Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A^∗ : V → V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned by

     

Diffusion and Brownian motion in Lagrangian coordinates

In this paper we consider the convection-diffusion problem of a passive scalar in Lagrangian coordinates, i.e., in a coordinate system fixed on fluid particles. Both the convection-diffusion partial differential equation and the Langevin equation are expressed in Lagrangian coordinates and are shown to be equivalent for uniform, isotropic diffusion. The Lagrangian diffusivity is proportional to the square of the relative change of surface area and is related to the Eulerian diffusivity through the deformation gradient tensor. Associated with the initial value problem, we relate the Eulerian to the Lagrangian effective diffusivities (net spreading), validate the relation for the case of linear flow fields, and infer a relation for general flow fields. Associated with the boundary value problem, if the scalar transport problem possesses a time-independent solution in Lagrangian coordinates and the boundary conditions are prescribed on a material surface/interface, then the net mass transport is proportional to the diffusion coefficient. This can be also shown to be true for large Peclet number and time-periodic flow fields, i.e., closed pathlines. This agrees with results for heat transfer at high Peclet numbers across closed streamlines.     

Anti-babesial compounds from Curcuma xanthorrhiza

Anti-babesial ingredients, (12R)- and (12S)-12,13-dihydro-12,13-dihydroxyxanthorrhizols, were isolated from Curcuma xanthorrhiza. The structures were established by the extensive NMR techniques. The assignments of (1)H NMR data of (12R)-12,13-dihydro-12,13-dihydroxyxanthorrhizol was revised, and (12S)-12,13-dihydro-12,13-dihydroxyxanthorrhizol was isolated as a pure form for the first time. The IC(50) of the active compounds were compared with that of commercial drug, diminazene aceturate (Ganaseg). IC(50) values of Ganaseg, (12R)-, and (12S)-12,13-dihydro-12,13-dihydroxyxanthorrhizols were 0.6 microg mL(-1), 8.3 microg mL(-1) and 11.6 microg mL(-1), respectively.     

Enhancement of nonlinear optical response of exciton–polaritons by controlling thickness of GaAs layer

We demonstrate a remarkable thickness dependence of nonlinear optical response of exciton–polaritons in GaAs thin films by degenerate four-wave mixing (DFWM) at low temperature (T=5 K). High-quality samples of layer thickness from 80 to 200 nm were grown by molecular beam epitaxy. Confined mode of the exciton–polariton dominantly contributed to the reflection spectra, which were examined by the calculation using a transfer matrix method. The DFWM intensity at exciton resonance was enhanced at a particular thickness (110 nm). This thickness dependence is in good agreement with the nonlocal theory, which shows a remarkable size dependence of the internal field relevant to the confined mode of the exciton–polaritons.     

Ethylene Polymerization Catalyzed by Titanium(IV) Complexes of a Triaryloxoamine Ligand [TiX{(OArCH2)3N}]

Summary: Titanium complexes containing a triaryloxoamine ligand, [TiX{(O‐2,4‐R₂C₆H₂‐6‐CH₂)₃N}] (R = Me, tBu; X = OiPr, O‐2,6‐iPr₂C₆H₃), exhibited notable catalytic activity for ethylene polymerization in the presence of MAO, especially at temperatures between 100 and 120 °C. Their activity increased upon the addition of a small amount of AlMe₃.

     

Nucleosides and nucleotides. 191 : Ring expansion reaction of 1-[2,3,5-tri-o-TBS-4α-formyl-β-D-ribo-pentofuranosyl]uracil by treating with (methylene)triphenylphosphorane to give a new nucleoside containing dihydrooxepine ring at the sugar moiety

When 1-[2,3,5-tri-O-TBS-4α-formyl-β-D-ribo-pentofuranosyl]uracil (5) was treated with (methylene)triphenylphosphorane in THF, an unusual ring-expansion reaction occurred to give a nucleoside (7) containing dihydrooxepine ring at the sugar moiety. A deuterium-label experiment showed that one carbon unit derived from the ylide was incorporated into the 5'-position of 7. A ring cleavage between the C-3' and C-4' of 5 during the reaction was suggested.     

The microenvironment of iron in (Ba,Ca)(Fe,Co)O3−δ catalyst system: A Mössbauer study

Undisturbed, non-fertilized woodland soil (“loamy sandy soil” type) from 1 m below surface was dry and wet sieved. Sieving fractions of <10–1000 μm were analyzed for total alpha-activity. Thorium and uranium contents were determined by alpha-spectrometry after radiochemical separation. Soluble and insoluble parts of thorium and uranium were determined in the sieved fractions indicating that the isotope distribution in soil correlates with the particle size distribution: The smaller the size fraction the higher the isotope content. Isotope ratios of²²⁸Th/²³²Th, and²³⁴U/²³⁸U are discussed.     

Kinetics and mechanisms of o/w microemulsion polymerization of styrene

To clarify the kinetics and mechanisms of o/w microemulsion polymerization, the o/w microemulsion polymerizations of styrene were carried out with different kinds and types of initiators and surfactants. The number of polymer particles and the molecular weight of polymer formed were not affected by the kinds and types of initiators and surfactants. The rate of polymerization with different kinds of azo-type oil-soluble initiators with widely different water-solubility was almost the same, but was much lower than that with KPS. A kinetic model proposed by us could mostly explain the kinetic behavior except for the difference in the polymerization rate with KPS and that with oil-soluble initiators.