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A Numerical Method for Solving Ill-Conditioned Equation Systems Arising from Radial Basis Functions
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作者 Edward J. Kansa 《American Journal of Computational Mathematics》 2023年第2期356-370,共15页
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ... Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers. 展开更多
关键词 Continuously Differentiable Radial Basis Functions Global Maxima and Minima Solutions of ill-conditioned Linear equations Block Gaussian Elimination Arbitrary Precision Arithmetic
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Novel method based on ant colony opti mization for solving ill-conditioned linear systems of equations 被引量:1
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作者 段海滨 王道波 朱家强 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2005年第3期606-610,共5页
A novel method based on ant colony optimization (ACO), algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from th... A novel method based on ant colony optimization (ACO), algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from the behavior of real ants. ACO algorithm is first introduced, a kind of positive feedback mechanism is adopted in ACO. Then, the solu- tion problem of linear systems of equations was reformulated as an unconstrained optimization problem for solution by an ACID algorithm. Finally, the ACID with other traditional methods is applied to solve a kind of multi-dimensional Hilbert ill-conditioned linear equations. The numerical results demonstrate that ACO is effective, robust and recommendable in solving ill-conditioned linear systems of equations. 展开更多
关键词 ill-conditioned linear systems of equations ant colony optimization condition number optimization.
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Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method 被引量:1
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作者 Tianmin Han Yuhuan Han 《Applied Mathematics》 2010年第3期222-229,共8页
In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improv... In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper. 展开更多
关键词 Nonlinear equations Ordinary Differential equations Numerical Integration Fixed Point ITERATION Newton’s Method STIFF ill-conditioned
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Note on the Number of Solutions of Cubic Diagonal Equations over Finite Fields 被引量:2
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作者 HU Shuangnian WANG Shihan +1 位作者 LI Yanyan NIU Yujun 《Wuhan University Journal of Natural Sciences》 CAS CSCD 2023年第5期369-372,共4页
Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of H... Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results. 展开更多
关键词 finite field rational point diagonal equations Jacobi sums
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THE ESTIMATION OF SOLUTION OF THE BOUNDARY VALUE PROBLEM OF THE SYSTEMS FOR QUASI-LINEAR ORDINARY DIFFERENTIAL EQUATIONS
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作者 黄蔚章 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1992年第8期745-754,共10页
This paper deals with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equationswhere x,f, y , h, A, B and C all belong to Rn , and g is an n×n matrix ... This paper deals with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equationswhere x,f, y , h, A, B and C all belong to Rn , and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder. 展开更多
关键词 systems of the quasi-linear ordinary differential equation singular perturbation diagonalIZATION asymptotic expansion
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Existence of Solution for a Coupled System of Fractional Integro-Differential Equations on an Unbounded Domain
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作者 Azizollah Babakhani 《Analysis in Theory and Applications》 2013年第1期47-61,共15页
We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arz... We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arzela-Ascoli theorem to show a fixed point theorem of Schauder. 展开更多
关键词 Fractional derivative/integral coupled system Volterra integral equation diagonalization method.
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Efficient Solutions of Coupled Matrix and Matrix Differential Equations
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作者 Zeyad Al-Zhour 《Intelligent Control and Automation》 2012年第2期176-187,共12页
In Kronecker products works, matrices are sometimes regarded as vectors and vectors are sometimes made in to matrices. To be precise about these reshaping we use the vector and diagonal extraction operators. In the pr... In Kronecker products works, matrices are sometimes regarded as vectors and vectors are sometimes made in to matrices. To be precise about these reshaping we use the vector and diagonal extraction operators. In the present paper, the results are organized in the following ways. First, we formulate the coupled matrix linear least-squares problem and present the efficient solutions of this problem that arises in multistatic antenna array processing problem. Second, we extend the use of connection between the Hadamard (Kronecker) product and diagonal extraction (vector) operator in order to construct a computationally-efficient solution of non-homogeneous coupled matrix differential equations that useful in various applications. Finally, the analysis indicates that the Kronecker (Khatri-Rao) structure method can achieve good efficient while the Hadamard structure method achieve more efficient when the unknown matrices are diagonal. 展开更多
关键词 MATRIX Products LEAST-SQUARES Problem Coupled MATRIX and MATRIX DIFFERENTIAL equations diagonal Extraction OPERATOR
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有限域上一类四次对角方程有理点的个数
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作者 胡双年 高继东 杜屹洋 《四川大学学报(自然科学版)》 CAS CSCD 北大核心 2024年第2期13-18,共6页
设p为素数,k为正整数,F_(q)是q=p^(k)的有限域.用F^(*)_(q)表示F_(q)的乘法群,即F^(*)_(q)=F_(q)/{0}.设f(x_(1),…,x_(n))是F_(q)上的多项式,用N(f(x_(1),…,x_(n))=0)表示f(x_(1),x_(2),…,x_(n))=0在F_(q)上的有理点个数.1981年,Myer... 设p为素数,k为正整数,F_(q)是q=p^(k)的有限域.用F^(*)_(q)表示F_(q)的乘法群,即F^(*)_(q)=F_(q)/{0}.设f(x_(1),…,x_(n))是F_(q)上的多项式,用N(f(x_(1),…,x_(n))=0)表示f(x_(1),x_(2),…,x_(n))=0在F_(q)上的有理点个数.1981年,Myerson给出了N(x^(4)_(1)+…+x^(4)_(n)=0)的递推公式.最近,赵等给出了N(x^(4)_(1)+x^(4)_(2)=c),N(x^(4)_(1)+x^(4)_(2)+x^(4)_(3)=c)和N(x^(4)_(1)+x^(4)_(2)+x^(4)_(3)+x^(4)_(4)=c)的精确公式,其中c∈F^(*)_(q).本文利用雅可比和以及一个类比Hasse-Davenport定理的结果给出了N(x^(4)_(1)+…+x^(4)_(n)=c)的精确公式,扩展了已有结果. 展开更多
关键词 有限域 有理点 对角方程 雅可比和
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两类五对角行列式的计算研究
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作者 韩摩西 张伟 胡卫敏 《长春师范大学学报》 2024年第8期1-7,共7页
五对角行列式的求解问题在偏微分方程的数值求解、图像处理、电路分析、生物信息学等方面具有非常广泛的应用.本文将两类五对角行列式通过Laplace展开,将各余子式与其自身表示为一阶线性递推方程组,在此基础上将行列式随阶数变化的值表... 五对角行列式的求解问题在偏微分方程的数值求解、图像处理、电路分析、生物信息学等方面具有非常广泛的应用.本文将两类五对角行列式通过Laplace展开,将各余子式与其自身表示为一阶线性递推方程组,在此基础上将行列式随阶数变化的值表示为系数矩阵的幂方与初值向量乘积的形式.并通过LU分解将两类行列式的值表示为二元递推方程组的解的连乘积的形式. 展开更多
关键词 五对角行列式 递推方程 LU分解
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Solving single-frequency phase ambiguity using parameter weights fitting and constrained equation ambiguity resolution methods 被引量:5
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作者 阳仁贵 欧吉坤 +3 位作者 袁运斌 张克非 闻德保 Ron Grenfell 《Journal of Central South University of Technology》 EI 2006年第1期93-98,共6页
Based on the structural characteristics of the double-differenced normal equation, a new method was proposed to resolve the ambiguity float solution through a selection of parameter weights to construct an appropriate... Based on the structural characteristics of the double-differenced normal equation, a new method was proposed to resolve the ambiguity float solution through a selection of parameter weights to construct an appropriate regularized matrix, and a singular decomposition method was used to generate regularization parameters. Numerical test results suggest that the regularized ambiguity float solution is more stable and reliable than the least-squares float solution. The mean square error matrix of the new method possesses a lower correlation than the variancecovariance matrix of the least-squares estimation. The size of the ambiguity search space is reduced and the search efficiency is improved. The success rate of the integer ambiguity searching process is improved significantly when the ambiguity resolution by using constraint equation method is used to determine the correct ambiguity integervector. The ambiguity resolution by using constraint equation method requires an initial input of the ambiguity float solution candidates which are obtained from the LAMBDA method in the new method. In addition, the observation time required to fix reliable integer ambiguities can he significantly reduced. 展开更多
关键词 global position system ill-conditioned state parameter weight fitting method constraint equation integer ambiguity
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ON THE PRIORI ESTIMATE OF MAXIMUM MODULUS OF SOLUTIONS TO A SYSTEM OF DIAGONALLY DEGENERATE ELLIPTIC EQUATIONS 被引量:1
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作者 WANG Xiangdong RONG Haiwu LIANG Xiting (Mathematics Department of Foshan University, Foshan 528000, China) 《Journal of Systems Science & Complexity》 SCIE EI CSCD 2003年第4期446-452,共7页
In this paper we first give an a priori estimate of maximum modulus ofsolutions for a class of systems of diagonally degenerate elliptic equations in the case of p > 2.
关键词 system of diagonally degenerate elliptic equations generalized solution priori estimate
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A New Approximation to the Linear Matrix Equation AX = B by Modification of He’s Homotopy Perturbation Method 被引量:1
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作者 Amir Sadeghi 《Advances in Linear Algebra & Matrix Theory》 2016年第2期23-30,共8页
It is well known that the matrix equations play a significant role in engineering and applicable sciences. In this research article, a new modification of the homotopy perturbation method (HPM) will be proposed to obt... It is well known that the matrix equations play a significant role in engineering and applicable sciences. In this research article, a new modification of the homotopy perturbation method (HPM) will be proposed to obtain the approximated solution of the matrix equation in the form AX = B. Moreover, the conditions are deduced to check the convergence of the homotopy series. Numerical implementations are adapted to illustrate the properties of the modified method. 展开更多
关键词 Matrix equation Homotopy Perturbation Method CONVERGENCE diagonally Dominant Matrix Regular Splitting
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Iterative convergence of boundary-volume integral equation method
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作者 Gengxin Yu Liyun Fu 《Earthquake Science》 CSCD 2011年第5期391-400,共10页
The boundary-volume integral equation numerical technique can be a powerful tool for piecewise heterogeneous media, but it is limited to small problems or low frequencies because of great computational cost. Therefore... The boundary-volume integral equation numerical technique can be a powerful tool for piecewise heterogeneous media, but it is limited to small problems or low frequencies because of great computational cost. Therefore, a restarted GMRES method is applied to solve large-scale boundary-volume scattering problems in this paper to overcome the computational barrier. The iterative method is firstly applied to responses of dimensionless frequency to a semicircular alluvial valley filled with sediments, compared with the standard Gaussian elimination method. Then the method is tested by a heterogeneous multilayered model to show its applicability. Numerical experiments indicate that the preconditioned GMRES method can significantly improve computational efficiency especially for large Earth models and high frequencies, but with a faster convergence for the left diagonal preconditioning. 展开更多
关键词 boundary-volume integral equation generalized Lipmann-Schwinger integral equation GMRES method diagonal preconditioner
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An Efficient Direct Method to Solve the Three Dimensional Poisson’s Equation 被引量:2
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作者 Alemayehu Shiferaw Ramesh Chand Mittal 《American Journal of Computational Mathematics》 2011年第4期285-293,共9页
In this work, the three dimensional Poisson’s equation in Cartesian coordinates with the Dirichlet’s boundary conditions in a cube is solved directly, by extending the method of Hockney. The Poisson equation is appr... In this work, the three dimensional Poisson’s equation in Cartesian coordinates with the Dirichlet’s boundary conditions in a cube is solved directly, by extending the method of Hockney. The Poisson equation is approximated by 19-points and 27-points fourth order finite difference approximation schemes and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The efficiency of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results. It is shown that 19-point formula produces comparable results with 27-point formula, though computational efforts are more in 27-point formula. 展开更多
关键词 Poisson’s equatION Finite DIFFERENCE METHOD Tri-diagonal Matrix Hockney’s METHOD Thomas Algorithm
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High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate 被引量:1
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作者 Alemayehu Shiferaw Ramesh Chand Mittal 《American Journal of Computational Mathematics》 2014年第2期73-86,共14页
In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved dire... In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly. The Poisson equation is approximated by fourth-order finite differences and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The accuracy of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results. 展开更多
关键词 Poisson’s equatION Tri-diagonal Matrix FOURTH-ORDER FINITE DIFFERENCE APPROXIMATION Hockney’s Method Thomas Algorithm
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Analysis of a Composition Operator’s Eigenvalue Equation on Unitary Spaces by the Krein-Rutman Theorem
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作者 A.N.M.Rezaul Karim 《Applied Mathematics》 2020年第2期76-83,共8页
The Krein-Rutman theorem is vital in partial differential equations that are non-linear and provides evidence of the presence of several significant eigenvalues useful in topological degree calculations, stability ana... The Krein-Rutman theorem is vital in partial differential equations that are non-linear and provides evidence of the presence of several significant eigenvalues useful in topological degree calculations, stability analysis, and bifurcation theory. Schr?der’s equation which has been used extensively in studies of turbulence is an equation with a single independent variable suitable for encoding self-similarity. The concept of Hilbert spaces has been an inner product space frequently used due to its convenience in countless dimensional vector analysis. This paper is aimed at proving a number of solutions through the Krein-Rutman theorem in unitary spaces especially in Hilbert spaces. It has been certainly observed that the whole Krein-Rutman theorem system has a fairly stable scope, and has strong regular features, and many non-linear elliptic operators need the most ethical principles to satisfy the comparison policy. 展开更多
关键词 Krein-Rutman Theorem Unitary Space Bounded Solutions diagonalIZATION Schroder equation
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模件式战术导弹舱内战斗部装配刚度辨识方法研究
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作者 商霖 张程 +2 位作者 周晓和 迟学谦 孟令涛 《导弹与航天运载技术(中英文)》 CSCD 北大核心 2023年第6期23-30,共8页
为了研究模件式战术导弹舱内战斗部装配刚度的辨识方法,以试验模态参数和有限元模型为基础,提出了一种通过模态特征方程反问题辨识模件式战术导弹舱内战斗部装配刚度参数的新方法。该方法以模态特征方程为基础,利用连接单元刚度矩阵的... 为了研究模件式战术导弹舱内战斗部装配刚度的辨识方法,以试验模态参数和有限元模型为基础,提出了一种通过模态特征方程反问题辨识模件式战术导弹舱内战斗部装配刚度参数的新方法。该方法以模态特征方程为基础,利用连接单元刚度矩阵的三对角线特征,推导得到了装配位置连接节点的内力和位移之间的关系方程。随后根据导弹的特点,采用三分段法进行质量和刚度分段,构建导弹动力学模型并进行模态分析。在数值试验中,将模态参数和导弹主体/分支结构的质量与刚度矩阵代入关系方程,最终辨识得到导弹主体结构和分支结构装配位置的刚度矢量值。数值试验结果验证了辨识方法的有效性和可靠性。 展开更多
关键词 辨识方法 连接单元 装配刚度 模态特征方程 三对角线矩阵
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非线性反应扩散方程的高精度有限差分方法
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作者 刘圣恩 葛永斌 祁应楠 《应用数学》 北大核心 2023年第3期747-755,共9页
首先,针对空间二阶导数,提出了一种五点六阶差分公式.然后,针对一维非线性反应扩散方程,空间导数项采用该差分公式离散,时间导数项采用Crank-Nicolson方法进行离散,再利用Richardson外推方法将时间精度提高到四阶,提出了一种时间四阶空... 首先,针对空间二阶导数,提出了一种五点六阶差分公式.然后,针对一维非线性反应扩散方程,空间导数项采用该差分公式离散,时间导数项采用Crank-Nicolson方法进行离散,再利用Richardson外推方法将时间精度提高到四阶,提出了一种时间四阶空间六阶精度的有限差分格式.由于每一个时间层上所形成的线性方程组是五对角形的,因此采用五对角追赶法进行计算,计算简单且高效.最后通过数值实验验证了格式的精确性和可靠性. 展开更多
关键词 非线性反应扩散方程 五点六阶差分公式 高精度 五对角追赶法 RICHARDSON外推
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有限域上对角方程ax^(d)+by^(2d)=c的可解性
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作者 陈燎 强诗瑗 陈龙 《四川大学学报(自然科学版)》 CAS CSCD 北大核心 2023年第6期40-46,共7页
有限域研究中的一个重要问题是所谓的幂和问题,即在充分大的有限域F_(q)中任意元素能否表示成一个元素的d1次幂和一个元素的d2次幂之和,其中d1和d2均为正整数.设a,b∈F_(q)^(*),c∈F_(q),d为正整数.在本文中,我们利用有限域上的概率测度... 有限域研究中的一个重要问题是所谓的幂和问题,即在充分大的有限域F_(q)中任意元素能否表示成一个元素的d1次幂和一个元素的d2次幂之和,其中d1和d2均为正整数.设a,b∈F_(q)^(*),c∈F_(q),d为正整数.在本文中,我们利用有限域上的概率测度,Cauchy-Schwarz不等式及广义Fourier变换等工具研究了某些子集的测度,由此证明当q>(4d^(6)-4d^(5)+d^(4)+d^(2)+√d^(6)(2d-1)^(2)(4d^(4)-4d^(3)+d^(2)+2))时方程axd+by2d=c在F_(q)上恒有解.进一步,我们证明当q≠5时方程ax^(2)+by^(4)=c在F_(q)上恒有解. 展开更多
关键词 有限域 对角方程 广义FOURIER变换 可解性
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Quantization of the Kinetic Energy of Deterministic Chaos
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作者 Victor A. Miroshnikov 《American Journal of Computational Mathematics》 2023年第1期1-81,共81页
In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic... In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic scalar dynamic structures, and the theoretical and experimental deterministic vector dynamic structures have been developed to compute the exact solution for deterministic chaos of the exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. To explore properties of the kinetic energy, rectangular, diagonal, and triangular summations of a matrix of the kinetic energy and general terms of various sums have been used in the current paper to develop quantization of the kinetic energy of deterministic chaos. Nested structures of a cumulative energy pulson, an energy pulson of propagation, an internal energy oscillon, a diagonal energy oscillon, and an external energy oscillon have been established. In turn, the energy pulsons and oscillons include group pulsons of propagation, internal group oscillons, diagonal group oscillons, and external group oscillons. Sequentially, the group pulsons and oscillons contain wave pulsons of propagation, internal wave oscillons, diagonal wave oscillons, and external wave oscillons. Consecutively, the wave pulsons and oscillons are composed of elementary pulsons of propagation, internal elementary oscillons, diagonal elementary oscillons, and external elementary oscillons. Topology, periodicity, and integral properties of the exponential pulsons and oscillons have been studied using the novel method of the inhomogeneous Fourier expansions via eigenfunctions in coordinates and time. Symbolic computations of the exact expansions have been performed using the experimental and theoretical programming in Maple. Results of the symbolic computations have been justified by probe visualizations. 展开更多
关键词 The Navier-Stokes equations Quantization of Kinetic Energy Deterministic Chaos Elementary Pulson of Propagation Internal Elementary Oscillon diagonal Elementary Oscillon External Elementary Oscillon Wave Pulson of Propagation Internal Wave Oscillon diagonal Wave Oscillon External Wave Oscillon Group Pulson of Propagation Internal Group Oscillon diagonal Group Oscillon External Group Oscillon Energy Pulson of Propagation Internal Energy Oscillon diagonal Energy Oscillon External Energy Oscillon Cumulative Energy Pulson
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