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1500 questions
63
votes
15 answers
Why learn to solve differential equations when computers can do it?
I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background.
One question is haunting me.
Why do I need to learn to do complex math operations on paper…
user60462
- 833
63
votes
7 answers
How to explain to the layperson what mathematics is, why it's important, and why it's interesting
A mathematician walks into a party.
No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. Somebody asks you:
"So, what do you do?"
What do…
Bruno Joyal
- 54,711
63
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
- 633
63
votes
9 answers
Does it ever make sense NOT to go to the most prestigious graduate school you can get into?
I'm a senior undergrad at a top-ish(say, top 15) math school. I'm a solid, not stellar, student. This year I'm taking the qualifying exam grad courses in algebra and analysis and have been taken aback by the "pressure cooker" atmosphere among grad…
anonymous
- 191
63
votes
3 answers
Expectation of Minimum of $n$ i.i.d. uniform random variables.
$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)?
I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to…
jet
- 633
63
votes
4 answers
Is the power set of the natural numbers countable?
Some explanations:
A set S is countable if there exists an injective function $f$ from $S$ to the natural numbers ($f:S \rightarrow \mathbb{N}$).
$\{1,2,3,4\}, \mathbb{N},\mathbb{Z}, \mathbb{Q}$ are all countable.
$\mathbb{R}$ is not countable.
The…
Martin Thoma
- 9,821
63
votes
6 answers
Area covered by a constant length segment rotating around the center of a square.
This is an idea I have had in my head for years and years and I would like to know the answer, and also I would like to know if it's somehow relevant to anything or useless.
I describe my thoughts with the following image:
What would the area of…
user136800
63
votes
15 answers
What is a good book for learning math, from middle school level?
Which books are recommended for learning math from the ground up and review the basics - from middle school to graduate school math?
I am about to finish my masters of science in computer science and I can use and understand complex math, but I…
user16974
63
votes
4 answers
How unique are $U$ and $V$ in the singular value decomposition $A=UDV^\dagger$?
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
- 1,265
63
votes
4 answers
What's the probability that a sequence of coin flips never has twice as many heads as tails?
I gave my friend this problem as a brainteaser; while her attempted solution didn't work, it raised an interesting question.
I flip a fair coin repeatedly and record the results. I stop as soon as the number of heads is equal to twice the number of…
Elliott
- 4,124
63
votes
3 answers
Why is it worth spending time on type theory?
Looking around there are three candidates for "foundations of mathematics":
set theory
category theory
type theory
There is a seminal paper relating these three topics:
From Sets to Types to Categories to Sets by Steve Awodey
But at this forum…
Hans-Peter Stricker
- 18,159
63
votes
18 answers
Which is larger? $20!$ or $2^{40}$?
Someone asked me this question, and it bothers the hell out of me that I can't prove either way.
I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas…
Alec
- 4,094
63
votes
4 answers
why geometric multiplicity is bounded by algebraic multiplicity?
Define
The algebraic multiplicity of $\lambda_{i}$ to be the degree of the root $\lambda_i$ in the polynomial $\det(A-\lambda I)$.
The geometric multiplicity the be the dimension of the eigenspace associated with the eigenvalue $\lambda_i$.
For…
Jack2019
- 1,525
63
votes
1 answer
Why are asymptotically one half of the integer compositions gap-free?
Question summary
The number of gap-free compositions of $n$ can already for quite small $n$ be very well approximated by the total number of compositions of $n$ divided by $2$. This question seeks to understand why.
The details
A composition of an…
Daniel R
- 3,199
63
votes
6 answers
Is $\{0\}$ a field?
Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms:
Closure under addition. If $x, y \in F$, then $x = y = I$, so $x + y = I +…
Dan
- 14,978