Article
Physics, Applied
Muhammad Zahid, Awais Younus, Mohamed E. Ghoneim, Mansour F. Yassen, Jamil Abbas Haider
Summary: Quaternion differential equations (QDEs) are a new type of differential equations that differ from ordinary differential equations. Finding the exponential matrices is important for solving the QDEs. The solution set of QDE, as a result of the noncommutativity of quaternions, is a right free module.
INTERNATIONAL JOURNAL OF MODERN PHYSICS B
(2023)
Article
Multidisciplinary Sciences
Aleksandr Krivoshein
Summary: This paper presents techniques and methods for constructing symmetric multiwavelets in the multivariate case. Wavelet matrix masks are constructed using a matrix extension algorithm, and their symmetry properties and other attributes are studied.
Article
Mathematics, Applied
Youssef El Haoui
Summary: This article extends the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform. Fundamental properties of this extension such as scaling, translation, rotation, and Parseval's identity are studied, and the associated Heisenberg-Pauli-Weyl uncertainty principle UP is derived. Additionally, the logarithmic uncertainty principle is generalized to the CQWT domain using the quaternion Fourier representation.
JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Jiangnan Wang, JinRong Wang, Rui Liu
Summary: This paper investigates the Hyers-Ulam stability of the first-order linear homogeneous quaternion matrix difference equation. Additionally, the Hyers-Ulam stability of the second-order linear homogeneous quaternion-valued forward and backward difference equation is proven by converting them into the first-order quaternion matrix difference equation. Finally, examples are provided to support the theoretical results.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2023)
Article
Computer Science, Theory & Methods
Hongzhi Wei, Ruoxia Li, Baowei Wu
Summary: This paper addresses the problems of stabilization and synchronization control of the fractional-order quaternion-valued fuzzy memristive neural networks by establishing conditions for the equilibrium point and proposing stability analysis with two suitable controllers. A vector ordering approach is developed to determine the magnitude of two different quaternions, and the effectiveness of the control method is demonstrated through simulation examples.
FUZZY SETS AND SYSTEMS
(2021)
Article
Automation & Control Systems
Lin Xiao, Sai Liu, Xin Wang, Yongjun He, Lei Jia, Yang Xu
Summary: This article extends the ZNN method to address dynamic quaternion-valued matrix inversion, proposing two QVZNN models and introducing a new nonlinear activation function to accelerate convergence.
IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS
(2022)
Article
Computer Science, Information Systems
Long H. Ngo, Nikolay M. Sirakov, Marie Luong, Emmanuel Viennet, Thuong Le-Tien
Summary: In this study, a novel sparse representation learning method in the Quaternion Wavelet (QW) domain for multi-class image classification is proposed. The method takes advantage of QW decomposition, PCA dimensionality reduction, and sparse representation to efficiently learn and capture meaningful and compact information from image data. Experimental results demonstrate that the proposed method achieves higher accuracy, sparsity, and robustness compared to contemporary methods including Neural Networks.
Article
Mathematics, Applied
Leping Suo, Michal Feckan, JinRong Wang
Summary: In this paper, the controllability and observability of linear quaternion-valued impulsive differential equations (QIDEs) are investigated. Sufficient and necessary conditions for state controllability and state observability of linear QIDEs are established. The theoretical results in the sense of complex-valued and quaternion-valued are equivalent to each other by the isomorphism between quaternion vector space and complex variables space as well as the adjoint matrix of quaternion matrix. Finally, the validity of theoretical results obtained is demonstrated by two shown instances.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Computer Science, Artificial Intelligence
Qiankun Song, Sihan Chen, Zhenjiang Zhao, Yurong Liu, Fuad E. Alsaadi
Summary: This paper considers the problem of passive filter design for fractional-order quaternion-valued neural networks (FOQVNNs) with neutral delays and external disturbance. By constructing Lyapunov-Krasovskii functional and using inequality technique, delay-independent and delay-dependent sufficient conditions are derived as linear matrix inequality (LMI) to confirm the stability and passivity of the augmented filtering dynamic system. A numerical example with simulations is provided to demonstrate the feasibility of the obtained theory results.
Article
Engineering, Electrical & Electronic
Zhao Zhang, Jiashu Zhang, Defang Li
Summary: Recently, a widely nonlinear quaternion recursive least square algorithm was proposed to enhance the performance of quaternion-valued second-order Volterra LMS algorithms. Additionally, a novel widely nonlinear quaternion Volterra recursive least square dichotomous coordinate descent filtering model was introduced to reduce computational complexity.
Article
Chemistry, Multidisciplinary
Dorota Majorkowska-Mech, Aleksandr Cariow
Summary: This paper presents fast algorithms for computing the discrete Fourier transform of real-valued sequences with lengths ranging from 3 to 9. The algorithms eliminate the redundancy of using complex-valued FFT for real-valued DFT by operating solely on real numbers. They are described in matrix-vector notation and their data flow diagrams are provided.
APPLIED SCIENCES-BASEL
(2022)
Article
Mathematics
Jun Guo, Yanchao Shi, Weihua Luo, Yanzhao Cheng, Shengye Wang, Antonio Lopes
Summary: This paper investigates the adaptive synchronization problem of quaternion-valued Cohen-Grossberg neural networks (QVCGNNs), both with and without known parameters. By constructing an appropriate Lyapunov function and utilizing parameter identification theory and decomposition methods, two effective adaptive feedback schemes are proposed to ensure global synchronization of CGQVNNs. The control gain of these schemes can be obtained using the Matlab LMI toolbox. The theoretical results presented in this work contribute to the literature on exploring the adaptive synchronization problem of quaternion-valued neural networks (QVNNs). Lastly, the reliability of the proposed theoretical schemes is demonstrated through two interesting numerical examples.
Article
Mathematics, Applied
Fengxia Zhang, Ying Li, Jianli Zhao
Summary: This article discusses the minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution of quaternion matrix equations. By converting the problems into real least squares problems and obtaining the solutions, the algorithms are simplified and more convenient to use, making them portable and efficient.
Article
Mathematics, Applied
Zhigang Jia, Qianyu Wang, Hong-Kui Pang, Meixiang Zhao
Summary: This study addresses the challenge of computing partial quaternion eigenpairs and proposes a new inverse quaternion iteration method with quaternion shifts. Through geometric explanation and numerical experiments, we demonstrate the efficiency and superiority of this approach.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Yifen Ke, Changfeng Ma, Zhigang Jia, Yajun Xie, Riwei Liao
Summary: A novel quasi non-negative quaternion matrix factorization (QNQMF) model is proposed to address the non-negativity dropout problem of quaternion models in color image processing. The quaternion projected gradient algorithm and the quaternion alternating direction method of multipliers are used to implement QNQMF by formulating it as non-convex constraint quaternion optimization problems. Experimental results show that encoding algorithms on quaternions outperform those on RGB channels in color image reconstruction and face recognition, especially when dealing with large facial expressions and shooting angle variations.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Engineering, Electrical & Electronic
A. T. Walden, D. Schneider-Luftman
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2015)
Article
Engineering, Electrical & Electronic
R. J. Wolstenholme, A. T. Walden
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2015)
Article
Engineering, Electrical & Electronic
D. Schneider-Luftman, A. T. Walden
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2016)
Article
Mathematics, Applied
Paul Ginzberg, Christiana Mavroyiakoumou
LINEAR ALGEBRA AND ITS APPLICATIONS
(2016)
Article
Engineering, Electrical & Electronic
L. Zhuang, A. T. Walden
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2017)
Article
Engineering, Electrical & Electronic
Swati Chandna, Andrew T. Walden
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2017)
Article
Engineering, Electrical & Electronic
A. T. Walden, Z. Leong
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2018)
Article
Engineering, Electrical & Electronic
P. Ginzberg, A. T. Walden
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2011)
Article
Engineering, Electrical & Electronic
Andrew T. Walden, E. A. K. Cohen
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2012)
Article
Engineering, Electrical & Electronic
P. Ginzberg, A. T. Walden
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2013)
Article
Engineering, Electrical & Electronic
Swati Chandna, A. T. Walden
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2013)
Review
Multidisciplinary Sciences
A. T. Walden
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2013)
