# All Questions

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### Any more generalization of Fermat's Little Theorem? [closed]

Fermat's Little Theorem: If $p$ is a prime and $\gcd(a,p)=1$ then $a^{p-1} \equiv1\pmod p$. Over the years, Fermat's Little Theorem have been generalized in several ways. I am aware of four different ...
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### Model for the (infinity,1)-category of (homotopy-)limit preserving functors

I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...
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### Functor of order $n$ in Mumford's abelian variety

Let $T$ be a contravariant functor on the category of complete varieties into the Category $\underline{\mathrm{Ab}}$ of abelian groups. Let $X_0,\ldots,X_n$ be any system of complete varieties, ...
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### Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
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### simplicial spaces without degeneracies

Suppose I have a simplicial space $X_{\bullet}$ without degeneracies (sometimes called semi-simplicial space or incomplete simplicial space). There still is a geometric realization $\lVert X \rVert$ ...
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### Closure of the graph of a function

Let $X_1, X_2$ be non-empty sets and let $R\subseteq X_1\times X_2$ such that for all $x\in X_1$ there is $y\in X_2$ such that $(x,y)\in R$. Are there topologies $\tau_i$ on $X_i$ for $i=1,2$ and a ...
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### Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...
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### 3 possible tensors in 2-categories?

let $\mathcal{A}$ be a 2-category, consider: $$\mathcal{A}(W \otimes_i A, X) \simeq_i \mathcal{C}at(W, \mathcal{A}(A, X)) \;\;\; i = 1, \: 2, \: 3.$$ where $W$ is a category, and $A$, ...
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### Groups with many vanishing elements

It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is What can be said about a finite group $G$ for which ...
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### On Thompson conjecture

Let $N(G)$ be the set of conjugacy class sizes of finite group $G$. Let $G$ be a finite group for which $N(G)=N(A_n)$, where $A_n$ is the alternating group of degree $n$. Let $p$ and $q$ be distinct ...
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### Signs and value of higher order terms in the Taylor expansion of a strongly convex function

Say that I have a function $f: S \rightarrow \mathbb{R}^+$ such that: $S \subset \mathbb{R}^n$ is a closed convex set such as $S=[-10,10]^n$ $f$ is continuous and infinitely differentiable at all ...
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### Dense subsets in tensor products of Banach spaces [migrated]

Assume $B_1$ and $B_2$ are Banach spaces of univariate functions. Moreover, assume that the sets $D_1 \subset B_1$ and $D_2 \subset B_2$ are dense with respect to the respective norms ...
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### solvability of linear elliptic pde on a torus

Consider a linear non-divergent form elliptic PDE on a flat torus $\mathbf{T}^n$, $$a_{ij}\partial_{ij}u+b_i\partial_iu=f$$ where all the coefficients and $f$ are smooth. What is the condition that ...
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### Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...
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### Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...
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### Saturation of null ideal

In ZFC, can we find more than continuum many non null sets of reals whose pairwise intersections are null?
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### Blow-up as polar coordinates?

While doing some explicit calculations involving a blow-up of the plane in a point, I realised what I was doing was basically writing things in polar coordinates. Somewhat astonished that I hadn't ...
### What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?
Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...