Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the mill spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares （M） solutions to a multilineax system and establish the relationship between the minimum-norm （N） leastsquares （M）solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.

Weighted Moore-Penrose Inverses and Weighted Core Inverses in Rings with Involution

In this paper,the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse,the e-core inverse and the f-dual core inverse in rings.Also,new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.

Screening biomarkers for spinal cord injury using weighted gene co-expression network analysis and machine learning

Immune changes and inflammatory responses have been identified as central events in the pathological process of spinal co rd injury.They can greatly affect nerve regeneration and functional recovery.However,there is still limited understanding of the peripheral immune inflammato ry response in spinal cord inju ry.In this study.we obtained microRNA expression profiles from the peripheral blood of patients with spinal co rd injury using high-throughput sequencing.We also obtained the mRNA expression profile of spinal cord injury patients from the Gene Expression Omnibus(GEO)database(GSE151371).We identified 54 differentially expressed microRNAs and 1656 diffe rentially expressed genes using bioinformatics approaches.Functional enrichment analysis revealed that various common immune and inflammation-related signaling pathways,such as neutrophil extracellular trap formation pathway,T cell receptor signaling pathway,and nuclear factor-κB signal pathway,we re abnormally activated or inhibited in spinal cord inju ry patient samples.We applied an integrated strategy that combines weighted gene co-expression network analysis,LASSO logistic regression,and SVM-RFE algorithm and identified three biomarke rs associated with spinal cord injury:ANO10,BST1,and ZFP36L2.We verified the expression levels and diagnostic perfo rmance of these three genes in the original training dataset and clinical samples through the receiver operating characteristic curve.Quantitative polymerase chain reaction results showed that ANO20 and BST1 mRNA levels were increased and ZFP36L2 mRNA was decreased in the peripheral blood of spinal cord injury patients.We also constructed a small RNA-mRNA interaction network using Cytoscape.Additionally,we evaluated the proportion of 22 types of immune cells in the peripheral blood of spinal co rd injury patients using the CIBERSORT tool.The proportions of naive B cells,plasma cells,monocytes,and neutrophils were increased while the proportions of memory B cells,CD8^(+)T cells,resting natural killer cells,resting dendritic cells,and eosinophils were markedly decreased in spinal cord injury patients increased compared with healthy subjects,and ANO10,BST1 and ZFP26L2we re closely related to the proportion of certain immune cell types.The findings from this study provide new directions for the development of treatment strategies related to immune inflammation in spinal co rd inju ry and suggest that ANO10,BST2,and ZFP36L2 are potential biomarkers for spinal cord injury.The study was registe red in the Chinese Clinical Trial Registry(registration No.ChiCTR2200066985,December 12,2022).

TOEPLITZ OPERATORS BETWEEN WEIGHTED BERGMAN SPACES OVER THE HALF-PLANE

In this paper,by characterizing Carleson measures,we investigate a class of bounded Toeplitz operator between weighted Bergman spaces with Békolléweights over the half-plane for all index choices.

一类树的邻接矩阵的Moore-Penrose广义逆

根据矩阵结构性质,利用分块矩阵技巧给出任意点数、任意长直径的毛毛虫树邻接矩阵的Moore-Penrose广义逆的具体形式,为进一步研究毛毛虫树的代数性质提供理论支撑.

Optimal investment based on relative performance and weighted utility

This paper studies the optimal portfolio allocation of a fund manager when he bases decisions on both the absolute level of terminal relative performance and the change value of terminal relative performance comparison to a predefined reference point. We find the optimal investment strategy by maximizing a weighted average utility of a concave utility and an Sshaped utility via a concavification technique and the martingale method. Numerical results are carried out to show the impact of the extent to which the manager pays attention to the change of relative performance related to the reference point on the optimal terminal relative performance.

The Weighted Basis for PHT-Splines

PHT-splines are defined as polynomial splines over hierarchical T-meshes with very efficient local refinement properties.The original PHT-spline basis functions constructed by the truncation mechanism have a decay phenomenon,resulting in numerical instability.The non-decay basis functions are constructed as the B-splines that are defined on the 2×2 tensor product meshes associated with basis vertices in Kang et al.,but at the cost of losing the partition of unity.In the field of finite element analysis and topology optimization,forming the partition of unity is the default ingredient for constructing basis functions of approximate spaces.In this paper,we will show that the non-decay PHT-spline basis functions proposed by Kang et al.can be appropriately modified to form a partition of unity.Each non-decay basis function is multiplied by a positive weight to form the weighted basis.The weights are solved such that the sum of weighted bases is equal to 1 on the domain.We provide two methods for calculatingweights,based on geometric information of basis functions and the subdivision of PHT-splines.Weights are given in the form of explicit formulas and can be efficiently calculated.We also prove that the weights on the admissible hierarchical T-meshes are positive.

Geographically and Temporally Weighted Regression in Assessing Dengue Fever Spread Factors in Yunnan Border Regions

Objective This study employs the Geographically and Temporally Weighted Regression(GTWR)model to assess the impact of meteorological elements and imported cases on dengue fever outbreaks,emphasizing the spatial-temporal variability of these factors in border regions.Methods We conducted a descriptive analysis of dengue fever’s temporal-spatial distribution in Yunnan border areas.Utilizing annual data from 2013 to 2019,with each county in the Yunnan border serving as a spatial unit,we constructed a GTWR model to investigate the determinants of dengue fever and their spatio-temporal heterogeneity in this region.Results The GTWR model,proving more effective than Ordinary Least Squares(OLS)analysis,identified significant spatial and temporal heterogeneity in factors influencing dengue fever’s spread along the Yunnan border.Notably,the GTWR model revealed a substantial variation in the relationship between indigenous dengue fever incidence,meteorological variables,and imported cases across different counties.Conclusion In the Yunnan border areas,local dengue incidence is affected by temperature,humidity,precipitation,wind speed,and imported cases,with these factors’influence exhibiting notable spatial and temporal variation.

An Improved Solov2 Based on Attention Mechanism and Weighted Loss Function for Electrical Equipment Instance Segmentation

The current existing problem of deep learning framework for the detection and segmentation of electrical equipment is dominantly related to low precision.Because of the reliable,safe and easy-to-operate technology provided by deep learning-based video surveillance for unmanned inspection of electrical equipment,this paper uses the bottleneck attention module(BAM)attention mechanism to improve the Solov2 model and proposes a new electrical equipment segmentation mode.Firstly,the BAM attention mechanism is integrated into the feature extraction network to adaptively learn the correlation between feature channels,thereby improving the expression ability of the feature map;secondly,the weighted sum of CrossEntropy Loss and Dice loss is designed as the mask loss to improve the segmentation accuracy and robustness of the model;finally,the non-maximal suppression(NMS)algorithm to better handle the overlap problem in instance segmentation.Experimental results show that the proposed method achieves an average segmentation accuracy of mAP of 80.4% on three types of electrical equipment datasets,including transformers,insulators and voltage transformers,which improve the detection accuracy by more than 5.7% compared with the original Solov2 model.The segmentation model proposed can provide a focusing technical means for the intelligent management of power systems.

Weighted multilinear p-adic Hardy operators and commutators on p-adic Herz-type spaces

In this paper,we introduce the weighted multilinear p-adic Hardy operator and weighted multilinear p-adic Ces`aro operator,we also obtain the boundedness of these two operators on the product of p-adic Herz spaces and p-adic Morrey-Herz spaces,the corresponding operator norms are also established in each case.Moreover,the boundedness of commutators of these two operators with symbols in central bounded mean oscillation spaces and Lipschitz spaces on p-adic Morrey-Herz spaces are also given.

Unsupervised Color Segmentation with Reconstructed Spatial Weighted Gaussian Mixture Model and Random Color Histogram

Image classification and unsupervised image segmentation can be achieved using the Gaussian mixture model.Although the Gaussian mixture model enhances the flexibility of image segmentation,it does not reflect spatial information and is sensitive to the segmentation parameter.In this study,we first present an efficient algorithm that incorporates spatial information into the Gaussian mixture model(GMM)without parameter estimation.The proposed model highlights the residual region with considerable information and constructs color saliency.Second,we incorporate the content-based color saliency as spatial information in the Gaussian mixture model.The segmentation is performed by clustering each pixel into an appropriate component according to the expectation maximization and maximum criteria.Finally,the random color histogram assigns a unique color to each cluster and creates an attractive color by default for segmentation.A random color histogram serves as an effective tool for data visualization and is instrumental in the creation of generative art,facilitating both analytical and aesthetic objectives.For experiments,we have used the Berkeley segmentation dataset BSDS-500 and Microsoft Research in Cambridge dataset.In the study,the proposed model showcases notable advancements in unsupervised image segmentation,with probabilistic rand index(PRI)values reaching 0.80,BDE scores as low as 12.25 and 12.02,compactness variations at 0.59 and 0.7,and variation of information(VI)reduced to 2.0 and 1.49 for the BSDS-500 and MSRC datasets,respectively,outperforming current leading-edge methods and yielding more precise segmentations.

PRODUCT TYPE OPERATORS ACTING BETWEEN WEIGHTED BERGMAN SPACES AND BLOCH TYPE SPACES

For analytic functions u,ψin the unit disk D in the complex plane and an analytic self-mapφof D,we describe in this paper the boundedness and compactness of product type operators T_(u,ψ,φ)f(z)=u(z)f(φ(z))+ψ(z)f'(φ(z)),z∈D,acting between weighted Bergman spaces induced by a doubling weight and a Bloch type space with a radial weight.

Algorithms and statistical analysis for linear structured weighted total least squares problem

Weighted total least squares(WTLS)have been regarded as the standard tool for the errors-in-variables(EIV)model in which all the elements in the observation vector and the coefficient matrix are contaminated with random errors.However,in many geodetic applications,some elements are error-free and some random observations appear repeatedly in different positions in the augmented coefficient matrix.It is called the linear structured EIV(LSEIV)model.Two kinds of methods are proposed for the LSEIV model from functional and stochastic modifications.On the one hand,the functional part of the LSEIV model is modified into the errors-in-observations(EIO)model.On the other hand,the stochastic model is modified by applying the Moore-Penrose inverse of the cofactor matrix.The algorithms are derived through the Lagrange multipliers method and linear approximation.The estimation principles and iterative formula of the parameters are proven to be consistent.The first-order approximate variance-covariance matrix(VCM)of the parameters is also derived.A numerical example is given to compare the performances of our proposed three algorithms with the STLS approach.Afterwards,the least squares(LS),total least squares(TLS)and linear structured weighted total least squares(LSWTLS)solutions are compared and the accuracy evaluation formula is proven to be feasible and effective.Finally,the LSWTLS is applied to the field of deformation analysis,which yields a better result than the traditional LS and TLS estimations.

The UNIFORM C^(0)ESTIMATE AND WEIGHTED ESTIMATE OF GENERALIZED CHRISTOFFEL-MINKOWSKI PROBLEMS

In this paper,we consider generalized Christo®el-Minkowski problems as followsσ_(k)(u_(ij)+uδ_(ij))/σ_(l)(u_(ij)+uδ_(ij))=|u^(p-1)f(x),x∈S^(n),where 0≤l≤k≤n,p-1>0 and f is positive,and we establish the weighted gradient estimate and uniform C^(0)estimate for the positive convex even solutions,which is a generalization of Guan-Xia[1]and Guan[2].

THE BOUNDEDNESS OF OPERATORS ON WEIGHTED MULTI-PARAMETER LOCAL HARDY SPACES

Though atomic decomposition is a very useful tool for studying the boundedness on Hardy spaces for some sublinear operators,untill now,the boundedness of operators on weighted Hardy spaces in a multi-parameter setting has been established only by almost orthogonality estimates.In this paper,we mainly establish the boundedness on weighted multi-parameter local Hardy spaces via atomic decomposition.

Large-Scale Multi-Objective Optimization Algorithm Based on Weighted Overlapping Grouping of Decision Variables

The large-scale multi-objective optimization algorithm(LSMOA),based on the grouping of decision variables,is an advanced method for handling high-dimensional decision variables.However,in practical problems,the interaction among decision variables is intricate,leading to large group sizes and suboptimal optimization effects;hence a large-scale multi-objective optimization algorithm based on weighted overlapping grouping of decision variables(MOEAWOD)is proposed in this paper.Initially,the decision variables are perturbed and categorized into convergence and diversity variables;subsequently,the convergence variables are subdivided into groups based on the interactions among different decision variables.If the size of a group surpasses the set threshold,that group undergoes a process of weighting and overlapping grouping.Specifically,the interaction strength is evaluated based on the interaction frequency and number of objectives among various decision variables.The decision variable with the highest interaction in the group is identified and disregarded,and the remaining variables are then reclassified into subgroups.Finally,the decision variable with the strongest interaction is added to each subgroup.MOEAWOD minimizes the interactivity between different groups and maximizes the interactivity of decision variables within groups,which contributed to the optimized direction of convergence and diversity exploration with different groups.MOEAWOD was subjected to testing on 18 benchmark large-scale optimization problems,and the experimental results demonstrate the effectiveness of our methods.Compared with the other algorithms,our method is still at an advantage.

用新激活函数加快新ZNN模型求解时变矩阵Moore-Penrose逆

基于梯度的神经网络(GNN)和张神经网络(ZNN)是两种可用于求解时变矩阵Moore-Penrose逆问题的递归神经网络。与GNN相比,ZNN的计算精度更高。此外,本文提出了一种新的ZNN模型。因此,本文主要利用带有新优化激活函数的ZNN模型来求解时变行满秩(或列满秩)矩阵Moore-Penrose逆问题。这种带有新优化激活函数的ZNN模型可以在有限时间内加速求解时变矩阵的Moore-Penrose逆。通过Lyapunov理论分析,得到了收敛时间的上限。仿真结果进一步证实了理论分析,并证明了采用新优化的激活函数的ZNN模型在求解时变矩阵Moore-Penrose逆时的有效性。

Operational optimization of copper flotation process based on the weighted Gaussian process regression and index-oriented adaptive differential evolution algorithm

Concentrate copper grade(CCG)is one of the important production indicators of copper flotation processes,and keeping the CCG at the set value is of great significance to the economic benefit of copper flotation industrial processes.This paper addresses the fluctuation problem of CCG through an operational optimization method.Firstly,a density-based affinity propagationalgorithm is proposed so that more ideal working condition categories can be obtained for the complex raw ore properties.Next,a Bayesian network(BN)is applied to explore the relationship between the operational variables and the CCG.Based on the analysis results of BN,a weighted Gaussian process regression model is constructed to predict the CCG that a higher prediction accuracy can be obtained.To ensure the predicted CCG is close to the set value with a smaller magnitude of the operation adjustments and a smaller uncertainty of the prediction results,an index-oriented adaptive differential evolution(IOADE)algorithm is proposed,and the convergence performance of IOADE is superior to the traditional differential evolution and adaptive differential evolution methods.Finally,the effectiveness and feasibility of the proposed methods are verified by the experiments on a copper flotation industrial process.

Hybrid Block Diagonalization Precoding for Multi-User Weighted Sum-Rate Maximization

This paper studies large-scale multi-input multi-output(MIMO)orthogonal frequency division multiplexing(OFDM)communications in a broadband frequency-selective channel,where a massive MIMO base station(BS)communicates with multiple users equipped with multi-antenna.We develop a hybrid precoding design to maximize the weighted sum-rate(WSR)of the users by optimizing the digital and the analog precoders alternately.For the digital part,we employ block-diagonalization to eliminate inter-user interference and apply water-filling power allocation to maximize the WSR.For the analog part,the optimization of the PSN is formulated as an unconstrained problem,which can be efficiently solved by a gradient descent method.Numerical results show that the proposed block-diagonal hybrid precoding algorithm can outperform the existing works.

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.