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1500 questions
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
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
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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
  • 3,300
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
  • 4,131
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
  • 7,032
64
votes
10 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
  • 5,765
64
votes
9 answers

If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.

Let $R$ be a ring, where $a^{3} = a$ for all $a\in R$ . Prove that $R$ must be a commutative ring. [ Historical note: this question does not meet current quality standards but since it has been heavily used as a canonical dupe target it should…
Aj I
  • 1,113
64
votes
20 answers

Coin flipping probability game ; 7 flips vs 8 flips

Your friend flips a coin 7 times and you flip a coin 8 times; the person who got the most tails wins. If you get an equal amount, your friend wins. There is a 50% chance of you winning the game and a 50% chance of your friend winning. How can I…
Jason
  • 3,707
64
votes
9 answers

Intuition behind "ideal"

To briefly put forward my question, can anyone beautifully explain me in your own view, what was the main intuition behind inventing the ideal of a ring? I want a clarified explanations in these points: Why is the name "ideal" coined?. In English…
IDOK
  • 5,498
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

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
64
votes
5 answers

A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function . The best I could do was to express it in terms of Euler Sums. Let $I$ denote…
Shobhit
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