/library/oar/handle/123456789/23565 2025-11-07T19:06:53Z /library/oar/handle/123456789/24501 Title: Crystallography and symmetry groups Abstract: Crystals are assemblages of very small basic units of matter repeated periodically in 3 dimensions. The connection with group theory is that each pattern can be characterized by its symmetry group. It turns out that there are only 230 of these so-called crystallographic space groups amongst which are 22, which crystallographers prefer to regard as distinct, but which, from an abstract point of view, form 11 pairs of isomorphic groups. Thus the space groups fall into 219 isomorphism classes. The enumeration of these space groups is built upon the 14 lattices determined by Bravais. Since the enumeration is quite complicated, we here look at some of the corresponding ideas involved in the analogous 2-dimensional problem where 17 groups, no two of which are isomorphic, arise. First recall that an isometry of the plane R2 is a distance- preserving mapping of R onto itself. Amongst such isometries are translations, rotations, reflections (in lines) and glide reflections. The latter being the result of an ordinary reflection in some line 1 followed by a translation parallel to 1. Figure 1 adequately describes these movements. 2003-01-01T00:00:00Z /library/oar/handle/123456789/24500 Title: Random numbers Abstract: It is very difficult to define 'randomness'. However a simple definition of Random Numbers would be as follows: Random numbers can be defined as a sequence of numbers which do not follow a regular pattern. Thus one cannot possibly guess the value of the next number in the sequence, as this may be bigger, equal or smaller than its previous values or set of values. An example of a random number sequence is: 1, 7, 13, 24, 8, 3, 6, 50, 39, 86, 87, 2, ... We describe how random sequences may be generated, and tested for randomness. 2003-01-01T00:00:00Z /library/oar/handle/123456789/24499 Title: On perfect numbers Abstract: A Perfect number is a number which is the sum of its proper divisors. A Mersenne Prime is a prime number of the form 2k - 1. We discuss a relation between these two types of numbers and deduce various properties of perfect numbers. 2003-01-01T00:00:00Z /library/oar/handle/123456789/24498 Title: Interlacing and carbon balls Abstract: The Interlacing Theorem gives a relation among the eigenvalues of a n x n matrix A and those of a (n -1) x (n -1) principal submatrix. We deduce the Generalized Interlacing theorem which interlaces the eigenvalues of a k x k principal submatrix of A with those of A. We apply this theorem to the hypothetical Carbon ball C40 which has two dodecahedral 6 pentagon caps. 2003-01-01T00:00:00Z