Parabola

This is then normalised to produce a, the parabola is then defined for any real values of u  

The Table 1.4 characterizes the curve represented by the equation. Example 1.9 

Parametric equations of the parabola. Let a and b be two lines in a plane with equations 

to obtain an equation of the curve, we eliminate the parameter t from the two equations. Eliminating t from Equation 1.67a and Equation 1.67b we get 

Equation cf tangent of a parabola. The slope of tire tangent line can be obtained as  

The slope of the tangent line at a point (x , y ) = 2a/y . The equation of the tangent line will be 

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Equation V-64 is that of a parabola, and electrocapillary curves are indeed approximately parabolic in shape. Because E ax tmd 7 max very nearly the same for certain electrolytes, such as sodium sulfate and sodium carbonate, it is generally assumed that specific adsorption effects are absent, and Emax is taken as a constant (-0.480 V) characteristic of the mercury-water interface. For most other electrolytes there is a shift in the maximum voltage, and is then taken to be Emax 0.480. Some values for the quantities are given in Table V-5 [113]. Much information of this type is due to Gouy [125], although additional results are to be found in most of the other references cited in this section. 

In classical mechanics, it is certainly possible for a system subject to dissipative forces such as friction to come to rest. For example, a marble rolling in a parabola lined with sandpaper will eventually lose its kinetic energy and come to rest at the bottom. Rather remarkably, making a measurement of E that coincides with... 

Since shallow-level impurities have energy eigenvalues very near Arose of tire perfect crystal, tliey can be described using a perturbative approach first developed in tire 1950s and known as effective mass theoiy (EMT). The idea is to approximate tire band nearest to tire shallow level by a parabola, tire curvature of which is characterized by an effective mass parameter m. ... 

HyperChein has two synch ron ons transit meth ods im piemen ted. The linear synchronous transit method (LST) searches for a maximum along a linear path between reactants and products. It may happen that this method will end up with a structure having two or more negative eigenvalues. The quadratic synchronous transit method (QSTlisan improvement of LST approach and searches for a maximum along a parabola connecting reactants and products, and for a minimum in all directions perpendicular to the parabola. 

In applying Simpson s rule, over the interval [a, i>] of the independent variable, the interval is partitioned into an even number of subintervals and three consecutive points are used to determine the unique parabola that covers the area of the first... 

The problem now is to find the best value of the intercept on the vertical axis. We can do this by fitting the experimental points to a parabola. 

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

In the case of drug design, it may be desirable to use parabolic functions in place of linear functions. The descriptor for an ideal drug candidate often has an optimum value. Drug activity will decrease when the value is either larger or smaller than optimum. This functional form is described by a parabola, not a linear relationship. 

Parabola or quadratic curve Cubic curve nth degree curve... 

The BQ term alone, wifh B positive, would give a pofenfial resembling fhe harmonic oscillator pofenfial in Figure 6.4 (dashed curve) buf wifh steeper sides. The inclusion of fhe AQ term, wifh A negative, adds an upside-down parabola af 0 = 0 and fhe resulf is a W-shaped pofenfial. The barrier heighf b is given by... 

Conic Sections The cui ves included in this group are obtained from plane sections of the cone. They include the circle, ehipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixea line, called the directrix. This ratio is the eccentricity e. lie = 0, the conic is a circle if 0 < e < 1, the conic is an ellipse e = 1, the conic is a parabola ... 

Second-Degree Least Squares with Five Points For five evenly spaced points x 2, X-, Xq, X, andi 2 (separated by distance / ) and their ordinates/ 2,/ i,/o,/i, and/2, assume a parabola is fit by least squares. Then the derivative at the center point is... 

This method approximates/(x) by a parabola on each subintei val. This rule is generally more accurate than the trapezoidal rule. It is the most widely used integration formula. 

Fig. 7. Cusp-shaped potential, made up of parabola and a vertical wall.

Consider a potential V x) having a single minimum separated from the continuous spectrum by a sufficiently large barrier satisfying (1.1), e.g., a cubic parabola (fig. 19)... 

Fig. 19. Cubic parabola potential. Turning points are shown. The dashed line indicates the stable potential with the same well frequency.

The identity of eqs. (2.6) (at T = 0) and (3.47) for the cubic parabola is also demonstrated in appendix A. Although at first glance the infinite determinants in (3.46) might look less attractive than the simple formulas (2.6) and (2.7), or the direct WKB solution by Schmid, it is the instanton approach that permits direct generalization to dissipative tunneling and to the multidimensional problem. 

For convenience of notation we accept from here on, that each frequency of the problem co has a dimensionless counterpart denoted by a capital Greek letter, so that co,- = coofl,. The model (4.28) may be thought of as a particle in a one-dimensional cubic parabola potential coupled to the q vibration. The saddle-point coordinates, defined by dVjdQ = dVjdq = 0, are... 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

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