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https://mathoverflow.net/questions/164939 | 4 | Put the following topology on $\mathbf{Z}\_{>0} \cup \infty$: the closed sets are the initial invervals $\{1,\dots,n\}$ for all $n$. If I understood Hochster's characterization of the underlying spaces of affine schemes, this space can be realized as Spec(R) for some commutative ring $R$. My question is how to give a c... | https://mathoverflow.net/users/321 | Constructing a ring whose spectrum is given by order ideals of Z with generic point | Consider the abelian group $\Gamma:=\mathbb{Z}^{(\mathbb{Z}\_{>0})}$ of sequences $f:\mathbb{Z}\_{>0}\to\mathbb{Z}$ with finite support. Order $\Gamma$ "inverse-lexicographically" by declaring a sequence $f:\mathbb{Z}\_{>0}\to\mathbb{Z}$ to be $>0$ if its *last* nonzero term is $>0$.
We get a totally ordered abelian ... | 5 | https://mathoverflow.net/users/7666 | 165200 | 86,296 |
https://mathoverflow.net/questions/165203 | 13 | Consider $GL\_2$ as the affine group scheme with coordinate ring
${\mathbb Z}[x\_1,x\_2,x\_3,x\_4,y]/(\det\left(\begin{array}{cc}x\_1& x\_2\\ x\_3& x\_4\end{array}\right)y-1)$.
The group scheme $PGL\_2$ is then given by the subring $S$ of $GL\_1$-invariants, which is the subring generated by all monomials of the form ... | https://mathoverflow.net/users/nan | Is GL2( R ) - > PGL2( R ) surjective? | The answer is no. Here is an explicit example. Let $R=\mathbb{Z}[\sqrt{-5}]$.
Consider the matrix $$\left(\begin{array}{cc}1+\sqrt{-5}& 2\\ 2& 1-\sqrt{-5}\end{array}\right).$$
It represents an element of $PGL\_2(R)$ that is not in the image of $GL\_2(R)$.
The motivation for this example is that the ideal $(2,1+\s... | 15 | https://mathoverflow.net/users/425 | 165208 | 86,297 |
https://mathoverflow.net/questions/165209 | 4 | I am consulting the recent paper ''On the Integrality of Modular Symbols and Kato's Euler system for Elliptic Curves'' by Chris Wuthrich. But I am confused regarding the definition of semistable reduction at a prime $p$ of an elliptic curve due to the following two definitions -
1) On page 196 of ''Arithmetic of Ell... | https://mathoverflow.net/users/44637 | Confusion regarding the definition of semistable reduction of an elliptic curve at a prime $p$ | The criteria given by Silverman and Husemoller are equivalent, and so determining the "correct definition" is a bit tricky. In particular, on page 361 of Arithmetic of Elliptic Curves, Silverman (the first edition) defines the conductor of an elliptic curve, and the exponent on $p$ in that conductor is $1$ if and only ... | 10 | https://mathoverflow.net/users/48142 | 165212 | 86,298 |
https://mathoverflow.net/questions/165185 | 26 | It is well known that size of the set of positive integers up to $n$ that can be written as $a^2+b^2$ is asymptotic to $C \frac{n}{\sqrt{\log n}}$. Here I'm interested mostly in the weaker fact that this set has density $0$.
The last can be proved by noticing that if $m$ is the sum of squares, it has no prime factor ... | https://mathoverflow.net/users/43383 | Distribution of $a^2+\alpha b^2$ | There is a positive proportion of such integers and this follows from work on the [Berry-Tabor conjecture](http://www.maths.bris.ac.uk/~majm/bib/3ecm.pdf) due to [Eskin, Margulis and Mozes](http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p03.pdf) (there is work by many others on this; see these paper... | 27 | https://mathoverflow.net/users/38624 | 165215 | 86,300 |
https://mathoverflow.net/questions/165210 | 16 | It is true that over a contractible manifold all differentiable vector bundles are trivial. However the method of proof does not apply in the holomorphic category.
It is also true that a contractible one dimensional complex manifold has no non-trivial line bundles. (Shown using the exponential sequence and a bit of t... | https://mathoverflow.net/users/45609 | Are all holomorphic vector bundles on a contractible complex manifold trivial? | No, even for line bundles. We have the short exact sequence of sheaves
$$0 \to \underline{\mathbb{Z}} \overset{2 \pi i}{\longrightarrow} \mathcal{O} \overset{\exp}{\longrightarrow} \mathcal{O}^{\ast} \to 0$$
where $\underline{\mathbb{Z}}$ is locally constant $\mathbb{Z}$ valued functions, $\mathcal{O}$ is holomorphic ... | 18 | https://mathoverflow.net/users/297 | 165217 | 86,301 |
https://mathoverflow.net/questions/165205 | 1 | Let K and T be the usual modal logical principles $\Box (\alpha \rightarrow \beta) \rightarrow (\Box \alpha \rightarrow \Box \beta)$ and $\Box \alpha \rightarrow \alpha$. Let U be the modal logical principle $\Box \Diamond \alpha \rightarrow \Box \alpha$. Let propositional modal logic KTU be the extension of classical ... | https://mathoverflow.net/users/37385 | Question on deriving $\alpha \rightarrow \Box \alpha$ in modal logic KTU | $\let\B\Box\let\D\Diamond\let\A\alpha$T gives $\B\D\neg\A\to\D\neg\A$, i.e., $\D\B\A\lor\D\neg\A$. This implies $\D(\A\to\B\A)$ in K, and then we derive $\B\D(\A\to\B\A)$ by necessitation, $\B(\A\to\B\A)$ by U, and $\A\to\B\A$ by T.
| 6 | https://mathoverflow.net/users/12705 | 165220 | 86,302 |
https://mathoverflow.net/questions/165227 | 10 | I have two finished articles (each about 25 pages long) but the second one uses results from the first one, none of which has been published yet. I would like to send them to some standard journal for consideration but I do not know exactly how to do so. Should I put them up on arxiv, refer back to the first article fo... | https://mathoverflow.net/users/50407 | How to publish two interdependent papers | Put both papers in the arxiv. Have the second cite the first, if it depends on the first. You may have the first cite the second as a "preprint" as a motivation, if it's useful. This can be updated by re-posting the first paper once you know the "coordinates" (arxiv number) of the second.
If the papers form a clear s... | 11 | https://mathoverflow.net/users/25355 | 165231 | 86,309 |
https://mathoverflow.net/questions/165241 | 25 | Firstly, the [help page](https://mathoverflow.net/help/on-topic) for Mathoverflow does not forbid asking such a question. Secondly I found a [similar question](https://mathoverflow.net/questions/114344/13-months-and-not-even-one-report-what-would-you-do) on Mathoverflow and thirdly as far as I know, waiting for two yea... | https://mathoverflow.net/users/22721 | Is two years without a referee report normal? | Two years without a report or decision is certainly too long. But this happens sometimes
for various reasons.
The journals do not have to show you the referee's reports. At least some journals
say that they don't have to. And that this is their policy. So a paper can be rejected
(or accepted) without a referee report... | 17 | https://mathoverflow.net/users/25510 | 165243 | 86,314 |
https://mathoverflow.net/questions/164935 | 11 | I'm working through McMullen's paper ["The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology"](http://www.math.harvard.edu/~ctm/papers/home/text/papers/alex/alex.pdf) and have a question concerning the following setup:
Given a link complement $(X, p)$ with $G = \pi\_1(X)$, the Alexander polynom... | https://mathoverflow.net/users/49911 | Relation between the Alexander module of a link and intermediate free abelian covers | Going from multivariable Alexander polynomials to onevariable Alexander polynomials is indeed a little awkward using the definition in terms of orders of modules.
The situation becomes much clearer if one translates the problem into Reidemeister torsions. The multi/onevariable Alexander polynomials are almost identica... | 9 | https://mathoverflow.net/users/2985 | 165256 | 86,317 |
https://mathoverflow.net/questions/165237 | 8 | It is well-known that, if $p$ is a Ramsey (selective) ultrafilter on $\omega$, then after adding a Sacks real $p$ remains an ultrafilter (well, it's really the upwards closure of $p$ the one that's an ultrafilter in the extension, but this is how they usually phrase it). The same result still holds if $p$ is "only" a P... | https://mathoverflow.net/users/13059 | Preservation of ultrafilters by Sacks forcing | Olga Yiparaki's 1994 thesis from the University of Michigan "On some tree partitions" characterizes the ultrafilters that are preserved by Sacks forcing as those that are "Ramsey" with respect to colorings related to the the Halpern-Lauchli Theorem. (She calls them "hlt-ultrafilters".)
This class of ultrafilters is b... | 10 | https://mathoverflow.net/users/18128 | 165260 | 86,319 |
https://mathoverflow.net/questions/160775 | 1 | I am reading the paper "Generating random bits from an arbitrary source: fundamental limits" by Vembu and Verdu. This paper is written in the language of information theory, however, I need to translate one quantity in this paper in terms of computer science.
*inf-entropy rate*, $\underline{H}(X)$, of an arbitrary so... | https://mathoverflow.net/users/41666 | Connection between inf-entropy rate and min-entropy | The proper of definition of inf-entropy is as follows:
$$
\underline{H}(\mathbf{X})=\text{p-}\liminf\_{n\to\infty}\frac{1}{n}\log\frac{1}{P\_{X^n}(X^n)}.\\
$$
where $\mathbf{X}=(X\_1,X\_2,\dots)$ and $\text{p-}\liminf\_{n\to\infty}$ is defined as follows:
$$
\text{p-}\liminf\_{n\to\infty}Z\_n=\sup\left\{\beta\mid \lim\... | 0 | https://mathoverflow.net/users/39212 | 165270 | 86,321 |
https://mathoverflow.net/questions/165266 | 9 | Let $G$ be a finite group acting on a complex vector space $V$ by pseudoreflections (i.e. every element of $G$ is a product of elements which fix hyperplanes in $V$). I would like to understand the abelianization of $G$.
If $r\_1$ and $r\_2 = gr\_1g^{-1}$ are conjugate pseudoreflections, then $r\_1r\_2^{-1} = r\_1gr\... | https://mathoverflow.net/users/1 | Can the difference of non-conjugate pseudoreflections lie in the commutator subgroup? | Sure: a slight enhancement of Geoff Robinson's comment will work. Let $s$ and $t$ be reflections (I'm going to write reflections instead of pseudoreflections) with the same image in the abelianization. Then for every one dimensional character $\chi$, we have $\chi(s)=\chi(t)$. Given a $G$-orbit of reflecting hyperplane... | 10 | https://mathoverflow.net/users/15933 | 165278 | 86,324 |
https://mathoverflow.net/questions/165027 | 6 | 15 hours and four up-votes but not a word from anybody. That's the result of [this question](https://math.stackexchange.com/questions/778010/reference-request-a-differential-equation-arising-in-geometry) to stackexchange.
My question is where the following differential equation arises naturally and where it appears i... | https://mathoverflow.net/users/6316 | Reference request: a differential equation in elementary geometry | Let us consider two light rays of the same frequency in the $x$-$y$ plane forming the angles $\alpha$ and $\alpha+d\alpha$ with the axis $x$. The four-momenta of the corresponding photons are
$$p\_1=\left (\frac{E}{c}, \frac{E}{c}\cos{\alpha}, \frac{E}{c}\sin{\alpha}
\right ),\;\;\;\mathrm{and}\;\;\;
p\_2=\left (\frac... | 5 | https://mathoverflow.net/users/32389 | 165286 | 86,328 |
https://mathoverflow.net/questions/165246 | 9 | Consider the preprojective algebra of type $A\_n$. It is well known that this algebra is of finite representation type when $n<5$, of tame representation type when $n=5$, and of wild representation type when $n>5$. In particular, this means that there exists a family of indecomposable representations of the preprojecti... | https://mathoverflow.net/users/50417 | One-Parameter Families of Indecomposable Representations of the Preprojective Algebra of type A5 | I only know the answer for type A5.
The smallest dimension-vector for a one-parameter family of indecomposable representations is (1, 2, 2, 2, 1). All the representations in the family have socle and head both equal to S\_2 \oplus S\_4. (Here S\_i is the one-dimensional representation attached to vertex i, with the sta... | 11 | https://mathoverflow.net/users/45000 | 165294 | 86,330 |
https://mathoverflow.net/questions/165147 | 3 | Let $C^\infty(X)$ denote the space of infinitely smooth functions on a compact manifold $X$ (at the beginning one may assume that $X$ is a circle, though I need a more general case). Let $\mathcal{D}(X)$ be the space of Schwartz distributions equipped with the $w^\*$-topology, namely the weak topology induced by the na... | https://mathoverflow.net/users/16183 | When sequentially continuous linear functional is continuous? | Thanks for the two previous answers. Both are very interesting. Meantime I have found another general result which seems to be particularly convenient for my purposes.
**Theorem. Let $E$ be a separable complete locally convex space. Then any sequentially continuous (in the weak\* topology) linear functional on $E'$ i... | 1 | https://mathoverflow.net/users/16183 | 165297 | 86,331 |
https://mathoverflow.net/questions/165288 | 3 | Noether's normalisation lemma says that if $R$ is an integral domain, finitely generated over a field $k$, with transcendence degree $n$ over $k$, then there exist elements $x\_{1}, x\_{2}, \ldots x\_{n} \in R$, algebraically independent over $k$, such that $R$ is integally dependent on the subring $k[x\_{1}, \ldots, x... | https://mathoverflow.net/users/15482 | A strengthened version of Noether's normalisation lemma? | Yes [when, as noted by Laurent Moret-Bailly, $\mathrm{Frac}(R)$ is separable over $k$]. See David Eisenbud 'Commutative Algebra With a View Toward Algebraic Geometry' Corollary 16.18 about this.
[Edit:] Here is the theorem: if $R$ is an integral domain, finitely generated over a field $k$, with transcendence degree $... | 7 | https://mathoverflow.net/users/50449 | 165299 | 86,333 |
https://mathoverflow.net/questions/165307 | 6 | Let $X$ be a compact connected manifold (with or without boundary) and let $H\_1(X)$ denote its first Cech integral cohomology group or, equivalently, its first cohomotopy group. Is it true that $H\_1(X)$ is finitely generated even if $X$ is not triangulable? Could you please also provide a reference?
| https://mathoverflow.net/users/50457 | First Cech cohomology of manifolds | Manifolds are ENRs and a compact ENR is a retract of a finite simiplicial complex. This is proved in corollaries A.8-A.9 of Hatcher's algebraic topology text. Retracts of finite simplicial complexes have finitely generated cohomology (of any flavor).
| 11 | https://mathoverflow.net/users/1573 | 165309 | 86,336 |
https://mathoverflow.net/questions/164987 | 17 | A theorem of Grothendieck states that any smooth reductive algebraic group over a field $k$ admits a maximal torus over $k$. My question concerns what happens for schemes.
>
> Let $S$ be a scheme and let $G$ be a smooth reductive group scheme over $S$. Does $G$ admit a maximal torus over $S$?
>
>
>
Given that ... | https://mathoverflow.net/users/5101 | Does every reductive group scheme admit a maximal torus? | Counterexamples exist even over $S = \mathrm{Spec} \mathbb{Z}$. See, for instance, Lemma 1.1 and Example 6.2 of [http://math.stanford.edu/~conrad/papers/redgpZ.pdf](http://math.stanford.edu/%7Econrad/papers/redgpZ.pdf) or other places of that paper. If $\mathscr{G}$ in Example 6.2 there had a maximal torus over $\mathb... | 13 | https://mathoverflow.net/users/5498 | 165315 | 86,340 |
https://mathoverflow.net/questions/165258 | 1 | Let $E$ and $F$ be two finite dimensional vector spaces. For every $k\in \mathbb{N}$, $E^{k}$ has a natural vector space structure and is isomorphic to $E\otimes \mathbb{R}^{k}$, in a natural way.
We denote by $L^{k}(E)$ $\;$($\Lambda ^{k}(E^{\*})$), the space of all $k$-linear maps (anti-symmetric $k$-linear maps) ... | https://mathoverflow.net/users/36688 | A multilinear question and its smooth version | I don't know whether such a classification exists, but the answer to the second multilinear question is "no":
Consider the case $E=F=\mathbb{R}^2$, $k=3$, and the map $T:(a,b,a',b',a'',b'')\mapsto (a,b,a,b,b',b'')$. This map is linear, but not of tensor form. However, it is antisymmetric tensorial, since $\alpha\circ... | 3 | https://mathoverflow.net/users/50465 | 165319 | 86,342 |
https://mathoverflow.net/questions/165303 | 21 | Given an alphabet with $n$ characters, what is the shortest sequence that contains all $n!$ permutations as subsequences?
A subsequence can be obtained from a sequence by deleting any characters, thus it's different from a substring, whose elements have to be contiguous in the original sequence. I say this because th... | https://mathoverflow.net/users/9211 | Shortest supersequence of all permutations of $n$ elements | This seems to be an open problem. It is listed in the OEIS as sequence [A062714](http://oeis.org/A062714), as noted by Ilya. Summarising the most important results:
Let $m$ be the length of such a sequence. Then [Newey](http://infolab.stanford.edu/pub/cstr/reports/cs/tr/73/340/CS-TR-73-340.pdf) (amongst others) descr... | 22 | https://mathoverflow.net/users/15934 | 165321 | 86,343 |
https://mathoverflow.net/questions/165302 | 15 | Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a simplicial version of Verdier's hypercovering theorem:
>
> Let $X$ be a locally fibrant simplicial presheaf on $\mathcal{C}$,... | https://mathoverflow.net/users/11640 | Modern versions of Verdier's hypercovering theorem? | Jardine has recent a paper called "The Verdier hypercovering theorem", based on an earlier paper called "Cocycle categories", which you should look at if you haven't.
He doesn't exactly say it this way, but it looks like the following is true: given presheaf such that $X$ is locally fibrant, then you can define the ... | 14 | https://mathoverflow.net/users/437 | 165324 | 86,344 |
https://mathoverflow.net/questions/165322 | 2 | Let $R = \prod\_{n\in\mathbf{N}}R\_n$ be an *infinite* direct product of discrete valuation rings $R\_n$. Why is $\mathrm{Pic}(R) = 0$?
| https://mathoverflow.net/users/nan | Picard group of infinite direct product of DVRs trivial | We aim to show that every $\mathbb{G}\_m$-torsor over $R$ is trivial. By descent, such a torsor is represented by an affine $R$-scheme $X$. Due to affineness, $X(R) = \prod\_n X(R\_n)$, so it remains to (invoke the axiom of countable choice and) note that each $X(R\_n)$ is nonempty because the pullback of $X$ to $\math... | 8 | https://mathoverflow.net/users/5498 | 165329 | 86,346 |
https://mathoverflow.net/questions/165312 | 10 | In 1995, de Jong proved the existence of regular alterations in arbitrary characteristic. I would like to have a little survey of important applications of this theorem, e.g. things you could do if you had resolution of singularities, but for which alterations are sufficient.
I am aware of the 1996 Bourbaki article b... | https://mathoverflow.net/users/44860 | Applications of alterations | There is a book, *Resolution of Singularities — A research textbook in tribute to Oscar Zariski*, edited by Hauser, Lipman, Oort and Quiros (Progress in Mathematics, Birkhäuser, 2000).
There, you'll get four papers related to alterations : one by Abramovich and Oort that explains De Jong's theorem, one by Geisser that ... | 8 | https://mathoverflow.net/users/10696 | 165330 | 86,347 |
https://mathoverflow.net/questions/24144 | 11 | We call a topological space $X$ a *Toronto space* if for any subspace $Y \subseteq X$ such that $Y$ and $X$ have the same cardinality it follows that $Y$ is homeomorphic to $X$.
Does anybody know what is known about the following question?:
Is there an uncountable, non-discrete, Hausdorff Toronto space?
It is no... | https://mathoverflow.net/users/5450 | Is there an uncountable, non-discrete, Hausdorff Toronto space? | You can read *"The Toronto problem"* by William Rea Brian (February 2014) to learn pretty much everything that is known about this problem. The article includes proofs of the folklore facts mentioned by Apollo and some other very interesting facts (e.g. Kunen´s result: An uncountable Hausdorff Toronto space contains no... | 8 | https://mathoverflow.net/users/17836 | 165331 | 86,348 |
https://mathoverflow.net/questions/165306 | 0 | I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that reason I was hoping to get an answer to the below question here.
Let $g:B(x, 2r) \to \mathbb R$ be real analytic and suppos... | https://mathoverflow.net/users/46298 | Harmonic extension in a ball $B(x, r) \subset \mathbb R^n$ | The answer is No. Consider the function $g(x,y)=\sum\_{k=0}^{\infty}r^{k}2^{-k}cos{n\_k\varphi}$, where $(r, \varphi)$ are the usial polar coordinates in the plane $(x,y)$. Then $g(x,y)$ is real analytic in the disk $B(0, 2)$. The harmonic extension of $g(x,y)$ from the unit circle is given by $u(x,y)=\sum\_{k=0}^{\inf... | 3 | https://mathoverflow.net/users/50471 | 165333 | 86,350 |
https://mathoverflow.net/questions/165072 | 8 | Take an $n \times n$ random matrix whose entries are i.i.d. with uniform distribution in $[0,1]$. Look at the sums of the elements of each row and then permute the rows so that these sums form an increasing sequence. Next, perform the analogous sorting of the columns.
*Obvious remarks:* Almost surely no two rows or c... | https://mathoverflow.net/users/1516 | Distribution of entries of a doubly-sorted random matrix | Let $X$ be the array after your operation. The law of $X\_{i,j}$ for $(i,j)$ chosen in a deterministic way from $[1,..,n]^2$ converges to the uniform law. The tilting due to your operations is very small, asymptotically - since the maximal row sum is of order $\sqrt{n}$ with a logarithmic correction, the distribution y... | 4 | https://mathoverflow.net/users/35520 | 165340 | 86,353 |
https://mathoverflow.net/questions/165341 | 4 | Given a birational proper morphism $f\colon X \rightarrow Y$
( Assume $X$ and $Y$ irreducible )
of complex algebraic varieties.
It is always true that $f^\* \colon \pi^{et}\_1 (X)\rightarrow \pi^{et}\_1(Y)$
is an isomorphism?
I think that this follows from (SGA.1 exp X. Corollary 1.4)
But im not totally sure.
Thanks... | https://mathoverflow.net/users/37338 | Isomorphism étale fundamental group | This is true if $X$ and $Y$ are smooth, see SGA I, exp. X, Cor. 3.4. But certainly not in general: if $Y$ is a plane cubic with one node and $f$ its normalization, $X$ is simply connected while $\pi \_1^{et}(Y)\cong \hat{\mathbb{Z}}$, cf. same SGA, exp. IX, example 5.5.
| 9 | https://mathoverflow.net/users/40297 | 165342 | 86,354 |
https://mathoverflow.net/questions/165304 | 20 | In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian one. The most known example, is Minkowski spacetime, which becomes Euclidean 4-space. Other example are Schwarzschild an... | https://mathoverflow.net/users/29850 | Obtain Lorentzian manifolds from Riemannian ones by Wick rotation | This is a somewhat different take on Igor's answer, and I offer it just in case you are interested.
First, one doesn't need to have any continuous symmetries in order to have this kind of 'Wick rotation' exist. For example, if $(M,g)$ is a real-analytic Riemannian manifold that admits a nontrivial isometric involutio... | 16 | https://mathoverflow.net/users/13972 | 165345 | 86,356 |
https://mathoverflow.net/questions/165338 | 28 | In a [previous MO question](https://mathoverflow.net/questions/164148/is-there-a-computable-ordinal-encoding-the-proof-strength-of-zf-is-it-knowable), I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha\_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ... | https://mathoverflow.net/users/2575 | Why isn't this a computable description of the ordinal of ZF? | **Theorem.** The following are equivalent.
1. The relation on $\mathbb{N}$ computed by your program P is a well-order.
2. ZF is $\Pi^1\_1$-sound.
Proof. You gave the argument for the reverse implication $(2\to 1)$, since you only needed to know that that whenever ZF proves that a certain computable relation is a w... | 22 | https://mathoverflow.net/users/1946 | 165347 | 86,357 |
https://mathoverflow.net/questions/165335 | 6 | Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main cardioid involving exp(2 pi i theta) but I could not find the formula.
I'm particularly interested in finding the exact locat... | https://mathoverflow.net/users/50473 | Precise location of the Mandelbrot Bulb Attachment to the main Cardioid | To elaborate a bit more on my comments:
Given a hyperbolic component $U\subset M$ of the Mandelbrot set -- i.e. a connected component of the interior of $M$ such that for all $c\in U$ the polynomial $p\_c(z)=z^2+c$ has an attracting cycle -- the length of attracting cycle of $p\_c(z)$ is constant for all $c\in U$. No... | 5 | https://mathoverflow.net/users/23444 | 165350 | 86,359 |
https://mathoverflow.net/questions/165352 | 3 | Suppose I have a system:
$$
Ax = b
$$
where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.
I'd like a combination of a minimum norm solution (for elements of $x$ which are not determined and a least squares solution for those that are deter... | https://mathoverflow.net/users/23064 | General method for under and over determined systems? | The [Moore-Penrose pseudoinverse](http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse) is probably what you're looking for. The *pseudoinverse solution* $A^+b$ is the smallest norm $x$ such that $\|Ax-b\|\_2$ is minimized. It can be computed using QR decomposition although you have to use rank-revealing QR... | 4 | https://mathoverflow.net/users/13832 | 165353 | 86,361 |
https://mathoverflow.net/questions/165349 | 5 | The standard definition is that a function $f:\mathbb{R}^n\to \mathbb{R}$ is *differentiable* at a point $x$ if there exists a linear map $\mathrm{d}f\_x: \mathbb{R}^n \to \mathbb{R}$ such that
$$f(x+h) = f(x) + \mathrm{d}f\_x(h) + \epsilon \|h\|$$
where $\epsilon\to 0$ as $h\to 0$. This is stronger than the existe... | https://mathoverflow.net/users/49 | Non-continuous higher differentiability | The funny thing is that I got almost the same question from one of my student last semester. The idea was to neglect both the linear and quadratic parts of the hypothetical expansion (without loss of generality, we may assume they are zero), and focus on $ \epsilon \|h\|^2$. We came to the following (obvious) one-dimen... | 8 | https://mathoverflow.net/users/13480 | 165355 | 86,362 |
https://mathoverflow.net/questions/165320 | 12 | For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with respect to this metric in the set of all measurable functions.
The question is to describe the completion of $C^1[0,... | https://mathoverflow.net/users/4312 | smooth Luzin theorem | This holds precisely if $f$ is approximately differentiable a.e. This condition implies that $f|\_A$ can be extended to a $C^1$ function for suitable sets $A$ of almost full measure by a Theorem from Federer's book, or see [here](http://books.google.com/books?id=EzkG1lf2G8kC&pg=PA53&lpg=PA53&dq=approximately%20differen... | 9 | https://mathoverflow.net/users/48839 | 165358 | 86,364 |
https://mathoverflow.net/questions/165346 | 6 | I need to compute a series of Tristram-Levine signatures for a family of torus knots. I was wondering if this has already been done or whether there is a good way to streamline the computation.
I am aware that Tristram's original paper on the signature invariants provides a formula for the $e^{\frac{2\pi i}{3}}$-sig... | https://mathoverflow.net/users/22431 | Closed formula or program for computing Tristram-Levine signatures of torus knots? | In general, closed formulas for such things tend to be rather slow, computationally. I would suggest that you look at the paper of Litherland: Signatures of iterated torus knots. Topology of low-dimensional manifolds, pp. 71–84, Lecture Notes in Math., 722, Springer, Berlin, 1979. Litherland starts with a formula, due ... | 6 | https://mathoverflow.net/users/3460 | 165365 | 86,367 |
https://mathoverflow.net/questions/165364 | 2 | Recently I read Dale Rolfsen's paper –A surgical view of Alexander’s polynomial. This is a good paper. But there is a lemma which I don’t know how to prove.The lemma is following:
Lemma:In the cover$R^{1}\times R^{2}\to S^{3}$- {trivial knot},let $\alpha,\beta$ be disjoint closed oriented curves downstairs which lift... | https://mathoverflow.net/users/42816 | linking number and covering | The kind of argument you'd see Rolfsen make in his textbook would be to consider how one constructs the abelian cover explicitly via the Seifert surface of the trivial knot. This is a disc. Ensure it intersects your curves transversely. This allows you to write a simplified diagram for your curves, consisting of a tang... | 1 | https://mathoverflow.net/users/1465 | 165369 | 86,369 |
https://mathoverflow.net/questions/164407 | 6 | I am reading papers about yamabe flow. I have a problem about how people derive it as a gradient flow.
Suppose we have $(M,g\_0)$, $g(t)=u^{\frac{4}{n-2}}(t)g\_0$ is another conformal metric. Let $R=R(t)$ be the scalar curvature and $s=\frac{\int\_M Rd\mu}{\int\_M d\mu}$ be the average scalar curvature at time $t$. I... | https://mathoverflow.net/users/22815 | Derivation of yamabe flow | The Yamabe flow is (up to a constant) the gradient flow of the Yamabe functional on the unit volume conformal class, as you expected. The comment by @Mark Peletier hints at your error: you aren't using the correct "inner product."
We briefly discuss the Ebin metric on the space of all metrics $Met$. Recall that $T\_... | 4 | https://mathoverflow.net/users/1540 | 165374 | 86,371 |
https://mathoverflow.net/questions/165202 | 5 | The Theorem 1.5 and 1.6 of
Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.
give a general answer for $H^d(BSO\_n,Z)$ and $H^d(BO\_n,Z)$, which are very complicated. I wonder what are $H^d(BSO\_\infty,Z)$ and $H^d(BO\_\infty,Z)$ for $d... | https://mathoverflow.net/users/17787 | What are the cohomology groups $H^d(BSO_\infty,Z)$ and $H^d(BO_\infty,Z)$? | The Theorem 1.5 and 1.6 you quote give the answer.
More precisely, for $SO$, in the range $d<6$, the only polynomial generators
are $p\_1$ which has degree 4, $\delta(w\_2)$ with degree 3 and $\delta(w\_4)$ with degree 5. The only relations are $2\delta(w\_{2i})=0$, which gives $$H^d(BSO\_{\infty};\mathbb{Z})\cong 0,... | 6 | https://mathoverflow.net/users/43326 | 165388 | 86,378 |
https://mathoverflow.net/questions/165384 | 3 | Suppose we have an algebraic group $G$ defined over a field $k$. Suppose we consider the fraction field of $k[G]$. Is it possible to get a situation where this field is not a separable extension of $k$? An example would be very helpful.
| https://mathoverflow.net/users/15482 | an algebraic group where the function field is not separable over the ground field | The answer depends on what you mean by "algebraic group". If you mean algebraic group in the sense of Borel/Humphreys/Springer, the field of fractions of $k[G]$ is always separably generated, more-or-less by definition. If you mean algebraic group scheme, which must be integral for the question to make sense, then it d... | 3 | https://mathoverflow.net/users/50505 | 165404 | 86,385 |
https://mathoverflow.net/questions/154042 | 3 | When I learned abstract algebra many years ago,I noticed the author deals with commutative ring say,$A$ has the proposition:$A^2=A$(without assuming it has identity).It seems that many proposition of commutative ring and module theory can be carried to the ring satisfying the condition above.Is there a name for this ki... | https://mathoverflow.net/users/41650 | On non-unital ring and algebraic geometry | Hope this helps
<http://webs.um.es/leandro/miwiki/lib/exe/fetch.php?id=curriculum&cache=cache&media=dssrttn.pdf>
If you remove the condition A^2=A things are much more complicated, but there is some literature about that.
Leandro
| 2 | https://mathoverflow.net/users/50507 | 165406 | 86,386 |
https://mathoverflow.net/questions/165403 | 4 | Let $X$ be a compact Riemann surface, Hitchin-Simpson correspondence over $X$ says that irreducible representations of $\pi\_1(X)$ one-to-one correspondend to stable Higgs bundles with vanishing Chern clssess over $X$.
Now suppose we fix a representation $\rho$ of $\pi\_1(X)$, then it corressponds to a stable Higgs b... | https://mathoverflow.net/users/4504 | Deformation of Hitchin-Simpson correspondence | I cannot say anything about the general case, but I think your example was considered already by Hitchin in his Self-duality paper (§11). Let me add some
comments because Hitchin somehow has an opposite point of view than asked in your question:
First note, that the solution of the self-duality eq's corresponding to ... | 4 | https://mathoverflow.net/users/4572 | 165411 | 86,388 |
https://mathoverflow.net/questions/165377 | 4 | To every natural number $n$, we can assign its Church numeral $\underline{n}.$ A formal definition would be:
>
> 1. $\underline{0}(f)=\mathrm{id}\_{\mathrm{dom}(f)}$
> 2. $\underline{n+1}(f) = \underline{n}(f) \circ f$
>
>
>
where each line is to be understood as implicitly universally quantified over every en... | https://mathoverflow.net/users/26080 | Class theory with support for self-application of class functions? | It seems to me that any of the usual class theories, such as GBC
or KM can handle this (and even ZFC since the relevant classes are
definable), and you have the key to the solution already in your remarks at the end of the question. Namely,
the operation of $\underline n$ on a class function $F$ is
determined by the op... | 5 | https://mathoverflow.net/users/1946 | 165412 | 86,389 |
https://mathoverflow.net/questions/165323 | 1 | Let $p$ be a nontrivial idempotent in a JB-algebra $A$ with Pierce decomposition $A = A\_1 \oplus A\_{1/2} \oplus A\_0$. Then the projection onto $A\_1$ (resp. $A\_0$) is given by $U\_p$ (resp. $U\_{p'}$). Let $x \in A\_{1/2}$ be a nonzero element. Then $x^2 \in A\_1 \oplus A\_0$.
The statement I am interested in is ... | https://mathoverflow.net/users/46472 | Square of Pierce 1/2 elements | After some more research I found a much easier proof: it follows immediately from Corollary 2.10 in "A Gelfand-Neumark Theorem for Jordan Algebras" by Alfsen, Schultz and Stormer. This corollary states that for positive $x \in A$, $x = U\_p x$ if and only if $U\_{p'} x = 0$.
| 2 | https://mathoverflow.net/users/46472 | 165416 | 86,392 |
https://mathoverflow.net/questions/165382 | 1 | Let $X$ be a random variable taking values in $\mathbb R^n$ with a probability distribution $\mathbb P$ that has a density $p$.
Consider further a linear mapping $\pi: \mathbb R^n \to \mathbb R^m$, i.e. $\pi$ is an $m \times n$ matrix. We assume $m<n$, i.e. the linear transformation is in general non-invertible!
No... | https://mathoverflow.net/users/nan | Push-forward density as surface integral | First of all, let us assume that $\pi$ is onto to avoid some unnecessary complications. Then we can find linear coordinates $x=(x\_1,\dotsc, x\_n)$ on $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$ and $y=(y\_1,\dotsc, y\_m)$ on $\bR^m$ such that, in these coordinates, $\pi$ is given by
$$ y\_j=x\_j,.\;\;\forall j=1,\dotsc, ... | 1 | https://mathoverflow.net/users/20302 | 165421 | 86,395 |
https://mathoverflow.net/questions/165075 | 1 | Assume we have an arbitrary high order polynomial $$f(L)=1-L\theta\_1-L^2\theta\_2-L^3\theta\_3-...-L^N\theta\_N$$ and we know all roots of this polynomial site outside the unit circle. It is obvious that the latter condition imposes some restrictions on $\theta\_1,\theta\_2,\theta\_3,...$. Then my question is whether ... | https://mathoverflow.net/users/48924 | Polynomial convex coefficients | The answer is that convexity doesn't hold for all $N$.
For a polynomial with degree $N$, you may define a new polynomial
$$ g(L) := L^N f(1/L). $$
We have that $f$ has all of its roots outside the unit circle if and only if $g$ has all of its roots inside the unit circle. This is also known in the literature as $g$... | 4 | https://mathoverflow.net/users/22389 | 165434 | 86,398 |
https://mathoverflow.net/questions/165359 | 13 |
>
> "Adleman refers to integers which factor completely into small primes as “smooth” numbers." (ME Hellman, JM Reyneri. *Advances in Cryptology*, 1983: [citation link](http://scholar.google.com/scholar?q=%22Adleman%20refers%20to%20integers%20which%20factor%20completely%20into%20small%20primes%20as%20smooth%20numbers... | https://mathoverflow.net/users/6094 | Why are smooth numbers called "smooth"? | I asked Ron Rivest, and he replied:
>
> Yes, I coined the term "smooth number" to refer to a number that
> has only small prime factors. I don't recall now much about the
> thinking process, except that smooth is rather the opposite of
> "lumpy"...
>
>
>
| 24 | https://mathoverflow.net/users/3106 | 165447 | 86,407 |
https://mathoverflow.net/questions/165405 | 10 | To put it short: In which active research areas of (pure) mathematics no (or only minimal) knowledge in category theory is required ?
To put it long: I know almost nothing about category theory - but I know that I do not like solving problems and providing arguments by diagram chasing and very "high-level" arguments.... | https://mathoverflow.net/users/43263 | 'Category-theory'-free areas of pure math, 'category-theory'-loaded areas of applied math | As a (slowly) recovering category-phobe, allow me to suggest that you change the way you think of category theory. Specifically, don't think of category theory as a "theory". A theory in mathematics generally consists of three components: a collection of related definitions, a collection of nontrivial theorems about th... | 25 | https://mathoverflow.net/users/4362 | 165457 | 86,410 |
https://mathoverflow.net/questions/165155 | 10 | Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha\_1 \geq \alpha\_2 \geq …\geq \alpha\_l)$, where $\sum\_k \alpha\_k = d$. Define a Laurent polynomial in $Q$ as follows:
$$
P\_\alpha(Q) = \sum\_{k=1}^\infty Q^{-\alpha\_k + k-1}(1-Q) \\
\quad \quad \quad \quad \quad \quad = Q^{l} + \sum\_{k=1}^{l} (Q... | https://mathoverflow.net/users/9617 | Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$ | Closely related infinite series appear in relation to the infinite wedge; this then gives a proof of your second question.
**Warnings**: I skimp out on the details of a slightly laborious Calculus I exercise at the end, but the answer was already getting long enough.
Also, it looks like I probably made a sign error... | 7 | https://mathoverflow.net/users/1102 | 165480 | 86,419 |
https://mathoverflow.net/questions/165470 | 2 | I've recently encountered the definition of a lattice polarized K3 surface. What is the idea behind the definition? Surely, there's something deeper to it than merely being a natural generalization of a polarized K3.
| https://mathoverflow.net/users/34884 | Lattice polarized K3 surfaces | I'm not sure how much detail you're looking for; the point is that you can think of a lattice polarized K3 surface as a K3 surface with several different line bundles. A very general algebraic K3 surface has Picard number 1 - for instance, a very general quartic surface in $\mathbf{P}^3$ has only the divisor class comi... | 4 | https://mathoverflow.net/users/2698 | 165486 | 86,421 |
https://mathoverflow.net/questions/165495 | 1 | A 2-dim complex manifold can be viewed as a 4-dim real manifold.
What is the relation between the Chern characteristic and the Pontryagin characteristic of the tangent bundle? It should be $p\_1=n\_1 c\_1^2 + n\_2 c\_2$. What are the values of $n\_1$ and $n\_2$?
| https://mathoverflow.net/users/17787 | Relation between Chern characteristic and Pontryagin characteristic | $p\_1=c\_1^2-2c\_2$.
This immediately follows from Theorem 4.5.1 in the book "Topological methods in algebraic geometry" by F. Hirzebruch.
This identity holds for any complex vector bundle of any rank over any reasonable space.
| 5 | https://mathoverflow.net/users/16183 | 165498 | 86,426 |
https://mathoverflow.net/questions/165426 | 1 | Let $X$ be a Banach space. We consider the evolution equation:
$$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$
where $A$ is a bounded operator.
I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, then every bounded nontrivial solution $x(t)$ on $\mathbb{R}$ satisfies $$\inf\_{t\in \mathbb{R}}|x(t)|>0.$$
I don't k... | https://mathoverflow.net/users/50511 | A property of one-parameter groups of operators | In the case of a general $C\_0$-group the assertion is false. Consider the weight $w(s):=\tfrac{1}{1+s^2}, s\in\mathbb{R}$ and the Banach space consisting of all functions $x:\mathbb{R}\to\mathbb{R}$ such that $wx$ is uniformly continuous and the norm $\|x\|:=\sup\_{s\in\mathbb{R}} |w(s)x(s)|$ is finite. Let $T(\cdot)$... | 3 | https://mathoverflow.net/users/50551 | 165512 | 86,431 |
https://mathoverflow.net/questions/165448 | 6 | Is there a name for the property of a sheaf $\mathcal F$ such that the restriction maps $\mathcal F(V) \to \mathcal F(U)$ are injective when $V$ is connected and $U$ is nonempty?
In other words, this is a sheaf which satisfies the [identity theorem](http://en.wikipedia.org/wiki/Identity_theorem) of complex analysis (... | https://mathoverflow.net/users/6779 | Is there a name for a "rigid" sheaf? | The problem is that your definition is well behaved only if there is enough open subsets $V$ such that $V$ is connected (if there is no such open subset, then your condition is empty) hence the notion as you defined is well behaved only on a locally connected space.
Once you have fixed this issue (either by slightly... | 10 | https://mathoverflow.net/users/22131 | 165513 | 86,432 |
https://mathoverflow.net/questions/165510 | 1 | Below let's work over coherent sheaves on a smooth projective algebraic curve.
We call a subsheaf $\mathcal{F'}$ of $\mathcal{F}$ saturated if it $\mathcal{F/F'}$ is locally free.
We call a locally free sheaf indecomposible if it cannot be written as a direct sum of two saturated subsheaves.
Suppose $\mathcal{E=... | https://mathoverflow.net/users/nan | Unique decomposition of locally free sheaf | This is proven by Atiyah (Theorem 3) in [On the Krull-Schmidt theorem with application to sheaves](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1956__84_/BSMF_1956__84__307_0/BSMF_1956__84__307_0.pdf). I think it's worth noting that techniques used are very broadly applicable; he really just uses that the algebra $A=\ma... | 4 | https://mathoverflow.net/users/66 | 165514 | 86,433 |
https://mathoverflow.net/questions/165490 | 1 | Which book about differential geometry will have these formula about torsion tensor?
$$\nabla\_{j}T^{i}\_{kl}+\nabla\_{k}T^{i}\_{lj}+\nabla\_{l}T^{i}\_{jk}=R^{i}\_{jkl}+R^{i}\_{klj}+R^{i}\_{ljk}$$
$$\left[\nabla\_i, \nabla\_j \right]X^k = R^k\_{lij}X^l+T^l\_{ij}\nabla\_l X^k $$
I read many books about differentia... | https://mathoverflow.net/users/43941 | Which book will discuss torsion tensor and affine connection in detail? | The second formula, but with the extra minus sign:
$$\left[\nabla\_i, \nabla\_j \right]X^k = -R^k\_{lij}X^l+T^l\_{ij}\nabla\_l X^k $$
can be found in <http://www.worldscientific.com/worldscibooks/10.1142/3812> (S.S. Chern, W.H. Chen and K.S. Lam, Lectures on Differential Geometry) formula 2.46 on page 121. The first id... | 2 | https://mathoverflow.net/users/32389 | 165516 | 86,434 |
https://mathoverflow.net/questions/165508 | 2 | I've recently started to learn about Chern and Segre classes, and it seems to me that they are very similar, sharing the same important properties and having closely related definitions.
Fulton's book starts by defining Segre classes, and then defines Chern classes.
My question is: what are the advantages of Chern c... | https://mathoverflow.net/users/50548 | Chern and Segre classes | Chern classes of a rank $r$ vector bundle vanish in degree greater that $r$ while Segre classes do not. On the other hand the definition of Segre classes easily generalizes to singular vector bundles such as cones (<http://math.stanford.edu/~vakil/245/245class14.pdf>). Normal cones are cental objects in deformation the... | 5 | https://mathoverflow.net/users/14514 | 165517 | 86,435 |
https://mathoverflow.net/questions/165526 | 4 | A complex vector bundle of rank $n$ can be viewed as a real vector bundle of rank $2n$. From nLab, we have that the second Stiefel-Whitney class of the real vector bundle is given by the first Chern class of the complex vector bundle mod 2: $w\_2=c\_1$ mod $2$. Do we have similar relations for other Stiefel-Whitney cla... | https://mathoverflow.net/users/17787 | Relation between Stiefel-Whitney class and Chern class | Yes; see problems 14B and 14E in Milnor-Stasheff, Characteristic classes. The main point is to verify this for the top Chern class/SW class by identifying both with the Euler class (mod 2). This in turn follows by comparing the integral and mod 2 Thom classes.
| 7 | https://mathoverflow.net/users/3460 | 165528 | 86,440 |
https://mathoverflow.net/questions/165535 | 4 | Let $G$ be a group of order $p(p^2+1)$, where $p$ is an odd prime number and $p>3$. Easily we can see that $G$ is solvable and so $G$ has a Hall subgroup $L$ of order $p^2+1$. Also we know that $P$, the Sylow subgroup of $G$ is a normal subgroup of $G$. Is it true in general that $L$ is a normal subgroup of $G$?
| https://mathoverflow.net/users/31045 | On the groups of order $p(p^2+1)$ | As Jeremy Rickard points out, the answer is no in general. However, $G$ has order divisible by $2,$ but not by $4.$ It follows that $G$ has a normal $2$-complement $K$ of order $p \left( \frac{p^{2}+1}{2} \right).$ Now $K$ has a normal $p$-complement by Burnside's transfer theorem, since ${\rm gcd}(p-1,\frac{p^{2}+1}{2... | 8 | https://mathoverflow.net/users/14450 | 165540 | 86,445 |
https://mathoverflow.net/questions/165499 | 5 | We say that a space $X$ *has a homology $p$-exponent* if some power of $p$ annihilates the $p$-torsion in $H\_\ast(X;\mathbb{Z})$.
I am interested in the homology exponents of the free infinite loop space $QX=\Omega^\infty\Sigma^\infty X$. Apparently, by examining the Bockstein spectral sequence, it should be possibl... | https://mathoverflow.net/users/8103 | Homology exponents for $QX$ | By May's Remark 2.6, the answer should be no. If $X$ is a Moore space with
$\tilde H\_\*(X; Z) = Z/p$ concentrated in an odd degree $2q-1$, and
$\tilde H\_\*(X; Z/p) = Z/p\{x, y\}$ with $\beta\_1(y) = x$, then in
$H\_\*(Q(X); Z/p)$ the $r$-th Bockstein $\beta\_r$ is
defined and nonzero on $y^{p^{r-1}}$ for each $r\ge1$... | 9 | https://mathoverflow.net/users/9684 | 165547 | 86,447 |
https://mathoverflow.net/questions/165546 | 2 | Suppose $C$ is a complete algebraic curve.
Define a coherent locally free sheaf $\mathcal{F}$ over $C$ to be stable if $\mu(\mathcal{E})<\mu(\mathcal{F})$ for any subsheaf $\mathcal{E}$, where $\mu(\mathcal{E})=\text{deg}(\mathcal{E})/\text{rank}(\mathcal{E})$.
Suppose $\tilde{C}\to C$is a degree $d$ covering, I ... | https://mathoverflow.net/users/nan | Example for pullback of stable sheaf not stable | Let $f \colon \tilde{E} \to E$ be an isogeny of degree $r$ between two elliptic curves, and let $\mathscr{L}$ be a degree $d$ line bundle on $\tilde{E}$, where $(r, \, d)=1$. Then $\mathscr{E} = f\_\* \mathscr L$ is stable vector bundle on $E$ of rank $r$ and degree $d$. Moreover
$$f^\* \mathscr{E} = \bigoplus\_{\sigma... | 1 | https://mathoverflow.net/users/7460 | 165554 | 86,449 |
https://mathoverflow.net/questions/165549 | 2 | I asked [here](https://mathoverflow.net/questions/164673/why-isnt-there-more-interest-in-large-powerset-axioms) about "large powerset axioms" and to my delight, learned that such axioms *are* being taken seriously. I've been toying with them ever since. My favourite is: "The continuum function is injective, and for all... | https://mathoverflow.net/users/26080 | Consistency of: "The continuum function is injective, and for all infinite cardinals $\kappa$ we have that $2^\kappa$ is weakly inaccessible." | If you make your requirement only for regular cardinals $\kappa$, then we can easily get an equiconsistency.
**Theorem.** The following theories are equiconsistent over ZFC:
1. There are unboundedly many inaccessible cardinals.
2. The continuum function $\kappa\mapsto 2^\kappa$ is injective and $2^\kappa$ is weakl... | 9 | https://mathoverflow.net/users/1946 | 165556 | 86,450 |
https://mathoverflow.net/questions/165555 | 2 | Assume that we know the mixed Hodge structure of a complex manifold $X$. Is it possible to compute the mixed Hodge structure of unramified covering $Y$ of $X$ if we know the deck transformation of the covering?
This is possible if we pass it to the Euler characteristics. If this is not true in general, what informat... | https://mathoverflow.net/users/50568 | Is it possible to compute the mixed Hodge structure of a unramified covering space? | If $p\colon Y\to X$ is the projection, the image $p\_\*\mathcal{O}\_Y$ splits into $\deg p$ line bundles $\mathcal{L}\_i$, roughly corresponding to the eigenvalues of $p$. Then the Hodge structure upstairs (split by the eigenvalues) is formed by $H^p(X;\Omega^q(X)\otimes\mathcal{L}\_i)$; you need these data.
If you w... | 2 | https://mathoverflow.net/users/44953 | 165557 | 86,451 |
https://mathoverflow.net/questions/165542 | 5 | Let $\mathfrak{M}$ be a countable transitive model of set theory.
Let $L$ be some countable language and $A$ be a countable (in $\mathfrak{M}$) $L$-structure.
My question is:
1. In $\mathfrak{M}$ can we carry the construction of Scott sentence of $A$ $\sigma(A)^\mathfrak{M}$?
2. Is $\sigma(A)^\mathfrak{M}$ identica... | https://mathoverflow.net/users/38200 | Scott sentence in models of set theory | 1. Of course.
2. The Scott sentence is not syntactically unique, it is only defined up to equivalence. Its defining property ($A\models\sigma$, and every countable model of $\sigma$ is isomorphic to $A$) is $\Pi^1\_2$, hence any sentence satisfying it in $\mathfrak M$ will also satisfy it in the real world by Shoenfiel... | 9 | https://mathoverflow.net/users/12705 | 165560 | 86,452 |
https://mathoverflow.net/questions/165562 | 7 | It is a (folklore?) fact that if $\kappa$ is a regular cardinal, and $\mathbb{P}$ is a $\kappa$-closed poset such that $\Vdash\_\mathbb{P} |\mathbb{P}| = \kappa$, then $\mathbb{P}$ is equivalent to $Col(\kappa,\mathbb{P})$, the collection of partial functions from $\kappa$ to $\mathbb{P}$ of size $<\kappa$, ordered by ... | https://mathoverflow.net/users/11145 | absorption of strategically closed posets | If $\mathbb{P}$ is $\kappa$-strategically closed then there is a projection from $Col(\kappa,\mathbb{P})$ onto $\mathbb{P}$.
The proof is very similar to the $\kappa$-closed case:
Let $\sigma$ be a winning strategy for the game of length $\kappa$.
For every condition in $Col(\kappa,\mathbb{P})$, $f:\eta \rightar... | 10 | https://mathoverflow.net/users/41953 | 165572 | 86,455 |
https://mathoverflow.net/questions/165573 | 9 | Let $X$ be a variety of general type.
Assume that $\dim X = 3$. In <https://eudml.org/doc/164223> it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal singularities and nef canonical bundle) is finite.
Is this statement now also known when $\dim X > 3$?
More preci... | https://mathoverflow.net/users/50572 | Is the number of minimal models finite | By
C. Hacon, J. McKernan, *"On the existence of flips"*, math.AG/0507597.
finiteness of minimal models in dimension $n-1$ implies the existence of flips in dimension $n$.
I guess the result you are looking for is Theorem B (pag 11) of BCHM (<http://www2.imperial.ac.uk/~pcascini/Papers/0610203.pdf>).
Perhaps Sec... | 4 | https://mathoverflow.net/users/14514 | 165577 | 86,457 |
https://mathoverflow.net/questions/165578 | 2 | The Birkhoff Ergodic Theorem states:
Let $(X,\mathcal{B},m)$ be a finite or sigma finite measure space. Suppose $T:(X,\mathcal{B},m)\to (X,\mathcal{B},m)$ is measure-preserving and $f\in L^1(m)$. Then
$$\lim\_{n\to \infty} \frac{1}{n} \sum\_{i=0}^{n-1} f(T^i(x))$$
converges a.e. to a function $g\in L^1(m)$.
C... | https://mathoverflow.net/users/30266 | Birkhoff Ergodic Theorem or Counterexample | The support $S:=\{f\neq0\}$ of an integrable function $f$ is in any case $\sigma$-finite, and so is the $T$-invariant set $$X':=\limsup\_{j\to\infty} T^{-j}(S):=\bigcap\_{k\ge0}\bigcup\_{j\ge k}T^{-j}(S). $$ Clearly the time average is zero in the complement of $X'$ (indeed, for these the points $f(T^j(x))$ eventually ... | 3 | https://mathoverflow.net/users/6101 | 165584 | 86,460 |
https://mathoverflow.net/questions/165588 | 5 | Assume that $M$ is a non-standard model of complete arithmetic, i.e. of the theory $Th(\mathbb{N})$. Suppose that $R$ and $S$ are proper cuts of $M$. (With a cut, I mean a subset of the universe of $M$ which is closed under successor and which is downward closed.) Let $\varphi(x,y)$ be a formula in the language of arit... | https://mathoverflow.net/users/nan | Overspill in models of arithmetic | No, this "double overspill" can fail; here's an example. Fix a non-standard element $q\in M$. Let $S$ be the cut consisting of just the standard numbers, and let $R$ consist of those elements $r\in M$ that are infinitely far below $q$ (i.e., $r+n<q$ for all standard $n$). Then $R$ is also a cut, and all elements $s\in ... | 7 | https://mathoverflow.net/users/6794 | 165589 | 86,463 |
https://mathoverflow.net/questions/165574 | 3 | For simplicity, call the Weil group of a local nonarchimedean field $F\_v$ to be the following extension:
$$1\longrightarrow F^\times\_v\longrightarrow W\_{F\_v}\longrightarrow\text{Gal}(F\_v/\mathbb Q\_v)\longrightarrow 1.$$
1. Finding this definition to be lacking in terms of describing $\ell$-adic representations,... | https://mathoverflow.net/users/48554 | Which Weil group over a $p$-adic field? | First, (2) and (3) are functionally equivalent. What matters are not the group themselves, but their categories of finite-dimensional complex representations (continuous in case (3), algebraic in the case (2)). But the two are the same: that's the simplest case of the famous Weyl's unitary trick. So working with (2) or... | 3 | https://mathoverflow.net/users/9317 | 165591 | 86,465 |
https://mathoverflow.net/questions/165595 | 0 | Is it possible to find $f,g\in L^2[(0,1)\times(0,1)]$ such that $$\log|x-y|=\int\_0^1f(x,t)g(t,y)dt\:\:\:\forall x,y\in(0,1).$$
| https://mathoverflow.net/users/48438 | Solvability of an integral equation | No. I'm elaborating on my comment: By CS, it would follow that $\left|\ln |x-y|\right| \le A(x)B(y)$, with $A,B\in L^2(0,1)$. So $\left| \ln |x-y|\right| \le C A(x)$ for a.e. $x$ and all $y$ from a set of almost full measure.
| 2 | https://mathoverflow.net/users/48839 | 165601 | 86,472 |
https://mathoverflow.net/questions/165602 | 3 | Page 17 of the following survey:
<http://arxiv.org/abs/1103.5380>
makes the claim that small resolutions, meaning resolutions such that the exceptional set is in codimension at least two, are automatically crepant.
Why is this true, or can someone point me to a reference?
Thanks.
| https://mathoverflow.net/users/6059 | Small resolutions are automatically crepant? | Let $X$ be a normal $\mathbb{Q}$-factorial variety, and let $f:Y\rightarrow X$ be a resolution. Then we can write
$$K\_Y = f^{\*}K\_X+\sum a(E\_i,X)E\_i,$$
where the $E\_i$'s are $f$-exceptional divisor. If $f$ is small then the contracted locus is in codimension at least two. That is there is no $E\_i$, and $K\_Y = f... | 1 | https://mathoverflow.net/users/14514 | 165626 | 86,479 |
https://mathoverflow.net/questions/165622 | 8 | Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda\_{\textrm{PF}}$.
>
> Does an upper bound on $\lambda\_{\textrm{PF}}$ give an upper bound on each of the entries of $M$? That is, does $\lambda\_{\textrm{... | https://mathoverflow.net/users/3121 | Does small Perron-Frobenius eigenvalue imply small entries for integral matrices? | This is true. Indeed, you can estimate the sum of *all $n^2$* elements of $A$ rather than individual elements. (Thanks to [thomashennecke](https://mathoverflow.net/users/37855/thomashennecke) for observing this, my original answer dealt with the row sums of $A$ instead of the sum of all $n^2$ elements of $A$.) Moreover... | 19 | https://mathoverflow.net/users/9924 | 165628 | 86,480 |
https://mathoverflow.net/questions/165630 | 5 | The *Seifert-van Kampen theorem* is the classical theorem of algebraic topology that the fundamental group functor $\pi\_1$ preserves pushouts; more often than not this is referred to simply as the *van Kampen theorem*, with no *Seifert* attached. Curious as to why, I tried looking up the history of the theorem, and (i... | https://mathoverflow.net/users/1481 | What was Seifert's contribution to the Seifert-van Kampen theorem? | According to Jahrbuch der Mathematik this theorem is in:
* Seifert, H.
Konstruktion dreidimensionaler geschlossener Räume. (German) JFM 57.0723.01
Berichte Leipzig 83, 26-66. Technische Hochschule Dresden, Diss (1931).
See the (long)
[review](http://zbmath.org/?q=an:57.0723.01) in zbMATH by Erika Pannwitz. The rel... | 6 | https://mathoverflow.net/users/26935 | 165632 | 86,481 |
https://mathoverflow.net/questions/156696 | 3 | Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In particular I would like two know if there is an isomorphism
$$\pi\_{\*}\mathcal{E}xt^{1}\_{\mathcal{X}}(\mathcal{F},... | https://mathoverflow.net/users/14514 | Grothendieck duality for stacks | By Corollary $2.10$ in <http://arxiv.org/pdf/0811.1955.pdf>
Let $f:\mathcal{X}\rightarrow \mathcal{Y}$ be a proper morphism of Deligne-Mumford stacks and $\mathcal{F}\in D^+\_c(\mathcal{X})$, $\mathcal{G}\in D^+(\mathcal{Y})$. The the morphsim
$$Rf\_{\*}R\mathcal{H}om\_{\mathcal{X}}(\mathcal{F},f^{!}\mathcal{G})\ri... | 1 | https://mathoverflow.net/users/14514 | 165639 | 86,485 |
https://mathoverflow.net/questions/165641 | 15 | This question was asked earlier on math.stackexchange: [click here](https://math.stackexchange.com/questions/737003/if-the-tensor-power-m-otimes-n-0-is-it-possible-that-m-otimes-n-1-i). See the comments and the answer by Jack Schmidt there.
Let $M$ be a module over a commutative ring $R$.
It is possible that $M \o... | https://mathoverflow.net/users/38068 | If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero? | My previous attempt was completely wrong, as Jason Starr politely pointed out.
But I think the idea I was grasping for does work, in this example:
Let $R=k[x,y]$ for a field $k$, and let
$$M=\frac{k[x,y,y^{-1}]}{k[x,y]}\oplus\frac{k[x,x^{-1},y]}{k[x,y]}.$$
[So that Jason's comment still makes sense, I'll leave m... | 10 | https://mathoverflow.net/users/22989 | 165643 | 86,487 |
https://mathoverflow.net/questions/165642 | 2 | I am working on a familly of toric varieties which seem to have the following property:
* the closure of the Kähler cone is a **simplicial** cone (and even a smooth cone with respect to the natural lattice).
This has the interesting aspect that it provides a natural basis of the $H^2$ cohomology group.
I was wond... | https://mathoverflow.net/users/10881 | Is the Kähler cone of a toric variety always simplicial? | The Kähler cone of a del Pezzo surface of degree 6 is not simplicial: see section 6 of [these notes](http://people.maths.ox.ac.uk/daviesr/resources/notes/dP6.pdf).
| 9 | https://mathoverflow.net/users/1797 | 165647 | 86,489 |
https://mathoverflow.net/questions/165625 | 4 | Assume $G$ be the fundamental group of a closed orientable hyperbolic 3-manifold. Let
$G\_{1} = \langle a\_{1},...,a\_{k} \rangle$ be a free subgroup of $G$, and let $G\_{2}=\langle a\_{k+1} \rangle$ be a free cyclic subgroup of $G$.
My question is:
If the set $\{a\_{1},...,a\_{k}, a\_{k+1}\}$ generates $G$,
does ... | https://mathoverflow.net/users/18496 | a question on rank of fundamental group | Suppose that $M$ fibers over circle; $G=\pi\_1(M)\cong P\_g \rtimes <a>$, where $P\_g$ is the geneus $g$ surface group. Then for each sufficiently large $k$, you can find a free rank $k$ subgroup $F\_k<P\_g<G$ such that $F\_k$, and $a$ generate $G$.
Edit: Let $D$ be a generating set of $P\_g$. Partition $D$ in two d... | 8 | https://mathoverflow.net/users/21684 | 165657 | 86,493 |
https://mathoverflow.net/questions/165603 | 10 | Consider the following integral:
$$
I\_k(\alpha)=\int\_{[0,1]^k}|x\_1-x\_2|^{\alpha}|x\_2-x\_3|^{\alpha}\ldots|x\_{k-1}-x\_k|^{\alpha}|x\_k-x\_1|^{\alpha}d\mathbf{x}.
$$
where $k=2,3,4,\ldots$
**The question is to find $\beta\_k=\inf\{\alpha\mid I\_k(\alpha)<\infty\}$.**
----------------------------------------------... | https://mathoverflow.net/users/37987 | When is this multiple integral finite? | The integral $I\_4(-1/2)$ is finite.
Write the integral as
$$I\_4(-1/2)=\int\_{[0,1]^4}\frac{dp\ dq\ dr\ ds}{\sqrt{\big|(p-q)(q-r)(r-s)(s-p)\big|}}$$
Assume wlog that $p$ is the largest, so
$$\frac{I\_4(-1/2)}{4} = \int\_{s<r<q<p} + \int\_{r<s<q<p} + \int\_{s<q<r<p} + \int\_{q<s<r<p} + \int\_{r<q<s<p} + \int\_{q<r<... | 3 | https://mathoverflow.net/users/nan | 165664 | 86,497 |
https://mathoverflow.net/questions/165672 | 3 | Let $k$ be an algebraically closed field. Let $X/k$ be a smooth projective variety. For a suitable embedding in $\mathbb{P}^{n}$ we can form a Lefschetz pencil $\widetilde{X} \to D = \mathbb{P}^{1}$.
**[Edit]:** In response to Jason Starr's comment: I assume that every singular fibre of a *Lefschetz pencil* has a sin... | https://mathoverflow.net/users/21815 | Bounding the number of critical points in a Lefschetz pencil | Using the Thom-Porteous formula, and assuming the standard definition of "Lefschetz pencil", the number of singular fibers is precisely $$c\_{n+1}(\Omega\_{X/k}\otimes\_{\mathcal{O}\_X}\mathcal{O}\_{\mathbb{P}^N}(1)|\_X) + c\_1(\mathcal{O}\_{\mathbb{P}^N}(1)|\_X)\cdot c\_n(\Omega\_{X/k}\otimes\_{\mathcal{O}\_X}\mathcal... | 7 | https://mathoverflow.net/users/13265 | 165674 | 86,501 |
https://mathoverflow.net/questions/165673 | 1 | I am writing code for solving linear equations of the form
$$A\_{n\times n}\cdot x=1\_n$$
where $n$ is on the order of $10^6$ and $A$ is a symmetric matrix with approx $10^3$ nonzero entries in each row. This makes its size barely manageable, but inverting it is infeasible, and I'm not sure which decomposition sui... | https://mathoverflow.net/users/38448 | Decompositions of sparse symmetric matrices and methods for solving large linear equations | I agree that this is a better question for scicomp.stackexchange.com.
1. Maybe. It depends on the sparsity structure of your particular matrix and the actual numerical values of the nonzero elements. You won't really know until you try.
2. Not in any way that I'm aware of.
3. Direct factorization (using a sparse LU ... | 3 | https://mathoverflow.net/users/9022 | 165687 | 86,505 |
https://mathoverflow.net/questions/165686 | 8 | To better understand what I'm asking about, let's immediately define some examples. Imagine that you are writing some paper which involves a lot of math narrative. And you have a term, say, *computing unit* (processor). Then at some point you introduce a mathematical variable, which you would use in formulas and text, ... | https://mathoverflow.net/users/50615 | Punctuation and Other Rules for Variables and Their Verbal Definitions in Math Narrative | I hope that this question stays open because it (question 1 in particular) deals with something that is a big problem in many math papers. Whether the problem is specific to mathematics, I can't say.
The answer to the first question is that alternative #1 is *almost always* preferable. The author should help the read... | 9 | https://mathoverflow.net/users/1682 | 165694 | 86,508 |
https://mathoverflow.net/questions/4965 | 11 | Are the derived categories of modular representations of algebraic groups compactly generated? (e.g. consider SL\_2 in characteristic 2). Note modular reps of finite groups are compactly generated (by the regular representation) - that's an example of compact generation of modules for an algebra. But here we're asking ... | https://mathoverflow.net/users/582 | Compact generation for modular representations | It appears this question is resolved in a definitive fashion in today's preprint
[Algebraic Groups and compact generation of
their derived categories of representations](http://arxiv.org/pdf/1405.1890v1.pdf) by Hall and Rydh. Their first theorem asserts that the quasicoherent derived category of the stack $BG$ for $G$ ... | 8 | https://mathoverflow.net/users/582 | 165701 | 86,512 |
https://mathoverflow.net/questions/165653 | 3 | Let $BG$ denote the classifying space of a (discrete) group and $BG\_+$ its disjoint union with a point.
**Question:** What is known about the stable homotopy groups $\pi^S\_\*(BG\_+)$ ?
If $G$ is finite (i.e., compact) there are results in terms of the Burnside ring and completion at primes (at least for $\pi\_0^S... | https://mathoverflow.net/users/14120 | Stable homotopy of classifying space for nilpotent groups | There is an Atiyah-Hirzebruch-Serre spectral sequence (or perhaps Atiyah-Hirzebruch-Lyndon-Hochschild-Serre SS in this context) of the form
$$E^2\_{pq} = H\_p(G/N, \pi\_q^S(BN\_+)) \implies \pi\_{p+q}^S(BG\_+)$$
for any normal subgroup $N \leq G$.
Take $G$ to be the Heisenberg group that you mention and $N = \ma... | 7 | https://mathoverflow.net/users/4649 | 165710 | 86,516 |
https://mathoverflow.net/questions/165713 | 5 | $\DeclareMathOperator\LG{LG}$In the paper [The - Conformal geometry of surfaces in the Lagrangian—Grassmannian and second order PDE](http://arxiv.org/abs/1009.1364) (published on [Proc. London Math. Soc.](https://doi.org/10.1112/plms/pdr023)), I've found an interesting statement:
>
> The Lie quadric $Q^3$, i.e., th... | https://mathoverflow.net/users/22606 | Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $\operatorname{LG}(2,4)$? | This, of course, is "classical". Here is a simple way to see the identification between the space of all circles (including point circles) on the two-sphere and the Lagrangian Grassmannian in $\mathbb{R}^4$ taken from Section 5 of the paper [Finsler surfaces with prescribed geodesics](https://arxiv.org/abs/1002.0243) b... | 9 | https://mathoverflow.net/users/21123 | 165717 | 86,517 |
https://mathoverflow.net/questions/165718 | 9 | Does it exist a computer program which calculates the cohomology of projective algebraic varieties ? For example, smooth surface in $\mathbb{P}^3$? like $\sum\_0^3 X\_i^3=0$
| https://mathoverflow.net/users/nan | Program for calculating cohomology | I will work out a couple of examples the Fermat cubic surface using MacAulay2. For instance we may compute the cohomology of the cotangent sheaf and of the tangent sheaf of the cubic surface $Z(x^3+y^3+z^3+w^3)\subset\mathbb{P}^3$ using MacAulay2.
Form this you can figure out how to compute sheaf cohomology for other... | 11 | https://mathoverflow.net/users/14514 | 165729 | 86,521 |
https://mathoverflow.net/questions/165725 | 1 | Let $M$ be an $n$-dimensional closed manifold. Choose $x \in M$. Using long exact sequence of pairs $(M,M - x)$, we have
$$H\_k(M - x, \mathbb{Z}) \cong H\_k(M, \mathbb{Z})$$
for $k<n-1$. For $k=n-1$, I see this is an isomorphism with $\mathbb{Q}$-coefficient. I am curious if there is an example that they are not isomo... | https://mathoverflow.net/users/11846 | homology of punctured manifolds | This is also an isomorphism with $\mathbb{Z}$-coefficients. One way to see it is to use the "open-closed" exact sequence
$$H^0(M,\mathbb{Z})\rightarrow H^0(\{x\} ,\mathbb{Z})\rightarrow H^1\_c(M-x,\mathbb{Z})\rightarrow H^1(M,\mathbb{Z})\rightarrow 0$$
and Poincaré duality which identifies the last (nontrivial) arrow w... | 1 | https://mathoverflow.net/users/40297 | 165731 | 86,522 |
https://mathoverflow.net/questions/165704 | 1 | In a comment on [this question](https://mathoverflow.net/questions/165349/non-continuous-higher-differentiability), Tom Goodwillie proposed a notion of higher differentiability that I elaborate to something like the following:
Let $f:\mathbb{R}^n \to \mathbb{R}$. Let's say that $f$ is *strongly twice differentiable* ... | https://mathoverflow.net/users/49 | Non-continuous higher differentiability, II | 1. No. Think of a $\mathbb Q$-linear map $f:\mathbb R\to \mathbb Q$.
2. Yes. For small nonzero $v$ and $w$ write
$$
E(v,w)=\frac{f(a+v+w)-f(a+v)-f(a+w)+f(a)-b(v,w)}{|v||w|}.
$$
By assumption you have bilinear $b$ such that the limit of $E(v,w)$ as $(v,w)\to (0,0)$ is zero. If you also assume that $f$ is differentiabl... | 2 | https://mathoverflow.net/users/6666 | 165733 | 86,523 |
https://mathoverflow.net/questions/165652 | 2 | Suppose we are given a category enriched over [**semi-simplicial**](http://ncatlab.org/nlab/show/semi-simplicial+set) sets, i.e. for the simplices in this category we have well-defined boundary maps, but no degeneracy maps. Suppose also that in this category all **inner** Kan conditions are satisfied, i.e. we have $\ma... | https://mathoverflow.net/users/4807 | Defining degeneracies for semi-simplicial sets with inner Kan conditions | The short answer is no. If you take the semi-simplicial set $X$ with one $0$-simplex and no higher simplices, then it has fillers for all inner horns (because it has no inner horns) but clearly does not admit degeneracies.
The question then becomes: what in addition to inner horn fillers could we assume to make this ... | 2 | https://mathoverflow.net/users/12547 | 165735 | 86,524 |
https://mathoverflow.net/questions/154274 | 4 | Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X})$. This sheaf is supported on $Y$.
I would like to relate $\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X})$ and $\mathcal{... | https://mathoverflow.net/users/14514 | Relating deformations of a scheme to deformations of its singular locus | Here is a partial answer.
Let $Y$ be a smooth variety over a field $k$ of characteristic zero. Let $G$ be a finite group acting on $Y$, and let $X = Y/G$ be the quotient. Assume that the set of points where the isotropy is not trivial is in codimension greater or equal than three, that is the singular locus of $X$ i... | 2 | https://mathoverflow.net/users/14514 | 165759 | 86,536 |
https://mathoverflow.net/questions/165758 | 6 | There are many results in number theory, where the existence
of some $B \subseteq \mathbb{N}$ with certain properties is proved by
a [probabilistic argument](https://en.wikipedia.org/wiki/Probabilistic_method) employing "random sets". One such example
would be the result of Erdős and Rényi, where they proved the existe... | https://mathoverflow.net/users/48408 | What makes a set random? | One set of answers to these questions is given by the theory of [algorithmic randomness](http://www.scholarpedia.org/article/Algorithmic_randomness).
| 8 | https://mathoverflow.net/users/47312 | 165761 | 86,537 |
https://mathoverflow.net/questions/165762 | 2 | Shoenfield absoluteness is well known for $\Pi\_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe.
But it is also known that $\Pi\_1^1$ statements are absolute between a countable transitive model of set theory and the universe.
My question is:
Where can I find a g... | https://mathoverflow.net/users/38200 | Absolutness of $\Pi_1^1$ statements | This is an immediate consequence of the fact that every $\Pi^1\_1$ statement is equivalent to the assertion that a certain relation is well-founded, and well-foundedness is absolute between transitive models. If a larger model thinks a relation is well-founded, then the smaller model must agree since it can have no inf... | 9 | https://mathoverflow.net/users/1946 | 165763 | 86,538 |
https://mathoverflow.net/questions/165751 | 2 | Suppose we have an elliptic fibration $f:X\to \mathbb{P}^1$, with a singular fiber $F$, can we construct an elliptic fibration over $\mathbb{P}^1$ with fiber $nF$?
| https://mathoverflow.net/users/nan | modify a fibration with a fiber of higher multiple | The answer is yes if $n$ is a prime power by a theorem of Liu-Lorenzini-Raynaud; see p. 497, Corollary 6.7 in <http://www.math.u-bordeaux1.fr/~qliu/articles/LLR.pdf> (This answered a question of Neron.) It's true in general if your fiber $F$ is multiplicative.
By the way: the answer is no if you are considering hones... | 2 | https://mathoverflow.net/users/4333 | 165773 | 86,543 |
https://mathoverflow.net/questions/165684 | 6 | There is a classical theorem about covering spaces and the actions of the fundamental group.
**Theorem 1:** *Let $B$ be a non-empty locally path-connected and path-connected space. The category of covering spaces over $B$ is equivalent to the category of left actions of $\pi\_1(B)$ on sets.*
There is a slight gener... | https://mathoverflow.net/users/1176 | A generalization of covering spaces to fiber bundles with totally path-disconnected fibers | The usual classification of covering spaces (stated in terms of a categorical equivalence) requires the conditions "locally path connected" and "semi-locally simply connected."
**Theorem 1:** If $X$ is a path-connected, locally path-connected and semi-locally simply connected space, then the category of covering spac... | 5 | https://mathoverflow.net/users/5801 | 165778 | 86,544 |
https://mathoverflow.net/questions/165783 | 9 | With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ continuous?
| https://mathoverflow.net/users/47958 | Existence of infinite groups that are too reluctant to be topological | There is a large literature about this, see "non-topologizable groups". These are, by definition, groups for which the only Hausdorff group topology is discrete. There are various examples, the first of which were obtained by Olshanskii and Shelah (see [here](http://arxiv.org/abs/math/0603513) for references) around 19... | 19 | https://mathoverflow.net/users/14094 | 165787 | 86,546 |
https://mathoverflow.net/questions/165791 | 4 | I'm looking for a maximum accuracy quadrature formula:
$$
\int\_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A\_1f(x\_1)+A\_2f(x\_2)+R(f)
$$
I don't know exactly if it's Trapezoidal rule which has the degree of accuracy one, or Simpson's rule with three or Gauss (these 3 I have studied deeply, but might be others), a... | https://mathoverflow.net/users/50132 | Quadrature formula max accuracy | I assume you mean maximum degree of accuracy for polynomials, that is, you require that the formula is exact for polynomial functions of degree up to $d$, and you look for the maximum possible $d$. This is the most usual requirement, although not necessarily the best one (see for instance1 <http://eprints.maths.ox.ac.u... | 5 | https://mathoverflow.net/users/1898 | 165801 | 86,550 |
https://mathoverflow.net/questions/165688 | 2 | Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c\_1(\wedge\_{\mathbb C}^{0,n}(TM, J))\in H\_{dR}^2(M)$$
is integral, for some almost complex structure $J$ on $M$ .Then why there exists a [$Spin^c$-structure](http://ncatlab.org/nlab/show/spin%5Ec+structure) $P\to M$ on... | https://mathoverflow.net/users/nan | A question on existence of $Spin^c$-structure $P\to M$ | You ask "why does a spin$^c$ structure exist on $(M,\omega)$?" It is a basic fact that every symplectic manifold admits a spin$^c$ structure. I will assume that you are instead interested in what Hochs is claiming in his comment on page 43 of his thesis: a spin$^c$ structure with determinant $L^{2\omega}$ exists on $(M... | 3 | https://mathoverflow.net/users/21375 | 165806 | 86,552 |
https://mathoverflow.net/questions/165794 | 0 | Where can we find a well developed material on direct limits of finite $p$-groups?
For instance, is there a characterization of such groups, which have a finite rank (that is every subgroup can be generated by a finite positive integer $k$?
Or, is it true that such a group is finite, provided that it is finitely ge... | https://mathoverflow.net/users/31883 | A question on direct limits of finite $p$-groups | (Too long for a comment) A finitely generated subgroup of a direct limit of groups is isomorphic to a quotient of a subgroup of one of the constituent groups.
Indeed, suppose that $\{G\_i, f\_{ij}\}\_{i\in I}$ is a directed family, $G$ a subgroup of $\lim\limits\_{\rightarrow} G\_i$, and that $G=\langle g\_1,\ldots,g... | 3 | https://mathoverflow.net/users/3959 | 165810 | 86,554 |
https://mathoverflow.net/questions/143599 | 12 | I'm working on solving the quartic Diophantine equation in the title. Calculations in maxima imply that the only integer solutions are
\begin{equation}
(r,s) \in \{(-3, -2), (-2, 3), (-1, 0), (0, -1), (0, 1), (1, 0), (2, -3), (3, 2)\}.
\end{equation}
Evidently, the set above are all solutions, and furthermore if $(r,s... | https://mathoverflow.net/users/19844 | Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$ | [Answered on stackexchange](https://math.stackexchange.com/questions/774956/a-pell-equation-inside-a-pell-equation/788692#788692)
where **Kieren MacMillian** linked to this MO question. Briefly:
This [Thue equation](http://en.wikipedia.org/wiki/Thue_equation)
is equivalent to Ljunggren's equation $X^2+1 = 2Y^4$.
Ljungg... | 7 | https://mathoverflow.net/users/14830 | 165815 | 86,558 |
https://mathoverflow.net/questions/165809 | 28 |
>
> Conjecture: There are constants $c,k$ such that every $(Z/nZ)^\*$ is generated by its elements smaller than $k (\log n)^c$.
>
>
>
Where $(Z/nZ)^\*$ is the multiplicative group of integers mod $n$. **My main question is: How "strong" is this conjecture relative to other unsolved conjectures? How "hard" do exp... | https://mathoverflow.net/users/29697 | How strong is this conjecture? $(Z/nZ)^*$ is generated by "small" elements | This problem seems to lie quite deep. To illustrate this consider the problem of estimating the least quadratic non-residue $\pmod p$ for a prime $p$ -- obviously if the numbers up to some point generate $({\Bbb Z}/p{\Bbb Z})^\*$ then they must contain a quadratic non-residue, so this problem should be ``easier." Vinog... | 35 | https://mathoverflow.net/users/38624 | 165816 | 86,559 |
https://mathoverflow.net/questions/165553 | 2 | A partition of $[n]$ is indecomposable if no subset of its blocks partitions $[k]$ with $k \in [n-1]$. Irreducible set partitions are defined at <http://oeis.org/A055105> . Both are counted by <http://oeis.org/A074664> . Is any bijection between them known?
| https://mathoverflow.net/users/29500 | Is there a bijection from indecomposable to irreducible set partitions? | Mike Zabrocki has kindly addressed me to the following note <http://www.billchen.org/publications/2011_P7_Unsplitable/2011_P7_Unsplitable.pdf> where the bijection is built (here "atomic"="indecomposable" and "unsplittable"="irreducible"). Note that the correct version of the definition of splittable or reducible partit... | 2 | https://mathoverflow.net/users/6101 | 165818 | 86,560 |
https://mathoverflow.net/questions/165777 | 3 | Assume that $E$ is a bundle of Lie Algebras. Let $g$ be an invariant metric on $E$, that is for all $p\in M$, $$g\_p([x,y],z)+g\_p(y,[x,z])=0,$$ where $x,y,z\in E\_p$ are arbitrary.
Is there a Riemannian connection $\nabla$ on $E$ such that:
$$\nabla\_U[X,Y]=[\nabla\_U X,Y]+[X,\nabla\_U Y]$$ holds for every $U\in \ma... | https://mathoverflow.net/users/40291 | Special Riemannian connections? | The answer is 'no, not always'.
Here's an example: Let $E\to M$ be an oriented Riemannian $3$-plane bundle over $M$, with inner product $g$. Then there is a well-defined bilinear cross-product operation on sections $\times :\Gamma E \times \Gamma E \to \Gamma E$ with the property that, if $X$ and $Y$ are unit-length... | 3 | https://mathoverflow.net/users/13972 | 165822 | 86,561 |
https://mathoverflow.net/questions/165780 | 7 | Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.
If $\mathcal X$ can be defined over $\overline{\mathbb Q}$, then $X$ can be defined over $\overline{\mathbb Q}$. This is because "the base-change of the coarse modul... | https://mathoverflow.net/users/4333 | Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is | No. Even if $X$ is "defined over $\overline{\mathbb{Q}}$" in the sense that $X$ is isomorphic to $X\_0\otimes\_{\overline{\mathbb{Q}}} \mathbb{C}$ for some variety $X\_0$ over $\overline{\mathbb{Q}}$, nonetheless the stack $\mathcal{X}$ may not be defined over $\overline{\mathbb{Q}}$.
For instance, let $X\_0$ be $\m... | 11 | https://mathoverflow.net/users/13265 | 165841 | 86,565 |
https://mathoverflow.net/questions/165825 | 3 | If $\mathcal{H}$ is the heart of a bounded t-structure in a triangulated category $\mathcal{T}$, then for every object $E$ in $\mathcal{T}$ there exists a finite sequence of integers $k\_1>k\_2>\dots >k\_n$ and a collection of triangles in $\mathcal{T}$
$$
E\_{i-1}\stackrel{\phi\_i}{\to}E\_i\to
A\_i \to E\_{i-1}[1]
$$
... | https://mathoverflow.net/users/8320 | Postnikov towers in bounded t-structures | Let $F$ be the fiber of $f:X \to Y$ (i.e. the cone shifted by $[-1]$) so that we have a distinguished triangle
$$
F \to X \to Y.
$$
Let $0 = F\_0 \to F\_1 \to \dots \to F\_n = F$ be its Postnikov tower. Let $E\_i$ be the cone of the map $F\_i \to X$ (the composition of $F\_i \to F$ and $F \to X$). Then $E\_0 = Cone(0 \... | 5 | https://mathoverflow.net/users/4428 | 165845 | 86,567 |
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