By H. S Hall
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Additional resources for A Short Introduction to Graphical Algebra
The Hasse diagram of the one 2-element lattice is seen in Fig. 2. Fig. 2. The one 2-element lattice. With n = 3 elements, one can also form just one lattice. The Hasse diagram of the one 3-element lattice is shown above in Fig. 1. Using n = 4 elements, one can form two lattices. The Hasse diagrams of the two 4-element lattices are pictured in Fig. 3. Fig. 3. The two 4-element lattices. The Hasse diagrams of the five 5-element lattices are illustrated in Fig. 4. Fig. 4. The five 5-element lattices.
The computation of the roots of an equation of third degree was done in different ways. For example, in the Middle Ages: • In China, a method based on polynomial division was used. • In Arabian mathematics, it meant finding the intersection points of a parabola with a hyperbola. The formula that allowed one to solve a third-degree equation by radicals is the result of the efforts of Italian mathematicians (Scipione del Ferro, Ludovico Tartaglia, Girolamo Cardano) in the sixteenth century. Cardano showed that (in today’s notation) any third-degree equation ax3 + bx2 + cx + d = 0 can be transformed into a simpler form y3 + py + q = 0 with the substitution x = y − b/(3a), where p = c/a − b2/(3a2), q = 2b3/(27a3) − (bc)/(3a2) + d/a.
It has long been known that the linear space is not generally an adequate mathematical framework for IR. In Chapter 9, the principle of invariance (PI) is described. According to PI, documents may or may not preserve their identities when looked at from different points of view. It is shown that PI together with the notion of fuzzy cardinality form a correct mathematical framework for the traditional vector space retrieval method, from which the latter can be formally (and hence correctly) obtained.
A Short Introduction to Graphical Algebra by H. S Hall