All Questions
15,614 questions
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Reference for computing cohomology of $Q_nS^0$ (the $n$-th component of infinite loop space)?
$Q_nS^0=\varinjlim_k(\Omega^kS^k)_n$, where $(\Omega^kS^k)_n$ refers to homotopy classes of maps $(S^k,\ast)\to (S^k,\ast)$ of degree $n$. I already know the case of $n=0$.
Is there any reference ...
3
votes
0
answers
133
views
Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
0
votes
1
answer
118
views
Nonstationary phase method for oscillatory integral
I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth.
The stationary phase method says that if $t_0\in [a,b]$ is such that ...
- 107
4
votes
1
answer
227
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Literature Request: The derived category is Krull-Schmidt
I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question
Literature request: $K^b(\text{...
4
votes
1
answer
376
views
Where to begin in Computational Group Theory?
I'm coding a small application that looks for periodic solutions to the gravitational n-body problem. I'm trying to better understanding the symmetries of solutions, which is made up of the product of ...
- 297
6
votes
0
answers
211
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Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck
In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
- 61
1
vote
1
answer
158
views
Comparison of special metrics on Riemann surfaces with the hyperbolic one
Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
- 31
1
vote
0
answers
111
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References on the partial trace
For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows :
$$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
- 1,389
0
votes
0
answers
22
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Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?
I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
- 698
5
votes
0
answers
156
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If $\omega_1$ is not inaccessible in $L$, how hard can it be to find a non-measurable $\Sigma^1_3$ set of reals?
In his wonderfully titled paper Can you take Solovay's inaccessible away? Shelah showed that if every $\mathbf{\Sigma}^1_3$ set of reals is Lebesgue measurable, then $\omega_1$ is an inaccessible ...
- 12.4k
14
votes
1
answer
556
views
What is the "schematic" point of view for regular polyhedra?
Last week, I read Wikipedia's article on Alexander Grothendieck. It lists his twelve greatest contributions to mathematics as accounted for in Grothendieck's own Récoltes et Semailles. The final item ...
4
votes
1
answer
92
views
Does an indexed functor $C \rightarrow \mathbb{B}$ extend to $\operatorname{Psh}(C) \rightarrow \mathbb{B}$?
This question concerns indexed categories and functors, as well as internal categories and functors.
$\newcommand{\Psh}{{\operatorname{Psh}}}$
$\newcommand{\Id}{{\operatorname{Id}}}$
$\newcommand{\...
- 285
1
vote
1
answer
90
views
Sobolev inequality with weight in the case $1<n\leq p$
Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
- 459
3
votes
0
answers
90
views
Reference request: étale local system on $\Gamma\backslash\mathcal H$ for $\Gamma$ non-congruence
Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an ...
- 775
4
votes
1
answer
197
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Solving a three-parameter recursive sequence
Consider the triple-indexed sequence of integers defined by
\begin{align} \label{coefficientsV} \nonumber
f(\alpha,\beta,\gamma)
&:=(2\alpha+8\beta+12\gamma-1)\cdot f(\alpha-1,\beta,\gamma)...
- 43.2k
3
votes
1
answer
118
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Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
1
vote
1
answer
240
views
The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.
is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?
if $a$ is ...
- 133
3
votes
0
answers
91
views
Are the reductions of the cuspidal characters of GL2(Fq) distinct?
Let $p$ be an odd prime and $q=p^n$ for some $n \geq 1$. If $\mathbb{F}_q$ is the unique, up to isomorphism, finite field with $q$ elements then the cuspidal representations of the group $\rm{GL}_2(\...
- 117
9
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0
answers
273
views
What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
- 1,802
4
votes
1
answer
174
views
Every elliptic surface contains only finitely many negative self-intersection rational curves?
By a properly elliptic surface, I mean an algebraic surface $X$ with Kodaira dimension $\kappa(X)=1$. It has a natural elliptic fibration $\pi\colon X\rightarrow S$.
According to section 5.2 of this ...
- 141
5
votes
1
answer
202
views
Turing degrees of lim infs of computable functions
The limit lemma gives a natural characterisation of functions $f : \mathbb{N} \to 2$ with Turing degree below $0'$: they are precisely those that can be written as $f(n) = \lim_k f_k(n)$ where $f_k : \...
- 4,378
1
vote
1
answer
177
views
Unitary representations of discrete (locally compact) groups
Let $\Gamma$ be a discrete (locally compact) subgroup of a locally compact Lie group. Let $H = L^2(\mathbb R^n, \mathbb C)$. Assume we have a complex unitary representation $$\Phi : \Gamma \to \...
- 357
7
votes
2
answers
242
views
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
2
votes
0
answers
61
views
Iwahori spherical representations of GL(n) with no nonzero fixed vectors under the fixator of a panel of the affine building
Let $G$ be the group $GL(n,F)$, where $F$ is a p-adic field, and $I$ its standard Iwahori subgroup. Let $\pi$ be an irreducible smooth representation of $G$ with nonzero $I$-fixed vectors. It is well ...
- 21
6
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0
answers
187
views
Is there a characterization of measurables in terms of indiscernibles?
There is a characterization of $\alpha$-Erdős cardinals in terms of sets of indiscernibles of order type $\alpha$. There is also a characterization of Ramsey cardinals in terms of sets of good ...
- 2,031
1
vote
0
answers
42
views
Sub-Gaussian analysis via bounded decomposition?
Let $\psi_\alpha(x) := \exp(x^\alpha)-1$.
The Sub-Gaussian Norm $\lVert X \rVert_{\psi_2}$ of a random variable $X$ is defined as
$$
\lVert X\rVert_{\psi_2} = \inf\{c>0\mid \mathbb{E}[\varphi_2(|X|/...
- 2,183
0
votes
0
answers
136
views
State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
0
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0
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61
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Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
2
votes
0
answers
66
views
Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
- 412
2
votes
1
answer
266
views
Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?
Let $E/K$ be a number field. For quadratic field extensions $L/K$, it is known that $\operatorname{Ш}(E/L)[2]$ can be arbitrarily large (cf. P. L. Clark and S. Sharif, "Period, index and ...
- 1,531
0
votes
0
answers
55
views
reference request: conditions for pointwise and operator-norm convergence of kernel projections
At a very high level, I’m interested in the following question. Suppose $X$ is a (separable) Hilbert space, and $T_n : X \rightarrow X$ is a sequence of finite rank self-adjoint maps that converges (...
- 101
4
votes
0
answers
126
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mod $p$ local Galois representation attached to elliptic curves
In the paper, lemma 4.4. The author gives the form of the representation of $G_p$ on $E[p]$ of the form
$$\begin{pmatrix} \varepsilon\chi & *\\0 & \chi^{-1} \end{pmatrix}.$$
Do they assumed ...
- 275
1
vote
1
answer
55
views
Positivity of caloric measure density on a cylinder
Let $u$ be a solution to the heat equation $u_t = \Delta u$ in the unit cylinder $B_1\times(-1,0) \subset \mathbb R^{n+1}$.
Then, it is well known (see for instance Chapter 2 in "Watson - ...
0
votes
1
answer
88
views
Singular continuous ergodic measures for the map $z \to z^2$
Where can I find the details of constructing singular continuous ergodic measures for the map $z \to z^2$ on the unit circle? I know that it was done by Furstenberg, but I could not find it explicitly ...
0
votes
0
answers
42
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Reference request: in Alexandrov geometry gradient flows preserve extremal subsets
It is mentioned in literature that in Alexandrov geometry gradient flows of semi-concave functions preserve each extremal subset.
I am looking for a proof of this fact.
- 21.8k
7
votes
3
answers
765
views
Implicit uses of Countable or Dependent Choice
What are instances of implicit reliance on countable or dependent choice in classic books? Two examples are
Introduction to Commutative Algebra by M.F. Atiyah and I.G. MacDonald
where it is claimed,...
6
votes
2
answers
319
views
Set theoretical foundations for derived categories
A modern approach to derived functors, that has been shown to be useful in a number of different circunstances is that of a derived category (see the book by Yakutieli, for example, here).
However, it ...
- 3,318
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votes
0
answers
104
views
Examples of function satisfying some bound
Let $f$ be an arithmetic function such that $f(n)\ll n^{\alpha}$ for some Real number $\alpha$ in $[0,1)$.
Can someone give me examples of such functions other than the sum of divisors function or is ...
- 1,247
5
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0
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176
views
Left Adjoint From the Category of Topological Groups to the Category of Condensed Groups
In Scholze's Lecture Notes on Condensed Sets, the author states that (Remark 1.8) the functor that takes a topological group $G$ to its condensation $\underline{G}$ has a left-adjoint, but we do not ...
- 241
0
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0
answers
52
views
Reference request for the determinant of a matrix constructed from Pascal's triangle
One can prove by induction that the matrix $M^{(n)}$ given by
$$ \begin{pmatrix}
1 & 1 & 1 & 1 & \dots & \binom{n}{0} \\
1 & 2 & 3 & 4 & \dots & \binom{n+1}{1} \...
- 5,546
13
votes
0
answers
331
views
Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
2
votes
0
answers
26
views
Reference for the biequivalence between the bicategory of distributors and the bicategory of two-sided discrete fibrations
It is well known that a distributor/profunctor $A \not\rightarrow B$, i.e. a functor $B^{\text{op}} \times A \to \mathrm{Set}$, is equivalent to a two-sided discrete fibration from $A$ to $B$. ...
- 10.6k
1
vote
1
answer
76
views
Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
- 20.2k
0
votes
0
answers
80
views
Relation between Chow groups and K theory
I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence
$$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
- 613
4
votes
0
answers
179
views
Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
1
vote
0
answers
150
views
What are alternative mathematical definitions of observers beyond Bennett and Hoffman's framework?
Motivation:
This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested.
In the study of information ...
- 3,054
4
votes
1
answer
269
views
Examples of discrete-space continuous-time dynamical systems
Something that I see occur repeatedly in my work is the need for formal notions of discrete-space continuous-time dynamics — these are generally realized as digital oscillators that are interact using ...
- 41
2
votes
0
answers
228
views
Any rigorous construction of $\phi^4$ theories without the mass term in the Lagrangian? (revised)
There are various papers on rigorous construction of massive $\phi^4$ theories in $2$ or $3$ Euclidean dimensions.
In 2D, there are in fact more general results such as this one by Glimm, Jaffe and ...
- 3,477
4
votes
0
answers
131
views
Ring theoretical aspects of the DAHA
The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively).
Nowdays there are many variations of the ...
- 3,318
11
votes
1
answer
883
views
Which paper is the "Taubes trick" from?
In symplectic geometry, the "Taubes trick" is an argument used to show that a moduli space $\mathcal{M}(J)$, depending on a parameter $J \in \mathcal{J}$, is cut out transversely for generic ...
- 243