A cusp

In order to find Eq we study first the auxiliary problem of a cusp-shaped harmonic potential with a wall placed at x = Xp (see fig. 7),... 

Curran [61C01] has pointed out that under certain unusual conditions the second-order phase transition might cause a cusp in the stress-volume relation resulting in a multiple wave structure, as is the case for a first-order transition. His shock-wave compression measurements on Invar (36-wt% Ni-Fe) showed large compressibilities in the low stress region but no distinct transition. 

The STO-3G wavefunction does not have a cusp at the nucleus. Very few molecular properties depend on the exact shape of the wavefunction at the nucleus ... 

The electron-electron repulsion operator has a singularity for r 12 = 0 which results in the exact wave function having a cusp (discontinuous derivative).- ... 

Under low-frequency excitation, the flame front is wrinkled by velocity modulations (Fig. 5.2.5). The number of undulations is directly linked to frequency. This is true as far as the frequency remains low (in this experiment, between 30 and 400 Hz). The flame deformation is created by hydrodynamic perturbations initiated at the base of the flame and convected along the front. When the velocity modulation amplitude is low, the undulations are sinusoidal and weakly damped as they proceed to the top of the flame. When the modulation amplitude is augmented, a toroidal vortex is generated at the burner outlet and the flame front rolls over the vortex near the burner base. Consumption is fast enough to suppress further winding by the structure as it is convected away from the outlet. This yields a cusp formed toward burnt gases. This process requires some duration and it is obtained when the flame extends over a sufficient axial distance. If the acoustic modulation level remain low (typically v /v < 20%),... 

Unlike the wave function, the electron density is an observable and can be measured experimentally, e. g. by X-ray diffraction. One of its important features is that at any position of an atom, p(r) exhibits a maximum with a finite value, due to the attractive force exerted by the positive charge of the nuclei. However, at these positions the gradient of the density has a discontinuity and a cusp results. This cusp is a consequence of the singularity ZA... 

The ipis wave function is therefore symmetrical along x through the origin, where it has a cusp. 

Lenzenweger (1999) performed MAXCOV using the three scales as taxon indicators. He used an interval size of. 50 SD and MRIN of 15. Two of the plots showed clear peaks, and one produced a cusp. The one incomplete peak was probably a consequence of the interval size being set too high, which allowed for only seven intervals on the input variable. A lower interval size would have produced a finer gradation and probably allowed the cusp to turn into a full peak. However, this may not have been possible due to the modest size of the sample. The base rates estimates were. 11,. 22, and. 13. The author did not report a base rate consistency test, but one can easily calculate the SD of the three estimates to be. 06, which is somewhat high but probably acceptable. 

We note in Fig. 7.6 that both M graphs pass smoothly through F = 0, whereas those for M exhibit a cusp (labelled as B), which is due to the asymmetry of the occupied levels when F / 0 (Table 7.2). States shifted above the band are also above the FL, and have only an indirect effect on AE, while states shifted below the band are filled, and act to produce a greater field effect. 

The singularity in the interelectron Coulomb potential r 1 creates a cusp in the exact solution of the Schrodinger equation,... 

These equations represent a parametric curve with parameter y. From a set of values of the parameter y it is possible to draw a curve in the xq — yo plane, so we obtain a bifurcation curve as a function of parameter y. This curve with a cusp point can be considered as the border that dividing the plane xq — yo into domains with one and three equilibrium states respectively. 

The differences between the single-configuration wavefunctions are more clearly illustrated by comparing their plots of the intracule function h(ri2), also shown in Fig. 1. This plot reveals the absence of an electron-electron cusp for both the closed and split-shell functions, but shows that the inclusion of exp( —yri2) causes the distribution to have a minimum at ri2=0, forming a cusp (of the correct sign) at that point. This feature will be important for the description of phenomena that depend upon the coincidence probability. 

Unfortunately, the A-representability constraints from the orbital representation are not readily generalized to the spatial representation. A first clue that the A-representability problem is more complicated for the spatial basis is that while every A-representable Q-density can be written as a weighted average of Slater determinantal Q-densities in the orbital resolution (cf. Eq. (54)), this is clearly not true in the spatially resolved formulation. For example, the pair density (Q = 2) of any real electronic system will have a cusp where electrons of opposite spin coincide but a weighted average of Slater determinantal pair densities,... 

Detonation Wave Cissoidal. Under this term is known a wave in the form of a curve having a cusp at the origin and point of infection at infinity. Such waves were investigated in France by Touvenin Argus. Following is an abstract of their papers (Ref 90)... 

Figure 3.18. The process of changes in morphology due to interface instability. (a)-(e) Starting from a smooth interface, it is seen that the morphology changes to a cusp array, to a rod structure, to a dendritic structure.

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