Spectral shift function and trace formula

The complete proofs of Krein’s theorem on the spectral shift function and the trace formula are given for a pair of self-adjoint operators such that either (i) their difference is trace-class or (ii) the difference of their resolvents is trace-class. The proofs, essentially due to Krein, is based on Herglotz’s theorem on the boundary value of the analytic functions whose imaginary part is non-negative on the upper half plane, and an almost optimal class of functions are obtained for which the trace formula is valid. Also an alternative method based on Weyl-von Neumann’s theorem for self-adjoint operators, avoiding the complex function theory and inspired by Voiculescu’s work, is given for the first case. Furthermore, some applications of the spectral shift function have been discussed.     

Kinetics and mechanism of the oxidation of acetaldehyde by chromic acid in perchloric acid

Summary The oxidation of MeCHO by chromium(VI) has been studied in HClO₄ medium over a wide range of experimental conditions and has been found to obey the rate law;v=k[MeCHO][HCrO ₄ ^– ][H⁺]. The calculated H and-S values for the reaction are 30±2kJ mol^–1 and 171±7J mol^–1deg^–1, respectively. The mechanism is discussed in terms of carbon-hydrogen bond cleavage.     

Use of a chiral surfactant for enantioselective reduction of a ketone

The influence of a chiral surfactant and a polymer-supported chiral additive on reduction of ketones using sodium borohydride will be described. Initial preparations involved methylation of (S)-leucinol to give (2S)-N , N-dimethyl-2-amino-4-methyl-1-pentanol (1) (67%). The chiral surfactant (2) was synthesized by reacting (1) with bromohexadecane (71%). The functionalized styrene for the polymer-supported chiral additive (5) was synthesized by reacting (1) with 4-vinylbenzyl chloride. Polymerization was carried out with 10% of the functionalized monomer (4), 5% cross-linking agent divinylbenzene, and 85% styrene with AIBN as the initiator. The activity of the chiral surfactant and polymeric additive were examined by using them as additives in a standard reduction of 2-pentanone with sodium borohydride to yield (R)- and (S)-2-pentanol (3) (20%). The resulting alcohol was analyzed by polarimetry (ee 9.5%) and also esterified with (2S)-methylbutyric acid prior to characterization by NMR. 13C NMR indicated an enantiomeric excess of 5.2% when the chiral surfactant was used, and 7% when the polymeric additive was used.     

Structural features of some small carbonyls in the 1,3nπ states: An mcscf study

Structural features of a series of tetraatomic carbonyls in the ^1,3nπ* states are studied by using a recently proposed MCSCF algorithm at the INDO/2 level. The computed geometrical parameters and the inversion barrier heights agree quite well with the available experimental or ab initio theoretical data. The calculated transition energies turn out to be somewhat lower than the experimental counterparts. The specific MCSCF algorithm used is found to have excellent convergence properties.     

Stop times in fock space stochastic calculus

Summary A stop time S in the boson Fock space over L ₂()₊ is a spectral measure in [0,] such that {S([0,t])} is an adapted process. Following the ideas of Hudson [6], to each stop time S a canonical shift operator U ^Sis constructed in . When S({}) has the vacuum as a null vector U ^Sbecomes an isometry. When S({})=0 it is shown that admits a factorisation _S]{S where _{S is the range of U ^Sand _S] is a suitable subspace of called the Fock space upto time S. This, in particular, implies the strong Markov property of quantum Brownian motion in the boson as well as fermion sense and the Dynkin-Hunt property that the classical Brownian motion begins afresh at each stop time. The stopped Weyl and fermion processes are defined and their properties studied. A composition operation is introduced in the space of stop time to make it a semigroup. Stop time integrals are introduced and their properties constitute the basic tools for the subject.     

Ternary copper complexes for photocleavage of DNA by red light: direct evidence for sulfur-to-copper charge transfer and d-d band involvement

A new class of ternary copper(II) complexes of formulation [Cu(L(n)B](ClO(4)) (1-4), where HL(n) is a NSO-donor Schiff base (HL(1), HL(2)) and B is a NN-donor heterocyclic base viz. 1,10-phenanthroline (phen) and 2,9-dimethyl-1,10-phenanthroline (dmp), are prepared, structurally characterized, and their DNA binding and photocleavage activities studied in the presence of red light. Ternary complex [Cu(L(3))(phen)](ClO(4)) (5) containing an ONO-donor Schiff base and a binary complex [Cu(L(2))(2)] (6) are also prepared and structurally characterized for mechanistic investigations of the DNA cleavage reactions. While 1-4 have a square pyramidal (4 + 1) CuN(3)OS coordination geometry with the Schiff base bonded at the equatorial sites, 5 has a square pyramidal (4 + 1) geometry with CuN(3)O(2) coordination with the alcoholic oxygen at the axial site, and 6 has a square planar trans-CuN(2)O(2) geometry. Binding of the complexes 1-4 to calf thymus DNA shows the relative order: phen > dmp. Mechanistic investigations using distamycin reveal minor groove binding for the complexes. The phen complexes containing the Schiff base with a thiomethyl or thiophenyl moiety show red light induced photocleavage. The dmp complexes are essentially photonuclease inactive. Complexes 5 and 6 are cleavage inactive under similar photolytic conditions. A 10 microM solution of 1 displays a 72% cleavage of SC DNA (0.5 microg) on an exposure of 30 min using a 603 nm Nd:YAG pulsed laser (60 mJ/P) in Tris-HCl buffer (pH 7.2). Significant cleavage of 1 is also observed at 694 nm using a Ruby laser. Complex 1 is cleavage inactive under argon or nitrogen atmosphere. It shows a more enhanced cleavage in pure oxygen than in air. Enhancement of cleavage in D(2)O and inhibition with sodium azide addition indicate the possibility of the formation of singlet oxygen as a reactive intermediate leading to DNA cleavage. The d-d band excitation with red light shows significant enhancement of cleavage yield. The results indicate that the phen ligand is necessary for DNA binding of the complex. Both the sulfur-to-copper charge transfer band and copper d-d band excitations helped the DNA cleavage. While the absorption of a red photon induces a metal d-d transition, excitation at shorter visible wavelengths leads to the sulfur-to-copper charge transfer band excitation at the initial step of photocleavage. The excitation energy is subsequently transferred to ground state oxygen molecules to produce singlet oxygen that cleaves the DNA.     

Spectroscopic and structural features of small thiocarbonyl molecules in low-lying excited states: Further applications of a variant of the orthogonal gradient method

Structural parameters of a set of five thiocarbonyl molecules in the lowest nπ* states are calculated by using a generalized orbital optimization algorithm (a variant of the orthogonal gradient method) in an INDO MCSCF framework. Transition energies, singlet-triplet splittings, planar inversion barriers, and dipole moments in nπ* states of different spin multiplicities are reported. Predicted structural features agree reasonably well with available experimental or theoretical data. Some interesting trends are noted in the computed inversion barrier heights, singlet-triplet splittings, and dipole moments in nπ* states.     

Oxidations with IBX: benzyl halides to carbonyl compounds, and the one-pot conversion of olefins to 1,2-diketones

A variety of benzyl halides were converted to the corresponding aldehydes/ketones in respectable yields by IBX in DMSO at 65 °C. The bromohydrin reaction of olefins using NBS-H₂O in DMSO can be nicely adapted to IBX-mediated oxidation of benzyl halides in such a way that olefins are converted to the corresponding 1,2-diketones in good isolated yields in one-pot.     

Cell separations using targeted monoclonal antibodies against overproduced surface proteins

Applied Biochemistry and Biotechnology - The capacity of polystyrene microspheres with immobilized antibodies against type 1 pili ofE. coli was measured. Using pure IgG-type monoclonal antibodies...