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Generalized Convolution and Product Theorems Associated With the Free Metaplectic Transformation and Their Applications

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SSRN 2023年

作者： Cui, Manjun Zhang, Zhichao School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing210044 China Center for Applied Mathematics of Jiangsu Province Nanjing University of Information Science and Technology Nanjing210044 China Jiangsu International Joint Laboratory on System Modeling and Data Analysis Nanjing University of Information Science and Technology Nanjing210044 China

Convolution theorems for the free metaplectic transformation (FMT) have great significance in the fields of fractional domain non-stationary signal processing theory and method. The existing results have found many applications in high-dimensional non-stationary signal sampling and filtering. However, a widely accepted closed-form expression has not yet been established. In this paper, we introduce the definitions of generalized convolution and product operators in the FMT domain, and use them to formulate the generalized convolution and product theorems for the FMT. The derived results include particular cases the existing convolution and product theorems for the FMT and also the convolution and product theorems for the Fourier transform, the fractional Fourier transform, and the linear canonical transform. We employ the generalized convolution theorem for the FMT in designing an optimal filter in the FMT domain. We also validate the correctness and effectiveness of the theoretical analysis through numerical experiments, demonstrating the superiority of the proposed optimal filtering method over the conventional ones. © 2023, The Authors. All rights reserved.

关键词： Convolution

ROTA-BAXTER OPERATORS, DIFFERENTIAL OPERATORS, PRE- AND NOVIKOV STRUCTURES ON GROUPS AND LIE ALGEBRAS

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arXiv 2024年

作者： Gao, Xing Guo, Li Han, Zongjian Zhang, Yi School of Mathematics and Statistics Lanzhou University Lanzhou730000 China Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics Lanzhou730070 China Department of Mathematics and Computer Science Rutgers University NewarkNJ07102 United States School of Mathematical Sciences Tonji University Shanghai200092 China School of Mathematics and Statistics Center for Applied Mathematics of Jiangsu Province Jiangsu International Joint Laboratory on System Modeling and Data Analysis NUIST Jiangsu Nanjing210044 China

Rota-Baxter operators on various structures have found important applications in diverse areas, from renormalization of quantum field theory to Yang-Baxter equations. Relative Rota-Baxter operators on Lie algebras are closely related to pre-Lie algebras and post-Lie algebras. Some of their group counterparts have been introduced to study post-groups, skew left braces and set-theoretic solutions of Yang-Baxter equations, but searching suitable notions of relative Rota-Baxter operators on groups with weight zero and pre-groups has been challenging and has been the focus of recent studies, by provisionally imposing an abelian condition. Arising from the works of Balinsky-Novikov and Gelfand-Dorfman, Novikov algebras and their constructions from differential commutative algebras have led to broad applications. Finding their suitable counterparts for groups and Lie algebras has also attracted quite much recent attention. This paper uses one-sided-inverse pairs of maps to give a perturbative approach to a general notion of relative Rota-Baxter operators and differential operators on a group and a Lie algebra with limit-weight. With the extra condition of limit-abelianess on the group or Lie algebra, we give an interpretation of relative Rota-Baxter and differential operators with weight zero. These operators motivate us to define pre-groups and Novikov groups respectively as the induced structures. The tangent maps of these operators are shown to give Rota-Baxter and differential operators with weight zero on Lie algebras. The tangent spaces of the pre-Lie and Novikov Lie groups are pre-Lie algebras and Novikov Lie algebras, fulfilling the expected property. Furthermore, limit-weight relative Rota-Baxter operators on groups give rise to skew left braces and then set-theoretic solutions of the Yang-Baxter *** Codes 22E60, 17B38, 17B40, 16W99, 45N05 Copyright © 2024, The Authors. All rights reserved.

关键词： Lie groups

Cohomology and Deformation Theory of O-Operators on Hom-Lie Conformal Algebras

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SSRN 2024年

作者： Asif, Sania Wang, Yao Mosbahi, Bouzid Basdouri, Imed School of Mathematics and Statistics Nanjing University of Information Science and Technology Jiangsu Nanjing210044 China Center for Applied Mathematics of Jiangsu Province Jiangsu International Joint Laboratory on System Modeling and Data Analysis Nanjing University of Information Science and Technology Jiangsu Nanjing210044 China University of Sfax Faculty of Sciences of Sfax BP 1171 Sfax3038 Tunisia University of Gafsa Faculty of Sciences of Gafsa Gafsa2112 Tunisia

In the present paper, we aim to introduce the cohomology of $\mathcal{O}$-operators defined on the Hom-Lie conformal algebra concerning the given representation. To obtain the desired results, we describe three different cochain complexes and discuss the interrelation of their coboundary operators. And show that differential maps on the graded Lie algebra can also be defined by using the Maurer-Cartan element. We further find out that, the $\mathcal{O}$-operator on the given Hom-Lie conformal algebra serves as a Maurer-Cartan element and it leads to acquiring the notion of a differential map in terms of $\mathcal{O}$-operator $\delta_\T$. Next, we provide the notion of Hom-pre-Lie conformal algebra, that induces a sub-adjacent Hom-Lie conformal algebra structure. The differential $\delta_{\b,\a}$ of this sub-adjacent Hom-Lie conformal algebra is related to the differential $\delta_\T$. Finally, we provide the deformation theory of $\mathcal{O}$-operators on the Hom-Lie conformal algebras as an application to the cohomology theory, where we discuss linear and formal deformations in detail. © 2024, The Authors. All rights reserved.

关键词： Algebra

Graph Linear Canonical Transform Based on CM-CC-CM Decomposition

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arXiv 2024年

作者： Li, Na Zhang, Zhichao Han, Jie Chen, Yunjie Cao, Chunzheng School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing210044 China Center for Applied Mathematics of Jiangsu Province Nanjing University of Information Science and Technology Nanjing210044 China Jiangsu International Joint Laboratory on System Modeling and Data Analysis Nanjing University of Information Science and Technology Nanjing210044 China School of Remote Sensing and Geomatics Engineering Nanjing University of Information Science and Technology Nanjing210044 China

The graph linear canonical transform (GLCT) is presented as an extension of the graph Fourier transform (GFT) and the graph fractional Fourier transform (GFrFT), offering more flexibility as an effective tool for graph signal processing. In this paper, we introduce a GLCT based on chirp multiplication-chirp convolution-chirp multiplication decomposition (CM-CCCM- GLCT), which irrelevant to sampling periods and without oversampling operation. Various properties and special cases of the CM-CC-CM-GLCT are derived and discussed. In terms of computational complexity, additivity, and reversibility, we compare the CM-CC-CM-GLCT and the GLCT based on the central discrete dilated Hermite function (CDDHFs-GLCT). Theoretical analysis demonstrates that the computational complexity of the CM-CC-CM-GLCT is significantly reduced. Simulation results indicate that the CM-CC-CM-GLCT achieves similar additivity to the CDDHFs-GLCT. Notably, the CM-CC-CM-GLCT exhibits better reversibility. Copyright © 2024, The Authors. All rights reserved.

关键词： Chirp modulation

Cohomology and Deformation Theory of O-Operators on Hom-Lie Conformal Algebras

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SSRN 2024年

作者： Asif, Sania Wang, Yao Mosbahi, Bouzid Basdouri, Imed School of Mathematics and Statistics Nanjing University of Information Science and Technology Jiangsu Nanjing210044 China Center for Applied Mathematics of Jiangsu Province Jiangsu International Joint Laboratory on System Modeling and Data Analysis Nanjing University of Information Science and Technology Jiangsu Nanjing210044 China University of Sfax Department of Mathematics Faculty of Sciences of Sfax BP 1171 Sfax3000 Tunisia University of Gafsa Faculty of Sciences of Gafsa Gafsa2112 Tunisia

关键词： Algebra

Multi-dimensional Graph Linear Canonical Transform

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arXiv 2024年

作者： Li, Na Zhang, Zhichao Han, Jie Chen, Yunjie Cao, Chunzheng School of Mathematics and Statistics Center for Applied Mathematics of Jiangsu Province Jiangsu International Joint Laboratory on System Modeling and Data Analysis Nanjing University of Information Science and Technology Nanjing210044 China Key Laboratory of System Control and Information Processing Ministry of Education Shanghai200240 China Key Laboratory of Computational Science and Application of Hainan Province Haikou571158 China School of Remote Sensing and Geomatics Engineering Nanjing University of Information Science and Technology Nanjing210044 China

Many multi-dimensional (M-D) graph signals appear in the real world, such as digital images, sensor network measurements and temperature records from weather observation stations. It is a key challenge to design a transform method for processing these graph M-D signals in the linear canonical transform domain. This paper proposes the two-dimensional graph linear canonical transform based on the central discrete dilated Hermite function (2-D CDDHFs-GLCT) and the two-dimensional graph linear canonical transform based on chirp multiplication-chirp convolution-chirp multiplication decomposition (2-D CM-CC-CM-GLCT). Then, extending 2-D CDDHFs-GLCT and 2-D CM-CC-CM-GLCT to M-D CDDHFs-GLCT and M-D CM-CC-CM-GLCT. In terms of the computational complexity, additivity and reversibility, M-D CDDHFs-GLCT and M-D CM-CC-CM-GLCT are compared. Theoretical analysis shows that the computational complexity of M-D CM-CC-CM-GLCT algorithm is obviously reduced. Simulation results indicate that M-D CM-CC-CM-GLCT achieves comparable additivity to M-D CDDHFs-GLCT, while M-D CM-CC-CM-GLCT exhibits better reversibility. Finally, M-D GLCT is applied to data compression to show its application advantages. The experimental results reflect the superiority of M-D GLCT in the algorithm design and implementation of data compression. Copyright © 2024, The Authors. All rights reserved.

关键词： Chirp modulation

Graph Chirp Signal and Graph Fractional Vertex-Frequency Energy Distribution

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arXiv 2025年

作者： Cui, Manjun Zhang, Zhichao School of Mathematics and Statistics The Center for Applied Mathematics of Jiangsu Province The Jiangsu International Joint Laboratory on System Modeling and Data Analysis Nanjing University of Information Science and Technology Nanjing210044 China School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing210044 China Key Laboratory of System Control and Information Processing Ministry of Education Shanghai Jiao Tong University Shanghai200240 China Key Laboratory of Computational Science and Application of Hainan Province Hainan Normal University Haikou571158 China

Graph signal processing (GSP) has emerged as a powerful framework for analyzing data on irregular domains. In recent years, many classical techniques in signal processing (SP) have been successfully extended to GSP. Among them, chirp signals play a crucial role in various SP applications. However, graph chirp signals have not been formally defined despite their importance. Here, we define graph chirp signals and establish a comprehensive theoretical framework for their analysis. We propose the graph fractional vertex-frequency energy distribution (GFED), which provides a powerful tool for processing and analyzing graph chirp signals. We introduce the general fractional graph distribution (GFGD), a generalized vertex-frequency distribution, and the reduced interference GFED, which can suppress cross-term interference and enhance signal clarity. Furthermore, we propose a novel method for detecting graph signals through GFED domain filtering, facilitating robust detection and analysis of graph chirp signals in noisy environments. Moreover, this method can be applied to real-world data for denoising more effective than some state-of-the-arts, further demonstrating its practical significance. Copyright © 2025, The Authors. All rights reserved.

关键词： Wiener filtering

A Skew Normal Mixed Models with Noise Estimation and Anisotropic Spatial Information Constraints

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SSRN 2022年

作者： Zhang, Kaili Ma, Lianhong Cao, Chunzheng Yang, Jianwei Zhang, Zhichao Chen, Yunjie School of Mathematics and Statistics Nanjing University of Information Science Nanjing210044 China Center for Applied Mathematics of Jiangsu Province Nanjing University of Information Science and Technology Nanjing210044 China Jiangsu International Joint Laboratory on System Modeling and Data Analysis Nanjing University of Information Science and Technology Nanjing210044 China

Accurate image segmentation facilitates image analysis. Gaussian mixture models (GMMs) are widely used for image segmentation due to their low complexity and high accuracy. However, the model assumes that the intensity distributions of images are symmetric, which makes it hard to obtain ideal results for images with asymmetric distributions. In addition, the model dose not consider any noise, which makes it difficult to obtain ideal distribution fitting results when images contain severe noises. And the model does not use any spatial information, which makes it sensitive to noise. To address these issues, we model the noise using a normal distribution and couple it into a skew normal mixture model to reduce the model's sensitivity to asymmetric distributions and noise. To further reduce the effect of noise, we propose a new anisotropic spatial information constraint term that preserves detailed information while reducing the effect of noise. Finally, an improved EM algorithm is proposed to solve the parameters of the model. Experimental results on synthetic and natural images illustrate that our model outperforms other state-of-the-art segmentation methods. © 2022, The Authors. All rights reserved.

关键词： Image segmentation

COHOMOLOGY AND DEFORMATION THEORY OF O-OPERATORS ON HOM-LIE CONFORMAL ALGEBRAS

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arXiv 2023年

In the present paper, we aim to introduce the cohomology of O-operators defined on the Hom-Lie conformal algebra concerning the given representation. To obtain the desired results, we describe three different cochain complexes and discuss the interrelation of their coboundary operators. And show that differential maps on the graded Lie algebra can also be defined by using the Maurer-Cartan element. We further find out that, the O-operator on the given Hom-Lie conformal algebra serves as a Maurer-Cartan element and it leads to acquiring the notion of a differential map in terms of O-operator δT . Next, we provide the notion of Hom-pre-Lie conformal algebra, that induces a sub-adjacent Hom-Lie conformal algebra structure. The differential δβ,α of this sub-adjacent Hom-Lie conformal algebra is related to the differential δT . Finally, we provide the deformation theory of O-operators on the Hom-Lie conformal algebras as an application to the cohomology theory, where we discuss linear and formal deformations in *** Codes 11R52, 15A99, 17B67, 17B10, 16G30 Copyright © 2023, The Authors. All rights reserved.

关键词： Algebra