Theorem mean-value

Using the mean value theorem for definite integrals... 

MD runs for polymers typically exceed the stability Umits of a micro-canonical simulation, so using the fluctuation-dissipation theorem one can define a canonical ensemble and stabilize the runs. For the noise term one can use equally distributed random numbers which have the mean value and the second moment required by Eq. (13). In most cases the equations of motion are then solved using a third- or fifth-order predictor-corrector or Verlet s algorithms. 

The function QG occurring in the integrand is represented by two straight line segments below the axis. Xf the mean value theorem is applied to the separate integrals, the result is... 

Then, applying the mean value theorem, the surface integral around the observation point p can be represented as... 

Note that since the mean value theorem involves an integration over all allowed values of the independent variable, the integration relevant to average speed is from zero to infinity. In terms of the mean value theorem stated in Eq. (25), the probability of a negative speed is zero, so the integral in Eq. (37) is only over positive values of v. The integral in Eq. (37) may be evaluated by making the... 

The average value of drift velocity, um, is given by integration of Eq. (79) over the radius of the tube according to the relevant mean value theorem,... 

The ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

Also, if j x) is any function which is intcgrable in the interval (— a, ) then, hv using the mean value theorem of the integrul calculus, we see that... 

Because the velocity u contains the random component u, the concentration c is a stochastic function since, by virtue of Eq. (2.2), c is a function of u. The mean value of c, as expressed in Eq. (2.5), is an ensemble mean formed by averaging c over the entire ensemble of identical experiments. Temporal and spatial mean values, by contrast, are obtained by averaging v ues from a single member of the ensemble over a period or area, respectively. The ensemble mean, which we have denoted by the angle brackets ( ), is the easiest to deal with mathematically. Unfortunately, ensemble means are not measurable quantities, although under the conditions of the ergodic theorem they can be related to observable temporal or spatial averages. In Eq. (2.7) the mean concentration (c) represents a true ensemble mean, whereas if we decompose c as... 

Provided that level shifts resulting from the configuration interaction are neglected we can apply the mean value theorem and factor out of the integrand of eq. (2-23) the average value of the matrix element... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

An examination of Eq. (230), which is similar in form to Eq. (81), shows that, for melting, d20i/dX2 > 0. From the mean value theorem it follows immediately that an upper bound for the temperature in melting is given by... 

We would have ten different values of sample variance. It can be shown that these values would have a mean value nearly equal to the population variance Ox. Similarly, the mean of the sample means will be nearly equal to the population mean [t. Strictly speaking, our ten groups will not give us exact values for Ox and p. To obtain these, we would have to take an infinite number of groups, and hence our sample would include the entire infinite population, which is defined in statistics as Glivenko s theorem [3]. 

J2iW= he total force acting on the electrons. The hypervirial theorem implies that the mean value of this force vanishes for an isolated system, = 0. 

According to a mean value theorem for integrals of the continuous functions, there is a mean vahie point r G for which... 

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