Variation definition

We shall use the principle of stationary action to obtain a variational definition of the force acting on an atom in a molecule. This derivation will illustrate the important point that the definition of an atomic property follows directly from the atomic statement of stationary action. To obtain Ehrenfest s second relationship as given in eqn (5.24) for the general time-dependent case, the operator G in eqn (6.3) and hence in eqn (6.2) is set equal to pi, the momentum operator of the electron whose coordinates are integrated over the basin of the subsystem 1. The Hamiltonian in the commutator is taken to be the many-electroii, fixed-nucleus Hamiltonian... 

Hendekovic, J. (1982) Novel variational definition of diabatic states. Chem. Phys. Lett., 90, 193. 

Since there is a definite phase relation between the fiindamental pump radiation and the nonlinear source tenn, coherent SH radiation is emitted in well-defined directions. From the quadratic variation of P(2cii) with (m), we expect that the SH intensity 12 will also vary quadratically with the pump intensity 1 ... 

Since the development of a method for polymer characterization has been spread over several sections and since the literature contains several variations in the manner data is displayed, a summary of some pertinent definitions and relationships will be helpful at this point ... 

Variations in the Force Due to Gravity. The mass of an object is the quantity of matter ia the object. It is a fundamental quantity that is fixed, and does not change with time, temperature, location, etc. The standard for mass is a platinum—iridium cylinder, called the International Kilogram, maintained at the International Bureau of Weights and Measures, ia Snvres, France. The mass of this cylinder is 1 kg by definition (9). AH national mass standards are traceable to this artifact standard. 

Estimates of the amount of natural gas available are made within the context of definitions and are subject to revision as definitions change, as additional information becomes available, as resources are consumed, or as undedyiag assumptions are altered. These definitions iaclude proved reserves where the resource is expected to be recoverable and marketable usiag known technology and prices probable reserves where a resource has been identified but not completely characterized and possible or potential gas where estimates are based on the available geological iaformation, historical trends, and previous successes. There are variations ia these definitions throughout the world. 

The advantage of this definition is that it does not depend on measuting the tangent of the response curve, although the variation ia the value of the blending octane number is greater. Typically, BONs are measured at an 80/20 mixture. This technique is also usehil when trying to measure the octane of a compound such as butane or methanol that is difficult or impossible to measure ia its pure state. 

The dominant crystal size is given by = 3Gr. This quantity is also the ratio mJwhich is often given the symbol 2-(J) Prom the definition of the coefficient of variation given by equation 41, cv = 50% for an MSMPR crystallizer. Such a cp may be too large for certain commercial products, which means either the crystallizer must be altered or the product must be screened to separate the desired fraction. 

Because of the close similarity in shape of the profiles shown in Fig. 16-27 (as well as likely variations in parameters e.g., concentration-dependent surface diffusion coefficient), a contrdling mechanism cannot be rehably determined from transition shape. If rehable correlations are not available and rate parameters cannot be measured in independent experiments, then particle diameters, velocities, and other factors should be varied ana the obsei ved impacl considered in relation to the definitions of the numbers of transfer units. 

Thermodynamic Properties The variation in solvent strength of a supercritical fluid From gaslike to hquidlike values may oe described qualitatively in terms of the density, p, or the solubihty parameter, 6 (square root of the cohesive energy density). It is shown For gaseous, hquid, and SCF CO9 as a function of pressure in Fig. 22-17 according to the rigorous thermodynamic definition ... 

At X-ray fluorescence analysis (XRF) of samples of the limited weight is perspective to prepare for specimens as polymeric films on a basis of methylcellulose [1]. By the example of definition of heavy metals in film specimens have studied dependence of intensity of X-ray radiation from their chemical compound, surface density (P ) and the size (D) particles of the powder introduced to polymer. Have theoretically established, that the basic source of an error of results XRF is dependence of intensity (F) analytical lines of determined elements from a specimen. Thus the best account of variations P provides a method of the internal standard at change P from 2 up to 6 mg/sm the coefficient of variation describing an error of definition Mo, Zn, Cu, Co, Fe and Mn in a method of the direct external standard, reaches 40 %, and at use of a method of the internal standard (an element of comparison Ga) value does not exceed 2,2 %. Experiment within the limits of a casual error (V changes from 2,9 up to 7,4 %) has confirmed theoretical conclusions. 

For exposure of reasons of observable discrepancy of results of the analysis simulated experiment with application synthetic reference samples of aerosols [1]. The models have demonstrated absence of significant systematic errors in results XRF. While results AAA and FMA depend on sort of chemical combination of an elements, method of an ashing of a material and mass of silicic acid remaining after an ashing of samples. The investigations performed have shown that silicic acid adsorbs up to 40 % (rel.) ions of metals. The coefficient of a variation V, describing effect of the indicated factors on results of the analysis, varies %) for Mn and Fe from 5 up to 20, for Cu - from 10 up to 40, for Pb - from 10 up to 70, for Co the ambassador of a dry ashing of samples - exceeds 50. At definition Cr by a method AAA the value V reaches 70 %, if element presences an atmosphere in the form of Cr O. At photometric definition Cr (VI) the value V is equal 40%, when the element is present at aerosols in the form of chromates of heavy metals. 

A major difficulty which has been encountered with these definitions (identified as a particular problem by EDSTAC) is the definition of the term adverse . For a chemical to be judged an ED, it is important to show that the response seen has an adverse effect on the health or reproductive capacity of affected organisms or populations and is not just a change which falls within the normal range of physiological variation. 

If classical Coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and the term for the Cl coefficients (C) is omitted, the solvated Eock operator is reduced to Eq. (6). The significance of this definition of the Eock operator from a variational principle is that it enables us to express the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule R ,... 

Suits and Bueche conclude their case-histories with a superb analysis of the sources, tactics and uses of applied research, and make the comment The case histories just summarised show, first of all, the futility of trying to label various elements of the research and development process as basic , applied or development . Given almost any definition of these terms, one can find variations or exceptions among the examples. ... 

Improvement in business performance is essential for growth and profit, but the ISO/TS 16949 requirements are not concerned with your growth and profits they are concerned with product quality, and one definition of product quality that signals improvement potential is freedom from defects . Achieving quality become a quest to eliminate defects and in so doing reduces variation in the operational processes, but even when there are no defectives, there will still be variation. One might well question the need to reduce variation when there are no defectives but by reducing variation you will have fewer breakdowns, fewer errors, less space allocated to inventory, less waste, etc. in fact fewer problems and increased profit as a result. 

Knowledge of the variation of stress and strain through the laminate thickness is essential to the definition of the extensional and bending stiffnesses of a laminate. The laminate is presumed to consist of per-... 

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