Numerical Evaluations

electric fields at the surface of nanometric structures were numerically calculated by the FDTD method. As shown in Fig. 2.16a, d, two types of calculation models were created to examine polarization dependencies in retrieving the 

As shown in Fig. 2.16a, the square-shaped structure was embedded in a periodic one-dimensional wire-grid structure, whose pitch was 150 nm, which models the typical structure of an embossed hologram. As shown in Fig.2.16d, on the other hand, the square-shaped structure, whose size was the same as that in Fig. 2.16a, was not provided with any grid structure. By comparing those two cases, we can evaluate the effect of the environmental structures around the nanophotonic code. Also, we chose the square-shaped structure that is isotropic in both the x and y directions to clearly evaluate the effects of environmental structures and ignore the polarization dependency originating in the structure of the nanophotonic code itself. Periodic-conditioned computational boundaries were located 1.5 im away from the center of the square-shaped structure. The wavelength was set to 785 nm. 

Second, from the viewpoint of facilitating recognition of the nanophotonic code embedded in the hologram, it is important to obtain a kind of higher visibility for the signals associated with the nanophotonic codes. To evaluate such visibility, here 

These mechanisms indicate that such nanophotonic codes embedded in holograms could also exploit these polarization and structural dependences, not only for retrieving near-mode information via optical near-field interactions. For instance, we could facilitate near-mode information retrieval using suitable input light polarization and environmental structures. 

---

A systematic comparison of two sets of data requires a numerical evaluation of their likeliness. TOF-SARS and SARIS produce one- and two-dhnensional data plots, respectively. Comparison of sunulated and experimental data is accomplished by calculating a one- or two-dimensional reliability (R) factor [33], respectively, based on the R-factors developed for FEED [34]. The R-factor between tire experimental and simulated data is minimized by means of a multiparameter simplex method [33]. 

In principle, this set of equations can be solved for the various constants, a through Q, just as a and b were obtained previously. In practice, however, the actual numerical evaluation involves considerable computation in all but the simplest examples. Computer solution by matrix techniques designed specifically to handle this type of data correlation problem is usually required. 

The acronym for chemical process quantitative risk analysis. It is the process of hazard identification followed by numerical evaluation of incident consequences and frequencies, and their combination into an overall measure of risk when applied to the chemical process industry. It is particularly applied to episodic events. It differs from, but is related to, a probabilistic risk analysis (PRA), a quantitative tool used in the nuclear industry... 

FTAP - Give it a fault tree or equivalent as linked Boolean equations, and it will find the cutsets (Section 2.2), but it does not numerically evaluate the probabilities. 

The first term is due to the irreversible expansion from V, to Vj, and the second term to the isentropic expansion from Vj to Vj. Adamczyk does not actually say how p3 should be chosen. A reasonable choice for seems to be the initial-peak shock overpressure, as calculated from Eq. (6.3.22). The equation presented above can be compared to the results of Guirao et al. (1979). They numerically evaluated the work done by the expanding contact surface. When the difference between... 

Chemical Process Quantitative Risk Analysis(CPQRA) The numerical evaluation of both incident consequences and probabilities or frequencies and their combination into an overall measure of risk. 

This criterion allows the computation of the radius of convergence p by means of convergent series. The numerical evaluation requires an estimate of the remainder terms of the series r(x). Relations (10) and (12) provide the inequalities... 

Numerical evaluation shows that inequality holds. 

Erom the previous two theorems, any stationary point of. /(p) yields the maximum of. /(p). Such a stationary point can often be found by using Lagrange multipliers or by using the symmetry of the channel. In many cases, a numerical evaluation of capacity is more convenient in these cases, convexity is even more useful, since it guarantees that any reasonable numerical procedure that varies p to increase. /(p) must converge to capacity. 

The combination of these equations gives forms that allow the readings to be expressed in a way that permits the numerical evaluation of the rate constants, such as... 

Moreover, it was established during the numerical evaluation of the unknown quantities in relations (22) and (29) that, while the definition of the two exponents T)t and r 2 in relation (22) is rather unstable, depending fraily on small variations of the value of the Ec-modulus, on the contrary, the single unknown 2r -exponent, defining relation (29), yields rather stable and reliable results. 

For the case where t8 = rp — tp, Equation 13 can be integrated directly to give ITto1al, the total amount of tertiary ion formed. For the other two cases, integration cannot be performed directly, and values of ITtotal were evaluated numerically on a KDF 9 computer, using a procedure for Simpson s rule. (Numerical evaluation of the directly integrable case provided a check on this procedure.) Ip and I8 are then given by... 

The resulting equations for heterogeneous polymers assume the same general form, but numerical evaluation of the second coefficient, A 2 or F2, involves formidable summations over the entire distribution. Molecular weights M occurring in the first term of the osmotic expressions must, of course, be replaced by number averages, Mn- Dilute solutions of two chemically different polymer species also have been treated. ... 

Exact analyses of experimental spectra from thick absorbers, however, have to be based on the transmission integral (2.26). This can be numerically evaluated following the procedures described by Cranshaw [10], Shenoy et al. [11] and others. 

Furthermore, since analytical derivatives are subject to user input error, numerical evaluation of the derivatives can also be used in a typical computer implementation of the Gauss-Newton method. Details for a successful implementation of the method are given in Chapter 8. 

Examine whether any of the estimated parameters follow an Arrhenius-type relationship. If they do, re-estimate these parameters simultaneously. A better way to numerically evaluate Arrhenius type constants is through the use of a reference value. For example, if we consider the death rate, kd as a function of temperature we have... 

Not only is the choice of a uniform prior-prejudice distribution not sensible it also exposes the calculation to two main sources of computational errors, both connected with the functional form of the MaxEnt distribution of scatterers, and with its numerical evaluation namely series termination ripples and aliasing errors in the numerical sampling of the exponential modulation of mix). The next two paragraphs will illustrate these issues in some detail. 

Power series have already been introduced to represent a function. For example, Eq. (1-35) expresses the function y = sin x as a sum of an infinite number of terms. Dearly, for x < 1, terms in the series become successively smaller and the series is said to be convergent, as discussed below. The numerical evaluation of the function is carried out by simply adding terms until the value is obtained with the desired precision. All computer operations used to evaluate the various irrational functions are based on this principle. 

The fast Fourier transform can be carried out by rearranging the various terms in the summations involved in the discrete Fourier transform. It is, in effect, a special book-keeping scheme that results in a very important simplification of the numerical evaluation of a Fburier transform. It was introduced into the scientific community in the mid-sixties and has resulted in what is probably one of the few significant advances in numerical methods of analysis since the invention of the digital computer. 

The numerical evaluation of definite integrals can be carried out in several ways. However, in all cases it must be assumed that the function, as represented by a table of numerical values, or perhaps a known function, is well behaved. While this criterion is not specific, it suggests that the functions haying pathological problems, e.g. singularities, discontinuities,..may not survive under the treatment in question. 

Relationship 23 provides a method for evaluating the parameter "a" that is defined by Equation 2A. The cumulative molar concentration of polymeric species PT0T was numerically evaluated via integration of population density distributions. The contribution of network molecules to the zeroth moment of the distribution is negligible. Results are presented by Figure A and show that... 

The overall response of a low-pass /-filter with N 13C pulses and N delays can be numerically evaluated using the function/LPN ... 

Here, 0 is the angle between the magnetic field vector and the unique symmetry axis. Any anisotropy in the g value is assumed to be small compared to the zero field splitting effects. For Cr3+, which is characterized by S = , and mB = f, , — J, — the polycrystalline spectrum has the shape indicated in Fig. 18 (39). An example of the polycrystalline spectrum for the S = 1 case in which both D and E are nonzero is shown in Fig. 19 (40). A numerical evaluation of D and E may be made from the structure indicated in the spectrum. 

Equation (5.15b) is the fundamental assumption underlying London s theory, which is essential both for numerical evaluation and for physical interpretation of the perturbative expressions. Whereas short-range intramolecular interactions in (5.16a) and (5.16b) must be described with properly antisymmetric eigenfunctions satisfying... 

The neglect of intermolecular exchange effects in (5.15b) greatly simplifies the numerical evaluation of the first-order correction... 

From the beginning, London s theory was recognized as an expedient, but somewhat arbitrary, device to simplify numerical evaluations and recover quasi-classical interpretations of selected long-range contributions to the total intermolecular interaction in the words of a classic text,25... 

For numerical evaluation, we use the simple trapezoidal rule, and a stepwise procedure similar to that in Example 12-5, which can be readily implemented by a spreadsheet program. For convenience, in equation (C), we let... 

Numerical Evaluation of Partial Digestions for Soil Analysis, Talbot VMS Cu-Zn Prospect, Manitoba, Canada... 

Visiers, I., Braunheim, B. B and Weinstein, H. (2000) Prokink a protocol for numerical evaluation of helix distortions by proline. Prot. Eng. 13,603-606. 

---