For n=2m\geqslant 4, let \Omega\in \mathbb{R}^{n} be a bounded smooth domain and N\subset \mathbb{R}^{L} a compact smooth Riemannian manifold without boundary. Suppose that \left \{ uk \right \}\in W^{m,2}\left ( \Omega ,N \right ) is a sequence of weak solutions in the critical dimension to the perturbed m-polyharmonic maps \frac{{\text{d}}}{{{\text{dt}}}}\left| {_{t = 0}{E_m}({\text{II}}(u + t\xi )) = 0} \right with Ωk → 0 in W^{m,2}\left( \Omega ,N \right )* and {u_k} \rightharpoonup u weakly in W^{m,2}\left( \Omega ,N \right ). Then u is an m-polyharmonic map. In particular, the space of m-polyharmonic maps is sequentially compact for the weak- W^{m,2} topology., Shenzhou Zheng., and Obsahuje seznam literatury
A new form of α-compactness is introduced in L-topological spaces by α-open L-sets and their inequality where L is a complete de Morgan algebra. It doesn’t rely on the structure of the basis lattice L. It can also be characterized by means of α-closed L-sets and their inequality. When L is a completely distributive de Morgan algebra, its many characterizations are presented and the relations between it and the other types of compactness are discussed. Countable α-compactness and the α-Lindelöf property are also researched.
Let $\varphi $ and $\psi $ be holomorphic self-maps of the unit disk, and denote by $C_\varphi $, $C_\psi $ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_\varphi -C_\psi $ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.
The follow up research into the IHDS process was carried out with a Couette device. The outcome of this study provides a comprehensive understanding of the effect that both the agitation intensity and the agitation time have on the kinetics and the mechanism of the aggregation process. The results obtained confirm the very favourable influence of high agitation intensity for the formation of more compact and dense aggregates than those formed by the accustomed flocculation conditions with low agitation intensity. This research also proved that the agitation intensity and time are the inherent means profoundly influencing the properties of the resultant aggregates such as their size, shape, density and homogeneity. Further, it was confirmed that the aggregation process passes through a minimum. Furthermore, it was verified that the aggregation process takes place in four consecutive phases, namely a) the phase of formation, b) the phase of compaction, c) the phase of a steady (equilibrium) state and d) most probably the phase of inner restructuring. The pattern of the aggregates development in these phases remains the same irrespective of the magnitude of the velocity gradient applied but the time at which these phases are completed is velocity gradient dependent. Last but not least this study proved that the dimensionless product Ca = G T = const. has no general validity. and Výzkum tvorby agregátů metodou IHDS pokračoval s Couettovým typem flokulačního zařízení. Výsledek této studie umožňuje porozumět vlivu intenzity a času míchání na kinetiku a mechanismus agregačního procesu. Získané výsledky potvrzují velmi příznivý vliv vysoké intenzity míchání na tvorbu kompaktnějších a hustších agregátů než jsou ty vytvořené běžnými flokulačními podmínkami s nízkou intenzitou míchaní. Tento výzkum také potvrdil, že intenzita míchání ve spojení s časem je přirozeným prostředkem výrazně ovlivňujícím vlastnosti výsledných agregátu jako je jejich rozměr, tvar, hustota a homogennost. Dále bylo potvrzeno, že agregační proces probíhá ve čtyřech následných fázích. Jedná se o: a) fázi tvorby, b) fázi zhutňování, c) fázi rovnovážného stavu a d) s největší pravděpodobností fázi vnitřní restrukturalizace. Struktura agregátů vytvořených v jednotlivých fázích je obdobná bez ohledu na velikost použitého rychlostního gradientu, ale čas potřebný k ukončení těchto fází je závislý na velikosti použitého rychlostního gradientu. V neposlední řadě tato studie potvrdila, že bezrozměrné kritérium Ca = G T = = konst. nemá všeobecnou platnost.
One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities.
We prove that compactness of the canonical solution operator to $\bar \partial $ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[\mathcal P,\bar M]$ defined on the whole $L^2_{(0,1)}(\Omega ),$ where $\bar M$ is the multiplication by $\bar z$ and $\mathcal P $ is the orthogonal projection of $L^2_{(0,1)}(\Omega )$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar \partial $-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by $z $ and $\bar z$.