Arithmetic procedures

Routine spectrometric analyses currently run in industry may involve mixtures of up to 20 or 30 components. The solution of systems of 20 or 30 simultaneous equations by hand calculations is so tedious an operation, and so subject to human error, that it is impractical to accomplish it on a routine basis. The necessary calculations are, however, a well-defined arithmetic procedure easily adapted to digital calculators. Analog computers can also be used in solving simultaneous equations but are subject to accuracy limitations. 

Ohlsson, S., Rees, E. An information processing analysis of the function of conceptual mderstartding in the learning of arithmetic procedures (Tech. Report No. KUL-88-03). Pittsburgh, PA University of Pittsburgh, Learning Research and Development Center. 1988. 

VanLehn, K. Arithmetic procedures are induced from examples. Li J. H. Hiebert (Ed.), Conceptual and procedural knowledge The case of mathematics (pp. 133-179). Hillsdale, NJ Erlbaum. 1986. 

ANOVA can also be used in situations where there is more than one source of random variation. Consider, for example, the purity testing of a barrelful of sodium chloride. Samples are taken from different parts of the barrel chosen at random and replicate analyses performed on these samples, in addition to the random error in the measurement of the purity, there may also be variation in the purity of the samples from different parts of the barrel. Since the samples were chosen at random, this variation will be random and is thus sometimes known as a random-effect factor. Again, ANOVA can be used to separate and estimate the sources of variation. Both types of statistical analysis described above, i.e. where there is one factor, either controlled or random, in addition to the random error in measurement, are known as one-way ANOVA. The arithmetical procedures are similar in the fixed- and random-effect factor cases examples of the former are given in this chapter and of the latter in the next chapter, where sampling is considered in more detail. More complex situations in which there are two or more factors, possibly interacting with each other, are considered in Chapter 7. 

We must guard against introduction of uncertainty by arithmetical procedures. The following rules will be helpful. 

Arithmetical Procedure—Deduct the observed distillation loss from each prescribed percentage evaporated in order to obtain the corresponding percentage recovered. C culate each required thermometer reading as follows ... 

Values obtained by the arithmetical procedure are affected by the extent to whidi the distillation graphs are nonlinear. Intervals between successive data points can, at any stage of the test, be no wider than the intervals indicated in 9.7. In no case shall a calculation be made that involves extrrmolation. 

Note 3—See Appendix XI for numerical examples illustrating these arithmetical procedures. 

Where simultaneous solution is indicated, a variety of direct arithmetic procedures may be used interchangeably. Where increased precision or error control has been specified in this test method, more complex calculations must be used. ... 

To estimate the computational time required in a Gaussian elimination procedure we need to evaluate the number of arithmetic operations during the forward reduction and back substitution processes. Obviously multiplication and division take much longer time than addition and subtraction and hence the total time required for the latter operations, especially in large systems of equations, is relatively small and can be ignored. Let us consider a system of simultaneous algebraic equations, the representative calculation for forward reduction at stage is expressed as... 

A procedure involving only the wall area and based on the cylindrical pore model was put forward by Pierce in 1953. Though simple in principle, it entails numerous arithmetical steps the nature of which will be gathered from Table 3.3 this table is an extract from a fuller work sheet based on the Pierce method as slightly recast by Orr and DallaValle, and applied to the desorption branch of the isotherm of a particular porous silica. 

Determine the heat duty by the usual procedures and define the boiling temperature on the shell side. Determine the arithmetic average of tube side temperature, t,. 

The long-interval method involves the calculation of k using the initial values of reactant concentrations successively with each of the other values of the measured concentrations and times. If there are (n + 1) measurements of the concentrations of interest (including the initial value), the procedure yields n values of k. The average value of k is then taken to be the arithmetic average of these computed values. 

Observe that we have in this procedure worked out some of the steps previously left to the THEOREM PROVER, The previous procedure involves having the progranmer select a set of inductive assertions and critical points, and then feed this into the computer parts a VERIFICATION CONDITION GENERATOR and a THEOREM PROVER. In this alternative construction we still need inductive assertions as the nature of the Rule of Iteration for WHILE statements shows. Now the inductive assertions are fed directly into the THEOREM PROVER which las been augmented by the special axioms and rules D0,D1,D2,D3 and D4 in addition to all of the usual arithmetic axioms, rules of inference, rules for handling identities and special axioms for the primitives in question (such as the factorial axioms in our example). In effect the THEOREM PROVER works backwards from the output condition and the various inductive assertions using DO - D3 to find what amounts to path verification conditions -... 

These CFAR procedures suffer from the fact that they are specifically tailored to the assumption of uniform and homogeneous clutter inside the reference window. Based on these assumptions, they estimate the unknown clutter power level using the unbiased and most efficient arithmetic mean estimator. Improved CFAR procedures should be robust with respect to different clutter background and target situations. Also in non-homogeneous situations CFAR techniques should remain able to provide reliable clutter power estimations. 

Censored CFAR. Richard and Dillard [19] have proposed a CFAR procedure which is based on CA-CFAR but is already close to the general OS-CFAR idea when they are calculating the largest m values inside the sliding window and excluding these values from the arithmetic mean calculation. This step makes modified CA-CFAR less sensitive in mul-... 

It can be shownthat asymptotically (i.e. in the limit where the number M of generated configurations tends to infinity) this procedure generates system configurations x, with a probability proportional to the Boltzmann weight, P,(x) = exp [— Jf(x)//cBr]/Z. Thus thermal averages are just calculated as simple arithmetic averages ... 

The calculation of a robust mean according to a method introduced by Huber and described in the relevant standards is shown in this slide. The method starts with median as the initial value m for an iterative procedure. All data outside the m + 1.5 STD are set to this border. Then a new value for m is calculated from the arithmetic mean of this new data. The procedure of... 

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