LINE-SEGMENT

I m = 1,.., m, viewed as two sets of planar parametric line segments. Figure 2 shows the schematic flow chart of the identification method for determining the parameters of an OSD. 

Figure C3.6.2 (a) The (fi2,cf) Poincare surface of a section of the phase flow, taken at ej = 8.5 with cq < 0, for the WR chaotic attractor at k = 0.072. (b) The next-amplitude map constmcted from pairs of intersection coordinates. ..,(c2(n-l-l),C2(n-l-2),C2(n-l-l)),...j. The sequence of horizontal and vertical line segments, each touching the diagonal B and the map, comprise a discrete trajectory. The direction on the first four segments is indicated.

Figure 12. Vibrational levels for the first-excited electronic state of H3 calculated [4] using Longuet-Higgins phase /4(R) = cp/2 Eq, (A.14) with a path-dependent phase A(R) =y(p,9,(p), The extra levels arising in one calculation but not in the other are indicated by longer line segments,...

Figure 13. Vibrational levels for the first-excited electronic state of HD2 calculated [8] using split basis (SB) technique with A(R) = tp/2 coordinate-transformation (CT) treatment with A(R) — tp/2 Eq. (A. 14) with A (R) — y(p, 9, tp). Shown by the longer line segments are the levels assuming different values in two sets of calculations.

Considering the line segment CD, for example, equations (11.110) reduce... 

To complete an approximate sketch of the titration curve, we draw separate straight lines through the two points before and after the equivalence point (Figure 9.35e). Finally, a smooth curve is drawn to connect the three straight-line segments (Figure 9.35f). 

Suppose that, near some fixed point G F, dT, the graph F, is a straight line segment parallel to the x axis. Let G (0, T) be an arbitrary fixed point and let Re C denote the ball of a sufficiently small radius with centre (a °,t°). First, we examine the smoothness of the function X = (IF, w). Let D stand for a first-order derivative and let (p denote an arbitrary smooth function in i 2s such that p = 0 outside Rzeji 0 < (> < 1, and dpidy = 0 on F. ... 

This method of optimization is known as the generalized reduced-gradient (GRG) method. The objective function and the constraints are linearized ia a piecewise fashioa so that a series of straight-line segments are used to approximate them. Many computer codes are available for these methods. Two widely used ones are GRGA code (49) and GRG2 code (50). 

Figure 8.24. Geometric fragmentation with randomly oriented line segments.

N. Ashgriz and J.Y. Poo, FLAIR Flux Line-Segment Model for Advection and Interface Reconstruction, J. Comput. Phys. 93 (1991). 

There are actually four turning points in each well. It does not matter from which one we start, because the straight-line segments between the turning points are incorporated into that part of g(E) which is responsible for the single well quantization. 

The Conjugate Peak Refinement (CPR) method may be eonsidered as a dynamical version of the chain method, where points are added or removed based on a sequenee of maximizations along line segments and minimizations along the conjugate directions. 

A quadrilateral is a four-sided polygon determined by four coplanar points (three of which are noncollinear), if the line segments thus formed intersect each other only at their end points, forming four angles. 

A chord of a circle (or sphere) is a line segment whose end points lie on the circle (or sphere). A line which intersects the circle (or sphere) in two points is a secant of the circle (or sphere). A diameter of a circle (or sphere) is a chord containing the center and a radius is a line segment from the center to a point on the circle (or sphere). 

For the Rankine cycle, the area enclosed by the line segments connecting points 1, 2, 3, 4, 1 on Figure 2-36 represents the net heat transferred into the system per unit mass, because... 

That Dfractai givcs the expected result for simple sets in Euclidean space is easy to see. If A consists of a single point, for example, we have N A, e) = 1, Ve, and thus that / fractal = 0. Similarly, if A is a line segment of length L, then N A,e) L/e so that Dfraciai = 1. In fact, for the usual n dinien.sional Euclidean sets, the fractal dimension equals the topological dimension. There are nrore complicated sets, however, for wliic h the two measures differ. 

Not "star" cf. Kbnig 1, p. 50. I emphasize that a corona contains a single point only. A corona is not a graph because its line segments are closed at one end only. 

The function QG occurring in the integrand is represented by two straight line segments below the axis. Xf the mean value theorem is applied to the separate integrals, the result is... 

Reference to Fig. 3-2 now makes it apparent that the value of the integral in (3-7) is equal to the length of the crosshatched line segment... 

The advantage of equation (A1.35) over equation (A1.34) is that it approximates the line segment with a curve (parabola) instead of a straight line, which generally allows the line segment to better follow the actual line, and increases the accuracy with which the area can be determined. 

Equations (7-29) and (7-32) both have the same form. It is easy to see that their temperature profiles are not linear. Their shapes are the same. Note that the temperature profile can be factored into two straight-line segments, one for each separate k. The composite will then be a line that curves upward in the usual plot. The tangent at any T can be used to obtain a value of an apparent activation enthalpy. The apparent activation enthalpy increases with temperature whenever the composite constant is a sum of the rate constants for elementary reactions. 

The structure of the [Fe2 / - 82) P o - 61148) )2] complex is presented in Figure 1. The structure we used was obtained from x-ray crystallography, symmetrized such that the environments of the two Fe atoms are the same. The local symmetry of each Fe atom is C3, the overall symmetry of the complex is Ci. To aid in the interpretation of the computed results we oriented the complex such that the z-axis passes through the center of mass and the projections of the two P — Fe- 8 line segments onto the z-axis are maximized. 

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