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1500 questions
120
votes
6 answers
Why don't analysts do category theory?
I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects.
Recently, I started taking some functional analysis courses…
gary
- 1,201
120
votes
26 answers
If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?
If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?
For example the square of $2$ is $2^2=2 \cdot 2=4 $ .
But square root of $2$ is not $\frac{2}{2}=1$ .
bluebellae
- 1,643
120
votes
18 answers
Fastest way to meet, without communication, on a sphere?
I was puzzled by a question my colleague asked me, and now seeking your help.
Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and the sphere is way bigger than you two. There are no…
Rob Audenaerde
- 1,245
119
votes
11 answers
Am I just not smart enough?
When I was doing math, let us say for example, introductory number theory, it seems to take me a lot of time to fully understand a theorem. By understanding, I mean, both intuitively and also rigorously (know how to prove or derive). However, I…
Kun
- 2,546
119
votes
16 answers
Do mathematicians, in the end, always agree?
I've been trying to study some different sciences in my life, ranging from biology to mathematics, and if I try to explain to people why I like mathematics above the others, I think the most important reason for me is that mathematicians, in the…
Kasper
- 13,940
119
votes
13 answers
In calculus, which questions can the naive ask that the learned cannot answer?
Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians' attempts to answer them.
Calculus is not known to be such a field, as far as I know. (For…
119
votes
3 answers
Why are infinitely dimensional vector spaces not isomorphic to their duals?
Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$).
I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the vector space $V=\mathbb F^{(\kappa)}$ (that is an…
Asaf Karagila
- 405,794
119
votes
9 answers
Why should I "believe in" weak solutions to PDEs?
This is a sort of soft-question to which I can't find any satisfactory answer. At heart, I feel I have some need for a robust and well-motivated formalism in mathematics, and my work in geometry requires me to learn some analysis, and so I am…
A. Thomas Yerger
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119
votes
1 answer
$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?
If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $2n-2$. A proof can be found here.
Two weeks and…
Lincoln Blackham
- 1,191
119
votes
7 answers
Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$)
Consider A $\Rightarrow$ B, A $\models$ B, and A $\vdash$ B.
What are some examples contrasting their proper use? For example, give A and B such that A $\models$ B is true but A $\Rightarrow$ B is false. I'd appreciate pointers to any tutorial-level…
user287424
- 1,309
119
votes
1 answer
Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$
Is there a way to assess the convergence of the following series?
$$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$
From numerical estimations it seems to be convergent but I don't know how to prove it.
Leonardo Massai
- 1,943
119
votes
4 answers
Element-wise (or pointwise) operations notation?
Is there a notation for element-wise (or pointwise) operations?
For example, take the element-wise product of two vectors x and y (in Matlab, x .* y, in numpy x*y), producing a new vector of same length z, where $z_i = x_i * y_i$ .
In mathematical…
levesque
- 1,469
119
votes
8 answers
Is the vector cross product only defined for 3D?
Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as
$$
\vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n
$$
It then mentions that $\vec n$ is the vector normal to the plane made by $\vec a$ and $\vec b$,…
VF1
- 2,063
119
votes
1 answer
Finding primes so that $x^p+y^p=z^p$ is unsolvable in the $p$-adic units
On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem:
Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\times}$ if and only if there exists an integer $a$…
ArtW
- 3,582
119
votes
21 answers
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives, by Leonard Mlodinow, Vintage Books, 2008. On p.107…
NotSuper
- 1,883