Scalar quantity multiplication

Multiplication of a complex number by a scalar (real number) is achieved by simply multiplying the real and imaginary parts of the complex number by the scalar quantity. Multiplication of two complex numbers is performed by expanding the expression (a + ih) c + id) as a. sum of terms, and then collecting the real and imaginary parts to yield a new complex number. 

We have already seen in the previous chapter that vectors may be added or subtracted. They may also be combined with each other or with scalar quantities by means of multiplication. 

As the name implies, the scalar product is a way of multiplying vectors which results in a scalar quantity. It is also known as the dot product, because the multiplication operation is represented by a dot. The scalar product of two vectors a and b is defined by... 

Tables S.l and 5.2 list the computational requirements for the new dynamic simulation algorithm, using the most efficient algorithms known for each calculation for different values of N. The computations are tabulated in toms of the matrix and vector quantities which are found in the first three stq>s of the algorithm. The requited scalar opoations (multiplications, additions) are given for an AT-link, serial manipulator with simple revolute and prismatic joints only. The efficient matrix transformations and oth simplifications described in Chapter 3 have been applied in each stq>, and the computations necessary to determine the individual link transformation matrices have also been included.

Table 3.3. The one-point scalar statistics and unclosed quantities appearing in the transport equations for inhomogeneous turbulent mixing of multiple reacting scalars at high Reynolds numbers. 

In this case, we again define a scalar objective function that measures our fit to the data. When we have different measured quantities, however, it often does not make sense to sum the squares of the residuals. The measured variables may differ in size from each other by orders of magnitude. The influence a measurement has on the objective also would be influenced by the arbitrary choice of the units of the measurement, which is obviously undesirable. The simplest way to address this issue is to employ weighted least squares. The reader should be aware that more general procedures are available for the multiple measurement case, including the maximum likelihood method [29,30, 31J. For simplicity of presentation, we focus here on weighted least squares. 

The result of this matrix multiplication is a single quantity (a scalar), which is the sole element in a 1 x 1 matrix. 

Reciprocal quantities may be calculated using formulae which are derived from multiple scalar and vector products ... 

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