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Infinity visualizing

To visualize an object with Dm symmetry, imagine a cylinder whose outside is covered with n slanted striations, as illustrated at the top of Figure 8. The two constructions shown (D symmetry) are enantiomorphs whose sense of chirality is related to the way in which the striations are slanted. As n approaches infinity, the symmetry of the constructions approaches Z) in the limit, infinitely many C2 axes are embedded in a plane perpendicular to the C axis. This is the symmetry of a stationary cylinder undergoing a twisting motion, as indicated by the arrows on the cylinders at the bottom of Figure 8, and of an axial tensor of the second rank.41 It is also the helical symmetry of a nonpolar object undergoing a screw displacement, that is, of an object whose enantiomorphism and sense of chirality are T-invariant. [Pg.19]

Taking derivatives of experimental data (i.e. for determining the coefficient of linear thermal expansion) is not quite as straightforward as taking derivatives of algebraic functions, since data tend to have some scatter. If, for example, a data set has a visually upward trend but two adjacent points are stacked on top of each other, the slope between these points is infinite. An improvement would be to average the slopes from a cluster of points, but if infinity is one of the values, the average value is still infinity. [Pg.95]

Figure 1.17 (a) Visual image of a 1 im-diameter polystyrene bead. The lower images show Raman images of the bead, recorded with a lOOx (NA = 0.95) infinity-corrected microscope objective at increments of (left to right) 1.0, 0.5 and 0.1 pm per step, respectively (b) Raman spectrum measured from the center of one of these beads. Reproduced with permission from Ref [41]. [Pg.33]

Another problem that can occur during the course of a minimization in redundant internal coordinates is the internal forces being reported as infinity or undefined [126]. As discussed in Section 10.2.3, the energy derivatives are typically computed in Cartesian coordinates and later converted to internals using Eq. (10). If the redundant internal coordinate definitions become ill defined and/or include linear dependences, then the conversion of forces and Hessians to internal coordinates can become problematic. The easy fix to this problem begins by inspecting the latest structure in the optimization using visualization software to ensure that the structure is reasonable. If all is well, start... [Pg.214]

Cash Award Visualizing Infinity Other Batrachions I The Crying of Fractal Batrachions... [Pg.342]

The recognition of self-similarity simplifies the description of the physical world, all in response of the vacuum to the curvature of space-time, but complicates the perception of three-dimensional beings of their four-dimensional environment, which they are physically unable to visualize. The perceived infinity of space is the illusion created in simply-connected tangent space by the multiply-connected cosmic reality. The large-scale structure of the imiverse is destined to remain unknown for a long time. [Pg.408]

The power n /f of a lens in visual optics is usually expressed in diopters (D), or reciprocal meters. That is, the power of a lens is in diopters if f is in meters. The power of the optical system of the human eye is approximately 60 D when the eye is focused at infinity. The corresponding focal length is 22 mm. [Pg.72]

Strictly speaking, an atomic orbital does not have a well-defined shape because the wave function characterizing the orbital extends from the nucleus to infinity. In that sense, it is difficult to say what an orbital looks like. On the other hand, it is certainly useful to think of orbitals as having specific shapes. Being able to visualize atomic orbitals is essential to understanding the formation of chemical bonds and molecular geometry, which are discussed in Chapters 8 and 9. In this section, we will look at each type of orbital separately. [Pg.216]

Logarithmic curve. Such curves can be identified from the visual and linguistically available knowledge as any curve that has asymptote to one of the axes, intercept on another axis perpendicular to the first one and then increases without any asymptote into the infinity for both axes values. This is a logarithmic curve of which the mathematical expression can be written as follows. [Pg.188]

Minimum Number of Plates. The slope of the operating line above the feed is On/Fn, and as this slope approaches unity the number of theoretical plates becomes smaller. When On/Vn is equal to 1, Or/D is equal to infinity, and only an infinitesimal amount of product can be withdrawn from a finite column. Frequently it is assumed that total reflux corresponds to the addition of no feed or to the removal of no products. If such is the case, the tower is not meeting the design conditions. It is better to visualize a tower with an infinite cross section, which is separating the feed at a finite rate into the desired... [Pg.128]

Visual inspection of their graphs is not enough to make sure the candidate dominates the unsealed target. Information about this may be hidden by the axis as both are going to 0 as parameter goes to infinity. [Pg.43]

To give a numerical example, a typical automotive camera has an imager width of 0.85 cm with a lens aperture A off/2 at a focal length/of 4 mm focused at 6 m. This setup generates from 1.5 m up to infinity a nearly uniform sharpness. On the upside, this removes the requirement of any mechanical parts for refocusing inside the camera. On the downside, the very supportive monocular depth cue of blur-from-defocus as described in Sect. 7.2 of chapter Human Visual Perception is not present. [Pg.499]

Many instruments designed to analyze spectral information, such as Fabry-Perot and Michelson interferometers, operate best in a collimated beam. Telescopes that produce such a beam are well-known for visual observations the Galilean and the astronomical telescope are examples. The first uses a convex objective lens and a concave eyepiece, while the latter a similar objective and an eyepiece of convex curvature. In either case, the focal points of both lenses must coincide. Implementations of both systems with mirrors and lenses are shown in Fig. 5.2.8. Afocal systems have the object as well as the image at infinity. The example shown... [Pg.161]


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