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https://mathoverflow.net/questions/146535 | 1 | I think the conformal mapping theory in the plane are quit interesting and useful in physics. I learned that there is very few conformal mappings in higher dimensions, is there any reason for that?
| https://mathoverflow.net/users/nan | Why there are so few conformal mappings in higher dimension? | Usually in mathematics, if an object enjoys too many good properties, then there will be some kind of rigidity result for it.
Conformal mappings are perfect in two dimension and they are solutions to the so-called Beltrami system, which is highly over-determined when n>3. Thus, it is very rigid. By a result of Liouvi... | 0 | https://mathoverflow.net/users/26608 | 146539 | 79,210 |
https://mathoverflow.net/questions/146528 | 3 | According to [this paper](http://arxiv.org/abs/1305.1965), a cube of a composition of matrix inverse, matrix elements' inverse, and matrix transposition modifies 3x3 matrix by multiplying on left and right side by diagonal matrices.
A square of such composition is identity for 2x2 matrices over a commutative ring be... | https://mathoverflow.net/users/38448 | What is the intuition behind Kontsevich-Iyudu-Shkarin result? | The answer is contained in [Kontsevich's paper](http://arxiv.org/abs/1109.2469) (where he conjectures this result).
| 2 | https://mathoverflow.net/users/11142 | 146540 | 79,211 |
https://mathoverflow.net/questions/146549 | 8 | I am reading some algebraic topological book, where they said that the p-th homology group tells us how many p-dimensional holes inside the set. How should I understand this? I know that in the planar case, the 1-th order homology group tells us exactly how many 1-holes the set has, but I do not have a good feeling abo... | https://mathoverflow.net/users/nan | What is the geometric meaning of homology group and cohomology group? | I don't think counting even 1-D holes is appropriate for intuitive understanding of homology. Take a torus for example: its 1st homology $H\_1$ is generated by a circle along the torus (the "hole" in this case it the dohnut hole), and a circle across the torus (now the "hole" is the void inside the dohnut surface). The... | 14 | https://mathoverflow.net/users/38448 | 146551 | 79,215 |
https://mathoverflow.net/questions/146555 | 0 | If two Riemannian manifolds $(M,g)$ and $(N,h)$ have the same constant curvature are they isometric?
| https://mathoverflow.net/users/42140 | Does identical curvature imply isometry | No. There are plenty of non-isometric compact surfaces of zero constant
curvature (tori). There are even more surfaces of constant curvature $-1$.
| 5 | https://mathoverflow.net/users/25510 | 146556 | 79,216 |
https://mathoverflow.net/questions/146339 | 2 | **Setup:** Let $\mathbb C$ be a category. Assume that the span $A \xleftarrow{a} X \xrightarrow{b} B$ has a pushout $A \xrightarrow{\mathsf{pinl}} A \sqcup\_X B \xleftarrow{\mathsf{pinr}} B$. Let $f : A \rightarrow C$ and $g : B \rightarrow C$ be such that $f \cdot a = g \cdot b$, which means that there exists a mediat... | https://mathoverflow.net/users/39351 | When does a pushout mediating arrow factor through the coproduct? | Taking $C$ to be the pushout, you're asking that the surjection
from the coproduct to the pushout be split.
There is no natural reason (in either the formal or informal sense)
why this should happen.
You don't say much about what sort of category $\mathbb C$ is supposed
to be, so we might as well switch the arrows ... | 2 | https://mathoverflow.net/users/2733 | 146558 | 79,217 |
https://mathoverflow.net/questions/146554 | 5 | 1) Apparently, physicist can calculate the GW invariants of quintic CY 3-fold up to genus 51.
I am looking for a reference that has a table of these number for some low degrees (say up to degree 5) and low genera (at least until g=3).
2) For each genus g, there is a lower bound $d(g)$ such that for every $d<d(g)$, al... | https://mathoverflow.net/users/5259 | Looking for a reference (on GW invariants of quintic) | 1) Huang, Klemm and Quackenbush computed the BPS invariants of the quintic 3-fold for low genera via the BCOV technique in <http://arxiv.org/abs/hep-th/0612125>. We can easily convert their data to get the GW invariants.
2) I think the bound is not a theorem, but an observation. We often assume such a vanishing cond... | 5 | https://mathoverflow.net/users/21014 | 146563 | 79,219 |
https://mathoverflow.net/questions/146483 | 2 | Let $n>3$ be a positive integer. We denote the symmetric group of $n$ elements by $S\_n$ and the identity mapping by $\mathrm{id}$. For every $f\in S\_n$, $f(1,2,\ldots,n)=(a\_1,a\_2,\ldots,a\_n)$, denote $a\_n$ by $m(f)$.
For any positive integer $1\leq k\leq n-2$, define $f\_k\in S\_n$ as follow:
$$f\_k:(1,\ldots... | https://mathoverflow.net/users/40096 | A conjecture about a specific subset of the symmetric group $S_n$ | The conjecture is true for $n=3$ or $4$, and false for $n>4$.
It is trivially true for $n=3$. For $n=4$, we can only use $f\_1$ and $f\_2$, so we simply check that $(f\_2\circ f\_1)^3 = id$, and that $m(f\_1)=2$ and $m((f\_2\circ f\_1)^2)=3$.
Assume now that $n>4$. We have that $f\_2\circ f\_3 \circ f\_2 \circ f\_1... | 4 | https://mathoverflow.net/users/39640 | 146564 | 79,220 |
https://mathoverflow.net/questions/146561 | 5 | The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here $\otimes$ denotes the component-wise tensor product of simplicial vector spaces (it gives to the category $sVect$ of simpl... | https://mathoverflow.net/users/32741 | Two definitions of modules in monoidal category | I will write $[B, C]$ instead of $\underline{\mathrm{Hom}}(B, C)$. Recall the tensor–hom adjunction:
$$\mathrm{Hom}(A \otimes B, C) \cong \mathrm{Hom}(A, [B, C])$$
Thus there is a canonical bijection between morphisms $\alpha : A \otimes V \to V$ and $\tilde{\alpha} : A \to [V, V]$. Let us show that $\alpha$ is an $A$-... | 9 | https://mathoverflow.net/users/11640 | 146565 | 79,221 |
https://mathoverflow.net/questions/146530 | 5 | Let $G$ be a reductive, linear algebraic group (variety) over an algebraically closed field $\Bbbk$ of characteristic zero. If $G$ is connected, I know from Humphrey's book that for any Borel subgroup $B\subseteq G,$ there is some opposite Borel subgroup $B^-$ such that $T=B\cap B^-$ is a Torus, and denoting by $U$ and... | https://mathoverflow.net/users/9947 | Bruhat decomposition for reductive groups in characteristic zero? | Maybe I can clarify some of the issues here. First, the distinction between "reductive" and "semisimple" is minor, since the difference between these lies in the maximal torus and doesn't really affect the Bruhat decomposition or the big cell $U^- T U$. But most books and survey articles do assume that $G$ is *connecte... | 5 | https://mathoverflow.net/users/4231 | 146570 | 79,223 |
https://mathoverflow.net/questions/146510 | 7 | Let $A$ be a Noetherian ring, $M$ a finite $A$-module and $I=(y\_1,\cdots,y\_n)$ an ideal of $A$ such that $M \neq IM$. Denote by $H\_i(y\_1,\cdots,y\_n;M)$ the homology at dimension $i$ of the augmented Koszul complex $K\_{\cdot}(y\_1,\cdots,y\_n) \otimes M$.
>
> Theorem: If $y\_1,\cdots,y\_n$ is a regular sequenc... | https://mathoverflow.net/users/32906 | zero homology of augmented Koszul complex implies the sequence is regular? | I don't think the corollary is true without further assumptions.
Take $R = k[x,y,z]$ and $I = (x(y-1), y,z(y-1)$. Since $x(y-1), y,z(y-1)$ is a regular sequence, hence depth $(I,R)$ = 3. But $x(y-1),z(y-1), y$ is not a regular sequence.
I believe that the statement is that $I$ can be generated by an $M$-sequence. ... | 4 | https://mathoverflow.net/users/22388 | 146576 | 79,227 |
https://mathoverflow.net/questions/146567 | 13 | Suppose $j:V\_\lambda \rightarrow V\_\eta$ is (elementary and) cofinal. Can $j$ be extended to all of $V$?
(Subsidiary question: What conditions are there on an ultrafilter/extender/whatever so that the induced $j:V \rightarrow M$ has $V\_\eta$ as an initial segment of $M$? Note that $j(\kappa)$ might well be less th... | https://mathoverflow.net/users/42160 | extending elementary embeddings from initial segments of V to all of V | At successor ordinals, the answer is no, not necessarily, assuming the consistency of a nontrivial instance of your hypothesis.
For a counterexample, let $\kappa$ be the least $1$-extendible cardinal. So there is $j:V\_{\kappa+1}\to V\_{\eta+1}$, and this map is elementary and cofinal, mapping $\kappa$ to $j(\kappa)... | 12 | https://mathoverflow.net/users/1946 | 146577 | 79,228 |
https://mathoverflow.net/questions/146463 | 100 | I first saw the term "entropy" in a chemistry course while studying thermodynamics.
During my graduate studies I encountered the term in many different areas of mathematics.
Can anyone explain why this term is used and what it means. What I am looking for is a few examples where the term "entropy" is used to describe s... | https://mathoverflow.net/users/8435 | What is entropy, really? | Here is a simple story one can tell about the entropy
$$H = -\sum\_{i=1}^n p\_i \log p\_i$$
of a discrete probability distribution. Suppose you wanted to describe how surprised you are upon learning that some event $E$ happened. Call your surprise upon learning that $E$ happened $s(E)$, the "surprisal." Here are ... | 124 | https://mathoverflow.net/users/290 | 146579 | 79,230 |
https://mathoverflow.net/questions/118689 | 7 | Suppose $$j:M\prec N$$ is a non-trivial elementary embedding. Under what conditions on the sets (classes?) $M$ and $N$ (or even the critical point of $j$) does $j$ extend to an elementary embedding $$k:L(M)\prec L(N)?$$
In the case where $M=N=V\_{\lambda+1}$, the existence of such a $k$ is a strictly stronger assump... | https://mathoverflow.net/users/5697 | Elementary Embeddings and Relative Constructibility | Not all such embeddings extend, as was mentioned at Bob Lubarsky's question on [Extending elementary embeddings from initial segments to all of $V$](https://mathoverflow.net/questions/146567/extending-elementary-embeddings-from-initial-segments-of-v-to-all-of-v).
Specifically, suppose that $\kappa$ is the least $1$-... | 4 | https://mathoverflow.net/users/1946 | 146585 | 79,233 |
https://mathoverflow.net/questions/146581 | 0 | By a *curve* I mean an integral one-dimensional scheme of finite type over a spectrum of a field.
Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is rational over $k$, if and only if $C$ is rational over any field extension $L/k$. I'm wondering, if there an elementary pr... | https://mathoverflow.net/users/40504 | Rationality of curve does not depend on base change | Well, why not turn my comment on Abhinav's answer into an answer?
$\newcommand{\ra}{\rightarrow}$
$\newcommand{\PP}{\mathbb{P}}$
$\newcommand{\F}{\mathbb{F}}$
This is a cut-and-paste from a passage in a paper I wrote earlier today.
>
> Lemma:
> Let $L/K$ be a purely transcendental field extension.
>
> a) Le... | 3 | https://mathoverflow.net/users/1149 | 146595 | 79,236 |
https://mathoverflow.net/questions/146600 | 2 | Let $X$ be a smooth projective variety and $Z$ is a closed subscheme in $X$ which is not a complete intersection in $X$. Assume the dimension of $X$ (resp. $Z$) is greater than $3$ (resp. $1$). Then,
1) For a general hyperplane $H$ in $X$, is it possible that $H.Z$ is complete intersection in $X$?
2) Suppose $X$ i... | https://mathoverflow.net/users/32151 | Which actions preserve non-complete intersections? | Yes for both questions. For (1) take $X = P^3$ and let $Z$ be a line with an embedded point. For (2) take $X = P^3$ and let $Z$ be a twisted rational cubic. Then $Z'$ is a nodal cubic curve on $X' = P^2$ which is a complete intersection.
| 4 | https://mathoverflow.net/users/4428 | 146602 | 79,238 |
https://mathoverflow.net/questions/146590 | 3 | Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code?
Is there an easy test? If so, could someone provide me a reference?
In short, if given a $[n,k,n-k+1]\_q$ code how do you show it is RS or not?
| https://mathoverflow.net/users/10035 | On MDS code property | I guess you exclude trivial MDS codes, generalized Reed-Solomon codes, and MDS codes that can be obtained by code extension.
If you exclude them all, there are still a bunch of MDS codes. In general, MDS codes of length $n$ and dimension $k$ over $\mathbb{F}\_q$ are equivalent to $n$-arcs in $\text{PG}(k-1,q)$. Gener... | 4 | https://mathoverflow.net/users/27829 | 146606 | 79,240 |
https://mathoverflow.net/questions/146619 | -2 | I want to analyze a discrete signal in time, that is sum of exponentials distributed around two distinct values (T1, T2). My goal is to calculate these two different distributions, T1, T2, weights and standard deviations.
I have found two MATLAB scripts doing this using Inverse Laplace transform:
<http://www.wolfso... | https://mathoverflow.net/users/42184 | Inverse Laplace transform in a signal that is a sum of exponentials - MATLAB | The [Padé-Laplace method](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1280453/) works well in many contexts in which you want to recover the coefficients of a sum of exponentials from sampled values with some noise. The basic idea is that if you apply the Laplace transform again (not the inverse Laplace transform), you... | 0 | https://mathoverflow.net/users/2954 | 146626 | 79,245 |
https://mathoverflow.net/questions/146628 | 7 | I am working with "usual" category theory, maybe over ZFC, and I have a functor $F : Set \to Set$. I'd like to apply Yoneda lemma to $F$, i.e. obtain:
$$ [Set, Set](h\_A, F) \cong F A $$
However, most of texts assume that the domain of $F$ should be a small category in order to apply it. I know this has to do with ... | https://mathoverflow.net/users/42193 | Yoneda on a not so small category | Zhen Lin states one direction of the Yoneda Lemma: given $x\in F A$, the natural transformation $\theta\_x:H\_A\to F$ on $f:A\to B$ is $\theta(f)=F f\cdot x$. The other direction is that every $\theta:H\_A\to F$ is of this form, where $x=\theta(\mathsf{id}\_A)$. I leave it as an exercise to show that $\theta\_x$ is nat... | 10 | https://mathoverflow.net/users/2733 | 146660 | 79,258 |
https://mathoverflow.net/questions/146650 | 2 | Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is independent of $m$ or $n$.
Question: how much computation is needed to find an $\epsilon$ accurate estimate of the largest... | https://mathoverflow.net/users/42202 | Dimension independent computational complexity of singular value decomposition | Have a look at the references cited in my [older answer here](https://mathoverflow.net/a/75382/8430), for the results of the kind you are looking for (complexity in there though is shown with a worst case O($\log n$) dependence on the dimension, but perhaps the analysis can be adapted to remove that dependence in your ... | 1 | https://mathoverflow.net/users/8430 | 146664 | 79,259 |
https://mathoverflow.net/questions/146653 | 15 | It's conjectured that, asymptotically, half of elliptic curves have rank 0, half have rank 1, and elliptic curves of rank $\geq 2$ have density 0. But what if we disregard elliptic curves of rank 0 or 1: Are there any conjectures about the average rank of elliptic curves of rank $\geq 2$?
More generally, for any inte... | https://mathoverflow.net/users/31308 | Average rank of elliptic curves, excluding those of low rank | A very simple random matrix heuristic says, based on the function field model, that the rank of a random elliptic curve is the dimension of the invariant subspace of a random element of $O(n)$ for large $n$.
We can easily compute the dimension of the space of matrices that preserve exactly $k$-dimensional subspace. W... | 11 | https://mathoverflow.net/users/18060 | 146665 | 79,260 |
https://mathoverflow.net/questions/146635 | 7 | Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\| = \min(\{x\}, 1 - \{x\})$, where $\lfloor x \rfloor$ is, as usual, the greatest integer $\le x$.
Let $x$ be a fixed ... | https://mathoverflow.net/users/16537 | Expected symmetry in the diophantine approximations of an irrational number | This can't be right in general. For example
$$
x = 0.101000001000000000000000001\ldots
= \sum\_{k=0}^\infty 10^{-3^k}
$$
has plenty of lower approximants with exponent just under 3
(the partial sums) but no upper ones.
| 5 | https://mathoverflow.net/users/14830 | 146668 | 79,263 |
https://mathoverflow.net/questions/146670 | 32 | Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega\_x : \Lambda^k T\_x M \to \mathbb{R}$ which is *linear*.
My suspicion is that this pointwise statement of linearity is essential... | https://mathoverflow.net/users/2362 | Converse to Stokes' Theorem | Yes $\omega$ and $\eta$ have to be linear.
We will need the following generalization of Minkowski's theorem on the existence of polyhedra with prescribed surface normals;
see Theorem 1, p. 475 in ["Gaussian images of surfaces and ellipticity of surface area functionals"](http://www.pdmi.ras.ru/~svivanov/papers/ellip.... | 21 | https://mathoverflow.net/users/1441 | 146673 | 79,264 |
https://mathoverflow.net/questions/146583 | 4 | Any finite extension of the rationals, along with its Galois group, can be interpreted in Peano arithmetic by straightforward means. For a fixed bound $n$ in the degree this is uniform in the coefficients of a defining polynomial so you can quantify in PA over degree (at most) $n$ extensions. As a convenient reference,... | https://mathoverflow.net/users/38783 | Interpreting the Galois theory of finite extensions of $\mathbb{Q}$ in PA | It looked straightforward to me, but proof theory can be tricky. Emil's comment convinces me I am not overlooking any pitfall here.
PA and EFA interpret the Galois groups of all finite extensions of the rationals, uniformly in the strings of coefficients of the defining polynomials. So PA and EFA can quantify over f... | 1 | https://mathoverflow.net/users/38783 | 146684 | 79,266 |
https://mathoverflow.net/questions/144858 | 5 | Let $k$ be a positive integer and $G=(V,E)$ be a $2$-connected simple graph.Suppose $v\in V(G)$ satisfy:
$(1)$there exists at least one vertex $u\in V(G)\backslash\{v\}$ such that $u$ is not adjacent with $v$;
$(2)$for any $u\in V(G)\backslash\{v\}$ such that $u$ is not adjacent with $v$,there is a $u$-$v$ path in ... | https://mathoverflow.net/users/40096 | A conjecture about odd path and odd cycle | False. As we want a counterexample, we naturally start with the Petersen graph, P. Note that for any vertex v of P and non-adjacent edge uw of P there is a Hamiltonian path from v to w that does not use the uw edge. On the other hand, there is no Hamiltonian cycle in P.
Our graph G will have 10t+15 vertices (where t ... | 5 | https://mathoverflow.net/users/955 | 146688 | 79,269 |
https://mathoverflow.net/questions/146681 | 6 | I want to know if there is proof of Borel Weil Bott theorem, that is as geometric as it can be.
Let $G$ be a semisimple compact Lie group and $T$ be a maximal torus. We know that $G/T$ is a projective manifold. A way to show this is to consider the complexification of $G$ and identify $G/T$ as $G\_{\mathbb C}/B$.
... | https://mathoverflow.net/users/41094 | Geometric structure of flag manifolds, Borel -Weil-Bott theorem | 1. Correct. You can be fairly explicit here. For each root $\alpha$, let $\omega\_\alpha \in \mathfrak g^\ast$ be a left-invariant form on $G$ that is dual to $\mathfrak g\_\alpha$. Then for $\lambda \in \mathfrak t^\ast$, we have $d\lambda = \sum\_{\alpha\in\Phi^+} \langle \lambda,\alpha\rangle \omega\_\alpha \wedge \... | 3 | https://mathoverflow.net/users/430 | 146690 | 79,270 |
https://mathoverflow.net/questions/146656 | 5 | Is there a classification for infinite finitely generated solvable groups all of whose abelian normal subgroups are finite?
I mean by classification something like presentation.
Edited: Is there an infinite finitely generated solvable group $G$ all of whose abelian normal subgroups are finite and $G$ is not residua... | https://mathoverflow.net/users/19075 | Finitely generated solvable groups all of whose abelian normal subgroups are finite | To your edited question:
>
> There is an infinite finitely generated solvable group with no infinite normal abelian subgroup.
>
>
>
The example is an extension of $\mathbf{Z}/p\mathbf{Z}$ by the lamplighter $(\mathbf{Z}/p\mathbf{Z})^2\wr\mathbf{Z}$.
Fix a prime $p$ congruent to -1 modulo 4 (so that -1 has no... | 7 | https://mathoverflow.net/users/14094 | 146711 | 79,275 |
https://mathoverflow.net/questions/146722 | 6 | A well known theorem by Scott says:
**If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then $2^{\kappa}=\kappa^{+}$.**
So:
**If $\text{GCH}$ is false in a measurable cardinal then there is a smaller cardinal which $\text{G... | https://mathoverflow.net/users/nan | The First Failure of GCH in Large Cardinals Smaller than Measurables | The statement is not true.
**Theorem.** If $\kappa$ is strongly unfoldable and the GCH holds below $\kappa$, then it holds at $\kappa$ also.
This is just because the strongly unfoldable cardinals are $\Sigma\_2$-reflecting, and the result also holds for any $\Sigma\_2$-reflecting cardinal. The strongly unfoldable c... | 9 | https://mathoverflow.net/users/1946 | 146724 | 79,282 |
https://mathoverflow.net/questions/146727 | 0 | Suppose $X$ is a proper closed subset of $\mathbb{P}^n\_k$, $P$ a point not in $X$ and a hyperplane $H$. Denote $m\_{P,H}$ the supremum of intersections of straight lines from $P$ to points in $H$ with $X$.
Is $m\_{P,H}$ finite? What is the relation between $m\_{P,H}$ and $X$?
| https://mathoverflow.net/users/nan | Intersection of lines and a closed variety in projective space | If you know that your $X$ is an intersection of hypersurfaces of degree $d$, then $m\_P\leq d$ for all $P$ (note that your $H$ does not play any role). Indeed there exists a hypersurface $V$ of degree $d$ containing $X$ but not $P$; then for any line $L$ passing through $P$, $\#X\cap L\leq \#V\cap L\leq d$. In particul... | 0 | https://mathoverflow.net/users/40297 | 146734 | 79,288 |
https://mathoverflow.net/questions/146715 | 0 | It's known that every cubic bridgeless graph has 1-factor ([Petersen](http://en.wikipedia.org/wiki/Petersen%27s_theorem)). But Does anybody know, how to prove that for every edge in a cubic bridgeless graph there exists a 1-factor, which contains it?
Because I found articles, where this is stated, but no proof of it... | https://mathoverflow.net/users/42232 | Union of perfect matchings in bridgeless cubic graphs | Petersen's theorem: A bridgeless cubic graph contains a one-factor.
This has been generalized by T. Schönberger [T. Schönberger, "Ein Beweis des Peterschen Graphensatzes" Acta Sci. Math. Szeged , 7 (1934) pp. 51–57], who proved that every edge of a bridgeless cubic graph lies in a one-factor.
Best regards,
Július Ko... | 4 | https://mathoverflow.net/users/41997 | 146751 | 79,295 |
https://mathoverflow.net/questions/146747 | 12 | The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(y)\rbrace$, i.e. for which
$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})\ x = \lbrace y : \phi(y)\rbrac... | https://mathoverflow.net/users/2672 | Ways to define "definability" | I am glad to see this question, Hans, which I believe gets right to the heart of the definability concept, on which some of your recent questions have focused. This is an excellent question.
First, let me say that I dispute your characterization of the OD sets. The claim that you state is not what is proved about OD,... | 12 | https://mathoverflow.net/users/1946 | 146754 | 79,296 |
https://mathoverflow.net/questions/146636 | 3 | We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^\*M×\mathbb{R}$.i.e,
($J^1M=T^\*M×\mathbb{R}$)
Is there something like this identity for higher jet bundle $J^kM$?
I editted my question after comment of Michael Murray
| https://mathoverflow.net/users/nan | looking for an identity for higher jet bundle $J^kM$? | First, note that there is a natural exact sequence of bundle maps
$$
Sym^kT^\*M \rightarrow J^kM \rightarrow J^{k-1}M
$$
but there is no natural splitting of this sequence. You can, however, do it by choosing a Riemannian metric and defining a map $J^{k-1}M \rightarrow J^kM$ by extending a $(k-1)$-jet to the $k$-jet wh... | 5 | https://mathoverflow.net/users/613 | 146760 | 79,300 |
https://mathoverflow.net/questions/146767 | 1 | Suppose $f:X \to Y$ is a finite morphism with $X$ and $Y$ being affine varieties, such that $X$ is unirational. In fact $X$ is more than unirational, it is the image of a morphism from a zariski open subset of $\mathbb P^n$. Are there any results that allows one to conclude the finite generation of the Chow ring $A^{\*... | https://mathoverflow.net/users/18129 | Chow groups of finite covers of unirational varieties | This is not true. Take $\bar{X}=$ a smooth cubic threefold in $\Bbb{P}^4$, $\bar{Y}=\Bbb{P}^3$, $f:\bar{X}\rightarrow \bar{Y}$ the projection from a general point of $\Bbb{P}^4$. Now let $H$ be a general hyperplane in $\Bbb{P}^3$, and $S:=f^{-1}(H)$; this is a smooth hyperplane section of $\bar{X}$. Put $Y:=\bar{Y}\sma... | 3 | https://mathoverflow.net/users/40297 | 146770 | 79,302 |
https://mathoverflow.net/questions/146768 | 0 | The system of diophantine equations $$\{x^2-y^2+z^2-u^2+q^2-t^2=0,\,xy+zt-uq=0 \}$$ is given. Do the formulas
$$x:=(j(p^2-4ps+3s^2)-(p-s)(3p^2-4ps+s^2))k^2+2(j-2(p-s))(p-s)kn+(j-p+s)n^2, $$
$$y:=(p-s)(4j(p-s)-3p^2+4ps-s^2)k^2+2(p-s)(j-2(p-s))kn-(p-s)n^2, $$
$$z:=(j(p^2-4ps+3s^2)+s(3p^2-4ps+s^2))k^2+2(p-s)(2s+j)kn+(j+s)... | https://mathoverflow.net/users/35959 | Solutions of system of diophantine equations | The equations $x^2-y^2+z^2-u^2+q^2-t^2=0$, $xy+zt-uq=0$
are the real and imaginary parts of $w\_1^2 + w\_2^2 + w\_3^2 = 0$
where $(w\_1,w\_2,w\_3) = (x+iy,z+it,q-iu)$. So we have a Pythagorean triple
over the Gaussian numbers (with the hypotenuse multiplied by $i$,
making it more symmetrical), and can just use the stan... | 12 | https://mathoverflow.net/users/14830 | 146776 | 79,305 |
https://mathoverflow.net/questions/146744 | 4 | All varieties are over $\mathbb{C}$.
Let $G$ be a connected reductive group, $B\subseteq G$ a Borel subgroup.
Let $O\_w$ be a $B$-orbit in $G/B$. I.e., $O\_w$ is a Bruhat cell. In particular, it is simply connected. Write $IC(O\_w, \underline{\mathbb{C}})$ for the minimal perverse extension of the (shifted) constant... | https://mathoverflow.net/users/23907 | Stalks of intersection cohomology complexes of Schubert varieties and Bruhat order | The stalk of the IC sheaf you're interested in looking at is the intersection cohomology of an actual space, given by the intersection of the Schubert variety $\bar{O}\_w$ with an orbit of an opposite Borel (through the unique fixed point of the torus given by the intersection of the Borels in $O\_v$). This intersectio... | 2 | https://mathoverflow.net/users/66 | 146778 | 79,306 |
https://mathoverflow.net/questions/146769 | 20 | The following *irreducible* trinomials are solvable:
$$x^5-5x^2-3 = 0$$
$$x^6+3x+3 = 0$$
$$x^8-5x-5=0$$
Their Galois groups are isomorphic to ${\rm D}\_5$, ${\rm S}\_3 \wr {\rm C}\_2$ and
$({\rm S}\_4 \times {\rm S}\_4) \rtimes {\rm C}\_2$, respectively.
**Question:** Is there an irreducible septic trinomial ... | https://mathoverflow.net/users/12905 | Is there an irreducible but solvable septic trinomial $x^7+ax^n+b = 0$? | If such polynomials exist, there will only be finitely many of them, up to composing on both sides with scalar polynomials $\alpha x$ with $\alpha\in\mathbf{Q}$. More generally, Guralnick and Shareshian proved that if $d=7$ or $d>8$ then there are only finitely many equivalence classes of irreducible degree-$d$ trinomi... | 19 | https://mathoverflow.net/users/30412 | 146780 | 79,307 |
https://mathoverflow.net/questions/146707 | 3 | This is a follow-up to [Question 146635](https://mathoverflow.net/questions/146635/expected-symmetry-in-the-diophantine-approximations-of-an-irrational-number), namely *Expected symmetry in the diophantine approximations of an irrational number*, which I will refer to for notation and terminology used here without expl... | https://mathoverflow.net/users/16537 | Numbers with balanced diophantine approximations | Yes, there are uncountably many examples (maybe I should have mentioned this variation with my previous answer). For example, $\sum\_{k=0}^\infty b\_k 10^{-3^k}$ with each $b\_k=1$ or $2$. There are uncountably many choices (cardinality of the continuum, even), and for each there are infinitely many lower approximants ... | 7 | https://mathoverflow.net/users/14830 | 146782 | 79,309 |
https://mathoverflow.net/questions/146779 | 2 | Let $f\_1$ and $f\_2$ be arbitrary self-mappings on $C([0,1])$ with $f\_2 > f\_1$. Define set $F = \{f \in (C[0,1])| f\_1 \leq f \leq f\_2 \mbox{ and } f \mbox{ is increasing}\}$. Is it true that every continuous self-mapping $\Phi: F \rightarrow F$ has a fixed point? For concreteness one could set $f\_1(x) = 0.5x$ and... | https://mathoverflow.net/users/42257 | Fixed point for a self-mapping on subset of C[0,1] | I think the following is an example of a map $\Phi$ that does not have a fixed point.
For arbitrary $f\_1<f\_2$ and $f \in F$ define
$$
\Phi(f)(x) = \begin{cases}
\cos(\pi x)f\_1(x) + (1-\cos(\pi x)) f(x) &{\rm for\;} x \le 1/2,\\
\cos(\pi (1-x))f\_2(x) + (1-\cos(\pi (1-x)))f(x) &{\rm for\;} x \ge 1/2.
\end{cases}... | 5 | https://mathoverflow.net/users/3928 | 146783 | 79,310 |
https://mathoverflow.net/questions/146786 | 16 | [Ackermann's function](http://en.wikipedia.org/wiki/Ackermann_function) is defined over integers $x$, $y$, $A(x,y)$,
with conditions for when $x=0$ or $y=0$, and otherwise uses recursive
definitions involving arguments $x-1$ and $y-1$.
>
> Is there a natural generalizations of $A(x,y)$ for $x,y \in \mathbb{R}$?
>
... | https://mathoverflow.net/users/6094 | Ackermann's function over the reals | It seems that we may extend the Ackermann function to the non-negative real plane with a continuous real-valued function obeying the Ackermann recursion.
Specifically, consider the Ackermann function defined on the Wikipedia page to which you link, which is equivalent to the following:
$$A(m,n)=\begin{cases} n+1 &\te... | 10 | https://mathoverflow.net/users/1946 | 146794 | 79,314 |
https://mathoverflow.net/questions/143510 | 4 | Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let
$$
C^1\_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|\_{\partial\Omega}=0\},
$$
and let $C^1\_c(\Omega)$ be the space of compactly supported $C^1$ functions in $\Omega$. We usually define $H^1\_0(\Omega)$ as the completion of $C^1\_c(\Omega... | https://mathoverflow.net/users/824 | Variation on the Sobolev space $H^1_0$ | Assume that $\Omega \subset \mathbb{R}^n$. I will prove that $C^1\_c (\Omega)$ is dense in $H^1 (\Omega) \cap C^1 (\Omega) \cap C\_0 (\Omega)$ endowed with the $H^1$ norm, where
$$
C\_0 (\Omega) = \{u: \Omega \to \mathbb{R} : \text{for every } \varepsilon > 0, \, u^{-1} (\mathbb{R} \setminus (-\varepsilon, \varepsilo... | 3 | https://mathoverflow.net/users/42047 | 146806 | 79,317 |
https://mathoverflow.net/questions/146799 | 16 | Is there an infinite group with exactly two conjugacy classes?
| https://mathoverflow.net/users/41807 | Is there an infinite group with exactly two conjugacy classes? | A detailed construction of a non-finitely generated version can be found at <https://math.stackexchange.com/questions/88980/infinite-group-with-only-two-conjugacy-classes>
as pointed out in the comments.
| 13 | https://mathoverflow.net/users/15934 | 146811 | 79,319 |
https://mathoverflow.net/questions/146813 | 11 | Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?
Note 1. Follow-up question: Jech's 1973 book on the axiom of choice seems to be cited as the source for the Feferman-Levy model. Can... | https://mathoverflow.net/users/28128 | Is sigma-additivity of Lebesgue measure deducible from ZF? | This depends on exactly how you define Lebesgue measure since some definitions incorporate countable additivity. However, there is a model of ZF, the Feferman-Levy model, where $\mathbb{R}$ is a countable union of countable sets which ensures that any countably additive measure on $\mathbb{R}$ has to be trivial.
| 11 | https://mathoverflow.net/users/2000 | 146814 | 79,320 |
https://mathoverflow.net/questions/146808 | 2 | It is known that the moduli space $\overline{M}\_{g}$ of genus $g$ curves is of general type for $g\geq 24$.
By Theorem 2.4 of
*Logan, Adam The Kodaira dimension of moduli spaces of curves with marked points. Amer. J. Math. 125 (2003), no. 1, 105–138.*
all but finitely many of the $\overline{M}\_{g,n}$ with $g>... | https://mathoverflow.net/users/14514 | Kodaira dimension of the moduli space of curves | So it seems your question is: if we know that $\overline M\_{g,n}$ is of general type, is the same true for $\overline M\_{g,n'}$ with $n' \geq n$? The answer is yes, and is a special case of the main theorem of: *J. Kollár, Subadditivity of the Kodaira dimension : fibers of general type*.
As far as I know there are ... | 3 | https://mathoverflow.net/users/1310 | 146819 | 79,323 |
https://mathoverflow.net/questions/146812 | 8 | A group $G$ is *Noetherian* (or *slender*) if all its subgroups are finitely generated. Does this imply that the minimal number of generators of subgroups of $G$ is bounded above?
For example, if $G$ is a polycyclic group that admits a polycyclic series of length $n$ then every subgroup of $G$ can be generated by $n$... | https://mathoverflow.net/users/12996 | Minimal number of generators of subgroups of Noetherian groups | No.
>
> For any countable family of countable involution-free groups $G\_1,G\_2,\dots$, there is a 2-generated group $H$ containing all $G\_i$ as proper subgroups such that each proper subgroup of $H$ is either cyclic or a conjugate of a subgroup of some $G\_i$.
>
>
>
This is [Obraztsov's embedding theorem](h... | 14 | https://mathoverflow.net/users/24165 | 146828 | 79,327 |
https://mathoverflow.net/questions/146817 | 7 | I already asked this question in MSE but did not get any answer/comment yet.
Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth function $f:E\to\mathbb{R}$ such that, when... | https://mathoverflow.net/users/30152 | how to obtain a generalized Morse function out of a fiber bundle? | It's basically this:
Associated with a manifold $N$ are two spaces, call them $F\_1$ and $F\_2$. The first is the space of generalized Morse functions on $N$ and the second is basically $\Omega^\infty S^\infty(BO\wedge N\_+)$. These are sufficiently canonical (functorial w.r.t. diffeomorphisms) that, if you have a f... | 6 | https://mathoverflow.net/users/6666 | 146829 | 79,328 |
https://mathoverflow.net/questions/146820 | 4 | Suppose we are given an uncountable set $\Gamma$ and a measure space $(\Omega, \mathcal{F}, \mu)$. I would like to know when the Banach space $\ell\_1(\Gamma)$ embeds into $L\_1(\mu)$. Of course, this is impossible when $\mu$ is a probability measure (or more generally, when $\mu$ is $\sigma$-finite). On the other hand... | https://mathoverflow.net/users/42273 | When does $\ell_1(\Gamma)$ embed into $L_1(\mu)$? | If $(X,\Sigma, \mu)$ is *semi-finite* (it has no infinite atoms) and non-$\sigma$-finite, by transfinite induction there is a family $\{E\_\alpha\}\_{\alpha \in \omega\_1}$ of disjoint measurable sets of positive finite measure (indeed, having found $E\_\beta$ for all $\beta < \alpha$, the complement of the countable u... | 4 | https://mathoverflow.net/users/6101 | 146832 | 79,330 |
https://mathoverflow.net/questions/146836 | -1 | The problem is:
Let $\vec{x}\in\mathbb{R}^d$ be the variable and $f(\vec{x})$ be a scalar function that is globally strictly convex in $\mathbb{R}^d$. We assume the unique optimum of $f$ to be finite(in all dimensions). Denote the optimum as $\hat{x}$. Suppose $f$ is smooth enough, can we know the sign of each dimensio... | https://mathoverflow.net/users/34309 | Determining the sign of each element of the optimal of a strict convex function | Try $d = 2$, $f(x\_1,x\_2) = (x\_1+1)^2 + (x\_1+1)(x\_2 + t) + (x\_2 + t)^2$ where $t$ is a parameter. The minimum is at $(-1,-t)$. The sign of the second component of the gradient changes at $t = -1/2$, not at $t=0$.
| 2 | https://mathoverflow.net/users/13650 | 146838 | 79,333 |
https://mathoverflow.net/questions/146772 | 5 | If $G$ is a connected Lie group acting on a vector $\mathbb{C}$-space $V$ then it is well known that the algebra of invariants $\mathbb{C}[V]^G$ coincides with the algebra of invariants $\mathbb{C}[V]^L$ of corresponding Lie algebra $L.$
**Question.** Let now $G$ be a finite group, $V$ be its representation. Is there... | https://mathoverflow.net/users/40637 | Linearisation of a group | Geometric considerations suggest the answer is always no: we expect $\text{Spec } \mathbb{C}[V]^G \cong V/G$ to have dimension $\dim V$, but we expect $\text{Spec } \mathbb{C}[V]^L \cong V/L$ to have dimension at most $\dim V - \dim L$, at least if the action is sufficiently nontrivial.
One proof is as follows. First... | 13 | https://mathoverflow.net/users/290 | 146844 | 79,335 |
https://mathoverflow.net/questions/146822 | 0 | Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its [complexification](http://en.wikipedia.org/wiki/Complexification_%28Lie_group%29).
I am looking for a symplectic structure (without use of coordinates) on
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here $G^{\mathbb{C}}=T^\*G$.(this equality is trivial by polar decomposi... | https://mathoverflow.net/users/nan | Symplectic structure on $Sym^kG^{\mathbb{C}} $ | Such a thing doesn't exist. The symmetric square of the cotangent bundle of a real $n$-dimensional manifold has dimension $n + {n+1 \choose 2}$, which is in particular odd whenever $n \equiv 2 \bmod 4$. So for example we can take $G = \text{SU}(2), G\_{\mathbb{C}} = \text{SL}\_2(\mathbb{C})$, which has real dimension $... | 2 | https://mathoverflow.net/users/290 | 146846 | 79,336 |
https://mathoverflow.net/questions/146020 | 11 | If $N$ and $N'$ are two closed hyperbolic 3-manifolds, then one would like to have an algorithm which determines whether or not $N$ and $N'$ are homeomorphic.
If $N$ and $N'$ are Haken, then such an algorithm is provided by Haken. The homeomorphism problem is solved in general by Sela's work on hyperbolic groups and ... | https://mathoverflow.net/users/2985 | The homeomorphism problem for hyperbolic 3-manifolds and the virtual Haken theorem | There is a bound on the size of the mapping class group of a hyperbolic manifold, namely the volume divided by the minimal volume of a hyperbolic 3-orbifold ([0.03905...](http://annals.math.princeton.edu/2012/176-1/p04)).
There is a (non-rigorous) algorithm to compute the isometry group by [Hodgson-Weeks](http://pro... | 10 | https://mathoverflow.net/users/1345 | 146851 | 79,339 |
https://mathoverflow.net/questions/146702 | 2 | You will have to excuse my lack of understanding of von Neumann algebras. I do not know if my question is trivial or nonsensical.
There are ITPFI factors of bounded type, and ITPFI factors of unbounded type. But is there anything in between? Specifically, is there a name for the class of factors which can be represen... | https://mathoverflow.net/users/8769 | ITPFI factors with restricted growth | See
Thierry Giordano and David Handelman, Matrix-valued random walks and variations on property AT, Münster J. Math 1 (2008) 15-72;
also the original paper by Giordano and Skandalis (cited in the one above), which I presume you are aware of. Bounded ITPF1 correspond to random walks corresponding to a sequence of P... | 3 | https://mathoverflow.net/users/42278 | 146855 | 79,341 |
https://mathoverflow.net/questions/146705 | 1 | Denote $f\_{\gamma}(x) =\frac { (1+\gamma)}{2} |x|^{\gamma}$. We consider:
$$I(\gamma) = \int\_{-1}^1\int\_{-1}^1 \ln (|x-y|) f\_{\gamma}(x) f\_{\gamma}(y) dx dy$$
I would like to know the limit of $I(\gamma)$ when $\gamma \to \infty$, and if this limit is infinite I would like to have an equivalent.
My first idea wa... | https://mathoverflow.net/users/16215 | limit of a singular integral | The exact result of the integral is:
$I(\gamma)=\frac{-2 \gamma -2 \pi (\gamma +1) \gamma \cot \left(\frac{\pi \gamma
}{2}\right)+(\gamma +1) \gamma \left(2 H\_{-\gamma -2}-H\_{-\frac{\gamma
}{2}-\frac{1}{2}}+H\_{-\frac{\gamma }{2}}-4 H\_{\gamma +1}+\log (4)\right)+2}{4
\gamma (\gamma +1)}$
where $H\_r$ are the ... | 1 | https://mathoverflow.net/users/42295 | 146857 | 79,342 |
https://mathoverflow.net/questions/146847 | 5 | Let $X\_k$ be the connected sum of $k$ projective planes. I am interested in necessary and sufficient conditions for the existence of a covering $\pi: X\_{k'} \to X\_k$, where $k$
and $k'$ are integers.
A necessary condition is that the Euler characteristic of $X\_{k'}$ is a multiple
of the Euler characteristic of $X... | https://mathoverflow.net/users/41156 | Sufficient condition for coverings between non-orientable surfaces | Your necessary condition, rephrased slightly, is sufficient. It should say that the Euler characteristic of $X\_{k'}$ is a positive multiple of $\chi(X\_k)$. This modification is to take care of the possibility that $X\_k$ is $RP^2$, which is not covered by any other non-orientable surface. If $\chi(X\_k)= \chi(X\_{k'}... | 8 | https://mathoverflow.net/users/3460 | 146858 | 79,343 |
https://mathoverflow.net/questions/146696 | 0 | Given a smooth projective variety $X \subset \Bbb{CP}^k$ why is it true that global sections of $O(l)|\_X, l >> 0 $ are just global sections of $O(l)$ on $\Bbb{CP}^k$ restricted to $X$?
Here $O(l)$ is just an appropriate power of the anti-tautological line bundle on $\Bbb{CP}^k$.
| https://mathoverflow.net/users/17965 | global sections of canonical line bundle of a projective variety | *This is an answer to the edited question*
Consider the short exact sequence
$$
0 \to \mathscr I\_X(l) \to \mathscr O\_{\mathbb P^k}(l) \to \mathscr O\_X(l) \to 0,
$$
and the long exact sequence of cohomology it induces:
$$
0 \to H^0(\mathbb P^k, \mathscr I\_X(l)) \to H^0(\mathbb P^k,\mathscr O\_{\mathbb P^k}(l)) \t... | 2 | https://mathoverflow.net/users/10076 | 146859 | 79,344 |
https://mathoverflow.net/questions/146527 | 1 | We know that the notion of [Jet bundle](http://ncatlab.org/nlab/show/jet+bundle) $J^kM×\mathbb{R}$, is generalization of cotangent bundle. What is the prequantization of $J^kM×\mathbb{R}$?
| https://mathoverflow.net/users/nan | pre-quantization of Jet bundle | In local covariant n-dimensional field theory, the jet bundle canonically supports a "pre n-plectic structure", namely that represented by the "pre-symplectic current density" which arises from the boundary term in the variation of the Lagrangian, in the sense of the variational bicomplex (for instance as introduced in... | 2 | https://mathoverflow.net/users/381 | 146866 | 79,347 |
https://mathoverflow.net/questions/141579 | 7 | Let $i:H \to G$ be a homomorphism of compact Lie groups. The induced representation $\iota\_\*V := \mathrm{Map}^H(G,V)$ of an $H$-representation $V$ does not give an element of the representation ring $R(G)$ of $G$ (because $\iota\_\*V$ is usually infinite-dimensional).
But $\iota\_\*V$ is an element of a larger grou... | https://mathoverflow.net/users/5206 | Representation ring and induced representation | I think that I have found an answer to my question. The definition ($\ast$) is not wrong, but not preferable. For a $G$-representation $M$, define $\mu^G\_M \in R.\!(G)$ by
$$\mu^G\_M(N) := \dim \mathrm{Hom}^G(N,M)\qquad(N\ \text{representation of }G)$$
By the universal property, $\mu^G\_M$ gives rise to a group homomo... | 1 | https://mathoverflow.net/users/5206 | 146872 | 79,348 |
https://mathoverflow.net/questions/146679 | 15 | It is well-known that expanders are hard to embed into Hilbert (or $\ell^p$) spaces - any embedding of an expander with $n$ vertices has distortion $\Omega(\log n)$.
Can anyone provide a reference (or a quick argument) that the same holds for any "nice" CAT(0) space (like the hyperbolic space) instead of Hilbert spac... | https://mathoverflow.net/users/2192 | Embedding expanders in CAT(0) spaces | Here is a class of Hadamard spaces in which, I think, no expander embeds:
Consider a Hadamard space $X$,
which is also "FDSCBB" in the sense
Burago, Gromov, and Perel'man (see [http://seven.ihes.fr/~gromov/PDF/3[86].pdf](http://seven.ihes.fr/~gromov/PDF/3%5B86%5D.pdf) section 7).
By Corollary 7.10 in [BGP] The tang... | 11 | https://mathoverflow.net/users/42307 | 146877 | 79,351 |
https://mathoverflow.net/questions/146885 | 0 | By the pointed canonical bundle formula the canonical bundle of $\overline{M}\_{g,n}$ is given by
$$K\_{\overline{M}\_{g,n}} = 13\lambda+\psi-2\delta-\sum\_{I}\delta\_{1,I}$$
where $\lambda$ is the Hodge class, $\psi = \sum\_{i=1}^{n}\psi\_{i}$ is the sum of the psi-classes and $\delta$ is total boundary divisor.
**A... | https://mathoverflow.net/users/14514 | Canonical bundle of the moduli space of curves | The cotangent bundle of $\overline M\_{g,n}$ is *never* ample. You can see this for instance by restricting it to the hyperelliptic locus.
There has been a lot of work in the last 5-10 years on running the (log) minimal model program on the moduli space of curves. You can read about this to get a feeling for how far... | 4 | https://mathoverflow.net/users/1310 | 146889 | 79,355 |
https://mathoverflow.net/questions/146617 | 0 | Let $(W,S)$ be a finite irreducible Coxeter-System of rank $n$ and $E$ be a real reflection representation of $W$. Let $x\in E$ and suppose that the isotropy group of $x$ is generated by one element in $S$. Now which are the subgroups of rank $n-1$ that do not stabilize $x$?
| https://mathoverflow.net/users/40054 | Subgroups of finite reflection groups that do not fix a point | You have to consider the Coxeter Graph: The subgroups you are looking for are those that you get by removing one edge...
| 0 | https://mathoverflow.net/users/42327 | 146916 | 79,363 |
https://mathoverflow.net/questions/146440 | 2 | Given a field $F$ of subsets of $\Omega$, we can define full conditional probabilities to be a function $P:F\times (F-\{ \varnothing \}) \to [0,1]$ such that:
1. $P(-|B)$ is a finitely-additive probability function for each $B\in F-\{\varnothing\}$
2. $P(A|B)P(B|C)=P(A|C)$ if $A\subseteq B\subseteq C$.
Suppose $G$ ... | https://mathoverflow.net/users/26809 | Two kinds of invariance of full conditional probabilities | The answer is negative and the counterexample is a lot easier than the two cases I considered in the question. (It would still be nice to have an answer to the two cases.)
Let $\Omega = \mathbb Z$, acting on itself by translation. Let $F$ be the algebra of finite unions of intervals $(a,b) \cap \mathbb Z$ for $a,b \i... | 0 | https://mathoverflow.net/users/26809 | 146917 | 79,364 |
https://mathoverflow.net/questions/146876 | 7 | Are there two models $M$ and $N$ for $\text{ZFC}$ such that:
**(1)** $M\subseteq N$
**(2)** $\aleph\_{1}^{N}=\aleph\_{1}^{M}$
**(3)** $\aleph\_{2}^{N}=\aleph\_{\omega +1}^{M}$
**Update:** According to Peter's useful answer it seems this problem is still open. So my question is about "partial" or "similar" resul... | https://mathoverflow.net/users/nan | A Special Pair of Models for ZFC (New Version) | As Péter indicates, this is open. It is related to a question that is a favorite of many. Shelah has conjectured that if $M\subseteq N$ are (proper class transitive) models of $\mathsf{ZFC}$ and $\kappa$ is a cardinal in $M$ with succesor $\lambda$, and $\lambda$ is a cardinal in $N$, then in $N$ we have that $\mathrm{... | 13 | https://mathoverflow.net/users/6085 | 146922 | 79,366 |
https://mathoverflow.net/questions/146914 | 8 | Is there any way to tell the number of distinct ways to factor $a\in\mathcal{O}\_k$ (up to units, of course) when $k$ is not a PID? A simple investigation in $\mathbb{Q}(\sqrt{-5})$ with integer ring $\mathcal{O}=\mathbb{Z}[\alpha]$ and Galois group $G=\langle\sigma\rangle$, with $\alpha = \sqrt{-5}$, gives, for exampl... | https://mathoverflow.net/users/4701 | Number of ways to write an integer as a product of irreducibles | You are asking for a formula for the number of decompositions $\mathfrak{m}=\mathfrak{a}\mathfrak{b}$, where $\mathfrak{a}$ and $\mathfrak{b}$ are principal ideals. Considering the group of characters $\widehat{\text{Pic}(\mathcal{O}\_k)}$, the number of these decompositions is
$$ \frac{1}{h\_k^2}\sum\_{\chi,\psi\in\wi... | 9 | https://mathoverflow.net/users/11919 | 146925 | 79,368 |
https://mathoverflow.net/questions/146923 | 6 | In 1934, Romanoff proved that the following set has positive lower density:
$$\displaystyle \mathcal{R}(x)= \{n \in \mathbb{N} : n \leq x, n = p + 2^k \}$$
where $p$ is a prime and $k \geq 0$ is a non-negative integer. In 2010 Lee proved the analogous result when powers of 2 are replaced by terms in the Fibonacci s... | https://mathoverflow.net/users/10898 | The density of numbers of the form $p + 2^k$ | The density is expected to exist, but this seems difficult.
As for question 2, the first explicit lower bound for $\liminf \frac{1}{x}\#\mathcal{R}(x)$ was established by Chen and Sun. I think the most up to date results are due (independently) to Pintz and to Habsieger and Roblot. The relevant references are
Pintz... | 9 | https://mathoverflow.net/users/16510 | 146926 | 79,369 |
https://mathoverflow.net/questions/146907 | 5 | Let $\mathcal{O}$ be the ring of integers in an algebraic number field and let $R \subset \mathcal{O}$ be an order. For instance, we might have $\mathcal{O} = \{\text{$x+i y$ $|$ $x,y \in \mathbb{Z}$}\}$ and $R = \{\text{$x+i y \in \mathcal{O}$ $|$ $y$ even}\}$. Question : Is $\text{SL}\_n(R)$ a finite-index subgroup i... | https://mathoverflow.net/users/42323 | Is SL_n of an order in a number ring finite-index in SL_n of the number ring? | Yes. If $f = (\mathcal O : R)$, then $f\mathcal O \subset R$ and consequently $\text{SL}\_n(R)$ contains $\ker (\text{SL}\_n(\mathcal O) \to \text{SL}\_n(\mathcal O/f\mathcal O))$, which has finite index because $\mathcal O/f\mathcal O$ is finite.
| 11 | https://mathoverflow.net/users/430 | 146929 | 79,370 |
https://mathoverflow.net/questions/146912 | 2 | I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
<http://www.unige.ch/~smirnov/papers/solenoid-j.pdf>
Smirnov says:
If T is an (n-1)-dimensional current with div(T)=0 ($\partial T=0$), t... | https://mathoverflow.net/users/42326 | (n-1)-dimensional normal currents and Smirnov's paper | Currents are distributions dual to smooth forms. I often get confused about the degrees, so I'm never sure what the standard meaning of an $m$-current. But according to Smirnov (Sec 1.4) an $m$-current is an element of the dual space of smooth $m$-forms. As is usual with distributions, differential operators on forms c... | 4 | https://mathoverflow.net/users/2622 | 146933 | 79,373 |
https://mathoverflow.net/questions/146908 | 2 | Is there a model of set theory in which:
1. Every projectively definable family of sets of reals has an OD or projectively definable member;
2. Every OD or projectively definable set of reals has the property of Baire.
PS: By a projectively definable family of sets of reals I mean:
There exists a formula $\varphi... | https://mathoverflow.net/users/38200 | Projectively definable family of sets of reals | Let me ignore the OD issue for a moment, and just prove that there is always a projectively definable family of sets of reals, with no projective member. Indeed, there is such a family consisting of a single set.
Namely, let $S$ be the full satisfaction relation on the reals for projective truth. Thus, $S$ consists ... | 6 | https://mathoverflow.net/users/1946 | 146961 | 79,384 |
https://mathoverflow.net/questions/146957 | 0 | Let's define a *discrete-analytic function* as a function that is equal to its Newton expansion:
$$f(x) = \sum\_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum\_{m=0}^{\infty} \binom {x}m \sum\_{k=0}^m\binom mk(-1)^{m-k}f(k)$$
Let's define a *natural function* such a function whose shift $f(x+n)$ is also des... | https://mathoverflow.net/users/10059 | Are all discrete-analytic funtions as defined here also natural? | **Updated** I think that for non-polynomials we need to restrict to non-negative $x$. With this restriction, are there discrete analytic functions which are **not** natural functions? I think not, at least under rather lax conditions.
Consider an arbitrary expansion $f(x)=\sum\_0^{\infty}a\_k \binom{x}{k}$ with the ... | 1 | https://mathoverflow.net/users/8008 | 146964 | 79,386 |
https://mathoverflow.net/questions/146497 | 1 | A **NEW DOUBT** arose in a previous question, so I am posting it again since the previous edits did not draw views.
Let $X\_n$ be random elements of $D$ (space of cad lag functions on $[0,1]$ as domain). $X\_n$ has asymptotically independents if $0\leq s\_1 \leq t\_1 \leq s\_2 \leq \ldots < s\_r \leq t\_r \leq 1$, th... | https://mathoverflow.net/users/nan | Billingsley Ch: $4$ Asymptotically Independent random elements | Take $s\_n:=s-1/n$: then $(X\_{s\_n},X\_t-X\_s)\to (X\_s,X\_t-X\_s)$ almost surely (by the property $\mathbb P\{X\in C\}=1$). Now we can check, uisng characteristic functions, that $X\_s$ and $X\_{t}-X\_s$ are independent.
A similar argument applies for more than two increments.
| 0 | https://mathoverflow.net/users/17118 | 146977 | 79,392 |
https://mathoverflow.net/questions/146973 | 5 | Consider the forcing notion(s) introduced by Friedman (or Mitchell or Neeman) for adding a club subset of $\omega\_2$ by finite conditions. In the generic extension CH fails, but I can't see the reals added by the forcing. Would you please give an explicit construction of $\aleph\_2$-many reals in the generic extension... | https://mathoverflow.net/users/11115 | An explicit construction of reals added after some forcing notions | Each of the posets you mention adds $\omega\_2$ many Cohen reals. Let $G$ be generic for any of the 3 posets you mentioned. The point is that any collection of $\omega\_1$-many reals in $V[G]$ can be captured by an intermediate extension of the form $V[G \cap M]$ where $M$ is a sufficiently elementary substructure of $... | 9 | https://mathoverflow.net/users/26319 | 146995 | 79,401 |
https://mathoverflow.net/questions/146902 | 0 | Let $\cal A$ be a Banach algebra with a bounded approximate identity, when a closed two sided ideal on it has a bounded approximate identity?
**In particular**, let $\cal B$ be a Banach algebra with a bounded approximate identity and $T:\cal A\to \cal B$ be an algebra homomorphism which is norm decreasing and surject... | https://mathoverflow.net/users/27066 | Bounded approximate identity and kernel of algebra homomorphism | **Let $\cal A$ be a Banach algebra with a bounded approximate identity, when a closed two sided ideal on it has a bounded approximate identity?**
This is a very nice problem that has been studied for special Banach algebras. I'm not sure if there is a "nice" characterization for a general Banach-algebra. Here is a li... | 2 | https://mathoverflow.net/users/17503 | 146996 | 79,402 |
https://mathoverflow.net/questions/146878 | 2 | Assuming that $p,q$ are probability distributions defined on the same support $\{x\_i\}\_{0 \leq i \leq n}$, $\epsilon$ a small real number, and $D\_{KL}$ the Kullback-Leibler divergence,
is there a method or an algorithm to find the set $\mathcal{P}\_{q, \epsilon}$ defined as :
$\mathcal{P}\_{q, \epsilon}= \{\ p\ ... | https://mathoverflow.net/users/29611 | Set of distributions that minimize KL divergence, | The cross-entropy method will easily allow you to approximate $\mathcal{P}\_{q,\epsilon}$ as an ellipsoid, which is likely reasonable if $\epsilon$ is small enough ($q$ is a global minimum so the hessian is semi definite positive around $q$)
The idea is to iteratively find a multivariate normal distribution that mini... | 0 | https://mathoverflow.net/users/8737 | 147001 | 79,404 |
https://mathoverflow.net/questions/147006 | 16 | While I was working on my paper I came across this question:
$\mathbf{Question}$. Suppose $(X,<)$ is an uncountable linear ordering of cardinality $\kappa$. Is there a subset $Y$ of $X$ such that $|Y|=\kappa$ and for any two elements $y\_{1},y\_{2}$ of $Y$ with $y\_{1}<y\_{2}$ there is an $x\in X$ such that $y\_{1}<x... | https://mathoverflow.net/users/27034 | Non-dense uncountable linear orderings | Define an equivalence relation $\sim$ on $X$ by $x \sim y$ if the interval between $x$ and $y$ is finite. Each $\sim$-equivalence class is either finite or countable. Therefore $|X/{\sim}| = \kappa$ since $\kappa$ is uncountable.
Any $Y \subseteq X$ that meets every ${\sim}$-class in exactly one point will be as req... | 18 | https://mathoverflow.net/users/2000 | 147011 | 79,409 |
https://mathoverflow.net/questions/147010 | 0 | Let $G$ be a reductive group defined over an algebraically closed field $k$ and $B$ be a fixed Borel subgroup of $G$. Suppose $X=Spec(R)$ is an affine scheme with $B$ rationally acting on it; hence $B$ acts on $R$. Then one can construct an induced $G$-algebra as follow:
$$ S=(k[G]\otimes R)^B. $$
Now, is it true that ... | https://mathoverflow.net/users/39715 | spectrum of an induced algebra | If $X=pt$, then $G\times^B X=G/B$ is the flag variety, which is not affine.
However we have $S=\Gamma(G\times^B X, \mathcal O)$ and hence $spec(S)$ is the affinization of $G\times^B X$.
| 4 | https://mathoverflow.net/users/2837 | 147014 | 79,410 |
https://mathoverflow.net/questions/147016 | 7 | Homotopy groups $\pi\_k$ were introduced before the homology groups $H\_k$ in the 1st topology book I read and the 1st topology course I took. Later on $\pi\_1$, $H\_k$, and $H^k$ appeared in numerous contexts in a variety of subjects.
On the other hand, I never encountered $\pi\_k$, $k\ge 2$ beyond the topology tex... | https://mathoverflow.net/users/38448 | Homotopy groups other than $\pi_1$ : what are they good for? | Here is a simple example: even if all you care about is computing $\pi\_1$, sometimes the easiest way to do it is to use the long exact sequence of a fibration, which requires knowing something about a $\pi\_2$ at least. For example, the standard (as far as I know) way to compute $\pi\_1(\text{SO}(n)), n \ge 3$ (an imp... | 15 | https://mathoverflow.net/users/290 | 147018 | 79,411 |
https://mathoverflow.net/questions/147003 | 2 | Consider an infinite integer lattice, or an infinite hexagonal lattice with unit length edges. Provided a set of $k$ possible vertex colors, is there a known largest possible minimum spacing that can be achieved between vertices of the same color? What about for three-dimensional integer lattice?
| https://mathoverflow.net/users/42372 | Achieving the largest possible minimum spacing between vertices of the same color in an integer lattice | Turning this question around, you can instead ask how many colours you need to ensure that no two points at distance at most $d$ receive the same colour. That is, add edges to your graph between all pairs of vertices at distance at most $d$: what is the chromatic number of this new graph?
If we join points at distanc... | 1 | https://mathoverflow.net/users/25485 | 147021 | 79,412 |
https://mathoverflow.net/questions/146974 | 5 | I apologize if this is too elementary for this site.
Given a closed subset, $X\subset \mathbb{R}^n$, given $X$, $X^C$ path-connected, show that any path-connected neighborhood of $X$, denoted $M$, has that $M-X$ is path-connected.
Both my professor and I are unable to solve it. It came up in the context of metric g... | https://mathoverflow.net/users/41103 | If a subset and its complement are path-connected, an neighborhood of the subset is path-connected | The answer seems to be yes.
Indeed, $M$ and $X^C$ are open subsets which cover $\mathbb{R}^n$ and whose intersection is $M \setminus X$. The Mayer-Vietoris sequence gives
$$H\_1(\mathbb{R}^n)\to H\_0(M\cap X^C)\to H\_0(M)\oplus H\_0(X^C) \to H\_0(\mathbb{R}^n)\to 0. $$
This sequence is isomorphic to
$$0 \to H\_0(M\s... | 6 | https://mathoverflow.net/users/39640 | 147025 | 79,416 |
https://mathoverflow.net/questions/147000 | 1 | Is the minimal resolution of Kleinian singularities of type $D\_k$ (i.e. the minimal resolution of singularities of the action of the binary dihedral group of order $4(k-2)$ on $ C^2$ simply connected? Is there a reference where their topology is described?
| https://mathoverflow.net/users/29850 | Simply connectedness of minimal resolution of Kleinian singularities | Yes it is simply connected. In general the retraction of $\mathbb C^2$ to $0$ will retract the resolution to the $0$ fiber, which is a tree of $\mathbb{CP}^1$s, hence homotopic to a wedge of $2$-spheres, in this case $k$ of them.
| 6 | https://mathoverflow.net/users/391 | 147035 | 79,422 |
https://mathoverflow.net/questions/104403 | 7 | I am looking for a book that studies the set of Hurwitz quaternions (HQ). In particular, I am interested in a connection between HQ and imaginary quadratic fields (IQF); quaternion orders $\mathcal{O}(\mu)$ and ideals of the form $n[a, b + \mu]$; information on similarities and differences between ideal theory in IQF a... | https://mathoverflow.net/users/22733 | Reference on ideal theory in Hurwitz quaternions | I feel bad for answering my own question, but after studying a topic for a while, my explorations resulted in a [research paper](http://arxiv.org/abs/1311.3379), which answers all of the questions above.
| 2 | https://mathoverflow.net/users/22733 | 147051 | 79,430 |
https://mathoverflow.net/questions/147055 | 11 | Given the *j-function*,
$$j(\tau)=\frac{1}{q}+744+196884q+21493760q^2+\dots$$
it is well-known that for $\tau=\tfrac{1+\sqrt{-d}}{2}$, positive integer $d$, then $j(\tau)$ is an algebraic integer of degree = *[class number](http://mathworld.wolfram.com/ClassNumber.html)* $h(-d)$. Thus,
$$j(\tfrac{1+\sqrt{-163}}{2... | https://mathoverflow.net/users/12905 | The j-function and Pell equations | So you're really asking when the ideal generated by $j(\tau)$ is the cube of an ideal. There's a famous paper of Gross and Zagier that describes the prime factorization of the ideal generated by the difference $j(\tau\_1)-j(\tau\_2)$ of two CM $j$-invariants. In your case $j(\tau\_2)=0$ is a CM value. This description ... | 11 | https://mathoverflow.net/users/11926 | 147059 | 79,433 |
https://mathoverflow.net/questions/147050 | 5 | The first element in the stable homotopy groups of a $K(\mathbb{Z}/2, n)$ (which is outside the range of the Freudenthal suspension theorem) is $\pi\_{2n} K(\mathbb{Z}/2, n) \simeq \mathbb{Z}/2$. In particular, there is a unique nontrivial stable map $S^{2n} \to K(\mathbb{Z}/2, n)$. This map is necessarily zero in coho... | https://mathoverflow.net/users/344 | The first element in the stable homotopy of a $K(\mathbb{Z}/2, n)$ | The mod 2 cohomology $H^\*(K(\mathbb{Z}/2,n))$ is freely generated over the Steenrod algebra by the canonical class $\iota\in H^n$ in degrees $\*\leq 2n$, and the only relation introduced in degree $2n+1$ is $Sq^{n+1}(\iota)=0$. Thus the Adams spectral sequence for $K(\mathbb{Z}/2,n)$ has classes in bidegrees $(0,n)$ (... | 8 | https://mathoverflow.net/users/75 | 147064 | 79,434 |
https://mathoverflow.net/questions/147048 | 4 | Start with an edgeless graph on $n$ labeled vertices, and note that the automorphism group is $\Sigma\_n$, the symmetric group on $n$ elements. Now imagine that we randomly start throwing in all of the $\binom{n}{2}$ edges one at a time.
>
> How does the automorphism group change?
>
>
>
The first edge that go... | https://mathoverflow.net/users/18263 | Tracking automorphism groups of graph processes | There is no simple exact answer to this question. The harmonic mean of the automorphism group sizes is related to the ratio between the labelled and unlabelled graph counts (since the number of graphs in the isomorphism class of $G$ is $n!/|\mathrm{Aut}(G)|$), so the theory of unlabelled enumeration (Polya, etc) provid... | 4 | https://mathoverflow.net/users/9025 | 147066 | 79,435 |
https://mathoverflow.net/questions/147057 | 3 | In the textbook from which I am teaching a Discrete Math course, the authors propose randomly generating an infinite sequence of decimal digits $d\_1, d\_2, \dots$. We are to think of this as the decimal expansion of a real number in the unit interval.
They propose that "it is overwhelmingly likely" that the resulti... | https://mathoverflow.net/users/468 | Can this informal argument (for the fact that almost all reals in the unit interval are irrational) be saved? | You can make sense of the uniform probability distribution on lots of infinite sets, notably any compact topological group $G$, where "uniform probability distribution" should mean "normalized [Haar measure](http://en.wikipedia.org/wiki/Haar_measure)." Here the group is an infinite direct product of copies of $\mathbb{... | 16 | https://mathoverflow.net/users/290 | 147072 | 79,437 |
https://mathoverflow.net/questions/146999 | 4 | This is taken from Timmermann's [Invitation to Quantum Groups and Duality](http://www.ems-ph.org/books/book.php?proj_nr=72).
Hi folks I am struggling a little with a small calculation in the above text.
I will just get right into it.
**Lemma 3.2.5**
*Let $(A,\Delta)$ be a Hopf $\*$-algebra with a normalised inte... | https://mathoverflow.net/users/35482 | Towards a quantum version of Schur's orthogonality relations | The formula should be
$$\begin{align}
Y^{-1}(R\otimes 1)X&=Y^{-1}(I\otimes h)(Y^{-1}(Q\otimes 1)X)X
\\&=(I\otimes h\otimes I)\left(Y\_{[13]}^{-1}Y\_{[12]}^{-1}(Q\otimes 1\otimes 1)X\_{[12]}X\_{[13]}\right)
\end{align}$$
i.e. the $Q$ in the first term should be an $R$.
The first equality is just the substitution o... | 3 | https://mathoverflow.net/users/36090 | 147074 | 79,438 |
https://mathoverflow.net/questions/147026 | 13 | If we extend the action of $\pi\_1(\Sigma\_g), g\geq 2,$ from $\mathbb{H}^2$ to its boundary $\partial\_{\infty}\mathbb{H}^2=S^1$, the surface bundle corresponding to this action of $\pi\_1(\Sigma\_g)$ on $S^1$ has a leaf that is isometric to $\mathbb{H}^2$. Hence $\mathbb{H}^2$ is even isometric to a leaf in a codimen... | https://mathoverflow.net/users/42384 | Is $\mathbb{H}^n$ quasi-isometric to a leaf of a codimension 1 foliation of a compact manifold? | One may generalize your 3-dimensional example to all dimensions in some sense.
There are compact hyperbolic $n$-manifolds which have orderable fundamental group, since the fundamental groups embed in a right-angled Artin group (which is in fact [bi-orderable](http://en.wikipedia.org/wiki/Linearly_ordered_group)), by... | 8 | https://mathoverflow.net/users/1345 | 147082 | 79,440 |
https://mathoverflow.net/questions/147090 | 7 | $\text{ZFC}$ proves that each $\aleph\_{n}$ for $n\in \omega$ is a regular cardinal. But it seems without the Axiom of Choice there are many consistent possible choices for cofinality of such cardinals.
**Please introduce some references for any known consistency result with $\text{ZF}$ about possible cofinalities of... | https://mathoverflow.net/users/nan | Possible Choices for Cofinality of $\aleph_n$ without Choice | Here are some partial answers:
1) By a result of Gitik, all $\aleph\_n$'s can have cofinality $\omega.$
2) the paper "Cofinality and measurability of the first three uncountable cardinals" by Apter, Jackson and Löwe completely solves the problem for $\aleph\_1, \aleph\_2$ and $\aleph\_3.$ The following is taken fro... | 7 | https://mathoverflow.net/users/11115 | 147092 | 79,443 |
https://mathoverflow.net/questions/147093 | 3 | In [HTT](http://www.math.harvard.edu/~lurie/papers/croppedtopoi.pdf).5.5.8.18 Lurie defines a projective object $P$ in a quasicategory $\bf C$ as an object such that its corepresented functor ${\rm Map}(P,-)$ "commutes with geometric realizations". I can catch the general idea that something like $\hom(P,|S|\_{X^\*}) \... | https://mathoverflow.net/users/7952 | Projective objects in HTT | The term geometric realization is used in HTT to refer to colimits indexed by $\Delta^{op}$. So an object $P \in \mathcal{C}$ is projective if and only if, for every simplicial object $X\_{\ast}$ in $\mathcal{C}$, the canonical map
$$ \varinjlim \text{Map}(P, X\_{\ast} ) \rightarrow \text{Map}(P, \varinjlim X\_{\ast} )... | 17 | https://mathoverflow.net/users/7721 | 147096 | 79,445 |
https://mathoverflow.net/questions/147110 | 7 | It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk.
I am pretty sure that the answer to the following is positive and well-known (to experts) and is somewhere in the literature:
Question. Are there (non-affine) surje... | https://mathoverflow.net/users/39654 | Surjective entire functions without critical points | Picards theorem will do the trick. Take $$h(z)=z\cdot e^{\int\_0^zf(t)dt}$$ where $f(z)=\frac{e^z-1}{z}.$ Then, clearly $$h'(z)=e^{\int\_0^zf(t)dt}(1+zf(z))=e^{\int\_0^zf(t)dt+z}\ne 0$$ and $h$ is locally univalent. If $h(z)$ is not surjective, then by Picard it omits just one complex value $A$ and we can write $h(z)=e... | 9 | https://mathoverflow.net/users/17503 | 147111 | 79,451 |
https://mathoverflow.net/questions/147098 | 17 | For any group $G$, cohomology can be viewed as a functor
$$
H^\ast(G,-): G{\sf\text{-}mod}\to {\sf GrAbGrp},
$$
where $G{\sf\text{-}mod}$ denotes the category of (left) $\mathbb{Z}[G]$-modules and ${\sf GrAbGrp}$ denotes the category of (non-negatively) graded abelian groups.
It seems that there are non-isomorphic gr... | https://mathoverflow.net/users/8103 | Do there exist non-isomorphic groups with the same cohomology? | The answer would seem to be yes if I understood your question properly. The paper <http://www.m-hikari.com/ija/ija-password-2009/ija-password9-12-2009/ladraIJA9-12-2009.pdf> shows that if $\mathbb ZG$ is isomorphic to $\mathbb ZH$, then one can choose an isomorphism which is augmentation preserving. They then show that... | 15 | https://mathoverflow.net/users/15934 | 147116 | 79,453 |
https://mathoverflow.net/questions/146935 | 7 | Does the equation $x^m-1=y^n+y^{n-1}+...+1$ have only finitely many solutions $(x,y,m,n)$ where $x,y$ are prime powers with $y>2$ and $m,n$ are integers with $m,n>1$?
This question arose in the study a group theory problem, where I'm examining a "natural" subgroup $G$ of $S\_d$ where $d:=x^m-1$. I can show that $G$ i... | https://mathoverflow.net/users/30412 | The equation $x^m-1=y^n+y^{n-1}+...+1$ in prime powers $x,y$ | Mike Bennett's comments gives good reason to suspect that my original question is beyond what can be proved with existing techniques. Here I record one thing that can be proved (again thanks to Mike's comment), namely that there are no solutions with $x=4$. That is:
the equation $4^m-1=y^n+y^{n-1}+...+1$ has no solutio... | 1 | https://mathoverflow.net/users/30412 | 147123 | 79,455 |
https://mathoverflow.net/questions/147102 | -2 | Suppose $w \in C^2 (S^{n-1}), \Lambda$ is Laplace-Beltrami operator on the sphere $S^{n-1}$, How can I prove follow Poincare inequality :
$\int\_{S^{n-1}} w\Lambda w d\sigma \leq (1-n) \int\_{S^{n-1}} |w|^2 d\sigma$
Remark: Set $w(x)=w(r,\theta), (r, \theta)$ are the polar coordinates in $R^n$, we get the following f... | https://mathoverflow.net/users/38739 | A Poincare inequality for the Laplace-Beltrami operator | Here is an elementary paper of Seeley that might help
<http://www.jstor.org/stable/2313760>
| 0 | https://mathoverflow.net/users/20302 | 147126 | 79,456 |
https://mathoverflow.net/questions/147112 | 7 | In knot theory, two link diagrams are equivalent if and only if they can be related by performing a finite number of Reidemeister moves. But sometimes it is so confusing that I don't know which type move should I perform on link to get desired result. Is there an efficient procedure to relate one link diagram to anothe... | https://mathoverflow.net/users/42411 | How do I efficiently find a sequence of Reidemeister moves between equivalent link diagrams? | In 2011 Coward-Lackenby proved the following, and they also provided an upper bound on the number of moves.
>
> There is a computable function F : N×N → N such that for any two connected
> diagrams D1 and D2 of a link with n1 and n2 crossings, there is a sequence of
> at most F(n1, n2) Reidemeister moves that ta... | 7 | https://mathoverflow.net/users/41219 | 147128 | 79,457 |
https://mathoverflow.net/questions/57529 | 4 | It is well known the existence of a T duality between the two heterotic string theories, $SO(32) \sim\_T E\_8 \times E\_8$. Beyond the trivial point that both groups have the same dimension (496, which actually is a prerequisite), is there some other mathematical relation between them?
I am thinking in other SO(N) gr... | https://mathoverflow.net/users/4037 | Does $SO(32) \sim_T E_8 \times E_8$ relate to some group theoretical fact? | The answer to this question can be found in Lubos Motl's answer to [this question of mine on Physics.SE](https://physics.stackexchange.com/q/65092/23119).
The key here are the weight lattices bosonic representations $\Gamma$ of these gauge groups.
As I understand it, the weight lattice of $E(8)$ is $\Gamma^8$, whe... | 6 | https://mathoverflow.net/users/36148 | 147129 | 79,458 |
https://mathoverflow.net/questions/147088 | 5 | Here is a power series, which looks a bit like a Hypergeometric function series, but I don't think that it is. Has anyone any idea what it is? Here $n,p,r$ are integers with $n\ge 0$ and $p\ge r\ge 0$:
$$
f\_{n,p,r}(x)\,=\,\sum\_{s=0}^p \frac{x^s}{s!}\ \frac{p!\,(2n+p+s+2)!\,(n+r+s+2)!}{(p-s)!\,(2n+r+s+3)!\,(n+s+2)!}... | https://mathoverflow.net/users/29625 | Identifying a special function from its power series | Henry is almost right; the sum can be written as
$$\frac { \left( 2n+p+2 \right) !\, \left( n+r+2 \right)!}
{ \left( 2n+r+3 \right) !\, \left( n+2 \right) !}
{}\_{3}F\_{2}\left({{2n+p+3,n+r+3,-p\,\,\,}\atop {2n+r+4,n+3}}\Bigm |-x\right)$$
For $x=-1$ (no need to assume $r\le p$) the sum can be evaluated by Saalschütz'... | 7 | https://mathoverflow.net/users/10744 | 147137 | 79,461 |
https://mathoverflow.net/questions/147032 | 12 | The geometrization theorem tells us:
>
> **Theorem (Thurston)** The mapping torus $M\_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S\_g \rightarrow S\_g$ from a genus $g$ surface to itself admits a complete hyperbolic metric of finite volume.
>
>
>
[Otal](http://books.google.ca/books?isbn=0821821539) has a c... | https://mathoverflow.net/users/42387 | Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$? | To address your last point, for a mapping torus of $S\_g$, there is an upper bound on the Heegaard genus of $2g+1$, which is sharp (in fact, [$M\_{\phi^n}$ will have rank $2g+1$ for $n$ large](http://www.msp.warwick.ac.uk/gtm/2008/14/p022.xhtml)), but is not always an equality. However, there is no upper bound on the v... | 6 | https://mathoverflow.net/users/1345 | 147142 | 79,465 |
https://mathoverflow.net/questions/147095 | 3 | I was wondering if there is any stationary distribution for bipartite graph? Can we apply random walks on bipartite graph? since we know the stationary distribution can be found from Markov chain, but we have two different islands in bipartite graph and connections occur between nodes from different groups.
| https://mathoverflow.net/users/42403 | Stationary distribution for bipartite graph | There is a distribution such that, if you start a random walk in that distribution, then it will remain there for all time: the usual formula, $\pi(v) = d(v)/2e(G)$, still works. What is no longer true is that the random walk will converge to the stationary distribution as time tends to infinity independent of the star... | 6 | https://mathoverflow.net/users/25485 | 147145 | 79,466 |
https://mathoverflow.net/questions/147141 | 4 | Are there any known non-trivial sufficient conditions, or full characterizations, of a totally right-preorderable group?
More precisely:
* **totally right-preorderable**: has a non-trivial total right-preorder
* **non-trivial total right-preorder**: transitive and symmetric relation ("preorder") $\le$ with $a\le b$... | https://mathoverflow.net/users/26809 | Totally right preorderable groups | A group $G$ admits a nontrivial preorder iff it admits a nontrivial order-preserving action on a totally ordered set (which can be chosen to be $\mathbf{Q}$ or its completion $\mathbf{R}$ if $G$ is countable).
Proof:
[I like left actions rather than right actions so I'll go ahead with left-invariant instead of right-... | 8 | https://mathoverflow.net/users/14094 | 147146 | 79,467 |
https://mathoverflow.net/questions/147136 | 5 | Let $f(x) = \exp(x)$ and $(\xi\_i)\_{i=0}^\infty, \, \xi\_i \in (0,1)$ be a sequence of points from the unit interval.
For $n \in \mathbb{N}$ let $P\_n$ be a polynomial of degree $n$ that interpolates $f$ at $\xi\_0, \ldots, \xi\_n$, i.e.
$$
f(\xi\_i)-P\_n(\xi\_i) = 0, \text{ for all } i=0,\ldots,n,
$$
**Question**... | https://mathoverflow.net/users/40117 | Do interpolation nodes have to be dense? | Unfortunately not. Just look at the (confluent) limit case when all $\xi\_i=0$: in this case $P\_n$ is just the $n$-th Taylor polynomial for $f$. which clearly converges to $f$ not only on $(0,1)$, but on the whole complex plane, but $\xi\_i$ are all at the origin.
A less extreme argument is: if we consider the sequ... | 2 | https://mathoverflow.net/users/7482 | 147157 | 79,473 |
https://mathoverflow.net/questions/147174 | 4 | On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ depending on sign convention. I know that under various curvature assumptions one can show that the $L^p$-spectrum is equal to t... | https://mathoverflow.net/users/nan | Spectrum of the Laplace-Beltrami operator on $L^p$: where is it? | The answer has a lot to do with the off-diagonal decay of the resolvent, (\Delta - \lambda)^{-1}. There are two rather different cases to consider, when M is R^n and when M = H^n. The former is [0,\infty), just as for L^2. This can be computed explicitly, but in fact there is a general theorem due to Sturm which states... | 6 | https://mathoverflow.net/users/17969 | 147175 | 79,479 |
https://mathoverflow.net/questions/147165 | -1 | which submodule of FG-module of a lie algebra $L$ will be determined I want to check that how we can find out a classical lie algebra like $D\_4$ and $E\_6$ are irreducible?
| https://mathoverflow.net/users/40491 | irreducible Classical Lie algebras | To clarify what is going on in such special cases, it will help to have two specific references:
G.M.D. Hogeweij, Almost-classical Lie algebras. I, II. Nederl. Akad. Wetensch. = Indag. Math. 44 (1982), no. 4, 441–452, 453–460.
Gerhard Hiss, Die adjungierten Darstellungen der Chevalley-Gruppen. [The adjoint represen... | 3 | https://mathoverflow.net/users/4231 | 147176 | 79,480 |
https://mathoverflow.net/questions/147160 | 2 | The usual axiomatization of a topological space (in the sense of Bourbaki) goes by declaring certain subsets as being open and such that a few axioms are fulfilled by the family of open subsets.
It is trivial to get an equivalent axiomatization of a topological space in terms of declaring certain subsets as being clo... | https://mathoverflow.net/users/1841 | Axiomatization of locally compact Hausdorff spaces via compact subspaces | This is as of now probably more of a really long comment than a complete answer.
There is a one-to-one correspondence between compact Hausdorff spaces with distinguished dense subspaces and structures called proximity spaces. Similarly, there is a more general correspondence between locally compact Hausdorff spaces w... | 2 | https://mathoverflow.net/users/22277 | 147177 | 79,481 |
https://mathoverflow.net/questions/147163 | 1 | There are many alternative compactifications of $M\_{g,n}$ which live naturally under the classical Deligne-Mumford compactification $\overline{M}\_{g,n}$.
For instace the moduli spaces of weighted curves $\overline{M}\_{g,A[n]}$ where $A=(a\_{1},...,a\_{n})$ is a vector of rational weights $0<a\_{i}\leq 1$. These spa... | https://mathoverflow.net/users/14514 | Moduli spaces admitting birational morphisms over moduli spaces of curves | Of course since every space is a moduli space, one can get a positive answer to this question by simply blowing up $\overline{M}\_{g,n}$.
But here is something more in the spirit of what you are asking. How about the moduli space of pairs $((C,x),p)$ consisting of a stable 1-marked, genus $g$ curve $(C,x)$ along wit... | 1 | https://mathoverflow.net/users/9617 | 147182 | 79,485 |
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