A matrix A in (max,min)-algebra (fuzzy matrix) is called weakly robust if Ak⊗x is an eigenvector of A only if x is an eigenvector of A. The weak robustness of fuzzy matrices are studied and its properties are proved. A characterization of the weak robustness of fuzzy matrices is presented and an O(n2) algorithm for checking the weak robustness is described.
We characterize Weilian prolongations of natural bundles from the viewpoint of certain recent general results. First we describe the iteration $F(EM)$ of two natural bundles $E$ and $F$. Then we discuss the Weilian prolongation of an arbitrary associated bundle. These two auxiliary results enables us to solve our original problem.
The σ-finiteness of a variational measure, generated by a real valued function, is proved whenever it is σ-finite on all Borel sets that are negligible with respect to a σ-finite variational measure generated by a continuous function.
During the last few decades the tree line has shifted upward on Mediterranean mountains. This has resulted in a decrease in the area of the sub-alpine prairie habitat and an increase in the threat to strictly orophilous moths that occur there. This also occurred on the Pollino Massif due to the increase in temperature and decrease in rainfall in Southern Italy. We found that a number of moths present in the alpine prairie at 2000 m appear to be absent from similar habitats at 1500-1700 m. Some of these species are thought to be at the lower latitude margin of their range. Among them, Pareulype berberata and Entephria flavicinctata are estimated to be the most threatened because their populations are isolated and seem to be small in size. The tops of these mountains are inhabited by specialized moth communities, which are strikingly different from those at lower altitudes on the same massif further south. The majority of the species recorded in the sub-alpine prairies studied occur most frequently and abundantly in the core area of the Pollino Massif. Species and, as a consequence, communities found at high altitudes are highly vulnerable to climate warming, and need further monitoring.
We prove that any infinite-dimensional non-archimedean Fréchet space $E$ is homeomorphic to $D^{\mathbb{N}}$ where $D$ is a discrete space with $\mathop {\mathrm card}(D)=\mathop {\mathrm dens}(E)$. It follows that infinite-dimensional non-archimedean Fréchet spaces $E$ and $F$ are homeomorphic if and only if $\mathop {\mathrm dens}(E)= \mathop {\mathrm dens}(F)$. In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field $\mathbb{K}$ is homeomorphic to the non-archimedean Fréchet space $\mathbb{K}^{\mathbb{N}}$.
In the class of all exact torsion theories the torsionfree classes are cover (precover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of finite type is presented.
We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T[(\mathcal M_k,\theta _k)_{k =1}^{l}]$ with index $i(\mathcal M_k)$ finite are either $c_0$ or $\ell _p$ saturated for some $p$ and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i(\mathcal M_k)$ and the parameter $\theta _k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T[(\mathcal A_k,\theta _k)_{k = 1}^\infty ]$ in terms of the asymptotic behaviour of the sequence $\Bigl \Vert \sum _{i=1}^n e_i\Bigr \Vert $ where $(e_i)$ is the canonical basis.