Arbitrary-precision computation

The great utility of moments is that, although the lineshape cannot be calculated analytically for an arbitrary configuration of nuclear spins, any moment may in principle be calculated to arbitrary precision from first principles (10). In practice, only the lowest moments are calculable because of computer time and precision constraints. In particular, the second moment M2 is the lowest moment containing spacial information... 

Historically, the general point of view in research until recently was that everything could be computed with an arbitrary precision, provided one has... 

However, not all of the experimentally observed absorption bands can be assigned to characteristic bonds or groups. Here, model calculations can be a useful tool to assign transitions to the involved vibrational states and may help to identify a variety of properties such as the molecular structure itself, the determination of reaction mechanisms, characterization of transition states, etc. The advantage of model calculations is that the quantum mechanical wavefunction can be inspected in detail and to arbitrary precision, only limited, of course, by the available computational capacities. 

Molecular level computer simulations based on molecular dynamics and Monte Carlo methods have become widely used techniques in the study and modeling of aqueous systems. These simulations of water involve a few hundred to a few thousand water molecules at liquid density. Because one can form statistical mechanical averages with arbitrary precision from the generated coordinates, it is possible to calculate an exact answer. The value of a given simulation depends on the potential functions contained in the Hamiltonian for the model. The potential describing the interaction between water molecules is thus an essential component of all molecular level models of aqueous systems. 

This information is supported by stress-strain behavior data collected in actual materials evaluations. With computers the finite element method (FEA) has greatly enhanced the capability of the structural analyst to calculate displacement, strain, and stress values in complicated plastic structures subjected to arbitrary loading conditions (Chapter 2). FEA techniques have made analyses much more precise, resulting in better and more optimum designs. 

This expectation that the results of a calculation that has been done correctly will themselves be correct also extends to the method by which the problem was solved. Unfortunately, because few equations in quantum mechanics can be solved exactly, scientists are forced to use numerical computational methods to calculate values. Strictly speaking, even though it is possible to calculate these values to arbitrary levels of precision, such solutions are only approximations of the exact value. Because of the difficulty involved in most of the calculations, a number of students expressed frustration with the notion that all of their hard work had only yielded them an approximate answer. 

In the above, we have given a computational scheme that allows us to define the connections and interactions between components in biochemical networks and to determine the dynamics in the resulting networks. For an arbitrary network, it is not possible to give a precise description of the dynamics without carrying out numerical simulations. However, all the networks obey certain dynamic rules that are set by the stmcture as embodied in the directed A-dimensional hypercube. Moreover, for networks that show certain structural feamres, such as cyclic attractors, it is possible to make precise statements about the dynamics even without further mathematical analysis or simulation. In other cases, analytical techniques are available to give insight into the dynamics observed—for example, in the cases in which it is possible to prove limit cycles... 

For better comparison of theoretical predictions for different-order processes, we have plotted the quantum Fano factors for both interacting modes in the no-energy-transfer regime with N = 2 — 5 and r = 5 in Fig. 7. One can see that all curves start from F w(0) = 1 for the input coherent fields and become quasistationary after some relaxations. The quantum and semiclassical Fano factors coincide for high-intensity fields and longer times, specifically for t > 50/(Og), where il will be defined later by Eq. (54). In Fig. 17, we observe that all fundamental modes remain super-Poissonian [F (t) >1], whereas the iVth harmonics become sub-Poissonian (F (t) < 1). The most suppressed noise is observed for the third harmonic with the Fano factor 0.81. In Fig. 7, we have included the predictions of the classical trajectory method (plotted by dotted lines) to show that they properly fit the exact quantum results (full curves) for the evolution times t > 50/(Og). The small residual differences result from the fact that the amplitude r was chosen to be relatively small (r = 5). This value does not precisely fulfill the condition r> 1. We have taken r = 5 as a compromise between the asymptotic value r oo and computational complexity to manipulate the matrices of dimensions 1000 x 1000. Unfortunately, we cannot increase amplitude r arbitrary due to computational limitations. 

As a general rule, the measurements yield relative intensities, i.e. integrated intensities with an arbitrary scale X, because it is difficult to know what part of the intensity of the primary beam passes through the crystal. The constant X is thus an unknown. The function g 6) is analytic and its values can be easily calculated. The calculation of the absorption factor A is carried out with a computer and, today, poses no major problem. The theory of extinction is still poorly understood, but the factor y is often close to 1. Thus, from the intensity measurements, structure amplitudes F hkl) are obtained on a relative scale, typically with a precision of the order of 1-5%. The values of F hkl) represent the experimental information about the distribution of the atoms in the unit cell. A discussion of this information forms the subject of this section. However, we will discuss neither the theory nor the practice of structure determination by diffraction. 

The thermal-stressed state of a polymer layer in the shape of a rectangle that is characterized by arbitrary internal structure was determined with the help of a computer. The configuration of the structure is represented with precision of up to the size of an individual element. Its components may differ in physical characteristics due to combining a number of these components into the munber of elements into which the rectangle is cross-cut. 

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