A note on the identity operators of fractional calculus

The application of an identity operator for Saigo’s fractional calculus operators is shown by evaluating the limit of an indeterminate form. Its special case yields the result which has been used as an infinitesimal generator in the semigroup theory. Also, an identity operator for the recently introduced multi-dimensional fractional operators (due to Srivastava and Raina [8]) is discussed.     

The gauge transformations of the Schwinger model

We determine the group of implementable local gauge transformations of massless quantum electrodynamics in two space-time dimensions in the covariant Landau gauge. It splits into an infinite discrete set of disjoint classes. The unitary operators representing the implementable gauge transformations are constructed explicitly. A subset of these operators does not reduce to multiples of the identity in the physical Hilbert space constructed according to the usual rules. The disappearance of the fermionic degrees of freedom is related to this fact. Combined with the properties of the global chiral transformations, it provides a better understanding of the model's vacuum structure.     

Error bound for an osculatory quadrature formula

Sharper estimates have been found for Donaldson's osculatory quadrature formula.     

A Laurent series expansion formula and its applications

This paper gives a certain Laurent series expansion for a generalized Rodrigues type formula. The main result finds many applications which are enumerated briefly.     

A class of convolution integral equations

A new class of convolution integral equations whose kernels involve an H-function of several variables, which is defined by a multiple contour integral of the Mellin-Barnes type, is solved. It is also indicated how the main theorem can be specialized to derive a number of (known or new) results on convolution integral equations involving simpler special functions of interest in problems of applied mathematics and mathematical physics.     

An algebraic geometry study of theb−c system with arbitrary twist fields and arbitrary statistics

We present an analysis of the generalb–c system (including the – system) on a compact Riemann surface of arbitrary genusg0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion (OPE) of the system. Studying a very (in fact optimally) general form of theb–c system, we prove rigorously that the standard practice of eliminating zero modes, and even the standard lagrangian, follow from the analyticity structure dictated by the OPE alone. We extend the analysis to consider the most general case of the presence of twist (e.g. spin) fields. We then determine all the possible correlation functions of theb–c system, with statistics unspecified, compatible with the OPE. On imposing Fermi and Bose statistics, we obtain the correlation functions of the fermionicb–c and – systems, respectively.     

Template synthesis,characterization and electrochemical studies of copper(II) macrocyclic complexes derived from 4-methyl-2,6-di(formyl/benzoyl) phenol and diamines

Copper(II) macrocyclic complexes [Cu(L)Cl2]·2H2O (1,2) have been synthesized from the copper(II)-directed condensation of 4-methyl-2,6-diformylphenol (1) or 4-methyl-2,6-dibenzoylphenol (2) with diamines [1,2-diaminopropane (a), 1,3-diaminopropane (b) and o-phenylenediamine (c)] in EtOH in high yields. These complexes are monomeric, non-electrolytes and paramagnetic, indicating a distorted octahedral geometry around copper, which is supported by i.r., electronic, e.s.r. and magnetic susceptibility measurements. Electrochemical studies of (1) and (2) using c.v. indicate an irreversible cathodic peak (ca. –0.65 to –0.76 V) corresponding to reduction of copper(II) to copper(I) and the diffusion-controlled nature of the electrode process.     

Aminoaryltellurium(IV) trihalides

The preparation and properties of some aminoaryltellurium(IV) trihalides are reported. Conductivity measurements show the compounds to behave as 1:1 electrolytes when in dilute solution. Cryoscopic measurements reflect the molecular, and sometimes polymeric, nature of the materials in more concentrated solutions. The ¹H nmr and infrared spectra are discussed.     

Heterogeneous nucleation of nitric acid trihydrate on clay minerals: relevance to type ia polar stratospheric clouds

Although critical to atmospheric modeling of stratospheric ozone depletion, selective heterogeneous nuclei that promote the formation of Type Ia polar stratospheric clouds (PSCs) are largely unknown. While mineral particles are known to be good ice nuclei, it is currently not clear whether they are also good nuclei for PSCs. In the present study, a high-vacuum chamber equipped with transmission Fourier transform infrared spectroscopy and a quadrupole mass spectrometer was used to study heterogeneous nucleation of nitric acid trihydrate (NAT) on two clay minerals-Na-montmorillonite and kaolinite-as analogs of atmospheric terrestrial and extraterrestrial minerals. The minerals are first coated with a 3:1 supercooled H2O/HNO3 solution prior to the observed nucleation of crystalline NAT. At 220 K, NAT formation was observed at low SNAT values of 12 and 7 on kaolinite and montmorillonite clays, respectively. These are the lowest SNAT values reported in the literature on any substrate. However, NAT nucleation exhibited significant temperature dependence. At lower temperatures, representative of typical polar stratospheric conditions, much higher supersaturations were required before nucleation was observed. Our results suggest that NAT nucleation on mineral particles, not previously treated with sulfuric acid, may not be an important nucleation platform for Type Ia PSCs under normal polar stratospheric conditions.     

Textural,thermal, optical and electrical properties of Iron nanoparticles dispersed 4′-(Hexyloxy)-4-biphenylcarbonitrile liquid crystal mixture

In this work, the effect of Iron nanoparticles (Fe NPs) dispersion in 4′-(Hexyloxy)-4-biphenylcarbonitrile (6OCB) nematic liquid crystal properties has been studied. Inclusion of Fe NPs (0.25 wt. %) in 6OCB liquid crystal (LC) on textures, isotropic–nematic transition temperature (T_I–N), electro-optical and dielectric properties have been investigated in planar aligned cell. The threshold voltage (V_th) and T_I–N decrease after dispersion of Fe NPs. Dielectric spectroscopy in nematic phase show that relaxation frequency (f_r) also decreases after dispersion of Fe NPs in 6OCB. The observed relaxation mode is due to the flip-flop motion of LC molecules about their short axis. The band gap and AC conductivity in case of 6OCB-Fe sample increase over pure 6OCB sample. A decrease in activation energy is also noticed.