parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/157502 | 2 | I'm studying some applications of small cardinals related to the Michael's Problem. Recall that we say that a space $X$ is a Michael space if X is a regular Lindelöf space such that $X\times \omega^\omega$ is not Lindelöf, and the question of whether such a space can be constructed without additional axioms is what we ... | https://mathoverflow.net/users/41407 | A question about small cardinals related to Michael's Problem | It is not true that if $\mathscr{F}$ is dominating then $\{K\_f:f\in\mathscr{F}\}$ is a cover of $\omega^\omega$. For example it is possible to have a dominating $\mathscr{F}$ such that $f(4)=6$ for every $f \in \mathscr{F}$ (the value of the functions at a single integer won´t change the fact that the family is or is ... | 3 | https://mathoverflow.net/users/17836 | 157505 | 83,286 |
https://mathoverflow.net/questions/157333 | 3 | Let us define a random variable $X$ with density function $p(x)$. We wish to calculate $\mathbb{E}[f(X)] = \int f(x)p(x)dx$. We can compute the expectation by Monte Carlo simulations as
$$\mathbb{E}[f(X)] =\frac{1}{N} \sum\_{i=1}^{N} f(x\_i)p(x\_i)$$
where $x\_i$ are sampled from $p(x)$.
Sometimes it is impossibl... | https://mathoverflow.net/users/46007 | How to perform Importance Sampling with Prior Information | Having explicit bounds on $\mathbb{E}[f(X)]$ does not appear to be useful. The variance can still be arbitrarily large, and the variance governs the rate of convergence for MC sampling. And if the variance is infinite, Monte Carlo estimation does not even converge to the expected value.
If you strengthen your assumpt... | 1 | https://mathoverflow.net/users/8938 | 157507 | 83,287 |
https://mathoverflow.net/questions/157501 | 2 | Let $f:X \to Y$ be a projective morphism of complex Noetherian schemes. Assume $Y$ is smooth and for all $y \in Y$, $f^{-1}(y)$ is of pure dimension $1$. Let $\mathcal{F}\_1, \mathcal{F}\_2$ and $\mathcal{F}\_3$ be coherent sheaves on $X$ flat over $Y$ satisfying the following short exact sequence:
$$0 \to \mathcal{F... | https://mathoverflow.net/users/46578 | A functorial property of higher right derived functors | Take the universal extension of $O$ by $O(-2)$ on $P^1$. By definition it is a vector bundle $E$ on $X = P^1\times A^1$ which is an extension
$$
0 \to p\_1^\*O(-2) \to E \to O \to 0.
$$
Let $Y = A^1$ and $f = p\_2$. Then after taking the pushforward you will get
$$
0 \to R^0f\_\*(E) \to O \stackrel{t}\to O \to R^1f\_\*... | 3 | https://mathoverflow.net/users/4428 | 157514 | 83,291 |
https://mathoverflow.net/questions/157481 | 5 | Take two points, $p\_0$ and $p\_k$, in $n$-dimensional Euclidean space, where $d(p\_0,p\_k)$ is the distance between the points. Now, draw an $n$-sphere of radius $r$ centered on $p\_0$ and uniformly select a new point, $p\_1$, in the volume of the sphere, and a new point $p\_2$ on the surface of the sphere.
Let $h\_... | https://mathoverflow.net/users/46908 | Probability distribution or the distance between two points in $n$-dimensional Euclidean space after a random perturbation of one point | Let us consider the case $n=2$. Assume $p\_k$ is at the origin and $p\_0$ is at the point $(1,0)$ on the $x$-axis, and that $r<1$.
Distribution of $h\_1$
----------------------
By subtracting a constant it suffices to find the distribution of $d:=d(p\_1,0)$.
The density for $d$, $f\_d(R)$, is proportional to $R\T... | 3 | https://mathoverflow.net/users/4600 | 157515 | 83,292 |
https://mathoverflow.net/questions/157529 | 3 | I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference.
Let $k$ be some (algebraically closed, if needed) field. There is a formal differentiation in the ring of formal power series $k[[x]]$. Let $F(x,y) \in k[[x,y]]$ be a formal series which ... | https://mathoverflow.net/users/2234 | existence and uniqueness of solutions for ODEs in formal power series? | Yes. No algebraicity assumption is necessary. Rewrite the desired condition as
$$y = \int\_0^x F(x, y) \, dx = \sum\_{n, m \ge 0} f\_{n, m} \int\_0^x x^n y^m \, dx = L(x, y).$$
We compute that
$$L(x, y\_0) - L(x, y\_1) = \sum\_{n, m \ge 0} f\_{n, m} \int\_0^x x^n (y\_0^n - y\_1^m) \, dx$$
hence that if $x^k | y... | 5 | https://mathoverflow.net/users/290 | 157530 | 83,297 |
https://mathoverflow.net/questions/157522 | 4 | Let $S$ and $R$ be two (not necessarily commutative) $k$-algebras for $k$ a field. If I have a $S$-$R$ bimodule $\_SM\_R$, I can form the functor $\_SM\_R\otimes\_R (-):R\text{Mod} \rightarrow S\text{Mod}$. Similarly, I can form $(-)\otimes\_S {\_SM\_R}$ to get a functor from $\text{Mod}S \rightarrow \text{Mod}R$. If t... | https://mathoverflow.net/users/17121 | Existence of a left adjoint to tensor product implies projectivity | I believe I have worked out an argument: We want the functor $F:R\text{Mod}\rightarrow S\text{Mod}$ by $\_S M\_R \otimes\_R (-)$. By tensor-hom duality, this functor is always a left adjoint so in particular it is right exact. If this functor is also a right adjoint, then it must be left exact which means that $M$ is f... | 1 | https://mathoverflow.net/users/17121 | 157544 | 83,304 |
https://mathoverflow.net/questions/157472 | 22 | What's the smallest absolute value possible of a non-zero eigenvalue of an $n$ by $n$ square matrix whose entries are either $0$ or $1$ (all operations are over $\mathbb{R}$)? I would be interested in estimates or bounds as I imagine an exact answer is tricky.
I asked this question previously at <https://math.stackex... | https://mathoverflow.net/users/45564 | Smallest non-zero eigenvalue of a (0,1) matrix | The smallest nonzero eigenvalue can decrease at least exponentially,
even for matrices that are sparse, symmetric, and invertible.
Explicitly, let $M\_n$ have $1$'s on the *anti*-diagonal, and also on the
first and third off-diagonals above it. For example,
here's the matrix for $n=13$:
```
0 0 0 0 0 0 0 0 0 1 0 ... | 21 | https://mathoverflow.net/users/14830 | 157554 | 83,306 |
https://mathoverflow.net/questions/157567 | 3 | Let $X$ be a K3 surface and $D$ an effective divisor such that $h^0(D)\geq2$ and $h^1(D)=0$.
Is this enough to show that $D$ is connected?
Any reference would also be appreciated (I looked in Saint-Donat' thesis but did not get the answer)
| https://mathoverflow.net/users/40038 | Sufficient conditions for a divisor to be connected on a K3 surface | I think the answer is yes. Here is a sketch proof.
Keep in mind that by Serre duality that $h^2(D)=0$.
Suppose that the divisor is not connected. Then there are two possibilities:
1) There is an isolated exceptional subdivsor in $D$ (by this I mean a divisor $E$ that is a connected collection of $-2$ curves with... | 2 | https://mathoverflow.net/users/13441 | 157574 | 83,311 |
https://mathoverflow.net/questions/157582 | 1 | Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int \_{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put,
$$A(\mathbb T):= \{f\in L^{1}(\mathbb T): \hat{f}\in \ell^{1}(\mathbb Z), \ \text {that is,} \ \sum\_{n\in \mathbb Z} |\hat{f}(n)| < \infty \}.$$
Resu... | https://mathoverflow.net/users/33018 | Result of Beurling concerning absolute convergence of Fourier series of |f| | This is Theorem V (page 16) from:
A. Beurling, [On the spectral synthesis of bounded functions](http://www.ams.org/mathscinet-getitem?mr=27891). Acta Math. 81 (1948).
In fact, Beurling proves the stronger statement:
>
>
> >
> > **Theorem** Let $f(x) = \sum\_{n=-\infty}^{\infty} a\_n e(nx)$ (with $a\_0=0$) hav... | 2 | https://mathoverflow.net/users/630 | 157588 | 83,315 |
https://mathoverflow.net/questions/157577 | 21 | One interesting fact in symmetric function theory is that the power symmetric function $p\_n$ can be written as an alternating sum of hook Schur functions $s\_{\lambda}$:
$$
p\_n = \sum\_{k+\ell = n} (-1)^\ell s\_{k, 1^\ell}.
$$
A priori, all that is known is that $p\_n$ can be expressed as a sum of Schur functions $... | https://mathoverflow.net/users/16002 | Why are the power symmetric functions sums of hook Schur functions only? | The coefficient of $s\_{\lambda}$ in $p\_{\mu}$ is (up to normalizing factors which I won't get right) the trace of the permutation of conjugacy class $\mu$ acting on the representation $Sp(\lambda)$ of $S\_n$. You are asking why the $n$-cycle acts with trace $0$ on every irrep except for the hooks (which are $\bigwedg... | 18 | https://mathoverflow.net/users/297 | 157592 | 83,316 |
https://mathoverflow.net/questions/108851 | 14 | For $G$ a graph, let $\alpha(G)$ be its independence number and $\Theta(G)=\lim\_n \sqrt[n]{\alpha(G^{\boxtimes})}$ its [Shannon capacity](https://en.wikipedia.org/wiki/Lov%C3%A1sz_number#Shannon_capacity_of_a_graph), where $\boxtimes$ denotes [strong product](https://en.wikipedia.org/wiki/Strong_graph_product).
Cons... | https://mathoverflow.net/users/27013 | graphs with independence number = Shannon capacity | By now, we have been able to resolve this question, and our [revised paper](http://arxiv.org/abs/1212.4084) contains a proof showing that the answer to all three questions is negative in a very strong sense. Let me provide a brief summary here and refer to the paper for more details.
What we show is this: there exist... | 5 | https://mathoverflow.net/users/27013 | 157595 | 83,319 |
https://mathoverflow.net/questions/157601 | 10 | In theoretical computer science, we classify problems according to their Turing degree. Is there any practical application of this?
**Edit:** Given that we cannot explicitly and mechanically understand the non-computable, what traces of non-computability (nonzero Turing degree) might we encounter in everyday life on ... | https://mathoverflow.net/users/nan | Is Turing degree actually useful in real life? | ### Application to everyday life
Any time you watch the "spinning beach ball" or "hour glass" on your computer, trying to decide whether it's time to reboot or just wait a little longer, you are doing something like trying to decide the Halting Problem which has Turing degree $\mathbf 0'$.
On the other hand, calcu... | 17 | https://mathoverflow.net/users/4600 | 157604 | 83,321 |
https://mathoverflow.net/questions/157603 | 12 | The $kth$ derivative of a function $f:\mathbb{R}^n \to \mathbb{R}$ can be thought of as a symmetric $k$-tensor. (Well, almost. It is not invariant under coordinate changes, and should really be thought of as part of a jet bundle. But we have nice coordinates on $\mathbb{R}^n$, so lets roll with it)
The exterior deriv... | https://mathoverflow.net/users/1106 | Are there more "types" of derivatives than "symmetric" or "alternating"? | This is a big subject. There are many, many different notions of derivative in different contexts, all of which derive in some way from jet bundles. This question was considered in great generality in the foundational works on so-called 'differential invariants' by Lie, Cartan, and their followers and was reconsidered ... | 14 | https://mathoverflow.net/users/13972 | 157607 | 83,324 |
https://mathoverflow.net/questions/157600 | 6 | If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim\_{\delta \to 0}\inf\left\{\sum\_{i=1}^\infty \left(\text{diam}(E\_i)\right)^d : \bigcup\_{i=1}^\infty E\_i \supseteq U, E\_i \text{ Borel}, \text{diam}(E\_i)<\delta\right\}
\end{equation}
We define the... | https://mathoverflow.net/users/23959 | Can Hausdorff dimension make sets into a Tropical Semiring? | Actually, the inequality for the Hausdorff dimension of product sets goes the other way
(fixed in the OP now):
$$
\dim\_{X\times Y}(U\_1\times U\_2) \ge \dim\_X(U\_1)+\dim\_Y(U\_2).
$$
And you probably need to assume something about the metric space, locally compact should do. This inequality is a consequence of F... | 8 | https://mathoverflow.net/users/11009 | 157618 | 83,329 |
https://mathoverflow.net/questions/157613 | 6 | Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of partial orders ($PO$).
>
> Is it possible to develope forcing for other first order theories? In fact I am asking f... | https://mathoverflow.net/users/nan | Forcing for Arbitrary First Order Theories | One of the most robust ways to understand forcing is via the method of Boolean ultrapowers, and this is a purely model-theoretic construction that makes sense to undertake with any first-order theory whatsoever. One may form the Boolean ultrapower of any graph, group, ring, field, partial order, and indeed of any struc... | 6 | https://mathoverflow.net/users/1946 | 157619 | 83,330 |
https://mathoverflow.net/questions/157551 | 3 | All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not twice-differentiable? In particular, is it the case that for every continuous function $f$ on $[0,1]$, the midpoint method o... | https://mathoverflow.net/users/3621 | Error of midpoint method for functions that are not twice-differentiable | If the function is monotone, then the error for the midpoint approximation with $n$ intervals is at most $\frac{f(1)-f(0)}{2n}$. Say we know $f$ is increasing and the values at the midpoints, $f(m\_i)$ for $i=1,\dots n$, as well as $f(0)$ and $f(1)$, are known. Then the $g\_L$ with the smallest integral that satisfies ... | 5 | https://mathoverflow.net/users/6649 | 157623 | 83,331 |
https://mathoverflow.net/questions/157624 | 2 | [Tennenbaums' theorem](http://arxiv.org/abs/1311.6375) proves neither addition nor multiplication can be recursive in any countable [non-standard model of arithmetic](http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic). Tennenbaum's proof applies to [theories much weaker than PA](http://www.math.cas.cz/~jera... | https://mathoverflow.net/users/26766 | Can a Decidable Theory Have Non-recursive Models? | A decidable theory can certainly have noncomputable models. For example, consider a theory with an infinite set of 0-ary relation symbols $A\_1, A\_2, \ldots$; no other relation, function, or constant symbols; and no axioms. This theory is decidable - it is basically just propositional logic - and its countable models ... | 8 | https://mathoverflow.net/users/5442 | 157627 | 83,332 |
https://mathoverflow.net/questions/157556 | 3 | The following came up in a problem on graph reconstruction. It isn't very important, but I thought some people here might find it interesting and not too trivial (I'm not a group theorist).
Take a set $\Omega\_n=\lbrace x\_1,\ldots,x\_n,y\_1,\ldots,y\_n\rbrace$ of $2n$ distinct atoms. A *partition of $\Omega\_n$ into... | https://mathoverflow.net/users/9025 | Permutation groups transitive on partitions into ordered pairs | The stabilizer of a partition in $C\_2 \wr S\_n$ is a subgroup $S\_n$ complementing the base group. So, we can attack the problem computationally by looking at the permutation action of $C\_2 \wr S\_n$ on the $2^n$ cosets of this subgroup. You are looking for its minimal transitive subgroups or, more specifically, for ... | 3 | https://mathoverflow.net/users/35840 | 157635 | 83,334 |
https://mathoverflow.net/questions/157620 | 4 | Define a full exceptional set in a triangulated category to be a partially ordered set of objects $\Delta\_i$ which generate the category and such that $\text{Ext}^\bullet(\Delta\_i, \Delta\_j) = 0$ unless $i \geq j$ and $\text{Ext}^\bullet(\Delta\_i, \Delta\_i) = k$.
Is there a known (nonequivariant) full exceptiona... | https://mathoverflow.net/users/6059 | Full exceptional set on flag variety | First, if a semisimple simply connected group $G$ acts on a variety $X$ then any exceptional object in $D^b(coh(X))$ has a $G$-equivariant structure (proved by Elagin and by Polishchuk).
Second, indeed, as abx mentioned Kapranov has constructed a full exceptional collection on all (partial) flag varieties of type $A... | 7 | https://mathoverflow.net/users/4428 | 157639 | 83,336 |
https://mathoverflow.net/questions/157647 | 2 | It seems like the sum $S(n)$ should be possible to upperbound by an expression of the form ${\mathcal O}(n^a\cdot \log^b(n))$ as $n \rightarrow \infty$:
$$
S(n)\stackrel{\triangle}{=}\sum\_{1\leq x \neq y\leq n} \frac{gcd(x,y)^2}{x y}.
$$
Any pointers, ideas, appreciated. Since $gcd(x,y) \leq \min(x,y)$ an upper bound ... | https://mathoverflow.net/users/17773 | An upper bound to sum of ratios of gcd's and products | Suppose that $x <y$. Write $x=ga$ and $y=gb$ where $g$ is the gcd of $x$ and $y$; thus $a$ and $b$ are coprime with $a<b$, and $g\le n/b$. So
$$
S(n) = 2 \sum\_{\substack{{a<b \le n} \\ {(a,b)=1}}} \frac{1}{ab} \sum\_{g\le n/b} 1 \le 2\sum\_{\substack{{a<b\le n} \\ {(a,b)=1}}} \frac{1}{ab} \frac{n}{b}.
$$
Now ignore ... | 8 | https://mathoverflow.net/users/38624 | 157660 | 83,347 |
https://mathoverflow.net/questions/157648 | 1 | Let $X$, $Y$ be Banach spaces with the same cardinality and $f: X \rightarrow Y$ be a surjection. Is it possible for $\varepsilon >0$ to find a bijection $g: X \rightarrow Y$ such that
$\|f-g\|\_{sup} < \varepsilon$?
| https://mathoverflow.net/users/46992 | Find a bijection near to a given surjection | Let $\kappa$ be the common cardinality of $X$ and $Y$. Fix well-orderings of $X$ and $Y$ of order-type $\kappa$ (i.e., each element has strictly fewer than $\kappa$ predecessors). Note that every open ball in either space also has cardinality $\kappa$ (because the space is covered by the countably many dilations, by in... | 5 | https://mathoverflow.net/users/6794 | 157663 | 83,349 |
https://mathoverflow.net/questions/157447 | 9 | It is a well-known elementary classical result in computability theory that there are computable infinite binary trees $T\subset 2^{<\omega}$ having no computable infinite branch. (One can build such a tree $T$ as follows: fix a computably inseparable pair of c.e. sets $A$ and $B$, and allow a binary string $t$ into th... | https://mathoverflow.net/users/1946 | Are there two computable binary trees such that each has a branch not computing any branch through the other? | An earlier reference for a stronger result: Jockusch and Soare gave an example of a pair of $\Pi^0\_1$ classes such that any pair of elements consisting of one set from each class forms a minimal pair in the Turing degrees. Here is the bibliographic data from MathSciNet.
```
@article {MR0282834,
AUTHOR = {Jockus... | 9 | https://mathoverflow.net/users/31026 | 157677 | 83,352 |
https://mathoverflow.net/questions/157672 | 9 | I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.
Specifically, from the general convergence rates stated in the Berry–Esseen theorem
<http://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem>
we know that, under certain conditions, the cu... | https://mathoverflow.net/users/44790 | Convergence rate of the central limit theorem near the center of the distribution | No, even in the most favorable case $(X\_i)\_{i\geqslant 0}$ iid with $\mathbb P(X\_i=1)=\mathbb P(X\_i=-1)=1/2$. Denoting $F\_n$ the cumulative distribution function of $n^{-1/2}S\_n$, we have by symmetry
$$F\_{2n}(0)=\frac 12(1+\mathbb P(S\_{2n}=0)).$$
Since $\mathbb P(S\_{2n}=0)=\binom{2n}n2^{-2n}$, denoting $\Phi$... | 14 | https://mathoverflow.net/users/17118 | 157679 | 83,353 |
https://mathoverflow.net/questions/157678 | 1 | I had been reading a couple of texts by J.P. Demailly, one of them titled "Effective bounds for very ample line bundles". In the introduction the author mentions a result due to I. Reider (stated in "Vector bundles of rank $2$ and linear systems on algebraic surfaces") which states that for a smooth projective complex ... | https://mathoverflow.net/users/46578 | A question on very ample line bundle on smooth projective surfaces | It is not stated as such in Reider's paper, but it is an easy consequence of his main theorem (thm. 1) : Reider proves that if $N$ is nef and $N^2\geq 10$, $K\_X+N$ is very ample unless there is an effective divisor $E$ on $X$ with $E.N\leq 2$ (+ some extra conditions). If $N=4L\ $ with $L$ ample, clearly such a diviso... | 2 | https://mathoverflow.net/users/40297 | 157680 | 83,354 |
https://mathoverflow.net/questions/157687 | 20 | I'm trying to motivate a bit of algebraic geometry in an abstract algebra course (while simultaneously trying to learn a bit of algebraic geometry), and I thought that it might be nice to present an example of the analogy between Riemann surfaces and algebraic integers (since one might say that the entire subject origi... | https://mathoverflow.net/users/33757 | Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$? | I think this will be a needlessly confusing example. In algebraic geometry over an algebraically closed field, there are two basic examples of nonnormal curves: the node and the cusp. Explicit equations are $y^2=x^2+x^3$ and $y^2 = x^3$. respectively.
If $X$ has a node, and $\tilde{X}$ is its normalization, then $X$ ... | 16 | https://mathoverflow.net/users/297 | 157698 | 83,361 |
https://mathoverflow.net/questions/157670 | 8 | In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.
Let's fix a representation $\bar{\rho}$ of a group $G$ and let $D\_{\bar{\rho}}$ be its deformation functor from the category $\hat{\mathcal{C}}$ of complete noetherian local $W(\mathbb{F})$-alg... | https://mathoverflow.net/users/33156 | Misunderstanding in the hypotheses of Schlessinger's criterion | I think the "mistake" is in the definition of the equivalence relation that defines a representation. Indeed, an $A$-valued representation (for some $A\in\hat{C}$) is a *conjugacy class* of homomorphisms $$
\rho:G\to \operatorname{GL}\_n(A)
$$
reducing to $\bar{\rho}$. This makes the equality
$$
D\_\bar{\rho}(A'\times\... | 9 | https://mathoverflow.net/users/18238 | 157700 | 83,363 |
https://mathoverflow.net/questions/157703 | 3 | Let $\pi:X\to\mathbb{P}^3$ be the blowing up at single point with $E$ be the exceptional divisor. Let $H=\pi^\ast\mathcal{O}\_{\mathbb{P}^3}(1)$.
In [Ample divisors on the blow up of $\mathbb{P}^3$ at points](http://www.ams.org/journals/proc/2002-130-09/S0002-9939-02-06488-2/S0002-9939-02-06488-2.pdf) , we know that ... | https://mathoverflow.net/users/39936 | Ample divisors on $\mathbb{P}^3$ blow-up along single point | Let $[x:y:z:w]$ be the homogeneous coordinates on $\mathbb{P}^3$ and suppose that the point blown up is given by the vanishing of the first three coordinates. Introduce a projective plane with coordinates $[X:Y:Z]$ then the blowup is a closed subset in $\mathbb{P}^3 \times \mathbb{P}^2$ given by the equations
$$
xY = y... | 4 | https://mathoverflow.net/users/11051 | 157712 | 83,367 |
https://mathoverflow.net/questions/157707 | 9 | Motivated by these following questions on tessellation:
[coloring in lattice](https://mathoverflow.net/questions/147374/coloring-in-lattice)
[Reference for Wang Tile](https://mathoverflow.net/questions/149565/reference-for-wang-tile)
[Computational approach deciding whether a set of Wang Tile could tile the space... | https://mathoverflow.net/users/nan | Conjecture on NP-completeness of tesselation of Wang Tile up to finite size | If one considers the anchor-tile tiling problem, where the tiling must include a specified anchor tile, then indeed this problem is NP-complete. To see this, suppose that we have a given NP problem, where there is a polynomial time computable Turing machine $M$, such that we want to know on input $x$ whether there is s... | 11 | https://mathoverflow.net/users/1946 | 157714 | 83,369 |
https://mathoverflow.net/questions/157697 | 0 | I'm a little embarassed that I can't answer this myself, so hopefully it will get answered very quickly.
Let $X$ be locally compact, Hausdorff. Consider $\text{C}\_\text{b}(X)$ the $C^\*$-algebra of bounded, complex-valued, continuous functions on it under the sup norm. The Gelfand transform takes $\text{C}\_\text{b}... | https://mathoverflow.net/users/23959 | What is the character that compactifies $\mathbb{R}$ through the Gelfand transform? | The Stone-Čech compactification of $\mathbb{R}$ is not its one-point compactification. The former is the largest compactification of a space, while the latter, if it exists, is the smallest compactification, and in general there will be many compactifications in between. $\mathbb{R}$ has, for example, a two-point compa... | 4 | https://mathoverflow.net/users/290 | 157715 | 83,370 |
https://mathoverflow.net/questions/157705 | 1 | Define Hilbert Transform (HT) as the convolution with the function $1/x$. E. Stein proves in his book
**Singular Integrals and Differentiability Properties of Functions**
that HT, when understood as a singular integral operator, is a bounded operator on $L^p(\mathbb{R})$ for $p\in (1, \infty)$.
I am wondering if HT ... | https://mathoverflow.net/users/31548 | Does Hilbert Transform commute with Function Multiplication modulo Compact on $L^p(R)$? | The problem reduces to the case of smooth functions with compact support, since they are norm dense in $C\_0$.
Now let $f$ be a smooth function with compact support. Then $[T,f]$ is an integral operator with smooth kernel $k(x,y) := (f(x)-f(y))/(x-y)$.
There is an [easy to check sufficient condition](http://www2.ma... | 1 | https://mathoverflow.net/users/22758 | 157720 | 83,371 |
https://mathoverflow.net/questions/157477 | 11 | Let $A$ be a rank 3 subgroup of the Euclidean plane, i.e. $A = \mathbb{Z} v\_1 + \mathbb{Z} v\_2 + \mathbb{Z} v\_3$, where $v\_1, v\_2, v\_3 \in \mathbb{R}^2$ are three $\mathbb{Q}$-linearly independent vectors.
Let $S^1 = \{v \in \mathbb{R}^2 ; \, |v|=1\}$ denote the unit circle in the plane.
Is it possible that $|A \... | https://mathoverflow.net/users/42355 | The intersection of a circle and a rank 3 subgroup of the plane | I am grateful to Edgardo whose answer supplied me with the Siegel theorem on integral points (I did not know it), and I am happy to see that my approach coincides with Edgardo's in many points.
The answer is negative: there is no abelian subgroup $A\subset{\mathbb R}^2$ of rank $3$ such that its intersection $I\_A:={... | 4 | https://mathoverflow.net/users/40352 | 157738 | 83,376 |
https://mathoverflow.net/questions/157740 | 5 | Let us call a bounded region $D$ in the plane maximal if the conditions $D\subset D'$ and
$\mathrm{diam} D'=\mathrm{diam} D$ imply $D'=D$.
Is it possible to describe all maximal regions?
The only examples I know are discs and Reuleaux triangles.
If a complete description is difficult, can one prove some properties of... | https://mathoverflow.net/users/25510 | Maximal regions with given diameter | I am pretty sure that the condition you state is equivalent to $D$ being of constant width, not just in the plane, but in every dimension. See, for instance <http://www.ciem.unican.es/encuentros/banach/2012/moreno1.pdf>
Other references:
Dalla, Leoni; Tamvakis, N. K. Sets of constant width and diametrically complet... | 4 | https://mathoverflow.net/users/36904 | 157745 | 83,380 |
https://mathoverflow.net/questions/157587 | 2 | I know the following is a well-known result.
Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup\_{z, w \in D} |f(z)-f(w)|$$
Furthermore, there is equality if and only if $f$ is linear.
I need some reference about the second part, i.e. there is equality if and onl... | https://mathoverflow.net/users/33122 | $ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear | This was first proved by Landau and Toeplitz in 1907. A reference for the proof (and for generalizations) is the *paper Area, capacity and diameter versions of Schwarz's lemma* by Burckel, Marshall, Minda, Poggi-Corradini and Ransford.
See Theorem 1.3 [here](http://arxiv.org/pdf/0801.3629v1.pdf)
| 3 | https://mathoverflow.net/users/1162 | 157747 | 83,381 |
https://mathoverflow.net/questions/157750 | 1 | Let $X$ be a smooth projective surface in $\mathbb{P}^n$ and $C$ be an effective curve. I know that the dualizing sheaf, $\omega\_C$ of $C$ is $\mathcal{E}xt^{n-1}\_{\mathbb{P}^n}(\mathcal{O}\_C,K\_{\mathbb{P}^n})$ where $K\_{\mathbb{P}^n}$ is the canonical sheaf on $\mathbb{P}^n$. As far as I have read (from some arti... | https://mathoverflow.net/users/46578 | On the dualizing sheaf of a curve | This of course can be found in any reference on duality (e.g. Hartshorne "Algebraic Geometry" chap. 3) but in Hartshorne's proof it's somehow mysterious where the $\mathscr{E}xt^{n-1}\_P(\mathscr{O}, \omega\_{P^n})$ comes from. You can explain it by studying relative duality for morphisms (e.g. Hartshorne "Residues and... | 6 | https://mathoverflow.net/users/3847 | 157752 | 83,383 |
https://mathoverflow.net/questions/157732 | 7 | Meta-matematical formulas of the language of set-theory (which are not sets, but just sequences of signs) should not be confused with mathematical ones (i.e. formulas coded as sets, e.g. finite sequences of natural numbers or even just natural numbers, if we wish). For instance, for each meta-mathematical formula $\phi... | https://mathoverflow.net/users/41274 | Is the forcing relation defined for mathematical formulas? | To answer Question 2, I think your intuition is right. Assume $N \prec H\_\lambda$, where $\lambda$ is a regular cardinal bigger than $2^\mathbb{P}$, where $\mathbb{P} \in N$ is a partial order. (Maybe Shelah has a more subtle argument where we assume less about $\lambda$, not sure.) Then $H\_\lambda$ is a model of $ZF... | 5 | https://mathoverflow.net/users/11145 | 157756 | 83,384 |
https://mathoverflow.net/questions/157755 | 4 | I have a question on Theorem 2.3 on page 34 of Hatcher's notes on 3-manifolds:
[Hatcher: Notes on Basic 3-Manifold Topology](http://www.math.cornell.edu/~hatcher/3M/3Mfds.pdf).
Regarding the class d), it follows from Proposition 2.1 on page 31, that $M(0,0;1/2,-1/2,\alpha/\beta)$ is fiber-diffeomorphic to $M(0,0;1/2,... | https://mathoverflow.net/users/27923 | Seifert fiberable manifolds with several Seifert fiberings | As explained on page 37 of the notes, a complete proof of the full classification of orientable Seifert manifolds (Theorem 2.2) is not given in the notes. What is missing is the classification of the manifolds that fiber over $S^2$ with exactly three multiple fibers. The statement here is that these Seifert manifolds a... | 14 | https://mathoverflow.net/users/23571 | 157766 | 83,386 |
https://mathoverflow.net/questions/157768 | 4 | Sorry if this is a completely stupid question (I'm a not a set-theorist, though I've been doing some reading in the subject), but I was wondering, specifically, about the exact provenance of the name. I know, of course, that it's after Woodin himself, but who coined it? Where is its first appearance in the literature? ... | https://mathoverflow.net/users/47047 | Origin of "Woodin cardinal" | The *notions* of Shelah cardinals and Woodin cardinals were introduced by Shelah and Woodin in their joint paper
>
> *Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable*. Israel J. Math., **70 (3)**, (1990), 381–394. [MR1074499 (92m:03087)](http://www.ams.org/mathscinet-getite... | 12 | https://mathoverflow.net/users/6085 | 157770 | 83,387 |
https://mathoverflow.net/questions/157741 | 13 | Let $D$ be a bounded simply connected region (open subset homeomorphic to the disc)
in the plane, containing the origin.
Suppose that for every line $L$ through the origin the intersection $L\cap\partial D$
consists of two points $z\_1$ and $z\_2$ such that
$|z\_1-z\_2|=\mathrm{diam} D$. Does it follow that $D$ is a ... | https://mathoverflow.net/users/25510 | Characterization of discs | Following the suggestion of Benoît Kloeckner, moderator, I have deleted my later answer, then edited and appended this (originally partial) answer to make it complete.
Pick an arbitrary direction. Then draw the line $L$ through the origin, perpendicular to the chosen direction. Since $L$ intersects the boundary of $D... | 7 | https://mathoverflow.net/users/36904 | 157780 | 83,393 |
https://mathoverflow.net/questions/157775 | 0 | Let $(H\_1 \subset G\_1)$ and $(H\_2 \subset G\_2)$ be [core-free](http://groupprops.subwiki.org/wiki/Core-free_subgroup) **maximal inclusions** of finite groups.
Their product, the inclusion $(H\_1 \times H\_2 \subset G\_1 \times G\_2)$, admits four obvious intermediate subgroups : $H\_1 \times H\_2$, $G\_1 \times ... | https://mathoverflow.net/users/34538 | Products of maximal inclusions of finite groups with a non-obvious intermediate | If I understand correctly, each $H\_i$ is maximal in $G\_i$ and core-free. In particular, unless $G\_i$ has prime order $p$, $H\_i$ is not normal in $G\_i$.
Let $X$ be a subgroup of $G\_1 \times G\_2$. For $i=1,2$ let $X^i$ be the projection of $X$ onto $G\_i$ and let $X\_i$ be the intersection of $X$ with $G\_i$. Th... | 3 | https://mathoverflow.net/users/36466 | 157785 | 83,395 |
https://mathoverflow.net/questions/157792 | 7 | Let $A$ and $B$ be two bounded symmetric positive operators in Hilbert space, such that $A-B$ is trace class. If needed, $A$ and $B$ may be assumed reasonably "small", let's say, Hilbert-Schmidt.
Does there exist another symmetric operator $C$, $0 \le C \le A$, $0 \le C \le B$, such that both $A-C$ and $B-C$ are trac... | https://mathoverflow.net/users/22758 | Is there a nice "minimum" of two symmetric operators? | Let $P=\left[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right]$ and $Q(\phi)=\left[\begin{matrix} \cos^2(\phi) & \cos(\phi)\sin(\phi) \\ \cos(\phi)\sin(\phi) & \sin^2(\phi) \end{matrix}\right]$. Then $P$ and $Q$ are orthogonal projections and if $\phi\neq 0$, then the only operator $T$ which satisfies $0 \leq T \leq P$... | 9 | https://mathoverflow.net/users/8176 | 157794 | 83,399 |
https://mathoverflow.net/questions/157806 | 1 | Can someone give me an example of a Banach space $X$ and contractive projection $P\in\mathcal{B}(X)$ such that $\ker P$ is not a range of any contractive projection $Q\in\mathcal{B}(X)$?
| https://mathoverflow.net/users/19593 | Contractively complemented subspaces without contractively complemented complement | The projection from the space $c$ of convergent sequences to the $1$-dimensional subspace of constant sequences with kernel $c\_0$. Any projection $c \to c\_0$ has norm at least $2$ (cf. exercises to chapter 2.5 in Albiac-Kalton).
| 2 | https://mathoverflow.net/users/22758 | 157809 | 83,400 |
https://mathoverflow.net/questions/157812 | -3 | Put, $C\_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim\_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\infty$) and $A:= L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C\_{0}(\mathbb R).$
>
> My Question: Given $|f|\in A$. C... | https://mathoverflow.net/users/33018 | $L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$? | No. A simple family of counter examples come from the Sobolev embeddings. Since
$$
W^{1,2}(\mathbb{R})\hookrightarrow C^{0,\frac{1}{2}}(\mathbb{R}),
$$
your identity would imply that continuous functions are always Hölder continuous.
| 3 | https://mathoverflow.net/users/40120 | 157813 | 83,402 |
https://mathoverflow.net/questions/157819 | 1 | The Stokes-Einstein rotational diffusion relation tells us that we can write down a rotational diffusion coefficient for a sphere as:
$D\_r \approx \frac{k\_B T}{\zeta\_f} \approx \frac{k\_B T}{(8 \pi \eta)(r)^3}$
Where $k\_B$ is Boltzmann's constant, $T$ is the temperature in Kelvin, $\zeta\_f \approx (8 \pi \eta)... | https://mathoverflow.net/users/47073 | Stokes-Einstein rotational diffusion and vector orientation time | If I have understood you correctly, your rotational diffusion problem is equivalent to the problem of a particle diffusing on the surface of a sphere (angular diffusion constant $D\_r$), starting at some polar angle $\theta\_0\in(0,\pi)$ and first crossing the angle $\theta\in(0,\theta\_0)$ at time $t$. This is a class... | 0 | https://mathoverflow.net/users/11260 | 157823 | 83,406 |
https://mathoverflow.net/questions/157820 | 1 | It is well-known that the centroid of a triangle is the intersection point of its three medians. The medians happen to be area bisectors, but it seems that most (all?) other lines through the centroid are not area bisectors. With other polygons there are area bisectors which pass through the centroid but not through a ... | https://mathoverflow.net/users/4362 | Extreme points and centroid | No, a line connecting a vertex to the centroid is not necessarily an area bisector. This follows easily from your observation that not every line through the centroid bisects area: take a line which goes through the centroid and is far from being an area bisector. You can deform C infinitesimally so that the intersecti... | 4 | https://mathoverflow.net/users/20186 | 157824 | 83,407 |
https://mathoverflow.net/questions/157851 | 2 | can someone point me to the direction how to calculate the derivatives of a sum of singular values of a matrix?
I am trying to minimize
$$\min\_A \parallel A \parallel\_\*+ \cdots $$ where $\parallel A \parallel\_\*=\sum\_i \sigma\_i$ is the sum of the singular values of $A$.
This is motivated by this [paper](https:/... | https://mathoverflow.net/users/47084 | derivative of sum of singular values | This function is not differentiable (consider $A=0$). If you are interested in learning about its subdifferential (and more on subdifferential of spectral functions), please refer to the excellent papers:
1. [A. Lewis. *Nonsmooth analysis of singular values: Part I*](http://link.springer.com/article/10.1007/s11228-00... | 2 | https://mathoverflow.net/users/8430 | 157854 | 83,417 |
https://mathoverflow.net/questions/157847 | 4 | I've seen some [previous questions](https://mathoverflow.net/q/44774) that show that the derivative operator on the set of smooth functions can be given by the Leibniz rule and/or chain rule and some other axioms.
Is there a similar characterization of the derivative $\mathcal{C}^1(\mathbb{R}) \to \mathcal{C}(\mathbb... | https://mathoverflow.net/users/nan | Algebraic characterization of real differentiation | Derivative on polynomials can be characterized as a linear map which satisfies
Leibniz rule, zero on constants and $1$ on the identity function.
This extends it uniquely to rational functions. Now in any space where rational functions
are dense, such an operator, if continuous, must be the derivative.
| 5 | https://mathoverflow.net/users/25510 | 157857 | 83,418 |
https://mathoverflow.net/questions/157863 | 12 | I know that the simple modules of $\mathbb{C}S\_n$ are called *Specht Modules*, and they are named after the German Mathematician [Wilhelm Specht](http://en.wikipedia.org/wiki/Wilhelm_Specht) because he studied them, but I think these modules were studied before him, for example by [Frobenius](http://en.wikipedia.org/w... | https://mathoverflow.net/users/37919 | Why are they called Specht Modules? | The question is interesting though perhaps not strictly "research-level". Terminology in mathematics develops a bit haphazardly, and sometimes things get misleading names. In this case the work of Specht around 1935 did place the representations of symmetric groups in the then-modern setting of module theory. But the n... | 18 | https://mathoverflow.net/users/4231 | 157869 | 83,421 |
https://mathoverflow.net/questions/157800 | 10 | In addition to the MO question [The Ramanujan Problems.](https://mathoverflow.net/questions/65226/the-ramanujan-problems) , I would like to ask the following.
Problem 754 from the list of the Ramanujan's problems ( <http://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/q754.htm> ) asks to show that
$$e^xx^{... | https://mathoverflow.net/users/32389 | Ramanujan's problem 754 still open? | Perhaps I should elevate my comment to an answer.
The problem is solved in Ekatherina A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comput. Appl. Math. 135 (2001), no. 2, 225–240, MR1850542 (2002i:33004).
A version of this paper is available [here](http://wis.kuleuven... | 17 | https://mathoverflow.net/users/3684 | 157870 | 83,422 |
https://mathoverflow.net/questions/157871 | 0 | I've recently come across the Frostmann Lemma (<http://en.wikipedia.org/wiki/Frostman_lemma>). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm primarily interested in $n=1$ and compact subsets).
I'm interested in finding similar results, i.e. starting with so... | https://mathoverflow.net/users/15002 | Constructing measures with support in a given set | I think the question is too broad. What are "specific properties"?
All sorts of equilibrium measures from potential theory quality.
Here is an example of a deep result: on every compact set in $R^n$ there exists a non-zero
measure satisfying the doubling condition, which means that the measure of every ball
is at mos... | 3 | https://mathoverflow.net/users/25510 | 157873 | 83,424 |
https://mathoverflow.net/questions/157793 | 4 | Let X be a compact symplectic manifold and $H\_t,J\_t$ a Floer regular pair of $\mathbb{S}^1$ dependent Hamiltonians and complex structures. The PSS maps are defined by considering $\mathbb{C}$ with a single marked point at the origin $\mathfrak{o}$ and a positive strip like end at $+\infty$. The maps satisfy Floer's e... | https://mathoverflow.net/users/36931 | Question about transversality for PSS map in Hamiltonian Floer cohomology | I recommend looking at M. Shwarz's thesis, <http://www.math.uni-leipzig.de/~schwarz/diss.pdf>, Section 4.2. A "generic" almost complex structure $J$ makes all non-constant maps regular. One first sets up a universal moduli space and shows that it is a Banach manifold. There are various ways of setting up the universal ... | 6 | https://mathoverflow.net/users/6223 | 157878 | 83,427 |
https://mathoverflow.net/questions/157880 | 2 | Is Every infinite c.e.language infinite or finite union of regular languages including at least one infinite regular language?
And is every infinite c.e.language that is not indexed language(that may generated by indexed grammar) infinite union of infinite regular languages?
Third:what class of languages may be finit... | https://mathoverflow.net/users/14024 | Every infinite c.e.language is infinite or finite union of regular languages including at least one infinite regular language? | A polynomial - time random language will not have any infinite regular subsets, so there's a counter example to the first question.
For a similar counterexample to the second question, we can increase the level of resource bounded randomness to something like an "EXPSPACE-random" language (since indexed languages ar... | 5 | https://mathoverflow.net/users/4600 | 157882 | 83,429 |
https://mathoverflow.net/questions/157795 | 6 | As is known, the set $\{1,\ldots,n\}$ has $2^n$ many subsets and $B\_n$ (the $n$th Bell number) many partitions, where clearly $B\_n<2^{2^n}$ and it is actually known that $B\_n<n^n$ for large $n$.
A napkin calculation suggests that $\{1,\ldots,n\}$ has about $n^2\log n$ many subsets that form arithmetic progression... | https://mathoverflow.net/users/4600 | Number of partitions whose blocks form arithmetic progressions | I will prove that
$$
A\_n = \Big(\frac{n}{e}\Big)^{n/2} \exp(O(\sqrt{n})).
$$
With more effort, one could probably even get an asymptotic formula. Moreover the argument suggests that
a typical such set partition consists of about $C\_1\sqrt{n}$ singleton sets, about $C\_2\sqrt{n}$ sets of size $3$, a bounded ... | 11 | https://mathoverflow.net/users/38624 | 157883 | 83,430 |
https://mathoverflow.net/questions/157833 | 0 | Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. We can define the algebraic entropy of $G$ by taking infimum of entropy of $G$ with respect to a generating set $S$. On the ... | https://mathoverflow.net/users/3922 | Entropy of Negatively pinched manifolds | There are closed hyperbolic 3-manifolds with arbitrarily large algebraic entropy. However, the critical exponent will always be $2$. So there isn't a relation, although there could possibly be an inequality.
**Addendum:** Sorry, I should have given justification for the statement above. The point is that for any fre... | 2 | https://mathoverflow.net/users/1345 | 157889 | 83,431 |
https://mathoverflow.net/questions/154913 | 82 | The [following identity on MathSE](https://math.stackexchange.com/questions/464769/how-to-prove-int-01-tan-1-left-frac-tanh-1x-tan-1x-pi-tanh-1)
$$\int\_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$
seems to be very ... | https://mathoverflow.net/users/18286 | A hard integral identity on MathSE | I have proved this equality by means of Cauchy’s Theorem
applied to an adequate function. Since my solution is too long to post it
here, I posted it in arXiv:
* Juan Arias de Reyna, *Computation of a Definite Integral*, arXiv:[1402.3830](http://arxiv.org/abs/1402.3830).
The function
$$G(z)=\frac{\log(1+(1+i)\,f(z)\... | 90 | https://mathoverflow.net/users/7402 | 157899 | 83,434 |
https://mathoverflow.net/questions/157897 | 5 | Five years ago, I made a [conjecture](http://list.seqfan.eu/pipermail/seqfan/2009-January/000524.html) on the number of correlation classes that are exhibited by pairs of words in an alphabet of a given size. I later [speculated](http://maths-people.anu.edu.au/~leopardi/ACCMCC2010-Leopardi-words-strings-talk.pdf) that ... | https://mathoverflow.net/users/12911 | Which automated theorem provers can address the combinatorics of periods in strings? | I think that
* Any of the standard theorem provers could formulate your statement without much trouble. There would be various choices about what kind of data structures to use (implicitly or explicitly), and the right choice might make a big difference to the ease of writing proofs.
* I had a quick look at the pape... | 3 | https://mathoverflow.net/users/10366 | 157908 | 83,438 |
https://mathoverflow.net/questions/157912 | 5 | I hope this is not too elementary!
Let $G$ be a algebraic reductive group over $\mathbb{C}$.
The group $G(\mathbb{C}[[t]])$ can be given the structure of a pro algebraic group as follows.
Let $l\in \mathbb{N}$ and $J^l:=\mathbb{C}[[t]] /t^l\mathbb{C}[[t]])$. Then $G(J^l)$ is known to be an algebraic group and $G(\m... | https://mathoverflow.net/users/32972 | About the pro-algebraic group structure of $G(\mathbb{C}[[t]])$ | As hinted by S. Carnahan, this is a particular case of a general construction, known as Weil restriction, Greenberg functor, or arc space, depending on the context.
Let $X$ be a scheme of finite type over a field $k$. Then there is a scheme $\mathcal L(X)$, which is a projective limit of schemes $\mathcal L\_m(X)$ of... | 13 | https://mathoverflow.net/users/10696 | 157918 | 83,441 |
https://mathoverflow.net/questions/157913 | 7 | I believe correctness about clique in even-hole-free graphs
of [graphclasses.org](http://graphclasses.org/classes/gc_547.html)
and the paper [Vertex elimination orderings for hereditary graph classes, Pierre Aboulker, Pierre Charbit, Nicolas Trotignon, Kristina Vuskovic](http://arxiv.org/abs/1205.2535) would imply $P=N... | https://mathoverflow.net/users/12481 | Seeming contradiction about P vs NP between graphclasses.org and at least two papers about clique in even-hole-free graphs | On <http://www.graphclasses.org>, an even hole means at least 6 vertices. If you open the details of the forbidden subgraphs list of the page for even-hole-free graphs and click on the link for even-hole, you'll see the definition.
In the articles you refer to, an even hole means at least 4 vertices. The even-hole-fr... | 14 | https://mathoverflow.net/users/47118 | 157919 | 83,442 |
https://mathoverflow.net/questions/157902 | 1 | I'm sure the answer to my question is well-known -- I'm mostly looking for a reference.
Suppose I have a nonsingular variety $X$ which fibers over $\mathbb{A}^1$. Moreover, suppose I have a stable map $C\_0 \stackrel{f\_0}{\to}X\_0$, where $X\_0$ is the fiber of $X$ over $0 \in \mathbb{A}^1$. When does this map defor... | https://mathoverflow.net/users/47095 | When does a stable map to a special fiber (locally) deform to a family of stable maps? | Asking for a deformation over $\mathbb{A}^1$ is quite restrictive. Even asking for formal deformations / deformations over an étale cover of $\mathbb{A}^1$ is nontrivial. The "standard" obstruction group for deforming a stable map is the hyper-Ext group $\mathbf{R}Hom^2\_{\mathcal{O}\_{C\_0}}(L^\bullet\_{f\_0},\mathcal... | 3 | https://mathoverflow.net/users/13265 | 157927 | 83,447 |
https://mathoverflow.net/questions/157758 | 0 | In all questions suppose $G$ metabelian p-group such that
* G is not regular ( so $cl(G) \geq p$ ), G is not a wreath product;
* $Z(G) \leq \phi(G)$.
1) Let $M$ normal abelian subgroup of $G$ such that $\frac{G}{M} \cong C\_{p^{n}}$ with $n \geq 2$. So it exists an element $g \in G - M$ such that $G=M\langle g\ran... | https://mathoverflow.net/users/40128 | Centralizer of derived subgroup | To summarize the material in the comments so that the question does not appear as unanswered...
(1) The first question has a negative answer as stated. For example, take
$$G = \langle x,y\mid x^8=y^8=[x,y]^2=[x,y,x]=[x,y,y]=1\rangle$$
a $2$-group of order 128 and class $2$. Take $M=\langle [x,y],x\rangle$, so that $... | 1 | https://mathoverflow.net/users/3959 | 157943 | 83,452 |
https://mathoverflow.net/questions/157942 | 4 | Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. Varopoulos proved that $\mu^{2n}(e) = O(n^{- \frac{d}{2}})$ and $n^{- \frac{d}{2}} = O(\mu^{2n}(e)) $ (we write $\mu^{2n}(e) \si... | https://mathoverflow.net/users/47125 | Connection between degree of growth and return probabilities of random walks on Lie groups | Every locally compact group of polynomial growth is QI to a simply connected nilpotent Lie group [Losert: *On the structure of groups with polynomial growth, Math. Z. 195(1) (1987) 109-117*], and probability of return is stable under quasi-isometries [Tessera *Large scale Sobolev inequalities on metric measure spaces a... | 3 | https://mathoverflow.net/users/14094 | 157946 | 83,453 |
https://mathoverflow.net/questions/157404 | 2 | Does any one know where one can find a reference about the following fact?
Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Fix two flat bundles $(L\_i,\nabla\_i)$ over $X$, (vector bundles with integrable connections) for $i=1,2$.
Define a new flat bundle $(M,\nabla\_M)$ with $M= L\_2... | https://mathoverflow.net/users/4504 | reference of extension of flat bundle | You are looking at extensions of $L\_2$ by $L\_1$ as $\mathcal{D}\_X$-modules, which are classified by $\mathrm{Ext}^1\_{\mathcal{D}\_X}(L\_2,L\_1)\cong \mathrm{Ext}^1\_{\mathcal{D}\_X}(\mathcal{O}\_X,M)$. Now it is well-known in the theory of $\mathcal{D}$-modules that there is a canonical isomorphism of $R\mathcal{H}... | 2 | https://mathoverflow.net/users/40297 | 157949 | 83,456 |
https://mathoverflow.net/questions/157938 | 33 | This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following situation: find a short exact sequence
$$ 0 \to X\_1 \to X\_2 \to X\_3 \to 0$$
(in some category of your choice), and a sec... | https://mathoverflow.net/users/37021 | (Short) Exact sequences with no commutative diagram between them | $\def\ZZ{\mathbb{Z}}$
This can happen in finitely generated abelian groups. Let $p$ be prime, set $G = \ZZ/p^2 \oplus \ZZ/p$ and set $H = \ZZ/p^3 \oplus \ZZ/p^2 \oplus \ZZ/p$. Then there are two non-isomorphic short exact sequences $0 \to G \to H \to G \to 0$. The first one is the sum of the extensions:
$$\begin{matri... | 43 | https://mathoverflow.net/users/297 | 157955 | 83,458 |
https://mathoverflow.net/questions/157868 | 8 | I'm studying the classic results on binary (integer) quadratic forms and I'm looking for a reference on the following result (maybe a book that contains a proof):
Let $O\_k$ be the ring of algebraic integers of $Q(\sqrt{d})$. In the set of ideals of $O\_k$, we define the equivalence relation $I \sim J \Leftrightarrow... | https://mathoverflow.net/users/37392 | Connection between quadratic forms and ideal class group | There is a concise account in the Appendix of these [notes](http://www.renyi.hu/~gharcos/heegner.pdf).
| 6 | https://mathoverflow.net/users/11919 | 157957 | 83,459 |
https://mathoverflow.net/questions/104138 | 5 | This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book *Elementary Induction on Abstract Structures.*
I am interested in Spector classes on $\mathbb{R}$, which are just a type of boldface pointclass. The smallest one is IND, the class of (boldface) inductive sets, which I... | https://mathoverflow.net/users/1682 | Companion of the pointclass of inductive sets | The relation $R$ can be empty; *i.e.*, if $\kappa$ is the least ordinal such that $L\_\kappa(\mathbb{R})$ is admissible, then the structure $(L\_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets. We can prove a somewhat more general statement.
Assume that $\kappa$ is an ordina... | 4 | https://mathoverflow.net/users/1682 | 157959 | 83,460 |
https://mathoverflow.net/questions/157903 | 8 | In
``Steel, John R. On Vaught's conjecture. Cabal Seminar 76–77, pp. 193–208''
the following is proved:
**Theorem.** Let $\phi\in L\_{\omega\_1,\omega}.$ If every model of $\phi$ is a tree, then $\phi$ has either $\leq \aleph\_0$ models or perfectly many countable models.
In page 206 of the paper the following is s... | https://mathoverflow.net/users/11115 | Vaught's conjecture for partial orders |
>
> This answer is an elaboration of the comment by Emil Jeřábek.
>
>
>
The reduction of Vaught's conjecture to the special case of partial orders is an immediate consequence of the fact that every structure in a finite language $L$ that has at least two elements is bi-interpretable with a special type of partia... | 8 | https://mathoverflow.net/users/9269 | 157961 | 83,461 |
https://mathoverflow.net/questions/157967 | 7 | In $\textit{Set Theory}$ by Jech 1978 edition, in the proof of Lemma 32.5 which you can hopefully see at the [Google book link](http://books.google.com/books?id=pLxq0myANiEC&pg=PA389).
In the course of the proof using the tree property, he produces from any weakly compact cardinal $\kappa$ a non principal $L\_\alpha$... | https://mathoverflow.net/users/43354 | Weakly Compact Cardinal and Iterability | This is a very nice question that reveals a subtle point about iterating ultrapowers. Namely, the issue is that in order to iterate the ultrapower construction, one needs the ultrafilter $U$ to be *weakly amenable* to the structure, which means that the structure has $\{\alpha\lt\kappa\mid X\_\alpha\in U\}$, whenever i... | 8 | https://mathoverflow.net/users/1946 | 157970 | 83,463 |
https://mathoverflow.net/questions/157973 | 16 | We know that there is an equivalence of categories between the two following categories:
$1)$ Classical varieties over $k$, where $k$ is an algebraically closed field. (Informally I mean locally ringed spaces formed by patching affine irriducible algebraic sets over $k$. This is the definition of algebraic variety pr... | https://mathoverflow.net/users/47136 | Classical algebraic varieties VS $k$-schemes VS schemes | Let's say $k=\mathbb{C}$ (although something like this should work over any algebraically closed field). Let $V\_1=\mathbb{P}^1-\{0,1,\infty,\pi\}$ and $V\_2 = \mathbb{P}^1-\{0,1,\infty,e\}$. One can see that $V\_1$ and $V\_2$ are not isomorphic as varieties/schemes over $\mathbb{C}$: such an isomorphism would extend t... | 33 | https://mathoverflow.net/users/5263 | 157979 | 83,466 |
https://mathoverflow.net/questions/156118 | 6 | (It is possible that an answer to this question can be found in the literature, but I couldn't find anything after searching for about an hour.)
Let $G$ be a compact, totally disconnected, second countable group. (Equivalently, a profinite group: it is the inverse limit of finite groups.) It makes sense to talk about... | https://mathoverflow.net/users/29566 | Uniqueness of composition series for profinite groups | Yes, the composition factors are unique up to permutation, and this can be derived from the Jordan-Hölder theorem for finite groups.
If $G$ is second countable, it has a composition series: one can obtain this by starting with some countable basis of identity neighbourhoods (which may be taken to consist of open norm... | 2 | https://mathoverflow.net/users/4053 | 158004 | 83,480 |
https://mathoverflow.net/questions/157783 | 6 | [Exact low rank matrix completion](http://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf) using nuclear norm minimization can be formulated as a semidefinite program (SDP). Following the notation in the paper, a convex problem for noisy matrix completion can be:
$$\text{minimize} \,\, \|X\|\_\* \quad \text{... | https://mathoverflow.net/users/46599 | SDP formulation of noisy low-rank matrix completion | The Frobenius norm does not cause a problem. Remember, the Frobenius norm of a matrix $X$ is actually nothing more than the 2-norm of the vector formed by stacking the columns of $X$ on top of each other. And for vectors, the vector 2-norm and the matrix 2-norm coincide. So if $\mathcal{Q}$ implements this "stacking" i... | 8 | https://mathoverflow.net/users/22079 | 158005 | 83,481 |
https://mathoverflow.net/questions/158000 | -1 | Consider a sequence of open sets in $R^n$: $\Omega\_1 \supset \Omega\_2 \supset\cdots$. Consider that this sets are bounded, convex with the boundary piecewise smooth .When i say smooth i mean $C^{\infty}$
I believe that the set $\operatorname{int}\left(\,\overline{\bigcap \Omega\_i}\,\right)$ have the boundary piece... | https://mathoverflow.net/users/47144 | The boundary of this set is piecewise smooth? | Since you can assume that $\Omega\_{n+1}$ is strictly contained in the interior of $\Omega\_n$ for all $n$, there is no reason to assume that the boundary of the limit has any interesting properties. Perhaps you are missing an hypothesis.
In any case, perhaps the following construction will give you food for thought.... | 1 | https://mathoverflow.net/users/47147 | 158008 | 83,482 |
https://mathoverflow.net/questions/158006 | 2 | Is the Ray-Singer analytic torsion for an arbitrary compact 3-manifold with finite Abelian fundamental group equivalent to the Ray-Singer analytic torsion of S^3 mod some direct product of Z\_N's? It seems like Thurston's elliptization conjecture implies the answer is yes, but my understanding in this area is very limi... | https://mathoverflow.net/users/47148 | Ray-Singer torsion of compact 3-manifolds with finite abelian fundamental group | A good reference for elliptic 3-manifolds is Thurston's book, "Three Dimensional Geometry and Topology, Vol 1." Theorem 4.4.14 classifies the possible fundamental groups of elliptic 3-manifolds. In particular, if the group is Abelian then the manifold is a lens space. To be explicit, this means the manifold is homeomor... | 4 | https://mathoverflow.net/users/27453 | 158009 | 83,483 |
https://mathoverflow.net/questions/157896 | 5 | Let $X$ be an elliptic K3 surface with $D\_{14}$ singular fiber. Do you know an explicit equation for such $X$? Also, how many disjoint sections such fibration admits? Any reference would be greatly appreciated.
| https://mathoverflow.net/users/40968 | K3 surface with $D_{14}$ singular fiber | [In comments **guest2014** amended the question to ask not for a
$D\_{14}$ fiber but for $I^\*\_{14}$, a.k.a. $\tilde D\_{18}$]
The elliptic surface
$$
X : y^2 = x^3 + (t^3+2t) x^2 - 2(t^2+1)x + t
$$
over ${\bf C}(t)$ has a $I^\*\_{14}$ fiber at $t=\infty$.
(Note that the right-hand side is a cubic in $x$ whose disc... | 6 | https://mathoverflow.net/users/14830 | 158010 | 83,484 |
https://mathoverflow.net/questions/158025 | 3 | Let $X\_1, X\_2$ be two smooth projective connected schemes of the same dimension in $\mathbb{P}^n$ such that $\dim(X\_1)=\dim(X\_2)\le n-2$. Assume futher that $X\_1 \cap X\_2$ is smooth. Is it possible that there exists a smooth projective hypersurface in $\mathbb{P}^n$ containing both $X\_1$ and $X\_2$? If not possi... | https://mathoverflow.net/users/46578 | Embedding a projective scheme is a smooth hypersurface | This is certainly not possible in general, since $X\_1$ itself cannot (in general) be embedded in a smooth hypersurface: for instance $\mathbb{P}^1\times \mathbb{P}^2$, embedded in $\mathbb{P}^5$ by the Segre embedding, is not contained in any smooth hypersurface, because then by Lefschetz theorem it would be a complet... | 5 | https://mathoverflow.net/users/40297 | 158026 | 83,488 |
https://mathoverflow.net/questions/158024 | 9 | We denote by $\mathcal{S}(\mathbb{R})$ the space of smooth and rapidly decreasing functions. We define on $\mathcal{S}(\mathbb{R})$ the family of semi-norms
$$\lVert \varphi \lVert\_{n,m} = \lVert (1+|\cdot|^m) \varphi^{(n)} \lVert\_\infty.$$
This family of semi-norms defines a topology $\tau$ on $\mathcal{S}(\mathbb{... | https://mathoverflow.net/users/39261 | Is the space of rapidly decreasing (non-smooth) functions nuclear? | Your space contains an isomorphic copy of $L^\infty([0,1])$ (consider the family of elements with support in the interval) and so is not nuclear.
By the way an explanation of the relationship between smoothness and nuclearity is that many spaces of test functions are generated in a natural way by differential
operat... | 12 | https://mathoverflow.net/users/45681 | 158027 | 83,489 |
https://mathoverflow.net/questions/158022 | 11 | The question is from the paper <http://arxiv.org/abs/1312.7115> (A curious formula related to the Euler Gamma function, by Bakir Farhi): is it possible to express the integral
$$\eta=2\int\limits\_0^1 \ln{(\Gamma(x))}\cdot \sin{(2\pi x)}\,dx= 0.7687478924\ldots$$
in terms of the known mathematical constants as $\pi,\,e... | https://mathoverflow.net/users/32389 | An integral related to the Euler gamma function | Let's introduce a notation
$$\alpha\_k := \intop\_0^1 \sin(2 \pi k z) \log \Gamma(z) dz$$
Let me also remind of the duplication formula:
$$\log \Gamma(2z) = \log \Gamma(z) + \log \Gamma(z + 1/2) + 2\log 2 \cdot z - \log(2 \sqrt \pi)$$
Now apply that to the calculation of $\alpha\_k$:
$$\alpha\_k = 2 \intop\_0... | 16 | https://mathoverflow.net/users/22758 | 158031 | 83,491 |
https://mathoverflow.net/questions/158029 | 5 | Let $F$ be a local field. Is there a reference for the following fact:
>
> No supercuspidal representation of $GL\_2(F)$ has an Iwahori-fixed vector?
>
>
>
I have a proof, by I'd prefer a reference, because it is not enlightening.
Rough sketch of proof: We can easily see that Iwahori-fixed vector implies dep... | https://mathoverflow.net/users/10400 | Supercuspidal with Iwahori fixed vector | I quote from one of my papers (On Bernstein's presentation of Iwahori-Hecke algebras and representations of split reductive groups over non-Archimedean local fields, Bulletin of the Kerala Mathematics Association, Special issue on Harmonic Analysis and Quantum Groups, December 2005, also available from <http://arxiv.or... | 4 | https://mathoverflow.net/users/9672 | 158035 | 83,492 |
https://mathoverflow.net/questions/157985 | 2 | I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $\delta\_{x\_0} f$ of a Lipschitz map $f$, due to F.H.Clarke in [On the inverse function theorem](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-64/issue-1/On-the-inverse-function-the... | https://mathoverflow.net/users/45138 | Radius of the ball where the inverse of Lipschitz maps exists | I have just found quite a good answer in [Marco Papi, On the Domain of the Implicit Function and Applications, Journal of Inequalities and Applications, Vol. 3, 2005. ISSN: 1025-5834.](http://downloads.hindawi.com/journals/jia/2005/373250.pdf?origin=publication_detail) It states and proves only the implicit function th... | 1 | https://mathoverflow.net/users/45138 | 158042 | 83,494 |
https://mathoverflow.net/questions/157717 | 4 | My question is pretty much as in the title. On page 100 of his thesis, Bourgeois gives a computation of the CZ(or I suppose more correctly this is the Robbin-Salamon index) index of the Reeb orbits and their multiple covers on a prequantization space.
<http://tel.archives-ouvertes.fr/docs/00/04/51/90/PDF/tel-00002421... | https://mathoverflow.net/users/6986 | How to compute Conley-Zehnder indices on prequantization spaces? | It seems to me that the computation in EGH or in Bourgeois's thesis only need the axioms that Oancea lists at the top of page 19. A pair of more recent references I like for a discussion of the Conley-Zehnder index are two papers by Jean Gutt:
* <http://arxiv.org/abs/1201.3728>
* <http://arxiv.org/abs/1307.7239>
Yo... | 3 | https://mathoverflow.net/users/477 | 158048 | 83,495 |
https://mathoverflow.net/questions/157958 | 5 | In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet spaces fails: The derivative of this map at the zero vector field is the identity, yet the map is not surjective in any n... | https://mathoverflow.net/users/16702 | Exponential mapping versus flow | You could definitely define a smooth structure on $\text{Diff}(M)$ in the vicinity of the identity by declaring your maps $X \mapsto \phi\_X$ or $X \mapsto e^{tX}|\_{t=1}$ to be diffeomorphisms onto. However, the topology induced by this procedure is not the natural topology which one would expect, i.e. the subspace to... | 6 | https://mathoverflow.net/users/17047 | 158056 | 83,496 |
https://mathoverflow.net/questions/158011 | 0 | Consider two smooth functions $f,g\in C^\infty(\Omega)$ with $\partial \Omega$ smooth and $\Omega\subset \mathbb{R}^3$. Assume that $f=g$ on $\partial \Omega$.
For any given $\varepsilon>0$, how to perturb $f$ to another smooth function $\tilde{f}$ such that $\|f-\tilde{f}\|\_{L^2}<\varepsilon$ and, if $g(x)\neq g(y)$... | https://mathoverflow.net/users/16881 | How to perturb a function to separate points | Okay, here's a counterexample. Let $\Omega \subset {\bf R}^3$ be the ball of radius 2 about the origin and observe that $\Omega$ contains the unit cube $[0,1]^3$. Define $f(x,y,z) = x$ and $g(x,y,z) = y$ on the unit cube and extend them to smooth functions on $\Omega$ which agree on $\partial \Omega$.
Now let $\tilde... | 3 | https://mathoverflow.net/users/23141 | 158057 | 83,497 |
https://mathoverflow.net/questions/158050 | 10 | I am arranging a weekly meeting of 2 hours with postgraduate students in ergodic theory (for a period of 3 weeks).
I am asking here for an advice of a book (or maybe a set of papers) to look at during our reading meetings. We would like to discuss some topic of current research (say not older than 5 years), but we do ... | https://mathoverflow.net/users/39115 | Ergodic theory and dynamical systems books references | For a beautiful overview, focusing on entropy and the variational principle, you can't beat Walters' book. I've given reading courses from it, and it is very well written and excellent for self-study.
The Einsiedler-Ward book (the first of a projected three volumes, parts of the second and third books are viewable a... | 8 | https://mathoverflow.net/users/8112 | 158061 | 83,498 |
https://mathoverflow.net/questions/158055 | 7 | Unlike algebraic K-theory, equivariant K-theory of affine space (over a field $k$) can be quite nontrivial, depending on the action of the group in question. For example, if one takes the standard action of $\mathbb G\_m$ on $\mathbb A^n$, then the $\mathbb G\_m$-equivariant K-theory of $\mathbb A^n$ can be computed us... | https://mathoverflow.net/users/42656 | Equivariant algebraic K-theory of affine space | In fact, just like in the case of non-equivariant algebraic $K$-theory, there is a homotopy-invariance for equivariant $K$-theory (of nonsingular varieties). Addressing your question specifically, for any action of an algebraic group $G$ on affine space, the answer is:
$$ K^G\_i({\Bbb A}^n) = R(G) \otimes K\_i(k), $$
w... | 7 | https://mathoverflow.net/users/5081 | 158062 | 83,499 |
https://mathoverflow.net/questions/157626 | 5 | The narrow Denjoy integral (which also goes by the names Henstock-Kurzweil integral, Perron integral, and Lusin integral) is a transfinite integration process defined by Denjoy in the early 20th century to generalize Lebesgue integration.
In Gordon's 1994 book, "The Integrals of Lebesgue, Denjoy, Perron and Henstock... | https://mathoverflow.net/users/6649 | Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$? | This is not a topic I know much about, but I do have quite a few references pertaining to real analysis and classical point set theory at home, and I looked through these this morning. For many decades the standard reference was Saks' book **Theory of the Integral**, so I looked there first. I'm fairly certain that eve... | 4 | https://mathoverflow.net/users/15780 | 158070 | 83,501 |
https://mathoverflow.net/questions/158058 | 10 | I would like to better understand the relationship between different notions of orientable sphere bundle. Let me say that a locally trivial fiber bundle $\pi\colon E\to M$ with fiber $S^n$ and structure group $G$ is a
topological sphere bundle if $G=\mathrm{Homeo}^+(S^n)$,
smooth sphere bundle if $G=\mathrm{Diffeo... | https://mathoverflow.net/users/6206 | Two questions about sphere bundles | Recall that linear, smooth, and topological $S^k$-bundles over a finite complex $X$ are classified by the sets of homotopy classes $[X, BO\_{k+1}]$, $[X, B\mathrm{Diff}(S^k)]$ and $[X, B\mathrm{Homeo}(S^k)]$. There are obvious maps between the sets, and we are interested in their cokernels.
For example, if $X=S^1$, t... | 10 | https://mathoverflow.net/users/1573 | 158072 | 83,502 |
https://mathoverflow.net/questions/158076 | 1 | On p. 205 of Katz's paper entitled "p-adic L-functions for CM fields" Katz says that
"Shimura's algebraicity theorem, in our context, is an easy consequence of the fact that Hodge decomposition of the $H\_{dR}^1$ of our CM abelian varieties is
purely algebraic (i.e. $H^{0,1}$ is defined over $\overline{\mathbb{Q}}$)... | https://mathoverflow.net/users/11765 | Algebraic Hodge decomposition of CM abelian varieties | Suppose, to simplify, that $A$ is defined over $\mathbb{Q}$. Then $H^1(A\_{\mathbb{C}},\mathbb{C})$ has two $\mathbb{Q}$-structures, one coming from singular cohomology, the other one from the algebraic de Rham cohomology; they are different. For instance, for an elliptic curve $A=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau ... | 6 | https://mathoverflow.net/users/40297 | 158081 | 83,504 |
https://mathoverflow.net/questions/157935 | 1 | Consider the usual model structure on Cat (category of small categories).
Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?
Actually, I wonder how can I compute homotopy limits of cosimplicial categories (considering them as diagrams in Cat)...
... | https://mathoverflow.net/users/18017 | Homotopy limit of a cosimplicial category | I will answer the title question about computing homotopy limits. Let $A : \mathcal{C} \to \mathbf{Cat}$ be a small (strict) diagram. By expanding the definitions of the cobar construction, one eventually discovers that $\operatorname{holim} A$ can be computed as the end
$$\int\_{[n] : \mathbf{\Delta}} \left[ \mathbf{I... | 4 | https://mathoverflow.net/users/11640 | 158092 | 83,506 |
https://mathoverflow.net/questions/157886 | 3 | Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$?
(Possibly by perturbing a rotation in the real-analytic topology?)
| https://mathoverflow.net/users/42388 | Real analytic ergodic diffeomorphisms of the two sphere | In $S^2$ it is possible to construct such an example by a quite different method (not as a limit of rotations) which is indeed Bernoulli with respect to Lebesgue.
The idea is to quotient the cat map of the torus by $\pm Id$ to get a homeomorphism of the sphere which is Bernoulli and then notice that the map is "stab... | 2 | https://mathoverflow.net/users/5753 | 158094 | 83,508 |
https://mathoverflow.net/questions/158090 | 0 | Suppose $A$ and $B$ are two algebras of the same signature, both having maximum condition on sub-algebras. Is it true that $A\times B$ has the same property?
| https://mathoverflow.net/users/44949 | Product of two algebras with maximum condition | Here is a semigroup example.
Take the semigroup $\mathbb Z\_+$ of positive integers. Each of its subsemigroups is finitely generated so it has the maximum condition on subsemigroups. But $\mathbb Z\_+\times \mathbb Z\_+$ is not finitely generated so it does not have the maximum condition.
Indeed, any element of the... | 1 | https://mathoverflow.net/users/15934 | 158095 | 83,509 |
https://mathoverflow.net/questions/158046 | 2 | This seemingly simple question stands unanswered on math.stackexchange.com for a couple of days (<https://math.stackexchange.com/questions/680211/a-question-about-the-universal-coefficient-theorem>), so I decided to repost it here, although it is hardly research related.
Let $X$ be some topological space, $R$ be a (u... | https://mathoverflow.net/users/19864 | A question about the universal coefficients theorems | Well, I seem to have sorted it all out, might as well write it all up.
**Statement.** For arbitrary abelian groups $A$, $B$ and $C$, a PID $R$ and an $R$-module G one has an isomorphism of $R$-modules $$\mathrm{Hom}\_\mathbb{Z}(A,G)\oplus\mathrm{Ext}\_\mathbb{Z}(B,G)=\\\mathrm{Hom}\_R(A\otimes R,G)\oplus\mathrm{Hom}\... | 1 | https://mathoverflow.net/users/19864 | 158101 | 83,512 |
https://mathoverflow.net/questions/158071 | 13 | I found 8 of them and believe there is no more:
$$2+3^2=3+2^3$$
$$2+6^2=6+2^5$$
$$6+15^2=15+6^3$$
$$3+16^2=16+3^5$$
$$3+13^3=13+3^7$$
$$2+91^2=91+2^{13}$$
$$5+280^2=280+5^7$$
$$30+4930^2=4930+30^5$$
I call the solution a principal pipe of order 2. Any idea to attack this equation?
If $n=x=2$, it becomes $2+y^2=y+... | https://mathoverflow.net/users/47184 | Integer Solutions of $x+y^n = y + x^m$ for $n < m$ | This problem was considered in a paper of Mignotte and Petho {Publ. Math. Debrecen 1999) and subsequently in one of mine [Canad. Math. J 2001], where there is a conjecture that the equation $a^x-b^y=c$ has, for fixed positive integers $a, b$ and $c$, with $a, b \geq 2$, at most one solution in positive integer exponent... | 11 | https://mathoverflow.net/users/7302 | 158110 | 83,516 |
https://mathoverflow.net/questions/158120 | 1 | In General linear group $GL(n,q)$, every matrix is conjugate with its rational canonical form. Using this fact we have the conjugacy classes of cyclic groups. My question is about the groups which are generated by two elements. I tried to find some results about conjugacy classes but it seems to be hard, any comment is... | https://mathoverflow.net/users/33209 | subgroups of General linear group with two generators | You would have an answer to your question if you could classify pairs of elements in $GL\_n(\mathbf F\_q)$ up to simultaneous conjugacy. This is the notorious matrix pair problem, which is the quintessential wild problem in representation theory (see [When is a classification problem "wild"?](https://mathoverflow.net/q... | 4 | https://mathoverflow.net/users/9672 | 158128 | 83,521 |
https://mathoverflow.net/questions/47672 | 45 | It is relatively easy to show (see below) that if we have two equilateral triangles of side 1 in $\mathbb R^3$,
such that their union has diameter $1,$ then they must share a vertex.
I wonder whether we have an analog of this in higher dimensions. To start with $4$ dimensions, the question is whether the following stat... | https://mathoverflow.net/users/5572 | two tetrahedra in $\mathbb R^4$ | One can read the proof of the conjecture here: <http://arxiv.org/abs/1402.3694>
Also in this paper there is the proof of Schur's conjecture. This conjecture states that any diameter graph in $\mathbb R^d$ on $n$ vertices may have at most $n$ cliques of size $d$.
| 12 | https://mathoverflow.net/users/40213 | 158129 | 83,522 |
https://mathoverflow.net/questions/133453 | -1 | Suppose we have a typical logdet function $\mathcal{L}$
$$
\mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q},
$$
where $\mathbf{S}$ is a Symmetric Positive Semi-Definite Matrix, $\mathbf{q}$ is a column vector, $\mathbf{I}$ is an identity matri... | https://mathoverflow.net/users/11273 | Regularized Gradient with respect to a matrix (with a specific structure) | @liubenyuan, some remarks. Necessarily $S>0$. On the other hand, your result is correct up a signum (have a look at the case $n=1$); infortunately that makes a big difference ! The derivative is $0$ if $A=S^{-1}(qq^T-S)S^{-1}$ and you want that $A\geq 0$, that is $qq^T\geq S$. Let $(\lambda\_i\geq 0)\_i$ be the spectru... | 0 | https://mathoverflow.net/users/9091 | 158134 | 83,523 |
https://mathoverflow.net/questions/135335 | 9 | Let $(N\_1 \subset M\_1)$ and $(N\_2 \subset M\_2)$ be two **[maximal](https://mathoverflow.net/questions/133651/is-there-a-maximal-finite-depth-infinite-index-irreducible-subfactor)** subfactors.
Their tensor product, the subfactor $(N\_1 \otimes N\_2 \subset M\_1 \otimes M\_2)$, admits four obvious intermediate su... | https://mathoverflow.net/users/34538 | What are the intermediate subfactors of the tensor product of two maximal subfactors? | This answer came after a discussion with [Feng Xu](http://www.math.ucr.edu/home/Faculty/Xu.htm). The following more general result is true:
*Theorem*:
Let $(N\_i \subset M\_i)$, $i=1,2$, be irreducible finite index subfactors. Then $$ \mathcal{L}(N\_1 \subset M\_1) \times \mathcal{L}(N\_2 \subset M\_2) \subsetneq \m... | 3 | https://mathoverflow.net/users/34538 | 158139 | 83,524 |
https://mathoverflow.net/questions/157239 | 5 | As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a more practical aspect. Say, if the program accidentally ("or systematically") find some "periodic structure", then it stops... | https://mathoverflow.net/users/40780 | Computational approach deciding whether a set of Wang Tile could tile the space up to some size | There are a few theory papers tackling issue involved in Wang Tile:
<http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.216.7285&rep=rep1&type=pdf>
Also recent development considering finite cases of Wang tile is considered here.
<https://arxiv.org/abs/1305.2796>
<https://arxiv.org/abs/1212.3380>
There ... | 4 | https://mathoverflow.net/users/nan | 158148 | 83,527 |
https://mathoverflow.net/questions/158165 | 2 | Suppose I have a curve $C$ over a field $k$ and that $p$ is a singular point of $C$. Let $f : X \to C$ be the desingularization of $C$ at $p$. Then for each $s \in f^{-1}(p)$ we have a map of local rings $\mathcal O\_{C,p} \to \mathcal O\_{X,p}$ and also of their completions: $\widehat{\mathcal O\_{C,p}} \to \widehat{\... | https://mathoverflow.net/users/23501 | Determining the desingularization from the complete local ring | $X$ is the normalization of $C$; the ring $\prod\_{f(s)=p}\widehat{\mathcal{O}\_{X,s}}$ is just the normalization of $\widehat{\mathcal{O}\_{C,p}}$ (that is, its integral closure in the total ring of fractions). It is completely determined by $\widehat{\mathcal{O}\_{C,p}}$.
| 5 | https://mathoverflow.net/users/40297 | 158169 | 83,531 |
https://mathoverflow.net/questions/158087 | 1 | For any abelian variety $A$, there is a dual abelian variety $\hat{A}$ which parametrizes degree zero line bundles.
Is it possible to expect similar duality for group varieties (suppose over $\mathbb{C}$ for simplicity)? I noticed that there is a notion of Cartier duality, but I feel this is not what I am looking for... | https://mathoverflow.net/users/29730 | Duality for group variety | The "degree" of a line bundle is not defined in general, it is better to study $\textrm{Pic}^0$, the connected component of the indentity of the Picard scheme. For abelian varieties this is exactly the dual abelian variety.
As noted in the comments, the Picard group of a linear algebraic group $G$ is finite, so $\tex... | 4 | https://mathoverflow.net/users/5101 | 158171 | 83,532 |
https://mathoverflow.net/questions/158166 | 6 | Let $S$ be an orientable surface with negative Euler characteristic. Can somebody provide a reference for the following well-known results:
1. the cyclic subgroup generated by a pseudo-Anosov element in the mapping class $MCG(S)$ is undistorted in the mapping class group.
2. the cyclic subgroup generated by a Dehn t... | https://mathoverflow.net/users/41219 | (Un)distorted subgroups in the mapping class group: reference required. | The following paper proves that all infinite cyclic subgroups of the mapping class group are undistorted (subsuming questions 1 and 2):
Farb, Benson; Lubotzky, Alexander; Minsky, Yair,
Rank-1 phenomena for mapping class groups.
Duke Math. J. 106 (2001), no. 3, 581–597.
For your third question, you don't specify wh... | 8 | https://mathoverflow.net/users/317 | 158177 | 83,534 |
https://mathoverflow.net/questions/158173 | 5 | It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there are weaker condition on $F$ which implies that the higher direct images to $S$ are trivial (but not necessarily the cohomol... | https://mathoverflow.net/users/11051 | Relative flasqueness? | Modulo quasi-compactness, the following seems to work:
**Definition.** Let $f:X\to S$ be a map of topological spaces and let $F$ be a sheaf on $X$. We call $F$ *flasque over $S$* if for any opens $U\subseteq V\subseteq X$, any $x\in U$, and any $\sigma\in\Gamma(U, F)$ there exists an open $W$ containing $f(x)$ such t... | 5 | https://mathoverflow.net/users/3847 | 158180 | 83,535 |
https://mathoverflow.net/questions/128429 | 1 | I'm reading the paper ['The big fundamental group, big Hawaiian earrings and the big free groups'](http://www.sciencedirect.com/science/article/pii/S0166864199001042). The authors state that the class of homotopy equivalences of loops in the space he dubs as the big Hawaiian earrings is not a set. I'm not really certai... | https://mathoverflow.net/users/33351 | When is the class of functions between sets a set? | To answer the question, putting together points made in the comments...
There is always a set of functions between two sets, unless you are using [predicative foundations](http://ncatlab.org/nlab/show/predicative+mathematics), where there may be no function sets, or the set theory [NF(U)](http://en.wikipedia.org/wiki... | 8 | https://mathoverflow.net/users/4177 | 158184 | 83,537 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.