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https://mathoverflow.net/questions/56599 | 5 | For any group character $\chi: G \to \mathbb{C}$, the function $\chi(g^k)$ is an integer-linear combination of other group characters in $\mathrm{Irr}(G)$. There doesn't seem to be a general way of finding the expansion in terms of the basic representations.
In the case of the symmetric group $S\_n$, the permutation... | https://mathoverflow.net/users/1358 | Exterior powers of the standard representation | Let $\chi$ be the character of some finite-dimensional $G$-representation $V$, for $G$ a finite group. The virtual character $\chi(g^n)$ can be found using a categorified version of the cycle index of the symmetric group $S\_n$:
$$\chi(g^n) = \frac{1}{n!} \sum\_{\lambda} \pm C\_{\lambda} \mathbb{S}\_{\lambda} V,$$
... | 6 | https://mathoverflow.net/users/9068 | 56607 | 35,342 |
https://mathoverflow.net/questions/56591 | 22 | Someone recently [asked](https://mathoverflow.net/questions/56564/what-are-the-epimorphisms-in-the-category-of-schemes) what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am often frustrated workin... | https://mathoverflow.net/users/9960 | What are the monomorphisms in the category of schemes? | In EGA IV, 17.2.6 the following characterization of monomorphisms is given:
>
> Let $f : X \to Y$ be a morphism locally of finite type. Then the following conditions are
> equivalent:
>
>
> a) $f$ is a monomorphism.
>
>
> b) $f$ is radicial and formally unramified.
>
>
> c) For every $y \in Y$, the fiber $f... | 52 | https://mathoverflow.net/users/2841 | 56608 | 35,343 |
https://mathoverflow.net/questions/56603 | 8 | The following questions might be trivial, however, I couldn't solve them:
Let $G$ be generated by a finite symmetric set $S$. Suppose that $\Gamma(G,S)$ is the corresponding right Cayley graph of $G$. Let $X$ be a metric space (or, maybe, a topological space with some nice structure).
(1) Is there a way to check ... | https://mathoverflow.net/users/9458 | Quasi-isometries vs Cayley Graphs | I guess that a star (a tree with $n$ infinite branches issued from a single vertex) should answer at least your first question. It should have $n$ ends, whatever meaningful definition you use, an we know that a group has $1$, $2$ or an infinity of ends.
Since quasi-isometry is an equivalence relation, you do not need... | 6 | https://mathoverflow.net/users/4961 | 56609 | 35,344 |
https://mathoverflow.net/questions/56593 | 5 | I realize that these objects were originally created by [Major Percy Macmahon](http://en.wikipedia.org/wiki/Percy_Alexander_MacMahon) and today have many applications but what was the original motivation for studying them?
| https://mathoverflow.net/users/12337 | Why were plane partitions invented? | MacMahon invented a technique which he called *partition analysis* to determine (multivariable) generating functions for many combinatorial objects and as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. This was introduced in his boo... | 8 | https://mathoverflow.net/users/2384 | 56611 | 35,346 |
https://mathoverflow.net/questions/56617 | 4 | It is easy to see from the construction of Bernstein polynomials that if $f$ is Lipschitz continuous on $[0,1]$ then the distance of $f$ from the subspace of polynomials of degree at most $n$ is $O(1/n)$ where the implied constant depends on the Lipschitz constant of $f$. More generally any information on the modulus o... | https://mathoverflow.net/users/3635 | Quantitative Weierstraß approximation | The answer is yes. The results which allow to infer the modulus of continuity of a function from the value of function's best approximation by polynomials are collectively known as Bernstein-type theorems.
>
> **Theorem** (S.N. Bernstein). Let $E\_n(f)$ be the best approximation to the function $f(.) \in C([a, b])$... | 7 | https://mathoverflow.net/users/5371 | 56619 | 35,350 |
https://mathoverflow.net/questions/56612 | 10 | Let $G$ be central extension of an abelian group $A$ by some group $H$.
Is it possible to characterize all irreducible representions of $G$
in terms of irreducible representations of $A$ and $H$?
| https://mathoverflow.net/users/4246 | Representations of central extensions | The question is somewhat unprecise. I assume you mean irreps over $\mathbb{C}$ and *finite* groups. Then the answer is "no" as long as you aren't more specific about the central extension in question. For example, four of the five nonisomorphic groups of order $p^3$ can be obtained as central extensions of $C\_p$ by $C... | 6 | https://mathoverflow.net/users/10266 | 56624 | 35,352 |
https://mathoverflow.net/questions/56569 | 6 | The following random construction is simple enough that I am guessing it must have been studied. Fix $d \ge 3$, and let $n > d$. For each of the $n$ vertices, pick exactly $d$ other vertices to connect it to, uniformly over all ${n-1 \choose d}$ possible choices, and making this choice independently over all $n$ vertic... | https://mathoverflow.net/users/4558 | Has the following kind of (minimum degree $d$) random graph been studied? | Yes, this model has been studied. For some early results, see
Fenner, T. I.; Frieze, A. M. On the connectivity of random $m$-orientable graphs and digraphs. Combinatorica 2 (1982), no. 4, 347–359.
So there it is called "random $m$-orientable graph"
| 5 | https://mathoverflow.net/users/12487 | 56626 | 35,354 |
https://mathoverflow.net/questions/56632 | 15 | A subset of $\mathbb{R}^n$ is
* $G\_\delta$ if it is the intersection
of countably many open sets
* $F\_\sigma$ if it is the union of countably many closed sets
* $G\_{\delta\sigma}$ if it is the union
of countably many $G\_\delta$'s
* ...
This process gives rise to the [Borel hierarchy](https://en.wikipedia.org/wi... | https://mathoverflow.net/users/1168 | A G-delta-sigma that is not F-sigma? | Any dense $G\_{\delta}$ with empty interior is of II Baire category, and cannot be $F\_\sigma$ by the Baire theorem (and of course it is in particular a $G\_{\delta\sigma}$).
| 9 | https://mathoverflow.net/users/6101 | 56636 | 35,358 |
https://mathoverflow.net/questions/56623 | 24 | I was asked (by myself) to give a proof of the following seemingly simple geometric statement, but after thinking a little I now suspect it could be less elementary than I thought (or am I being silly?). Does anybody know it, and can give an answer or a reference to it? Of course, I'm quite sure it should fit within a ... | https://mathoverflow.net/users/6101 | A puzzle about finding three points $(x,y)$, $(x,z)$ and $(y,z)$ in a subset of a square. | Let $S(x)=\{y\mid(x,y)\in S\}$ and $S^{-1}(y)=\{x\mid(x,y)\in S\}$, and let $\lambda\_n$ denote the Lebesgue measure on $[0,1]^n$. We have
$$\begin{align\*}
\int\_S(\lambda\_1S(x)+\lambda\_1S^{-1}(y))\,dx\,dy
&=\int\_S\lambda\_1S(x)\,dx\,dy+\int\_S\lambda\_1S^{-1}(y)\,dx\,dy\\
&=\int(\lambda\_1S(x))^2\,dx+\int(\lambd... | 30 | https://mathoverflow.net/users/12705 | 56646 | 35,364 |
https://mathoverflow.net/questions/56642 | 0 | The following statement seems to be taken as given in papers I'm reading:
>
> Let $\mathcal{M}^n$ be a compact, embedded hypersurface in $\mathbb{R}^{n+1}$. Then $\mathcal{M}$ is the boundary of some compact domain $\Omega \subset \mathbb{R}^
> {n+1}$.
>
>
>
Is this an elementary result? I feel there must be s... | https://mathoverflow.net/users/11266 | Compact Hypersurfaces Bounding Compact Domains | There are several ways to see this fact, which is a simple instance of [Alexander duality](https://secure.wikimedia.org/wikipedia/en/wiki/Alexander_duality).
Here is the simplest I know.
Let $H$ be a compact smooth hypersurface in $\mathbb{R}^n$, whithout boundary, and $x\in \mathbb{R}^n\setminus H$.
Then the radial ... | 4 | https://mathoverflow.net/users/6451 | 56654 | 35,369 |
https://mathoverflow.net/questions/56655 | 9 | I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly find a point inside an arbitrary geometry?
| https://mathoverflow.net/users/10306 | Get a point inside a polygon | See question 3.6 in the Comp.Graphics.Algorithms FAQ: <http://apodeline.free.fr/FAQ/CGAFAQ/CGAFAQ-3.html>
| 6 | https://mathoverflow.net/users/532 | 56659 | 35,372 |
https://mathoverflow.net/questions/42799 | 12 | Let us take an N-dimensional (N odd) irreducible representation V of SL(2,R).
It is known that (e.g., Lie groups and Lie algebras III by Vinberg and Onischik, 1994 p. 94) in V there is an invariant symmetric bilinear form $b$ for the action of SL(2,R). Thus, SL(2,R) is embedded into $O(V,b)$ - the orthogonal group o... | https://mathoverflow.net/users/979 | A decomposition of the "spin representation" of SL(2) | Yes this question is a bit old, but I can never resist some fun character theory. Maybe you have already figured out a satisfactory answer to your original question, but in case not, here is a purely computational method for finding the decomposition you seek.
By general weight theory, the weights of the representati... | 11 | https://mathoverflow.net/users/12301 | 56662 | 35,374 |
https://mathoverflow.net/questions/56661 | 1 | Does anyone know any undecidable problems in the dynamics of functions (not necessarily monoid homomorphisms or anything) from \Sigma^\* to \Sigma^\* where \Sigma is a finite set? In particular, I'm wondering if there is an algorithm that takes as input a (computable) function f:Sigma^\* \rightarrow \Sigma^*, a w \in \... | https://mathoverflow.net/users/8434 | undecidability in the dynamics of functions $f: \Sigma^* \rightarrow \Sigma^*$ | There is no such algorithm as you request, since if there were, you could solve the halting problem. To see this, consider that the operation of a Turing machine can be viewed as a function $f$ to be iterated on the finite strings representing Turing machine configurations. So, for any TM program and input, let $f$ be ... | 1 | https://mathoverflow.net/users/1946 | 56664 | 35,375 |
https://mathoverflow.net/questions/56663 | 2 | In the paper "On the Erdős distinct distance problem in the plane" by Larry Guth and Nets Hawk Katz,
<http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf>
they define the functions $d(P)$, (distinct distances between $N$ points), and $Q(P)$, (quadruples on $N$ points where the distances between the first two p... | https://mathoverflow.net/users/5068 | Determining the vector space for application of Cauchy Schwarz | Let $d(P)=\{d\_1,\dots,d\_r\}$ and write $P^2=\cup\_{i=1}^r S\_i$, where $(p\_1,p\_2)\in S\_i$ if $|p\_1,p\_2|=d\_i$. Now you can see that $Q(P)=\cup (S\_i\times S\_i)$ and so the inequality is only saying
$$r(s\_1^2+\cdots+s\_r^2)\geq (s\_1+\cdots+s\_r)^2$$
because $s\_1+\cdots+s\_r=N^2$.
| 4 | https://mathoverflow.net/users/2384 | 56666 | 35,377 |
https://mathoverflow.net/questions/56627 | 20 | Let $g$ be a directed, connected multigraph, on $n$ vertices, without loops.
Define
$$P\_g(x\_1,\dots,x\_n) := Sym\left[ \prod\_{(i,j) \in g} (x\_i-x\_j) \right]$$
where $(i,j)$ is the directed edge from $i$ to $j,$ and $Sym$ denotes the symmetrization,
that is, sum of all permutations of variables in the argume... | https://mathoverflow.net/users/1056 | Symmetric polynomial from graphs | I am afraid this innocuous-looking question is in fact extremely hard, and I would be surprised if one could find a necessary and sufficient criterion which is more useful than the definition itself. Here is why:
The question pertains to the classical invariant theory of binary forms.
Firstly, suppose your graph is $... | 20 | https://mathoverflow.net/users/7410 | 56672 | 35,380 |
https://mathoverflow.net/questions/56647 | 4 | Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma\_{\sigma \in \mathrm{Gal}(K/\mathbb{Q})} M^\sigma$. What can one say about the rank of $M'$? It seems like both zero and full rank can occur... | https://mathoverflow.net/users/11108 | Rank of sum of Galois conjugates of a matrix | Here is one way to get a theorem about this situation:
Although $M'$ may not be invertible, you can always adjust by a scalar $\lambda \in K^\times$ to ensure that $(\lambda M)'$ is invertible. This can be seen as follows: let $\{\alpha\_1, \dots, \alpha\_n\}$ be a basis for $K/\mathbb{Q}$. Write $M = \alpha\_1 M\_1 ... | 1 | https://mathoverflow.net/users/11108 | 56676 | 35,383 |
https://mathoverflow.net/questions/56658 | 3 | Is there any work describing (indecomposable)projectives, injectives in category of D-modules on some flag variety?
I wonder whether someone has used quivers(say Auslander-Reiten sequences)to describe the homological properties for category of D-modules
Thanks!
| https://mathoverflow.net/users/1851 | Descriptions of projectives (injectives) in category of D-modules | The following paper of Tim Hodges may be of interest:
$K$-theory of $\mathcal{D}$-modules and primitive factors of enveloping algebras of semisimple Lie algebras. Bull. Sci. Math. (2) 113 (1989), no. 1, 85–88.
He proves that the Quillen $K$-groups of the abelian category of coherent $\mathcal{D}\_X$-modules on any... | 4 | https://mathoverflow.net/users/6827 | 56685 | 35,387 |
https://mathoverflow.net/questions/56671 | 4 | It's well known that we can exhibit the comma category as a particular type of 2-limit in Cat. When working with 2-categories, there is a naïve comma object given by by boosting up the ordinary diagram for a comma category, but this fails to have many of the useful formal properties we might expect.
The answer is th... | https://mathoverflow.net/users/1353 | Can we exhibit the 2-category of Grothendieck fibrations as a 2 (or 3)-limit? | I'm not sure whether you're working in Cat or 2-Cat; I'll assume the latter.
If B is an object of a finitely (2-)complete 2-category K, then the 2-category Fib(B) of fibrations over B is monadic over K/B, where 'fibration' is meant in Street's sense (see [nLab](http://ncatlab.org/nlab/show/fibration+in+a+2-category) ... | 5 | https://mathoverflow.net/users/4262 | 56687 | 35,388 |
https://mathoverflow.net/questions/56675 | 5 | Is it true that on a smooth Stein manifold (or smooth affine variety or smooth complete intersection in $\mathbb{C}^{n}$), every Kahler form is exact outside a compact set?
| https://mathoverflow.net/users/3566 | Is every Kahler form on a Stein manifold exact outside a compact set? | This seems extremely false to me: I can't think of any example of a positive dimensional Stein manifold $W$ and a compact subset $K$ such that $H^2(W) \to H^2(W \setminus K)$ has nontrivial kernel. That means there should be an abundance of counter-examples: Just take any Kahler form representing a nontrivial class in ... | 9 | https://mathoverflow.net/users/297 | 56691 | 35,391 |
https://mathoverflow.net/questions/56679 | 3 | For an infinite dimensional Hopf algebra $H$, a non-degenerate dually pairing Hopf algebra $H'$, and a choice of basis $e\_i$ of $H$, is the dual basis $e^i$ (defined of course by $e^i(e\_j) = \delta\_{ij}$) contained in $H'$?
I am interested in the specific case of $SU\_q(N)$ and the dually paired Hopf algebra $\mat... | https://mathoverflow.net/users/12653 | Dual of a Basis for a Hopf Algebra Conatined in all Dually Paired Hopf Algebras | No. Let $k$ be a field of characteristic $0$. Consider the symmetric algebra on one generator $k[x]$, with comultiplication $x \mapsto x\otimes 1 + 1\otimes x$. It has a Hopf pairing with itself, given by $\langle x^m,x^n\rangle = n! \hspace{.5ex} \delta\_{m=n}$. Then consider the bases of $k[x]$ given by expansion aro... | 5 | https://mathoverflow.net/users/78 | 56692 | 35,392 |
https://mathoverflow.net/questions/56686 | -2 | Hi,
I'm learning about relations on sets, and I'm trying to figure out what exactly antisymmetric means.
The way we represent a relation is like a adjacency matrix. In my textbook I see that symmetric relation is symmetrical with respect to the diagonal (of the adj. matrix) and that is logical to me, but they also m... | https://mathoverflow.net/users/5938 | Graphic representation of an antisymmetric relation on a set | The "A beats B in roshambo" relation is antisymmetric in your sense. There does not exist a permutation of the set {rock, paper, scissors} for which the adjacency matrix is supported above the diagonal. So you are correct and your textbook is wrong.
For future reference, this question and related ones are probably be... | 2 | https://mathoverflow.net/users/78 | 56694 | 35,393 |
https://mathoverflow.net/questions/56677 | 111 |
>
> *"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."*
>
>
> Felix Klein
>
>
>
What notions are used but not clearly defined in modern mathematics?
---
To clarify further what is the purpose of the question f... | https://mathoverflow.net/users/3811 | What notions are used but not clearly defined in modern mathematics? | One of the most important contemporary mathematical concepts without a rigorous definition is
quantum field theory (and related concepts, such as Feynman path integrals).
Note: As noted in the comments below, there is a branch of pure mathematics --- constructive field theory --- devoted to making rigorous sense of... | 96 | https://mathoverflow.net/users/2874 | 56710 | 35,399 |
https://mathoverflow.net/questions/56713 | 2 | Let $G$ be a simply connected semisimple algebraic $K$-group and $K$ be a finite extension of $k$.
Is $R\_{K/k}G$ still a simply connected algebraic group?
We say $G$ is simply connected if for any central isogeny $G'\to G$ is in fact an isomorphism of algebraic groups.
| https://mathoverflow.net/users/11056 | Is restriction of scalars of simply connected algebraic groups still SC? | May I assume that $K/k$ is separable?
Let $T$ be a maximal torus in $G$. Since $G$ is simply connected, the weight lattice and character lattice for $T$ are the same. This remains true if we replace $G$ by a direct product of $[K:k]$ copies of $G$, and $T$ by a corresponding product of tori. Over the algebraic closur... | 3 | https://mathoverflow.net/users/4494 | 56714 | 35,402 |
https://mathoverflow.net/questions/56701 | 8 | I'm studying the structure of the Specht module for $S\_n$ and I would like to know if there is some generalizations of this structure for Weyls groups or Coxeter groups.
Also, I'm interest to know about category-theoretics way of study this module. I just know one article about this subject:
<http://www.math.uni-b... | https://mathoverflow.net/users/13251 | What is a Specht module? | The short answer to your question about Specht modules for other types than the symmetric group is "yes". The long answer is that you have to dig into the extensive literature built up around cyclotomic Hecke algebras and the like. One place to look is the arXiv, where the papers of Andrew Mathas (Sydney) and his colla... | 7 | https://mathoverflow.net/users/4231 | 56729 | 35,413 |
https://mathoverflow.net/questions/56743 | 1 | Let $Y$ be real valued random variable on probability space $(\Omega,
\mathcal{F}, P)$, such that $Y>0$ almost surely. Suppose $(X^a: a\in
\Lambda)$ be a set of random
variables in the same probability space with the same distribution as
$Y$.
[Q.] Is the following true?
$$\inf\_a (X^a) >0, \quad a.s.-P$$
| https://mathoverflow.net/users/5656 | infimum of a set of positive r.v. with the same distribution | No, certainly not. Let $Y \sim U(0,1)$, so $Y > 0$ a.s. If $\{X^a : a \in \mathbb{N}\}$ are iid $U(0,1)$, then it is easy to see that $\inf\_a X^a = 0$ a.s. In fact this will be true for any $Y$ with essential infimum $0$.
| 3 | https://mathoverflow.net/users/4832 | 56744 | 35,421 |
https://mathoverflow.net/questions/56755 | 3 | All the Turing machines we consider have (1) a two-way infinite tape (2) one and only one halting
state (3) an alphabet of exactly two symbols-"1" and " "(or "blank"). Let n be any positive integer.
Let H(n) be the smallest number of active states that such a Turing machine needs in order that,
starting with one of its... | https://mathoverflow.net/users/4423 | A question about the "information-content" of a very simple type of Turing machine. | Suppose your function were computable. First observe that the function $H$ cannot be bounded, because there are only finitely many programs of a given size. Consider now the algorithm that searches for a number $n$ for which $H(n)$ is very large. For example, we could design such a program using some fixed $r$ number o... | 3 | https://mathoverflow.net/users/1946 | 56759 | 35,430 |
https://mathoverflow.net/questions/56767 | 3 | Is there a "canonical" set of generators for a given coxeter group?
If so, is there a method for going from an arbitrary set of generators of the group to the canonical?
(The "textbook" definition doesn't include this in the definition of these groups, although it certainly seems to use them.)
Thanks.
| https://mathoverflow.net/users/13249 | Coxeter group generators | There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example... | 13 | https://mathoverflow.net/users/nan | 56771 | 35,437 |
https://mathoverflow.net/questions/56737 | 15 | A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a neighbourhood $ U$ such that $\pi ^{-1} (U) $ is $S$-diffeomorphic (=fiber preserving diffeomorphism) to $U\times F$ for some ... | https://mathoverflow.net/users/450 | When is a holomorphic submersion with isomorphic fibers locally trivial? | Here is a variant of Jason's example with a proof that it is not even topologically locally trivial. Let $T$ be a (complex) manifold that admits a morphism $\phi$ onto $\mathbb P^1=\mathbb P^1\_{\mathbb C}$ (or $S^2$ if you prefer) and there exists a point $a\in T$ with $b=\phi(a)\in \mathbb P^1$ such that $\{a\}=\phi^... | 5 | https://mathoverflow.net/users/10076 | 56773 | 35,438 |
https://mathoverflow.net/questions/56770 | 7 | I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this question is that my [category number](http://en.wikipedia.org/wiki/N-category_number) has been hovering somewhere around ... | https://mathoverflow.net/users/78 | How equivalent are the theories of reduced and groupal $\infty$-groupoids? | To answer your first set of questions in order (I'm going to use the word "space" for "$\infty$-groupoid"):
Yes, this is all correct.
You seem to be familiar with how to construct $BG$ as a simplicial space via a bar construction. To turn this into an actual space, you just have to form the [geometric realization](... | 8 | https://mathoverflow.net/users/75 | 56775 | 35,440 |
https://mathoverflow.net/questions/56746 | 2 | Does anyone know a closed form or a good approximation of the cumulative distribution function of hypergeometric distribution?
| https://mathoverflow.net/users/8379 | Cumulative distribution function of hypergeometric distribution | Look at:
An accurate computation of the hypergeometric distribution function
By Trong Wu (ACM TOMS, 1993).
| 5 | https://mathoverflow.net/users/11142 | 56785 | 35,446 |
https://mathoverflow.net/questions/56753 | 30 | From which sources would you learn about crystalline cohomology and the de-Rham-Witt complex?
| https://mathoverflow.net/users/nan | learning crystalline cohomology | With enough enthusiasm, I would try to learn about crystalline cohomology and the de-Rham-Witt complex from the homonymous article by Illusie:
Illusie, Luc. Complexe de deRham-Witt et cohomologie cristalline. (French) Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501--661. MR0565469 (82d:14013)
Fortunately, it... | 20 | https://mathoverflow.net/users/11928 | 56786 | 35,447 |
https://mathoverflow.net/questions/56613 | 2 | I am trying to find some general properties of the zeros of
$P(z) = \sum\_{i=1}^n \frac{\alpha\_i}{z+z\_i}$,
with $\sum\_{i} \alpha\_i = 0$, $z\_i \in [-M\; 0], i=1,\ldots,n$ and all $\alpha\_i$ and $z\_i$ are real, and $M<\infty$.
Actually, what I really hope to find is that the real part of the zeros lie in t... | https://mathoverflow.net/users/4677 | Zeros of linear partial fractions | $${1\over z+1}+{-2\over z+10}+{1\over z+20}={170-z\over(z+1)(z+10)(z+20)}$$ seems to satisfy your conditions with $n=3$ and $M=20$ but the zero at 170 is not in the same interval as the poles.
| 2 | https://mathoverflow.net/users/3684 | 56788 | 35,449 |
https://mathoverflow.net/questions/56651 | 7 | [**EDITED** by Y. Choi - I have attempted to paraphrase the original question into something a bit terser and more precise; if this is not what the original poster intended, they should make corrections themselves.]
Let $G$ be a locally compact abelian (LCA) group and let $f\in L^1(G)$. Can we always find $g\in L^2(G... | https://mathoverflow.net/users/13244 | On a decomposition of L^1(G) | I'm answering Yemon's version of the question.
The answer is trivially yes for discrete $G$ since $\ell^1(G) \subset \ell^2(G)$, so let me focus on the non-discrete case.
The first observation to make is that $B(G)$ is contained in the bounded (and uniformly continuous) functions of $G$. So the question asks in par... | 5 | https://mathoverflow.net/users/11081 | 56791 | 35,451 |
https://mathoverflow.net/questions/56778 | 10 | It is known that all amenable groups do not contain free subgroups (of rank>1). But there are amenable groups containing free semigroups. Which amenable groups cannot contain free semigroups?
| https://mathoverflow.net/users/7307 | Amenable groups not containing free semigroups | This is the answer to the question asked by Henry. The wreath product $\mathbb Z\_2 {\rm wr} G$, where $G$ is the Grigorchuk (torsion) group of subexponential growth, obviously has exponential growth and is amenable and torsion. In particular, it has no free subsemigroups.
For elementary amenable (in particular, sol... | 18 | https://mathoverflow.net/users/10251 | 56793 | 35,452 |
https://mathoverflow.net/questions/56794 | 5 | Existence of non-measurable sets requires the use of Axiom of choice. And one can construct models eg. Solovay model without using the Axiom of choice where there are no non-measurable sets.
So, Can one start with the the hypothesis that non-measurable sets exist and then 'prove' the Axiom of choice?
| https://mathoverflow.net/users/13069 | Existence of Non-measurable sets | The answer to your question is "no".
Using the technique of forcing one can easily violate the axiom of choice in a variety of ways (essentially, by focusing on "large enough sets") without affecting the fact that there are non-measurable sets of reals. This includes leaving a bit of choice to make sense of the cons... | 8 | https://mathoverflow.net/users/6085 | 56796 | 35,453 |
https://mathoverflow.net/questions/56795 | 5 | Suppose $\mu$ is a fixed partition of $n$ of length $l(\mu)$, and I was encountered with the following sum, namely
$\sum\_{\nu} \chi\_{\nu}(\mu)$.
I did some calculation using the character table that I can find (mainly Fulton & Harris's book, they have the character table up to $S\_5$), and found that the sum does n... | https://mathoverflow.net/users/3569 | sum of the character of the symmetric group | Your conjecture is correct; as a matter of fact, it is possible to compute these sums $C(\mu):=\sum\_\nu\chi\_\nu(\mu)$ exactly for any $\mu$:
Suppose $\mu$ has $m\_i$ parts of length $i$, for $i=1,2,\ldots$. Then $C(\mu)=\prod\_{i>0} c\_{i,m\_i}$, where $c\_{i,m\_i}$ is the coefficient of $t^{m\_i}/({m\_i}!)$ in $\e... | 10 | https://mathoverflow.net/users/730 | 56797 | 35,454 |
https://mathoverflow.net/questions/56789 | 7 | For equivalence of unbranched coverings of topological spaces, there is a criteria:
Two coverings (unbranched) $p\_1\colon Y\_1\rightarrow X$ and $p\_2\colon Y\_2\rightarrow X$ are equivalent iff for some $q\in X$ and $\bar{q\_1}\in p\_1^{-1}(q)$ and $\bar{q\_2}\in p\_2^{-1}(q)$, the induced subgroups $p\_\*\pi\_1(Y\... | https://mathoverflow.net/users/12484 | Equivalence of Branched Coverings | The answer is **yes**: the equivalence class of the covering is detected by the monodromy representation of the fundamental group of the base minus the branch locus, up to conjugacy.
More precisely, let $f \colon X \to Y$ be a (possibly branched) covering of degree $d$ of Riemann surfaces. Choosing a point $y\_0 \in ... | 8 | https://mathoverflow.net/users/7460 | 56806 | 35,459 |
https://mathoverflow.net/questions/56807 | 17 | Hello,
I wanted to know that how researchers in mathematics keep updates in their field of interest.
(original question by Rahul Gupta)
| https://mathoverflow.net/users/13279 | How to be updated with current advances in mathematics | There are several ways, and it depends on the precise meaning of 'in their field of interest'.
With a narrow definition of this an important and classical way to keep up to date is personal correspondence, people in the same field often know each other and thus people actively inform people of which they know that th... | 17 | https://mathoverflow.net/users/nan | 56810 | 35,460 |
https://mathoverflow.net/questions/56802 | 6 | Let $P(x)=x^{n}+a\_{1}x^{n-1}+\cdots+a\_{n-1}x+a\_{n}$, where $a\_1, a\_2, \dots, a\_n$ are intergers.
Question 1. When does the polynomial $P(x)$ have its zeros all being pure imaginary or zero(here 0 is a root of the given polynomial)?
Question 2. Does there exist a characterization that $P(x)$ have its zeros all... | https://mathoverflow.net/users/13277 | When does a polynomial have all pure imaginary roots? | A necessary and sufficient condition is that $P(x)$ is a power of $x$ times a product of terms $x^2+c$ with $c$ real and positive. Hence $P(x)$ is $x^{n-2m}Q(x^2)$ where $m\ge0$ and $Q(t)$ is a unitary polynomial in $t$ of degree $m$ with integer nonnegative coefficients. Finally, a necessary condition is that $a\_{k}=... | 11 | https://mathoverflow.net/users/4661 | 56812 | 35,462 |
https://mathoverflow.net/questions/56819 | 1 | Is the Rudin-Keisler order of ultrafilters linear?
Is it a well ordering?
| https://mathoverflow.net/users/4086 | Is the Rudin-Keisler order of ultrafilters linear? | This would be more suitable as a comment, but I do not have enough reputation points.
Among the first results that google spits out for Rudin Keisler is the paper
Anatoly Gryzlov: On the Rudin-Keisler order on ultrafilters; <http://dx.doi.org/10.1016/S0166-8641(96)00109-5>
It claims that there are incomparable ultrafi... | 10 | https://mathoverflow.net/users/8250 | 56825 | 35,470 |
https://mathoverflow.net/questions/56831 | 0 | I met projective space via a recent class on perspective drawing, believe it or not, but I didn't know that this was the "space" we were using. I came across a more detailed description trawling the net.
In a book on point-set topology that I bought, it describes Euclidean n-space as a field made of (sorry I don't kn... | https://mathoverflow.net/users/13147 | Random question: Is there a set-theoretic description of projective space? | One definition is that the $n$-dimensional (real) projective space is the space of lines through the origin in $R^{n+1}$. Technically it is a topological quotient of $R^{n+1}$ minus the origin, by the equivalence relation $x\sim \lambda x$ for all (nonzero) vectors $x$ in $R^{n+1}$ and all (nonzero) scalars $\lambda$.
... | 1 | https://mathoverflow.net/users/12260 | 56834 | 35,474 |
https://mathoverflow.net/questions/56836 | 8 | I'm following the open courseware content on Machine Learning from Stanford University. In the [lecture notes](http://www.stanford.edu/class/cs229/notes/cs229-notes1.pdf), it is given that
$$\Delta\_A \ tr(ABA^TC) = CAB + C^TAB^T$$
which I tried but couldn't prove easily. It is not required to follow the course con... | https://mathoverflow.net/users/5287 | Proof of a fact about traces | I guess $\Delta\_A$ denotes the derivative with respect to the elements of the matrix $A$ (more conventionally denoted by $\partial\_{A}$).
To evaluate the derivative with respect to $A\_{ij}$, write out the trace in terms of components and then use $\partial\_{A\_{ij}} A\_{mn} = \delta\_{im} \delta\_{jn}$,
$$\partia... | 14 | https://mathoverflow.net/users/49924 | 56838 | 35,477 |
https://mathoverflow.net/questions/56839 | 0 | Let $n=a\_ka\_{k-1}\ldots a\_1a\_0$ be a $(k+1)$ digit number.
**Step 1:** Find $a\_k \times a\_{k-1} \times \ldots \times a\_1 \times a\_0= c\_tc\_{t-1}\ldots c\_1c\_0$.
**Step 2:** Find $a\_k + a\_{k-1} + \ldots + a\_1 + a\_0= d\_ld\_{l-1}\ldots d\_1d\_0$.
Then form $n\_1=c\_tc\_{t-1}\ldots c\_1c\_0d\_ld\_{l-1}... | https://mathoverflow.net/users/6770 | In the sense of Collatz | Yes, the only way that you can get a sequence which doesn't converge to a repeating cycle is if you find a sequence with unbounded elements, but it is easy to show that this is not the case. If you start with a number $n$, with $k$ digits, then the number of digits in $f(n)$ (after you perform both steps) is at most $2... | 5 | https://mathoverflow.net/users/2384 | 56846 | 35,482 |
https://mathoverflow.net/questions/56823 | 5 | Suppose given a finite extension $L/K$ of number fields. I would like to develop a better intuition for the Verlagerung giving an embedding $Ver : Gal(K^{ab}/K) \to Gal(L^{ab}/L)$.
For example, let $L'/L$ be a finite abelian (Galois) extension. Define $\phi:Gal(K^{ab}/K)\to Gal(L'/L)$ to be the composition of the Ve... | https://mathoverflow.net/users/5831 | Verlagerung made "explicit" | I am no expert, but let me share an idea. For a number field $M$ let us denote by $C\_M$ the idele class group of $M$. By class field theory, $K' \subset L'$ is the same as $N\_{L'/K}(C\_{L'})\subset N\_{K'/K}(C\_{K'})$, where $N$ stands for the norm map. Let us denote by $U$ the open subgroup $N\_{L'/L}(C\_{L'})$ of $... | 5 | https://mathoverflow.net/users/11919 | 56866 | 35,490 |
https://mathoverflow.net/questions/56801 | 3 | How to determine whether an intersection of a convex polytope and a simplex in $R^{n}$ is not empty?
The polytope is given in a halfspace representation.
I'm aware that there are some algorithms which work in $R^{2}$ and $R^{3}$, but I don't know any that would work in $R^{n}.$
| https://mathoverflow.net/users/3794 | An intersection of a convex polytope and a simplex | I find @Harald's comment a little mysterious, but an obvious algorithm for the OP is to
observe that a simplex is the intersection of $n+1$ halfspaces $h\_1, \dotsc, h\_{n+1},$ and then check that $Q=P \cap h\_1 \cap h\_2 \dots \cap h\_{n+1}$ [where $P$ is your polytope) is nonempty. This last problem is a standard li... | 2 | https://mathoverflow.net/users/11142 | 56870 | 35,494 |
https://mathoverflow.net/questions/56698 | 2 | A question came up on MSE and it generated, for me, the following question:
When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic interpretation? Are they cofibrant objects in some model structure on the arrow category of topological spaces? (feel free to replac... | https://mathoverflow.net/users/3901 | Model categories and cellular maps | The answer to your question is yes in the case of cellular maps of topological spaces.
(I think a similar argument works in the simplicial case, but I doubt the result is true in the CW case.)
There is a model structure on the arrow category in which a square (i.e., a morphism)
$$
X\_0 \quad \to \quad X\_1
$$
$$
\dow... | 2 | https://mathoverflow.net/users/8032 | 56873 | 35,497 |
https://mathoverflow.net/questions/56865 | 10 | There is a very basic theorem for the Zariski topology.
Let X = Spec(R) and Y=Spec(R/I) for I some reduced ideal. Y obtains a topology two ways, one is the subspace topology as a subset of X and another as the spectrum of a ring. These topologies are the same by the correspondence between ideals of R containing I and... | https://mathoverflow.net/users/12129 | The etale site of a closed subscheme and its etale Grothendieck subtopology | The statement is at least true Zariski locally. That is, given an étale map $V\to Y$, there exists a Zariski open cover $X=\bigcup X\_i$ so that the pullback of $V$ to $Y\cap X\_i$ is the restriction of an étale neighborhood of $X\_i$.
To see this, use the structure theorem for étale morphisms: [Theorem 34.11.3](http... | 8 | https://mathoverflow.net/users/1 | 56874 | 35,498 |
https://mathoverflow.net/questions/56860 | 14 | Does anyone know of a reference or have any idea for an explicit description of the ring of differential operators for polynomial algebras over the integers? I'm hoping there is something analogous to the case of a polynomial ring over a field $K$, where if the field has characteristic $0$ an explicit description is gi... | https://mathoverflow.net/users/13288 | Explicit ring of differential operators for polynomial algebras over the integers? | The answer to your first question is "yes". You can find a calculation of the full ring of differential operators on a suitably nice scheme here : Theoreme 16.11.2 on page 54 of [EGA 4 IV, PIHES 32 (1967)](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1967__32_/PMIHES_1967__32__5_0/PMIHES_1967__32__5_0.pdf). Generall... | 17 | https://mathoverflow.net/users/6827 | 56886 | 35,504 |
https://mathoverflow.net/questions/43382 | 3 | Let $C$ be a coalgebra and $\Delta: C \to C\otimes C$ a co-multiplication map. Then, due the co-associative property we can consider $\Delta^m$. But how is defined $\Delta^{m}: C \to C^{\otimes m}$?
Given $f,g \in C$ and $1\leq k \leq m$ can we have
$$\begin{align\*}\Delta^{m-1}(fg)&=\Delta^{k-1}(fg) \otimes \opera... | https://mathoverflow.net/users/40886 | How to work with co-multiplication? | To add to the answer above, I'd like to advertise the so-called *sumless Sweedler notation* here.
This notation works as follows: let $C$ be a coalgebra; then if $c \in C$ then we write $\Delta(c) = c\_1 \otimes c\_2$ as an abbreviation of the more precise $\Delta(c) = \sum\_{i=1}^k c\_{i1} \otimes c\_{i2}$ for some... | 10 | https://mathoverflow.net/users/6827 | 56888 | 35,505 |
https://mathoverflow.net/questions/56889 | 6 | The *homotopy dimension* $h \dim X$ of a space $X$ is the minimal dimension of a CW-complex $Z$ homotopy equivalent to $X$.
I am interested in the generalisation to maps $f\colon X\to Y$. Here's what I think it should be:
>
> The *homotopy dimension* of $f\colon X\to Y$ is the smallest $k$ such that $f$ factorise... | https://mathoverflow.net/users/8103 | Homotopy dimension of a mapping | Regarding Question 1: No, I do not think that's correct. In my opinion, the
definition should be one of the following:
The *relative homotopy dimension* of $f: X \to Y$ is $\le k$ if and only if there is a factorization
of $f$ as
$$
X \overset{f'}\to Y' \overset{g} \to Y
$$
in which $f'$ is an inclusion, $Y'$ is ob... | 8 | https://mathoverflow.net/users/8032 | 56895 | 35,508 |
https://mathoverflow.net/questions/56891 | 6 | Given a (finite, simple, undirected) graph $\mathcal{G} = (V, E)$, an *edge binning* associates each $e\_{ij} \in E$ with one or the other of its vertices $v\_i, v\_j \in V$. Let $c\_i$ be the number of edges associated with vertex $v\_i$ in a given edge binning. Find an edge binning such that $\max\_{v\_i \in V}(c\_i)... | https://mathoverflow.net/users/5029 | Unidentified Combinatorial Problem | This problem is equivalent to the *graph orientation problem* also known as the *graph balancing problem*. One is given an undirected graph and has to give an orientation of the edges which minimizes the maximum out-degree. If this value is $k$, then the graph is called $k$-orientable. Here are some articles on the top... | 10 | https://mathoverflow.net/users/2384 | 56896 | 35,509 |
https://mathoverflow.net/questions/56884 | 8 | Suppose I have a finite set of points in $\mathbb{P}^1$ (over the complex numbers), and suppose that at each point, I am given a [**Edit:** quasi-unipotent] conjugacy class in $Sp(2g,\mathbb{Z})$ for $g$ a fixed positive integer. Then near each point, I have an analytic neighborhood where I can construct a family of co... | https://mathoverflow.net/users/121 | Does there exist a family of curves (or abelian varieties) on the punctured line with specified monodromy on H^1? | There are several restrictions. First, the existence of potential semistable reduction (Grothendieck-Mumford) implies that your representative $\sigma$ at every puncture must be quasi-unipotent of level 2, i.e, there exists a positive integer $N$ such that $(\sigma^N-1)^2=0$. Second, the Zariski closure $G\subset Sp\_{... | 12 | https://mathoverflow.net/users/9658 | 56899 | 35,511 |
https://mathoverflow.net/questions/56900 | 6 | Let $G$ be an algebraic group (not necessarily linear) defined over an
algebraically closed field $k$,
acting on a smooth integral $k$-variety $X$.
Let $x\_0\in X(k)$ and let $\pi\_1(X,x\_0)$ denote the étale
(Grothendieck's) fundamental group of $X$.
Assume that either $G$ fixes $x\_0$ or the group $\pi\_1(X,x\_0)$ is... | https://mathoverflow.net/users/4149 | A characteristic-free proof that the action of a connected algebraic group $G$ on the fundamental group of a $G$-variety is trivial | This is false as stated in positive characteristic. For example, suppose that the characteristic of $k$ is $p$; take $X = \mathbb A^1\_k = \mathop{\rm Spec} k[t]$ and $G = \mathbb G\_{\rm m}$. If $a \in k^\*$ and $E$ is the standard étale cover $E = \mathop{\rm Spec} k[x,t]/(x^p - x - t) \to \mathop{\rm Spec} k[t]$ of ... | 11 | https://mathoverflow.net/users/4790 | 56907 | 35,516 |
https://mathoverflow.net/questions/56902 | 2 | Trying to think a bit about an MO question\*, and not being experienced in category theory, I happened to ask myself the following question that I'm quite sure would be pretty elementary for the non-layman in the field.
Of course feel free to close in case it's too localized or anyway not suited for MO.
>
> Given c... | https://mathoverflow.net/users/4721 | Equivalent functors (elementary question) | Let $\mathcal{A}$ have objects $0$ and $1$, with morphisms $u\_i:i\to i$ for $i\in\{0,1\}$ satisfying $u\_i^2=1\_i$, and no other non-identity morphisms. Let $\mathcal{B}$ be a group $G$, regarded as a category with one object. Then for each pair $g=(g\_0,g\_1)\in G^{2}$ with $g\_0^2=g\_1^2=1$, we have a functor $F\_g:... | 6 | https://mathoverflow.net/users/10366 | 56911 | 35,517 |
https://mathoverflow.net/questions/56920 | 6 | Guo and Qi recently discovered sharp bounds for the harmonic numbers (*qq.v.* [doi:10.1016/j.amc.2011.01.089](http://www.google.com/search?q=doi%3A10.1016/j.amc.2011.01.089)). For example, they show that $$H\_n < \ln(n) + \frac{1}{2n} + \gamma - \frac{1}{12n^2+\frac{6}{5}},$$ where $H\_n$ is the $n$th harmonic number a... | https://mathoverflow.net/users/3951 | Bounds for generalized harmonic numbers | Let $f(x)=1/\sqrt{x}$. Then by Euler-Maclaurin summation,
$$ \sum\_{2\leq k\leq n} \frac{1}{\sqrt{k}} = \int\_1^n \frac{dx}{\sqrt{x}}+\sum\_{r=0}^m\frac{(-1)^{r+1}B\_{r+1}}{(r+1)!} \big(f^{(r)}(n)-f^{(r)}(1)\big)+ R$$
where $R$ is a remainder term of the form
$$R=\frac{(-1)^m}{(m+1)!}\int\_1^n B\_{m+1}(x)f^{(m+1)}(x)dx... | 10 | https://mathoverflow.net/users/3659 | 56924 | 35,523 |
https://mathoverflow.net/questions/56925 | 2 | Is it possible to place non-trivial probability measures on sets of cardinality strictly greater than the continuum -- in particular, on sets of cardinality 2^c? (Any references would be appreciated.)
| https://mathoverflow.net/users/13307 | Probability Measures and Cardinality > c | There's even more. If your set can be considered as a product of probability spaces, its measure will have some good properties. This is a theorem due to Alexandra Ionescu Tulcea: given an infinite (with arbitrary cardinality) family of probability spaces
$$\left\{(\Omega\_n, \mathscr{B}\_n, \mu\_n)\right\}\_{n \in I}$... | 1 | https://mathoverflow.net/users/12563 | 56927 | 35,525 |
https://mathoverflow.net/questions/56930 | 6 | What does the classical proof of the proposition "there exists irrational numbers a, b such that $a^b$ is rational" want to reveal? I know it has something to do with the difference between classical and constructive mathematics but am not quite clear about it. Materials I found online does not give quite clear explana... | https://mathoverflow.net/users/13012 | About the proof of the proposition "there exists irrational numbers a, b such that a^b is rational" | Presumably, the proof you have in mind is to use $a=b=\sqrt2$ if $\sqrt2^{\sqrt2}$ is rational, and otherwise use $a=\sqrt2^{\sqrt2}$ and $b=\sqrt 2$. The non-constructivity here is that, unless you know some deeper number theory than just irrationality of $\sqrt 2$, you won't know which of the two cases in the proof a... | 26 | https://mathoverflow.net/users/6794 | 56931 | 35,527 |
https://mathoverflow.net/questions/56932 | 13 | This is just a general curiosity question:
In the standard textbook treatments of characteristic classes, and in particular the treatment of universal Pontrjagin classes, it's standard to consider $H^\ast(BSO,Z[1/2])$ (or $H^\ast(BSO(n),Z[1/2])$) in order to kill the 2-torsion. But I'm curious about that 2-torsion, s... | https://mathoverflow.net/users/6646 | What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$? | The basic fact is that the 2-torsion all has order exactly 2, so it injects into the mod 2 cohomology, forming a subalgebra of the polynomial algebra on the Stiefel-Whitney classes. This subalgebra can also be described as the image of the mod 2 Bockstein homomorphism, so it is computable although the answer is not eas... | 32 | https://mathoverflow.net/users/23571 | 56945 | 35,533 |
https://mathoverflow.net/questions/56693 | 14 | Is there an algorithm for generating (some or all) subfields of a certain genus of a given function field (even a random one,I mean for example generating a random elliptic subfield of a certain given function field). I did a quick search and it seems to me that the problem is heavily treated in the case of cyclic and ... | https://mathoverflow.net/users/6776 | Subfields of a function field | The algorithm to embed function fields, ie. to test if a function field E can be embedded into a function field F has been developed (and implemented in Magma) by a student of Florian Hess: Gerriet Möhlmann as
part of his Diploma work. His thesis (in German) can be found at
<http://www.math.tu-berlin.de/~kant/publicati... | 5 | https://mathoverflow.net/users/13311 | 56952 | 35,538 |
https://mathoverflow.net/questions/56938 | 36 | Terms like "in the natural way" or "the natural X" are used frequently in mathematical writing. While it is certainly clear most of the time what is meant, on occasion, I have been confounded. The word "natural" seems to be one of the most ambiguous terms used in formal mathematics. I have never seen anyone actually de... | https://mathoverflow.net/users/6137 | What does the adjective "natural" actually mean? | Actually, there is an exact meaning, but it is not always used in that sense. For two functors $\mathsf F,\mathsf G:\mathscr A\to \mathscr B$ a *natural transformation* is a morphism of functors $\eta:\mathsf F\to\mathsf G$ that is compatible with the functors in the obvious (sic!) way.
For instance if $\mathsf F={\r... | 36 | https://mathoverflow.net/users/10076 | 56956 | 35,540 |
https://mathoverflow.net/questions/56955 | 7 | I was recently trying to think of a simple example that demonstrates that the natural inclusion of an abelian group $G$ into
$$\widehat{\widehat{G}}=\text{Hom}\\_{\mathsf{Ab}}(\text{Hom}\_{\mathsf{Ab}}(G,\mathbb{C}^\times),\mathbb{C}^\times)$$
is not necessarily an isomorphism. Note that I'm looking at all homomorphis... | https://mathoverflow.net/users/1916 | The natural inclusion of an infinite abelian group $G$ into $\widehat{\widehat{G}}$ | Since $\mathbb{C}^\times \cong \mathbb{R}\\_{+}\times S^1$ by polar coordinates, it suffices to show that $\text{Hom}({\mathbb R}\\_+,{\mathbb R}\\_+)$ is uncountable. But for any real number $a$, $x\mapsto x^a$ gives an endomorphism of ${\mathbb R}\_+$. Explicitly, there are uncountably many group homomorphisms from $... | 11 | https://mathoverflow.net/users/75 | 56959 | 35,542 |
https://mathoverflow.net/questions/56953 | 1 | Based on some experiments, I find that the following two statements are correct. But I can not prove this. At the same time, I still can not find the counterexmaples.
Let $p(x)=x^{n}+a\_{2}x^{n-2}+a\_{3}x^{n-3}+\dots+a\_{n-1}x+a\_{n}$ be a polynomial with interger coefficients, where $a\_{k}\geq0$ for every even $k$ ... | https://mathoverflow.net/users/13277 | Two problems about the zeros of a prescribed polynomial | $(x^3-1)(x^3-2)(x^2+a^2)=x^8+a^2x^6-3x^5-3a^2x^3+2x^2+2a^2$ has both two different non-zero real roots and $\pm ai$ as roots.
**EDIT** polynomial adjusted for adjusted conditions. If $a\_2$ needs to be at least $b$, set $a$ so $a^2\geq b$. :)
| 7 | https://mathoverflow.net/users/10076 | 56960 | 35,543 |
https://mathoverflow.net/questions/56962 | 37 | Stacks *qua* moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The classic example is principal bundles/torsors, the whole category of which is actually a groupoid. But what about objects ... | https://mathoverflow.net/users/4177 | What about stacks of categories in algebraic geometry? | The category theoretical definition of stacks (as given for instance in Giraud: Cohomologie non-abélienne) allow for arbitrary categories as targets (the stack condition only involves isomorphisms however). A natural example is the category of quasi-coherent sheaves (which has the category of vector bundles as a subcat... | 20 | https://mathoverflow.net/users/4008 | 56974 | 35,551 |
https://mathoverflow.net/questions/56947 | 7 | The standard reference for the statement that "any abstract variety is an open subscheme of a complete variety" is Nagata's 1962 paper *Imbedding of an abstract variety in a complete variety*. Unfortunately, this paper was apparently written before the language of schemes became standard, and uses Nagata's own language... | https://mathoverflow.net/users/5094 | Scheme-theoretic account of why every variety embeds in a complete variety | Apart from Brian's, published as:
Deligne's notes on Nagata compactifications. J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257.
there are:
Lütkebohmert, On compactification of schemes. Manuscripta Math. 80 (1993), no. 1, 95–111.
and
Vojta: Nagata's embedding theorem, arXiv:0706.1907
and, finally
Deligne... | 10 | https://mathoverflow.net/users/6348 | 56988 | 35,559 |
https://mathoverflow.net/questions/56990 | 8 | Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
All the best,
Sebastian
| https://mathoverflow.net/users/13321 | Probabilities independent of ZFC? | There are several issues.
On the one hand, any set can be made countable by forcing, and this process will certainly affect the measure of the set, if it did not have measure zero in the ground model.
But in the context of the Lebesgue measure on the reals, say, it is natural to consider not the set itself, but t... | 19 | https://mathoverflow.net/users/1946 | 56992 | 35,561 |
https://mathoverflow.net/questions/56912 | 21 | Let $V$ be a real vector space. It is well known that a subset $B\subset V$ is the unit ball for some norm on $V$ if and only if $B$ satisfies the following conditions:
1. $B$ is convex, i.e. if $v,w\in B$ and $\lambda\in[0,1]$ then $\lambda v+(1-\lambda)w \in B$.
2. $B$ is balanced, i.e. $\lambda B \subset B$ for al... | https://mathoverflow.net/users/6514 | Can you tell whether a space is Banach from the unit ball? | A sufficient (additional) condition is that $B$ *be compact for some Hausdorff vector topology for* $V$. The proof goes as follows.
Letting $\langle x\_i\rangle$ be a Cauchy sequence, it is contained in some $nB$, and so it has some cluster point $y$ there. Given $\varepsilon>0$, there is $i\_0$ such that $x\_i-x\_j\... | 8 | https://mathoverflow.net/users/12643 | 56996 | 35,564 |
https://mathoverflow.net/questions/56984 | 2 | I'm learning about de Jong's theory of resolution of singularities and the following fact is used numerous times: an alteration of varieties $h: X \rightarrow Y$ factors as $X \xrightarrow{\pi} Z \xrightarrow{f} Y$, where $\pi$ is a modification and $f$ is a finite morphism. From an exercise in Hartshorne I know I can ... | https://mathoverflow.net/users/13139 | Alterations factor as modification + finite map | As Sandor points out, this is Stein factorization.
Let $X$, $Y$ be varieties over a field $K$. Let $h: X\to Y$ be a proper morphism. Then
$h\_\*(O\_X)$ is coherent and
$Y':=Spec(h\_\*(O\_X))$ (cf. EGA II.1.3 for the definition of $Spec$ of a sheaf of quasicoherent algebras) is finite over $Y$. Consider the Stein facto... | 4 | https://mathoverflow.net/users/8680 | 56998 | 35,565 |
https://mathoverflow.net/questions/56997 | 3 | Hello, all!
Could somebody draw a proof-sketch of next expression from tensor algebra on matrices over finite fields:
determinant of tensor product $A~ \times ~B$ of $n \times n$-matrix $A$ over finite field $GF(q)$ on $m \times m$-matrix $B$ over finite field $GF(q)$ is $\det(A)^m \cdot \det(B)^n$.
Please, give me a... | https://mathoverflow.net/users/nan | tensor product of matrices | Darij's first comment could be made into an answer as follows.
Darij advised to write
$$A \otimes B = (A \circ I\_n) \otimes (I\_m \circ B) = (A \otimes I\_m) \circ (I\_n \otimes B)$$
where the second equation follows from functoriality of the tensor product. Here both $A \otimes I\_m$ and $I\_n \otimes B$ are ... | 7 | https://mathoverflow.net/users/2926 | 57004 | 35,570 |
https://mathoverflow.net/questions/56978 | 5 | Let us consider a boolean hypercube $C = \{-1, 1\}^n$. Let $S = \{x \in C \mid |\{i \mid x\_i = -1\}| = \varepsilon n\}$ be a Hamming sphere in $C$ (here $\varepsilon$ stands for the fixed parameter from $(0, 1/2)$). Let us sample $X \in S$ uniformly at random. And we are interested in estimating $\mathrm{E}[X\_1 X\_2 ... | https://mathoverflow.net/users/3448 | Product of coordinates of a random point from Hamming sphere | If I am not mistaken, this expectation equals the coefficient of $x^{\varepsilon n}$ in $(1-x)^{\alpha n}(1+x)^{(1-\alpha) n}$ divided by the corresponding coefficient in $(1+x)^n$ (which equals, of course, $\binom{n}{\varepsilon n}$). Such coefficient may be represented as integrals over unit circle (and so over $[0,2... | 5 | https://mathoverflow.net/users/4312 | 57006 | 35,571 |
https://mathoverflow.net/questions/56939 | 7 | Let $X$ be a linearly ordered topological space with a countable dense subset. Does it necessarily follow that $X$ is metrizable?
EDIT: Apollo's comment int he answers implies the answer is negative. Let $X$ be the open unit interval $(0,1)$ and adjoin to every real number $x$ a "ghost number" $x'$ such that $x'$ is ... | https://mathoverflow.net/users/4903 | Must a linearly ordered, separable space be metrizable? | You already found a (classical) counterexample: the double arrow ($[0,1] \times \{0,1\}$, ordered lexicographically), which is even compact and separable. There is however a nice metrization theorem for linearly ordered spaces (due to [Lutzer](http://www.ams.org/journals/proc/1969-022-02/S0002-9939-1969-0248761-1/S0002... | 8 | https://mathoverflow.net/users/2060 | 57022 | 35,580 |
https://mathoverflow.net/questions/57015 | 23 | If $H$ is a closed subgroup of a topological group $G$, then the orbit map $G\to G/H$ is a principal bundle, yet somewhat surprisingly, it need not be locally trivial.
In the wikipedia [article on fiber bundles](http://en.wikipedia.org/wiki/Fiber_bundle#Quotient_spaces) it is claimed that if $H$ is a Lie group, then $... | https://mathoverflow.net/users/1573 | Which principlal bundles are locally trivial? |
>
> ...if $H$ is a Lie group, then $G \to G/H$ is locally trivial. Is the claim true, and if so, what is the reference?
>
>
>
Yes, it is true. See the Corollary in section 4.1 of: "On the Existence of Slices for Actions of Non-compact Lie Groups", which you can download here: <http://vmm.math.uci.edu/ExistenceOf... | 25 | https://mathoverflow.net/users/7311 | 57023 | 35,581 |
https://mathoverflow.net/questions/57031 | 14 | Felix Hausdorff was of course a great mathematician, who had major effects on several branches of mathematics. However he also wrote literature and philosophy and was affiliated with important German musicians. When Nazism came to power, Hausdorff failed to escape in time, lost his job, and finally committed suicide in... | https://mathoverflow.net/users/4087 | Biography of Felix Hausdorff | There is the "Hausdorff edition" project (E. Brieskorn, F. Hirzebruch, W. Purkert, R. Remmert and E. Scholz) which will entail all collected works and is supposed to have a decent biography as well. Out of the planned nine volumes only four have been published. It seems like the first volume where the biography is supp... | 13 | https://mathoverflow.net/users/2384 | 57036 | 35,588 |
https://mathoverflow.net/questions/57038 | 12 | Let M be a smooth manifold. Can every k-form $\omega$ on M be written as a sum of k-forms, that are wedge products of 1-forms, i.e. $\omega = \sum\_{i=0}^n \alpha\_1^{(i)} \wedge \ldots \wedge \alpha\_k^{(i)} $, where $\alpha\_l^{(i)} \in \Omega^1(M) $ ? If M is compact, one can cover M be finitely many charts and use ... | https://mathoverflow.net/users/13338 | k-form: sum of wedge products of 1-forms? | The answer is yes and you need transversality in some form. Here we use Whitneys embedding theorem, but a weaker statement would suffice. Embed $M \subset \mathbb{R}^m$. Thus there is a monomorphism $TM \to M \times \mathbb{R}^m$ of vector bundles, dualizing to an epimorphism $M \times \mathbb{R}^m \to T^{\ast} M$. Let... | 11 | https://mathoverflow.net/users/9928 | 57047 | 35,593 |
https://mathoverflow.net/questions/57017 | 11 | Suppose I have n sets $X\_1,\dots,X\_n$ consisting of $k$ points each, where all $nk$ points are i.i.d. uniform random samples in the unit square $[0,1]\times[0,1]$. Consider the shortest path that goes through at least one element of each set $X\_i$. Is the asymptotic behavior of this, a la the Beardwood-Halton-Hammer... | https://mathoverflow.net/users/11828 | Generalized Euclidean TSP | You should be able to get $O(\sqrt{n/k})$ by choosing a smaller square of area $1/k$, which will contain one point from most of the point sets, and use the BHH theorem to find a TSP tour of this. Now, you have to show that adding the points from the point sets you left out doesn't increase the length of the tour much. ... | 10 | https://mathoverflow.net/users/2294 | 57051 | 35,595 |
https://mathoverflow.net/questions/57025 | 46 | I'm soon giving an introductory talk on de Rham cohomology to a wide postgraduate audience. I'm hoping to get to arrive at the idea of de Rham cohomology for a smooth manifold, building up from vector fields and one-forms on Euclidean space. However, once I've got there I'm not too sure how to convince everyone that it... | https://mathoverflow.net/users/12653 | Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness | The motivation that most appeals to me is very simple and can come up in a freshman vector calculus course.
We say that a vector field $F$ in $\mathbb{R}^3$ is *conservative* if $F = \nabla f$ for some scalar-valued function $f$. This has natural applications in physics (e.g. electric fields). It's easy to see this h... | 43 | https://mathoverflow.net/users/4832 | 57054 | 35,596 |
https://mathoverflow.net/questions/57027 | 12 | Can anyone point me to a classification/construction of the irreducibles for $U\_q(\mathfrak{sl}\_n)$, or the associated small quantum groups, when the parameter $q$ is a root of unity and $n>2$? Neither Jantzen or Lusztig's quantum group books seem to help.
Edit: perhaps the best way to clarify what I mean when I re... | https://mathoverflow.net/users/6481 | Simple modules for $U_q(\mathfrak{sl}_n)$ at roots of unity | I'm interpreting your quantum group as the quantized enveloping algebra studied by Lusztig and many others, starting with the divided power version of the usual enveloping algebra of a semisimple Lie algebra. There is a different version based on the usual enveloping algebra, studied especially by DeConcini, Kac, Proce... | 9 | https://mathoverflow.net/users/4231 | 57056 | 35,597 |
https://mathoverflow.net/questions/57037 | 15 | I am not sure if this is an appropriate question, but I was asked this by a colleague today and do not know how to answer it.
1) Are there any rational solutions to the following equation:
$$x^3-8x^2+5x+1 = -7y^2(x-1)x$$
2) Is it possible that this is an elliptic curve in disguise? I have noticed that after project... | https://mathoverflow.net/users/3659 | Are there any rational solutions to this equation? | [Complete revamp of answer. It is based on the one before, but is better!]
In Tim's hyperelliptic equation, make the change of variables $y$ to $y/(-7)^2$, and $x$ to $x/(-7)$, to get:
$$y^2=x(x+7)(x^3+56x^2+245x-343)$$
For every prime $p$ with $v\_p(x)<0$, the valuation must in fact be even, thus appears to an eve... | 19 | https://mathoverflow.net/users/2024 | 57059 | 35,598 |
https://mathoverflow.net/questions/56936 | 2 | Hello all,
could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem
$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$
$u = g \;,\; x \in \Gamma = \partial \Omega$
to the smoothness of $f$, $q$ and $g$?
$\Omega$ is a convex polygonal domain in $\... | https://mathoverflow.net/users/12163 | smoothness of solution for second order elliptic problem | Grisvard's book is a standard reference for elliptic problems in domains with corners.
| 2 | https://mathoverflow.net/users/12120 | 57069 | 35,604 |
https://mathoverflow.net/questions/57067 | 4 | Preface: I am fairly new to the concept of Hausdorff dimension, so I don't know how interesting a question this is.
Identify walks on $\mathbb{Z}$ with infinite binary sequences (say $0$ means moving left, $1$ means moving right). It is then well-known that, in the Cantor space $2^\mathbb{N}$ under the Lebesgue measu... | https://mathoverflow.net/users/13344 | Hausdorff dimension of non-recurrent walks | First of all the dimension of the space. You didn't give the metric you want to use, so I'll use my favourite one: two points are at distance $2^{-n}$ if they first disagree in the $n$th symbol. The 1-Hausdorff measure agrees with coin-flipping measure so that the full space has Hausdorff dimension 1.
Now for the non-... | 9 | https://mathoverflow.net/users/11054 | 57074 | 35,606 |
https://mathoverflow.net/questions/57078 | 2 | Suppose I have a function $f$, and positive integers $x$ and $y$ such that $x$ is a
square, $x \ne y$ and $y \ne \sqrt{x}$.
Now, assume that:
$|\frac{f(x)}{y} - \frac{f(y)}{x}| > 2$
$|\frac{f(\sqrt{x})}{y} - \frac{f(y)}{\sqrt{x}}| > 1$
for all $x, y$ in the domain of $f$.
(Note that $|N|$ is the absolute valu... | https://mathoverflow.net/users/10365 | A question on a special type of function | I'm assuming the first inequality needs absolute values also; if not then exchanging the roles of $x$ and $y$ forces a number and its opposite to both be greater than two.
In this case, there is such an $f$ which is injective:
Let $f(1)=1$.
Assume $f$ has been defined and satisfies the two properties on $\{1,\ldo... | 2 | https://mathoverflow.net/users/13353 | 57087 | 35,613 |
https://mathoverflow.net/questions/57084 | 2 | Suppose $p$ is a point in $\mathbb{R}^n$ so that among the set $S$ of polynomials in $\mathbb{Z}[x\_1,\ldots,x\_n]$ which equal zero at $p$, $p$ is the only point in some neighborhood of $p$ at which all of them equal zero.
Is there necessarily a finite set $S\_2\subseteq S$ of polynomials so that $p$ is the only poi... | https://mathoverflow.net/users/13353 | A unique zero of a system of polynomials is a zero of a finite system. | The existence of a finite $S\_2$ follows from [Hilbert's basis theorem](http://en.wikipedia.org/wiki/Hilbert%27s_basis_theorem), as $\mathbb{R}$ is obviously Noetherian. Just take a finite generating system of the ideal generated by $S$.
About your auxilliary questions:
a) This is wrong. Just take $f = x^2 + y^2$; t... | 2 | https://mathoverflow.net/users/7001 | 57088 | 35,614 |
https://mathoverflow.net/questions/56113 | 7 | What is the status of Ihara's lemma for Shimura curves over totally real fields?
In particular, why is it not implicit in the exact sequence of Rajaei, "On the levels of mod $l$ Hilbert modular forms" (Crelle 2001), Theorem 3 (3.18)?
To be a little bit more precise (or at least to give the main idea), let $F$ be a t... | https://mathoverflow.net/users/6121 | Status of Ihara's lemma for Shimura curves over totally real fields? | Okay, so the answer to the naive question is most likely no for several reasons. For instance, the identifications explained in the second paragraph show that the injection in Rajaei is really an analogue of Ihara's lemma for totally definite quaternion algebras (via Carayol's description of supersingular points). As w... | 3 | https://mathoverflow.net/users/6121 | 57095 | 35,619 |
https://mathoverflow.net/questions/57072 | 35 | In an interview (at <http://www.alainconnes.org/docs/Inteng.pdf>) Connes remarks that
>
> I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point is that as soon as you have a non-standard number, you get a non-measurable set. And in Choquet’s circle, having we... | https://mathoverflow.net/users/10946 | A remark of Connes on non-standard analysis |
>
> ...as soon as you have a non-standard number, you get a non-measurable set.
>
>
>
Every nonstandard natural number $N$ gives rise to a nonprincipal ultrafilter $U$ on $\mathbb{N}$, by saying that a set $X\subset\mathbb{N}$ is in $U$ if and only if $N\in X^\*$, the nonstandard analogue of $X$. In other words,... | 65 | https://mathoverflow.net/users/1946 | 57108 | 35,624 |
https://mathoverflow.net/questions/57106 | 1 | Hi there,
I know there are fairly straightforward ways to write down the schemes of infinite dimensional projective spaces (not restricting myself to only countable dimensions), but what happens with infinite dimensional *general linear groups* ? Is there a well-known, standard way to define its group scheme, just as i... | https://mathoverflow.net/users/12884 | Group scheme of infinite dimensional linear groups ? | You can write an infinite dimensional general linear group as a filtered colimit of finite dimensional general linear groups, where the filtering is over finite subsets of an infinite basis, ordered by inclusion. This process produces an ind-scheme, since each inclusion is a closed embedding. The basis can be uncountab... | 3 | https://mathoverflow.net/users/121 | 57111 | 35,626 |
https://mathoverflow.net/questions/57103 | 2 | Waterhouse in his thesis (Abelian varieties over finite fields, Ann. scient. \'Ec. Norm. Sup., t. 2, 1969, p 521-560) seems to use without comments the following fact:
Let $k$ be a finite field, and let $A$, $B$ be two abelian varieties over $k$ that are $k$-isogenous. Consider the set $I(A,B)$ of all the $k$-isogeni... | https://mathoverflow.net/users/4800 | Do isogenies between AVs over finite fields separate finite subgroups? | Let me describe a natural straightforward generalization of Chris Wutrich's counterexample.
Let $B$ be a $g$-dimensional abelian variety over a field $k$ and assume that $End\_k(B)$ is a principal ideal domain. Let $A$ be another abelian variety over $k$ that is not $k$-isomorphic to $B$ but $k$-isogenous to it. Then... | 7 | https://mathoverflow.net/users/9658 | 57112 | 35,627 |
https://mathoverflow.net/questions/57062 | 4 | Hello, given graph $G=(V,E)$ with $n=|V|$ and $k=|E|$, what is the probability that it does not contain any cycle $C\_l$ for $l\geq3?$
**The requested clarification**:
My intention was to form the question in such a way, that there is no information about any distribution of the edges, and *n* and *k* are parameters.... | https://mathoverflow.net/users/13343 | Probability, that a graph G does not contain a cycle | I will assume the uniform distribution on all (labelled) graphs with $n$ vertices and $k$ edges. An acyclic graph on $n$ vertices has at most $n-1$ edges, so let me further assume $k< n$. More precisely, $k=n-c$, where $c$ is the number of connected components, i.e., the graph is acyclic iff it is union of $c$ disjoint... | 19 | https://mathoverflow.net/users/12705 | 57114 | 35,629 |
https://mathoverflow.net/questions/56315 | 5 | I'm solving the following Schroedinger equation in the domain $r>0$
$\psi''(r) + \left(E-\frac{a}{r^b}\right)\psi(r)=0 $,
where $0 < b < 2$ and $a, E$ are positive constants. Primarily I'm interested in the asymptotical power behavior of the solution as $r\to 0$. To be complete in the description of the problem, I ... | https://mathoverflow.net/users/5550 | Approximate analytic solution of Schroedinger equation with arbitrary power potential | As $r$ approaches zero, the coefficient of $\psi$ becomes dominated by the contribution of $-a/r^b$. This means that in the vicinity of zero your solution is dominated by solution of the following equation
$$
\psi''-\frac{a}{r^b}\psi=0.
$$
This can be demonstrated more rigorously (and, also, refined to the higher accur... | 4 | https://mathoverflow.net/users/8670 | 57117 | 35,631 |
https://mathoverflow.net/questions/57120 | 0 | Does anyone have an example of two spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?
is it possible?
| https://mathoverflow.net/users/10466 | Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings? | A very standard example would be $S^2\vee S^4$ and $\mathbf{C}P^2$.
| 8 | https://mathoverflow.net/users/10366 | 57121 | 35,633 |
https://mathoverflow.net/questions/57129 | 25 | Obvious necessary condition is that the center must be a cyclic group. Is it sufficient (doubt here)? If not, is there any nice characterization in terms of group structure, without appealing to representations?
| https://mathoverflow.net/users/4312 | Which finite groups have faithful complex irreducible representations? | A finite abelian group has a faithful irreducible representation if and only if it is cyclic. The case of finite groups was solved by Gaschütz in
W. Gaschütz, *Endliche Gruppen mit treuen absolut-irreduziblen Darstellungen.* Math. Nach. 12 (1954)
From Mathematical Reviews:
"In the present formulation the author c... | 28 | https://mathoverflow.net/users/8176 | 57133 | 35,640 |
https://mathoverflow.net/questions/57128 | 1 | Let $M$ denote a finite von Neumann algebra with trace $\tau$, and $L^{2}(M)$ denote the standard (trivial) M-M correspondence (binormal bimodule). The coarse correspondence is $L^{2}(M) \overline{\otimes} L^{2}(M)$ with commuting left and right actions of $M$ given by $x(\xi \otimes \eta)y=(x \xi) \otimes (\eta y)$ fo... | https://mathoverflow.net/users/6269 | How coarse is the coarse correspondence? | The commutant of the action of $M \otimes M^{op}$ on $L^2(M) \otimes\_2 L^2(M)$ is $M^{op} \bar \otimes M$ with the obvious action, whereas the commutant of the action on $L^2(M) \otimes\_2 L^2(M) \otimes\_2 L^2(M)$ is $M^{op} \bar \otimes B(L^2(M)) \bar \otimes M$ (again with the obvious action). The first algebra is ... | 1 | https://mathoverflow.net/users/8176 | 57135 | 35,642 |
https://mathoverflow.net/questions/57132 | 9 | In the same vein of [this](https://mathoverflow.net/questions/57120/spaces-which-have-the-same-homology-groups-the-same-cohomology-groups-but-have) MO question, one can ask:
>
> If two spaces $X$, $Y$ have isomorphic generalized cohomology rings $\mathrm{h}^{\bullet}(X)\cong \mathrm{h}^{\bullet}(Y)$ for *every* mul... | https://mathoverflow.net/users/4721 | Are generalized cohomology theories a set of complete homotopy invariants for spaces ? | No, they don't have to be homotopically equivalent. In fact:
* There is a map of CW-complexes $X \to Y$ which is an isomorphism on (co)homology, full stop, for every generalized (co)homology theory $h$, multiplicative or not.
* This is, in fact, equivalent to the map $X \to Y$ being an isomorphism on integral homolog... | 15 | https://mathoverflow.net/users/360 | 57136 | 35,643 |
https://mathoverflow.net/questions/53932 | 11 | As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}\_q$ is defined to be the invariant subalgebra of $SU\_q(N)$ under the coaction $\Delta\_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the comultiplication of $SU\_q(N)$, and $\pi: SU\_q(N) \to U\_q(N-1)$ is the Hopf algebra surjection defined by setting, f... | https://mathoverflow.net/users/12653 | Generators of the Odd Dimensional Quantum Spheres | This is shown in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. The result you ask for is Proposition 63 in Chapter 11. I'd expand more upon this but I have to give a talk shortly, and anyway it's all there in the book.
Edit: I guess I should say that the point here is the representation ... | 5 | https://mathoverflow.net/users/703 | 57147 | 35,651 |
https://mathoverflow.net/questions/57127 | 11 | What's the strategy for computing the Picard group of a variety with more than one irreducible components?
For instance, consider the simple case where $X$ has two components $C$ and $D$, meeting transversely at one point. Then it seems that $\text{Pic}(X)=\text{Pic}(C)\times\text{Pic}(D),$ but I don't know how to p... | https://mathoverflow.net/users/370 | Picard group of reducible varieties | I will only consider the case of connected projective (this is not really necessary) curves $X, C, D$. over an algebraically closed field $k$. The canonical injection $O\_X\to O\_C\times O\_D$ induces an exact sequence of sheaves on $X$
$$ 1 \to O\_X^\* \to O\_C^\* \times O\_D^\* \to F \to 1 $$
where $F$ is a skyscrap... | 16 | https://mathoverflow.net/users/3485 | 57153 | 35,656 |
https://mathoverflow.net/questions/57118 | 12 | Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V)\_{0}$
be the group of codimension-$j$ homologically trivial $K$-rational cycles on $V$, modulo rational equivalence. A conjecture of Beilinson and Bloch predicts that the dimension of $\mathrm{Ch}^j(V)\_{0} \otimes \mathbf{Q}$ as a $\mathbf{Q}$-vecto... | https://mathoverflow.net/users/1464 | Compatibility of Bloch-Kato and Beilinson-Bloch | The Bloch-Kato conjecture is actually more precise than that. As mentioned by Hunter Brooks, there is indeed an $\ell$-adic Abel-Jacobi map
$$\phi : Ch^j(V)\_0 \to H^1(G\_K,H^{2j-1}\_{\mathrm{et}}(V\times\_{K}\overline{K},\mathbf{Q}\_{\ell})(j))$$
The map $\phi$ is also called the cycle class map and is defined for... | 9 | https://mathoverflow.net/users/6506 | 57154 | 35,657 |
https://mathoverflow.net/questions/57079 | 4 | Consider $(M^{n},g)$ to be a Riemannian manifold and suppose that $X$ is a smooth non-trivial Killing vector field on $M$. Away from the zeros of $X$ we have a natural distribution $D$ of $(n-1)$-planes defined so that $D\_p$ is orthogonal to $X\_p$. If the distribution $D$ is (completely) integrable then it is straigh... | https://mathoverflow.net/users/26801 | Special Killing Vector Fields | I now think that my comment might indeed be the complete answer in the case when $X$ has no zeroes.
Guiseppe's answer has been a sort of Socratic catalyst.
Indeed, in that case the distribution $D$ defined by $\omega$ and $X^\flat$ agree. So $D$ is integrable if and only if the ideal generated by either $\omega$ o... | 2 | https://mathoverflow.net/users/394 | 57170 | 35,664 |
https://mathoverflow.net/questions/57140 | 0 | Let $a$, $b$, and $c$ be positive numbers. Consider the cubic equation $x^2(ax+b) = c$. Are there any useful bounds (upper and lower) I can put on the unique root of this equation for $x>0$? For example, we know that $x^\* <(c/a)^{1/3}$ and $x^\* < \sqrt{c/b}$. This equation can be solved explicitly, but that expressio... | https://mathoverflow.net/users/12263 | Approximate roots of a cubic | Note that, since your $f(x):=x^2(ax+b)-c$ is convex on $[0,+\infty)$ the [Newton's iteration](http://en.wikipedia.org/wiki/Newton_method) with initial point $x\_0>0$ produces a sequence $x\_n$ which is decreasing for $n\ge1$ and converges to $x^\*$.
From the equation, $y\_n:=c/ax\_n^2\, -\, b/a$ also converges to $x^\... | 6 | https://mathoverflow.net/users/6101 | 57171 | 35,665 |
https://mathoverflow.net/questions/57166 | 32 | I was wondering if there is any good software out there that allows you to do specific computations in algebraic topology. For example:
* Create a simplicial complex/set and ask questions about its homology, cohomology;
* Build manifolds using handle decompositions;
* Calculate homotopy limits, colimits.
Something ... | https://mathoverflow.net/users/4239 | Computational software in Algebraic Topology? | There are several programs that answer to your first demand whilst the others, as Ryan says, are a bit more vague. There are books written on computational homology (and its applications) for instance, see <http://chomp.rutgers.edu/> and the computational homology project. For simplicial complexes, the Plex routines wr... | 12 | https://mathoverflow.net/users/3502 | 57173 | 35,667 |
https://mathoverflow.net/questions/57101 | 5 | I should say that I'm not a category theorist or an abstract algebraist, so maybe this will be very pedestrian. I have the following, somewhat vague question:
>
> I have categories C and D, a forgetful functor $U:C\rightarrow D$. This has a left adjoint, but does not have a right adjoint. Are there other situations... | https://mathoverflow.net/users/406 | Looking for substitutes for co-free modules in a topological setting | I did say I would respond, so here are some thoughts. I'll repeat the caveat that I am not a functional analyst by any means.
There's something in category theory called the [Chu construction](http://ncatlab.org/nlab/show/Chu+construction), which has very nice categorical properties and which seems to be a natural p... | 5 | https://mathoverflow.net/users/2926 | 57179 | 35,669 |
https://mathoverflow.net/questions/57184 | 8 | Let $(\Omega,P,\mathcal{F})$ be a probability space with filtration $\mathbb{F} = (\mathcal{F}\_t), t \in [0,T]$, where $T$ can be finite or infinite. Let $M$ be a cadlag (local) martingale with respect to $\mathbb{F}$, and let $\mathbb{F}^M$ be the filtration generated by $M$ and then completed with respect to $P$.
... | https://mathoverflow.net/users/2310 | Filtrations generated by cadlag martingales. | No, that is not true. Consider the following, defined on a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}\_t\}\\_{t\in[0,T]},\mathbb{P})$.
1. $W$ is a standard Brownian motion.
2. $U$ is an $\mathcal{F}\_0$-measurable Bernoulli random variable independent of $W$, with $\mathbb{P}(U=0)=\mathbb{P}(U=1)=1... | 11 | https://mathoverflow.net/users/1004 | 57187 | 35,672 |
https://mathoverflow.net/questions/57186 | 13 | The classifying space of a group $G$ is given by taking a contractible space $E$ equipped with a free $G$-action, and looking at the quotient, which we dub $BG$. The homotopy type of this space (and thus its cohomology) depend only on $G$, and this gives us one definition of group cohomology.
Now, we can also look at... | https://mathoverflow.net/users/1703 | Why isn't the orbifold cohomology of $pt/G$ equal to the cohomology of $BG$? | Orbifold cohomology is not a model for the cohomology of the orbifold (or more generally stack) itself, but for that of its inertia stack (a.k.a. derived loop space), which parametrizes points of the stack together with automorphisms. The inertia of $BG$ is $G/G$ (which is also the correct homotopy type for the free lo... | 34 | https://mathoverflow.net/users/582 | 57191 | 35,674 |
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