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1500 questions
65
votes
8 answers
What is Cauchy Schwarz in 8th grade terms?
I'm an 8th grader. After browsing aops.com, a math contest website, I've seen a lot of problems solved by Cauchy Schwarz. I'm only in geometry (have not started learning trigonometry yet). So can anyone explain Cauchy Schwarz in layman's terms, as…
Schneider
- 749
65
votes
1 answer
Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure?
This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there.
$\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism…
user98602
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
3 answers
Is "A New Kind of Science" a new kind of science?
A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, I realise that many ideas of NKS seem to be not…
Andrea
- 2,483
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
- 23,137
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…
jnm2
- 3,260
64
votes
2 answers
Computation with a memory wiped computer
Here is another result from Scott Aaronson's blog:
If every second or so your computer’s
memory were wiped completely clean,
except for the input data; the clock;
a static, unchanging program; and a
counter that could only be set to 1,
2,…
Casebash
- 9,437
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
5 answers
Picking random points in the volume of sphere with uniform probability
I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of the sphere?
Since I'm unable to answer my own…
MHK
- 643
64
votes
3 answers
What does "∈" mean?
I have started seeing the "∈" symbol in math. What exactly does it mean?
I have tried googling it but google takes the symbol out of the search.
Locke
- 813
64
votes
6 answers
Midpoint-convexity and continuity implies convexity
Assume that function $f$ is continuous on an interval $(a,b)$. Given that
$$ f \left( \frac{x+y}{2} \right) \leqslant \frac{f(x) + f(y)}{2} \,,$$
how can I show that $f$ is convex?
Jack
- 843
64
votes
6 answers
Why do differential forms have a much richer structure than vector fields?
I apologize in advance because this question might be a bit philosophical, but I do think it is probably a genuine question with non-vacuous content.
We know as a fact that differential forms have a much richer structure than vector fields, to name…
Jia Yiyang
- 1,113
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
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
- 5,735