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1500 questions
87
votes
7 answers
Poisson Distribution of sum of two random independent variables $X$, $Y$
$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{P}( \lambda + \mu)$ but I don't understand how to…
user31280
87
votes
1 answer
$\frac{1}{n}$ as a difference of Egyptian fractions with all denominators $
Is there a good characterization of the set $S$ of positive integers $n$ such that $\frac{1}{n}$ can be represented as a difference of Egyptian fractions with all denominators $< n$? For example, $44 \in S$ because $$ \dfrac{1}{44} = \left(…
Robert Israel
- 473,652
87
votes
5 answers
Do Diagonal Matrices Always Commute?
Let $A$ be an $n \times n$ matrix and let $\Lambda$ be an $n \times n$ diagonal matrix. Is it always the case that $A\Lambda = \Lambda A$ ? If not, when is it the case that $A \Lambda = \Lambda A$ ? If we restrict the diagonal entries of $\Lambda$…
Jeff Scott
- 879
87
votes
5 answers
Is there a known mathematical equation to find the nth prime?
I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
thyrgle
87
votes
5 answers
Limit of the nested radical $x_{n+1} = \sqrt{c+x_n}$
(Fitzpatrick Advanced Calculus 2e , Sec. 2.4 #12) For $c \gt 0$ , consider the quadratic equation $x^2 - x - c = 0, x > 0$ . Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been…
cnuulhu
- 1,015
87
votes
5 answers
Why is a circle in a plane surrounded by 6 other circles?
When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm looking for an answer deeper than "there are…
John Smith
- 2,290
86
votes
10 answers
Best Algebraic Geometry text book? (other than Hartshorne)
Lifted from Mathoverflow : I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion per answer please.…
user218
86
votes
1 answer
Can one deduce Liouville's theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?
As you probably know, the classical proof of the non-emptiness of the spectrum for an element $x$ in a general Banach algebra over $\mathbb{C}$ can be proven quite easily using Liouville's theorem in complex analysis: every bounded, entire function…
Mark
- 6,002
86
votes
8 answers
The myth of no prime formula?
Terence Tao claims : For instance, we have an exact formula for the $n^\text{th}$ square number – it is $n^2$ – but we do not have a (useful) exact formula for the $n^\text{th}$ prime number $p_n$! “God may not play dice with the universe, but…
user177691
- 893
86
votes
2 answers
Cardinality of Borel sigma algebra
It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, and actually hard to find a proof of it. Can…
Syang Chen
- 3,576
86
votes
3 answers
What algorithm is used by computers to calculate logarithms?
I would like to know how logarithms are calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that logarithms are calculated directly from the hardware. So the question is: what…
zar
- 4,776
86
votes
1 answer
How to solve fifth-degree equations by elliptic functions?
From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$ ) by elliptic functions. How to get that? Update: How to transform a general higher degree five or higher…
ziang chen
- 8,009
86
votes
3 answers
Is there a function with a removable discontinuity at every point?
If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions to get students to think more deeply about the…
Pete L. Clark
- 101,276
86
votes
3 answers
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
Consider the sequence defined as $x_1 = 1$ $x_{n+1} = \sin x_n$ I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't too happy about. (I think I did this by showing that…
Aryabhata
- 84,362
86
votes
11 answers
Proof of triangle inequality
I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
ivan
- 3,405