Equation (1.1) also acts as a possible starting point for the derivation of Boltzmann-like equations for wave systems. If the deterministic evolution prescribed by (1.1) is coupled with random initial data, then the time-evolved field ψt becomes a random field. It is conjectured that the co...
It can be derived based on a trajectory point of view [59], but similar results can also be derived by a pure PDF point of view. This can be done by writing a simple gain-loss equation of the probabilities of the separate states and generalizing the result by a class of continuous ...
Curve of a Baseball Have you ever seen a baseball pitcher deliver a curveball? When the baseball takes a curved trajectory as it passes the plate, in most cases, the batter would not be able to judge the path of the ball and miss the ball. One of the crucial aspects of curveball can...
If a rotational motion (and the axis defining it) are viewed through a mirror whose reflection plane is parallel to the rotational axis, the handedness of the trajectory will be reversed, but the axial direction (the axial vector) will remain the same. Because axial vectors do not satisfy ...
the construction of a suitable extension of the unit normal vector field of the interface of the strong solution in the vicinity of its space-time trajectory is required. furthermore, the length of this vector field is required to decrease quadratically fast as one moves away from the interface...
Claude-Louis Navier put forward the component of viscosity (friction) for a more realistic and difficult problem of viscous fluids. During the entire middle period of the 19th century, George Gabriel Stokes refined this work even though entire solutions were found only in the case of basic two-...
The main problem in the case α∈ (0, 1 2 ) is derivation of control of L5([0, T ]; L10(Ω)) norm to solutions of the linear problem (1.2) ∂t2u + γ(−∆x)α∂tu − ∆xu = h(t), u|t=0 = u0, ∂tu|t=0 = u1, u|∂Ω = 0, where (u0, u1) ...
Fig. 1. The red dashed circle u2 + v2 = 1 is approached by solutions of αe from all points except the origin, e.g., the blue trajectory from ( 1 , 1 ) spirals towards the circle. The red circle, 88 green region u2 ≤ v 2 + 9 2 , and origin are invariants of the system...
46.1 Derivation of the Pendulum Equation d2θdt2+gLSin[θ]=0 The force acting on the pendulum can be broken into two components, one in the direction of the rod and the other in the direction of the pendulum's motion. If the rod does not stretch, the component directed along the rod...
Consequently, the problem of planning an exciting trajectory is greatly simplified. The main drawback of this method is the possible accumulation of errors from one step to the next. To overcome this problem, a weighted least-squares solution can be used [Gautier 97]. View chapterExplore book ...