# All Questions

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### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and ...
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### Exponential Sum Bound

In http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf, Bruedern and Wooley mention the following fact on the bottom of page 6: Let ...
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### Estimating the volume of a union of balls

Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$. If I want to estimate $$\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}$$ where ...
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### Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*} f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...
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### Weighted Kaplan-Meier estimator

Let two samples $(T_1, \ldots ,T_n)\sim F$ and $(C_1, \ldots ,C_n)\sim G$ are given, but not observed. Instead we observe $\tilde T_i = \min (T_i, C_i)$ and $\Delta _i = \mathbf{1}(T_i \leq C_i)$, ...
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### What does a Turing machine compute? [on hold]

I suspect that it might be necessary to define for a Turing machine how its inputs and outputs are to be interpreted in order to be able to say e.g. that a Turing machine $T$ computes an arithmetical ...
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### Focal points for the exponential map and Jacobi fields

It is known that in a Riemannian manifold $(M,g)$, if there is a closed geodesic and a non-zero, periodic, non-constant Jacobi field along it, then M has a focal point. Is the converse true? That is ...
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### Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...
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### PhD, vector calculus [on hold]

Are there published papers showing symmetric versions of the gradient and the Laplacian on the surface of a sphere? Let X, Y and Z be mutually perpendicular unit vectors in three space. A sphere of ...
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### An angle-doubling trick of Kirillov and Berenstein [on hold]

Kirillov and Berenstein, in their article "Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux" (available at math.uoregon.edu/~arkadiy/bk1.pdf), present a ...
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### Probability of matching under cyclic permutations

In A conjecture about the entropy of matrix vector products I asked a conjecture relating to the entropy of a matrix-vector product. This conjecture is as yet unproven. domotorp then made another ...
Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...