In this paper, we investigate the dynamical behaviors of the rational difference equation $x_n=\left(a{x}_{n-1}\right)/\left(b+c{x}_n{x}_{n-1}\right)$ with arbitrary initial conditions, where a, b, and c are real numbers. A general solution is obtained. The asymptotic stability of the equilibrium points is investigated, using a nonlinear stability criterion combined with basin of attraction analysis and simulation to determine the stability regions of the equilibrium points. The existence of the periodic solutions is discussed. We investigate the codim-1 bifurcations of the equation. We show that the equation exhibits a Neimark–Sacker bifurcation. For this bifurcation, the topological normal form is computed. To confirm our theoretical results, we performed a numerical simulation as well as numerical bifurcation analysis by using the Matlab package MatContM.

In this paper, we consider the explicit solution of the following system of nonlinear rational difference equations: $x_{n+1}=x_{n-1}/\left(x_{n-1}+r\right)$ , $y_{n+1}=x_{n-1}{y}_n/\left(x_{n-1}{y}_n+r\right)$ , with initial conditions $x_{-1}$ , $x_0$ and $y_0$ , which are arbitrary positive real numbers. By doing this, we encounter the hypergeometric function. We also investigate global dynamics of this system. The global dynamics of this system consists of two kind of bifurcations.

To find a solution of unconstrained optimization problems, we normally use a conjugate gradient (CG) method since it does not cost memory or storage of second derivative like Newton’s method or Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. Recently, a new modification of Polak and Ribiere method was proposed with new restart condition to give a so-call AZPRP method. In this paper, we propose a new modification of AZPRP CG method to solve large-scale unconstrained optimization problems based on a modification of restart condition. The new parameter satisfies the descent property and the global convergence analysis with the strong Wolfe-Powell line search. The numerical results prove that the new CG method is strongly aggressive compared with CG_Descent method. The comparisons are made under a set of more than 140 standard functions from the CUTEst library. The comparison includes number of iterations and CPU time.

Given a locally convex algebra E, we shall construct a vector space A(E) on which multiplication is defined on it. We will show the multiplication so defined on A(E) is associative and that there is a natural embedding of the sets E and its dual space E^* in to the vector space A(E). This embedding allows to carryover the operations(linear and bilinear) defined on E and E* to the algebra A(E). Finally we discussed and derive some results about A(E) when E=S(R), the space of functions of rapid decay.