This Thesis expands the theoretical research done in the area of evolutionary algorithms. The (1+1)EA is a simple algorithm which allows to gain some insight in the behaviour of these randomized search heuristics. This work shows ways to possible improve on existing bounds. The general good runtime of the algorithm on linear functions is also proven for classes of quadratic functions. These classes are defined by the relative size of the quadratic and the linear weights. One proof of the paper looks at a worst case algorithm which always shows a worst case behaviour than many other functions. This algorithm is used as an upper bound for a lot of different classes. / Department of Computer Science

We call two increasing sequences of positive integers {aᵢ}, {b(j)} a "Ramanujan Pair" if the following identity holds: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). The goal of this investigation is to determine all Ramanujan Pairs. Although this goal was not completely reached, we have determined all pairs for which the first term a₁ ≥ 5 and have proved that any Ramanujan Pair which begins with a₁ = m, where 1 ≤ m ≤ 4, aside from the known pairs, would have to branch off the first Euler identity with {aᵢ} = {i + m - 1}, {b(j)} = {j m}. A great deal of computing was done to discover the proofs given here. The search methods used and their programs are discussed in detail. Beyond these results, we have found all finite Ramanujan Pairs. Finally, modular Ramanujan Pairs (where the coefficients in the identity are reduced modulo n) are also examined.

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