Fractional derivatives: probability interpretation and frequency response of rational approximations

The theory of fractional calculus (FC) is a useful mathematical tool in many applied sciences. Nevertheless, only in the last decades researchers were motivated for the adoption of the FC concepts. There are several reasons for this state of affairs, namely the co-existence of different definitions and interpretations, and the necessity of approximation methods for the real time calculation of fractional derivatives (FDs). In a first part, this paper introduces a probabilistic interpretation of the fractional derivative based on the Grünwald-Letnikov definition. In a second part, the calculation of fractional derivatives through Padé fraction approximations is analyzed. It is observed that the probabilistic interpretation and the frequency response of fraction approximations of FDs reveal a clear correlation between both concepts.

Simultaneous decisions or leadership in an international competition

In this paper, we study an international market with demand uncertainty. The model has two stages. In the first stage, the home government chooses an import tariff to maximize the revenue. Then, the firms engage in a Cournot or in a Stackelberg competition. The uncertainty is resolved between the decisions made by the home government and by the firms. We compare the results obtained in the three different ways of moving on the decision make of the firms.

Writing and living on the stage of history

This article studies the intercultural trajectory of a Portuguese female aristocrat of the eighteenth to nineteenth centuries. Her trajectory of intercultural transition from a Portuguese provincial lady into an independent owner of a sugar mill in tropical Bahia is documented through family letters, which provide a polyphonic representation of a movement of personal, family, and social transculturation over almost two decades. Maria Bárbara began her journey between cultures as a simple spectator-reader, progressively becoming a commentator-actor-protagonist-author in society, in politics, and in history. These letters function as a translation that is sometimes consecutive, other times simultaneous, of the events lived and witnessed. This concept of intercultural translation is based on the theories of Boaventura de Sousa Santos (2006, 2008), who argues that cultural differences imply that any comparison has to be made using procedures of proportion and correspondence which, taken as a whole, constitute the work of translation itself. These procedures construct approximations of the known to the unknown, of the strange to the familiar, of the ‘other’ to the ‘self’, categories which are always unstable. Likewise, this essay explores the unstable contexts of its object of study, with the purpose of understanding different rationalities and worldviews.