Nonzero numbers

If an equation appears in more than one loop, one tear can break all of the loops in which the equation appears if in each of these loops the equation included in both loops has exactly the same equation feeding it in the loop. For example, column 2 of the occurrence matrix in Fig. 16a contains nonzero elements in rows 2 and 4. Upon inspection of the corresponding loops, B and D, in the tree of Fig. 15 it is seen that f2 is fed by f3 in both loops (/3 appears immediately after f2 in paths involved in loops B and D). Therefore, loops B and D can be broken by tearing the output variable of f3 from the equation f2. Now instead of putting l s as entries in row i and column j of the occurrence matrix to indicate that equation j appears in loop i, we insert instead the number of the equation that feeds equation j in loop i as in Fig. 16b. Then if the same nonzero number appears more than once in the same column, all of the loops represented by the rows in which that number appears... 

The inverse A" of a matrix A serves the same role in matrix algebra that the reciprocal of a number serves in ordinary algebra. That is, for a nonzero number a in ordinary algebra,... 

Zeros to the right of a decimal point and precede nonzero numbers are not significant, e.g., 0.001. 

Zeros to the left of a decimal point that follow nonzero numbers, e.g., 100, are usually not significant, unless one of the zeros is identified by the reporter as significant. This is done by underlining the zero or placing a bar over that zero, e.g., 10. 

When the leftmost digit to be dropped is 5 followed by a nonzero number, that digit and any digits that follow are dropped. The last digit in the rounded number increases by one. For example, 8.7519 rounded to 2 significant figures equals 8.8. 

The fluxes were estimated simply by giving two small, nonzero numbers. A negative sign was put in front of the estimate of the flux of methanol because methanol is more volatile than water and will transfer from the liquid to the vapor phases. The mole fractions at the interface are given values of 1/n, where n is the number of components. We will see later that it would have been more logical to estimate the interfacial mole fractions at their corresponding bulk values. The temperature is somewhere between the boiling point of methanol and that of water. 

The definition of the vaporization point efficiency is axiomatic. For let a and b be any two nonzero numbers which are finite and positive. Then if a is unequal to b, there exists a positive number c such that a = cb. Similarly, if the fugacity of component / in the vapor phase at any point above the liquid on a plate is unequal to its fugacity f[ in the liquid, then there exists a multiplier called the vaporization point efficiency, such that... 

Lets look at these two numbers more closely. In each case, the number 10 raised to an exponent replaced the zeros that preceded or followed the nonzero numbers. For numbers greater than 1, a positive exponent is used to indicate how many times the coefficient must be multiplied by 10 in order to obtain the original number. Similarly, for numbers less than 1, a negative exponent indicates how many times the coefficient must be divided by 10 in order to obtain the original number. 

Rule 2. Zeros between nonzero numbers are always significant. 

Count all nonzero numbers, zeros between nonzero numbers, and final zeros to the right of the decimal place. (Rules 1,2, and 3 ... 

Here, A and /t are arbitrary nonzero numbers. To ensure the Hermiticity of the left-hand sides of Eqs. (14.31) and (14.32), we define Fc andFv as... 

A, IX, and a are arbitrary nonzero numbers and set to 0.1 in the present study. Thus, Fock operator for occupied orbitals is rewritten as follows ... 

Since the numerator and denominator of the approximant can be haiiiilessly multiplied by any nonzero number, we may set, without losing anything, the following normalization eondition ... 

The smallest nonzero number of electron exchanges in Aabc equals 1. Such an exchange may take place only between two molecules-eg., AB. This results in terms proportional to in the three-body expression, where S means the overlap integral between the molecular orbitals of A and B. The third molecule does not participate in the electron exchanges, but is not just a spectator in the interaction. If it were, the interaction wonld not be three-body (see Fig. 13.12). 

Proof. We give the proof for the case 5=1, i.e., for the system (2.1). Since the matrix A has the real canonical form, one of its parameters is an arbitrary nonzero number. Assume that is a positive fixed number. Then, according to Lemma 6.1, we have for the function A"x... 

To get the decimal into the standard position, move it 1 place to the left. This changes the exponent from -4 to -3, so the final result is 2.3 X 10 (This number in decimal form, 0.0023, can be written correctly as either 23 X 10 " or 2.3 X 10 but scientific notation requires that the decimal point be to the right of the first nonzero number.)... 

Zeros located between nonzero numbers or trailing zeros located at the end of numbers will be considered significant. Thus, 2050 pL, 2.050 mL, and 0.002050 L all represent the same volume measurement, and all contain four significant figures. 

Zeros not preceded by nonzero numbers are not significant figures. These zeros are sometimes called leading zeros. 

Reciprocals of above If activity i has one of the above nonzero numbers assigned to it when compared with activity j, then j has the reciprocal value when compared with i A comparison mandated by choosing the smaller element as the unit to estimate the larger one as a multiple of that unit... 

Trailing zeros (zeros to the right of a nonzero number) that fall after a decimal point are significant. 

Leading zeros (zeros to the left of the first nonzero number) are not significant. They only serve to locate the decimal point. 

Zeros coimt as significant digits when they are interior zeros (zeros between two numbers) and when they are trailing zeros (zeros after a decimal point). Zeros are not significant digits when they are leading zeros, which are zeros to the left of the first nonzero number. 

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