Trending questions
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4
votes
1
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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....
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 ...
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 $\...
4
votes
0
answers
168
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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 ...
1
vote
0
answers
75
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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 $\mathscr{M}$...
1
vote
0
answers
69
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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 \...
4
votes
1
answer
270
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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 ...
16
votes
3
answers
782
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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 ...
1
vote
0
answers
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 $d$ ...
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 ...
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{...
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 ...
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?
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 ...
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|\...
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 $\mu_1, \dots, \mu_n$ ...
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
$$ G = \bigsqcup_{b\in B} bA.$$
...
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_{[\...
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 ...
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 ...
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, ...
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 $\...
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 ...
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 ...
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 ...
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 ...
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)$$
How could I perform this summation? Or is there any good analytical/closed-form ...
3
votes
0
answers
91
views
Reference request: étale local system on $\Gamma\backslash\mathcal H$ for $\Gamma$ non-congruence
Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an ...
2
votes
1
answer
161
views
Smallest dimensional faithful complex representation of $\mathrm{PSL}(k,q)$
For given $k>1$ and $q$ a prime power, what is the minimal dimension, as a function of $(k,q)$, for which a faithful complex representation of the projective special linear group over $\mathbb{F}_q$...
1
vote
1
answer
118
views
Lower bound for a commutator trace
I have this Hilbert space of square-integrable complex-valued functions on a square, $\mathbb{L}^2([0,1]^2)$. And let $M_x$, $M_y$, and $M_{x+y} = M_x+M_y$ be the operators of multiplication by the ...
0
votes
0
answers
95
views
Class multiplication coefficients of symmetric groups
My question is that I was working with some counting problems, and finally the answer should be
$$
\nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
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 ...
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. ...
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 ...
-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 ...
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 ...
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 ...
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 \...
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, ...
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 ...
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 ...
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 ...
-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)?
...
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{...
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 $\mu(H)=0$ ...
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,
...
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^...
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,...
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 \...
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 ...