Form, closed

The decoupled set of equations in system (20) can be solved for all the Qi and associated velocities V<.i by closed-form formulas that depend on the eigenvalues [71]. The harmonic position and velocity vectors at time nAt can then obtained from the expressions ... 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

What furnace engineers most need is a closed-form solution of the problem, theoretically sound in structure and therefore containing a minimum number of parameters and no empirical constants and, preferably, physically visuaHzable. They can then (1) correlate data on existing furnaces, (2) develop a performance equation for standard design, or (3) estimate performance of a new furnace type on which no data are available. 

The variables are separable, but an integration in closed form is not possible because of the odd exponent. Numerical integration followed by substitution into (4) will provide both A and B as functions of t. The plots, however, are of solutions of the original differential equations with ODE. 

FreundlicVi-Type Relations A binary Freundlich isotherm, obtained from the ideal adsorbed solution theoiy in loading-explicit closed form [Crittenden et al.. Environ. Sci. TechnoL, 19, 1037 (1985)], is... 

Typically, the random values, x, from a particular distribution are generated by inverting the closed form CDF for the distribution F x) representing the random variable, where ... 

Many distributions can be represented in closed form except for the Normal and Lognormal types. The CDF for these distributions can only be determined numerically. For example, the 3-parameter Weibull distribution s CDF is in closed form, where ... 

The benefit using the approach over other methods is that it can be used to solve the general reliability equation for combination of distribution, including when there are multiple load applications. This is only possible when the loading stress is described in closed form. 

Flanged Rectangular Opening TTie flow field for an infinitely flanged rectangular opening was first solved in closed form by Tyaglo and Shepelev. 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

The presence of D g 26 governing differential equation and the boundary conditions renders a closed-form solution impossible. That is, in analogy to both bending and buckling of a symmetric angle-ply (or anisotropic) plate, the variation in lateral displacement, 5vy, cannot be separated into a function of x alone times a function of y alone. Again, however, the Rayleigh-Ritz approach is quite useful. The expression... 

Eqs. (77 and (78) are transcendental and we cannot obtain the solutions in a closed form. Nevertheless, they are convenient for obtaining some exact results. 

There is no general explicit mathematical treatment of complicated rate equations. In Section 3.1 we describe kinetic schemes that lead to closed-form integrated rate equations of practical utility. Section 3.2 treats many further approaches, both experimental and mathematical, to these complicated systems. The chapter concludes with comments on the development of a kinetic scheme for a complex reaction. 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

Although a closed-form solution can thus be obtained by this method for any system of first-order equations, the result is often too cumbersome to lead to estimates of the rate constants from concentration-time data. However, the reverse calculation is always possible that is, with numerical values of the rate constants, the concentration—time curve can be calculated. This provides the basis for a curve-... 

The salts of some enamines crystallize as hydrates. In such cases it is possible that they are derived from either the tautomeric carbinolamine or the amino ketone forms. Amino ketone salts (93) ( = 5, 11) can serve as examples. The proton resonance spectra of 93 show that these salts exist in the open-chain forms in trifluoroacetic acid solution, rather than in the ring-closed forms (94, n = 5, 11). The spectrum of the 6-methylamino-l-phenylhexanone cation shows a multiplet at about 2.15 ppm for phenyl, a triplet for the N-methyl centered at 7.0 ppm and overlapped by signals for the methylene protons at about 8.2 ppm. The spectrum of 93 ( = 11) was similar. These assignments were confirmed by determination of the spectrum in deuterium oxide. Here the N-methyl group of 93 showed a sharp singlet at about 7.4 ppm since the splitting in —NDjMe was much reduced from that of the undeuterated compound. 

In the infinite sum each successive term is smaller than the previous by a constant factor ( -hujVT which is <1), and can therefore be expressed in a closed form. Only the vibrational frequency is needed for calculating the vibrational partition function for a harmonic oscillator, i.e. only the force constant and the atomic masses are required. 

Another transformation of this type is a conversion of the ring-closed form 393 to the derivative of aroylhydrazone of acetylacetone 394 (Scheme 145) (81KK1165 92KK1184). 

Ring-opening is not a mandatory direction of the complexation reaction. Thus, the ring-closed form of the benzothiazoline 395 ligand was found to be preserved in its complex 396 with BuSnCla, whose structure was determined with X-rays (Scheme 146) [93JOM(454)67]. 

The previous discussion has shown us how to calculate the total number of possible cyclic states. We also know, from Lemma 2, that all cycle lengths must divide the maximal cycle length Hiv obtain the exact number of distinct cycles and their lengths takes a little bit more work. If flw prime, we know that the only possible cyclic lengths are 1 and It can then be shown that only the null configuration is a fixed point unless N is some multiple of 3, it which case there are exactly four distinct cycles of length one. If Hat i ot prime, there can exist as many cycles as there are divisors of Although there is no currently known closed form... 

Each 2" -order LST class is a strict refinement of some mean field class. There are no simple closed forms for either the total number of 2" -order class or the the number of rules within a given 2 -order class. 

As alluded to earlier, there are no closed form expressions either for the total number of clas.ses or rules within a given class. It is generally true, however, that for a given radius-r rule, classes both increase in number and decrease in size as the order N of the LST increases for fixed order, on the other hand, the number and size of cltLSses increases with radius. 

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