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1500 questions
65
votes
4 answers
Difference between supremum and maximum
Referring to this lecture , I want to know what is the difference between supremum and maximum. It looks same as far as the lecture is concerned when it explains pointwise supremum and pointwise maximum
user31820
- 943
- 2
- 10
- 12
65
votes
6 answers
If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?
Question: Suppose we have one hundred seats, numbered 1 through 100. We randomly select 25 of these seats. What is the expected number of selected pairs of seats that are consecutive? (To clarify: we would count two consecutive selected seats as a…
David Steinberg
- 1,450
65
votes
8 answers
What is Cauchy Schwarz in 8th grade terms?
I'm an 8th grader. After browsing aops.com, a math contest website, I've seen a lot of problems solved by Cauchy Schwarz. I'm only in geometry (have not started learning trigonometry yet). So can anyone explain Cauchy Schwarz in layman's terms, as…
Schneider
- 749
65
votes
3 answers
Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?
I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it?
Are there any positive irrational numbers $a\ne e$ such that…
july
- 1,261
65
votes
15 answers
Choice of $q$ in Baby Rudin's Example 1.1
First, my apologies if this has already been asked/answered. I wasn't able to find this question via search.
My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2. In the second version of…
Rachel
- 2,874
65
votes
1 answer
Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure?
This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there.
$\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism…
user98602
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…
Mr Bingley
- 908
65
votes
14 answers
Express 99 2/3% as a fraction? No calculator
My 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is appreciated:
The possible answers are:
$1…
lara400
- 866
65
votes
5 answers
Jacobi identity - intuitive explanation
I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as an axiom for defining a Lie algebra). Could…
aelguindy
- 2,698
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
- 22,531
65
votes
3 answers
Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.
Let
$$
\text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction}
$$
So for example, the terms of the sum $\text{S}_6$…
Simon Parker
- 4,303
64
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
- 60,569
64
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
- 1,259
64
votes
19 answers
What are some good ways to get children excited about math?
I'm talking in the range of 10-12 years old, but this question isn't limited to only that range.
Do you have any advice on cool things to show kids that might spark their interest in spending more time with math? The difficulty for some to learn…
David McGraw
- 865
64
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
- 2,078