By Kostomarov

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This is very fortunate, as most of the important actions in football such as long throw-ins, corners, and free kicks take place within the low-drag regime after this switch occurs. 2 Variations in the drag coefficient Cd with different ball speeds. The typical speed ranges for long throws and free kicks are shown in the shaded regions. dimples, grooves, and knuckleballs | 23 As already noted, surface roughness is responsible for this important transition, and it was not realised for many years that the stitched seams between the ball’s panels produced just the right degree of aerodynamic roughness to do the trick.

Imagine all the curves C that enclose a given area A, and then choose the one that minimises the perimeter. The solution remains a circle but now we have a minimisation problem. When mathematicians look to maximise or minimise a quantity, they are solving an extremal problem. The solution to Dido’s problem is intuitively a circle. However, it was a very long time after the foundation of Carthage before mathematicians had developed the right tools to prove the solution is a circle. 1 Queen Dido’s problem: find the maximum area A for a given perimeter length C.

Children (circles) are signiﬁcantly more strongly connected to other children than adults (squares) are to other adults, and children and adults are only loosely connected to each other. We can improve the realism of the standard SIR model by splitting the population into children and adults and creating two linked submodels: one for the progression of the infection through children and one for adults. Each of these submodels is dependent on the mixing within and between the subpopulations. We allow the contact rates to fall during the summer holidays (indicated by the shaded region in Fig.

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