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1500 questions
106
votes
30 answers

What is the single most influential book every mathematician should read?

If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
c4il
  • 223
106
votes
9 answers

What are good "math habits" that have improved your mathematical practice?

I currently feel like I am not doing maths the best way I could; that is, I'm not making the most out of my time when I'm working on maths problems. The main thing I feel is that I'm not organizing my mind and my derivations as clear as I could,…
user56834
  • 12,925
106
votes
40 answers

Theorems with an extraordinary exception or a small number of sporadic exceptions

The Whitney graph isomorphism theorem gives an example of an extraordinary exception: a very general statement holds except for one very specific case. Another example is the classification theorem for finite simple groups: a very general statement…
Hans-Peter Stricker
  • 18,159
106
votes
8 answers

Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$

Compute $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
user 1591719
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106
votes
3 answers

Cardioid in coffee mug?

I've been learning about polar curves in my Calc class and the other day I saw this suspiciously $r=1-\cos \theta$ looking thing in my coffee cup (well actually $r=1-\sin \theta$ if we're being pedantic.) Some research revealed that it's called a…
meiji163
  • 3,959
106
votes
5 answers

How to get a reflection vector?

I'm doing a raytracing exercise. I have a vector representing the normal of a surface at an intersection point, and a vector of the ray to the surface. How can I determine what the reflection will be? In the below image, I have d and n. How can I…
Nick Heiner
  • 1,231
105
votes
8 answers

Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations?

Moderator Notice: I am unilaterally closing this question for three reasons. The discussion here has turned too chatty and not suitable for the MSE framework. Given the recent pre-print of T. Tao (see also the blog-post here), the continued…
electronp
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105
votes
2 answers

Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers?

Our algebra teacher usually gives us a paper of $20-30$ questions for our homework. But each week, he tells us to do all the questions which their number is on a specific form. For example, last week it was all the questions on the form of $3k+2$…
CODE
  • 4,921
105
votes
23 answers

Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals , ... One of my students just rose and asked me: Why do we assume so much in math? Is math…
Anz Joy
  • 1,420
105
votes
20 answers

What are some examples of mathematics that had unintended useful applications much later?

I would like to know some examples of interesting (to the layman or young student), easy-to-describe examples of mathematics that has had profound unanticipated useful applications in the real world. For my own purposes, the longer the gap between…
Eric Tressler
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105
votes
18 answers

What seemingly innocuous results in mathematics require advanced proofs?

I'm interested in finding a collection of basic results in mathematics that require rather advanced methods of proof. In this list we're not interested in basic results that have tedious simple proofs which can be shorted through more advanced…
mark
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105
votes
1 answer

Simplicial Complex vs Delta Complex vs CW Complex

I am a little confused about what exactly are the difference(s) between simplicial complex, $\Delta$-complex, and CW Complex. What I roughly understand is that $\Delta$-complexes are generalisation of simplicial complexes (without the requirement…
yoyostein
  • 19,608
104
votes
8 answers

Is there any difference between mapping and function?

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to $\mathbb R$. But I am not ok with this answer. I…
Hassan Muhammad
  • 4,282
104
votes
9 answers

Congruence Arithmetic Laws, e.g. in divisibility by $7$ test

I have seen other criteria for divisibility by $7$. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. $323$) is interesting, but could not prove it. Let $n = (a_ka_{k-1}\ldots a_2a_1a_0)_{10} =…
Mathsource
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104
votes
8 answers

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to how to prove it is irreducible.…
MathTeacher
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