  Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent centered stationary
Gaussian processes with unit variance and almost surely continuous sample
paths. For given positive constants $u,T$, define the set of conjunctions
$C_{[0,T],u}:=\{t\in [0,T]: \min_{1 \le i \le n} X_i(t) \ge u\}.$ Motivated by
some applications in brain mapping and digital communication systems, we obtain
exact asymptotic expansion of $ P(C_{[0,T],u} \not=\varphi)$ as $u\to\infty$.
Moreover, we establish the Berman sojourn limit theorem for the random process
$\{\min_{1 \le i \le n} X_i(t), t\ge0\}$ and derive the tail asymptotics of the
supremum of each order statistics process.
