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.     

NDMAP: the best material to study the Haldane's prediction

As predicted by Haldane, spin, S=1 one-dimensional (1D) Heisenberg antiferromagnet (HAF) has an energy gap between the singlet ground state and first excited triplet. On application of magnetic field, the triplet state Zeeman splits and the energy of one of the triplet state becomes zero at a critical field, H_c. Above H_c the system recovers magnetism. Then, we expect that a quasi-1D HAF will show a magnetic long-range ordering (LRO) at low temperatures due to the inter-chain coupling. This field-induced LRO has not been observed before due to complication of the crystal structure in the materials studied so far and/or technical difficulty.From a heat capacity measurement on a single crystal of an S=1 quasi-Q1D HAF, Ni(C₅H₁₄N₂)₂N₃(PF₆), we found an anomaly at a temperature in finite fields indicating a field-induced phase transition. A magnetic LRO is confirmed by a neutron diffraction measurement on the same sample. The temperature versus magnetic field phase diagram of this compound is constructed and discussed.     

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     

Ruthenium-catalyzed one-pot hydroformylation of alkenes using carbon dioxide as a reactant

A new hydroformylation of alkenes using carbon dioxide as a reactant is shown to take place in the presence of ruthenium cluster complexes and halide salts. Similar or even better yields of alcohols were formed as compared to the conventional hydroformylation with CO under the same reaction conditions. The reaction proceeded in three steps: CO₂ is first converted to CO; then it is used as a reagent for hydroformylation to give aldehyde; subsequently, it is hydrogenated to alcohol. ESI-mass spectrometric analyses of the reaction solutions indicated formation of four kinds of ruthenium anionic complexes including tetra-, tri-, and mononuclear species. On the basis of experimental findings, possible roles of these complexes are discussed.     

Time evolution of density fluctuation in the supercritical region. 2. Comparison of hydrogen- and non-hydrogen-bonded fluids

The time evolution of the density fluctuation of molecules is investigated by dynamic light scattering in six neat fluids in supercritical states. This study is the first to compare the dynamics of density inhomogeneity between hydrogen- and non-hydrogen-bonded fluids. Supercritical methanol and ethanol are used as hydrogen-bonded fluids, whereas four non-hydrogen-bonded fluids were used: CHF(3), C(2)H(4), CO(2), and Xe. We measure the time correlation function of the density fluctuation of each fluid at the same reduced temperatures and densities and investigate the relationship between the dynamic and static density inhomogeneities of those supercritical fluids. In all cases, the profile of the time correlation function of the density fluctuation is characterized by a single-exponential function, whose decay is responsible for the dynamics characterized by hydrodynamic conditions. We obtain correlation times from the time correlation function and discuss dynamic and static inhomogeneity using the Kawasaki theory and the Landau-Placzek theory. While the correlation times in the six fluids show noncoincidence, those values agree well with each other except for the supercritical alcohols when scaled to a dimensionless parameter. Although the principle of corresponding state is observed in the non-hydrogen-bonded fluids, both the supercritical methanol and ethanol deviate from that principle. This deviation is attributed to the presence of hydrogen bonding among alcohol molecules at high temperature and low density. The average cluster size of each fluid is estimated under the same thermodynamic conditions, and it is shown that the clusters of supercritical alcohols are on average 1.5-1.7 times larger than those of the four non-hydrogen-bonded fluids. Moreover, the thermal diffusivity of each neat fluid is obtained over wide ranges of density and temperature.     

Determination of the specific interaction of disubstituted aromatic compounds on hydrous zirconium oxide-loaded porous polymer resin

Summary A porous polymer resin loaded with hydrous zirconium oxide (Zr-gel) was evaluated as a stationary phase for the liquid chromatographic separation of disubstituted aromatic compounds. The prepared Zr-gel was used to separate disubstituted phenols andortho-substituted benzoic acid derivatives in reversed-phase mode. The retention time of catechol was greater than those of other disubstituted phenols; this implies that the Zr-gel has specificity for the phenolic hydroxyl group. The retention behavior ofortho-substituted benzoic acid derivatives on the Zr-gel was also evaluated in reversedphase mode using buffer. Phthalic acid was specifically retained in the vicinity of pH 6.0, which has been regarded as the equal adsorbic point. Similar behavior was also observed for salicylic acid, although the retention time of salicylic acid was less than that of phthalic acid. It seems that the specific retention behavior of hydrous zirconium oxide is a result of complexation with ligands such as hydroxyl and carboxyl groups. The results of this study have revealed that the retention mechanism of hydrous zirconium oxide is a combination of ion exchange and interaction based on complexation with ligands.