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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/74073 | 11 | While the Kronecker-Weber theorem —that every finite abelian extension of $\mathbb Q$ is contained in a cyclotomic field— is always attributed to, well, Leopold Kronecker and Heinrich Martin Weber, most sources I've seen that care to go into such details observe that their proofs were incomplete and were later fixed by... | https://mathoverflow.net/users/1409 | The first complete proof of the Kronecker-Weber theorem | The correct reference is
* Olaf Neumann, *Two proofs of the Kronecker-Weber theorem "according to Kronecker, and Weber"*, J. Reine Angew. Math. 323 (1981), 105-126
This is also the source that Schappacher relies on. Neumann analyses Weber's first proofs
(there's not much of a proof in Kronecker) and points out his... | 12 | https://mathoverflow.net/users/3503 | 74098 | 45,067 |
https://mathoverflow.net/questions/74086 | 2 | Given a projective surface $S$, and a smooth projective curve $C\subset S$ over $\mathbb{C}$. Furthermore we have a locally free sheaf $E$ of rank $r$ on $S$.
Then for any $l\geq 1$, the projective scheme $\operatorname{Quot}(E,l)$ classifies quotients $E\to T$, such that $\operatorname{dim}\_{\mathbb{C}}(H^0(S,T))=l... | https://mathoverflow.net/users/3233 | Is restricting the support of an Artinian sheaf a closed condition? | The equation of the curve $C$ gives a chain of maps
$$
H^0(S,T) \stackrel{C}\to H^0(S,T(C)) \stackrel{C}\to H^0(S,T(2C)) \stackrel{C}\to \dots
$$
It is clear that $supp(T) \subset C$ iff $C^N = 0$ (as a map $H^0(S,T) \to H^0(S,T(NC))$) for $N$ sufficiently large. In fact it is enough to take $N = l$. Thus $X$ is the z... | 1 | https://mathoverflow.net/users/4428 | 74099 | 45,068 |
https://mathoverflow.net/questions/74104 | 0 | Hi all,
I have been looking for the volume on Asterisque about Kummer surfaces but I do not have all the information. Dose any one know which volume/year is it?
Thanks.
| https://mathoverflow.net/users/17495 | Asterisque on Kummer surfaces | Perhaps, Asterisque 126 (1985):
mathscinet/search/publications.html?pg1=ISSI&s1=12429
| 4 | https://mathoverflow.net/users/17497 | 74110 | 45,071 |
https://mathoverflow.net/questions/74111 | 3 | Is there any classification result(s) regarding how many symplectic structures on CP^n?
| https://mathoverflow.net/users/16750 | How many Symplectic Structures on CP^n? | One of the headline consequences of Taubes' work on Seiberg-Witten theory on symplectic four- manifolds was that the standard symplectic form on $\mathbb{C}P^2$ is the unique one (up to scale, of course); see Theorem B of [this paper](http://www.mrlonline.org/mrl/1995-002-002/1995-002-002-010.html). Of course for $\mat... | 5 | https://mathoverflow.net/users/424 | 74120 | 45,074 |
https://mathoverflow.net/questions/74117 | 20 | For beginers, any suggestions?
| https://mathoverflow.net/users/16750 | Any good reference for Tits Building? | A good reference is Ken Brown and Peter Abramenko's "Buildings" (there is also the first edition freely available at <http://www.math.cornell.edu/~kbrown/buildings/>). Otherwise, you may want to look at Tits' "Reductive groups over local fields" from the Corvallis 1979 proceedings (see [Corvallis 1979 proceedings](http... | 11 | https://mathoverflow.net/users/13027 | 74122 | 45,075 |
https://mathoverflow.net/questions/74092 | 1 | I would like to understand why bubbling of disks are said to be co-dimension 1 phenomena and bubbling of spheres co-dimension 2 phenomena.
| https://mathoverflow.net/users/17492 | codimension of bubbling of disk and sphere | Moduli spaces of psuedoholomorphic curves have an associated expected dimension, given by the Fredholm index of the appropriate Cauchy-Riemann operator; if an appropriate transversality condition holds then the moduli space will (at least at non-multiply-covered curves) be a smooth manifold of actual dimension equal to... | 4 | https://mathoverflow.net/users/424 | 74126 | 45,077 |
https://mathoverflow.net/questions/74130 | 1 | Let $R$ be a Dedekind domain with quotient field $K$, let $L$ be a finite separable extension of $K$, and let $S$ be the integral closure of $R$ in $L$. If $\mathfrak{p}$ is a nonzero prime ideal of $R$ that is contained in the union of the prime ideals of $R$ that split completely in $S$, does it follow that $\mathfra... | https://mathoverflow.net/users/17218 | The union of the totally split primes | If the class group is finite, then writing $\mathfrak{p}^h = (\alpha)$, it follows that
$\alpha$ is contained in a prime ideal $\mathfrak{q}$ which splits completely in $S$, and thus $\mathfrak{p}^h \subseteq \mathfrak{q} \Rightarrow \mathfrak{p} = \mathfrak{q}$ (because $R$ has dimension one). It sounds like that suff... | 5 | https://mathoverflow.net/users/17277 | 74139 | 45,085 |
https://mathoverflow.net/questions/74106 | 2 | Is there any well written introduction for the modular space of complex structures on $T^2$?
| https://mathoverflow.net/users/16750 | Any introduction to Teichmuller Space of $T^2$? | As requested, I'm promoting this comment to an answer:
McKean and Moll's book "Elliptic curves" is a basic introduction to 2-tori with complex structure from the function theoretic, geometric, and arithmetic perspectives. What's closest to what you want is discussed in section 2.6 (on moduli of elliptic curves) and ... | 4 | https://mathoverflow.net/users/353 | 74144 | 45,090 |
https://mathoverflow.net/questions/73830 | 60 | Hello,
Recently, a colleague of mine pointed me to a MathSciNet review of one of my papers that is completely off the mark - it is not negative or anything like that, but it grossly misrepresents the contents of the paper (when describing the origins of the techniques and questions in the paper, for instance, as well... | https://mathoverflow.net/users/17422 | A question about MathSciNet etiquette | Thanks to everyone for the advice. It seems that the question has stopped attracting new reactions, so I'll try to summarize what I took from the discussion.
* There is no way (nor should there be) or hiding the fact that I'm the one complaining about the review's quality - anyway, the reviewer signed his/her review ... | 11 | https://mathoverflow.net/users/17422 | 74148 | 45,091 |
https://mathoverflow.net/questions/68659 | 2 | When X = pts, we know that the index of [D] equal to 0. What about X is not a point. Thanks
| https://mathoverflow.net/users/15970 | In K-homology K(X), if the Dirac operator D is invertible, does [D] represent zero element? | This question is answered, e.g., in *Higson, Roe: Analytic K-Homology*.
Proposition 12.2.4 states that if X is $Spin^c$, then the K-homology fundamental class [X] is not equal to zero.
Since the K-homology fundamental class of X is defined by the K-homology class of any Dirac operator on any complex spinor bundle i... | 5 | https://mathoverflow.net/users/13356 | 74151 | 45,094 |
https://mathoverflow.net/questions/74080 | 5 | Dear community,
i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them.
To explain my question i will use an example:
Let $V^k$ be the space of twice differentiable functions $U:[0,2\pi] \to \mathbb{R}^k$
with periodic boundary conditions.
Consider the ... | https://mathoverflow.net/users/17482 | Diagonalization of a matrix of differential operators | Another approach, for this particular example is to try to solve the equation $AMA^{-1} - m I = 0$, where $A$ is an invertible $2$-by-$2$ matrix of functions and $m$ is a scalar differential operator. There are a number of solutions to this. For example,
$$
A = \begin{pmatrix}\cos(\tfrac12x) & \sin(\tfrac12x)\cr
-\si... | 4 | https://mathoverflow.net/users/13972 | 74155 | 45,098 |
https://mathoverflow.net/questions/74119 | 4 | I'm familiar with container functors and older work by Dybjer on categorical models for W-types in the extensional theory, but I was looking for some similar semantics in the intensional case.
| https://mathoverflow.net/users/14899 | What are categorical models of W-types in intensional type theory? | Unless I've not understood your question correctly (sometimes people mean different things by the distinction between *intensional* and *extensional*), then I think the answer is: the semantics for W-types in intensional type theory are exactly the same as the semantics for W-types in extensional type theory.
A model... | 4 | https://mathoverflow.net/users/6485 | 74157 | 45,099 |
https://mathoverflow.net/questions/74102 | 17 | I just had to make use of an elementary rational function identity (below). The proof is a straightforward exercise, but that isn't the point. First, "my" identity is almost surely
not original, but I don't have a reference for it. Perhaps someone knows it (like a lost cat without a collar) or, more likely, could spot ... | https://mathoverflow.net/users/5045 | rational function identity | I have seen a cat of a similar breed in the representation theory of symmetric groups. Out of habit, let me quote a lemma attributed to Littlewood in
Donald Knutson, *$\lambda$-rings and the Representation Theory of the Symmetric Group*, Springer 1973 (LNM #308), Chapter III, section 2, p. 149:
$\sum\limits\_{\sigm... | 16 | https://mathoverflow.net/users/2530 | 74160 | 45,101 |
https://mathoverflow.net/questions/74095 | 18 | Is there a set convention for which name (maiden name or married name) a female married mathematician should use?
While this question addresses women's maiden name it applies equally to men's maiden name when it differs from their married name. The question seeks for an advice for the dilemma: whether to use the mai... | https://mathoverflow.net/users/17489 | Maiden Names vs. Married Names | Like all questions involving names and marriage, there is no set convention (at least in the US). I know a male mathematician who publishes under his wife's last name which he took at marriage and I know people who have started publishing under a new name before they took it legally. As Ben says, there's also no rules ... | 22 | https://mathoverflow.net/users/22 | 74166 | 45,103 |
https://mathoverflow.net/questions/74121 | 3 | Let $M$ be a closed manifold, $m$ be the normalized volume measure on $M$, and $f:M\to M$ be a $C^2$ transitive Anosov diffeomorphism. Consider the pushforward $f^km$ defined by
----------$f^km(A):=m(f^{-k}A)$ for all measurable subset $A\subset M$.
Then the Birkhoff averages $\nu\_k=\frac{1}{k}\sum\_{j=0}^{k-1}f^j... | https://mathoverflow.net/users/11028 | Accumulation points of the Birkhoff average of $m$ | Yes. In fact, we have $\mathcal{V}(m) = \{\mu\_+\}$ whenever $m$ is a probability measure on $M$ that is absolutely continuous with respect to volume. This is shown in (0.4) of "[A measure associated with Axiom-A attractors](http://www.jstor.org/pss/2373810)" by David Ruelle, *American Journal of Mathematics* **98** (1... | 5 | https://mathoverflow.net/users/5701 | 74169 | 45,105 |
https://mathoverflow.net/questions/74168 | 3 | Does there exist a compact Riemannan manifold $M^n$ and an $L > 0$ such that the number of homotopy classes of simple closed curves $\gamma$ on $M^n$ whose shortest representatives have length at most $L$ is infinite? For surfaces ($n=2$) with constant curvature metrics, this is impossible. Thanks!
| https://mathoverflow.net/users/17512 | Geodesics of bounded length on a Riemannian manifold | This cardinality is always finite, for any compact locally simply connected metric space. If there were infinitely many non-homotopic curves of length $\le L$, they would have a converging subsequence (by Arzela-Ascoli). In a locally simply connected space, any two sufficiently close curves are homotopic, so curves in ... | 18 | https://mathoverflow.net/users/4354 | 74178 | 45,111 |
https://mathoverflow.net/questions/74164 | 1 | This is a question I've had for a while and really don't know how to go about finding an answer:
Does there exist a pair of binary operations, $\boxplus$ and $\boxtimes$, other than the usual $+$ and $\times$, such that $(\mathbb{Q}, \boxplus, \boxtimes)$ forms a ring?
I realize that there's probably some "axiom of... | https://mathoverflow.net/users/3400 | Other Ring Structures on $\mathbb{Q}$ | Given any bijection $f : \mathbb{Q} \to R$ where $(R,\oplus,\otimes)$ is some (necessarily countable) ring, you'll be able to get a new ring structure $(\mathbb{Q},\boxplus,\boxtimes)$ isomorphic to $(R,\oplus,\otimes)$, by setting:
$a \boxplus b = f^{-1}(f(a)\oplus f(b))$
$a \boxtimes b = f^{-1}(f(a)\otimes f(b))... | 12 | https://mathoverflow.net/users/7521 | 74189 | 45,118 |
https://mathoverflow.net/questions/73579 | 9 | Hi,
I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn.
I do not understand why the analytic index map $\operatorname{ind}\colon K\_{cpt}(T^\ast X) \to Z$, as defined in chapter III.$13 in equation (13.8), agrees with th... | https://mathoverflow.net/users/13356 | Understanding the analytic index map of the Atiyah-Singer index theorem | The Fredholm index of an elliptic operator only depends on the symbol class. Here is the proof (which I memorize from Lawson-Michelsohn and Atiyah-Singer).
If $D: \Gamma(E\_0) \to \Gamma(E\_1)$ has order $k \neq 0$, pick a connection $\nabla$ on $E\_0$. Then $A=(1+\nabla^{\ast}\nabla)$ is a self-adjoint invertible op... | 7 | https://mathoverflow.net/users/9928 | 74192 | 45,120 |
https://mathoverflow.net/questions/74185 | 10 | I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces. Note that by "bundle", I mean a holomorphic fiber bundle, which means it is locally holomorphically trivial.
I'm really just looking for information about holomorphic disc bu... | https://mathoverflow.net/users/1116 | Classification of holomorphic disc bundles | There's a paper of H. L. Royden
["Holomorphic fiber bundles with hyperbolic fiber", Proc. AMS, Volume 43, Number 2, April 1974](http://www.ams.org/journals/proc/1974-043-02/S0002-9939-1974-0338465-0/S0002-9939-1974-0338465-0.pdf)
which proves
"Theorem: The holomorphic fiber bundles with hyperbolic fiber
M and ... | 7 | https://mathoverflow.net/users/1116 | 74193 | 45,121 |
https://mathoverflow.net/questions/74135 | 8 | Hello,
I am a PhD student who does not have extensive computational experience seeking advice from those experienced with computational modelling as to which method would be most appropriate for solving my particular problem.
**Background**
*Physical Scenario*
The Salvinia is a small floating fern. Its leaves h... | https://mathoverflow.net/users/12069 | Computational methods for dealing with geometrically complicated solid boundaries in fluid-air interface problems | In addition to level set methods, there a couple other things you might want to look into.
One possibility is isogeometric analysis (T.J.R. Hughes and collaborators), which is designed for taking complicated smooth surfaces and discretizing them for finite-element-like computations.
Another thing to think about for... | 5 | https://mathoverflow.net/users/17113 | 74194 | 45,122 |
https://mathoverflow.net/questions/74181 | 1 | I have a pretty basic question:
given is a complex manifold of dimension n or a smooth projective variety over $k$ (char 0, algebraically closed).
Then people are often speaking of "the cohomology class of the diagonal".
Can you explain me how you realize this class, i.e. in which cohomology group does it land, how d... | https://mathoverflow.net/users/16876 | Cohomology class of the diagonal | Let $\Delta : M \to M \times M$ be the diagonal map. Since $M$ is a complex manifold, say of complex dimension $n$, it has a canonical orientation class $[M] \in H\_{2n}(M, \mathbb{Z})$. Then you can take the pushforward in homology to get $\Delta\_\ast [M] \in H\_{2n}(M \times M, \mathbb{Z})$. If $M$ is compact then $... | 6 | https://mathoverflow.net/users/83 | 74199 | 45,126 |
https://mathoverflow.net/questions/73992 | 5 | Let $E$ be a (complete) topological vector space, and $u:E\to E$ be continuous. Is it always true that if ${\rm Im}(u)$ is of finite codimension in $E$, then it is closed in $E$ or do we have to assume something on $E$? (It is OK if $E$ is Frechet by the open mapping theorem applied to ${\rm id}\oplus u:F\oplus E\to E$... | https://mathoverflow.net/users/17467 | finite codimension implies closed? | No. For $E$ take $X\oplus \ell\_2$, where $X$ is the direct sum of continuum many copies of the scalar field under the direct sum topology. This is the largest locally convex topology on $X$ and any linear mapping from $X$ into a locally convex space is continous. Write $X=X\_1 \oplus X\_2$ with each $X\_i$ isomorphic ... | 5 | https://mathoverflow.net/users/2554 | 74200 | 45,127 |
https://mathoverflow.net/questions/74202 | 4 | Quoting English Wikipedia: 'a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group'.
I'd like to ask if there is any nice criterion to characterize groups that can be embedded just in a finitely generated group?
| https://mathoverflow.net/users/17519 | Characterization of subgroups of finitely generated groups | A group embeds in a finitely (even 2-) generated group iff it is countable. It was proved by Higman, B. Neumann and H. Neumann (Higman, Graham; Neumann, B. H.; Neumann, Hanna
Embedding theorems for groups.
J. London Math. Soc. 24, (1949). 247–254.)
| 10 | https://mathoverflow.net/users/nan | 74205 | 45,128 |
https://mathoverflow.net/questions/74201 | 2 | I have a general question, and then the specific version of that question I need for research. All vector spaces over $\mathbb{C}$.
Grassmanians of planes
----------------------
The $(2,n)$-Grassmannian, denoted $Gr(2,n)$, is the space of all 2 dimensional subspaces of $n$ dimensional space. This is naturally a com... | https://mathoverflow.net/users/750 | Smoothness of hypersurfaces in Grassmannians | Here is the answer to your specific question (and, in fact, something a little more general):
The hypersurface defined by $f=x\_{ij}-x\_{kl}$ (assuming that $i$, $j$, $k$, and $l$ are distinct) is smooth if $n\le 5$ and singular for $n>5$. The singular locus consists of the $2$-planes that lie in the codimension $4$ ... | 11 | https://mathoverflow.net/users/13972 | 74207 | 45,130 |
https://mathoverflow.net/questions/74212 | 29 | Suppose we have integer matrices $A\_1,\ldots,A\_n\in\operatorname{GL}(n,\mathbb Z)$. Define $\varphi:F\_n\to\operatorname{GL}(n,\mathbb Z)$ by $x\_i\mapsto A\_i$. Is there an algorithm to decide whether or not $\varphi$ is injective?
| https://mathoverflow.net/users/35353 | Is it decidable whether or not a collection of integer matrices generates a free group? | For $n=1, 2$ the answer is "yes" since the group is virtually free, for $n\ge 3$ the answer is not known (an open problem).
**Edit.** In fact even for two $n\times n$-matrices the problem is open. Moreover the solution of the following ``easier" problem is not known: for which algebraic integers $\lambda$ the matr... | 27 | https://mathoverflow.net/users/nan | 74217 | 45,134 |
https://mathoverflow.net/questions/74208 | 20 | What's the meaning of the adjective 'alternating' in the name of the 'alternating group' ?
| https://mathoverflow.net/users/10194 | Meaning of 'alternating' group ? | Evidence for Brendan McKay's suggestion that it comes from the older concept of an [alternating polynomial](http://en.wikipedia.org/wiki/Alternating_polynomial) may be found in Burnside's classic book, *Theory of Groups of Finite Order*. When defining the symmetric and alternating groups, he puts an asterisk next to th... | 21 | https://mathoverflow.net/users/3106 | 74220 | 45,137 |
https://mathoverflow.net/questions/74113 | 4 | Is it true that every involution $\sigma$ (i.e., $\sigma^2=identity$) of an Enriques surface $X$ acts trivially on $K\_X^{\otimes 2}$ i.e., for any $\omega\in K\_X^{\otimes 2}$ we have $\sigma^\* \omega=\omega$, where by $K\_X^{\otimes 2}$ we mean the tensor 2 of the conanical bundle of $X$.
| https://mathoverflow.net/users/13559 | every involution of an Enriques surface is | Let me try an argument different from Christian's: $\sigma$ does not act freely as $\chi(\mathcal O\_X)=1$ and hence not divisible by $2$. At a fixed point $x$, $\sigma$ acts by $\pm1$ on the fibre of $\omega\_X$ and hence acts by $1$ on the fibre of $\omega\_X^{\otimes2}$. It also acts by a scalar on a global non-zero... | 9 | https://mathoverflow.net/users/4008 | 74225 | 45,142 |
https://mathoverflow.net/questions/74229 | 4 | Let $X$ be a smooth projective manifold, $L$ is a holomorphic vector bundle over $X$ such that $mL$ ($m$ is some positive integer) is base point free. If $\iota\\_m$ is the map into the projective space associated to the complete system and each fiber of this map has dimension zero. Then $L$ is ample.
I see this stat... | https://mathoverflow.net/users/15882 | Ampleness of a kind of line bundle | This is indeed rather standard (where I assume that you mean that $L$ is a *line* bundle). It follows from the following general facts:
1. Let $m$ be a positive integer. Then a line bundle $L$ is ample if and only if $mL$ is ample.
2. Let $X$ be a proper (e.g. a projective) scheme over a field $k$ (or in fact over an... | 7 | https://mathoverflow.net/users/13302 | 74234 | 45,147 |
https://mathoverflow.net/questions/74245 | 4 | Consider the following diagram of regular local rings
$\begin{matrix}
\hat{A} & \xrightarrow{\quad\hat\varphi\quad} & \hat{B} \\
\ \uparrow\scriptstyle\alpha & \circlearrowleft & \ \uparrow\scriptstyle\beta \\
A & \xrightarrow{\quad\varphi\quad} & B
\end{matrix}$
where $\widehat{\\,\dot\\,}$ denotes the completion ... | https://mathoverflow.net/users/9947 | Pulling back roots from the Completion | Non. Let $A$ be the localization of $\mathbb C[x]$ at the maximal ideal $x\mathbb C[x]$ and let $B=A[z]/(z^2-x(1+x))$. Then $1+x$ is the square of a unit in $\hat{A}$. Hence $\hat{B}=\hat{A}[y]/(y^2-x)$. But $x$ is not a square in $B$ because otherwise $1+x$ would be a square in $B$, but one can check directly that thi... | 6 | https://mathoverflow.net/users/3485 | 74248 | 45,156 |
https://mathoverflow.net/questions/74257 | 6 | Can anyone give me an example of:
>
> * An infinite abelian but non-cyclic group whose automorphism group is cyclic.
>
>
>
| https://mathoverflow.net/users/1483 | Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic | Let $G$ be the (non-finitely generated) additive group of rational numbers with square-free denominators. Then the only automorphism of $G$ is negation.
From [here](https://groupprops.subwiki.org/wiki/Aut-cyclic_not_implies_cyclic).
| 17 | https://mathoverflow.net/users/460 | 74260 | 45,161 |
https://mathoverflow.net/questions/74203 | 1 | Let $K$ be a quadratic number field, and let $E\_1$ and $E\_2$ be two isogeneous elliptic curves over $K$. Assume we know that $j(E\_1)^\sigma=j(E\_2)$ where $\sigma$ is the generator of the Galois group of $K/Q$.
Can we then say that some twist of $E\_1$ is a $Q$-curve? If so, is there a good way of describing the ne... | https://mathoverflow.net/users/92 | What can we say if E^sigma is isogeneous to a twist of E? | Yes -- a Q-curve is one whose geometric isogeny class is preserved by Galois, and that's evidently the case here. Of course there is no guarantee that E\_1 and its Galois conjugate are isogenous over K. Is that the question you're asking? If so, I think there's a cohomological criterion for this due to Quer -- at least... | 2 | https://mathoverflow.net/users/431 | 74261 | 45,162 |
https://mathoverflow.net/questions/69677 | 2 | Suppose I have a higher category, say a simplicial category $C$, and I want to invert a certain type of morphism $W$. Simplicial localization (for example hammock localization), gives a localized simplicial categetory $W^{-1}C$. Then I truncate this category by passing to $\pi\_0$ of hom-sets to obtain the 1-category $... | https://mathoverflow.net/users/4709 | truncation commutes with localization? | This is Denis-Charles Cisinski's answer, given in the comments:
Yes, the map $\pi\_0(W^{-1}C)\to W^{-1}\pi\_0(C)$ is always an equivalence of categories (this follows immediately by comparing the corresponding universal properties). The same remains true (for formal reasons as well) if you look at the truncations of ... | 1 | https://mathoverflow.net/users/4709 | 74268 | 45,166 |
https://mathoverflow.net/questions/74272 | 7 | I'm a Mathematics masters student currently
studying some aspects of TQFT. I'm interested in Langlands, mainly
because it sounds oppressive! Is anyone familiar with any links between
CFT and Langlands, or more importantly any readable intros to these
sorts of areas.
How do Hecke Eigensheaves come into things? What do... | https://mathoverflow.net/users/17532 | Conformal Field Theory and Langlands | There's a review article by Edward Frenkel which exactly fits your need: see ["Lectures on the Langlands Program and Conformal Field Theory"](http://arxiv.org/abs/hep-th/0512172). It's posted on hep-th, but as I (as a string theorist) had difficulties reading it, it should be written for mathematicians :)
| 8 | https://mathoverflow.net/users/5420 | 74273 | 45,168 |
https://mathoverflow.net/questions/74191 | 6 | (previous title " Zero sum of binomials coefficients - a stronger version ")
This is a stronger version of [another question](https://mathoverflow.net/questions/74045/zero-sum-of-binomial-coefficients).
Is there an $N\in \mathbb N$ and a sequence of non-constant functions $ \left\{ p\_n:[n] \to \{ 1,-1 \} \right\}\_{... | https://mathoverflow.net/users/17476 | What is the degree of a symmetric boolean function? | It was already pointed out in the comments that determining for which $n$ one can find a non-constant $p\_n$ is an open problem. I thought I'd give a bit of context and my understanding on what is known so far. The problem as stated has a negative answer because when $n+1$ is prime, $p\_n$ must be constant.
The sums ... | 6 | https://mathoverflow.net/users/2384 | 74277 | 45,171 |
https://mathoverflow.net/questions/74283 | 3 | I post again a question I asked in the post by Descartes:
Since this is the topic on diagonal, I like to ask a question: Let $X$ be a compact Kahler manifold of complex dimension $n$, and let $\Delta \_X\subset X\times X$ be the diagonal of $X$. We denote by $\{\Delta \_X\}$ its cohomology class in $H^{2n}(X\times X)... | https://mathoverflow.net/users/17326 | de Rham cohomology class of diagonal | Take $X$ a smooth compact curve of genus $>0$. The diagonal can be written as $\sum \pi\_1^\* e\_i \smile \pi\_2^\* e'\_i$ where $e\_i$ form a basis of the total cohomology and $e\_i'$ form a Poincar\'e dual basis. So the $\pi\_1^\*H^1\smile\pi\_2^\* H^1$-component of the diagonal is nonzero, and so the diagonal does n... | 3 | https://mathoverflow.net/users/2349 | 74285 | 45,176 |
https://mathoverflow.net/questions/74271 | 4 | Hello, all!
I have a big sum of log-normal (with location parameter $\mu$ and scale parameter $\sigma$) random variables $X\_i$ $\sum\_{i=1}^N X\_i$ with $N \gg 1$.
How could I estimate convergence rate to a gaussian distribution relative to $\mu$ and $\sigma$?
Thank you.
| https://mathoverflow.net/users/nan | CLT convergance rate for sum of log-normals | Log normal distribution has finite variance, so if you subtract the mean, the magic words are "Berry-Esseen theorem". If you don't subtract the mean, the sum diverges.
| 4 | https://mathoverflow.net/users/11142 | 74286 | 45,177 |
https://mathoverflow.net/questions/74237 | 9 | In the works of Cisinski, Tabuada, and Schlichting certain non-connective K-theory groups for a differential graded category $C$ are defined. As far as I understand, $K\_i(C)$ is not necessarily zero for $i<0$. Yet are there any sufficient conditions on $C$ that ensure that this condition is fulfilled? Are $K\_i(C)$ ju... | https://mathoverflow.net/users/2191 | The vanishing of non-connective K-theory in negative degrees | This abstract non-connective $K$-theory, when restricted to schemes, is known from a long time: this is the Bass $K$-theory functor $K^B$ considered by Thomason and Trobaugh (the article of Thomason and Trobaugh is a classic on the subject which must be read anyway). The comparison of the abstract construction of non-c... | 11 | https://mathoverflow.net/users/1017 | 74290 | 45,179 |
https://mathoverflow.net/questions/74289 | 2 | Can anyone tell me where hyperchaotic system is used in real world, its applications? It'd be great if you could provide a literature to back any application as well. Thanks.
| https://mathoverflow.net/users/16883 | Whats the application of Hyperchaotic system? | <http://www.scholarpedia.org/article/Hyperchaos#Experimental_hyperchaotic_behaviors>
and some more pointers:
* C. Stan, C.P. Cristescu, and D. Alexandroaei, Chaos and hyperchaos in a symmetrical discharge plasma: experiment and modelling, University Politehnica Of Bucharest Scientific Bulletin-Series A-Applied Math... | 2 | https://mathoverflow.net/users/11260 | 74292 | 45,181 |
https://mathoverflow.net/questions/74302 | 7 | Suppose $V\models ZFC$ and $P\in V$ is a poset of forcing conditions.
* It is a basic theorem in forcing that $V[G]\models ZFC$ for any generic extension by a $V$-generic filter $G$.
* It is also known that if $V\subseteq M\subseteq V[G]$, and $M\models ZFC$ then $M$ is a forcing extension of $V$ and $V[G]$ is a for... | https://mathoverflow.net/users/7206 | Symmetric extensions and class forcing | These intermediate model questions were thoroughly investigated by Serge Grigorieff in *Intermediate Submodels and Generic Extensions in Set Theory* [Annals of Mathematics 101 (1975), 447-490].
The basic result of the kind you are looking for is due to Solovay (according to Grigorieff):
>
> Let $P$ be a poset in ... | 7 | https://mathoverflow.net/users/2000 | 74307 | 45,189 |
https://mathoverflow.net/questions/74298 | 7 | Hi all,
Could you give me a suggestion of suitable book about Banach Manifolds for someone that have background in functional analysis at the level of Conway's book and Do Carmo's book on Riemannian Geometry ?
To help the indication the problems I am facing in my research are for example, what are the usual condit... | https://mathoverflow.net/users/nan | A book on Banach Manifold for a Dynamicist | Especially if your interested in dynamical systems, I highly recommend [Abraham--Marsden--Ratiu, Manifolds, tensor analysis, and applications](http://www.ams.org/mathscinet-getitem?mr=960687).
For a more Riemannian-geometric/global-analytic focus, you might want to try [Klingenberg, Riemannian Geometry](http://www.am... | 3 | https://mathoverflow.net/users/2063 | 74311 | 45,192 |
https://mathoverflow.net/questions/74252 | 20 | Kronecker. Nuff said. Even the numbers themselves historically started
as positive integers and were subsequently generalized to hell and back.
Here are some other well known concepts that "should" involve $\mathbb{N}$
but were generalized to $\mathbb{Q}$, $\mathbb{R}$ or even $\mathbb{C}$:
1. Dimension $\rightarro... | https://mathoverflow.net/users/11504 | Things that should be positive integers...really? | The [writhe](http://en.wikipedia.org/wiki/Writhe) is the fundamental differential geometric invariant of a closed space curve. I think it is the most useful topological invariant outside mathematics- biologists use it to study circular DNA molecules, and chemists use it in the study of long polymers. For space curve $C... | 16 | https://mathoverflow.net/users/2051 | 74314 | 45,194 |
https://mathoverflow.net/questions/73989 | 3 | Does any one know the Jacobson radical of the path algebra of the following quiver?
$$\bullet \leftrightarrows \bullet$$
How many simplerepresentations of it are there?
Is there any software that computes the Jacobson radicals of infinite dimensional non-commutative algebras?
| https://mathoverflow.net/users/9096 | The Jacobson radical of an infinite dimensional algebra | As there seem to be some differing opinions in the comments as to whether all
irreducible representations are finite-dimensional let me give the argument I
had in mind. A module over the path algebra is the same thing as a
representation of the quiver, which in this case means two vector spaces $U$ and
$V$ and linear m... | 3 | https://mathoverflow.net/users/4008 | 74331 | 45,205 |
https://mathoverflow.net/questions/74338 | 3 | Suppose I have a $C^\infty$ smooth function $f$ defined on the reals.
I can apply Taylor's formula and get the local expression
$$
f(x) = \sum\_{i=0}^l\frac{f^{(i)}(0)}{l!}x^i+ f^{(l+1)}(\xi(x))x^{l+1}.
$$
Question: Is the function $\xi $ smooth? The function $f$ can in principle be as nice as you want.
| https://mathoverflow.net/users/6035 | Taylor Series Remainder | Note that the point $\xi$ in the expression of the remainder is not unique in general (as it is clear already for $l=0$). According to a common phenomenon, lack of unicity may cause a lack of continuity. For an example where there is no continuous $\xi$ (again for $l=0$) think of a smooth function $f$ which is positive... | 6 | https://mathoverflow.net/users/6101 | 74339 | 45,208 |
https://mathoverflow.net/questions/53924 | 9 | Let $H$ be a Hopf algebra over a field $k$, and $I$ be a biideal of $H$. I am looking for conditions that guarantee that $I$ is a Hopf ideal (that means $S\left(I\right)\subseteq I$).
One condition that definitely works is that $\dim\left(H / I\right) < \infty$ (where $\dim$ means dimension as a $k$-vector space). Th... | https://mathoverflow.net/users/2530 | Is a biideal of a Noetherian Hopf algebra automatically Hopf? | The answer is yes.
In ["Quotients of Hopf Algebras"](https://doi.org/10.1080/00927877808822321 "zbMATH review at https://zbmath.org/0391.16006"), Warren D. Nichols, Comm. Algebra 6(1978), 1789-1800, proves that if $I$ is a bi-ideal then it will be a Hopf ideal under any of the following conditions:
* $H/I$ is finit... | 6 | https://mathoverflow.net/users/2384 | 74340 | 45,209 |
https://mathoverflow.net/questions/74293 | 2 | So How does one prove (rigorously) that
$$
Frac(\mathbb{C}[x,y,t]/(y^2-x^3-t)) \not\simeq Frac(\mathbb{C}[t][x,y]/(y^2-x^3-1))?
$$
So here $Frac$ denotes the fraction field of an integral domain.
Note that this gives an example of (a non-trivial) isotrivial family over $\mathbf{C}^{\times}$.
| https://mathoverflow.net/users/11765 | (non-trivial) isotrivial family of elliptic curves over C^{\times} | There is a slightly different proof which works over any field $k$ (of characteristic different from 2 and 3). The first field is just $k(x,y)$. As Rita indicated, if it is isomorphic to the second field as $k$-extension, than there exists a birational map $f: \mathbb P^2\to \mathbb P^1\times E$. This birational map in... | 5 | https://mathoverflow.net/users/3485 | 74343 | 45,210 |
https://mathoverflow.net/questions/74341 | 2 | My question concerns the notion of a generically finite morphism
$f: X \rightarrow Y$ of "nice" schemes, say integral and noetherian.
I want to define $f$ gen. finite if the generic fibre is finite.
Can I characterize this property somehow by the relation of the dimensions of $X$ and $Y$?
For example, can I con... | https://mathoverflow.net/users/16876 | Generically finite morphisms | If $f$ is locally of finite type **EDIT: and dominant**, then your inequality holds. ~~First replace $Y$ by the Zariski closure of $f(X)$ and we are reduced to the case when $f$ is dominant~~. For all $x\in X$ and $y=f(x)$, we have the dimension formula
$$ \dim O\_{X,x}+ \dim (\overline{\lbrace x \rbrace}\cap X\_y)\le ... | 6 | https://mathoverflow.net/users/3485 | 74344 | 45,211 |
https://mathoverflow.net/questions/74247 | 4 | Consider a morphism $f: Y \to X$ between two varieties and consider the stacks parametrizing coherent sheaves on them $\mathcal{M}\_X, \mathcal{M}\_Y$.
Does one have for free an induced pullback morphism $f^\*: \mathcal{M}\_X \to \mathcal{M}\_Y$?
I guess the question reduces to: if $S$ is a base scheme and $E$ is a... | https://mathoverflow.net/users/16857 | can one define the pullback between stacks of coherent sheaves for non-flat morphisms? | (I had to delete my earlier answer; the following answer is based on a comment by ulrich which vanished when I deleted the answer.)
The answer to your question is *no*.
*Counterexample:* Let $X$ be the plane and $Y$ the blow-up of $X$ at the origin. Consider the tautological family of length 1 skyscraper sheaves o... | 2 | https://mathoverflow.net/users/4709 | 74347 | 45,213 |
https://mathoverflow.net/questions/74348 | 0 | I'm trying to find a set of uniform measure 1/2 over $ \{ -1,1 \} ^n \times \{-1,1\}^n$ such that the inner product of $(x,y)\in\{ -1,1 \} ^n \times \{-1,1\}^n$ will hold $|\langle x,y\rangle|< \frac{\sqrt(n)}{c}$ for some constant $c$.
I believe that a better way to look at it is saying I have a simple random walk. ... | https://mathoverflow.net/users/nan | Radius of random walk on Z | By simple counting argument one has
$$P(X\_1+\cdots+X\_n=r)=\frac{1}{2^n}\binom{n}{\frac{n-r}{2}}$$
ignoring parity conditions. Stirling approximation implies that if $r=o(n)$ we have
$$\binom{n}{\frac{n-r}{2}}=O\left(n^{-\frac{1}{2}}2^n\right).$$
So that
$$P(|X\_1+\cdots+X\_n|\le r)=\sum\_{k=-r}^r \frac{1}{2^n}\binom{... | 1 | https://mathoverflow.net/users/2384 | 74352 | 45,215 |
https://mathoverflow.net/questions/74213 | 1 | $M$ is an $n\times n$ matrix. Consider the submatrices $M(P;Q)$ formed from $P$ rows and $Q$ columns of $M$ where $P$ and $Q$ are disjoint indices.
Is there some way to encode the various determinants such as $\det M((1,2,4);(3,5,6))$, $\det M((1,2);(3,5))$, and $\det M((1,3);(2,4))$ in a single object much like chro... | https://mathoverflow.net/users/16557 | Encoding information about submatrix determinants | If you consider $M\colon V\to V$ as a linear map of an $n$-dimensional vector space $V$ into itself (with some canonical basis $e\_i$ used to express $M$ as a matrix), then the $k$-minors (which you called subdeterminants) of $M$ are the components of the induced linear map $\bigwedge^k M\colon \bigwedge^k V \to \bigwe... | 1 | https://mathoverflow.net/users/2622 | 74355 | 45,217 |
https://mathoverflow.net/questions/74320 | 17 | For a homotopy sheaf $\mathcal{F}$ of ring spectra over some space (/ site / whatever) $X$ with a cover $U\_i$, we can build a "descent spectral sequence" with signature $$E^1\_{p, q} = \pi\_{p+q} \mathcal{F}\left(\coprod\_{|I| = q} U\_I \right) \Rightarrow \pi\_{p+q} \mathcal{F}(X).$$ This comes about by building the ... | https://mathoverflow.net/users/1094 | Multiplicativity in the descent spectral sequence | Using your $X$ and $Y$, you get an augmented simplicial object as follows:
$$
\cdots Y \times\_X Y \times\_X Y \Rrightarrow Y \times\_X Y \Rightarrow Y
$$
Applying $\cal F$ to this diagram, you get a coaugmented cosimplicial ring spectrum. The spectral sequence for the homotopy groups of Tot of this which realizes you... | 9 | https://mathoverflow.net/users/360 | 74367 | 45,224 |
https://mathoverflow.net/questions/70419 | 11 | Suppose we have a tower of fibrations of spectra $\{X\_k\}\_{k\in\mathbb{N}}$ with inverse limit $X\_\infty$, and let $F\_k$ be the fibre of the map $X\_k\to X\_{k-1}$. There is then a spectral sequence $E^1\_{jk}=\pi\_j F\_k \Longrightarrow \pi\_{j+k} X\_\infty$. If we instead have a tower of fibrations of based space... | https://mathoverflow.net/users/10366 | Unbased spectral sequences | As requested, I am reposting this comment as an answer.
Bousfield covers this material in "Homotopy spectral sequences and obstructions," Israel J. Math 66. The discussion is specific to cosimplicial objects (e.g. the discussion of obstruction cocycles in Section 5) and the general method of obtaining "partially" def... | 7 | https://mathoverflow.net/users/360 | 74368 | 45,225 |
https://mathoverflow.net/questions/74349 | 2 | Can all nonzero degree map between compact Riemann surfaces (both genus >1 ) be deformed to holomorphic maps, if we can change the conformal structures on them? The simplest case: does there exist holomorphic map of degree one from \Sigma\_m to \Sigma\_n (m>n>1)?
| https://mathoverflow.net/users/17547 | Degree of holomorphic maps between compact Riemann surfaces | The basic obstruction here is the [Riemann-Hurwitz formula](http://en.wikipedia.org/wiki/Riemann%25E2%2580%2593Hurwitz_formula): If there is a degree $d$ map from a Riemann surface of genus $g$ to one of genus $h$, with branch points of orders $e\_1$, $e\_2$, ..., $e\_r$, then
$$2(g-1) = 2d (h-1) + \sum (e\_i -1).$$
As... | 2 | https://mathoverflow.net/users/297 | 74375 | 45,228 |
https://mathoverflow.net/questions/74373 | 2 | Let K be an imaginary quadratic field, A(K) its p-class group, and H(K) its p-Hilbert class field. If rk(A(K))=2, a result due to Arrigoni tells us that p^3 divides the order of the class group of H(K). Are there any explicit non-trivial lower bounds in the case that rk(A(K))>2 ?
| https://mathoverflow.net/users/17115 | Lower bound on the class group of the p-Hilbert class field of an imaginary quadr. field | If $G$ is the Galois group of the $p$-class field tower over $K$, then $A(H(K))=G'/G''$ is a quotient of $G\_2/G\_4$, where $G\_i$ denotes the lower central series. By Arrigoni's calculation that $G\_2/G\_4$ has $p$-rank exactly $\frac{d(d-1)(2d+5)}{6}$, this serves as a lower bound for the $p$-rank of $A(H(K))$. When ... | 3 | https://mathoverflow.net/users/35575 | 74377 | 45,230 |
https://mathoverflow.net/questions/74371 | 0 | Let $R$ be a commutative Noetherian ring, $M$ is a finitely generated $R$-module. If $F: Mod \to Mod$ is a left exact functor and $R^iF(E)=0$ where $E$ is injective module. Assume that $F(-) \cong Hom(M,-)$, can we infer the $i-th$ right devired functors $R^iF(-)\cong Ext^i(M,-)$?
| https://mathoverflow.net/users/9141 | Equivalent functors | Yes. For example, if you compute right derived functors by injective resolutions, then naturality of the isomorphism between $F$ and $\text{Hom}(M,-)$ will ensure that you have an isomorphism between the two complexes whose cohomology groups give you the two derived functors.
| 1 | https://mathoverflow.net/users/6794 | 74378 | 45,231 |
https://mathoverflow.net/questions/74381 | 0 | For e.g. any range of number 0 - n
>
> 0 1 2 3 4 5 6
>
>
>
to:
>
> 0 2 4 6 4 2 0
>
>
>
Is there a name for this kind of formula or calculation?
| https://mathoverflow.net/users/17553 | Is there a name for a formula to calculate ascending numbers to a quadratic-like sequence? | The [tent map](http://en.wikipedia.org/wiki/Tent_map). In your case it'd be scaled to $f(x) = 2 \min(x, 6-x)$.
| 2 | https://mathoverflow.net/users/3400 | 74383 | 45,234 |
https://mathoverflow.net/questions/74388 | 4 | Let $A$ and $B$ be two Hermitian matrices. The famous [Golden-Thompson inequality](http://en.wikipedia.org/wiki/Golden-Thompson_inequality) states that
$$\text{tr}(e^{A+B}) \le \text{tr}(e^Ae^B)$$
However, for determinants we have equality
$$\det(e^{A+B}) =\det(e^Ae^B)$$
I was wondering if similar results can ... | https://mathoverflow.net/users/8430 | Extensions to the Golden-Thompson inequality? | This is theorem IX.3.5 in "Matrix analysis" by R. Bhatia (Graduate Texts in Mathematics, 169). See also corollary IX.3.6 and theorem IX.3.7. The Golden-Thompson inequality holds when $Tr$ is replaced with a function $f$ which satisfies $f(XY)=f(YX)$ and $|f(X^{2m})|\le f(|XX^{\ast}|^m)$ for all $m\geq 1$. Such function... | 9 | https://mathoverflow.net/users/2384 | 74390 | 45,237 |
https://mathoverflow.net/questions/68288 | 7 | I am wondering about the following:
>
> Suppose that $S$ is a non-compact
> hyperbolic surface of finite area.
> Suppose that $\lambda \subset S$ is a
> non-trivial, geodesic,
> measured lamination. Forget the transverse measure. Is there a
> (non-compact) geodesic lamination $\lambda'$
> containing $\lambda... | https://mathoverflow.net/users/8183 | Laminations as a limit of ideal triangulations | It turns out that the answer to the question is "yes". Saul Schleimer and I needed this result for a paper that we just finished writing, so we ended up sorting it out. The full argument is written down in [Lemma A.6 in the Appendix of this paper](https://arxiv.org/abs/1108.5748), so what follows below is an outline.
... | 5 | https://mathoverflow.net/users/8183 | 74391 | 45,238 |
https://mathoverflow.net/questions/74399 | 2 | So let us define the generalized disc of degree $n$ as
$$
\mathbb{D}\_n:=\{w\in M\_{n\times n}(\mathbb{C}):w=w^t, I\_n-w\overline{w}>0\}.
$$
For a Hermitian matrix $A$, the notation $A>0$ means that it is positive definite.
**Q**: So how do you prove cleanly that $\mathbb{D}\_n$ is bounded?
| https://mathoverflow.net/users/11765 | Looking for a simple proof that the generalized disc is bounded | The diagonal entries of a positive definite matrix are real and non-negative. If we let the rows of $w$ be $v\_1,\dots,v\_n$, then the diagonal entries of $I\_n-w\overline{w}$ are $1-v\_i\overline{v\_i}=1-\sum\_{j=1}^n|w\_{ij}|^2$. So in particular $|w\_{ij}|\le 1$. This implies that $\mathbb D\_n$ is bounded.
| 2 | https://mathoverflow.net/users/2384 | 74401 | 45,244 |
https://mathoverflow.net/questions/74136 | 22 | I have a few questions on the history of PDE.
1. Who first wrote down the formula for the solution of the Cauchy problem for the heat equation involving the heat kernel? I have seen it called Poisson's formula. If it is true Poisson has a formula for each of the heat, wave, and Laplace equations.
2. Who is the discov... | https://mathoverflow.net/users/824 | History of fundamental solutions | Question 2 is getting clearer now. My sources are Parseval's [article](http://books.google.ca/books?id=SDMVAAAAQAAJ&pg=PA515) from 1800, Poisson's [memoire](http://books.google.com/books?id=TZ8AAAAAYAAJ&pg=PA121) from 1819, Hadamard's [Lectures on Cauchy's Problem in Linear Partial Differential Equations](http://books.... | 12 | https://mathoverflow.net/users/824 | 74403 | 45,246 |
https://mathoverflow.net/questions/74370 | 6 | Suppose we have an invertible matrix q in a finite subgroup $Q$ of
$Gl(n,\mathbb Z)$, the group of all invertible integer matrices. Now I want to
find all $x\; mod\; \mathbb Z^n$ for which
$(q+q^2+q^3+...+q^m).x = 0\quad mod\; \mathbb Z^n$
where $m$ is the order of $q$ in the finite subgroup $Q$ of $Gl(n,\mathbb Z)... | https://mathoverflow.net/users/17551 | Finite subgroup of $Gl(n,\mathbb Z)$ and congruences | **Edit:** I couldn't resist my predilection for generalizations: Using darij grinberg's simplification, the proof below shows:
Let $k$ be a field, $q \in GL\_n(k)$ a matrix of finite exponent $m$ with char$(k) \nmid m$ and $M \subseteq k^n$. Futhermore, let $E$ be the eigenspace of $q$ corresponding to the eigenvalu... | 5 | https://mathoverflow.net/users/10194 | 74404 | 45,247 |
https://mathoverflow.net/questions/74297 | 5 | BACKGROUND
----------
Assume a poset $\langle P, \le \rangle$. For two points $a,b \in P$
with $a \le b$, then $I = [a,b] = \{ x : a \le x \le b \}$ is the
interval between $a$ and $b$.
When $P$ is a chain (e.g. ${\mathbb Z}, {\mathbb R}$), then the $I$
are just standard intervals. Two real intervals $I=[a,b],J=[c,... | https://mathoverflow.net/users/17333 | Intervals in posets: how to extend interval orders, Allen's algebra, and interval graphs to intervals of posets? | The subset order on intervals of a finite poset has received some
attention. See for example Exercises 3.10, 3.76(b), 3.138, and 3.158(c)
of <http://math.mit.edu/~rstan/ec/ec1.pdf>.
| 4 | https://mathoverflow.net/users/2807 | 74411 | 45,248 |
https://mathoverflow.net/questions/74415 | 11 | Let $f : X \to X/\sim$ be a quotient map from a topological space $X$ to the quotient space $X/\sim$ for $\sim$ some equivalence relation. Let $S \subseteq X/\sim$. Is it true that $f^{-1}(\overline{S}) = \overline{f^{-1}(S)}$?
The specific case I have in mind is a Borel subgroup $B$ of a Chevalley group $G$ acting ... | https://mathoverflow.net/users/12402 | Is the preimage of the closure the closure of the preimage under a quotient map? | Yes, this is true for any open continuous map $f: X \to Y$.
Since $\bar{B} = \neg int (\neg B)$ where $\neg$ denotes complementation and $int$ denotes interior, and since the preimage map $f^{-1}(-): P(Y) \to P(X)$ preserves complementation, we just need to check that $f^{-1}$ preserves the interior operation. The i... | 8 | https://mathoverflow.net/users/2926 | 74416 | 45,250 |
https://mathoverflow.net/questions/74427 | 13 | This is just a random question I was thinking of. There are lot of cases of things in algebraic geometry unifying and generalizing geometric and arithmetic ideas. For example, the etale fundamental group putting together both Galois theory and covering spaces, so that the etale covers are just field extensions.
I wa... | https://mathoverflow.net/users/12402 | Does the name divisor in algebraic geometry relate to divisor in the basic arithmetic or ring theory sense? | This answers address the part of the question asking what the divisors above correspond to in number theory and ring theory. I also include some vague remarks on the naming.
The relevance of Dedekind domains in the geometric context was already mentioned in a comment. And, the notion of Dedekind domains in some sens... | 12 | https://mathoverflow.net/users/nan | 74434 | 45,259 |
https://mathoverflow.net/questions/74425 | 9 | Let *G* be a nontrivial finite group. Is it true that the sum of the orders of all elements of *G* is not divisible by the order of *G*?
| https://mathoverflow.net/users/17565 | A question on the sum of element orders of a finite group | It is false in general, for instance there's a group of order $3\cdot 5\cdot 7=105$ with sum of orders equal to $1785=3\cdot 5\cdot 7\cdot 17$. (In Magma, it is the first of the two groups of order 105 in the "small groups" database).
However it is true for all groups of even order, because the sum of orders of eleme... | 20 | https://mathoverflow.net/users/20038 | 74441 | 45,263 |
https://mathoverflow.net/questions/74406 | 2 | Suppose we have a map $f:X\to Y$ and we form the mapping cylinder $M\_f$. Hatcher claims that it is obvious that the pair $(M\_f, X \cup Y)$ satisfies the homotopy extension property. Equivalently we could find a retraction of $M\_f \times I$ to $M\_f\times \{0\} \cup (X \cup Y)\times I$. I don't see how we can get thi... | https://mathoverflow.net/users/17559 | Homotopy Extension Property involving mapping cylinder | Neil has given an explicit retraction. But it may be useful to note that you can obtain results like this from a combination of some "easier" facts:
* The pair $(I,\{0,1\})$ has the HEP.
* If $(L,K)$ has the HEP where $K$ and $L$ are locally compact Hausdorf, and if $Z$ is any space, then $(Z\times L, Z\times K)$ has... | 8 | https://mathoverflow.net/users/437 | 74443 | 45,264 |
https://mathoverflow.net/questions/74450 | 2 | We say that an action of a (discrete) group G on a locally compact space X is called proper if the map from $G\times X$ to $X\times X$ defined by $(g,x)\mapsto (gx,x)$ is proper. Why is a proper action amenable? (see On the Baum-Connes assembly map for discrete groups-Alain Valette, proof of lemma 2.13). If this is a c... | https://mathoverflow.net/users/9401 | proper action and amenable action | Look at that paper by C. Anantharaman-Delaroche: <http://www.univ-orleans.fr/mapmo/membres/anantharaman/publications/Exactness02.pdf>
In Prop. 2.2, point (2), you find an equivalent condition for amenability of the $G$-action on $X$, in terms of the existence of a net $(g\_i)$ of continuous, non-negative functions on... | 5 | https://mathoverflow.net/users/14497 | 74453 | 45,267 |
https://mathoverflow.net/questions/74459 | 9 | An alternative ring is an algebraic structure where all the field axioms are true except for the commutativity and associativity of multiplication, but it is alternative, i.e. for all a,b $a(ba)=(ab)a$ and $(aa)b=a(ab)$. If I prove in such a structure that for all a,b $ab=ba$ holds, does it follow that $a(bc)=(ab)c$?
... | https://mathoverflow.net/users/16425 | Are commutative alternative rings associative? | No. There are commutative alternative rings that are not associative. An example due to Kaplansky is a commutative alternative algebra over the $\mathbb F\_3$ with basis $\lbrace x,y,z,u,v,w\rbrace$ and relations $xy=u, yz=v, xv=w, uz=-w$ (the other products are zero).
---
It turns out you are interested in commu... | 13 | https://mathoverflow.net/users/2384 | 74462 | 45,270 |
https://mathoverflow.net/questions/74386 | 10 | I've seen a bunch of definitions of spectra in the literature, and the fanciest seems to be the $(\infty, 1)$-category of spectra obtaining by "stablizing" the higher category of spaces, as in DAG I. I don't really understand this stabilization procedure yet and would like to connect this idea to the more concrete noti... | https://mathoverflow.net/users/344 | How should I think of the $\infty$-category of spectra? | Basically you want to know what the space of maps between two spectra $X$ and $Y$ is. Well, each map from $X$ to $Y$ is a sequence of maps from $X\_n$ to $Y\_n$ and thus $\mathrm{map}(X,Y)$ is a subspace of the product of the $Y\_n^{X\_n}$. And you can build an entire spectrum of maps from $X$ to $Y$, whose nth space i... | 9 | https://mathoverflow.net/users/644 | 74465 | 45,271 |
https://mathoverflow.net/questions/74410 | 3 | I've chatted informally with some folks about this question before and gotten some very nice insights, but I thought I'd toss it out to a wider audience because it is a continuing curiosity of mine.
Roughly, here's what I have in mind: Let $\mathcal{E}$ denote the eigencurve of some tame level $N$. At classical point... | https://mathoverflow.net/users/12107 | What is the nature of the locus in the eigencurve associated to some conditions on the associated automorphic representation (at $p$)? | Since the eigencurve only sees classical modular forms with non-zero $U\_p$-eigenvalue, there are strong restrictions on the local factor at $p$ for the attached automorphic representations (e.g. you won't ever see supercuspidals). I think the situation is that if you fix a Nebentypus, with conductor $p^r$ at $p$, then... | 4 | https://mathoverflow.net/users/1979 | 74467 | 45,273 |
https://mathoverflow.net/questions/74468 | 3 | Assume that $\alpha\_1,\ldots,\alpha\_n$ are algebraic numbers. Assuming that
$\sum\_{i=1}^n \alpha\_i^k \in \mathbb{Z}$
for all $k\in\mathbb{N}$. Does this imply that the $\alpha\_i$ are actually algebraic integers? I know that if these $\alpha\_i$ are the conjugates of some algebraic number $\alpha$, then the rel... | https://mathoverflow.net/users/14371 | Implications of a relation on algebraic numbers | Proved by grobber ([Alexandru Chirvasitu](https://mathoverflow.net/users/9151/alexandru-chirvasitu)) on AoPS: <http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=157732>
**EDIT:** Also, <http://www.artofproblemsolving.com/Forum/viewtopic.php?f=38&t=335001> might contain a solution (posts #4, #7 and #8).
| 6 | https://mathoverflow.net/users/2530 | 74470 | 45,274 |
https://mathoverflow.net/questions/74433 | 11 | I've been looking at some definitions of Grothendieck rings. However I've not found a good definition that I've understood. Any recommendations?
I'm referring to the definition in tensor categories, more specifically I've discovered there are some structure coefficients in a Grothendieck Ring, I understand the mathema... | https://mathoverflow.net/users/17532 | Definition of a Grothendieck ring | I'll expand my comments into an answer. Since I'm not quite sure what parts bother you,
I'll assume it's everything! The Grothendieck construction is actually
a family of related constructions, which is brilliant in its simplicity.
Whenever, you have a collection of things (e.g. finite sets) that can split
into parts, ... | 16 | https://mathoverflow.net/users/4144 | 74474 | 45,276 |
https://mathoverflow.net/questions/74454 | 2 | This question, like all of my previous questions regarding Langlands, is very naive.
All $g\geq 1$ curves come from quotients of the upper half plane. The curves $X\_0(N)$ come from quotients of special subgroups of the group of automorphisms of the upper half plane. This might imply that they are easier to work with... | https://mathoverflow.net/users/5756 | Are the $L$-functions of $X_0(N)$ automorphic? | Langlands, in his [Antwerp II article](http://sunsite.ubc.ca/DigitalMathArchive/Langlands/shimura.html#antwerp), was the first to show that the zeta function of a modular curve is exactly the product (well, some of the $L$-functions are in the denominator) of $L$-functions of modular forms (previous results of Eichler,... | 7 | https://mathoverflow.net/users/1021 | 74481 | 45,280 |
https://mathoverflow.net/questions/74488 | 1 | I am trying to find a (smooth) compact complex surface $X$ so that the set of irreducible curves $C$ on $X$ for which $C.C<0$ is infinite. Do any of you know of an example. Thanks.
| https://mathoverflow.net/users/17326 | Compact complex surfaces having infinitely many negative curves? | Blow up $\mathbb P^2$ at 9 points. See e.g. Hartshorne exercise 5.4.15e) and the reference there.
| 4 | https://mathoverflow.net/users/17580 | 74490 | 45,285 |
https://mathoverflow.net/questions/74471 | 8 | Consider the $\mathbb{R}$-vector space of sufficiently nice real-valued functions on the unit square $I^2$, where "sufficiently nice" could be taken to mean any one of a number of things - say continuous for now.
In analogy with matrix multiplication, we can define the product of two such functions $F$ and $G$ as
$... | https://mathoverflow.net/users/6779 | Multiplying functions on the unit square as generalized matrices | This is a partial answer concerning the determinants of such objects. While Will Jagy is right that such objects will have an uncountable spectrum, there still may be some hope for assigning a determinant in some cases as follows. First there is one obvious candidate of a trace for such objects: $Tr(F) = \int\_0^1 F(t,... | 3 | https://mathoverflow.net/users/12301 | 74493 | 45,288 |
https://mathoverflow.net/questions/74485 | 2 | Let $M$ be a compact smooth manifold and $m$ be a normalized volume induced by some Riemannian metric on $M$. Let $f\in\mathrm{Diff}^1(M)$ and $R\_f$ be the set of recurrent points of $f$ (A point $x$ is recurrent if $x\in\omega(f,x)\cap \alpha(f,x)$, or equally $\liminf\_{n\to\pm\infty}d(f^nx,x)=0$).
The Hopf decom... | https://mathoverflow.net/users/11028 | Hopf decompostion for diffeomorphisms | Let me first notice that the proof of the Poincare recurrence theorem you are referring to is actually valid for any action of a countable group with an invariant measure. Now, if instead of $\mathbb Z$ we are talking about general groups, then there are, for instance, examples of Fuchsian groups such that their action... | 2 | https://mathoverflow.net/users/8588 | 74504 | 45,293 |
https://mathoverflow.net/questions/74503 | 13 | There appear to be two definitions of the word ergodic.
The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is *ergodic* if
>
> the only $T$-invariant sets have measure 0 or 1.
>
>
>
However, a Markov chain is ergodic if
... | https://mathoverflow.net/users/8769 | Different uses of the word "ergodic" | Unfortunately, the way the term "ergodic" is used in the theory of (finite) Markov chains is completely misleading from the point of view of general ergodic theory. To be consistent, one should have called "ergodic" the chains whose state space does not admit a decomposition into non-trivial non-communicating subsets. ... | 15 | https://mathoverflow.net/users/8588 | 74505 | 45,294 |
https://mathoverflow.net/questions/74508 | 10 | **Question.** Let $V$ be a complex projective manifold of general type (we can even assume that the canonical bundle of $V$ is ample). Suppose $\varphi: V\to V$ is a non-identical automorphism. Can $\varphi$ be isotopic to the identity map (i.e. $\varphi\in Diff\_0(V)$)?
I hope the answer is no, and this can be easi... | https://mathoverflow.net/users/943 | Finite order automorphisms of complex projective manifolds isotopic to identity | The answer to your question is unknown already for surfaces $S$ of general type.
Note that, if $S$ is simply connected, by a result of Quinn (see "Isotopy of 4-manifolds", Journal of Differential Geometry 1986) every automorphism acting trivially on rational cohomology must be *topologically* isotopic to the identit... | 13 | https://mathoverflow.net/users/7460 | 74510 | 45,298 |
https://mathoverflow.net/questions/74398 | 2 | Let $V$ be a representation of some torus $T$. It is then well-known that the Duistermaat-Heckman measure for $P(V)$ is the weak limit of the properly rescaled distribution of multiplicities of weights in $\mathrm{Sym}^n(V)$.
I've seen many allusions to the fact that the analogous statement is true for general compac... | https://mathoverflow.net/users/98 | Non-Abelian Duistermaat-Heckman Measure (not just a reference request) | Hello again. Yes, it's true.
The more general statement you want is, let $X$ be projective with a
$K$-equivariant ample line bundle ${\mathcal O}(1)$. For each $n$, let $\mu\_n$
be a measure on ${\mathfrak t}^\*\_+$,
$$ \mu\_n := \sum\_{\lambda \in {\mathfrak t}^\*\_+}
\frac{\dim Hom\_K(V\_{n\lambda}, \Gamma(X;{\m... | 7 | https://mathoverflow.net/users/391 | 74514 | 45,301 |
https://mathoverflow.net/questions/74496 | 23 | Consider infinite digraphs whose vertices are the integers $\mathbb Z$, with the property that there are exactly two arcs coming out of each vertex. (There is no restriction on the number of in-coming arcs.)
The question: is there such a digraph such that for $m,n\in\mathbb Z$, with $m<n$, there is a directed path fr... | https://mathoverflow.net/users/9025 | Jumping in the integers | Yes, such a directed graph exists, with a "quickly" computable path of length $O(\log|n-m|)$ from $m$ to $n$ for any distinct integers $m,n$.
I'll first construct a digraph with this property but with each vertex having out-degree at most $8$, then show how to get from it a digraph of out-degree $2$.
Denote by $v(x... | 27 | https://mathoverflow.net/users/14830 | 74517 | 45,303 |
https://mathoverflow.net/questions/74525 | 7 | Let $f,g: I \to I := [0,1]$ be continous functions satisfying $f \circ g = g \circ f$. Does
there exist $x\_0 \in I$ such that $f(x\_0) = g(x\_0)$ ?
Background: In a homework the problem was posed with $g=\operatorname{id}$ (where
it can easily be solved with the help of the intermediate value theorem). The
lectu... | https://mathoverflow.net/users/17588 | Have commuting functions a common value ? | Yes. The set of fixed points of $g$ is closed, nonempty, and is mapped into itself by $f$. Letting $a\le b$ be, respectively, the minimum and maximum fixed points of $g$, we have $f(a)\ge a=g(a)$ and $f(b)\le b=g(b)$. So, by the intermediate value theorem, there is an $x\in[a,b]$ with $f(x)=g(x)$.
Also, to reiterate ... | 13 | https://mathoverflow.net/users/1004 | 74530 | 45,307 |
https://mathoverflow.net/questions/74393 | 18 | I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I can access.
[Lax and Richtmyer](http://www.mathe.tu-freiberg.de/~ernst/Lehre/IVP/Literatur/laxRichtmyer1956.pdf) (195... | https://mathoverflow.net/users/12898 | Who introduced the notion of "stability" in numerical analysis? | John von Neumann is credited as having pioneered the stability analysis of finite difference schemes. Crank and Nicholson [1] acknowledge Von Neumann when they demonstrate the stability of their scheme in 1947, and a few years later the method was applied in a meteorological context in a paper co-authored by Von Neuman... | 18 | https://mathoverflow.net/users/11260 | 74536 | 45,313 |
https://mathoverflow.net/questions/74522 | 1 | Hello,
Can anybody explain to me how, in model theory, a type $p$ in a theory $T$ and language $L$ implies a type $p'$ in theory $T'$ and language $L'$, with $T \subset T'$ and $L \subset L'$. \
Also in the same context, how strong orthogonality of two definable sets $D$ and $D'$ is equivalent to the condition: If $A'$... | https://mathoverflow.net/users/nan | Type implication | For the first one, elements of your expanded language might be definable in the smaller language. As a very simple example, consider $T = Th((\omega, S)$, the theory of the natural numbers with the successor function, and $T' = T = Th((\omega, S, 0)$, the same but with $0$ in the language. $p(x) = \{ \neg \exists y ( S... | 1 | https://mathoverflow.net/users/15713 | 74537 | 45,314 |
https://mathoverflow.net/questions/74538 | 7 | Countable models of PA fall into two categories: the standard one $(\omega, S)$ and the nonstandard ones (all the rest). The only way I've seen to construct a nonstandard model is through taking an ultraproduct or, equivalently, using the compactness theorem. My question is wether or not these are all the models there ... | https://mathoverflow.net/users/15713 | Are all countable, nonstandard models of arithmetic given by ultrapowers? | An ultrapower will never yield a countable nonstandard model of PA --- either you will recover the standard model or the result will be uncountable.
As far as the construction of a model is concerned, due to Tennenbaum's theorem (see <http://en.wikipedia.org/wiki/Tennenbaum's_theorem>) you will never see a recursive ... | 25 | https://mathoverflow.net/users/5147 | 74539 | 45,315 |
https://mathoverflow.net/questions/74521 | 3 | We have a given positive martingale $\rho\_t$, with the dynamics:
$$\textrm{d} \rho\_t = \lambda\_t \rho\_t \textrm{d} W\_t$$
where $W\_t$ is a standard Brownian motion. Now we have a "dumped" process p\_t:
$$\textrm{d} p\_t = a\_t \lambda\_t \rho\_t \textrm{d} W\_t$$
where $0 \leq a\_t \leq 1$.
(Silly) question: ar... | https://mathoverflow.net/users/3160 | Stochastic integrals as honest martingales -- comparison criterion | No, $\rho$ need not be a proper martingale. To guarantee that $p\_t=\int\_0^ta\_sd\rho\_s$ is a martingale for all predictable $0\le a\_t\le 1$ you need the additional property that $\sup\_{s\le t}\rho\_s$ is integrable. In fact, for any cadlag martingale $\rho$, the following are equivalent.
1. $\sup\_{s\le t}\vert\... | 6 | https://mathoverflow.net/users/1004 | 74540 | 45,316 |
https://mathoverflow.net/questions/74458 | 18 | Is there some known way to create the Mandelbrot set (the boundary),
with an iterated function system (IFS)?
Julia sets can be formed by iterating the two functions $z \mapsto \pm \sqrt{z-c},$
and this quickly converges to the Julia set. Formally, the two functions form a Hutchinson operator,
and the Julia set is the... | https://mathoverflow.net/users/1056 | Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system? | I know very little about the Mandelbrot set or complex dynamics, but there are several features which strongly suggest that its boundary is not the attractor of an iterated function system, at least a reasonable one.
For example.
* Around [Misiurewicz point](https://en.wikipedia.org/wiki/Misiurewicz_point)s (which ... | 19 | https://mathoverflow.net/users/11009 | 74541 | 45,317 |
https://mathoverflow.net/questions/74545 | 4 | Johnstone and Silverman (2005) claimed that for large x
$\frac{1-\Phi(x)}{\phi(x)} \approx \frac{1}{x}$
where $\Phi(x)$ and $\phi(x)$ are the CDF and PDF for a normal random variable.
I was able to verify the claim numerically. Q: But how would I show this analytically? This seems like it should be easy, but I... | https://mathoverflow.net/users/17591 | Approximation to the ratio of a Gaussian CDF to PDF | If you interpret this as the existence of the limit
$$
\lim\_{x \rightarrow \infty} \frac{x(1-\Phi(x))}{\phi(x)}
$$
then it is easy to verify using l'Hopital's rule.
| 5 | https://mathoverflow.net/users/613 | 74551 | 45,324 |
https://mathoverflow.net/questions/74561 | 4 | The cone of symmetric positive semidefinite $n\times n$ matrices is the convex hull of rank $1$ matrices. That is, every symmetric positive semidefinite matrix is a convex combination of rank 1 matrices.
>
> Does this property generalize to solutions of linear systems of semidefinite matrices?
>
>
>
Let me be... | https://mathoverflow.net/users/nan | Is a solution of a linear system of semidefinite matrices a convex combination of rank 1 solutions? | No. Try $n=k=2$ with $A\_1 = \pmatrix{1 & 0\cr 0 & -1\cr}$ and $A\_2 = \pmatrix{1 & 1\cr 1 & -1\cr}$. The only symmetric matrices $X$ with $(A\_1,X) = (A\_2,X) = 0$ are multiples of $I$, so there are no rank 1 solutions.
| 4 | https://mathoverflow.net/users/13650 | 74563 | 45,329 |
https://mathoverflow.net/questions/74556 | 4 | For a permutation of a finite set $X$, define its *supporting set* as the complement in $X$ of its *fixed-point set*. The term *support* describes the size of a supporting set. For example, a $k$-cycle has support $k$.
Consider a subgroup $H$ of $S\_n$, such as $H=Aut(G)$ for an $n$-vertex graph $G$.
We are intereste... | https://mathoverflow.net/users/17596 | Generator sets of a subgroup of $S_n$ with $O(n)$ total support - do they always exist ? | About Q1, this is standard. Take a strong generating set. List the strong generators in inverse order of the length of their stabiliser. I.e., first come the generators that fix all but the last point in the base, and last come the generators that move the first element of the base. Now delete every generator whose cyc... | 4 | https://mathoverflow.net/users/9025 | 74564 | 45,330 |
https://mathoverflow.net/questions/74558 | 3 | For topological space $X$ (connected, path connected etc.), there is classification of coverings of $X$ : for fixed $x\_0\in X$, consider $\pi\_1(X,x\_0)$. Then there is a $1-1$ correspondance between conjugacy class of subgroups of $\pi\_1(X,x\_0)$ and covering spaces of $X$ (upto isomorphism).
We define universal cov... | https://mathoverflow.net/users/17456 | Coverings of a graph of groups | The universal cover of a graph of groups *is* the Bass--Serre tree. This is described in Serre's book *Trees*, to which you refer. I don't have a copy to hand, so I can't give you the precise reference, but it's the main object of study throughout the book.
Let me also add that, when trying to think about coverings o... | 1 | https://mathoverflow.net/users/1463 | 74569 | 45,333 |
https://mathoverflow.net/questions/74576 | 1 | Is there infinite number of prime pairs $(p\_k,p\_n)$ that satisfy equality $p\_n=2p\_k-3$
| https://mathoverflow.net/users/17600 | infinite number of prime pairs | This is surely open. The usual conjecture is that, given a collection of linear equations in primes, if there is no "local" obstacle to it having infinitely many solutions, then it does. Here local obstacles would mean either (1) the size of the real solutions is bounded or (2) there is some prime number $q$ such that ... | 18 | https://mathoverflow.net/users/297 | 74579 | 45,336 |
https://mathoverflow.net/questions/74544 | 27 |
>
> "One cannot walk to infinity on the real line if one uses steps of bounded
> length and steps on the prime numbers. This is simply
> a restatement of the classic result that there are arbitrarily
> large gaps in the primes."
>
>
>
So begins the paper by
Gethner, Wagon, and Wick,
["A Stroll Through the Gaussi... | https://mathoverflow.net/users/6094 | The quaternion moat problem | Having an infinite walk of bounded step length in the quaternions (or in $\mathbb Z^k$ in Gerry's version), gives us a sequence of primes $p\_1,p\_2\dots$ with $p\_{k+1}-p\_k=O(\sqrt{p\_k})$. However the best unconditional result we have so far on prime gaps is $O(p\_k^{0.525})$ by Baker, Harman and Pintz. So these pro... | 14 | https://mathoverflow.net/users/2384 | 74582 | 45,338 |
https://mathoverflow.net/questions/74594 | 6 | Disclaimer: When I say fastest growing set, I mean set with the fastest growing get-the-nth-member function. I don't know the technical term for this property and my math vocabulary is limited.
The Goldbach conjecture states that every even number can be expressed as the sum of two primes. Let's only concern ourselve... | https://mathoverflow.net/users/17604 | Fastest growing set of odd numbers such that any even number can be expressed as the sum of two elements. | Let the set consist of all numbers whose binary expansion has the form $1+\sum\_{i\geq 1} a\_i 2^{2i}$ or $1+\sum\_{i\geq 1}b\_i 2^{2i-1}$. Then $x\_k\approx k^2$, which is the largest exponent possible.
| 19 | https://mathoverflow.net/users/2807 | 74599 | 45,348 |
https://mathoverflow.net/questions/74326 | 2 | Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein constructs a linear operator $E\_K$ continuously mapping the Sobolev space $W\_{p,k}(K)$ into the Sobolev space $W\_{p,k... | https://mathoverflow.net/users/11211 | Stein's extension operator and wave front sets | The answer is NO already in one dimension. In fact, if $K$ is the interval $[-1, 1] \subset \mathbb R$ then the Schwartz kernel $K(x, y)$ of the operator $\tilde E\_K: C^{\infty}(\mathbb R) \to C^{\infty}(\mathbb R)$ must be equal to $\delta(x-y)$ in the square $(-1, 1) \times (-1, 1)$, and its support must be containe... | 7 | https://mathoverflow.net/users/16546 | 74606 | 45,352 |
https://mathoverflow.net/questions/74270 | 2 | What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 of his book, Renormalization and 3-Manifolds which Fiber over the Circle?
He provides this value for the orbifold bu... | https://mathoverflow.net/users/17531 | How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups? | Let me first make a comment on Jorgensen's work. His Annals paper that you refer to was based on computations he first made in the cusped case, but did not appear in [print until recently](http://www.ams.org/mathscinet-getitem?mr=2044551). He then did "orbifold [Dehn filling](http://en.wikipedia.org/wiki/Dehn_filling)"... | 5 | https://mathoverflow.net/users/1345 | 74607 | 45,353 |
https://mathoverflow.net/questions/74588 | 5 | Let $X \to S$ be a morphism of schemes. What properties of morphisms satisfy: if all fibers $X\_s \to k(s)$ over closed points $s \in S$ satisfy the property, then the morphism $X \to S$ satisfies the same property?
For example, is this true (maybe under additional conditions; connected base et cetera) for proper mor... | https://mathoverflow.net/users/1107 | Does property P for fibers over closed points of a morphism of schemes imply property P globally? | It might be helpful to split your question into two parts:
1. When is it true that if the fibers $X\_s \to {\rm Spec} \kappa (s)$ satisfy a certain property for all $s \in S$, $X \to S$ satisfies this property?
2. When is it true that if the fibers $X\_s \to {\rm Spec} \kappa (s)$ satisfy a certain property for all c... | 13 | https://mathoverflow.net/users/13302 | 74609 | 45,355 |
https://mathoverflow.net/questions/74570 | 2 | A morphism of set-valued functors $\eta: F \to G$ on $\mathcal{C}$ is called smooth if for all epimorphisms $B \to A$, the natural morphism $F(B) \to F(A) \times\_{G(A)} G(B)$ is surjective.
Obviusly "smooth => formally" smooth for $\mathcal{C} = \mathrm{Sch}$.
Now my question: Does the converse hold?
My thoughts... | https://mathoverflow.net/users/12832 | formally smooth => smooth | In the case where $F\to G$ is representable by an epimorphism $C\to D$ of rings, then your smoothness condition implies that $C\to D$ has a section. (Take $A\to B$ to be the given map $C\to D$.) But it is not hard to find a formally smooth epimorphism that does not admit a section. Any localization will do. For instanc... | 5 | https://mathoverflow.net/users/1114 | 74623 | 45,364 |
https://mathoverflow.net/questions/74626 | 5 | Let $F\_0 \subset F\_1 \subset F\_2 \subset \cdots$ and $K\_0 \subset K\_1 \subset K\_2 \subset \cdots$ be two towers of fields. Also, let $F = \cup\_{i=0}^\infty F\_i$ and $K = \cup\_{i=0}^\infty K\_i$.
Now suppose for each $i$ we have injective homomorphisms from $F\_i$ to $K\_{\sigma(i)}$ and from $K\_i$ to $F\_{\... | https://mathoverflow.net/users/17263 | Given 2 towers of fields, when are these fields isomorphic? | This is false even in the case you describe as it would imply Cantor-Schroeder-Bernstein for fields. That said, we can take the standard counter example for the claim of CSB for fields as a counter example for your claim. Let $F\_i = \mathbb{C}(X)$ and $K\_i = \mathbb{C}$ for all $i.$ Then $K\_i$ injects into $F\_i$ an... | 13 | https://mathoverflow.net/users/13816 | 74627 | 45,367 |
https://mathoverflow.net/questions/74604 | 3 | Can one give an example of non-compact space $X$ which satisfies the following conditions:
* the countable union of compact subsets is relatively compact,
* for every closed noncompact subset $A$ of $X$ there is a positive lower semicontinuous function on $X$ which is bounded on every compact subset of $X$ but unboun... | https://mathoverflow.net/users/16872 | Topological space with some conditions | I believe such a space cannot exist for the following reason:
Suppose it does. By the second requirement, there should be an unbounded function $f: X \to (0, \infty)$, which is bounded on every compact set. This means we have countably many points $x\_1, x\_2, x\_3, \dots$ such that for each $n$ the inequality $f(x\_... | 4 | https://mathoverflow.net/users/16447 | 74628 | 45,368 |
https://mathoverflow.net/questions/74124 | 8 | By Magnus-Moldavansky theorem (see Wilhelm Magnus, Abraham Karrass, Donald Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations, Reprint of the 1976 second edition, Dover Publications, Inc., Mineola, NY, 2004 and <http://arxiv.org/PS_cache/math/pdf/0608/0608635v3.pdf>), ever... | https://mathoverflow.net/users/nan | Magnus hierarchy of 1-related groups | You can reverse engineer distortion. Consider
the group $\langle a\_0, a\_1, a\_2 |a\_1^{a\_2a\_0}= a\_1^2\rangle$. By the [Freiheitssatz](http://en.wikipedia.org/wiki/Freiheitssatz), the subgroups generated by $\langle a\_0, a\_1\rangle$ and $\langle a\_1, a\_2\rangle$ are free. But both are distorted, since $a\_1$ is... | 6 | https://mathoverflow.net/users/1345 | 74629 | 45,369 |
https://mathoverflow.net/questions/74592 | 7 | Let me denote $X\_n$ the set of transpositions in $n$ elements. Equivalently, $X\_n$ is the set of doubletons in $[1,n]\times[1,n]$. The cardinality of $X\_n$ is $N=\frac{n(n-1)}{2}$.
If $f:{\mathbb Z}/N{\mathbb Z}\rightarrow X\_n$ is a bijection, let us denote
$$r(f):=\min\{|\ell-m|;\ell\ne m\quad\hbox{and}\quad f(... | https://mathoverflow.net/users/8799 | Well-balanced covering of transpositions in $n$ elements | Close to $n/2$ is possible. I'll do odd $n$ and leave even $n$ for someone else's pleasure.
Let $m=(n-1)/2$. For $i=0,\ldots,n-1$ and $j=1,\ldots,m$, let $M(i,j)$ be the pair $\{i-j,i+j\}$ (all values taken mod $n$, of course). The solution is
$$ M(0,1).\ldots,M(0,m),M(1,1),\ldots,M(1,m),\ldots,M(n-1,1),\ldots,M(n-1,... | 11 | https://mathoverflow.net/users/9025 | 74631 | 45,370 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.