Invariance properties

Kleier D A and Binsch G 1970 General theory of exchange-broadened NMR line shapes. II. Exploitation of invariance properties J. Magn. Reson. 3 146-60... 

If there is no approximate Hessian available, then the unit matrix is frequently used, i.e., a step is made along the gradient. This is the steepest descent method. The unit matrix is arbitrary and has no invariance properties, and thus the... 

Table 8-4 summerizes the most important invariance properties of different structure desaiptors. 

The actual invariants in invariant properties of a lamina include not only U., U4, and U5 because they are the constant terms in Equation (2.93) but functions related to U-), U4, and U5 as shown in Problem Set 2.7. The terms U2 and U3 are not invariants. The only invariants of an orthotropic lamina can be shown to be... 

Stephen W. Tsai and Nicholas J. Pagano, Invariant Properties of Composite Materials, in Composite Materials Workshop, S. W. Tsai, J. C. Haipin, and Nicholas J. Pagano (Editors), St. Louis, Missouri, 13-21 July 1967, Technomic, Westport, Connecticut, 1968, pp. 233-253. Also AFML-TR-67-349, March 1968. 

Quasi-isotroplc laminates do not behave like Isotropic homogeneous materials. Discuss why not, and describe how they do behave. Why is a two-ply laminate with a [0°/90°] sacking sequerx and equat-thickness layers not a quasi-isotropic laminate Determine whether the extensional stiffnesses are the same irrespective of the laminate axes for the two-ply and three-ply cases. Hint use the invariant properties In Equation (2.93). 

Stephen W. Tsai and Nicholas J. Pagano, Invariant Properties of Composite Materials, AFML-TR-67-379, March 1968. 

G. W. Trucks and M. J. Frisch, Rotational Invariance Properties of Pruned Grids for Numerical Integration, in preparation (1996). 

It turns out that this choice is correct and that L and L" can be so chosen that (11-144) is satisfied. We shall return to this question in our discussion of the asymptotic condition in Section 11.5 of the present chapter. As preparation for these considerations, we turn in the next few seotions to a discussion of the invariance properties of quantum electrodynamics and their consequences. 

Invariance Properties.—Before delving into the mathematical formulation of the invariance properties of quantum electrodynamics, let us briefly state what is meant by an invariance principle in general. As we shall be primarily concerned with the formulation of invariance principles in the Heisenberg picture, it is useful to introduce the concept of the complete description of a physical system. By this is meant at the classical level a specification of the trajectories of all particles together with a full description of all fields at all points of space for all time. The equations of motion then allow one to determine whether the system could, in fact, have evolved in the way... 

Spectral Representation.—As an application of the invariance properties of quantum electrodynamics we shall now use the results obtained in the last section to deduce a representation of the vacuum expectation value of a product of two fermion operators and of two boson operators. The invariance of the theory under time inversion and more particularly the fact that... 

Invariance principle, 664 Invariance properties of quantum electrodynamics, 664 Inventory problem, 252,281,286 Inverse collisions, 11 direct and, 12 Inverse operator, 688 Investment problem, 286 Irreducible representations of crystallographic point groups, 726 Isoperimetric problems, 305 Iteration for the inverse, 60... 

Quantization of radiation field in terms of field intensity operators, 562 Quantum electrodynamics, 642 asymptotic condition, 698 gauge invariance in relation to operators inducing inhomogeneous Lorentz transformations, 678 invariance properties, 664 invariance under discrete transformations, 679... 

For comparatively high repetition-rates (period T < 5t) fluorescence decays could also overlap between adjacent pulses. Thanks to the scale-invariant properties of the exponentials, no error is introduced when the decay is a pure single-exponential. Conversely, the preexponential factors can be altered when multiple lifetime decays... 

Also global fitting techniques, where the space invariance (or any other invariance property) of one or more fitting parameters is exploited, have been successfully used to analyze fluorescence lifetime images [45, 46]. When applicable, global analysis techniques provide more homogeneous SNRs and reduce the number of fitted parameters. 

Comparison of Equations (1) and (2) shows that the chemical potentials are intensive quantities, that is, they do not depend on the amount of each species, because if all the nt are increased in the same proportion at constant T and p, the /x, must remain unchanged for G to increase in the same rate as the nt. This invariance property of the /x, is of the utmost importance in restricting the possible forms that the /x- may take. 

Let us consider now processes where intermediate stationary Hamiltonians are mediating the interconversion. In these processes, there is implicit the assumption that direct couplings between the quantum states of the precursor and successor species are forbidden. All the information required to accomplish the reaction is embodied in the quantum states of the corresponding intermediate Hamiltonian. It is in this sense that the transient geometric fluctuation around the saddle point define an invariant property. 

A desired property (linear invariance property) of QQ-plots is that when the two distributions involved in the comparison are possibly different only in location and/or scale, the configuration of the QQ-plots will still be linear, with a nonzero intercept if there is a difference in location, and/or a slope different from unity if there is a difference in scale. 

Note that Eq. (6) includes thermodynamic equilibrium (v° = 0) as a special case. However, usually the steady-state condition refers to a stationary nonequilibrium state, with nonzero net flux and positive entropy production. We emphasize the distinction between network stoichiometry and reaction kinetics that is implicit in Eqs. (5) and (6). While kinetic rate functions and the associated parameter values are often not accessible, the stoichiometric matrix is usually (and excluding evolutionary time scales) an invariant property of metabolic reaction networks, that is, its entries are independent of temperature, pH values, and other physiological conditions. 

The history of the study of symmetry properties of Eq. (3) goes back to the beginning of the twentieth century. Invariance properties of Maxwell equations have been studied by Lorentz [40] and Poincare [41,42]. They have proved that Eq. (3) are invariant with respect to the transformation group named by the Poincare suggestion the Lorentz group. Furthermore, Larmor [43] and Rainich [44] have found that equations (3) are invariant with respect the singleparameter transformation group... 

Many problems appear to be ripe for a more quantitative discussion. What is the error involved in the introduction of unstable states as asymptotic states in the frame of the 5-matrix theory 16 What is the role of dissipation in mass symmetry breaking What is the consequence of the new definition of physical states for conservation theorems and invariance properties We hope to report soon about these problems. We would like, however, to conclude this report with some general remarks about the relation between field description and particles. The full dynamical description, as given by the density matrix, involves both p0 and the correlations pv. However, the particle description is expressed in terms of p (see Eq. (50)). Now p has only as many elements as p0. Therefore the... 

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