Space parabolic

Rosato, D. V, Outer Space Parabolic Reflector Energy Converters, SAMPE, June 1963. 

Figure 6.7 Parts of equally spaced parabolic cyclide surfaces are shown in the upper illustration. These were obtained from the parametrisation introduced at equations (6.114) to (6.117) with the mutually orthogonal parabolic defects (6.111) and (6.112) indicated by the two bold curves. The lower illustration shows a cross-section of the upper figure in the plane z = 0 the parabolic defect is evident and the corresponding smectic layers are arranged as shown with the other parabola being perpendicular to the page and passing through the point (0,0,0). By symmetry, there is a similar cross-section in the plane y — 0.

By following Section II.B, we shall be more specific about what is meant by strong and weak interactions. It turns out that such a criterion can be assumed, based on whether two consecutive states do, or do not, form a conical intersection or a parabolical intersection (it is important to mention that only consecutive states can form these intersections). The two types of intersections are characterized by the fact that the nonadiabatic coupling terms, at the points of the intersection, become infinite (these points can be considered as the black holes in molecular systems and it is mainly through these black holes that electronic states interact with each other.). Based on what was said so far we suggest breaking up complete Hilbert space of size A into L sub-Hilbert spaces of varying sizes Np,P = 1,..., L where... 

All Np states belonging to the Pth sub-space interact strongly with each other in the sense that each pair of consecutive states have at least one point where they form a Landau-Zener type interaction. In other words, all j = I,... At/> — I form at least at one point in configuration space, a conical (parabolical) intersection. 

One further effect of the formation of bands of electron energy in solids is that the effective mass of elecuons is dependent on the shape of the E-k curve. If dris is the parabolic shape of the classical free electron tlreoty, the effective mass is the same as tire mass of the free electron in space, but as tlris departs from the parabolic shape the effective mass varies, depending on the curvature of tire E-k curve. From the dehnition of E in terms of k, it follows that the mass is related to the second derivative of E widr respect to k tlrus... 

Eq. (87) really describes a needle crystal which, without noise, has no side branches. The corresponding star structure then cannot fill the space with constant density and the amount of material solidified in parabolic form increases with time, only fike rather than like P for a truly compact (initially finite) object in two dimensions. 

It has been discovered recently that the spectrum of solutions for growth in a channel is much richer than had previously been supposed. Parity-broken solutions were found [110] and studied numerically in detail [94,111]. A similar solution exists also in an unrestricted space which was called doublon for obvious reasons [94]. It consists of two fingers with a liquid channel along the axis of symmetry between them. It has a parabolic envelope with radius pt and in the center a liquid channel of thickness h. The Peclet number, P = vp /2D, depends on A according to the Ivantsov relation (82). The analytical solution of the selection problem for doublons [112] shows that this solution exists for isotropic systems (e = 0) even at arbitrary small undercooling A and obeys the following selection conditions ... 

The canonical form of a grid equation of common structure. The maximum principle is suitable for the solution of difference elliptic and parabolic equations in the space C and is certainly true for grid equations of common structure which will be investigated in this section. 

In this chapter we study the stability with respect to the initial data and the right-hand side of two-layer and three-layer difference schemes that are treated as operator-difference schemes with operators in Hilbert space. Necessary and sufficient stability conditions are discovered and then the corresponding a priori estimates are obtained through such an analysis by means of the energy inequality method. A regularization method for the further development of various difference schemes of a desired quality (in accuracy and economy) in the class of stability schemes is well-established. Numerous concrete schemes for equations of parabolic and hyperbolic types are available as possible applications, bring out the indisputable merit of these methods and unveil their potential. 

A locally one-dlinensional scheme (LOS) for the heat conduction equation in an arbitrary domain. The method of summarized approximation can find a wide range of application in designing economical additive schemes for parabolic equations in the domains of rather complicated configurations and shapes. More a detailed exploration is devoted to a locally one-dimensional problem for the heat conduction equation in a complex domain G = G -f F of the dimension p. Let x — (sj, 2,..., a- p) be a point in the Euclidean space R. ... 

What is the probability density as a function of the momentum p of an oscillating particle in its ground state in a parabolic potential well (First find the momentum-space wave function.)... 

A non-invasive infrared (IK) method has been developed for the measurement of temperatures of small moving fuel droplets in combustion chambers. 7111 The IR system is composed of two coupled off-axis parabolic mirrors and a MCT LWIR detector. The system was used to measure the temperature variations in a chain of monosized droplets generated with equal spacing and diameter (200 pm), moving at a velocity of >5 m/s and evaporating in ambient air. The system was also evaluated for droplet temperature measurements in flames under combustion conditions. 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

Let us now consider the overlap between the spherical and the Stark basis. For the latter, the momentum space eigenfimctions, which in configuration space correspond to variable separation in parabolic coordinates, are similarly related to alternative hyperspherical harmonics [2]. The connecting coefficient between spherical and 5 torA basis is formally identical to a usual vector coupling coefficient (from now on n is omitted from the notation) ... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

A study of the effeets of space and time dependent frietion was presented in Ref 68. One finds a substantial reduction of the rate relative to the parabolic barrier estimate when the friction is stronger in the well than at the barrier. In all eases, the effeets beeome smaller as the redueed barrier height beeomes larger. Comparison with moleeular dynamics simulations shows that the optimal planar dividing surfaee estimate for the rate is usually quite aeeurate. 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

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