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1500 questions
137
votes
5 answers
Incremental averaging
Is there a way to incrementally calculate (or estimate) the average of a vector (a set of numbers) without knowing their count in advance?
For example you have a = [4 6 3 9 4 12 4 18] and you want to get an estimate of the average but you don't…
Ali
- 1,791
136
votes
8 answers
Why “characteristic zero” and not “infinite characteristic”?
The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we define it to be $0$.
But why characteristic zero? Why…
Srivatsan
- 26,761
136
votes
5 answers
Difference between "≈", "≃", and "≅"
In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"?
The Unicode standard lists all of them inside the Mathematical Operators Block.
≈ : ALMOST EQUAL TO (U+2248)
≃ :…
GOTO 0
- 2,052
136
votes
4 answers
How to show that a set of discontinuous points of an increasing function is at most countable
I would like to prove the following:
Let $g$ be a monotone increasing function on $[0,1]$. Then the set of points where $g$ is not continuous is at most countable.
My attempt:
Let $g(x^-)~,g(x^+)$ denote the left and right hand limits of $g$…
AKM
- 1,461
136
votes
20 answers
Real life applications of Topology
The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here.
What are the various real life applications of topology?
Bhargav
- 3,007
136
votes
5 answers
Why does L'Hopital's rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$?
$$\lim_{x \to \infty} \frac{x}{x+\sin(x)}$$
This is of the indeterminate form of type $\frac{\infty}{\infty}$, so we can apply l'Hopital's…
Paul R
- 2,413
135
votes
3 answers
Why is an average of an average usually incorrect?
Can someone explain why taking an average of an average usually results in a wrong answer? Is there ever a case where the average of the average can be used correctly?
As an example, let's say that an assessment is given to three schools and I…
O.O
- 1,571
135
votes
10 answers
what is expected from a PhD student?
As a PhD student in applied mathematics or mathematics in general, are you expected to be able to prove every problem, for example, in an elementary real analysis book? I know it sounds silly but I am wondering if I have high expectations of…
Tomas Jorovic
- 4,053
135
votes
8 answers
When to learn category theory?
I’m an undergraduate student eager to learn category theory, but so far I have only a basic background in linear algebra and set theory. I’ve also taken a short number-theory course that introduced elementary group concepts and modular arithmetic.…
Vicfred
- 2,957
135
votes
5 answers
How to find solutions of linear Diophantine ax + by = c?
I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?
I tried to play around with it:
$x = (c - by)/a$, hence $a|(c - by)$.
$a$, $c$…
roxrook
- 12,399
134
votes
8 answers
Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.
Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi…
bryn
- 10,045
134
votes
8 answers
How to prove that eigenvectors from different eigenvalues are linearly independent
How can I prove that if I have $n$ eigenvectors from different eigenvalues, they are all linearly independent?
Corey L.
- 1,359
134
votes
13 answers
Why study linear algebra?
Simply as the title says. I've done some research, but still haven't arrived at an answer I am satisfied with. I know the answer varies in different fields, but in general, why would someone study linear algebra?
Aaron
- 1,379
134
votes
4 answers
What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?
My friend gave me this puzzle:
What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges?
I tried to draw the picture and I drew a smaller (concentric)…
terrace
- 2,087
- 2
- 17
- 27
133
votes
9 answers
Prove that $C e^x$ is the only set of functions for which $f(x) = f'(x)$
I was wondering on the following and I probably know the answer already: NO.
Is there another number with similar properties as $e$? So that the derivative of $ e^x$ is the same as the function itself.
I can guess that it's probably not, because…
Timo Willemsen
- 1,657