All Questions
153,441
questions
3
votes
2
answers
299
views
Recursive formula for inverse Kazhdan-Lusztig polynomials
Let $(W,S)$ be a Coxeter group. Then the Kazhdan-Lusztig polynomials $P_{x,w}$ are known to satisfy the following recursive identity:
For $x < w$ and $s \in S$ satisfying $\ell(sw) < \ell(w)$, ...
- 185
1
vote
0
answers
100
views
Trace embedding and unbounded domain
Let $D\subset\mathbb{R}^d$ be an open domain and let consider the open cylinder $D\times (0,T)\subset\mathbb{R}^{d+1}$ where $T\in (0,+\infty)$ arbitrary. Let $H^{1}(D\times (0,T))$ be the Sobolev ...
- 33
3
votes
1
answer
836
views
Cohomology of weighted projective spaces
Suppose that $a,b,c$ are positive integers such that $gcd(a,b,c) = 1$. Let $X$ be the complex weighted projective space $\mathbb{C} \mathbb{P}(1,a,b,c)$. How to compute the group $T(H^{3}(X,\mathbb{Z})...
- 6,933
0
votes
0
answers
55
views
Continuity of a composite function
Let $n=2$ or 3 and let $\Omega$ be a bounded domain of $\mathbb{R}^n$. Let $T>0$ and $f \in L^2([0,T],H^1(\mathbb{R}^n))$.
Is the mapping
\begin{equation}
\begin{array}{rcl}
C^0([0,T],C^1(\bar{\...
- 133
0
votes
0
answers
72
views
Discrete primal-duality in optimization
I would like to inquire about the existence of something perhaps similar to the duality theorem in the convex analysis or convex programming in the discrete setting. Here is a concrete example.
Let $...
- 2,169
1
vote
1
answer
334
views
Steady Euler flows with compact support?
What is known about (3D) steady incompressible Euler flows with compact support?
(Looking for results in a field you are not familiar with sure is tough.
I had a hope to find clues ...
- 6,861
5
votes
1
answer
389
views
Relative Steenrod's problem
Thom's theorem states that for every homology class $\alpha \in H_{*}(X)$ there exists an integer $k = k(\alpha)$ such that the class $k\, \alpha$ comes from the fundamental class of an orientable ...
- 251
2
votes
0
answers
472
views
Absolute Hodge cycles over $\mathbf{Q}$
In the 1986 notes by Milne "Hodge cycles on abelian varieties", Deligne defines the notion of absolute Hodge cycles.
For a smooth projective variety defined over $k\subset\mathbf{C}$ non ...
user120812
2
votes
2
answers
127
views
Determining when two biquadratic polynomials generate the same field
Consider the family of monic biquadratic polynomials given by $f_{a,b}(x) = x^4 + 2ax^2 + b$ with $a,b$ integers. Let $K_{a,b}$ denote the isomorphism class of quartic fields obtained by adjoining any ...
- 25.5k
2
votes
0
answers
75
views
Exponential decay for wave equation in even dimensions
Consider the wave equation
$$
u_{tt} = \Delta_x u - q(x)u, \quad x \in\mathbb R^d, \; t > 0,\tag{1}\\
u(0,x) = u_0(x) \in H^1_\text{comp}(\mathbb R^d),\\
u_t(0,x) = u_1(x) \in L^2_\text{comp}(...
- 1,269
0
votes
1
answer
213
views
Express as Meijer-G function
I want to express $e^{x^2}$ as MeijerG function?
it would be possible? or what?
can i use $e^x$ MeijerG expression for this one?
2
votes
1
answer
115
views
From socle of quotients to socle of ring itself
Let $I_1, \dots , I_n$ be ideals of a ring $R$ with identity having zero intersection. Assume that for some $x\in R$, $x+I_ i$ is an element of the right socle of $R/I_ i$, for each $ i=1,\dots , n$....
- 323
2
votes
1
answer
42
views
Is the difference sequence of the maximum size of $k$-colorable subgraph non-increasing?
Given a simple graph $G=(V,E)$, we use $A_k$ to denote the vertex set of a maximum $k$-colorable subgraph in $G$ when $k\ge 1$, and $A_0=0$.
Will the sequence $|A_1|-|A_0|,|A_2|-|A_1|,\cdots,|A_{\...
- 45
4
votes
2
answers
708
views
When Kan extensions don't exist
Kan extension are an incredibly useful concept when they exist. My question is: can we still derive information about a functor when Kan extensions don't exist, and if so, what information and in what ...
- 753
1
vote
1
answer
79
views
What are examples of a second order operational tangent vector on an infinite dimensional Hilbert space
In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space.
http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf
...
- 155
1
vote
0
answers
185
views
Cutting a circle from the hyperbolic plane
Let D be the Poincare' disk its natural hyperbolic metric and with at least 1 marked point on $\partial D$. Suppose I cut an hyperbolic circle of radius $r$ away from it, then I get a Riemann surface ...
- 1,335
-1
votes
1
answer
223
views
Group isomorphism $R^{\times} \simeq C_n\times C_2$? [closed]
Let $n$ be an positive integer and $R:=\mathbb{Z}[X]/(X^2,nX)$.
Do we have $R^{\times} \simeq C_n\times C_2$ (the group of units of $R$) as it seems to be the case for small values of $n$.
If so, do ...
- 2,752
18
votes
1
answer
727
views
Is a field that never embeds twice in another field necessarily a prime field?
Call a field $k$ unrepeatable$^1$ if for every field $L$ there are either zero or one field homomorphisms $k \to L$. Then the prime fields $\mathbb{Q}$ and $\mathbb{F}_p$ for $p$ prime are clearly ...
- 5,219
1
vote
0
answers
42
views
Quantitative error control in Minkowski-Stein formula
Let $K\subseteq\mathbb R^d$ be a compact convex body with non-empty interior, and $E$ be a $(d-1)$-dimensional linear subspace of $\mathbb R^d$. Let $\theta\in\mathbb R^d$ be the unit vector such that ...
- 373
3
votes
0
answers
283
views
Prerequisites to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups
What is the minimal background one must have to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups?
- 1,831
9
votes
3
answers
251
views
Looking for generalization of projective model structure
If $\mathcal{M}$ is a cofibrantly generated model category and $\mathcal{C}$ is a small category, then we can give $\mathcal{M}^\mathcal{C}$ the projective model structure, in which weak equivalences ...
- 2,052
6
votes
1
answer
402
views
Example of a locally presentable locally cartesian closed category which is not a topos?
The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a ...
- 61.6k
10
votes
2
answers
2k
views
Why the circle for Pontryagin duality? [duplicate]
For a locally compact group $G$, we define the Pontryagin dual as $\hat G = Hom(G,\mathbb T)$ where $\mathbb T$ is the circle group and the homomorphisms are continuous group maps. This duality has a ...
- 7,646
3
votes
1
answer
727
views
Navier-Stokes equations and machine learning
I am looking for a reference explaining how to solve the Navier-Stokes equations numerically using machine learning algorithms .
Thank you in advance for your help .
- 143
27
votes
3
answers
2k
views
Defining computable functions categorically
Computable functions may be defined in terms of Turing machines or recursive functions, or some other model of computation. We normally say that the choice doesn't matter, because all models of ...
- 1,316
4
votes
2
answers
307
views
Generalizations of tangent $\infty$-topos
If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\...
- 4,410
13
votes
1
answer
342
views
Cartography of the duals of GL, PGL, SL, etc
A short version of this question could be
What are the duals of $PGL(2,\mathbf{Q}_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?
I should obviously add some precisions.
there are different ...
- 5,314
3
votes
1
answer
120
views
Are isotopic transversal curves on a foliated surface transversally isotopic?
Let $F$ be an orientable surface (possibly with boundary) with a foliation $\cal F$ with $k$-prong saddle singularities only for $k\geq 3,$ (as in figure borrowed from Farb-Margalit book). Suppose ...
- 2,370
1
vote
1
answer
166
views
Lower bound of the spectrum of a Schrodinger operator on a bounded domain
I am trying to look for references on estimate of the lower bound of the spectrum of a Schrodinger operator $-\Delta + V$ on a bounded domain in three-dimensional space. For simplicity, we can take ...
- 91
12
votes
0
answers
858
views
Conditions for the second homotopy group to be Abelian
What is the weakest set of assumptions on a pair of spaces $X\subset M$ for which the second homotopy group $\pi_2(M,X) $ is guaranteed to be Abelian?
Naively, I expected that Abelian $\pi_1(X)$ ...
- 393
3
votes
1
answer
484
views
Reference request: Irreducible operators
I had asked this question on MSE but did not get any response.
I would like some reference to books that talk about irreducible operators on Banach lattices and its properties. A quick google search ...
- 343
2
votes
0
answers
67
views
evolution of Grassmannians along geodesic line
Let $p_0$, $p_1$ be two $n \times 2$ orthonormal matrices that represent two points on the real $Gr_{2,n}$, i.e. two 2-d subspaces in $\mathbb{R}^n$. Let $p(t): [0,1] \rightarrow Gr_{2,n}$ be a ...
- 181
5
votes
1
answer
1k
views
Reference request: The resolvent is analytic in the resolvent set
I am busy reading through Taylor's paper Spectral Theory of Closed Distributive Operators.
On page 192, he defines the resolvent and spectrum of $T$:
Later on in the paragraph, he then proceeds by ...
- 289
2
votes
1
answer
191
views
Total variation and relative $\ell_\infty$ metric
Let
$$D_{tv}(P,Q) = \frac{1}{2}\sum_{a \in A}|P(a)-Q(a)|$$
and
$$D_{\infty}(P,Q) = \sup_{a \in A} \log \max\{\frac{P(a)}{Q(a)}, \frac{Q(a)}{P(a)}\},$$
where $P$ and $Q$ denote probability measures on ...
- 43
2
votes
1
answer
2k
views
Design a Galton Board to simulate a uniform distribution
This link http://mathworld.wolfram.com/GaltonBoard.html suggests that a certain specific placement of pegs can be used to simulate a binomial or a normal distribution. Is there a specific peg ...
- 121
4
votes
1
answer
368
views
torsion free modules $M$ over Noetherian domain of dimension $1$ for which $l(M/aM) \le (\dim_K K \otimes_R M) \cdot l(R/aR), \forall 0 \ne a \in R$
Let $R$ be a Noetherian domain of Krull-dimension $1$ (i.e. every non-zero prime ideal maximal). Let $M$ be a torsion free $R$-module . Let $K$ be the fraction-field of $R$ and let $r=\dim_K S^{-1}M=\...
user111524
2
votes
1
answer
501
views
The Euler's totient function and the product of distinct primes dividing $n$ versus the Heronian means
For integers $n\geq 1$ with $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ we denote the squarefree kernel or radical of an integer $n$ (see if you want this Wikipedia). And $\...
user75829
5
votes
0
answers
145
views
Characters on $PGL(2)$
I am interested in computations with characters on $PGL(2, F)$ and not in $GL(2, F)$, and have some issues concerning definitions.
The notion of conductor is standard for characters $\chi$ of a $p$-...
- 5,314
1
vote
0
answers
533
views
Blow-ups in étale cohomology
If $f:X\to\text{Spec}(k)$ is a smooth projective variety over a separably closed field, with $i:Z\subset X$ a closed smooth sub variety, with complement $j: U\to X$, and $f':X'\to \text{Spec}(k)$ is ...
user92332
0
votes
1
answer
529
views
Definition of Elliptic pseudodifferential operators
A symbol $p \in S^m(\Omega)$ where $\Omega \subset \mathbb{R^n}$, or its corresponding operator $p(x,D) \in \Psi^m(\Omega)$ is said to be elliptic of order m if for every compact $A\subset \Omega$ ...
5
votes
1
answer
257
views
When is the etale cohomology of a variety simple if its Hodge structure is simple?
Suppose $X$ is a smooth projective variety defined over $\mathbb{Q}$, and the pure Hodge structure on $H^{2n}(X)$ is of a very simple form,
\begin{equation}
\mathbb{Q}(-n)^{b^{2n}}
\end{equation}
...
- 2,961
0
votes
0
answers
133
views
Operations on étale sheaves
Which of the following operations on étale sheaves $A$ commute with tensor powers? (eg. for instance $i^*(A^{\otimes n})=(i^*(A))^{\otimes n}$?)
$i^*(A)$, $i$ closed immersion.
$i_*(A)$
$i^!(A)$
$i_!(...
user120812
5
votes
0
answers
331
views
Kac's theorem for quiver representations over an arbitrary ground field
Let $Q$ be a quiver without loops (cycles of length 1). Kac proved that if $K$ is algebraically closed, the dimension vectors of indecomposable representations of $Q$ over $K$ are exactly the ...
- 6,125
7
votes
1
answer
265
views
Can the subobject classifier be covered by an object in a subtopos?
Let $\mathcal{E}$ be a topos and $\Omega$ its subobject classifier.
Is it possible to have a nonidentity local operator (a.k.a Lawvere-Tierney topology) $j\colon\Omega\to\Omega$, a $j$-sheaf $X\in\...
- 8,559
2
votes
0
answers
67
views
Support of étale sheaves
Let $X$ be a scheme, $i: Z\to X$ a closed subscheme, $j: U\to X$ its complement in $X$. Assume the codimension of $Z$ in $X$ is large (at least $2$).
Let $A$ be an étale sheaf on $U$, $B$ an étale ...
user120812
3
votes
1
answer
355
views
Estimate for the composition of two Hilbert-Schmidt operators
Let
$U$, $H$, $\tilde H$ be infinite-dimensional separable $\mathbb R$-Hilbert spaces
$Q$ be a self-adjoint and nonnegative nuclear linear operator on $U$
$\Psi$ be a Hilbert-Schmidt operator from$^1$...
- 161
5
votes
1
answer
649
views
linear independence of exponentials
Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.
Is any finite subset of $X$ linearly independent? If yes, why? If no, is the ...
- 3,771
2
votes
1
answer
480
views
Cardinality of fractal without CH? [closed]
This appears unanswered on reddit.
What are possible choices for the cardinality of plane or line fractal
without assuming CH?
Are there fractals for which the answer is not easy?
(Sieprinski ...
- 24.2k
3
votes
1
answer
1k
views
Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables
Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...
- 1,016
3
votes
1
answer
558
views
What is a half cusp in hyperbolic geometry?
I already asked this question on math.stackexchange, but it was suggested that I post it here as well.
The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and ...
- 1,335