Numerical Expressions

The most common type is a numerical expression involving the algebraic operators as follows  

Parentheses can be used (and should be used liberally) to reflect the exact hierarchy of operations. The operational hierarchy is the same as for standard algebraic expressions as follows  

When operations are at the same hierarchical level, they are performed from left to right. 

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The legal basis for the sale of fertilizers throughout the world is laboratory evaluation of content as available nitrogen, phosphorus, and potassium. By convention, numerical expression of the available nutrient content of a fertilizer is by three successive numbers that represent the percent available of N, P20, and K O, respectively. Thus, for example, a 20—10—5 fertilizer contains available nitrogen in the amount of 20% by weight of N, available phosphoms in amount equivalent to 10% of P2O5, and available potassium in amount equivalent to 5% K O. The numerical expression of these three numbers is commonly referred to as the analysis or grade of the fertilizer. Accepted procedures for laboratory analysis are fixed by laws that vary somewhat from country to country. 

Numerical measures of polarizability analogous to electronegativity have been defined. Hardness t] is numerically expressed as... 

Stress analysis involves using the descriptions of the product s geometry, the applied loads and displacements, and the material s properties to obtain closed-form or numerical expressions for internal stresses as a function of the stress s position within the product and perhaps as a function of time as well. Tlie term engineering formulas refers primarily to those equations reviewed previously and given in engineering handbooks by which the stress analysis can be accomplished. 

It is not essential, however, that the unit of heat should be defined in terms of the rise of temperature produced when heat is absorbed by a standard body, say unit mass of water. Any effect of heat absorption which is capable of measurement and numerical expression might be used, and the method of measurement would in all cases be consistent with the axiom that if two identical systems are acted upon by heat in the same way so as to produce two other identical systems, the quantities of heat supplied to the systems are equal. Lavoisier and Laplace (1780-84) took as unit that quantity of heat which must be absorbed by unit mass of ice in order to convert it completely into water. This unit is of course different from the one we adopted, but if a quantity of heat A has been found to raise from lo ° to 16 ° twice as much water as another quantity of heat B, then A will also melt twice as much ice as B. 

Table II. Numerical Expressions Used to Represent the 957. Confidence Limits of the Viscoelastic Quantities...

Note that for the calculations of precision and standard deviation (equations 38-1 through 38-4), the numerator expression is given as 2(n — 1). This expression is used due to the 2 times error contribution from independent errors found in each independent set (i.e., X and Y) of results. 

A quantitative study of the cholesteric induction and of the chiral transfer from dopant to phase requires the definition of the helical twisting power 3. This quantity expresses the ability of a chiral dopant to twist a nematic phase and can be numerically expressed in Eq. (2) where p is the cholesteric pitch, c the dopant molar fraction, and r its enantiomeric excess its sign is taken to be positive or negative for right-handed (P), or left-handed (M) cholesterics, respectively. This relation holds for molar fractions <0.01-0.05 ... 

The relations between the intrinsic pressure and other physical constants developed in the foregoing paragraphs have been found from theoretical considerations based on Laplace s theory, that is, on the assumption of cohesive forces acting over very small distances. They are of interest to us inasmuch as there is a necessary connection between intrinsic pressure and surface tension. While no numerical expression has so far been found for this, it is obvious that high intrinsic pressures must be accompanied by high surface tensions, since the surface tension is a manifestation of the same cohesive force as causes intrinsic pressure. (See, however, equation 3, p. 27, for an empirical relation between the two.)... 

All methods of numerically expressing the concentration of a solution contain a term in the denominator referring to the entire solution except... 

Resolution, on the other hand, is a more technical term. It refers to the distance between adjacent bands relative to their bandwidths and acknowledges the fact that proteins are distributed in Gaussian profiles with overlapping distributions. The numerical expression for resolution is obtained by dividing the distance between the centers of adjacent bands by some measure of their average bandwidths. It expresses the distance between band centers in units of bandwidth and gives a measure of the overlap between two adjacent bands. For preparative applications, when maximal purity is desired, two proteins to be isolated should be separated by at least a bandwidth. In many applications it is sufficient to be able to simply discern that two bands are distinct. In this case bands can be less than a bandwidth apart. 

Substitution of values for the physical constants yields the numerical expression... 

An appropriate formalism for Mark-Houwink-Sakurada (M-H-S) equations for copolymers and higher multispecies polymers has been developed, with specific equations for copolymers and terpolymers created by addition across single double bonds in the respective monomers. These relate intrinsic viscosity to both polymer MW and composition. Experimentally determined intrinsic viscosities were obtained for poly(styrene-acrylonitrile) in three solvents, DMF, THF, and MEK, and for poly(styrene-maleic anhydride-methyl methacrylate) in MEK as a function of MW and composition, where SEC/LALLS was used for MW characterization. Results demonstrate both the validity of the generalized equations for these systems and the limitations of the specific (numerical) expressions in particular solvents. 

The rate constant fc°rai°H is determined from a plot of (Rtooh/RoH)2 against CH [TOOH] at several different [TOOH] concentrations. In a similar way, fctran H can be obtained by the addition of COOH to Tetralin. The numerical expression at 30 °C. is... 

Filled circle Ti4+, small circle O2-, large circle tsa2+. The arrows show dipole moment. The numericals express the Ti-O bond length in nm. 

There is another characteristic point in the voltammogram known as the isopoint [25, 26]. At the isopoint, the current is zero regardless of the scan rate (see black dot in Fig. 5.4). In reference [26], the following numerical expression was reported to determine the difference between the peak potential of the forward scan, FpeXf, ar,d the potential of the isopoint, F 1 6, in terms of the switching potential Ffinai ... 

Measurement endpoint Numerical expression of a specific assessment endpoint or effect criterion (e.g., IC50, NOEC, LOEC). Volume 1(3,10),... 

What is the importance of the null and the alternative hypotheses They enable us to link the baseline and alternative condition statements to statistical testing and to numerically expressed probabilities. The application of a statistical test to the sample data during data quality assessment will enable us to decide with a chosen level of confidence whether the true mean concentration is above or below the action level. If a statistical test indicates that the null hypothesis is not overwhelmingly supported by the sample data with the chosen level of confidence, we will reject it and accept the alternative hypothesis as a true one. In this manner we will make a choice between the baseline and the alternative condition. 

The value is the numerically expressed magnitude of a quantity either in a material or indicated by a calibrated instrument (see Fig. 2). An uncertainty must be associated with every value. Its important estimation is discussed in detail in the section The uncertainty below (see also [6, 8]). 

Numerical expression of eqs. (12b) and (13) used in the literature for various ions in water solutions (kobs expressed... 

At last, in his model, the ACF playing the role of [GD iv(Vj] in Eq. (309), is extracted from the two other models playing the role of [G rmi(f)] and [O0(t)IP]T. They give a numerical expression, which after Fourier transform, is assumed to give the peeled-off spectral density, that is, the SD of the H-bonded centrosymmetric cyclic dimer that would be observed if the Fermi resonances were missing. 

The quality of enantioselective reactions can be numerically expressed using the enantiomer ratio or the enantiomeric excess (ee). The former is equal to the ratio of the yield of enantiomers normalized to 100. The latter is the difference of this ratio, expressed as a per-... 

Here, T is temperature and C, s are concentrations of reactants, intermediates and products. Very occasionally, in the entire field of solid—gas reactions, intermediates are formed in such a way that reaction rate is not proportional to surface area and eqn. (1) is violated. This has been observed in halide—halogen reactions (Sect. 4) but not in reductions. The specific rate, r, may be controlled by chemical reaction at the reaction interface, or by diffusion of reactants and products through either the gas phase or the solid reacted zone. Therefore, numerous expressions for r are... 

The quality of enantioselective reactions is numerically expressed as the so-called enantiomeric excess (ee). It is equal to the yield of the major enantiomer minus the yield of the minor enantiomer in the product whose total yield is normalized to 100%. For example, in the Sharpless epoxidation of allyl alcohol (see Figure 3.7). S - and R-glycidol are formed in a ratio of 19 1. For a total glycidol yield standardized to 100%, the S-glycidol fraction (95% yield) thus exceeds the /f-glycidol fraction (5% yield) by 90%. Consequently, S-glycidol is produced with an ee of 90%. 

Density can be expressed in terms of weight per volume (e.g. pounds per cubic foot or grams per cubic centimeter). In geology, it is expressed in terms of specific gravity, which is a numerical expression of the ratio between the weight of a substance and the weight of an equal volume of water ... 

Specific gravity a numerical expression of the ratio between the weight of a substance and the weight of an equal volume of water a measure of a substance s density. 

To measure Divisions Symbol Numerical expression Parts of basic unit English equivalent... 

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