Sensitivity analysis in interval-valued trapezoidal fuzzy number linear programming problems

The aim of this article is to introduce a formulation of fuzzy linear programming problems involving the level (h^L,h^U)$(h^L\text{,}{h}^U)$-interval-valued trapezoidal fuzzy numbers as parameters. Indeed, such a formulation is the general form of trapezoidal fuzzy number linear programming problems. Then, it is demonstrated that study of the sensitivity analysis for the level (h^L,h^U)$(h^L\text{,}{h}^U)$-interval-valued trapezoidal fuzzy number linear programming problems gives rise to the same expected results as those obtained for trapezoidal fuzzy number linear programming problems.     

The Drazin inverse of a singular,unreduced tridiagonal matrix

For a tridiagonal, singular matrix A   we present a method for the computation of the polynomial p(λ)$p(\lambda )$ such that A^D=p(A)$A^D=p(A)$ holds, where A^D$A^D$ is the Drazin inverse of A. The approach is based on the recursion of characteristic polynomials of leading principal submatrices of A.     

Limit properties of monotone matrix functions

The basic objects in this paper are monotonically nondecreasing $n\times n$ matrix functions $D(·)$ defined on some open interval $?=(a\text{,}b)$ of $\mathbb{R}$ and their limit values $D(a)$ and $D(b)$ at the endpoints a and b which are, in general, selfadjoint relations in ${\mathbb{C}}^n$. Certain space decompositions induced by the matrix function $D(·)$ are made explicit by means of the limit values $D(a)$ and $D(b)$. They are a consequence of operator inequalities involving these limit values and the notion of strictness (or definiteness) of monotonically nondecreasing matrix functions. This treatment provides a geometric approach to the square-integrability of solutions of definite canonical systems of differential equations.     

Simulation and evaluation of dual fully fuzzy linear systems by fuzzy neural network

In this paper, a novel hybrid method based on fuzzy neural network for approximate solution of fuzzy linear systems of the form Ax = Bx + d, where A and B are two square matrices of fuzzy coefficients, x and d are two fuzzy number vectors, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate solution, a simple and fast algorithm from the cost function of the fuzzy neural network is proposed. Finally, we illustrate our approach by some numerical examples.     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.     

Improved Newton’s method with exact line searches to solve quadratic matrix equation

In this paper, we study the matrix equation AX²+BX+C=0$A{X}^2+BX+C=0$, where A,B$A,B$ and C$C$ are square matrices. We give two improved algorithms which are better than Newton’s method with exact line searches to calculate the solution. Some numerical examples are reported to illustrate our algorithms.     

Block tridiagonal reduction of perturbed normal and rank structured matrices

It is well known that if a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ solves the matrix equation f(A,A^H)=0$f(A,A^H)=0$, where f(x,y)$f(x,y)$ is a linear bivariate polynomial, then A is normal; A   and A^H$A^H$ can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of A is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.     

Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of a graph $G$, respectively. The matrix $D(G)+A(G)$ is called the signless Laplacian matrix of $G$. The spectrum of the matrix $D(G)+A(G)$ is called the Q-spectrum of $G$. A graph is said to be determined by its Q-spectrum if there is no other non-isomorphic graph with the same Q-spectrum. In this paper, we prove that all starlike trees whose maximum degree exceed $4$ are determined by their Q-spectra.     

Reducibility of steady-state bifurcations in coupled cell systems

A general theory for coupled cell systems was formulated recently by I. Stewart, M. Golubitsky and their collaborators. In their theory, a coupled cell system is a network of interacting dynamical systems whose coupling architecture is expressed by a directed graph called a coupled cell network. An equivalence relation on cells in a regular network (a coupled cell network with identical nodes and identical edges) determines a new network called quotient network by identifying cells in the same equivalence class and determines a quotient system as well. In this paper we develop an idea of reducibility of bifurcations in coupled cell systems associated with regular networks. A bifurcation of equilibria from subspace where states of all cells are equal is called a synchrony-breaking bifurcation. We say that a synchrony-breaking steady-state bifurcation is reducible in a coupled cell system if any bifurcation branch for the system is lifted from those for some quotient system. First, we give the complete classification of codimension-one synchrony-breaking steady-state bifurcations in 1-input regular networks (where each cell receives only one edge). Second, we show that under a mild condition on the multiplicity of critical eigenvalues, codimension-one synchrony-breaking steady-state bifurcations in generic coupled cell systems associated with an n  -cell coupled cell network with _Dn$D_n$ symmetry, a regular network, is reducible for n>2$n>2$.