49,774 reputation
464146
bio website users.utu.fi/lahtonen/…
location Rusko, Finland
age 50
visits member for 3 years, 2 months
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.


2s
revised Question about geometric sequence problem
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58s
revised Geometry and algebra stacked
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3m
revised Find the side length of shortest equilateral triangle with given conditions.
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5m
revised False Acceptance Rate
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9m
revised Reducing a fraction by dividing top and bottom
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12m
revised Prove or find a counter example to the claim that for all sets A,B,C if A ∩ B = B ∩ C = A ∩ C = Ø then A∩B∩C ≠ Ø
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1h
comment What math have I missed as an Engineeering graduate?
I think it is safe to say that this question is either too broad or primarily based on opinion. Given that an average math PhD is still missing at least 95% of known math (I think I'm flattering myself and a few others with that estimate), it is safe to say that you have missed nearly all. But you aren't alone in that!
2h
comment Equivalent definitions of a lattice in a real vector space of finite dimension
If $\dim V=n$, then any $n+1$ elements of $\Gamma$ must be linearly dependent over $\Bbb{Z}$. For otherwise a big ball of radius = $R$ contains at least a constant $\times R^{n+1}$ points of $\Gamma$. As the volume of that ball is a constant times $R^n$, letting $R\to\infty$ leads to contradiction with $\Gamma$ being discrete. Thus $\Gamma$ being finitely generated, and obviously torsion free, implies that $\Gamma$ is free. The rest is relatively easy (see Adam's answer for pointers).
3h
comment Characterization of the transcendentals over a field
@user42912: Study the Wikipedia article on transcendence degree. Hopefully that makes it clear. The concept is often not met in a first course on field extensions.
3h
comment Characterization of the transcendentals over a field
Furthermore, it is also assumed that $F/K$ is finitely generated. This prevents the possibility of a transcendental element $z$ such that $[F:K(z)]$ is infinite (even though it is algebraic by virtue of $F/K$ having transcendence degree one).
4h
comment Characterization of the transcendentals over a field
Not surprisingly I own a copy of this book. Keenan's got it right. An umbrella assumption of transcendence degree one is made very early, because the author does not want to repeat it, and does not consider other kinds of function fields.
4h
reviewed Reject suggested edit on The number of real roots of $2x^5+8x-7$
4h
reviewed Reject suggested edit on Determining Total Assets, Total Liabilities From a Financial Statement with Missing Values
13h
revised Calculate the inequality
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13h
revised Every increasing function from a certain set to itself has at least one fixed point
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13h
revised How do you work out $\sqrt[4]{16^3}$ without a calculator.
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13h
revised Find the following limit: $L=\lim_{n\to \infty}\frac{\left(2\sqrt[n]{n}-1\right)^n}{n^2}$
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13h
revised Math Puzzle - Rock Problem
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13h
revised Equality of real numbers whose squares sum to 1
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13h
revised How do I find the inverse of a $\cos^2 \theta$?
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