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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2
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0answers
98 views

$\mathscr Coh_{X|S} $ is algebraic and of finite type

Let $S$ be a Noetherian scheme and $X$ a projective $S$-scheme. Reading Laumon-Moret--Bailly's "Champs Algebriques", Theorem 4.6.2.1: $ \mathscr Coh_{X|S} $ and $ \mathscr Fib_{X|S,r} $ are ...
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243 views

A measure of non-uniformity of a vector/probability distribution?

In the course of a research project about discrete probability distributions, my coauthors and I keep seeing some quantity appear, and I would like to understand whether it has been studied or has a ...
1
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1answer
71 views

References about transfinite socle series

I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series. Let $R$ be an associative unital ring and $...
0
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1answer
76 views

Definition of center of ternary ring of operators

Let $H$ and $K$ be Hilbert spaces and $B(H,K)$ denotes the space of bounded operators from $H$ to $K$. Recall that a ternary ring of operators (TRO) $V$ is a closed subspace of $B(H,K)$ which is ...
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40 views

What is the maximal weight submodule of $\text{Hom}_{\mathfrak{g}}(M,N)$?

Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. For ...
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2answers
2k views

Why is the Vandermonde determinant harmonic?

It can be checked that the Vandermonde determinant defined as $$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$ is a harmonic function, that is $\Delta V = 0$ where ...
1
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1answer
193 views

A paper by W. Ljunggren

I am looking for the following paper by Ljunggren, Wilhelm: "Zur Theorie der Gleichung $x^2 + 1 = Dy^4$", Avh. Norske Vid. Akad. Oslo. I., 1942 (5): 27 The main result of this paper which I am ...
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0answers
31 views

Outer-regular product of $\tau$-additive measures

Due to the deficiencies of the simple product measure defined on measurable rectangles, there have been many different constructions of product measures in more specialized circumstances. Originally, ...
5
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2answers
130 views

Properties of measures that are not countably additive but have countably additive null ideals

This is a very naive question, maybe more of a reference request than anything else. Let $(X, \mathcal X)$ be a measurable space. If $m$ is a real-valued function on $\mathcal X$, we say that $m$ has ...
3
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0answers
52 views

Hales' generalization of the stacked bases theorem (seeking a proof)

In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
1
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1answer
90 views

Survey/references on random geometric $K$-NN – $K$-nearest-neighbour graphs?

[Edit:] Some related info on number of connected components of NN-graphs can be found here: https://cstheory.stackexchange.com/a/47037/2408 Sample $N$ points in $\mathbb{R}^d$ from some distribution $...
2
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1answer
40 views

Do continuous motions of the vertices of convex polyhedra that maintain local convexity imply global convexity? (Reference request)

A convex polyhedron has all of its internal dihedral angles in $(0, \pi)$. However, if I start with an abstract polyhedron $P$, let's say a triangulated one, so I don't have to worry about planarity ...
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0answers
57 views

Outer automorphisms of incidence algebras of distributive lattices

Let $L$ be a finite distributive lattice with incidence algebra $I(L)$ over a field $K$. Then there is the result that any automorphism of $I(L)$ is given by the composition of an inner automorphism ...
2
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0answers
46 views

Milnor fibration for non-isolated singularities

Can someone provide a precise reference/statement of the fibration theorem for non-isolated singularities? So given $f:\mathbb{C}^n\to \mathbb{C}$, and an $x\in Sing(f)$, what exactly does the ...
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0answers
60 views

how to construct a finite energy map

In the construction of harmonic maps by Eells and Sampson, one needs to start with a map with finite energy and use the heat equation to deform it into a harmonic map. The construction of such a ...
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52 views

Need reference of books which deals with ideal theory of tensor product

Is there any book which deals with Ideal theory of tensor product of $C^{\ast}-$ algebras
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2answers
417 views

A conjecture of De Giorgi on weighted Sobolev spaces

Let $\mu$ be a probability measure on $\mathbb{R}^d$ which is absolutely continuous with respect to the Lebesgue measure with density $\rho$. Assume that, for all $t>0$, \begin{align*} \exp \left(...
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2answers
372 views

An observation on the Riemann $\xi$ function

Anyone seen these conclusions about the Riemann xi function or see any errors here? With $\xi(s)$ the entire Landau Riemann xi function defined by the Hadamard product representation $$\xi(s) = (1/...
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65 views

Prerequisites/Preparation for understanding a research paper - global solutions to Einstein field in Bondi Coordinates

I would like to read this paper: João L. Costa, Filipe C. Mena, Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi ...
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34 views

Poisson reduction in odd/graded Poisson geometry?

I would like to know whether there is any literature on Poisson reduction of $\mathbb Z$- or $\mathbb Z_2$-graded Poisson algebras. A $\mathbb Z$-graded Poisson algebra with degree $p\in\mathbb Z$ ...
3
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1answer
155 views

Algebraic vector bundles on the punctured spectrum: an exact reference for a result

Let $(R, \mathfrak m)$ be a Noetherian local ring of depth at least $2$. Let $X=Spec(R)$ denote the affine- scheme with structure sheaf $\mathcal O_X$ and $U=Spec(R)\setminus \{\mathfrak m\}$ be the ...
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59 views

Moments of Dirichlet $L$-functions on the critical line

I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line, $$ M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt, $$ where $\chi$ is a ...
2
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0answers
89 views

Representability result

Let $X$ and $S$ be schemes over a field $k$. Reading this paper, there is a result on the representability of a morphism (proposition 3.1, page 4). Which result or reference on representability is ...
2
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0answers
69 views

Reference for Rellich Kondrachov theorem on bounded domains and spaces with finite measure

I read at the top of the page $580$ of this paper that the imbedding $W^{1,2}(\Gamma,d\mu) \hookrightarrow L^2(\Gamma,d\mu)$ is compact for a bounded domain $\Gamma \subset \mathbb{R}^n$ and a measure ...
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2answers
1k views

Categorification of probability theory: what does a “probability sheaf” tell us (if anything) about probability theory?

Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with. So, my understanding is that category theory and related fields of higher mathematics ...
4
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1answer
283 views

Further developments of Cartier–Gabriel–Kostant–Milnor–Moore Structure Theorem for cocommutative Hopf algebras

A very well-known theorem in Hopf algebra theory (see, for example, Lorenz - A tour of representation theory or the EGNO book (Etingof, Gelaki, Nikshych, and Ostrik - Tensor categories)) states that ...
2
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1answer
158 views

Cohomology of derived tensor product of complexes and Künneth spectral sequence

Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive ...
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0answers
44 views

Expected order of magnitude of character sums under GRH

Let $\chi$ be a nonprincipal character with modulus $q$. Under GRH, what is the expected order of magnitude of $\sum_{n \le x} \chi(n)$, where I think of $x$ and $q$ as growing, but $x$ is smaller ...
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0answers
78 views

Invariant subspace of a nonlinear map

First please see this very simple fact: Fact: $\ $ Any linear map $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ has a proper invariant linear subspace. By an invariant subspace we mean a space $M$ ...
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155 views

Basic questions and reference on Grothendieck ring of varieties

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Spec{Spec}$I am looking for a comprehensive reference on the theory of the Grothendieck ring of varieties over a field $K$, denoted $K_0(K)$ here, ...
3
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0answers
146 views

What is known about products of zeta values?

A couple of years ago, I asked this MSE question on the evaluation of the product of even zeta values: $$ \prod_{n=1}^\infty \zeta(2n) \approx 1.82 \quad .$$ While it can be shown that the product ...
3
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0answers
52 views

Almost quadratic difference sets

Does there exist a characterization of sets $S$ such that $|S-S|$ is "almost quadratic" in $|S|$? For instance, what are some examples of sets such that $|S-S|$ is on the order of $\frac{{|S|}^2}{\log ...
3
votes
1answer
299 views

Derivatives of Riemann $\xi$ and traces of zeros

Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
2
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0answers
53 views

Initial-boundary value problem for systems of conservation laws

For the Euler equations in a bounded domain $$ \begin{cases} \rho_t + q_x = 0 \\ q_t + (q^2/\rho + \rho)_x = - q \\ u|_{t=0} = u_0 \\ u|_{x=0} = g_0(t), \quad u|_{x=1} = g_1(t) \end{cases} $$ in which ...
3
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0answers
69 views

A good reference for Bochner spaces

I am looking for some references on Bochner spaces containing basic stuff such as measurability, convergence and $L^p$ theory. I already have the Analysis in Banach Spaces: Volume I book which covers ...
2
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0answers
42 views

$C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation

Consider a Fokker-Planck equation: $$ \partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0, $$ with initial condition ...
1
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1answer
87 views

Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$

Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$. Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of $$p(w)=n+\sum_{j=1}^{m}\frac{...
3
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1answer
473 views

Schlessinger's thesis

In Deligne-Mumford's "The irreducibility of the space of curves of given genus", the authors use the "Schlessinger's theory", and refer his "thesis". Where can I read it? It seems to be different from ...
1
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1answer
76 views

Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes

I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ...
6
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1answer
199 views

An unpublished note by Bloch-Kato on p-divisible groups and Dieudonné crystals

I wonder if anyone could find the following unpublished paper of Bloch-Kato: Spencer Bloch and Kazuya Kato, $p$-divisible groups and Dieudonné crystals, unpublished. A similar question is here ...
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0answers
92 views

Coverings of (DM) stacks and categories of descent data

If $X$ is a DM stack, we know that there is a surjective étale morphism $U \to X$ with $U$ a scheme. Combining Lemma 4.5 of these notes and Proposition 12.7.4 of Champs Algébriques one should be able ...
2
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1answer
198 views

Seeking a combinatorial proof for the invariance of a $q$-series

Start with some notations: $(a,q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$, shortened by $(a)_n$, and $(a)_{\infty}=\prod_{k=0}^{\infty}(1-aq^k)$. It's easy to verify (using algebraic means) that, for each $...
10
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0answers
115 views

How much smaller is the Čech complex than the Vietoris-Rips complex?

The Čech complex is a subcomplex of the Vietoris-Rips complex. The V-R complex includes as a simplex a set of points with pairwise distances at most $\epsilon$, whereas the Č complex includes as a ...
4
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1answer
282 views

How many trees have $n$ nodes with fewer than three neighbors?

We want to know how many trees have $n$ nodes with fewer than three neighbors. For $n=1$, the only possibility is a single node. For $n=2$, the only possibility is two connected nodes. For $n=3$, ...
5
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0answers
64 views

Reference request: sufficiently smooth functions on the plane belong to the projective tensor square of $L^2$ of the line

Let $\newcommand{\ptp}{\widehat{\otimes}}\ptp$ denote the projective tensor product of Banach spaces. Some back of the envelope calcuations, using the Fourier transform and Plancherel/Parseval, ...
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0answers
27 views

Polynomial approximations of the vector field and distance between generated flows

Let $\textbf{h} = (h_1,...,h_n)$ be a $C^1$ system of ODEs defined everywhere on on some compact subspace $\mathbb{X} \subset \mathbb{R}^n$. Suppose we have a polynomial approximation $\textbf{p} = (...
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4answers
671 views

What is known about ordinary character values at involutions?

Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$. Let $x$ be an involution in $G$. I'd like to ask the following Question 1: ...
18
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1answer
503 views

Subgroup $\mathrm{E}_6$ generated by $\mathrm{Spin_7}$ and $\mathrm{SL}_3$

Let $\mathbb{O}$ be the octonion algebra (say over $\mathbb{R}$) and let $J_{3}(\mathbb{O})$ be the set of $3 \times 3$ hermitian matrices with octonion coefficients, that is: $$ J_3(\mathbb{O}) = \...
3
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0answers
46 views

Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries

Could anyone please suggest related papers or article about the topic related to my one question below? Reduce PDE to ODE by dilation symmetry I also cite a paper in the link above. We know that ...
0
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1answer
65 views

Maximum in solution set to a Diophantine equation related to unit fractions

Some time ago, Kellogg communicated to Carmichael a result with an incomplete proof, which was soon after verified as correct. I do not recall the source but recall the result. Define $$S_n = \{ (x_1,...

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