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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 $\...
Lin Chen's user avatar
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117 views

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 ...
Matan Tal's user avatar
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 ...
Zhiyu's user avatar
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3 votes
1 answer
128 views

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 ...
Atom's user avatar
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3 votes
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191 views

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 ...
hofnumber's user avatar
19 votes
0 answers
478 views

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 $\mathrm{C}^\ast$-...
Narutaka OZAWA's user avatar
1 vote
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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 ...
Alexander Chervov's user avatar
3 votes
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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 $...
Siya's user avatar
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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 ...
zarathustra's user avatar
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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 ...
G. Fougeron's user avatar
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101 views

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 ...
Hadrian Heine's user avatar
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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}(...
Notamathematician's user avatar
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 ...
ah--'s user avatar
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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 ...
Sean's user avatar
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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. $\textbf{Logic:}$ ...
Zuhair Al-Johar's user avatar
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 ...
Rellw's user avatar
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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 $$ \...
user82261's user avatar
  • 357
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 ...
user197402's user avatar
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 ...
Martin Brandenburg's user avatar
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 ...
Ken's user avatar
  • 2,292
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 ...
Akira's user avatar
  • 825
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]...
Ben Marlin's user avatar
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 ...
Sarah Brooks's user avatar
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 ...
dicemaster666's user avatar
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 ...
M.González's user avatar
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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/\...
E. KOW's user avatar
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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 ...
Joseph O'Rourke's user avatar
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 ...
naahiv's user avatar
  • 391
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 ...
K. Strong's user avatar
  • 423
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{...
Sebastian Pozo's user avatar
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 ...
Cob's user avatar
  • 331
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}\\ ...
Federico Poloni's user avatar
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 ...
Turbo's user avatar
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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 ...
AccidentalFourierTransform's user avatar
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). $...
Tommaso Scognamiglio's user avatar
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 \...
Lee White's user avatar
-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 ...
Helmut Ruhland's user avatar
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,...
Yessir03's user avatar
  • 683
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 ...
John Doe's user avatar
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 \...
DenOfZero's user avatar
  • 113
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 ...
Notamathematician's user avatar
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 ...
Serge the Toaster's user avatar
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 \...
H A Helfgott's user avatar
  • 20.2k
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 ...
Vicent Miralles's user avatar
-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 ...
charlie_beck's user avatar
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 ...
Lave Cave's user avatar
  • 293
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 ...
kindasorta's user avatar
  • 2,907
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{\...
Niemero's user avatar
  • 137
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 ...
MathematicalMind1618's user avatar
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 ...
Peter LeFanu Lumsdaine's user avatar

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