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

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Название Mathematics for Enzyme Reaction Kinetics and Reactor Performance
Автор произведения F. Xavier Malcata
Жанр Химия
Серия
Издательство Химия
Год выпуска 0
isbn 9781119490333



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++ xm; the generalized formula looks indeed like

      where the summation in the right‐hand side is taken over all sequences of (nonnegative) integer indices k1 through km, such that the sum of all ki ’s is n. The multinomial coefficients are given by

      – and count the number of different ways an n‐element set can be partitioned into disjoint subsets of sizes k1, k2, , km . For the example above, one would have been led to

      Trigonometry is the branch of mathematics that studies relationships involving lengths of sides and amplitudes of angles in triangles. This field emerged in the Hellenistic world during the third century, by the hand of Euclid and Archimedes – who studied the properties of chords and inscribed angles in circles, while proving theorems equivalent to most modern trigonometric formulae; Hipparchus from Nicaea (Asia Minor) produced, however, the first tables of chords in 140 BCE – analogous to the current tables of sine values, which were completed in the second century CE by Greco‐Egyptian astronomer Ptolemy from Alexandria (Egypt). By those times, it was realized that the lengths of the sides of a right triangle and the angles between those sides satisfy fixed relationships; hence, if at least the length of one side and the amplitude of one angle is known, then all other angles and lengths can be algorithmically determined.

      2.3.1 Definition and Major Features

      Consider a unit vector u, i.e.

      that degenerates to

      which may be rewritten as

      in view again of Eq. (2.287); therefore, sin θ is given by the distance, images, of the extreme point, B, of u to the horizontal axis. Sine and cosine constitute the basic trigonometric functions; they are also known as circular functions, owing to the loci of the extreme points of u describing a circle upon full rotation – as per Fig. 2.10a.

Image described by caption and surrounding text.

      (2.292)