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https://mathoverflow.net/questions/22369 | 7 | I would just like a clarification related to closed subschemes.
If $(X,{\cal O}\_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}\_X$ will be a sheaf of rings on $A$ where $i:A\rightarrow X$ is the inclusion map. (Recall that the inverse image $i^{-1}{\cal O... | https://mathoverflow.net/users/1148 | Closed subschemes and pulling back the structure sheaf via the inclusion map | It might help to consider the extreme case when $x$ is a closed point of $X$,
and $i$ is the inclusion $\{x\} \hookrightarrow X$. The pullback $i^{-1}\mathcal O\_X$
is then the stalk of $\mathcal O\_X$ at $x$, i.e. the local ring $A\_{\mathfrak m}$,
if Spec $A$ is an affine n.h. of $x$ in $X$, and $\mathfrak m$ is the ... | 25 | https://mathoverflow.net/users/2874 | 22383 | 14,767 |
https://mathoverflow.net/questions/22392 | 1 | Consider the following polynomial series:
$S(x) = \sum\_{i=1}^{\infty}(-1)^{i+1}x^{i^{2}}$
Between 0 and 1, this looks like a well-behaved function - is there any way to write this function in this interval without using a series?
Given $0 < S(x) < 1$, I need to solve the equation for $x$ (in the 0 to 1 interval)... | https://mathoverflow.net/users/5579 | Polynomial series | I don't know what you mean by "polynomial series" since your function doesn't seem to have much to do with polynomials (perhaps you could elaborate?).
$S(x) = -\frac12 \theta(\frac12, \frac{\log x}{\pi i }) - 1$, where $\theta$ is [Jacobi's theta function](http://en.wikipedia.org/wiki/Theta_function). I don't know of... | 4 | https://mathoverflow.net/users/121 | 22394 | 14,775 |
https://mathoverflow.net/questions/22398 | 11 | In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which are categories of modules by Freyd-Mitchell). This raises the natural question:
>
> What is meant by a "block" in an... | https://mathoverflow.net/users/4231 | What is a "block" in an abelian category? | Here's a definition of blocks taken from [Comes-Ostrik](http://arxiv.org/abs/0910.5695) (which just happened to be the first paper that came to mind that I knew talks about blocks, it's not a standard reference for this):
>
> Let A denote an arbitrary F-linear category. Consider the weakest equivalence relation on ... | 12 | https://mathoverflow.net/users/22 | 22400 | 14,777 |
https://mathoverflow.net/questions/22390 | 2 | Is there a general theory of embeddings of the (total variety of) the tangent bundle on a (nonsingular) projective variety into projective space? I suppose what I really mean is (and to be more precise about the projective space), if $\mathbb{P}$ is the *projective completion* of $T\_X\rightarrow X$, then what can be s... | https://mathoverflow.net/users/5395 | How far is the tangent bundle from projective space? | The simplest way to get a "projective completion" is to consider the projectivization on $X$ of $T\_X \oplus L$ for some line bundle $L$ on $X$. In this case the complement will be the projectivization of $T\_X$ and will have codimension 1. Sometimes you can contract this completion to get smaller complement, e.g. if $... | 7 | https://mathoverflow.net/users/4428 | 22407 | 14,783 |
https://mathoverflow.net/questions/22410 | 3 | I'm trying to find out more about geometry of surfaces and, in particular, Gaussian curvature. I understand that it can be defined in terms of the principal curvatures (extrinsically) and also intrinsically, and that the result that it can be equivalently defined in these two ways was a significant result. Does anyone ... | https://mathoverflow.net/users/5431 | Equivalent definitions of Gaussian curvature | I like the presentation of the Theorema Egregium in *A Comprehensive Introduction to Differential Geometry* (Volume 2) by Michael Spivak. A translation of the original paper by Gauss and the historical background can be found in *General investigations of curved surfaces of 1827 and 1825* by James Caddall Morehead and ... | 5 | https://mathoverflow.net/users/5371 | 22414 | 14,789 |
https://mathoverflow.net/questions/22339 | 7 | Baer's criterion can be generalized as follows: Let $A$ be an abelian category satisfying (AB3-5) with a generator $R$ and let $T : A^{o} \to Set$ be a continuous functor such that $T(R) \to T(I)$ is surjective for all subobjects $I \subseteq R$. Then for every monomorphism $M \to N$ the map $T(N) \to T(M)$ is surjecti... | https://mathoverflow.net/users/2841 | Baer's criterion for functors | Regarding the original question (with $A=R$-$\mathbf{Mod}$), I think that
by SAFT *any* continuous functor $A^{\mathrm{op}}\to \mathbf{Set}$ is representable,
and hence the assertion in the original question does not generalize
Baer's theorem.
In detail (with $A=R$-$\mathbf{Mod}$):
(\*) $R$ is a generator in $A$, a... | 2 | https://mathoverflow.net/users/2734 | 22421 | 14,793 |
https://mathoverflow.net/questions/22423 | 2 | Let's say a morphism $f:X\to Y$ is **compactifiable** if it admits a factorization $f = pj$ with $j:X\to P$ an open immersion and $p:P\to Y$ proper.
In SGA 4 Exp. XVII, Deligne says that Nagata proved that any morphism of separated integral northerian schemes is compactifiable but that he didn't understand the proof... | https://mathoverflow.net/users/1985 | Compactifiable morphisms | Brian Conrad has written up a proof of Nagata's theorem, starting from notes of Deligne: <http://math.stanford.edu/~conrad/papers/nagatafinal.pdf>. About the analytic case, I have no idea.
| 5 | https://mathoverflow.net/users/4790 | 22424 | 14,795 |
https://mathoverflow.net/questions/22075 | 12 | This is not my field, a friend needs the answer for the following question. Suppose we have a decreasing probability function, $p: N \rightarrow [0,1]$ such that $sum\_n p(n) = \infty$. Take the graph where we connect two integers at distance d with probability $p(d)$. Will this graph be connected with probability one?... | https://mathoverflow.net/users/955 | Connectedness of random distance graph on integers | All right. Here goes, as promised. We shall work with a big circle containing a huge number $N$ of points and a sequence of probabilities $p\_1,\dots,p\_L$ such that $\sum\_j p\_j=P$ is large (so we never connect points at the distance greater than $L$ but connect points at the distance $d\le L$ with probability $p\_d$... | 9 | https://mathoverflow.net/users/1131 | 22425 | 14,796 |
https://mathoverflow.net/questions/22393 | 9 | Is the following statement true?
>
>
> >
> > Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S\_{\mathfrak{p}}$ are formally smooth for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally smooth.
> >
> >
> >
>
>
>
I'... | https://mathoverflow.net/users/1353 | Is formal smoothness a local property? | I think this works. Suppose we have a ring $R$ and an $R$-module $M$ all of
whose localisations are projective and consider $S=S^\ast\_RM$, the symmetric algebra
on $M$. Then $R \rightarrow S$ is formally smooth precisely when $M$ is
projective. Let $\mathfrak p$ be a prime ideal of $S$ and $\mathfrak q$ the
prime idea... | 16 | https://mathoverflow.net/users/4008 | 22432 | 14,803 |
https://mathoverflow.net/questions/22453 | 3 | I'm trying to understand the proof of the Oppenheim conjecture using Ratner's theorem, and I don't immediately see why $SO(2,1)$ is generated by unipotents. Why is $SO(2,1)$ generated by unipotents? More generally, are there equivalent conditions to being generated by unipotents that are easier to check?
| https://mathoverflow.net/users/5598 | How Can I Tell when A Subgroup of a Lie Group is Generated by Unipotents? | Groups like SO$(2,1)$ have been studied in several frameworks: geometry and generation of classical groups over various ground fields, where unipotent elements tend to appear as transvections; real Lie groups, where the structure theory of groups like this is well developed (as in Helgason's old book, for example, repu... | 6 | https://mathoverflow.net/users/4231 | 22458 | 14,818 |
https://mathoverflow.net/questions/22459 | 17 | Let $p$ be an odd prime and $G := \langle x,y : x^p = y^p = (xy)^p = 1 \rangle$. I want to show that $G$ is infinite and wonder if there is a good way to prove this. I'm familiar with the basics of geometric group theory, but don't know if they can be used here.
Obviously $G^{ab} = \mathbb{Z}/p \times \mathbb{Z}/p$. ... | https://mathoverflow.net/users/2841 | 〈x,y : x^p = y^p = (xy)^p = 1〉 | Many techniques discussed here: [group-pub](http://lists.maths.bath.ac.uk/sympa/arc/group-pub-forum/2010-01/msg00005.html)
EDIT: Some of the ideas, in the above thread, I like the most:
If $q$ is a prime congruent to $1$ mod $p$, then consider the Frobenius group $H\rtimes K$ with $H$ cyclic of order $q$ and $K$ cy... | 8 | https://mathoverflow.net/users/1446 | 22460 | 14,819 |
https://mathoverflow.net/questions/22465 | 13 | Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators?
| https://mathoverflow.net/users/5036 | Free division rings? | If you had a "free division ring" $F$ on a set $X$, and any division ring $R$, then any set theoretic map $X\to R$ would correspond to a unique division ring homomorphism $F\to R$. If $X$ has at least two elements $x\neq y$, let $R=\mathbb{Q}$. Then you cannot extend both a set theoretic map $f\colon X\to R$ that sends... | 18 | https://mathoverflow.net/users/3959 | 22469 | 14,826 |
https://mathoverflow.net/questions/22472 | 4 | As the title says, is there a mathematical object referred to as "ivy" or "ivy type" or similar?
I have a type of graph where this name fits perfectly, but I don't want it to clash with something already defined.
(I could in this paper call it a "reduced graph" or "contracted graph" but the above definition would m... | https://mathoverflow.net/users/1056 | Is there a mathematical object called "ivy"? | Ivy does not appear to be a common term in mathscinet. Integrable vector Young functions are called IVY-functions in [MR2055989](http://www.ams.org/mathscinet-getitem?mr=2055989) and related papers. Otherwise all occurrences are the plant or a person.
| 14 | https://mathoverflow.net/users/3710 | 22474 | 14,829 |
https://mathoverflow.net/questions/22477 | 12 | I'm looking for an example of a set S, and a sigma algebra on it, which has no atoms.
Motivation: It seems to me that a lot of definitions in probability and stochastic processes - conditional probability, filtrations, adapted processes - become a lot simpler if phrased in terms of a sample space partitioned into ato... | https://mathoverflow.net/users/4279 | Sigma algebra without atoms ? | In your second question, you are asking merely for an [atomless Boolean algebra](http://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined), of which there are numerous examples. One easy example related to the one given on the Wikipedia page is the collection of periodic subsets of the natural numbers N. That ... | 18 | https://mathoverflow.net/users/1946 | 22482 | 14,832 |
https://mathoverflow.net/questions/22484 | 4 | The title says is all.
To motivate the problem, here is a theorem for finite sets.
Theorem: If S is a finite set, then it can be proved that the atoms of any sigma algebra on S form a partition of S.
I am trying to extend this theorem to a countable set.
* It is easy to show that the atoms must be disjoint - th... | https://mathoverflow.net/users/4279 | Is there a sigma algebra without atoms on a countably infinite set ? | There is an old result of Tarski which says that any algebra A of sets which is
1. κ-complete (i.e. A is closed under unions and intersections of fewer than κ sets from A), and
2. satisfies the κ-chain condition (i.e. there is no family of κ many pairwise disjoint nonempty sets in A),
then A is necessarily atomic.... | 5 | https://mathoverflow.net/users/2000 | 22489 | 14,835 |
https://mathoverflow.net/questions/22452 | 8 | Encouraged by
[Does anyone have an electronic copy of Waldspurger's "Sur les coefficients de Fourier des formes modulaires de poids demi-entier"?](https://mathoverflow.net/questions/16354/)
I realized I could ask for this rare item here.
Again, it is Yury G. Teterin's 1984 (Russian) preprint "Representation of numbers... | https://mathoverflow.net/users/3324 | Does anyone have access to a copy of Yury G. Teterin's 1984 (Russian) preprint "Representation of numbers by spinor genera" | Yura Teterin left mathematics and POMI but may be he is reading his POMI e-mail <http://www.pdmi.ras.ru/~yuri/>
also one can try to ask for a scan at POMI editorial dept (Vera Simonova simonova@pdmi.ras.ru) or at the library lib@pdmi.ras.ru
| 9 | https://mathoverflow.net/users/2702 | 22491 | 14,836 |
https://mathoverflow.net/questions/22454 | 9 | Let $m,n$ be positive integers with $m \leqslant n$, and denote by $\mu\_M$ the minimal polynomial of a matrix.
Do we know for which $m$ the set $E\_m$ of $M \in \mathfrak{M}\_n(\mathbb{R})$ such that $\deg(\mu\_M) = m$ is connected?
| https://mathoverflow.net/users/3958 | Connected subset of matrices ? | The answer is always yes. Indeed the set is path-connected.
Let $C(f)$ denote the companion matrix associated to the monic
polynomial $f$. Every matrix $A$ is similar to a matrix in rational
canonical form:
$$B=C(f\_1)\oplus C(f\_1 f\_2)\oplus\cdots\oplus C(f\_1 f\_2,\cdots f\_k)$$
where here $\oplus$ denotes diagona... | 5 | https://mathoverflow.net/users/4213 | 22496 | 14,841 |
https://mathoverflow.net/questions/22399 | 2 | For any integer $n\ge 3$, let $P(x)=\sum\limits\_{i(=2k)\ge 0}^{n}\binom{n}{2k}(1-x)^k$, $Q(x)=\sum\limits\_{i(=2k+1)\ge 0}^{n}\binom{n}{2k+1}(1-x)^{k+1}$. Define $\frac{P(x)}{Q(x)}\equiv\sum\limits\_{i=0}^{\infty}c\_ix^i $. I like to know $c\_0> c\_1> c\_2>\cdots$, but I don't know how to show it.
| https://mathoverflow.net/users/3818 | Decreasing coefficients? | The polynomials $P\_n$ and $Q\_n$ can be written as
$$
P\_n(x)=(1+\sqrt{1-x})^n+(1-\sqrt{1-x})^n,
\qquad
Q\_n(x)=\sqrt{1-x}\bigl((1+\sqrt{1-x})^n-(1-\sqrt{1-x})^n\bigr).
$$
In particular, they are both solutions to the linear difference equation
$P\_{n+1}(x)=2P\_n(x)-xP\_{n-1}(x)$ implying that their quotient
$f\_n(x)=... | 10 | https://mathoverflow.net/users/4953 | 22498 | 14,843 |
https://mathoverflow.net/questions/22499 | 0 | I want to show, that $GL\_n(\mathbb{Z}/b\mathbb{Z})$ operates transitively on
$X = \{ (v\_1, \ldots, v\_n) \in (\mathbb{Z}/b\mathbb{Z})^n \ | \ v\_1\mathbb{Z}/b\mathbb{Z} + \ldots + v\_n\mathbb{Z}/b\mathbb{Z} = \mathbb{Z}/b\mathbb{Z}\} $
($b$ is an integer.)
IOW
1. $A \in GL\_n(\mathbb{Z}/b\mathbb{Z}), x \in X ... | https://mathoverflow.net/users/5607 | Operation of GL_n(Z/bZ) | Any transformation
$$(v\_1,\ldots,v\_n)\mapsto (v\_1,\ldots,v\_{j-1},v\_j+av\_k,v\_{j+1},\ldots,v\_n)$$
for $j\ne k$ is achievable by means of some such matrix. It suffices to reduce
an admissible vector to $(1,0,\ldots,0)$ by means of a sequence of such reductions.
I would do it in three stages
1. Make $v\_n$ into a... | 1 | https://mathoverflow.net/users/4213 | 22500 | 14,844 |
https://mathoverflow.net/questions/22478 | 73 | Take, for example, the Klein bottle K. Its de Rham cohomology with coefficients in $\mathbb{R}$ is $\mathbb{R}$ in dimension 1, while its singular cohomology with coefficients in $\mathbb{Z}$ is $\mathbb{Z} \times \mathbb{Z}\_2$ in dimension 1. It is in general true that de Rham cohomology ignores the torsion part of s... | https://mathoverflow.net/users/4362 | Can analysis detect torsion in cohomology? | You can compute the integer (co)homology groups of a compact manifold from a Morse function $f$ together with a generic Riemannian metric $g$; the metric enters through the (downward) gradient flow equation
$$ \frac{d}{dt}x(t)+ \mathrm{grad}\_g(f) (x(t)) = 0 $$
for paths $x(t)$ in the manifold.
After choosing furthe... | 43 | https://mathoverflow.net/users/2356 | 22506 | 14,850 |
https://mathoverflow.net/questions/22473 | 11 | Hi, everyone:
For the sake of context, I am a graduate student, and I have taken classes in
algebraic topology and differential geometry. Still, the 2 proofs I have found
are a little too terse for me; they are both around 10 lines long, and each line
seems to pack around 10 pages of results. Of course, I am consider... | https://mathoverflow.net/users/5566 | Request: intermediate-level proof: every 2-homology class of a 4-manifold is generated by a surface. | Torsten's answer was good, but there are also more elementary answers. Here's one, which is essentially a big transversality argument, followed by a mild de-singularization.
Let me consider $M$ to be a triangulated 4-manifold, and represent a class in $H\_2(M)$ as a sum of 2-faces of the triangulation. For each 2-fac... | 14 | https://mathoverflow.net/users/5010 | 22509 | 14,851 |
https://mathoverflow.net/questions/22523 | 1 | If we have function $y=L(x\_1,x\_2,x\_3,...,x\_n)$, and function $z=R(x\_1,x\_2,x\_3,...,x\_n)$. How to compute the derivative $\frac{dy}{dz}$?
Shall I do $\frac{dy}{dz} = \sup\_{g\in \Re^n}\frac{\bigtriangledown\_x L \cdot g}{\bigtriangledown\_x R \cdot g}$?
Is there any mathematical term associated with this kind... | https://mathoverflow.net/users/2013 | Implicit derivative? | There is no reason to expect that such a thing as $dy/dz$ exists. If you rephrase everything in terms of differentials, you have $$dy=\sum\_{i=1}^n\frac{\partial L}{\partial x\_i}dx\_i,\quad dz=\sum\_{i=1}^n\frac{\partial R}{\partial x\_i}dx\_i$$ where $dx\_1,\ldots,dx\_n$ are linearly independent, and so you find $dy=... | 8 | https://mathoverflow.net/users/802 | 22526 | 14,861 |
https://mathoverflow.net/questions/22522 | 1 | In the paper on MinWise independent permutations ([MinWise independent permutations](http://www.cs.princeton.edu/courses/archive/spr04/cos598B/bib/BroderCFM-minwise.pdf)), the authors say that it is often convenient to consider permutations rather than hash functions (Pg-3).
While I understand that for a set X to be ... | https://mathoverflow.net/users/5611 | How is a permutation taken as an equivalent of a hash function in MinWise independent permutations? | For the purpose of min-wise hashing, the only goal of the permutation (or hash function) is to map the initial bag of numbers into another bag, so that we can then pick off the min as the 'hash value'. So for this purpose, you should think of the permutation as merely a function that takes [1..n] and maps it to [1..n] ... | 1 | https://mathoverflow.net/users/972 | 22527 | 14,862 |
https://mathoverflow.net/questions/22462 | 81 | This is a question I've asked myself a couple of times before, but its appearance on MO is somewhat motivated by [this thread](https://mathoverflow.net/questions/22359/why-havent-certain-well-researched-classes-of-mathematical-object-been-framed-by), and sigfpe's comment to Pete Clark's answer.
I've often heard it cl... | https://mathoverflow.net/users/4367 | What are some examples of interesting uses of the theory of combinatorial species? | Composition of species is closely related to the composition of symmetric collections of vector spaces ("S-modules"), which is a remarkable example of a monoidal category everyone who had ever encountered operads necessarily used. Applying ideas coming from this monoidal category interpretation has various consequences... | 30 | https://mathoverflow.net/users/1306 | 22537 | 14,867 |
https://mathoverflow.net/questions/22530 | 6 | This is probably really easy, but I just need someone to help me get mentally unstuck. As part of a description of the [McKay correspondence](http://www.valdostamuseum.org/hamsmith/McKay.html), I want to show that if $G$ is a finite subgroup of $SU(2)$ and $V$ the corresponding 2-dimensional representation, then $\dim ... | https://mathoverflow.net/users/290 | For a representation V of a finite group G, when is Hom(W, W⊗V) trivial for all irreps W? | One necessary condition is that the center of $G$ needs to act trivially in $V$ for $\mathrm{Hom}(W, W \otimes V)$ to ever be non-trivial. The character of the center justs multiplies in a tensor product, and so we can't have a map from $W$ to $V \otimes W$ if $V$ has a non-trivial central character. (This also follows... | 5 | https://mathoverflow.net/users/5010 | 22540 | 14,869 |
https://mathoverflow.net/questions/22552 | 6 | The golden ratio $\phi=\frac{1+\sqrt5}2$ is [sometimes said](http://en.wikipedia.org/wiki/Continued_fraction#A_property_of_the_golden_ratio_.CF.86) to be one of the most difficult numbers to approximate with rational numbers, because its continued fraction development $$\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{... | https://mathoverflow.net/users/1409 | Numbers characterized by extremal properties | Certainly there is one type of study of this question, coming under the heading of the Markoff Spectrum, going back to the German tranliteration of Markov. There is a very nice book called "The Markoff and Lagrange Spectra" by Thomas W. Cusick and Mary E. Flahive. It is this topic that lead to the Markoff Numbers, whic... | 13 | https://mathoverflow.net/users/3324 | 22555 | 14,877 |
https://mathoverflow.net/questions/22549 | 27 | Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if and only if $\lambda < \mu$. A beautiful theorem of Roth asserts that algebraic numbers have irrationality measure $2$. For ... | https://mathoverflow.net/users/1464 | How often are irrational numbers well-approximated by rationals? | $\mathcal{Q}(x,\lambda)$ has positive relative density if and only if $\lambda\le 1$.
This follows from Weyl's Theorem on Uniform Distribution. (There is a nice concise proof in Cassels' "Diophantine Approximation".)
Weyl's Theorem: Let $I\subset \mathbb{R}$ be an interval of length $\epsilon \le 1$. Let $S\_N(I)$ b... | 23 | https://mathoverflow.net/users/5229 | 22566 | 14,885 |
https://mathoverflow.net/questions/22553 | 15 | A Theorem of Cartier (e.g. [Mumford](http://books.google.com/books?id=-5weWX_YD6sC&printsec=frontcover&dq=curves+on+an+algebraic+surface&source=bl&ots=1r4O86OtJy&sig=PIH5b5EipbZYJWXrWy77NvltSP4&hl=en&ei=4PjUS9uMKMOC8gbbi_mCDA&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBIQ6AEwAg#v=onepage&q&f=false), Lecture 25) stat... | https://mathoverflow.net/users/5337 | Are group schemes in Char 0 reduced? (YES) | The answer is yes - *every* group scheme over a field of characterstic zero is reduced: see [Schémas en groupes quasi-compacts sur un corps et groupes henséliens](http://www.math.u-psud.fr/~biblio/numerisation/docs/P_PERRIN-109/pdf/P_PERRIN-109.pdf) (especially Thm. 2.4 in part II and Thm. 1.1 and Cor. 3.9 in part V of... | 12 | https://mathoverflow.net/users/781 | 22573 | 14,888 |
https://mathoverflow.net/questions/22579 | 47 | I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.
| https://mathoverflow.net/users/4692 | What are the reasons for considering rings without identity? | The reason is simple: There are many non-unital rings which appear quite naturally.
If $X$ is a locally compact space (in the following every space is assumed to be Hausdorff), then $C\_0(X)$, the ring of continuous complex-valued functions on $X$ vanishing at infinity, is a $C^\ast$-algebra which is unital if and on... | 71 | https://mathoverflow.net/users/2841 | 22584 | 14,895 |
https://mathoverflow.net/questions/22582 | 4 | 1) I was wondering if there exists a criterion (of (a) combinatorial or (b) geometric nature) for a sum of simple vectors $V\in\wedge^k(\mathbb R^n)$,
$V=e\_{a\_{11}}\wedge\cdots\wedge e\_{a\_{1k}} + \cdots + e\_{a\_{m1}}\wedge\cdots\wedge e\_{a\_{mk}}$, where $e\_1,\cdots, e\_n$ is a basis of $\mathbb R^n$
to be a... | https://mathoverflow.net/users/5628 | Criterion for being a simple vector | You are looking for the Plucker relations, a collection of quadratic polynomials in the $c\_m$'s which vanish if and only if $V$ can be written as $v\_1 \wedge v\_2 \wedge \cdots \wedge v\_k$. You should be able to read about them in most books on algebraic geometry. For example, Griffiths-Harris or Miller-Sturmfels wi... | 13 | https://mathoverflow.net/users/297 | 22588 | 14,896 |
https://mathoverflow.net/questions/22514 | 5 | What can be said and done about the "SIGN-Gordon equation"?
$$\varphi\_{tt}- \varphi\_{xx} + \text{sgn}(\varphi) = 0.$$
It came up [here](https://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion/).
| https://mathoverflow.net/users/1837 | Sign-Gordon Equation | Turns out it appears in literature as the "signum-Gordon equation". For example the paper
[Signum-Gordon wave equation and its self-similar solutions](http://th-www.if.uj.edu.pl/acta/vol38/pdf/v38p3099.pdf)
| 10 | https://mathoverflow.net/users/2384 | 22590 | 14,898 |
https://mathoverflow.net/questions/22211 | 35 | Some time ago I mentioned a certain open question in an MO answer, and Pete Clark suggesting posting the question on its own. OK, so here it is:
First, the setup. Let $X$ be a projective scheme over a field $k$. By Grothendieck, there is a locally finite type $k$-scheme $A = {\rm{Aut}}\_ {X/k}$ representing the funct... | https://mathoverflow.net/users/3927 | Finiteness property of automorphism scheme | I wanted to add some things to the comments I had already made but the list of
comments have become very large and the comments I have already made are
becoming more and more difficult to follow so I'll put everything (including the
things I have already said) here instead even though it is not an answer to the
questio... | 16 | https://mathoverflow.net/users/4008 | 22596 | 14,903 |
https://mathoverflow.net/questions/22587 | 2 | This is a modification of the somewhat naive question that I asked below.
Suppose $X$ is a real Banach space of cotype-2, and $u\_1, u\_2, ... u\_n$ are unit vectors in this space. For $\gamma = ((\gamma\_1, \gamma\_2, ..., \gamma\_n))$ an arbitrary element of $\mathbb{R}^n$, suppose that $\|\gamma\_1 u\_1 + \gamma\_... | https://mathoverflow.net/users/5621 | A bound on linear functionals over cotype 2 spaces | Did you mean to have a factor of ${\sqrt n}$ on the right hand side? Consider $u\_i$ orthonormal in a Hilbert space.
In fact, the best constant is of order $n$, but if you restrict the $u\_i$ to be C-unconditional, then obviously the sup is bounded by C times the cotype 2 constant times ${\sqrt n}$.
Further comme... | 3 | https://mathoverflow.net/users/2554 | 22605 | 14,910 |
https://mathoverflow.net/questions/22607 | 0 |
>
> **Possible Duplicate:**
>
> [solving f(f(x))=g(x)](https://mathoverflow.net/questions/17614/solving-ffxgx)
>
>
>
Here is a nice interview question for computer science people:
Write a unary function `f` such that
`f(f(x)) = -x`
Constraints:
1. The function should be pure (i.e. it should have no s... | https://mathoverflow.net/users/5633 | Interview Question | Divide the domain into separate sets of four, each set having the form {x,y,-x,-y}. Now, let f simply rotate within these sets one step. That is, map x to y, and y to -x, and -x to -y, and -y to x. Thus, doing it twice maps every x to -x, as desired. (Also let f(0)=0.)
| 10 | https://mathoverflow.net/users/1946 | 22611 | 14,914 |
https://mathoverflow.net/questions/22616 | 1 | If X is a set and A is a subset of X containing at least two elements, then certainly for any element $a \in A$, the principal ultrafilter of $a$ contains the principal filter of A (which is NOT an ultrafilter). Are all ultrafilters containing the principal filter of A of this form, or are there non-principal examples?... | https://mathoverflow.net/users/4528 | Ultrafilters containing a principal filter | If a filter $X$ contains any set $A$, then it contains the principal filter of $A$. Thus you are really asking: for which subsets $A$ of a set $X$ can a free ultrafilter contain $A$?
It is a standard exercise to see that such free ultrafilters exist iff $A$ is infinite.
Let me briefly sketch the proof:
1. If an u... | 7 | https://mathoverflow.net/users/1149 | 22617 | 14,916 |
https://mathoverflow.net/questions/22624 | 62 | I am working on my [zero knowledge proofs](https://en.wikipedia.org/wiki/Zero-knowledge_proof) and I am looking for a good example of a real world proof of this type. An even better answer would be a Zero Knowledge Proof that shows the statement isn't true.
| https://mathoverflow.net/users/5601 | Example of a good Zero Knowledge Proof | The classic example, given in all complexity classes I've ever taken, is the following: Imagine your friend is color-blind. You have two billiard balls; one is red, one is green, but they are otherwise identical. To your friend they seem completely identical, and he is skeptical that they are actually distinguishable. ... | 164 | https://mathoverflow.net/users/658 | 22628 | 14,923 |
https://mathoverflow.net/questions/22619 | 12 | I keep running across papers that refer to a set of lecture notes by Robert MacPherson at MIT during the fall of 1993 on Perverse Sheaves. There might also be a set of notes from lectures in Utrecht in 1994 taken by Goresky. For references to these elusive, unpublished notes see the work of Maxim Vybornov and A. Polish... | https://mathoverflow.net/users/1622 | Perverse Sheaves - MacPherson Lecture Notes | The notes on Friedman's page are great -- they were very helpful when I was learning about perverse sheaves as a graduate student, since they explain how to think about middle perversity perverse sheaves on complex analytic spaces with complex stratifications (the case of most interest for most people) without having t... | 16 | https://mathoverflow.net/users/3889 | 22649 | 14,932 |
https://mathoverflow.net/questions/22635 | 36 | **Disclaimer**
Of course not, I'm aware of Gödel's second incompleteness theorem. Still there is something which does not persuade me, maybe it's just that I've taken my logic class too long ago. On the other hand, it may turn out I'm just confused. :-)
**Background**
I will be talking about models of set theory;... | https://mathoverflow.net/users/828 | Can we prove set theory is consistent? | Would you accept it if Set1 just proved the existence of a model for Set2 (in the same way that Set1 proves the consistency of formalized Peano arithmetic by providing a model of it)?
If so, and if you accept in Set1 that there is an inaccessible cardinal κ, then the set Vκ is a model of ZFC, provably in Set1. Most s... | 21 | https://mathoverflow.net/users/5442 | 22656 | 14,937 |
https://mathoverflow.net/questions/22658 | 15 | Why are inverse images of functions more central to mathematics than the image?
I have a sequence of related questions:
1. Why the fixation on continuous maps as opposed to open maps? (Is there an epsilon-delta definition of open maps in metric spaces?)
2. Is there an inverse-image characterization of homomorphisms... | https://mathoverflow.net/users/5651 | Why are inverse images more important than images in mathematics? | Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. (In other words, if you believe that the essence of what it means to be a topological space has to do with functions out of the space, you are privileging inverse images over images. A related ... | 12 | https://mathoverflow.net/users/290 | 22666 | 14,943 |
https://mathoverflow.net/questions/22642 | 21 | ### Background on why I want this:
I'd like to check that suspension in a simplicial model category is the same thing as suspension in the quasicategory obtained by composing Rezk's assignment of a complete Segal space to a simplicial model category with Joyal and Tierney's "first row functor" from complete Segal spa... | https://mathoverflow.net/users/3413 | Is there a combinatorial way to factor a map of simplicial sets as a weak equivalence followed by a fibration? | I don't know if this does what you want, but it's one thing I know how to do.
If $X\to Y$ is a map between *Kan complexes*, then you can build a factorization using the path space construction. Thus, $P=Y^I\times\_Y X$, and $P\to Y$ is given by evaluation, while $X\to P$ is produced using the constant path. Here $I=\... | 16 | https://mathoverflow.net/users/437 | 22667 | 14,944 |
https://mathoverflow.net/questions/22131 | 4 | I want to know if the following fact has a standard name and/or reference
>
> Let $X$ be a subset of $\mathbb R^2$ and $B$ be a disc of the same area as $X$.
> Set $X\_\epsilon$ to be the $\epsilon$-neigborhood of $X$. Then $$area\\,X\_\epsilon\ge area\\,B\_\epsilon.$$
>
>
>
| https://mathoverflow.net/users/5524 | Name for an inequality of isoperimetric type | This inequality is essentially equivalent to the [Classical isoperimetric inequality](http://en.wikipedia.org/wiki/Isoperimetric_inequality). If you have a measurable body $X$ in $\mathbb{R}^n$ and a ball $B\subset \mathbb R^n$ of same volume then you have the following:
$$Area(X)=\lim\_{\epsilon \to 0} \frac{\operator... | 3 | https://mathoverflow.net/users/2384 | 22671 | 14,948 |
https://mathoverflow.net/questions/22676 | 5 | Both in abstract algebra and measure theory is there term ring/algebra, but their definition are different and we can not deduce one from the other, the only requirement in definition they share is that they be closed under two operations: addition and multiplication in abstract algebra while difference and union in me... | https://mathoverflow.net/users/5072 | terminology about ring/algebra in abstract algebra and measure theory | A [ring of sets](http://en.wikipedia.org/wiki/Ring_of_sets) is a ring(usual definition) with the operations intersection(multiplication) and symmetric difference(addition). A [sigma ring](http://en.wikipedia.org/wiki/Sigma_ring) is a special kind of a ring of sets. Now a [sigma algebra](http://en.wikipedia.org/wiki/%CE... | 9 | https://mathoverflow.net/users/2384 | 22679 | 14,952 |
https://mathoverflow.net/questions/22638 | 7 | There exists information on the Picard (and Brauer) group of a reductive algebraic group over a number field k. For example, Sansuc shows (in his big Crelle paper of 1980) that if G is connected and semisimple over a number field k, then Pic G is the group of k-rational points of the character group of the fundamental ... | https://mathoverflow.net/users/5641 | Picard groups of reductive group schemes | Assuming you know that Pic of the generic fibre is trivial, this seems to follow immediately from the localisation sequence: since G is a group there is a section, so the map Pic U to Pic G is an injection. On the other hand (since G is smooth so Pic = Cl) there is an exact sequence:
$\ \ \ \oplus\_x \mathbb{Z}\_x \t... | 3 | https://mathoverflow.net/users/519 | 22684 | 14,956 |
https://mathoverflow.net/questions/22659 | 25 | (Alternate title: *Find the Adjoint: Hopf Algebra edition*)
I was chatting with Jonah about his question [Hopf algebra structure on $\prod\_n A^{\otimes n}$ for an algebra $A$](https://mathoverflow.net/questions/22536/hopf-algebra-structure-on-prod-n-a-otimes-n-for-an-algebra-a). It's very closely related to the foll... | https://mathoverflow.net/users/1 | Does the forgetful functor {Hopf Algebras}→{Algebras} have a right adjoint? | The article [here](http://arxiv4.library.cornell.edu/pdf/0905.2613) proves that the forgetful functor from $k$-Hopf algebras to $k$-algebras has a right adjoint. The main tool they use is the Special Adjoint Functor Theorem.
| 18 | https://mathoverflow.net/users/4517 | 22685 | 14,957 |
https://mathoverflow.net/questions/22694 | 4 | In coordinate-free language, my question is as follows. Let $M$ be an $n$-dimensional manifold with volume form, and let $\mathcal D$ be a smooth (integrable, if necessary) distribution with constant rank $k\leq m$ (by which I mean that $\mathcal D$ is a subbundle of the tangent bundle, spanned by $k$ many every-linear... | https://mathoverflow.net/users/78 | Does a (smooth, constant-rank, integrable) distribution have a (local) basis of divergence-free vector fields? | For any non-zero vector field $v$ and a volume form $\omega$ there exists (locally) a positive function $f$ such that $L\_{fv}\omega=0$ (Indeed, one can take coordinates such that $v=\frac{\partial}{\partial x\_1}$, $\omega=A(x\_1,...,x\_n)dx\_1\wedge ...\wedge dx\_n$. In these coordinates the condition $L\_{fv}\omega=... | 8 | https://mathoverflow.net/users/2823 | 22696 | 14,962 |
https://mathoverflow.net/questions/22673 | 22 | In my mind, [algebraic topology](http://en.wikipedia.org/wiki/Algebraic_topology) is comprised of two components:
1. [Chain complex](http://en.wikipedia.org/wiki/Chain_complex) information, which is **intrinsic** information concerning how your object may be built up out of simple "lego blocks".
2. [Characteristic cl... | https://mathoverflow.net/users/2051 | Natural setting for characteristic classes? | Here is a perspective that might help to put characteristic classes into a more general framework. I like to think that there are two levels of the theory. One is geometric and the other is about extracting information about the geometry through algebraic invariants.
Bear with me if this sounds to elementary and obviou... | 7 | https://mathoverflow.net/users/4910 | 22704 | 14,967 |
https://mathoverflow.net/questions/22702 | 4 | ***Question*** : are the continuous characters of the form
* $\eta : \mathbb{Z}\_p^\* \to \mathbb{Z}\_p^\*$, or
* $\eta : (1+p\mathbb{Z}\_p)^{\times} \to \mathbb{Z}\_p^\*$ (i.e., on the principal units in $\mathbb{Z}\_p^\*$)
well understood? Can such characters be classified in either case ?
I'm hoping to find a... | https://mathoverflow.net/users/4399 | Classifying continuous characters $X \to \mathbb{Z}_p^*$, $X=\mathbb{Z}_p^*$ or $(1+p\mathbb{Z}_p)^{\times}$ ? | Yes, these are very well-understood! Here's what they are. If $p$ is odd then $\mathbf{Z}\_p^\times$ is a direct product of $\mu$, the subgroup of $p-1$th roots of unity, and $1+p\mathbf{Z}\_p$, the principal units. A continuous character of the product is a product of continuous characters, so that reduces the first p... | 8 | https://mathoverflow.net/users/1384 | 22705 | 14,968 |
https://mathoverflow.net/questions/22707 | 5 | Suppose $M$ and $M'$ are two $R$-moules (I am most interested in the case of $R$ a DVR). If $M\otimes M'$ is a semi-simple module (i.e., every submodule is a direct summand) then is it true that the tensor factors $M$ and $M'$ are semi-simple ?
I.e., if this is not true then it would be nice to see a counter-example.... | https://mathoverflow.net/users/4399 | If a tensor product of modules is semi-simple, are the tensor factors semi-simple ? | No; if $R$ is a DVR with uniformizing paramenter $t$, the $R$-module $(R/tR) \otimes (R/t^2R) = R/tR$ is semisimple, but $R/t^2R$ is not.
| 19 | https://mathoverflow.net/users/4790 | 22709 | 14,970 |
https://mathoverflow.net/questions/22706 | 2 | In my Analysis class, we started to prove a theorem that said:
>
> Let a > 1. So there is a unique increasing function $f:(0,\infty)\to\mathbb{R}$ so that:
>
>
> 1. $f(a) = 1$
> 2. $f(xy) = f(x) + f(y)\quad\forall x, y > 0$
>
>
> First we supposed its existence. Then through 2 it follows that f is a group homom... | https://mathoverflow.net/users/5660 | Uniqueness of the logarithm function | Suppose that at some point $x$ the function $f(x)$ assumes a different value $f(x)\ne\sup A\_x$.
If $f(x)>\sup A\_x$, take a rational number $m/n$ in the interval $(\sup A\_x,f(x))$. It does not belong to $A\_x$ (otherwise it's at most $\sup A\_x$), so $a^m>x^n$ implying $m>nf(x)$, hence $f(x)$ is less than $m/n$, a ... | 5 | https://mathoverflow.net/users/4953 | 22710 | 14,971 |
https://mathoverflow.net/questions/22722 | 20 | I am looking for a proof of the following fact:
*If $R$ is a principal artin local ring and $M$ a finitely generated $R$-module, then $M$ is a direct sum of cyclic $R$-modules.*
(Apparently such rings $R$ are called, e.g. in Zariski-Samuel, special principal ideal rings.)
I almost didn't ask this question for fea... | https://mathoverflow.net/users/4351 | Why are finitely generated modules over principal artin local rings direct sums of cyclic modules? | Let $I$ be the annihilator of $M$, by assumption $I=(\pi^i)$ for some $i$. One can view $M$ as an $R/I$ module and furthermore, embed $0 \to R/I \to M$. But $R/I$ is also principal artin local, so it is a quotient of a DVR by an element (by Hungerford's [paper](https://projecteuclid.org/journals/pacific-journal-of-math... | 15 | https://mathoverflow.net/users/2083 | 22729 | 14,979 |
https://mathoverflow.net/questions/22629 | 48 | Are there primes of every Hamming weight? That is, for every integer $n \in \mathbb{Z}\_{>0}$ does there exist a prime which is the sum of $n$ distinct powers of $2$?
In this case, the [Hamming weight](https://en.wikipedia.org/wiki/Hamming_weight) of a number is the number of $1$s in its binary expansion.
Many prob... | https://mathoverflow.net/users/5597 | Are there primes of every Hamming weight? | Fedja is absolutely right: this has been proven, for sufficiently large $n$, by Drmota, Mauduit and Rivat.
Although it looks at first sight as though this question is as hopeless as any other famous open problem on primes, it is easy to explain why this is not the case. Of the numbers between $1$ and $N := 2^{2n}$, t... | 56 | https://mathoverflow.net/users/5575 | 22738 | 14,984 |
https://mathoverflow.net/questions/22735 | 4 | Let K denote either the field of real numbers or the complex fields. An analytic function over $K^n$ is a function that can be represented locally by a convergent power series in n variables with coefficients in K.
My question is that can we take K to be other fields? It seems that such a field K should satisfy some ... | https://mathoverflow.net/users/nan | Analytic Functions over Fields other than Real or Complex Numbers | There are several rich theories of analysis on non-archimedian theories. Neal Koblitz' book on $p$-adic analysis is a good introduction. Non-archimedian analysis by Bosch, Güntzer and Remmert is more encyclopedic. Berkovich's Spectral Theory and Analysis over Non-archimedian Fields introduces his beautiful theory of an... | 13 | https://mathoverflow.net/users/5147 | 22746 | 14,991 |
https://mathoverflow.net/questions/22634 | 3 | I have a problem wherein I have defined a function $I\_r(t) = \int e^{(2r-1)at} \int e^{(2r-3)at} \cdots \int e^{at} dt\cdots dt$, and $I\_r(0) = 0$, for $r = 1,2,3,\ldots$.
I find that $e^{-ar^2t} I\_r(t) = \left(1-e^{-at}\right)^r q(t)$, where $q(exp(-at))$ is a polynomial of $e^{-at}$. Is there a general technique... | https://mathoverflow.net/users/5640 | "Nice" Solution to repeated integral | I don't think you'll get a closed form but you get get an expression that involves products and sums that isn't too horrible and is easy to evaluate for particular cases. It'll take me forever to type it up here with nice formatting but I'll describe the approach in a way that's easy to reproduce.
We're starting with... | 2 | https://mathoverflow.net/users/1233 | 22757 | 14,998 |
https://mathoverflow.net/questions/22745 | 3 | A special case says it all ... Let $ w\_1 < w\_2 < \ldots < w\_{12} $ be an increasing sequence of $12$ integers ("weights") such that the total weight $W=\sum\_{k=1}^{12}w\_k$ is even.
Say that $I \subseteq \lbrace 1,2, \ldots ,12 \rbrace$ is an exact subset iff
the sum $\sum\_{k \in I}w\_k$ equals $\frac{W}{2}$. M... | https://mathoverflow.net/users/2389 | Unique way to partition into two parts of equal weight | The answer is **yes**. Consider the sequence
$100, 200, 201, 202, 500, 601, 700, 701, 801, 1000, 1194, 1200.$
It is easy to see that $X=\{1,2,5,7,10,12\}$ and $Y=\{3,4,6,8,9,11\}$ are exact. Moreover, we claim that they are the only exact subsets. To see this, note that for every subset $X'$ of $X$, the sum of the... | 10 | https://mathoverflow.net/users/2233 | 22764 | 15,002 |
https://mathoverflow.net/questions/22753 | 6 | Either the following is a really stupid question or it is a really really stupid question, but here goes:
Does there exist a classification of $\ell$-adic 2-dimensional representations of $\mathrm{Gal}(\bar{\mathbb{Q}}\_p/\mathbb{Q}\_p)$, where $\ell\neq p$?
I did a quick search of the internet that came up rather... | https://mathoverflow.net/users/2147 | Classification of l-adic representations | When $\ell \neq p,$ these are rather straightforward to classify (except when $p = 2$);
see Tate's article in the second volume of Corvallis, for example.
The idea is that if $\rho$ is irred., then (unless $p = 2$), it must be induced from a character of a quadratic extension; thus the classification is given by loca... | 11 | https://mathoverflow.net/users/2874 | 22765 | 15,003 |
https://mathoverflow.net/questions/22767 | 11 | Consider a (non-stellated) polygon in the plane. Imagine that the edges are rigid, but that the vertices consist of flexible joints. That is, one is allowed to move the polygon around in such a way that the vertices stay a fixed distance from their adjacent neighbors. Such a system is called a **polygonal linkage**.
... | https://mathoverflow.net/users/750 | Is the area of a polygonal linkage maximized by having all vertices on a circle? | This is a theorem of Cramer. See "[On the impossibility of one rule-and-compass construction](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.980)" by Vladimir Jankovic.
For the quadrilateral case the quickest proof is using [Brahmagupta's formula](http://en.wikipedia.org/wiki/Brahmagupta%27s_formula#Exte... | 5 | https://mathoverflow.net/users/2384 | 22775 | 15,010 |
https://mathoverflow.net/questions/22678 | 0 | In a ring **R** (nonempty class of sets closed under difference and finite union), any sequence (here means a function on natural numbers $\mathbb N$) {$E\_i$} in **R** can be disjointlized to a disjoint sequence {$F\_i$} such that $\bigcup E\_i=\bigcup F\_i$ by traditional induction using the equation $F\_i=E\_i-\bigc... | https://mathoverflow.net/users/5072 | disjointlize an arbitrary sequence in a ring? | Let's take the usual ring of finite unions $E$ of half-open rectangles $[a,b)\times [c,d)$ on the plane. The closed half-plane $x+y\ge 0$ is a union of continuum of such rectangles (all possible rectangles contained in that closed half-plane) but, since each $E$ contained in this half-plane can intersect the boundary l... | 6 | https://mathoverflow.net/users/1131 | 22778 | 15,012 |
https://mathoverflow.net/questions/22768 | 4 | I'm trying to read chapter XIII of SGA1, and I'd appreciate some help about a few issues I'm having.
1. Definition 2.1.1. is of tamely ramified sheaves. The definition is as such: if $U$ is an open subscheme of $X$ (which is a scheme over $S$), $F$ a sheaf of sets on $U$, and $Y$ the reduced induced closed subscheme ... | https://mathoverflow.net/users/2665 | SGA1 Chapter XIII (tamely ramified sheaves) | If you look in the text, $y$ is not a closed point but a *maximal* point, meaning a maximally generic point. Then $O\_{X\_{\bar{s}y}}$ is a discrete valuation ring, and so it makes sense to talk about tameness of extensions. Also, surely she's working in the etale topology, otherwise it would be kind of silly to try an... | 2 | https://mathoverflow.net/users/1114 | 22787 | 15,019 |
https://mathoverflow.net/questions/22771 | 4 | Suppose $F/\mathbb{Q}$ is a totally real field of degree $d$ and class number one. Fix an ordering $\sigma\_1, \dots, \sigma\_d$ on the embeddings of $F$. Is
$\sum\_{\alpha \in \mathcal{O}\_F}e^{2\pi i (z\_1 \sigma\_1 (\alpha^2)+ \dots +z\_d \sigma\_d(\alpha^2))}$
a Hilbert modular form of parallel weight $1/2$?
... | https://mathoverflow.net/users/1464 | Jacobi's theta function over totally real fields | Yep -- though I have never thought through any technicalities regarding definition of half-integral weight Hilbert modular forms; I'm comfortable saying, at least, that the square of that theta function is a Hilbert modular form of weight 1. Harvey Cohn wrote several papers about this: see e.g.
MR0113855 (22 #4686)
C... | 5 | https://mathoverflow.net/users/431 | 22793 | 15,023 |
https://mathoverflow.net/questions/22719 | 11 | Given a real manifold $M$ with symplectic $2$-form $\omega$,
one can ask whether the cohomology class $[\omega] \in H^2(M;{\mathbb R})$ lies in the image of
$H^2(M;{\mathbb Z})$. If so, one can ask for a line bundle ${\mathcal L}$
with $c\_1({\mathcal L}) = [\omega]$ (or even better,
a connection $\alpha$ on $\mathca... | https://mathoverflow.net/users/391 | What are the implications of torsion in H^2 for geometric quantization? | If L\_1 and L\_2 are two line bundles on a manifold $M$ that differ by torsion, then their Chern characters
$$ch(L\_1) = 1 + c\_1(L\_1) + \frac{1}{2}c\_1(L\_1)^2 + \cdots$$
$$ch(L\_2) = 1 + c\_1(L\_2) + \frac{1}{2}c\_1(L\_2)^2 + \cdots$$
agree, if only because the right-hand sides of these formulas are taking place in ... | 8 | https://mathoverflow.net/users/1048 | 22794 | 15,024 |
https://mathoverflow.net/questions/22800 | 7 | Does the existence of (left) Haar measure on a locally compact topological group require that the group be Hausdorff?
| https://mathoverflow.net/users/1148 | Must a locally compact group be Hausdorff in order to possess a Haar measure? | No. Simon Rubinstein-Salzedo's ["On the existence and uniqueness of invariant measures on locally-compact groups"](http://simonrs.com/HaarMeasure.pdf) presents a proof of existence (and uniqueness up to a multiplicative strictly positive constant) of a left Haar measure given a locally-compact, not necessarily Hausdorf... | 7 | https://mathoverflow.net/users/441 | 22802 | 15,026 |
https://mathoverflow.net/questions/22801 | 10 | Let $X$ be a closed, orientable surface of genus at least 2, and let $\phi: \pi\_1(X) \to \pi\_1(X)$ be a surjective homomorphism. Is $\phi$ necessarily injective?
| https://mathoverflow.net/users/5499 | Self-homomorphisms of surface groups | Yes. Surface groups are [Hopfian](http://en.wikipedia.org/wiki/Hopfian_group). More generally, all [residually finite](http://en.wikipedia.org/wiki/Residually_finite) groups are Hopfian -- see Theorem IV.4.10 in Lyndon and Schupp's book "Combinatorial Group Theory".
| 10 | https://mathoverflow.net/users/317 | 22803 | 15,027 |
https://mathoverflow.net/questions/13291 | 10 | Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta function associated to $X(\overline{\mathbb{F}\_p})$ for $p$ a prime of good reduction. This is because one can identify the fa... | https://mathoverflow.net/users/290 | Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function? | Having resolved my [ignorance](https://mathoverflow.net/questions/22801/self-homomorphisms-of-surface-groups) concerning surface groups I can now answer question 1 negatively (or at least some formulation thereof). It is impossible if $Y$ is an oriented surface of genus at least $2$.
Suppose that $f: Y \to Y$ is a se... | 4 | https://mathoverflow.net/users/5499 | 22804 | 15,028 |
https://mathoverflow.net/questions/22806 | 3 | All varieties are defined over $\mathbb{C}$. Let $\pi : X \to C$ be an elliptic surface with $X$ and $C$ smooth. Then there is a Jacobian surface $\overline{\pi}: J(X) \to C$ (with a section) associated to $X$. Do we also have a morphism $\varphi : X \to J(X)$ such that $\overline{\pi}\circ \varphi = \pi$? (If yes, a p... | https://mathoverflow.net/users/5197 | Jacobian surface associated to an elliptic surface | I think the following at least gives a rational map: Choose a curve $D$ in $X$ which maps dominantly to $C$ with degree $n$. For a smooth fibre $F$ of $\pi$ define a map to the correspoding fibre of $\bar{\pi}$ by sending a point $p$ to the class of $n[p] - [F\cdot D]$. (This is a cycle of degree 0.) With a little work... | 3 | https://mathoverflow.net/users/519 | 22810 | 15,031 |
https://mathoverflow.net/questions/22799 | 5 | Let $E$ be an arbitrary Banach space and let $T:E^{\*}\rightarrow\ell^{2}$
be a linear continuous operator. Is it true that $T$ must be the
$so$-limit (i.e., limit w.r.t. the strong operator topology) of a
net $(S\_{d})^{\*}$ $\left(d\in\mathcal{D}\right)$ of adjoint operators,
with $S\_{d}$:$\ell^{2}$ $\rightarrow$ $E... | https://mathoverflow.net/users/2508 | Is any continuous linear operator from a dual Banach space to a separable Hilbert space the strong-operator limit of a net of adjoint operators of less or equal norm ? | You have that $T^\*:(\ell^2)^\* \rightarrow E^{\*\*}$, so using that $(\ell^2)^\*$ is isomorphic to $\ell^2$ (just in the linear sense, as we already have a co-ordinate system), we can regard $T^\*$ as a map $\ell^2\rightarrow E^{\*\*}$.
By the Principle of Local Reflexivity (I've used a paper of Behrends in the past... | 7 | https://mathoverflow.net/users/406 | 22822 | 15,038 |
https://mathoverflow.net/questions/22814 | 18 | A group $G$ is said to be linear if there exists a field $k$, an integer $n$ and an injective homomorphism $\varphi: G \to \text{GL}\_n(k).$
Given a short exact sequence
$1 \to K \to G \to Q \to 1$ of groups where $K$ and $Q$ are linear (over the same field), is it true that $G$ is linear too?
Background: [Arithmet... | https://mathoverflow.net/users/3380 | Are extensions of linear groups linear? | The universal central extension $\widetilde{\text{Sp}\_{2n}}\mathbb{Z}$ is the preimage of $\text{Sp}\_{2n}\mathbb{Z}$ in the universal cover of $\text{Sp}\_{2n}\mathbb{R}$, and fits into the sequence
$$1\to \mathbb{Z}\to \widetilde{\text{Sp}\_{2n}}\mathbb{Z}\to \text{Sp}\_{2n}\mathbb{Z}\to 1.$$
Deligne proved that... | 25 | https://mathoverflow.net/users/250 | 22823 | 15,039 |
https://mathoverflow.net/questions/19410 | 26 | This is a more focused version of [Summation methods for divergent series](https://mathoverflow.net/questions/19201/summation-methods-for-divergent-series).
>
> Let $a\_n$ be a sequence of real
> numbers such that $\lim\_{x \to 1^{-}}
> > \sum a\_n x^n$ and $\lim\_{s \to 0^{+}}
> > \sum a\_n n^{-s}$ both exist. (I... | https://mathoverflow.net/users/297 | Do Abel summation and zeta summation always coincide? | I think the answer is 'yes.' I don't have a suitably general reason why this is the case, although surely one exists and is in the literature somewhere.
At any rate, for the problem at hand, we have for $s > 0$
$$\sum \frac{a\_n}{n^s} = \frac{1}{\Gamma(s)}\int\_0^\infty \sum a\_n e^{-nt} t^{s-1} dt.$$
*Edit: the ... | 11 | https://mathoverflow.net/users/5621 | 22825 | 15,041 |
https://mathoverflow.net/questions/22821 | 8 | I'm studying the deformation theory of compact complex manifolds as developed by Kodaira and Spencer. On the side I'm reading as much about deformation theory in general as I can get my hands on (and understand), and I've been wondering about the relationship between the basic definitions in the analytic and algebraic ... | https://mathoverflow.net/users/4054 | Complex analytic vs algebraic families of manifolds | The standard situation in Kodaira-Spencer's work is the following:
If you're on the algebraic side and you have a smooth ("smooth" in the sense of algebraic geometry) and proper (proper in the sense of algebraic geometry) map $\pi: X \to Y$ , then when you translate this to the analytic side, "smooth" turns into "sub... | 4 | https://mathoverflow.net/users/83 | 22829 | 15,044 |
https://mathoverflow.net/questions/22830 | 5 | The Baire category theorem implies that a nonempty complete metric space without isolated points must be uncountable. In many situations I have encountered, the "natural examples" of complete metric spaces without isolated points (of a certain type, or possibly with some additional structure) in fact have at least cont... | https://mathoverflow.net/users/1149 | Improvements of the Baire Category Theorem under (not CH)? | A complete space without isolated points has at least continuum cardinality. At least if you agree to use (some form of) Axiom of Choice.
Choose two disjoint closed balls $B\_1$ and $B\_2$. Inside $B\_1$, choose disjoint closed balls $B\_{11}$ and $B\_{12}$. Inside $B\_2$, choose disjoint closed balls $B\_{21}$ and $... | 15 | https://mathoverflow.net/users/4354 | 22831 | 15,045 |
https://mathoverflow.net/questions/22839 | 3 | Let X be locally compact and Hausdorff, and let $f:X\rightarrow\mathbb R$ be a function. Suppose that for all finite regular (positive) Borel measures $\mu$, we know that $f$ is $\mu$-measurable. Does it follow that $f$ is Borel? If not, what's a good counter-example?
Definitions: The Borel sigma-algebra is generated... | https://mathoverflow.net/users/406 | Borel vs measure for all Borel measures | Let $f: \mathbb{R} \to \mathbb{R}$ be the characteristic function of a subset $A \subseteq \mathbb{R}$ which is analytic but not Borel. Then $f$ is universally measurable but not Borel.
| 11 | https://mathoverflow.net/users/4706 | 22840 | 15,050 |
https://mathoverflow.net/questions/22838 | 89 | I am interested in how to select interesting yet reasonable problems for students to work on, either at Honours (that is, a research-based single year immediately after a degree) or PhD.
By this I mean a problem that is unsolved but for which there is a good chance that a student can solve it either completely or pa... | https://mathoverflow.net/users/1492 | How do you select an interesting and reasonable problem for a student? | Let me first answer a slightly different question, how to organize one's thoughts about such problems. I simply maintain a list of suitable projects, with ideas on how to approach them, and put them in a file "Dissertation Problems.tex". Some of these are projects that I might like to carry out myself, but many are pro... | 59 | https://mathoverflow.net/users/1946 | 22844 | 15,053 |
https://mathoverflow.net/questions/22818 | 8 | The following is a somewhat vague question concerning logic, but with ideas from algebraic geometry (see in particular the example at the end). The vagueness is in the notion of "language".
Let $A$ be a set and fix some language in which to write propositions which assign a value of true or false to each element of A... | https://mathoverflow.net/users/2811 | Ideals of statements? | Yes, of course, the algebraic aspects of logic have been very well studied. There is a lot to say about this, but since I am supposed to be on a voluntary MO hiatus until the end of the semester, I will only mention a few things.
You might want to ask for your "ideals" to be closed under logical equivalence too. Othe... | 9 | https://mathoverflow.net/users/2000 | 22854 | 15,062 |
https://mathoverflow.net/questions/22851 | 2 | Is there a neat way to use something like the Brier score to score an infinite set of forecasts/outcomes?
| https://mathoverflow.net/users/4076 | How would one extend the Brier score to an infinite number of forecasts? | A general form of the Brier score, for an essentially arbitrary outcome space $\cal X$, is as follows.
Let $q(\cdot)$ be your quoted density for a random quantity $X$, with respect to a dominating measure $\mu$ over $\cal X$. Then your score, when outcome $X=x$ is realised, is
$$S(x, q(\cdot)) = \int q(t)^2 d\mu(t) ... | 5 | https://mathoverflow.net/users/5689 | 22856 | 15,063 |
https://mathoverflow.net/questions/22828 | 8 | I am confused and don't get the big picture concerning the connection between
* [Ito integral](http://en.wikipedia.org/wiki/Ito_integral)
* [Stratonovich integral](http://en.wikipedia.org/wiki/Stratonovich_integral)
* Standard results in probability theory concerning skewed distributions.
**Example:** Take e.g. the... | https://mathoverflow.net/users/1047 | Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory | It's not quite clear where exactly you have a difficulty. Of course, using stochastic calculus and Ito's integral (which is central to the modern theory of stochastic processes) to derive properties of the log-normal distribution is an overkill, but it might be a nice exercise.
Some random quick points on why Ito's i... | 10 | https://mathoverflow.net/users/2968 | 22858 | 15,064 |
https://mathoverflow.net/questions/22857 | 2 | Is there any introduction to abelian varieties of CM type?any reference?Like how to construct a abelian varieties given a CM field E?What is the properites of the Mumford Tate group of the abelian varieties of CM type?
| https://mathoverflow.net/users/3945 | Abelian varieties of CM type? | IIRC I learnt a lot from Katz' papers from the 1970s. Of course the basic construction is the same as the elliptic curve case: you take C^g, quotient out by the lattice coming from E via its g embeddings into C, and then you have to prove that the quotient is an abelian variety, which involves writing down a non-degene... | 3 | https://mathoverflow.net/users/1384 | 22863 | 15,067 |
https://mathoverflow.net/questions/22855 | 7 | A $\Delta$-set is a contravariant functor from the category $\Delta'$ of order-preserving injections to the category of sets (this is essentially what Allen Hatcher calls a $\Delta$-complex).
A main reason for working with simplicial sets instead of $\Delta$-sets should be that they allow quotients (see e.g. Allen Ha... | https://mathoverflow.net/users/4676 | Why do Delta-sets not allow quotients? | The basic issue is that not every function that we would like to describe between $\Delta$-complexes can be realized by a natural transformation between functors. The lack of degeneracy maps means that no map $X \to Y$ of $\Delta$-complexes that sends any simplex down to a degenerate simplex can be realized by a natura... | 6 | https://mathoverflow.net/users/360 | 22864 | 15,068 |
https://mathoverflow.net/questions/22865 | 4 | Does there exist a two variable analogue of the Weil conjecture?
What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V\_n = V ( GF(p^n))$ of points of a smooth algebraic variety over finite fields of characteristic $p$. Is it possible to have a sensible two-paramet... | https://mathoverflow.net/users/5301 | 2d Weil conjecture | The theory of arithmetic motivic integration (see, for example, the paper by Denef and Loeser in the proceedings of the 2002 ICM) takes into account the numbers you wish to encode in your two variable zeta function.
As Kevin Buzzard rightly points out, if the scheme is smooth, then the counts along powers of primes ... | 6 | https://mathoverflow.net/users/5147 | 22875 | 15,072 |
https://mathoverflow.net/questions/22860 | 22 | I need to give a lot of quite basic background to this question because I think (at least from conversing with fellow graduate students) that most mathematicians have not *really* thought about fractions for a long time. I think that there is an interesting germ of an idea in here somewhere, but I cannot exactly pinpoi... | https://mathoverflow.net/users/1106 | Do rational numbers admit a categorification which respects the following "duality"? | I don't claim to have a full answer, but I wanted to put it here so that others might help elaborate it.
I think that the "classical" (aside: at a recent seminar on Heegaard Floer homology, the word "classical" was defined as "posted on the arXiv") categorification of the positive rationals is to the world of finite ... | 12 | https://mathoverflow.net/users/78 | 22879 | 15,075 |
https://mathoverflow.net/questions/22884 | 1 | I have an graph with the following attributes:
* Undirected
* Not weighted
* Each vertex has a minimum of 2 and maximum of 6 edges connected to it.
* Vertex count will be < 100
* Graph is static and no vertices/edges can be added/removed or edited.
I'm looking for **all** subgraphs between a random subset of the ve... | https://mathoverflow.net/users/5692 | Graph algorithm to find all subgraphs that connect N arbitrary vertices | It looks like the paper
*Generating all the Steiner trees and computing Steiner intervals for a fixed number of terminals*
by Costa Dourado, de Oliveira, and Protti is what you want (available from ScienceDirect). I think the paper gives an algorithm for generating all the *minimal* (under subgraph inclusion) sub... | 4 | https://mathoverflow.net/users/2233 | 22888 | 15,079 |
https://mathoverflow.net/questions/22881 | 7 | Let $S$ be a locally noetherian scheme, $C$ the category of proper smooth $S$-schemes with geometrical connected fibres and $C\_\*$ the category of pointed objects of $S$, i.e. objects of $C$ together with a morphism $S \to C$. Also denote $A$ the category of abelian schemes over $S$. There is a well-known rigidity res... | https://mathoverflow.net/users/2841 | scheme-theoretic description of abelian schemes | How about smooth proper morphisms $X \to S$ with connected fibers, a section $S \to X$, such that the sheaf of Kähler differentials $\Omega\_{X/S}$ is a pullback from $S$, and such that the group scheme $\underline{\rm Aut}\_S X$ acts transitively on the fibers? The essential point is that the hypothesis on the differe... | 6 | https://mathoverflow.net/users/4790 | 22892 | 15,082 |
https://mathoverflow.net/questions/22883 | 14 | Let $U$ be an open subscheme of $\textrm{Spec} \ \mathbf{Z}$. The complement of $U$ is a divisor $D$ of $\textrm{Spec} \ \mathbf{Z}$.
**Q**. Can we classify the etale coverings of $U$ of a given degree?
Given a finite etale morphism $\pi:V\longrightarrow U$, the normalization of $X$ in the function field $L$ of $V... | https://mathoverflow.net/users/4333 | Etale coverings of certain open subschemes in Spec O_K | As Kevin points out, $V$ is indeed $\mathcal{O}\_K[\frac{1}{2}]$ in your example. Your link to the fundamental group is also correct. $\pi\_1(U)$ is the Galois group of the maximal extension of $\mathbb{Q}$ unramified outside 2 (since you can restrict attention to the *connected* etale $\mathbb{Q}$-algebras)\*. More ge... | 5 | https://mathoverflow.net/users/35575 | 22893 | 15,083 |
https://mathoverflow.net/questions/22891 | 5 | For a projective variety $X$, Serre's vanishing theorem says that $H^i(X, \mathcal{F}(n))=0$ for any coherent sheave, $i>1$ and sufficiently large $n$. I am wondering, is there a similar type of vanishing theorem on quasi-projective varieties, namely, let $Y$ be a quasi-projective variety, what can we say about the van... | https://mathoverflow.net/users/2348 | Serre type vanishing theorem of coherent sheaves on quasi-projective variety? | Even for quasi-affine variety you don't have a vanishing theorem except Grothendieck's vanishing theorem on a noetherian topological space of finite dimension.
Consider, for example, affine plane without point $\mathbf{A}^2\backslash 0.$ Then the structure sheaf is ample but the first cohomology $H^1(\mathbf{A}^2\ba... | 11 | https://mathoverflow.net/users/2464 | 22894 | 15,084 |
https://mathoverflow.net/questions/22897 | 42 | Is there a nice characterization of fields whose automorphism group is trivial? Here are the facts I know.
1. Every prime field has trivial automorphism group.
2. Suppose *L* is a separable finite extension of a field *K* such that *K* has trivial automorphism group. Then, if *E* is a finite Galois extension of *K* c... | https://mathoverflow.net/users/3040 | Fields with trivial automorphism group | As Robin as pointed out, for all primes $p$, $\mathbb{Q}\_p$ is **rigid**, i.e., has no nontrivial automorphisms. It is sort of a coincidence that you ask, since I spent much of the last $12$ hours writing up some material on multiply complete fields which has applications here:
Theorem (Schmidt): Let $K$ be a field ... | 32 | https://mathoverflow.net/users/1149 | 22903 | 15,090 |
https://mathoverflow.net/questions/22906 | 1 | Equivalently, is there a graph that contains an infinite simple path that has a start and an end point? My intuition is that there is no such graph, but I'm finding it hard to articulate why.
| https://mathoverflow.net/users/800 | Is it possible to define a graph that has two vertices that are infinitely far apart? | One reasonable definition of "simple path" is a connected acyclic subgraph with vertex degree at most two. But, to avoid circularity, we need a definition for being connected. I think the standard definition is that a subgraph is connected iff every two vertices are at a finite distance from each other, so with this de... | 8 | https://mathoverflow.net/users/440 | 22912 | 15,094 |
https://mathoverflow.net/questions/22868 | 5 | Let $f(x)$ be a polynomial with real coefficients, and let $||\cdot||$ be the distance-from-the-nearest-integer function. It is known that for any $ \epsilon > 0 $, the set $S$ of positive integer solutions of the inequality $||f(x)|| < \epsilon$ has bounded gaps. This means that if $x\_1 < x\_2 < \ldots$ are the eleme... | https://mathoverflow.net/users/5229 | Wanted: A constructive version of a theorem of Furstenberg and Weiss | Dear RJS,
I think Tim Gowers is right - the problem seems too hard. Reasonably good bounds are known on (for example) the *least* $n \geq 1$ for which $\Vert n^2 \sqrt{2} \Vert \leq \epsilon$; one can find such an $n$ with $n \leq \epsilon^{-7/4 + o(1)}$. This is a result of Zaharescu [Zaharescu, A; Small values of $... | 5 | https://mathoverflow.net/users/5575 | 22913 | 15,095 |
https://mathoverflow.net/questions/22914 | 13 | Let A be an Abelian category.
From this category, we can form the chain complex category Ch(A). The objects of Ch(A) are chain complexes of objects of A. The morphisms of Ch(A) are chain maps. Ch(A) is an Abelian category for every Abelian category A.
Now from Ch(A), we can form the chain homotopy category K(A). Th... | https://mathoverflow.net/users/5698 | Is the chain homotopy category, K(Ab), an Abelian category? By Ab, I mean the category of Abelian groups. | You might want to have a look at [my answer](https://mathoverflow.net/questions/15658/how-do-i-know-the-derived-category-is-not-abelian/15662#15662) to this question.
| 11 | https://mathoverflow.net/users/310 | 22916 | 15,097 |
https://mathoverflow.net/questions/22907 | 9 | This question is inspired by, but is independent of: [Sheaf Description of G-Bundles](https://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles)
Line bundles are classified by $H^1(X,\mathcal{O}^\times\_X)$. We also know that in general that $H^1(X,G)$, where $G$ is a sheaf from open sets in $X$ to $Grps... | https://mathoverflow.net/users/5309 | Confusion about how the first cohomology classifies torsors | The general principle is: if you have some objects which are locally trivial but globally possibly not trivial then the isomorphism classes of such objects are classified by $H^1(X,\underline{Aut})$, where $\underline{Aut}$ is the sheaf of automorphisms of your objects.
So, if your objects locally are $U\times \mathb... | 14 | https://mathoverflow.net/users/1784 | 22921 | 15,101 |
https://mathoverflow.net/questions/22929 | 12 | A generic elliptic curves over C has automorphism group of order 2. The elliptic curves with extra automorphisms are C/Z[i] (automorphism group of order 4) and C/Z[w] where w is a primitive 3rd root of unity (automorphism group of order 6). One can use their extra automorphisms to prove that they can be defined over Q
... | https://mathoverflow.net/users/683 | Extra automorphisms of curves and definability over \bar Q | If the group has order 84(g-1), then the curve is a Galois cover of the (2,3,7) orbifold. By Belyi's theorem, it is defined over a number field.
| 9 | https://mathoverflow.net/users/121 | 22933 | 15,109 |
https://mathoverflow.net/questions/22927 | 242 | As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't see the point. Yes, one can prove some pretty disturbing things, but I just don't feel like losing any sleep over it if ... | https://mathoverflow.net/users/4362 | Why worry about the axiom of choice? | The best answer I've ever heard --- and I think I heard it here on MathOverflow from Mike Shulman, which suggests that this question is roughly duplicated somewhere else --- is that you should care about constructions "internal" to other categories:
1. For many, many applications, one wants "topological" objects: top... | 151 | https://mathoverflow.net/users/78 | 22938 | 15,113 |
https://mathoverflow.net/questions/22923 | 39 | Does there exist an algorithm which computes the Galois group of a polynomial
$p(x) \in \mathbb{Z}[x]$? Feel free to interpret this question in any reasonable manner. For example, if the degree of $p(x)$ is $n$, then the algorithm could give a set of permutations $\pi \in Sym(n)$ which generate the Galois group.
| https://mathoverflow.net/users/4706 | Computing the Galois group of a polynomial | There is an algorithm described in an ancient and interesting book on Galois Theory by Leonard Eugene Dickson. Here is a brief sketch in the case of an irreducible polynomial $f\in \mathbb{Q}[x]$.
Suppose that $z\_1\ldots z\_n$ are the roots of $f$ in some splitting field of $f$ over $\mathbb{Q}$. (We don't need to c... | 35 | https://mathoverflow.net/users/5229 | 22946 | 15,118 |
https://mathoverflow.net/questions/22941 | 4 | For example, deciding whether or not the following is a category seems to depend on the above question (from Awodey's Category Theory, pg. 6):
> "What if we take sets as objects and as arrows, those $f : A \rightarrow B$ such that for all $b \in B$, the subset $f$-1$(b) \subseteq A$ is finite?"
Define for each $n ... | https://mathoverflow.net/users/5700 | Is an "infinite compositions of arrows" meaningful? | This is probably not what the OP is looking for, but there is a notion of "infinite composition of arrows" which often appears for example in categorical homotopy theory:
If $f\_0 : X\_0 \to X\_1$, $f\_1 : X\_1 \to X\_2$, $\ldots$ are morphisms in a category $C$ and the colimit of the diagram $X\_0 \to X\_1 \to X\_2 ... | 10 | https://mathoverflow.net/users/126667 | 22947 | 15,119 |
https://mathoverflow.net/questions/22950 | 17 | When I hear the phrase "line bundle" the first thing that pops into my head is a mobius band. But this is a bad picture from an algebraic point of view since any line bundle on an affine variety is trivial. Anyway, my question is: is there a way of seeing more concretely what "goes wrong" when you try to construct the ... | https://mathoverflow.net/users/2615 | why isn't the mobius band an algebraic line bundle? | Consider the real algebraic line bundle $\mathcal{O}(-1)$ over the real algebraic variety $\mathbb{R}\mathbb{P}^1$. It is nontrivial hence continuously isomorphic to the "Moebius" line bundle (there are only 2 line bundles on the circle, up to continuous isomorphism), so its total space is homeomorphic o the "Moebius s... | 27 | https://mathoverflow.net/users/4721 | 22954 | 15,121 |
https://mathoverflow.net/questions/22078 | 48 | It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}\_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freeness of... | https://mathoverflow.net/users/3927 | Smooth linear algebraic groups over the dual numbers | This is not a direct answer to the question for a general group scheme $G \to S$ and I am not an expert in this area. However, I would like to point out that the resolution property of stacks is a natural condition that appears in this context of Hilbert's 14th problem by work of R. W. Thomason:
*Equivariant resoluti... | 11 | https://mathoverflow.net/users/4101 | 22955 | 15,122 |
https://mathoverflow.net/questions/22953 | 12 | Let $0 < a < 1$ be fixed, and integer $n$ tends to infinity. It is not hard to show that the number of integers $k$ coprime to $n$ such that $1\leq k\leq an$ asymtotically equals $(a+o(1))\varphi(n)$.
The question is: what are the best known estimates for the remainder and where are they written?
Many thanks!
| https://mathoverflow.net/users/4312 | distribution of coprime integers | Vinogradov, I. M. An introduction to the theory of numbers, Ch. 2, problem N 19. It gives error term $O(\tau(n))$. But direct application of inclusion-exclusion principle gives $O(2^{\omega(n)})$ (where $\tau$ is the number of divisors, and $\omega$ is the number of prime divisors. Standart solution with Mobius functio... | 10 | https://mathoverflow.net/users/5712 | 22959 | 15,125 |
https://mathoverflow.net/questions/22943 | 3 | This question comes from reading Washington's proof of Iwasawa's theorem, and wanting to learn the commutative algebra version of the classification of finitely-generated $\Lambda$-modules. I went to the reference in Serre, and am now hung up on a point in his classification.
---
Let $A$ be a regular local ring o... | https://mathoverflow.net/users/5473 | Reflexive modules over a 2-dimensional regular local ring | Hello,
I guess that $\mathrm{codh}$ actually means $\mathrm{depth}$, that is the length of a maximal regular
sequence on a module.
Then $\mathrm{codh}(N/M)\geq 1$ is a consequence of the fact that the maximal ideal of $A$ does not annihilate the module due to reflexivity.
The next inequality is Auslander-Buchsbaum.
F... | 2 | https://mathoverflow.net/users/3556 | 22961 | 15,127 |
https://mathoverflow.net/questions/22908 | 36 | A few months ago, I was curious about some properties of Maass cusp forms, of nonabelian arithmetic origin. As a result, I went through a somewhat predictable process of finding a totally real $A\_4$ extension of $Q$, lifting the resulting projective Galois representation to an honest Galois representation, and writing... | https://mathoverflow.net/users/3545 | Does anyone want a pretty Maass form? | [these are comments, not an answer, but there were too many for the comments box]
Hey---I wrote that code too! I did it to teach myself "practical Maass forms". I wrote in pari, not sage. I didn't do the example you did. Here's what I did, for what it's worth. First I tried a dihedral example. I used the Hilbert clas... | 13 | https://mathoverflow.net/users/1384 | 22963 | 15,128 |
https://mathoverflow.net/questions/22975 | 14 | A quote from Wikipedia's article on the [Rotation group](http://en.wikipedia.org/wiki/Rotation_group):
>
> Consider the solid ball in $\mathbb{R}^3$ of
> radius $\pi$ [...].
> Given the above, for every point in
> this ball there is a rotation, with
> axis through the point and the origin,
> and rotation angl... | https://mathoverflow.net/users/5716 | How to demonstrate $SO(3)$ is not simply connected? | A loop is homotopically trivial if it can be continuously deformed to the constant loop. This means that at every step of the deformation (every "instant in time") you still have a loop. There are two kinds of loops on the unit ball with antipodal identifications in the boundary: either it's also a loop in the ball (wi... | 12 | https://mathoverflow.net/users/394 | 22977 | 15,138 |
https://mathoverflow.net/questions/22985 | 0 | The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let's consider the natural embedding $GL\_n(\mathbb C) \to \mathbb C^{n^2} \backslash \{0\}$. As was discussed [in this que... | https://mathoverflow.net/users/2260 | Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology | $\mathbb C^{n^2} \smallsetminus {0}$ is homotopy equivalent to $S^{2n^2-1}$, not $S^{n^2-1}$, and $\mathrm H^{2n^2-1}(GL\_n(\mathbb C)) = 0$.
| 11 | https://mathoverflow.net/users/4790 | 22988 | 15,144 |
https://mathoverflow.net/questions/22984 | 18 | Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H\_g$ of isotopy classes of diffeomorphisms of $K$ is called the *handlebody group*. (It embeds as a subgroup of the mapping class group of the genus $g$ surface $\partial K$.) The fundamental group of $K$ is a free group of rank $g$, so th... | https://mathoverflow.net/users/4910 | The kernel of the map from the handlebody group to Outer automorphisms of a free group | This result is due to Luft; see "Actions of the homeotopy group of an orientable 3-dimensional handlebody".
McCullough, in "Twist groups of compact 3-manifolds", proves that the twist group is not finitely generated and gives further references.
EDIT. Ninja'ed (well, at least the first sentence).
| 12 | https://mathoverflow.net/users/1650 | 23006 | 15,152 |
https://mathoverflow.net/questions/23003 | 5 | These are parametrized by $H^1(Gal(\mathbb{Q}), Aut X)$, where X is *some* $\mathbb{Q}$-model of the curve.
It was established in [Confusion about how the first cohomology classifies torsors](https://mathoverflow.net/questions/22907/confusion-about-how-the-first-cohomology-classifies-torsors)
that fiber bundles over ... | https://mathoverflow.net/users/5309 | Understanding different Q-models of a curve over C | Yes.
Roughly, the idea is that once your base object $X/B$ is fixed then for any object $Y$ you can consider the sheaf $Iso(X,Y)$ of isomorphisms from $X$ to $Y$. This has an action of the sheaf $Aut(X)$ and on any open cover $U$ where $Y|\_U \cong X|\_U$, making a choice of such an isomorphism gives an isomorphism o... | 4 | https://mathoverflow.net/users/360 | 23015 | 15,158 |
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