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159,093 questions
3
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0
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605
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Diffeomorphically vs holomorphically trivial canonical bundle [duplicate]
Possible Duplicate:
Two definitions of Calabi-Yau manifolds
Given a compact kahler manifold M with diffeomorphically trivial canonical bundle. Is it true that the canonical bundle is also ...
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4
votes
0
answers
188
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Asymptotics of infinite Gauss sums
I'm looking for a description of the error term in the asymptotics of
$\sum_{k\in \mathbb{Z}^n} \exp(-(|k|^2+|k+a|^2)/(2 T))$
as $T \to \infty$, which should be uniform in the parameter $a \in \...
- 9,366
31
votes
7
answers
5k
views
Why is it useful to classify the vector bundles of a space?
It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...
- 29.1k
1
vote
2
answers
338
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Orbit maximin problems
I'm interested in answering questions such as: find a permutation group of n elements restricted to the class of solvable groups for which the minimum of the sizes of orbits of subsets of size 2 is ...
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- 1
14
votes
2
answers
2k
views
Semi-linear operators
If $V_1$ and $V_2$ are finite-dimensional vector spaces over a field $E$, each equipped with an $E$-linear operator $\phi$, we can tell if $V_1$ and $V_2$ are isomorphic as $\phi$-modules by comparing ...
- 7,108
2
votes
2
answers
3k
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Periodicity of data
I have some real data (data packets in a router). When I plot it I can see there is a clear periodicity on the dataset (24hours+-).
But how can I discover the periodicity of the data without being by ...
- 389
6
votes
1
answer
700
views
What notions of universe does predicative type theory admit?
Palmgren (1997), On universes in type theory, discusses work of several theorists that provide what we might call a family of Large Universe Axioms (LUAs) for predicative type theory, culminating in ...
- 2,283
5
votes
3
answers
2k
views
Binomial distribution parity
Let $X \text{~} \text{Binomial}(n, p)$.
What is $\text{P}[X \mod 2 = 0]$? Is it of the form $1/2 + O(1/2^n)$?
- 265
14
votes
2
answers
1k
views
The maximum order of finite subgroups in $GL(n,Q)$
For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders.
It appear in many articles a ...
- 10.7k
28
votes
4
answers
5k
views
Complexity of testing integer square-freeness
How fast can an algorithm tell if an integer is square-free?
I am interested in both deterministic and randomized algorithms. I also care about both unconditional results and ones conditional on GRH ...
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23
votes
2
answers
2k
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Five Front Battle
Two generals are fighting a five front battle. Each general has 1 unit of army, which he divides into five separate armies that he sends to the five fronts. If one general sends more army to a front ...
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20
votes
8
answers
3k
views
What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?
In modern valuation theory, one studies not just absolute values on a field, but also Krull valuations. The motivation is easy enough:
If $k$ is a field, a valuation ring of $k$ is a subring $R$ ...
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1
vote
2
answers
2k
views
Why is solving a MILP w/o an objective function so much faster?
When solving a MILP (mixed integer linear program) using a linear relaxation, the solver finds a feasible solution much faster if there is no objective function. The same problem with an objective ...
- 383
13
votes
2
answers
4k
views
Example of connected-etale sequence for group schemes over a Henselian field?
Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this ...
- 7,108
7
votes
2
answers
1k
views
Elementary proof that projective space is a quotient
Fix an algebraically closed † ground field $k$ of any characteristic. I want to use the classical definition of projective $n$-space $\mathbb{P}^n$ as set quotient of $\mathbb{A}^{n+1}\...
- 11.3k
5
votes
2
answers
357
views
Truncated exact sequence of homotopy groups
This is a question about a name of a very useful lemma,
that permits one in particular to show that smooth birational complex projective
varieties have isomorphic fundamental groups.
If this lemma ...
- 3,376
36
votes
3
answers
7k
views
What is the difference between PSL_2 and PGL_2?
Let $K$ be a field and $G:=SL_2(K)$, then $G$ is a $K-$split reductive group (to use some big words). These groups are classified by a based root datum $(X,D,X',D')$. Let $G'$ be group associated to $(...
- 7,108
9
votes
4
answers
2k
views
Hilbert scheme of points on a complex surface
I don't know about schemes and every definition of a Hilbert scheme (quite naturally!) involves schemes. But, the Hilbert scheme of points on a complex surface is known to be smooth (Fogarty). So is ...
- 156k
1
vote
2
answers
819
views
what is summation in the sense of a principal value?
In one paper I saw this equality:
$$\sum_{\eta=-\infty}^{\infty}\frac{z}{(z+\eta)}=\pi z\cot(\pi z)$$
which is the same as
$$\sum_{\eta=-\infty}^{\infty}\frac{1}{(z+\eta)}=\pi \cot(\pi z)$$
where ...
- 267
4
votes
1
answer
583
views
reference for a result on thick subcategories and t-structures
A thick subcategory of a triangulated category $C$ is essentially one that one can get away with declaring to be zero, i.e. it is the subcategory which sent to 0 when declares that all maps whose ...
- 1,355
4
votes
1
answer
559
views
Approximation of stacks / algebraic spaces
Let $B$ be a ring which is the colimit of rings $B_\lambda$. Let $X_\lambda$ be a stack (not necessarily algebraic) over $B_\lambda$ such that $X_\lambda \times_{B_\lambda} B_\mu = X_\mu$ and let $X =...
- 5,039
7
votes
2
answers
2k
views
Intersection form in twisted homology (homology with local coefficients)
The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
- 57.6k
0
votes
1
answer
201
views
Compare standard deviations from medians?
Let $G = (U, V, E)$ be a bipartite graph with $|U| = |V|$, $|U|$ large. If the median degree of a node in U is 4, and the median degree of a node in V is 7, is there a way to tell whether the degree ...
- 61.9k
20
votes
3
answers
5k
views
Moduli space of K3 surfaces
It is known that there exists a fine moduli space for marked (nonalgebraic) K3 surfaces over $\mathbb{C}$. See for example the book by Barth, Hulek, Peters and Van de Ven, section VIII.12. Of course ...
- 1,806
17
votes
1
answer
2k
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What is the recent development of D-module and representation theory of Kac-Moody algebra?
I just started to collect the papers of this field and know little things. So if I make stupid mistake, please correct me.
It seems that there are several approaches to localize Kac-Moody algebra(in ...
CommunityBot
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13
votes
8
answers
21k
views
What to look for in applicants to graduate programs (in mathematics)?
Hello,
I was thinking about what should be looked at when deciding on the admission of applicants to graduate programs in mathematics and I thought MO would be a good place to get opinions. What do ...
- 65.4k
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
CommunityBot
- 1
3
votes
2
answers
183
views
Find Vandermonde data to satisfy V*1=p
I would like to state something about the existence of solutions $x_1,x_2,\dots,x_n \in \mathbb{R}$ to the set of equations
$\sum_{j=1}^n x_j^k = np_k$, $k=1,2,\dots,m$
for suitable constants $p_k$...
- 50.8k
3
votes
2
answers
682
views
Change of coordinates introduced through dx
Hi,
I have a superspace spanned by 4 commuting coordinates + 2 anti-commuting ones $\{x^\mu,\theta^\alpha\}$, I have to do the change of coordinates $dx^\mu\to dy^\mu= dx^\mu+d\theta^\alpha \eta_\...
- 733
2
votes
0
answers
466
views
Math quote, who is this attributed to?
So there is a semi famous quote on 4 manifold theory, someone said 4 manifold theory is difficult because it is like the phase transition of matter. Anyone know who this quote is attributed to? I ...
2
votes
1
answer
693
views
When is the restriction map on global sections an embedding
Given a scheme $X$ with generic point p and a quasi-coherent sheaf $F$ on $X$.
Viewing $X$ as a scheme over $Spec(\mathbb{Z})$, let us assume
$f: X \rightarrow Spec(\mathbb{Z})$ is a proper map.
...
- 11.1k
5
votes
2
answers
752
views
Is there a name for this algebraic structure?
I found myself "naturally" dealing with an object of this form:
X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and ...
- 47.3k
19
votes
2
answers
7k
views
Generalization of the shakehands/condom puzzle?
The classic handshake puzzle goes something like this:
"Given that everyone has a different skin disease, how can you safely shake hands with 3 people when you have only 2 gloves?"
Its common ...
- 27.7k
14
votes
2
answers
578
views
Algebraic data and purity associated to codimension greater than 2
Consider the following statement: Let $X$ be a smooth and geometrically integral variety over a field $k$ and let $U$ be any open subset of $X$ whose complement is of codimension greater or equal to $...
- 2,976
2
votes
0
answers
517
views
When deRham curve is bijection?
Motivation: Suppose we have deRham curve. From wikipedia:
Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings on M:
$d_0:\ M \...
- 65.4k
1
vote
8
answers
2k
views
What is "rich structure", actually? [closed]
An ubiquitous claim in mathematics is that such-and-such mathematical entity has a rich structure or more structure than another one. Most oftenly the entity is a structure - a set explicitly equipped ...
CommunityBot
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2
votes
0
answers
438
views
Is it always possible to choose two subsets with the same sum?
Given two positive integers $n, m$, let $A$ be a multiset of $n$ integers taken from { $ 1,2,\cdots, m$ }, and $B$ be a multiset of $m$ integers taken from { $1,2,\cdots,n$ }.
Is it always possible ...
- 537
9
votes
1
answer
391
views
Suppose C and D are Morita equivalent fusion categories, can you say anything about R I: C->Z(C)=Z(D)->D?
If C and D are (higher) Morita equivalent fusion categories, then the Drinfel'd centers Z(C) and Z(D) are braided equivalent. Given any fusion category C we have a restriction functor Z(C)->C (by ...
- 3,176
18
votes
1
answer
2k
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What's the Hilbert class field of an elliptic curve?
My question points in a direction similar to Qiaochu's, but it's not the same (or so I think). Let me provide you with a little bit of background first.
Let E be an elliptic curve defined over some ...
CommunityBot
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1
vote
1
answer
1k
views
how to compute a highest weight vector [closed]
Do you know how to compute highest weight vectors in practice?
- 156k
4
votes
1
answer
1k
views
Combinatorics journals processing time
This is a spin-off question from How to select a journal?. Is there is any data available regarding processing time (acceptance time, time from submission to publication, or similar) specifically for ...
CommunityBot
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1
vote
1
answer
488
views
Do separable and normal have topological meanings for fields?
The terminology would suggest that a separable field extension is so because the resulting field extension has some sort of separable topology, and that a normal extension corresponds to one with a ...
- 41.1k
2
votes
1
answer
690
views
How to build the principal SU(2) bundles on surfaces?
Is there a way to classify (and build) the principal SU(2) bundles over a given topological surface up to homeomorphism? In the end, I would like to examine the associated bundle whose fiber is a ...
- 4,424
3
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1
answer
460
views
Can the étale topology ever be realized as an "honest" topology?
In general, the étale topology does not form a topology in the strict sense. However, is there any subcategory of $Sch$ where we can realize the étale topology as an honest topology on some scheme?
- 904
0
votes
1
answer
962
views
Quadratic Twist of Legendre Form
What is the quadratic twist of an elliptic curve in Legendre Form?
How do you show an elliptic curve and its quadratic twist is isomorphic when they are in Legendre Form?
- 4,646
5
votes
4
answers
2k
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Galois cohomology of linear groups over local fields
Let $F$ be a local field of characteristic zero (for simplicity), $\overline{F}$ an algebraic closure of $F$ and $L/F$ a fixed finite Galois extension. If $G$ is a linear algebraic group defined over $...
- 7,108
3
votes
0
answers
404
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Wolff's application of CS to analysis
In the foreword of Tom Wolff's "Lectures on Harmonic Analysis", C. Fefferman writes "[Wolff made] (as far as I know) the first serious application of theoretical computer science to analysis." What ...
- 13k
19
votes
2
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1k
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What manifold has $\mathbb{H}P^{odd}$ as a boundary?
This question is motivated by What manifolds are bounded by RP^odd? (as well as a question a fellow grad student asked me) but I can't seem to generalize any of the provided answers to this setting.
...
CommunityBot
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1
vote
1
answer
783
views
Probability of n k-sided dice showing exactly m different faces
I found the following closed form solution for the abovementioned problem:
$${1\over k^n}\cdot{k!\over (k-m)!}\cdot{\{{n\over m}\}}$$ with ${\{{n\over m}\}}$ being the Stirling Number of the second ...
- 28k
13
votes
10
answers
3k
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How do you become a good listener?
Often I find myself just taking notes during lectures and not really following what is said. This has always been an issue for me, I do not seem to learn anything in the classroom. Instead the ...
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