  Let $u_t = u_{xx} - q(x) u, 0 \leq x \leq 1$, $t>0$, $u(0, t) = 0, u(1, t) =
a(t), u(x,0) = 0$, where $a(t)$ is a given function vanishing for $t>T$, $a(t)
\not\equiv 0$, $\int^T_0 a(t) dt < \infty$. Suppose one measures the flux $u_x
(0,t) := b_0 (t)$ for all $t>0$. Does this information determine $q(x)$
uniquely? Do the measurements of the flux $u_x (1,t) := b(t)$ give more
information about $q(x)$ than $b_0 (t)$ does?
  The above questions are answered in this paper.
