Most Popular
1500 questions
64
votes
17 answers
Interesting "real life" applications of serious theorems
As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like:
"You have a great circular pizza with $n$…
user149890
64
votes
4 answers
Examples of morphisms of schemes to keep in mind?
What are interesting and important examples of morphisms of schemes (especially varieties) to keep in mind when trying to understand a new concept or looking for a counterexamples?
Examples of what I'm looking for:
The projection from the hyperbola…
user115940
- 1,969
64
votes
4 answers
How to calculate the pullback of a $k$-form explicitly
I'm having trouble doing actual computations of the pullback of a $k$-form. I know that a given differentiable map $\alpha: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ induces a map $\alpha^{*}: \Omega^{k}(\mathbb{R}^{n}) \rightarrow…
Tony Burbano
- 641
- 1
- 6
- 3
64
votes
4 answers
How to tell if a set of vectors spans a space?
I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, c_3, c_4$ such that $(x, y, z) = c_1(1, 1, 1) +…
Javier
- 7,298
64
votes
7 answers
What is the antiderivative of $e^{-x^2}$
I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$
So, I of course didn't know what this $\text{erf}$ was…
Phaptitude
- 2,249
64
votes
3 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
- 1,100
64
votes
4 answers
Why are two permutations conjugate iff they have the same cycle structure?
I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
Han Solo
- 641
64
votes
3 answers
What does "isomorphic" mean in linear algebra?
My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have the same vector structure. I'm not sure what that…
Sujaan Kunalan
- 10,944
64
votes
7 answers
If eigenvalues are positive, is the matrix positive definite?
If the matrix is positive definite, then all its eigenvalues are strictly positive.
Is the converse also true?
That is, if the eigenvalues are strictly positive, then matrix is positive definite?
Can you give example of $2 \times 2$ matrix with…
user957
- 3,387
64
votes
14 answers
What is a proof?
I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra).
Mathematics is a system of axioms which you choose yourself for a set of undefined entities, such that those entities satisfy certain basic…
user78743
64
votes
2 answers
Why is $\varphi$ called "the most irrational number"?
I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio of integers. What is meant by most irrational?…
Christopher King
- 10,365
64
votes
8 answers
Why do the French count so strangely?
Today I've heard a talk about division rules. The lecturer stated that base 12 has a lot of division rules and was therefore commonly used in trade.
English and German name their numbers like they count (with 11 and 12 as exception), but not…
Martin Thoma
- 9,821
64
votes
5 answers
What does it mean to integrate with respect to the distribution function?
If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as:
$$E(X) = \int x f(x) dx$$
where the bounds of integration are implicitly $-\infty$ and…
Jeromy Anglim
- 877
64
votes
6 answers
Is there a reason it is so rare we can solve differential equations?
Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously complicated solutions or be unsolvable. Is there some…
novawarrior77
- 790
64
votes
4 answers
A "new" general formula for the quadratic equation?
Maybe the question is very trivial in a sense. So, it doesn't work for anyone. A few years ago, when I was a seventh-grade student, I had found a quadratic formula for myself. Unfortunately, I didn't have the chance to show it to my teacher at that…
lone student
- 14,709