Hi I am at a loss here -- I am stuck (and have been) on a particular question, and have n clue about what to do to get some help about it https://math.stackexchange.com/questions/2286468/k-functional-between-ell-1-and-ell-2-for-a-specific-sequence
Basically, I have already put two bounties on it to "draw attention" (the second expires tomorrow), but in vain.
Is it just that no one knows, or is it that bounties are inherently useless given the number of questions appearing on Math.SE?
(I have considered Mathoverflow, but I am not sure it fits the bill there -- even though is is a research question. Plus, I don't have much reputation there)
In favor of MO for your problem is that 1) you've already had it on MSE for quite some time, to no effect, and 2) you've got references which you cite.
Against it is the fact that I dunno what the threshold really is.
Yup. Most of the MO posts are a variation of "I am not sure I can parse the title, and when by chance I can it gets answered in a fraction it takes me to read the question"...
But what's not obvious to me: Given a particular linear combination of them, can I always find vectors $u,v$ such that said combination equals $uv^T-vu^T$?
Hmm. For $n$ sufficiently large this definitely doesn't work, since $n(n-1)/2$ grows faster than the $2n$-dimensional space of matrices of the form $uv^T-vu^T$.
I should admit at this point that what I specifically have in mind is the case of $n=3$.
@BalarkaSen In fiber bundles, horizontal lifts of paths provide diffeomorphisms between fibers. Is it right to say that this is essentially due to existence, uniqueness and smooth dependence of solutions to ODEs?
After I discussed with my supervisors tmr, I might consider asking on the main CSE site about how to handle optimisation where the hessian end up singular
More precisely, if your antisymmetric matrix is of the form $a_1 E_1+a_2 E_2+a_3 E_3$, where $E_1=e_1 e_2^T-e_2 e_1^T$ and so forth cyclically, then $a^2=a_1^2+a_2^2+a_3^2$.
(At least that's what I seem to be getting from Mathematica)
Which dovetails nicely with the point re: all eigenvalues being imaginary.
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in number theory to compute series.
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I just realized I can produce fantastic pieces of stream-of-consciousness from normally written passages by translating between various languages in Google Translate
@BalarkaSen I had wonderful works under a variety of languages, Google Translate can create a sense of a known flow path that included translation and writing
In many languages, including translation, I have a wonderful book, gives a sense of flow to Google
And one last one : Some are language and translation, and with the arrival of Google
@BalarkaSen After some English->Bengali->Spanish->English cycles, this becomes "Secret? This brave new strategy to spend the people. This should be more"
So take a fiber bundle $\pi:M\to B$ with fibers $F$. I pick a horizontal complement $H$ to the vertical subbundle and call a vector field $X\in \Gamma(TM)$ basic if it is horizontal and $\pi$-related to a vector field on the base.
If I consider a path in the base I have horizontal lifts $\tilde\gamma_p$ for any $p\in F_{\gamma(0)}$
They define the diffeomorphism $\tau_\gamma:F_{\gamma(0)}\cong F_{\gamma(t)}$ via $\tau_\gamma(p)=\tilde\gamma_p(t)$.
This is a one-parameter family of diffeomorphisms I guess
I want to understand the following claim: "$\tau_\gamma$ is part of a (local) one-parameter group of diffeomorphisms associated with a basic vector field"
The obvious basic vectors to look at are the derivatives of the $\tilde\gamma_p$, but those don't give a global vector field in $M$ in any obvious way to me. Just as the vector field $\dot\gamma$ is only defined along $\gamma$, I only see how to define an associated basic vector fields on the fibers over the image of $\gamma$. Is this somehow enough?
I imagine being associated to a basic vector field means exactly what you think it does; it's associated to $\gamma'$. Sure it extends to all of $M$; bump it up outside a nbhd of $\gamma$ in $B$
No, self-intersection is a dumb issue. But I see; I was thinking of $\pi$ being the tangent bundle; what does it mean to compute "rate" in an arbitrary vector bundle? $\gamma'$ is not the right thing to look at
$\gamma'$ lives in $TB$, not $M$. I suspect you do something with the connection?
Then why does that not work? You get $\gamma'$ below is a section of $TM$ over $\gamma$, right? Extend that to a section around a tubular nbhd $U$ of $\gamma$/
I have a sequence of derivatives of injective analytic functions on a compact set that converges uniformly to the analytic function $g$. Is $g$ a derivative of an injective function?
Similarly, I get a family of sections $\tilde\gamma_p'\in\tilde\gamma_p^*TM$ for each $p\in F_{\gamma(0)}$ so basically "a vector field on the fibers above the image of $\gamma$"
But that doesn't extend to a vector field on all of $M$
@BalarkaSen $\gamma'$ does not live in $TB$ naturally. It lives in $\gamma^*TB$. But you can think of it as living of the part of $TB$ that lies over the image of $\gamma$, sure.
I am too tired to do this. I give up. I suspect thinking about it all carefully should easily tell you why a path in $B$ gives a horizontal section of $TM$
So take a fiber bundle $\pi:M\to B$ with fibers $F$. I pick a horizontal complement $H$ to the vertical subbundle and call a vector field $X\in \Gamma(TM)$ basic if it is horizontal and $\pi$-related to a vector field on the base.
@Danu Then why can you not use a partition of unity to get a global section? if $X$ is a section of $E/B$ along a path $\gamma$, then you can always extend that to a section of $E$. That is what I want to understand
"According to Bézout's theorem, two different cubic curves over an algebraically closed field which have no common irreducible component meet in exactly nine points (counted with multiplicity). The Cayley–Bacharach theorem thus asserts that the last point of intersection of any two members in the family of curves does not move if eight intersection points (without seven co-conic ones) are already prescribed."
They mention that it serves to unite Pappus's hexagon theorem, Pascal's theorem, and the associativity of the group of elliptic curves. That's pretty neat.
_Dear Sir or Madam, will you read my book?_ _It took me years to write, will you take a look?_
bah
@BalarkaSen Yeah. At least that person edited their post to include more info. (A pity that info doesn't include their attempt to actually put it together.)
if you pick three complex numbers $z_1,z_2,z_3$ and compute $z_1z_2\overline{z_3}+z_1\overline{z_2}z_3 + \overline{z_1}z_2z_3$ you get a complex number whose argument tells you the direction of the euler line of a triangle whose sides have the directions of $z_1,z_2,z_3$
Hi. Suppose that $B_1$ is a basis for $V$. $B_2$ is also a set of independent vectors, but it's not a subset of $V$, and $V \subset span(B_2)$. I suppose that there is no name for $B_2$ in such a case.. Anyway, I am looking for some analogies of a "minimal" option and "bigger" ones.. For example, the coarsest topology and the finest topology, trivial and discrete $\sigma$-algebras.. what else? Preferably in statistical/probabilistic context
and iirc, if you resize them so that $z_1+z_2+z_3=0$, the magnitude of the thing is proportional to the area of the triangle times GO (or whatever other distance you measure on the euler line)
wait no
it shuold be area squared
sweats uncomfortably as something is not homogeneous
Yeah, I've just been waiting so long. 4 years to get into school. I used to work full-time nights at a hotel, and all night I'd do physics homework just to get it done. I know I'm sort of acting entitled to enrollment...
"Listen kid. I've been hearing that crap ever since I was at UCLA. I'm out there busting my buns every night. Tell your old man to drag Walton and Lanier up and down the court for 48 minutes!"
