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1500 questions
87
votes
6 answers
Prove that $i^i$ is a real number
According to WolframAlpha, $i^i=e^{-\pi/2}$ but I don't know how I can prove it.
Isaac
- 1,109
87
votes
3 answers
What is the Tor functor?
I'm doing the exercises in "Introduction to commutive algebra" by Atiyah&MacDonald. In chapter two, exercises 24-26 assume knowledge of the Tor functor.
I have tried Googling the term, but I don't find any readable sources. Wikipedia's explanation…
Fredrik Meyer
- 20,228
87
votes
4 answers
Volumes of n-balls: what is so special about n=5?
The volume of an $n$-dimensional ball of radius $1$ is given by the classical formula
$$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$
For small values of $n$, we have
$$V_1=2\qquad$$
$$V_2\approx 3.14$$
$$V_3\approx 4.18$$
$$V_4\approx…
Andrey Rekalo
- 7,804
87
votes
1 answer
Is there an easy way to show which spheres can be Lie groups?
I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it involved de Rham cohomology, but I don't really know…
Aaron Mazel-Gee
- 8,039
87
votes
3 answers
Do you prove all theorems whilst studying?
When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my studies -- sequence two of Calculus -- but what I'm…
user24383
87
votes
27 answers
Easy example why complex numbers are cool
I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary.
The best example would be possible to explain rigourously and also be clearly important in a daily day sense.
I.e. complex…
htd
- 1,764
87
votes
10 answers
100 Soldiers riddle
One of my friends found this riddle.
There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75
lose a left arm, 70 lose a right arm. What is the minimum number of
soldiers losing all 4 limbs?
We can't seem to agree on a way to approach…
86
votes
9 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
3 answers
Does every Abelian group admit a ring structure?
Given some Abelian group $(G, +)$, does there always exist a binary operation $*$ such that $(G, +, *)$ is a ring? That is, $*$ is associative and distributive:
\begin{align*}
&a * (b * c) = (a*b) * c \\
&a * (b + c) = a * b + a * c \\
&(a + b) * c…
Mikko Korhonen
- 25,224
86
votes
11 answers
Does $G\cong G/H$ imply that $H$ is trivial?
Let $G$ be any group such that
$$G\cong G/H$$
where $H$ is a normal subgroup of $G$.
If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In what kind of group could I search for a…
Klaus
- 4,105
86
votes
11 answers
Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints
I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented:
The King has $1000$ wines, $1$ of which is poisoned. He needs to identify the poisoned wine as soon as possible, and…
Justin L.
- 14,532
86
votes
4 answers
If $G/Z(G)$ is cyclic, then $G$ is abelian
Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer):
Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. [If $G/Z(G)$ is cyclic with…
Altar Ego
- 5,282
86
votes
12 answers
How to show that $\det(AB) =\det(A) \det(B)$?
Given two square matrices $A$ and $B$, how do you show that $$\det(AB) = \det(A) \det(B)$$ where $\det(\cdot)$ is the determinant of the matrix?
Learner
- 2,696
86
votes
9 answers
Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$
I found the following formula
$$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$
and it is cited that Euler proved the formula above , but how ?
Do there exist other proofs…
Zaid Alyafeai
- 14,343
86
votes
4 answers
A path to truly understanding probability and statistics
I'm embarrassed to say that I have a PhD and hold an asst professorship, but get tripped up when reading statistics research. I am in a field of Business that is similar to IO Psychology or Social Psych. I spend too much time reading applied stats…
user27634
- 961