Can anyone think of a nice application of the fact deformation retracts are closed under pushouts?
The formal statement is that if $$\begin{matrix}A &\xrightarrow{f}& B \\ \ \ \downarrow i& & \ \ \downarrow j \\ C &\xrightarrow{g}& D \end{matrix}$$ is a pushout square with $A$ a deformation retract of $C$, then $B$ is a deformation retract of $D$.

can you give me an example of an element P in $ \mathbb{R} \left[ X \right]$ such that for all $ U \in \mathbb{R} \left[ X \right] $ $ U^{"}-XU^{'}\neq P $
help

if (A-I)^2 = 0 prove that A is invertible
(A-I)(A-I)=0 => A-I = 0 => A=I => det(A)=1

@DanielFischer I want to solve the recurrence relation $T(n)=7T\left( \frac{n}{3} \right)+n^2$.
I have tried the following:
$a=7 \geq 1, b=3>1, f(n)=n^2$
$n^{\log_b a}=n^{\log_3 7} \approx n^{1.77}$
So $f(n)=n^2 = \Omega(n^{\log_b a+ \epsilon})$, for $\epsilon=0.23>0$
$a f \left( \frac{n}{b}\right)=7 \left( \frac{n}{3}\right)^2=\frac{7}{9}n^2\leq cf(n)$ for any $c \in [\frac{7}{9},1)$.
So from the case 3 of Matser theorem, we have that $f(n)=\Theta(f(n))=\Theta(n^2)$.

A ok.. @DanielFischer So you mean that I could write it as follows, right?
$n^{\log_b a}=n^{\log_3 7}<n^2 (\log_3 7 \approx 1.77)$

So $f(n)=n^2 = \Omega(n^{\log_b a+ \epsilon})$, for $\epsilon=0.23>0$

$a f \left( \frac{n}{b}\right)=7 \left( \frac{n}{3}\right)^2=\frac{7}{9}n^2\leq cf(n)$ for any $c \in [\frac{7}{9},1)$.
So from the case 3 of Matser theorem, we have that $f(n)=\Theta(f(n))=\Theta(n^2)$.