parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/122725 | 13 | I believe that the following questions are very basic, but I don't know how to get a reference.
Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is
unibranch at zero (i.e. analytically irreducible). Then I guess one should be able to define "arithmetic genus defect" of the c... | https://mathoverflow.net/users/13441 | "Arithmetic genus" of a plane curve singularity. | The difference between the geometric genus of the singularity and the geometric genus of a
smoothing (this one being called the arithmetic genus of the singularity) is often called
the *delta invariant*. If $A$ is the local ring of the singularity, $B$ its normalization, then the delta invariant is the dimension of the... | 13 | https://mathoverflow.net/users/25309 | 122738 | 68,921 |
https://mathoverflow.net/questions/122728 | 6 | Suppose that $\mathcal{C}$ is a cartesian closed category. When is the category of abelian group objects $\mathcal{Ab}(\mathcal{C})$ a symmetric monoidal closed with respect to something substituting tensor product in the classical case $\mathcal{C}=\mathcal{Set}$?
I need reference or sketch of an argument in the cas... | https://mathoverflow.net/users/30916 | abelian group objects category | I [previously](https://mathoverflow.net/questions/114703/why-does-tensor-product-in-abv-require-colimits-in-v/114708#114708) indicated how to construct the tensor product for $\textbf{Ab}(\mathcal{V})$ when $\mathcal{V}$ is a sufficiently nice cartesian closed category. (The comments are probably more helpful than the ... | 5 | https://mathoverflow.net/users/11640 | 122742 | 68,923 |
https://mathoverflow.net/questions/122736 | 3 | For the projective line $CP^1$, its cohomology ring has a single generator. Moreover, this generator is given by the cohomology class of the fundamental form associated associated to the Fubini--Study Kahler metric on $CP^1$. If I have understood correctly this also holds for all the higher projective spaces.
Now the... | https://mathoverflow.net/users/31696 | Kahler Metric Fundamental Forms and Cohomology Ring Generators | Antonio,
The answer to first question is no. Let $F$ be the variety of full flags in $\mathbb{C}^3$. This can be viewed as variety of
pairs $F=\lbrace (p,\ell)\in \mathbb{C}\mathbb{P}^2\times \mathbb{C}\check{\mathbb{P}}^2\mid p\in \ell\rbrace$. There are two generators for cohomology given by pulling back the Kaehle... | 5 | https://mathoverflow.net/users/4144 | 122743 | 68,924 |
https://mathoverflow.net/questions/122346 | 4 | The best rank $r$ approximation to a given matrix $M$ in Frobenius norm, according to Eckart-Young theorem, is truncated SVD - just keep $r$ largest singular values. What if I need to construct best rank $r$ approximation in a different matrix norm, for example quadratic $\|M\|^2 = \sum\_{i,j}S\_{ij}M\_{ij}^2$, for som... | https://mathoverflow.net/users/31589 | best rank r approximation for non-Frobenius norm | Your problem is what is commonly known as **weighted low-rank matrix approximation.** This problem has not received as much interest as it deserves. A good starting point on this problem is the ["Weighted low-rank approximations"](http://www.yaroslavvb.com/papers/srebro-weighted.pdf) paper by Srebro and Jaakola. They o... | 4 | https://mathoverflow.net/users/8430 | 122745 | 68,926 |
https://mathoverflow.net/questions/122748 | 18 | There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$,
* a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$
* or an almost complex structure is a section $J$ of $\textrm{End}(TM)$ which is everywhere an anti-involution (*i.e.* ... | https://mathoverflow.net/users/18974 | what is a spinor structure? | Chapter 9 of *Elements of Noncommutative Geometry,* by Gracia-Bondia, Varilly, and Figueroa, has this perspective on spin$^c$ and spin structures.
The way to think about this algebraically is that the module of (continuous, say) sections of a spinor bundle over a (compact, Riemannian) manifold $M$ is Morita equivale... | 11 | https://mathoverflow.net/users/703 | 122750 | 68,928 |
https://mathoverflow.net/questions/122723 | 1 | What is the number of families of $4t+1$ subsets of order $2t$ of the set $\{1,2, \ldots ,p\}$, where $p$ is a prime number which is equal to $4t+1$ and the order of intersection of each pair of the subsets is $t$ or $t-1$ and each subset has $t$-intersection with $2t$ subsets and $t-1$-intersection with the other $2t$... | https://mathoverflow.net/users/31179 | The number of specific structure | Yes, equivalently, the problem can be stated as follows: We are looking for the number of $p\times p$ 0-1-matrices $A$ with all row sums $2t$ such that $AA^T=tJ+tI-B$, where $B$ a 0-1-matrix with also all row sums equal to $2t$. (Here, $J$ denotes the all-1-matrix).
It is not hard to find *circulant* matrices that fu... | 2 | https://mathoverflow.net/users/29783 | 122753 | 68,929 |
https://mathoverflow.net/questions/122752 | 3 | I'm attempting a proof by induction and, for the inductive step, it would be very useful for me to have some control on a Folner sequence. Indeed, let $G$ be a finitely generated amenable group, fix a finite symmetric set of generators for $G$ and denote by $B\_n$ the ball of radius $n$ in the Cayley graph of $G$ with ... | https://mathoverflow.net/users/24891 | Existence of nice Folner sequences | This is not possible for $\mathbb{Z}\wr \mathbb{Z}$ and (most) other groups. For the structure of Folner sets there, see, in particular, Erschler, Anna
On isoperimetric profiles of finitely generated groups. (English summary)
Geom. Dedicata 100 (2003), 157–171. and references there.
| 6 | https://mathoverflow.net/users/nan | 122754 | 68,930 |
https://mathoverflow.net/questions/122551 | 3 | I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization.
The Statement
-------------
Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local homeomorphism (this context uses local homeomorphisms in place of open subset sets and covers of open sets). Given an ... | https://mathoverflow.net/users/19926 | Twisting an object P by an H-Torsor I | I am going to put as much detail as possible in this answer without writing any diagrams (too many diagrams!).
In the Sketch of the Proof as outlined above, I was using local homeomorphisms instead of open subsets and open covers. I will switch to open (sets/covers) in this solution for simplicity and variety.
Let ... | 3 | https://mathoverflow.net/users/19926 | 122757 | 68,931 |
https://mathoverflow.net/questions/122759 | 6 | In the classic paper [On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field](http://www.jstor.org/discover/10.2307/1970825?uid=3737864&uid=2129&uid=2134&uid=2&uid=70&uid=4&sid=21101716697401), Quillen proved (Theorem 6):
>
> $H^i(GL(n,p^d),\mathbb{F}\_p)=0$ for $0 < i < d(p-1)$ and all $n... | https://mathoverflow.net/users/27895 | Mod-p cohomology of $GL(n,p^d)$ | There are some results known in this direction; see the paper [*On the vanishing ranges for the cohomology of finite groups of Lie type*](http://dx.doi.org/10.1093/imrn/rnr130), by Christopher Bendel, Daniel Nakano, and Cornelius Pillen (Int. Math. Res. Not. (2012), 2817-2866). In particular, they show that if $p \geq ... | 6 | https://mathoverflow.net/users/7932 | 122761 | 68,932 |
https://mathoverflow.net/questions/122764 | 4 | Hello all.
Suppose $X$ is a Polish space, $\mu$ is a Borel probability measure on $X$, and $T:X \to X$ is a continuous $\mu$-preserving map which is *not* ergodic.
Does there necessarily exist a Borel set $A \subset X$ such that
* $\mu(A) \in (0,1)$;
* $\mu(A \ \triangle \ T^{-1}(A)) = 0$;
* $A$ has non-empty int... | https://mathoverflow.net/users/15570 | Characterising ergodicity of continuous maps | Let $T \colon X \to X$ be a minimal transformation of a compact metric space which is not uniquely ergodic, let $\mu$ be a non-ergodic $T$-invariant measure on $X$, and let $A$ be a set with nonempty interior such that $\mu(A \triangle T^{-1}A)=0$. I claim that necessarily $\mu(A)=1$, contradicting the above conjecture... | 5 | https://mathoverflow.net/users/1840 | 122769 | 68,938 |
https://mathoverflow.net/questions/122765 | 3 | On page 156 of Milne's Class field theory notes available online [here](http://www.jmilne.org/math/CourseNotes/cft.html), he claims that the Hilbert class field of $K = \mathbb Q(\sqrt{-6})$ is the splitting field of $x^2+3$ but I don't believe so.
The prime 5 does not ramify in $K$ but does so in $L = \mathbb Q(\sq... | https://mathoverflow.net/users/2720 | Exercise in Milne's CFT notes | How did you determine that the index $(\mathcal O\_L : M)$ is $8$? It seems to me that it's actually $160$, which is divisible by $5$.
Incidentally, a quick way to see that Milne is correct is to note that the discriminant of $K$ is $-4\cdot 6$ and $6$ is an idoneal number: this means that the Hilbert class field of ... | 7 | https://mathoverflow.net/users/430 | 122771 | 68,939 |
https://mathoverflow.net/questions/122760 | 10 | In a well known 1973 paper, [Fischer and Marsden](http://www.numdam.org/numdam-bin/fitem?id=AIHPA_1980__33_2_147_0) pointed out (with similar, contemporary remarks made in the physics literature by [Brill and Deser](http://projecteuclid.org/euclid.cmp/1103859178)) that the space of solutions of some non-linear partial ... | https://mathoverflow.net/users/2622 | Linearization instability and singular points of algebraic varieties | Maybe the best answer I can give is ''learn about deformation theory''. But I will take a shot at writing the dictionary you request, in the case where $E(u)$ is a polynomial function
* linearization stability <--> smoothness / instability <--> singularity
* linearization stable/unstable point <--> singular / smooth ... | 7 | https://mathoverflow.net/users/4707 | 122775 | 68,940 |
https://mathoverflow.net/questions/122776 | 4 | Let $\Gamma$ be a lattice in a (real or p-adic) Lie group.
Is it true that for a given natural number $n$ there exists a finite index subgroup $\Sigma\subset\Gamma$ such that each $\sigma\in\Sigma$ is an $n$-th power of some element of $\Gamma$?
In other words, is it true that for given $\sigma\in\Sigma$ there exists... | https://mathoverflow.net/users/nan | Property of lattices in Lie groups | The answer is no, even in higher rank groups. For example, take $\Gamma = SL\_3({\mathbb Z})$. If such a $\Sigma $ existed for any $n$, then its completion in the profinite (same as congruence) completion of $\Gamma$ would have this property that every element in $\Sigma$ would be an $n$-th power. But the completion of... | 8 | https://mathoverflow.net/users/23291 | 122791 | 68,946 |
https://mathoverflow.net/questions/122650 | 3 | I would like to know if there can be some kind of classification of normal rational surfaces with Gorenstein singularities, such that their canonical divisor is effective.
**Additional question.** Are there such surfaces at all?
I could imagine constructing such a surface by blowing up several points on an ellipti... | https://mathoverflow.net/users/13441 | A classification of rational surfaces with effective $K$ | Here is an example (I hope!).
Take $X$ a double cover of $\mathbb P^2$ branched over a normal sextic $B$. It is a normal Gorenstein surface and the standard formulae for double covers give $K\_X=0$.
Now assume that $B$ has an ordinary quadruple point $P$ and is smooth elsewhere, so that $X$ has an elliptic Gorenstei... | 8 | https://mathoverflow.net/users/10610 | 122799 | 68,951 |
https://mathoverflow.net/questions/122800 | 4 | These are really two questions, but the second presupposes the first.
First, let $( B\_i )\_{i\in I}$ be an arbitrary family of Boolean algebras. I want to directly form a product of them that is like the product topology on the product of their Stone spaces, ideally without using AC (or BPI, pun not intended). I hav... | https://mathoverflow.net/users/26809 | Products of Boolean algebras and probability measures thereon | This is called the "direct sum" or "internal sum" of Boolean algebras. See *Introduction to Boolean Algebras* by Givant and Halmos, p. 427 for the abstract definition and p. 432 for the concrete description you want (the elements of the internal sum are finite joins of finite meets of elements and complements of elemen... | 3 | https://mathoverflow.net/users/23141 | 122802 | 68,952 |
https://mathoverflow.net/questions/122818 | 2 | Let $A$ be a $k \times k$ invertible matrix over complex numbers.
If it possible to write its nth root as an analytic function (i.e. power series in $A$)?
EDIT: Complex coefficients can be functions of $A$.
Notes
-----
If a matrix $A$ has only one eigenvalue $\lambda$, then it is simple. We take
$$B = \exp\le... | https://mathoverflow.net/users/9093 | Nth root of a matrix as an analytic function? | If I am reading this correctly, you are fine with a power series whose (scalar) coefficients depend on the matrix $A$. In this case, it suffices to take a polynomial $p$ that interpolates $\sqrt[n]{x}$, such that for each eigenvalue $\lambda$ with multiplicity $k\_\lambda$, the first $k\_\lambda-1$ derivatives of $p$ c... | 3 | https://mathoverflow.net/users/1898 | 122823 | 68,963 |
https://mathoverflow.net/questions/122807 | 2 | The following is from Stein and Shakarchi's *Complex Analysis*:
>
> For each $a>0$ we denote by ${\mathcal F}\_a$ the class of all functions $f$ that satisfy the following two conditions:
>
>
> 1. The function $f$ is holomorphic in the horizontal strip
> $$S\_a=\{z\in{\Bbb C}:|\Im(z)|<a\}$$
> 2. There exists a co... | https://mathoverflow.net/users/nan | Class of functions in which the Fourier inversion holds | This space has no name (no that I know), and it is probably invented by Stein and Shakarchi for
some pedagogical purpose, or for convenience of exposition.
As Serge says it is not contained in the Schwartz space $S$ (of test functions). Neither it contains the Schwartz space $S$.
It is contained in the space of Sch... | 5 | https://mathoverflow.net/users/25510 | 122836 | 68,973 |
https://mathoverflow.net/questions/122770 | 2 | Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices $A$ for which there is a diagonal matrix $D=\mathrm{diag}(d\_1,\ldots,d\_n)$ with integer entries such that $DA$ is symmetric).
It is sometimes useful to insist that $\gcd(d\_1,\ldots,d\_n... | https://mathoverflow.net/users/4366 | A question on Lusztig's `graph with automorphism' construction? | I think what you're looking for is the observation that if you do have a non-trivial gcd, then you can replace $\Gamma$ and $a$ with another pair that give the same answer and have gcd 1.
After all, let $\ell$ be the quotient $\operatorname{lcm}(d\_1,\dots, d\_n)/\operatorname{gcd}(d\_1,\dots, d\_n)$, and consider t... | 1 | https://mathoverflow.net/users/66 | 122848 | 68,979 |
https://mathoverflow.net/questions/122859 | 6 | Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind which - by $M$. More precisely, for a function $f$ on $X$, define
$$ Mf(x) := \sup\_{B \ni x} \frac{1}{\mu(B)} \int\_B f(y) ... | https://mathoverflow.net/users/nan | For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)? | Such counterexamples are known. For instance, [P. Sjögren has shown](http://www.jstor.org/stable/2374340) that the associated uncentered maximal to the $2$-dimensional Gaussian measure on $\mathbb{R}^2$ isn't weak type $(1,1)$. In a later paper, [Forzani, Scotto, Sjögren and Urbina](http://www.ams.org/journals/proc/200... | 3 | https://mathoverflow.net/users/630 | 122862 | 68,985 |
https://mathoverflow.net/questions/122795 | 1 | Hi,
Suppose that $E/F$ is a unramified extension of local fields of characteristic zero. Let
$G = GL\_n$. Then it is well-known (due to Clozel?) that base change of tempered representations from $G(F)$ to $G(E)$ holds.
Question: does the same result hold in the case of characteristic $p > 0$?
Thanks!
EDIT: As O... | https://mathoverflow.net/users/36285 | unramified base change in characteristic p > 0? | Maybe I should reproduce my comment as an answer, in order for MO not to treat the question has unanswered.
The full local Langlands correspondence is a theorem of G.Laumon, M.Rappoport and U.Stuhler in $\mathcal D$-elliptic sheaves and the Langlands correspondence (Invent. Math. 113).
Here is the Mathscinet review... | 1 | https://mathoverflow.net/users/2284 | 122867 | 68,988 |
https://mathoverflow.net/questions/122819 | 0 | I am trying to understand a particular case of this [question](https://mathoverflow.net/questions/122655/what-is-the-structure-of-the-stack-of-complexes-supported-in-dimension-less-than).
Let $T$ be an affine scheme of finite type over the ground field C. Let $X \to T$ be a morphism. I'll assume this to be the base c... | https://mathoverflow.net/users/25442 | The locus where a sheaf is supported in a certain dimension | Maybe I misunderstand something, but by Nakayama, the support of $E\_t$ is just the support of $E$ intersected with $X\_t$, and the support of $E$ is closed in $X$. Let $Z$ be the suppor of $E$. By Chevalley's semi-continuity theorem (EGA, IV.13.1.3), the set of $x\in X$ such that $\dim\_x Z\_t\le r$ ($x\mapsto t$) is ... | 2 | https://mathoverflow.net/users/3485 | 122873 | 68,989 |
https://mathoverflow.net/questions/122871 | 5 | Hi
Let $V$ be an affine or projective variety. Recall that $V$ is a local complete intersection (l.c.i) if all its local rings are complete intersection. Also recall that $V$ is a complete intersection (c.i) if $I(V)$ is generated by a regular sequence of length $codim(V)$.
I do not know a single example of a l.c.i ... | https://mathoverflow.net/users/31532 | Example of locally complete intersection varieties which are not smooth and not complete intersection | If $X\subset \mathbb P^N$ is an l.c.i projective subvariety that linearly spans $\mathbb P^N$, and if $p\in\mathbb P^n\setminus X$ is a point s.t. the projection of $X$ from $p$ into $\mathbb P^{N-1}$ is an isomorphism onto its image $X'$, then $X'$ will not be a complete intersection (since the linear system of its hy... | 6 | https://mathoverflow.net/users/29992 | 122875 | 68,991 |
https://mathoverflow.net/questions/122870 | 4 | Let be $f$ a $2\pi-$periodic function and $\hat{f}(k)=\frac{1}{2\pi}\int\_0^{2\pi}f(x)e^{-ikx}dx$. Consider the operator:
\begin{equation}
Tf(x)=\sum\_{k\in\mathbb{Z}}sign(k)\ \hat{f}(k)\ e^{ikx}.
\end{equation}
I would like to know if the operator $T:L^p(0,2\pi)\rightarrow L^p(0,2\pi)$ is bounded.
| https://mathoverflow.net/users/31551 | About the boundedness of a multiplication operator. | When $p=2$, boundedness is a triviality and it is the only trivial case. It is not true for $p=1$ nor for $p=\infty$, although the Fourier multiplier $sign(D\_x)$ sends $L^1$ into $L^1\_w$ and the Marcinkiewicz interpolation theorem implies boundedness in $L^p$ for all $p\in]1,+\infty[$.
The operator $sign(D\_x)$ is ... | 5 | https://mathoverflow.net/users/21907 | 122885 | 68,995 |
https://mathoverflow.net/questions/122863 | 2 | Consider the family $\mathbb{T}$ of compact oriented surfaces homeomorphic to the torus $\mathcal{T} = S^1 \times S^1$. Consider arbitrary continuous mappings $k: \mathcal{T} \rightarrow \mathbb{R}$ which obey the condition $\int\_\mathcal{T} k = 0$.
>
> For which curvature-like mappings $k$
> does exist a surfac... | https://mathoverflow.net/users/2672 | Possible curvatures of the topological torus | The answer is known for all smooth surfaces:
M. S. Berger, Riemannian structure of prescribed Gaussian curvature for compact 2-manifolds, J. Differential Geom. 5 (1971), 325-332.
J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. 99 (1974), 14-47.
(This deals with the case of torus).
... | 7 | https://mathoverflow.net/users/25510 | 122886 | 68,996 |
https://mathoverflow.net/questions/122855 | 5 | In the theory of automorphic representations one says that G satisfies a "multiplicity one" property if every cuspidal representation occurs with multiplicity one in $L^2(G(F)\backslash G(A))$.
One also says that G satisfies "strong multiplicity one" if a cuspidal representation is uniquely determined up to isomorphi... | https://mathoverflow.net/users/8080 | Is strong multiplicity one (obviously) stronger than multiplicity one? | I'm not sufficiently privileged to leave comments yet, but in support of the semantics I'll mentioned the following paper:
<http://www2.math.ou.edu/~rschmidt/papers/sqfree.pdf>
In this paper, Schmidt refers to his Corollaries 3.16 and 3.17 as "strong multiplicity one theorems". This would agree with Tom's definition ... | 4 | https://mathoverflow.net/users/12055 | 122887 | 68,997 |
https://mathoverflow.net/questions/122858 | 3 | Bogomolov's paper "Sur l'algébricité des représentations l-adiques" proves that the image of the $\ell$-adic Galois representation associated to an abelian variety over a number field $K$ is open in the $\mathbb{Q}\_
{\ell}$ points of its $\mathbb{Q}\_{\ell}$-Zariski closure.
The Comptes-Rendus announcement has only... | https://mathoverflow.net/users/27736 | An abelian Hodge-Tate representation lands in a torus | What Bogomolov proved is the algebraicity of the corresponding $\ell$-adic representation. What you are trying to do is to prove its semisimplicity, which was done by Faltings 4 years later. The presence of additive factors $G\_a$ in the Zariski closure of the Galois image is not an issue as far as you are interested o... | 4 | https://mathoverflow.net/users/9658 | 122892 | 68,998 |
https://mathoverflow.net/questions/122682 | 1 | **I'VE COMPLETELY REVISED MY QUESTION**
I wish to take a simple undirected graph (i.e. the complete graph K\_4)
Arbitrarily direct said graph, and then create a line graph from the directed version of the graph.
However, in Sage it appears to create a line graph that shows a connection between two edges (that are ... | https://mathoverflow.net/users/31684 | Ihara zeta function (graph theory) coefficients using a line graph | I received an answer (from fidelbc on sagemath) elsewhere and this is what I had wanted:
```
G = graphs.CompleteGraph(4)
D = G.to_directed()
L = D.line_graph()
L.delete_edges([((x,y,None), (y,x,None)) for x,y in G.edges( labels=None ) ])
L.delete_edges([((x,y,None), (y,x,None)) for y,x in G.edges( labels=None ) ])
... | 1 | https://mathoverflow.net/users/31684 | 122896 | 68,999 |
https://mathoverflow.net/questions/122872 | 14 | In GTM82, I read a model for the relative cohomology of (M,N) with N a submanifold of M.
And in the page:
[Relative De Rham cohomologies](https://mathoverflow.net/questions/111059/relative-de-rham-cohomologies/111063#111063),
I got to know that there is another model for relative cohomology by using differential f... | https://mathoverflow.net/users/31731 | de rham model for relative cohomology | Suppose that $C$ is a closed subset of $M$. Denote by $\newcommand{\eO}{\mathscr{O}}$ $\eO$ its complement.
The DeRham cohomology of $M$ is in fact the cohomology associated to a particular soft resolution of the constant sheaf $\newcommand{\ur}{\underline{\mathbb{R}}}$ $\ur$ on $M$.
To any sheaf $\newcommand{\eS}... | 13 | https://mathoverflow.net/users/20302 | 122900 | 69,001 |
https://mathoverflow.net/questions/122894 | 3 | Let $U(\mathcal{H})$ be the group of unitary operators on a Hilbert space with the norm topology. Let $H\subset U(\mathcal{H})$ be a closed subgroup.
Under which condidions (on the subgroup) is there a local cross section for the projection $p:U(\mathcal{H})\to U(\mathcal{H})/H$.
| https://mathoverflow.net/users/21985 | Local cross sections for Unitary group in a hilbert space | You are basically asking: When is $p \colon U(\mathcal{H}) \to U(\mathcal{H})/H$ a principal $H$-bundle. Equipped with the norm topology $U(\mathcal{H})$ is a Banach-Lie group. There is a theorem for quotients of Banach-Lie groups by Glöckner and Neeb in a paper called [Banach-Lie Quotients, Enlargibility and Universal... | 5 | https://mathoverflow.net/users/3995 | 122902 | 69,002 |
https://mathoverflow.net/questions/122901 | 17 | Hey all,
I just ran into a question of Mazur from his 1978 paper **'Rational isogenies of Prime Degree'** and I wonder what is the status of his question:
"Are there examples of elliptic curves $E/\mathbf{Q}$, $E'/\mathbf{Q}$ which are not isogenous over $\mathbf{Q}$ such that $E[N] \cong E'[N]$ (as $\text{Gal}(\over... | https://mathoverflow.net/users/12668 | Elliptic curves over QQ with isomorphic n-torsion | This question comes down to a question about a certain Hilbert modular surface X\_N, which parametrizes pairs of elliptic curves with isomorphic N-torsion -- you can think of it as
X(N) x X(N) / PSL\_2(Z/NZ)
where the group acts diagonally on the two factors.
This surface, like all Hilbert modular surfaces, has s... | 17 | https://mathoverflow.net/users/431 | 122903 | 69,003 |
https://mathoverflow.net/questions/122898 | 1 | Let $X$ be an anabelian curve over a number field $K$ and let $p:Y\rightarrow X$ be a finite etale cover. Then is anything known (or has anything been conjectured) about the field of definition of $Y$? In particular, should it be an abelian extension of $K$?
| https://mathoverflow.net/users/16858 | Field of definition of a finite etale cover of an anabelian curve | Certainly not, unless I totally misunderstand your question. For instance, an unramified double cover of X is going to be defined over the field obtained by adjoining a 2-torsion point of the Jacobian of X, and that field is going to be non-abelian as all get out.
| 3 | https://mathoverflow.net/users/431 | 122904 | 69,004 |
https://mathoverflow.net/questions/122891 | 20 | I would like to know the smallest-index subgroups of ${\rm SL}(3,\mathbb{Z})$.
The smallest I could find has even entries $a\_{3,1}$ and $a\_{3,2}$,
along the bottom row. I could not figure out whether there are
subgroups of index $2$ or $3$.
A search found lots of information about ${\rm SL}(2,\mathbb{Z})$,
but no... | https://mathoverflow.net/users/19964 | Small-index subgroups of SL(3,Z) | To fill in the comments, there are basically two serious issues involved.
1) You want to know that every subgroup of finite index in $\mathrm{SL}(3,\mathbb{Z})$ contains some *congruence kernel*: the kernel of the natural reduction homomorphism induced by $\mathbb{Z} \rightarrow \mathbb{Z}/n\mathbb{Z}$. In other word... | 20 | https://mathoverflow.net/users/4231 | 122907 | 69,007 |
https://mathoverflow.net/questions/122897 | 6 | Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function.
Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex in $p$?
Related questions:
[Binomial Expectation of Convex Function](https://mathoverflow.net/questions/120636/binomi... | https://mathoverflow.net/users/31735 | Is the Binomial Expectation of Convex Function Convex in p? | Here is my original answer (see below for a better one):
Writing down
\[
g(p) = \sum\_{k=0}^n h(k)\binom{n}{k}p^k(1-p)^{n-k}
\]
and differentiating twice gives
\[
g''(p) = \sum\_{k=0}^n h(k)\binom{n}{k} \\
\cdot \left[ k(k-1)p^{k-2}(1-p)^{n-k} - 2k(n-k)p^{k-1}(1-p)^{n-k-1} + (n-k)(n-k-1)p^k(1-p)^{n-k-2}\right].
\]
... | 7 | https://mathoverflow.net/users/5963 | 122909 | 69,008 |
https://mathoverflow.net/questions/122923 | 0 | Let $\Gamma \subseteq PSL\_2(\mathbb{R})$ be a Fuchsian group, possibly containing elliptic elements. Is it true that $N(\Gamma) / \Gamma$, where $N(\Gamma)$ the normalizer of $\Gamma$ in $PSL\_2(\mathbb{R})$, is isomorphic to the automorphism group of $\Gamma \backslash \mathcal{H}^\*$?
Here, $\mathcal{H}^\*$ is the... | https://mathoverflow.net/users/7313 | Fuchsian groups and automorphisms of Riemann surfaces | The compact Riemann surface $\Gamma\backslash\mathcal H^\*$ may have authomorphisms that do not preserve the set of points corresponding to the cusps of $\Gamma$, so I believe the answer is no.
| 3 | https://mathoverflow.net/users/29992 | 122925 | 69,016 |
https://mathoverflow.net/questions/122920 | 2 | Does the Čech cohomology always give rise to a long exact sequences given a short exact sequence of sheaves?
Clearly that cannot occur for sheaves on a paracomact (perhaps also Hausdorff, I'm not sure about that) space, but the argument one uses there relies on the topological assumptions in a crucial way and cannot ... | https://mathoverflow.net/users/16504 | Does the Čech cohomology always yield long exact sequences from short ones? | It should be no. Sorry, I don't have the time to write anything detailed, so I'll just leave assembly instructions. Take a look at my answer here:
[equivalence of Grothendieck-style versus Cech-style sheaf cohomology](https://mathoverflow.net/questions/4214/equivalence-of-grothendieck-style-versus-cech-style-sheaf-coh... | 7 | https://mathoverflow.net/users/4144 | 122928 | 69,019 |
https://mathoverflow.net/questions/122930 | 12 | Let $X$ be a connected, based CW complex. Then the James splitting
of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra
$$
\Sigma^{\infty} \Omega\Sigma X\_+ \quad \simeq \quad \Sigma^{\infty} (S^0 \vee X \vee X^{[2]} \vee X^{[3]} \vee \cdots ) ,
$$
where $X^{[n]}$ denotes the $n$-fold smash pro... | https://mathoverflow.net/users/8032 | On the stable splitting of loops on a suspension | Hi John, this is not a complete proof, but it should give the idea of one. There is a clever proof of the splitting of suspension spectra of $\Omega^n\Sigma^n X$ due to Ralph Cohen in his
paper "Stable proofs of stable splittings". His proof is also given in section VII.5 of
LMS "Equivariant stable homotopy theory". It... | 12 | https://mathoverflow.net/users/14447 | 122933 | 69,021 |
https://mathoverflow.net/questions/122919 | 10 | Consider the triangular array $X\_{n,k}$ such that, for each $n>0$, the variables $(X\_{n,1},\cdots,X\_{n,n})$ have the following properties:
1. For any given $1 \le L \le n$, all
subsets of
$(X\_{n,1},\cdots,X\_{n,n})$ of size
$L$ have the same joint distribution
(even after applying an arbitrary permutation).
2. Ea... | https://mathoverflow.net/users/29510 | Central Limit Theorem (and Berry-Esseen theorem) for non-independent variables | These types of conditions (exchangeability and $O(1/n)$ covariance) do not even ensure that $S\_n$ is approximately normal.
Ignore odd $n$. (I believe similar examples can be constructed for odd $n$.) Let $n=2m$. Let each $X\_{n,k} \in \lbrace -1,1 \rbrace$. Let each set of $m$ positive signs have probability ${2m \c... | 7 | https://mathoverflow.net/users/2954 | 122935 | 69,022 |
https://mathoverflow.net/questions/122945 | 0 | Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following
Theorem: Fix $k \geq 0, 0<\alpha<1$. Let $f\in C^{k,\alpha}(S^1)$. Then its harmonic extension $H(f)$, which is the solution to... | https://mathoverflow.net/users/6953 | Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data) | It follows from the Schauder theory. You can also establish Kellog's theorem directly. One approach is given in DiBenedetto's PDE book, where he uses Kellog's theorem in the proof of Schauder estimates.
| 1 | https://mathoverflow.net/users/824 | 122955 | 69,030 |
https://mathoverflow.net/questions/122952 | 16 | I am trying to solve exercise in Huybrechts's book 'Complex geometry'
While solving problems, one problem kept me from going forward.
That is,
The surface $\Sigma\_n=\mathbb{P}$ $(\mathcal{O}\_ {\mathbb{P}^1}\oplus \mathcal{O}\_{\mathbb{P}^1} (n))$ is n-th Hirzebruch surface.
Show that $\Sigma\_n$ is isomorphic... | https://mathoverflow.net/users/29326 | On a Hirzebruch surface | The first thing I would like to point out is that the variety we care about is actually
{$([x\_0 , x\_1 ],[y\_0,y\_1,y\_2]):{x\_0}^n y\_1 - {x\_1}^n y\_2 =0$}$ =V \subset \mathbb{P}^1 \times \mathbb{P}^2$
(the defining equations you gave are not usually homogenous, and this is what Huybrechts gives in his book: <ht... | 20 | https://mathoverflow.net/users/17630 | 122956 | 69,031 |
https://mathoverflow.net/questions/122918 | 7 | A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and antipode then follows.
The real forms of the Drinfeld-Jimbo quantum groups $U\_q(\mathfrak{g})$ have been classified. This re... | https://mathoverflow.net/users/703 | Real forms of Drinfeld-Jimbo quantum groups | Two references I recall are
E. Twietmeyer, Real forms of Uq (g), Lett. Math. Phys. 24, 49-58, 1992.
V. Lyubashenko, Real and imaginary forms of quantum groups, Lecture Notes in Math. 1510, 1992, pp 67-78.
| 3 | https://mathoverflow.net/users/30364 | 122960 | 69,033 |
https://mathoverflow.net/questions/122958 | 3 | This is a stronger version to Szemerédi's theorem.
Let $C : \mathbb{N}\rightarrow 2^{\mathbb{N}}$ be a choice function such that $C(n)$ is a subset of $\{1,...,n\}$ with size at least $\frac{n}{M}$ for some nonzero constant $M$ that only depends on $C$. Can we guarantee an arithmetic progression in $C(n)$ of length ... | https://mathoverflow.net/users/19078 | existence of arithmetic progression of nonzero density | This is far too ambitious. Recall that the [van der Waerden number](http://en.wikipedia.org/wiki/Van_der_Waerden_number) W(k,c) is the smallest $N$ such that if we color $[N]$ with $c$ colors we are guaranteed to have a $k$ term arithmetic progression. [Gasarch and Haeupler have given](http://arxiv.org/abs/1005.3749) a... | 4 | https://mathoverflow.net/users/630 | 122967 | 69,036 |
https://mathoverflow.net/questions/122941 | 2 | OK, then I read Frobenius method in mathworld (I learned when I took ODE 2):
<http://mathworld.wolfram.com/FrobeniusMethod.html>
My question is:
Are there any ODEs where the solution is given by full Laurent series?, i.e its negative indexed coeffecients are not zero starting from some negative integer.
Thanks in a... | https://mathoverflow.net/users/13904 | Laurent series expansion for ODE. | Every analytic function in a ring has a Laurent expansion. Thus if your differential equation
has a solution analytic in a ring it has a Laurent expansion. See the examples given in other answers.
The difference between a full Laurent
series (I understand that "full" means infinitely many coefficients in both direct... | 1 | https://mathoverflow.net/users/25510 | 122975 | 69,040 |
https://mathoverflow.net/questions/122976 | 5 | Loosely speaking the question is: If an element of a commutative ring is infinitely divisible by $x$, is it the product of $x$ with an infinitely divisible element?
More preceisely:
For a fixed element $x$ of a commutative unital ring $R$ let $I(x)=\bigcap \lbrace x^nR :n\in \mathbb N \rbrace$ be the ideal of element... | https://mathoverflow.net/users/21051 | A question of Allan on infinite divisibility | What's to stop you from just freely building a counterexample? That is, let the ring be generated by elements $x,a,b\_1,b\_2,\dots,b\_n,\dots$ subject to (only) the relations $a=x^nb\_n$ for all positive integers $n$. Then $a$ is in $I(x)$. Unless I'm making a stupid mistake, the only solutions $q$ of $a=xq$ are finite... | 4 | https://mathoverflow.net/users/6794 | 122980 | 69,042 |
https://mathoverflow.net/questions/122985 | 3 | Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P\_X(x)$ and $P\_Y(y)$ on these sets. Assume each value of $P\_X(x)$ and $P\_Y(y)$ is rational.
For each $n$ such that $nP\_X(x)$ and $nP\_Y(y)$ are integer-valued, let $(X\_1,\cdots,X\_n)$ be a random sequence with $nP\_X(x)$ occure... | https://mathoverflow.net/users/29510 | Central Limit Theorem for additive function of permutations of sequences | I think I found the answer actually - in the paper "A Combinatorial Central Limit Theorem" (Hoeffding, 1951)
The above setting seems to be recovered by letting $c\_n(i,j)$ equal $f(x,y)$ for $P\_X(x)P\_Y(y)n^2$ values in the $(i,j)$ grid.
Then the function $d\_n(i,j)$ in Eq. (8) can be written as a function of $(x,... | 1 | https://mathoverflow.net/users/29510 | 122995 | 69,048 |
https://mathoverflow.net/questions/122982 | 7 | Let $H$ be a Hilbert space, and $T:D(T)\subset H\rightarrow H$ and $S:D(S)\subset H\rightarrow H$ be unbounded self-adjoint operators.
Is $T+S:D(T)\cap D(S)\rightarrow H$ self-adjoint?
| https://mathoverflow.net/users/31760 | Sum of two self-adjoint unbounded operators | The answer is no. $\newcommand{\bR}{\mathbb{R}}$ Consider the Hilbert space of $L^2$-functions
$$ u:[0,1]\to \bR^2,\;\;u(t)=(x(t), y(t)) . $$
Define
$$ D(T)=\Bigl\lbrace u\in H;\;\int\_0^1 \bigl\vert u'(t)\bigr\vert^2 dt <\infty,\;;\;x(0)=x(1)=0\;\Bigr\rbrace, $$
$$ D(S)=\Bigl\lbrace u\in H;\;\int\_0^1 \bigl|u... | 7 | https://mathoverflow.net/users/20302 | 122998 | 69,049 |
https://mathoverflow.net/questions/122932 | 4 | Let $G$ be a topological group, and let $F$ be the Markov free topological group over $G$. We define an action of $G$ on $F$ as follows: $G\times F\rightarrow F$, $^{g}(g\_{1}^{\epsilon\_{1}}...g\_{n}^{\epsilon\_{n}})$=$(^{g}g\_{1})^{\epsilon\_{1}}...(^{g}g\_{n})^{\epsilon\_{n}}$.
**Question:** Is the above-defined act... | https://mathoverflow.net/users/31207 | a free topological group as a topological module | I don't know the answer to the general question but the answer seems to be "yes" when $G$ is compact. Here is a straightforward argument.
The corresponding action of $G$ on the free topological monoid $M=\coprod\_{n\geq 0}(G\sqcup G^{-1})^n$ is certainly continuous. When $G$ is compact $F(G)$ is the topological quoti... | 1 | https://mathoverflow.net/users/5801 | 123008 | 69,056 |
https://mathoverflow.net/questions/123009 | 13 | Before asking the question I should say that I don't know much about algebraic groups and I'm not sure if the question has the right level for MO. If not, please let me know and I will delete the question that emerged when I was trying to understand some theorems in a paper on the vanishing range of the cohomology of f... | https://mathoverflow.net/users/27895 | Simply connected simple algebraic groups | If $G$ is an adjoint algebraic group then it is always simple as an abstract group, (EDIT: This is because any proper normal subgroup of a simple algebraic group must be finite and lie in the centre). In general assume $G$ is connected reductive algebraic group and let $\pi : G \to G\_{ad}$ be an adjoint quotient of $G... | 16 | https://mathoverflow.net/users/22846 | 123010 | 69,057 |
https://mathoverflow.net/questions/122990 | 5 | A version of this question on stackexchange got a few comments from one person and no answers.
Let $e\_k$ be the $k$th-degree elementary symmetric polynomial in variables $x\_1,\ldots,x\_n$ (so in particular $e\_k=0$ if $k>n$). I don't know much about these polynomials.
Consider the rational function
$$
\frac{e\_1+... | https://mathoverflow.net/users/6316 | Reference request on symmetric polynomials | The fact that the binary operation defined by (2) is associative can be found in the Wikipedia article on formal groups (section 2) <http://en.wikipedia.org/wiki/Formal_group>. Presumably one might find more in the literature of formal groups.
| 2 | https://mathoverflow.net/users/10744 | 123019 | 69,062 |
https://mathoverflow.net/questions/122915 | 11 | Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in 'Espaces analytiques relatives et theorem de finitude' by Houzel it is assumed that the valuation is non-discrete and that... | https://mathoverflow.net/users/3396 | bornological vector spaces over a non-archimedean field | I try to reply to your questions:
1) In the first part of "Espaces analytiques relatives et theorem de finitude" Houzel says that the field under consideration is supposed to be maximally compact. I don't see any point where he use this hypothesis in his article neither in the result that he recall from the "Seminari... | 7 | https://mathoverflow.net/users/31767 | 123028 | 69,066 |
https://mathoverflow.net/questions/123042 | 2 | Oddball question: say I want to travel from $(a, b)$ where $b > 0$ to $(c, d)$ where $d < 0$ using the shortest path, where I can travel at velocity $v\_1$ in the upper half-plane and at velocity $v\_2$ in the lower half-plane. (E.g., a light ray would take this path.) Where should I cross the $x$ axis? I want an expli... | https://mathoverflow.net/users/22485 | Computing a point of refraction | By scaling and translation, you can assume WLOG that $a=0$ and $c=1$. Also let $r = (V\_2/V\_1)^2$. You are left with a quartic polynomial in $x$ having $3$ independent parameters. Yes, a computer algebra system can solve it, but the result will be
"wallpaper": a very complicated expression that I won't even try to wr... | 1 | https://mathoverflow.net/users/13650 | 123044 | 69,074 |
https://mathoverflow.net/questions/123041 | 6 | Consider the usual sine function $\mathbb{R}\rightarrow \mathbb{R}$. Is there some (single) group structure we can put on $\mathbb{R}$ with respect to which sine becomes a homomorphism?
I suspect the answer is either no for a trivial reason, or yes by a simple set-theoretic argument (probably providing a great many s... | https://mathoverflow.net/users/21857 | Can sine be made into a homomorphism? | If you require the group structure to be continuous, this is impossible. Indeed, in that case, the image $[-1,1]$ would have to be a group as well. But $[-1,1]$ is not homogeneous, so it cannot be a topological group.
Without requiring continuity, it is possible. Let me first give a construction where you restrict fr... | 6 | https://mathoverflow.net/users/75 | 123046 | 69,075 |
https://mathoverflow.net/questions/123027 | 2 | Problem:
You are given a sample of size $m$ from $n$ independent normally distributed random variables. Expectations and standard deviations of the random variables are unknown. Estimate, which random variable has the largest expectation.
Discussion:
The interesting case is when $n > 2$ and the expectations of th... | https://mathoverflow.net/users/31775 | Estimate which random variable has highest expectation | For the case of $3$ random variables, choose two of them (at random) and compare the sample means of the first half of the samples. Then compare the sample means of the second half of the samples
for the winner of the first round and the third random variable. You'll have probability
greater than $1/3$ of the result be... | 0 | https://mathoverflow.net/users/13650 | 123047 | 69,076 |
https://mathoverflow.net/questions/123048 | 0 | Hi all,
on the forum page <http://www.groupsrv.com/science/about506648.html> one can read the following (i cut out nonimportant parts):
Question: Does anyone know any condition (non trivial) that ensure that the
global sections of the tensor product of two sheaves is the tensor
product of the global sections?
A... | https://mathoverflow.net/users/29657 | Spectral sequence for composition of global sections and tensor product of sheaves | [Grothendieck spectral sequences](http://en.wikipedia.org/wiki/Grothendieck_spectral_sequence) deal with the derived functor of a composition. If you're familiar with derived categories, then it takes a very simple form. It says that under the reasonable conditions that make sure that all derived functors in question e... | 1 | https://mathoverflow.net/users/10076 | 123049 | 69,077 |
https://mathoverflow.net/questions/123056 | 9 | I read that one example of k3 surface is a double cover of $\mathbb{P}^2\mathbb{C}$ ramified over a sextic. My question is why a sextic? i believe that the sextic is isomorphic to the ramification divisor, but why is this? also how can i see that k3 surfaces realized this way are deformation equivalent?
thank you
| https://mathoverflow.net/users/20483 | k3 surface as ramified double cover of $\mathbb{P}^2$ | **Q1:** Hurwitz formula + canonical divisor of $\mathbb P^2$.
**Q2:** Move the curve in $\mathbb P^2$.
| 22 | https://mathoverflow.net/users/10076 | 123059 | 69,080 |
https://mathoverflow.net/questions/123068 | 12 | A well-known result of Kunen says that there is no non-trivial elementary embedding $j: V \rightarrow V.$ There are several proofs of this theorem (see Kanamori, The higher infinite). I wonder to know if the following argument works:
Assume on the contrary that such an embedding exists. Let $\Phi(\alpha)$ be the foll... | https://mathoverflow.net/users/11115 | Kunen's inconsistency result | Your $\Phi(\kappa)$ uses an existential quantifier over *classes* (since $j$ is a class), whereas $j$ is only assumed to preserve *first order* formulae, so there is no reason to conclude that $j(\kappa)$ is also the least such that $\Phi(j(\kappa))$,
| 17 | https://mathoverflow.net/users/9269 | 123070 | 69,082 |
https://mathoverflow.net/questions/123063 | 0 | Given an integer d, is there a way of finding max(n^m), where d=n\*m, except brute-forcing it ?
n and m don't have to be primes.
For example:
The number 12 can be factorized into two factors as: 1\*12, 2\*6, 3\*4, 4\*3, 6\*2 and 12\*1.
Raising the first factor to the second gives: 1, 64, 81, 64, 36 and 12.
The maxi... | https://mathoverflow.net/users/31782 | How can an integer be factorized as n*m so that n^m has the highest value. | If $n$ is divisible by three than the required split is $(3,n/3)$. If $n$ is not divisible by 3 then the required split is $(p,n/p)$ where $p$ is the smallest prime divisor of $n$.
| 5 | https://mathoverflow.net/users/23388 | 123072 | 69,083 |
https://mathoverflow.net/questions/123061 | 1 | What are the easiest brute force algorithms for solving closest and shortest vector problems?
I want to find an arbitrary, but small ($\lesssim 20$) number of lattice vectors closest to a given point. Since I consider low spatial dimension of four or less, the hardness of the closest vector problem is not an importan... | https://mathoverflow.net/users/10712 | Brute force lattice problems | I know this in the language of quadratic forms. Once you have expressed your form in any somewhat reduced form, you have $f(x) = (1/2) x^T A x$ where $A$ is a symmetric positive and integral matrix. To find all $f(x) \leq M$ what I do is find the largest possible value of each $x\_i$ by Lagrange multipliers. This surro... | 1 | https://mathoverflow.net/users/3324 | 123078 | 69,084 |
https://mathoverflow.net/questions/115409 | 2 | If two torsion theories on a ring localize the ring to the same extension ring, I can find no reason that their "meet" in the lattice of torsion theories must also localize to the same ring. I cannot find anything in Golan's encyclopedia that addresses questions like this.
Does anyone have a counter-example?
Here i... | https://mathoverflow.net/users/4994 | torsion theories localizing the base ring to the same ring | I also assume that by torsion theory we mean hereditary torsion theory. I use Bo Stenström's book *Rings of Quotients* (especially chapter IX §2 later on). And today I work with right modules.
If by "the same" you really mean nothing more than that they are isomorphic as rings, the following should be an easy counte... | 1 | https://mathoverflow.net/users/27465 | 123094 | 69,092 |
https://mathoverflow.net/questions/123102 | 3 | As far as I know, it is an open problem to give a formula counting transitive relations on an $n$-element set. Is it easier to count the idempotent relations, that is relations that are both transitive and interpolative? (A relation $\rho$ is interpolative when $x\rho y\implies((\exists z)\ x\rho z \wedge z\rho y).$)
... | https://mathoverflow.net/users/20803 | How many idempotent relations are there on an $n$-element set? | The answer seems to be in Butler, K. K.-H.,The number of idempotents in (0,1)-matrix semigroups, Linear Algebra and Its Applications 5 (1972), 233–246. I will see if I have access to the journal and will tell you more.
**Edit.** The paper is [here](http://ac.els-cdn.com/0024379572900055/1-s2.0-0024379572900055-main.p... | 7 | https://mathoverflow.net/users/15934 | 123103 | 69,094 |
https://mathoverflow.net/questions/123101 | 0 | Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19).
Let $\Lambda\_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda\_{\mathbb{R}}$ a positive subspace of dimention 3.
I have found this proposition:
$W^\perp\cap\Lambda=0$ if and only if $\exists w \in W$ such as $w^\perp \cap \La... | https://mathoverflow.net/users/20483 | orthogonality in a lattice | For each $\lambda \neq 0$ in $\Lambda$, consider the subspace $H\_\lambda$ of $W$ given by $\langle \lambda, x \rangle = 0$. Saying $W^\perp \cap \Lambda = 0$ means these are all proper subspaces of $W$. Now, since $\mathbb{R}$ is uncountable, it's not too hard to show that a countable union of proper subspaces can't b... | 1 | https://mathoverflow.net/users/2698 | 123110 | 69,098 |
https://mathoverflow.net/questions/123118 | 5 | Conjectured by Mordell and later proved by Faltings, a non-singular algebraic curve of genus $g$ over $\mathbb{Q}$ has finitely many rational points if $g > 1$. Since the genus of the Fermat curve $x^{n} + y^{n} = 1$ is $\frac{n(n-1)}{2}$ by the degree formula, Faltings' Theorem implies that it can only have finitely m... | https://mathoverflow.net/users/10280 | Topology in Arithmetic | Dear user02138, you may be interested in the **Bombieri-Lang conjecture**. It has a weak form and a strong form. The latter is:
>
> If $X$ s a smooth variety over a number field $K$ which is of general type over a number field, then there are only finitely many curves of (geometric) genus $\leq 1$ contained in $X$,... | 6 | https://mathoverflow.net/users/9317 | 123121 | 69,102 |
https://mathoverflow.net/questions/123130 | 0 | im looking for a non-noetherian ring with infinite krull dimension.would you help?
| https://mathoverflow.net/users/31803 | krull dimension | A polynomial ring $k[X\_1,X\_2,\ldots]$ with an infinite number of variables will do. Finding a *noetherian* ring with infinite Krull dimension is harder (and first accomplished by Nagata).
| 2 | https://mathoverflow.net/users/1148 | 123132 | 69,105 |
https://mathoverflow.net/questions/122310 | 9 | In their paper "A barren extension", Henle, Mathias, and Woodin considered the forcing whose conditions are the sets of natural numbers ordered by inclusion up to finite error.
If we force over $L(\mathbb{R})$ and assume that this is a model of determinacy, they show that this will add a free ultrafilter U on the natur... | https://mathoverflow.net/users/21255 | A linear order obtained by forcing with P(omega)/fin | The answer to the first question is no, at least if one is willing to assume large cardinals in $V$. I'd have to research what the optimal hypothesis is. It is shown in the two Neeman-Zapletal papers that if there are infinitely many Woodin cardinals below a measurable cardinal and above some cardinal $\kappa$, then fo... | 8 | https://mathoverflow.net/users/31807 | 123133 | 69,106 |
https://mathoverflow.net/questions/119565 | 3 | How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E\_7$.
| https://mathoverflow.net/users/nan | degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ | You don't really need Vinberg theory to find the degrees of the invariants here. This representation of $SL\_8=SL(V)$ can be constructed by considering a maximal rank involution of an algebraic group of type $E\_7$, which has fixed points isomorphic to $SL(V)/\{\pm I\}$ and such that the $(-1)$ eigenspace in the Lie al... | 3 | https://mathoverflow.net/users/26635 | 123137 | 69,108 |
https://mathoverflow.net/questions/122973 | 2 | I seem to recall that a general *homogenous* real polynomial $P$ of even degree in $n$ variables, $n\geq 3,$
cannot always be expressed as $P(x\_1,\dotsc,x\_n)=\sum\_j a\_j Q\_j^2(x\_1,\dotsc,x\_n)$ where $a\_j \in \mathbb{R},$ and the $Q\_j^2$ are *homogenous of the same degree* as $P.$
(Please, correct me if I am wro... | https://mathoverflow.net/users/1056 | Homogenous polynomials as sum or differences of squares and symmetric polynomials | My college provided me with a simple example:
$$x^2 + xy + y^2$$ cannot be expressed as a sum/difference of symmetric, homogenous polynomials of degree 1, for obvious reasons.
(Each homogenous, symmetric polynomial of degree 1 in two variables are of the form a(x+y).
Thus, all possible $Q\_j$ are of the form $a'(x+y)... | 2 | https://mathoverflow.net/users/1056 | 123143 | 69,111 |
https://mathoverflow.net/questions/123136 | 4 | I understand why the set of natural numbers $\mathbb N = \{ 0, 1, 2, \cdots \}$ is equipped with a total order. Indeed, every monoid has a [pre-order](http://ncatlab.org/nlab/show/preorder), where $$n' \succeq n \quad \mathrm{if~and~only~if} \quad n' = n + m \quad \mathrm{~for~some~} m.$$ In the case of $\mathbb N$, th... | https://mathoverflow.net/users/238 | Why do we choose the standard total order on the integers? | I assume (your) monoids are cancellative.
Then the pre-order you define is a (partial) order if and only of the monoid $M$ is reduced (i.e. has no invertible elements besides the neutral one).
For getting an order on the Grothendieck group $G$ say the element in $M$ are "positive elements" and define on $G$ the r... | 6 | https://mathoverflow.net/users/nan | 123145 | 69,113 |
https://mathoverflow.net/questions/123117 | 9 | Let $E$ be the elliptic curve $x^3+y^3+z^3=0$.
**Question.** Are there injective morphisms $E\to \mathbb CP^2$ of arbitrary high degree?
*Comments.* 1) There are injective morphisms $E\to \mathbb CP^2$ of degree $3$ (the obvious one) and of degree $6$ - whose image is the curve dual to the cubic. 2) Clearly for $\... | https://mathoverflow.net/users/13441 | Injective morphism from an elliptic curve to $\mathbb CP^2$. | The answer is yes. And more generally, one has the following result:
**Lemma:** for any elliptic curve $X$, the degree of injective morphisms $X\to \mathbb{P}^2\_{\mathbb{C}}$ (or $\mathbb{C}P^2$ if you want) is not bounded.
**Proof:** The curve $X$ is isomorphic to a plane cubic of equation $X^3+Y^3+Z^3=\lambda X... | 11 | https://mathoverflow.net/users/23758 | 123155 | 69,120 |
https://mathoverflow.net/questions/123159 | 5 | For this question we will consider the Zariski site of affine schemes and a stack $\mathcal{M}$ over it. I don't know what a substack is, but I have a guess. The stack $\mathcal{M}$ has an underlying path-component sheaf $\pi\_0\mathcal{M}$, and we have a map $\mathcal{M}\rightarrow \pi\_0\mathcal{M}$. A stack $\mathca... | https://mathoverflow.net/users/23122 | When is a substack closed? | No need to guess, just look it up. <http://ens.math.univ-montp2.fr/~toen/cours8.pdf> Definition 1.1.
| 3 | https://mathoverflow.net/users/25442 | 123165 | 69,122 |
https://mathoverflow.net/questions/123096 | 2 | Inspired by the result of Schinzel and Smyth that a totally real number other than $0$ and $\pm 1$ has height at least $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.240659\ldots$, Bombieri and Zannier discovered that totally $p$-adic algebraic numbers which are not roots of $1$ likewise have height bounded away... | https://mathoverflow.net/users/26522 | Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero? | In my paper *Mesures et équidistribution sur les espaces de Berkovich*, Crelle, 2006,
I had proved an equidistribution theorem in the good reduction case. The limit measure is a Dirac mass at a single point of the Berkovich space (the one whose reduction is the generic point of the special fiber). Since the Galois orbi... | 3 | https://mathoverflow.net/users/10696 | 123166 | 69,123 |
https://mathoverflow.net/questions/123138 | 1 | Let $\Gamma \leq PSL\_2(\mathbb{R})$ be a Fuchsian group. What is the relation between $N(\Gamma) = \{ \alpha \in PSL\_2(\mathbb{R}) \mid \alpha \Gamma \alpha^{-1} = \Gamma \}$ and $Aut(\Gamma \backslash \mathcal{H})$, the group of automorphisms of the Riemann surface $\Gamma \backslash \mathcal{H}$?
Here $\mathcal{H... | https://mathoverflow.net/users/7313 | Fuchsian groups and their normalizers | Unfortunately, as far as I know, nobody really explains such things as they are considered to be "too elementary". The most basic, briefest and down-to-earth reference I know for the needed background in Fuchsian groups is S.Katok's book "Fuchsian groups." Here are some proofs:
A Fuchsian group is called *elementary*... | 8 | https://mathoverflow.net/users/21684 | 123169 | 69,125 |
https://mathoverflow.net/questions/123160 | 2 | I need a proof for this proposition:
>
> If a uniformity $\mathfrak U$ on $X$ has a
> countable fundamental system of
> entourages, then it can be defined by
> a pseudometric on $X$.
>
>
>
which is the proposition 2 on page 142 of [this book](http://books.google.com/books?id=bQwhdmL6IjUC&pg=PA142&lpg=PA142&... | https://mathoverflow.net/users/31813 | A uniformity with a countable base is a pseudometric uniformity. | First, find a fundamental system of entourages $U\_0 \supseteq U\_1 \supseteq \cdots$ such that
1. $U\_0 = X \times X$
2. $(x\_0,x\_1) \in U\_i$ iff $(x\_1,x\_0) \in U\_i$.
3. If $(x\_0,x\_1),(x\_1,x\_2),(x\_2,x\_3) \in U\_{i+1}$ then $(x\_0,x\_3) \in U\_i$.
Define $e(x\_0,x\_1) = \inf\lbrace 2^{-i} : (x\_0,x\_1) \... | 5 | https://mathoverflow.net/users/2000 | 123171 | 69,126 |
https://mathoverflow.net/questions/123172 | 1 | Let $r\_{s,k}(l)$ be the number of representation of a natural number $l$ as a sum of $s$ positive $k$-th powers. Then the circle method of Hardy and Littlewood gives the asymptotic formula
\begin{equation}
r\_{s,k}(l) = \frac{\Gamma(1+1/k)^s}{\Gamma(s/k)} \mathfrak{S}\_{s,k}(l) l^{s/k-1} + o(l^{s/k-1}),
\end{equatio... | https://mathoverflow.net/users/27855 | Upper bounds on the number of representation of a natural number as a sum of $s$ positive $k$-th powers. | There are $O(x^{1/k})$ $k$-th powers of size at most $x$, so there are $O(x^{s/k})$ sums of $s$ $k$-th powers of size at most $x$ but only $x$ integers of size at most $x$ so some integer at most $x$ needs to be represented at least $\gg x^{s/k -1}$ times.
| 2 | https://mathoverflow.net/users/2290 | 123177 | 69,128 |
https://mathoverflow.net/questions/123178 | 1 | Let $M$ be a Kahler manifold, with Kahler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $L\_{X} g = 0$, where $L\_{X}$ is the Lie derivative along $X$. Let $R$ be the Riemannian curvature tensor of $g$. Is $L\_{X} R = 0$?
| https://mathoverflow.net/users/7035 | Lie derivative of curvature | $g\mapsto R(g)$ is a natural (non-linear) operation: it commutes with pullbacks by diffeomorphisms.
This is just a transcription of: curvature transforms correctly under chart changes.
Thus we get
$L\_X R(g) = dR(g)(L\_X g)$ which implies "Yes"; two earlier comments also said this.
| 6 | https://mathoverflow.net/users/26935 | 123187 | 69,131 |
https://mathoverflow.net/questions/123135 | 21 | Are there any major fundamental results in finite-dimensional linear algebra discovered after early XX century? Fundamental in the sense of non-numerical (numerical results, of course, are still interesting and important); and major in the sense of something on the scale of SVD or Jordan normal form.
(EDIT) As sever... | https://mathoverflow.net/users/31589 | Modern developments in finite-dimensional linear algebra | Definitely, some items on the top of my list are:
1. Random matrix theory --- both asymptotic and non asymptotic; including things like semi-circular law, circular law, and so on. Check out Terry Tao's blog for very nice summaries.
2. The resolution of Horn's conjecture (see this [nice summary article by R. Bhatia](h... | 12 | https://mathoverflow.net/users/8430 | 123190 | 69,132 |
https://mathoverflow.net/questions/123015 | 4 | Let $(R, \mathfrak{m})$ be a local ring and $X = Spec(R)$. Let $Y = V(I)$ be a closed subscheme of $X$, defined by an ideal $I \subset R$, and let $P \in X$ (in fact, $P \in Y$) be the closed point. Let $(\hat{X}, \mathcal{O}\_{\hat{X}})$ be the formal completion of $X$ along $Y$ ($\hat{X} = Y$ as a topological space, ... | https://mathoverflow.net/users/31771 | Comparison for formal local cohomology | Dear Nick: If your local ring $R$ is Gorenstein, then $H^i\_p(\hat{X},\mathcal{O}\_{\hat{X}})\cong\varprojlim H^i\_{\mathfrak{m}}(R/I^n)$ for all $i$, as you want. This is proved, for instance, in Proposition 2.2, page 334 of A. Ogus' *[Local cohomological dimension of algebraic varieties](http://www.jstor.org/discover... | 3 | https://mathoverflow.net/users/16046 | 123193 | 69,133 |
https://mathoverflow.net/questions/123060 | 3 | Consider a random walk on the integers where the probability of transitioning from $n$ to $n+1$ is $p\_n$ (and of course, the probability of transitioning from $n$ to $n-1$ is $1-p\_n$); we assume all $p\_n$ are strictly less than $1$. Note that the probability of going right is position-dependent, and we certainly do ... | https://mathoverflow.net/users/31781 | How does changing the transition probabilities affect the concentration of a position-dependent random walk? | There's no reason to believe that the new speed will be $2 \epsilon$ more then the old speed. To give a concrete example, take $\epsilon=0.01$ and let the environment be $p\_n=0.01$ when $n$ is a multiple of 100, and $p\_n=0.98$ otherwise.
How does $X\_t$ behave? for most $n$'s (but not most of the time!) it has a sp... | 4 | https://mathoverflow.net/users/1061 | 123196 | 69,134 |
https://mathoverflow.net/questions/103619 | 29 | This question is about *sophistication*, a way of measuring the amount of "interesting, non-random information" in a binary string, which was proposed by Kolmogorov and others in the 1980s. I'll define all the needed concepts below, but for further reading, I recommend [this paper](http://people.cs.uchicago.edu/~fortno... | https://mathoverflow.net/users/2575 | Can a string's sophistication be defined in an unsophisticated way? | Hi Scott,
are you asking whether CSoph(x) may be much more NSoph\_c(x) for every constant c? This question looks strange as NSoph\_c(x) increases, as c decreases. Thus the question reduces to the case c=0 (or may be you allow negative values?). Which question precisely did you have in mind? Note also that Kolmogorov ... | 9 | https://mathoverflow.net/users/31823 | 123200 | 69,136 |
https://mathoverflow.net/questions/123141 | 7 | The following is perhaps a standard question, but i could not find a plain enough answer
by simply searching online.
Q: Given a knot $K$ and its $(p,q)$-cable $K\_{p,q}$ what is a relation
between the Vassiliev invariants of $K$ and $K\_{p,q}$?
In particular, I would be happy with a formula for the 2nd coefficient... | https://mathoverflow.net/users/31808 | Vassilliev invariants of knots and their cables | As I mentioned in a comment, for the degree $2$ invariant $v\_2$ which is the coefficient of $z^2$ in the Conway Polynomial, we have that $v\_2(K\_{p,q})=av\_2(K)+b$. If $K$ is the unknot, this implies that $b=v\_2(T\_{p,q})$, where $T\_{p,q}$ is the $(p,q)$-torus knot (assuming here $p,q$ are relatively prime.) Alvare... | 9 | https://mathoverflow.net/users/9417 | 123220 | 69,145 |
https://mathoverflow.net/questions/122986 | 2 | I am interested in complexity of algorithms which have access to the following peculiar sort of oracle:
Suppose that an invocation of an algorithm f with an input of size n has access to an oracle for f which works only when given input of size n/2 or less. For definiteness, let's say that f(x) for x less than n/2 ca... | https://mathoverflow.net/users/11026 | Size-limited oracles | Computational problems that can be efficiently (i.e., polynomial-time) computed from solutions of the problem on shorter instances are known as *(downward) self-reducible*. A classical example is SAT: given a CNF $\phi$ in variables $x\_0,\dots,x\_n$, let $\phi\_0$ and $\phi\_1$ be the CNFs in variables $x\_0,\dots,x\_... | 3 | https://mathoverflow.net/users/12705 | 123226 | 69,147 |
https://mathoverflow.net/questions/123225 | 3 | Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles over $X$ with fibre $F$ and isomorphism classes of fibre bundles over $Y$ with fibre $F$. I am interested in properties of ... | https://mathoverflow.net/users/25358 | Homotopy Equivalences and Induced Correspondences between Fibre Bundles | Well, if your base map $f\colon X \to Y$ is a homotopy equivalence, then the induced map $\tilde{f}\colon f^{\*}E \to E$ will also be a homotopy equivalence.
First, the restriction of $\tilde{f}$ to any fibre is the identity map. Then write down the sequence of homotopy groups for a fibration for both the fibre bundl... | 2 | https://mathoverflow.net/users/24221 | 123233 | 69,149 |
https://mathoverflow.net/questions/123213 | 8 | Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space of all polynomial functions on $E$. It is well-known (see 9.7 in Claudio Procesi's book on Lie Groups) that as a $G$-mod... | https://mathoverflow.net/users/9947 | Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module | You can give a formula for these numbers in terms of plethysm and internal product and inner product.
The starting point is Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley which is a formula for the Schur functors of the defining representation of the symmetric group in terms of plethysm and inner pr... | 2 | https://mathoverflow.net/users/3992 | 123240 | 69,153 |
https://mathoverflow.net/questions/123224 | 2 | Let $\Gamma=Cay(G,S)$ be a Cayley graph over a group $G$, $H$ be a proper subgroup of $G$ and $\Sigma=Cay(H,T)$ where $S$ and $T$ are inversed-closed subsets of $G$ and $H$ not containing idendity, respectively. Is there any relation between $Aut(\Gamma)$ and $Aut(\Sigma)$ in general? When $T=H\cap S$, what can we say?... | https://mathoverflow.net/users/27831 | Is there any relation between automorphism group of a Cayley graph over a group and over its subgroup? | I am reasonably confident that the real answer to your question is "No". For example,
if $T=H\setminus 1$ then the automorphism group of $\mathrm{Cay}(H,T)$ is the full
symmetric group. For another, take $S=T$; then $\Gamma$ will have a large automorphism
graph, a wreath product. (If you want your Cayley graphs connect... | 1 | https://mathoverflow.net/users/1266 | 123258 | 69,161 |
https://mathoverflow.net/questions/123198 | 15 | Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$:
$$Y=\operatorname{Hom}(\pi\_1(M^3),\operatorname{SL}(2,\mathbb C))$$
$$X=\operatorname{Hom}(\pi\_1(M^3),\operatorname{SL}(2,\mathbb C))//\operatorname{SL}(2,\mathbb C)$$
I would like to think of these as schemes. The... | https://mathoverflow.net/users/35353 | Can the SL_2 character variety of a three-manifold be nonreduced? | There is actually an old (ca 1986) example of nonreduced $SL(2, {\mathbb C})$-representation scheme of a 3-manifold group. Take an oriented Seifert manifold $M$ which fibers over the $S^2(3,3,3)$ orbifold (sphere with three cone points of order 3). The fundamental group of the base-orbifold is von Dyck group with prese... | 15 | https://mathoverflow.net/users/21684 | 123259 | 69,162 |
https://mathoverflow.net/questions/123252 | 8 | Let $X$ be a metric space, $\Sigma\_{1}$ the borel sigma algebra and
$\Sigma\_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma\_{2}$ can it be extended to a
measure on $\Sigma\_{1}?$
| https://mathoverflow.net/users/18384 | Extension of measures from the ball sigma-algebra to the borel sigma-algebra | Take a set $X$ of power $\aleph\_1$, with the discrete metric where two distinct points have distance $1$. The balls are singletons and the whole space. The ball sigma-algebra is the countable and co-countable sets. Let countable sets have measure zero, co-countable sets have measure 1.
Now all subsets are open, so ... | 11 | https://mathoverflow.net/users/454 | 123260 | 69,163 |
https://mathoverflow.net/questions/123140 | 4 | 1. In [Artin1968](http://www.ams.org/mathscinet-getitem?mr=232018) he considers $\underline{analytic}$ equations, but over the ring $R=k\{x\_1,..,x\_n\}$. In [Artin1969](http://www.ams.org/mathscinet-getitem?mr=268188) he works with $R=k\{x\_1,..,x\_n\}/I$, not necessarily regular, but considers $\underline{polynomial}... | https://mathoverflow.net/users/2900 | Artin approximation theorems over non-regular rings/non-Noetherian rings | Concerning (2), here are some references:
For certain subrings of $R[[T\_1,\dots,T\_N]]$ where $R$ is a complete valuation ring of rank 1, see:
H. Schoutens: Approximation properties for some non-Noetherian local rings. Pac. J. Math. 131(2), 331–359 (1988).
For any henselian valuation ring, with fraction field ... | 6 | https://mathoverflow.net/users/7666 | 123263 | 69,165 |
https://mathoverflow.net/questions/123264 | 0 | I suppose this is really an economics question, but I'm posting here for want of a more appropriate forum. My question concerns an *aggregate demand system* in which we have $n$ variants of a product, and the demand function of variant $i$ is given by $$D\_i(p\_1,\dots,p\_n)$$ where $p\_i$ is the price of variant $i$. ... | https://mathoverflow.net/users/31842 | Market-clearing price vector in an "aggregate demand system" | You can find this in an standard graduate-level microeconomics textbook, such as Mas Colell-Whinston-Green. A solution exists with or without the assumption of gross substitutes, and does indeed use a fixed-point theorem, such as the Brouwer or Kakutani fixed point theorem. Abraham Wald has a proof for a special case t... | 2 | https://mathoverflow.net/users/3711 | 123265 | 69,166 |
https://mathoverflow.net/questions/123286 | 9 | It is well-known that log terminal singularities are both rational and log
canonical. It is also well-known that log canonical singularities are not necessarily
log terminal and also not necessarily rational. However, it seems that the usual
example(s) given to show that show this at the same time. What I mean is that ... | https://mathoverflow.net/users/31847 | Rational and log canonical singularity that is not log terminal | This is a good question and it is not entirely trivial, because there is no such example using a simple cone construction. More generally, if the resolution graph of the (normal) singularity is a chain of rational curves, then if the singularity is log canonical, then it is necessarily log terminal. (I won't include th... | 14 | https://mathoverflow.net/users/10076 | 123287 | 69,177 |
https://mathoverflow.net/questions/123290 | 2 | The totient sum function has an identity:
$\displaystyle\sum\_{k=1}^{N}\varphi(k) = \sum\_{k=1}^N {\rm M}(\lfloor\frac{N}{k}\rfloor)k $
$\varphi(k)$ is the Euler totient function, and $M$ is the Mertens function $\displaystyle M(N)=\sum\_{k=1}^N \mu (k)$ where $\mu$ is the Moebius function.
My question: What is ... | https://mathoverflow.net/users/31853 | How to rewrite this totient summation in terms of Mertens? | Since $\varphi(n)=n\sum\_{d|n}\frac{\mu(d)}{d},$ by switching the order of summation we have that for fixed $l$ $$\sum\_{n\leq x}\varphi(n)n^{l}=\sum\_{kd\leq x}\mu(d)k^{l}d^{l}k$$
$$=\sum\_{k\leq x}k^{l+1}\sum\_{d\leq\frac{x}{d}}\mu(d)d^{l}.$$ Now, $$\sum\_{d\leq y}\mu(d)d^{l}=\int\_{0}^{y}t^{l}d\left(M(t)\right)=M... | 1 | https://mathoverflow.net/users/12176 | 123297 | 69,184 |
https://mathoverflow.net/questions/121856 | 5 | I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few answers which satisfy the first requirement, but not the second.
1. **Give a topological space $X$.** The fundamental gro... | https://mathoverflow.net/users/35353 | How to specify a finite group up to inner automorphism? | You can give a monoidal category with a tensor functor to the category of finite sets where tensor is Cartesian product that is equivalent to the category of permutation representations of the group. By Grothendieck's Galois theory, one can recover the group from this. This improves on linear representations because yo... | 2 | https://mathoverflow.net/users/18060 | 123299 | 69,185 |
https://mathoverflow.net/questions/123209 | 2 | Hi everybody.
In "M. B. Nathanson - Elementary Methods in Number Theory" is shown (Theorem 7.14) that if $A$ is a set of positive integers such that $\sum\_{a \in A} 1 / a$ converges then the set of multiples of $A$ has a natural density (a set of positive integers $S$ has a natural density if exists $\lim\_{x \to \i... | https://mathoverflow.net/users/21706 | Natural density of a set of positive integers not in certain congruence classes | Here is an amplification of @Greg Martin's answer.
Let $S$ be any set whose upper density and lower density differ. Let the upper and lower densities be $\alpha$ and $\beta$ respectively. Write $S(N)$ for $|S\cap[1,N]|$. Let $a\_1 < a\_2 < \ldots $ be the increasing enumeration of $S^c$. Let $\delta<\alpha-\beta$.
... | 4 | https://mathoverflow.net/users/11054 | 123300 | 69,186 |
https://mathoverflow.net/questions/123283 | 3 | I suppose this is a question with a well known answer. Suppose $A$ and $B$ are two algebras over some field and there is a map
$$
f: \operatorname{K\_0}(A) \to \operatorname{K\_0}(B)
$$
is it necessarily induced by a tensor product with some bimodule? If not in general, is it true for some reasonable $A$, $B$ and $f$? ... | https://mathoverflow.net/users/21029 | Morphisms between $K_0$ | I'd say no to both questions.
1) If $k$ is a field then $K\_0(k)= \mathbb Z$ generated by $[k]$ and the class of any $k$-module is positive, so $-n\colon K\_0(k)\rightarrow K\_0(k)$ cannot be induced by a bimodule, $n>0$.
2) If $k$ is a field of positive characteristic and $k'$ is a field of characteristic $0$ th... | 10 | https://mathoverflow.net/users/12166 | 123304 | 69,187 |
https://mathoverflow.net/questions/122953 | 5 | Is there a classification theory of smooth projective varieties with Picard number 1?
By Lefschetz theorem, if $X$ is a complete intersection variety of dimension at least $3$, then the Picard number $\rho(X)=1$.
Can some one give an example of smooth projective variety with picard number 1 which is not a complete... | https://mathoverflow.net/users/2348 | Smooth projective varieties of Picard number one | Picard number $1$ is the most frequent case among all varieties, so you cannot expect a classification. It's quite the opposite, you might stand a chance to classify those (within some class) that have Picard number larger than $1$. For instance a general $K3$ surface has Picard number $1$ and the locus of those with a... | 7 | https://mathoverflow.net/users/10076 | 123307 | 69,189 |
https://mathoverflow.net/questions/123306 | 7 | There has been much work done on the kissing number problem (of determining the greatest number of congruent spheres which can touch a single sphere in a packing) in Euclidean space for dimensions $1$ to $24$ and even some asymptotic work. My question is whether or not the kissing number problem has been studied in Non... | https://mathoverflow.net/users/20343 | Kissing Number of Spheres in Non-Euclidean Geometry | To understand the cases of spherical or hyperbolic geometry, it is helpful to think in terms of spherical codes. A Euclidean kissing configuration is equivalent to an arrangement of points on a sphere with all angles between them at least 60 degrees (these are the points of tangency with the surrounding spheres). The s... | 12 | https://mathoverflow.net/users/4720 | 123308 | 69,190 |
https://mathoverflow.net/questions/123219 | 2 | Let $K$ be a field. For a matrix $A\in GL\_n(K)$ we can find the Jordan normal form $A'$ in $GL\_n(\overline{K})$, where $\overline{K}$ is the algebraic closure of $K$. We write $j\_\alpha(A)$ for the number of $1\times 1$ Jordanblocks with eigenvalues $\alpha$ in $A'$. Now consider $\alpha\notin K$ and the correspondi... | https://mathoverflow.net/users/43085 | Multiple eigenvalues over imperfect fields | To draw a conclusion from the comments, assume $\alpha$ is an eigenvalue of $A$, where $K$ has characteristic $p$. If $K(\alpha):K$ is inseparable, the minimal polynomial of $\alpha$ over $K$ is inseparable, hence $f'=0$. On the other hand, if $\alpha$ is the only root of $f$, $f$ divides the characteristic polynomial ... | 2 | https://mathoverflow.net/users/43085 | 123311 | 69,191 |
https://mathoverflow.net/questions/123318 | 0 | Let's consider a directed graph with positive edge weights. For every vertex we determine the difference
**D = (summary weight of edges directed FROM this vertex)-(summary weight of edges directed INTO this vertex).**
We are given the difference D for every vertex. Sum of all the D's is always 0. What is the best ... | https://mathoverflow.net/users/31860 | Minimum number of edges - directed graph with given sums of weights | First, divide the vertex weights into a maximum number of portions that each sum to 0 and handle them separately.
For each set of vertices whose weights sum to 0, you can can implement them as a path with edges forward or backward. Just arrange them in arbitrary order and decide one edge at a time. For example for $a... | 2 | https://mathoverflow.net/users/9025 | 123320 | 69,195 |
https://mathoverflow.net/questions/123191 | 2 | Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K\_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. Also assume that $A$ is an effective Cartier divisor, then $\tau(X, \Delta)=\tau(X, \Delta+\varepsilon A)$ for $0<\varepsilon... | https://mathoverflow.net/users/26003 | On a Strongly F-regular Pair (X, \Delta) | It isn't true as stated unfortunately. For example, take $X$ to be an ordinary elliptic curve, $\Delta = 0$ and $M = 0$. Then $S^0(X, \tau(X) \otimes O(M)) = H^0(X, O\_X)$. However, for any effective Cartier $A > 0$ and any $\varepsilon > 0$, we have $S^0(X, \tau(X, \varepsilon A) \otimes O\_X(M)) = 0$ (this can be che... | 1 | https://mathoverflow.net/users/3521 | 123324 | 69,197 |
https://mathoverflow.net/questions/123310 | 4 | Let $S^2$ be the 2-sphere and $g$ some metric on it. Is it possible
1. to construct embedding $\iota:S^2\rightarrow \mathrm{R}^3$ s.t. $g=\iota^\* g\_{\mathrm{R}^3}$ and
2. decide when given $(S^2,g)$, there is such an embedding as above?
With $\iota: S^1\rightarrow \mathrm{R}^n$ ($n=2,3$) constant metrics generate... | https://mathoverflow.net/users/31760 | Construct embedding given metric | Although, as jc says, this question was addressed in earlier questions, the answer is scattered among the accepted answer as well as the comments. So here is a summary:
1. By the Nash-Kuiper theorem (which I believe is one of the original uses of the so-called h-principle), there always exists a global $C^1$ isometri... | 10 | https://mathoverflow.net/users/613 | 123325 | 69,198 |
https://mathoverflow.net/questions/123278 | 5 | Can anyone tell me if there is a Mayer-Vietoris sequence for an arbitrary homotopy pushout (hence homotopy pullback) of spectra and an arbitrary (co)homology theory. If this comes from some easy way of writing down a pushout/pullback as a fiber sequence, it'd be really cool to see that spelled out (as if I were a baby)... | https://mathoverflow.net/users/11546 | Mayer-Vietoris Sequence for Arbitrary Bicartesian Square of Spectra | Mayer-Vietoris sequences can be obtained from excision isomorphisms.
Anything worthy of the name "homology theory" will give a long exact sequence
$$\dots \to h\_n(A)\to h\_n(X)\to h\_n(A\to X)\to h\_{n-1}(A)\to \dots$$
for each morphism $A\to X$. And for a square, a.k.a. map of morphisms $(C\to B)\to (A\to X)$, it... | 7 | https://mathoverflow.net/users/6666 | 123330 | 69,202 |
https://mathoverflow.net/questions/123319 | 2 | Assume $\Gamma$ is the Cayley graph of an amenable$^{\*}$ group and that the simple random walk has non-trivial Poisson boundary$^{\*\*}$. Is there a spanning connected subgraph $\Gamma'$ of some $k$-fuzz$^{\*\*\*}$ of $\Gamma$ such that the Poisson boundary of $\Gamma'$ is trivial but the growth of balls in $\Gamma'$ ... | https://mathoverflow.net/users/18974 | Spanning subgaph with trivial Poisson boundaries | I don't think much is currently known. Concerning $(\ast\ast)$, there are also finitely presented groups with this property (although they are essentially obtained from wreath products) - see Erschler, Liouville property for groups and manifolds, Invent. Math. 155 (2004). As for $(\ast\ast\ast)$, it is a particular cas... | 3 | https://mathoverflow.net/users/8588 | 123332 | 69,203 |
https://mathoverflow.net/questions/123331 | 16 | Is it algorithmically decidable if two finitely presented amenable groups are isomorphic?
Or slightly different:
Does there exist a family of amenable groups (indexed by natural numbers) for which one cannot algorithmically decide if two elements of the family are isomorphic?
| https://mathoverflow.net/users/31864 | Is the isomorphism problem for amenable groups decidable? | **EDIT:** The isomorphism problem for finitely presented solvable groups in the variety of all solvable groups of derived length $\le 7$ is undecidable. This was proved by Kirkinskiĭ and Remeslennikov (Kirkinskiĭ, A. S.; Remeslennikov, V. N.
`The isomorphism problem for solvable groups.' (Russian)
Mat. Zametki 18 (1975... | 26 | https://mathoverflow.net/users/7644 | 123338 | 69,206 |
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