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 |
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https://mathoverflow.net/questions/123942 | 20 | Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how smooth-manifold intuition leads me astray. I know very little algebraic geometry, and so please forgive and correct me if a s... | https://mathoverflow.net/users/78 | How many flat connections has a line bundle in algebraic geometry? | I presume that your variety $X$ is smooth.
Consider the additive map $\mathrm d\log \colon \mathscr O\_X^\*\to \Omega^1\_X$ that sends $f$ to $\mathrm df/f$. It induces a map $c\_1$ in cohomology from $H^1(X,\mathscr O\_X^\*)$
to $H^1(X,\Omega^1\_X)$ — a coherent avatar of the first Chern class.
By Hodge Theory, $H^1... | 29 | https://mathoverflow.net/users/10696 | 123945 | 69,486 |
https://mathoverflow.net/questions/123924 | 2 | Assume we have a probability measure $\mu$ on $\mathbb R^n$. Assume it satisfies
$$
\mu(||x|| > u) \le Ce^{-au} \ \ \forall u > 0
$$
In other words, its tail is dominated by an exponential function.
Then, how to prove Poincare inequality?
$$
Var\_{\mu}f \le K\int||\nabla f||^2d\mu
$$
| https://mathoverflow.net/users/27698 | Sub-exponential tail implies Poincare inequality | As Mark Meckes already pointed out, this is not enough to proof a Poncaré inequality.
If you are interested in the case where $\mu$ is absolutely continuous, then there are conditions formulated in terms of the log-density of $\mu$, i.e. $\mu$ is of the form
$$
\mu(\mathrm{d}x) = \exp(-H(x))
$$
The minus sign is j... | 3 | https://mathoverflow.net/users/13400 | 123954 | 69,489 |
https://mathoverflow.net/questions/123793 | 4 | Consider a Coxeter group presentation $< s\_1, \ldots s\_n \mid (s\_i s\_j)^{m\_{ij}}>$ with $m\_{ii}=1$ for all $i$. I can prove (using <http://arxiv.org/abs/1011.4255>) that if for each $i$ there are at most two $j\neq i$ with $m\_{ij}< \infty$, then the corresponding Cayley graph is planar. Is this fact known?
In ... | https://mathoverflow.net/users/26286 | Planar Cayley graphs/complexes for coxeter groups | Let $\Gamma$ be the Coxeter group in question. Order the $s\_i$ so that if $m\_{ij}\neq\infty$ then $|i-j|\leq 1$. Now let $P$ be a polygon with edges $e\_i$ and angles $\pi/m\_{i,i+1}$ between consecutive edges. Note that some vertices are 'ideal', meaning that they have interior angle zero.
The polygon $P$ can be r... | 5 | https://mathoverflow.net/users/1463 | 123957 | 69,490 |
https://mathoverflow.net/questions/123960 | 3 | The following question came up while I was working through an example:
>
> Does there exist an $\ell^1$ sequence of complex numbers $a\_n$, not all zero, such that $\sum\_n a\_n n^{-p} = 0$ for all $p \in [0,1]$?
>
>
>
| https://mathoverflow.net/users/2206 | Does there exist a sequence of complex numbers such that... | Let $U\subset \Bbb C$ be the set of complex number with positive real part and $f\colon U\to\Bbb C$ given by $f(z):=\sum\_{n=1}^{+\infty}a\_n\exp(-z\log n)$. Since there is uniform convergence on compact subsets of $\Bbb C$, $f$ is holomorphic. We have $f(z)=0$ if $z\in (0,1]$, hence $f$ vanishes identically on the con... | 9 | https://mathoverflow.net/users/17118 | 123970 | 69,493 |
https://mathoverflow.net/questions/123964 | 5 | Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the answer to the question have an effect on the PSPACE proof by J. Canny? For example, ETR is not in PSPACE, if numbers in ... | https://mathoverflow.net/users/32039 | The existential theory of the reals | In order for this to be a computational problem in the first place, you have to fix a representation of the coefficients by finite strings (which in particular implies that the field is countable). The answer will in general depend on the representation.
For the most obvious case, if the coefficients are taken from t... | 4 | https://mathoverflow.net/users/12705 | 123972 | 69,495 |
https://mathoverflow.net/questions/123974 | 1 | For a continuous function $u: \mathbf{R^2}\to \mathbf{R}$, consider the zero level set $E:=\{x\in \mathbf{R^2}:u(x)=0\}$, write $Int(E)$ as the collection of the interior points of the set $E$, my question is then, if $L^2(E\setminus Int(E))$ equals to zero or not. By $L^2$ I mean the 2 dimensional Lebesgue measure.
| https://mathoverflow.net/users/32045 | Lebesgue measure of level set of continuous functions | Depending on $u$ it may or may not have zero measure. To have nonzero, take $E \subset \mathbb{R}^2$ to be a closed set of positive measure without interior points (for example a fat Cantor set) and $u(x) := \textrm{dist}(x,E)$.
| 2 | https://mathoverflow.net/users/11716 | 123976 | 69,497 |
https://mathoverflow.net/questions/123932 | 4 | Let $S$ be a compact submanifold of $X$ smooth manifold. I know that $T\_X|\_S=T\_S\oplus N\_{S/X}$ where $N\_{S/X}$ is the normal bundle. I have read that the euler class $e(N\_{S/X})$ corresponds (via integration over S, i suppose) to the self intersection number $S\cdot S$. I've thought about it, but i don't know ho... | https://mathoverflow.net/users/32032 | euler class of the normal bundle and self intersection number | I've just taught this in my graduate class. Check these [notes on intersection theory.](http://www.nd.edu/~lnicolae/TopicsFall2011-Intersect.pdf)
The result you want is contained in Thm. 4.7. Again, you need $2\dim S=\dim X$.
| 7 | https://mathoverflow.net/users/20302 | 123977 | 69,498 |
https://mathoverflow.net/questions/123966 | 3 | Let $X$ be a topological space and $U\subset X$ an open subset. Let's work in the category of sheaves of abelian groups on $X$. Consider the constant sheaf on $U$, $\mathbb{Z}\_U$, given by $\mathbb{Z}\_U(V)=\{\text{continuous maps }U\cap V\rightarrow \mathbb Z\}$, where $\mathbb Z$ is given the discrete topology. I've... | https://mathoverflow.net/users/12166 | Constants sheaves on an open subset | This is not true; for example, take $X = \mathbb R^2$, $U = \mathbb R^2 \smallsetminus \{(0,0)\}$. Then your $\mathbb Z\_U$ coincides with $\mathbb Z\_X$, and $Hom(\mathbb Z\_U, F)$ is $F(X)$, not $F(U)$.
For the formula to hold you have to take as $\mathbb Z\_U$ the extension of the constant sheaf by $0$, which is a... | 4 | https://mathoverflow.net/users/4790 | 123978 | 69,499 |
https://mathoverflow.net/questions/123967 | 3 | I'm interested in knowing classes of topological spaces $X$ which admit a basis of open sets $\{U\_i\}\_{i\in I}$ such that $U\_i\cap U\_j$ is connected for all $i,j\in I$. Do manifolds have this property? Riemannian ones maybe? If not, what if we relax the condition by saying that $U\_i\cap U\_j$ has a finite number o... | https://mathoverflow.net/users/12166 | Bases of open sets with connected intersections | I reckon you consider the empty subspace to be connected (since for a Hausdorff space, at least one such intersection must be empty). In that case, for a Riemannian manifold, you can get not just connectedness, but contractibility for all (nonempty) finite intersections of basis elements, by taking a neighborhood basis... | 5 | https://mathoverflow.net/users/2926 | 123979 | 69,500 |
https://mathoverflow.net/questions/123948 | 20 | What is the Kodaira dimension of symmetric products of curves? That is, given a projective smooth, connected complex curve $C$, what is the Kodaira dimension of $C^{(d)}=C^d/\mathfrak S\_d$?
When $d> g$, the genus of $C$, then $C^{(d)}$ is a bundle in projective spaces over the Jacobian of $C$, hence all pluriforms o... | https://mathoverflow.net/users/10696 | Kodaira dimension of symmetric products of curves | Let $C$ be a smooth projective connected complex curve of genus $\geq 2$. Let me show that $C^{(d)}$ is of general type if $1\leq d\leq g-1$.
Equivalently, one needs to show that the image $W\_d$ of $C^{(d)}$ in the jacobian $J(C)$ is of general type, because $C^{(d)}\to W\_d$ is birational. If $W\_d$ were not of gen... | 25 | https://mathoverflow.net/users/2868 | 123980 | 69,501 |
https://mathoverflow.net/questions/123989 | 6 | Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function.
>
> **Question:** Is the space $GVect(M,f)$ of all gradient-like vector-fields for $f$ contractible or even convex?
>
>
>
To eliminate any misunderstanding, this is the definition of gradient-like vector fie... | https://mathoverflow.net/users/29827 | Is the space of gradient-like vector fields contractible? | $GVect(M,f)$ is a convex set in the space of all vector fields. Thus it is contractible.
Edit, since you asked for details: I assume that the Morse charts $\Phi$ are the same for all $X$.
Let $X,Y$ be gradient-like for $f$, then:
1. $df(q)(t.X(q)+(1-t).Y(q)) = t.df(q)(X(q)) + (1-t).df(q)(Y(q)) < 0$ for $0\le... | 6 | https://mathoverflow.net/users/26935 | 123991 | 69,504 |
https://mathoverflow.net/questions/124006 | 1 | So I asked this on maths SE because I don't truly consider it to be a research level question. This question mostly arises out of my completely limited understanding of perverse sheaves. However I do think it is much more likely to receive an answer here, so I am posting it here as well.
---
Assume $\mathbf{G}$ i... | https://mathoverflow.net/users/22846 | Decomposing Semisimple Perverse Sheaves | $\operatorname{Hom}\_{\mathcal A}(E,K)$, as a functor from the category of $\mathcal A$-modules to the category of groups, can be expressed in terms of generators of relations. Specifically, you sum one copy of $K$ for each generator of $E$, then you sum one copy of $K$ for each relation of $E$, and you form a map from... | 1 | https://mathoverflow.net/users/18060 | 124009 | 69,511 |
https://mathoverflow.net/questions/124010 | 0 | Let $n\geq 1$ be an integer, $P$ the set of rational prime numbers. I am interested in upper bounds for $\prod\_{p\in P; \ p\mid n}(1+1/p)$ in terms of n.
I would like to find explicit real numbers $a,b$ such that for any integer $n\geq 1$ it holds $\prod\_{p\in P; \ p\mid n}(1+1/p)\leq a\log(n)^b$.
By the answer ... | https://mathoverflow.net/users/32050 | Estimating $\prod_{p\mid n}(1+1/p)$ in terms of n | If you want a weaker bound that is completely explicit, the following method is pretty simple. Let $\omega(n)$ denote the number of distinct prime factors of $n$. Since the function $1+1/x$ is a decreasing function of $x$, and the smallest prime equals 2, we certainly have
$$
\prod\_{p\mid n} \bigg( 1+\frac1p \bigg) \l... | 3 | https://mathoverflow.net/users/5091 | 124015 | 69,513 |
https://mathoverflow.net/questions/124011 | 32 | **Remark:** I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated 4/15/63, followed by the parenthetical note: *Thus Kreisel saw an analogy between forcing and Friedberg's [1957] prio... | https://mathoverflow.net/users/22971 | Similarities between Post's Problem and Cohen's Forcing | **In this edit, a historical note is added at the end.**
The quote below from Kunen's classic text *Set Theory* maybe of interest to you. Note that Kunen is pointing out that certain classical constructions in recursion theory can be viewed as precursors to forcing; but he does not claim that *priority arguments* fal... | 23 | https://mathoverflow.net/users/9269 | 124018 | 69,514 |
https://mathoverflow.net/questions/124013 | 1 | I wonder what is known about a fundamental region for SO($n,1$) modulo its integer points? is there only one cusp? and if one writes a Siegel set in the form of
$K A\_\tau N\_c$, where $N\_c$ is compact and $A\_\tau$ is the set of elements diag$(a,1,...,1,1/a)$ for $a < \tau$, then what is an estimate for $\tau$? in S... | https://mathoverflow.net/users/32051 | Siegel set in SO(n,1) modulo integer points? | The number of cusps could be more than 1, see [here](http://www.ma.utexas.edu/users/areid/allflat_new.pdf), remark on page 294.
| 2 | https://mathoverflow.net/users/21684 | 124019 | 69,515 |
https://mathoverflow.net/questions/122811 | 8 | If $\Sigma\_g$ is a genus-$g$ surface, $g \geq 2$, then let $\mathcal{M}(\Sigma\_g)$ be its twisted SU(2) representation variety, i.e. $$\mathcal{M}(\Sigma\_g) := \{ (A\_1, B\_1, \ldots, A\_g, B\_g) \in SU(2)^{2g} \:|\: [A\_1,B\_1]\cdots[A\_g,B\_g] = -I \}/SO(3),$$ where $SO(3) = SU(2)/\{\pm 1\}$ acts on these tuples b... | https://mathoverflow.net/users/20391 | How many "elementary" characterizations of twisted SU(2) representation varieties are known? | I think Nate might know this by now, but in case anyone else is curious, there is a generalization of the intersection-of-quadrics-in-$\mathbb{P}^5$ picture to general genus, proved first in this paper, I believe
>
> Classification of Vector Bundles of
> Rank 2 on Hyperelliptic Curves, U.V.
> Desale and S. Ramana... | 3 | https://mathoverflow.net/users/492 | 124022 | 69,516 |
https://mathoverflow.net/questions/123998 | 3 | Let $X$ be a smooth projective curve over a field $k$. In chapter 8 of the book *Neron Models* by Bosch et al., there is a general result (namely Proposition 4) which implies that if $X$ admits a rational point, then the Picard functor which sends a $k$-scheme $T$ to $\text{Pic}(X \times T)/\text{Pic}(T)$ is already a ... | https://mathoverflow.net/users/3544 | Line bundles on a pointless curve | There are such examples over some fields with trivial Brauer group, but not over finite fields. Let me explain why.
Over a finite field $k$, any geometrically integral variety $X$ has a zero-cycle of degree $1$. Indeed, by the Lang-Weil estimates, it has a rational point over the field extensions of $k$ of degrees $2... | 6 | https://mathoverflow.net/users/2868 | 124026 | 69,518 |
https://mathoverflow.net/questions/123926 | 4 | Let $E$ and $F$ be vector bundles on a smooth projective variety, say.
A. Lascoux ("Classes de Chern d'un produit tensoriel", *C. R. Acad. Sci. Paris Sér. A-B* **286** (1978), no. 8, A385–A387) gave formulas for the Chern classes of $E \otimes F$, $Sym^2 E$ and $\bigwedge^2 E$ in terms of the Chern classes of $E$ an... | https://mathoverflow.net/users/nan | Reference request: Lascoux's formulas for Chern classes of tensor products and symmetric powers | You say you don't have access to Lascoux's paper, but it is actually available online at the BNF:
>
> [http://gallica.bnf.fr/ark:/12148/bpt6k62341359/f397.image](http://gallica.bnf.fr/ark%3A/12148/bpt6k62341359/f397.image)
>
>
>
| 5 | https://mathoverflow.net/users/19276 | 124027 | 69,519 |
https://mathoverflow.net/questions/124021 | 2 | I'm trying to solve a very practical optimization problem and I think I hit a dead-end.
There are $N$ products ($N \sim 50$). Each product can have a price $p\_i$ in range between 1 and 40 dollars. For each product there's a non-decreasing demand function $d\_i(p)$, $a > b \Rightarrow d(a) >= d(b)$. Demand functions ... | https://mathoverflow.net/users/32054 | Optimization problem | I would recommend first thinking carefully about the trivial case $N=1$ and the nontrivial case $N=2$.
For $N=1$, the constraint is simply $pd(p) \le Cvd(p)$, so it's pretty clear that you'll maximize $pd(p)$ when $p=Cv$ (since $d$ is non-decreasing).
For $N=2$, the constraint is
$$pd\_1(p) + qd\_2(q) \le Cv\_1d... | 0 | https://mathoverflow.net/users/15837 | 124031 | 69,521 |
https://mathoverflow.net/questions/124032 | 5 | Cross-posted from math.stackexchange:
Let C be a simple closed plane curve and let D be its interior. Recall that the width of C in a direction θ is the distance between two supporting lines for D which are perpendicular to θ. A curve is said to have constant width if its width is the same in every direction; see the... | https://mathoverflow.net/users/4362 | How do you prove that every curve of constant width is convex? | First, observe that for a closed convex region $R$ of constant width $w$, there can be no interval $[a,b]\subset \partial R$. For take the support line $l$ to $R$ containing $[a,b]\subset l$, and let $c$ be a point in the other parallel support line for $R$ realizing the width $w$. Then the height of the triangle $abc$... | 6 | https://mathoverflow.net/users/1345 | 124036 | 69,522 |
https://mathoverflow.net/questions/124025 | 5 | Hi, I've read somewhere that a compact (Abelian) topological semigroup has at least one idempotent and if it contains identity and is not a group then it has one extra idempotent apart from the identity. Could you please give me some reference, where I could find this? Thanks in advance for any help.
| https://mathoverflow.net/users/29706 | Idempotents in compact semigroups | Let me supplement Boris's answer since the result you ask for is easier than the result Boris cites.
Let $T$ be a (nonempty) semigroup. Then an easy exercise is the following: $T$ is a group iff $tT=T=Tt$ for all $t\in T$.
Suppose now that $S$ is a compact semigroup. Then by Zorn's lemma and compactness it contains... | 7 | https://mathoverflow.net/users/15934 | 124041 | 69,525 |
https://mathoverflow.net/questions/124050 | 0 | We need to Find a non constant map $f:\mathbb{C}^3\to \mathbb{C}$ such that for any three distinct complex numbers $z\_1,z\_2,z\_3$ and any automorphism $\phi$ of $\mathbb{C}$, we have
$f(z\_1,z\_2,z\_3)= f(\phi(z\_1),\phi(z\_2),\phi(z\_3))$
Thank you for help and discussion.
| https://mathoverflow.net/users/32065 | to find a function with a property | If the $z\_i$s are all distinct, take
$f(z\_1,z\_2,z\_3) = \frac{z\_1-z\_2}{z\_3-z\_2}$
This is the cross-ratio. It can take any complex value except for $0$ and $1$. It is clearly invariant. You can get many other functions by composing this with some function from $\mathbb C - \{0,1\}$ to $\mathbb C$. Indeed, the... | 1 | https://mathoverflow.net/users/18060 | 124051 | 69,530 |
https://mathoverflow.net/questions/124034 | 1 | I suppose that Beilinson's compact generator (and, in fact, tilting object) $\mathcal{O} \oplus \mathcal{O}(1)$ in $D(\mathbb{P}^1)$ is the most well known example. I have the following simple construction for compact generator (but not a tilting object) on projective line. Lets take any point $z \in \mathbb{P}^1$ and ... | https://mathoverflow.net/users/21029 | Compact generator of $D(\mathbb{P}^1)$ | It is correct. Taking the cone of a map $k(z) \to O[1]$ you get $O(1)$. Thus both $O$ and $O(1)$ are contained in the subcategory generated by $C$, and since these two generate $D(P^1)$, the same is true for $C$.
| 3 | https://mathoverflow.net/users/4428 | 124052 | 69,531 |
https://mathoverflow.net/questions/124057 | 1 | I was recently reading one of the papers of Kapranov on $\mathcal{M}\_{0,n}$ and he says that one can see it as a subvariety of the Hilbert scheme parametrizing all subschemes of a certain projective space. What is the precise definition of this Hilbert scheme? Why it is sure it exists? Is it in EGA somewhere or does a... | https://mathoverflow.net/users/4096 | one "big" Hilbert scheme? | The construction is done in Geometry of Algebraic Curves by Arbarello et al.
| 0 | https://mathoverflow.net/users/27840 | 124062 | 69,535 |
https://mathoverflow.net/questions/124059 | -1 | Can anyone explain me what is a Ramsey Graph with a simple example?
What are its properties?
| https://mathoverflow.net/users/32067 | What is a Ramsey Graph? | The form in which you might be familiar with the result is this:
>
> For a pairs of parameters $(r,b)$ there exists an $n$ such that for every (edge-)coloring of the complete graph on $n$ vertices with colors r(ed) and b(lue) there will exist a complete subgraph on $r$ vertices colored red or a complete subgraph on... | 4 | https://mathoverflow.net/users/nan | 124069 | 69,540 |
https://mathoverflow.net/questions/124066 | 6 | Good afternoon.
Can anybody give me an example of a continuous map $T:X\to X$ defined on a Polish space $X$ which admits an invariant Borel probability measure but no ergodic Borel probability measure? Typically, the space $X$ could be a separable infinite-dimensional Banach space, or the space $\mathbb N^{\mathbb N}... | https://mathoverflow.net/users/37371 | Non-existence of ergodic measures | The existence of such an example is prevented by the ergodic decomposition theorem, which asserts that every $T$-invariant measure on a standard probability space $(X,\mathcal{B},m)$ can be expressed as a (possibly uncountably infinite) convex combination of ergodic $T$-invariant measures by means of an integral over t... | 12 | https://mathoverflow.net/users/1840 | 124077 | 69,543 |
https://mathoverflow.net/questions/124000 | 4 | Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of
$$F(\gamma) = \int\_a^b \| \dot{\gamma}(s) \|\_g ds$$
The so-called "null curves" are defined as curves for which the integrand $\| \dot{\gamma}(s) \|\_g$ is zero, i.e. they have zero "speed" measured b... | https://mathoverflow.net/users/nan | Are all null curves of a Lorentzian metric extrema? | Actually, your notation is causing some confusion. In one very real sense (probably not your intended one) the answer to your question is *yes*, not *no* which is probably the answer to the question that you intended to ask.
It depends on what you mean by $\|\dot\gamma(s)\|\_g$. If you borrow this notation directly ... | 6 | https://mathoverflow.net/users/13972 | 124080 | 69,545 |
https://mathoverflow.net/questions/124084 | -1 | I am trying to prove that if $u:(0,1)\to\mathbb{R}$ lies in $W^{1,1}(0,1)$, then $u\in C(0,1)$. Is there any help anybody can offer?
Thanks.
| https://mathoverflow.net/users/32073 | Every function in W^{1,1}(0,1) is continuous on (0,1) | Since $u'\in L^1(0,1)$, you find from the Lebesgue differentiation theorem that
$$
\int\_{1/2}^x u'(t) dt=u(x)+Cst,\quad x\in(0,1).
$$
As a result $u$ is a continuous function and the constant above is $-u(1/2).$
| 0 | https://mathoverflow.net/users/21907 | 124086 | 69,547 |
https://mathoverflow.net/questions/124095 | 1 | We are given a permutation group G acting on a finite set X. Finite set $\Omega$ contains all mappings $\omega$ from the set X to finite set K. We define mapping $\hat{g}$ on the set $\Omega$ in the following way
$(\hat{g}(\omega))(x)= \omega(g(x))$
Question: Is the statement below true?
If a,b are in G then $((\... | https://mathoverflow.net/users/32074 | A yes no question concerning induced group | The "right" way to make $G$ act on $\Omega$ is $\hat g(\omega)(x)=\omega(g^{-1}(x))$. Not only does this give you a left action (rather than a right action), but it fits nicely with the "set of ordered pairs" view of functions. If you regard $\omega$ as set of ordered pairs $(x,\omega(x))$, then the action of any $g\in... | 1 | https://mathoverflow.net/users/6794 | 124096 | 69,550 |
https://mathoverflow.net/questions/124079 | 2 | In [this paper 4.7](http://arxiv.org/pdf/0808.3647.pdf), the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log resolution $\tilde{X}$, with logarithmic poles along the exceptional locus. Roughly speak... | https://mathoverflow.net/users/10083 | Lifting vector fields to its resolution in char $p$ | Dear CX, Taking a short break from revising: I think this fails in positive characteristic. Let $k$ be a field of characteristic $p>0$. Let $m>1$ be an integer.
Let $\mathbb{A}^6\_k$ have coordinates $x\_0,x\_1,x\_2,y\_0,y\_1,y\_2$. Consider the affine hypersurface $X$ with defining equation $f = x\_0^py\_0+x\_1^py\... | 3 | https://mathoverflow.net/users/13265 | 124097 | 69,551 |
https://mathoverflow.net/questions/118965 | 5 | This is a question on geometric tilting theory. On smooth projective variety it is possible to define in general tilting object as perfect complex that satisfy some properties, but are there examples of tilting objects that are actually complexes? All examples I know are vector bundles obtained as sums of exceptional s... | https://mathoverflow.net/users/21029 | Examples of tilting objects that don't come from exceptional sequences | The answer to your more specific question is yes, there are tilting objects that involve complexes not quasi-isomorphic to any vector bundle. Consider the blowup $X$ of $\mathbb{P}^2$ at a single point $p$. Then, Orlov showed that there is a semiorthogonal decomposition $D^b(X)=\langle e,O\_X,O\_X(1),O\_X(2)\rangle$, w... | 6 | https://mathoverflow.net/users/100 | 124100 | 69,554 |
https://mathoverflow.net/questions/124103 | 9 | As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow *any* variety with a (mixed) Hodge structure.
Again, AFAIU, the idea is to resolve the variety, then take intersections of the exceptional pieces, resolve those and so on.
As you can se... | https://mathoverflow.net/users/25442 | what is Deligne's cohomological descent (and what are some examples) | As you said, it is a generalization of Cech theory. The standard example to understand first is a divisor $D=\bigcup D\_i$ with simple normal crossings. A resolution of singularities is obtained by simply taking a disjoint union of components $X\_0= \coprod D\_i$. Let $\pi\_0:X\_0\to D$ be the obvious map. The cohomolo... | 17 | https://mathoverflow.net/users/4144 | 124107 | 69,556 |
https://mathoverflow.net/questions/121499 | 2 | I am interested in whether the transgression maps for group cohomology and group homology are related via a version of the universal coefficient theorem.
Let $G$ be a group, $H$ a normal subgroup of $G$ and let $A$ and $B$ finite rank free $\mathbb{Z}$-modules equipped with actions of $G$ and a $G$-equivariant perfec... | https://mathoverflow.net/users/8891 | Transgression maps in group cohomology and group homology / duality of spectral sequences | We have the following general result:
>
> **Lemma:** Let $K$ be a filtered complex of ablian groups and let $I$ be an ablian group. Filter the cocomplex $K^\ast := \text{Hom}(K,I)$ by the dual filtration (def. in proof). Then there is a homomorphism $\phi\_r: E\_r(K^\ast) \to E^r(K)^\ast$ of abelian groups such th... | 5 | https://mathoverflow.net/users/10194 | 124114 | 69,558 |
https://mathoverflow.net/questions/124116 | 6 | What is the maximum number of edges a planar subgraph of $K\_n$ can have? Is there a simple way to calculate this if not, are there some values of n for which this is easier to find out?
| https://mathoverflow.net/users/24478 | Maximum number of edges in a planar graph | Suppose the graph has $n$ vertices, $m$ edges, and $f$ faces. By Euler's formula we know that
$$n - m + f = 2$$
Now presume there are at least three vertices. Every face must be a triangle, otherwise you can increase the number of edges by dividing a face with an edge. Since every edge borders two faces, $2m=3f$. There... | 11 | https://mathoverflow.net/users/30994 | 124118 | 69,559 |
https://mathoverflow.net/questions/124113 | 5 | (I ran into this question while thinking about the (Strong) Inner Model Hypothesis: see <http://www.jstor.org/stable/4093051> or <http://arxiv.org/abs/0711.0680>.)
A *measurable cardinal* is a cardinal $\kappa$ such that there is an elementary embedding $j: V\rightarrow M\subseteq V$ with $M$ an inner model of $V$ an... | https://mathoverflow.net/users/8133 | Versions of large cardinals with target model in a generic extension | If there is a precipitous ideal $I$ on $\omega\_1$---a hypothesis equiconsistent over ZFC with the existence of a measurable cardinal---then after forcing with $P(\omega\_1)/I$, we get an elementary embedding $j:V\to M\subset V[G]$ with critical
point $\omega\_1$. Thus, on this hypothesis, $\omega\_1$ is outer measurab... | 3 | https://mathoverflow.net/users/1946 | 124120 | 69,560 |
https://mathoverflow.net/questions/124115 | 1 | What is the effective 2-form corresponding to the equation
$det Hess v=(v-q\_1v\_{q\_1}-q\_2v\_{q\_2})^4$
you can find the definition of effective forms [here](http://matematicas.uniandes.edu.co/~mikarm/TALLER_7/pdf/kushner.pdf)
| https://mathoverflow.net/users/nan | finding effective 2-form corresponding to an equation | Unless I'm mistaken, all the second derivatives of $v$ are isolated in the Hessian on the left-hand side. The notes you linked to already give the effective form that generates the Hessian. To get the effective form on for the right-hand side, simply multiply by $dq\_1 \wedge dq\_2$. So, I believe, the effective form y... | 1 | https://mathoverflow.net/users/2622 | 124121 | 69,561 |
https://mathoverflow.net/questions/120924 | 6 | **Background:**
For any commutative ring $R$, let $\mathbf{Symm}\_R$ be the ring of symmetric functions in countably many variables $x\_1$, $x\_2$, $x\_3$, ... over $R$. ("Symmetric functions" really means symmetric power series of bounded degree.) It is known that $\mathbf{Symm}\_R$ is generated by the elementary sy... | https://mathoverflow.net/users/2530 | Is the "renormalized third comultiplication" on $\mathbf{Symm}$ integral? | I have a proof of the integrality of $\Delta\_3$ using Dwork's lemma (which tells when a given vector $\left(v\_1,v\_2,v\_3,...\right)\in A^{\left\lbrace 1,2,3,...\right\rbrace}$ over some commutative ring $A$ is the vector of ghost components of a big Witt vector -- note that this is equivalent to the existence of a r... | 3 | https://mathoverflow.net/users/2530 | 124123 | 69,562 |
https://mathoverflow.net/questions/124135 | 2 | How is Hodge theory of harmonic forms related to maxwell's equations.Atiyah says that Hodge was directly motivated by considerations of maxwell's equations while commenting on donaldson.
| https://mathoverflow.net/users/30081 | maxwell's equations and hodge theory | In the absence of electric current, Maxwell's equations say precisely that the electromagnetic potential is a harmonic 1-form; [see Wikipedia](http://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field#Differential_forms_approach). This is in a space-time manifold, so it isn't the usual Hodge t... | 4 | https://mathoverflow.net/users/13268 | 124141 | 69,570 |
https://mathoverflow.net/questions/124119 | 4 | I want to know if $C^{\infty}[0,1]$ or $S$ (Schwartz function space) is separable. Can somebody offer me some results or references?
Thank you!
| https://mathoverflow.net/users/23657 | Is $C^{\infty}[0,1]$ or $S$ separable? | We need the following extension of Weierstrass approximation theorem:
>
> Let $f$ be a smooth function on $[0,1]$. Then we can find a sequence $\{P\_n\}$ of polynomials such that $(P\_n^{(d)})$ converges uniformly to $f^{(d)}$ for each integer $d$.
>
>
>
To see that, we first notice that for each integer $d$,... | 8 | https://mathoverflow.net/users/17118 | 124143 | 69,572 |
https://mathoverflow.net/questions/124151 | 2 | On page 134, Weil divisors, example 6.5.2, he said:
>
> The divisor of $y$ is $2Y$, because
> $y=0$ implies $z^2=0$, and $z$
> generate the maximal ideal of the
> local ring at the generic point of
> $Y$.
>
>
>
I was stupid and can not figure this out. Can someone give a down to earth computation what is t... | https://mathoverflow.net/users/27083 | A question on generic point and a question on Hartshorne | If $X=Spec(R)$ is an affine scheme (as it is in the example you refer to) and $Y$ is the subscheme defined by a prime ideal $P$, then the generic point of $Y$ is the point $[P]\in Spec(R)$. The local ring at that point is the localization $R\_P$.
In the Hartshorne example, $R={\mathbb C}[x,y,z]/(xy-z^2)$, and $P$ is ... | 6 | https://mathoverflow.net/users/10503 | 124154 | 69,575 |
https://mathoverflow.net/questions/124155 | 1 | I was wondering if there are some necessary and sufficient conditions for the quotient space to be Haussdorf. I have been trying a little for a while, but I only got very restrictive sufficient conditions.
| https://mathoverflow.net/users/32097 | If X is a Haussdorf topological space and R and equivalence relation on X, when is X/R Haussdorf? | If $X$ is Hausdorff *and* the quotient map $X\to X/R$ is open, then $X/R$ is Hausdorff if and only if $R\subseteq X\times X$ is closed (see <https://math.stackexchange.com/questions/91639/x-sim-is-hausdorff-if-and-only-if-sim-is-closed-in-x-times-x>).
| 5 | https://mathoverflow.net/users/35353 | 124163 | 69,579 |
https://mathoverflow.net/questions/124144 | 6 | Let $X$ be a set and $u$ be a free ultrafilter on $X$. We can consider a topology on $X$ by declaring every element of $u \cup \{\emptyset \}$ to be open.
El'kin's original motivation for looking at this topology is that $X$ is a space without isolated points which can't be split into disjoint dense subspaces. Anoth... | https://mathoverflow.net/users/11647 | Periodic point-free maps and free ultrafilters. | Your question has been answered by Joseph with several references,
but since this nice result has several attractive proofs, let me
try to provide one.
**Theorem.** If $\mu$ is an ultrafilter on a set $X$ and
$f:X\to X$ has the property that $A\in\mu\leftrightarrow
f^{-1}A\in\mu$, then $f(x)=x$ for $\mu$-almost all $... | 9 | https://mathoverflow.net/users/1946 | 124176 | 69,586 |
https://mathoverflow.net/questions/124152 | 0 | Assume $F$ is a field of characteristic $\neq 2$. Let $(V,q)$ be a quadratic space such that $\rm dim~ q\geq 3$. When $q$ is irreducible it is known that
there exist a purely transcendental field extension $K/F$ such that $[F(q):K]=2$. Here $F(q)$ is a function field of a scheme $X\_q:=\rm Proj (S(V^\*)/q)$ associate... | https://mathoverflow.net/users/31747 | Quadratic subextension of the function field of quadric. | Let $v \in V$ be a non-isotropic vector, then the natural projection $\operatorname{Proj}(S (V^\*)/q)\to \operatorname{Proj}(S (v^\perp)) $ is a finite surjective map of algebraic varieties of degree $2$, so it corresponds to a degree $2$ extension of function fields. The second algebraic variety is just $\mathbb P^{d-... | 2 | https://mathoverflow.net/users/18060 | 124181 | 69,589 |
https://mathoverflow.net/questions/124186 | 5 | Let $R$ be a commutative ring (with $1$) such that every non-zero divisor in $R$ is a unit (see [Rings in which every non-unit is a zero divisor](https://mathoverflow.net/questions/42647/rings-in-which-every-non-unit-is-a-zero-divisor) for various stabs at what these are called). Let $M$ be a finitely generated $R$-mod... | https://mathoverflow.net/users/17263 | When does End(M) consist entirely of zero, zero divisors, and units? | Here is a small positive result: If $R$ is Artinian, $End\_R(M)$ is an $R$-algebra which is f.g. as $R$-module, hence Artinian, hence classical (as I call these rings, following Lam).
On the other hand, as soon as the Krull dimension of $R$ is $\geq 1$, $M = R/p$ for a non-maximal prime ideal $p$ will give a countere... | 7 | https://mathoverflow.net/users/27465 | 124191 | 69,595 |
https://mathoverflow.net/questions/123788 | 14 | I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO.
Let $S$ be a finite set of integers. For $P$ a subset of $S$, I note
$$s\_P = \sum\_{s \in P} s$$
the sum of elements in $P$.
I assume that $S$ satisfies the following property
$... | https://mathoverflow.net/users/9317 | On the $L^1$-norm of certain exponential sums | As Noam correctly mentioned, the upper bound has to be exponential. However, the exponent can, indeed, be improved.
One (fairly cheap) trick is to look at $|f(x)|^2=2^{|S|}\prod\_S(1+\cos 2\pi sx)$. If we can show that the product is bounded from above by $e^{-c|S|}$ outside a set of measure $e^{-c|S|}$, we can get ... | 10 | https://mathoverflow.net/users/1131 | 124193 | 69,597 |
https://mathoverflow.net/questions/124083 | 1 | Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta\_x$ are the Gaussian measure with zero covariance and mean $x \in X$
When does a topological affine space $X$ admit a non-trivial Gaussian measure? Namely, one ... | https://mathoverflow.net/users/238 | Gaussian measures on non-separable spaces | There is a paper by H. Sato, <http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118797795>
proving that 1) a Gaussian measure on a reflexive Banach space is always concentrated on a separable subspace, and 2) the canonical Gaussian cylinder measure of a nonseparable Huber... | 3 | https://mathoverflow.net/users/12205 | 124205 | 69,604 |
https://mathoverflow.net/questions/124224 | 11 | Coding is obviously a fundamental tool in reverse mathematics, and practitioners take care to both demonstrate the correctness of their coding mechanisms and point out their limitations. Harvey Friedman posted on the FOM mailing list in 2004 (see [1] and [2]), prefacing the mathematical part of his first posting with t... | https://mathoverflow.net/users/10384 | New research on coding in reverse mathematics? | I can offer a computational perspective. In computable mathematics we are interested in "computing with mathematical objects" such as integers, finite sets, real numbers, infinite-dimensional Banach spaces, compact subsets of $\mathbb{R}^n$, and many other "infinite" things. Some of these are pretty complicated, so the... | 10 | https://mathoverflow.net/users/1176 | 124237 | 69,615 |
https://mathoverflow.net/questions/124232 | 3 | We have $2n\times 2n$ binary matrix with $k$ of its elements are $1$. We are searching for an $n\times n$ submatrix full of $1$s.
What is the least $k$ such that we can always find one?
What is the least $k$ for $n=10$?
Definition of submatrix: <http://en.wikipedia.org/wiki/Submatrix>
| https://mathoverflow.net/users/27805 | Square submatrix | We can represent this matrix as a bipartite graph, with rows as left vertices and columns as right vertices. This problem is then a special case of extremal problems with forbidden graphs, in particular, special case of [Zarankiewicz problem](http://en.wikipedia.org/wiki/Zarankiewicz_problem). The answer to this specia... | 7 | https://mathoverflow.net/users/32083 | 124241 | 69,617 |
https://mathoverflow.net/questions/124255 | 2 | Let T be a first order set theory formalized in the language L(ZF) of ZF, which has "membership"
and "=" as its only atomic predicates. For each positive integer n, let P(n) be the sentence which
expresses that "there exists a set having exactly n elements". P(n) can be formalized in L(ZF) and
is an axiom of T for each... | https://mathoverflow.net/users/4423 | A question about formalized theories that may be both consistent and w-consistent | If we replace your axiom $Q$ by the stronger-seeming assertion
$Q^+$ that asserts: "there is a largest set, which contains all
other sets as subsets, but which is Tarski finite", then the
corresponding theory $T$, asserting every $P(n)$ and also $Q^+$,
is consistent, but not $\omega$-consistent.
To see this, simply c... | 3 | https://mathoverflow.net/users/1946 | 124262 | 69,622 |
https://mathoverflow.net/questions/124260 | 12 | Let $G$ be a locally compact subgroup, $\mu$ a Haar-measure.
For $f \in L^1(G)$, and for $\pi$ a unitary, topology irreducible, representation of $G$ on
an Hilbert space $H\_\pi$, it is customary to define a continuous operator $\pi(f)$ on $H\_\pi$ by $$\pi(f) v = \int\_G f(g) \pi(g)(v) d\mu(g).$$
Let us say here th... | https://mathoverflow.net/users/9317 | Trace Class Functions on locally compact groups | *The following is a suggestion how to prove a weak variant of the OP's original question affirmative.*
>
> **Theorem:** Let $G$ be a unimodular, seperable, type-I group.
> For every element $\phi$ in $C\_c^\infty(G)$ (in the sense of Bruhat), the operator $\pi(\phi)$ is trace class for almost every unitary represe... | 5 | https://mathoverflow.net/users/10400 | 124269 | 69,623 |
https://mathoverflow.net/questions/124278 | 6 | **Background.** I'm trying to compute some homology groups using a Mayer-Vietoris argument, but I really need local coefficients.
**Question 1.** What does the Mayer-Vietoris sequence look like when using local coefficients?
Consider an open cover $X = U \cup V$ with inclusion maps
$$\begin{array}{ccccc}
& & U \cap... | https://mathoverflow.net/users/16109 | Mayer-Vietoris sequence in homology with local coefficients | Whitehead's ["Elements of Homotopy Theory"](http://rads.stackoverflow.com/amzn/click/0387903364), in particular chapter VI, seems to have everything you ask for. (Note that the Mayer-Vietoris sequence is a formal consequence of excision and the long exact sequence of a pair; see section 2.3 of Hatcher's book).
| 7 | https://mathoverflow.net/users/8103 | 124279 | 69,628 |
https://mathoverflow.net/questions/124252 | 1 | Please help me prove the following identity
$$
a(J\_1(a)Y\_0(a)-J\_0(a)Y\_1(a))=\frac{2}{\pi}
$$
for any $a$.
$J$ and $Y$ are bessel functions of the first and second kind respectively.
Thank you.
| https://mathoverflow.net/users/31251 | Bessel identities | This is equivalent to computing the Wronskian of $J\_0$ and $Y\_0$, since $J'\_0 = -J\_1$ and $Y'\_0 = -Y\_1$.
$$
W(x) =
\begin{vmatrix}
J\_0 & Y\_0 \\\
J'\_0 & Y'\_0
\end{vmatrix} =
\begin{vmatrix}
J\_0 & Y\_0 \\\
-J\_1 & -Y\_1
\end{vmatrix} =
J\_1 Y\_0 - J\_0 Y\_1.
$$
But the zeroth-order Bessel equation is:
$$
\frac... | 4 | https://mathoverflow.net/users/32130 | 124284 | 69,630 |
https://mathoverflow.net/questions/121749 | 5 | This question is [cross-posted](https://math.stackexchange.com/questions/300026/number-of-permutations-with-k-inversions-and-with-a-single-clamped-value) from math.stackexchange because it might be too technical.
Let $S\_n$ be the symmetric group. Recall that the number of inversions of a permutation $\sigma\in S\_n$... | https://mathoverflow.net/users/934 | Number of Permutations with k-inversions and with a single clamped value | It follows from the Knuth-Netto formula that the asymptotics of $I\_n(k)$, for $k$ fixed, is $n^k/k!$ to first order.
I claim the asymptotic behaviour of $I\_n^{\sigma(y)=x}(k)$ is as $n^{k-|x-y|}/(k-|x-y|)!$.
For convenience I am going to assume that $x\geq y$. The other case can be treated the same way.
Firs... | 1 | https://mathoverflow.net/users/468 | 124287 | 69,632 |
https://mathoverflow.net/questions/124208 | 4 | This question did not get anything from math.stackexchange. Therefore, and although probably very elementary, I pose it here. It is about an example in an
[article](https://eudml.org/eudml/doc/183526) of Dekking and Mendez France:
For an integer $n$ let $s(n)$ be the number of ones in the binary expansion,
so that $... | https://mathoverflow.net/users/21051 | Exponential sums and binary expansions | Always look for the counterexample would be a worthwhile motto it seems.
For $\alpha = 1/3$ (so that $e^x$ is a complex cube root of unity), $n = 6$ and $m = 9$ provide one. This is because $s(6) = s(9)$ and $s(6), s(7), s(8)$ are all distinct modulo 3.
For the record the first search for a counterexample was wit... | 4 | https://mathoverflow.net/users/1907 | 124288 | 69,633 |
https://mathoverflow.net/questions/124265 | 4 | Let $U$ be a smooth quasi-projective variety. Does there always exist a smooth compactification of $U$? If not always when can we have smooth compactification?
In particular, suppose $X$ is a singular projective variety and $U$ is the smooth locus. The question is does there always exist a smooth projective variety $Y... | https://mathoverflow.net/users/9164 | Non-uniqueness of smooth compactification | The main idea is this: You can always find an $X$ such that $X\supseteq U$ and $X\setminus U\supseteq \Sigma := \mathrm{Sing} X$. Then in characteristic zero apply Hironaka's resolution theorem, which says that there exists a resolution of singularities $\pi:Y\to X$ such that $\pi$ is an isomorphism over $X\setminus \S... | 8 | https://mathoverflow.net/users/10076 | 124290 | 69,634 |
https://mathoverflow.net/questions/124289 | 3 | Let $k$ be a commutative ring, a $R$-Bimodule $M$ over a $k$-algebra $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this being $R$ itself with left and right multiplications as the actions. A $R$-Bimodule $M$ can also be considered a left $R \otimes R^{op}$-mo... | https://mathoverflow.net/users/21136 | Let $M$ be a $R$-Bimodule that happens to be projective, is its associated left $R \otimes R^{op}$-module projective too? | No. A ring $R$ for which $R$ is projective as a left $R\otimes\_{\mathbb Z}R^{op}$-module is sometimes called separable over ${\mathbb Z}$.
This is equivalent to the splitting, as a surjection of $R\otimes\_{\mathbb Z}R^{op}$-modules, of the multiplication map $R\otimes\_{\mathbb Z}R\to R$. So applying $-\otimes\_R M... | 4 | https://mathoverflow.net/users/22989 | 124304 | 69,638 |
https://mathoverflow.net/questions/124147 | 10 | We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.
If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for $H^k(M)$? References/more detail would be appreciated. Thanks.
(Crossposted from <https://math.stackexc... | https://mathoverflow.net/users/32094 | Do eigenfunctions of elliptic operator form basis of $H^k(M)$? | The answer is no, if the Fredholm index of $L$, which is the integer
$\mathrm{ind}\, L=\dim\mathrm{ker}\, L- \dim \mathrm{ker} \, L^\ast$,
is negative. (Regard $L$ as an unbounded operator on $L^2(M)$ with domain $D(L)=H^{2k}(M)$.) A proof by contradiction goes as follows. If the eigenvectors formed a basis of $H^k(M)$... | 15 | https://mathoverflow.net/users/nan | 124307 | 69,639 |
https://mathoverflow.net/questions/124253 | 2 | Where i can find any proof of next theorem?
Theorem of Ostrowski: Let $\sum\_{n=0}^{\infty}a\_nz^n$ be a power series with radius of convergence 1, which is analytically continuable beyond unit disk. Then the power series is overconvergent if it possesses Hadamard-Ostrowski gaps.
| https://mathoverflow.net/users/32118 | Where i can find the proof of Ostrowski theorem | In English, in the book by P. Dienes, The Taylor series, Dover, NY, 1957.
In German: in the original A. Ostrowski's paper.
| 3 | https://mathoverflow.net/users/25510 | 124313 | 69,642 |
https://mathoverflow.net/questions/124310 | 3 | Consider the following block matrix
$A=\pmatrix{A\_1 & A\_2\cr kA\_2^\top & A\_3}$
where $A\_1$ is a symmetric matrix, $A\_3$ is diagonal matrix and all entries of $A$ are real and non-negative.
How can we show that all eigenvalues of $A$ are real?
Note: $A\_2$ is not a square matrix.
Thanks in advance
| https://mathoverflow.net/users/32136 | Non symmetric matrices with real eigenvalues | $$
\begin{pmatrix}1&0\\\0&k^{-1/2}\end{pmatrix}
\begin{pmatrix}A\_1&A\_2\\\ kA\_2^T&A\_3\end{pmatrix}
\begin{pmatrix}1&0\\\0&k^{1/2}\end{pmatrix}
= \begin{pmatrix}A\_1& k^{1/2}A\_2\\\ k^{1/2}A\_2^T&A\_3\end{pmatrix}
$$
Note that $A\_3$ only needs to be symmetric.
| 7 | https://mathoverflow.net/users/1266 | 124315 | 69,643 |
https://mathoverflow.net/questions/124311 | 16 | My question is: **Does the forgetful functor F:(Mfd) $\to$ (Top) preserve pullbacks?**
Detailed explanation is following.
A pullback is defined as a manifold/topological space satisfying a universal property: a manifold/topological space $Z$ (more precisely a diagram $Y \overset{g}{\leftarrow}Z\overset{g'}{\rightar... | https://mathoverflow.net/users/5206 | Pullbacks as manifolds versus ones as topological spaces | Here's a counterexample. Let $X=Y=\mathbb{R}$, and let $Y'$ be a point. Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth map such that $f^{-1}(\{0\})=\{1,1/2,1/3,\dots\}\cup\{0\}$, and let $f'$ map $Y'$ to $0$. Then the topological pullback is just the space $W=\{1,1/2,1/3,\dots\}\cup\{0\}$. But since manifolds are locally ... | 18 | https://mathoverflow.net/users/75 | 124319 | 69,644 |
https://mathoverflow.net/questions/122781 | 4 | Hi everyone! **Answered to my satisfaction in the comments - thanks nosr and Jacob Bell! :)**
Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are a variety of definitions of soft sheaf, let me emphasize my definition (taken from Gelfand-Manin, Methods of Homolo... | https://mathoverflow.net/users/30971 | Restricting a Soft Sheaf to an Open is again Soft? | This is more of a comment than an answer, but it won't fit in the comment section.
Gelfand-Manin's definition of sections over a closed set looks suspicious. In GM, for any subset $Z\subseteq X$, $$\mathcal{F}(Z):=\varinjlim\_{V\supseteq Z}\mathcal{F}(V).$$ In general, this is different from $\Gamma(Z, \mathcal{F}\m... | 3 | https://mathoverflow.net/users/5738 | 124322 | 69,645 |
https://mathoverflow.net/questions/124028 | 28 | For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the *celebrated Sobolev inequalities*:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$,
If $k > n/p$ then $u$ lies in a particular Hölder space.
It is also known that ... | https://mathoverflow.net/users/12310 | What goes wrong for the Sobolev embeddings at $k=n/p$? | I'll take a stab. In the following we consider the case $W^{1,n}$ in $\mathbb{R}^n$. My short answer is that under rescaling by factor $\lambda$, derivatives scale by $\lambda$ and volumes by $\lambda^{-n}$, so integrating derivatives to the $n$ won't change under rescaling. The following examples illustrate how this a... | 19 | https://mathoverflow.net/users/16659 | 124331 | 69,651 |
https://mathoverflow.net/questions/121573 | 1 | I wish to show that
$|j\_n(x)| < \frac{1}{\sqrt{x}}$
for $n=0,1,2,\ldots$ and $x>0$, where $j\_n$ is the spherical Bessel function of the first kind.
Experimenting with Matlab I am sure that this is the case by I am unable to prove it analytically.
Since
$j\_n(x) = \frac{1}{\sqrt{x}}\sqrt{\frac{\pi}{2}}J\_{n+... | https://mathoverflow.net/users/31372 | Spherical Bessel functions | In Abramowitz and Stegun, Handbook of Mathematical Functions, you will find the formula 10.1.52
$$\sum\_0^\infty j\_n^2(x)={Si(2x)\over 2x}.$$
Consequently
$$|j\_n(x)|\le {1\over\sqrt{x}}\sqrt{{\max\_{[0,\infty)}|Si(x)|\over 2}}\sim {0.962\over\sqrt{x}}.$$
| 4 | https://mathoverflow.net/users/12120 | 124340 | 69,655 |
https://mathoverflow.net/questions/124339 | 4 | Let $K$ be a finitely generated extension of an algebraically closed field of characteristic zero, and $A,B$ abelian varieties over $K$.
Then is $Hom\_K(A,B)\otimes \mathbb{Z\_l} \cong Hom\_{Gal(\bar{K} / K)}(T\_l(A),T\_l(B))$, where $T\_l(A)$ is the $l$-adic Tate module of $A$?
| https://mathoverflow.net/users/16858 | Tate conjecture for abelian varieties over a finitely generated extension of an algebraically closed field | The answer is no: an easy (not interesting) counterexample is provided by "constant" abelian varieties, i.e., when $A$ and $B$ are defined over an algebraically closed field $k$ of characteristic zero while $K$ is finitely generated over $k$.
So, let's assume that the $K/k$-traces of $A$ and $B$ are zero, i.e., both... | 12 | https://mathoverflow.net/users/9658 | 124351 | 69,659 |
https://mathoverflow.net/questions/122972 | 1 | Given a sequence $A\_n(k)$ defined as follows:
$A\_0(0)=1$, $A\_0(k)=0$ for all nonzero integers $k$ and
$$A\_{n}(k)=(n+1-k)^2A\_{n-1}(k-1)+2(n(n+1)-k^2)A\_{n-1}(k)+(n+1+k)^2A\_{n-1}(k+1)$$
for all positive integers $n$ and (positive or negative) integers $k$. Does there exist an explicit formula for $A\_n(k)$? This ... | https://mathoverflow.net/users/31756 | A recursive Double sequence related to uniform Cardinal B-spline | Edit: ok, now that I have more than 5 minutes to spare I can clean this up a bit and add a wikipedia reference.
I'm going to write A(n,k) for $A\_n(k)$. First of all, note that it's easy to see that A(n,k) = A(n,-k) by induction on n, and that the A(n,k) are zero unless -n <= k <= n. So we may as well just start comp... | 3 | https://mathoverflow.net/users/20281 | 124358 | 69,662 |
https://mathoverflow.net/questions/124354 | 3 | Let $Q$ be a smooth manifold, and let $G$ be a Lie group which acts smoothly on $Q$ on the left.
* **Question 1:** does the group $G$ act naturally on the tangent bundle $TQ \to Q$?
My motivation here is $Q = \mathbb R^d$ with group of symmetries the Euclidean group $G = \operatorname{Euc}(d) \cong \mathbb R^d \rt... | https://mathoverflow.net/users/238 | Symmetry group for the frame bundle of a G-space | The chain rule for manifolds states that if $f : M \to N$ and $g : N \to Q$ are smooth, then the derivative $Df : TM \to TN$ and $Dg : TN \to TQ$ satisfy
$$D(g \circ f) = Dg \circ Df$$
So if $g \in G$, let the action of $g$ on $Q$ be denoted $L\_g : Q \to Q$. So $L\_g \circ L\_h = L\_{gh}$, therefore $D(L\_{gh}) = D(... | 3 | https://mathoverflow.net/users/1465 | 124359 | 69,663 |
https://mathoverflow.net/questions/124355 | 3 | I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are welcome.
I have some PDE literature and some SDE literature, but need to figure out SPDEs... Thanks!
| https://mathoverflow.net/users/31856 | Good books on stochastic partial differential equations? | On the analytical side, I like a lot the book [A Concise Course on Stochastic Partial Differential Equations](http://books.google.com/books?id=Q-JDooJbjCAC&pg=PP4&dq=prevot+roeckner&hl=en&sa=X&ei=9dc_Ub_zIK214AOI9YHwDw&ved=0CDAQ6AEwAA) by Prevot and Roeckner. It is a very well written introduction to SPDEs.
Besides t... | 3 | https://mathoverflow.net/users/20026 | 124368 | 69,665 |
https://mathoverflow.net/questions/124352 | 14 | The Thom spectrum MO is a module over the ring spectrum π≤0S=H**Z**, where S is the sphere spectrum.
In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ\*(MO).
On the other hand, MU and MSO are not modules over π≤0S, because they have nontrivial k-invariants.
I wonder if the above result about MO can... | https://mathoverflow.net/users/402 | Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum? | For any $k$ and any $0<n<\infty$, $K(n)\wedge \pi\_{\leq k}S=0$. Indeed, this is true for any spectrum with finitely many homotopy groups, since $K(n)\wedge H\mathbb{Z}=0$ and any such spectrum has a finite filtration into Eilenberg-MacLane spectra. If a spectrum $M$ admits a $\pi\_{\leq k}S$-module structure, then $M$... | 24 | https://mathoverflow.net/users/75 | 124372 | 69,668 |
https://mathoverflow.net/questions/124367 | 2 | Its known ( see " The birational geometry of degenerations") that there exist a smooth one parameter family (i.e. total space is smooth) of two dimensional complex toris over unit disk whose central fiber is a normal crossing union of (say four for example) some copies of $\mathbb{P}^2$ blown up at 3 points of a triang... | https://mathoverflow.net/users/5259 | Toroidal embedding | Watch out: *toroidal* $\neq$ *toric* !
It is not possible to realize this situation in a toric variety, at least not so that $\pi$ is a toric morphism, because toric varieties are rational by definition and complex tori are not. (I am assuming that by *complex torus* you mean a compact quotient of $\mathbb C^g$).
A... | 3 | https://mathoverflow.net/users/10076 | 124375 | 69,670 |
https://mathoverflow.net/questions/124384 | 1 | Let $r\_1, r\_2, \ldots, r\_k$ be positive integers with or without repetition such that $1\le r\_i \le n$ for $i = 1, 2, \ldots, k$. Let $f$ be a continuous multivariate function with the property that the value of $f(r\_1, r\_2, \ldots, r\_k)$ is independent of the order the arguments $r\_1, r\_2, \ldots, r\_k$. The ... | https://mathoverflow.net/users/23388 | Multivariate functions whose value is independent of the order of the arguments | You could symmetrize an arbitrary function by averaging it over the symmetric group.
Explicitly: take your favourite function $h$ in $n$ variables, and set $f(r\_1, ..., r\_n) = \frac{1}{n!}\sum\_{\sigma} h(r\_{\sigma(1)}, ..., r\_{\sigma(n)})$ where the sum is taken over all permutations $\sigma$ of the numbers $1,.... | 4 | https://mathoverflow.net/users/20281 | 124387 | 69,674 |
https://mathoverflow.net/questions/124376 | 2 | Say $X \subset \mathbb{P}^n$ is monomial, by which I mean that it's cut out by a homogenous ideal generated by monomials, or in other words it's a fixed point on the Hilbert scheme for the action of the big torus. What is known about the following question:
>
> When is $X$ in the closure of the locus in the Hilbert... | https://mathoverflow.net/users/4707 | Which monomial subschemes are limits of smooth subschemes? | In the zero-dimensional case, the scheme will certainly be supported only on the $n+1$ fixed points in $\mathbb P^n$ of the torus action, so we can work locally in a neighborhood of each point. Thus we can consider an affine monomial ideal.
Replace the monomial $ \prod\_i x\_i^{d\_i}$ with $\prod\_i (x\_i)(x\_i-t)(x\... | 4 | https://mathoverflow.net/users/18060 | 124389 | 69,676 |
https://mathoverflow.net/questions/124396 | 5 | Given G, a fuchsian group and a finite sub set A of G. Does there exist a finite index subgroup H in G such that inter section of A with H is empty?
| https://mathoverflow.net/users/23358 | finite index subgroup of a fuchsian group | The answer is yes. The group is residually finite. This means that if $A$ is taken to be a finite set of elements of the Fuchsian group $G$ $not \quad containing\quad identity$, then there exists a finite quotient of $G$ where no element of $A$ is trivial. The kernel of this quotient map is a finite index subgroup$H$ n... | 7 | https://mathoverflow.net/users/23291 | 124397 | 69,679 |
https://mathoverflow.net/questions/124328 | 4 | If the jacobian of a curve splits then the L-function splits as well simply because isogenous Abelian varieties have the same L-function. Is the converse true? Or under what condition is it true?
**Edit:** According to Faltings' theorem pointed out by Francois, if the L-function split into product of L functions of A... | https://mathoverflow.net/users/8932 | Does split L-function imply split jacobian | Faltings proved that two abelian varieties $A$ and $B$ defined over a number field are isogenous if and only if they have the same $L$-function. See Korollar 2 p. 361 in
[Faltings, G. *Endlichkeitssätze für abelsche Varietäten über
Zahlkörpern.* Invent. Math. 73 (1983), no. 3, 349--366.](http://dx.doi.org/10.1007/BF... | 3 | https://mathoverflow.net/users/6506 | 124403 | 69,681 |
https://mathoverflow.net/questions/124408 | 2 | I define the following wheighted Sobolev spaces
$$L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int\_{\mathbb{R}^3}|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$$
and
$$H^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg| D^\alpha u\in L^{2,s}(\mathbb{R}^3),|\alpha|\leq 2\bigg\rbrace$$
I know that the classical Sobolev space $H^2(\mathbb{... | https://mathoverflow.net/users/32163 | Embedding of weighted Sobolev spaces | Continuity is a local property: functions which are in your $L^{2,s}$ are locally in $L^2$, so functions in $H^{2,s}$ are locally in the Sobolev space $H^2(\mathbb R^3)$, thus are continuous functions (even Hölder 1/2-$\epsilon$).
| 5 | https://mathoverflow.net/users/21907 | 124413 | 69,687 |
https://mathoverflow.net/questions/124393 | 0 | Let $\mathfrak{m}$ be a Lie sub-algebra of the Lie algebra $\mathfrak{g}$. Is there a name for the smallest ideal of $\mathfrak{g}$ containing $\mathfrak{m}$? It certainly exists and coincides with the intersection of all the ideals containing $\mathfrak{m}$. The analogy with normal closure in group theory would sugges... | https://mathoverflow.net/users/1049 | Name for ideal generated by Lie subalgebra | In analogy with the notion of normal closure in group theory, the smallest ideal of ${\mathfrak g}$ containing the subalgebra ${\mathfrak m}$ is indeed called the ideal closure of ${\mathfrak m}$ and denoted by ${\mathfrak m}^{\mathfrak g}$. This terminology is rather diffused in the literature: for instance, you can f... | 3 | https://mathoverflow.net/users/14653 | 124415 | 69,688 |
https://mathoverflow.net/questions/124412 | 1 | Let $\mathcal{S}\_g$ denote the fundamental group of an oriented surface of genus $g\ge 2$.
Does $\mathcal{S}\_g$ contain subgroups $A$ and $B$ of finite index such that $A\cap B = \lbrace e\rbrace$?
| https://mathoverflow.net/users/8103 | Intersections of subgroups of surface groups | As Misha says in a comment, for any group $G$ with subgroups $A,B$, we have
$|G:A\cap B|\leq |G:A||G:B|$
(exercise). In particular, if $G$ is infinite (as here) then $A\cap B$ is non-trivial.
| 4 | https://mathoverflow.net/users/1463 | 124416 | 69,689 |
https://mathoverflow.net/questions/124421 | 0 | ***Introduction:***
As a quick reminder, the Gel'fand Yaglom theorem uses the generalized zeta-function approach to compute functional determinants of differential operators. Given a differential operator on $x\in [0,1]$
$$O\_1=\frac{d^2}{dx^2}+V(x).$$
The functional determinant can be obtained from a solution $... | https://mathoverflow.net/users/26176 | Gel'fand Yaglom functional determinant of non-diagonal operator? | Not an expert, but you caught my interest. Looks like you may need to be a little careful in higher dimensions, but there are versions of the theorem which work. See, for example:
<http://library.msri.org/books/Book57/files/70kirsten.pdf>
| 0 | https://mathoverflow.net/users/20281 | 124424 | 69,691 |
https://mathoverflow.net/questions/124297 | 8 | Suppose you are on a manifold. Suppose you have a trivial bundle and a trivial subbundle of it. If you divide this trivial bundle with its trivial subbundle, do you get a trivial bundle as a quotient? The answer is generally no. What are the conditions on the manifold (s.c. etc.) and bundles (codim e.g.) to get a trivi... | https://mathoverflow.net/users/25455 | Quotient of trivial bundles | If the dimension of the base $X$ of the bundles is less than the difference $n-k$ of the fiber dimensions then the quotient bundle is trivial. To see this it is enough to consider the case $k=1$ and then induct. An embedding of a trivial rank one bundle in a trivial rank $n$ bundle amounts to a map $X\to S^{n-1}$. If t... | 6 | https://mathoverflow.net/users/6666 | 124427 | 69,692 |
https://mathoverflow.net/questions/124418 | 1 | Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one conclude that $H$ is a linearly reductive complex Lie group? I would appreciate any and all references and suggestions.
Tha... | https://mathoverflow.net/users/25358 | A Criterion for Reductivity of Lie Subgroups | As some of the comments suggest, "semisimple" correlates well for the groups and their Lie algebras in characteristic 0, but "reductive" doesn't work so well. For a Lie algebra in characterisic 0, Bourbaki *Groupes et algebres de Lie* (Chapter I, section 6.4) defines it to be *reductive* just when its adjoint represent... | 2 | https://mathoverflow.net/users/4231 | 124429 | 69,693 |
https://mathoverflow.net/questions/124401 | 4 | Many [stochastic processes](http://en.wikipedia.org/wiki/Stochastic_process) that you encounter are kind of well-behaved, i.e. have infinite variation, yet finite [quadratic variation](http://en.wikipedia.org/wiki/Quadratic_variation).
My question revolves around stochastic processes that have infinite variation, inf... | https://mathoverflow.net/users/1047 | Ito formulae for stochastic processes with finite cubic, quartic ... n-tic variation | Thank you Didier for giving references to my paper with Errami.
Other later related reference focusing on fractional Brownian motion are the following.
Gradinaru, Mihai; Nourdin, Ivan; Russo, Francesco; Vallois, Pierre $m$-order integrals and generalized Itô's formula: the case of a fractional Brownian motion with ... | 7 | https://mathoverflow.net/users/32175 | 124433 | 69,694 |
https://mathoverflow.net/questions/124391 | 4 | Let $\mathcal M$ be a premouse.
$\mathcal M$ is said to be $\omega$-small if and only if whenever we have that $\kappa=crit(E)$ for some extender $E$ on the sequence of $\mathcal M$, then $\mathcal J^{\mathcal M}\_{\kappa} \not \models$ There are $\omega$-many Woodins.
$\mathcal M$ is said to be properly small if and o... | https://mathoverflow.net/users/3859 | $\omega$-small and properly small premice. | If $\mathcal M$ is $\omega$-small, then many $\mathcal J^{\mathcal M}\_\beta$ may think that (a large fragment of $\mathsf{ZF}$ holds and) there are plenty of Woodin cardinals. What matters is that for any such $\beta$ there is a larger $\tau$ where we see that this is no longer the case.
This may happen for a varie... | 4 | https://mathoverflow.net/users/6085 | 124441 | 69,699 |
https://mathoverflow.net/questions/124432 | 3 | Let $C$ be a projective curve (over an algebraically closed field) of genus $\geq 1$. Let $S = C \times C$. By normalisation we have a ramified cover $C \to \mathbb{P}^1$ and so a map $p: S \to \mathbb{P}^1 \times \mathbb{P}^1$. I am interested in finding families of curves on $S$ passing through a given point $P \in S... | https://mathoverflow.net/users/2234 | families of curves on surfaces which are products of curves | $\mathrm{Pic}(\mathbb P^1\times \mathbb P^1)\simeq \mathbb Z\oplus\mathbb Z$, while $\mathrm{rank}\ \mathrm{Pic} (C\times C)\geq 3$ (the diagonal has self-intersection $2-2g$ and its intersection with both $\{q\}\times C$ and $C\times \{q\}$ is $1$, so the determinant of the intersection matrix of these three curves is... | 4 | https://mathoverflow.net/users/10076 | 124444 | 69,700 |
https://mathoverflow.net/questions/124451 | 4 | Suppose $X$ is a smooth variety over $\mathbb{C}$. Let $K^{b}(X)$ be the homotopy category of bounded complex of coherent sheaves, and $D^{b}(X))$ be the derived category of bounded complex of coherent sheaves. One can define:
$$Hom^{\cdot}(A^{\cdot}, B^{\cdot}): K^{b}(X) \to K(Ab)$$ as
$$Hom^{i}(A^{\cdot},B^{\cdot}):=... | https://mathoverflow.net/users/29730 | Calculate $Hom$ in derived category | For computations you it is better to replace the first argument by a locally free resolution. This allows to compute the local $R{\mathcal H}om$. Then you can use the local-to-global spectral sequence to compute the global $RHom$. In your particular example, as everything happens in a neighborhood of $x$ and since $x$ ... | 6 | https://mathoverflow.net/users/4428 | 124454 | 69,704 |
https://mathoverflow.net/questions/124453 | 2 | Let $X,Y$ be smooth varieties defined over $k$. Suppose $P$ is a coherent sheaf on $X \times Y$ flat over $X$, considering the Fourier-Mukai transform
$$\Phi\_P : D^{b}(X) \to D^{b}(Y)$$
which is defined as
$$F \mapsto q\_\* (P \otimes p^{\*}F).$$
Suppose $k(x)$ is the skyscraper sheaf on the closed point $x \in X$... | https://mathoverflow.net/users/29730 | Skyscraper sheaf under Fourier-Mukai transform | $p^\*$ for smooth $p$ is exact, so $p^\*$ of a skyscraper sheaf in the derived and regular senses are identical. So now we just have a pullback of a skyscraper sheaf - the structure sheaf of a single fiber and zero elsewhere.
We want to check that the higher tensor products vanish, so that the derived tensor product ... | 2 | https://mathoverflow.net/users/18060 | 124460 | 69,706 |
https://mathoverflow.net/questions/124461 | 0 | I want to study about Symplectic group actions and moment map, especially Hamiltonian Group Actions. Can you help me with some concretely example of Hamiltonian group actions ? Where can I find some exemple?
| https://mathoverflow.net/users/16444 | Hamiltonian group actions - examples | I think these <http://www.math.ist.utl.pt/~acannas/Books/lsg.pdf> are very good notes, and freely available.
| 2 | https://mathoverflow.net/users/15155 | 124467 | 69,710 |
https://mathoverflow.net/questions/124466 | 8 | [Toen-Vaquié](http://arxiv.org/abs/math/0509684v4) construct a category of schemes relative to some complete cocomplete closed symmetric monoidal category $C$. Affine schemes correspond by definition 1:1 to commutative monoid objects in $C$. For $C=\mathsf{Ab}$ we get the usual category of schemes, where affine schemes... | https://mathoverflow.net/users/2841 | What about schemes built up out of graded rings? | It is my impression (without being very careful about it) is that the C-geometry (for C the category of graded abelian groups) is the same as geometry over $BG\_m$, i.e., geometry of schemes with
an action of the multiplicative group. This should follow from the identification of $C$ with representations of $G\_m$. (I... | 10 | https://mathoverflow.net/users/582 | 124469 | 69,712 |
https://mathoverflow.net/questions/124462 | 6 | I'm interested in knowing/collecting some properties of epimorphisms of rings (with identity) that are true over commutative rings but are false in the non-commutative case.
Example: I learned from MO that if $R \hookrightarrow S$ is an epimorphism of commutative rings then $S/R$ is a torsion left $R$-module. But the... | https://mathoverflow.net/users/18571 | Properties of ring epimorphisms that are true only over commutative rings | A commutative finitely generated ring is Hopfian (proved by Malcev) i.e. every surjective endomorphism is an automorphism. That is not true for non-commutative finitely generated rings.
| 4 | https://mathoverflow.net/users/nan | 124471 | 69,713 |
https://mathoverflow.net/questions/124474 | 3 | Let $\rho : G \to GL(V)$ be an irreducible representation of a finite group. Schur's lemma says if $\pi:GL(V) \to GL(V)$ intertwines with $\rho$, that is, $\pi \rho(g) = \rho(g) \pi$ for every $g\in G$, then $\pi = \lambda I$ for some $\lambda \in \mathbb{C}$.
Is there a similar lemma for $\rho = m\_1 \rho\_1 \oplus... | https://mathoverflow.net/users/26659 | Generalization of Schur's Lemma | This is an easy exercise. If $\rho \_i$ are "different" (i.e. inequivalent) then by Schur's lemma, $Hom \_G(\rho \_i,\rho \_i)={\mathbb C}I$ and $Hom \_G(\rho \_i, \rho \_j)=0$. Hence
the commutant of $G$ in $End(\rho)$ is easily seen to be the product
$$M\_{m\_1}({\mathbb C})\times \cdots \times M\_{m\_k}({\mathbb ... | 13 | https://mathoverflow.net/users/23291 | 124479 | 69,714 |
https://mathoverflow.net/questions/124473 | 3 | Is the space of Riemannian metrics, over a compact manifold, complete when endowed with the $C^k$-topology of metrics?.
Is there a good reference for this?
| https://mathoverflow.net/users/31521 | $C^k$ topology of metrics | No it is not complete. It is an open convex cone of the Banach space
(Frechet space if $k=\infty$)
$\Gamma\_{C^k} S^2T^\*M$
of $C^k$-sections of the vector bundle $S^2T^\*M$. 0 is always in the closure of this cone, and many more things.
The norm on this Banach space depends on many choices (charts, metric, etc.), but... | 7 | https://mathoverflow.net/users/26935 | 124492 | 69,719 |
https://mathoverflow.net/questions/124487 | 0 | Let M be a complete Riemannian manifold.Suppose there are two non-parabolic ends on M with respect to $M\backslash {B\_p}\left( {{R\_0}} \right)$Then there is a harmonic function f on M.Is it right that $\int\_{{B\_p}\left( R \right)} {f \le CR} $ for $R \ge {R\_0}$ and a constant C independent of R?
| https://mathoverflow.net/users/24637 | Integral of a harmonic function on a manifold with two non-parabolic ends | What made you expect this? Since the ends are non-parabolic, their volume growth is at least quadratic. On the other hand, the harmonic functions arising from two non-parabolic ends are linear combinations of the constant and the function whose value at a point $x$ is the probability that the Brownian motion started at... | 1 | https://mathoverflow.net/users/8588 | 124497 | 69,721 |
https://mathoverflow.net/questions/124493 | 3 | If I consider the Sobolev space $H^2(\mathbb{R}^3)$ I have the norm
$$\Vert u\Vert\_{H^2(\mathbb{R}^3)}=\bigg(\sum\_{|\alpha|\leq 2}\Vert D^\alpha u\Vert^2\_{L^2(\mathbb{R}^3)}\bigg)^\frac{1}{2}.$$
Is this norm is equivalent to
$$\Vert u\Vert=\left(\Vert u\Vert\_{L^2}^2+\Vert\Delta u\Vert\_{L^2}^2\right)^{\frac{1}{2}}... | https://mathoverflow.net/users/32195 | Doubt on norm of the Sobolev space $H^2(\mathbb{R}^3)$ | You should write:
$$\Vert u\Vert=\left(\Vert u\Vert\_{L^2}^2+\Vert\Delta u\Vert\_{L^2}^2\right)^{\frac{1}{2}}.$$
Then the two norms are equivalent. You can see this by either using interpolation inequalities or by going to the Fourier transform where the estimates are easy:
$$\Vert u\Vert\_{H^2(\mathbb{R}^3)}=\bigg(\su... | 3 | https://mathoverflow.net/users/26935 | 124499 | 69,722 |
https://mathoverflow.net/questions/124484 | 11 | In textbooks Stokes' theorem is usually formulated for orientable manifolds (at least I couldn't find any version not using orientability). Is Stokes theorem: $\int\limits\_{M}d\omega=\int\limits\_{\partial M} \omega$ also true for non-orientable manifolds?
Are there any references, you can tell me?
Regards.
| https://mathoverflow.net/users/21870 | Stokes theorem for manifolds without orientation? | This is just more information on Jose's comment.
Look at section 10 (pages 122 to 128) in:
Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008. [(pdf)](http://www.mat.univie.ac.at/~michor/dgbook.pdf)
On the orientable double cov... | 11 | https://mathoverflow.net/users/26935 | 124501 | 69,723 |
https://mathoverflow.net/questions/124504 | 13 | Let $X$ be an algebraic variety. I have read the following definitions:
1. $X$ is *factorial* if every Weil divisor on $X$ is Cartier.
2. $X$ is *locally factorial* if all its local rings are unique factorisation domains.
I am comfortable with each of these. However, I have also seen it stated ([in this question](h... | https://mathoverflow.net/users/22975 | Factoriality: local or global? | It's a standard result in commutative algebra that every noetherian integral domain is a UFD if and only if every prime ideal of height 1 is principal. When applied to the local rings of X this gives exactly the equivalence above.
| 16 | https://mathoverflow.net/users/4790 | 124510 | 69,726 |
https://mathoverflow.net/questions/124247 | 1 | This seems like the kind of thing an expert should be able to answer off the top of their head:
Recall that a valuation ring is an integral domain $A$ such that for every $a \in Frac(A)$ we have either $a \in A$ or $a^{-1} \in A$. One of the many equivalent definitions of a Hensel valuation ring, is: a valuation ring... | https://mathoverflow.net/users/12914 | Are hensel valuation rings N2? | The following counterexample is very similar to exercise 2 to Bourbaki's *Algèbre commutative* ch. VI §8 and works as well. Actually it is example 4.9 in Scholze's *[Perfectoid Spaces](http://www.math.uni-bonn.de/people/scholze/PerfectoidSpaces.pdf)*; there, the main point is that $B$ is an "almost finitely generated" ... | 2 | https://mathoverflow.net/users/27465 | 124525 | 69,730 |
https://mathoverflow.net/questions/124526 | 8 | Let $G$ be a Lie group (possibly disconnected). Consider the natural short-exact sequence $$1\rightarrow G\_0\rightarrow G\rightarrow\pi\_0(G)\rightarrow 1,$$ where $G\_0$ is the identity component of $G$, and $\pi\_0(G)\cong G/G\_0$ is the component group. Under what conditions does this sequence admit a splitting, so... | https://mathoverflow.net/users/25358 | Splitting of a Short-Exact Sequence of Lie Groups | See pages 177 to 190 of:
Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008.
[(pdf)](http://www.mat.univie.ac.at/~michor/dgbook.pdf)
| 5 | https://mathoverflow.net/users/26935 | 124532 | 69,733 |
https://mathoverflow.net/questions/124527 | 5 | I need to understand some of the theory of smooth four manifolds. Eventually I might be interested in learning about Donaldson theory. For the moment I am mainly interested in question such as: if you have a compact, connected, simply-connected four manifold, in how many different ways can I embed a smooth, compact 2-s... | https://mathoverflow.net/users/29850 | Understanding four manifolds (more details inside) | My recommendation would be the book of Freedman and Quinn, [Topology of 4-manifolds](http://rads.stackoverflow.com/amzn/click/0691085773). It's hands-on, very very good, and suitable I think for a reader of your background. Indeed, I would strongly recommend it to anyone interested in embeddings and immersions of surf... | 4 | https://mathoverflow.net/users/2051 | 124537 | 69,736 |
https://mathoverflow.net/questions/124522 | 1 | The intersection of 3 quadrics in $P^5$ is a K3 surface $S$.
There is a natural map $S^{[2]} \to G(1,5)$ well defined everywhere, because a generic K3 doesn't contain any line and this family is maximal.
This is moreover injective, because a line can meet $S$ in at most 2 points (otherwise it is contained in each of t... | https://mathoverflow.net/users/27125 | Is this an embedding of $S^{[2]}$? | The embedding of $S$ into $P^5$ induces an embedding $S^{[2]} \to (P^5)^{[2]}$. On the other hand, it is easy to check that $(P^5)^{[2]}$ is isomorphic to $P\_{G(1,5)}(S^2U)$, the projectivization of the symmetric square of the tautological rank 2 bundle on the Grassmannian. This gives the map which you want.
EDIT. S... | 3 | https://mathoverflow.net/users/4428 | 124548 | 69,741 |
https://mathoverflow.net/questions/124529 | 3 | Let $F$ be a global field. What is the measure of $PGL\_2(F) \backslash PGL\_2(\mathbb{A})$?
This depends of course on the normalizations of the Haar measures on $PGL\_2(F)$ and $PGL\_2(\mathbb{A})$. Probably its most likely available in the literature for $PGL\_2(F)$ admits the discrete measure and $PGL\_2(\mathbb{A... | https://mathoverflow.net/users/10400 | Volume of PGL(2,F) \ PGL(2, A) | If you take the Tamagawa measure, the measure is 2. It is equal to the index of the universal covering, so for a simply connected group $G$ the volue of $G({\mathbb A})/G(F)$ is one. This is Kottwitz's Theorem, formerly known as the Tamagawa number conjecture. A proof is in
<http://www.jstor.org/discover/10.2307/200700... | 6 | https://mathoverflow.net/users/nan | 124553 | 69,744 |
https://mathoverflow.net/questions/124555 | 6 | I am interested in the following problem : given a finite field $F$ and two unipotent elements $g\_1,g\_2\in\mathrm{SL}\_2(F)$ which do not commute, what can we say about the subgroup they generate? More specifically, is there an additional condition under which it is known that they generate all of $\mathrm{SL}\_2(F)$... | https://mathoverflow.net/users/32210 | Subgroups of $SL_2(F)$ generated by unipotent elements | You should look up Dickson's Theorem. It deals pretty comprehensively with the two dimensional case when $F$ is finite, at least in odd characteristic. The proof may be found in Gorenstein's 1968 book "Finite Groups". I don't remember the statement precisely, but I think it's that $\langle \left( \begin{array}{clcr} 1 ... | 9 | https://mathoverflow.net/users/14450 | 124561 | 69,747 |
https://mathoverflow.net/questions/124557 | 5 | Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it is possible to characterize them using inner model theory. For instance the $HOD^{L(\mathbb{R})}$ is a certain mouse, $\m... | https://mathoverflow.net/users/3859 | Models of Determinacy | Certainly, $L(\mathbb R)$ is not a mouse (rather, a weasel) over a countable set, and the only way I see of thinking of it as a mouse and still capturing all the reals is making it a mouse over $\mathbb R$, in which case the answer is trivial. There is interesting work on $\mathbb R$-premice, but I do not think this is... | 6 | https://mathoverflow.net/users/6085 | 124564 | 69,749 |
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