Transposable elements, defined

Aa° is the column vector containing the changes of concentrations of all the reactants as elements. The stoichiometric vector v corresponds to its transpose v° defined above. Eq. (2.2) is explicitly written as follows ... 

A linear coordinate transformation may be illustrated by a simple two-dimensional example. The new coordinate system is defined in term of the old by means of a rotation matrix, U. In the general case the U matrix is unitary (complex elements), although for most applications it may be chosen to be orthogonal (real elements). This means that the matrix inverse is given by transposing the complex conjugate, or in the... 

If a matrix is equal to its transpose, it is said to be a symmetric matrix. If the elements of A are complex numbers, the complex conjugate of A is defined as... 

The transposed matrix A1 is defined as the interchange of rows and columns of A. This can also be seen as the reflection of all elements of A at its main diagonal (along ay, i=j), according to... 

If the matrix 4 is defined as the transpose of the matrix 4, each element in is given by... 

To begin with, there are a total of 24 permutations on this diagonal element. These can be minimized by eliminating all the permutations that transpose spin—orbitals of a different spin variety (a and 3), because these permutations have zero contributions to all the title quantities. We can therefore group the spin—orbitals of the determinant into two subsets and define elementary permutations that act exclusively within the subsets. From there, we can build up more complex permutations, until all the contributing permutations are included. 

Furthermore, traditional notation for scalars, vectors and variables will be adopted. A scalar of fixed value, e.g., the number of factors in a model, is represented by an italicized capitol, A. An italicized lowercase letter, e.g., the nth factor, represents a scalar of arbitrary value. All vectors are column vectors designated by lowercase bold, e.g., x. Matrices are given by uppercase bold, e.g., X, and cubes (third-order tensors by uppercase open-face letters, e.g., R. Transposes of matrices and vectors, defined by switching the row and column indices, is designated with a superscript T, e.g., xT. The transpose of a cube need not be defined for this chapter. Subscripts designate a specific element of a higher-order tensor, where the initial order is inferred by the number of subscripts associated with the scalar. 

Alternatively to the keyword SOLUTION, SOLUTIONSPREAD can be used for the input of solution. The input is transposed compared to the input for SOLUTION, i.e. the rows of input for SOLUTION become the columns of input for SOLUTION SPREAD. It is especially convenient to define more than one aqueous solution composition using this tab-limited format. Data obtained e g. from a laboratory spreadsheet format can be copied directly into the PHREEQC input file. SOLUTION SPREAD is compatible with the format of many spreadsheet programs, as e.g. Excel. The column headings are element names, element valence state names or isotope names. One subheading can be used to define speciation (e.g. as S04 , or as N03 ), specify element specific units, redox couples, phase names and saturation indices. All succeeding lines are the data values for each solution, with one solution defined on each line. 

T and U are matrices of score vectors, P and C are the transposed matrices of the loading vectors, E and F are matrices of residuals. The inner relation which describes the correlation between the scores is defined by D which is a diagonal matrix in which the elements, dH, are the correlation coefficients of the linear relation between the scores. H is the matrix of residuals from the correlation fit. 

A square matrix with zero non-diagonal elements is called a diagonal (or quasi-scalar) matrix it is defined by a[Pg.403]

The composition vector in the (n — l)-dimensional space defined by the straight line reaction paths The composition vector g at time t The composition vector g at time t = 0 An arbitrary vector The jth element of the vector y A vector in the orthogonal system of coordinate The transpose of the vector y A small interval of time An interval of time... 

The position of each element in a matrix is defined by its row and column number. So in row i and column j of the matrix X, there is the element dk On exchanging rows and columns, we obtain the transpose matrix X. Thus, the element x , found in column i and row j of the transpose matrix, is equal to the element of the original matrix. An example is as follows ... 

The N singular values of a real N xN matrix (i.e., all elements of the matrix are real numbers) are defined as the square root of the eigenvalues of the matrix formed by multiplying the original matrix by its transpose. 

The molecules most commonly encountered generally possess a number of symmetry elements. Every molecule is associated with a point group, where the point group is defined as the collection of symmetry operations which when applied about a point leave the molecular configuration unchanged in space, while the point itself remains at rest, or more precisely, transposes into itself. There is a... 

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