scholarly journals Matrix Diophantine equations over quadratic rings and their solutions

2020 ◽  
Vol 12 (2) ◽  
pp. 368-375
Author(s):  
N.B. Ladzoryshyn ◽  
V.M. Petrychkovych ◽  
H.V. Zelisko
Keyword(s):  
Matrix Equation ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Standard Form ◽  
Matrix Equations ◽  
System Of Linear Equations ◽  
Structure Of Solutions ◽  
Linear Diophantine Equations ◽  
The Matrix ◽  
Solution Methods

The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

scholarly journals The Matrix Linear Unilateral and Bilateral Equations with Two Variables over Commutative Rings

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
N. S. Dzhaliuk ◽  
V. M. Petrychkovych
Keyword(s):  
Linear Equations ◽  
Standard Form ◽  
Commutative Rings ◽  
Matrix Equations ◽  
General Solutions ◽  
Particular Solutions ◽  
The Matrix ◽  
Bezout Domains

The method of solving matrix linear equations and over commutative Bezout domains by means of standard form of a pair of matrices with respect to generalized equivalence is proposed. The formulas of general solutions of such equations are deduced. The criterions of uniqueness of particular solutions of such matrix equations are established.

scholarly journals Fuzzy Symmetric Solutions of Fuzzy Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaobin Guo ◽  
Dequan Shang
Keyword(s):  
Matrix Equation ◽  
Symmetric Solution ◽  
Fuzzy Numbers ◽  
Linear Equations ◽  
Nonsingular Matrix ◽  
Matrix Equations ◽  
System Of Linear Equations ◽  
Fuzzy Matrix ◽  
Fuzzy Linear System ◽  
Product Of Matrices

The fuzzy symmetric solution of fuzzy matrix equationAX˜=B˜, in whichAis a crispm×mnonsingular matrix andB˜is anm×nfuzzy numbers matrix with nonzero spreads, is investigated. The fuzzy matrix equation is converted to a fuzzy system of linear equations according to the Kronecker product of matrices. From solving the fuzzy linear system, three types of fuzzy symmetric solutions of the fuzzy matrix equation are derived. Finally, two examples are given to illustrate the proposed method.

scholarly journals Matrix and vector sequence transformations revisited

1995 ◽  
Vol 38 (3) ◽  
pp. 495-510 ◽  
Author(s):  
C. Brezinski ◽  
A. Salam
Keyword(s):  
Linear Equations ◽  
Convergence Acceleration ◽  
Difference Operators ◽  
Scalar Case ◽  
Extrapolation Methods ◽  
System Of Linear Equations ◽  
Vector Sequence ◽  
The Matrix ◽  
Sequence Transformations

Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation difference operators. In this paper, these two approaches are generalized to the matrix and the vector cases.

scholarly journals A novel interpretation of least squares solution

1992 ◽  
Vol 15 (1) ◽  
pp. 41-46
Author(s):  
Jack-Kang Chan
Keyword(s):  
Least Squares ◽  
Source Localization ◽  
Convex Combination ◽  
Linear Equations ◽  
Singular Values ◽  
Statistical Interpretation ◽  
System Of Linear Equations ◽  
Overdetermined System ◽  
The Matrix ◽  
Cramer’S Rule

We show that the well-known least squares (LS) solution of an overdetermined system of linear equations is a convex combination of all the non-trivial solutions weighed by the squares of the corresponding denominator determinants of the Cramer's rule. This Least Squares Decomposition (LSD) gives an alternate statistical interpretation of least squares, as well as another geometric meaning. Furthermore, when the singular values of the matrix of the overdetermined system are not small, the LSD may be able to provide flexible solutions. As an illustration, we apply the LSD to interpret the LS-solution in the problem of source localization.

scholarly journals MATRIX EQUATIONS AND HILBERT'S TENTH PROBLEM

2008 ◽  
Vol 18 (08) ◽  
pp. 1231-1241 ◽  
Author(s):  
PAUL BELL ◽  
VESA HALAVA ◽  
TERO HARJU ◽  
JUHANI KARHUMÄKI ◽  
IGOR POTAPOV
Keyword(s):  
Algebraic Number ◽  
Matrix Equation ◽  
Restricted Problem ◽  
Algebraic Number Field ◽  
Matrix Equations ◽  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem ◽  
Integral Matrices ◽  
The Matrix ◽  
Matrix Semigroup

We show a reduction of Hilbert's tenth problem to the solvability of the matrix equation [Formula: see text] over non-commuting integral matrices, where Z is the zero matrix, thus proving that the solvability of the equation is undecidable. This is in contrast to the case whereby the matrix semigroup is commutative in which the solvability of the same equation was shown to be decidable in general. The restricted problem where k = 2 for commutative matrices is known as the "A-B-C Problem" and we show that this problem is decidable even for a pair of non-commutative matrices over an algebraic number field.

scholarly journals MATHEMATICAL MODELING OF THE 3(Н)-QUINAZOLIN-4-ONЕ SYNTHESIS PROCESS

2021 ◽  
pp. 51-57
Keyword(s):  
Mathematical Modeling ◽  
Reaction Rate ◽  
Linear Equations ◽  
Product Yield ◽  
Farm Animals ◽  
Synthesis Process ◽  
Optimal Conditions ◽  
System Of Linear Equations ◽  
Molar Ratios ◽  
The Matrix

The aim is to optimize the conditions for the synthesis of 3(H)-quinazolin-4-one by the method of mathematical modeling to develop a technology for producing the substance of a new domestic drug used in the treatment of farm animals from helminths. In mathematical modeling, the method of a small number of squares was used. Analytical dependences of the product yield on temperature, reaction time, and molar ratios of the starting materials were determined. A system of linear equations has been compiled. The system of linear equations was performed by the matrix method (A, B, C).The average reaction rate was determined. Based on the results obtained, a 3(H)-quinazolin-4-one diagram using the Maple 18 program and an icon diagram of the reaction duration, temperature, and reaction rate are shown. Based on the results of mathematical modeling, a highly efficient technological scheme for obtaining 3(H)-quinazolin-4-one has been developed. Based on this technology, compound 3(H)-quinazolin-4-one was synthesized in quantitative products at the Institute of Plant Chemistry, at a pilot production plant.The results obtained confirmed the found optimal conditions

scholarly journals HERMITE POLYNOMIAL APPROACH FOR SOLVING SINGULAR PERTURBATED DELAY DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 20 (4) ◽  
pp. 845-854
Author(s):  
SUAYIP YUZBASI ◽  
NURCAN BAYKUS SAVASANERIL
Keyword(s):  
Algebraic System ◽  
Hermite Polynomial ◽  
Linear Equations ◽  
Hermite Polynomials ◽  
System Of Linear Equations ◽  
Collocation Points ◽  
The Matrix ◽  
Collocation Approach ◽  
Delay Differential ◽  
Derivatives Of

In this study, a collocation approach based on the Hermite polyomials is applied to solve the singularly perturbated delay differential eqautions by boundary conditions. By means of the matix relations of the Hermite polynomials and the derivatives of them, main problem is reduced to a matrix equation. And then, collocation points are placed in equation of the matrix. Hence, the singular perturbed problem is transformed into an algebraic system of linear equations. This system is solved and thus the coefficients of the assumed approximate solution are determined. Numerical applications are made for various values of N.

scholarly journals Eigenvalues and Eigenvectors for 3×3 Symmetric Matrices: An Analytical Approach

2020 ◽  
pp. 106-118
Author(s):  
Abu Bakar Siddique ◽  
Tariq A. Khraishi
Keyword(s):  
Analytical Solutions ◽  
Analytical Approach ◽  
Numerical Solutions ◽  
Linear Equations ◽  
Symmetric Matrices ◽  
Matrix Equations ◽  
Eigenvalues And Eigenvectors ◽  
Engineering Problems ◽  
The Matrix ◽  
Research Problems

Research problems are often modeled using sets of linear equations and presented as matrix equations. Eigenvalues and eigenvectors of those coupling matrices provide vital information about the dynamics/flow of the problems and so needs to be calculated accurately. Analytical solutions are advantageous over numerical solutions because numerical solutions are approximate in nature, whereas analytical solutions are exact. In many engineering problems, the dimension of the problem matrix is 3 and the matrix is symmetric. In this paper, the theory behind finding eigenvalues and eigenvectors for order 3×3 symmetric matrices is presented. This is followed by the development of analytical solutions for the eigenvalues and eigenvectors, depending on patterns of the sparsity of the matrix. The developed solutions are tested against some examples with numerical solutions.

scholarly journals Improved Heuristics for Short Linear Programs

2019 ◽  
pp. 203-230
Author(s):  
Quan Quan Tan ◽  
Thomas Peyrin
Keyword(s):  
Random Matrices ◽  
Selection Criteria ◽  
Block Cipher ◽  
State Of The Art ◽  
Linear Equations ◽  
Linear Programs ◽  
System Of Linear Equations ◽  
New Methods ◽  
The Matrix

In this article, we propose new heuristics for minimising the amount of XOR gates required to compute a system of linear equations in GF(2). We first revisit the well known Boyar-Peralta strategy and argue that a proper randomisation process during the selection phases can lead to great improvements. We then propose new selection criteria and explain their rationale. Our new methods outperform state-of-the-art algorithms such as Paar or Boyar-Peralta (or open synthesis tools such as Yosys) when tested on random matrices with various densities. They can be applied to matrices of reasonable sizes (up to about 32 × 32). Notably, we provide a new implementation record for the matrix underlying the MixColumns function of the AES block cipher, requiring only 94 XORs.

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