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1500 questions
137
votes
5 answers
Help find hard integrals that evaluate to $59$?
My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$. I want to try and make a definite integral that equals $59$. So far I can only think of ones that are easy to…
Argon
- 26,059
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 have…
Ali
- 1,791
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
28 answers
What are some examples of notation that really improved mathematics?
I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational conventions that simplify problem statements and ideas,…
Patrick
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136
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.
- 1,379
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 rule:…
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 want…
O.O
- 1,571
135
votes
9 answers
Produce an explicit bijection between rationals and naturals
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such…
Alex Basson
- 4,401
135
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
- 10,135
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 $a|(c - by)$. $a$, $c$ and…
roxrook
- 12,457
134
votes
8 answers
What are differences between affine space and vector space?
I know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by Paul Bamberg and Shlomo Sternberg. In Chapter 1…
user41451
- 1,445
134
votes
13 answers
Advantages of Mathematics competition/olympiad students in Mathematical Research
Everyone in this community I think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries participating from around the world. What's…
user9413