Backward

Another factor that can be important in the design of evaporators is the condition of the feed. If the feed is cold, then the backward-feed arrangement has the advantage that a smaller amount of liquid must be heated to the higher temperatures of the second and first stages. 

Finally, the evaluation can be performed by rolling back the tree, starting at the leaves, and working backwards towards the trunk of the tree. 

The principle of the acquisition system is to translate the probe into a tube (including hemispherical drilled holes) step by step, every 0.04 mm, after a forwards and backwards 360 rotation of the tube trigging every 0.2° angular step a 360° electronic scanning of tube with the 160 acoustic apertures. During the electronic scanning the tube is assumed to stay at the same place. The acquisition lasts about 30 minutes for a C-scan acquisition with a 14 kHz recurrence frequency. 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

Maier M, Kaiser W and Giordmaine J A 1969 Backward stimulated Raman scattering Phys. Rev. 177 580-99... 

EIOs), backward wave oscillators (BWOs) or magnetrons are available. Their spectral characteristics may be favourable however, they typically require highly stabilized high-voltage power supplies. Still higher frequencies may be obtained using far-infrared gas lasers pumped for example by a CO- laser [49]. 

Dissociative recombination and associative ionization are represented by the forward and backward directions of... 

For pure S-wave scattering, the difFerential cross section (DCS) is isotropic. For pure P-wave scattering, tlie DCS is symmetric about 0 = n/2, where it vanishes the DCS rises to equal maxima at 0 = 0, ti. For combined S- and P-wave scattering, the DCS is asynnnetric with forward-backward asynnnetry. 

Symmetry oscillations therefore appear in die differential cross sections for femiion-femiion and boson-boson scattering. They originate from the interference between imscattered mcident particles in the forward (0 = 0) direction and backward scattered particles (0 = 7t, 0). A general differential cross section for scattering... 

Thompson K and Makri N 1999 Influence functionals with semiclassical propagators in combined fonA/ard-backward time J. Chem. Phys. 110 1343... 

Thompson K and Makri N 1999 Rigorous fonA/ard-backward semiclassical formulation of many-body dynamics Phys. Rev. E 59 4729... 

The paradigmatical binding reaction (equation (C2.l4.22)) is generally analysed as a second order forward reaction and a first order backward reaction, leading to the following rate law ... 

In hyperspherical coordinates, the wave function changes sign when <]) is increased by 2k. Thus, the cotTect phase beatment of the (]) coordinate can be obtained using a special technique [44 8] when the kinetic energy operators are evaluated The wave function/((])) is multiplied with exp(—i(j)/2), and after the forward EFT [69] the coefficients are multiplied with slightly different frequencies. Finally, after the backward FFT, the wave function is multiplied with exp(r[Pg.60]

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

Backward Analysis In this type of analysis, the discrete solution is regarded as an exact solution of a perturbed problem. In particular, backward analysis of symplectic discretizations of Hamiltonian systems (such as the popular Verlet scheme) has recently achieved a considerable amount of attention (see [17, 8, 3]). Such discretizations give rise to the following feature the discrete solution of a Hamiltonian system is exponentially close to the exact solution of a perturbed Hamiltonian system, in which, for consistency order p and stepsize r, the perturbed Hamiltonian has the form [11, 3]... 

In applications, one is often interested in approximating time averages over a time interval [0, T] via associated mean values of a , k = 1. ..Tfr. For T (or r) small enough, the above backward analysis may lead to much better error estimates than the worst case estimates of forward analysis. 

Summarizing, from a mathematical point of view, both forward and backward analysis lead to the insight that long term trajectory simulation should be avoided even with symplectic discretizations. Rather, in the spirit of multiple as opposed to single shooting (cf. Bulirsch [4, 18]), only short term trajectories should be used to obtain reliable information. 

E. Hairer. Backward analysis of numerical integrators and symplectic methods. Annals of Numerical Mathematics 1 (1994)... 

E. Hairer and Ch. Lubich. The life-span of backward error analysis for numerical integrators. Numer. Math. 76 (1997) 441-462... 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

Symplectic integration methods replace the t-flow (pt,H by the symplectic transformation which retains Hamiltonian features of They poses a backward error interpretation property which means that the computed solutions are solving exactly or, at worst, approximately a nearby Hamiltonian problem which means that the points computed by means of symplectic integration, lay either exactly or at worst, approximately on the true trajectories [5]. 

S. Reich. Backward error analysis for numerical integrators. SIAM J. Numer. Anal., to appear, 1999. 

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