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/57851 | 5 | Given a Shimura variety $S$, is it possible to imbed $S$ as a special Subvariety
of the Siegel modular variety $A\_{g,N}$, for some $g$ and level $N$? I expect that the answer is yes, essentially since every
semisimple group over $\mathbb{Q}$ should imbed into $GL\_n$ via its adjoint representation,
and $GL\_n$ imbeds... | https://mathoverflow.net/users/4181 | Are all Shimura Varieties Special Subvarieties of the Siegel modular Variety? | The answer is no, in general. The problem is to find an embedding so that the minuscule character corresponding to the Shimura datum for $S$ induces the minuscule character of $GSp\_{2n}$ corresponding to a decomposition into Lagrangians.
In the affirmative direction, for most classical, simply connected groups (and... | 4 | https://mathoverflow.net/users/7868 | 57883 | 36,058 |
https://mathoverflow.net/questions/57859 | 7 | Stated simply, the question is:
-------------------------------
>
> Consider two elementary symmetric polynomials $\sigma\_{k}$ and $\sigma\_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U\_{k}$ and $U\_{k+1}$. Let $V\_{i\_{1}i\_{2}\dotsb i\_{j}}$ be the coordinate linear space $\{x\in\mathbb{R}^{n}: x\_{i\_{1}} = \do... | https://mathoverflow.net/users/13508 | Characterizing intersection of zero sets of elementary symmetric polynomials on R^n | To establish that $U\_k \cap U\_{k+1} \subset W\_{n-k+1}$:
Consider the polynomial $P(t)=\prod (t-x\_i)$. The elementary symmetric polynomials $\sigma\_i$ are its coefficients, up to the sign.
Suppose that $\sigma\_k=\sigma\_{k+1}=0$. This means that $0$ is a multiple root of the derivative of order $(n-k-1)$ of $P... | 7 | https://mathoverflow.net/users/6768 | 57887 | 36,061 |
https://mathoverflow.net/questions/57904 | 27 | Recall that the usual definition of a triangulated category is an additive category equipped with a self equivalence called $[1]$ in which certain diagrams, of the form $X \to Y \to Z \to X[1]$ are called "exact", satisfying certain axioms. Two of these axioms are that
(1) Given $X \to Y$, it can be extended to an e... | https://mathoverflow.net/users/297 | Why not define triangulated categories using a mapping cone functor? | Let's take $C$ to be the category of chain complexes of abelian groups, and homotopy classes of maps. What do you want the cone functor to be? An object of $Ar(C)$ consists of a pair of chain complexes $X$ and $Y$, together with a homotopy class of chain maps between them. To construct the mapping cone in the usual way... | 21 | https://mathoverflow.net/users/10366 | 57905 | 36,069 |
https://mathoverflow.net/questions/57902 | 4 | I believe the following statement is true, and I've even seen it referenced [here](https://mathoverflow.net/questions/31904/closed-3-manifolds-with-free-abelian-fundamental-groups). Could someone point me to a proof?
>
> The fundamental group of a closed hyperbolic 3-manifold is not a free product.
>
>
>
| https://mathoverflow.net/users/6429 | Fundamental groups of closed hyperbolic 3-manifolds are freely indecomposable | If $M$ is a closed $3$-manifold and $\pi\_1(M) \cong A \ast B$ with $A$ and $B$ nontrivial, then Kneser's conjecture (which is a theorem -- the proof can be found in Hempel's book on 3-manifolds) says that we can write $M = M' \sharp M''$ where $M'$ and $M''$ are closed 3-manifolds with $\pi\_1(M')=A$ and $\pi\_1(M'')=... | 20 | https://mathoverflow.net/users/317 | 57907 | 36,070 |
https://mathoverflow.net/questions/57890 | 2 | It's just a wording question:
>
> How does one tell - by a simple
> adjective - that a collection is "of
> the size of a **proper** class"?
>
>
>
Their might be several sizes of proper classes, but on the other side, it's not a problem that there are several sizes of infinite/uncountable classes to call all... | https://mathoverflow.net/users/2672 | Simple adjective for "of the size of a proper class"? | **Proper-class-many**.
>
> “We show that if there exist proper class many Woodin cardinals, then the set of reals x for which there is exists an ordinal α with {a ∈ Pω1 (α) | x ∈ L[a]} stationary is countable.” —Paul Larson, *Reals constructible from many countable sets of
> ordinals*.
>
>
>
It’s grammatic... | 12 | https://mathoverflow.net/users/2273 | 57910 | 36,072 |
https://mathoverflow.net/questions/57908 | 3 | I have a large data set, A, containing 100 x/y pairs. I've divided it into two smaller data sets, B and C, containing 30 and 70 x/y pairs respectively.
I have [Pearson's product-moment correlation r](http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient) for each of the two smaller data sets, B ... | https://mathoverflow.net/users/13519 | Combining Correlation Coefficients | You can't do that, as Gerry Myerson has pointed out.
If you want a way to break down the computation, though, go back to one of the formulas for it:
$$ r\_{xy} = {n \sum\_i x\_i y\_i - \sum\_i x\_i \sum\_i y\_i \over \sqrt{n \sum\_i x\_i^2 - (\sum x\_i)^2} \sqrt{n \sum\_i y\_i^2 - (\sum\_i y\_i)^2}}. $$
(See the ... | 7 | https://mathoverflow.net/users/143 | 57914 | 36,076 |
https://mathoverflow.net/questions/57915 | 0 | In many textbooks, it is said that a set is countable if we can list the elements as $a\_1, a\_2, \dots$.
My question is: is it true that a set is countable if and only if there exists a Turing machine to enumerate (without termination) all the elements in the set?
| https://mathoverflow.net/users/13520 | Is it true that a set is countable if and only if there exists a Turing machine to enumerate all the elements in the set? | **No.**
Consider the set of all pairs $(x,n,i)$, where: $x$ is the code for a Turing machine $T\_x$; $n$ is a natural number; and either $i=1$ and $T\_x$ halts on input $n$, or $i=0$ and $T\_x$ diverges on $n$.
This set is certainly countable (it’s isomorphic to $\mathbb{N}^2$, just by forgetting the $i$-component)... | 4 | https://mathoverflow.net/users/2273 | 57918 | 36,078 |
https://mathoverflow.net/questions/57924 | 1 | Consider the action of $GL\_n(R)$ on $M\_{n \times n}(R)$ by conjugation, where $R$ is a ring (or field)? How can we classify the orbits? To what extent does the characteristic polynomial and the trace classify the orbits?
| https://mathoverflow.net/users/10400 | Conjugation orbits in the square matrices | For a field, this is given by the rational canonical form (see Section 7.2 of Hoffman and Kunze's Linear Algebra, for example). Even in this case, the trace and characteristic polynomial are quite weak as invariants. What you need are the *invariant factors*. For general rings, this is very hard and often a *wild class... | 9 | https://mathoverflow.net/users/9672 | 57928 | 36,083 |
https://mathoverflow.net/questions/57922 | 7 | Is there an invariant, which encodes the failure of the Bruhat decomposition to hold for a reductive group with coefficients in a local ring like the p-adic integers or the ring $\mathbb{Z}/\mathfrak{p}^r$?
Example $G =GL\_2$: Fix a Borel subgroup $B$, e.g. the upper diagonal matrixes. The coset space $B \backslash G... | https://mathoverflow.net/users/10400 | Bruhat decomposition for G(R), R local ring or R=Z/p^r | Bruhat decomposition over $\mathbf Z/p^r\mathbf Z$ is precisely the problem we looked at in [this paper](http://dx.doi.org/10.1080/00927870600876250). We defined several invariants of double cosets, and classified the pairs $(n,k)$ for which, when $G=GL\_n(\mathbf Z/p^k\mathbf Z)$, the cardinality of $B\backslash G/B$ ... | 10 | https://mathoverflow.net/users/9672 | 57931 | 36,086 |
https://mathoverflow.net/questions/57900 | 7 | A Riemannian manifold is *hyperkähler*, if there are three complex structures $I,J,K$, which are all compatible with the Riemannian metric (i.e., $(v,Iw)$ defines a symplectic form and similarly for $J$ and $K$). Furthermore, we also need the complex structures to satisfy the quaternionic relations $I^2=J^2=K^2=-1$ and... | https://mathoverflow.net/users/13518 | Existence of closed manifolds with more than 3 linearly independent complex structures? | A manifold admitting a triple of complex structures satisfying quaternionic
relations also admits a torsion-free connection (called "Obata connection") preserving the quaternionic structure. Such a connection is unique. For a hyperkaehler manifold, the Obata connection coinsides with the Levi-Civita. Therefore, the man... | 7 | https://mathoverflow.net/users/3377 | 57932 | 36,087 |
https://mathoverflow.net/questions/57801 | 9 | Is there a short way to prove that for each irreducible polynomial $f$ in $k[x\_1,...,x\_n]$ the principal ideal $(f)$ is radical without using unique factorization of polynomials? A short proof of this statement (contained, for example, in the "Primer on CA of Milne") uses the fact that polynomial ring is an UFD, but ... | https://mathoverflow.net/users/13441 | If a polynomial f is irreducible then (f) is radical, without unique factorization? | Here are some thoughts on why such a proof might be hard to find (and interesting). They are not totally rigorous, but I think they give a different perspective, so might be worth something.
I believe the existence of an irreducible element whose ideal is not radical might be related to *non-trivial torsions* in the... | 7 | https://mathoverflow.net/users/2083 | 57935 | 36,089 |
https://mathoverflow.net/questions/57941 | 15 | Consider a commutative square in a category $\mathcal{C}$
$$\begin{array}{ccc}
A&\rightarrow&B\\\
\downarrow&&\downarrow\\\
C&\rightarrow&D
\end{array}$$
Suppose $\mathcal{C}$ is abelian. If this square is a pull-back and $B\rightarrow D$ or $C\rightarrow D$ is an epimorphism, then this square is also a push-out square... | https://mathoverflow.net/users/12166 | Pull-backs which are push-outs | Yes! [Pretoposes](http://ncatlab.org/nlab/show/pretopos) (and in particular toposes) also have this property. It is a remarkable fact that pretoposes (which you can think of as having the first-order exactness properties of toposes or $Set$-like categories) have "most" of the same exactness properties as abelian catego... | 18 | https://mathoverflow.net/users/2926 | 57950 | 36,099 |
https://mathoverflow.net/questions/57952 | 13 | Fix $p>0$ a rational prime, and $K$ an algebraic number field with Galois group $G\_K:=Gal(\bar{\mathbb{Q}}/K) $. The Fontaine-Mazur conjecture predicts that if $\rho:G\_K\rightarrow GL(V)$ is a finite dimensional $\mathbb{Q}\_p$-representation, then it comes from a motive over $K$ (like a subquotient of $H^i\_c(X\time... | https://mathoverflow.net/users/9246 | how irregular can a $p$-adic Galois representation be? | Here are two things that can occur:
1) If V is the representation attached to an overconvergent modular form f, then V will be unramified at almost every prime but will not be de Rham at p (unless f is classical).
2) Ramakrishna has written an article "Infinitely ramified Galois representations". Here's part of the... | 22 | https://mathoverflow.net/users/5743 | 57955 | 36,101 |
https://mathoverflow.net/questions/57951 | 1 | Let $R$ be a valuation ring. We don't assume it to be discrete or have a finite residue field. Let $T$ be a split torus over $R$, so $T\cong {\mathbb G}\_m^r$ for some $r$.
Left there be given a homomorphism of group schemes $T\to {\rm PGL}\_n$, defined over $R$ ,does there exist a lift to a morphism $T\to {\rm GL}\_n$... | https://mathoverflow.net/users/nan | Lifting of projective representations of a torus over a valuation ring | The obstruction to lifting is given by a central extension $E$ of $T$ by $\mathbb{G}\_m$. Such an extension must be commutative because the commutator induces a bi-homomorphism $T\times T\to\mathbb{G}\_m$, and every such map is trivial. So $E$ is a torus, and the extension is dual to an extension of the constant group ... | 2 | https://mathoverflow.net/users/7666 | 57956 | 36,102 |
https://mathoverflow.net/questions/57965 | 5 | I can understand why $P^A = NP^A$ does not imply $P=NP$, $A$ can "contain" the powers of NP.
However, why does $P^B \neq NP^B$ not imply $P \neq NP$? It seems like if $P$ and $NP$ denote the same classes; then we should be able to arbitrarily substitute one for the other (as long as the only thing of interest is the ... | https://mathoverflow.net/users/3609 | Oracle, Relativization, and P vs NP, [Philosophical] | The notation is deceptive. $P^A$ is not something constructed from objects $P$ and $A$, but rather something *analogous* to $P$. In fact, $P$ is a special case of $P^A$, namely $P=P^\varnothing$. The same holds, mutatis mutandis, for $NP$. Removing the contrapositive for extra clarity, the proper way of stating your qu... | 13 | https://mathoverflow.net/users/12705 | 57967 | 36,108 |
https://mathoverflow.net/questions/57894 | 8 | Hi. I have a question.
Let $(M,\omega)$ be a closed symplectic 4-manifold equipped with a free circle action which preserves $\omega$ (symplectic circle action).
My question is , is there an example of $M$ which is not homeomorphic to $S^1 \times N$? ($N$ is a closed oriented 3-manifold)
Thank you in advance.
... | https://mathoverflow.net/users/11705 | symplectic 4-manifolds with free circle action | Here's an example, using a construction of Fernandez, Gray and Morgan (1991):
Take a closed surface $S$ with area form $\omega$, let $\phi$ be an area-preserving diffeomorphism, and $p\colon S\_\phi \to S^1$ its mapping torus. This carries a closed 2-form $\omega\_\phi$ induced by $\omega$, and a closed 1-form $p^\a... | 11 | https://mathoverflow.net/users/2356 | 57968 | 36,109 |
https://mathoverflow.net/questions/57973 | 2 | It appears I am profoundly confused in the following nice argument of Ginibre and Mehta and beautifully presented in Djalil Chafai's blog <http://blog.djalil.chafai.net/2010/11/02/aspects-of-the-complex-ginibre-ensemble/>
In particular look at theorem 3 and 4.
To summarize the argument, we know that the joint eigenv... | https://mathoverflow.net/users/4923 | Why doesn't the argument of circular law convergence of Ginibre spectrum give the same result for GUE? | Dear John, to my knowledge, for the GUE, we have basically the same story with a determinantal kernel and nice orthogonal polynomials, which are the Hermite polynomials. Nothing goes wrong (see e.g. the article in EJP by Ledoux). The polynomials of the complex Ginibre ensemble are just particularly simple. So simple th... | 3 | https://mathoverflow.net/users/12065 | 57991 | 36,119 |
https://mathoverflow.net/questions/57953 | 5 | What is the precise relationship between the Selmer group of an abelian variety and that of its dual? For instance, does the vanishing of one not imply the same for the other?
To fix ideas, let $A$ be an abelian variety defined over a number field $K$, with $A^t$
denote the corresponding dual abelian variety. Fix a ... | https://mathoverflow.net/users/6121 | Selmer of an abelian variety versus that of its dual. | Let $\varphi:A\to A^t$ be a polarization. Then $\varphi$ is an isogeny.
In order to study the difference between the Selmer groups of $A$ and of $A^t$ you need to study the torsion subgroups of $A(K)$ and $A^t(K)$, and to study the difference between the Tate-Shafarevich groups of $A$ and $A^t$. The comparison of the t... | 5 | https://mathoverflow.net/users/8621 | 57995 | 36,122 |
https://mathoverflow.net/questions/57329 | 0 | Let $\Gamma (G;(G\_i)\_{i \in I})$ be a coset geometry (in the sense of Buekenhout) firm, residually connected and flag transitive with Borel subgroup $B$. Consider $H$ any subgroup of $B$. I want to find all the elements $\rho\_i$ in the minimal parabolics subgroups $G\_{I-\{i\}}$ such that $[ \langle H ,\rho\_i \rang... | https://mathoverflow.net/users/12039 | Find elements $\rho_i$ such that $H < B : [\langle H, \rho_i \rangle : H ] = 2$ | First compute the normalizer $N\_{P\_i}(H)$ of $H$ in each minimal parabolic subgroup $P\_i$ of $\Gamma$. Then consider the quotient $N\_{P\_i}(H)/H$ and the natural homomorphism $\varphi\_i:N\_{P\_i}(H) \to N\_{P\_i}(H)/H$. Now define the set
$$S\_i := \{ \rho \in N\_{P\_i}(H) \mid (\varphi\_i(\rho))^2 = 1 \} $$
of ... | 1 | https://mathoverflow.net/users/12039 | 57996 | 36,123 |
https://mathoverflow.net/questions/57997 | 9 | Let $G$ be a linear algebraic group over some field, and let $V$ and $W$ be two simple rational representations of $G.$ Is $V\otimes W$ semi-simple?
I was trying to convince myself that if $G$ has a faithful semi-simple representation, then $G$ is linearly reductive, and was reduced to the question above. The problem... | https://mathoverflow.net/users/370 | Tensor product of simple representations | If $G$ is a(ny) group, if $k$ is a field of characteristic 0, and if $V$ and $W$ are semisimple finite dimensional $kG$ modules, then $V \otimes\_k W$ is indeed semisimple as a $kG$-module. This is due to Chevalley, and (I think I'm not off-base in saying this) inspired the characteristic $p>0$ result of Serre mentione... | 12 | https://mathoverflow.net/users/4653 | 58018 | 36,132 |
https://mathoverflow.net/questions/58010 | 5 | In Mitchell's book "Theory of Categories", Corollary I.16.8 (page 24) states that the following holds in any exact category:
>
>
> Let
> $$
> 0 \to A \to B \to C \to 0
> $$
> $$
> 0 \to B^' \to B \to B^{''} \to 0
> $$
> be short exact sequences. Then $B^' \to B \to C$ is epi iff $A \to B \to B^{''}$ is epi.
>... | https://mathoverflow.net/users/10194 | On a corollary in Mitchell's book | I don't remember what an exact category is in this context, but in any case this seems to be true in any pointed category. Name the arrows $i:A\to B$, $p:B\to C$, $j:B'\to B$, and $q:B\to B''$, and suppose that $pj$ is epi. If
$fqi=0$ then by the universal property of $p:B\to C$ we have
$fq=gp$ for a unique $g$. Now $... | 6 | https://mathoverflow.net/users/10862 | 58029 | 36,139 |
https://mathoverflow.net/questions/57972 | 7 | Can anybody talk about how Higher K theory and algebraic cycles play roles in representation theory? I am more interested in how they play roles in Kazhdan-Lusztig conjectures.
Of course K\_0 plays important roles in representation theory,but I don't know whether Higher K theory is also useful in representation theor... | https://mathoverflow.net/users/1851 | Higher K theory and algebraic cycles in representation theory? | The $K\_2$ functor has an important role in representation theory, starting from
works of Bloch in which he identified the Kac-Moody central extension of loop algebras in terms of $K\_2$ - a more updated form of this is the IHES paper of Deligne-Brylinski about $K\_2$ central extensions of reductive groups. The univer... | 9 | https://mathoverflow.net/users/582 | 58032 | 36,142 |
https://mathoverflow.net/questions/58035 | 1 | An interesting fact was relayed to me in [another question of mine](https://mathoverflow.net/questions/57902/fundamental-groups-of-closed-hyperbolic-3-manifolds-are-freely-indecomposableBlockquote) that
> If $M$ is any closed manifold with universal cover homeomorphic to $R^n$ for $n>1$ then $\pi\_1(M)$ is freely ... | https://mathoverflow.net/users/6429 | Sufficient Conditions for Free Indecomposability | There are some more-or-less equivalent conditions:
* Bass--Serre theory says that a group is freely indecomposable if and only if it acts on a tree with trivial edge stabilizers and no global fixed point.
* More deeply, Stallings' Ends Theorem asserts that a finitely generated group splits over a *finite* subgroup if... | 3 | https://mathoverflow.net/users/1463 | 58038 | 36,145 |
https://mathoverflow.net/questions/58004 | 36 | I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated things (at least for me!).
My question is, how does one really motivate the functional equation of the zeta function? ... | https://mathoverflow.net/users/11286 | How does one motivate the analytic continuation of the Riemann zeta function? | You do not try to motivate it! Even Riemann didn't see a nice argument right away.
Riemann's first proof of the functional equation used a contour integral and led him to a yucky functional equation expressing $\zeta(1-s)$ in terms of $\zeta(s)$ multiplied by things like $\Gamma(s)$ and $\cos(\pi{s}/2)$. Only after ... | 38 | https://mathoverflow.net/users/3272 | 58039 | 36,146 |
https://mathoverflow.net/questions/58009 | 13 | Let $f: X \to Y$ be a morphism of varieties such that its fibres are isomorphic to $\mathbb{A}^n$. Since the definition of a vector bundle stipulates that $f$ be locally the projection $U \times \mathbb{A}^n \to U$, it is likely that there exist morphisms that are not locally of that form, but I can't come up with an e... | https://mathoverflow.net/users/2234 | non-locally trivial A^n bundles | In Jack's example the fiber is not scheme-theoretically $\mathbb A^1$. You can get a counterexample by taking $Y$ to be a nodal curve, $Y'$ its the normalization, with one of the two points in the inverse image of the node removed, and $X = Y' \times \mathbb A^1$.
If we assume that the map is smooth, this becomes qui... | 17 | https://mathoverflow.net/users/4790 | 58046 | 36,149 |
https://mathoverflow.net/questions/57469 | 15 | Suppose $D$ is an explicitly given rank 9 central simple algebra over ${\mathbb Q}$ (or a number field). For example it could be specified by two cubic subfields ${\mathbb Q}(a), {\mathbb Q}(b)\subset D$ and one relation $ba=\sum a\_i b\_j$. What is the best way to determine whether $D$ is a matrix algebra or a divisio... | https://mathoverflow.net/users/3132 | How to distinguish division algebras from matrix algebras? | This may be repeating what others have said as it essentially follows the maximal order approach, but have you looked at Nebe, Gabriele; Steel, Allan, Recognition of division algebras, J. Algebra 322 (2009), no. 3, 903–909?
<http://dx.doi.org/10.1016/j.jalgebra.2009.04.026>
Preprint version and magma code available h... | 8 | https://mathoverflow.net/users/7443 | 58065 | 36,159 |
https://mathoverflow.net/questions/58060 | 16 | I have been looking at Church's Thesis, which asserts that all intuitively computable functions are recursive. The definition of recursion does not allow for randomness, and some people have suggested exceptions to Church's Thesis based on generating random strings. For example, using randomness one can generate string... | https://mathoverflow.net/users/9896 | Can randomness add computability? | I asked [a related question at CS Theory](https://cstheory.stackexchange.com/questions/1263/truly-random-number-generator-turing-computable), which ended with this question:
>
> Is it the case that a TM [Turing Machine] with access to a pure source of randomness (an oracle?), can compute a function that a classical... | 20 | https://mathoverflow.net/users/6094 | 58066 | 36,160 |
https://mathoverflow.net/questions/57969 | 4 | Let $\pi:X \rightarrow Y$ be a finite morphism of schemes and $\mathfrak{F}$ be an etale sheaf on $X$. Then for a $y \in Y$ we have the stalk $(\pi\_{\*}\mathfrak{F})\_{y}=\prod\_{\pi(x)=y}\mathfrak{F}\_{x}^{d(x)}$ where $d(x)$ is the separable degree of the field extension $k(x)/k(y)$ (Corollary 3.5.(c) in Chapter II.... | https://mathoverflow.net/users/12847 | Stalks of etale sheaves | It is enough to prove this in the case where $Y$ is $Spec$ of a strictly Henselian ring. I think, one sees the main point in the argument already in the following special case:
Let $Y=Spec(K)$ and $X=Spec(E)$ where $E/K$ is a finite separable extension of fields. Denote
$\pi: X\to Y$ the canonical map. Let $\overlin... | 3 | https://mathoverflow.net/users/8680 | 58068 | 36,162 |
https://mathoverflow.net/questions/58047 | 17 | This is an idle question. Is there an example of a number field $K$ for which the maximal everywhere-unramified extension $M|K$ is of infinite degree, and yet the group $\mathrm{Gal}(M|K)$ has been completely determined ?
| https://mathoverflow.net/users/2821 | Explicitly describable maximal unramified extension of a number field | No, I'm pretty sure not.
In general, the theory is much more developed for the maximal *pro-p-quotient* of the groups you're asking about, and even in this more explored setting, not a single explicit presentation of an infinite such group is known (to me, for sure, but I think to anyone -- **Edit:** Nigel Boston app... | 10 | https://mathoverflow.net/users/35575 | 58075 | 36,166 |
https://mathoverflow.net/questions/57876 | 5 | I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what the exact correspondence is here? Is the homotopy space some kind of moduli space for the flat connections?
| https://mathoverflow.net/users/1867 | Flat Principal Connections and Homotopy Groups? | The comments have addressed relating fundamental group and a flat connection. Something can be said about the moduli space of flat connections. Goldman, in <http://www.springerlink.com/content/g468047131514211/>, considers the moduli space of flat connections over a surface genus $>1$. He defines a symplectic form and ... | 1 | https://mathoverflow.net/users/8358 | 58088 | 36,175 |
https://mathoverflow.net/questions/58091 | 8 | If we take Peano Arithmetic and restrict induction to formulas over various fragments of the arithmetic hierarchy, say to the $\Sigma^0\_n$ formulas for various $n$ or some other interesting fragments, how does the proof theoretic ordinal for the theory vary?
| https://mathoverflow.net/users/4085 | Ordinal Analysis of Peano Arithmetic with Restricted Induction | The proof-theoretic ordinal of $I\Sigma^0\_n$ (for $n > 0$) is well-known to be $\omega\_{n+1}$, where $\omega\_1:=\omega$, $\omega\_{n+1}:=\omega^{\omega\_n}$. See e.g. [Avigad & Sommer](http://www.jstor.org/stable/421195).
| 7 | https://mathoverflow.net/users/12705 | 58092 | 36,178 |
https://mathoverflow.net/questions/58096 | 7 | Hello,
Given a finitely presentable group $G$, I'm interested in the cup-product from $H^1$ to $H^2$ with real coefficients. I want to know if this is explicitly computable (with a computer) with a presentation of the group.
More precisely, I want a program that takes the generators and relations as entries and retur... | https://mathoverflow.net/users/12517 | Computations in group cohomology | EDIT: I was explaining this to a grad student today, and I realized that I didn't give any references. The result I describe below was first stated by Sullivan in
Sullivan, Dennis
On the intersection ring of compact three manifolds.
Topology 14 (1975), no. 3, 275-277.
He claims it is true for a 3-manifold, but all ... | 29 | https://mathoverflow.net/users/317 | 58098 | 36,180 |
https://mathoverflow.net/questions/58040 | 40 | In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to [classical hypergeometric functions](http://en.wikipedia.org/wiki/Generalized_hypergeometric_function), [basic (q-) hypergeometric functions](http://en.wikipedia.org/wiki/Basic_hypergeometric_series),... | https://mathoverflow.net/users/2384 | Groups, quantum groups and (fill in the blank) | That's a wonderful question, but I think there's a fundamental confusion here about two possible roles of the rational/trigonometric/elliptic trichotomy -- the one asked in the question and the one that leads to elliptic quantum groups -- which are in some sense "Fourier dual". (Everything I understand about this I lea... | 35 | https://mathoverflow.net/users/582 | 58107 | 36,183 |
https://mathoverflow.net/questions/57641 | 4 | I was just sent a 'phone tree' by a fellow parent in my kids class.
This is a way for emergency messages to be sent to everyone in the class. They way it works is that you are given a call by the person above you in the list, you are then supposed to call the person below you.
The particular tree in question looks ... | https://mathoverflow.net/users/13467 | Understanding efficiency vs reliability trade offs in a phone tree design | Spreading information is a very important topic in computer science and has many applications, in particular also in network design, as you already mentioned. One well-studied way of doing this is called "Randomized Rumor Spreading". Just use google...
Popular results in this context show that the routing time is log... | 1 | https://mathoverflow.net/users/9386 | 58117 | 36,189 |
https://mathoverflow.net/questions/57715 | 0 | I've coded up the FFT for a dataset I'm working with. My intent is to create a waterfall plot of the result, but the problem I'm running into is if I change my input data size, then I get a different number of frequency bins. Currently I'm just making my input dataset twice the size of the number of pixels I need to ma... | https://mathoverflow.net/users/13479 | Remap FFT frequency bin distribution | Pick a large number of points to discretize the frequency domain with. When you have a time signal with less points zero pad until you hit that number. This is sometimes called "spectral interpolation" <https://ccrma.stanford.edu/~jos/st/Zero_Padding_Theorem_Spectral.html> and does a nice job of interpreting the freque... | 0 | https://mathoverflow.net/users/6360 | 58118 | 36,190 |
https://mathoverflow.net/questions/58113 | 8 | There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer.
Recently I found that the same result is true if we replace $\alpha n$ by $\alpha n^2$ or any polinomial p such that $p(0)=0$.
... | https://mathoverflow.net/users/13566 | Kronecker Approximation theorem and Fibonacci numbers | Well, as Gerry has pointed out, this is certainly not true for all $\alpha$. On the other hand, this is true for a.e. $\alpha$. More precisely, the sequence $2^n\alpha$ is equidistributed mod 1 for a.e. $\alpha$.
I believe this result is due to H. Weyl and can be found in Cornfeld, Fomin and Sinai `Ergodic Theory'. (... | 12 | https://mathoverflow.net/users/8131 | 58119 | 36,191 |
https://mathoverflow.net/questions/57994 | 0 | Hi, I would like some help on something that in some ways has already been touched in the past here. I saw these relevant questions being answered, but I am still unable to understand some things. I would be grateful if someone could help me.
Let $M \models$ ZFC that is countable and let $p(x) = {0 \in x, \ldots, n \... | https://mathoverflow.net/users/13533 | Saturated extensions of ZFC models | I'm addressing both the question and the comments, but possibly this question should be closed. First let's be clear what we mean by a model of set theory. A model is a set $M$ (or perhaps a class) with a binary relation $E$. $(M,E)$ satisfies *Foundation* if for every $x$ in $M$, there is an $E$-minimal $y$
in $M$ suc... | 4 | https://mathoverflow.net/users/10774 | 58123 | 36,194 |
https://mathoverflow.net/questions/54799 | 5 | Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement.
Without going through the machinery of the sheaves, at least now, is it possible to bypass the Godement's result?
And, if ye... | https://mathoverflow.net/users/12617 | On a proof of the existence of tubular neighborhoods. | In the finite-dimensional setting, it's possible to construct tubular neighborhoods without anything like Godement's lemma. Many sources simply rely on a point-set topology argument that's based on the same idea as Godement's lemma (to be precise, I'm talking about the argument on p. 109 of Lang's book Differential and... | 19 | https://mathoverflow.net/users/4042 | 58124 | 36,195 |
https://mathoverflow.net/questions/58094 | 27 | **Setup:** Let $k$ be a field, let $n$ be a positive integer, and let $R := k[[x\_1,\ldots,x\_n]]$ denote the commutative ring of formal power series over $k$ in $x\_1,\ldots,x\_n$. We know that there is exactly one maximal ideal for $R$, namely $\langle x\_1,\ldots,x\_n \rangle$.
By localizing at the multiples of t... | https://mathoverflow.net/users/13556 | Maximal Ideals in Formal Laurent Series Rings? | Your ring $L$ is a localization of the power series rings $R= k[[x\_1,\cdots,x\_n]]$ at the multiplicative set $M$ of monomials in $R$.
So the prime ideals of $L$ correspond to prime ideals in $R$ which do not meet $M$. Clearly, the maximal primes are the set of biggest primes which do *not* contain any variable. Si... | 15 | https://mathoverflow.net/users/2083 | 58126 | 36,197 |
https://mathoverflow.net/questions/58100 | 7 | What is the minimal resolution of singularities of the surface
$S^2(X^3+Y^3+Z^3)-3(S^2+T^2)XYZ=0$ which is a subset of $\mathbb{P}^1\times\mathbb{P}^2$
Please note that in this equation $[S:T]\in{\mathbb{P^1}}$ and $[X:Y:Z]\in{\mathbb {P^2}}$ and by $\mathbb{P^n}$ we mean n-dimensional complex projective space.
| https://mathoverflow.net/users/13559 | minimal resolution of singularities | Let us start by writing down the computation of the singular points in the chart $S=1$.
Writing $\lambda:=T/S$, in the chart $S=1$ we can rewrite the equation of the surface as
$$X^3+Y^3+Z^3-3(1+\lambda^2)XYZ=0.$$
This is an elliptic fibration over $\mathbb{C}$ (with coordinate $\lambda$), whose fibres are the curve... | 10 | https://mathoverflow.net/users/7460 | 58142 | 36,203 |
https://mathoverflow.net/questions/58148 | 5 | When students are first learning about groups, a classic example of a group that is *not* defined as a set of functions is the group whose underlying set is $\mathbb{R}\setminus-1$, and whose operation is $x\*y=x+y+xy$. This naturally leads one to wonder about what other polynomials in two variables give rise to a grou... | https://mathoverflow.net/users/6856 | Polynomial group Laws on $\mathbb{R}^2$ | Emil's basic observation can be extended. Polynomials are smooth (i.e., infinitely differentiable) functions. What can you say about smooth group laws on the set of real numbers?
In different language: a Lie group is a manifold which comes equipped with group operations which are smooth with respect to the manifold ... | 8 | https://mathoverflow.net/users/2926 | 58160 | 36,211 |
https://mathoverflow.net/questions/58159 | 6 | Let $G\_1$ be a finite-index subgroup of $G\_2$. Let $i : H^{\ast}(G\_2) \rightarrow H^{\ast}(G\_1)$ be the induced map of rings. There is then a transfer homomorphism $\tau : H^{\ast}(G\_1) \rightarrow H^{\ast}(G\_2)$ whose key property is that $\tau(i(x)) = [G\_2:G\_1] \cdot x$ for all $x \in H^{\ast}(G\_2)$. I have ... | https://mathoverflow.net/users/13577 | Cup products and the transfer map | What is true is that $\tau$ is a map of *modules*; that is,
$$\tau(i^\*(x)\cup y) = x\cup \tau(y)$$
for $x\in H^\*(G\_2)$ and $y\in H^\*(G\_1)$.
In particular, the kernel of $\tau$ is a sub-$i^\*(H^\*(G\_2))$-module of $H^\*(G\_1)$.
For an example, consider $G\_1=C\_p$ (cyclic group) and $G\_2=\Sigma\_p$ (symmetric... | 9 | https://mathoverflow.net/users/437 | 58171 | 36,218 |
https://mathoverflow.net/questions/58162 | 5 | Hi All,
I would like to know the Automorphism group of some simple classical groups, such as PSL(n,q) or some PSU or PSp groups. Could you please give me some recommended books or papers then? I could not find one.
Thanks in advance.
| https://mathoverflow.net/users/11228 | Automorphism Group of some Classical groups | As Tom points out, the Springer survey by Dieudonne (in French) is a standard source, with lots of references to the primary literature since at least the older work of Schreier and
van der Waerden. Later work by O'Meara and others has mainly been concerned with more general settings over fields and commutative rings, ... | 9 | https://mathoverflow.net/users/4231 | 58172 | 36,219 |
https://mathoverflow.net/questions/55383 | 15 | One of the first things we usually learn when we study iterated forcing is that we can force over a model of ZFC + GCH to make the continuum function ($\lambda \mapsto 2^{\lambda}$) restricted to some set $S$ of regular cardinals follow any nondecreasing pattern satisfying [König's Theorem](http://en.wikipedia.org/wiki... | https://mathoverflow.net/users/11318 | Impact of discrepancy between Kunen's and Jech's definition of iterated forcing on full support iterations | I hope I can answer this question in a reasonable way. The natural way to define the iteration $\mathbb P\*\dot{\mathbb Q}$ is to consider all pairs $(p,\dot q)$ with
$p\in\mathbb P$ and $p\Vdash\dot q\in\dot{\mathbb Q}$.
Unfortunately this is a proper class. (The definition in Jech's book also gives a proper class... | 12 | https://mathoverflow.net/users/7743 | 58174 | 36,221 |
https://mathoverflow.net/questions/58156 | 3 | Let $K$ be a field and $n \geq 0$. Serre proved that $\text{Qcoh}\_f(\mathbb{P}^n\_K)$ is equivalent to the localization of $\text{grMod}\_f(K[x\_0,...,x\_n])$, in which the inclusions $M\_{\geq a} \to M$ become inverted. Here the index $f$ means "finitely presented".
What happens if we replace $K$ by an arbitrary ri... | https://mathoverflow.net/users/2841 | Coherent sheaves on projective space over a general ring | Serre's proof can indeed be generalized to noetherian rings - or even noetherian schemes - once you have proved that for a projective morphism between (locally) noetherian schemes the higher direct images of coherent sheaves are again coherent. You find all the arguments in EGA III, Section 2.3, in particular Scholie (... | 8 | https://mathoverflow.net/users/13302 | 58185 | 36,228 |
https://mathoverflow.net/questions/58175 | 2 | I compute well-known "sample Pearson correlation coefficient" of two vectors:
$r(X,Y) = \frac{\sum ^n \_{i=1}(X\_i - \bar{X})(Y\_i - \bar{Y})}{\sqrt{\sum ^n \_{i=1}(X\_i - \bar{X})^2} \sqrt{\sum ^n \_{i=1}(Y\_i - \bar{Y})^2}}$ . So far so good.
I need to generalize it as follows.
I need to add "weights", $w\_i\in[... | https://mathoverflow.net/users/12330 | Generalize Pearson | I think that the correct thing to do is treat the values as if they were repeated $w\_i$ times. That is, assume first that the weights $w\_i$ are non-negative integers, the data actually comes in the form such that the pair $(x\_i, y\_i)$ appears $w\_i$ times, and compute the standard Pearson correlation for such (repe... | 2 | https://mathoverflow.net/users/1778 | 58191 | 36,231 |
https://mathoverflow.net/questions/58193 | 11 | Where can I find a calculus textbook that emphasizes differentials?
Is there such a book that I could realistically require my calculus students to use?
I want a textbook that supports me when I tell my students something like:
$\Delta((x^2+1)^5)\approx5(x^2+1)^4\Delta(x^2+1)\approx5(x^2+1)^4(2x\Delta x)$
$d((x^2... | https://mathoverflow.net/users/12106 | Leibnizian calculus textbook | There is a marvelous old book (19th Century if I recall correctly) where I learned Calculus the first time, called "Calculus Made Easy" by Silvanus P. Thompson, and subtitled "What one fool can do another can". He explains that dx means a "little bit of x" and shows a square with sides x and x + dx and you can see why ... | 26 | https://mathoverflow.net/users/7311 | 58196 | 36,234 |
https://mathoverflow.net/questions/53467 | 9 | In Jech one can find a lower bound for the consistency strength of PFA in terms of large cardinals. I don't have my copy of Jech in front of me at the moment, but as I recall the presentation of this fact goes something like this:
* It's stated and proven that PFA implies the failure of $\square \_{\kappa}$ for all $... | https://mathoverflow.net/users/7521 | What is the consistency strength of the failure of square, in terms of large cardinals | My understanding is that the weaker form of square was initially needed to get a better lower bound on the consistency strength of PFA but that, with improvements in our understanding of the fine structure of the current inner models, the original form of square now suffices to give the best lower bounds.
There are t... | 7 | https://mathoverflow.net/users/10774 | 58198 | 36,235 |
https://mathoverflow.net/questions/57520 | 54 | It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider mirror symmetry as science, what are some examples there, that can be understood?
I would like to explain a bit this quest... | https://mathoverflow.net/users/13441 | Examples in mirror symmetry that can be understood. | Here is my biased view of a simple example: the two-torus.
Everything I know about homological mirror symmetry
stems from this example.
Because the example is one-dimensional, a symplectic form
is just an area form, and Lagrangians are simply
curves, and the holomorphic maps which are part of the Fukaya
category are ... | 32 | https://mathoverflow.net/users/1186 | 58199 | 36,236 |
https://mathoverflow.net/questions/58186 | 3 | Let A, B be two C\*-Algebras and $A\otimes B$ denote their minimal tensor product(I don't know whether C\*-norm matters or not, but for simplicity we can assume that one of them is nuclear so all C\*-norm coincide). Let x be a non-zero positive element in $A\otimes B$, can we always find a single tensor $0\neq x\_1\oti... | https://mathoverflow.net/users/9858 | positive element in C* tensor product | The same answer as before, the matrix
$$
a=\begin{bmatrix}
1 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1
\end{bmatrix}
$$
in $M\_2(\mathbb{C})\otimes M\_2(\mathbb{C})$, also works here since it is twice a rank one projection and so any smaller positive matrix must be a scalar multiple of $a$.
| 10 | https://mathoverflow.net/users/6460 | 58200 | 36,237 |
https://mathoverflow.net/questions/56295 | 8 | Does anyone have a copy of the unpublished preprint of Karen Uhlenbeck *A priori estimates for Yang-Mills fields* from around 1986?
It appears to have circulated for some time, and it is quoted in several papers in the field (Uhlenbeck-Yau, Daskalopoulos-Wentworth, De Bartolomeis-Tian,...) sometimes with precise refe... | https://mathoverflow.net/users/13168 | K.Uhlenbeck's preprint "A priori estimates for Yang-Mills fields" | Sorry it took so long. The scanned version of Uhlenbeck's preprint is available here:
[<http://www.math.uwaterloo.ca/~karigiannis/uhlenbeck-preprint.pdf>](http://www.math.uwaterloo.ca/~karigiannis/uhlenbeck-preprint.pdf)
I can leave it there for a few months, at least.
| 11 | https://mathoverflow.net/users/6871 | 58202 | 36,238 |
https://mathoverflow.net/questions/58204 | 13 | Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ such that $y = f(x)$. Thus for $n = 1$, $f(\mathbb{C}) = \mathbb{C}$ if and only if $f$ is a non-constant polynomial. Can w... | https://mathoverflow.net/users/13569 | When are complex polynomial maps almost surjective? | Being algebraically independent is indeed a necessary and sufficient condition for the image of $f$ to be dense.
As $f\colon\mathbb{C}^n\to\mathbb{C}^n$ is regular, its image is *constructible* and, in particular, contains a non-empty open subset of its closure (under the Zariski topology). See Theorem 10.2 of [J.S. ... | 22 | https://mathoverflow.net/users/1004 | 58205 | 36,239 |
https://mathoverflow.net/questions/58203 | 15 | The recent paper On the Erdos distinct distance problem in the plane
Authors: Larry Guth, Nets Hawk Katz prodded me to get a non-trivial example. Here is what I cannot find: an example of 12 distinct points in the plane with only 5 different distances between points. The regular 12-polygon has 6 different lengths but I... | https://mathoverflow.net/users/13584 | Erdos distance problem n=12 | Here goes my poor explanation:
Take a regular hexagonal lattice with distance 1 between nearest neighbors, and choose a 15-point equilateral triangle in this lattice (15 is a triangular number). Remove the 3 vertices of the triangle. You'll be left with 12 points and 5 distinct distances.
Edit: Just checked the OEI... | 17 | https://mathoverflow.net/users/12722 | 58207 | 36,241 |
https://mathoverflow.net/questions/58179 | 3 | Hello!
**Background:** I am trying to understand a proof in
>
> Béla Bollobás, Random Graphs, 2nd edition, Ch. 11.1
>
>
>
about the clique number of G(n,p) random graphs.
**The problem:** At some point, the following sequence is defined:
$F\_\ell := \frac{\binom{r}{\ell} \binom{n-r}{r - \ell} } {\binom{n... | https://mathoverflow.net/users/13579 | Proof of the clique-number for random graphs: How to bound this binomial-like expression? | If I remember correctly, I convinced myself of a very similar statement in the Alon/Spencer book (but there even the sum of the intermediate $F\_l$ had to be dominated by $F\_3+F\_{r-1}$) by looking at the quotient
$$\frac{F\_{l+1}}{F\_l}=\frac{(r-l)^2}{(l+1)(n-2r+l+1)}\frac{\left(p^{-\binom{l+1}{2}}-1\right)}{\left... | 1 | https://mathoverflow.net/users/12674 | 58208 | 36,242 |
https://mathoverflow.net/questions/58215 | 2 | I'm well aware of the fact that the number of Chen primes between $N/2$ and $N$ for large enough $N$ is at least
$$\frac{c\_1N}{\ln^2(N)}$$
(Green and Tao). My question is: is there possibly an upper bound for chen primes between $N/2$ and $N$? What I am eventually trying to prove is that there are infinitely many int... | https://mathoverflow.net/users/10920 | Upper bound on Chen primes in an interval? | Yes. Sieve methods are *much* better at proving upper bounds than lower bounds. By the Selberg sieve, or alternatively using the combinatorial sieve, you can prove that the number of Chen primes is bounded above by $C\_2 N/\log^2 N$ for some particular value of $C\_2$. (ed: Please see Terry Tao's important caveat below... | 4 | https://mathoverflow.net/users/1050 | 58220 | 36,248 |
https://mathoverflow.net/questions/57611 | -1 | I feel stupid for having to ask this, but does anybody have any idea how to handle
$$\frac{d}{x}\sum\_{n=k}^{g(x)}f(n,x)?$$
Example:
$$\frac{d}{dx}\sum\_{n=6}^{i^2+2i} \frac{1}{\ln{(i^2)}-\ln{\ln n}}.$$
If we were able to separate the summand into two functions, one with only $i$ as a variable, and one with only $n$ as... | https://mathoverflow.net/users/10920 | Derivative of Sum over Variable of derivative | This reminds me uncomfortably about a remark that Terry Tao made in [this answer](https://mathoverflow.net/questions/40082/why-do-we-teach-calculus-students-the-derivative-as-a-limit/44572#44572) about the importance of teaching the derivative as a limit: at least one person tried to prove Fermat's Last Theorem by diff... | 1 | https://mathoverflow.net/users/8212 | 58222 | 36,249 |
https://mathoverflow.net/questions/58221 | 1 | Page 121 of Computational Complexity, A Modern Approach states:
6.11 (Open Problem) Suppose make a stronger assumption than $NP \subset P/poly$: every langauge in NP has linear size circuits. Can we show something stronger than PH = $\Sigma\_2^p$.
Context: earlier in the chapter, it is shown that $NP \subset P/poly... | https://mathoverflow.net/users/3609 | "NP has linear circuits" --> something interesting? [soft, philosophical, open] | [Lance Fortnow, Rahul Santhanam and I](http://www.cs.cmu.edu/~ryanw/circuit.pdf) have shown some nontrivial results in this direction. For example, $NP$ has $O(n^c)$ size circuits for some fixed $c$ if and only if $P^{NP[n]}$ has $O(n^k)$ size circuits for some fixed $k$. The paper has several results along these lines... | 7 | https://mathoverflow.net/users/2618 | 58224 | 36,250 |
https://mathoverflow.net/questions/58201 | 7 | For example, "P=NP implies PH=P" is interesting ... because most of us don't believe PH=P, so it provides strong evidence P != NP.
On other hand, "P=NP implies EXP has circuit of $2^n/n$ size" seems fairly weak for the following reasons:
(1) we know that there are circuits of size $2^n/n$
(2) we know that most ra... | https://mathoverflow.net/users/3609 | Why is "P = NP implies EXP has circuit of $2^n/n^" interesting? [Soft, Philosophical] | Everybody believes that EXP contains problems of exponential circuit complexity, but we are very far from proving it, and in fact we know that any proof of such a result cannot be a relativizing argument, cannot be an ''algebraizing'' argument in the sense of Aaronson and Wigderson, and cannot be a ``natural proof'' in... | 13 | https://mathoverflow.net/users/12595 | 58225 | 36,251 |
https://mathoverflow.net/questions/58231 | 2 | Let x be a complex number.
What is the Stirling formula for x(x+1)(x+2)...(x+n-1) when n goes to infinity?
| https://mathoverflow.net/users/2666 | What is the Stirling formula for x(x+1)(x+2)...(x+n-1)? | It looks like you want a formula for the asymptotics of the Pochhammer symbol $(x)\_n$ as $n \to \infty$. One such formula is provided about halfway down [Wolfram's page](http://functions.wolfram.com/GammaBetaErf/Pochhammer/introductions/FactorialBinomials/ShowAll.html):
$$(x)\_n \sim \frac{2\pi}{\Gamma(x)} e^{-n}n^{... | 11 | https://mathoverflow.net/users/121 | 58235 | 36,255 |
https://mathoverflow.net/questions/57094 | 1 | At the references section of the wikipedia article for Definable set, one finds the following entry:
Slaman, Theodore A. and W. Hugh Woodin. Mathematical Logic: The Berkeley Undergraduate Course. Spring 2006.
What kind of material is it? Manuscripted lecture notes? Is it available somehow? I'm highly curious about ... | https://mathoverflow.net/users/6466 | Slaman and Woodin on Mathematical logic | The version of the notes I have is from 2006, they are organized in the form of a short book. It is my understanding they have been updated since, and I believe the current version has new material on model theory, computability, and incompleteness. In particular, I think that Woodin's proof of the second incompletenes... | 3 | https://mathoverflow.net/users/6085 | 58239 | 36,257 |
https://mathoverflow.net/questions/58249 | 17 | I was wondering whether anyone knows of any good non-technical or even popular expositions of the Birch—Swinnerton-Dyer conjecture, for someone with minimal background in elliptic curves. I was thinking of something along the lines of du Sautoy's excellent book '*The Music of the Primes*' on the Riemann hypothesis, tho... | https://mathoverflow.net/users/12160 | A non-technical account of the Birch—Swinnerton-Dyer Conjecture | Try Zagier's MPI preprint MPI/89-48 with the title *The
Birch and Swinnerton-Dyer conjecture from a naive point
of view*.
**Addendum** (2012/02/23). An even more elementary introduction is [this one](http://people.mpim-bonn.mpg.de/zagier/files/tex/BSDwHarder/fulltext.pdf) by Harder and Zagier.
| 16 | https://mathoverflow.net/users/2821 | 58250 | 36,262 |
https://mathoverflow.net/questions/48953 | 0 | Hi, I apologize if there is already an (obvious) answer to my question, but please bear with me for the moment as I find it hard to see a good way to answer this question:
In the same way that the Mersenne primes uniquely determine the even perfect numbers (i.e. in the "bijective" sense), do the Euler primes likewise... | https://mathoverflow.net/users/10365 | Perfect Numbers - On Mersenne and Euler Primes | I tried to get in touch with Dean Hickerson and here's what he got to say regarding the problem of determining the status of squares with respect to solitude or friendliness:
>
> Notice that all of the squares from $1$ to $121$ are
> solitary (since they satisfy $gcd(n, \sigma(n)) = 1$). (See [OEIS - A014567](http... | 0 | https://mathoverflow.net/users/10365 | 58264 | 36,271 |
https://mathoverflow.net/questions/58033 | 8 | Recently I started reading some articles about the
degree of commutativity of finite groups. I have some questions:
1. In "Subgroup commutativity degrees of finite groups" Tarnauceanu
proposes the following formula for calculating the degree of
commutativity of subgroups of a finite group G:
$$ \mathrm{sd}(G) = \fr... | https://mathoverflow.net/users/nan | Degree of commutativity of finite groups and subgroups | This is in answer to your second question. There is a note by Gustafson:
* MR0327901 (48 #6243) Gustafson, W. H.
What is the probability that two
group elements commute? Amer. Math.
Monthly 80 (1973), 1031–1034.
where he proves the result Ben mentions, viz. if $G$ is a finite nonabelian group, then $d(G) \leq 5/8$.... | 7 | https://mathoverflow.net/users/2114 | 58266 | 36,272 |
https://mathoverflow.net/questions/58267 | 12 | Let $K$ be a finite field and $G$ be a discrete group.
>
> Is it true that for every $a,b\in K[G]$ the condition $ab=0$ implies $ba=0$?
>
>
>
It does not seem to be related to zero divisor problem, any ideas if this can be true and for which fields?
| https://mathoverflow.net/users/8699 | Group ring and left zero divisor | Let G be non-abelian of order 6, with x of order 2 and y of order 3. In such a group yxy = x, since both x and xy have order 2. Let K be a field with 2 elements. Then (x+y)⋅(1+xy) = x+y + y+yxy = x+y + y+x = 0, but (1+xy)⋅(x+y) = x+y + xyx+xyy = x+y + yy + xyy ≠ 0.
You may be thinking of the property: if a⋅b = 0 then... | 12 | https://mathoverflow.net/users/3710 | 58272 | 36,275 |
https://mathoverflow.net/questions/58265 | 8 | It is somehow a general principle that the (infinitesimal) local behavior of a representable moduli functor $X$ at some point $x$ is closely related to the deformation problem of the structure parameterized by $x$: the deformation functor should be pro-represented by the formal completion of $X$ at $x$ ($X$ representab... | https://mathoverflow.net/users/9246 | About the Serre-Tate theorem | Concerning the first question, the answer is yes.
Given an abelian variety $A$ over $k$ of genus $g$, we have that the first cohomology of its tangent space is of dimension $g^2$.
Moreover, by a theorem of Grothendieck, we know that the deformations of $A$ are unobstructed. This yields that the base ring of the unive... | 7 | https://mathoverflow.net/users/5273 | 58274 | 36,277 |
https://mathoverflow.net/questions/58279 | 3 | Is the corestriction map from a subgroup H to a group G, on the first Tate cohomology group H^1, dependent on the chosen transversal of H is G?
| https://mathoverflow.net/users/13600 | corestriction and transversals | No, because the map on cochains is $f\mapsto \sum\_{g\in G/H}gfg^{-1}$ where $f$ is $H$-linear, so that any other coset representative $gh$ would not have any effect.
This shows that $cor^G\_H$ is independent of the choice of transversal on the level of cohomology. But in that paper of Eunmi Choi, a $\textit{differen... | 2 | https://mathoverflow.net/users/12310 | 58281 | 36,279 |
https://mathoverflow.net/questions/58270 | 11 | Are there any historical articles about the origins of zeta functions of curves over global fields (undoubtedly starting with $\mathbb{Q}$)? In particular who (and when did this happen) first have the idea of creating such a thing? Was it by analogy with the zeta functions of number fields?
| https://mathoverflow.net/users/2784 | Historical Articles about zeta functions of curves | On the last page of chapter 18 of Ireland and Rosen's book, they say that Weil defined zeta-functions of smooth varieties over number fields in his 1954 paper "Abstract versus Classical Algebraic Geometry". It's in Weil's Collected Works, vol. II, pp. 550--558.
Edit: In an article by Lang about the history of the Tan... | 12 | https://mathoverflow.net/users/3272 | 58284 | 36,281 |
https://mathoverflow.net/questions/58276 | 14 | The motivation for asking this question is a passage (3.2) in an [article](http://books.google.co.in/books?id=adfidZ9Ojg8C&lpg=PP1&dq=in%20scope%20of%20logic&pg=PA108#v=onepage&q&f=false) by Greg Hjorth where he said that "...it is also an attractive feature of the theory of Borel cardinalities and of the theory of $L(... | https://mathoverflow.net/users/3462 | Parts of Set Theory immune to independence | It is a theorem of Woodin that if there is a proper class of Woodin cardinals, then the theory of $L(\mathbb{R})$ can not be changed by forcing. Since forcing and large cardinals are essentially our only means for establishing independence results, this can be interpreted as saying that the theory of $L(\mathbb{R})$ is... | 24 | https://mathoverflow.net/users/10774 | 58285 | 36,282 |
https://mathoverflow.net/questions/58288 | 3 | In section III.3.4 of Eisenbud & Harris's "The Geometry of Schemes," we/they construct an infinite family of double structures on $\mathbb{P}^1 \subset \mathbb{P}^3$ that are distinguished from each other by their genus. Here is the construction (let's just worry about $\mathbb{C}$ for now):
Let $d$ be a non-negative... | https://mathoverflow.net/users/1231 | Classification of fat projective lines? | (Note: I don't assume $C$ is a curve, or even that it is smooth.)
Suppose $C'$ is a 2-fold thickening of $C$. Then $C'$ has the same underlying topological space as $C$. On that space, we have a short exact sequence of $\newcommand{\O}{\mathcal O}\O\_{C'}$-modules
$$\newcommand{\L}{\mathcal L}\tag{$\dagger$}
0\to \L\... | 6 | https://mathoverflow.net/users/1 | 58295 | 36,286 |
https://mathoverflow.net/questions/42803 | 8 | A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand, a quasi-Lie algebra is defined by replacing the axiom $[x,x]=0$ with the antisymmetry axiom $[x,y]+[y,x]=0$ for all $x... | https://mathoverflow.net/users/9417 | Quasi-Lie algebras in nature? | Tom Goodwillie:
The homotopy groups of a (say, simply connected) space $X$ form a graded Lie algebra under Whitehead product, in which the even-dimensional part (which is actually the $\pi\_n$ for odd $n$ ) can have elements $x$ such that $[x,x]$ has order $2$ -- for example $\pi\_n(S^n)$ for most odd values of $n$.
... | 2 | https://mathoverflow.net/users/9417 | 58303 | 36,290 |
https://mathoverflow.net/questions/58301 | 0 | Hello,
I've recently write down some measure for sets and now I wonder how it is called or where it is described?
The measure itself is the following:
Let $A$ & $B$ -- two sets of values from a single space (real line for example)
Let:
$d(A, B) = \sum\_{a \in A} \sum\_{b \in B} \frac{dist(a, b)}{len(A) \* len(B)}... | https://mathoverflow.net/users/13608 | How the distance between sets is called? | This seems related to cluster analysis, and your d(A, B) is used in "average linkage clustering."
<http://en.wikipedia.org/wiki/Cluster_analysis>
| 3 | https://mathoverflow.net/users/13611 | 58309 | 36,293 |
https://mathoverflow.net/questions/58283 | 10 | In the very beginning of the book "Introduction to Invariants and Moduli" Shigeru Mukai
proves Molien's formula for the Hilbert series of the invariant ring of a finite group action on $\mathbb C^n$. For example, in the case of the standard action of Quaternions on $\mathbb C^2$ the Hilbert series is $\frac{1-t^{12}}{(... | https://mathoverflow.net/users/13441 | A question about an application of Molien's formula to find the generators and relations of an invariant ring | As pointed out by Richard Stanley, there are some subtleties in finding the invariant generators.
The Molien series only tells you the Hilbert series of the ring of invariants up to *reduced* rational forms. So some information is lost, as illustrated in Richard's answer.
One can gain some information by looking ... | 10 | https://mathoverflow.net/users/2083 | 58311 | 36,294 |
https://mathoverflow.net/questions/58307 | 0 | Following is a doubt I got when I was reading Csiszar's Annals of Statistics paper "Why least squares and maximum entropy: An axiomatic approach to inference for linear inverse problems".
He comes up with a set of axioms to justify $\ell\_2$-norm minimisation in $\mathbb{R}^n$.
He also argues that $\ell\_2$-norm mi... | https://mathoverflow.net/users/7699 | Nonexistence of projection | HI Ashok,
I think $\mathbb{R}^n\_+$ requires strictly positive components. Then it's easy to come up with the situation above, namely consider $L$ to be the part of the line $x+y = 1$ in $\mathbb{R}\_+^2$. And $u = (2,1)$. Then there is no closest point in $L$ to $u$, because it wants to be $(1,0)$, which has nonposit... | 2 | https://mathoverflow.net/users/4923 | 58315 | 36,298 |
https://mathoverflow.net/questions/58259 | 2 | I'm trying to find the following maximum: $\max\_{\gamma}\sum\_{|\alpha|=q}\binom{\alpha}{\gamma}$. Here $\alpha=(\alpha\_1,\ldots, \alpha\_n),\gamma=(\gamma\_1,\ldots, \gamma\_n)$ are multi-indices. The binomial coefficient is defined as $\binom{\alpha}{\gamma}=\frac{\alpha !}{\gamma! (\alpha-\gamma)!}=\prod\_{i=1}^n ... | https://mathoverflow.net/users/nan | How do i maximize $\max_{\gamma}\sum_{|\alpha|=q}\binom{\alpha}{\gamma}$? | I claim that $$\sum\_{|\alpha|=q} \binom{\alpha}{\gamma}=\binom{n+q-1}{|\gamma|+n-1}$$
and so the answer is simply all $\gamma$ with $|\gamma|=\lfloor\frac{n+q-1}{2}\rfloor -n+1$. A quick proof comes from the following generating function
$$\frac{x^l}{(1-x)^{l+1}}=\sum\_{p=0}^{\infty} \binom{p}{l}x^p$$ and looking at ... | 4 | https://mathoverflow.net/users/2384 | 58319 | 36,301 |
https://mathoverflow.net/questions/58320 | 2 | Let $G$ be a closed, connected Lie group, and let $H$ be a closed (and therefore Lie) subgroup. There is a natural action of $G$ on the space of left cosets $G/H$, for which the stabiliser of $aH$ is the conjugate subgroup ${}^aH:=aHa^{-1}$.
Now let $G$ act diagonally on $G/H\times G/H$. The stabiliser of $(aH,bH)$ i... | https://mathoverflow.net/users/8103 | Intersections of conjugates of Lie subgroups | Suppose G=SL(2,C) and let H be the stabilizer of a line (so a Borel subgroup). The matrix $$\begin{pmatrix}a&b\\\\c&d\end{pmatrix}$$
in G acts on projective space by
$$ z \mapsto \frac{az+b}{cz+d}$$
The stabilizer of ∞ is the matrices with c=0, a Borel subgroup. The stabilizer of both ∞ and 0 is the matrices with b=c=0... | 4 | https://mathoverflow.net/users/3710 | 58322 | 36,302 |
https://mathoverflow.net/questions/58324 | 8 | [Sloane's A077463](http://oeis.org/A077463/) obviously suggests that for any positive integer $n$ there exist $n$ consecutive primes and only them in between $m$ and $2m$ for some natural number $m$.
For instance, for
$n=1$, take $m=2$; $\hspace{.2in}$$2<3<4$;
$n=2$, take $m=7$; $\hspace{.2in}$$7<11,13<14$;
$n=... | https://mathoverflow.net/users/5627 | A partial converse to Bertrand's Postulate | Consider the function $f(m):=\pi(2m)-\pi(m)$ which counts the number of primes $m< p \leq 2m$. It is easy to see that $f(m+1)-f(m)$ equals $-1$ or $0$ or $1$ depending whether $m+1$ and $2m+1$ are primes. On the other hand, by simple estimates for prime numbers, it can be seen that $f(m)$ tends to infinity. Therefore $... | 18 | https://mathoverflow.net/users/11919 | 58326 | 36,303 |
https://mathoverflow.net/questions/58327 | 4 | Let $A$ be a compact subset of $R^n$ and $d\_S(\bullet, A)$ be the
*signed distance function* of $A$. Namely, $d\_S(p,A) =
d\left({p,\partial A} \right)$ for p in A, and $d\_S(p,A) =
-d\left({p,\partial A} \right)$ for p not in A. Here $d$ denotes the
usual Euclidean distance from a point to a set. .
**Question 1:** ... | https://mathoverflow.net/users/13616 | Jordan measurability of the level sets | *I was not able to determine if this is a homework problem or not. In any case, here are some observations:*
For the first question there are to cases:
1. When $r = 0$ the set $A\_r = A$ and the question then is if the boundary of $A$ has Lebesgue measure zero.
2. When $r \ne 0$ the boundary of the set $A\_r$ even ... | 2 | https://mathoverflow.net/users/11716 | 58328 | 36,304 |
https://mathoverflow.net/questions/58323 | 5 | Let $X$ be an arbitrary scheme. A quasi coherent sheaf $\cal F$ is said to be injective if $Hom\_{ O\_X}(-, \cal F)$ is exact. We can also regard a quasi coherent sheaf $\cal G$ on $X$ such that for all open subset $U$ of $X$, $\cal G(U)$ is an injective $\cal O\_X$-module. So we can ask a question that
1) Is there a... | https://mathoverflow.net/users/13615 | componentwise injective quasi coherent sheaves | The condition you want is $X$ a locally noetherian scheme. Then by Hartshorne's "Residues & Dualities," Proposition 7.17, $\cal{F}$ is an injective ${\cal O}\_X$-module if and only if for each $x \in X$, the stalks ${\cal F}\_x$ are injective ${\cal O}\_x$-modules. If the sections are injective ${\cal O}\_X(U)$-modules... | 3 | https://mathoverflow.net/users/13151 | 58331 | 36,305 |
https://mathoverflow.net/questions/58341 | 10 | Suppose $X$ is a path connected space such that every connected covering space of $X$ is trivial (1-fold.) Must $X$ be simply connected?
Intuitively, the answer seems to be no (imagine taking a disk, cutting out a square, and gluing in $T\times T$ where $T$ is the topologist's sine curve.) But this is a rather weak i... | https://mathoverflow.net/users/9455 | Is a space with no covering spaces simply connected? | No, the [harmonic archipelago](https://arxiv.org/abs/math/0501426) (An illustration is on pg 7 of [W. A. Bogley and A. J. Sieradski, Universal path spaces, preprint](https://web.archive.org/web/20210506162111/http://people.oregonstate.edu/%7Ebogleyw/research/ups.pdf)) is a locally path connected subspace of $\mathbb{R}... | 18 | https://mathoverflow.net/users/5801 | 58344 | 36,312 |
https://mathoverflow.net/questions/58305 | 6 | The category of $\mathbb{Z}$-graded abelian groups is equivalent to the category of comodules over the commutative Hopf algebra (over $\mathbb{Z}$) $A=\mathbb{Z}[t,t^{-1}]$, with comultiplication $t\mapsto t\otimes t$ and counit $t\mapsto 1$. Explicitly, the correspondence is given as follows: given an $A$-comodule $M$... | https://mathoverflow.net/users/75 | Is there an interpretation of the "anticommutative" symmetric monoidal structure on $\mathbb{Z}$-graded abelian groups in terms of $\mathbb{G}_m$ actions? | As Qiaochu points out, you can include a little extra data in your Hopf algebra $\mathbb G\_m$, and use it to realize the braiding. Here's the best way to say this, since you've decided to work with comodules over $\mathbb Z[t,t^{-1}]$. The point is that the comultiplication on $\mathbb Z[t,t^{-1}]$ makes the space of ... | 5 | https://mathoverflow.net/users/78 | 58345 | 36,313 |
https://mathoverflow.net/questions/58298 | 3 | Hi
There's Markov random field (MRF) which, by my Wikipedia-based knowledge, is an extension of Markov chain. I'd like to think of it as going from 1D to higher dimensional spaces. Inherent in its definition though, the space a MRF lives on (i.e., the index set of the stochastic process) is a discrete graph. So it's ... | https://mathoverflow.net/users/9504 | Markov random field with continuous index set | Essentially the Markov property in higher dimensions means that for any index set $D$ with nice boundary, the conditional distribution of the restriction of the field to indices in $D$ conditioned on the realization outside of $D$ coincides with the same thing conditioned on the realization on the boundary of $D$.
Fr... | 1 | https://mathoverflow.net/users/2968 | 58350 | 36,316 |
https://mathoverflow.net/questions/58339 | 63 | These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably the concrete notion of (complete) *flag of subspaces* $0 \subset V\_1 \subset \dots \subset V\_n =V$ with $\dim V\_k = k$... | https://mathoverflow.net/users/4231 | Origin of terms "flag", "flag manifold", "flag variety"? | Armand Borel's Bourbaki Seminar 121 *Groupes algébriques* is from 1955, and uses "drapeau" (page 7). (It's [online at archive.numdam.org](http://www.numdam.org/item/SB_1954-1956__3__229_0/).) This may not be the earliest occurrence, but there is a good reason for attention to the full flag variety in this context (the ... | 26 | https://mathoverflow.net/users/6153 | 58352 | 36,318 |
https://mathoverflow.net/questions/58129 | 17 | Consider a game of cops and robbers on a finite graph. The robber, for reasons left to the imagination, moves entirely randomly: at each step, he moves to a randomly chosen neighbour of his current vertex. The cop's job is to catch the robber as quickly as possible:
>
> How do we find a strategy for the cop which m... | https://mathoverflow.net/users/785 | Cops and drunken robbers | I think that I have an example where the optimal strategy for a random robber is
different from the normal winning strategy.
Let me specify the graph first. We have 5 points A,B,C,D and E forming a cycle. So
an edge connects A to B, and an edge connects B to C etc.
points A and C are also connected by an edge. We al... | 10 | https://mathoverflow.net/users/1098 | 58356 | 36,319 |
https://mathoverflow.net/questions/58357 | 6 | How many $n$-th roots of unity does a finite field $\mathbb{GF}\left(p^k\right)$ have? ($p$ is prime). And then kind of related to that, is there a finite field with exactly $n\_1,\ldots,n\_N$ $n\_1$-th,...$n\_N$-th roots of unity?
| https://mathoverflow.net/users/2763 | Number of n-th roots of unity over finite fields | I can quickly answer your first question. The multiplicative group of $\mathbb{GF}(p^k)$ is cyclic, let $g$ be a generator. For an element $x$ of the group $x^n=1$ holds iff $x=g^m$ with $nm$ divisible by $p^k-1$. The latter is equivalent to $m$ divisible by $(p^k-1)/d$, where $d:=\gcd(n,p^k-1)$, hence the $n$-th roots... | 10 | https://mathoverflow.net/users/11919 | 58359 | 36,321 |
https://mathoverflow.net/questions/58329 | 26 | Consider the polynomial $(1+x)(1+x^2)\dots (1+x^n)=1+x+\dots+x^{n(n+1)/2}$, which enumerates subj. How to prove that it's coefficients increase up to $x^{n(n+1)/4}$ (and hence decrease after this)? Or maybe this is false?
This problem was proposed long ago on some Russian high school competition, but nobody managed ... | https://mathoverflow.net/users/4312 | Partitions to different parts not exceeding $n$ | In fact, you *cannot* prove unimodality of the coefficients of $P\_n(x)=(1+x)\cdots (1+x^n)$ using the result about $B\_m/G$ mentioned by Qiaochu. The $P\_n(x)$ result is implicit in work of Dynkin on the principal sl(2) subalgebra of a complex semisimple Lie algebra. Hughes was the first to realize that a special case... | 34 | https://mathoverflow.net/users/2807 | 58367 | 36,326 |
https://mathoverflow.net/questions/58353 | 2 | I am having trouble verifying the following claim in Van Vu's 2000 paper "On a refinement of Waring's problem". First we define a few things.
Let $m \in \mathbb{N}\_0$ and $r \geq 2, r \in \mathbb{N}$ be fixed. Choose $P\_j \in $ {$2, 4, \cdots, 2^t$} where $t$ is chosen so that $2^t$ is the smallest integer power of... | https://mathoverflow.net/users/10898 | Sum over diadic boxes | As each $P\_A$ is a product of at most $l$ elements of {$2,4,\dots,2^t$} counted with multiplicity, we have for any $\lambda>0$,
$$\sum\_{A \in \mathcal{P}} P\_A^\lambda\leq(2^\lambda+2^{2\lambda}+\cdots+2^{t\lambda})^l<(2^{(t+1)\lambda}/(2^\lambda-1))^l.$$
Assuming $\lambda$ and $l$ are fixed, we obtain
$$\sum\_{A \in... | 4 | https://mathoverflow.net/users/11919 | 58368 | 36,327 |
https://mathoverflow.net/questions/58363 | 5 | Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}\_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there be between the local Tamagawa number $c\_v(A) = [A(K\_v): A\_0(K\_v)]$ and the cardinality of the $p$-primary subgroup... | https://mathoverflow.net/users/6121 | Tamagawa numbers of abelian varieties and torsion. | There is no general relation between the local $p$-primary torsion and the Tamagawa numbers. I believe one can have $p$-torsion points that map to non-trivial or to the trivial element in the group of components of the Neron model.
This should indicate you that, in your Euler characteristic formula, you can not hope ... | 4 | https://mathoverflow.net/users/5015 | 58376 | 36,332 |
https://mathoverflow.net/questions/58242 | 4 | Consider a surface $\Gamma$ in $R^3$. The surface $\Gamma$ is a graph, i.e. $\Gamma = (x,y, h(x,y))$, for $x \in R^2$ and some smooth function $h$, where $h$ and all its derivatives are periodic on [0,1]^2.
I'm trying to estimate the eigenfunctions and eigenvalues of the Laplace-Beltrami operator for this surface. Of... | https://mathoverflow.net/users/4047 | Estimating laplace-beltrami spectra for a graph surface in $R^3$ | Some suggestions:
1. To show there is a spectral gap, one can use the following approach: The variational characterization of the first eigenvalue
$$
E\_1 = \inf\_{\|\psi\| = 1} \langle \psi, H \psi \rangle
$$
can be used to obtain an upper bound on the first eigenvalue. Then Temple's inequality (I think it's in Re... | 3 | https://mathoverflow.net/users/3983 | 58381 | 36,333 |
https://mathoverflow.net/questions/46168 | 10 | Let $FM\_2=\langle a,b\rangle$ be the free monoid of rank 2. If we add a formal inverse to the word $aba$, we get the free group $F\_2$ (because both $a$ and $b$ will have inverses).
**Question:** For which other words $w=w(a,b)$, adding a formal inverse to $w$ turns the free monoid into the free group?
I need a ... | https://mathoverflow.net/users/nan | Adding a formal inverse of an element to a free monoid | OK, I will move my partial answer here as an answer to my question. If anybody can improve that answer, it would be good.
**A possible solution.** I think I found a solution but it is not very explicit, so a more explicit description is welcome. Let $w$ be a word. Let $S\_0=\{w\}$ . We shall construct sets of word... | 2 | https://mathoverflow.net/users/nan | 58382 | 36,334 |
https://mathoverflow.net/questions/58330 | 9 | If $X$ is a compact Kahler manifold, then Hodge theory says that its cohomology decomposes as a direct sum
$$ H^{p+q}(X,\mathbb C) = \bigoplus\_{p,q} H^{p,q}(X,\mathbb C) $$
where $H^{p,q}(X,\mathbb C) = H^q(X,\Omega\_X^p)$ are the Dolbeault cohomology groups and $H^{p,q}$ and $H^{q,p}$ are conjugate isomorphic. On... | https://mathoverflow.net/users/4054 | Hodge theory on complex spaces | Let me add to Donu's mentioning Du Bois's Hodge decomposition. First of all, many feel that part of the credit is due to Deligne as Du Bois built heavily on his ideas. Then again that is probably true for many things in Hodge theory.
Anyway, Du Bois's main idea was that one can do Deligne's construction "one step ear... | 14 | https://mathoverflow.net/users/10076 | 58384 | 36,335 |
https://mathoverflow.net/questions/58304 | 2 | I'm new to sieve theory, and I'm trying desperately to understand Selberg's sieve. I would like to apply the sieve to give me a nice upper bound on primes of the set
$$A^D(N)= \{ Dq-2 : q\in P, N/2 < q \leq N \} $$
But basically, for a fixed N, I would like $A^D(N)$ to be the set of elements of the form $Dq-2$ for a fi... | https://mathoverflow.net/users/10920 | Selberg sieve on a certain Set. | Getting an upper bound here is, at the level of research, a simple exercise. To do it the quickest way with least background, I'd suggest using the large sieve. Here you are essentially looking at estimating the number of integers between $N/2$ and $N$ which, modulo primes $r\leq \sqrt{N}$, are neither $0$ nor $2\bar{D... | 0 | https://mathoverflow.net/users/20038 | 58387 | 36,336 |
https://mathoverflow.net/questions/29811 | 19 | I was going through [this article (Who Is Alexander
Grothendieck?)](https://web.archive.org/web/20110323070731/http://www.ihes.fr/document?id=1723&id_attribute=48)(Wayback Machine) which appeared in the *Notices of the AMS*, and in it, there's a picture(page 936) which shows a mathematical diagram drawn by Grothendieck... | https://mathoverflow.net/users/7144 | Grothendieck's mathematical diagram | First of all, I believe that according to Grothendieck's definition of dessins d’enfants the picture (if I am looking at the right one) indeed seems to show one. At the same time you have a point that this is not one of the more interesting ones.
On the other hand it might be the very first one Grothendieck ever dre... | 6 | https://mathoverflow.net/users/10076 | 58391 | 36,338 |
https://mathoverflow.net/questions/58379 | 9 | [Bott periodicity](http://en.wikipedia.org/wiki/Bott_periodicity) implies that $\Omega(SU)\simeq G(\infty)$. Here, by $G(\infty)$, I mean the direct limit $\underset{m\to \infty}{\lim} G\_m(\mathbb{C}^{2m})$ where $G\_m(\mathbb{C}^{2m})\subset G\_{m+1}(\mathbb{C}^{2m+2})$ by stabilization (or some similar nice model fo... | https://mathoverflow.net/users/1345 | Does there exist a contractible fiber bundle with fiber $G(\infty)$ and base $SU(\infty)$? | I'm not completely sure that I understand your notation, so this may be not what you want, but in case it is close enough, I'll have a go. The bit I'm assuming is that $G\_m$ is the Grassmannian of $m$-places in $\mathbb{C}^{2m}$. That seems fairly safe, but my brain is refusing to check the homotopy types of everythin... | 7 | https://mathoverflow.net/users/45 | 58396 | 36,341 |
https://mathoverflow.net/questions/58395 | 3 | Let $(A,m)$ be a local commutative associative algebra over the field of complex numbers, $m^n\ne 0$, $m^{n+1}=0$ for some $n>0$, and
(1) $A$ is finite dimensional as vector space
(2) for any nonzero ideal $I$ of $A$, we have $m^n\subset I$
What can we say about such an $A$? For example, whether it is always a qu... | https://mathoverflow.net/users/13634 | A problem for finite dimensional commutative algebra | Such a ring has simple socle, namely $\mathfrak m^n$. It follows easily that $A$ is self-injective, Gorenstein and, of course, of dimension $0$. You can construct them using Macaulay's method of *inverse system*; this is explained in Eisenbud's book on commutative algebra, if I recall correctly.
If $A\_4=\mathbb C[x,... | 4 | https://mathoverflow.net/users/1409 | 58399 | 36,342 |
https://mathoverflow.net/questions/58397 | 27 | Intuitively, the Galois group (of a splitting field over $\mathbb Q$ of) a polynomial $f\in\mathbb Q[X]$ taken at random is most probably the full permutation group on the roots of $f$. This intuition can be made precise as follows.
*Elementary*: For inegers $d \geq 1$ and $N \geq 1$, consider the set $P\_{d,N}$ of p... | https://mathoverflow.net/users/5952 | The Galois group of a random polynomial | As Torsten remarks, the answer is yes and the result is well-known. P. X. Gallagher proved the following: If $E\_d(N)$ denotes the set of degree $d$ polynomials with coefficients $|a\_i|\le N$ with Galois group not equal to $S\_d$, then
$$
|E\_d(N)|\ll N^{d-1/2}\log N
$$This bound gives you what you want. There is a di... | 15 | https://mathoverflow.net/users/3996 | 58403 | 36,344 |
https://mathoverflow.net/questions/57747 | 4 | Is there any result known for the number of different solutions of $1 = \sum\limits\_{k=0}^n \frac{1}{x\_k}$ in dependency of the length $n$ of this partition?
All I know, up to now, is that there are for every $n$ only finitely many different solutions and the maximal $x\_k$ is given by the $k$th element of the Sylves... | https://mathoverflow.net/users/13488 | Function or bounds for the number of solutions of $\sum_{i=0}^k \frac{1}{x_i} = 1$ | An upper bound of about $c\_0^{2^k}$ follows by elementary induction, (from your comment $x\_k$ is bounded by the Sylvester sequence).
Here $c\_0 \approx 1.264$ is $\lim u\_n^{\frac{1}{2^n}}$, ($u\_n$ the $n$-th term of the
Sylvester sequence).
An upper bound of $c\_0^{(1+\epsilon) 2^{k-1}}$ is in a paper by C.
Sándo... | 5 | https://mathoverflow.net/users/7673 | 58404 | 36,345 |
https://mathoverflow.net/questions/58378 | 14 | I am currently reading an article about TFTs (DW - Group Cohomology and TFTs), and I have
a few questions:
(1) Let $M$ be a 3-man., then we know there exists some 4-man. $B$ such that $\partial B = M$. Now, let $E$ be a $G$-bundle over $M$, when can we extend $E$ to a $G$-bundle over $B$ and a connection $A$ over $E... | https://mathoverflow.net/users/13132 | Questions from Chern-Simons theory | (1) Obstruction theory: there is a sequence of obstructions to extending a bundle On $M$ to all of $B$. For homotopy theoretic purposes, a bundle is the same as a map to $BG$. The obstructions live in $H^{n}(B,M;\pi\_{n-1}(BG))$. These groups are trivial for a $4$-manifold $B$ when $BG$ is $3$-connected, which happens ... | 6 | https://mathoverflow.net/users/9928 | 58407 | 36,347 |
https://mathoverflow.net/questions/58401 | 1 | We have the following result:
Let $R=\mathbb{C}[t]\_f$, with $f=(t-a\_1)(t-a\_2)\cdots (t-a\_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix (not necessary pointwise) the set $\{a\_1,\ldots,a\_n,\infty\}$.
Is it a known result or a direct consequence of s... | https://mathoverflow.net/users/13634 | A problem on Moebius transformations | Since $\textrm{Spec}(\mathbb{C}[t]\_f)=\mathbb{A}^1-\{a\_1, \ldots, a\_n\}$, we are reduced to compute $\textrm{Aut}(\mathbb{A}^1-\{a\_1, \ldots, a\_n\})$.
Every automorphism
$\phi \colon \mathbb{A}^1-\{a\_1, \ldots, a\_n\} \longrightarrow \mathbb{A}^1-\{a\_1, \ldots, a\_n\}$
gives rise to a birational map
$\b... | 3 | https://mathoverflow.net/users/7460 | 58409 | 36,348 |
https://mathoverflow.net/questions/56323 | 7 | In *Antoine Henrot Michel Pierre* -
**Variation et optimisation de formes, Une analyse geometrique**, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of the problem are pretty easy, and I solved them. The 4-th seems a little harder. The only indication I get is to us... | https://mathoverflow.net/users/13093 | Baire Category Theorem Application | Choose $\varepsilon > 0$ and consider sets defined by
$$\Sigma\_k = \{E:\ \left|\int\limits\_{E} (f\_n-f\_m) \right| \leqslant \varepsilon, \textrm{ if } n,m \geqslant k \}$$
Since for any measurable set a limit of integrals exists, we have
$\Sigma = \bigcup\limits\_{k} \Sigma\_k$. Note, that given an integrable functi... | 5 | https://mathoverflow.net/users/13099 | 58421 | 36,354 |
https://mathoverflow.net/questions/58427 | 2 | True or false: If G is special 2-group, then |G:Z(G)| is a square.
Recall that G is a special p-group if Z(G) = G' = Frat (G) be elementary abelian. The above assertion is true, whenever |Z(G)| = 2, that is, G is an extra-special 2-group.
Thank you.
| https://mathoverflow.net/users/13641 | On special 2-groups | Being systematic a $2$-special group is specified completely by two $\mathbb Z/2$-vector
spaces $U=G/Z(G)$ and $V=Z(G)$ together with the induced square map $\Gamma^2U\to V$
which is non-degenerate in the following sense: We have an injective map $\Lambda^2U\to\Gamma^2U$
mapping $u\land v\mapsto\gamma\_1(u)\gamma\_1(v)... | 6 | https://mathoverflow.net/users/4008 | 58430 | 36,359 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.