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