New answers tagged reference-request
7
votes
Accepted
Fibers of generic smooth maps between manifolds of equal dimension
Yes, the claim is true, and here's a reference. For $M$ compact, your condition is satisfied by a "finite mapping." Such finite mappings form a residual set when $\dim M \leq \dim N$. See pp....
- 6,571
2
votes
Vector bundles over a Stein space are projective
Corollary 2.6.5 in Forstnerič "Stein Manifolds and Holomorphic Mappings". (2nd Edition)
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4
votes
Accepted
Vector bundles over a Stein space are projective
I suppose you were given essentially a reference in the comment. Alternatively, to expand a little on the proof of this fact that I mention in the answer above the linked comment, to me this argument ...
- 1,457
1
vote
Coradical filtration for comodules is exhaustive
Exhaustiveness of the coradical filtration of a coalgebra is proven in Theorem 5.2.2 of Montgomery's book "Hopf algebras and their actions on rings." This is not as general as what you asked ...
- 1,335
6
votes
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Grothendieck purity for Brauer groups of stacks
The relevant result is now in the literature.
It first appeared as Proposition 8.1 in On Brauer groups of tame stacks by Anchenjang. Actually he proves the result for all algebraic stacks smooth over ...
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8
votes
Theory of $n$-truncated $A_\infty$ categories/functors?
$A_n$-spaces are already discussed in the original paper of Stasheff, Homotopy associativity of H-spaces, I and II.
In the linear setting, $A_n$-algebras are discussed e.g. in A∞-algebras, spectral ...
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4
votes
On a probabilistic integer factorization algorithm given bounds for one prime factor
If $D\le 3$, it is known how to do this in polynomial time by Coppersmith's algorithm [Cop96]: note that $q \approx N^{\frac{D}{D+1}}$ and therefore $B_2-B_1 \approx q^{\frac{D-1}{D}} \approx N^{\frac{...
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3
votes
In search of a $q$-analogue of a Catalan identity
If the identity were to be rewritten as:
\begin{equation}
\sum_{k=0}^n 2 C_k \binom{2(n-k)}{n-k} = \binom{2n + 2}{n+1},
\end{equation}
here is a $q$-analog:
\begin{equation}
\label{eq:q-cbc-cat}
\sum_{...
- 31
3
votes
Comments and reference-request on books for KK-theory
Here is a very rough outline of the proof of the index theorem using KK-theory:
Define $KK_G(A, B)$, where $G$ is a Lie group and $A$ and $B$ are
[adjectives] C*-algebras, and the Kasparov product ...
- 29.2k
5
votes
Accepted
Reference request for elementary convex geometry property
Indeed, this can be proved more simply, and in greater generality -- assuming only that the support of $P$ is contained in $C$ (rather than in $\mathcal X$).
Indeed, without loss of generality the ...
- 128k
8
votes
Accepted
Reference request: The non-productivity of Lindenbaum numbers
Karl-Heinz Diener proved in On the transitive hull of a κ‐narrow relation that for all class relations $R$, if $R$ is $\kappa$-narrow in the sense that $\aleph^\ast(R^{-1}[\{x\}])\leqslant\kappa$ for ...
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3
votes
Laplacian on manifolds and random matrix theory
This is somewhat different since the metrics are not generic, but there is a natural way to define the space of hyperbolic surfaces of a fixed genus. There has been a large amount of research studying ...
- 6,001
0
votes
References on hyperbolic harmonics
It seems you are seeking the homogeneous angular eigenfunctions of the hyperbolic Laplacian $\nabla^2_{\mathbb{H}^n} u = 0$ defined on a real Lobachevsky space of dimension $n$. In that case, the ...
- 195
0
votes
Intensity and compensator for a jump process
You can refer to the Theorem 2.3 in this paper https://arxiv.org/pdf/2407.21651. Hope this can help you find the solution.
7
votes
Accepted
Reference request: $\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \Lambda_+$
One reference is Donaldson and Sullivan's paper "Quasiconformal 4-manifolds". See Lemma 2.3. They prove a a little more. A traceless symmetric tensor is an infinitesimal change of metric ...
- 3,409
6
votes
Accepted
Reference request: Algebras over monoid objects in a monoidal category
I believe the original reference for this fact is Theorem 2 of Maranda's 1966 On Fundamental Constructions and Adjoint Functors, although the terminology is not modern. A more readable reference is ...
- 10.6k
4
votes
Accepted
Lower bound in the singularity of random Bernoulli matrices
On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions
derives the lower bound (theorem 1)
$$\mathbb{P}\{ \text{det}(A_n) = 0\} \geq 2^{2-n} \binom{n}...
- 188k
0
votes
What are the finite étale coverings of a quasi-hyperelliptic surface?
By the Enriques classification in positive characteristic of Bombieri and Mumford, quasi-hyperelliptic surfaces should be regarded as quotients of a product of an elliptic curve and the rational ...
- 103
4
votes
Condition for a functor to induce a cartesian closed functor between categories of presheaves
Here is an observation you already almost made: since $F^*$ has a left adjoint $F_!$ given by left Kan extension, $F^*$ preserves exponential objects if and only if $F_!$ preserves products.
Note that ...
- 189
6
votes
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
This is by no means complete, but too long for a comment, so I am posting some observations as a partial answer. I give some necessary conditions on $f$ for the existence of a continuous map $\phi$ ...
- 716
1
vote
Baer sums of extensions
Here is my own try for an answer, assuming further that multiplication by $n$ on $\mathcal{A}$ is invertible for all $n\neq 0$, and specializing to a category of modules for simplicity. I wish to show ...
- 2,907
1
vote
Unitary representations of discrete (locally compact) groups
I don't have enough reputation to comment but I wanted to add the following :
Like it was said in the comments in general this isn't possible but if you only care about the operator norm then this is ...
- 11
3
votes
About Grothendieck and special cases
An idea is discussed in Colin Mclarty's https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathbiographies/mclarty1.pdf which could be where your paraphrase of Grothendieck's method ...
- 9,597
20
votes
Accepted
Infinite series and sum of two squares
Expanding on my comments: $a(n)$ is congruent mod 5 to the number of
two-square representations $n = r^2 + s^2$
(allowing each of the integers $r,s$ to be positive, negative, or zero).
Thus if $n$ has ...
- 79.9k
3
votes
Accepted
Is the evolution family self-adjoint?
No, $U(t,s)$ is, in general, not self-adjoint. If we, for ease of formulation, denote the parameters $t,s$ as "times", then $U(t,s)$ is given by the time-ordered exponential
$$
U(t,s) = T\...
- 6,696
2
votes
Possible refinements of the large sieve inequality
Though a bit late --
In general, look at Theorem 2.1 of
https://ramare-olivier.github.io/Maths/Eigenvalues-JTNB.pdf
This improves a bit on the c in (N+cQ^2).
For a generic sifted set, look at Theorem ...
- 341
5
votes
Accepted
Representing positive integers $n$ by binary forms $n=ax^2+by^2$, $a\geq 0$, $b\geq 0$
There is no such finite set of pairs $(a_k,b_k)\in\mathbb{Q}_{\geq 0}^2$. Indeed, because the problem concerns rational representations, we can assume without loss of generality that $(a_k,b_k)\in\...
- 105k
2
votes
Accepted
Bounding a number-theoretic integral
The integral looks something like $$\sum _{n=1}^\infty \frac {\Lambda (n)}{n^c}\int _1^Tt^{1/2-c}\cdot e(t-t\log (X/nt))\cdot dt\hspace {10mm}e(z)=e^{2\pi iz}.$$
The derivative of the phase is ...
- 1,381
12
votes
Accepted
Moments of a random variable related to uniform distribution on sphere
Note that the random vector $u=(u_1,u_2,\ldots u_n)$, uniformly distributed on the unit sphere, can be replaced by the ratio $u=y/|y|$, with $y=(y_1,y_2,\ldots y_n)\sim N(0,I_n)$ a multivariate ...
- 188k
7
votes
Accepted
What's the deal with De Morgan algebras and Kleene algebras?
Per request of the OP, I’m reposting my comments as an answer. This is a series of observations without any references; most of these things are well known/easily shown.
Normal forms.
Using De Morgan’...
- 47.1k
3
votes
Accepted
Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?
This follows from Theorem 1.5 of Alex Smith's paper "The distribution of $\ell^\infty$-Selmer groups in degree $\ell$ twist families I" which states
Suppose $A/\mathbb Q$ is an elliptic ...
- 148k
17
votes
What's the deal with De Morgan algebras and Kleene algebras?
There are a lot of questions bundled together here. I will give some references for some of the questions.
An early paper on these topics is:
Lattices with involution
J. A. Kalman
Trans. Amer. Math. ...
- 14.6k
0
votes
A good introduction to the study of the Thue Equation
In the case that the Mordell-Weil rank of the Jacobian of the curve is less than the genus, Lorenzini and Tucker used the Chabauty-Coleman method to give an upper bound on the number of rational ...
- 704
0
votes
A good introduction to the study of the Thue Equation
You may like this survey article of Prof. Michel Waldschmidt https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ProcHRI2017ThueEquations.pdf. It is relatively newer.
- 260
5
votes
Where to begin in Computational Group Theory?
As it was made quite clear in the comments, you are not at the stage where you can ask a sensible question. Thus, I am treating your question as a reference request. The first issue is that there is ...
- 12.2k
-1
votes
Proofs without words
Countable union of countable sets is one of my all-time favorites (and surely one of the all-time best):
here
It's a picture of Cantor's pairing function.
6
votes
Literature Request: The derived category is Krull-Schmidt
I do not know a source showing this directly, but you can get it by combining two sources.
Let us write $k$ for the base field. By a theorem of Balmer and Schlichting (Thm. 2.8 here), the category $K^...
- 2,928
2
votes
Accepted
Nonstationary phase method for oscillatory integral
For stationary phase, you usually consider the integral
$$I(\lambda)=\int_a^b f(t) e^{i\lambda g(t)}\,dt$$
with $\lambda>0$ a large parameter. If there are no stationary points inside $[a,b]$, then ...
- 840
0
votes
Looking for review of delay differential equations involving $f(x)$ and $f(x/k)$
In spite of offering a bounty, this question did not get an answer, so it may be that there is no such review.
The closest thing that I could find is the book Polyanin et al., Delay Ordinary and ...
- 3,065
4
votes
Comparison of special metrics on Riemann surfaces with the hyperbolic one
First of all, the constants $c_i$ will have to depend on the complex structure of $X$ since without prescribing a complex structure one cannot talks about dependence on a basis of the space of ...
- 12.2k
11
votes
Accepted
What is the "schematic" point of view for regular polyhedra?
Too long for a comment. I did a search in French and German but it didn't help much. I think this might have to do with the topic of "polyhedra over finite fields".
Someone asked the same ...
- 498
0
votes
The Paley-Wiener theorem and exponential decay.
If you assume that $A=\hat{\omega}$ decays exponentially in both directions, i.e. $\exists\epsilon>0:\,\lim_{t\to \pm \infty} \hat{\omega}(t)e^{\epsilon t}=0$... or else you assume $\omega$ to be ...
- 1,461
4
votes
Accepted
Does an indexed functor $C \rightarrow \mathbb{B}$ extend to $\operatorname{Psh}(C) \rightarrow \mathbb{B}$?
There is definitely discussion of internal presheaves – the whole of section B2.5 is about them!
In particular, the result you seek is Corollary 2.5.8:
[Let $\mathcal{S}$ be a cartesian category with ...
- 15.9k
6
votes
Solving a three-parameter recursive sequence
I assume a typo as suggested in my comments, such that
\begin{align}
f(\alpha,\beta,\gamma)
&=(2\alpha+8\beta+12\gamma-1) \, f(\alpha-1,\beta,\gamma) \\
{}&- 2(\alpha+1) \, f(\...
- 3,671
1
vote
Accepted
The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
This problem should really be posed over a general finite field (reductions of elements in $\mathcal O_L$ modulo $P$ can be computed in polynomial time).
Over any finite field $F$, the equation $ax^2+...
- 651
3
votes
Sobolev inequality with weight in the case $1<n\leq p$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\phi}{\varphi}$
I will answer it for all $0 < q < \infty$, $1\le n\le p$ and all non-negative measures $\mu$. The desired inequality holds if and only ...
- 6,476
3
votes
Accepted
Reference request: ray class group as quotient of finite ideles
The ray class group is a more general object than the one considered in the original post. It is defined as
$$K^\times \backslash \mathbb{A}_{K}^\times /(U_\infty U_{K,I}),$$
where $U_\infty$ is an ...
- 105k
6
votes
Accepted
Every elliptic surface contains only finitely many negative self-intersection rational curves?
I am just posting my comment as one answer. Let $\varpi:Z\to \mathbb{P}^1$ be the rational elliptic surface obtained by blowing up the projective plane along base locus of a general pencil of plane ...
3
votes
Polarities for intuitionistic linear logic formulas inside classical linear logic (without linear implication)
This restricted grammar is discussed here:
François Lamarche. Proof nets for intuitionistic linear logic: Essential nets. Research report
, INRIA, 2008.
https://inria.hal.science/inria-00347336
...
2
votes
Online References for Cartan Geometry
The videos of the Training School on Cartan Geometry, held in Brno, Czechia, September 4-8, 2023 are here on YouTube. They include talks on:
Boris Doubrov (Minsk), Cartan geometry via exterior ...
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