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comment How to know a number is divisible by a given number without using a calculator?
If you know modular arithmetic, this is explained in math.uconn.edu/~kconrad/blurbs/ugradnumthy/universaldivtest.pdf, where it is extended to a similar test for divisibility by any integer relatively prime to 10.
Oct
18
comment $\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does
If $\int_X|f|\,d\mu < \infty$ it is not automatically true that $\int_X f^+\,d\mu$ and $\int_Xf^-\,d\mu$ are finite, since perhaps $f = \pm 1$ (so $|f| = 1$) with $f = 1$ on a non-measurable subset of $X$ and $f = -1$ on the complement.
Oct
18
comment $\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does
It is false that existence of $\int_X |f(x)|\,d\mu$ implies existence of $\int_X f(x)\,d\mu$ in general. You must assume that $f$ is measurable and then there is an equivalence as you wrote. Without assuming $f$ is measurable, you could let $f = 1$ on a non-measurable subset $A$ of $X$ and $f = -1$ on the complement of $A$ in $X$. Then $|f(x)| = 1$ for all $x$, so $\int_X |f(x)|\,d\mu$ exists, but $\int_X f(x)\,d\mu$ does not exist.
Oct
18
comment Friends and Enemies of Infinities
Presenting this as some kind of continuing dispute among mathematicians reminds me of the intelligent design movement claiming that there is a real uncertainty among scientists over the role of evolution. The proportion in favor of one side (acceptance of infinity and related objects) is so overwhelming that to suggest there is still a contest is a misuse of the word "contest".
Oct
16
comment Theorems in number theory whose first proofs were long and difficult
The result you are describing is about the positive rational numbers and involves prime factorization, but whether or not number theory is equivalent in some sense to studying finite sets is irrelevant to it.
Oct
16
comment Theorems in number theory whose first proofs were long and difficult
Number theory was most definitely not just about the integers in 1937. For instance, the basic facts of class field theory were already known by then. I looked in Stillwell's book on the history of math where he discusses the result of Ackermann, and it relies on declaring that combinatorics is just a name for the theory of finite sets and elementary number theory is just a name for the theory of the positive integers. Neither description is a fair reflection of what the subjects are really about.
Oct
16
comment Theorems in number theory whose first proofs were long and difficult
That is using a strange definition of "number theory". Just because a result involves the positive integers does not make it number theory, and not everything in number theory is about the positive integers (or prime numbers).
Oct
16
answered Theorems in number theory whose first proofs were long and difficult
Oct
16
comment Theorems in number theory whose first proofs were long and difficult
Number theory is not equivalent to finite set theory. Where did you get that idea?
Oct
16
comment Reference request: Galois descent
I don't see what's wrong with citing a recent text. The texts by Silverman on elliptic curves are often cited even if it's not for a theorem due to Silverman himself, simply because his books are a convenient source to learn about the material covered by them.
Oct
15
comment Reference request: Galois descent
How old is "very old (and still useful)": 50 years? And why are more recent discussions in textbooks not suitable?
Oct
13
comment Contraction mapping theorem with respect to the supremum norm.
Why not actually calculate the integral? Then remember that $0 \leq x \leq 1$.
Oct
12
comment “Stick it to the man!” Mathematical discoveries that resulted in persecution.
See the story by Anna Varvak (currently the top-ranked one) among the answers to mathoverflow.net/questions/53122/mathematical-urban-legends
Oct
12
comment “Stick it to the man!” Mathematical discoveries that resulted in persecution.
From Weil's autobiography I learned that Lie had been arrested in Paris in 1870 soon after the start of the Franco-Prussian war for possibly being a spy because his work habit of taking notes in the forest was suspicious. See the end of ams.org/journals/bull/1899-05-07/S0002-9904-1899-00628-1/… for more on this episode.
Oct
12
comment “Stick it to the man!” Mathematical discoveries that resulted in persecution.
@studiosus: I reread Weil's autobiography. He wrote that he was brought to jail in Finland simply for looking like a foreigner while standing near some military equipment. Later the police searched his apartment and found among the incriminating evidence a letter in Russian from Pontryagin, not Kolmogorov. Considering Weil's interest in topological groups, it makes sense that he corresponded with Pontryagin rather than Kolmogorov.
Oct
12
answered Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )
Oct
12
comment “Stick it to the man!” Mathematical discoveries that resulted in persecution.
German mathematicians who were Jewish were forced out of their positions by the Nazis. Landau in particular was boycotted by his students under the guise that his particular "style" of mathematics was unsuited to the classroom in Germany.
Oct
12
comment “Stick it to the man!” Mathematical discoveries that resulted in persecution.
Galileo was persecuted by the Roman Catholic Church for the discoveries he made from his telescope, though this was not a mathematical discovery so it does not fit your request.
Oct
12
comment “Stick it to the man!” Mathematical discoveries that resulted in persecution.
@studiosus: neither of those examples are people being persecuted for their work, as the original question had asked.
Oct
6
comment Converse of the Fundamental Theorem of Calculus
You need to say something about the kind of functions $f$ you are considering when you are integrating them. Since you don't want to assume $f$ is continuous, what are you willing to assume about $f$ so that $\int_a^x f(t)\,dt$ makes sense? Also mention your mathematical background: one or two semesters of calculus? Measure theory? Something in between?