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Alexander Shapiro 《Numerische Mathematik》1982,39(2):239-245
Summary We shall in this paper consider the problem of determination a row or column scaling of a matrixA, which minimizes the condition number ofA. This problem was studied by several authors. For the cases of the maximum norm and of the sum norm the scale problem was completely solved by Bauer [1] and Sluis [5]. The condition ofA subordinate to the pair of euclidean norms is the ratio /, where and are the maximal and minimal eigenvalue of (A
H
A)1/2 respectively. The euclidean case was considered by Forsythe and Strauss [3]. Shapiro [6] proposed some approaches to a numerical solution in this case. The main result of this paper is the presentation of necessary and sufficient conditions for optimal scaling in terms of maximizing and minimizing vectors. A uniqueness proof for the solution is offered provided some normality assumption is satisfied. 相似文献
58.
A. I. Shatenshtein N. N. Magdesieva Yu. I. Ranneva I. O. Shapiro A. I. Serebryanskaya 《Theoretical and Experimental Chemistry》1969,3(3):194-196
Deuteration measurements have been made with t-C4H9OK and with 4 moles of CH3COOH mixed with one mole of CF3COOH against 2D-selenophen, 3D-selenophen, 3CH3, 2D-selenophen and 5CH3, 2D-selenophen. Reactivity in the selenohpen-thiopen series is examined via protophilic and electrophilic isotopic-exchange reactions. The factors for the partial exchange rates with acid and base for heterocyclics with D at position 2 and CH3 at position 3, 4, or 5 are compared with those factors for isomers of monodeuterotoluene; it is found that the electronic effect of the CH3 group is transmitted similarly for these heterocyclics and for the benzene ring. Reasonably good agreement is found between the relative constants for deuterium exchange in thiophen, selenophen, and methyl derivatives of these as catalyzed by alkali-metal t-butylates in dimethylsulfoxide (DMSO) and in t-butanol mixed with diethylene glycol dimethyl ether. This shows that the results with DMSO correctly characterize the reactivity in protophilic hydrogen exchange. 相似文献
59.
A. I. Kokorin L. S. Bogach A. B. Shapiro É. G. Rozantsev 《Russian Chemical Bulletin》1976,25(9):1874-1878
1. | Nitroxyl biradicals of the triazine, N,N-bis(triazinyl)ethylenediamine, N,N-bis(triazinyl)piperazine, and s-nitropyrimidine series have been prepared. |
2. | The ESR spectra of liquid and frozen solutions of these biradicals have been obtained and N. - O group separations calculated. |
3. | Toluene solutions of (II), (V), (VI), and (X) biradicals contain at least four different conformations, each with its own value of the exchange integral. |
4. | In one of the biradical (V) conformations, spin interaction exchange is through the chain of atoms linking the N.-O fragments. |
60.
Gerbasi D Brumer P Thanopulos I Kral P Shapiro M 《The Journal of chemical physics》2004,120(24):11557-11563
An application of a recently proposed [P. Kral et al., Phys. Rev. Lett. 90, 033001 (2003)] two step optical control scenario to the purification of a racemic mixture of 1,3 dimethylallene is presented. Both steps combine adiabatic and diabatic passage phenomena. In the first step, three laser pulses of mutually perpendicular linear polarizations, applied in a "cyclic adiabatic passage" scheme, are shown to be able to distinguish between the L and D enantiomers due to their difference in matter-radiation phase. In the second step, which immediately follows the first, a sequence of pulses is used to convert one enantiomer to its mirror-imaged form. This scenario, which only negligibly populates the first excited electronic state, proves extremely useful for systems such as dimethylallene, which can suffer losses from dissociation and internal conversion upon electronic excitation. We computationally observe conversion of a racemic mixture of dimethylallene to a sample containing approximately 95% of the enantiomer of choice. 相似文献