The notion of a bilattice was introduced by Shulman. A bilattice is a subspace analogue for a lattice. In this work the definition of hyperreflexivity for bilattices is given and studied. We give some general results concerning this notion. To a given lattice L we can construct the bilattice \sum {_L}. Similarly, having a bilattice Σ we may consider the lattice \mathcal{L}_\Sigma . In this paper we study the relationship between hyperreflexivity of subspace lattices and of their associated bilattices. Some examples of hyperreflexive or not hyperreflexive bilattices are given., Kamila Kliś-Garlicka., and Obsahuje seznam literatury
The smoothed aggregation method has became a widely used tool for solving the linear systems arising by the discretization of elliptic partial differential equations and their singular perturbations. The smoothed aggregation method is an algebraic multigrid technique where the prolongators are constructed in two steps. First, the tentative prolongator is constructed by the aggregation (or, the generalized aggregation) method. Then, the range of the tentative prolongator is smoothed by a sparse linear prolongator smoother. The tentative prolongator is responsible for the approximation, while the prolongator smoother enforces the smoothness of the coarse-level basis functions., Jan Brousek, Pavla Fraňková, Petr Vaněk., and Obsahuje seznam literatury
We consider the class H0 of sense-preserving harmonic functions f = h + \bar g defined in the unit disk |z| < 1 and normalized so that h(0) = 0 = h′(0) − 1 and g(0) = 0 = g′(0), where h and g are analytic in the unit disk. In the first part of the article we present two classes PH0(α) and GH0(β) of functions from H0 and show that if f \in PH0(α) and F \in GH0(β), then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial sums) {s_{n,n}}\left( f \right)\left( z \right) = {s_n}\left( h \right)\left( z \right) + \overline {{s_n}\left( g \right)\left( z \right)} , where f = h + \bar g \in H0, sn(h) and sn(g) denote the n-th partial sums of h and g, respectively. We prove, among others, that if f = h + \bar g \in H0 is a univalent harmonic convex mapping, then sn,n(f) is univalent and close-to-convex in the disk |z| < 1/4 for n ≥ 2, and sn,n(f) is also convex in the disk |z| < 1/4 for n ≥ 2 and n ≠ 3. Moreover, we show that the section s3,3(f) of f \in CH0 is not convex in the disk |z| < 1/4 but it is convex in a smaller disk., Liulan Li, Saminathan Ponnusamy., and Obsahuje seznam literatury
Let $\theta\in(0,1)$, $\lambda\in[0,1)$ and $p,p_0,p_1\in(1,\infty]$ be such that ${(1-\theta)}/{p_0}+{\theta}/{p_1}=1/p$, and let $\varphi, \varphi_0, \varphi_1 $ be some admissible functions such that $\varphi, \varphi_0^{p/{p_0}}$ and $\varphi_1^{p/{p_1}}$ are equivalent. We first prove that, via the $\pm$ interpolation method, the interpolation $\langle L^{p_0),\lambda}_{\varphi_0}(\mathcal{X}), L^{p_1),\lambda}_{\varphi_1}(\mathcal{X}), \theta\rangle$ of two generalized grand Morrey spaces on a quasi-metric measure space $\mathcal{X}$ is the generalized grand Morrey space $L^{p),\lambda}_{\varphi}(\mathcal{X})$. Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces., Yi Liu, Wen Yuan., and Obsahuje bibliografické odkazy
We investigate the invariant rings of two classes of finite groups $G\leq{\rm GL}(n,F_q)$ which are generated by a number of generalized transvections with an invariant subspace $H$ over a finite field $F_q$ in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with different invariant subspaces which have the same dimension. We provide a explicit classification of these groups and calculate their invariant rings., Xiang Han, Jizhu Nan, Chander K. Gupta., and Obsahuje bibliografické odkazy
We investigate isometric composition operators on the weighted Dirichlet space {D_\alpha } with standard weights {(1 - {\left| z \right|^2})^\alpha },\alpha > - 1 . The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space D. We solve some of these but not in general. We also investigate the situation when {D_\alpha } is equipped with another equivalent norm., Shi-An Han, Ze-Hua Zhou., and Obsahuje seznam literatury
Let L := −Δ + V be a Schrödinger operator on\mathbb{R}^{n} with n\geqslant 3 and V\geqslant 0 satisfying \Delta ^{-1}V\in L^{\infty }(\mathbb{R}^{n}). Assume that φ: {R}^{n} × [0,∞) → [0,∞) is a function such that φ(x,,) is an Orlicz function, φ(•, t) \in A_{\infty }({R}^{n}) (the class of uniformly Muckenhoupt weights). Let w be an L-harmonic function on {R}^{n} with 0< C_{1}\leq w\leq C_{2}, where C_{1} and C_{2} are positive constants. In this article, the author proves that the mapping H_{\phi ,L} (\mathbb{R}^n ) \mathrel\backepsilon f \mapsto wf \in H_\phi (\mathbb{R}^n ) is an isomorphism from the Musielak-Orlicz-Hardy space associated with L,H_{\phi ,L} (\mathbb{R}^n ), to the Musielak-Orlicz-Hardy space H_\phi (\mathbb{R}^n ) under some assumptions on φ. As applications, the author further obtains the atomic and molecular characterizations of the space H_{\phi ,L} (\mathbb{R}^n ) associated with w, and proves that the operator {( - \Delta )^{ - 1/2}}{L^{1/2}} is an isomorphism of the spaces H_{\phi ,L} (\mathbb{R}^n ) and H_\phi (\mathbb{R}^n ). All these results are new even when φ(x, t) ≔ t^{p}, for all x \in \mathbb{R}^{n} and t \in [0,∞), with p ∞ (n/(n + μ_{0}), 1) and some μ_{0} \in (0, 1]., Sibei Yang., and Obsahuje seznam literatury
The maximum nullity over a collection of matrices associated with a graph has been attracting the attention of numerous researchers for at least three decades. Along these lines various zero forcing parameters have been devised and utilized for bounding the maximum nullity. The maximum nullity and zero forcing number, and their positive counterparts, for general families of line graphs associated with graphs possessing a variety of specific properties are analysed. Building upon earlier work, where connections to the minimum rank of line graphs were established, we verify analogous equations in the positive semidefinite cases and coincidences with the corresponding zero forcing numbers. Working beyond the case of trees, we study the zero forcing number of line graphs associated with certain families of unicyclic graphs., Shaun Fallat, Abolghasem Soltani., and Obsahuje seznam literatury
We give a classification of all linear natural operators transforming p-vectors (i.e., skew-symmetric tensor fields of type (p, 0)) on n-dimensional manifolds M to tensor fields of type (q, 0) on TAM, where TA is a Weil bundle, under the condition that p ≥ 1, n ≥ p and n ≥ q. The main result of the paper states that, roughly speaking, each linear natural operator lifting p-vectors to tensor fields of type (q, 0) on TA is a sum of operators obtained by permuting the indices of the tensor products of linear natural operators lifting p-vectors to tensor fields of type (p, 0) on TA and canonical tensor fields of type (q − p, 0) on TA., Jacek Dębecki., and Obsahuje seznam literatury