parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/139690 | 2 | By "Steinitz's theorem" I mean his theorem that a graph is the graph of a three-dimensional convex polyhedron iff it is of genus 0 and has a vertex connectivity of at least three. Can a similar characterization be made for genus 1, or beyond?
| https://mathoverflow.net/users/26327 | Generalizing Steinitz's theorem | Well, no direct generalization is known. One related result (which seems to have come out of an attempt to generalize Steinitz) is [this paper by D. Eppstein and E. Mumford.](http://arxiv.org/abs/0912.0537) However, since Steinitz' theorem is a simple consequence of the circle packing theorem, one can reasonably argue ... | 1 | https://mathoverflow.net/users/11142 | 139755 | 76,453 |
https://mathoverflow.net/questions/139421 | 13 | The alpha invariant $\alpha(X)$ of a Fano manifold $X$ of dimension $n$ is defined as the infimum of log canonical thresholds of (effective) $\mathbb{Q}$-divisors $D\sim\_{\mathbb{Q}}-K\_X$. Similarly, for $G\subset Aut(X)$ a compact subgroup of the automorphism group, one defines $\alpha\_G(X)$ considering only $G$-in... | https://mathoverflow.net/users/22294 | Are there examples of Fano manifolds such that Tian's alpha invariant satisfies $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric? | I see. THis is more subtle. There is no known example.
I think it will be impossible or very hard to create one.
Vanya
| 7 | https://mathoverflow.net/users/38631 | 139765 | 76,458 |
https://mathoverflow.net/questions/138978 | 6 | Is anything known about minimum dilatation pseudo-anosovs on non-orientable surfaces?
More specifically it is known whether the asymptotic behavior for log(minimal dilatation) is 1/genus? (The lower bound follows from the lower bound for pseudo-anosovs on orientable surfaces, but I do not know of any construction tha... | https://mathoverflow.net/users/11214 | Minimum dilatation pseudo-anosovs on non-orientable surfaces | One may construct upper bounds in the non-orientable case the same way as McMullen does [in his paper](http://www.ams.org/mathscinet-getitem?mr=1832823) (see p. 523 for a short description of his "renormalization" procedure, or Section 10).
| 3 | https://mathoverflow.net/users/1345 | 139767 | 76,459 |
https://mathoverflow.net/questions/139758 | 6 | Let $\kappa$ be a cardinal (I'm most interested in $\kappa=\aleph\_{\omega+1}$ but I suspect a general answer is known). What is the cofinality of $(P(\kappa)/NS,\subseteq)$? By this I mean the least cardinal $\lambda$ such that there exists a subcollection $X\subseteq P(\kappa)/NS$ of size $\lambda$ such that for any ... | https://mathoverflow.net/users/38814 | cofinality of $(P(\kappa)/NS,\subseteq)$ | I'm going to assume that when you write $\subseteq$, you really mean $\subseteq\_{NS}$. We say $A \subseteq\_{NS} B$ when $A$ is contained in $B$ except for a nonstationary set, i.e. $A \setminus B \in NS$. If we don't do this, then as Joel said, $\lambda = \kappa$, since the collection of "coatoms" is cofinal.
If we... | 10 | https://mathoverflow.net/users/11145 | 139772 | 76,461 |
https://mathoverflow.net/questions/139774 | 3 | It seems to be a folklore that for any genus $g$, there is a number field $K$ and a curve $X$ over $K$, such that $X$ has good reduction at all the places of $K$. Are any simple proofs of this?
| https://mathoverflow.net/users/38767 | curves with good reduction everywhere | One way of seeing this is by appealing to Rumely's general local-global principle over $\bar{\mathbb{Z}}$, applied here to the moduli stack: an algebraic scheme over the algebraic integers $\bar{\mathbb{Z}}$ has a solution (point) in $\bar{\mathbb{Z}}$ if and only if it does in all $v$-adic completions $\bar{\mathbb{Z}... | 3 | https://mathoverflow.net/users/26522 | 139775 | 76,463 |
https://mathoverflow.net/questions/139706 | 2 | I made an incremental algorithm which I would like to evaluate the complexity. The algorithm works with a sliding window of size n.
To study the complexity, the window is considered full and the data present are ${x\_{1},...,x\_{n}}$ . They are assumed to be random, i.i.d, but with no assumption regarding their distr... | https://mathoverflow.net/users/38528 | Computation of the mean of a random variable to estimate algorithm complexity | It seems to me that $P(d>k)=\tfrac1{k+1}$ for $k\in[0,n]$, so that $$E[d]=\sum\_{k\ge 0}P(d>k)=H\_{n+1}\simeq \ln n.$$ The reason is that the last $k+1$ terms of the sequence are in random order, so that the last one is the smallest among the last $k+1$ terms of the sequence with probability $P(d>k)=\tfrac1{k+1}$. When... | 1 | https://mathoverflow.net/users/34435 | 139788 | 76,466 |
https://mathoverflow.net/questions/139724 | 2 | The answers given to the question whether [all zeros in the critical strip of $\zeta(s)\pm\zeta(1-s)$ lie on the critical line](https://mathoverflow.net/questions/89518/are-the-semi-trivial-zeros-of-zetas-pm-zeta1-s-all-on-the-critical-li), suggest that this can indeed be proven, however only for those zeros where $s \... | https://mathoverflow.net/users/12489 | Are the zeros of the sum/difference of these integrals all on the critical line? | **EDIT**
Since $I(s)$ is hard to compute using the integral, searched
for zeros in the critical strip expressing $I(s)$ with zeta.
Zeros of $I(s)-I(1-s)$:
```
(0.542373937181871507937660440099538246914 -/+ 169.9110890356259176158839120274631129439j)
```
Zeros of $I(s)+I(1-s)$:
```
(0.789829041872580107037... | 3 | https://mathoverflow.net/users/12481 | 139789 | 76,467 |
https://mathoverflow.net/questions/139790 | 1 | Let $X$ be an infinite dimensional topological space such that :
$ \forall n \in \mathbb{N}$, $ \exists X\_{n} \subset X$, $n$-dimensional subspaces verifying :
* $\forall r<n$, the homotopy groups $\pi\_{r}(X\_{n})$ are trivial.
* $X\_{n} \subset X\_{n+1}$
* $\bigcup\_{n \in \mathbb{N}} X\_{n}$ is dense in $X$.
... | https://mathoverflow.net/users/34538 | Homotopy problem for infinite dimensional topological space | Of course not. Let $X = \mathbf R^\infty \times S^1$, and let $X\_n = \mathbf R^{n-1} \times I\_n$ where $\{I\_n\}$ is an increasing sequence of intervals in $S^1$ whose union is $S^1 \setminus \{\mathrm{point}\}$.
| 6 | https://mathoverflow.net/users/1310 | 139792 | 76,468 |
https://mathoverflow.net/questions/139802 | 1 | Can someone give me some references to read where existence/uniqueness of nonlinear parabolic PDE are treated via the Galerkin method or fixed point methods or something like that (anything but semigroups, which I am not faimilar with)?
I always come across H. Amann's work when I search on this topic but his work app... | https://mathoverflow.net/users/35613 | Nonlinear parabolic PDEs existence with Galerkin method? | Well, it is difficult to have something without semigroups, since [Lunardi](http://books.google.de/books/about/Analytic_Semigroups_and_Optimal_Regulari.html?id=mWojiHzg9bEC&redir_esc=y) is still one of the most comprehensive references.
Try [Krylov](http://www.ams.org/bookstore-getitem/item=gsm-96) then, it is an ex... | 3 | https://mathoverflow.net/users/12898 | 139808 | 76,472 |
https://mathoverflow.net/questions/139794 | 3 | Given a smooth Fano variety $X$ over $\mathbb{C}$, we can define the index, $I(X)$, as the divisibility of $-K\_X$ inside of $Pic(X)$. There is a theorem which states that $I(X) \leq n+1$, where $n$ is the dimension of the variety. Moreover varieties of index $n$ are quadrics and varieties of index $n+1$ are projective... | https://mathoverflow.net/users/6986 | Question about a variant of the index of a Fano manifold | Keyword: "pseudo-index". See, for instance, the work of Jiung-Cheng Chen and Cho - Miyaoka - Shepherd-Barron.
| 3 | https://mathoverflow.net/users/13265 | 139815 | 76,476 |
https://mathoverflow.net/questions/111582 | 7 | I'll begin with the question, which is intrinsically interesting:
>
> Let *M* be a manifold with some submanifold *Y*. Suppose that $W \rightarrow M$ is a smooth, proper map. Does there exist another map $W \rightarrow M$ homotopic to the original that is ALSO proper and transverse to the submanifold *Y*?
>
>
>
... | https://mathoverflow.net/users/6936 | Proper maps and transversality | Regarding the coboundary map in complex cobordism, I think that this was done by Dold in "Geometric Cobordism and the Fixed Point Transfer" (at least for oriented cobordism) in 2.10.
| 2 | https://mathoverflow.net/users/16628 | 139816 | 76,477 |
https://mathoverflow.net/questions/139801 | 9 | Shape? At the usual mathematical literature when we can discuss about the shape of a "space" that we have a kind of "topography" on it. For example a topology, metric, geometry, etc.
Note that for example: we can define an "unformal" topology on proper class of $Ord$ which any continuous map $f:Ord\longrightarrow Ord$ ... | https://mathoverflow.net/users/nan | What is the shape of mathematical universe? | There is an answer to this for the universe of constructive mathematics. According to Martín Escardó, a universe of Martin-Löf type theory has indiscrete topology, constructively. He has a [draft](http://www.cs.bham.ac.uk/~mhe/papers/universe-indiscrete-and-rice.pdf) about it, and [some slides](http://www.cs.bham.ac.uk... | 8 | https://mathoverflow.net/users/35779 | 139820 | 76,480 |
https://mathoverflow.net/questions/139810 | 6 | I have already asked a similar question at
<https://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra>
but have received no answer.
Sorry, I ask a similar question second time, as it is very important for my research.
Is it true that every Brouwerian lattice (=loc... | https://mathoverflow.net/users/4086 | Embedding a Brouwerian lattice into a Boolean lattice | $\def\pw{\mathcal P}$The answer to all the questions is yes. Here are some relevant embedding results:
**Proposition 1:** Every partial order $(P,\le)$ can be embedded in the powerset lattice $(\pw(P),\subseteq)$, i.e., an atomic complete Boolean algebra. The embedding can be made to preserve all existing meets, or a... | 12 | https://mathoverflow.net/users/12705 | 139828 | 76,482 |
https://mathoverflow.net/questions/139737 | 3 | In any partial order $(P,\leq)$ it is easy to see that every chain generates (i.e., by taking the upwards closure) a filter, and any filter built as a result of the Rasiowa-Sikorski lemma in forcing is of this form. When can we reverse this?
More specifically, I am interested in the partial order of all closed subset... | https://mathoverflow.net/users/16107 | When is a filter generated by a (countable) chain? | $\mathbf{Theorem}$ Let $X$ be a $T\_{1}$-space. Let $\mathcal{Z}$ denote the lattice of all closed subsets of $X$. Let $M\subseteq\mathcal{Z}$ be a maximal filter. Then
$M=\{C\in\mathcal{Z}|x\in C\}$ for some $x\in X$ or $M$ is not countably generated.
$\mathbf{Proof}$ Assume that $M$ is countably generated, and $M\... | 4 | https://mathoverflow.net/users/22277 | 139834 | 76,484 |
https://mathoverflow.net/questions/139835 | 3 | Is it possible to approximate an inverse of a sparse matrix with a sparse matrix?
The problem comes up in numerical non-linear quasi-Newton optimization: given a sparse Hessian a good starting point for L-BFGS would be a sparse approximation of the Hessian's inverse, with the emphasis on sparsity of the matrix rather t... | https://mathoverflow.net/users/38448 | Sparse approximation of the inverse of a sparse matrix | Not in general. An explicit and elementary counterexample
is the sparse triangular matrix with $1$'s on the diagonal
and $-1$'s just above it: the inverse is the triangular matrix with
**every** entry on or above the diagonal equal $1$.
| 11 | https://mathoverflow.net/users/14830 | 139837 | 76,485 |
https://mathoverflow.net/questions/139747 | 8 | Let $S=k[x\_1,...,x\_n]$ be a polynomial ring over field $k$ with maximal ideal $m=(x\_1,...,x\_n)$. I wanna make a $3$-dimensional $S$-module $M$ such that $H^0\_m(M)=H^1\_m(M)=0$ and $H^2\_m(M)\neq 0$ be finitely generated (or in general case: $H^i\_m(M)$ be finite for all $i=0,1,2$ ). Is there a simple way to create... | https://mathoverflow.net/users/36801 | example of Local cohomology | Take $M$ to be the second syzygy of $k$ over $S=k[x\_1,x\_2,x\_3]$. Then a graded version of local duality tells us that $H^2\_m(M)$ is dual to $Ext^1(M,R)= Ext^3(k,R)$, the last one is $k$ either by direct computation or duality again.
One can easily generalize this, the $j$ syzygy of $k$ in $n$ variables will have... | 12 | https://mathoverflow.net/users/2083 | 139841 | 76,487 |
https://mathoverflow.net/questions/139226 | 1 | I have the following optimal control problem
$$
J=\int\_0^TF(t,y\_1(t),y\_2(t))dt \to \min,
$$
subject to
\begin{align}
&\dot y\_1(t) = f(t,y\_1(t),y\_2(t)) + g(t)\nu(t),\\
&\dot y\_2(t) = \nu(t),
\end{align}
where $\nu$ is an impulsive control, and $y\_i$ are phase coordinates. Results on necessary conditions (maxim... | https://mathoverflow.net/users/19988 | On impulsive optimal control with functions of not bounded variation | In case someone else is interested, there is a paper by Arutyunov, Karamzin and Pereira, "Pontryagin’s maximum principle for constrained impulsive control problems", which presents a version of maximum principle for problems with non-smooth data. Here is a [link](http://www.sciencedirect.com/science/article/pii/S036254... | 0 | https://mathoverflow.net/users/19988 | 139843 | 76,489 |
https://mathoverflow.net/questions/139836 | -3 | Is it known whether siegel upper half plane is dense in the space of nonsingular matrices of same dimension .$.<http://en.wikipedia.org/wiki/Siegel_upper_half-space>.
Actually the question i have in my mind is to show that almost all 2 diml complex tori are not abelian varieties.I think I have solved the problem by usi... | https://mathoverflow.net/users/30081 | a question on Siegel Upper Half Space | There are symmetric matrices whose imaginary part is negative definite, like $-I$ for example, and these form an open set which the Siegel upper half space does not enter. On the other hand, since the Siegel upper half space is open in the symmetric matrices, it is somewhere dense, for example near $I$. But if you look... | 4 | https://mathoverflow.net/users/13268 | 139846 | 76,490 |
https://mathoverflow.net/questions/139826 | 13 | Let $R$ be the ring of integers in a number field. While studying the congruence subgroup property for $\text{Sp}\_{2g}(R)$ in
Bass, H.; Milnor, J.; Serre, J.-P.
Solution of the congruence subgroup problem for SLn(n≥3) and Sp2n(n≥2).
Inst. Hautes Études Sci. Publ. Math. No. 33 1967 59–137.
they quote a theorem of ... | https://mathoverflow.net/users/317 | Bass's paper "Symplectic groups and modules", used in proof of the congruence subgroup property for Sp | Your question is essentially about surjective stability for *relative* symplectic $K\_1$. The latter follows from the usual (absolute) surjective stability for $K\_1$, which in symplectic case starts at $2n\geq \mathop{\mathrm{sr}}(R)$. To prove this, one can use so-called "Stein relativization", as described in M. Ste... | 7 | https://mathoverflow.net/users/5018 | 139847 | 76,491 |
https://mathoverflow.net/questions/139849 | 0 | For some reason I am having trouble locating a transparent explanation of precisely what are the morphisms in the category of zig-zags. The objects of this category are specified completely by triples $t:=(n, t\_+, t\_-)$ where $t\_+, t\_-$ form a partition of the set $[n]:=\{1, 2,\dots, n\}$ for any positive integer $... | https://mathoverflow.net/users/8157 | What are the morphisms in the category of zig-zags? | A monotone map $f:[n]\to[m]$ is partition preserving if for all $i\in[n]$, $i\in t\_+$ implies $f(i)\in t\_+$ and $i\in t\_-$ implies $f(i)\in t\_-$. More simply, this means that the inverse image of every point in $[m]$ is an interval in $[n]$ on which all the arrows are pointing the same way. You should think of $f$ ... | 2 | https://mathoverflow.net/users/75 | 139851 | 76,493 |
https://mathoverflow.net/questions/139817 | 8 | Studying stability of certain non-autonomous dynamical systems on Lie groups I have come across the following question: Exactly which finite-dimensional, real Lie groups have adjoint representations that are bounded away from zero?
Edit: by "bounded away from zero" I mean that the image of the adjoint representation ... | https://mathoverflow.net/users/24389 | Which Lie groups have adjoint representations that are bounded away from zero? | The adjoint rep is always bounded away from $0$. Let $\mathfrak{g}\_0$ be a simple quotient of $\mathfrak{g}$. (I consider the $1$-dimensional Lie algebra to be simple, so there is always a simple quotient.) Let $\mathfrak{h}$ be the kernel of $\mathfrak{g} \to \mathfrak{g}\_0$ and let $H = \exp(\mathfrak{h})$.
The ... | 6 | https://mathoverflow.net/users/297 | 139862 | 76,497 |
https://mathoverflow.net/questions/139824 | 7 | A consequence of the famous Jørgensen inequality is that there is a lower bound for the distance between closed geodesics in hyperbolic three-manifolds: for any $R>0$ there is a c>0 such that for any such manifold $N$ and any two distinct closed geodesics loops $c\_1,c\_2$ on $M$ both of length less than $R$ we have
$... | https://mathoverflow.net/users/32210 | How close can closed geodesics be? | I think such a bound exists (depending only on pinched curvature constant $\kappa$, dimension $n$, and $R$). Suppose one has an infinite sequence of pinched negatively curved manifolds where geodesics of length $\leq R$ have Hausdorff distance approaching $0$. By the generalized Margulis Lemma (see Ballmann-Schroeder),... | 8 | https://mathoverflow.net/users/1345 | 139871 | 76,503 |
https://mathoverflow.net/questions/139799 | 2 | I have a $N$ person game where each person has a set of $M$ discrete strategies. I know from the theory that at least one mixed strategy Nash Equilibrium exists.
Can someone please tell me how do I find one of those equilibrium points by numerical simulation?
I can not find in the book any explanation of how to si... | https://mathoverflow.net/users/38825 | Simulating Mixed Nash Equilibria | First, an exact equilibrium may not be computable in general, so usually the idea is to specify an error parameter $\epsilon$ and look for an $\epsilon$-equilibrium.
Finding an $\epsilon$-Nash equilibrium is considered a hard problem (it is complete for the complexity class $\mathsf{PPAD}$). So there are no known "fa... | 3 | https://mathoverflow.net/users/29697 | 139882 | 76,506 |
https://mathoverflow.net/questions/139888 | 2 | (caveat: I'm not a number-theorist or Langlands-programme-er, and I don't expect to understand all the answers to this question, but I figured they might be useful to someone besides me).
I've been making videos of symmetries of Klein's $j(\tau)$:
<https://math.stackexchange.com/questions/466975/elements-of-sl2-mat... | https://mathoverflow.net/users/4672 | The relationship between SL(2,Z) and Gal(Qbar,Q) | The answer to the question as stated is no. The reason is that $\text{SL}\_2(\mathbb{Z})$ contains an element of order $4$, while $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ does not. In fact the following more general claim holds.
**Proposition:** Let $K$ be a field and let $g \in \text{Gal}(\overline{K}/K)$ have... | 7 | https://mathoverflow.net/users/290 | 139890 | 76,508 |
https://mathoverflow.net/questions/139511 | 5 | A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such coupling $\Bbb P$ we have
$$
2\cdot \Bbb P(X^2\setminus\Delta\_X)\geq\|P - \tilde P\| \tag{1}
$$
where $\Delta\_X = \{(x,x):x\... | https://mathoverflow.net/users/11768 | Coupling of non-probability/sub-probability measures | I thought, I could turn the comments into an answer…
The approach by couplings does not work without modifications and the reason is that couplings do not exist if the measures have different total mass: If $P$ and $Q$ are two measures on $X$ with different total masses which were coupled by $\mu$, then
$$
\int\_x\in... | 4 | https://mathoverflow.net/users/9652 | 139894 | 76,511 |
https://mathoverflow.net/questions/139884 | 2 | I have a reductive group $G$, sitting inside $GL\_n$, everything over some algebraically closed field of characteristic 0. Let $\mu$ be a cocharacter (of a maximal torus in $G$) such that the induced filtration on the vector space is a 1-step filtration, with the filtered piece having dimension $n/2$ (I'm assuming that... | https://mathoverflow.net/users/38861 | Levi action on the unipotent radical | The adjoint action of $L$ on ${Lie}(U)$ is very rarely irreducible, but this does happen in your case provided that the derived subgroup of $G$ is simple. If $Lie(L)$ intersects properly with at least two simple ideals of $Lie(G)$ then intersecting ${Lie}(U)$ with those ideals we'll obtain some non-trivial proper $L$-s... | 3 | https://mathoverflow.net/users/24386 | 139899 | 76,514 |
https://mathoverflow.net/questions/87430 | 21 | Assuming the axiom of choice it is very easy to see that $\aleph\_1$ is a regular Joe of a successor cardinal. It is not very large in any way except the fact that it is the first uncountable cardinal.
If however we begin with a model of ZFC+Inaccessible, we can construct models of ZF in which $\aleph\_1$ is somewhat... | https://mathoverflow.net/users/7206 | What sort of large cardinal can $\aleph_1$ be without the axiom of choice? | *-----edited to include corrections, thanks Joel and Tanmay-----*
From a model of "ZFC + $large(\kappa)$" you can get a model of "ZF + $large(\kappa)$ + $\kappa=\omega\_1$" if $large(\ )$ is a large cardinal property that is preserved under small forcing and can be written in the form
"for every set of ordinals $X... | 21 | https://mathoverflow.net/users/17176 | 139907 | 76,519 |
https://mathoverflow.net/questions/99454 | 6 | This question is partly motivated by my answer to [this question](https://math.stackexchange.com/questions/95570/does-constant-modulus-on-boundary-of-annulus-imply-constant-function/95581#95581) on math.stackexchange.
Let $\Omega$ be a bounded $n$-connected domain in the plane, bounded by $n$ pairwise disjoint Jordan... | https://mathoverflow.net/users/1162 | On holomorphic branched coverings of a domain in the plane to the unit disk | I now know a lot more about this question than I did when I asked it, so for the sake of completness let me add :
By applying the Riemann mapping theorem $n$ times, we can assume that $\Omega$ is bounded by *analytic* Jordan curves. In this case, we have the following theorem due to Bieberbach, which says that there ... | 2 | https://mathoverflow.net/users/1162 | 139916 | 76,521 |
https://mathoverflow.net/questions/139910 | 8 | It seems that usually smooth structures on compact 4-manifolds are distinguished by Seiberg-Witten/Donaldson invariants. And I don't know another direct way to do that. But in the case when $b\_2^+$ is even the invariants vanish identically.
Does there exist an example of smooth compact exotic 4-manifold with $b\_2^... | https://mathoverflow.net/users/38637 | Exotic 4-manifolds with even positive partial betti number | An often-overlooked invariant of a closed oriented 4-manifold $X$ is the closed oriented 4-manifold $\overline{X}$: the orientation-reversed manifold. Taking the Seiberg-Witten invariants of $\overline{X}$, one can distinguish smooth structures on manifolds $X$ with $b^->0$ odd, regardless of the parity of $b^+$. For i... | 12 | https://mathoverflow.net/users/2356 | 139921 | 76,524 |
https://mathoverflow.net/questions/139863 | 5 | For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr\_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of square-integrable functions on the chosen manifold). Also one now defines the generalized heat-kernel corresponding to this as... | https://mathoverflow.net/users/38852 | Mellin transform between heat kernel and zeta-function | Being of Laplacian type means that $D$ is a second order p.d.o. whose principal symbol coincides with that of a Laplacian of a metric $g$ on the manifold. Being positive signifies that that for any smooth functions with compact support $\newcommand{\bR}{\mathbb{R}}$ $f\_0, f\_1: M\to \bR$, $f\_0\neq 0$, you have
$$ \... | 9 | https://mathoverflow.net/users/20302 | 139928 | 76,528 |
https://mathoverflow.net/questions/139883 | 4 | Say you have some kind of "algebraic" category $A$ with a forgetful functor $U : A \to \mathbf{Set}$ which has a left adjoint $F : \mathbf{Set} \to A$. The natural transformations $U \to U$ can be interpreted as the terms of the algebraic theory. For example, if $A=\mathbf{CRing}$, these are just polynomials. Suppose w... | https://mathoverflow.net/users/21655 | Terms of an algebraic theory that act as homomorphisms | An algebraic theory in which every term is a homomorphism is called **commutative**: [ncatlab.org/nlab/show/commutative+algebraic+theory](http://ncatlab.org/nlab/show/commutative+algebraic+theory). I'm not sure that's *quite* what you were asking, because here "term" means "term in any number of variables", that is, na... | 6 | https://mathoverflow.net/users/586 | 139930 | 76,529 |
https://mathoverflow.net/questions/139861 | 7 | Consider the smallest Weyl algebra $A\_1=\{q,p; qp-pq=1\}$. It is known that there exist pairs of commuting elements, say $L$ and $M$, that obey various polynomial relations, e.g. elliptic curves. I wonder how big the commutative subalgebra of the Weyl algebra can be? (of course, I mean how big the generating set of co... | https://mathoverflow.net/users/2052 | How big can a commutative subalgebra of Weyl algebra be? | The key papers on this seem to be by Krichever: I skimmed "[Commutative rings of ordinary linear differential operators](http://link.springer.com/article/10.1007/BF01681429)" and "[Integration of nonlinear equations by the methods of algebraic geometry](http://link.springer.com/article/10.1007/BF01135528)". I am not ce... | 7 | https://mathoverflow.net/users/297 | 139931 | 76,530 |
https://mathoverflow.net/questions/139933 | 3 | Is every first countable profinite group actually second countable?
| https://mathoverflow.net/users/38889 | Is every first countable profinite group, second countable? | Although this question has been answered already by Masked Avenger, it seems like a good idea for me to explain why every first-countable Hausdorff group is metrizable. If $G$ is a Hausdorff topological group and $U$ is an open neighborhood of the identity $e$, then let $R\_{U}$ be the binary relation on $G$ where $(x,... | 6 | https://mathoverflow.net/users/22277 | 139947 | 76,536 |
https://mathoverflow.net/questions/139900 | 0 | I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can be what he states it to be.
Let
$\beta':S^1 \rightarrow I \vee S^1$ be defined by $(2t,\ast)$ if $t \leq 1/2$ and $(\ast... | https://mathoverflow.net/users/38870 | Misprint in Switzer's algebraic topology? | I think that what you wrote is not precisely what Switzer defined, but perhaps you made some simplifications that I did not notice. I'm going to give up for now on combing through the details of the formulas and just say what the big picture is.
Look at the schematic on page 47. We see that what $\beta':(D^{n+1},s\_0... | 7 | https://mathoverflow.net/users/6005 | 139950 | 76,537 |
https://mathoverflow.net/questions/139946 | 3 | Suppose that T is a set theory with global choice (for example ZF+ global choice).
Question: Does T prove the existence of a well-order on the whole universe V whose restriction to the class On of ordinals is the natural well-order on ON ?
Gérard Lang
| https://mathoverflow.net/users/30395 | Does global choice allow to extend the natural well-order on On to a well-order of the whole universe | Yes, use global choice to choose, for each ordinal $\alpha$, a well-ordering $<\_\alpha$ of the sets of rank $\alpha$ in the cumulative hierarchy. Then well-order all the sets by putting $x$ before $y$ if either $x$ has lower rank than $y$ or they have the same rank $\alpha$ and $x<\_\alpha y$.
| 6 | https://mathoverflow.net/users/6794 | 139951 | 76,538 |
https://mathoverflow.net/questions/139942 | 4 | It seems that Gauss states the following theorem in his first paper on biquadratic residues(Werke vol. II pp. 67-92). I cannot read Latin, but I have a Japanese translation of the paper. However, it is difficult to decipher the paper even in Japanese. Is the theorem right? If yes, how do you prove it?
**Theorem** Let... | https://mathoverflow.net/users/37646 | The biquadratic character of $2$ mod $p$ for a prime of the form $p=4n+1$ | Yes, this theorem is correct.
From quartic reciprocity we obtain $(2/p)\_4 = i^{ab/2}$, or in other words, $2^{(p-1)/4} \equiv f^{ab/2} \pmod p$. (There is an elementary proof of this fact due to Dirichlet: see Exercise 4.24 in Cox, "Primes of the Form $x^2+ny^2$.) So if $2 \equiv g^\lambda \pmod p$, then $f^{ab/2} \... | 10 | https://mathoverflow.net/users/430 | 139954 | 76,541 |
https://mathoverflow.net/questions/139937 | 8 | In §1.12 of the [Homotopy type theory book](http://homotopytypetheory.org/book/), it is mentioned that indiscernibility of identicals is a consequence of path induction. More precisely, for each type $C$ dependent over a type $A$, there is a term
$$\mathsf{transport} : \prod\_{a\_0, a\_1 : A} \prod\_{p : a\_0 =\_A a\_1... | https://mathoverflow.net/users/11640 | The independence of path induction | The problem with postulating only $\mathsf{transport}$ is that it is too weak to characterize the identity type up to equivalence. Let me define another type, called $\mathsf{doubleId}$, which has a $\mathsf{doubleRefl}$ and a $\mathsf{doubleTransport}$ satisfying the same rules as $\mathsf{refl}$ and $\mathsf{transpor... | 7 | https://mathoverflow.net/users/1176 | 139971 | 76,547 |
https://mathoverflow.net/questions/29280 | 14 | This question is inspired by my inability to make any progress on [Will Jagy's question](https://mathoverflow.net/questions/23943).
Giving a positive answer to this question should be strictly easier than proving Jagy's conjectures.
Suppose that $K/\mathbb{Q}$ is an imaginary quadratic extension. Let $\chi$ be the c... | https://mathoverflow.net/users/297 | Achieving consecutive integers as norms from a quadratic field | The answer to Speyer's question as stated is no, this need not be
the case. To see this let $p\equiv 3\pmod 4$ be a prime and consider
the associated imaginary quadratic field ${\Bbb Q}(\sqrt{-p})$. Note that
the associated quadratic character is simply the Legendre symbol $(\frac{n}{p})$.
Then as observed by Dav... | 7 | https://mathoverflow.net/users/38624 | 139976 | 76,549 |
https://mathoverflow.net/questions/139978 | 10 | If $f:{\mathbb{R}}^n\to{\mathbb{R}}^n$ $(n\ge2)$ is a bijection such that the image of every line is a line (continuity of $f$ *not assumed*), must $f$ be an affinity?
Assuming continuity would certainly suffice, even assuming that $f$ is order-preserving on each line. Is there a counterexample if we drop the assumpt... | https://mathoverflow.net/users/36904 | Line-preserving bijection of ${\mathbb{R}}^n$ onto itself | Yes $f$ must be an affinity – this is called the fundamental theorem of affine geometry and is found e.g. on page 52 of M. Berger's [*Geometry*](http://books.google.com/books?id=5W6cnfQegYcC&pg=PA52). (For other treatments and history, see [this question](https://mathoverflow.net/questions/191817/who-first-proved-the-f... | 14 | https://mathoverflow.net/users/19276 | 139979 | 76,551 |
https://mathoverflow.net/questions/131221 | 68 | The recent sensational news on bounded gaps between primes made me wonder: what is the status of Yitang Zhang's earlier arXiv preprint [On the Landau-Siegel zeros conjecture](https://arxiv.org/abs/0705.4306)? If this result is correct, then (in my opinion) it is even bigger news for analytic number theory. Has anyone c... | https://mathoverflow.net/users/11919 | Yitang Zhang's 2007 preprint on Landau–Siegel zeros | This is not an answer regarding the paper, but I think should be helpful. During a recent [interview](https://web.archive.org/web/20140831055114/http://blog.sina.com.cn/s/blog_c24597bf0101ctdp.html) (in Chinese), he commented:
>
> 问：前几天我去北京遇到葛立明，他说当时你在做个大问题，快做出来了。所以找你去新罕布什尔大学。
>
>
> 答： **那是关于Siegel零点的工作**，**我有一篇网... | 26 | https://mathoverflow.net/users/18850 | 139982 | 76,554 |
https://mathoverflow.net/questions/66457 | 11 | The two-variable elliptic genus is a topological invariant of almost-complex manifolds that takes values in power series. These power series turn out to describe weak Jacobi forms when the manifold is Calabi-Yau. It is defined by constructing a formal power series in K-theory out of exterior and symmetric powers of tan... | https://mathoverflow.net/users/121 | Is there a canonical map from the cohomology of orbifold chiral de Rham on an orbifold to the cohomology of chiral de Rham on a crepant resolution? | The short answer is no, one does not expect such isomorphism. What one does expect is some kind of "flat" family of super-vertex algebras that interpolates from one to the other. The base of the family should be the Kahler parameters of the model, i.e. complex parameter of the mirror, if such exists. In the case of CY ... | 4 | https://mathoverflow.net/users/38468 | 139985 | 76,556 |
https://mathoverflow.net/questions/139983 | 2 | Suppose I have sets of points $Z\_1,\dots,Z\_N$, such that $|Z\_i|=m$ for all $i$, and where all $m\times N$ points are independently distributed uniformly at random in the unit square. Can someone give me a lower bound within the right order of magnitude of the expected length of a "nearest-neighbor graph" of these po... | https://mathoverflow.net/users/38914 | Expected length of a certain kind of nearest-neighbor graph | Consider $N$ fixed and $m \to \infty$. Partition the unit square into $k^2$ squares of side $1/k$, where $k \approx \sqrt{m}$. Then the probability that
no square contains both a member of $Z\_i$ and a member of $\overline{Z\_i}$ should, I think, be on the order of $e^{-cm}$ for some positive constant $c$
(and I suspec... | 0 | https://mathoverflow.net/users/13650 | 139994 | 76,558 |
https://mathoverflow.net/questions/139992 | 11 | I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d\_1<d\_2<\dots<d\_k = n$, I would like to find examples of an affine variety space $V/\mathbb{C}$, a reductive group G acting on V, and a linearization $L$ such th... | https://mathoverflow.net/users/33895 | Partial (or complete) flag varieties as GIT quotients of affine spaces | If you're willing to quotient by a nonreductive group, then $M\_n//B$ will get you the $GL(n)$ flag manifold. (People are usually afraid to do so, worrying that the ring of invariants won't be Noetherian, but this one is.)
That flag manifold is also available reductively. Let $V\_0,V\_1\ldots,V\_n$ be a list of vecto... | 15 | https://mathoverflow.net/users/391 | 139995 | 76,559 |
https://mathoverflow.net/questions/139823 | 2 | hi I posted this question on mathematics stackexhange ( <https://math.stackexchange.com/questions/468855/what-are-the-rosser-turquette-axioms-of-lukasiewicz-3-valued-propositional-logic> ) but did not get an helping answer (but did get two down votes) hope on this site maybe somebody can help me.
I am trying to get m... | https://mathoverflow.net/users/38835 | what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic? | I found the book in downloadable form at <http://www.uni-leipzig.de/~logik/gottwald/treatise.pdf> , and I quickly glanced through the material up to and including the page you mention. The first observation is that the axiom you quoted as $AX\_{RT}5$ appears in this version of the book as $Ax\_{RT}6$, so, unless there ... | 4 | https://mathoverflow.net/users/6794 | 139997 | 76,560 |
https://mathoverflow.net/questions/139968 | 4 | Let $X$ be a scheme. Is the category of quasi-coherent (commutative) $\mathcal{O}\_X$-algebras cocomplete?
**Remark**.
The same question was asked in [MSE](https://math.stackexchange.com/questions/253807/is-the-category-of-quasi-coherent-mathcalo-x-algebras-cocomplete) last year. Since nobody has answered it, I post ... | https://mathoverflow.net/users/37646 | Is the category of quasi-coherent $\mathcal{O}_X$-algebras cocomplete? | It is well-known that the category of commutative algebras over a commutative ring is cocomplete, for example since it's algebraic over the category of sets. For a diagram of quasi-coherent algebras on a scheme, we can construct the colimit on each affine piece and then glue the resulting algebras together. That this w... | 9 | https://mathoverflow.net/users/2841 | 140003 | 76,562 |
https://mathoverflow.net/questions/139530 | 4 | Let S be a singular complex algebraic surface. Let $\psi: \chi \rightarrow \Delta$ be a QQ-Gorenstein smoothing of S (central fibre is isomorphic to S). Let G be a finite group acting on the fibres of above deformation with no fixed points. What can we say about the new deformation family (where fibres are quotients of... | https://mathoverflow.net/users/12969 | QQ-Gorenstein smoothing of a surface and a finite group action | I assume that by a "$\mathbb Q$-Gorenstein smoothing" you mean that $\chi$ is $\mathbb Q$-Gorenstein and probably $\Delta$ is the unit disk.
I also assume that by "$\mathbb Q$-Gorenstein" you probably mean something like
1. $\chi$ is normal
2. $K\_\chi$ is $\mathbb Q$-Cartier,
but you're probably not requiring t... | 2 | https://mathoverflow.net/users/10076 | 140004 | 76,563 |
https://mathoverflow.net/questions/139987 | 16 | Since $\pi\_4 (PU(2)) = \pi\_4 (SO(3)) = {\mathbb Z}\_2$, the two-element group,
we know that half of the two-sphere bundles over the 5-sphere $S^5$ are trivial
and the other half are non-trivial and all isomorphic. Can you write an explicit concrete realization for this non-trivial bundle? I have in mind something alo... | https://mathoverflow.net/users/2906 | A concrete realization of the nontrivial 2-sphere bundle over the 5-sphere? | Is this concrete enough? Recall that $\mathrm{SU}(3)$ fibers over $S^5$, with fibers equal to $\mathrm{SU}(2)$ and that this fibration is nontrivial. Let $S^1\subset \mathrm{SU}(2)$ be (any) subgroup and let $B = \mathrm{SU}(3)/S^1$. Then $B$ fibers over $S^5$ with fibers $S^2$. If $B$ were trivial, then $\mathrm{SU}(3... | 28 | https://mathoverflow.net/users/13972 | 140008 | 76,564 |
https://mathoverflow.net/questions/139543 | 2 | Let $E$ be locally convex topological vector space. Let $c^\infty E$ denote the same
vector space equipped with the $c^\infty$-topology (i.e. the finest topology on it, s.t.
all smooth curves $\mathbb{R} \rightarrow E$ are continous, see "The convenient setting
of global analysis" by A.Kriegl and P.Michor, I.2.12). If ... | https://mathoverflow.net/users/33600 | $c^\infty$-topologies on spaces of compactly supported sections and their products | $\def\Gmc#1{\Gamma\_{\rm c}(#1)}\def\ci{{\rm c}^\infty}
$Generally the linear isomorphism $\ci\Gmc{V\_1\oplus V\_2}\to\ci\Gmc{V\_1}\times\ci\Gmc{V\_2}$ is **not** a homeomorphism. For example, when taking $E=\mathcal D(\mathbb R)$, the identity map $\iota:c^\infty E\times c^\infty E\to c^\infty(E\times E)$ is not conti... | 2 | https://mathoverflow.net/users/12643 | 140009 | 76,565 |
https://mathoverflow.net/questions/140010 | 4 | Let $Y$ be an oriented manifold of dimension three, and let $X=Y\times S^1$. We have
$$H^2(X,\mathbb{Z}\_2)=H^2(Y,\mathbb{Z}\_2)\oplus H^1(Y,\mathbb{Z}\_2).$$
Pick an element $m\oplus n\in H^2(Y,\mathbb{Z}\_2)\oplus H^1(Y,\mathbb{Z}\_2)$, and consider $\mathfrak{P}(m\oplus n)$, where $\mathfrak{P}$ is the Pontryagi... | https://mathoverflow.net/users/5420 | Pontryagin square on $Y\times S^1$ where $Y$ is three-dimensional | I think $[X] \frown \mathfrak{P}(m \oplus n) = 2\langle m \cdot n + n^3, [Y] \rangle$.
The class $m \oplus n$ is better thought of as $m \otimes 1 + n \otimes x$ under the Kunneth decomposition, where $x \in H^1(S^1;\mathbb{Z}/2)$ is the nontrivial element. Then the quadratic property of $\mathfrak{P}$ and naturality... | 3 | https://mathoverflow.net/users/318 | 140016 | 76,571 |
https://mathoverflow.net/questions/140029 | 4 | Let $G(k, n)$ be the Grassmannian of $k$-dimensional subspaces of $K^{n}$, $K$ a field, embedded in $\mathbb{P}^{N}$ by the Plücker embedding. In Harris' Algebraic Geometry, A First Course, Theorem 10.19 states that
$$\mathrm{Aut}(G(k, n)) = \mathrm{Aut}(G(k, n), \mathbb{P}^{N}),$$
where $\mathrm{Aut}(G(k, n), \mathb... | https://mathoverflow.net/users/19252 | Picard group of $G(k, n)$ saying about automorphisms | In fact, one has the following result.
>
> **Proposition 1.** Let $X \subset \mathbb{P}^n$ be a smooth subvariety such that $\textrm{Pic}(X)$ is generated by the hyperplane section $\mathscr{O}\_X(1)$. Then every automorphism of $X$ is induced by an automorphism of $\mathbb{P}^n.$
>
>
>
The proof is very easy.... | 6 | https://mathoverflow.net/users/7460 | 140031 | 76,579 |
https://mathoverflow.net/questions/140024 | 0 | Let $g:\mathbb{R}^n \to \mathbb{R}$ be a smooth real-valued function that decays like some positive power of $|x|^{-1}$ at infinity. Define the first order distribution $\mu $ by
$$
\langle \mu , \phi \rangle = \int \limits \_{\mathbb{R}^n} \big ( \phi (g(x)) - \phi (0) \big ) \,dx , \qquad \phi \in C\_0^\infty (\math... | https://mathoverflow.net/users/19433 | Support of a distribution | I think so.
Let $t\_0\ne 0$ be in that interval and note that the interval also equals the image of $g$. Assume $t\_0$ does not lie in the support of $\mu$.
Then there exists a neighborhood $U$ of $t\_0$ such that $\langle \mu,\phi\rangle=0$ for every $\phi$ supported in $U$.
By shrinking $U$ we can assume that there e... | 1 | https://mathoverflow.net/users/nan | 140032 | 76,580 |
https://mathoverflow.net/questions/140037 | 8 | Spanier-Whitehead stabilization provides a way to extend a category $\bf E$ to a bigger one $\mathcal{SW}\_\Omega(\bf E)$ where a given endofunctor $\Omega$ is invertible. The category $\mathcal{SW}\_\Omega(\bf E)$ is constructed with
1. Objects the pairs $(A,n)\in Ob(\mathbf C)\times\mathbb Z$;
2. The set of morphis... | https://mathoverflow.net/users/7952 | Stabilization of $\infty$-categories versus SW stabilization | I think it's more typical to use this construction to invert the suspension functor, rather than the loop functor. So let me write $\Sigma$ where you wrote $\Omega$.
1) The category $SW\_{\Sigma}(\mathcal{C})$ can be identified with the direct limit of the sequence
$$ \cdots \rightarrow \mathcal{C} \stackrel{\Sigma}{... | 12 | https://mathoverflow.net/users/7721 | 140043 | 76,586 |
https://mathoverflow.net/questions/140041 | 3 | According to Wikipedia <http://en.wikipedia.org/wiki/Geodesic>, a geodesic " *is a generalization of the notion of a "straight line" to "curved spaces* " and further " *In the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. If... | https://mathoverflow.net/users/31310 | Geodesics in Graphs:Shortest Paths vs Going as Straight Ahead as Possible | An interesting class of graphs are the median graphs ( see here <http://en.wikipedia.org/wiki/Median_graph>). If you fill in all (maximal) subgraphs which are Hamming cubes you will get a $CAT(0)$ cubical complex. The $CAT(0)$ distance is given by putting the euclidian metric in each cube and then taking the shortest p... | 3 | https://mathoverflow.net/users/33828 | 140048 | 76,588 |
https://mathoverflow.net/questions/139822 | 5 | This post here is a specification of this [post](https://mathoverflow.net/questions/139790/homotopy-problem-for-infinite-dimensional-topological-space).
Let $(X\_{n},d\_{n})\_{n \in \mathbb{N}}$ be a sequence of [intrinsic metric](http://en.wikipedia.org/wiki/Intrinsic_metric) spaces verifying :
* $X\_{n}$ have [to... | https://mathoverflow.net/users/34538 | Homotopy problem for infinite dimensional topological space II | The intristic metric assumption seems to be added to exclude $S^1$ from the previous post. But what about $S^d$? It can be approximated with bigger and bigger disks
$\{x\in S^d\ |\ x\_1\leq 1-\frac{1}{n} \}$ (with the natural intristic metric) and in the limit you get the sphere. (Of course dimension of such a disk is ... | 2 | https://mathoverflow.net/users/38944 | 140050 | 76,589 |
https://mathoverflow.net/questions/139668 | 34 | Let $X$ be a non-compact metric space (though if the answer to the question is positive, then it probably also holds for more general spaces like, e.g., paracompact Hausdorff) and $E \to X$ a vector bundle over it.
>
> Suppose that over every compact subset $K \subset X$ the restricted bundle $E|\_K$ is trivial. Ca... | https://mathoverflow.net/users/13356 | vector bundle trivial over every compact subset, then it is globally trivial | As Igor Belegradek showed in the comments, one could find an example by finding a CW-complex $X$ and a map $X \to BO(n)$ which is not nullhomotopic, but where the restriction to every finite subcomplex is nullhomotopic. Such a map is called a phantom map. The question "is this map nullhomotopic?" has the same answer wh... | 27 | https://mathoverflow.net/users/360 | 140051 | 76,590 |
https://mathoverflow.net/questions/140059 | 3 | Let $S$ be a set of $k$ distinct natural numbers, each
from the interval $[2,n]$, with least common multiple $\mathop{lcm}$.
What fraction $\rho$ of the numbers $2,3,4,\ldots,\mathop{lcm}$
are divisible by a member of $S$?
For example, if
$S=(4,6,8,9)$, then $|S|=k=4$, $n=9$, and $\mathop{lcm}=72$.
The numbers $2,\ld... | https://mathoverflow.net/users/6094 | Divisibility properties of a stream of numbers | Let $L$ denote the lcm of your $k$ numbers. If a number $\ell$ below $L$ is divisible
by one of the $k$ numbers, then $\ell$ must have some common factor $>1$ with $L$. Therefore
the number of elements you want is bounded above by $L-\phi(L)$ (the numbers that are not
coprime to $L$).
So if the $k$ numbers are al... | 5 | https://mathoverflow.net/users/38624 | 140062 | 76,592 |
https://mathoverflow.net/questions/140046 | 15 | $\newcommand{\Spec}{\mathrm{Spec}\ }$
Let $(P)$ be a property of rings. I call $(P)$ local when $(P)$ satisfy these two
conditions:
* if $A$ is a ring satisfying $(P)$, then the distinguished rings $A\_f$ also
satisfy $(P)$;
* If $\Spec A$ is covered by distinguished open $\Spec A\_i$ with the $A\_i$ having
$(P)$, th... | https://mathoverflow.net/users/26737 | What are the local properties of schemes preserved under global sections? | ### Normality
For an integral scheme, being normal (integrally closed in ones own fraction field) satisfies this property. Indeed, suppose that $a, b \in A = \Gamma(X, O\_X)$ are such that $a/b$ satisfy some polynomial $p(x) \in A[x]$. Then $a|\_U, b|\_U$ satisfy the same polynomial after restriction to each (affine) ... | 7 | https://mathoverflow.net/users/3521 | 140063 | 76,593 |
https://mathoverflow.net/questions/140071 | 6 | I'm having trouble sorting out some basic definitions concerning Chevalley groups. The groups I'm interested in are the simply connected groups of type $C\_n$, so the groups $\text{Sp}\_{2n}$. The roots in $C\_n$ are $\{\pm 2 \epsilon\_i \text{ $|$ } 1 \leq i \leq n\} \cup \{\pm \epsilon\_i \pm \epsilon\_j \text{ $|$ }... | https://mathoverflow.net/users/38956 | One-parameter subgroups of symplectic group associated to roots | The actual matrices depend of course on the particular alternating bilinear form you use to define $\mathrm{Sp}\_{2n}$. A standard choice is to use the form whose matrix relative to a basis $(e\_1,\dots,e\_n,e\_{-n},\dots,e\_{-1})$ is
$$
J=
\begin{pmatrix}
& & & & & 1\\
& & & & \cdots\\
& & & 1\\
& & -1\\
& \cdots\\
-1... | 4 | https://mathoverflow.net/users/19276 | 140073 | 76,594 |
https://mathoverflow.net/questions/140019 | 6 | Improvement after J-M Schlenker's comment below :
*This post has been divided into two parts, the second part is [here](https://math.stackexchange.com/questions/477438/uniquely-geodesic-spaces).*
>
> **Question** : Is a finite dimensional metric space, [uniquely geodesic](http://en.wikipedia.org/wiki/Glossary_of... | https://mathoverflow.net/users/34538 | Uniquely geodesic and CAT(0) spaces? | The CAT(0) asumption is sufficient: it implies that any two points are connected by a unique geodesic segment. This is well-known and follows from the definitions.
However, as pointed out by HenrikRüping, the CAT(0) asumption is not necessary, you can for instance perturb the hyperbolic plane by putting a small lump ... | 8 | https://mathoverflow.net/users/9890 | 140082 | 76,598 |
https://mathoverflow.net/questions/139744 | 2 | Let
$$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$
where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper [incomplete gamma function](https://en.wikipedia.org/wiki/Incomplete_gamma_function) $\Gamma(s,x)=\int\_x^\infty t^{s-1}e^{-t}dt$ regularized by gamma... | https://mathoverflow.net/users/18910 | Limit involving regularized gamma function and its inverse | Below I give a sketch of a proof.
First an approximation for the inverse incomplete gamma function, $Q^{-1}$, is needed.
Henceforth I assume that
$$
g(x)\sim \gamma x^s
$$
for large $x$ with $s>0$. Then ($0<b<1$)
$$
1-b^{1/g(x)}\sim (\gamma x^s)^{-1}\ln\left(\frac{1}{b}\right),
$$
which is small for large $x... | 1 | https://mathoverflow.net/users/37436 | 140085 | 76,600 |
https://mathoverflow.net/questions/140081 | 5 | Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.
| https://mathoverflow.net/users/38961 | Cover of a n-simplex with balls | Yes.
If $P$ is our point, then it will be contained in the ball corresponding to edge $(i,j)$ if and only if the angle $\angle iPj$ is greater than or equal to $\frac{\pi}{2}$. If there is no such edge $(i,j)$, then every vertex $j$ is on a fixed side of the hyperplane through $P$ orthogonal to the line connecting ve... | 6 | https://mathoverflow.net/users/2363 | 140087 | 76,601 |
https://mathoverflow.net/questions/140091 | 1 | The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that are supposed to verify the analogue of the Riemann Hypothesis?
Thanks in advance.
| https://mathoverflow.net/users/13625 | Has a universality theorem been proved for the Davenport-Heilbronn L function? | <http://siauliaims.su.lt/pdfai/2008/laurincikas-08.pdf>
>
> **Abstract.** In the paper, the joint universality in the Voronin sense for
> Hurwitz zeta-functions with parameters $a\_1; \dots ; a\_r$ such that the system
> $\{ \log(m + a\_j) : m = 0;1;2; \dots ; j = 1; \dots ; r \}$ is linearly independent over
> ... | 4 | https://mathoverflow.net/users/10400 | 140097 | 76,603 |
https://mathoverflow.net/questions/139010 | 5 | I have been trying to understand the proof of the following result, which is considered well-known.
Theorem: Fix a compact metric space $X$, a homeomorphism $T:X \to X$, and a continuous map $ A : X \to \mathrm{SL}\_2(\mathbb{R}) $. Define the skew-product
$$
(T,A): X \times \mathbb{R}^2 \to X \times \mathbb{R}^2,... | https://mathoverflow.net/users/38023 | Uniform hyperbolicity decay estimate | I nearly asked this exact question earlier this year, after coming to the same series and being confounded by the problems which arise when the decay of $\|A\_m(x)\|^{-2}$ is too irregular. I discussed this result this summer with an expert in the field and we agreed that behind Yoccoz's "easily checked" lurks perhaps ... | 2 | https://mathoverflow.net/users/1840 | 140099 | 76,605 |
https://mathoverflow.net/questions/140090 | 3 | I would like to compute the expected number of vectors in $\mathbb{F}\_2^n$ we need to draw (following a uniform distribution) so that they form a basis of $\mathbb{F}\_2^n$, i.e., that we have $n$ linearly independent vectors.
| https://mathoverflow.net/users/38965 | Expected number of random binary vectors so that the form a basis | This is a $q$-analogue of the coupon collector problem.
If the first few vectors span a subspace of codimension $k$, then the probability of drawing a vector outside this subspace is $1-2^{-k}$ and the expected number of draws to get a vector outside this subspace is the reciprocal $\frac{2^k}{2^k-1}=1+\frac{1}{2^k-... | 4 | https://mathoverflow.net/users/2954 | 140105 | 76,607 |
https://mathoverflow.net/questions/139239 | 2 | I have recently come accross the star product of copulas, that is if $A$ and $B$ are 2-copulas and $\{C\_t\}\_{t\in[0,1]}$ is a family of copulas, then $C(x,y,z) = \int\_0^y C\_t(\frac{\partial}{\partial t} A(x,t),\frac{\partial}{\partial t} B(t,z))dt$ is the star product of $A$ and $B$, and $C$ itself is a copula.
A... | https://mathoverflow.net/users/38521 | star-product of copulas | Below is an example where the inequality does not hold:
For each $t$ we choose $C\_t$ to be the independent copula $C\_t(u,v)=u\cdot v$ and denote the corresponding star-product of $A$ and $B$ by
$A\star B$.
For copulas $A$ and $B$ with probability density functions (pdf) $a$ and $b$, resp., we obtain from
$$
A\... | 0 | https://mathoverflow.net/users/38971 | 140107 | 76,608 |
https://mathoverflow.net/questions/140106 | 3 | It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a elementary proof for it?
Any comments and reference will be appreciated
| https://mathoverflow.net/users/11966 | Fixed points on Riemann surface | This theorem is an immediate consequence of a result by B. Maskit, which states that one may associate with $\Omega$ another domain $\Omega'$, conformally equivalent to $\Omega$, such that all conformal self-maps of $\Omega'$ are Möbius transformations. For a proof of Maskit's theorem, I suggest you look at the followi... | 8 | https://mathoverflow.net/users/1162 | 140111 | 76,610 |
https://mathoverflow.net/questions/140110 | 3 | Let $\pi : X \to B$ be a proper, surjective holomorphic submersion, where both $X$ and $B$ are compact Kahler manifolds. Assume that all the fibers $X\_b = \pi^{-1}(b)$ are smooth. Is the family $\pi : X \to B$ then locally trivial?
The answer is "no" when formulated like this: Let $\pi : X \to B$ be the family where... | https://mathoverflow.net/users/4054 | Is a holomorphic family whose fibers are all smooth locally trivial? | By Grauert-Fischer Theorem, a smooth family of compact complex manifold is locally trivial if and only if all the fibers are analytically isomorphic.
However, there exist smooth families $\pi \colon X \to B$ such that the fibres are *not* isomorphic, and so they are *not* locally trivial.
The easiest examples occur... | 16 | https://mathoverflow.net/users/7460 | 140112 | 76,611 |
https://mathoverflow.net/questions/135021 | 4 | One can define cochain complexes of (combinatorial) graphs, where each term is a vector space of linear combinations of certain (isomorphism classes of) graphs, and where the differential $d$ is a signed sum of edge contractions. One can understand the kernel of $d$ by understanding the image (and hence the cokernel) o... | https://mathoverflow.net/users/27972 | Reference for the image of the adjoint to the differential in graph cohomology (which yields STU & IHX)? | One just needs to check that with your definition of pairing, one always gets $\langle d^\*\Gamma\_1,\Gamma\_2\rangle=\pm\langle\Gamma\_1,d\Gamma\_2\rangle$. Notice that the right-hand side counts the number of homeomorphisms (counted with signs) between $\Gamma\_1$ and $\Gamma\_2/e$, where $e$ runs through the set of ... | 3 | https://mathoverflow.net/users/9800 | 140116 | 76,613 |
https://mathoverflow.net/questions/140115 | 1 | I want to find the most compact representation of a vector as a linear combination of a set of vectors B. B has more elements (on purpose) that is needs to have to describe the subspace.
For example
```
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 1 1 0
```
Given
```
0 2 2 1
```
I want to obtain:
```
0 0 0 1 2
``... | https://mathoverflow.net/users/38972 | Finding the most compact representation of a vector in an "overdetermined base" | This problem and various related problems are known to be NP-hard to solve exactly, but there has been a lot of work on efficient approximations. See [this wikipedia page](http://en.wikipedia.org/wiki/Sparse_approximation) or try googling things like "sparsest vector", "LASSO", "Orthogonal Matching Pursuit".
None of ... | 3 | https://mathoverflow.net/users/5963 | 140119 | 76,616 |
https://mathoverflow.net/questions/140124 | 1 | Consider a metric space $(M,d)$ and consider a collection of points $X\_n := \{x\_1,\dots,x\_n\} \subset M$. Let
$$
N\_\epsilon(y;X\_n) := | \{ x \in X\_n: d(x,y) \le \epsilon \}|
$$
where the RHS is the cardinality of a set. For a set $Y \subset M$, define
$$
N\_\epsilon(Y;X\_n) = \inf\_{y \in Y} N\_\epsilon(y;X\_n... | https://mathoverflow.net/users/36687 | A measure of closeness to a discrete set in a metric space | In $\mathbb{R}^n$, the map $y \mapsto N\_\epsilon(y,X\_n)$ is the convolution of the empirical measure of the sample $X\_n$ with the indicator function $I\_\epsilon$ of the $\epsilon$-ball around $0$. Therefore, if you take for $X\_n$ a sample of i.i.d. random variables with law $\mu$, the law of large numbers implies ... | 5 | https://mathoverflow.net/users/38566 | 140127 | 76,619 |
https://mathoverflow.net/questions/140007 | 1 | I need a space with countable tightness which is not a Fréchet space. In this space, I am searching for a point with **no** deleted neighborhood consisting entirely of P-points.
(A P-point is a point $x \in X$ such that for every $G\_\delta$ set $O$ containing $x$, $x \in \operatorname{int}(O)$ or equivalently $M\_x ... | https://mathoverflow.net/users/38926 | A space with countable tightness which is not a Fréchet space? | First note that in a space with countable tightness: $P$-point $\Leftrightarrow$ weak $P$-point $\Leftrightarrow$ Isolated point. So you are looking for a point for which every deleted neighborhood contains a non-isolated point. You can do a lot better:
Let $X$ be a countable *maximal space* (i.e. such that the topol... | 2 | https://mathoverflow.net/users/17836 | 140132 | 76,622 |
https://mathoverflow.net/questions/140120 | 5 | What is the $\varepsilon\to 0$ limit of the following double integral
$$\int\limits\_{-1}^1d\tau\;\sqrt{1-\tau^2}\;\tau\int\limits\_0^\infty dq\;q^2e^{iq(\tau+i\varepsilon)}\;?$$
I was asked about this integral by my friend who got it in a physics research project. In fact this is just $n=1$ case of a more general inte... | https://mathoverflow.net/users/32389 | Limit of a double integral | I believe these integrals can be evaluated directly in the $\varepsilon=0$ limit by interpreting the result of the inner integral as a distribution.
$$
\int\limits\_0^\infty dq \, q^{n+1} \, e^{iq(\tau + i\varepsilon)}
\to (-i)^{n+1}\frac{\partial^{n+1}}{\partial\tau^{n+1}} \left(\pi\delta(\tau) + \mathcal{P}\frac{i}{... | 6 | https://mathoverflow.net/users/2622 | 140148 | 76,627 |
https://mathoverflow.net/questions/140145 | 1 | Given two algebraic varieties $X,Y$ with finitely generated class group,
such that exist a small modification $\phi : X \rightarrow Y$,
i.e. there exists open subsets $U \subset X, V\subset Y$ such that
$codim (X-U), codim (Y-V) \geq 2$,
$\phi (U) \subset V $ and $\phi |\_U : U \rightarrow V$ is an isomorphism.
Then... | https://mathoverflow.net/users/37338 | Small birational maps on algebraic varieties | If $X$ and $Y$ are smooth and proper, then this looks a lot like the $K$-equivalence, which states that for some (=any) smooth proper $Z$ that maps to both $X$ and $Y$ the exceptional divisors are the same, with multiplicities. You condition seems a bit weaker in that the exceptional divisors for the maps must only hav... | 1 | https://mathoverflow.net/users/38468 | 140154 | 76,628 |
https://mathoverflow.net/questions/139199 | 3 | Given a pseudo Anosov mapping class $f:S\_{g,n}\rightarrow S\_{g,n}$ is the Lefschetz number for $f^m$ negative for some $m$ depending only on $(g,n)$?
The Lefschetz number of a mapping class $f$ can be defined as $2-Tr(f^{\star})$ where $f^{\star}$ is the induced map on $H\_1(S\_{g,n},\mathbb{Z})$.
If the leading... | https://mathoverflow.net/users/38496 | Iterated Lefschetz numbers | Your question may be recast as:
Is there an $N(n)$ such that for any integral matrix $A\in SL(n,\mathbb{Z})$, there is $k\leq N(n)$ with $tr(A^k)>2 $ (with a few small exceptional cases)?
(this is not quite equivalent, since the image of $Mod(S\_{g,n})$ is not
all of $SL(H\_1(S\_{g,n}))$, but it certainly implies wh... | 2 | https://mathoverflow.net/users/1345 | 140161 | 76,631 |
https://mathoverflow.net/questions/140158 | 17 | The question is in the title, but here is some background:
I previously asked for a general criterion to decide [which colimits commute with which limits in the category of sets](https://mathoverflow.net/questions/93262/which-colimits-commute-with-which-limits-in-the-category-of-sets) and received encouraging answers... | https://mathoverflow.net/users/644 | Which limits commute with filtered colimits in the category of sets? | Categories $J$ such that limits of shape $J$ commute with filtered colimits in sets are called L-finite. There are several known characterization of them: see [the nLab page about it](http://ncatlab.org/nlab/show/L-finite+category). The page refers to Robert Paré, *Simply connected limits* ([pdf](http://cms.math.ca/cjm... | 19 | https://mathoverflow.net/users/20233 | 140164 | 76,632 |
https://mathoverflow.net/questions/140144 | 4 | For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int\_0^1 q(x)dx=1$ and
$$\int\_{0}^1 \int\_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- \frac{q(t)q((s-t)/(1-t))}{1-t}\right|dt ds<\varepsilon$$
Somehow the question does not look hard, but still I don't see if it is true ... | https://mathoverflow.net/users/8699 | Approximation of an integral over the unit ball of L_1 | There does not exist such a function $q$ if $\varepsilon<1/2$.
Indeed, if $q$ is a positive measurable function on $[0,1]$ of integral $1$, pick $a \in [0,1]$ such that $\int\_0^a q(x) dx = 1/2$.
Then
$$\int\_{0}^a \int\_{0}^{s} \frac{q(s)q(t/s)}{s} dt ds = 1/2,$$
whereas
$$\int\_{0}^a \int\_{0}^{s} \frac{q(t)q((... | 6 | https://mathoverflow.net/users/10265 | 140167 | 76,635 |
https://mathoverflow.net/questions/139331 | 9 | Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by parallel-plus, where the series sum of $x$ and $y$ is $x+y$ and the parallel sum is $xy/(x+y)$? (See the related posts [cho... | https://mathoverflow.net/users/3621 | Generalizing detropicalization | Those are all.
Given a function $r$, by restricting to a particular value of $y$ (barring finitely many), we get a rational function, hence a map $\mathbb P^1 \to \mathbb P^1$. For all but finitely many values of $y$, this map will have the same degree, $d$. Assuming $r$ is nonconstant, let $y\_1$ and $y\_2$ be two s... | 6 | https://mathoverflow.net/users/18060 | 140182 | 76,642 |
https://mathoverflow.net/questions/133098 | 3 | Edited after mistake in the first version.
It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a set of asymptotic density one (Selberg's result is actually more precise: one can take $x+O(f(c) \log^2 x)$ where $f(x)$ i... | https://mathoverflow.net/users/9317 | Primes in short intervals with a preassigned frobenius | One reference where a Hoheisel type result (right number of such
primes in every interval $(x,x+x^{1-\delta})$ for some $\delta>0$) is proved unconditionally is the paper by Balog and Ono "The Chebotarev density theorem and
some questions of Serre" (see <http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/062.pdf>)... | 3 | https://mathoverflow.net/users/38624 | 140183 | 76,643 |
https://mathoverflow.net/questions/140163 | 5 | Let $F : R^k \to R^k$ be a smooth map whose Jacobian
$J(F): R^k \to R$ vanishes on a discrete set $S$, so that if
$O$ is the complement of $S$, then $f: O \to R^k$, the
restriction of $F$, is a local diffeomorphism, and in particular
there are only finitely many points $x$ inside the unit ball of
$R^k$ such that $... | https://mathoverflow.net/users/7311 | Algorithm for finding inverse images of a local diffeomorphism | With a Lipschitz bound on the derivative, Newton's method gives an algorithm to efficiently approximate $F^{-1}$. [Hubbard's calculus textbook](http://www.math.cornell.edu/~hubbard/vectorcalculus.html) has a write-up using this perspective, viewing the result as a consequence of Kantorovich's Theorem. There's a fairly ... | 3 | https://mathoverflow.net/users/1465 | 140186 | 76,645 |
https://mathoverflow.net/questions/140139 | 9 | I'm looking for a more general Feynman-Kac formula that works in the case of jump-diffusion processes.
I know that, given a pure diffusion process like
$$dS\_t=\mu\_tdt+\sigma\_tdW\_t,$$ if $u(t,s)$ satisfies the PDE
$$f\_t(t,s)+\mu\_tf\_s(t,s)+\frac{\sigma\_t^2}{2}f\_{ss}(t,s)-V(s)f(t,s)=0$$ with terminal condition $... | https://mathoverflow.net/users/38982 | Feynman-Kac for jump-diffusion | Hi it is possible to get some Feynman-Kac formula in this case. The proof only use the martingale property and Itô's formula for jump-diffusion processes.
So let's have $X$ s.t. (I took the compensated version of your sde):
$dX\_t=[\mu(t,X\_t)+\lambda(t)\gamma(t,X\_t)]dt + \sigma(t,X\_t)dW\_t+ \gamma(t,X\_{t-})d\t... | 7 | https://mathoverflow.net/users/2642 | 140199 | 76,650 |
https://mathoverflow.net/questions/140200 | 8 | A graph $G$ is called *super connected* if for every connected subgraph $H\subset G$ the graph $G-H$ obtained from $G$ after deletion of all vertices from $H$ is also connected.
>
> **Conjecture**: The only super connected graphs are $K\_{n}$ and $C\_{n}$.
>
>
>
**ADDED 1:** Some related results can be found h... | https://mathoverflow.net/users/39026 | On "super connected" graphs | I think that conjecture is true. Let $a$, $b$ be two non-adjacent vertices. I claim that $H = G \setminus \{a, b\}$ must contain exactly two connected components. Indeed, if it has more than two, we can take any of these, say $H\_1$, and $H\_1 \cup \{a, b\}$ will be connected while the rest of $G$ not. On the other han... | 14 | https://mathoverflow.net/users/25905 | 140211 | 76,654 |
https://mathoverflow.net/questions/140207 | 0 | Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuously differentiable mapping. We assume that the set $$\{x\in\mathbb{R}^{n};j(f)(x)=0\}$$ is a hypersurface of $\mathbb{R}^{n}$, where $j(f)(x)$ denotes the jacobian determinant of $f$ at $x$.
I wonder if it is possible to apply a weak variant of the Inverse theore... | https://mathoverflow.net/users/24060 | A question about the inverse theorem function in $\mathbb{R}^n$ | The problem is fold singularities. Take $\mathbb{R}^n$, fold it in half and project. The function $x^2$ as a function from the reals to the reals has a fold singularity at 0. You need some additional hypotheses about the derivatives of your map along the singular set to get the conclusion you want. If you have a gradua... | 1 | https://mathoverflow.net/users/4304 | 140217 | 76,657 |
https://mathoverflow.net/questions/140220 | 1 | Let $M$ be a Riemannian manifold, and let $x, y \in M$ be non-conjugate points.
Let $r, R>0$ be two numbers. I am looking for a bound on the number of geodesics between $x$ and $y$ of Length between $r$ and $R$, like "there are at most $N(r, R)$ geodesics between $x$ and $y$", depending on the curvature of $M$.
It... | https://mathoverflow.net/users/16702 | Number of geodesics of certain length | I think that no such bounds exist. For example, you can take $M$ to be a (complete) surface of revolution in Euclidean $3$-space that has negative curvature bounded from below by $-1$ and that is asymptotic to the classic pseudospherical surface (i.e., a 'spike' going off to infinity). On such a surface, you can find a... | 7 | https://mathoverflow.net/users/13972 | 140232 | 76,661 |
https://mathoverflow.net/questions/140230 | 8 | Can somebody give me a nice example of blow-up of a smooth algebraic variety along a singular subvariety? Something I can do some exercise on and check the differences with a smooth blow-up. Thanks!
| https://mathoverflow.net/users/4096 | blow-up along singular variety | Perhaps the easiest is to blow up a plane along a fat point:
Blow up $\mathbb A^2\_k=\mathrm{Spec} k[x,y]$ at the ideal $(x^2,y^2)$. You should get a pinch point (Whitney's umbrella). This itself is an interesting singularity. It's simple normal crossing away from the pinch point, but not so simple there. If you want... | 16 | https://mathoverflow.net/users/10076 | 140241 | 76,666 |
https://mathoverflow.net/questions/131341 | 7 | This is a question I asked at Math.SE but got no answers: <https://math.stackexchange.com/q/397164/7110/>
Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \mathrm{H}^\ast(X; \mathbb{Q})$, where $\mathrm{H}^\ast$ denotes singular coho... | https://mathoverflow.net/users/13356 | Chern Character Isomorphism for non-finite CW complexes, resp. for non-CW complexes | I have found a reference:
>
> "M. Karoubi, *Les isomorphismes de Chern et de Thom-Gysin en K-theorie*, Seminaire Henri Cartan, vol. 16, no. 2, Expose no. 16, 1963-1964".
>
>
>
There the isomorphism $\mathrm{K}^0(X) \otimes \mathbb{Q} \cong \mathrm{\check{H}}^{ev}(X; \mathbb{Q})$ is shown for every compact Haus... | 2 | https://mathoverflow.net/users/13356 | 140245 | 76,669 |
https://mathoverflow.net/questions/140103 | 7 | Ribet proved the Serre epsilon conjecture using $p$-adic Galois representations (<http://math.berkeley.edu/~ribet/Articles/invent_100.pdf>). Can someone show how to replace all use of $p$-adics in this (or some other) proof of the theorem by arithmetic modulo some finite number of specified powers?
A word on the mot... | https://mathoverflow.net/users/38783 | Has Ribet's theorem been proved using only finite powers of primes? | This is a partial answer regarding the use of the use of the absolute Galois group of the rationals. In our paper [Reverse Mathematics and Algebraic Field Extensions](http://arxiv.org/abs/1209.4944), Jeff Hirst, Paul Shafer and I analyze what is needed to have Galois theory of infinite extensions work as expected in th... | 6 | https://mathoverflow.net/users/2000 | 140246 | 76,670 |
https://mathoverflow.net/questions/140208 | 5 | Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2.
Let $Q=Q(D)$ denote the root lattice of $D$.
Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is generated by reflections $r\_\alpha$ for $\alpha\in D$.
The automorphism $\nu$ acts on $Q$.
Let $W\_0$ denote the centraliz... | https://mathoverflow.net/users/4149 | A subgroup of the Weyl group | As indicated in my comments, the 1968 book *Simple Groups of Lie Type* by R.W. Carter has a good elementary treatment of your question (to which the answer is yes) in Chapter 13. All of this goes back pretty far in the history of Lie theory, with the Weyl group and root system arising from a simple Lie algebra (or Lie ... | 4 | https://mathoverflow.net/users/4231 | 140247 | 76,671 |
https://mathoverflow.net/questions/83161 | 16 | I am trying to locate a copy of J. T. Condict's senior thesis on odd perfect numbers:
J. Condict, On an odd perfect number's largest prime divisor, Senior Thesis, Middlebury College (1978).
I am sure a soft copy would have been archived somewhere. Would anybody know where (in the Internet) that archive is?
| https://mathoverflow.net/users/10365 | On J. T. Condict's Senior Thesis on Odd Perfect Numbers | This is Jim (Condict) Grace, the author. I just saw this post. I'm sorry to hear that Middlebury may have lost their copy. I'm not sure where mine is, but I'll keep an eye out for it. (It may be at a family house 1,000 miles away but I'll look for it at Christmas when I visit.) If I find it, I'll scan and post it somew... | 68 | https://mathoverflow.net/users/39046 | 140248 | 76,672 |
https://mathoverflow.net/questions/140250 | 3 | The complex cohomology H^\* of the manifold of flags in C^n is a quotient of C[x1,...,xn] by the ideal generated by symmetric polynomials with no constant term. In particular it has an action of the symmetric group, by permuting the variables x. If you ignore the grading on H^\*, this is the regular representation of S... | https://mathoverflow.net/users/29980 | How does the grading on the cohomology of a flag variety break up the regular representation of W? | I'm unsure if this would be termed a 'simple rule' but perhaps this will be of some use: the *Kostka-Foulkes polynomials* $K\_{\lambda\mu}(q)$ ($\lambda,\mu$ partitions of $n$) determine the change of basis matrix in $\Lambda(q)$, the one variable ring of symmetric functions, between the Hall-Littlewood polynomials and... | 5 | https://mathoverflow.net/users/9970 | 140256 | 76,675 |
https://mathoverflow.net/questions/140197 | 2 | Let $X$ be an integral (and singular) curve over the complex field and let $A$ be a torsion free sheaf (not necessarily locally free) of rank $1$ on $X$. We denote by $\mathrm{Quot}^n\_A$ the Quot scheme parametrizing quotient $q:A\to Q$ of length $n$. Is this Quot scheme always isomorphic to the Hilbert scheme $\mathr... | https://mathoverflow.net/users/33841 | Quot schemes and Hilbert schemes on a singular curve | That is definitely not true. Let $X$ be an integral curve that has a single ordinary double point $p$ (and no other singular points). Let $\nu:\widetilde{X}\to X$ be the normalization. The fiber of $\nu$ over $p$ consists of two closed points, $p\_1$ and $p\_2$.
Let $A$ be $\nu\_\*\mathcal{O}\_{\widetilde{X}}$. Let $... | 4 | https://mathoverflow.net/users/13265 | 140257 | 76,676 |
https://mathoverflow.net/questions/139874 | 14 | An object $X$ of a category $C$ is said to be *projective* if the hom-functor $C(X,-)$ preserves epimorphisms (or, in general, some restricted class of epimorphisms such as the regular or effective ones). The axiom of choice is equivalent to the assertion that all objects of **Set** are projective. In general, a "proje... | https://mathoverflow.net/users/49 | Pullback-stability of internally projective objects | For toposes, the stability property of your first question does hold. Suppose $X$ is internally projective in $\mathbb{C}$. And suppose $q: B \to A$ is an epimorphism from $v : B \to I$ to $u: A \to I$ in $\mathbb{C}/I$, hence an epimorphism in $\mathbb{C}$. Write $X^\*$ for the object $X \times I$ of $\mathbb{C}/I$. T... | 8 | https://mathoverflow.net/users/13506 | 140262 | 76,677 |
https://mathoverflow.net/questions/140129 | 9 | Let $T$ be a game tree and $T\in N\in M$, where $N,M$ are the two least admissibles containing $T$. Let $A$ be a boolean combination of two lightface open sets in $[T]$, or alternatively, a boolean combination of two open sets that happens to be a $\Delta\_1$-definable class of $N$ and $M$. (There may be further work n... | https://mathoverflow.net/users/18628 | $\Sigma^0_1\wedge\Pi^0_1$-Determinacy holds in the second admissible above the game | (As the absentee supervisor am I allowed to pitch in?) To my mind this is least confusing if we take $A$ to be $B\cap C$ the intersection of a lightface open with a lightface closed set in the full Baire space. Here then the `game tree' for our purposes can be defined to be the set of finite partial plays in $\omega^{<... | 9 | https://mathoverflow.net/users/6942 | 140269 | 76,680 |
https://mathoverflow.net/questions/139235 | 1 | The following is a theorem of Elkik from 1978 describing how rational singularities behave under flat morphisms.
Let $f:X\rightarrow S$ be a flat morphism of schemes of finite type over $\mathbb{C}$ such that both $S$ and the fibres of $f$ have rational singularities.
I am confused about one step in the proof and I... | https://mathoverflow.net/users/18013 | Rational singularities under flat morphisms | I think you are. That is, making it more complicated than it is.
You were done right after saying that "by flat base change the natural map $$f^{\ast}R^i\varphi\_{\ast}\mathcal{O}\_{S'}\rightarrow R^i\varphi'\_\*f'^{\ast}\mathcal{O}\_{S'}$$ is an isomorphism."
*By definition* $f^\*\mathscr O\_S=\mathscr O\_X$ and ... | 2 | https://mathoverflow.net/users/10076 | 140271 | 76,681 |
https://mathoverflow.net/questions/140218 | 0 | The Frostman Shift of an inner function at the value which assumed infinitely and is not an asymptomatic value is an infinite Blaschke Product. But how to charaterize it when it is an asymptotic value?
By Fatou's theorem, the radial limit function
$$\phi^{\ast}(\zeta):=\lim\_{r\rightarrow1^{-}}\phi(r\zeta),$$
for a b... | https://mathoverflow.net/users/30754 | How to characterize the value which assumed by an inner function infinitely often but is an asymptotic value？ | If $a$ is an asymptotic value, the inner function $\phi\_a$ may be a Blaschke product or not. It is not clear what do you mean by "how to chracterize it". In what terms?
It is an inner function
for which zero is an asymptotic value. (This characterizes it. Is this what you want?).
In a special case when all zeros of ... | 2 | https://mathoverflow.net/users/25510 | 140274 | 76,682 |
https://mathoverflow.net/questions/140266 | 2 | Let $H$ be a bounded symmetric domain.
What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?
| https://mathoverflow.net/users/39058 | Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain | The Bergman metric on any bounded symmetric domain is equal to the biholomorphism invariant Kaehler-Einstein metric, which makes it a Hermitian symmetric space. See Mok, **Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds**, p. 59, proposition 3.
| 3 | https://mathoverflow.net/users/13268 | 140279 | 76,684 |
https://mathoverflow.net/questions/140277 | -1 | I am essentially looking for a book that would hold my hand through basic concepts to more complicated ones. I am coming from physics. I am looking to make some connections with Classical mechanics and or quantum field theory. I have been told that QFT is really just geometry and have only seen a few manifest aspects o... | https://mathoverflow.net/users/nan | Regarding understanding differential geometry | I think the books "Topology, Geometry, and Gauge Fields: Foundations" and "Topology, Geometry, and Gauge Fields: Interactions" are exactly what you're looking for. They are not short, but you don't actually want a short book anyway because the only way to write a short differential geometry book is to give few terse ex... | 5 | https://mathoverflow.net/users/4362 | 140296 | 76,686 |
https://mathoverflow.net/questions/140294 | 5 | I'm just starting to learn about sheaves, and I'm confused about a certain matter:
I've just learned, to my delight, that every sheaf $S$ on a space $X$ is the sheaf of sections of a particular bundle (by bundle I mean a surjective map $E\to X$), specifically the bundle of stalks of $S$.
So, for example, the sheaf ... | https://mathoverflow.net/users/39079 | Does the bundle of germs of functions $f:X\to \mathbb R$ have the same sheaf of sections as $X\times \mathbb R$? | These bundles have the same sections. But which is simpler depends on your point of view. $X\times \mathbb R\to X$ is simpler set theoretically but is `more complicated' topologically in the the sense that it is not a local homeomorphism. The sheaf, taken as a bundle, is a local homeomorphism.
| 7 | https://mathoverflow.net/users/38783 | 140299 | 76,688 |
https://mathoverflow.net/questions/139672 | 4 | Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) boundary? Currently I am trying to read the original Jaco-Shalen paper and a Scott's survey paper, buy they do not look... | https://mathoverflow.net/users/38749 | Geometrization & JSJ decomposition with boundary | The work of Jaco-Shalen and Johannson actually handles manifolds with boundary. The theorem they prove (in the language of Johannson's Book) is:
An irreducible, boundary-irreducible 3-manifold with useful boundary pattern possesses a characteristic submanifold, unique up to isotopy.
The complement of the character... | 9 | https://mathoverflow.net/users/39082 | 140301 | 76,689 |
https://mathoverflow.net/questions/140303 | 5 | I am wondering if there is only one unique Yoneda isomorphism, that is a natural isomorphism (natural in C and P, that is) between Hom(yC,P) and PC.
The Yoneda lemma says that there exists at least one, and its common proof is constructive, so we have an example. But is it the only one?
If it is unique, how do we p... | https://mathoverflow.net/users/29853 | Unicity of Yoneda isomorphism | If there is an isomorphism $\mathrm{Hom}(y C, P) \cong P (C)$ natural in $P$ and $C$, then by restricting to the representable functors one obtains an automorphism of the identity functor of the category $C$ comes from. There are categories for which the identity functor has a non-trivial automorphism group, e.g. $\mat... | 8 | https://mathoverflow.net/users/11640 | 140304 | 76,691 |
https://mathoverflow.net/questions/140295 | 10 | This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal left ideal which is not a direct summand (as a left ideal).
| https://mathoverflow.net/users/nan | Countable Maximal Ideals | Here is an example. Choose an uncountable algebraically independent set $S\subset \mathbb{R}$, and let $k=\mathbb{Q}(S)$. For each $x\in S$, choose a sequence $(q\_n(x))$ of rationals converging to $x$. Given any $y\in k$, write $y=f(x\_1,\dots,x\_m)$ for some rational function $f$ and some $x\_i\in S$ and define $q\_n... | 13 | https://mathoverflow.net/users/75 | 140307 | 76,692 |
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