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comment Does a convergent sequence in theory ever reach its limit?
To actually reach the limit, one must often speed up toward the end. That's what Achilles did in real life in order to catch the tortoise. (And, in several regards, I'm not just kidding around here...)
Apr
29
comment Is the Hodge star an $SO(n)$-equivariant isomorphism to the dual representation?
@SaalHardali, $g\cdot (v_1\wedge\ldots\wedge v_m)=gv_1\wedge \ldots \wedge gv_m$.
Apr
23
comment Dilation of Fourier transform
For several reasons, the dilation factor should not be $\delta$... (E.g., there is Dirac $\delta$, in such a context.)
Apr
21
comment units of the quotient ring of the integers over a prime power $[{\Bbb Z}/P^e\Bbb Z]^*$ is cyclic multiplicative group
Maybe I'm not understanding, but "proof of existence of primitive roots..." should be in any elementary number theory text, and maybe on Wiki, and maybe somewhere on this site, too.
Apr
21
comment units of the quotient ring of the integers over a prime power $[{\Bbb Z}/P^e\Bbb Z]^*$ is cyclic multiplicative group
Search on "primitive roots": this phrase refers to the generators of those cyclic groups (and their existence).
Apr
19
comment Can the real vector space of all real sequences be normed so that it is complete ?
Also, I guess a person should say that this "answer" is mostly unrelated to issues about "algebraic/Hamel" bases, axiom of choice, etc. The point was to argue that a natural structure (complete metrizable topological vector spaces) is Frechet, but not Banach.
Apr
19
revised Solve a hypothetical mass-level extinction scenario
edited tags
Apr
19
comment If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$.
It's not completely clear what "Lebesgue measure" on a sphere is... unless one intends exactly "group-invariant measure"... but perhaps then the questioner is asking how this could be computed.
Apr
19
comment If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$.
Some quibbles: $S^2$ would be better referred-to as the two-dimensional unit sphere in $\mathbb R ^3$, and (since it's not a group in its own right) the measure is the $O(3,\mathbb R)$-invariant one. (Yes, there is a relation to the actual Haar measure on this orthogonal group, but maybe that's not the point here.) So: rotation-invariant distribution supported on $S^2$. It's compactly-supported, so you can meaningfully apply it to exponentials, to correctly compute its Fourier transform..
Apr
19
comment Prove that $e^x$ is not a tempered distribution on $\mathbb{R}$
As @Aaron speculates, smoothing functions at discontinuous cut-offs can change the Schwartz semi-norms ... significantly. I guess the operational question is whether it's simpler (in whatever context you find yourself) to see whether you can be sufficiently subtle in doing smooth truncations... versus taking a somewhat different approach to the question. It's true that we seem not to collectively have well-known examples of Schwartz functions that decay (much) more slowly than exponentials! :)
Apr
19
comment Newton potential for Neumann problem on unit disk
I think the notion that there are strict rules about usage, etc., is an artifact of lower-division math, and it might be good to anticipate that not many people actually adhere to whatever rules one may declare. E.g., in fact, $\Delta$ is standard: it is what mathematicians often write. If anything, the use of $\delta^2$ for $\delta$-on-$\mathbb R^2$ is pretty non-standard, and arguably un-necessary. Anyway, it's pretty hard to maintain a prescriptivist world-view in mathematical notation, and kind-of pointless to recommend it.
Apr
19
comment Prove that $e^x$ is not a tempered distribution on $\mathbb{R}$
A little problem: those truncated versions of exponentials are not smooth, so are not Schwartz...
Apr
19
comment Newton potential for Neumann problem on unit disk
Actually, in quite a few reasonable contexts, it is not at all obligatory to write $\ln$, nor to put arrows over names of vectors, and it is widely accepted to write $\Delta$ for Laplacians and Laplace-Beltrami operators...
Apr
19
revised Can the real vector space of all real sequences be normed so that it is complete ?
deleted 4 characters in body
Apr
19
answered Can the real vector space of all real sequences be normed so that it is complete ?
Apr
19
comment Find the Subextensions of $\Bbb Q(\sqrt2, \sqrt3, \sqrt5)/\Bbb Q$
Your observation that $2^3=8$ does not imply that there are $8$ subgroups...
Apr
17
comment $\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert space
After your first two lines, you are done. The rest that you wrote is a repeat of definitions/characterizations of tensor products of finite-dimensional spaces. "Sequences on" a finite set are just numbers indexed by it...
Apr
17
comment $\int_0^\infty {x^a\over (x^2+1)^2} dx$ where $0<a<1$.
Use the so-called Hankel or keyhole-contour, to exploit the fact that if we go around 0 the $a$th power of $x$ changes by $e^{2\pi i a}\not=1$, for $0<a<1$, then evaluate by residues.
Apr
17
comment Finding polynomial in modular? // $(n!)+1$ prime
Use Sun-Ze's theorem (also known as "Chinese Remainder Theorem") applied to each of the coefficients individually.
Apr
17
comment Determine the complex contour integral $\oint \limits_{C} \frac{2}{z^3+z}dz$ without using Residue Theorems
Ah, indeed! And after doing an example or two like this, we see that the proof of the residue theorem is ... obvious. :)