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The main contribution of this paper is to establish a number of equivalent conditions for the core–EP inverse of an operator, to be n -potent. We prove that the core–EP inverse of an operator is n -potent if and only if the Drazin inverse of the same operator is n -potent. Thus, we present new characterizations for n -potency of the Drazin inverse. Consequently, we get many characterizations for the core–EP inverse (and Drazin inverse) to be an idempotent. We observe that the core–EP inverse of an operator is idempotent if and only it is the orthogonal projector. Furthermore, we show that the n -potency of an operator implies n -potency of its core–EP inverse and develop the condition for the converse to hold. Applying these results, we obtain necessary and sufficient conditions for the n -potency and idempotency of the core inverse. Notice that the core inverse of an operator is n -potent (or idempotent) if and only if the given operator is n -potent (idempotent).

Per- and polyfluoroalkyl substances (PFAS) contamination has drawn considerable attention in the rivers of Shandong, China. However, their seasonal variations have not been well addressed due to the geographic scale. Here, surface waters were collected seasonally at estuaries of 13 major rivers in Shandong to investigate the spatiotemporal variations, mass discharges and ecological risks of PFAS. Two novel perfluoroalkyl ether carboxylic acids, hexafluoropropylene oxide-dimer acid (HFPO-DA) and hexafluoropropylene oxide-trimer acid (HFPO-TA), and 10 legacy PFAS were all detectable and their total concentrations (∑PFAS) ranged largely from 23 to 25,400 ng/L. Elevated levels of all target compounds were found in the Xiaoqing River but Majia and Jiaolai Rivers were also of concern. Perfluorooctanoic acid (PFOA) was dominant in all the rivers. Its concentration in the Xiaoqing River estuary increased 6-fold from 2011 to 2014 and reached a plateau during 2014–2020. The nonsignificant seasonality of the PFAS concentrations in rivers with seasonal flows may indicate their different emission patterns (i.e., higher emission in the wet season and lower emission in the dry season). The ratio of maximum to minimum concentration ( C max / C min ) was proposed to characterize the influence of local point sources, with C max / C min ranging from 10 to 14,000. Xiaoqing and Yellow Rivers discharged ∼98% of ∑PFAS flux to the sea, which was estimated to be 28.0 t in 2020. This study raises attention to the industrial emission patterns of PFAS. Further investigations with higher time resolution were needed to understand the emission patterns of PFAS in Shandong.

Our main goal is to present new representations and properties of the m -weak group inverse which presents a generalization of the weak group inverse, the generalized group inverse, the group inverse and the Drazin inverse. Consequently, we obtain new properties of the generalized group inverse and recover some properties of the weak group inverse and the Drazin inverse. Our results are illustrated by numerical examples. An equivalent condition for the continuity for the m -weak group inverse is studied too. Applying the m -weak group inverse, solvability of several linear equations and their solutions are considered.

This paper presents a novel and fast EMD-based (empirical mode decomposition-based) image fusion approach via morphological filter. Firstly, we develop a multi-channel bidimensional EMD method based on morphological filter to conduct image fusion. It uses the morphological expansion and erosion filters to compute the upper and lower envelopes of a multi-channel image in the sifting processing, and can decompose the input source images into several intrinsic mode functions (IMFs) with different scales and a residue. It significantly improves the computation efficiency of EMD for multi-channel images while maintaining the decomposition quality. Secondly, we adopt a patch-based fusion strategy with overlapping partition to fuse the IMFs and residue instead of the pixel-based fusion way usually used in EMD-based image fusion, where an energy-based maximum selection rule is designed to fuse the IMFs, and the feature information extracted by IMFs is used as a guide to merge the residue. Such strategy can extract the salient information of the source images well and can also reduce the spatial artifacts introduced by the noisy characteristics of the pixel-wise maps. A large number of comparative experiments on the fusion of several commonly used image data sets with multi-focus and multi-modal images, show that our newly proposed fusion method can obtain much better results than the existing EMD-based image fusion approaches. Furthermore, it is very competitive with the state-of-the-art image fusion methods in visualization, objective metrics, and time performance. The code of the proposed method can be downloaded from: https://github.com/neepuhjp/MFMBEMD-ImageFusion .

The rising use of 3D digital products has increased the demand for copyright protection. In this paper, we propose a novel and robust 3D watermarking method with high imperceptibility based on EMD (empirical mode decomposition) on surfaces. It first defines a normalized modulus signal on a 3D host model so as to involve EMD into 3D watermarking effectively. And then, it extracts different scale features of the defined signal by using EMD to locate the proper embedding positions. After this, the watermark signal is embedded repeatedly and cyclically into the 3D host model to enhance the robustness. The embedding strength is optimized according to a predefined fidelity parameter to control the imperceptibility of the watermark. Many experiment results show that the proposed method can obtain good results against various attacks while maintaining high invisibility, such as pseudo-random noise, Laplacian smoothing, simplification, subdivision, cropping, and similarity transformation. Furthermore, it is very competitive with the current state-of-the-art 3D watermarking techniques in terms of robustness and invisibility.

This paper is devoted to studying the Drazin inverse of certain structured matrices under newly introduced restrictive conditions. Specifically, we focus on the Drazin inverse of two types of anti-triangular block matrices. Moreover, several special cases of these results are also discussed. New representations for the Drazin inverse of an arbitrary block matrix are provided under certain conditions, extending recent results in the literature. Furthermore, a numerical example is presented to illustrate the theoretical findings.

In this paper, we give expressions for the generalized Drazin inverse of a (2,2,0) block matrix over a Banach algebra under certain circumstances, utilizing which we derive the generalized Drazin inverse of a 2 × 2 block matrix in a Banach algebra under weaker restrictions. Our results generalize and unify several results in the literature.

In this paper, we provide original representation for the Drazin inverse of P + Q under the conditions P²Q = 0, Q(PQ)² = 0, Q²PQ² = 0, QPQ3 = 0 and QPQ²PQ = 0. Then, we apply our results to derive some new expressions for the Drazin inverse of a 2 × 2 complex block matrix M = [ A B C D ] ∈ ℂ n × n (where A and D are square matrices but not necessarily of the same size). Finally, several illustrative numerical examples are given to demonstrate our results.

In this paper, we analyze the index relation of anti-triangular block matrices and their entries to separately obtain new and strict expressions for the Drazin inverse of them under certain circumstances. As applications, we utilize the relationship between the anti-triangular block matrix and a 2 × 2 block matrix to establish several formulae. Our results generalize and unify a series of results in the literature.

The expressions for the Drazin inverse of two kinds of anti-triangular block matrices are developed under new and weaker assumptions relative to those already used recently in this subject. Applying our results concerning the Drazin inverse and anti-triangular block matrices, we propose some characterizations and representations of the Drazin inverse of a 2 × 2 block matrix. In this way, we expand some notable achievements in characterizing and representing generalized inverses of partitioned matrices.