Series Exponential

This is simply a special case of Taylors series when h is set to zero. Exponential Series... 

The temperature is then raised to 72 °C, and Taq polymerase catalyzes the addition of further nucleotides to the two primed DNA strands. When replication of each strand is finished, two copies of the original DNA now exist. Repeating the denature-anneal-synthesize cycle a second time yields four DNA copies, repeating a third time yields eight copies, and so on, in an exponential series. 

Let write other form (33) assigning the condition (p <(p 1. Then factorizing ln(l — (p) in exponential series we will obtain... 

Although satisfactory criteria for deciding whether data are better analyzed by distributions or multiexponential sums have yet to established, several methods for determining distributions have been developed. For pulse fluorometry, James and Ware(n) have introduced an exponential series method. Here, data are first analyzed as a sum of up to four exponential terms with variable lifetimes and preexponential weights. This analysis serves to establish estimates for the range of the preexponential and lifetime parameters used in the next step. Next, a probe function is developed with fixed lifetime values and equal preexponential factors. An iterative Marquardt(18) least-squares analysis is undertaken with the lifetimes remaining fixed and the preexponential constrained to remain positive. When the preexponential... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

The number e emerges from the exponential function, which is the sum of the exponential series, such that... 

In the expression for the reaction rate constant, expansion in an exponential series is most appropriately done near the theoretical temperature... 

It is obvious here that it is the rate of the reaction and of heat release at temperatures close to the temperature of the hot wall that is the essential factor. Expanding the rate close to this temperature (T ) into an exponential series using the method systematically developed by Frank-Kamenetskii, we find... 

To establish the relationship between a and the force constants of eq. (11), the Morse function is expanded as an exponential series, using the same zero point in energy ... 

Extension of the Debye-Hiickel Theory.—In the calculation of the electrical density in the vicinity of an ion (p. 82), it was assumed that ZiGp/kT was negligible in comparison with unity, so that all terms beyond the first in the exponential series could be neglected. According to calculations made by Muller (1927), the neglect of the additional terms is justifiable provided that... 

On the basis of this general theory (35, 38), the initial region of the isotherm is described by an exponential series with virial coefficients ... 

M. Dubinin (Academy of Sciences of the USSR, Moscow, USSR) The survey paper considers the fundamental problems of adsorption of vapors on zeolites within the framework of a general theory of molecular adsorption on nonporous and porous adsorbents. Among other things, it quotes examples of description of adsorption equilibria on zeolites over temperature ranges not exceeding 50° for initial and intermediate regions of isotherms by an exponential series with virial coefficients taking into account both adsorbent—adsorbate and adsorbate—adsorbate interactions. It is assumed that these equations can be used for practical calculations of adsorption equilibria. 

The solution to (6.11) with the conditions (a)-(c) of Eq. (2.5)] can be obtained from (6.6) by use of Duhamel s theorem (Bartels and Churchill, 1942). Several simplifications will first be made. Since in A(t), To > T, it follows from successive use of the binomial and exponential series expansion that... 

The term in brackets is clearly an exponential series (page 285), equivalent to... 

Thompson 2002). In the pyrolysates, the slope break in the -alkane series occurs at Q, indicating that nQ should be included in the light end exponential progression. The absence of n-alkane data in PVT analyses precludes the measurement of SF(C2- nQ). Thus, the best-developed light end exponential series in oils, SF(C3- C5), is compared here with that best developed in the pyrolysates, SF(C2-mC5). 

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