Conjugate direction definition

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

There are different ways of defining the coefficients /3 . The basic idea is to make directions l(m,j ]) and l(m,j) conjugate in some geometrical sense, which would guarantee much faster convergence of the iterations. We will introduce a definition of the conjugate directions below. 

The nonzero vectors dj,. . . , dj are said to be conjugate with respect to the positive definite matrix H if they are linearly independent and d Hdy = 0 for i A j. A method that generates such directions when applied to a quadratic function with Hessian matrix H is called a conjugate direction method. These methods will locate the minimum of a quadratic function in a finite number of iterations, and they can also be applied iteratively to optimize nonquadratic functions. 

By definition, every carbanion possesses an unshared pair of electrons and is therefore a base. When a carbanion accepts a proton, it is converted to its conjugate acid (see Chapter 8). The stability of the carbanion is directly related to the strength of the conjugate acid. The weaker the acid, the greater the base strength and the lower the stability of the carbanion. Here, by stability we mean stability toward a proton donor the lower the stability, the more willing the carbanion is to accept a... 

In general, a set of n linearly independent directions of search s°, s, s" are said to be conjugate with respect to a positive-definite square matrix Q if... 

Structures 1 and 1, with identical relative configuration at C-2. C-3, are not covered by Masamune s very scant directions which do not include a guideline on the definition of the backbone. Thus, an allylmetal expert , having prepared 1 as one member of a series with the substituent of the ally] group varied, will probably describe 1 as syn, but investigators working on aldol reactions or conjugate additions will certainly prefer the term anti for 1. ... 

In the Bronsted definition, an acid donates a proton and a base accepts a proton. The strengths of acids and bases are measured by the extent to which they lose or gain protons, respectively. In these reactions acids are converted to their copjugate bases and bases to their conjugate acids. Acid-base reactions go in the direction of forming the weaker acid and the weaker base. 

Although no free epinephrine is excreted under the conditions of those experiments just described, it is permissible to question whether they are representative of the fate of epinephrine as it is secreted in the body, or whether conjugation is a mode of inactivation of catechol pressor amines administered as drugs. The relatively minute amounts of adrenaline or noradrenaline secreted normally and the limitations of present analytical methods definitely handicap a direct approach to the problem. However, it has been possible to study the fate of adrenaline secreted by the body in a more or less physiological type of experiment. 

The first iteration in a CG method is the same as in SD, with a step along the current negative gradient vector. Successive directions are constructed differently so that they form a set of mutually conjugate vectors with respect to the (positive-definite) Hessian A of a general convex quadratic function. 

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