ironic [ahy-ron-ik] Spell Syllables Examples Word Origin See more synonyms on Thesaurus.com adjective 1. using words to convey a meaning that is the opposite of its literal meaning; containing or exemplifying irony : an ironic novel; an ironic remark. 2. of, relating to, or tending to use irony or mockery; ironical. 3. coincidental; unexpected: It was ironic that I was seated next to my ex-husband at the dinner.
@Semiclassical I am more interested in what one can say about the distribution given by kernel of $\phi$. I am pretty sure it's going to be integrable.
In any case, the issue is there's no reason to believe $d\phi = \omega \wedge \phi$ for a universal $\omega$ for all $\phi$. I think if there is such a universal $\omega$ that corresponds to integrability of an appropriate distribution (kernel of $\phi$?) or something.
Anyways, bottom line is that my initial thought was too simplistic. But once you go up to 2-forms then there's a chance for more interesting possibilities.
I don't really see a way to make these things vector spaces rather than O_X modules (modules over the ring of smooth functions) which is wear this issue comes from.
so I don't have time to share it. I'll think about it some more while I take my shower, return, write some notes and possibly have a decent target plan for collatz.
"Open questions with this level of notoriety can lead to what Richard Lipton calls “mathematical diseases” (and what I termed an unhealthy amount of obsession on a single famous problem). (See also this xkcd comic regarding the Collatz conjecture.) As such, most practicing mathematicians tend to spend the majority of their time on more productive research areas that are only just beyond the range of current techniques.
Nevertheless, it can still be diverting to spend a day or two each year on these sorts of questions, before returning to other matters; so I recently had a go at the problem. Needless to say, I didn’t solve the problem, but I have a better appreciation of why the conjecture is (a) plausible, and (b) unlikely be proven by current technology, and I thought I would share what I had found out here on this blog."
You left out a lot of context; I'm tempted to think you're just cherry-picking bits of that conversation to make others look bad while ignoring your own behavior and how it contributed to the situation.
Here's something I would consider a really critical bit of context:
(for casual readers: "...
Small question - if $A$ is compact, I know that $ker(I-A)$ is finite. Is there a neat way to show, from that, that $ker(\alpha I-A)^n$ is finite for each $n\in N$?
But I think in order to prove the later part, I'm just gonna have to say that if it were of infinite dimension, I could've shown some partial mapping that yields that $dim(ker(I-A))$ is infinite...
Cheers anyhow. I will try a few methods. It's a supposedly easy question so I am just looking for a neat way to do it.. I also hate when they say a question is easy. So intimidating
Alternatively - using direct sums.. the proof is rather long. There is a shorter proof for a private case - where we can find some $A_0$ of finite degree "close enough" to $A$
Basically, at most a $k$ dimensional subspace of your whole Banach space is killed, so in particular, at most a $k$ dimensional subspace of the image of $I-A$ is killed as well, you know what I mean?
That's the heuristic reasoning which I'm pretty sure can be made rigorous
Okay so, if you consider $(\alpha I - A)^n$, its kernel (I believe) should be $\ker(\alpha I - A)^{n-1} + \{x : (\alpha I - A)x \in \ker(\alpha I - A)\}$, right?
So, by induction hypothesis, the first should be finite dimensional, and I think if the second were infinite dimensional, you would be able to show that the kernel couldn't be finite dimensional, because you just collapsed an infinite dimensional space to a finite dimensional one like come on QED
First is that the kernel is as I said it is, second is that if an operator has finite nullity, the image of an infinite dimensional subspace is also infinite dimensional
haha Induction too. But gonna try that one by myself for now - if you're curious, we know that $Im(I-A)$ is closed, and we wanna show that $Im(\alpha I - A)^n$ is closed.
I've only seen spectral theorem for compact operators on Hilbert spaces
What does it even mean to be self-adjoint on a Banach space though?
(Also @Steamy do you know algebraic topology at all? I'm having a bit of trouble understanding the most recent lecture and I'm wondering if I could run it by you just to be sure)
(Or anyone for that matter)
Hey @s.harp! Are we finally on chat at the same time?
[Dream from a train nap] Define $$\alpha^n=\int f((forgot))dx$$ Now $$\int f((forgot))^n dx = \alpha^{n+1}$$
So basically, you have an integro-recurence relation. The dream however never showed what the base case is thus there is no way to make further sense of it
@Daminark Well, there's a part of the spectral theorem that says that for any nonzero eigenvalue $\lambda$ of a compact operator $A$, there is some $n$ such that $\operatorname{Ker}(\lambda -A)^n = \operatorname{Ker}(\lambda -A)^{n+1}$, and this subspace is finite dimensional. I didn't really follow the discussion, but it reminded me of that.
However, after some random pondering and free association about the dream itself after I woke up, I then came up with something that might be a little bit more sensible:
Let $I=\int f(x)dx$. There exists functions $f(x)$ such that the following holds for all $n \in \Bbb{N}$
Lol yeah, Peter's trying to develop things that more closely matches how modern algebraic topologists think about stuff, apparently the first time he's trying this way. We'll see how it goes
Apparently the based version agrees with the unbased version for CW complexes up to weak homotopy equivalence, but that's quite a bit beyond me right now
So his book defines a crash product $X\land Y = X\times Y/X\lor Y$ where $X\lor Y = x_0\times Y \cup X\times y_0$
This lets you express $\Sigma X = X\land S^1$
On the other hand there's this property of loop spaces or something
It's basically this Haskell-type reading of functions
You have a bijection from, I think the idea is that the set of continuous maps from $X$ to the set of continuous maps from $Y$ to $Z$ is bijective to the space of continuous maps from $X\times Y$ to $Z$, and the compact-open topology makes it a homeomorphism, said that's about what we need to know
Perhaps. His endgame seems to be using this stuff to define cohomology groups via Eilenberg-MacLane spaces
But yeah an Eilenberg-MacLane space of class $K(G,n)$ is a connected space whose homotopy groups are all trivial except for the $n^{th}$ one, which is $G$
$$\int I^n dx =I^{n+1}$$ Ok it turns out we can do something about this: Differentiate both sides to get $$I^n =(n+1)I^n f(x)$$ We can rearrange this equation to get $$I^n (1- (n+1) f(x))= 0$$ This give us two conditions: $$I^n=0\text{ or } f(x) = \frac{1}{n+1}$$ Now consider the base case: $$f(x)=0\text{ or } f(x) = 1$$
Therefore the only consistent solution is the zero function
I'm not an expert on the subject, but If it is worth I'd like to start on getting some grasp on the subject.
Assuming geometric algebra framework, Is there somewhere a list of formulas where for example the intersection between a line and a plane, line and sphere, if the point belong to a plane ...
I see many seemly common patterns across different integrals from the chat crawling and the recent chat messages, which is why I just sorta-reflexive action, analysing them
For example, user3133165's integral somehow reminds me of this complex analysis integral from the chat crawling, and I then noticed there are some similarities, which is then confirmed after a change of variables
I am not sure what I can make use of these results through, so I just decided to lay them around... (though I agree I need to massively shorten spacings and unecessary typing else it is going to flood the chat)
I think it is safe to say that my mind is currently (involuntarily?) stuck at thinking about ordinals and integrals, and I guess it will be more or less like that until I computed the main results in the ordinal collapse function
...hmmm, I think I might trying digging MSE to see whether there's a theorem for conditions on integrals that can only be done by complex analysis methods. Such No-go theorem will greatly help characterising complex and real integrals
Complex analysis can only do definite integrals, and it's usually only possible if either side goes to $\infty$, preferrably both.
And we use complex analysis when the function doesn't have a primitive in terms of elementary functions, like in the case of that integrand with exponentials
Which should also be a hint that your substitutions won't get you anywhere - one function has a primitive in terms of elementary functions, the other one doesn't.
I see, hmm in that case, that's weird, because I get $-\frac{(1-u)^{a-1}}{u}$, which is easily reducible to a simple polynomial if $a$ is an integer. Perhaps I have made a careless mistake somewhere...
Hello,I am trying to finding the minimum moves to make a string palindrome I have come up with the following formula is it correct? We should make a matrix to keep the results .if first=last characher then $T(i,j)= T(i-1,j-1) $ and if first is different from last then $ T(i,j) = min ( T(i-1,j) , T(i,j-1) ) $
O wait, I miss out $1>a>0$, that explains it. If $a$ is not an integer, the $(1-u)^{a-1}$ cannot be expanded into a polynomial. Instead, one will get an infinite power series
thus consistent with the nonelementary nature of this integral
@IvanIvanov what is i,j, are they two different letters in an input string?
It can be looked as if the first is equal to the last then we look at the string from 2 to the end-1 .If they are different the we might remove the first or the last character and search in depth again ,and since we look at the minimum steps I add 1
How does the +1 work, so clearly here I have the substring accdbda or ccdbdab. Sicne you said T[i,j] is a nxn matrix, I am guessing +1 is the identity matrix thus has the effect of translating the whole string to the left from 0-7 to 1-8?
Hello,I am trying to finding the minimum moves to make a string palindrome I have come up with the following formula is it correct? We should make a matrix to keep the results .if first=last characher then $T(i,j)= T(i-1,j-1) $ and if first is different from last then $ T(i,j) = min ( T(i-1,j) , T(i,j-1) ) $
You left out a lot of context; I'm tempted to think you're just cherry-picking bits of that conversation to make others look bad while ignoring your own behavior and how it contributed to the situation.
Here's something I would consider a really critical bit of context:
(for casual readers: "...
