329
votes
0answers
16k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
188
votes
0answers
6k views

The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
164
votes
0answers
4k views

Does there exist 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, $f(S)\subset Y$ ...
106
votes
0answers
3k views

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
90
votes
0answers
2k views

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
79
votes
0answers
2k views

Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
75
votes
0answers
1k views

Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
74
votes
0answers
878 views

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's ...
73
votes
0answers
3k views

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $...
70
votes
0answers
3k views

Making Friends around a Circular Table

I have $n$ people seated around a circular table, initially in random order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every ...
69
votes
0answers
2k views

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...
66
votes
0answers
1k views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
58
votes
0answers
3k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\...
55
votes
0answers
2k views

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty ...
48
votes
0answers
572 views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
46
votes
0answers
1k views

Application of Hilbert's basis theorem in representation theory

In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set ...
46
votes
0answers
1k views

Is $ f_n=\frac{(x+1)^n-(x^n+1)}{x}$ irreducible over $\mathbf{Z}$ for arbitrary $n$?

In this document on page $3$ I found an interesting polynomial: $$f_n=\frac{(x+1)^n-(x^n+1)}{x}$$ Question is whether this polynomial is irreducible over $\mathbf{Q}$ for arbitrary $n \geq 1$ ? In ...
45
votes
0answers
860 views

The Right Triangle Game

I am looking for a deeper understanding, particularly the optimum strategy and the maximum score as a function of grid size, of the following (single-player) game played with an $n$ by $m$ grid: ($6 ...
41
votes
0answers
2k views

Does this sequence have any mathematical significance?

Take the sequence 001 and repeatedly append its second half to itself, using the larger half if the length is odd. This gives you ...
40
votes
0answers
1k views

Computing the Chern-Simons invariant of SO(3)

I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
38
votes
0answers
1k views

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
38
votes
0answers
989 views

What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
36
votes
0answers
652 views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
34
votes
0answers
416 views

Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{...
33
votes
0answers
1k views

Irrationality of $e$

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^...
33
votes
0answers
2k views

Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f: X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\mathcal{...
32
votes
0answers
557 views

On Ramanujan's curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $

In Ramanujan's Notebooks, Vol IV, p.20, there is the rather curious, $$\sqrt{2\,\Big(1-\frac{1}{3^2}\Big) \Big(1-\frac{1}{7^2}\Big)\Big(1-\frac{1}{11^2}\Big)\Big(1-\frac{1}{19^2}\Big)} = \Big(1+\frac{...
32
votes
0answers
1k views

Is this *really* a categorical approach to *integration*?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
31
votes
0answers
340 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 $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
31
votes
0answers
544 views

What is that thing that keeps showing in papers on different fields?

A few months ago, when I was studying strategies for the evaluation of functional programs, I found that the optimal algorithm uses something called Interaction Combinators, a graph system based on a ...
30
votes
0answers
779 views

How to solve this two simultaneous “divisibilities” : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I can not progress anymore......
28
votes
0answers
495 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 $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
27
votes
0answers
695 views

Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of ...
26
votes
0answers
534 views

How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is $M$...
25
votes
0answers
800 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
25
votes
0answers
527 views

How prove $ \sum (a+b+c+d)\sum\frac{ab+ac+ad+bc+bd+cd}{a+b+c+d}\sum\frac{abc+abd+bcd+cda}{ab+ac+ad+bc+bd+cd}\leq \sum a\sum b\sum c\sum d$

let $a_{i}>0,b_{i}>0,c_{i}>0,d_{i}>0,i=1,2,\cdots,n $ show that $$\sum_{i=1}^{n}(a_{i}+b_{i}+c_{i}+d_{i})\sum_{i=1}^{n}\dfrac{a_{i}b_{i}+b_{i}c_{i}+c_{i}d_{i}+d_{i}a_{i}+a_{i}c_{i}+b_{i}...
25
votes
0answers
322 views

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
25
votes
0answers
663 views

When are nonintersecting finite degree field extensions linearly disjoint?

Let $F$ be a field, and let $K,L$ be finite degree field extensions of $F$ inside a common algebraic closure. Consider the following two properties: (i) $K$ and $L$ are linearly disjoint over $F$: ...
25
votes
0answers
1k views

On Ramanujan's Question 359

In JIMS 4, p.78, Question 359 was asked by Ramanujan. (See The Problems Submitted by Ramanujan to the Journal of the Indian Mathematical Society, p. 9, by Bruce Berndt, et al.) If, $$\sin(x+y) = 2\...
24
votes
0answers
502 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} \...
23
votes
0answers
529 views

Numbers $n$ such that the digit sum of $n^2$ is a square

Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. About a month ago, a friend of mine taught me the following: $$S\left(9\color{red}{^2}\right)=S(81)=8+1=3\color{red}{^2}$$ $$S\...
23
votes
0answers
496 views

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

Question : For every even $k\ge 4$, is the following $(\star)$ true? $$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$.}\...
23
votes
0answers
381 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 ...
23
votes
0answers
626 views

Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some ...
22
votes
0answers
638 views

Formulae of the Year 2016

Decode the following limits to welcome the new year! This is my love limits (Created by me). I hope you Love it. Let $$A_{n}=\dfrac{n}{n^2+1}+\dfrac{n}{n^2+2^2}+\cdots+\dfrac{n}{n^2+n^2}$$ show that ...
22
votes
0answers
335 views

Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
22
votes
0answers
462 views

Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?

This question is inspired by my answer to the question "How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?". The sums $f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ (for positive integer $...
21
votes
0answers
175 views

Painting the plane red and blue: Is it possible for each unit circumference to contain exactly $n$ blue points?

I recently stumbled upon the following problem: Consider the plane: You may color each point either red or blue. Is there a way to color it such that each unit circumference (centred anywhere) ...
21
votes
0answers
308 views

Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? Context: ...
21
votes
0answers
210 views

How smooth can non-nice associative operations on the reals be?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is "no" -- a ...

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