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1500 questions
94
votes
14 answers
I roll a die repeatedly until I get 6, and then count the number of 3s I got. What's my expected number of 3s?
Consider the following experiment. I roll a die repeatedly until the die returns 6, then I count the number of times 3 appeared in the random variable $X$. What is $E[X]$?
Thoughts: I expect to roll the die 6 times before 6 appears (this part is…
nettle
- 1,269
94
votes
7 answers
Intuition in algebra?
My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) MacLane and Birkhoff's Algebra on my own.
The…
Michael Benfield
- 1,835
94
votes
5 answers
Root Calculation by Hand
Is it possible to calculate and find the solution of $ \; \large{105^{1/5}} \; $ without using a calculator? Could someone show me how to do that, please?
Well, when I use a Casio scientific calculator, I get this answer: $105^{1/5}\approx "…
Kerim Atasoy
- 859
94
votes
4 answers
A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum
Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma\left(A^T\right)$ where $A^T$ is the transpose matrix of $A$.
Zizo
- 1,841
94
votes
1 answer
Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc... & Representation Theory of Special Functions
Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question here. The gist of the theory is as follows:
The…
bolbteppa
- 4,389
94
votes
8 answers
A map is continuous if and only if for every set, the image of closure is contained in the closure of image
As a part of self study, I am trying to prove the following statement:
Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$, where…
Holdsworth88
- 8,818
94
votes
8 answers
Why are groups more important than semigroups?
This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me for quite some time and I'm not entirely certain…
user23211
93
votes
10 answers
Surprise exam paradox?
I just remembered about a problem/paradox I read years ago in the fun section of the newspaper, which has had me wondering often times. The problem is as follows:
A maths teacher says to the class that during the year he'll give a surprise exam, so…
sashoalm
- 1,379
93
votes
1 answer
The Right Triangle Game
I am looking for a deeper understanding, particularly the optimum strategy and the maximum score as a function of grid size, of the following (single-player) game played with an $n$ by $m$ grid:
($6 \times 6$ example)
Rules
Start with a grid made…
user139000
93
votes
2 answers
Sheaf cohomology: what is it and where can I learn it?
As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter.
Question 1. What is sheaf cohomology? I have a…
Zhen Lin
- 90,111
93
votes
5 answers
What's going on with "compact implies sequentially compact"?
I've seen both counterexamples and proofs to "compact implies sequentially compact", and I'm not sure what's going on.
Apparently there are compact spaces which are not sequentially compact; quick googling and wikipedia checks will turn up examples…
matt
- 2,125
93
votes
0 answers
Probability for an $n\times n$ matrix to have only real eigenvalues
Let $A$ be an $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has only real eigenvalues?
The answer cannot be $0$ or $1$, since the set of matrices with distinct real…
Exodd
- 10,844
93
votes
2 answers
Do most numbers have exactly $3$ prime factors?
In this question I plotted the number of numbers with $n$ prime factors. It appears that the further out on the number line you go, the number of numbers with $3$ prime factors get ahead more and more.
The charts show the number of numbers with…
SmallestUncomputableNumber
- 1,631
- 10
- 15
93
votes
4 answers
A continuous, nowhere differentiable but invertible function?
I am aware of a few example of continuous, nowhere differentiable functions. The most famous is perhaps the Weierstrass functions
$$W(t)=\sum_k^{\infty} a^k\cos\left(b^k t\right)$$
but there are other examples, like the van der Waerden functions, or…
levitopher
- 2,655
93
votes
11 answers
Results that came out of nowhere.
Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. Nevertheless, there is this big romantic idea in pop…
Alexander Gruber
- 26,963