scholarly journals The Hermitian -Conjugate Generalized Procrustes Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Xia Chang ◽  
Xue-Feng Duan ◽  
Qing-Wen Wang
Keyword(s):  
Sufficient Conditions ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Complex Matrix ◽  
Approximation Solution ◽  
Hermitian Conjugate ◽  
Procrustes Problem ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Conjugate Solution

We consider the Hermitian -conjugate generalized Procrustes problem to find Hermitian -conjugate matrix such that is minimum, where , , , and (, ) are given complex matrices, and and are positive integers. The expression of the solution to Hermitian -conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian -conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian -conjugate solution to the linear system of complex matrix equations , , ( and are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.

scholarly journals On the HermitianR-Conjugate Solution of a System of Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Chang-Zhou Dong ◽  
Qing-Wen Wang ◽  
Yu-Ping Zhang
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Approximation Problem ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Solvability Conditions ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient ◽  
Conjugate Solution

LetRbe annbynnontrivial real symmetric involution matrix, that is,R=R−1=RT≠In. Ann×ncomplex matrixAis termedR-conjugate ifA¯=RAR, whereA¯denotes the conjugate ofA. We give necessary and sufficient conditions for the existence of the HermitianR-conjugate solution to the system of complex matrix equationsAX=C and XB=Dand present an expression of the HermitianR-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares HermitianR-conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.

scholarly journals (Anti-)Hermitian Generalized (Anti-)Hamiltonian Solution to a System of Matrix Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Juan Yu ◽  
Qing-Wen Wang ◽  
Chang-Zhou Dong
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Optimal Approximation ◽  
Matrix Equations ◽  
Approximation Solution ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient ◽  
The Given

We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equationsAX=B,XC=Dare derived, respectively. Secondly, the optimal approximation solutionmin⁡X∈K⁡∥X^-X∥is obtained, whereKis the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system andX^is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.

scholarly journals The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jinrong Shen ◽  
Wei Liu ◽  
Baiyu Wang ◽  
Xiangyang Peng
Keyword(s):  
Real Number ◽  
Explicit Expression ◽  
Number Field ◽  
Sufficient Conditions ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
General Representation ◽  
Solvable Case ◽  
Necessary And Sufficient ◽  
Real Number Field

In this paper, a kind of constrained inverse eigenproblem and optimal approximation problem for centrosymmetric matrices are considered. Necessary and sufficient conditions of the solvability for the constrained inverse eigenproblem of centrosymmetric matrices in real number field are derived. A general representation of the solution is presented for a solvable case. The explicit expression of the optimal approximation problem is provided. Finally, a numerical example is given to illustrate the effectiveness of the method.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

On Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices

2013 ◽  
Vol 860-863 ◽  
pp. 2727-2731
Author(s):  
Kai Fu Liang ◽  
Ming Jun Li ◽  
Ze Lin Zhu
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Design Automation ◽  
Sufficient Conditions ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Linear Quadratic ◽  
Real Representation ◽  
The Matrix ◽  
Necessary And Sufficient

Hamiltonian matrices have many applications to design automation and autocontrol, in particular in the linear-quadratic autocontrol problem. This paper studies the inverse problems of generalized Hamiltonian matrices for matrix equations. By real representation of complex matrix, we give the necessary and sufficient conditions for the existence of a Hermitian generalized Hamiltonian solutions to the matrix equations, and then derive the representation of the general solutions.

scholarly journals Oscillatory and Asymptotic Behavior of a First-Order Neutral Equation of Discrete Type with Variable Several Delay under Δ Sign

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Radhanath Rath ◽  
Chittaranjan Behera
Keyword(s):  
Difference Operator ◽  
Sufficient Conditions ◽  
Neutral Equation ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
First Order ◽  
Forward Difference ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Delay Difference

We obtain necessary and sufficient conditions so that every solution of neutral delay difference equation Δyn-∑j=1kpnjyn-mj+qnG(yσ(n))=fn oscillates or tends to zero as n→∞, where {qn} and {fn} are real sequences and G∈C(R,R), xG(x)>0, and m1,m2,…,mk are positive integers. Here Δ is the forward difference operator given by Δxn=xn+1-xn, and {σn} is an increasing unbounded sequences with σn≤n. This paper complements, improves, and generalizes some past and recent results.

On the Prime Ideals in a Commutative Ring

2000 ◽  
Vol 43 (3) ◽  
pp. 312-319 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Prime Ideals ◽  
Preceding Result ◽  
Finite Commutative Ring ◽  
The Axiom Of Choice ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

scholarly journals A class of constrained inverse eigenvalue problem and associated approximation problem for symmetrizable matrices

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1903-1909
Author(s):  
Xiangyang Peng ◽  
Wei Liu ◽  
Jinrong Shen
Keyword(s):  
Eigenvalue Problem ◽  
Symmetric Matrix ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Optimal Approximation ◽  
Original Matrix ◽  
Approximation Solution ◽  
Solvability Conditions ◽  
Inverse Eigenvalue ◽  
Symmetrizable Matrices

The real symmetric matrix is widely applied in various fields, transforming non-symmetric matrix to symmetric matrix becomes very important for solving the problems associated with the original matrix. In this paper, we consider the constrained inverse eigenvalue problem for symmetrizable matrices, and obtain the solvability conditions and the general expression of the solutions. Moreover, we consider the corresponding optimal approximation problem, obtain the explicit expressions of the optimal approximation solution and the minimum norm solution, and give the algorithm and corresponding computational example.

On Sums of Progressions of Positive Integers

1970 ◽  
Vol 54 (388) ◽  
pp. 113-115
Author(s):  
R. L. Goodstein
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Common ◽  
Necessary And Sufficient ◽  
Positive Integers

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.

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