Select Sparse recovery in probability via $l_q$-minimization with Weibull random matrices for 0 < $q$ ≤ 1 GAO Yi, PENG Ji-gen, YUE Shi-gang Applied Mathematics-A Journal of Chinese Universities. 2018, (1)   DOI: 10.1007/s11766-018-3430-2 摘要( 8 )   Although Gaussian random matrices play an important role of measurement matrices in compressed sensing, one hopes that there exist other random matrices which can also be used to serve as the measurement matrices. Hence, Weibull random matrices induce extensive interest. In this paper, we first propose the $l_{2,q}$ robust null space property that can weaken the $D$-RIP, and show that Weibull random matrices satisfy the $l_{2,q}$ robust null space property with high probability. Besides, we prove that Weibull random matrices also possess the $l_q$ quotient property with high probability. Finally, with the combination of the above mentioned properties, we give two important approximation characteristics of the solutions to the $l_q$-minimization with Weibull random matrices, one is on the stability estimate when the measurement noise $e \in \mathbb{R}^n$ needs a priori $\|e\|_2\leq \epsilon$, the other is on the robustness estimate without needing to estimate the bound of $\|e\|_2$. The results indicate that the performance of Weibull random matrices is similar to that of Gaussian random matrices in sparse recovery.
 Select Waiting times and stopping probabilities for patterns in Markov chains ZHAO Min-zhi, XU Dong, ZHANG Hui-zeng Applied Mathematics-A Journal of Chinese Universities. 2018, (1)   DOI: 10.1007/s11766-018-3522-z 摘要( 2 )     PDF(0KB)( 2 ) Suppose that $\mathcal C$ is a finite collection of patterns. Observe a Markov chain until one of the patterns in $\mathcal C$ occurs as a run. This time is denoted by $\tau$. In this paper, we aim to give an easy way to calculate the mean waiting time $E(\tau)$ and the stopping probabilities $P(\tau=\tau_A)$ with $A\in\mathcal C$, where $\tau_A$ is the waiting time until the pattern $A$ appears as a run.
 Select On the strong convergence properties for weighted sums of negatively orthant dependent random variables DENG Xin, TANG Xu-fei, WANG Shi-jie, WANG Xue-jun Applied Mathematics-A Journal of Chinese Universities. 2018, (1)   DOI: 10.1007/s11766-018-3423-1 摘要( 2 )     PDF(0KB)( 0 ) In the paper, the strong convergence properties for two different weighted sums of negatively orthant dependent (NOD) random variables are investigated. Let $\{X_n, n\geq1\}$ be a sequence of NOD random variables. The results obtained in the paper generalize the corresponding ones for i.i.d. random variables and identically distributed NA random variables to the case of NOD random variables, which are stochastically dominated by a random variable $X$. As a byproduct, the Marcinkiewicz-Zygmund type strong law of large numbers for NOD random variables is also obtained.
 Select Nichols algebras over weak Hopf algebras WU Zhi-xiang Applied Mathematics-A Journal of Chinese Universities. 2018, (1)   DOI: 10.1007/s11766-018-3327-0 摘要( 2 )     PDF(0KB)( 1 ) In this paper, we study a Yetter-Drinfeld module $V$ over a weak Hopf algebra $\mathbb{H}$. Although the category of all left $\mathbb{H}$-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules $V$, we construct Nichols algebra $B(V)$ over the weak Hopf algebra $\mathbb{H}$, and a series of weak Hopf algebras. Some results of [8] are generalized.}