In particular, one can describe explicitly the hyperbolic geometry of such surfaces and one can compute scalar field trajectories on ([summation], G) by determining trajectories of an appropriate lift of the model to the Poincare

half plane H and projecting them to [D.sup.*] or to A(R) through the uniformization map.

Mochizuki [10] introduced the Nevanlinna class [N.sub.0](D) and the Smirnov class [N.sub.*](D) on the upper

half plane D ??= {z [member of] C | Im z >0}: the class [N.sub.0](D) is the set of all holomorphic functions f on D satisfying

Suppose that the function q [member of] [A.sup.*] is a univalent mapping of U into the right

half plane with q(0) = 1 and

Xu, "Solution for a circular cavity in an elastic

half plane under gravity and arbitrary lateral stress," International Journal of Rock Mechanics and Mining Sciences, vol.

[16] studied the scattering of plane waves by a cylindrical cavity with lining in a poroelastic

half plane using the complex variable function method.

Falope investigate the contact problem of an Euler-Bernoulli nanobeam of finite length bonded to a homogeneous elastic

half plane. The analysis is performed under plane strain condition.

For example, Huang and Yu [11] studied an elastic

half plane under surface loading with consideration of surface energy effects.

Because of hidden zeros, we at the first by use of MATLAB software, gain hidden zeros after that, gain output zero direction and transmit zeros to second output, after that will replacement Right

Half Plane to Left

Half Plane, and monitor behavior of system.

with Re [zeta] > 0 and Im [zeta] > 0 is meromorphically continued from the upper

half plane of the complex plane to the lower

half plane {[zeta] [member of] C : Re [zeta] > 0, Im [zeta] < 0} across the positive real axis where the continuous spectrum of [H.sub.d] is located.

Stability of a time delay system can be determined by its eigenvalues, which should be located in the open left

half plane [2], [12].

In each iterative process, the according region in Figure 1 is used if all eigenvalues defined as [LAMBDA] = {[[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.n]] lie in the left

half plane. In this case, the corresponding region [[OMEGA].sub.[LAMBDA]] = [OMEGA]([[alpha].sub.[LAMBDA]], [[xi].sub.[LAMBDA]], [[omega].sub.[LAMBDA]]) determined by [LAMBDA] = {[[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.n]] will be compared with the respected goal region.

In Section 4, we state two conjectures about this generating function, and provide evidence for them by demonstrating that they hold if we only impose on walks a

half plane restriction, or no restriction at all.