Riemann—Lebesgue theorem

Пусть?(x) — ограниченн ая функция на отрезке [0,1] и ее функция распределен ияΦ(t) удовлетворяет услов ию $$\Phi \left( t \right) + \Phi \left( { - t} \right) = 1.$$ Еслиf(x) — конечная поч ти всюду функция, то дл яF _n (t) — функции распределе ния произведенияf(x)?(nx) — вы полнены соотношения и В частности, еслиf(x) — и нтегрируемая функци я, то из (1) следует, что $$\mathop {\lim }\limits_{n \to \infty } \mathop \smallint \limits_0^1 f\left( x \right)\varphi \left( {nx} \right)dx = 0 $$      

A convenient synthesis of 3-formyl-2-oxo-1,2-dihydroquinoxaline chlorophenylhydrazones: Novel tautomeric equilibria between hydrazone imine and diazenyl enamine forms with long-range pSSrototropy

3-Formyl-2-oxo-1,2-dihydroquinoxaline chlorophenylhydrazones 4a-c were synthesized from the reactions of 3-methyl-2-oxo-1,2-dihydroquinoxaline 3 with chlorophenyl diazonium salts, and 4b,c were found to exhibit the tautomeric equilibria between the hydrazone imine and diazenyl enamine forms in the dimethylsulfoxide solutions.     

Reversible and regioselective reaction of La@C82 with cyclopentadiene

The reversible and regioselective reaction of La@C82 with cyclopentadiene was carried out. The activation and thermodynamic parameter of the retro-reaction was estimated from the Arrhenius plots of the rate constants.     

Effect of molecular packing on adsorption and micelle formation of a homologous cationic surfactant mixture of hexadecyltrimethylammonium bromide and dodecyltrimethylammonium bromide

The surface tension of an aqueous solution of a hexadecyltrimethylammonium bromide (HTAB) and dodecyltrimethylammonium bromide (DTAB) mixture was measured as a function of the total molality and the composition of DTAB at 298.15 K under atmospheric pressure. The phase diagrams of adsorption and micelle formation were constructed and the excess Gibbs energy was evaluated by analyzing the phase diagrams thermodynamically. Both the excess Gibbs energy in the adsorbed film and the excess surface area are negative; therefore the mutual interaction between HTAB and DTAB is said to be stronger than that between the same species and is enhanced with increasing adsorption. By combining the results with those obtained in previous studies, we claimed that DTAB molecules can use effectively the space among the hydrocarbon chains of HTAB molecules and their polar head groups take a staggered arrangement at the surface so as to reduce the electrostatic repulsion. Consequently the dispersion force between hydrophobic chains becomes stronger. Furthermore, the comparison of the excess Gibbs energy in the adsorbed film with that in the micelle shows that the staggered arrangement of molecules is not necessary in the spherical micelle.     

Molecular structure and electrochemical properties of the Pt complex of 1,1′-bis(methylthio)ferrocene and the Pd complexes of 1,4,7-trithia[7]- and 1,5,9-trithia[9] (1,1′)ferrocenophanes

Molecular structures of (triphenylphosphine) [1,1′-bis-(methylthio)ferrocene-S,S′,Fe]Pt(BF₄)₂ (1), (1,5,9-trithia[9]ferrocenophane-S,S′,S″,Fe)Pd(BF₄)₂ (2), and (acetonitrile)(1,4,7-trithia[7]ferrocenophane-S,S′,S″,Fe)Pd(BF₄)₂ (3) were determined by X-ray analyses. The Pt in 1 and the Pd atom in 2 have a somewhat distorted square-planar geometry including the Fe atom of the ferrocene moiety, while the Pd atom in 3 is coordinated by one equivalent of acetonitrile and takes a distorted tetragonal-pyramidal geometry. The distances of the Fe---M bond (M = Pd, Pt) in 1–3 are 2.851(2), 2.827(2), and 3.0962(8) Å, respectively. Cyclic voltammetry of 1–3 gave no reversible wave, but afforded some information supporting the presence of a dative bond.     

A Rapid in Situ Preconcentration Method for Phosphorus-Containing Anions in Natural Waters Using Zirconium Loaded Sep-Pak Accell CM

A simple and rapid preconcentration of orthophosphate and condensed phosphate from natural waters using a Sep-Pak CM Cartridge (Accell^TM Plus CM,Waters) loaded with zirconium (Zr-CM Cartridge) method was developed.     

Super structure formation in a wholly aromatic copolyester fiber

The structure of a wholly aromatic copolyester fiber containing 60 mol% p‐hydroxybenzoic acid (PHB), 20 mol% 4−4′‐dihydroxydiphenyl, 15 mol% terephthalic acid and 5 mol% isophthalic acid was studied by means of electron microscopy and wide‐angle X‐ray diffraction. The X‐ray diffraction pattern of the wholly aromatic copolyester fibers showed two sets of diffractions: one set was composed of sharp spots which arose from relatively high crystalline phase. The crystal structure analyzed by these spots was orthorhombic and the lattice dimensions were a = 0.869 nm, b = 0.510 nm, c = 1.20 nm and ρ = 1.50 g/cm³. Another set was characteristic of streaks on the meridian extending parallel to the equator. X‐ray scattering intensity distribution on the meridian was calculated as the square of Fourier transform of random chain model. Comparison of this intensity distribution with the observed meridional maxima concluded that the streaks were due to rather disordered chains with a PHB content of less than 50%. Dark field image (DFI) taken from the meridional 002 reflection showed that slender crystallites were distributed over the whole visual field, oriented parallel to the fiber axis. On DFI from the equatorial 200 reflection, some of these crystallites were also observable, forming groups that distributed randomly in the field. All crystallites belonging to the same group co‐oriented in a*‐ and c*‐axis directions, though disordered parts intervened among the crystallites. This is attributed to the fact that, though the content of PHB in the segments of disordered parts was only 50%, these PHB held the co‐orientation among the slender crystallites within one group. Heat treatment induced the development of block segment and subsequent crystallite growth with fiber. This reorganization improved the thermostability and the mechanical properties. Copyright © 2000 John Wiley & Sons, Ltd.     

Constraints on four-Fermi contact interactions from low-energy electroweak experiments

We investigate the constraints on four-Fermi contact interactions from low-energy lepton-quark and lepton-lepton scattering experiments – polarization asymmetries in electron (muon)-nucleon scattering experiments, cesium and thallium atom parity violation measurements, neutrino-nuclei and neutrino-electron scattering experiments. These constraints are then combined by assuming the lepton and quark universalities and gauge invariance of the contact interaction, which leave independent six lepton-quark and three pure-leptonic interactions. Impacts of these constraints on models with an additional -boson are briefly discussed. We also present updates of the low-energy constraints on the and parameters. Received: 28 November 1997 / Published online: 26 February 1998     

The limiting normality of the test statistic for the two-sample problem induced by a convex sum distance

Critchlow (1992, J. Statist. Plann. Inference, 32, 325–346) proposed a method of a unified construction of a class of rank tests. In this paper, we introduce a convex sum distance and prove the limiting normality of the test statistics for the two-sample problem derived by his method.     

Analysis of the hysteresis curve of ferroelectric polymers by a phenomenological relaxation theory

A simple relaxation theory for the displacement versus electric field hysteresis of ferroelectric polymers is developed in which the relaxation time is assumed to be a function of electric field, as has been experimentally evidenced by polarization-reversal switching. The theory gives an analytical expression for the hysteresis curve. The coercive field E_c predicted by the theory agrees well with data on E_c as a function of temperature for poly(vinylidene fluoride) from ?60 to 20°C and with data on E_c as a function of frequency for vinylidene fluoride/trifluoroethylene copolymer (73/27 molar ratio) at 20°C over the range 0.01–0.7 Hz.