Teiji Takagi, Founder of the Japanese School of Modern Mathematics

This article is a brief historical report on Teiji Takagi which was prepared at the commencement of ‘Takagi Lectures’ of The Mathematical Society of Japan. The first of its two purposes is to give some informations on the circumstances of education and research of mathematics in Japan surrounding Takagi who could finally established himself as the founder of the Japanese school of modern mathematics. The other is a brief overview on Takagi’s works of mathematics some of which are still attractive to and influential on especially ambitious students of mathematics. The author hopes that careful readers may find some hints for the questions how and why Takagi was able to establish his class field theory. At the end of this article the readers will find an English translation of the preface of his book Algebraic theory of numbers (in Japanese) which is the only thing that he left for us to see his total view over class field theory after the establishment of Artin’s reciprocity law.     

High pressure Raman study on the local structure of 1-ethyl-3-methylimidazolium tetrafluoroborate

We have investigated the pressure-induced Raman spectral changes of 1-ethyl-3-methylimidazolium tetrafluoroborate ([emim][BF₄]). We found that [emim][BF₄] did not crystallize up to 1.2 GPa. The Raman CH stretching spectra arising from the CH₃ groups of the ethyl-chain and the CH₃ group adjacent to the imidazolium-ring in [emim]⁺ cation largely changed against pressure. Moreover, the Raman intensity of the CH₂ (N) bending band arising from the alkyl-chain drastically changes with increasing pressure, but that of the imidazolium-ring in-plane bending band arising from the imidazolium-ring is independent of pressure. Our results show that the environment around the alkyl-chain of [emim][BF₄] is largely perturbed rather than that around the imidazolium-ring upon compression.     

Production and characterization of refractive index gratings in high-birefringence fibre optics

In this work, we report production and modelling of both Bragg gratings and rocking filters in high birefringence fibre optics. Bragg gratings are produced by UV (257.0 nm) external interferometric exposition of a frequency-doubled Ar⁺ ion laser, whereas for rocking filters formation the visible Ar⁺ ion laser lines in 488.0 and 514.5 nm are used in an internal method. The spectral characteristics due to the birefringence properties are studied through numerical methods and experimental techniques. The spectral responsivity of the structures under temperature changes and stress application is presented.     

Some families of Mordell curves associated to cubic fields

We introduce some Mordell curves of two different natures both of which are associated to cubic fields. One set of them consists of those elliptic curves whose rational points over the rational number field are described by or closely related to cubic fields. The other is a one-parameter family of Mordell curves which gives all (cyclic) cubic twists and all quadratic twists of the Fermat curve X³+Y³+Z³=0.     

Photorefractive effect in the relaxor ferroelectric material 0.91Pb(Zn1/3Nb2/3)O3-0.09PbTiO3

The photorefractive effect in a nominally undoped 0.91Pb(Zn1/3Nb2/3)O3-0.09PbTiO3 single crystal is measured. We report what is, to our knowledge, the first observation of the photorefractive effect in Pb-based relaxor ferroelectric crystals. The crystal is grown by the flux solution method. Then it is cut into a 2 mm x 4 mm x 8 mm piece and electrically poled along the [111] direction. The coupling constant of the two-wave mixing is 17 cm(-1), and the normalized time constant under 1-W/cm(2) illumination is 12 s at a wave-length of 476 nm. The effective trap density is calculated as 5 x 10(16) cm(-3) from the Debye screening length under the assumption of Kukhtarev's band-transport model. The dominant carrier is identified to be holes from the direction of two-wave mixing energy transfer.     

Tuning membership functions of kernel fuzzy classifiers by maximizing margins

We propose two methods for tuning membership functions of a kernel fuzzy classifier based on the idea of SVM (support vector machine) training. We assume that in a kernel fuzzy classifier a fuzzy rule is defined for each class in the feature space. In the first method, we tune the slopes of the membership functions at the same time so that the margin between classes is maximized under the constraints that the degree of membership to which a data sample belongs is the maximum among all the classes. This method is similar to a linear all-at-once SVM. We call this AAO tuning. In the second method, we tune the membership function of a class one at a time. Namely, for a class the slope of the associated membership function is tuned so that the margin between the class and the remaining classes is maximized under the constraints that the degrees of membership for the data belonging to the class are large and those for the remaining data are small. This method is similar to a linear one-against-all SVM. This is called OAA tuning. According to the computer experiment for fuzzy classifiers based on kernel discriminant analysis and those with ellipsoidal regions, usually both methods improve classification performance by tuning membership functions and classification performance by AAO tuning is slightly better than that by OAA tuning.     

Circular autocorrelation of stationary circular Markov processes

The stationary Markov process is considered and its circular autocorrelation function is investigated. More specifically, the transition density of the stationary Markov circular process is defined by two circular distributions, and we elucidate the structure of the circular autocorrelation when one of these distributions is uniform and the other is arbitrary. The asymptotic properties of the natural estimator of the circular autocorrelation function are derived. Furthermore, we consider the bivariate process of trigonometric functions and provide the explicit form of its spectral density matrix. The validity of the model was assessed by applying it to a series of wind direction data.     

Crystal structure and spin state of Fe0.5Co0.5(NCS)2(bpa)2

Fe(NCS)₂(bpa)₂, Co(NCS)₂(bpa)₂, and Fe_0.5Co_0.5(NCS)₂(bpa)₂ were synthesized. X-ray structural analysis revealed that the structure was the same among the three complexes. Although the metal-ligand distance in the mixed crystals was shorter than the distance in Fe(NCS)₂(bpa)₂ and was similar to that in Co(NCS)₂(bpa)₂, the spin state remained temperature-independent Fe^II high-spin state in ⁵⁷Fe M?ssbauer spectra.     

Paterno-Buchi photocyclization of 2-siloxyfurans and carbonyl compounds. Notable substituent and carbonyl (Aldehyde vs ketone and singlet- vs triplet-excited state) effects on the regioselectivity (Double-bond selection) in the formation of bicyclic exo-oxetanes

Paterno-Buchi coupling, photochemical [2 + 2] cycloaddition, of carbonyl compounds 2a-f with 2-siloxyfurans 1a-d has been investigated in detail. The stereoselective formations of exo-oxetanes 3 and 4 were observed in high yields. The regioselectivity (double-bond selection, 3 vs 4) was found to be largely dependent upon the carbonyls, the substituents at the furan ring, and the excited state of the carbonyls (singlet vs triplet). The photoreaction of aldehydes 2a-c gave bicyclic exo-oxetanes 3 and 4 at regio-random, independent upon their excited states and the substituents at furan ring. However, the photoreaction of the triplet state of ketones 2d-f was found to give regioselectively exo-oxetanes 4, except for the 4-methyl-2-siloxyfurane 1d case. The singlet-excited state of acetone 2f gave both oxetanes 3 and 4 at regio-random. For the singlet-state photochemistry, the approach direction of the electrophilic oxygen of the excited carbonyls to the furan ring is proposed to be an important factor for the exo-stereoselection. The Griesbeck model can rationalize the regio- and exo-selective formation of oxetanes in the triplet-state photoreaction.     

Syntheses of linear ethoxysiloxanes by the oxidative condensation of triethoxysilane

Linear ethoxysiloxanes were synthesized by the oxidative condensation of hydrosilane. Triethoxysilane was subjected to oxidation to form triethoxysilanol, and pentaethoxydisiloxane was formed by the condensation reaction of triethoxysilane with triethoxysilanol. Pentaethoxydisiloxane was also subjected to oxidative condensation to form a mixture of nona- and decaethoxytetrasiloxanes. In contrast, pentaethoxydisiloxane, heptaethoxytrisiloxane, and nonaethoxytetrasiloxanes were subjected to the reaction with ethanol in the presence of zinc to isolate hexaethoxydisiloxane, octaethoxytrisiloxane, and decaethoxytetrasiloxane, respectively.