Matrix equations (AX = B,XC = D) is a class of important matrix equations. The problem of the solution of the matrix equations (AX = B,XC = D) arise in engineering and in some special matrix inverse problems [1-3]. Many authors have been devoted to the study of this problem, and a series of useful results have been obtained. For example, Mitra [4] gave the common solution of minimum possible ra...

and Applied Analysis 3 Moreover, using the condition (iii), we obtain z (t) ≤ h (t) h −1 (τ 0 ) ( 󵄩󵄩󵄩󵄩y0 − x0 󵄩󵄩󵄩󵄩 + 󵄨󵄨󵄨󵄨η 󵄨󵄨󵄨󵄨) . (12) Then from (12), we get 󵄩󵄩󵄩󵄩x (t, τ0, y0) − x (t − η, t0, x0) 󵄩󵄩󵄩󵄩 ≤ [ 󵄩󵄩󵄩󵄩y0 − x0 󵄩󵄩󵄩󵄩 + 󵄨󵄨󵄨󵄨τ0 − t0 󵄨󵄨󵄨󵄨] h (t) h −1 (τ 0 ) . (13) So by Definition 1 with c = 1, the solution x(t, τ 0 , y 0 ) of (2) is ITDhS with respect to the solution x(t − η, t 0 , x 0 ). T...

Chiral Schiff bases (1H to 4H) and a series of their Group 4 metal alkoxide complexes [(R-1)2Ti(O(i)Pr)2, (R-2)2Ti(O(i)Pr)2, (R-1)(2)Zr(O(i)Pr)2, (R-2)2Zr(O(i)Pr)2, (R-3)2Zr(O(i)Pr)2, (R-4)2Zr(O(i)Pr)2, (S-1)2Zr(O(i)Pr)2 and (rac-1)2Zr(O(i)Pr)2] have been prepared and characterised by 1H, and 13C NMR spectroscopy. In solution, both Lambda and Delta isomers were observed, suggesting a low degree...

The problem of indexing multidimensional objects is considered. First, a classification of existing methods is given along with a discussion of the major issues involved in multidimensional data indexing. Second, a variation to Guttman’s R-trees (R+-trees) that avoids overlapping rectangles in intermediate nodes of the tree is introduced. Algorithms for searching, updating, initial packing and ...

This paper considers a biobjective transportation problem with various fuzzy objective functions coefficients. Fuzzy coefficients can be of different types such as triangular, trapezoidal, (semi) $L-R$, or flat (semi) $L-R$ fuzzy numbers. First, we convert the problem to a parametric interval biobjective transportation problem using $gamma$-cuts of fuzzy coefficients. Then, we consider a fix $g...

We study the singularly perturbed state-dependent delay-differential equation εẋ(t) = −x(t)− kx(t− r), r = r(x(t)) = 1 + x(t), (∗) which is representative of a broader class of such equations of the form εẋ(t) = g(x(t), x(t− r)), r = r(x(t)). It is known that for every sufficiently small ε > 0, equation (∗) possesses at least one so-called slowly oscillating periodic solution, and moreover, the...

In this paper, an analytical method is developed to obtain the solution for the one dimensional transient thermal and mechanical stresses in a hollow cylinder made of functionally graded material (FGM) and piezoelectric layers. The FGM properties are assumed to depend on the variable r and they are expressed as power functions of r but the Poisson’s ratio is assumed to be constant. Transient te...