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Volume 3, 2020

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Mathematical Foundations of Computing

May 2020 , Volume 3 , Issue 2

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Unconstrained $ \ell_1 $-$ \ell_2 $ minimization for sparse recovery via mutual coherence
Pengbo Geng and Wengu Chen
2020, 3(2): 65-79 doi: 10.3934/mfc.2020006 +[Abstract](165) +[HTML](83) +[PDF](509.47KB)

The paradigm of compressed sensing is to exactly or stably recover any sparse signal \begin{document}$ x\in \mathbb{R}^n $\end{document} from a small number of linear measurements \begin{document}$ b = Ax+e $\end{document}, where \begin{document}$ A\in\mathbb{R}^{m\times n} $\end{document} with \begin{document}$ m\ll n $\end{document} and \begin{document}$ e\in \mathbb{R}^m $\end{document} denotes the measurement noise. \begin{document}$ \ell_1 $\end{document}-\begin{document}$ \ell_2 $\end{document} minimization has recently become an effective signal recovery method. In this paper, a mutual coherence based signal recovery guarantee by the unconstrained \begin{document}$ \ell_1 $\end{document}-\begin{document}$ \ell_2 $\end{document} minimization model is given to achieve the stable recovery of any sparse signal \begin{document}$ x $\end{document} in the presence of the Dantzig Selector (DS) type noise or the \begin{document}$ \ell_2 $\end{document} bounded noise, respectively. To the best of our knowledge, this is the first mutual coherence based sufficient condition to achieve sparse signal recovery via the unconstrained \begin{document}$ \ell_1 $\end{document}-\begin{document}$ \ell_2 $\end{document} minimization.

A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertainty
Nasim Ullah and Ahmad Aziz Al-Ahmadi
2020, 3(2): 81-99 doi: 10.3934/mfc.2020007 +[Abstract](169) +[HTML](95) +[PDF](1352.69KB)

This research work proposes a novel triple mode sliding mode controller for a nonlinear system with measurement noise and uncertainty. The proposed control has the following goals (1) it ensures the transient and steady state robustness of the system in closed loop (2) it reduces chattering in the control signal with measurement noise. Fuzzy system is used to tune the appropriate order of the fractional operators for the proposed control system. Depending on the tuned range of the fractional operators, the proposed controller can operate effectively in the following three modes (1) classical sliding mode (SMC) (2) fractional order sliding mode (FSMC) (3) Integral sliding mode control (ISMC). With the noisy feedback, the performance of the classical SMC and SMC with boundary layer degrades significantly while ISMC shows better performance. However ISMC exhibits large transient overshoots.The proposed control method optimally selects the appropriate mode of the controller to ensure performance(transient and steady state) and suppresses the effect of noisy feedback. The proposed scheme is derived for the permanent magnet synchronous motor, s (PMSM) speed regulation problem which is subject to uncertainties, measurement noise and un-modeled dynamics as a case study. The effectiveness of proposed scheme is verified using numerical simulations.

Weaving K-fusion frames in hilbert spaces
Hanbing Liu, Yongdong Huang and Chongjun Li
2020, 3(2): 101-116 doi: 10.3934/mfc.2020008 +[Abstract](168) +[HTML](131) +[PDF](390.18KB)

\begin{document}$ K $\end{document}-fusion frames are generalizations of fusion frames in frame theory. In this paper, based on the weaving frames and \begin{document}$ K $\end{document}-fusion frames, we propose the notion of weaving \begin{document}$ K $\end{document}-fusion frames and conduct relevant research. First, we give some characterizations of weaving \begin{document}$ K $\end{document}-fusion frames. Then, by means of operator theory and frame theory, we present several novel construction approaches of weaving \begin{document}$ K $\end{document}-fusion frames. Finally, we discuss transitivity of weaving \begin{document}$ K $\end{document}-fusion frames.

Multivariate weighted kantorovich operators
Ana-Maria Acu, Laura Hodis and Ioan Rasa
2020, 3(2): 117-124 doi: 10.3934/mfc.2020009 +[Abstract](169) +[HTML](90) +[PDF](326.7KB)

Recently, some weighted Durrmeyer type operators were used as research tools in Learning Theory. In this paper we introduce a class of multidimensional weighted Kantorovich operators \begin{document}$ K_n $\end{document} on \begin{document}$ C(Q_d) $\end{document} where \begin{document}$ Q_d $\end{document} is the \begin{document}$ d $\end{document}-dimensional hypercube \begin{document}$ [0,1]^d $\end{document}. We show that each \begin{document}$ K_n $\end{document} has a unique invariant probability measure and determine this measure. Then, using results from approximation theory and the theory of ergodic operators, we find the limit of the iterates of \begin{document}$ K_n $\end{document} and give rates of convergence of the iterates toward the limit. Finally, we show that some Kantorovich type operators previously investigated in literature fall into the class of operators introduced in our paper. Other properties and applications, involving Learning Theory, will be presented in a forthcoming paper, where we will consider also operators on spaces of Lebesgue integrable functions on the hypercube \begin{document}$ Q_d $\end{document}.

Homography estimation along short videos by recurrent convolutional regression network
Yang Mi, Kang Zheng and Song Wang
2020, 3(2): 125-140 doi: 10.3934/mfc.2020014 +[Abstract](172) +[HTML](100) +[PDF](3671.01KB)

Many moving-camera video processing and analysis tasks require accurate estimation of homography across frames. Estimating homography between non-adjacent frames can be very challenging when their camera view angles show large difference. In this paper, we propose a new deep-learning based method for homography estimation along videos by exploiting temporal dynamics across frames. More specifically, we develop a recurrent convolutional regression network consisting of convolutional neural network (CNN) and recurrent neural network (RNN) with long short-term memory (LSTM) cells, followed by a regression layer for estimating the parameters of homography. In the experiments, we evaluate the proposed method on both the synthesized and real-world short videos. The experimental results verify that the proposed method can estimate the homographies along short videos better than several existing methods.



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