![$f:[0,1]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/9/d/0/9d00bb0f11e69022dded4a8baba1b496f66d272e.png)
be a continuous function. Prove that 
is a continuous function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
such that 
,
such that:![\[
F(c) = \frac{1}{c} \int_0^c F(x) \,dx
\]](http://latex.artofproblemsolving.com/8/f/e/8fea4d84bc98b6f4953e19976cfa554945a37ea6.png)
,
is the greatest prime divisor of 
.
![$f(x)\in C[0,3]$](http://latex.artofproblemsolving.com/8/e/2/8e2d8e1cf446975c3e3aa894610ac8b6371737b4.png)
satisfy 
for all 
and
Show that
(with multiplicity 3),
(with multiplicity 2),
(with multiplicity 1),
(with multiplicity 3), 
(with multiplicity 2), and 
(with multiplicity 1)?
Y by
Let 
be an 
square matrix with eigenvalues 
(each 
having algebraic multiplicity 
, so that 
). Let 
be a polynomial. It is known that if 
is an eigenvalue of then 
is an eigenvalue of 
.
The question is: Are all the eigenvalues of of the form 
(counting multiplicities)?
Furthermore, is there any relation in the nuclear space corresponding to each of those eigenvalues? (equation
vs. equation 
)
be an 
square matrix with eigenvalues 
(each 
having algebraic multiplicity 
, so that 
). Let 
be a polynomial. It is known that if 
is an eigenvalue of then 
is an eigenvalue of 
.The question is: Are all the eigenvalues of
of the form 
(counting multiplicities)?Furthermore, is there any relation in the nuclear space corresponding to each of those eigenvalues? (equation
vs. equation 
)This post has been edited 2 times. Last edited by Mathloops, Mar 26, 2025, 4:07 PM
is a Jordan block then one can explicitly calculate 
and see what happens on the diagonal.
can be brought into upper triangular form hence 
for some invertible 
and 
upper triangular. The eigenvalues of 
(with alg mpl) are the eigenvalues of 
.
then 
is upper diagonal and has diagonal elts 
.
is upper diagonal and has diagonal elements 
,
since 
is upper diagonal with diagonal elts 
.
