Ž . We classify minimal pairs X, G for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus X G and the quotient surface Y X G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Mori’s extremal ray theory and recent results of Alexeev and also Ambro on the existence of good anti-canonica...

The main goal of this paper is to investigate the periodic character, invariant intervals, oscillation and global stability and other new results of all positive solutions of the equation$$x_{n+1}=frac{alpha+beta x_{n}}{A + Bx_{n}+ Cx_{n-k}},~~ n=0,1,2,ldots,$$where the parameters $alpha$, $beta$, $A$, $B$ and $C$ are positive, and the initial conditions $x_{-k},x_{-k+1},ldots,x_{-1},x_{0}$ are...

The Status of Spontaneous Order in von Hayek’s Theory of Freedom and Its Educational Implications M. Moosivand Y. Ghaa’edi, Ph.D. S. Zarghaami, Ph.D. A. R. Mahmoodniyaa, Ph.D. The history of human thought shows that the concept of freedom and its use in different frameworks has been a major topic of discussion in philosophy. Within the educational system, freedom ...

This paper presents a method for designing a ®xed controller, which is able to control in a stable fashion a family of linear time-invariant n-th order plants with arbitrary relative degree. This plant family is de®ned in terms of a nominal transfer function of rational type and bounded variations of the coef®cients of numerator and denominator polynomials. The design method is based on the alg...

In this paper, some properties of all positive solutions are considered for a higher order rational difference equation, mainly for the existence of eventual prime period two solutions, the existence and asymptotic behavior of nonoscillatory solutions and the global asymptotic stability of its equilibria. Our results show that a positive equilibrium point of this equation is a global attractor ...

* Correspondence: [email protected] Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia Full list of author information is available at the end of the article Abstract In this article we study the global convergence result, boundedness, and periodicity of solutions of the difference equation xn+1 = axn−s + bxn−l + cxn−k dxn−l + ...

We investigate in this paper the global behavior of the following difference equation: xn+1 = (Pk(xn i0 ,xn i1 , . . . ,xn i2k ) + b)/(Qk(xn i0 ,xn i1 , . . . ,xn i2k ) + b), n = 0,1, . . ., under appropriate assumptions, where b [0, ), k 1, i0, i1, . . . , i2k 0,1, . . . with i0 < i1 < < i2k, the initial conditions xi 2k ,xi 2k+1, . . . ,x0 (0, ). We prove that unique equilibrium x = 1 of that...