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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35 views

Tangent coalgebras and hyperalgebras

Takeuchi's On coverings and hyperalgebras of affine algebraic groups references Tangent coalgebras and hyperalgebras. I and II, with II being listed as submitted to MAMS. On coverings and ...
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On the Hochschild cohomology of the minimal model of an $A_\infty$ algebra

Suppose $(A, (\mu_k))$ is a (curved) $A_\infty$ algebra, and let $(\tilde A, (\tilde\mu_k))$ be its minimal model. Now, we have two Hochschild cohomology rings $HH^*(A)$ and $HH^*(\tilde A)$. (It may ...
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What is the covering density of a very thin annulus? Is it $\frac{\pi\sqrt{51\sqrt{17}-107}}{16}$?

Take some very small $\epsilon>0$, and consider the annulus/ring given by the set $\{(r,\theta)\ |\ 1-\epsilon\le r\le1\}\subset \mathbb{R}^2$. We wish to place translated copies of this annulus ...
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Which non-compact quaternion-Kähler spaces are Kähler?

The list of quaternion-Kähler compact symmetric spaces can be found here. I am curious to know which of the non-compact versions of these spaces are Kähler. If the answer is known also for non-...
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Stability properties of essential geometric morphisms

Notation. $\mathsf{Topoi}$ is the bicategory of topoi, geometric morphisms and natural transformations between left adjoints. $\mathsf{Topoi}_{\text{ess}}$ is the bicategory of topoi, essential ...
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Weil numbers and Weil cohomologies

I read in the literature (I cannot track down the reference) that if, for smooth projective varieties over finite fields, Grothendieck's standard conjectures are true, then there is an algebraic ...
2
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4answers
861 views

Proving a binomial sum identity

QUESTION. Let $x>0$ be a real number or an indeterminate. Is this true? $$\sum_{n=0}^{\infty}\frac{\binom{2n+1}{n+1}}{2^{2n+1}\,(n+x+1)}=\frac{2^{2x}}{x\,\binom{2x}x}-\frac1x.$$
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1answer
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Bijection from “black-white balanced” partitions to pairs of partitions

Definition Call a partition $\lambda$ of an even integer $2n$ "black-white balanced" if the following equivalent conditions are satisfied: In the usual (Ferrers-)Young diagram of $\lambda$, ...
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About derived divided power envelope

Assume $A$ is a $\mathbb{Z}_{(p)}$-algebra with ideal $I$ and $A,A/I$ are $p$-torsionfree. In this survey, Akhil Mathew defines the derived divided power envelope $LD_I(A)$ in Construction 7.15, after ...
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2answers
899 views

Geometric interpretations of the exponential of entropy

Question: Might there be a natural geometric interpretation of the exponential of entropy in Classical and Quantum Information theory? This question occurred to me recently via a geometric inequality ...
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Number of homeomorphisms between spaces [closed]

I need help with the following. We know that 2 spaces are homeomorphic if there is a continuous and bijective function with a continuous inverse, there are also other theorems that assure us that ...
4
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1answer
71 views

A narrower dichotomy for the quadratic variation of differentiable functions?

$\newcommand\P{\mathcal P}$A "partition" $P$ (of the interval $[0,1]$) is a finite sequence $(t_0,\dots,t_n)$ such that $0=t_0<\cdots<t_n=1$; then the mesh of $P$ is $\|P\|:=\max_{1\le ...
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A way to bound $\sum_{1 \leq n \leq X} \min ( \| \alpha n \|^{-1} , X/n)$?

Let $\alpha$ be a real number and $|| \cdot ||$ be the distance to the nearest integer. I want to find a non-trivial upper bound for $$ \sum_{1 \leq n \leq X} \min ( || \alpha n ||^{-1} , X/n), $$ ...
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Is there a good algorithm to divide two integers without using division directly? [migrated]

I am wondering whether this question is appropriate for MathOverflow, but I have asked elsewhere and gotten no satisfactory answer. Problem. Given positive integers $a$ and $b$, obtain $\frac{a}{b}$ ...
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Applications of ZFA-Set Theory

The set theory with atoms (ZFA), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements. ...
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Class groups and zeta functions for maximal orders in CSAs

I'm looking for references for certain algebraic objects in the context of maximal orders in (finite dimensional) central simple algebras over algebraic number fields. Does anyone know of any good ...
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45 views

Binary relational structures in which all binary relations are equivalence relations

I’m interested in (finite) binary relational structures (i.e. relational structures with only unary and binary relations) in which all binary relations are equivalence relations. Is there a name for ...
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1answer
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Is there essentially unique notion of module over monoidal stable $\infty$-categories?

There is this (folklore?) fact: for a commutative ring $R$, the category of $R$-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/...
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4answers
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Integrality of a sum

Consider the following sequence defined as a sum $$a_n=\sum_{k=0}^{n-1}\frac{3^{3n-3k-1}\,(7k+8)\,(3k+1)!}{2^{2n-2k}\,k!\,(2k+3)!}.$$ QUESTION. For $n\geq1$, is the sequence of rational numbers $a_n$ ...
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Results about Schrödinger equations

Does anyone know any paper or book that deals with Schrodinger equations, specifically on asymptotic properties like blowup or limitation of solution when time goes to infinity using Schrödinger ...
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Groups associated with infinite dimensional Lie algebras

There is a classical correspondence between Lie algebras (over $\mathbb{R}$ or $\mathbb{C}$) and Lie groups in the finite dimensional case: to every Lie group $G$ there is an associated Lie algebra $\...
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In percolation on a lattice, how is “infected” status correlated for points in a region around the origin?

Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster....
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1answer
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uniform convergence of $H^r$ projectors on compact sets?

Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\...
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118 views

The number of ideals in a number ring below a given norm

I'm doing some number theory, and my deficient knowledge of algebraic number theory shows itself; I have to ask for help. Suppose $K$ is a number field, and we want to count the ideals in $\mathcal ...
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145 views

Li-Yau inequality on $\mathbb R^2$ for functions that are somewhat close to $1$

Let $u:\mathbb R^2\times \mathbb R_{>0}\to \mathbb R_{>0}$ be a positive solution to the heat equation on $\mathbb R^2$ ($u_{xx}+u_{yy}=u_t$, no constants). The Li-Yau inequality in this case ...
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1answer
96 views

Approximation of $C^1$-smooth equivariant maps by infinitely smooth ones

Let $M,N$ be smooth closed manifolds acted by a finite group $G$. Let $f\colon M\to N$ be a $C^1$-smooth $G$-equivariant map. Is it true that for any $\varepsilon>0$ there exists a $C^\infty$-...
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3answers
484 views

In search of an alternative proof of a series expansion for $\log 2$

We all know the series expansion $$\log 2=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}n. \tag1$$ I also am able to use the method of Wilf-Zeilberger to the effect that $$\log 2=3\sum_{n=1}^{\infty}\frac{(-1)^{...
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68 views

Counting number of distinct eigenvalues

Let $\Omega$ be a Lipschitz domain in $\mathbb{R}^n$, and let $N(\lambda)$ be the number of Dirichlet Laplacian eigenvalues less than or equal to $\lambda$. The famous Weyl's law says that as $\...
8
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1answer
243 views

A dichotomy for the quadratic variation of differentiable functions?

For a real-valued function $f$ on $[0,1]$, define its quadratic variation by the formula $$[f]:=\limsup\sum_{j=1}^n(f(t_j)-f(t_{j-1}))^2,$$ where the $\limsup$ is taken over all "partitions" ...
6
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3answers
506 views

Exact formula for $\chi(X, \, S^n \Omega^1_X)$

I have a smooth, compact complex surface $X$, and I need an explicit formula for the Euler characteristic $$\chi(X, \, S^n \Omega^1_X),$$ where $S^n$ denotes the symmetric product, in terms of $c_1(X),...
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88 views

Cochains with multilinear differentials

Let $G$ be a group and let $M$ be a $G$-module. We denote by $(C^*(M,G),d)$ the complex of inhomogeneous cochains, i.e. $C^n(G,M)=M^{G^n}$. We say that a cochain $a\in C^n(G,M)$ is multilinear if it ...
2
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1answer
298 views

An identity for the elliptic theta function

For real $s>0$, let $$S(s):=\sum_{n=-\infty}^\infty e^{-n^2/(2s^2)} =\vartheta _3\left(0,e^{-1/(2 s^2)}\right),$$ where $\vartheta$ is the elliptic theta function. Plotting suggests that the ...
2
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1answer
306 views

Have others explored Mendelson's approach to comprehension?

Mendelson, in Introduction to Mathematical Logic, 4th ed, 1997, had a more elegant approach to comprehension than predecessors, in my opinion. With $x\in\mathbf{V}$ short for $\exists y(x\in y)$, and ...
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0answers
122 views

Classifying spaces of odd dimensional $p$-spheres

By a statement on page 68 of Kane's book "The homology of Hopf spaces"(1988), I know that the classifying space $BS^{2n-1}_{(p)}$ of a $p$-local $(2n-1)$-dimensional sphere exists if $n$ ...
4
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1answer
84 views

Regular nilpotents and minimal parabolic subalgebras in real semisimple Lie algebras

Let $\mathfrak{g}$ be a real semisimple Lie algebra. A subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ is parabolic if its complexification is parabolic in $\mathfrak{g}_\mathbb{C}$, meaning it contains a ...
6
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1answer
192 views

Hypergeometric function evaluation 4F3

I need to show that for $m$ being non-negative integer, the hypergeometric function ${}_4F_3$ below evaluates to $-1/2$ independent of $m$. This is Mathematica notation, but we have 4 and 3 sets of ...
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1answer
97 views

Word for two morphisms that are equivalent up to right-composition with isomorphism

Let $f:A\to C$, $g:B\to C$ be morphisms in some category. I call $f,g$ "equivalent" iff there exists an isomorphism $h$ such that $f\circ h=g$ (and consequently $g\circ h^{-1}=f$). Question:...
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Question about terminology for a class of “self-modular” mappings between rings

(In the scenario I have in mind, rings need not be unital.) The following notion has come up in some joint work that is being written up. Let $R$ and $S$ be rings, and let $D$ be a subring of $R$. Is ...
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1answer
58 views

Modular S-matrix of (p,q) minimal model

What is the expression for the modular S-matrix of (p,q) minimal model? The Wiki https://en.wikipedia.org/wiki/Minimal_model_(physics) does not provide S-matrix
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1answer
84 views

Intuition/references for understanding bound states/discrete spectrum relationship

I am trying to form intuition for the following `well-known' facts about spectrum of unbounded operators (Schrodinger/wave etc.) $L$ on $\mathbb{R}^n$. Let $\lambda\in\mathbb{R}$ satisfy $Lf=\lambda f$...
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64 views

On $q$, $k$ modulo $3$ and the residue of $\sigma(n^2)$ modulo $x$ when $q^k n^2$ is an odd perfect number with special prime $q$

Chen and Luo (now published) proved in Theorem 3.3, page 7 that if $m = q^k n^2$ is an odd perfect number with special prime $q$, then we have the biconditionals $$\sigma(n^2) \equiv 1 \pmod 4 \iff q \...
4
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2answers
197 views

Hardy-Littlewood circle method for non-diagonal quadratic forms

In short, the question is for any references describing how to use the Hardy-Littlewood circle method to find an asymptotic for the number of solutions to $F(x_1, ..., x_s) = k$ for $(x_1, ..., x_s) \...
4
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0answers
148 views

Motivic cohomology of rigid analytic spaces

There is a satisfactory theory of B1-homotopy theory for rigid analytic spaces defined by Ayoub in the style of Voevodsky, and I'm aware of some work about the corresponding theory of motives, e.g. ...
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0answers
83 views

Polynomial invariant relating the circumradius and sides of a cyclic polygon

This question deals with the polynomial invariant which relates the circumradius and the squares of the sides of a cyclic polygon. This invariant is discussed briefly in the seminal paper On the Areas ...
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95 views

Fully explicit Linnik's Theorem

Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < ...
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0answers
99 views

Invariant on C*-algebras-number of closed unbounded derivation it admitted

In working of the unbounded derivation of C*-algebras. I observed the following: For topological manifold $M$, the number of closed, linear independent, unbounded derivation it admitted on $C(M)$ is ...
6
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1answer
346 views

Cohomology of quotient by free action

Let $G$ be a finite group. Let $G$ act freely on a CW-complex $X$. I heard that the following fact is true. Claim. The canonical map $H^*(X/G,F)\to H^*(X,F)^G$ is an isomorphism, where $F$ is a field ...
2
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1answer
218 views

“Radical” Catalan numbers?

Let $C_n=\frac1{n+1}\binom{2n}n$ be the well-known Catalan numbers. Here is a curiosity. QUESTION. Are there infinitely many $C_n$ that are "radical", i.e. that are square-free?
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76 views

Isoperimetric inequality for general metric space

Consider some space $\mathcal{S}$ with metric $d$ and measure $\mu$. For arbitrary set $H$ denote the $v$-bound of $H$ by $\delta_v(H):= \{x \mid x \notin H: \exists y \in H \text{ s.t. } d(x,y) \le v ...
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1answer
2k views

“The boat is not longer than it is.”

Bertrand Russell, I believe, somewhere presents a joke (if I remember correctly). Someone is shown the boat of another, and the first says: "I thought that your boat is longer than it is." ...

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