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1500 questions
79
votes
7 answers
Is "The empty set is a subset of any set" a convention?
Recently, I learned that for any set $A$, we have $\varnothing \subset A$. I found some explanation of why it holds.
$\varnothing\subset A$ means "for every object $x$, if $x$ belongs to the empty set, then $x$ also belongs to the set $A$". This is…
Searene
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- 8
79
votes
8 answers
Why do books titled "Abstract Algebra" mostly deal with groups/rings/fields?
As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem to 'only' cover groups, rings and fields. Why is…
GeorgW
- 799
79
votes
20 answers
Proof of the hockey stick/Zhu Shijie identity $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
After reading this question, the most popular answer use the identity
$$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1},$$
or, what is equivalent,
$$\sum_{t=k}^n \binom{t}{k} = \binom{n+1}{k+1}.$$
What's the name of this identity? Is it the identity of…
hlapointe
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79
votes
10 answers
Highest power of a prime $p$ dividing $N!$
How does one find the highest power of a prime $p$ that divides $N!$ and other related products?
Related question: How many zeros are there at the end of $N!$?
This is being done to reduce abstract duplicates. See
Coping with *abstract* duplicate…
user17762
79
votes
13 answers
Why there is no sign of logic symbols in mathematical texts?
Either in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, ...), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, etc. Instead, I just see their "read as" or…
MKR
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78
votes
5 answers
Help me solve my father's riddle and get my book back
My father is a mathteacher and as such he regards asking tricky questions and playing mathematical pranks on me once in a while as part of his parental duty.
So today before leaving home he sneaked into my room and took the book I am currently…
user161516
78
votes
2 answers
Why absolute values of Jacobians in change of variables for multiple integrals but not single integrals?
If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is:
$$ \int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt.
\qquad\text{(1)}$$
If $T=\{x=f(u,v); y=g(u,v)\}$ is a change of 2D coordinates, then the formula…
ShungChing
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78
votes
32 answers
Interesting and unexpected applications of $\pi$
$\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$
Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$
and the generalization of $\zeta (2k)$, my…
MT_
- 19,971
78
votes
6 answers
Matrix is conjugate to its own transpose
Mariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose.
I spent quite a bit of time on it now, and still could not prove it. At the risk of devaluing myself, might I ask someone else…
George
- 1,997
78
votes
4 answers
Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$
Consider the following integral:
$$\mathcal{I}=\int_1^\infty\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}\,,$$
where $\operatorname{arccsc}$ is the inverse…
Vladimir Reshetnikov
- 32,650
78
votes
5 answers
What is the geometry in algebraic geometry?
Coming from a physics background, my understanding of geometry (in a very generic sense) is that it involves taking a space and adding some extra structure to it. The extra structure takes some local data about the space as its input and outputs…
d_b
- 1,075
78
votes
10 answers
Why is the Traveling Salesperson Problem "Difficult"?
The Traveling Salesperson Problem is originally a mathematics/computer science optimization problem in which the goal is to determine a path to take between a group of cities such that you return to the starting city after visiting each city exactly…
stats_noob
- 4,107
78
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
- 1,651
78
votes
4 answers
What is the difference between a Ring and an Algebra?
In mathematics, I want to know what is indeed the difference between a ring and an algebra?
user70795
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78
votes
5 answers
Can someone explain the Yoneda Lemma to an applied mathematician?
I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood everything until we covered the Yoneda Lemma,…
Chris Taylor
- 29,755