′? OCH₃ was unsuccessful. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2694–2706, 2006     

Electrode kinetics of the metal complexes with aminopolycarboxylic acids: Part II. Cadmium-ethylenediamine-N,N,N′,N′-tetraacetate (EDTA) complexes

The d.c. polarographic current-potential curves of Cd(II)-EDTA complexes were examined in the pH range 0.5–10.0, to elucidate the mechanism of their electrode processes and to determine the relevant electrochemical kinetic parameters. It was shown that the first wave observed below pH 3 at ?0.58 to ?0.65 V vs. SCE is the reversible reduction wave of Cd(II) aquo-ion with kinetically-controlled limiting current, and the second wave observed above pH 1.5 at ?0.75 to ?1.21 V vs. SCE corresponds to the simultaneous irreversible reduction of four complex species, CdH₃L⁺, CdH₂L, CdHL^? and CdL^2?, where CdH_pL^(p?2)+ and L^4? denote the protonated complex species with p protons and the unprotonated EDTA ion, respectively. Analysis of the dependence of limiting current on the hydrogen ion concentration led to the conclusion that the preceding reaction determining the behaviour of limiting current is CdH₃L⁺?Cd²⁺+H₃L^? with k₃^d=6.3×10² s^?1 and k₃^f=3.3×10⁶ s^?1M^?1, where k₃^d and k₃^f are the dissociation and formation rate constants, respectively. On the other hand, from analysis of the dependence of half-wave potentials of the second wave on the hydrogen ion concentration, the kinetic parameters of the four complex species were evaluated, and are given in Table 1. Further, it was shown that the cathodic rate constants of these four charge transfer processes at some reference potential together with those of Cd(II)-HEDTA complexes fulfil the linear free energy relationship.