Uncorrelated

Correlated events are related in time and this time relation can be measured either with respect to an external clock or to the events themselves. Random or uncorrelated events bear no fixed time relation to each other but, on the other... 

In the classical picture developed above, the wavepacket is modeled by pseudo-particles moving along uncorrelated Newtonian trajectories, taking the electrons with them in the form of the potential along the Uajectory. In this spirit, a classical wavepacket can be defined as an incoherent (i.e., noninteracting) superposition of confignrations, X/(, t)tlt,(r, t)... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

I Principal Component Analysis (PCA) transforms a number of correlated variables into a smaller number of uncorrelated variables, the so-called principal components. 

As the number of steps fj, in each block increases, so it would be expected that the block averages become uncorrelated. When this is the case, then the Vciriance of the block averages, will become inversely proportional to is calculated as follows ... 

A number of simulation methods based on Equation (7.115) have been described. Thess differ in the assumptions that are made about the nature of frictional and random forces A common simplifying assumption is that the collision frequency 7 is independent o time and position. The random force R(f) is often assumed to be uncorrelated with th particle velocities, positions and the forces acting on them, and to obey a Gaussiar distribution with zero mean. The force F, is assumed to be constant over the time step o the integration. 

The Microstate Cl Method lowers the energy of the uncorrelated ground state as well as excited states. The Sngly Btcited Cl Method is particularly appropriate for calculating UV visible spectra, and does not affect the energy of the ground state (Brillouin s Theorem). 

These equations assume that the measurements A and B are uncorrelated that is, Sa is independent of Sb. 

Consider the problem of assessing the accuracy of a series of measurements. If measurements are for independent, identically distributed observations, then the errors are independent and uncorrelated. Then y, the experimentally determined mean, varies about E y), the true mean, with variance C /n, where n is the number of observations in y. Thus, if one measures something several times today, and each day, and the measurements have the same distribution, then the variance of the means decreases with the number of samples in each day s measurement, n. Of course, other fac tors (weather, weekends) may make the observations on different days not distributed identically. 

If the variables are uncorrelated and have the same variance, then... 

The covariance is zero when the variables are uncorrelated. The variance of i> can be obtained from eijualinn 2.1-11. E(Q ) may be found by squaring equation 2.7-13 and taking the expectation to give equation 2 7-18,... 

Equation 2.7-18 for the system variance is equation 2.7-21. Since the covariance of uncorrelated variables is zero, this becomes equation 2.7-23 (i.e., the system variance is the sum of the component variances). 

If the system distribution is the product of the component distributions (equation 2.7-23), the mean is given by equation 2.7-24 which, for uncorrelated variables becomes equation 2.7-25... 

When Hartree-Fock theory fulfills the requirement that 4 be invarient with respect to the exchange of any two electrons by antisymmetrizing the wavefunction, it automatically includes the major correlation effects arising from pairs of electrons with the same spin. This correlation is termed exchange correlation. The motion of electrons of opposite spin remains uncorrelated under Hartree-Fock theory, however. 

Of these, all are experimentally observable except the Svaience state level which is a calculated value for a carbon atom with 4 unpaired and uncorrelated electron spins this is a hypothetical state, not amenable to experimental observation, but is helpful in some discussions of bond energies and covalent bonding theory. 

The correlation function is a number between -1 and 1, where 1 indicates that the two quantities are completely correlated, -1 that they are (completely) anti-correlated and 0 means that they are independent (uncorrelated). 

Pi()2i )2i, )i2, 3. ..) and Pf are projection operators on a specific state k. For the purpose of configuration averaging the previous theorem for uncorrelated disorder still holds good with Mid replaced by... 

Figures 3.10-3.15 present some qualitative evidence for the self-organization of space-time patterns emerging out of initial configurations of uncorrelated sites. In this Section we introduce some of the quantitative characterizations of selforganization in elementary r = 1, k = 2 rules by examining these systems from two different points of view.

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