Spectra of some invertible weighted composition operators on Hardy and weighted Bergman spaces in the unit ball

In this paper, we investigate the spectra of invertible weighted composition operators with automorphism symbols, on Hardy space $H^2(\mathbb{B}_N)$ and weighted Bergman spaces $A_\alpha^2(\mathbb{B}_N)$, where $\mathbb{B}_N$ is the unit ball of the $N$-dimensional complex space. By taking $N=1$, $\mathbb{B}_N=\mathbb{D}$ the unit disc, we also complete the discussion about the spectrum of a weighted composition operator when it is invertible on $H^2(\mathbb{D})$ or $A_\alpha^2(\mathbb{D})$.Comment: 23 Page

Alternative mechanism of avoiding the big rip or little rip for a scalar phantom field

Depending on the choice of its potential, the scalar phantom field $\phi$ (the equation of state parameter $w<-1$) leads to various catastrophic fates of the universe including big rip, little rip and other future singularity. For example, big rip results from the evolution of the phantom field with an exponential potential and little rip stems from a quadratic potential in general relativity (GR). By choosing the same potential as in GR, we suggest a new mechanism to avoid these unexpected fates (big and little rip) in the inverse-\textit{R} gravity. As a pedagogical illustration, we give an exact solution where phantom field leads to a power-law evolution of the scale factor in an exponential type potential. We also find the sufficient condition for a universe in which the equation of state parameter crosses $w=-1$ divide. The phantom field with different potentials, including quadratic, cubic, quantic, exponential and logarithmic potentials are studied via numerical calculation in the inverse-\textit{R} gravity with $R^{2}$ correction. The singularity is avoidable under all these potentials. Hence, we conclude that the avoidance of big or little rip is hardly dependent on special potential.Comment: 9 pages,6 figure

Classification under Streaming Emerging New Classes: A Solution using Completely Random Trees

This paper investigates an important problem in stream mining, i.e., classification under streaming emerging new classes or SENC. The common approach is to treat it as a classification problem and solve it using either a supervised learner or a semi-supervised learner. We propose an alternative approach by using unsupervised learning as the basis to solve this problem. The SENC problem can be decomposed into three sub problems: detecting emerging new classes, classifying for known classes, and updating models to enable classification of instances of the new class and detection of more emerging new classes. The proposed method employs completely random trees which have been shown to work well in unsupervised learning and supervised learning independently in the literature. This is the first time, as far as we know, that completely random trees are used as a single common core to solve all three sub problems: unsupervised learning, supervised learning and model update in data streams. We show that the proposed unsupervised-learning-focused method often achieves significantly better outcomes than existing classification-focused methods