Is it safe to say the largest order of any element in $S_5$ is 6? Since, the largest length cycles of a product of disjoint cycles is 3 and 2, the lcm of which is 6. The 5-cycle has order 5.
@EricStucky We know that from one side. To see what could go wrong, take $u(x) = x^2$. Then $Eu$ is $x|x|$, whose derivatives are $|x|$, $\text{sign}(x)$, and then a delta function.
In order to deduce if it belongs there, don't we check if it holds $\langle Eu, \phi' \rangle=-\langle (Eu)', \phi \rangle ,\langle Eu, \phi'' \rangle=\langle (Eu)'', \phi \rangle, \langle Eu, \phi''' \rangle=-\langle (Eu)''', \phi \rangle $ ?
Good morning friends :) I was wondering if you guys could point me to any good resources regarding characteristic polynomials' relationship with eigenvalues.
@EricStucky Hi Eric, I'm using "Linear Algebra: 5th edition" by Otto Bretscher
I was doing their textbook problems and I was confused as to how to prove the characteristic polynomials between 2 similar matrices are the same - and that's when I realized my fundamentals for this idea isn't very strong
Yup - that's how they described it. And since they didn't go over the fact(yet) that AS = SV diagonalization where S = eigenvectors and V = eigenbasis, I can't really find a way to explain it.
That's a completely formal property. You always have that in the sense of distributions. But $(Ex^2)'''$ is a distribution with no representation as an $L^p$ function, at least for the construction of $E$ we gave.
@EricStucky "Smash the heck out of it" - you mean just take a 2x2 and say "After computing this matrix of variables S^-1 * A * S, we can see that the characteristic polynomial's the same" after some computation using arbitrary variables?
Yeah. You can be a little less terrible than that by showing chi(A) = x^2 - (trA)x + (detA) and then showing those coefficients are invariant under similarities.
It's a good book, though not necessarily one I pedagogically agree with. I like to encourage students to read Axler's "Linear algebra done right" along with Treil's "Linear algebra done wrong".
It's used in vectorizing operations because of cache coherence and speed-ups. I'm sure you all know about least-squares and universal approximators and some data refinement steps with principal component analysis with SVD
In general it's useful in the data science field mostly, but it also does wonders for optimizing a function - in fact I experienced a 200x speedup in writing a vectorized matrix multiplication in C vs naïve implementation in python
Interesting - I've been meaning to learn that stuff for a while on Coursera or some MOOC(since our uni doesn't offer a course like that until upper div).
So I have a finite group and a representation $\varphi: \Gamma \to SU(2)$. (It's injective, if that matters.) What does it mean to talk about the Dynkin diagram of this representation? What is a root system of this representation?
Let's define a sort of faulty integral. For the purposes of this question we shall assume that this is the regular integral. This integral integrates all functions properly however it's gets confused when it sees floor. It has a delusion that floor is an arbitrary constant and just holds it fixed...
@AlexClark The special unitary group. It's one of the classical Lie groups. It's 3-dimensional, and the Lie algebra is the same as $\Bbb R^3$ with the cross product.
@PVAL Do you know an example of a 3-mfd that doesn't bound a definite 4-mfd? note I don't care about the sign of the bounding 4-mfd. (is it even possible with current technology to prove that this is true of some 3fold?)
also, what happened to the contractible stein manifold thing?
It seems so simple. Like if it wasn't 3n+1 and just +1, it is easy to see you would eventually reach a power of 2. the 3n skips a lot of those powers and makes it harder to land on one
Duck, I'm starting to become more and more convinced that there is a rigorous foundation for your line of inquiry. But it's all still very confusing to me.
if you have any suggestions for rewording it feel free to suggest away. :p
user147690
@MikeMiller Where did this come up? I don't think you'll be able to talk about Dynkin diagrams here, maybe Coxeter graphs? I doubt the associated Lie algebra is semisimple if the Lie bracket is the cross product?
i know the second portion of it probably isn't the strongest answer or the most rigorous. That's probably something that could classified as "sporadic", but that's just me.
"sporadic" isn't really a math term so I guess it's a matter of opinion
My main mental block is that you keep wanting to treat floor(x) like a "thing" but it's not really a "thing", it's just a function and functions behave in certain ways. You keep saying "I don't care about the ordinary integral" but the fact is that you are modifying your foundations much deeper than just the definition of the integral.
But in the past, this was never clear to me, and with this question it became so.
It's a simple Lie algebra. Yes, the word was "Dynkin diagram". Some quick geometry shows that $\Bbb R^3$ with the cross product is neither abelian nor has any nontrivial ideals: such a thing would be 2-dimensional, $V$. Now pick a vector $w$ neither in $V$ nor perpendicular to it. Crossing with this vector gets us something again out of $V$.
@AlexClark Wikipedia assures me that the Dynkin diagram of $SU(2)$ is $A_1$ (a single vertex, no edges). But I'm looking for the "Dynkin diagram of a representation", as opposed to that of $SU(2)$. In any case, it's no big deal, the authors provide an alternative way of understanding all this.
No worries, @AlexClark. Don't waste too much of your time on this garbage; I know how to use their computation without the Dynkin diagrams. I was just interested in the alternate approach.
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@EricStucky This notion of random knot that you mention is pretty common. It's due to Tutte in the 60s, and I think it's Nathan Dunfield's preferred method of picking a random knot, and Nathan is the best.
There's a really nice book on surfaces in 4-space that I haven't had a chance to really dig into, but one day hope to. It's exciting that there are various ways people might be able to access this part of knot theory, and the fact that 4 dimensions are already a bit wild means one might expect it has "special" difficulties that higher-dimensional knot theory doesn't.
For instance, there's a classification of the groups that appear as the complement of an $n$-sphere in $S^{n+2}$ for $n>2$. But this is unknown for $n=1,2$.
The thing is that a lot of knot theory was inaccessible for a long time. We had no idea how to get bounds on, say, the genus of a knot, or the unknotting number (number of times you need to allow the knot to cross over itself to unknot it).
For special kinds of knots, the Jones polynomial made progress on some parts of that. I think that was the 80s? After that, Khovanov homology in the 90s was a good tool. But one of the current kings of knot theory is Heegaard Floer knot homology, which Lipschitz talked at length about (ok, stated results at length).
The invariant $\text{HFK}^\infty(K)$ is a bigraded abelian group (it has two gradings $s$ and $t$ instead of the one in regular singular homology). Using this, you can... detect the genus of the knot (completely calculable from HFK!!! Not even just bounds!) Get bounds on the four-dimensional knot genus. Get bounds on the unknotting number. Prove that knots aren't concordant (though Tim Cochran and his collaborators have their own program for studying concordance).
You can tell whether the complement of the knot fibers over $S^1$ (that is, can you decompose it into a circle's worth of punctured surfaces?) and many other cool questions.
Do you mind if I just wholesale quote you on this stuff? I think it would be good to do a correction post anyway, and knowing very little I'm not sure how well I could paraphrase. XD
If you know that that map is continuous, @Vrouvrou, you're done, since $T = \tilde{T}\circ \pi$, where $\pi$ is the canonical projection onto $E/\ker T$.
Feel free. I am not really a knot theorist, so I am out of date about what we can do now and what we can't. I think some of the currently favorite questions are "What is the 4-dimensional genus of a knot?" and "What is the unknotting number of a knot?", though there are many many many many more.
It's the minimal genus of a surface it bounds in $B^4$.
Thinking of $S^3$ as the boundary of $B^4$.
I should really be careful, because there are two different kinds of 4-dimensional genus: smooth and topological. I'm thinking about smooth, though topological I think is also hard to access.
One of the postdocs at UCLA told me this charming fact. There are things we still don't know about the trefoil! In fact, there are two trefoils: the left-handed one, and the right-handed one. You can do something called the "Whitehead double" to a knot $K$. Kristen Hendricks told me that the Whitehead double of the left-handed trefoil is known to have smooth 4-ball genus zero. (That's known as being "smoothly slice" - it bounds a smooth disc in the 4-ball.)
This is open for the whitehead double of the right-hand trefoil!
I do believe that it's known to be topologically slice, but not yet known to be smoothly so.
Hm, perhaps I have a skewed impression because I mostly know about knots from talks, and at talks it makes sense that they would advertise the topology/geometry applications.
Actually I don't think we even have a knot theorist at UMN :/
I know Tyler Lawson wrote a great paper with someone moving to UCLA about knot theory last year :) But, glancing through your directory, I believe with you.
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