Matrix Sequences Research Articles

A Study on Matrix Sequence with Generalized Guglielmo Numbers Components

We know that the diverse applications of matrix sequences in fields such as physics, engineering, architecture, nature, and art. Numerous authors have delved into the study of these matrix sequences in existing literature. In this study, we define and investigate the generalized Guglielmo matrix sequence. For this aim we explore four specific cases of that sequence that are called triangular matrix sequences, Lucas-triangular matrix sequences, oblong matrix sequences, and pentagonal matrix sequences. Next, we present Binet's formulas, generating functions, the summation formulas and some elementary identities for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there always exist some relationship between generalized triangular, Lucas-triangular, oblong and pentagonal matrix sequences.

The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence

In this study we investigate a generalization of modified Pell sequence, which is called (s,t)-modified Pell sequence. By considering this sequence, we define the matrix sequence whose elements are (s,t)-modified Pell numbers. Furthermore, we define various binomial transforms for modified (s,t)-Pell matrix sequence. Finally, we give some relationships for (s,t)-modified Pell matrix sequences such as Binet formulas, the generating functions and some sum formulas...

Matrix periods and competition periods of Boolean Toeplitz matrices II

This paper is a follow-up to the paper of Cheon et al. (2023) [2]. Given subsets S and T of {1,…,n−1}, an n×n Toeplitz matrix A=Tn〈S;T〉 is defined to have 1 as the (i,j)-entry if and only if j−i∈S or i−j∈T. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices A=Tn〈S;T〉 satisfying the condition (⋆) max⁡S+min⁡T≤n and min⁡S+max⁡T≤n are d+/d and 1, respectively, where d+=gcd⁡(s+t|s∈S,t∈T) and d=gcd⁡(d,min⁡S). In this paper, we claim that even if (⋆) is relaxed to the existence of elements s∈S and t∈T satisfying s+t≤n and gcd⁡(s,t)=1, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy (⋆) but the relaxed condition. For example, for any positive integers k,n with 2k+1≤n, it is easy to see that Tn〈k,n−k;k+1,n−k−1〉 does not satisfy (⋆) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence {Am(AT)m}m=1∞ is Tn〈d+,2d+,…,⌊n/d+⌋d+〉.

Novel quadriphase ZCZ sequences for QS-CDMA systems

In this article, a novel method for constructing quadriphase Zero Correlation Zone (ZCZ) sequence sets based on the mutually orthogonal complementary sets (MOCS) matrix and polyphase perfect sequence is presented. Based on the obtained MATLAB results in terms of correlation functions, it can be mentioned that the resultant ZCZ set is suitable for quasi-synchronous Code Division Multiple Access (QS-CDMA) systems. The obtained sequence sets are near-optimal concerning mathematical bound, and their construction is more flexible than other polyphase ZCZ constructions. Furthermore, their parameters can be chosen flexibly.

EEG emotion recognition based on differential entropy feature matrix through 2D-CNN-LSTM network

Emotion recognition research has attracted great interest in various research fields, and electroencephalography (EEG) is considered a promising tool for extracting emotion-related information. However, traditional EEG-based emotion recognition methods ignore the spatial correlation between electrodes. To address this problem, this paper proposes an EEG-based emotion recognition method combining differential entropy feature matrix (DEFM) and 2D-CNN-LSTM. In this work, first, the one-dimensional EEG vector sequence is converted into a two-dimensional grid matrix sequence, which corresponds to the distribution of brain regions of the EEG electrode positions, and can better characterize the spatial correlation between the EEG signals of multiple adjacent electrodes. Then, the EEG signal is divided into equal time windows, and the differential entropy (DE) of each electrode in this time window is calculated, it is combined with a two-dimensional grid matrix and differential entropy to obtain a new data representation that can capture the spatiotemporal correlation of the EEG signal, which is called DEFM. Secondly, we use 2D-CNN-LSTM to accurately identify the emotional categories contained in the EEG signals and finally classify them through the fully connected layer. Experiments are conducted on the widely used DEAP dataset. Experimental results show that the method achieves an average classification accuracy of 91.92% and 92.31% for valence and arousal, respectively. The method performs outstandingly in emotion recognition. This method effectively combines the temporal and spatial correlation of EEG signals, improves the accuracy and robustness of EEG emotion recognition, and has broad application prospects in the field of emotion classification and recognition based on EEG signals.

Modeling Excitable Cells with the EMI Equations: Spectral Analysis and Iterative Solution Strategy

In this work, we are interested in solving large linear systems stemming from the extra–membrane–intra model, which is employed for simulating excitable tissues at a cellular scale. After setting the related systems of partial differential equations equipped with proper boundary conditions, we provide its finite element discretization and focus on the resulting large linear systems. We first give a relatively complete spectral analysis using tools from the theory of Generalized Locally Toeplitz matrix sequences. The obtained spectral information is used for designing appropriate preconditioned Krylov solvers. Through numerical experiments, we show that the presented solution strategy is robust w.r.t. problem and discretization parameters, efficient and scalable.

Matrix‐less methods for the spectral approximation of large non‐Hermitian Toeplitz matrices: A concise theoretical analysis and a numerical study

AbstractIt is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non‐Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non‐Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix‐size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non‐Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high‐precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.

Medical image encryption algorithm based on a new five-dimensional multi-band multi-wing chaotic system and QR decomposition

In this study, we propose a medical image encryption algorithm based on a new five-dimensional (5D) multi-band multi-wing chaotic system and QR decomposition. First, we construct a new 5D multi-band multi-wing chaotic system through feedback control, which has a relatively large Lyapunov exponent. Second, we decompose the plaintext image matrix and chaotic sequence into an orthogonal matrix and upper triangular matrix using QR decomposition. We multiply the orthogonal matrix decomposed from the original image by the orthogonal matrix decomposed from the chaotic sequence. In this process, we use the chaotic sequence to control left and right multiplication. Simultaneously, we chaotically rearrange the elements in the upper triangular matrix using the improved Joseph loop and then multiply the two resulting matrices. Finally, we subject the product matrix to bit-level scrambling. From the theoretical analysis and simulation results, we observed that the key space of this method was relatively large, the key sensitivity was relatively strong, it resisted attacks of statistical analysis and gray value analysis well, and it had a good encryption effect for medical images.

A hardware friendly and meaningful image encryption strategy based on a binary measurement matrix and de Bruijn sequence

A hardware friendly and meaningful image encryption strategy based on a binary measurement matrix and de Bruijn sequence

Detectability of Boolean networks: A finite-time convergent matrix approach

This paper investigates the detectability of Boolean networks (including Boolean control networks (BCNs) and probabilistic Boolean networks (PBNs)) and reveals the relationship between detectability and observability from the perspective of decompositions. Firstly, the single-experiment detectability and the arbitrary-experiment detectability of BCNs are converted into the consistent reachability and the any-input sequence reachability of a logica control network through an augmented approach. Then these two kinds of detectability are testified via two defined non-augmented matrix sequences. Moreover, some algorithms are given to obtain the current state and suitable input sequences satisfying corresponding detectability. Subsequently, the non-augmented approach is used to discuss the strong detectability of PBNs, and is generalized to analyze detectability and observability decompositions of BCNs. Finally, several examples are presented to illustrate these results.

Computational ghost imaging encryption using RSA algorithm and discrete wavelet transform

Ghost imaging technology has been widely used in the field of optical image encryption owing to its unique non-local imaging characteristics. However, there are some problems such as long encryption and decrypting time, poor reliability and large number of keys. To solve these problems, this paper proposed a computational ghost imaging encryption scheme based on RSA algorithm and wavelet transform. Firstly, the representation relationship between measurement matrix and random integer sequence is established to reduce the transmission of key quantity. Secondly, RSA encryption algorithm is used to improve the security of the system. Finally, using wavelet decomposition for packet transmission. In addition, the cake cutting matrix is used as the measurement matrix to reduce the sampling times and improve the quality of reconstructed images. The simulation results show that the encryption system has strong information transmission security and good robustness. It provides some reference for image encryption and information security transmission.

Silicon infused Molybedenum di-selinide (MoSe2) nanosheets for enhanced hydrogen evolution reaction (HER) and lithium-ion battery (LIB) applications

The systematized design of two-dimensional (2D) nanocatalysts for sensible energy-efficient reactions is quite a puzzling prospective aspect in energy-related applications. In this current work, we used a top-down strategy to synthesize silicon (Si) nanoparticles using a high-energy mechanical ball milling technique. In hydrothermal reaction, these Si nanoparticles were introduced into the 2D MoSe2 matrix sequence based on varying the percentages of silicon (0–15%) vs Selinium content. The high crystalline phases of the synthesized 10% Si:MoSe2 exhibited the sheet-like morphology and silicon inserted layered structure. The pristine MoSe2@CC & Si:MoSe2@CC electrodes were prepared by tape casting method. During HER reaction, the miniscule insertion of Si into MoSe2 matrix amplified the current density until 10% addition and moved the reduction curves towards lower onset overpotential (76.23 mV) in addition to the significant Tafel value (112.3 mV/decade). The potential of the better performed electrode in HER was explored as an anode for Lithium-ion battery (LIB) applications. The electrode delivered a discharge capacity of 406.7 mAh g−1 at a current density of 100 mA/g after 100 charge-discharge cycles. Furthermore, the rationally constructed bifunctional electrode (Si:MoSe2@CC) showed strong physicochemical stability, which can be used in future sustainable energy generation and storage devices.

A stronger form of Yamamoto's theorem on singular values

For a matrix T∈Mm(C), let |T|:=T⁎T. For A∈Mm(C), we show that the matrix sequence {|An|1n}n∈N converges to a positive-semidefinite matrix H whose jth-largest eigenvalue is equal to the jth-largest eigenvalue-modulus of A (for 1≤j≤m). In fact, we give an explicit description of the spectral projections of H in terms of the eigenspaces of the diagonalizable part of A in its Jordan-Chevalley decomposition. This gives us a stronger form of Yamamoto's theorem which asserts that limn→∞⁡sj(An)1n is equal to the jth-largest eigenvalue-modulus of A, where sj(An) denotes the jth-largest singular value of An. Moreover, we also discuss applications to the asymptotic behaviour of the matrix exponential function, t↦etA.

On the quasi-stability criteria of monic matrix polynomials

This paper is a continuation of a recent investigation by Zhan and Dyachenko (2021) on the Hurwitz stability of monic matrix polynomials with algebraic techniques. By improving an inertia formula for matrix polynomials with respect to the imaginary axis, we show that, under some conditions, the quasi-stability of a monic matrix polynomial can be tested via the Hermitian nonnegative definiteness of two block Hankel matrices built from its matricial Markov parameters. Moreover, for the so-called doubly monic matrix polynomials, the quasi-stability criteria can be formulated in a much simpler form. In particular, the relationship between Hurwitz stable matrix polynomials and Stieltjes positive definite matrix sequences established in Zhan and Dyachenko (2021) is included as a special case.

Controllability Analysis of Linear Time-Varying T-H Equation with Matrix Sequence Method

A satellite is considered to be moving relative to a nominal elliptical orbit, whose dynamics are usually described by the Tschaunner-Hempel equation (T-H equation). In this paper, we propose to transform the second-order time-varying system represented by the linear T-H equation with a second-order matrix form into a first-order time-varying system. Then, the controllability of the first-order time-varying system is investigated with the matrix sequence method including e = 0 . Meanwhile, we study the observability of the first-order time-varying system with a specific form of measurement. The advantages of the matrix sequence method for controllability and observability analysis are tested by numerical examples, respectively. Dual theory is used to investigate the controllability and observability of the corresponding dual system of the T-H equation. The corresponding conclusions are obtained.

Symbol based convergence analysis in block multigrid methods with applications for Stokes problems

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied.However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated symbols. In this paper, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f, which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies.

On the limit of the sequence [formula omitted] for a multipartite tournament [formula omitted

For an integer k≥2, let A be a Boolean block matrix with blocks Aij for 1≤i,j≤k such that Aii is a zero matrix and Aij+AjiT is a matrix with all elements 1 but not both corresponding elements of Aij and AjiT equal to 1 for i≠j.Jung et al. (2023) studied the matrix sequence {Am(AT)m}m=1∞. This paper, an extension of the one above, was initiated by the observation that {Am(AT)m}m=1∞ converges if A has no zero rows. We compute the limit of the matrix sequence {Am(AT)m}m=1∞ if A has no zero rows. To this end, we take a graph theoretical approach: noting that A is the adjacency matrix of a multipartite tournament D, we compute the limit of the graph sequence Cm(D)m=1∞ when D has no vertices of outdegree 0.

Multi-Contrast Jones-Matrix Optical Coherence Tomography—The Concept, Principle, Implementation, and Applications

Jones-matrix optical coherence tomography (JM-OCT) is an extension of OCT that provides multiple types of optical contrasts of biological and clinical samples. JM-OCT measures the spatial distribution of the Jones matrix of the sample and also its time sequence. All contrasts (i.e., multi-contrast OCT images) are then computed from the Jones matrix. The contrasts obtained from the Jones matrix include the conventional and polarization-insensitive OCT intensity, cumulative and local phase retardation (birefringence), degree-of-polarization uniformity quantifying the polarization randomness of the sample, signal attenuation coefficient, sample scatterer density, Doppler OCT, OCT angiography, and dynamic OCT. JM-OCT is a generalized version of OCT because it measures the generalized form of the sample information; i.e., the Jones matrix sequence. This review summarizes the basic conception, mathematical principle, hardware implementation, signal and image processing, and biological and clinical applications of JM-OCT. Advanced technical topics, including JM-OCT-specific noise correction and quantity estimation and JM-OCT's self-calibration nature, are also described.

Guaranteed root-mean-square estimates of the forecast of matrix observations under conditions of statistical uncertainty

We investigate the problem of linear estimation of unknown mathematical expectations based on observations of realizations of random matrix sequences. Constructive mathematical methods have been developed for finding linear guaranteed RMS estimates of unknown non-stationary parameters of average values based on observations of realizations of random matrix sequences. It is shown that such guaranteed estimates are obtained either as solutions to boundary value problems for systems of linear differential equations or as solutions to the corresponding Cauchy problems. We establish the form and look for errors for the guaranteed RMS quasi-minimax estimates of the special forecast vector and parameters of unknown average values. In the presence of small perturbations of known matrices in the model of matrix observations, quasi-minimax RMS estimates are found, and their guaranteed RMS errors are obtained in the first approximation of the small parameter method. Two test examples for calculating the guaranteed root mean square estimates and their errors are given.

Comparison of circulation patterns of mumps virus in the Netherlands and Spain (2015-2020).

Mumps is a viral infection mainly characterized by inflammation of the parotid glands. Despite of vaccination programs, infections among fully vaccinated populations were reported. The World Health Organization (WHO) recommends molecular surveillance of mumps based on sequencing of the small hydrophobic (SH) gene. The use of hypervariable non-coding regions (NCR) as additional molecular markers was proposed in multiple studies. Circulation of mumps virus (MuV) genotypes and variants in different European countries were described in the literature. From 2010 to 2020, mumps outbreaks caused by genotype G were described. However, this issue has not been analyzed from a wider geographical perspective. In the present study, sequence data from MuV detected in Spain and in The Netherlands during a period of 5 years (2015- March 2020) were analyzed to gain insights in the spatiotemporal spread of MuV at a larger geographical scale than in previous local studies. A total of 1,121 SH and 262 NCR between the Matrix and Fusion protein genes (MF-NCR) sequences from both countries were included in this study. Analysis of SH revealed 106 different haplotypes (set of identical sequences). Of them, seven showing extensive circulation were considered variants. All seven were detected in both countries in coincident temporal periods. A single MF-NCR haplotype was detected in 156 sequences (59.3% of total), and was shared by five of the seven SH variants, as well as three minor MF-NCR haplotypes. All SH variants and MF-NCR haplotypes shared by both countries were detected first in Spain. Our results suggest a transmission way from south to north Europe. The higher incidence rate of mumps in Spain in spite of similar immunization coverage in both countries, could be associated with higher risk of MuV exportation. In conclusion, the present study provided novel insights into the circulation of MuV variants and haplotypes beyond the borders of single countries. In fact, the use of MF-NCR molecular tool allowed to reveal MuV transmission flows between The Netherlands and Spain. Similar studies including other (European) countries are needed to provide a broader view of the data presented in this study.