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https://mathoverflow.net/questions/146622 | 9 | There's a natural map $f:\overline{\mathcal{M}}\_{1,1}\to \overline{M}\_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence $f^\*:\mathbb{Z}\to\mathbb{Z}$ is an homomorphism. What homomorphism? My guess is: $x \mapsto 24 x$ since the generator of the s... | https://mathoverflow.net/users/4096 | $Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space | I believe the number is 12.
I will assume the characteristic of the base field is not 2 or 3 so that I can use $\overline{\mathcal{M}}\_{1,1} \simeq \mathbf{P}(4,6)$. Recall that $\mathbf{P}(4,6)$ is constructed by dividing $V = \mathbf{A}^2 \smallsetminus \{ (0,0) \}$ by the weight $(4,6)$-action of $\mathbf{G}\_m$.... | 7 | https://mathoverflow.net/users/32 | 147188 | 79,487 |
https://mathoverflow.net/questions/147172 | 8 | Since "cubes" with higher dimension than three exist I think it's natural to ask for higher dimensional Rubik's cubes. These so called hypercubes don't seem to have been described from a group theoretic point of view.
Are there any papers on this? Is the group of the $3\times 3\times 3 \times 3$ cube a subgroup of a ... | https://mathoverflow.net/users/42261 | Higher dimensional Rubik's cube group | The 4-dimensional, i.e. $3 \times 3 \times 3 \times 3$, equivalent of the
Rubik's cube has 8 three-dimensional sides, each of which consists of
$3^3 = 27$ three-dimensional colored "squares".
Of these 27 "squares", the one in the center is fixed. Thus in total our
4-dimensional Rubik's cube has $8 \cdot 26 = 208$ movab... | 4 | https://mathoverflow.net/users/28104 | 147214 | 79,496 |
https://mathoverflow.net/questions/146946 | 9 | A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces.
Are homotopy fully faithful functors preserved under (homotopy) pushout?
More precisely, if $C\to D$ is fully faithful, and $C\to E$ is an arbitrary functor, is the canonical map $E\to E\sqcup\_C... | https://mathoverflow.net/users/9581 | Do Homotopy Fully Faithful Functors Push-out? | The answer is yes, fully-faithful functors are stable under co-base change.
This is a model independent statement and so we can in particular take $\infty$-category to mean Segal categories. Then this follows directly from Cor. 16.6.2 in the arXiv version of Carlos Simpson's book "[Homotopy theory of higher categori... | 11 | https://mathoverflow.net/users/184 | 147220 | 79,497 |
https://mathoverflow.net/questions/147217 | 12 | I am currently interested in the following sequence:
$$\begin{cases}u\_0 & = & \alpha\\u\_n & = & u\_{n-1}^2-n\end{cases}$$ where $\alpha > C \approx 1.75793275... $ with $C$ being the [Nested Radical Constant](http://mathworld.wolfram.com/NestedRadicalConstant.html).
I'm interested to get the behaviour of this seque... | https://mathoverflow.net/users/42473 | Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$ | Let us just consider the case $\alpha=2$ where there is an elegant answer: There exists a
constant $\lambda$ such that for all $n$ we have $u\_n = \lceil \lambda^{2^n}\rceil$. First note that by induction it is easy to see that $u\_n \ge (n+2)$ for all $n\ge 0$. Define
$$
\lambda= 2 \prod\_{n=1}^{\infty} \Big(1-\fra... | 14 | https://mathoverflow.net/users/38624 | 147234 | 79,500 |
https://mathoverflow.net/questions/147228 | 2 | Let $a\_n$ be a linear recurrence with integer constant coefficients
and initial values.
Is it possible $a\_n$ to satisfy all of these:
1. $a\_n = 0$ infinitely often.
2. if $a\_n \ne 0$, $ | a\_n |$ is of exponential growth
(to avoid cases like $\dots 0 , n , 0 ,n+1, \dots$).
3. $ | a\_n |$ is conjectured to be pr... | https://mathoverflow.net/users/12481 | Is there a linear recurrence with infinitely many zeros, conjecturally infinitely many primes and non-zero terms of exponential growth? | Yes. $a\_n=a\_{n-1}+a\_{n-2}-a\_{n-3}+a\_{n-4}-a\_{n-5}$ has a solution containing zeros and Fibonacci numbers: $0,1,0,1,0,2,0,3,0,5,0,8\ldots$.
| 4 | https://mathoverflow.net/users/nan | 147237 | 79,503 |
https://mathoverflow.net/questions/147232 | 1 | Does anybody know of a text (doesn't matter which form - article, book etc. - doesn't matter which language) which makes the transition from analysis in $\mathbb{R}^n$ to analysis in Banach or Hilbert spaces ? (Either space is fine, but both would be a plus.)
I'm thinking of a text that goes through the basic key co... | https://mathoverflow.net/users/14101 | From Calculus in $\mathbb{R}^n$ to Calculus in $\mathcal{H}$ | Banach space is the standard setting for the French Analysis courses of the second half of XX century, for example
H. Cartan, Calcul differentiel. Formes differentielles, Herman, Paris, 1967.
It does not have Lebesgue integral.
The course of Schwartz has everything, including a careful discussion of
differences between... | 3 | https://mathoverflow.net/users/25510 | 147248 | 79,510 |
https://mathoverflow.net/questions/146097 | 2 | This question is migrated from [math.stackexchange](https://math.stackexchange.com/questions/538261/choosing-the-order-of-tikhonov-regularization-of-an-inverse-problem).
Let me first describe the problem I am trying to solve and then the question I have. I greatly appreciate anyone who can shine some light on it.
... | https://mathoverflow.net/users/41802 | Choosing the order of Tikhonov regularization of an inverse problem | To start with: What you call Tikhonov regularization, is usually called Lavrentiev regularization (in the case of self-adjoint, non-negative definite $M$). The idea there is to shift the spectrum of $M$ away from zero. In the case of non-self-adjoint $M$, Tikhonov regularization is
$$
b\_\lambda = (M^\*M + \lambda I)^{... | 5 | https://mathoverflow.net/users/9652 | 147250 | 79,511 |
https://mathoverflow.net/questions/147117 | 2 | Let $E/K$ be an elliptic curve over a number field $K$ and let $E[n]$ denote the full $n$-torsion of $E$, for a positive integer $n$. With $End\_K(E)$ we denote the endomorphisms of $E/K$ which are defined over $K$ and similarly we define $End\_K(E[n])$ to be the endomorphisms of $E[n]$ which are defined over $K$. Ther... | https://mathoverflow.net/users/12668 | When are K-automorphisms of the n-torsion of an elliptic curve E/K liftable to K-endomorphisms of E? | There exists a positive integer $n\_0$ (that depends only on $E$ and $K$) such that the map $\phi$ is surjective if $n$ and $n\_0$ are relatively prime. The same is true not only for elliptic curves but for arbitrary abelian varieties over a finitely generated field. See Inv. Math. 79 (1985), 309-322; arXiv 1301.5594 .... | 2 | https://mathoverflow.net/users/9658 | 147256 | 79,516 |
https://mathoverflow.net/questions/147259 | 5 | The question is in the title, I haven't been able to locate a discussion of these kind of properties.
| https://mathoverflow.net/users/8887 | Can an open manifold with positive Ricci curvature be non simply connected at infinity? | For interiors of compact manifolds [simply connected at infinity](http://en.wikipedia.org/wiki/Simply_connected_at_infinity) is equivalent to assuming that each boundary component is simply-connected. There are many such manifolds of positive Ricci curvature, e.g. the product of a circle and a high dimensional Euclidea... | 9 | https://mathoverflow.net/users/1573 | 147262 | 79,518 |
https://mathoverflow.net/questions/147261 | 4 | Let, $C$ be a projective curve (not necessarily reduced), $i:C \to \mathbb{P}^n$ be a closed immersion. Does there exist a bound on/geometric interpretation of the dimension of $H^0(i^\*(\mathcal{O}\_{\mathbb{P}^n}(1))$?
| https://mathoverflow.net/users/38832 | Dimension of the global sections of the Serre twisting sheaf on a curve | Thinking about $i^\ast \mathcal O\_{\mathbb{P}^n}(1)$ as $\mathcal O\_C(1)$, there is a natural exact sequence
$$0 \to I\_C(1) \to \mathcal O\_{\mathbb{P}^n}(1) \to \mathcal O\_C(1) \to 0.$$
Assuming $C$ is non-degenerate, we obtain an exact sequence in cohomology
$$0\to H^0(\mathcal O\_{\mathbb{P}^n}(1))\to H^0(... | 5 | https://mathoverflow.net/users/7399 | 147268 | 79,519 |
https://mathoverflow.net/questions/147263 | 2 | I'm looking to learn about integral structures (or models?) on classical algebraic groups.
To begin with I have been learning about algebraic groups, quadratic forms and lattices. And also looking at this paper by Jonathan Hanke called "Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over... | https://mathoverflow.net/users/15566 | What should I read if I want to learn about integral structures on classical algebraic groups? | not sure what you mean by good. The standard books on quadratic forms over number fields are in Hanke's references. I would add that Hanke studied under Shimura, so you should take a look at [shimura\_2010](http://www.springer.com/mathematics/algebra/book/978-1-4419-1731-7) and [shimura\_2012](http://www.springer.com/m... | 4 | https://mathoverflow.net/users/3324 | 147269 | 79,520 |
https://mathoverflow.net/questions/143867 | 0 | Fix $n$ and let
$0\leftarrow \mathcal{F}\leftarrow \bigoplus \mathcal{O}\_{\mathbb{P}^n}(a\_i)\leftarrow \bigoplus \mathcal{O}\_{\mathbb{P}^n}(b\_i)\leftarrow \cdots$
be an exact sequence.
Then we can say that $\mathbb{P}(\mathcal{F})\hookrightarrow \mathbb{P}(\bigoplus \mathcal{O}\_{\mathbb{P}^n}(a\_i))$ but what m... | https://mathoverflow.net/users/13803 | Projective bundles | According to the long exact sequence, the sheaf $\mathcal{F}$ is coherent and it is over ${P}^n$, so there is a graded module $M$ such that the coherent sheaf $\tilde{M}$ is just the $\mathcal{F}$, the one may construct $\mathbb{P}(\mathcal{F})$ by using this $M$. The way to construct this $M$ can be found in GTM52 ALG... | 2 | https://mathoverflow.net/users/42491 | 147273 | 79,522 |
https://mathoverflow.net/questions/147278 | 4 | It has been proved by Radziszewski in this paper
K. Radziszewski. Sur une probleme extremal relatif aux gures inscrites et circonscrites aux gures convexes. Ann. Univ. Mariae Curie-Sklodowska, Sect. A6, pages 5-18, 1952.
that the area of the largest inscribed rectangle (LIR) inside a convex polygon (C) is at least ... | https://mathoverflow.net/users/42499 | Largest inscribed rectangle inside a convex polygon | The result also appeared in a shorter paper, Constantin Radziszewski, Sur un problème extrémal relatif aux figures inscrites dans les figures convexes, C. R. Acad. Sci. Paris 235 (1952) 771–773, MR0054268 (14,896f). The review by W Gustin summarizes the proof.
There is also a proof in Wilhelm Süss, Ueber Parallelogr... | 2 | https://mathoverflow.net/users/3684 | 147285 | 79,524 |
https://mathoverflow.net/questions/147187 | 4 | If $A$ is a Noetherian ring, $M$ is a finitely generated module,
$I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$,
then we know that $\hat{A}\otimes\_{A}M\cong\hat{M}$.
Also in Atiyah&Macdonald, there is a remark on Page 109 that the functor $M \mapsto \hat{M}$ is not exact without assuming $M$... | https://mathoverflow.net/users/42455 | On exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$ | I know that this was answered at the comments, but I want to emphasize that actually, the completion functor is not exact from either side, not from the left and not from the right. The only thing that is true (over noetherian or non-noetherian rings) is that completion preserves surjection:
Let $k$ be a field, $A=k[... | 9 | https://mathoverflow.net/users/3759 | 147295 | 79,527 |
https://mathoverflow.net/questions/147282 | 2 | Assume that $D\_n\subset D$, where $D$ is the unit disk is an increasing sequence of Jordan domains with smooth boundaries such that $\cup\_{n=1}^\infty D\_n=D$ and let $f\_n: D \to D\_n$ be conformal mapping such that $f(0)=0$ and $\arg(f\_n'(0))=0$. Is $f\_n$ uniformly convergent to $Id$.
| https://mathoverflow.net/users/36162 | Uniform convergence of conformal mappings | The answer is no. Consider the following sequence $D\_n$. Take the disc $|z|<1-1/n=r\_n$,
and remove from it the arc $\{ z:|z|=r\_{n-1},|\arg z|<\pi-1/n\}$ and the interval
$(r\_{n-1},r\_n)$. The result is a simply connected region $D\_n$.
This $D\_n$ contains $|z|<r\_{n-1}$, so the union of $D\_n$ is the whole unit di... | 5 | https://mathoverflow.net/users/25510 | 147301 | 79,529 |
https://mathoverflow.net/questions/147291 | 2 | How many elements of $LU$ decomposition of a symmetric matrix change after adding a sparse symmetric matrix? Is it more efficient to recompute $LU$ decomposition after adding a sparse matrix comparing to computing it from scratch for the sum?
More formally, let $M$ be a symmetric matrix, $L\_M\cdot U\_M$ its pivoted ... | https://mathoverflow.net/users/38448 | Updating $LU$ decomposition after adding a sparse matrix | As far as I know, there is theory available only for updating decompositions under *low-rank* perturbations: see [Golub, Van Loan, *Matrix Computations* 3rd ed., ch 12] for details on updating QR decompositions, and references to other types of updates. QRs are easier to update because there is no pivoting involved, bu... | 2 | https://mathoverflow.net/users/1898 | 147317 | 79,535 |
https://mathoverflow.net/questions/147321 | 2 | Given a finite collection of embedded $C^\infty$ curves which pass through the origin in $\mathbb{R}^2$ with different tangent directions and never again intersect, is there a clean way of prescribing a Riemmannian metric whose geodesics include those curves?
| https://mathoverflow.net/users/36931 | Prescribing finitely many unparameterised planar geodesics | Sorry, I misread your question and answered a local version instead. I'll leave it here (under the horizonal line) in case it turns out to be useful.
If your finite set of curves is rectifiable (can be made into lines by a diffeomorphism of the plane), then do that and then take the Euclidean metric. So the question:... | 3 | https://mathoverflow.net/users/21123 | 147324 | 79,537 |
https://mathoverflow.net/questions/147327 | 6 | Q1. What is the consistency strength of the failure of square on singular cardinals?
Q2. What are known as partial results in this direction?
| https://mathoverflow.net/users/nan | Consistency Strength of the Failure of Square on Singular Cardinals | One place to start reading is this:
>
> James Cummings and Sy-David Friedman, "**[$\square$ On the Singular Cardinals](http://www.jstor.org/stable/27590333)**". *The Journal of Symbolic Logic* Vol. **73**, No. 4 (Dec., 2008), pp. 1307-1314.
>
>
>
| 5 | https://mathoverflow.net/users/7206 | 147328 | 79,538 |
https://mathoverflow.net/questions/147284 | 6 | Suppose G is a connected semi-simple Lie group with finite center, and A, B are one parameter subgroups of the same Cartan subgroup. If the connected components of the identity of the centralizers of A and B are equal does it follow that the centralizers of A and B are equal ?
| https://mathoverflow.net/users/42506 | Centralizers of one parameter subgroups in semi-simple Lie groups | The answer is affirmative, and it is not necessary to assume that the center of $G$ is finite; discreteness (which is a consequence of semisimplicity of the Lie algebra) is sufficient. The crux of the matter is to carefully turn the analytic problem into an algebraic one (keeping track of connectedness issues on the an... | 6 | https://mathoverflow.net/users/39487 | 147337 | 79,542 |
https://mathoverflow.net/questions/147338 | 2 | The [nlab page](http://ncatlab.org/nlab/show/operad#definition_as_monoid_23) says
>
> A (Set-based) operad is a monoid in the monoidal category
> $(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup\_{n\ge 0} S\_n$.
>
>
>
The monoidal structure is given by the so called complicated **substitution product** $... | https://mathoverflow.net/users/7341 | A question on the definition of operad | The answer is no, because any Day convolution product $F \ast G$ on presheaves $F, G: \mathbb{P}^{op} \to Set$ is cocontinuous in each of the separate arguments $F, G$, and yet the substitution product $F \circ G$ is not separately cocontinuous (only $- \circ G$ is cocontinuous, not $F \circ -$).
---
In case it ... | 6 | https://mathoverflow.net/users/2926 | 147340 | 79,543 |
https://mathoverflow.net/questions/147031 | 11 | I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might be able to iron out for me.
If I take a field $k$ of characteristic $\neq\ell$, then if I have understood this
<http... | https://mathoverflow.net/users/13647 | Pro-algebraic versus continuous Galois cohomology, and schematic homotopy types | This answer is due to Jon Pridham.
While we might not expect $H^i(G\_k,V)=H^i(G\_k^\mathrm{alg},V)$ for every finite dimensional, continuous $G\_k$-representation $V$, there are certain results from the motivic theory that suggest that this might be true if $k$ is a number field (or local field of char $0$), and that... | 2 | https://mathoverflow.net/users/13647 | 147351 | 79,547 |
https://mathoverflow.net/questions/147329 | 9 | While reading a paper [*An Arithmetic Proof of John’s Ellipsoid Theorem*](http://dmg.tuwien.ac.at/schuster/john1.pdf) by Gruber and Schuster, I have a question on their proof.
---
Consider an $n\times n$ real symmetric and [positive definite](http://en.wikipedia.org/wiki/Positive-definite_matrix) matrix $\mathbf ... | https://mathoverflow.net/users/11361 | Set of Positive Definite matrices with determinant > 1 forms a convex set | Here is a textbook level description of the above. I assume you know what a convex set and convex function on this set are. Given that, let us know prove that the determinant is strictly log-concave on hermitian positive definite matrices.
**Claim.** Let $A, B > 0$. Then, $\det\left(\frac{A+B}{2}\right) \ge \sqrt{\de... | 11 | https://mathoverflow.net/users/8430 | 147352 | 79,548 |
https://mathoverflow.net/questions/147270 | 10 | Is there either a closed form (in terms of the moments of $X\_1$, say) or good bounds on
$$
\mathbb{E} \sup\_{k \leq n} \frac{1}{k} \sum\_{i=1}^k X\_i,
$$
where $X\_i$ are iid and arbitrarily nice? (In my specific application, $X\_i$ are given by $(B\_i - p)^2$, where $B\_i$ are iid Bernoulli variables with mean $p$.) ... | https://mathoverflow.net/users/17883 | Expected supremum of average? | You're asking about maximal inequalities. These are known in more generality for measure-preserving transformations. As has already been pointed out, in your special case, you can expect to get a constant bound. The averages very quickly approach the limit, so you're looking at the average of the max of the first few t... | 7 | https://mathoverflow.net/users/11054 | 147360 | 79,553 |
https://mathoverflow.net/questions/147359 | 0 | Consider a torus $T^n$. An differential operator $\mathcal{O}$ acts on differential forms on $T^n$. Let $R$ be a smooth vector field on $T^n$, whose orbits are dense on $T^n$ (for instance, irrational flow on $T^2$).
My question is: if $\mathcal{L}\_R$ and $ \mathcal{O}$ commutes, does it imply $\mathcal{O}$ also com... | https://mathoverflow.net/users/15884 | Does such an operator commutes with the whole torus action? | No. The assumption is coordinate-independent (i.e., preserved by self-diffeomorphisms) but the desired conclusion is not.
Begin with $R$ being the standard irrational flow and $\mathcal O$ a coordinate differentiation. Apply a generic self-diffeomorphism of the torus. The image of the coordinate field no longer commu... | 3 | https://mathoverflow.net/users/4354 | 147362 | 79,555 |
https://mathoverflow.net/questions/147320 | 0 | Assume that $f\_n$ is a sequence of conformal injective mappings of the unit disk $D$ onto the nested smooth Jordan domains $D\_n\subset D$, such that $\cup\_{n=1}^\infty D\_n=D$ and $D\_n$ are images of $n/(n+1) D$ under a fixed diffeo q.c. mapping of the unit disk onto itself. Assume as well that $f\_n\to id$ uniform... | https://mathoverflow.net/users/36162 | Integral and conformal mappings II | The answer is no. All these integrals can be infinite.
Let us fix $a,b$, $0<a<b<1$.
I will first construct a Jordan region $G$
containing $|z|<a$, contained in $|z|<b$, and such that
for $f$ mapping conformally $D$ onto $G$, the integral is infinite.
Consider a smooth (except at the endpoint $b$) simple curve beginni... | 1 | https://mathoverflow.net/users/25510 | 147367 | 79,558 |
https://mathoverflow.net/questions/144998 | 3 | I have a probability distribution over $\{0,1\}^n$ but instead of knowing the full joint distribution $p(x\_1,\dots,x\_n)$, I only know $p(x\_i=x\_j)$ for each $i,j$. How could I draw a random binary vector $x$ from *some* distribution that has these marginals? Since $n$ is large I would rather not search over all dist... | https://mathoverflow.net/users/39485 | Drawing random variates from a partially described probability distribution | Joris,
So we assume we know only the marginals $p(x\_i)$ and the probabilities that $p(x\_i=x\_j)$. In terms of the physics' "spin" notation, $s\_i=\pm 1$, this means that we know $\left<s\_i \right> \equiv \sum\_{s\_i} s\_i p(s\_i)$ and $\left<s\_i s\_j \right> \equiv \sum\_{s\_i s\_j} s\_i s\_j p(s\_i,s\_j)$.
Th... | 4 | https://mathoverflow.net/users/42545 | 147386 | 79,563 |
https://mathoverflow.net/questions/147385 | 4 | Does anyone know an expression (in terms of simpler functions) for the following Epstein Zeta function:
$\sum \frac{1}{(m^2+m n+n^2)^s}$
I know an expression (in terms of the Dirichlet Beta function) exist for $\sum \frac{1}{(m^2+n^2)^s}$. I did look hard into the literature (being not an expert) but couldn't find ... | https://mathoverflow.net/users/41940 | Epstein zeta functions | I had thought the question meant to refer to Dedekind *zeta*, not *Beta*, since that Epstein zeta is ($4\times$) the Dedekind zeta function of the Gaussian integers. The case of the question gives ($6\times$) the Dedekind zeta function of the "Eisenstein integers", namely, $\mathbb Z[\rho]$, where $\rho$ is a cube root... | 8 | https://mathoverflow.net/users/15629 | 147389 | 79,564 |
https://mathoverflow.net/questions/147398 | 4 | The following problem is likely too special for MO.
However I have no clue how to deal with it, so I'll just try. Nevertheless
it is a combinatorial problem and a discussion about general methods
in this context will be interesting and helpful.
How can one determine whether the following equation
$\sum\_{p=2}^n\fr... | https://mathoverflow.net/users/21965 | Combinatorial Technique Needed | One technique which is very helpful in proving identities like this one is to express that expression as a coefficient in a generating function.
Consider the expression
$$\sum\_{p\geq 2} \frac{(-1)^p}{p!}\binom{p}{2}\left(x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots\right)^{p-2}\left(x^2+2x^3+3x^4+\cdots\right)$$
You can che... | 16 | https://mathoverflow.net/users/2384 | 147402 | 79,570 |
https://mathoverflow.net/questions/147379 | 6 | I am confused about the following. I know that for two line bundles $L\_1, L\_2$ on an algebraic curve $C$ the vector space ${\rm Ext}^1(L\_1,L\_2)$ classifies isomorphism classes of rank two vector bundles on $C$ which are extensions of $L\_2$ by $L\_1$. My question is, does this mean that there is a "universal" rank ... | https://mathoverflow.net/users/42066 | ${\rm Ext}^1$ and extensions of line bundles on a curve | I wanted to work this out for myself anyways, so here's a summary of the argument in Le Potier.
Suppose $X$ is a projective variety, and $E,G$ are locally free sheaves on $X$. Put $S = {\rm{Ext}}^1(G,E)$, and let ${\bf E},{\bf G}$ be the constant families on $S\times X$, namely ${\bf E} = p\_2^\ast E$ and ${\bf G} = ... | 9 | https://mathoverflow.net/users/7399 | 147404 | 79,571 |
https://mathoverflow.net/questions/147400 | 8 | Konrad Waldorf shows in [his paper](http://arxiv.org/abs/0911.3212) one may realize a Grothendieck topology on the category of diffeological spaces. Is there any work exploring stacks over the category of diffeologies?
| https://mathoverflow.net/users/42563 | Stacks over diffeologies | I will show that stacks over diffeological spaces are "the same" (in the sense of equivalence of 2-categories) as ordinary stacks on manifolds.
The Grothendieck **pre**-topology in question is the Grothendieck topology of "subductions". A map $f:X \to Y$ between diffeological spaces is a subduction if for every map $... | 9 | https://mathoverflow.net/users/4528 | 147425 | 79,577 |
https://mathoverflow.net/questions/147118 | 4 |
>
> Given a positive integer $n$, is there an algorithm (or even better a closed formula) that provides me with a hyperelliptic curve $C/\mathbb Q$, such that its Jacobian $J:=Jac(C)$ possesses a $\mathbb Q$-rational torsion point $P$ of exact order $n$?
>
>
>
The genus of $C$ may very well vary with different v... | https://mathoverflow.net/users/12668 | Is there a formula for a hyperelliptic curve over QQ, such that its Jacobian contains a rational torsion point of extact order n, for any given n? | One can take
$$
\begin{array}{llllll}
y^2 + (a\_g x^g+...+a\_0) y &=& x^{2g+1} &&& \text{ ($n=2g+1$ odd)} \cr
y^2 + (2cx\_{g+1} +a\_g x^g+...+a\_0) y &=& -c^2x^{2g+2} &&& \text{ ($n=2g+2$ even).} \cr
\end{array}
$$
The divisor of the $y$ function is
$$
\text{div}(y) = n(0,0)-n(\infty),
$$
and so $D=(0,0)-(\infty)$ ... | 7 | https://mathoverflow.net/users/3132 | 147429 | 79,579 |
https://mathoverflow.net/questions/140093 | 8 | I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.
| https://mathoverflow.net/users/34063 | Configuration spaces of the torus | The calculation for even-dimensional manifolds, and in particular the torus, is carried out by Felix-Thomas in their paper "Rational Betti numbers of configuration spaces."
| 7 | https://mathoverflow.net/users/34063 | 147434 | 79,583 |
https://mathoverflow.net/questions/130976 | 14 | Cobordism genera can often be refined to $E\_\infty$-orientations in the sense of Ando-Blumberg-Gepner-Hopkins-Rezk:
1) the mod 2 Euler characteristic $MO\to H\mathbb{F}\_2$;
2) the $\widehat A$-genus $MSpin\to KO$ (Ando-Hopkins-Rezk, Joachim);
3) the Todd genus $MSpin^c\to K$ (Joachim);
4) the Witten genus $MS... | https://mathoverflow.net/users/34063 | Does the signature admit a homotopy coherent refinement? | [Since my comment above appears to have been helpful, I am repeating it here.]
I must admit I am unfamiliar with L-theory. Nevertheless, I came across a recent article on the arXiv which is related: [*Commutativity properties of Quinn spectra*](http://arxiv.org/abs/1304.4759) by Gerd Laures and James McClure. It stat... | 7 | https://mathoverflow.net/users/21095 | 148434 | 79,584 |
https://mathoverflow.net/questions/107467 | 8 | Given $q = e^{2\pi i \tau}$ and the Eisenstein series $E\_{2k}(\tau)$, i.e.,
$$E\_2(\tau) = 1-24\sum\_{n=1}^\infty \frac{n q^n}{1-q^n}$$
$$E\_4(\tau) = 1+240\sum\_{n=1}^\infty \frac{n^3 q^n}{1-q^n}$$
and so on. Define the function,
$$F\_{2k}(\tau) = \frac{E\_{2k}(\tau)}{\left(E\_2(\tau)-\frac{3}{\pi\; \Im(\tau)... | https://mathoverflow.net/users/12905 | Eisenstein series and 163? | I inadvertently came across the partial answer to my own question. It turns out Ramanujan had already explored something similar. Let $q = e^{2\pi i \tau}$, $\tau=\tfrac{1+\sqrt{-n}}{2}$, and,
$$P\_n = 1-24\sum\_{k=1}^\infty \frac{k q^k}{1-q^k}$$
$$Q\_n = 1+240\sum\_{k=1}^\infty \frac{k^3 q^k}{1-q^k}$$
$$R\_n = 1... | 3 | https://mathoverflow.net/users/12905 | 148444 | 79,588 |
https://mathoverflow.net/questions/147381 | 1 | Is there a name for a graph very similar to a hypercube, but generalized from $2^d$ vertices to $k^d$? Alternatively, similar to a 2-dimensional grid, but generalized to higher dimensions?
In 2 dimensions, it would be a square grid with a side length of $k$. In 3 or more dimensions the graph would look like a subdivi... | https://mathoverflow.net/users/42549 | Proper name for hypergrid or subdivided hypercube? | Here is a (incomplete) list of names that people use for the object (with examples):
* "generalized grid graph" ([mathematica](http://mathworld.wolfram.com/GridGraph.html))
* "hypergrid graph" ([matlab](http://www.mathworks.com/matlabcentral/fileexchange/10922-matlabbgl/content/matlab_bgl/grid_graph.m))
* "multi-dime... | 2 | https://mathoverflow.net/users/39495 | 148447 | 79,589 |
https://mathoverflow.net/questions/147427 | 6 | Let $S := \{A\_0, A\_1, \dots, A\_d\}$, where $A\_k \in \mathbb{C}^{n \times n}$, be a set of (generally noncommuting) matrices. I am interested in finding a nonsingular $X \in \mathbb{C}^{n \times n}$ such that the elements of
$$SX = \{A\_kX \colon k=0,1,\dots,d\}$$
commute. In other words, I want a nonsingular $X... | https://mathoverflow.net/users/36450 | For a set of matrices $S$, find $X$ such that the elements of $SX$ commute | This is not a complete solution by any means, but here are some ideas.
If one of $A\_j$ (or their linear combinations) is invertible, then one can get
a necessary and sufficient condition. Namely, if $B\_i=A\_iX$ commute then so do $B\_iB\_j^{-1}=A\_iA\_j^{-1}$. So one can take $A\_iA\_j^{-1}$ and see if it commutes ... | 2 | https://mathoverflow.net/users/38468 | 148474 | 79,597 |
https://mathoverflow.net/questions/148458 | 22 | An important piece of Monstrous moonshine is the j-function,
$$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$
In the paper "*[Umbral Moonshine](http://arxiv.org/pdf/1204.2779v3.pdf)*" (2013), page 5, authors Cheng, Duncan, and Harvey define the function,
$$H^{(2)}(\tau)=2q^{-1/8}(-1 + 45q + \color{... | https://mathoverflow.net/users/12905 | Monstrous moonshine for $M_{24}$ and K3? | I can answer your first question. In [arXiv:1208.4074](https://arxiv.org/abs/1208.4074) by Dabholkar, Murthy and Zagier you can find a formula that implies
$H^{(2)}(\tau)= \frac{48 F\_2^{(2)}(\tau)- 2 E\_2(\tau)}{\eta(\tau)^3}$
where $E\_2(\tau)$ is the quasi modular Eisenstein series and
$F\_2^{(2)}(\tau)= \sum\_{r>s>... | 11 | https://mathoverflow.net/users/10475 | 148477 | 79,598 |
https://mathoverflow.net/questions/146703 | 3 | Can $\Pi^m\_n$ indescribable cardinal be the first one where $\text{GCH}$ fails?
Hauser showed in
Hauser,K.: Indescribable cardinals and elementary embeddings.
J. Symb. Logic 56, 439457 (1991)
that the answer is positive for $m=1$. But for $m \ge 2$, he only violated $\text{GCH}$ at indescribable while violating i... | https://mathoverflow.net/users/42227 | Failure of GCH at indescribable cardinals | The answer is that this is impossible. The GCH cannot fail for the first time at a $\Pi^2\_1$-indescribable cardinal. To see this, take any $\Pi^2\_1$-indescribability embedding $j:M\to N$, meaning that $M$ is a transitive model of ZFC of size $\kappa$ with $M^{\lt\kappa}\subset M$, and $N$ is transitive, with $\text{c... | 2 | https://mathoverflow.net/users/1946 | 148478 | 79,599 |
https://mathoverflow.net/questions/148473 | 1 | $\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$
where matrice $A \in \mathbb{R}\_{n \times n}$ is a positive semi-definite with eigenvectors $x = (x\_{1},x\_{2},....,x\_{n})\in \mathbb{R}\_{n \times n}$ with $x\_i=(x\_i(1),x\_i(2),...,x\_i(n))^{T}$ and eigenvalues $ \lambda = (\... | https://mathoverflow.net/users/41233 | Eigenvalue problem with quadratic constraints | since you say that $A$ is positive semi-definite, you're restricting yourself to real symmetric matrices $A$, so the matrix $x$ of eigenvectors is an $n\times n$ orthogonal matrix, with $\sum\_{i=1}^{n}x\_{i}^{2}(j)=1$ for all $j=1,2,\ldots n$. --- your constraint is therefore satisfied automatically.
| 1 | https://mathoverflow.net/users/11260 | 148485 | 79,603 |
https://mathoverflow.net/questions/147189 | 3 | So, if we have an infinite dimensional Hilbert space $H$ then the way you put a ring structure on $F(H)$ is by taking the isomorphism $H\oplus H \to H$ we can define the sum of two Fredholm operators as
$$ H \to H \oplus H \to H \oplus H \to H$$
where the middle map is the sum of the two operators.
What is the equiv... | https://mathoverflow.net/users/17260 | Ring structure on K-theory modeled on fredholm operators | The formula is $$A \cdot B = \begin{bmatrix}
A \otimes I & -I \otimes B^\* \\
I \otimes B & A^\* \otimes I
\end{bmatrix}$$
the sign $- I \otimes B^\*$ is to make associativity work out.
I found it in here: Klaus Janich. Vektorraumbundel und der Raum der Fredholm-Operatoren. Math. Ann., 161:129–142, 1965
| 2 | https://mathoverflow.net/users/17260 | 148491 | 79,606 |
https://mathoverflow.net/questions/148479 | 6 | Consider a ctm $\mathfrak{M}$ of $ZF+AD^+$. Is it possible to force over $\mathfrak{M}$ to get a model of ZFC which satisfies further the following:
1. Every projectively definable family of sets of reals has an $OD\_a$ member;
2. Every $OD\_a$ set of reals has the property of Baire;
Where $a$ is any real parameter.
... | https://mathoverflow.net/users/38200 | Forcing over models of determinacy | $\newcommand\R{\mathbb{R}}\newcommand\OD{\text{OD}}$
The answer is no.
First, notice that the family consisting of all subsets $R\subset\R^2$ that are a well-ordering of the reals is projectively definable in your sense, since we can say that $R$ is a well-ordering by quantifying only over countable objects: $R$ i... | 10 | https://mathoverflow.net/users/1946 | 148499 | 79,609 |
https://mathoverflow.net/questions/148486 | 6 | In this question, a **graph** is a finite, undirected graph without loops or multiple edges, and a **colouring** of a graph is a proper vertex colouring. The **product** $G \times H$ of graphs $G$ and $H$ is the graph whose vertex-set is the product of the vertex-sets of $G$ and $H$ and whose edge-set is the product of... | https://mathoverflow.net/users/586 | What is the status of this strong form of Hedetniemi's conjecture? | The answer is no. In the comments, Gil Kalai suggested looking at products of two odd cycles, and indeed, this yields a counterexample.
Consider $C\_3 \times C\_5$, where $C\_n$ denotes the $n$-cycle. This has chromatic number $3$, and there is a 3-colouring given by
$$
\begin{pmatrix}
1&3&1&2&3\\
1&2&1&2&3\\
1&2&1&... | 5 | https://mathoverflow.net/users/586 | 148503 | 79,611 |
https://mathoverflow.net/questions/120609 | 0 | Patrick D. Baier in his [Ph.D. thesis](https://people.maths.ox.ac.uk/hitchin/hitchinstudents/baier.ps.gz) for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14):
Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ and $\Omega\in\wedge^3 V^\ast$. Then we can find uni... | https://mathoverflow.net/users/nan | about decomposition of three forms | I have no idea what the second half of the statement is, but here's what I think the first half says:
If $X \in V$ and $\theta \in V^\*$ such that $\langle \theta, X\rangle \ne 0$, then given any nonzero $\Omega \in \Lambda^3V^\*$, there exists unique elements $\psi \in \Lambda^2V^\*$ and $\phi \in \Lambda^3V^\*$ suc... | 1 | https://mathoverflow.net/users/613 | 148506 | 79,612 |
https://mathoverflow.net/questions/148497 | 0 | Suppose we have a curve $X$ (of genus $\geq 3$), and we know that $\{\phi\_i : X \to E\_i\ \textrm{ for } i = 1, ..., r\}$ are covers of degrees $d\_i$ (with the $d\_i$'s not necessarily all equal), where the $E\_i$ are (genuinely) distinct elliptic curves. Suppose further that we know, for example, that $J = Jac(X)$ i... | https://mathoverflow.net/users/nan | Isogeny of abelian varieties | OK, assume $\alpha \_i=1$ for all $i$ (this is the only case where the question makes sense). You are given an isogeny $JC\sim A=E\_1\times \ldots \times E\_g$, with the $E\_i$ in different isogeny classes. Poincaré complete reducibility theorem tells you that this decomposition is unique (up to isogeny), so any nontri... | 4 | https://mathoverflow.net/users/40297 | 148507 | 79,613 |
https://mathoverflow.net/questions/148471 | 2 | This question is a follow up to my comment to [Sum of the reciprocal of perfect numbers](https://mathoverflow.net/questions/99227/sum-of-the-reciprocal-of-perfect-numbers). I would like to know which results have been published about the possible irrationality of the sum of reciprocals of perfect numbers. For example, ... | https://mathoverflow.net/users/13625 | Any results towards the irrationality of the sum of reciprocals of perfect numbers? | As Stanley Yao Xiao commented, a definite answer to this question would be equivalent to solving an open problem. If we assume two reasonable conjectures, however, then the sum $\sigma$ of the reciprocals of the perfect numbers is irrational:
* The Lenstra-Pomerance-Wagstaff conjecture that there are asymptotically $... | 8 | https://mathoverflow.net/users/39521 | 148516 | 79,617 |
https://mathoverflow.net/questions/148523 | -7 | As I was playing around with Mersenne numbers, and discovered the notion of Wagstaff prime going off Wikipedia, I started considering the sequence, for a given $odd$ prime number $p$, defined as follows:
$2^{p}-1, \frac{2^{p}+1}{3}, \frac{2^{p}+3}{5}, \cdots \frac{2^{p}+p}{p+2}$.
It seems that the first term of this ... | https://mathoverflow.net/users/13625 | Is $2^{p}-1$ prime iff for $\frac{p-1}{2}$ odd positive integers $n$ below $p$, $(n+2)\vert (2^{p}+n)$? | $2^{19}-1$ is prime, but neither $(2^{19}+3)/5$ nor $(2^{19}+9)/11$ is an integer.
| 6 | https://mathoverflow.net/users/3684 | 148528 | 79,621 |
https://mathoverflow.net/questions/148517 | 6 | Let $f:X \to Y$ be a projective morphism between irreducible Noetherian schemes. If a fiber over a closed point of $Y$ is reduced is the generic fiber reduced?
| https://mathoverflow.net/users/32151 | Reduced special fiber implies reduced generic fiber for a projective morphism? | Let's impose a flatness hypothesis (whose necessity is explained in Jason Starr's answer). The answer is still negative, but to explain the context for the counterexample it is instructive to first record some necessary features of any counterexample, so we know where to look.
In view of my above comment about geomet... | 19 | https://mathoverflow.net/users/39487 | 148535 | 79,624 |
https://mathoverflow.net/questions/148536 | 15 | I hope the questions are not too vague.
1. Is the mapping class group of an orientable punctured surface $CAT(0)$ ?
2. Is any of the remarkable simplicial complexes (curve complex, arc complex...) built on a punctured surface $CAT(0)$?
3. Is there any "nice" action (say, proper or cocompact) of the mapping class gro... | https://mathoverflow.net/users/41219 | Mapping class group and CAT(0) spaces | (1) [Bridson showed that](https://arxiv.org/abs/0908.0685) if a mapping class group of a surface (of genus at least 3) acts on a CAT(0) space, then Dehn twists act as elliptic or parabolic elements. This implies that the mapping class groups of genus $\geq 3$ are not CAT(0) (**Edit:** as pointed out by Misha in the com... | 17 | https://mathoverflow.net/users/1345 | 148543 | 79,626 |
https://mathoverflow.net/questions/147167 | 4 | I usually consider a cyclic extension $K$ of degree an odd prime $p$ over the rational field $\mathbf{Q}$.
In this case, there is a well-known result that "every ambiguous class in the class group $\operatorname{Cl}\_K$, which is a class fixed by the Galois group of $\operatorname{Gal}(K/\mathbf{Q})$, becomes trivial... | https://mathoverflow.net/users/42443 | About principal ideal theorem in number fields | I have been trying to prove the result for a couple of days, without success, so I post what I got in the meanwhile. **Let me suppose throughout that $\operatorname{Gal}(E/K)\cong(\mathbb{Z}/p)^2$** (the case $E/K$ cyclic is solved by Hilbert 94).
As Franz Lemmermeyer noticed, the answer is clear when $E$ is the Hilb... | 3 | https://mathoverflow.net/users/18238 | 148571 | 79,636 |
https://mathoverflow.net/questions/145934 | 9 | The $2$-category of topoi and geometric morphisms is not locally small. For example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is equivalent to the category of all abelian groups, which is not small. This problem of course persists for higher topoi, since ... | https://mathoverflow.net/users/4528 | Local smallness and (higher) topoi | Consider a theory which has no models in $\mathrm{Set}$, but has a model in $\mathrm{Sh}(L)$ for some locale $L$. For example, the theory $\mathcal{CLF}$ of complete linearly ordered fields with more than $\sharp \mathbb{R}$ number of elements will do. For $ L $ we can take Barr or Diaconescu covering of $\mathbb {B}\m... | 2 | https://mathoverflow.net/users/10605 | 148576 | 79,638 |
https://mathoverflow.net/questions/148575 | 1 | The Frobenius number of a set of coprime integers is the largest number that not can be written as the sum of integer multiples of numbers in that set.
I'm looking for a general reference on Frobenius numbers.
| https://mathoverflow.net/users/15684 | Reference request for Frobenius numbers | There's a book by Jorge L. Ramírez Alfonsín called The Diophantine Frobenius Problem.
| 6 | https://mathoverflow.net/users/3684 | 148577 | 79,639 |
https://mathoverflow.net/questions/148476 | 3 | For an additive $A$ and any morphism $f:X\to Y$ in $C(A)$ one has the following distinguished triangle in the homotopy category $K(A)$: $X\to Y\to Cone(f)\to X[1]$.
1. What is the closest analogue of this construction for a (more or less) general pointed homotopy category? My problem here is that we do not have to p... | https://mathoverflow.net/users/2191 | Analogues of 'cone' distinguished triangles for pointed model categories? | 1) Mikhail, mapping cones etc, are defined for arbitrary maps. The problem is that they are not homotopy invariant *unless your model category is left proper*. Therefore, in general you must take cofibrant replacements etc. Complexes form a left proper model category with the projective and with the injective model str... | 6 | https://mathoverflow.net/users/12166 | 148582 | 79,641 |
https://mathoverflow.net/questions/148550 | 5 | My question is the following:
>
> Is there a small complex, perhaps analogous to the Chevalley-Eilenberg complex, computing the (co)homology of a restricted Lie algebra over a field of characteristic $p>0$?
>
>
>
Even if the answer is no, I would appreciate any references dealing with such computations.
| https://mathoverflow.net/users/34063 | Computation of restricted Lie algebra (co)homology | That was a large part of the subject of my 1964 PhD thesis.
While the motivation came from algebraic topology, the
relevant algebra was published separately in the paper
<http://www.math.uchicago.edu/~may/PAPERS/3.pdf>.
It is very obvious from the case of abelian restricted
Lie algebras with zero restriction what the... | 5 | https://mathoverflow.net/users/14447 | 148588 | 79,645 |
https://mathoverflow.net/questions/148559 | 4 | Fix a conductor. Then
1) Do the elliptic curves in the same isogeny class have the same reduction type at a prime of bad reduction of the curve ?
2) Do the elliptic curves belonging to two different isogeny classes corresponding to the fixed conductor, have the same reduction type at a prime of bad reduction of th... | https://mathoverflow.net/users/30999 | Isogeny classes and reduction types of elliptic curves at primes of bad reduction | Let $E$ be an elliptic curve over a $p$-adic field $k$. Let $\varphi: E \to E'$ be an isogeny defined over $k$. Write minimal Weierstrass equations with integer coefficients for both curves. Write $\varphi$ with fractions of polynomials with integer coefficients and consider the reduction of that map on the reduced equ... | 6 | https://mathoverflow.net/users/5015 | 148590 | 79,647 |
https://mathoverflow.net/questions/148437 | 13 | We know that the alternating group of degree $n>7$ has an irreducible character of degree $n-1$. The latter number is the smallest nontrivial one for each the alternating group has an irreducible character of that degree.
Does the Alternating group of degree $n>7$ have exactly one irreducible character of degree $n-1... | https://mathoverflow.net/users/19075 | Does the Alternating group of degree $n>7$ have exactly one irreducible character of degree $n-1$? | **Old answer:** You know already that the answer is ``yes.'' For a reference, see result 2 of
>
> Rasala, Richard *On the minimal degrees of characters of $S\_n$*. J. Algebra 45 (1977), no. 1, 132–181.
>
>
>
This gives the answer for $n\geq 9$. The *Atlas of Finite Groups* then gives the result for $n=7,8,9$. ... | 16 | https://mathoverflow.net/users/801 | 148592 | 79,648 |
https://mathoverflow.net/questions/147124 | 0 | In rational homotopy theory there are two concepts,formal and coformal spaces.I want to know example of a space which is
1)not formal and coformal,
2)rationally hyperbolic and not coformal.
| https://mathoverflow.net/users/33699 | Rational formal and coformal space | In this paper you will find examples that fits with your first question:
<http://www.sciencedirect.com/science/article/pii/S0040938303000053>
"Elliptic rational spaces whose cohomologies and homotopies are isomorphic"
Tetsu Nishimotoa, Hiroo Shiga, Toshihiro Yamaguchi.
The authors consider a family of elliptic spaces... | 0 | https://mathoverflow.net/users/27816 | 148593 | 79,649 |
https://mathoverflow.net/questions/148574 | 14 | Accounts of modular forms say that they were studied in the early 19th century, but then define modular forms using terminology that didn't exist until the 20th century. How did the earliest mathematicians to investigate modular forms define them? What motivated their exploration?
| https://mathoverflow.net/users/136 | How did Gauss and contemporaries think of modular forms? | There is also the book of F. Klein, Development of Mathematics in XIX century, vol. I, which has a large chapter on Gauss which describes his work on modular forms. This was written in XX century, but Klein was essentially a XIX century mathematician, so you can see from this book "how did they think".
| 13 | https://mathoverflow.net/users/25510 | 148594 | 79,650 |
https://mathoverflow.net/questions/148598 | 3 | Let $M$ be a compact complex surface, i.e., a complex manifold with complex dimension two. Also, let us denote its Kähler metric by $g$ and its Kähler form by $K$. Let us denote the Kähler class by $[K] \in H^2(M)$. Since $K$ is harmonic with respect to $g$, the Hodge dual of $K$ with respect to $g$, $\*\_g K$ is also ... | https://mathoverflow.net/users/13731 | Questions on the Hodge Dual of the Kähler Class | (1) Yes. If $M$ is $n$-dimensional and $K$ is the Kahler form of a Kahler metric $g$ (or just the Kahler form of a hermitian metric) we have
$$
\*\_g \frac{K^p}{p!} = \frac{K^{n-p}}{(n-p)!}
$$
for any $p = 0,\ldots,n$. For a surface this gives $\*\_g K = K$. This seems to answer your (2) and (3) also.
This is actuall... | 9 | https://mathoverflow.net/users/4054 | 148605 | 79,657 |
https://mathoverflow.net/questions/146061 | 7 | I'm trying to interpret the join of $(\infty,1)$-category in a more conceptual way. Let me try to explain what I have in mind.
In the classical setting it is almost a triviality to express the join of two categories as a [collage](http://ncatlab.org/nlab/show/cograph+of+a+profunctor) along the terminal profunctor: g... | https://mathoverflow.net/users/7952 | Joins of, and limits in, $(\infty,1)$-categories via profunctors | I am not sure I can say anything useful about the weighted limit question; I want to remark about the other two however.
The join of simplicial (properly speaking, simplicially enriched, which I still denote as $sCat$) categories is indeed preserved by the coherent nerve construction. However, the functor $\mathfrak ... | 3 | https://mathoverflow.net/users/42658 | 148612 | 79,660 |
https://mathoverflow.net/questions/148549 | 6 | What is the Hilbert space compression exponent of the standard lamplighter group $\mathbb{Z\_{2}} \wr \mathbb{Z}$? For $\mathbb{Z} \wr \mathbb{Z}$ it is known to be $2/3$ by work of Austin, Naor and Peres, but I couldn't find a reference for the $\mathbb{Z\_{2}}$ case.
| https://mathoverflow.net/users/2192 | Hilbert space compression of the lamplighter group | This is [here](http://arxiv.org/abs/math/0603138): (Tessera 2006, published in CMH 2011): note that lamplighters are among those groups in the class $(\mathcal{L})$ introduced page 3.
| 6 | https://mathoverflow.net/users/14094 | 148620 | 79,664 |
https://mathoverflow.net/questions/148624 | 4 | Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has ***odd*** [class number](http://mathworld.wolfram.com/ClassNumber.html) $h(-d)$. Is it true that one can always solve the diophantine equation,
$$x^2+dy^2 = 2^{2+h(-d)}\tag{1}$$
with ***od... | https://mathoverflow.net/users/12905 | On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$ | Because $d=-1\pmod{8}$, $2$ splits in $K=\mathbb{Q}(\sqrt{-d})$. Let $\mathfrak{p}$ be a prime above $2$. Then $\mathfrak{p}^{h(-d)}$ is principal and has norm $2^{h(-d)}$. Let $z=x+\frac{1+\sqrt{-d}}{2}y$ be a generator, so $z$ has norm $2^{h(-d)}$. But this can be rewritten as $(2x+y)^2+dy^2 = 4\cdot 2^{h(-d)}$. Sinc... | 3 | https://mathoverflow.net/users/40821 | 148625 | 79,665 |
https://mathoverflow.net/questions/147371 | 1 | It is a theorem of Auslander that if $G< GL(2,\mathbb C)$ is a finite subgroup without pseudo-reflections, then the Auslander-Reiten quiver of $K[x,y]^G$ coincides with the McKay quiver of $G$ with the canonical representation. I would like to know what is known for arbitrary finite subgroups of $GL(2,\mathbb C)$, even... | https://mathoverflow.net/users/42547 | Do Auslander-Reiten quivers coincide with the McKay quivers for arbitrary subgroups of GL(2,C)? | This is not true.
First up, you need to take the completion of $k[x,y]^G$ before the statement even makes sense; for an AR sequence to exist you need your endomorphism rings to be local. Also, you might want a Krull-Schmidt category too, and this for these reasons Auslander studies $k[[x,y]]^G$.
Now in the case of ... | 4 | https://mathoverflow.net/users/40472 | 148637 | 79,671 |
https://mathoverflow.net/questions/148561 | 8 | Let $\mathbb F$ be a field and let $a, b, c, d$ be fixed elements in the field $\mathbb F$.
Consider the formulas
1) $\exists\;x\;\;:\;\;x^2=-1.$
2) $\exists\;x\;\;:\;\;(xa=c\land xb=d).$
Formula $(1)$ can be false in $\mathbb F$ but true in a field extension of $\mathbb F$. For exemple, formula $(1)$ is false i... | https://mathoverflow.net/users/24864 | Formulas in a Field and in a Field Extension | This is a major clean-up of my previous argument, thanks to the very helpful comments of Emil Jerabek.
I interpret the question as follows: Characterize all sentences in the language of fields that are preserved under field extensions.
Claim: A formula $\phi$ is preserved under field extensions if and only if the ... | 5 | https://mathoverflow.net/users/5229 | 148638 | 79,672 |
https://mathoverflow.net/questions/147376 | 3 | Setup:
Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}\_n$ denote the $\mathcal{O}$-ideal in $\mathcal{K}$ generated by $t^{-n}$, $\mathcal{O}^l$ the ideal in $\mathcal{O}$ generated by $t^l$. Now $G(\mathcal{O}/\m... | https://mathoverflow.net/users/32972 | A question on algebraic loop groops | The answer is yes. See for example
Pressley, Andrew; Segal, Graeme (1986), Loop groups, Oxford Mathematical Monographs. Oxford Science Publication
| 2 | https://mathoverflow.net/users/42600 | 148649 | 79,676 |
https://mathoverflow.net/questions/148657 | 0 | Let $F$ be an infinite field and let $f \in F[x\_{11},x\_{12},...,x\_{nn}]$ be an **arbitrary** polynomial in $n^2$ variables. Consider the function $\phi : M\_n(F)\longrightarrow F$ defined by $\phi((a\_{ij})) = f(a\_{11},a\_{12}, ..., a\_{nn})$ and suppose that $\phi(I\_n) = 1$ and $\phi(AB)= \phi(A)\phi(B)$, for any... | https://mathoverflow.net/users/41303 | Multiplicative functions $\phi : M_n(F) \longrightarrow F$ with $\phi(I) = 1$ | The answer is yes. Classically, there is a group endomorphism
$g$ of $F^\*$ such that $\phi(M)=g(\det M)$ for all non-singular $M$
(this is obvious if $|F|=2$, otherwise one uses the fact that $[GL\_n(F),GL\_n(F)]=SL\_n(F)$).
Then, $g$ is a polynomial map from $F$ to itself that satisfies $g(XY)=g(X)g(Y)$. From there,... | 1 | https://mathoverflow.net/users/34951 | 148662 | 79,680 |
https://mathoverflow.net/questions/148667 | 2 | It is obvious that for any given $f(x)$ there exists $g(x)$ such that $f(x)=0 \Leftrightarrow g(x)=x$. We could use this fact to solve any root finding problem using [fixed point iteration](http://en.wikipedia.org/wiki/Fixed_point_method) method, only if $g(x)$ promises convergence of the fixed point iteration method.
... | https://mathoverflow.net/users/39492 | Convergence of fixed point iteration algorithm | You do not specify your class of $f$, and most importantly, do not specify where the convergence should hold, for which $x\_0$.
In general, the answer on your question is
Newton's method: $g(x)=x-f(x)/f'(x)$. The iterates usually converge when $x\_0$ is sufficiently close to the root, and if $f'(x)\neq 0$ at the root... | 4 | https://mathoverflow.net/users/25510 | 148672 | 79,684 |
https://mathoverflow.net/questions/148664 | 5 | Assuming consistency of $\text{ZFC}$ (with some large cardinal axiom), is the following statement consistent with $\text{ZFC}$?
Any $\Sigma\_1$ statement with parameters $\omega\_1,\omega\_2$ which holds in a $\omega\_1,\omega\_2$ preserving forcing extension holds in $V$.
| https://mathoverflow.net/users/nan | $\Sigma_1$ Statements and Forcing Extensions | The answer is yes. Your theory holds in any model of the Maximality Principle for $(\omega\_1,\omega\_2)$-preserving forcing. So indeed, one can get a stronger result.
Specifically, this version of the maximality principle asserts that whenever a statement $\sigma$ is forceable by $(\omega\_1,\omega\_2)$-preserving f... | 9 | https://mathoverflow.net/users/1946 | 148675 | 79,686 |
https://mathoverflow.net/questions/148674 | 2 | Is there a pair of continuous surjective functions say $f\_1$ and $f\_2$ from $\mathbb{Q}$ to itself such that for every $x, y \in \mathbb{Q}$, $f\_1^{-1}(x) \cap f\_2^{-1}(y)$ is non-empty?
| https://mathoverflow.net/users/29256 | Continuous Functions | Take the two projections from ${\mathbb Q}\times{\mathbb Q}$ to ${\mathbb Q}$ and use the [fact](https://mathoverflow.net/a/148680/10503) that ${\mathbb Q}\times{\mathbb Q}$ is homeomorphic to ${\mathbb Q}$.
| 8 | https://mathoverflow.net/users/10503 | 148678 | 79,687 |
https://mathoverflow.net/questions/148604 | 5 | For a Kahler manifold $M$, and a smooth vector bundle $E$ over $M$, let us denote by $A^{(p,q)} := \Omega^{(p,q)} \otimes E$ the bundle of forms with values in $E$. Now with respect to a choice of connection on $E$, we can extend $d$ to a mapping d$ \_E: A^{(p,q)} \to A^{(p+1,q+1)}$. Moreover, we can extend $\partial$,... | https://mathoverflow.net/users/42100 | Are the Kahler Identities for a Holomorphic Vector Bundle Actually Interesting? | They can be used to prove the vanishing theorems (and more). Kodaira-Nakano vanishing theorem and Kodaira embedding theorem follow from these identities. One proves that the difference of the $\partial$ and $\bar\partial$-Laplacians is a commutator of Hodge
$\Lambda$ operator and the curvature. This commutator happens ... | 6 | https://mathoverflow.net/users/3377 | 148685 | 79,691 |
https://mathoverflow.net/questions/148681 | 3 | Given a graph $G$, it is possible to construct a sequence $H\_1, H\_2, \dotsc, H\_k$ of graphs each a set of disjoint edges (i.e., every vertex of degree at most $1$) with $G = \bigcup H\_i$.
For example:
For the complete bipartite graph $K\_{n,n}$, only $n$ graphs are needed, which can be seen by taking one perfec... | https://mathoverflow.net/users/42691 | Building graphs as unions of disjoint edges | Well, in some sense, there is no known way of doing it effectively for arbitrary graphs. (It is NP complete to determine whether $k=\Delta(G)$ or $k=\Delta(G)+1$.) You are actually interested in the edge chromatic number of a graph. See:
<http://en.wikipedia.org/wiki/Edge_coloring>
| 5 | https://mathoverflow.net/users/38267 | 148689 | 79,694 |
https://mathoverflow.net/questions/148539 | 9 | This question is related to this one: [Local smallness and (higher) topoi](https://mathoverflow.net/questions/145934/local-smallness-and-higher-topoi) which has not yet been answered.
The $2$-category of topoi and geometric morphisms is not locally small. (As I mentioned in the question above, for example, if $A$ is ... | https://mathoverflow.net/users/4528 | Are topoi and etale geometric morphisms locally small? | There is only a set of isomorphism classes of étale geometric morphisms between any two Grothendieck toposes. In fact, the same is true for essential geometric morphisms.
Recall that Grothendieck toposes are locally presentable categories, and that the left adjoint of a $\kappa$-accessible functor between $\kappa$-ac... | 8 | https://mathoverflow.net/users/11640 | 148697 | 79,698 |
https://mathoverflow.net/questions/148659 | 2 | Let $D$ be a divisor on a (complex) K3 surface.
Suppose $D^2\geq0$. In general, $D$ is *nef* if $D\cdot C\geq0$ for all irreducible curves on the surface.
Is it sufficient in our case to check this for smooth rational curves (i.e. the (-2) curves) ?
| https://mathoverflow.net/users/40038 | Nefness on a K3 surface | First off, you obviously have to assume something about $-D$ not being effective, because otherwise you could take a negative ample class.
The cone of curves of a K3 surface is pretty well described in [this paper](http://link.springer.com/article/10.1007/BF01450509). And there is a newer version of it that works in... | 5 | https://mathoverflow.net/users/10076 | 148699 | 79,700 |
https://mathoverflow.net/questions/148712 | 3 | How can I simply prove the following fact:
Let $A := \{1, \dots n \}$ and $B := \{1, \dots, \lfloor \frac{n}{4} \rfloor \}$. Let $d \in (0,1)$ and let $R$ be a randomly choosen (with uniform distribution) subset of cardinality $\lfloor n^{d} \rfloor$ from all subsets of that cardinality. Then for any $d' < d$:
$$\m... | https://mathoverflow.net/users/42695 | Cardinality of intersection of a random subset with a fixed subset | I think a reasonably quick way to do it is something like this:
Write $C = A \setminus B$. If we can show that
\begin{equation} \tag{\*}
\Pr[|C \cap R| \geq .8 \lfloor n^d \rfloor] = o\_n(1)
\end{equation}
then we're done because $.2 \lfloor n^d \rfloor > n^{d'}$ for sufficiently large $n$.
For each $1 \leq i \l... | 6 | https://mathoverflow.net/users/658 | 148732 | 79,710 |
https://mathoverflow.net/questions/148728 | 1 | I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes\_n\mathbb{Z}$, say
$$\mathbb{Z}^2\rtimes\_n\mathbb{Z}=\langle x,y,z: xz=zx, zy=yz, xy=yz^nx\rangle$$ for some natural number $n>0$.
For example, $G$ could be the discrete Heise... | https://mathoverflow.net/users/9305 | $Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$ | Let $G\_n$ the group generated by $x$, $y$, $z$ with $z$ central and $yx=xyz^n$; this is your group up to renaming; I like this one better :-)
The only finite conjugacy classes of the group $G\_n$ are those of the powers of $z$. It follows that the center of the group algebra $\def\ZZ{\mathbb Z}\ZZ G$ is $\ZZ Z$ wit... | 2 | https://mathoverflow.net/users/1409 | 148733 | 79,711 |
https://mathoverflow.net/questions/148731 | 36 | Let $f(n)$ denote the number of (isomorphism classes of) groups of order $n$. A couple easy facts:
1. If $n$ is not squarefree, then there are multiple *abelian* groups of order $n$.
2. If $n \geq 4$ is even, then the dihedral group of order $n$ is non-cyclic.
Thus, if $f(n) = 1$, then $n$ is a squarefree odd numbe... | https://mathoverflow.net/users/31308 | For which $n$ is there only one group of order $n$? | $f(n)=1$ if and only if $\gcd(n,\phi(n))=1$, where $\phi$ is the Euler phi-function. These $n$ are tabulated at <http://oeis.org/A003277>
The result is found in Tibor Szele, Über die endichen Ordnungszahlen, zu denen nur eine Gruppe gehört, Comment. Math. Helv. 20 (1947) 265–267, MR0021934 (9,131b).
| 54 | https://mathoverflow.net/users/3684 | 148738 | 79,715 |
https://mathoverflow.net/questions/148726 | 3 | So, I'm reading through some notes on the etale fundamental group (mostly Murre, but also some other notes I have), and I find it confusing how in a galois category $\mathcal{C}$ with fundamental functor $F$, the automorphism group $\text{Aut}(X)$ of an object $X\in\mathcal{C}$ can act on $F(X)$ ``on the right''. I kno... | https://mathoverflow.net/users/15242 | how do automorphisms act on the right in grothendieck's galois theory | Yes.
First notice that the "restriction map" you describe in Galois theory is actually kind of subtle. You first take the subgroup of automorphisms of the larger field that send the smaller field to itself. You then have to mod out by the subgroup that fix the smaller field. So if $L$ is the Galois closure of $K$, th... | 1 | https://mathoverflow.net/users/18060 | 148743 | 79,718 |
https://mathoverflow.net/questions/148736 | 7 | Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics.
When studying Feller processes, or in general semigroups of operators defined on a Banach space, it seems the only notion of dynamical... | https://mathoverflow.net/users/42714 | Why aren't operator semigroups studied from a dynamical perspective? | Jack Hale had important work in generalizing concepts of dynamical systems to the infinite dimensional setting, see his monographs:
[Asymptotic behavior of dissipative systems](http://books.google.hu/books?id=fSzscCu37ygC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false)
It turns out that, under compactness assumtions, it ... | 6 | https://mathoverflow.net/users/12898 | 148747 | 79,719 |
https://mathoverflow.net/questions/136607 | 16 | The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in closed form have been of interest.
It seems that the classical programme of the PDE community, i.e., (i) existence (ii... | https://mathoverflow.net/users/2082 | Einstein field equations in perspectives from PDE and functional analysis | The statement
>
> It seems that the classical programme of the PDE community, i.e., (i) existence (ii) uniqueness (iii) regularity, heavily employing concepts from functional analysis, has not found prominent application in general relativity.
>
>
>
is just plain wrong. You are overlooking quite a lot of stuff... | 18 | https://mathoverflow.net/users/3948 | 148763 | 79,723 |
https://mathoverflow.net/questions/148767 | 6 | From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed 3-manifold $M$, I can get a link on which performing surgery gives $M$. Now I want to get a surgery link from a given explic... | https://mathoverflow.net/users/36445 | Getting surgery link from Heegaard splitting | Yes- it's easy. There's the "digging the trench" construction, nicely described in [A simple proof of the fundamental theorem of Kirby calculus on links](http://www.ams.org/journals/tran/1992-331-01/S0002-9947-1992-1065603-2/S0002-9947-1992-1065603-2.pdf) by Ning Lu, for example.
In short and with all details suppres... | 6 | https://mathoverflow.net/users/2051 | 148775 | 79,728 |
https://mathoverflow.net/questions/148642 | 6 | A paper I'm currently reading uses the following fact. If $A$ is a unital $C^\*$-algebra, $P=P^2\in A$, then there are $T, F\in A$ s.t. $F$ is an orthogonal projection ($F=F^\*=F^2$) and
$$P=F+FT(1-F).\tag{1}$$
**The question:**
>
> Since no proof or further reference is given, I assume, that this fact
> is obvi... | https://mathoverflow.net/users/8134 | Expression of a non-orthogonal projection in a $C^*$ algebra via an orthogonal one | Just a few remarks in addition to David Handleman's answer and your work. To make sense of ranges and nullspaces below, you can assume your $C^\*$-algebra is sitting in $B(H)$. I just added that hoping it makes things more "obvious". But the whole thing works of course in an abstract $C^\*$-algebra. Even the notion of ... | 3 | https://mathoverflow.net/users/35324 | 148776 | 79,729 |
https://mathoverflow.net/questions/148760 | 4 | Let $k$ be a field. Suppose we have an exact sequence of $k$-group schemes (not finite-type)
$$
1\to H\to G\to K\to 1
$$
In other words, the sheaf quotient $G/H$ is representable by a $k$-group scheme. Now, suppose that we have a finite group $\Gamma$ acting on the groups, functorially and compatible with the morph... | https://mathoverflow.net/users/14379 | Representability of a certain group scheme quotient | The question is imprecise concerning the topology involved and quasi-compactness conditions on the group schemes, so in view of the motivation let's first stick to the affine case before we venture beyond that.
Rather generally, consider any left-exact sequence
$$1 \rightarrow G' \rightarrow G \rightarrow G''$$
of af... | 7 | https://mathoverflow.net/users/39487 | 148785 | 79,733 |
https://mathoverflow.net/questions/148768 | 2 | I asked the following question (slightly paraphrased) about a week ago in Stack exchange but no one knew the answer to the particular question. I was hoping someone here might be able to help me.
"Does anyone know of a reference that explains the concept of forcing by fixing a forcing language that has a (I believe u... | https://mathoverflow.net/users/39939 | Forcing Language | $\newcommand\P{\mathbb{P}}\newcommand\B{\mathbb{B}}$
Now I understand what you want.
The usual account of forcing has an explosion in the size of the official language, by adding all the $\P$-names as official terms to the language. And not only does this make the language a proper class, but as you point out it a... | 3 | https://mathoverflow.net/users/1946 | 148792 | 79,734 |
https://mathoverflow.net/questions/147325 | 1 | The role of Reynolds operator in GIT has always been a little mystery to me. Actually I see that in some proofs it gets used in an efficient way, but what I cannot grasp is the general philosophy. I see of course it is a projector onte the invariant subspace of a G-representation. But, as I asked, when do I want to use... | https://mathoverflow.net/users/4096 | when does one want to use the Reynolds operator in GIT? | It is used to prove that a uniform categorical quotient by the action of a reductive group on an affine scheme exists and that the quotient inherits basic properties of the original scheme such as affine, algebraic, noetherian. In particular in that proof it is used to prove that the ring of invariants of a quotient by... | 2 | https://mathoverflow.net/users/10076 | 148799 | 79,737 |
https://mathoverflow.net/questions/148781 | 1 | Let $S$ be an orientable closed 2-surface with genus at least 2 and $C$ be a non-separating essential simple closed curve in $S$. Denote $S\_{C}=S-N(C)$. Let $f$ be a pseudo anosov map of $S\_{C}$. Hence $f$ induces a natural homeomorphism of $S$ which fix the $C$ pointwise, still denoted by $f$.
My question are：
1... | https://mathoverflow.net/users/18496 | Some questions on partial pseudo anosov maps | In answering all of your questions I am going to assume that $S$ has a hyperbolic structure, and that the stable and unstable laminations are geodesic laminations. This point of view is explained in the book of Casson and Bleiler. The answers given here can be derived from what one learns in that book.
Question 1: No... | 4 | https://mathoverflow.net/users/20787 | 148800 | 79,738 |
https://mathoverflow.net/questions/148790 | 2 | Given a $p \times p$ positive definite matrix $\Sigma$, why eigenvectors of $\Sigma$, stacked as columns of a matrix $R \equiv [r\_1 \, r\_2 \, \ldots \, r\_p]$, optimize the following orthogonally constrained minimization problems?
$$
\mathrm{minimize}~~~~\log \det\big( I \odot (R^\mathsf{T}\Sigma R) \big)~~~~~\text{s... | https://mathoverflow.net/users/19394 | Why eigenvectors optimize this orthogonally constrained nonlinear minimization problem? | Your minimization problem is equivalent to
\begin{equation\*}
\min\_{R^TR=I}\quad\prod\_{i=1}^p r\_i^T\Sigma r\_i,
\end{equation\*}
and it can be shown (using *Hadamard's determinant inequality* and some more argumentation) that this minimum overall $p$ orthonormal tuples is achieved by choosing the $r\_i$ correspondin... | 3 | https://mathoverflow.net/users/8430 | 148804 | 79,741 |
https://mathoverflow.net/questions/148655 | 7 | May I have some clarification about original proof of Gödel's Completeness Theorem compared to "standard" Henkin's proof based on Model Existence Lemma ?
My understanding of Gödel's original proof is this :
1) Perform some syntactical transformation, in order to reduce the general problem to a particular class of w... | https://mathoverflow.net/users/42676 | Original proof of Gödel's completeness theorem compared to Henkin's proof | With regard to the amount of set theory required to prove the completeness theorem, the wikipedia page on [The completeness theorem](http://en.wikipedia.org/wiki/Godel%27s_completeness_theorem#Relationship_to_the_compactness_theorem) asserts:
>
>
> >
> > When considered over a countable language, the completeness... | 4 | https://mathoverflow.net/users/1946 | 148805 | 79,742 |
https://mathoverflow.net/questions/148756 | 18 | Define $N\_n$ as $n$ th natural number: $N\_0=0, N\_1=1, N\_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N\_n}=N\_{2^{n}}$.
(Which says: For all finite cardinal $n$ we have: $2^{n~\text{th finite cardinal}}=2^{n}~\text{th finite cardinal}$).
What this means?
The *gap* b... | https://mathoverflow.net/users/nan | A New Continuum Hypothesis (Revised Version) | In the following answer, by Foreman-woodin model, I mean the model constructed by them in the paper "The generalized continuum hypothesis can fail everywhere.
Ann. of Math. (2) 133 (1991), no. 1, 1–35. "
**Questions 1 and 3 have positive answer:** In Foreman-Woodin model for the total failure of GCH the following hol... | 10 | https://mathoverflow.net/users/11115 | 148806 | 79,743 |
https://mathoverflow.net/questions/147060 | 4 | I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions $u\in W^{1,p}(\mathbb{R}^d)$. More precisely:
Assume $u\in W^{1,p}(\mathbb{R}^d)$, fix a constant $M\in \mathbb{R}$, and let
$$
E\_M:=\{x\in \mathbb{R}^d,\hspace{1cm}u(x)>M\... | https://mathoverflow.net/users/33741 | sub and super-levelset regularity for Sobolev functions | One positive answer is that this set is $p$-quasi-open, see some resource about capacity theory, e.g., here: <https://math.stackexchange.com/questions/48776/capacity-theory-beginner-resources>.
| 3 | https://mathoverflow.net/users/32507 | 148814 | 79,747 |
https://mathoverflow.net/questions/148779 | 7 | **Statement** Given a finite abelian group $G$ and two independent random variables $X,Y$ taking values in $G$ and satisfying $d\_{TV}(X,U\_G)\leqslant \delta$ and $d\_{TV}(Y,U\_G)\leqslant \delta$ (where $U\_G$ denotes a uniformly distributed over $G$ and $d\_{TV}$ is the total variation distance), we ask how close is... | https://mathoverflow.net/users/13099 | Convergence rate of the convolution of almost uniform measures on $\mathbb{Z}_p$ | I think in general you can't expect anything better than $2$ regardless of the group.
Saying $d\_{TV}(X, U\_G) \leq \delta$ is equivalent to saying we can write the measure corresponding to $X$ as
$$\mu\_X=\mu\_U + (\mu\_1 - \mu\_2),$$
where $\mu\_U$ is the uniform measure, and $\mu\_1$ and $\mu\_2$ are positive me... | 2 | https://mathoverflow.net/users/405 | 148817 | 79,749 |
https://mathoverflow.net/questions/146971 | 8 | Let $(X,f)$ be a Belyi pair, i.e. a Riemann surface $X$ together with a morphism $f: X \to \mathbb{P}^1$, ramified only in $0,1, \infty$. Grothendieck's dessin d'enfant is the pre-image $G$ of the interval $[0,1]$ considered as a graph embedded in $X$ (a dessin d'enfant has more information than that, but let us concen... | https://mathoverflow.net/users/2234 | Is there an algorithm to compute efficiently the dessin d'enfant from a Belyi pair? | This question is addressed in the recent preprint arXiv:13112529, in section 7. This is a survey of computing Belyi maps from the designs, but section 7 addresses the inverse problem.
| 5 | https://mathoverflow.net/users/25510 | 148823 | 79,752 |
https://mathoverflow.net/questions/147415 | 8 | I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity:
>
> Let $X=(X\_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ and $r$ be a positive integer such that $2\leq r\leq n-1$. For any $1\leq \ell\_1<\ell\_2<\cdots< \ell\_r\leq n$, we... | https://mathoverflow.net/users/42572 | A Problem on Linear Algebra | The following proof is due to Yeping ZHANG:
Let $V$ be a $n$ dimensional Euclidean vector space. Let $(e\_1,\cdots,e\_n)$ be an orthogonal basis.
For any $k=1,\cdots,n$, let $\Lambda^k V$ the $k$-th exterior product of $V$. We equip $\Lambda^k V$ with a metric ${\lVert \cdot \rVert}\_{\Lambda^k V}$ such that $(e\_{... | 3 | https://mathoverflow.net/users/42572 | 148831 | 79,755 |
https://mathoverflow.net/questions/148833 | 9 | My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the computations and I noticed that the computations works over any commutative $\mathbb{Z}$-algebra. Why these groups are not introd... | https://mathoverflow.net/users/8419 | Chevalley Groups over an arbitrary ring. | The clean definition of adjoint Chevalley groups can be given in the spirit of what you are trying to do, but there is a hidden subtlety because it is only the torus in the *simply connected* case that is literally generated by the coroot groups (i.e., the simple positive coroots are a basis of its cocharacter group) w... | 16 | https://mathoverflow.net/users/39487 | 148838 | 79,757 |
https://mathoverflow.net/questions/148834 | 0 | Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x\_{\max}$ of the polynomial of the form $x\_{\max} \geq f(k)$.
So far I was able to show that $x\_{\max} \geq k$ which is quite close to t... | https://mathoverflow.net/users/22051 | Bounds on the largest root of a polynomial | I would write this as a comment, but as I'm new here, it doesn't allow me to do so.
Anyway, one can make your bound better by replacing $x\_{max}\ge k$ by $x\_{max}\ge k+\frac{1}{2k}$. This can be seen by considering $q(y)=p(y+k)=y^3+(2k+1)y^2+(k^2+1)y-k+1=0$. Definitely $q(\frac{1}{k})>0$ and actually, it's not diff... | 4 | https://mathoverflow.net/users/21124 | 148839 | 79,758 |
https://mathoverflow.net/questions/148840 | 4 | Let $T$ be a real torus, and let $X$ and $Y$ be $T$-spaces. Under what conditions (if any) will the existence of graded $H^\*\_T$-algebra isomorphism between the $T$-equivariant cohomologies of $X$ and $Y$ (say over the rationals) imply the existence of a $T$-equivariant homotopy equivalence between $X$ and $Y$?
| https://mathoverflow.net/users/25358 | Is there a Whitehead-type theorem in T-equivariant cohomology? | One set of sufficient conditions may be obtained if your map plays nicely with respect to subspaces fixed by closed subgroups. Let $X$ and $Y$ be $G$-spaces for any $G$ (not only the torus) and assume that you have an equivariant map $f:X \to Y$. If for any closed subgroup $H < G$ the induced map $X^H \to Y^H$ of $H$-s... | 4 | https://mathoverflow.net/users/18263 | 148841 | 79,759 |
https://mathoverflow.net/questions/148835 | 6 | Let $M$ be the moduli stack of ordinary but possibly nodal elliptic curves over the field $\overline{\mathbf{F}\_p}$. Then $M$ has a $\mathbb{Z}\_p^{\times}$-torsor over it, given by the moduli scheme of "trivialized" elliptic curves: that is, elliptic curves equipped with an isomorphism between the formal group and $\... | https://mathoverflow.net/users/344 | Fundamental group of the moduli stack of ordinary generalized elliptic curves | For $p>13$, there are at least two supersingular $j$ invariants, say $a$ and $b$, and adjoining $\sqrt[N]{\frac{j-a}{j-b}}$ is always an etale cover for $N$ prime to $p$. This gives an additional portion of the fundamental group.
To compute the full fundamental group, we can first take an etale cover that kills the e... | 5 | https://mathoverflow.net/users/18060 | 148842 | 79,760 |
https://mathoverflow.net/questions/148777 | 6 | Assume ZF. Consider the claim:
(1) For any infinite set $\Omega$, there is a finitely additive probability measure $\mu:2^\Omega\to[0,1]$ with $\mu(A) = 0$ whenever $|A|<|\Omega|$.
Then (1) is implied by Hahn-Banach and the claim that the union of two sets of cardinality less than $|\Omega|$ has cardinality less th... | https://mathoverflow.net/users/26809 | Strength of some claims about finitely additive measures on infinite sets? | Assertion (1) is equivalent to the Axiom of Choice. This is because it implies that for every infinite set $\Omega$, the set $\{A \subseteq \Omega : |A| \lt |\Omega|\}$ is an ideal. It follows from this that every infinite cardinal number $\mathfrak{m}$ is indecomposable — there are no cardinals $\mathfrak{p},\mathfrak... | 7 | https://mathoverflow.net/users/2000 | 148845 | 79,762 |
https://mathoverflow.net/questions/144752 | 4 | I posed this question to [another user](https://mathoverflow.net/users/8320/domenico-fiorenza), but we weren't able neither to make a precise guess, nor to give a precise proof. Is it a "not-so-well" well-known fact?
Consider a connected non-finite CW complex $X$; denote its Poincare' series as $p\_X(T)=a\_0+a\_1T+\d... | https://mathoverflow.net/users/7952 | The geometric meaning of an inverse Poincare' series | The short answer is "morally, $Y$ should be the loop space of $X$," and we'll see how far this answer gets towards justifying that.
---
First some words that are not about spaces. My go-to example of two power series, both of which are the Hilbert / Poincare series of something, whose product is $1$ is
$$\lef... | 5 | https://mathoverflow.net/users/290 | 148846 | 79,763 |
https://mathoverflow.net/questions/146791 | 15 | Let $X$ be a finite spectrum. Say that $X$ has **characteristic two** if multiplication by two on $X$ is nullhomotopic.
Does there exist a noncontractible finite spectrum of characteristic two?
(This is mentioned as an open problem in Barratt's 1959 [paper](http://www.maths.ed.ac.uk/~aar/papers/barratt1.pdf%E2%80... | https://mathoverflow.net/users/344 | Finite spectrum annihilated by multiplication by two | This doesn't seem very hard. Am I missing something?
Let $X$ be a non-trivial finite spectrum of characteristic $2$. Then let $R=\mathrm{Hom}(X,X)$, the function spectrum of maps $X$ to $X$. This $R$ is an associative $S$-algebra, with $0=2$ in $\pi\_0R$; furthermore, $R$ is finite.
If $X\neq0$, then $H\_\*(X;F)\ne... | 15 | https://mathoverflow.net/users/437 | 148847 | 79,764 |
https://mathoverflow.net/questions/148850 | 5 | It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R\_M$ is also hereditary. Is the Noetherian assumption necessary?
| https://mathoverflow.net/users/41303 | Localizations of hereditary rings | You do not need to suppose your ring to be noetherian. In fact, the following stronger statement is true:
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> If $R$ is a hereditary commutative ring and $S\subseteq R$ is a subset, then the ring $S^{-1}R$ is hereditary.
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>
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In order to prove this we use the following two facts:
1) A commutative ring $R$ i... | 5 | https://mathoverflow.net/users/11025 | 148855 | 79,766 |
https://mathoverflow.net/questions/148852 | 1 | I'm looking for an example of a recurrence relation where it provably does not have a closed form.
Does such a thing exist?
For example, I believe $S(n, k)$, or the Stirling numbers of the second type, do not have a closed form -- but I'm not sure if they *don't* or if they *can't*.
| https://mathoverflow.net/users/40801 | Does there exist a recurrence relation which cannot be written in a closed form? | I'm sure you're aware that linear recurrent sequences are well understood and can be solved exactly by a "closed formula". Once you enter the world of non-linear recurrences, even when restricting to constant coefficients, you can get all sorts of strange behavior even with some very simple looking recurrences. One exa... | 8 | https://mathoverflow.net/users/2384 | 148856 | 79,767 |
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