In this paper, we consider the global existence, uniqueness and $L^{\infty }$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type $u_{t}-\mathop {\rm div}(|\nabla u|^{m-2}\nabla u)=u|u|^{\beta -1}\int _{\Omega } |u|^{\alpha } {\rm d} x$ in $\Omega \times (0,\infty )$ with zero Dirichlet boundary condition in $\partial \Omega $. Further, we obtain the $L^{\infty }$ estimate of the solution $u(t)$ and $\nabla u(t)$ for $t>0$ with the initial data $u_0\in L^q(\Omega )$ $(q>1)$, and the case $\alpha +\beta < m-1$.
In this paper $LJ$-spaces are introduced and studied. They are a common generalization of Lindelöf spaces and $J$-spaces researched by E. Michael. A space $X$ is called an $LJ$-space if, whenever $\lbrace A,B\rbrace $ is a closed cover of $X$ with $A\cap B$ compact, then $A$ or $B$ is Lindelöf. Semi-strong $LJ$-spaces and strong $LJ$-spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.
We characterize statistical independence of sequences by the $L^p$-discrepancy and the Wiener $L^p$-discrepancy. Furthermore, we find asymptotic information on the distribution of the $L^2$-discrepancy of sequences.
We show for $2\le p<\infty $ and subspaces $X$ of quotients of $L_{p}$ with a $1$-unconditional finite-dimensional Schauder decomposition that $K(X,\ell _{p})$ is an $M$-ideal in $L(X,\ell _{p})$.
We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $\mathcal K(X,Y)$ is an $M(r_1 r_2, s_1 s_2)$-ideal in the space of all continuous linear operators $\mathcal L(X,Y)$ whenever $\mathcal K(X,X)$ and $\mathcal K(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $\mathcal L(X,X)$ and $\mathcal L(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $\mathcal K(X,Y)$ in $\mathcal L(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results on $M$-ideals.
In analogy with effect algebras, we introduce the test spaces and $MV$-test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between $MV$-algebras and $MV$-test spaces.
In this paper, we study the existence of the $n$-flat preenvelope and the $n$-FP-injective cover. We also characterize $n$-coherent rings in terms of the $n$-FP-injective and $n$-flat modules.
We introduce the notion of weak dually residuated lattice ordered semigroups (WDRL-semigroups) and investigate the relation between $R_0$-algebras and WDRL-semigroups. We prove that the category of $R_0$-algebras is equivalent to the category of some bounded WDRL-semigroups. Moreover, the connection between WDRL-semigroups and DRL-semigroups is studied.
We observe that a separable Banach space $X$ is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if $\mathcal L(X,Y)$ is not reflexive for reflexive $X$ and $Y$ then $\mathcal L(X_1, Y)$ is is not reflexive for some $X_1\subset X$, $X_1$ having a basis.
The study deals with political activities of the Soviet Army in Czechoslovakia after the intervention on August 21, 1968, and its sympathizers from the ranks of the Communist Party of Czechoslovakia. The authoress examines the topic in the early stage of the so-called normalization (until the spring of 1970), focusing on the local level; however, she sets her research into a broader period context and derives general conclusions from its results. Although the offi cial agreement on the temporary stay of Soviet troops in the territory of Czechoslovakia declared that the Soviet Army should not interfere with domestic affairs of the Czechoslovak state, the Soviet leadership kept devising plans how to make use of the presence of Soviet troops for political purposes. Soviet offi cers participated in the dissemination of Soviet propaganda, established contacts with local anti-reform party offi cials, spoke at their forums, complained about hostile attitudes of Czechoslovak political bodies, and thus kept pressing for a legitimization of the political arrangements. The authoress shows that local pro-Soviet activists, who had maintained contacts with the Soviet Army from the very beginning and been taking over its political agenda, were playing a crucial role in the success of these efforts. In line with Soviet intentions, they were implementing the normalization process ''from below'',initiating purges in various organs, demanding the dismissal of offi cials protesting against presence of the Soviet Army, participating in the subsequent political vetting. They were actively pushing through a change of the offi cial approach to the Soviet Army and helped break its boycott by the Czechoslovak society, which had initially been almost unanimous. In doing so, they were making use of their personal contacts to organize manifestation ''friendship'' meetings and visits of Soviet soldiers to Czechoslovak schools and factories. The authoress analyzes the reasons of the attitude of these activists, most of whom came from the ranks of ''old'' (pre-war) and ''distinguished'' members of the Communist Party of Czechoslovakia, and illustrates the development outlined above by specifi c examples. By way of conclusion, she notes that, although different forms of the CzechoslovakSoviet ''friendship'' since 1968 are often viewed as mere formalistic acts without any deeper meaning at the level of ''lived'' experience, they were, from the viewpoint of the Soviet policy, well thought-out and centrally planned propagandistic activities which contributed to the promotion of the Soviet interpretation of the Prague Spring and the Soviet invasion and discredited its opponents. and Přeložila Blanka Medková