Point, representative

Such an ensemble of systems can be geometrically represented by a distribution of representative points m the F space (classically a continuous distribution). It is described by an ensemble density fiinction p(p, q, t) such that pip, q, t)S Q is the number of representative points which at time t are within the infinitesimal phase volume element df p df q (denoted by d - D) around the point (p, q) in the F space. 

Fig. 5. A representation of ranitidine displaying four layers of the Connolly solvent-accessible dot surface normally color-coded in this process to correspond with the energies of electrostatic potential (color not shown here). Thus, the highest charge density would be indicated by red dots representing points where the attraction to an atom is strongest, and conversely, purple points would signify regions of maximal positive charge.

Traversing for Mean Velocity Mean velocity in a duct can be obtained by dividing the cross section into a number of equal areas, finding the local velocity at a representative point in each, and averaging the results. In the case of rectangular passages, the cross section is usually divided into small squares or rectangles and the velocity is found at the center of each. In circular pipes, the cross section is divided into several equal annular areas as shown in Fig. 10-7. Read-... 

Some tight formations can be seen at 9,250 and 9,400 in. The representative point at 8,900-ft plots in the porous area of the graph. The other points seem to be mostly shale points. Confirmation should be sought with logging data. 

Phase Plane Singular Points.—We shall define the plane of the variables (x,y = x) as the phase plane and investigate the behavior of integral curves (or characteristics) in that plane by means of Eq. (6-2). In case we wish to associate with these curves the motion of the representative point R(x,y), we shall rather speak of them as trajectories and in this case one has to use Eq. (6-1). 

If, however, one communicates to such a system an impulse capable of transferring the representative point JR to a point A outside the unstable cycle (point A, Fig. 6-6), it is clear that from that point the self-excitation can start and JR will approach the outer stable cycle. 

We have thus the following representation of the phenomenon (Fig. 6-10) on the arc AB the motion is slow at the point B the representative point ceases to follow the curve F(x) and its horizontal velocity becomes very large, bringing it to the point C at which a slow motion begins on the arc CD, followed by another jump DC, and so on. 

If C is an orbit, we call it orbitaJly stable if, given e > 0, there is 7) > 0 such that if B is a representative point following a neighboring trajectory, and B is at a distance i) from C at time t0, then B remains within a distance e from 0 for t > t0. If no such rj exists, O is orbitally unstable... 

Thus, if B (the representative point) was initially inside some surface F = c, it will continue to remain inside that surface. With the stronger condition, TF < 0, B will cross successively all surfaces V = c from outside to inside and will thus approach the origin asymptotically. Theorems on instability can be interpreted analogously. 

The principle is to consider a differential equation as an operator which after some time (which will be assumed to be 2n) transfers the representative point JR from a point A of the phase plane to some other point B. This process can be written conventionally as... 

Curve C of Fig. 6-15(a) corresponds to /a 0.1, and curve C of Fig. 6-15(b) to p = 10. It is observed that in the first case the curve O is almost a circle (as in a harmonic oscillator) while in the second case it has very special features. In fact from very high velocity (y — sc), the solution drops very rapidly to zero (at the point where it cuts the x-axis). At this point the relaxation interval begins, the representative point moving near the x-axis with a very small velocity after that its velocity increases rapidly to a considerable value, and soon. 

Now suppose the representative point had passed from 1 to 2, not along the line of heterogeneous states, but along some path on the surface itself this will be a continuous transition of state. Then, from (1) ... 

The contour lines represent points of relative density 1.0, 0.9, 0.8,..0.1 for a hydrogen atom. This figure, with the added proton 1.06 A from the atom, gives the electron distribution the hydrogen molecule-ion would have (in the zeroth approximation) if the resonance phenomenon did not occur it is to be compared with figure 6 to show the effect of resonance. 

The optimize command maximizes a statistical "likelihood function". The higher this function, the more likely is the parameter to be the correct one. In the figure below, the symbols represent points calculated by the program Topaz (the full model), and the solid lines are the values calculated from the reduced-order model using the parameters determined by the program. 

Fig. 1.—Amino Acid Sequence for Glycophorins and A. [Diamonds represent points of 0-glycosylation. The single N-glycosylation point occurs at Asn-26. The data were obtained from Refs. 8 and 19.]...

The number of detailed studies on these last systems is nowadays sufficiently large to generalize the results, and to project the conclusion to more complex (o "perverse" according to Coulson) systems. The traditional view of a reaction occurring on a well defined surface, with a flux of representative points passing the transition state region is unteiiable. The separation between static and dynamic aspects of a problem, so often exploited for studies an isolated molecule must be reconsidered. 

Chernoff H (1973) The use of faces to represent points in k-dimensional space graphically. J Am Statist Assoc 68 361... 

Kleiner B, Hartigan JA (1981) Representing points in many dimensions by trees and castles. J Am Stat Assoc 76 260... 

With due regard to the lateral variations in composition which can arise as a consequence of source geometry and positioning (discussed in Section II), it is vise to analyze the alloy film at a number of representative points. For example, if a catalytic reaction was carried out over an alloy film deposited inside a spherical vessel maintained at a constant temperature over its entire area, then the mean alloy composition (and the uniformity of composition) is required. A convenient procedure is to cut glass reaction vessels carefully into pieces at the end of the experiment and to determine the composition by X-ray fluorescence analysis of a number of representative pieces. Compositions of Pd-Ag alloy films (40) determined at 12 representative parts of a spherical vessel from the intensities of the AgK 12 and PdKau fluorescent X-ray emissions are shown in Table V mean compositions are listed in the first column. (The Pd and Ag sources were separate short concentric spirals.) In other applications of evaporated alloy films to adsorption and catalytic studies, as good or better uniformity of composition was achieved. Analyses of five sections of a cylindrical... 

Mean composition (% Pd) % Pd analyzed at representative points on vessel surface... 

Figure 27. Seven possible cases of the polygonal surface representation in a single pyramid. The Euler characteristic is calculated as a sum of the number of faces and the number of vertices minus the number of edges of the polygons. The black and white circles represent points with higher and lower values relative to the threshold one. The gray area is the schematic representation of the surface inside a pyramid [225].

Although an electrophilic reagent should and probably does form -complexes with the aromatic substrate, it is not useful to consider such complexes as intermediates. They represent points on a possible path, but only one of several possible paths, to the a-bonded transition state. If there is a complete equilibrium between the ground state, the 7r-complex, and the transition state, then the particular path taken by the... 

Fig. 9 shows comparisons of CFD results with experimental data at a Reynolds number of 986 at three of the different bed depths at which experiments were conducted. The profiles are plotted as dimensionless temperature versus dimensionless radial position. The open symbols represent points from CFD simulation the closed symbols represent the points obtained from experiment. It can be seen that the CFD simulation reproduces the magnitude and trend of the experimental data very well. There is some under-prediction in the center of the bed however, the shapes of the profiles and the temperature drops in the vicinity of the wall are very similar to the experimental case. More extensive comparisons at different Reynolds numbers may be found in the original reference. This comparison gives confidence in interstitial CFD as a tool for studying heat transfer in packed tubes. 

The orbit described by the representative point in the phase plane (pq) is given by the relationship between q and p. From... 

The value of the phase constant a does not affect the size or shape of the orbit, but it does affect the position of the orbit as a function of time. If at time t = t number of different identical oscillators have q = 0 but values of q lying between v and — v the representative points will lie on a straight line coincident with the p axis. In time, these points will describe their individual orbits but remain collinear. 

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