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https://mathoverflow.net/questions/61520 | 9 | Perhaps this is a bit naïve, but I was wondering if it possible to (at least formally) represent [Bundle Gerbes](http://en.wikipedia.org/wiki/Bundle_gerbe) as Characteristic Classes. Disclaimer: My understanding of Bundle Gerbes is limited to [this](http://arxiv.org/abs/math.dg/0508618) paper of Hitchin so perhaps I'm ... | https://mathoverflow.net/users/14157 | Bundle Gerbes as Characteristic Classes | The fact that you are dealing with *compact* and/or *finite dimensional* Lie groups is completely irrelevant. The fact that these group are *Lie* is also partially irrelevant (unless you care about putting connections on your bundle gerbes, in which case it becomes very relevant).
More relevant is whether the groups *a... | 8 | https://mathoverflow.net/users/5690 | 61560 | 38,127 |
https://mathoverflow.net/questions/61569 | 8 | (I apologize in advance for the vagueness of my question). Let $G$ be a reductive algebraic group over $\mathbb C$ with Lie algebra $\frak g$ and Borel $B$. I have seen casual references to the fact that relative Lie algebra cohomology (sometimes also called cohomology of a pair) with coefficients in a $B$-module $V$ i... | https://mathoverflow.net/users/1528 | Relative Lie Algebra cohomology and sheaf cohomology | The statement you're looking for is that for a representation $V$ of $G$, the sheaf cohomology of the vector bundle $G\times\_B V$ on $G/B$ coincides with the Lie algebra cohomology for $\mathfrak{b}$ acting on $V$. This goes back to [Kostant](http://scholar.google.com/scholar?cluster=1752913425771819379&hl=en&as_sdt=0... | 11 | https://mathoverflow.net/users/66 | 61571 | 38,134 |
https://mathoverflow.net/questions/61555 | 1 | Let $A,B$ be two local rings and put $\mathfrak{m}\_A, \mathfrak{m}\_B$ their maximal ideals. Now suppose that we have an injection $0 \to A \to B$ and put $\mathfrak{n} := A \cap \mathfrak{m}\_B $. It seems to me quite obvious that it should be $\mathfrak{m}\_A = \mathfrak{n}$ (at least in the geometric way of thinkin... | https://mathoverflow.net/users/14339 | maximal ideal in local subrings | Daniel Litt is exactly right, it is not necessary in general.
However, in the setting of algebraic geometry you are fine. Suppose now that $A$ and $B$ are obtained by localizing finitely generated $k$-algebras at a maximal ideal. Then $B/{\mathfrak{m}}\_B$ is a finite extension field of $k$ (if $k$ is algebraically c... | 6 | https://mathoverflow.net/users/3521 | 61572 | 38,135 |
https://mathoverflow.net/questions/61530 | 0 | Let $\mathcal{A}$ be an abelian category and let $\mathcal{C}$ be the (abelian) category of complexes of objects in $\mathcal{A}$. Suppose we have a small indexing category $\mathcal{I}$ and a functor $F:\mathcal{I}\to \mathcal{C}$, which defines a diagram indexed by $\mathcal{I}$ which gives a complex $C\_i$ for each ... | https://mathoverflow.net/users/14341 | Boundary operator in the colimit of complexes. | For each integer $n$, the $n$th term $C[n]$ in the colimit complex $C$ is the colimit of the $n$th terms $C\_i[n]$. For each $i \in \mathcal{I}$ and each integer $n$, the $n$th boundary map in $C\_i$ is $\partial: C\_i[n] \to C\_i[n-1]$, so you can compose with your chain map $f\_i$ (restricted to $C\_i[n-1]$) to get $... | 1 | https://mathoverflow.net/users/4194 | 61582 | 38,140 |
https://mathoverflow.net/questions/61334 | 12 | Consider [nerve](http://en.wikipedia.org/wiki/Nerve_of_a_covering) $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$.
[Helly theorem](http://en.wikipedia.org/wiki/Helly_theorem) says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N\_n$.
It seems that not all finite simplicial... | https://mathoverflow.net/users/10330 | Helly theorem + Nerve | ### Nerves of convex sets in general
There is quite a bit known about nerves of families of convex sets in $R^n$. Indeed Helly's theorem asserts that the $n$-skeleton determines the entire complex. In fact, considerably more is known beyond Helly's theorem. It follows, for example that all homology group of the nerve... | 12 | https://mathoverflow.net/users/1532 | 61583 | 38,141 |
https://mathoverflow.net/questions/61573 | 4 | Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}\_m)$, and want to know if this is part of a systematic collection of objects. For example, is there a "free ring scheme on $\mathbb{G}$"?
... | https://mathoverflow.net/users/9509 | Tensor and Hom objects for finite flat group schemes | The answer to your question is 'no'. If there existed a tensor product for finite flat group scheme, there would exist one for $p$-divisible groups (over rings of integers of $p$-adic fields, for example), and for the corresponding (crystalline) Galois representations. Yet there are certain weights for these representa... | 4 | https://mathoverflow.net/users/2191 | 61586 | 38,142 |
https://mathoverflow.net/questions/61585 | 8 | A bounded operator $A$ in a Hilbert space is called nilpotent if there exists $n$ such that $A^{n}=0$. An operator is called quasi-nilpotent iff
$$
\limsup\_{n\to\infty}{ \|A^{n}\|^{1/n}}=0.
$$
Every nilpotent operator is clearly quasi-nilpotent. The family of quasi-nilpotent operators is very important for the hyper... | https://mathoverflow.net/users/13825 | Limits of Nilpotent and Quasi-nilpotent Operators in a $\mathrm{II}_1$-factor | The definition of quasi-nilpotent is that the spectrum of $A$ is zero, which is the same as $\|A^n\|^{1/n} \to 0$.
See Apostol's paper On the norm-closure of nilpotents, Rev. Roumaine Math. Pures Appl. 19 (1974), 277-282 or the later paper Apostol, C.; Foiaş, C.; Pearcy, C.
That quasinilpotent operators are norm-limi... | 4 | https://mathoverflow.net/users/2554 | 61589 | 38,145 |
https://mathoverflow.net/questions/60727 | 4 | In $\mathbb{R}^n$, given an unit $L\_1$ sphere $\mathcal{B}\_n: |x\_1|+|x\_2|+\ldots+|x\_n|\leq 1$ and a hyperplane $\mathcal{P}: a\_1x\_1+a\_2x\_2+\ldots+a\_nx\_n=0$. Does there always exist a rotation such that $\mathcal{B}\_n\cap\mathcal{P}$ is embedded into the $(n-1)$ dimensional unit $L\_1$ sphere: $\mathcal{B}\_... | https://mathoverflow.net/users/14127 | Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere. | Thanks for the comments above. I have just proofed this problem is only true when $n\leq 4$.
When $n\leq 4$, without loss of generality, assume $|a\_n|\geq |a\_1|, \ldots, |a\_{n-1}|$. Let $Q$ be the rotation on the plane spanned by $(a\_1, \ldots, a\_n)$ and $(0,\ldots, 0, 1)$ such that applying $Q$ to $(a\_1, \ldot... | 1 | https://mathoverflow.net/users/14127 | 61594 | 38,147 |
https://mathoverflow.net/questions/61593 | 4 | Why are the Killing fields on a complete Riemannian manifold themselves complete (that is, the integral curves of the Killing fields are defined for all time)?
| https://mathoverflow.net/users/14359 | Geodesic completeness and complete Killing fields | The corresponding flow, say $\Phi^t: M\to M$ preserves the metric and the field.
Thus, for any $x\in M$, the curve $\alpha\_x\colon t\mapsto \Phi^t(x)$ has constant speed.
Therefore it can not escape to infinity in finite time.
More precisely:
if $\alpha\_x$ is defined on a bounded interval $(a,b)$
then the restric... | 9 | https://mathoverflow.net/users/1441 | 61598 | 38,148 |
https://mathoverflow.net/questions/61596 | 6 | Let $X\subset \mathbb{P}^n$ be a smooth projective variety with ideal sheaf $I\_X$. The conormal sequence is given by
$$
0\to I\_X/I\_X^2\to \Omega\_{\mathbb{P}^n}|\_X\to \Omega\_{X}\to 0.
$$
For which varieties $X$ is the sequence above split?
If I'm not mistaken, if $X$ a hypersurface, the sequence is split if a... | https://mathoverflow.net/users/3996 | When does the conormal bundle sequence split? | You are right. This is a result due to Van de Ven.
[A. Van de Ven, A property of algebraic varieties in complex projective spaces. In: Colloque Géom. Diff. Globale (Bruxelles, 1958), 151–152, Centre Belge Rech. Math., Louvain 1959. MR0116361 (22 #7149) Zbl 0092.14004]
Even more is true. Recently [Ionescu and Repet... | 10 | https://mathoverflow.net/users/605 | 61609 | 38,155 |
https://mathoverflow.net/questions/61597 | 10 | Background: As a result of teaching recurrence relations in various courses over the years, I am working on a paper on periodic integer sequences generated by second-order, linear, homogeneous recurrences with constant coefficients.
Briefly, if the recurrence generates a sequence of integers, must the coefficients be... | https://mathoverflow.net/users/14321 | Integer sequences and integer coefficients in recurrence relations. | This is *Fatou's lemma*. One reference is Exercise 4.1(a) of my book *Enumerative Combinatorics*, vol. 1. This is repeated as Exercise 4.2(a) at <http://math.mit.edu/~rstan/ec/ec1.pdf>.
| 14 | https://mathoverflow.net/users/2807 | 61611 | 38,156 |
https://mathoverflow.net/questions/61614 | 4 | I am split between two graduate schools in my decision. One of them is ranked about 25 places higher on the latest NRC rankings. The higher-ranked school emphasizes research, and over half its graduates get postdocs, while only around 25% of its graduates go straight into tenure-track positions. I spoke to the graduate... | https://mathoverflow.net/users/14363 | As a student split very closely between seeking a research university career and a liberal arts career after getting a math Ph.D., should I choose a school that prioritizes research or teaching? | You should go to the higher-ranked school. You are likely to be surrounded by stronger students and learn more math there. Unless the higher-ranked school is one of a very few VERY highly-ranked places, the faculty will be very friendly to and supportive of your ambitions towards LAC teaching, should that be the direct... | 10 | https://mathoverflow.net/users/431 | 61617 | 38,160 |
https://mathoverflow.net/questions/61615 | 22 | In [Cube-free infinite binary words](https://mathoverflow.net/questions/61373) it was established that there are infinitely many cube-free infinite binary words (see the earlier question for definitions of terms). The construction given in answer to that question yields a countable infinity of such words. In a comment ... | https://mathoverflow.net/users/3684 | Are there uncountably many cube-free infinite binary words? | Denote by $\mu$ the mapping from the Thue-Morse sequence, $\mu(0)=01$ and $\mu(1)=10$. Now define a sequence of maps from binary words to binary words, $g$, so that $g\_{\emptyset}(w)=w$, $g\_{0b}(w)=\mu^2(g\_{b}(w))$ and $g\_{1b}(w)=0\mu^2(g\_{b}(w))$. Now given an infinite binary sequence $B=b\_1b\_2\dots$ define $w\... | 12 | https://mathoverflow.net/users/2384 | 61622 | 38,163 |
https://mathoverflow.net/questions/61621 | 0 | We view subsets of the natural numbers as their characteristic functions, which are elements of the Cantor space $2^\mathbb{N}$. We take the uniform probability measure on the Cantor space. Under this view, what is the measure of the family of all productive sets (in the sense of computability theory)? Immune sets? Set... | https://mathoverflow.net/users/14024 | What is the measure of productive and immune sets in the Cantor space? | Isn't this revised question just the one Andrej Bauer suggested and I answered in the comments on your earlier continued-fraction version? In the uniform measure on subsets of $\mathbb N$ (equivalent, if you replace sets with their characteristic functions, to the product measure on $\{0,1\}^{\mathbb N}$ arising from t... | 4 | https://mathoverflow.net/users/6794 | 61626 | 38,165 |
https://mathoverflow.net/questions/61623 | 9 | As with [this older related question](https://mathoverflow.net/questions/56200/proper-way-to-deal-with-papers-youve-already-refereed), question anonymous for obvious reasons.
If I have been asked to review the same paper twice, is it OK to acknowledge in my review that I am the same person as one of the referees from... | https://mathoverflow.net/users/14365 | Is it OK for a referee to acknowledge identity with a previous referee? | I have done this as a reviewer, and have had it done to me as an author. I have no problem with it. I do think you should disclose to the editor that you've reviewed the paper before. (This can create the awkward situation of having to say "I reviewed this for journal A, and didn't think the theorem was interesting eno... | 12 | https://mathoverflow.net/users/431 | 61630 | 38,168 |
https://mathoverflow.net/questions/61606 | 9 | It is well-known that every closed surface in $\mathbb R^3$ having constant Gauss curvature is a round sphere. Inspired by [this question](https://mathoverflow.net/questions/61143), I'd like to ask whether a similar rigidity holds for centro-affine curvature.
More precisely, let $M\subset\mathbb R^3$ be a smooth clos... | https://mathoverflow.net/users/4354 | surfaces of constant centro-affine curvature | This is a classical question that was probably posed by Blaschke and partially solved by Jorgens and Calabi but I think was finally solved completely by Cheng and Yau in "Complete Affine Hypersurfaces. Part I. The Completeness of Affine Metrics" in CPAM 1986, if you assume sufficient regularity of the boundary. These b... | 8 | https://mathoverflow.net/users/613 | 61633 | 38,170 |
https://mathoverflow.net/questions/48944 | 7 | I'm hoping to learn something about planar algebras while attacking a planar algebra question with an undergrad research student. I'm thinking about reading [this paper](http://arxiv.org/PS_cache/arxiv/pdf/0808/0808.0764v3.pdf), as Kuperberg's program seems like the sort of thing I'm looking for, but maybe there are be... | https://mathoverflow.net/users/6269 | What are some natural and attractive open problems in Jones's theory of planar algebras? | One important kind of question in planar algebras is given generators and relations for a planar algebra can you:
* Find an algorithm which takes an arbitrary closed diagram and evaluates it to give a number
* Show that any two ways of evaluating a closed diagram gives the same answer
* Find an explicit basis for eve... | 8 | https://mathoverflow.net/users/22 | 61637 | 38,173 |
https://mathoverflow.net/questions/61570 | 7 | Is the reduction $X\_{red}$ of a flat, finite, surjective scheme $X$ over an integral base $S$ still flat?
I could possibly add that I am already aware we can assume the base $S$ to be local and complete, and we can assume $X$ is local (and henselien). So far this hasn't seemed to help me.
My intuation is that I sh... | https://mathoverflow.net/users/12914 | Is the reduction of a flat, finite, surjective scheme over an integral base still flat? | In analytic geometry, you can see to the Douady's example in Fischer book (p.151) where $X:=\lbrace{(x,s,t)\in {\Bbb C}^{3}: x^{3}+sx +t=0; 27t^{2}+4s^{3}=0}\rbrace$, $S:=\lbrace{(s,t): 27t^{2}+4s^{3}=0\rbrace}$ and $f:X\rightarrow S$ is a finite, surjective and flat map (induced by the canonical projection). Then, it ... | 5 | https://mathoverflow.net/users/14378 | 61663 | 38,186 |
https://mathoverflow.net/questions/61664 | 1 | Let's say that I have a loaded $d$-sided die where the relative probabilities for the die landing on a particular side, $(p\_1, ..., p\_d)$, are known. How many times must I roll the die to, on average, to sample all possible outcomes at least once?
I hope this isn't too low level a question...
Edit: Didier just ni... | https://mathoverflow.net/users/14324 | Number of required trials to sample all possible states of a 'd'-sided loaded die | For the symmetric case when $p\_i=1/d$ for every $i$, see [here](https://math.stackexchange.com/questions/28905). The answer is
$$
E(T)=\sum\_{k=1}^d\frac{d}{k},
$$
where $T$ is the number of times the die has been rolled at the first instant when all the possible outcomes have been sampled at least once.
In the gen... | 5 | https://mathoverflow.net/users/4661 | 61668 | 38,190 |
https://mathoverflow.net/questions/60873 | 2 | In Selick's very pretty paper "Odd primary torsion in $\pi\_\*(S^3)$" he makes use of an automorphism which was established by Toda in his paper "On the double suspension $E^2$".
Unfortunately, Selick just refers to [10], but the paper is about 40 pages and dense reading; after skimming it several times, I fear that ... | https://mathoverflow.net/users/3634 | Reference for an automorphism in a paper of Toda | Do you mean the argument in the last page of Selick's paper?
If so, he does not make use of an "automorphism". What he wanted to say is the difference between the map
$$(H'\circ\Omega\gamma)\_\* : H\_\*(\Omega S^{2p+1}\{p\}) \longrightarrow H\_\*(\Omega S^{2p+1}\_{(p)})$$
and the map $(\Omega i)\_\*$ is given by a no... | 2 | https://mathoverflow.net/users/377 | 61672 | 38,193 |
https://mathoverflow.net/questions/61632 | 61 | The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless there were a good conceptual reason for the contrary. The typical situation would be for a proof in finite combinatorics ... | https://mathoverflow.net/users/2051 | What is the high-concept explanation on why real numbers are useful in number theory? | The [Gödel Speedup Theorem](http://math.ucsd.edu/~sbuss/ResearchWeb/godelone/index.html) provides some explanation why real numbers (and variants) are useful in proving statements in number theory.
Real numbers, complex numbers, and $p$-adic numbers are second-order objects over the natural numbers. Thus a proof of a... | 79 | https://mathoverflow.net/users/2000 | 61677 | 38,195 |
https://mathoverflow.net/questions/61678 | 36 | Algebraic geometry is quite new for me, so this question may be too naive. therefore, I will also be happy to get answers explaining why this is a bad question.
I understand that the basic philosophy begins with considering an abstract commutative ring as a function space of a certain "geometric" object (the spectrum... | https://mathoverflow.net/users/14379 | What is the general geometric interpretation of modules in algebraic geometry? | Roughly a module can be thought of as a vector bundle on the spectrum, where the dimension of fibers may vary. Let me give some examples and facts:
* A free modules corresponds to trival vector bundles, or more generally projective modules correspond to vector bundles as you already pointed out.
* Let $R$ be the coor... | 40 | https://mathoverflow.net/users/2837 | 61680 | 38,197 |
https://mathoverflow.net/questions/61661 | 6 | Hempel, in his 1987 article "Residual Finiteness for 3-Manifolds", shows that if $M$ is a compact Haken 3-manifold, then $\pi\_1(M)$ is residually finite. The outline of the proof is basically:
1. Reduce to the case $M$ is closed and irreducible;
2. Use the JSJ decomposition to split $M$ along tori, getting pieces wh... | https://mathoverflow.net/users/1446 | Residual Finiteness of Fundamental Group of Compact 3-Manifold | For the first question, the answer is yes. Geometrization implies that the only non-Haken manifolds irreducible manifolds are compact hyperbolic manifolds (with no cusps), and there again $\pi\_1$ is residually finite.
For the second part, I'm not aware of other arguments, and I think this is still the standard way t... | 7 | https://mathoverflow.net/users/5010 | 61681 | 38,198 |
https://mathoverflow.net/questions/61692 | 2 | Recall that the analytic rank $r^{\rm an}(E)$ of a (modular)
elliptic curve $E$ is defined to be the order of vanishing of its
Hasse-Weil $L$-function $L(E,s)$ at $s=1$. A conjecture due to
Ralph Greenberg in [MR1260957 (95a:11059)] implies in particular
the following:
Basic fact: Let $\Sigma$ be a finite set
of prim... | https://mathoverflow.net/users/11928 | Elliptic curves with arbitrarily large conductor | The conductor is divisible by the same primes as the discriminant. The discriminant of $y^2=x^3+b$ is $-432b^2$. So you can certainly make the conductor divisible by an arbitrarily large number of distinct primes, and by whichever primes you want (as long as you want 2 and 3).
| 5 | https://mathoverflow.net/users/3684 | 61696 | 38,204 |
https://mathoverflow.net/questions/61695 | 11 | It is well known that for *rings*, Artinian implies Noetherian (the famous Hopkins–Levitzki theorem) and it is also well known that there are Artinian *modules* which are not Noetherian. A simple example can be found in
<http://en.wikipedia.org/wiki/Artinian_module#Relation_to_the_Noetherian_condition>
Since rings... | https://mathoverflow.net/users/14379 | Are there any finitely generated artinian modules that are not Noetherian? | Suppose you have an Artinian but not Noetherian finitely generated $R$ module $M$. Let $0\leq M\_1\leq M\_2\leq \cdots \leq M\_n=M$ be a finite chain of $R$-modules such that each composition factor $M\_i/M\_{i-1}$ is cyclic for each $i$.
Certainly each composition is Artinian since subquotients of Artinian modules ... | 21 | https://mathoverflow.net/users/345 | 61700 | 38,205 |
https://mathoverflow.net/questions/61610 | 6 | Does there exist a Borel measure or any valid measure on an infinite dimensional Banach space such that a bounded open set in this space has a positive measure ?
| https://mathoverflow.net/users/14362 | Measures on infinite dimensional Banach spaces | The negative result mentioned in the comment by Zen Harper, is about invariant measures, non-existent on infinite-dimensional Banach spaces. If one does not require the invariance, there is no problem. See, for example, the calculation of the Gaussian measure of a ball in the paper
<http://titan.math.udel.edu/~wli/pa... | 11 | https://mathoverflow.net/users/12205 | 61714 | 38,212 |
https://mathoverflow.net/questions/61697 | 4 | The hamiltonian flow box theorem, as stated in Abraham and Marsden's Foundations of Mechanics, says that:
Given an hamiltonian system $(M,\omega,h)$ with $dh(x\_0)\neq 0$ for some $x\_0$ in $M$, there is a symplectic chart $(U,\phi)$ on $M$ centered at $x\_0$ such that $\phi\_{\ast}h(x)=h(x\_0)+\omega\_0(\phi\_{\ast}... | https://mathoverflow.net/users/12617 | On the proof of the hamiltonian flow box theorem | My guess would be that you can find such a proof in the literature, since the Moser trick is such a powerful tool, though I don't know where.
Instead let me sketch a proof of the fact that any two Hamiltonian systems $(M\_i,\omega\_i,h\_i)$ are locally isomorphic around non deg. points $x\_i\in M\_i, i=0,1$ using th... | 5 | https://mathoverflow.net/users/745 | 61719 | 38,214 |
https://mathoverflow.net/questions/61701 | 8 | Let $G$ be a semisimple linear algebraic group, $P$ be a parabolic subgroup and $w$ be an element of the Weyl group of $G$. I want to calculate the Picard group of the Schubert variety $X\_P(w):=\overline{BwP/P} \subset G/P$. I'm particularly interested in the case of a maximal parabolic subgroup, but I suppose the gen... | https://mathoverflow.net/users/14385 | Picard group of Schubert varieties | Alex Yong and I work this out in our paper for the case of the Borel subgroup, but I'm pretty sure it's the same for every parabolic.
When is a Schubert variety Gorenstein?, Advances in Math. 207 (2006), 205-220.
Please note our conventions in that paper are backwards from yours in that our Schubert varieties are $... | 5 | https://mathoverflow.net/users/3077 | 61721 | 38,215 |
https://mathoverflow.net/questions/61644 | 1 | This is the mirror of [previous post](https://math.stackexchange.com/questions/32757/invariance-of-walls-self-intersection-under-the-regular-homotopy).
For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point s... | https://mathoverflow.net/users/13453 | Regular homotopy invariance of Wall's self-intersection form. | The double points of a generic immersion $f:P^p \to Q^q$ of closed manifolds without triple points constitute a closed $(2p-q)$-dimensional manifold $S\_2[f]$ (defined as in the question). This is also true for a generic immersion $(f,\partial f):(P,\partial P) \to (Q,\partial Q)$ of manifolds with boundary: in this ca... | 3 | https://mathoverflow.net/users/732 | 61723 | 38,217 |
https://mathoverflow.net/questions/61720 | 5 | Let $f:\mathbb R^2\rightarrow\mathbb R$ be a symmetric function: $f(y,x)=f(x,y)$. It can therefore be written has a function of the elementary symmetric polynomials, here $f(x,y)=F(x+y,xy)$, where $F(\sigma,\mu)$ is defined over $4\mu\le\sigma^2$.
Let me assume that $f$ is of class $\mathcal C^r$. It is clear from th... | https://mathoverflow.net/users/8799 | Symmetric functions and regularity | Just note that $F(\sigma,\mu)=f\left(\frac{\sigma+\sqrt{\sigma^2-4\mu}}{2}, \frac{\sigma-\sqrt{\sigma^2-4\mu}}{2}\right)$.
| 7 | https://mathoverflow.net/users/7666 | 61727 | 38,220 |
https://mathoverflow.net/questions/61733 | 2 | Suppose that $x\sim N(0, V)$ is $p$ dimensional with $V$ diagonal having elements $v\_i^2$. Then
\begin{align}
f\_x(x) & \propto \left(\prod\_p v\_i\right)^{-1} \exp\left(-\frac{1}{2}\sum\_p \frac{x^2\_i}{v\_i^2}\right) \\[10pt]
& \propto \left(\prod\_p v\_i\right)^{-1} \exp\left(-\frac{1}{2}\sum\_p \frac{\|x\|\_2^2x... | https://mathoverflow.net/users/12064 | Dependence between direction and magnitude of multivariate normal random vector | Your reasoning looks right, although I'm not that familiar with the exact notation you're using, except that the $v\_i$ should be in the denominator, not the numerator.
In the second case the answer is yes. In general, say you have any norm $\| \cdot \|$ on $\mathbb{R}^p$. There is a measure $\mu$ on the boundary of ... | 5 | https://mathoverflow.net/users/1044 | 61736 | 38,225 |
https://mathoverflow.net/questions/61729 | 9 | Hello,
My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$:
$c \neq 0$, $c \neq 1$, $c \neq 1+1$, $c \neq 1+1+1$, etc...
and $c=ab \implies a=1~~ou~~b=1$. We obtain a system of axioms $S$.
$S$ is consistent, by... | https://mathoverflow.net/users/12806 | Non-Standard Prime | First, you haven't actually specified a particular field,
since the field $F$ that you have will depend on your
choice of $c$ and of $M$. For example, different
nonstandard models can seriously affect even the
cardinality of the field $F$ that you produce, so they are
not all the same. (A Lowenheim-Skolem argument show... | 14 | https://mathoverflow.net/users/1946 | 61738 | 38,226 |
https://mathoverflow.net/questions/61741 | 4 | I asked this [question](https://math.stackexchange.com/questions/31675/how-to-prove-an-extension-of-zfc-is-conservative) on math.stackexchange.com and got some great comments from Andres Caicedo, but no answers. His comments pointed me toward class forcing, but that's way out of my field. I'm basically an infant in adv... | https://mathoverflow.net/users/10828 | How to prove an extension of ZFC is conservative | If all the axioms are indeed of the indicated form, then they'd be satisfied if you interpret $R$ as the universally true relation. Of course, that makes your axioms conservative. But I conjecture that what it really shows is that you have, at least implicitly, some additional axioms in mind that are not of the indicat... | 4 | https://mathoverflow.net/users/6794 | 61747 | 38,231 |
https://mathoverflow.net/questions/61718 | 13 | I don't seem to be able to find anything written about Spin TQFT's in dimension (1+1). Does anyone know any references? Or is there some reason it is uninteresting?
| https://mathoverflow.net/users/318 | Spin TQFT's in dimensions (1+1) | This is covered in Moore and Segal ["D-branes and K-theory in 2D topological field theory"](http://www.physics.rutgers.edu/~gmoore/Dbranes_Ktheory_Final.pdf). In particular on around page 16 there is a characterization analogous to "1+1 TQFTs = Commutative Frobenius algebras".
| 9 | https://mathoverflow.net/users/184 | 61748 | 38,232 |
https://mathoverflow.net/questions/61634 | 11 | Does anyone know how to estimate (as $n$ tends to infinity) the number of solutions of
$$n=x+y+z$$
where $x,y,z$ are positive integers with $x$ coprime to $y$ and to $z$?
Computer experiments suggest that there are roughly $cn^2$ solutions, where $c>0$ is an absolute constant.
| https://mathoverflow.net/users/14367 | Number of certain positive integer solutions of n=x+y+z | For $n$ prime my heuristics tells me that $c=\frac{1}{2}\prod\_p\left(1-\frac{2}{p^2}\right)$, the product being over all primes. Is this supported by computer experiments? If yes, I will share more details.
| 6 | https://mathoverflow.net/users/11919 | 61757 | 38,238 |
https://mathoverflow.net/questions/61759 | 3 | Suppose $G$ has the presentation $\langle t, x\_1, x\_2, ... | R \rangle$ where each relator in $R$ has the form $t^{-1}x\_it = x\_j$ for some $i,j$. Does $G$ have an element of order 2?
This is an HNN extension of a free group, if that changes anything.
| https://mathoverflow.net/users/6429 | Does the group given by this presentation have an element of order 2? | By the torsion theorem for HNN-extensions, every element of finite order is conjugate to an element of the base, which in your case is a free group, so the answer is no.
See Lyndon and Schupp's Combinatorial Group Theory.
| 13 | https://mathoverflow.net/users/1335 | 61760 | 38,240 |
https://mathoverflow.net/questions/61761 | 0 | Given a partial order of elements, one can use topological sorting to produce a sorted list of elements. For example, if we have the partial order A->B and A->C, then the possible topological sort results are [A,B,C] and [A,C,B].
I am interested in producing a sorted list of **sets** [$S\_1, \ldots, S\_k$] that sati... | https://mathoverflow.net/users/14397 | Topological sort of partial order into sorted sets | EDIT: This answer was for a previous version of the question.
There is usually no such list: consider the case where some element is incomparable to everything else.
| 0 | https://mathoverflow.net/users/644 | 61762 | 38,241 |
https://mathoverflow.net/questions/61764 | 5 | According to Wikipedia, if $0^{\sharp}$ exists, then every uncountable cardinal in $V$ satisfies every large cardinal property in $L$ that can be realized in $L$, e.g. weak compactness, total ineffability, etc. It's easy enough to see why every uncountable in $V$ will be inaccessible, or even Mahlo, in $L$.
**How can... | https://mathoverflow.net/users/7521 | If $0^{\sharp}$ exists, then every uncountable cardinal in $V$ is as large as it can be in $L$. | If $0^\sharp$ exists, then every uncountable cardinal
$\kappa$ of $V$ is one of the Silver indiscernibles in $L$,
and this implies that $L\_\kappa$ is an elementary substructure of $L$. This implies
that $\kappa$ is a limit cardinal in $L$ and therefore,
since some of the indiscernibles are regular, that $\kappa$
is in... | 9 | https://mathoverflow.net/users/1946 | 61765 | 38,242 |
https://mathoverflow.net/questions/61752 | 9 | I know Vandermonde's convolution for binomial coefficients:
$$\sum\_{j=0...k} \binom{n}{j} \binom{m}{k-j} = \binom{n+m}{k}$$
Is there a similar multiplicative convolution? More precisely, is there a simple formula for the coefficients $a(k\_j,k'\_j,k)$ in the following
identity?
$$\sum\_{j} a(k\_j,k'\_j,k) \binom... | https://mathoverflow.net/users/14395 | Multiplicative Convolution for Binomial Coefficients | Riordan and Stein, in "Arrangements on Chessboards" (*Journal of Combinatorial Theory, Series A*, **12** 72-80, 1972) consider the numbers $A(r,s,k)$ defined by
$$\sum\_{r,s} \binom{n}{r} \binom{m}{s} A(r,s,k) = \binom{nm}{k},$$ or, as others have pointed out, the number of $r \times s$ $(0,1)$-matrices with $k$ $1$'s... | 10 | https://mathoverflow.net/users/9716 | 61770 | 38,246 |
https://mathoverflow.net/questions/61750 | 5 | Given an Lie Algebra $L$ (of finite dimension and over an algebraically closed field with zero characteristic) and an ideal $I$, is it truth that
$rad\left(\dfrac{L}{I}\right)= \pi(rad(L))$,
where $\pi$ is the projection?
What happens if $L$ isn't finite dimensional?
And if the field has positive characteristic?... | https://mathoverflow.net/users/14312 | Radical of projection equals projection of radical? | A counter-example for the infinite dimensional case:
Let $V$ be a vector space with a basis $\{e\_i\mid i\in \mathbb{Z}\_+\}$ and $V\_p=\langle e\_i\mid i=1,\dots,p\rangle$. Consider
$$\mathfrak{t}\_\infty:= \{ x \in \mathfrak{gl}(V)\mid x(V\_i)\subset V\_i \} .$$
as a lie subalgebra of $\mathfrak{gl}(V)$.
Let $E\_... | 2 | https://mathoverflow.net/users/3061 | 61778 | 38,248 |
https://mathoverflow.net/questions/61781 | 19 | I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of his proposed ideas was a "homotopy theory of categories." What does this mean, precisely?
I know that any category corre... | https://mathoverflow.net/users/344 | What is the homotopy theory of categories? | I am not knowledgeable enough to have much to say I have not writen in my answer to [a previous question of yours](https://mathoverflow.net/questions/58497/is-there-a-high-concept-explanation-for-why-simplicial-leads-to-homotopy-theor/58512#58512), and I think that David Roberts's answer (or, rather immodestly, my prev... | 18 | https://mathoverflow.net/users/5587 | 61790 | 38,254 |
https://mathoverflow.net/questions/61788 | 13 | This question is inspired by [this one](https://mathoverflow.net/questions/61615/). In that thread, it's established that there are uncountably many cube-free infinite binary strings (where $x \in 2^{\omega}$ is *cube-free* iff $\forall \sigma \subset x,\ \sigma \sigma \sigma \not \subset x$). Here $\subset$ denotes "s... | https://mathoverflow.net/users/7521 | Is the set of cube-free binary sequences perfect? | I found an article of J.D. Currie and R.O. Shelton that answers the first question. ["The set of k-power free words over Σ is empty or perfect"](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WDY-48CFJJV-2&_user=10&_coverDate=07%2F31%2F2003&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=... | 12 | https://mathoverflow.net/users/2384 | 61799 | 38,256 |
https://mathoverflow.net/questions/61798 | 2 | Let $M$ be a rank-$2$ vector bundle on a $K3$ surface $S$ such that $h^0(M)\geq 2$ and $h^2(M)=0$. Is it possible that $h^2(\det M)>0$? If yes, can you give me some examples?
| https://mathoverflow.net/users/33841 | Determinant bundles of rank $2$ sheaves on $K3$ surfaces. | Choose a section $s \in H^0(M)$, let $Z$ be the zero locus of $s$, then we have a right exact sequence
$$
M \to \det M \to (\det M)\_{|Z} \to 0.
$$
Note that $\dim Z \le 1$, hence $H^2((\det M)\_{|Z}) = 0$.
On the other hand, the functor $H^2$ is right exact (since $S$ is a surface), hence $H^2(M)$ surjects onto $H^2(\... | 3 | https://mathoverflow.net/users/4428 | 61804 | 38,259 |
https://mathoverflow.net/questions/61773 | 22 | Basic question: Is it correct that the $p$-adic Langlands correspondence is known for $GL\_2$ only over $Q\_p$ but not other $p$-adic fields? If so, I would like to request some light to be shed on this restriction, i.e., why only $Q\_p$ but not its extensions. Any reference for this (and prospects) would be much appre... | https://mathoverflow.net/users/11786 | $p$-adic Langlands correspondence | Yes, this is correct.
The problem is that when you replace $Q\_p$ by an extension, the dimension of $GL\_2(F)$ as a $p$-adic analytic group increases. This also means that the cohomological dimension of its open subgroups increases. This leads to representation theory of $GL\_2(F)$
of being much more complicated th... | 22 | https://mathoverflow.net/users/13024 | 61808 | 38,262 |
https://mathoverflow.net/questions/61820 | 2 | This question is related to this question [link](https://mathoverflow.net/questions/61818/about-properties-of-groupoid-c-algebras).
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^\*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. Are they nice criteria known which e... | https://mathoverflow.net/users/5831 | Nuclearity of certain semigroup crossed product C*-algebras | At least in the case that $S$ is the positive integers, this is discussed in the paper by G. Murphy, "Crossed products of $C^\ast$-algebras by endomorphisms", Int. Eq. and Operator Th. Volume 24, Number 3, 298-319, DOI: 10.1007/BF01204603. His result is that the crossed product is nuclear iff $A$ is nuclear.
| 4 | https://mathoverflow.net/users/12660 | 61830 | 38,272 |
https://mathoverflow.net/questions/61829 | 6 | I'm not sure if this stuff all falls under what most would just term "probability", but I'm researching applied macroeconomics and need to get a handle on the following concepts ASAP:
* probability spaces and sigma algebras
* Borel sets
* convergence
* stationarity/ergodicity
* martingales
* laws of large numbers
* s... | https://mathoverflow.net/users/11281 | Best introduction to probability spaces, convergence, spectral analysis | There is no royal road to probability. The closest is W. Feller's book, which has many (but not all) of the topics you mention, but I strongly advise reading (at least parts of) it first. Otherwise, you will go through life hopelessly confused.
| 7 | https://mathoverflow.net/users/11142 | 61832 | 38,274 |
https://mathoverflow.net/questions/61824 | 4 | Hello everyone,
The complex numbers of modulus 1 may be seen as the real points of the scheme
$\mathrm{Spec}\ \mathbb{Q}[X,Y] / (X^2+Y^2-1)$.
Can elements of norm $1$ in $\mathbb{C}\_p$ be seen as the set of $\mathbb{Q}\_p$-points of an affine scheme?
A related question is: what are the $\mathbb{Q}\_p$-points ... | https://mathoverflow.net/users/14422 | p-adic numbers of norm 1 | The difference is that $\mathbb{C}/\mathbb{R}$ is a finite extension. This means that you can write a general element of $\mathbb{C}$ as $x+iy$ with $x,y \in \mathbb{R}$, take its norm to get $x^2+y^2$, and set that equal to 1; the resulting equation defines an affine variety $T$ over $\mathbb{R}$, whose $\mathbb{R}$-p... | 6 | https://mathoverflow.net/users/3753 | 61834 | 38,276 |
https://mathoverflow.net/questions/61840 | 5 | Do you believe P=NP?
I've seen some mathematicians say that if P=NP their work would be worthless and restricted to enunciating theorems. They seem to believe that there exist an almost philosophical impediment to P=NP. Do you agree with that? Does the possibility of P=NP bother you?
| https://mathoverflow.net/users/14312 | Do you believe P=NP? | Contrary to a popular misunderstanding: if P = NP, then the proof of any statement $A$ can be found by an algorithm in time polynomial *in the length of the shortest proof of $A$*, not in the length of $A$ itself. Moreover, the exponent of the polynomial could easily be so large as to make this algorithm practically wo... | 31 | https://mathoverflow.net/users/12705 | 61845 | 38,279 |
https://mathoverflow.net/questions/61843 | 2 | Consider the function $\sigma(n)/n$, where $\sigma$ is the usual sum-of-divisors function. I read somewhere that it is unknown what rational numbers are in fact values of this function (or at any rate that characterizing them is an open question). Well, that was a while ago, and I suspect it was in one of my older refe... | https://mathoverflow.net/users/12192 | Which rationals are sum-of-divisor function quotients | This paper may help
MR2346095 (2008i:11005)
Stanton, William G.; Holdener, Judy A.
Abundancy "outlaws'' of the form $\frac{\sigma(N)+t}{N}$. (English summary)
J. Integer Seq. 10 (2007), no. 9, Article 07.9.6, 19 pp. (electronic).
11A25 (11Y55 11Y70)
PDF Clipboard Journal Article Make Link
The abundancy index of a p... | 4 | https://mathoverflow.net/users/11016 | 61849 | 38,281 |
https://mathoverflow.net/questions/61839 | 2 | Consider real numbers $a\_i$ and $b\_i$ for $i=1\dots n$ and define a function by
$f(x) = \max\_i ( a\_i + b\_i x )$
We desire to find $\min\_x f(x)$. Obviously this occurs at an intersection of two lines:
$x = - \frac{a\_i - a\_j}{b\_i - b\_j}$
for $b\_i\neq b\_j$ and there are at most $n(n-1)/2$ such intersec... | https://mathoverflow.net/users/11727 | Efficient algorithm for finding the minima of a piecewise linear function | Looking at a figure one is lead to the following algorithm which traces out the graph of $f$:
One may assume $a\_1< a\_2 < \ldots < a\_n$. If $0 < a\_1$ or $a\_n < 0$ then $m:=\min\_x f(x)=-\infty$. Otherwise for $x$ large negative one has $f(x)=a\_1 x + b\_1$ and for $x$ large positive one has $f(x)=a\_n x+b\_n$. Th... | 4 | https://mathoverflow.net/users/8050 | 61855 | 38,286 |
https://mathoverflow.net/questions/61340 | 9 | Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ has a harmonic majorant on $\Omega$, i.e. there is a function $u$ harmonic on $\Omega$ such that
$$|f(z)|^p \leq u(z) $$
for al... | https://mathoverflow.net/users/1162 | Removable sets for harmonic functions and Hardy spaces of general domains | Yes, it is true that $H^p(\Omega)$ consists only of the constant functions for all $1\le p\le\infty$. This is because all nonnegative harmonic functions $f\colon\Omega\to\mathbb{R}$ are constant.
You can prove this using the properties of Brownian motion. If $\Omega$ is a connected open subset of the plane and $B\_t$... | 7 | https://mathoverflow.net/users/1004 | 61868 | 38,292 |
https://mathoverflow.net/questions/61859 | 7 | The units with norm $+1$ in a pure cubic number field $K$ generated
by a cube root of $m = ab^2$, where $a$ and $b$ are coprime and
squarefree integers, correspond to integral points on the torus
$$ R\_{K/\mathbb Q}^{(1)}:
X\_1^3 + ab^2X\_2^3 + a^2bX\_3^2 - 3abX\_1X\_2X\_3 = 1. $$
According to Voskresenskii (Algebrai... | https://mathoverflow.net/users/3503 | Parametrization of 2-dimensional torus | I decided to have a go at finding a parametrisation by following the instructions of Coray and Tsfasman (reference in my comment above), using Magma. Amazingly enough, it works, even working generically with $a,b$ variables.
Here's what I did. Working over the field $K(\omega, \sqrt[3]{a^2b})$, find the three singula... | 6 | https://mathoverflow.net/users/3753 | 61870 | 38,293 |
https://mathoverflow.net/questions/61704 | 3 | I hope this isn't too narrowly focused. I have a question concerning the inverse of the Bott map as defined in Atiyah's paper, *Bott Periodicity and the Index of Elliptic Operators.* On page 122 he defines it as the composition
$$K^{-2}(X) \to K(S^2 \times X) \stackrel{\text{index} \bar\partial}{\longrightarrow} K(X)... | https://mathoverflow.net/users/4622 | Defining the inverse of the Bott map | Your argument against your definition is not convincing. You say you are writing down an exact sequence for your first map, but in fact you write down one for your second map. The second map is not necessarily injective. We only need it to be injective on the image of the first map.
The definition you write looks fi... | 2 | https://mathoverflow.net/users/4648 | 61872 | 38,295 |
https://mathoverflow.net/questions/61861 | 1 | Let $A \in \mathbb{Z}^{n \times n}$ be a positive semidefinite *sparse* matrix. I am looking for asymptotically fast algorithms for computing the nullspace basis of $A$ (or just random elements in the nullspace). I wonder whether there are methods that can exploit the fact that $A$ is positive semidefinite to achieve b... | https://mathoverflow.net/users/4415 | Fast algorithms for computing nullspace of a positive semidefinite matrix over Z | Presuming you are concerned with ease of implementation, and since you know that the matrix in question is positive semi-definite, one possibility would be to proceed as follows: Given the sparse representation, it is fast to perform a matrix-vector multiply. Thus you can use the use power iteration (see <http://en.wik... | 1 | https://mathoverflow.net/users/14424 | 61873 | 38,296 |
https://mathoverflow.net/questions/61857 | 10 | It's known that if $L\subset gl(V)$, with $V$ finite dimensional, is a semisimple Lie algebra, then the abstract and usual Jordan decompositions in $L$ coincide. Is it possible to provide a counter-example if $L$ isn't semisimple?
Remark: The underlying field is algebraically closed of characteristic $0$ .
| https://mathoverflow.net/users/14312 | Abstract Jordan Decomposition different from usual Jordan Decomposition | I'm tempted to amplify Ben's precise short answer by emphasizing how subtle the notion of abstract (or intrinsic) Jordan decomposition really is. You start with a matrix Lie algebra which is (1) required to be isomorphic to its image under the adjoint representation (in other words, centerless). Then it makes sense to ... | 17 | https://mathoverflow.net/users/4231 | 61877 | 38,299 |
https://mathoverflow.net/questions/61878 | 24 | Given a linear connection on a Riemmanian manifold $M$ and $\phi^1,...,\phi^n$ a local frame for $T^\*M$ we can define the connection 1-forms $\omega^j\_i$. We define the curvature 2-forms by $\Omega\_i^j=\frac{1}{2}R\_{klij}\phi^k \wedge \phi^l$.
We have the following identities also known as Cartan's first and seco... | https://mathoverflow.net/users/5450 | Geometric interpretation of Cartan's structure equations | The $1$-forms $\omega^i\_j$ define an affine connection on the tangent bundle, and the first structure equation gives the formula for the torsion tensor.
It is equivalent to the equation
$$
\nabla\_X Y - \nabla\_Y X - [X,Y] = \tau(X,Y),
$$
where the connection $\nabla$ is defined using the $1$-forms $\omega^i\_j$. I th... | 16 | https://mathoverflow.net/users/613 | 61886 | 38,304 |
https://mathoverflow.net/questions/61814 | 5 | I studied mathematical logic using a book not written in English. I would now like to study it again using a textbook in English. But I hope I can read a text that is similar to the one I used before, so I ask here for recommendations. Any recommendation will be appreciated. The characters of the mathematical logic boo... | https://mathoverflow.net/users/5072 | Ask for recommendations for textbook on mathematical logic | I was going to recommend the English translation of the two volume sequence by Cori and Lascar. But after reading again your message it is highly possible that this is the text you used. I really like these two introductory books.
| 2 | https://mathoverflow.net/users/5450 | 61887 | 38,305 |
https://mathoverflow.net/questions/61869 | 6 | I wonder if I can make an algorithm to check if a given graph $G=(V,E)$ is acyclic or not with the complexity of $O(|V|)$.
I modified the BFS algorithm to do this, but the complexity seems to be $O(|V|+|E|)$.
| https://mathoverflow.net/users/14433 | Is it possible to check a graph for acyclicity in $O(|V|)$ time? | If a graph on $n$ nodes has $n$ or more edges then it has a cycle (since trees are the acyclic graphs with the greatest number of edges and have exactly $n-1$ edges). So if your BFS ever traverses $n$ or more edges you may immediately stop and report that the graph has a cycle. Otherwise BFS terminates in $O(n)$ time a... | 11 | https://mathoverflow.net/users/4401 | 61889 | 38,307 |
https://mathoverflow.net/questions/61715 | 5 | First, I have to admit that I don't have much knowledge of Spin Geometry and Index Theory, the question could be too simple or naive and secondly there may be too many questions.
Let $D$ be the Dirac Operator for standard metric and $S$ be the Spin bundle on $S^{2}$.There is a unique Spin structure on $S^{2}$. How d... | https://mathoverflow.net/users/9534 | equivariant index of Dirac Operator on $S^{2}$ | As Sebastian said, the Dirac operator can be identified with the $\bar{\partial}$-operator on the line bundle of degree $-1$ (the inverse of the Hopf bundle) and so its (equivariant) index is trivial.
Now you ask for the equivariant index of the twisted Dirac. The reasonable group to study equivariance is $SU(2)$, beca... | 3 | https://mathoverflow.net/users/9928 | 61909 | 38,316 |
https://mathoverflow.net/questions/61876 | 8 | I've been told many times that the Torelli map $J:\mathcal{M}\_g\to \mathcal{A}\_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard groups. On the level of rational picard groups, they're both generated by the determinant of the Hodge bundle, but why ... | https://mathoverflow.net/users/622 | Pullback along the Torelli map is an isomorphism | I see two ways interpreting "on the nose":
1) At the level of the orbifold Picard group, or that is at stack level, and in my view this is the correct framework, both Pic\_{orb}(M\_g) and Pic\_{orb}(A\_g) are generated by the Hodge class. (The Hodge class on M\_g is by definition the pull-back of the Hodge class
on A\... | 9 | https://mathoverflow.net/users/14444 | 61910 | 38,317 |
https://mathoverflow.net/questions/61885 | 8 | This is a basic question about Grothendieck's conjectural category $M\_k$ of pure motives (over a field $k$). This construction first produces a category (the "false category of motives") which need not be Tannakian; the category undergoes a modification of the commutativity constraints which produces a category (the "... | https://mathoverflow.net/users/11786 | commutativity constraint in Grothendieck's motives | I don't know any alternate geometric construction of the commutativity contraint, but there is a natural way to explain this, as a simple fact of homological algebra (and assuming very optimistic conjectures about mixed motives).
Let $DM\_{gm}(k)$ be Voevodsky's triangulated category of geometric (=constructible) mot... | 18 | https://mathoverflow.net/users/1017 | 61915 | 38,321 |
https://mathoverflow.net/questions/61897 | 8 | Consider some lattice in R^n.
Take some point "P" in R^n (which does not belong to this lattice in general).
What are the algorithms to find some nearest lattice point to "P" ?
"Nearest" - means in the sense of the standard Eucleadian distance.
Lattice - means the standard thing - takes some vectors h\_1 ... h\_n a... | https://mathoverflow.net/users/10446 | How to find nearest lattice point to given point in R^n ? Is it NP ? | This problem is often called the "closest vector problem" for lattices (especially by people in theoretical computer science). The real issue is what sort of lattice you have. For example, for the integer lattice $\mathbb{Z}^n$ the problem is easy, and it's not too hard to do it for other famous lattices, for example r... | 12 | https://mathoverflow.net/users/4720 | 61919 | 38,323 |
https://mathoverflow.net/questions/61921 | 5 | So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that it is essential to take into account the group structure since the Cantor set is a closed totally disconnected subset... | https://mathoverflow.net/users/11765 | On closed totally disconnected subgroups of connected real Lie groups | Let's just give a quick argument:
Let $G$ be a Lie group (with Lie algebra $\mathbb g$) and $H \subset G$ be a closed subgroup. If $H$ is not discrete, then there exists a sequence $(g\_n)\_{n \in \mathbb N}$ of elements of $H$, which converges to $1\_G$. Write $g\_n = \exp( \alpha\_n \xi\_n)$ (for $n$ large enough),... | 10 | https://mathoverflow.net/users/8176 | 61930 | 38,326 |
https://mathoverflow.net/questions/61922 | 1 | can anyone tell me the reason behind the failure of hasse-local global principle for selmer curves,
i think that adding some more will fix it,but is there any established reason behind the failure,
thanking you ,
may god bless you
| https://mathoverflow.net/users/nan | failure of hasse-local global principle | I'm not sure exactly what you mean by Selmer curves, perhaps you mean those curves modeled by the diophantine equation originally studied by Selmer
$$ax^3 + by^3 + cz^3 = 0.$$
Today, Selmer's examples are thought of in a more general context, relating to the study of the group of rational points on elliptic curves or... | 11 | https://mathoverflow.net/users/4872 | 61936 | 38,328 |
https://mathoverflow.net/questions/61935 | 9 | Can someone give me an example of a non-quasi-compact morphism of schemes which arises naturally in the field of Algebraic Number Theory?
| https://mathoverflow.net/users/5031 | Quasi-compact maps in Number Theory | A typical non-Noetherian ring that would arise in algebraic number theory would be the
ring $\mathbb Z\_p \otimes\_{\mathbb Z\_{(p)}} \mathbb Z\_p$, where I am writing $\mathbb Z\_{(p)}$ to denote the localization of $\mathbb Z$ at the prime ideal $(p)$, and $\mathbb Z\_p$ to denote its completion (the usual ring of $p... | 10 | https://mathoverflow.net/users/2874 | 61942 | 38,332 |
https://mathoverflow.net/questions/61948 | 4 | Hello again! More of the same bumbling down the road of algebraic topology. This time, I am trying to figure out exactly how much information the face poset of a CW complex encodes. It has often been remarked that a CW complex has much more data in it than a simplicial complex due to the fact that keeps track of charac... | https://mathoverflow.net/users/4642 | Are there two non-homeomorphic finite regular CW complexes with isomorphic face posets? | It is not hard to reconstruct a finite regular CW complex from its face poset. This is essentially the order complex construction. One just has to understand the order complex $\Delta(P)$ of the poset $P$ not as a mere simplicial complex, but rather as the barycentric subdivision of a "cone complex" $C(P)$, where a con... | 12 | https://mathoverflow.net/users/10819 | 61961 | 38,343 |
https://mathoverflow.net/questions/61937 | -2 | Can someone point me to an article concerning the "inversion" of formulas?
| https://mathoverflow.net/users/5031 | Localization of Formulas | Although the question mentions inversion of formulas and the comment mentions inversion of an operation ("or"), the example leads me to guess that the proposer actually wants neither of these but rather to invert elements of some algebraic structure, probably Boolean algebras. In case my guess is correct, let me point ... | 3 | https://mathoverflow.net/users/6794 | 61963 | 38,345 |
https://mathoverflow.net/questions/61953 | 0 | I often think about the phase space with quite deep interpretations. For example, contraction of phase space means losing energy. But, some of the energy is easily restored (free energy?) while some other is hard to restore (enthalpy, internal energy). This is like dividing the phase space to flexible and rough?
Are ... | https://mathoverflow.net/users/13343 | Deeper meanings of Phase Space -- any books? | For your kind of reflections I think a useful reference could be the fantastic ["The structure of dynamical systems. A symplectic view of physics"](http://books.google.it/books?hl=it&lr=&id=4tBrbryIKQAC&oi=fnd&pg=PR17&dq=the+structure+of+dynamical+systems&ots=tVBfnMVnPn&sig=Pzi0dGmzfTSXkEh8wOFomIp3ngA#v=onepage&q&f=fal... | 2 | https://mathoverflow.net/users/12617 | 61967 | 38,347 |
https://mathoverflow.net/questions/44226 | 12 | Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h\_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed the disc $D$ real-analytically and isometrically into real Euclidean space ${\mathbb{R}}^n$ for some large $n$. (I thin... | https://mathoverflow.net/users/nan | Can the unit complex 1-dimensional disc be embedded isometrically into complex euclidean space? | The answer is 'no', there is no holomorphic curve in $\mathbb{C}^n$ (for any $n$) such that the induced metric has constant negative curvature. To my knowledge, this was first proved by E. Calabi many many years ago, essentially using the structure equations for holomorphic curves in $\mathbb{C}^n$. The proof is easy, ... | 13 | https://mathoverflow.net/users/13972 | 61972 | 38,349 |
https://mathoverflow.net/questions/61959 | 10 | I've just read on Wikipedia that the original Taniyama conjecture about L-functions of elliptic curves over an arbitrary number field was still unproven.
This made me want to know more about this conjecture, but after a quick glance at the first results displayed by Google, I couldn't find out anything else than the ... | https://mathoverflow.net/users/13625 | Taniyama's original conjecture | There is [an article by S. Lang](http://www.ams.org/notices/199511/forum.pdf) on this subject that appeared in the Notices of the AMS (11/1995); it contains the exact statements of Taniyama's problem(s) on this subject. I reproduce two below (this is taken from a longer list of problems, not all are relevant to the que... | 13 | https://mathoverflow.net/users/nan | 61974 | 38,350 |
https://mathoverflow.net/questions/60307 | 1 | I have three related questions:
(1) How does one describe the possible Riemannian metrics that are compatible with a conformal structure on a two dimensional surface?
(2) Can all conformable structures be realized through embeddings of the surface in Euclidean 3-space?
(3) How does one understand solutions to the... | https://mathoverflow.net/users/13827 | metrics compatible with conformal structures | I don't have much to say about (1) or (2), but (3) is, of course, a classical equation and the usual interpretation is this:
I'm assuming that you are talking about a domain in the $z$-plane, also known as the $xy$-plane, and that, by $\Delta f$ you mean the classical $f\_{xx} + f\_{yy}$, not the negative of this or ... | 5 | https://mathoverflow.net/users/13972 | 61978 | 38,353 |
https://mathoverflow.net/questions/61954 | 11 | Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed as a quotient of the tensor algebra $T\left(L\right)$ of $L$. Let $U\left(L\right)$ denote the universal enveloping algeb... | https://mathoverflow.net/users/2530 | PBW theorem over a Q-algebra, without freeness or flatness | I liked the Cohn paper, although it's been a while since I read it, and I read it more to understand the counterexample in the second half than the proof in the first half. A proof in the case when your $k$ is a commutative algebra over $\mathbb Q$ is available in:
* Deligne, Pierre; Morgan, John W. Notes on supersym... | 9 | https://mathoverflow.net/users/78 | 61981 | 38,355 |
https://mathoverflow.net/questions/61980 | 0 | Problem:
Given: $\phi(s) = \sum\_p \frac{\log p}{p^s}$
(summation is only over primes)
Why is: $\lim\_{e \rightarrow 0} e \phi(1+e)$ = 1 ?
Context: <http://mathdl.maa.org/images/upload_library/22/Chauvenet/Zagier.pdf>
Section IV, Page 706
Some silly analysis by me:
So here's my dumb approach, which fails:... | https://mathoverflow.net/users/3609 | Given: $\phi(s) = \sum_p \frac{\log p}{p^s}$ Why is: $\lim_{e \rightarrow 0} e \phi(1+e)$ = 1 ? | I think you're approaching the question in the wrong way. The whole point is that you can show that for $\Re(s) > 1$,
$$\Phi(s) = \sum\_{p}{\frac{\log p}{p^s}} = \frac{1}{s - 1} + E(s),$$
where the function $E(s)$ is meromorphic on the open half-plane $\Re(s) > 1/2$ with poles possibly at the zeroes of $\zeta(s)$; in f... | 4 | https://mathoverflow.net/users/3803 | 61988 | 38,358 |
https://mathoverflow.net/questions/61985 | 4 | Let a number $x = \sqrt[a\_1]{p\_1} + \sqrt[a\_2]{p\_2} + \ .. \ + \sqrt[a\_n]{p\_n}$ be a number such that all $a\_n$ are integers and all $p\_n$ are rational. I've been noticing that for every number x, the degree of its minimal polynomial is seemingly always equal to $\prod\_{1}^n \ a\_n$.
Is that valid for all va... | https://mathoverflow.net/users/14456 | Finding the degree of minimal polynomials | No. Some conditions are needed on the $a\_i$ and $p\_i$. For instance, take n=2, $a\_1 = a\_2 = 2$, $p\_1 = p\_2 = 2$. Then $x = 2 \sqrt{2}$, which has minimal polynomial $x^2 - 8$. As an even simpler example, n=1, $a\_1 = 2$, $p\_1 = 4$, then $x$ is rational.
For a less trivial example, take $a\_1= 4$, $a\_2 = 6$, $... | 6 | https://mathoverflow.net/users/12722 | 61990 | 38,359 |
https://mathoverflow.net/questions/61995 | 2 | The following was recently on my algebraic geometry homework:
>
> Let $k$ be an algebraically closed field, $f\in B=k[x\_1,\ldots,x\_n]$, and $A=B/(f)$. Show that $\Omega\_{A/k}$ is locally free of rank $n-1$ $\iff$ $\nexists\\, p\in k^n$ such that $f(p)=0$ and all $\frac{\partial f}{\partial x\_i}(p)=0$.
>
>
>
... | https://mathoverflow.net/users/1916 | Kahler differentials of a hypersurface over a non-algebraically closed field | For $k$ alg. closed you can phrase the statement as $\Omega\_{A/k}$ is loc. free iff Spec$(A)$ is smooth. 'Spec$(A)$ smooth iff $\Omega\_{A/k}$ is loc free' should be true without requiring $k = \bar{k}$. But if $k \ne \bar{k}$ then the condition on the derivatives is not the same as smoothness. For example if $C$ is a... | 2 | https://mathoverflow.net/users/7 | 61997 | 38,362 |
https://mathoverflow.net/questions/61998 | 9 | Let $K$, $L$ be finite extensions of the $p$-adic numbers. Suppose $\chi:G\_K\rightarrow L^{\times}$ is crystalline. It is my understanding that if either $K$ or $L=\mathbb{Q}\_p$, then $\chi$ must be a Tate twist of an unramified character. Is there a classification of crystalline characters in general?
| https://mathoverflow.net/users/5513 | Crystalline Characters | Hi Kevin,
1. Go to <http://math.stanford.edu/~conrad/>
2. Download "Grunwald--Wang for global character groups"
3. Read appendix A, especially prop A.3
The answer (note that $K$ and $L$ are switched in Brian's paper) is that once you've identified your character as a character of $K^\times$ via local class field th... | 17 | https://mathoverflow.net/users/5743 | 62004 | 38,365 |
https://mathoverflow.net/questions/57251 | 2 | A group G is called completely reducible if it is a direct product of simple groups. It is known that a Krull-Schmidt Remak type unicity for the decomposition in the direct product of simple groups hold. The proof s of this facts can be found for example in Chapter 3 of D. Robinson's book [A Course in the Theory of Gro... | https://mathoverflow.net/users/13384 | Finite completely reducible groups-reference request | Well, it depends what you are prepared to accept as a reasonable answer. A finite group is
completely reducible if and only if the intersection of all its maximal normal subgroups
is trivial. This is presumably a lot of work to check for any reasonably sized group.
On the other hand, you can regard the intersection of ... | 2 | https://mathoverflow.net/users/14450 | 62006 | 38,366 |
https://mathoverflow.net/questions/61994 | 15 | This question is motivated by the classical fact from differential geometry :
*Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ acts transtively on the configuration space of $n$-points in $M$ or equivalently it acts $n$-transitively on $M$.*
As I... | https://mathoverflow.net/users/1993 | How transitive are the actions of symplectomorphism groups ? | Dear Somnath Basu, the answer to question is that, for any $k\geq 2$, the k-fold transitivity of the action of $\mathrm{Sympl}(M,\omega)$ on $M$ has only one obstruction, the trivial one, i.e. connectivity of $M$.
But has you proposed there is even more.
In particular [Theorem A in a paper of W. Boothby](http://ww... | 19 | https://mathoverflow.net/users/12617 | 62018 | 38,374 |
https://mathoverflow.net/questions/61996 | 3 | This is a [question](https://math.stackexchange.com/questions/29272/counting-some-vanishing-polynomials-in-mathbbz-nx) I asked on Math.SE and got only a partial answer. I hope I will have better chances here.
Given the ring of polynomials $\mathbb{Z}\_n[X]$, consider $$\mathbb{P}\_n = \lbrace a\_0 +a\_1x+a\_2x^2+\cd... | https://mathoverflow.net/users/6770 | Counting some polynomials that have a zero in $\mathbb{Z}_n[X]$ | I think there's a simple answer when $n$ is prime. Count instead the polynomials that don't have a zero. Such a polynomial must map $\lbrace0,1,\dots,n-1\rbrace$ to $\lbrace1,\dots,n-1\rbrace$. There are $(n-1)^n$ such maps. But each of those maps corresponds to a unique polynomial, since Lagrange interpolation works o... | 4 | https://mathoverflow.net/users/3684 | 62020 | 38,375 |
https://mathoverflow.net/questions/61984 | 20 | Motivation:
Let $\mathbb{Q}\_{\infty,p}$ be the field obtained by adjoining to $\mathbb{Q}$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal cyclotomic extension $\mathbb{Q}^\text{cycl}$ of $\mathbb{Q}$. By Kronecker–Weber, $\mathbb{Q}^\text{cycl}$ is also the maxi... | https://mathoverflow.net/users/11786 | Rational points on algebraic curves over $\mathbb Q^\text{ab}$ | Actually Ken Ribet proved that if $K$ is a number field and $K(\mu\_{\infty})$ is its infinite cyclotomic extension generated by all roots of unity then for every abelian variety $A$ over $K$ the torsion subgroup of $A(K(\mu\_{\infty}))$ is finite:
<http://math.berkeley.edu/~ribet/Articles/kl.pdf> .
On the other han... | 14 | https://mathoverflow.net/users/9658 | 62026 | 38,378 |
https://mathoverflow.net/questions/61939 | 5 | On a smooth surface of positive Gauss curvature that is embedded in Euclidean 3 space the principal curvatures seem to define a smooth field of ellipses whose major and minor axes are lined up with the directions of curvature and whose ratio of lengths is k1/k2, the ratio of the principal curvatures.
My first questi... | https://mathoverflow.net/users/13827 | Conformal structure determined by principal curvatures | When you define an actual ellipse field, you get more than a conformal structure, you get a Riemannian metric.
There are two natrual and reciprocal ways to define an ellipse field from the principle curvatures of a smooth convex surface
such that the ratio of major and minor axes is $k\_1/k\_2$.
If $x\_1$ and $x\_2$ ... | 8 | https://mathoverflow.net/users/9062 | 62027 | 38,379 |
https://mathoverflow.net/questions/62016 | 12 | Suppose a group G acts on a chain complex K and induced action on H(K) is trivial. What "secondary operations" on H(K) can be defined in this situation?
---
**Example.** If $G=\langle\sigma\rangle/\sigma^n$ acts trivially on H(K) then $x-\sigma x=dl(x)$ (for some function $l$) and a secondary operation $x\mapsto ... | https://mathoverflow.net/users/1556 | "Secondary operations" for a group acting on a chain complex | One family of secondary operations that arise come from trying to find the difference between "classes that look like they are acted on trivially" and "classes that are genuinely (or coherently) acted on trivially".
Let $\cdots \to F\_2 \to F\_1 \to F\_0 \to \mathbb{Z} \to 0$ be a free resolution of $\mathbb{Z}$. We ... | 11 | https://mathoverflow.net/users/360 | 62037 | 38,387 |
https://mathoverflow.net/questions/62010 | 1 | I want to learn about Niemeier lattice and Leech lattice in it. I will be pleased if some one could introduce some books or Lecture notes to me.
| https://mathoverflow.net/users/13684 | Niemeier Lattice | The standard reference is Conway and Sloane. If this is too much, you could try
Wolfgang Ebeling's book
"Lattices and codes
A course partially based on lectures by F. Hirzebruch." Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, ISBN: 3-528-06497-8. This covers Venkov's classification of Niemeier... | 7 | https://mathoverflow.net/users/51 | 62038 | 38,388 |
https://mathoverflow.net/questions/58794 | 8 | What are bounds on number of conjugacy classes in terms of number of elements of a group ?
(I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and Geoff Robinson. Below is original text of the question. (Alexander Chervov) ).
---
It's about the first step to find ... | https://mathoverflow.net/users/3898 | Bounds on number of conjugacy classes in terms of number of elements of a group ? | There is a theorem of E. Landau which proves that if you fix a positive integer h,
there are only finitely many finite groups with h conjugacy classes. This proof
is more number theory than group theory, in fact. More recently, one person who has
worked more extensively on this question using more group theory is L. Py... | 11 | https://mathoverflow.net/users/14450 | 62040 | 38,389 |
https://mathoverflow.net/questions/62025 | 12 | Let $M$ be a Riemannian manifold, $x$ and $y$ are two points in $M$.
Assume that $x$ is not in the cut locus of $y$. Does there exist a neighborhood $U$ of $x$ and a neighborhood $V$ of $y$ such that for every point $u$ in $U$ and for every point $v$ in $V$ we have that $u$ is not in the cut locus of $v$?
| https://mathoverflow.net/users/14462 | An elementary question about the cut locus | For a unit tangent vector $u$ with footpoint $p$ let $t(u)$ be the supremum of positive numbers such that the geodesic $t\to \exp\_p(tu)$ is minimizing on $[0,t(u)]$. The *cut locus* at $p$ is the set of points $\exp\_{p}(t(u) u)$ of $M$ for which $t\_u$ is finite.
A basic result is that $u\to t(u)$ defines a conti... | 8 | https://mathoverflow.net/users/1573 | 62041 | 38,390 |
https://mathoverflow.net/questions/54434 | 106 | As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion-free connection $\nabla\_g$, the Levi-Civita connection, that is compatible with the metric.
I was wondering if one can reverse this situation: Given a manifold with $M$ with connection $\nabla$, when does there exist a Riemannian metric... | https://mathoverflow.net/users/1648 | When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection? | Bill and Willie have (of course) given correct answers in terms of the holonomy of the given torsion-free connection $\nabla$ on the $n$-manifold $M$. However, it should be pointed out that, practically, it is almost impossible to compute the holonomy of $\nabla$ directly, since this would require integrating the ODE t... | 81 | https://mathoverflow.net/users/13972 | 62042 | 38,391 |
https://mathoverflow.net/questions/62021 | 11 | There are some well know formulas of Abramov about derived systems.
Firstly let $(X,\mu,f)$ be a probability preserving system and $A\subset X$ is measurable such that $\bigcup\_{n\ge0}f^nA=X$. Let $\mu\_A$ be conditional probability of $(\mu,A)$ and $f\_A:A\to A$ be the first return map with respect to $A$, that is... | https://mathoverflow.net/users/11028 | Entropy of first return map and suspension flows | What may seems intuitive to you depends on your mental model[s] for entropy. This is not static, since we change our mental models and learn new models with time and experience.
One good intuitive way to think about entropy is in terms of [information theory](http://en.wikipedia.org/wiki/Information_theory), as develop... | 13 | https://mathoverflow.net/users/9062 | 62045 | 38,393 |
https://mathoverflow.net/questions/61890 | 11 | Suppose $G$ is a finite simple group and $|G|$ is a multiple of $60$. Does it follow that $G$ has a subgroup isomorphic to $A\_{5}$? If so, can this be proven without using the Classification?
| https://mathoverflow.net/users/12610 | (A very limited instance of) Lagrange's Theorem's converse and A_5 | I think the answer to the question is yes, but it is very unlikely that it can be proved without using the classification of finite simple groups.
Note that $A\_5$ of order 60 is the only simple group order for which this statement is true, because for all higher order simple groups $G$, there will be groups $L\_2(p)... | 10 | https://mathoverflow.net/users/35840 | 62053 | 38,397 |
https://mathoverflow.net/questions/62059 | 4 | Is it known whether there is a wellfounded model of ZFC, containing all reals, in which CH fails? In which it obtains?
| https://mathoverflow.net/users/8547 | CH and wellfounded models of ZFC containing all reals | Cole:
Suppose first that $V=L$. Then certainly there is no well founded model as you want, because any well-founded model $M$ is correct about $L$, meaning $L^M=L\_{ORD^M}$, and if $M$ contains all the reals, then this model must contain $L\_{\omega\_1}$ and must therefore satisfy CH.
(The assumption of $V=L$ is ... | 5 | https://mathoverflow.net/users/6085 | 62065 | 38,404 |
https://mathoverflow.net/questions/62070 | 12 | I have an extremely elementary question. Let's say someone randomly hands you a theta function associated to a Niemeier lattice (unimodular even, n=24). What can you say about which Niemeier lattice gave you this theta function in the first place?
| https://mathoverflow.net/users/5730 | Niemeier lattices and theta functions | The theta series for a Niemeier lattice determines the lattice in most cases, but there are five ambiguous pairs.
The theta series of an even unimodular lattice must be a polynomial in the theta series of $E\_8$ and $\Lambda\_{24}$ (this is a modular forms calculation). Thus, for Niemeier lattices, it must be a linea... | 22 | https://mathoverflow.net/users/4720 | 62071 | 38,407 |
https://mathoverflow.net/questions/62069 | 4 | What is the defining formula for sectional curvature?
$K\_1(X,Y) = \frac{ \langle R(X,Y)Y, X \rangle} {\langle X,X \rangle \langle Y,Y \rangle - \langle X,Y \rangle} $
as in <http://en.wikipedia.org/wiki/Sectional_curvature>
OR
$K\_2(X,Y) = \frac{ \langle R(X,Y)X, Y \rangle} {\langle X,X \rangle \langle Y,Y \ra... | https://mathoverflow.net/users/12573 | What is the defining formula for Sectional Curvature | There is no standard sign convention for the sign curvature tensor.
Depending on author,
the same thing is denoted as
$$\langle R(X,Y)Y, X \rangle\ \ \text{or}\ \ \langle R(X,Y)X, Y \rangle.$$
But the sectional curvature is always positive for sphere and always negative for Lobachevky space. That makes you to choose ... | 14 | https://mathoverflow.net/users/1441 | 62072 | 38,408 |
https://mathoverflow.net/questions/62047 | 18 | Let $K$ be the field of fractions of
$\mathbb{C}[[z]]\otimes\_{\mathbb{C}}\mathbb{C}[[w]]\subset \mathbb{C}[[z, w]].$ Given
a formal power series in $t, f\in \mathbb{C}[[t]],$ is there any simple criterion which
will conclude that $f(zw)$ does not belong to $K?$ I suspect that
$f(zw)\in K$ if
and only if $f(t)$ is a ... | https://mathoverflow.net/users/8257 | How to prove that $e^{zw}$ can not be written as a rational expression in functions in $z$ and in $w$? | **Theorem 1.** Let $f\in\mathbb{C}[[t]]$. Then $f(zw)\in \mathbb{C}[[z]]\otimes\_{\mathbb{C}}\mathbb{C}[[w]]$ if and only if $f\in\mathbb{C}[t].$
**Proof.** The "if" part is obvious. For the "only if" part assume that $f\notin\mathbb{C}[t]$ but
$$f(zw) = \sum\_{i=1}^n g\_i(z)h\_i(w),$$
where $g\_i\in\mathbb{C}[[z]]$ ... | 15 | https://mathoverflow.net/users/11919 | 62073 | 38,409 |
https://mathoverflow.net/questions/62074 | 6 | Margulis' normal subgroup theorem states that any normal subgroup of a higher rank lattice is either finite or of finite index. The obvious question is: can one classify finite normal subgroups of such lattices? (even $SL(n, \mathbb{Z})$ and $Sp(2n, \mathbb{Z})$ would be a good start).
| https://mathoverflow.net/users/11142 | Margulis normal subgroup theorem | These are the central subgroups, see <http://www.mathematik.uni-regensburg.de/loeh/seminars/normal_subgroup_thm.pdf> . It is proved that every non-central normal subgroup has finite index (page 7).
| 15 | https://mathoverflow.net/users/nan | 62077 | 38,411 |
https://mathoverflow.net/questions/62080 | 3 | This question seems easy and I feel like it should be true, but as of yet I have not been able to convince myself one way or the other, so I figure someone around here knows for sure.
>
>
> >
> > **Question:** Let $\alpha = U+iV$ be a root of some $P\in\mathbb{Q}[z]$ so that $\alpha$ is algebraic. Is it true that... | https://mathoverflow.net/users/12301 | On the real and imaginary parts of algebraic numbers | Algebraic numbers form a field and they are closed under conjugation, therefore $U=(\alpha+\bar\alpha)/2$ and $V=(\alpha-\bar\alpha)/(2i)$ are algebraic.
Reference: Lang's Algebra (Springer 2002), Chapter V, Proposition 1.6
| 15 | https://mathoverflow.net/users/11919 | 62081 | 38,413 |
https://mathoverflow.net/questions/62061 | 3 | First, apologies for the title. This is an odd question, and I couldn't come up with a simple title for it.
I'm trying to come up with a good algorithm for the following, and it's giving me a headache.
I have (not necessarily disjoint) sets $S\_1,\ldots,S\_T$, each $S\_t$ contains a subset of $1,\ldots,N$ for some ... | https://mathoverflow.net/users/14473 | Generating sets of tuples from possible candidate lists (or finding perfect matchings in uniform hypergraphs) | Here is a possible title **matchings in hypergraphs** This is just a restatement but it might help with finding references. For $T=2$ one has the usual graphs and there are good algorithms for finding maximum size matchings. For larger $T$ finding matchings is (in general) NP-complete even for $T=3$. It may be that the... | 2 | https://mathoverflow.net/users/8008 | 62082 | 38,414 |
https://mathoverflow.net/questions/62092 | 2 | Hello,
as I'm not an analyst, I'm having difficulties with the following, certainly well-known problem: one is given the PDE $\Delta u(x,y)=\sqrt{x^2+y^2}$ in the "region" $x^2+y^2\leq1$ with the boundary coundition $u(x,y)=0$ whenever $x^2+y^2=1$. The most obvious "answer" would be $u(x,y)=\sqrt{x^2+y^2}$, but the p... | https://mathoverflow.net/users/13462 | Poisson equation in the plane | The usual way to approach this would be to first search for radial solutions, i.e. look for functions $v$ such that $v(|(x,y)|)=u(x,y)$ and $v$ solves the equation. This should reduce the problem to an ODE in this special case, from which it is easy to find a solution.
| 2 | https://mathoverflow.net/users/11266 | 62102 | 38,425 |
https://mathoverflow.net/questions/62100 | 4 | I'm interested in efficiently generating (or iterating over) sets of all monomials of a degree $n$ over $r$ variables,, up to relabeling variables; this can be identified with the set of partitions of $n$ into at most $r$ parts.
More generally, I need to efficiently generate (or iterate over) the set of all sets of ... | https://mathoverflow.net/users/7725 | Is there a good known algorithm for generating sets of monomials? (Alternatively, what is the fastest known algorithm for generating integer partitions?) | See Chapter 7.2.1.4 of Knuth "The Art of Computer Programming" Vol. 4 Fascicle 3. On page 38 he gives algorithms to efficiently enumerate all partitions of an integer $n$ (see Algorithm P), or to enumerate all partitions of $n$ into exactly $m$ parts (see Algorithm H). Running Algorithm H repeatedly with $m$ varying fr... | 7 | https://mathoverflow.net/users/4757 | 62103 | 38,426 |
https://mathoverflow.net/questions/62088 | 21 | The conjugacy classes of the permutation group $S\_n$ are indexed by partitions like $[6]$ and $[2,2,2] = [2^3]$ describing the cycle type. What happens when you take products of two whole conjugacy classes? I saw in a paper,
$$[6][2^3] = 6[3,1^3] + 8[2^2,1^2]+5[5,1]+4[4,2]+3[3^2]$$
Which I take to mean if you multiply... | https://mathoverflow.net/users/1358 | Products of Conjugacy Classes in S_n | Short answer: Yes, on Hurwitz spaces.
Let's set these numbers up as the structure constants of $Z\_d=Z(\mathbb{C}[S\_d])$, the center of the group ring of the symmetric group $S\_d$. The ring $Z\_d$ has basis $K\_{\mu}$, where $\mu$ is a partition of $d$, and $K\_\mu$ represents the sum of all permutations of cycle t... | 24 | https://mathoverflow.net/users/1102 | 62106 | 38,429 |
https://mathoverflow.net/questions/62078 | 7 | Let $L$ be a Lie algebra. It is known that $L$ admits a Levi decomposition (possibly non unique):
$L=S\oplus rad(L) $,
where $rad(L)$ is the solvable radical and $S$ is a semisimple subalgebra.
If $L$ is reductive, this is, if
$rad(L)=Z(L)$,
where $Z(L)$ is the center, then $S$ is an ideal.
Is the converse... | https://mathoverflow.net/users/14312 | When is the Levi subalgebra an ideal? | To answer the original question: Certainly it's possible for $L$ to be non-reductive (and non-solvable) while a Levi subalgebra $S$ is an ideal. Just form the direct sum as Lie algebras of your favorite semisimple Lie algebra and a nonabelian solvable Lie algebra. I think the additional edited question about when $S$ c... | 6 | https://mathoverflow.net/users/4231 | 62107 | 38,430 |
https://mathoverflow.net/questions/62095 | 13 | Let $m$ be a secret message that needs to be sent to $n >1$ recipients. Let each recipient $r\_i$ have a public key $p\_i$ and private key $s\_i$. Is there a scheme such that we can encrypt the message $m$ using the $n$ public keys and produce a encrypted message $E(m)$ such that only the $n$ intended recipients can de... | https://mathoverflow.net/users/9468 | Encrypting a message for multiple recipients | This problem is often called broadcast encryption: how can you set up a system that will enable transmission of an encrypted message to an arbitrarily chosen subset of the people involved? There's a trade-off between two difficulties. If you just give each user an individual key in a generic public key cryptosystem, wi... | 20 | https://mathoverflow.net/users/4720 | 62111 | 38,432 |
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