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159,054 questions
9
votes
2
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Quotient of a Hausdorff topological group by a closed subgroup
Sorry if this question is below the level of this site: I've read that the quotient of a Hausdorff topological group by a closed subgroup is again Hausdorff. I've thought about it but can't seem to ...
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11
votes
2
answers
2k
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Maximum degree in maximal triangle free graphs
It's easy to see that in bipartite maximal triangle free graphs (n vertices), the maximum degree is at least $\lceil n/2 \rceil$. What about mtf graphs in general? Must there always be some vertex ...
- 56.6k
4
votes
1
answer
945
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Interpolation by a function whose second derivative is bounded
I don't know if this is an easy question for specialists in the field. Consider
the following interpolation problem : let $\varepsilon >0$, $X$ be a finite
set of real numbers and $g$ be a real-...
- 3,595
17
votes
4
answers
2k
views
Representations of surface groups via holomorphic connections
EDIT: Tony Pantev has pointed out that the answer to this question will appear in forthcoming work of Bogomolov-Soloviev-Yotov. I look forward to reading it!
Background
Let $E \to X$ be a ...
- 131
8
votes
1
answer
506
views
Residual finiteness for graph manifold groups
Is there a simple proof that 3-dimensional graph manifolds have residually finite fundamental groups?
By "simple" I mean the proof that does not use any hard 3d topology. I care because I wish to ...
- 7,800
5
votes
5
answers
3k
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Field structure for R^n
Hi!
Is it possible to define a product on R^n for n>2 such that R^n can be made into a field?
R is a field in its own right with the standard operations and R^2 can be made into a field by ...
13
votes
3
answers
1k
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Flat SU(2) bundles over hyperbolic 3-manifolds
Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold?
The literature on such bundles over 3-manifolds is huge and my naive searches don'...
- 68.9k
5
votes
5
answers
3k
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Tetrad postulate: Implies or results from the metricity of the connection?
Hi,
I see that the tetrad postulate:
$\nabla_{\mu}e_{\nu}^{I}=\partial_{\mu}e_{\nu}^{I}-\Gamma_{\mu\nu}^{\rho}e_{\rho}^{I}+\omega_{\mu J}^{I}e_{\nu}^{J}=0$
Can be merely derived from writing a ...
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10
votes
2
answers
1k
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Does a universal Frobenius map exist?
For any prime p, one has the Frobenius homomorphism Fp defined on rings of characteristic p.
Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ...
CommunityBot
- 1
1
vote
1
answer
316
views
intersection cohomology when the resolution is not semi-small
When we have a variety and a resolution of singularities, but it is not semi-small (i.e. the dimensions of the fibres do not satisfy the right conditions), then what can we say about the intersection ...
- 2,706
5
votes
2
answers
682
views
What are natural transformations in 1-categories?
It's well-known that, for lots of concrete categories (but by no means all), we can think of the objects as themselves being small categories, and morphisms are the functors between these categories. ...
CommunityBot
- 1
7
votes
2
answers
627
views
Probability vertices are adjacent in a polygon
With regard to my original question:
A subset of k vertices is chosen from the vertices of a regular N-gon. What is the probability that two vertices are adjacent?
I suppose that the responses ...
CommunityBot
- 1
2
votes
1
answer
432
views
Is line bundle determined by the parameter space and fiber?
f:X-->Y is flat and projective map between integral varieties over k, an algebraically closed field. Suppose every fiber at closed points of Y is still an integral variety. L is a line bundle on X, if ...
- 6,239
5
votes
2
answers
516
views
Contracting maps of hyperbolic manifolds
Is it possible for SOME positive $c$, $c<1$ to find a pair of COMPACT hyperbolic manfiolds $M^3$ and $N^3$
with a positive degree map $$f: M^3 \to N^3,$$ such that $f$ is contacting with constant $...
- 68.9k
0
votes
2
answers
3k
views
Stability analysis of a system of 2 second order nonlinear differential equations
How does one linearize and analyze such a system?
Just noticed I could edit this, so from my comment below:
I am trying to get a feel for what analysis us used beyond the introduction I have had. ...
- 309
1
vote
1
answer
359
views
finding the closure when blowing a variety at a singularity
I'm having trouble finding the closure in the definition of a blow-up. For example, take the following example, with a node at $(0,0)$ (and at some other points) (it's not a homework question, just a ...
- 16k
11
votes
2
answers
738
views
Building elliptic curves into a family
Suppose $E/ \mathbb{Q}$ is an elliptic curve whose Mordell-Weil group $E(\mathbb{Q})$ has rank r. When can we realize E as a fiber of an elliptic surface $S\to C$ fibered over some curve, with ...
- 156k
7
votes
1
answer
508
views
Is there a way to check if a relative Hilbert Scheme is reduced?
More specifically, suppose I have a rational curve on a complete intersection, and I know that the relative Hilbert Scheme is not smooth at the point corresponding to my rational curve. Is there any ...
- 2,955
4
votes
2
answers
2k
views
Probability Question
You have $N$ boxes and $M$ balls. The $M$ balls are randomly distributed into the $N$ boxes. What is the expected number of empty boxes?
I came up with this formula:
$\sum_{i=0}^{N}i\binom{N}{i}\...
- 85.6k
2
votes
2
answers
1k
views
Result of repeated applications of the binomial distribution?
What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?
To clarify, an example.
Suppose that a bunch of people are playing a game with k (to ...
- 1,491
2
votes
1
answer
265
views
Hausdorff Derived Series
There is a short section in the book Locally Compact Groups by Markus Stroppel (Chapter B7) on the notion of a "Hausdorff Solvable Group", which he defines as a topological group with a descending ...
- 56.6k
1
vote
1
answer
820
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Comparing Iwahori Decompositions
Let G be a p-adic group, U a (n appropriate) unipotent subgroup and I an Iwahori subgroup. Then there are Iwahori decompositions I\G/I=U\G/I=W where W is the affine Weyl group. I suspect that
$$...
- 44.7k
7
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1
answer
1k
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Elementary questions in arithmetic geometry
In many theories there is a rough divide between elementary problems that can be solved with "one's hands", and "deep results that require powerful tools". For example, I am told that Hodge theory is ...
- 33.3k
4
votes
1
answer
721
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"Transient" of the discrete-time Riccati equation
It is a well-known result that, if the pair $(A,Q^{1/2})$ is stabilizable and the pair $(A, C)$ is detectable, the solution to the discrete-time Riccati recursion
$P(t+1) = A P(t) A^T - A P(t) C^T\ (...
- 4,577
1
vote
2
answers
218
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Sheaf isomorphism.
Suppose you have a curve $C$ such that deg$K_C =0$ and $\Gamma(C,\Omega_C^1) \neq 0$. Does this automatically imply that $\vartheta_C \equiv \Omega_C^1$? My thought is yes, I've seen a proposition (...
- 3,396
4
votes
2
answers
545
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Is every group object in TopMan a Lie group?
Recall that a Lie group is a group object in the category of C∞ manifolds.
If I have a group object in the category of topological manifolds, can I necessarily equip it with a smooth structure ...
- 6,556
6
votes
2
answers
1k
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Closed, complemented subspaces of $l^1(X)$ when $X$ is uncountable
... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in ...
- 31.5k
29
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3
answers
2k
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Embeddings of $S^2$ in $\mathbb{CP}^2$
Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line?
Note: I suspect ...
- 13.2k
0
votes
2
answers
1k
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What is the localization of Q[x]/(x) at 0
Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??
I think the result is
Q[x]/...
- 1,033
4
votes
3
answers
323
views
Approximately known matrix
What linear algebraic quantities can be calculated precisely for a nonsingular matrix whose entries are only approximately known (say, entries in the matrix are all huge numbers, known up to an ...
- 1
11
votes
3
answers
500
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Local-globalism for similar matrices?
My background on number theory is very weak, so please bear with me...
Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in $\...
- 156k
2
votes
1
answer
510
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Are the C(S^n, S^n)'s homeomorphic ?
Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ?
[both endowed with the sup metric (or equivalently the compact-open topology)]
Generally, C(S^n, S^n), with n >= 1, is a ...
- 29.1k
11
votes
2
answers
416
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Subfields joining an algebraic element to another
Let $\alpha$ and $\beta$ be two algebraic numbers over $\mathbb Q$. Say that a subfield $\mathbb K$ of $\mathbb C$ joins $\alpha$ to $\beta$ iff $\beta \in {\mathbb K}[\alpha]$ but $\beta \not\in {\...
- 3,595
1
vote
4
answers
541
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Does the Golden Ratio Apply to Timing as Well? [closed]
I've seen the golden section applied to art, but does it apply to sound/timing as well? Just curious.
- 25.6k
3
votes
0
answers
677
views
Sebastiani-Thom isomorphism for D-modules
Considering $f:X\to \mathbb{C}$, $g:X\to \mathbb{C}$ and $f\oplus g:(x,y)\mapsto f(x)+g(y)$.
The Sebastiani-Thom isomorphism is an isomorphism $\Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes \...
- 7,527
2
votes
3
answers
416
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Nicest coset representatives of the symplectic group in the general linear group
What is a "nice" way of choosing coset representatives for the symplectic group $Sp_{2k}(\mathbb{C})$ in the general linear group $GL_{2k}(\mathbb{C})$?
- 33.3k
1
vote
1
answer
243
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asymptotic families of ramanujan near-integers?
This is a follow-up to the question on the Ramanujan constant.
Can one find an asymptotic family of shockingly near-integers obtained by raising e to some simple algebraic number, in the spirit of ...
- 45.7k
4
votes
1
answer
321
views
What functorial topologies are there on the space of linear maps between LCTVS?
Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the Hahn-...
- 47.3k
0
votes
2
answers
736
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Points on algebraic stacks
I'm a bit confused concerning a definition in Laumon--Moret-Bailly. Perhaps someone could shed some light on the following.
It concerns the definition of (closed) point in Chapter 5. More precisely, ...
- 1,645
5
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2
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550
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Reps of $U(n)$ for the bundles of holomorphic and antiholomorphic forms of projective space
What are the representations of $U(n)$ that induce (see link text) the bundles of holomorphic $\Omega ^{(1,0)}$ and antiholomorphic $\Omega ^{(0,1)}$ forms of $\mathbb{CP}^n$ (recalling the well-known ...
CommunityBot
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11
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1
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410
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An "existence contra partition of unity" statement for integer matrices?
While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
- 2,600
4
votes
4
answers
1k
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Near Trivial Quiver Varieties
So, today I started learning the definition of a quiver variety, and wanted to make sure I'm understanding things right, so first, my setup:
I've been looking at the simplest case that didn't look ...
- 44.7k
3
votes
3
answers
2k
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Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
4
votes
0
answers
497
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A Local CLT with large variance
For n an even integer, $0 \leq i \leq$ ${n} \choose{j}$, $1 \leq j \leq n$ let $X_{i,j}$ be a
random variable taking values $\frac{n}{2}-j,0,j - \frac{n}{2}$ with equal probability. Let $S_{n}$ be ...
- 263
11
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2
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2k
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Characterization of Riemannian metrics
This is probably an insanely hard question, but given an abstract metric space, is there some way to determine whether it's a manifold with a Riemannian, or more generally a Finslerian, metric? If ...
- 47.3k
2
votes
1
answer
466
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Semiclassical explanation of "Structured" spaces [closed]
We have our general notions of manifolds, schemes, et cetera, and other geometric "spaces", and we realize that a lot of these look like topological spaces with structure sheaves i.e. structured ...
- 45.7k
3
votes
1
answer
256
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Generation of object-internal structure in a strict 2-category
Suppose we're given a strict and small 2-category $C$, and an object of $C$ called $A$. Can we produce an internal category structure on $A$ in some canonical way (maybe by some sort of argument ...
- 66.8k
1
vote
1
answer
205
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Introductions to Disease- and Price-Modeling
I'm looking for resources (anything from short articles to books) about building mathematical models or computer simulations of 'things that happen' in populations.
Specifically, I'm curious about 1)...
- 101
8
votes
3
answers
2k
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Holomorphic and antiholomorphic forms of projective space
For $\mathbb{CP}^1$ the bundles of holomorphic and antiholomorphic forms are equal to the $\mathcal{O}(-2)$ and $\mathcal{O}(2)$ respectively. Do the holomorphic and antiholomorphic bundles of $\...
- 21k
-3
votes
1
answer
358
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Ordering of tuples equivalent to mapping to R?
Suppose we have a non-strict total ordering on tuples of real numbers (the ordering includes tuples of differing lengths). Any tuple, t2, generated from another tuple, t1, by increasing one or more ...
- 386