Sina
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Best Algebraic Topology book/Alternative to Allen Hatcher free book?
10 votes

If you want a more rigorous book with geometric motivation I reccomend John M. Lee`s topological manifolds where he does a lot of stuff on covering spaces homologies and cohomologies. As a supplement ...

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Pushforward of Lie Bracket
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7 votes

To do this computation in coordinates without using functions and points you have to adopt the physicist way of writing things which is messy and unplesant :-) However chain rule takes care of all the ...

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Volume form on Hamiltonian level surface
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3 votes

Given a hypersurface $S$ in a manifold $M$ with volume form $\omega$ you can construct the volume element on $S$ induced by $\omega$ as follows: Take a field of unit normal vectors $N$ along $S$ and ...

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Is there relationship between magnitude of matrix-vector multiplication and determinant of that matrix?
3 votes

If $A$ is an invertible operator of an n dimensional vector space to itself then $$\frac{1}{det(A^{-1})^{1/n}} \leq |A|\leq \frac{|A^{-1}|^{n-1}}{det(A^{-1})}$$ This you can get by the relation $$...

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Why is the topological pressure called pressure?
3 votes

Please see, Page 55 of Prof. Oliveira's notes: http://cdsagenda5.ictp.trieste.it/askArchive.php?base=agenda&categ=a11165&id=a11165s16t18/lecture_notes There he partially answers this ...

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Wiener measure on continuous paths
2 votes

As Ian explained above, the Wiener measure of a path is indeed $0$. In general, whenever you have uncountable number of elements in your space, your probability measure can be only non-zero on a ...

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Can't we consider the curve $t\to(\gamma(t),X(\gamma(t))$ instead of the covariant derivative $\nabla_{\gamma(t)}X$?
2 votes

I was able to give a partial answer to my question, the part where I ask about the relation between linear connection on $M$ and Ehresmann connections on $TM$. It goes as follows. Given a linear ...

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Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.
2 votes

Simply because you pull back the equation $\omega(X,\cdot) = dH$ to an equation on $T^*(TM)$ using the Legendre transformation. Indeed denote $\phi_L: TM \rightarrow T^*M$ the Legendre transformation ...

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Proofs of properties of a measureable and Lebesgue integrable function
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2 votes

a) $\int f d\mu = \int_{X_a} f d\mu + \int_{(X_a)^c} f d\mu $. But $\int_{X_a} f d\mu > a\mu(X_a)$ and since $f$ is positive $\int_{(X_a)^c} f d\mu>0$. Therefore $\int f d\mu > a\mu(X_a)$ ...

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Image of $\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$
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2 votes

Let $T = \{D \in Q^{2 \times 2} | D^t = D\}$ Let $B \in im(\phi)$. Then $B = A + A^t$ so $B^t = A^t + A = B$. So $im(\phi) \subset T$. Let $A \in T$ so that $A^t = A$. Then write $A = \frac{A + A}{2}...

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KAM theory and the Ergodic hypothesis
1 votes

Ergodic hypothesis more or less states the the Hamiltonian flow restricted to each energy surface is ergodic for the Liouville measure. Now if you take an integrable Hamiltonian system with a 2n ...

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Manifolds, charts and coordinates
1 votes

By definition a coordinate system is a collection of charts $\{U_i,\phi_i\}_i$ where $U_i$ are open sets in your manifold that cover your whole manifold and $\phi_i:U_i \rightarrow \mathbb{R}^n$ are ...

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Geodesic flows and Curvature
1 votes

Okay, I found the answer for the second question in Kobayashi's book: it says isometries of a riemannian manifold (with some additional conditions) forms a lie group whose lie algebra is the space ...

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Some Questions on Construction of Wiener Measure/Wiener Process
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0 votes

Okay, I think I got the answer. For $t$ irrational, let $t_n$ be an increasing sequence of rationals converging to $t$ and define the sets $C'_n = C'(t_n,U)$. If $\gamma \in C'(t,U)$ (defined using ...

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Given a closed simple curve in $\mathbb{R}^3$, find a disk $S$ bounding $C$ so that its geodesic curvature remains the same in $S$
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0 votes

Motivated by the answer above it seems it is easy to put the answer into a form (following Cartan) which relates the Frenet frame and the Darboux frame of the curve. I assume the curve $C(t)$i s arc ...

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Show a limit of piecewise function does not exist as x tends to 0
0 votes

If the limit exists then for all $x_n \rightarrow 0$ it must be that $f(x_n) \rightarrow c$ where $c$ is the limit. But for every number $r$ in $[-1,1]$ you can find a sequence $x_n$ such that $f(x_n) ...

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Knowing the hypotenuse and the direction of the adjacent, how would I get the length of the adjacent
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0 votes

Let the left top corner of the rectangle be P5. Call a=|P5-P3| (distance between P5 and P3) and b = |P5-P1|. Then $$H^2 = b^2 + (X+a)^2$$ $$A^2 = b^2 + a^2 $$ Solve the second one for b, i.e $b^2 = ...

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