Recently Active Questions
159,065 questions
4
votes
2
answers
2k
views
Brauer-Manin obstruction and Tate-Shafarevich group of an Abelian variety
I read that the Brauer-Manin obstruction $A(\mathbb{A}_K)^{\mathbf{Br}}$ of an Abelian variety $A$ over a number field $K$ equals (naturally?) its Tate-Shafarevich group $\mathrm{III}(A)$.
Is this ...
- 18.7k
1
vote
3
answers
5k
views
how to get a feasible solution to dual program from a feasible solution to primal program?
If a feasible solution to a linear programming is known, and the corresponding value of the objective function is close to the optimum, can we get a feasible solution to the dual programming which ...
- 1,638
11
votes
3
answers
3k
views
Looking for reference on gauge fields as connections.
Can anyone give me references where I would see a detailed exposition of how to translate gauge field theory as known to physicists into the language of connections. I am looking for a detailed ...
- 341
18
votes
1
answer
753
views
Balancing problem
There was a problem in an Olympiad selection test, which went as follows: Consider the set $\{1,2,\dots,3n \}$ and partition it into three sets A, B and C of size n each. Then, show that there exist x,...
- 85.6k
3
votes
1
answer
977
views
Notation for "the inclusion map is a homotopy equivalence"
It's sometimes convenient to have different notations for "$A$ is a subset of $B$" depending on what the inclusion map does:
If it's non-surjective, $A\subsetneq B$ or $A\subset B$, depending on your ...
- 47.3k
3
votes
1
answer
5k
views
? A graph is four colorable if and only if it is planar.
? A graph is four colorable if and only if it is planar.
Is this true, I know that if a graph is planar it is four colorable, but is it true that if a graph is four colorable it must be a planar ...
- 191
9
votes
4
answers
1k
views
Strict Class Numbers of Totally Real Fields
In their paper Computing Systems of Hecke Eigenvalues Associated to Hilbert Modular Forms, Greenberg and Voight remark that
...it is a folklore conjecture that if one orders totally real fields by ...
- 19.2k
1
vote
1
answer
191
views
Monoidal operations on categories where the maps on Aut, End are injective
Suppose we have a monoidal category $(\mathcal{C},\otimes)$. I am interested in the conditions or situations where the following do and do not hold:
For any objects A, B of $\mathcal{C}$, the induced ...
- 7,320
12
votes
1
answer
4k
views
Collection of subsets closed under union and intersection
Suppose A is a set and S is a collection of subsets closed under arbitrary unions and intersections. Can we find a collection F of functions from A to itself such that a subset B of A is in S if and ...
- 236k
22
votes
3
answers
5k
views
what is an Euler system and the motivation for it?
I tried to read the definition of it on Rubin's book "Euler systems" but it looks highly technical. Can anyone shed some light on it? In particular, is there some starting examples?
The wiki entry is ...
- 4,713
4
votes
1
answer
687
views
A categorical question [closed]
Hi everyone, I want to know the descriptive translation and explanation of a paragraph of EGAI, (chapter 0, 3.1.3)
Supposons que $K$ soit la catégorie définie par une « espèce de structure
avec ...
- 1,376
3
votes
2
answers
7k
views
Is the product of first $n$ prime numbers $+1$ another prime number? [closed]
Hi,
I know that the answer is no, yet I dont know how to prove it wrong. Finding a counterexample is not a good solution because it is a past written exam question with no calculators allowed. The ...
CommunityBot
- 1
16
votes
2
answers
1k
views
Why the similarity between Hodge theory for compact Riemannian and complex manifolds?
I'm aware to varying extents of the existence of certain decompositions of the space of $k$-forms on a compact complex or compact Riemannian manifold that split into closed, co-closed, and harmonic ...
- 25.6k
18
votes
3
answers
2k
views
A binomial sum is divisible by p^2
This is a question I have since longer time, but I have absolutely no idea how to proceed on it.
Let $p>3$ be a prime. Prove that $\displaystyle\sum\limits_{k=1}^{p-1}\frac{1}{k}\binom{2k}{k}\...
CommunityBot
- 1
4
votes
2
answers
648
views
Gaps in nx (mod 1)
It is known that if you choose n point at random on S1 = [0,1], the nearest neighbor spacings between the points are exponentially distributed with mean 1.
For example, two of our n points could be ...
- 22.8k
1
vote
2
answers
579
views
Interpreting Euler's Criterion for Idoneal Numbers
(I am a very, very new to mathematics, so I apologise in advance for posing a question so basic, but am out of ideas).
In Idoneal Numbers and some Generalizations, pp. 15, Ernst Kani quotes Euler's ...
CommunityBot
- 1
11
votes
1
answer
1k
views
Extension of induced reps over Z: is it a sum of induced reps?
Let $G$ be a finite group. If $L$ is a finite free $\mathbf{Z}$-module with an action of $G$, say $L$ is induced if it's isomorphic as a $G$-module to $Ind_H^G(\mathbf{Z})$ with $H$ a subgroup of $G$ ...
- 44.7k
3
votes
1
answer
378
views
functorial meaning of irreducibility of a moduli space
Iasked me the question what the interpretation of the irreducibility of a moduli space is for the functor it represents. For proper, there is the valuative criterion and for (formally) smooth, there ...
- 1,990
4
votes
1
answer
734
views
Parametric polynomial solution of a single polynomial equation
Let $P$ be a polynomial in $n$ variables with rational coefficients,
$P \in {\mathbb Q}[Z_1,Z_2, \ldots ,Z_n]$, and consider the algebraic
set
$Z=\lbrace (z_1,z_2,z_3, \ldots ,z_n) \in {\mathbb Q}^n |...
- 30.5k
3
votes
1
answer
484
views
Do all correlation coefficients induce a pseudometric?
The Kendall tau distance was originally defined as a correlation coefficient. It seems clear to me that every metric function $d$ that is bounded by $b$, induces a correlation coefficient. That is:
...
- 4,577
8
votes
2
answers
738
views
A nice variety without a smooth model
Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that
--- $X(K)\neq\emptyset$,
--- the $l$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_l)$ is ...
CommunityBot
- 1
3
votes
2
answers
740
views
Nonnegative polynomial in two variables
What can be said about the polynomials $f\in\mathbb Q[x, y]$ which are nonnegative on $\mathbb R\times \mathbb R$?
Motivation: this may lead to progress in the question about polynomial onto map $\...
CommunityBot
- 1
8
votes
2
answers
741
views
elementary equivalence of infinitary symmetric groups
Two questions:
Suppose a and b are two uncountable cardinals. Consider the symmetric groups on sets of sizes a and b respectively (the symmetric group on a set is the group of all bijections from the ...
- 236k
6
votes
0
answers
532
views
Can one calculate Ext's between microlocalized perverse sheaves/D-modules using topology?
So, I know one really good technique for calculating Ext's between perverse sheaves/D-modules using topology: the convolution algebra formalism, worked out in great detail in the book of Chriss and ...
- 44.7k
2
votes
1
answer
614
views
Derived Functors in arbitrary triangulated categories
Let ${\mathcal D}$ be a triangulated category, ${\mathcal C}$ a triangulated subcategory and $Q: {\mathcal D}\to {\mathcal D}/{\mathcal C}$ the corresponding Verdier-localization. Now suppose we have ...
- 15.6k
37
votes
6
answers
4k
views
Examples of applications of the Borel-Weil-Bott theorem?
In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes:
A representation Ri of a group G should be seen as a quantum object. This ...
- 44.7k
3
votes
2
answers
625
views
Continuation up to zero of a Dirichlet series with bounded coefficients
Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a ...
- 65.4k
4
votes
3
answers
2k
views
Homotopy groups of smooth manifolds?
For a fixed $d$, is there a relationship between the homotopy groups of smooth $d$-manifolds?
The $d=1$ case is trivial, but I already don't know how to approach $d=3$ (I should have said that the ...
- 44.4k
3
votes
1
answer
849
views
cardinality of product modulo direct sum
let $(X_i)_{i \in I}$ be an infinite family of sets with $|X_i| \geq 2$. we define an equivalence relation on $X = \prod_{i \in I} X_i$ by $x \sim y \Leftrightarrow \{i : x_i \neq y_i\}$ is finite. ...
- 44.4k
4
votes
3
answers
780
views
geometry of triangulated category and D-modules theory
This question was motivated by the answers On noncommutative algebraic geometry there, he mentioned, there are some people taking category of modules as category of coherent sheaves on non-existence ...
CommunityBot
- 1
3
votes
1
answer
2k
views
Conditions that allow unique solutions for Linear Diophantine equations
(This posting became very long, so I should note that there are two alternative but nearly equivalent formulations of the same question being given. The first one asks for the optimal strategy for ...
CommunityBot
- 1
15
votes
3
answers
3k
views
H-space structure on infinite projective spaces
Any Eilenberg-MacLane space $K(A,n)$ for abelian $A$ can be given the structure of an $H$-space by lifting the addition on $A$ to a continuous map $K(A\times A,n)=K(A,n)\times K(A,n)\to K(A,n)$.
Does ...
- 998
1
vote
1
answer
294
views
Recentering a Spherical Coordinate Sytem
How do you recenter a spherical coordinate system. For example, if the center were at $\left (0, 0, 0 \right )$ and I wanted to move the center of the spherical coordinate system to $\left (\rho_{1}, \...
- 47.3k
12
votes
0
answers
816
views
Lower bounds for linear forms of logarithms (a la Baker)?
Let $\lambda_1$, $\lambda_2$, and $a$ be three fixed complex algebraic
numbers.
For a given integer $n$, write
$\Theta(n) = \arg(a \lambda_1^n + \lambda_2^n)$.
Assuming $\Theta(n)$ is not zero, I am ...
- 121
5
votes
2
answers
3k
views
A telegram by Grothendieck to Serre
In an opinion piece which appeared in the AMS Notices of January 2010, John Wermer tells us that he once heard about a seminar given by Grothendieck which was described as "a telegram by Grothendieck ...
- 8,296
5
votes
3
answers
2k
views
Undecidable graph problems?
Can anyone name a undecidable problem that is genuinely graph-related? (Genuine means: not a standard one in graph's disguise.)
- 12.1k
7
votes
0
answers
213
views
Decomposition of certain projectives for cyclotomic q-Schur algebras
In representation theory, a very popular set of finite dimensional algebras are the $q$-Schur algebras, which are given by looking at the endomorphisms of $V^{\otimes d}$ where $V$ is the standard ...
- 44.7k
20
votes
2
answers
1k
views
Are non-empty finite sets a Grothendieck test category?
A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, ...
- 13.6k
-1
votes
1
answer
502
views
Name for probabilistic version of Pascal's identity and differentiation formula for binomial distribution
I'm trying to find a standard name or standard reference for two simple-to-prove relations involving binomial distributions.
Define:
$b(n,r,p) := \binom{n}{r}p^r(1 - p)^{n-r}$
i.e., it is the ...
- 14k
16
votes
2
answers
2k
views
"synthetic" reasoning applied to algebraic geometry
A hyperlinked and more detailed version of this question is at
nLab:synthetic differential geometry applied to algebraic geometry.
Repliers are kindly encouraged to copy-and-paste relevant bits of ...
- 19.9k
4
votes
4
answers
822
views
How can we formalize the naturality of certain characteristic subgroups?
I'm trying to get a better handle on characteristic subgroups, and many nice examples are given with some sort of "natural" definition. For example, it's clear that the center, torsion subgroup, and ...
- 7,320
1
vote
0
answers
222
views
Classification of properties of structures
Is there a sensible classification of the properties of structures with a given signature $\sigma$, e.g. graphs with $\sigma = \lbrace R \rbrace$?
For example like this:
properties defined by first-...
- 9,720
63
votes
1
answer
7k
views
Smooth proper scheme over Z
Does every smooth proper morphism $X \to \operatorname{Spec} \mathbf{Z}$ with $X$ nonempty have a section?
EDIT [Bjorn gave additional information in a comment below, which I am recopying here. -- ...
- 65.4k
15
votes
2
answers
2k
views
Lifting the p-torsion of a supersingular elliptic curve.
Let $K$ be a finite extension of $\mathbf{Q}_p$, with integer ring $R$ and residue field $k$. Say $G/R$ is a finite flat (commutative) group scheme of order $p^2$, killed by $p$. Say the special fibre ...
CommunityBot
- 1
2
votes
3
answers
571
views
How does the Dirichlet process work?
Hi, i'm looking to get into nonparametric bayesian techniques but I'm having problem understanding what's going on in the definition of the Dirichlet process or how it works. So what does P ~ DP(&...
CommunityBot
- 1
2
votes
1
answer
168
views
Local supporting points of Lipschitz functions
Let X be a separable reflexive Banach space and f:X\to\mathbb{R} be a
Lipschitz function. Say that a point x in X is a local supporting point
of f if there exist x^* in X^* and an open neighborhood U ...
- 45k
13
votes
2
answers
912
views
Is there an analogue of the Lefschetz fixed point theorem for discrete dynamical systems?
Background/Motivation
Let $(X, f)$ be a discrete dynamical system. For now, $X$ is just a set and $f$ is just a function $f : X \to X$. Suppose that $f^n$ has a finite number of fixed points for ...
- 118k
6
votes
2
answers
605
views
Complexity class of problems solvable using Q&A site
Motivation
We will be trying to find what is the complexity class of problems solvable by a polynomial time algorithm (poster) that has access to a certain oracle (Q&A site) formalizing certain ...
CommunityBot
- 1
2
votes
3
answers
3k
views
Is there any random variable which has unbounded fourth moment? [closed]
In many statements in probability, there is an assumption like bounded fourth moment. So is there any random variable which has unbounded fourth moment?
- 2,955
3
votes
1
answer
997
views
What roles do "base change" play in algebraic geometry?
It might be a not very specific problem. I just wanna know how much do we rely on the property of "base change closed". In the definition of Grothendieck pretopology, we require a collection of ...
- 19.6k