Extended kac-akhiezer formulae and the fredholm determinant of finite section hilbert-schmidt kernels

This paper deals with some results (known as Kac-Akhiezer formulae) on generalized Fredholm determinants for Hilbert-Schmidt operators on L2-spaces, available in the literature for convolution kernels on intervals. The Kac-Akhiezer formulae have been obtained for kernels which are not necessarily of convolution nature and for domains in R(n).

Trace Formulae for Curvature of Jet Bundles over Planar Domains

For a domain Omega in C and an operator T in B-n(Omega), Cowen and Douglas construct a Hermitian holomorphic vector bundle E-T over Omega corresponding to T. The Hermitian holomorphic vector bundle E-T is obtained as a pull-back of the tautological bundle S(n, H) defined over by Gr(n, H) a nondegenerate holomorphic map z bar right arrow ker(T - z), z is an element of Omega. To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are rather intricate. They have given a set of invariants to determine if two rank n Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle J(k)(L-f), we have shown that the curvature of the line bundle L-f completely determines the class of J(k)(L-f). In case of rank Hermitian holomorphic vector bundle E-f, We have calculated the curvature of jet bundle J(k)(E-f) and also obtained a trace formula for jet bundle J(k)(E-f).

Trace Formulae in Operator Theory

In this article, we survey several kinds of trace formulas that one encounters in the theory of single and multi-variable operators. We give some sketches of the proofs, often based on the principle of finite-dimensional approximations to the objects at hand in the formulas.

Lanthanide chelate complexes of salicyloyl hydrazide-salicylaldehyde schiff base (SHSASB) and anthranilic acid-salicylaldehyde schiff base (AASASB)

New lanthanide complexes of salicylaldehyde-Schiff bases with salicyloyl hydrazide and anthranilic acid, were synthesized by a novel method consisting of refluxing the mixtures of Schiff base ligands and lanthanide trichloroacetate in acetone. Solid complexes of formulae Ln(SHSASB)s*2Hz0 and Ln2(AASASB)s*2Hz0 where Ln = La-Yb and Y, were isolated. Proton NMR and IR spectra for the complexes reveal the bidentate binding of both the Schiff base ligands to the lanthanide ion. Electronic spectra along with the conductance data for the complexes indicate a coordination number of six for the lanthanide ion in the complexes of both the Schiff bases.

Synthesis and Spectroscopic Investigations of Complexes of Lanthanide Nitrates with N-(4-Hethyl-2-Pyridyl)-Acetamide

New complexes of Lanthanide nitrates with N-(4-methyl-2-pyridyl)-acetamide (4-me-aapH) of the general formulae. [Ln(4-me-aapH)2] [NO3] (where Ln=La=La-Yb and Y)have been synthesized and haracterised by chemical analysis, molar conductivity and physical methods such as infrared, 13C NMR an electronic spectra in the visible region. Molar conductance and infrared data point to the presence to the coordinated nitrates groups. Infrared and 13C NMR data have been interpreted in terms of the coordination of the legand to the metal ion through the oxygen of the secondary amide and nitrogen of the hetrocyclic ring, in a bidentate fashion. Coordination number of ten seems probable for the complexes.

Enumeration of hypergraphs

Formulae for the generating functions for hypergraphs, dihypergraphs, oriented hypergraphs, selfcomplementary directed hypergraphs and self complementary hypergraphs are presented here.

Complexes of lanthanide nitrates with 2-N-(6-picoIyl)-benzamide

New complexes of lanthanide nitrates with 2-N-(6-picolyl)-benzamide of the formulae Ln2[6-pic-BA], [NO3l6 (Ln = Y and La-Yb) have been prepared and characterised by chemical analysis, infrared, molar conductance and electronic spectral data. Molar conductance data along with IR data point to the presence of co-ordinated nitrate groups. IR spectra prove the bidentate co-ordination of the ligand to the metal ion, through the oxygen of the secondary amide and the nitrogen of the heterocyclic ring. Electronic spectral studies in the visible region suggest an eight co-ordinate geometry around the metal ions.

On a linear steady-state system of equations in microcontinuum fluid mechanics

Galerkin representations and integral representations are obtained for the linearized system of coupled differential equations governing steady incompressible flow of a micropolar fluid. The special case of 2-dimensional Stokes flows is then examined and further representation formulae as well as asymptotic expressions, are generated for both the microrotation and velocity vectors. With the aid of these formulae, the Stokes Paradox for micropolar fluids is established.

Complexes of Lanthanide Perchlorates with N-(2-Pyrimidyl)benzamide

New complexes of lanthanide perchlorates with N-(2-pyrimidyl)benzamide (BApymH) of the general formulae [Ln(BApymH)4](ClO4)3 (where Ln = La-Yb and Y) have been synthesised and characterised by chemical analysis, molar conductivity and physical methods such as infrared and electronic spectra in the visible region. Molar conductance and infrared data point to the ionic nature of the per-chlorate groups in the complexes. IR data unequivocally proves that the coordination of the ligand to the metal ion takes place in a bidentate fashion through the oxygen of the secondary amide and nitrogen of the pyrimidine ring. From a comparison of the visible electronic spectral shapes of the Nd3+ and Ho3+ complexes with those reported in the literature, an eight coordinate geometry around the metal ion has tentatively been assigned in all the complexes.

Complexes of Rare-Earth Metals with Meconic Acid

Two series of complexes of meconic acid (H3 Mec) with rare-earths have been prepared by varying the preparative procedure. The compounds have the general formulae, [Ln(Mec) (H2O)2]·3 H2O (whereLn=La, Ce, Pr, Nd, Sm, Ho and Y) and [Ln(HMec) (H2 Mec) (H2O)2]·4 H2O (whereLn=La, Pr, Nd and Sm). The infrared spectral data indicate that the carboxylate groups are bound to the rare-earth metal in a bidentate fashion. Thermal studies indicate that two water molecules are coordinated in each case. The complexes are probably polymeric.

Lanthanide perchlorate complexes of 4-cyano pyridine-N-oxide, 4-chloro 2-picoline-N-oxide and 4-dimethyl amino-2-picoline-N-oxide

Complexes of lanthanide perchlorates with 4-cyano pyridine-1-oxide, 4-chloro 2-picoline-1-oxide and 4-dimethyl-amino 2-picoline-1-oxide have been isolated for the first time and characterized by analysis, conductance, infrared, NMR and electronic spectra. The complexes of 4-cyano pyridine-1-oxides have the composition Ln(CyPO)6(ClO4)3. 2H2O (Ln=La, Sm, Dy and Ho); Ln(CyPO)7 (ClO4)3. 2H2O (Ln=Pr, Nd, Er and Yb); and Ln(CyPO)5 (ClO4)3. 2H2O (Ln=Gd and Tb). The complexes of 4-chloro 2-picoline-1-oxide analyse for the formulae Ln(CpicO)6 (ClO4)3 (Ln=La, Pr, Nd and Ho); and Ln (CpicO)5 (ClO4)3 (Ln=Er and Yb), and those of 4-dimethylamino 2-picoline-1-oxide for Ln(DMPicO)6 (ClO4)3 (Ln=La and Nd); Ln(DMPicO)7 (ClO4)3 (Ln=Gd, Er and Yb); and Ln(DMPicO)8 (ClO4)3 (Ln=Dy and Ho).

On the unsteady flow of a fairly concentrated suspension due to contracting or expanding walls of a tube

Consummating our earlier work [1], the unsteady flow of a fairly concentrated suspension due to a single contraction or expansion of the walls of a tube is studied. A comparison of the results obtained by using two different formulae for the additional drag terms in the governing equations has been made. A region of circulation in the flow field is observed when the volume fraction Z greater-or-equal, slanted 0.3, the Schmidt number Sc < 1 and the density ratio (density of the particulate phase/density of the fluid phase) > 1.

Natural frequencies of polar orthotropic annular plates

The classical Rayleigh-Ritz method with simple polynomials as admissible functions has been used for obtaining natural frequencies of transversely vibrating polar orthotropic annular plates. The method in conjunction with transformations introduced in the analysis has been found to be quite effective, particularly for large hole sizes. Estimates of natural frequencies corresponding to modes with one as well as two nodal diameters are obtained for the nine combinations of clamped, simply supported and free edge conditions and for various values of rigidity ratio and hole sizes. Based on the variation of eigenvalue parameter with rigidity ratio, the frequencies of these modes as well as those of axisymmetric modes have been expressed by means of simple formulae in terms of rigidity ratio and the frequencies of corresponding modes in the isotropic case. These formulae have been used in determining the fundamental frequencies of orthotropic plates.

Unsteady laminar boundary layers in a compressible stagnation flow

The unsteady laminar compressible boundary-layer flow in the immediate vicinity of a two-dimensional stagnation point due to an incident stream whose velocity varies arbitrarily with time is considered. The governing partial differential equations, involving both time and the independent similarity variable, are transformed into new co-ordinates with finite ranges by means of a transformation which maps an infinite interval into a finite one. The resulting equations are solved by converting them into a matrix equation through the application of implicit finite-difference formulae. Computations have been carried out for two particular unsteady free-stream velocity distributions: (1) a constantly accelerating stream and (2) a fluctuating stream. The results show that in the former case both the skin-friction and the heat-transfer parameter increase steadily with time after a certain instant, while in the latter they oscillate thus responding to the fluctuations in the free-stream velocity.

Complexes of lanthanide iodides with 4-methylpyridine-1-oxide and 2-methylpyridine-1-oxide

Complexes of lanthanide iodides with 4-methylpyridine-1-oxide and 2-methylpyridine-1-oxide of the formulae Ln(4-MePyO)8I3.xH2O (x=0 or 2) and Ln(2-MePyO)5I3.xH2O (x=0, 1 or 3) have been prepared and characterized by analyses, conductance, infrared and proton NMR data. Infrared spectra of the complexes indicate that the coordination of the ligand to the metal ion takes place through the oxygen of the N-O group of the ligand. Proton NMR data for the paramagnetic complexes indicate that both contact and pseudocontact interactions are responsible for the isotropic shifts. Proton NMR spectra of the 2-methylpyridine-1-oxide complexes indicate a restricted rotation of the ligand about the N-O group.