Vector Operations Research Articles

Stability criteria for nonlinear Volterra integro-dynamic matrix Sylvester systems on measure chains

In this paper, we establish sufficient conditions for various stability aspects of a nonlinear Volterra integro-dynamic matrix Sylvester system on time scales. We convert the nonlinear Volterra integro-dynamic matrix Sylvester system on time scale to an equivalent nonlinear Volterra integro-dynamic system on time scale using vectorization operator. Sufficient conditions are obtained to this system for stability, asymptotic stability, exponential stability, and strong stability. The obtained results include various stability aspects of the matrix Sylvester systems in continuous and discrete models.

Two-Dimensional Broad Learning System for Data Analytic

Recently a novel learning algorithm called the broad learning system (BLS) has been established by the theory of pseudo-inverse and compressed sensing technology. As an alternative to deep learning, BLS has gained extensive attention in data analytics. Unlike the existing deep structures based learning tools (such as deep neural networks), it is able to rapidly achieve incremental learning, and does not require tedious retraining to remodel the system. However, given image data modeling tasks, one-dimensional BLS requires a conventional vectorization operation, the dimension of input will be quite large for high resolution image. This operation may result in huge computational complexity, thereby increasing the time of model construction. To resolve this issue, a two-dimensional version of BLS, namely 2D-BLS, is first proposed to fast construct randomized learner models by using matrix inputs. Specifically, the proposed method employs left and right projecting vectors to replace the usual high dimensional input weight. Moreover, the theoretical analyses on the superiority of 2D-BLS against BLS are presented in terms of the algorithm complexity and the difference of random parameter space. Empirical results based on MNIST dataset, NORB 3D object image recognition dataset, ORL dataset, YaleB dataset and Handwritten Digit dataset demonstrate that the proposed 2D-BLS outperforms the original BLS, some extensions of BLS and some 2-D machine learning methods in terms of the efficiency of model construction. This result also shows good potential of 2D-BLS for image data analytics.

General Fractional Vector Calculus

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

New governing equations for fluid dynamics

The difference in the governing equation between inviscid and viscous flows is the introduction of viscous terms. Traditional Navier–Stokes (NS) equations define stress based on Stokes’s assumptions. In NS equations, stress is supposedly proportional to strain, and both strain and stress tensors are symmetric. There are several questions with NS equations, which include the following: 1. Both symmetric shear terms and stretching terms in strain and stress are coordinate-dependent and thus not Galilean invariant. 2. The physical meaning of both diagonal and off-diagonal elements is not clear, which is coordinate-dependent. 3. It is hard to measure strain and stress quantitatively, and viscosity is really measured by vorticity, not by symmetric strain. 4. There is no vorticity tensor in NS equations, which plays an important role in fluid flow, especially for turbulent flow. The newly proposed governing equations for fluid dynamics use the vorticity tensor only, which is anti-symmetric. The advantages include the following: 1. Both shear and stress are anti-symmetric, which are Galilean invariants and independent of coordinate rotation. 2. The physical meaning of off-diagonal elements is clear, which is anti-symmetric shear stress. 3. Viscosity coefficients are obtained by experiments, which use vorticity. 4. The vorticity term can be further decomposed into rigid rotation and anti-symmetric shear, which are important to turbulence research. 5. The computation cost for the viscous term is reduced to half as the diagonal terms are all zero and six elements are reduced to three. Several computational examples are tested, which clearly demonstrate both NS and new governing equations have exactly the same results. As shown below, the new governing equation is identical to NS equations in mathematics, but the new one has lower cost and the several advantages mentioned above, including the possibility to study turbulent flow better. It is recommended to use the new governing equation instead of NS equations. The unique definition and operation of vectors and tensors by matrix and matrix operation are also discussed in this paper.

Analysis of the vector hidden-charm tetraquark states without explicit P-waves via the QCD sum rules

In the present work, we adopt the scalar, pseudoscalar, vector, axialvector and tensor (anti)diquark operators as the elementary building blocks to construct vector and tensor local four-quark currents without introducing explicit P-waves, and explore the mass spectrum of the vector hidden-charm tetraquark states via the QCD sum rules comprehensively, and revisit the interpretations of the existing Y states in the scenario of vector tetraquark states. We resort to the energy scale formula to enhance the pole contributions and improve the convergent behaviors of the operator product expansion, and we should bear in mind that the predictions are rather sensitive to the particular energy scales which obey the uniform/same constraint. The predicted vector hidden-charm tetraquark states can be confronted to the experimental data in the future.

Generalized Kinematics Analysis of Hybrid Mechanisms Based on Screw Theory and Lie Groups Lie Algebras

Advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms, and the generalized analysis method and concise kinematics transfer matrix are obtained. In this study, first, according to the kinematics analysis of serial mechanisms, the basic principles of Lie groups and Lie algebras are briefly explained in dealing with the spatial switching and differential operations of screw vectors. Then, based on the standard ideas of Lie operations, the method for kinematics analysis of parallel mechanisms is derived, and Jacobian matrix and Hessian matrix are formulated recursively and in a closed form. Then, according to the mapping relationship between the parallel joints and corresponding equivalent series joints, a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are examined. A case study is performed to verify the calculated matrices wherein a humanoid hybrid robotic arm with a parallel-series-parallel configuration is considered as an example. The results of a simulation experiment indicate that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practically feasible.

Probing new physics effects in Λ b → Λ(→pπ −)ℓ + ℓ − decay via model-independent approach

In the last few decades, the FCNC transitions played a pivotal role in the ongoing efforts to test the Standard Model (SM) parameters. These transitions are also in the limelight to constrain the various New Physics (NP) scenarios. Regarding this, both the rare meson and baryon decays are pertinent as they occur through FCNC transitions and contain information about the heavy to light quark transition sector, where the SM has not yet examined totally. Therefore, it is interesting to analyze a phenomenological rich four-body decay channel . Here, we incorporate the NP through a model-independent approach, which modifies both the SM Wilson coefficients (WCs), and introduces the new vector, axial-vector, scalar, pseudo-scalar, and tensor operators. To check the SM and to explore the possibilities of the NP, several physical observables such as the branching ratio , forward-backward asymmetries , and , longitudinal (transverse) polarization fractions , asymmetry parameters , the angular moments, and some other angular observables, extracted from certain foldings, have been examined to compare our results with LHCb. To see which new coupling among VA, SP, and T can accommodate the available experimental data, we have examined the influence of these couplings separately to the values of these observables. For the angular moments, most of the LHCb observations lie close to the SM, and the current constraints on new VA couplings do not change the SM results significantly. For angular coefficients, , and the agreement with the experimental measurement gets improved if we invoke the tensor couplings along with SM WCs. We hope that when experimental measurements become more accurate, together with B meson decays, the will help us to put constraints on the parametric space of these WCs.

The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains

Using the spectral theory on the S-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form T=∑ℓ=13eℓaℓ(x)∂xℓ,x=(x1,x2,x3)∈Ω¯,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} T=\\sum _{\\ell =1}^3e_\\ell a_\\ell (x)\\partial _{x_\\ell }, \\ \\ \\ x=(x_1,x_2,x_3)\\in \\overline{\\Omega }, \\end{aligned}$$\\end{document}where, Omega can be either a bounded or an unbounded domain in mathbb {R}^3 whose boundary partial Omega is considered suitably regular, overline{Omega } is the closure of Omega and e_ell , for ell =1,2,3 are the imaginary units of the quaternions mathbb {H}. The operators T_ell :=a_ell (x)partial _{x_ell }, for ell =1,2,3, are called the components of T and a_1, a_2, a_3: overline{Omega } subset mathbb {R}^3rightarrow mathbb {R} are the coefficients of T. In this paper we study the generation of the fractional powers of T, denoted by P_{alpha }(T) for alpha in (0,1), when the operators T_ell , for ell =1,2,3 do not commute among themselves. To define the fractional powers P_{alpha }(T) of T we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo S-resolvent operator of T. In this paper we consider two different boundary conditions. If Omega is unbounded we consider Dirichlet boundary conditions. If Omega is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of T represented by the operator sum _{ell =1}^3a_ell ^2(x)n_ell (x) partial _{x_ell }+a(x)I, for xin partial Omega , where I is the identity operator, a:partial Omega rightarrow mathbb {R} is a given function and n=(n_1,n_2,n_3) is the outward unit normal vector to partial Omega . The Robin-type boundary conditions associated with the generation of the fractional powers of T are, in general, different from the Robin boundary conditions associated to the heat diffusion problem which leads to operators of the type sum _{ell =1}^3a_ell (x)n_ell (x) partial _{x_ell }+b(x)I, xin partial Omega . For this reason we also discuss the conditions on the coefficients a_1, a_2, a_3: overline{Omega } subset mathbb {R}^3rightarrow mathbb {R} of T and on the coefficient b:partial Omega rightarrow mathbb {R} so that the fractional powers of T are compatible with the physical Robin boundary conditions for the heat equations.

Improving graph neural network via complex-network-based anchor structure

The technique of graph/network embedding in artificial intelligence, which embeds graph data into a low-dimensional vector space in the form of machine learning, can help computer to efficiently process and analyze the complex graph data by using the vector operations. Graph Neural Network (GNN) and neural network, an end-to-end graph embedding technique based on Graph Signal Processing (GSP), which aggregates the topological information of the neighborhoods of each node in a graph, has attracted wide attention. However, most of the existing GNN models are limited to local structure information, and the location differences of nodes in the global topology are not sufficiently considered. This leads to that many nodes with the similar local topology are very difficult to distinguish. To address this problem, we propose an anchor-structure-aware GNN (AS-GNN) model to implement more accurate node distinguishment by capturing the global topology information based on the characteristics of complex networks. Anchor structure is defined as a key sub-graph composed of key nodes and edges in a graph. Taking it as a location reference, we can get the location information of each node in the global topology of graph and carry it into the embedding of nodes. By this way, the node vectors including richer topology information of graph are produced by GNN, and thus the nodes with similar local topologies can be distinguished well. To evaluate the performance of AS-GNN, we compare AS-GNN with some existing baseline models by the experiments of the classic GNN application tasks of link prediction and pairwise node classification on five real-world datasets. The experimental results have confirmed the above claims.

A quantum search algorithm of two-dimensional convex hull

Despite the rapid development of quantum research in recent years, there is very little research in computational geometry. In this paper, to achieve the convex hull of a point set in a quantum system, a quantum convex hull algorithm based on the quantum maximum or minimum searching algorithm (QUSSMA) is proposed. Firstly, the novel enhanced quantum representation of digital images is employed to represent a group of point set, and then the QUSSMA algorithm and vector operation are used to search the convex hull of the point set. In addition, the algorithm is simulated and compared with the classical algorithm. It is concluded that the quantum algorithm accelerates the classical algorithm when the value of the convex hull point is under a certain condition.

On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

A comprehensive and practical guide to the Hess and Smith constant source and dipole panel

ABSTRACT In a series of landmark reports and papers, J.L. Hess and A.M.O. Smith of Douglas Aircraft Company, Inc. introduced the quadrilateral constant source panel to solve three-dimensional nonlifting potential flow problems. Later a panel with constant dipole (doublet) distribution was added for lifting flow computations. Hess and Smith's publications provide equations for the computation of the velocities induced by the singularity distributions along with required geometric properties of the panel. Equations are presented considering an implementation in Fortran (Versions II and IV), the commonly used programming language for numerical methods at the time. The present paper builds on Hess and Smith's groundbreaking work, restating equations with modern programming languages in mind capable of fast vector operations like Fortran 95, Python or Julia. Formulas are provided for the computation of geometric properties, coordinate transformations, as well as first and second-order potential derivatives. Example input and output data allow readers to test and validate their own implementation.

Two-Dimensional ISAR Fusion Imaging of Block Structure Targets

The range resolution and azimuth resolution are restricted by the limited transmitting bandwidth and observation angle in a monostatic radar system. To improve the two-dimensional resolution of inverse synthetic aperture radar (ISAR) imaging, a fast linearized Bregman iteration for unconstrained block sparsity (FLBIUB) algorithm is proposed to achieve multiradar ISAR fusion imaging of block structure targets. First, the ISAR imaging echo data of block structure targets is established based on the geometrical theory of the diffraction model. The multiradar ISAR fusion imaging is transformed into a signal sparse representation problem by vectorization operation. Then, considering the block sparsity of the echo data of block structure targets, the FLBIUB algorithm is utilized to achieve the block sparse signal reconstruction and obtain the fusion image. The algorithm further accelerates the iterative convergence speed and improves the imaging efficiency by combining the weighted back-adding residual and condition number optimization of the basis matrix. Finally, simulation experiments show that the proposed method can effectively achieve block sparse signal reconstruction and two-dimensional multiradar ISAR fusion imaging of block structure targets.

Design a Hybrid FPGA Architecture for Visible Digital Image Watermarking in Spatial and Frequency Domain

Hybrid field programmable gate array (FPGA) implementation is proposed to improve the performance of visible image watermarking systems. The visible watermarking process is implemented as pixel by pixel operation under a spatial domain or vector operation in the frequency domain. The proposed approach is mainly designed for watermarking the images taken from digital cameras of various sizes. The padding technique is used for unequal sizes of the watermark image and original host image. The architecture data path consists of eight and six stages of pipeline capable of watermarking on the pixel-based operation and vector-based operation, respectively. The dual image watermarking architecture data path consists of a 13-stage pipeline. Pipeline and parallelism mechanisms are used to improve throughput. To improve the performance in discrete cosine transform operations at the frequency domain, the shift-add technique replaces the conventional multipliers. The clock gating technique is employed to reduce the power by preventing unnecessary switching in a path. Hardware implementation of the algorithm is tested in Intel Cyclone FPGA with the device of EP4CGX22CF19C6, with which the throughput achieved is 1.27[Formula: see text]Gbits/s with a total area utilization of 35[Formula: see text]digital signal processing (DSP) blocks, 378 look-up tables (LUTs) and 486 registers.

A Set of Integral Grid-Coding Algebraic Operations Based on GeoSOT-3D

As the amount of collected spatial information (2D/3D) increases, the real-time processing of these massive data is among the urgent issues that need to be dealt with. Discretizing the physical earth into a digital gridded earth and assigning an integral computable code to each grid has become an effective way to accelerate real-time processing. Researchers have proposed optimization algorithms for spatial calculations in specific scenarios. However, a complete set of algorithms for real-time processing using grid coding is still lacking. To address this issue, a carefully designed, integral grid-coding algebraic operation framework for GeoSOT-3D (a multilayer latitude and longitude grid model) is proposed. By converting traditional floating-point calculations based on latitude and longitude into binary operations, the complexity of the algorithm is greatly reduced. We then present the detailed algorithms that were designed, including basic operations, vector operations, code conversion operations, spatial operations, metric operations, topological relation operations, and set operations. To verify the feasibility and efficiency of the above algorithms, we developed an experimental platform using C++ language (including major algorithms, and more algorithms may be expanded in the future). Then, we generated random data and conducted experiments. The experimental results show that the computing framework is feasible and can significantly improve the efficiency of spatial processing. The algebraic operation framework is expected to support large geospatial data retrieval and analysis, and experience a revival, on top of parallel and distributed computing, in an era of large geospatial data.

Linear functional organization of the omic embedding space.

MotivationWe are increasingly accumulating complex omics data that capture different aspects of cellular functioning. A key challenge is to untangle their complexity and effectively mine them for new biomedical information. To decipher this new information, we introduce algorithms based on network embeddings. Such algorithms represent biological macromolecules as vectors in d-dimensional space, in which topologically similar molecules are embedded close in space and knowledge is extracted directly by vector operations. Recently, it has been shown that neural networks used to obtain vectorial representations (embeddings) are implicitly factorizing a mutual information matrix, called Positive Pointwise Mutual Information (PPMI) matrix. Thus, we propose the use of the PPMI matrix to represent the human protein–protein interaction (PPI) network and also introduce the graphlet degree vector PPMI matrix of the PPI network to capture different topological (structural) similarities of the nodes in the molecular network.ResultsWe generate the embeddings by decomposing these matrices with Nonnegative Matrix Tri-Factorization. We demonstrate that genes that are embedded close in these spaces have similar biological functions, so we can extract new biomedical knowledge directly by doing linear operations on their embedding vector representations. We exploit this property to predict new genes participating in protein complexes and to identify new cancer-related genes based on the cosine similarities between the vector representations of the genes. We validate 80% of our novel cancer-related gene predictions in the literature and also by patient survival curves that demonstrating that 93.3% of them have a potential clinical relevance as biomarkers of cancer.Availability and implementationCode and data are available online at https://gitlab.bsc.es/axenos/embedded-omics-data-geometry/.Supplementary information Supplementary data are available at Bioinformatics online.

New physics in b → sℓℓ decays with complex Wilson coefficients

We perform a data-driven analysis of new physics (NP) effects in exclusive b→sℓ+ℓ− decays in a model-independent effective theory approach with dimension six operators considering scalar, pseudo-scalar, vector and axial-vector operators with the corresponding Wilson coefficients (WC) taken to be complex. The analysis has been done with the most recent data while comparing the outcome with that from the relatively old data-set. We find that a left-handed quark current with vector muon coupling is the only one-operator (O9) scenario that can explain the data in both the cases with real and complex WC with a large non-zero imaginary contribution. We simultaneously apply model selection tools like cross-validation and information-theoretic approach like Akaike Information Criterion (AIC) to find out the operator or sets of operators that can best explain the available data in this channel. The O9 with complex WC is the only one-operator scenario which survives the test. However, there are a few two and three-operator scenarios (with real or complex WCs) which survive the test, and the operator O9 is common among them.

DESIGN AND SIMULATION OF NEURON-EQUIVALENTORS ARRAY FOR CREATION OF SELF-LEARNING EQUIVALENT-CONVOLUTIONAL NEURAL STRUCTURES (SLECNS)

In the paper, we consider the urgent need to create highly efficient hardware accelerators for machine learning algorithms, including convolutional and deep neural networks (CNN and DNNS), for associative memory models, clustering, and pattern recognition. We show a brief overview of our related works the advantages of the equivalent models (EM) for describing and designing bio-inspired systems. The capacity of NN on the basis of EM and of its modifications is in several times quantity of neurons. Such neural paradigms are very perspective for processing, clustering, recognition, storing large size, strongly correlated, highly noised images and creating of uncontrolled learning machine. And since the basic operational functional nodes of EM are such vector-matrix or matrix-tensor procedures with continuous-logical operations as: normalized vector operations “equivalence”, “nonequivalence”, and etc. , we consider in this paper new conceptual approaches to the design of full-scale arrays of such neuron-equivalentors (NEs) with extended functionality, including different activation functions. Our approach is based on the use of analog and mixed (with special coding) methods for implementing the required operations, building NEs (with number of synapsis from 8 up to 128 and more) and their base cells, nodes based on photosensitive elements and CMOS current mirrors. Simulation results show that the efficiency of NEs relative to the energy intensity is estimated at a value of not less than 1012 an. op. / sec on W and can be increased. The results confirm the correctness of the concept and the possibility of creating NE and MIMO structures on their basis.

Mapping out the space for new physics with leptonic and semileptonic B_{(c)} decays

Decays of B mesons with leptons in the final state offer an interesting laboratory to search for possible effects of physics from beyond the Standard Model. In view of puzzling patterns in experimental data, the violation of lepton flavour universality is an interesting option. We present a strategy, utilising ratios of leptonic and semileptonic B decays, where the elements |V_{ub}| and |V_{cb}| of the Cabibbo–Kobayashi–Maskawa (CKM) matrix cancel, to constrain the short-distance coefficients of (pseudo)-scalar, vector and tensor operator contributions. The individual branching ratios allow us then to extract also the CKM matrix elements, even in the presence of new-physics contributions. Bounds on unmeasured leptonic and semileptonic decays offer important additional constraints. In our comprehensive analysis, we give also predictions for decays which have not yet been measured in a variety of scenarios.

Algebraic Characteristics of the Matrix Conversions Class and Its Hardware Implementation

This paper derives the algebraic characteristic of the matrix transformations class by the method of isomorphic mappings on the algebraic characteristic of the class of vector transformations using the primitive program algebras. The paper also describes the hardware implementation of the matrix operations accelerator based on the obtained results. The urgency of the work is caused by the fact that today there is a rapid integration of computer technology in all spheres of society and, as a consequence, the amount of data that needs to be processed per unit time is constantly increasing. Many problems involving large amounts of complex computation are solved by methods based on matrix operations. Therefore, the study of matrix calculations and their acceleration is a very important task. In this paper, as a contribution in this direction, we propose a study of the matrix transformations class using signature operations of primitive program algebra such as multi place superposition, branching, cycling, which are refinements of the most common control structures in most high-level programming languages, and also isomorphic mapping. Signature operations of primitive program algebra in combination with basic partial-recursive matrix functions and predicates allow to realize the set of all partial-recursive matrix functions and predicates. Obtained the result on the basis of matrix primitive program algebra. Isomorphism provides the reproduction of partially recursive functions and predicates for matrix transformations as a map of partially recursive vector functions and predicates. The completeness of the algebraic system of matrix transformations is ensured due to the available results on the derivation of the algebraic system completeness for vector transformations. A name model of matrix data has been created and optimized for the development of hardware implementation. The hardware implementation provides support for signature operations of primitive software algebra and for isomorphic mapping. Hardware support for the functions of sum, multiplication and transposition of matrices, as well as the predicate of equality of two matrices is implemented. Support for signature operations of primitive software algebra is provided by the design of the control part of the matrix computer based on the RISC architecture. The hardware support of isomorphism is based on counters, they allow to intuitively implement cycling in the functions of isomorphic mappings. Fast execution of vector operations is provided by the principle of computer calculations SIMD.