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bio website math.ntnu.no/~hanche
location Trondheim, Norway
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visits member for 2 years, 5 months
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Jun
25
comment Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.
And what I did, in my comment on @Hayden's answer.
Jun
25
comment Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.
It would be easier to start with taking all the odd numbers. Next, take all odd multiples of 2. Then take all odd multiples of 4. In the $n$th step, take all odd multiples of $2^n$.
Jun
25
comment Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.
No, he's saying take every other odd number. Take 1, leave 3 for later, take 5, leave 7 for later, take 9, …
Jun
25
comment Can an empty array be useful?
Haskell has tuples of any nonnegative arity (except 1, as a 1-tuple is just conflated with the corresponding member). In particular, there is a zero-tuple, denoted by (). It is of a type also denoted () – this type has only one member. Is it useful? I haven't learned enough Haskell yet to judge.
Jun
25
comment Can an empty array be useful?
@Eric: Your point being? (I think you were making a joke, but I am too dense to get it.)
Jun
25
comment Can an empty array be useful?
@jpmc26 I think Eric explains it well. That is what I had in mind. To expand a bit on it, don't confuse the “elements” (we should say coordinates) of an array with the elements of a vector space. A one-dimensional array of length one has one element, but it corresponds to a vector in a vector space with infinitely many elements.
Jun
24
comment Can an empty array be useful?
Classical vector spaces include the Euclidean spaces, set of matrices, or higher dimensional tensor spaces – involving multidimensional arrays, if you wish. But also subspaces of these, which are the solution sets of systems of linear (homogeneous) equations. Sorry, this space is too small for a whole introduction to vector spaces. Try wikipedia, perhaps? Anyhow, the trivial vector space has only one member: The zero vector.
Jun
24
comment Can an empty array be useful?
Perhaps more to the point, the trivial vector space.
Jun
24
comment Prove that the set $A$ is measurable and find its Lebesgue measure.
Mapping the real number $x\in[0,1]$ to the $n$th digit in the decimal expansion of $x$.
Jun
24
answered Prove that the set $A$ is measurable and find its Lebesgue measure.
Jun
24
comment Measure theory for $L^2$
(On a side note, unless you know that $F$ is positive definite, you could have trouble with some of your square roots.)
Jun
24
comment Measure theory for $L^2$
Ah. Then the point is most likely to use Claim1. If you can show the inequality whenever you replace $u$, $v$, $w$ by $1_{K_n}u$, $1_{K_n}v$, $1_{K_n}w$, then you can use a convergence theorem to extend the inequality to the general case.
Jun
24
comment Measure theory for $L^2$
Claim2 requires a purpose for which you may make the stated assumptions. What is this purpose?
Jun
24
comment What is $T^nf(t)$? (Question on integrals)
Induction is a good plan, but so is your idea of trying it with $T^2$ first.
Jun
24
answered Asymptotic expansion of $z^{-x}$
Jun
24
comment Please someone help with this nearly impossible integral
Argh, you're right: In my calculation, $a+bx$ morphed into $ax+b$ while its derivative remained as $b$. It seems that $ax+b$ is the canonical first order polynomial in my head. It is interesting though, that Maple came up with a complicated answer containing two sums of logarithms, summed over the roots of two different third degree polynomials. These sums must cancel somehow. I guess it shows that CASes can be deceptive.
Jun
24
comment Please someone help with this nearly impossible integral
Did you actually try it yourself? It doesn't seem to work out nicely, and the answer given by Maple doesn't agree either. It looks rather messy, in fact.
Jun
24
comment $ ∑_{x∈X}|f(x)|<∞ \quad ⇔ \quad ∑_{i=1}^∞|f(x_i)|<∞ $
The sum on the left, is that defined as the supremum for all sums over finite subsets of $X$? Your notation is garbled up. Do the two different styles of $X$ mean something? What do sum over in the last sum, $x\notin X$? And where does non-countability enter the picture, when you started with a countable set? Finally, I don't know what you mean by “this Process” and the odd looking inequalities ($\cdots\ge\infty$) in the final box.
Jun
24
answered These two definitions of a hyperplane are equivalent?
Jun
24
comment Giving some hints
@CiaPan good catch. Voting to close. To the poster: Reposting a question is considered extremely bad form. Please stop doing it.