Number of exponential functions

The method given in Section 2.2.2 for the calculation of concentrationtime laws turns out to be successful, if the eigenvalues are different and real. The number e,- gives the number of exponential functions, which are necessary to determine the concentration the compound A,- in dependence on time 

There is another way to determine this number of exponential functions. It uses simple rules. They are derived from the fact that differential equations which describe the time dependence of concentrations have to be always linear, their coefficients being constant. If these differential equations are not homogeneous, a formal interference function can be introduced. It is represented by a sum of exponential functions. Then, the solution of the differential equations is a linear combination of exponential functions. They consist of powers of the eigenvalues of the homogeneous system and of powers of the interference functions. 

In practice, first the differential equation or the total system of differential equations have to be set up. Next the number of eigenvalues of the homogeneous differential equations have to be determined. Finally the sum of exponential functions is given by this value and the number of interference functions. The principle of this calculation is explained in the following. It is assumed that the compound A,- reacts according to the mechanism 

This scheme has the following meaning reactant A,- is formed by just one or 

Backward reactions are allowed in principle. However, they are not allowed to form A ( or to start from A,-. Under these assumptions the differential equation (with respect to a,) is given by 

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A power function can be related to the sum of an infinitely large number of exponential functions ... 

In order to obtain a more realistic description of reorientational motion of intemuclear axes in real molecules in solution, many improvements of the tcf of equation Bl.13.11 have been proposed [6]. Some of these models are characterized in table Bl.13.1. The entry number of terms refers to the number of exponential functions in the relevant tcf or, correspondingly, the number of Lorentzian terms in the spectral density function. 

In this section the properties of linear systems relevant to kinetic analysis are discussed. Using this approach information can be extracted from the characteristics of diverse graphical representations. By looking for extrema and points of inflection in the concentration-time diagrams one can draw conclusions with respect to the mechanism - a prerequisite to determine the number of exponential functions of the overall rate law. 

In the equation above, the a (0 represent the interference functions ij = 1-1, i -1,...). They exhibit the dependence on time for those reactants which form A,. It is assumed that they are known. Therefore the result is the number of exponential functions... 

Under these conditions the number of exponential functions ei equals two. The concentrations depend on time according to... 

Two-dimensional semiclassical studies described in section 4 and applied to some concrete problems in section 6 show that, when no additional assumptions (such as moving along a certain predetermined path) are made, and when the fluctuations around the extremal path are taken into account, the two-dimensional instanton theory is as accurate as the one-dimensional one, and for the tunneling problem in most cases its answer is very close to the exact numerical solution. Once the main difficulty of going from one dimension to two is circumvented, there seems to be no serious difficulty in extending the algorithm to more dimensions that becomes necessary when the usual basis-set methods fail because of the exponentially increasing number of basis functions with the dimension. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

This equation ultimately expresses no more than the fact that the number of terminal groups increases as a function of the number of (former) functional groups of the core (core multiplicity) and that of the branching units (branching multiplicity) rises exponentially with the number of generations. 

Here cw- is the coefficient with which the /-th basis function Xt(r Q (usually a Slater-type function, or STF for short) enters the jU-th orbital of the a-th configuration, Q is an adjustable parameter (usually the STF s exponential parameter), and TV /the fixed but otherwise arbitrary number of basis functions. 

Bruker uses the command EM (exponential multiplication) to implement the exponential window function, so a typical processing sequence on the Bruker is EM followed by FT or simply EE (EF = EM + FT). Varian uses the general command wft (weighted Fourier transform) and allows you to set any of a number of weighting functions (lb for exponential multiplication, sb for sine bell, gf for Gaussian function, etc.). Executing wft applies the window function to the FID and then transforms it. 

Suppose a number of experiments are conducted with measurements taken of heat-transfer rates of various fluids in turbulent flow inside smooth tubes under different temperature conditions. Different-diameter tubes may be used to vary the range of the Reynolds number in addition to variations in the mass-flow rate. We wish to generalize the results of these experiments by arriving at one empirical equation which represents all the data. As described above, we may anticipate that the heat-transfer data will be dependent, on the Reynolds and Prandtl numbers. An exponential function for each of these parameters is perhaps the simplest type of relation to use, so we assume... 

Despite the interest to obtain AO integral algorithms over cartesian exponential orbitals or STO fimctions [43] in a computational universe dominated by GTO basis sets [2], this research was started as a piece of a latter project related to Quantum Molecular Similarity [44], with the concurrent aim to have the chance to study big sized molecules in a SCF framework, say, without the need to manipulate a huge number of AO functions. 

The rate constants were derived from multi-exponential fits to open and closed time probability density functions (25). The number of open and closed states represents the number of exponential components required to fit each of the probability density functions, e.g. fitting the open and closed time distributions for the control data (lO M L-glutamate alone) required 3 and 4 components respectively. Numbers in brackets are percentages of channel populations occupying given open or closed states. 

Description of ion-channel kinetics via admittance analysis provides a framework within which linear kinetic models can be compared to macroscopic data (from a population of channels) in a membrane. Analysis of conduction via driving-point-function determinations also provides proper data (from a true linear analysis) for comparison with the relaxation times obtained from microscopic data from one or a small number of channels in a membrane patch isolated by a micropipette (4). In Markov modeling, the open- and closed-time distributions are fitted to sums of exponential functions (15). 

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