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/74017 | 8 | It is "well-known" that the Hecke algebra $\mathcal{H}$ can be thought
of as the Grothendieck group for the category of perverse sheaves on
$G/B$, where the product in $\mathcal{H}$ corresponds to convolution
of sheaves by the Borel subgroup. This means, given perverse sheaves
$X$ and $Y$ on $G/B$ and their classes $[X... | https://mathoverflow.net/users/3077 | Parabolic convolution of perverse sheaves in terms of the Hecke algebra | Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and fix a Borel subgroup $B \subset G$. One can consider the 2-category with objects parabolic subgroups $P \supset B$ and 1-morphisms $P \to Q$ given by $D^b\_{P\times Q}(G)$ (the $P \times Q$-equivariant derived category of $G$ with respect to the a... | 3 | https://mathoverflow.net/users/919 | 76539 | 46,347 |
https://mathoverflow.net/questions/75734 | 4 | If $S$ is the Zariski-Riemann space of a noetherian subring $k$ of a field $K$, Zariski-Samuel prove that $S$ is quasi-compact. If $S'$ is the subspace of valuations that are discrete (i.e. that valuation group is isomorphic to $\mathbb{Z}^n$ with the lexicographical ordering), is $S'$ still quasi-compact?
Is $S'$ de... | https://mathoverflow.net/users/12914 | Is the subspace of DVR's of the Zariski-Riemann space still quasi-compact? | This is only a partial answer: $S'$ is dense in $S$. (As mentioned in the comments, assuming that $K$ is finitely generated over $k$ as a field.)
In fact, a stronger result is true: Let $S\_{\mathrm{DVR}}$ be the subspace of all DVRs (i.e. value group isomorphic to $\mathbb{Z}$), then $S\_{\mathrm{DVR}}$ is dense in ... | 1 | https://mathoverflow.net/users/7150 | 76544 | 46,351 |
https://mathoverflow.net/questions/76546 | 6 | I am just wondering, how to prove the Hahn-Banach theorem constructively for a *finite dimensional* normed vector space.
Thanks in advance for any helpful answers.
| https://mathoverflow.net/users/11757 | How to prove the Hahn-Banach constructively | Same way as for the infinite dimensional case, except you avoid Zorn's lemma by counting dimensions.
| 3 | https://mathoverflow.net/users/11142 | 76548 | 46,353 |
https://mathoverflow.net/questions/76520 | 0 | I have been asked to add to an existing linear programming model several constraints dealing with ratios among continuous decision variables. An example ratio constraint would be like:
$x\_1\*x\_2 - x\_3\*x\_4 = 0$
They told me to preserve the lineal nature of the problem, so I am trying to find an equivalent linea... | https://mathoverflow.net/users/18098 | Nonlinear constraint and product of variables | $x\_1 x\_2 - x\_3 x\_4 = 0$ is inherently nonlinear, and maybe more importantly non-convex: e.g. the midpoint of two feasible solutions may not be (in fact, hardly ever is) a feasible solution. So in general there is no such thing as an equivalent formulation that is linear. However, in special cases something might be... | 2 | https://mathoverflow.net/users/13650 | 76554 | 46,354 |
https://mathoverflow.net/questions/76585 | 15 | Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied?
| https://mathoverflow.net/users/5259 | Moduli space of genus 2 curves | Genus 2 curves are hyperelliptic and so their coarse moduli space is just the Riemann-Hurwitz space $(\mathbb{P}^1)^6/(SL\_2 \cdot S\_6)$. So the description of $M\_2$ is closedly linked with the invariants of binary sextic forms. The classic reference is the paper
J. Igusa, [Arithmetic Variety of Moduli for Genus T... | 25 | https://mathoverflow.net/users/3996 | 76589 | 46,371 |
https://mathoverflow.net/questions/76530 | 2 | I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or suggest a better reference.
Let $\Omega$ be defined by the open domain in $\mathbb{H} $ bounded by the lines $\Re(\tau)=0,... | https://mathoverflow.net/users/6953 | A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane | 1. Ahlfors explains that $\lambda(\Omega) = \mathbb H$, $\lambda(\Omega^\ast) = \mathbb H^\ast$ and $\lambda(\overline{\Omega} \backslash \Omega) = \mathbb R\backslash \{0,1\}$ (here $\overline{\Omega}$ is the closure of $\Omega$ in $\mathbb H$). Thus $\lambda$ maps $\overline{\Omega} \cup \Omega^\ast$, which is a fund... | 2 | https://mathoverflow.net/users/430 | 76590 | 46,372 |
https://mathoverflow.net/questions/76565 | 9 | Hi,
Is there an example of a proper smooth map of schemes $f:X\to Y$ and a vector bundle $E$ on $X$
such that $f\_\*E$ is not locally free on $Y$?
Thanks
| https://mathoverflow.net/users/36285 | pushforward of locally free sheaf is locally free? | Here is a an example, albeit with $Y$ non reduced:
Let $C$ be a smooth projective curve of genus $g > 0$ over a field $k$ and let $C\_{\epsilon} = C \times\_{k} Spec(k[\epsilon])$ where $k[\epsilon] = k[x]/(x^2)$ is the ring of dual numbers. Let $\mathcal{L}$ be a non-trivial line bundle on $C\_{\epsilon}$ such that ... | 5 | https://mathoverflow.net/users/519 | 76599 | 46,374 |
https://mathoverflow.net/questions/76608 | 3 | In his paper "Categories and cohomology theories" Graeme Segal considers the category of finite length chain complexes of finite dimensional vector spaces: Let $n = (n\_i)\_{i \in \mathbb{Z}}$ be a sequence of positive integers almost all zero. Then he claims that the space $K\_n$ of chain complexes $E$ with $E^i = \ma... | https://mathoverflow.net/users/3995 | Chain complexes of vector bundles | The space $K\_n$ sits inside the space of sequences of linear maps
$$L\_n = \Pi\_i Hom(E^i,E^{i+1}).$$
This is just a space of sequences of matrices, so it is a real vector space of dimension $\sum\_i (n\_i \cdot n\_{i+1})$. We give it the usual euclidean topology for real vector spaces.
The subspace $K\_n$ consists... | 7 | https://mathoverflow.net/users/4910 | 76610 | 46,379 |
https://mathoverflow.net/questions/76264 | 2 | Hello,
I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might relate to certain special functions.
Let
$$
G(t,x)=\frac{e^{-\frac{x^2}{2t}}}{\sqrt{2\pi t}}.
$$
The problem is
$$
\... | https://mathoverflow.net/users/36814 | A definite integral | Happy Birthday to Mathoverflow. Wish it flourish and thank many warmhearted people here for their helps! :-)
Here is one solution. Let
$$
G\_\sigma(t,x)=\frac{\exp(-\frac{x^2}{2\sigma t})}{\sqrt{2\pi \sigma t}}
$$
Clearly,
$$
\int\_0^t \frac{G\_\sigma(t-s,x)}{\sqrt{s}} d s =
\int\_0^t \frac{e^{-\frac{x^2}{2\... | 3 | https://mathoverflow.net/users/36814 | 76611 | 46,380 |
https://mathoverflow.net/questions/76616 | 4 | Let $k$ be a number field. Is it possible that $k$ has an infinite (non-abelian) extension that is unramified *everywhere*?
Thank you!
| https://mathoverflow.net/users/18116 | Unramified extensions of number fields | Two things:
1) Yes, certainly. By class field theory and the finiteness of the class group, the maximal *abelian* unramified extension of *any* number field is of finite degree. Thus any infinite unramified extension is non-abelian -- in particular, any infinite class field tower. The Golod-Shafarevich examples and ... | 15 | https://mathoverflow.net/users/35575 | 76617 | 46,382 |
https://mathoverflow.net/questions/76603 | 5 | Is it possible to build such an objective function for a given set of constraints, so that there will be **only one** optimal solution?
My general problem is to get **any vertex** of a polytope formed by a set of given linear constraints. I need this in polynomial time.
If I use the ellipsoid method, I'll get an op... | https://mathoverflow.net/users/10609 | Linear programming - uniqueness of optimal solution | A random objective will work. I don't think there is any (cheap) deterministic way of doing this. On the other hand, I don't really understand your issue with the ellipsoid method. Your solution will be on a lower-dimensional face of your polytope, so iterating your ellipsoid method at most $d$ times you will get a ver... | 5 | https://mathoverflow.net/users/11142 | 76618 | 46,383 |
https://mathoverflow.net/questions/76620 | 31 | ***A geometric way of looking at differential equations***
In the literature for the h-principle (for example Gromov's *Partial differential relations* or Eliashberg and Mishachev's *Introduction to the h-principle*), we often see the following (all objects smooth):
>
> Give a fibre bundle $\pi:F\to M$ over some ... | https://mathoverflow.net/users/3948 | Jet bundles and partial differential operators | If $\mathcal{R}\subset J^rX$ is closed, then there's a smooth function $f:J^rX\to\mathbb R$ with $\mathcal{R}=f^{-1}(0)$. So you can construct a differential operator $H:J^kX\to M\times \mathbb{R}$ by $H(\theta):=(\pi\_X(\pi^r\_0(\theta)),f(\theta))$ and the equation $\mathcal{R}$ will be given by $H(j^r\phi)=0$.
So ... | 12 | https://mathoverflow.net/users/745 | 76627 | 46,387 |
https://mathoverflow.net/questions/76636 | 1 | Let ${\cal P}(\mathbb{F}\_q^n)$ be the set of all subspaces of the vector space $\mathbb{F}\_q^n$ (where $q$ is a prime power).
Fix a $Z \in {\cal P}(\mathbb{F}\_q^n)$. Define a relation ~ on ${\cal P}(\mathbb{F}\_q^n)$ as follows:
$A$ ~ $B$ iff $A+Z = B+Z$
It is easy to show that this is an equivalence relatio... | https://mathoverflow.net/users/18080 | Counting the number of equivalence classes of subspaces | 1) It equals the number of subspaces of $\mathbb F\_q^n/Z$.
2) Fix a subspace $A$, denote its equivalence class by $[A]$. Consider the map $[A]\to\mathcal P(Z)$, $B\mapsto B\cap Z$. Its fiber over some $U\subseteq Z$ consists of all $B$ satisfying $B\cap Z=U$ and $B+Z=A+Z$. This set may be identified via $B\mapsto B/... | 2 | https://mathoverflow.net/users/2035 | 76645 | 46,395 |
https://mathoverflow.net/questions/76638 | 4 | Given a class function $f: G \to \mathbb Q$, where $G$ is a finite abelian group, is there an easy way to decide whether $f$ is an element of the rational representation ring $R\_{\mathbb Q}(G)$, i.e. whether $f$ is a virtual character of some representations of $G$?
If it makes things easier you could also assume th... | https://mathoverflow.net/users/18122 | When is a class function on a group G (finite abelian) into the rational numbers Q an element of the rational representation ring of G? | This essentially boils down to the case of a cyclic group. For a cyclic group of order n, the irreducible representations correspond to the action on $\mathbb Q[\omega\_d]$ where $\omega\_d$ is a primitive $d^{th}$-root of unity where $d$ divides $n$. So one can easily produce the rational character table and check if ... | 3 | https://mathoverflow.net/users/15934 | 76652 | 46,400 |
https://mathoverflow.net/questions/76600 | 1 | The group of three dimensional rotations $SO(3)$ is a subgroup of the Special Euclidean Group $SE(3) = \mathbb{R}^3 \rtimes SO(3)$. The manifold of $SO(3)$ is the three dimensional real projective space $RP^3$. Does $RP^3$ cause a separation of space in the manifold of $SE(3)$?
(edit) Sorry about lack of clarity. My... | https://mathoverflow.net/users/18115 | Does the manifold of the three dimensional group of rotations SO(3) cause a separation of space in the group of rigid motions SE(3)? | Okay, now I think I understand your question. This is the question I will answer:
* Question: Let $X$ be a connected $4$-dimensional subspace of $SE(3)$ that contains $SO(3)$. Is it possible for $X \setminus SO(3)$ to be connected? Disconnected?
The answer to both questions is yes. So there is no Jordan separation ... | 3 | https://mathoverflow.net/users/1465 | 76661 | 46,407 |
https://mathoverflow.net/questions/29995 | 3 | Let $pcf(a)$ denote the set of regular cardinals such that $J\_{\leq \lambda} - J\_{<\lambda} \neq \emptyset$ and let $maxpcf(a)$ denote the maximum of $pcf(a)$. The $J\_{\leq \lambda}$ are the usual ideals built up inductively in which you throw in all set on which you can have a scale mod $J\_{<\lambda}$
There is a... | https://mathoverflow.net/users/3859 | Some Pcf Theory | Hi. I know this is over a year late, but I the proof you're looking at matches that given on page 61-62 of "Cardinal Arithmetic". The argument can be finished along the following lines:
(1) First, we may as well assume $|\mathfrak{a}|^+<\min(\mathfrak{a})$, as we can derive the result you want if we get it in this mo... | 3 | https://mathoverflow.net/users/18128 | 76665 | 46,408 |
https://mathoverflow.net/questions/76667 | 2 | I was looking at the formula to compute the schubert class of a grassmanian in terms of a more elementary schubert cycles via giambelli's formula, and on the other hand Porteous formula tells us how to compute the determinacy locus of a homomorphism between two vector bundles i.e. when does thier rank drop from maximal... | https://mathoverflow.net/users/18129 | Giambelli and Porteous Formula | William Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math J. 65 (1992) 381--420
| 3 | https://mathoverflow.net/users/3077 | 76668 | 46,410 |
https://mathoverflow.net/questions/76670 | 14 | Consider the "infinite chessboard" on the plane. Think of it as the lattice $X\_1:=\mathbb{Z}^2$, and also finer chessboards $X\_n$ corresponding to $\frac{1}{n}\cdot \mathbb{Z}^2$, $n\geq 1$. Given two squares (i.e. vertices) $u,v$ of $X\_n$ one can define the "knight distance" $d\_n(u,v)$ as the minimum number of mov... | https://mathoverflow.net/users/4721 | What is the limit of the "knight" distance on finer and finer chessboards? | Let $(x,y)$ and $(x+2a,y+a)$ be points in space. Then clearly the distance between these two points is $a$. Therefore, the unit ball around 0 must contain the octagon with vertices $(2,1)$, $(1,2)$, $(-1,2)$ and so on.
I argue that this is all it contains. To see this, construct linear invariants showing how far you ... | 13 | https://mathoverflow.net/users/18060 | 76673 | 46,413 |
https://mathoverflow.net/questions/76663 | 5 | Let $X$ be a scheme. What technical hypotheses must be imposed on $X$ to assure that a point $p \in X$ is closed if and only if the 1-point set $\{p\}$ is constructible?
| https://mathoverflow.net/users/18127 | When are constructible points closed? | Let $X$ be locally noetherian. Then $\{x\}$ is constructible if and only if $\{x\}$ is locally closed (for non-noetherian schemes the notion of constructibility is more complicated and all kind of terrible things can happen, e.g. there exist closed points $x$ such that $\{x\}$ is not constructible). Moreover it is a ni... | 12 | https://mathoverflow.net/users/13302 | 76674 | 46,414 |
https://mathoverflow.net/questions/76631 | 12 | Classicly, for a spin Riemannian manifold $M$, the $\hat{A}(M)$ genus will be $0$, if the scalar curvature is positive.
The proof is to use the Lichnerowicz formula. we have the index of the Dirac operator will be $0$, i.e.,
$$Ind(D\_+)=0.$$
On the other hand, by the index theorem of Atiyah and Singer, we have
$$In... | https://mathoverflow.net/users/16326 | Vanishing of $\hat{A}$ genus and positive scalar curvature | After thinking about this question for two hours, my belief is strengthened that there cannot be a proof of the vanishing of the A-genus
for spin manifolds with positive scalar curvature without using both the index theorem for the Dirac operator and the Lichnerowicz formula (or other analytic techniques).
In the com... | 24 | https://mathoverflow.net/users/9928 | 76695 | 46,431 |
https://mathoverflow.net/questions/76691 | 3 | Today I came across the integral
$\int\_a^\infty e^{-bx} I\_n(x) dx$
where $I\_n$ is the modified Bessel function of the first kind. There is a solution for $a=0$, provided in Gradshteyn and Ryzhik, but I am afraid no closed-form solution exists otherwise. Correct me if I am wrong!
| https://mathoverflow.net/users/18134 | A Bessel integral | I suspect your fears are justified. You are basically trying to integrate the integrand from $0$ to $a$ (since, as you have noted, and Mathematica confirms, the integral from $0$ has a closed form) You are thus trying to evaluate the indefinite integral of a Bessel function times an exponential, which does not exist in... | 1 | https://mathoverflow.net/users/11142 | 76702 | 46,436 |
https://mathoverflow.net/questions/76709 | 5 | Hello;
We know that the space of riemannian metrics on a compact manifold is an open cone in the space of symmetric 2-tensors.
Is it reasonable to think that metrics with positive sectional curvature (even positive at a specific point $x \in M$) also form a convex cone?
This is a question about the local behaviour... | https://mathoverflow.net/users/12019 | Space of metrics with positive sectional curvature | No, the formula for curvature is nonlinear with respect to metric tensor in a very essential way.
In particular,
a convex combination of two positively curved metrics can have negative curvature.
In fact, arbitrary large negative sectional curvature.
**For example,** the induced metric on any embedding $\mathbb{S}^... | 18 | https://mathoverflow.net/users/1441 | 76716 | 46,442 |
https://mathoverflow.net/questions/76625 | 4 | Consider $d$ random variables. For each set of $k$ variables, we are given a joint probability distribution. We want to know that whether these distributions correspond to a valid joint probability distribution of all $d$ variables. We can assume that each variable has a finite domain.
I think a necessary condition i... | https://mathoverflow.net/users/17661 | There are d random variables. Given all k-D joint probability distributions with some k<d, what is the necessary and sufficient condition for these distributions to be feasible? | I asked myself the very same question some time ago. First, let me show that the obvious necessary condition is not sufficient.
Let $X\_1,Y\_1,Y\_2,Z\_2,Z\_3,X\_3$, be six random variables having the same non-deterministic law such that: $X\_1=Y\_1$, $Y\_2=Z\_2$ and $(Z\_3,X\_3)$ are independent. Then there cannot ex... | 1 | https://mathoverflow.net/users/4961 | 76740 | 46,454 |
https://mathoverflow.net/questions/76595 | 4 | The statement is:
($u$ is a fixed node in a fixed graph $G$)
$G$ is 3-connected
if and only if
the set of u-cycles span $\mathbb{R}^{E(G)}$.
A u-cycle is a simple (no vertex repetitions) cycle in G that contains the given node u.
A cycle is identified with its characteristic vector in $\mathbb{R}^{E(G)}$ that is ... | https://mathoverflow.net/users/11541 | Is this statement about the real edge space of a graph known or trivial? | (RESTART) Here is a complete solution.
Call a graph *3-edge-connected* if the number of components cannot be increased by removing only 1 or 2 edges. (This allows for the graph to be disconnected already.)
Let $R~$ be a ring that has an identity and a left inverse of 2, which I'll call $\frac12$. (For example, a fi... | 3 | https://mathoverflow.net/users/9025 | 76741 | 46,455 |
https://mathoverflow.net/questions/76671 | 1 | Given is a locally finite countable connected poset which satisfies further the following properties:
1. Let $C$ be any maximal chain ( i.e. inextendible chain) and $A$ be any antichain. Then $A$ is covered by both the sets ${\rm Past}(x)$ and ${\rm Future}(x)$ for $x$ running over $C$ i.e. $A \subset \bigcup\_{x \i... | https://mathoverflow.net/users/nan | Automorphisms of locally finite countable posets-2 | If I've understood the hypotheses correctly, the covering relation gives a connected locally finite directed graph: starting from a point $x$, the maximal elements underneath it form a finite antichain, as do the elements covering it. The automorphism group is therefore a second-countable totally disconnected locally c... | 2 | https://mathoverflow.net/users/4053 | 76744 | 46,457 |
https://mathoverflow.net/questions/76733 | 10 | Let $G$ be a finite group of order $n$ and denote by $\pi\_e(G)$ the set of element orders of $G$. What can be said about $G$ if $\pi\_e(G)$ forms a sublattice of the lattice of divisors of $n$?
| https://mathoverflow.net/users/17565 | A question on the set of element orders of a finite group | Let $G$ be a finite group, $n(G)$ the l.c.m. of orders of elements in $G$. Here are some obvious observations. A group $G$ belongs to your class $\mathcal C$ iff $G$ contains the cyclic group of order $n(G)$. Every $p$-group belongs to $\mathcal C$. The class is closed under direct products of groups with co-prime orde... | 13 | https://mathoverflow.net/users/nan | 76749 | 46,459 |
https://mathoverflow.net/questions/76722 | 3 | I am working with surfaces in Euclidean 3-space. If we let $X = X(u,v)$ denote a parameterization of such a surface, then the mean curvature, $H = H(u,v)$, can be computed in terms of the coefficients for the first and second fundamental forms.
My question is this: Is it possible to express the mean curvature, $H(u,... | https://mathoverflow.net/users/18145 | Support Function and Mean Curvature | The following holds in any dimension: If $h$ is the support function, then the quadratic form given by $\nabla^2h + hg$, where $g$ is the Riemannian metric on the unit sphere, is the inverse to the second fundamental form. Its eigenvalues (with respect to an orthonormal basis) are the principal radii (reciprocals of th... | 10 | https://mathoverflow.net/users/613 | 76753 | 46,462 |
https://mathoverflow.net/questions/76750 | 7 | I know that all compact Riemann surfaces with the same genus are topologically equivalent. Moreover they are diffeomorphic. But are they biholomorphic, too?
In other words, is the complex structure conserved?
| https://mathoverflow.net/users/18041 | Classification compact Riemann Surfaces | Some magic words for this question are "moduli space" or "moduli stack". In the early days, one was interested in a variety or variety-like object which would classify projective complex curves (compact Riemann surfaces) of given genus $g$, i.e., whose points correspond to isomorphism classes of curves (or biholomorphi... | 14 | https://mathoverflow.net/users/2926 | 76758 | 46,465 |
https://mathoverflow.net/questions/76754 | 2 | This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ (we can assume that $T$ is affine). Let $\mathbb{T}$ be a smooth lift of $T$ over the ring $W\_{2}(k)$, the ring of witt vec... | https://mathoverflow.net/users/18000 | Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius | This is parallel transport. Note that you only get an explicit formula for the isomorphism between $F\_1^\*H(\mathbb{T})$ and $F\_2^\*H(\mathbb{T})$. To get explicit formulas over other thickenings of $\mathbb{T}$, you would need lifts of both $\mathbb{T}$ and also of the Frobenius lifts. For convenience, set $H\_1=H(\... | 4 | https://mathoverflow.net/users/7868 | 76759 | 46,466 |
https://mathoverflow.net/questions/76763 | 2 | Hi. What type of 2n dimensional real symmetric matrices can be diagonalized with symplectic transformations (meaning M->SMS^T, S^T means transpose and S is an element of the 2n dimensional real symplectic group. Usually normal forms of the literature are given as representatives of orthogonal group orbits, but I need t... | https://mathoverflow.net/users/18157 | Families of quadratic Hamiltonians | A $2n\times 2n$ dimensional Hermitian matrix that can be diagonalized by a symplectic transformation can be viewed as an $n\times n$ matrix with elements consisting of $2\times 2$ blocks of the quaternion real form
${\bar{z}\;-\bar{w}}\choose{w\;\; z}$
so if you choose real $z$ and $w$ you have constructed a real ... | 0 | https://mathoverflow.net/users/11260 | 76765 | 46,468 |
https://mathoverflow.net/questions/76747 | 7 | There is a classic result of Baumslag which states,
Thm: If $G$ is residually finite then so is $\operatorname{Aut}(G)$.
While Grossman proved the (essentially) analogous result for $\operatorname{Out}(G)$,
Thm: If $G$ is conjugacy separable and every conjugating automorphism is inner then $\operatorname{Out}(G)$... | https://mathoverflow.net/users/6503 | Linking the residual finiteness of $G$ with $Aut(G)$ or $Out(G)$ | First, Baumslag's result is for finitely generated groups only. HW already says that essentially the Out of a residually finite group can be ``arbitrary". Now if you take a Tarski monster with trivial Out, then the direct product of it with the residually finite group above gives a non-residualy finite group with an ar... | 10 | https://mathoverflow.net/users/nan | 76770 | 46,472 |
https://mathoverflow.net/questions/76772 | 11 | Let $P$ be a prime ideal in the polynomial ring $K\left[x\_1,...,x\_m\right]$ and $Q$ be a prime ideal in the polynomial ring $K\left[y\_1,...,y\_n\right]$.
Is $P+Q$ a prime ideal in $K\left[x\_1,...,x\_m,y\_1,...,y\_n\right]$ ?
For example for $Q=\left(y\_1,...,y\_n\right)$, it is easy to prove that. But how abou... | https://mathoverflow.net/users/18161 | Is the sum of two prime ideals in different polynomial rings, K[X_i] and K[Y_i] a prime ideal in K[X_i Y_i]? | We have $K\left[x\_1,...,x\_m,y\_1,...,y\_n\right] \cong K\left[x\_1,...,x\_m\right] \otimes K\left[y\_1,...,y\_n\right]$ (where all tensor products are over $K$), and under this isomorphism, the ideal of $K\left[x\_1,...,x\_m,y\_1,...,y\_n\right]$ generated by $P+Q$ corresponds to $P\otimes K\left[y\_1,...,y\_n\right]... | 28 | https://mathoverflow.net/users/2530 | 76775 | 46,474 |
https://mathoverflow.net/questions/76778 | 1 | I'm reading Yao's unpredictability -> pseudorandomness construction
and Goldreich/levin's pseudorandom permutation -> pseudorandom generator construction.
My question is:
is there a direct way to show that:
given a pseudorandom function, we can construct a pseudorandom permutation out of it?
[or is this questio... | https://mathoverflow.net/users/18163 | Pseudorandom Functions / Pseudorandom Permutations | That would be the celebrated Luby Rackoff result.
| 2 | https://mathoverflow.net/users/18168 | 76789 | 46,482 |
https://mathoverflow.net/questions/76779 | 11 | In an interesting article (available [here](http://www-math.mit.edu/~tchow/closedform.pdf)), Timothy Chow proposes that a closed-form number be defined as an element of the smallest subfield of $\mathbb{C}$ that is closed under $\exp$ and a chosen branch of $\log$. It is fun to check that pretty much any number that yo... | https://mathoverflow.net/users/2926 | New results on Chow's notion of closed-form numbers? | There is also the recent paper by Borwein and Crandall, [Closed Forms: What they are and why we care"](http://carma.newcastle.edu.au/jon/closed-form.pdf), to appear in the *Notices of the AMS*. He gives 7 different methods via which one can approach closed forms. Chow's notion is #4. For some strange reason, I am rathe... | 6 | https://mathoverflow.net/users/3993 | 76796 | 46,485 |
https://mathoverflow.net/questions/76791 | 6 | The classical [de Moivre-Laplace theorem](http://en.wikipedia.org/wiki/De_Moivre-Laplace_theorem) states that we can approximate the normal distribution by discrete binomial distribution:
$${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)^2 / (2npq)}.$$
My question is: are there more precise, ... | https://mathoverflow.net/users/12898 | Quanitative de Moivre–Laplace theorem (reference request) | Firstly, I think by "qualitative" you mean "quantitative". Secondly, while there is a huge literature on the quantitative versions of the central limit theorem, the canonical results can be found in Feller's Vol 2. For the center of the distribution there is the Berry-Esseen theorem, for the tails there is the large de... | 3 | https://mathoverflow.net/users/11142 | 76799 | 46,486 |
https://mathoverflow.net/questions/76795 | 7 | One knows that many models of set theory exist. In topos theory,"the" category of sets is to play the role of the point. Since many models of set theory are around, I believe one of the following to be true.
1. There is one category of sets and the model determines what is true about the category of sets.
2. There ar... | https://mathoverflow.net/users/16801 | What is a category of sets? | Here is a thoroughly Platonist answer to your question: Both 1 and 2 are true. There is one category of sets. Its objects are all of the sets, and its morphisms are all of the functions between them. But there are many other categories that can be (and in fact have been) called the category of sets (by abuse of languag... | 16 | https://mathoverflow.net/users/6794 | 76803 | 46,489 |
https://mathoverflow.net/questions/76781 | 2 | for the Verma module $M(\lambda)$, it has a dual $M(\lambda)^{\vee}$,
also as $n^{-}$ module, $M(\lambda)$ isomorphic to $U(n^{-})$
so it is very nature to ask for the dual Verma module $M(\lambda)^{\vee}$ whether
there is such similar property, like as $n^{+}$ module ,whether there exists an iso??
and how to understa... | https://mathoverflow.net/users/15797 | about dual Verma module in BGG category O | As an $\mathfrak n ^+$ modules, all dual Vermas are isomorphic to $U(\mathfrak n^-)$ where we use the coadjoint action, identifying $\mathfrak n^-$ with the dual of $\mathfrak n ^+$ using the killing form.
| 6 | https://mathoverflow.net/users/66 | 76809 | 46,495 |
https://mathoverflow.net/questions/76588 | 7 | Here's a question from Shelah's book *Classification Theory*. Given an order $I$, we consider cuts of the form $(A,B)$ where for all $a \in A, b \in B$ we have $a < b$ and $A \cup B = I$. The cofinality of a cut $(A,B)$ is the pair $(\lambda, \kappa)$ where $\lambda$ is the cofinality of $A$ and $\kappa$ is the cofinal... | https://mathoverflow.net/users/18111 | Linear orders with only short cuts | I don´t think the statement is true as it is. If $J=\omega\_1+1$ then for any $I \supseteq J$ (of whatever cardinality) you can define a cut as: $x \in A$ iff $x \in I$ and $\exists \alpha \in \omega\_1 (x<\alpha) $ and $B=I \setminus A$. This cut can't have cofinality $(\omega, \omega)$.
| 4 | https://mathoverflow.net/users/17836 | 76812 | 46,496 |
https://mathoverflow.net/questions/76818 | 5 | Hi;
By a celebrated theorem of J.Nash, we know that any riemannian manifold with smooth enough metric tensor can be realised as an embedded submanifold of $\mathbb{E}^N$ for some $N$.
Can one hope to be able to embed (compact) manifolds into some space with constant curvature? If the answer is yes and we have the i... | https://mathoverflow.net/users/12019 | Embedding of riemannian manifolds into space forms | There is no bound on curvature, only on the dimension.
(The bound for the diameter of ambient space gives no bound for the intrinsic diameter of embedded space.)
In fact, it is straightforward to modify the proof of Nash's theorem to prove the following:
>
> *Any $n$-dimensional Riemannian manifold can be embedde... | 8 | https://mathoverflow.net/users/1441 | 76819 | 46,501 |
https://mathoverflow.net/questions/76419 | 1 | Let $F \rightarrow E\_i \rightarrow X\_i$ be a bundle with fibre $F$ for i=1,2.
Let $f:E\_1 \rightarrow E\_2$ be a bijective continuous map and $h: X\_1 \rightarrow X\_2$ a homeomorphism.
Is f then also a heomeomorphism?
If not, what further properties are needed for f to be a hemeomorphism?
| https://mathoverflow.net/users/18074 | When is a bijective map between bundles a homeomorphism? | It has been pointed out in the comments that this sort of thing cannot hold for arbitrary fiber bundles.
To follow up on euklid345's comment regarding vector bundles, there is a statement of this type for arbitrary principal bundles, assuming the appropriate definitions. There's a detailed discussion of this in my co... | 1 | https://mathoverflow.net/users/4042 | 76828 | 46,507 |
https://mathoverflow.net/questions/76838 | 6 | If a finite group G acts on a smooth variety X over complex number field and the fixed locus of G is smooth subvariety of codimension 1, will the resulting quotient variety be smooth? What will happen if the fixed locus has lager codimension ? thanks.
| https://mathoverflow.net/users/18181 | A question about quotient singularity | In general, for a finite group $G$ acting faithfully on a smooth variety $X$, whether or not the quotient is smooth is determined by the
[Chevalley-Shephard-Todd theorem:](http://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem)
For $x \in X$, let $G\_x\subset G$ be the stabilizer of $x$. Then a... | 9 | https://mathoverflow.net/users/519 | 76847 | 46,513 |
https://mathoverflow.net/questions/76835 | 1 | When thinking in terms of Dynkin diagrams, I am naively used to see that the diagram for a subgroup can be extracted from the diagram of the group by removing some roots. Now, I noticed that for SO(10) and its subgroup SU(4)xSU(2)xSU(2), no such removal happens, the Dynkin diagrams have the same number of nodes.
How ... | https://mathoverflow.net/users/4037 | subgroups with the same number of roots that the group. | Removing a random edge and keeping the nodes do not give you a subgroup. The subgroups you mention are obtained by adding one node (and a few edges) and then by removing an inner node. Perhaps you should see Borel-Siebenthal's paper on `maximal subgroups of maximal rank in compact Lie groups' for more on this process. ... | 9 | https://mathoverflow.net/users/9991 | 76848 | 46,514 |
https://mathoverflow.net/questions/76849 | 9 | A group $G$ is *locally cyclic* if whenever $H \le G$ is a finitely generated subgroup then $H$ is cyclic. If $G$ is a locally cyclic group then $G$ is isomorphic to a quotient of a subgroup of the rational numbers under addition. An online proof of this fact appears at [groupprops](http://groupprops.subwiki.org/wiki/L... | https://mathoverflow.net/users/7709 | Reference request: a locally cyclic group is isomorphic to a section of the rational numbers | For torsion-free groups it is proved in Kurosh, "Group theory", See 3d edition, Chapter VIII, Section 30 (of course the result can be found in the 1st edition as well). Oroginally it was proved in Reinhold Baer, "Abelian groups without elements of finite order". Duke Math J. 3 (1): 68–122, 1937. For groups with torsion... | 9 | https://mathoverflow.net/users/nan | 76856 | 46,520 |
https://mathoverflow.net/questions/76727 | 13 | I have been dabbling in learning basic things about probability theory and (of course) being of the school of abstract nonsense I have tried to understand things in its language. I apologize if this question is therefore somehow obvious.
As I understand it, if $X$ is a probability space with measure $\mu$ and $f \col... | https://mathoverflow.net/users/6545 | What are two independent, uniformly distributed random variables on the unit interval? | Let us consider more closely the question about space-filling curves. The [Peano curve](http://en.wikipedia.org/wiki/Space-filling_curve) and the [Hilbert curve](http://en.wikipedia.org/wiki/File%3aHilbert_curve.svg), and several other variations of them, have parametrizations $[0,1]\to[0,1]^2$ that actually take the 1... | 8 | https://mathoverflow.net/users/6101 | 76865 | 46,522 |
https://mathoverflow.net/questions/76864 | 5 | Hi,
It seems to be a common knowledge that the polynomials $x^n$ are dense in $L^2$ spaces with various probability weights, such as the gamma distribution weight $x^{\alpha-1}e^{-x}/\Gamma(\alpha)\;dx$.
>
> Is there any reference to this fact preferrably including the condition which property of the weight impl... | https://mathoverflow.net/users/979 | Polynomials are dense in weighted $L^2$ space | Sure, the beautiful book by N.I.Akhiezer *The Classical Moment Problem and Some Related Questions in Analysis*, where in particular you can find a thorough discussion of the property of the density of the polynomials in $L^1$ and $L^2$ for measures with finite moments of all order (together with sufficient conditions a... | 8 | https://mathoverflow.net/users/6101 | 76866 | 46,523 |
https://mathoverflow.net/questions/76844 | 0 | Hi,
just a short question on the theory of D-Modules:
if one has $p:X\times Y \rightarrow Y$ the projection of two smooth projective complex varieties to the second factor, then what is a $p^{\*}D\_Y$-Module?
Is this the same as a vectorbundle on $X\times Y$ with an integrable connection relative (!) to $X$?
| https://mathoverflow.net/users/18183 | Pullback of D-Modules | This was basically answered in the comments. A $p^\*D\_Y$-module is a vector bundle with an integrable relative connection on $X \times Y$ if and only if the underlying $\mathcal{O}\_{X \times Y}$-module is locally free, i.e., a vector bundle. In other words, if you are given a vector bundle on $X \times Y$, an action ... | 0 | https://mathoverflow.net/users/121 | 76867 | 46,524 |
https://mathoverflow.net/questions/76640 | 11 | Let D be an ample R-divisor, is the round down [mD] very ample for any sufficiently divisible number m?
I think it's true. But I do not know how to arrange an argument.
| https://mathoverflow.net/users/18119 | Is [mD] very ample if D is ample? | I am not sure if this is the shortest proof, but I think that it is a proof.
Let $A=$ very ample line bundle. After replacing D by a multiple, you may assume that
$$C=D - K\_X - (n+1) A$$ is ample where $n=\dim X$.
By Angehrn and Siu, we know that $$K\_X+(n+1)A + \text{(ample line bundle)}$$ is very ample.
No... | 6 | https://mathoverflow.net/users/15642 | 76887 | 46,531 |
https://mathoverflow.net/questions/76882 | 4 | Let $\zeta \in \overline{\mathbb{Q}}$ be a primitive 8th root of unity and let $K = \mathbb{Q}(\zeta)$. Let $N\_{K/\mathbb{Q}}$ be the corresponding norm.
Consider the three-dimensional $\mathbb{Q}$-torus defined by $N\_{K/\mathbb{Q}}(x) = 1$ for $x \in K^\times$. Is it rational over $\mathbb{Q}$?
I think the answe... | https://mathoverflow.net/users/1107 | Rationality of three-dimensional torus | The torus is not rational.
This is due to B.E. Kunyavskii, *On Tori with a biquadratic splitting field,* Izv. Akad. Nauk SSSR Ser. Mat 42 (1978), 580-587; English translation Math USSR-IZV. 12(1978), 536-542.
An easier to find reference is V.E. Voskresenskii's book *Algebraic Groups and their Birational Invariants... | 11 | https://mathoverflow.net/users/339 | 76891 | 46,533 |
https://mathoverflow.net/questions/76894 | 3 | If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its [injective tensor norm](http://en.wikipedia.org/wiki/Topological_tensor_product) to be
$$\|A\|\_{\rm inj} := \max\_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1}
|\langle A, x\otimes y\otimes z\rangle|.$$
Here the norm on vec... | https://mathoverflow.net/users/7718 | injective tensor norms for real tensors | I just found [this paper](http://dmle.cindoc.csic.es/pdf/RRACEFN_2000_94_04_05.pdf), which gives an example in which the real and complex version of the norm *are* different. The tensor in this example is also symmetric, which provides an example for part 2 as well.
| 2 | https://mathoverflow.net/users/7718 | 76902 | 46,538 |
https://mathoverflow.net/questions/76903 | 12 | In pondering
[this](https://mathoverflow.net/questions/58800/period-integrals-of-the-fiber-of-elliptically-fibered-k3-manifolds)
MO question and in particularly its 1st answer, and answers to
[this one](https://mathoverflow.net/questions/76307/a-k3-over-p1-with-six-singular-a-1-fibers) recently posed, I realized the... | https://mathoverflow.net/users/2906 | Dodecahedral K3? | There's a pretty K3 with icosahedral symmetry and 12 singular fibers each of type II (so a double root of the discriminant but with additive reduction); would that do? It's $y^2 = x^3 + P(t)$ where $P$ has icosahedral symmetry. Explicitly one can take $P(t) = t^{11} - 11 t^6 - t$. This surface is isotrivial (i.e. with ... | 17 | https://mathoverflow.net/users/14830 | 76905 | 46,540 |
https://mathoverflow.net/questions/76470 | 4 | I have a question regarding harmonic maps from all of ${\Bbb R}^2$ into a domain in ${\Bbb R}^2$. Before stating my question in full generality, let me ask a special case of the question first. Is it possible to find two non-constant harmonic functions $u$ and $v$ on ${\Bbb R}^2$ such that $u>v^3$ at every point? My gu... | https://mathoverflow.net/users/15234 | Harmonic functions on the plane | The answer is, indeed, negative. WLOG $v(0)=0$. Take the intersection of the region $-A<v<2A$ with a huge disk (more precisely, take the connected component $\Omega$ of this intersection containing the origin). It is simply connected by the maximum principle. The nice thing about the plane is that once we have a curve ... | 8 | https://mathoverflow.net/users/1131 | 76933 | 46,549 |
https://mathoverflow.net/questions/76206 | 4 | Let $K$ be a number field with ring of integers $O\_K$. Is there a section of $\mathbf{P}^1\_{O\_K}$ over $O\_K$ whose image is disjoint from $0$, $1$ and $\infty$? If $K=\mathbf{Q}$ this is not possible because any integer $n>1$ is divisible by a prime number. What if $K \neq \mathbf{Q}$?
| https://mathoverflow.net/users/18023 | Is there a section disjoint from 0, 1 and infinity on the projective line | Such sections are tantamount to solutions of the unit equation $u + u' = 1$ in $O\_K^\*$. This is indeed impossible for $K = {\bf Q}$, when $O\_K = {\bf Z}$ and the only units are $\pm 1$; but there can be such solutions for other number fields $K$, though it is known that in each $K$ there are only finitely many solut... | 15 | https://mathoverflow.net/users/14830 | 76963 | 46,567 |
https://mathoverflow.net/questions/76971 | 17 | Ralph Cohen (professor at Stanford) is teaching a class on algebraic topology and moduli spaces this quarter, beginning by reviewing his perspective of Morse theory. He defined "nice" metrics, proved that they are dense in the $L^2$ space of metrics on $\mathbb R^n$, proved one result using that, and doesn't need them ... | https://mathoverflow.net/users/1198 | ``Nice'' metrics for a Morse gradient field: counterexample request | Assuming Giuseppe's suggested correction is right, here's what you have to worry about: Consider the function $f(x,y) = \tfrac12 x^2 + y^2 + x^2 y$ and the metric $g = (1+2y)\ dx^2 + dy^2$ on the half-plane $y > -\tfrac12$. You can compute that
$$
\nabla f = x\ \frac{\partial\ }{\partial x} + (2y+x^2)\ \frac{\partial\... | 25 | https://mathoverflow.net/users/13972 | 76983 | 46,573 |
https://mathoverflow.net/questions/76978 | 2 | In an excerpt of an article by Bernd Sturmfels, I found:
**Theorem 5.5.** The tropical Grassmannian $G^{′}\_{2,n}$ is a simplical complex known as the space of phylogenetic trees.... It is denoted by $T\_n$ and is defined as follows. The vertex set consists of all unordered pairs $\left \{ A,B \right \}$ where $A$ a... | https://mathoverflow.net/users/12178 | "Face" numbers for tropical Grassmannian G′_2,7 simplical complex ? | You have the right numbers. $\mathrm{Trop} \ G(2,n)$ is the space of phylogenetic trees studied in [Billera-Holmes-Vogtman.](http://www-stat.stanford.edu/~susan/papers/lap.pdf) It has $1 \times 3 \times 5 \times \cdots \times (2n-5)$ maximal faces and $2^{n-1} - n-1$ vertices, matching your $945$ and $56$.
The first... | 7 | https://mathoverflow.net/users/297 | 76984 | 46,574 |
https://mathoverflow.net/questions/76985 | 6 | If G is a discrete cofinite volume subgroup of PSL(2,C),then G acts on H3, H3/G is a 3-dim hyperbolic orbifold N with finite volume, my question is : Is it right in most situations that we can find a hyperbolic 3 manifold M as a finite covering space of N?
This question is equivalent to the following :
do most dicrete ... | https://mathoverflow.net/users/18181 | Can most 3 dimensional hyperbolic orbifolds with finite volume be covered by a hyperbolic manifold? | Yes, this is true for all of them. Any finitely generated matrix group has a torsion-free subgroup of finite index; this is the so-called "Selberg's lemma". A canonical source is Ratcliffe's Hyperbolic Manifolds book (you can probably find the relevant section on google books for free, or on gigapedia.com if you are so... | 8 | https://mathoverflow.net/users/11142 | 76987 | 46,576 |
https://mathoverflow.net/questions/77029 | 11 | Hello,
Here is an interesting problem. It looks elementary, but it has taken me some efforts without solving it. Let
$$
h(x) = e^{x^2/2} \Phi(x),\qquad \text{with}\quad \Phi(x):=\int\_{-\infty}^x \frac{e^{-y^2/2}}{\sqrt{2\pi}} dy.
$$
The question is whether the function $h(x)$ is monotone increasing over $R$? Are... | https://mathoverflow.net/users/36814 | Is the function $e^{x^2/2} \Phi(x)$ monotone increasing? | We can write $h(x)=\frac 1{\sqrt{2\pi}}\int\_{-\infty}^x \exp\left(\frac{x^2-y^2}2\right)dy$. Now put $t=x-y$. We get
\begin{align}
h(x)&=\frac 1{\sqrt{2\pi}}\int\_0^{+\infty}\exp\left(\frac{x^2-(x-t)^2}2\right)dt\\\
&=\frac 1{\sqrt{2\pi}}\int\_0^{+\infty}\exp\left(xt-\frac{t^2}2\right)dt.
\end{align}
We can different... | 35 | https://mathoverflow.net/users/17118 | 77030 | 46,598 |
https://mathoverflow.net/questions/77032 | 1 | We know that $GF(p^c)$ is a subfield of $GF(p^{cn})$. Also we know that elements in $GF(p^c)$ can be represent by degree $c$ polynomials with coefficients in $\mathbb Z\_p$, where multiplication is done by usual polynomial multiplication modulo a degree $c$ irreducible polynomial $p$.
The question is, given the repre... | https://mathoverflow.net/users/17016 | Finding an embedding efficiently in field extension of finite field | H. Lenstra Finding isomorphisms between finite fields, Math. Comp. 56 (1991), 329–347.
BTW $O(polylog\ {p^{cn}})$ because the answer typically won't be sparse.
| 3 | https://mathoverflow.net/users/2290 | 77033 | 46,600 |
https://mathoverflow.net/questions/77037 | 4 | Let $G$ be a group (usually infinite), $R$ a ring and $\rho: G \rightarrow Gl\_n(\mathbb{Z})$ a finite-dimensional representation of $G$. Then we can define a functor from the category of projective $RG$-modules to itself by sending a projective module $P$ to $\mathbb{Z}^n \otimes\_{\mathbb{Z}} P$ with the diagonal act... | https://mathoverflow.net/users/18256 | Unitary representation acting on the K-theory of the reduced group $C^*$-algebra | Even more is true: denote by $R(G)$ the ring of homotopy classes of finite-dimensional representations of $G$ (not necessarily unitary ones); then there is a module action of $R(G)$ on $K\_\*(C\_r^\*G)$. See my memoir: ``Les fibr\'es en th\'eorie de Kasparov'', Acad. Royale de Belgique, M\'emoire Classe des Sciences, 2... | 6 | https://mathoverflow.net/users/14497 | 77041 | 46,603 |
https://mathoverflow.net/questions/77043 | 8 | Suppose that $\kappa$ is a regular cardinal and let $NS$ be the ideal of its nonstationary subsets. One can consider the Boolean algebra $P(\kappa) /NS$ and say that (if $\lambda$ is another cardinal) $NS$ is $\lambda$ saturated iff there are no antichains in $P(\kappa) / NS$ of length $\lambda$. It is an elegant resul... | https://mathoverflow.net/users/4753 | A result of Shelah about the nonstationary ideal | Try also Chapter XVI of "Proper and Improper Forcing" (entitled "Large ideals on $\aleph\_1$ from smaller cardinals"). It's hard to tell exactly what's in there, but he does say in the chapter he will "keep old promises from 84-85 mentioned in [Sh:253]", where [Sh:253] is the paper Michael mentions, and he does claim t... | 7 | https://mathoverflow.net/users/18128 | 77049 | 46,605 |
https://mathoverflow.net/questions/77040 | 3 | Let us say that I have a complex abelian variety $A$, an ample line bundle on $A$, $L$, and an effective divisor $E\in|L|$. It is well known that it exists an isogeny $\varphi:A\rightarrow B$ and a principal polarization $M$ on $B$ such that $L\simeq \varphi^\*M$. My question is: can we say something about the divisor ... | https://mathoverflow.net/users/6949 | The image of divisor under isogeny | There are different possible behaviours.
Here's an example: let $B$ be a ppav, $\Theta$ an effective theta divisor, let $f\colon A\to B$ be a connected e'tale double cover given by an element $\eta\in Pic^0(B)[2]$ and set $E=f^\*\Theta$.
Of course $E\to \Theta$ is 2-to-1 by construction.
By the projection formula ... | 6 | https://mathoverflow.net/users/10610 | 77053 | 46,607 |
https://mathoverflow.net/questions/77044 | 8 | Needed for [this](https://arxiv.org/abs/1204.6506) paper:
Here is a possibly more clear version of my question. A Turing machine (with $1$ tape) has sets of tape letters $Y$, state letters $Q$, two symbols $\alpha$ and $\omega$ that mark the ends of the tape and a set of commands $\Theta$. A configuration is any word... | https://mathoverflow.net/users/nan | Turing machines that always halt | Jean-Camille Birget answered my question. These are called universally halting Turing machines.
The oldest reference is:
Martin Davis (1956). A note on universal Turing machines. In Shannon,
C. E., McCarthy, J., eds, Automata Studies, pp. 167-175. Princeton
University Press.
Birget proved a complexity version o... | 10 | https://mathoverflow.net/users/nan | 77076 | 46,614 |
https://mathoverflow.net/questions/77024 | 2 | Let $a$ be a closed point in $\mathbf{P}^1\_{\overline{\mathbf{Q}}}$.
Let $Y \cong \mathbf{P}^1\_{\overline{\mathbf{Q}}} $ and let $\pi:Y\to \mathbf{P}^1\_{\overline{\mathbf{Q}}}$ be a finite morphism which is unramified above $a$. Let $b$ be a point in $\pi^{-1}(a)$.
Can one effectively bound the height of $b$ in... | https://mathoverflow.net/users/18253 | Comparing heights of rational points on curves through covers | Let me reformulate the problem in less fancy terms. Let $\pi(x)\in\overline{\mathbb{Q}}$ be a rational function of degree $d\ge1$. As ACL noted, there is a standard estimate
$$
dh(y) - c\_1(\pi) \le h(\pi(y)) \le dh(y) + c\_2(\pi), \quad (\*)
$$
where it is relatively easy to give explicit formulas for $c\_1$ and $c\_... | 3 | https://mathoverflow.net/users/11926 | 77087 | 46,620 |
https://mathoverflow.net/questions/77085 | 0 | I am not a mathematician but out of curiosity I am trying to implement the [SIS epidemic model](http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIS_model) when the nodes have mobility to understand how it will change the results. I understand how to perform this simulation in an analytical fashion... | https://mathoverflow.net/users/3560 | How are epidemic models simulated in case of mobility? | You are free to define the parameters as you wish, as long as you document it. But in the standard (non-spatial) SIS model the rate at which new infections occur is $\beta I S$, i.e. each infective individual infects each susceptible individual at a rate $\beta$. This would correspond in your model to a situation where... | 2 | https://mathoverflow.net/users/13650 | 77090 | 46,621 |
https://mathoverflow.net/questions/77065 | 1 | I seem to recall that the construction of Gal representations associated to eigenforms with CM was done much before the general cases due to Eichler-Shimura, Deligne-Serre and Deligne. Was this done by Hecke? Can someone give the exact reference? Thanks.
| https://mathoverflow.net/users/5310 | Reference for "Gal represenations attached to CM eigenforms" | The idea of Galois representations attached to modular forms is one which evolved over the
course of the 1960s, as far as I know. One should look at various papers of Serre from that
time (available e.g. in his collected works), as well as his book on abelian $\ell$-adic representations, to get a feeling for how the s... | 7 | https://mathoverflow.net/users/2874 | 77095 | 46,624 |
https://mathoverflow.net/questions/77100 | 1 | Suppose $f\colon X \rightarrow Y$ is a continuous map of topological spaces and $s\colon Y \rightarrow X$ is a continuous section to $f$, i.e., $f\circ s = 1$. If $f$ is proper does this mean that $s$ is proper as well? (A continuous map is proper if the preimage of any compact set is compact.)
This is true for schem... | https://mathoverflow.net/users/18271 | Is a section of a proper map proper? | Not in general. As Mariano stated in the comments, it's true if $X$ is $T\_2$. Here's a counterexample when $X$ is only $T\_1$:
Let $X$ be the space obtained by gluing two closed unit intervals (call them $I\_1,I\_2$) along some open subinterval. Let $Y$ be a closed unit interval, $f:X\rightarrow Y$ the obvious quoti... | 7 | https://mathoverflow.net/users/5513 | 77104 | 46,627 |
https://mathoverflow.net/questions/77070 | 9 | Here are some questions about the earthquake deformation of hyperbolic surface that I can't answer or find references.
I briefly recall the settings. Let's fix a closed surface $S$ with genus $g\geq 2$. A point $h$ in the Teichmuller space $\mathscr{T}$ of $S$ may be thinked of
either (a) as a marked hyperbolic stru... | https://mathoverflow.net/users/17294 | Questions on Thurston's earthquake flow | On Q1 I would guess that you can find explicit deformations corresponding to an earthquake path on a non-simple lamination in the quite special case of the punctured torus. You might find some help for instance in
MR0697067 (85d:32047)
Waterman, Peter; Wolpert, Scott
Earthquakes and tessellations of Teichmüller spac... | 5 | https://mathoverflow.net/users/9890 | 77105 | 46,628 |
https://mathoverflow.net/questions/77057 | 5 | Hi All,
Suppose I've a symmetric matrix $A\_{N \times N} = (A\_{ij})$ which has a eigen value decomposition $A = UDU'$. I would like to know under what conditions $\frac{\partial U}{\partial A\_{ij}}$ exists for all $i,j = 1,2, \ldots, N$. I found the following paper which talks about estimating the Jacobian of the S... | https://mathoverflow.net/users/18036 | partial Derivatives of Eigen value decomposition or Singular value decomposition | To make Igor's more precise, Kato's book tells us that
1. if an eigenvalue of a matrix $A$ is simple, then it extends as an analytic function $M\mapsto\lambda(M)$ defined in a neighbourhood of $A$, such that $\lambda(M)$ is an eigenvalue of $M$.
2. if $s\mapsto A(s)$ is an analytic, one-parameter, family of real symm... | 5 | https://mathoverflow.net/users/8799 | 77108 | 46,629 |
https://mathoverflow.net/questions/77071 | 5 | Assume we are workling on $\mathbb{P}^n$ for some $n\geq 1$ and we have a coherent sheaf $F$ on it.
Then there are two (well known?) spectral sequences $E\_r^{p,q}$ with $E\_1$-term:
$E\_1^{p,q}=H^q(\mathbb{P}^n,F(p))\otimes \Omega^{-p}(-p)$
$E\_1^{p,q}=H^q(\mathbb{P}^n,F\otimes \Omega^{-p}(-p))\otimes O\_{\math... | https://mathoverflow.net/users/3233 | Generalized Beilinson spectral sequences | The answer depends very strongly on your algebra $R$. For example, a particular case is when $R = O + L$ (where $L$ is a line bundle) and the multiplication is given by a map $L^2 \to O$ (given by a divisor $D$) the category $Coh(P^n,R)$ is equivalent to $Coh(X)$, where $X$ is the double covering of $P^n$ ramified in $... | 2 | https://mathoverflow.net/users/4428 | 77109 | 46,630 |
https://mathoverflow.net/questions/77111 | 2 | Assume one has a finitedimensional vector space $V$ over the complex numbers and a discrete subgroup $G$ of translations on $V$, so that the quotient is a complex manifold $X=V/G$.
My question: can one express the cohomology group $H^1(X,\mathcal{O\_X})$ in terms of the data $V$ and $G$?
| https://mathoverflow.net/users/18275 | Cohomology of a quotient manifold | This answer has been corrected per Torsten's remark.
If $V/G = Y$ is a complex torus, i.e. $G$ has maximal rank, then Hodge symmetry says $H^1(Y,\mathcal O) = \overline{H^0(Y,\Omega)}$. But the cotangent bundle of $Y$ is trivial by translation on the group, so $H^0(Y,\Omega) = H^0(Y,\mathcal O) \otimes (T\_0 Y)^\ast$... | 2 | https://mathoverflow.net/users/1310 | 77115 | 46,632 |
https://mathoverflow.net/questions/77089 | 10 | Fix a field $k$. For a singular variety $X$, I understand that the Grothendieck group $K^0(X)$ of vector bundles on $X$ is not necessarily isomorphic to the Grothendieck group $K\_0(X)$ of coherent sheaves on $X$.
I am curious to learn what is known about these two groups in one family of examples: $\mathbb P^n\_{D}... | https://mathoverflow.net/users/4 | Grothendieck group for projective space over the dual numbers | If $X$ is a noetherian separated scheme and $X\_{red}$ its reduction , we have $K\_0(X)=K\_o(X\_{red})$: in other words $K\_o$ doesn't see nilpotents .
Much more generally and profoundly, Quillen has proved that for all his $K$-theory groups, $K\_i(X)=K\_i(X\_{red})$.
In your particular case you thus have (in the... | 9 | https://mathoverflow.net/users/450 | 77125 | 46,637 |
https://mathoverflow.net/questions/77046 | 5 | Is it possible to embed de Rham cohomology of a two-dimensional closed surface of genus $g\geq 2$ into the differential graded algebra of differential forms (with de Rham differential and wedge product) on the surface as a differential graded subalgebra (in a way that is compatible with canonical projection from closed... | https://mathoverflow.net/users/18260 | Formality of de Rham algebra for two-dimensional closed surfaces | The answer is *no*. Suppose that $\alpha\_1,\ldots,\alpha\_g,\beta\_1,\ldots,\beta\_g$ were closed $1$-forms on $M$ such that their cohomology classes were a basis of $H^1(M)$ and they satisfied $\alpha\_i\wedge\alpha\_j = \beta\_i\wedge\beta\_j = 0$ while $\alpha\_i\wedge\beta\_j = \delta\_{ij}\ \gamma$ where $\gamma$... | 14 | https://mathoverflow.net/users/13972 | 77127 | 46,639 |
https://mathoverflow.net/questions/77119 | -1 | For SU2 and even SU2(q) the triangle condition is, well, the triangle
condition (conveniently, all irreps are described by (half)integer J completely).
Additionally, all three J of a triple must add to integer.
But what is the analogue to that for some arbitrary (quantum) group?
As usual, I don't have a clue but an... | https://mathoverflow.net/users/11504 | Triangle condition for quantum 6j symbols? | As you probably guessed the answer is "its complicated." I guess you mean what is the rule for when the tensor product of two irreps $V\_\lambda \otimes V\_\gamma$ contains an irrep $V\_\delta$. The answer involves identifying the irrep with its highest weight vector $\lambda$ and viewing it in the Weyl chamber. Humphr... | 6 | https://mathoverflow.net/users/18283 | 77134 | 46,640 |
https://mathoverflow.net/questions/77130 | 3 | Let $f(x,y)$ be a complex degree $d$ polynomial that has this particular
form.
$$ f = \frac{f\_{02}}{2} y^2 + \frac{f\_{21}}{2} x^2 y +
\frac{f\_{12}}{2} x y^2 + \frac{f\_{03}}{6} y^3 + \frac{f\_{40}}{24} x^4+ \ldots $$
This polynomial $f$ can be thought of as an element of $\mathbb{C}^{M\_d}$, where
$M\_d = \fra... | https://mathoverflow.net/users/4463 | Closure of singular points | If I understand correctly, you ask what can be the results of the collision of two singular points, of $A\_4$ and $A\_1$ types. In general there does not seem to exist an ultimate effective method to treat such questions. Only in some simple cases, for example in this case.
A somewhat simpler question is: given a poi... | 4 | https://mathoverflow.net/users/2900 | 77144 | 46,644 |
https://mathoverflow.net/questions/77152 | 6 | Let X be CW complex. I'm trying to prove (using Zorn's lemma) that there is maximal contractible subcomplex. Problem is that I'm not able to show that increasing union of contractible subcomplexes has to be contractible itself.
| https://mathoverflow.net/users/18291 | Increasing union of contractible CW complexes | By various standard lemmas, a CW complex $X$ is contractible if and only if every map $u:S^{n-1}\to X$ (for any $n>0$) can be extended over $B^n$. In this context $u(S^{n-1})$ is compact and therefore (by another standard lemma) contained in some subcomplex with only finitely many cells. If $X$ is the union of some tot... | 14 | https://mathoverflow.net/users/10366 | 77155 | 46,647 |
https://mathoverflow.net/questions/77007 | 5 | Questions
---------
1. Is there a version of the Furstenberg-Zimmer Theorem for
non-invertible measure preserving systems?
2. Where can I find it?
3. What is the precise statement?
Background
----------
In many works that reference the Furstenberg-Zimmer Theorem,
the theorem itself is not stated.
Authors usually ... | https://mathoverflow.net/users/18191 | Furstenberg-Zimmer theorem: non-invertible systems | Posted as requested - consult the book by Manfred Einsiedler and Tom Ward - "Ergodic Theory with a view towards number theory" - published in GTM, especially in ch 7.
| 3 | https://mathoverflow.net/users/8857 | 77160 | 46,648 |
https://mathoverflow.net/questions/77122 | 1 | Let $L$ be a pseudo-effective divisor, we may define its numerical fixed part $N\_{\sigma}(L)$. How to prove it is a divisor? I know there is a proof in Nakayama's book, but I can't find this book.
| https://mathoverflow.net/users/18119 | How to prove the existence of divisorial Zariski decomposition? | By definition, you first define $\sigma\_\Gamma(L)$ for big divisors and then you take the limit.
In other words, if $L$ is big, then clearly $\sigma\_\Gamma(L)$ is non zero for only finitely many divisors. Indeed, $L=A+B$ with $A$ ample $\mathbb Q$-divisor, and $B\ge 0$. Thus $\sigma\_\Gamma(L)>0$ implies $\Gamma$ i... | 3 | https://mathoverflow.net/users/15642 | 77164 | 46,651 |
https://mathoverflow.net/questions/77088 | 1 | Let $X = \{x\_1, \dots, x\_n\}$ denote a finite set of $n$ points in the unit square $S$, and let's center $S$ at the origin. Let $F(X) = \sum\_{i=1}^n \| x\_i \| $ and let $G(X) = \iint\_S \min\_i \|x - x\_i\|~ dA $ be the average distance between a uniformly sampled point in $S$ and its nearest neighbor in $X$. Clear... | https://mathoverflow.net/users/17860 | Distances between and among points in a region | One can show that $F(X)> c\cdot G(X)^{-2}$ for some $c>0$, provided that $G(X)$ is sufficiently small.
Let $G(X)=\varepsilon$. Divide the square into $\approx(100\varepsilon)^{-2}$ small squares of size $100\varepsilon\times 100\varepsilon$. At least 9/10 of these squares must contain points of $X$. Indeed, if a $100... | 3 | https://mathoverflow.net/users/4354 | 77166 | 46,653 |
https://mathoverflow.net/questions/77107 | 6 | Let $G$ be a connected simply-connected Lie group correspondence to simple Lie algebra $g$,consider the loop algebra $g((t))$ and so called "Natural Borel subalgebra" $n[t,t^{-1}]\oplus h[t]$,denoted by $\mathfrak{b}$ and consider ind-group $G((t))$ associated to $g((t))$ and ind group $N\_{-}((t))$ and $H[[t]]$ corres... | https://mathoverflow.net/users/1851 | Several questions on semi infinite flag manifold | Until you get a better answer this may help you. As far as I understand the semi-infinite flag manifold appeared in Feigin-Frenkel's paper [1] where they weren't really defined as an algebro-geometric object but rather they constructed what morally should be called some sheaves on them. Since both $G(K)$ and $N\_-(K)\c... | 9 | https://mathoverflow.net/users/17980 | 77168 | 46,654 |
https://mathoverflow.net/questions/77175 | 66 | In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation never really translates into a formal definition - we instead approximate the tangent line by secant lines.
I seem to have... | https://mathoverflow.net/users/1106 | Taking "Zooming in on a point of a graph" seriously | In algebraic geometry, this construction is known as the *tangent cone* to the graph. More generally, suppose we have the zero set of any polynomial $f(x,y) = 0$, and assume $f(0,0)=0$. Then we can write
$f(x,y) = a\_m (x,y) + a\_{m+1}(x,y) +a\_{m+2}(x,y) +\cdots$
where $a\_i(x,y)$ is a homogeneous polynomial of de... | 33 | https://mathoverflow.net/users/7399 | 77188 | 46,661 |
https://mathoverflow.net/questions/77194 | 2 | Consider the automorphism of the algebra $U\_q(\widehat{\mathfrak{sl}}\_n)$ induced by the obvious diagram automorphism of the extended type A Dynkin diagram. More precisely, if the vertices of the Dynkin diagram are labelled $0,1,\ldots,n-1$, define $\overline{i}$ to be the number which is congruent to $-i$ modulo $n$... | https://mathoverflow.net/users/15632 | Fixed points of quantised enveloping algebra for affine $\mathfrak{sl}_n$ | I'm not sure that I know a proof that there's no way to write that algebra as a quantized universal enveloping algebra, but it's definitely not the QUEA of $(\mathfrak{\widehat{sl}}\_n)^\sigma$, the fixed points on the Lie algebra (since it's not generated by the elements you expect). Even worse, it doesn't even contai... | 2 | https://mathoverflow.net/users/66 | 77198 | 46,664 |
https://mathoverflow.net/questions/77128 | 9 | **Problem**
The Weierstrass function $W(x)$ is given by
$W(x)=\sum\_{n\geq 0} a^n \cos(b^n \pi x)$
where $0< a <1$ and $b$ is an odd integer such that $ab > 1+3\pi/2$.
A function $f:\mathbb{R}\rightarrow \mathbb{R}$ is said to have a point of increase if there exists a $t \in \mathbb{R}$ and $\delta>0$ such tha... | https://mathoverflow.net/users/18279 | Does the Weierstrass function have a point of increase? | The original proof of Weierstrass (see pages 4 to 7 in Elgar (ed.): Classics on Fractals, Westview Press, 2004) constructs, for any $x\_0\in\mathbb{R}$, two sequences $(x'\_n)$ and $(x''\_n)$ such that
$$x'\_n < x\_0 < x''\_n,\qquad x'\_n\to x\_0,\qquad x''\_n\to x\_0,$$
but
$$\frac{W(x'\_n)-W(x)}{x'\_n-x}\qquad\text... | 10 | https://mathoverflow.net/users/11919 | 77200 | 46,665 |
https://mathoverflow.net/questions/77211 | 5 | I would like to explicitly compute the limit of a family of stable maps in $\overline{M}\_{0,n}(\mathbb{P}^r,d)$. I know in principle how this works for families of curves without maps as I found a lot of literature on stable reduction for this case, like Harris and Morrison's "Moduli of Curves". Is there any literatur... | https://mathoverflow.net/users/18305 | Stable reduction for maps | I don't know any literature for how to do this, but I think I can do this example by hand in an ad hoc way.
As you say you have to blow up $\mathbb C \times \mathbb P^1$ in the two points you gave, so the special fiber becomes a chain of three $\mathbb P^1$s, with a marked point on each component. On the middle one o... | 4 | https://mathoverflow.net/users/1310 | 77215 | 46,670 |
https://mathoverflow.net/questions/77196 | 3 | Suppose $T: X \to X$ is a continuous map
and $\mu$ a $T$-ergodic probability measure over the
Borel sets of $X$.
Now, suppose $K \subset \mathrm{Hom}(X)$ is a compact group
of measure-preserving homeomorphisms of $X$ commuting with $T$.
Consider the factor $Y = K \backslash X = \{Kx | x \in X\}$.
Then, defining $\hat{T... | https://mathoverflow.net/users/18191 | Compact group extension of a zero entropy system. | [I'm assuming G means K]
In case the elements of K are not measure preserving, I doubt that the question is correct.
In general, one can easily show that entropy decrease by moving into factors.
Here's an example.
Take $T^{1}$ to be Z\R, and the times 2 map.
Take a nice measure supported on suitable Cantor set with... | 1 | https://mathoverflow.net/users/8857 | 77221 | 46,674 |
https://mathoverflow.net/questions/76723 | 11 | Let $[n]:=\lbrace 1, \dots, n \rbrace$. We define a partial ordering on the set of subsets of $[n]$ as follows. We say that $X \preceq Y$ if there is an injective map $f:X \to Y$ such that $x \leq f(x)$ for all $x \in X$. This is a pretty standard construction in poset theory.
The motivation for this question comes ... | https://mathoverflow.net/users/2233 | What is the size of a largest antichain in this poset? | Calling this poset $M(n)$, the fact that it has the [Sperner property](http://en.wikipedia.org/wiki/Sperner_property_of_a_partially_ordered_set) was conjectured in B. Lindström, "A conjecture on a theorem similar to Sperner's", Combinatorial Structures and Their Applications, p. 241.
It turns out that $M(n)$ has the ... | 16 | https://mathoverflow.net/users/2384 | 77222 | 46,675 |
https://mathoverflow.net/questions/77242 | 8 | What are the necessary conditions for two of the terms in the Pythagorean triplet $a^2 = b^2 + c^2$ to be prime numbers?
| https://mathoverflow.net/users/18229 | prime numbers and Pythagorean triplets | There is a well-known parametrization of Phythagorean triples as $k(m^2 - n^2)$, $2kmn$ , $k(m^2 + n^2)$ with positive integers $k,m,n$ and $m$ greater $n$.
Now, if two are prime we get $k=1$. And also the middle term is never prime.
So the question is when are $m^2 - n^2$ and $m^2 + n^2$ both prime.
The former fac... | 20 | https://mathoverflow.net/users/nan | 77247 | 46,687 |
https://mathoverflow.net/questions/77244 | 8 | Hi,
In Mumford's "Red Book of Varieties and Schemes", Chapter III, Paragraph 10 (entitled "Flat and smooth morphisms"), the following property is stated:
Let $M$ be a $B$-module, and $B$ an algebra over $A$. Let $f\in B$ have the property that for all maximal ideals $m \subset A$, multiplication by $f$ is injective... | https://mathoverflow.net/users/2095 | Quotient of flat module is flat - a property in Mumford's Red book | This is false without finiteness conditions: let $k$ be a field, $A=k[X,Y]$, $B=A\_{(X)}$, $M=B$, $f=X$.
| 6 | https://mathoverflow.net/users/2035 | 77250 | 46,689 |
https://mathoverflow.net/questions/57057 | 7 | I've made this a new question, rather than expanding the [first one](https://mathoverflow.net/questions/56962/what-about-stacks-of-categories-in-algebraic-geometry).
Torsten gives a good [answer](https://mathoverflow.net/questions/56962/what-about-stacks-of-categories-in-algebraic-geometry/56974#56974), and it partia... | https://mathoverflow.net/users/4177 | What about stacks of categories in algebraic geometry? II | I think the right thing to do with category-valued stacks is to keep the notion of "representability" the same (that is, use (pseudo) 2-pullbacks rather than comma objects), but to replace representability of the diagonal by representability of $X^2 \to X\times X$, where $X^2$ is the power (cotensor) of X by the free-l... | 7 | https://mathoverflow.net/users/49 | 77262 | 46,696 |
https://mathoverflow.net/questions/76972 | 3 | How can i proove that on the space of circle diffeomorphisms with the C-r topology, Morse Smale diffeomorphisms and Structurally stable diffeomorphisms are the same set?
| https://mathoverflow.net/users/18231 | Structural stability on the circle | It is not hard to show that to be $C^r$-structurally stable, the diffeomorphism must be Morse-Smale: Indeed, if it has irrational rotation number one can easily perturb (by composing with a small rotation) in order to change the rotation number (conjugacy invariant, see for example Proposition 11.1.9 in Katok-Hasselbla... | 2 | https://mathoverflow.net/users/5753 | 77271 | 46,702 |
https://mathoverflow.net/questions/77277 | 6 | classification of irreducible admissible (g,K)-module for GL(3,R)
Is there a classification of irreducible admissible (g,K)-module for GL(3,R)?
For GL(2,R) we have principal series, discrete series and etc. Is there such a result for GL(3,R) or GL(n,R)?
| https://mathoverflow.net/users/2666 | classification of irreducible admissible (g,K)-module for GL(3,R) | For a general real reductive group, all irreducible admissible $({\mathfrak g},K)$-modules are quotients of parabolically-induced discrete series (or limits thereof) representations (where we allow "trivial" parabolic induction ($P=G$) for discrete series on the group). See Theorem 14.92 in Knapp's Representation Theor... | 9 | https://mathoverflow.net/users/6753 | 77297 | 46,718 |
https://mathoverflow.net/questions/77288 | 9 | Bruns/Herzog "Cohen-Macaulay-Rings" has a note in the notes for Chapter 1, saying roughly that after the influx of homological algebra into commutative ring theory, modules became popular objects (instead of ideals). They cite Gröbner's 1949 book "Moderne algebraische Geometrie" as the birthplace of "Vektormoduln", whi... | https://mathoverflow.net/users/5495 | When and where did the term "module" enter commutative algebra? | From the website Chris Dionne mentioned in the comments:
MODULE. A JSTOR search found the English term in E. T. Bell’s “Successive Generalizations in the Theory of Numbers,” American Mathematical Monthly, 34, (1927), 55-75. Bell was describing the work of Dedekind, basing his account on Dedekind’s French article, “Su... | 8 | https://mathoverflow.net/users/14167 | 77305 | 46,726 |
https://mathoverflow.net/questions/77270 | 9 | It is well known that a generalized Cantor space $2^A$ is separable if and only if $|A| \leq 2^{\aleph\_0}$. This means that one cannot decide in $ZFC$ whether the space $2^{\omega\_2}$ is separable or not. On the other hand it is also well known that every $2^A$ has the ccc.
>
> Does anyone know a topological spac... | https://mathoverflow.net/users/17836 | A topological space for which having the ccc is independent of ZFC? | Let $T$ be the $L$-least Suslin tree of $L$, with the usual cone topology. Thus, in $L$, this is a Suslin tree, a tree of height $\omega\_1$ wtih all countable levels, and satisfying the countable chain condition. As a topological space, it is c.c.c. in $L$.
This tree is absolutely definable, in the sense that the d... | 8 | https://mathoverflow.net/users/1946 | 77312 | 46,729 |
https://mathoverflow.net/questions/77278 | 41 | Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" and "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques".
| https://mathoverflow.net/users/16858 | Introductory text on Galois representations | Kevin Ventullo's suggestion of Silverman's book is a very good one. The first examples
of Galois representations in nature are Tate modules of elliptic curves, and if you haven't
read about them in Silverman's book, you should.
If you have read Silverman's book, a nice paper to read is Serre and Tate's "On the good r... | 61 | https://mathoverflow.net/users/2874 | 77328 | 46,737 |
https://mathoverflow.net/questions/77325 | 4 | Recently I have met an interesting problem $\rho$: $G \rightarrow SL(2,R)$ be a faithful representions of a finite group by real 2\times 2 matrices of determinant 1, then we can get this group is cylic.
what is more, how can we determine all finite groups wich have a faithful real two dimensional representation? I fe... | https://mathoverflow.net/users/11966 | finite groups with faithful real two dimensional representation | Since it's not clear, I assume the question you're asking is the following. "Fix an $n\geq2$. Which finite groups have a faithful real $n$-dimensional representation? Equivalently, what are the finite subgroups of $\operatorname{GL}\_n \mathbb R$?" (If you just want to know which finite groups admit faithful real repre... | 6 | https://mathoverflow.net/users/430 | 77329 | 46,738 |
https://mathoverflow.net/questions/77261 | 3 | For a orientable three manifold M with totally geodesic boundary, this inequality is true. Because the rank of (fundemantal group of boundary M)=rank (homology group of boundary M )
then we use the "half live half die" theorem to get the theorem.
But in the orbifold case, we do not have such good things.
After passing... | https://mathoverflow.net/users/18301 | In hyperbolic 3-orbifold with totally geodesic boundary case, is it true: rank(the fundamental group of boundary M)< or equal 2 rank(fundmental group of M)? | One may obtain an estimate improving your factor of 4 to a factor of 3.
The ranks of hyperbolic 2-orbifolds were computed by [Zieschang et al](http://www.ams.org/mathscinet-getitem?mr=382457). If $\partial\mathcal{O}$ has genus $g$ and $p$ cone points, then they show that $rank(\pi\_1\partial\mathcal{O})\leq 2g+p-1$... | 3 | https://mathoverflow.net/users/1345 | 77330 | 46,739 |
https://mathoverflow.net/questions/76797 | 1 | Let $U\subset \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be the Zariksi open set of ordered quadruple of distinct points in the projective line. The quotient of $U$ by the projective transformation group $PSL(2)$ can be identified to $\mathbb{P}^1$ by cross-ratio.
Motivated by a paper of Fock... | https://mathoverflow.net/users/17294 | Configuration space of flags | There are many GIT quotients, since to define one requires a choice of $G$-line bundle, so a pair of naturals for each $F$.
There's an obvious democratic choice -- $(a,b) = (1,1)$ for every $F$ -- but I think this will in general lead to properly semistable reduction, which is a little icky. (Instead of just taking ... | 1 | https://mathoverflow.net/users/391 | 77344 | 46,743 |
https://mathoverflow.net/questions/77063 | 3 | Let $X$ be a Polish space. Let $J\in\mathbb{N}$.
Let $\lbrace a^n\_1\rbrace\_n,\dots,\lbrace a^n\_J\rbrace\_n$ be $J$ sequences of reals.
Let $\lbrace \mu^n\_1\rbrace\_n,\dots,\lbrace \mu^n\_J\rbrace\_n$ be $J$ sequences of probability measures in $\Delta(X)$.
For each $j\leq J$, let $\mu^n\_j$ weakly converge to... | https://mathoverflow.net/users/12713 | Sequences of linear combinations of measures | The generalization you suggest is true in any real topological vector space $X$ (Hausdorff or not), under the further assumption that the limit family $(\mu^\infty \_ 1,\dots,\mu^\infty \_ J)$ be linearly independent. The natural generalization to net convergence is also true (with essentially the same proof) .
**Fac... | 1 | https://mathoverflow.net/users/6101 | 77347 | 46,745 |
https://mathoverflow.net/questions/77334 | 2 | A bivariate polynomial of degree $m+n$ is,
$ p(x,y) = \sum\_{k=1}^n\sum\_{j=1}^m a\_{jk}x^ky^j$
where $a\_{mn}\neq0$ and $a\_{jk}\in\mathbb{R}$ for $1\leq j\leq m$, $1\leq k\leq n$.
I would like to understand how the roots of a bivariate polynomial behave. It is clear that the roots cannot form patches (unless $... | https://mathoverflow.net/users/2011 | Roots of bivariate polynomials | I don't think most of your questions are appropriate for MO, you should try math.stackexchange.com if you have any follow-up questions, as it sounds as if your knowledge and questions are at advanced undergraduate/beginning graduate level, but I'll try to answer some of them. A good reference is Fulton "Algebraic Curve... | 5 | https://mathoverflow.net/users/2290 | 77352 | 46,747 |
https://mathoverflow.net/questions/77238 | 2 | Let $X$ be a smooth projective variety over the complex numbers. One has the Hodge-Decomposition
$H^1\_{DR}(X) \simeq \Omega^1(X) \oplus H^1(X,\mathcal O\_X)$ (here consider the underlying manifold).
With $H^1\_{DR}(X)$ I denote the first De-Rham cohomology group, which is also the first hypercohomology group of the ... | https://mathoverflow.net/users/18183 | Ext groups with connection and Hodge Decomposition | Sorry, my comment yesterday was rushed and this will be only slightly less so. Thus it is more
of a hint. It is perhaps easier to view $E$ etc. as a $C^\infty$ bundle equipped with a $\bar\partial$ operator. Then the exact sequence
$$0\to \mathcal{O}\_X\to E\to \mathcal{O}\_X\to 0$$
admits a $C^\infty$ splitting, with ... | 2 | https://mathoverflow.net/users/4144 | 77359 | 46,750 |
https://mathoverflow.net/questions/77363 | 0 | Let $SL\_{n+1}$ act on $\mathbb{P}^n$ in the natural way. Suppose I take two linear subspaces $\mathbb{P}^m$ and $\mathbb{P}^{n-m}$, with $m < n$, that intersect in one point. Is the action of $SL\_{n+1}$ transitive on the set of such couples of linear subspaces?
| https://mathoverflow.net/users/4096 | Action of $SL_{n+1}$ on couples of linear spaces in $\mathbb{P}^n$. | Sure it does. Denote your variety by $X$. Consider this action on the corresponding vector space $V$. Then your projective subspaces correspond to linear subspaces $V\_1$ and $V\_2$ of dimension $m+1$ and $n-m+1$ that intersect transversely. There is an obvious projection from the variety $Y$ of all frames in $V$ to $X... | 4 | https://mathoverflow.net/users/10941 | 77368 | 46,753 |
https://mathoverflow.net/questions/77377 | 0 | Let n>3. Is there any way to generate all integer solutions of linear diophantine equation in n variables, or at least to determine number of such solutions?
Thanks in advance.
| https://mathoverflow.net/users/18340 | Linear diophantine equation in n variables | The number of solutions is either zero or infinite. As for the ways to generate them, yes there are many ways. Look at Morris Newman's "Integer matrices", and check out "Hermite Normal Form".
| 4 | https://mathoverflow.net/users/11142 | 77378 | 46,758 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.