Paraboloid shapes

As an effect of the linear and quadratic vibronic integrals the adiabatic potential surface stays no longer paraboloid-shaped. It exhibits an additional warping with several local minima and saddle points out of the reference high-symmetry configuration Q0. 

To avoid these aberrations a surface would have to be ground to fixed curvature with ellipsoidal or paraboloidal shape (Heald 1982). This is extremely difficult to achieve with any accuracy with the very large values needed for i v at X-ray reflection grazing incident angles (equation (5.24)). This approach is more feasible for longer wavelengths where 6C (equation (5.20)) and <3V (equation (5.25)) are larger and Rv (equation (5.24)) is smaller. 

Experimentally, a roughly paraboloid shape for the coarse-mixing region (Fig. 2) is observed. 

The paraboloidal antenna derives its name from the paraboloidal shape of the reflecting metallic surface. A source is placed at the feed point, like a horn, and the reflected wave flluminates the circular aperture. For the simplistic case of a uniform field distribution over the circular aperture, the useful parameters for this case are given in Table 13.5. 

After the transport vehicle is fixed on the centrifuge with the filled casting mould, the top heating device is raised a few centimetres above the casting mould. The centrifuge is set into rotation (5.6 revolutions per minute). The paraboloid-shaped upper side of the meniscus is generated by the centrifugal force. 

Figure 4.5 Example of Taylor dispersion in a microchannel with a steady PoiseuiUe flow (a) initial flat concentration of the solute, (b) stretching of the solute to a paraboloid-shaped plug neglecting diffusion, and (c) solute plug with diffusion indicated by vertical arrows...

When a paraboloidal-shaped tip is pressed into the film with a loading force F, the contact area s radius a is given by... 

The first observation is that the cured shape of an unsymmetric cross-ply laminate is often cylindrical, whereas we would predict it to be a saddle shape (hyperbolic paraboloid) from classical lamination theory (the curvatures can be shown to be = - Ky or - = Ky). A thick lami-... 

Unlike the curves you may have seen in geometry books (such as bullet-shaped paraboloids and saddle surfaces) that are simple functions of x and y, certain surfaces occupying three dimensions can be expressed by parametric equations of the form x = f(u,v), y = g(u,v), z = h(u,v). This means that the position of a point in the third dimension is determined by three separate formulas. Because g, and h can be anything you like, the remarkable panoply of art forms made possible by plotting these surfaces is quite large. For simplicity, you can plot projections of these surfaces in the x-y plane simply by plotting (x,y) as you iterate u and in a... 

In the case of energy minimization, the goal of the added term should be to make what was a local minimum flat, or slightly convex, thus causing the system to roll away to another minimum. The obvious term to do this is a paraboloidal mound complementary in shape to the harmonic neighborhood of the local minimum ... 

The curvature of a wavefront appears transformed into the curvature of a mirror surface shaped so that it would focus the total wavefront into the point of ohservation.The reason is that a focusing mirror reflects light in such a way that the total wavefront arrives to the focal point at one point of time. Thus, a small flat wavefront that passes by will appear tilted at 45°. A larger flat wavefront will not only appear tilted but will also be transformed into a paraboloid whose focal point is the point of observation. A spherical wavefront appears transformed into an ellipsoid, where one focal point is the point source of light (A) and the other is the point of observation (B). This configuration represents one of the ellipsoids of the holodiagram. 

Van Landigham et al. reviewed nanoindentation of polymers, [40, 41] including a summary of the most common analyses of load-indentation data. Chief among these methods is an analysis of indentation load-penetration curves according to the Oliver-Pharr method. [42] This method is based on relationships developed by Sneddon for the penetration of a flat elastic half space by different probes with particular axisymmetric shapes (e.g., aflat-ended cylindrical punch, a paraboloid of revolution, or a cone) [43], More recently, Withers and Aston discussed indentation in the context of plasticity and viscoelasticity [44]. 

Equation 5.163 can be generalized for smooth surfaces of arbitrary shape (not necessarily spheres). Eor that purpose, the surfaces of the two particles are approximated with paraboloids in the vicinity of the point of closest approach Qi = h. Let the principal curvatures at this point be Cl and c[ for the hrst particle, and C2 and c for the second particle. Then the generalization of Equation 5.163 reads ... 

Our early work on the hydrogen atom confined inside prolate spheroidal boxes also dealt with the molecular hydrogen H+ and molecular HeH++ ions [18]. The investigation of the hydrogen molecular ion was extended recently for confinement in boxes with the same shape with penetrable walls [40]. On the other hand, more than ten years ago, we investigated the ground state of the helium atom confined in a semi-infinite space [46] and inside boxes [47] with paraboloidal boundaries. 

Figure 5 Ground-state energy evolution of He - located at the focal position D = 0.5 au - as a function of cage volume for hard-wall spheroidal confinement. Dashed curve results of this work. Continuous line values obtained by Ley-Koo et al. [49] for He confinement by symmetric paraboloidal boxes corresponding approximately to the same shape and volume as the prolate spheroidal box considered here [51].

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