Uniform probability

In a shooting move a trial path is generated from an old path as follows. First, one of the L states of the old path z(° 8) is selected at random with a uniform probability, i.e., all states on the path have the same probability to be selected.3 The selected state consists of the positions qsmall perturbation (ip to the old momenta... 

The exponential distribution with parameter X is the distribution of waiting times ( distance in time) between events which take place at a mean rate of X. It is also the distribution of distances between features which have a uniform probability of occurrence (Poisson process), such as the simplest model of faults on a map. The gamma distribution with parameter n and X l, where n is an integer is the distribution of the waiting time between the first and the nth successive events in a Poisson process. Alternatively, the distribution /(t), such as... 

The nucleation law for a uniform probability with time t of conversion of a site on the metal electrode into nuclei is given by... 

Time-Dependent Escape Rate (Reaction Rate).186 To explain the meaning of P (t)—the probability of finding the particle still in the well at time t, recalling that it was generated within it with uniform probability—let the random variable T be the time at which the particle crosses the barrier (suffers a reaction). It is obvious that the survival probability defined as the probability of the particle being in the well after time t satisfies... 

The assumption about a uniform probability for any distribution of the energy between the harmonic oscillators may now be used to determine the probability Pet >e (E). It can be expressed as the ratio between the density of states corresponding to the situation where the energy exceeds the threshold energy in the reaction coordinate and the total density of states at energy E, that is, N(E) of Eq. (7.36). 

To obtain an estimate of E(x,g), we use a simple formula that assumes a uniform probability, 1/p, for elimination of trait x ... 

Each phase is characterized by a mean duration D and a variability V. As soon as the prescribed duration of a given phase is reached, the transition to the next phase of the cell cycle occurs. The time at which the transition takes place varies in a random manner according to a distribution of durations of cell cycle phases. In the case of a uniform probability distribution, the duration varies in the interval [D(l - V), D( 1 + V)]. 

It is easy to extend this model so as to create a physical condition closer to BQD physics. This extended dynamic model has two rather than only one laminar spatial region, divided by a border [100]. The particle in the left and right spatial laminar regions move toward the right and the left, respectively. Thus in both cases they move toward the border. When the particle reaches the border, it is injected back either to the right or to the left, with uniform probability. Let us write the corresponding density equation in the generic form... 

Fig. S. Illustration of the interaction of the vector e(t) with the unit square, (a) The slope is rational and equal to i e visits the two segments in a periodic manner, (b) The slope is irrational e visits the whole square with a uniform probability.

For irrational values of , the terminus of the e vector visits all accessible positions in the unit square with uniform probability. Hence, the time average [Pg.46]

Using Equations (11.32) and (11.33), it is straightforward to verify that the random time T follows the exponential probability distribution if r follows a uniform probability distribution ... 

Diffusion on a spherical surface. Introducing the usual spherical polar co-ordinates we obtain the uniform probability density in solid angle 0) = (47t) - when 6 ranges from 0 to tt and from 0 to In. In equilibrimn with a field E... 

Gy (1992, p. 374) calls this the equiprobabilistic model, where each of the. .. particles composing the lot. .. is submitted to the selecting process independently with a uniform probability. .. of being selected. ... 

To select the optimal subset of compounds, that is, a set of compounds having the maximal chemical diversity, a probabilistic search algorithm is applied, which consists in selecting a subset of compounds based on a probability assigned to each compound. This algorithm optimizes the joint entropy (/H) of the subset of selected compounds. The task is performed iteratively, assigning each ith compound an initial uniform probability Pi = 1 fn, then calculating the score S that is added to the previous compound probability as... 

The dependence of the variances (90) on a+ at fixed a is shown in Fig. 6. The qualitative explanation of this effect of strong increase of quantum fluctuations in the low-intensity limit is based on the consideration of the probability to have a given phase difference A+. At a+ =0, there is a uniform probability distribution in the system, causing the limit relations (91). Creation of very few photons of the mode m I 1 leads to the formation of some domains with almost equal probabilities having phase differences A+ and A + n. So, it looks like a phase bunching (for the bunching of photons, see Refs. 14 and 15). Further increase of a+ leads to formation of a more or less sharp probability distribution that cannot reach the 8-function shape because the variances (90) achieve the saturation described by the expression... 

Let s consider that under the current operating conditions the values of x fall within a rectangle Xcurrent = x e [200 4000] Ax2 e [10 20]. Furthermore, we will assume that the two decision variables (xj and x2) are independent and have uniform probability distributions. Using the available model, fix), we computed the current average total manufacturing cost, (y Xcurrent) = 743.5, a reference value that can be used to estimate the savings achieved with the implementation of any uncovered final solutions, X. ... 

The next possible separation of two particles, rj, is generally limited by their ionic radii. The separation at which they can enter into interaction with each other is double the lattice constant. The energetic minimum, as can be seen in Fig. 21, is exactly at the lattice constant and its size is given by the above mentioned 5/3 ys. Now, only particles which arrive between r and 2a are considered, and a uniform probability is assumed for every position between these two limits. Then, it only rarely arises that a particle occupies the energetically favourable position at a. However, in an effort to nevertheless achieve this energetically favourable position, an expansion of the film takes place if a particle occupies a place between r and a, and a contraction occurs if a position between a and 2a is occupied. However, in both cases the energy difference arises in the form of deformation energy. 

Note that by choosing to change 8(cos 0) with a uniform probability, we cancel the potentially troublesome sin 6 terms. 

Instead of the steady state of (quasi) homogeneous nucleation, during which nuclei appear at constant rate, both the total number of critical nuclei and the duration of the nucleation process are confined by the maximum number of preexisting sites Mo. If there is a uniform probability with time of converting these sites into critical nuclei S — Mno one obtains the following first-order nucleation law as an approximate solution of the reaction sequence (17)... 

The information entropy attains a maximum for uniform probability distribution (7) ... 

For transverse isotropy of the sample, we may average over the range 0 27t with a uniform probability function ... 

Step (1) A frequency is sampled across [t min) maxl (PAR) according to the uniform probability density function Pviv)=-—. This fre-... 

A probability sample is one for which every unit in a finite population has a positive probability of selection, but not necessarily equal to that of other units, like random (or uniform probability) samples [15]. 

Apparently, the core of the procedure is the mutation operation, because this routine folds new structures, derived from those previously formed. Also, it is clear that this is the part of the algorithm, which allows to implement kinetic effects into the simulation. As mentioned above, disruption and formation of stems may be considered as elementary reactions in multistep RNA folding. Similar to the Monte Carlo procedure, at any given step of simulation a choice between different possible reactions can be made using their probabilities that depend on rate constants. Therefore addition and removal of stems with non-uniform probability distributions can mimic dynamic RNA folding, and different approximations may increase the efficiency of the algorithm. We consider separately some of these approximations, because they represent essential features of RNA dynamics. 

Calculating Possibilities. Consider the sequence-structure relation in Figure 2. A uniform probability of is assigned to each sequence. Note that the uniform distribution is not a requirement of the mathematical model. It was assumed that all sequences could adopt the same conformation. The possibility for each structural hypothesis to contain the conformation was computed using Zadeh s theory of possibility. 

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