parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/51464 | 6 | It's known that if $A$ is an abelian variety of totally multiplicative reduction over a p-adic field K, then, after taking a finite field extension, it becomes isomorphic, as a rigid analytic group, to $G\_m^g/\prod\_{i=1}^g q\_i^{\mathbb{Z}}$ where $q\_i$ are points (after a finite field extension) of $G\_m^g$ which g... | https://mathoverflow.net/users/7108 | Tate models for semistable algebraic varieties with mixed reduction over a local field | If $A$ has semi-abelian reduction, then $A$ is uniformized by a semi-abelian variety $G\_A$,
namely there is an exact sequence
$$0 \to \Gamma\_A \to G\_A \to A \to 0,$$
where $\Gamma\_A$ is free of finite rank, $G\_A$ is semi-abelian, and the maximal abelian variety quotient of $G\_A$ has good reduction.
This is due... | 10 | https://mathoverflow.net/users/2874 | 51465 | 32,274 |
https://mathoverflow.net/questions/51474 | 5 | Let $G$ be a finite abelian group. Is it true that the following element of the group ring ${\mathbb Z}[G]$:
$$
\prod\_{g\ne 1}(1-g)
$$
is non-zero?
| https://mathoverflow.net/users/nan | Group ring computation | Counterexample : $G=(\mathbb{Z}/2)^2$. The product is $1-a-b+ab -ab +a^2b+ab^2 -a^2b^2$, with $a^2=b^2=1$, $ab=ba$.
PS: on the other hand, this is true if (and only if?) $G$ cyclic, since then you have an injective character $\chi : G\to \mathbb{C}^\times$, whose linear extension to $\mathbb{Z}[G]$ has nonzero value ... | 15 | https://mathoverflow.net/users/6451 | 51475 | 32,278 |
https://mathoverflow.net/questions/51294 | 23 | I'm not quite sure if this is the right place to ask, and if this is the right way to ask, but I dare.
In philosophy (of mind, e.g.) the concept of [supervenience](http://en.wikipedia.org/wiki/Supervenience) is used:
>
> "Supervenience [is] used to describe
> relationships between sets of
> properties in a mann... | https://mathoverflow.net/users/2672 | Supervenience in mathematics | I want to try another answer, not because I think you will necessarily accept it, but because if you don't then I think your reasons for not doing so will clarify the question.
One definition of supervenience is that A supervenes on B if you can't have a change to A without a change to B. For instance, some people ho... | 16 | https://mathoverflow.net/users/1459 | 51487 | 32,282 |
https://mathoverflow.net/questions/51499 | 4 | Is there a finite set of primes, $S \subset Spec(\mathbb{Z})$ such that if $K$ over $\mathbb{Q}$ is a number field such that every $p \in S$ is completely split in $K$ then $K=\mathbb{Q}$? If so, what do we know about such $S$'s?
| https://mathoverflow.net/users/5309 | Is there a finite set of primes such that if K over Q is completely split at all those primes then it is Q? | This is not even true for quadratic extensions. Given primes $p\_1, p\_2, ... p\_n$ find a prime $q \equiv 1 \bmod 4p\_1 ... p\_n$, which exists by Dirichlet's theorem, and consider $K = \mathbb{Q}(\sqrt{q})$.
| 15 | https://mathoverflow.net/users/290 | 51500 | 32,288 |
https://mathoverflow.net/questions/51494 | 41 | It is well known that a separable space is a topological space that has a countable dense subset. I am wondering how is this related to the name 'separable'? Any intuition where the name come from?
| https://mathoverflow.net/users/1992 | Why the name 'separable' space? | As far as I know the word *separable* was introduced by M. Fréchet in *Sur quelques points du calcul fonctionnel*, Rend. Circ. Mat. Palermo **22** (1906), 1-74. The paper can be obtained via [this link](http://dx.doi.org/10.1007/BF03018603) (Springer). It's the famous paper in which he introduced metric spaces. He cons... | 40 | https://mathoverflow.net/users/11081 | 51501 | 32,289 |
https://mathoverflow.net/questions/51520 | 7 | In the presence of Godel's Completeness Theorem, the 2nd Incompleteness Theorem has
the following strictly model theoretic interpretation: if there exists any model at all of (say) ZFC, there also exists a model M of ZFC such that M contains no (internal) model of ZFC.
That is the direct model-theoretic translation of:... | https://mathoverflow.net/users/10909 | 2nd Incompleteness and Model Theory | Hi David. The following is ("probably", he says) not what you want, but I am leaving it here, as it may explain some of the context. It is an argument of Woodin, similar to one of Jech, which though mathematically alike, is more "proof theoretic" in nature. I wrote some notes on it; they are [here](http://caicedoteachi... | 7 | https://mathoverflow.net/users/6085 | 51521 | 32,302 |
https://mathoverflow.net/questions/51512 | 0 | My question is quite simple, but I was unable to find an answer by googling, since you can't exactly google syntax. What does the $\in \cdot$ mean in:
$$\lim\_{n\to\inf}||P(S\_n\in\cdot)-P(S\_n+k\in\cdot)||\_{tv}=0$$
This is from coupling lectures in probability theory.
| https://mathoverflow.net/users/12111 | "X \in \cdot" in Probability Measure | If $S\_n$ is a real-valued random variable, then I would read $P(S\_n \in \cdot)$ as denoting the probability measure $P \circ S\_n^{-1}$ on $\mathbb{R}$, i.e. the set function $B \mapsto P(S\_n \in B)$ where $B \in \mathcal{B}\\_{\mathbb{R}}$, the Borel $\sigma$-algebra on $\mathbb{R}$. The quantity in norm bars is th... | 1 | https://mathoverflow.net/users/4832 | 51523 | 32,303 |
https://mathoverflow.net/questions/51171 | 21 | There are two questions here, an explicit one, and another (more vague) one that motivates it:
I am pretty certain the following should have a negative answer, but at the moment I'm not seeing how to argue about this and cannot locate an appropriate reference.
>
> In set theory without choice, suppose
> $X$ is ... | https://mathoverflow.net/users/6085 | Splitting infinite sets | Define a permutation model of ZFA as follows. Starting as usual (Ch 4 of Jech's *Axiom of Choice*) from a well-founded model $\mathcal M$ of ZFAC with infinite set $A$ of atoms, let $G$ be the group of all permutations of $A$; so $G$ can be identified with the group of all automorphisms of $\mathcal M$. For each finite... | 8 | https://mathoverflow.net/users/12109 | 51524 | 32,304 |
https://mathoverflow.net/questions/51525 | 1 | hello, I need a book where i can find the proof for the classification of 1-dimensional topological manifolds.
(i already have Milnor's for the classification of 1-dimensional smooth manifolds)
thank you
| https://mathoverflow.net/users/10820 | need a reference (topological manifolds) | If I remember well, Introduction to topological manifolds (Lee) proves the classification theorem for 1-dim manifolds.
| 3 | https://mathoverflow.net/users/1049 | 51528 | 32,307 |
https://mathoverflow.net/questions/51386 | 6 | I wonder what the prerequisites for learning the representation theory of reductive groups over a p-adic field are? Can someone recommend me any book or article for learning this theory if I wanna clearly know what this theory is about and what kind of applications this theory has?
I just know some fundamental conce... | https://mathoverflow.net/users/1930 | Prerequisites for P-adic Representations | Hello, I think a good first step is to learn the theory of admissible representations of p-adic groups and for this Godement's notes on Jacquet-Langlands theory and then Casselman's unpublished book on p-adic groups (available from his website) are good starting points. A good way to read Casselman's notes is to rewrit... | 9 | https://mathoverflow.net/users/10458 | 51537 | 32,310 |
https://mathoverflow.net/questions/51534 | 10 | We know that the resultant of two polynomials can be computed as the determinant of their Sylvester matrix ( <http://en.wikipedia.org/wiki/Sylvester_matrix> ). How do we compute the resultant of more than two polynomials? Can we generalise the use of the Sylvester matrix for this case?
| https://mathoverflow.net/users/12117 | Multipolynomial resultants | To speak of a (single) resultant of several polynomials one must allow for several variables as well and the situation becomes trickier that in the one variable case.
Let $f=(f\_0,\ldots, f\_k)$ be homogeneous polynomials (say over an algebraically closed field) of degrees $d\_0,\ldots, d\_k$ in $k+1$ variables. One ... | 9 | https://mathoverflow.net/users/2349 | 51545 | 32,317 |
https://mathoverflow.net/questions/27255 | 8 | The probability mass function for the [Skellam distribution](http://en.wikipedia.org/wiki/Skellam_distribution) for a count difference $k=n\_1-n\_2$ from two [Poisson-distributed](http://en.wikipedia.org/wiki/Poisson_distribution) variables with means $\mu\_1$ and $\mu\_2$ is given by:
$$
f(k;\mu\_1,\mu\_2)= e^{-(\m... | https://mathoverflow.net/users/1047 | Skellam distribution: Deep connection between Poisson distributions and Bessel function? | A mathematical reason is as follows.
On the one hand, the Laurent series for the modified Bessel functions of the first kind $I\_k$ can be deduced from the Laurent series for the Bessel functions of the first kind $J\_k$ given [here](http://en.wikipedia.org/wiki/Bessel_function). It reads
$$
\sum\_{k\in\mathbb{Z}}I\... | 12 | https://mathoverflow.net/users/4661 | 51549 | 32,319 |
https://mathoverflow.net/questions/11812 | 4 | Let $X$ and $Y$ be two positive random variables with $Y < X$; these may be highly correlated. I would like a reasonable condition on $X$ and $Y$ so that the ratio $X/Y$ has a finite moment-generating function. By this I mean that $\mathbb E e^{r X/Y} < \infty$ for all $r \in \mathbb R$.
Here's one answer. Suppose th... | https://mathoverflow.net/users/238 | When does the ratio X/Y of two random variables have a finite moment-generating function? | Let $D$ denote a set of probability distributions of positive random variables and $r$ a positive real number. Consider the following properties:
1. For every random variables $X$ and $T$ with probability distributions in $D$, $E(\mathrm{e}^{rXT})$ is finite.
- For every random variable $X$ with probability distribut... | 4 | https://mathoverflow.net/users/4661 | 51553 | 32,322 |
https://mathoverflow.net/questions/51530 | 17 | There are multiple definitions of when points $P$ and $Q$ on schemes $X$ and $Y$ are equisingular. One can require that either $P$ and $Q$ have isomorphic analytic (i.e. complex analytic), or formal, or étale neighborhoods. My main question is how are these three definition related. I know that for simple singularities... | https://mathoverflow.net/users/3822 | Analytic vs. formal vs. étale singularities | All these notions are equivalent; Mike Artin proved that. For the étale topology this is in "Algebraic approximation of structures over complete local rings", Inst. Hautes Études Sci. Publ. Math. No. 36 1969, 23–58; the analytic case it treated in "On the solutions of analytic equations", Invent. Math. 5 1968, 277–291.... | 18 | https://mathoverflow.net/users/4790 | 51560 | 32,329 |
https://mathoverflow.net/questions/50713 | 13 | I'm trying to get an idea of [Drinfeld's Zastava space](http://arxiv.org/abs/alg-geom/9707010). It seems to be an infinite-dimensional version of the flag variety, for affine Lie algebras.
But, first of all, why is it called Zastava (Застава)? Sadly I don't understand Russian and I don't understand the connotation. ... | https://mathoverflow.net/users/5420 | Why is Drinfeld's Zastava space called Zastava? | The term was coined by one Michael de Finkelberg during his visit to Croatia. The word is indeed Croatian and means ``flag''. I was happy to have a Croatian word in mathematics.
The strategy of giving a new notion an old name but in a different language is not perfect.
| 31 | https://mathoverflow.net/users/12125 | 51567 | 32,334 |
https://mathoverflow.net/questions/51581 | 4 | Hi,
Does somebody know a proof (or a reference) for the following statement:
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function. Suppose that for all $x$, $f^n(x)=0$ for almost all $n$ (i.e. for all but finitely many $n$), $f^n$ being the $n$-th derivative of $f$. Then $f$ is a polyn... | https://mathoverflow.net/users/12126 | An application of Baire category theorem | For a proof of the much stronger result indicated above, see Page 53 [here](http://books.google.co.il/books?id=P30Y7daiGvQC&pg=PA53&lpg=PA53&dq=%252522function+is+a+polynomial%252522+donoghue&source=bl&ots=CUDjO7Sd8m&sig=ahp6yFK0fO7kUkhgLCiFOH6OgZ8&hl=iw&ei=fSEqTbHeNsTFswby5JjhAg&sa=X&oi=book_result&ct=result&resnum=1&... | 4 | https://mathoverflow.net/users/10227 | 51584 | 32,344 |
https://mathoverflow.net/questions/51589 | 0 |
>
> **Possible Duplicate:**
>
> [Derivate Bessel Function with respect to order](https://mathoverflow.net/questions/4019/derivate-bessel-function-with-respect-to-order)
>
>
>
Dear colleagues,
I have a question about the modified Bessel function of the second kind, $I\_\nu(x)$ and $K\_\nu(x)$. I want to know... | https://mathoverflow.net/users/11936 | Results on derivatives with respect to the parameter of Modified Bessel Function | See, for example, [here](http://dlmf.nist.gov/10.38). As noted in this link, further details can be found in [this paper](http://www.informaworld.com/smpp/content~db=all?content=10.1080/10652460410001727572).
| 0 | https://mathoverflow.net/users/10227 | 51591 | 32,349 |
https://mathoverflow.net/questions/51572 | 2 | It is a theorem that every commutative monoid is inversible, i.e. is isomorphic with a submonoid of a(commutative) group. It is also clear that a group contains all submonoids generated by any subset of its underlying set. but it is also known that non every monoid is inversible, i.e. cannot be isomorphically imbedded ... | https://mathoverflow.net/users/30395 | Non-inversible monoids | A complete description of monoids embeddable into groups was given by Malcev more than 70 years ago. The description is in a form of infinitely many quasi-identities, the easiest of those are the two cancelative laws. See Chapter 12 of Clifford, A. H.; Preston, G. B. The algebraic theory of semigroups. Vol. II.
Mathema... | 6 | https://mathoverflow.net/users/nan | 51606 | 32,356 |
https://mathoverflow.net/questions/51588 | 7 | Let x be an element in a C\*-algebra A, is it true that if x approximately commute with every element in A, then x is near the centre of A? More precisely, I want to know whether the following is true： Let x be an element in a C\*-algebra A with norm 1. Then for any $\epsilon>0$, there exist a $\delta>0$ such that the ... | https://mathoverflow.net/users/9858 | Perturbation in C*-Algebra | The present question seems very close to the following one, which has been well-studied in the literature:
>
> Let $A$ be a C\*-algebra. Does there exist a constant $K>0$ such that, for each $x\in A$, the distance from $x$ to $Z(A)$ is bounded above by
> $$ K \sup\{ \Vert xy-yx \Vert \;:\; y\in A, \Vert y\Vert \le... | 4 | https://mathoverflow.net/users/763 | 51608 | 32,357 |
https://mathoverflow.net/questions/51593 | 9 | Let $M$ be a smooth $m$-dimensional manifold, $TM$ its tangent bundle and $SM$ its unit sphere bundle.
Are there some simple examples where $SM$ is fibrewise homotopy-equivalent to the trivial bundle $S^{m-1} \times M$, yet $TM$ is not trivial as a vector bundle? Does it ever happen for $M$ a sphere?
Via classifyi... | https://mathoverflow.net/users/1465 | Fibrewise homotopy-equivalence of unit sphere bundles vs isomorphism of tangent bundles | It is proved in [Kaminker, J., Proc. Amer. Math. Soc. 41 (1973), 305–308] that the tangent sphere bundle of a closed smooth H-manifold is (unstably) fibre homotopy trivial. On other other hand, surgery theory allows to construct H-manifolds with non-trivial rational Pontrjagin classes, see e.g. [Victor Belfi, Pacific J... | 9 | https://mathoverflow.net/users/1573 | 51610 | 32,358 |
https://mathoverflow.net/questions/51604 | 9 | Here is what I mean by "Cartan's semisimplicity criterion":
Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Assume that the center of $\mathfrak g$ is trivial. Then, the following three assertions are equivalent:
(1) The Killing form on $\mathfrak g\times \mathfrak g$ is no... | https://mathoverflow.net/users/2530 | Can Lie algebra cohomology prove Cartan's Semisimplicity Criterion? | My favorite proof is showing that the trivial module is projective in the category $\mathcal R$ of finite dimensional reps; you can do this using cohomology and the quadratic Casimir (so here the Killing form comes in). After that, showing that all fin. dim. reps. are semisimple amounts to showing that they are all pro... | 6 | https://mathoverflow.net/users/1409 | 51617 | 32,363 |
https://mathoverflow.net/questions/51600 | 0 | Can anyone point me to an explanation and a proof of this theorem?
For reference, it is mentioned in Kolmogorov's almost everywhere divergent function in $L$ as given in Zygmund, volume I. In the third edition it can be found at the top of page 306. A google search leads to several references to this theorem (the Ma... | https://mathoverflow.net/users/9909 | Lebesgue's Majorized Convergence Theorem | I also believe that this just is the dominated convergence theorem. The relevant line on the top of page 306 seems to be:
"The majorized convergence implies that S[f] is obtained by adding formally the Fourier series of the individual terms on the right of (3.4), that is by writing out in full the successive polynomi... | 2 | https://mathoverflow.net/users/630 | 51618 | 32,364 |
https://mathoverflow.net/questions/51607 | 7 | Extensive googling (and searching here) has yielded nothing, unfortunately.
I knew a language genius once who offered to translate it for me as a favor, but I turned him down because it seemed like too much to ask someone to do for free.
At some point I suppose I'm going to just have to learn french.
| https://mathoverflow.net/users/11830 | Shot in the dark: Is there an english translation of Deligne-Rapoport "Les schemas de modules..." anywhere? | <https://math.stackexchange.com/questions/8854/has-deligne-rapoport-been-translated>
| 2 | https://mathoverflow.net/users/12132 | 51633 | 32,377 |
https://mathoverflow.net/questions/51615 | 2 | This may be a silly question, but I have no intuition in this direction. Every category internal to a [Mal'cev category](http://ncatlab.org/nlab/show/Mal%27cev+category) is a groupoid (this is why categories internal to $Grp$ are groupoids). If this was true it would put restrictions on generalising algebraic stacks (w... | https://mathoverflow.net/users/4177 | Do affine schemes form a Mal'cev category? | (This is a repeat of an above comment.) The category of affine schemes is not Mal'cev. This can be disproven by producing an reflexive, non-symmetric relation on an affine scheme $X$ whose graph is a closed subscheme of $X\times X$.
Take $X=\mathbb{A}^1=Spec(\mathbb{C}[x])$. The relation $(x,x)$ and $(x,0)$ (as $x$ r... | 7 | https://mathoverflow.net/users/750 | 51651 | 32,390 |
https://mathoverflow.net/questions/51582 | 3 | Suppose we have an abelian extension of Hopf algebras,
$$k \rightarrow k^G \rightarrow A \rightarrow kF \rightarrow k.$$
According to the general theory there is a left action of $F$ on $G$ and a $2$-cocycle $\sigma:F\times F \rightarrow k^G$ such that $A=k^G$ #${}\_{\sigma} kF$ as algebras.
1)Is it true that $\... | https://mathoverflow.net/users/2805 | Cocyles for abelian extensions | Sebastian,
Suppose that $k$ is an algebraic closed field of characteristic zero.
Let $G$ be a finite group and $X$ a finite right $G$-set, so $k^X$ is left $G$-module. We want to see that $H^2(G,k^X)$ is a finite group. Let $X=\cup\_{i=1}^n X\_i$ where each $X\_i$ is a transitive $G$-set, then as $G$-module $k^X= \... | 1 | https://mathoverflow.net/users/6517 | 51653 | 32,392 |
https://mathoverflow.net/questions/51658 | 11 | Everybody knows the effect of pixelated objects (e.g. faces) on TV. Is there a way - and which mathematical method lies behind it - to un-pixelate the region? Beware: I am not talking about smoothing it out (unblur) image by image - but about using the information of a whole sequence of pixelated pictures where the obj... | https://mathoverflow.net/users/1047 | How to un-pixelate pixelated regions in films? | This is a problem that has been addressed by the computer vision community in the past few years. You may want to look into upsampling and super-resolution techniques as well as vectorization methods. Here is a recent paper on the topic in ICCV 2009 which gives a good summary of the recent efforts in this field: <http:... | 14 | https://mathoverflow.net/users/37597 | 51660 | 32,395 |
https://mathoverflow.net/questions/51661 | 9 | A Morse function $f: M \rightarrow \mathbb R$ on a connected closed manifold $M$ is called $\mathit{perfect}$ with respect to the field $\mathbb F$ if all of the Morse inequalities are equalities, i.e. the number of critical points of $f$ with index $k$ coincides with the $k$-th Betti number of $M$ with respect to $\ma... | https://mathoverflow.net/users/12137 | Restrictions of perfect Morse functions to submanifolds | Isn't there a counterexample with $(M,N)=(\mathbb CP^2,\mathbb RP^2)$?
| 8 | https://mathoverflow.net/users/6666 | 51666 | 32,398 |
https://mathoverflow.net/questions/51657 | 3 | Consider two context-free languages $L\_1, L\_2$. Of course, $L\_1 - L\_2, L\_1\cap L\_2, \bar{L}\_1$, etc. are not necessarily context-free, but they are context-sensitive (the second is easy, the other two I think follow from Immerman-Szelepcsenyi (if I spelled that right)). However, there's no nice structure to cont... | https://mathoverflow.net/users/12138 | 'Closure' of CFLs under complementation and intersection | Grammars involving the usual context-free operations plus intersection are called [conjunctive grammars](https://en.wikipedia.org/wiki/Conjunctive_grammar). Adding negation (in addition to intersection) gives [boolean grammars](https://en.wikipedia.org/wiki/Boolean_grammar).
[Alexander Okhotin](http://users.utu.fi/al... | 5 | https://mathoverflow.net/users/2361 | 51678 | 32,405 |
https://mathoverflow.net/questions/51674 | 5 | Given a set $A$ of positive integers, set $r\_{A,h}(N)$ to be the number of $h$-tuples $(a\_1, \cdots, a\_h)$ such that $N = a\_1 + \cdots + a\_h$. Set $f(z) = \sum\_{a \in A} z^a$. Then by Cauchy's Theorem we have that $r\_{A,h}(N) = \frac{1}{2\pi i} \int\_{|z| = r} \frac{f(z)^h}{z^{N+1}}dz$ for $0 < r < 1$. This can ... | https://mathoverflow.net/users/10898 | A question on the singular series and singular integral in Hardy-Littlewood Circle Method | $A$ should have a fairly regular distribution in arithmetic progressions to yield a reasonable singular series and singular integral. This is why Vinogradov published his solution to the ternary Goldbach problem only after the Siegel-Walfisz theorem was available. The idea is that the relevant trigonometric polynomial ... | 2 | https://mathoverflow.net/users/11919 | 51690 | 32,411 |
https://mathoverflow.net/questions/51685 | 19 | To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to complete the proof of L'Hôpital). I'm assuming that Cauchy was responsible for his MVT, which means that Bernoulli didn't know ... | https://mathoverflow.net/users/4194 | How did Bernoulli prove L'Hôpital's rule? | L'Hôpital's rule was first published in [*Analyse des Infiniment Petits*](https://archive.org/details/infinimentpetits1716lhos00uoft).
According to [*The Historical Development of The Calculus*](https://books.google.com/books?id=D2SWE_iZjYsC&printsec=frontcover&dq=historical+development+of+calculus+books&hl=en&ei=R2o... | 28 | https://mathoverflow.net/users/5371 | 51691 | 32,412 |
https://mathoverflow.net/questions/51684 | 7 | Let $p$ be prime (of size roughly $100$, say). Suppose that $M$ is a matrix with coefficients in $\mathbf{F}\_p$ with roughly $An$ rows and $n$ columns, where $A>1$ is some fixed small constant. Suppose, moreover, that
1. Every row of $M$ has at most $B$ non-zero entries, which are all $\pm 1$, for some small $B$.
2.... | https://mathoverflow.net/users/nan | Row reduction of sparse matrices | The [LinBox project](http://www.linalg.org/) provides a C++ library which can effectively compute the rank of sparse matrices over finite fields.
You could also look at [Sage](http://www.sagemath.org/) which should also allow this, but I am not sure whether they have effective implementations for this particular case... | 4 | https://mathoverflow.net/users/8338 | 51697 | 32,417 |
https://mathoverflow.net/questions/51683 | 2 | Hi, I am interested in the distribution of return times in simple random walks on finite graphs.
Let $G$ be a connected finite graph with, with two independent random walks. If both random walks start are at time $t\_0$ on the same node in the graph, how long does it take until they meet again? I have not found pape... | https://mathoverflow.net/users/12142 | Exist closed forms of the distribution of return time in markov chains? | I assume you mean the distribution of the time of first return, and I assume you're talking
about finite graphs.
I'll volunteer the naive answer: let $M$ be the matrix of transition probabilities for your random walk, and let $M\_i$ be the matrix obtained by modifyiong $M$ to make the $i$th column all 0's except for ... | 3 | https://mathoverflow.net/users/9062 | 51702 | 32,421 |
https://mathoverflow.net/questions/51662 | 1 | Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$.
What is a set of representatives of the orbits of $\Delta$ under $f$ ?
Thanks!
| https://mathoverflow.net/users/40886 | orbit of a Dynkin diagram automorphism | In a bunch of important cases, the orbits (of the group $\langle f \rangle$) acting on $\Delta$, correspond to roots in another root system. This is the procedure called "folding", which gives a way to reduce non simply-laced root systems to simply-laced ones. For example, $A\_{2n-1}$ folds to $C\_n$, $D\_{n+1}$ folds ... | 4 | https://mathoverflow.net/users/468 | 51705 | 32,422 |
https://mathoverflow.net/questions/51698 | 12 | I hope this question is not too soft for MO.
The Wikipedia says about *finitism* that it is an extreme form of *constructivism*. See <http://en.wikipedia.org/wiki/Finitism>. I doubt that this is correct.
As I understand, there were different approaches to solve the crisis of the foundation of mathematics. One was c... | https://mathoverflow.net/users/5917 | Is finitism an extreme form of constructivism? |
>
> this is not a rejection of infinitism methods, but just an attempt to find a better foundation. While, constructivism is really a rejection of certain arguments.
>
>
>
I think that's arguable. There do exist (or, at least, there have existed) people who are *philosophically* finitist, in that they really do ... | 6 | https://mathoverflow.net/users/49 | 51709 | 32,424 |
https://mathoverflow.net/questions/51718 | 17 | Following section 1.4, which is entitled "Constructions Using Amalgams" is the innocent-enough looking exercise:
Show that the group defined by the presentation (presumably on the generators $x\_1, x\_2, x\_3$)
$x\_2x\_1x\_2^{-1}=x\_1^2 $
$ x\_3x\_2x\_3^{-1} = x\_2^2$
$ x\_1x\_3x\_1^{-1} = x\_3^2$
is trivial.... | https://mathoverflow.net/users/11818 | About an exercise in Serre's "Trees" | I don't have the book next to me but I'm quite sure I remember the exercise and, as far as I was able to determine, your first guess is the correct one. This isn't an amalgam exercise in any natural way (or any way at all), but rather an example to put the case of 4 generators in perspective.
| 9 | https://mathoverflow.net/users/25 | 51719 | 32,430 |
https://mathoverflow.net/questions/51671 | 9 | Let $A$ be a $C^\*$-algebra. The group $Aut(A)$ of $\ast$-automorphisms of $A$ is usually equipped either with the pointwise norm topology, i.e. the topology generated by the semi-norms $\lVert \varphi \rVert\_a = \lVert \varphi(a) \rVert$ or with the uniform topology, i.e. the one which it earns by inclusion into the ... | https://mathoverflow.net/users/3995 | topology on the automorphism group of a C* algebra | If $Aut(A)$ carries the uniform topology but $U(A)/Z(U(A))$ the topology induced by the strict one, then the bijection is not continuous.
For example, let $A=C\_0(N,M\_2(C))$ be the $C^\*$-algebra of all $2\times 2$-matrix-valued functions on the naturals vanishing at infinity, and define for each $n \in N \cup \{\i... | 6 | https://mathoverflow.net/users/12136 | 51741 | 32,440 |
https://mathoverflow.net/questions/51727 | 7 | On [nlab](http://ncatlab.org/nlab/show/sheafification) it says that a presheaf is locally isomorphic to a sheaf. What do they mean by locally isomorphic? Their definition of locally isomorphic is given in terms of Grothendieck topologies which i think is overkill.
When I first read the nlab page, I thought that it m... | https://mathoverflow.net/users/4002 | Presheaves are locally sheaves? | Dear Daniel, the reason you couldn't find a proof of your statement nor locate one in the literature is that it is false ; so you were quite right to "have doubts now" ! Here are two (essentially equivalent) statements that hopefully clarify the situation.
**I) Given a presheaf $\mathcal F$ on a topological space, it... | 14 | https://mathoverflow.net/users/450 | 51743 | 32,442 |
https://mathoverflow.net/questions/51704 | 4 | Let $M$ be an $n$-dimensional hypersurface in $\mathbb R^{n+1}$, such that principal curvatures are bounded from below by a constant $\delta$. Is there any lower bound on the curvature of the curves on $M$? Curves should be intersection of a two plane and the manifold.
| https://mathoverflow.net/users/12145 | Lower bound on the curvature of the curves on $M$ | Let $P$ be a 2-plane in $\mathbb R^{n+1}$ and $\gamma=P\cap M$. Choose a parametrisation of $\gamma$ by arc-length. Then the curvature of $\gamma$ is $K=||\nabla\_{\dot\gamma}\dot\gamma||$, where $\nabla$ is the flat covariant derivative in $P$ (or in $\mathbb R^{n+1}$ as $P$ is totally geodesic). Now, the Levi-Civita ... | 5 | https://mathoverflow.net/users/10675 | 51748 | 32,444 |
https://mathoverflow.net/questions/51721 | 8 | I'm looking for references on the proof (due to B. Moishezon, I guess) that any Moishezon space becomes a projective smooth complex variety after a finite number of blow-ups (called a modification?) His own articles on this that I could find are mainly in Russian, except a few survey papers (in English) without too muc... | https://mathoverflow.net/users/370 | References on Moishezon spaces in English/French | Dear Shenghao,
I) I am happy to report that according to Ueno's *Classification of Algebraic Varieties and Compact Complex Spaces*, Springer LNM 439, 1975, Moishezon's papers have been translated into English : AMS Translation Ser.2, *63* (1967), 51-177.
II) Here is the precise statement you request. Given a Moishe... | 9 | https://mathoverflow.net/users/450 | 51752 | 32,448 |
https://mathoverflow.net/questions/51761 | 4 | I'm going to be clear about definitions before I start so there's no ambiguity. Let D be a subset of the complex numbers and let $f: D \to \mathbb{R}^{+}$ be a positive real-valued map defined on D. We will write $f(x) = O(g(x))$ if $g: D \to \mathbb{R}^{+}$ and there exists a positive constant A such that:
$\display... | https://mathoverflow.net/users/12160 | Does f(x)~g(x) imply $f(x) \asymp g(x)$? | The result is false if $D$ is not closed: take a boundary point $a\not\in D$ and let $f(x)=|x-a|$ and $g(x)=1+f(x)$.
If $D$ is closed, it is true. There is some $R>0$ so that $\tfrac12 f(x) < g(x) < 2f(x)$ whenever $|x| > R$. On the other hand, on the compact set $\{x\in D\colon |x|\le R\}$ there are bounds $0 < m \l... | 9 | https://mathoverflow.net/users/802 | 51764 | 32,452 |
https://mathoverflow.net/questions/51766 | 11 | When performing induction on say a graph $G=(V,E)$, one has many choices for the induction parameter (e.g. $|V|, |E|$, or $|V|+|E|$). Often, it does not matter what choice one makes because the proof is basically the same. However, I just read the following ingenious proof of König's theorem due to Rizzi.
**König's t... | https://mathoverflow.net/users/2233 | Easier induction proofs by changing the parameter | Cauchy's proof by induction of the inequality between the arithmetic and geometric means (written in his 1821 *Cours d'Analyse*).
Of course, the base of the induction, for $n=2$, immediately comes from $(\sqrt x\_1-\sqrt x\_2)^2\ge0$, but then, although it is actually possible to follow the natural induction steps, m... | 15 | https://mathoverflow.net/users/6101 | 51772 | 32,455 |
https://mathoverflow.net/questions/51715 | 12 | Does the category of complete Boolean algebras have binary coproducts?
Note that this category does *not* have countable coproducts. Indeed, the coproduct of countably many copies of the four element complete Boolean algebra would be the free complete Boolean algebra on countably many generators, and such an object d... | https://mathoverflow.net/users/2206 | Coproducts of complete Boolean algebras | Chris Heunen's comment under the OP can be turned into a proof. Suppose the category of compact Hausdorff extremally disconnected spaces has binary products. Let $X \times Y$ denote the product in that category. If $|X|$ denotes the underlying set, then of course the canonical map
$$|X \times Y| \to |X| \times |Y|$$... | 16 | https://mathoverflow.net/users/2926 | 51773 | 32,456 |
https://mathoverflow.net/questions/51790 | 4 | I am interested in families of finite groups arising from direct products of other groups. For instance, abelian groups are a simple example of such families (direct products of cyclic groups), but there are more. The family of hamiltonian groups is another such family; a hamiltonian group is a group where all the subg... | https://mathoverflow.net/users/11134 | Families of finite groups arising from direct products... | A finite group is nilpotent iff it is a direct product of $p$-groups.
| 3 | https://mathoverflow.net/users/11771 | 51791 | 32,467 |
https://mathoverflow.net/questions/51794 | 5 | Can anyone give a simple example of a sequence that converges, but there's no computable function that gives $N$ as a function of $\epsilon$, i.e., the [modulus of convergence](http://en.wikipedia.org/wiki/Modulus_of_convergence) is not computable?
In the literature, all I could find were aesthetically unpleasant exa... | https://mathoverflow.net/users/9211 | Simple example of a sequence without computable modulus of convergence | Because of the relaxed requirements, the following seems to work. Let $f$ be an increasing function from natural numbers to natural numbers that grows faster than any computable function. (For example, define $f(n)$ to be $n$ plus the largest number obtainable by giving any input $\leq n$ to any Turing machine with Göd... | 9 | https://mathoverflow.net/users/6794 | 51796 | 32,469 |
https://mathoverflow.net/questions/51801 | 7 | Looking over the treatment of the Eilenberg-Steenrod axioms in a few of my favorite introductory algebraic topology texts, I see that some include an "axiom of compact support", while others do not. Whether or not one needs such an axiom for the homology theory to be uniquely defined (assuming it satisfies the dimensio... | https://mathoverflow.net/users/6646 | When do we need the axiom of compact support for a homology theory to be uniquely defined? | The axioms that Hatcher uses (on page 160 of the current online version of his book) include an axiom about arbitrary wedges going to direct sums. This has the same effect (in the presence of his other axioms and in the CW setting) as an axiom about compact support (by which I assume you mean a statement that the homol... | 7 | https://mathoverflow.net/users/6666 | 51807 | 32,475 |
https://mathoverflow.net/questions/51798 | 5 | This question is related to
[Intuition behind the Eichler-Shimura relation?](https://mathoverflow.net/questions/19390/intuition-behind-the-eichler-shimura-relation)
and
[L-functions and higher-dimensional Eichler-Shimura relation](https://mathoverflow.net/questions/50004/l-functions-and-higher-dimensional-eichle... | https://mathoverflow.net/users/2260 | Generalizing Eichler-Shimura to higher dimension, again | Dear Evgeny,
The paper *Congruence relations on some Shimura varieties* of Wedhorn (and some follow-up papers) proves what is (I think) the state of the art on Eichler--Shimura relations on higher dimensional Shimura varieties. Namely, he establishes the congruence relation for a wide class of PEL Shimura varieties. ... | 6 | https://mathoverflow.net/users/2874 | 51816 | 32,482 |
https://mathoverflow.net/questions/51795 | 4 | Suppose $(X,\mathcal{B},\mu)$ is a measure space, and let $B\subseteq X^2$ be an arbitrary set.
1) Is there a nice characterization of the circumstances under which there is a $\sigma$-algebra $\mathcal{C}\supseteq\mathcal{B}$ and a measure $\nu$ on $\mathcal{C}$ extending $\mu$ such that $B$ is measurable with respe... | https://mathoverflow.net/users/8991 | Measurability of sets of pairs | Without regularity at least, such pathologies can happen. In fact, taking products is not necessary to get them. It can happen that with $A\subset X$ we have $\nu\_1(A)\neq \nu\_2(A)$. Taking the product of $A$ with $X$ yields counterexamples like the one sought in (2).
First, it is easy to do this in a trivial way b... | 5 | https://mathoverflow.net/users/5963 | 51820 | 32,484 |
https://mathoverflow.net/questions/42999 | 6 | Let $\mathcal C$ be a [monoidal category](http://ncatlab.org/nlab/show/monoidal+category). Recall that the *(Drinfel'd) center* of $\mathcal C$ is the braided monoidal category $Z(\mathcal C)$ with:
* Objects: pairs $M \in \mathcal C$ and $\mu: M\otimes(-) \overset\sim\to (-)\otimes M$ a natural iso, which is require... | https://mathoverflow.net/users/78 | Which monoidal categories are equivalent to their centers? | Some of these questions are addressed (in the derived setting) in my paper [Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry](http://arxiv.org/abs/0805.0157) with John Francis and David Nadler --- for the underived setting you might also want to look at Hinich's [Drinfeld double for orbifolds](htt... | 8 | https://mathoverflow.net/users/582 | 51821 | 32,485 |
https://mathoverflow.net/questions/51793 | 5 | For a positive integer $n$ let
$$
\sigma\_2(n) = \sum\_{d \mid n} d^2.
$$
There are many positive integers $n$ for which
$$
n \mid \sigma\_2(n).
$$
But, when $n$ has the particular form
$$
n=pq^2
$$
where $p, q$ are distinct odd prime numbers and
$$
p \equiv 1 \pmod{4}
$$
there seems to be none.
Questio... | https://mathoverflow.net/users/11016 | Positive integers $n$ that divide $\sigma_2(n)$ | As already mentioned in other posts, $p\equiv1\bmod3$ and $q\equiv1\pmod4$.
Since $p\mid q^4+q^2+1=(q^2+q+1)(q^2-q+1)$, the prime $p$ divides (at least)
one of the two factors; in particular, $p\le 1+q+q^2<(1+q)^2$ implying
$q>\sqrt p-1$. It happens very rare that $p^2+1$ is divisible by a square
$>(\sqrt p-1)^2$. Excl... | 4 | https://mathoverflow.net/users/4953 | 51825 | 32,488 |
https://mathoverflow.net/questions/51802 | 15 | It is very well known that if $\alpha : \mathscr{F} \to \mathscr{G}$ is a morphism of sheaves, then it induces homomorphisms on the stalks. I have been wondering for a while if given a collection of homomorphisms between the stalks of two sheaves and some suitable patching condition, can we construct a homomorphism of ... | https://mathoverflow.net/users/4002 | Under what circumstances do morphisms on the stalks of a sheaf induce a sheaf morphism | There is a simple condition. Stalk maps $\alpha\_p : F\_p \to G\_p$ yield a map of sheaves $\alpha : F \to G$ if and only if for all open $U \subseteq X$ and sections $s \in F(U)$ there is an open covering $U = \cup\_i U\_i$ and sections $t\_i \in G(U\_i)$ such that $\alpha\_p(s\_p) = (t\_i)\_p$ for all $p \in U\_i$ an... | 14 | https://mathoverflow.net/users/2841 | 51827 | 32,489 |
https://mathoverflow.net/questions/51835 | 4 | Consider a smooth manifold $M$ and a bundle $\pi\colon E\to M$ over it, where each fibre of $E$ is an algebraic variety. Is there a special name for this kind of bundle? The idea I have is that questions about the topology of the fibres can then be turned into algebraic ones. One example that I have in mind is of an $S... | https://mathoverflow.net/users/2622 | Name for bundle of algebraic varieties over a smooth manifold | I do not know names for these kind of structures, but here are some hints. There are two cases you might want to consider:
1.) all the fibres are isomorphic as algebraic varieties. Let $Aut(V)\subset Diff(V)$ be the group of all automorphisms of a variety $V$, with the $C^{\infty}$-topology and $G \subset Aut(V)$ be ... | 4 | https://mathoverflow.net/users/9928 | 51843 | 32,495 |
https://mathoverflow.net/questions/51842 | 5 | Does a compact semilocally simply connected geodesic space have the homotopy type of a compact CW complex? Actually what I'd like to know is whether the fundamental group of such a space is finitely presented.
Edit: As Bruno Martelli notes, this is obviously false, but the question of whether the fundamental group i... | https://mathoverflow.net/users/9417 | Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex? | The bouquet of infinitely many shrinking 2-spheres in $\mathbb R^3$ centered in $(0,0,n)$ and of radius $n$ is a compact simply connected geodesic space, which is not homotopic to a compact CW complex.
| 8 | https://mathoverflow.net/users/6205 | 51845 | 32,497 |
https://mathoverflow.net/questions/51739 | 12 | Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a base scheme $S$. Then the $S$-schemes $\mathbb{P}(\mathcal{F}), \mathbb{P}(\mathcal{G})$ are defined by a universal property: to map from an $S$-scheme $T$ (with structure map $g: T \to S$) into $\mathbb{P}(\mathcal{F})$ is the same thing as giving a line bu... | https://mathoverflow.net/users/344 | Functorial way of showing that the Segre (or Plucker) morphism is a closed embedding? | I think this is a very interesting question. It gives another proof of the Segre embedding, makes it somehow more natural and puts it into a more general context, which I explain below.
I will just talk about the Segre embedding $\mathbb{P}^n \times \mathbb{P}^m \to \mathbb{P}^{n+m+nm}$. The general case of $\mathbb{... | 4 | https://mathoverflow.net/users/2841 | 51847 | 32,499 |
https://mathoverflow.net/questions/51860 | 10 | If $\mathcal{C}$ is a category, then surely the category of simplicial
objects $s\mathcal{C}$ is not automatically a model category. What conditions
must $\mathcal{C}$ satisfy in order for $s\mathcal{C}$ to have a reasonable model structure?
| https://mathoverflow.net/users/3634 | Model categories of simplicial objects | It always has a model structure using Kan's theory of Reedy categories. For a proof, see Hirschhorn *Model Categories and their Localizations* 15.3.
This is because $\Delta$ and $\Delta^{op}$ are both Reedy categories.
I will address the more general question as well:
If $C$ is any small category, the condition f... | 2 | https://mathoverflow.net/users/1353 | 51862 | 32,509 |
https://mathoverflow.net/questions/51348 | 8 | According to the article of Hauser:
The Hironaka theorem on resolution of singularities <http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/home.html>
The existence of resolution of complex analytic varieties was proven in:
* Aroca, J.-M., Hironaka, H., Vicente, J.-L.: The theory of the maximal co... | https://mathoverflow.net/users/943 | Simplified treatment of resolutions of complex analytic varieties? | I had a chance to check with Jarek Włodarczyk, who points out
that the analytic case is not much harder (to him), but that it does require some extra care.
In addition to the above references, the resolution of singularities of
analytic spaces is treated in
* Bierstone, Milman, Canonical desingularization in charact... | 7 | https://mathoverflow.net/users/4144 | 51880 | 32,518 |
https://mathoverflow.net/questions/51879 | 2 | Suppose that $f$ is a [class 1 Baire function](http://en.wikipedia.org/wiki/Baire_function) defined on ${}[0,1]$ such that $\int\_0^1 f(x)p(x)dx=0$ for all polynomials $p$.
Of course, $f$ must be 0 almost everywhere. Can we conclude that in fact $f$ is zero except at countably many points? If yes, how high up in the... | https://mathoverflow.net/users/11449 | Baire class 1 functions for which $\int_0^1 fp=0$ for all polynomials $p$ | No.
Define $g : \mathbb{N} \times [0,1] \to \mathbb{R}$ by $f(n,x) := \operatorname{max}(0,1+((-n)\cdot \operatorname{inf}(\{|x+(-y)| : y\in (\operatorname{Cantor} \operatorname{set})\}))).$
$g$ is continuous, so it is in Baire class 0.
Define $f : [0,1] \to [0,1]$ by $f(x) := \displaystyle\lim\_{n\to \in... | 4 | https://mathoverflow.net/users/nan | 51883 | 32,519 |
https://mathoverflow.net/questions/51863 | 18 | I've always taken this for granted until recently:
In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations $\gamma\_1:S^1 \rightarrow C$.
Is it true that for all $\varepsilon >0$, there exists $\delta >0$ s.t. any Jordan curve ... | https://mathoverflow.net/users/9322 | Does Riemann map depend continuously on the domain? | Here's a conceptual proof why this is true, up to things which are intuitively obvious and not hard to prove:
In the unit disk, almost every Brownian path hits the boundary. The hitting measure equals
proportional to arc length. In two dimensions, a conformal map takes trajectories of Brownian paths to trajectories o... | 24 | https://mathoverflow.net/users/9062 | 51893 | 32,525 |
https://mathoverflow.net/questions/51887 | 15 | Are there constructive examples of doubly stochastic matrices (whose rows and columns all sum up to $1$ and contain only non-negative entries) that are not diagonalizable?
| https://mathoverflow.net/users/1837 | Non-diagonalizable doubly stochastic matrices | Sure. For example:
$$A = \begin{pmatrix}
5/12 & 5/12 & 1/6 \\
1/4 & 1/4 & 1/2 \\
1/3 & 1/3 & 1/3
\end{pmatrix}$$
Note that
$$A \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} = \begin{pmatrix} 1/4 \\ -1/4 \\ 0 \end{pmatrix} \ \mbox{and} \ A^2 \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} = 0.$$
This shows that $A$ is no... | 32 | https://mathoverflow.net/users/297 | 51897 | 32,528 |
https://mathoverflow.net/questions/51905 | 64 | I hope this is appropriate for mathoverflow. Understanding $\mathbb{C}\_p$ has always been something of a stumbling block for me. A standard thing to do in number theory is to take the completion $\mathbb{Q}\_p$ of the rationals with respect to a $p$-adic absolute value. The resulting field is then complete, but has no... | https://mathoverflow.net/users/9960 | How to picture $\mathbb{C}_p$? | 1. You do whatever works for you. Some people think more algebraically, others more geometrically. I certainly don't know what "to picture" means in this context, but then, I am a more algebraic person, so maybe others will be able to say more. Can you picture $\mathbb{Q}^{ab}$, say? I can't.
2. A typical element is, b... | 19 | https://mathoverflow.net/users/35416 | 51907 | 32,534 |
https://mathoverflow.net/questions/51909 | 6 | This problem arose in the study of Latin squares with a large number of subsquares, although it appears interesting in its own right.
>
> **Question**: What is the maximum cardinality of a family $F \subseteq 2^{[n]}$ of subsets of $[n]:=\{1,2,\ldots,n\}$ for which any two distinct $A,B \in F$ satisfy $|A \cap B| \... | https://mathoverflow.net/users/2264 | What is the largest family F of subsets of [n] for which any two distinct sets A and B in F have an intersection of size at most min(|A|,|B|)/2? | For an affine space over $\mathbb{z}\_2$ with $n=2^k$ we have $\binom{n}{2}=(2^{k}-1)(2^{k-1})$ however there are $\binom{n}{3}/4=\frac{2^{k}(2^{k}-1)(2^{k}-2)}{24}$ 2 dimensional flats of which any pair intersect in at most two points.
Details: Consider the $n=2^k$ binary vectors of length $k$. Among the sets of 4 v... | 6 | https://mathoverflow.net/users/8008 | 51915 | 32,540 |
https://mathoverflow.net/questions/51900 | 7 | Suppose we have a monoidal category equipped with additional data that almost makes it a braided monoidal category except that only one of the hexagon identities
is satisfied.
Can we then prove the other hexagon identity?
If not, is there an explicit counterexample and can we prove the other identity under the addition... | https://mathoverflow.net/users/402 | Does one of the hexagon identities imply the other one? | Consider the category of $A-$graded vector spaces (here $A$ is an abelian group) with
obvious tensor product and trivial associator. Then each isomorphism $a\otimes b\to b\otimes a$ can be specified as a nonzero complex number $B(a,b)$. Now the first hexagon axiom says that the function $B(a,b)$ is linear in the first... | 12 | https://mathoverflow.net/users/4158 | 51923 | 32,547 |
https://mathoverflow.net/questions/51898 | 10 | I owe the idea of asking this question to Max Muller and
[his curiosity](https://mathoverflow.net/questions/26035/).
*What is the set of $\alpha$ in the interval $0\le\alpha < 1$ for which
the alternating sum*
$$
\sum\_{n=1}^\infty\frac{(-1)^{n+[n^\alpha]}}n
$$
*converges*? Here $[\ \cdot\ ]$ denotes the integral par... | https://mathoverflow.net/users/4953 | Convergence of alternating harmonic sums | It converges for all $0\le\alpha<1$. Define the $k$-*block* to be the set of $n$ such that $[n^\alpha]=k$ (it ranges from $\lceil k^{1/\alpha}\rceil$ to $\lceil (k+1)^{1/\alpha}\rceil-1$).
The absolute value of the contribution to the sum from the $k$-block is at most the reciprocal of its left endpoint (the terms f... | 11 | https://mathoverflow.net/users/11054 | 51926 | 32,550 |
https://mathoverflow.net/questions/51193 | 9 | We've recently seen this question: [Can the number of solutions $ab(a+b+1)=n$ for $a,b,n \in \mathbb{Z}$ be unbounded as $n$ varies?](https://mathoverflow.net/questions/50479) It appears initially plausible that the answer is yes, but evidently there are good reasons to believe that the answer might be no. The same que... | https://mathoverflow.net/users/8008 | Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z[t]$ be unbounded as n varies? | The $\mathbb{Z}[t]$ example above with 3 (pairs of) positive solutions for $ab(a+b-1)=n$ has 5 in $\mathbb{Z}[u,v]/(u^2-2v^2+1)$ if $t=\frac{(2v+u)(u-v)}{2}$. This gives us a parametric family of curves in $ab(a+b-1)=n$ in $\mathbb{Z}$ with 5 pairs of positive integer points . We need $u>v$ so first cases are $(u,v,n)=... | 2 | https://mathoverflow.net/users/8008 | 51938 | 32,560 |
https://mathoverflow.net/questions/51948 | 2 | we know that a complex torus,which is algebraic,is called abelian variety.Recently i see another definition.Let $\mathbb{C}^n/\Lambda$ be a complex torus,where $\Lambda$ is a lattice of $\mathbb{C}^n$.$\mathbb{C}^n/\Lambda$ is abelian variety when there is a real skew-symmetric bilinear $E$ form on $\mathbb{C}^n$,satis... | https://mathoverflow.net/users/11901 | A equivalent definition of abelian variety | In a complex torus it is (in principle) simple to relate Dolbeaut and deRham (with values in $\Z$ ) cohomologies. Therefore, under your assumptions, one can write down explicitly a holomorphic bundle (the theta bundle) which is postive. This bundle can be written down if only condition 1) and 3) are satisfies, but, it ... | 4 | https://mathoverflow.net/users/4572 | 51953 | 32,567 |
https://mathoverflow.net/questions/51942 | 18 | Let $R$ be a principal ideal domain and $A \in M\_n R$. It is well known that there exist invertible matrices $Q$ and $S$ and a diagonal matrix $D= {\rm diag}(a\_1,\dots,a\_n)$ such that
* $a\_i \mid a\_{i+1}$ for all $1 \leq i \leq n-1$, and
* D=QAS.
The matrix $D$ is called Smith normal form of $A$ and is unique... | https://mathoverflow.net/users/8176 | Analogue of Smith normal form for matrices over $\mathbb Z[t]$ | For Q1 the problem is that one invariant of the matrix is the (isomorpism class
of the) cokernel and any $\mathbb Z[t]$-module generated by $n$ elements and
$n$-relations appears as such an invariant. There simply are too many modules
over a $2$-dimensional ring such as $\mathbb Z[t]$.
As for Q2 you cannot really hop... | 19 | https://mathoverflow.net/users/4008 | 51956 | 32,569 |
https://mathoverflow.net/questions/51950 | 5 | Are there any example of $II\_1$-factor $M$ with maximal abelian von Neumann subalgebra $A$ and non-zero derivation $\delta:M\rightarrow B(H)$ such that $\delta(a)=0$ for every $a\in A$?
| https://mathoverflow.net/users/8699 | Derivation of von Neumann algebra which is zero on MASA | Let $M= L \mathbb F\_2$ and $H = \ell^2 \mathbb F\_2$, where $\mathbb F\_2 = \langle a,b \rangle$ is the free group on two generators and $B(H)$ is a bimodule via the left and right multiplication with the left-regular representation $\lambda \colon L \mathbb F\_2 \to B(\ell^2 \mathbb F\_2)$.
Define $\delta(x) = [x,\... | 3 | https://mathoverflow.net/users/8176 | 51958 | 32,571 |
https://mathoverflow.net/questions/51935 | 6 | I have the following naive (and inexpert) question about the
reduction of Shimura curves at primes dividing the discriminant
of the underlying quaternion algebra. It requires some background
to state. That is, let $F$ be a totally real field of degree $d$.
Fix a real place $\tau\_1$ in the set of real
places $\lbrace \... | https://mathoverflow.net/users/6121 | "Bad" reduction of Shimura curves via dual graphs | [inkspot](https://mathoverflow.net/questions/51935/bad-reduction-of-shimura-curves-via-dual-graphs/51947#51947) is indeed correct that the component graphs are indeed not generally trees.
As you seem to have deduced for yourself, Cerednik–Drinfeld uniformization is a highly nontrivial concept, and it really helps to ... | 9 | https://mathoverflow.net/users/3384 | 51961 | 32,574 |
https://mathoverflow.net/questions/51964 | 3 | Let $X$ be a rationally connected smooth projective variety defined over $\mathbb C$.
(1) Can we find a surface $S \subset X$ such that $ (-K\_X)^2 \cdot S > 0 ? $
If yes, can we assume that $S$ intersects properly a given codimension $2$ subvariety $V \subset X$ only at isolated points ? The answer is obviously **n... | https://mathoverflow.net/users/605 | Positivity of the anticanonical bundle of a rationally connected manifold | It seems to me that the answer to your question is **no**, because of the following example (which came to my mind after reading Artie Prendergast-Smith's comment).
Consider a pencil $\lambda Q\_1 + \mu Q\_2$ of quartic surfaces in $\mathbb{P}^3$, and let $Z$ be its base locus, that in general will be a smooth curve ... | 5 | https://mathoverflow.net/users/7460 | 51979 | 32,584 |
https://mathoverflow.net/questions/51978 | 13 | The title says it all. I am looking for an explanation or reference for why the homology of the ribbon graph complex computes the cohomology of the mapping class groups of surfaces.
I've seen explanations of this using operads, but my understanding is that the operad viewpoint is more recent and not how the above que... | https://mathoverflow.net/users/5323 | Why does (Ribbon) Graph (co)Homology Compute (co)Homology of MCG? | Chapter 2 of Harer's paper "The cohomology of the moduli space of curves" is good.
The point is that there is the "arc complex", a simplicial complex which gives a suitable triangulation of Teichmüller space which is compatible with the action of the mapping class group. The simplices of the simplicial complex corres... | 11 | https://mathoverflow.net/users/83 | 51984 | 32,588 |
https://mathoverflow.net/questions/51644 | 3 | Given a subset $S$ of a [Hadamard manifold](http://en.wikipedia.org/wiki/Hadamard_manifold) $M$. Is there a curvature criterion (for $\partial S$) to decide whether $S$ is convex.
I am looking for a ganeralization of the following statement:
A connected subset of $\mathbb{R}^2$ bounded by a smooth curve $\gamma$ wi... | https://mathoverflow.net/users/3969 | Convex subsets of Hadamard manifolds | Thank you. After some googling I found the paper "Locally convex hypersurfaces of negatively curved spaces" by S. Alexander (ams.org/journals/proc/1977-064-02/…), which adresses precisely this question.
| 1 | https://mathoverflow.net/users/3969 | 51990 | 32,591 |
https://mathoverflow.net/questions/51995 | 7 | Can the circle group $S^1$ act smoothly and freely on the Klein bottle? I'm sure there is some obvious reason why the answer is no, which eludes me right now.
We can view $K$ as the quotient of $S^1\times S^1\subset\mathbb{C}\times\mathbb{C}$ by the involution $(z\_1,z\_2)\to (-z\_1,z\_2^{-1})$. Then we get an almost... | https://mathoverflow.net/users/8103 | Is there a smooth free circle action on the Klein bottle? | No. If you had a smooth free action, the quotient would be a compact connected 1-manifold, so a circle. So the Klein bottle would be an orientable circle bundle over the circle, but there's only one and that is a torus.
So the tools I'm using are (1) when the quotient of a manifold by a free action of a compact Lie ... | 14 | https://mathoverflow.net/users/1465 | 51997 | 32,594 |
https://mathoverflow.net/questions/51989 | 2 | I'm a beginner to this. Can anyone please point me to any resources for studying about equational logic, preferably with some example proofs to wet my feet in?
Thanks in advance!
| https://mathoverflow.net/users/12203 | Equational logic | Beyond what is taught in (American) high school algebra courses, I don't really know any beginner level treatments on equational logic. The link Ricky Demer provides a brief bibliography which includes a book on Universal Algebra and one on Mathematical Logic; browsing through your local university math library should ... | 3 | https://mathoverflow.net/users/3402 | 51998 | 32,595 |
https://mathoverflow.net/questions/51987 | 18 | Is there a list somewhere of which types of Diophantine equations are solvable, which types are not solvable, and which types are not known to be solvable or not? (When I say solvable, I mean that we can determine in a finite number of steps whether or not there exist solutions.) For instance, we know that linear Dioph... | https://mathoverflow.net/users/7089 | Which types of Diophantine equations are solvable? | Craig:
For a while, there was some research on improving bounds on the number of variables or degree of unsolvable Diophantine equations. Unfortunately, I never got around to cataloging the known results in any systematic way, so all I can offer is some pointers to relevant references, but I am not sure of what the c... | 32 | https://mathoverflow.net/users/6085 | 52005 | 32,600 |
https://mathoverflow.net/questions/51991 | 2 | In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an **elementary correspondance** from *X* (Smooth connected scheme over $k$) to *Y* (separated scheme over $k$) as an irreducible closed subset $W$ of $X\times Y$ whose associated integral subscheme is finite and surjective over $X$."
As far as I kno... | https://mathoverflow.net/users/12204 | Are finite correspondances flat? | I believe the answer to your first question is no. Here's an example sketch: let $X$ be $A^2$, and let $W$ be two copies of $A^2$ glued at the origin (realized as the union of two transverse linear subspaces of $A^4$, say), mapping to $X$ by the "fold" map (projection to a third linear subspace, say). Actually that's n... | 7 | https://mathoverflow.net/users/3931 | 52012 | 32,603 |
https://mathoverflow.net/questions/51783 | 14 | This question is about (not necessarily symmetric) monoidal categories enriched over a symmetric monoidal category $\mathcal{V}$. Assume that $\mathcal{V}$ is closed. You may also assume that $\mathcal{V}$ is (co)complete if you wish.
If $k$ is a commutative ring, a $k$ algebra can be defined in two ways. Either as ... | https://mathoverflow.net/users/12166 | Enriched monoidal categories | There is a theorem in category theory, generally regarded as folklore, which says that for a symmetric monoidal closed category $V$, the following structures are equivalent:
1. a category $C$ with an action $V\times C\to C$ of the monoidal category $V$ on $C$, which we may write as $(v,c)\mapsto v\*c$, for which $-\*... | 13 | https://mathoverflow.net/users/10862 | 52016 | 32,605 |
https://mathoverflow.net/questions/52023 | 17 | Does Poincare-Hopf index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}\_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field considered (maybe it should vanish ou... | https://mathoverflow.net/users/11521 | Is there a Poincare-Hopf Index theorem for non compact manifolds? | Every noncompact manifold admits nonzero vector fields, or more generally,
vector fields with any specified set of isolated zeros along with the behavior near
that zero.
However, if you have information of the behavior of a vector field
near infinity, or just in a neighborhood of the boundary of a compact set, there... | 24 | https://mathoverflow.net/users/9062 | 52026 | 32,613 |
https://mathoverflow.net/questions/51936 | 7 | EDIT: Question solved.
---
Let me explain what I mean.
The classical formulation of Galois descent, e. g. in [Crawley-Boevey's "Cohomology and central simple algebras"](http://www.amsta.leeds.ac.uk/~pmtwc/cohom.pdf), uses the following notion:
**Definition:** A $K$-vector space *with additional structure* is ... | https://mathoverflow.net/users/2530 | What kind of structures allow Galois descent? | If you allow tensor powers of duals of $X$, then all the things you are talking about are just elements of certain vector spaces ($X\to X$ is just an element of $X^∗\otimes X$, for example). And any Galois invariant element of $V\otimes L$ is an element of $V$ (where $V$ is a vector space over $K$).
| 3 | https://mathoverflow.net/users/7868 | 52030 | 32,616 |
https://mathoverflow.net/questions/52032 | 36 | I want to know some examples of topological spaces which are not metrizable. Of course one can construct a lot of such spaces but what I am looking for really is spaces which are important in other areas of mathematics like analysis or algebra. I know most spaces arising naturally in other areas of mathematics are metr... | https://mathoverflow.net/users/12213 | Examples of non-metrizable spaces | I'm not an expert, but I believe the space of Distributions (in any number of variables), (a.k.a. generalised functions, including the Dirac delta, its derivatives, etc.) as used in PDE theory, is a topological vector space, but *non-metrisable*; even though sequences *are* sufficient to do everything.
However, the s... | 14 | https://mathoverflow.net/users/6651 | 52045 | 32,624 |
https://mathoverflow.net/questions/52043 | 6 | The $abc$-conjecture implies that the equation $a+b=c$ has only finitely many primitive solutions in the multiplicative semigroup generated by any particular finite set of primes.
I would appreciate any information about the status of this a priori weaker statement,
and citations to the literature if any exist.
| https://mathoverflow.net/users/10909 | Weak form of the $abc$-conjecture? | The fact that the [S-unit equation](http://en.wikipedia.org/wiki/S-unit) has finitely many solutions is due to Siegel and Mahler. The statement has been generalized quite a bit (for example to the fact that $u\_1+\cdots +u\_n=1$ has finitely many solutions, which uses some generalization of Schmidt's subspace theorem).... | 10 | https://mathoverflow.net/users/2384 | 52046 | 32,625 |
https://mathoverflow.net/questions/52015 | 7 | Is there a simple description of a Chow ring of a blow-up of a point on a smooth projective variety? Or at least of successive blow-ups of $\mathbb{P}^n$?
Maybe something like $A(\tilde{X})=f^\*(A(X))\oplus\mathbb{Z}(E)$, where $f\colon\tilde{X}\to{}X$ is a blow-up, E is an exceptional divisor, with multiplication... | https://mathoverflow.net/users/12208 | Simple description of a Chow ring of blow-ups. | The general formula about the intersection ring of blow-ups is discussed in Fulton's book. In your case you want to study the intersection ring of a smooth algebraic variety $V$ blown up at a point $Z$. There is a simple formula for this situation by Keel. You can find it in his paper: Intersection Theory of Moduli Spa... | 7 | https://mathoverflow.net/users/5286 | 52056 | 32,632 |
https://mathoverflow.net/questions/52063 | 0 | Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure of $k$; yet one can assume that the extension is finite). Then the scheme $X\_{k'}$ is not necessarily reduced. Yet wha... | https://mathoverflow.net/users/2191 | An inseparable lift of a regular variety. | Even if $X\_{k'}$ is reduced, it is not necessarily regular: let $p>2$ denote the caracteristic, take $t$ in $k$ which is not a $p$th power, and consider the plane curve $X$ defined by $y^2=x^p-t$. It is geometrically reduced, and over $\overline{k}$ it has a unique singular point, namely $(t^{1/p},0)$. This correspond... | 3 | https://mathoverflow.net/users/7666 | 52066 | 32,637 |
https://mathoverflow.net/questions/52079 | 3 | I'm in doubt about the topology of maps between fibres of vector bundles.
Consider $E$ and $F$ vector bundles and the set of all linear maps from a fibre of $E$ to a fibre of $F$, ie, the set of all linear maps $T:E\_x \rightarrow F\_y$, where $E\_x$ is the fiber over $x$ and $F\_y$ is the fiber over $y$.
I want to... | https://mathoverflow.net/users/12223 | Topology of maps between fibers of vector bundles | If both bundles were trivial, say $X\times \mathbb{R}^n\rightarrow X$ and $Y\times \mathbb{R}^n\rightarrow Y$ one could just take $X\times Y\times M(m,n,\mathbb{R})$.
Different choices of trivializations should give homeomorphic spaces, so this topology seems to be right.
In general the bundles are just locally trivi... | 3 | https://mathoverflow.net/users/3969 | 52084 | 32,647 |
https://mathoverflow.net/questions/52060 | 6 | 1) It is well known that between a prime $p$ and $p^2$ always exist a prime, but what is the shortest proof of that (by elementary methods or not)?
(One can say that we can have it as a collorary of Bertrand's postulate, but it is a stronger result.)
2) I ask it as an example, are there any characteristic example... | https://mathoverflow.net/users/14726 | What is the shortest proof of the existence of a prime between $p$ and $p^2$ ? other examples? | It is possible to shorten Bertrand's Postulate's proof so it proves only the above. We can throw away the usually-proven upper bound on the primoral. Explicitly, following [Wikipedia's "Proof of Bertrand's postulate"](http://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate):
Lemma 1: $$\frac{4^{\lfloor n^2/2 \rf... | 6 | https://mathoverflow.net/users/2024 | 52085 | 32,648 |
https://mathoverflow.net/questions/52035 | 8 | Let $P\_n(t)$ be polynomials with integer coefficients with $d\_n = \deg(P\_n(t))$ going to infinity when $n$ goes to infinity
and with nonzero discriminants $disc(P\_n(t)) \neq 0$.
Question: Is
$$
\lbrace disc({P\_n(t)})\rbrace ^{\frac{1}{d\_n}}
$$
bounded when $n$ goes to infinity ?
| https://mathoverflow.net/users/11016 | Old question of Serre on discriminants of a sequence of polynomials | Some people seem a little confused by the wording of this question.
The better way of phrasing the question is: Is there a *lower* bound on
the root discriminants of *polynomials* of degree $d$ as $d \rightarrow \infty$.
The survey paper by Odlyzko (noted by Gerry in the comments above):
<http://archive.numdam.org/AR... | 10 | https://mathoverflow.net/users/nan | 52092 | 32,652 |
https://mathoverflow.net/questions/52090 | 6 | Is it true that the Hausdorff dimension of the limit set of a Kleinian group is the supremum of Hausdorff dimensions of finitely generated subgroups [perhaps under addtional hypotheses?]I can't seem to find a reference which proves (or disproves) this fact. This is related to my previous question on commutator of $\Gam... | https://mathoverflow.net/users/11142 | Infinitely generated Kleinian groups | No. Here's how you can construct, given $\epsilon > 0$, an infinitely generated Kleinian group whose limit set has Hausdorff
dimension 2, but every finitely generated subgroup has Hausdorff dimension $< \epsilon$.
The easiest examples are infinite free groups. First choose a sequence of bounds for the target Hausdorf... | 12 | https://mathoverflow.net/users/9062 | 52098 | 32,657 |
https://mathoverflow.net/questions/51944 | 2 | Well-known and useful facts are:
* any symmetric matrix over $\mathbb R$ is semi-simple (i.e. diagonalizable), and
* any hermitean matrix over $\mathbb C$ is semi-simple.
I will loosely speak about the shape of a matrix and mean the existence of some (linear) relations between matrix-entries (or functions of the m... | https://mathoverflow.net/users/8176 | Semi-simple matrices over fields of finite characteristic | This is only a hint, not an answer.
There is a simple characterization of semisimple matrices over finite fields. Namely, if $A\in M\_n(F\_q)$, its eigenvalues lie in $F\_{q^m}$, $m=lcm(2,\dots,n)$, and there is $P\in GL\_n(F\_{q^m})$ such that $P^{-1}AP$ is a diagonal of Jordan blocks $\lambda\_i I + N\_i$, $i=1,\do... | 4 | https://mathoverflow.net/users/6451 | 52100 | 32,658 |
https://mathoverflow.net/questions/52107 | 11 | Is there any concept in monoids that is similar to the concept "conjugate class" in groups? For example, are there any such similar concept in symmetric inverse monoids? Thank you very much.
| https://mathoverflow.net/users/11877 | The concept "conjugate class" in monoids. | There are several different definitions of conjugacy for semigroups. For inverse semigroups the best, in my opinion, definition is this: $a$ is conjugate to $b$ if there exists $t$ such that $t^{-1}at=b$, $tt^{-1}=e, t^{-1}t=f$, $ae=ea=a$, $bf=fb=b$. In this case the conjugacy relation is an equivalence relation and th... | 9 | https://mathoverflow.net/users/nan | 52112 | 32,666 |
https://mathoverflow.net/questions/52006 | 2 | An old problem (recently ``reheated" by Casas-Alvero, etc.) consists of trying to determine if degree $n$ polynomials
$P(t)$ that are $n$-th powers
are characterized by the following condition:
$$
\deg( gcd(P(t),P^{(r)}(t))) >0
$$
for all positive integers $r$ and $r \neq n,$
where $P^{(r)}(t)$ is the $r$-th formal d... | https://mathoverflow.net/users/11016 | Characterizing degree $n$ polynomials that are $n$-th powers in $GF(p)[t]$ | $x^{p-1}-x^{(p-1)/2}$ for $p \equiv 1 \mod 4$.
| 4 | https://mathoverflow.net/users/2290 | 52117 | 32,670 |
https://mathoverflow.net/questions/52116 | 2 | A question which, I suppose, is as easy as abc for an expert. For a given finite field $F$ of odd characteristic (if needed, the characteristic is $3$ and the size of $F$ is large), do there exist two quadratic forms on $F^5$ without non-trivial common zeroes? More generally, what should be the relation between $n$ and... | https://mathoverflow.net/users/9924 | Quadratic forms without common zeroes | Chevalley-Warning: If you have a system of forms of degrees $d\_1,...,d\_k$ in $n$ variables, they will have a common non-trivial zero if $n > \sum d\_i$. For $d\_i=2$, the condition is $n > 2k$.
There is a full proof in the wikipedia page: <http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem>
| 8 | https://mathoverflow.net/users/2290 | 52118 | 32,671 |
https://mathoverflow.net/questions/52099 | 17 | Is there a cohomology theory for algebraic groups which captures the variety structure and restricts to the ordinary group cohomology under certain conditions.
| https://mathoverflow.net/users/12226 | cohomology theory for algebraic groups | For a full treatment of the foundations it's best to consult Part I of the book *Representations of Algebraic Groups* by J.C. Jantzen (2nd ed., AMS, 2003) even though it's not easily available online. Rational (or Hochschild) cohomology has been well developed, including the broader scheme framework (Demazure-Gabriel b... | 16 | https://mathoverflow.net/users/4231 | 52120 | 32,673 |
https://mathoverflow.net/questions/52129 | 3 | Let $K$ be a number field, with ring of integers $O\_K$, and let $\alpha\in O\_K$ be a primitive element for the extension $K/Q$, with minimal monic polynomial $f(x)\in Z[x]$. If $p$ is a prime number, then the factorisation of the principal ideal $(p)$ in the Dedekind ring $O\_K$ can be described in terms of the reduc... | https://mathoverflow.net/users/4800 | Can the factorisation of (p) in a number field K be described by the minimal polynomial of a primitive element? | The discriminant of the polynomial is equal to the discriminant of the field times the square of the index $[\mathcal{O}\_K: \mathbf{Z}[\alpha]]$. Hence, if the $p$-adic valuation of the discriminant of $f(x)$ is $0$ or $1$, then one already has an integral basis $p$-adically.
Yet already the examples $x^2 - p^2$ and ... | 4 | https://mathoverflow.net/users/nan | 52139 | 32,685 |
https://mathoverflow.net/questions/52126 | 13 | This may be a fairly simple question. Suppose *G* is a (T0) topological group. Assume that *G* is path-connected, locally path-connected, and semilocally simply connected, so that covering space theory applies.
Question: Is it true that for any element of $\pi\_1(G,e)$ (where *e* is the identity element of *G*), ther... | https://mathoverflow.net/users/3040 | Each element of fundamental group of a topological group represented by homomorphism? | No. A continuous homomorphism $S^1\to G$ yields a map $BS^1\to BG$. The space $BS^1$ is homotopy equivalent to $\mathbb CP^\infty$. There is a topological group $G$ such that $BG$ is homotopy equivalent to the sphere $S^2$. A map corresponding to a generator of $\pi\_1G=\pi\_2BG=H\_2S^2$ would give an isomorphism $H^2B... | 24 | https://mathoverflow.net/users/6666 | 52140 | 32,686 |
https://mathoverflow.net/questions/52135 | 9 | Let $A$ be a DVR and let $X/A$ be a smooth, proper scheme with geometrically integral fibers. Is there an easy way to see that the Picard group of $X$ is isomorphic to the Picard group of the generic fiber $X\_\eta$ of $X$?
| https://mathoverflow.net/users/1107 | Picard group of scheme over DVR | Here is an argument, which hopefully is not too dodgy:
Firstly, if $D\_{\eta}$ is an effective divisor on $X\_{\eta}$, then the Zariski closure $D$
of $D\_{\eta}$ will be an effective divisor on $X$. Since $Pic(X\_{\eta})$ is generated
by the classes of effective divisors, this shows that the restriction map from $Pi... | 6 | https://mathoverflow.net/users/2874 | 52148 | 32,688 |
https://mathoverflow.net/questions/51875 | 2 | I'm recently encountering such a problem, which I think is very intuitive but I have no background knowledge on this field:
Given a signal with certain frequency distribution, e.g. we know that the signal has only high frequency part, we can expect that the number of zero-crossings of the signal has to be large. Is t... | https://mathoverflow.net/users/12187 | Estimate on zero-crossings of band passing signals | In the discrete frequency setting (periodic signal) everything is nice and clean: since we can find a real-valued trigonometric polynomial $P$ of degree $\le n$ with given $2n$ or fewer roots on the circle, the real-valued signal $f=\sum\_k a\_k z^k$ that changes sign at most than $2n$ times, should have some non-zero ... | 2 | https://mathoverflow.net/users/1131 | 52150 | 32,690 |
https://mathoverflow.net/questions/52146 | 5 | Let $\mathbb{Z}\_n$ denote the ring of the $n$-adic integers. I recently read a paper which used the fact that the Baumslag-Solitar groups BS($\pm$1,n) and BS(n,$\pm$1) can be realized as functions $\mathbb{Z}\_n \rightarrow \mathbb{Z}\_n$. Can BS(m,n) (for m and n arbitrary) be realized as a group of functions $\mathb... | https://mathoverflow.net/users/8434 | Realizing Baumslag-Solitar groups as functions of the $n$-adic integers | If you mean action by automorphisms, then the answer is "no" since the Baumslag-Solitar groups $BS(m,n)$, $|m|\ne |n|\ge 2$ are not residually finite. The groups $BS(m,n)$ do act nicely on the products of a tree and the Hyperbolic space: <http://www.emis.de/journals/JLT/13-2/galpl.ps.gz> .
| 6 | https://mathoverflow.net/users/nan | 52156 | 32,693 |
https://mathoverflow.net/questions/52081 | 13 | Cipolla's algorithm [http://en.wikipedia.org/wiki/Cipolla's\_algorithm](http://en.wikipedia.org/wiki/Cipolla%27s_algorithm) is an efficient algorithm for finding a square root modulo a prime number. Is there an efficient algorithm for finding a square root modulo a prime power?
| https://mathoverflow.net/users/7089 | Is there an efficient algorithm for finding a square root modulo a prime power? | Joe Silverman's comment gives the method. (if the square root of A mod p is 0 you have any easy first step.... let $\gcd(A\ ,p^n)=p^j.$ If $j$ is odd, give up, otherwise let $A=p^{2k}B$ and find the $\mod p \ $ square root of $B$ (if it is a quadratic residue.)
I ascertained this by looking at the modular square root... | 6 | https://mathoverflow.net/users/8008 | 52159 | 32,695 |
https://mathoverflow.net/questions/52164 | 11 | Where can you find Grothendieck's "Récoltes et Semailles"?
Is it available anywhere?
| https://mathoverflow.net/users/7626 | Where can you find Grothendieck's "Récoltes et Semailles"? | [http://people.math.jussieu.fr/~leila/grothendieckcircle/RetS.pdf](http://people.math.jussieu.fr/%7Eleila/grothendieckcircle/RetS.pdf)
archived at the [Wayback Machine](https://web.archive.org/web/20141029175552/http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/RetS.pdf)
| 6 | https://mathoverflow.net/users/11260 | 52166 | 32,699 |
https://mathoverflow.net/questions/52039 | 9 | I'm wondering about the question
>
> "If we have a finitely presented \_\_, is it necessarily finitely presented with respect to any finite generating set for it?"
>
>
>
I know this is true for groups and for [$R$-modules](https://mathoverflow.net/questions/1788/does-finitely-presented-mean-always-finitely-pr... | https://mathoverflow.net/users/5583 | Does "finitely presented" mean "always finitely presented", considered in general | Yes in general.
See Adámek and Rosicky, Locally Presentable and Accessible Categories Cambridge University Press, Cambridge, (1994).
T. 3.12 p. 143.
Of course "in general" I mean: every "algebraic theory" (many sorted) Set models.
For topological algebraic structures, (like profinite groups) this equivalence i... | 9 | https://mathoverflow.net/users/6262 | 52167 | 32,700 |
https://mathoverflow.net/questions/52075 | 6 | Hi,
Perhaps it's a stupid question, in that case i'll delete it.
Let M be a compact orientable smooth (Kahler if changes things) manifold of dimension $dim\_{\mathbb{R}}(M)=2n$ with $n\geq1$, let $\gamma$ be a simple smooth closed curve that lies in a (holomorphic) coordinate chart and that can be taken as small as n... | https://mathoverflow.net/users/4971 | Solution of Plateau Problem for a simple, smooth closed curve on a Riemannian Manifold (Kahler) gives a surface that can be parametrized by a closed disk? | I've been waiting for someone with more expertise than me to answer, but since they haven't (so far) I'll say something.
There are different versions of Plateau-like problems; I'm not sure if there's a specific single one that's generally accepted as "the Plataeu problem". One can ask for a mapped-in disk with minima... | 12 | https://mathoverflow.net/users/9062 | 52168 | 32,701 |
https://mathoverflow.net/questions/51940 | 9 | If $R$ is a commutative ring (with $1$), then an $R$-algebra $A$ is said to be *separable* if $A$ is projective as an $A$-$A$-bimodule. (The notion of an "$A$-$A$-bimodule" includes the requirement that $R$ acts the same way from the left and from the right. The notion of "projective $A$-$A$-bimodule" is defined in the... | https://mathoverflow.net/users/2530 | Is there any transitivity for separable algebras? | You do not even need that the intermediate algebra be commutative:
>
> Let $R$ be commutative and let $A\subseteq C\subseteq B$ be a chain of $R$-algebras. If $B$ is a separable extension of $C$ and $C$ of $A$, then $B$ is a separable extension of $A$.
>
>
>
This is one of the exercises in Pierce's *Associativ... | 3 | https://mathoverflow.net/users/1409 | 52173 | 32,703 |
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