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10h
comment Question about the limits of definite integrals
I think the assertion following the "we all know that" is not true, to begin with. The sine function is odd, so what we can easily see is that the integral is $0$... not that the integrand is an "even" function, as that first displayed equality would suggest.
17h
revised Sum: $1-2+3-4+5-6+…$
added 1320 characters in body
18h
comment Sum: $1-2+3-4+5-6+…$
@Thomas, it is not clear to me that a "point of view" must remain static, and thereby be unable to adapt to phenomena that fall outside its initial or original scope. It is surely a matter of taste, but the way I'd describe my own preferences is that I'd prefer to "make sense of" more things, rather than fewer (by too-aggressively declaring things non-sensical). Thus, very often I find myself spending time trying to ascertain "the rules" (though that term makes me nervous) that would accurately reflect "things going as desired". Not external rules, but partly contingent on their effects.
18h
answered Sum: $1-2+3-4+5-6+…$
18h
revised Sum: $1-2+3-4+5-6+…$
edited tags
Jul
3
answered Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?
Jun
27
comment Four questions about finite fields
Oops! :) Sorry! Bad arithmetic here! :) I'll delete my comment shortly...
Jun
27
comment Non real complex in metric completions of $\mathbb Q$
@LuisGomezSanchez, I don't think there is good evidence for any such thing. Sure, "it makes a better story" that way, but by that standard arguably almost all academic mathematicians "hate" all others, because they would have like to prove all the theorems themselves, etc.
Jun
26
comment Non real complex in metric completions of $\mathbb Q$
@AsafKaragila, it may mislead the questioner to say "isomorphic to $\mathbb C$"... and to say "metric completion" ... which requires similar understanding of "what metric?". Although non-canonical (algebraic) isomorphisms among these things have been used in important work (Grothendieck-Deligne-etal), it is sort-of "over-clocking" the ideas, I think.
Jun
26
comment Non real complex in metric completions of $\mathbb Q$
It's not "pathology" in $\mathbb Q_p$, but simply "different from $\mathbb R$-ness". As in the answer(s), the problem is "judging" p-adic completions in terms of real and complex numbers... while, in reality, there is no genuine comparison.
Jun
25
comment Mathematics is not a spectator's sport?
I forgot to mention the "machismo" aspect of the pop-culture mythology of mathematics... that, I claim, is manifest in most quips about "if you've not A, then you can't B"... Sure, we can create self-referential situations in which not playing by rule X entails Y, but that doesn't even being to mean that X causally entails Y. The artifactual rules set by "teachers" no matter how honorable their intentions, are not facts of nature.
Jun
25
answered Adjoint map is Lie homomorphism
Jun
25
answered Mathematics is not a spectator's sport?
Jun
25
revised Partial fraction of $\frac 1{x^6+1}$
edited body
Jun
25
comment Partial fraction of $\frac 1{x^6+1}$
@Ian, oop, thx, I'll repair that...
Jun
24
answered Partial fraction of $\frac 1{x^6+1}$
Jun
23
comment How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?
What is $T[-]$?
Jun
21
revised Can the Dirac Delta, $\delta $, be obtained by taking the limits of a rectangular pulse?
added 811 characters in body
Jun
21
comment Dirac Delta Properties
While the "bottom line" is as @M.Wind says, that $x^2$ times a distribution $u$ really means $(x^2 u)(f)=u(x^2f)$ for test function $f$, this very thing does make sense of "multiplication of distributions by smooth functions". That is, those left-hand sides really do have sense as distributions, as does the right-hand side... but using the argument "x" potentially confuses things, yes! Might be better to write just "$f$" rather than $f(x)$, and $\delta_a$ rather than $\delta(x-a)$. The usual notation for $x\to x^2$ as being just "$x^2$" may be inescapable...
Jun
21
comment Error in or another way of calculating $\frac{1}{2} \in \mathbb{Q}_3$
That sum you wrote is a legitimate, convergent $3$-adic series. If you mean to require rearranging to have coefficients all in the range $0,1,2$, then, yes, there's work to do. But there is no such requirement. Your first computation was fine, as was your second (apart from mistakenly thinking $-4=2$ $3$-adically). There are many correct sums-of-series in the $3$-adic numbers, just as in the real numbers...