All Questions
362,793
questions with no upvoted or accepted answers
596
votes
0
answers
23k
views
Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
- 71.9k
171
votes
0
answers
4k
views
Sorting of prime gaps
Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$
If we rearrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$, if the gaps are arranged from smallest to largest, we have a new ...
- 10k
171
votes
1
answer
4k
views
Does every ring of integers sit inside a ring of integers that has a power basis?
Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form
$$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$
...
- 2,087
137
votes
0
answers
10k
views
Pullback and Pushforward Isomorphism of Sheaves
Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\...
- 7,278
124
votes
0
answers
4k
views
Mondrian Art Problem Upper Bound for defect
Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles?
...
- 20.6k
123
votes
0
answers
3k
views
If polynomials are almost surjective over a field, is the field algebraically closed?
Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
- 323k
113
votes
0
answers
2k
views
On the Constant Rank Theorem and the Frobenius Theorem for differential equations.
Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
- 14.4k
110
votes
0
answers
4k
views
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^{\...
- 3,465
109
votes
0
answers
3k
views
Ring structure on the Galois group of a finite field
Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
- 155k
107
votes
0
answers
2k
views
Classification of local Artin (commutative) rings which are finite over an algebraically closed field
A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(\mathcal{A})\rightarrow X$ where $\mathcal A$ is a local Artin ring finite over $k$ can be extended to every $Y'\...
user16544
101
votes
0
answers
3k
views
A question about divisibility of sum of two consecutive primes
I was curious about the sum of two consecutive primes and after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question:
Find the least natural ...
- 4,801
93
votes
1
answer
2k
views
What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?
Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with a unit perimeter?
A reasonable first guess would be the regular tetrahedron of ...
- 15.7k
88
votes
1
answer
2k
views
Why is a PDE a submanifold (and not just a subset)?
I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold.
Let $\pi: E \to M$ be a smooth locally trivial fibre bundle.
In Gromovs words a partial differential ...
- 1,448
82
votes
0
answers
2k
views
Probability for an $n\times n$ matrix to have only real eigenvalues
Let $A$ be an $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has only real eigenvalues?
The answer cannot be $0$ or $1$, ...
- 10.6k
81
votes
0
answers
1k
views
Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
- 15.4k
77
votes
0
answers
2k
views
Complete, Finitely Axiomatizable, Theory with 3 Countable Models
Does there exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models?
A few relevant comments:
There is a classical example of a complete theory ...
- 5,026
73
votes
0
answers
2k
views
Is there a "ping-pong lemma proof" that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
Let $f,g\colon \mathbb R \to \mathbb R$ be the permutations defined by $f\colon x \mapsto x+1$ and $g\colon x \mapsto x^3$, or maybe even have $g\colon x \mapsto x^p$, $p$ an odd prime.
In the book, ...
user29123
72
votes
0
answers
802
views
Does the $32$-inator exist?
Background
It is common popular-math knowledge that as we extend the real numbers to complex numbers, quaternions, octonions, sedenions, $32$-nions, etc. using the Cayley-Dickson construction, we lose ...
- 5,631
68
votes
0
answers
911
views
Dedekind Sum Congruences
For $a,b,c \in \mathbb{N}$, let $a^{\prime} = \gcd(b,c)$, $b^{\prime} = \gcd(a,c)$, $c^{\prime} = \gcd(a,b)$ and $d = a^{\prime} b^{\prime} c^{\prime}$. Define $\mathfrak{S}(a,b,c) = a^{\prime} \...
- 16.9k
65
votes
0
answers
2k
views
Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form.
Introduction:
We know that:
$$\sum_{x=0}^\infty \frac{1}{x!}=e$$
But what if we replaced $x!$ with $!x$ also called the subfactorial function also called the $x$th derangement number? This ...
- 9,987
59
votes
0
answers
2k
views
Determinant of a matrix that contains the first $n^2$ primes.
Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix
$$
\left(\begin{matrix}
p_1 & p_2 & \cdots & p_n \\
p_{n+1} & p_{n+2} & \...
- 752
57
votes
1
answer
2k
views
Does the average primeness of natural numbers tend to zero?
Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
- 18.2k
55
votes
0
answers
2k
views
What Rubik's Twist configuration has the lowest visible surface area?
The Rubik's Twist has been a fun time sink. From the wiki page,
[It] is a toy with twenty-four wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that ...
- 101
53
votes
1
answer
2k
views
$2^n$th decimal place of $\sqrt{2}.$
Someone on Art of Problem Solving claims to know how to calculate the $2^{2020}$th decimal place of $\sqrt{2},$ and will tell us if everyone gives up. Brute force will not work, nor will a BBP style ...
- 4,914
50
votes
0
answers
2k
views
A Nice Problem In Additive Number Theory
$\color{red}{\mathbf{Problem\!:}}$ $n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...
- 5,899
49
votes
0
answers
1k
views
How many ways can I arrange the numbers $1$ to $N$ with this divisibility condition?
For the numbers $1, \ldots, N$, how many ways can I arrange them such that either:
The number at $i$ is evenly divisible by $i$, or
$i$ is evenly divisible by the number at $i$.
Example: for $N = 2$,...
- 599
45
votes
0
answers
923
views
How are topological invariants obtained from TQFTs used in practice?
Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places:
Atiyah, Topological quantum field theory
Lurie, Topological Quantum Field Theory ...
- 571
44
votes
0
answers
2k
views
Pattern in Pascal's triangle
Updated question
This "reverse" pattern can be plotted as a function of a triangle, read
by rows:
$$ T(n,k) = (\delta)^k F\binom{n}{k} \left\lfloor f(t(k)) \right\rfloor ,\delta\in\{1,-1\}.$$...
- 12.4k
43
votes
0
answers
5k
views
Arithmetic and geometric genus
There are two notion of genus in algebraic geometry, namely arithmetic genus $p_a=(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)$ and geometric genus $p_g=\dim H^0(X,\Omega^{\dim X})$. I keep forgetting ...
- 4,951
42
votes
0
answers
1k
views
On the equivalence relation $(a,b) \sim (c,d)\iff a+b=c+d$
Setup
Let $A=\{a_1<a_2<\cdots<a_p\}$ and $B=\{b_1<b_2<\cdots<b_q\}$ be two finite sets of real numbers. Define an equivalence relation $\sim_{A,B}$ on $R_{p,q}=\{1,\dots,p\}\times\{1,...
- 19.9k
40
votes
0
answers
9k
views
Regarding metrizability of weak/weak* topology and separability of Banach spaces.
Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known
Theorem. If $X$ is separable, then $\...
user160185
39
votes
0
answers
917
views
Does there exist a polynomial $P(x,y)$ which detects all non-squares?
Problem. Does there exist a two-variable polynomial $P(x, y)$ with integer coefficients such that a positive integer $n$ is not a perfect square if and only if there is a pair $(x, y)$ of positive ...
- 10.7k
39
votes
0
answers
722
views
$A$ and $B$ commute on a dense set but $e^{iA}$ and $e^{iB}$ do not
Let $A$ and $B$ be unbounded, symmetric operators on a Hilbert space $H$ with a common domain $D$. If $AB = BA$ on $D$, is it necessarily that case that $e^{iA}$ and $e^{iB}$ also commute? If $A$ and $...
- 11.6k
38
votes
0
answers
930
views
Using the Brun Sieve to show very weak approximation to twin prime conjecture
I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly.
I don't really know much about ...
- 89.8k
38
votes
0
answers
726
views
Continued fraction with prime reciprocal entries
We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
- 30.8k
38
votes
0
answers
928
views
Closed model categories in the sense of Quillen [1969] vs the modern sense
The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition:
Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
- 88.7k
37
votes
0
answers
820
views
The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms
PROBLEM. Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms.
It suffices to show that the terms of the sequence
$$\,b_n=\mathrm{e}^...
- 82.6k
37
votes
0
answers
3k
views
This should be a piece of cake...... right?
You probably know the following problem:
We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only cut ...
- 7,600
36
votes
0
answers
737
views
Grasshopper jumping on circles
Can we characterize the grasshopper sequence?
Let $n\in\mathbb N$ be the number of stones $s\in\{0,1,2\dots,n-1\}=S$ on a circle that the grasshopper can jump on. Let $v(s)$ be the number of times ...
- 12.4k
36
votes
0
answers
608
views
Is every finite list of integers coprime to $n$ congruent $\pmod n$ to a list of consecutive primes?
For example the list $(2, 1, 2, 1)$ is congruent $\pmod 3$ to the consecutive primes $(5, 7, 11, 13)$. But how about the list $(1,1,1,1,1,1,1,1,2,3,4,3,2,3,1) \mod 5$?
More generally, we are given ...
- 1,948
36
votes
0
answers
1k
views
Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?
Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
- 1,001
35
votes
0
answers
973
views
Does any uncountable complete theory have exactly two countable models?
The following is a theorem by Vaught.
Theorem. Let $T$ be a complete theory in a countable language.
Then, $T$ cannot have exactly two countably infinite models (up to isomorphism).
A proof can ...
- 615
35
votes
0
answers
1k
views
Can you make the triviality of $\langle a,b,c \mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 \rangle$ more trivial?
I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.)
Let $G$ be a group, and let $a,b,c\...
- 72.3k
35
votes
0
answers
1k
views
Constructing an infinite chain of subfields of 'hyper' algebraic numbers?
This has now been cross posted to MO.
Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn ...
- 3,760
34
votes
0
answers
1k
views
Exponential map on the ellipsoid.
Consider the ellipsoid $M \subset \mathbb{R}^3$ defined by
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$
where $0 < a < b < c$, equipped with the usual Riemannian induced ...
- 3,867
34
votes
0
answers
537
views
Does this Condition on Exit Times imply $X_t$ is a Local Supermartingale?
Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $$\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}.$$ We can interpret $\tau_{s,a,b}$ as the ...
- 4,562
33
votes
0
answers
534
views
An iterative logarithmic transformation of a power series
Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion:
$$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$
Then, at each step ...
- 46.8k
33
votes
0
answers
840
views
Colossally abundant numbers and the Riemann hypothesis
[This question has lead me to ask a follow up on MathOverflow.]
A recent tweet by John Baez has reminded me of the astonishing fact$^1$ that the Riemann hypothesis (RH) can be disproved by finding a ...
- 73
33
votes
0
answers
546
views
Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations
Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$
...
- 66.8k
33
votes
0
answers
674
views
Visualizing the Partition numbers (suggestions for visualization techniques)
So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
- 5,648