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A subalgebra of the full matrix algebra M n (K) , K a field, satisfying the identity [ x 1 , y 1 ] [ x 2 , y 2 ] ⋯ [ x q , y q ] = 0 is called a D q subalgebra of M n (K). In the paper we deal with the structure, conjugation and isomorphism problems of maximal D q subalgebras of M n (K). We show that a maximal D q subalgebra A of M n (K) is conjugated with a block triangular subalgebra of M n (K) with maximal commutative diagonal blocks. By analysis of conjugations, the sizes of the obtained diagonal blocks are uniquely determined. It reduces the problem of conjugation of maximal D q subalgebras of M n (K) to the analogous problem in the class of commutative subalgebras of M n (K). Further examining conjugations, in case A is contained in the upper triangular matrix algebra U n (K) , we prove that A is already in a block triangular form. We consider the isomorphism problem in a certain class of maximal D q subalgebras of M n (K) which contain all D q subalgebras of M n (K) with maximum dimension. In case K is algebraically closed, we invoke Jacobson's characterization of maximal commutative subalgebras of M n (K) with maximum (K -)dimension to show that isomorphic subalgebras in this class are already conjugated. To illustrate it, we invoke results from [19] and find all isomorphism (equivalently conjugation) classes of D q subalgebras of M n (K) with maximum possible dimension, in case K is algebraically closed. [ABSTRACT FROM AUTHOR]

We describe all Rota–Baxter operators R of weight zero on the algebra U 3 (F) of upper-triangular matrices of order three over a field of characteristic 0. For this, we apply the following three ingredients: properties of R(1), conjugation with suitable (anti)automorphisms of U 3 (F) , and computation with the help of Singular of the Gröbner basis for the system of equations arisen on the coefficients of R. [ABSTRACT FROM AUTHOR]

Let A be a unital algebra with nontrivial idempotents. We considered A as a superalgebra according to Ghahramani and Zadeh’s method. We provided a description of supercommuting maps on A. As a consequence, we gave a description of supercommuting maps on matrix algebras, which is different from the result on commuting maps of matrix algebras. Finally, we proved that every supercommuting map on triangular algebras is a commuting map. [ABSTRACT FROM AUTHOR]

We give a classification of the group gradings on M p (K) , here p is a prime number and K is a field that contains a primitive p-th root of unity if char K ≠ p . The gradings are isomorphic to an elementary grading, to a division grading with support isomorphic to Z p or to a division grading with support isomorphic to Z p × Z p . [ABSTRACT FROM AUTHOR]

Let f = f (x 1 , ... , x m) be a multilinear polynomial over a field F. An F -algebra A is said to be f -zpd (f -zero product determined) if every m -linear functional φ : A m → F which preserves zeros of f is of the form φ (a 1 , ... , a m) = τ (f (a 1 , ... , a m)) for some linear functional τ on A. We are primarily interested in the question whether the matrix algebra M d (F) is f -zpd. While the answer is negative in general, we provide several families of polynomials for which it is positive. We also consider a related problem on the form of a multilinear polynomial g = g (x 1 , ... , x m) with the property that every zero of f in M d (F) m is a zero of g. Under the assumption that m < 2 d − 3 , we show that g and f are linearly dependent. [ABSTRACT FROM AUTHOR]

We solve for quantum geometrically realised pre-spectral triples or ‘Dirac operators’ on the noncommutative torus C θ [ T 2 ] and on the algebra M 2 (C) of 2 × 2 matrices with their standard quantum metrics and associated quantum Riemannian geometry. For C θ [ T 2 ] , we obtain a standard even spectral triple but now uniquely determined by full geometric realisability. For M 2 (C) , we are forced to a particular flat quantum Levi-Civita connection and again obtain a natural fully geometrically realised even spectral triple. In both cases there is an odd spectral triple for a different choice of a sign parameter. We also consider an alternate quantum metric on M 2 (C) with curved quantum Levi-Civita connection and find a natural 2-parameter family of Dirac operators which are almost spectral triples, where fails to be antihermitian. In all cases, we split the construction into a local tensorial level related to the quantum Riemannian geometry, where we classify the results more broadly, and the further requirements relating to the pre-Hilbert space structure. We also illustrate the Lichnerowicz formula for which applies in the case of a full geometric realisation. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution.

We consider cyclic algebras, Milnor's symbol algebras, and certain graded algebra structures on them. We classify these gradings with respect to both isomorphism and equivalence relations. Some of them induce gradings on matrix algebras, which we also classify. As an application, we obtain the classification of all group gradings on the algebra M p (F) , where p is a prime number and F is an arbitrary field. [ABSTRACT FROM AUTHOR]

This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution. [ABSTRACT FROM AUTHOR]

Analytical thinking is dominated by binary ideas. From pairwise interactions to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this paper, we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that there is a rich universe of higher arity ideas beyond binarity waiting to be explored. To illustrate this, we discuss several higher-order phenomena appearing in a wide range of research areas, paying special attention to instances of ternary interactions. From the point of view of formal sciences and mathematics, higher arity thinking opens up new paradigms of algebra, symbolic calculus and logic. In particular, we delve into the special case of ternary structures, as that itself reveals ample surprises: new notions of associativity (or lack thereof) in ternary operations of cubic matrices, ternary isomorphisms and ternary relations; the integration problem of 3-Lie algebras; and generalizations of adjacency in 3-uniform hypergraphs. All these are open problems that strongly suggest the need to develop new ternary mathematics. Finally, we comment on potential future research directions and remark on the transdisciplinary nature of higher arity science. [ABSTRACT FROM AUTHOR]

Abstract Physics systems are becoming increasingly complex and require more and more computing time. Quantum computing, which has shown its efficiency on some problems, such as the factorisation of a number with Shor's algorithm, may be the solution to reduce these computation times. Here, the authors propose two quantum numerical schemes for the simulation of physics phenomena, based on the finite difference method. The aim is to see if quantum versions of standard numerical schemes offer an advantage over their classical counterparts, either in accuracy, stability or computation time. First, the authors will present the different phenomena studied as well as the classical solution methods chosen. The authors will then describe the implementation of the quantum numerical schemes and present some results obtained on the different physics phenomena beforehand and then compare both approaches, classical and quantum.

Abstract The joint design problem for the coexistence of multiple‐input multiple‐output (MIMO) radar and multi‐user multiple‐input‐single‐output (MU‐MISO) communication is investigated. Different from the conventional design schemes, which require defining the primary function, we consider designing the transmit waveform, precoding matrix and receive filter to maximize the radar SINR and the minimal SINR of communication users, simultaneously. By doing so, the promising overall performance for both sensing and communication is achieved without requiring parameter tuning for the threshold of communication or radar. However, the resulting optimization problem which contains the maximin objective function and the unit sphere constraint, is highly nonconvex and hence difficult to attain the optimal solution directly. Towards this end, the epigraph‐form reformulation is first adopted, and then an alternating maximisation (AM) method is devised, in which the Dinkelbach’s algorithm is used to tackle the nonconvex fractional‐programing subproblem. Simulation results indicate that the proposed method can achieve improved performance compared with the benchmarks.

Abstract It is more and more important to analyse and process complex data for gaining more valuable knowledge and making more accurate decisions. The multigranulation decision theory based on conditional probability and cost loss has the advantage of processing decision‐making problems from multi‐levels and multi‐angles, and the neighbourhood rough set model (NRS) can facilitate the analysis and processing of numerical or mixed type data, and can address the limitation of multigranulation decision‐theoretic rough sets (MG‐DTRS), which is not easy to cope with complex data. Based on the in‐depth study of hybrid‐valued decision systems and MG‐DTRS models, this study analysed neighbourhood MG‐DTRS (NMG‐DTRS) deeply by fusing MG‐DTRS and NRS; a matrix‐based approach for approximation sets of NMG‐DTRS model was proposed on the basis of the matrix representations of concepts; the positive, boundary and negative domains were constructed from the matrix perspective, and the concept of positive decision recognition rate was introduced. Furthermore, the authors explored the related properties of NMG‐DTRS model, and designed and described the corresponding solving algorithms in detail. Finally, some experimental results that were employed not only verified the effectiveness and feasibility of the proposed algorithm, but also showed the relationship between the decision recognition rate and the granularity and threshold.

The pure state evaluation map from M n (C) to C (C P n − 1) is a completely positive map of C * -algebras intertwining the U n symmetries on the two algebras. We show that it extends to a cochain map from the universal calculus on M n (C) to the holomorphic ∂ ¯ calculus on C P n − 1 . The method uses connections on Hilbert C * -bimodules. [ABSTRACT FROM AUTHOR]

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix-variables over a ground polynomial ring in not necessary commuting variables. These matrix subalgebras of full matrix rings over polynomial rings show up in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, which are interesting noncommutative algebras arising from the study of Gl--varieties. [ABSTRACT FROM AUTHOR]

Abstract In order to deal with coarse‐grained and multi‐grained calculation problems, as well as granularity transformation problems in information system, quotient space theory is introduced in rough set theory. The main idea of this research is to try to maintain the important properties of the original space into the quotient space. Aimed to preserve the micro properties and the macro properties, two pairs of approximation operators on the quotient space are defined. When it comes to the composite of quotient spaces, the idea of these operators shows greater advantages. Examples are cited to illustrate possible applications of these operators, and their matrix representations are also given to make the calculations easy. Finally, all approximation operators on the quotient space involved so far are compared and their relationships are shown through a diagram.

In the present paper we describe Jordan matrix algebras over a field by generators and relations. We prove that the minimun number of generators of some special Jordan matrix algebras over a field is 2.

Abstract This study proposes a novel multi‐view soft block diagonal representation framework for clustering complete and incomplete multi‐view data. First, given that the multi‐view self‐representation model offers better performance in exploring the intrinsic structure of multi‐view data, it can be nicely adopted to individually construct a graph for each view. Second, since an ideal block diagonal graph is beneficial for clustering, a ‘soft’ block diagonal affinity matrix is constructed by fusing multiple previous graphs. The soft diagonal block regulariser encourages a matrix to approximately have (not exactly) K diagonal blocks, where K is the number of clusters. This strategy adds robustness to noise and outliers. Third, to handle incomplete multi‐view data, multiple indicator matrices are utilised, which can mark the position of missing elements of each view. Finally, the alternative direction of multipliers algorithm is employed to optimise the proposed model, and the corresponding algorithm complexity and convergence are also analysed. Extensive experimental results on several real‐world datasets achieve the best performance among the state‐of‐the‐art complete and incomplete clustering methods, which proves the effectiveness of the proposed methods.

This paper aims to investigate the screw motion in generalized space. For this purpose, firstly, the change in the screw coordinates is analyzed according to the motion of the reference frames. Moreover, the special cases of this change, such as pure rotation and translation, are discussed. Matrix multiplication and the properties of dual numbers are used to obtain dual orthogonal matrices, which are used to simplify the manipulation of screw motion in generalized space. In addition, the dual angular velocity matrix is calculated and shows that the exponential of this matrix can represent the screw displacement in the generalized space. Finally, to support our results, we give two examples of screw motion, the rotation part of which is elliptical and hyperbolic. [ABSTRACT FROM AUTHOR]

We derive, using matrices, new classes of linear recurrence identities linking the general term of a degree three linear recurrence sequence with those of an associated ‘cohort’ sequence that differs only in its initial values. The class description is seen to recover that previously established for the so called Horadam sequence and its cohort sequence set. [ABSTRACT FROM AUTHOR]

The joint design problem for the coexistence of multiple‐input multiple‐output (MIMO) radar and multi‐user multiple‐input‐single‐output (MU‐MISO) communication is investigated. Different from the conventional design schemes, which require defining the primary function, we consider designing the transmit waveform, precoding matrix and receive filter to maximize the radar SINR and the minimal SINR of communication users, simultaneously. By doing so, the promising overall performance for both sensing and communication is achieved without requiring parameter tuning for the threshold of communication or radar. However, the resulting optimization problem which contains the maximin objective function and the unit sphere constraint, is highly nonconvex and hence difficult to attain the optimal solution directly. Towards this end, the epigraph‐form reformulation is first adopted, and then an alternating maximisation (AM) method is devised, in which the Dinkelbach's algorithm is used to tackle the nonconvex fractional‐programing subproblem. Simulation results indicate that the proposed method can achieve improved performance compared with the benchmarks. [ABSTRACT FROM AUTHOR]

Abstract A simultaneous Diophantine approximation (SDA) algorithm takes instances of the partial approximate common divisor (PACD) problem as input and outputs a solution. While several encryption schemes have been published and their securities depend on the presumed hardness of variant of the PACD problem, fewer studies have attempted to extend the SDA algorithm to be applicable to these variants. In this study, the SDA algorithm is extended to solve the general PACD problem. In order to proceed, first the variants of the PACD problem are classified and how to extend the SDA algorithm for each is suggested. Technically, the authors show that a short vector of some lattice used in the SDA algorithm gives an algebraic relation between secret parameters. Then, all the secret parameters can be recovered by finding this short vector. It is also confirmed experimentally that this algorithm works well.

Abstract The authors address the problem of tensor completion from limited samplings. An improved generalized tubal Kronecker decomposition is first proposed to reveal the tensor structure of the targeted data, and the improved generalized tensor tubal‐rank and multi‐rank are also introduced. The tensor completion problem is then formulated as the improved multi‐rank minimisation problem. Instead of using the singular value decomposition(SVD), this work proposes an alternating optimisation method to update all of the subspaces of the data tensor for the recovery process, and thus it is computationally inexpensive. A rank‐decreasing method is derived to reveal the tensor rank in order to avoid the troublesome user defined hyper‐parameter selection. Experiments are carried out on both the synthetic data and the real datasets, and it is shown that the proposed approach can get a better performance than the state‐of‐the‐art approaches with moderate computational complexity.

Abstract The issue of aircraft tracking using passive sensors with poor measurement precision and a variable update rate is investigated. First, the need for a time‐dependent Markov chain transition matrix for the Interacting Multiple Model (IMM) is thoroughly evaluated using simulation and real data from an experimental multilateration system. Three different methods for computing the time‐dependent matrix are compared, showing their superiority for highly aperiodic sensors. Second, the Variable Structure Interacting Multiple Model (VS‐IMM) for passive sensors with poor geometry is proposed, in which the mode set update is based on local dilution of precision or distance root mean square error. The performance of the estimator is compared with the IMM using real data. The VS‐IMM results in the best performance for areas of both high and low measurement precision.

Abstract A novel approach to de‐embed the frequency response of filters connected to a 2‐channel multiplexer without the need for unplugging is proposed. The technique to manifold coupled multiplexers with arbitrary number of filters is also theoretically generalised. In multiplexer tuning, there are practically important circumstances in which it is advantageous to de‐embed the frequency response of filters in multiplexers without detaching them. The authors show that using a novel technique and reasonable approximations, all filters of a multiplexer can be modelled by simple microwave networks, and then, the equivalent rational function of the channel filters can be estimated appropriately by the rational H2 approximation; extraction of this model facilitates tuning of the filters using the coupling matrix method. The impact of mechanical tolerances on the accuracy of the proposed technique is also studied. Next, this method is examined on a 2‐channel multiplexer and its application in de‐embedding and tuning of its connected filters is verified. Provided that the mechanical tolerances of the structure are sufficiently small, this technique is a powerful engineering solution towards tuning of manifold‐coupled multiplexers.

For two given graphs, we construct an augmentation graph. We obtain an injective algebra homomorphism from the Leavitt path algebra of the augmentation graph to the tensor product of the Leavitt path algebras of the given graphs, and give a necessary and sufficient condition on the surjectivity of the homomorphism. This gives rise to a realization of the tensor product of two Leavitt path algebras as a Leavitt path algebra. [ABSTRACT FROM AUTHOR]

It is more and more important to analyse and process complex data for gaining more valuable knowledge and making more accurate decisions. The multigranulation decision theory based on conditional probability and cost loss has the advantage of processing decision‐making problems from multi‐levels and multi‐angles, and the neighbourhood rough set model (NRS) can facilitate the analysis and processing of numerical or mixed type data, and can address the limitation of multigranulation decision‐theoretic rough sets (MG‐DTRS), which is not easy to cope with complex data. Based on the in‐depth study of hybrid‐valued decision systems and MG‐DTRS models, this study analysed neighbourhood MG‐DTRS (NMG‐DTRS) deeply by fusing MG‐DTRS and NRS; a matrix‐based approach for approximation sets of NMG‐DTRS model was proposed on the basis of the matrix representations of concepts; the positive, boundary and negative domains were constructed from the matrix perspective, and the concept of positive decision recognition rate was introduced. Furthermore, the authors explored the related properties of NMG‐DTRS model, and designed and described the corresponding solving algorithms in detail. Finally, some experimental results that were employed not only verified the effectiveness and feasibility of the proposed algorithm, but also showed the relationship between the decision recognition rate and the granularity and threshold. [ABSTRACT FROM AUTHOR]

Abstract Considered is the plane‐wave scattering from and absorption by a thin circular dielectric disk. The analysis uses a set of the singular integral equations for the effective electric and magnetic currents, derived using the generalized boundary conditions on the disk median section. Following the recently developed analytical preconditioning procedure, these equations are discretized by the Galerkin technique with judiciously chosen expansion functions, which provide for the Fredholm second‐kind nature of the resulting matrix equations. This guarantees the code convergence and high efficiency. It is demonstrated that the developed technique delivers the most important features of thin dielectric disks–the resonances on the natural modes. In the resonances on the slab modes, the disk can be well‐transparent and the shadow is created only by its rim; in the whispering gallery mode resonances, the scattering occurs mainly in the disk plane.

Abstract Scattering and absorption of the H‐polarized plane wave by the infinite grating of flat graphene strips are considered in the environment met most frequently—on or at the surface of a dielectric‐slab substrate. The full‐wave meshless code is based on the analytical semi‐inversion using the Riemann–Hilbert problem solution. This leads to a Fredholm second‐kind matrix equation for the Floquet harmonic amplitudes that guarantees code convergence and provides easy control of computational error, which can be reduced to machine precision. The matrix elements are combinations of elementary functions, and therefore, the code is accurate and quite economical. This enables computation of the reflectance, transmittance, and absorbance as a function of the frequency in the wide band from static case to 10 THz. Numerical results show that such a metasurface with micrometre‐sized strips is a composite periodic open resonator. It is highly frequency‐selective thanks to the interplay of three types of natural modes—low‐Q slab, moderate‐Q plasmon strip, and ultra‐high‐Q lattice—that do not exist in the absence of the substrate. Varying the chemical potential of graphene, one can manipulate the electromagnetic characteristics of the metasurface at a fixed frequency from almost total transmission to almost total reflection.

Abstract We consider two‐dimensional (2‐D) thin dielectric parabolic reflector, covered with graphene from both sides, illuminated symmetrically by an E‐polarized electromagnetic plane wave. Our aim is to estimate the focussing ability of such a composite reflector depending on the graphene parameters. We use a version of the two‐side generalized boundary condition, modified for a thin multilayer case. The scattering is formulated as an electromagnetic boundary‐value problem; it is cast to a set of two coupled singular integral equations that are further subjected to analytical regularisation based on the known Riemann–Hilbert problem solution. Thanks to this procedure, the numerical results are computed from a Fredholm second‐kind matrix equation that guarantees convergence and provides easily controlled accuracy. In the lower part of the THz range, high values of the focusing ability are observed even for a thin reflector; they are greater than for a purely dielectric reflector and a free standing graphene reflector. On the other hand, a regime of almost full transparency, intrinsic for the dielectric layer, can spoil focusing ability. Novel aspect is that the location in frequency of this effect can be controlled, in wide range, by changing the chemical potential of graphene.

Abstract Penalized least squares (PLS) is a popular data smoothing technique. However, existing PLS smoothing algorithms behave as low‐pass filters (LPF), and, hence, they may introduce distortions to bandpass signals. These algorithms also have difficulties associated with the selection of the smoothing parameter and the order of the difference matrix they utilize. A new PLS smoothing algorithm is developed to overcome these limitations. In the proposed algorithm, two difference matrices and two regularisation parameters are utilized. As a result, the proposed algorithm can act as either an LPF or a bandpass filter. The particular shape of the filter is determined by the values of the regularisation parameters and the orders of the two difference matrices. The spectral characteristics of the smooth signal are utilized to derive closed‐form expressions for the optimum values of these four parameters. The derived algorithm is further developed to handle the case of having missing and/or outlier samples in the measured data. The proposed algorithms are completely data driven, and they do not require any special optimisation tools. In addition, the proposed algorithms are also shown to produce significantly improved results compared to some existing smoothing techniques.

Abstract Tensor singular value decomposition (t‐SVD) provides a novel way to decompose a tensor. It has been employed mostly in recovering missing tensor entries from the observed tensor entries. The problem of applying t‐SVD to recover tensors from limited coefficients in any given ortho‐normal basis is addressed. We prove that an n × n × n3 tensor with tubal‐rank r can be efficiently reconstructed by minimising its tubal nuclear norm from its O(rn3n log2(n3n)) randomly sampled coefficients w.r.t any given ortho‐normal basis. In our proof, we extend the matrix coherent conditions to tensor coherent conditions. We first prove the theorem belonging to the case of Fourier‐type basis under certain coherent conditions. Then, we prove that our results hold for any ortho‐normal basis meeting the conditions. Our work covers the existing t‐SVD‐based tensor completion problem as a special case. We conduct numerical experiments on random tensors and dynamic magnetic resonance images (d‐MRI) to demonstrate the performance of the proposed methods.

Abstract Next‐generation sequencing techniques have dramatically increased the amount of genomic data being sequenced, which calls for the acceleration of the alignment algorithms. This article proposes an FPGA‐based accelerated implementation of the seed extension kernel of the Burrows–Wheeler alignment genomic mapping algorithm. The well‐known Smith–Waterman algorithm is used during the seed extension to find the optimum alignment between the sequences. The state‐of‐the‐art architectures in the literature use one‐dimensional (1‐D) systolic arrays to fill a similarity matrix, based on the best score out of all match combinations, mismatches and gaps are computed. The cells on the same anti‐diagonal are computed in parallel in these architectures. We propose a novel 2‐dimensional architecture in which all the cells on the same rows and the same columns are computed in parallel and, thereby, significantly accelerated the process. The similarity matrix cells are computed in two phases: (1) the calculation phase and (2) error compensation phase. The calculation phase roughly approximate the cell values and the approximation error is fixed up during the error compensation phase. Our simulation results show that the proposed architecture can be up to 718x and 1.7x faster than the software execution and the 1‐D systolic arrays, respectively.

The implementations of matrix multiplication on contemporary, vector-oriented, and multicore-oriented computer hardware are very carefully designed and optimized with respect to their efficiency, due to the essential significance of that operation in other science and engineering domains. Consequently, the available implementations are very fast and it is a natural desire to take advantage of the efficiency of those implementations in other problems, both matrix and nonmatrix. Such an approach is often called a black box matrix computation paradigm in the literature on the subject. In this article, we gathered a broad series of algorithms taking advantage of the efficiency of fast matrix multiplication algorithms in other mathematical and computer science operations. [ABSTRACT FROM AUTHOR]

We classify all Rota–Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which do not arise from the decompositions of the entire algebra into a direct vector space sum of two subalgebras. [ABSTRACT FROM AUTHOR]

In this article, we dispute about the characterization of Cellular automata with restricted vertical neighborhood and Von neumann neighborhood of null boundary conditions over the field Z2 in uniform cellular automata and hybrid cellular automata. Transition rule matrix for uniform and hybrid cellular automara with null boundary con- dition is obtained and the reversibility of the uniform cellular automata studied. [ABSTRACT FROM AUTHOR]

Abstract A design technique is developed for the probing signals of a Multi‐Input Multi‐Output (MIMO) radar. The concentration of the energy of the signal in its essential duration and essential bandwidth is achieved through the use of a class of time‐frequency concentrated functions called the WLJ functions as the synthesizing signal set. The goal is to design a signal vector having a pre‐specified desired covariance (CoV) matrix while ensuring that the side‐lobes of the ambiguity functions are small. Since CoV matrices are structurally constrained, they form a manifold in the signal space. Hence, we argue that the difference between these matrices should not be measured in terms of the conventional Euclidean distance (ED); rather, the distance should be measured along the surface of the manifold, that is, in terms of a Riemannian distance (RD). In either case, the signal optimisation problem is non‐convex in the design variables, involving, respectively, a quartic and a square‐root objective function. An efficient algorithm based on successive convex approximation is developed in which the original non‐convex problems are transformed so that they can be approximated by a convex quadratically constrained quadratic problem at each stage, resulting in good approximate solutions. Comparing the designs using ED and RD, we find that the convergence of the algorithm can be significantly faster when optimising over the manifold (RD) than when optimising over the whole space (ED). More importantly, for tight constraints, the use of RD yields solutions which satisfy the constraints far better than the use of ED.