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  • Applied Mathematics-A Journal of Chinese Universities
      2018年, 第1期 刊出日期:2018-03-01   
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    Sparse recovery in probability via $l_q$-minimization with Weibull random matrices for 0 < $q$ ≤ 1 收藏
    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.
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    Waiting times and stopping probabilities for patterns in Markov chains 收藏
    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.
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    On the strong convergence properties for weighted sums of negatively orthant dependent random variables 收藏
    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.
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    Modeling stochastic mortality with O-U type processes 收藏
    ZHENG Jing, TONG Chang-qing, ZHANG Gui-jun
    Applied Mathematics-A Journal of Chinese Universities. 2018, (1)   DOI: 10.1007/s11766-018-3349-7
    摘要( 1 )     PDF(0KB)( 0 )
    Modeling log-mortality rates on O-U type processes and forecasting life expectancies are explored using U.S. data. In the classic Lee-Carter model of mortality, the time trend and the age-specific pattern of mortality over age group are linear, this is not the feature of mortality model. To avoid this disadvantage, O-U type processes will be used to model the log-mortality in this paper. In fact, this model is an AR(1) process, but with a nonlinear time drift term. Based on the mortality data of America from Human Mortality database (HMD), mortality projection consistently indicates a preference for mortality with O-U type processes over those with the classical Lee-Carter model. By means of this model, the low bounds of mortality rates at every age are given. Therefore, lengthening of maximum life expectancies span is estimated in this paper.
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    2D-3D registration for 3D analysis of lower limb alignment in a weight-bearing condition 收藏
    SHIM Eungjune, KIM Youngjun, LEE Deukhee, LEE Byung Hoon, WOO Sungkyung, LEE Kunwoo
    Applied Mathematics-A Journal of Chinese Universities. 2018, (1)   DOI: 10.1007/s11766-018-3459-2
    摘要( 1 )     PDF(0KB)( 0 )
    X-ray imaging is the conventional method for diagnosing the orthopedic condition of a patient. Computerized Tomography(CT) scanning is another diagnostic method that provides patient's 3D anatomical information. However, both methods have limitations when diagnosing the whole leg; X-ray imaging does not provide 3D information, and normal CT scanning cannot be performed with a standing posture. Obtaining 3D data regarding the whole leg in a  standing posture is clinically important because it enables 3D analysis in the weight bearing condition. Based on these clinical needs, a hardware-based bi-plane X-ray imaging system has been developed; it uses two orthogonal X-ray images. However, such methods have not been made available in general clinics because of the hight cost. Therefore, we proposed a widely adaptive method for 2D X-ray image and 3D CT scan data. By this method, it is possible to three-dimensionally analyze the whole leg in standing posture. The optimal position that generates the most similar image is the captured X-ray image. The algorithm verifies the similarity using the performance of the proposed method by simulation-based experiments. Then, we analyzed the internal-external rotation angle of the femur using real patient data. Approximately 10.55 degrees of internal rotations were found relative to the defined anterior-posterior direction. In this paper, we present a useful registration method using the conventional X-ray image and 3D CT scan data to analyze the whole leg in the weight-bearing condition.
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    Existence and uniqueness results for mild solutions of random impulsive abstract neutral partial differential equation over real axis 收藏
    P Indhumathi, A Leelamani
    Applied Mathematics-A Journal of Chinese Universities. 2018, (1)   DOI: 10.1007/s11766-018-3449-4
    摘要( 1 )     PDF(0KB)( 0 )
    In this paper, we discuss the existence and uniqueness of mild solutions of random impulsive abstract neutral partial differential equations in a real separable Hilbert space. The results are obtained by using Leray-Schauder Alternative and Banach Contraction Principle. Finally an example is given to illustrate our problem.
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    Data driven composite shape descriptor design for shape retrieval with a VoR-Tree 收藏
    WANG Zi-hao, LIN Hong-wei, XU Chen-kai
    Applied Mathematics-A Journal of Chinese Universities. 2018, (1)   DOI: 10.1007/s11766-018-3536-6
    摘要( 1 )     PDF(0KB)( 0 )
    We develop a data driven method (probability model) to construct a composite shape descriptor by combining a pair of scale-based shape descriptors. The selection of a pair of scale-based shape descriptors is modeled as the computation of the union of two events, i.e., retrieving similar shapes by using a single scale-based shape descriptor. The pair of scale-based shape descriptors with the highest probability forms the composite shape descriptor. Given a shape database, the composite shape descriptors for the shapes constitute a planar point set. A VoR-Tree of the planar point set is then used as an indexing structure for efficient query operation. Experiments and comparisons show the effectiveness and efficiency of the proposed composite shape descriptor.
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    Nichols algebras over weak Hopf algebras 收藏
    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.}
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