Symmetrical Matrices Research Articles

Some Limiting Measures Related to Free and Boolean Convolutions

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability.

Symmetrical Multidimensional Matrices and Their Inversion

The article discusses the development of the theory of the multidimensional matrices in the part regarding the symmetrical multidimensional matrices. The symmetry property is considered in terms of the structure of the multidimensional matrix. The so called kq-dimensional symmetrical matrices are considered along with the multidimensional symmetrical matrices, i. e. the matrices symmetrical to the multi-indices containing q indices. Unit, unit symmetrical and unit kq-symmetric matrices are considered. The matrices inverse to the multidimensional matrices with respect to the unit, unit symmetrical, and unit kq-symmetrical matrices are defined. It is proven that the matrices, inverse to the multidimensional matrices with respect to the unit symmetrical and unit kq-symmetrical matrices are the Moore–Penrose matrices. The distinct instances are given.

AN INNOVATIVE TECHNIQUE FOR CLASSIFICATION AND DETECTION OF SKIN CANCER USING IMAGE PROCESSING AND SVM ALGORITHM

Skin cancer is a serious health issue that affects millions of people worldwide. One in every three cancers is a skin cancer and, most people fail to identify and get a diagnosis which is why early detection is essential for effective treatment and improving patient outcomes. In recent years, computer vision and machine learning have become important tools for the automatic detection of skin cancer. One of the commonly used machine learning algorithms for this task is Support Vector Machines (SVMs). SVMs are a type of supervised learning algorithm that is used for classification tasks. In skin cancer detection, the SVM classifier is trained on a dataset of dermatoscopic images of skin lesions. The first step in the process is feature extraction, where relevant information is extracted from the images to serve as input to the SVM classifier. This information can include color, texture, and shape features, among others.
 The training and testing of the SVM classifier is then performed using a portion of the dataset, with the remainder being used to evaluate its performance. During the testing phase, the SVM classifier is used to predict the class label of each image, which can be malignant or benign. The accuracy of the classifier is evaluated by comparing its predictions to the actual class labels of the images in the evaluation dataset.
 The results of using SVMs in skin cancer detection have been promising, with high accuracy rates being achieved. This highlights the potential of SVMs as a useful tool for skin cancer screening and early detection. In conclusion, the use of SVMs in skin cancer detection provides a fast, automatic, and reliable method for detecting skin cancer, which can help to improve patient outcomes. Currently it is really very important to watch and analyse the cancer disease automatically at intervals the first stages. Irregular streaks square measure one in every of the foremost very important features (included in most of dermoscopy algorithms) that show high association with carcinoma and basal cell malignant growth malady. The diagnostic test technique for the detection is most painful and harmful so we have a tendency to tend to square measure going for the machine-driven detection. Here we have a tendency to tend to square measure practice the GLCM choices for the detection the choices of skin lesions square measure extracted normalized symmetrical grey Level Co-occurrence Matrices GLCM. GLCM based texture choices square measure extracted from each of the four classes and given as input to the Multi-Class Support vector machine that's utilized for classification purpose.

SYMMETRICAL MATRICES CHARACTERISTIC OF INTEGERS WITH INTEGER EIGEN VALUE

Estes (1992) stated that the set of eigen values from symmetrical matrices of Z is a set of totally real algebraic integers. Estes was not able to ensure that eigen values of a symmetrical matrices are integers. Mckee and Smyth (2007) observed more about the eigen value of symmetrical integer matrices. James and Chris proved that symmetrical integer matrices have eigen value with interval ranging in [-2,2]. Contrary to that, Martin and Wong (2009), stated that almost all integer matrices have no integer eigen value. Previous studies that could not show the characteristic of the eigen value made Cao and Koyunco studied and tried to determine the characteristic of symmetrical integer matrices for rank 2 and rank 3. The result shows that they have integer eigen value. In accordance to Cao and Kuyonco study, this article elaborates the characteristic of a symmetrical integer matrices for rank 4, and 5 to show the characteristic of a symmetrical integer matrices with integer eigen value for rank 1, 2, 3 and for rank 4 and 5.

A Consensus Model for Group Decision Making with Hesitant Fuzzy Information

This article presents a more improved consensus-based method for dealing with multi-person decision making (MPDM) that uses hesitant fuzzy preference relations (HFPRís) that arenít in the usual format. We proposed a Lukasiewicz transitivity (TL-transitivity)-based technique for establishing normalised hesitant fuzzy preference relations (NHFPRís) at the most essential level, after that, a model based on consensus is constructed. After that, a transitive closure formula is created in order to build TL -consistent hesitant fuzzy preference relations (HFPRís) and symmetrical matrices. Afterwards, a consistency analysis is performed to determine the degree of consistency of the data given by the decision makers (DMs), as a result, the consistency weights must be assigned to them. After combining consistency weights and preset(predeÖned) priority weights, the Önal priority weights vector of DMs is obtained (if there are any). The consensus process determines either data analysis and selection of a suitable alternative should be done directly or externally. The enhancement process aims to improve the DMís consensus measure, despite the implementation of an indicator for locating sluggish points, in the circumstance that an unfavorable agreement is achieved. Finally, a comparison case demonstrates the relevance and e§ectiveness of the proposed system. The conclusions indicate that the suggested strategy can provide insight into the MPDM system.

METHOD FOR WEIGHTS CALCULATION BASED ON INTERVAL MULTIPLICATIVE PAIRWISE COMPARISON MATRIX IN DECISIONMAKING MODELS

Context. The pairwise comparison method is a component of several decision support methodologies such as the analytic hierarchy and network processes (AHP, ANP), PROMETHEE, TOPSIS and other. This method results in the weight vector of elements of decision-making model and is based on inversely symmetrical pairwise comparison matrices. The evaluation of the elements is carried out mainly by experts under conditions of uncertainty. Therefore, modifications of this method have been explored in recent years, which are based on fuzzy and interval pairwise comparison matrices (IPCMs).
 Objective. The purpose of the work is to develop a modified method for calculation of crisp weights based on consistent and inconsistent multiplicative IPCMs of elements of decision-making model.
 Method. The proposed modified method is based on consistent and inconsistent multiplicative IPCMs, fuzzy preference programming and results in more reliable weights for the elements of decision-making model in comparison with other known methods. The differences between the proposed method and the known ones are as follows: coefficients that characterize extended intervals for ratios of weights are introduced; membership functions of fuzzy preference relations are proposed, which depend on values of IPCM elements. The introduction of these coefficients and membership functions made it possible to prove the statement about the required coincidence of the calculated weights based on the “upper” and “lower” models. The introduced coefficients can be further used to find the most inconsistent IPCM elements.
 Results. Experiments were performed with several IPCMs of different consistency level. The weights on the basis of the considered consistent and weakly consistent IPCMs obtained using the proposed and other known methods have determined the same rankings of the compared objects. Therefore, the results using the proposed method on the basis of such IPCMs do not contradict the results obtained for these types of IPCMs using other known methods. Rankings by the proposed method based on the considered highly inconsistent IPCMs are much closer to rankings based on the corresponding initial undisturbed IPCMs in comparison with rankings obtained using the known FPP method. The most inconsistent elements in the considered IPCMs are found.
 Conclusions. The developed method has shown its efficiency, results in more reliable weights and can be used for a wide range of decision support problems, scenario analysis, priority calculation, resource allocation, evaluation of decision alternatives and criteria in various application areas.

Distributed Model Predictive Control for Two-Dimensional Electric Vehicle Platoon Based on QMIX Algorithm

In this paper, an improved distributed model predictive control (IDMPC) method for the platoon consisting of electric vehicles is put forward. And the motion of the platoon is performed in two dimensions, which contains longitudinal motion and lateral motion. Firstly, a platoon model is built based on the car-following model for a single following vehicle. Then, the IDMPC strategy is designed with the consideration of multiple objectives. The symmetrical weight matrices in the IDMPC are important for the final control effect. To control each following vehicle in the platoon coordinately, the weights for the IDMPC are optimized based on the QMIX algorithm in multi-agent reinforcement learning. The QMIX can fully consider the global information in the multi-vehicle following process; therefore, the IDMPC can get optimal control variables. Finally, the simulation and experimental results verify the IDMPC. Compared to the comparison strategies, the IDMPC has the better lane tracking, stability in lateral direction and economic performance.

A Behavior-Simulated Spherical Fuzzy Extension of the Integrated Multi-Criteria Decision-Making Approach

Since its inception in 1965, fuzzy sets have been developed for many years and are widely used in multi-criteria decision making (MCDM) problems. Recently, spherical fuzzy sets (SFS), one of the most recent fuzzy sets, have been applied to extend and reinforce MCDM methods. To contribute to this development, the aim of this study is to propose a novel SFS extension of the integrated MCDM method that takes into account the psychological behavior of decision makers. In the proposed approach, the evaluation criteria are first weighted by the spherical fuzzy Decision-Making Trial and Evaluation Laboratory (SF DEMATEL) method based on symmetrical linguistic comparison matrices. Another notable advantage of this process is determining the interrelationship between the evaluation criteria. In the next stage, the spherical fuzzy Interactive Multi-Criteria Decision-Making method in the Monte Carlo simulation environment (SF TODIM’MC) was applied to evaluate the alternatives. This method allows the process of evaluating alternatives to be performed continuously with different psychological behavioral parameters, which are considered as asymmetric information. As a result, the influence of the decision maker’s psychological behavior on the evaluation results is analyzed comprehensively. The robustness of the proposed approaches is verified through their application to prioritizing post-COVID-19 operational strategies in the Vietnam logistics sector. Numerical results have provided a cause-and-effect relationship between the negative effects of the pandemic and their weights. Furthermore, the results of prioritizing the operational strategies in the simulated environment provide rankings corresponding to different levels of risk aversion. Based on the results, the proposed spherical fuzzy approach is promising for expert-based decision-making problems under psycho-behavioral influence.

Post‐fault operation strategy for single open‐phase fault in three‐phase induction motor drives

Single open-phase fault (SOPF) in three-phase induction motor (TPIM) drives leads to undesired performances and in some cases shutdown of the drive system. This research proposes a post-fault operation strategy based on direct rotor field-oriented control (RFOC) technique for wye-connected TPIM drives during SOPF. In the proposed control system, symmetrical and asymmetrical transformation matrices are utilized for stator voltage and current variables, respectively. In addition, an extended Kalman filter (EKF) is used for the rotor flux estimation, filtration of stator currents, and obtaining their time derivatives. The performance of the introduced technique including steady and transient states during different reference speeds is experimentally validated for a wye-connected TPIM drive considering SOPF. Additionally, the performance of the proposed strategy is experimentally compared with conventional RFOC strategies.

A high gain and wide bandwidth Grooved AML loaded Vivaldi Antenna design for imaging and communication applications

AbstractIn this paper, a Vivaldi antenna with a grooved, tapered slot loaded with an artificial material lens (AML) is designed and analyzed. The unique feature of the proposed antenna is to provide excellent gain (more than 5.0 dBi) over the entire ultra‐wide bandwidth within the compact area, i.e., 40 × 45 mm2. For improving the gain, symmetrical slot matrices were etched at the flare and AML unit cells were imprinted at the extended dielectric portion. Measured results confirm its operating frequency range from 4.65 to 12.65 GHz, having a peak gain of 9.2 dBi to a minimum of 5 dBi. As compared to conventional antenna, the proposed Vivaldi antenna shows an increase in gain of 2.2 dBi. Furthermore, time‐domain analysis suggests very good isolation (∼35 dB), linear phase response, and negligible group delay (∼1 ns). The compact size, wide bandwidth, good gain, stable radiation, and transient features make it a suitable choice for imaging, and communication applications.

A New Method for Numerical Integration of Higher-Order Ordinary Differential Equations Without Losing the Periodic Responses

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

Sustainable Decision Making Using a Consensus Model for Consistent Hesitant Fuzzy Preference Relations—Water Allocation Management Case Study

This paper presents an improved consensus-based procedure to handle multi-person decision making (MPDM) using hesitant fuzzy preference relations (HFPRs) which are not in normal format. At the first level, we proposed a ukasiewicz transitivity (TL-transitivity) based scheme to get normalized hesitant fuzzy preference relations (NHFPRs), subject to which, a consensus-based model is established. Then, a transitive closure formula is defined to construct TL-consistent HFPRs and creates symmetrical matrices. Following this, consistency analysis is made to estimate the consistency degrees of the information provided by the decision-makers (DMs), and consequently, to assign the consistency weights to them. The final priority weights vector of DMs is calculated after the combination of consistency weights and predefined priority weights (if any). The consensus process concludes whether the aggregation of data and selection of the best alternative should be originated or not. The enhancement mechanism is indulged in improving the consensus measure among the DMs, after introducing an identifier used to locate the weak positions, in case of the poor consensus reached. In the end, a comparative example reflects the applicability and the efficiency of proposed scheme. The results show that the proposed method can offer useful comprehension into the MPDM process.

Spectral-Based Directed Graph Network for Malware Detection

As a kind of behavioral-feature based malware detection approach, spectral graph-based deep learning has attracted considerable research efforts with the fast growth of threats of malicious programs. However, previous spectral based graph neural networks can hardly be applied to directed graphs due to the asymmetrical nature of the graph adjacency matrix. In order to address the issues of existing techniques, we propose a Spectral-based Directed Graph Network (SDGNet) architecture to classify directed graphs. In SDGNet, the weighted graph matrix normalization methods transform the graph adjacency matrix into three symmetrical graph matrices that describe different aspects of node information interaction. Then, the SDGNet extracts graph representations with different layers of multi-aspect directed GCN. On each layer, three node embeddings learned from the symmetrical graph matrices are fused together for a graph representation. The multi-layer graph representations are further concatenated together to form a comprehensive representation for classification with a combined loss function. We evaluate the proposed algorithm on a public benchmark data, and the experimental results show that it outperforms state-of-the-art algorithms.

Accounting for the effect of correlations by modulo averaging as part of neural network integration of statistical tests for small samples

The Aim of the paper is to demonstrate the advantages of taking into consideration real correlations by means of their symmetrization, which is significantly better than completely ignoring real correlations in cases of statistical estimation using small samples. Methods . Instead of real correlation numbers different in sign and modulo, identical values of correlation numbers moduli are used. It is shown that the equivalence of transformation to symmetrization is subject to the condition of identical probabilities of errors of the first and second kind for asymmetrical and equivalent symmetrical correlation matrices. The authors examine the procedure of accurate calculation of equal data correlation coefficients by trial and error and procedure of approximate calculation of symmetrical coefficients by averaging the moduli of real correlation numbers of an asymmetrical matrix. Results . The paper notes a practically linear dependence of equal probabilities of errors of the first and second kind from the dimension of the symmetrized problem being solved under logarithmic scale of the variables taken into consideration. That ultimately allows performing the examined calculations in table form using low-bit, low-power, inexpensive microcontrollers. The examined transformations have a quadratic computational complexity and come down to using pre-constructed 8-bit binary tables that associate the expected probability of errors of the first and second kind with the parameter of equal correlation of data. All the table calculations are correct and do not accumulate input data round-off errors. Conclusions . The now widely practiced complete disregard of the correlations when performing statistical analysis is very detrimental. It would be more correct to replace the matrices of real correlation numbers with symmetrical equivalents. The approximation error caused by simple averaging of the moduli of coefficient of asymmetrical matrices decreases as the square of their dimension or the square of the number of neurons that generalize classical statistical tests. When 16 and more neurons are used, the approximation error becomes negligible and can be disregarded.

Physics-preserving averaging scheme based on Grünwald-Letnikov formula for gas flow in fractured media

The heterogeneous natures of rock fabrics, due to the existence of multi-scale fractures and geological formations, led to the deviations from unity in the flux-equations fractional-exponent magnitudes. In this paper, the resulting non-Newtonian non-Darcy fractional-derivatives flux equations are solved using physics-preserving averaging schemes that incorporates both, original and shifted, Grünwald-Letnikov (GL) approximation formulas preserving the physics, by reducing the shifting effects, while maintaining the stability of the system, by keeping one shifted expansion. The proposed way of using the GL expansions also generate symmetrical coefficient matrices that significantly reduces the discretization complexities appearing with all shifted cases from literature, and help considerably in 2D and 3D systems. Systems equations derivations and discretization details are discussed. Then, the physics-preserving averaging scheme is explained and illustrated. Finally, results are presented and reviewed. Edge-based original GL expansions are unstable as also illustrated in literature. Shifted GL expansions are stable but add a lot of additional weights to both discretization sides affecting the physical accuracy. In comparison, the physics-preserving averaging scheme balances the physical accuracy and stability requirements leading to a more physically conservative scheme that is more stable than the original GL approximation but might be slightly less stable than the shifted GL approximations. It is a locally conservative Single-Continuum averaging scheme that applies a finite-volume viewpoint.

Quantum solution of coupled harmonic oscillator systems beyond normal coordinates

Normal coordinates can be defined as orthogonal linear combinations of coordinates that remove the second order couplings in coupled harmonic oscillator systems. In this paper we go further and explore the possibility of using linear although non-orthogonal coordinate transformations to get the quantum solution of coupled systems. The idea is to use as non-orthogonal linear coordinates those which allow us to express the second-order Hamiltonian matrix in a block diagonal form. To illustrate the viability of this treatment, we first apply it to a system of two bilinearly coupled harmonic oscillators which admits analytical exact solutions. The method provides in this case, as an extra mathematical result, the analytical expressions for the eigenvalues of a certain type of symmetrical tridiagonal matrices. Second, we carry out a numerical application to the Barbanis coupled oscillators system, which contains a third order coupling term and cannot be solved in closed form. We demonstrate that the non-orthogonal coordinates used, named oblique coordinates, are much more efficient than normal coordinates to determine the energy levels and eigenfunctions of this system variationally.

Mathematical Modeling of Building Constructions Using Hybrid Computing Systems

Consideration is given to algorithm for solving on computers of hybrid architecture (with multi-core processors and graphics accelerators) the systems of linear algebraic equations with band symmetrical positive definite matrices arising in the numerical modeling of 3D building constructions using the finite element method. Some results of the proposed hybrid algorithms testing for solving problems of static strength analysis of some building objects and individual constructions are presented.

A new direct time integration method for the semi-discrete parabolic equations

A direct time integration method is presented for the solution of systems of first order ordinary differential equations, which represent semi-discrete diffusion equations. The proposed method is based on the principle of the analog equation, which converts the N coupled equations into a set of N single term uncoupled first order ordinary differential equations under fictitious sources. The solution is obtained from the integral representation of the solution of the substitute single term equations. The stability and convergence of the numerical scheme is proved. The method is simple to implement. It is self-starting, unconditionally stable, accurate, while it does not exhibit numerical damping. The stability does not demand symmetrical and positive definite coefficient matrices. This is an important advantage, since the scheme can solve semi-discrete diffusion equations resulting from methods that do not produce symmetrical matrices, e.g. the boundary element method. The method applies also to equations with variable coefficients as well as to nonlinear ones. It performs well when long time durations are considered and it can be used as a practical method for integration of stiff parabolic equations in cases where widely used methods may fail. Numerical examples, including linear as well as non linear systems, are treated by the proposed method and its efficiency and accuracy are demonstrated.

Preconditioned Conjugate Gradient Method for Solution of Large Finite Element Problems on CPU and GPU

In this article the preconditioned conjugate gradient (PCG) method, realized on GPU and intended to solution of large finite element problems of structural mechanics, is considered. The mathematical formulation of problem results in solution of linear equation sets with sparse symmetrical positive definite matrices. The authors use incomplete Cholesky factorization by value approach, based on technique of sparse matrices, for creation of efficient preconditioning, which ensures a stable convergence for weakly conditioned problems mentioned above. The research focuses on realization of PCG solver on GPU with using of CUBLAS and CUSPARSE libraries. Taking into account a restricted amount of GPU core memory, the efficiency and reliability of GPU PCG solver are checked and these factors are compared with data obtained with using of CPU version of this solver, working on large amount of RAM. The real-life large problems, taken from SCAD Soft collection, are considered for such a comparison.

Stochastic Identification of Stability of Competitive Interactions in Ecosystems.

The problem of finding an optimum within a set of possibilities that represent the varying successfulness of numerous subjects competing with one another is highly relevant in the field of ecosystem interactions. We propose a method for solving this problem by the application of the Nash equilibrium concept, which is frequently used in ecology. The proposed model is based on the transformation of the initial payoff vectors of subjects that interact in different situations into a statistical set of symmetrical game matrices that consist of permutations of payoff values. The equilibrium solution is expressed as values of the probability of Nash equilibrium occurrence with uniform distribution over all possible permutations based on uncertainty of positions of payoff values in the matrix. We assume that this equilibrium solution provides information on the distribution of the degree of stability among individual situations and interacting subjects. In this paper, we validate this assumption and demonstrate its application to a dataset that represents interspecies interactions in plant ecology. We propose that the use of the Nash equilibrium in the analysis of datasets formalized according to the Pareto optimality scheme is applicable in numerous other contexts.