This paper describes an attempt at developing an evolutionary algorithm capable of solving non-trivial cases of integer factorisation, which are at the heart of security behind the modern public key cryptography systems. After reviewing previous work in integer factorisation and describing the developed genetic algorithm the paper addresses issues of convergence to a local maxima associated wit...

Let Qn(x) = ∑n i=0 Aix i be a random algebraic polynomial where the coefficients A0, A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆j = Aj − Aj−1, j = 0, 1, 2, · · · are independent, A−1 = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We study the asymptotic behaviour of the expected number of lo...

In this paper, we investigate the geometry of a general class of gradient flows with multiple local maxima. we decompose the underlying space into disjoint regions of attraction and establish the adjacency criterion. The criterion states a necessary and sufficient condition for two regions of attraction of stable equilibria to be adjacent. We then apply this criterion on a specific type of grad...

In the analysis of dynamic financial quantities such as stock prices, equity prices, etc., reasonable results are often obtained if we only consider local maxima (“peaks”) and local minima (“troughs”) and ignore all the other values. The empirical success of this strategy remains a mystery. In this paper, we provide a possible explanation for this success. 1 Formulation of the Problem Peak-and-...

CHE based architecture for GA, on a single COTS FPGA, designed to perform optimization of single variable functions is presented in this chapter. The presented architecture’s ability to optimize is tested and validated with three different single variable functions. They are, a linear function, a quadratic function with only one global maximum, and a cubic function with one global maxima and lo...

In 1965, Motzkin and Straus related global maximaof a certain quadratic program to the maximum clique size in a certain graph. We extend this result to relate strict local maxima of this program to certain maximal cliques, and certain maxima to non cliques. Our results are useful to a companion paper which employs this QP in a neural net model to nd large cliques in graphs.