49,730 reputation
464146
bio website users.utu.fi/lahtonen/…
location Rusko, Finland
age 50
visits member for 3 years, 1 month
seen 4 mins 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.


3h
awarded  Nice Answer
22h
comment A question about algebraically closed fields
+1 Well done. This works.
2d
revised multiplication in Galois Fields
added 231 characters in body
2d
revised multiplication in Galois Fields
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2d
answered multiplication in Galois Fields
2d
comment multiplication in Galois Fields
@Hurkyl: In that sense, yes. I am more worried about unsuspecting programming students taking the identification of the coset $x+\langle p(x)\rangle\in\Bbb{F}_2[x]/\langle p(x)\rangle$ with the integer $2$ way too seriously. The mapping $q(x)\mapsto q(2)$ for polynomials $q\in\Bbb{F}_2[x]$ is not without merit, but it does lead to several misunderstandings such as the one witnessed here. I've seen (more than) my share, trust me.
2d
comment multiplication in Galois Fields
Some people (programmers?) cherish an internal presentation of polynomials in $\Bbb{F}_2[x]$ as strings of bits, and try to interpret those bit-fields as integers. This is all fine for writing elements of $GF(2^n)$ compactly (I do it that way in my programs as well!). But, it comes with the cost. Namely it hides the fact that addition in $GF(2^n)$ is the bitwise XOR of those bitfields, and multiplication should be done as polynomials modulo the defining polynomial.
2d
comment multiplication in Galois Fields
No, it is not correct. $4$ and $2$ are not elements of $GF(8)$ in a very natural way, and the multiplication is not that of integers modulo $8$. See this Q&A pair for some examples. Other examples on the site abound.
2d
comment Linear Algebra, Quadric form
A pedant's remark: The set $Z$ that you wrote down is not a vector space. You usually cannot write $\phi_1\cdot\psi_1+\phi_2\cdot\psi_2$ in the form $\phi\cdot\psi$. This is why question 2 specifically talks about sums of quadratic forms of the type $\phi\cdot\psi$. This may or may not be related to your difficulties. BTW, what's the definition of a quadratic form you are using?
Jul
23
revised Find angle of incomplete rotation matrix
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Jul
23
answered Find angle of incomplete rotation matrix
Jul
23
comment Prove $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are finite.
$K$ non-empty and $KK\subseteq K$ is enough. Because $KK\subseteq K$ all the powers of all the elements of $K$ will be in $K$. Because $H$ is finite, all the elements of $K$ have finite order. Consequently the identity and the inverses are positive powers. NOTE: It is essential that $H$ is finite for this argument to work.
Jul
23
comment How to find $[A_n,A_n]$
@Koenraad: Is it clear how you get that ALL the 3-cycles are in $[A_n,A_n]$, if $n\ge5$?
Jul
23
comment Prove $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are finite.
The finiteness of $H_1,H_2$ may have been listed as an assumption to enable the use of a simplified subgroup criterion: If $H$ is a finite group and $K\subseteq H$ is a non-empty subset such that $KK\subseteq K$, then $K$ is a subgroup of $H$.
Jul
23
revised A question from the mod p irreducibility test's proof
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Jul
23
comment A question from the mod p irreducibility test's proof
Good, @VHP! But this was a good question (+1)! Well done in spotting this possibility!
Jul
23
answered A question from the mod p irreducibility test's proof
Jul
23
comment How prove $\frac{1}{\sqrt{2006}}<a_{2006}<\frac{2}{\sqrt{2006}}$ in sequence?
My suggestion is to check whether you can prove the inequalities $$a_n<\frac2{\sqrt n}$$ and $$\frac1{\sqrt n}<a_n$$ by induction using the inequality that you were given in the inductive step. Remember that sine is an increasing function in a relevant interval. As is square root. The inductive step may only work for a range of values of $n$, so I don't know what would be a good base case for the two inductions.
Jul
23
comment How prove $\frac{1}{\sqrt{2006}}<a_{2006}<\frac{2}{\sqrt{2006}}$ in sequence?
Looks like that (or something similar) will be helpful here. Try it out!
Jul
23
comment How prove $\frac{1}{\sqrt{2006}}<a_{2006}<\frac{2}{\sqrt{2006}}$ in sequence?
But you could have searched the site also. See this related answer to get an idea of what this entails. That does use Taylor series (and may be only heuristic), but there are alternatives. Anyway, you need to estimate the difference between $x$ and $\sin x$, when $x$ is relatively small. See how that compares with similar difference for $f(x)=1/\sqrt x$ and $2/\sqrt x$. Or if that doesn't work, then you need to ask someone else for advice :-)