**1**

vote

**0**answers

56 views

### Is Carlos Simpson's Descent available online?

I am not sure whether this question is suitable for MO. Is the paper "Descent" by Carlos Simpson in the book "Alexandre Grothendieck: A Mathematical Portrait" page 83-142 (or a similar version of that ...

**0**

votes

**0**answers

5 views

### Generate harmonic polynomials for a finite group

Let $G$ be a finite group and $H_G$ be the set of harmonic polynomials. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. ...

**3**

votes

**2**answers

153 views

### Are there other nontrivial integer solutions to the equation $9x^3 -1 = y^3$ besides $(x,y) =(1,2)$?

Does the above Diophantine equation have other integer solutions besides $(x,y)=(1,2)$ and $(x, y) = (0, -1)$?

**0**

votes

**1**answer

27 views

### References about Hasse diagrams of root systems

This is to ask about references of Hasse diagrams of irreducible root systems. I found here and there nice pictures of root systems of type $E$. I would like to ask for Hasse diagrams of classical ...

**1**

vote

**1**answer

20 views

### Construction of $n$ makes $s_2(nk)<s_2(n)$

$s_2(n)$ denotes the sum of the standard base-2 digits of $n$.
For a fixed odd number $k>1$, can we construct $n\in \mathbb{Z}^+$, to make $s_2(nk)<s_2(n)$?
To clarify, that's not $s_2(nk) \lt ...

**9**

votes

**2**answers

248 views

### What are the implications of the simple loop conjecture?

Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol.
Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow ...

**4**

votes

**0**answers

32 views

### Kernels of $SL(2,\mathbb{Z})$ representations from modular tensor categories

It is well known that if $\mathcal{C}$ is a modular tensor category then one may construct a representation of $SL(2,\mathbb{Z})$ using the $S$ and $T$ matrices of $\mathcal{C}$. This representations ...

**6**

votes

**1**answer

148 views

### Is it possible to make an algorithm that could predict the likelihood that a program will halt? [on hold]

Today I began to read about computability theory. I do not even have an elementary understanding of the topic but it certainly got me thinking. I know there is there is no 'one-for-all' algorithm that ...

**7**

votes

**0**answers

57 views

### Fundamental units with norm $-1$ in real quadratic fields

If we have distinct primes $p \equiv q \equiv 1 \pmod 4,$ with Legendre $(p|q) = (q|p) = -1,$ there is a solution to $u^2 - pq v^2 = -1$ in integers and the fundamental unit of $O_{\mathbb ...

**0**

votes

**0**answers

14 views

### A variance-preserving Boolean function

Let a random variable $X$ be given with $P_X$ supported over $\mathcal{X}$. What are the necessary conditions for the existence of a boolean function $f:\mathcal{X}\to \{0,1\}$ such that ...

**3**

votes

**3**answers

338 views

### Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it ...

**7**

votes

**2**answers

161 views

### Spectrum of Ring of Smooth Functions on $\mathbb{R}^n$

When we define smooth manifold, we starting with topological space $M$ which localy homeomorphic to $\mathbb{R}^n$ and setting up sheaf $\mathscr{F}(M)$ of functions on it which localy isomorphic to ...

**11**

votes

**2**answers

174 views

### “Fractally self-similar” numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at ...

**-1**

votes

**0**answers

23 views

### Examples of real orbits with irrational period

Im looking for examples of periodic orbits on the real line given by piecewise analytic functions $g$.
More specific at most 2 pieces.
Im talking about integer iterations starting at $f(0)=0$ and with ...

**2**

votes

**0**answers

25 views

### Positivity of semiclassical pseudodifferential operators

Let me first give some background. (My reference is Martinez's book An introduction to semiclassical and microlocal analysis)
Let $m\in\mathbb{R}$, and ...

**0**

votes

**0**answers

64 views

### Problem related to Frobenius coin problem

Call a pair of integers $a,b\in\Bbb N$ with $\mathsf{gcd}(a,b)=1$ a good coprime pair if
$$ax+by=rs\mbox{ and }ax'+by'=tu\mbox{ for some }r,s,t,u,x,y,x',y'>0$$
$$\implies \mbox{} av+bw=ru\mbox{ and ...

**3**

votes

**1**answer

116 views

### Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition
\begin{equation}
...

**-1**

votes

**0**answers

47 views

### Does this product over primes converge for all non-principal Dirichlet Characters for $\Re(s) > \frac12$?

I like to expand on this this question:
Numerical evidence suggests that the following product over primes ($p$):
$$\displaystyle F_n(s):= \prod_{p}^\infty ...

**3**

votes

**1**answer

465 views

### A generalized theorem of Hall's marriage theorem

We all know Hall's marriage theorem as following:
A bipartite graph $G$ with bipartition $\{ A,B \}$ contains a matching of $A$ if and only if $|N(S)|\geq |S|$ for all $S\subseteq A$.
And I am ...

**3**

votes

**2**answers

45 views

### Affine hull of a set of non-negative matrices with fixed row-sums

Fix any non-negative matrix $M \in \mathbb{R}_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column.
Further, fix any positive vector $r \in \mathbb{R}_{> 0}^m$.
With $nz(M) := ...

**4**

votes

**1**answer

99 views

### geometric interpretation of derivation between two algebras

Given a smooth manifold $M,$ it is well known that any derivation of the algebra of smooth functions $C^{\infty}(M)$ can be seen (or it is associated to) a smooth vector field on $M.$ I am looking for ...

**0**

votes

**0**answers

34 views

### Random process & probability problem met in wireless communication

A random process r obeys the following distribution:
$p(r,ṙ)=rb_0\exp(−\frac{r^2}{2b_0})\sqrt{2πb_2}exp(−\frac{\dot{r}^2}{2b_2})$, where $\dot{r}$ is the derivative of r in the time domain.
You can ...

**3**

votes

**0**answers

121 views

### Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth.
Let's ...

**-1**

votes

**0**answers

79 views

### On the Theory of Infinite Step Processes of Sequential Decision Making [on hold]

On the border of Set Theory and Game Theory there is a broad class of Infinite Step Processes of Sequential Decision Making that can be characterized by the main following property: a Future ...

**0**

votes

**0**answers

45 views

### Regarding a new divergence function of two probability distributions

Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and any constant $0 \leq \alpha \leq 1$, ...

**0**

votes

**0**answers

72 views

### Hodge metric on pair

I am looking for the definition of Hodge metric, like definition 2.2 here
https://hal.archives-ouvertes.fr/hal-00322845/document
But instead of Vector bundle if we have divisor $D$ with conic ...

**2**

votes

**0**answers

36 views

### Geometric unfolding of a difference equation

Does anyone know a link to the (superb) slides of a talk given by Zeeman in 1996, I think (with same title as this question)? It used to be at www.math.utsa.edu/ecz/gu.html .

**2**

votes

**2**answers

202 views

### Fourier transform localisation (still unanswered, but apparently off-topic?) [on hold]

In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...

**7**

votes

**0**answers

94 views

### Convergence rate of Fagin's 0-1 law for first-order properties of random graphs

Fagin's 0-1 law for first-order properties of random graphs states that, for every first-order sentence in the logic of graphs, the probability that a uniformly random $n$-vertex graph models the ...

**1**

vote

**2**answers

279 views

### Polynomials which always assume perfect power values

Let $f(x)$ be a non-constant polynomial with integer coefficients. It is a well-known result that if $f(n)$ is a square for all integers $n$, then $f$ must in fact be the square of a polynomial (see, ...

**14**

votes

**1**answer

339 views

### Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction),
as defined by Grayson in Higher algebraic K-theory: II (page 219),
takes as an ...

**5**

votes

**1**answer

130 views

### How to determine whether a power of eta function is a eigenform? [on hold]

I find that it is complicated to do this from the definition. In fact, I know that $\eta^k(m z)$ is a eigenform for $k=1,2,3,4,6,8,12,24$ and $m=\frac{24}{gcd(k,24)}$. What I want to know is the cases ...

**7**

votes

**0**answers

95 views

### Adding minimal subsets to $\aleph_\omega$

Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all
$\alpha < \kappa.$
Question. Is it consistent that there ...

**3**

votes

**0**answers

125 views

### Many-sorted nominal sets as sheaves

The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...

**1**

vote

**1**answer

65 views

### concentration inequality for $d$-dimensional martingale

Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...

**2**

votes

**0**answers

72 views

### Integral representation of the complex homogeneous polynomial $z_1\cdots z_n$

Consider the transform (see e.g., (5.1) in this paper):
\begin{equation*}
\Lambda_\mu(q)(z) := \int_{\Delta_n} ...

**0**

votes

**0**answers

31 views

### Complexity theory and closed form formulas in analysis

My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...

**3**

votes

**0**answers

112 views

### On the relationship between two lesser-known recurrence relations

On January 2004, in his work Integer-valued polynomials on prime numbers and
logarithm power expansion, Jean-Luc Chabert showed that
\begin{equation} \left(-\frac{\ln(1-x)}{x}\right)^m = \sum_{n = ...

**2**

votes

**1**answer

355 views

### Is there a shorter proof of Fermat's Last Theorem for $n=4$ than that of infinite descent? [on hold]

Out of curiosity, i'm wondering whether there exists a shorter proof of FLT for $n=4$ with respect to the one of infinite descent ?
The Wikipedia article on this subject states that more proofs were ...

**34**

votes

**1**answer

542 views

### Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : number of groups of order $n$
Does $N(n)=n$ hold for some $n>1$ ?
I checked the OEIS-sequence https://oeis.org/A000001 as well as the squarefree numbers in the range ...

**2**

votes

**1**answer

126 views

### Can we Characterise Rings of Continuous Functions?

Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...

**6**

votes

**2**answers

216 views

### non commutative polynomial which is zero for all matrix evaluation

I want to work on $K$ an algebraic closed (commutative) field of characteristic zero (even if it seems to be more general).
We can define the free K-algebra of polynomials in non commutative ...

**15**

votes

**1**answer

223 views

### What is higher equivariant homotopy?

In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is ...

**8**

votes

**1**answer

89 views

### Compute the index of the Dirac operator on $C_0(R^2)$ to obtain Bott element in $K_0$

I am studying the paper of Baum-Connes-Higson to understand the Connes-Kasparov conjecture. In example 4.23, they discuss the case $G=\mathbb{R}^2$. I have constructed the Dirac operator, but I’m ...

**0**

votes

**0**answers

22 views

### Existance of a fixed point

Let $[a,b]$, $[c,d]$ be two intervals of $\mathbb{R}$ with $a\lt c\lt b\lt d$ and $f:[a,b]\to[c,d]$ be a bijective, smooth function.
My question is:
If $[a,b]\not=[c,d],$ which condition(s) ...

**15**

votes

**3**answers

876 views

### Number theory and physics

I was following some lectures by Edward Frenkel about Langlands correspondence. He was describing some analogies between number theory and theoretical physics (Mirror symmetry). At some point ( my ...

**34**

votes

**29**answers

7k views

### Most intriguing mathematical epigraphs

Good epigraphs may attract more readers. Sometimes it is necessary.
Usually epigraphs are interesting but not intriguing.
To pick up an epigraph is some kind of nearly mathematical problem: it ...

**3**

votes

**1**answer

122 views

### a property implying co-circularity

Let $A_1, A_2,\ldots, A_n$ be $n$ distinct points in the plane.
For every $1\le i\le n$, let $D_i$ be the sum of the distances from point $A_i$ to all the other points.
Suppose that $D_i=D_j$ for ...

**2**

votes

**0**answers

63 views

### A construction of the fundamental solution for Schroedinger equations

Does someone know some book or lecture notes useful for the reading of the paper
"A construction of the fundamental solution for the Schrödinger equation", Fujiwara, Daisuke, J. Analyse Math. 35 ...

**35**

votes

**4**answers

4k views

### Are dagger categories truly evil?

Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the ...