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1500 questions
67
votes
3 answers
What use is the Yoneda lemma?
Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse concepts. Many of the tools of category theory…
Alex Becker
- 62,289
67
votes
3 answers
Dual norm intuition
The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it to solve problems, but in reality I still lack…
trembik
- 1,309
67
votes
7 answers
Why does the median minimize $E(|X-c|)$?
Suppose $X$ is a real-valued random variable and let $P_X$ denote the distribution of $X$. Then $$ E(|X-c|) = \int_\mathbb{R} |x-c| dP_X(x). $$ The medians of $X$ are defined as any number $m \in \mathbb{R}$ such that $P(X \leq m) \geq \frac{1}{2}$…
Tim
- 49,412
67
votes
7 answers
Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $
I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$
A is for Ambition
- 2,178
67
votes
6 answers
Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?
In this wikipedia article for improper integrals, $$ \int_0^{\infty}\frac{\sin x}{x}dx $$ is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) Lebesgue integral. Here are my questions : Why does…
user9464
67
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
- 2,089
67
votes
4 answers
How unique are $U$ and $V$ in the singular value decomposition $A=U\Sigma V^\dagger$?
According to Wikipedia: A common convention is to list the singular values in descending order. In this case, the diagonal matrix $\Sigma$ is uniquely determined by $M$ (though the matrices $U$ and $V$ are not). My question is, are $U$ and $V$…
capybaralet
- 1,315
67
votes
1 answer
Effect of elementary row operations on determinant?
1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results in the determinant scaling by that constant. Using…
dfg
- 4,121
67
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
- 671
67
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
- 7,484
67
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 and…
Phaptitude
- 2,317
67
votes
8 answers
Is there a way to get trig functions without a calculator?
In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator? Sometimes I don't feel right when I can't do…
Jonathan Lam
- 853
67
votes
10 answers
Is it not effective to learn math top-down?
By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving basic results. Eventually you'll get there but is…
EffingLoveMath
- 22,779
67
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 $2$…
user957
- 3,477
67
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
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