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Grothendieck construction on fibred categories/stacks

This question is related to a previous question of mine, which has so far gone unanswered. For a fixed site $\mathcal C$, the fibred categories over $\mathcal C$ form a (strict) $2$-category, see here....
gimothytowers's user avatar
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 ...
GURI920826's user avatar
6 votes
0 answers
75 views

What are the algebras of the powerset intersection (oplax) monad?

The assignment $X\mapsto\mathcal{P}(X)$ and $f\mapsto f_*$ (direct images) defines a functor $\mathcal{P}\colon\mathsf{Sets}\to\mathsf{Sets}$. This functor has a monad structure whose multiplication $\...
Emily's user avatar
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4 votes
0 answers
168 views

How to prove the following equation (which involves binomials and determinant of 2×2 matrices)?

I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...
tongjun's user avatar
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1 vote
0 answers
75 views

Parametrized moduli spaces of semistable bundles by varying Kähler classes

Inspired by Liviu Nicolaescu's answer here, I was eager on trying to come up with examples of parameter spaces $\Lambda$, configuration spaces $\mathscr{C}$ and parametrized moduli spaces $\mathscr{M}$...
Niemero's user avatar
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1 vote
0 answers
69 views

Descent of $G$-invariant formal system of parameters using GAGF

Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
user267839's user avatar
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4 votes
1 answer
270 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 ...
Thomas Pluck's user avatar
16 votes
3 answers
782 views

Show there is no positive r.v. $U$ such that $\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge (k+1)/2 }]}{\mathbb{E}[U^k]}, \, \forall k \in \mathbb{N}_0$

Let $U$ be a non-negative random variable such that for all $k \in \mathbb{N}_0$ \begin{align} \frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge \frac{k+1}{2} }]}{\mathbb{E}[U^k]}. \end{align} In ...
Boby's user avatar
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115 views

How can I derive $V$ from the following equation?

$$ [(U-V)^{-1}X^{T}Y]^{T}\;[(U-V)^{-1}X^{T}Y] = I. $$ Here $U$ and $V$ are symmetric $d \times d$ matrices; $X=[x_1,x_2,...,x_n]$ is an $n \times d$ data matrix ($n$ is the number of samples and $d$ ...
dawei's user avatar
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8 votes
4 answers
1k views

Counting with trees

Let $\mathcal{U}_n$ denote the set of unrooted unlabelled trees with $n$ edges. For $T\in\mathcal{U}_n$, let $1^{u_1}2^{u_2}\cdots n^{u_n}$ be its degree distribution, that is, $u_i=\#$ of vertices ...
T. Amdeberhan's user avatar
2 votes
2 answers
189 views

Compute corestriction map on group cohomology in Magma

I am trying to use Magma to compute the corestriction of a second cohomology class, but I’m not sure how to interpret the output. The details and code are given below. Consider the group $G = \mathrm{...
Jef's user avatar
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5 votes
1 answer
622 views

Non-atomic probability measures on N

One can intuitively imagine picking a random natural number and ask to what extent the intuition can be axiomatized. Using the axiom of choice, there is a total finitely additive (monotonic) averaging ...
Dmytro Taranovsky's user avatar
1 vote
2 answers
188 views

Non-compact surfaces with non-negative Gauss curvature

Is there a topological classification of non-compact complete connected 2-dimensional Riemannian manifolds with non-negative Gauss curvature?
asv's user avatar
  • 21.8k
0 votes
1 answer
74 views

Lower Bound on the Probability for the Sum of IID Random Variables

Let $X_1,\ldots,X_n$ be $n$ iid normalized random variables (with finite variance, possibly sub-Gaussian). Suppose further that $\mathbb{P}(X_1 > 0 ) > 1/2$, implying a positive skew in the ...
xabialgebra's user avatar
2 votes
1 answer
107 views

Elliptic regularity with negative Sobolev space on bounded or unbounded domains

I am looking for some reference which deals with the existence and regularity of solution to $ -\Delta u = f $ in bounded or unbounded domain $\Omega$ and with Dirichlet boundary condition, $u|\...
pde's user avatar
  • 121
5 votes
2 answers
532 views

Which coupling of uniform random variables maximises the essential infimum of the sum?

Recall that a coupling of probability measures $\mu_i$ is a set of random variables $X_i$ defined on the same probability space $\Omega$ such that $X_i \sim \mu_i$. Question: Let $\mu_1, \dots, \mu_n$ ...
Nate River's user avatar
  • 6,313
9 votes
0 answers
292 views

Tilings in finite (not necessarily Abelian) groups

Let $G$ be a finite (not necessarily abelian) group. We call $A \subseteq G$ a right-tiling (for simplicity, a tiling) of $G$ if there exists a $B \subseteq G$ so that $$ G = \bigsqcup_{b\in B} bA.$$ ...
Anurag Sahay's user avatar
  • 1,354
2 votes
0 answers
85 views

Formula for sum involving products of (symplectic) Schur functions

This question is a continuation of a question asked yesterday which had a very nice answer. Consider the summation $$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\...
Rellek's user avatar
  • 553
7 votes
1 answer
974 views

a claim for a proof of the invariant subspace problem [closed]

Recently four mathematicians claimed to have proven the invariant subspace problem, which is the problem that states Does every bounded operator on a separable Hilbert space have a non-trivial ...
euleroid's user avatar
17 votes
1 answer
414 views

Is $MU/I_\infty$ an $E_\infty$ ring?

Fix a prime $p$, and suppose that $p>2$ for simplicity, although many things should also work for $p=2$. Let $F$ be the usual formal group law defined over $MU_*$, and let $I_\infty$ be the ideal ...
Neil Strickland's user avatar
18 votes
2 answers
2k views

When did the distinction between "pure" and "applied" mathematics become common?

Some ages ago, there was no difference between chemistry, physics, mathematics, and perhaps even philosophy. These were not further distinguished and largely practiced by the same people. Obviously, ...
shuhalo's user avatar
  • 5,327
4 votes
0 answers
238 views

Jacobian of exponential map

I am playing around with the coarea formula and came across the problem of finding the Jacobian of the exponential map. Let $G$ be a compact, semisimple Lie group with associated Lie algebra $\...
DarkViole7's user avatar
2 votes
1 answer
108 views

Discrete isoperimetric inequality involving the diameter of an n-gon

I am interested in discrete isoperimetric-type inequalities that allow one to bound the perimeter of an $n$-gon from above (as opposed to below, as in the classical case when one bounds the perimeter ...
Anton's user avatar
  • 1,625
1 vote
1 answer
224 views

Proper smooth pushforward of vector bundle is a vector bundle?

Suppose $X$ and $Y$ are algebraic varieties over a field $k$, and $f:X \to Y$ is proper smooth. Then for a vector bundle $E$, and any $i\ge 0$, do we have $R^if_*(E)$ locally free? I know we need the ...
Richard's user avatar
  • 785
23 votes
3 answers
3k views

Why believe in the existence of large cardinals rather than just their consistency?

Large cardinal hypotheses and related hypotheses like projective determinacy are well-known to be gauges of the consistency strength of various theories. What reasons are there to believe in their ...
Jesse Elliott's user avatar
6 votes
0 answers
188 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 ...
C7X's user avatar
  • 2,031
0 votes
1 answer
114 views

Sum of an infinite series

I have the following infinite series $$S=\sum_{n=1}^\infty \frac{x^n}{n!}e^{\frac{c}{n}}, ~(x>0,c>0)$$ How could I perform this summation? Or is there any good analytical/closed-form ...
AD Le's user avatar
  • 19
3 votes
0 answers
91 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 ...
Richard's user avatar
  • 785
2 votes
1 answer
161 views

Smallest dimensional faithful complex representation of $\mathrm{PSL}(k,q)$

For given $k>1$ and $q$ a prime power, what is the minimal dimension, as a function of $(k,q)$, for which a faithful complex representation of the projective special linear group over $\mathbb{F}_q$...
Fetchinson0234's user avatar
1 vote
1 answer
118 views

Lower bound for a commutator trace

I have this Hilbert space of square-integrable complex-valued functions on a square, $\mathbb{L}^2([0,1]^2)$. And let $M_x$, $M_y$, and $M_{x+y} = M_x+M_y$ be the operators of multiplication by the ...
Chilperic's user avatar
  • 121
0 votes
0 answers
95 views

Class multiplication coefficients of symmetric groups

My question is that I was working with some counting problems, and finally the answer should be $$ \nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
user545662's user avatar
7 votes
2 answers
841 views

Why is $\mathbb R^{\mathbb N}$ not high-dimensional enough?

In this paper [1], the authors consider the limiting distribution of $$S_{n,p}:=\frac{1}{\sqrt n}\sum_{k=1}^nX_k$$ for $p\rightarrow\infty$ as $n\rightarrow\infty$, where $X_1, X_2,\dots, X_n$ are ...
Quertiopler's user avatar
5 votes
2 answers
218 views

Smooth toric variety which is a cube is a bott tower (reference request)

According to Lee, Masuda and Park (page 3), the following result is "well-known in toric topology". I've found a proof, but I would like a published reference. Let $X$ be a toric variety. ...
David E Speyer's user avatar
0 votes
1 answer
96 views

On the behaviour of individual random walks of a Markov Chain

My current research (on Probabilistic Automaton) brought me to the following question regarding Markov Chains. I state the definitions for the sake of clarity. Let $M$ be a discrete-time finite Markov ...
santi cifu's user avatar
-2 votes
0 answers
54 views

Density of squared bessel process

I was trying to find a transition density function for a squared Bessel process. In the book "Continuous martingale and Brownian motion" by Revuz and Yor, I find a Corollary on page 441 that ...
LOREY CHU's user avatar
4 votes
1 answer
257 views

Approximating an $L^1$ function with Riemann sums

Note: Here all functions are genuine functions, i.e. pointwise defined measurable functions instead of defined only a.e. Let $f: [0, 1] \to \mathbb R$ be an arbitrary $L^1$ function. Of course, $f$ is ...
Nate River's user avatar
  • 6,313
2 votes
0 answers
43 views

Distributions and time-kernels

Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
G. Blaickner's user avatar
  • 1,429
1 vote
1 answer
69 views

Exhausting sequences contain a $\pi$ lift of a subset with a $(1-\delta)$ factor

Let $\pi : Y \to X$ be a measurable map between the $\sigma$-finite measure spaces $(Y, \mathcal{B}, \nu)$ and $(X, \mathcal{A}, \mu)$. Suppose there exists $c \in (0, \infty)$ such that for all $A \...
abcdmath's user avatar
  • 105
1 vote
0 answers
238 views

Claimed proofs of graph labelling conjectures [closed]

The following recent series of arXiv papers claims to prove several of the most famous graph labelling conjectures. Edinah Gnang is the common author, none of the papers seem to be published further, ...
David Wood's user avatar
  • 1,319
1 vote
1 answer
188 views

Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?

This question is related to This question. When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
Haidara's user avatar
  • 178
26 votes
1 answer
943 views

What are the integer solutions to $x^4+2y^4=3z^4$?

This equation has the obvious integer solution $(x,y,z)=(\pm 1,\pm 1,\pm 1)$. By Faltings's theorem, the equation has finitely many primitive integer solutions (those with $\gcd(x,y,z)=1$). What is ...
Ashleigh Wilcox's user avatar
1 vote
0 answers
88 views

Equivariant resolution of singularity making a pullback of a line bundle admit a root

I am considering the following situation. Let $X$ be an irreducible normal projective variety with an action of a linear algebraic group $H$, and we have a $H$-equivariant line bundle $L$ over $X$. We ...
Ji Woong Park's user avatar
-1 votes
0 answers
73 views

Analog of ceil and floor of $\sqrt{a(a+1)}$ in modular arithmetic

If we take ceil and floor of $\sqrt{x(x+1)}$ (when it exists) we get $x$ and $x+1$ respectively. Is there an analog of this assuming roots exist in modular arithmetic (at least modulo primes)? ...
Turbo's user avatar
  • 13.9k
4 votes
1 answer
250 views

Galois action on the pro-algebraic completion of the singular fundamental group

Let $X$ be a smooth variety over a field $K \subset \mathbb{C}$. The singular fundamental group $\pi_1(X^{\text{an}}, x)$ generally does not carry an action of the absolute Galois group $\operatorname{...
HJK's user avatar
  • 199
0 votes
1 answer
64 views

Transitive map on a profinite group

Let $f$ be a continuous endomorphism of a compact Hausdorff totally disconnected topological group $G$ and let $H$ be a closed normal subgroup of G such that $f(H)\subseteq H$ and with $\mu(H)=0$ ...
Nick Belane's user avatar
0 votes
0 answers
91 views

Studying flows on $L^2(\mathbb{R}^2)$ given by vector fields using unitary operators

Background: In the $xy$-plane (or a 2-sphere if you are concerned about compactness), we can think about three different types of flow given by vector fields. (1) pushes everything towards the origin, ...
Edwin Beggs's user avatar
  • 1,143
0 votes
1 answer
236 views

Solving a 0-1 quadratic matrix inequality

I am working on a binary optimization problem. So far I have derived the following constraint functions. \begin{align} \begin{bmatrix} \left( P + \sum_{i=1}^n (\sum_{j=1}^n x_{i, j} \alpha_j) e_i e_i^...
zycai's user avatar
  • 11
2 votes
1 answer
189 views

Uniqueness of differences of roots of polynomials over finite field

Let $f$ be a polynomial over a finite field $\mathbf{F}_p$ with $p \neq 2$. Let $R$ be the roots of $f$ in some extension field. I am interested in the multiset of differences $R - R = \{ r - s \mid r,...
darko's user avatar
  • 269
7 votes
1 answer
213 views

Existence of asymptotic sequence in ergodic measure-preserving transformations

Let $(X,\mathcal{F},\mu)$ be a measure space and let $T:X\to X$ be an ergodic measure-preserving transformation. We assume that $T$ satisfies the property that if $B \in \mathcal{F}$ and $T^{-1}B \...
DenOfZero's user avatar
  • 113
0 votes
2 answers
364 views

Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?

I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
Haidara's user avatar
  • 178

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