user149792
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Why is the tangent bundle orientable?
25 votes

Sorry to resurrect, but we leave a $($detailed$)$ proof here that $TM$ has the structure of an oriented $2n$-manifold, even if the $n$-manifold $M$ is non-orientable. Let $\{(U_\alpha, \phi_\alpha)\}...

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Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space
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19 votes

Sorry for reviving such an old problem... Anyways, what is important here is that $\text{arctan}$ is a bijection from $\mathbb{R}$ to $( -\pi/2, \pi/2 )$, and it is an isometry if we give $\left(-\pi/...

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Embeddings are precisely proper injective immersions.
16 votes

This is not quite true, since, for example, the open unit disk is embedded in $\mathbb{R}^2$ by the inclusion, and the inverse image of the closed unit disk is the open unit disk. What we will show is ...

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Is an open $n$-ball homeomorphic to $\mathbb{R}^{n}$?
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16 votes

We will show when we have a normed space $E$ and $r>0$, $E$ and its subset $B_r(0) = \{x \text{ }|\text{ }\|x\| < r\}$ are homeomorphic. $($Wikipedia gives a pretty okay primer on normed spaces, ...

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Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?
12 votes

Sorry to resurrect such an old post, but I would like to supply a proof of this without invoking Jordan normal form or Schur decomposition. So we want to show the following statement. Theorem. Let $...

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What are examples of parallelizable manifolds, and why does parallelizable correspond to $TM$ being trivial?
11 votes

Some good $($introductory$)$ sources, in general, for all things smooth manifolds: Topology from the Differentiable Viewpoint, by Milnor Differential Topology, by Guillemin-Pollack Differential Forms ...

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Show $f$ is analytic if $f^8$ is analytic
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11 votes

Suppose $f(z) \neq 0$. Then $${{f(z+h)^8 - f(z)^8}\over{h}} = {{f(z+h) - f(z)}\over{h}}\left(f(z+h)^7 + f(z+h)^6f(z) + \dots + f(z)^7\right).$$ Using continuity, we see the expression in parentheses ...

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Factoring $x^n+n$ over $\mathbb{Z}[x]$
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10 votes

It turns out $p_n(x)$ is reducible only when it is "obvious." More precisely, polynomial $p_n(x)$ is reducible over $\mathbb{Z}$ if and only if at least one of the following two conditions hold. ...

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Show that the outward unit normal is smooth vector field
8 votes

We know that if $X \subset \mathbb{R}^N$ is a $k$-dimensional manifold, then $T_x(X)$ is a $k$-dimensional linear subspace of $\mathbb{R}^N$. Moreover, $\partial X$ is a $(k-1)$-dimensional manifold ...

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Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series
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8 votes

1. We first talk a bit about the underlying method of the proof. We want look for $(I + \epsilon T)^{-1}$ as an infinite series, similar to the way we would expand $1/(1+x)$ into a power series for $|...

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Abstract Nonsingular Curves
7 votes

Sorry for resurrecting such an old post, but Section §1.6 of Hartshorne annoys me in the sense that the concept of "abstract nonsingular curve" is not used very much (if at all) in algebraic geometry. ...

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Variant of strong approximation.
7 votes

Let $K$ be a global field with a distinguished place $w$. Let $\mathbb{A}^w$ be the restricted direct product of all the $K_{v\ne w}$ with respect to the subgroups $\mathcal{O}_v$ whenever $v$ is ...

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Congruent quadrilaterals in a tri-colored $72$-gon
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7 votes

This is essentially Problem #7 from Tournament of the Towns, Fall 2011, Junior A-Level and Problem #2 from USAMO 2012. Tournament of the Towns, Fall 2011, Junior A-Level, Problem #7. Each vertex of ...

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How do ideal quotients behave with respect to localization?
7 votes

Going off user26857's comment, we provide a counterexample for Proposition 3.14 in the case that $M$ is not finitely generated. Hopefully you can use this to construct a counterexample for Corollary 3....

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Measure - exercise 22 from Folland
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7 votes

Have $\mu$ be a $\sigma$-finite measure on the $\sigma$-algebra $\mathscr{M}$. Have $\mu^*$ be the outer measure induced by $\mu$, $\mathscr{M}^*$ the $\mu^*$-measurable sets and $\overline{\mu}$ the ...

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Find all positive integers $n$ such that $\phi(n) + \tau(n) > n$.
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7 votes

The answer is $n = 1$, $n = 4$, or $n$ is any prime number. Let $S$ be the set of integers less than or equal to $n$ that are relatively prime to $n$, and let $T$ be the set of integers that are ...

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Subring of $M_7(\mathbb{Z}_2)$ isomorphic to $\mathbb{F}_{128}$?
6 votes

The first sentence of the problem's statement says that $(\mathbb{Z}/2\mathbb{Z})^7$ is a simple $A$-module; hence, by Schur's Lemma, the ring $A' = \text{End}_A((\mathbb{Z}/2\mathbb{Z})^7)$ is a skew ...

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Every countably generated sigma algebra is generated by a random variable
6 votes

In the interests of completeness, we provide a complete solution. Lemma. Let $X_1, X_2, \dots$ be a sequence of Bernoulli random variables $($not necessarily independent, not necessarily with same $p$...

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Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?
6 votes

We claim that given the conditions set forth in the problem, $\text{Ind}_{Z(G)}^G \rho$ is always not irreducible. Set $Z = Z(G)$. Let $W$ be an irreducible $Z$-representation. We would like to ...

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Every immersion is locally an embedding?
6 votes

Pick coordinate charts around $p \in M$ and $f(p) \in N$; assume that $M$ and $N$ are of dimension $m$, $n$ respectively, and $($by continuity$)$ that the points of $M$ corresponding to our chart ...

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Irreducible representations of $S_3$
5 votes

We have $$x^3 = y^2 = (xy)^2 = 1,$$ so $$yx = (yx)^{-1} = x^2y,\text{ }yx^2 = x^2yx = x^4y = xy.$$ a. Fix a nonzero $u\in V$, and let $$v = u+xu+x^2u = (1+x+x^2)u,\text{ }w=u+yu = (1+y)u.$$ Since $\{1,...

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Number of roots the degree of the map?
5 votes

A simpler solution than the one of Lee Mosher is as follows. Let $z_1, \dots, z_n$ be the roots of $p$ inside the unit circle, and let $w_1, \dots, w_m$ be the roots of $p$ outside the unit circle. So$...

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Geometrically, why do line bundles have inverses with respect to the tensor product?
5 votes

If I had to give a tl;dr to my incoherent drivel above, it would probably be as follows. $V \otimes V^*$ has a canonical basis if and only if $V$ is $1$-dimensional. We have a circle. We draw the ...

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Laplace-de Rham operator
5 votes

b. We compute $\nabla(\alpha)$ for an arbitrary $1$-form $\alpha(x) = \sum_{i=1}^n \alpha_i(x)\,dx_i$. We have$$\partial \hspace{.5mm} d\alpha = \partial\left(\sum_{i \neq j} D_j\alpha_i\,dx_j \wedge ...

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Symmetry group of hypercube in $\mathbb{R}^4$
5 votes

First of all, it is a worthwhile exercise to identify the set $H$ of hyperfaces $($which are cubes of side length $2)$ and verify that $|H| = 8$. Choose any hyperface of $T$. This hyperface can be ...

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Open subset in Irreducible Topological Space is dense.
4 votes

Let $U$ be a nonempty open subset of an irreducible topological space $X$. Denote by $\overline{U}$ the closure of $U$ in $X$. Then $(X - U, \overline{U})$ is a decomposition of $X$. Because $X$ is ...

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No nonconstant coprime polynomials $a(t)$, $b(t)$, $c(t) \in \mathbb{C}[t]$ where $a(t)^3 + b(t)^3 = c(t)^3$.
Accepted answer
4 votes

We give a completely elementary proof simply working in the ring $\mathbb{C}[t]$ and using that it is a UFD. Suppose there are some solutions of$$a(t)^3 + b(t)^3 = c(t)^3$$in $\mathbb{C}[t]$. Choose a ...

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Visualizing a generator of $H^2(S^1 \times S^1)$
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4 votes

Here is a visualization for the regular singular homology theory with coefficients in $\mathbb{R}$. The generator for $H_2$ is going to be the cycle that is "the entire manifold" (this is called the ...

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Fundamental group of $X = \{(p, q)|p \neq −q\}\subset S^n \times S^n$
4 votes

Solution 1. Let$$g: X \to S^n, \text{ }(p, q) \mapsto p.$$We see that $g \circ f = \text{Id}_{S^n}$. Now, let $(p, q) \in X$. There is a well-defined shortest path $\lambda_{p, q}: [0, 1] \to S^n$ ...

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Inclusion of rings and induced map, fibers?
4 votes

The prime ideals of $\mathbb{R}[x]$ have the form $\langle f\rangle$, where $f$ is one of: $f(x) = 0$; $f(x) = x - \lambda$, where $\lambda \in \mathbb{R}$; $f(x) = x^2 + bx + c$, where $b, c \in \...

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