Hardy type derivations on fields of exponential logarithmic series

We consider the valued field \mathds{K}:=\mathbb{R}((\Gamma)) of formal series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma>$). We investigate how to endow \mathds{K} with a logarithm $l$, which satisfies some natural properties such as commuting with infinite products of monomials. In the article "Hardy type derivations on generalized series fields", we study derivations on \mathds{K}. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In her monograph "Ordered exponential fields", the first author described the exponential closure \mathds{K}^{\rm{EL}} of (\mathds{K},l). Here we show how to extend such a log-compatible derivation on \mathds{K} to \mathds{K}^{\rm{EL}}.Comment: 25 page

Exponentiation in power series fields

We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable

Revision of the South African endemic bee genus Redivivoides Michener, 1981 (Hymenoptera: Apoidea: Melittidae)

The South African endemic bee genus Redivivoides Michener, 1981 is revised and redefined. The genus comprises seven species, six of which are described here as new: Redivivoides capensis sp. nov. ♀♂, R. eardleyi sp. nov. ♀, R. kamieskroonensis sp. nov. ♀, R. karooensis sp. nov. ♀♂, R. namaquaensis sp. nov. ♀♂ and R. variabilis sp. nov. ♀♂. A key to species is provided

Mechanisms in Dynamically Complex Systems

In recent debates mechanisms are often discussed in the context of ‘complex systems’ which are understood as having a complicated compositional structure. I want to draw the attention to another, radically different kind of complex system, in fact one that many scientists regard as the only genuine kind of complex system. Instead of being compositionally complex these systems rather exhibit highly non-trivial dynamical patterns on the basis of structurally simple arrangements of large numbers of non-linearly interacting constituents. The characteristic dynamical patterns in what I call “dynamically complex systems” arise from the interaction of the system’s parts largely irrespective of many properties of these parts. Dynamically complex systems can exhibit surprising statistical characteristics, the robustness of which calls for an explanation in terms of underlying generating mechanisms. However, I want to argue, dynamically complex systems are not sufficiently covered by the available conceptions of mechanisms. I will explore how the notion of a mechanism has to be modified to accommodate this case. Moreover, I will show under which conditions the widespread, if not inflationary talk about mechanisms in (dynamically) complex systems stretches the notion of mechanisms beyond its reasonable limits and is no longer legitimate