F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

The second volume ends with a brief coverage of statistics – starting with continuous probability functions and statistical descriptors, and proceeding to discussion in depth of the normal distribution; such other continuous distributions as lognormal, chi‐square, Student's t ‐, and Fisher's F ‐distributions are reviewed next – spanning from mathematical derivation, through calculation of major descriptors, to discussion of most relevant features (including generation of distinct continuous probability functions). Statistical hypothesis testing is addressed next, complemented with the alternative approach of parameter and prediction inference – resorting to linear regression analysis as germane mode of parameter estimation.

F. Xavier Malcata Professor of Chemical Engineering University of Porto (Portugal)

Reading maketh a full man, conference; a ready man, and writing; an exact man.

Francis Bacon

Quantification of any entity or concept requires association to a numerical scale, so as to permit subsequent abstract reasoning and objective comparability; hence, every measurement carried out in the physicochemical world leads to a number, or scalar. Such numbers may be integer, rational (if expressible in the form p / q , where p and q denote integer numbers), or irrational (if not expressible in the previous form, and normally appearing as an infinite, nonrecurring decimal). If considered together, rational and irrational numbers account for the whole of real numbers – each one represented by a point in a straight line domain.

Departing from real numbers, related (yet more general) concepts have been invented; this includes notably the complex numbers, z – defined as an ordered pair of two real numbers, say, z ≡ a + ιb , where a and b denote real numbers and ι denotes , the imaginary unit. Therefore, z is represented by a point in a plane domain. In the complex number system, a general n th degree polynomial equation holds exactly (and always) n roots, not necessarily distinct though – as originally realized by Italian mathematicians Niccolò F. Tartaglia and Gerolamo Cardano in the sixteenth century; many concepts relevant for engineering purposes, originally conceived to utilize real numbers (as the only ones adhering to physical evidence), may be easily generalized via complex numbers.

The next stage of informational content is vectors – each defined by a triplet ( a,b,c ), where c also denotes a real number; each one is represented by a point in a volume domain and is often denoted via a bold, lowercase letter (e.g. v). Their usual graphical representation is a straight, arrowed segment linking the origin of a Cartesian system of coordinates to said point – where length (equal to , as per Pythagoras’ theorem), coupled with orientations (as per tan{ b / a } and tan{ c / }) fully define the said triplet. An alternative representation is as [ a b c ] or – also termed row vector or column vector, respectively; when three column vectors are assembled together, say, , , and , a matrix results, viz. , termed tensor – which may also be obtained by joining three row vectors, say, [ a 1 a 2 a 3], [ b 1 b 2 b 3], and [ c 1 c 2 c 3]. The concept of matrix may be generalized so as to encompass other possibilities of combination of numbers besides a (3 × 3) layout; in fact, a rectangular ( p × q ) matrix of the form , or [ a i,j; i = 1, 2 ,…, p ; j = 1, 2 , …, q ] for short, may easily be devised.

Matrices are particularly useful in that they permit algebraic operations (and the like) be performed once on a set of numbers simultaneously – thus dramatically contributing to bookkeeping, besides their help to structure mathematical reasoning. In specific situations, it is useful to design higher order number structures, such as arrays of (or block) matrices; for instance, may be also represented as , provided that, say, A 1,1≡ [ a 1], A 1,2≡ [ a 2 a 3], , and represent, in turn, smaller matrices. An issue of compatibility arises in terms of the sizes of said blocks, though; for a starting ( p × q ) matrix A, only ( p 1× q 1) A 1,1, ( p 1× q 2) A 1,2, ( p 2× q 1) A 2,1, and ( p 2× q 2) A 2,2matrices are allowed – obviously with p 1 + p 2 = p and q 1 + q 2 = q .

One of the most powerful applications of matrices is in solving sets of linear algebraic equations, say,

(1.1)