Equality of Ultradifferentiable Classes by Means of Indices of Mixed O-regular Variation

We characterize the equality between ultradifferentiable function classes defined in terms of abstractly given weight matrices and in terms of the corresponding matrix of associated weight functions by using new growth indices. These indices, defined by means of weight sequences and (associated) weight functions, are extending the notion of O-regular variation to a mixed setting. Hence we are extending the known comparison results concerning classes defined in terms of a single weight sequence and of a single weight function and give also these statements an interpretation expressed in O-regular variation.

Ограниченные композиционные операторы на весовых функциональных пространствах на единичном круге

We introduce a general class of weighted spaces $\calH(\beta)$ of holomorphic functions in the unit disk $\mathbb{D}$, which contains several classical spaces, such as Hardy space, Bergman space, Dirichlet space. We~characterize boundedness of composition operators $C_{\varphi}$ induced by affine and monomial symbols $\varphi$ on these spaces $\calH(\beta)$. We also establish a sufficient condition under which the operator $C_{\varphi}$ induced by the symbol $\varphi$ with relatively compact image $\varphi(\mathbb{D})$ in $\mathbb{D}$ is bounded on $\calH(\beta)$. Note that in the setting of $\calH(\beta)$, the characterizations of boundedness of composition operators $C_{\varphi}$ depend closely not only on functional properties of the symbols $\varphi$ but also on the behavior of the weight sequence $\beta$.

THE TRAPPING PROBLEM OF WEIGHTED (2,2)-FLOWER NETWORKS WITH THE SAME WEIGHT SEQUENCE

Intuitively, edge weight has an effect on the dynamical processes occurring on the networks. However, the theoretical research on the effects of edge weight on network dynamics is still rare. In this paper, we present two weighted network models by adjusting the matching relationship between weights and edges. Both network models are controlled by the weight factor [Formula: see text]. They have the same connection structure and weight sequence when [Formula: see text] is fixed. Based on their self-similar network structure, we study two types of biased walks with a trap. One is standard weight-dependent walk, while the other is mixed weight-dependent walk including both nearest-neighbor and next-nearest-neighbor jumps. For both weighted scale-free networks, we obtain exact solutions of the average trapping time (ATT) measuring the efficiency of the trapping process in both network models. Analyzing and comparing the obtained solutions, we find that the ATT is related to the walking rule and the spectral dimension of the fractal network, and not all ATT for the weighted networks are affected by the weight factor [Formula: see text]. In other words, not all weight adjustments can change the trapping efficiency in the network. We hope that the present findings could help us get deeper understanding about the influence factor of biased walk in complex systems.

RANDOM WALKS IN HETEROGENEOUS WEIGHTED PSEUDO-FRACTAL WEBS WITH THE SAME WEIGHT SEQUENCE

Intuitively, link weight could affect the dynamics of the network. However, the theoretical research on the effects of link weight on network dynamics is still rare. In this paper, we present two heterogeneous weighted pseudo-fractal webs controlled by two weight parameters [Formula: see text] and [Formula: see text] ([Formula: see text]). Both graph models are scale-free deterministic graphs, and they have the same weight sequence when [Formula: see text] and [Formula: see text] are fixed. Based on their self-similar graph structure, we study the effect of heterogeneous weight on the random walks in graph with scale-free characteristics. We obtain analytically the average trapping time (ATT) for biased random walks in graphs with a trap located at a fixed node. Analyzing and comparing the obtained solutions, we find that in the large graph limit, the ATT for both graph models all grow as a power function of the graph size (number of nodes) with the exponent [Formula: see text] dependents on the ratio of parameters [Formula: see text] and [Formula: see text], but their exponents [Formula: see text] are not the same, one gets the minimum when [Formula: see text], while the other gets the maximum. Furthermore, the average weighted shortest path length (AWSPL) to the trap is calculated for both graph models, respectively. We show that when the graph size tends to infinity, their AWSPL grows unbounded with the graph size for most parameters. We hope that these results could help people understand the impact of heterogeneous weight on network dynamics.

Dynamics of Operator-Weighted Shifts

This paper investigates the dynamics of bilateral operator-weighted shifts on [Formula: see text] with a weight sequence of positive diagonal operators on a Hilbert space [Formula: see text]. Necessary and sufficient conditions for the bilateral weighted shifts to be hypercyclic (subspace-hypercyclic, frequently hypercyclic, Devaney chaotic, respectively) are provided. As a consequence, it is shown that for any [Formula: see text]-set [Formula: see text] of positive numbers which is bounded and bounded away from zero, there exists an invertible bilateral operator-weighted shift [Formula: see text] such that [Formula: see text]. Furthermore, the (hereditary) Cesàro-hypercyclicity of the bilateral weighted shifts is characterized.

Generalised distances of sequences II: B-distances with weight sequences

In this paper, we investigate the weighted B-distances of infinite sequences. The general neighbourhood sequences were introduced for measuring distances in digital geometry (Zn), and the theory was recently extended for application to sequences. By assigning various weights to the elements of the sequences the concept is further generalized. An algorithm is presented which provides a shortest path between two sequences. Formula is also provided to calculate the weighted B-distance of any two sequences with a neighbourhood sequence B and a weight sequence. There are several neighbourhood sequences, which do not generate metrics. We prove a necessary and sufficient condition for a B-distance to define a generalized metric space above the sequences. Moreover, our results can be applied if the elements of the sequences used with various weights. In case of weight functions used B-distances we present also the metric conditions.