This publication is a presentation of my fresh study paintings into the speculation and functions of mathematical kinfolk among finite units. the ensuing effects and ideas are expressed in 'conjunction' with the belief of a union of simplicial complexes and the algorithms and total technique became often called Q-Analysis. The mathematical rules are primarily comhinatorial and are built in Section-A of the e-book, while Section-R comprises an software to a group learn (that of the IJniversity of Rssex). the entire paintings has been released in different places, within the ordinary type of papers, and a entire creation to the complete process can be present in my prior publication, Mathematical constitution in Human Affairs (Heinemann, London 1974). a lot of the learn paintings noted during this current hook shouldn't have been attainable with no the co-operation of my examine workforce, of whom i might rather prefer to thank J.II. Johnson, and its puhlication owes much to the secretarial aid supplied via ~Iary Low. R.H. Atkin IJniversity of Rssex U.K.

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Thus the change of velocity, what is generally and loosely written as 6v, is already a measure of some force (or possibly i~ulse) experiencing in the backcloth SeN). which we claim the "particle" is It is of course common, particularly in an elementary education, to speak of the force as causing the change in the velocity. But this, on closer examination, tends to lead us in a circular path, so that it is more realistic to regard the change of pattern as a description of the same thing as the word force is a description of.

AnV corresponds to the grading of the augmented K. viz •• K = K- l Gl KO @ ... , w = a_ l cr_ l + E a Oi cr o + ••• + This is a polynomial, with coefficients in J, in the algebraic representation AV. When we wish to distinguish strictly between w as a map and w as a polynomial we shall refer to the latter as a pattern polynomial, w. The complex K itself, by the result in [2], thereby becomes represented by a pattern Wo in which each aPi = 0 or 1, depending whether the simplex crp i is absent/present in K.

At a time have dimensions qij' qijk' etc .. The canonical representation of K now induces an overall pattern generator of the form L eCK) i

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