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159,100 questions
11
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Is any true sentence in the second-order Peano Axioms provable
Forgive the elementary nature of the question. I understand that the second order Peano Axioms are categorical in the sense that all their models are isomorphic. This equivalence class of models is ...
25
votes
4
answers
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A Peculiar Model Structure on Simplicial Sets?
I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
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42
votes
2
answers
3k
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Commutative rings to algebraic spaces in one jump?
Typically, in the functor of points approach, one constructs the category of algebraic spaces by first constructing the category of locally representable sheaves for the global Zariski topology (...
- 9,418
9
votes
4
answers
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alternative construction of the quotient group
The background for this question is that I know that many students starting to learn algebra focus on the standard construction of the quotient group $G/N$ instead of working with the universal ...
- 57.6k
7
votes
3
answers
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The continuous as the limit of the discrete
Reading this documment: www.math.ucla.edu/~tao/preprints/compactness.pdf, I got interested in the following thing: "One can also use compactifications to view the continuous as the limit of the ...
- 57.6k
0
votes
1
answer
415
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If the 4-genus of a link is zero, is it a slice link?
An n-component slice link is a link that bounds n disjoint discs in B^4. And the 4-genus of a link is defined to be the minimal genus of orientable surfaces bounded by it in B^4.
My question is: if ...
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18
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2
answers
2k
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Galois representations attached to newforms
Suppose that $f$ is a weight $k$ newform for $\Gamma_1(N)$ with attached $p$-adic Galois representation $\rho_f$. Denote by $\rho_{f,p}$ the restriction of $\rho_f$ to a decomposition group at $p$. ...
- 4,807
3
votes
1
answer
367
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Homotopy classes of complex bundle maps and isotropic immersions into contact manifolds
This is a follow-up question to my previous one where I was trying to understand the classes of Legendrian immersions of circles into contact manifolds.
I'm interested in classifying isotropic ...
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6
votes
1
answer
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The inverse limit of locally free module
A is an I-adic complete Noetherian ring. M is a finitely generated A module. For any n>0, $M/I^nM$ is a finitely generated locally free A/I^n-module. Is M necessarily a locally free A-module?
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vote
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answer
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Extension of a holomorphic vector bundle
Let $E$ be a holomorphic vector bundle over $\mathbb{P}^n\setminus\begin{Bmatrix}[1,0,0,\cdots,0]\end{Bmatrix}$. Let $D$ be a connection on $E$. Let $\widetilde{E}$ be an extension of $E$. Since $\...
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1
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The (n+1)-st cohomology of K(Z/p,n).
I was looking through my notes for a homotopy theory course and found the following mysterious statement (K is of course the Eilenberg-Maclane space):
$$H^{n+1}(K(\mathbb Z_p,n);\mathbb Z_p) \cong \...
- 4,424
-2
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1
answer
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calculate percentiles from a histogram [closed]
Hi,
Could someone explain to me or point out some documentation on how to compute a given percentile from a histogram ?
0
votes
2
answers
4k
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Convergence of iterative algorithm.
For quite a long time I'm trying to prove convergence of an iterative algorithm in case of a particular system of nonlinear equations.
Here are some characteristics of this system:
It consists of n ...
CommunityBot
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9
votes
2
answers
1k
views
Borsuk pairs of Banach spaces
Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$
$\in$ $X$ ;...
CommunityBot
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1
vote
2
answers
907
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Principal Bundles over Complex Projective Varieties
For various reasons, I'm interested in working with complex projective varieties that are also principal bundles. I began by looking at projective spaces themselves $\mathbb{CP}^n = SU(n+1)/U(n)$, ...
- 44.7k
11
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2
answers
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What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?
In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...
- 3,131
6
votes
2
answers
2k
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groups as categories and their natural transformations
If one views a group as a one object category with the elements of the group as morphisms then a natural transformation between functors of such categories is an inner automorphism, i.e. if we have ...
- 4,519
11
votes
3
answers
1k
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Does Ribet's level lowering theorem hold for prime powers?
I often use the following theorem (that one can state more generally) in my research.
Let E/Q be an elliptic curve of conductor N corresponding to a modular form f(E), l a prime of good or ...
- 818
4
votes
2
answers
551
views
Normality of an affine semigroup
An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is ...
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10
answers
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views
How can I really motivate the Zariski topology on a scheme?
First of all, I am aware of the questions about the Zariski topology asked here and I am also aware of the discussion at the Secret Blogging Seminar. But I could not find an answer to a question that ...
- 12.3k
1
vote
2
answers
175
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Freeness of the Canonical $SU(n)$ Action
I have another question about $SU(n)$, again I hope it's not too basic. For $n=2$, the action of $SU(2)$ on $C^2$ is free since it's equal to the group of rotations. In general, the group of rotations ...
- 28k
7
votes
1
answer
918
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Integral expression for zeta(2)
By computing the sum of all Bernoulli numbers via Borel summation (I learned this technique from Varadarajan's excellent book Euler through time. A new look at old themes, 2006) I found that $$\sum ...
- 4,485
17
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2
answers
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Why does the Gamma function satisfy a functional equation?
In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...
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10
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1
answer
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Is the Burnside ring a lambda-ring? + conjecture in Knutson p. 113
Warning: I'll be using the "pre-$\lambda$-ring" and "$\lambda$-ring" nomenclature, as opposed to the "$\lambda$-ring" and "special $\lambda$-ring" one (although I just used the latter a few days ago ...
- 33.8k
7
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1
answer
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Cyclic extensions coming from E[p] \equiv F[p],
Let p be a prime and let K be a field containing the p'th roots of unity. Let E be an elliptic curve over K. We consider the the moduli problem $Y_E(p)$, which sends L to set of elliptic curves F/L, ...
- 23.8k
7
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2
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684
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Yet more on distortion
I would like to elaborate a little bit on my previous question which can be found
here.
Firstly, let me recall that a separable Banach space $(X, \| \cdot \|)$ is said to be
arbitrarily distortable ...
CommunityBot
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3
votes
2
answers
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Indexing the Line Bundles of a Flag Manifold
Following on from this question link text, how are the line bundles of a complex flag variety indexed? Are they still labeled by the integers? If so, why? A representation theory explanition in terms ...
CommunityBot
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4
votes
5
answers
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Is reflexivity of equality an axiom or a theorem?
Everybody knows that equality is reflexive: $\forall(x)(x=x)$. But should reflexivity of equality be taken as an axiom of logic or as a theorem of set theory?
If you choose the former then you ...
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3
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What are the right categories of finite-dimensional Banach spaces?
This is inspired partly by this question, especially Tom Leinster's answer.
Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
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9
votes
1
answer
996
views
Topological "Interpolation" ?
Let E be a normed space, and let $T$:E * $\rightarrow$ E * be
a nonlinear operator.
Suppose that :
1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous).
and
2) $T$ is ...
- 25.8k
3
votes
2
answers
416
views
Which Banach spaces have categorical duals?
I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces ...
CommunityBot
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6
votes
0
answers
639
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Hilbert subspaces of indefinite inner product spaces
Let $E$ be a real linear space, endowed with a non-degenerate symmetric
bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$
satisfies the following [sequential] ...
- 25.8k
1
vote
1
answer
365
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Naturally definable sets of natural numbers (3)
[This shall be the last of a series of questions, see Naturally definable sets of natural numbers (2)]
I cannot explain why I have been so stubborn not to see the most straight-forward definition for ...
CommunityBot
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4
votes
2
answers
292
views
Goedelizability and decidability of a property of Peano formulas
Sorry for not knowing the answers to these elementary questions:
Is the property of formulas of the first-order language of Peano arithmetic of "defining a finite set of natural numbers" goedelizable?...
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-1
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1
answer
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Naturally definable sets of natural numbers (2): Can the circle be broken?
(follow-up to: Naturally definable sets of natural numbers)
Every formula $\Psi(x)$ in the first-order language of Peano arithmetic defines a set of natural numbers. Some of these sets are finite, ...
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0
votes
1
answer
198
views
An integral arising in statistics(2)
The integral I am interested in is:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 one can use contour integration.
So for K>1 we have :
$$\pi/2-\...
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0
votes
1
answer
412
views
An integral arising in statistics
The integral I need:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 this integral is
$$\pi/2-\int_{Arc}\frac{\exp(ixy)}{1+y^{2}}dy $$
Where Arc ...
- 267
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votes
3
answers
1k
views
What's algebraic approach to QM good for?
The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional ...
- 21.6k
5
votes
4
answers
447
views
Is every monomorphism of commutative Hopf algebras (over a field) injective?
Is it true that any monomorphism of commutative Hopf algebras over a field is injective? Moreover, is it true that any epimorphism of commutative Hopf algebras over a field is surjective?
- 3,929
8
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3
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randomness in nature [closed]
What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...
- 24.7k
11
votes
1
answer
813
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Approximation to divergent integral
Hi everyone,
I'm a physicist working on stochastic processes and I've come up against an integral that I'm not able to approximate using steepest descent (I don't have a large or small parameter), ...
CommunityBot
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2
votes
2
answers
395
views
When do primes lift uniquely (provided they lift at all)?
Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for the induced map $Spec(S) \...
- 19.6k
2
votes
5
answers
2k
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CM of elliptic curves
This question is related to this one.
Tate module of CM elliptic curves
There seem to be several versions of "complex multiplication".
Fact 1: We say $E/\mathbb{C}$ has CM if $End_C(E) \supsetneq Z$. ...
CommunityBot
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2
votes
0
answers
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modification of singlestart in global optimization
When minimizing a nonconvex function $f : \Omega \rightarrow \mathbb{R}$ that may have multiple minima, there are some very simple strategies to improve the odds of finding the global minimum point. ...
- 25.8k
5
votes
2
answers
864
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Hilbert $C^*$-modules and approximate units
Hi,
Given a $\sigma$-unital $C^*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon_i\}, i\in I$ in $A$ such that each $\epsilon_i$ is of the ...
CommunityBot
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9
votes
1
answer
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how do you evaluate the p-adic modular form E_p-1 in the region |j|<1
background/motivation
let Ek denote the modular form of level one and weight k with q-expansion given by $E_k(q)=1- \frac{2k}{b_k}\sum_n \sigma_{k-1}(n)q^n$ where σi is the divisor sum and bk ...
- 57.6k
11
votes
3
answers
1k
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Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 4,060
-1
votes
3
answers
1k
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Naturally definable sets of natural numbers
(This is a follow-up question from over there: Natural models of graphs.)
(And it has a follow-up question over there: Naturally definable sets of natural numbers (2): Can the circle be broken?)
...
CommunityBot
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3
votes
4
answers
658
views
A specific branched cover of S^2 as a subgroup of Pi_1
This is a follow-up question to: Degree 2 branched map from the torus to the sphere
This is a silly computation, but for whatever reason this is taking me much, much longer than it should. So ...
CommunityBot
- 1
0
votes
3
answers
1k
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When is the radical of the extension of a prime ideal prime?
(All rings assumed to be commutative and unital)
Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial ...
- 156k