Corresponding elements

Both tables, the atom and the bond lists, are linked through the atom indices. An alternative coimection table in the form of a redundant CT is shown in Figure 2-21. There, the first two columns give the index of an atom and the corresponding element symbol. The bond list is integrated into a tabular form in which the atoms are defined. Thus, the bond list extends the table behind the first two columns of the atom list. An atom can be bonded to several other atoms the atom with index 1 is connected to the atoms 2, 4, 5, and 6. These can also be written on one line. Then, a given row contains a focused atom in the atom list, followed by the indices of all the atoms to which this atom is bonded. Additionally, the bond orders are inserted directly following the atom in-... 

Therefore a = /nxl = hi, fl2i = hi, etc. (elements in the first column of a a,re the same as the elements in the first column of /) similarly multiplying rows of / by columns of u and equating the result with the corresponding element of a all of the elements of lower and upper triangular matrices are found. The general formula for obtaining elements of / and u can be expressed as... 

It should be evident that there must be the same number of elements in two matrices to be added and that the elements must be ananged in the same way, so that there is a match of one element in matrix A with its corresponding element in matrix B. Such matrices are said to be conformable to addition. 

The differences between corresponding elements in ordered sets (vectors) in, for example, columns 2 and 3 of output 3-4 give an ordered set of residuals. In this example, the residual for the first data point is 0.2256 — 0.2255161 = 8.39 x 10 . SSE is sometimes written... 

The value of a determinant is not changed if to the elements of any row (or column) are added a constant multiple of the corresponding elements of any other row (or column). 

Equality of Matrices Let A = ( 7 ), B = hf). Two matrices A and B are equal (=) if and only if they are identical that is, they have the same number of rows and the same number of columns and equal corresponding elements (ay = by for all i andJ). 

Addition and Subtraction The operations of addition (-h) and subtraction (—) of two or more matrices are possible if and only if they have the same number of rows and columns. Thus A B = (ay by) i.e., addition and subtraction are of corresponding elements. 

The triple product of three noncolinear line elements in the reference configuration provides a material element of volume dV. Another well-known theorem in tensor analysis provides a relation with the corresponding element of volume dv in the current spatial configuration... 

Two matrices, [A] and [B], can be added only if they have the same number of rows and columns, respectively. Then, the sum [C] is obtained by adding the corresponding elements of [A] and [B] ... 

The lines in the spectrum from any element always occur in the same positions relative to each other. When sufficient amounts of several elements are present in the source of radiation, each emits its characteristic spectrum this is the basis for qualitative analysis by the spectrochemical method. It is not necessary to examine and identify all the lines in the spectrum, because the strongest lines will be present in definite positions, and they serve to identify unequivocally the presence of the corresponding element. As the quantity of the element in the source is reduced, these lines are the last to disappear from the spectrum they have therefore been called the persistent lines or the rates ultimes (R.U. lines), and simplify greatly the qualitative examination of spectra. 

This is the fundamental differential equation. The reader who is acquainted with the rules for transforming the variables in a surface integral will observe that it has the geometrical interpretation that corresponding elements of area on the (v, p) and (s, T) diagrams are equal (cf. 43). 

STRATEGY Divide each mass percentage by the molar mass of the corresponding element to obtain the number of moles of that element found in exactly 100 g of the compound. Divide the number of moles of each element by the smallest number of moles. If fractional numbers result, then multiply by the factor that gives the smallest whole numbers of moles. 

Corresponding elements in the two vectors of means are subtracted, and the differences are squared and added. The square root of the sum (15.21) is equal to the Euclidean distance in 15 dimensions separating the two points that represent the group means. This distance forms the base line in Fig. 4.20. 

Corresponding elements in the vector representing one particular sample and in the appropriate vector of means are worked up as in 2) to find the Euclidean distance between point i and its group mean (see lines marked with an asterisk ( ) in Table 4.16) this forms the second side of the appropriate triangle in Fig. 4.20. 

Instead of a conjunction of preconditions, as used by the STRIPS and conditional operators, the functional operator has a set of conjunctions of preconditions (Fig. 2c). Each element in the set describes some possible situation that might exist before the operator is applied. For each element of the set of preconditions, there is a corresponding element in the set of postconditions. The functional operator is a more flexible model than the STRIPS or conditional operators. It comes closer to the modeling needs for the synthesis of operating procedures for chemical processes, but as we will see in the next section, we need to introduce additional aspects in order to capture the network-like structure of chemical processes. 

Before examining selenium in some detail, as an illustration of the phenomena under consideration, let us note here that although oxidative UPD of sulfur can be achieved from sulfide solutions, the oxidative UPDs of Se and Te are complicated by the fact that, unlike sulfides, the Se and Te species are not very stable in aqueous solutions they tend to react with even traces of O2, to form the corresponding element. 

Different from sole combinations of micro devices, this refers to a total system with many functional elements and flow-distribution and, recollecting zones, typically composed of 2-D plate-type architecture. Each of these plates usually has a separate fimction, comprising imit operations and reaction. Frequently, micro mixing and micro heat exchange fimctions and corresponding elements are employed. Often, the system can be composed of different elements resulting in different process flow combinations. Such an approach may be termed a construction kit. 

For the example discussed here, the points A and B are found as follows. First, we calculate m and n by dividing all elements of v, by the corresponding elements of Vj, which gives ... 

The value of a determinant is unchanged if, to each element of one row (column) is added a constant A times the corresponding element of another row (column). Thus, we have, for example... 

The height of a given X-ray peak is a measure of the amount of the corresponding element in the sample. The X-ray production cross-sections are known with good accuracy, the beam current can be measured by, for example, a Faraday cup (Figure 4.1) and the parameters of the experimental set-up are easily determined so that the sample composition can be determined in absolute terms. 

The elements in groups 15-18 in period 3 and beyond also form multiple bonds with each other. But because these elements have smaller electronegativities than the corresponding elements of period 2, they are appreciably weaker than the multiple bonds between... 

If the free element is less active than the corresponding element in the compound, no reaction will take place. A short list of metals in order of their reactivities and an even shorter list of nonmetals are presented in Table 7-1. The metals in the list range from very active to very stable the nonmetals listed range from very active to fairly active. A more comprehensive list, a table of standard reduction potentials, is presented in general chemistry textbooks. 

In each of these cases, the free element is less active than the corresponding element in the compound, and cannot replace that clement from its compound. 

As the adjustments at the winning node move each of its weights slightly toward the corresponding element of the sample vector, the node learns a little about this sample pattern and, thus, is more likely to again be the... 

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