初始渐近(Initial Asymptotic):假设 X 足够平滑,使得 \lim_{t\to 0} \ddot X(t) 存在。中值定理保证,存在 \xi\in(0, t) ,满足 \dot X(t)/t = (\dot X(t) - \dot X(0))/t = \ddot X(\xi) 。因此,从 ODE 我们推导出 \ddot X(t) + 3\ddot{X}(\xi) + \nabla f(X(t))...
Initial angle of 180°, unstable equilibrium Physically, we can see that these two equilibrium positions are quite distinct from each other. While the pendulum can indeed rest at the (0, 0) equilibrium point, it can hardly maintain rest at the (π, 0) equilibrium point because an infinites...
However, in some cases, the initial temperature distribution is nonuniform due to the residual heat that is retained after previous technological process (i.e., self-tempering and induction hardening with interrupted quenching). For most induction heating problems, thermal boundary conditions represent ...
The equation for displacement is then rather simple. $$\Delta{x} = x_f - x_i $$ Where {eq}x_f {/eq} = final position {eq}x_i {/eq} = initial position Say that Fran walks 4 meters to the right and then walks 7 meters to the left. We can visualize this with vectors drawn...
We prove maximal speed estimates for nonlinear quantum propagation in the context of the Hartree equation. More precisely, under some regularity and integrability assumptions on the pair (convolution) potential, we construct a set of energy and space localized initial conditions such that, up to ...
When the ray intersects a leaf node, set initial guess for (u0, v0) using the parametric ranges stored with the bounding box. A good choice might be to use the midpoint of the range: u0 = (umin + umax)/2, and similarly for v0. c. Repeatedy apply the Newton iteration step until...
This is a first-order ode dealing with t, n, dN/dt. We want to solve for N. First, to define a function returning dy/dt even though this ODE does not explicitly depend on t. Then, set the initial condition, arrange an array of time and call the odeint function ...
For local optimization, most methods in NLopt, like :LN_BOBYQA, solve the problem very fast but require a good initial condition. The different backends options available for Bayesian method offer some tradeoffs between time, accuracy and control. It is observed that sufficiently high accuracy can...
A great number of studies have also been published on the improvement of the initial condition. 5.6.4 Error estimates The errors made on the computation of the sound field are first caused by the use of the parabolic approximation. In [14], F. Tappert gives for a simple case a ...
The choice of global versus local optimization make a huge difference in the timings. BBO tends to find the correct solution for a global optimization setup. For local optimization, most methods in NLopt, like :LN_BOBYQA, solve the problem very fast but require a good initial condition. ...