Rows and Columns

Select a row on the left-hand side and choose Insert/Row. A new row is placed above the one you selected. The cells are renumbered, and all the formulas are changed, too, to reflect the new numbering system. Use this same method to insert a column the new colunm will appear to the left of the one you have selected. 

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The scan area is recognized as a sequence of points set out in rows and columns and detected in a raster-like marmer under adjustable computer control [3]. 

Longuet-Higgins [7] also reinforces the discussion by tbe following qualitative demonstration of a cyclic sign change for the LiNaK like system subject to Eq. (3), in which rows and columns are labeled by the basis functions... 

For this algorithm, each processor is assigned atoms, so the force calculation time is O( ). Using the communication scheme mentioned above, each processor communicates with ( /P — 1) processors in each row and column. Thus the total number of terms being communicated per step is (-/P — 1)( ). Therefore, 0(N) CPU time is required in communicating the net force per step. Therefore,... 

A major disadvantage of a matrix representation for a molecular graph is that the number of entries increases with the square of the number of atoms in the molecule. What is needed is a representation of a molecular graph where the number of entries increases only as a linear function of the number of atoms in the molecule. Such a representation can be obtained by listing, in tabular form only the atoms and the bonds of a molecular structure. In this case, the indices of the row and column of a matrix entry can be used for identifying an entry. In essence, one has to distinguish each atom and each bond in a molecule. This is achieved by a list of the atoms and a list of the bonds giving the coimections between the atoms. Such a representation is called a connection table (CT). 

The molecule is built up of 1b amino acids, but the file also contains the positions of the oxygen atoms of 12 water molecules contained within the unit cell. To keep the example simple, only the most important parts of the file are presented and discussed here. Each part of the file is annotated with corresponding row and column numbers. The complete file can be obtained from the PDB [53] or from this book s website [153],... 

The characteristic of a relational database model is the organization of data in different tables that have relationships with each other. A table is a two-dimensional consti uction of rows and columns. All the entries in one column have an equivalent meaning (c.g., name, molecular weight, etc. and represent a particular attribute of the objects (records) of the table (file) (Figure 5-9). The sequence of rows and columns in the tabic is irrelevant. Different tables (e.g., different objects with different attributes) in the same database can be related through at least one common attribute. Thus, it is possible to relate objects within tables indirectly by using a key. The range of values of an attribute is called the domain, which is defined by constraints. Schemas define and store the metadata of the database and the tables. 

A matrix can be defined as a two-dimensional arrangement of elements (numbers, variables, vectors, etc.) set up in rows and columns. The elements a are indexed as follows ... 

We shall often encotmter square matrices, which have the same number of rows and columns. A diagonal matrix is a square matrix in which all the elements are zero except for those on the diagonal. The unit or identity matrix I is a special type of diagonal matrix in which all the non-zero elements are 1 thus the 3x3 unit matrix is ... 

In general, the imposition of boundary eonditions is a part of the assembly process. A simple procedure for this is to assign a eode of say 0 for an unknown degree of freedom and 1 to those that are specified as the boundary conditions. Rows and columns corresponding to the degrees of freedom marked by code 1 are eliminated from the assembled set and the other rows that contain them are modified via transfer of the product of the specified value by its corresponding coefficient to the right-hand side. The system of equations obtained after this operation is determinate and its solution yields the required results. 

MODIFY - addressing of members of the coefficient matrix are adjusted to allocate their row and column index in the banded matrix. 

For each M , Ml combination for which one can write down only one product function (i.e., in the non-equivalent angular momentum situation, for each case where only one product function sits at a given box row and column point), that product function itself is one of the desired states. For the p2 example, the piapoot and piap.ia (as well as their four other Ml and Ms "mirror images") are members of the 3p level (since they have Ms = 1) and IpiapiPI and its Ml mirror image are members of the level (since they have Ml... 

For a sample of polystyrene in benzene, experimental values of Kcj/R are entered in the body of Table 10.2. The values are placed at the intersection of rows and columns labeled c and 9, respectively. In the following example... 

In Table 7.7, all the transition wavenumbers have been arranged in rows and columns so that the differences between wavenumbers in adjacent columns correspond to vibrational level separations in the lower (ground) electronic state and the differences between adjacent rows to separations in the upper electronic state. These differences are shown in parentheses. The variations of the differences (e.g. between the first two columns), are a result of uncertainties in the experimental measurements. 

The value of a determinant lAl is not changed if the rows and columns are interchanged. 

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. 

Transposition The matrix obtained from A by interchanging the rows and columns of A is called the transpose of A, written A or A. ... 

Since the t distribution relies on the sample standard deviation. s, the resultant distribution will differ according to the sample size n. To designate this difference, the respec tive distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus I. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant). 

Row and Column Names F2C1 Feed 2 at conversion level 1... 

Equation (8.90) is non-singular since it has a non-zero determinant. Also the two row and column vectors can be seen to be linearly independent, so it is of rank 2 and therefore the system is controllable. 

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] ... 

For a more complicated [B] matrix that has, say, n columns whereas [A] has m rows (remember [A] must have p columns and [B] must have p rows), the [C] matrix will have m rows and n columns. That is, the multiplication in Equations (A.21) and (A.22) is repeated as many times as there are columns in [B]. Note that, although the product [A][B] can be found as in Equation (A.21), the product [B][A] is not simultaneously defined unless [B] and [A] have the same number of rows and columns. Thus, [A] cannot be premultiplied by [B] if [A][B] is defined unless [B] and [A] are square. Moreover, even if both [A][B] and [B][A] are defined, there is no guarantee that [A][B] = [B][A]. That is, matrix multiplication is not necessarily commutative. 

A matrix is a rectangular array of numbers, its size being determined by the number of rows and columns in the array. In this context, the primary concern is with square matrices, and matrices of column dimension 1 (column vectors) and row dimension 1 (row vectors). 

While the rows and columns of A obviously depend on a particular choice of vertex labels, the generic structural j)roperties of G must remain invariant under a permutation of rows and columns. Much of this structural information can in fact be extracted from the spectrum of G the spectrum of a graph G,... 

Just as was the case for one-diinensional majority rules considered in the previous section, we again recognize that the two-dimensional majority rule is but a special case of the generalized threshold rule defined in equation 5.121. Intuitively, the idea is simply to let aij represent a two-dimensional lattice that is built out of our a-priori structureless set of sites, i = 1, 2,..., A. Suppose we arrange these N sites into n rows with rn sites per row, so that N = n x rn. Then the site positioned on the row and column, can be identified with the original site... 

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