Recently Active Questions
159,066 questions
12
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References for a certain generalization of Hochschild cohomology?
Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \...
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3
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2
answers
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Probability distribution of the median
Suppose we have $2k + 1$ points $a_1, ..., a_{2k+1}$. Each point is uniformly distributed between 0 and N. What is the distribution of the median (i.e. of the k+1-th point) ?
What happens if $a_1, ......
- 156k
10
votes
1
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Sites which are stacks over themselves
A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hom-presheaves are sheaves). Is there a ...
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26
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Is every curve birational to a smooth affine plane curve?
Is every curve over $\mathbf{C}$ birational to a smooth affine plane curve?
Bonnie Huggins asked me this question back in 2003, but neither I nor the few people I passed it on to were able to answer ...
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7
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1
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282
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Can you construct a mapping space from local data? (looking for reference)
I'd to know if/where there is a reference for the following construction.
Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
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2
answers
2k
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Canonical topology on the category of schemes?
Every category admits a Grothendieck topology, called canonical, which is the finest topology which makes representable functor into sheaves.
Is there a concrete description of the canonical topology ...
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4
votes
1
answer
335
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"Eigenvalue characters"
This question is an addition to my question on simultaneous diagonalization from yesterday and it is probably also obvious but I just don't know this: Let $G$ be a commutative affine algebraic group ...
CommunityBot
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2
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1k
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Difference between Alexander polynomial and Blanchfield pairing
For a Seifert matrix $V$ of a knot $K$, the Alexander module has presentation matrix $V-tV^T$. The determinant of this matrix is the Alexander polynomial, which is the order of the Alexander module. ...
- 28.1k
1
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1
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334
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Singular matrix and wedge product
If I have a singular matrix $X$ with components $X_{\mu\nu}$:
$t^{\nu}X_{\mu\nu}=0$
By considering now $X_{\mu\nu}$'s as components of a 2-form can I say that:
$X\wedge X=0$ ?
If yes, how?
- 56.6k
8
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6
answers
1k
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Combinatorial distance ≡ Euclidean distance
Definition: A polytope has property X iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds:
disteuclidean(vi, vj) = f(distcombinatorial(vi, vj))
with ...
- 9,720
0
votes
1
answer
280
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Thousands of rays intersections with Triangles in 3D space [closed]
Hi
There are thousands of rays and triangles. We need get all the intersection points. If we use the normal two level loops,we need O(m*n) time complexity.Is there any way to low the time complexity ...
- 3,059
15
votes
3
answers
6k
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Simultaneous diagonalization
I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ ...
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4
votes
1
answer
838
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Relating Deligne-Lusztig virtual representation characters to Green functions
I have 2 questions - the first is what the title refers to, and the second is something I want a reference on (I thought I'd include them in one post since they are very strongly related). Sorry this ...
- 3,131
3
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2
answers
2k
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Concentration of measure for gaussian inner products
There exists extensive theory for the concentration of Gaussian measure. Through that, it can be easily proved that the square of the $\ell_2$ norm of a length $n$ zero mean Gaussian vector ${\bf x}$ ...
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4
votes
3
answers
4k
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Range of binomial probability, given a certain number of observations?
Let's say I am given $n$ flips of a coin, $k$ of which are heads.
These are iid flips.
Can I say, with probability $p > 1/2$, that the true probability of heads is in range $[p_1, p_2]$ ?
What is ...
- 4,577
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vote
2
answers
800
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How many dimensions I need to embed a graph? [duplicate]
Possible Duplicate:
What is the max number of points in R^3, interconnected by generic curves?
Given a set of points connected by edges lying on an euclidean plane,
I'd like to find which is the ...
CommunityBot
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3
votes
1
answer
299
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disagreement between two definitions of the singular boundary map
Hi everyone, I have a little problem with the definition of singular boundary map in singular homology theory. It appears to be some disagreement between two authors. The first one is Hatcher in his '...
- 47.3k
1
vote
1
answer
415
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Is this an identity in Lie bialgebras?
Perhaps this will be a trivial question. For this post, everything is over your favorite field of characteristic $0$.
Definitions and notation
Recall that a Lie algebra is a vector space $\mathfrak ...
- 47.3k
0
votes
2
answers
144
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Random values and their probability of reoccuring [closed]
I have a web application that prompts users to answer a question when the computer they are using is not recognized. A user complained today saying she is always prompted for the same question. I ...
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11
votes
4
answers
958
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Geometry of the multilagrangian Grassmannian
Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$.
Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
- 28.9k
0
votes
2
answers
408
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How to construct matrices with periodicity [closed]
Suppose I want to construct an $n\times n$ matrix ${\bf A}$ such that ${\bf A}^n={\bf I}$. Matrices that have period $n$ and admit such property are permutation matrices. However, I was wondering if ...
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5
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1
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Inverting a covariance matrix numerically stable
Given an $n\times n$ covariance matrix $C$ where $n$ around $250$, I need to calculate $x\cdot C^{-1}\cdot x^t$ for many vectors $x \in \mathbb{R}^n$ (the problem comes from approximating noise by an $...
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4
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527
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How can we describe the splitting of nilpotent orbit for "very even" partitions in the special orthogonal group?
I understand if a partition $\lambda$ has all parts even and all multiplicities even, then the nilpotent orbit corresponding to $\lambda$ splits up into two orbits. By the nilpotent orbit ...
CommunityBot
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2
votes
1
answer
221
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Embedding group algebra $F[S_m X S_n]$ into a group algebra $F[S_{m+n}]$
Here's a question I've been thinking about, it's a curiosity that I don't know how to answer. There could be a simple counterexample, or it could be true and I don't know how difficult it would be to ...
- 2,216
8
votes
2
answers
1k
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Is every subgroup of an algebraic group a stabilizer for some action?
Suppose G is an algebraic group (over a field, say; maybe even over ℂ) and H⊆G is a closed subgroup. Does there necessarily exist an action of G on a scheme X and a point x∈X such that ...
- 2,270
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2
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1k
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"Requires axiom of choice" vs. "explicitly constructible"
I think I'm a bit confused about the relationship between some concepts in mathematical logic, namely constructions that require the axiom of choice and "explicit" results.
For example, let's take ...
CommunityBot
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8
votes
0
answers
443
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Lifting sections of bundles
Assume that $X$ is a quasiprojective scheme with a fixed imbedding into a smooth scheme M given by an ideal sheaf $\mathcal{I}$. Assume further that there is a locally free sheaf $E_X$ on $X$ that is ...
- 16k
22
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3
answers
6k
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Satisfiability of general Boolean formulas with at most two occurrences per variable
(If you know basics in theoretical computer science, you may skip immediately to the dark box below. I thought I would try to explain my question very carefully, to maximize the number of people that ...
- 4,729
8
votes
3
answers
404
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Searching the symmetric group
You want to design a set of yes/no questions for quickly searching the symmetric group. The questions have to be of the form "Does your permutation move $a_1$ to $b_1$ or $a_2$ to $b_2$ or ... or $a_k$...
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23
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4
answers
4k
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Curriculum reform success stories at an "average" research university
Greetings all,
There's a never-ending story that many of us have sunk our teeth into. How do we go about teaching subjects like calculus and analysis "well?" Most universities that I'm familiar ...
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4
votes
3
answers
2k
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resolution of singularities on surfaces
Let |V| be a (incomplete) linear series on a nonsingular projective surface. Hironaka says that there is a resolution of the singularities of |V| along smooth centers. If the base locus of |V| is just ...
- 6,239
3
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1
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1k
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plane hyperelliptic curves
I'm aware that h/w problems are frowned upon (understandably) here. However - this really is just inspired by some h/w related confusion, so hopefully that's ok.
Anyway, can one have a smooth ...
- 65.4k
18
votes
1
answer
943
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Do chains and cochains know the same thing about the manifold?
This question was inspired by Poincaré quasi-isomorphism
Let $M$ be a closed oriented $n$-manifold. The cap product with the fundamental class of $M$ induces an isomorphism $H^i(M,\mathbf{Z})\to ...
CommunityBot
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39
votes
3
answers
6k
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Why do finite homotopy groups imply finite homology groups?
Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\...
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4
votes
3
answers
2k
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Most important domains, extension theorems, and functions in several complex variables
For a new learner of several complex variables, the many domains (eg holomorphically convex, pseduconvex, Stein) and the many extension theorems (eg Riemann) and the many functions (plurisubharmonic) ...
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-2
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2
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931
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Can topologies induce a metric?
Let {X,T} be a topology, T the set of open subsets of X.
Definition: Three points x, y, z of X are in relation N (Nxyz, read "x is nearer to y than to z") iff
there is a basis B of T and b in B ...
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9
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1
answer
723
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Presentation for the double cover of A_n
The wikipedia page Covering groups of the alternating and symmetric groups gives explicit presentations for the double covers of the symmetric group Sn (n ≥ 4). Can someone provide a similar ...
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10
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1
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836
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what was Hilbert's geometric construction in his 17th problem?
Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
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2
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3k
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Is the Torelli map an immersion?
The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see this question for a description which works for families....
CommunityBot
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11
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3
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394
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Are the “identity object axioms” in the definition of a braided monoidal category needed? (Answered: No)
I am asking because the literature seems to contain some inconsistencies as to the definition of a braided monoidal category, and I'd like to get it straight. According Chari and Pressley's book ``A ...
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4
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3k
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How to do Computations Using the Decomposition Theorem for Perverse Sheaves
This is a follow-up to this post on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.
My question is how does one use the ...
CommunityBot
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2
votes
2
answers
849
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Curves on elliptic ruled surfaces?
Let $S\overset{\pi}{\to} E$ be a ruled surface over an elliptic curve over complex field. Clearly, there are rational curves and elliptic curves on $S$. Is there any higher genus curves on $S$. Are ...
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1
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weight space for a Lie group representation
I understand how weights are defined for a Lie algebra representation.
How are weight spaces defined for a Lie group action (with respect to a fixed torus)?
I know this is a very embarrassing ...
- 4,577
7
votes
2
answers
541
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(Co-) Homology associated to Waldhausen K-Theory
Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this ...
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2
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1
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324
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Descend finite etale algebras
Let $\pi:X\to\mathcal X$ be a presentation of an Artin stack $\mathcal X$ of finite type over a field $k,$ and let $f:Y\to X$ be a finite \'etale covering. Does there exist a finite \'etale covering $...
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4
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What is the max number of points in R^3, interconnected by generic curves?
The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of it....
- 118k
2
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1
answer
160
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Lower bound for characteristic variety
Let K be an algebraically closed field of char. 0, let A_n(K) be the Weyl algebra. Let I in A_n(K) be a left ideal generated by p elements. Set M := A_n / I.
Does the following then hold?
dim Ch(M) \...
- 3,131
2
votes
5
answers
894
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Models of the reals which have no unmeasurable sets
I recall being told -- at tea, once upon a time -- that there exist models of the real numbers which have no unmeasurable sets. This seems a bit bizarre; since any two models of the reals are ...
- 7,329
2
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2
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356
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Is there a specific name for matrices with nonsingular principal submatrices?
Is there a specific name for matrices with nonsingular principal submatrices?
- 7,329
1
vote
1
answer
340
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for a natural exponential family, A is the cumulant function of h?
Reading "Monte Carlo Statistical Methods" by Robert and Casella, they mention that if
$f(x) = h(x) \exp(\langle \theta, x \rangle - A(\theta))$
defines a family of distributions for $X$, parametrized ...
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