Change of time

Substituting this into the second equation of (6-185), and making some intermediate calculations (and change of time scale) (Reference 6, p. 457) one obtains ultimately the differential equation ... 

It can be easily shown that Eqs. (2.7) and (2.10) are identical. Indeed, the three-dimensional integration domain in (2.7) can be divided into six subdomains, each corresponding to a term of (2.10) upon a proper change of time variables ... 

Firstly, the time scales phenomena in which the molecular aspect of the solute-solvent interactions is the determinant aspect (a subject central to this book) span about 15 orders of magnitude, and such a sizeable change of time scale implies a change of methodology. Secondly, the variety of scientific fields in which the dynamical behaviour of liquids is of interest to give an example friction in hydrodynamics and in biological systems has to be treated in different ways. 

Nevertheless, let us rewrite this equation, with a change of time integration variables, in the totally equivalent form... 

The quantities a and have units of concentration, so if all concentrations (nutrients, organisms, and Michaelis-Menten constants) are measured in units of then S may be scaled out of system (2.1). Similarly, the units of m , and D are reciprocal time, so with an appropriate change of time scale D may also be scaled out of the system. Moreover, the conversion factors 7 and 7 can be incorporated into m, and f,. This is essentially the scaling that has been used in all of the previous models. With these changes of scale, the new system takes the form... 

The eigenfunctions are often chosen to depend on time, while the operators are chosen time independent. This choice is referred to as the Schrodinger representation. The change of time of a quantity is called evolution. The evolution of the wave function towards equilibrium is determined by the Schrodinger equation. 

Changes of time of the treatment and frequency in the considered limits (5-10 min. and 50-150 Hz, respectively) lead to less prononced effects (the confidential interval is 10 %). On the basis of the results gained, optimal compositions of latex mixtures and conditions of vibrowave stirring (ratio of latexes poly(butadiene-styrene) poly(butadieneacrylate) = 75 25 vol., duration of vibrowave stirring - 7,5 min., frequency-150 Hz) have been chosen. 

To measure the rate of change of any two variables, we fix upon one variable as the standard of reference. When x is the standard of reference for the rate of change of the variable y, we call dy/dx the. x-rate of y. In practical work, the rate of change of time, t, is the most common standard of reference. If desired we can interpret (4) or (5) to mean... 

Because the time or frequency scales over which the compliance changes are very large, it is usual to use logarithmic scales for t or co. When this is done it is necessary to shift the curves for different temperatures by a constant amount log a or — log aj along the t or co axes, respectively, in order to get superposition with the curve for T. The quantity aj is therefore called the shift factor. Figure 7.14 shows data for a set of measurements before and after the shifts have been applied. The idea that the same effect can be produced on the compliances or moduli either by a change of temperature or by a change of time-scale is called time-temperature equivalence. 

Theta measures the change of the option value to the change of time to maturity. 

In other words, for a reversible vector field, the coordinate transformation z Rz is equivalent to the change of time 1 -t. 

As can be seen from Figure 4 a wide scatter of convective heat flux gives relatively little change of time-to-loss-of-strength. As for example, time-to-loss-of-strength of 4 minutes may be caused by convective heat flux between 40 and 95 kW/m. ... 

This change of time variables, first used in polymerization analysis by Dostal and Mark [26], is known as the eigenzeit transformation. It allows for a much easier solution of equation (3.14) since the monomer concentration is a continuously changing quantity in a batch reactor. If the temperature control is poor, the reaction rate constant would also change. 

Change of time scale and change of time unit can be defined analogously to the previous case. 

Every change of time scale (including every change of time unit) preserves the character of polynomial differential equations in the sense above. 

If (IR ) is an invariant set of a differential equation then the change of scale and the change of time scale preserves this property. 

Despite the change of time interval amplitude is big, the change of dynamic stability safety coefficient is very small. Considering the relation of strength reduction ratio with displacement (Fig. 15), the result is as shown in Table 4. This shows that the seismic slope isn t sensitive to the time interval. The relationship curves on strength reduction ratio with... 

Because concentration is proportional to time (cone. c t), setting the rate of change in concentration equal to the rate of change of time requires a proportionality constant, k (for time changing from t = 0 to t = some measured time). In this case, cone. = k t. The proportionality constant (k) is defined as the rate constant. Note that the small k is used to represent the rate constant, whereas a capital K is used to indicate an equilibrium constant. 

To get a kinetic expression, one makes use of the first-order kinetics of Fig. 2.8 and writes Eq. (3), with N (T) being the equilibrium number of holes, and N its instantaneous value. The rate constant, which was called k in Fig. 2.8, is replaced by 1/r, the relaxation time. By introduction of the heating rate q - (dr/dt), the change of time dependence to temperature dependence at constant heating rate is accomplished. Equation (3) describes the nottiso-thermal kinetics of the glass transition. A full solution of the equation is... 

The virtual instrument system based on LabviewS.O for ion analyzer, which has advantages of easy operation and friendly interface, can determine pX, realize real-time displaying and drawing a curve of pX or mV with the change of time t,... 

After the change of time Rdt —> dt the system reduces to the form... 

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