53,981 reputation
467154
bio website users.utu.fi/lahtonen/…
location Rusko, Finland
age 51
visits member for 3 years, 5 months
seen 2 hours ago

General non-sense:

Mostly I teach here. I want to encourage beginning students to think for themselves, so I use a lot of hints and comments. More advanced questions I often just answer.

However, I don't do 1-on-1 chat. Don't ask. My answers and comments are for all to see, so I don't like to hide them. Also I feel that entering a chatroom carries with it an obligation to continue an on-line discussion, and then I would not be using my time on my own terms.

Relevant personal history:

PhD from Notre Dame in '90.

Drifted from representation theory of algebraic groups to applications of algebra into telecommunications, mostly coding theory, and lately mostly teaching at college level.

3 graduate students with awarded PhDs

I have mostly worked at our local University at Turku, Finland. At one point I tried working for Nokia Research Center. It was ok, but an old dog didn't learn all the tricks, and then they downsized, so I returned to the Uni as a tenured lecturer.


10h
answered How to calculate this $\sin\frac{\pi}{9}\sin\frac{2\pi}{9}\sin\frac{4\pi}{9}$?
10h
revised Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?
edited body
10h
comment Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?
If this is too broad in general, I think the 3D case is interesting enough. Of course, if I'm wrong about 2D, that is even more interesting!
10h
asked Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?
13h
comment The galois group of the polynomial $x^9+x^3+1$
Yeah. The product of the roots of $y^3+y+1$ is $-1$, so adjoining the cubic roots of two of them does give those of the the third. By your calculation we then get an upper bound of 108.
13h
comment The galois group of the polynomial $x^9+x^3+1$
@AdamHughes: Are you sure that if you adjoin the cube roots of one of the zeros of $y^3+y+1$ you get the cube roots of the other zeros free of charge?
14h
comment How to find all irreducible polynomials in Z2 with degree 5?
A good, generic way of doing this. You can generalize this to higher degrees by building a list of irreducibles as you go. Not unlike the sieve of Eratosthenes.
14h
answered How to find all irreducible polynomials in Z2 with degree 5?
19h
comment Show that four codewords is the maximal size for a code in V^8 = {(a1,…a8) | ai is in {0,1}} that corrects 2 errors
Probably something simpler is out there. My coffee break is over.
19h
answered Show that four codewords is the maximal size for a code in V^8 = {(a1,…a8) | ai is in {0,1}} that corrects 2 errors
1d
comment A problem about splitting field & irreducibility of a polynomial
Are you familiar with the argument showing that the splitting field of a polynomial of degree $m$ has degree at most $m!$? Use that for both factors.
1d
answered A problem about splitting field & irreducibility of a polynomial
1d
comment For finite group $G$ when is $|Aut(G)| < |G|$?
The automorphism group of $\Bbb{Z}_2\times\Bbb{Z}_2$ has order six. The automorphism group of $\Bbb{Z}_2^3$ has order 168...
1d
comment What is the axis and the rotation angle for the transformation that takes $(x_1,x_2,x_3)$ to $(x_3,x_1,x_2)$
No. The direction of the axis is uniquely determined by the mapping. Remember that the rotation does not change the axis at all. That's why you should first figure which vectors don't change at all under $R$.
1d
comment Why are randomly drawn vectors nearly perpendicular in high dimensions
@user161825: I had already upvoted your answer among others :-)
1d
comment What is the axis and the rotation angle for the transformation that takes $(x_1,x_2,x_3)$ to $(x_3,x_1,x_2)$
What was your axis? Is $(2,1,0)$ orthogonal to that?
2d
comment What is the axis and the rotation angle for the transformation that takes $(x_1,x_2,x_3)$ to $(x_3,x_1,x_2)$
The same process works (sans the reality check part) for all rotations of $\Bbb{R}^3$.
2d
answered What is the axis and the rotation angle for the transformation that takes $(x_1,x_2,x_3)$ to $(x_3,x_1,x_2)$
2d
comment Why are randomly drawn vectors nearly perpendicular in high dimensions
Others are using more rigour, but I think it simply as follows. By rotational symmetry of the distribution you might as well look at the inner product of a random vector and $(1,0,0,\ldots,0)$. That inner product is zero-mean, but its variance will be $1/n$. So when $n$ is in the hundreds, you need quite a few SDs off the mean to have a significant inner product (for a suitable value of "significant").
2d
comment A $2$-Sylow subgroup of $A_5$ has five conjugates?
It implies that $A_4\subseteq N_{A_5}(P)$, so $n=|N_{A_5}(P)|\ge12.$ Therefore the number of Sylow $2$-subgroups of $A_5$ is at most five. And also a factor of five, because five is a prime. After all $n$ is a factor of $60$ that is divisible by $12$, so either $n=12$ or $n=60$. Thus $60/n$ is either one or five. So if you can exhibit more than one Sylow $2$-subgroups of $A_5$, you will have exactly five of them. Can you do that? Hint: There are five natural copies of $A_4$ sitting as subgroups of $A_5$, each with there own Sylow $2$-subgroup.