Generalized Power Function Research Articles

An ab initio study of the influence of substituents and intramolecular hydrogen bonding on the carbonyl bond length, and the harmonic and anharmonic stretching force constants. I. Monosubstituted carbonyl compounds

The CO bond length and the quadratic, cubic and quartic stretching force constants, calculated ab initio using the unscaled 4–31G basis set with full geometry optimization, are reported for three series of monosubstituted carbonyl compounds in which the atom directly bonded to the carbonyl carbon is another carbon, a nitrogen, or an oxygen atom, respectively. The data are analyzed in terms of the In ƒ versus In r e relationship, and also the generalized power functions and exponential functions proposed by Herschbach and Laurie. Not only does the atom directly bonded to the carbonyl carbon affect the magnitude of r e and the force constants, but the rest of the substituent group is found to be capable of exerting an even greater influence. Within each series of compounds the overall progression from the shortest to the longest CO bonds is tentatively attributed to a diminishing electron density in the bonding region.

A new approach to performance-oriented flow control

Flow control is proposed as a means of obtaining an "optimal tradeoff" between low delay and high throughput in computer networks. Several versions of "optimal tradeoff" are defined based on network power. A class of algorithms which attempt to optimize network performance are investigated. These algorithms operate on the design principles of dynamic, distributed execution and use of local information. These design principles force the algorithms to be suboptimal, and we thus investigate the relative performance of each in different network configurations. Several properties of power as a network performance objective function are examined. In certain configurations, two variations of network power are unfair to certain users by not permitting them to send any messages. A version of network power ("product of powers") corrects this deficiency. Other properties discussed include the nonconvexity of the generalized power function.

On the General Power Function

On the General Power Function

Constrained regression in chemical engineering practice

Absolute equality constraints in multiple regression are shown to introduce only minor modifications in the normal equations by a subset of the regression parameters. It is essential that these are ordered such that the subset is observable through the constraints. The constraints are shown to give greater residuals about the regression, but the precision in the parameter estimation is improved. In general power function curve fitting, a quadratically convergent iterative computation of the optimal exponents uses the normal equations as equality constraints. The correlations between the coefficients and the exponents in a sum of power functions become very high, when the number of terms increases such that their individual observability is being greatly reduced. The effect is shown to be a very poor precision in the parameter estimation and a heavy oscillation in the coefficients. The algorithms suggested are illustrated by curve fitting of some binary equilibrium data, where the relative volatility model gives a superior fit to the data.

Alternative power laws for ratio scaling.

To take account of the observed lack of fit of the power law near thresh-old intensities, two different modifications of the power law have been proposed by various investigators. In this paper, both of these two laws are derived as a special case of a generalized power function for ratio scaling. A method is presented for discriminating between the special laws which provides (i) a prescription for the manipulation of independent variables, and (ii) specification of theoretical curves to which empirical curves are to be compared. Maximum-likelihood estimators are derived for the exponents of the special laws under the assumption that the observed subjective ratios are log normal.

Low-energy photons in elementary particle collisions

General equations of local field theory (including quantum electrodynamics in all orders in e) have been used to obtain the first two terms in the series expansion of the bremsstrahlung matrix element in the momenta of the soft photons. For bremsstrahlung of one photon with momentum k the first term of this expansion is of order k−1, the second one is of order e 2lnk, and the third term is independent of k. The coefficients of the first and second terms are proportional to the matrix element for the corresponding process without emission of the soft photons. With the aid of this expansion the first two terms in the series expansion of the bremsstrahlung cross section in the radiated energy are obtained; a term of order k 2lnk in the series expansion of the total cross section for scattering of a photon with momentum k by a charged particle is found. It is shown that the imaginary part of the elastic Compton scattering amplitude at fixed momentum transfer is a generalized power function of the energy s near the threshold s = m 2; the first term in the expansion of the amplitude in powers of s − m 2 is found and the next term is estimated. It is shown that the factorization of infrared divergences of any scattering amplitude follows from simple physical considerations and the spectral condition.

Some Remarks on the General Power and Exponential Functions

The mathematics teacher has from time to time to make a decision as to how much he should say to a class about a topic which is too difficult to discuss with full mathematical rigour. This situation may arise with students who are too young or too immature in their mathematical development to appreciate a rigorous discussion, or with older students who are interested in mathematics purely as a tool which they require in the study of other subjects such as engineering and have not time to spare even if they would be interested in a rigorous development of the particular topic.