While Rafa Benítez might have made an initial balls-up of his half-time reorganisation – at one point, due to a chalkboard malfunction, he was going to field 12 men, then at another only 10 – he was wise enough to feature the substitute Dietmar Hamann in both of those line-ups.
The initial ball forward for the first had come from Jan Vertonghen and when Rose jinked inside Ahmed Elmohamady – all too easily – and rolled back to Eriksen, he crashed his shot high past Marshall.
With Begovic out of his goal having come for the initial ball, Dempsey turned the ball into the empty net from distance.
The latter is iteratively found by deflating an initial ball until it cannot shrink anymore without losing points in (see [16] for more details).
In the rest of the proof all the balls will be centred at the point x, starting from an initial ball B ( x, R ) ⊂ Ω with R ≤ 1, which is not restrictive of course.
To investigate the influence of friction on the dynamic behavior of the system, different friction coefficients and initial ball positions were considered and the equations were solved numerically with a commercial software package, Matlab.
Several Dortmund players looked to be offside as the initial ball was played into the Malaga area and, seconds later, Santana was also in an offside position when he scored the winner, touching home a goal-bound effort from substitute Julian Schieber.
The real source of the goal came from a poor decision made by David De Gea, as he left his line without any chance of saving the initial ball crossed into his six-yard box.
Initial ball position was also varied, but not analyzed statistically.
Initial ball speed was randomized to make descent duration unpredictable from trial to trial.
For each trajectory, we optimized the initial ball velocity vector to ensure that the ball passed at the appropriate time, height and lateral position.
