A mathematical definition of the financial bubbles and crashes

We check the validity of the mathematical method of detecting financial bubbles or crashes, which is based on a data fitting with an exponential function. We show that the period of a bubble can be determined nearly uniquely independent of the precision of data. The method is widely applicable for stock market data such as the Internet bubble.     

Noncolliding Brownian Motion and Determinantal Processes

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin–McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin–McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.     

On the continuity of the map ϑ → ¦ϑ¦ from the predual of a W1-algebra

Let ?, ψ be elements in the predual of a W¹-algebra. For their absolute value parts ¦?¦, ¦ψ¦, the estimate $\text{∥¦?¦ ? ¦ψ¦∥ ? (2 ∥? + ψ∥ ∥? ? ψ∥)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is obtained.     

Dynamic helicity inversion by achiral anion stimulus in synthetic labile cobalt(II) complex

The helical chirality of a Co(II) complex with a chiral tetradentate ligand is completely inverted from Lambda to Delta by the addition of achiral NO3- anion as an external stimulus.     

Kinetic resolution of rac-phenylalanine by stereoselective complexation to a chiral cobalt complex through pi-pi stacking interaction

A cobalt(III) complex with chiral ligand, H2cpel (N-carboxymethyl-N-pyridylethyl-l-leucine), was prepared for chiral recognition of amino acids. Through the competitive coordination of racemic phenylalanine to the chiral cobalt complex, [Co(cpel)(CO(3))](-) (1), enantioselective recognition was achieved on the ternary complex, which was determined on the basis of HPLC analysis with a chiral column. The formation rate for the [Co(cpel)(l-phe)] complex (2) was 6-times superior to that of [Co(cpel)(d-phe)] (3). The preferential formation of 2 might be illustrated by the interligand pi-pi stacking interaction. Crystal structural analysis for 2 and 3 revealed that aromatic rings, pyridine ring of CPEL and phenylalanine sidechain, in 2 were very close each other but those in 3 were far apart. Such interligand aromatic interaction in 2 was also examined by the use of (1)H NMR spectra.     

An accurately-constructed structural model for an active site of Fe-containing superoxide dismutases (Fe-SODs)

A novel structural model, [Fe(bpga)(N(3))(OCH(3))] (1) (BPGA: bis(6-pivalamido-2-pyridylmethyl)(carboxy-methyl)amine), for an active site of Fe-containing SOD enzyme with N(3)(-) has been accurately designed and prepared, which has been characterized by X-ray structure analysis and UV-vis, ESR, and IR spectroscopies. Reaction of 1 with acid/base in methanol has shown that the superoxide-accessible site is the N(3)(-)-coordination site in the equatorial plane.     

Induction of optical activity in oligosilanes within the internal cavity of cyclodextrins

The first example of induced optical activity of oligosilanes within the internal cavity of gamma-cyclodextrins is reported.     

Supercritical fluid chromatography with post-column addition of supporting electrolyte solution for electrochemical determination of tocopherol and tocotrienol isomers

A supercritical fluid chromatography with electrochemical detection system was developed for the simultaneous determination of tocopherol and tocotrienol isomers. The supercritical fluid chromatography with electrochemical detection system was connected with an additional pump to create a flow path to add a supporting electrolyte solution. The supporting electrolyte solution was mixed with a mobile phase in a post-column fashion, enabling the independent control of the separation and detection. After optimization of the measurement conditions, vitamin E isomers and an internal standard substance (2,2,5,7,8-pentamethyl-6-hydroxychroman) were separated within 30 min using a mixture of supercritical carbon dioxide and methanol (99:1, v/v) as a mobile phase and a cyanopropyl column (4.6 mm inner diameter × 250 mm length, 5 μm). For the electrochemical detection, methanol containing 1.0 mol/L ammonium acetate was used as a supporting electrolyte solution, and the applied potential was set at +0.8 V. This analytical method showed good linearity (5–100 μg/mL) and repeatability (less than 2.5% relative standard deviation, n = 6) and was applicable to the determination of tocopherol and tocotrienol isomers in nutrition supplements.