I'm a bit confused by a hint on a homework problem. Given an embedding $f: T^2 \to S^3$ and a closed 2-form on $S^3$, I want to show that $\int_{T^2} f^* \omega = 0$ and the hint is to homotope $f$ to a constant map and then use Stoke's theorem. But if we homotope $f$ to a constant map then the image of $f$ is just a single point in $S^3$ and I'm not sure how to apply Stoke's theorem to that setup
Somehow this has to become the integral of $d \omega$ over $S^3$ or some part of $S^3$ so the thing I homotope to should be the boundary of that corresponding thing.
Maybe I can think of the point as being the boundary of some $S^1 - \{p\}$ sitting in $S^3$
Oh I think I need that the form is closed because if $f \sim g$ then $f^*$ and $g^*$ are equal as maps.... on the appropriate DeRham Cohomology which only include closed forms
We've got a few people working with Soug, like he's teaching literacy in PDE but since enough (3-4) of us were thinking of a reading course with him, he was like yeah do literacy and I'll just focus on you guys
It seems that recently the activity in functional analysis chatroom is bigger than before. And there is no shortage of users who ask questions there. It would be nice to get more users with good knowledge of functional analysis who could help with the question which remained unanswered.
Of course, there are many other chat rooms which could have more activity that they currently have: List of chatrooms
@Sadhu Maybe posting the question here in chat in the format which includes the title might increase the chances that somebody have a look.
If the title is included, users in chat can have at least a basic idea what the question is about without having to click the link. And if they see that it's a topic they are interested in, they might have a look.
In this case you might also try to make the title more descriptive. The phrase "following expression" does not say much about the question, maybe something like "modular computation with Fibonacci numbers and factorials" would be better.
Or you could simply include the expression $(\frac{\sum_{i=N}^M F_i\cdot i!}{K}) \mod p$ in the title.
It's probably too algebraic for your taste, but group cohomology is really dope. You can do really awesome stuff, in particular when the group is finite
When the group is finite, you can patch together homology and cohomology to get one long exact sequence which is infinite in both directions
The construction uses a particular description of the usual $H^0$ and $H_0$ which works only for finite groups
There's this awesome thing about Tate cohomology called dimension shifting $\hat{H}^n$ is the tate cohomology which is a regular cohomology group for $n\geq 1$ and a regular homology group (i.e. a $\operatorname{Tor}$ group). Then for any $G$-module $A$, you can define other $G$-modules $A^{\star}$ and $A_{\star}$ such that $\hat{H}^{n+1}(G,A)=\hat{H}^n(G,A^\star)$ and $\hat{H}^{n-1}(G,A) = \hat{H}^n(G,A_\star)$
This means that if you want to prove something for all $G$-modules, you can just say without loss of generality $n=0$ (and for $n=0$ it's often obvious)
I am hoping someone can corroborate my chain of reasoning. Let $P=[p_{ij}] \in M_n(\Bbb{C})$ be a gram matrix. The there exists a inner product space $(V,\langle, \rangle_V)$, which is possibly infinite dimensional, and vectors $v_1,...,v_n \in V$ such that $p_{ij} = \langle v_i, v_j \rangle_V$. ...
I am confused by a passage in the wikipedia article Universal $C^*$-algebra, particularly these two sentences
This means that depending on the generators and relations, a universal C*-algebra may not exist. In particular, free C*-algebras do not exist.
My confusion arises in connection with...
What are the conditions in terms of $h,k$ to be able to draw three
distinct normals to the parabola $y^2= 4ax$ ?
Normal to the parabola from $(h,k)$ is given by: $am^3 +(2a-h)m+k=0$. This equation can yield three distinct slopes $m_1,m_2,m_3$ if $\Delta>0$ (source).
The $\Delta$ of the eq...
Let $\pi_1(x)$ be the number of primes less than equal to $x$ which are $1 \pmod 4$, and similarly define $\pi_3(x)$. Littlewood proved that the inequality between these switches back and forth infinitely. Can someone provide a link for that one, or any modern source discussing this?
Alright so looks like you have so difficulties about factoring, it's a fairly essential concept if you want to solve your problem so I suggest you get more familiar with it before coming back to your problem
What are the conditions in terms of $h,k$ to be able to draw three
distinct normals to the parabola $y^2= 4ax$ ?
Normal to the parabola from $(h,k)$ is given by: $am^3 +(2a-h)m+k=0$. This equation can yield three distinct slopes $m_1,m_2,m_3$ if $\Delta>0$ (source).
The $\Delta$ of the eq...
