# Most Popular

1500 questions

**57**

votes

**4**answers

### Are functions of independent variables also independent?

It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed.
If I have two independent random variables, $X_1$ and $X_2$, then I define two other random variables $Y_1$ and $Y_2$, where $Y_1$…

LLS

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**57**

votes

**5**answers

### Picking random points in the volume of sphere with uniform probability

I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of the sphere?
Since I'm unable to answer my own…

MHK

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**57**

votes

**3**answers

### What does "∈" mean?

I have started seeing the "∈" symbol in math. What exactly does it mean?
I have tried googling it but google takes the symbol out of the search.

Locke

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**57**

votes

**9**answers

### Best Algebraic Topology book/Alternative to Allen Hatcher free book?

Allen Hatcher seems impossible and this is set as the course text?
So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online.
Any good intro to Algebraic topology books?
I can…

simplicity

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**57**

votes

**8**answers

### How is the Gödel's Completeness Theorem not a tautology?

As a physicist trying to understand the foundations of modern mathematics (in particular Model Theory) $-$ I have a hard time coping with the border between syntax and semantics. I believe a lot would become clearer for me, if I stated what I think…

Lurco

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**57**

votes

**8**answers

### Is it possible that "A counter-example exists but it cannot be found"

Then otherwise the sentence "It is not possible for someone to find a counter-example" would be a proof.
I mean, are there some hypotheses that are false but the counter-example is somewhere we cannot find even if we have super computers.
Sorry,…

ThePortakal

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**57**

votes

**12**answers

### Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $

I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges, but what about the alternating harmonic series…

Isaac

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**57**

votes

**4**answers

### Would a proof to the Riemann Hypothesis affect security?

If a solution was found to the Riemann Hypothesis, would it have any effect on the security of things such as RSA protection? Would it make cracking large numbers easier?

Freeman

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**57**

votes

**1**answer

### What's the value of this Viète-style product involving the golden ratio?

One way of looking at the Viète (Viete?) product
$${2\over\pi} = {\sqrt{2}\over 2}\times{\sqrt{2+\sqrt{2}}\over 2}\times{\sqrt{2+\sqrt{2+\sqrt{2}}}\over 2}\times\dots$$
is as the infinite product of a series of successive 'approximations' to 2,…

Steven Stadnicki

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**57**

votes

**1**answer

### Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

It is well-known that the function
$$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$
is smooth everywhere, yet not analytic at $x = 0$. In particular, its Taylor series exists there, but it equals $0 + 0x +…

The_Sympathizer

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**57**

votes

**5**answers

### Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.

rupa

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**57**

votes

**12**answers

### I almost quit self-studying mathematics, but should I continue?

Before I move on to the main idea of this post, I need to tell you some background information about myself. Hopefully, it proves useful for you in giving me advice. I'm a 16 year old high school student who just recently got interested in…

user93971

**57**

votes

**6**answers

### Do harmonic numbers have a “closed-form” expression?

One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq n} \frac{1}{k}$, and failed. But should we…

Srivatsan

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**57**

votes

**3**answers

### Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and geometric motivation for groups, except hastily…

ante.ceperic

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**57**

votes

**2**answers

### Wild automorphisms of the complex numbers

I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could explain in the simplest possible way (please) how…

Gerard

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