If f preserves orientation, then (varphi_{theta}> 0) a.e.
f is a preserving orientation circle homeomorphism on (S^{1}).
If f preserves orientation, then (rho_{r}> 0) a.e.
As f preserves orientation, then (J_{f}=u _{x}v_{y}>0) a.e.
If T preserves orientation and is not the identity, then there are no fixed points.
(i) f is a preserving orientation circle homeomorphism on (S^{1}).
Let (f:Omegato mathbb {C}) be an ACL (absolute continuous on lines) homeomorphism in a domain (Omegasubset mathbb {C}) that preserves orientation.
Let (a, b, c, din S^{1}) be the four points of the circle which preserve orientation; that is, (aprec bprec cprec dprec a) on the circle.
Let f be a preserving orientation circle homeomorphism with infinite number of break points, i.e., the points at which the derivative of f has jumps, and finite number of singular points, i.e., the points (x_{i}), (i=1,2,ldots,n), such that (f'(x_{i})=infty), (i=1,2,ldots,n).
Phase of the face images was scrambled in the Fourier domain via random permutation, a procedure known to preserve orientation content (Dakin et al. 2002).
For density determination the left anterior dorsal cerebrum and right posterior dorsal cerebrum were placed flat on aluminum foil to preserve orientation of the white matter and immediately submerged in liquid nitrogen.
