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https://mathoverflow.net/questions/81611 | 7 | Let us work in the "nice" situation where $X,Y,Z$ are smooth complex algebraic varieties, not necessarily compact. Assume that the fiber product $W:= X \times\_Z Y$ is also smooth. What can we say about the Picard group of $W$?
More precisely, assume that $Pic(X)=Pic(Z)=0$.
1) Can we deduce that $Pic(W)=Pic(Y)$?
... | https://mathoverflow.net/users/19452 | Picard groups of (fiber) products | It can happen that $Pic(X) = Pic(Y) = Pic(Z) = 0$ but $Pic(W) \neq 0$!
For example, let $f: \mathcal{E} \to Z$ be a non-isotrivial family of elliptic curves, where $Z$ is a smooth rational curve. Then $Pic(\mathcal{E})$ is finitely generated, so by removing a finite number of (images of) sections of $f$ we obtain a s... | 11 | https://mathoverflow.net/users/519 | 81617 | 48,918 |
https://mathoverflow.net/questions/81576 | 2 | Hi,
I am looking for a closed-form expression for the finite sum of the product of an exponential function with a polynomial function --- that is, the sum
```
N
∑ a^n * n^k
n=0
```
where N and k (but not a) are non-negative integers.
Now, for my problem I don't necessarily need the exact answer as lon... | https://mathoverflow.net/users/19442 | Sum of products of exponentials and polynomials | Let $S(N,a,k)= \sum\_{n=0}^N a^n n^k$. Multiplying by $x^k/k!$ and summing on $k$ gives the exponential generating function
$$\sum\_{k=0}^\infty S(N,a,k) \frac{x^k}{k!}=
\frac{(ae^x)^{N+1}-1}{ae^x-1}.$$
From this formula, it is easy to calculate $S(N,a,k)$ for small values of $k$ using Maple, Mathematica, or Sage, etc.... | 10 | https://mathoverflow.net/users/10744 | 81621 | 48,921 |
https://mathoverflow.net/questions/81638 | 5 | Let $F \leqslant E$ be a field extension.
If $a, b \in E$ are algebraic over $F$ then $a+b$ and $ab$ are algebraic as well. There is an short proof of this using the extension $E(a,b)$:
$[E(a,b):E]$ is finite so all elements are algebraic otherwise powers non-algebraic would form an infinite linearly independent se... | https://mathoverflow.net/users/nan | Constructive proof of algebraic elements forming a subfield | Let me see if I remember this right. Let $f$ be the minimal polynomial of $a$ and $g$ the minimal polynomial of $b$ over $F$. Consider the polynomial $h(x)$ obtained as the resultant with respect to the variable $y$ of the polynomials $f((x+y)/2),g((x-y)/2)$. A root $c$ of $h$ is therefore an element of the algebraic c... | 3 | https://mathoverflow.net/users/2290 | 81640 | 48,929 |
https://mathoverflow.net/questions/81632 | 0 | Let $A$ be a symmetric positive matrix, and let $B$ be invertible. Is
$$BAB^{-1} + B^{-1}AB$$
always positive?
---
Let $C$ be a real matrix with real positive spectrum. Is
$$C + C^T$$
positive?
---
Are these two problems the same?
| https://mathoverflow.net/users/3441 | Positivity and symmetrization | *EDIT:* I changed the counterexample to ensure that $B$ is also *symmetric*
**First question**: NO, not even if $B$ is also (symmetric) positive definite
Take
$$ A = \begin{pmatrix}
5 & 2\\\\
2 & 4\end{pmatrix}\qquad
B = \begin{pmatrix}
13 & 15\\\\
15 & 18\end{pmatrix}
$$
Here, both $A$ and $B$ are (symmetric) ... | 2 | https://mathoverflow.net/users/8430 | 81646 | 48,934 |
https://mathoverflow.net/questions/81586 | 0 | Any arbitrary ellipse in the x-y plane can be described with five parameters -- usually the center’s x and y coordinate positions, x0 and y0; the distance between focal points, d; the eccentricity, $\epsilon$; and the counterclockwise tilt angle, $\psi$, of the line through the center and foci with respect to the horiz... | https://mathoverflow.net/users/19443 | Can an ellipse's center be determined from a perimeter point's coordinates? | To elaborate on J.M.'s comment, your parameters $d$ and $\epsilon$ determine the size and shape of an ellipse, and $\psi$ determines its orientation in the plane, but once those are set you're still free to rigidly translate the ellipse (without rotating it), which means the point $(x\_1,y\_1)$ can be anywhere along it... | 3 | https://mathoverflow.net/users/15837 | 81650 | 48,936 |
https://mathoverflow.net/questions/81215 | 3 | If $p<1$ and $X$ is a random variable distributed according to the geometric distribution $P(X = k) = p (1-p)^{k-1}$ for all $k\in \mathbb{N}$, then it is easy to show that $E(X) = \frac 1p$, $\mathop{Var}(X)=\frac{1-p}{p^2}$ and $E(X^2) = \frac{2-p}{p^2}$.
Now consider a "conditional" geometric distribution, define... | https://mathoverflow.net/users/5701 | Conditional geometric distributions | I'm not sure what you really want but here is a couple of simple minded inequalities that can serve as a baseline.
Below $g=\sum\_{k\in J}\gamma^k$, $M=\sum\_{k\in J}k\gamma^k$, so $\mu=\frac Mg$. We'll need the counting function $F(n)=\#\{k\in G: k\le n\}$ of the set $J$. I will assume that $F$ is extended as a cont... | 3 | https://mathoverflow.net/users/1131 | 81657 | 48,939 |
https://mathoverflow.net/questions/81654 | 1 | Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}\_p^n$?
Recall that semialgebraic subsets are obtained from $p$-adic algebraic varieties using boolean connectives and coordinate projection.
The results I am interested in are a sort of alge... | https://mathoverflow.net/users/nan | Etale cohomology in the $p$-adic setting | 1) A ball (of finite radius) is compact, and a punctured ball is not, so they cannot be diffeomorphic.
2) A theorem of Serre (*Topology*, 1965, vol. **3**, p. 409-412) classifies all compact $p$-adic manifolds.
The result is:
>
> a) Any such (non-empty) manifold is isomorphic to a finite disjoint union of balls.... | 7 | https://mathoverflow.net/users/10696 | 81664 | 48,944 |
https://mathoverflow.net/questions/81651 | 14 | I am trying to understand the following introductory passage in an early lecture by the philosopher/mathematician Gottlob Frege because I am interested in how Frege conceived of the role of geometric intuition in mathematical reasoning.
>
> One of the most far-reaching advances made by analytic geometry in more rec... | https://mathoverflow.net/users/2833 | Vocabulary of 19th Century analytic projective geometry: What are "order" and "dimension"? | As Francesco Polizzi remarks, the idea is to identify $\text{Sym}^2(\mathbb{P}^2)$ with a cubic hypersurface in $\mathbb{P}^5$. Frege's lecture (see link in comments) actually goes on to explain how this is done. The cubic hypersurface is (in the six variables $s\_1,s\_2,s\_3,t\_1,t\_2,t\_3$),
$\text{det} \begin{pmat... | 6 | https://mathoverflow.net/users/2819 | 81668 | 48,947 |
https://mathoverflow.net/questions/81626 | 24 | Recently a colleague and I needed to use the fact that the natural map $SL\_2(\mathbb{Z}) \rightarrow SL\_2(\mathbb{Z}/N\mathbb{Z})$ is surjective for each $N$. I happily chugged my way through an elementary proof, but my colleague pointed out to me that this is a consequence of strong approximation.
After browsing t... | https://mathoverflow.net/users/1050 | Is strong approximation difficult? | There are two very-different questions here: the best arguments for surjectivity of the natural maps $SL(n,R)\rightarrow SL(n,R/I)$, and about Strong Approximation. While it is true that Strong Approximation more-than implies these surjectivities in situations that are not quite elementary, it is serious overkill, I th... | 23 | https://mathoverflow.net/users/15629 | 81670 | 48,949 |
https://mathoverflow.net/questions/81325 | 6 | Hi,
i have a sequence of immersed disc $u\_n: \mathbb{D} \rightarrow \mathbb{R}^3$ which converge to a singular cover of the disc: $z^k$ for $k\geq 2$, moreprecisely $u\_n \rightarrow z^k$ in $C^2(\mathbb{D})$. Of course the Gauss curvature of the image $\Sigma\_n=u\_n(\mathbb{B})$ blows up thanks to Gauss-Bonnet formu... | https://mathoverflow.net/users/9253 | degenerating surface | When $k$ is odd, there does exist such a family of immersions satisfying Paul's requirements. (When $k$ is even, Vitali Kapovich has shown, using a clever topological argument, that it's not possible to have such a family of degenerating immersions. Please see his answer for the details.)
Set $k=2m+1$, and consider t... | 9 | https://mathoverflow.net/users/13972 | 81671 | 48,950 |
https://mathoverflow.net/questions/81662 | 3 | Given a smooth affine symplectic variety $V$ with an action of a connected algebraic $G$. If $\mu$ is the moment map, the define the affine quotient to be :
$X = \mu^{-1}(0)// G = \text{Spec}\mathbb{C}[\mu^{-1}(0)]^{G}$
This is an algebraic Variety (may be singular).
The Hamiltonian reduction of $V$ is defined ... | https://mathoverflow.net/users/9534 | Hamiltonian Reduction and Affine quotient | We can assume that $G$ is affine, since an abelian variety must act trivially on any affine variety. The closed points of $X$ are exactly the closed $G$-orbits on $\mu^{-1}(0)$. On an affine variety, closed orbits can be distinguished by invariant global functions, and if one orbit is in the closure of another they wil... | 2 | https://mathoverflow.net/users/66 | 81686 | 48,957 |
https://mathoverflow.net/questions/81696 | 2 | I believe the following nice statement is true, but I cannot find a reference or proof it myself.
---
In a 2-category(i.e., bicategory), the composition of composable 2-cells is unambiguously defined.
--------------------------------------------------------------------------------------------------
Where 2-cell... | https://mathoverflow.net/users/7341 | Composition of composable 2-cells in a 2-category is unambiguously defined? | A well-known reference:
>
> John Power, A 2-categorical pasting theorem, *Journal of Algebra* Volume 129, Issue 2, March 1990, Pages 439-445.
>
>
>
See the [nLab page on pasting diagrams](http://ncatlab.org/nlab/show/pasting+diagram) for more.
Power's paper just deals with *strict* 2-categories, if I rememb... | 5 | https://mathoverflow.net/users/586 | 81698 | 48,963 |
https://mathoverflow.net/questions/81655 | 17 | Let $\mathbb{HP}^2$ denote the quaternionic projective plane. According to
[A note on $\mathcal{E}(\mathbb{HP}^n)$ for $n\leq 4$](http://escholarship.lib.okayama-u.ac.jp/mjou/vol33/iss1/17/), N. Iwase, K-I. Maruyama, S. Oka, Math. J. Okayama Univ. 33 (1991) , 163-176.
any homotopy self equivalence of $\mathbb{HP}^2... | https://mathoverflow.net/users/318 | Signs in the unstable homotopy groups of spheres | OK, after having made so many stupid comments, I felt obligated to remember what I knew about unstable homotopy theory in order to try to say something meaningful.
Recall that $\pi\_7(S^4\vee S^4)\cong\pi\_7S^4\oplus\pi\_7 S\_4\oplus \mathbb{Z}$, where the third summand is the kernel of the homomorphism
$$\pi\_7(S^4\... | 12 | https://mathoverflow.net/users/12166 | 81708 | 48,967 |
https://mathoverflow.net/questions/81681 | 9 | I know of only 2 main techniques to create a model of $ZFC$. The first one is creating a model which is an extension of $V$: this is forcing. The second technique is that of inner model theory and looking at subclasses of $V$. Do all methods to generate models of $ZFC$ fall in the these 2 categories or are there other ... | https://mathoverflow.net/users/3859 | Creating Models of $ZFC$ | As Carl Mummert mentioned, the usual ways of constructing models — completeness and compactness — are also available for the construction of models of set theory. The models constructed in this way are usually not (externally) wellfounded but they have found some interesting uses.
For example, Joel David Hamkins prov... | 14 | https://mathoverflow.net/users/2000 | 81711 | 48,969 |
https://mathoverflow.net/questions/81645 | 3 | An epimorphism $f$ is said to be *extremal*, if for any decomposition $f=i\circ p$ with $i$ a monomorphism, the morphism $i$ is automatically an isomorphism. (This is from the textbook by F.Borceux.)
Let us say that $f$ is *weakly extremal*, if for any decomposition $f=i\circ p$ with $i$ a monomorphism *and $p$ an e... | https://mathoverflow.net/users/18943 | Weakened notion of extremal epimorphism? | A counterexample:
Consider the monoid $\langle a,b,c\mid ac=bc\rangle$ as a category with one object. Then $a,b$ are monics, $bc$ is an epic and $c$ is not an epic. So $bc$ is not an extremal epic, but it is easily to see weakly extremal.
| 4 | https://mathoverflow.net/users/18814 | 81712 | 48,970 |
https://mathoverflow.net/questions/81714 | 23 | Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a *compactification* of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-dimensional manifold, such that $M$ is the interior of $\bar{M}$. I understand that not every manifold $M$ has such a com... | https://mathoverflow.net/users/2622 | Uniqueness of compactification of an end of a manifold | Suppose $\overline{M}\_i$, $i=0,1$, are compact smooth manifolds with boundary whose interiors are diffeomorphic: let $\psi$ be such a diffeomorphism, and $M$ for either interior (identified via $\psi$). For both manifolds one can find a smooth collar $c\_i : \partial \overline{M}\_i \times [0, 1) \hookrightarrow \over... | 19 | https://mathoverflow.net/users/318 | 81724 | 48,974 |
https://mathoverflow.net/questions/81717 | 11 | Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum\_{i=0}^n a\_i x^i$ where $a\_i\in GF(2)$ and $a\_n=1$.
Can we always find such an irreducible polynomial $p(x)$ of degree $n$ where $\textrm{degree}(p(x)-x^n)\leq n/2$?
Example: $p(x)=1+x^2+x^3+x^5+x^{400}$ is a... | https://mathoverflow.net/users/6154 | Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2? | I think (but I could easily be wrong), that this follows, at least for many degrees (in the stronger form also conjectured by Noam and Gjergji) from the results of S. D. Cohen as in
MR2092633 (2005g:11245)
Cohen, Stephen D.(4-GLAS)
Primitive polynomials over small fields. Finite fields and applications, 197–214,
Lec... | 6 | https://mathoverflow.net/users/11142 | 81725 | 48,975 |
https://mathoverflow.net/questions/81732 | 26 | **The question**
Let $f$ be a nonconstant polynomial over $\mathbb{C}$. Let's say that a point $c \in \mathbb{C}$ is **unusual** for $f$ if every root $x$ of $f(x) - c$ is repeated. Can $f$ have more than one unusual point?
**Short remarks**
* There can be exactly one unusual point, e.g. if $f(x) = x^2$. There ca... | https://mathoverflow.net/users/586 | Given a polynomial f, can there be more than one constant c such that every root of f(x)-c is repeated? | This is impossible by the [Mason-Stothers theorem](https://en.wikipedia.org/wiki/Mason%E2%80%93Stothers_theorem) (which holds over any algebraically closed field of characteristic zero).
We want to find $f, g, h$ such that $f + g = h$ where $g$ is a constant and $f, h$ have all of their roots repeated. If $g$ is non... | 50 | https://mathoverflow.net/users/290 | 81736 | 48,979 |
https://mathoverflow.net/questions/81721 | 21 | By a "homotopy commutative diagram," I mean a functor $F: \mathcal{I} \to \mathrm{Ho}(\mathrm{Top})$ to the homotopy category of spaces. By a "strictification," I mean a lifting of such a functor to the category $\mathrm{Top}$ of topological spaces. I am curious about simple instances where such a "strictification" doe... | https://mathoverflow.net/users/344 | A homotopy commutative diagram that cannot be strictified | EDIT: Tom Goodwillie, in the comments, points out (interpreted in a quite charitable way) that there are two mistakes with the following argument. The $\pi\_1$-obstruction does exist. However:
* It misreads the question and assumes that there is some portion of an actual diagram which is homotopy commutative.
* Even ... | 12 | https://mathoverflow.net/users/360 | 81749 | 48,986 |
https://mathoverflow.net/questions/81652 | 10 | Let $H$ be a Hilbert space and $U$ a closed subspace of $H\times H$ .
Does then exist closed subspaces $V$ and $W$ of $H$ such that $H\times H =
U \oplus (V\times W)$ ?
See also [Perturbations of an operator that disconnect the spectrum](https://mathoverflow.net/questions/24672/perturbations-of-an-operator-that-disc... | https://mathoverflow.net/users/17261 | Complement of a subspace which is a cartesian product | Let me expand my comment: the answer is yes, and in general $V$ and $W$ can be constructed as follows: let $p\_1,p\_2:U \to H$ denote the restrictions to $U$ of the two coordinate projections. Then $V$ is the image of the spectral projection $1\_{[0,1/2]}(p\_1p\_1^\*)$ and $V$ is the image of the spectral projection $1... | 8 | https://mathoverflow.net/users/10265 | 81754 | 48,990 |
https://mathoverflow.net/questions/24672 | 9 | The following question came to me while working on a technical matter about transversality in infinite dimension, and I'm really curious to know whether it has an affirmative answer at least under extra hypotheses.
>
> Let A be a bounded linear operator on
> a Banach space E. Does it exist a
> bounded linear oper... | https://mathoverflow.net/users/6101 | Perturbations of an operator that disconnect the spectrum | For Hilbert spaces, the conjecture follows from fact 4 and the answer to question [Complement of a subspace which is a cartesian product](https://mathoverflow.net/questions/81652/complement-of-a-subspace-which-is-a-cartesian-product) applied to the kernel of the map $H\times H\ni (v,w) \mapsto Av + (I-A)w \in X$ .
| 6 | https://mathoverflow.net/users/17261 | 81760 | 48,992 |
https://mathoverflow.net/questions/81705 | 1 | I am interesting in the notion of cobordism.
In particular, I want to understand Smale's proof of Poincare conjecture
in higher dimention.
In a webpage, I knew that the proof is followed by h-cobordism shortly.
However, I do not notice that proof.
Please tell me how to proof it.
I like simple proof diagramatic a... | https://mathoverflow.net/users/16516 | Generalized Poincare conjecture from h-cobordism | In order to prevent this question from resurfacing periodically due to the lack of an answer, here's a summary of the answers that are in the comments.
* Milnor's "Lectures on the h-cobordism theorem"
* Kosinski's "Differential Manifolds"
| 3 | https://mathoverflow.net/users/284 | 81767 | 48,997 |
https://mathoverflow.net/questions/81775 | 2 | In a [recent question](https://mathoverflow.net/questions/81446/for-which-algebras-does-differential-operators-satisfy-a-pbw-like-theorem), I recalled the notion of *differential operator*, *polyderivation*, and *principal symbol* for a commutative algebra $A$ over some fixed commutative ring $k$. (I will not repeat th... | https://mathoverflow.net/users/78 | What's an example of a commutative algebra over $\mathbb Q$ that fails to satisfy this version of the "PBW theorem" | Let me try to give a naive answer. Consider the following symmetric triderivation on $A=k[x]/x^2$:
$$
x\partial\_x\otimes\partial\_x\otimes\partial\_x:(x,x,x)\mapsto x
$$
How could it be in the image of $s\_3$, considering that any differential operator on $A$ is of order $\leq 2$?
| 5 | https://mathoverflow.net/users/7031 | 81786 | 49,008 |
https://mathoverflow.net/questions/81772 | 5 | I'm very curious about this and would be really grateful for any help or comments in this direction. If we consider any of the following number-theoretical constants:
1)The various singular series arising from any given system $\Psi: \mathbb{Z}^{d}\rightarrow \mathbb{Z}^t$,$d,t \geq 1$ in the Green-Tao paper "Linear... | https://mathoverflow.net/users/18494 | How are these number-theoretical constants actually distributed? | These are Euler products which are convergent (though only conditionally, usually), and such Euler products quite often have a limiting distribution when taken in "reasonable" families. Once something like this is proved, one can conclude that the values are dense in the support of the limiting measure. These limit mea... | 5 | https://mathoverflow.net/users/20038 | 81787 | 49,009 |
https://mathoverflow.net/questions/81684 | 2 | I have a conjecture concerning how "tightly" two equivalent $n$-fold extensions of modules over an algebraic group over a field might
be "linked". I suspect that the
question has been already studied in at least some depth, and a
positive or negative answer (or partial positive or negative answer) would either way be ... | https://mathoverflow.net/users/19048 | Can the 'linkages' between equivalent extensions of modules of an algebraic group be taken to have bounded length? | The linkage bound is 2.
If the algebraic group is simple, say over an algebraically closed $k$,
then one has the following lemma.
Lemma. If $V$, $W$ are finite dimensional,
there is an $m$ depending on $V$, $W$, so that if $St\_n$
is the $n$-th Steinberg module with $n\geq m$ the natural map
$Ext^i(V,W)\to Ext^i... | 5 | https://mathoverflow.net/users/4794 | 81788 | 49,010 |
https://mathoverflow.net/questions/81792 | 4 | Let $G=\langle H, t; A^t=B\rangle$ by an $HNN$-extension of $H$, $A$ and $B$ isomorpic subgroups of $H$ where conjugation by $t$ induces the isomorphism.
Assuming $H$ is a finite group it is a well-known consequence of Britton's Lemma that if $\hat{H}$ is a subgroup of $G$ with $\hat{H}\cong H$ then $\hat{H}$ is conj... | https://mathoverflow.net/users/6503 | When are isomorphic copies of the base group in an $HNN$-extension subgroups of the base group (up to conjugacy)? | If H is a finitely generated torsion group, or more generally has Serre's property FA (fixed point property on trees) then what you want is true. Indeed H will have a fixed point in the Bass-Serre tree of the HNN extension. Since all vertex stabilizers are conjugate on this case, you are done.
| 6 | https://mathoverflow.net/users/15934 | 81795 | 49,011 |
https://mathoverflow.net/questions/81793 | 1 | I am trying to understand how estimates on sublevel integrals imply estimates on oscillatory integrals. Specifically in [this](http://arxiv.org/abs/0804.1579) article by M. Greenblatt it says on page 7:
>
> By well-known methods relating sublevel integrals to oscillatory integrals, the above results about [some sub... | https://mathoverflow.net/users/12688 | Oscillatory integral decay & sublevel set growth | The author supplies copious references -- Stein's book seems particularly relevant...
| 1 | https://mathoverflow.net/users/11142 | 81796 | 49,012 |
https://mathoverflow.net/questions/81690 | 11 | Three points $z\_1$, $z\_2$, $z\_3$ on the complex plane are given by the coefficients $a\_k$'s of the cubic polynomial $f(z)=(z-z\_1)(z-z\_2)(z-z\_3)=\sum\_{k=0}^3 a\_k z^k$. How does one express the (signed) area $V$ of the triangle with vertices $z\_1$, $z\_2$, $z\_3$ in terms of $a\_k$'s and $\overline{a}\_k$'s? On... | https://mathoverflow.net/users/11100 | area of triangle from coefficients of its cubic? | NB: Note that my $a\_k$ have different signs from those defined in the question. For me,
$$
(z - z\_1)(z-z\_2)(z-z\_3) = z^3 - a\_1\ z^2 + a\_2\ z - a\_3,
$$
so that $a\_k$ is the $k$-th elementary symmetric function of the $z\_i$. This doesn't really affect the answer in any significant way.
While I don't think that... | 13 | https://mathoverflow.net/users/13972 | 81803 | 49,013 |
https://mathoverflow.net/questions/81811 | 2 | I heard this example was given in Whitehead's paper A CERTAIN OPEN MANIFOLD WHOSE GROUP IS UNITY.( <http://qjmath.oxfordjournals.org/content/os-6/1/268.full.pdf> ) But I was confused by his term. Thus I'm looking for an explanation in more standard terms about this example.
But since my aim is to know about an exampl... | https://mathoverflow.net/users/10333 | Contractible noncompact 3-manifold without boundary not homeomorphic to $\Bbb R^3$ | It's discussed in Kirby's "The topology of 4-manifolds", around page 80, and at a glance the argument looks "modern".
| 4 | https://mathoverflow.net/users/13119 | 81813 | 49,019 |
https://mathoverflow.net/questions/81257 | 7 | The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,
$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta A}$
where $YM(A)$ is the Yang Mills 'action' the integral of the curvature square,
$YM(A) = \int d^4x Tr F\_{\mu\n... | https://mathoverflow.net/users/4526 | Yang Mills gradient/heat flow on 4-torus | Donaldson and Kronheimer wrote their book by 1990. There were some further developments about long time behaviour of Yang-Mills flow on four manifolds by, among others, Struwe and collaborators. You may try starting with [Schlatter's dissertation](http://www.ams.org/mathscinet-getitem?mr=1443269).
Crawling through M... | 4 | https://mathoverflow.net/users/3948 | 81818 | 49,022 |
https://mathoverflow.net/questions/81791 | 12 | Recall that if $\Gamma$ is a finite-index subgroup of $\operatorname{SL}\_2(\mathbf{Z})$, then a *cusp* of $\Gamma$ is an orbit of $\Gamma$ on the set $\mathbf{P}^1\_{\mathbf{Q}}$. If $-1\notin \Gamma$, then for any cusp $c$, the stabilizer of $c$ in $\Gamma$ (well-defined up to conjugacy) is an infinite cyclic group g... | https://mathoverflow.net/users/2481 | Does there exist a finite-index subgroup of SL2Z with all cusps irregular? | The thrice-punctured sphere can be represented as a quotient of the upper half-plane by $\Gamma(2)$. One may take as a fundamental domain the region contained in the geodesics between $i\infty, 0, 1, -1$, where $1$ and $-1$ are identified by the "translation by 2" operator. The fundamental group of the punctured sphere... | 8 | https://mathoverflow.net/users/121 | 81822 | 49,024 |
https://mathoverflow.net/questions/81820 | 3 | How "frequent" are smooth projective varieties $X$ with trivial canonical bundle $\omega\_X = \bigwedge^d \Omega^1\_{X/k}$?
E.g. for curves $C/k$, the canonical bundle is trivial iff the genus $g(C) = 1$ (elliptic curves). What is the situation like in the higher dimensional case?
| https://mathoverflow.net/users/nan | How "frequent" are smooth projective varieties with trivial canonical bundle? | If $X$ is a complex projective manifold with trivial canonical bundle, then by a theorem of Bogomolov, there is a finite unramified cover $\tilde X$ of $X$ which decomposes into a product $A\times X\_1\times\ldots\times X\_n\times Y$. Here $A$ is an abelian variety; $X\_i$ are irreducible holomorphic symplectic manifol... | 14 | https://mathoverflow.net/users/6107 | 81824 | 49,025 |
https://mathoverflow.net/questions/81551 | 0 | Let $X=(X\_1,X\_2,X\_3)\sim \text{Dirichlet}(a\_1,a\_2,a\_3)$ and $Y=(Y\_1,Y\_2,Y\_3)\sim \text{Dirichlet}(a\_1+b\_1,a\_2+b\_2,a\_3)$, where all $a\_i$ and $b\_i$ are positive. Is there a natural coupling between $X$ and $Y$ such that $X\_1\geq Y\_1$ and $X\_2\geq Y\_2$ with probability 1?
The following coupling doe... | https://mathoverflow.net/users/19438 | Is there a monotone coupling of Dirichlet random variables? | This is not always possible.
Fix $a\_1,a\_2,a\_3,b\_1$. As $b\_2\to\infty$, we have $Y\_1\to0$ in probability, so it is not stochastically larger than $X\_1$.
A necessary condition is domination of the expectations, namely $\frac{a\_i+b\_i}{\sum a\_j+b\_j} \ge \frac{a\_i}{\sum a\_j}$ for $i=1,2$, but this is not s... | 1 | https://mathoverflow.net/users/9422 | 81829 | 49,027 |
https://mathoverflow.net/questions/81815 | 13 | Short version: Is the axiom of union independent of the rest of axioms of ZF?
NO) [Tourlakis (2003)](http://books.google.es/books?id=zwv0RgAACAAJ&dq=tourlakis+set+vol+2&hl=es&ei=vWbOTu6YHYmb8QP8-MzODw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA) says in p. 177 that the axiom of union can be derived from th... | https://mathoverflow.net/users/19498 | Is the Axiom of Union independent of the rest of ZF? | The usual version of collection is the second one in Kaveh's answer. It, with the remaining axioms, won't give the axiom of union. A counterexample is given in the question, except for a slightly unusual definition of $H\_\kappa$ for singular $\kappa$; it should be the collection of those sets $x$ such that each member... | 15 | https://mathoverflow.net/users/6794 | 81831 | 49,029 |
https://mathoverflow.net/questions/81799 | 2 | I have considered about Lehmer's conjecture that φ(n)|(n-1) if and only if n is a prime. I generalized the conjecture onto genernal finite commutative ring. The idea is below.
Set $A$ is a finite commutative ring, and $B$ is its unit group. Then there is a partition of $A$ by $B$ in multipication. Choose the represen... | https://mathoverflow.net/users/19037 | A question on finite commutative rings | Suppose that $|A^\times|$ divides $|A| -1$, where $A^\times = B$ is the group of units.
Since a finite ring has a unique factorization into local rings, we can write
$A = \prod\_{i=1}^m A\_i$ with local rings $(A\_i,\mathfrak{m}\_i)$ and find for the unit group
$$A^\times = \prod\_{i=1}^m \hspace{1pt} A\_i^\time... | 3 | https://mathoverflow.net/users/10194 | 81832 | 49,030 |
https://mathoverflow.net/questions/81827 | 2 | Let $X = (f=0) \subset \mathbb{C}^3$
be an isolated hypersurface singularity and
$\mu: \tilde{X} \rightarrow X$ be a resolution of a singularity whose exceptional locus $E$ is simple normal crossing. Then
**Question** Is $H^1(\tilde{X}, \mathcal{O}\_{\tilde{X}}(- E)) =0$ ?
I think it's OK if it is Du Bois. How ... | https://mathoverflow.net/users/12390 | $H^1$ of a certain line bundle on the resolution of a 2-dimensional hypersurface singularity | If $X$ is a seminormal surface, then actually this vanishing is equivalent to $X$ being Du Bois:
1. If $X$ is Du Bois, the vanishing is a direct consequence of Cor.6.2 of [this paper](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-51/issue-1/Du-Bois-pairs-and-vanishing-theorems/10.1215/0023608... | 4 | https://mathoverflow.net/users/10076 | 81833 | 49,031 |
https://mathoverflow.net/questions/81092 | 2 | Hi everyone!
This is my first post, apologies if I made any mistakes anywhere.
Here goes the question:
Consider all length 7 binary sequences.
Let $X$ be the set of sequences with hamming weight 3 and let $Y$ be the set of sequences with hamming weight 4.
For each vertex in $X$, connect an edge to each element in $... | https://mathoverflow.net/users/16054 | Enumerating all Hamiltonian Cycles in a Bipartite Vertex Transitive Graph | The case n=7 is pretty similar to counting the Hamiltonian cycles in the 6-cube (which has 64 vertices but more edges). That problem has been open for a long time, but was recently settled by Harri Haanpaa and myself. The approach is general and we just submitted a paper with the title "A Dynamic Programming Approach t... | 3 | https://mathoverflow.net/users/19500 | 81835 | 49,032 |
https://mathoverflow.net/questions/81800 | 5 | In some texts on classical mechanics and not only, the Euler--Lagrange equations of motion are directly obtained as solution of variational problems.
On the other side, sometimes reading about hamiltonian mechanics, one find the expression that this latter formulation is preferred to the lagrangian one because of it... | https://mathoverflow.net/users/12617 | The Lagrangian formulation of mechanics without going through variational principles. | The Lagrangian (or variational) formulation of the Euler-Lagrange equations and the Hamiltonian formulation are equivalent. This equivalence can be made quite explicit and goes a bit deeper than the standard treatments show. The equivalence can be established in several steps, which I'll try to outline below with refer... | 10 | https://mathoverflow.net/users/2622 | 81846 | 49,037 |
https://mathoverflow.net/questions/80637 | 8 | Hi,
I am reading about p-adic representations from Fontaine's book which can be found at <http://staff.ustc.edu.cn/~yiouyang/research.html>. On page 145
where they prove Proposition 5.24 which is essentially the theorem
of Tate-Sen, they show $H^{n}(Gal(L/K\_{\infty},C(i)^{G\_L})=0$ and
the argument is essentially sam... | https://mathoverflow.net/users/2081 | P-adic representations | The whole thing is done with more details in Tate's original article "p-divisible groups", section 3.2. Tate proves that one can approximate a cocyle in $C\_p(i)$ by cocyles with values in $Q\_p^{alg}(i)$ and this is how he reduces the computation to the "discrete case".
I would suggest that it's better to prove the... | 7 | https://mathoverflow.net/users/5743 | 81861 | 49,043 |
https://mathoverflow.net/questions/81859 | 3 | For $m,n,c\in\mathbb{N}$ let $S(m,n;c)$ be the Kloosterman sum
$$S(m,n;c)=\sum\_{a=1, \gcd (a,c)=1}^ce\left(\frac{ma+n\overline{a}}{c}\right).$$
The Kuznetsov Trace Formula allows us to obtain bounds for sums of the form
$$\sum\_{c\le x,~ c=0\mathrm{mod} q}\frac{S(m,n;c)}{c},$$
which are better than those obtained by s... | https://mathoverflow.net/users/19368 | Sums of Kloosterman sums over primes | No, this is a well-known open problem. One doesn't even know that the sign of $S(1,1;p)$ changes infinitely often... The best that has been achieved are estimates restricted to moudli $c$ with a bounded number of prime factors, by combining sieve methods with automorphic forms (and some average forms of Sato-Tate), see... | 5 | https://mathoverflow.net/users/20038 | 81862 | 49,044 |
https://mathoverflow.net/questions/81850 | 5 | What is the cardinality of the set $F$ of all [normal functions](https://en.wikipedia.org/wiki/Normal_function) $f \colon \omega\_1 \to \omega\_1$, where $\omega\_1$ is the first uncountable ordinal?
What is the least cardinality of a subset of $F$ such that every function in $F$ is bounded by some element of the subse... | https://mathoverflow.net/users/9550 | Cardinality of cofinal set of normal functions $f \colon \omega_1 \to \omega_1$ | For both questions, the answer does not change when you remove the word "normal" from the question.
For the first question: There is a 1-1 map $f\mapsto N(f)$ that assigns to each function $f$ a normal function $N(f)$. $N(f)(\alpha)$ just adds up all values of $f$ below $\alpha$. (Or better: of $f+1$, to make it stri... | 9 | https://mathoverflow.net/users/14915 | 81865 | 49,046 |
https://mathoverflow.net/questions/81805 | 9 | The hyperoctahedral group $H\_n$ has several descriptions; as a wreath product; as signed permutation matrices; as the Weyl group of type $B\_n$ or $C\_n$. In all these descriptions it is apparent that the symmetric group $S\_n$ is a subgroup.
I would like to know the Frobenius characters of the restrictions of the i... | https://mathoverflow.net/users/3992 | Restriction of characters of hyperoctahedral groups. | Even though you found the answer, I'd like to point to the following reference, where I learned this result, at a time when the book by Macdonald (note the captitalization) did not yet have an appendix B to chapter I (i.e., before its second edition):
A. V. Zelevinsky, *Representations of Finite Classical Groups* A H... | 4 | https://mathoverflow.net/users/19077 | 81879 | 49,053 |
https://mathoverflow.net/questions/81877 | 2 | Let (M,g) be a negatively curved manifold , let p be any point of M and denote by G=π1(M,p) . the minimal representative (by minimal i mean the smallest length representative ) of every α in G is a simple closed geodesic loop at p . my question is why it should be simple ?
| https://mathoverflow.net/users/nan | Minimal representative of the elements of the fundamental group of a negatively curved manifold | For a negatively curved manifold, there is a unique geodesic in a free homotopy class, and a unique geodesic broken loop in a homotopy class, and it is the shortest curve. In neither case is the curve necessarily simple. For references, almost any book on differential geometry will work (Cheeger/Ebin, Ballmann/Gromov/S... | 4 | https://mathoverflow.net/users/11142 | 81886 | 49,055 |
https://mathoverflow.net/questions/81858 | 6 | Let $\Phi$ be an irreducible root system, with positive roots $\Phi^+$ relative to the base $\Delta$.
If $W$ is the Weyl group, how can I determine if $-I$ belongs to $W$? Equivalently how can I see if the (unique) longest element in $W$ is $-I$?
| https://mathoverflow.net/users/19518 | Does -I belong to Weyl group? | As Koen S points out, the longest element of an irreducible Weyl group is treated in an earlier question (in fact, it comes up in several questions). The question asked here presupposes a standard linear realization of the Weyl group, as occurs in the structure theory of a semisimple Lie algebra over $\mathbb{C}$ for i... | 11 | https://mathoverflow.net/users/4231 | 81898 | 49,062 |
https://mathoverflow.net/questions/81873 | 2 | Suppose $M$ is a closed 3-manifold, $C\subset M$ is a simple closed curve which represent identity in $\pi\_{1}(M)$. Then $C$ bounds an immersed disk in $M$.
My question is:
When does it bound an imbedded disk in $M$?
I don't know about it at all. If you have any reference, please tell me.
Thank you!
| https://mathoverflow.net/users/18496 | what is the meaning of a curve $C$ representing Identity in fundamental group? | Suppose that $K$ is a simple closed curve in $M^3$. I'll assume that $M$ is orientable, compact, and without boundary. Let $V$ be a closed regular neighborhood of $K$; so $V \cong S^1 \times D^2$ is a solid torus. Let $X$ be the closure of $M - V$; so $X$ is the *exterior* of $K$. Let $T = X \cap V$; so $T$ is a two-to... | 4 | https://mathoverflow.net/users/1650 | 81908 | 49,069 |
https://mathoverflow.net/questions/81913 | 8 | How "frequent" are smooth projective varieties $X$ with (anti-)ample canonical bundle $\omega\_X = \bigwedge^d \Omega^1\_{X/k}$?
E.g. for curves $C/k$, the canonical bundle is ample iff the genus $g(C) > 1$ and anti-ample iff $g(C) = 0$. What is the situation like in the higher dimensional case?
(This question is i... | https://mathoverflow.net/users/nan | How "frequent" are smooth projective varieties with (anti-)ample canonical bundle? | Just as in the case of curves, there is in general a trichotomy of cases:
Let $X$ be a smooth projective variety. Then $X$ is *built* from pieces $Y$ with
* $\kappa(Y)<0$
* $\kappa(Y)=0$
* $\kappa(Y)=\dim Y$.
Here $\kappa$ denotes the [Kodaira dimension](http://en.wikipedia.org/wiki/Kodaira_dimension). If the anti... | 27 | https://mathoverflow.net/users/10076 | 81920 | 49,075 |
https://mathoverflow.net/questions/81910 | 20 | Write $X\_N$ for this blow up. Place the N points in 'general position' as needed. Then $X\_6$ embeds in $CP^2$ as a smooth cubic surface. (See, eg, Griffiths and Harris.) But there is no other $N$ (except $N=0$)
for which $X\_N$ embeds in $CP^3$.
(Proof: The topology of the blow-up
disagrees with that of a smooth surf... | https://mathoverflow.net/users/2906 | When does the blow-up of $CP^2$ at N points embed in $CP^4$? | For $N=1$ the answer is yes: the embedding into ${\mathbb P}^4$ is given by the linear system of conics through the blown up point (the image has degree $d=3$).
For $N=5$, the system of cubics through the 5 points gives an embedding ($d=4$).
ADDED: here are 2 slightly less obvious examples:
For $N=8$ one can take qua... | 26 | https://mathoverflow.net/users/10610 | 81922 | 49,076 |
https://mathoverflow.net/questions/81740 | 28 | I am looking for a modern introduction to spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what has been learned since the notes were written). In particular I'm interested in a source that covers some of the variations on Sp... | https://mathoverflow.net/users/2203 | Modern source for spectra (including ring spectra) | [I'm a novice, and this got posted out of order: it answers Bak's question below.]
Sure, I can provide that. The cited reference was published in 1995, which
was well before details of symmetric or orthogonal spectra were available,
so it gives a fair amount of background but only refers to EKMM spectra for
a modern ... | 23 | https://mathoverflow.net/users/14447 | 81930 | 49,079 |
https://mathoverflow.net/questions/81937 | 5 | Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}\_q^n$ with addition componentwise where $\mathbb{Z}\_q$ is the integers mod $q$. Let $V$ be a submodule of $W$. Let $V^{\perp} = \{w \in W : \forall v \in V \quad w \cdot v = 0 \}$ where $w\cdot v = w\_1v\_1 + \ldots + w\_nv\_n$. Is it true that ${(V^{... | https://mathoverflow.net/users/19538 | "Orthogonal complement" in $\mathbb{Z}_q^n$ | Yeah, it's true. Since $\mathbb{Z}/q$ is a principal ideal ring, there is an extension of the Euclidean algorithm to matrices that puts any matrix in Smith normal form. It means that after an automorphism of $(\mathbb{Z}/q)^n$, any submodule $V$ can be put into a standard form in which it is generated by vectors of the... | 12 | https://mathoverflow.net/users/1450 | 81938 | 49,082 |
https://mathoverflow.net/questions/81789 | 9 | Let $F$ be a local field and $G= GL(n,F)$.
Assume that $\gamma$ is an element of $G$ and $G\_\gamma$ is its centralizer.
The orbital integral is defined as
$$ O\_\gamma^G( \phi) = \int\limits\_{G\_\gamma \backslash G} \phi( g^{-1} \gamma g) d g.$$
We can assume wlog that $\gamma$ is elliptic. Can we lift the ellip... | https://mathoverflow.net/users/10400 | Is this a subcase of the fundamental lemma? | I will offer some words on this, but only because no-one else has; I was holding out hoping that one of the more automorphic people would chip in. It might be worth taking much of the below with a pinch of salt.
So I've been trying to penetrate the "fundamental lemma" literature myself, and let me begin by showing my... | 15 | https://mathoverflow.net/users/1384 | 81940 | 49,083 |
https://mathoverflow.net/questions/80177 | 12 | There is a combinatorial question posed to me (or rather, posed near me) by my adviser. I am having quite a lot of difficulty proving it. It goes:
>
> For any NIP theory $T$ (complete with infinite models as usual) and any partitioned formula $\phi(x; y)$, there are natural numbers $k$ and $N$ such that for any fin... | https://mathoverflow.net/users/15735 | A Model-Theoretic Helly's Theorem | First, let me point out that the proof of udtfs is joint work with Artem Chernikov.
Hunter Johnson's answer is correct. The reference is Matousek's paper. I don't know of any proof not using probabilty theory. I tried a little bit to look for a model theoretic proof, but did not find any.
However, Matousek's $(p,k)... | 6 | https://mathoverflow.net/users/19534 | 81941 | 49,084 |
https://mathoverflow.net/questions/80971 | 8 | Let $S$ be the spectrum of a discrete valuation ring with generic point $\eta$. Let $C/\eta$ be a smooth connected curve with an $\eta$-valued point, and let $\mathcal{C}/S$ be the smooth locus of the minimal proper regular model of $C$ over $S$. Let $N/S$ denote the Neron model of the Jacobian of $C\_\eta$, and let $\... | https://mathoverflow.net/users/4710 | Is the extension of the Abel-Jacobi map to the smooth locus of the minimal regular model of a curve an immersion? | See my article:
<http://arxiv.org/abs/math/9806173>
Best regards, Bas Edixhoven.
P.S. Thanks to Liu Qing who drew my attention to this.
| 10 | https://mathoverflow.net/users/19540 | 81943 | 49,085 |
https://mathoverflow.net/questions/81508 | 13 | I originally posted this question on math.stackexchange (<https://math.stackexchange.com/questions/83182/modulus-of-continuity-take-2>), but it's been a few days and I haven't received any correct answers.
Let $\rho: \mathbb{R}^+ \to \mathbb{R}^+$ be a continuous nondecreasing function such that $\rho(t) = 0$ if and ... | https://mathoverflow.net/users/4362 | Modulus of Continuity | All right, I don't think that anything I'll say below will be new to Paul. However it'll address Vaughn's concerns (to the extent he expressed them in the comments by the moment of this writing).
Of course, the ultimate purpose is to show that for a given modulus of continuity $\rho$, the set $U$ of continuous functi... | 10 | https://mathoverflow.net/users/1131 | 81952 | 49,089 |
https://mathoverflow.net/questions/81967 | 1 | Consider the $n$-dimensional sphere $S^n$. I'm especially interested in the $n=4$ case. The Hilbert space $L^2(S^n)$ can be decomposed into a direct sum of eigenspaces of the Laplacian, which are finite dimensional. I'm looking for non-isometric conformal transformations
$$f: S^n \to S^n$$
s.t. for some $\lambda, \... | https://mathoverflow.net/users/11146 | Conformal transformations and harmonic analysis on the sphere | The only possibility is the trivial one when $\lambda$ is so small that the only eigenfunctions with eigenvalue less than $\lambda$ are constants (eigenvalue zero). Otherwise the eigenfunctions with eigenvalue less than $\lambda$ span the space of polynomials of degree at most $d$ for some positive integer $d$, and the... | 2 | https://mathoverflow.net/users/14830 | 81969 | 49,094 |
https://mathoverflow.net/questions/81977 | 3 | Let's assume $M$ is a symplectic manifold with the group action $G$. If $Lie(G)$ is semi simple then why the Hamiltonian condition, which requires the existence of linear map $Lie(G)\to C^{\infty}(M,R)$ is always satisfied?
| https://mathoverflow.net/users/13559 | If Lie(G) is semi simple then the moment map exists! | Let $\mathfrak{g}=Lie(G)$. The action of $G$ on $M$ gives a morphism of Lie algebras $a:\mathfrak{g}\rightarrow Vect\_{symp}(M)$.
Since $\mathfrak{g}$ has trivial abelianization, $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$, i.e. any element can be decomposed into commutators. An easy computation shows that a commutato... | 7 | https://mathoverflow.net/users/18512 | 81981 | 49,100 |
https://mathoverflow.net/questions/81982 | 6 | I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, connections). I know that cubical sets behave more nicely when one has an extra set of degeneracies called connections. The questi... | https://mathoverflow.net/users/14167 | What is the intuition of connections for cubical sets? | A list of precise references for connections on cubical sets has to start with :
R. Brown, P. J. Higgins and R. Sivera, 2011, Nonabelian Algebraic Topology: Filtered spaces, crossed complexes, cubical homotopy groupoids , volume 15 of EMS Monographs in Mathematics , European Mathematical Society.
as in there Brown,... | 7 | https://mathoverflow.net/users/3502 | 81987 | 49,103 |
https://mathoverflow.net/questions/81990 | 1 | It is probably a trivial question. But I don't see the answer (and I didn't find anywhere).
Given a (complete and cocomplete) category X and an object A of X, we can define the "undercategory" A/X. See <http://ncatlab.org/nlab/show/under+category>
I have already noticed that the coproduct of a set {i\_l: A to X} i... | https://mathoverflow.net/users/18017 | Product in undercategory | For a family of objects $A \to B\_i$ in $A/X$, their coproduct is usually called the [*pushout*](http://ncatlab.org/nlab/show/pushout) of the morphisms $A \to B\_i$ in $C$. It represents the functor $X \to \mathrm{Set}$, which maps $P$ to the set of families of morphisms $B\_i \to P$ which "coincide" on $A$. Pushouts i... | 5 | https://mathoverflow.net/users/2841 | 81996 | 49,107 |
https://mathoverflow.net/questions/81998 | 6 | Could any tell me if a multivariate polynomial generated from the sum of irreducible single variable polynomial is irreducible?
For example, f(x)=x^2+2x+2, g(x)=x^2+3x+3, h(x)=x^3+2x^2+2x+2 all of them are irreducible, then what about f(x,y,z) = f(x)+g(y)+h(z)?
| https://mathoverflow.net/users/19557 | Irreducibility for multivariate polynomial polynomial generated from sum of irreducible polynomial in one variable | $(x^2+1)-(y^2+1)=(x+y)(x-y)$
| 7 | https://mathoverflow.net/users/18814 | 82000 | 49,110 |
https://mathoverflow.net/questions/82003 | 10 | It is well-known that the fundamental group of a twice-punctured torus is a free group of rank three.
I see that there is no one-to-one correspondence between the homotopy classes of essential simple loops on twice-punctured torus and the conjugacy classes of primitive elements in a free group of rank three.
Do w... | https://mathoverflow.net/users/19558 | Primitive elements in a free group of rank three | A classification of primitive elements in a free group of rank greater than two is a hard problem, and there is no really satisfactory classification known. I am pretty sure [this paper of Shpilrain](http://www.sci.ccny.cuny.edu/~shpil/countprim.ps) is pretty close to the last word. As for elements representing simple ... | 5 | https://mathoverflow.net/users/11142 | 82008 | 49,113 |
https://mathoverflow.net/questions/81965 | 1 | I'm trying to understand the composition product (Tall-Wraith monoid) for $R$-rings (aka commutative $R$-algebras). Multivariable polynomials over $R$ are examples $R$-rings. I think that $R[X\_1,...,X\_n] ⊙\_R R[Y\_1,...,Y\_m]$ is (isomorphic to) $R[Z\_{1,1},...,Z\_{n,m}]$, and that the composition product of $p \in R... | https://mathoverflow.net/users/4085 | Composition Product of Multivariable Polynomials | Yes, this is correct, except that the plethystic monoidal product refers not to a monoidal product on the category of $R$-rings, but to a monoidal product on the category of co-$R$-ring objects in the category of $R$-rings. (What one might call an $R$-biring.) Or, it could also refer to a right action
$$\odot \colon... | 4 | https://mathoverflow.net/users/2926 | 82009 | 49,114 |
https://mathoverflow.net/questions/81989 | 25 | This question has no justification other than a bit of fun.
We all know that the cubic is solvable by "radicals" ($\root2\of{}$ and $\root3\of{}$) in characteristics $\neq2,3$. The formula was discovered by the Italians in the 16th century [(see here)](http://en.wikipedia.org/wiki/Cubic_equation).
In characteristic... | https://mathoverflow.net/users/2821 | Solving the cubic by "radicals" in characteristics 2 and 3 | I asked an undergraduate (Dubravka Bodiroga at Hood College) to work these results out last summer. Here is her cubic formula in characteristic 3 (paraphrasing from something she sent me):
Consder the polynomial
$$
x^3 - a\_1 x^2 + a\_2 x - a\_3,
$$
where the coefficients belong to a commutative ring in which $3=0$. ... | 16 | https://mathoverflow.net/users/2490 | 82012 | 49,117 |
https://mathoverflow.net/questions/82007 | 4 | Let $F$ be a local field. Consider the group extension (split)
$$ PSL(n,F) \rightarrow PGL(n,F) \rightarrow F^\times / (F^\times)^n.$$
What knowledge about $PGL(n)$ is necessary in order to understand the representation of $PSL(n)$ from this?
| https://mathoverflow.net/users/10400 | How to understand the representation theory of $SL(n)$ from $GL(n)$? | In the case of a non-Archimedean local field $F$, one may reduce the representation theory of $H={\rm SL}(n)$ to that of $G={\rm GL}(n)$. For instance supercuspidal representations of $H$ are obtained as constituents of the restriction to $H$ of the supercuspidal representations of $G$ (these restrictions are semisimpl... | 5 | https://mathoverflow.net/users/4767 | 82013 | 49,118 |
https://mathoverflow.net/questions/82014 | 3 | For a homomorphism of rings $R \to S$, the following are equivalent:
a) $(-) \otimes\_R S : \mathrm{Mod}(R) \to \mathrm{Mod}(S)$ reflects isomorphisms
b) $R \to S$ satisfies effective descent with respect to modules.
c) $R \to S$ is *pure*: For every $R$-module $M$ the natural map $M \to M \otimes\_R S$ is a mono... | https://mathoverflow.net/users/2841 | Noetherian descent extension for a given ring | I am afraid that this is not true for any non-noetherian ring.
Let $R \to S$ be a descent extension, and assume that $S$ is noetherian. Let $I$ be an ideal of $R$; there will be a finitely generated ideal $J \subseteq I$ of $R$ such that $JS = IS$. Hence $R/I$ and $R/J$ become isomorphic when tensored with $S$, so th... | 8 | https://mathoverflow.net/users/4790 | 82017 | 49,121 |
https://mathoverflow.net/questions/81979 | 4 | I know that: Let X be a uniformly convex Banach space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf\_{x\in C}\left\Vert a-x \right\Vert$.
Moreover I know that: let $X$ be a Banach space, such that for every $a\in X$ and $C\subset X$ closed and conve... | https://mathoverflow.net/users/19550 | Projection exists ⇒ Uniformly convex? | Your modified question has an affirmative answer. An equivalent form, in view of the Eberlein-Smulian theorem, is whether the Banach space $X$ must be reflexive if every closed bounded non empty set admits best approximations. If $X$ is not reflexive, then by R. C. James' famous characterization of reflexivity, there i... | 8 | https://mathoverflow.net/users/2554 | 82018 | 49,122 |
https://mathoverflow.net/questions/81960 | 166 | The question briefly:
Can one explain the "Dzhanibekov effect" (see youtube videos from space station or comments below) on the basis of the standard rigid body dynamics using Euler's equations? (Or explain that this is impossible and that would be yet another focus ... )
Here are more details.
See these curious... | https://mathoverflow.net/users/10446 | The "Dzhanibekov effect" - an exercise in mechanics or fiction? Explain mathematically a video from a space station | One can see this effect qualitatively from Newtonian first principles such as $F=ma$ (as opposed to Hamiltonian or Lagrangian principles, such as conservation of energy and angular momentum) by looking at a degenerate case, when one moment of inertia is very small and the other two are very close to each other.
More ... | 171 | https://mathoverflow.net/users/766 | 82020 | 49,124 |
https://mathoverflow.net/questions/81349 | 2 | My current research took me to the realm of PDE's (which for the long time used to be terra incognita for me as I'am a probabilist). Equations that I'am working with are mostly of second order or Hamilton-Jacobi equations. It's no suprise that I'am dealing with various notions of weak solutions (viscosity solutions mos... | https://mathoverflow.net/users/19379 | Regularity theory for "nice" differential equations | Just to make everything clear: the answer for my question is "yes, it is possible".
Appropriate example is in paper under link provided by pgassiat in comments.
| 0 | https://mathoverflow.net/users/19379 | 82033 | 49,130 |
https://mathoverflow.net/questions/81942 | 4 | In C. McMullen's [Uniformly Diophantine numbers in a fixed real quadratic field](http://www.math.harvard.edu/~ctm/papers/home/text/papers/cf/cf.pdf)
generalized Fibonacci sequence are defined as follows:
$f\_0=0,f\_1=1,f\_m=tf\_{m-1}-nf\_{m-2}$ where some fixed $t\in \mathbb Z$ and $n$ is $+1$ or $-1$ and $t^2-4n>0$... | https://mathoverflow.net/users/6836 | Primes in generalized fibonacci sequences | These are Lucas Sequences (<http://en.wikipedia.org/wiki/Lucas_sequence>), of which the Fibonacci Sequence is a specific case, and they share with the usual Fibonacci Sequence the following characteristics which, unless there are further reasons preventing what I'm about to say from being true, would allow the same sor... | 2 | https://mathoverflow.net/users/18494 | 82042 | 49,133 |
https://mathoverflow.net/questions/81995 | 2 | If manifolds have sectional curvature lower bound, then those manifolds has subsequence convergent to Alexandrov space. Is there similar results for manifolds with Ricci lower bound?
| https://mathoverflow.net/users/19555 | any compactness theorem for manifolds which has Ricci lower bound? | This is getting too long for a comment so I'm posting this as an answer...
@guoyi xu No you can not claim in any sense that $|\nabla\_y dist\_x|> 1- \delta$ for any $y \in B\_{b(x)}(x)\backslash \{x\}$ for limits of spaces with just lower Ricci bounds. That is indeed different from Alexandrov spaces where you do have... | 9 | https://mathoverflow.net/users/18050 | 82043 | 49,134 |
https://mathoverflow.net/questions/82057 | 1 | Hi! I ran into this PDE working on a question in cake cutting. Here it is:
$x\partial\_1f(x,y)-(1-x)\partial\_2f(y,x)=0$
for all $(x,y) \in [0,1]\times[0,1]$.
Thanks!
| https://mathoverflow.net/users/19579 | Help with what is most likely an easy PDE | The updated PDE is simpler than the original one. The same argument I gave in the comments still applies, but now you don't need to solve a wave equation, just integrate along one of the coordinates.
Namely, give $f(x,y)$ any sufficiently regular value on $(x,y)\in[0,1]\times[0,1]$ with $y\ge x$. On the other half of... | 3 | https://mathoverflow.net/users/2622 | 82061 | 49,139 |
https://mathoverflow.net/questions/81939 | 19 | I am curious about how much descriptive set theory is involved in inner model theory.
For instance Shoenfield's absoluteness result is based on the construction of the Shoenfield tree which projection is $\aleph\_1$-Suslin. Also the Schoenfield tree is homogeneous, meaning the direct limit $M\_x$ of the ultrapowers b... | https://mathoverflow.net/users/3859 | Why does inner model theory need so much descriptive set theory (and vice versa)? | I think that in some ways you have answered your question yourself: we see that to prove properties about sets, say within the projective hierarchy, we need representations of those sets of reals as *trees*, but moreover nice trees with certain properties (and the homogeneous trees you mention in particular with measur... | 23 | https://mathoverflow.net/users/6942 | 82066 | 49,143 |
https://mathoverflow.net/questions/82046 | 14 | I have the following question for which I haven't been able to find any reference or proof.
Suppose we know that a univariate polynomial $P(X)$ with integer coefficients is the sum of squares of two polynomials with rational coefficients.
Is it true that $P(X)$ must also be the sum of squares of two polynomials wi... | https://mathoverflow.net/users/11134 | About integer polynomials which are sums of squares of rational polynomials... | Yes. Suppose $n\in \mathbb N$ is minimal so that $P(x)=f\_1^2+f\_2^2$, where $nf\_1$ and $nf\_2$ are in $\mathbb Z[x]$.
Let $p$ be a prime with $p^\alpha||n$. Since $P\in \mathbb Z[x]$ we have $p^{2\alpha}| (p^\alpha f\_1)^2+(p^\alpha f\_2)^2$. Denoting $p^\alpha f\_i$ by $g\_i$, and letting $\beta$ be square root of... | 24 | https://mathoverflow.net/users/2384 | 82073 | 49,146 |
https://mathoverflow.net/questions/45482 | 3 | Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a compact subset K is holomorphically convex (see [Rossi, "Holomorphically convex sets in several complex variables," Ann. Ma... | https://mathoverflow.net/users/3566 | Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex? | The answer to this question is yes, $K$ is necessarily holomorphically convex. See p.161 of "Topological Algebras Selected Topics" by A. Mallios,
| 2 | https://mathoverflow.net/users/3566 | 82077 | 49,148 |
https://mathoverflow.net/questions/82072 | 6 | QUESTION RETRACTED - My original argument was fundamentally mistaken (mixing up lower and upper semi-continuity). Sorry (and thanks for the useful comments)
I need, and (unless I am seriously mistaken) can prove, the following:
`Let $E \subseteq F$ be an (isometric) inclusion of Banach spaces, and let $E^*_1$, $F^*... | https://mathoverflow.net/users/16722 | Continuous choice of Hahn-Banach extensions | If I understand the claims of the OP correctly, I don't think that such a section can actually exist (if there is a misunderstanding on my part, I will happily retract this answer!).
Upon reading the question, I immediately thought of topological vector space versions of the Michael continuous selection theorem (for ... | 8 | https://mathoverflow.net/users/848 | 82081 | 49,149 |
https://mathoverflow.net/questions/82027 | 8 | Let's say a quiver $Q$ is covered by cycles if each of it’s arrows can be included in an oriented cycle.
It's easy to prove that if a path-algebra with relations $KQ/I$ (where $I$ is an admissible ideal) is selfinjective then $Q$ is covered by cycles.
The following question occured to me: is the opposite true?
Is it... | https://mathoverflow.net/users/19564 | Quivers of selfinjective algebras. | One way to prove a finite-dimensional algebra is self-injective is to find an isomorphism of modules $A\cong A^\*$, that is a non-degenerate bilinear form such that $(ab,c)=(a,bc)$. Of course, such a form is uniquely determined by the linear map $t(a)=(a,1)$ (you can recover the form by $(a,b)=t(ab)$).
Thus, for any... | 7 | https://mathoverflow.net/users/66 | 82085 | 49,152 |
https://mathoverflow.net/questions/82062 | 5 | Ax & Kochen [1] proved that for every $d\in\mathbb{N}$ there exists a finite set $A(d)$ such that for every prime $p\not\in A(d),$ every homogeneous polynomial of degree $d$ over $\mathbb{Q}\_p$ in at least $d^2+1$ variables has a nontrivial zero.
1. Is there an effective procedure for determining $A(d)$?
2. For what... | https://mathoverflow.net/users/6043 | Determining the exceptional set in the theorem of Ax & Kochen | Scott Brown (Mem. AMS, 1978) gave a bound for the largest prime $p\_0(d)$ lying in $A(d)$. So we know that for every $d\in {\Bbb N}$, one has
$$p\_0(d)\le 2^{2^{2^{2^{2^{d^{11^{4d}}}}}}}.$$
Good! In addition, one knows that $A(d)$ is empty for $d=1$ (no prizes), $d=2$ (classical) and $d=3$ (Demyanov and Lewis, independ... | 15 | https://mathoverflow.net/users/19588 | 82088 | 49,154 |
https://mathoverflow.net/questions/82001 | 23 | Imagine that you are in the following situation: You write up a proof which eventually gets published. There you need a result which is not so well-known but it is contained in another paper P; therefore you just cite it. You read P and come to the conclusion: It's awful. You need plenty of time to insert the details o... | https://mathoverflow.net/users/2841 | Citation of a paper with a proof you would like to improve | Improving existing proofs is an important and undervalued part of mathematics. We don't just want to know *whether* something is true; we want to know *why* it's true. So I think that if you have a better proof of something, you should find a way to share it with the world.
Here are a couple of thoughts about the pra... | 25 | https://mathoverflow.net/users/586 | 82093 | 49,156 |
https://mathoverflow.net/questions/82097 | 3 | Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum\_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum\_{k|n}^{n}1.
$$
One can observe the following pattern in the values of $D(n)$,
$$
\lbrace{D(n)\rbrace}=\lbrace \overbrace{1,3,5,\;}^{3,odd}\overbrace{8,10,14,16,20,}^{5,even}\overbrace{2... | https://mathoverflow.net/users/6842 | Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function | If $B(s)$ is any Dirichlet series which factors as an Euler product
$\prod\_{p\in\mathbb{P}}B\_p(s)$ satisfying
$$ B\_p(s)=1-\frac{1}{p^s}+O\left(\frac{1}{p^{2\sigma}}\right),\qquad s=\sigma+it, $$
then $B(s)=A(s)/\zeta(s)$, where $A(s)$ is given by an Euler product which is absolutely and locally uniformly convergent ... | 8 | https://mathoverflow.net/users/11919 | 82100 | 49,161 |
https://mathoverflow.net/questions/82117 | 2 | I have a [finite] lattice enriched with additional operations. I would like either:
1. find a pair of binary operations
(and constants) satisfying semiring
laws, or
2. prove that no such operations exist
As the structure is finite, one can approach problem #1 via computerized search.
For illustration purpose if I... | https://mathoverflow.net/users/11632 | Prove that algebraic structure is not semiring? | I will interpret your question to mean:
>
> Is there a small (closed-form) expression in the existing symbols in my algebraic structure which can be interpreted as $\oplus$ and $\otimes$ for a semiring?
>
>
>
Of course, this also assumes you have candidates for $0$ and $1$. I would use the [finite model verifi... | 4 | https://mathoverflow.net/users/3993 | 82129 | 49,174 |
https://mathoverflow.net/questions/82130 | 3 | Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the local cohomology module $\mathrm{H}^i\_{\mathfrak{m}}(R)$ is of finite length. If we write $\ell(\cdot)$ for length of a mod... | https://mathoverflow.net/users/16046 | Frobenius functor and length of local cohomology | Let $p=2$, $R=k[[x,y]]/(x^2,xy)$. Then $H^0\_m(R) \cong R/m= k$. $F(k) = R/m^{[2]}= k[[x,y]]/(x^2,xy,y^2)$ has length $3 \neq 2\times 1$.
In general for a finite length module $M$, the condition that $\ell(F(M))=p^d\ell(M)$ is pretty restrictive. For $M=k$ this forces $R$ to be regular (Kunz). Also, when $R$ is a com... | 6 | https://mathoverflow.net/users/2083 | 82132 | 49,175 |
https://mathoverflow.net/questions/82090 | 10 | Let $I$ be a filtered poset, which you should think of as being huge. Let $A\_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume that all structure maps $A\_i \rightarrow A\_j$ are inclusions. Given a nuclear $C^{\star}$-algebra $N$, is it then true... | https://mathoverflow.net/users/18256 | When do tensor products of C*-algebras commute with colimits? | If $N$ is exact and the tensor products are minimal then $A\otimes N$ is the colimit of the $A\_i\otimes N$'s. Say the connecting maps are $\phi\_{i,j}$. Then to check that $A\otimes N$
is the colimit of $\{A\_i\otimes N,\phi\_{i,j}\otimes\mathrm{id}\_N\}$ two properties must be verified:
(1) the union of the ranges ... | 7 | https://mathoverflow.net/users/13381 | 82135 | 49,178 |
https://mathoverflow.net/questions/81776 | 12 | Background: I found [this](http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf) interesting translation of Godel's *On formally undecidable propositions of Principia Mathematica and related systems I* that, along with translating it into English, uses more modern and understandable symbols, includes s... | https://mathoverflow.net/users/6342 | Up-to-date version of Principia Mathematica? | In his Stanford Encyclopedia of Philosophy article "[The Notation of Principia Mathematica](http://plato.stanford.edu/entries/pm-notation/)", Bernard Linsky makes the following claim:
>
> This translation is offered as an aid to learning the original notation, which itself is a subject of scholarly dispute, and emb... | 13 | https://mathoverflow.net/users/5442 | 82137 | 49,179 |
https://mathoverflow.net/questions/82136 | 10 | What is the least [recursive ordinal](http://en.wikipedia.org/wiki/Recursive_ordinal) $\alpha$ such that there is no algorithm in [complexity class $\mathsf{P}$](http://en.wikipedia.org/wiki/P_(complexity)) which implements a well-ordering of $\mathbb{N}$ with [order type](http://en.wikipedia.org/wiki/Order_type) $\alp... | https://mathoverflow.net/users/9550 | Ordinals and complexity classes | There is no such recursive ordinal, because in fact every computable ordinal is the order type of a polynomial time computable relation on $\mathbb{N}$. In other words, the least ordinal not describable by a polynomial time relation on $\mathbb{N}$ is $\omega\_1^{ck}$, the same as the least ordinal not describable by a... | 17 | https://mathoverflow.net/users/1946 | 82144 | 49,183 |
https://mathoverflow.net/questions/82148 | 5 | Let $\Gamma$ be a nonelementary hyperbolic group. Then is it true that $\Gamma^{p}:=<\gamma^{p}| \gamma \in \Gamma>$ is a finite index subgroup of $\Gamma$? Here $p$ is a prime number. What is known about this problem? What can we say when $p$ is just an odd number(not necessarily prime) or just a postive integer?
| https://mathoverflow.net/users/9552 | Burnside problem for hyperbolic groups? | A. Yu. Olshanskii in the paper "Periodic quotient groups of hyperbolic groups." ((Russian) Mat. Sb. 182 (1991), no. 4, 543--567; translation in Math. USSR-Sb. 72 (1992), no. 2, 519–541) proved that for every torsion-free non-elementary hyperbolic group $G$ there is a number $N \in \mathbb{N}$ such that for any odd $n \... | 14 | https://mathoverflow.net/users/7644 | 82156 | 49,187 |
https://mathoverflow.net/questions/82159 | 1 | Is there a known algorithm which runs in close to linear time, which receives a k-partite directed graph and outputs all pairs (s,t) where s is a vertex in the 'first' part of the graph and t is a vertex in the 'k-th' part?
EDIT: I wasn't specific enough. The assumption is that there are only edges between G\_i and G... | https://mathoverflow.net/users/19601 | Reachability in a k-partite graph | Firstly, the number of such pairs might be worse than linear in the number of edges. Consider
three equal parts with the first two parts having vertices of out-degree $n^{1/2}$. Then the number of edges is $O(n^{3/2})$ but the number of connected $(s,t)$ pairs can be $\Theta(n^2)$.
Secondly, the connection of each pa... | 1 | https://mathoverflow.net/users/9025 | 82165 | 49,192 |
https://mathoverflow.net/questions/82039 | 2 | Assume that $2^{\omega\_1}=2^\omega=\mathfrak{c}$. Let $D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$={$\xi<\mathfrak{c}:y(\xi)=1$}, the *support* of $y$, and let $X$={$x\in Y:0<|\operatorname{supp}(x)|\le\omega\_1$}; $|X|=\mathfrak{c}^{\omega\_1}=(2^{\omega\_1})^{\omega\_1}=\mathf... | https://mathoverflow.net/users/18465 | Does X have any diagonal properties? | For a space $X$ to have a $G\_\delta$ diagonal, it needs (although this is not enough) to have countable pseudo-character (i.e. every point must be a $G\_\delta$). This is clearly not the case for your space.
| 4 | https://mathoverflow.net/users/17836 | 82171 | 49,198 |
https://mathoverflow.net/questions/82177 | 32 |
>
> Is there a profinite group $G$ which is not its own profinite completion?
>
>
>
Surely not, I thought. But upon looking into it, I found that there is a special name given to a $G$ which *is* its own profinite completion, namely "strongly complete". And a recent (2003) hard theorem (which according to Wikipe... | https://mathoverflow.net/users/13741 | A profinite group which is not its own profinite completion? | Example taken from Ribes and Zalesskii's book "Profinite groups". Take an infinite set $I$ and a finite group $T$. You can let $G$ be the profinite group $\prod\_I T$. Denote its elements by $(g\_i)\_{i\in I}$. Let $\mathcal F$ be an ultrafilter which contains the filter of all cofinite subsets of $I$. If you denote $H... | 28 | https://mathoverflow.net/users/2384 | 82180 | 49,203 |
https://mathoverflow.net/questions/82181 | 2 | The proof assistants Coq and Isabelle give conflicting formal proofs about $a \mod 0 \qquad \forall a \in \mathbb{Z}$.
According to Coq
$$ a \mod 0 = 0$$
and Isabelle proves
$$ a \mod = a$$
`mod` is the function, not a congruence.
>
> Which way is it?
>
>
>
All the computer algebra systems I tried give an ... | https://mathoverflow.net/users/12481 | How to interpret conflicting formal proofs about "a mod 0 = ? " | If $a$ mod 0 is to be defined at all (and I'm not entirely convinced that it should be), then it ought to differ from $a$ by a multiple of 0, which means to me that it ought to be $a$. But it's asserted in the question that the computer systems have a strange notion of division by 0, so they might think that everything... | 8 | https://mathoverflow.net/users/6794 | 82182 | 49,204 |
https://mathoverflow.net/questions/81243 | 1 | Let $(X, Vertex(X))$ be a Polyhedral surface (defined like in Polthier) , $x\_0 \in X$ a vertex. Let $B\_\epsilon(x\_0)$ the euclidean ball centred at $x\_0$ with radius $\epsilon$, $\epsilon > max length(e), e \in edge(X) $. Define $\mathcal{B}\_ \epsilon(x\_0)$ the intersection of $B\_\epsilon(x\_0)$ with $X$ to be t... | https://mathoverflow.net/users/19354 | Euclidean neighborhoods on Polyhedral surface | My hunch is that it is difficult to exploit the structure of the 1-skeleton of your
polyhedral surface $\partial P$ to gain efficiency,
especially in view of your $n$ only being on the order of $10^3$. I suspect efficiencies
might only kick in for much larger $n$.
If you nevertheless want to explore options, I recom... | 0 | https://mathoverflow.net/users/6094 | 82187 | 49,207 |
https://mathoverflow.net/questions/81946 | 1 | The group of automorphisms of S(5,8,24), M\_{24}, is 5-transitive.
Other than Symmetric groups are there any other 5-transitive groups?
If not, would it be correct to say S(5,8,24) is the most symmetric object (not counting trivially obvious objects like the graph K\_{n}) in existence?
| https://mathoverflow.net/users/18648 | The Symmetry of Steiner System S(5,8,24) | If you're only interested in finite permutation groups, then Koen S has given you the answer you needed. If you allow infinite objects, then there are much more symmetric objects than S(5,8,24).
In fact, there is a notion of "highly transitive permutation groups": these are permutation groups (acting on an infinite s... | 4 | https://mathoverflow.net/users/12858 | 82199 | 49,210 |
https://mathoverflow.net/questions/82191 | 3 | Suppose $(S,d)$ is a Polish space, and $X$, $(X\_n)$ are random variables such that $X\_n \to X$ in probability in $(S,d)$. Now suppose $d'$ is another metric on $S$, giving the same topology. Does $X\_n \to X$ in probability in $(S,d')$?
I believe the answer is yes, and the following is a sketch proof. Since $S$ is ... | https://mathoverflow.net/users/3676 | Convergence in probability only depends on topology? | A sequence converges in probability to, say, $f$ iff every subsequence has a further subsequence that converges almost surely to $f$. This second condition is independent of the metric that gives the topology, hence so is the first.
| 11 | https://mathoverflow.net/users/2554 | 82204 | 49,212 |
https://mathoverflow.net/questions/82108 | 5 | I'm writing up some notes on the nLab about things like embedding spaces and infinite spheres and similar things (can't link to them yet as I haven't put them up yet). One aspect that crops up time and time again is the contractibility of some big space, such as an infinite sphere, and this almost always boils down to ... | https://mathoverflow.net/users/45 | Is there a standard notation for a "shift space" in functional analysis? | I don't know of any standard notation for such things, Andrew. $V\oplus V$ is often called the square of $V$, so for the second one I would just say that $V$ is isomorphic to its square.
Being shiftable looks much stronger for separable Banach spaces than being isomorphic to hyperplanes. Is it really stronger? Is $\... | 2 | https://mathoverflow.net/users/2554 | 82205 | 49,213 |
https://mathoverflow.net/questions/82202 | 3 | How many non-isomorphic classes of regular graphs on $(2n+1)^{m}$ vertices with $m(2n+1)^{m}$ edges with vertex degree $2m$, where $n,m \in \mathbb{N}$ are there? Is there a classification known? Can there can be more than one such class (that is are they all isomorphic)?
Is there an example of such non-isomorphic gr... | https://mathoverflow.net/users/16007 | Isomorphic regular graphs | The asymptotic number of $m$-regular graphs on $N$ vertices is well understood and can be found, for example, in Bollobas' Random Graphs (the argument uses Bollobas' "configuration model"). With probability $1$ a graph has no automorphisms, so this is also the number of isomorphism classes as long as $N$ is large. In y... | 7 | https://mathoverflow.net/users/11142 | 82210 | 49,215 |
https://mathoverflow.net/questions/82216 | 1 | I'm looking for a textbook reference of the following elementary fact (a reference for an excercise in a textbook is also welcome):
Let $R$ be a commutative ring and let $\mathfrak{p}$ be a prime ideal of $R$. An ideal $P \trianglelefteq R[X]$ with $P \cap R = \mathfrak{p}$ is prime iff $P = \mathfrak{p}R[X]$ or if ... | https://mathoverflow.net/users/10194 | Prime ideals in univariate polynomial rings | I am pretty sure this is an exercise in Atiyah-McDonald.
**EDIT** I just looked, and Exercises 2-5 in the first chapter have very similar statements, but not exactly the statement you are looking for...
| 1 | https://mathoverflow.net/users/11142 | 82217 | 49,218 |
https://mathoverflow.net/questions/82213 | 2 | Hi there.
Assume $(M,g)$ is a Riemanian manifold and $E\to M$ is a
vector bundle with a bundle metric $\langle\cdot,\cdot\rangle$. We then have the pre-Hilbert space $H\_0:=\Gamma\_c^\infty(E)$ of compactly supported smooth sections with $(s\_1,s\_2):=\int\_M\langle s\_1,s\_2\rangle dV\_g$ . In a paper i'm currently ... | https://mathoverflow.net/users/19617 | Completing The Space Sections in a Vectorbundle | This is true in general. I don't know a reference for the statement, but it is pretty simple just to work it out. The point is that $L^2(M,E)$ is a Hilbert space which contains $H\_0$ as a dense linear subspace, so it must be the completion.
| 2 | https://mathoverflow.net/users/703 | 82218 | 49,219 |
https://mathoverflow.net/questions/82113 | 4 | Can we get a closed form for the following contour integral?. Let us assume that n is a non-negative integer,
$\frac{1}{2\pi i}\int^{c+i\infty}\_{c-i\infty}\frac{\Gamma(n-s)\Gamma(s)\Gamma(k-s)}{\Gamma(1+n-s)}{}\_1F\_1(1+n-s,n+1,\frac{\alpha}{2b})\, b^s\,\mathrm{d}s$
and also how to choose the value of c in order s... | https://mathoverflow.net/users/19493 | Integral with confluent hypergeometric function | Remmy,
**Here's some additional information:**
I've recently encountered a very similar integral in my own work, with a ratio of gamma functions and a hypergeometric function and a Bessel function. The orders of both the hypergeometric function and the Bessel function depended on $s$. I was still able to produce a ... | 3 | https://mathoverflow.net/users/8955 | 82237 | 49,228 |
https://mathoverflow.net/questions/82236 | 7 | What is the largest ~~cardinal number~~ integer $\kappa$ such that every $\kappa$-coloring of $\mathbb{R}^2$ contains a triangle with area 1 and all vertices of the same color?
| https://mathoverflow.net/users/9550 | $\kappa$-coloring of $\mathbb{R}^2$ and triangle with area 1 | R. Graham in "On Partitions of $\mathbb E^n$" , J. Combinatorial Theory Ser. A, 28 (1980), 89–97, proves that for any finite coloring of $\mathbb R^n$ and any positive $a\in \mathbb R$ there is a monochromatic simplex of volume $a$.
The question is originally due to Gurevich from the 70's. Graham's proof has been si... | 10 | https://mathoverflow.net/users/2384 | 82242 | 49,231 |
https://mathoverflow.net/questions/82243 | 1 | How to calculate the infinite sum of the following series, related to binomial expansion for rational number, $r$:
$$1-\frac{r}{1!}\cdot\frac{1}{3}+\frac{r(r-1)}{2!}\cdot\frac{1}{5}-\frac{r(r-1)(r-2)}{3!}\cdot\frac{1}{7}+\ \dots$$.
I know the limit:
$$1-1/3+1/5-1/7+\ \dots = \pi/4$$
and I can calculate:
$$... | https://mathoverflow.net/users/19624 | Limit of an infinite series, related to (generalised Newton's) binomial expansion | You need
$$\sum\_{k\geq 0} (-1)^{k}\binom{r}{k}\frac{1}{2k+1}=\int\_0^1 \sum\_{k\geq 0}(-1)^{k}\binom{r}{k} x^{2k}dx$$
$$=\int\_0^1 (1-x^2)^r dx$$
and this is a beta integral
$$\int\_0^1 (1-x^2)^r dx=\frac{1}{2}\int\_0^1 x^{-1/2}(1-x)^r dx=\frac{1}{2}B(\frac{1}{2},r+1)=\frac{\sqrt{\pi}\Gamma(r+1)}{2\Gamma(r+\frac{3}{2... | 6 | https://mathoverflow.net/users/2384 | 82245 | 49,232 |
https://mathoverflow.net/questions/82214 | 8 | Dear MO\_World,
I have (another) question about relaxing the assumptions in the sub-additive ergodic theorem. Apologies if this is something I should know already...
There are a number of statements of the Kingman sub-additive ergodic theorem and its extensions. Here is a fairly typical version:
Let $T\colon X\to... | https://mathoverflow.net/users/11054 | non-integrable subadditive ergodic theorem | I think I have a counterexample, though it turned out to be more intricate than I first expected.
The space $X$ is the 2-adics ${\bf Z}\_2$ with the usual probability Haar measure and the shift $Tx := x+1$. To describe the functions $f\_n$, we will put a weighted directed graph on $X$, that is to say a number of edge... | 8 | https://mathoverflow.net/users/766 | 82249 | 49,233 |
https://mathoverflow.net/questions/7678 | 7 | Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}\_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf{T}$ is finitely generated over a regular ring $\Lambda$ of dimension 3. Let $\mathfrak{m}$ be a maximal non-Eisentein ... | https://mathoverflow.net/users/2284 | Free subquotient of Galois representations coming from Hida theory | $\newcommand\T{\mathbf{T}\_{\mathfrak{m}}}$
$\newcommand\Q{\mathbf{Q}}$
$\newcommand\m{\mathfrak{m}}$
$\newcommand\F{\mathbf{F}}$
$\newcommand\Frob{\mathrm{Frob}}$
$\newcommand\rhobar{\overline{\rho}}$
$\newcommand\eps{\epsilon}$
First, as Professor Emerton mentions, the construction of $L^{+}$ you gave
is not necess... | 8 | https://mathoverflow.net/users/nan | 82252 | 49,234 |
https://mathoverflow.net/questions/82225 | 4 | Let $C$ be a smooth projective curve and let $C^{(n)}$ be its $n$th symmetric power.
Let $E$ be a $S\_n$-equivariant vector bundle over the Cartesian power $C^n$. Suppose that $E$ descends to a vector bundle $\tilde{E}$ over the symmetric power $C^{(n)}$.
Then there are two things I can do:
1. I can compute the $... | https://mathoverflow.net/users/83 | Index vs. equivariant index (and then taking invariant part)? | If I interpret your question correctly, the answer should be positive. In stack-theoretic terms, you have a bundle on the quotient stack $[C^n/S\_n]$. This gives an element of the equivariant K-theory $K\_{S\_n}(C^n)$. In case 1, you push forward from $K([C^n/S\_n]) = K\_{S\_n}(C^n)$ to the representation ring $K([\mat... | 2 | https://mathoverflow.net/users/4790 | 82254 | 49,235 |
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