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https://mathoverflow.net/questions/114471 | 2 | Let $X$ be a variety (i.e. a reduced scheme of finite type over a field) and let $G$ be an abstract group, finitely generated, acting of $X$ algebraically freely. The example I have in mind is $\mathbb Z$ acting by shifts on $\mathbb A^1$. The quotient in this example clearly does not exist as a noetherian scheme since... | https://mathoverflow.net/users/2234 | quotients of varieties as non-noetherian schemes? | The quotient exists as an algebraic space (if we ignore the literature that defines algebraic spaces to be separated). The equivalence relation is given by the standard action groupoid. This is étale, since both the projection and the action map from $X \times G$ to $X$ are locally finitely presented and formally étale... | 1 | https://mathoverflow.net/users/121 | 114537 | 64,912 |
https://mathoverflow.net/questions/114492 | 6 | I've been thinking about the following propagation of singularities result:
Let $X$ be a compact manifold, and let $P$ be a differential operator (of, say, order $m$) on $X$ whose principal symbol $\sigma\_m(P)$ is real-valued. Suppose that $Pu=0$. Then the wavefront set of the solution $u$ is a union of maximally ex... | https://mathoverflow.net/users/29413 | propagation of singularities & the Schrodinger equation | There are propagation of singularities results for the Schr\"odinger operator, but they are usually stated on asymptotically Euclidean manifolds. Early results appear in a paper in CMP by Zelditch in around 82 or 83 exhibit the typical weird behaviour where singularities disappear then reappear at later discrete points... | 7 | https://mathoverflow.net/users/17969 | 114538 | 64,913 |
https://mathoverflow.net/questions/114529 | 10 | Let $F\_n=F\ast F'$ be a free splitting of the free group $F\_n$ and $\phi\in Aut(F\_n)$. The free factor $F$ is said to be *invariant under $\phi$* if $\phi(F)\subseteq F$.
I recently wondered if this already implies that $\phi(F)=F$, and I found a positive answer:
An exercise in Magnus-Karrass-Solitar's book on c... | https://mathoverflow.net/users/12996 | Invariant free factor of a free group | There is a proof attributed to Peter Scott in Lemma 6.0.6 of "The Tits alternative for Out(F\_n) I: Dynamics of exponentially growing automorphisms", MR1765705. The proof uses the Kurosh subgroup theorem, although the proof is then carried out more generally for any finite rank subgroup $F$ of $F\_n$ using the LERF pro... | 10 | https://mathoverflow.net/users/20787 | 114541 | 64,914 |
https://mathoverflow.net/questions/114512 | 10 | I would like to apologize in advance if my question is too simple for mathematical community here: I am physicist by education.
It is well known that for a topological group $G$ acting transitively on a space $X$ and its subgroup $H \subset G$ one can construct a principal bundle whose fibers are homeomorphic to the ... | https://mathoverflow.net/users/29418 | Fibrations of $SU(4)$ | From the homotopy exact sequence of the fibration
$$
\mathrm{SU}(2)\times \mathrm{SU}(2) \longrightarrow \mathrm{SU}(4)\longrightarrow
\frac{\mathrm{SU}(4)}{\mathrm{SU}(2)\times \mathrm{SU}(2)} = Q
$$
and standard facts about $\pi\_i\bigl(\mathrm{SU}(k)\bigr)$, one sees that $\pi\_i(Q)=0$ for $i = 0, 1, 2, 3$ and that... | 10 | https://mathoverflow.net/users/13972 | 114546 | 64,916 |
https://mathoverflow.net/questions/113977 | 7 | Let $A \to B$ be a proper morphism of $C^\*$-algebras. A nondegenerate representation of $B$ induces a nondegenerate representation of $A$. Does the converse hold?
I.e.: let $A \to B$ be a morphism of $C^\*$-algebras such that every nondegenerate representation of $B$ induces a nondegenerate representation of $A$. Do... | https://mathoverflow.net/users/22789 | Proper morphisms of C*-algebras / Nondegenerate representations | This is true. Factoring by the kernel of the homomorphism, we may assume that $A$ is a C\*-sub-algebra of $B$ and the homomorphism is just the inclusion.
So assume that $A\subseteq B$ and
(a) Every non-degenerate representation of $B$ restricted to $A$ is non-degenerate.
Then $A$ cannot be contained in the kernel ... | 6 | https://mathoverflow.net/users/13381 | 114547 | 64,917 |
https://mathoverflow.net/questions/114540 | 13 | Let $d \geq 3$ and suppose that $K \subset \mathbb{R}^{d}$ is a convex body (compact, convex, non-empty interior). Is the following true?
>
> The boundary $\partial K$ is a $C^1$-manifold if and only if for each projection $\pi:K\rightarrow H$ to a hyperplane $H$ has the property that $\partial \pi(K)$ is a $C^1$-m... | https://mathoverflow.net/users/18279 | Can you see smoothness of the boundary of a convex body from its shadow? | The statement for $C^1$ regularity is true, but with "dimension-2 projections" instead of "codimension-1 projections''. This is even stronger, if $d\ge 3$. On the other hand, for $d=2$ the statement with "hyperplane projections" fails, since $1$ dimensional projections are just closed intervals, whose boundary is certa... | 14 | https://mathoverflow.net/users/6101 | 114550 | 64,918 |
https://mathoverflow.net/questions/114542 | 3 | My question will be very short.
*Suppose we have a Boolean algebra $B$ which admits an uncountable independent family. Does it follow that there is an uncountable chain of elements in $B$?*
Manifestly, this is the case for (infinite) complete Boolean algebras, although the proofs of existence of uncountable indepe... | https://mathoverflow.net/users/29433 | Independent families and chains | The free Boolean algebra in any number of generators contains no uncountable chain. This can be seen as follows.
If $X$ is the set of generators, we can identify elements of the algebra with functions $f\colon2^X\to2$ which only depend on finitely many variables. Let $d(f)$ be the set of variables $f$ depends on. Ass... | 3 | https://mathoverflow.net/users/12705 | 114552 | 64,919 |
https://mathoverflow.net/questions/114555 | 121 | Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is certainly Taylor expansion. They have a quantity (function) that they need to approximate: they expand it in Taylor serie... | https://mathoverflow.net/users/4721 | Does Physics need non-analytic smooth functions? | As a physicist "in nature" perhaps I can give a few examples that illustrate how non-analytic functions
can appear in physics and counter the idea that physicists do not worry about the justification
of these procedures.
Example 1 involves one of the most precise comparisons between experiment and theory known to
phy... | 105 | https://mathoverflow.net/users/10475 | 114563 | 64,923 |
https://mathoverflow.net/questions/114517 | 1 | Let $G$ be a finite group of Lie type. Let $H$ be a subgroup of $G$ which contains unipotent elements. I want to find a 'nice' subgroup of $G$ that contains $H$, for example a Levi subgroup of $G$ which is minimal with this property. Do you have an idea how we can do this?
My motivation comes from the following theo... | https://mathoverflow.net/users/25202 | How we characterize a subgroup of finite group of Lie type with unipotent elements. | To attempt an answer, I'll replace your notation with my own. One method is to work inside a corresponding semisimple (or reductive) algebraic group $G$ over an algebraically closed field, relative to which your finite group of Lie type is constructed. There is a 1971 paper by Borel and Tits
[here](http://gdz.sub.uni-g... | 2 | https://mathoverflow.net/users/4231 | 114567 | 64,926 |
https://mathoverflow.net/questions/114463 | 7 | Background
----------
Suppose $X$ is a compact metric space, and that $\varphi: X\to X$ is a homeomorphism of $X$.
We say a subset $A$ of $X$ is $\varphi$-invariant if $\varphi(A) = A$. A $\varphi$-invariant set is minimal if it is closed, $\varphi$-invariant, nonempty and the smallest of all such sets. We say $(X,... | https://mathoverflow.net/users/29404 | When does a homeomorphism split into essentially minimal homeomorphisms? | The answer is no.
For each pair $(n,k) \in \mathbb{Z}^+ \times \mathbb{Z}$ we define a point $p(n,k) \in \mathbb{R^2}$ as: $$p(n,k)=\left(1+\frac{1}{n},\frac{k}{n^2} \right)$$ if $|k| \leq n$, and $$p(n,k)=\left( \frac{1}{k}\cos(1/n), \frac{1}{k} \sin(1/n) \right)$$ otherwise.
We let $$X= \left\{ p(n,k): (n,k) \in ... | 3 | https://mathoverflow.net/users/17836 | 114577 | 64,934 |
https://mathoverflow.net/questions/82917 | 11 | I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of quantum groups. I am very new to crystal bases, so I would also appreciate corrections if my questions are not well-formu... | https://mathoverflow.net/users/703 | R-matrices, crystal bases, and the limit as q -> 1 | I never found a precise reference for the statement about the R-matrix, so I ended up writing it up myself. The precise statements and proofs can be found in $\S 4.1$ of my paper with Alex Chirvasitu, *Remarks on quantum symmetric algebras*, available [here](http://arxiv.org/abs/1206.1614).
| 4 | https://mathoverflow.net/users/703 | 114584 | 64,938 |
https://mathoverflow.net/questions/114579 | 0 | I am looking for reference on Casselman-Shalika formula for GL(n) and PGL(n) at finite place p.
| https://mathoverflow.net/users/2666 | Reference on Casselman-Shalika formula for GL(n) and PGL(n)? | In [this paper](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pja/1195518347), Shintani proves the Casselman-Shalika(-Shintani) formula for GL(n). This preceded Casselman-Shalika's [paper](http://www.numdam.org/item?id=CM_1980__41_2_207_0) by a few years. Several of [Cogdell's](http:/... | 3 | https://mathoverflow.net/users/6753 | 114592 | 64,941 |
https://mathoverflow.net/questions/114590 | 6 | If we want to add one real number the simplest way to do it is to use Cohen forcing. The poset is $\lbrace p\colon n\to 2\mid n\in\omega\rbrace$ which is a countable set. We can think of this as approximating a new set by finite sets.
If we want to add $\omega$-many pairwise generic real numbers we can do it by takin... | https://mathoverflow.net/users/7206 | Forcing with product vs. box product | Your third notion of forcing, with conditions $p:\omega\times n\to 2$ ordered by extension as $n$ increases, is the same as the forcing to add a function $\omega$ to the reals of the ground model. This forcing is equivalent to the forcing $\text{Coll}(\omega,\mathbb{R})$, to collapse the ground model continuum to be co... | 9 | https://mathoverflow.net/users/1946 | 114594 | 64,943 |
https://mathoverflow.net/questions/114589 | 1 | The Shannon capacity of a graph is defined as
$$\Theta(G) = \sup\_k \sqrt[k]{\alpha(G^k)}.$$
So, $\alpha(G) \leq \Theta(G)$ but $\Theta(G)$ can be strictly greater than $\alpha(G)$. I am wondering if there is any upper bound based on the independence number itself? Specifically, are there graphs where $\Theta(G) \geq... | https://mathoverflow.net/users/13886 | Upper bound on Shannon capacity based on independence number | Self-complementary vertex-transitive graphs have Shannon capacity $\sqrt n$, so if this number is far from $\alpha$, then you have what you're looking for.
Paley graphs have this property, and as you can see here, there are examples for which $\alpha$ is indeed much less than the Shannon capacity.
<http://www.resea... | 3 | https://mathoverflow.net/users/4580 | 114602 | 64,945 |
https://mathoverflow.net/questions/114607 | 6 | Assuming the axiom of choice, is there a well-ordering of the reals such that every initial segment is closed for the usual topology? If the continuum hypothesis helps, we can also assume it.
An initial segment is a set of the form $\{x : x<y\}$ for some $y$, according to the well-ordering $<$.
| https://mathoverflow.net/users/21059 | Well-ordering with a topological property | No. There cannot be a strictly increasing $\omega\_1$-sequence of closed sets in a topological space with a countable base. Their complements would be unions of open sets from the basis, and it is impossible to drop elements of a countable set uncountably many times.
| 11 | https://mathoverflow.net/users/14915 | 114609 | 64,948 |
https://mathoverflow.net/questions/114606 | 2 | For a group $G$ there is a well-defined map $\operatorname{Irr}(G) \to \operatorname{Lin}(G)$ which sends $\chi \mapsto \det \chi$, where $\det \chi$ the linear character of $G$ given by taking the determinant of the representation affording $\chi$.
In general, is there a good way to go about computing $\det \chi$ wi... | https://mathoverflow.net/users/14469 | Computing determinants of characters | If you know $\chi$ then you can write down $\det \chi$ using [Newton's identities](http://en.wikipedia.org/wiki/Newton%27s_identities). This is simply the observation that one can express the determinant of a matrix $\rho(g)$ in terms of traces of powers $\rho(g)^k=\rho(g^k)$ of that matrix.
| 3 | https://mathoverflow.net/users/430 | 114616 | 64,953 |
https://mathoverflow.net/questions/114620 | 8 | I have not read anything thoroughly about higher categories. I am only aware that in higher categories, we have higher dimensional cells, after adjusting for the intuition that 0-, 1-, 2-dimensional cells are respectively, objects, morphisms, and commuting squares. This terminology is similar to what we from *arrow con... | https://mathoverflow.net/users/25165 | Is it possible to approach higher categories from the point of view of the arrow functor? | The construction you describe yields, for any category, an $n$-fold category ([nLab](http://ncatlab.org/nlab/show/n-fold+category)), a concept originally introduced by Ehresmann. This can be defined iteratively using the language of [internal categories](http://ncatlab.org/nlab/show/internal+category), namely a 1-fold ... | 8 | https://mathoverflow.net/users/4177 | 114622 | 64,956 |
https://mathoverflow.net/questions/114591 | 4 | Suppose you have an alphabet with countably many letters. Every letter has a particular weight (for instance, as in the game of Scrabble). There are a total of $n^2$ letters that have weight $n$.
Given any word in this alphabet, let the weight of that word be the sum of the weights of its letters (again, as in Scrabb... | https://mathoverflow.net/users/8183 | Generic words of given weight | Here is a back of envelope computation. There will be no rigor whatsoever, just a cookbook approach that may be acceptable to a physicist but which every self-respecting mathematician should frown upon. It can give a plausible (but not guaranteed) answer to some of your questions if we just want the general order of ma... | 8 | https://mathoverflow.net/users/1131 | 114627 | 64,960 |
https://mathoverflow.net/questions/114626 | 18 | I have 3 general abstract reasons to care about complex analysis in a single variable:
1. The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, and so it is abstractly very important. Holomorphic functions are intimately related to harmonic functions, so holomorphic functions are ... | https://mathoverflow.net/users/27968 | Fundamental motivation for several complex variables | On my opinion, there are two very general reasons why analytic functions are important.
1. Solutions of many (almost all) important differential (and functional) equations are analytic.
For example,
all elementary and special functions arise in this way. Moreover, they are also usually
analytic functions of parame... | 15 | https://mathoverflow.net/users/25510 | 114630 | 64,962 |
https://mathoverflow.net/questions/114558 | 7 | Given an nonisotrivial elliptic fibration $f:X\rightarrow P^1$, where $X$ is smooth and $P^1$ is a projective line. Could anybody provide some information on the restriction of the cotangent bundle $\Omega^1\_{X}$ to the smooth fibers of $f$, e.g. does it split to line bundles, (semi)stable...?
| https://mathoverflow.net/users/5661 | restriction of the cotangent bundle of an elliptic surface | It's semi-stable, but not stable on every fiber.
Assuming that the characteristic is zero this sheaf does not split.
This may be true in positive characteristic, but as *Damian Rössler* points out the proof below requires characteristic zero.
>
> **Claim 1.**
> Let $V\subseteq \mathbb P^1$ be an arbitrary non-e... | 9 | https://mathoverflow.net/users/10076 | 114633 | 64,963 |
https://mathoverflow.net/questions/114605 | 6 | I want to solve a matrix $\Omega$ from a equation $\sum\_k (\Omega + \Theta\_k)^{-1} = Q$. The $Q$ and $\Theta, \forall k=1,\ldots,K$ are known, and are positive definite matrices. $\Omega$ also has to be positive definite. all matrices are large (a few thousands of columns and rows). My questions are:
(1) Is there a... | https://mathoverflow.net/users/29442 | Solve equation with matrix variable | Here is a partial solution to the first question in the original post. Let's look at the equation
\begin{equation}\label{1}\tag{1}
\sum\nolimits\_{i=1}^m (X+ \Theta\_i)^{-1} = Q.
\end{equation}
**Lemma (Existence).** If all $\Theta\_i$ are (strictly) positive definite, then \eqref{1} has a positive semidefinite solu... | 6 | https://mathoverflow.net/users/8430 | 114636 | 64,966 |
https://mathoverflow.net/questions/114640 | 1 | Put in other words, given an even-dimensional sphere $S^{2k}$: is there a manifold $M$ such that $T^\* M$ is diffeomorphic to $S^{2k}$?
| https://mathoverflow.net/users/29449 | Can a sphere be a phase space? | Of course, the spheres are compact while cotangent bundles are noncompact (unless in dimension 0). Nevertheless, a bit more interesting is the question whether the even dimensional spheres can be phase spaces in the sense of symplectic manifolds. There the $\mathbb{S}^2$ is an example: the volume form is non-degenerate... | 17 | https://mathoverflow.net/users/12482 | 114641 | 64,970 |
https://mathoverflow.net/questions/114494 | 0 | Is there any ring $R$ with essential right ideal $I$ such that
$(I:I)\cap \{ t\in R \mid t(I:I)t \subseteq I \} \neq 0 $ and
for every non-zero $ x,y\in R$,
$\{ r\in R \mid xr\in(I:I)\}\cap\{ r\in R\mid xry\notin I \}\neq \emptyset$ ?
where $(I:I)=\{ r\in R\mid rI\subseteq I\}$
| https://mathoverflow.net/users/29414 | ring with a condition | After answering, the question changed so I adapt my answer to the new question.
As I was remarking in the comments to your question, it is impossible to construct such ring. In fact, as you want that, for all $x,y\in R\setminus\{0\}$, the intersection $\{r\in R:xr\in (I:I)\}\cap\{r\in R:xry\notin I\}\neq \emptyset$, ... | 1 | https://mathoverflow.net/users/24891 | 114649 | 64,974 |
https://mathoverflow.net/questions/114585 | 5 | I'm currently in need an explicit formula in classical cohomology which I'm pretty sure is well known, but which I've been unable to find in the references I am aware of.
Let $X$ be a smooth manifold and let $\mathcal{U}=\{U\_\alpha\}$ be a fixed open cover of $X$ such that all the finite intersections $U\_{\alpha\_... | https://mathoverflow.net/users/8320 | An explicit homotopy equivalence between the de Rham complex and the Cech-de Rham total complex | Thanks to an email by Chris Rogers, I now know that my question above is precisely the subject of Proposition 9.5 in Bott-Tu, Differential Forms in Algebraic Topology., where an explicit formula for the homotopy operator in terms of a partition of unit subordinate to the given open cover is given.
They also write "Th... | 6 | https://mathoverflow.net/users/8320 | 114659 | 64,978 |
https://mathoverflow.net/questions/106948 | 20 | The [Erdős–Gallai theorem](http://en.wikipedia.org/wiki/Erd%25C5%2591s%25E2%2580%2593Gallai_theorem) gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph.
In particular $d\_1 \ge d\_2 \ge \dots \ge d\_n$ is the degree sequence of a graph on $n... | https://mathoverflow.net/users/4558 | Is there an analogue of the Erdős–Gallai theorem for simplicial complexes? | Very little is known about the question (and even about the easier case of vertex degrees), and it contains as a special case some notoriously hard questions: For example the case that all $d\_i$s are equal to 1 (or to some $\lambda$ is the question on the existence of combinatorial designs of certain parameters. I sup... | 6 | https://mathoverflow.net/users/1532 | 114668 | 64,981 |
https://mathoverflow.net/questions/114660 | 11 | I'm trying to understand the 2 spin structures on the circle. Since the frame bundle for the circle is just the circle itself, Spin structures on $S^1$ correspond to double covers of $S^1$. There are two choices: the connected double cover and the disconnected double cover.
From the point of view of Spin cobordism, w... | https://mathoverflow.net/users/22781 | Spin structures on $S^1$ and Spin cobordism | As Fabian pointed out in the comments, you have to be more careful about how you trivialize $SO(D^2)$. I'm going to use the standard coordinates $(x,y)$ on $\mathbb{R}^2$ (note that these are not global coordinates on $D^2$, but they still trivialize the frame bundle). Thus we have a global section of $SO(D^2)$ which a... | 12 | https://mathoverflow.net/users/4362 | 114669 | 64,982 |
https://mathoverflow.net/questions/89159 | 4 | Hello, considering that for real numbers, the intersection of intervals defined by simple inequalities has a quite simple form as
$$
\bigcap\_i\{x|x\leq a\_i\}=\{x|x\leq\min\_i\{a\_i\}\}
$$
However, what is the case if the variables are chosen as Hermitian matrices, and the interval defined by inequality is replaced ... | https://mathoverflow.net/users/19399 | What is the geometry of the intersection of some cones defined by generalized inequalities? | T. Ando has some papers on the structure of the intersection of these cones:
Extreme points of an intersection of operator intervals, (1994)
Parameterization of minimal points of some convex sets of matrices, Acta Sci Math Szeged 57 (1993), 3-10.
Not sure if this will solve your problem, but it is yet another way for... | 3 | https://mathoverflow.net/users/29459 | 114670 | 64,983 |
https://mathoverflow.net/questions/114647 | 12 | Lubin and Tate show in their paper *Formal moduli for one-parameter formal Lie groups* that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of deformations is represented by a formal scheme isomorphic to $\mbox{Spf } \mathbb{W}(k)[[u\_1,\ldots,u\_{h-1}]]$. Modulo lower ... | https://mathoverflow.net/users/303 | What is the universal deformation of the formal additive group $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$? | $\DeclareMathOperator{\Ext}{Ext} \newcommand{\G}{\hat{\mathbb{G}}} \DeclareMathOperator{\Maps}{Maps} \renewcommand{\phi}{\varphi}$ The analysis of the infinitesimal deformation space of the Honda formal groups $H\_n$ uses three calculations which govern the existence of square-zero deformations. These can be phrased in... | 18 | https://mathoverflow.net/users/1094 | 114695 | 64,995 |
https://mathoverflow.net/questions/114560 | 4 | In the paper [http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf](http://www.mat.univie.ac.at/%7Eschachermayer/pubs/preprnts/prpr0154.pdf)
you can find a trajectorial version of Doob's inequality. It is given by:
>
> $$\bar{s}^2\_T+4\sum\_{k=0}^{T-1}\bar{s\_k}(s\_{k+1}-s\_k)\le 4s^2\_T$$
>
>
... | https://mathoverflow.net/users/28002 | Trajectorial version of Doob's $L^2$ inequality | I think induction over $T$ will be hard. But you can prove the inequality by considering the times $n$ at which the maximum $\bar s\_n$ increases. For example, let
$
1 = k\_1, \ldots, k\_r \le T
$
be the different times at which $s\_k = \bar s\_k$ attains a maximum (with respect to all previous times). Then it holds
$$... | 6 | https://mathoverflow.net/users/25062 | 114699 | 64,996 |
https://mathoverflow.net/questions/114704 | 4 | My intuition is that the answer is yes:
Let $G$ be the original group, and let $H$ be a subgroup of $G$.
Let $\mu$ be a Haar measure on $G$ that is both right- and left-invariant.
I think that if we restrict $\mu$ to $H$ and restrict the translation
to translations by elements of $H$, then invariance must be preserved.... | https://mathoverflow.net/users/29469 | Is every subgroup of a connected unimodular (matrix) Lie group also unimodular? | Take the so-called ax+b group, i.e. the connected component of the affine group of the real line. Or, even more concretely,
$$\left\{ \left( \matrix{ a & b \\ 0 & 1 } \right) \colon a>0, b\in{\mathbb R} \right\}.$$
This is not unimodular.
So I think the "proof by handwaving'' has a mistake somewhere. Probably your co... | 11 | https://mathoverflow.net/users/763 | 114705 | 64,999 |
https://mathoverflow.net/questions/114703 | 3 | In Tom Leinster's [book on operads](http://arxiv.org/abs/math/0305049), he gives Ab(V), the category of abelian groups in a symmetric monoidal category V, as an example of a multicategory that doesn't arise from a monoidal category, since Ab(V) will not generally have a tensor product. Example 2.1.5 on page 37.
I can... | https://mathoverflow.net/users/19860 | Why does tensor product in Ab(V) require colimits in V? | It is easy to *define* the tensor product as being the object that represents the bilinear maps functor, but to prove that tensor products exist requires something extra. If you have free abelian groups, then it is enough to have coequalisers of abelian groups to construct the tensor product, but even the construction ... | 7 | https://mathoverflow.net/users/11640 | 114708 | 65,001 |
https://mathoverflow.net/questions/114715 | 25 | Is a domain $D$, all of whose localizations $D\_P$ for $P \in Spec(D)$ are noetherian, itself noetherian ?
The question is motivated by proposition 11.5 of Neukirch's Algebraic Number Theory:
>
> Let $\mathfrak{o}$ be a noetherian integral domain. $\mathfrak{o}$ is a Dedekind domain if and only if, for all prim... | https://mathoverflow.net/users/22217 | Is a domain all of whose localizations are noetherian itself noetherian ? | I had the exact same question not too long ago. Apparently if you drop the noetherian precondition in Neukirch's definition of "Dedekind domain" then you get what some people call an "almost Dedekind domain". There are indeed examples of almost Dedekind domains that aren't Dedekind (i.e. aren't noetherian). The first o... | 20 | https://mathoverflow.net/users/430 | 114719 | 65,004 |
https://mathoverflow.net/questions/114724 | 3 | $\textbf{Question: }$We know that the depth of a noetherian local ring is at most the dimension. Do there exist noetherian local rings with high dimension but zero depth? If not, what's the smallest possible depth for a noetherian local ring of dimension $n$?
| https://mathoverflow.net/users/25854 | Depth zero, high dimension | Let $S = k[x\_1, \ldots, x\_n, y]/(x\_1 y, x\_2 y, \ldots, x\_n y, y^2)$ and let $R$ be the local ring of $S$ at $0$. Then $\dim R = \dim S = n$, but there are no regular elements, since $y$ annihilates the maximal ideal of $R$ - so $R$ has depth zero.
| 9 | https://mathoverflow.net/users/3847 | 114729 | 65,010 |
https://mathoverflow.net/questions/114716 | 1 | Is I consider an ind scheme such as $G(k((t)))$ for a reductive connected group over $k=\bar{k}$
I have the conjugacy action of $G(k[[t]])$.
In what category can I make the quotient $[G(k((t))/ad(G(k[[t]])]$?
In the category of presheaves? The category of fppf sheaves?
And if I want to make the fiber product of... | https://mathoverflow.net/users/27398 | quotient of ind scheme | Not sure what you are really asking. You can always do quotients in fppf sheaves, as explained by ancient greeks. Just quotient presheaves and sheafify around.
The point about the affine grassmanian is that it is turning out to be an ind-scheme, which is not true for quotients of general ind-schemes.
I am not sure ... | 1 | https://mathoverflow.net/users/5301 | 114736 | 65,015 |
https://mathoverflow.net/questions/114745 | 25 | There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example.
Now, for a monic polynomial of degree $n$, this is impossible (I think).
**So, my question is,** given a monic polynomial with integer coefficients of degree $n$... | https://mathoverflow.net/users/1056 | Monic polynomial with integer coefficients with roots on unit circle, not roots of unity? | There exist irreducible monic polynomials such that all their roots apart from two lie on the unique circle (and are not roots of unity). Such polynomials can be chosen among Salem polynomials and they exit in arbitrary high degree. By definition a Salem polynomial $S(x)\in \mathbb Z[x]$ is a monic irreducible reciproc... | 29 | https://mathoverflow.net/users/943 | 114748 | 65,022 |
https://mathoverflow.net/questions/114747 | 7 | Let $V$ be a smooth vector field on $\mathbb{R}^n$. Assume that the maximal solution to the Cauchy problem $x'=V(x), x(0)=x\_0$ exist only for $t\in [0,T)$, where $T$ is finite, denote this time by $T(x\_0)$. Is $T$ continuous with respect to $x\_0$ ? Is it $C^1$ ?
| https://mathoverflow.net/users/8887 | Dependence of the blow-up time of existence of an ODE with respect to initial condition. | No, but you can say it is *lower semicontiuous*, even wrto the initial time (that is, the optimistic situation: perturbing a little the initial data the existence is ensured almost up to $T$ , and could even be much greater) .
Precisely, given a Banach space $E$, an open set $\Omega\subset \mathbb{R}\times E$ and $f:... | 6 | https://mathoverflow.net/users/6101 | 114753 | 65,027 |
https://mathoverflow.net/questions/114755 | 7 | Suppose we are given a map $f:A \rightarrow B$ between two dg-algebras which are formal.
Is the map $f$ also "formal" in some sense?
More precisely can we find isomorphisms $\phi\_A:A\rightarrow H^\bullet(A)$ and $\phi\_B:B\rightarrow H^\bullet(B)$ in the derived category of dg-algebras such that
$$H^\bullet(f) \c... | https://mathoverflow.net/users/2837 | Morphisms between formal dg-algebras | Let me give you an example which has a topological flavour. Let us consider, the De-Rham complex of differential forms on a sphere $S^n$, we denote it $A^\*(S^n)$, it is a formal commutative differential graded algebra. Let us look at the morphisms of commutative differential graded algebras between $A(S^2)$ and $A(S^3... | 7 | https://mathoverflow.net/users/27816 | 114764 | 65,033 |
https://mathoverflow.net/questions/114758 | 11 | Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic numbers are widely believed to be normal, this implies the transcendentality of the irrationals in the Cantor set. Now, I a... | https://mathoverflow.net/users/29481 | Transcendentality of all irrationals in the Cantor set | This question was asked by Mahler ("Some suggestions for further research", Bull. Austral. Math. Soc. 29 (1984), no. 1, 101–108).
See Adamczewski, Bugeaud, "On the decimal expansion of algebraic numbers" (2005) for some things that are known. I have confirmed with a colleague that this is still very much a (widely) ... | 14 | https://mathoverflow.net/users/3651 | 114765 | 65,034 |
https://mathoverflow.net/questions/114531 | 6 | **Idea**
Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) continuous. If you have better regularity away from one isolated point, say you are $C^1$ on the puncutered ball, can the... | https://mathoverflow.net/users/4281 | Divergence form Elliptic PDE Removable Singularity/Regularity Question | I believe the following is a counterexample:
Let $N=1$, $B\_1(0)=(-1,1)$, $u(x)=|x|$, then $|u^\prime(x)| = 1$, $u^{\prime \prime}(x) = 2\delta\_0$,
and
$u^\prime(x) = 2H(x)-1$,
where $H$ is the Heaviside function, $H \in L^\infty \cap W^{1,2}\_{loc}((-1,1)\setminus \{0\})$.
Thus $u$ solves $(\frac{1}{2} u^... | 2 | https://mathoverflow.net/users/28090 | 114767 | 65,035 |
https://mathoverflow.net/questions/114770 | 3 | The complexity class $PR$ is the set of all formal languages that can be decided by a primitive recursive function. Is there any language $l$ known to be complete for this class, i.e., for every language in $PR$ there is a primitive-recursive reduction from it to $l$?
Perhaps this notion is trivial, or it trivially d... | https://mathoverflow.net/users/29482 | How would one characterize a PR-complete language? | Let me consider another reformulation of the question which makes it nontrivial: is there a PR-complete language under *polynomial-time* reductions?
The answer is no, since (apart from the first few levels) each level of the Gregorczyk hierarchy is closed under polynomial reductions. Thus, if the alleged complete lan... | 3 | https://mathoverflow.net/users/12705 | 114773 | 65,038 |
https://mathoverflow.net/questions/114539 | 2 | Hello,
Assume we have $(n+1)$ isometries $S\_1,...,S\_{n+1}$ in the separable Hilbert space $H$ with the properties that $\sum\_{i=1}^{n+1}S\_iS\_i^\*=I, S\_i^\*S\_j=0$ (i.e. $S\_i$ are the generators of the Cuntz algebra $O\_{n+1}$). In the $C^\* $ algebra $C^\*(I, S\_1,..., S\_n)$ consider the closed ideal generate... | https://mathoverflow.net/users/29432 | Ideal spanned by matrix units isomorphic to compact operators | Well, the argument goes roughly as follows. You can define the universal C\*-algebra $K$ generated by elements $( e\_{i,j} ), i,j\in\mathbb{N}$ with relations $e\_{i,j}^\*=e\_{j,i}$ and $e\_{i,j}e\_{k,l}=\delta\_{j,k}e\_{i,l}$. That is, $K$ is a C\*-algebra generated by such elements and whenever $A$ is another C\*-alg... | 4 | https://mathoverflow.net/users/29404 | 114780 | 65,043 |
https://mathoverflow.net/questions/114787 | 40 | What is "Teichmüller Theory"? What part has been worked out / foreseen by O. Teichmüller himself and what is further development? Is there some current work which might be considered as continuation/completion of this theory?
**Background** The question might be seen as too naive and can be answered by google or [Wi... | https://mathoverflow.net/users/10446 | What is "Teichmüller Theory" and its history? | First of all, let me recommend a book: J. Hubbard, **Teichmüller theory**, vol. 1.
Let me try to list briefly Teichmüller's own contribution to Teichmüller theory.
Bers's papers of 1960-s are good primary sources. The few papers of Teichmüller himself
that I read are also exciting, but my poor knowledge of German does ... | 48 | https://mathoverflow.net/users/25510 | 114792 | 65,046 |
https://mathoverflow.net/questions/114789 | 1 | I'm trying to find information about a specific lattice, which is proving difficult since I am not sure what its standard name is.
Consider the regular $n$-Simplex embedded in $\mathbb{R}^n$ with one of the $(n+1)$ vertices centered at the origin. Does the lattice generated by the position vectors of the remaining ve... | https://mathoverflow.net/users/17546 | Help with (Coxeter?) lattice identification. | The regular $n$-simplex has $n+1$ vertices. Other than that typo, you are right.
The lattice $A\_n$ can be embedded into $\mathbb{Z}^{n+1}$, where $\mathbb{R}^{n+1}$ has the standard norm, and $A\_n$ is the rank $n$ sublattice where the coordinates sum to zero.
Now, inside $\mathbb{Z}^{n+1}$, the $n+1$ basis vecto... | 3 | https://mathoverflow.net/users/297 | 114794 | 65,047 |
https://mathoverflow.net/questions/114775 | 5 | Hi
I have a function $F:\mathbb{R} ^ n\rightarrow \mathbb{R}^n$ for which I know there exist a unique fixed point $x ^ \*$ (say). I also know that the Jacobian of $F$ at each point $x$ in $\mathbb{R} ^ n$ has all of its eigenvalues in $[0,1)$ (but they are different for each $x$). Are these facts enough for me to say... | https://mathoverflow.net/users/29483 | fixed point of a particular vector valued function | Your question is stated as a Conjecture just before Theorem 2.1.5 in the book
MR1015711
Belitskiĭ, G. R. and Lyubich, Yu. I.
Matrix norms and their applications.
Birkhäuser Verlag, Basel, 1988.
| 1 | https://mathoverflow.net/users/25510 | 114797 | 65,048 |
https://mathoverflow.net/questions/114801 | 3 | Let $G=A\ast \mathbb{Z}$ be the free product of a group $A$ and the cyclic group $\mathbb{Z}$ and suppose $K$ is a subgroup of $G$. By Kurosh Subgroup Theorem we know that $K=F\ast (\ast\_{i\in I}(K\cap A^{u\_i}))$, where $F$ is free group and $u\_i$ are some representatives of double cosets $KxA$ in $G$.
Now suppose... | https://mathoverflow.net/users/44949 | Normal Subgroups of Free Products | Set $A$ equal to $\mathbb{Z}$, which satisfies the ascending chain condition ("ACC", every strictly ascending chain of (normal) subgroups eventually terminates). Then $G=\mathbb{Z}\ast\mathbb{Z}=F\_2$ and $F\_2$ contains normal subgroups that are not finitely generated.
Examples:
1) The commutator subgroup is norma... | 8 | https://mathoverflow.net/users/12996 | 114802 | 65,051 |
https://mathoverflow.net/questions/114790 | 0 | Let $J$ be a Jacobian variety defined over a field $k$ and let $\Theta$ be a symmetric theta-divisor on $J$.
It's shown (for instance) in the book Complex Abelian Varieties by Lange and Birkenhake that the linear system of $2\Theta$ is base point free if $k=\mathbb{C}$ and that it gives an embedding of the Kummer va... | https://mathoverflow.net/users/26576 | Reference request: base point freeness of $2\Theta$ | If $D$ is an ample divisor on an abelian variety, then $2D$ is base point free and $3D$ is very ample. One reference for this is Mumford's book "Abelian Varieties", II 6 and III 17.
| 3 | https://mathoverflow.net/users/3847 | 114804 | 65,052 |
https://mathoverflow.net/questions/114795 | 2 | Let $\{z\_n\}$ be an infinite sequence of complex numbers. Under which conditions on these numbers does there exist an entire function $f$ such that the $z\_n$ are the zeros of $f$ and $|f(z)|< C \exp(c|\Im z|)$ for some constants $C,c>0$?
| https://mathoverflow.net/users/23753 | Growth in imaginary direction of an entire function with prescribed zeros | It is unlikely that a simple explicit necessary and sufficient condition exists.
But some complicated condition is given in the paper
MR2411971
Favorov, S. Yu.
Zero sets of entire functions of exponential type with additional conditions on the real line.
Algebra i Analiz 20 (2008), no. 1, 138--145 (Russian). Translat... | 5 | https://mathoverflow.net/users/25510 | 114809 | 65,054 |
https://mathoverflow.net/questions/114800 | 1 | I am looking for a good starting point (book or articles) for studying Toeplitz matrices. Specifically as mentioned in the title, I am most interested in the case where they are of the form
$$A = \{\phi(i-j)\}\_{i,j\in\mathbb{Z}}$$
where $\phi:\mathbb{R}\to\mathbb{R}$ and $\phi(i-j)=\phi(j-i)$.
I so far have looked a... | https://mathoverflow.net/users/22500 | Infinite Real Symmetric Toeplitz Matrix Reference | I strongly recommend the following book
Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators
Lloyd N. Trefethen & Mark Embree
The first chapter is partly devoted to Toeplitz matrices, although their interest is focused on the non-selfadjoint case. Anyhow, the book is pleasant to read and very... | 0 | https://mathoverflow.net/users/21907 | 114811 | 65,056 |
https://mathoverflow.net/questions/114816 | 6 | Suppose $G$ is a finitely generated Hausdorff topological group. Must $G$ be first countable (or perhaps a sequential space)? What if we restrict to the abelian case?
I wonder if this is even true for the additive group of integers $\mathbb{Z}$. There certainly are non-discrete, Hausdorff group topologies on $\mathbb... | https://mathoverflow.net/users/5801 | Hausdorff group topologies on finitely generated groups | No. The Bohr topology on $\mathbb{Z}$ is not first countable, in fact the least size of a local base at $0$ is $2^{\aleph\_0}$. It is also known that this topology is not sequential (because there are no non-trivial convergent sequences).
| 11 | https://mathoverflow.net/users/17836 | 114821 | 65,057 |
https://mathoverflow.net/questions/114819 | 1 | Let X a smooth projective curve over $\mathbb{C}$.
We fix $d$ distinct closed points $x\_{1},\dots,x\_{d}$.
Can we find a finite surjective morphism $\pi:X\rightarrow\mathbb{P}^{1}$
and local uniformizers on $\mathbb{P}^{1}$, $t\_{1},\dots,t\_{d}$
such that $\forall i, k[t\_{i}^{-1}]\subset k[X-x\_{i}]$
---
| https://mathoverflow.net/users/27398 | finite surjective morphism to the projective line | No. This condition implies that $t\_i$, pulled back to $X$, vanishes only at $x\_i$. Thus, $x\_i$ is the entire fiber of the point where $t\_i$ vanishes, so if $n$ is the degree of the map, then $\mathcal O(1) = \mathcal O(n x\_i)$.
Thus $\mathcal O(nx\_i)=\mathcal O(nx\_j)$, so $x\_i-x\_j$ is an $n$-torsion divisor ... | 6 | https://mathoverflow.net/users/18060 | 114823 | 65,058 |
https://mathoverflow.net/questions/114814 | 2 | This question is related to my previous question [about the regularity of the Maxwell equations](https://mathoverflow.net/questions/112086/regularity-of-the-maxwell-equations).
Assume we are working on a space where there are only electric point charges, $(q\_i)$, and a blob of matter $M$ which is a bounded, closed a... | https://mathoverflow.net/users/5295 | Distributional limits concerning the regularity of Maxwells equations | Yes, there is a way of interpreting this integral. Decompose E into a component $E\_t$ tangential to level sets of $\epsilon$ and a component $E\_n$ normal to them. Moreover, let us set $D=\epsilon E$, and decompose it in the same way. You can write your integrand as
$E\_t^2\nabla \epsilon-D\_n^2\nabla(1/\epsilon)$. No... | 3 | https://mathoverflow.net/users/12120 | 114831 | 65,061 |
https://mathoverflow.net/questions/114849 | 3 | Hallo,
It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught of as some open neighbourhood of the zero section of the cotangent bundle $T^{\*}M$. This complex manifold is far from b... | https://mathoverflow.net/users/22073 | Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold | I think the answer to your question should be positive and below is a sketch of what should work (I think).
Any real analytic manifold can be realised as the real part of a complex projective manifold. I.e. one should embedd the real analytic manifold smoothly in $\mathbb R^n$ and then approximate by a real algebraic... | 2 | https://mathoverflow.net/users/943 | 114859 | 65,072 |
https://mathoverflow.net/questions/114850 | 6 | 1. Is it true that if a Fibonacci number $F\_{n}$ divides the product of two Fibonacci numbers, then it must divide at least one of them?
2. Is it true that for all $n \ne 1,2,6,12$, there exists a prime divisor $p$ of $F\_{n}$ such that the entry point (first appearance as a divisor in the Fibonacci sequence) of $p$ i... | https://mathoverflow.net/users/29500 | Is every Fibonacci number "Fibonacci-prime"? | As François Brunault said in his comment, your #2 is true: Carmichael's theorem, which establishes the existence of a *primitive prime divisor* of $F\_n$ for every $n\ne1,2,6,12$, together with [$\gcd(F\_m,F\_n) = F\_{\gcd(m,n)}$](http://en.wikipedia.org/wiki/Fibonacci_number#Primes_and_divisibility), shows that no Fib... | 15 | https://mathoverflow.net/users/5091 | 114866 | 65,075 |
https://mathoverflow.net/questions/114875 | 23 | Original Problem
----------------
If $f$ is an entire function such that
$$ f(z+1)-f(z)=f'(z) $$
for all $z$.
Is there a non-trivial solution? ($f(z)=az+b$ is trivial)
And here is something uncertainty
---------------------------------
If we use Fourier transform, how to define it to ensure any entire function ha... | https://mathoverflow.net/users/22954 | On equation $f(z+1)-f(z)=f'(z)$ | Linear functional equations can be solved with Fourier transform.
Let $\lambda\_k$ be the roots of the equation $e^\lambda-1=\lambda$. There are infinitely many
such roots. Then
$$f(z)=\sum\_k a\_ke^{\lambda\_k z}$$
is a solution.
Here the sum can be finite, and $a\_k$ arbitrary, or the sum can be infinite, and $a... | 56 | https://mathoverflow.net/users/25510 | 114878 | 65,081 |
https://mathoverflow.net/questions/114881 | 2 | Let $f:\mathbb{R}\to\mathbb{R}$ a *convex decreasing* function. Let $x\_0 < x\_1 < x\_2$.
Studying the behaviour of the difference quotient, it is clear that
$$f(x\_0)-f(x\_2) \leq M (f(x\_0)-f(x\_1))$$
with $M=\frac{x\_2-x\_0}{x\_1-x\_0}>0$.
Now take $F:\mathbb{R}^2\to\mathbb{R}$ *convex* and *decreasing* with respe... | https://mathoverflow.net/users/22980 | A consequence of convexity | You can use the triangle inequality to solve this by looking at each coordinate separately.
$F(x\_0,y\_0)-F(x\_2,y\_2)=F(x\_0,y\_0)-F(x\_2,y\_0)+F(x\_2,y\_0)-F(x\_2,y\_2)$ $\leq M\_1(F(x\_0,y\_0)-F(x\_1,y\_0))+M\_2(F(x\_2,y\_0)-F(x\_2,y\_1))$. Replacing $F(x\_1,y\_0)$ with $F(x\_1,y\_1)$ only increases the right side... | 2 | https://mathoverflow.net/users/27933 | 114883 | 65,082 |
https://mathoverflow.net/questions/114887 | 3 | Let HALTS-IN-N be the canonical $EXPTIME$-complete language {<$M$,$n$> | the deterministic Turing machine (DTM) encoded by $M$ halts in $n$ or fewer steps, with $n$ encoded in binary}. Since HALTS-IN-N is accepted in $EXPTIME$ by simulating $M$ for $n$ steps (or until it halts, whichever comes first), its complement is... | https://mathoverflow.net/users/29482 | If NP=EXPTIME, does every DTM have a succinct "execution proof"? | Q1: Yes (except that the certificates you get may have size polylogarithmic in $n$, not just logarithmic, and you need to apply the argument to both HALTS-IN-N and its complement, as pointed out by Andreas).
Q2: Well, NP = EXP contradicts all kinds of conjectures from complexity theory: it makes the polynomial hierar... | 7 | https://mathoverflow.net/users/12705 | 114892 | 65,084 |
https://mathoverflow.net/questions/114895 | 4 | Let $Y$ be a projective scheme. The naive definition of a Hilbert scheme of subschemes $X$ of $Y$ would require us to projectively embed $Y$, then ask that $X$ have a fixed Hilbert polynomial $p$.
However, this space is usually disconnected. Cheap example: $Y$ is two points, $p=1$, and the Hilbert scheme is $Y$ itsel... | https://mathoverflow.net/users/391 | Disconnectedness of Hilbert schemes of projective schemes | Take a generic quintic in $\mathbb CP^4$ and consider lines on it
| 5 | https://mathoverflow.net/users/13441 | 114896 | 65,086 |
https://mathoverflow.net/questions/114889 | 4 | I am trying to understand some things related to elliptic curves and finite flat group schemes but I am a little bit confused.
Let $A$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$.
Let $F: A \rightarrow A^{(p)}$ be the Frobenius isogeny. Then $\ker F$ is as a finite f... | https://mathoverflow.net/users/29513 | Kernel of powers of Frobenius on supersingular elliptic curves | The kernel of $F^2$ is the same as the kernel of $[p]$, once $A^{(p^2)}$ and $A$ are identified, and is a non-trivial extension of $\alpha\_p$ by $\alpha\_p$, whose class can be described in terms of the supersingular modular form $B$. See Ulmer, p-descent in characteristic p. Duke Math. J. 62 (1991), 237–265, section ... | 4 | https://mathoverflow.net/users/2290 | 114900 | 65,088 |
https://mathoverflow.net/questions/114901 | 0 | Let $S$ be a smooth projective surface and $C$ a smooth, irreducible curve contained in $S$. Let $E\_1$ and $E\_2$ be two vector bundles on $S$ having the same rank and assume they lie in a short exact sequence
$$
0\to E\_1\stackrel{f}{\to} E\_2\to A\to 0, $$
with $A\in\mathrm{Pic}(C)$. In this situation $E\_1$ is cal... | https://mathoverflow.net/users/33841 | Elementary transformations and determinant maps. | There is no relation. In fact, if $E\_2$ is fixed then $\det E\_2\otimes O\_C$ is fixed, while $A$ can be taken to be any invertible quotient of $E\_{2|C}$.
| 0 | https://mathoverflow.net/users/4428 | 114908 | 65,091 |
https://mathoverflow.net/questions/114905 | 6 | I am curious to know if the following number is irrational or transcendental:
$$\displaystyle A = \sum\_p 2^{-p},$$
where the sum is over all positive primes. A similar question can be asked for any number $k$ other than 2.
Edit: In retrospect and as pointed out by some comments below that it is trivial that $A$ is... | https://mathoverflow.net/users/10898 | Are these numbers irrational and/or transcendental? | The number $A$ is irrational, since the characteristic function of the set of prime numbers is not eventually periodic.
It definitely should be transcendental as it is not a base $2$ normal number, ie the asymptotic frequency of digits is not the same (and algebraic numbers are conjectured to be normal). But this sh... | 11 | https://mathoverflow.net/users/nan | 114910 | 65,092 |
https://mathoverflow.net/questions/114893 | 14 | A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.
I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous map $\phi$ from the unit disk $|z|\le 1$ to $\mathbb C^1$ such that
$\phi$ is holomorphic for $|z|<1$ and the image of t... | https://mathoverflow.net/users/13441 | A "holomorphic" Peano curve? | Here it is:
MR0015154
Salem, R.; Zygmund, A.
Lacunary power series and Peano curves.
Duke Math. J. 12, (1945). 569–578.
| 17 | https://mathoverflow.net/users/25510 | 114914 | 65,093 |
https://mathoverflow.net/questions/112069 | 7 | (Sorry for the [crossposting](https://math.stackexchange.com/questions/234108/a-fibrant-objects-structure-on-bf-top), but I'm really interested in this question).
One can [define](http://arxiv.org/abs/1011.2926) (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological... | https://mathoverflow.net/users/7952 | A fibrant-objects structure on Top | Your question was addressed in the following paper:
Carmen Elvira-Donazar and Luis-Javier Hernandez-Patricio. [Closed model categories for the $n$-type of spaces and simplicial sets](http://www.sci-prew.inf.ua/v118/1/S0305004100073485.pdf). Math. Proc. Camb. Phil. Soc. (1995), 118, 93.
Allow me to define an $n$-fib... | 6 | https://mathoverflow.net/users/11540 | 114916 | 65,094 |
https://mathoverflow.net/questions/114913 | 1 | Assume that $A$ is an arbitrary positive integrable function on $[0,1]$. Whether exists a convex function $f\_A(x)=x g(x)$ of $(0,+\infty)$ into itself (depending on $A$) such that $\lim\_{x\to +\infty} g(x)=+\infty $ and $$\int\_0^1 A(x) g(1/x^2) dx <+\infty.$$ This question is related to membership of $g$ to some Din... | https://mathoverflow.net/users/26543 | Dini condition and integrability condition | The answer is yes. First define inductively a sequence $x\_k>0$ such that $x\_{k+1} < x\_k/2$, and
$$\int\_0^{x\_k}A(x)dx<2^{-k}.$$
This is possible because $A$ is integrable.
Then define a continuous function $[0,1]$ by $h(x\_k)=k$
and $h$ is linear on each interval $[x\_{k+1},x\_k]$. It is easy to see that
this fu... | 2 | https://mathoverflow.net/users/25510 | 114920 | 65,096 |
https://mathoverflow.net/questions/114909 | 6 | What is known about normal subgroups of $SL\_2(\mathbb{C}[X])$? Can one hope for a congruence subgroup property, i.e. that every (non-central) normal subgroup contains the kernel of the reduction modulo some ideal of $\mathbb{C}[X]$?
| https://mathoverflow.net/users/14497 | Normal subgroups of $SL_2$ of a polynomial ring | [EDIT] These groups have been studied for a long time from various viewpoints, so there is a long paper-trail. I'd emphasize however that working over the complex numbers is usually similar to working over an arbitrary infinite field.
Finite fields on the other hand occur more often in arithmetic contexts.
Concernin... | 9 | https://mathoverflow.net/users/4231 | 114921 | 65,097 |
https://mathoverflow.net/questions/114925 | 5 | McDuff proved that there exist continuum many non-isomorphic (separable) II${}\_1$ factors. I would like to politely ask whether it is known/open if one can find $2^{\mathfrak{c}}$ (or at least $\mathfrak{c}^+$) many such factors.
My feeling is that this is not possible to construct more than $\mathfrak{c}$ separable... | https://mathoverflow.net/users/29433 | Number of II${}_1$ factors | Your argument is correct. An alternate and more "intrinsic" argument is to look at the predual, which is a separable Banach space. There are only continuum many separable Banach spaces, since they are determined by the metric on a countable dense subset that is a $\mathbb Q[i]$-vector space. Thus there are only continu... | 6 | https://mathoverflow.net/users/75 | 114931 | 65,101 |
https://mathoverflow.net/questions/114824 | 4 | How do you solve this recurrence (or multivariate recurrences in general)? Note that $p\in[0,1]$ and $n\in\mathbb{N}$ are given constants, where $np\leq 1$.
$$f:(\mathbb{N}\cup\{0\})\times(\mathbb{N}\cup\{0\})\rightarrow[0,1]$$
Base case: $f(0,b)=(1-np)^b$ $\forall$ $b\geq 0$.
$f(a,b)=f(a-1,b-1)[n-(a-1)]p+f(a,b-1)[... | https://mathoverflow.net/users/29495 | How to solve a specific multivariate recurrence relation (or general ones) | I get $$f(a,b) = \frac{ (1-np)^b n!}{(n-a)!} \sum\_{j=0}^b \binom{b}{j} \left\{ j \atop a \right\} \left(\frac{p}{1-np}\right)^j,$$ where $\left\{ j \atop a \right\}$ is a [Stirling number of the second kind](http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind). I tested the formula in Mathematica against ... | 6 | https://mathoverflow.net/users/9716 | 114941 | 65,107 |
https://mathoverflow.net/questions/114938 | 3 | Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell\_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not very important, I just want to exclude some trivial cases, like the algebra of finite/cofinite subsets on some uncounta... | https://mathoverflow.net/users/29433 | Algebras with countable chains only | The second requirement is too strict: it makes the first one impossible.
The commutative von Neumann algebra $\ell^\infty(\mathbb{N})$ has as Gelfand spectrum the Stone-Cech compactification of $\mathbb{N}$ (with the discrete topology). This, in turn, is the Stone space of the Boolean algebra $\mathcal{P}(\mathbb{N})... | 1 | https://mathoverflow.net/users/10368 | 114944 | 65,108 |
https://mathoverflow.net/questions/114948 | 8 | While dealing with $BO(n)$, $BSO(n)$ and $BSpin(n)$ with the universal coefficient theorem and Künneth formula, I came to have the following question:
The universal coefficient says $H^n(X;M)\cong \hom(H\_{n}(X;\mathbb{Z}),M)\oplus {\rm Ext}^{1} (H\_{n-1}(X;\mathbb{Z},M))$ for a $\mathbb{Z}$-module $M$.
When $X=BSp... | https://mathoverflow.net/users/12744 | Homology of classifying space of spin group BSpin(n) | Consider the classifying space $EG$ of a given group $G$. In your case, $G={Spin}(n)$. We have a fibration $G\to EG\to BG$ where $EG$ is contractible and $G$ acts freely on it. Therefore, the long exact sequence in homotopy groups tells you that $\pi\_j(BG)\cong \pi\_{j-1}(G)$. But $G={Spin}(n)$ is a Lie group which is... | 8 | https://mathoverflow.net/users/1993 | 114953 | 65,112 |
https://mathoverflow.net/questions/114879 | 2 | I have two tables at my disposal, one work dataset and one reference dataset. Each dataset has got two columns, lets say these are fields A and B. I would like the rows in reference dataset with the rows in work dataset that are 'closest' w.r.t. some distance. I have more rows in work dataset than in reference dataset,... | https://mathoverflow.net/users/29509 | An optimization problem, non complete bipartite graph and hungarian algorithm | There's a standard trick to convert the min cost matching problem on a balanced bipartite graph to one on an unbalanced bipartite graph. Let $G = (X \cup Y, E, w)$ be the bipartite graph where $E \subset X \times Y$ and $|X| \le |Y|$.
Now create a copy of $G$ and add it "reversed", so that in the new graph both side... | 5 | https://mathoverflow.net/users/972 | 114954 | 65,113 |
https://mathoverflow.net/questions/114927 | 3 | It is well known that any plane cubic curve can be obtained as the discriminant locus of a conic bundle (actually even just of a net of conics). Does this hold true also for all nodal cubics (with double lines over the nodes)? How does one see this?
| https://mathoverflow.net/users/29522 | plane cubics and conic bundles | Yes, at least if we're not in characteristic 2.
Since all nodal cubics are projectively equivalent, it is enough to find one example. Trying a few symmetric $3 \times 3$ determinants soon turns up the matrix
$$
M = \left[ \begin{array}{ccc}
x & x & y \cr
x & z & 0 \cr
y & 0 & x
\end{array} \right]
$$
with determina... | 7 | https://mathoverflow.net/users/14830 | 114958 | 65,115 |
https://mathoverflow.net/questions/114955 | 5 | Given $M$ a finite von Neumann algebra with trace $\tau$, $T\in M$ invertible.
The Fuglede-Kadison determinant is defined as
$\Delta(T)=e^{\tau(log|T|)}$,
where $|T|=(T^\*T)^{\frac{1}{2}}$, and $\tau(log|T|)=\int\_{0}^{||T||}log(t)d\mu\_{|T|}(t)$, and the probability measure $\mu\_{|T|}$ is defined on spectrum(... | https://mathoverflow.net/users/9305 | Fuglede-Kadison determinants in $L(\mathbb{F}_2)$ | The spectral measures for self-adjoint elements in $\mathbb C F\_2$ are very special. In particular, it is known that non of the elements in $\mathbb C F\_2$ has a kernel when acting via the left-regular representation on $\ell^2 F\_2$. This was shown by Peter Linnell using index-theoretic methods in
Linnell, Peter,
... | 7 | https://mathoverflow.net/users/8176 | 114968 | 65,119 |
https://mathoverflow.net/questions/114828 | 5 | Let $R$ be a $k$-algebra ($k$ a field) and a domain of finite Krull dimension. In
$\quad$ [Krull dimension less or equal than transcendence degree?](https://mathoverflow.net/questions/79959/krull-dimension-transcendence-degree/79974#79974)
it is shown that
$$\text{Krull-dim}(R) \le \text{trans.deg}\_k Quot(R).\t... | https://mathoverflow.net/users/10194 | Is this height-transcendence-degree inequality true without AC ? | Here is a proof of (\*\*) by induction on the height of $P$.
If $P=0$, the inequality (\*\*) is obvious. Let $P$ be a prime ideal of $R$ of height $d \geq 1$, and consider a chain of prime ideals $0=P\_0 \subset P\_1 \subset \ldots \subset P\_d = P$ of length $d$ in $R$. The domain $R/P\_1$ has finite Krull dimension... | 3 | https://mathoverflow.net/users/6506 | 114978 | 65,122 |
https://mathoverflow.net/questions/114974 | 5 | I have probably a stupid question about representations of algebraic groups:
Let $G$ be an algebraic group and $L$ be a Lie algebra of $G$. What is the connection between
categories of representations of $G$ and $L$ (are they equivalent)?
Now, let $V$ be an irreducible representation of $G$, how to prove that $V$ i... | https://mathoverflow.net/users/29536 | algebraic groups and their Lie algebras | I suggest the following lecture notes of Bruhat:
www.math.tifr.res.in/~publ/ln/tifr14.pdf
Chapter 3 & 4 should answer most of your questions.
For example, there are statements like this :
Proposition 1(pg.19). To every analytic representation h : G −→ G′ there
corresponds a map dh : U(G) → U(G′) which is a rep... | 2 | https://mathoverflow.net/users/10400 | 114980 | 65,123 |
https://mathoverflow.net/questions/114979 | 1 | Hi,
We consider subspaces of $\mathbb{R}^N$.
Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of $\mathbb{R}^N$ into $\{0,1\}$.
The variables $X\_1,\dots,X\_N$ are gaussian taking their values in $\mathbb{R}^N$. The... | https://mathoverflow.net/users/29537 | Equivalence between choosing a subspace and choosing its orthogonal | The Grassmannian $\mathrm{Gr}(n,N)$ is a compact manifold with a well defined canonical probability measure, its "uniform measure", characterized by the fact that it is invariant by the action of the orthogonal group $O(N)$. Starting from those $(X\_i)\_ i$, both $\mathrm{Span}(X\_1,\dots,X\_n)$ and $\mathrm{Span}(X\_1... | 2 | https://mathoverflow.net/users/6101 | 114983 | 65,125 |
https://mathoverflow.net/questions/114548 | 5 | Given two random sets $A,B$ in a finite field (say $x\in A$ independently and with probability $1/2$), what is known about the additive energy $E(A,B)=|\{(a,a',b,b')\in A\times A\times B\times B: a+b=a'+b'\}|$?
Equivalently, what is the distribution of the random variable
$\|1\_A\*1\_B\|\_2$? I'm mostly interested i... | https://mathoverflow.net/users/18698 | Additive energy of random sets | Expanding on my earlier comment, the concentration has to be really quite good: at least exponential in $N$, the order of the additive group. (Edit: and you can't get better concentration because the state space is only exponentially large.)
Recall the following variant of the Hoeffding-Azuma martingale concentration... | 5 | https://mathoverflow.net/users/25485 | 114987 | 65,128 |
https://mathoverflow.net/questions/113980 | 6 | I'm working through the details of Deligne and Mumford's 69' paper, "The Irreducibility of the Space of Curves of Given Genus", and I had a few quick questions:
1) On p. 77, they claim that for $x$ a node, $\pi:C'\rightarrow C$ the blowing-up of $x$ and $x\_1,x\_2\in C'$ the two points of $\pi^{-1}(x)$. Then supposed... | https://mathoverflow.net/users/13139 | Questions on theorem in Deligne-Mumford's '69 Paper: $\omega_C^n$ is very ample $n\geq 3$ | Since no one has written an official answer to this question and the bounty is ending in a few hours, I'll provide an answer of my own to the question, though see ulrich's comments for another approach.
1) The crucial observation here is that $\mathfrak m\_x=\pi\_\*(\mathcal O\_{C'}(-x\_1-x\_2))$ where $x\in C$ is a ... | 3 | https://mathoverflow.net/users/13139 | 114990 | 65,130 |
https://mathoverflow.net/questions/114988 | 3 | Let $\mathbb{F}\_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$ and let $D$ be the very ample divisor $3C\_0+5f$ on $\mathbb{F}\_1$ (notation as in [Hartshorne, Algebraic geometry, p. 373]). Then $|D|$ gives an embedding of $\mathbb{F}\_1$ in $\mathbb{P}^{17}$ as a surface of degree $21$.
... | https://mathoverflow.net/users/15606 | Equations of the Hirzebruch surface embedded in a large space. | Denote by $T\_0,T\_1$ a basis for the global sections of $\mathcal{O}\_{\mathbb{F}\_1}(f)$. Denote by $X$ a basis for the global sections of $\mathcal{O}\_{\mathbb{F}\_1}(C\_0)$. Denote by $Y$ a global section of $\mathcal{O}\_{\mathbb{F}\_1}(C\_0+f)$ that is linearly independent from $T\_0X,T\_1X$. Then the "total coo... | 10 | https://mathoverflow.net/users/13265 | 114991 | 65,131 |
https://mathoverflow.net/questions/114996 | 5 | Suppose that $(S,\*)$ is a finite set equipped with a binary operation. Extend the binary operation to the vector space $V$ with basis $S$.
The set of probability measures on $S$, viewed as a compact convex subset of $V$ is closed under $\*$
and, since $\*$ is continuous, there are idempotent measures on $S$.
Must tw... | https://mathoverflow.net/users/10774 | Do distinct idempotent measures on finite binary systems have distinct supports? | No if I understood. Take the two element left zero semigroup. All measures are idempotent.
**Added** a left zero semigroup is one satisfying the identity xy=x
**Added** A finite semigroup $S$ satisfies that distinct idempotent measures have distinct support iff for all idempotents $e,f\in S$ one has $SeS=SfS$ imp... | 4 | https://mathoverflow.net/users/15934 | 114999 | 65,134 |
https://mathoverflow.net/questions/114950 | 5 | Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wavefront sets of $u$ is given by:
$WF\_h(u)=(\mathrm{supp}(u)\times\{0\})\cup WF(u)$.
Would someone please explain this relationship? Than... | https://mathoverflow.net/users/29413 | C^\infty versus semiclassical wavefront sets | Heuristically, this result follows because the semiclassical wavefront set measures oscillations at frequency $\frac{\xi}{h}$. To understand this, observe that since $u$ does not depend on $h$, if $u$ is smooth, it has small high frequency oscillations, and hence, for $\xi\neq 0$, and hence points where $u$ is smooth a... | 2 | https://mathoverflow.net/users/29549 | 115008 | 65,138 |
https://mathoverflow.net/questions/115000 | 5 | I should be tarred and feathered for not knowing at least the status of the following question.
>
> **Question:** Let $\Gamma$ be a discrete amenable group. If $\pi:\Gamma \rightarrow B(\mathcal{H})$ is a unitary representation of $\Gamma$ on a separable Hilbert space $\mathcal{H}$, is the von Neumann algebra $\pi... | https://mathoverflow.net/users/6269 | Is the von Neumann algebra associated to a unitary representation of an amenable group always injective? | This is because you get a representation of the maximal(=reduced by amenability) C\*-algebra $C^\ast(\Gamma)$ on $H$, and the image has to be a nuclear algebra because it's a quotient of nuclear one. Then, $\pi(\Gamma)''$, being a weak\*-closure of a nuclear C\*-algebra, has to be injective.
| 8 | https://mathoverflow.net/users/9942 | 115009 | 65,139 |
https://mathoverflow.net/questions/114977 | 5 | I asked this question previously on math.stackexchange.com, where it had little traction.
Consider the symmetric random walk on $\{0,1,…,n\}$ with transition probabilities $P(j→j±1)=1/2$ for $0 < j < n$ and $P(0→0)=P(0→1)=P(n→n)=P(n→n−1)=1/2$. I am interested in the spectrum of the transition matrix (which is symmet... | https://mathoverflow.net/users/7352 | Spectrum of transition matrix for symmetric random walk | The previous answer of Pablo Lessa seems to be related to a different problem:
with periodic
boundary conditions. Your conditions are not periodic.
Your matrix is a special Jacobi matrix, and the characteristic polynomial
can be found explicitly.
Let $A$ be your matrix, $x=(x\_0,\ldots,x\_n)$ an eigenvector with ei... | 2 | https://mathoverflow.net/users/25510 | 115014 | 65,140 |
https://mathoverflow.net/questions/111948 | 10 | I am currently reading the 1998 article *Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators* by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following inequality: if $f$ is a holomorphic function defined in an open ball $U$, $\Gamma$ is a circular contour contained in ... | https://mathoverflow.net/users/1840 | Modern version of an inequality of R. M. Gabriel for contour integrals | This has been called "Gabriel's problem" by Ana Granados, who has reviewed it in [Michigan Math. J. **46**, 461-487 (1999)](http://projecteuclid.org/euclid.mmj/1030132475). I think you'll find all the background and recent developments, together with open problems, that you might want in her overview. There are several... | 3 | https://mathoverflow.net/users/11260 | 115019 | 65,144 |
https://mathoverflow.net/questions/114982 | 8 | The Riemann-Hilbert correspondence, as proved by Kashiwara and Mebkhout, says that for X a smooth algebraic variety over $\mathbb{C}$ there is an equivalence of triangulated categories
$D^b\_c(X,\mathbb{C})\cong D^b\_\mathrm{rh}(\mathcal{D}\_X)$
between the bounded derived category of complexes of $\mathbb{C}$-modu... | https://mathoverflow.net/users/13647 | Tensor product of $\mathcal{D}$-modules and constructible sheaves | This is correct. Verdier duality does not preserve tensor products in general. Another point of view is that each of these categories has two versions of tensor product, $\otimes ^\ast$ and $\otimes ^!$, which are interchanged by Verdier duality. It just happens that for $D$-modules the shriek version (or a shift of it... | 10 | https://mathoverflow.net/users/7762 | 115026 | 65,150 |
https://mathoverflow.net/questions/114922 | 3 | Let $p\in (1,\infty)$ and let $q$ be conjugate to $p$. Is there a subspace of $\ell\_1(\ell\_p)$ isomorphic to $\ell\_q$? Of course, I am uninterested in the case $p=2$.
| https://mathoverflow.net/users/29433 | Embedding of $\ell_p$ into infinite direct sums | The answer is no. Let $P\_n$ be the natural projection from $Z\_{1p} :=\ell\_1(\ell\_p)$ onto the sum of the first $n$ copies of $\ell\_p$. Let $Z$ be any subspace of $Z\_{1p}$ that contains no isomorphic copy of $\ell\_p$. Then the restriction of $P\_n$ to $Z $ is strictly singular, so there is a norm one vector $x\_n... | 3 | https://mathoverflow.net/users/2554 | 115033 | 65,152 |
https://mathoverflow.net/questions/115034 | 2 | Suppose we are given a thick subcategory of the compact objects in the homotopy category of modules over a ring spectrum $R$. Are there conditions we can place on $R$, or on the category (compact) $R$-modules to ensure that every thick subcategory has a single generator? It seems that, given the existence of finite Bou... | https://mathoverflow.net/users/11546 | Generators of Thick Subcategories | When $R$ is the Eilenberg-MacLane spectrum of a Noetherian ring, thick subcategories are in bijection with specialization-closed subsets of $\mathrm{Spec}\ \pi\_0(R)$. Such a thick subcategory is generated by a single compact object iff the specialization-closed subset is actually Zariski-closed (and in that case a gen... | 4 | https://mathoverflow.net/users/75 | 115042 | 65,158 |
https://mathoverflow.net/questions/115046 | 9 | In June 2012, Bill Press and Freeman Dyson published a [remarkable paper](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387070/) on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that I feel I should have been explicitly aware of all my life but wasn't: If $M$ is a s... | https://mathoverflow.net/users/3106 | Determinantal formula for the nullspace of a singular matrix | If the rank of $M$ is $k$ then $\Lambda^kM$ is a ${n\choose k}\times{n\choose k}$ matrix (consisting of al $k\times k$ minors of $M$) of rank $1$. Any of its nonzero rows is a point on $Gr(k,n)$ in its Plucker embedding. This point gives equations of the nullspace.
For example, if $k=1$ then any nonzero row of $M$ gi... | 11 | https://mathoverflow.net/users/4428 | 115048 | 65,162 |
https://mathoverflow.net/questions/115030 | 2 | Introduction
------------
Let $k$ be a local field. Let $C$ be the spectrum of $\mathcal{O}\_{k}$. Let $X/k$ be a smooth projective curve with a semistable model $\mathcal{X}/C$.
Let $J$ be the Jacobian of $X$. The identity component of the reduction $\tilde{J}$ fits into an exact sequence
`\[ 1 \to L \to \tilde{J}... | https://mathoverflow.net/users/21815 | Relating the toric rank of a semistable curve and the first Betti number of its reduction graph | Your "reduction" $\widetilde{J}$ is really the identity component of the reduction. Also, I think you should assume $\mathcal{X}$ is regular (as may be arranged). The big theorem that is relevant here is due to Raynaud (see 9.5/4 in "Neron Models"): the relative identity component of the Neron model is the separated op... | 5 | https://mathoverflow.net/users/29283 | 115055 | 65,167 |
https://mathoverflow.net/questions/115001 | 32 | Rencently a breakthrough was made in the context of the **Minimal Model Program** by the work of Birkar-Cascini-Hacon-McKernan. They proved that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.
Since I'm a master student and so I have no a wide view of the subject ... | https://mathoverflow.net/users/11927 | Open problems in Birational Geometry, after BCHM | [Just 'cause Artie asked:] :)
Many parts of the mmp are not know for log canonical pairs. There are many results in that direction, but also many questions are open. In some sense log canonical is a more natural class than klt or even dlt and it is very important from the point of view of applications to moduli theor... | 18 | https://mathoverflow.net/users/10076 | 115057 | 65,169 |
https://mathoverflow.net/questions/115061 | 25 | Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
Not every profinite group is an absolute galois group of a field (the only finite ones have order $1$ or $2$ by Artin-Sc... | https://mathoverflow.net/users/2841 | Profinite groups as étale fundamental groups | [Edit:] The answer should be positive, that is, every profinite group appears as the fundamental group of a scheme. Here is a sketch of proof.
First of all, I claim that for any finite group $G$ there exists a complex affine simply connected variety $X$ with a free action of $G$. Start from a faithful finite-dimensio... | 33 | https://mathoverflow.net/users/4790 | 115064 | 65,172 |
https://mathoverflow.net/questions/115051 | 4 | (the title got out of hand)
Say I have a surface $X$, then I also have M, the Hilbert scheme of curves and points on X.
This can be seen as a moduli space of quotients $O\_X \to O\_Z$.
If $I\_Z$ is the kernel of that map, I would like to impose the condition that $Ext^2(I\_Z,O\_Z) = 0$. I imagine this is badly beha... | https://mathoverflow.net/users/25442 | is there some condition I can impose on families of curves on a surface such that the second Ext between the ideal sheaf and the structure sheaf is zero? | If I am not mistaken, we *always* have $\mathrm{Ext}^2(\mathcal I\_Z,\mathcal O\_Z)=0$ for any closed subscheme $Z\subset X$, so the question is somewhat vacuous.
We have an exact sequence
$$0\to \mathcal I\_Z \to \mathcal O\_X \to \mathcal O\_Z\to 0,$$
and applying $\mathrm{Hom}( -,\mathcal O\_Z)$ gives a surj... | 5 | https://mathoverflow.net/users/7399 | 115066 | 65,173 |
https://mathoverflow.net/questions/114972 | 7 | Let $\{x\_1,\dots,x\_n\}$ be pairwise distinct complex numbers and $l\_1+l\_2+\dots+l\_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as
$$V=
\begin{bmatrix}
v\_{1,0}&v\_{2,0}&\dots&v\_{n,0}\\\\
v\_{1,1}&v\_{2,1}&\dots&v\_{n,1}\\\\
\vdots\\\\
v\_{1,N-1}&v\_{2,N-1}&\dots&v\_{n,N-1}
\end{bmatrix}$$ where $... | https://mathoverflow.net/users/23862 | Norm of inverse confluent Vandermonde matrix | For $j=1,\dots,n$ and $k=0,1,\dots,l\_j-1$ denote by $u\_{j,k}$ the row with index $l\_1+\dots +l\_{j-1}+k$ of the matrix $V^{-1}$. By using a generalization of the Hermite interpolation formula (see [3]), in [2] it is shown that the elements of $u\_{j,k}$ are the coefficients of the polynomial
$$ {1\over k!} \sum\_{t=... | 2 | https://mathoverflow.net/users/23862 | 115094 | 65,184 |
https://mathoverflow.net/questions/115090 | 7 | Suppose we have two stochastic differential equations with the same initial conditions:
$$d X\_t^1= b\_1(t,X\_t^1)dt + dW\_t$$
$$d X\_t^2= b\_2(t,X\_t^2)dt + dW\_t,$$
$X\_0^1=X\_0^2=x\_0$; $W\_\cdot$ is a standard one-dimensional Brownian motion, the functions $b\_1$ and $b\_2$ are uniformly bounded and Lipschitz.
Us... | https://mathoverflow.net/users/7646 | total variation distance between two solutions of SDE | Such a bound can be derived with Girsanov's theorem and Pinsker's inequality. Let $X\_t = x\_0 + W\_t$. Supposing $b$ has linear growth in $x$, we may define measures $P\_i$, $i=1,2$ by
$\frac{dP\_i}{dP} = \exp\left(\int\_0^Tb\_i(t,X\_t)dW\_t - \frac{1}{2}\int\_0^T|b\_i(t,X\_t)|^2dt\right)$.
Then, under $P\_i$, $W^... | 6 | https://mathoverflow.net/users/26459 | 115095 | 65,185 |
https://mathoverflow.net/questions/115091 | 15 | In ZFC there is no set that is the set of all sets, for this we introduce the notion of class. But then what is the 'class' of all classes, is it actually a class? Do we apply the same idea again? But then at what stage do we stop? Does this show that classes are not the right notion to go beyond sets, but more of an a... | https://mathoverflow.net/users/22002 | What notion captures the 'class' of all classes? | Since the question is rather philosophical (e.g., "right notion"), I'll use it as an excuse to record my philosophical opinions on this topic. The intuition underlying ZFC, i.e., the intuition of the cumulative hierarchy of sets, contains two quite vague notions, (1) the notion of "arbitrary subset" of an infinite set,... | 19 | https://mathoverflow.net/users/6794 | 115096 | 65,186 |
https://mathoverflow.net/questions/115092 | 6 | Hallo,
Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G\_{\mathbb{C}}$. Does there exist a Kähler structure on $G\_{\mathbb{C}}$ which is $G$-invariant (maybe in a neighbourhood of $G$ in $G\_{\mathbb{C}}$)?
hapchiu
| https://mathoverflow.net/users/22073 | Kähler form on complex Lie group | Yes, such a Kähler form always exists: Embed $G$ as a matrix group in $\mathrm{SU}(n)$ for some $n$ and then let $G\_\mathbb{C}\subset \mathrm{SL}(n,\mathbb{C})\subset M\_n(\mathbb{C})$ be the complexification. Choose a Kähler form on this latter vector space, pull it back to $G\_\mathbb{C}$ and then, using the compact... | 13 | https://mathoverflow.net/users/13972 | 115103 | 65,189 |
https://mathoverflow.net/questions/115101 | 6 | Let $A$ be a unital $C^\*$-algebra, let $G$ be a compact group, let $\alpha:G\to\mbox{Aut}(A)$ be a continuous action, and let $H$ be a closed subgroup of $G$. Is there any relationship between the crossed products $A\rtimes\_\alpha G$ and $A\rtimes\_{\alpha|\_H}H$?
I really only need this for $G=\mathbb{T}$ the unit... | https://mathoverflow.net/users/29566 | Crossed product of a C*-algebra by a subgroup | You always have an injective $\*$-homomorphism from $A\rtimes H$ into the multiplier algebra of $A\rtimes G$ (the reason is that you can view functions on $H$ as measures on $G$ which are supported on $H$). If $H$ is open in $G$ (a rather unfrequent situation, as you know), then $A\rtimes H$ sits as a $C^\*$-subalgebra... | 5 | https://mathoverflow.net/users/14497 | 115108 | 65,191 |
https://mathoverflow.net/questions/115104 | 2 | A typical formal statement of the Axiom of Replacement is (leaving out some technical details about extra "parameter variables" that $\phi$ sometimes takes.)
$[\forall x,y,z\ (\phi(x,y) \wedge \phi(x,z) \implies y=z)] \implies [\forall X \exists Y \forall y\ (y \in Y \iff (\exists x \in X)\ \phi(x,y))]$.
The hypoth... | https://mathoverflow.net/users/9961 | Weakest Hypothesis Needed for Axiom of Replacement | Basically, if one is trying axiomatize ZFC, then all these versions of replacement are equivalent modulo the other ZFC axioms, and there is really nothing going on here. The axiom is quite robust and is invariant under these kind of changes. There aren't really any hidden technicalities that cannot be easily overcome. ... | 3 | https://mathoverflow.net/users/1946 | 115111 | 65,193 |
https://mathoverflow.net/questions/115068 | 2 | In A. Mann's paper: Enumerating finite groups and their defining relations (1998, J. group theory), that can be found [Here](http://www.cecm.sfu.ca/~jborwein/ex2.pdf) ,
Mann's says (see pages 62-63):
" Let $F$ be a free group of rank d, and let $F\_i$ be the lower p-central series of $F$ . Let $ H= F/F\_{c+1} $ fo... | https://mathoverflow.net/users/25272 | Varieties Of Groups & Enumeration Of Size of Isomorphic Factor Groups | This property is not true in general for free groups. I have finally remembered a counterexample! There are 19 normal subgroups $N$ of the free group $F\_2$ of rank 2 with $F\_2/N \cong A\_5$. It was proved in
B.H. Neumann, and H. Neumann. "Zwei Klassen
charakteristischer Untergruppen und ihrer Faktorgruppen". Math. ... | 3 | https://mathoverflow.net/users/35840 | 115116 | 65,195 |
https://mathoverflow.net/questions/115099 | 11 | Grothendieck universes are equivalent to ZFC+a strongly inaccessible cardinal. This is low on the large cardinal axiom list. Is it enough to place category theory on a firm foundational basis, and how about higher category theory, does it remain enough?
| https://mathoverflow.net/users/22002 | Are grothendieck universes enough for the foundations of category theory? | Mike Shulman wrote a nice expository paper on set theoretical foundations for category theory
<http://arxiv.org/abs/0810.1279>
In Section 6 he explains the difficulties of working with large categories using just ZFC, and he discusses various ways to deal with these size issues. Some of these do not assume the exis... | 15 | https://mathoverflow.net/users/1649 | 115123 | 65,200 |
https://mathoverflow.net/questions/115112 | 73 | I hope this question is appropriate for MO. It comes from a genuine desire to understand the big picture and ground my own studies "morally".
I'm a graduate student with interest in number theory. I feel like I'm in danger of losing the big picture as I venture a bit deeper and reflect on where I am at. My fundamenta... | https://mathoverflow.net/users/13542 | How does "modern" number theory contribute to further understanding of $\mathbb{N}$? | You can find the answer in the history of the subject. For brevity let us consider the following two genuinely number theoretic questions that were of great interest already to Gauss (and Fermat, Euler, Lagrange, Legendre, Jacobi, Dirichlet, Eisenstein):
(1) For which primes is a given integer a quadratic residue?
... | 38 | https://mathoverflow.net/users/11919 | 115142 | 65,207 |
https://mathoverflow.net/questions/115141 | 3 | **The short:** Let
$X= \{(x,y,z) \in \mathbb{C}^\* \times \mathbb{C} \times \mathbb{C} \, |\, yz-x\neq 0\}$
Compute $H^\*\_c(X)$ (say with $\mathbb{C}$-coefficients).
**The long:** Unless I messed something up, the answer should be
$H\_c^3(X) = \mathbb{C}$, $H\_c^4(X) = \mathbb{C}^2$, $H\_c^5(X) = \mathbb{C}^2$... | https://mathoverflow.net/users/23907 | A cohomology computation request. | Reladenine -- your $X$ is the complement in the affine 3-space of the union of two hypersurfaces $Y$ and $Z$, the first given by $x=0$, the second by $yz=x$. The intersection $Y\cap Z$ is the union of two intersecting affine lines. Moreover, both $Y$ and $Z$ are isomorphic to $\mathbb{C}^2$ (note that $Z$ is the graph ... | 8 | https://mathoverflow.net/users/2349 | 115143 | 65,208 |
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