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1500 questions
62
votes
5 answers
Is there a definition of determinants that does not rely on how they are calculated?
In the few linear algebra texts I have read, the determinant is introduced in the following manner;
“Here is a formula for what we call $\det A$. Here are some other formulas. And finally, here are some nice properties of the determinant.”
For…
AnonymousCoward
- 5,548
62
votes
3 answers
Review of my T-shirt design
I'm a graphics guy and a wanna-be mathematician. Is the T-shirt design below okay? Or if there's a bone headed error, I'd appreciate a heads up.
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HopDavid
- 833
62
votes
7 answers
how to be good at proving?
I'm starting my Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc.
I know how the proving system works and I can understand the…
uohzxela
- 1,637
62
votes
2 answers
On Ramanujan's curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $
In Ramanujan's Notebooks, Vol IV, p.20, there is the rather curious relation for primes of form $4n-1$,
$$\sqrt{2\,\Big(1-\frac{1}{3^2}\Big) \Big(1-\frac{1}{7^2}\Big)\Big(1-\frac{1}{11^2}\Big)\Big(1-\frac{1}{19^2}\Big)} =…
Tito Piezas III
- 54,565
62
votes
7 answers
Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.
Let $k$ be a natural number. Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.
I tried to prove this by supposing one of them is a square number and by substituting the corresponding $k$ value. But I failed to prove it.
If we ignore…
Angelo Mark
- 5,954
62
votes
1 answer
Penrose's remark on impossible figures
I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the group operation is - composition of…
anon
- 151,657
62
votes
5 answers
Difference between topology and sigma-algebra axioms.
One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under countable union. It is very clear mathematically…
Creator
- 3,128
62
votes
3 answers
Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$
Does $$~\displaystyle{\int}_0^1\frac{\text{arctanh }x}{\tan\left(\dfrac\pi2~x\right)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$$ possess a closed form expression ?
This recent post, in conjunction with my age-old interest in…
Lucian
- 48,334
- 2
- 83
- 154
62
votes
10 answers
Why are primes considered to be the "building blocks" of the integers?
I watched the video of Terence Tao on Stephen Colbert the other day (here), and he stated, like many mathematicians do, that the primes are the building blocks of the integers.
I've always had trouble with this idea, because I don't think it's a…
Matt Gregory
- 2,027
61
votes
3 answers
Areas versus volumes of revolution: why does the area require approximation by a cone?
Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. However, if we want to find the surface area,…
Eric O. Korman
- 19,137
61
votes
2 answers
Category-theoretic limit related to topological limit?
Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...?
To be more precise, is there a category-theoretic setting of some non-trivial…
Rafael Mrden
- 2,199
61
votes
3 answers
What is Haar Measure?
Is there any simple explanation for Haar Measure and its geometry?
how do we understand analogy Between lebesgue measure and Haar Measure?
How to show integration with respect to Haar Measure?
what do we mean by integrating with respect to…
Milan Amrut Joshi
- 1,109
61
votes
13 answers
What is the solution of $\cos(x)=x$?
There is an unique solution with $x$ being approximately $0.739085$. But is there also a closed-form solution?
corto
- 1,045
61
votes
3 answers
Geometrical interpretation of Ricci curvature
I see the scalar curvature $R$ as an indicator of how a manifold curves locally (the easiest example is for a $2$-dimensional manifold $M$, where the $R=0$ in a point means that it is flat there, $R>0$ that it makes like a hill and $R<0$ that it is…
Daniel Robert-Nicoud
- 29,795
61
votes
6 answers
What are some examples of infinite dimensional vector spaces?
I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.
emDiaz
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