gnometorule
  • Member for 9 years, 11 months
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11 answers
26 votes
6k views
Gap year to study math
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26 votes

You're self-motivated; the kids you mention as a general rule pushed by parents trying to compensate for what they feel is a lack in their lives. Some will excel; some will flame out as a lot that ...

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2 answers
18 votes
8k views
Fundamental group is abelian iff the fundamental group isomorphisms (a-hat) coincide
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20 votes

Hint: note that $f \ast \alpha  := \gamma$ is a path from $x$ to $y$, and calculate $\bar{\gamma}$ by its definition. Then you are given:   $\hat{\gamma} ([g])= \hat{\alpha} ([g])$,   which should ...

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7 answers
46 votes
2k views
Struggling with "technique-based" mathematics, can people relate to this? And what, if anything, can be done about it?
17 votes

I can actually relate fairly well to this. Please don't take this the wrong way, but my main advice would be: Stop worrying! By this I don't mean that you should not be concerned about your grades ...

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3 answers
3 votes
13k views
Computing limits which involve square roots, such as $\sqrt{n^2+n}-n$
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15 votes

$\begin{align} \sqrt{n^2 + n} - n & = (\sqrt{n^2 + n} - n) \cdot \frac{\sqrt{n^2 + n} + n}{\sqrt{n^2 + n} +n} \\ & = \frac{n^2+n-n^2}{\sqrt{n^2 + n} + n} \\ & = \frac{1}{\frac{\sqrt{n^2 + ...

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3 answers
3 votes
6k views
Inverse of sum of two functions
15 votes

Let $f(x) = x, g(x) = -x$, both obviously invertible. Then $(f+g)(x) == 0$, which is not invertible.

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4 answers
32 votes
28k views
I want to get good at math, any good book suggestions?
13 votes

The classic book reference is probably Polya's "How to solve it". Maybe have a look at it.

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3 answers
6 votes
3k views
Complete induction: $\sum^n_{i=1}\frac{1}{(2i-1)(2i+1)}=\frac{n}{2n+1}$
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9 votes

For a full solution, proceed like this: $n=1$: $$\sum_{i=1}^1 \frac{1}{(2i-1)(2i+1)} = \frac{1}{(2-1)(2+1)} = \frac{1}{3} = \frac{1}{2 \cdot 1 +1},$$ so it holds for $n=1$. Assume next that it ...

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2 answers
6 votes
3k views
How to show that $[0,1]^{\omega}$ is not locally compact in the uniform topology?
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9 votes

Use the equivalent definition of local compactness: if $x \in U \subset C$, $U$ open, $C$ compact, then there must be a ball $B_{\epsilon}(x)$ whose closure $\bar{B}$ is contained in $U$. Apply this ...

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28 answers
259 votes
43k views
Too old to start math
8 votes

This popped as a newsletter question, and it seems appropriate to share personal experience. It is true that most significant math research seems to be done before 30. I remember Hirzebruch from my ...

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1 answers
0 votes
212 views
Odd matrix notation: $||\mathbf{A}||$
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8 votes

It could be a determinant, but more likely it is a matrix norm (in particular engineering rings a bell as it could relate to some numerical analysis; but I don't know of course for sure).

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2 answers
8 votes
2k views
Connection between connected and compact spaces?
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7 votes

There are no such connections.

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2 answers
15 votes
4k views
Prerequisites for Algebraic Topology
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7 votes

While Munkres is arguably self-contained, there are further topics (on top of the above mentioned) you'll soon run into: (1) Free Groups (abelian and not). (2) Commutator subgroups, and (from ...

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3 answers
1 votes
116 views
Dividing polynomials
6 votes

Yes, the 1 cannot vanish. In your case, using the distributive law, $$\begin{align} \frac{9x + 8x^2 +1}{x} & = \frac{9x}{x} + \frac{8x^2}{x} + \frac{1}{x} \\ & = 9 + 8x + \frac{1}{x} \\ & \...

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4 answers
16 votes
1k views
How did the ancients view *infinitesimals*?
6 votes

Your question got me reading (googling). This paper: Leibniz's Infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russel and beyond (Katz & Sherry, ...

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7 answers
55 votes
17k views
What does it take to get a job at a top 50 math program in the U.S.?
5 votes

I can add anecdotal evidence for my former program, which wasn't math proper but attracted a fair number of math undergrads, if not people with Ph.D.s in physics prior to even entering this particular ...

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1 answers
4 votes
4k views
Expected value of $xx^{T}$ for multidimensional Gaussian
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5 votes

I think your central question is how to write $z$ as $$z = \sum_{j=1}^D (u_j^t z) \, u_j = \sum_{j=1}^D \langle u_j, z \rangle u_j \quad (1)$$ This, you can see as follows. As you have a complete, ...

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2 answers
2 votes
3k views
Lower limit topology is a Hausdorff space $T_2$
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5 votes

Hint: Assume without loss of generality $x < y$. You need to find one interval $I_1 = [a, b)$ in which $x$ lies, and one $I_2 = [c, d)$ in which $y$ lies, such that $a < b \leq c < d.$ These ...

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5 answers
6 votes
4k views
If G is a finite group with an even number of elements, then binary product of two distinct elements is identity.
5 votes

Elementary way: For $g$ simply take the identity $e$. To find another, assume that each element $h$ has an inverse $h^{-1}$ that is not $h$ ($h \neq h^{-1}$). Summing the elements $\{h, h^{-1} \}$ ...

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5 answers
6 votes
2k views
How did Newton invent calculus before the construction of the real numbers?
5 votes

In Leibnitz' case (who co-invented) by (quite wrongly) assuming that the Greek atomos (indivisible) really was that: a smallest indivisible part he called "Monaden", and which also explains that his ...

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2 answers
1 votes
170 views
How is this equation called?
5 votes

It's the binomial coefficient, and you should google combination and permutation as a starter.

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4 answers
2 votes
209 views
the relationship between $f^{-1}(x)$ and $x$
4 votes

Apply your condition $f(y) < y$ to $y:=f^{-1}(x)$, and you immediately get $$x = f(f^{-1}(x))<f^{-1}(x)$$

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4 answers
2 votes
160 views
Can the definition of continuity be said both of these ways?
4 votes

Let $$f(x) := \begin{cases} 1, & x\in \mathbb{Q} \\ 0, & x\in \mathbb{R} - \mathbb{Q}, \\ \end{cases}$$ a very discontinuous function. Then $\epsilon := 2$ will do, for any $x, t, \delta >0$...

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2 answers
2 votes
131 views
Let $A \subset \mathbb{R}$ be connected and $f:A\to \mathbb{R}$ be continuous. Show $f(A)$ is connnected
4 votes

If you use $f^{-1}(U \cup V) = f^{-1}(U) \cup f^{-1}(V)$, which is always true, your idea will show a contradiction in the next step. Do you see it?

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3 answers
3 votes
837 views
Wouldn't each addition take time $O(n)$?
3 votes

To reconcile this with the general case, note that each block matrix addition adds two matrices of dimension $\frac{n}{2} \times \frac{n}{2}$, so of $\frac{n^2}{4}$ elements, which is the complexity ...

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1 answers
3 votes
229 views
Cauchy inequality proof
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3 votes

No, you're entirely right, and you probably just copied it down incorrectly. Here's the one point you're currently missing: $$<x, x> = ||x||^2,$$ so if it weren't the square as you say it ...

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3 answers
1 votes
678 views
Relatively Prime
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3 votes

If $$1 = u(n/d) + v(s/d) := ua + vb,$$ and for some integer $c$ you have $c|a$ and $c|b$, then obviously $$c|ua+vb = 1.$$ Edit (in more detail per a former comment): If $c|a$, then $a = cx$ (not $c ...

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3 answers
3 votes
2k views
find the derivative of $e^{-2t} \cos(4t)$
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3 votes

It should be $$ e^{-2t}(-\sin(4t) \cdot 4)+ \cos(4t)(e^{-2t}(-2)).$$ Your logic is right, but $$\frac{d}{dt}(\cos(4t)) = -4\sin(4t),$$ not $-4\sin(4t)\cos(4t).$

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3 answers
3 votes
474 views
Show that $G/C$ is abelian if $C$ is normal closure of subgroup generated by elements $aba^{-1}b^{-1}$
3 votes

If $x, y \in G$, then $xyx^{-1}y^{-1} \in C$. Hence, $$Cxyx^{-1}y^{-1} = C,$$ or $$Cxy = Cyx,$$ which means that $G/C$ is abelian. Edit: corrected a typo pointed out by Loki Clock.

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1 answers
8 votes
1k views
Ito integral almost sure and $L^2$ limit
3 votes

The key advantage of Ito integrals is that they are martingales. For this, a.s. convergence would not be enough in general. You don't quite need $\mathcal{L^2}$ convergence, but something almost as ...

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2 answers
3 votes
1k views
Prerequisites for learning kalman filtering
3 votes

If you just want to implement Kalman Filtering, and by this mean 'coding it up' and already have the coding skills, nothing really other than a good reference book. Personally, I learned Kalman ...

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