Original Matrices Research Articles

Fast Type-II Hartley Transform Algorithms for Short-Length Input Sequences

This paper presents the type-II fast discrete Hartley transform (DHT-II) algorithms for input data sequences of lengths from 2 to 8. The starting point for developing the eight algorithms is the representation of DHT-II as a matrix–vector product. The underlying matrices usually have a good block structure. These matrices must then be successfully factorized to obtain a computational procedure that reduces the number of operations in computing the matrix–vector product. In some cases, it is necessary to pre-decompose the original matrices into submatrices and rearrange the rows and/or columns of the resulting matrices to find the factorizations that would substantially save the arithmetic operations. As a result of applying the pointed transformations, we synthesized the final algorithms with reduced computational complexity. The correctness of the obtained algorithmic solutions was theoretically justified using the rigorous mathematical background of each of them. Then, the complex algorithms were further tested using the MATLAB R2023b software to confirm their performance. Finally, an evaluation of the computational complexity for each obtained solution was compared with the computational complexity of the direct calculation of the matrix–vector product and existing fast DHT-II algorithms. The obtained factorizations of the DHT-II transformation matrices on average reduce the number of additions by 5% and the number of multiplications by 73% compared with the direct calculation of the matrix–vector product.

The Effects of the Incorporation of Luminescent Vanadate Nanoparticles in Lithium Borate Glass Matrices by Various Methods

The glass-ceramic materials studied in this work are designed using combinations of lithium vanadate borate glass matrices and lanthanum/rare earth (RE) vanadate nanoparticles. Three different techniques of sintering of the glass matrix and vanadate nanoparticles are investigated. The morphological characteristics and spectral properties of the glass-ceramic samples obtained by different techniques are investigated and analyzed in comparison with the properties of the original glass matrices and nanoparticles. The luminescence spectra of all glass-ceramic samples consist of a wideband glass matrix emission and the characteristic line emission of the RE ions that are incorporated into the glass matrices as nanoparticles. The RE luminescence of these glass-ceramics is promising for various optoelectronic applications.

EEG FEATURE EXTRACTION AND RECOGNITION BASED ON MULTI-FEATURE FUSION

In response to the issue of insufficient information and low classification accuracy in single-channel feature extraction algorithms for EEG signals, the paper proposed a method based on Adaptive Noise-assisted Complete Ensemble Empirical Mode Decomposition (CEEMDAN) and multi-feature fusion. Firstly, we acquired multi-channel motor imagery EEG data and preprocessed it. Next, utilized CEEMDAN with adaptive noise to adaptively decompose the channel data. Secondly, extracted features from the Intrinsic Mode Functions (IMFs) which we obtained after CEEMDAN decomposition for different feature layers and fused them, obtaining the original feature matrices for each feature layer. Finally, we fused and reduced the dimensionality of the original feature matrices from each layer, input the resulting matrix into a classification model for classification, and output the classification results. We achieve an average classification accuracy of 88.59% on the BCI Competition II Data Set III, and validate the results on self-collected single-channel motor imagery EEG data, achieving a classification accuracy of 89.72%. Experimental results indicate that multi-feature fusion outperforms single features, and the combination of CEEMDAN decomposition with multi-feature fusion achieves higher classification accuracy.

An efficient physics-dimension-based reduction method for computing frequency response functions of viscoelastically damped systems

The frequency response functions (FRFs) are of critical interest to dynamic problems, however, they suffer from computational challenges for viscoelastically damped systems. In this paper, a physics-space-based reduction method is proposed for predicting the FRFs of large-scale viscoelastically damped systems involving the standard linear solid model. A physics-dimension subspace is constructed based on original system matrices and viscoelastic parameters, which can be easily generated by using a recursive manner. A projection basis generation algorithm is then developed to generate a standard orthonormal basis within the physics-dimension subspace. With the help of the standard orthonormal basis and the moment-matching-based reduction method, a physics-space-based reduction method is proposed for efficiently predicting the FRFs of large-scale viscoelastically damped systems. Unlike the widely used state-space reduction method, the reduced system of the proposed method can preserve system’s physical structure so that the physical meaning can be captured. Using both theoretical and numerical analyses, the proposed physics-space-based method is more accurate and efficient than the state-space-based reduction method.

A tolerance index based non-cooperative behaviour managing method with minimum cost in social network group decision making

This paper introduces a novel consensus theoretical framework designed to effectively manage non-cooperative behaviour in social network group decision making (SNGDM). It addresses the challenge by considering both individuals’ willingness to adjust preferences and the associated costs of achieving consensus. To deal with this issue, the personalised individual semantics (PIS) model is employed to handle original evaluation matrices by converting linguistic terms into numerical values based on experts’ personalised opinions. Subsequently, a tolerance index (TI) is defined to reflect the willingness of experts to adjust their preferences. An improved minimum cost (MC) feedback model based on TI is established. The novelty of the proposed approach is that its integration of individual preference adjustment willingness and consensus efficiency, effectively preventing groupthink. In addition, a maximum group consensus degree optimisation model is proposed to detect non-cooperative behaviour of experts. To ensure an optimal solution for the minimum cost feedback model, a weight update method is proposed, considering the trust relationship between experts. A detailed analysis regarding the selection of tolerance thresholds to prevent over-penalisation of weights of non-collaborators is reported. Finally, comprehensive numerical and comparative analyses are presented to validate the proposed method.

Raman imaging spectroscopic solutions for microplastics advanced analysis: Insights from Choqueyapu river basin (La Paz, Bolivia)

Microplastics (MPs) are causing global concern due to their role as vectors of environmental contaminants. Evaluating their impact on environmental compartments, particularly in sediment and freshwaters, remains challenging due to difficulties in gathering chemical and morphological data. In fact, the analytical process can vary depending on the matrix considered, the non-homogeneous characteristics of MPs, and the targeted size range. Sample treatment is crucial for sediments and waters, requiring a balance between matrix removal and preservation of the MPs. Consequently, MPs often remain embedded in significant amounts of the original matrices, compromising their characterisation. In this regard, Raman spectroscopy shows promise for their comprehensive molecular analysis. However, overcoming the drawbacks associated with fluorescence from organic matter, feldspar, or clays requires considerable effort. Effective signal acquisition necessitates fine-tuning parameters, including background reduction and signal-to-noise ratio amplification. Moreover, data handling involved in chemical scanning large surfaces at high resolution is a challenging task. To overcome these drawbacks, chemometrics have demonstrated high efficacy in processing and extracting targeted information. The application of chemometrics could be relevant in environmental studies due to the large number of samples, the complexity of signal acquisition, and the dataset volumes managed. As such, this study proposes spectroscopic analytical solutions, augmented by chemical imaging and algorithmic processing, for advanced MPs analysis. A spectroscopic working approach was devised and tested through a real case study conducted in the Choqueyapu River basin (La Paz, Bolivia). This methodology allowed the morphological, molecular and quantitative identification of over 44 particles/L and 91 MPs particles/kg, in water and sediment, respectively, consisting of PE, PET, PP, PS and PMMA. MP abundance varied significantly across studied areas, spanning 2 to 4 orders of magnitude. PET fibres predominated in freshwaters, while Lipari Sector sediments were hotspots for PE and PS fragments.

Melt breccias in parautochthonous basement rocks of the Morokweng impact structure, South Africa – Evidence for a friction origin

Archaean granitoid gneisses and younger metadolerite and dolerite intrusions in a drill core from the late-Jurassic Morokweng impact structure, host mm- to m-wide aphanitic dykes that are spatially and geometrically associated with cataclasite-bearing reverse-slip faults. The dykes contain lithic and mineral clasts derived from their wallrocks. Petrographic and BSEM analysis indicates that the original dyke matrices are completely altered to a smectite-zeolite-magnetite paragenesis. However, a melt precursor is argued based on evidence for low-viscosity flow, high dyke aspect ratios and intrusive relationships into both the wallrocks and cataclasite, and the sharply contrasting matrix hydrothermal mineral grain size and textures relative to those found in the cataclasites. A subset of cm-wide pink-orange monomictic dykes have a felsic composition closely resembling their granitic wallrocks. More voluminous, polymictic, red dykes show compositions intermediate between the mafic and felsic wallrock endmembers in the drill core, which both occur in close proximity to the dykes. These hybrid compositions are interpreted to have developed either through mixing of mafic and felsic melts along faulted contacts between the doleritic and gneissic rocks, or by mechanical mixing, with or without assimilation, of a dolerite-derived melt with incohesive granitoid-derived cataclasite within the fault zones. The structural, petrographic and geochemical evidence support the dykes originating by friction melting (pseudotachylite, s.s.) caused by high-strain-rate faulting of the crater basement during the excavation and/or modification stages of the Morokweng impact, rather than by contamination of impact melt intrusions. Based on their spatial context relative to the impact melt sheet and the moderately high shock levels of their host rocks, these pseudotachylite dykes are interpreted as part of the parautochthonous peak ring of the Morokweng impact structure.

Preconditioned discontinuous Galerkin method and convection‐diffusion‐reaction problems with guaranteed bounds to resulting spectra

AbstractThis paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection‐diffusion‐reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection‐)diffusion‐reaction problems.

Spectral Tensor Layers for Communication-Free Distributed Deep Learning.

In this article, we propose a novel spectral tensor layer for communication-free distributed deep learning. The overall framework is as follows: first, we represent the data in tensor form (instead of vector form) and replace the matrix product in conventional neural networks with the tensor product, which in effect imposes certain transformed-induced structure on the original weight matrices, e.g., a block-circulant structure; then, we apply a linear transform along a certain dimension to split the original dataset into multiple spectral subdatasets; as a result, the proposed spectral tensor network consists of parallel branches where each branch is a conventional neural network trained on a spectral subdataset with ZERO communication cost. The parallel branches are directly ensembled (i.e., the weighted sum of their outputs) to generate an overall network with substantially stronger generalization capability than that of each branch. Moreover, the proposed method enjoys a byproduct of decentralization gain in terms of memory and computation, compared with traditional networks. It is a natural yet elegant solution for heterogeneous data in federated learning (FL), where data at different nodes have different resolutions. Finally, we evaluate the proposed spectral tensor networks on the MNIST, CIFAR-10, ImageNet-1K, and ImageNet-21K datasets, respectively, to verify that they simultaneously achieve communication-free distributed learning, distributed storage reduction, parallel computation speedup, and learning with multiresolution data.

A non-intrusive data-based reformulation of a hybrid projection-based model reduction method

We present a novel data-driven reformulation of the iterative SVD-rational Krylov algorithm (ISRK), in its original formulation a Petrov-Galerkin (two-sided) projection-based iterative method for model reduction combining rational Krylov subspaces (on one side) with Gramian/SVD based subspaces (on the other side). We show that in each step of ISRK, we do not necessarily require access to the original system matrices, but only to input/output data in the form of the system's transfer function, evaluated at particular values (frequencies). Numerical examples illustrate the efficiency of the new data-driven formulation.

SOFTWARE IMPLEMENTATION OF DECOMPOSITION METHODS FOR DETERMINING COMPLEX ONE-PARAMETER GENERALIZED INVERSE MOORE-PENROSE MATRICES (I)

Software implementations of analytical and numerical-analytical decomposition methods for determining complex one-parameter generalized inverse Moore-Penrose mat-rices are presented. Analytical methods are based on the first two Moore-Penrose con-ditions: in both cases the same solution to the problem is obtained, as it should be. Numerical-analytical methods are based on the obtained analytical relations and differential Pukhov transformations. Computational methods are implemented using modern infor-mation technologies, in particular Python programming language, NumPy and SymPy libraries. Software implementations consist of programs for entering the di-mensions and elements of a given matrix, implementation of an analytical method, de-termination of matrix elements derivatives and their values, calculation of matrix discretes and restoration of original matrices, as well as determining pseudo-inverse matrix. In addi-tion, the software implementation of the numerical-analytical method also contains a cus-tom implemented functions for determining the given matrix elements derivatives and for calculating the matrix discretes at the approximation center with a given scaling coeffi-cient. Algorithm flowcharts and Python source codes for all listed programs are presented. The software implementation is tested on a model example with a square matrix, for which an analytical solution is obtained using two approaches - the determination of the usual inverse one-parameter matrix and the decomposition method for determining the same matrix. This matrix is defined in exactly the same way as numerical-analytical decomposition methods.

Switched linear singular systems in discrete time: Solution theory and observability notions

We study the solution theory of linear switched singular systems. In a recent paper by Anh et al. (2019), it was highlighted that the common assumption that each mode of the switched system is index-1 is not sufficient to guarantee existence and uniqueness of solutions of the corresponding switched system and the notion of “jointly index-1” was introduced. However, until now it was not clear what conditions are actually required to guarantee existence and uniqueness of solutions if the switching signal is not considered arbitrary. In particular, we study the two relevant situations where the mode sequence is fixed (and the switching times are arbitrary) and where the whole switching signal is fixed. In both cases, we provide conditions in terms of the original system matrices which ensure existence and uniqueness of solutions. We also extend the idea of the one-step map introduced by Anh et al. (2019) to these two cases. It turns out that in the case of a fixed switching signal, the index-1 condition for the individual modes is also not necessary (in addition to not being sufficient). Furthermore, we utilize the established solution theory to provide characterizations of observability and determinability of switched singular systems.

Dirac: A command-line γ-matrix calculator

A software for simplification of Dirac matrix polynomials that arise in particle physics problems is implemented. Program summaryProgram Title: DiracCPC Library link to program files:https://doi.org/10.17632/s5w3zyhy58.1Developer's repository link:https://github.com/skutnii/diracLicensing provisions: MITProgramming language: C++20Nature of problem: Particle physics computations require simplification of Dirac matrix polynomials. Non-commutativity and tensorial nature of Dirac matrices makes the task nontrivial. Complexity of the expressions grows quickly with the number of factors making manual computations beyond a few simplest cases unfeasible and highly error-prone. Existing software solutions are either proprietary and highly expensive or have poor if any support for Dirac matrix algebra.Solution method: The problem is solved by considering a representation of Dirac matrix algebra with 5×5pseudo-matrices - matrices made of the algebra's structure constants. In this representation, multiplication of the original matrices corresponds to multiplication of pseudo-matrices and Lorentz index contraction. The problem then gets reduced to simplification of polynomials in Minkowski metric, Kronecker, and Levi-Civita symbols.Additional comments including restrictions and unusual features: The input and output have ▪-like syntax with single-expression command line input and interactive shell modes. Scripting can be used for batched input processing and advanced ▪ formatting of the results. Example scripts demonstrate this feature and provide automated correctness tests. The software supports both rational and floating-point arithmetic.

Synthetic Mueller Imaging Polarimetry

The transformation of the state of polarization of a light beam via its linear interaction with a material medium can be modeled through the Stokes–Mueller formalism. The Mueller matrix associated with a given interaction depends on many aspects of the measurement configuration. In particular, different Mueller matrices can be measured for a fixed material sample depending on the spectral profile of the light probe. For a given light probe and a given sample with inhomogeneous spatial behavior, the polarimetric descriptors of the point-to-point Mueller matrices can be mapped, leading to respective polarimetric images. The procedure can be repeated sequentially using light probes with different central frequencies. In addition, the point-to-point Mueller matrices, consecutively measured, can be combined synthetically through convex sums leading to respective new Mueller matrices, in general with increased polarimetric randomness, thus exhibiting specific values for the associated polarimetric descriptors, including the indices of polarimetric purity, and generating new polarimetric images which are different from those obtained from the original Mueller matrices. In this work, the fundamentals for such synthetic generation of additional polarimetric images are described, providing a new tool that enhances the exploitation of Mueller polarimetry.

Enhanced Antioxidant Activity and Reduced Cytotoxicity of Silver Nanoparticles Stabilized by Different Humic Materials.

The current article describes the biological activity of new biomaterials combining the "green" properties of humic substances (HSs) and silver nanoparticles. The aim is to investigate the antioxidant activity (AOA) of HS matrices (macroligands) and AgNPs stabilized with humic macroligands (HS-AgNPs). The unique chemical feature of HSs makes them very promising ligands (matrices) for AgNP stabilization. HSs have previously been shown to exert many pharmacological effects mediated by their AOA. AgNPs stabilized with HS showed a pronounced ability to bind to reactive oxygen species (ROS) in the test with ABTS. Also, higher AOA was observed for HS-AgNPs as compared to the HS matrices. In vitro cytotoxicity studies have shown that the stabilization of AgNPs with the HS matrices reduces the cytotoxicity of AgNPs. As a result of in vitro experiments with the use of 2,7-dichlorodihydrofluorescein diacetate (DCFDA), it was found that all HS materials tested and the HS-AgNPs did not exhibit prooxidant effects. Moreover, more pronounced AOA was shown for HS-AgNP samples as compared to the original HS matrices. Two putative mechanisms of the pronounced AOA of the tested compositions are proposed: firstly, the pronounced ability of HSs to inactivate ROS and, secondly, the large surface area and surface-to-volume ratio of HS-AgNPs, which facilitate electron transfer and mitigate kinetic barriers to the reduction reaction. As a result, the antioxidant properties of the tested HS-AgNPs might be of particular interest for biomedical applications aimed at inhibiting the growth of bacteria and viruses and the healing of purulent wounds.

An operator inference oriented approach for linear mechanical systems

Model order reduction techniques allow the construction of low-dimensional surrogate models that can accelerate engineering design processes. Often, these techniques are intrusive, meaning that they require direct access to underlying high-fidelity models. Accessing these models is laborious or may not even be possible in some cases. Therefore, there is an interest in developing non-intrusive model order reduction techniques to construct low-dimensional models directly from simulated or experimental data. In this work, we focus on a recent data-driven methodology, namely operator inference, that aims at inferring the reduced operators using only trajectories of high-fidelity models. We present an extension of operator inference for linear mechanical systems, preserving the underlying second-order structure. Furthermore, we study a particular case in which complete information about the external forces is available. In this formulation, the reduced operators, having certain properties inspired by the original system matrices, are enforced by adding constraints to the inference problem. We illustrate the proposed methodologies using three numerical examples.

Single phase multi-view clustering using unified graph learning and spectral representation

The most widely used multi-view clustering approaches can be classified into different groups: Subspace multi-view clustering algorithms, multi-view kernel approaches, matrix factorization approaches, and spectral clustering algorithms. Most approaches rely on combining predefined individual similarity matrices from multiple views, and then estimate the common similarity matrix. Therefore, their performance can be severely affected by noisy original similarity matrices. Moreover, most of these methods integrate different spectral projection matrices of each view together, which can also affect the clustering results.To address these shortcomings, we propose a single-phase multiview clustering method with Consensus Graph Learning and Spectral Representation (MCGLSR) in this paper. Instead of directly integrating the similarity matrices of the different views, which could introduce noise, our proposed method jointly generates similarity graphs of the views and their common similarity matrix (graph matrix) using a unified global objective function. At this stage, the similarity matrices of the different views are enforced to be sufficiently similar, which eliminates the problem of noise and promotes a clearer unified data structure. Moreover, our proposed objective function is able to recover the common spectral projection and soft cluster assignments based on the common graph structure. The proposed method takes as input a kernelized representation of the views features and directly provides the individual graphs, the common graph, the common spectral representation, and the cluster assignments. Our technique allows us to obtain the final cluster assignments without requiring an external clustering algorithm. Several real-world datasets are used to evaluate our technique and demonstrate its efficacy. The code of the proposed method will be available at the following link: https://github.com/SallyHajjar/MCGLSR.git.

Multi-omics fusion analysis models with machine learning predict survival of HER2-negative metastatic breast cancer: a multicenter prospective observational study.

Multi-omics fusion analysis models with machine learning predict survival of HER2-negative metastatic breast cancer: a multicenter prospective observational study.

Schur decomposition of several matrices

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it.

Dynamic instability mechanisms of offshore pipelines and risers

Dynamic response of offshore pipelines and risers under cross-flow (e.g. ocean currents, vessel motion etc.) has successfully been analysed. Although the pipelines are used for fluid transportation, sufficient attention has not been paid to the dynamic instability under axial flow. In this study, axial flow-induced instability mechanisms of offshore pipelines and risers are investigated. To this end, the interaction of the axial flow forces of a fluid element with the forces of the corresponding pipe element during pipeline flexural motion is analysed. Interpretation of the terms of the motion equation is provided and analogies with the model of dynamic buckling of a Euler beam are detected. Original transfer matrices are developed for the dynamic control of pipelines under axial flow. The critical values of the flow velocity causing dynamic instability and the corresponding axial flow-induced buckling modes are determined. Implementation of the method to a representative case study is provided and discussed.