Most Popular
1500 questions
67
votes
2 answers
Explain this mathematical meme (Geometers bird interrupting Topologists bird)
My knowledge of geometry is just a little bit above high school level and I know absolutely nothing about topology. So, what is the point of this meme?
(Original unedited webcomic: “Juncrow” by False Knees)
Hanlon
- 1,749
67
votes
4 answers
Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?
Does the following series converge or diverge? I would like to see a demonstration.
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}.
$$
I can see that:
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} \leqslant \sum_{n=1}^{\infty} \frac{1}{n^{1…
user55114
67
votes
26 answers
What are some mathematically interesting computations involving matrices?
I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving matrices.
Preferably these examples would be…
providence
- 4,408
- 1
- 32
- 45
67
votes
10 answers
Is memory unimportant in doing mathematics?
The title says it all.
I often heard people say something like memory is unimportant in doing mathematics. However, when I tried to solve mathematical problems, I often used known theorems whose proofs I forgot.
EDIT
Some of you may think that…
Makoto Kato
- 42,602
67
votes
3 answers
In combinatorics, how can one verify that one has counted correctly?
This is a soft question, but I've tried to be specific about my concerns. When studying basic combinatorics, I was struck by the fact that it seems hard to verify if one has counted correctly.
It's easiest to explain with an example, so I'll give…
Stephen
- 1,013
67
votes
3 answers
Approximating a $\sigma$-algebra by a generating algebra
Theorem. Let $(X,\mathcal B,\mu)$ a finite measure space, where $\mu$ is a positive measure. Let $\mathcal A\subset \mathcal B$ an algebra generating $\cal B$.
Then for all $B\in\cal B$ and $\varepsilon>0$, we can find $A\in\cal A$ such that…
Davide Giraudo
- 172,925
67
votes
19 answers
What are the names of numbers in the binary system?
The names we use are very much related to the radix we use
$0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9$
zero - one - two - three - four - five - six - seven - eight -nine
We repeat the names
$21$ twenty one, $22$ twenty two .. and so on.
This is not…
user37421
- 917
67
votes
29 answers
How to explain for my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$?
I was really upset while I was trying to explain for my daughter that $\frac 23$ is greater than $\frac 35$ and she always claimed that $(3$ is greater than $2$ and $5$ is greater than $3)$ then $\frac 35$ must be greater than $\frac 23$.
At this…
Akam
- 639
- 1
- 6
- 11
67
votes
2 answers
When can we interchange the derivative with an expectation?
Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Is it true in general that $ \frac{d} {dt} \mathbb{E}(Y_t) = \mathbb{E}(f(X_t)) $? If not, under what conditions would we be allowed to…
Jonas
- 2,423
67
votes
17 answers
Good "history of mathematical ideas" book?
All too often, mathematical history books include far too much material on the private lives of the personalities involved and not enough information on the actual ideas. Mathematics is a living subject in itself, which is not enhanced by knowing…
67
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
- 8,773
67
votes
2 answers
Period of the sum/product of two functions
Suppose that period of $f(x)=T$ and period of $g(x)=S$, I am interested what is a period of $f(x) g(x)$? period of $f(x)+g(x)$? What I have tried is to search in internet, and found following link for this.
Also I know that period of $\sin(x)$ is…
dato datuashvili
- 9,194
67
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
- 1,016
67
votes
7 answers
Is the sum and difference of two irrationals always irrational?
If $x$ and $y$ are irrational, is $x + y$ irrational? Is $x - y$ irrational?
Miguel Mora Luna
- 3,197
67
votes
6 answers
Is it possible to formulate category theory without set theory?
I have never understood why set theory has so many detractors, or what is gained by avoiding its use.
It is well known that the naive concept of a set as a collection of objects leads to logical paradoxes (when dealing with infinite sets) that can…
Matt Calhoun
- 4,404