# Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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### Who classified varieties that are commutative groups?

Who are the authors of the theorems asserting that connected varieties/manifolds which are abelian groups are isomorphic to ${\bf R}^k \times {\bf T}^n$? In the smooth setting, I guess this is due to ...
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### If $M$ and $N$ are closed and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic?

If $M$ and $N$ are closed smooth manifolds, and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic?
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### Group structure for distributive lattices

On the (finite) Boolean lattice there is a group structure given by the symmetric difference and this group is an elementary abelian 2-group. Question: Does there exist a natural group structure on ...
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### Density function approximation with respect to $L^1$ distance

Given iid samples $X_1,...,X_N$ drawn from some unknown distribution with not necessarily continuous density function $f(x)$ are there any theorems/papers where based on the data $X_1,...,X_N$ an ...
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### Average value of a fractional part of a function

Let $f(x): \mathbb{R} \to \mathbb{R}_{\geq 0}$ be a smooth function. I am interested in estimating sums of the form $$\sum_{ A < n \leq B } \{ f(n)\}$$ where $\{ c \}$ denotes the fractional part ...
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### Dirichlet's unit theorem in finite characteristic

I'm looking for a source of the following analog of Dirichlet's unit theorem for finite characteristic fields: Let $\mathbb{F}_p$ be a finite field and denote $K=\mathbb{F}_p(x)$ to be the field of ...
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### Polynomial isometries of $\mathbb{A}^2_\mathbb{C}$

I have the following question, which I'm sure must be explored somewhere. Consider a group of polynomial automorphisms of $\mathbb{A}^2_\mathbb{C}$ preserving a standard hermitian metric. Is there any ...
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### versal deformation ring of a p-divisible group with some tensors

I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with ...
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### References for quivers and derived categories of coherent sheaves for a string theory student

I'm a student mostly from physics knowledge hoping to learn about the math involved the string theory paper Topological Quiver Matrix Models and Quantum Foam. Context: The topological string theory ...
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### Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks

A version of Brauer's second main theorem is as follows: Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$. If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
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### A modern reference to the Zsygmondy Theorem

I need to cite the classical Zsigmondy Theorem, which was proved in 1892. Is there any modern reference to this theorem? I mean some standard textbook in Number Theory containing this theorem together ...
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### A generalization of metric spaces

Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i....
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### Necessity of conditions $N$ odd, square-free and $\chi$ quadratic in Kohnen's plus space - modular forms of half-integral weight

Kohnen introduced the "plus" space as a subspace of the space of modular forms of half integral weight, first in his 1980 paper and then generalized the work in a later 1982 paper. Why is ...
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### Lower bound for diagonal Ramsey numbers —- reference request

Using the first moment method, in 1947 Erd\H{o}s gave a lower bound on the diagonal Ramsey numbers $R(k,k)$: $$R(k,k) \geq (1+o(1))\frac{k}{e\sqrt{2}} 2^{k/2}.$$ In 1975 Spenser used the Lov\’asz ...
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### How to describe the compact real forms of the exceptional Lie groups as matrix groups?

I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe ...
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### Is there a better reference for existence/regularity for parabolic PDEs (and systems) than the book of Ladyzenskaja, Solonnikov, Uralceva?

The book of Ladyzenskaja, Solonnikov, Uralceva contains almost everything most people need yet the typesetting and notation is disgusting to the eye. Is there any better text that covers the same type ...
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