Nonsingular Research Articles

Establishing the Relative Importance of Challenges in Early-stage Entrepreneurship using Analytical Hierarchy Process

Entrepreneurship is of utmost importance in a country’s economic development. Starting one’s own business is not easy and it is even more difficult to sustain in this highly competitive environment. This research helps understand the relative importance of challenging factors that hinders the growth of an enterprise resulting in an entrepreneur deciding whether to continue the business or exit it in the early phase itself. The analytic hierarchy process (AHP) is the technique used here to determine the relative position of these factors in persuading an entrepreneur’s decision to continue or exit the business. A thorough literature review identified the key challenging factors in early-stage entrepreneurship related to, Finance, Administration, Personal, Product & Market, Environment, and Human Resources. Pairwise comparison is done by collecting the opinions of eight specialists from the industry and from academia, for formulating the reciprocal matrix and analysis using AHP. The finding of this study helps entrepreneurs identify the major hurdles and their impact on the successful launch or growth of their enterprise. On the other side, this research can also be useful for the Government and other support agencies in framing policies, to wipe out hurdles that pave way for a conducive entrepreneurial environment.

Approximate Fuzzy Inverse Matrix Calculation Method using Scenario-based Inverses and Bisection

In this paper, we introduce a numerical method to construct the inverse of a square matrix whose elements are trapezoidal or triangular fuzzy numbers (FNs). A set of fuzzy linear equations is required to be solved in order to determine the fuzzy inverse matrix. The proposed technique first iteratively searches the possible solution intervals and then narrows those too-wide estimated intervals via bisection. Using interval arithmetic in left and right matrix multiplication, we aim to approximate the identity matrix as a result of product operations. The dissimilarity of the endpoints of intervals belonging to multiplication matrices with the identity matrix is considered to be an error function to be minimized. In this way, even if the entries of a matrix are uncertain, the fuzzy inverse matrix containing all inverse matrices can be found quickly with the use of computer technology. The method is explained, and comparisons are drawn with inverse stable examples from the literature.

Characterizations of the Generalized MPCEP Inverse of Rectangular Matrices

In this paper, we introduce a new generalized inverse, called the G-MPCEP inverse of a complex matrix. We investigate some characterizations, representations, and properties of this new inverse. Cramer’s rule for the solution of a singular equation A x = B is also presented. Moreover, the determinantal representations for the G-MPCEP inverse are studied. Finally, the G-MPCEP inverse being used in solving appropriate systems of linear equations is established.

Analysis of an image noise sensitivity mechanism for matrix-operation-mode-decomposition and a strong anti-noise method.

Mode decomposition (MD) based on the matrix operation (MDMO) is one of the fastest mode decomposition methods in fiber laser which has great potential for optical communications, nonlinear optics and spatial characterization applications. However, we found that the image noise sensitivity is the main limit to the accuracy of the original MDMO method, but improving the decomposition accuracy by using conventional image filtering methods is almost ineffective. By using the norm theory of matrices, the analysis result shows that both the image noise and the coefficient matrix condition number determine the total upper-bound error of the original MDMO method. Besides, the greater the condition number, the more sensitive of MDMO method is to noise. In addition, it is found that the local error of each mode information solution in the original MDMO method is different, which depends on the L2-norm of each row vector of the inverse coefficient matrix. Moreover, a more noise-insensitive MD method is achieved by screening out the information corresponding to large L2-norm. In particular, selecting the higher accuracy among the original MDMO method and such noise-insensitive method as the result in a single MD process, a strong anti-noise MD method was proposed in this paper, which displays high MD accuracy in strong noise for both near-filed and far-filed MD cases.

Magic fermions: Carroll and flat bands

The Carroll algebra is constructed as the c → 0 limit of the Poincare algebra and is associated to symmetries on generic null surfaces. In this paper, we begin investigations of Carrollian fermions or fermions defined on generic null surfaces. Due to the availability of two different (degenerate) metrics on Carroll spacetimes, there is the possibility of two different versions of Carroll Clifford algebras. We consider both possibilities and construct explicit representations of Carrollian gamma matrices and show how to build higher spacetime dimensional representations out of lower ones. Actions for Carroll fermions are constructed with these gamma matrices and the properties of these actions are investigated.We show that in condensed matter systems where the dispersion relation becomes trivial i.e. the energy is not dependent on momentum and bands flatten out, Carroll symmetry generically appears. We give explicit examples of this including that of twisted bi-layer graphene, where superconductivity appears at so called magic angles and connect this to Carroll fermions.

On the Inverse of a Square Matrix Using Logarithm

Assume that Ais an nnmatrix with real numbers as entries and the determinant of A≠ 0. In this study, we established the results of getting the inverse of a matrixA using thelogarithmic methodand used some examples to discuss the necessity of getting the inverse of a matrix using the logarithm for both positive and negative determinants in terms of a given determinant of a matrix A. If the det A > 0, we have()1Alog det Aadj A−=, where =detAdetA. However, if the det A < 0, we have()1detAAadjAlog−=−, wheredetdetAA=.

English

In next-generation high throughput satellite (NGHTS), precoding (a preprocessing technique at the gateway (GW) can mitigate the interbeam interference with no additional resource at the user terminal. However, the calculation of precoding requires matrix inversion, which involves huge complexity in terms of hardware resources with real-time calculation for near error-free results. This paper proposes a novel approach for precoding matrix calculation using the Neumann series. In this way, the algorithm does not directly compute the matrix inversion. Hence the computational complexity is highly reduced. The proposed method uses the 4th-order iterative series with the initial guess as an inverse of the diagonal matrix of the input to construct the Neumann series. This leads to fast convergence of the matrix inversion process with fewer resources. The algorithm is claimed to be generic as it is seamlessly applied to any linear precoding scheme. The paper examines the resource complexity, convergence probability and Bit Error Rate (BER) performance for the proposed method with Zero Forcing (ZF) and Regularized Zero Forcing (RZF) based linear precoding scheme. Experiment results demonstrate that the proposed algorithm accomplishes superior performance with fewer resources compared to the Neumann series and Joint iterative Newton/Chebyshev-based Neumann series methods.

Data-driven supply chain monitoring using canonical variate analysis

The monitoring of supply chain operations and the ability to detect abnormal operation in a timely manner is important to the functioning and economics of a supply chain system. This paper presents a data-driven supply chain monitoring method based on canonical variate analysis (CVA). A sparse CVA algorithm is used to address singular covariance matrices. In addition, a fault impact prediction method that utilizes the time-dependent relationships inherent in the CVA model is proposed. The proposed monitoring scheme is validated on two case studies — the classical beer distribution game, and a more complicated supply chain that has material flow in forward and reverse directions. The performance of CVA in fault detection is examined and compared against dynamic principal component analysis (DPCA) under cross-correlated and autocorrelated demand. Results show that CVA is effective in detecting abnormal supply chain operations, and achieves comparable performance to DPCA in a lower-dimensional latent space.

Spectral techniques and mathematical aspects of K 4 chain graph

The resistance distance between any two vertices of a connected graph is defined as the net effective resistance between them. An electrical network can be constructed from a graph by replacing each edge with a unit resistor. Resistance distances are computed by methods of the theory of resistive electrical networks (based on Ohm’s and Kirchhoff’s laws). The standard method to compute resistance distance is via the Moore-Penrose generalized inverse of the Laplacian matrix of the underlying graph G. In this article, we used the electric network approach and the combinatorial approach, to derive the exact expression for resistance distances between any two vertices of the ring model. By employing a recursive connection, firstly we determined all the eigenvalues and their multiplicities in relation to the corresponding Laplacian and normalised matrices. Secondly, we obtained the Kirchhoff index and multiplicative degree Kirchhoff index for . Finally, we calculated the mean first passage time and Kemeny constant of . Our computed results provide a comprehensive approach for exploring random walks on complex networks, especially biased random walks, which may also help to better understand and tackle some practical problems such as search and routing on networks.

The Moore-Penrose inverse of the distance matrix of a helm graph

In this paper, we give necessary and sufficient conditions for a real symmetric matrix and, in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank-one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86-104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n)^{-1}$. Precisely, if $n$ is odd, we find a symmetric positive semi-definite Laplacian-like matrix $L$ of order $2n-1$ and a vector $\mathbf{w}\in \mathbb{R}^{2n-1}$ such that\begin{eqnarray*}D(H_n)^{\dagger} = -\frac{1}{2}L +\frac{4}{3(n-1)}\mathbf{w}\mathbf{w^{\prime}},\end{eqnarray*}where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.

Distributed finite-time optimization algorithms with a modified Newton–Raphson method

In this paper, we propose a distributed finite-time optimization protocol for single-integrator continuous-time multi-agent systems, which utilizes a modified Newton–Raphson method. The inverse of Hessian matrix, sign function and the gradient are adopted for the design of the algorithms. The proposed algorithms make agents converge to the network optimizer under any initial state and finite time, respectively. Lyapunov method and the properties of sign function are employed to verify the convergence of the proposed algorithms. Besides, the adaptive method is also considered to avoid complex parameter conditions and centralized parameters. Finally, numerical simulations are provided to testify the presented results.

PEvaChain: Privacy-preserving ridge regression-based credit evaluation system using Hyperledger Fabric blockchain

Secure, compliant and authentic multiparty data sharing and collaborative modelling are of great significance to the accuracy of credit evaluation systems. Homomorphic encryption has the feature of supporting ciphertext calculation without sacrificing the accuracy of the model. However, the credibility and security of the existing centralized data homomorphic encryption sharing-aggregation mode have brought great hidden dangers and exacerbated the risk of private data being disclosed. Ensuring the authenticity and controllability of data are also difficulties faced by homomorphic encryption technology. To solve these problems, we propose a novel decentralized privacy-preserving credit evaluation system with trustworthy data content and calculations named PEvaChain based on Hyperledger Fabric blockchain. The PEvaChain consists of three main components: identity management, off-chain encrypted data uploading, and on-chain data security sharing-aggregation. With the help of Hyperledger Fabric’s special member access mechanism, incorporating the ciphertext-policy attribute-based encryption (CP-ABE) access control scheme avoids unauthorized access. The original data are transformed by invertible random matrices off the chain, which meets data transfer agreements requirements when data uploading and eliminates the privacy disclosure concerns of data providers to a certain extent. Paillier homomorphic encryption-based data security sharing-aggregation on the chain ensures the security of multiparty data sharing and aggregation while realizing the minimum output and utilization of the original data. Security analysis demonstrates the security and compliance of PEvaChain in terms of data access, encrypted data uploading, sharing-aggregation, and storage. The experimental results show that the proposed approach is feasible, safe and efficient.

A regularised fast recursive algorithm for fraction model identification of nonlinear dynamic systems

The fraction model has been widely used to represent a range of engineering systems. To accurately identify the fraction model is however challenging, and this paper presents a regularised fast recursive algorithm (RFRA) to identify both the true fraction model structure and the associated unknown model parameters. This is achieved first by transforming the fraction form to a linear combination of nonlinear model terms. Then the terms in the denominator are used to form a regularisation term in the cost function to offset the bias induced by the linear transformation. According to the structural risk minimisation principle based on the new cost function, the model terms are selected based on their contributions to the cost function and the coefficients are then identified recursively without explicitly solving the inverse matrix. The proposed method is proved to have low computational complexity. Simulation results confirm the efficacy of the method in fast identification of the true fraction models for the targeted nonlinear systems.

An algorithm for searching optimal variance component estimators in linear mixed models

Estimation of variance components is essential in data analysis based on a linear mixed model. Among different estimation methods, the most commonly used in practice are likelihood-based approaches: maximum likelihood and restricted maximum likelihood. These methods do not guarantee positive-unbiased estimators, however. The method of moments ensures unbiasedness but may yield negative estimates, which can cause problems when estimating fixed effects by generalized least squares. The paper develops an algorithm to derive strictly positive unbiased estimators with minimum variance in a well-defined class. The key idea is to take the method-of-moments estimator, given by a quadratic form of a symmetric matrix A, as a starting point and modify it using the class of square non-singular regularization matrices Q while preserving unbiasedness in addition to ensuring positivity. Different subclasses of structured Q are possible for convenience instead of all possible Q matrices. A search algorithm then finds a local or global optimal matrix A depending on Q and the corresponding optimal variance component estimator by minimizing the variance, a function of the unknown variance components and kurtosis parameters in the model. Availability of an optimal matrix A means that the theoretical and numerical properties of the estimator can be conveniently studied, also in small samples. The proposed method further allows finding matrices A leading to quadratic forms closely approximating the corresponding numerical values of the likelihood-based estimates. The paper investigates the dependence of variance functions on the unknown model parameters. Using two illustrative examples, Examples I and II, the paper also illustrates the use of the matrix Q and the determination of the kurtosis parameters in the search for the optimal variance component estimators by keeping the bias zero and the variance trim for the bias–variance trade-offs.

Second-Order Side-Channel Analysis Based on Orthogonal Transform Nonlinear Regression

In recent years, side-channel analysis technology has been one of the greatest threats to information security. SCA decrypts the key information in the encryption device by establishing an appropriate leakage model. As one of many leakage models, the XOR operation leakage proposed by linear regression has typical representative significance in side-channel analysis. However, linear regression may have the problem of irreversibility of a singular matrix in the modeling stage of template analysis and the problem of poor data fit in the template analysis after the cryptographic algorithm is masked. Therefore, this paper proposes a second-order template analysis method based on orthogonal transformation nonlinear regression. The irreversibility of a singular matrix and the inaccuracy of the model are solved by orthogonal transformation and adding a negative direction to the calculation of the regression coefficient matrix. In order to verify the data fitting effect of the constructed template, a comparative experiment of template analysis based on regression, Gaussian, and clustering was carried out on SAKURA-G. The experimental results show that the second-order template analysis based on orthogonal transformation nonlinear regression can complete key recovery without sacrificing the performance of regression estimation. Under the condition of high noise and high order template analysis, the established template has good universality.

Multistep Methods for Second-Order Differential-Algebraic Equations

We consider the initial value problem for linear systems of second-order ordinary differential equations with an identically singular matrix multiplying the principal part. Sufficient conditions for the existence of a unique solution are given in terms of matrix polynomials. For such problems, multistep difference schemes are proposed. An analysis of their stability and calculations of a model example are carried out.

On geometric circulant matrices with geometric sequence

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is ( g , gq , g q 2 , … , g q n − 1 ) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples.

Von Staudt constructions for skew-linear and multilinear matroids

This paper compares skew-linear and multilinear matroid representations. These are matroids that are representable over division rings and (roughly speaking) invertible matrices, respectively. The main tool is the von Staudt construction, by which we translate our problems to algebra. After giving an exposition of a simple variant of the von Staudt construction we present the following results:Undecidability of several matroid representation problems over division rings. An example of a matroid with an infinite multilinear characteristic set, but which is not multilinear in characteristic \(0\). An example of a skew-linear matroid that is not multilinear.  Mathematics Subject Classifications: 05B35, 52B40, 14N20, 52C35, 20F10, 03D40Keywords: Matroids, division ring representations, subspace arrangements, \(c\)-arrange\-ments, multilinear matroids, von Staudt constructions, word problem, Weyl algebra, Baumslag-Solitar group

Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices

The characterization of inverses of symmetric tridiagonal and block tridiagonal matrices and the development of algorithms for finding the inverse of any general non-singular tridiagonal matrix are subjects that have been studied by many authors. The results of these research usually depend on the existence of the LU factorization of a non-sigular matrix A, such that A = LU. Besides, the conditions that ensure the nonsingularity of A and its LU factorization are not promptly obtained. Then, we are going to present in this work two extremely simple sufficient conditions for existence of the LU factorization of a Toeplitz symmetric tridiagonal matrix A. We take into consideration the roots of the modified Chebyshev polynomial, and we also present an analysis based on the parameters of the Crout’s method.

A novel dynamic load identification approach for multi-source uncertain structures based on the set-theoretical wavelet transform and layered noise reduction

This paper presents a dynamic load identification method based on wavelet decomposition. This method can identify multiple types of dynamic loads. Firstly, the parametric matrix is simplified to a diagonal matrix requiring the number of modal truncations by taking advantage of the orthogonality of the principal mode shapes and the mass, stiffness and damping matrices. By using the excellent characteristics of the wavelet space, such as the integrity of the wavelet space and the multi-resolution analysis, the generalized matrix of each parameter is transformed into the wavelet transform, which can effectively express the response history with the wavelet coefficients, further improve the computational efficiency, reduce the singularity of the inverse matrix, and enhance the anti-noise property of load recognition. In order to get closer to the practical engineering application, the interval analysis method of Taylor series expansion is proposed, and the propagation analysis of uncertain variables is carried out. Finally, the wavelet semi-soft threshold method is used to filter the signal to improve the accuracy of load recognition. The applicability of the proposed method is verified by numerical examples and simulation test cases.