Trending questions
159,035 questions
8
votes
2
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360
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Is the unbounded derived $\infty$-category of a general abelian category stable?
Let $A$ be any abelian category. Consider the stable $\infty$-category $N_{dg}(Ch(A))$ defined as the differential graded nerve of the category of chain complexes $Ch(A)$ in $A$. We can define an $\...
4
votes
0
answers
117
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Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
2
votes
3
answers
181
views
Stabilizers of the action of Levi on abelianization of nilpotent radical
$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
3
votes
1
answer
128
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Comparing two different principles of premeasure-to-measure extension
It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for ...
3
votes
0
answers
191
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Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification
$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
19
votes
0
answers
478
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On C*-rigidity problem for torsion-free groups
I'd like to address the $\mathrm{C}^\ast$-rigidity problem for
torsion-free groups (see
this paper),
which asks for non-isomorphic torsion-free groups with isomorphic
(reduced) group $\mathrm{C}^\ast$-...
1
vote
0
answers
126
views
Growth polynomial of the Associahedron graph ? (Is it approximately Gaussian ?)
Consider Associahedron, consider graph build from its vertices and edges. Choose some vertex. Let us count the number of vertices on distances $k$ from the selected vertex. Write a generating ...
3
votes
1
answer
197
views
Fibrations of categories with terminal objects
Fix $\mathbf{B}$ for a category with pullbacks.
Proof: $(\Longrightarrow)$ Suppose that $P$ is a fibration of categories with terminal objects. Then for every $I$ in $\mathbf{B}$ there is an object $...
0
votes
0
answers
109
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Calculi of pseudodifferential operators and K-theory
I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of ...
4
votes
1
answer
379
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 ...
4
votes
0
answers
101
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Full subcategories of stable $\infty$-categories closed under all shifts
Is there a name for an $\infty$-category C that admits a zero object and suspensions such that the suspension functor $C \to C$ is an equivalence but which does not necessarily admit cofibers and ...
0
votes
1
answer
131
views
Closed form for $\sum\limits_{k=0}^{n} [\operatorname{wt}(k) = m]$ where $\operatorname{wt}(n)$ is the binary weight of $n$
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$).
Let $a(n,m)$ be the family of integer sequences such that
$$
a(n,m) = \sum\limits_{k=0}^{n} [\operatorname{wt}(...
1
vote
1
answer
85
views
Hilbert symbol of a quaternion algebra given ramified places
I am reading the paper: https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-3/Derived-Arithmetic-Fuchsian-Groups-of-Genus-Two/em/1227121388.full
in order to find an explicit ...
2
votes
1
answer
83
views
Causality of Killing vector fields in a Lorentzian Ricci flat spacetime
In a connected Lorentzian spacetime that is Ricci flat and also nice (say smooth, global hyperbolic, etc), can a global Killing vector field be null in an open subset and timelike (or spacelike) in ...
1
vote
0
answers
90
views
About synonymy relationships around these two theories?
The following question is about patterns of synonymy relationships around two theories, $T^+$ and $\sf PA$.
For purposes of self inclusiveness I'll re-iterate $T$ and its extensions.
$\textbf{Logic:}$ ...
3
votes
1
answer
114
views
Selmer complex and total complex
Thanks for your reading. I'm studying Selmer complex book by Jan Nekovar. For the definition of Selmer complex I meet a problem.
In the introduction(page 9, 0.8.0) the author gives us a definition of ...
1
vote
0
answers
100
views
Unitary representations of Fuchsian and Kleinian groups
Let $\Gamma$ be a discrete group that is either Fuchsian ($\Gamma \subseteq \text{PSL}(2,\mathbb R)$) or Kleinian ($\Gamma \subseteq \text{PSL}(2,\mathbb{C})$).
I have a unitary representationL
$$
\...
2
votes
1
answer
130
views
Question about maps on cofibers being zero
I have the following problem, I am skeptical that the solution is as easy as the one I wrote so would be grateful if some expert can enlighten me.
Let $G\colon \mathcal{C}\to \mathcal{D}$ be a functor ...
6
votes
1
answer
244
views
Is it possible that the number of $\mathcal{T}$-algebras is an arithmetic progression?
For a single-sorted algebraic theory $\mathcal{T}$ denote by $t_n$ the number of $\mathcal{T}$-algebras with $n$ elements (up to isomorphism). Is there an example for $\mathcal{T}$ such that ...
6
votes
0
answers
147
views
Are cofibrant objects flat with respect to Day convolution?
Question
Let $\mathcal{C}$ be a small symmetric monoidal category. The category $\mathsf{sSet}^{\mathcal{C}}$ of simplicial precosheaves on $\mathcal{C}$ is a symmetric monoidal model category with ...
0
votes
1
answer
171
views
Is the evolution family self-adjoint?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
\newcommand{\qtext}[1]{\quad\text{#1}}
\newcommand{\qtextq}[1]{\quad\text{#1}\quad}
$
I am reading Roland Schnaubelt's survey ...
10
votes
2
answers
460
views
Is the number of similarity classes of integer matrices with given minimal and characteristic polynomial finite?
Latimer and MacDuffee proved that there is a bijection between similarity classes of integer matrices with irreducible characteristic polynomial $\chi$ and the ideal class monoid of $\mathbb{Z}[\alpha]...
5
votes
0
answers
114
views
Is transverse intersection continuous?
Let $S_0$ be a smooth compact $k$-dimensional manifold with boundary and $\mathcal{E}_{\rm p}(S_0, \mathbb{D}^n)$ be the space of smooth proper embeddings into the unit disk in $\mathbb{R}^n$ with the ...
5
votes
1
answer
184
views
Localizing spaces at stable homotopy equivalences
According to Bousfield's theory of localization, given any homology theory $E_*$ one can produce a reflective localization of the category of (pointed) CW complexes, in the sense that any such CW ...
3
votes
0
answers
160
views
Gowers' dichotomy for quotients
Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable.
A ...
5
votes
1
answer
301
views
Operations on filtered / graded vector spaces via $\mathbb{A}^1 / \mathbb{G}_m$
A well known fact (e.g. Moulinos - The geometry of filtrations) is that one can describe the category of filtered vector spaces as $\operatorname{FilVect} \simeq \operatorname{QCoh}(\mathbb{A}^1/\...
2
votes
1
answer
235
views
Tiling with one of each shape
Q. Is there a tiling of the plane by one each of simple polygons of $n$ vertices:
one triangle, one quadrilateral, one pentagon, $\ldots$ ,
one simple polygon of $n$ vertices, $\ldots$ ?
Here a ...
6
votes
2
answers
238
views
Is the (inverse) Dold-Kan functor fully faithful on chain complexes of commutative monoids?
My question
The background/notation for all of the content of this post is in Lurie, Higher Algebra [HA], Ch. 1.2.3. Everything is purely 1-categorical.
Let $\mathcal{A}$ a semiadditive category (with ...
4
votes
0
answers
116
views
Categories enriched in $(m,n)\text{-Cat}$ with Crans–Gray tensor product
(All higher categories will be strict unless otherwise noted):
It is a somewhat folklorish fact that if you equip the category of $(m,n)$ categories with the cartesian monoidal structure, then a ...
4
votes
1
answer
228
views
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{...
3
votes
2
answers
297
views
Are degeneracy loci of general morphisms always locally complete intersections?
Let $X$ be a smooth irreducible complex variety of dimension $n \ge 6$. Let $E$ be a globally generated rank $r \ge 2$ vector bundle on $X$ and let $\varphi : {\mathcal O}_X^{\oplus (r-1)} \to E$ be a ...
3
votes
1
answer
202
views
Matrix-tree theorem for inverse matrices
Let $L$ be the Laplacian of a directed weighted graph on $n$ nodes, e.g., for $n=4$:
$$
L = \left(\begin{array}{cccc} w_{1,1}+w_{1,2}+w_{1,3}+w_{1,4} & -w_{1,2} & -w_{1,3} & -w_{1,4}\\ ...
3
votes
1
answer
597
views
Primes which are safe and Sophie Germain
If $p$ is a Sophie Germain prime then $2p+1$ is safe prime.
If $2p+1$ is safe prime then $p$ is Sophie Germain prime.
What is their conjectured distribution of primes $p$ which are both Sophie ...
8
votes
1
answer
534
views
The cars problem, again
Consider the following simple problem: We are given $2n$ parking spots, labelled from 0 to $2n-1$. There are $n$ cars on the first $n$ spots, and the remaining $n$ spots are free. At every step, every ...
2
votes
1
answer
156
views
$\mathbb{C}^*$-action on moduli space of Higgs bundles
Let $M_{r,d}$ be the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ over a compact Riemann surface. Over $M_{r,d}$ we have a $\mathbb{C}^*$-action $$t \cdot (E,\phi)=(E, t \phi). $...
1
vote
3
answers
162
views
Upper Bound for $\mathbb{E}\left[\max_{j \in \mathcal{N}} h_{j}\right]$
Assume $\{h_j\}_{j\in \mathcal{N}}$ are independent Gamma random variables, each with potentially different distributions and parameters. I am looking for an upper bound for $\mathbb{E}\left[\max_{j \...
-1
votes
0
answers
49
views
$s^2 + 1$ has no prime factors $p \equiv 3 \mod 4$? [migrated]
$s^2 + 1$ is the squared length of a vector in a square lattice for every integer $s$. The number of integer
solutions in $a, b$ for $a^2 + b^2 = s^2 + 1$ is of course $> 0$, e.g. $a = s, b = 1$ is ...
21
votes
3
answers
2k
views
Trigonometric inequality
For odd and coprime positive integers $p$ and $q$, the following inequality holds:
$$\sum_{m=1}^{p} \sum_{n=1}^{q} \frac{2}{\cos(\frac{2m\pi}{p})+\cos(\frac{2n\pi}{q})} \le pq(|p-q|+1)$$
Unfortunately,...
0
votes
0
answers
151
views
Compactification of the Jacobian of singular curves via parabolic modules
I would like to better understand a certain compactification of the Jacobian variety of a singular algebraic plane curve as described in Cook's Ph.D. 1993 thesis Local and Global Aspects
of the Module ...
4
votes
0
answers
122
views
Projection behavior under the automorphism $\Phi$ on a diffuse semi-finite von Neumann algebra
Let $\mathcal{M}$ be a diffuse semi-finite von Neumann algebra acting on a Hilbert space $\mathcal{H}$, equipped with a faithful normal semi-finite trace $\varphi$. Let $\Phi : \mathcal{M} \to \...
0
votes
1
answer
170
views
Partial sums of binomial coefficients and related family of polynomials
Let $a(n)$ be A302117. Here
$$
a(n) = 4(n-1)a(n-1) - \frac{1}{3}\prod\limits_{k=0}^{n-1}(2k-3), \\
a(0) = 0.
$$
Let
$$
T(n,k) = \sum\limits_{i=0}^{k} \binom{n}{i}.
$$
Let $P_n(z)$ be the family of ...
0
votes
0
answers
96
views
Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
2
votes
0
answers
194
views
Functions such that the *integral* of the Fourier transform is non-negative?
Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that
$$\int_{-\infty}^x \widehat{f}(t) dt \...
0
votes
0
answers
83
views
Exercise on bounded sets in locally convex metrizable spaces [migrated]
I'm stuck with the following exercise:
Let $(E,\mathcal{P})$ be a metrizable locally convex space. Show that for each sequence $(A_{n})_{n}$ of bounded subsets in $E$ there is a sequence of positive ...
-3
votes
1
answer
195
views
Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
4
votes
2
answers
367
views
Notion of prime congruences
We have the idea of a prime ideal in a commutative ring $R$ but in universal algebra, we generalize the notion of ideal to that of a congruence. I’ve thought over the question of what a prime ...
2
votes
1
answer
153
views
Baer sums of extensions
Apologies in advance if this question is too basic, I looked briefly through Weibel and the stacks project and couldn't find any relevant reference.
Let $\mathcal{A}$ denote an abelian category, and ...
1
vote
1
answer
418
views
Uses of the Mukai vector
Let $X$ be say a smooth projective variety. For $\mathcal{E}^\bullet \in D^b(X)$ the so-called Mukai vector is defined as $$v(\mathcal{E}^\bullet) = \operatorname{ch}(\mathcal{E}^\bullet)\sqrt{\...
2
votes
1
answer
65
views
On the stationarity of Gaussian processes
I am trying to understand and prove the statement:
The normal (or Gaussian) process is stationary in the wide sense if and only if it is strictly stationary.
I know the following:
A strictly ...
4
votes
0
answers
277
views
Derek the Differentiable Dinosaur
I’ve come across several fond references to some semi-published lecture notes from Warwick in the 80s, by Bill Breckon (and, in some mentions, I. Harrison), Differentiating functions of lots of ...