Sampling of insect communities is very challenging and for reliable interpretation of results the effects of different sampling protocols and data processing on the results need to be fully understood. We compared three different commonly used methods for sampling forest beetles, freely hanging flight-intercept (window) traps (FWT), flight-intercept traps attached to trunks (TWT) and pitfall traps placed in the ground (PFT), in Scots pine dominated boreal forests in eastern Finland. Using altogether 960 traps, forming 576 sub-samples, at 24 study sites, 59760 beetles belonging to 814 species were collected over a period of a month. All of the material was identified to species, with the exception of a few species pairs, to obtain representative data for analyses. Four partly overlapping groups were used in the analyses: (1) all, (2) saproxylic, (3) rare and (4) red-listed species. In terms of the number of species collected TWTs were the most effective for all species groups and the rarer species the species group composed of (groups 1-2-3-4) the larger were the differences between the trap types. In particular, the TWTs caught most red-listed species. However, when sample sizes were standardized FWTs and TWTs caught similar number of species of all species groups. PFTs caught fewer species of all species groups, whether the sample sizes were standardized or not. In boreal forests they seem to be unsuitable for sampling saproxylic, rare and red-listed species. However, the PFTs clearly sampled different parts of species assemblages than the window traps and can be considered as a supplementary method. The abundance distribution of saproxylic species was truncated lognormal in TWT and pooled material, whereas unclassified material failed to reveal lognormal distribution in all the trap types and pooled material. The results show that even in boreal forests sample sizes of at least thousands, preferably tens of thousands of individuals, collected by a high number of traps are needed for community level studies. Relevant ecological classification of material is also very important for reliable comparisons. Differences in the performance of trap types should be considered when designing a study, and in particular when evaluating the results.
The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper presents two methods to handle the algebraic condition: the first one is based on iterative minimization of a cost functional defined as the integral of the square of the algebraic expression to be equal to zero. The second method converts the algebraic-differential equation into a singularly perturbed system of partial differential equations only. Both methods are compared and the simulation results are presented including on-line control implementation to some practically motivated laboratory models.
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.
We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence $\mathbb{L}_1$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/(p)$. Further, we show that $(\mathbb{Z}, \mathbb{L}^\ast _1)$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.