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A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?

Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't ...
Martin Brandenburg's user avatar
617 votes
0 answers
24k 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, $f(S)\subset Y$ ...
Willie Wong's user avatar
  • 73.8k
183 votes
0 answers
5k 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 ...
daniel's user avatar
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177 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 $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ...
Eins Null's user avatar
  • 2,177
146 votes
0 answers
11k 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 $\...
Matt's user avatar
  • 7,428
135 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 $K$. ...
Eric Wofsey's user avatar
131 votes
0 answers
5k 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? ...
Ed Pegg's user avatar
  • 21.4k
121 votes
0 answers
3k 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 ...
Elias Costa's user avatar
  • 14.8k
115 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 ...
Martin Brandenburg's user avatar
114 votes
0 answers
4k 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 ...
CODE's user avatar
  • 4,991
110 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'\...
user avatar
108 votes
0 answers
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 ...
RavenclawPrefect's user avatar
99 votes
0 answers
3k 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 $1$, ...
Exodd's user avatar
  • 11.4k
95 votes
0 answers
3k 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 ...
hase_olaf's user avatar
  • 1,528
85 votes
0 answers
2k 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 $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
Chain Markov's user avatar
  • 15.7k
85 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 ...
Primo Petri's user avatar
  • 5,234
81 votes
0 answers
1k 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 ...
pregunton's user avatar
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77 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, ...
user avatar
75 votes
0 answers
1k views

If $f(n)$ is the number of groups of order $n$, then is $f(a)\cdot f(b)\leq f(a\cdot b)$?

Let $f(n)$ be the number of groups of order $n$ up to isomorphism. We want to prove that: $$f(a) \cdot f(b) \leq f(a \cdot b)$$ for all nonnegative integers $a$ and $b$. Our progress: If $a \cdot b \...
Jorge Rael's user avatar
71 votes
0 answers
942 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} \...
user02138's user avatar
  • 17.1k
63 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} & \...
Rofl Ukulus's user avatar
58 votes
0 answers
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 ...
Nilotpal Sinha's user avatar
56 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 ...
Elle's user avatar
  • 111
53 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 ...
ShBh's user avatar
  • 6,134
50 votes
0 answers
1k 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 ...
Melissa's user avatar
  • 621
47 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,...
Olivier Bégassat's user avatar
46 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: $$ T(n,k) = (\delta)^k F\binom{n}{k} \left\lfloor f(t(k)) \right\rfloor ,\delta\in\{1,-1\}.$$...
Vepir's user avatar
  • 12.5k
42 votes
0 answers
1k 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 ...
Prism's user avatar
  • 11.3k
42 votes
0 answers
1k 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}^...
Yiorgos S. Smyrlis's user avatar
42 votes
0 answers
10k 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 $\...
user avatar
42 votes
1 answer
3k views

Cutting $n$ circular cakes of different radii but equal heights into $p>n$ equal shares

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 ...
Hippalectryon's user avatar
41 votes
0 answers
987 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 ...
davidlowryduda's user avatar
  • 92.6k
41 votes
0 answers
780 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 $...
user15464's user avatar
  • 11.8k
40 votes
0 answers
781 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. ...
G Tony Jacobs's user avatar
39 votes
0 answers
1k views

Dividing a polyhedron into two similar copies of itself

The paper Dividing a polygon into two similar polygons provides that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original). Right ...
Numeral's user avatar
  • 1,860
39 votes
0 answers
977 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 ...
Zhen Lin's user avatar
  • 90.9k
38 votes
0 answers
865 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 ...
Vepir's user avatar
  • 12.5k
37 votes
0 answers
656 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 ...
user avatar
37 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 ...
Rodrigo's user avatar
  • 1,043
36 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 ...
Mason's user avatar
  • 3,813
35 votes
0 answers
2k views

Are these generalizations known in the literature?

By using $$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$ and $$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(...
Ali Shadhar's user avatar
  • 25.8k
35 votes
0 answers
1k views

If $K\cong K(X)$ then must $K$ be a field of rational functions in infinitely many variables?

If $k$ is any field, then the field $K=k(X_0,X_1,\dots)$ of rational functions in infinitely many variables satisfies $K(X)\cong K$ (by mapping $X$ to $X_0$ and $X_n$ to $X_{n+1}$). My question is, ...
Eric Wofsey's user avatar
35 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 ...
student's user avatar
  • 3,907
34 votes
0 answers
596 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 ...
Vladimir Reshetnikov's user avatar
34 votes
0 answers
578 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 $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
goblin GONE's user avatar
  • 68.1k
34 votes
0 answers
1k views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $$V= \begin{bmatrix} ...
trion's user avatar
  • 353
34 votes
0 answers
575 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 ...
Ben Derrett's user avatar
  • 4,602
34 votes
0 answers
710 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 ...
graveolensa's user avatar
  • 5,698
34 votes
2 answers
2k views

How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and four ...
Yuriy S's user avatar
  • 31.6k
33 votes
0 answers
901 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 ...
ntessore's user avatar

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