TABULATE procedure

The Reaction-Diffusion Manifolds (REDIM) approach represent an extension to the formulation of the standard ILDM. Where the ILDM is in fact a relaxation of a set of ordinary differential equations (ODE s) describing a homogenous system, the REDIM formulation generalizes for a set of partially differential equation (PDE s) where also the coupling between the reaction and diffusion processes are accounted for. Bykov and Maas (Bykov Maas, 2007) have performed the full derivation of this generalized system in the framework of ILDM and an optimized tabulation procedure of generalized coordinates. They present the method in... 

I.P.P.D and its relatives have become standard procedures for the characterization of the structure of both clean surfaces and those having an adsorbed layer. Somoijai and co-workers have tabulated thousands of LEED structures [75], for example. If an adsorbate is present, the substrate surface structure may be altered, or reconstructed, as illustrated in Fig. VIII-9 for the case of H atoms on a Ni(llO) surface. Beginning with the (experimentally) hypothetical case of (100) Ar surfaces. Burton and Jura [76] estimated theoretically the free energy for a surface transition from a (1 x 1) to a C(2x 1) structure as given by... 

It is instructive for the student to construct a rough melting point diagram (compare Section 1,13 and Fig. 1,12, 1) for mixtures of cinnamic acid and urea. Weigh out 1 00 g. each of the two finely powdered components, and divide each into ten approximately equal portions on a sheet of clean, smooth paper. Mix 4 portions of cinnamic acid (A) with 1 portion of urea B) intimately with the aid of a spatula on a glass slide, and determine the melting point (the temperature at which the mixture just becomes completely fluid is noted). Repeat the procedure for 3 parts of A and 2 parts oiB 2 parts of A and 3 parts of B and 1 part of A and 4 parts of B. Tabulate your results as follows —... 

Interplanar Spacings. Diffractometer alignment procedures require the use of a well-prepared polycrystalline specimen. Two standard samples found to be suitable are silicon and a-quartz (including Novaculite). The 26 values of several of the most intense reflections for these materials are listed in Table 7.6 (Tables of Interplanar Spacings d vs. Diffraction Angle 26 for Selected Targets, Picker Nuclear, White Plains, N.Y., 1966). To convert to d for Ka or to d for Ka2, multiply the tabulated d value (Table 7.6) for Ka by the factor given below ... 

The reactivity ratios of a copolymerization system are the fundamental parameters in terms of which the system is described. Since the copolymer composition equation relates the compositions of the product and the feedstock, it is clear that values of r can be evaluated from experimental data in which the corresponding compositions are measured. We shall consider this evaluation procedure in Sec. 7.7, where it will be found that this approach is not as free of ambiguity as might be desired. For now we shall simply assume that we know the desired r values for a system in fact, extensive tabulations of such values exist. An especially convenient source of this information is the Polymer Handbook (Ref. 4). Table 7.1 lists some typical r values at 60°C. 

The tabulation of plant nutrient costs, by product, ia Table 16 shows the principal reasoa for the popularity of anhydrous ammonia as a fertilizer ia the United States. The fob price per ton of nitrogen in the form of ammonia is less than half that for any other nitrogen product. Also, ammonia s relatively high nitrogen content of 82.2% favors low transportation costs, in spite of the need for specialized handling equipment and procedures. 

A well-known example of blocking arises in the comparison of wear for different types of automobile tires. Tire wear may vary from one automobile to the next, irrespective of the tire type, because of differences among automobiles, variabiUty among drivers, and so forth. Assume, for example, that for the comparison of four tire types (A, B, C, and D), four automobiles (1, 2, 3, and 4) are available. A poor procedure would be to use the same type of tire on each of the four wheels of an automobile and vary the tire type among automobiles, as in the following tabulation ... 

As in previous volumes of Organic Syntheses unchecked procedures are tabulated at the end of this volume. Of the preparations received between May 16, 1976, and June 30, 1977, only those that have been accepted by the Board of Editors for checking are listed. These unchecked procedures are available from the Secretary s office for a nominal fee. 

If the thermodynamic data for a compound of interest have not been determined and abulated, it may be possible to estimate AHf or AGj from tabulated data pertaining to dividual structural units. Procedures have been developed for estimating thermodynamic characteristics of hydrocarbons and derivatives by summing the contributions of the constituent groups. The group increments are derived from experimental thermochemical iata and therefore depend on the existence of reliable data for the class of compounds of merest. 

Three major themes have been emphasized in this chapter. The first is that an effective data collection system is one of the most powerful tools available to minimize human error. Second, data collection systems must adequately address underlying causes. Merely tabulating accidents in terms of their surface similarities, or using inadequate causal descriptions such as "process worker failed to follow procedures" is not sufficient to develop effective remedial strategies. Finally, a successful data collection and incident investigation system requires an enlightened, systems oriented view of human error to be held by management, and participation and commitment from the workforce. 

The C1E color coordinates are obtained by a similar procedure, repeating the convolution for each of the tabulated color matching functions that describe the response of the ideal observer s blue, green, and red cones (b, rA). 

Thermodynamic Functions of the Gases. To apply Eqs. (1-10), the free energies of formation, Ag , for all gaseous species as a function of temperature are required. Tabulated data were fit by a least-squares procedure to derive an analytical equation for AG° of each vapor species. For the plutonium oxide vapor species, the data calculated from spectroscopic data (3 ) were used for 0(g) and 02(g), the JANAF data (.5) were used and for Pu(g), data from the compilation of Oetting et al. (6) were used. The coefficients of the equations for AG° of the gaseous species are included in Table I. 

Repeating this procedure at the various temperature points selected, the heat-exchanger area may then be obtained as the area under a plot of Y.q vs. 1/6 A7 , or as A — Hq/UAT according to the following tabulation ... 

Procedure The two algorithms FF050 and FF025 permit tabulated F-val-ues for the confidence levels p = 0.05 and 0.025 to be approximated with high accuracy. The strong curvature of the F - f f j2,p) surface militates against simple and flexible functions like polynomials only two confidence levels are available. 

In this table you can see that clinical success is summarized for each treatment by visit. The key here is by visit. If the data set to be summarized is simply sorted by visit, then PROC FREQ, PROC TABULATE, or some other procedure can be executed with a BY VISIT statement. If the data set were denormalized, then the task of producing the required summary would be more difficult. 

Note that for generating tables there are single SAS procedures such as PROC TABULATE and PROC REPORT that can perform all of the preceding four steps. We will examine PROC TABULATE and PROC REPORT as possible reporting methods after you see what a typical clinical trial table looks like. 

PROC MEANS, PROC SUMMARY, and PROC TABULATE are other SAS procedures that you can use to get descriptive statistics and place them into output data sets. However, those procedures do not offer any descriptive statistical variables that you cannot get from PROC FREQ or PROC UNIVARIATE. 

Animal infectivity methods Some viruses do not cause recognizable effects in cell cultures but cause death in the whole animal. In such cases, quantification can only be done by some sort of titration in infected animals. The general procedure is to carry out a serial dilution of the unknown sample, generally at ten-fold dilutions, and samples of each dilution are injected into numbers of sensitive animals. After a suitable incubation period, the fraction of dead and live animals at each dilution is tabulated and an end point dilution is calculated. This is the dilution at which, for example, half of the injected animals die. Although such serial dilution methods are much more cumbersome and much less accurate than cell culture methods, they may be essential for the study of certain types of viruses. 

The relationship between flow rate, pressure drop, and pipe diameter for water flowing at 60°F in Schedule 40 horizontal pipe is tabulated in Appendix G over a range of pipe velocities that cover the most likely conditions. For this special case, no iteration or other calculation procedures are required for any of the unknown driving force, unknown flow rate, or unknown diameter problems (although interpolation in the table is usually necessary). Note that the friction loss is tabulated in this table as pressure drop (in psi) per 100 ft of pipe, which is equivalent to 100pef/144L in Bernoulli s equation, where p is in lbm/ft3, ef is in ft lbf/lbm, and L is in ft. 

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