Finding the values

When creating a table or chart, designate a variable to represent a part of the problem, and see what the results are as you systematically change that variable. For example, if you re trying to find two numbers the product of which is 60 and the sum of which is as small as possible, let the first number be x. Then the other number is Add the two numbers together to see what you get. Table 1-2 shows the different values for the two numbers and the sum — if you stick to whole numbers. 

I stop here, because I m just going to get the same pairs of numbers in the opposite order. If the rule is that the numbers can t be fractions, then the two numbers with a product of 60 and with the smallest possible sum are 6 and 10. 

When making a table or chart, you want to be as systematic as possible so you don t miss anything - especially if that anything is the correct answer. After you ve determined a variable to represent a quantity in the problem, you need to go up in logical steps — by ones or twos or halves or whatever is appropriate. In Table 1-2, in the preceding section, you can see that I went up in steps of 1 until I got to the 6. One more than 6 is 7, but 7 doesn t divide into 60 evenly, so I skipped it. Even though the work isn t shown here, I mentally tried 7, 8, and 9 and discarded them, because they didn t work in the problem. When you re working with more complicated situations, you don t want to skip any steps — show them all. 

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These expressions define Pn-2n for number of 1-2 shells in series in terms of R and Xp in each shell. The expressions can be used to define the number of 1-2 shells in series required to satisfy a specified value of Xp in each shell for a given R and Pjv 2n- Hence the relationship can be inverted to find the value of N which satisfies Xp exactly in each 1-2 shell in the series ... 

The minimisation problem can be formally stated as follows given a function/which depends on one or more independent variables Xi,X2,..., Xj, find the values of those variables where/ has a minimum value. At a minimum point the first derivative of the function with respect to each of the variables is zero and the second derivatives are all positive ... 

Finding the values of G allows the determination of the frequency-domain spectrum. The power-spectrum function, which may be closely approximated by a constant times the square of G f), is used to determine the amount of power in each frequency spectrum component. The function that results is a positive real quantity and has units of volts squared. From the power spectra, broadband noise may be attenuated so that primary spectral components may be identified. This attenuation is done by a digital process of ensemble averaging, which is a point-by-point average of a squared-spectra set. 

Then find the value of the feedback capacitor C. The designer knows the value of the input resistor (R). It is the upper resistor in the voltage divider responsible for the voltage feedback to the error amplifier. One then performs Equation B.15. 

To find the value of the feedback capacitor, one evaluates the following ... 

Find the value of the eritieal damping eoeffieient Q spring-mass-damper system shown in Figure 3.17. 

Find the value of K to give the system a elosed-loop time eonstant of one seeond. What is the steady-state value of ujoit) when V[ t) has a value of 10 V. 

Find the value of the proportional eontroller gain K to make the eontrol system shown in Figure 5.4 just unstable. 

For the system given in Figure 7.14 (i.e. Example 7.4) find the value of the digital eompensator gain K to make the system just unstable. For Example 7.4, the ehar-aeteristie equation is... 

Find the values of K and h if a is seleeted to eaneel the non-unity open-loop pole. 

Using a GA with a population of 10 members, find the values of the controller gain K and the tachogenerator constant that maximizes the fitness function... 

When run, the program invites the user to seleet a point in the graphies window, whieh may be used to find the value of A" when ( = 0.25. If the last line of examp59.m is typed at the MATLAB prompt, the eursor re-appears, and a further seleetion ean be made, in this ease to seleet the value of K for marginal stability. This is demonstrated below... 

Classicists believe that probability has a precise value uncertainty is in finding the value. Bayesians believe that probability is not precise but distributed over a range of values from heterogeneities in the database, past histories, construction tolerances, etc. This difference is subtle but changes the two approaches. 

This is a technique developed during World War II for simulating stochastic physical processes, specifically, neutron transport in atomic bomb design. Its name comes from its resemblance to gambling. Each of the random variables in a relationship is represented by a distribution (Section 2.5). A random number generator picks a number from the distribution with a probability proportional to the pdf. After physical weighting the random numbers for each of the stochastic variables, the relationship is calculated to find the value of the independent variable (top event if a fault tree) for this particular combination of dependent variables (e.g.. components). 

The small lattice can be enlarged to the desired size by changing the number of points from N to IN — 1 and finding the values of 0/ y in the new lattice sites by interpolation. The interpolation done to enlarge the lattice has no influence on the results. It may speed up the calculations but only if it is done appropriately. 

After inserting (71) into (66) and some combinatorics we find the value of LO X) = where by X we denote one of the phases studied... 

For phosphoric acid at 0°C we find the value J = 0.2056 electron-volt. 

The smallest value of K for any acid in Table 9 is 2.36 X 10-11 for carbonic acid at 0°C. From (132) in this case we find the value J = 0.6708 electron-volt. The values of K for the self-dissociation of water, given in Table 9, are still smaller. To calculate J we use (129) instead of (91), and at 60°C obtain the value J = 1.092 electron-volts. 

From the data given in Table 9 find the value of J for the proton transfer (125) in water at 10°C. 

Taking from Tables 9 and 11 the values for the equilibrium constants for the proton transfers from the acetic acid molecule and from the anilinium ion, and using (145), find the value of —kT In K, for the proton transfer... 

Taking the values for the Ag+ ion and the Cl" ion from Table 25, and adjusting the cratic term, find the value of the partial molal entropy of silver chlorido in an aqueous solution having a molality equal to 10 8. 

Knowing the two rates and the ratio of the two concentrations, we can readily find the value of m. 

Find the value of fCb for the conjugate base of the following organic acids. 

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