Covariance equilibria

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

In a CFD calculation, one is usually interested in computing only the reacting-scalar means and (sometimes) the covariances. For binary mixing in the equilibrium-chemistry limit, these quantities are computed from (5.154) and (5.155), which contain the mixture-fraction PDF. However, since the presumed PDF is uniquely determined from the mixture-fraction mean and variance, (5.154) and (5.155) define mappings (or functions) from (I>- space ... 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

Another factor to be taken into account is the degree of over determination, or the ratio between the number of observations and the number of variable parameters in the least-squares problem. The number of observations depends on many factors, such as the X-ray wavelength, crystal quality and size, X-ray flux, temperature and experimental details like counting time, crystal alignment and detector characteristics. The number of parameters is likewise not fixed by the size of the asymmetric unit only and can be manipulated in many ways, like adding parameters to describe complicated modes of atomic displacements from their equilibrium positions. Estimated standard deviations on derived bond parameters are obtained from the least-squares covariance matrix as a measure of internal consistency. These quantities do not relate to the absolute values of bond lengths or angles since no physical factors feature in their derivation. 

Equilibrium vaporization ratio for component i on stagej Specific rate constant Variance-covariance matrix of measurement adjustments Heat transfer Variance of adjustment to measurementj... 

Much of what is currently understood about the Cenozoic history, of deep-sea temperature, carbon chemistry, and global ice volume, has been gleaned from the stable isotope ratios of benthic foraminifera. Benthic foraminifera extract carbonate and other ions from seawater to construct their tests. In many species, this is achieved near carbon and oxygen isotopic equilibrium. Kinetic fractionation effects tend to be small and constant (Grossman, 1984, 1987). As a result, shell fi C and strongly covary with the isotopic... 

Capital asset pricing model An economic model of equilibrium in capital markets which predicts rates of return on all risky assets as a function of their correlation (or covariance) with the overall market portfolio. 

As demonstrated in the definition of Mi, M2 and M3, kinetic parameters preset as constants for kinetic analysis of GST reaction curve should have strong covariance. Except Kiq as an unknown kinetic parameter for optimization, other kinetic parameters are those reported (Kunze, 1997 Pabst, et al, 1974). To optimize Kiq, two criteria are used. The first is the consistency of predicted Am at a series of GSH concentrations using data of 6.0-min reaction with that by the equilibrium method after 40 min reaction (GST activity is optimized to complete the reaction within 40 min). The second is the resistance of Vm to reasonable changes in data ranges for analyses. After stepwise optimization, Kiq is fixed at 4.0 pmol/L Am predicted for GSH from 5.0 pmol/L to 50 pmol/L is consistent with that by the equilibrium method (Zhao, L.N., et al. 2006) the estimation of Vm is resistant to changes of data ranges (Fig. 5). Therefore, Kiq is optimized and fixed as a constant at 4.0 pmol/L. 

However, bringing the pH of the test solution to equilibrium with the atmosphere, in the brass tests 3 and 4, or carrying out the test with the standard solution in an enclosed vessel as in brass test 5, did not reduce what was admittedly an already low covariance in the results. A second explanation of large scatter could be the (random) presence of particulate matter (e.g. oxides) which is dissolved by the acidification of the test solution after the sample has been removed at the end of the test. 

Define equilibrium market weights and variance-covariance matrix for all assets Get inputs for calculating equilibrium expected returns... 

Chemical reaction equilibrium constant Gas-particle fluctuation covariance (m /s )... 

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