- u ″ + q u ∈ λ F ( ⋅, u ), u ′ ( 0 ) = 0, u ′ ( 1 ) = 0, where F is a "set-valued representation" of a function with jump discontinuities along the line segment [0, 1] × {0}, and λ ∈ [0, ∞) is a parameter. The proof of our main result relies on an approximation procedure. Mathematics Subject Classification 2000: 34B16; 34B18.
Whittle's solution to the problem of estimating the parameters and the proof of consistency of parameters of the above sine wave used arguments which are not mathematically rigorous.
Fix a > λ 1 and take c as the bifurcation parameter, then by the proof of Theorem 3.2, we can obtain a supercritical bifurcating branch from the point ( λ 1 ( − d ( 1 − e − γ θ a ) ) ; θ a, 0 ).
Finally, from (2.15) we have begin{aligned} y^{(3)}(t)= {lambdaover sqrt{pi}} t^{-3/2} + lambda^{2} y(t), quad0< t< 1, end{aligned} (2.16) so the singular boundary value problems (1.1) holds for any real parameter (lambda>0), thus the proof of the theorem is complete.
In the appendix, we do not implement such three-parameter representation since the proof will become more cumbersome and prolix by employing the new one.
Finally, in the case where this sum is reduced to one element, which is the case for non-characteristic points, we use modulation theory coupled with a nonlinear argument to show the exponential convergence (in the self-similar time variable) of the various parameters and conclude the proof.
The admissible values of the parameter γ used in the proof of Theorem 2 are (gamma>1).
Experiments were performed at the optimized parameters to proof the validity of the analysis.
The selection procedure for any system parameters, such as learning rates and some constant parameters, is represented by the proof of proposed theorems.
By the well known theorem on continuity of the roots of an equation as a function of parameters (see [33], 9.17.4), the proof for Theorem 1 is completed.
For the interpolation formula, which is used in proving the master theorem, Hardy proved it using the domain (0, π) for the parameter A that appeared in the proof.
