Note that $F$ is the product over the integers $m$ in $B$.
Hence $V(x)$ is the maximum value of $J_x(v)$ over all controls $v$.
......where $E$ runs over the family $B$.
To show the greater simplicity of our method over Brown's, let us ......
Over the past ten years the isomorphic structure of spaces of weighted holomorphic functions has been largely [= almost completely] determined.
Firms employing over 1000 people accounted for 50% of total employment. [= represented 50%]
The method of proof carries over to domains satisfying ......
The equation $PK=0$ then goes over to $QK=0$.
Here $s$ takes over the role of the time parameter.
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