In this answer I have a long equation. I've artificially broken it up into three terms t = A (B + C) so it doesn't boggle the mind when transcribing.
However when I tried to display it completely, it runs off the right margin and into the hot network questions area.
I'm using MacOS and Chrome,...
In the definition of a semi regular continued fraction, $$a_{0} + \frac{\varepsilon_{1}}{a_{1}+}\frac{\varepsilon_{2}}{a_{2}+}\frac{\varepsilon_{3}}{a_{3}+} \cdots \frac{\varepsilon_{n}}{a_{n}+} \cdots $$ what is the intution behind the property $\varepsilon_{n+1} + a_{n} \ge 1$
quick question about the minimal polynomial of a matrix. $m_A(t) = \prod_{j}(t-a_j)^{d_j}$ where $a_j$ are all the eigenvalues of $A$. Question is, why shouldn't all but one $d_j$ be $1$? Where the one that's left is the minimal index of one of the linear operators $A-a_jI$?
@MikeMiller I typed up the thing I was asking about yesterday here: physics.stackexchange.com/questions/460476/…. I've finished my problem, but there is still a question, the "add-on" at the bottom, that I'm quite interested in. Give it a look if you're interested
@N.Maneesh I have mentioned this also in the general topology chat room. Maybe somebody might notice the question there and they add something useful. (However, the room is not very active, so the likelihood is not too high.)
I surely knew some, whether I remember it is another matter, but just ask your question and someone will be able to help most likely even if it's not me
Let $X$ be a compact topological space. Suppose that for any $x, y ∈ X$ with $x \ne y$ there exist open sets $U_x$ and $U_y$ containing $x$ and $y$, respectively, such that $$U_x \cup U_y = X$$ and $$U_x \cap U_y = Ø$$Let $V ⊆ X$ be an open set. Let $x ∈ V$. Show that there exists a set $U$ which is both open and closed and $x ∈ U ⊆ V$.
user131753
Can anyone give me any hint regarding this problem?
user131753
It is easy to see that $X$ is compact Hausdorff and hence $T_4$. Which would imply that there exists an open set $U$ such that $x\in U\subseteq \overline{U}\subseteq V$. But I can't proceed from here.
Can someone please have a look at it? I posted it a few minutes ago and it didn't get any attention. Implicit differentiation to obtain expression value -
Question and attempt at the question
I've been trying to evaluate the following expression, I'm not sure if I'm heading into the right direction though. Could someone kindly let me know what the correct approach to tackle questions like this?
Let $X$ be some topological space. Given a subspace $A \subseteq X$, does there exist an open set $U$ such that $A \subseteq U$ and $U$ deformation retracts onto $A$? How about if $A$ is closed? If not, what if we further assume that $X$ Hausdorff? Heck, assume it is a metric space if necessary.
I ask this question because I think my professor is presupposing that this is true in a hint he gave me for a problem I'm working on.
If $S \subset M$ is a compact oriented $k$-submanifold of an oriented $n$-manifold $M$, then it has a Poincaré dual $\eta_S \in H_{dR,c}^{n-k}(M)$ in the de Rham cohomology group with compact support defined by $$ \int_M \eta_s \wedge \omega = \int_M i^*\omega \quad \text{ for all } \omega \in H^k_{dR}(M) $$ and a fundamental class $[S] \in H_k(M;\mathbb{R})$. Is it true that $[S] \mapsto \eta_S$ under the composition $$ H_k(S;\mathbb{R}) \xrightarrow{\text{i}_*} H_k(M;\mathbb{R}) \xrightarrow{\text{PD}} H^{n-k}_{c}(M;\mathbb{R}) \xrightarrow{\text{DR}} H_{dR,c}^{n-k}(M), $$
@user193319 it can fail even under all those assumptions, look at the bad point of the Hawaiian earring as A. It probably holds in the situation you're interested in if that's your professor's suggestion though
Let X be a metric space, and we have a collection of compact sets in X. theorem sais if the intersection of any finite subcollection is non empty then so is the intersection of all of them
Let $X := (S^1 \times S^1) \cup (D^2 \times \{x_0\}$, where $x_0 \in S^1$. Is the fundamental group of $X$ isomorphic to $\Bbb{Z}^2/\langle (m,n) \rangle$, or did I screw something up?
a screenshot would be enough, though I'm no expert for complicated integrals. Also afaik such internals of mathematica are usually hardly accessible. If they are most likely people on the "mathematica stack exchange" forum can tell you how.
So the alternative left is to do a proper literature search I fear. But for that it would be good to know the specific expression you are talking about.
In case you can specifically formulate your question you could go here mathematica.stackexchange.com alternatively you could ask for a reference with a MSE question. I know there are some incredibly able integral crunchers around, who might be able to help as well.
This question is a simplification of a previously asked question: Polylogarithmic integrals
Consider the following type of function:
\begin{equation}
\int \frac{\prod_{i=1}^N \log(x-\beta_i)}{x-\alpha} dx
\end{equation}
For the simple case of $N=2$ the integral is already highly complicated:
\b...
Let $X := (S^1 \times S^1) \cup (D^2 \times \{x_0\}$, where $x_0 \in S^1$. Is the fundamental group of $X$ isomorphic to $\Bbb{Z}^2/\langle (m,n) \rangle$, or did I screw something up?
@MikeMiller More details: Let $A_\alpha = S^1 \times S^2$, $A_\beta = D^2 \times \{x_0\}$. Then $A_\alpha \cap A_\beta \simeq S^1$, so $\pi_1(A_\alpha \cap A_\beta)$ is a cyclic group. By VK theorem, we have $\pi_1(X) = \pi_1(A_\alpha) \ast \pi_1(A_\beta)/N$, where $N$ is the normal subgroup of $\pi_1(A_\alpha) \ast \pi_1(A_\beta) \simeq \pi_1(\alpha) \simeq \Bbb{Z}^2$ generated by $i_{\alpha \beta)}(\omega) i_{\beta \alpha}(\omega)^{-1}$ with $\omega \in \pi_1(A_\alpha \cap A_\beta)$...
...continued...
But $N = \langle i_{\alpha \beta}(\omega) \rangle$ where $\omega$ is the generator of $\pi_1(A_\alpha \cap A_\beta)$.
Note that $i_{\alpha \beta} : \pi_1(A_\alpha \cap A_\beta) \to \pi_1(A_\alpha)$ is the hom. induced by the inclusion map.
$i_{\beta \alpha}$ is just the map into the trivial.
Since I don't know what $i_{\alpha \beta}$ does to $\omega$, I just said that $N$ isomorphic to some cylic group $\langle (n,m) \rangle$ in $\Bbb{Z}^2$.
You need to know that to answer this problem; in fact, you need to know what that map is to prove the claimed fact: that $\pi_1(S^1 \times S^1) \cong \Bbb Z^2$.
It is, but you're hung up because you don't know what that map is (which you should; it would be defined in the first line of whatever that proof was). The professor assumes you do.
Well, $\pi_1(S^1) \simeq \Bbb{Z}$, so let $\varphi$ denote the isomorphism. Then $\varphi \times \varphi$ would be the isomorphism from $\pi_1(S^1 \times S^1)$ to $\Bbb{Z}^2$.
But what is the map $\pi_1(S^1) \times \pi_1(S^1) \to \pi_1(S^1 \times S^1)$? That's the part we really care about - we're trying to see where $(\omega,0)$ goes in this map (using your notation). What does it do at the level of loops?
Maybe it will help if I say something more generally. There is a natural map (which is an isomorphism) $\pi_1(X, x) \times \pi_1(Y, y) \to \pi_1(X \times Y, (x,y))$. What is this map?
What does it do to a pair $\gamma: S^1 \to X$, $\gamma': S^1 \to Y$? What loop does it spit out in the product?
@abenthy That's true. $\eta_S$ basically integrates to $0$ over smooth simplices normal to $S \subset M$. By the de Rham isomorphism, it gives rise to a cochain $\psi$ representing an element of $H^{n-k}(M; \Bbb R)$ which, upon eating an $(n-k)$-simplex normal to $S$, gives $0$. The PD isomorphism $H^{n-k}(M) \to H_k(M)$ is just $\alpha \mapsto [M] \frown \alpha$.
Triangulate $M$ in a way that $S \subset M$ is a subcomplex and $(n-k)$-simplices near the zero section of a tubular neighborhood are normal to $S$ (tangent to the normal fibers). This chain $c$ obtained from the triangulation represents $[M] \in H_n(M)$. $c \frown \varphi$ then just leaves you with the simplices that lie on $S$, which is a triangulation of $S$, hence represents $[S] \in H_k(M)$.
@MikeMiller Okay. $\pi_1(S^1 \times S^1) \simeq \Bbb{Z}$ by $[\omega_{pq}] \mapsto (p,q)$, where $\omega_{pq} = (\omega_p, \omega_q)$. So $(1,0) \mapsto [\omega_{10}]$ and $(0,1) \mapsto [\omega_{01}]$ are the generators of $\pi_1(S^1 \times S^1)$. If $[\omega]$ is the generator of $\pi_1(S^1 \times S^1 \cap D^1 \times \{x_0\})$, where do I send $[\omega]$ by $i_{\alpha \beta}$? Do I send it to $[\omega_{10}]$ or $[\omega_{01}]$?
Does anyone know the term for this type of hypothesis test? H_0 : \mu_1 + \mu_2 + \mu_3 - 3 \mu_4 = 0. Or could you point me in the right direction for figuring out how to conduct it?
@topologicalmagician That doesn't even make any sense, to start with. Or is $y=y(x,u)$ a function of both variables and these are partial derivatives? Then it holds if the function is continuously twice differentiable.
An orientation of a vector space $V$ is a function $\varphi$ from the set of ordered bases of $V$ to the set $\{-1,1\}$ so that for any $B = (b_1, \cdots, b_n)$ and $C = (c_1, \cdots, c_n)$, $\varphi(B) \varphi(C) = \operatorname{sgn}([\operatorname{id}]_{BC})$
In this answer I have a long equation. I've artificially broken it up into three terms t = A (B + C) so it doesn't boggle the mind when transcribing.
However when I tried to display it completely, it runs off the right margin and into the hot network questions area.
I'm using MacOS and Chrome,...
Question and attempt at the question
I've been trying to evaluate the following expression, I'm not sure if I'm heading into the right direction though. Could someone kindly let me know what the correct approach to tackle questions like this?
This question is a simplification of a previously asked question: Polylogarithmic integrals
Consider the following type of function:
\begin{equation}
\int \frac{\prod_{i=1}^N \log(x-\beta_i)}{x-\alpha} dx
\end{equation}
For the simple case of $N=2$ the integral is already highly complicated:
\b...
