Gas-phase DFT single point energy calculations were performed with the Gaussian 03 package(42) using Becke's three-parameter hybrid functional (B3LYP)(43) and fixed saddle-shape conformation of the AuTPP as obtained from our VASP calculations.
For standard definitions of attracting fixed point, saddle point, stable manifold, and related notions, see [8, 24].
This new type of implant was designed to wrap around the upper part of the alveolar bone like a saddle, and at the center of saddle, the implant served the purpose of fixing the saddle structure onto the alveolar bone and connecting the implant with prosthodontics.
The implant used with the SI was the same as the CI in shape, but its length was only 5.6 mm, and its diameter was 4.0 mm for the purpose of fixing the saddle structure to the underlying alveolar bone.
This is used to obtain the necessary starting data for an established algorithm for computing one-dimensional (1D) unstable manifolds of an associated saddle fixed point of a suitable Poincaré map.
Above this value of g Ca, the branch of periodic orbits terminates in a homoclinic bifurcation HC involving the saddle fixed point.
Since the well-known problems were variational problems, complementary problems, fixed point problems, saddle point problems and other problems proposed from the equilibrium problem, it has become the most attractive topic for many mathematicians [6 8].
It is well known that equilibrium problems include many important problems in nonlinear analysis and optimization such as the Nash equilibrium problem, variational inequalities, complementarity problems, vector optimization problems, fixed point problems, saddle point problems, and game theory.
These problems are models whose formulation includes optimization, variational inequalities, (vector) optimization problems, fixed point problems, saddle point problems, Nash equilibria and complementarity problems as particular cases (see [1, 5, 9]).
For large and negative I, the fixed points with (rho=1) are a source and a sink, and there exists another unphysical saddle fixed point with (rho>1) (Fig. 13(a)).
Equilibrium problems (in short, EP) by Blum and Oettli [1] provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems and noncooperative games as particular cases.
