# All Questions

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### 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 ...
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### 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. ...
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### 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 $(x, y) = (0, -1)$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $0 \leq \alpha \leq 1$, ...
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### 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 ...
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### 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 .
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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} ...
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### 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 $(q(t),p(t))$ ...
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### 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 \left(-\frac{\ln(1-x)}{x}\right)^m = \sum_{n = ...
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### 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 ...
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### 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 ...
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### 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 $C$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...