Matrix Operations Research Articles

Mathematical modeling and problem solving: from fundamentals to applications

The rapidly advancing fields of machine learning and mathematical modeling, greatly enhanced by the recent growth in artificial intelligence, are the focus of this special issue. This issue compiles extensively revised and improved versions of the top papers from the workshop on Mathematical Modeling and Problem Solving at PDPTA'23, the 29th International Conference on Parallel and Distributed Processing Techniques and Applications. Covering fundamental research in matrix operations and heuristic searches to real-world applications in computer vision and drug discovery, the issue underscores the crucial role of supercomputing and parallel and distributed computing infrastructure in research. Featuring nine key studies, this issue pushes forward computational technologies in mathematical modeling, refines techniques for analyzing images and time-series data, and introduces new methods in pharmaceutical and materials science, making significant contributions to these areas.

An Invitation to Higher Arity Science

Analytical thinking is dominated by binary ideas. From pairwise interactions to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this paper, we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that there is a rich universe of higher arity ideas beyond binarity waiting to be explored. To illustrate this, we discuss several higher-order phenomena appearing in a wide range of research areas, paying special attention to instances of ternary interactions. From the point of view of formal sciences and mathematics, higher arity thinking opens up new paradigms of algebra, symbolic calculus and logic. In particular, we delve into the special case of ternary structures, as that itself reveals ample surprises: new notions of associativity (or lack thereof) in ternary operations of cubic matrices, ternary isomorphisms and ternary relations, the integration problem of 3-Lie algebras, and generalizations of adjacency in 3-uniform hypergraphs. All these are open problems that strongly suggest the need to develop new ternary mathematics. Finally, we comment on potential future research directions and remark on the transdisciplinary nature of higher arity science.

The direct RBF-based partition of unity method for solving nonlinear fractional parabolic equations

This paper aims to analyze a novel localized radial basis function method known as the ’direct RBF-based partition of unity method’ for solving nonlinear fractional parabolic equations. In the proposed method, the weight functions are not operated on by the differential operators, resulting in a decrease in computational cost and algorithmic complexity. Another advantage of the direct RBF-based partition of unity method compared to the standard RBF-based partition of unity method is that in the direct method, it is possible to use discontinuous weights, which is not applicable in the standard method. Furthermore, we conduct a convergence analysis of the method and derive an error bound for the local approximation. This error bound is determined by considering conditions on the eigenvalues of the Laplacian operator matrix. To ensure the accuracy and reliability of our proposed approach, we conducted a numerical simulation and provided two numerical examples for comparison. We evaluated the consistency between the theoretical and numerical results. Our comparison between the standard RBF-based partition of unity and our proposed framework shows that the direct method is more efficient computationally, but both versions have relatively the same accuracy.

On the number of components of the essential spectrum of one 2х2 operator matrix

In this paper, a 2х2 block operator matrix H is considered as a bounded and self-adjointoperator in a Hilbert space. The location of the essential σess(H) of operator matrix H is described via the spectrum of the generalized Friedrichs model, i.e. the two- and three-particle branches of the essential spectrum σess(H) are singled out. We prove that the essential spectrum σess(H) consists of no more than six segments (components).

Time-varying graph learning from smooth and stationary graph signals with hidden nodes

Learning graph structure from observed signals over graph is a crucial task in many graph signal processing (GSP) applications. Existing approaches focus on inferring static graph, typically assuming that all nodes are available. However, these approaches ignore the situation where only a subset of nodes are available from spatiotemporal measurements, and the remaining nodes are never observed due to application-specific constraints, resulting in time-varying graph estimation accuracy declines dramatically. To handle this problem, we propose a framework that consider the presence of hidden nodes to identify time-varying graph. Specifically, we assume that the graph signals are smooth and stationary on the graphs and only a small number of edges are allowed to change between two consecutive graphs. With these assumptions, we present a challenging time-varying graph inference problem, which models the influence of hidden nodes in terms of estimating the graph-shift operator matrices that have a form of graph Laplacian. Moreover, we emphasize similar edge pattern (column-sparsity) between different graphs. Finally, our method is evaluated on both synthetic and real-world data. The experimental results demonstrate the advantage of our method when compared to existing benchmarking methods.

A direct transcription method for solving distributed-order fractional optimal control problems

This paper considers the numerical solution of distributed-order Fractional Optimal Control Problems (FOCPs). The common approaches for solving these problems involve approximating the problem with a multi-term FOCP and subsequently discretizing the resulting multi-term problem using either a direct or indirect method. However, this paper bypasses the need for approximation with a multi-term FOCP by introducing innovative approximation formulas for distributed-order fractional derivatives. These approximation formulas are directly derived from the conventional Grünwald–Letnikov, L1, and trapezoidal formulas. Building upon these formulas, we develop the relevant fractional-order distributed-order operational matrices. Utilizing the operational matrix, we easily discretize the distributed-order FOCP to a Non-Linear Programming (NLP), which can be solved by a suitable NLP-solver. Additionally, to enhance the efficiency of solving the resulting NLP, we determine the closed-forms of both the Jacobian of constraints and the gradient of the objective function. The presented method is characterized by its speed, simplicity, and ease of implementation. Moreover, it can be used to solve a wide range of distributed-order FOCPs, such as those involving nonlinear dynamics, free final times, free terminal conditions, and path constraints. By means of ample numerical tests, the accuracy and efficiency of the proposed method are assessed.

Application of flatlet oblique multiwavelets to solve the fractional stochastic integro‐differential equation using Galerkin method

This study introduces a new numerical approach named flatlet oblique multiwavelets (FOMW) to solve fractional‐order stochastic integro‐differential equation (FSI‐DE). The FOMW is used to create an operational matrix of the stochastic integral, which helps transform the FSI‐DE into a linear system of algebraic equations. This method requires only a small number of basis functions to achieve satisfactory results. The paper also includes a comprehensive discussion on error estimation and convergence analysis, which provides an upper bound of error for the proposed method. Furthermore, numerical experiments have been conducted and compared with other methods, demonstrating the superiority of FOMW in terms of accuracy and efficiency.

Total positivity and least squares problems in the Lagrange basis

SummaryThe problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included.

New Robust Model for Stability and H∞ Analysis for Interconnected Embedded Systems

This paper presents a novel approach to analyzing the robust stability of interconnected embedded systems. The paper starts by discussing the challenges associated with designing stable and robust embedded systems, particularly in the context of interconnected systems. The proposed approach combines the H∞ control theory with a new model for interconnected embedded systems, which takes into account the effects of communication delays and data losses. The paper provides a detailed mathematical analysis of the new model and presents several theorems and proofs related to its stability. The effectiveness of the proposed approach is demonstrated through several practical examples, including a networked control system and a distributed sensor network. The paper also discusses the limitations of the proposed approach and suggests several directions for future research. The proposed filter design method establishes a sufficient condition for the asymptotic stability of the error system and the satisfaction of a predefined H∞ performance index for time-invariant bounded uncertain parameters. This is achieved through the use of the strict linear matrix inequalities (LMI) approach and projection lemma. The design is formulated in terms of linear matrix inequalities (LMI). Numerical examples are provided to demonstrate the effectiveness of the proposed filter design methods.

Matrix methods for some new covering-based multigranulation fuzzy rough set models under fuzzy complementary β-neighborhoods

Fuzzy complementary β-neighborhoods (FCNs) are used to find information relevant to the target in data mining. Based on FCNs, there are six types of covering-based multigranulation fuzzy rough set (CMFRS) models have been constructed, which can be used to deal with the problem of multi-criteria information systems. These CMFRS models are calculated by set representations. However, it is time-consuming and error-prone when set representations are used to compute these CMFRS models in a large multi-criteria information system. Hence, it is important to present a novel method to compute them quickly, which is our motivation for this paper. In this paper, we present the matrix representations of six types of CMFRS models on FCNs. Firstly, some new matrices and matrix operations are given in a multi-criteria information system. Then, matrix representations of three types of optimistic CMFRSs on FCNs are proposed. Moreover, matrix approaches are also used for computing three types of pessimistic CMFRSs on FCNs. Finally, some experiments are presented to show the effectiveness of our approaches.

Sustainable cycling and mechanism of Co(II)/Co(IV)/Co(III) with S(IV) highly triggered by Co(IV) in E/Co(II)/S(IV) for enhanced removal of reactive brilliant red X-3B

The massive discharge and inadequate treatment of dye wastewater have led to the extensive accumulation of azo dyes in natural water bodies, which is a severe threat to human health. In this study, a novel homogeneous electrochemical system, denoted E/Co(II)/S(IV), was developed by the incorporation of an electric field and cobalt/sulfite. Notably, the decontamination efficiency of the E/Co(II)/S(IV) system for reactive brilliant red X-3B (X-3B) was 95.14 % within 10 min, which was accomplished by utilizing only a trace amount of Co(II). The entire process exhibited an exceptionally low energy consumption of only 0.331 kWh/m3. Single-factor experiments were conducted to explore the influence of various operational parameters and water matrix components on the system decontamination efficiency. By combining the chelator EDTA and chemical probe PMSO, the presence of Co(III) and Co(IV) was detected. The results showed that Co(IV) mainly triggered the chain reaction. The activation products of S(IV), including SO3•−, SO4•−, SO5•− and HO∙, played a dominant role in the degradation of X-3B, with SO4•− having the most substantial contribution. Under anodic oxidation, Co(II) underwent double-electron transfer, leading to the conversion of Co(II) into Co(IV). Subsequently, Co(IV) activated S(IV) via a single-electron transfer mechanism, initiating a free radical chain reaction. During the reaction, Co(IV) transformed into Co(III) and Co(II), thus achieving the Co(II)/Co(IV)/Co(III) cycle. The generation of oxygen as a result of the electrochemical process offers a viable solution for addressing the issue of hypoxia stemming from the rapid activation of S(IV). The active sites in the X-3B molecule were elucidated through density functional theory predictions, and three degradation pathways for X-3B were identified via mass spectrometry analysis. The toxicological characteristics of both X-3B and its degradation products were comprehensively assessed using toxicity estimation software tool and algal culture experiments. This research offers valuable insights into the refinement of cobalt/sulfite systems and introduces innovative concepts for electrochemical applications, which have broad-reaching implications for dye wastewater purification.

Matrices and Galois Theory

Summary This note shows how to use matrix operations to solve problems students encounter in Galois theory. In particular, we show how eigenspaces can be used to find fixed fields for automorphisms. We illustrate the methods with an example, and we also give a brief description of the general procedure. Adding this approach to a standard course in Galois theory allows students to see another example of how linear algebra appears naturally in more advanced courses.

Mersenne Matrix Operator and Its Application in $p-$Summable Sequence Space

In this study, it is introduced the regular Mersenne matrix operator which is obtained by using Mersenne numbers and examined sequence spaces described as the domain of this matrix in the space of $p$-summable sequences for $1\leq p \leq \infty$. After that, it investigated some properties and inclusion relations, established the Schauder basis, and stated $\alpha-$, $\beta-$, and $\gamma-$duals of the aforementioned spaces. Additionally, it is characterized by the matrix classes from newly described spaces to classical sequence spaces. Finally, we studied the compactness of matrix operators on related sequence spaces.

Numerical approximation of fractional SEIR epidemic model of measles and smoking model by using Fibonacci wavelets operational matrix approach

In the present article, we have considered two essential models (The epidemic model of measles and the smoking model). Across the globe, the primary cause of health problems is smoking. Measles can be controlled in infectious populations using the mathematical model representing the direct transmission of infectious diseases. The Caputo fractional derivative operator of order α∈[0,1] is used to determine the solution of the system of fractional differential equations in both models. We employed the Fibonacci wavelets operational matrix of integration and developed the Fibonacci wavelet collocation method (FWCM) to solve a nonlinear coupled system of fractional differential equations (NCSFDEs). The proposed approach transforms the given model into a system of algebraic equations. The Newton-Raphson method is considered to extract the unknown coefficient values. Numerical outcomes are obtained to illustrate the simplicity and effectiveness of the proposed method. Graphs and tables show how consistently and effectively the developed strategy works. Mathematical software called Mathematica has been used to perform all calculations. Theorems explain the convergence of this approach.

A verifiable EVM-based cross-language smart contract implementation scheme for matrix calculation

The wide application of smart contracts allows industry companies to implement some complex distributed collaborative businesses, which involve the calculation of complex functions, such as matrix operations. However, complex functions such as matrix operations are difficult to implement on Ethereum Virtual Machine (EVM)-based smart contract platforms due to their distributed security environment limitations. Existing off-chain methods often result in a significant reduction in contract execution efficiency, thus a platform software development kit interface implementation method has become a feasible way to reduce overheads, but this method cannot verify operation correctness and may leak sensitive user data. To solve the above problems, we propose a verifiable EVM-based smart contract cross-language implementation scheme for complex operations, especially matrix operations, which can guarantee operation correctness and user privacy while ensuring computational efficiency. In this scheme, a verifiable interaction process is designed to verify the computation process and results, and a matrix blinding technology is introduced to protect sensitive user data in the calculation process. The security analysis and performance tests show that the proposed scheme can satisfy the correctness and privacy of the cross-language implementation of smart contracts at a small additional efficiency cost.

Quantum algorithms for matrix operations and linear systems of equations

Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations. Using the ‘sender-receiver’ model, we propose quantum algorithms for matrix operations such as matrix-vector product, matrix-matrix product, the sum of two matrices, and the calculation of determinant and inverse matrix. We encode the matrix entries into the probability amplitudes of the pure initial states of senders. After applying proper unitary transformation to the complete quantum system, the desired result can be found in certain blocks of the receiver’s density matrix. These quantum protocols can be used as subroutines in other quantum schemes. Furthermore, we present an alternative quantum algorithm for solving linear systems of equations.

Learning physics-based reduced-order models from data using nonlinear manifolds.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Convergence of power sequences of B-operators with applications to stability

The B-operators (abbreviation for Brownian-type operators) are upper triangular 2 × 2 2\times 2 block matrix operators that satisfy certain algebraic constraints. The purpose of this paper is to characterize the weak, the strong and the uniform stability of B-operators, respectively. This is achieved by giving equivalent conditions for the convergence of powers of a B-operator in each of the corresponding topologies. A more subtle characterization is obtained for B-operators with subnormal ( 2 , 2 ) (2,2) entry. The issue of the strong stability of the adjoint of a B-operator is also discussed.

Bernoulli wavelet application to the numerical solution of Jeffery–Hamel flow problem

In this study, we introduced the Bernoulli wavelet method (BWM) to address the nonlinear Jeffery–Hamel flow problem. Utilizing a newly devised operational matrix of integration with the Bernoulli wavelet, we apply the Bernoulli wavelet expansion in conjunction with the collocation method to convert the given differential equation into a series of nonlinear equations. These equations are subsequently solved using an appropriate iterative technique. The BWM demonstrates superior accuracy when compared to existing methods like differential transformation, homotopy perturbation, variational iteration, and Runge–Kutta (R–K). Our results indicate the effectiveness of BWM in providing more precise solutions for the Jeffery–Hamel flow, a significant phenomenon with wide applications in various engineering fields, including chemical, aerospace, civil, biomechanical, mechanical, and environmental engineering. The study also delves into the influence of Reynolds number and convergent/divergent channel configurations on Jeffery–Hamel flow applications.

The Galerkin Bell method to solve the fractional optimal control problems with inequality constraints

In this manuscript, we adopt the Caputo fractional derivative approach and employ the Galerkin-Bell method to tackle fractional optimal control problems (FOCPs) with equality and inequality constraints in multi-dimensional settings. We derive the Riemann–Liouville (RL) operational matrix for Bell polynomials to facilitate our analysis. By leveraging these matrices and utilizing the Galerkin method, we transform the FOCP into a system of algebraic equations that can be readily solved. We delve into the Bell polynomials’ convergence analysis and error estimation and introduce a residual correction procedure for error estimation. To assess the effectiveness and applicability of our proposed method, we conduct experiments on four different examples and compare our results with those previously reported in the literature. Our findings reveal that the obtained results are highly satisfactory, and in some instances, we achieve the exact solution.