Trending questions
159,029 questions
1
vote
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49
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Square-integral involving Brownian bridge
Let $B(t)$ be a standard Brownian bridge on $[0,1]$. Let $x>0$ be a (small) parameter. What is the distribution of
$$
\int_0^{1-x} \left( B(t + x) - B(t) \right)^2 dt?
$$
As noted I am interested ...
- 2,217
2
votes
1
answer
145
views
Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$
Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring.
There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
- 542
-3
votes
0
answers
75
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
1
vote
0
answers
72
views
Hasse principle for Brauer groups of fields of transcendence degree 2
In his paper "A Hasse principle for function fields over PAC fields" (DOI link), Ido Efrat proves the following result: Let $F$ be an extension of a perfect PAC field $K$ of relative ...
- 41
4
votes
2
answers
350
views
$K_2$ over finite fields and polynomials over finite fields
I am interested in presentations of the group $SL_n(\mathbb{F}_q)$ (and eventually $SL_n(\mathbb{F}_q[t])$).
The standard "Chevalley" presentation of $SL_n(R)$ for a ring $R$ has generators $...
- 173
3
votes
1
answer
180
views
Can one say that there are equal numbers of sets satisfying formulas in Second Order Arithmetic?
Is there a way of saying in second order arithmetic that the number of sets $X$ such that $\phi$ equals the number of sets $X$ such that $\psi$, where $\phi$ and $\psi$ are formulas with $X$ free, and ...
- 2,555
2
votes
0
answers
124
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Derived tensor products and regular sequences
Let $R \to A$ be a homomorphism of commutative rings, and let $x\in R$ be an element (or a sequence of elements in $R$, if you prefer) that is both $R$-regular and $A$-regular. Then we have
$$
A\...
- 125
2
votes
0
answers
55
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+50
Local Lipschitz continuity of signature map $S:C^{1-\text{var}}([0,T],\mathbb{R}^d) \to \mathcal{H}$
Just came across the claim that the signature map (between path space and tensor space) is locally Lipschitz continuous with respect to the $1-$variation norm (see section A.2.1).
More specifically, ...
- 161
2
votes
1
answer
142
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Does this result above six points follow have a name?
Does this result above six points follow have a name?
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points in the plane and $AB, CF, ED$ are concurrent and $BC, DA, FE$ are concurrent then $CD, EB, AF$ ...
- 1,776
2
votes
0
answers
99
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+100
Inequalities for norm of centered Gaussian and uncentered Gaussian
Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm.
Let $x \in \mathbb{R}^n$ and define
$$
F(x) = \mathbb{E}[\|x + g\| - \|g\|].
$$
I am wondering if it is possible to ...
- 380
3
votes
0
answers
28
views
Given a metric space $X$, is there a natural way to view the quasi-isometry group $QI(X)$ as a topological group?
Given a metric space $(X,d)$, we define $QI(X)$ as the set of quasi-isometries $f : X \to X$, modulo the equivalence relation
$$
f \sim g \ \ \ \ \text{ if and only if } \ \ \ \sup_{x \in X} \ d(f(x)...
- 639
7
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answers
117
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Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights
$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
- 6,622
6
votes
1
answer
498
views
NBG, ZFC+I, and Global Choice
In Shulman's 2008 paper 'Set Theory for Category Theory', he includes amongst the axioms of $\sf NBG$ the axiom of limitation of size. Being well known to imply the axiom of global choice, it seems to ...
- 63
1
vote
1
answer
80
views
A contradiction involving HoTT function extensionality in higher inductive types
I'm having difficulty resolving what seems to be a contradiction between two lemmas in the HoTT Book and am seeking help to determine where I'm going wrong. Below, all types may be higher inductive ...
- 13
8
votes
0
answers
206
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Can $\mathbb{C}$ have a "doppelganger" in $L(\mathbb{R})$ with countable automorphism group?
Working in $\mathsf{ZFC}$ + large cardinals (a proper class of Woodins, to be precise), is there a field $F\in L(\mathbb{R})$ such that $V\models F\cong\mathbb{C}$ and $L(\mathbb{R})\models\vert\...
- 20.5k
1
vote
0
answers
34
views
Vertex coloring of the Rado graph
Is there a reference for the following fact about the Rado graph (the random countable graph) which came up in an answer to this question?
If the vertices of the Rado graph $G=(V,E)$ are colored with ...
- 13.4k
1
vote
0
answers
63
views
Reference request: Proof theory in $W_1^1$
Buss defined $V_2^1$ as a second-order bounded arithmetic corresponding to $\mathsf{PSPACE}$.
Later, Skelley introduced $W_1^1$, a third-order bounded arithmetic of $\mathsf{PSPACE}$.
Since the ...
- 11
0
votes
1
answer
105
views
If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality
\begin{equation*}
\int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
- 481
2
votes
1
answer
109
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Neumann problem for the Laplacian with Dirac delta functions
I have encountered a problem while dealing with the adjoint method in potential flow that is also described, in a similar fashion, in (eq. 39) of this paper. The problem is essentially this:
$$\begin{...
- 21
9
votes
1
answer
302
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
5
votes
1
answer
128
views
Algebras over the trivial $\infty$-operad
I'm learning the concept of algebras over $\infty$-operads, following Higher Algebra. The simplest case is when the operad being the trivial operad $\mathrm{Triv}^\otimes$, defined as the 1-full ...
- 165
18
votes
13
answers
2k
views
When is 4 qualitatively different than $n\leq 3$?
Inspired by When is 2 qualitatively different from 3?
Also similar to Are there mathematical concepts that exist in dimension 4, but not in dimension 3? (Math SE), but with the restriction of being ...
-3
votes
0
answers
157
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
-1
votes
0
answers
26
views
Estimate the value of the PDF $P(f)$ at the minimal $f_0$ of the random-variable function $f(\mathbf{x})$
Let $f(\mathbf{x})=f(x_1,x_2,\dotsc,x_N)$ with $N>2$ be a real and continuous function and $f(\mathbf{x})\ge f_0$ for any $\mathbf{x}\in\mathbb{R}^N$. Now let $x_1,x_2,\dotsc,x_N$ be the i.i.d. ...
- 375
1
vote
1
answer
136
views
About dimensions of quotients of quasi projective varieties
This question is related to this one. If I have an locally closed, quasi projective scheme $X$ contained in an affine space, and a linearly reductive group $G$ acting freely on $X$, are there examples ...
- 558
2
votes
0
answers
66
views
Consistency of Sigma-V-2 uniformization with AD
Is ZF + AD consistent with: For every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(r)$ example?
DC is provable in ZF + every true $Σ^V_2$ statement has a $Δ^V_2$ example (i.e. witness). ...
- 7,545
5
votes
1
answer
286
views
Codimension zero embeddings and maps with small fibers
Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here.
...
- 501
7
votes
2
answers
341
views
Reference request: $\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \Lambda_+$
I am looking for proof of the "well-known" result that for a $4$-dimensional Riemannian manifold $(M, g)$, we have an isomorphism
$$
\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \...
- 113
-1
votes
0
answers
115
views
Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
3
votes
1
answer
224
views
Finite generativity of algebra with valuation
Let $C$ be a commutative finitely generated algebra with no zero divisors. If necessary, we can assume it to be graded and a unique factorization domain. Let $a\in C$ be a prime element.
Let's also ...
0
votes
0
answers
67
views
$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable
Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported.
I am wondering if we can ...
- 91
-2
votes
1
answer
61
views
On the correspondence between infinitesimal and integral description of connections
It is the title of an article by Petko Nikolov Triste Sissa 1981. I cannot access this pdf yet I remember that it was once avaliable on libgen and now I cannot find it. Please help.
- 19
2
votes
1
answer
466
views
About the number of critical points of a function
Suppose that $f$ is a totally monotone function on $(0,\infty)$, so that $(-1)^n f^{(n)}\ge0$ for all $n=0,1,\dots$, $f(0+)\in(0,\infty)$, and $f(t)\sim\frac{1}{t^{\frac{3}{2}}}$ as $t\to\infty$. Can ...
- 49
4
votes
1
answer
281
views
A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ ...
- 954
0
votes
0
answers
73
views
Co-boundary crossed homomorphism & "sign" preserving. Why 2-valued components is special?
Suppose $h_{g}: \mathbb{R}^n \to \mathbb{R}^{n-1}$ be a coboundary crossed homomorphism with action $g$ as a cyclic permutation of coordinates on $\mathbb{R}^n$ vectors. So, the acting group is a ...
- 11
3
votes
1
answer
167
views
Theory of $n$-truncated $A_\infty$ categories/functors?
One can define certain higher categorical truncations. For example, a discussion from the quasi-categories point of view can be found in HTT 5.5.6.
On the other hand, as a model of linear $\infty$-...
- 169
10
votes
2
answers
387
views
Iteration of $\aleph_2$-properness
Let us say a forcing $P$ is proper for a class of models $\mathcal C$, if for large enough regular $\theta$ and $M \prec H_\theta$ in $\mathcal C$ with $P \in M$, every $p \in P \cap M$ can be ...
- 18.6k
3
votes
0
answers
81
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
13
votes
0
answers
304
views
What is the status of this conjecture on symplectic forms "standard-at-infinity" on $\mathbb{R}^{2n}$?
In McDuff and Salamon's Introduction to Symplectic Topology, the following open problem is mentioned.
Problem 50 (Standard-at-infinity)
Let $n \ge 3$ and let $\omega$ be a symplectic form on $\mathbb{...
0
votes
0
answers
78
views
Chow moving lemma with additional property
All varieties are over algebraically closed field of characteristic zero. Let $S$ be a smooth projective surface. Let $D$ be an irreducible divisor on $S$, $H$ be another divisor and $Z\subset S$ be a ...
- 219
3
votes
1
answer
305
views
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$ as $\operatorname{GL}(n,\mathbb{C})$-modules
Consider the natural representations of $\operatorname{GL}(n,\mathbb{C})$ in the spaces
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$.
Is it known how to ...
- 21.8k
1
vote
0
answers
137
views
What is a quantum condensed space?
Due to a categorical equivalence involving compact topological spaces and unital commutative $\mathrm{C}^*$-algebras, there is a practise involved in so-called quantum mathematics where a ...
- 1,037
12
votes
1
answer
397
views
Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\int_S ...
- 24.2k
2
votes
1
answer
103
views
Number of disjoint set triplets in a union-closed family
Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is allowed to contain multiple instances of the same set. Let $U(\mathcal{F})$ be the ...
- 1,000
3
votes
2
answers
147
views
Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
- 1,109
7
votes
0
answers
132
views
On the optimal strength of Goodstein's theorem
Goodstein's theorem is a famous example of an arithmetical statement that is unprovable in $\mathsf{PA}$ but provable in a stronger theory. It is well-known that Goodstein's theorem implies the ...
- 3,042
2
votes
1
answer
71
views
Coradical filtration for comodules is exhaustive
It is a standard fact in the theory of coalgebras and comodules that, given a coalgebra $C$ and a comodule $M$ over $C$, the coradical filtration
$$M_n := \Delta^{-1}(M\otimes C_0 + M_{n-1}\otimes C)$$...
- 518
6
votes
1
answer
257
views
Strict versus weak Gray tensor product
For two strict $(\infty,\infty)$-categories $C$, $D$
one can consider the strict lax Gray tensor product $C \otimes_\text{strict} D$.
Similarly, for two weak $(\infty,\infty)$-categories $C$, $D$
one ...
- 1,361
2
votes
0
answers
61
views
Wieferich primes and identities for the Euler quotients of $2^n+1$ and $\frac{2^n+1}{3}$
Let $n>1$ be odd integer.
Define the Euler quotient $a(n)=\frac{2^{\varphi(n)}-1 \bmod n^2}{n}$.
Number $n$ with $a(n)=0$ is Wieferich number and if it is prime
it is Wieferich prime.
It is open ...
- 25.4k
2
votes
0
answers
116
views
Induced homology map zero implies zero in cobordism?
I had asked this in math stackexchange, but got no reply. Hence, I'm asking here.
[I'm no expert in (co)bordism theory, and I've been struggling with it for the past few weeks. Any good references on ...
- 21