Nonzero Elements Research Articles

THE SUFFICIENT AND NECESSARY CONDITIONS FOR A MODULE TO BE A WEAKLY UNIQUE FACTORIZATION MODULE

T A torsion-free module over an integral domain is called Unique Factorization Module (UFM) if satisfied some conditions: (1) Every non-zero element has an irreducible factorization, that is , with are irreducible in and is irreducible in , and (2) if are two irreducible factorizations of , then in , and we can rearrange the order of the ’s so that in for every . The definition of UFM is a generalization of the concept of factorization on the ring which is applied to the module. In this study, we will discuss another definition that is a generalization of UFM, namely by the Weakly Unique Factorization Module (w-UFM). First, some concepts that play an important role in defining w-UFM are given. After that, the definition and characterization of w-UFM is also given. The results of this study will provide the sufficient and necessary conditions of the w-UFM.

Commutative Rings and Corresponding V-Graphs

Algebraic graph theory, which applies algebraic techniques to graph problems, is a pivotal area of study. Many ring algebraic properties have representations in graph theory. In this paper, we introduce an innovative type of graph related to rings that we call the V-graph. Let T be a ring with identity. The V-graph of T, denoted by V(T), consists of a set of vertices equal to all non-zero elements of T. Two different vertices u and v are adjacent if and only if uv is a regular element in T. We present several examples to demonstrate how this form of graph differs from the known ring-based graphs, such as zero-divisor graphs, unit graphs, and quasi-regular graphs. We calculate the diameter and independent number, as well as the domination number for the V-graph of the ring of integers Zn.

A 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 $K\in O(ploy\log n)$.

Controlling formal fibers of countably many principal prime ideals

Let [Formula: see text] be a complete local (Noetherian) ring. For each [Formula: see text], let [Formula: see text] be a nonempty countable set of nonmaximal, pairwise incomparable prime ideals of [Formula: see text], and suppose that if [Formula: see text], then either [Formula: see text] or no element of [Formula: see text] is contained in an element of [Formula: see text]. We provide necessary and sufficient conditions for [Formula: see text] to be the completion of a local integral domain [Formula: see text] satisfying the condition that, for all [Formula: see text], there is a nonzero prime element [Formula: see text] of [Formula: see text] such that [Formula: see text] is exactly the set of maximal elements of the formal fiber of [Formula: see text] at [Formula: see text]. We then prove related results where the domain [Formula: see text] is required to be countable and/or excellent.

On existence of primitive pairs (ξ + ξ −1, f(ξ)) with arbitrary traces

Let F q n be a finite extension of the finite field F q of degree n, where q is a prime power and n is a positive integer. Suppose a , b ∈ F q . In this article we search for pairs (q, n) such that there exists a primitive pair ( ξ + ξ − 1 , f ( ξ ) ) with Tr F q n / F q ( ξ ) = a and Tr F q n / F q ( ξ − 1 ) = b , where ξ is a non-zero element of F q n and f is any non-exceptional rational function in F q n ( x ) .

E-EXACT SEQUENCE AND SOME RESULTS

Let R be a commutative ring with identity, M be a R-module and N be a submodule of M. N is called to be essential (large) in M if N∩Rm≠0 for any nonzero element m∈M and we showed by N≤_e M. A sequence of R-modules and R-morphisms …→┴ M_(i-1) □(→┴f_(i-1) M_i →┴f_i ) M_(i+1) →┴f_(i+1) … is called exact at M_i if Im(f_(i-1) )=Ker (f_i). Also this sequence is called e-exact at M_i if Im(f_(i-1))≤_e Ker(f_i) and it is called e-exact if it is e-exact at each M_i. In this note, we present the concept of the characterization of E-homotopy and E-resolution with some results such as chain map for e-exact sequence and comparing theorem for e-exact sequence.

The unit fractions from a Euclidean domain generate a DVR

AbstractLet D be a Euclidean domain, with fraction field K. Let R(D) be the subring of K generated by the reciprocals of the nonzero elements of D. The main theorem states that if $$R(D) \ne K$$ R ( D ) ≠ K , then R(D) is a rank 1 discrete valuation ring that contains a field consisting of the units of D along with 0. Along the way, it is shown that R(D) contains all fractions of the form a/b, where $$a,b \in D \setminus \{0\}$$ a , b ∈ D \ { 0 } and a has Euclidean value no larger than that of b. Connections are also made to ideas from medieval Italian mathematics.

Least angle sparse principal component analysis for ultrahigh dimensional data

AbstractPrincipal component analysis (PCA) has been a widely used technique for dimension reduction while retaining essential information. However, the ordinary PCA lacks interpretability, especially when dealing with large scale data. To address this limitation, sparse PCA (SPCA) has emerged as an interpretable variant of ordinary PCA. However, the ordinary SPCA relies on solving a challenging non-convex discrete optimization problem, which maximizes explained variance while constraining the number of non-zero elements in each principal component. In this paper, we propose an innovative least angle SPCA technique to address the computational complexity associated with SPCA, particularly in ultrahigh dimensional data, by sequentially identifying sparse principal components with minimal angles to their corresponding components extracted through ordinary PCA. This sequential identification enables solving the optimization problem in polynomial time, significantly reducing computational challenges. Despite its efficiency gains, our proposed method also preserves the main attributes of SPCA. Through comprehensive experimental results, we demonstrate advantages of our approach as a viable alternative for dealing with the computational difficulties inherent in ordinary SPCA. Notably, our method emerges as an efficient and effective solution for conducting ultrahigh dimensional data analysis, enabling researchers to extract meaningful insights and streamline data interpretation.

PSpMv: precision-based sparse matrix partition and SpMV optimization

AbstractThe new generation of computing devices tends to support multiple floating-point formats and different computing precision. Besides single and double precision, half precision is embraced and widely supported by new computing devices. Low-precision representations have compact memory size and lightweight computing strength, and they also bring opportunities to the optimization of BLAS routines. This paper proposes a new sparse matrix partition approach based on IEEE 754 standard floating-point format. An input sparse matrix in double precision is partitioned and transformed into several sub-matrices in different precision without loss of accuracy. Most non-zero elements can be stored in half or single precision, if the most significant bits of exponent and the least significant bits of mantissa are zeros in double-precision representation. Based on this mixed-precision representation of sparse matrix, we also present a new SpMV algorithm pSpMV for GPU devices. pSpMV not only reduces the memory access overhead, but also reduces the computing strength of floating-point numbers. Experimental results on two GPU devices show that pSpMV achieves a geometric mean speedup of 1.39x on Tesla V100 and 1.45x on Tesla P100 over double-precision SpMV for 2,554 sparse matrices.

STUDY OF PARAMETER SPACE OF MULTIDIMENSIONAL SYSTEM WITH RELAY HYSTERESIS AND PERTURBATION

The object of research is an 𝑛-dimensional system of ordinary differential equations that contains a constant diagonal matrix with real eigenvalues, two-position relay nonlinearity of hysteresis type with a parameter and a continuous periodic perturbation function with a parameter. In the case of a special type of feedback vector (one non-zero element), we obtain conditions for system parameters that ensure the existence of a unique two-point oscillatory periodic solution with a period multiple of the perturbation function period. We establish a functional dependence of the nonlinearity parameter on the perturbation function parameter. Influence of perturbation function parameter values on the existence of the solution is investigated. We offer an algorithm for seeking system parameters and the instant of the first relay switching in the case when the solution period is given. Theoretical results, including the proposed algorithm, are illustrated by the example of a three-dimensional system.

A Novel, Direct Matrix Solver for Supersonic Boundary Element Method Systems

For problems with very fine surface meshes, typically the most time-consuming step of a boundary element method (BEM, also called a panel method) is solving the final linear system of equations. Many have already studied how to efficiently solve the dense, asymmetric systems which arise in elliptic BEMs. However, this has not been studied for a supersonic aerodynamic BEM, for which the governing PDE is hyperbolic. Due to this hyperbolic character, the matrix equation which arises from a supersonic BEM has a large number of identically zero elements. But the resulting linear system of equations is also not sparse in the standard sense. Hence, the efficient solution of the linear system of equations arising in a supersonic BEM is here considered. A novel sorting algorithm is developed whereby the non-zero elements may be arranged into a useful structure with minimal cost. A novel direct solution method is developed here based on fast Givens rotations and the QR decomposition. This novel solver leverages the unique matrix structure to solve the supersonic system of equations more quickly than traditional direct methods. This novel method is then compared to other direct and iterative matrix solvers and is shown to be more robust than iterative solvers and more efficient than other direct solvers, with a computational time complexity of approximately O(N2.5).

Operational Support Estimator Networks.

In this work, we propose a novel approach called Operational Support Estimator Networks (OSENs) for the support estimation task. Support Estimation (SE) is defined as finding the locations of non-zero elements in sparse signals. By its very nature, the mapping between the measurement and sparse signal is a non-linear operation. Traditional support estimators rely on computationally expensive iterative signal recovery techniques to achieve such non-linearity. Contrary to the convolutional layers, the proposed OSEN approach consists of operational layers that can learn such complex non-linearities without the need for deep networks. In this way, the performance of non-iterative support estimation is greatly improved. Moreover, the operational layers comprise so-called generative super neurons with non-local kernels. The kernel location for each neuron/feature map is optimized jointly for the SE task during training. We evaluate the OSENs in three different applications: i. support estimation from Compressive Sensing (CS) measurements, ii. representation-based classification, and iii. learning-aided CS reconstruction where the output of OSENs is used as prior knowledge to the CS algorithm for enhanced reconstruction. Experimental results show that the proposed approach achieves computational efficiency and outperforms competing methods, especially at low measurement rates by significant margins.

GPU Based Virtual Machine for Sleptsov Net Computing

Novel paradigm of Sleptsov Net Computing (SNC) mends imperfections of modern HPC architecture with computing memory implementation and provides a vivid graphical language, fine granulation of concurrent processes, and wide application of formal methods for reliable software design. IDE and VM for SNC have been developed and described in early papers. In the present paper, we considerably reduce GPU memory and thread consumption of the previous prototype implementation, introducing a matrix with condensed columns (MCC), and enhance performance, using the first fireable transition choice on the transition sequence reordered according to the lattice of priorities. MCC takes into consideration procedures of Sleptsov net (SN) arcs processing to enhance performance with rather little overhead compared to traditional sparse matrix formats. We represent a matrix of SN arcs by a pair of considerably smaller matrices, having the same number of columns, with the number of rows equal to the maximal number of nonzero elements over the source matrix columns. The first matrix contains the indexes within the source matrix, the second matrix contains the values within the source matrix. To run an SN, we use the corresponding rectangular matrix of GPU threads, with very few indirect memory accesses via MCC. As a result, we increase VM performance in 2–4 times, and reduce by hundred times GPU memory and thread consumption on programs from the SN software collection.

LO-SpMM: Low-cost Search for High-performance SpMM Kernels on GPUs

As deep neural networks (DNNs) become increasingly large and complicated, pruning techniques are proposed for lower memory footprint and more efficient inference. The most critical kernel to execute pruned sparse DNNs on GPUs is Sparse-dense Matrix Multiplication (SpMM). To maximize the performance of SpMM, despite the high-performance implementation generated from advanced tensor compilers, they often take a long time to iteratively search tuning configurations. Such a long time slows down the cycle of exploring better DNN architectures or pruning algorithms. In this article, we propose LO-SpMM to efficiently generate high-performance SpMM implementations for sparse DNN inference. Based on the analysis of nonzero elements’ layout, the characterization of the GPU architecture, and a rank-based cost model, LO-SpMM can effectively reduce the search space and eliminate possibly low-performance candidates. Besides, rather than generating complete SpMM implementations for evaluation, LO-SpMM constructs simplified proxies to quickly estimate performance, thereby substantially reducing compilation and execution costs. Experimental results show that LO-SpMM can reduce the search time by 281× at most, while the performance of generated SpMM implementations is comparable to or better than the state-of-the-art sparse tensor compiling solutions.

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 tradeoff between redundant computations and data transfer overhead, this article introduces a series of SpMV optimization techniques tailored for multi-GPU environments and effectively enhances the execution efficiency of iterative algorithms on multiple GPUs. First, 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 2.00× and 1.85× on platforms equipped with two RTX 3090 and two Tesla V100-SXM2, respectively. The average speedup of 2.65× is achieved on a platform equipped with four Tesla V100-SXM2.

Classification of three dimensional anti-dendriform algebras

This article is devoted to the classification of anti-dendriform algebras that are associated with associativity. In particular, the paper is devoted to classifying anti-dendriform algebras associated with null-filiform associative algebras and three-dimensional algebras. It is known that any finite-dimensional associative algebra without a nonzero idempotent element is nilpotent. If an associative algebra has a nonzero idempotent element, then there does not exist a compatible anti-dendriform algebra structure associated with associative algebras.

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.

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.

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.

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.