A phytosociological synthesis of weed vegetation of southern Moravia (Czech Republic) was performed using the Braun-Blanquet approach. Gradsect sampling, i.e. a priori stratified selection of sampling sites, was used for the field survey. Using this method, 115 quadrants of the Central European mapping grid (6 × 5.6 km) were chosen. Three hundred and ten relevés recorded in 1997–2002 were classified, based on the Cocktail method, which defines sociological species groups and then creates formal definitions of vegetation units. In total, nine associations of the class Stellarietea mediae were distinguished in southern Moravia. Three associations were included in the alliance Caucalidion lappulae (Lathyro-Adonidetum, Euphorbio-Melandrietum, Veronicetum hederifoliotriphylli) and three in the alliance Scleranthion annui (Aphano-Matricarietum, SperguloScleranthetum, Erophilo-Arabidopsietum). For each of the alliances Veronico-Euphorbion, Spergulo-Oxalidion and Panico-Setarion one association was distinguished, respectively, SetarioFumarietum, Panico-Chenopodietum polyspermi and Echinochloo-Setarietum pumilae . Species composition of these associations is documented in a synoptic table. Their structure, ecology, and distribution are commented.
Given $\alpha $, $0<\alpha <n$, and $b\in {\mathrm BMO}$, we give sufficient conditions on weights for the commutator of the fractional integral operator, $[b,I_\alpha ]$, to satisfy weighted endpoint inequalities on $\mathbb{R}^n$ and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on $\mathbb{R}^n$.
Let $m$ be a positive integer, $0<\alpha <mn$, $\vec {b}=(b_{1},\cdots ,b_{m})\in {\rm BMO}^m$. We give sufficient conditions on weights for the commutators of multilinear fractional integral operators $\Cal {I}^{\vec {b}}_{\alpha }$ to satisfy a weighted endpoint inequality which extends the result in D. Cruz-Uribe, A. Fiorenza: Weighted endpoint estimates for commutators of fractional integrals, Czech. Math. J. 57 (2007), 153–160. We also give a weighted strong type inequality which improves the result in X. Chen, Q. Xue: Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl., 362, (2010), 355–373.
First, some classic properties of a weighted Frobenius-Perron operator P u ϕ on L 1 (Σ) as a predual of weighted Koopman operator W = uUϕ on L∞(Σ) will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of P u ϕ under certain conditions
Generalised halfspace depth function is proposed. Basic properties of this depth function including the strong consistency are studied. We show, on several examples that our depth function may be considered to be more appropriate for nonsymetric distributions or for mixtures of distributions.
In this paper we study integral operators of the form \[ Tf(x)=\int | x-a_1y|^{-\alpha _1}\dots | x-a_my|^{-\alpha _m}f(y)\mathrm{d}y,
\] $\alpha _1+\dots +\alpha _m=n$. We obtain the $L^p(w)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\ge 1$ such that $w( a_ix) \le cw( x)$ for a.e. $x\in \mathbb R^n$, $1\le i\le m$. Moreover, we prove $\Vert Tf\Vert _{{\mathrm BMO}}\le c\Vert f\Vert _\infty $ for a wide family of functions $f\in L^\infty ( \mathbb R^n)$.