Newest Questions
159,101 questions
2
votes
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Reference request : A SPDE model
Let $\Omega_0\subset\mathbb R^d$ be open and bounded with sufficiently smooth boundary $\partial\Omega_0$. Let $O\subset \Omega_0$ be a random open subset. Set $\Omega:=\Omega_0\setminus O$. Consider ...
- 1,409
2
votes
1
answer
184
views
Gorenstein projective module over commutative local algebras
Let $A$ be a local commutative finite dimensional algebra over a field $K$.
An $A$-module $M$ is called Gorenstein projective if $M$ is reflexive, $Ext_A^i(M,A)=0=Ext_A^i(M^{*},A)$ for all $i>0$ ...
- 26.5k
1
vote
0
answers
106
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Benjamini-Schramm convergence: convergence on metric balls implies weak convergence?
In the famous paper by Benjamini-Schramm 2001, they consider the space of rooted graphs with uniformly bounded degrees, this space modulo rooted graph isomorphism is denoted by $\mathcal X$.
This ...
- 502
1
vote
0
answers
133
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Find the number of tuples $(x_0,x_1,\dots)$ such that $n=\sum\limits_{j=0}^\infty x_j2^{j}$ and $m=\sum\limits_{j=0}^\infty x_j(2^{j}-1)$
While studying the homology mod-2 of a certain family of groups $\lbrace G_n\rbrace$, where $n$ is the number of generators of the group, I was able to prove the following:
The homology group $H_m(G_n,...
- 911
2
votes
1
answer
245
views
Markov property for groups?
My question again refers to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...
- 309
3
votes
0
answers
147
views
Derived categories and minimal model program
I read Kawamata's note, and the goal of this note is to construct an equivalence
$$ \Phi:D^{b}(X)\to D^{b}(Y)$$
for a flop $X\to Z\leftarrow Y$, and I wonder to the moduli approach and it seems to me ...
- 392
2
votes
1
answer
165
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$K_0((k[x]/(x^2))[y])$
Let $K_0(R):= K_0(P(R))$ where $P(R)$ is the category of finitely generated projective $R$-modules, where $R$ is a commutative ring with unity. Now if $R = k[x]/(x^2)$, $R$ is a local ring thus all ...
- 167
2
votes
0
answers
51
views
Estimating the Hausdorff distance of parallel facets of convex polytopes
Background
Let $\mathcal{K}_P^n$ denote the class of open, convex, $n$-dimensional polytopes in $\mathbb{R}^n$ containing the origin. For each $K\in \mathcal{K}_P^n$, its gauge function $f_*:\mathbb{R}...
- 21
1
vote
0
answers
103
views
Freidlin Wentzell for stochastic differential inclusions
Consider the SDI
$$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$
Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?
- 1,944
2
votes
0
answers
126
views
What is the colimit of the punctured $k$-cube $\{X^{\epsilon_1 + \dotsb + \epsilon_k} \mid \epsilon_i \in \{0,1\} \text{ not all } 1\}$?
Let $\mathcal C$ be a stably monoidal $\infty$-category, and let $I \xrightarrow f X \to Y$ be a fiber sequence where $I$ is the unit. Then for each $k \in \mathbb N$, we can form a canonical cubical ...
- 64k
19
votes
2
answers
851
views
The discriminant of the Okada algebra
The Okada algebra $\mathfrak{O}_n$ over a field $K$ has generators
$E_1,\dots,E_{n-1}$ and relations $E_i^2=x_iE_i$,
$E_{i+1}E_iE_{i+1}=y_i E_{i+1}$, and $E_iE_j=E_jE_i$ for $|i-j|\geq
2$, where $x_i,...
- 50.8k
6
votes
0
answers
345
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Why can't a Lie group act transitively on a finite volume hyperbolic manifold?
In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?",
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...
- 4,081
4
votes
0
answers
226
views
Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?
Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...
- 4,081
1
vote
0
answers
65
views
Banach tori: classification up to Fréchet homeomorphisms
Consider the set $T$ in $l_p$ defined as closure of
\begin{equation}
T = \{ (x_1,\dotsc,x_n,\dotsc): x_j = \frac{1}{2^{(j/p)}} e^{it_j}, t_j \in \mathbb{R}/\mathbb{Z} \}.
\end{equation}
This seems to ...
- 593
2
votes
0
answers
78
views
Analogy between quasi-injective modules & extensible Banach spaces
Let $X$ be a module. $X$ is said to be quasi-injective if every homomorphism $h:A\to X$ from any submodule $A\subseteq X$ has an extension to an endomorphism $\tilde{h}:X\to X$.
A module $X$ is quasi-...
- 2,605
4
votes
0
answers
219
views
Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$
EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect.
Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
- 7,320
2
votes
0
answers
308
views
A question on Giles Gardam counter example to the Unit conjecture of Kaplansky
The unit version of the Kaplansky conjecture is about units in $FG$ where $F$ is a field and $G$ is a torsion free group. In a recent counter example by Giles Gardam, it is given an ...
- 366
3
votes
1
answer
313
views
Another possible strengthening of the union-closed sets conjecture
Consider a union-closed family $\mathcal{F}=\{A_1,…,A_n\}$ of $n$ finite sets. Let $\mathcal{A}(\mathcal{F})$ be the set of elements that belong to at least half of the sets of $\mathcal{F}$.
Another ...
- 700
9
votes
1
answer
370
views
G-topological spaces and locales
Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...
user513306
1
vote
0
answers
60
views
Necessary and/or sufficient conditions for LR coefficients of Schubert polynomials to be zero
There is a simple condition for determining whether LR coefficients for Schur polynomials are $0$, without invoking the Littlewood-Richardson rule. Is there anything similar, possibly weaker, for ...
- 2,168
1
vote
0
answers
49
views
Size of conformal factor under uniformisation
Consider closed orientable surfaces whose metrics are hyperbolic (i.e., $K=-1$) except in a region which is a hemisphere of a unit sphere attached to the hyperbolic region along a closed geodesic (of ...
- 16.6k
1
vote
0
answers
93
views
The curvature of the induced connection on the antidual bundle
Let $E\to M$ be a complex vector bundle over a (real, smooth) manifold and $\nabla$ a connection on $E\to M$ whose curvature is $R$. From Section 1.5 of "Differential Geometry of Complex Vector ...
- 1,173
6
votes
1
answer
273
views
Consistency in pure type systems
Summary
My question is about how (i) a certain presentation of pure type systems in the $\lambda$-cube, bears on (ii) a standard definition of consistency in pure type systems. In short, I'm ...
- 63
4
votes
2
answers
350
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Is there a consistent, unsound, $\omega$-inconsistent, effective theory that doesn't prove its own inconsistency?
Can we have a consistent and effective (fulfilling Godel's criteria) first order theory $T$, that is both arithmetically unsound and $\omega$-inconsistent, and yet doesn't prove its own inconsistency (...
- 11.3k
-1
votes
1
answer
294
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Must 'special' $u,v \in \mathbb{C}[x,y]$ be symmetric polynomials?
The idea for the following question came from Joachim König's last comment appearing
here, namely, the example with $u=x+y^3,v=x^3+y$.
Let $u,v \in \mathbb{C}[x,y]-\mathbb{C}$. Denote by $\alpha$ the ...
- 2,837
17
votes
1
answer
1k
views
Why is the bicategory viewpoint useful?
In ring theory one often wants to think about bimodules as being morphisms between rings using tensor product as composition. However, this composition is only associative if one uses isomorphism ...
- 171
2
votes
1
answer
116
views
Sufficient condition for pair of real quadrics to have real intersection
In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero.
Let $ q $ be a real quadric. Then $ q $ has a real zero if and only if $ q $ has indefinite ...
- 4,081
18
votes
2
answers
1k
views
What notable theorems cannot be automatically proven without choice using Shoenfield absoluteness?
There have been a couple of recent questions, here and here, regarding the role of the axiom of choice in real-analytic results with applicability to general relativity. This lead me to look at some ...
- 13.2k
4
votes
1
answer
183
views
Find an order-embedding of $S_3\times{\bf2}\times{\bf2}$ into ${\mathbb Z}^4$
A function $f:P\to Q$ from a poset $(P,\le_P)$ to a poset $(Q,\le_Q)$ is an order-embedding if, for all $p,p'\in P$, $p\le_P p'$ if and only if $f(p)\le_Q f(p')$.
We partially order the Cartesian ...
- 1,644
4
votes
1
answer
185
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Canonicality of group of integers for reductive groups over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group over a non-Archimedean local field $K$ and $\mathcal{O}_K$ be ...
- 6,006
4
votes
0
answers
63
views
Possible questions about the Tate-Shafarevich subgroup of a Galois hypercohomology group?
$\newcommand{\wt}{\widetilde}$
Let $n=1,2$. There are infinite torsion abelian groups $H^1$, $H^2$ killed by some natural number $m$.
There are finite subgroups
$$ {\rm Sha}^1 \subset H^1,\quad ...
- 14.2k
8
votes
2
answers
1k
views
Can we effectively axiomatize a theory that proves the negation of its own Gödel's sentence?
For any consistent and effective theory $T$ fulfilling Gödel's criteria, let $G_T$ be the Gödel sentence of $T$, that is: $$ G_T \iff \neg \exists x: \operatorname{Proof}_T(x,\ulcorner G_T \urcorner)$$...
- 11.3k
1
vote
1
answer
148
views
Non-absolutely continuous foliation
What is a simple example of a (continuous) foliation of a manifold that is not absolutely continuous?
(A foliation is said to be absolutely continuous if holonomy maps between smooth transversals send ...
- 133
0
votes
0
answers
137
views
Convexity of an equivalent norm
Let $X=l_2$ with usual norm $\|\cdot\|_2$. We define a subspace of $X$ as $D=conv (B_{l_2} \cup B),$ where $B = \{ (x_n) \in l_2 : \sum_{n=1}^\infty \frac{n}{2} x_n^2 \leq 1\}$, conv is the convex ...
- 85
7
votes
1
answer
434
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Do the exceptional root systems arise in the real world?
I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...
6
votes
1
answer
131
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Finding the continued fraction for the "tails" of $\eta(3)$
I am interested in the continued fractions for the "tails" or "correction term" of the series sum of specific constants. For example, the Madhava's correction term for $\pi/4$:
$$
\...
- 161
5
votes
0
answers
255
views
Confusion in identification of quasicoherent sheaves on BG and G -representations
I asked this question on MSE a few days ago, but didn't get a response and also managed to confuse a senior colleague with it since then. This is probably a stupid question, so please bear with me.
...
- 1,071
0
votes
1
answer
164
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Upper bounds estimates of Minkowski sum
Let $A,B \subset \{0,...,d\}^n$, do we have any result that says $|A+B| \leq \mathcal{O}_d(|A|\cdot |B|)^\tau$ for $\tau < 1$. The case $\tau = 1$ is trivial, and due to the restricted setting, I ...
- 343
2
votes
0
answers
85
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Turning cocycles in cobordism into an inclusion or a fibering
By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega^n_U(X)$ is represented by cocyles
$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$
where $M$ is a ...
- 171
4
votes
0
answers
267
views
Reference for Riemann–Zariski space
I am currently studying an article of Koji SEKIGUCHI "Ringed Spaces of Valuation Rings and Projective Limits of Schemes", and in the last paragraph the author proves that the Riemann–Zariski ...
- 41
1
vote
0
answers
113
views
Inclusion of finite dimensional C*-algebras and relative commutants of subfactors
Given a subfactor $N\subset M$ with finite Jones index, the inclusion of relative commutants $N^{\prime}\cap M\subset N^{\prime}\cap M_1$ (here, $M_1$ is the basic construction of $N\subset M$) is a ...
- 393
1
vote
0
answers
124
views
Regularity of minimizing harmonic maps with no topological obstructions
So during (not really) my research I stumbled upon the following question, for which I could not find results in literature in any direction. It is not stated super precisely mathematically speaking, ...
- 646
2
votes
1
answer
123
views
Generalize the conception of 'stable' solution and 'stable outside a compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold
I want to talk about generalizing the conception of 'stable' solution and 'stable outside the compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold.
I'm reading $\Delta u +e^u=0$...
- 809
2
votes
1
answer
247
views
Linear system with matrix as a variable
I have the following two linear systems:
$$\begin{bmatrix} u_{11} & u_{12} \end{bmatrix} A = 0$$
$$\begin{bmatrix} u_{21} & u_{22} \end{bmatrix} B = 0$$
Both $A,B$ are $2 \times 2$ matrices ...
- 129
3
votes
1
answer
235
views
Classification of curves whose tangent spaces are large
Let $Z\subset \mathbb{P}^n$ be a pure 1-dim closed subscheme and $n\geq 2$. We assume that $\deg Z:=\mathbb{P}^{n-1}.Z=d$ and $\dim T_p Z=d$ for any closed point $p\in Z$.
Is there any classification ...
- 565
2
votes
1
answer
105
views
Numerical method with rational nodes and weights to compute exact value of definite integral?
Description
Let $p(x)$ be a polynomial of degree $n$ and rational coefficients.
I'm interested in computing numerically the exact value of the integral $I$, which is also rational
$$I = \int_{a}^{b} p(...
- 253
1
vote
0
answers
66
views
Parameter estimation of a Taylor expansion
Let $a,b$ two real numbers, $\theta$ a real parameter and suppose that you have an analytic function of the form:
$$
f_\theta(x)\triangleq \sum_{k\in\mathbb{N}}a_k(\theta)x^k \quad\forall x\in[a,b],
$$...
- 393
2
votes
0
answers
87
views
Number of minimal generators for a the group algebra of a p-group
Let $G$ be a finite $p$-group and $K$ a field of characteristic $p$.
Then the group algebra $KG$ is local and thus the quotient of a non-commutative polynomial ring $K\langle x_i\rangle$ by an ...
- 26.5k
1
vote
0
answers
154
views
Intermediate Jacobian for small resolution of a singular Fano threefold?
I am mainly interested in the nodal Gushel-Mukai threefold. Let $X$ be a Gushel-Mukai threefold with one node, then by page 21 of the paper https://arxiv.org/pdf/1004.4724.pdf there is a short exact ...
- 1,982
2
votes
1
answer
194
views
Is a simple J-holomorphic curve injective everywhere except for finitely many points?
Let $(M^{2n},J)$ be an almost complex manifold and $(\Sigma,j)$ a closed Riemann surface. Suppose $u: \Sigma \to M$ is a simple, nonconstant, $J$-holomorphic curve. Can we prove that the set
$$
Z:=\{z\...
- 1,441