Matrix Algebra Research Articles

A simplex approach for binary linear programming

This article is a research, design, and development effort, on an algorithm for Binary Linear Programming (BLP). The idea of the algorithm is based on the concepts borrowed from the Simplex method of Linear Programming (LP). The Simplex method has both column selection and row selection logics for generating a pivot. In this article we select both logics of LP to use as only the column selection logic for the BLP. The article provides the complete implementable C-language source code of this version of the BLP algorithm to show all the details of the logics. In addition, it provides a simple software requirement for embedded engineering products based on the laws of nature. It helps to design, modify, and maintain a simpler code for the algorithms. Therefore, the article uses a multidisciplinary approach. It integrates optimization theory, matrix algebra, computer programming, software engineering design requirements, social science, laws of nature, etc. This BLP algorithm will shed some light on the P-NP problem and its solution.

Extending Abelian Rings: A Generalized Approach

We introduce a novel framework for assessing the centrality of idempotents within a ring by presenting a general concept that assigns a degree of centrality. This approach aligns with the previously established notions of semicentral and q-central idempotents by Birkenmeier and Lam. Specifically, we define an idempotent $e$ in a ring $R$ to be $n$-central, where $n$ is a positive integer, if $[e, R]^ne=0$, where $[x,y]$ represents the additive commutator $xy-yx$. If every idempotent in a ring $R$ is $n$-central, we refer to $R$ as $n$-Abelian. Our study lays the groundwork by presenting foundational results that support this concept and examines key features of $n$-central idempotents essential for appropriately categorizing $n$-Abelian rings among various generalizations of Abelian rings introduced in prior literature. We provide examples of $n$-central idempotents that do not fall under the categories of semicentral or $q$-central. Furthermore, we demonstrate that the ring of upper matrices $\mathbb{T}_n(R)$, where $R$ is Abelian, is an $n$-abelian. We also prove that a ring where all of its idempotents are $n$-central is an exchange ring if and only if the ring is clean.

Residue maps, Azumaya algebras, and buildings

The goal of this note is to give an explicit formula for the residues of twists of the matrix algebra in terms of the twisting cocycle. Combined with the Fixed Point Theorem for actions of finite groups on affine buildings, this leads to a quick proof of the well-known characterization of unramified algebras in terms of Azumaya algebras.

A matrix variant of the Erdős-Falconer distance problems over finite field

We study a matrix analog of the Erdős-Falconer distance problems in vector spaces over finite fields. We provide lower bounds for the cardinality of a subset of the matrix algebra over a finite field such that its distance set is the whole matrix algebra. There arises an interesting analysis of certain quadratic matrix Gauss sums.

Energy efficiency and performance analysis of a legacy atomic scale materials modeling simulator (VASP)

This work tackles the performance and energy consumption analysis of a legacy scientific application, the VASP (Vienna Ab-initio Simulation Package), an application commonly used by physicists and chemists for modeling materials at the atomic scale. Many of these scientific applications have been implemented in Fortran, where energy metrics instrumentation is not straightforward. We obtained performance figures (execution time and energy consumption) by instrumenting the source code using EML. This energy measurement library has been modified to introduce Fortran interfaces for these metrics. The analysis was carried out using different matrix algebra libraries, parallelization techniques, and hardware platforms, emphasizing on the MPI, OpenMP, and CUDA parallel implementations of the algorithms used in VASP. We employ various material specifications (atomic structures) and molecular sizes of a silicon-based crystal to create a set of benchmarks for these specifications, leading to some recommendations for final users regarding performance improvements. The proposed benchmarking technique assists the user in selecting the right combination of problem size, compilers, and parallelization options available in VASP. For a given system platform, the user will be able to determine not only the architecture to use (GPU or multicore processors), but also the appropriate library and parallelization according to the atomic structure and molecular size.

Very good gradings on matrix rings are epsilon-strong

We investigate properties of group gradings on matrix rings M n ( R ) , where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on M n ( R ) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that M n ( R ) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when M n ( R ) is an epsilon-crossed product. Our results are illustrated by several examples.

Congruence-simple matrix semirings

It is well known that the full matrix ring over a skew-field is a simple ring. We generalize this theorem to the case of semirings. We characterize the case when the matrix semiring [Formula: see text], of all [Formula: see text] matrices over a semiring [Formula: see text], is congruence-simple, provided that either [Formula: see text] has a multiplicatively absorbing element or [Formula: see text] is commutative and additively cancellative.

Some Properties of Rings of the Type f(x),g(x)-clean

In [1], the authors introduced a property of rings called the f(x),g(x)-clean property. We investigate some new results about this property. In particular, we prove that the ring of matrices over a ring R is f(x),g(x)-clean if the ring R is f(x),g(x)-clean as well. We demonstrate, among many other things, that a ring's f(x),g(x)-cleanness may always be passed to R[[x]], but not always to its polynomial ring R[x].

Ring isomorphic to a structural matrix ring over a division ring

Ring isomorphic to a structural matrix ring over a division ring

Parametrical synthesis of various radio devices with the set quantity of cascades of type «the mixed two-port network – nonlinear part»

Background. Presence of possibility of analytical definition of a part of parametres of various radio devices, optimum by criterion of maintenance of preset values of modules and phases of transfer functions on necessary quantity of frequencies, considerably reduces time of numerical optimisation of other part of parametres by criterion of formation demanded PFC and FFC in a strip of frequencies. Till now such problems dared concerning radio devices only with one cascade of type «a nonlinear part - the coordination the device» or «the coordination the device - a nonlinear part». In quality coordination devices were used the jet, resistive, complex or mixed two-port networks. The problem of multicascade radio devices with jet two-port networks is solved also. Change of basis for the coordination two-port networks and a place of inclusion of a nonlinear part leads to change of area of a physical realizability.
 Aim. Working out of algorithms of parametrical synthesis of radio devices with any quantity of identical and unequal cascades of type «the coordination the mixed two-port network – a nonlinear part» by criterion of maintenance of the set frequency characteristics. Nonlinear parts are presented in the form of a nonlinear element and parallel either consecutive on a current or pressure of a feedback.
 Methods. The theory of two-port networks, matrix algebra, a decomposition method, a method of synthesis of actuation microwave devices, numerical methods of optimisation.
 Results. In interests of achievement of the specified purpose systems of the algebraic equations are generated and solved. Models of optimum two-port networks in the form of mathematical expressions for definition of interrelations between elements of their classical matrix of transfer and for search of dependences of mixed of two-poles from frequency are received. It is shown, that at certain parities between quantity of identical cascades and values of resistance of a source of a signal and loading of the one-cascade radio device frequency characteristics of one-cascade and multicascade radio devices appear identical or similar. Such schemes are named by equivalent. Use of unequal cascades leads to substantial growth of a working strip of frequencies.
 Conclusion. The comparative analysis of theoretical results (PFC and FFC radio devices, value of parametres), received by mathematical modelling in system «MathCad», and the experimental results received by scheme modelling in systems «OrCad» and «MicroCap», shows their satisfactory coincidence.

Semiassociative algebras over a field

An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.

The kernels of powers of linear operator via Weyr characteristic

The adjoint of a matrix in the Lie algebra associated with a matrix algebra is a fundamental operator, which can be generalized to a more general operator φ AB : X → AX − XB by two matrices A and B. The kernel of the operator is very well-known and it can be found in Gantmacher's book. The formulas for the dimensions of the kernels of arbitrary powers of the operator φ AB were given in terms of the Segre characteristics of these two matrices by the second and third authors in this paper and their collaborators. This paper provides an alternative approach to this problem via the Weyr characteristic in a more essential method. We obtain formulas for the dimensions of the kernels of arbitrary powers of the operator in terms of the Weyr characteristics. Furthermore, the basis for kernel of each power of the operator is described explicitly. As a consequence, for arbitrary square matrices A and B over an algebraically closed field, the dimension of the kernel of each power of the operator φ A − λI , B for eigenvalues λ of φ AB can be viewed as a similarity invariant of the operator φ AB , so we characterize the operator within similarity, which should be of interest to a number of people (including physicists).

К выбору периода дискретизации нелинейных гибридных систем управления

Hybrid systems are often used in various technical applications, such as robotics, aviation, space, energy, etc. The emergence of hybrid control systems is due to the use of computers to implement control laws, including nonlinear ones. Digital means cannot implement continuous control laws. However, the known methods of design, especially of nonlinear systems, lead precisely to continuous controls, which caused the need to discretize the continuous control with the largest possible period. Solving the problem of determining the maximum permissible sampling period of a nonlinear control system is a rather complex stage of its creation. In this article the problem of definition of the maximum allowed sample period of the nonlinear hybrid system control and its dependence on the module of the maximum eigenvalue of the functional matrix of quasilinear model of the continuous system is also considered. The nonlinear hybrid system is created by discretizing the control of the nonlinear continuous system. This continuous system is synthesized using the algebraic polynomial and matrix method of the nonlinear control systems design in which quasilinear models are used. It has been is established that the value of the maximum permissible sample period depends not only on the module of the maximum eigenvalue, but also on initial conditions and external influences. These dependences are complex and it is difficult to find them theoretically. Experimentally, based on the example of the specific nonlinear hybrid control system it is shown that eigenvalues smaller on the module lead to larger values of the maximum of the permissible sample period. The problem of choosing the sample period of control laws of the real hybrid control systems can be solved in the similar way based on a compromise between the system high-speed performance and the sample period.

Guest Column: Structure in Communication Complexity and Constant-Cost Complexity Classes

Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction is typically easy as the structural implications of small complexity often imply a bound on some matrix parameter. The challenge lies in establishing the reverse direction, which requires understanding the structure of Boolean matrices for which a given matrix parameter is small or large. We will discuss several research directions that align with this overarching theme.

Amitsur's theorem, semicentral idempotents, and additively idempotent semirings

Abstract The article explores research findings akin to Amitsur’s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In most results related to rings and semirings, Birkenmeier’s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.

Multiplicative maps on generalized \(n\)-matrix rings

Let \(\mathfrak{R}\) and \(\mathfrak{R}'\) be two associative rings (not necessarily with identity elements). A bijective map \(\varphi\) of \(\mathfrak{R}\) onto \(\mathfrak{R}'\) is called an \textit{\(m\)-multiplicative isomorphism} if {\(\varphi (x_{1} \cdots x_{m}) = \varphi(x_{1}) \cdots \varphi(x_{m})\)} for all \(x_{1}, \dotsc ,x_{m}\in \mathfrak{R}.\) In this article, we establish a condition on generalized matrix rings, that assures that multiplicative maps are additive. And then, we apply our result for study of \(m\)-multiplicative isomorphisms and \(m\)-multiplicative derivations on generalized matrix rings.

POAS: a framework for exploiting accelerator level parallelism in heterogeneous environments

In the era of heterogeneous computing, a new paradigm called accelerator level parallelism (ALP) has emerged. In ALP, accelerators are used concurrently to provide unprecedented levels of performance and energy efficiency. To reach that there are many problems to be solved, one of the most challenging being co-execution. In this paper, we present a new scheduling framework called POAS, a general method for providing co-execution to applications. Our proposal consists of four steps: predict, optimize, adapt and schedule. With POAS, an unseen application can be executed concurrently in ALP with little effort. We evaluate POAS on a heterogeneous environment consisting of CPUs, GPUs (CUDA cores), and XPUs (Tensor cores) on two different fields, namely linear algebra (matrix multiplication benchmark) and deep learning (convolution benchmark). Our experiments prove that POAS provides excellent performance and completes the tasks within a time very close to the optimal time for the hardware and applications used, with a negligible execution time overhead. Moreover, the POAS predictor performed exceptionally well, achieving very low RMSE values for both use cases. Therefore, POAS can be a valuable tool for fully exploiting ALP and improving overall performance over offloading in heterogeneous settings.

Abelian model structures on comma categories

UDC 512.64 Let A and B be bicomplete Abelian categories, which both have enough projectives and injectives and let T : A → B be a right exact functor. Under some mild conditions, we show that hereditary Abelian model structures on A and B can be amalgamated into a global hereditary Abelian model structure on the comma category ( T ↓ B ) . As an application of this result, we give an explicit description of a subcategory that consists of all trivial objects of the Gorenstein flat model structure on the category of modules over a triangular matrix ring.

Graphs from matrices - a survey

Let R be a commutative ring with identity. For a positive integer n ≥ 2 , let M n ( R ) be the set of all n × n matrices over R and M n ( R ) * be the set of all non-zero matrices of M n ( R ) . The zero-divisor graph of M n ( R ) is a simple directed graph with vertex set the non-zero zero-divisors in M n ( R ) and two distinct matrices A and B are adjacent if their product is zero. Given a matrix A ∈ M n ( R ) , Tr(A) is the trace of the matrix A. The trace graph of the matrix ring M n ( R ) , denoted by Γ t ( M n ( R ) ) , is the simple undirected graph with vertex set { A ∈ M n ( R ) * : there exists B ∈ M n ( R ) * such that Tr ( AB ) = 0 } and two distinct vertices A and B are adjacent if and only if Tr ( AB ) = 0. For an ideal I of R, the notion of the ideal based trace graph, denoted by Γ I t ( M n ( R ) ) , is a simple undirected graph with vertex set M n ( R ) ∖ M n ( I ) and two distinct vertices A and B are adjacent if and only if Tr ( AB ) ∈ I . In this survey, we present several results concerning the zero-divisor graph, trace graph and the ideal based trace graph of matrices over R.

A resilience-based methodology to assess the degree of interdependency between infrastructure

Seismic vulnerability of urban regions depends on the infrastructure systems that need to remain operative during and after earthquakes. In modern societies, the increasing interdependence of the various infrastructures may significantly affect the community resilience because a damage or a failure on a system may propagate to other interconnected ones. In this regard, the original concept of seismic resilience needs to be expanded to a more general definition that may consider the dependence of multiple infrastructures. In particular, this paper aims to develop a formulation of the seismic resilience by proposing an algebraic matrix form for the calculation of the seismic resilience. The presented formulation has the main advantages to be easily implemented in numerical platforms and to calculate the degree of interdependency between infrastructure.