On products of consecutive arithmetic progressions. III
作者：
Zhang, Y.*
期刊：
Acta Mathematica Hungarica ,2021年163(2):407-428 ISSN：0236-5294
通讯作者：
Zhang, Y.
作者机构：
[Zhang, Y.] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Peoples R China.;[Zhang, Y.] Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Peoples R China.
通讯机构：
[Zhang, Y.] C;[Zhang, Y.] H;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Peoples R China.;Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Peoples R China.
关键词：
consecutive arithmetic progression;Diophantine equation;Pell equation;positiveinteger solution
摘要：
Let
$$f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)$$
be a polynomial with
$$k \geq 2, d \geq 1$$
. We consider the Diophantine
equation
$$\prod_{i=1}^{r} f(x_i, k_i, d) = y^{2}, r \geq 1$$
. Using the theory of Pell
equations, we affirm a conjecture of Bennett and van Luijk [3]; extend some results of this Diophantine equation for
$$d=1$$
, and give a positive answer to Question 3.2 of Zhang [19].
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$\theta$-triangle and $\omega$-parallelogram pairs with areas and perimeters in certain proportions
作者：
Li, Yangcheng* ;Zhang, Yong
期刊：
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS ,2020年50(3):1059-1071 ISSN：0035-7596
通讯作者：
Li, Yangcheng
作者机构：
[Li, Yangcheng; Zhang, Yong] Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha, Peoples R China.
通讯机构：
[Li, Yangcheng] C;Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha, Peoples R China.
关键词：
theta-triangle;omega-parallelogram;area;perimeter;elliptic curve
摘要：
By the theory of elliptic curves, we show that given a convex angle theta, there exist, except for finitely many exceptions, infinitely many pairs of rational theta-triangle and omega-parallelogram with areas and perimeters in fixed proportions (alpha, beta) respectively, satisfying that sin omega is a previously fixed rational multiple of sin theta, where alpha and beta are positive rational numbers.
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Stationary splitting iterative methods for the matrix equationAXB=C
作者：
Liu, Zhongyun;Li, Zhen;Ferreira, Carla;Zhang, Yulin*
期刊：
Applied Mathematics and Computation ,2020年378:125195 ISSN：0096-3003
通讯作者：
Zhang, Yulin
作者机构：
[Li, Zhen; Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Peoples R China.;[Ferreira, Carla; Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Zhang, Yulin] U;Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
关键词：
Hermitian positive definite;H-matrices;Stationary splitting iteration;Induced splitting;Curves fitting
摘要：
Stationary splitting iterative methods for solving AXB = C are considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A = M - N by a matrix H such that (I - H)(-1) exists. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants. In particular, for certain surface fitting applications our method is much more efficient than the progressive iterative approximation (PIA), a conventional iterative method often used in computer aided geometric design (CAGD). (C) 2020 Elsevier Inc. All rights reserved.
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On inexact alternating direction implicit iteration for continuous Sylvester equations
作者：
Liu, Zhongyun* ;Zhou, Yang;Zhang, Yulin
期刊：
Numerical Linear Algebra with Applications ,2020年27(5):e2320- ISSN：1070-5325
通讯作者：
Liu, Zhongyun
作者机构：
[Liu, Zhongyun; Zhou, Yang] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Peoples R China.;[Zhang, Yulin] Univ Minho, Ctr Matemat, Braga, Portugal.
通讯机构：
[Liu, Zhongyun] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Peoples R China.
关键词：
ADI iteration;continuous Sylvester equations;convergence;inexact ADI iteration;positive definite matrices
摘要：
In this paper, we study the alternating direction implicit (ADI) iteration for solving the continuous Sylvester equation AX + XB = C, where the coefficient matrices A and B are assumed to be positive semi-definite matrices (not necessarily Hermitian), and at least one of them to be positive definite. We first analyze the convergence of the ADI iteration for solving such a class of Sylvester equations, then derive an upper bound for the contraction factor of this ADI iteration. To reduce its computational complexity, we further propose an inexact variant of the ADI iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer ADI iteration. The convergence is also analyzed in detail. The numerical experiments are given to illustrate the effectiveness of both ADI and inexact ADI iterations. © 2020 John Wiley & Sons, Ltd.
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Fast solvers for tridiagonal Toeplitz linear systems
作者：
Liu, Zhongyun;Li, Shan;Yin, Yi;Zhang, Yulin*
期刊：
Computational and Applied Mathematics ,2020年39(4):1-10 ISSN：2238-3603
通讯作者：
Zhang, Yulin
作者机构：
[Liu, Zhongyun; Li, Shan; Zhang, Yulin] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Peoples R China.;[Yin, Yi] Hunan Coll Informat, Dept Basic Courses, Changsha 410200, Peoples R China.;[Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Zhang, Yulin] U;Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
关键词：
Tridiagonal Toeplitz matrices;Diagonally dominant;Schur complement;Block LU factorization;Pivoting;15A23;15B05;65F05;65F10
摘要：
Let A be a tridiagonal Toeplitz matrix denoted by A= Tritoep (β, α, γ). The matrix A is said to be: strictly diagonally dominant if | α| > | β| + | γ| , weakly diagonally dominant if | α| ≥ | β| + | γ| , subdiagonally dominant if | β| ≥ | α| + | γ| , and superdiagonally dominant if | γ| ≥ | α| + | β|. In this paper, we consider the solution of a tridiagonal Toeplitz system Ax= b, where A is subdiagonally dominant, superdiagonally dominant, or weakly diagonally dominant, respectively. We first consider the case of A being subdiagonally dominant. We transform A into a block 2 × 2 matrix by an elementary transformation and then solve such a linear system using the block LU factorization. Compared with the LU factorization method with pivoting, our algorithm takes less flops, and needs less memory storage and data transmission. In particular, our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency. Then, we deal with superdiagonally dominant and weakly diagonally dominant cases, respectively. Numerical experiments are finally given to illustrate the effectiveness of our algorithms. © 2020, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.
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Arithmetic properties of polynomials
作者：
Zhang, Yong* ;Shen, Zhongyan
期刊：
Periodica Mathematica Hungarica ,2020年81(1):134-148 ISSN：0031-5303
通讯作者：
Zhang, Yong
作者机构：
[Zhang, Yong] Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Peoples R China.;[Shen, Zhongyan] Zhejiang Int Studies Univ, Dept Math, Hangzhou 310012, Peoples R China.
通讯机构：
[Zhang, Yong] C;Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Peoples R China.
关键词：
Diophantine system;Integer solution;Parametric solution;Pellian equation;Elliptic curve
摘要：
First, we prove that the Diophantine system f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)has infinitely many integer solutions for f(X) = X(X+ a) with nonzero integers a≡0,1,4(mod5). Second, we show that the above Diophantine system has an integer parametric solution for f(X) = X(X+ a) with nonzero integers a, if there are integers m,n,k such that {(n2-m2)(4mnk(k+a+1)+a(m2+2mn-n2))≡0(mod(m2+n2)2),(m2+2mn-n2)((m2-2mn-n2)k(k+a+1)-2amn)≡0(mod(m2+n2)2),where k≡0(mod4) when a is even, and k≡2(mod4) when a is odd. Third, we get that the Diophantine system f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)=f(r)f(s)has a five-parameter rational solution for f(X) = X(X+ a) with nonzero rational number a and infinitely many nontrivial rational parametric solutions for f(X) = X(X+ a) (X+ b) with nonzero integers a,b and a≠ b. Finally, we raise some related questions. © 2020, Akadémiai Kiadó, Budapest, Hungary.
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A Diophantine equation with the harmonic mean
作者：
Zhang, Yong* ;Chen, Deyi
期刊：
Periodica Mathematica Hungarica ,2020年80(1):138-144 ISSN：0031-5303
通讯作者：
Zhang, Yong
作者机构：
[Zhang, Yong] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha, Hunan, Peoples R China.;[Zhang, Yong] Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.;[Chen, Deyi] Zhejiang Univ, Sch Math Sci, Hangzhou 310027, Zhejiang, Peoples R China.
通讯机构：
[Zhang, Yong] C;[Zhang, Yong] H;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha, Hunan, Peoples R China.;Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
关键词：
Diophantine equation;Pell’s equation;Integer solutions;Rational parametric solutions
摘要：
Let f∈ Q[x] be a polynomial without multiple roots and deg f≥ 2. We give conditions for f= x2+ bx+ c under which the Diophantine equation 2 f(x) f(y) = f(z) (f(x) + f(y)) has infinitely many nontrivial integer solutions and prove that this equation has infinitely many rational parametric solutions for f= x2+ bx with nonzero integer b. Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials. © 2019, Akadémiai Kiadó, Budapest, Hungary.
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On the Diophantine equations z2= f(x) 2± f(y) 2 involving quartic polynomials
作者：
Zhang, Yong* ;Zargar, Arman Shamsi
期刊：
Periodica Mathematica Hungarica ,2019年79(1):25-31 ISSN：0031-5303
通讯作者：
Zhang, Yong
作者机构：
[Zhang, Yong] Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
通讯机构：
[Zhang, Yong] C;Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
关键词：
Diophantine equation;Quartic polynomial;Rational solution;Elliptic curve
摘要：
By the theory of elliptic curves, we prove that the Diophantine equations z2= f(x) 2± f(y) 2 have infinitely many rational solutions for some quartic polynomials, which gives a positive answer to Question 4.3 of Ulas and Togbé (Publ Math Debrecen 76(1–2):183–201,2010) for quartic polynomials. © 2018, Akadémiai Kiadó, Budapest, Hungary.
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The eigen-structures of real (skew) circulant matrices with some applications
作者：
Liu, Zhongyun;Chen, Siheng;Xu, Weijin;Zhang, Yulin*
期刊：
Computational and Applied Mathematics ,2019年38(4):1-13 ISSN：2238-3603
通讯作者：
Zhang, Yulin
作者机构：
[Chen, Siheng; Xu, Weijin; Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.;[Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Zhang, Yulin] U;Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
关键词：
Real Schur form;Real circulant matrices;Real skew-circulant matrices;Real Toeplitz matrices;CSCS iteration
摘要：
The circulant matrices and skew-circulant matrices are two special classes of Toeplitz matrices and play vital roles in the computation of Toeplitz matrices. In this paper, we focus on real circulant and skew-circulant matrices. We first investigate their real Schur forms, which are closely related to the family of discrete cosine transform (DCT) and discrete sine transform (DST). Using those real Schur forms, we then develop some fast algorithms for computing real circulant, skew-circulant, and Toeplitz matrix-real vector multiplications. Also, we develop a DCT-DST version of circulant and skew-circulant splitting (CSCS) iteration for real positive definite Toeplitz systems. Compared with the fast Fourier transform (FFT) version of CSCS iteration, the DCT-DST version is more efficient and saves a half storage. Numerical experiments are presented to illustrate the effectiveness of our method. © 2019, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.
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Some remarks on Jacobi and Gauss–Seidel-type iteration methods for the matrix equation AXB=C
作者：
Liu, Zhongyun* ;Zhou, Yang;Zhang, Yuelan;Lin, Lu;Xie, Dongxiu
期刊：
Applied Mathematics and Computation ,2019年354:305-307 ISSN：0096-3003
通讯作者：
Liu, Zhongyun
作者机构：
[Liu, Zhongyun; Zhou, Yang; Zhang, Yuelan] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.;[Lin, Lu] Xiamen Univ, Sch Math Sci, Xiamen 361005, Peoples R China.;[Xie, Dongxiu] Beijing Informat Sci & Technol Univ, Sch Sci, Beijing 100192, Peoples R China.
通讯机构：
[Liu, Zhongyun] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.
关键词：
Matrix equation;Classical splitting;Convergence;Norm
摘要：
Tian, et al. proposed in [5] several Jacobi and Gauss-Seidel-type iterative methods for solving matrix equation AXB = C. Those methods were demonstrated to be effective by the given numerical experiments. However, we find that there is a technical error in the proof of the main theorem (Theorem 3.3). In this note we first show this erratum by an example. Then we establish a new convergence theorem which contains the Theorem 3.3 in [5] as a special case. (C) 2019 Elsevier Inc. All rights reserved.
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HERON TRIANGLES WITH FIGURATE NUMBER SIDES (vol 157, pg 478, 2019)
作者：
Peng, J.;Zhang, Y.*
期刊：
Acta Mathematica Hungarica ,2019年159(2):689-689 ISSN：0236-5294
通讯作者：
Zhang, Y.
作者机构：
[Peng, J.; Zhang, Y.] Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
通讯机构：
[Zhang, Y.] C;Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
摘要：
It was discovered that the original online version of the paper contained a mistake, on page 481, lines -4 to -1. It should be changed into the following: (Formula presented.). © 2019, Akadémiai Kiadó, Budapest, Hungary.
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HERON TRIANGLES WITH FIGURATE NUMBER SIDES
作者：
Peng, J.;Zhang, Y.*
期刊：
Acta Mathematica Hungarica ,2019年157(2):478-488 ISSN：0236-5294
通讯作者：
Zhang, Y.
作者机构：
[Peng, J.; Zhang, Y.] Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
通讯机构：
[Zhang, Y.] C;Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
关键词：
Heron triangle;figurate number;polygonal number;binomial coefficient;Pellian equation
摘要：
By the theory of Pellian equation and the method of undetermined coefficients, we show that there exist infinitely many isosceles Heron triangles whose sides are polygonal numbers and binomial coefficients.
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ON THE DIOPHANTINE EQUATION f(x) f(y) = f(z)(n )INVOLVING LAURENT POLYNOMIALS, II
作者：
Zhang, Yong;Zargar, Arman Shamsi*
期刊：
COLLOQUIUM MATHEMATICUM ,2019年158(1):119-126 ISSN：0010-1354
通讯作者：
Zargar, Arman Shamsi
作者机构：
[Zhang, Yong] Changsha Univ Sci & Technol, Sch Math & Stat, Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.;[Zargar, Arman Shamsi] Univ Mohaghegh Ardabili, Dept Math & Applicat, Fac Sci, Ardebil 5619911367, Iran.
通讯机构：
[Zargar, Arman Shamsi] U;Univ Mohaghegh Ardabili, Dept Math & Applicat, Fac Sci, Ardebil 5619911367, Iran.
关键词：
Diophantine equation;Laurent polynomial;Rational parametric solution
摘要：
We investigate the non-trivial rational parametric solutions of the Diophantine equation f(x)f(y) - f(z)n, where f = xk +axk-1 + b/x, k ≥ 2, x2 +a/x+b/x2 for n = 1, and f = x2 +ax+b+a3/(27x), x2 +ax+b+a3/(16x)+a4/(256x2) for n = 2. © Instytut Matematyczny PAN, 2019.
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The general inner-outer iteration method based on regular splittings for the PageRank problem
作者：
Tian, Zhaolu* ;Liu, Yong;Zhang, Yan;Liu, Zhongyun;Tian, Maoyi*
期刊：
Applied Mathematics and Computation ,2019年356:479-501 ISSN：0096-3003
通讯作者：
Tian, Zhaolu;Tian, Maoyi
作者机构：
[Tian, Zhaolu] Shanxi Univ Finance & Econ, Coll Appl Math, Taiyuan 030006, Shanxi, Peoples R China.;[Liu, Yong] Shanxi Univ, Inst Loess Plateau, Taiyuan 030006, Shanxi, Peoples R China.;[Tian, Maoyi; Zhang, Yan] Shandong Univ Sci & Technol, Geomat Coll, Qingdao 266590, Shandong, Peoples R China.;[Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.
通讯机构：
[Tian, Zhaolu; Tian, Maoyi] S;Shanxi Univ Finance & Econ, Coll Appl Math, Taiyuan 030006, Shanxi, Peoples R China.;Shandong Univ Sci & Technol, Geomat Coll, Qingdao 266590, Shandong, Peoples R China.
关键词：
PageRank;Inner-outer iteration;Regular splitting;Preconditioner;Convergence
摘要：
In this paper, combined the regular splittings of the coefficient matrix I - alpha P with the inner-outer iteration framework [9], a general inner-outer (GIO) iteration method is presented for solving the PageRank problem. Firstly, the AOR and modified AOR (MAOR) methods for solving the PageRank problem are constructed, and several comparison results are also given. Next, the GIO iteration scheme is developed, and its overall convergence is analyzed in detail. Furthermore, the preconditioner derived from the GIO iteration can be used to accelerate the Krylov subspace methods, such as GMRES method. Finally, some numerical experiments on several PageRank problems are provided to illustrate the efficiency of the proposed algorithm. (C) 2019 Elsevier Inc. All rights reserved.
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Trigonometric transform splitting methods for real symmetric Toeplitz systems
作者：
Liu, Zhongyun* ;Wu, Nianci;Qin, Xiaorong;Zhang, Yulin
期刊：
Computers & Mathematics with Applications ,2018年75(8):2782-2794 ISSN：0898-1221
通讯作者：
Liu, Zhongyun
作者机构：
[Liu, Zhongyun; Wu, Nianci; Qin, Xiaorong] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.;[Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Liu, Zhongyun] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.
关键词：
Sine transform;Cosine transform;Matrix splitting;Iterative methods;Real Toeplitz matrices
摘要：
In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A. Theoretical analyses show that if the generating function f of the n×n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method. © 2018 Elsevier Ltd
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On products of consecutive arithmetic progressions. II
作者：
Zhang, Y.*
期刊：
Acta Mathematica Hungarica ,2018年156(1):240-254 ISSN：0236-5294
通讯作者：
Zhang, Y.
作者机构：
[Zhang, Y.] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.;[Zhang, Y.] Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
通讯机构：
[Zhang, Y.] C;[Zhang, Y.] H;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.;Hunan Prov Key Lab Math Modeling & Anal Engn, Changsha 410114, Hunan, Peoples R China.
关键词：
Diophantine equation;consecutive arithmetic progression;Pellian equation
摘要：
Let
$${f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)}$$
be a polynomial with
$${k \geq 2}$$
,
$${d \geq 1}$$
. We consider the Diophantine equation
$${\prod_{i = 1}^{r} f(x_i, k_i, d) = y^2}$$
, which is inspired by a question of Erdős and Graham [4, p. 67]. Using the theory of Pellian equation, we give infinitely many (nontrivial) positive integer solutions of the above Diophantine equation for some cases.
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ON THE DIOPHANTINE EQUATION f(x)f(y) = f(z)(n) INVOLVING LAURENT POLYNOMIALS
作者：
Zhang, Yong*
期刊：
COLLOQUIUM MATHEMATICUM ,2018年151(1):111-122 ISSN：0010-1354
通讯作者：
Zhang, Yong
作者机构：
[Zhang, Yong] Changsha Univ Sci & Technol, Coll Math & Comp Sci, Changsha 410114, Hunan, Peoples R China.;[Zhang, Yong] Zhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R China.
通讯机构：
[Zhang, Yong] C;[Zhang, Yong] Z;Changsha Univ Sci & Technol, Coll Math & Comp Sci, Changsha 410114, Hunan, Peoples R China.;Zhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R China.
关键词：
Diophantine equations;Laurent polynomials;Rational parametric solutions
摘要：
Using the theory of elliptic curves, we investigate nontrivial rational parametric solutions of the Diophantine equation f(x)f(y) = f(z)n, where n = 1, 2 and f(X) are some simple Laurent polynomials. © Instytut Matematyczny PAN, 2018.
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Robust parameter identification using parallel global optimization for a batch nonlinear enzyme-catalytic time-delayed process presenting metabolic discontinuities
作者：
Yuan, Jinlong;Zhang, Yuduo;Ye, Jianxiong;Xie, Jun;Teo, Kok Lay;...
期刊：
Applied Mathematical Modelling ,2017年46:554-571 ISSN：0307-904X
通讯作者：
Zhu, Xi
作者机构：
[Yin, Hongchao; Yuan, Jinlong] Dalian Univ Technol, Sch Energy & Power Engn, Dalian 116024, Liaoning, Peoples R China.;[Yuan, Jinlong; Feng, Enmin] Dalian Univ Technol, Sch Math Sci, Dalian 116024, Liaoning, Peoples R China.;[Xiu, Zhilong; Yuan, Jinlong] Dalian Univ Technol, Sch Life Sci & Biotechnol, Dalian 116024, Liaoning, Peoples R China.;[Zhang, Yuduo] Dalian Nationalities Univ, Coll Sci, Dalian 116600, Liaoning, Peoples R China.;[Ye, Jianxiong] Fujian Normal Univ, Sch Math & Comp Sci, Fuzhou 350007, Fujian, Peoples R China.
通讯机构：
[Zhu, Xi] T;Tianjin Univ Technol, Sch Management, Tianjin 300384, Peoples R China.
关键词：
Nonlinear time-delayed switched system;Biological robustness;Parallel optimization;Batch culture;Convergence analysis
摘要：
In this paper, a nonlinear enzyme-catalytic time-delayed switched dynamical system is considered to describe batch culture of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumoniae. This system can not only predict the exponential growth phase but also the lag and the stationary growth phases of batch culture since it contains two switching times for representing the starting moment of lag growth phase and the time when the cell specified growth rate reaches the maximum. The biological robustness is expressed in terms of the expectation and variance of the relative deviation. Our aim is to identify the switching times. To this end, a robust parameter identification problem is formulated, where the switching times are decision variables to be chosen such that the biological robustness measure is optimized. This problem, which is governed by the nonlinear system, is subject to a quality constraint and continuous state inequality constraints. Using a hybrid time-scaling transformation to parameterize the switching times into new parameters, an equivalently robust parameter identification problem is investigated. The continuous state inequality constraints are approximated by a conventional inequality constraint, yielding a sequence of approximate robust parameter identification subproblems. The convergence analysis of this approximation is also investigated. Owing to the highly complex nature of these subproblems, a parallel algorithm, based on simulated annealing, is proposed to solve these subproblems. From an extensive simulation study, it is observed that the obtained optimal switching times are satisfactory. © 2017 Elsevier Inc.
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On the Diophantine system f(z)=f(x)f(y)=f(u)f(v)
作者：
Zhang, Yong* ;Shen, Zhongyan
期刊：
Periodica Mathematica Hungarica ,2017年75(2):295-301 ISSN：0031-5303
通讯作者：
Zhang, Yong
作者机构：
[Zhang, Yong] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.;[Shen, Zhongyan] Zhejiang Int Studies Univ, Dept Math, Hangzhou 310012, Zhejiang, Peoples R China.
通讯机构：
[Zhang, Yong] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.
关键词：
Diophantine system;Positive integer solution;Rational solution;Elliptic curve
摘要：
We show that the Diophantine system f(z)=f(x)f(y)=f(u)f(v)has infinitely many nontrivial positive integer solutions for f(X) = X2- 1 , and infinitely many nontrivial rational solutions for f(X) = X2+ b with nonzero integer b. © 2017, Akadémiai Kiadó, Budapest, Hungary.
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The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices
作者：
Liu, Zhongyun* ;Qin, Xiaorong;Wu, Nianci;Zhang, Yulin
期刊：
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES ,2017年60(4):807-815 ISSN：0008-4395
通讯作者：
Liu, Zhongyun
作者机构：
[Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.;[Wu, Nianci; Qin, Xiaorong; Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Liu, Zhongyun] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.
关键词：
15A23;65F10;65F15;Hermitian positive definite;CSCS splitting;Gauss-Seidel splitting;iterative method;Toeplitz matrix
摘要：
<jats:title>Abstract</jats:title><jats:p>It is known that every Toeplitz matrix <jats:italic>T</jats:italic> enjoys a circulant and skew circulant splitting (denoted CSCS) i.e., <jats:italic>T = C−S</jats:italic> with <jats:italic>C</jats:italic> a circulantmatrix and <jats:italic>S</jats:italic> a skew circulantmatrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive deûnite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss–Seidel (GS) iterative methods if the CSCS is convergent, and that there is always a constant α such that the shifted CSCS iteration converges much faster than the Gauss–Seidel iteration, no matter whether the CSCS itself is convergent or not.</jats:p>
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