Random values

Equation (B3.4.6) is solved by starting at a small value of R, denoted by where the potential is high and the wavefiinction is exponentially vanishing, and picking random values for and Then, the... 

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

We have encountered oscillating and random behavior in the convergence of open-shell transition metal compounds, but have never tried to determine if the random values were bounded. A Lorenz attractor behavior has been observed in a hypervalent system. Which type of nonlinear behavior is observed depends on several factors the SCF equations themselves, the constants in those equations, and the initial guess. 

We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose iron content was 77.60% so that p = 77.60 and the confidence interval does not include the known value. Since the FeO was a standard, a one-tailed test should have been used since only random values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for —to05) should have been used, and now the confidence interval becomes 77.11 0.23. A systematic error is presumed to exist. 

Equation (9-72) can be integrated between limits to determine the probability that a random value hes between the selected limits. Extensive tables of/( r) and the associated integral are available (see Sec. 3). 

A frequency distribution cui ve can be used to plot a cumulative-frequency cui ve. This is the cui ve of most importance in business decisions and can be plotted from a normal frequency distribution cui ve (see Sec. 3). The cumulative cui ve represents the probability of a random value z having a value of, say, Z or less. 

The ciimnlative prohahility of a normally distributed variable lying within 4 standard deviations of the mean is 0.49997. Therefore, it is more than 99.99 percent (0.49997/0.50000) certain that a random value will he within 4<3 from the mean. For practical purposes, <3 may he taken as one-eighth of the range of certainty, and the standard deviation can he obtained ... 

Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution cui ve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probabihty of occurrence. The cumulative probabihty of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. 

In the same way, random values of the other factors can be obtained. These can then be combined to give random values of (DCFRR) and (NPV) and, in turn, used to plot cumulative-probabihty cui-ves for (DCFRR) and (NPV). The computer may be required to perform some 10,000 to 50,000 calculations. 

In practice, it may not be possible to use conjugate prior and likelihood functions that result in analytical posterior distributions, or the distributions may be so complicated that the posterior cannot be calculated as a function of the entire parameter space. In either case, statistical inference can proceed only if random values of the parameters can be drawn from the full posterior distribution ... 

For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

Typically, the random values, x, from a particular distribution are generated by inverting the closed form CDF for the distribution F x) representing the random variable, where ... 

In general, structural behaviors emerging from random value states under typical Fs can be grouped into four representative categories ... 

Step 1 Initialize weights Wi(t = 0) to be small random values and choose the threshold r. 

Step 1 Initialize all weights to small random values. 

The orientation is defined by the distribution of a single unique axis (the OX3 axis) in the unit with respect to the sample axes. The v / variable therefore takes random values so the Plmn are non-zero only for n = 0. The coefficients up to fourth order are therefore P2oo and P400 as in (i) together with... 

Initialize the weights with small random values in a range around 0 (e.g. -0.3 to +0.3). 

Initialize the weight vectors of all units with random values. 

We should note that expressions (2.21) and (2.27) were obtained in application to a specific bridge of the open type characterized by thickness h and initial concentration of superstoichiometric metal [Me ]o- In real polycrystal with dominant fraction of bridges of this very type there is a substantial spread with respect to the thickness of bridges and to concentration of defects. Therefore, the local electric conductivity of the material in question is a random value of statistical ohmic subgrid formed by barrier-free contacts of microoystals. 

One of the criteria for acceptance of the order of a polynomial least squares fit is that the deviations between calculated and random values be distributed randomly over the range of conditions covered by the data. The concept of randomness for this purpose probably cannot be defined rigorously. However, the following test for randomness is used whenever the original data set contains seven or more values. 

Since the node weights are initially seeded with random values, at the start of training no node is likely to be much like the input pattern. Although the match between pattern and weights vectors will be poor at this stage, determination of the winning node is simply a competition among nodes and the absolute quality of the match is unimportant. 

As seen above, the angle ij/ takes random values (n — 0) for structural units with cylindrical symmetry. The ODF is then defined by two angles and reduces to the following expansion in surface spherical harmonics... 

Verifying the nature of the curve for at least two sets of variances, calculated from different numbers of random values, was necessary in light of the larger values of... 

Of course, the role of the artificially introduced stochastics for mimicking the effect of all eddies in a RANS-based particle tracking is much more pronounced than that for mimicking the effect of just the SGS eddies in a LES-based tracking procedure. In addition, the random variations may suffer from lacking the spatial or temporal correlations the turbulent fluctuations exhibit in real life. In RANS-based simulations, these correlations are not contained in the steady spatial distributions of k and e and (if applicable) the Reynolds stresses from which a typical turbulent time scale such as k/s may be derived. One may try and cure the problem of missing the temporal coherence in the velocity fluctuations by picking a new random value for the fluid s velocity only after a certain period of time has lapsed. 

We know from statistics the moment of distribution of random values ... 

The procedure consists in producing 500 normal deviates u i.e., random numbers normally distributed with zero mean and unit variance. We then compute 500 random values x of CSr normally distributed with mean /i = 70ppm and variance [Pg.233]

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