Browsing by Browse by FOR 2008 "010101 Algebra and Number Theory"

This data set include raw and analysed data for each thesis chapter. Data set files are broken down by chapter, and the thesis chapters with data are Chapter 3 to chapter 7.

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every 'n × n (n > 2)' singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.

Kappraff described the panels in the pavement of the Reading Room of the Library on the second floor of the San Lorenzo church complex in Florence. Work on the library was begun in 1523 by Pope Clement VII, Giulio di Medici, as a monument to his uncle, Lorenzo di Medici. The library was one of the few successes of Clement's disastrous reign, characterized as it was by bad political decisions. In the Timaeus panel of the library, Michelangelo, the designer of the library, used the number relations (the scale) of the lambda figure which had previously been used as the musical system studied by Pythagoras. Kappraff used the 'lambda triangle' in Table 1 'found in Plato's 'Timaeus' and referred to there as the World Soul'. Strictly speaking, the lambda diagram displayed in Table 1 is that given in Taylor but with the empty space between the two slanting lines / (hence the designation 'lambda') filled in a methodical and obvious way. Plato himself does not appear to have used the lambda figure as such though he used the two generating scales 1, 2, 4, 8 and 1, 3, 9, 27 shown by the slanting lines to describe the creation by the Demiurge of the World Soul. These scales are represented linearly (essentially in one line) in the commentary on the 'Timaeus'.

It is well-known how to compute the structure of the second homotopy group of a space, X, as a module over the fundamental group π₁X, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, π₃X as a module over π₁X. Moreover, we determine π₃X as an extension of π₁X-modules derived from Whitehead's Certain Exact Sequence. Our method is based on the theory of quadratic modules. Explicit computations are carried out for pseudo-projective 3-spaces X=S¹Ue²Ue³ consisting of exactly one cell in each dimension ≤ 3.