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https://mathoverflow.net/questions/79132 | 2 | I'm trying to get the gist of the proof of the Weil conjectures. Let $X$ be a variety over $\mathbb{F}\_{p^n}$. A priori $Z(X,t)\in \mathbb{Q}[[t]]$. Due to the Grothendieck-Lefschetz fixed point theorem, $Z(X,t)=\prod P\_i(t)^{(-1)^{i+1}}$, where $P\_i(t)$ is the characteristic polynomial of the Frobenius acting on $H... | https://mathoverflow.net/users/5309 | Is the integrality of the zeta function easy? | It is not necessarily obvious that this suffices, though it is true (fortunately). You can find a proof in Milne's [Lectures on Etale Cohomology](http://jmilne.org/math/CourseNotes/lec.html). (I happened to just be looking at this while thinking about [this question](https://mathoverflow.net/questions/79115/in-what-way... | 12 | https://mathoverflow.net/users/6753 | 79133 | 47,628 |
https://mathoverflow.net/questions/79113 | 26 | I wondered whether there were an infinite number of
palindromic primes written in binary (11, 101, 111, 10001, 11111, 1001001, 1101011, ...)
and quickly discovered that it is unknown
([OEIS A117697](http://oeis.org/A117697)).
Indeed, even though almost all palindromes in any base are composite,
whether there are an inf... | https://mathoverflow.net/users/6094 | Why so difficult to prove infinitely many restricted primes? | To give a vague answer, I think these questions are difficult because they mix multiplicative conditions (being prime) and additive conditions (as in the twin prime case).
Basically all results about primes that I can think of come down to unique factorization of the integers. For example, the zeta function is given ... | 26 | https://mathoverflow.net/users/1050 | 79134 | 47,629 |
https://mathoverflow.net/questions/79084 | 8 | Let $X$ be a topological space (say a manifold). A result of R. Thom states that the pushforwards of fundamental classes of closed, smooth manifolds generate the rational homology of $X$. This work of Thom predates the development of bordism. Is there now a more elementary proof of this result that does not rely on spe... | https://mathoverflow.net/users/18632 | Representing rational homology by manifolds | A nice, direct combinatorial construction was given by Gaifullin, see his [papers](https://arxiv.org/abs/0712.1709) [on the](https://arxiv.org/abs/0806.3580) [arXiv](https://arxiv.org/abs/0912.3933) (equivalently: *[Explicit construction of manifolds realizing the prescribed homology classes](https://arxiv.org/abs/0712... | 10 | https://mathoverflow.net/users/10819 | 79140 | 47,633 |
https://mathoverflow.net/questions/79152 | 1 | Hello?
I have a simple question about combinatorial group theory.
For a group $G$, I saw $[G\_k, G\_m] \subset G\_{k+m}$ and these two subgroups need not be equal.
Then is there any known condition that they can be equal?
How about the case that $G$ is (f.g.) free?
Actually, my first question was that:
for a f.g. f... | https://mathoverflow.net/users/15728 | When $[G_k,G_m] = G_{k+m}$? | Presumably $G=G\_1 \ge G\_2 \ge G\_3 \ge \cdots$ is the lower central series of $G$?
So, for a free group $F$ of rank $r$, $F\_2$ = $[F,F]$, and $F/[F\_2,F\_2]$ is the free $r$-generator metabelian group.
$F\_m < [F\_2,F\_2]$ would imply that $F/[F\_2,F\_2]$ is nilpotent of class at most $m$, but since there are 2-... | 8 | https://mathoverflow.net/users/35840 | 79155 | 47,641 |
https://mathoverflow.net/questions/79166 | 4 | Let us consider the group $G:=\mathbb{Z}\_N^a$ (the product of the cyclic group with $N$ elements with itself $a$ times). Suppose we are given a number $m$ that divides $N$.
I would like to know how many elements $x$ in $G$ have the property that $(N/m)x$ has order precisely $m$ (and not any number dividing $m$).
F... | https://mathoverflow.net/users/18800 | Cardinality of the set of elements of fixed order. | Let $Ord(m)$ be the number of elements of order $m$ in $\mathbb{Z}\_N^a$, where $m|N$. A nice way to compute $Ord(m)$ is to start from the observation
$$\sum\_{m|N} Ord(m) = N^a$$
and to apply [Möbius inversion](http://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula):
$$Ord(m) = \sum\_{d|m} \mu(m/d)d^a.$$ ... | 7 | https://mathoverflow.net/users/2926 | 79171 | 47,647 |
https://mathoverflow.net/questions/79174 | 10 | The induction schema of Peano Arithmetic is standardly given as the universal closure of $\phi(0)\land \forall x (\phi(x)\rightarrow \phi(x+1)) \rightarrow \forall x\phi(x)$. However, since the language of arithmetic has a name for every standard number, it is not obvious (to a beginner like me) why parameters are nece... | https://mathoverflow.net/users/18805 | Parameters in arithmetic induction axiom schemas | The two theories are equivalent. To see this, let's assume that we have the parameter-free induction, and suppose that $\phi(x,y)$ is a formula with two free variables. Suppose we have a model $M$, satisfying the parameter-free induction, and there is a $b\in M$ such that $\phi(x,b)$ obeys the hypothesis of the inducti... | 7 | https://mathoverflow.net/users/1946 | 79179 | 47,649 |
https://mathoverflow.net/questions/79178 | 2 | My question concerns PL-homeomorphism and shellability.
I see from [a previous question here](https://mathoverflow.net/questions/62958/do-bistellar-flips-preserve-shellability) that bistellar flips do not preserve shellability. However, the following two results are corollaries of Pachner's Theorem:
1. Every simpli... | https://mathoverflow.net/users/18807 | How does Pachner's Theorem preserve shellability in simplicial spheres? | My understanding is that the proof of shellability uses Pachner moves in one dimension down: if $S$ is a $d$-dimensional simplicial PL-sphere, then its boundary can be partitioned into two $(d-1)$-dimensional balls (e.g. one of the simplices is a ball itself, and all of the other simplices also form a ball), there exis... | 3 | https://mathoverflow.net/users/440 | 79180 | 47,650 |
https://mathoverflow.net/questions/79182 | 1 | Recently I just learned the Kobayashi distance on complex manifolds and wants to get some feeling of how it looks like on exmaples of manifolds with positive Ricci curvature. I have a feeling that the Kobayashi distance on those manifolds should vanish since those are not very "hyperbolic".
A simple example is extend... | https://mathoverflow.net/users/12904 | How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature? | Positive Ricci curvature Kaehler manifolds are Fano, and therefore rationally connected. See Janos Kollar's book [Rational Curves on Algebraic Varieties](http://books.google.com/books?hl=en&lr=&id=oqW3GabJLjgC&oi=fnd&pg=PA1&dq=janos+kollar+rational+curves&ots=_a9ah5mhdV&sig=-xKWuKioqlb6J_U_L_bxP8Kd-Wg#v=onepage&q=janos... | 5 | https://mathoverflow.net/users/13268 | 79185 | 47,653 |
https://mathoverflow.net/questions/79177 | 1 | Let $f\_n \in C^2(\bar{\Omega})$ be a sequence satisfying
$\Delta f\_n - f\_n^3 \to 0 \ \ {\rm in} \ \ L^2(\Omega)$
where $\Omega \subset {\mathbb R}^2$ is bounded and open with a smooth boundary. Is it necessarily true that $\|f\_n^3\|\_{L^2(\Omega)}$ and $\|\Delta f\_n\|\_{L^2(\Omega)}$ are uniformly bounded? If ... | https://mathoverflow.net/users/18406 | $L^2$ boundeness of a sequence | A counterexample in one dimension: take $\Omega:=(0,1)$ and $f\_n(x):=\frac{\sqrt 2}{x+\frac{1}{n}}$. Then $f''\_n(x)-f\_n^3(x)=0$ while $\| f''\_n \| \_{2,\Omega}=\|f^3\_n\|\_{2,\Omega}=O(n^{5/2}) \, .$
| 3 | https://mathoverflow.net/users/6101 | 79191 | 47,656 |
https://mathoverflow.net/questions/79200 | 2 | The *orientation sheaf* of an $n$-manifold $M$ is $\mathcal{O}\_n=Sheaf(U\mapsto H\_n(M,M-U;\mathbb{Z}))$, with stalks given by $(\mathcal{O}\_n)\_x = lim H\_n(M,M-U)=H\_n(M,M-x)=\mathbb{Z}$ (the limit is over neighborhoods $U$ containing $x\in M$).
Suppose $M$ is orientable and closed, so that $H\_n(M)=\mathbb{Z}$. ... | https://mathoverflow.net/users/12310 | Orientation Sheaf and Double Cover | An elaboration.
$\mathcal O\_n$ is a bundle over $M$ with fiber $\mathbb Z$. There is an action of the integers on it, because the integers act on homology. Earlier I said this was a principal bundle, I was too tired! The action is of course not free. In particular, this bundle $\mathcal O\_n \to M$ has a section.
... | 3 | https://mathoverflow.net/users/1465 | 79210 | 47,662 |
https://mathoverflow.net/questions/79215 | 9 | **Note:** By an "analytic non-algebraic" surface below I mean a two dimensional compact analytic variety $X$ (over $\mathbb{C}$) which is not an algebraic variety.
A property of Nagata's example (see the end of the post for the construction) of a non-algebraic normal analytic surface $X$ is the following:
($\sta... | https://mathoverflow.net/users/1508 | Pathologies of analytic (non-algebraic) varieties. | The answer is **no**.
A counterexample is provided by the so-called *Hopf surfaces* (they were actually constructed by Kodaira, see Donu Arapura's comment).
A Hopf surface of type $\alpha=(\alpha\_1, \, \alpha\_2)$, where $0 < |\alpha\_1| \leq |\alpha\_2| < 1$, is the compact complex surface $H\_{\alpha}$ obtained... | 13 | https://mathoverflow.net/users/7460 | 79216 | 47,665 |
https://mathoverflow.net/questions/79217 | 38 | How to find an integer part of $10^{10^{10^{10^{10^{-10^{10}}}}}}$? It looks like it is slightly above $10^{10^{10}}$.
| https://mathoverflow.net/users/9550 | How to calculate [10^10^10^10^10^-10^10]? | I think the number in question is $10^{10^{10}}+10^{11}\ln^4(10)$ plus a tiny positive number. That is, it starts with a digit $1$, followed by $10^{10}-13$ zeros, then by the string $2811012357389$, then a decimal point, and then some garbage (which starts like $4407116278\dots$).
To see this let $x:=10^{-10^{10}}$,... | 81 | https://mathoverflow.net/users/11919 | 79219 | 47,666 |
https://mathoverflow.net/questions/79197 | 9 | As is well known, the group $PSL(2,\mathbb Z)$ is isomorphic to the free product $C\_2 \ast C\_3$ of cyclic groups of order $2$ and $3$. Call the generators of the cyclic groups $S$ and $T$.
>
> **Problem:** Given a prime number $p$ and a natural number $n$, write a presentation of the quotient $PSL(2, \mathbb Z/p^... | https://mathoverflow.net/users/2631 | Presentations of PSL(2, Z/p^n) | I usually use [Sunday's](http://dx.doi.org/10.4153/CJM-1972-118-x) presentation: see [MR0311782](https://mathscinet.ams.org/mathscinet-getitem?mr=0311782). His T has order 2 but your S will be what he denotes ST.
| 7 | https://mathoverflow.net/users/6355 | 79221 | 47,668 |
https://mathoverflow.net/questions/79229 | 6 | I've been reading Hatcher's survey "A Short Exposition of the Madsen-Weiss Theorem". In it, he discusses the Barratt-Priddy-Quillen theorem, which says that the homology of the infinite symmetric group is the same as the homology of one component of $\Omega^{\infty} S^{\infty}$. My question : what space exactly is bein... | https://mathoverflow.net/users/18822 | What space does "Loop infinity of the infinite dimensional sphere" refer to? | Your first statement is correct: it is taking a colimit of $\Omega^n S^n$. The inclusion from one to the next is called the "suspension" map. The points of $\Omega^n S^n$ are basepoint-preserving functions $S^n \to S^n$, and the suspension map takes such a function $f$ and sends it to $f \wedge id: S^n \wedge S^1 \to S... | 15 | https://mathoverflow.net/users/360 | 79235 | 47,678 |
https://mathoverflow.net/questions/79239 | 3 | Indeed I am now trying to read a series of papers written by Einsiedler, Lindenstrauss, Michel and Venkatesh that study distribution of periodic torus orbits on homogeneous spaces. They make heavy use of automorphic L-functions, and Eisenstein series on adelic homogeneous spaces, but I almost know nothing about that.
A... | https://mathoverflow.net/users/18823 | Looking for concise books on automorphic L-functions, Eisenstein series on adelic homogeneous spaces | This is not exactly what you've asked for, but I'll address this article directly, because it is not related to automorphic L-functions "directly" but more to homogeneous dynamics.
You can actually read it with minimal knowledge about those stuff, if you believe some "black boxes", or some technical gadgets.
First of... | 4 | https://mathoverflow.net/users/8857 | 79243 | 47,681 |
https://mathoverflow.net/questions/74888 | 3 | For a quasihereditary algebra $A$, we have a partially ordered set $\Lambda$ parameterizing the simples $L(\lambda)$/projectives indecomposables $P(\lambda)$. It also parameterize a set of special modules called the standard modules $\Delta(\lambda)$ such that $\Delta(\lambda) \to P(\lambda)$ is a surjection such that ... | https://mathoverflow.net/users/1041 | Is Ext algebra of standard modules of quasihereditary algebras directed? | Everything you need is written here
<http://www.math.uni-bielefeld.de/~sek/select/K-L.pdf>
(see Lemma 3). I couldn't find the originial source, but it must be one of the references listed.
Let $B=\text{Ext}\_A^\ast (\Delta,\Delta)$ with idempotents $e\_i=\text{id}\_{\Delta(i)}$. Then from Lemma 3 it follows that
... | 2 | https://mathoverflow.net/users/18756 | 79246 | 47,683 |
https://mathoverflow.net/questions/79250 | 0 | Hello.
Yesterday we proofed, that $\mathbb{R}$ is not quasi-isometric to $\mathbb{R}^2$ (both endowed with the standard Euclidean metric).
Step 1.: $\mathbb{R}$ is q.i. $\mathbb{Z}$ and $\mathbb{R}^2$ is q.i. to $\mathbb{Z}^2$, so we only need to show $\mathbb{Z}$ is q.i. to $\mathbb{Z}^2$.
This is clear.
Step 2... | https://mathoverflow.net/users/17255 | Question about the proof of the fact that IR is not quasi-isomtric to IR^2 | 1. It is the circle, or the elements of $\mathbb{Z}^2$ in that circle.
2. It is the union of the circles around the points of $f(N\_{\lambda(r+C)}(0))$.
3. This is because $f$ is a map and therefore there are less or equal points $f(x)$ than $x$.
| 1 | https://mathoverflow.net/users/15887 | 79251 | 47,687 |
https://mathoverflow.net/questions/76629 | 3 | I found the following definition of domain of holomorphy in several places.
Def1: A connected open set $\Omega$ in the n-dimensional complex space ${\mathbb{C}}^n$ is called a domain of holomorphy if there do not exist non-empty open sets $U \subset \Omega$ and $V \subset {\mathbb{C}}^n$ where $V$ is connected, $V \... | https://mathoverflow.net/users/11395 | Domain of Holomorphy | Further to Ben's answer, it might be useful to picture the situation in $\mathbb{C}$. (Of course in $\mathbb{C}$ every domain is a domain of holomorphy, but we can still exhibit the same phenomenon that causes us to need the more complicated definition.)
The principal branch of the logarithm $f := \operatorname{Log}$... | 5 | https://mathoverflow.net/users/3651 | 79252 | 47,688 |
https://mathoverflow.net/questions/79253 | 5 | It is sometimes emphasized that a "concrete category" is not a property of a category $C$, but rather a structure, i.e. a faithful functor from $C$ to $Set$. Thus, When people talk about a concrete category $C$ they *really* mean $C$ together with some implicitly defined and naturally understood functor from $C$ to $Se... | https://mathoverflow.net/users/14379 | two essentially different concretizaions | One of course has many examples. To define the terms, a *concretization* of a category $C$ should be a faithful functor $C \to \mathrm{Set}$. In examples, concretizing functors are never full, so I will not ask this, but I do not mind asking, say, that the concretizing functor reflect isomorphisms. Maybe you want moreo... | 7 | https://mathoverflow.net/users/78 | 79259 | 47,691 |
https://mathoverflow.net/questions/79256 | 3 | **Short version:** has anyone done geometry on something that is the formal filtered colimit of Frechet manifolds?
**Longer version:** A colleague and I came up with a concept today that seems like we require generalising smooth Frechet manifolds to something that is a filtered colimit of such manifolds: Ind-Frechet ... | https://mathoverflow.net/users/4177 | Ind-Frechet manifolds? | **Short version:** Yes, me.
**Slightly Longer Version:**
*Added in Edit*: What I write below is concerned with the space of *continuous piecewise-smooth* paths or loops. That is, continuous maps which are smooth except on a finite subset of the interval (or circle). Dropping the continuity requirement is only a co... | 7 | https://mathoverflow.net/users/45 | 79263 | 47,694 |
https://mathoverflow.net/questions/79269 | 1 | Consider a $p$-adic field $K$ with the standard topology inherited from the usual $p$-adic norm $\mid \cdot \mid$. Consider the ultrametric space $X=K^n$ with the topology inherited from the norm $\| \cdot \|$ defined as $\|x\|=\max\_{i=1}^n (|x\_1|,\dots,|x\_n|)$ with $x=(x\_1,\dots,x\_n) \in K^n$. Now we have two que... | https://mathoverflow.net/users/nan | Partitioning a compact open set into balls in an ultrametric space | The answers are yes and yes.
1. is true because the sets $U\_{\epsilon} = \{(x\_1,\dots,x\_n),\
|x\_i| \leq \epsilon\}$ for a basis of neighborhood of $0$ in $X$ for your norm by definition,
but also a basis of neighborhood of $X$ for the product topology since any neighborhood of $0$
in the product topology contain... | 2 | https://mathoverflow.net/users/9317 | 79274 | 47,696 |
https://mathoverflow.net/questions/79276 | 3 | Recall that in a commutative ring $A$ an ordered pair of elements (a,b) is said to form a regular sequence if the ideal $\langle a,b\rangle $ is strictly included in $A$ ,if $a$ is not a zero-divisor in $A$ and if the class of $b$ is not a zero-divisor of $A/\langle a\rangle$.
A friend of mine has asked me if in tha... | https://mathoverflow.net/users/450 | Does a regular pair of elements in a noetherian domain remain regular if their order is switched? | The only part to be shown is that $a$ is not a zero-divisor on $A/(b)$. Consider some $s\in A$ such that $as\in(b)$, say $as=bt$. Since $b$ is not a zero-divisor on $A/(a)$, we conclude that $t$ maps to zero in $A/(a)$, i.e., $t=au$. Since $A$ is a domain, it follows that $s=bu$, so $s$ maps to zero in $A/(b)$.
| 8 | https://mathoverflow.net/users/2035 | 79280 | 47,700 |
https://mathoverflow.net/questions/79282 | 1 | Does anyone know groups which admit presentations with two more generators than relators and are not residually finite? If so, do we know anything about the finite residual of such groups?
Any examples of finitely presented or finitely generated groups which are not residually finite but whose inner automorphism grou... | https://mathoverflow.net/users/18834 | Non-residually finite groups | P. Deligne has constructed non-residually finite central extensions of some arithmetic groups, so for these the inner automorphism group is residually finite, see: P. Deligne. Extensions centrales non r ́esiduellement finies de groupes arithm ́etiques. CR Acad. Sci. Paris, s ́erie A-B, 287, 203–208, 1978.
See also th... | 10 | https://mathoverflow.net/users/14497 | 79283 | 47,701 |
https://mathoverflow.net/questions/79264 | 1 | Let $k$ be a field, and $R$ a $k$-algebra. Suppose that $R\_{\mathfrak{p}}$ is a finitely generated $k$-algebra for all prime ideals $\mathfrak{p}$ of $R$. Does this imply that $R$ is also a finitely generated $k$-algebra? (I think this is false, but couldn't find a counterexample.)
| https://mathoverflow.net/users/18830 | If all localizations of an algebra at primes are of finite type over a field | Take any compact Hausdorff space $X$ and let $R$ consist of the locally constant functions $X\to k$. Then, $R$ is a $k$-algebra. Its prime ideals are all of the form $\mathfrak{p}=\left\{f\in R\colon f(x)=0\right\}$ for $x\in R$, so $R\_{\mathfrak{p}}\cong k$ is trivially finitely generated as a $k$-algebra. If $X$ has... | 4 | https://mathoverflow.net/users/1004 | 79284 | 47,702 |
https://mathoverflow.net/questions/78939 | 28 | I'm trying to read Quillen's paper "Rational homotopy theory" and am a little confused about the construction. As I understand, he associates a dg-Lie algebra over $\mathbb{Q}$ to every 1-reduced simplicial set via a somewhat long series of Quillen equivalences. But the construction that I had heard before makes spaces... | https://mathoverflow.net/users/344 | Two questions on rational homotopy theory | I'm not sure if this will still be helpful, but here is my understanding of the Quillen model. Everything correct that I write below, I learned from John Francis. (Probably in the same lecture that Theo mentioned in his comment above.) Any mistakes are not his fault---more likely an error in my understanding.
**Befor... | 35 | https://mathoverflow.net/users/3593 | 79296 | 47,707 |
https://mathoverflow.net/questions/79293 | 9 | It is known that the rational homotopy theory of spaces (e.g. simplicial sets) is equivalent in some sense to the homotopy theory of cdgas over $\mathbb{Q}$. This has been expressed in various forms in the literature. For instance, Felix-Halperin-Thomas show that
homotopy classes of maps between simply connected ratio... | https://mathoverflow.net/users/344 | Is the polynomial de Rham functor a Quillen equivalence? | The answer to question 1 is no. For one thing, the functor to the category of commutative DGAs is not essentially surjective. For any space $X$, $H^0(X)$ is a product of copies of $\mathbb Q$, whereas it is relatively easy to rig up CDGAs that don't have this property.
More seriously, nonnegatively (cohomologically!)... | 7 | https://mathoverflow.net/users/360 | 79297 | 47,708 |
https://mathoverflow.net/questions/79295 | 3 | Just for curiosity I have done a quick web-search and I have seen that some people are studying manifolds with amenable fundamental group. On the other hand, any finitely presented group and then, in particular, Thompson's group arises as the fundamental group of a compact 4-manifold. So I have a series of questions:
... | https://mathoverflow.net/users/13809 | Amenability of Thompson's group looking at a 4-manifold having it as the fundamental group | Thompson group has a Wirtinger-like presentation $x\_i^{x\_j}=x\_{i+1}, 0\le i < j\le 4$, a $C$-group in the terminology of Kuzʹmin, Yu. V. Groups of knotted compact surfaces, and central extensions. Mat. Sb. 187 (1996), no. 2, 81--102; translation in Sb. Math. 187 (1996), no. 2, 237–257 . Hence by a result of Kuzmin (... | 8 | https://mathoverflow.net/users/nan | 79300 | 47,709 |
https://mathoverflow.net/questions/79299 | 0 | For S ,any given c.e.set,does there exist a M (integer) and a partially computable function outputing every element of S the c.e.set ,such that $\forall x\in S,\exists n x=f(n)$ and $x=f(n)\leq M^n$?.
| https://mathoverflow.net/users/14024 | any given c.e.set has number M whose power bounds the corresponding elements of S? | Yes, even with $M=2$. Start with any partial recursive function that enumerates $S$ and "slow it down" so that it won't produce output $x$ until after $\log\_2 x$ steps. With more delay, you can ensure, for example, that $f(n)\leq n$ for all $n$.
More formally, if $S=\{x:(\exists y)\ R(x,y)\}$, and if $\langle,\rang... | 1 | https://mathoverflow.net/users/6794 | 79303 | 47,710 |
https://mathoverflow.net/questions/79298 | 13 | For forcing notions $\mathbb{P}$ and $\mathbb{Q}$ let us write $\mathbb{P}\triangleleft\mathbb{Q}$ if forcing with $\mathbb{Q}$ always adds a $\mathbb{P}$-generic filter over $V$. In other words, $\mathbb{P}\triangleleft\mathbb{Q}$ holds if there is a $\mathbb{Q}$-name $\tau$ such that $\tau[H]$ is a $\mathbb{P}$-gener... | https://mathoverflow.net/users/2436 | Cantor-Bernstein for notions of forcing | Here is a counterexample, which works in ZFC without any
additional large cardinal or other extra hypothesis. This
argument, which verifies the guess I made in my original answer, is the result of a conversation I had with Arthur
Apter.
The example involves the forcing $\mathbb{S}$ to add a
stationary non-reflecting ... | 9 | https://mathoverflow.net/users/1946 | 79323 | 47,721 |
https://mathoverflow.net/questions/79317 | 7 | This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. Start with two finite groups $A,B$ and their product $G:=A \times B$, working over a splitting field $K$ for the groups involved... | https://mathoverflow.net/users/4231 | Reference for projective covers of direct products of finite groups? | The paper Representations of direct products of finite groups.
Burton Fein
Source: Pacific J. Math. Volume 20, Number 1 (1967), 45-58.
[Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-20/issue-1/Representations-of-direct-products-of-finite-groups/pjm/1102992967.full)
has what you are ... | 7 | https://mathoverflow.net/users/15934 | 79325 | 47,723 |
https://mathoverflow.net/questions/79267 | 1 | If we need to find vector in R^n which is orthogonal to given (n-1) vectors,
this is basically solving linear system of equations and can be done in O(n^3) operation.
I wonder is there some simplification to do it if it is additionally known that vectors
are v\_i are orthonormal ?
Probably NO.
But may be I am mis... | https://mathoverflow.net/users/10446 | Find vector in R^n which is orthogonal to given (n-1) vectors v_i under condition v_i are orthonormal. | You want to find the last row of an orthogonal matrix given $n-1$ rows, right? Since the sum of squares in each column is $1$, you can find the absolute values of the entries in about $n^2$ operations. So, it remains to determine the sign pattern for non-zero entries. That can be also done quickly: add two columns with... | 14 | https://mathoverflow.net/users/1131 | 79339 | 47,733 |
https://mathoverflow.net/questions/79342 | 24 | The answer of following classical problem is surely known, but I can't find a reference
>
> For which positive integer $n$ is the set $S\_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) determined by congruences?
>
>
>
A set of prime $S$ is said *determined by congruences* if there is a positive integ... | https://mathoverflow.net/users/9317 | Primes of the form $x^2+ny^2$ and congruences. | You want the idoneal numbers,
<http://oeis.org/A000926> and <http://en.wikipedia.org/wiki/Idoneal_number>
See also pages 81-82 in Duncan A. Buell, *Binary Quadratic Forms*
Depending on what you mean by 32, the primes represented by $x^2 + 8 y^2$ are, in fact, given by congruences. However, half of those same primes... | 16 | https://mathoverflow.net/users/3324 | 79343 | 47,734 |
https://mathoverflow.net/questions/79158 | 3 | In corollary 4.3.13 of Rosenberg's book Algebraic K-theory and its applications, it's proved that $K\_2(F)=0$ if $F$ is a finite field.
The last sentence in the proof says that the central extension $\varphi:St(F)\rightarrow SL(F)$ is trivial over $N(F)$, then it is concluded that p-primary part of $K\_2(F)$ vanishes... | https://mathoverflow.net/users/8932 | A question about a proof in Rosenberg's algebraic K-theory book | In the following all references refer to Rosenberg's book.
Let $U(n,F) \le Sl(n,F)$ be the group of upper triangular matrices with 1's in the diagonal and let $N(n,F) \le St(n,F)$ be the group in the proof of 4.2.3. Furthermore let $K\_2(n,F)$ be the kernel of $\varphi\_n: St(n,F) \to Sl(n,F)$ and denote by $E \le S... | 6 | https://mathoverflow.net/users/10194 | 79344 | 47,735 |
https://mathoverflow.net/questions/79245 | 1 | Suppose there is a function $f:\mathbb{R}\_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not?
For example, there is a function $f(x,y)=-x^2+2y\log(1+x)-y\log(y),x>0,y>0$. How can we prove that it is bounded above (as indicated in the simulation results)?
| https://mathoverflow.net/users/18827 | How to determine whether a multivariate function is bounded or not | There isn't any general method. What you usually need to do in practice is like Robert illustrated. The function is bounded on compact sets on which the function is continuous, so you need to focus on the neighbourhoods of points of non-continuity and on trajectories going to infinity. Note that it is easy to make a co... | 2 | https://mathoverflow.net/users/9025 | 79347 | 47,738 |
https://mathoverflow.net/questions/79273 | 1 | Let $X$ be a smooth complete variety over an algebraically closed field of dimension $\geq3$. Given a divisor $D\_1$ on $X$ with $D\_1 \cdot C>0$ for every curve $C \subset X$, and a divisor $D\_2$ on $X$ satisfying $D\_2^2 \cdot S>0$ for every surface $S \subset X$, does there exist a divisor $D$ on $X$ satisfying $D ... | https://mathoverflow.net/users/11661 | Intersection positivity for curves and surfaces | I am not sure whether your original question is true, but it seems to me that your proposed solution does not work. Here is why:
The main problem is that you know very little about $D\_2$. Given what we know, it does not even have to be effective!
**Example** Let $X$ be a smooth complete variety with divisors $A,B$... | 3 | https://mathoverflow.net/users/10076 | 79359 | 47,743 |
https://mathoverflow.net/questions/79356 | 5 | What is the difference between regular singularities and logarithmic singularities? Could someone give me a reference where the distinction is clearly explained? I apologize in advance if this question is very trivial (and inappropriate for the level of MO), but I am a bit confused because I feel, in some papers, peopl... | https://mathoverflow.net/users/18848 | regular singularities and logarithmic singularities | My guess is that the OP is thinking of singularities of integrable connections rather than singularities of varieties. If so logarithmic singularities usually refer to solutions of $y'=1/x$ whose (many-valued) solution is exactly $\log x$. It may possibly also refer to higher order equations whose solutions are powers ... | 6 | https://mathoverflow.net/users/4008 | 79362 | 47,745 |
https://mathoverflow.net/questions/79350 | 8 | Hi,
Let $G$ be a reductive, connected group, $T$ a maximal torus, and $B$ a Borel subgroup containing $T$ with unipotent radical $U$. Then it turns out that the functions on the algebraic variety $G/U$ give a representation of $G$ where each irreducible representation appears exactly once. Geometrically, $G/U$ is a $... | https://mathoverflow.net/users/10580 | T-bundles and the Borel-Weil-Bott theorem | I'm very skeptical about the possibility of getting the full Borel–Weil–Bott theorem just by studying $G/U \to G/B$. Probably the closest thing I can think of is Bott's original proof of his theorem, which involves studying certain $\mathbb P^1$-bundles $G/B \to G/P$. On the other hand, you can prove the Borel–Weil the... | 11 | https://mathoverflow.net/users/430 | 79364 | 47,747 |
https://mathoverflow.net/questions/79373 | 4 | Consider the following Markov process: Start with an integer $N = N\_0$. Now repeatedly choose an $N\_i$ uniformly at random in the range $[1...N\_{i-1}]$ until $N\_i = 1$ at which point one terminates the process. This produces a nonincreasing integer sequence $\{N\_0,N\_1,\ldots,N\_{k-1},N\_k = 1\}$.
Experimental e... | https://mathoverflow.net/users/12301 | Some questions concerning a random number process | For the main question, your process is "roughly" the same as the one starting at $N\_0=N$, and defined by $N\_{i+1} = U\_{i+1} N\_{i}$, where $U\_i$ are iid uniform in $[0,1]$. I guess that the "roughly" can be easily made more precise.
For this modified process, your question becomes: what is $k(N)$, the smallest $k... | 9 | https://mathoverflow.net/users/10265 | 79381 | 47,751 |
https://mathoverflow.net/questions/79376 | 2 | I would like to know why for a smooth projective variety $X$ over an algebraically closed field $k$, numerical and homological equivalence coincide for divisors. Here by homological equivalence I mean that we have chosen a Weil cohomology theory with coefficients in a field $L$, in particular, there is no torsion.
Wh... | https://mathoverflow.net/users/11395 | Reference for Numerical vs Homological equivalence | There's a quick proof in Yves André's book "Une introduction aux motifs" (proposition 3.4.6.1).
Note that a stronger result is true : actually, algebraic equivalence coincides with numerical equivalence for divisors on $X$ (\*)("Matsusaka's theorem"). The only reference I know for this result is Matsusaka's original... | 3 | https://mathoverflow.net/users/12336 | 79382 | 47,752 |
https://mathoverflow.net/questions/79366 | 25 | Peter Sarnak believes that integer factorization is in $P$. It is a well-known open problem in TCS to identify the real complexity class of integer factorization. Take a look at this link for Peter Sarnak's lectures where he mentions that he does [not believe factoring is not in $P$](http://www.austms.org.au/tiki-downl... | https://mathoverflow.net/users/16007 | Evidence for integer factorization is in $P$ | I don't think there is any compelling evidence that integer factorization can be done in polynomial time. It's true that polynomial factoring can be, but lots of things are much easier for polynomials than for integers, and I see no reason to believe these rings must always have the same computational complexity. (Stra... | 43 | https://mathoverflow.net/users/4720 | 79389 | 47,755 |
https://mathoverflow.net/questions/79388 | 6 | Let $P$ be a set of $n$ points.
Assuming I know the pairwise distances for each pair of points.
What would be the minimum dimension of the space in which I could place those $n$ points with respect to the different pairwise distances.
The idea would be to set a first point at random coordinates in a multi-dimensional... | https://mathoverflow.net/users/18857 | Minimum space dimension to place n-points knowing pairwise distances | For the basic result, start with
[Wikipedia](https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma)
or Google "Johnson-Lindenstrauss Lemma".
| 13 | https://mathoverflow.net/users/2554 | 79395 | 47,758 |
https://mathoverflow.net/questions/79393 | 4 | Suppse $n > m$ are positive integers. Then it is an elementary exercise to show that the number of $m$-tuples of non-negative integers $(x\_1, \cdots, x\_m)$ such that $x\_1 + \cdots + x\_m = n$ (order matters) is $\binom{n+m-1}{m-1}$. My question is a variant of this, where instead of considering the sum $x\_1 + \cdot... | https://mathoverflow.net/users/10898 | A weighted sum of non-negative integers | The quick and dirty way is to regard the weighted sum as counting the number of lattice points $(x\_2, ... x\_m)$ satisfying $w\_2 x\_2 + ... + w\_m x\_m \le n$ and $w\_1 | (n - w\_2 x\_2 - ... - w\_m x\_m)$. The second condition is satisfied $\frac{1}{w\_1}$ of the time. The number of lattice points satisfying the fir... | 10 | https://mathoverflow.net/users/290 | 79396 | 47,759 |
https://mathoverflow.net/questions/79399 | 7 | Take a rational function of a single complex variable. View it as a continuous function from the Riemann sphere to itself. Is there a nice way to compute which element of $\pi\_2(S^2)$ this corresponds to?
| https://mathoverflow.net/users/18858 | What element of $\pi_2(S^2)$ do rational functions represent? | It's the cardinality of the preimage of a generic point, because generically the local degree of a complex analytic function is always +1. If the rational function is $a(x)/b(x)$, then the number of solutions in $x$ to $a(x)/b(x) = y$ for generic $y$ is $\max(\deg a,\deg b)$.
| 16 | https://mathoverflow.net/users/1450 | 79400 | 47,761 |
https://mathoverflow.net/questions/79394 | 5 | Let $\kappa$ be an infinite cardinal and let $B$ be the random forcing for adding $\kappa$-many random reals.
**Question:** What are the elements of $B$. More precisely given a condition $p \in B$, what are the domain and the range of $p$, if there are any?. What does it mean "a coordinate of $p$"?.
| https://mathoverflow.net/users/11115 | Random real forcing | There are different ways of representing $B$. One possibility is to let $B$ consist of all closed subsets of $2^\kappa$ of positive measure; another possibility is to allow all positive Borel sets. Two conditions are equivalent if their symmetric difference has measure zero.
Each basic clopen set has a finite set of... | 4 | https://mathoverflow.net/users/14915 | 79402 | 47,763 |
https://mathoverflow.net/questions/79380 | 2 | Let $X$, $Y$ be Banach spaces, and $T\colon X\to Y$ be linear and bijective ($D(T)=X, R(T)=Y)$. Can one infer directly that $T$ is continuous? If not, is there a counterexample?
| https://mathoverflow.net/users/11291 | linear bijective operator | The question was simply answered in the comments. If you choose algebraic bases (which are also called Hamel bases) for $X$ and $Y$, then it is elementary to make a bijective linear operator which is unbounded and therefore discontinuous. You only have to check that the algebraic bases have the same cardinality, but no... | 2 | https://mathoverflow.net/users/1450 | 79404 | 47,764 |
https://mathoverflow.net/questions/79390 | 8 | Suppose I have a grouplike $A\_{\infty}$-space $G$ that carries an *additional structure* as a topological group (which does not coincide with the $H$-space structure of the $A\_{\infty}$-part). Denote the $H$-space multiplication by $\*$ and the group multiplication by $\cdot$ and suppose that
$$
(a\*b) \cdot (c \*d)... | https://mathoverflow.net/users/3995 | Eckmann-Hilton for $A_{\infty}$-spaces? | EDIT: Here is a *counterexample* to the stated question.
I'm going to start with a topological group $G$ which is a product of Eilenberg-Mac Lane spaces. Specifically, I'm going to choose $G \simeq K(\mathbb{Z},2) \times K(\mathbb{Z},7)$. The Dold-Thom theorem allows me to actually construct this as an abelian topolo... | 15 | https://mathoverflow.net/users/360 | 79408 | 47,767 |
https://mathoverflow.net/questions/79397 | 0 |
>
> **Possible Duplicate:**
>
> [What is the equivariant cohomology of a group acting on itself by conjugation?](https://mathoverflow.net/questions/20671/what-is-the-equivariant-cohomology-of-a-group-acting-on-itself-by-conjugation)
>
>
>
Let $G$ be a compact Lie group. Where can one read about the equivaria... | https://mathoverflow.net/users/18632 | Reference request for equivariant cohomology of G | This question has already been asked (and answered) here
[What is the equivariant cohomology of a group acting on itself by conjugation?](https://mathoverflow.net/questions/20671/what-is-the-equivariant-cohomology-of-a-group-acting-on-itself-by-conjugation)
| 1 | https://mathoverflow.net/users/3891 | 79413 | 47,770 |
https://mathoverflow.net/questions/79374 | 4 | I hope someone can help me, although this question is rather specific.
I am reading John Roe's chapter on Getzler symbols in "Elliptic operators, topology and asymptotic methods" to understand the proof of the Atiyah-Singer index theorem.
* So, for the differential operators on functions of $M$ (say, $D(M)$), there... | https://mathoverflow.net/users/16702 | Symbol map in Getzler calculus | Actually you can imitate the Getzler calculus using your symbol map $D(\Sigma M) \to C(TM) \otimes \bigwedge TM$ and you will still prove an interesting theorem, just not the Atiyah-Singer index theorem. In fact the theorem you will prove is basically Weyl's asymptotic formula for the eigenvalues of the Laplacian. The ... | 4 | https://mathoverflow.net/users/4362 | 79416 | 47,773 |
https://mathoverflow.net/questions/79417 | 4 | I am reading Helgason's book. In Chapter 3 he proved the existence of
Cartan subalgebra for a semisimple Lie algebra $\mathfrak g$
(definition: a Cartan subalgebra
is a maximal abelian subalgebra all whose element $H$ satisfies $\text{ad}\_H$
is semisimple).
It seems to me the proof is quick: if $H\in {\mathfrak... | https://mathoverflow.net/users/18863 | Existence of Cartan subalgebra | This argument does not work because the Killing form is not generally positive definite, so the orthogonal of a subspace wrt the Killing form is not necessary a complement of the subspace.
| 12 | https://mathoverflow.net/users/2481 | 79421 | 47,775 |
https://mathoverflow.net/questions/79423 | 2 | Is there a quick formula to find the ratios; $\displaystyle\frac{r(k)}{rtotal}$ for $k=1 \dots n$ without calculating the numerator and the denominator where;
$\displaystyle r(k) = \sum\limits\_{i=1}^{n} {{n^2 \choose (k-1)\*n+i}}$ and $\displaystyle rtotal = \sum\limits\_{k=0}^{n} {r(k)}$.
I know the denominator ... | https://mathoverflow.net/users/10062 | Sum of combinations | If you are interested in approximations to your ratios, you may find the accepted answer (and some comments of mine) to this MathOverflow post useful: [Sum of 'the first k' binomial coefficients for fixed n](https://mathoverflow.net/questions/17202/sum-of-the-first-k-binomial-coefficients-for-fixed-n) .
Essentially,... | 2 | https://mathoverflow.net/users/3402 | 79425 | 47,777 |
https://mathoverflow.net/questions/79383 | 8 | I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson metric $h$, show that it is Kahler, and obtain results on the holomorphic sectional curvature by heroic calculations.
Fr... | https://mathoverflow.net/users/4054 | Calculating a curvature tensor by polarization | The explicit polarization formula is the following, taken from [this paper of Bishop and Goldberg](http://www.ams.org/journals/tran/1964-112-03/S0002-9947-1964-0163271-8/home.html).
Working with real tangent vectors (instead of $(1,0)$ vectors, but it's easy to switch from one point of view to the other) the holomor... | 10 | https://mathoverflow.net/users/13168 | 79429 | 47,778 |
https://mathoverflow.net/questions/49388 | 57 | If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The existence of a non-zero linear functional can be shown by taking a basis of $V$ and specifying the values of the functio... | https://mathoverflow.net/users/932 | Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice? | To add the proof for my claim in Todd's answer, which essentially repeats Läuchli's original [1] arguments with minor modifications (and the addition that the resulted model satisfies $DC\_\kappa$).
We will show that it is consistent to have a model in which $DC\_\kappa$ holds, and there is a vector space over $\math... | 19 | https://mathoverflow.net/users/7206 | 79437 | 47,781 |
https://mathoverflow.net/questions/79353 | 3 | Consider this curve $f(x,y)=0$ given by
$$ f(x,y) := y^3 + y^2 x + x^4 =0.$$
Is it obvious that after a change of coordinates near the origin, this
curve is equivalent to
$$ \hat{y}^2 \hat{x} + \hat{x}^4 = 0 $$
I think, these are both $D\_5$ singularities. It seems like the change of
coordinates that would ach... | https://mathoverflow.net/users/4463 | What is the simplest way to represent a $D_5$ singularity? | For classifying plane curves singularities, the "coordinate approach" is not always the better one\*. For your question, the general case is in table 1, page 3, C.T.C Wall article" "sextic curves and quartic surfaces with higher multiplicity". For general methods, I recommend, C.T.C Wall article: "Notes in the classifi... | 3 | https://mathoverflow.net/users/16409 | 79439 | 47,782 |
https://mathoverflow.net/questions/79432 | 6 | Let $K\_1$ be a perfect field. Let $K\_2/K\_1$ and $K\_3/K\_2$ be quadratic extensions. Let $K\_4/K\_3$ be the Galois closure of $K\_3$ over $K\_1$. Is it true that either $K\_3 = K\_4$ or $K\_4/K\_3$ is quadratic such that the Galois group of $K\_4$ over $K\_1$ is isomorphic to $\mathcal{D}\_4$, the dihedral group wit... | https://mathoverflow.net/users/1107 | Quadratic extension of quadratic extension | Yes. Let $a\_1$ be a generator of $K\_3$ over $K\_1$ and let $a\_2$, $a\_3$, and $a\_4$ be the other three roots of the polynomial of $a\_1$. Then say that $a\_2$ is the other root of the polynomial of $a\_1$ over $K\_2$. Then The Galois group of $K\_4/K\_1$ acts on the partitioned set $\{\{a\_1,a\_2\},\{a\_3,a\_4\}\}$... | 10 | https://mathoverflow.net/users/1450 | 79440 | 47,783 |
https://mathoverflow.net/questions/79430 | 4 | Let $f, \hat{f}, g,$ and $\hat{g}$ be continuous probability densities. Define probability densities $p \propto fg$ and $\hat{p} \propto \hat{f}\hat{g}$. Is it true that
\begin{align\*}
||p - \hat{p}||\_{1} \le || f - \hat{f}|| \_{1} + ||g - \hat{g}||\_1
\end{align\*}
| https://mathoverflow.net/users/18867 | total variation distance of product of measures | Here is a real counterexample, as verified by mathematica: so we take the same discrete two point space as below, and let $f=(u\_1,u\_2), \hat{f} = (v\_1,v\_2), g = (a\_1,a\_2), \hat{g}=(b\_1,b\_2)$, with the relation that $x\_1 + x\_2 = 1$ where $x \in \{u,v,a,b\}$. So your claim becomes in this special case
$$ |\fr... | 1 | https://mathoverflow.net/users/4923 | 79445 | 47,786 |
https://mathoverflow.net/questions/79442 | 19 | For what positive x's the number of distinct values taken by x^x^...^x with parentheses inserted in all possible ways is not represented by the sequence [A000081](http://oeis.org/A000081)? ~~Is it exactly the set of positive algebraic numbers?~~ Is it a superset of positive algebraic numbers? Is it countable? Is $2^{\s... | https://mathoverflow.net/users/9550 | Number of distinct values taken by x^x^...^x with parentheses inserted in all possible ways | The answer to the second question is "no". Consider the unique solution $x > 0$ to the equation $x^x = 3$. By the Gelfond-Schneider theorem, this number is transcendental. But we have
$$((x^x)^x)^x = x^{x^3} = x^{(x^{(x^x)})}$$
so that two of the parenthesizations coincide. So evidently this set contains transcen... | 16 | https://mathoverflow.net/users/2926 | 79446 | 47,787 |
https://mathoverflow.net/questions/79435 | 7 | Hello everybody,
I'm stuck with proving (or disproving) the following statement.
**Statement**:
For every $0$-dimensional Polish space $(X,\mathcal{T}\ )$, and a countable basis of clopen sets $\mathcal{B}$ for $\mathcal{T}$, every open set is the *disjoint* union of clopen sets in $\mathcal{B}$.
Every open set i... | https://mathoverflow.net/users/11618 | Question about 0-dimensional Polish spaces | Disjointify from the bottom up instead of from the top down, as you do in Real Analysis 1.
The question is at the level of homework, but I give a hint because you identify yourself and explain what you have tried, including looking at the literature, and I see why you are stuck on the problem.
EDIT 11.1.11: First ... | 9 | https://mathoverflow.net/users/2554 | 79449 | 47,790 |
https://mathoverflow.net/questions/79262 | 9 | I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying morphism of topological spaces is surjective.
I believe this is the same as saying that for a morphism $f: X \to Y$ an... | https://mathoverflow.net/users/16857 | surjective morphism of schemes or epimorphism of sheaves? | I think (what currently is) Lemma 5.9 (tag 05VM) of the Algebraic Spaces chapter of the stacks project is exactly what we want.
Let F,G fppf sheaves and let $F \to G$ be a schematic, flat, locally of finite presentation and surjective (as in: surjective on fields). Then it is an epimorphism of sheaves.
| 3 | https://mathoverflow.net/users/16857 | 79456 | 47,796 |
https://mathoverflow.net/questions/78707 | 4 | A little background: As far as I know there is no standard definition of a quantum cellular automaton in the literature. Different authors use different definitions. Here I propose my own definition (though I probably consider it my own out of ignorance rather than originality). This definition seems very natural but i... | https://mathoverflow.net/users/11146 | Are all quantum cellular automata invertible & representable? | [Niel de Beausdrap](https://cstheory.stackexchange.com/users/248/niel-de-beaudrap) found [this](http://arxiv.org/abs/quant-ph/0405174%20) article. The authors (Schumacher and Werner) propose a definition of quantum cellular automata and prove they are always invertible (Corrolary 7 on p. 10). Nominally their definition... | 2 | https://mathoverflow.net/users/11146 | 79462 | 47,798 |
https://mathoverflow.net/questions/79458 | 5 | 1-Let $P=Add(\omega\_1, \kappa)$, and let $D$ be a dense open subset of $P$. Then there is a dense subset $S$ of $D$ such that for every $f \in S$ and any $g \in P$, if $domf=domg$ and the set $\{ \beta: f(\beta) \neq g(\beta) \}$ is finite then $g \in D$.
Why is this true?
2-Assume $0^{\sharp}$ exists, and let $\l... | https://mathoverflow.net/users/11115 | Some questions from the paper "Forcing the failure of CH by adding a real" by Shelah and Woodin | Let me answer question 1. Conditions in $P$ are partial
functions $p$ from $\omega\_1\times\kappa\to 2$, with
countable domain, ordered by inclusion. For any condition
$p$, since $\text{dom}(p)$ is countable, there are only countably
many finite modifications of $p$ on this domain. For each
such finite modification $p^... | 8 | https://mathoverflow.net/users/1946 | 79466 | 47,799 |
https://mathoverflow.net/questions/79428 | 0 | Suppose we are in characteristic $p$, and that the field, $K$, that we are working over is imperfect. We have a map $\Theta: K \to K^{\delta}$ where each coordinate function $\Theta\_i$ is an additive function. Suppose further that at least one of the $\Theta\_i$ has non-zero derivative.
Is the Zariski closure of the... | https://mathoverflow.net/users/18866 | Zariski closures of one parameter additive maps in positive characteristic | The image of a map $\Theta(x) = (\Theta\_1(x),\ldots,\Theta\_{\delta}(x))$ where the $\Theta\_i$ are polynomials is always an algebraic variety of dimension at most one, and of dimension one if one of these polynomials is non constant. This follows from general facts. Now, there is no tangent space to the image, what y... | 2 | https://mathoverflow.net/users/2290 | 79470 | 47,801 |
https://mathoverflow.net/questions/79455 | 3 | Let $S=k[x\_1,\dots,x\_n]$ be a polynomial ring, and $A:=k[x^{u^{(1)}}, \dots x^{u^{(l)}}]$ a monomial subalgebra, generated by monomials $x^{u^{(i)}} = \prod\_{j=1}^n x\_j^{u^{(i)}\_{j}}$ with $u^{(i)} \in \mathbf{N}^n$. What are sufficient criteria for the inclusion $A \to S$ to split? What is the splitting?
Since S ... | https://mathoverflow.net/users/5495 | Which monomial subalgebras are direct summands of polynomial rings | **This part answers when $A$ (the monomial algebra) is a direct summand of the polynomial ring as an $A$-module.**
Normality is also sufficient, also this doesn't depend on the field $k$. This is essentially exercise 6.1.10 in "Cohen-Macaulay rings" by Bruns and Herzog.
Let $S\subset T$ be affine semigroups. Call $... | 2 | https://mathoverflow.net/users/2384 | 79474 | 47,803 |
https://mathoverflow.net/questions/79459 | 1 | We have studied about volume of solid of revolution of plane regions in undergraduate Calculus classes. we might have observed that volume of solid of revolution varies whenever we allow to vary the line of revolution. It raised questions! like, Does there exist a point in every simply connected compact plane region su... | https://mathoverflow.net/users/18875 | Existence of point of Volume invariance. | [Note: This is an answer to the original version of the question, without the convexity condition.]
No. Consider two disks of small radius $\epsilon$, joined by a very thin ($<< \epsilon$) filament. The volume of the solid of revolution will be approximately proportional to the sum of lengths of the circles of revolu... | 1 | https://mathoverflow.net/users/284 | 79476 | 47,804 |
https://mathoverflow.net/questions/79494 | 7 | This question has been frustrating me for a while, because I'm pretty sure the solution is simple and I'm just not seeing it. I have found the claim (e.g., in Segal's Polycyclic Groups) that if $G$ is polycyclic and $H\leq G$, then $h(G) = h(H)$ if and only if $[G:H]<\infty$. The "if" direction is trivial, but I'm havi... | https://mathoverflow.net/users/10937 | Infinite index subgroup of polycyclic group has strictly lower Hirsch length? | Consider the series $A\_0 < A\_1 < ... < A\_n=G$ with cyclic factors. Suppose $G/A\_{n-1}$ is infinite (if it is finite, replace $G$ by $A\_{n-1}$). So $h(G)=h(A\_{n-1})+1$. Let $H$ be your subgroup of $G$. Consider its intersection $H'=H\cap A\_{n-1}$. Then $H'$ is normal in $H$ and $H/H'$ is cyclic. Assume $h(H)=h(G)... | 10 | https://mathoverflow.net/users/nan | 79495 | 47,814 |
https://mathoverflow.net/questions/79493 | 18 | Is it true that for any finite nontrivial group *G*, there exist two inequivalent irreducible representations of *G* over the complex numbers that have the same degree.
If so, is there an easy proof? If not, what is the smallest counterexample?
Note: Any counterexample group must be perfect, because if the abeliani... | https://mathoverflow.net/users/3040 | Does every finite nontrivial group have two distinct irreducible representations over the complex numbers of equal degree? | It seems that the answer is yes. A MathSciNet search brought up the paper
>
> Y. Berkovich, D. Chillag, and M. Herzog, *[Finite groups in which the degrees of the nonlinear irreducible characters are distinct](http://www.ams.org/journals/proc/1992-115-04/S0002-9939-1992-1088438-9/home.html)*, Proc. Amer. Math. Soc.... | 23 | https://mathoverflow.net/users/430 | 79496 | 47,815 |
https://mathoverflow.net/questions/79498 | 2 | Suppose $M\_{g}$ is the mapping torus $\Sigma\_{g} \times [0, 1]/ (x, 0) \equiv (\tau x, 1)$, where $\Sigma\_{g}$ is the hyperbolic space with genus $g,$ and $\tau : \Sigma\_{g} \to \Sigma\_{g}$ is an isometry map.
Now suppose $\gamma$ is a simple closed geodesic in $M\_{g}$, could we find a tubular neighborhood of ... | https://mathoverflow.net/users/13054 | The Tubular Neighborhood of a Closed Geodesic | No, it is flat only in case if the geodesic is horisontal; i.e., locally we have $\gamma(t)=(x,t)$.
In general, the torus will have an $\mathbb S^1$-invariant metric which is not flat.
In the simplest case of vertical geodesic the metric tensor in coordinates $(t,\theta)$
will look like this
$$
\begin{pmatrix}
1&0
... | 3 | https://mathoverflow.net/users/1441 | 79499 | 47,817 |
https://mathoverflow.net/questions/79503 | 3 | I'm reading Dwyer and Fried's paper "Homology of free abelian covers, I". In it, they make the following claim, which I'm having trouble verifying.
Let $F$ be a field and $A = F[x\_1^{\pm 1},\ldots,x\_{\beta}^{\pm 1}]$ be a ring of Laurent polynomials over $F$. Let $M$ be a finitely-generated module over $A$. Recall ... | https://mathoverflow.net/users/18884 | Modules of finite support | If $M$ is finite-dimensional, the ring $A/\mathrm{ann}(M)$ is as well, so it is Artinian, hence it has only finitely many prime ideals, and we have $\mathrm{supp}(M)=\mathrm{Spec}(A/\mathrm{ann}(M))$.
Conversely, suppose the support of $M$ is finite. For any $\mathfrak p\in\mathrm{supp}(M)$, the subset $\mathrm{Spec}... | 4 | https://mathoverflow.net/users/2035 | 79506 | 47,820 |
https://mathoverflow.net/questions/79480 | 2 | Hello,
What is the standard reference (including proofs) for a $\sigma>1-\frac{A}{\log t}$ type zero-free region for the Dedekind zeta-function and also, order estimates for $\zeta\_K(s)$ and $\frac{1}{\zeta\_K(s)}$ as $t\to\infty$ in such regions?
Thanks.
| https://mathoverflow.net/users/18881 | Zero-free regions for the Dedekind zeta-function | You can find a proof in the paper of Lagarias and Odlyzko on the effective Chebotarev density theorem.
| 4 | https://mathoverflow.net/users/327 | 79509 | 47,822 |
https://mathoverflow.net/questions/79512 | 3 | $\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a smooth projective variety over an algebraically closed field $k$. Let $\NS(X)$ denote the group $[\Pic(X)/\Pic^{\text a}(X)]\otimes\_{\mathbb{Z}}\mathbb{Q}$, where $\Pic^{\text a}(X)$ denotes the divisors which are algebraically equivalent to 0. If ... | https://mathoverflow.net/users/11395 | Non-degenerate pairing on Néron–Severi Group | $\DeclareMathOperator\NS{NS}$I don't have know a reference off hand. So let me just give you a proof when
$\operatorname{char} k=0$. I expect it's true in general.
We can assume without loss of generality that $k=\mathbb{C}$.
Then via the exponential sequence and the Lefschetz $(1,1)$ theorem, $\NS(X)\_\mathbb{Q}$
(y... | 7 | https://mathoverflow.net/users/4144 | 79517 | 47,826 |
https://mathoverflow.net/questions/79510 | 5 | The question I want to ask is related to the Boone-Higman conjecture (see
[Embedding in f.p. simple groups](https://mathoverflow.net/questions/73568/embedding-in-f-p-simple-groups) for the details).
We discussed recently with Ievgen Bondarenko this conjecture and he noticed that it would follow if the answer to the ... | https://mathoverflow.net/users/13070 | F.p. groups where all elements of the same order are conjugate | I think the answer to the main question is negative. That is, there exists a finitely presented group $G$ with decidable WP that does not embed into any f.p. group where all elements of the same order are conjugate.
Here is the construction. Let us start with a recursively presented group $S$ such that the WP in $S$... | 7 | https://mathoverflow.net/users/10251 | 79539 | 47,836 |
https://mathoverflow.net/questions/79536 | 2 | Landau's Theorem for Dirichlet series with real coefficients ($c\_n$) states that if the coefficients are of fixed sign for all sufficiently large $n$, then the point $\sigma\_0$ on the abscissa of convergence of the series is itself a singularity of the function represented by the series in the half plane $\sigma>\sig... | https://mathoverflow.net/users/10980 | Converse to a theorem of Landau on Dirichlet series | I'm not sure you can hope for much. For example consider the case $c\_n=1$ if $n$ is not a square, and $c\_n=-1$ otherwise. The associated Dirichlet series has a pole at $s=1$, but of course the terms are not of fixed sign for sufficiently large $n$.
In general, knowing analytic properties of a Dirichlet series (such... | 4 | https://mathoverflow.net/users/5101 | 79540 | 47,837 |
https://mathoverflow.net/questions/79538 | 1 | This is again a request for references. I'd appreciate a pointer to any published proof of the following:
>
> **Proposition.** Given $n \in \mathbb{N}^+$, let
> $\Phi$ be a function $\mathbb{C}^n
> > \to \mathbb{C}^n$. Then $\Phi$ is an
> isometry of
> $(\mathbb{C}^n,\|\cdot\|)$ into itself
> if and only if the... | https://mathoverflow.net/users/16537 | All the isometries of $\mathbb{C}^n$ into itself are made like these | The proposition of the posting can't be true in general, I'm afraid: e.g., the isometry group of $\mathbb{C}^2$ has (real) dimension $4+6=10$, while $\dim U(2)+\dim O(2)+4=9$. However, the isometries of $\mathbb{R}^n$ (and in particular, of $\mathbb{C}^n$) are easy to describe.
Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be ... | 5 | https://mathoverflow.net/users/2349 | 79545 | 47,840 |
https://mathoverflow.net/questions/79524 | 4 | Consider the heat equation
$$\dot{u} = \Delta u$$
with initial conditions
$$u\_0 = \delta(x)$$
for some point $x$ in the domain $\Omega$ of the problem. If $\Omega$ is $\mathbb{R}^n$, then this problem has a closed-form solution given by the Euclidean heat kernel
$$k\_t(x,y) = \frac{1}{(4\pi t)^{n/2}}e^{-|x-y... | https://mathoverflow.net/users/1557 | How do you solve linear systems whose solutions decay exponentially? | This subject has been studied in literature --- it turns out that for some class of matrices (such as [M-matrices](http://en.wikipedia.org/wiki/M-matrix)) you can obtain componentwise accurate solutions. A good starting point is the section on "accurate floating point computation" on [Jim Demmel's home page](http://www... | 6 | https://mathoverflow.net/users/1898 | 79547 | 47,841 |
https://mathoverflow.net/questions/79546 | 8 | Let's consider algebraic curves over a fixed algebraically closed field $K$.
It's well known, that every smooth elliptic curve (genus $g = 1$) can be embedded in a quadric surface in $\mathbb{P}^3$. This fact follows simply from the Riemann–Roch theorem.
More generally, for smooth hyperelliptic curves of higher gen... | https://mathoverflow.net/users/18730 | Can any smooth hyperelliptic curve be embedded in a quadric surface? | Yes.
Let $C$ be a hyperelliptic curve of genus $g$, and let $L$ be a general line bundle of degree $g+1$. By Riemann-Roch, $\dim|L| = 1$ and $|L|$ is base-point free, so the complete series $|L|$ gives a degree $g+1$ map to $\mathbb{P}^1$. Then the product of this map and the degree $2$ map $C\to \mathbb{P}^1$ gives ... | 26 | https://mathoverflow.net/users/7399 | 79551 | 47,845 |
https://mathoverflow.net/questions/79543 | 11 | Let $R$ be any ring, let $\text{Mod}\_R$ be the category of right $R$-modules and let $\text{Ab}$ be the category of abelian groups. There is a classical theorem of Eilenberg (I think) which says that for any right exact functor $F:\text{Mod}\_R \to \text{Ab}$ which preserves direct sums, there exists a left module str... | https://mathoverflow.net/users/1107 | Incarnations of a theorem of Eilenberg | More generally, the Theorem of Eilenberg-Watts says the following: The category of cocontinuous functors $\mathrm{Mod}(R) \to \mathrm{Mod}(S)$ is equivalent to the category of $(R,S)$-bimodules. A bimodule ${}\_R M \_{S}$ corresponds to the functor $- \otimes\_R M$. There is a recent paper by A. Nyman which deals more ... | 15 | https://mathoverflow.net/users/2841 | 79552 | 47,846 |
https://mathoverflow.net/questions/45948 | 4 | I am trying to understand (Saito's?) category of mixed Hodge modules as a category (i.e. I am not interested in its construction, just in properties of objects and morphisms). I would be grateful for any nice references for this (though I have some texts already); yet currently I am interested in the following question... | https://mathoverflow.net/users/2191 | Is there a $k$-structure for Hodge modules over a $k$-variety? | I think that the answer is "yes". If you denote by $MFW(X)$ (resp. $MFW(X\_\mathbb{C})$) the category of regular holonomic $D$-modules on $X$ (resp. $X\_\mathbb{C}$) with a good filtration $F$ and a finite filtration $W$, then there are obvious functors $MFW(X)\rightarrow MFW(X\_\mathbb{C})$ and $MHM(X\_\mathbb{C})\rig... | 2 | https://mathoverflow.net/users/12336 | 79553 | 47,847 |
https://mathoverflow.net/questions/79554 | 3 | I know two generalizations of Hilbert 90, but I don't if there is a statement that contains both:
### The first statement
Let $K$ be a field, then $H^1(Gal(K), GL\_n(K^{sep}))=0$.
### The second statement
Let $X$ be a scheme, then $H^1(X,\mathbb{G}\_m)=Pic(X)$.
### Question
Is there a nice characterization ... | https://mathoverflow.net/users/5309 | What is the general statement of Hilbert 90? | $H^1$ computed via sheaf cohomology coincides with the Cech $H^1$, which can be interpreted as giving transition functions. In particular, $H^1(X, GL\_n)$ is in bijection with the set of rank $n$ vector bundles on $X$ in the Zariski topology (Theorem 11.4 in Milne's notes on etale cohomology).
| 5 | https://mathoverflow.net/users/18512 | 79556 | 47,848 |
https://mathoverflow.net/questions/55790 | 10 | Could one define $\mathbb{Q}\_l$-perverse etale sheaves over more or less general (excellent, separated) base scheme by combining the results of Gabber and Ekedahl? Would their functoriality properties (i.e. left and right $t$-exactness of various image functors, mostly those coming from affine morphisms) be similar to... | https://mathoverflow.net/users/2191 | Bad behaviour of perverse sheaves over 'general' bases? | The answer is very likely "yes", but you will need to put together some technical articles (and unpublished results) that may not have yet been put together. Here are the key ingredients, as I see it :
(1) General definition of perverse t-structure, not using stratifications (using stratifications is probably a bad i... | 6 | https://mathoverflow.net/users/12336 | 79560 | 47,850 |
https://mathoverflow.net/questions/79542 | 11 | In ordinary category theory, the notion of limit in a category $C$ is usually formulated with a category (of indices) $J$ and a functor $F:J\to C$ (a diagram in $C$), and a limit of this diagram is something satisfying some universal property.
In the context of quasi-categories (I looked in the nLab and in *Higher To... | https://mathoverflow.net/users/10217 | Limits in an $(\infty,1)$-category | **Definitions**
I think there is a definition that should fit into most models of $(\infty,1)$-categories. If you want an "elevator speech" answer, it's:
**Definition.** *A limit of a diagram $\mathcal D \to \mathcal C$ is a terminal object in the $(\infty,1)$-category of objects living over $\mathcal D$.*
This i... | 22 | https://mathoverflow.net/users/3593 | 79571 | 47,856 |
https://mathoverflow.net/questions/79566 | 3 | The question was a bit long for the title, so let me explain what I mean here. Let $k$ be some field (ideally, a number field). Let $X$ be a curve that is defined over $k$, such that it has a $k$-point. Let $K$ be an algebraic extension of $k$ which is *not* finitely generated. Does this imply that $X$ has infinitely m... | https://mathoverflow.net/users/5756 | Does a curve have infinitely many $K$-rational points under these hypotheses? | Let $X$ be the plane curve defined by the equation $x^2+y^2=0$. Let $k=\mathbb Q$ and $K=\mathbb R$.
The curve has exactly one point in both.
| 5 | https://mathoverflow.net/users/18060 | 79574 | 47,857 |
https://mathoverflow.net/questions/77229 | 5 | Let $V$ be a real affine variety in $\mathbb R^n$, i.e. the zero set of a real polynomial $p(x\_1,\dots,x\_n)$. Consider the following three definitions of the dimension of $V$, $dim(V)$.
>
> ***Definition 1***: if $I$ is the ideal of
> polynomials vanishing on $V$, then
> $dim(V)$ is the maximum dimension of a
>... | https://mathoverflow.net/users/5572 | dimension of a real affine variety | They are all equivalent, including definition 1, to the Krull dimension of $S/I$, where $S=\mathbb{R}[x\_1,\ldots,x\_n]$ is the polynomial ring $I$ lives in. This is very good news for people like me who want to apply algebraic geometry to statistics, where numbers are mostly real.
Here's how it goes:
Definition 1... | 4 | https://mathoverflow.net/users/84526 | 79579 | 47,860 |
https://mathoverflow.net/questions/79120 | 5 | Let $X=\mathrm{Spec}(A)$ be an affine variety, $Z\subseteq X$ a closed, reduced subscheme. Let
$$\beta:Y=\mathrm{Bl}\_Z(X)\to X$$
be the blow-up of $X$ in $Z$. In other words, $Y=\mathrm{Proj}(A[IT])$ for $I:=I(Z)$. Let $E:=\beta^{-1}(Z)$ be the exceptional divisor. For a point $Q\in E$, I am now wondering how the... | https://mathoverflow.net/users/9947 | Completion of local rings in the exceptional divisor of a blow-up | We can write $A = K[X\_1, \ldots, X\_n]/\mathfrak a$.
Suppose that $I = (a\_1, \ldots, a\_m)$; then
$A[It] \cong A[T\_1, \ldots, T\_m]/\mathfrak b$ for some ideal
$\mathfrak b$. A description for $\mathfrak b$ is given in Section 1.1 of
W. Vasconcelos, \textit{Integral Closure, Rees algebras, multiplicities and
Algorit... | 2 | https://mathoverflow.net/users/14895 | 79580 | 47,861 |
https://mathoverflow.net/questions/41457 | 4 | MacMahon in the paper [Divisors of Numbers and their Continuations in the Theory of Partitions](http://plms.oxfordjournals.org/cgi/reprint/s2-19/1/75.pdf) defines several generalized notions of the sum-of-divisors function; for example, if we write $a\_{n,k}$ for the sum
$$
\sum s\_1 \cdots s\_k
$$
where this sum is ta... | https://mathoverflow.net/users/1703 | What literature is known about MacMahon's generalized sum-of-divisors function? | For anyone who comes by this later, it turns out that the following relationship is true for the functions $A\_k(q)$:
$$
A\_k(q) = \frac{1}{(2k+1)2k}\Big(\big(6A\_1(q) + k(k-1)\big)A\_{k-1}(q) - 2q\frac{d}{dq}A\_{k-1}(q)\Big)
$$
After contacting George Andrews (as suggested), he and I wrote a joint paper proving this r... | 2 | https://mathoverflow.net/users/1703 | 79586 | 47,864 |
https://mathoverflow.net/questions/79583 | 10 | Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex. Or, equivalently, every interior point lies along a straight line segment which intersects two non-adjacent edges.
When is this pro... | https://mathoverflow.net/users/17193 | When does every point in a polytope lie along a chord between its edges? | The question asks whether every point $v$ in the interior of a 3-polytope $P \ $ is on an interval between two edge-points. This is easy. Project the edges of $P$ onto a unit sphere centered at $v$. Call the resulting graph $G$ *blue*. Take the opposite $-G$ and call this graph *red*. Clearly red and blue graphs inters... | 26 | https://mathoverflow.net/users/4040 | 79587 | 47,865 |
https://mathoverflow.net/questions/79591 | 0 | Let
* $f: X\to Y$ be a quasi compact separated morphism
* $\{U\_i\}$ be an open affine covering of $X$
* $V$ be an open affine subset of $Y$
In Hartshorne's AG book chapter III, proposition 8.7 uses ;
$\{U\_i \cap f^{-1}(V)\}$ forms an open **affine** cover of $f^{-1}(V)$
Is it really true?
| https://mathoverflow.net/users/18854 | Question about affine open covering | No, this is not true, but it is not what Hartshorne uses. You need $Y$ separated, then the cartesian square $$\begin{matrix} U\_i\cap f^{-1}(V) & \to & U\_i\times V \\\\ \downarrow && \downarrow \\\\ Y & \to & Y\times Y \end{matrix}$$ exhibits $U\_i\cap f^{-1}(V)$ as a closed subscheme of the affine scheme $U\_i\times ... | 1 | https://mathoverflow.net/users/2035 | 79593 | 47,869 |
https://mathoverflow.net/questions/76253 | 2 | Let $k$ be a field and $A\subset \mathbb{N}^d$ a vector configuration. Let $R,S$ be commutative $k$-algebras, both graded by the affine semigroup $\mathbb{N}A$. Is the 'multidgraded Segre product' $R \otimes\_{\mathbb{N} A} S := \bigoplus\_{\mathbb{N} A} R\_a \otimes\_k S\_a$ known in the literature?
| https://mathoverflow.net/users/5495 | Where did the multigraded Segre product appear in the literature? | I found a precursor of the notion in the paper "On unmixedness theorem" by Chow (American Journal of Mathematics, Vol. 86, No. 4, Oct., 1964). He considers a Segre-type product of the form $\bigoplus\_i R\_{id} \otimes S\_{ie}$ for given bi-degree $(d,e)$. This captures the idea of giving linear relations on the genera... | 0 | https://mathoverflow.net/users/5495 | 79611 | 47,877 |
https://mathoverflow.net/questions/79612 | -4 | I've been considering this sequence:
$$1,2,3,6,12,24,48,96,192,...$$
I've generated the sequence from the rule
$$V\_n=\sum\_{0\leq i \lt n} V\_i$$
$$V\_0=1; V\_1=2V\_0=V\_0+V\_0$$
What interests me most, is that this sequence - with its rule requiring the sum of a finite, but unbounded, number of components is... | https://mathoverflow.net/users/18913 | An interesting, simple, sequence - surprised to find little material. | Of course, yes. Take some fraction $\frac{f}{g}$ ($f$ and $g$ are polynomials) and build its recurrent sequence. Further, take $\frac{f}{g}+h$ ($h$ is a polynomial of degree 4) and do the same.
| 0 | https://mathoverflow.net/users/18814 | 79614 | 47,878 |
https://mathoverflow.net/questions/79601 | 6 | Let $G$ be an infinite group such that the centraliser of any non central element is finite (and bounded).
Is there any structure theorem known about $G$ ?
Such a group seems to be at the other extreme of an FC-group (whose centralisers all have finite index). I can add the following requirements alltogether if nee... | https://mathoverflow.net/users/18583 | Groups whose centralisers are finite | Free Burnside groups of sufficiently large odd exponents $p$ ($p\ge 665$) have all centralizers of nontrivial elements cyclic of order $p$ by a result of Adian. By a result of S. Ivanov, there are groups with this property and finite number ($p$) of conjugacy classes, provided $p$ is big enough and odd. All these (and ... | 12 | https://mathoverflow.net/users/nan | 79615 | 47,879 |
https://mathoverflow.net/questions/79608 | 4 | Hi everyone!
Given a finite simple group $S$, is it always possible to find two elements $x,y \in S$ with the property that for every $a,b \in S$ we have $\langle x^a,y^b \rangle = S$?
More in general, given a finite almost simple group $X$ with socle $S$, is it possible to find two elements $x,y \in X$ with the pr... | https://mathoverflow.net/users/5710 | Conjugation stable generating sets in almost simple groups | This property is known as "being invariably generated by $x$ and $y$" in the literature (though it has not been so extensively studied). For a nonabelian finite simple group, Kantor, Lubotzky and Shalev have shown that, indeed, such $x$ and $y$ exist. (This is Theorem 1.3 in <https://arxiv.org/abs/1010.5722> , which co... | 5 | https://mathoverflow.net/users/20038 | 79616 | 47,880 |
https://mathoverflow.net/questions/79609 | 6 | Consider the group $G=H\rtimes{}C$, where $H$ has order prime with $p$ and $C$ is cyclic of order $p^k$, and $C\rightarrow{}\mathrm{Aut}(H)$ is faithful (or equivalently $G$ has trivial $p$-core). Assume $V$ to be an irreducible (not just indecomposable) faithful representation of $G$ over $\mathbb{F}\_p$.
I need a r... | https://mathoverflow.net/users/3680 | Irreducible mod-p representation of a semidirect product with trivial p-core | The following is a direct proof that any extension of $G$ by $V$ splits. It is taken from [a joint paper of mine with Tim Dokchitser](http://arxiv.org/abs/1103.2047), where the proof starts in the last paragraph of page 12.
First, note that for any $n\in\mathbb{N}$, $H^n(H,V)=0$, since it is killed by $|H|$ and $|V|$... | 6 | https://mathoverflow.net/users/35416 | 79625 | 47,885 |
https://mathoverflow.net/questions/79622 | 16 | It is usual in algebraic geometry to represent morphisms by vertical arrows pointing downwards, like that :
$$\begin{matrix} X \\\\ \downarrow \\\\ S \end{matrix}$$
I suppose this stemmed from Grothendieck's amazingly original idea that a morphism of schemes should *always* be considered as some sort of fibre bundl... | https://mathoverflow.net/users/450 | Did Grothendieck introduce vertical arrows that denote morphisms? | In Hasse's school of number theory, it was quite common to represent an extension of fields by writing the bigger field above the smaller one and drawing a line segment between them, without an arrowhead. This is one possible source of Grothendieck's notation.
| 9 | https://mathoverflow.net/users/2821 | 79629 | 47,888 |
https://mathoverflow.net/questions/79624 | 3 | Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$, and let $N$ be a nilpotent orbit of $\mathfrak{g}$. What is the equivariant cohomology of its closure, $H^\*\_G(\overline{N})$, with respect to the group $G$ for $\mathfrak{g}$?
Also, $H^\*\_G(\overline{N})$ has a natural "inner product" which takes value ... | https://mathoverflow.net/users/5420 | Equivariant cohomology of nilpotent orbits | First, since $\overline N$ is contractible its equivariant cohomology is the same as for $pt$. The Poincare pairing
is uniquely determined by $\int\_{\overline N} 1$ (since it is linear with respect to
$H^\*\_G(pt)$).
More precisely, any cohomology class
of $\overline N$ has the form $\alpha\cdot 1$ where $\alpha$ ... | 2 | https://mathoverflow.net/users/3891 | 79632 | 47,890 |
https://mathoverflow.net/questions/79481 | 2 | For an affine scheme $Spec R$ and a scheme $X$ we know that $Hom(X,Spec R) = Hom(R,\Gamma(X,\mathcal{O}\_X))$.
Does it still hold when we replace $X$ with an algebraic stack?
My guess is yes as $Hom(-,Spec R)$ glues like a sheaf and given an algebraic stack we can consider an atlas and the hypercovering given by it... | https://mathoverflow.net/users/16857 | Does Hom(X,Spec R) = Hom(R, O(X)) hold for algebraic stacks? | I think this is true.
Take a smooth atlas $U\to X$, and notice that sections of $\mathcal{O}\_X$ (which you can see as morphisms of quasi-coherent sheaves $\mathcal{O}\_X \to \mathcal{O}\_X$) correspond to sections of $\mathcal{O}\_U$, such that the two restrictions to $U\_1:=U\times\_X U$ by means of the two project... | 3 | https://mathoverflow.net/users/5516 | 79638 | 47,896 |
https://mathoverflow.net/questions/79640 | 4 | Hi, these are three questions regarding extendability of holomorphic vector fields on complex projective space to its blow up along a subvariety.
Let $\mathbb{P}^n$ be the complex projective space, with homogeneous coordinates of a point $p\in\mathbb{P}^n$ $$p=[z\_0:\ldots:z\_n]$$
1) suppose we blow up the point $p\_... | https://mathoverflow.net/users/4971 | Holomorphic vector fields on $\mathbb{P}^n$ that extend to the blow up | **1)** No. There are many more vector fields. The vector fields you are looking for are precisely those which vanish at $p\_0$. Since $h^0( \mathbb P^n, T \mathbb P^n) = (n+1)^2 -1$ and you are imposing $n$ linearly independent conditions you should get $n^2 + n - 1$ vector fields. In homegeneous coordinates they can b... | 3 | https://mathoverflow.net/users/605 | 79643 | 47,897 |
https://mathoverflow.net/questions/79641 | 2 | Hi,
Let $k$ be a field and $A$ a local noetherian $k$-algebra. If its completion is a UFD,
is it true that $A$ is a UFD? Proof?
Thanks
| https://mathoverflow.net/users/36285 | UFD property descends? | I think this is simpler than having to quote a "Theorem".
Let $\mathfrak a\subset A$ be an ideal such that $\hat{\mathfrak a}=\mathfrak a\hat A$ is principal, generated by $\hat f\in\hat A$. Let $f\in \mathfrak a$ be an element with the same residue mod $\hat{\mathfrak m}$, i.e., such that $\hat f\ \mathrm{mod}\ \hat... | 11 | https://mathoverflow.net/users/10076 | 79649 | 47,898 |
https://mathoverflow.net/questions/73967 | 12 | In my PhD thesis I am studying the Auslander-Reiten theory of a particular class of wild algebras. My question is:
Are there instances of algebras/categories, where the Auslander-Reiten theory of a wild algebra is understood (in the sence that one knows which shapes of components occur) beside the following:
* Here... | https://mathoverflow.net/users/15887 | Auslander-Reiten theory of wild algebras known in examples? | For self-injective Koszul algebras of Loewy length greater than three such that the Yoneda algebra is Noetherian, all (stable) components of the Auslander-Reiten quiver for *graded* modules are of the form $\mathbb Z A\_{\infty}$ (Martinez-Villa, Zacharia: Approximation with modules having linear resolution).
It shou... | 2 | https://mathoverflow.net/users/18756 | 79653 | 47,899 |
https://mathoverflow.net/questions/79621 | 8 | Prove that if we have two rectangular parallelepiped (cuboids) such that one of them is placed inside the other then the sum of the three lengths of the inner parallelepiped is at most the sum of the three lengths of the exterior parallelepiped.
In 2 dimensions the problem is trivial.
Does this hold in higher dimension... | https://mathoverflow.net/users/6140 | Two rectangular parallelepiped | This version works for all parallelepipeds, not only rectangular ones:
If you replace each parallelepiped by all points that have distance at most $\varepsilon$ to a point in the parallelepiped, you can still place the smaller inside the bigger one. In particular, the smaller object has a smaller volume.
We divide... | 10 | https://mathoverflow.net/users/14102 | 79655 | 47,901 |
https://mathoverflow.net/questions/79658 | 3 | My knowledge is very limited for complex geometry. I have the following question:
If we have two complex vector bundles $E\to X$ and $F\to X$ such that we have an isomorphism $\mathcal O\left(E\right) \cong \mathcal O\left(F\right)$ between the sheaf of holomorphic sections, do we have an isomorphism $E \cong F$ ?
| https://mathoverflow.net/users/17492 | Does isomorphisms of sheaf of holomorphic sections implies isomorphisms of two holomorphic vector bundles over the same complex space ? | No. On $\mathbb P^1=\mathbb P^1(\mathbb C)$ we have $\Gamma(\mathbb P^1,\mathcal O\_{\mathbb P^1}(-1))=\Gamma(\mathbb P^1,\mathcal O\_{\mathbb P^1}(-2)=0$, but $O\_{\mathbb P^1}(-1)$ and $O\_{\mathbb P^1}(-2)$ are not isomorphic.
However on an *affine algebraic* variety $X$, the answer is "yes". There is an amazing ... | 5 | https://mathoverflow.net/users/450 | 79660 | 47,903 |
https://mathoverflow.net/questions/79645 | 4 | Let $f(x\_1,\ldots, x\_n)\in\Bbbk [x\_1,\ldots,x\_n]$ be a given polynomial (assume $\Bbbk$ algebraically closed if you want). Suppose that we are given $n$ polynomials $v\_1,\ldots v\_n \in\Bbbk[x\_1,\ldots, x\_n]$. Suppose that we *know* that there exists a polynomial $P(t\_1,\ldots,t\_n)\in\Bbbk[t\_1,\ldots,t\_n]$ s... | https://mathoverflow.net/users/4721 | Simplifying a polynomial | This is the (multivariate) *functional decomposition* problem. It has a long history, going back to 1922 work by Ritt and 1941 work by Engstrom. See the introduction to [Algorithms for the Functional Decomposition of Laurent Polynomials](http://www.csd.uwo.ca/~watt/pub/reprints/2009-calculemus-laurent.pdf) by Stephen W... | 4 | https://mathoverflow.net/users/3993 | 79669 | 47,907 |
https://mathoverflow.net/questions/79663 | 7 | Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ a $\sigma$-algebra. Assume $\nu : \mathcal{A} \to [0,\infty]$ is a finitely additive measure. If $f : X \to [0,\infty]$ is a measurable function, we can define $$ \int\_{X}f\,d\nu$$ in the standard way. If $f,g :X \to [0,\infty]$ are simple measurable functions then it ... | https://mathoverflow.net/users/4002 | Does integrating with respect to a finitely additive measure respect addition? | As mentioned in the question, the inequality
$$
\begin{align}\int(f+g)\, d\nu\ge\int f\,d\nu+\int g\,d\nu&&{\rm(1)}\end{align}
$$
follows easily from the definition of the integral $\int f\,d\nu$ (as the supremum of the integrals of nonnegative simple functions bounded by $f$). So, I'll just show the reverse inequality... | 5 | https://mathoverflow.net/users/1004 | 79672 | 47,908 |
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