1. Given the function $F$ applied to a four-digit positive integer $\overline{a b c d}$ yields the integer $F(\overline{a b c d})=a^{4}+b^{3}+c^{2}+d^{1}$. For example, $F(2019)=2^{4}+0^{3}+1^{2}+9^{1}=26$.
Then $F(2019)-F(2018)+$
$$
F(2017)-\cdots-F(2000)=
$$
$\qquad$
$$
-1.10
$$

When $0 \leqslant d \leqslant 8$,
$$
F(\overline{a b c(d+1)})-F(\overline{a b c d})=d+1-d=1 \text {. }
$$

Therefore, the original expression is
$$
\begin{aligned}
= & (F(2019)-F(2018))+ \\
& (F(2017)-F(2016))+\cdots+ \\
& (F(2001)-F(2000)) \\
= & \underbrace{1+1+\cdots+1}_{10 \uparrow}=10 .
\end{aligned}
$$