Basic Algebra Research Articles

A high-performance implementation of atomistic spin dynamics simulations on x86 CPUs

Atomistic spin dynamics simulations provide valuable information about the energy spectrum of magnetic materials in different phases, allowing one to identify instabilities and the nature of their excitations. However, the time cost of evaluating the spin-spin correlation functions in real space increases quadratically as the number of spins N, leading to significant computational effort, making the simulation of large spin systems very challenging. In this work, we propose to use a highly optimized general matrix multiply (GEMM) subroutine to calculate the dynamical spin-spin correlation function that can achieve near-optimal hardware utilization. Furthermore, we fuse the element-wise operations in the calculation of S(q,t) into the in-house GEMM kernel, which results in further performance improvements of 44% - 71% on several relatively large lattice sizes when compared to the implementation that uses the GEMM subroutine in OpenBLAS, which is the state-of-the-art open source library for Basic Linear Algebra Subroutine (BLAS).

Left co-Köthe rings and their characterizations

Köthe’s classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe’s problem for basic finite-dimensional algebras. But, so far, Köthe’s problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe’s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them.

GPS Experiment to Detect the Speed of Light Greater than C

More than 100 years ago, Einstein proposed a synchronisation method to measure the speed of light with the movement of the beam "there" and "back". This method, in principle, does not allow measuring the speed of movement in one direction: if light travels in one direction at a speed C + V greater than C, and in the opposite direction at a speed C-V less than C, and covers a distance L for different time intervals t1 and t2, the average speed L / (t1 + t2) in accordance with elementary algebra you always get less than C. As the only argument for using this method of measurement, relativists still cal impossibility to accurately synchronize clocks separated by some distance, since "the speed of the clock changes with a change in the gravitational potential or their speed of movement”. The experiment we proposed with two GPS satellites will allow us to measure the travel times of signals between satellites both in the forward and backward directions and prove that the distance between the satellites the signal travels in one direction at a speed less than C, and in the other at a speed greater than C, which unambiguously proves the fallacy of the basic postulate of SRT - the postulate of the invariance of the speed of light.

App 4LG3BR4 a tool for learning basic Algebra in students of Information and Communication Technologies

Currently, technology has evolved and has been adopted in education, this is where another field opens up to analyze academic performance, now students do not interact with books or notebooks, they can do it with a Tablet, with a PC, a Smartphone or some other device that allows them to be interconnected; in this sense, the educational experience has different dimensions. Therefore, this research focused on the use of a mobile application for learning basic algebra, with the students of the Division of Information and Communication Technologies of the Technological University of La Selva. The study was carried out under a quasiexperimental quantitative approach, using tests designed with dichotomous answers (pre-test and post-test) as data collection tools. The results show that using the App 4LG3BR4 in the subject of Linear Algebra in the teaching and learning process, significantly increases the academic performance of students, compared to the traditional teaching of these subjects in the aforementioned subject.

A matrix theory introduction to seaweed algebras and their index

The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based on the connected components of a “meander”: a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This said, we present several results that do not appear elsewhere and do appeal to more advanced language in the Introduction to provide added context.

Symmetric homology and representation homology

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0 0 , the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras.

Synthesis of quasi-optimal fast filters by the least square criterion

The subject of the article research is special signal processing methods based on the optimal discrete filtering theory. The goal is to increase the efficiency of model-based methods for processing information signals by reducing computational costs and increasing the speed of optimal discrete filtering algorithms. Applied methods: description of dynamic processes in terms of state space using elements of vector-matrix algebra, weighted least squares method, elements of Kalman's theory of optimal discrete filtering, basic concepts of the O'Reilly–Luenberger theory of functional observers, elements of probability theory, statistical modeling by the Monte Carlo method. Results: a new method for reducing computational costs is proposed, which uses the approximation of the Kalman filter transfer matrix time dependence by given piecewise linear functions according to the least squares criterion. The effectiveness of the method was evaluated on the example of a second-order dynamical system. On the basis of a comparative analysis, several acceptable variants of the considered approximation are proposed. The practical significance of the work lies in the further development of methods for the synthesis of quasi-optimal high-speed filters. The operability of the proposed modifications is confirmed by the example of a second-order linear dynamic system. The efficiency of the algorithms was evaluated by the statistical modeling method according to the criterion "accuracy-computational costs". It is shown that the total savings in the number of multiplication and addition operations can reach tens of times due to insignificant losses in the accuracy of the filtering process.

Computation of minimal covariants bases for 2D coupled constitutive laws

We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and — possibly coupled — laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any n-uplet of bidimensional constitutive tensors.

An efficient heterogeneous parallel algorithm of the 3D MOC for multizone heterogeneous systems

Recent innovations in powerful high-performance computing (HPC) architecture have enabled high-fidelity whole-core neutron transport simulations in a reasonable amount of time. Among high-fidelity numerical methods, the one-step 3D Method of Characteristic (MOC) is becoming increasingly popular in the reactor design community due to its numerical accuracy, geometrical flexibility and high degrees of parallelism. In this work, we propose a sophisticated heterogeneous parallel algorithm of the 3D MOC based on a supercomputer prototype consisting of brand-new multizone heterogeneous processors called the MT-3000, which has a peak double precision performance of 11.6 TFLOPS when operating at 1.2 GHz. In the 3D MOC calculation procedure, the total run time is dominated by the so-called MOC transport sweep where large amounts of tracks traverse the mesh grids along different directions. We significantly improve the performance of the MOC transport sweep through a three-level parallel algorithm with multiple hiding latency techniques. We have also implemented heterogeneous level-1 basic linear algebra subprograms (BLAS) that's the second computational hot spot. The 3D C5G7 benchmark problem is employed to verify the correctness and efficiency of the heterogeneous parallel 3D MOC algorithm. Numerical results show that it achieves very high performance on the MT-3000-based supercomputer prototype. Specifically, the three-layer parallel algorithm has proven to be highly scalable for each layer, and the Quarter-MT-3000 performance of the MOC transport sweep has achieved 23.6% of the peak performance, which is close to the theoretical upper limit.

Finite-dimensional algebras not arising as blocks of group algebras

Abstract We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and Sambale. In particular, we fully classify blocks with 16-dimensional basic algebra.

A symmetric version of the Euler equations by using Generalized Bernoulli Method

The aim of this article is to show a way to extend the usefulness of the Generalized Bernoulli Method (GBM) with the purpose to apply it for the case of variational problems with functionals that depend explicitly of all the variables. Moreover, after expressing the Euler equations in terms of this extension of GBM, we will see that the resulting equations acquire a symmetric form, which is not shared by the known Euler equations. We will see that this symmetry is useful because it allows us to recall these equations with ease. The presentation of three examples shows that by applying GBM, the Euler equations are obtained just as well as it does the known Euler formalism but with much less effort, which makes GBM ideal for practical applications. In fact, given a variational problem, GBM establishes the corresponding Euler equations by means of a systematic procedure, which is easy to recall, based in both elementary calculus and algebra without having to memorize the known formulas. Finally, in order to extend the practical applications of the proposed method, this work will employ GBM with the purpose to apply it for the case of solving isoperimetric problems.

A new type restricted quantum group

In this paper, we define a new type restricted quantum group Ūq(sl2*) and determine its Hopf Poincaré-Birkhoff-Witt-deformations Ūq(sl2*,κ) in which Ūq(sl2*,0)=Ūq(sl2*) and the classical restricted Drinfeld–Jimbo quantum group Ūq(sl2) is included. We show that Ūq(sl2*) is a basic Hopf algebra, then uniformly realize Ūq(sl2*) and Ūq(sl2) via some quotients of (deformed) preprojective algebras corresponding to the Gabriel quiver of Ūq(sl2*). Moreover, we obtain a uniform tensor-categorical realization of Ūq(sl2*) and Ūq(sl2), which is consistent with the above-mentioned Hopf-algebraic realization.

Basic Algebra Ability Analysis of Junior High School students with the Rasch Model Approach

The purpose of this study was to determine students' basic algebraic abilities based on the analysis of the items (items) and the analysis of students' abilities (person) using the Rasch modeling. The instrument used is a test instrument in the form of multiple choices about algebraic material. This test was given to 662 students of class VIII and IX at SMP Negeri 5 Kendari. Data analysis was performed by RASCH modeling using the Winstep application. The results of the person analysis stated that the overall ability of the students was greater than the difficulty level of the questions. This shows that the consistency of students' answers in solving multiple choice questions on algebraic material according to the level of difficulty is strong or good, but the quality of the items used in collecting data on students' algebraic abilities is of moderate quality.

Generalized fuzzy neighborhood system-based multigranulation variable precision fuzzy rough sets with double TOPSIS method to MADM

Multigranulation fuzzy rough sets (MFRSs), variable precision fuzzy rough sets (VPFRSs) and their combinations with traditional decision-making methods are fundamental in the research of multi-attribute decision-making (MADM) problems. However, the existing theories and methods have two deficiencies: (1) many existing MFRSs are based on fuzzy relations or fuzzy coverings, which may not effectively solve some MADM problems; (2) VPFRSs usually do not meet the inclusion property, which hinders their further development in theory and application. Noted that both fuzzy relations and fuzzy coverings can generate fuzzy neighborhood systems in a natural way, so the fuzzy neighborhood system provides a broader multigranulation framework. Inspired by this, by using the generalized fuzzy neighborhood system, we propose three multigranulation variable precision fuzzy rough set models, namely optimistic, pessimistic and compromise, respectively. As expected, these models contain many existing models as special cases, thus broadening the framework of rough set. Additionally, we verify that these models fulfill many basic algebra properties, particularly the generalized inclusion properties, which overcome the limitation of VPFRSs. Finally, by combining the proposed fuzzy rough set models with the TOPSIS method, we develop a new method to MADM. A series of experiments show that our method has some obvious advantages.

Equivalent Conditions of Normal Autometrized L-algebra

In this paper, we discuss the properties of some elements in autometrized algebra. We also introduce some properties of normal autometrized algebra. Furthermore, we obtain normal autometrized l-algebras among the normal autometrized semialgebras satisfying several conditions and discussing some of its properties. We also prove that a normal autometrized semialgebra satisfying certain conditions is a lattice and distributive. Finally, we establish an equivalent condition between normal autometrized l-algebra and representable autometrized algebra with some conditions. We also prove that in a normal autometrized l-algebra satisfying some conditions, there exists a subset which is a convex normal subalgebra.

An additive framework for kirigami design.

We present an additive approach for the inverse design of kirigami-based mechanical metamaterials by focusing on the empty (negative) spaces instead of the solid tiles. By considering each negative space as a four-bar linkage, we identify a simple recursive relationship between adjacent linkages, yielding an efficient method for creating kirigami patterns. This allows us to solve the kirigami design problem using elementary linear algebra, with compatibility, reconfigurability and rigid-deployability encoded into an iterative procedure involving simple matrix multiplications. The resulting linear design strategy circumvents the solution of a non-convex global optimization problem and allows us to control the degrees of freedom in the deployment angle field, linkage offsets and boundary conditions. We demonstrate this by creating a large variety of rigid-deployable, compact, reconfigurable kirigami patterns. We then realize our kirigami designs physically using two simple but effective fabrication strategies with very different materials. Altogether, our additive approaches present routes for efficient mechanical metamaterial design and fabrication based on ori/kirigami art forms.

Parallel Optimization of BLAS on a New-Generation Sunway Supercomputer

The new-generation Sunway supercomputer has ultra-high computing capacity. But due to the unique heterogeneous architecture of the supercomputer, the open-source versions of basic linear algebra subprograms (BLAS) are insufficient for performance or compatibility. In addition, due to the update of the architecture, BLAS based on the previous Sunway could not fully exploit the performance of the successor. To address the challenges, we propose an optimized BLAS on the new-generation Sunway supercomputer in this paper. Specially, for achieving efficient computation, a parallel optimization method based on the new-generation Sunway for the Level-1 BLAS computing between vectors and the Level-2 BLAS computing between vectors and matrices is first proposed. Then, an adaptive scheduling algorithm for various data sizes is proposed, which is used to balance the tasks of core groups. Finally, to achieve highly efficient general matrix multiplication (GEMM) kernels, a parallel optimization method based on the new-generation Sunway for the Level-3 BLAS computing between matrices is proposed, which includes source-level optimization as well as assembly-level optimization. Experimental results show that the memory bandwidth utilization of the optimized Level-1/2 BLAS exceeds 95%, and the computational efficiency of the optimized GEMM kernel exceeds 94%.

Fast matrix multiplication via compiler‐only layered data reorganization and intrinsic lowering

AbstractThe resurgence of machine learning has increased the demand for high‐performance basic linear algebra subroutines (BLAS), which have long depended on libraries to achieve peak performance on commodity hardware. High‐performance BLAS implementations rely on a layered approach that consists of tiling and packing layers—for data (re)organization—and micro kernels that perform the actual computations. The algorithm for the tiling and packing layers is target independent but is parameterized to the memory hierarchy and register‐file size. The creation of high‐performance micro kernels requires significant development effort to write tailored assembly code for each architecture. This hand optimization task is complicated by the recent introduction of matrix engines by 's (Matrix Multiply Assist—MMA), (Advanced Matrix eXtensions—AMX), and (Matrix Extensions—ME) to deliver high‐performance matrix operations. This article presents a compiler‐only alternative to the use of high‐performance libraries by incorporating, to the best of our knowledge and for the first time, the automatic generation of the layered approach into LLVM, a production compiler. Modular design of the algorithm, such as the use of LLVM's matrix‐multiply intrinsic for a clear interface between the tiling and packing layers and the micro kernel, makes it easy to retarget the code generation to multiple accelerators. The parameterization of the tiling and packing layers is demonstrated in the generation of code for the MMA unit on IBM's POWER10. This article also describes an algorithm that lowers the matrix‐multiply intrinsic to the MMA unit. The use of intrinsics enables a comprehensive performance study. In processors without hardware matrix engines, the tiling and packing delivers performance up to (Intel)—for small matrices—and more than (POWER9)—for large matrices—faster than PLuTo, a widely used polyhedral optimizer. The performance also approaches high‐performance libraries and is only slower than OpenBLAS and on‐par with Eigen for large matrices. With MMA in POWER10 this solution is, for large matrices, over faster the vector‐extension solution, matches Eigen performance, and achieves up to of BLAS peak performance.

Products of unipotent elements in certain algebras

Abstract Let F be an algebraically closed field and let R be a locally finite algebra over F. This paper aims to show that any element of R is a product of at most three unipotent elements from R if and only if the element lies in the first derived subgroup of the unit group of R. In addition, this necessary and sufficient condition is applied to twisted group algebras of locally finite groups over a field of either zero characteristic or characteristic p ≠ 0 {p\not=0} for some prime p. Moreover, we explore some crucial properties satisfied by certain algebras like the conditions concerned with the connections between unipotent elements of index 2 and commutators, as well as we investigate the unipotent radical of some subgroups of a finite-dimensional algebra R over a field F with at least four elements. In particular, we again apply these results to twisted group algebras.

Colored planar algebras for commuting squares and applications to Hadamard subfactors

We define a colored planar algebra associated to a non-degenerate commuting square and identify the biunitary of the square as an element of the planar algebra. We prove a variation of the graph planar algebra embedding theorem and use the biunitary to construct representations of annular algebras and quantum groups from the commuting square. When the commuting square subfactor is amenable we can compute elements in the spectrum of the principal graph of the subfactor. This leads to two criteria which imply non-flatness of the biunitary and infinite depth of the subfactor. Computations with these criteria are performed with a continuous family of biunitaries on the 3311 principal graph, Petrescu's continuous family of 7×7 complex Hadamard matrices, and type II Paley Hadamard matrices. We conclude that all of Petrescu's 7×7 complex Hadamard matrices and all type II Paley Hadamard matrices yield infinite depth subfactors.