Let $\tilde{f}$, $\tilde{g}$ be ultradistributions in $\mathcal Z^{\prime }$ and let $\tilde{f}_n = \tilde{f} * \delta _n$ and $\tilde{g}_n = \tilde{g} * \sigma _n$ where $\lbrace \delta _n \rbrace $ is a sequence in $\mathcal Z$ which converges to the Dirac-delta function $\delta $. Then the neutrix product $\tilde{f} \diamond \tilde{g}$ is defined on the space of ultradistributions $\mathcal Z^{\prime }$ as the neutrix limit of the sequence $\lbrace {1 \over 2}(\tilde{f}_n \tilde{g} + \tilde{f} \tilde{g}_n)\rbrace $ provided the limit $\tilde{h}$ exist in the sense that \[ \mathop {\mathrm N\text{-}lim}_{n\rightarrow \infty }{1 \over 2} \langle \tilde{f}_n \tilde{g} +\tilde{f} \tilde{g}_n, \psi \rangle = \langle \tilde{h}, \psi \rangle \] for all $\psi $ in $\mathcal Z$. We also prove that the neutrix convolution product $f \mathbin {\diamondsuit \!\!\!\!*\,}g$ exist in $\mathcal D^{\prime }$, if and only if the neutrix product $\tilde{f} \diamond \tilde{g}$ exist in $\mathcal Z^{\prime }$ and the exchange formula \[ F(f \mathbin {\diamondsuit \!\!\!\!*\,}g) = \tilde{f} \diamond \tilde{g} \] is then satisfied.
In this paper we use a duality method to introduce a new space of generalized distributions. This method is exactly the same introduced by Schwartz for the distribution theory. Our space of generalized distributions contains all the Schwartz distributions and all the multipole series of physicists and is, in a certain sense, the smallest space containing all these series.
With climatic warming there is an expectation that phytophagous insects will increasingly use alien (non native) plants as nectar sources and larval host plants. Alien plant use is investigated in British butterflies. Butterflies are considered to be larval host plant specialists relative to their use of nectar plants. Supporting this view, use of alien plants as nectar sources (50.1%, 27 novel plant families) is almost twice that of their use as larval host plants (21.6%; three novel plant families). Some 80% of the variation in percent alien nectar plant use is accounted against 30% of that for percent alien host plant use. The key variable accounting for alien plant use is butterfly mobility. Other prominent variables that facilitate access to alien nectar plants are southern distributions, longer adult life span, host plants in garden biotopes. A different set of variables additionally account for alien host plant exploitation (% alien host plant use: woody host plants; number of alien host plants: polyphagy; greater abundance of native host plants in gardens). Although threatened butterfly species do not depend on alien plants, this may well reflect on specialisation in resource use accompanying habitat fragmentation and an inability to use novel resources that are becoming increasingly available. Detailed study of alien resources is advocated to assess the importance of alien plant resources for phytophagous insects.