Matrix Operations Research Articles

Numerical Solution of the Forward Kolmogorov Equations in Population Genetics Using Eta Functions

This paper introduces a new numerical method for solving forward Kolmogorov equations in population genetics. Since there's no simple analytical expression for the distribution of allele frequencies (DAF), we use these equations to derive it. The accuracy of solving these equations depends on the choice of base functions, so we use Eta-based functions for better approximations. By employing the operational matrix of integral, we simplify the partial differential equation to an algebraic one. The method's error bounds, stability, and validity are demonstrated through numerical examples. Finally, we apply this method to analyze the behavior of forward Kolmogorov equations under various evolutionary forces.

Application of fibonacci wavelet frame operational matrix for the analysis of arrhenius-controlled heat transfer flow in a microchannel

The effects of free convective Arrhenius-controlled heat transfer fluid in a microchannel encased by two parallel vertical plates has been examined in the current work. A new functional integration matrix has been developed by implementing Fibonacci Wavelet Frame known as Fibonacci wavelet Frame operational matrix. Primary goal of this study is to provide a unified method via FWF to compute an approximate solution to a non-linear differential equation systems for the fluid flowing in a microchannel. The leading nonlinear coupled differential equations has been tackled by this novel approach. The derived results are discussed visually and displayed with the help of various plots. Additionally, thermophysical parameters of engineering interest, such as the nusselt number and sheer stress are estimated and depicted in the graphs. It is worthwhile to note that fluid velocity and fluid temperature rises as the chemical reaction parameter value enhances. A remarkable agreement comes to light when the numerical data generated through the current method is compared with the findings from the previous work.

Self-adjusted graph based semi-supervised embedded feature selection

Graph-based semi-supervised feature selection has aroused continuous attention in processing high-dimensional data with most unlabeled and fewer data samples. Many graph-based models perform on a pre-defined graph, which is separated from the procedure of feature selection, making the model hard to select the discriminative features. To address this issue, we exploit a self-adjusted graph for semi-supervised embedded feature selection method (SAGFS), which learns an optimal sparse similarity graph to replace the pre-defined graph to alleviate the effect of data noise. SAGFS allows the learned graph itself to be adjusted according to the local geometric structure of the data and the procedure of selecting features to select the most representative features. Besides that, we introduce l2,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_{2,p}$$\\end{document}-norm to constrain the projection matrix for efficient feature selection. An efficient alternating optimization algorithm is presented, together with analyses on its convergence. Systematical experiments on several publicly datasets are performed to analyze the proposed model from several aspects, and demonstrate that our approaches outperform other comparison methods.

Temperature rise of high-speed bearing in gearbox of 5 MW wind turbine based on Bayesian-LightGBM and improved PSO-SVM troubleshooting

5MW wind turbine gearbox high-speed bearing temperature rise failure is one of the important factors affecting the stable operation of the wind turbine, accurate prediction and timely diagnosis can effectively improve the efficiency of the wind turbine. In this paper, a combined modelling wind turbine gearbox high-speed bearing temperature rise prediction method based on Bayesian-LightGBM and improved PSO-SVM is proposed with a 5 MW wind turbine as the research object. Firstly, the initial dimensionality reduction of SCADA data is performed by sparse random projection matrix, which reduces the redundant data. Secondly, feature selection is performed on the remaining data using Bayesian-LightGBM to identify 13 key input feature parameters. Then, the hyperparameters of the PSO algorithm are optimised using Bayesian algorithm and further, the optimised PSO algorithm is applied to identify the SVM parameters. Finally, a simulation experiment platform is established based on MATLAB to verify the temperature rise of high-speed bearings in gearboxes of wind turbines by example calculation and comparative analysis. The results show that the model established in this paper is effective, the prediction results are accurate and the performance is stable, and then compared with the algorithms such as PSO-SVM, SVM, BP, etc, the coefficient of determination of the algorithm is greater than 0.994 in both the training set and the test set, and the average decision percentage error is around 1.88% in both.

Modeling the Two-Strain Dynamics of COVID-19 in Ghana Using a Logistic Growth Model

Through mutation, viruses constantly change, bringing into existence new variants; SARS-CoV-2 is no different. In December 2020, variants with different characteristics that could affect transmissibility and death emerged around the world of which Ghana is not an exception. To address this new phenomenon, a two-strain mathematical model of SARS-CoV-2 was formulatedto analyzed thetransmission dynamicsin Ghana. Thedisease-free equilibriumwas calculated. The basicreproduction number, <I>R</I><sub>0</sub>= max{<I>R</I><sub>0A</sub>, <I>R</I><sub>0B</sub>} = max(0.9957945674, 1.109170840), associated with the model is computed using the next generation matrix operator. The disease-free equilibrium is found to be locally asymptotically stable when both <I>R</I><sub>0<I>A</I></sub>, <I>R</I><sub>0<I>B</sub> <sub></I></sub>< 1, but unstable otherwise. In addition to the disease-free, the boundary equilibrium for strain <I>A</I> and strain <I>B</I> was also calculated. Using the Gershgorin’s circle theorem, it was shown that the boundary equilibrium is locally asymptotically stable when both <I>R</I><sub>0<I>A</I></sub>, <I>R</I><sub>0<I>B</sub> <sub></I></sub>> 1, but unstable when otherwise. Simulations of the model were carried out. Results indicate that the government should intensify its efforts to vaccinate a larger proportion of the population and also recommends implementing comprehensive control measures, such as the use of face masks, social distancing, and contact tracing, to mitigate the spread of the disease.

Decomposition of official population projections in Brazil

Brazil’s population has continually changed over time. However, the effects of each demographic component on official population projections are still unclear. In this research, we replicate the Population Projections Revision 2018 of the Brazilian Institute of Geography and Statistics using a projection matrix model and the cohort-component method by sex, age and geographic regions. Thereafter, we established four alternative scenarios to decompose the official benchmark scenario into four demographic effects (fertility, age structure, mortality, and migration) and one residual effect. The results reveal that fertility and age structure effects have been the main drivers of the population dynamics between 2010 and 2060. On the one hand, the negative effect of fertility below replacement level has been mostly offset by the still positive effect of the age structure, even across regions in the country. On the other hand, age structure effect has been decreasing over time. By 2048, the effects of age structure, mortality and migration will no longer counterbalance fertility effects, resulting in a population decline.

On some geometric properties of sequence spaces of generalized arithmetic divisor sum function

Recently, some new sequence spaces ℓp(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p}(\\mathfrak{A}^{\\alpha })$\\end{document}(0<p<∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(0< p<\\infty )$\\end{document}, c0(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{0}(\\mathfrak{A}^{\\alpha })$\\end{document}, c(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c(\\mathfrak{A}^{\\alpha })$\\end{document}, and ℓ∞(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{\\infty }(\\mathfrak{A}^{\\alpha })$\\end{document} have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138) as matrix domains of Aα=(an,vα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{A}^{\\alpha }=(a_{n,v}^{\\alpha })$\\end{document}, where am,vα={vαρ(α)(m),v∣m,0,v∤m,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ a_{\\mathfrak{m},v}^{\\alpha }=\\left \\{ \ extstyle\\begin{array}{c@{\\quad}c@{\\quad}c} \\dfrac{v^{\\alpha }}{\\rho ^{(\\alpha )}(\\mathfrak{m})} & , & v\\mid \\mathfrak{m}, \\\\ 0 & , & v\ mid \\mathfrak{m},\\end{array}\\displaystyle \\right . $$\\end{document}and ρ(α)(m):=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho ^{(\\alpha )}(\\mathfrak{m}):=$\\end{document} sum of the αth\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha ^{\ ext{th}}$\\end{document} power of the positive divisors of m∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{m}\\in \\mathbb{N}$\\end{document}. They obtained their duals, matrix transformations and associated compact matrix operators for these matrix classes.This article deals with some geometric properties of these sequence spaces.

In-Memory modelling and simulations

A mechanism for modeling faults as addresses on smart data structures is proposed, which excludes the algorithm for modeling input test sets to obtain a test map of logical functionality. Smart data structures are represented by a logical vector and its derivatives in the form of truth tables and matrices. The test map is a matrix whose coordinates are defined by the combinations of all logical faults that are tested on the binary sets of the exhaustive test. The construction of the test map is focused on the architecture of in- memory computing based on read-write transactions, which makes the simulation mechanism economical in terms of simulation time and energy consumption due to the absence of a central processor. A logical vector as a single component of input data does not require synthesis into a technologically permitted structure of elements. Synthesis of smart data structures based on four matrix operations creates a fault test map like addresses for any logic. The proposed mechanism is focused on the service of SoC IP-cores under the control of the IEEE 1500 standard. The proposed mechanism has no analogues in the design and test industry in terms of simplicity and predictability of data structure sizes and the absence of a test set modeling algorithm.

Fractional logarithmic Chebyshev cardinal functions: Application for Caputo–Hadamard fractional optimal control problems

This study investigates a specific type of fractional optimization problems, subject to a dynamical system including the Caputo–Hadamard fractional differentiation. In this way, a novel class of basis functions known as the fractional logarithmic Chebyshev cardinal functions is introduced. An operational matrix regarding the Hadamard fractional integral of these fractional functions is derived and employed to develop an efficient numerical method for the provided problem. The established algorithm converts the solution of the primary fractional optimization problem into the solution of an algebraic system of equations by representing the state and control variables via the introduced fractional functions. Two test problems are considered to confirm the effectiveness of the suggested method. The derived outcomes of solving these examples highlight the satisfactory accuracy and efficiency of the designed method.

On the $q$-Cesaro bounded double sequence space

In this article, the new sequence space $\tilde{\mathcal{M}}_u^q$ is acquainted, described as the domain of the 4d (4-dimensional) $q$-Cesaro matrix operator, which is the $q$-analogue of the first order 4d Cesaro matrix operator, on the space of bounded double sequences. In the continuation of the study, the completeness of the new space is given, and the inclusion relation related to the space is presented. In the last two parts, the duals of the space are determined, and some matrix classes are acquired.

Efficient Convolutional Neural Networks Utilizing Fine-Grained Fast Fourier Transforms

Convolutional Neural Networks (CNNs) are among the most prevalent deep learning techniques employed across various domains. The computational complexity of CNNs is largely attributed to the convolution operations. These operations are computationally demanding and significantly impact overall model performance. Traditional CNN implementations convert convolutions into matrix operations via the im2col (image to column) technique, facilitating parallelization through advanced BLAS libraries. This study identifies and investigates a significant yet intricate pattern of data redundancy within the matrix-based representation of convolutions, a pattern that, while complex, presents opportunities for optimization. Through meticulous analysis of the redundancy inherent in the im2col approach, this paper introduces a mathematically succinct matrix representation for convolution, leading to the development of an optimized FFT-based convolution with finer FFT granularity. Benchmarking demonstrates that our approach achieves an average speedup of 14 times and a maximum speedup of 17 times compared to the regular FFT convolution. Similarly, it outperforms the Im2col+GEMM approach from NVIDIA’s cuDNN library, achieving an average speedup of three times and a maximum speedup of five times. Our FineGrained FFT convolution approach, when integrated into Caffe, a widely used deep learning framework, leads to significant performance gains. Evaluations using synthetic CNNs designed for real-world applications show an average speedup of 1.67 times. Furthermore, a modified VGG network variant achieves a speedup of 1.25 times.

A New 3D reconstruction method for pressure-sensitive paint measurements under limited optical access

Reconstruction of 3D pressure field is crucial for pressure-sensitive paint measurements in turbomachinery experiments. Existing methods impose high requirements on feature points in terms of quantity, spatial distribution, and positional accuracy, and is difficult to apply under limited optical access, such as in planar cascade wind tunnels. We propose a novel direct linear transformation method based on contour matching (CM-DLT). This method leverages model contours captured from a tilted camera perspective, which provides the basis for identifying the projection matrix linking image pixels to 3D spatial points. The subsequent optimization of the projection matrix significantly enhances the accuracy and robustness of 3D pressure field reconstruction. In a benchmark experiment, the efficiency and reprojection accuracy of 3D reconstruction are investigated. Subsequently, the CM-DLT method is successfully applied in the cascade wind tunnel experiment. The obtained 3D pressure fields of two surfaces of adjacent blades are analyzed.

Multi-view clustering with adaptive anchor and bipartite graph learning

Multi-view subspace clustering optimizes and integrates the graph structure information of each view. At present, the subspace clustering methods based on the abstract graph have better performance and improve the clustering results. Although the existing clustering algorithms have achieved excellent results, there are still four deficiencies: 1) Expensive time overhead, with most algorithms having a high time complexity; 2) Using fixed anchor points and the separation of anchor graph learning from subsequent graph construction, resulting in inadequate use of underlying graph structure between views; 3) Multi-view data have inevitable noise and in the original high-dimensional space data fusion may cause the loss of important information; 4) Most algorithms require additional post-processing to obtain clustering labels. To solve the above problems, we innovate a novel anchor-adaptive multi-view clustering algorithm based on a bipartite graph . Specifically, anchor graph learning and subspace graph construction are adaptively combined into a unified optimization framework. The projection matrix, consensus anchor matrix, and similarity matrix optimize each other to promote the clustering effect and structure a constrained bipartite graph to describe the relationship between representative points and sample points. At the same time, connectivity constraints are used to ensure that the connected components represent clusters directly. Our method has linear time complexity about sample size. Comparison experiments with the state-of-the-art algorithms demonstrate the effectiveness of the proposed method on various benchmark datasets. Moreover, our method is robust to noisy data and suitable for large-scale datasets.

Gustafson, Jeribi, Weidmann and Wolf essential spectra and generator properties for one-sided coupled operator matrices

This paper is concerned with the closedness (closability), essential spectra and generator properties for a one-sided coupled operator matrix A 0 acting in a product of Banach spaces X and Y. By combining the Browder Resolvent with the Schur factorization, some basic properties related to the operator entries of A 0 are collected, including perturbations, closedness and spectra. Utilizing these results, we present a necessary and sufficient condition for A 0 to be closed (closable), and the essential spectra as well as generator properties of A 0 are further described.

On disease and healing: a theoretical sketch

The onset and progression of a neurological disease can often be explained in terms of brain-network alteration. They can be formalized as the action of an operator representing the disease, the so-called K-operator, acting on the network. The healing process can thus be seen as the inverse of the disease mechanism. However, perfect healing is often impossible to achieve. Here, we formalize the ideal healing in terms of perturbative variation of the possible partial healing. The modeling and analytical strategy is based on techniques from theoretical physics, with the language of matrix operators. In addition, using the language of category theory, we also formalize the progressive abstraction from the reality of diseased patients to the definition of a disease and the comparison between different diseases as a natural transformation between colimits. This theoretical presentation can provide a new, interdisciplinary perspective on neurological investigation and possibly foster new theoretical-experimental developments.

Data-Driven Modeling of Tire-Soil Interaction with POD-Based Model Order Reduction

Abstract A data-driven model capable of predicting time-domain solutions of a high-fidelity tire-soil interaction model is developed to enable quick prediction of mobility capabilities on deformable terrain. The adaptive model order reduction based on the proper orthogonal decomposition (POD), for which the high-dimensional equations are projected onto the reduced subspace, is utilized as the basis for predicting the time-domain tire-soil interaction behavior. The projection-based model order reduction, however, requires many online matrix operations due to the successive updates of the nonlinear functions and Jacobians at every time step, thereby hindering the computational improvement. Therefore, a data-driven approach using a Long Short Term Memory (LSTM) neural network is introduced to predict the reduced order coordinates without the projection and time integration processes for computational speedup. With this model, a hybrid data-driven/physics-based off-road mobility model is proposed, where four separate LSTM-POD data-driven tire-soil interaction models are integrated into the physics-based multibody dynamics (MBD) vehicle model through a force-displacement coupling algorithm. By doing so, the individual data-driven tire-soil interaction model can be constructed efficiently, and the MBD and LSTM models are assembled as a single off-road mobility model and analyzed with existing off-road mobility solvers. The predictive ability and computational benefit of the proposed data-driven tire-soil interaction model with the POD-based model order reduction are examined with several numerical examples.

A feasible numerical computation based on matrix operations and collocation points to solve linear system of partial differential equations

In this work, a system of linear partial differential equations with constant and variable coefficients via Cauchy conditions is handled by applying the numerical algorithm based on operational matrices and equally-spaced collocation points. To demonstrate the applicability and efficiency of the method, four illustrative examples are tested along with absolute error, maximum absolute error, RMS error, and CPU times. The approximate solutions are compared with the analytical solutions and other numerical results in literature. The obtained numerical results are scrutinized by means of tables and graphics. These comparisons show accuracy and productivity of our method for the linear systems of partial differential equations. Besides, an algorithm is described that summarizes the formulation of the presented method. This algorithm can be adapted to well-known computer programs.

Low computational demand stray light correction method for hyperspectral imaging spectrometers

Stray light correction in hyperspectral imaging spectrometers has long been restricted by high computational requirements. This paper presents a low computational demand method, based on the matrix operations for spectrometer stray light correction, using an iterative approach to efficiently correct stray light across both spectral and spatial dimensions. The efficacy of this method is demonstrated through its application to simulated and real images, achieving an overall reduction of stray light by over 50%, with significantly reduced computation time and memory usage compared to the method by Zong et al. based on a sparse matrix [Appl. Opt. 45, 1111 (2006)10.1364/AO.45.001111APOPAI2155-3165]. By enabling stray light correction on general-purpose computers, this method enhances affordability and accessibility, promoting broader use and reducing measurement uncertainties in various hyperspectral imaging applications.

Monthly Characteristics and Source–Receptor Relationships of Anthropogenic Total Nitrate in Northeast Asia

The complex nonlinear characteristics of atmospheric chemistry necessitate the development of new methods for calculating source–receptor (S–R) relationships for secondary air pollutants. In this study, the monthly characteristics and S–R relationships of anthropogenic total nitrate (i.e., the sum of N from nitric acid, inorganic nitrate, and peroxyacetyl nitrate) in Northeast Asia were simulated and analyzed. The Community Multiscale Air Quality (CMAQ), Fifth-Generation NCAR/Penn State Mesoscale (MM5), and Sparse Matrix Operator Kernel Emissions (SMOKE) models were employed for air quality modeling, meteorological fields, and emissions processing, respectively. The study area encompassed Republic of Korea, Japan, and most of China. Five source/receptor regions were defined to derive the S–R relationships: three in China, one in Republic of Korea, and one in Japan. To produce data for the calculation of the S–R relationship, several experiments were conducted with a 20% reduction in NOx emission sources. As a result of the S–R relationships, China was rarely impacted by the other two countries. The total depositions in other countries were significantly dominated by China (i.e., 43.5% and 40.7% in Republic of Korea and Japan, respectively, and up to 82.3% in December for Republic of Korea).

Identification of fractional order time delay system with measurement noise using variable period integration operational matrix

This paper proposes a parameter identification method for fractional-order time-delay systems with measurement noise, based on the Legendre wavelet variable-period integration operational matrix. A variable-period integration operational matrix and a variable-period time-delay integration operational matrix are constructed by variable-period sampling principle. For the impulse noise, global Gaussian noise and local Gaussian noise present in the identification data, the variable-period sampling, operational matrix global coefficient decomposition and matrix local coefficient decomposition methods are used to reconstruct the variable-period integration operational matrix to separate the noise from the data; the fractional-order time-delay system is expressed as an integral equation by using the reconstructed variable-period operational matrix, this reduces the influence of measurement noise on system identification accuracy. To further improve the parameter identification accuracy, the integral equation is solved by the multi-innovation least squares algorithm. The proposed method’s effectiveness is verified through several simulations and an experimental study of a constant-temperature system.