We consider in this article the analysis of multicomponent signals, defined as superpositions of modulated waves also called modes. More precisely, we focus on the analysis of a variant of the ...second-order synchrosqueezing transform, which was introduced recently, to deal with modes containing strong frequency modulation. Before going into this analysis, we revisit the case where the modes are assumed to be with weak frequency modulation as in the seminal paper of Daubechies et al. 8, to show that the constraint on the compactness of the analysis window in the Fourier domain can be alleviated. We also explain why the hypotheses made on the modes making up the multicomponent signal must be different when one considers either wavelet or short-time Fourier transform-based synchrosqueezing. The rest of the paper is devoted to the theoretical analysis of the variant of the second order synchrosqueezing transform 16 and numerical simulations illustrate the performance of the latter.
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The notion of the core inverse of tensors with the Einstein product was introduced, very recently (Sahoo et al. Comp Appl Math 39:9, 2020). In this paper we establish a few sufficient and necessary ...conditions for reverse-order law of this inverse. Further, we discuss mixed-type reverse-order law for core inverse. In addition to these, we discuss core inverse solutions of multilinear systems via the Einstein product. The prowess of the inverse is demonstrated to solve the Poisson problem in the multilinear system framework.
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Core and core-EP inverses of tensors Sahoo, Jajati Keshari; Behera, Ratikanta; Stanimirović, Predrag S. ...
Computational & Applied Mathematics,
03/2020, Volume:
39, Issue:
1
Journal Article
Peer reviewed
Open access
Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The ...results are verified using specific algebraic approach, based on proposed definitions and previously verified properties. The approach used here is new even in the matrix case.
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Within the field of multilinear algebra, inverses and generalized inverses of tensors based on the Einstein product have been investigated over the past few years. The notion of the weighted ...Moore–Penrose inverses of even-order tensors in the framework of the Einstein product was introduced recently (Ji and Wei in Front Math China 12(6):1319–1337, 2017). In this article, we introduce the weighted Moore–Penrose inverse of an arbitrary-order tensor. We also investigate the singular value decomposition and full-rank decomposition of arbitrary-order tensors using
reshape
operation. Derived representations are used for two purposes: (1) to obtain a few new characterizations and representations of weighted Moore–Penrose inverse of arbitrary-order tensors; (2) to explore various necessary and sufficient conditions for the reverse-order law for the inverse to hold. In addition to these, we discuss applications of singular value decomposition and the Moore–Penrose inverse of an arbitrary-order tensor to a few 3D color image processing.
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Mosić and Djordjević introduced the notation of the gDMP inverse for a linear operator on Hilbert space in Mosić and Djordjević (J Spectr Theory 8(2):555–573, 2018) by considering generalized Drazin ...inverse with the Moore-Penrose inverse. This article introduces two new classes of inverses: GD1 (generalized Drazin and inner) inverse and 1GD (inner and generalized Drazin) inverse for Banach space operators. The existence and uniqueness of the GD1 (also 1GD) inverse are discussed along with some properties through core-quasinilpotent decomposition and closed range decomposition operator. We further establish a few explicit representations of the GD1 inverse and their interconnections with generalized Drazin inverse. In addition, we discuss a few properties of GD1 (also 1GD) inverse through binary relation.
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We introduce, investigate, and apply two types of generalized inverses for square complex matrices using an arbitrary inner inverse of the input matrix and its core-EP inverse. Introduced ...combinations of inner inverses and the core-EP inverse are termed inner core-EP (ICEP) and core-EP inner (CEPI) inverse. Additionally, we extend the notion of the P-core inverse for square matrices with arbitrary indices. A few equivalent characterizations and effective representations of the introduced generalized inverses are suggested. In addition, representations of these inverses are established via core-EP and HS decompositions. The induced binary relations for these inverses are introduced and their properties are considered. An application of the proposed inverses in solving linear systems is also discussed.
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In this paper, we introduce new representations and characterizations of the outer inverse of tensors through QR decomposition. Derived representations are usable in generating corresponding ...representations of main tensor generalized inverses. Some results on reshape operation of a tensor are added to the existing theory. An effective algorithm for computing outer inverses of tensors is proposed and applied. The power of the proposed method is demonstrated by its application in 3D color image deblurring.
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The notion of the Moore-Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of ...different generalized inverses of tensors. A new method to compute the Moore-Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.
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In this paper, we introduce the notion of outer generalized inverses, with predefined range and null space, of tensors with rational function entries equipped with the Einstein product over an ...arbitrary field, of characteristic zero, with or without involution. We assume that the involved tensor entries are rational functions of unassigned variables or rational expressions of functional entries. The research investigates the replacements in two stages. The lower-stage replacements assume replacements of unknown variables by constant values from the field. The higher-order stage assumes replacements of functional entries by unknown variables. This approach enables the calculation on tensors over meromorphic functions to be simplified by analogous calculations on matrices whose elements are rational expressions of variables. In general, the derived algorithms permit symbolic computation of various generalized inverses which belong to the class of outer generalized inverses, with prescribed range and null space, over an arbitrary field of characteristic zero. More precisely, we focus on a few algorithms for symbolic computation of outer inverses of matrices whose entries are elements of a field of characteristic zero or a field of meromorphic functions in one complex variable over a connected open subset of
C
. Illustrative numerical results validate the theoretical results.
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A family of varying-parameter finite-time zeroing neural networks (VPFTZNN) is introduced for solving the time-varying Sylvester equation (TVSE). The convergence speed of the proposed VPFTZNN family ...is analysed and compared with the traditional zeroing neural network (ZNN) and the finite-time zeroing neural network (FTZNN). The behaviour of the proposed neural networks under various activation functions is proved theoretically and verified experimentally. In addition, the stability and noise resistance of the proposed VPFTZNN family are discussed. Further, the proposed VPFTZNN models are applied in the computation of current flows in an electrical network.
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