Pseudo-time

Table 2.6-2 (upper) shows that a gamma prior (equation 2,6-11) updated with exponential data produces a gamma posterior (equation 2.6-12) by adding rto t andM to Becau.se the prior is derived from other than test data, ris called pseudo-time and (ppseudo-failure. The mean, E(A) and standard deviation, a, of the prior and [losierior are given by Table 2.6-2 (lower). 

Before reviewing the results of pseudo-time-dependent models of homogeneous cloud interiors in detail, it is worthwhile to mention five successful predictions/ex-... 

Generalization and Numerical Implementation of Pseudo-Time Schroedinger Equation for Quantum Scattering Calculations. 

The full development of these mechanism analysis tools has not been achieved at this time, but some of the basic features have been set fbrth[18]. The key concept is to introduce further modulation into the optimal control field e(f) to reveal the dominant quantum pathways. This task calls for special care, as the control field already is temporally modulated in a delicate fashion to manipulate the quantum system. A way around this difficulty is feasible by introducing a new pseudo time-like variable s such that e(r) — In practice, s is a... 

The process response is recorded such that a time constant Tc) and deadtime Td) may be determined from the data. For non-selfregulating processes, a pseudo time constant may be determined from the total time it takes the process to move an amount equivalent to the controller output minus the deadtime before the process begins to respond. 

Fig. 27.—Rate curves for the periodate oxidation of the Schardinger dextrins V, amylodextrin , 7 A, 0 0,a. The pseudo-time B has been placed on a logarithmic scale in order to facilitate kinetic analysis and to compress a rather wide range of real times ...

For nonlinear elliptic partial differential equations, successive relaxation or finite difference approximations can be used in both the coordinates.[7] [12] [13] (Constantinides Mostoufi, 1999 Davis, 1984, Finlayson, 1980) As illustrated by Schiesser (1991),[2] a method of lines was used for 2D and 3D steady state problems by adding a pseudo time derivative, applying finite differences in all the... 

Leung, C.M. Herbst, E. Huebner, W.F. Synthesis of complex molecules in dense interstellar clouds via gas-phase chemistry — A pseudo time-dependent calculation, Astrophys. J. Suppl. 1984, 56, 231-256. 

To obtain the solution at the new time step in the implicit method, several outer iterations are performed. If the time step is small, only a few outer iterations per time step are necessary. For steady problems, the time step may be infinite and the under-relaxation parameter acts like a pseudo-time step. 

Introducing a difference in pseudo-time 0 by choice but constant and symbolised by A one finds... 

Since absorbances can only be measured in a certain dynamic range, the dilute solutions used in fluorescence measurements cannot be taken. Therefore the assumption is necessary that the photochemical reaction is governed by the same mechanism for 10 and 10 M solutions. Furthermore the higher concentration forbids an approximation according to eq. (5.111). However, in Section 3.3.11 a principle was introduced substituting the normal time scale by a transformation into a pseudo time (f) In consequence the three equations of eq. (5.147) have to be substituted by... 

S. Zhao. Pseudo-time-coupled nonlinear models for biomolecular surface representation and solvation analysis. Int J. Numerical Methods Biomed. Eng., 27 1964-1981, 2011. 

S. Zhao. Operator splitting adi schemes for pseudo-time coupled nonlinear solvation simulations. J. Comput Phys., 257 1000-1021, 2014,... 

The summation is performed over the whole solution domain. The desired mass of the reference phase is calculated as the initial mass of the reference phase. In Eq. 10a, t is a pseudo time. The term sign(lref) refers to the sign of for the reference phase. The droplet is chosen as the reference phase in this entry. The steady-state values of ij/ are used as the initial condition for Eq. 10a. 

Let us consider a material undergoing stress relaxation. The shift factor aj (T) describes the change of material time-scale with temperature. When temperature rises, the material time-scale shortens so that the relaxation proceeds faster, as if the material has an internal time clock —one unit of material time is equivalent to (7) units of observe time t. That is, the relaxation process of the material is calculated in a pseudo-time tlor (7). Consider a material undergoing stress relaxation. The relaxation speed is determined by an internal time-scale (or clock) within the material. As temperature rises, so does the amount of molecular motion occurring in one unit of an observation time, and the material s time scale shortens so that the relaxation proceeds faster. The shift factor aj gives a shift of material time-scale with temperature. 

Morland and Lee (1960) showed how to incorporate tune-temperature shifting into linear viscoelastic boundary-value problems. For the purpose a pseudo-time i(t) is introduced such that the amount of time that passes during an interval d< is given by dt/flj-. Then we have... 

We associate a (pseudo-) time variable t = v/N with the label v of successively generated configurations, where is the total number of... 

Smaller molecules tend in general to show less of a temporal effect with abundwces at early time being closer to steady-state abundances. Another key result of pseudo-time-dependent models is that large abundances of neu atomic carbon are present at early time. Pseudo-time-dependent models represent an improvement over steady-state models because the chemist can be investigated at eras before steady state is reached. Early time abundances are particularly interesting because it is important to realize that large abundances of complex molecules can be produced fairly quickly. 

All of these types of chemical models - steady-state, pseudo-time-dependent, and fully time-dependent - are themselves dependent on rate coefficients which are often highly uncertain and which need to be determined b experimental and/or theoretical techniques before the models can be put on a sound basis. Tlie approach in this review will be to... 

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