Discrete points

During the inspection of an unknown object its surface is scanned by the probe and ultrasonic spectra are acquired for many discrete points. Disbond detection is performed by the operator looking at some simple features of the acquired spectra, such as center frequency and amplitude of the highest peak in a pre-selected frequency range. This means that the operator has to perform spectrum classification based on primitive features extracted by the instrument. 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

The number of discrete points of /cicc( ) determines the resolution of the chirality code is a smoothing factor which in practice controls the width of the peaks obtained by a graphical representation versus u. An example of a chir-... 

By including characteristic atomic properties, A. of atoms i andj, the RDF code can be used in different tasks to fit the requirements of the information to be represented. The exponential term contains the distance r j between the atoms i andj and the smoothing parameter fl, which defines the probability distribution of the individual distances. The function g(r) was calculated at a number of discrete points with defined intervals. 

See Refs. 198, 218, and 256. The z -transform is useful when data is available at only discrete points. Let... 

In most aqueous systems, the corrosion reaction is divided into an anodic portion and a cathodic portion, occurring simultaneously at discrete points on metallic surfaces. Flow of electricity from the anodic to the cathodic areas may be generated by local cells set up either on a single metallic surface (because of local point-to-point differences on the surface) or between dissimilar met s. 

Numerical simulations offer several potential advantages over experimental methods for studying dynamic material behavior. For example, simulations allow nonintrusive investigation of material response at interior points of the sample. No gauges, wires, or other instrumentation are required to extract the information on the state of the material. The response at any of the discrete points in a numerical simulation can be monitored throughout the calculation simply by recording the material state at each time step of the calculation. Arbitrarily fine resolution in space and time is possible, limited only by the availability of computer memory and time. 

In the finite-difference appntach, the partial differential equation for the conduction of heat in solids is replaced by a set of algebraic equations of temperature differences between discrete points in the slab. Actually, the wall is divided into a number of individual layers, and for each, the energy conserva-tk>n equation is applied. This leads to a set of linear equations, which are explicitly or implicitly solved. This approach allows the calculation of the time evolution of temperatures in the wall, surface temperatures, and heat fluxes. The temporal and spatial resolution can be selected individually, although the computation time increa.ses linearly for high resolutions. The method easily can be expanded to the two- and three-dimensional cases by dividing the wall into individual elements rather than layers. 

Radiation heat flux is strongly time dependent, because both the flame surface area and the distance between the flame and intercepting surfaces vary during the eourse of a flash fire. The path of this curve ean be approximated by calculating the radiation heat flux at a sufficient number of discrete points in time. 

By this discretization in the polar coordinates, one can allow an atom to be displaced to one of the 121 (=1+8+16+24+32+40) discrete points around each reference lattice point. 

For a function f(x) given only as discrete points, the measure of accuracy of the fit is a function d(x) = f(x) - g(x) where g(x) is the approximating function to f(x). If this is interpreted as minimizing d(x) over all x in the interval, one point in error can cause a major shift in the approximating function towards that point. The better method is the least squares curve fit, where d(x) is minimized if... 

Now, to be sure, McCulloch-Pitts neurons are unrealistically rendered versions of the real thing. For example, the assumption that neuronal firing occurs synchronously throughout the net at well defined discrete points in time is simply wrong. The tacit assumption that the structure of a neural net (i.e. its connectivity, as defined by the set of synaptic weights) remains constant over time is known be false as well. Moreover, while the input-output relationship for real neurons is nonlinear, real neurons are not the simple threshold devices the McCulloch-Pitts model assumes them to be. In fact, the output of a real neuron depends on its weighted input in a nonlinear but continuous manner. Despite their conceptual drawbacks, however, McCulloch-Pitts neurons are nontrivial devices. McCulloch-Pitts were able to show that for a suitably chosen set of synaptic weights wij, a synchronous net of their model neurons is capable of universal computation. This means that, in principle, McCulloch-Pitts nets possess the same raw computational power as a conventional computer (see section 6.4). 

In 1991, Warnock [127] described the surface as a set of discrete points. For each point i," the polish rate is ... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

An example of integrated heat-transfer modehng and reactor design is shown in Figure 11.6. A predicted thermal profile for the reactor section of a combined reactor-heat exchanger is the solid line, while the discrete points are experimentally measured temperatures along the reactor length. The thermal profile is controlled... 

The conclusion is that for every particular set of basis functions and given data, there exists an appropriate size of G that can approximate both accurately and smoothly this data set. A decisive advantage would be if there existed a set of basis functions, which could probably represent any data set or function with minimal complexity (as measured by the number of basis functions for given accuracy). It is, however, straightforward to construct different examples that acquire minimal representations with respect to different types of basis functions. Each basis function for itself is the most obvious positive example. A Gaussian (or discrete points... 

The non-destructive nature of MRI is critical to undertaking long duration experiments that require measurements to be made at discrete points in time. 

In a point release attack, the agent would be released in a plume from a single point (or perhaps several discrete points simultaneously). Individuals near the point of release would be exposed to agent at high levels, whereas those farther away would be exposed at lower doses. Initially, the spread of agent would be confined to the space in which it was... 

Gas phase molecules are adsorbed at discrete points of attachment on the surface that... 

Only few discrete points of each correlation function are actually chosen for the evaluation, namely the points32 located at... 

See Ogunnaike, Babatunde A., and W. Harmon Ray, Process Dynamics, Modeling, and Control, Oxford University Press (1994) Seborg, D., T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York (2003). The z-transform is useful when data is available at only discrete points. Let... 

The region between the walls is first divided into bins, and the density at the midpoint of each bin is treated as an independent variable. If the density is desired at M discrete points, then the numerical method reduces to simultaneously solving M equations in M unknowns ... 

A discrete-event simulation tool considers - nomen est omen - discrete events at discrete points in time. Typically, in a discrete-event simulator items such as parts are moving through the modeled system changing their state, e.g., when they enter or leave a machine. A reactor in the process industry continuously produces a certain output. This is something a discrete-event simulator is not really made... 

The equilibrium line for the liquid-liquid transition given in Figure 5.12(a) is also given in Figure 5.12(b). This line goes through the intersections of all pairs of lines that satisfy xj g + x g =1, e.g. the intersection of the lines for xsi.B = 0.1 and 0.9 and that for xsi.b = 0.2 and 0.8. These two specific examples represent two discrete points on the equilibrium curve. 

Let us return for the moment to Eq. (2.2). In atmospheric problems it is impossible to solve the equations of motion analytically. Under these conditions information about the instantaneous velocity field u is available only from direct measurements or from numerical simulations of the fluid flow. In either case we are confronted with the problem of reconstructing the complete, continuous velocity field from observations at discrete points in space, namely the measuring sites or the grid points of the numerical model. The sampling theorem tells us that from a set of discrete values, only those features of the field with scales larger than the discretization interval can be reproduced in their entirety (Papoulis, 1%5). Therefore, we decompose the wind velocity in the form... 

Dynamic matrix control uses time-domain step-response models (called convolution models). As sketched in Fig. 8.18, the response (x) of a process to a unit step change in the input (Ami = ) made at time equal zero can be described by the values of x at discrete points in time (the fc, s shown on the figure). At r nTJ, the value of X is h r,. If Affii is not equal to one, the value of x at f = n7 is b j Aibi, The complete response can be described using a finite number (NP) values of b coefficients. NP is typically chosen such that the response has reached 90 to 95 percent of its final value. 

An example of this procedure is shown in Fig. 1. This example shows the build-up of the 2D potential of Ti2S projected along the short c axis, but the principle is the same for creating a 3D potential. The potential is a continuous function in real space and can be described in a map (Fig. 1). On the other hand, the structure factors are discrete points in reciprocal space and can be represented by a list of amplitudes and phases (Table 1). In this Fourier synthesis we have used the structure factors calculated from the refined coordinates of Ti2S °. 

SWV experiments are usually performed on stationary solid electrodes or static merciuy drop electrodes. The response consists of discrete current-potential points separated by the potential increment AE [1,20-23]. Hence, AE determines the apparent scan rate, which is defined as AE/t, and the density of information in the response, which is a number of current-potential points within a certain potential range. The currents increase proportionally to the apparent scan rate. For better graphical presentation, the points can be interconnected, but the fine between two points has no physical significance, as there is no theoretical reason to interpolate any mathematical function between two experimentally determined current-potential points. The currents measured with smaller A are smaller than the values predicted by the interpolation between two points measured with bigger AE [3]. Frequently, the response is distorted by electronic noise and a smoothing procedure is necessary for its correct interpretation. In this case, it is better if AE is as small as possible. By smoothing, the set of discrete points is transformed into a continuous current-potential curve. Care should be taken that the smoothing procedttre does not distort the square-wave response. 

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