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https://mathoverflow.net/questions/161078 | 4 | Let $G$ be a finite group. Define $d:G\times G\longrightarrow\mathbb{N}$ by $d(x,y)=o(xy^{-1})-1, \forall\, x,y\in G$. Then $d$ is a metric on $G$ if and only if
$$(\*)\hspace{5mm}o(ab)<o(a)+o(b), \forall\, a,b\in G.$$It is easy to see that an abelian group satisfies $(\*)$ if and only if it is a $p$-group. Moreover, i... | https://mathoverflow.net/users/17565 | On a metric associated to certain finite groups | The group ${\rm PSL}(2,7)$ seems to be a $CP$ group according to your definition, but it contains an element $a$ of order $2$ and an element $b$ of order $3$ whose product $ab$ has order $7,$ so you do not have a metric in this group.
In fact, as I write, I realise that in a group $G$ where $d$ defines a metric, the p... | 6 | https://mathoverflow.net/users/14450 | 161086 | 84,683 |
https://mathoverflow.net/questions/159655 | 2 | Why does the following diagram commute?$\require{AMScd}$
\begin{CD}
H^0(X,\mathscr{A}) \times \mathrm{Ext}^2\_X(\mathscr{A},\mu\_{\ell^n}) @>>> H^2(X,\mu\_{\ell^n}) \\ @VVV @| \\
H^1(X,\mathscr{A}[\ell^n]) \times H^1(X,\mathscr{E}xt\_X^1(\mathscr{A},\mu\_{\ell^n})) @>>> H^2(X,\mu\_{\ell^n}).
\end{CD}
($\mathscr{A}/X$... | https://mathoverflow.net/users/nan | commutative diagram with Yoneda pairing, Weil pairing and edge morphism | 1. Show that (in the lower diagram) $H^1(X,\mathscr{H}om(\mathscr{A}[\ell^n],\mu\_{\ell^n}))$ embeds via the first edge homomorphism into $\mathrm{Ext}^1(\mathscr{A}[\ell^n],\mu\_{\ell^n})$ (by the local-to-global Ext spectral sequence) and that the lower horizontal and left vertical arrows factor through this.
2. Now ... | 0 | https://mathoverflow.net/users/nan | 161093 | 84,685 |
https://mathoverflow.net/questions/161071 | 0 |
>
> Is there an example of a real analytic (compact) manifold $M$ such that the following two lie algebras are isomorphic Lie algebras:
>
>
> 1. $\chi^{\infty}(M)$, the Lie algebra of all smooth vector fields on $M$.
> 2. $\chi^{\omega}(M)$, the Lie algebra of all analytic vector fields on $M$.
>
>
>
(Before a... | https://mathoverflow.net/users/36688 | Comparison of two infinite dimensional Lie Algebras | The Lie algebra of analytic vector fields on a compact real analytic manifold is simple —this was proved in [Grabowski, J. Isomorphisms and ideals of the Lie algebras of vector fields. Invent. Math. 50 (1978/79), no. 1, 13--33. MR0516602 (80g:57036)]—while the algebra of smooth vector fields has lots and lots of ideals... | 5 | https://mathoverflow.net/users/1409 | 161109 | 84,692 |
https://mathoverflow.net/questions/161107 | 0 | I need to solve the following system of equations. solve([
a\*c\*e=a2\*c2\*e2,
b\*d\*f=t\*b2\*d2\*f2,
t\*a1\*c1\*e1=a3\*c3\*e3,
b1\*d1\*f1=b3\*d3\*f3,
a\*b\*a1\*b1=a2\*b2\*a3\*b3,
c\*d\*c1\*d1=c2\*d2\*c3\*d3,
e\*f\*e1\*f1=e2\*f2\*e3\*f3,
b+c=b2+c2,
d+e=d2+e2,
b1+c1=b3+c3,
d1+e1=d3+e3,
a1\*b1\*(b + c) + c\*d\*(b... | https://mathoverflow.net/users/11877 | Good software for solving a system of algebraic equations | The Maple command $$sol := SolveTools:-PolynomialSystem({a\*c\*e = a2\*c2\*e2, b\*d\*f = t\*b2\*d2\*f2, b1\*d1\*f1 = b3\*d3\*f3, a\*b\*a1\*b1 = a2\*b2\*a3\*b3, c\*d\*c1\*d1 = c2\*d2\*c3\*d3, e\*f\*e1\*f1 = e2\*f2\*e3\*f3, t\*a1\*c1\*e1 = a3\*c3\*e3, b+c = b2+c2, b1+c1 = b3+c3, d+e = d2+e2, d1+e1 = d3+e3, a1\*b1\*(b+c)+... | 5 | https://mathoverflow.net/users/35959 | 161111 | 84,693 |
https://mathoverflow.net/questions/161105 | 3 | Suppose I have a random variable $Z$ defined on a probability space built on a $\sigma$-algebra $\mathcal{F}$ (with $\mathbb{E}[Z^2] < \infty$). Moreover, suppose that I have two filtrations $\mathcal{A}$, $\mathcal{B} \subseteq \mathcal{F}$. My question is: If we define the random variables
$$ X = \mathbb{E}[Z | \math... | https://mathoverflow.net/users/48584 | Positive correlation of conditional expectations | Of course not. Let $X,Y$ be negatively correlated centered standard Gaussian (say with correlation -1/2). Let $Z=X+Y$ and let ${\cal A}=\sigma(X)$, ${\cal B}=\sigma(Y)$. Then
$E(Z|{\cal A})=X+E(Y|{\cal A})=X-X/2=X/2$, and $E(Z|{\cal B})=Y/2$.
| 7 | https://mathoverflow.net/users/35520 | 161113 | 84,694 |
https://mathoverflow.net/questions/161129 | 1 | Assume that $X$ and $Y$ are two Banach spaces and $T:X\to Y$ is a bounded surjective linear operator.
A consequence of the Michael selection theorem is that:"There is a continuous function $g:Y\to X$ such that $T\circ g=Id\_{Y}$".
>
> Can we always find a linear map $g$ as above?
>
>
>
| https://mathoverflow.net/users/36688 | A linear consequence of the Michael selection theorem | Answer No. Just to add that the example that there is no bounded linear 1-1 mapping
$\ell^\infty/c\_0 \to \ell^\infty$ is due to Phillips in 1940 (as commented in the link provided by Yemon Choi above, discussing the complementary subspace problem,
also credited to Phillips in the link to pdf I found and posted).
| 2 | https://mathoverflow.net/users/48481 | 161134 | 84,701 |
https://mathoverflow.net/questions/161121 | 17 | The following definition is due to Donald J. Newman:
A connected open subset $D$ of the plane $\mathbb C$
is *simply connected*
if and only if its complement $\widetilde D = \mathbb C \setminus D$
is ``connected within $\varepsilon$ to $\infty$'',
that is, if for any $z\_0 \in \widetilde D$ and $\varepsilon > 0$
... | https://mathoverflow.net/users/48481 | Who first defined _simply connected_, reference? | It seems indeed pretty clearly to have been **Riemann**, in [§6 of his Inaugural Dissertation](http://books.google.com/books?id=EzIPAAAAIAAJ&pg=PA9) (1851). There he defines *zusammenhängend* as well as *einfach*, *zweifach* and *mehrfach zusammenhängend* without citing any prior sources.
Yet a possible antecedent is... | 17 | https://mathoverflow.net/users/19276 | 161139 | 84,702 |
https://mathoverflow.net/questions/154480 | 3 | In "Meyer Sets and their Duals" Moody proves that any Meyer set union a finite number of points is again a Meyer set. Additionally, any Meyer set is contained in a finite union of model sets whose internal space (the space you are projecting from) is $\mathbb{R}^d.$ Thus it is clear that any finite collection of points... | https://mathoverflow.net/users/22781 | Is any finite collection of points contained in a cut and project set with $\mathbb{R}^d$ as internal space? | The answer is no.
There are two big issues here. The first is the following: If $\Lambda$ is a subset of a model set $\Lambda(W)$, then it is a subset of the model sets $\Lambda(K)$ for all $W \subset K$. So it is always trivial to construct bigger sets.
Now, if by an unique cut and project set you mean an unique ... | 2 | https://mathoverflow.net/users/11552 | 161140 | 84,703 |
https://mathoverflow.net/questions/161117 | 1 | Consider the pushout of a diagram $A\leftarrow B\rightarrow C$ of categories and assume that at least one of the arrows is an embedding, i.e. injective on objects and arrows. When applying the nerve functor to the pushout diagram, do we get a homotopy pushout in simplicial sets?
| https://mathoverflow.net/users/27923 | Pushout of categories along embeddings gives homotopy pushout? | This is wrong. Let $B=\mathbb{N}$, $A$ the cone over $B$ and $C$ the cocone over $B$. The pushout category is contractible, since it has an initial and a terminal object. But the homotopy pushout on the nerve level is the homotopy type of a $2$-sphere.
| 5 | https://mathoverflow.net/users/27923 | 161161 | 84,706 |
https://mathoverflow.net/questions/161151 | 3 | Over number fields, two quadratic forms are equivalent iff they have the same dimension, signature, discriminant and Hasse invariant.
How is the situation like over finitely generated fields?
| https://mathoverflow.net/users/nan | equivalence of quadratic forms over finitely generated fields | In general, quadratic forms are not determined by their Stiefel-Whitney invariants, see Scharlau, Winfried: Quadratische Formen und Galois-Cohomologie. In: Invent.
Math., 4 (1967), 238–264, p. 251 f.
However, if $\dim{q} \leq 3$, $q$ is determined by dimension, discriminant and Hasse invariant, see Lam, Tsit-Yuen: In... | 3 | https://mathoverflow.net/users/nan | 161170 | 84,710 |
https://mathoverflow.net/questions/161163 | 5 | I consider the following two situations:
1. Let $B$ be a simply connected space, and $F\to E\to B$, $F'\to E'\to B$ two fibrations with a map $f:E'\to E$ sending fibers to fibers and inducing the identity on $B$ (in my case $f$ is injective but I don't think it matters).
2. Let $f:B\to B'$ be a map between simply con... | https://mathoverflow.net/users/3465 | Relative Serre spectral sequences? | These relative versions of the Leray-Serre spectral sequence appear as Exercises 5.5 and 5.6 in McCleary's "A User's Guide to Spectral Sequences". Your case 2 is also mentioned on p17 of Allen Hatcher's "Spectral Sequences in Algebraic Topology" book project, available [here.](http://www.math.cornell.edu/~hatcher/#SSAT... | 6 | https://mathoverflow.net/users/8103 | 161173 | 84,712 |
https://mathoverflow.net/questions/161159 | 10 | Suppose $n$ is a big number and $k\geq 2$. How many sets $S\_1,\dots,S\_m\subset [n]$ can we find such that
(1) $|S\_i| = k$ for all $i$,
(2) $|S\_i\cap S\_j| \leq 1$ for all $i\ne j$.
What's the maximum possible value of $m$?
(I just need to know the growth order of $m$ depending on $n$ and $k$. For instance, when ... | https://mathoverflow.net/users/48609 | maximum size of intersecting set families | I noted the simple upper bound $m\le \binom{n}{2}/\binom{k}{2}= n(n-1)/(k(k-1))$ in my comment above. It seems that Wilson proved that if $k-1$ divides $n-1$, and $\binom{k}{2}$ divides $\binom{n}{2}$ then for large $n$ this upper bound is attained. See page 1424 of this interesting ICM article of Péter Frankl's ICM ar... | 15 | https://mathoverflow.net/users/38624 | 161184 | 84,716 |
https://mathoverflow.net/questions/161176 | 6 | In *stochastic equations in infinite dimensions* written by Da Prato **reproducing kernel Hilbert space** is defined
>
> Let $\mu$ be a symmetric Gaussian measures on a separable Banach space $E$. A linear subspace $H \subset E$ equipped with a Hilbert norm $|\cdot|\_{H}$ is said to be a *reproducing kernel Hilber... | https://mathoverflow.net/users/44175 | A strange definition about reproducing kernel Hilbert spaces for symmetric Gaussian measures on a separable Banach space | The two notions coincide in the particular case when $E = C(X)$ consists of the continuous functions from $X$ to $\mathbf{R}$ with $X$ playing the same role as in the Wikipedia article. (Say $X$ is a compact space to avoid technicalities.) In this case, if $H$ is defined as in Da Prato, since $H \subset C(X)$, the poin... | 5 | https://mathoverflow.net/users/38566 | 161185 | 84,717 |
https://mathoverflow.net/questions/161183 | 1 | I am unclear how the inverse mean curvature flow starts with a minimal surface. If there is some point $p$ with mean curvature $H(P)=0$, how should we treat this problem and let the flow start?
| https://mathoverflow.net/users/48621 | How does the inverse mean curvature flow start with minimal surface? | It isn't clear exactly what sort of initial conditions you're requiring.
The difficulty with minimal initial conditions is part of the reason why it was an amazing result when Huisken--Ilmanen [constructed](http://projecteuclid.org/euclid.jdg/1090349447) a "weak inverse mean curvature flow" which
(1) Can start at... | 6 | https://mathoverflow.net/users/1540 | 161186 | 84,718 |
https://mathoverflow.net/questions/161150 | 17 | I was reading a review article [arXiv:1310.7644](http://arxiv.org/abs/1310.7644) and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be triangulated, using the technique of the Seiberg-Witten monopole equation.
As a physi... | https://mathoverflow.net/users/5420 | How can gauge theory techniques be useful to study when topological manifolds can be triangulated? | The very short answer is that there is no direct connection between gauge theory (which is living on some perhaps hypothetical smooth 4-manifold) and triangulation of some high-dimensional topological manifold. Here are some remarks to justify that statement.
The question addressed in Quinn's expository article is no... | 27 | https://mathoverflow.net/users/3460 | 161189 | 84,719 |
https://mathoverflow.net/questions/161116 | 5 | From the work of [Lott--Villani](http://math.berkeley.edu/~lott/LottVillani.pdf) and [Sturm](http://link.springer.com/article/10.1007%2Fs11511-006-0002-8), I know that the following fact holds:
>
> (\*) Suppose that $(M\_k,g\_k,dvol\_{g\_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci curva... | https://mathoverflow.net/users/1540 | Ricci curvature under rough convergence | Let me try to prove it.
Let $M$ be an $m$-dimensional Riemannian manifold.
Set
$$f\_p(z)=\frac{|p-z|\_M^2}2,$$
where $|p-z|\_M$ denotes the distance from $p$ to $z$ in $M$.
Note that $M$ has nonnegative Ricci curvature if and only if
$$(\Delta f\_p)(x)\le m$$
for any $p$ and $x\in M$.
The later means that for th... | 5 | https://mathoverflow.net/users/1441 | 161194 | 84,721 |
https://mathoverflow.net/questions/161187 | 9 | Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with **nonnegative** coordinates we have
$$\sum\_{i,j} K\_{ij} f\_i f\_j \le \sum\_{ij} K^\prime\_{ij} f\_i f\_j$$
Can someone prove the following inequality?
$$ \sum\_{i,j} \exp K\_{ij} \le \sum\_{i,j} \exp K^\pri... | https://mathoverflow.net/users/22758 | An inequality for positive definite matrices | Here is a simple proof (thanks to the OP for a proof that the Hadamard exponential is CP).
I'll write $K$ and $H \equiv K'$ to keep the notation simpler.
As $K \succeq 0$, it follows that its Schur exponent $[e^{k\_{ij}}]$ is also psd. Thus, in particular, we can write the following factorization:
\begin{equation\*... | 7 | https://mathoverflow.net/users/8430 | 161197 | 84,722 |
https://mathoverflow.net/questions/161192 | 2 | Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that $[C]\cdot K\_X < 0$, then one can contract this extreme ray to get a morphism $f: X \to Y$.
I want to know if there is any re... | https://mathoverflow.net/users/29730 | Properties of extreme rays | Let $X$ be a smooth projective variety, and let $L(R)$ be the locus of a $K\_X$-nagative extremal ray $R$. Any irreducible component $Z$ of $L(R)$ is uniruled. If the contraction associated to $R$ is divisorial then $L(R)$ is irreducible and uniruled. You can find this, for instance, in Proposition $6.10$ of Debarre's ... | 3 | https://mathoverflow.net/users/14514 | 161199 | 84,723 |
https://mathoverflow.net/questions/161145 | 7 | Does the height of a real symmetric matrix with non-negative entries control the height of its Perron-Frobenius eigenvector, under some reasonable definition of heights?
Just as an example of what kind of heights I would be happy with. For the matrix, consider the model case where all of its entries are integer, and ... | https://mathoverflow.net/users/9924 | The height of the Perron-Frobenius eigenvector | Grrr, I am not used to this yet:
"Birkhoff norm" places the problem in a less elementary setting:
[What is the Birkhoff norm of a Perron vector?](https://mathoverflow.net/questions/101234/what-is-the-birkhoff-norm-of-a-perron-vector)
Second edit: I just found
[concentration for eigenvectors](https://mathoverflo... | 5 | https://mathoverflow.net/users/37855 | 161206 | 84,725 |
https://mathoverflow.net/questions/161169 | 11 | Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher.
Even in the simplest case, the infinite tensor product of Hilbert spaces, we have to be patient to find a good references (apart from the origi... | https://mathoverflow.net/users/48614 | Infinite tensor product of states | If you're working at the level of Hilbert spaces, I think the more usual procedure nowadays is to fix a distinguished unit vector $u\_i$ in each Hilbert space $H\_i$, then define the tensor product to be the Hilbert space generated by all elementary tensors $\bigotimes v\_i$ such that $v\_i \in H\_i$ for all $i$ and $v... | 10 | https://mathoverflow.net/users/23141 | 161209 | 84,726 |
https://mathoverflow.net/questions/161208 | 6 | Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the Zariski-closed $G$-orbits on the variety of complete flags in $\mathbb{C}^n$ for these two cases?
I believe the whole flag var... | https://mathoverflow.net/users/34464 | Closed orbits of complete flags in $\mathbb{C}^n$ | The whole flag variety is definitely not a single $O(n)$-orbit. You can see this already in the case of $n=2$, where the flag variety is just $\mathbb P^1$, and there are two orbits: $\{\pm i\}$ and everything else. More generally, by Witt's theorem, the $O(n)$-orbit on the grassmannian of $r$-planes through any given ... | 9 | https://mathoverflow.net/users/430 | 161215 | 84,729 |
https://mathoverflow.net/questions/160872 | 10 | Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the oriented triangle
$$ z\_1 \to z\_2 \to z\_3 \to z\_1 $$
is non-degenerate and is oriented counterclockwise.
If we write $z\_... | https://mathoverflow.net/users/3621 | Axiomatizing orientation in the complex plane | A too long waste of bits in three parts. The core of the answer is the summary at the end of part 2 section 2.
Part 1. Orientation.
====================
Before treating the aspects on which I agree, let me say one aspect where I disagree (but quite probably only about terminology).
I would not say that orientatio... | 2 | https://mathoverflow.net/users/46855 | 161221 | 84,730 |
https://mathoverflow.net/questions/161222 | 3 | Let $X\_1,X\_2,\ldots,X\_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows:
$$\hat{F}\_n(x)=\frac{1}{n}\sum\_{i=1}^n\mathbf{1}\_{\{X\_i\leq x\}}(x)\tag{1}$$
where $\mathbf{1}\_A(x)$ is an indicator fun... | https://mathoverflow.net/users/18910 | What is known about the distribution of the errors in empirical approximation of a CDF? | I believe your two empirical approximations are the same: aren't they both the total fraction of observations less than or equal to $x\_0$?
As for the question of asymptotic normality, [Donsker's Theorem](http://en.wikipedia.org/wiki/Donsker%27s_theorem) gives the main result in this direction, namely that the proces... | 5 | https://mathoverflow.net/users/5963 | 161245 | 84,737 |
https://mathoverflow.net/questions/161247 | 8 | This question (which is more a curiosity than a research problem) originates from these two:
1. <https://math.stackexchange.com/questions/720254/is-there-a-nonempty-open-bounded-subset-of-plane-whose-boundary-contains-no-1-di>
2. [complement of a totally disconnected closed set in the plane](https://mathoverflow.net/... | https://mathoverflow.net/users/40911 | Must a closed totally path-disconnected subset of the sphere have connected complement? | A circular version of the [pseudo-arc](http://en.wikipedia.org/wiki/Pseudo-arc) (where you construct it out of "circular chains" whose ends connect up to each other) is a counterexample. It is connected and totally path-disconnected, and its complement has two components. This example seems to be due to Bing (Example 2... | 10 | https://mathoverflow.net/users/75 | 161253 | 84,738 |
https://mathoverflow.net/questions/161040 | 10 | This is a geometric puzzle though it might conceivably
also define a special class of Pythagorean triples.
A *perfect squared square* PSS is a square (as a plane figure)
partitioned into smaller squares, each of a different size.
There are other types of squared squares or
squared rectangles that have been studi... | https://mathoverflow.net/users/48481 | Could a perfect squared square be split into two perfect squared squares? | Nice question. This is not (any longer) an answer, but a strategy.
First, try to construct 25 mutually disjoint squared squares of the same order. Then arrange them according to a 3,4,5 template.
I initially thought that the 25 mutually disjoint squares should be an easy construction and followed from varying a pa... | 5 | https://mathoverflow.net/users/45255 | 161259 | 84,741 |
https://mathoverflow.net/questions/160799 | 9 | Consider the following question:
Let $K\subset S^{3}$ be a nontrivial knot, and let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ vanishes?
The answer seems to be no. Indeed by Gordon-Luecke's theorem, $Y$ is not $S^{3}$, while as far as I know... | https://mathoverflow.net/users/44651 | Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$? | Kronheimer and Mrowka first proved that 1-surgery on a knot has non-trivial instanton Floer homology using Witten's conjecture (see their paper <http://arxiv.org/abs/math/0311489>). They later proved this using sutured instanton Floer homology as you suggested (see Section 7 of <http://arxiv.org/abs/0807.4891>).
As f... | 4 | https://mathoverflow.net/users/3405 | 161263 | 84,743 |
https://mathoverflow.net/questions/161239 | 8 | Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X\_i$ of $X$. It is well known that the etale cohomology of $\eta$ (i.e. the corresponding cohomology of the Galois group of $\eta$) with $\mathbb{Z}/l^n\mathbb{Z}$-coeffici... | https://mathoverflow.net/users/2191 | Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties? | You may consider $X=\mathbf{P}^1\_k$, and look at the first cohomology group $\varinjlim H^1\_{et}(X\_i,\mathbf{Z}\_l)$. By the toposic version of Hurewicz theorem, this colimit is the same as $\varinjlim Hom\_{cont}(\pi\_1(X\_i),\mathbf{Z}\_l)$ (where $Hom\_{cont}$ means continous morphisms of topological groups). Eac... | 7 | https://mathoverflow.net/users/1017 | 161264 | 84,744 |
https://mathoverflow.net/questions/161269 | 1 | Let $(X,\tau)$ be a Banach space and $L\subset X$ an arbitrary finite dimensional subvector space. Let $f:L\rightarrow X$ be continuous.
1) Is the set convex hull $cx (f(L))$ also finite-dimensional?
2) If not, would it help to assume that $(X,\leq, \tau)$ is a Banach lattice and that $f$ is monotone
( i.e. $x\l... | https://mathoverflow.net/users/48667 | Convex hull of a continuous image | Certainly not: take for instance $X:=L^2[0,1]$ and $L$ one dimensional, which we can identify with $\mathbb{R}$. Define $f(t)$ to be the characteristic function of the interval $\{s\in[0,1]: 0\le s\le t\}$.
| 6 | https://mathoverflow.net/users/6101 | 161274 | 84,747 |
https://mathoverflow.net/questions/160753 | 8 | Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable arrows in $C$. Define a $X$-component of $\sigma$ to be a maximal substring of the form $X\rightarrow X\rightarrow...\righta... | https://mathoverflow.net/users/27923 | Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid | I found a proof for this statement. But the reason is another one that one might think. Let $C^-$ be the full subcategory generated by the objects of $C$ minus the $X$. I claim: If there is a simplex with more than one $X$-component at all, then both the inclusions $NC^-\rightarrow NC$ and $NC^-\rightarrow N^rC$ are ho... | 1 | https://mathoverflow.net/users/27923 | 161275 | 84,748 |
https://mathoverflow.net/questions/161271 | 9 | Inspired by [this](https://math.stackexchange.com/questions/716182/consecutive-quadratic-residues) recent question, I wondered if a similar result is true for quadratic non-residues, namely, if **it is true that for every $k \in \mathbb{N}$ there exists a prime $p$ such that exists $k$ consecutive quadratic non-residue... | https://mathoverflow.net/users/37392 | Consecutive non-quadratic residues | The answer to both your questions is positive and indeed, every given pattern of quadratic residues and non-residues of fixed length appears among consecutive elements of ${\mathbb F}\_p$, for all $p$ large enough; moreover, it appears about the expected number of times. This is non-trivial, but fairly standard.
Fix... | 23 | https://mathoverflow.net/users/9924 | 161279 | 84,750 |
https://mathoverflow.net/questions/161282 | 2 | Let $X\_1,X\_2,\ldots,X\_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows:
$$\hat{F}\_n(x)=\frac{1}{n}\sum\_{i=1}^n\mathbf{1}\_{\{X\_i\leq x\}}(x)\tag{1}$$
where $\mathbf{1}\_A(x)$ is an indicator fun... | https://mathoverflow.net/users/18910 | Estimating the variance of error in empirical approximation to a distribution | If you're only interested in estimating $F$ at a single point $x\_0$, then you are really just estimating the probability $p\_0 = \mathbb{P}\left(X\in (-\infty,x\_0]\right)$. The indicator variables $B\_i = [X\_i\leq x\_0]$ are i.i.d. Bernoulli with (potentially unknown) probability $p\_0$, so you can apply the usual C... | 5 | https://mathoverflow.net/users/5963 | 161294 | 84,752 |
https://mathoverflow.net/questions/161292 | 2 | If $K$ is a number field, a result from Bach tells us that the primes in $K$ of norm smaller that $12 (\log |\mathrm{Discriminant}(K)|)^2$ generate the ideal class group $\mathrm{Cl}\_K$. Is there any known bound if we require the generating primes to be unramified in $K/\mathbb Q$? (i.e. a result of the form "prime id... | https://mathoverflow.net/users/48499 | Small generators of the ideal class group | First, do not forget that Bach's result is conditional to GRH.
What Bach proves is that for every nontrivial character $\chi$ of the ray class group of $K$ of conductor $\mathfrak{f}$, there is a prime $\mathfrak{p}$ of norm less or equal to $3\log(\Delta\_K^2N\mathfrak{f})^2$ such that $\chi(\mathfrak{p})\neq 1$. In... | 8 | https://mathoverflow.net/users/40821 | 161297 | 84,754 |
https://mathoverflow.net/questions/161349 | 4 | I thought ZFC proved the existence of an inductive well-ordering that is itself a set for any stage of V. NBG with only the regular AC should then prove/assert the existence of a class R of ordered pairs **(a,b)** such that either:
a has lesser rank than b, or:
they both have rank α and **(a,b)** exists in the ZFC... | https://mathoverflow.net/users/48688 | Why doesn't choice imply global choice (in NBG)? | Your first sentence is true (modulo the word "inductive"), but not in the way you mean: $ZFC$ proves the existence of *many* set well-orderings of each $V\_\alpha$. Now, under some further assumption - say, $V=L$ - there might be a sequence of somehow canonical well-orderings of the $V\_\alpha$, in which case we can in... | 6 | https://mathoverflow.net/users/8133 | 161351 | 84,764 |
https://mathoverflow.net/questions/161306 | 3 | Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet differentiable?
| https://mathoverflow.net/users/36688 | A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable | From <http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/viewFile/285/416>
Fr\'echet differentiability implies G\^ateaux differentiability, but the converse is true only for finite-dimensional Banach spaces, in general. As an example, the mapping $f:L^1[0,\pi]\rightarrow\mathbb{R}$ defined by $f(x)=\int\_0^\pi... | 6 | https://mathoverflow.net/users/46855 | 161373 | 84,772 |
https://mathoverflow.net/questions/161298 | 2 | I want to get some advice from you about the existence (and the uniqueness if possible) of a strong solution on my SDE. In fact, due to the structure of the problem that I consider, both the drift (usually denoted by mu) and diffusion (usually denoted by sigma) coefficients are discontinuous. More specifically, both co... | https://mathoverflow.net/users/48678 | Strong solutions on SDE (stochastic differential equations) with discontinuous drift and diffusion coefficients | A partial but quick answer. Put the equation in the natural scale, namely
take $\varphi(x)=2\frac{x-1}{x+1}$ and set $Y\_t = \varphi(X\_t)$, then
$$
dY=\frac14(4-Y^2)f\Bigl(\frac{2+Y}{2-Y}\Bigr)\,dB,
$$
where $f(x)=a+b$ if $x\leq k$ and $f(x)=a$ otherwise.
Feller's test tells us that any solution starting in $y\in(-2... | 0 | https://mathoverflow.net/users/8966 | 161377 | 84,775 |
https://mathoverflow.net/questions/161386 | 6 | **Question:** Can we have a set theory in which there exists a $\kappa$-Suslin tree with $\kappa$ larger than the least measurable cardinal?
A $\kappa$-Suslin tree is a tree with levels indexed by $\kappa$, the cardinality of each level is less than $\kappa$, and all chains and antichains have cardinality less than $... | https://mathoverflow.net/users/16678 | Can we have a $\kappa$-Suslin tree where $\kappa$ is above a measurable cardinal? | Yes, one can add Suslin trees above a measurable cardinal, by forcing that adds no subsets to the measurable cardinal and therefore preserves its measurability.
For example, if $\delta$ is measurable and $\kappa=\delta^+$, then one can add a $\kappa$-Suslin tree by forcing with the conditions consisting of ${\lt}\del... | 9 | https://mathoverflow.net/users/1946 | 161388 | 84,780 |
https://mathoverflow.net/questions/161280 | 15 | This is related to [this question.](https://mathoverflow.net/questions/136012/multiply-transitive-groups) It is well-known that $S\_n$ and $A\_n$ are the only six transitive permutation groups, and it is likewise well-known that the proof of this requires the classification of finite simple groups. The question is: is ... | https://mathoverflow.net/users/11142 | Multiply transitive groups, continued | *Since @Igor asked for some references, I'll make this an answer and summarise some of the things mentioned above.*
No one knows how to prove this result without resort to CFSG. It would be a huge result if someone could manage it. There are a number of results that head in the same general direction, as follows:
1... | 13 | https://mathoverflow.net/users/801 | 161392 | 84,782 |
https://mathoverflow.net/questions/161384 | 3 | The seminal theorem of Faltings confirms Mordell's conjecture: that is, curves of genus at least 2 have at most finitely many rational points. The proof of Faltings' theorem is not effective, meaning there is no way to estimate the number of rational points on a given curve.
What is the conjectured truth for these c... | https://mathoverflow.net/users/10898 | Curves of high genus with many rational points | Apparently it is a conjecture of Lang that there is a bound depending on the genus only. Check out [Lucia Caporaso's paper](http://www.mat.uniroma3.it/users/caporaso/CountingPoints.pdf) (I am sure there are more recent references).
| 2 | https://mathoverflow.net/users/11142 | 161394 | 84,783 |
https://mathoverflow.net/questions/161402 | 4 | Let $1(n)$ be the number of digits $1$ in binary representation of number $n$.
For example, $13=1101\_2$ so $1(13)=3\\$
Is there explicit form of $\,\,\sum{1(i)x^i} $?
I checked OEIS and didn't find information about this sequence (1,1,2,2,2,3,1,2,2,3,...)
| https://mathoverflow.net/users/41522 | Number of 1 in binary representation of n | The generating function you ask about, more typically written as $\sum\_{i\ge 0} s\_2(i) x^i$, can be expressed as
$$\frac{1}{1-x} \sum\_{m\ge 0} \frac{x^{2^m}}{1+x^{2^m}}$$
The number of 1's in the binary expansion is just the sums of digits; there also exists a generalization for sums of digits function in other ba... | 19 | https://mathoverflow.net/users/nan | 161404 | 84,786 |
https://mathoverflow.net/questions/161318 | 7 | Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}\_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G\_{(r)}$ denote the $r$th Frobenius kernel of $G$, that is, the kernel of the Frobenius endomorphism which sends each entry of a matrix to its $p^r$th power. The c... | https://mathoverflow.net/users/32261 | Complexity of rational $\mathrm{GL}_{n(r)}$-modules | I don't recall seeing an explicit answer to your question in the literature, but the basic outline starts with the (restricted) Lie algebra or first Frobenius kernel: here an upper bound on the complexity of any finite dimensional module is given by the dimension of the "restricted nullcone", which for $p$ at least the... | 3 | https://mathoverflow.net/users/4231 | 161410 | 84,789 |
https://mathoverflow.net/questions/161390 | 5 | Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}\_{dg}(\mathcal{A})$.
Assume further there exists a $dg$-algebra $\mathcal{B}$ and an isomorphism $f:\mathcal{A}\to \mathcal{B}$, then you can find a $G$-action on $\mathcal... | https://mathoverflow.net/users/32972 | Equivariant Formality | I believe your second statement is true when $\mathcal{A}$ is the minimal Sullivan model of a $G$-space $X$. Namely, if the $G$-space $X$ is rationally formal, then it is rationally $G$-formal, in the sense that it has a rational $G$-minimal model which can be constructed from $H^\ast(X;\mathbb{Q})$ with it's $G$-actio... | 3 | https://mathoverflow.net/users/8103 | 161422 | 84,795 |
https://mathoverflow.net/questions/161281 | 3 | Let $u\_1, u\_2, u\_3 \in \mathbb{Z}$ such that $u\_1^2 + u\_2^2 = u\_3^2$.
Is $(u\_3 + \frac{u\_1 + u\_2}{\sqrt{2}})^2$ bounded from below?
The irrationality of $\sqrt{2}$ certainly precludes zero, but can we say more?
How should I think about such problems in general? It looks like an interplay between equations ... | https://mathoverflow.net/users/17408 | Is this bounded from below? | This will answer your questions in the comments, and a question that combines them into something else.
First, the most trivial one: to minimize $(u\_4 + \frac{u\_1 + u\_2+u\_3}{\sqrt{2}})^2$, given that you already have solutions for fewer variables, the solution is to set $u\_3=0$ and use your previous answer.
Se... | 3 | https://mathoverflow.net/users/18086 | 161440 | 84,802 |
https://mathoverflow.net/questions/161441 | 10 | Let $A$ be a set of all irrational numbers $\rho \in (0, 1)$ represented as a continued fraction $\rho=[a\_{1}, a\_{2},...,a\_{n},...],$ such that $a\_{n}\leq \text{const}\cdot n^{\epsilon}$ for some $\epsilon \in (0, 1/2).$ ($\epsilon$ and const are fixed.)
Question: Is the Lebesgue measure of the set $A$ positive? ... | https://mathoverflow.net/users/46144 | Measure of a set of irrational numbers | Your set has measure zero by theorems of Khinchin. First a theorem of Khinchin shows that for almost all real numbers $x$ (i.e. outside a set of measure zero) one has
$$
\lim\_{n\to \infty} \frac{\log q\_n}{n}= C
$$
for a positive constant $C$. Here $p\_n/q\_n$ are the convergents of $x$. So almost surely, the denom... | 22 | https://mathoverflow.net/users/38624 | 161444 | 84,803 |
https://mathoverflow.net/questions/161458 | 3 | Let $\mathcal{A}\subset \mathcal{B}$ be two categories with $\mathcal{A}$ full and reflective in $\mathcal{B}$. Let $R:\mathcal{B}\to\mathcal{A}$ be the reflection. That $R$ is the left adjoint to the inclusion $\mathcal{A}\subset \mathcal{B}$ is equivalent to the fact that every object of $\mathcal{A}$ is orthogonal t... | https://mathoverflow.net/users/24563 | About reflective full subcategories and small-orthogonality classes | Since $Z$ is right orthogonal to every component of the adjunction unit, it is in particular right orthogonal to $\eta\_Z : Z \to R Z$. Thus $\eta\_Z : Z \to R Z$ admits a retraction, say $r : R Z \to Z$. Let $a = \eta\_Z \circ r$. It is an idempotent endomorphism of $R Z$. But $\eta\_Z \circ r = R r \circ \eta\_{R Z}$... | 3 | https://mathoverflow.net/users/11640 | 161463 | 84,809 |
https://mathoverflow.net/questions/161300 | 1 | Suppose $X$, $Y$, $Z$ are projective varieties in $\mathbb{P}^n\_K$ of dimension $n-1$, where $K$ is a field. $X$, $Y$, $Z$ intersect properly, and $P$ is one of their intersection irreducible components of dimension $n-3$. Suppose their intersection multiplicity at $P$ is $i(P; X,Y,Z; \mathbb{P}^n)$. Do we always have... | https://mathoverflow.net/users/37096 | Relation between intersection multiplicity and Hilbert-Samuel multiplicities | It's a result in "méthodes d'algèbre abstraite en géométrie algébrique" written by Samuel.
| 1 | https://mathoverflow.net/users/37096 | 161464 | 84,810 |
https://mathoverflow.net/questions/136692 | 3 | Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F\_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ideal of $\mathcal{O}\_K$. I want to get a upper bound of the local Hilbert-Samuel function of $X$ at the point $\xi$, t... | https://mathoverflow.net/users/37096 | How to estimate a local hilbert samuel funcion | $H\_\xi(k)={n+k\choose k}-{n-\mu\_\xi+k\choose k}$, where $\mu\_\xi$ is the multiplicity of $\xi$, and $H\_\xi(k)$ is its Hilbert-Samuel function.
| 0 | https://mathoverflow.net/users/37096 | 161465 | 84,811 |
https://mathoverflow.net/questions/161449 | 15 | Hilbert's 20th problem concerns the existence of solutions to the fundamental problem in the calculus of variations. I understand that Hilbert, Lebesgue and Tonelli were pioneers in this area.
In particular, I believe that Hilbert answered his problem soon but there were some gaps. Tonelli pioneered the idea of weak... | https://mathoverflow.net/users/44981 | Who first resolved Hilbert's 20th problem? | A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this [AMS bulletin](http://www.ams.org/journals/bull/1926-32-04/S0002-9904-1926-04231-2/S0002-9904-1926-04231-2.pdf). The original work was published in Italian, *Fondamenti di Calcolo delle Variazioni* (Bologna, 1921 & 1923) -... | 12 | https://mathoverflow.net/users/11260 | 161468 | 84,812 |
https://mathoverflow.net/questions/161459 | 5 | I have a fully connected graph $G=(V,E)$ with $n$ vertices. The edge weights $w(e)$ with $e\in E$ are non-negative and form a metric space (e.g. Hamming distance), thus for vertices $v,u,y \in V$, we have $w(v,y) \leq w(v,u)+w(u,y)$.
However, it is *expensive* to calculate $w(\cdot)$.
My question is, is there an ... | https://mathoverflow.net/users/11987 | Minimum Spanning Tree of Graph with Unknown Weights | In general the triangle inequality will not help. To see this, recall that the minimum spanning tree depends only on the relative order of the weights. Take an arbitrary MST problem and scale all the weights to lie in $[2,3]$. Now the triangle inequality is satisfied but the MST is the same.
To put it another way, if... | 6 | https://mathoverflow.net/users/9025 | 161469 | 84,813 |
https://mathoverflow.net/questions/161461 | 2 | It is well-known that the number of trailing zeros in the factorial $k!$ is given by the nice function $$ z(k) := \sum\_{i \ge 1} \left\lfloor \frac{k}{5^i} \right\rfloor. $$
Now assume that we want to count how many nonnegative integers up to $n$ have an even number of trailing zeros in their factorial. We could jus... | https://mathoverflow.net/users/8460 | Integers up to n having an even number of trailing zeros in their factorial | You can use a very similar reasoning to the one commonly used to prove [Lucas's lemma](http://en.wikipedia.org/wiki/Lucas%27_theorem). The method below is not very short, but I think it's transparent, and it's straightforward to adapt it to trailing zeros in any base $b$ congruent to any remainder
$r$ mod $m$ (10, 0, ... | 4 | https://mathoverflow.net/users/48772 | 161478 | 84,816 |
https://mathoverflow.net/questions/161480 | 2 | The inflation of graph $G$ is a graph $I(G)$
which is obtained by replacing each vertex $x$ by a complete graph
$K\_{\deg(x)}$ and joining each edge to a different vertex of $K\_{\deg(x)}$.
Let $G$ be a connected cubic graph.
$I(G)$ is line graph (and claw-free).
Experimental data on up to $14$ vertices suggests... | https://mathoverflow.net/users/12481 | The edge chromatic number and pefectness of inflation of cubic graph | I believe that both (1) and (2) are true.
Let us denote the vertices of $I(G)$ as follows:
$V(I(G)) = \{ u^v : uv \in E(G) \}$, so that
$E(I(G)) = \{ u^vv^u: uv \in E(G) \} \cup \{ u^vu^w: v\neq w; uv, uw \in E(G) \}$
As for (1), both $G$ and $I(G)$ are cubic, so by Vizing's theorem we have $\chi'(G), \chi'(I(G)) \in... | 1 | https://mathoverflow.net/users/48772 | 161484 | 84,818 |
https://mathoverflow.net/questions/161486 | 6 | Let $n\in\mathbb N$ and $p$ be any prime. Denote by $\mathbb Q\_p$ the $p$-adic numbers. For a field extension $L/K$ denote by $Tr\_{L/K}$ the corresponding trace function.
Let $\zeta\_n$ be a primitve $n$-th root of unity and $\xi\_n$ any $n$-th root of unity. I would like to know the following:
What is $Tr\_{\mat... | https://mathoverflow.net/users/39270 | Trace of n-th root of unity in cyclotomic extension of p-adic rationals | Certainly not in general: If $p \equiv 1 \bmod n$ (but $n \neq \pm 1$) then $\mathbb{Q}\_p(\zeta\_n) = \mathbb{Q}\_p$) so $Tr(\zeta\_n) = \zeta\_n$ which is not in $\mathbb{Q}$. To give another example, take $n=5$ and $p \equiv 4 \bmod 5$. Then $\mathbb{Q}\_p$ contains a square root of $5$ and $x^4+x^3+x^2+x+1 = (x^2 +... | 6 | https://mathoverflow.net/users/297 | 161489 | 84,821 |
https://mathoverflow.net/questions/161483 | 4 | Why, in functional analysis, is so important to calculate best constant in an embedding inequality?
Cross-posted from "<https://math.stackexchange.com/questions/727690/understanding-reasons-for-best-constant-in-inequalities>".
| https://mathoverflow.net/users/45729 | Understanding reasons for best constants in inequalities | A simple reason, among many others: for instance, you would like to know whether a certain functional is bounded below, because you are looking for a minimizer of it, and you can prove some inequality of the form $f(u)\ge \|u\|\_1^\alpha-c\|u\|\_2^\alpha +b $.
| 6 | https://mathoverflow.net/users/6101 | 161490 | 84,822 |
https://mathoverflow.net/questions/161493 | 4 | Let $P$ be a finite projective space of order $q$ and dimension $d$. I am interested in finding the least $k$ such that for any set $S$ of $k$ points of $P$, and for any set $S'$ of $k$ hyperplanes of $P$, some point of $S$ is incident to some hyperplane of $S'$.
I believe I can prove $k=O( q^{\frac{d+1}2})$ fairly e... | https://mathoverflow.net/users/45855 | Point-Hyperplane incidence in finite projective spaces | $q^{\frac{d}{2}}$ is a lower bound. Assume $d=2e$, a $d$-dimensional projective space has $d+1=2e+1$ variables $x\_0,x\_1,\dots,x\_e,x\_{e+1},\dots,x\_{2e}$. Let the points be all vectors with $x\_0$ coordinate $1$, $x\_1$ through $x\_e$ arbitrary, and $x\_{e+1}$ through $x\_{2e}$ zero. Similarly, let the hyperplanes b... | 3 | https://mathoverflow.net/users/18060 | 161495 | 84,824 |
https://mathoverflow.net/questions/161501 | 3 | I would be thankful for a reference to any result that says "how often" an equation of the form $$c\_1x\_1 + c\_2x\_2 + ... + c\_nx\_n = 0,$$
where $n$ is fixed, $c\_1, ..., c\_n \in \mathcal{O}\_K$ are arbitrarily given algebraic integers of a fixed number field $K$ and $x\_1, ..., x\_n$ vary in the set of units of ... | https://mathoverflow.net/users/10591 | Linear polynomials in units of number fields | There's a paper of Evertse, Gyory, Stewart, and Tijdeman in which they prove that for "most" (in an appropriate sense) $a,b\in K$, the equation $ax+by=1$ has at most 2 soutions in units $x,y\in\mathcal{O}\_K^\*$. The proof is to assume that there are 3 solutions $(x\_i,y\_i)$, $1\le i\le 3$, use those three equations t... | 6 | https://mathoverflow.net/users/11926 | 161507 | 84,828 |
https://mathoverflow.net/questions/161432 | 15 | I'm curious about geometric quantization.
Of course, I know the procedure:
Start with a classical phase space $T^{\*}X$, $X$ is the configuration space, then do prequantization by creating a prequantum (complex) line bundle (of course, the symplectic structure must satisfy the Bohr-Sommerfeld condition). The space of... | https://mathoverflow.net/users/nan | Geometric Quantization | Perhaps we can approach something like an answer, by following the lines set out in Ritter's [exposition](http://arxiv.org/abs/math-ph/0208008) of geometric quantization (2002). Geometric quantization works because the Heisenberg equations of motion in quantum mechanics have the same structure as Hamilton's equations i... | 4 | https://mathoverflow.net/users/11260 | 161516 | 84,830 |
https://mathoverflow.net/questions/161522 | 3 | Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map.
Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id\_{Y}$?
| https://mathoverflow.net/users/36688 | A differentiable version of the Michael selection theorem | No. Consider the case of a surjective bounded linear operator $T:X\to Y$ which is not a (top-linear) left inverse (that is, $\operatorname{ker}(T)$ does not split in $X$). However by classical selection theorems a surjective bounded linear operator $T$ has a continuous right inverse $g$, even $1$-homogeneous; but it ca... | 3 | https://mathoverflow.net/users/6101 | 161530 | 84,835 |
https://mathoverflow.net/questions/161104 | 1 | Let $A$ be a finite dimensional algebra over finite field (not necessarily associative). Then $A$ is said to be homogeneous if $Aut(A)$ acts transitively on the one-dimensional subspace of A. If A is homogeneous then either $A^2=0$ or $\text{dim}A=1$. Now I want to check this property for a finite dimensional simple Li... | https://mathoverflow.net/users/40491 | homogeneous algebras | The simple Lie algebra $W(1,2)^{(2)}$ of dimension 3 over $GF(2)$ obviously contains 7 subspaces of dimension 1, but its automorphism group has order 6 (see e.g. section 5.3 of the paper "B. Eick: Some new simple Lie algebras in characteristic 2: J. Symbol. Comput. 45, 943 -- 951 (2010)"). It is then clear that this Li... | 2 | https://mathoverflow.net/users/14653 | 161539 | 84,840 |
https://mathoverflow.net/questions/161526 | 1 | Formal series over a field (or ring) k can be composed in the following sense: given $y \in k[[x]]$, $y$ lying in the maximal ideal, there exists a unique map of topological rings $k[[x]] \to k[[x]]$ mapping $x$ to $y$.
Is there a similar statement for Puiseux series, $\cup\_n k((x^{1/n}))$? I am not sure what a corr... | https://mathoverflow.net/users/2234 | composition of Puiseux series? | Yes, if the second series is in the maximal ideal of $k[[x^{1/n}]]$ for some $n$. Simply write it as $x^{a/n}$ times a constant times a series whose leading term is $1$. Then we can take the $m$th root of this series for any integer $m$, with the $m$th root of $x^{a/n}$ equal to $x^{a/(mn)}$, the $m$th root of the cons... | 3 | https://mathoverflow.net/users/18060 | 161542 | 84,842 |
https://mathoverflow.net/questions/161515 | 3 | Let A be the disk algebra, of continuous functions on the closed disk holomorphic on the interior, with sup-norm denoted || . || . Let x be an interior point of the disk. Does there exist a proper A-ideal J such that the sup over nonzero elements of J of | f(x) | / || f || is 1 ?
| https://mathoverflow.net/users/35486 | ideals in the disk algebra | How about the following: let $M\_1=\{f\in A : f(1)=0\}$, which is a maximal ideal in $A$, and let $f\_n(z) = (1-z)^{1/n} = \exp (n^{-1}\log(1-z))$ where we take $(-\infty,0]$ as our branch cut for $\log$ (the point being that $z\mapsto 1-z$ maps the open unit disc into the open right-half-plane).
Then $(f\_n)$ is a s... | 3 | https://mathoverflow.net/users/763 | 161543 | 84,843 |
https://mathoverflow.net/questions/160599 | 18 | Let $X\_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:
$$dX\_t = f(X\_t)\,dt + dW\_t,$$
where $W\_t$ is an $n$-dimensional Wiener process and $f$ is of class $C^2\_b$, i.e., it is twice differentiable and the function and all its derivatives up to order 2 are bounded. For ... | https://mathoverflow.net/users/24041 | Fictitious density of paths of diffusion processes outside the Cameron--Martin space | Here's a proof of the statement for $f=0$, so that $X=W$ is a Wiener process. (The proof with general $f$ is a bit more involved, and I give this further below). I'll base the proof on the following simple result. Here, I am using $B\_\epsilon=\left\{\omega\in\mathcal{W}^n\colon\sup\_{t\in[0,1]}\lVert\omega(t)\rVert\le... | 9 | https://mathoverflow.net/users/1004 | 161544 | 84,844 |
https://mathoverflow.net/questions/161552 | 1 | Let $P$ be certain property. Let $S \subset \mathbb{C}^n\times \mathbb{C}^m$ be a set of closed points such that for any point in $S$, it satisfies the property $P$. I know for any $x \in \mathbb{C}^n$, the set $(x, \mathbb{C}^m) \cap S$ is an open set (in Zariski topology, but may be empty) and for any $y \in \mathbb{... | https://mathoverflow.net/users/29730 | constructible set and fibre product | No, you need other assertions. For example, let $f\colon \mathbb{C}\to \mathbb{C}$ be any bijective function. Then, the complement of the graph of $f$ in $\mathbb{C}^2$ satisfies your property: the intersection with each fiber of each projection is just $\mathbb{C}$ minus one point, so is open in the Zariski topology.
... | 2 | https://mathoverflow.net/users/23758 | 161557 | 84,849 |
https://mathoverflow.net/questions/161572 | 1 | In this question all objects are real analytic.(manifolds, differential forms..)
Assume that $M$ is a compact manifold and $\alpha \in \Omega^{1}(M)$ is a one form.
We define a map $\phi:\Omega^{\*}(M)\to \Omega^{\*+1}(M)$ with wedge product; $\phi(\beta)=\alpha \wedge \beta$. Then $\phi \circ \phi=0$. Then we have... | https://mathoverflow.net/users/36688 | A cohomology associated to a 1- form | Suppose that $\alpha$ is nowhere zero. A differential form $\beta$ satisfies $\alpha\wedge \beta=0$ just when $\beta=\alpha \wedge \gamma$ for some $\gamma$ by Cartan's lemma. So the cohomology vanishes, finite dimensional. On the other hand, take $\alpha=0$. Then the kernel is everything, the image nothing, so the quo... | 5 | https://mathoverflow.net/users/13268 | 161579 | 84,854 |
https://mathoverflow.net/questions/161563 | 4 | Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the area and my questions are not clear-enough).
1)What do we mean when we say two contact structures $(M,\xi); (M,\xi')$ ... | https://mathoverflow.net/users/48796 | Comparing Contact Structures: What do we Mean when we Say that two Contact Structures are Homotopic/Eliashbergs Class. of OT structures | I hope I'm not mucking up some definitions below, as I am by no means an expert, but let me attempt to answer your questions. No doubt you have a copy, but just for others, [here](http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf) is a link to the relevant paper of Eliashberg, by the way.
You might also ... | 4 | https://mathoverflow.net/users/353 | 161584 | 84,858 |
https://mathoverflow.net/questions/161586 | 19 | (I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...)
>
> Are there examples of results in "classical" [\*] graph theory that have
> been achieved by using simplicial complex methods?
>
>
>
Another terse way of putting that question is perhaps:
>
> Can methods from 05E45 be u... | https://mathoverflow.net/users/22051 | What have simplicial complexes ever done for graph theory? | There are quite a few examples where simplicial complexes, more general complexes, and algebraic topology in general had important impact on graph theory. (Usually, the applications are indirect and by looking at the graph as a 1-dimensional simplicial complex.) Some such applications are even mentioned in the book you... | 31 | https://mathoverflow.net/users/1532 | 161588 | 84,860 |
https://mathoverflow.net/questions/161072 | 35 | By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ (h^{11}-240,h^{21}+240)$. This shift, together with mirror symmetry accounts for a certain symmetry in the top portion of the ... | https://mathoverflow.net/users/38468 | What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds? | In the mathoverflow answer
[Today's world record on the Betti numbers of Calabi-Yau three-folds.](https://mathoverflow.net/questions/110535/todays-world-record-on-the-betti-numbers-of-calabi-yau-three-folds/110547#110547),
the paper <http://arxiv.org/abs/arXiv:1207.4792> is cited. In this paper, the shift of Hodge numb... | 16 | https://mathoverflow.net/users/25309 | 161601 | 84,864 |
https://mathoverflow.net/questions/161603 | 1 | Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a mapping:
$$
hol: L\mapsto hol(L)\in Aut(V),
$$
the holonomy map, which gives us the (linear) transformation of vectors a... | https://mathoverflow.net/users/30366 | Possible homotopy-theoretical approach to Gauss-Bonnet | Considering your second display, you probably mean $\mathrm{hol}^1\colon\pi\_2(A)\to\pi\_1(S^1)$. In any case, for all closed surfaces $A$ except $S^2$ and $\mathbb{R}\mathrm{p}^2$, you have $\pi\_k(A)=0$ for all $k\ge2$, so there's no hope. For proof, just consider the uniformization.
| 4 | https://mathoverflow.net/users/44953 | 161609 | 84,869 |
https://mathoverflow.net/questions/161604 | 6 | Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G\_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us assume that $k$ is perfect. In general, $G\_{\operatorname{red}}$ is not normal in $G$ (See SGA 3, VI$\_A$, 0.2). The Wiki... | https://mathoverflow.net/users/21885 | Is $G_{\operatorname{red}}$ normal in $G$? | The claim is false. Over a field of characteristic $p>0$, take for $G$ the semidirect product $\alpha\_p\rtimes\mathbb{G}\_\mathrm{m}$ where $\mathbb{G}\_\mathrm{m}$ acts on $\alpha\_p$ by scaling. Then $G$ is connected but $\mathbb{G}\_\mathrm{red}=\{0\}\times \mathbb{G}\_\mathrm{m}$ is not normal in $G$.
| 10 | https://mathoverflow.net/users/7666 | 161613 | 84,871 |
https://mathoverflow.net/questions/161556 | 5 | Let $(i\_{n})$ be a strictly increasing sequence of natural numbers,
$(v\_{n})$ be an unbounded sequences of natural numbers
and $M\geq 2$. Denote by $\mathcal{I}(i\_{n}, v\_{n}, M)$ the set of all irrational numbers $\alpha=[a\_{1}, a\_{2},...,a\_{s},...)\in (0,1)$ which is $a\_{i\_{n}}\leq v\_{n}$ and
$a\_{s}\leq ... | https://mathoverflow.net/users/46144 | Lebesgue measure of some set of irrational numbers | The continued fraction expansion is related to the Gauss transformation $T:(0,1)\to(0,1)$, defined by
$$ Tx:=\frac{1}{x} \mod 1. $$
(Indeed, if $x=[a\_1,a\_2,\ldots)$, then $Tx=[a\_2,a\_3,\ldots)$.)
It is well known that $T$ admits an absolutely continuous invariant probability measure $\mu$, given by
$$ \mu(A):=\fra... | 10 | https://mathoverflow.net/users/19603 | 161617 | 84,872 |
https://mathoverflow.net/questions/160385 | 0 | Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th Semester). For me, besides different ways to solve such particular equations (linear or not, constant coefficients or not , or... | https://mathoverflow.net/users/34304 | Teaching profession:Differential Equations and Mean Value Theorems | (This is an answer to my secondary question above.)
A very good reference I came to the use of which in my teaching is :
Ordinary Differential Equations by Wolfgang Walter , translated by Russel Thompson (Graduate Texts in Math., Springer 1998). That is a living and lively teaching book. It contains many explicit ex... | 0 | https://mathoverflow.net/users/34304 | 161622 | 84,875 |
https://mathoverflow.net/questions/161027 | 0 | Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form
$$u\_t + (-\Delta)^s u = f$$
where the nonlocal operator is the fractional Laplacian)
in the setting of Sobolev spaces (as opposed to Hoelder spaces, which I hope to avoid completely if possible)? I am only interested in the well-p... | https://mathoverflow.net/users/48222 | Nonlocal (parabolic) PDEs in the Sobolev space setting | Basically the well-posedness is very much the same. I recommend the book by Majda and Bertozzi "Vorticity and incompressible flow". For nonlinear equations in different contexts you may check <http://arxiv.org/pdf/1306.6197>, <http://arxiv.org/pdf/1211.5392>, <http://arxiv.org/pdf/0806.1180>, <http://arxiv.org/abs/math... | 1 | https://mathoverflow.net/users/33135 | 161631 | 84,879 |
https://mathoverflow.net/questions/161450 | 4 | Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal B}$.
Assume the following:
1. $f(X)$ is independent of $Y$
2. $g(Y)$ is independent of $X$
Can we say that there e... | https://mathoverflow.net/users/47772 | Joint probability distribution as functions | No. Here is a counterexample. Let $\mathcal{A} = \{1,2,3,4\}$, $\mathcal{B} = \{1,2\}$, and $f(x) = g(x) = \lceil\frac{x}{2}\rceil$. Let the joint probability mass function of $X$ and $Y$ be given by the matrix
\[
P = \frac{1}{8}\begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{bmatri... | 3 | https://mathoverflow.net/users/5963 | 161633 | 84,880 |
https://mathoverflow.net/questions/161517 | 14 | Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a complete bibliography of work done in ergodic theory that has been directly motivated by this conjecture. What I have is thi... | https://mathoverflow.net/users/39115 | Furstenberg $\times 2 \times 3$ conjecture, bibliography | Well that will be some lengthy answer.
The first article that was published after the famous disjointness paper is another paper by Hillel called "Intersections of Cantor sets", it's related to the motivating question to the $\times 2,\times 3$ which arise from normality of numbers and fractals.
This can be seen as s... | 11 | https://mathoverflow.net/users/8857 | 161636 | 84,881 |
https://mathoverflow.net/questions/161637 | 0 | I see many wonderful papers where the authors include some script written in Maple, Macaulay or other software that are needed for their proof. How do you insert that in your tex file?
| https://mathoverflow.net/users/4096 | Inserting maple or macaulay script in a paper | For a lot of languages, [lgrind](http://www.ctan.org/pkg/lgrind) gives nice results, but it is no longer maintained and a bit long in the tooth. The [listings](http://en.wikibooks.org/wiki/LaTeX/Source_Code_Listings) package is a bit more modern.
| 2 | https://mathoverflow.net/users/5734 | 161639 | 84,883 |
https://mathoverflow.net/questions/161600 | 1 | Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least $C$. Is $u$ a constant function? ($u(Rx)$ is a rotation of $u$ on $S^2$)
| https://mathoverflow.net/users/45673 | Symmetry on a sphere | The answer is no. One may consider the function $u(x,y,z)=x$ on the sphere $x^2+y^2+z^2=1$. If $R(1,0,0)=(1,0,0)$ then your set is empty, otherwise it is an open hemisphere.
One may replace $u=x$ by any strictly monotonous function in $x$.
| 3 | https://mathoverflow.net/users/17581 | 161647 | 84,887 |
https://mathoverflow.net/questions/161571 | 3 | The theory $\mathrm{ACA}\_0$ is not reflexive (because it is finitely axiomatisable and cannot prove its own consistency). So how, if at all, is it possible to prove that $\mathrm{Q+Con(PA)}$ cannot be interpreted in $\mathrm{ACA}\_0$?
(I am assuming that for an interpretation of the second-order language of arithmet... | https://mathoverflow.net/users/15482 | How do you prove that Q+Con(PA) can't be interpreted in ACA_0? | Firstly, your proposed reasoning does not work since $Con(PA)$ is a $\Pi^0\_1$ statement.
Secondly, your parenthetical assumption is not in accordance with the general meaning of interpretations, and the result is provable without this assumption.
>
> The fact that $Q + Con(PA)$ is not interpretable in $ACA\_0$ i... | 8 | https://mathoverflow.net/users/9269 | 161662 | 84,890 |
https://mathoverflow.net/questions/161659 | 3 | Does any one know how to calculate the following integration?
$$
\int\_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question [here](https://mathoverflow.net/questions/161068/has-anyone-seen-this-series) , based on which one can have a upper bound. Can any ... | https://mathoverflow.net/users/36814 | How to integrate an exponential function of an exponential function? | You can expand the integrand in powers of z. The coefficient of $z^k$ is
$$e^{-ky^2}\sum\_{j=1}^{k-1} {1\over j!(k-j)!}=e^{-ky^2}(2^k-2)/k!.$$
This yields the following series expression for the integral:
$$\sum\_{k=1}^\infty \sqrt{\pi\over k}(-2+2^k)z^k/k!.$$
| 8 | https://mathoverflow.net/users/12120 | 161668 | 84,892 |
https://mathoverflow.net/questions/161683 | 3 | Is the following problem undecidable?
>
> Given a sequence of $n$ gaps $d\_1,d\_2,...,d\_n$, does there exist a sequence of $n+1$ primes $p\_1,p\_2,...,p\_{n+1}$ such that $p\_{i+1} - p\_i = d\_i$ ?
>
If not, is the problem still undecidable if we pick a fixed $n$, e.g. only one gap (n=1), or only two gaps (n=2)... | https://mathoverflow.net/users/35419 | Decidability of prime gap sequences | This is probably unknown. The problem is that we know very little about the distribution of prime numbers. In general, decidability questions only become interesting when we "have a good grasp" about the object we are manipulating. It is easy to generate decidability questions which are only hard because of our limited... | 6 | https://mathoverflow.net/users/12126 | 161684 | 84,899 |
https://mathoverflow.net/questions/161676 | 13 | Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}\_{\omega\_1\omega}$ - at least, assuming the models in question have all the relevant infinitary formulas. In both cases, the easiest way to prove this i... | https://mathoverflow.net/users/8133 | The (non-)absoluteness of second-order elementary equivalence | **In this edit I have modified my previous argument, and added the appropriate conditions to Harrington's theorem.**
>
> I will explain below why it is possible to arrange a model $M$ of $ZFC$ that:
>
>
> **(a)** contains two nonisomorphic countable structures $\cal{A}$ and $\cal{B}$ (in a finite vocabulary) that... | 15 | https://mathoverflow.net/users/9269 | 161694 | 84,902 |
https://mathoverflow.net/questions/161678 | 5 | Assume a process with Itô dynamics of the generic form
$$dX\_t=\mu(t,X\_t)dt+\sigma(t,X\_t)dW\_t$$
and let $f:\mathbb{R}\to\mathbb{R}$ be borel-measurable. Is the following function smooth ?
$$g(t,x)=\mathbb{E}[f(X\_T)|\mathcal{F}\_t]$$
I remember comming upon the proof of above once but I cannot find it any longe... | https://mathoverflow.net/users/24196 | Smoothness of $g(t,x)=\mathbb{E}[f(X_T)|\mathcal{F}_t]$ | This depends on what you assume of $\mu$, $\sigma$, and $f$. I'll provide a few examples. Let's assume $f$ is bounded throughout, since what I'll say below is still true in the unbounded case under suitable growth assumptions.
First, as long the SDE is well-posed (in the sense of weak existence and uniqueness in law)... | 5 | https://mathoverflow.net/users/44169 | 161716 | 84,909 |
https://mathoverflow.net/questions/161715 | 4 | On a Riemannian manifold $M$ with riemann curvature tensor $R\_{\mu\nu\rho\sigma}$ written as (endomorphism valued) curvature two-tensor of the Levi-Civita connection $R=R\_{\mu\nu}dx^\mu\wedge dx^\nu$, consider the following term.
$$\int\_M tr(R\wedge \*R)=2\int\_M tr(R\_{\mu\nu}R^{\mu\nu})dvol\_g=2\int\_M tr(R\_{\m... | https://mathoverflow.net/users/19325 | What is the meaning of Yang-Mills action evaluated on Levi-Civita connection? | You can find information about functionals that are quadratic in the curvature in Besse's book *Einstein Manifolds*. In particular, see Chapter 4, Section H and the references cited therein. Your questions about topological interpretations and possible connections with GR, etc., are addressed in some of those reference... | 7 | https://mathoverflow.net/users/13972 | 161720 | 84,911 |
https://mathoverflow.net/questions/161726 | 2 | Let M be a non-commutative monoid. It is possible that all representation of M are one dimensional ??
(for groups the answer is negative. Take a non zero x=[a,b]. Take a representation where x does not act trivially. This rep can not be one dimensional. But on monoids I can not take the commutator)
| https://mathoverflow.net/users/48866 | Rep of Non-Commutative Monoids | The question is vague. I am assuming you want finite monoids over the complex field, although I could answer over any field, and I am assuming you want irreducible reps. There are noncommutative monoids whose irreducible representations are all 1-dim. I characterized with Almeida, Margolis and Volkov all such monoids (... | 5 | https://mathoverflow.net/users/15934 | 161729 | 84,915 |
https://mathoverflow.net/questions/158799 | 9 | Define a pointclass to be:
* *boldface inductive-like* if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and
* *lightface inductive-like* if it is $\omega$-parameterized, has the scale p... | https://mathoverflow.net/users/1682 | Obtaining a lightface pointclass from a boldface one | See Lemma 3.4 and preceding paragraph of my paper "A characterization of jump operators", JSL vol. 53, 1988.
Howard Becker
| 4 | https://mathoverflow.net/users/48871 | 161742 | 84,919 |
https://mathoverflow.net/questions/161732 | 4 | Let $G$ be a finite group of odd order. Suppose that $G$ has a *real* 4-dimensional faithful representation. Is it true that $G$ should be abelian in this case?
| https://mathoverflow.net/users/43459 | Real representation of group of odd order | We can assume $G\leq O(4)$ wlog. Because $|G|$ is odd, we have in fact $G\leq SO(4)$ ($g\mapsto g^2$ is a bijection and hence every $g$ must have positive determinant) and a monomorphism $G\to PSO(4) \cong SO(3)\times SO(3)$. Now lets look at the two projections $\pi\_1(G), \pi\_2(G)\leq SO(3)$. We consult the [list of... | 8 | https://mathoverflow.net/users/3041 | 161743 | 84,920 |
https://mathoverflow.net/questions/161713 | 4 | I'm not sure if this is research level, so feel free to vote to migrate.
Suppose we have a complete boolean algebra $A$, with a *dense*, $\sigma$-complete subalgebra $B$, and a $\sigma$-complete homomorphism $h : B \to C$, where $C$ is complete. Does $h$ have a $\sigma$-complete extension $h' : A \to C$?
EDIT: I fo... | https://mathoverflow.net/users/11145 | extending $\sigma$-complete boolean homomorphism | If the Boolean algebra $A$ is c.c.c., then of course every
$\sigma$-complete dense subalgebra $B$ is all of $A$, and so in this case the
answer is trivially affirmative.
If $A$ is not c.c.c., however, then fix an uncountable maximal antichain
$X\subset A$, which we may assume has size $\aleph\_1$, and proceed along J... | 4 | https://mathoverflow.net/users/1946 | 161750 | 84,924 |
https://mathoverflow.net/questions/161621 | 5 | Let us first review the usual construction of a Mukai flop. Suppose $M$ is a smooth $2m$-dimensional projective variety over $\mathbb C$ containing a closed $m$-dimensional subvariety $W$, and suppose we have a projective birational morphism $\nu:M\to\overline M$ such that $\nu$ is an isomorphism away from $W$, and suc... | https://mathoverflow.net/users/2177 | Recognizing a Mukai flop | If you consider your flop a $(K+D)$-flip, then you might find the additional condition you need. Any flop is actually a $(K+D)$-flip for a well chosen $D$. (For the definition of a $(K+D)$-flip see 3.33 of [Birational Geometry of Algebraic Varieties
By Kollár-Mori](http://books.google.com/books?id=YrysxvPbBLwC&printse... | 4 | https://mathoverflow.net/users/10076 | 161753 | 84,925 |
https://mathoverflow.net/questions/161701 | 19 | Is there a finite group $G$ and a divisor $d$ of $|G|$ so that $G$ contains exactly two subgroups of order $d$?
The motivation for this question is an old qual problem (see <http://www.math.wisc.edu/~passman/qualjan01.pdf>, problem 1). If $G$ is such a group and $A$ and $B$ are the two subgroups of order $d$, then it... | https://mathoverflow.net/users/48142 | Finite groups $G$ so that $G$ has exactly two subgroups of a given order | I asked Bob Guralnick this question in class this morning. I expected he would have something to contribute. I did not expect (although perhaps I should have) that the existence of such a group $G$ and such a divisor $d$ of $|G|$ is his own result dating back to 2002.
The title of the article he sent me is "Groups wi... | 25 | https://mathoverflow.net/users/32261 | 161754 | 84,926 |
https://mathoverflow.net/questions/161757 | 1 | This question was asked in
<https://math.stackexchange.com/questions/729648>. Since I did not get any answer I am asking it here.
In an answer in Mathoverflow I see an answer but I could not understand it. May anybody help me for Understanding it. The question and its answer are:
[On the character degrees of a fin... | https://mathoverflow.net/users/48877 | On the Complement of a subgroup | If a finite group $G$ has an Abelian Sylow $p$-subgroup $P,$ then by transfer, we have $P \cap G^{\prime} \cap Z(G) = 1.$ Your group $N = O\_{p}(N) \times H.$ Since $G/N \cong {\rm PSL}(2,p)$ it now follows that $O\_{p}(N) \leq Z(G).$ Furthermore, $O\_{p}(N)$ is complemented in $G$ by Gaschutz's theorem ($G$ can't have... | 3 | https://mathoverflow.net/users/14450 | 161760 | 84,929 |
https://mathoverflow.net/questions/161731 | 5 | Inspired by:
[Conjecture on NP-completeness of tesselation of Wang Tile up to finite size](https://mathoverflow.net/questions/157707/conjecture-on-np-completeness-of-tesselation-of-wang-tile-up-to-finite-size)
And the practicality of this topic (solving tessellation on a lattice):
[coloring in lattice](https://ma... | https://mathoverflow.net/users/40780 | practical algorithms for np complete problems | This is a very general question and difficult to answer. The kind of approach people will use is dependent on a lot of factors like the following:
* how much work can i spend on building my solver?
* what do i expect from my solutions (e.g. proven guarantees)?
* what kind of problem do i have (e.g. discrete search-sp... | 9 | https://mathoverflow.net/users/48880 | 161762 | 84,931 |
https://mathoverflow.net/questions/161768 | 27 | The mapping class group of a manifold is the group $\pi\_0 Diff(M)$ of components of the diffeomorphism group. There are several variations: oriented manifolds and orientation preserving diffeomorphisms, etc.
I am interested in finding out if we know any sorts of general properties of these groups. Are they finitely... | https://mathoverflow.net/users/184 | Are the mapping class groups of manifolds finitely presentable? | These things can be pretty wild. For instance, for $n \geq 5$ the mapping class group of the $n$-torus is not even finitely generated : it is a split extension of $\text{GL}\_n(\mathbb{Z})$ by an infinite rank abelian group. See Theorem 4.1 of
A. E. Hatcher, Concordance spaces, higher simple-homotopy theory, and appl... | 30 | https://mathoverflow.net/users/317 | 161775 | 84,939 |
https://mathoverflow.net/questions/161538 | 7 | I am reading Uri Abraham's chapter on Proper Forcing in the *Handbook of Set Theory* and I have a quite trivial question on the definition of $\alpha$-proper forcing. Since there are many equivalent definitions and characterizations, let me recall those related to my question.
If $\lambda$ is a fixed regular cardinal... | https://mathoverflow.net/users/41274 | On the definition of $\alpha$-proper poset | Does a proof by contradiction work here?
Your (\*) is stated as a theorem and not a hypothesis, so I am assuming by $(\*)$ you really mean the right-hand side. Since it depends on $\alpha$, I'll call it $(\*)\_\alpha$.
Given $P$, we want to see that $(\*)\_\alpha$ implies $\alpha$-proper for all $\alpha<\omega\_1$.... | 4 | https://mathoverflow.net/users/18128 | 161779 | 84,940 |
https://mathoverflow.net/questions/161767 | 4 | Suppose that we have a Banach space $X$ together with some locally convex Hausdorff topology on $X$, weaker than the one given by the norm, which makes the unit ball of $X$ compact. Is $X$ (Banach-space) isomorphic to $Y^\*$ for some Banach space $Y$?
I am aware of a related result of Kajser which says that if $X$ is... | https://mathoverflow.net/users/15129 | Introducing a dual space structure | Yeah, this is true. Let ${\cal T}$ be an LCH topology on $X$ which makes the unit ball compact. Then every ${\cal T}$-continuous linear functional takes the unit ball to a compact subset of the scalar field and hence is bounded for the norm. We also know from general theory that the ${\cal T}$-continuous linear functio... | 4 | https://mathoverflow.net/users/23141 | 161782 | 84,942 |
https://mathoverflow.net/questions/161756 | 15 | Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of certain strength.
For instance, weakly compact cardinals are precisely the cardinals $\kappa$ where the $\kappa$-compactnes... | https://mathoverflow.net/users/22277 | Does there exist a supercompactness theorem? | The answer is yes. In the paper "[Compactness for omitting of types](http://dx.doi.org/10.1016/0003-4843%2878%2990007-4)" (Annals of Mathematical Logic, vol 14(1) (1978), pp. 39-56), Benda gives such a characterization. He proves the following:
**Theorem.** The following are equivalent:
1) $\kappa$ is supercompact,... | 16 | https://mathoverflow.net/users/11115 | 161786 | 84,946 |
https://mathoverflow.net/questions/161752 | 2 | Consider the following language:
$L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$.
($G$ is undirected).
[This answer](https://stackoverflow.com/questions/10825249/find-all-vertices-on-all-simple-paths-be... | https://mathoverflow.net/users/47499 | Is undirected short-simple-path-through-3-vertices decidable in polynomial time? | I think this reduction to min-cost flow has a chance of working. First we change the graph into a network with directed edges. Each vertex $x$ other than $s,t,v$ is replaced by a pair of vertices $x^+,x^-$ joined by an edge $x^+{\to}x^-$ with capacity 1 and cost 1. Each edge of the original is replaced by two directed ... | 2 | https://mathoverflow.net/users/9025 | 161801 | 84,948 |
https://mathoverflow.net/questions/160665 | 4 | I would like to know the proof of the following theorem:
>
> Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f\_n)\_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ be such that $f\_{n}(t)\rightharpoonup f(t)$ in $Y$ for a.a. $t\in I$. If $f\_n$ is bounded in $L^p(I... | https://mathoverflow.net/users/45729 | A useful criterion in vector integration | Here is an answer when $p<\infty$ and $I$ bounded. The argument might be adapted to $p=\infty$ by replacing some weak $L^p$ convergences below by the weak-\* $L^{\infty}$ one, and I really believe $I$ bounded is not an issue (otherwise argue locally on any $J\subset I$)
Just a few obvious remarks first: if $X$ is ref... | 4 | https://mathoverflow.net/users/33741 | 161802 | 84,949 |
https://mathoverflow.net/questions/161703 | 12 | In *Rudin - Real & Complex Analysis* we have the following
**Lemma 6.3.** If $z\_1, \ldots, z\_n \in \mathbb{C}$ then there is a subset $S \subseteq \{1,\ldots,n\}$ for which
$$\left|\sum\_{k \in S} z\_k\right| \geq \frac1{\pi} \sum\_{k=1}^n |z\_k|.$$
This is used by Rudin to prove that a complex measure $\mu$ has ... | https://mathoverflow.net/users/nan | Subset of vectors whose sum has a large norm | Finally, I found that a stronger result is known:
**Theorem.** For any integer $d \geq 1$ and $\delta \in [0,1]$, let
$$C\_{d,\delta} = \frac{\Gamma(\tfrac{d}{2})(1-\delta^2)^{(d-1)/2}}{2\sqrt{\pi}\,\Gamma(\tfrac{d+1}{2})}.$$
Then for all $v\_1, \ldots, v\_n \in \mathbb{R}^d$ there exists $u \in \mathbb{R}^d$ with $\... | 2 | https://mathoverflow.net/users/nan | 161804 | 84,950 |
https://mathoverflow.net/questions/161800 | 1 | Let $f:ℝ²→ℝ$ be an arbitrary harmonic function. **A level curve in two dimensions is a curve on which the value of a function $f(x,y)$ is a constant**. My **question** is: Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$ (this mean that the... | https://mathoverflow.net/users/25947 | Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$ | Modulo an elementary transformation you are asking for harmonic functions which are zero on a line and nowhere else. Such a function is linear. This is Theorem I in [*Short proofs of three theorems on harmonic functions*](http://www.ams.org/journals/proc/1988-102-04/S0002-9939-1988-0934865-6) by H.P. Boas and R.P. Boas... | 5 | https://mathoverflow.net/users/12120 | 161808 | 84,953 |
https://mathoverflow.net/questions/161576 | 1 | For a general plane curve of degree $\ge 4$, the number of bitangent lines is known. Also, I found that the number of tritangent planes have been worked out for some space curves given by intersection of two surfaces (e.g., quadric and cubic).
I have the following related questions:
(1) Is it true that every smooth... | https://mathoverflow.net/users/47552 | multi-tangent space for algebraic curves | This is an answer to the second question, for which integers $r$ does a "general" complete intersection curve has an $r$-tangent hyperplane. The computation below show that for every integer $r>n$, for a sufficiently "general" complete intersection curve $C$, there exists no $r$-tangent hyperplane $H$. For $r=n$, the a... | 2 | https://mathoverflow.net/users/13265 | 161814 | 84,956 |
https://mathoverflow.net/questions/161811 | 4 | For a (commutative) CY manifold of dimension $n$, Serre duality implies that there are ifunctorial isomorphisms
$$\operatorname{Hom}\_{D^b(X)}(E,F)\cong\operatorname{Hom}\_{D^b(X)}(F,E[n])^\*$$
in the bounded derived category of coherent sheaves and $[n]$ denotes the $n$-fold shift.
Kontsevich and Soibelman define a... | https://mathoverflow.net/users/5323 | Why should noncommutative CYs be dgas? | For any reasonable smooth scheme the category $D^b(X)$ is equivalent to perfect DG-modules over a DG-algebra. So, the world of schemes is embedded into the world of DG-algebras. This is why we think about DG-algebras as about noncommutative spaces.
| 5 | https://mathoverflow.net/users/4428 | 161816 | 84,957 |
https://mathoverflow.net/questions/161815 | 6 | Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not $2$-transitive, i.e., the action of $G$ on pairs of distinct elements of $X$ is not transitive. Because $G$ is not $2$-transitive, t... | https://mathoverflow.net/users/398 | Primitive, non-2-transitive groups with very large orbitals? | Let $G=S\_n$, acting on the ${{n} \choose {2}}$ $2$-sets from $[n]$. The stabilizer of a $2$-set $X$ is maximal in $S\_n$ and has two orbits on the remaining $2$-sets, one consisting of those that intersect $X$ nontrivially. The other orbit has size ${{n-2} \choose {2}}$. Now let $n \rightarrow \infty$.
| 12 | https://mathoverflow.net/users/36466 | 161819 | 84,959 |
https://mathoverflow.net/questions/161036 | 3 | The constant term of the Eisenstein series (for an adele group $GL\_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for example,
$$E\_N(\varphi(s),g)=\varphi(s)(g)+M(s)\varphi(s)(g).$$
The resulting the trace formula involves the logarithmic d... | https://mathoverflow.net/users/48554 | What is the logarithmic derivative of an (intertwining) operator? | The functional equation gives $M(s)M(-s)=1$, so yes $M(-s)=M^{-1}(s)$. Note that $M(s)'$ is not an intertwiner, only intertwines the compact group $K$ in $GL\_2(A)$. The irreducible $K$-isotypes of $GL\_2(A)$-reps have all multiplicity one, meaning when you restrict to $K$ irreducible representations of $GL\_2(A)$ deco... | 1 | https://mathoverflow.net/users/10400 | 161822 | 84,960 |
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