Algebra notation

In the present study, matrix algebra (see [42] for a more detailed description) is used as a shorthand for otherwise tedious formulae. In matrix algebraic notation, small boldface letters denote vectors (for example, a) and capital boldface letters denote matrices (for example, X). Vectors can be either row vectors or column vectors (see Figure 6.31, in which some operations are... 

Sometimes we must combine two or more units to describe a quantity. For instance, we might express the speed of a car as 60 mi/h (also mph). Recall that the algebraic notation means 1/r applying this notation to units, we see that h means 1/h, or per hour. So the unit of speed could also be expressed as mi-h . ... 

I have endeavored to introduce each topic by taking the simplest example that brings out the characteristic features. At times, the analysis may seem painfully long and heavy-handed and the student who is able to do so should jump ahead. I hope that this will temper the wind of generality for those who find an algebraic notation obscure. At the same time, I make no apology for using the reactions 2 = 0, nor even the time-honored A — B... 

These expansion terms will subsequently be called configuration state functions (CSFs). The actual construction of these expansion terms is discussed later in this section. It is assumed here that such an expansion exists and some of the algebraic notation that will be used in this review is introduced. The expectation value of an operator may be written as... 

Employing the standard tensor algebra notations for tensor spherical components... 

The calculated value for the variance of our new variable X2> confirms that there is an increased spread of the data on the new axis. As well as this algebraic notation, it is worth pointing out that the coefficients of the... 

There are now modeling languages that allow the user to express a model in a very compact algebraic form, with whole classes of constraints and variables defined over index sets. Models with thousands of constraints and variables can be defined in a couple of pages, in a syntax that is very close to standard algebraic notation. The algebraic form of the model is kept separate from the actual data for any particular instance of the modeL The computer takes over the responsibility of trans-... 

Classes may have operator functions, so that objects can be manipulated using algebraic notation. 

The following class declaration describes a data type that represents complex numbers, which are numbers defined as having both real and imaginary parts. In algebraic notation the letter i indicates the imaginary part of a number, thus 50 + loot represents a complex with real part of 50 and imaginary part of 100. A more realistic complex number class would have many more facilities the simplifications imposed in this simple example are for clarity. 

Expected Profit. In general algebraic notation including ii )ut and fixed costs a firm s expected profit is... 

As a multipole can be considered simultaneously as a set of poles and a set of dipoles, the algebraic notation must distinguish three organization levels (no problem with Formal Graphs). This is done in numbering poles with a single number and dipoles with pairs of subsequent numbers. 

In general throughout the chapter, although some of the algebraic notation may appear unfriendly, in fact the algebra itself is rudimentary. 

The working expressions derived below for the various real-world circumstances necessarily involve a complex algebraic notation, so a list of the symbols used is added in Appendix 8.7. Note that the following treatment is based upon an earlier discussion (Boyd 1993) but that some of the symbols used are different to avoid confusion with other notations used in this book. 

In the presence of FSSEs, they would be written with an algebraic notation as in Eqs. (67). 

The Mie scattering efficiency database can be reorganised into a matrix (a block of numbers) in which each row is an individual Q curve. This matrix hereafter is denoted as Q, where the bold font indicates a matrix according to standard linear algebra notation. If our raw spectrum to be corrected comprises 2000 absorbance values, then the 1000 Q curve matrix will have 1000 rows and 2000 columns (a 1000x2000 matrix). 

The HP calculator used a different sequence called Reverse Polish Notation (RPN). Jan Lukasiewicz (1878-1956), a Polish mathematican, developed a parenthesis-free system of entering numbers that places die operator before the numbers to be manipulated. This is known as Polish Notation (PN). Conventional algebraic notation places operators between the munbers, as in Eq. (15.3). Hewlett-Packard (HP) calculators employ a memory stack and an entry procedure where the operator follows the number (the reverse of PN). Hence, the term Reverse Polish Notation (RPN). This provides a parenthesis-free procedure that requires fewer keystrokes, in complex algebraic applications, than the conventional algebraic approach. Equation (15.3) would be carried out on the HP 32S as ... 

It is often convenient to manipulate the discretely sampled array of data points in terms of the matrix algebra notation [9]. A set of spectra recorded as a function of wavenumber and... 

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