Ellipse, defined

From equations (8), (9) and (10) it is evident that the path of the electron in the ith region is a segment of the KepleT ellipse defined by the segmentary radial and the azimuthal quantum numbers n and k, so that it can be described by the known equations... 

Figure F5.1.15 A representation of a CMC ellipse defines the volume of color acceptability. Figure courtesy of GretagMacbeth. This black and white facsimile of the figure is intended only as a placeholder for full-color version of figure go to http //www.currentprotocols.com/colorfigures...

Thus, the original ellipse defined by A has transformed to an ellipse defined by B = PAP as a result of the permanent biaxial stretch defined by Q. 

Egan Egg Ellipse defined in the ClogP and PSA space. Estimate whether a compound s absorption and membrane permeation are good enough to be orally bioavailable (10)... 

Figure 2. Schematic representation of the elliptic phase diagram characteristic ofproteins. In this Pressure-temperature diagram, the ellipse defines pressure and temperature range inside -which the protein is stable and not denatured (Suzukf, Ha-wley ).

Now consider the hypothetical problem of trying to teach the physics of space flight during the period in time between the formulation of Kepler s laws and the publication of Newton s laws. Such a course would introduce Kepler s laws to explain why all spacecraft proceed on elliptical orbits around a nearby heavenly body with the center of mass of that heavenly body in one of the focal points. It would further introduce a second principle to describe course corrections, and define the orbital jump to go from one ellipse to another. It would present a table for each type of known spacecraft with the bum time for its rockets to go from one tabulated course to another reachable tabulated course. Students completing this course could run mission control, but they would be confused about what is going on during the orbital jump and how it follows from Kepler s laws. 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

This equation defines an ellipse with semiaxes A and muA, determined by the initial values of p and q ... 

ASP is called the true anomaly of the planet. It is found that, in nstronoinieal calculations, the true anomaly is not a very convenient angle with which to deal. Instead we use the mean anomaly, which is defined to be 271 times the ratio of the area of the elliptic sector ASP to the area of the ellipse. Another angle of significance is the eccentric anomaly, u, of the planet defined to be the angle ACQ where Q is the point in which the ordinate through P meets the auxiliary circle of the ellipse. 

This is the general equation for an ellipse in two independent variables, as shown in Fig. 21. If a new coordinate system is defined that has its center at the point S and axes directed along Xt and X2 of Fig. 21, then Eq. (Ill) reduces to... 

Figure 2. Equipotential sections through the potential energy surface for an exchange reaction. The sections define ellipses if the surfaces are parabolic the top left set refer to the initial state (precursor complex) and the bottom right set refer to the final state (successor complex). The dashed line indicates the reaction coordinate. Parameters P and Pa reflect the state of polarization of the solvent, and coordinates d2 and da reflect the inner-shell configurations of the two reactants...

With regard to comprehensive LC data elaboration, the acquired data is commonly elaborated with dedicated software that constructs a matrix with rows corresponding to the duration of the second-dimension analysis and data columns covering all successive second-dimension chromatograms. The result is a bidimensional contour plot, where each component is represented as an ellipse-shaped peak, defined by double-axis retention time coordinates. When creating a 3D chromatogram, a third axis by means of relative intensity is added. The colour and dimension of each peak is related to the quantity of each compound present in the sample. Figure 4.9 illustrates an example of data elaboration in comprehensive LC. 

Although a strictly monochromatic wave, one for which the time dependence is exp( — itot), has a well-defined vibration ellipse, not all waves do. Let us consider a nearly monochromatic, or quasi-monochromatic beam ... 

Figure B3.5.3 The relation of ellipticity to the differential absorption of circularly polarized radiation. The oscillating radiation sine wave, 01, is proceeding out of the plane of the paper towards the viewer. (A) Plane-polarized radiation is made up of left- and right-handed circularly polarized components, OL and OR, respectively. Absorption by a chromophore in a nonchiral environment results in an equal reduction in intensity of each component, whose resultant is a vector oscillating only in the vertical plane—i.e., plane-polarized radiation. (B) Interaction of the radiation with achiral chromophore leads to unequal absorption, so that combination of the emerging vectors, OL and OR, leads to a resultant that describes an elliptical path as it progresses out of the plane of the paper. The ratio of the major and minor axes of the ellipse is expressed by tan 0, thus defining ellipticity. The major axis of the ellipse makes an angle (q) with the original plane, which defines the optical rotation. This figure thus demonstrates the close relation between optical rotation and circular dichroism.

Figure 12. (a) Index ellipsoid defined by ne and no with a ray of light propagating in an arbitrary direction OP (b) an ellipse that is formed by the intersection of the plane normal to OP and the index ellipsoid. The principal axes of this ellipse are the angularly dependent index ne (9) and the angularly independent index nQ. 

The diarylhydrazine 7i-system formally consists of two cross-conjugated fragments (ellipses Ax and A2, Scheme 2.25), but most of the molecular orbitals have well-defined delocalized character and these parts of the molecule cannot be considered as quasiautonomous. 

Figure 1.15 Tilted collision frame at the sample. The photon beam direction defines the z-axis the x- and y-axes are aligned with the major (a) and minor (b) axes of the polarization ellipse which lies in the plane perpendicular to the direction of the photon beam. X is the tilt angle between the x-axis and the plane of the storage ring. The direction of the emitted electron is described by the polar and azimuthal angles 0 and measured in the tilted...

It is convenient to make these substitutions, i.e, to replace the general Stokes parameters and S2 by St and a tilt angle A (which implies S2 = 0). S, is then defined as the greatest possible value of the Stokes parameter for linear polarization or, alternatively, by the excess of linear polarization intensity along the major (a) and minor (b) axes of the polarization ellipse, i.e.,... 

---