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1.
2.
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. 相似文献
3.
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. 相似文献
4.
Kalyan Kumar Das D. Mukherjee S. P. Bhattacharyya 《International journal of quantum chemistry》1986,29(6):1745-1754
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. 相似文献
5.
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. 相似文献
6.
Kalyan K. Das D. Mukherjee S. P. Bhattacharyya 《International journal of quantum chemistry》1989,35(4):483-494
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. 相似文献
7.
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... 相似文献
8.
Summary Rate constants for the oxidation of thiosulphate by hexachloroplatinate(IV) have been measured. The kinetics of the oxidation of thiosulphate follow a second-order rate law, first order with respect to thiosulphate and first order with respect to platinum(IV). The influence of pH is small. The rates are found to depend on the nature and concentration of the cations and follow the order: Cs+>Rb+>K+>Na+>Li+. The activation parameters calculated from the temperature studies are: H=42.9 k J mol–1 and S=–102 JK–1 mol–1. A mechanism of the reaction in terms of intermediate formation of free radicals followed by the formation of tetrathionate is postulated to explain the kinetic behaviour. 相似文献
9.
The oxidation of glycolic, lactic, malic, and a few substituted mandelic acids by 2,2′‐bipyridinium chlorochromate (BPCC) in dimethylsulphoxide leads to the formation of corresponding oxoacids. The reaction is first order each in BPCC and the hydroxy acids. The reaction is catalyzed by the hydrogen ions. The hydrogen ion dependence has the form: kobs = a + b [H+]. The oxidation of α‐deuteriomandelic acid exhibited a substantial primary kinetic isotope effect (kH/kd = 5.29 at 303 K). Oxidation of p‐methylmandelic acid was studied in 19 different organic solvents. The solvent effect has been analyzed by using Kamlet's and Swain's multiparametric equations. A mechanism involving a hydride ion transfer via a chromate ester is proposed. © 2002 Wiley Periodicals, Inc. Int J Chem Kinet 34: 248–254, 2002 相似文献
10.
Kalyan Kali Sen Gupta Ambikesh Mahapatra Balaram Ghosh 《Transition Metal Chemistry》1991,16(5):500-502
Summary The reactivities of - and -glycerophosphates towards MnO
4
–
in HClO4 medium have been examined. The reactions between glycerophosphates and MnO
4
–
are first order with respect to [glycerophosphate], [MnO
4
–
] and [H+]. There is no evidence for cleavage of the C–O–P bond rather than the terminal –CH2OH group being oxidized to the respective phosphoglyceraldehyde. 相似文献