A note on unit and class number of real quadratic fields

Letp be an odd prime withp1 (mod 4) andQ(p) be the real quadratic field. Also let andh denote the fundamental unit and the class number of Q(p), respectively. The main purpose of this paper is to study the explicit expressions of ^h and ^2h , and to discuss the problems related to the conjecture of Ankeny-Artin-Chowla.     

Highly regio- and stereocontrolled SN2′ reactions of gem-difluorinated vinyloxiranes with monoalkylcopper reagents

Reaction of gem-difluorinated vinyloxiranes with RCu(X)Li allowed us to introduce the R group regioselectively at the fluorine-attached terminal carbon atom in an S_N2′ manner to afford (E)-allylic alcohols exclusively, while homoallylic alcohols with anti stereochemical relationship were found to be obtained selectively from higher-ordered cuprates derived from CuCl and RMgBr in a ratio of 1:3.     

Fluorescence spectra,fluorescence quantum yields and fluorescence lifetimes of polymers having trans-1,2-dicarbazolylyclobutane or carbazolyl group

Quantum yields, lifetimes and shapes of fluorescence from polymers containing the trans-1,2-dicarbazolylcyclobutane (DCZB) or carbazolyl structures were studied in N,N-dimethylformamide. No sandwich-type excimer formation was observed for DCZB polymers. The so-called second excimer observed in poly(9-vinylcarbazole) might also be produced in poly(9-ethyl-3-vinylcarbazole).     

Electrochiroptical systems based on biphenyl-2,2′-diyl-type dicationic dyes: strong chiroptical signals through the transmission of point chirality to axial chirality

Huge CD amplitude (A = +538) was attained by diastereomeric preference (50% de) for the title dication with two bis[4-(R)-sec-butoxyphenyl]methylium units in benzene at 23 °C. Intramolecular π-π stacking is the origin for effective transmission of point chirality to axial chirality to attain the chiroptical enhancement of 400-times. Thanks to the strong CD signaling, chiroptical changes upon electrolysis could be readily detected, thus realizing a new class of electrochiroptical response systems. Chiroptical properties could also be modified by solvent polarity.     

Thermodynamic Insight into the Origin of a Calix[n]arene-[60]fullerene Interaction and its Application to a Porphyrin-[60]fullerene Energy Transfer System

The calixarene-fullerene interaction,which causes only a slight change in the absorptionspectra, has been substantiated by calorimetricmeasurements: the H° values arecomparable with those obtained from the associationconstants estimated by a spectroscopic method. Furthermore, we determined the association constantbetween homooxacalix[3]arene and [60]fullerenederivatives with a porphyrin moiety by fluorescencespectra.     

Synthesis of 3-substituted 4H-1-benzothiopyran-4-ones

3-Substituted 2-phenyl-4H-1-benzothiopyran-4-ones (thioflavones) were prepared to test antimicrobial activity. It was found that 3-(phenyl)thiochromone derivatives (isothioflavones) were prepared by the Meerwein reaction of thiochromone with p-nitrobenzenediazonium ion. 3-(Formyl)thioflavone exhibits weak antimicrobial activity against Trichophytons and Candida.     

Formation of non-racemic E- and Z-olefins based on discrimination of enantiotopic carbonyl groups in α-diketones by a chiral phosphonate reagent

A chiral HWE reagent reacted with an alternative carbonyl group of meso-α-diketones of bicyclo[2.2.1] system to give non-racemic (Z)- and (E)-olefins, respectively.     

A rigorous control of logarithmic corrections in four-dimensional ϕ4 spin systems

Using Gawedzki and Kupiainen's rigorous block spin transformation method, we study critical phenomena in ⁴ spin systems in four dimensions. In Part I of this work we investigate in detail the renormalization group trajectory of the system not exactly at the critical point.