Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
12,082
questions
3
votes
0answers
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 ...
2
votes
0answers
67 views
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 ...
14
votes
0answers
117 views
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 ...
3
votes
0answers
45 views
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-...
3
votes
0answers
38 views
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 ...
3
votes
0answers
134 views
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
votes
4answers
861 views
Proving a binomial sum identity
QUESTION. Let $x>0$ be a real number or an indeterminate. Is this true?
$$\sum_{n=0}^{\infty}\frac{\binom{2n+1}{n+1}}{2^{2n+1}\,(n+x+1)}=\frac{2^{2x}}{x\,\binom{2x}x}-\frac1x.$$
3
votes
1answer
108 views
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 $\lambda$, ...
1
vote
0answers
103 views
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 ...
13
votes
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 ...
-2
votes
0answers
95 views
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
votes
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 ...
0
votes
0answers
62 views
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
$$
\sum_{1 \leq n \leq X} \min ( || \alpha n ||^{-1} , X/n),
$$
...
0
votes
0answers
96 views
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 $\frac{a}{b}$ ...
3
votes
2answers
234 views
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.
...
1
vote
0answers
39 views
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 ...
0
votes
0answers
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 ...
4
votes
1answer
126 views
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}/...
14
votes
4answers
2k views
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 $a_n$ ...
0
votes
0answers
63 views
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 ...
8
votes
2answers
252 views
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 $\...
0
votes
0answers
49 views
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....
1
vote
1answer
61 views
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=\...
1
vote
0answers
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 ...
7
votes
0answers
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 ...
2
votes
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 $C^\infty$-...
6
votes
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
$$\log 2=3\sum_{n=1}^{\infty}\frac{(-1)^{...
1
vote
0answers
68 views
Counting number of distinct eigenvalues
Let $\Omega$ be a Lipschitz domain in $\mathbb{R}^n$, and let $N(\lambda)$ be the number of Dirichlet Laplacian eigenvalues less than or equal to $\lambda$. The famous Weyl's law says that as $\...
8
votes
1answer
243 views
A dichotomy for the quadratic variation of differentiable functions?
For a real-valued function $f$ on $[0,1]$, define its quadratic variation by the formula
$$[f]:=\limsup\sum_{j=1}^n(f(t_j)-f(t_{j-1}))^2,$$
where the $\limsup$ is taken over all "partitions" ...
6
votes
3answers
506 views
Exact formula for $\chi(X, \, S^n \Omega^1_X)$
I have a smooth, compact complex surface $X$, and I need an explicit formula for the Euler characteristic
$$\chi(X, \, S^n \Omega^1_X),$$
where $S^n$ denotes the symmetric product, in terms of $c_1(X),...
0
votes
0answers
88 views
Cochains with multilinear differentials
Let $G$ be a group and let $M$ be a $G$-module. We denote by $(C^*(M,G),d)$ the complex of inhomogeneous cochains, i.e. $C^n(G,M)=M^{G^n}$.
We say that a cochain $a\in C^n(G,M)$ is multilinear if it ...
2
votes
1answer
298 views
An identity for the elliptic theta function
For real $s>0$, let
$$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
votes
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 ...
2
votes
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
votes
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
votes
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 ...
1
vote
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:...
4
votes
0answers
73 views
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 ...
1
vote
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
2
votes
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$...
0
votes
0answers
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
votes
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
votes
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. ...
2
votes
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 ...
0
votes
0answers
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 < ...
1
vote
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
votes
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
votes
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?
3
votes
0answers
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 ...
22
votes
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." ...
