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1500 questions
68
votes
11 answers
Average Distance Between Random Points on a Line Segment
Suppose I have a line segment of length $L$. I now select two points at random along the segment. What is the expected value of the distance between the two points, and why?
Kenshin
- 2,230
68
votes
6 answers
$n$th derivative of $e^{1/x}$
I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula
$$\frac{\mathrm d^n}{\mathrm dx^n}f(x)=(-1)^n e^{1/x} \cdot \sum _{k=0}^{n-1} k!…
Listing
- 14,087
68
votes
7 answers
How do we know the ratio between circumference and diameter is the same for all circles?
The number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same for every circle?
How do we know that $\pi =…
Happy
- 1,443
68
votes
6 answers
Proving the existence of a proof without actually giving a proof
In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. the celebrated probabilistic method and many things…
Damian Reding
- 8,894
- 2
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- 40
68
votes
15 answers
Why do I get one extra wrong solution when solving $2-x=-\sqrt{x}$?
I'm trying to solve this equation:
$$2-x=-\sqrt{x}$$
Multiply by $(-1)$:
$$\sqrt{x}=x-2$$
power of $2$:
$$x=\left(x-2\right)^2$$
then:
$$x^2-5x+4=0$$
and that means:
$$x=1, x=4$$
But $x=1$ is not a correct solution to the original equation.
Why…
TheLogicGuy
- 1,036
68
votes
17 answers
Why does the derivative of sine only work for radians?
I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm discovering that I've been wrong all along! I'm guessing…
Kyle Delaney
- 1,441
68
votes
5 answers
Sum of two closed sets in $\mathbb R$ is closed?
Is there a counterexample for the claim in the question subject, that a sum of two closed sets in $\mathbb R$ is closed? If not, how can we prove it?
(By sum of sets $X+Y$ I mean the set of all sums $x+y$ where $x$ is in $X$ and $y$ is in…
ro44
- 683
68
votes
9 answers
Do commuting matrices share the same eigenvectors?
In one of my exams I'm asked to prove the following
Suppose $A,B\in \mathbb R^{n\times n}$, and $AB=BA$, then $A,B$ share the same eigenvectors.
My attempt is let $\xi$ be an eigenvector corresponding to $\lambda$ of $A$, then $A\xi=\lambda\xi$,…
Vim
- 13,905
68
votes
3 answers
Is Stokes' Theorem natural in the sense of category theory?
Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation
$$\int_M d\omega = \int_{\partial M} \omega.$$
Doesn't that look like…
Martin Brandenburg
- 181,922
68
votes
1 answer
Semi-direct v.s. Direct products
What is the difference between a direct product and a semi-direct product in group theory?
Based on what I can find, difference seems only to be the nature of the groups involved, where a direct product can involve any two groups and the…
retro
- 801
67
votes
3 answers
Dual norm intuition
The dual of a norm $\|\cdot \|$ is defined as:
$$\|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
- 49,162
67
votes
4 answers
Similar matrices have the same eigenvalues with the same geometric multiplicity
Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities.
Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices over $\mathbb R$ or $\mathbb C$. We say $A$ and $B$ are…
user2723
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
- 2,049
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