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https://mathoverflow.net/questions/62108 | 6 | Hurwitz' theorem states that for a finite separable morphism $f : X \to Y$ of curves of degree $n$ and with ramification divisor $R$, we have
$2 g(X) - 2 = n (2 g(Y) - 2) + \deg(R)$.
Besides, we have $\deg(R)=\sum\_{p \in X} (e\_p - 1)$ if $f$ has only tame ramification [Hartshorne, IV, Cor. 2.4]. One of the conseq... | https://mathoverflow.net/users/2841 | Algebraic applications of Hurwitz' theorem | As far as I remember the Riemann-Hurwitz-formula is used to prove the inequality
$|\mathrm{Aut}(F|K)|\leq 84(g-1)$
for the number of automorphisms of an algebraic function field $F$ of one variable over $K$,
where $K$ has characteristic $0$ and $g\geq 2$ holds for the genus of $F|K$.
| 4 | https://mathoverflow.net/users/3556 | 62112 | 38,433 |
https://mathoverflow.net/questions/62113 | 2 | Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. Then Frobenius direct image $F\_\*(E)$ is also an ample vector bundle on $X^{(1)}$?
| https://mathoverflow.net/users/14252 | Ample bundle under Frobenius morphism | Definitely no. Take $X=\mathbb{P}^1$, $E=\mathcal{O}(1)$. Write
$$F\_\*(E)= \bigoplus\_{i=1}^p \mathcal{O}(a\_i)$$
If $a\_i\ge 0$, for all $i$, then
$$2=h^0(E)=h^0(F\_\*(E)) \ge \sum (a\_i+1) \ge p$$
so it's not even semipositive when $p>2$.
Note this works for any finite map $F:\mathbb{P}^1\to \mathbb{P}^1$,
so in... | 8 | https://mathoverflow.net/users/4144 | 62117 | 38,436 |
https://mathoverflow.net/questions/62116 | 6 | Consider some lattice in R^n. Take some point "P" in R^n (which does not belong to this lattice in general). The problem is to find "nearest" lattice point. The problem is known NP-hard in general it is named "closest vector problem".
Is it correct that "hardness" comes from the "lattice reduction" step ?
I mean if... | https://mathoverflow.net/users/10446 | Closest vector problem (=nearest lattice point) is trivial for "reduced lattice" ? | It actually remains hard, although not as hard. See the paper *Hardness of approximating the closest vector problem with pre-processing* by Alekhnovich, Khot, Kindler, and Vishnoi (FOCS 2005, <http://www.math.ias.edu/~misha/papers/cvpp.pdf>). They show that even if you are allowed to do arbitrary pre-processing after h... | 12 | https://mathoverflow.net/users/4720 | 62121 | 38,439 |
https://mathoverflow.net/questions/62125 | 38 | I have heard the following statement several times and I suspect that there is an easy and elegant proof of this fact which I am just not seeing.
>
> **Question**: Why is it true that an invertible nxn matrix with non-negative integer entries, whose inverse also has non-negative integer entries, is necessarily a p... | https://mathoverflow.net/users/184 | Invertible matrices of natural numbers are permutations... why? | Proof: The condition that $M$ has nonnegative integer entries means that it maps the monoid $\mathbb{Z}\_{\geq 0}^n$ to itself. The condition that $M^{-1}$ is likewise means that $M$ is an automorphism of this monoid.
The basis elements $(0,0,\ldots,0,1,0,\ldots, 0)$ in $\mathbb{Z}\_{\geq 0}^n$ are the only elements ... | 108 | https://mathoverflow.net/users/297 | 62126 | 38,443 |
https://mathoverflow.net/questions/62043 | 2 | This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask.
1) Let $\sigma = (A; \{0,1\}; +, \times)$ be a signature. Form the language $L(\sigma)$ over $\sigma$. Let $T$ be the theory of commutative rings and let $M$ ... | https://mathoverflow.net/users/5031 | Model Theoretic Localization | EDIT: At the bottom, I'll explain how, if you have a ring $R$ and a subset $A$ of $R$, how you can construct a theory such that any model of that theory contains a copy of $R$ such that the copies of elements of $A$ are units.
First, let me comment on a couple issues with the approach you're taking. Then I'll try to ... | 2 | https://mathoverflow.net/users/7521 | 62133 | 38,447 |
https://mathoverflow.net/questions/62137 | 1 | Let $X$ be a projective variety and let $L$ be a line bundle on $X$. Suppose for all locally free sheaves $M$ on $X$,
$
H^i(X,{L^\*}^{\otimes r} \otimes M)=0
$ for $i<\dim X$ and $r$ sufficiently big.
Does it follow that $L$ is an ample line bundle? Here $L^\*$ denotes the dual of $L$.
This is of course clear if $... | https://mathoverflow.net/users/14199 | Ample line bundle and Duality | As was pointed out in the comment above by Laurent Moret-Bailly, you can't choose any $M$. Let me try to answer a slightly different question which I hope is close to what you intended.
**Edit:** This should also answer the revised question, in the negative.
The answer is no, ample line bundles do not satisfy the c... | 7 | https://mathoverflow.net/users/3521 | 62142 | 38,455 |
https://mathoverflow.net/questions/61971 | 2 | From "Lectures on the combinatorics of free probability" by Nica and Speicher we have a necessary sufficient criteria for an element of $M\_n(M)$ being free from $M\_n(\mathbb{C})$ for a non-commutative probability space (NCPS) $M$. Do we have a similar result for such an element being $\ast$-free from $M\_n(\mathbb{C}... | https://mathoverflow.net/users/13670 | when is an element of $M_n(M)$ $\ast$-free from $M_n(\mathbb{C})$ for a $\ast$-non-commutative probability space $M$. | Yes, one can get a similar characterization, only you will need to talk about $\ast$-cumulants instead of cumulants (in other words, you will need to allow both $x\_{ij}$'s and $x\_{ji}^\ast$'s in the formula involving the cyclic cumulants above, and require that all other cumulants vanish).
Let me explain the sourc... | 4 | https://mathoverflow.net/users/12660 | 62151 | 38,459 |
https://mathoverflow.net/questions/62144 | 11 | I am an inexperienced logician, so I may be completely missing something major in this question. I also may be misconstruing the idea of decidability. However, I was wondering if all 6 of the remaining Millennium Prize Problems are decidable in the sense of Gödel.
If any of the associated theories were not decidable, w... | https://mathoverflow.net/users/14484 | Are the Millennium Prize Problems all decidable? | There are very few results which allow us to know that a mathematical claim will be provable or disprovable within ZFC without actually proving or disproving it. To the best of my knowledge, the only exceptions are theories which have [quantifier elimination](http://en.wikipedia.org/wiki/Quantifier_elimination). Few1 o... | 42 | https://mathoverflow.net/users/297 | 62153 | 38,461 |
https://mathoverflow.net/questions/62159 | 8 | Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. So please indulge me even if the question is well-known, or trivial.
So suppose we have a projective variety $Y \subse... | https://mathoverflow.net/users/14449 | Relationship between Hilbert schemes and deformation spaces | When you say "deformation functor", you have to be careful to specify exactly which functor you are thinking about. There are two relevant deformation functors at play here.
One is the functor $D$ of abstract deformations of $Y$. If $A$ is an Artin local $k$-algebra, then $D(A)$ is the set of isomorphism classes of f... | 10 | https://mathoverflow.net/users/332 | 62165 | 38,464 |
https://mathoverflow.net/questions/62086 | 5 | Recall that a spectrum is called connective if it is $(-1)$-connected (that is, its homotopy is concentrated in nonnegative degrees).
However, this left me scratching my head a bit. Why "connective"? Is there some geometric intuition behind it that I'm missing?
| https://mathoverflow.net/users/1353 | Why are connective spectra called "connective"? | Maybe just because (-1)-connected is a mouthful? And connected would not be the right term, since that would imply trivial $\pi\_0$. Other than that, I don't know.
| 9 | https://mathoverflow.net/users/4042 | 62175 | 38,473 |
https://mathoverflow.net/questions/62156 | 29 | This question comes to me via a friend, and apparently has something to do with quantum physics. However, stripped of all physics, it seems interesting enough on its own. I assume someone has asked this question before, but I have no idea what to search for:
Suppose we have points $P$ and $Q$ on $S^2$, and two availa... | https://mathoverflow.net/users/35336 | Maneuvering with limited moves on $S^2$ | \**More comments at end on finding group elements with small word length \**
If you're willing to accept an element of the group (as distinguished from a word expressing an element) there is an algorithm that will produce such an element moving $P$ to within $\epsilon$ of $Q$ that is polynomial as a function of the n... | 39 | https://mathoverflow.net/users/9062 | 62176 | 38,474 |
https://mathoverflow.net/questions/62163 | 36 | Hopefully, MathOverflow is the correct place for this.
I had a student approach me and ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any topology involved, but I'd appreciate it if I could give her a more full answer. So, topology answers are the best but o... | https://mathoverflow.net/users/10206 | Math and Wormholes | **A bit of General Relativity and Causality theory**
One feature of general relativity is that the space-time is modelled as a Lorentzian manifold. The Lorentzian metric on the manifold has signature (-+...+), and thus distinguish between *time-like*, *space-like*, and *null* directions. A $C^1$ curve in your manifol... | 104 | https://mathoverflow.net/users/3948 | 62181 | 38,478 |
https://mathoverflow.net/questions/60562 | 12 |
>
> Let $X=Spec(A)$ be a reduced normal affine scheme over an algebraically closed field $k$ of characteristic $0$, with an action of a connected reductive group $G$. Suppose
>
>
> * $x\in X$ is a $G$-invariant $k$-point,
> * $X$ contains a dense open stabilizer-free $G$-orbit (i.e. a dense open copy of $G$), and
... | https://mathoverflow.net/users/1 | Is an affine "G-variety" with reductive stabilizers a toric variety? | Vera Serganova showed me the following (affirmative) answer. I'll use the setup from Remark 1. Note that Remark 2 shows that $X$ cannot contain a *positive* highest weight vector. So the following result does the job.
>
> **Proposition:** Let $V$ be a representation of a reductive group $G$ and let $X$ be the closu... | 7 | https://mathoverflow.net/users/1 | 62182 | 38,479 |
https://mathoverflow.net/questions/62033 | 1 | It is known that cos(N) spans a countable dense set in [-1,1].
(N: any natural number)
As far as I know generally, for any continuous function f defined in [a,b],
f is Riemann integrable where its domain is a countable dense set in [a,b].
**My question: will cos[t\_n\*Log(p)] Spans a countable dense set in [-1,1]... | https://mathoverflow.net/users/14464 | Cos[Im[zetazero(n)]Log(prime)] spans a countable dense set in [-1,1]? | For any fixed real $\alpha$, the fractional parts of the numbers $\alpha \gamma$, where $\beta+i\gamma$ runs over all zeros of $\zeta(s)$ in the critical strip with $0<\gamma < T$, become uniformly distributed in $\mathbf{R}/\mathbf{Z}$ as $T\to \infty$. This is a theorem of Fujii; see his paper "On the zeros of Dirich... | 4 | https://mathoverflow.net/users/1464 | 62187 | 38,481 |
https://mathoverflow.net/questions/62185 | 10 | I've read that every finitely generated one relator group embeds in a two generator one relator group, and that this fact follows from the Freiheitssatz.
Unfortunately, the only proof I can find of this fact applies [B.H. Neumann's proof](http://www.jstor.org/stable/91573) for denumerable n-relator groups, and it do... | https://mathoverflow.net/users/6429 | Freiheitssatz implies a finitely generated one relator group embeds in a two-generator one relator group? | Yes, this follows from the Freiheitssatz. Assume that the 1-relator group is defined by $G=\langle g\_1,\ldots, g\_n | R(g\_1,\ldots,g\_n)\rangle$, such that the relator $R(g\_1,\ldots, g\_n)$ is cyclically reduced, and involves the generators $g\_1$ and $g\_n$ non-trivially. By the [Freiheitssatz](http://en.wikipedia.... | 12 | https://mathoverflow.net/users/1345 | 62195 | 38,487 |
https://mathoverflow.net/questions/62197 | 9 | I need to understand the representation theory of $S\_n$ (symmetric group on $n$ letters) and so could someone suggest a good reference for this. There is a similar question on mathoverflow here
[A learning roadmap for Representation Theory](https://mathoverflow.net/questions/2755/a-learning-roadmap-for-representati... | https://mathoverflow.net/users/11395 | Representation theory of $S_n$ | "The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions" by Bruce Sagan might be a good place to start.
| 15 | https://mathoverflow.net/users/9481 | 62200 | 38,491 |
https://mathoverflow.net/questions/62155 | 29 | The cohomology of Shimura varieties and Drinfeld shtukas is conjectured to realize the representations sought for in the Langlands programme/conjectures, the cohomology of Deligne-Lusztig varieties realizes representations of the classical groups over finite fields: How did people find those varieties?
| https://mathoverflow.net/users/nan | how to find the varieties whose cohomology realizes certain representations? | Regarding Shimura varieties:
One has to first consider the case of modular curves, which has served throughout as an impetus and inspiration for the general theory.
The study of modular curves (in various guises) goes back to the 19th century, with the work
of Jacobi and others on *modular equations* (which from a ... | 23 | https://mathoverflow.net/users/2874 | 62211 | 38,495 |
https://mathoverflow.net/questions/62218 | 46 | There are quite a few german mathematical theorems or notions which usually are not translated into other languages. For example,
[Nullstellensatz](http://en.wikipedia.org/wiki/Nullstellensatz), [Hauptvermutung](http://en.wikipedia.org/wiki/Hauptvermutung), [Freiheitssatz](http://en.wikipedia.org/wiki/Freiheitssatz),... | https://mathoverflow.net/users/2841 | German mathematical terms like "Nullstellensatz" | Führerdiskriminantenproduktformel.
| 132 | https://mathoverflow.net/users/2821 | 62221 | 38,498 |
https://mathoverflow.net/questions/62234 | 1 | Could someone please tell me what I am missing in the following argument. Either my understanding of the exact statement of Ado's theorem is wrong, or there is a flaw in my argument below.
For a finite dimensional Lie algebra ${\bf g}$ consider the adjoint representation $\phi:{\bf g}\rightarrow {\bf gl}\left({\bf g... | https://mathoverflow.net/users/14510 | Ado's Theorem Proof | Your argument fails because the bracket can (sometimes) take pairs of elements into the centre. Therefore the direct sum as vector spaces isn't necessarily a direct sum of Lie algebras. For nilpotent Lie algebras, such as the Heisenberg algebra, you can't make the claim about block-diagonal form.
| 13 | https://mathoverflow.net/users/6153 | 62238 | 38,510 |
https://mathoverflow.net/questions/61812 | 17 | Let $H$ be a finite index subgroup of a finitely generated group $G$. Assume that $Out(H)$ is finite. Can $Out(G)$ be infinite?
| https://mathoverflow.net/users/3969 | Outer automorphisms of finite extensions | I think the answer is yes. Let $A$ be a f.g. group. Assume that $\text{Out}(A)=1$ ($\text{Out}(A)$ finite would probably be enough) and that $A$ contains a central copy of $C^{(\infty)}$, the direct sum of countably many copies of the cyclic group $C$ of prime order $p$. Then $\text{Out}(A\times C)$ is infinite: indeed... | 9 | https://mathoverflow.net/users/14094 | 62243 | 38,512 |
https://mathoverflow.net/questions/62239 | 9 | In 'Buildings and Finite $BN$-Pairs', Jacques Tits gives the following statement which is left as an easy exercise.
Let $G\_1,G\_2,G\_3$ be three subgroups of a group $G$. Then the following conditions are equivalent.
1. $G\_2G\_1 \cap G\_3G\_1 = (G\_2 \cap G\_3) G\_1$
2. $(G\_1 \cap G\_2) \cdot (G\_1 \cap G\_3) = ... | https://mathoverflow.net/users/12039 | Proof of an 'easy' exercise in a book of Tits | (1) implies (2):
Indeed, the left side of (2) is obviously contained in the right side. So pick an arbitrary $h=g\_2g\_3=g\_1 \in (G\_2G\_3)\cap G\_1$. Now, $g\_3=g\_2^{-1}g\_1$ is contained in $G\_2G\_1$ but also in $G\_3G\_1$ and hence by (1) in $(G\_2\cap G\_3)G\_1$. So there is $g\_{23}\in G\_2\cap G\_3$ and $\tild... | 8 | https://mathoverflow.net/users/8338 | 62248 | 38,513 |
https://mathoverflow.net/questions/62253 | 0 | Here is the fact in measure theory:
FACT : Let $E$ be a Lebesgue measurable subset of $\mathbb{R}^n$. Almost every $x\in E$ satisfies $\lim\limits\_{m(B)\to 0,~x\in B}\frac{m(B\cap E)}{m(B)}=1$ i.e. limit is taken over the ball $B$ containing $x$ with shrinking it.
Using this fact, I want to prove that
If a Lebes... | https://mathoverflow.net/users/13453 | Question on measure theory | Clearly, you can only have this for $alpha\le 1$. Let $F:=[0, 1]\setminus E$, and take a covering $\{I\_k:k=1,..., n\}$ of $F$ of subintervals of $[0, 1]$ such that $\sum\_{k=1}^n m(I\_k)\le m(F)-\epsilon$ for some $\epsilon>0$. By assumption $m(F\cap I)\le (1-\alpha)m(I)$. Then
$(1-\alpha)(m(F)-\epsilon)\ge(1-\alpha... | 0 | https://mathoverflow.net/users/1969 | 62257 | 38,521 |
https://mathoverflow.net/questions/62206 | 1 | Let $A\to B$ be a morphism of (commutative) algebras and $M$ a $B$-module. The $A$-bilinear map $B\times M\to M$ given by $(b,m)\mapsto bm$ induces a surjective homomorphism $B\otimes\_{A}M\to M$.
Give sufficient and necessary (or at least sufficient) conditions for this mapping to be an isomorphism of $B$-modules.
... | https://mathoverflow.net/users/66825 | Criteria for Preservation of a Module Structure under Extension of Scalars. | [Frankild, Sather-Wagstaff, and Wiegand](http://www.math.unl.edu/~rwiegand1/ExtVanish/paper.pdf): As long as ${}\_BM$ is finitely generated, $A \longrightarrow B$ is flat local, $\mathfrak{m}\_A B = \mathfrak{m}\_B$, and the extension of residue fields is an isomorphism, this happens if and only if $\mathrm{Ext}\_A^i(B... | 2 | https://mathoverflow.net/users/460 | 62262 | 38,524 |
https://mathoverflow.net/questions/62265 | 9 | This semester, I teach an introduction to probability course tailored for students with no science background and so with very *very* little prerequisites. We started with the basics of analytic combinatorics then moved on to random variables and the study of common laws (binomial, hypergeometric, geometric, Poisson). ... | https://mathoverflow.net/users/2284 | Characterization of the Poisson law | What about the characterization of Poisson point processes ?
Let us consider a counting process $(N(t))\_{t \ge 0}$. That is, $N(0)=0$, $N(t)$ only increases by jump of height $1$, and is right continuous. You can see $N(t)$ as the number of points of a random set in $]0,t]$.
Then $N(t)\_t$ is a homogeneous poisson... | 6 | https://mathoverflow.net/users/12088 | 62270 | 38,528 |
https://mathoverflow.net/questions/62291 | 2 | What are the relations between the notation in Lusztig's book [introduction to quantum groups](http://books.google.com/books?id=HKPjCUiOUQ0C&printsec=frontcover&dq=introduction+to+quantum+groups+lusztig&hl=en&ei=UrWtTeeNKIfZgQewic3sCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false) and the u... | https://mathoverflow.net/users/11877 | notation in Lusztig's book: introduction to quantum groups | Yes, the $E$, $F$, and $K$ are standard generators of $U\_\nu(\mathfrak{sl}\_2)$. Sometimes $$q=\nu^{-1}.$$
I don't know what is standard. More generally, the standard (Chevalley) generators for $U\_\nu$ are $E\_i,F\_i,K\_i$ ($i\in I$).
The algebra $\mathbf{f}$ (generated by $\theta\_i,i\in I$, say) is isomorphic (... | 5 | https://mathoverflow.net/users/4366 | 62301 | 38,547 |
https://mathoverflow.net/questions/58987 | 10 | **Background/motivation**
I'm working on contact topology (in dimension three): a fundamental theorem of Giroux gives us a bijection between contact structures (up to isotopy) and open books (up to negative stabilisation).
Moreover, a stabilisation is a "hands on" operation both at the abstract level (*i.e.* surfac... | https://mathoverflow.net/users/13119 | Is there a table of (fibred knot) monodromies? | I've produced a table of monodromies for about 63% of the hyperbolic, fibred knots listed on knotinfo. This is available at: <http://surfacebundles.wordpress.com/knot-complements/> <https://bitbucket.org/Mark_Bell/bundle-censuses/overview> (this link now contains significantly more data - for the origional data look un... | 13 | https://mathoverflow.net/users/3121 | 62306 | 38,551 |
https://mathoverflow.net/questions/62246 | 6 | There is a definition that has always left nagging questions in my mind. To set it up, let $(\Omega, \cal{A}, (\cal{F}\_t)\_{t\geq 0}, P)$ be a filtered probability space. From Comets & Meyre's *Calcul stochastique et modèles de diffusions*,
**Definition 3.2.** A real-valued function $\phi$ on $\mathbb{R}^+\times \Om... | https://mathoverflow.net/users/8353 | What does progressively measurable actually entail? | The formula
$$\mathbb{E} \int\_{R+} \phi^{2}(t,\omega)dt$$
is a double integral a la Fubini-Tonelli. And if you did back there is probably a condition on the filtration saying that $\mathcal{F}\_t \subset \mathcal{A}$ for all $t \ge 0$. So that progressive measurability does imply that $\phi^{2}(t,\omega)$ is $\ma... | 6 | https://mathoverflow.net/users/11332 | 62311 | 38,555 |
https://mathoverflow.net/questions/62300 | 6 | Given a representable, surjective morphism of Artin stacks $\phi:\mathcal{F}\to\mathcal{G}$, is it true (like it happens for schemes) that $\dim\mathcal{G}\leq\dim\mathcal{F}$?
| https://mathoverflow.net/users/33841 | Morphism of Artin stacks | No. $B\mathbb{G}\_m\to\mathrm{pt}$ is surjective with source of dimension -1.
As commented below this is nonsense, since this is not representable.
Let me try to make amends for my flip response to the question. Of course what I am about to write could be equally stupid. However, $\phi$ surjective means that the i... | 4 | https://mathoverflow.net/users/10849 | 62316 | 38,559 |
https://mathoverflow.net/questions/62312 | 16 | A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is a **p-point** (or **weakly selective**) iff for every partition $\omega = \bigsqcup \_{n < \omega} Z\_n$ into null sets, i.e each $Z\_n \not \in \mathcal{U}$, there exists a measure one set $S \in \mathcal{U}$ such that $S \cap Z\_n$ is finite for each $n$.
A n... | https://mathoverflow.net/users/7521 | Is every p-point ultrafilter Ramsey? | Amit:
The converse is not true, not even under MA. This is a result of Kunen, and the paper you want to look at is "Some points in $\beta{\mathbb N}$", Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 3, 385–398.
There is a related notion, called **$q$-point**. These are ultrafilters such that any finite-to-one ... | 22 | https://mathoverflow.net/users/6085 | 62317 | 38,560 |
https://mathoverflow.net/questions/62319 | 9 | If you wanted to learn $\mathbf{A}^1$-homotopy theory, which sources in which order would you use?
| https://mathoverflow.net/users/nan | learning $\mathbf{A}^1$-homotopy theory | A book that might be helpful, that is probably mentioned on website above is [http://www.amazon.com/Motivic-Homotopy-Theory-Nordfjordeid-Universitext/dp/3540458956/ref=sr\_1\_1?ie=UTF8&qid=1303257360&sr=8-1](http://rads.stackoverflow.com/amzn/click/3540458956)
Also, people now call it Motivic instead of $\mathbb{A}^1... | 2 | https://mathoverflow.net/users/3901 | 62353 | 38,584 |
https://mathoverflow.net/questions/62339 | 6 | Suppose I have a finitely presented group $G,$ and a subgroup $H$ of $G$ given by its finite generating set (given as words in the generators of $G.$ I want to know whether $H/[H, H]$ is finite. Is this question tractable (for your favorite definition of "tractable" -- decidable would be a good start...)
| https://mathoverflow.net/users/11142 | computing abelianizations | It is undecidable even when $G$ is a direct product of two free groups. Look at Corollary C on page 2 in [this paper.](http://people.maths.ox.ac.uk/bridson/papers/BMiller07/BMiller08.pdf)
| 10 | https://mathoverflow.net/users/nan | 62362 | 38,588 |
https://mathoverflow.net/questions/62360 | 5 | I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and that they commute. I know that, since they generate a commutative $C^\*$ algebra, on the one hand they can be viewed as ... | https://mathoverflow.net/users/7193 | Is independence meaningful for commutative $C^*$-algebras? | For your first question, yes, you can view two commuting normal operators as functions on the same measure space. Let us for simplicity assume that $T\_i : i=1,2$ are self-adjoint acting on a Hilbert space $H$. Then there exists a measure $\mu$ defined on $\mathbb{R}^2$ with values in projections oh $H$, so that: $\mu(... | 6 | https://mathoverflow.net/users/12660 | 62370 | 38,592 |
https://mathoverflow.net/questions/62375 | 1 | For integer $k>3$, is something known about the rational points on
$$ \frac{ x^k-y^k }{ x-y } - (x-y)^{k-2} = 0 $$
It is genus 0 curve for $3 < k < 100$.
Coprime integer solutions are unlikely because of the Fermat-Catalan conjecture.
Rational solutions with coprime numerators and denominator of suitable size a... | https://mathoverflow.net/users/11847 | Rational points on $ \frac{ x^k-y^k }{ x-y } - (x-y)^{k-2} = 0 $ , $k>3$, genus 0 | The projective closure is given by $\frac{X^k - Y^k}{X - Y} - Z(X - Y)^{k-2} = 0$. There is an obvious isomorphism from $\mathbb{P}^1$ given by
$$(R : S) \mapsto \left( R(R - S)^{k-2} : S(R - S)^{k-2} : \frac{R^k - S^k}{R - S} \right).$$
From here it is straightforward to parameterize the rational points on the aff... | 6 | https://mathoverflow.net/users/290 | 62377 | 38,597 |
https://mathoverflow.net/questions/62380 | 6 | Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$.
Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ .
What conditions on the vector space would or would not force $f$ to be linear?
examples: finite dimensional, complete... | https://mathoverflow.net/users/nan | When do 0-preserving isometries have to be linear? | If you assume $f$ to be surjective then $f$ has to be linear without any assumptions on $V$ by the [Mazur-Ulam theorem](http://en.wikipedia.org/wiki/Mazur%2DUlam_theorem). Wikipedia doesn't offer much more information than a link to the beautiful recent [proof](https://web.archive.org/web/20180516125105/http://www.hels... | 14 | https://mathoverflow.net/users/11081 | 62382 | 38,598 |
https://mathoverflow.net/questions/62366 | 4 | Let $R$ be a regular local ring of dimension at least 2, and let $U$ be the complement of the closed point in $\mathrm{Spec} R$. Given a polarized abelian scheme over $U$, under what hypotheses can it be extended over the entire base?
In the mixed characteristic or equicharacteristic $p$ setting, some conditions are... | https://mathoverflow.net/users/14541 | Extending abelian schemes | A false theorem was given in Faltings and Chai in the end of the section V of their book, but as you said Raynaud found a counter-example. Vasiu and Zink have since worked on the subject :
<http://www.mathematik.uni-bielefeld.de/~zink/ValidusJ.pdf>
| 1 | https://mathoverflow.net/users/14546 | 62387 | 38,600 |
https://mathoverflow.net/questions/62385 | 10 | I am an optical engineer, so please forgive any ignorance my questions betoken. I am interested in whether one can tear down the manifold of a finite dimensional Lie group,
leaving an abstract group, and then give the group another manifold structure that makes it again into a Lie group and get anything essentially di... | https://mathoverflow.net/users/14510 | Can a Lie group as an abstract group be given more than one topology making it a Lie group? | To answer the question of the title: given a group, there may be infinitely many different topologies that render it a Lie group: $\mathbb{R}^n$ and $\mathbb{R}^m$ (where $m$ and $n$ are positive integers) are isomorphic as groups since they are $\mathbb{Q}$-vector spaces of the same dimension, hence they are isomorphi... | 19 | https://mathoverflow.net/users/2349 | 62388 | 38,601 |
https://mathoverflow.net/questions/62379 | 4 | Provided two diagonal real matrix which has positive entries, $V$ and $U$.
Find a real matrix $A$, satisfying $A^TA=a^2I$ for some scalar $a$, to minimise
$\left|A^TVA-U\right|\quad\quad(\*)$
where the matrix norm could be an induced one, or in form of $|M|^2\_{F}=\mathrm{tr}(M^TM)$.
I believe the problem is... | https://mathoverflow.net/users/6526 | An optimization problem in numerical linear algebra | The set of matrices $A$ is a cone, smooth away from the origin. With the Frobenius norm $\sqrt{{\rm Tr}(M^TM)}$, you can use differential calculus. The minimum is achieved at some $A$. If $A\ne0\_n$, that is $a\ne0$, the admissible variations are $\delta A=\rho A+AB$ with $\rho\in\mathbb R$ and $B$ skew-symmetric. When... | 2 | https://mathoverflow.net/users/8799 | 62395 | 38,604 |
https://mathoverflow.net/questions/62396 | 11 | Let $X$ be a smooth projective variety and $Z$ a closed subscheme. Let $X'$ be the blow-up of $X$ with center $Z$.
>
> Under what conditions on $Z$ is $X'$
> Cohen-Macaulay?
>
>
>
In the case $Z$ is non-singular, the blow-up $X'$ will also be non-singular, so in particular CM. At the other extreme, any birat... | https://mathoverflow.net/users/3996 | When is a blow-up Cohen-Macaulay? | Since Cohen-Macauleyness is a local property, we can restrict ourselves to the affine case.
So, let $R$ be a Noetherian ring, $I \subset R$ be an ideal and let us consider the so called *Rees algebra*
$\mathcal{R}:= \oplus\_{n=0}^{\infty} I^n=R[It]\subset R[t]$,
together with the associated graded ring
$\mathca... | 3 | https://mathoverflow.net/users/7460 | 62398 | 38,606 |
https://mathoverflow.net/questions/61081 | 1 | I am playing with some questions concerning connections between
certain poset partitions and their linear extensions. This is not
my usual playground, I just happened to stumble upon something.
When I was playing with these things, I came up with a very
simple construction that I cannot find anywhere.I suspect that ... | https://mathoverflow.net/users/4814 | Actions of $Z_n$ and actions of $Z_{n-1}$ | I might be missing something, but it seems to me that there is not much going on in your construction. In fact, your original action of $Z\_n$ on $X$ does nothing more than putting a cyclic ordering on your set $X$, and your complicated-looking construction creates a new cyclically ordered set $Y$ obtained from $X$ by ... | 2 | https://mathoverflow.net/users/12858 | 62399 | 38,607 |
https://mathoverflow.net/questions/62282 | 19 | In his chapter about Hurwitz' theorem for curves, Hartshorne shows that $\mathbb{P}^1$ is simply connected, i.e. every finite étale morphism $X \to \mathbb{P}^1$ is a finite disjoint union of $\mathbb{P}^1$s. In an exercise the reader is invited to show that $\mathbb{P}^n$ is simply connected, using the result for $\ma... | https://mathoverflow.net/users/2841 | $\mathbb{P}^n$ is simply connected | Here is a sketch of an argument which directly uses simple connectedness of $\mathbb P^1$, and is related to the simple connectedness of rationally connected smooth varieties mentioned by Sandor in one of his answers.
The idea is to treat the $\mathbb P^1$s in $\mathbb P^n$ as analogous to arcs in a topological spac... | 15 | https://mathoverflow.net/users/2874 | 62405 | 38,608 |
https://mathoverflow.net/questions/62368 | 8 | It is known that if $f:M\rightarrow N$ is a homotopy equivalent, then the the process of pullback gives a one-one correspondence between bundles over $N$ and $M$ up to isomorphism. Is the converse( that if $f$ gives a 1-1 correspondence between them, then $f$ is a homotopy equivalence)true? Or any counterexample?
By ... | https://mathoverflow.net/users/10333 | How does all of the bundles over a certain manifold characterize the homotopy class of the base manifold? | If you are liberal enough as to what you consider to be "a bundle", then this is true. The reason is that the symmetric product
$$ G := SP^\infty(S^{n-1}) \simeq K(\mathbb{Z},n-1)$$
is a topological abelian group, and principal $G$-bundles are classified by maps to $BG \simeq K(\mathbb{Z}, n)$. Thus if $f : X \to Y$ in... | 10 | https://mathoverflow.net/users/318 | 62411 | 38,612 |
https://mathoverflow.net/questions/62393 | 10 | I've heard assertions of the sort:
1. Let there be a Riemann metric (not very smooth, say of class $C^1$ or $C^2$ or maybe $C$?) in a neighbourhood of a point on a manifold. Then it is possible to choose coordinates so that the metric is $C^\infty$ or even analytic in them.
2. In case of 3-dimensional manifolds it is... | https://mathoverflow.net/users/14551 | Questions on smoothness of Riemann metrics | 1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. The atlas with harmonic functions as coordinates is the best [proved by Samuil Shefel (1979) and rediscovered by Dennis DeTurck and Jerry Kazdan (1981)]. But, the obtained metric might be worse than $C^\infty$.
2. There is... | 14 | https://mathoverflow.net/users/1441 | 62421 | 38,619 |
https://mathoverflow.net/questions/62354 | 1 | I'm currently in a theory of computing class and as such I have been looking up information about P vs NP and other complexity classes out of curiosity. In the process I cam across a blog post discussing a recent "proof" (it turned out to be wrong of course) of P not equal NP and in it they discussed the FO(LFP) comple... | https://mathoverflow.net/users/14534 | FO complexity class | In complexity theory, it is customary to talk about the complexity of languages, i.e., of sets of finite strings of symbols from some alphabet. When one regards FO as a complexity class, the context is broader; one talks about the complexity of sets of finite structures. By a (finite) structure is meant a (finite) set ... | 5 | https://mathoverflow.net/users/6794 | 62423 | 38,621 |
https://mathoverflow.net/questions/62427 | 0 | I am computer scientist, not a mathematician, I've been reading some papers on argumentation in AI that uses the term 'maximal' set without defining it. I think it's left undefined because it's a term used widely in mathematics? The paper at the end of this post. I hope this question isn't too simple for this forum and... | https://mathoverflow.net/users/10814 | What is a maximal set in the context of argumentation in AI | Typically, the term maximal means that a set $A$ has a property but $A \cup \{x \}$ does not have this property for any $x \notin A$. (For example, a maximal matching is a set of edges which is a matching, but if any new edge is added it is no longer a matching.) This is weaker than a *maximum* set, which is the larges... | 0 | https://mathoverflow.net/users/9896 | 62433 | 38,627 |
https://mathoverflow.net/questions/62426 | 1 | Hi,
I have a set of N objects randomly distributed in a 2D physical space. Each object (i) generates a bernoulli random number (0 or 1) based on a marginal probability Pr(xi = 1) = p. These objects a correlated by physical distance. The closer the objects are, the larger their correlation is.
E.g. If objects i and ... | https://mathoverflow.net/users/14558 | Generating Bernoulli Correlated Random Variables with Space Decaying Correlations | Here's a suggestion:
Define a non-negative decreasing function $w(r)$ measuring interaction strength. Given each
object its own independent $N(0,1)$ random variable $N\_i$. Now set
$$
Y\_i=\frac{\sum\_{j}w(\|x\_i-x\_j\|)N\_j}{\sqrt{\sum\_j w(\|x\_i-x\_j\|)^2}},
$$
where $x\_i$ denotes the location of the $i$th objec... | 2 | https://mathoverflow.net/users/11054 | 62443 | 38,634 |
https://mathoverflow.net/questions/62446 | 7 | I want to know whether the principal congruence subgroups of $SL(n, \mathbb{Z})$ are characteristic? please suggest me a reference.
| https://mathoverflow.net/users/13835 | Principal congruence subgroups of $SL(n, \mathbb{Z})$ | Yes, they are caracteristic.
Let $\rho: GL(n, \mathbb{Z}) \to GL(n, \mathbb{Z}/m\mathbb{Z})$ be reduction mod $m$ and $\Gamma\_n(m) := ker(\rho) \cap SL(n, \mathbb{Z})$ be a congruence sugroup.
According to the discussion in [Automorphisms of $SL\_n(\mathbb{Z})$](https://mathoverflow.net/questions/57235/automorph... | 8 | https://mathoverflow.net/users/10194 | 62457 | 38,641 |
https://mathoverflow.net/questions/62404 | 2 | Let $FO$ be first-order logic and $FO^k$ be $k$-variable segment of $FO$, i.e. $FO^k$ has only $k$ variables.
To my understanding, for every sentence $\varphi\in FO$ there exists a sentence $\psi\in FO^2$ such that for all finite structures $\mathfrak{A}$ with linear order it is the case that $\mathfrak{A}\vDash\varp... | https://mathoverflow.net/users/nan | $2$-variable segment of FO over ordered, finite structures | It is not the case. As an example, the following paper:
Kouck´y, M., Lautemann, C., Poloczek, S., Th´erien, D.: Circuit lower bounds via
Ehrenfeucht-Fraiss´e games, 2006.
shows that, over words, FO[+, $\times$] with 2 variables is equivalent to AC$^0$ with *linear size* circuits --- while FO[+, $\times$] is equival... | 4 | https://mathoverflow.net/users/7687 | 62461 | 38,644 |
https://mathoverflow.net/questions/62464 | 10 | Is there a classification of the commutative rings (with unit) such that each module over the ring is projective ?
| https://mathoverflow.net/users/10194 | Rings with all modules projective ? | They're called "semisimple artinian" rings. Prove that a ring $R$ (no commutativity is required) is semisimple artinian iff (equivalently)
0) (definition is most books in Ring Theory) $R$ is right artinian and has no nonzero nilpotent right ideals.
1) Any right R-module is projective.
2) Any right R-module is inj... | 17 | https://mathoverflow.net/users/7952 | 62465 | 38,647 |
https://mathoverflow.net/questions/62456 | 16 | Let $H$,$K$ be closed connected subgroups of a compact Lie group $G$. Let $L:=\langle H,K \rangle$ be the subgroup they generate, *ie*, the smallest subgroup of $G$ containing them both. Must $L$ be closed?
Notes:
1. False if $G$ is not compact.
2. False if $H$, $K$ not connected: consider two $\mathbb{Z}/2\mathbb{... | https://mathoverflow.net/users/14566 | In a compact lie group, can two closed connected subgroups generate a non-closed subgroup? | Maybe it's helpful to formulate a somewhat different version of Mikhail's answer, from the viewpoint of the Borel-Tits development of reductive algebraic groups over an arbitrary field. When the field is $\mathbb{R}$, much of the basic structure of Lie groups is recaptured this way including the structure of compact co... | 12 | https://mathoverflow.net/users/4231 | 62470 | 38,651 |
https://mathoverflow.net/questions/62383 | 1 | For handle decomposition of surface, suppose I have a twisted 1-handle(twisted only once) adjacent to an isolated pair of linked handles, is the handle slide operation enough to move the previous one to become 3 twisted 1-handle(each is the same as the previous twisted 1-handle), which proves that these two are homeomo... | https://mathoverflow.net/users/1956 | Handle slides homeomorphism | I'm too lazy to draw this right now, but I think I can describe it anyway. Consider a twisted band attached right next to a dual pair of untwisted bands. I'll write this $A\bar{A}BCBC$. Here the overbar represents a twist and the bands attach to the same letters on each end. Now slide the B band into the middle of the ... | 3 | https://mathoverflow.net/users/9417 | 62484 | 38,659 |
https://mathoverflow.net/questions/61487 | 11 | Assume I have some set $P$ with $||P|| = N$ unique elements. I also have $S$ multisets, $(m\_1, ..., m\_S)$, of cardinality $L$, consisting of elements in $P$ chosen with uniform probability. We call a multiset, $m\_i$, 'distinguishable' if it contains at least $k$ elements, though not necessarily distinct elements, th... | https://mathoverflow.net/users/14324 | Probability of unique elements in each of 'S' multisets sampled with uniform probability | **Edit:** Changed the formulas slightly, making it (hopefully) correct, but even less useful.
---
I think I can write down an expression for the probability, but I'm not sure how useful it is. My interpretation of the random experiment is that a random $(S\times L)$-matrix over $\{1,\ldots,N\}$ is generated by ch... | 1 | https://mathoverflow.net/users/12674 | 62486 | 38,660 |
https://mathoverflow.net/questions/62475 | 5 | I ran across the following statement in a paper, and it seems fishy to me:
### Lemma: If $A$ is any Hopf algebra, and if $U$ is an $\mathbb{N}\_0$-filtered $A$-module algebra, then $U$ and $\mathrm{gr} (U)$ are isomorphic as $A$-modules.
There is no proof of the lemma, it just states that it is a well-known fact.
... | https://mathoverflow.net/users/703 | Associated graded of filtered module-algebra over a Hopf algebra | The statement is definitely false. For example, let $A = \mathbb k[x]$ be the group algebra of $\mathbb Z$. Let $U$ be the two-dimensional module in which $x$ acts by $\bigl( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \bigr)$. The span of $\bigl( \begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\bigr)$ is a submod... | 2 | https://mathoverflow.net/users/78 | 62488 | 38,661 |
https://mathoverflow.net/questions/62471 | 4 | Given a minimal parabloic subgroup we know that conjugation by the longest element in the weyl group takes it to the opposite parabolic.
Can we do the same thing if we choose a standard parabolic subgroup? Can we always find an element in the weyl group such that conjugation by this element takes it to the opposite ... | https://mathoverflow.net/users/8974 | Can every parabolic subgroup be conjugated to its opposite by an element of the Weyl group? | If I understood your question correct, then the answer is no. I will assume for simplicity that you are talking about parabolic subgroups of complex simple Lie groups. Then your question translates to the corresponding question about closed subsystems of root systems. Recall that the standard parabolic subgroups biject... | 8 | https://mathoverflow.net/users/2164 | 62489 | 38,662 |
https://mathoverflow.net/questions/62483 | 1 | Let G be an non-Abelian locally compact group. What is the set af all multiplicative functionals of L1(G)? (When G is abelian the answer is the dual group)
| https://mathoverflow.net/users/13230 | multiplicative functionals | Multiplicative functionals on the group algebra correspond to one-dimensional unitary representations of the group. In the non-commutative case such (nontrivial) representations can be absent or constitute a small part of the set of all irreducible unitary representations. See, for example,
M. A. Naimark, Normed ring... | 4 | https://mathoverflow.net/users/12205 | 62493 | 38,665 |
https://mathoverflow.net/questions/62516 | 13 | Is every holomorphic line bundle on the - say - punctured unit disc $\dot{\Delta} \subseteq \mathbf{C}$ trivial? Griffiths-Harris (p. 39) prove that $H^{p,q}\_{\overline{\partial}}(\Delta) = 0$ (for $q \geq 1$), and mention that by replacing discs by annuli that one could prove also $H^{p,q}\_{\overline{\partial}}(\dot... | https://mathoverflow.net/users/3824 | Holomorphic line bundles on a punctured disc | Yes, every holomorphic vector bundle of any rank is trivial on the punctured disk $\dot{\Delta}$ . Indeed, since $\dot{\Delta}$ is a Stein manifold ( like any non-compact Riemann surface ! ) the Oka meta-principle (here a Theorem of Grauert ) says that the classification of holomorphic vector bundles on that manifold i... | 14 | https://mathoverflow.net/users/450 | 62521 | 38,678 |
https://mathoverflow.net/questions/62525 | 1 | I found the following the following differential equation in the context of a Game Theory problem. I was wondering if this is related to any known family of equations or whether there is any hint about properties it might have. I am looking for functions $f:[0,1]^2 \rightarrow [0,1]^2$ satisfying:
$x\_1 \frac{\partia... | https://mathoverflow.net/users/8130 | Structure of solutions of a PDE from a game theory problem | Apply $\partial\_2$ to the first, $\partial\_1$ to the second and sum. You find
$$(x\_1+x\_2-1)\partial\_1\partial\_2(f\_1+f\_2)=0.$$
Away from the line $L:\,x\_1+x\_2=1$, you have $\partial\_1\partial\_2(f\_1+f\_2)=0$. Insert this in the original equations and you obtain $\partial\_1\partial\_2f\_j=0$. Whence
$$f\_i(x... | 5 | https://mathoverflow.net/users/8799 | 62529 | 38,682 |
https://mathoverflow.net/questions/62528 | 4 | I am reading Milne's *Étale cohomology*, III.4.
A twisted form of an object $Y$ (a scheme, a sheaf of modules, of algebras...) over a scheme $X$ is an object $Y'$ such that there exists a covering in some topology (let's say étale to fix the ideas), $(U\_i \to X)$ such that $Y \times\_X U\_i \cong Y' \times\_X U\_i$ ... | https://mathoverflow.net/users/2234 | Twisted forms and $\check{H}^1$ | For your question 2, note that the fiber product $U\_0\times\_XU\_0$ can be non-trivial (unlike the case of an open sub-set $U\_0\subset X$, where it would be $U\_0$ again) with two different projections $p\_1$ and $p\_2$ onto $U\_0$ giving two different structures to it as a $U\_0$-scheme. The isomorphism $\alpha\_{0,... | 5 | https://mathoverflow.net/users/7868 | 62536 | 38,686 |
https://mathoverflow.net/questions/62531 | 13 | Let $\mathbb{C}^n = V + \mathbb{C}$ be the defining representation of the symmetric group. Is there a nice formula for how $\Lambda^i V \otimes \Lambda^j V$ splits into irreps?
| https://mathoverflow.net/users/4707 | Symmetric group irreps in tensor products of exterior products of the standard representation | Congratulations, you have asked one of the few questions of this type for which there is a positive answer. Such a formula is the main result (Theorem 2.1) of [Remmel's paper "A formula for the Kronecker products of Schur functions of hook shapes"](http://dx.doi.org/10.1016/0021-8693%2889%2990191-9).
A few points of ... | 21 | https://mathoverflow.net/users/297 | 62537 | 38,687 |
https://mathoverflow.net/questions/62495 | 9 | Given N, what is a finite non-abelian (and preferably non-solvable) group G in whose subgroup lattice Sub[G] there is an interval that is a chain of length at least N?
Since N can be arbitrarily large (but fixed), perhaps there is no easy answer. In that case, can someone suggest which sorts of groups to look at to f... | https://mathoverflow.net/users/9124 | Which finite nonabelian groups have long chains of subgroups as intervals in their subgroup lattice? | You can construct groups with chains of arbitrary length with the help of wreath products. Since you want to assume that the subgroup $H$ is corefree, you get a *faithful* action of $G$ on the cosets of $H$, so one may identify $G$ with a permutation group. Now assume that $G$ is a transitive permutation group on the s... | 7 | https://mathoverflow.net/users/10266 | 62546 | 38,692 |
https://mathoverflow.net/questions/62532 | 3 | The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.
Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is said to be defined over $k$ if $f(V\_k)\subset W\_k$ and these are elements of $Hom\_K(V,W)\_k\subset Hom\_K(V,W)$. This... | https://mathoverflow.net/users/11395 | $k$ structures on $K$ vector spaces | The statement appears to be wrong. What you need is for $V$ to be finitely generated. It is a general theorem of commutative algebra that, if $R\rightarrow S$ is a flat map of commutative rings, and $M$ and $N$ are $R$-modules with $M$ finitely presented over $R$, then the natural map
$$Hom\_R(M,N)\rightarrow Hom\_S(S\... | 4 | https://mathoverflow.net/users/7868 | 62549 | 38,693 |
https://mathoverflow.net/questions/51848 | 15 | This question is related to [that](https://mathoverflow.net/questions/51732) of Thurston. However, I am not interested in algebraic integers, and I wish to focus on random matrices instead of random polynomials.
When considering (entrywise) non-negative matrices $M$, a natural probability measure seems to be
$$\prod\... | https://mathoverflow.net/users/8799 | Distribution of the spectrum of large non-negative matrices | This is a non-centered iid random matrix whose entries have mean one and variance one (and decay exponentially at infinity), and as such, is subject to the circular law with one outlier. Thus, there will be one eigenvalue roughly near n, and the rest will be more or less uniformly distributed in the complex disk of rad... | 11 | https://mathoverflow.net/users/766 | 62553 | 38,695 |
https://mathoverflow.net/questions/62510 | 3 | Maple seems to suggest that for any real $a\ge 1$ and positive integer $K$ and $n$ with $K\le n/(a+1)$ one has
$$ a^n + na^{n-1} + \binom{n}{2}a^{n-2} +...+ \binom{n}{K}a^{n-K} \le a^{n-K} e^{nH(K/n)}, $$
where $H(x)=-x\log(x)-(1-x)\log(1-x)$ is the entropy function.
An essentially equivalent form is as follows. Sup... | https://mathoverflow.net/users/9924 | A large deviation / binomial coefficients bound | After a little thinking, there is a strikingly simple proof, running as follows.
Dividing through both sides of the inequality by $a^{n-K}$, we see that the larger is $a$, the stronger the estimate is. Thus, the general case will follow from that where $K=n/(a+1)$. For brevity we write $p:=K/n$ and $q:=1-p$, so that ... | 3 | https://mathoverflow.net/users/9924 | 62556 | 38,697 |
https://mathoverflow.net/questions/62557 | 9 | I would be shocked if the following were not true, but I can't seem to see a proof.
Claim:
Let $R$ be an integral domain containing an AC (uncountable if you wish) field $k$. Let $a, b \in R$, and suppose that the ideal $(a, b)$ is height 2.
Then for general $\alpha, \beta \in k$, the element $\alpha a + \beta b... | https://mathoverflow.net/users/2579 | irreducibility of generic linear combination of polynomials? | This is not true even over $\mathbb C$. Take $\mathbb C[x,y]$ and $x^2, y^2$. You need general combination of a regular sequence of length at least $3$. Search for "local Bertini theorem" and "Flenner".
ADDED: the relevant reference is Satz 4.9 and 4.10 (*Die Sätze von Bertini für lokale Ringe* by H. Flenner, Mathem... | 11 | https://mathoverflow.net/users/2083 | 62566 | 38,704 |
https://mathoverflow.net/questions/62571 | 4 | For a smooth variety $X$ when $Pic^0(X)$ is trivial, we get an isomorphism between $N^1(X)$ and Picard group and life become easier.
My question is whether in general there are theorems, criteria ... which results in triviality of $Pic^0$ in some useful examples ?? Like for hyeprsurfaces in projective space, or in to... | https://mathoverflow.net/users/5259 | Vague question on $Pic^0$ | Working over $\mathbb{C}$:
We have the short exact sequence of sheaves $0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^\* \to 0$ and we thus have
$$H^1(X, \mathbb{Z}) \to H^1(X, \mathcal{O}) \to \mathrm{Pic}(X) \to H^2(X, \mathbb{Z}) \to H^2(X, \mathcal{O}).$$
Note that the kernel of the last map is discrete, while... | 8 | https://mathoverflow.net/users/297 | 62572 | 38,707 |
https://mathoverflow.net/questions/62492 | 6 | Is there an explicit example of non-Kahler manifold $M$ such that $M$ satisfies the dd^c lemma ?
| https://mathoverflow.net/users/14576 | Non-Kahler manifolds and the dd^c-lemma | Here is an example of a Moishezon manifold which is easy to visualize. Take a high degree (e.g. a quintic) hypersurface $Z$ in $\mathbb{P}^{4}$ which has a single ordinary double point. Let $X$ be a small resolution of $Z$. Explicitly, a small analytic neighborhood of the singularity can be identified with the vertex o... | 12 | https://mathoverflow.net/users/439 | 62573 | 38,708 |
https://mathoverflow.net/questions/62544 | 14 | Let $X=\mathbb C^2$, let $X^{[n]}$ be the Hilbert scheme of length $n$ 0-cycles in $X$, and let $X^{[n]}\_0$ be the closed subscheme formed by the 0-cycles supported at 0. As far as I know $X^{[n]}\_0$ and $X^{[n]}$ have the same homotopy type. Can anybody suggest a proof? (according to Nakajima this can be proved by a... | https://mathoverflow.net/users/14559 | Homotopy type of Hilbert schemes of points of $\mathbb C^2$ | It is a general folklore result, that if ${\mathbb C}^\times$ acts on a smooth complex variety $X$ so that the fixed point set $X^{{\mathbb C}^\times}$ is proper and the limit
${{\rm lim}\_{\lambda \to 0} \lambda z } $
exists for every $z\in X$, then the downward flow $$D:=\{ z\in X | {\rm lim}\_{\lambda \to \infty} ... | 20 | https://mathoverflow.net/users/1583 | 62578 | 38,710 |
https://mathoverflow.net/questions/62501 | 0 | Thanks to S. Carnahan for suggestion:
Question: I would like to know if there are results in the literature concerning the prime numbers of the form
$$
p = 2q^n-1
$$
where
$$
q
$$
is an odd prime number.
Moreover, conjectural asymptotics are of special interest.
Furthermore, there seems not to be an entry in the ... | https://mathoverflow.net/users/11016 | Prime solutions of $\sigma(p)/2$ equal a power of a prime number | Each fixed $n$ is a special case of Schinzel's "Hypothesis H" (also known as the Bateman-Horn conjecture), which conjectures that there are infinitely many pairs of primes $p,q$ related by your equation $p=2q^n-1$. (For $n=1$ this is a special case of the Hardy-Littlewood prime $k$-tuples conjecture.) So it's conjectur... | 3 | https://mathoverflow.net/users/5091 | 62590 | 38,718 |
https://mathoverflow.net/questions/62594 | 1 | Let $G$ be a group satisfying
$H\_1(G;\mathbb{Z})$ is free abelian group and $H\_i(G;\mathbb{Z})=0$ for $i\geq 2$.
Is it true that $G$ is free group?
| https://mathoverflow.net/users/13453 | On the group homology | Perfect, locally free groups exist. Such a thing has vanishing $H\_1(G,\mathbb Z)$, has $H\_p(G,M)=0$ for all $p\geq2$ and all $M$, and is not free.
A. J. Berrick constructs an explicit example [here](http://www.math.nus.edu.sg/~matberic/ch1.ps). If you prefer an example where $H\_1(G,\mathbb Z)$ is free and *non-zer... | 8 | https://mathoverflow.net/users/1409 | 62595 | 38,722 |
https://mathoverflow.net/questions/62608 | 3 | Let $1 \to N \to G \to H \to 1$ be a short exact sequence of topological groups. Such an exact sequence is said to be topologically split if $G$ is $N \times H$ as a
topological space.
Can someone give me an example of a topologically split short exact sequence of non-discrete connected topological groups. Of course ... | https://mathoverflow.net/users/14595 | Topologically split extensions of topological groups | I think something like the real $ax+b$ group (a.k.a. the affine group of ${\mathbb R}$) ought to do the trick.
The following is not the full $ax+b$ group but may be easier to handle here. Take
$$ G = \left\{\left( \matrix{ a & b \\ 0 & 1 } \right) \colon a >0, b \in {\mathbb R} \right\} $$
and take $N$ to be those ... | 4 | https://mathoverflow.net/users/763 | 62609 | 38,727 |
https://mathoverflow.net/questions/62610 | 4 | Hurwitz proved in 1893 that the number of automorphisms of a Riemann surface of genus $g \geq 2$ is bounded by $84(g-1)$. See [Wikipedia](http://en.wikipedia.org/wiki/Hurwitz%2527s_automorphisms_theorem) for some references. I want to understand the proof in the language of algebraic geometry, namely for a complete alg... | https://mathoverflow.net/users/2841 | Proof of Hurwitz's automorphisms theorem | Here is an extremely useful fact: if $X$ is a regular variety and $f \colon X \dashrightarrow \mathbf{P}^n$ is a rational map defined on a dense open $U \subset X$, then $\mathcal{f}$ extends uniquely to a maximal open subset $U'$ of $X$ and the complement of $U'$ has codimension at least two. In particular, when $X$ i... | 4 | https://mathoverflow.net/users/1310 | 62611 | 38,728 |
https://mathoverflow.net/questions/62612 | 3 | Hello everybody!
I'm interested in $\Delta\_{2}^{1}$ subsets of Polish spaces, i.e. those sets that are both $\Pi\_{2}^{1}$ and $\Sigma\_{2}^{1}$ in the [boldface hierarchy of Polish spaces.](http://en.wikipedia.org/wiki/Projective_hierarchy)
There is a notion of "being complicated" for a subset of a Polish space.
... | https://mathoverflow.net/users/11618 | $\Delta_{2}^{1}$-hard set? | There can't be a $\Delta^1\_2$-complete set. Suppose that $B$ is any $\Delta^1\_2$ set, and let $A$ consist of the points $b$ with $f\_b(b)\notin B$, where $f\_b$ is the continuous function canonically indexed by $b$. Thus, the set $A$ is also $\Delta^1\_2$, but it cannot be that $A=f^{-1}B$ for any continuous function... | 10 | https://mathoverflow.net/users/1946 | 62615 | 38,729 |
https://mathoverflow.net/questions/62618 | 2 | (Perhaps a not very well defined question)
Let $(S\_t)\_t$ be a (flat) family of compact complex surfaces. Assume the generic member is smooth while $S\_0$ has isolated singularities. As the simplest case consider a degenerating family of surfaces in $\Bbb P^3$. Let $\tilde{S}\to S\_0$ be the minimal resolution of si... | https://mathoverflow.net/users/2900 | Can one obtain surfaces with interesting invariants as resolutions of singular surfaces? | The answer to your first question is definitely **yes**. In fact, many interesting examples of smooth, complex algebraic surfaces are given by desingularization of singular ones. The subject is too broad to be fully treated in a MO post. However, let me just recall the paper by Stephan Endraß, Ulf Persson and Jan Steve... | 5 | https://mathoverflow.net/users/7460 | 62620 | 38,731 |
https://mathoverflow.net/questions/62617 | 5 | I am looking for the reference for the following fact (used, for example, in the proof of theorem 4.4. in Breuil's expose about local-global compatibility at Bourbaki):
For $f$ a modular cuspidal form of weight $k \geq 2$, let $\rho \_f$ be the associated Galos representation and let $\pi \_p (\rho \_{f, |Gal(\overli... | https://mathoverflow.net/users/10701 | Galois representation associated to a modular form is crystalline iff... | As Keerthi writes in his comment above, this follows from T. Saito's results. The statement in one direction, that non-trivial $GL\_2(\mathbb Z\_p)$-invariants implies that the associated $p$-adic Galois rep'n is crystalline at $p$, is in fact easier, and goes back to Scholl, and in principle goes back further than tha... | 7 | https://mathoverflow.net/users/2874 | 62625 | 38,732 |
https://mathoverflow.net/questions/62624 | 8 | Does there exist a complex analytic Lie group which doesn't have faithful representations in $GL(N,\mathbb R)$, viewed as a real Lie group?
There are examples of complex Lie groups which do not allow faithful complex representations, like tori $\mathbb C^n/\mathbb Z^{2n}$, but such tori have many faithful real repres... | https://mathoverflow.net/users/2164 | Complex Lie group without faithful real representations? | Take the complex Heisenberg group of 3 by 3 upper triangular unipotent complex matrices, and mod out by a subgroup $Z\times Z$ in the center.
| 17 | https://mathoverflow.net/users/51 | 62631 | 38,734 |
https://mathoverflow.net/questions/62628 | 1 | I have a weighted sum,
weighted sum = w1\*mu1 + (1-w1)\*mu2
with
variance weighted sum = (w1^2)\*var1 + ((1-w1)^2)\*var2 + 2\*w1\*(1-w1)\*cov
in which
mu1 = mean 1;
mu2 = mean 2;
var1 = variance for mean 1;
var2 = variance for mean 2;
cov = covariance;
w1 = weight (ranging from 0 to 1)
Even though I can c... | https://mathoverflow.net/users/14602 | Maximizing the ratio of (weigthed sum)/sqrt(variance_weighted_sum) | It's easy enough to take the derivative and solve for that = 0. The result I get is $w\_1 = \frac{\mu\_2 cov - \mu\_1 var\_2}{(\mu\_1+\mu\_2) cov - \mu\_1 var\_1 - \mu\_2 var\_2}$. This critical point is not necessarily in the interval $[0,1]$ and might be a minimum or inflection rather than a maximum, so the maximum m... | 0 | https://mathoverflow.net/users/13650 | 62636 | 38,737 |
https://mathoverflow.net/questions/62638 | -1 | I have 2 ways of defining a chain complex on a manifold, one of which is the cellular chain complex, $C^{CW}\_\*$. I know that a cellular map $f: X^n \rightarrow Y^n $ such that $ f(X^n) \subset Y^n $ for all $ n $ induces a map on the cellular chain complexes $ f\_\* : C^{CW}\_\* (X^n) \rightarrow C\_\*^{CW}(Y^n) $ th... | https://mathoverflow.net/users/14603 | How does a chain map induce another chain map on an isomorphic chain complex? | Assuming $I\colon C^{CW}\to D$ is a natural isomorphism between functors from CW-complexes to chain complexes, just set
$$f\_D = I\circ f\_\*\circ I^{-1}.$$
This should work.
| 1 | https://mathoverflow.net/users/8103 | 62640 | 38,740 |
https://mathoverflow.net/questions/62642 | 26 | Does any one know why $d\_3: H^\* (X, K^0(point))\rightarrow H^{\*+3}(X,K^0(point))$ is actually extended $Sq^3$ to $\mathbb{Z} $ coefficient.
| https://mathoverflow.net/users/14354 | Third differential in Atiyah Hirzebruch spectral sequence | This follows from the following considerations:
1. This differential in the Atiyah-Hirzebruch spectral sequence must be a stable cohomology operation for general nonsense reasons (the first nonvanishing differential always is, no matter what the generalized cohomology theory is).
2. There are exactly two stable cohom... | 30 | https://mathoverflow.net/users/360 | 62644 | 38,743 |
https://mathoverflow.net/questions/62630 | 20 | I am very curious about this remark in Lesson Four of Rota's talk, [Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations](http://www.ega-math.narod.ru/Tasks/GCRota.htm):
>
> "For second order linear differential equations, formulas for changes of dependent and independent variables are ... | https://mathoverflow.net/users/14601 | What theorem of Liouville's is Gian-Carlo Rota referring to here? | See E. Hille, Ordinary differential equations in the complex domain, Wiley, New York, 1976. The Liouville transformation is given on Page 179. The invariant mentioned by Rota is the function $Q(z)$ appearing as a coefficient of the equation in the canonical form.
| 14 | https://mathoverflow.net/users/12205 | 62649 | 38,747 |
https://mathoverflow.net/questions/62607 | 32 | In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups
$H^i\_{\mathrm{et}}(\operatorname{Spec} O\_{K, S}, M),$
seem to come up often, where $K$ is a number field, $S$ is a finite set of places of $K$, and $M$ is... | https://mathoverflow.net/users/2481 | How is etale cohomology of integer rings related to Galois cohomology? | By usual (sometimes not so trivial) homological arguments, one can reduce to the case where $M$ is a finite discrete module over an artinian ring of residual characteristic $p$. In that case, I think you want $S$ to contain places above $p$ as well, even if your $M$ is unramified at $p$, so let me assume this.
The m... | 15 | https://mathoverflow.net/users/2284 | 62658 | 38,751 |
https://mathoverflow.net/questions/62577 | 1 | Let $Q$ be the generator of a well-behaved (not necessarily reversible) Markov process $X$ on $[n] = \{1,\dots,n\}$ and let $Q^\otimes = \sum\_{m=1}^N I^{\otimes(m-1)} \otimes Q \otimes I^{\otimes(N-m)}$ be the tensor sum generating $N$ instances of $X$ (see [this MO answer](https://mathoverflow.net/questions/58567/spe... | https://mathoverflow.net/users/1847 | Rate of decay of variance for a tensor product Markov process (100 pt bounty for good answer by 1800 EST Fri) | Working with a single copy of the Markov chain for now, you can simplify the expression for the Dirichlet form as $\mathcal D\_Q(f)=-\sum\_{j,k}p\_jQ\_{jk}f\_jf\_k$ (you obtain this from the previous expression just by noting that the row sums of the $Q$ are 0).
Let $e^{(0)},\ldots,e^{(d-1)}$ be the eigenvectors of ... | 2 | https://mathoverflow.net/users/11054 | 62659 | 38,752 |
https://mathoverflow.net/questions/62307 | 3 | I present a simplified version of the problem, and if that's easy to answer then someone can try answering the more complicated version.
Simplified version
------------------
Suppose $X$ is a tangent vector field on a complex manifold. Then we have the Dolbeault operator $\bar\partial$, and a well-defined $\bar\par... | https://mathoverflow.net/users/14528 | If $X$ fails to be holomorphic, what is $\mathcal{L}_X \bar\partial - \bar\partial \mathcal{L}_X$? | I'll answer the simpler question. Since $J^2=-1$, it follows that for any vector field $X$, the endomorphism $L\_X(J)$ anti-commutes with $J$. In other words, $L\_X(J)$ is a section of $\bar{T}^\* \otimes T$. This is the same space where $\bar{\partial}X$ lives and in fact they are equal, at least up to a constant fact... | 2 | https://mathoverflow.net/users/380 | 62670 | 38,758 |
https://mathoverflow.net/questions/62683 | 13 | A very simple question, I just totally forgot how it was called, and Google is not helping.
There's a pair of functions $f:X\to Y$, $g:Y\to X$.
$fgf = f$, $gfg = g$, but $fg$ and $gf$ don't need to be identities (and usually are not in interesting cases).
A simple example would be $f(a,b,c)=(a,b)$, $g(a,b)=(a,b,0... | https://mathoverflow.net/users/14613 | $fgf = f$, $gfg = g$, $fg$ not necessarily identity, what is this called? | It is called "generalized inverse". In that case $fg$ and $gf$ are idempotents. In particular, if you have a semigroup of maps $X\to X$ (i.e. a set of maps closed under composition) such that every $f$ has a generalized inverse, the semigroup is called *regular*. If the generalized inverse is unique, the semigroup is ... | 16 | https://mathoverflow.net/users/nan | 62687 | 38,768 |
https://mathoverflow.net/questions/62689 | 10 | This is a reference request.
The phrase in the title is, if I remember correctly, how Eli Stein described the following set (the definition may be faulty, but I think it is right):
>
> There exists a set $S$, which is a subset of the unit square $[0,1]^2\subset\mathbb{R}^2$ with **full Lebesgue measure**, with ... | https://mathoverflow.net/users/3948 | "A sea-side town where every house can see the sea" | This is a [Nikodymn set](http://en.wikipedia.org/wiki/Nikodym_set). I haven't seen a citation to Nikodymn's original paper, but the history is breifly discussed (with references) in Stein's survey article [Singular Integrals: The Roles of Calderón and Zygmund](http://www.ams.org/notices/199809/stein.pdf).
Edit: Stein... | 12 | https://mathoverflow.net/users/630 | 62691 | 38,771 |
https://mathoverflow.net/questions/62627 | 4 | Reading on the tropical approach to enumerative geometry I have come across the claim:
given a projective toric surface from a polygon P, we can consider a tautological bundle of algebraic / holomorphic functions with newton polygon contained in P (i.e. take functions that in the dense torus can be written as a combina... | https://mathoverflow.net/users/14105 | Intersection of curves on projective toric surface and some enumerative questions | The answer to your question B is yes *and* no :) You see: replacing $P$ by $kP$ for any $k \geq 1$ gives rise to the same toric surface (the latter is precisely the $k$-uple Veronese embedding of the former). And, the defining polynomial of any curve will fit (up to translation) in $kP$ for a sufficiently large $P$.
... | 2 | https://mathoverflow.net/users/1508 | 62707 | 38,779 |
https://mathoverflow.net/questions/61852 | 13 | Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps
$$ K\_1(\mathcal{E})^{(2)} \to K\_1(E)^{(2)} \to H\_D^3(E\_{/ \mathbb{R}} , \mathbb{R}(2) )$$
from (an Adams eigenspace of) K-theory (with rational coefficients) to Deligne cohomology of $E$. ... | https://mathoverflow.net/users/349 | A question on K_1 of an elliptic curve | Let me explain why Beilinson's conjecture implies that $\iota$ is the zero map (thus your first question has conditionally a negative answer).
Let $\mathcal{E}$ be a proper regular model of $E$ over $\mathbf{Z}$. The morphism $E \to \mathcal{E}$ induces a $\mathbf{Q}$-linear map $\iota : K\_1(\mathcal{E})^{(2)} \to K... | 8 | https://mathoverflow.net/users/6506 | 62712 | 38,781 |
https://mathoverflow.net/questions/62334 | 5 | Let's say I have a bag with $A$ red balls, $B$ blue balls, and a total number of balls $N = A + B$. With uniform probability, and sampling without replacement from the $N$ balls, I fill an integer number of bins, $S$, with exactly $L$ balls each, where $N = S\*L$. As such, the bag of all $N$ balls should be empty by th... | https://mathoverflow.net/users/14324 | Probability of having a bounded ratio of two types of balls in each of 'S' bins after random partitioning of a fixed number of balls | For some parameters, you will not see much of a difference between the exact value and assuming that the balls are independent, or that the bins are independent. The independent approximations are much easier to calculate exactly.
You can estimate the probability using inclusion-exclusion. Whether the computable esti... | 4 | https://mathoverflow.net/users/2954 | 62731 | 38,788 |
https://mathoverflow.net/questions/62708 | 10 | What do we know about the height of the minimal (transitive) model of ZFC, that is, about the least α such that Lα is a model of ZFC? Call this ordinal μ. It is countable, since otherwise we could build L inside the collapse of a countable elementary submodel of Lμ to obtain a lesser such α. Moreover, as I learned [her... | https://mathoverflow.net/users/8547 | Height of minimal model of ZFC | I claim that the ordinal $\mu$, if it exists, is actually
$\Pi^1\_1$ definable, which improves on your $\Delta^1\_2$
claim. What I mean by this is that the set of reals coding
a relation on $\omega$ having order type $\mu$ is a
$\Pi^1\_1$ set of reals. Even more, I claim that it is a
$\Delta^1\_1$ property about reals,... | 11 | https://mathoverflow.net/users/1946 | 62732 | 38,789 |
https://mathoverflow.net/questions/62729 | 1 | Let $A$ be a C\*-algebra and let $\alpha$ be an action of the circle group $S\_1$ on $A$ (Gauge action).
We define the following map:
$$E:A\rightarrow A;\quad E(a):=\int\alpha\_t(a)\textrm{d} t.$$
My question is why $E$ is a conditional expectation into the fixed point algebra for the gauge action?
For example if we ... | https://mathoverflow.net/users/14621 | Why is this a conditional expectation into the fixed point algebra? | For your first question, you can check that $E(a)$ is invariant with respect to the action, by using that the Haar measure is left-invariant. The other properties are also straightforward to check.
With respect to your second question, note that $\alpha\_t(S\_j^\*) = e^{-it} S\_j^\*$. Consider an element $S\_{i\_1} \... | 2 | https://mathoverflow.net/users/9545 | 62736 | 38,793 |
https://mathoverflow.net/questions/62715 | 6 | Let $A$ be an abelian variety over a number field $K$ and consider the Neron model $\mathcal{A}$ of $A$ over $X=Spec{\mathcal{O}\_K}$. If $\mathcal{A}^0$ is the identity component of $\mathcal{A}$, then $\mathcal{A}^0$ is an open subgroup scheme of $\mathcal{A}$ that fits into a short exact sequence
$$0 \rightarrow \m... | https://mathoverflow.net/users/13628 | Global Sections of the Identity Component of Neron model | First, a remark: the free part of $A(K)$ is a quotient, not a sub, and so it is possible that a point of infinite order in $A(K)$ could have non-trivial image
in $\Phi\_A(X)$. Probably what you mean is that $\mathcal A^0(X)$ and $A(K)$ have the same
free rank.
Regarding torsion, my interpretation of your question is... | 7 | https://mathoverflow.net/users/2874 | 62737 | 38,794 |
https://mathoverflow.net/questions/62720 | 8 | Hi All,
I'd like to solve some math puzzles, especially in the context of probability theory, but I'm open to other areas too. The kind of problems that does not require much knowledge of mathematics, except, perhaps, for a basic background.
I found this one helpful
[http://www.amazon.com/Fifty-Challenging-Pro... | https://mathoverflow.net/users/13822 | probability and math puzzle books/references | [Problems and Snapshots from the World of Probability](http://books.google.ca/books?id=KCsSWFMq2u0C&printsec=frontcover&dq=blom+holst+sandell&source=bl&ots=R8Dj0_qoQg&sig=Evx4thEukrKZ0BZXfyVIzS-9TYI&hl=en&ei=CtqyTbKVJ460sAP_i53kCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBkQ6AEwAA#v=onepage&q&f=false) by Gunnar Blo... | 5 | https://mathoverflow.net/users/nan | 62743 | 38,800 |
https://mathoverflow.net/questions/62747 | 1 | Let $X$ be a separable metric space and $p$ a probability measure on the Borel Sets of $X$.
Denote $S\_p$ the support of $p$, i.e. the set of points which have positive measure for any ball around them
How to prove that the support of p is of full measure, i.e. $p(S\_p)=1$?
Thanks,
Shlomi
| https://mathoverflow.net/users/14623 | Support of Probability Measures on Separable Metric Spaces | A separable metric space is [strongly Lindelof](http://en.wikipedia.org/wiki/Lindelof_space), that is, every open cover of an open subset has a countable subcover. The complement of the support is the union of all open balls with zero measure. By reducing to a countable subcover, we see that this set has measure zero. ... | 7 | https://mathoverflow.net/users/nan | 62751 | 38,805 |
https://mathoverflow.net/questions/62722 | 2 | Speaks of my own ignorance, but only a few days ago (thanks to MO), I discovered the journal called [Positivity](http://www.springer.com/birkhauser/mathematics/journal/11117)
I really liked many of the articles in that journal, and wanted to know if the better informed MO folks could provide me a list of other journa... | https://mathoverflow.net/users/8430 | Journals similar to "Positivity"? | Operators and Matrices, Journal of Operator Theory
| 6 | https://mathoverflow.net/users/8699 | 62752 | 38,806 |
https://mathoverflow.net/questions/62633 | 4 | The Beauville-Siu theorem states that for a compact Kahler manifold the following two statements are equivalent:
1. $M$ admits a surjective holomorphic map with connected fibers to a compact Riemann surface of genus at least $2.$
2. $M$ admits a surjective homomorphism to the fundamental group of a compact Riemann su... | https://mathoverflow.net/users/11142 | Orbifold Beauville-Siu | Actually, I'm not sure the references I gave above are going to help that much. So let me just
do it here, since it's useful to have a record somewhere. The key is to have an equivariant
form the the standard Castelnuovo-de Franchis (CdF). This follows by looking at the usual proof.
>
> Theorem. Let $M$ be a compac... | 3 | https://mathoverflow.net/users/4144 | 62755 | 38,809 |
https://mathoverflow.net/questions/62695 | 7 | I think people have some general strategy to do matrix multiplication fast. But what about for the finite field of $p$ elements? (e.g. when $p=2$, one should have some faster way.)
So my question is, given two integer entry matrix, $A$ and $B$. Is there a fast way to compute $AB \mod p$?. And is there a fast way of c... | https://mathoverflow.net/users/11286 | Is there a fast way to compute matrix multiplication mod $p$? | For any field, we can define the exponent of matrix multiplication over that field to be the smallest number $\omega$ such that $n \times n$ matrix multiplication can be done in $n^{\omega + o(1)}$ field operations as $n \to \infty$. Schönhage proved that it is invariant under taking field extensions, so it depends onl... | 20 | https://mathoverflow.net/users/4720 | 62759 | 38,810 |
https://mathoverflow.net/questions/62052 | 9 | I am interested in a polyhedral/combinatorics problem that arises in algebraic geometry in the context of geometric invariant theory (GIT).
**Algebro-geometric background**: Consider the natural diagonal action of $\text{SL}\_{d+1}$ on $(\mathbb{P}^d)^n$. For each linearization $L$ one can consider the GIT quotient ... | https://mathoverflow.net/users/10930 | Finding the vertices of a polyhedral complex coming from a GIT wall and chamber decomposition | This is not a real answer. However, I'd seen something closely related to this before, but they were tangential to what I cared about and I didn't sit and think about it. This question made me stop and think about it again (for way to long now), and I'm posting some of what I've learned as it may help, and that so I ca... | 2 | https://mathoverflow.net/users/1102 | 62763 | 38,812 |
https://mathoverflow.net/questions/62710 | 2 | In Shafarevich Basic Algebraic Geometry I, there is a theorem (p.39 Theorem 5) that state:
Any Irreducible closed set is birational to a hypersurface of some affine space $\mathbb{A}^n$
I wonder why it should be irreducible? is the result false for a reducible closed set?
| https://mathoverflow.net/users/14585 | Irreducible closed set birational to a hypersurface | I usually define birational, for reduced schemes of finite type, to mean that there exists an isomorphism on a *dense* open set. With this definition, the answer to this question is
* If and only if the closed subset is equidimensional (in this context I mean all irreducible components have the same dimension).
Sup... | 3 | https://mathoverflow.net/users/3521 | 62769 | 38,816 |
https://mathoverflow.net/questions/62797 | 3 | The set $S$ of even perfect numbers $n$ such that $n+1$ is a prime number contains
$$
6,28,33550336,137438691328
$$
Latter number found by Joerg Arndt, corresponds to $M\_{19}$ (mersenne)
Question: Is $S$ reduced to these $4$ numbers.
New: Joerg Arndt checked up to exponent $110503$ that the corresponding number ... | https://mathoverflow.net/users/11016 | Even Perfect numbers $n$ with $n+1$ prime | There's a conjecture (for which I can't find a source now) that the number of Mersenne primes $2^n-1$ with $n < x$ is $c \log x$ for some constant $c$. Differentiating this, the "probability" that $2^n-1$ is prime is about $c/n$. (This is unconditional; that is, I'm not assuming $n$ is prime.)
The even perfect numbe... | 12 | https://mathoverflow.net/users/143 | 62798 | 38,830 |
https://mathoverflow.net/questions/62781 | 10 | In algebraic geometry, it is a sad fact of life that pushforward doesn't preserve being a coherent sheaf; for example, the pushforward by the complement of a divisor of the structure sheaf (or more generally a line bundle) has essentially no hope of being again coherent. On the other hand, on a smooth variety, if I pul... | https://mathoverflow.net/users/66 | Is the pushforward of a line bundle on the smooth locus of a terminal singularity again a line bundle? | You probably know this, but it warrants pointing out. Suppose that $X$ is a normal variety. Set $U = X \setminus \text{Sing X}$ with the natural inclusion $i : U \to X$, and pushforward the structure sheaf, then $i\_\* \mathcal{O}\_U = \mathcal{O}\_X$. It just doesn't work for arbitrary line bundles.
If your variety... | 11 | https://mathoverflow.net/users/3521 | 62800 | 38,832 |
https://mathoverflow.net/questions/62779 | 2 | hello
Let (A, d) be an n-point metric space
for $t \geq 1$,the task it to find an integer $m$ and an embedding $f : A \rightarrow R^m$ s.t.
$\forall x,y \in A$ : $d(x,y) \leq d\_1(f(x), f(y)) \leq t\*d(x,y)$ where $d\_1$ denotes the $l\_1$ norm.
Is it possible to check if f exists, given t using linear programin... | https://mathoverflow.net/users/44243 | existence of l1 embedding using LP feasibility | The question is slightly ambiguous, since it doesn't specify how large the linear program can be or how much preprocessing can be devoted to producing it. However, if everything is required to run in polynomial time, then the answer is no, assuming ${\rm P} \ne {\rm NP}$. Specifically, take $t=1$, so we are trying to d... | 4 | https://mathoverflow.net/users/4720 | 62802 | 38,834 |
https://mathoverflow.net/questions/62804 | 7 | In their recent (2009) paper Eventually Different Functions and Inaccessible Cardinals, Brendle and Löwe consider a 'tree version' of the Hechler forcing. This forcing $\mathbb{D}$ consists of nonempty trees $T\subseteq\omega^{<\omega}$ with the property that there is a unique *stem* $s\in T$ so that for every $t\in T$... | https://mathoverflow.net/users/2436 | Tree Version of Hechler Forcing | This forcing is a special case of forcing with trees that branch into a filter, the filter in this case being the co-finite sets. (This, in turn, can be viewed as a special case of Shelah's creature forcing.) So the example has certainly implicitly appeared in the literature. An early reference to forcing with trees br... | 9 | https://mathoverflow.net/users/13878 | 62808 | 38,838 |
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