It is proved that a Köthe sequence space is weakly orthogonal if and only if it is order continuous. Criteria for weak property (${\mathbf \beta }$) in Orlicz sequence spaces in the case of the Luxemburg norm as well as the Orlicz norm are given.
The problem of pole assignment by state feedback in the class of non-square linear systems is considered in the paper. It is shown that the problem is solvable under the assumption of weak regularizability, a newly introduced concept that can be viewed as a generalization of the regularizability of square systems. Necessary conditions of solvability for the problem of pole assignment are established. It is also shown that sufficient conditions can be derived in some special cases. Some conclusions and prospects for further studies are drawn in the last section.
It is proved that for a zero-dimensional space $X$, the function space $C_p(X,2)$ has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if $X$ is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space $E$, the function space $C_p(X,E)$ is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial positive answer to a question posed by van Mill and Wattel.
In this paper we consider the following Dirichlet problem for elliptic systems: $$ \begin {aligned} \overline {DA(x,u(x),Du(x))}=&B(x,u(x),Du(x)),\quad x\in \Omega ,\cr u(x)=&0,\quad x\in \partial \Omega , \end {aligned} $$ where $D$ is a Dirac operator in Euclidean space, $u(x)$ is defined in a bounded Lipschitz domain $\Omega $ in $\mathbb {R}^{n}$ and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned systems in the space $W_{0}^{1,p(x)}(\Omega , {\rm C}\ell _{n})$ under appropriate assumptions.
Existence of a weak solution to the $n$-dimensional system of stochastic differential equations driven by a fractional Brownian motion with the Hurst parameter $H\in (0,1)\setminus \{\frac 12\}$ is shown for a time-dependent but state-independent diffusion and a drift that may by split into a regular part and a singular one which, however, satisfies the hypotheses of the Girsanov Theorem. In particular, a stochastic nonlinear oscillator driven by a fractional noise is considered.
In this paper we prove that the collection of all weakly distributive lattice ordered groups is a radical class and that it fails to be a torsion class.
The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces Lp(Rd) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x\in Rd > 1) on the variable Lebesgue spaces Lp(·)(Rd), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator Msγδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or
p- ≥ s), then the maximal operator Msγδ is bounded on the space Lp(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·)))., Kristóf Szarvas, Ferenc Weisz., and Obsahuje seznam literatury
The main results of this paper are that (1) a space $X$ is $g$-developable if and only if it is a weak-open $\pi $ image of a metric space, one consequence of the result being the correction of an error in the paper of Z. Li and S. Lin; (2) characterizations of weak-open compact images of metric spaces, which is another answer to a question in in the paper of Y. Ikeda, C. liu and Y. Tanaka.
Some sufficient conditions are provided that guarantee that the difference of a compact mapping and a proper mapping defined between any two Banach spaces over $\mathbb {K}$ has at least one zero. When conditions are strengthened, this difference has at most a finite number of zeros throughout the entire space. The proof of the result is constructive and is based upon a continuation method.
A dominating set $D\subseteq V(G)$ is a {\it weakly connected dominating set} in $G$ if the subgraph $G[D]_w=(N_G[D],E_w)$ weakly induced by $D$ is connected, where $E_w$ is the set of all edges having at least one vertex in $D$. {\it Weakly connected domination number} $\gamma _w(G)$ of a graph $G$ is the minimum cardinality among all weakly connected dominating sets in $G$. A graph $G$ is said to be {\it weakly connected domination stable} or just $\gamma _w$-{\it stable} if $\gamma _w(G)=\gamma _w(G+e)$ for every edge $e$ belonging to the complement $\overline G$ of $G.$ We provide a constructive characterization of weakly connected domination stable trees.