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The development of industrial agriculture has enabled a sharp increase in food trade at the global scale. Worldwide trade underpins food security by distributing food surpluses to food deficient countries. The study of agricultural product flows can provide insights on the complex interactions between exporting and importing countries and the resulting network structures. Commercial partnerships between countries can be modelled using a complex network approach. Based on the detailed trade matrices from FAO covering the period from 1986 to 2013, we present an analysis of the world cereal trade in terms of weighted and directed networks. The network nodes are the countries and the links are the trades of agricultural products in mass. We reveal the changing topology and degree distribution of the world network during the studied period. We distinguish three entangled subnetwork structures when considering the temporal stability of the trades. The three subnetworks display distinct properties and a differential contribution in total trade. Trades of uninterrupted activity over the 28-year study period compose the backbone network which accounts for two thirds of all traded mass and is scale-free. Inversely, two thirds of the trades only have one or two consecutive years of activity and define the transient subnetwork which displays random growth and accounts for very little traded mass. The trades of intermediate duration display an exponential growth both in numbers and in traded mass and define the intermediate subnetwork. The topology of each subnetwork is a time invariant. The identification of invariant structures is a useful basis for developing prospective agri-food network modelling to assess their resilience to perturbations and shocks. [ABSTRACT FROM AUTHOR]

Black and Scholes (1973) proved that under certain assumptions about the market place, the value of a European option, as a function of the current value of the underlying asset and time, verifies a Cauchy problem. We give new conditions for the existence and uniqueness of the value of a European option by using semigroup theory. For this, we choose a suitable space that verifies some conditions, what allows us that the operator that appears in the Cauchy problem is the infinitesimal generator of a C0-semigroup T(t). Then we are able to guarantee the existence and uniqueness of the value of a European option and we also achieve an explicit expression of that value. [ABSTRACT FROM AUTHOR]

Recent patterns of fertility in Europe show marked differences between countries. Recent United Kingdom and Irish fertility curves show 'distortions' in terms of a 'bulge' in early age fertility, distinct from the smoother curves of other European countries. These patterns may not be adequately described by mathematical functions used by previous studies to model fertility curves. A mixture model with two component distributions may be more appropriate. The suitability of the simple and mixture Hadwiger functions is examined in relation to the fertility curves of a number of European countries. While the simple Hadwiger model fits recent period age-specific fertility distributions for some countries, others which display a 'bulge' in early age fertility require a mixture Hadwiger model. Some of the parameters of the Hadwiger models appear to be related to familiar demographic indices. The simple and mixture Hadwiger models appear useful in describing and comparing fertility patterns across European countries. [ABSTRACT FROM AUTHOR]

New Zealand is young in length of European settlement. Although the sites of one or two towns are more than 100 years old, all urban development has taken place in less than a century. Because of the apparent uniformity among New Zealand towns an analysis of the functions which they perform becomes of particular importance as a means of differentiating dearly among them. As a result this aspect of urban geography probably takes on an even greater significance in New Zealand than in most other countries. It is therefore of interest to distinguish the kinds of New Zealand towns on the basis of their functions and to classify the urban areas of this small country.