Real or complex number

Unfortunately, it is not possible to automatically generalize the Abelian Stokes theorem [e.g., Eq. (4)] to the non-Abelian one. In the non-Abelian case one faces a qualitatively different situation because the integrand on the l.h.s. assumes values in a Lie algebra g rather than in the field of real or complex numbers. The picture simplifies significantly if one switches from the local language to a global one [see Eq. (5)]. Therefore we should consider the holonomy (7) around a closed curve C ... 

Is it possible to apply linear operations to the space These are defined as scaling an element by a real or complex number, and addition of two elements to obtain a third. The following rules must be satisfied Let a,/ ,... be scalars (real or complex) and x, y.. g S. Then... 

Here the root finder can be any one of the algorithms from the vast literature. Assuming that the chosen algorithm works properly, it takes the n real or complex coefficients a for i = 0 to i = n — 1 of p as input and produces n real or complex numbers Xj, the roots of p, as output. To do so, we have tacitly assumed that the leading coefficient an of p is normalized to equal 1 since dividing p of degree n by a constant (an 0) does not alter its roots. 

In quantum mechanics we often encounter associative algebras of operators and matrices which are noncommutative. For example, the set of all n x n matrices over the real or complex number fields is an n2-dimensional vector space which is also an associative, noncommutative algebra whose multiplication is just the usual matrix multiplication. Also, the subset of all diagonal n x n matrices is a commutative algebra. 

The eigenvalues are also known as the natural frequencies of the circuit, which are the reciprocals of the circuit response time constant. The eigenvalues of Equation 2.155 could be real or complex numbers. If the natural frequencies are complex,... 

For motion in the x direction the analysis is more involved because X can exist as either a real or complex number, depending on whether < > > l/(4a) or < > < 1/(4a). For the case < > > l/(4a) the plot is of the stable focus type (Fig. 7.11). The right-hand side of Figure 7.11 has been lightly shaded since this is an imaginary zone—this is the space behind the infinite plane. 

A mapping M is a prescription that assigns elements of a set A to elements of a set F. The set A is called the domain of the mapping, the set B is called the co-domain. If A and B are sets of real or complex numbers, then the mapping M can sometimes be expressed conveniently with the help of a simple function /. The function / is then called a... 

The square brackets are used to denote a matrix. The are the elements of the matrix. They may be real or complex numbers. Quite often the elements of a matrix will be functions or operators. The matrix [yl] above has p rows and m columns and is said to be a p X m matrix, alternatively, the matrix [ /11 is said to be of order p X m. (It is common to use the letter n to denote the number of rows a matrix has. To avoid confusion with the rest of this book, we have reserved the letter n for denoting the number of components in a mixture.) The order of the matrices we will encounter in this book usually is obvious from the context but, in the event that it becomes necessary to distinguish between matrices of different order, this will be done by appending a subscript to the bracket notation. Thus, the p Xm matrix [A] would be written... 

In the first chapter, we saw that if we wanted to rotate the 2px function, we automatically also needed its companion 2py function. If this is extended to out-of-plane rotations, the 2/ function will also be needed. The set of the three p-orbitals forms a prime example of what is called a linear vector space. In general, this is a space that consists of components that can be combined linearly using real or complex numbers as coefficients. An n-dimensional linear vector space consists of a set of n vectors that are linearly independent. The components or basis vectors will be denoted as fi, with I ranging from 1 to n. At this point we shall introduce the Dirac notation [1] and rewrite these functions as / >, which characterizes them as so-called kef-functions. Whenever we have such a set of vectors, we can set up a complementary set of so-called fera-functions, denoted as /t I The scalar product of a bra and a ket yields a number. It is denoted as the bracket fk fi). In other words, when a bra collides with a ket on its right, it yields a scalar number. A bra-vector is completely defined when its scalar product with every ket-vector of the vector space is given. 

A very important and frequently studied type of bundles is that of the vector bundles. In this case, the fiber is required to have the structure of a vector space (usually over the real or complex numbers), and the homeomorphism = U X F is required to be a vector space isomorphism for each x U. 

Comment. In Mathcad the variable keeping a massive (vector or matrix) can include text, scalar (real or complex number) and vice versa. A variable in Mathcad is not connected with one or another variable type that exists in most part of traditional programming languages. 

The numeric data type is used to represent the numbers. It can be used to represent the numbers in various types, such as signed or unsigned integer, double or single precision number, and real or complex numbers. The type of numeric control and indicator can be chosen using the property window. 

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