Nonzero Elements Research Articles

On nonnil S-SFT rings

Let [Formula: see text] be a commutative ring with nonzero identity element and S a multiplicative subset of [Formula: see text]. In this paper, we introduce and investigate the notion of nonnil S-SFT rings. The ring [Formula: see text] is said to be nonnil S-SFT, if for each nonnil-ideal [Formula: see text] of [Formula: see text], there exist [Formula: see text], [Formula: see text] and a finitely generated ideal [Formula: see text] such that [Formula: see text] for all [Formula: see text], in that case, the ideal [Formula: see text] is called S-SFT. It is shown that the ring [Formula: see text] is nonnil S-SFT ring if and only if each nonnil-prime ideal (disjoint with S) is S-SFT. The transfert of the nonnil S-SFT concept to flat overrings and trivial extension is investigated. We give several characterizations of nonnil S-SFT rings. Also, we give a characterization of the nonnil-SFT variant in terms of the nonnil S-SFT variant.

ApSpGEMM: Accelerating Large-scale SpGEMM with Heterogeneous Collaboration and Adaptive Panel

The Sparse General Matrix-Matrix multiplication (SpGEMM) is a fundamental component for many applications, such as algebraic multigrid methods (AMG), graphic processing, and deep learning. However, the unbearable latency of computing high-dimensional, large-scale sparse matrix multiplication on GPUs hinders the development of these applications. An effective approach is heterogeneous cores collaborative computing, but this method must address three aspects: (1) irregular non-zero elements lead to load imbalance and irregular memory access, (2) different core computing latency differences reduce computational parallelism, and (3) temporary data transfer between different cores introduces additional latency overhead. In this work, we propose an innovative framework for collaborative large-scale sparse matrix multiplication on CPU-GPU heterogeneous cores, named ApSpGEMM. ApSpGEMM is based on sparsity rules and proposes reordering and splitting algorithms to eliminate the impact of non-zero element distribution features on load and memory access. Then adaptive panels allocation with affinity constraints among cores improves computational parallelism. Finally, carefully arranged asynchronous data transmission and computation balance communication overhead. Compared with state-of-the-art SpGEMM methods, our approach provides excellent absolute performance on matrices with different sparse structures. On heterogeneous cores, the GFlops of large-scale sparse matrix multiplication is improved by 2.25 to 7.21 times.

Lippmann-Schwinger equation representation of Green's function and its preconditioned generalized over-relaxation iterative solution in wavelet domain

The calculation of Green's function is the core of seismic forward and inverse methods based on integral operators. When the Lippmann-Schwinger (L-S) equation is used to calculate Green's function in strongly scattering media, both the Born scattering series and the numerical iterative method encounter issues of slow convergence or divergence. Although the renormalization method derived from quantum mechanics can effectively address the convergence problem of Born scattering series in strong scattering problems, it is acknowledgeed that the convergence conditions and rates of convergence of different reformulation series may vary, and no universal convergence reformulation scattering series exists. Numerical methods for solving integral equations tend to be more general and mathematically robust. In this work, we focus on the numerical solution method of L-S equations. By using a wavelet-domain preconditioner to a reformulated or equivalent Lippmann-Schwinger (L-S) equation, we present an iterative method for numerically solving the equivalent L-S equation aimed at improving the rate of convergence and iteration efficiency in strongly inhomogeneous media. Following Jakobsen et al. (2020), we first introduce a small imaginary component into the background wave number，then rewrite the L-S equation to derive the equivalent complex wave number L-S equation. This reformulation ensures that the coefficient matrix exhibits a banded structure after numerical discretization, allowing the wavelet coefficient matrix to maintain good sparsity. We employ a multi-level fill-in incomplete LU (ILU) factorization method along with a block ILU-based algebraic recursive multilevel solve (ARMS) method in the wavelet domain to generate sparse approximate inverses as preconditioning operators, thereby accelerating the convergence of the generalized successive over-relaxation (GSOR) iterative method. This method is applied to compute numerical Green's functions in strongly inhomogeneous media. Numerical results demonstrate that our method yields simulation outcomes consistent with those obtained from the direct method for solving the original real wave number L-S equation. By testing various preconditioners, we find that the ARMS preconditioner offers significant advantages in operator generation efficiency and non-zero element filling ratio, effectively accelerating the convergence of the GSOR iterative method while achieving higher computational efficiency.

ZONAL LABELING OF EDGE COMB PRODUCT OF GRAPHS

Given a plane graph $G=(V,E)$. A zonal labeling of graph $G$ is defined as an assignment of the two nonzero elements of the ring $\mathbb{Z}_3$, which are $1$ and $2$, to the vertices of $G$ such that the sum of the labels of the vertices on the border of each region of the graph is $0\in\mathbb{Z}_3$. A graph $G$ that possess such a labeling is termed as zonal graph. This paper will characterize edge comb product graphs that are zonal. The results show that $P_m\trianglerighteq_eC_n$, $C_n\trianglerighteq_e C_r$, $S_p\trianglerighteq_e C_n$, and $S_p\trianglerighteq_e F_t$ are zonal in some cases, but not in others.

Variational quantum algorithm for the Poisson equation based on the banded Toeplitz systems

Abstract For solving the Poisson equation it is usually possible to discretize it into solving the corresponding linear system $Ax=b$. Variational quantum algorithms (VQAs) for \textcolor{red}{the discreted} Poisson equation have been studied before. We give a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix $A$. In detail, we decompose the matrix $A$ and $A^2$ into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the $d$-dimensional Poisson equation with Dirichlet boundary conditions, \textcolor{red}{the number of decomposition terms is less than the work in [Phys. Rev. A108, 032418 (2023)]}. Based on the decomposition of the matrix, we design quantum circuits that \textcolor{red}{evaluate efficiently} the cost function. \textcolor{red}{Additionally, numerical simulation verifies the feasibility of the proposed algorithm. In the end,} the VQAs for linear systems of equations and matrix–vector multiplications with $K$-banded Teoplitz matrix $T_n^K$ \textcolor{red}{are} given, where $T_n^K\in R^{n\times n}$ and $K\in O(ploy\log n)$.

Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings

The following problem has been known since the 80s. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(\Gamma)|\neq 1$, where $I(\Gamma)$ is the set of involutions in $\Gamma$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.

Modules with reduced endomorphism rings

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including [Formula: see text]-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings [Formula: see text] for which every faithful multiplication [Formula: see text]-module is endo-reduced is precisely that of reduced rings; while the class of rings [Formula: see text] for which every multiplication [Formula: see text]-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

Light scattering by Möbius particles

Using the Discrete-Dipole Approximation (DDA) we study scattering-angle dependences of the orientationally averaged Mueller matrix elements for small Möbius particles with different chirality. A Möbius particle is a strip of a certain width and thickness, the ends of which are attached to each other after twisting one of them at an angle by a multiple of □. The Mueller matrices Mik for such particles have 16 non-zero elements. The scattering-angle dependences of the intensity M11, linear polarization degree (-M12/M11), and Circular Intensity Differential Scattering (CIDS = -M14/M11) show their sensitivity to the refractive index and number of the strip twisting. For instance, the CIDS depends significantly on the complex part of the refractive index. Particles with more twists show a decrease in the maximum value of the positive branch of linear polarization and an increase in the width and minimum of the negative branch. The twist parameter's influence on CIDS is non-monotonic, initially increasing and then approaching zero with further twisting.

Parallel cyclic reduction of padded bordered almost block diagonal matrices

The solution of linear systems is a crucial and indispensable technique in the field of numerical analysis. Among linear system solvers, the cyclic reduction algorithm stands out for its natural inclination to parallelization. So far, the cyclic reduction has been applied primarily to linear systems with almost block diagonal matrices. Some of its variants widen the usage to almost block diagonal matrices with a last block of rows introducing a set of non-zero elements in the first group of columns. In this work, we extend cyclic reduction to matrices with additional non-zero elements below and to the right without any limitations. These matrices, called padded bordered almost block diagonal, arise from the discretization of optimal control problems featuring arbitrary boundary conditions and free parameters. Nonetheless, they also appear in two-point boundary value problems with free parameters. The proposed algorithm is based on the LU factorizations, and it is designed to be executed in parallel on multi-thread architectures. The algorithm performance is assessed through numerical experiments with different matrix sizes and threads. The computation times and speedups obtained with the parallel implementation indicate that the suggested algorithm is a robust solution for solving padded bordered almost block diagonal linear systems. Furthermore, its structure makes it suitable for the use of different matrix factorization techniques, such as QR or SVD. This flexibility enables tailored customization of the algorithm on the basis of the specific application requirements.

SpEpistasis: A sparse approach for three-way epistasis detection

Epistasis detection is a fundamental application in the areas of bioinformatics and biomedicine, providing important insights regarding the relationship between the human genome and the occurrence of certain diseases. Exhaustive epistasis detection approaches are employed to achieve an accurate and deterministic solution, at the cost of high computational complexity, especially when targeting high-order epistasis. While recent works employ vectorization and cache-blocking techniques to alleviate this burden, these solutions are now limited by the maximum performance of the functional units of computing systems. Thus, to further improve the performance of epistasis detection it is necessary to reduce its number of memory transfers and computations. To tackle this issue, this work proposes SpEpistasis, which performs three-way epistasis detection by relying on sparse features, which by only storing the non-zero elements of the dataset, allows for reducing the number of operations needed for epistasis detection. To achieve this goal, a new hybrid format to represent the input dataset is proposed, which stores a subset of the data in the compressed sparse row format. Moreover, new sparse-aware algorithmic approaches are also proposed in order to leverage both the hybrid format and the vector capabilities of current CPUs from Intel, AMD, and ARM. The experimental results show that SpEpistasis provides a speedup up to 3.7× and average speedups of around 1.8× and 1.33× when compared with other state-of-the-art works.

On fusible rings and related notions

A ring is called left fusible if each of its nonzero elements is the sum of a left zero-divisor and a non-left zero-divisor. This paper aims to extend the existing theory of left fusible rings and related notions such as being left unit-fusible, being regular left fusible, and being uniquely left fusible. New examples of these rings are presented. A new notion of left generalized unit-fusible rings is introduced and discussed. The asymmetry of these notions is addressed.

Bulk photovoltaic effect in ferroelectric nematic liquid crystals.

The bulk photovoltaic (BPV) effect in ferroelectric liquid crystals is of increasing scientific interest owing to its great potential for light-energy conversion. The ferroelectric nematic phase exhibits a huge spontaneous polarization that can be aligned to a preferred direction. In this Letter, we investigate the tensorial properties of the BPV effect in the planarly aligned ferroelectric nematic phase of the liquid crystalline material RM734. A steady-state short-circuit photocurrent of ~160 pA and an open-circuit photovoltage of ~50 mV were observed in a cell with a thickness of 5.5 µm under the illumination of ultraviolet light without any bias voltage. Based on the photocurrent measurements in different electrode configurations, the non-zero elements of the BPV tensor were obtained. The BPV effect is attributed to the combination of the spontaneous polarization and the asymmetric distribution of photoinduced charge carriers. This study not only provides an understanding of the bulk PV mechanism in soft ferroelectrics but also promises a wide range of unprecedented, to the best of our knowledge, benefits for light harvesting to engineer marketable photovoltaic devices.

Leibniz algebras in which all centralizers of nonzero elements are zero algebras

This paper is concerned with generalizing the results for Lie CT-algebras to Leibniz algebras. In some cases our results give a generalization even for the case of a Lie algebra. Results on A-algebras are used to show every Leibniz CT-algebra over an algebraically closed field of characteristic different from 2,3 is solvable or is isomorphic to s l 2 ( F ) . A characterization is then given for solvable Leibniz CT-algebras. It is also shown that the class of solvable Leibniz CT-algebras is factor closed.

Hyperpolyadic Structures

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley–Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the “imaginary tower”), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call “half-quaternions” and “half-octonions”. The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary “half-octonions” is unitless and totally associative.

A novel weighted sparse classification framework with extended discriminative dictionary for data-driven bearing fault diagnosis

Dictionary learning has emerged as an effective approach for data-driven fault diagnosis due to its strong sparse representation ability. Nevertheless, the gathered vibration signals always exhibit a time-shift phenomenon, greatly diminishing the representation capability of dictionary learning methods. Besides, in real-world industrial scenarios, where heavy noise as common features inevitably cover the discriminative features, the decline in the recognition performance of data-driven fault diagnosis methods is a critical challenge. In this paper, a novel weighted sparse classification framework with extended discriminative dictionary (WSC-EDD) is proposed. First, an extended discriminative dictionary design strategy is developed to construct generalized-domain extended dictionary model with strong representation and discrimination ability, in which a novel generalized-domain dictionary fusion method is developed to reduce the effect of time-shift phenomenon and enhance the representation ability of the dictionary. Further, generalized-domain discriminative sub-dictionaries are optimized by the K-singular value decomposition in a data-driven fashion and then the whole extension dictionary are designed adaptively. Second, a weighted optimization method is developed to highlight the contribution of non-zero elements in the correct projection region in sparse codes, in which weighted diagonal matrices are designed. Finally, a sparse recognition method is established for bearing fault classification. The experiment results on three challenging bearing datasets demonstrate that the proposed framework yields average diagnosis accuracies of 99.75%, 99.93% and 100%, respectively. Moreover, the superiority and robustness of WSC-EDD are further confirmed by comparing with several competing methods.

Implementation and optimization of SpMV algorithm based on SW26010P many-core processor and stored in BCSR format

The irregular distribution of non-zero elements of large-scale sparse matrix leads to low data access efficiency caused by the unique architecture of the Sunway many-core processor, which brings great challenges to the efficient implementation of sparse matrix–vector multiplication (SpMV) computing by SW26010P many-core processor. To address this problem, a study of SpMV optimization strategies is carried out based on the SW26010P many-core processor. Firstly, we design a memorized data storage transformation strategy to transform the matrix in CSR storage format into BCSR (Block Compressed Sparse Row) storage. Secondly, the dynamic task scheduling method is introduced to the algorithm to realize the load balance between slave cores. Thirdly, the LDM memory is refined and designed, and the slave core dual cache strategy is optimized to further improve the performance. Finally, we selected a large number of representative sparse matrices from the Matrix Market for testing. The results show that the scheme has obviously speedup the processing procedure of sparse matrices with various sizes and sizes, and the master–slave speedup ratio can reach up to 38 times. The optimization method used in this paper has implications for other complex applications of the SW26010P many-core processor.

On a class of constacyclic codes of length 4ps over 𝔽pm[u] 〈u3〉

Let [Formula: see text] be a prime such that [Formula: see text], where [Formula: see text] is a positive integer. For any nonzero element [Formula: see text] of [Formula: see text], we determine the algebraic structure of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text], where [Formula: see text] and [Formula: see text] is a positive integer. If the unit [Formula: see text] is a square, [Formula: see text], each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of an [Formula: see text]-constacyclic code and an [Formula: see text]- constacyclic code of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that any nonzero polynomial of degree at most [Formula: see text] over [Formula: see text] is invertible in the ambient ring [Formula: see text]. It is also proven that the ambient ring [Formula: see text] is a local ring with the unique maximal ideal [Formula: see text], where [Formula: see text]. Such [Formula: see text]-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each [Formula: see text]-constacyclic code are obtained. The non-existence of self-dual and isodual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text], when the unit [Formula: see text] is not a square, is likewise proved.

Optimization of Large-Scale Sparse Matrix-Vector Multiplication on Multi-GPU Systems

Sparse matrix-vector multiplication (SpMV) is one of the important kernels of many iterative algorithms for solving sparse linear systems. The limited storage and computational resources of individual GPUs restrict both the scale and speed of SpMV computing in problem-solving. As real-world engineering problems continue to increase in complexity, the imperative for collaborative execution of iterative solving algorithms across multiple GPUs is increasingly apparent. Although the multi-GPU-based SpMV takes less kernel execution time, it also introduces additional data transmission overhead, which diminishes the performance gains derived from parallelization across multi-GPUs. Based on the non-zero elements distribution characteristics of sparse matrices and the trade-off between redundant computations and data transfer overhead, this paper introduces a series of SpMV optimization techniques tailored for multi-GPU environments and effectively enhances the execution efficiency of iterative algorithms on multiple GPUs. Firstly, we propose a two-level non-zero elements-based matrix partitioning method to increase the overlap of kernel execution and data transmission. Then, considering the irregular non-zero elements distribution in sparse matrices, a long-row-aware matrix partitioning method is proposed to hide more data transmissions. Finally, an optimization using redundant and inexpensive short-row execution to exchange costly data transmission is proposed. Our experimental evaluation demonstrates that, compared with the SpMV on a single GPU, the proposed method achieves an average speedup of 2.00x and 1.85x on platforms equipped with two RTX 3090 and two Tesla V100-SXM2, respectively. The average speedup of 2.65x is achieved on a platform equipped with four Tesla V100-SXM2.

The specifics of the Galois field GF(257) and its use for digital signal processing

An algorithm of digital logarithm calculation for the Galois field GF(257)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GF(257)$$\\end{document} is proposed. It is shown that this field is coupled with one of the most important existing standards that uses a digital representation of the signal through 256 levels. It is shown that for this case it is advisable to use the specifics of quasi-Mersenne prime numbers, representable in the form p=2n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p=2}^{n}+1$$\\end{document}, which includes the number 257. For fields GF(2n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GF({2}^{n}+1)$$\\end{document}, an alternating encoding can be used, in which non-zero elements of the field are displayed through binary characters corresponding to the numbers + 1 and − 1. In such an encoding, multiplying a field element by 2 is reduced to a quasi-cyclic permutation of binary symbols (the permuted symbol changes sign). Proposed approach makes it possible to significantly simplify the design of computing devices for calculation of digital logarithm and multiplication of numbers modulo 257. A concrete scheme of a device for digital logarithm calculation in this field is presented. It is also shown that this circuit can be equipped with a universal adder modulo an arbitrary number, which makes it possible to implement any operations in the field under consideration. It is shown that proposed digital algorithm can also be used to reduce 256-valued logic operations to algebraic form. It is shown that the proposed approach is of significant interest for the development of UAV on-board computers operating as part of a group.

A Study on the Classification of All Simple Commutative Near-Rings upto Isomorphism

This research paper aims to classify all simple commutative Near-Rings up to isomorphism, which are rings with no zero divisors and no ideals containing a nonzero element that is not prime ideal. The authors use various techniques from algebraic number theory and representation theory of semi simple algebras to construct representations of these rings over the integers or rational numbers. They also provide examples showing how these classes can be distinguished by their algebraic properties, such as whether they are PID (prime idempotent domain), zero-divisorless, or have ideals containing elements with prime index. The study contributes new results and insights into the classification of near-rings, which has implications for other areas of abstract algebra and number theory.