Most Popular
1500 questions
71
votes
9 answers
Introductory texts on manifolds
I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of.
I was wondering if someone can recommend to me some introductory texts…
user38268
71
votes
7 answers
How to straighten a parabola?
Consider the function $f(x)=a_0x^2$ for some $a_0\in \mathbb{R}^+$. Take $x_0\in\mathbb{R}^+$ so that the arc length $L$ between $(0,0)$ and $(x_0,f(x_0))$ is fixed. Given a different arbitrary $a_1$, how does one find the point $(x_1,y_1)$ so that…
sam wolfe
- 3,335
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
- 31,356
71
votes
5 answers
What does proving the Riemann Hypothesis accomplish?
I've recently been reading about the Millennium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even fully grasp the problem, but seeing the hypothesis and the other problems I wonder: what practical use will a…
Mythio
- 929
71
votes
2 answers
What is the difference between the Jacobian, Hessian and the Gradient?
I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me.
From my understanding, The gradient is the slope of the most rapid descent. Modifying your position…
Pluviophile
- 949
71
votes
8 answers
Using Gröbner bases for solving polynomial equations
In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean algorithm and Gaussian elimination". I've tried to…
J. M. ain't a mathematician
- 75,051
71
votes
2 answers
A comprehensive list of binomial identities?
Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do.
user813
71
votes
7 answers
Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?
I self-realized an interesting property today that all numbers $(a,b)$ belonging to the infinite set $$\{(a,b): a=(2l+1)^2, b=(2k+1)^2;\ l,k \in N;\ l,k\geq1\}$$ have their AM and GM both integers.
Now I wonder if there exist distinct real numbers…
Gaurang Tandon
- 6,449
71
votes
2 answers
Is there an intuitive reason for a certain operation to be associative?
When I read Pinter's A Book of Abstract Algebra, Exercise 7 on page 25 asks whether the operation
$$x*y=\frac{xy}{x+y+1}$$
(defined on the positive real numbers) is associative. At first I considered this to be false, because the expression is so…
Zirui Wang
- 1,417
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
- 2,382
71
votes
14 answers
What isn't a vector space?
I'm really confused about vector spaces. We're learning about them in Linear Algebra, and my book doesn't give good examples of what a vector space is. I understand sets and vectors, but I don't understand vector spaces. From the definitions they've…
Aleksandr Hovhannisyan
- 2,983
- 4
- 34
- 59
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
- 4,274
71
votes
3 answers
First-Order Logic vs. Second-Order Logic
Wikipedia describes the first-order vs. second-order logic as follows:
First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables…
Sadeq Dousti
- 3,291
71
votes
2 answers
Let, $A\subset\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$.
A challenge problem from Sally's Fundamentals of Mathematical Analysis.
Problem reads: Suppose $A$ is a subset of $\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$ with the usual…
David Bowman
- 5,572
71
votes
11 answers
Why is the complex plane shaped like it is?
It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this?…
user64742
- 2,207