The problem is here.
Consider the polynomial $P(z)=a_0+a_1z+...+a_{n-1}z^{n-1}+z^n $. Show that, for all sufficiently large values of $r$, there must exist points $z_0, z_1$ on the circle $|z|=r$ such that $|e^{P(z_0)}|=e^{(1/2)r^n}, |e^{P(z_1)}|=e^{-(1/2)r^n} $.
I tried to show that there e...
What is more computationally efficient for computing the inverse of a matrix?
1) Using the gaussian reduction algorithm?
2) Using the characteristic polynomial?
Both have a mechanically labor intensive process. So I'm curious what has been found to be the case when we are dealing with larger values. I'm guessing characteristic because it doesn't involve as much "division" through the process.
if you have integer or rational coefficients and want to do exact arithmetic that is some overhead but a computer doesn't really care if it's multiplying or dividing in that case.
thorgott, i'm guessing cayley hamilton theorem, coefficient of I will be nonzero, solve for A [polynomial in A] = I
@barista my thinking is a little clouded this morning (well, more that just this morning :-)), but i think we have $\lim_{r \to \infty}{ |e^{f(r)}| \over e^{{1 \over 2}r^n}} \to \infty$ and $\lim_{r \to \infty}{ |e^{f(\alpha r)}| \over e^{-{1 \over 2}r^n}} \to 0$ (with $\alpha = e^{i { \pi \over n}}$), which you can use along with the IVT to get the desired result.
We have an integer matrix, we can do stuff related to its rank by reducing it $\bmod p$, the rank of the original matrix is larger than the rank of the reduced matrix.
If we reduce it $\bmod 2$ we get the symmetric matrix in which $1$ is outside the diagonal and $0$ in it.
Finding the rank of thi...
What I want to use is the following
we have an integer matrix and we know that if we reduce it mod_p then the rank is k (in the field mod p)
we want to say the rank of the original matrix is at least k
So, in terms of the logic of the proof that, for any $A, B \subset \mathbb{R}$, we have $\sup (A+B) = \sup A + \sup B$. If we produce the following steps in the proof:
1. Let $a,b$ respectively be elements of any subsets $A,B$ of $\mathbb{R}$. 2. Then, $a+b \leq \sup(A+B)$. 3. But, $a+b \leq \sup A + \sup B$.
Is the legitimate next step, in terms of the logic of the proof, $\sup(A+B) \leq \sup A + \sup B$ or $\sup A + \sup B \leq \sup(A+B)$?
in general, from $x \leq y$ and $x \leq z$ it's hard to deduce an order relation between $y$ and $z$, so you have to dig a little deeper and have a goal in mind.
the fact that 3 holds for arbitrary a in A and b in B shows you that $\sup A + \sup B$ is an upper bound of $A+B$. that tells you something about the relation between $\sup A + \sup B$ and $\sup(A+B)$, doesn't it?
Right, it tells us $\sup(A+B) \leq \sup A + \sup B$, since $\sup(A+B)$ is the least upper bound of $A+B$ and up to that point $\sup A + \sup B$ is only some upper bound
@shintuku I think you are making more work than necessary. Since $a+b \le \sup A + \sup B$ you get $\sup (A+B) \le \sup A + \sup B$. since $a+b \le \sup (A+B)$, you get $\sup A + b \le \sup (A+B)$ and then $\sup A + \sup B \le \sup (A+B)$.
I am using a result in a problem but I am not sure how to write or support it correctly.
We have an integer matrix $A$ over the reals.
We can define a matrix $A'$ over $\mathbb F_p$ by just taking each entry $\bmod p$.
I want to say that the rank of $A'$ is less than or equal to the rank of $A$.
...
I have to find all the permutations from $S_8$ that commute with $(1234)(5678)$. Counting how many there are is easy using group actions and so. But I can't find a way to briefly describe the group of $32$ elements that commute with this guy.
conmuting by $\sigma$ will give you permutation $(\sigma(1),\sigma(2),\sigma(3),\sigma(4))(\sigma(5),\sigma(6),\sigma(7),\sigma(8))$ So if you decide what $\sigma(1)$ is all of the $\sigma(i)$ with $i\in \{2,3,4\}$ are determined, and then you have $4$ options for $\sigma(5)$ and the rest are determined again.
thinking more constructively, the elements (1234)^a (5678)^b for 0 <= a, b < 3 will commute with that and be distinct elements. that gives you 16 elements. i think the involution sending 1 <-> 5, 2 <-> 6, 3 <-> 7, 4 <-> 8 or something similar to that might also commute. and maybe those generate everything.
They are generated by $\sigma$ and $\theta$ and $\gamma$, where $\sigma$ is the first cycle, $\theta$ is the second cycle, and $\gamma$ is the involution that transposes $I$ and $4+i$.
Hello, I wanted to ask the fellow math students, and preferably researchers on how a "mathematical day" should look like. I have always the feeling that I can not find a good balance between solving exercises, studying lecture notes, studying from books and so on. Especially when I attempt a problem and can not figure it out, I often feel like I would have been better off just attempting easier exercises or study more from the books, to advance.
cornman it varies a lot from person to person. everybody has different strengths and weaknesses. i think it is common to want to study theorems and theory maybe a little too much at the expense of understanding specific problems, key examples, and so on.
with the caveat that particularly at the undergraduate level and particularly with the internet being a firehose of mushy nonsense, it is sometimes difficult to find good sources of problems and sometimes easy to confuse yourself with problems that presuppose different definitions or a different set of working tools that the solver has access to.