# Questions tagged [reference-request]

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

10,929
questions

**2**

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### $\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 ...

**9**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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 ...

**33**

votes

**2**answers

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**

vote

**1**answer

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 ...

**1**

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**0**answers

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**

votes

**2**answers

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**0**

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**0**answers

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

**11**

votes

**2**answers

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(...

**0**

votes

**2**answers

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/...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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 ...

**1**

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**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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 ...

**13**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 $M$ ...

**5**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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**

vote

**1**answer

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**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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, ...

**0**

votes

**0**answers

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} = (...

**15**

votes

**4**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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,...