Lyapunov exponents for products of complex Gaussian random matrices

The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = \Sigma^{1/2} G_i^{\rm c}$, where $\Sigma$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$ complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.Comment: 15 page

Can genetically-modified cotton contribute to sustainable development in Africa?

Genetically modified (GM) crops and sustainable development remain the foci of much media attention, especially given current concerns about a global food crisis. However, whilst the latter is embraced with enthusiasm by almost all groups, GM crops generate very mixed views. Some countries have welcomed GM, but others, notably those in Europe, adopt a cautious stance. This article aims to review the contribution that GM crops can make to agricultural sustainability in the developing world. Following brief reviews of both issues and their linkages, notably the pros and cons of GM cotton as a contributory factor in sustainability, a number of case studies from resourcepoor cotton farmers in Makhathini Flats, South Africa, is presented for a six-year period. Data on expenditure, productivity and income indicate that Bacillus thuringiensis (Bt) cotton is advantageous because it reduces costs, for example, of pesticides, and increases income, and the indications are that those benefits continued over at least the six years covered by the studies. There are repercussions of the additional income in the households; debts are reduced and money is invested in children's education and in the farms. However, in the general GM debate, the results show that GM crops are not miracle products which alleviate poverty at a stroke, but nor is there evidence that they will cause the scale of environmental damage associated with indiscriminate pesticide use. Indeed, for some GM antagonists, perhaps even the majority, such debates are irrelevant – the transfer of genes between species is unnatural and unethical. For them, GM crops will never be acceptable despite the evidence and pressure to increase world food production