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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6
votes
0answers
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
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1answer
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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0answers
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
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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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0answers
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
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0answers
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 ...
4
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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
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1answer
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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2answers
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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1answer
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
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1answer
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 ...
-1
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0answers
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
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1answer
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 ...
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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 ...
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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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1answer
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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0answers
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
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1answer
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
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1answer
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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0answers
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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1answer
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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0answers
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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1answer
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
1answer
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
1answer
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
1answer
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
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0answers
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
1answer
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 ...
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0answers
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
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0answers
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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0answers
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
2answers
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
0answers
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
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1answer
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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0answers
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
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1answer
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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0answers
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
0answers
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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1answer
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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1answer
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
1answer
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 ...
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0answers
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
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1answer
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
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1answer
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
1answer
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
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0answers
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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0answers
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
votes
0answers
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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1answer
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.