Variational estimate

The eigenvalue Ei gives the variational estimate for the energy of the i state, and the entries in the eorresponding eigenveetor Ci k give the eontribution of the CSF to the wavefunetion Fi in the sense that... 

In the probabilistic design calculations, the value of Kt would be determined from the empirical models related to the nominal part dimensions, including the dimensional variation estimates from equations 4.19 or 4.20. Norton (1996) models Kt using power laws for many standard cases. Young (1989) uses fourth order polynomials. In either case, it is a relatively straightforward task to include Kt in the probabilistic model by determining the standard deviation through the variance equation. 

The calculated loading stress, L, on a component is not only a function of applied load, but also the stress analysis technique used to find the stress, the geometry, and the failure theory used (Ullman, 1992). Using the variance equation, the parameters for the dimensional variation estimates and the applied load distribution, a statistical failure theory can then be formulated to determine the stress distribution, f L). This is then used in the SSI analysis to determine the probability of failure together with material strength distribution f S). 

The formulations for the failure governing stress for most stress systems can be found in Young (1989). Using the variance equation and the parameters for the dimensional variation estimates and applied load, a statistical failure theory can be formulated for a probabilistic analysis of stress rupture. 

Air flows at 12 m/s through a pipe of inside diameter 25 mm. The rate of heat transfer by convection between the pipe and the air is 60 W/m2K. Neglecting the effects of temperature variation, estimate the pressure drop per me Ire length of pipe. 

Table III. Total Sample Size Required Based on the Preliminary Sample Coefficient of Variation Estimate ...

Fig. 4.10. Atmo.spheric CO2 variation estimated by modified BLAG model including CO2 flux related to mantle plume activity (Ishikawa, 1996). t c02 = 202/ 002 - prescut-day PcOi)-...

Fig. 4.11. Atmospheric CO2 variation estimated by modified GEOCARB II model including volcanic eruption rate of circum-Pacific region by Kennett et al. (1977) (Kashiwagi et al., 2000). y represents the contribution of the flux from back arc basin to that from subduction zones at present. Rco = PcOi/PcOi 02 Pfesent-day PC02)-...

The simple variational estimate (2.29b) compares reasonably with the actual d-rich sp0 01d5-48 metal NHO found in this case (i.e., predicted 90% versus actual 84% d character). Similarly, for the mid-series Mn—F species the corresponding (3-spin matrix elements have the values... 

The perturbative estimates of Table 3.21 may be compared with a more direct variational estimate of the pi-type non-Lewis correction (i NL(7t)) to the localized Lewis structure,... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

We do not discuss here variational estimates of the for details see the review article [33]. Note here only that radius estimates show that for a typical small-radius electron defect, the activator atom A0 and begin to deviate from the Onsager radius L at very small D values only (low temperature) when the applicability of binary approximation itself is questionable. It comes from the fact that due to small values of tq 0.5 A and e 5 the Onsager... 

Let us consider now several variational estimates of the effective radius taking into account annihilation, tunnelling and an elastic interaction. If tunnelling term in equation (4.2.25) is large in comparison with others in brackets, we can use for the upper estimate equation (4.2.15) as a trial function y(r) which leads to... 

Arrhenius law, K oc exp(—Ees/(k T)), does not hold here. (The same is true for the Coulomb interaction [39, 46, 68].) Variational estimates of the effective radius at high temperatures, when the recombination is controlled predominantly by annihilation, are discussed in [60]. Variational estimates of the effective radius taking into account annihilation, tunnelling and an elastic interaction were discussed in detail in [33]. 

Fig. 4.7. Temperature dependence of the effective radius of H, A0 recombination in KBr controlled by an elastic interaction, diffusion and tunnelling. Curve 1 - exact result, 2 - effect of tunnelling and annihilation, 3 - isotropic attraction and annihilation, 4 - pure annihilation. Variational estimates upper bound when (i) tunnelling dominates (equation (4.2.32) - curve 5) or an elastic interaction dominates (equation (4.2.34) - curve 6). Curve 7 - lower bound estimate, equation (4.2.36), when an elastic interaction is a predominant factor.

In table III, the "crystallinity" variations estimated after multiple irradiations are shown. In spite of the degradation which is clearly noticeable (gradual fading of the weaker interference) the estimated crystallinity often shows very limited decrease (the standard deviations are approximately 8 %). 

Davies16) has given a rigorous mathematical analysis of the variational estimate of the ground state of the Hamiltonian K based on trial functions consisting of the tensor product x of a molecular wavefunction 0, and a coherent state 0 for the boson field, i.e. one searches for the minimum value of... 

Fig. 2. Ne = Np = 16, Vs = 1.31. Dependence of total energy, variance and energy difference for a pair of proton configurations S, S ) on the RQMC projection time. The study is performed for Te = 0.02Dotted lines represent the variational estimates with their error bars. In panel b) and c) the lines are exponential fits to data and in panel d) the continnons line is a linear fit in the region < 0.005. Black circles (3BF-A) are resnlts obtained with the analitical three-body and backflow trial wave functions discnssed earlier, the red triangle is a variational resnlt with a Slater-Jastrow trial function with simple plane wave orbitals and the blue squares are results from a trial function with LDA orbitals and an optmized two-body Jastrow...

The variational estimate is consistent with our previous development because from the two first terms of the Laurent series (3.32) we obtain... 

The variational estimate E(D) given by (3.50) is a single-valued function of D confined to the energy window... 

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