Exponential multiplier

Functional fonns used for the repulsion include the simple exponential multiplied by a linear combmation of powers (possibly non-mteger) of r, a generalized exponential function exp(-h(r)), where b r) is typically a polynomial in r, and a combination of these two ideas. 

Accuracy - gold + 5%, silver = 10% Exponential multiplier Second significant figure... 

If you remember from Chap. 6, repeated roots of the characteristic equation yielded time functions that contained an exponential multiplied by time. 

D. EXPONENTIAL MULTIPLIED BY TIME. In the Laplace domain we found that repeated roots l/(s -I- a) occur when we have the exponential multiplied by time. We can guess that similar repeated roots should occur in the z domain. Let us consider a very general function ... 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

Inserting exponential multiplier exp(—wcp) into Eq. (270a) before cos bip, we obtain using transformations analogous to those which yield Eq. (273) ... 

The main reason of appearance of exponential multiplier eq in Eqs. (2.7) and (2.8) for the rate of interfacial electron transfer from semiconductor particles to dissolved oxygen molecules or protons seems to the electric charge of these electrons. This causes considerable changes in the potential of the nanoparticles double layer, which, in turn, affects directly the rate of electron transfer [12-15]. 

Prefix Abbreviation Exponential multiplier Meaning Example using length... 

Here T is the temperature of acoustic phonons (thermostat), T is the temperature of optical phonons (284), the anharmonicity constant k is much less than 1, and flq is the frequency of acoustic phonons it is possible to assume ilq = (10 2 to 10 1)flDebye. The coupling between the optical and acoustic phonons is strongest near f>Debye 1 and because of this, for sufficiently large anharmonicity, k > 10 2 even at T = T", the last exponential multiplier can be approximated by the exponent below, with the dispersion being neglected ... 

Thus, the relative error of the result is equal to the relative error of numbers to be exponentiated, multiplied by the exponent. 

A variety of pulse sequence programs were employed. A conventional cross polarization, single contact program was used to obtain spectra of Intact sediments. Contact times were varied from 200-3000 psec with a three sec recycle time. For dephaslng delay experiments, a 50 psec delay was Inserted prior to data collection. This delay consisted of two 25 ysec Intervals separated by a 10 psec, 180° refocusing pulse. Data was collected In 2K of memory, exponentially multiplied with 50 Hz of line broadening, and expanded to 8 K prior to Fourier transformation. All spectra are the result of 5000 accumulations. 

Equation [A2] contains the product of a complex exponential multiplied by a complex error function. Thus, the product of Eq. [A7] and [A 16] must be separated into its real and imaginary parts as well as the product of Eq. [A8] and [A 16]. The product of Eq. [A7] and [A 16] produces... 

In contrast, the liquid interface can be considered homogeneous which is a precondition for a non-localised adsorption. Any free space at the liquid interface is available for ion adsorption. At the transition from a localised to a non-localised adsorption the exponential term of the Stem-Langmuir equation can be preserved and the pre-exponential multiplier must be changed. This was done by Martynov (1979). The basis of this approach is the notation of a homogeneous potential well along the liquid interface as a whole. 

Exponential / 7.2.5 Exponential Multiplied by Time / 7.2.6 Impulse (Dirac Delta Function 8 d) Inversion of Laplace Transforms Transfer Functions... 

The appropriate SVD-derived spectral and temporal eigenvectors were selected and the temporal vectors were modeled. Ideally, the temporal vectors are the kinetic traces of individual components, each one being associated with a spectrum of a pure component Le., the spectral vector). Once the temporal vectors had been modeled the pure component spectra were reconstructed as a function of the pre-exponential multiplier obtained from the analysis, SVD determined spectral eigenvectors, and the corresponding eigenvalues. After the spectra of the component species were determined, the extinction profile was calculated and used along with the calculated decay times to construct a linear combination of the pure component species contributions to the observed... 

Figure 1.1. 600 MHz NMR spectrum of potassium hydrogen phthalate (KHP) and maleic acid dissolved in D2O. The KHP is a primary standard by which the maleic acid concentration can be quantitated. The data represents 8 FTDs coadded into 28,800 points (zero-filled to 32K points) across a spectral width of 7200.1Hz. An exponential multiplier equivalent to 0.5Hz...

Fig. 6. Effects of methotrexate + uridine (arrows) on cell multiplication, DNA, RNA, and protein synthesis in exponentially multiplying cells. Each unit on the ordinates corresponds to a doubling of the parameter considered. The triangles represent an untreated control population, the circles a population incubated with 0.05 mM methotrexate (M), plus 20 mM uridine (U) from the time indicated by the arrows. (From Villadsen and Zeuthen. 1970. Exp. Cell / es., 61 302-310.)...

Synchronization of DNA Replication in Exponentially Multiplying Cell Populations... 

In bacterial and mammalian cell systems in exponential growth new DNA synthesis occurs in synchrony when the blockage of DNA synthesis by selective thymine starvation is released after a period. The same is true in Tetrahymena cultures. This is seen from Figure 7, in which 2 mM thymidine was added after an exponentially multiplying population had been treated with methotrexate + uridine for 3 hours (3 hours is nearly equal to one generation time). Synthesis of DNA began immediately after thymidine was added, and cell division occurred 110 minutes later. 

Control experiments showed that synchronous divisions, as reflected in changes of the division index, can be induced normally with heat shocks in the presence of either uridine, thymidine, or methotrexate. With uridine or with thymidine added before the shocks the pattern of synchronous divisions is similar to what is seen in the control, but with methotrexate the first division is slightly delayed and the second division much delayed, resembling what was observed with exponentially multiplying cells. On the other hand, when the inhibitor complex of DNA synthesis, methotrexate + uridine, is added just before the seven heat shocks, all later division activity is practially eliminated. When it is added during the period of heat shocks, more and more cells will take part in division 1 the later the agent is supphed. When the inhibitor is added in the interval between shocks 6 and 7 there is extensive participation in the first division, but no cells ever divide a second time. When the inhibitor is added at EH7 there is nearly full participation in division 1, and a few calls can take part also in the second division. From such... 

After transformation and subsequent integration of the exponential multiplier in Eq. (1.29), the average adhesive force can be obtained. The average adhesive force, in contrast to the median force, represents the average magnitude of adhesion of all particles, from F in to F ax The average adhesive force, which takes into account all changes in particle distribution with respect to adhesive force, includes not only the median force F but also the standard deviation a, and hence provides the fullest characterization of particle adhesion. 

Hence we see that / cr should decrease rather rapidly with increasing 0 due to the negative exponential multiplier in Eqn. 368. 

Harmonic oscillator wavefunctions 17 (7) consist of a Gaussian exponential multiplied by one of the tabulated Hermite polynomials. 

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