# Questions tagged [reference-request]

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

9,753
questions

**6**

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165 views

### Complex factorization of the angular part of the Laplacian

Some time ago some research led me to the following equality:
\begin{equation}
\frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\...

**3**

votes

**1**answer

171 views

### Reference request: quantifier elimination test

I'm having difficulty finding this result in the standard texts.
Theorem. Let $T$ be a theory in a language $\mathcal{L}$. TFAE:
1) $T$ has quantifier elimination,
2) Whenever $M, N$ are $...

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44 views

### Posets with two partial (self-)distributive operations

Let $(X, {\sqsubset}, {\circ}, {\ast})$ be a set $X$ with a strict partial order $\sqsubset$ and two partial binary operations $\circ$ and $\ast$ such that for any $a, b, c \in X$:
$a \circ b$ and $a ...

**7**

votes

**0**answers

187 views

### Adequate equivalence relations and algebraic $K$-theory

I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow ...

**2**

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21 views

### Convergence rate of cardinal series (Whittaker-Shannon interpolant)

Given $f \in C^{k}_{0}[a, b]\cap L^{2}(\mathbb{R})$, what can we say about the convergence rate of the cardinal series
$$
s(t) = \sum_{j=0}^{n-1} f(a+jh) \mathrm{sinc}\left(\pi\left(\frac{t-a}{h} -j \...

**2**

votes

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64 views

### Partial regularity of harmonic maps into spheres

Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...

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112 views

### Flat base change in the complex analytic setting

On page 255 of Hartshorne's Algebraic Geometry, it is shown that "cohomology commutes with flat base extension":
Proposition III.9.3: Let $f : X \to Y$ be a separated morphism of finite type of ...

**5**

votes

**1**answer

101 views

### ASD connection for Line bundle over $4$-manifold

Let $(M,g)$ be an oriented closed Riemannian $4$ manifold.
Let $L\to M$ be a complex line bundle.
Q Under what condition, we can find an ASD connection of $L$, i.e. a connection $A$ such that $F^+...

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96 views

### Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?

Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...

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**2**answers

113 views

### Monotonicity of $M$-sequence

Consider the following definition in the second page of this article:
For any two integers $k,n\ge 1$, there is a unique way of writing
$$n=\binom{a_k}{k}+\binom{a_{k-1}}{k-1}+\dots+\binom{a_i}{i}...

**1**

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**1**answer

174 views

### Further study of “Elementary geometry” in the sense of Tarski

Tarski in the article "WHAT IS ELEMENTARY GEOMETRY" describes four candidates ($\mathscr{E}_2,\mathscr{E}'_2,\mathscr{E}''_2,\mathscr{E}'''_2$) to be called "Elementary geometry". Here the name "...

**4**

votes

**1**answer

136 views

### Representation of iterated generic embedding

I'm looking for a reference (if there is one) for a representation theorem for iterated generic embeddings. What I mean by representation is a generalization of the following:
If $U$ is an ...

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**0**answers

28 views

### Correlated tree interval and existence of unary subtree

We have a collection of random intervals $\{I_{k}:=(X_{k},Y_{k})\}_{k=1}^{\infty}\subset [0,1]$ s.t.
For deterministic $l_{k}\to 0$ we have $0<l_{k}^{a_{1}}\leq Y_{k}-X_{k}\leq l_{k}^{a_{2}}$.
The ...

**5**

votes

**1**answer

98 views

### Are proper subspaces of Banach spaces which are isomorphic to the ambient Banach space necessarily complemented?

I had the following little question pop up, but I cannot seem to find any reference to it.
Let $X$ be a Banach space and $E\subseteq X$ a proper subspace with $E$ isomorphic to $X$ itself. Is the ...

**1**

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**0**answers

40 views

### $G$-analogues of symmetric functions (reference request)

Let $G$ be a simple graph with vertex set $V$. Stanley defined the $G$-analogs of the symmetric function as follows:
For $i \ge 0$, define $$e^G_i = \sum_S \big(\prod_{v \in S}v\big)$$ where the sum ...

**2**

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**0**answers

41 views

### Finitistic dimension conjecture for quadratic algebras

The finitistic dimension of a finite dimensional algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension. The finitistic dimension conjecture says ...

**10**

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**1**answer

147 views

### Verbal description, or terminology, for the ${\mathcal L}_p$-spaces of Lindenstrauss and Pelczynski

This question is intended for Banach-space specialists and so I will not repeat all the definitions here. My aim is to find out how the Banach space community refers to such spaces in discussions, and ...

**7**

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135 views

### Correspondence between matrix multiplication and a graph operation of Lovasz

In his book "Large networks and graph limits" (available online here: http://web.cs.elte.hu/~lovasz/bookxx/hombook-almost.final.pdf), Lovasz describes a multiplication operation (he calls it ...

**2**

votes

**1**answer

164 views

### Filling $1..mn$ into a $m×n$ rectangle such that every number $<mn$ is dominated

This is a problem from my professor, who claimed that it's open:
Combinatorial problem.
Fill $1,2,...,mn$ into a rectangle of size $m\times n$, such that for every number other than $mn$, ...

**9**

votes

**1**answer

367 views

### What do absolute neighborhood retracts look like?

In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. ...

**4**

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93 views

### Open problems about Morita and derived invariants

Are there properties of rings of which one does not know whether they are Morita or derived invariances?
For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/...

**3**

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**1**answer

67 views

### Continuity of a differential of a Banach-valued holomorphic map

Originally posted on MSE.
Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...

**4**

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**0**answers

223 views

### Kaczorowski's Paper on Distribution of Primes

I am looking for a digital copy of the following paper by Jerzy Kaczorowski: ON THE DISTRIBUTION OF PRIMES (mod4)
https://www.degruyter.com/view/j/anly.1995.15.issue-2/anly.1995.15.2.159/anly.1995.15....

**3**

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**1**answer

55 views

### Exponential Deconvolution Using the First Derivative

There is an interesting observation using the first derivative to deconvolve an exponentially modified Gaussian:
The animation is here, https://terpconnect.umd.edu/~toh/spectrum/...

**3**

votes

**1**answer

226 views

### Asymptotic formula for the number of connected graphs

It can be shown that the set of graphs with $N$ vertices $G_N$ has cardinality:
\begin{equation}
\lvert G_N \rvert = 2^{N \choose 2} \tag{1}
\end{equation}
Recently, I wondered how much bigger $\...

**2**

votes

**1**answer

98 views

### Reference for the nearby Lagrangian conjecture for $T^*S^1$

I am looking for a reference for the proof of the nearby Lagrangian conjecture of $T^*S^1$, that is, that every exact and compact Lagrangian submanifold of the cylinder is Hamiltonianly isotopic to ...

**4**

votes

**1**answer

111 views

### Eigenvalues and Domain of the Laplace-Beltrami Operator

Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...

**0**

votes

**0**answers

61 views

### Research Request for a Paper of A.M. Leontovich

I am looking for a digital copy of a the English version of the paper "The Number of Mappings of Graphs, Ordering of Graphs, and Muirhead's Theorem" by A.M. Leontovich. The math.ru link to the paper ...

**2**

votes

**1**answer

63 views

### Continuous embedding between parabolic Sobolev spaces

I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct?
Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider ...

**1**

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**0**answers

37 views

### Schrödinger equation: well-posedness with Hartree potential and Yukawa potential

Consider the Schrödinger equation of Hartree type (HT):
$$i\partial_tu +\Delta u + (V\ast |u|^2)u=0, u(x,0)=u_0$$
with $(x,t)\in \mathbb R^d \times \mathbb R.$
where $V$ is some potential.
(1) when $...

**2**

votes

**0**answers

116 views

### Algebra of meromorphic functions on a Riemann surface

Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...

**3**

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**0**answers

105 views

### Generalization of normal subgroup

I am wondering whether the following concept appears in the group theory literature under some (perhaps different) name. Let $G$ be a group and let $A,B$ be subgroups of $G$.
Definition. Say that $(...

**3**

votes

**2**answers

136 views

### Uniform convergence of averages for stationary ergodic process

Let $\{X_t, t\in\mathbb R\}$ be a well-behaved$^*$ stationary ergodic process.
I'm interested in the uniform convergence of averages:
$$
\sup_{|x|\le R_n} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \...

**3**

votes

**0**answers

101 views

### Weak convergence for generic periodic points, which obtains from specification

Let $X$ be a compact metric space and let $T:X\rightarrow X$ be a Anosov and transitive map.
Let $x$ be a generic point so that $$ \mu_{n_{i}}:=\frac{1}{n_{i}}\sum_{j=0}^{n_{i}-1} \delta_{T^{j}(x)} \...

**3**

votes

**1**answer

108 views

### Two directed colimits of same spaces with different inclusions

For any natural number $n$, let $i_{n},j_{n}:X_{n}\rightarrow X_{n+1}$ be a pair of monomorphisms of simplcial sets.
Define $$X=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{i_n} X_{n+1}\cdots \}...

**5**

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79 views

### Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat?

Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying:
$$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$
Where $(r,\theta,\phi)$ ...

**8**

votes

**1**answer

212 views

### When does $BG \to BA$ loop to a homomorphism?

If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) ...

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105 views

### Paper “Tetrahedron rolled onto a plane”

I am looking for the article Charles W. Trigg, "Tetrahedron rolled onto a plane", J. Recreational Mathematics, 3(2):82–87, 1970.
It is from @Joseph O'Rourke comment in the previous post "Die-rolling ...

**5**

votes

**0**answers

98 views

### Asymptotic expansion for the average of $\omega(n)^2$

Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...

**1**

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**1**answer

69 views

### Shortest path on graphs

I would like to now if there has been any work on related problems, that is, shortest path problem in dynamically evolving graphs.

**0**

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**1**answer

96 views

### Reference request: for a proof of a reflection [from transitive sets] based axiomatization of ZF\Reg.?

The following system is quoted from Harvey Friedman's lecture notes. The language is first order logic with membership $\in$.
Axioms:
Extensionality. $(\forall x)(x \in y \leftrightarrow x \in z) \...

**4**

votes

**1**answer

73 views

### Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields

The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis.
In ...

**0**

votes

**0**answers

108 views

### Reference request: smooth affine curves are planar

Let $X\rightarrow\mathrm{Spec}\:\mathbb{C}$ be an affine smooth morphism of relative dimension$\leq 1$. What is a reference for the fact that there exists a $\mathbb{C}$-locally closed immersion $X\...

**11**

votes

**1**answer

293 views

### What system suffices to show the strength of PRA is $\omega^\omega$?

Russell O'Connor wrote in 2009 (link):
PRA has consistency strength equivalent to the well-foundness of $\omega^\omega$, which can be stated again as the termination of some other program on all ...

**2**

votes

**1**answer

79 views

### Jensen's Formula for Arbitrary Neighborhoods

The Jensen's formula says the following: Let $f$ be analytic on the disc $D$ of radius $R$ centered at the origin such that $f(0)\neq 0$, then
\begin{align}
\log(|f(0)|)+ \sum_{i=1}^n \log \left(\...

**3**

votes

**1**answer

130 views

### Weak enrichment and bicategories

I'm trying to find examples where the following perspective on bicategories is developed.
We can define a 2-category as being enriched in Cat, where Cat is treated as a monoidal category using the ...

**3**

votes

**0**answers

83 views

### Extensive survey of computations of equivariant stable stems

Where can I find a comprehensive survey of computations of equivariant stems?
To my knowledge, the status is:
Classical Work of Araki and Iriye, Osaka J. Math. 19 (1982). ...

**2**

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41 views

### Picard group of the moduli space of semistable rank 2 parabolic vector bundles over smooth complex projective curves with trivial determinant

I am looking for the Picard group of the moduli space of semistable rank 2 parabolic vector bundles over smooth complex projective curves with trivial determinant.
Having determinant trivial, I ...

**5**

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**0**answers

75 views

### Bound on the sum of projective and injective dimension

Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category.
In proposition 1.2. of https://link.springer.com/article/10....

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**1**answer

76 views

### Reference request: metric spaces with curvature bounded from below (CBB) spaces

What is the/a main reference book for spaces with curvature bounded from below (CBB spaces/spaces with curvature $\geq \kappa$ in the sense of Alexandrov)? Looking for an up to date reference.