Direct problem

We need to point out that, if the wavelengths of laser radiation are less than the size of typical structures on the optical element, the Fresnel model gives a satisfactory approximation for the diffraction of the wave on a flat optical element If we have to work with super-high resolution e-beam generators when the size of a typical structure on the element is less than the wavelengths, in principle, we need to use the Maxwell equations. Now, the calculation of direct problems of diffraction, using the Maxwell equations, are used only in cases when the element has special symmetry (for example circular symmetry). As a rule, the purpose of this calculation in this case is to define the boundary of the Fresnel model approximation. In common cases, the calculation of the diffraction using the Maxwell equation is an extremely complicated problem, even if we use a super computer. 

Now, we can make the comparison beween the real defect signal and the simulated one which have been computed by solving the linearized direct problem. The measurements were made at 300,150,50 kHz. The flaw is a notch of 8mm length, 1mm width, and 1mm depth. Representative data (300 kHz) for the notch-shaped flaw are shown in Fig. 3. 

Central to systems analysis is the attempt to deyelop a predictiye model for the system s response to a giyen disturbance or perturbation. If a model can be proposed in adyance, then experimental input/output measurement and signal analysis are used to yerify that the proposed model is correct. This approach is called the "direct problem" of systems analysis. 

The Direct Problem. Given a particle of specified shape, size, and composition, which is illuminated by a beam of specified irradiance, polarization, ami... 

Figure 1.5 (a) The direct problem Describe the tracks of a given dragon. (b) The inverse problem Describe a dragon from its tracks. 

A direct kinetic problem consists of calculating multi-component reaction mixture compositions and reaction rates on the basis of a given kinetic model (both steady-state and unsteady-state) with the known parameters. Reliable solution for the direct problem is completely dependent on whether these parameters, obtained either on theoretical grounds or from special experiments, have reliable values. Modern computers can solve high-dimensional problems. Both American and Soviet specialists have calculated kinetics for the mechanisms with more than a hundred steps (e.g. the reac-... 

A reverse kinetic problem consists in identifying the type of kinetic models and their parameters according to experimental (steady-state and unsteady-state) data. So far no universal method to solve reverse problems has been suggested. The solutions are most often obtained by selecting a series of direct problems. Mathematical treatment is preceded by a qualitative analysis of experimental data whose purpose is to reduce drastically the number of hypotheses under consideration [31]. 

Fig. 15. The possible mechanisms by which a strong electric field can affect cells in suspension or adherently growing. Most of the heat is produced near the electrodes and, therefore, tends not to be a direct problem as it can be easily dissipated into the substrate. This heating can, however, induce convection currents which, in turn, may impose mechanical stress on an adherent cell. There is also some heating between the electrodes. At low frequencies, this occurs only in the medium although it may be concentrated in regions surrounding the cell. At high frequencies, this heating becomes more uniform but, because high frequency currents can flow inside the cell, there is some internal heat production. The total amount of heat evolved depends on the conductivity of the medium and on the square of the applied voltage.

Directional problem. Suppose that you have a protein synthesis system that is actively synthesizing a protein designated A. Furthermore, you know that protein A has four trypsinsensitive sites, equally spaced in the protein, that, on digestion with trypsin, yield the peptides Aj, A2, A3, A4, and A5. Peptide Aj is the amino-terminal peptide, and A5 is the carboxyl peptide. Finally, you know that your system requires 4 minutes to synthesize a complete protein A. At t = 0, you add all 20 amino acids, each carrying a label. 

Today, the theory of plastic foams is concerned with the solution of a problem which the author termed as a direct or physical problem. For any material including foamed materials, the direct problem is formulated as follows in which way are the properties... 

When assessing the merits of the diode array, the user should consider the various steps that are involved in methods development. It will become clear that for every hurdle one encounters, diode-array detection renders the task less laborious. It is most often than not that in developing new procedures you are confronted with drawbacks such as contaminants, peak overlap, artifacts, sample cleanup, and baseline noise. For instance, the presence of a shoulder on a major chromatographic peak raises questions as to its origin. With the aid of the diode array, a spectral profile of the shoulder peak will quickly determine its existence. A flat line of low absorbance throughout its spectra indicates that it is an artifact. The user can then go back to his procedure and ascertain if the root of the problem is systematic. The direction problem-solving takes is crucial and if gone astray, vast amounts of time are wasted. The... 

For all the three schemes of ECM, the direct problems are solved in a similar way. The solutions of the inverse problem are known only for the first scheme, because for the other ECM schemes, the shape and dimensions of the machined surface do not enable one to determine, uniquely, the shape of the TE working surface. 

The quasi-steady-state approximation may be used, because the rates of the transfer processes in the I EG (meters per second) are considerably higher than the rate of the variation of the WP surface (millimeters per minute). Within the framework of the quasi-steady-state approximation, it is possible to divide the initial problem into two subproblems (1) Calculation of the transfer processes in the I EG and the determination of the ECM rate field Va. (2) Calculation of the WP surface evolution for the direct problem or correction of the TE surface for the inverse problem. However, even under this simplification, solving the direct and inverse ECM problems, especially for sculptured WPs, involves great difficulties. 

Within the quasi-steady-state approximation, the most popular method is the embedding method involving the successive solving of a series of direct problems and corrections of the TE working surface until a desired shape of the WP is assured [45, 46, 50, 51]. Several methods of TE correction have been proposed they have a lot in common and can be presented in the following generalized form ... 

The subsystem for the automatic design of ECM operations in the dialog mode enables the following problems to be solved (1) Analysis (the direct problem) the shape and size of the WP surface are determined by the given TE surface and (2) synthesis (the inverse problem) the working surface of the TE is determined in order to produce the desired WP surface. 

The solution of the inverse problem is of principal commercial importance. However, the exact solution of this problem is difficult, because it requires the solution of a problem with unknown boundaries. Therefore, for the automation of the design of ECM operations, the following approach is used the inverse problem is reduced to the solution of a series of direct problems and, by solving them, the TE working surface and the machining parameters are corrected. 

The direct problem is not simply the calculation of the integral (17) for given (r), but also requires the specification of this function for the assigned surface preparation conditions. 

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

The approach here employed in the direct problem solution for forced convection in micro-channels, is borrowed from a recent work on diffusion in heterogeneous media, with arbitrarily space variable thermophysical... 

The direct problem solution is first validated by direct comparison with the benchmark results provided in Ref. [6], as illustrated in Table 1 below, for the case of a parallel-plates under prescribed uniform wall temperature... 

The constructed algorithm for the inverse analysis was then also validated for this same benchmark problem with Bi = co, from a theoretical perspective, assuming the fluid temperature at the wall to be measurable. Simulated experimental results were produced with 50 terms in the eigenfunction expansion provided in Ref. [6], and the direct problem solution in the inverse analysis was implemented with just 10 terms in the expansion here proposed to avoid the so called inverse crime [28]. A total of 1,000 measurements are provided, with white noise considered normally distributed... 

A total of 1,000 uniformly distributed points along the dimensionless channel length, Z/= 5, was initially adopted. The simulated measurements were considered normally distributed with averages at the simulated values and 1% standard deviation. They were obtained with 50 terms in the eigenfunction expansions, while the direct problem solution within the inverse problem procedure was handled with 20 terms only, again in order to avoid the inverse crime. 

Under certain circumstances vibration may cause direct problems with the product, independent of the pack, i.e. segregation of powders, separation of emulsions, surface dulling of tablets, settling down of product, etc. Vibration will also act in combination with compression forces and thereby possibly accelerate the weakening or distortion of board outers, cartons, etc. It can also lead to the distortion of flexible packs such as strips or sachets. 

METHODS FOR THE SOLUTION OF DIRECT PROBLEMS OF INDUCTION LOGGING... 

Thus a solution of the direct problem of induction logging consists of determination of a function which satisfies the Helmholtz equation inside every area and the boundary conditions ... 

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