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1500 questions
65
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
- 156,297
65
votes
9 answers
Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?
The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What motivates this is the following example. The…
Jacob Garber
- 918
65
votes
2 answers
Global sections of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$, understanding structure sheaves and twisting.
In chapter 2 section 7 (pg 151) of Hartshorne's algebraic geometry there is an example given that talks about automorphisms of $\mathbb{P}_k^n$ . In that example Hartshorne states that $\mathcal{O}(-1)$ has no global sections. However, we know that…
MJoszef
- 1,105
65
votes
13 answers
Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.
Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and the $x$-axis between $x=0$ and $x=1$ is equal to…
Daniel W. Farlow
- 23,258
65
votes
9 answers
Are there other kinds of bump functions than $e^\frac1{x^2-1}$?
I've only seen the bump function $e^\frac1{x^2-1}$ so far. Where could I find examples of functions $C^∞$ on $\mathbb{R}$ that are zero everywhere except on $(-1,1)$? Are there others that do not involve the exponential function? Are there any with…
jnm2
- 3,260
64
votes
6 answers
Geometric interpretation of mixed partial derivatives?
I'm looking for a geometric interpretation of this theorem regarding the symmetry of mixed partial derivatives : Equality of Mixed Partial Derivatives (Clairaut’s Theorem) Assume that $f$ is defined an open set $D$ of $\mathbb{R}^2$ and $f_{xy}$ and…
dfg
- 4,121
64
votes
6 answers
What should an amateur do with a proof of an open problem?
Assuming that somebody is not an employee of a university, just a math amateur, and makes a proper proof of some well known open math problem, what should he do with it? Publish on the internet for verification or send to some authorities? Is there…
Ztalloc
- 647
64
votes
12 answers
Very good linear algebra book.
I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), linear maps and their matrix representation and…
Mike
- 641
64
votes
8 answers
Probability that random moves in the game 2048 will win
I have recently played the game 2048 , created by Gabriele Cirulli, which is fun. I suggest trying if you have not. But my brother posed this question to me about the game: If he were to write a script that made random moves in the game 2048, what…
N. Owad
- 7,022
64
votes
9 answers
How are the Taylor Series derived?
I know the Taylor Series are infinite sums that represent some functions like $\sin(x)$. But it has always made me wonder how they were derived? How is something like $$\sin(x)=\sum\limits_{n=0}^\infty \dfrac{x^{2n+1}}{(2n+1)!}\cdot(-1)^n =…
Anonymous Computer
- 5,745
64
votes
4 answers
Why is the Laplacian important in Riemannian geometry?
As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. Fundamentally, I would like to know why the Laplacian is…
Jesse Madnick
- 33,049
64
votes
5 answers
Is there a 3-dimensional "matrix" by "matrix" product?
Is it possible to multiply A[m,n,k] by B[p,q,r]? Does the regular matrix product have generalized form? I would appreciate it if you could help me to find out some tutorials online or mathematical 'word' which means N-dimensional matrix product.…
danny_23
- 773
64
votes
5 answers
Every nonzero element in a finite ring is either a unit or a zero divisor
Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
rupa
- 723
64
votes
2 answers
Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$
Using a numerical search on my computer I discovered the following inequality: $$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$ where $\rho$ is the positive root of the polynomial equation…
HWﾠ
- 1
- 7
- 9
64
votes
2 answers
Why does the spectral norm equal the largest singular value?
This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right \| _2$ of a complex matrix $A$ is defined as…
mathemage
- 850