Marek
  • Member for 11 years, 2 months
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Why does this expression equal $\pi$?
14 votes

One way to see this is by noting that $$\arctan(x)' = \cos^2(\arctan(x)) = {1 \over 1 + \tan^2(\arctan(x))} = {1 \over 1 + x^2}$$ where we used that $\tan(x)' = {1 \over \cos^2(x)}$, the rule ...

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Top homology of an oriented, compact, connected smooth manifold with boundary
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12 votes

Just to complete the details of Dan's hint. The top homology of a non-compact manifold vanishes, e.g. by Poincare duality $H_n(M) \cong H^0_c(M)$ where $H^*_c(M)$ is cohomology with compact support. ...

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What is the relationship between the Boltzmann distribution and information theory?
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12 votes

In both information theory and physics there is a fundamental quantity called entropy associated with every measure. For the sake of simplicity, let me assume that the number of possible events is ...

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Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$
11 votes

This is a partial answer but too long for comment. First plug in $x = 1$ to get $$f(y + {1 \over y}) = yf(1 + y^2)$$ and $y = 1$ to get $$f(f(x) + x) = xf(x^2 + 1)$$ Observe that the RHS of both ...

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Is the inverse of a linear transformation linear as well?
10 votes

Use the relation $$T^{-1} \circ T = {\rm Id}$$ and linearity of $T$ and $\rm Id$ to obtain $$T^{-1} (a T(v) + b T(w)) = av + bw.$$ Now write $v' = T(v)$ and $w' = T(w)$. We get $$T^{-1} (a v' + b w') =...

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Proving $1 > 0$ using only the field axioms and order axioms
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10 votes

Suppose $1 < 0$. Adding $(-1)$ to both sides we'd also have $0 < -1$ (addition axiom). But if $0 < a$ then it must also hold that $0 < a^2$ (multiplication axiom). For $a = -1$ this means $...

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Integral question: $\displaystyle\int \frac{x^{n-2}}{(1 + x)^n} {\rm d}x$
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9 votes

$$\int {x^{n-2} \over (1+x)^n} {\rm d} x = \int (1+x)^{-2} \left({x \over 1+x}\right)^{n-2} {\rm d} x =\int y^{n-2} {\rm d} y$$ using the substitution $y = {x \over 1 + x}$.

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concrete examples of tangent bundles of smooth manifolds for standard spaces
9 votes

I think operations other than $\times$ are irrelevant here because bundles are locally product structures. So let's restrict just to this case. Then you are in fact asking whether the tangent bundle ...

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Poincare dual of unit circle
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8 votes

EDIT 2: Oh, I think I finally understood your reasoning. You argued as follows: assume $\eta_S$ vanishes. Then the integral also has to always vanish. But for your $\omega$ it does not. The problem ...

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Please help me, Group Theory. Prove $b^{33}=e$.
8 votes

$$b^{49} = (b^7)^7 = (ab^4a)^7 = a (b^7)^4 a = a (a b^4 a)^4 a = b^{16}.$$

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Riemann surface with punctures corresponds to a hyperbolic surface with cusps
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8 votes

First suppose that $n=0$ on a genus $g \geq 2$ surface. Then the surface can be endowed with a hyperbolic metric (this is essentially the uniformization theorem). Note that $g \geq 2$ is necessary, e....

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Projection, canonical immersion/submersion - are they equivalent, and are they open maps?
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7 votes

Well, projections are open since basic open sets in the product topology are $U \times V$ with $U$ and $V$ open, so projecting down leaves you with $U$ and you're done. Now, since submersion is ...

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Morphisms in the category of natural transformations?
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7 votes

They are not uninteresting. Just look up the term n-category (e.g. Baez's introduction). Nevertheless, it's true that functors and natural transformations already suffice for most common ideas and ...

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Why is the fundamental group of the projective plane $C_{2}$?
6 votes

Consider the map $z \mapsto z^2$ of the unit circle in the complex plane. This gives us a fibration $S^1 \to S^1$ of degree two. If you travel around the target circle once then this will take you to $...

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Inner product of De Rham cohomology classes
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6 votes

No, this is not possible because the integration depends heavily on the representative you choose. Here's an illustration: take a circle $S^1$ and a bump form $b_1$ around $1$ (I am thinking of the ...

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Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value?
6 votes

The short answer is no. Take constant function $f(x) \equiv C \in \mathbb{C}$. Disregarding normalization, we have $\hat{f} = C \delta$ (in the sense of distributions). Clearly, there is no way to ...

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"Proof" that $g(t) = (t,t,t,...)$ is not continuous with uniform topology
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5 votes

$B(x,r)$ is not open in uniform topology. For example take $y = (1, 1/2, 1/3, \dots)$. Then $z := (1, 1, 1, \dots) \in B(y, 1)$ but clearly $B(z, \epsilon) \not \subset B(y, 1)$ for every $\epsilon &...

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intersection form of $CP^2$
5 votes

All of these operations are only up to sign. Once you pick the sign though, everything is fixed. It is similar to integration: you pick a generator $\omega \in H^n(M, \mathbb R)$ and this gives you a ...

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Connection between Euler characteristic and degree of the Gauss map
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5 votes

I think you are just mixing up two definitions of Gauss map. One is intrinsic to the manifold (and is equal half of the Euler characteristic) while the other depends on the embedding into a Euclidean ...

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Multiple Choice question about a continuous function
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5 votes

EDIT: upon your request, I'll add a further discussion. I'll start with (B), which is both true and also harder than the rest of the cases. Note that after the substitution I mentioned we have $$n \...

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On order of elements of a infinite group
5 votes

No, we can't. Take for example the modular group. This group can be generated by elements $S^2 = 1$ and $(ST)^3 = 1$. Yet $S(ST) = S^2 T = T$ is a translation which generates (a group isomorphic to) $\...

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Partial differential equations in "pure mathematics"
5 votes

Calabi conjecture stated that on compact Kähler manifold there exists a unique Kähler metric with prescribed Ricci form. This problem can be reformulated in terms of a (non-linear) PDE and has been ...

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Introductory texts on manifolds
5 votes

I enjoyed Fecko's Differential Geometry and Lie Groups for Physicists. It doesn't contain complete bottom-up theory building and omits hard proofs but it is a very neat general introduction to the ...

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Why isn't $SO(3)\times SO(3)$ isomorphic to $SO(4)$?
5 votes

On the level of Lie algebras we have that ${\mathfrak {so}}(n)$ are just antisymmetric matrices $n \times n$. It turns out that the six-dimensional space of such $4\times 4$ matrices decomposes into ...

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How to find an explicit formula from a generating function
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5 votes

If you know what Taylor expansion is, then you should know that $S_n(s)$ in e.g.f. is just its $n$th derivative at zero. For the ordinary g.f., it's almost the same, except that you have to divide the ...

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What's the quickest way to see that the subset of a set of measure zero has measure zero?
4 votes

If a measurable set $A$ is a subset of a measurable set $B$ then $B \setminus A$ is also measurable (by axioms of the $\sigma$-algebras) and since $A$ and $B\setminus A$ are disjoint, by additivity of ...

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How a group represents the passage of time?
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4 votes

It's hard to say what Terry meant exactly unless you ask him. Nevertheless, it seems quite probable that he is talking about flows on manifolds (or more general kind of spaces but let's stick with ...

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The Hairy ball theorem and Möbius transformations
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4 votes

As Daniel mentioned in comments, there is an intimate relation between flows, integral curves and vector fields (a collection of arrows at every point of the sphere). Given a flow, one can follow ...

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What is $\pi_2 (S^2/ S^1 \vee S^1)$?
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4 votes

It's not that hard to prove that $S^2/X$ is homeomorphic to $S^2 \vee S^2 \vee S^2$ directly. If you can draw the picture with the three disks then you know where to send every point (think about the ...

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Probability question: $100$ balls with $r$ red balls.
4 votes

Consider all the possible results of drawing the balls. Each result can be represented by a binary number having 1 when we draw a red ball, so that the number of 1s in the number is $r$. Your question ...

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