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1500 questions
71
votes
24 answers
Theorems' names that don't credit the right people
The point of this question is to compile a list of theorems that don't give credit to right people in the sense that the name(s) of the mathematician(s) who first proved the theorem doesn't (do not) appear in the theorem name. For instance the…
Git Gud
- 31,732
71
votes
7 answers
How do you show monotonicity of the $\ell^p$ norms?
I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
user1736
- 9,083
71
votes
7 answers
Proof that the largest eigenvalue of a stochastic matrix is $1$
The largest eigenvalue of a stochastic matrix (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$. Wikipedia marks this as a special case of the Perron-Frobenius theorem , but I wonder if there is a simpler (more direct)…
koletenbert
- 4,170
71
votes
2 answers
What is the difference between minimum and infimum?
What is the difference between minimum and infimum? I have a great confusion about this.
Manoj
- 1,837
71
votes
18 answers
Unsolved Problems due to Lack of Computational Power
I was recently reading up about computational power and its uses in maths particularly to find counterexamples to conjectures. I was wondering are there any current mathematical problems which we are unable to solve due to our lack of computational…
user671231
71
votes
4 answers
Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$
One of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \leq x \quad\text{for all $x > -1$}.$$ The upper bound is…
Federico Magallanez
- 1,988
71
votes
28 answers
Non-associative operations
There are lots of operations that are not commutative. I'm looking for striking counter-examples of operations that are not associative . Or may associativity be genuinely built-in the concept of an operation? May non-associative operations be of…
Hans-Peter Stricker
- 18,829
71
votes
10 answers
There are 4 cups of liquid. Three are water and one is poison. If you were to drink 3 of the 4 cups, what is the probability of being poisoned?
In Season 5 Episode 16 of Agents of Shield, one of the characters decides to prove she can't die by pouring three glasses of water and one of poison; she then randomly drinks three of the four cups. I was wondering how to compute the probability of…
student
- 952
71
votes
3 answers
Necessity/Advantage of LU Decomposition over Gaussian Elimination
I am reading the book "Introduction to Linear Algebra" by Gilbert Strang and couldn't help wondering the advantages of LU decomposition over Gaussian Elimination! For a system of linear equations in the form $Ax = b$ , one of the methods to solve…
71
votes
9 answers
Given real numbers: define integers?
I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following: Integer numbers are just special cases (a subset) of real numbers. Imagine a world where you know only real numbers. How…
Daniel A.A. Pelsmaeker
- 1,139
71
votes
5 answers
What is the 'implicit function theorem'?
Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why they use Jacobian matrix to illustrate this…
Flying pig
- 927
71
votes
12 answers
Will assuming the existence of a solution ever lead to a contradiction?
I'm reading Manfredo Do Carmo's differential geometry book. In section 1-7, he discusses the "Isoperimetric Inequality" which is related to the question of what 2-dimensional shape maximizes the enclosed area for a closed curve of constant length.…
Geoffrey
- 2,425
71
votes
6 answers
Projection is an open map
Let $X$ and $Y$ be (any) topological spaces. Show that the projection $\pi_1$ : $X\times Y\to X$ is an open map.
Alisha
- 759
- 1
- 6
- 3
71
votes
11 answers
Is it technically incorrect to write proofs forward?
A question on an assignment was similar to prove: $$2a^2-7ab+2b^2 \geq -3ab.$$ and my proof was: $$2a^2-4ab+2b^2\geq0$$ $$a^2-2ab+b^2\geq0$$ $$(a-b)^2\geq0$$ which is true. However, my professor marked this as incorrect and the "correct" way to do…
mtheorylord
- 4,350
71
votes
1 answer
Difference between simplicial and singular homology?
I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and maybe this doesnt't help my intuition), but I am…
user48168
- 1,365