Nonsingular Research Articles

Higher‐order inverse mass matrices for the explicit transient analysis of heterogeneous solids

AbstractNew methods are presented for the direct computation of higher‐order inverse mass matrices (also called reciprocal mass matrices) that are used for explicit transient finite element analysis. The motivation of this work lies in the need of having appropriate sparse inverse mass matrices, which present the same structure as the consistent mass matrix, preserve the total mass, predict suitable frequency spectrum and dictate sufficiently large critical time step sizes. For an efficient evaluation of the reciprocal mass matrix, the projection matrix should be diagonal. This condition can be satisfied by adopting dual shape functions for the momentum field, generated from the same shape functions used for the displacement field. A theoretically consistent derivation of the inverse mass matrix is based on the three‐field Hamilton principle and requires the projection matrix to be evaluated from the integral of these shape functions. Unfortunately, for higher‐order FE shape functions and serendipity FE elements, the projection matrix is not positive definitive and can not be employed. Therefore, we study several lumping procedures for higher order reciprocal mass matrices considering their effect on total‐mass preserving, frequency spectra and accuracy in explicit transient simulations. The article closes with several numerical examples showing suitability of the direct inverse mass matrix in dynamics.

Hybrid synchronisation method based on inverse generalised and inverse projected high dimensional discrete chaotic systems

High-dimensional discrete chaotic systems have a wide range of engineering applications, while the chaotic synchronization method is the key to confidential communication applications. Based on the proposed discriminant theorem for high-order polynomial chaotic mapping, in this paper, a hybrid inverse generalization and inverse projection synchronization method for high-dimensional discrete chaotic systems is constructed. The method increases the flexibility of synchronization control by designing to enable the coexistence of inverse projection synchronization and inverse generalized synchronization in chaotic systems, by specifically using an invertible and adjustable constant diagonal matrix to extend the control capability of the error system. Meanwhile, this paper proposed the validity of this hybrid synchronization scheme based on the multi-stationary chaotic system for the first time, through comparison, it is certified that the multi-stationary discrete chaotic synchronization system introduced in this paper not only has complex dynamics behavior but also has a faster synchronization speed. Meanwhile, this paper proposed the validity of this hybrid synchronization scheme based on the multi-stationary chaotic system for the first time, through comparison, it is certified that the multi-stationary discrete chaotic synchronization system introduced in this paper not only has complex dynamics behavior but also has a faster synchronization speed. Finally, in this paper, the hybrid synchronization-based encryption system is also constructed, in which the transmitter switches the chaotic system between different attractors by changing the initial conditions, and then uses different chaotic attractors to mask the plaintext information. The experimental results show that the system has higher security and larger key space.

The new notion of Bohl dichotomy for non-autonomous difference equations and its relation to exponential dichotomy

In A. Czornik et al. [Spectra based on Bohl exponents and Bohl dichotomy for non-autonomous difference equations, J. Dynam. Differ. Equ. (2023)] the concept of Bohl dichotomy is introduced which is a notion of hyperbolicity for linear non-autonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set BD of systems with Bohl dichotomy and the set ED of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that ED ⊆ BD . Using a counterexample we show that the closure of ED is not contained in BD . The main result of this paper is the characterization int ⁡ BD = ED . The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of non-autonomous difference equations as a universal tool.

C-S and Strongly C-S Orthogonal Matrices

In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if ASB=0 and BAS=0, where AS is the generalized core inverse of A. The characterizations of C-S orthogonal matrices and the C-S additivity are also provided. And, the connection between the C-S orthogonality and C-S partial order has been given using their canonical form. Moreover, the concept of the strongly C-S orthogonality is defined and characterized.

Randomized Block Kaczmarz Methods for Inner Inverses of a Matrix

In this paper, two randomized block Kaczmarz methods to compute inner inverses of any rectangular matrix A are presented. These are iterative methods without matrix multiplications and their convergence is proved. The numerical results show that the proposed methods are more efficient than iterative methods involving matrix multiplications for the high-dimensional matrix.

Detection of Multiplicative False Data Injection Cyberattacks on Process Control Systems via Randomized Control Mode Switching

A fundamental problem at the intersection of process control and operations is the design of detection schemes monitoring a process for cyberattacks using operational data. Multiplicative false data injection (FDI) attacks modify operational data with a multiplicative factor and could be designed to be detection evading without in-depth process knowledge. In a prior work, we presented a control mode switching strategy that enhances the detection of multiplicative FDI attacks in processes operating at steady state (when process states evolve within a small neighborhood of the steady state). Control mode switching on the attack-free process at steady-state may induce transients and generate false alarms in the detection scheme. To minimize false alarms, we subsequently developed a control mode switch-scheduling condition for processes with an invertible output matrix. In the current work, we utilize a reachable set-based detection scheme and use randomized control mode switches to augment attack detection capabilities. The detection scheme eliminates potential false alarms occurring from control mode switching, even for processes with a non-invertible output matrix, while the randomized switching helps bolster the confidentiality of the switching schedule, preventing the design of a detection-evading “smart” attack. We present two simulation examples to illustrate attack detection without false alarms, and the merits of randomized switching (compared with scheduled switching) for the detection of a smart attack.

Energy demand forecasting of remote areas using linear regression and inverse matrix analysis

Efficient energy demand forecasting is pivotal for addressing energy challenges in remote areas of Bangladesh, where reliable access to energy resources remains a concern. This study proposes an innovative approach that combines linear regression analysis (LRA) and inverse matrix calculation (IMC) to forecast energy demand accurately in these underserved regions. By leveraging historical energy consumption data and pertinent predictors, such as meteorological conditions, population dynamics, economic indicators, and seasonal patterns, the model provides reliable forecasts. The application of the proposed methodology is demonstrated through a case study focused on remote regions of Bangladesh. The results showcase the approach's effectiveness in capturing the intricate dynamics of energy demand and its potential to inform sustainable energy management strategies in these remote areas. This research contributes to the advancement of energy planning and resource allocation in regions facing energy scarcity, fostering a path towards improved energy efficiency and development. These techniques can be applied to estimate short-term electricity demand for any rural or isolated region worldwide.

Overall signature of the metric and the cosmological constant

We consider a little known aspect of signature change, where the overall sign of the metric is allowed to change, with physical implications. We show how, in different formulations of general relativity, this type of classical signature change across boundaries with a degenerate metric can be made consistent with a change in sign (and value) of the cosmological constant Λ. In particular, the separate “mostly plus” and “mostly minus” signature sectors of Lorentzian gravity are most naturally associated with different signs of Λ. We show how this general phenomenon allows for classical solutions where the open dS patch can arise from a portion of AdS space time. These can be interpreted as classical “imaginary space” extensions of the usual Lorentzian theory, with a 2 < 0.

Determinants of Arrowhead Matrices over Finite Commutative Chain Rings

Arrowhead matrices have attracted attention due to their rich algebraic structures and numerous applications. In this paper, we focus on the enumeration of n × n arrowhead matrices with prescribed determinant over a finite field Fq and over a finite commutative chain ring R. The number of n × n arrowhead matrices over Fq of a fixed determinant a is determined for all positiveintegers n and for all elements a ∈ Fq. As applications, this result is used in the enumeration of n × n non-singular arrowhead matrices with prescribed determinant over R. Subsequently, some bounds on the number of n × n singular arrowhead matrices over R of a fixed determinant are given. Finally, some open problems are presented.

An efficient cheating-detectable secret image sharing scheme with smaller share sizes

As a new approach to image protection, polynomial-based secret image sharing (PSIS) has attracted a lot of attention from many researchers in recent decades. When SIS technology is applied in practice, it is inevitably subject to various types of attacks, with cheating being the most likely to occur. To deal with the cheating problem in the process of secret image reconstruction, several effective (k,n)-PSIS schemes have been proposed. However, most previous schemes need to sacrifice the size of the share to achieve cheating detection capability. To address this issue, an efficient cheating-detectable (k,n)-PSIS scheme with smaller share sizes is designed based on the singularity of square matrix in this paper. In comparison with previous schemes, the proposed one can not only detect the complicity of up to k−1 cheaters, but also reduce the size of shares to 1k times of the secret image. Moreover, the secret image can be recovered without loss, and it does not need to be scrambled before sharing, which could reduce the complexity of the scheme. Theoretical proofs and experiments show that the contrast and security conditions of the proposed scheme are satisfied, and the cheating detection capability is available as well.

Least Squares Minimum Class Variance Support Vector Machines

In this paper, we propose a Support Vector Machine (SVM)-type algorithm, which is statistically faster among other common algorithms in the family of SVM algorithms. The new algorithm uses distributional information of each class and, therefore, combines the benefits of using the class variance in the optimization with the least squares approach, which gives an analytic solution to the minimization problem and, therefore, is computationally efficient. We demonstrate an important property of the algorithm which allows us to address the inversion of a singular matrix in the solution. We also demonstrate through real data experiments that we improve on the computational time without losing any of the accuracy when compared to previously proposed algorithms.

Euclidean and circum-Euclidean distance matrices: characterizations and interlacing property

Motivated by the inverse formula of the distance matrix of a tree and the Moore–Penrose inverse of a circum-Euclidean distance matrix (CEDM), in this paper, we study a general real square matrix M whose Moore–Penrose inverse can be expressed as the sum of a Laplacian-like matrix L and a rank one matrix. In particular, for a symmetric hollow matrix M, under an assumption, we show that M is a Euclidean distance matrix if and only if L is positive semidefinite. Based on this, we obtain a new characterization for CEDMs involving their Moore–Penrose inverses. As an application, we show that the distance matrices of block graphs and odd-cycle-clique graphs are CEDMs. Finally, we establish an interlacing property between the eigenvalues of a Euclidean distance matrix M (including the singular case) and its associated Laplacian-like matrix L, which generalizes the interlacing property proved for the distance matrices of trees.

Singular Value and Matrix Norm Inequalities between Positive Semidefinite Matrices and Their Blocks

In this paper, we obtain some inequalities involving positive semidefinite 2×2 block matrices and their blocks.

Tridiagonal Interval Matrix: Exploring New Perspectives and Application

Tridiagonal interval matrices are relevant in diverse applications, especially in dealing with parameter estimation, optimization and circuit analysis uncertainties. This research paper aims to improve the computational efficiency of obtaining the inverse of a general tridiagonal interval matrix. This matrix is pivotal in electric circuit analysis. We achieve this by employing interval arithmetic operations in the LU decomposition process, enabling effective handling of circuit parameter uncertainties. This approach generates an inverse interval matrix that addresses uncertainties in circuit analyses.

Outcome for the group SL(2,57)

The set of all (n×n) non-singular matrices over the field F this set forms a group under the operation of matrix multiplication. This group is called the general linear group of dimension over the field F, denoted by . The determinant of these matrices is a homomorphism from into F* and the kernel of this homomorphism was the special linear group and denoted by Thus is the subgroup of which contains all matrices of determinant one. The rational valued characters of the rational representations written as a linear combination of the induced characters for the groups discuss in this paper and find the Artin indicator for this group after study the rational valued characters of the rational representations and the induced characters.

Key analysis of the hill cipher algorithm (Study of literature)

The security of the data (message) sent is very important in maintaining the confidentiality of the message. Many algorithms can be used to secure messages. Among them is the Hill Cipher algorithm. The Hill Cipher algorithm is a classic cryptographic algorithm. This algorithm uses a square matrix key. In connection with the Hill Cipher algorithm key, this literature aims to examine the matrix that can be used as a key in the encryption and decryption process in the Hill Cipher algorithm. In this literature, we take a square matrix of order 3x3 and use 36 characters (A-Z, 0-9). To produce cipher text, the method used is to carry out an encryption process based on the Hill Cipher algorithm. The encryption process is carried out by multiplying the key matrix by the plaintext. The decryption process to get the plain text back is done by multiplying the ciphertext by the inverse modulo matrix of the key matrix. The encryption process can always be carried out, but not all decryption processes can be carried out. Because not all matrices can be used as key matrices. The results of this literature show that the key matrix must be a matrix of order mxm. A matrix with determinant value = 0 cannot be used as a key. Likewise, a matrix whose determinant value is not relatively prime with the number of characters to be encrypted or decrypted cannot be used as a key matrix

Determining the Inverse of a Matrix over Min-Plus Algebra

Linear algebra over the semiring R_ε with ⊗ (plus) and ⨁ (maximum) operations which is known as max-plus algebra. One of the isomorphic with this algebra is a min-plus algebra. Min-plus algebra that is the set R_(ε^' )=R∪{ε'}, with ⊗^' (plus) and ⨁' (minimum) operations. Given a matrix whose components are elements of R_(ε^' ) is called min-plus algebra matrices. Any matrix can be connected by an inverse. In conventional algebra, a square matrix is said an invertible matrix if the det⁡〖(A)〗≠0. In contrast to max-plus algebra, a matrix is said to have inverse condition if it meets certain conditions. Some concepts from the max-plus algebra can be transformed to the min-plus, because of their structural similarity. This means that the inverse matrix concept in max-plus can be constructed into a min-plus version. Thus, this study will explain the inverse of a matrix over the min-plus algebra, property of multiplying two invertible matrices, and connection between invertible matrix and linear mapping used the literature study method, with literature sources such as books, journals, articles, and theses. The data analysis technique used in this research is qualitative data analysis technique. Then, this article has a principal result that is matrix A∈R_(ε^')^(n×n) has a right inverse if and only if there are permutations of σ and the value of λ_i&lt;ε', i∈{1,2,3,…,n} such that A=P_σ ⊗^' D(λ_i ) which is the inverse of matrices. Furthermore, if B is the correct inverse that satisfies A⊗^' B=E then B⊗^' A=E and B is uniquely determined by A.

S-Box Generation Algorithm by Constructing the Non-Singular Adjacency Matrix Using the Genetic Algorithm

S-Box Generation Algorithm by Constructing the Non-Singular Adjacency Matrix Using the Genetic Algorithm

Parametrizing spectral filters for nonsingular polynomial matrices

Two polynomial matrices are spectrally equivalent if they have the same finite and infinite elementary divisors. Spectral equivalence can be characterized using very special polynomial matrices called spectral filters. The main results in this paper concern the parametrization of spectral filters for any given pair of nonsingular spectrally equivalent polynomial matrices of arbitrary degree d. First we parametrize the set of their spectral filters of degree less than d when the given spectrally equivalent matrices have nonsingular leading coefficients. The parameter space is the subset of invertible matrices of the centralizer of any of their linearizations. When the leading coefficients are singular, we provide a parametrization of the set of spectral filters of degree d−1 and show how to obtain the spectral filters of degree less than d−1. Now the parameter space is the subset of invertible matrices of the centralizer of any linearization of the reversals with respect to a scalar that is not an eigenvalue of the given matrices. Ideas from the Theory of Linear Control Systems are used to introduce left and right realizations of matrix polynomials. These are the building blocks on which the parameterization of spectral filters is constructed. The results are illustrated with examples.

A note on products of idempotents in the ring of upper triangular infinite matrices

It is known that every singular matrix of a finite degree defined over a field can be written as a product of a finite number of idempotent matrices. Insipired by this result we investigate the same problem for N × N matrices. It turns out that in this case the analogous theorem does not hold.