In this paper we prove that the lateral completion of a projectable lattice ordered group is strongly projectable. Further, we deal with some properties of Specker lattice ordered groups which are related to lateral completeness and strong projectability.
A reflexive topological group $G$ is called strongly reflexive if each closed subgroup and each Hausdorff quotient of the group $G$ and of its dual group is reflexive. In this paper we establish an adequate concept of strong reflexivity for convergence groups. We prove that complete metrizable nuclear groups and products of countably many locally compact topological groups are BB-strongly reflexive.
This paper introduces the notion of a strong retract of an algebra and then focuses on strong retracts of unary algebras. We characterize subuniverses of a unary algebra which are carriers of its strong retracts. This characterization enables us to describe the poset of strong retracts of a unary algebra under inclusion. Since this poset is not necessarily a lattice, we give a necessary and sufficient condition for the poset to be a lattice, as well as the full description of the poset.
An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A \oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\begin{pmatrix} a&b\\ *&* \end{pmatrix} \in M_2(S)$. The dual assertions are also proved.
We study singular boundary value problems with mixed boundary conditions of the form (p(t)u ' ) ' + p(t)f(t, u, p(t)u ' ) = 0, lim t→0+ p(t)u ' (t) = 0, u(T) = 0, where [0, T] ⊂ . We assume that D ⊂ R 2 , f satisfies the Carathéodory conditions on (0, T) × D, p ∈ C[0, T] and 1/p need not be integrable on [0, T]. Here f can have time singularities at t = 0 and/or t = T and a space singularity at x = 0. Moreover, f can change its sign. Provided f is nonnegative it can have even a space singularity at y = 0. We present conditions for the existence of solutions positive on [0, T).
Let $(X,\Vert \cdot \Vert _X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma $-finite measure space $(Ø,\Sigma ,\mu )$. Let $(X)$ be the space of all strongly $\Sigma $-measurable functions $f\: Ø\rightarrow X$ such that the scalar function ${\widetilde{f}}$, defined by ${\widetilde{f}}(ø)=\Vert f(ø)\Vert _X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E(X)$. In particular, the strong topology $\beta (E(X), E(X)^\sim _n)$ ($E(X)^\sim _n=$ the order continuous dual of $E(X)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.
In max-min algebra the standard pair of operations plus and times is replaced by the pair of operations maximum and minimum, respectively. A max-min matrix A is called strongly robust if the orbit x,A⊗x,A2⊗x,… reaches the greatest eigenvector with any starting vector. We study a special type of the strong robustness called the strong \textit{\textbf{X}}-robustness, the case that a starting vector is limited by a lower bound vector and an upper bound vector. The equivalent condition for the strong \textit{\textbf{X}}-robustness is introduced and efficient algorithms for verifying the strong \textit{\textbf{X}}-robustness is described. The strong \textit{\textbf{X}}-robustness of a max-min matrix is extended to interval vectors \textit{\textbf{X}} and interval matrices \textit{\textbf{A}} using for-all-exists quantification of their interval and matrix entries. A complete characterization of AE/EA strong \textit{\textbf{X}}-robustness of interval circulant matrices is presented.
Let $\mathcal {W}$ be a self-orthogonal class of left $R$-modules. We introduce a class of modules, which is called strongly $\mathcal {W}$-Gorenstein modules, and give some equivalent characterizations of them. Many important classes of modules are included in these modules. It is proved that the class of strongly $\mathcal {W}$-Gorenstein modules is closed under finite direct sums. We also give some sufficient conditions under which the property of strongly $\mathcal {W}$-Gorenstein module can be inherited by its submodules and quotient modules. As applications, many known results are generalized.
We call a sequence $(T_n)$ of measure preserving transformations strongly mixing if $P(T_n^{-1}A\cap B)$ tends to $P(A)P(B)$ for arbitrary measurable $A$, $B$. We investigate whether one can pass to a suitable subsequence $(T_{n_k})$ such that $\frac{1}{K} \sum _{k=1}^K f(T_{n_k}) \longrightarrow \int f \mathrm{d}P$ almost surely for all (or “many”) integrable $f$.