Newest Questions
159,030 questions
0
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142
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Does weak $H^1$ convergence imply $L^2$ convergence when multiplied with an exponentially decaying function?
I'm trying to see if given a sequence $\{f_n\}_n\in H^1$ which converges weakly in $H^1$ to a function $f_*$, the $L^2$ norm $\|R^2f_n\|_{L^2}^2$ converges to $\|R^2f_*\|^2_{L^2}$, where $R$ is a ...
4
votes
1
answer
292
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Strengthening the direct integral decomposition of von Neumann algebas
Let $M$ be a von Neumann with separable predual. It well known that one can write $M$ as a direct sum $M=M_I\oplus M_{II} \oplus M_{III}$ of von Neumann algebras of types $I$, $II$ and $III$.
It is ...
- 769
1
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0
answers
152
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Reference for application of local cohomology to complex manifolds with points removed
Reference request - I am looking at Dolbeault cohomology on compact complex manifolds (not Riemann surfaces) with points removed. I have been told that the key to doing this is to look at Local ...
- 1,143
4
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2
answers
583
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Compactification of a product of manifolds
Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
- 319
4
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1
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417
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A result on symmetric closed monoidal categories
In this discussion from the categories mailing there is mention of the following result by Robin Houston, supposedly proved in 2006:
Theorem. Let $\mathcal{C}$ be a symmetric closed monoidal category,...
- 764
2
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1
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245
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Non-example to PBW theorem
I am interested in a (simple) example of an associative algebra $A$ with 1 generated by $x_1, \ldots, x_n$ which is quadratic (i.e. all relations between $x_1, \ldots, x_n$ have degree $\leqslant 2$) ...
- 31
2
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0
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98
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Understanding simple point processes (part 2)
This is a follow up of this previous question. I'm trying to understand the following proposition from An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods
by Daley ...
- 181
2
votes
1
answer
143
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Trace morphism for projective morphism on differentials forms
Let $k$ be a field, $X$ and $Y$ two connected $k$-varieties, and $f:X\rightarrow Y$ a dominant projective morphism of relative dimension $d$.
I would like to know under which condition there is a ...
- 728
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282
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Do invariant open sets generate the $\sigma$-algebra of invariant sets?
Let $X$ be a Polish space with Borel $\sigma$-algebra $B(X)$. Let $G$ be a locally compact group. $T:G\times X\to X$ be a continuous action of $G$ on $X$.
The $\sigma$-algebra of invariant sets is ...
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3
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375
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Convex optimization without Slater's condition
In nearly all convex optimization methods that I read about, it is assumed that the problem satisfies Slater's condition, that is, there is a point that strictly satisfies all constraints (the ...
- 3,559
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2
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603
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Structure theorem for artinian modules?
Let $K$ be a field and let $A$ be a $K$-algebra which is finite dimensional as $K$-vector space. Then the nice structure theorem for artinian rings says that we can write $A$ as the direct product of ...
- 3,031
9
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139
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Locally presentable and accessible categories without the axiom of choice?
Is there a good reference for the study of locally presentable and accessible categories without the axiom of choice? For instance, it seems one will need to understand:
What is a good notion of $\...
- 64k
1
vote
1
answer
117
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Fitting a simplex to set of almost orthogonal vectors
I am trying to solve the following question, that I guess (hope?) has been solved before but I couldn't find any reference.
Let $S$ be a set of $d$ unit vectors in a $d$-dimensional Euclidean space ...
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1
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0
answers
59
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Distribution of joint Gaussian conditional on their sum of squares
Given a random gaussian matrix $\mathbf{X}$ with zero mean matrix and covariance matrix $\mathbf{\Sigma}$, and two deterministic matrices $\mathbf{A}$ and $\mathbf{B}$. If I know the value of $\|\...
- 11
2
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1
answer
240
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Exact calculations with Moyal product by "Bopp Shift"
I'm now working on my Phd thesis on the area of deformation quantization and field theory. After doing all the "ground work" (definitions, motivations, basics of the theory etc) I have now ...
- 480
2
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0
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342
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What's the definition of a mouse in Mitchell's handbook article "the covering lemma"?
In the book "handbook of set theory", in the chapter "the covering lemma", definition 3.24, Mitchell defines what is mouse.
However he did not give any definition of $\mathcal{U}_\...
- 1,490
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1
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95
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Aggregate function of concave functions
I have two functions $f,g$ form $[0,1]^n$ to $R_{\geq 0 }$ that are concave and monotone.
Given a point $x =(x_1,\ldots,x_n)\in [0,1]^n$, I define $\operatorname{cube}(x) = [0,x_1]\times \ldots \times ...
- 107
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votes
0
answers
55
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Quadratic surjective map between spheres
The quadratic function $f:\mathbb R^4\to\mathbb R^3$
$$f(a,b,c,d)=\begin{bmatrix} 2(ac + bd)&2(ad - bc)&a^2 + b^2 - c^2 - d^2\end{bmatrix}$$
surjectively maps the sphere $S^3$ to the sphere $S^...
- 479
3
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1
answer
176
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Stochastic representation of Laplace equation with Neumann boundary condition
Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$.
What if ...
- 1,904
1
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0
answers
257
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Confusion regarding the invariant rational functions
I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below)
It says that "every invariant rational function can ...
- 839
13
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2
answers
2k
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Is GCH useful in proving theorems?
By GCH I mean the Generalized Continuum Hypothesis. Let me give some context before presenting my question.
When the axiom of choice was introduced by Zermelo in his 1904 proof of Well-Ordering ...
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2
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0
answers
102
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Alternative construction for the loop space (?)
There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...
- 2,152
4
votes
2
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312
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Injective hulls of metric spaces
In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...
- 41
2
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0
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19
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Reference request for dinatural transformations arising from free Cartesian closed categories
Let $g_n$ be a discrete graph with $n$ nodes and $\operatorname{F}$ the free functor of the adjunction between the category of graphs and the category of Cartesian closed categories and functors, as ...
6
votes
0
answers
122
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Linear equation among divisors of a positive integer
Let $a,b,c$ be co-prime positive integers and let $\theta \in (1/4, 1/3)$ be a real number. For each positive integer $k$, does there exist a positive integer $N$ such that the linear diophantine ...
- 26.9k
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104
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Local systems on $\mathbb P^1$ and on the formal punctured disc
Consider the projective curve $\mathbb P^1$ over a finite field $k$.
Consider $\ell$-adic local systems $E$ on $\mathbb P^1\backslash \{0,\infty\}$ such that
a) $E$ is tame at $\infty$
b) The ...
- 7,037
5
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1
answer
300
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3-functoriality of the lax Gray tensor product
In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For ...
- 10.7k
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214
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Kan extensions in Grothendieck school
Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early ...
- 894
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0
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107
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Knot invariants in WZW CFT via Holographic Principle
In the physics literature the Holographic Principle relates
theories in the bulk and the theories in the asymptotic boundary.
While the bulk theory is the 3D Chern-Simons theory, the
corresponding ...
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7
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1
answer
289
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Is the Rado graph the unique countable graph that has all finite graphs as induced subgraphs?
I understand that since the Rado graph is the Fraïsse limit of the class of finite graphs, it is the unique homogeneous graph with this property. Is there another graph not isomorphic to the Rado ...
- 526
3
votes
1
answer
247
views
How to generate a random function with conditions?
The background is as follows:
I consider the following differential equation
$$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$
where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...
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4
votes
1
answer
289
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Is there a category theoretic definition of a cryptographic commitment scheme?
I'm trying to come up with a composeable framework for cryptographic commitment schemes, where inclusion proofs can be combined in different ways. I'm thinking this can be done with category theory, ...
- 153
1
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0
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118
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A question about cohomology with local coefficient
Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
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2
votes
1
answer
131
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derived completion and flat base change
Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings.
We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete.
For a ...
- 349
0
votes
0
answers
51
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Kernel perfection in some powers of cycles
Suppose I orient the edges of the power of cycle graph $G=C_n^k$ where $n=16$ and $k=4$ in such a way that all the generated cycles by the elements $1,2,3,4$ are given the standard lexical orientation....
- 2,089
1
vote
1
answer
359
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Three-dimensional analogues of Hirzebruch surfaces
There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
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2
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0
answers
126
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Random walk with same directions and different step sizes
Let $X\sim e^{iU}$, where $U$ is uniformly distributed on $(0, 2\pi]$. Define $\chi_1, \cdots, \chi_t$ as i.i.d. random variables with the same distribution as $X$.
Consider the following two random ...
- 499
3
votes
2
answers
622
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Non-simple groups $G$ with only non-trivial quotient isomorphic to $G$
If $G$ is a group such that every non-trivial subgroup is isomorphic to $G$ itself, then $G= \mathbb{Z}$ is the only infinite group with that property (up to isomorphism). Amongst the finite groups we ...
- 48.9k
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94
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Understanding simple point processes
Background
I'm studying the basic theory of Random Finite Sets (RFS), which is the name that is used in my field to denote simple point processes.
A simple point process is a random variable whose ...
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2
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349
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Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation
$\DeclareMathOperator\GL{GL}$A few months ago, the following discussion took place on AoPS, concerning matrices that have nonzero determinant under any permutation of their entries: https://...
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1
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1
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145
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Zero loci of sections of wedge product of bundles
Let $V$ be a $\mathbf{C}$-vector space of dimension $n$, and consider the Grassmannian $G:=Gr(2, V)$ of 2-dim subspaces of $V$. Then we have the tautological subbundle $E\subset V\otimes \mathcal{O}_G$...
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1
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211
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Macroscopic sets - a notion of largeness for Lebesgue null sets
Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
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209
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Extending representations of $\mathrm{GL}(n,\mathbb{F}_p) \times \mathrm{GL}(m,\mathbb{F}_p)$ to $\mathrm{GL}(n+m,\mathbb{F}_p)$
$\DeclareMathOperator\GL{GL}$In this question, representations means finite-dimensional complex representations.
Fix some $n,m \geq 2$ and some prime $p$. I'm interested in representations $V$ of $\...
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0
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169
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Is there a sharper Golden–Thompson inequality?
For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality
$$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$
holds, and it is known to be a strict inequality whenever $[A, B] \...
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21
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Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points
The following question resisted attacks at Math SE, so I thought I would try posting it here.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least ...
- 3,567
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votes
1
answer
40
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How to handle the evaluation of functions on staggered ghost nodes?
I have a convection-diffusion-reaction steady state PDE in the form
$$
\frac{\partial C}{\partial x} = \frac{1}{u_0(x)}\left(\frac{\partial}{\partial z} \left( \mathcal{D}(z) \frac{\partial C}{\...
- 111
1
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0
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110
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Moser iteration epsilon-regularity for non-linear system in general dimension
I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...
6
votes
5
answers
953
views
Two arcs in the complement of a disc must intersect?
Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$.
Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...
- 3,317
0
votes
0
answers
55
views
Sum of Skellam-distributed number of random variables
Suppose $X_i$ are i.i.d, and $N \sim \text{Skellam}(\mu_1$, $\mu_2$). Is it possible to find a closed form for the p.d.f of $S_N$, defined by $S_N = X_1 + \cdots X_N$ when $N \ge 0$, and $S_{-N} = -...
- 11
2
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0
answers
144
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Examples of counting holomorphic curves in cylindrical reformulation of Heegaard Floer
In 2005, Robert Lipshitz reformulated Heegaard Floer in a "cylindrical setting" by counting holomorphic curves in $\Sigma \times [0,1] \times \mathbb{R}$ where $\Sigma$ is a Heegaard surface ...
- 21