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https://mathoverflow.net/questions/236192 | 1 | I have the following recurrence equation:
$$(\mu\ n + \nu) f\_{n} + J\Phi^{\*} \sqrt{n+1}f\_{n+1} + J\Phi\ \sqrt{n}f\_{n-1} = 0$$
for complex numbers $f\_{n}$ where $n = 0,1,2,3,...,\infty$ and complex $\Phi$ and real $\mu, \nu, J$. Is there a way to find a general expression for $f\_{n}$ in terms of $f\_0$?
A few fi... | https://mathoverflow.net/users/57699 | find solution of complex number recurrence equation | Multiply the equation with $x^n/sqrt(n!)$. Define a generating function $g(x)$ with coefficients $g\_n:=f\_n/sqrt(n!)$.
You will get a homogeneous differential equation for $g(x)$.
| 2 | https://mathoverflow.net/users/90355 | 236218 | 109,338 |
https://mathoverflow.net/questions/236090 | 4 | By a $k$-variety, we will mean a separated scheme of finte type over a field $k$. Let $k$ be of characteristic 0. Given a smooth quasi-projective $k$-variety $X$, there is a projective $k$-variety $\bar{X}$ containing $X$. Since char$(k)=0$, we can take a resolution of singularities $\pi:\widetilde{X}\to\bar{X}$ where ... | https://mathoverflow.net/users/39193 | Uniqueness of smooth compactification upto a smooth morphism | This desire fails on a much more basic level. A smooth morphism is flat, and you can't even have a flat morphism between two smooth compactifications.
A flat morphism has equidimensional fibers. If it is also proper and birational, then it is finite and a finite birational morphism mapping onto something normal is a... | 9 | https://mathoverflow.net/users/10076 | 236227 | 109,340 |
https://mathoverflow.net/questions/235399 | 2 | Basic discrete control theory mostly studies systems which can be represented as $x\_n=A(n)x\_{n-1}+B(n)u\_n$.
I wonder if optimal control of specific discrete systems of the type $x\_n = A(n,u)\cdot x\_{n-1}$ have been described in any paper?
Or maybe there is a control theory handbook where different systems and ... | https://mathoverflow.net/users/89951 | Specific discrete system $x_n = A(n,u)\cdot x_{n-1}$ control papers | If you are willing to consider the special case that $A(n,u)$ is linear in control $u$, then this becomes a bilinear control system (see for example, [the book by David Elliott](http://www.springer.com/us/book/9781402096129?token=prtst0416p)), for which optimal control results have appeared in literature.
| 2 | https://mathoverflow.net/users/18526 | 236228 | 109,341 |
https://mathoverflow.net/questions/236236 | 7 | Given a d-dimensional polytope P with n points, then what is the minimum number of simplices that are spanned by vertices of P? This question led my research to matroids and so my question is: what is the minimum number of bases af a matroid, that comes from a fulldimensional convex set with n vertices in dimension d? ... | https://mathoverflow.net/users/90363 | minimum number of bases of a matroid, that comes from a convex polytope | It looks that David Speyer strengthening holds. Namely, if a matroid $M$ on a set $E$, $|E|=n$, has rank $k\leqslant n$ and minimal circuit size at least $p$, $k\geqslant p-1$, than $M$ has at least $\binom{n-k+p-1}{p-1}$ bases. (In our situation $k=d+1$, $p=4$.)
Proof. Use induction. Cases $n=k$ and $k=p-1$ are clea... | 4 | https://mathoverflow.net/users/4312 | 236239 | 109,342 |
https://mathoverflow.net/questions/236245 | 7 | Let $\mathcal{A}$ be a subcategory of $\mathcal{C}$. Let $D(\mathcal{A})$ and $D(\mathcal{C})$ be the associated derived categories. We can define $D\_\mathcal{A}(\mathcal{C}) = \{X \in \mathcal{C}\hspace{2pt} |\hspace{2pt} H^i(X) \in \mathcal{A}\}$. We have natural inclusions $$D(\mathcal{A}) \to D\_\mathcal{A}(\mathc... | https://mathoverflow.net/users/57044 | Equivalence between a derived subcategory and a subcategory of the derived category | Take $C$ to be sheaves of abelian groups on the sphere, and let $A$ be the abelian subcategory of locally constant abelian groups. Then $A$ is equivalent to the category of abelian groups and so $Hom(\mathbb Z,\mathbb Z[2])$ is different depending on whether you take it in $D(A)$ or $D(C)$
| 4 | https://mathoverflow.net/users/52918 | 236249 | 109,344 |
https://mathoverflow.net/questions/236243 | 7 | Let $G$ be a transitive permutation group on a finite set $\Omega$. It is clear
that if $G$ is regular, then every proper subgroup of $G$ is intransitive. Is
there any other class of groups with this property? I mean that which transitive groups has no proper transitive subgroup?
| https://mathoverflow.net/users/27831 | Transitive permutation groups which all of their proper subgroups are intransitive | Consider the symmetric group $S$ on $n$ symbols in its action on the $k$-subsets of $\{1,\ldots,n\}$. If $k\ne2,4$ and $n=2k+1$, then the only transitive subgroups of $S$ are $S$ itself and the alternating group. Hence in this case the alternating group acts transitively but all its proper subgroups are intransitive.
S... | 13 | https://mathoverflow.net/users/1266 | 236260 | 109,352 |
https://mathoverflow.net/questions/236246 | 1 | Let $\gamma$ be a smooth planar curve. Assume that $\gamma$ divides the plane into two domains and, it addition, that one of these domains is unbounded and convex. What can be said about the behavior of the curvature of $\gamma$ at infinity: does it tend to zero, if yes, is there a more precise asymptotics which holds ... | https://mathoverflow.net/users/75826 | Unbounded convex domains in 2D | No, the curvature does not need to go to $0$. Consider starting with a polygonal curve with infinitely many corners (e.g. the line segments joining $[n,n^2]$ to $[n+1, (n+1)^2]$ for integers $n$), and smoothing out the corners. If the angle at the $x=n$ corner changes by $\epsilon\_n$ and the piece of curve where this ... | 4 | https://mathoverflow.net/users/13650 | 236262 | 109,353 |
https://mathoverflow.net/questions/236229 | 4 | Let $\mathcal{S}=S\_2(\Gamma\_0(N) \cap \mathbf{Z} [[ q ]]$ be the set of cusp forms of weight $2$ on $\Gamma\_0(N)$ with integral coefficients.
Let $f \in \mathcal{S}$ be a normalized newform, so it has an associated elliptic curve $E\_f$ by Eichler-Shimura, and denote by $L$ the orthogonal complement of $f$ in $\ma... | https://mathoverflow.net/users/10547 | Congruence Primes and Modular Degrees | This, and more, is all in Theorem 2.1 of Agashe-Ribet-Stein:
<http://www.math.fsu.edu/~agashe/math/moddeg3.pdf>
| 4 | https://mathoverflow.net/users/90375 | 236263 | 109,354 |
https://mathoverflow.net/questions/236268 | 1 | There is a claim from a paper which I do not understand:
>
> Let $D$ be a domain in $\mathbb{R}^d$. Let $(p^{\eta})\_{\eta >0}$ be a family of densities for random variables on $(C[0,T], \mathbb{R}^d)$. Suppose that
>
>
> (i) For any bounded open domain $\mathcal{O}$ in $D$, the family of densities $(p^{\eta}|\_... | https://mathoverflow.net/users/62049 | A diagonalisation argument applied to density functions | Suppose $\mathcal O\_1 \subseteq \mathcal O\_2$. If a sequence $\eta\_k$ converges in $L^2([0,T], L^2(\mathcal O\_2))$ to $\eta$, then the restrictions
$R(\eta\_k) = \left.\eta\_k(t)\right|\_{\mathcal O\_1}$ also converge to $R(\eta) = \left. \eta(t)\right|\_{\mathcal O\_1}$ in $L^2([0,T], L^2(\mathcal O\_`))$, with $... | 1 | https://mathoverflow.net/users/13650 | 236272 | 109,358 |
https://mathoverflow.net/questions/236281 | 6 | Metric on a Riemannian manifold $(M,g)$ is *Einstein*, if for some function $\lambda\colon M\to \mathbb R$
$$
Ric(g)=\lambda g.
$$
It is well know, that such $\lambda$ is, in fact, a constant.
The notion of Einstein metric fits perfectly into the world of K\"ahler manifolds, in this case such a metric is called *K\"a... | https://mathoverflow.net/users/40950 | Chern-Einstein metrics on complex Hermitian manifolds | Let me write your equations instead as
$$\Theta^{(1)}\_{i\bar{j}}=\lambda\_1 g\_{i\bar{j}},$$
$$\Theta^{(2)}\_{i\bar{j}}=\lambda\_2 g\_{i\bar{j}},$$
where $\lambda\_1,\lambda\_2$ are real-valued functions. The Chern-Ricci form $\Theta^{(1)}=\sqrt{-1}\Theta^{(1)}\_{i\bar{j}}dz^i\wedge d\bar{z}^j$ is closed.
The case $... | 7 | https://mathoverflow.net/users/13168 | 236285 | 109,363 |
https://mathoverflow.net/questions/236286 | 3 | Is the canonical module of a Buchsbaum ring a Buchsbaum module?
| https://mathoverflow.net/users/23240 | Canonical module of a Buchsbaum ring | Yes. See Theorem 4.9 on page 138 of [Buchsbaum Rings and Applications](https://books.google.com/books?id=iVHooQEACAAJ) by Jürgen Stückrad and Wolfgang Vogel.
| 4 | https://mathoverflow.net/users/10076 | 236287 | 109,364 |
https://mathoverflow.net/questions/236288 | 5 | I want to understand compact complex manifolds $(M^{2n}, J)$ with the following property: there exists a collection $\{X\_i\}\_{i=1}^L$ of holomorphic vector fields (sections of $(T^{1,0}\_{\mathbb C} M)$) such that for all $p \in M$, $\{X\_i(p)\}\_{i=1}^L$ spans $(T^{1,0}\_{\mathbb C} M)\_p$. For instance, it is satis... | https://mathoverflow.net/users/40460 | Complex manifolds with spanning sets of holomorphic vector fields | On any compact complex manifold, the set of all global holomorphic vector fields is a finite dimensional Lie algebra. They span the tangent space at every point just when each component of the manifold is a homogeneous for the action of its biholomorphism group.
Some points in the proof: By compactness, all holomorp... | 7 | https://mathoverflow.net/users/13268 | 236290 | 109,365 |
https://mathoverflow.net/questions/235244 | 1 | Please, I need a small help with a reference.
Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and bounded.
I have heard, that the generalized cantor's theorem is the way to prove the existence, but I am unfortunate... | https://mathoverflow.net/users/89875 | Extremal of an L^1 continuous functional on a compact bounded set | Found it out.
There might have been a mistake, the proof should not be Cantors generalized theorem, but just a generalization of the basic fact, that a continuous function on a compact closed set attains extremas.
| 0 | https://mathoverflow.net/users/89875 | 236295 | 109,367 |
https://mathoverflow.net/questions/236264 | 6 | I'm trying to piece together a proof of the projective bundle formula from several incomplete sources. Here's the statement I'd like to prove:
>
> **Projective bundle formula:** Let $\pi: E \to X$ be a vector bundle of rank $r$ over a compact space $X$. Let $\mathbb{P}(E)$ be the projective bundle of $E$ with disti... | https://mathoverflow.net/users/22810 | (Geometric) Proof for the projective bundle formula in K-theory | First, for any bundle $V$ of dimension $d$ over $Y$ put
$$ \lambda(V)(t) = \sum (-1)^k[\Lambda^k(V)]t^{d-k} \in K^0(Y)[t]. $$
This is a monic polynomial of degree $d$ over $K^0(Y)$. It satisfies $\lambda(L)(t)=t-[L]$ if $L$ is a line bundle, and $\lambda(A\oplus B)(t)=\lambda(A)(t)\lambda(B)(t)$. Thus, if $L$ is isomo... | 5 | https://mathoverflow.net/users/10366 | 236300 | 109,368 |
https://mathoverflow.net/questions/236291 | 8 | Suppose $\mathsf{AD}\_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.)
I believe under $\mathsf{AD}\_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is homogeneous Suslin. Let $T$ be some tree so that $A = p[T]$. Suppose $\mathbb{P}$ is a small forcing in $V\_{\omega +... | https://mathoverflow.net/users/43354 | $\mathsf{AD}_\mathbb{R}$ and Elementary Embeddings | Yes, you can show this using your assumption that every set of reals in $L(A,\mathbb{R})$ is $\delta$-weakly homogeneously Suslin (Woodin's Pmax book, theorem 2.30). In that case $(A,\mathbb{R})^{\#}$ exists by closure of pointclass under countable unions. Alternatively you can assume that all sets of reals in $L(A,\ma... | 4 | https://mathoverflow.net/users/3859 | 236305 | 109,369 |
https://mathoverflow.net/questions/236266 | 33 | In this [this google+ post](https://plus.google.com/+UrsSchreiber/posts/NpBTyHu5ojM) of Urs Schreiber, he says: **"Grading over the sphere spectrum is supersymmetry"** and then he redirect us to the abstract idea of superalgebra (in nLab).
Are there some references (other than Kapranov and nLab) about these ideas?
| https://mathoverflow.net/users/83957 | What is the relation between the sphere spectrum and supersymmetry? | Let's agree that whatever "supersymmetry" means it has something to do with working in the symmetric monoidal category of super vector spaces (e.g. we might want to consider Lie algebras or commutative algebras in this category), or something like it. The question is what, if anything, this has to do with the sphere sp... | 22 | https://mathoverflow.net/users/290 | 236314 | 109,370 |
https://mathoverflow.net/questions/236317 | 3 | A semigroup $(A,\cdot)$ that is idempotent (i.e. $a^2=a$ for every element $a\in A$) is naturally generated by its subgroups (every element on itself constitutes a trivial group). I would like to know what is known about semigroups that are generated by (cyclic) groups? Again, such groups do not need to share a common ... | https://mathoverflow.net/users/18376 | What is known about semigroups that are generated by (cyclic) subgroups? | In the non-commutative case you can't say much. The monoid of all maps on n letters is generated by the symmetric group and an idempotent. In fact semigroups generated by idempotents can be quite wild. The singular nxn matrices over a field are generated by idempotents.
In the commutative case much more can be said.... | 3 | https://mathoverflow.net/users/15934 | 236318 | 109,372 |
https://mathoverflow.net/questions/236313 | 3 | Let $g$ be a finite dimensional real Lie algebra and $(,)$ be a nondegenerate invariant symmetric bilinear form on $g$. Let $r\in g\bigotimes g$ be a skew-symmetric solution of the MCYBE. We may regard $r$ as a linear map $g^{\star}\mapsto g$, and hence as a linear map $\rho: g\rightarrow g$,identifying $g^{\star}$ wit... | https://mathoverflow.net/users/83554 | The Jacobi identity of a Lie algebra? | Write the modified CYBE as $B\_R(x,y)+\lambda [x,y]=0$ with
$$
B\_R(x,y):=[R(x),R(y)]-R([R(x),y]+[x,R(y)]),
$$
and $\lambda\in \mathbb{R}$. Then
note that the bracket $[x,y]\_R:=[R(x),y]+[x,R(y)]$ satisfies the Jacobi identity if and only if
$$
[B\_R(x,y),w]+[B\_R(y,w),x]+[B\_R(w,x),y]=0
$$
for all $x,y,w\in \mathfrak... | 4 | https://mathoverflow.net/users/32332 | 236320 | 109,374 |
https://mathoverflow.net/questions/236326 | 5 | The [Gessel sequence](https://oeis.org/A135404) is known for Ira Gessel's Lattice Path Conjecture of $2001$, which has been proved by Kauers, Koutschan and Zeilberger in $2009$ with the aid of a computer. Later, other proofs were found ("human proofs"), e.g., by using Weierstrass elliptic functions (see [here](https://... | https://mathoverflow.net/users/32332 | Are the Gessel sequence integers composite for all $n\ge 3$? | Yes. Towards a contradiction, suppose $n \geq 2$ is such that $a\_{n+1}$ is prime. Say, $a\_{n+1}=p$. Then, $(3n+5)(n+2)p=4(6n+5)(2n+1)a\_n$. Evidently, $p$ does not divide $a\_n$ since $a\_{n+1} > a\_n$. Thus, $p$ divides $6n+5$ or $2n+1$. This implies that $6n+5 \geq p=a\_{n+1}$, which is a contradiction for all $n \... | 6 | https://mathoverflow.net/users/2233 | 236329 | 109,377 |
https://mathoverflow.net/questions/236299 | 3 | Let $N$ be an integer $\geq 3$ and $X(N)\rightarrow \mathrm{Spec } \mathbb{Z}[1/N]$ is the projective smooth modular curve defined in Deligne-Rappoport. Is there an exemple of $N$ for which the special fibre $X\_{\mathbb{F}\_p}(N)$ of $X(N)$ at some prime $p \nmid N$ is isomorphic to the projective $\mathbb{P}^{1}\_{\m... | https://mathoverflow.net/users/46460 | Special fibre of the modular curve $X(N)$ | (1) The Riemann-Hurwitz applied to the covering $X(N)\to X(1)$ (over the complex numbers) furnishes a number theoretical formula for the genus of $X(N)$. (The degree of this covering is the cardinality of $\mathop{\rm SL}(2,\mathbf Z/N\mathbf Z)/\{\pm \mathrm I\_2\}$, the ramification can be analysed.) Unless $N$ is sm... | 4 | https://mathoverflow.net/users/10696 | 236335 | 109,379 |
https://mathoverflow.net/questions/234926 | 6 | The *generalized Cantor space* is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.
A space is *$\kappa$-compact* if every open cover has a subcover of cardinality (strictly) smaller than $\kappa$.
**Problem.** For which cardinals $\kapp... | https://mathoverflow.net/users/2415 | When does the generalized Cantor space embed in a $\kappa$-compact space | 1. Let $\kappa$ be an uncountable regular cardinal that is not weakly compact and let $C$ be a $\kappa$-compact subset of ${}^\kappa 2$. Assume, towards a contradiction, that there is a continuous injection of ${}^\kappa 2$ into $C$. Since $C$ is closed in ${}^\kappa\kappa$, Lemma 2.9 of [this paper](http://arxiv.org/a... | 7 | https://mathoverflow.net/users/90412 | 236346 | 109,384 |
https://mathoverflow.net/questions/236309 | 2 | I was reading the following paper
<http://scitation.aip.org/docserver/fulltext/aip/journal/jmp/4/7/1.1704018.pdf?expires=1460721373&id=id&accname=2112043&checksum=607EDBF852A9E209F384DB6DA228B416>
about the Taub-NUT metric, and I don't understand the origin of equation (2.55). Here is the (more or less self-contain... | https://mathoverflow.net/users/51137 | Choosing a coordinate transformation | I will add here some more details to expand my comment. Any two functions $Y\_1(x^3,x^4)$ and $Y\_2(x^3,x^4)$ give local coordinates on any open domain of the $(x^3,x^4)$-plane where their Jacobian determinant is non-vanishing, $\frac{\partial(Y\_1,Y\_2)}{\partial(x^3,x^4)}$. Moreover, if you already have the function ... | 2 | https://mathoverflow.net/users/2622 | 236359 | 109,386 |
https://mathoverflow.net/questions/236341 | 7 | I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt.
At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense.". The reason should be that in the coordinates for the moduli space of translation surfaces given by the period map (... | https://mathoverflow.net/users/90420 | Are square tiled surfaces dense in the moduli space of translation surfaces? |
>
> Is Hubert and Schmidt's assertion wrong?
>
>
>
No, they are correct. They allow squares where the sidelength is not equal to one.
If you like, we can say that the two translation surfaces $(X, \omega)$ and $(X, r\omega)$ (for $r$ positive and real) are "scalar multiples" of each other. Then under any defin... | 3 | https://mathoverflow.net/users/1650 | 236361 | 109,387 |
https://mathoverflow.net/questions/236345 | 3 | For any $n \geq 1$, let $\Sigma\_n$ denote the closed orientable surface of genus n. In <http://arxiv.org/abs/1202.6302>, the authors showed that for any $n$, there is a degree *two*, $\pi\_1$-surjective, map $ f: S^1 \times \Sigma\_n \to \#\_n S^1 \times S^2$ (the $n$-fold connected sum).
Question: For any $n \geq ... | https://mathoverflow.net/users/90424 | Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n | You can find such a map which is degree one. The cut number $n$ of a closed 3-manifold $M$ is the maximal number of 2-sided disjointly embedded surfaces $\Sigma\_1,\ldots,\Sigma\_n$ which do not separate $M$. For each connected surface $\Sigma\_i$, we can make a degree one map to the 2-sphere by taking a spine of the s... | 4 | https://mathoverflow.net/users/1345 | 236364 | 109,390 |
https://mathoverflow.net/questions/236251 | 3 | I was watching Tom LaGatta's [talk on category theory](https://www.youtube.com/watch?v=o6L6XeNdd_k&t=56m15s) recently, and one of the audience brought up a statement that caught my attention (around the 56:15 mark):
>
> I was gonna say, there was a book I read, and they claim that some fella discovered there was a ... | https://mathoverflow.net/users/41261 | Differential equations → predicate logic mapping | I'm not sure what "book" prompted them to say this, but the ideas sound like the ideas from [Synthetic Differential Geometry](https://en.wikipedia.org/wiki/Synthetic_differential_geometry) (SDG) (see also [Synthetic Geometry of Manifolds](http://home.math.au.dk/kock/SGM-final.pdf)) and [Fractional Exponential Functors ... | 6 | https://mathoverflow.net/users/86291 | 236368 | 109,392 |
https://mathoverflow.net/questions/222893 | 8 | Let $K, K'$ be knots in $S^3$, and $T, T'$ the boundaries of their tubular neighborhoods.
Recall that by theorems of Waldhausen, and Gordon and Luecke, one knows the following: an isomorphism $[\pi\_1(T) \to \pi\_1(S^3 \setminus K)] \cong [(\pi\_1(T') \to \pi\_1(S^3 \setminus K')]$ implies that $K$ and $K'$ are isot... | https://mathoverflow.net/users/4707 | What does the representation category of the knot group know? | So long as I allow myself infinite size representations, and writing $\mathbb{Z}[G]-mod$ for representations of $G$ in $\mathbb{Z}$-modules, then so long as I have the forgetful functor
$$\mathbb{Z}[G]-mod \to \mathbb{Z}-mod$$
I can recover $\mathbb{Z}[G]$ as endomorphisms of this functor.
In general, one cannot... | -1 | https://mathoverflow.net/users/4707 | 236377 | 109,393 |
https://mathoverflow.net/questions/236380 | 4 | What is the closed form of
$$\sum\_{k=0}^n \frac{x^k}{k!}$$
as a function of $x$ and $n$?
Knowing that it converges to $e^x$ when $n\to \infty$.
| https://mathoverflow.net/users/90443 | Is there a closed form for $\sum_{k=0}^n \frac{x^k}{k!}$? | Denoting this by $f\_n(x)$, we get $f\_n'-f\_n=-x^n/n!$, solving this differential equation we have $(f\_n(x)e^{-x})'=-x^n e^{-x}$, thus taking into account initial condition $f\_n(0)=0$ we get integral representation $$f\_n(x)=e^x-\frac1{n!}e^x\int\_0^x t^n e^{-t}dt.
$$
This may be further rewritten as $$f\_n(x)=\frac... | 10 | https://mathoverflow.net/users/4312 | 236381 | 109,395 |
https://mathoverflow.net/questions/236370 | 6 | I recently had the need to appeal to some complex geometry in my research and have been trying to unravel the various relationships surrounding the *Koszul-Malgrange theorem*.
According to nlab, the theorem goes as follows.
>
> Theorem 1. (Koszul-Malgrange theorem)
>
>
> Holomorphic vector bundles over a comple... | https://mathoverflow.net/users/43687 | Confusion surrounding the Koszul-Malgrange theorem | Since I was rather surprised that *those* authors would make such a claim, I looked up the original reference [Koszul, Malgrange, Sur certaines structures fibrées complexes.] Here is a rough translation of the relevant theorem:
>
> Theorem 2. Let $G$ be a complex Lie group, $V$ a complex manifold, $P$ a principal b... | 10 | https://mathoverflow.net/users/4144 | 236391 | 109,397 |
https://mathoverflow.net/questions/210208 | 6 | Let $S\_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces on $S\_g\}$
Given a collection of $n$ vectors $v\_1,\dots,v\_n$ in $\mathbb{R}^2$ there is a natural way to construc... | https://mathoverflow.net/users/60675 | How to get a polygon from a translation surface $(X,\omega)$ |
>
> How do I determine the base $\{\rho\_j\}\_j$?
>
>
>
There are (infinitely) many possibilities for the base. You will need to review the notion of "natural coordinates" for an abelian differential to see why there is at least one base (see, for example, page 25 of "Flat surfaces" by Zorich). Once you have on... | 1 | https://mathoverflow.net/users/1650 | 236397 | 109,399 |
https://mathoverflow.net/questions/236398 | 4 | Let's will write $K\_n$ for the Eilenberg-MacLane space $K(\mathbb{Z},n)$. I remind that $K\_n$ is equivalent to the loop space of $K\_{n+1}$.
Let’s consider the map $\smallsmile:K\_n\times K\_m \to K\_{n+m}$ corresponding to the cup product.
Given two elements $x:K\_n$ and $y:K\_m$, we can see $x$ as a loop in $K\... | https://mathoverflow.net/users/10217 | Formula relating the cup product in dimensions n and n+1 | It is a bit of a roundabout way of proving it, but what you're looking for is immediately implied by the fact that the Eilenberg-Mac Lane spectrum is a commutative ring spectrum, as proven for example in in Example 1.14 of Stefan Schwede's [book on symmetric spectra](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&sour... | 3 | https://mathoverflow.net/users/43054 | 236401 | 109,400 |
https://mathoverflow.net/questions/236413 | 3 | I have a simple question: transitivity of $Spin(7)$ in triples of orthogonal vectors. Let $Spin(7)\subset SO(8)$ act on $\mathbb{R}^8$, and $e\_1,e\_2,e\_3$, $v\_1,v\_2,v\_3$ be two triples of mutually orthogonal vectors.
**Is there a (unique?) transformation in $Spin(7)$ that takes $e\_i$ to $v\_i$, $i=1,2,3$?**
... | https://mathoverflow.net/users/62367 | Transitivity of $Spin(7)$ in triples of vectors | We know that $\operatorname{Spin}(7)$ acts transitively on unit vectors, with stabilizer $G\_2$, so we need only prove that $G\_2$ acts transitively on pairs of orthogonal unit vectors. But then apply the same argument, since $G\_2$ acts transitively on the unit sphere in $\mathbb{R}^7$, with stabilizer $SU(3)$, which ... | 4 | https://mathoverflow.net/users/13268 | 236416 | 109,407 |
https://mathoverflow.net/questions/236423 | 2 | I want to show the following theorem in a lecture:
Let $F \in C^{\infty}(\mathbb{C}^{k}, \mathbb{C})$ such that $F(0)=0.$
Let $G: \mathbb{R}^n \rightarrow \mathbb{C}^{k}$, $x \mapsto (f\_1(x),..,f\_k(x))$ where $f\_1,...,f\_k \in \mathcal{S}(\mathbb{R}^n;\mathbb{C}).$ (the space of rapidly decaying functions.)
Th... | https://mathoverflow.net/users/90461 | Simplify proof for rapidly decaying functions | This outline may help. After multiplying $F$ by a smooth cutoff, you may assume $F$ and all of its partial derivatives are bounded in a neighborhood of the image of $G$ without changing the composition $F(G(x))$. This operation does not change the composition $F\circ \tilde{G}$ for $\tilde{G}$ that are $C^0$ close to $... | 1 | https://mathoverflow.net/users/7193 | 236427 | 109,410 |
https://mathoverflow.net/questions/236383 | 9 | I am looking for the name and notation of the following separation axiom , temporarily denoted by $T\_i$ (where $i=\sqrt{-1}$ is the imaginary unit):
***Axiom $T\_i$***: For any point $x$ of a topological space $X$ and any neighborhood $O\_x$ of $x$ there is a closed subset $F$ in $X$ that contains $x$ and is contain... | https://mathoverflow.net/users/61536 | New separation axiom? | According to the [Wikipedia article about ${\mathrm T}\_1$ spaces](https://en.wikipedia.org/wiki/T1_space) your ${\mathrm T}\_i$-spaces are called $\it symmetric$ or ${\mathrm R}\_0$-spaces. There are several equivalent conditions, my personal favorite being that point closures are antidiscrete.
Unfortunately I was n... | 12 | https://mathoverflow.net/users/41291 | 236428 | 109,411 |
https://mathoverflow.net/questions/236395 | 5 | I feel experts might be able to answer this question immediately.
Let $G$ be a connected $\mathbb Q$-simple and $\mathbb Q$-isotropic algebraic group.
Let $S$ be a maximal $\mathbb Q$-split torus of $G$ and
let $T\supset S$ be a maximal torus defined over $\mathbb Q$. Let $T\_a$ be the maximal anisotropic subtorus ... | https://mathoverflow.net/users/11056 | structure of maximal tori in semisimple algebraic groups | The Weil-restriction construction suggested by user89334 is $\mathbf{Q}$-simple but not absolutely simple. To give absolutely simple examples, consider $G = {\rm{SL}}\_n(D)$ for a central division algebra $D$ over $\mathbf{Q}$ with dimension $d^2>1$.
A maximal split torus in $G$ is given by the diagonal torus $S$ in... | 5 | https://mathoverflow.net/users/81332 | 236433 | 109,413 |
https://mathoverflow.net/questions/236405 | 3 | Setting
-------
Let $I\subseteq\mathbb C[x\_0,\ldots,x\_n]=:S$ be a homogeneous ideal and $X\subseteq\mathbb P^n$ the scheme defined by $I$. Consider the action of the symmetric group $\mathfrak S\_{n+1}$ on $S$ by permuting the variables. Assume that $I$ is invariant under under some subgroup $G\subseteq\mathfrak S\... | https://mathoverflow.net/users/9947 | Checking smoothness of the components of a highly symmetric scheme via quotient? | Here is an example where $X/G$ is singular and the components of $X$ are smooth.
In $\mathbb{P}^3$ with coordinates $x,y,z,w$, consider the smooth conics
$$\begin{array}{lll}X\_1:& x^2-y^2=yw,&z=w\\ X\_2:& z^2-w^2=yw,&x=y.\end{array}$$
They are exchanged by the involution $\sigma:(x:y:z:w)\mapsto(z:w:x:y)$ and me... | 3 | https://mathoverflow.net/users/7666 | 236454 | 109,419 |
https://mathoverflow.net/questions/236452 | 3 | Recently, when I was working with Cayley graphs, I faced up with a special group. The original group is as follows:
$$G:=<a,b,c|ab=ba,a^{10}=cbc^{-1}>.$$
We can show that this group can be rewrite as follows:
$$G=<x,y|xy^{-1}x^{10}yx^{-1}y^{-1}x^{-10}y=1>.$$
In general, there is not any special things about the pow... | https://mathoverflow.net/users/19885 | Properties of a special finitely presented groups | Your group $G$ is *not* solvable since it has a quotient isomorphic
to ${\rm S}\_5$. You can see this with [GAP](http://www.gap-system.org) as follows:
```
gap> F := FreeGroup("a","b","c");
<free group on the generators [ a, b, c ]>
gap> AssignGeneratorVariables(F);
#I Assigned the global variables [ a, b, ... | 1 | https://mathoverflow.net/users/28104 | 236457 | 109,420 |
https://mathoverflow.net/questions/236451 | 5 | Let $X$ be an infinite set and let $C(X)$ denote the collection of connected Hausdorff topologies on $X$. Suppose $N\subseteq C(X)$ has the property that whenever $\tau\neq\sigma \in N$ then $(X,\tau)$ and $(X,\sigma)$ are not homeomorphic. In terms of $|X|$, how large can $|N|$ be at most?
| https://mathoverflow.net/users/8628 | Cardinality of connected Hausdorff topologies | Let $\kappa:=|X|$. Then $2^{2^\kappa}$ is an obvious upper bound for the number of topologies on $X$.
Every ultrafilter on $\kappa$ will give you a Hausdorff topological space on $\kappa+1$; these are $2^{2^\kappa}$ many spaces. Some of them might be homeomorphic, but there are only $2^\kappa$ many bijections, so yo... | 10 | https://mathoverflow.net/users/14915 | 236458 | 109,421 |
https://mathoverflow.net/questions/236453 | 7 | Nlab introduces the [globular category](https://ncatlab.org/nlab/show/globe) as a geometrical model to construct certain higher categorical structures (e. g. strict $\omega$-categories), just as quasi-categories, for example, are modelled on simplices. However, I didn't find much information on the geometric intuition ... | https://mathoverflow.net/users/37059 | What do globes (used to construct globular sets, $\omega$-categories, etc.) actually look like? | The paper on the fundamental [globular groupoid](http://groupoids.org.uk/pdffiles/globularhha.pdf) of a filtered space has some pictures, and a definition of the simplicial nerve of a globular $\omega$-groupoid. Relations with other areas are in the paper M. Kapranov, "Membranes and higher groupoids" arxiv 1602.06166 .... | 5 | https://mathoverflow.net/users/19949 | 236459 | 109,422 |
https://mathoverflow.net/questions/179203 | 10 | For a strong limit cardinal $\kappa$ the notion of *$\kappa$-Kurepa tree* is trivial: the full binary tree is a $\kappa$-Kurepa tree. Accordingly, we consider the following strengthening:
A *slim $\kappa$-Kurepa tree* is a tree $T$ of height $\kappa$ such that for every infinite $\alpha < \kappa$ the $\alpha$-th leve... | https://mathoverflow.net/users/1682 | Slim Kurepa tree at a singular strong limit cardinal of uncountable cofinality | The following is proved by Erdos-Hajnal-Milner in ``[On sets of almost disjoint subsets of a set](http://bolyai.hu/~p_erdos/1968-08.pdf). Acta Math. Acad. Sci. Hungar 19 1968 209–218'', from which the required result follows
**Theorem.** ssume $\aleph\_0 < cf(\kappa) < \kappa$ and $\forall \theta< \kappa, \theta^{cf(... | 5 | https://mathoverflow.net/users/11115 | 236460 | 109,423 |
https://mathoverflow.net/questions/236450 | 4 | Assume that $M$ is a simply connected closed Riemannian manifold with no boundary and nonnegative sectional curvaure Assume that ${\bf Z}\_n=(g),\ n\geq 3$ acts on $M$ isometrically. Then if $gx=x$, i.e., it is a fixed point, clearly $g$ acts on ${\rm cut}\ x$ Here I have a question : $g\cdot x \in {\rm cut}\ x$ can ha... | https://mathoverflow.net/users/36572 | How isometric action on Riemannian manifold acts on cut locus | It can happen that $g.x\in cut(x)$ for some $x$. This is what happens for $S\_{n+1}$ acting by permutation of homogeneous coordinates on $\mathbb{CP}^n$.
| 4 | https://mathoverflow.net/users/28128 | 236461 | 109,424 |
https://mathoverflow.net/questions/236473 | 10 | Alice and Bob each secretly chooses an integer between 1 and 10, `a` and `b`. They want to know (with high probability) whether or not `a` equals `b`, without revealing any other information. Can they?
| https://mathoverflow.net/users/416 | Zero knowledge proof of equality | In typical mathematician fashion, let me explain how to reduce your problem to a harder one ☺, namely homomorphic encryption. (**Edit**: I should have made it clear that the problem of constructing homomorphic encryption schemes, at least to the extent required here, is indeed solved.)
Assume Charlie is an "honest bu... | 6 | https://mathoverflow.net/users/17064 | 236482 | 109,430 |
https://mathoverflow.net/questions/233201 | 2 | It is known that an invertible *mpt* $S$ is weakly mixing if and only if $S \times T$ is ergodic for any ergodic invertible *mpt* $T$. Is it more generally true that the invariant $\sigma$-field of $S \times T$ is the product of the trivial $\sigma$-field times the invariant $\sigma$-field of $T$ whenever $S$ is weakly... | https://mathoverflow.net/users/21339 | Invariant $\sigma$-field of a product with a weakly mixing transformation | The answer is yes (I am assuming here the spaces are standard and the measures are probability measures).
For this kind of questions I find that it is convenient to think about quotient spaces rather then sub-$\sigma$-algebras. In particular, instead of considering the $\sigma$-algebra of invariants you better consid... | 2 | https://mathoverflow.net/users/89334 | 236491 | 109,433 |
https://mathoverflow.net/questions/236238 | 6 | Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero.
If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' fourier transform matrix:
$$F\_N = \frac{1}{\sqrt{N}} \begin{bmatrix}
1&1&1&1&\cdots &1 \\
1&\omega&\omega^2&\omega^3&\cd... | https://mathoverflow.net/users/41654 | What is this Lie algebra? | I can't see your question having a completely general answer. In the 'generic' case, I think the subalgebra generated by $A$ and $\tilde{B}$ is just $\mathfrak{su}(N)$.
To see this, first of all note that $\tilde{B}$ is of the form $$\begin{bmatrix} 0 & b\_1 & b\_2 & \dots & b\_{N-1} \\ b\_{N-1} & 0 & b\_1 & \dots & ... | 2 | https://mathoverflow.net/users/26635 | 236492 | 109,434 |
https://mathoverflow.net/questions/236493 | 2 | Is there an algorithm for finding minimal covers of a set of sets in which each element of the universe appears in exactly 2 sets? I realize that LP relaxation approximates this to within a factor of 2.
| https://mathoverflow.net/users/89475 | Minimum cover for sets in which each element appears in exactly 2 sets? | This is the well-known problem of [Minimum Vertex Cover](https://en.wikipedia.org/wiki/Vertex_cover). It is conjectured to be NP-hard to approximate within $2-\epsilon$ for any $\epsilon > 0$. Many $2$-approximation algorithms exist.
| 2 | https://mathoverflow.net/users/7732 | 236495 | 109,436 |
https://mathoverflow.net/questions/236476 | 5 | Let $X$ be a smooth, quasi-projective variety, $G$ be a finite group which acts freely and properly on $X$. Denote by $\alpha:X \to X/G$ the quotient. Is $\alpha$ generically etale?
Also, as I am new to this topic (of group action on varieties) can someone suggest a good reference for the topic? I am mainly interest... | https://mathoverflow.net/users/58203 | Finite group action on quasi-projective varieties | Let $X$ be a normal variety over an algebraically closed field of characteristic 0 with a finite group $G$ acting effectively. Since $G$ is finite it is reductive and a geometric quotient $X/G$ exists. The quotient map $X\to X/G$ is étale at a point $x\in X$ if and only if the stabilizer at $x$ is trivial. Since the eq... | 10 | https://mathoverflow.net/users/12218 | 236507 | 109,440 |
https://mathoverflow.net/questions/236508 | 22 | Motivated by the central limit theorem, one expects that
$$\binom{n}{k} \approx \frac{2^n}{\sqrt{\pi n/2}} \exp\left(-\frac{(k-n/2)^2}{n/2}\right).$$
Computations suggest that the ratio of the two sides approaches 1 only for $|k-n/2| < 2\sqrt{n}$, and presumably this will follow from some version of the CLT.
In the ... | https://mathoverflow.net/users/935 | Are there good bounds on binomial coefficients? | Let $h(x)=-x\ln x-(1-x)\ln (1-x)$ be the binary entropy function in nats, then for $k\in [1,n-1]\cap \mathbb{Z}$ we have
$$
\sqrt{\frac{n}{8k(n-k)}}\exp\{nh(k/n)\} \leq \binom{n}{k} \leq \sqrt{\frac{n}{2\pi k(n-k)}}\exp\{nh(k/n)\}
$$
where the upper bound approaches equality if $k$ and $n-k$ are both large. This is obt... | 28 | https://mathoverflow.net/users/17773 | 236511 | 109,443 |
https://mathoverflow.net/questions/236402 | 2 | Let $R = \mathrm{GF}(q)$, $S = \mathrm{GF}(q^n), \ n\geq 2$ be extension of $R$, $h$ be a primitive element of $S$. I want to count or estimate the number $N$ of bases of the following form.
Let $$\vec{\beta} = (\beta\_0,\beta\_1,\ldots,\beta\_{n-1})$$ be a basis of the space $\_RS$ with the property that there exists ... | https://mathoverflow.net/users/85489 | Bases of the special form | Let me estimate the number $N\_k$ of such bases for a fixed $k$. *Aside remark.* The number of bases for $k=k\_0$ equals the number of those for $k=n-k\_0$, as the bijection $(\beta\_0,\dots,\beta\_{n-1})\to (\beta\_{n-1},\dots,\beta\_{k\_0},h^{-1}\beta\_{k\_0-1},\dots,h^{-1}\beta\_0)$$ shows.
So let us fix any $k$. ... | 2 | https://mathoverflow.net/users/17581 | 236531 | 109,449 |
https://mathoverflow.net/questions/236060 | 17 | I recently went to a talk of [Oleg Viro](http://www.math.stonybrook.edu/~oleg/) where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed [here](https://mathoverflow.net/questions/14877/how-much-of-differential-geometry-can-be-developed-entirely-without... | https://mathoverflow.net/users/13960 | Foundations of topology | Reading section 5 in Grothendieck's essay *Esquisse d'un programme* it becomes clear that with regard to topology Grothendieck was bothered by some artificial foundational problems introduced by the fact that the foundations of topology were created by analysts rather than by geometers and topologists. Specifically he ... | 14 | https://mathoverflow.net/users/28128 | 236541 | 109,450 |
https://mathoverflow.net/questions/236538 | 8 | Is there an elementary proof of this Banach space fact?
>
> If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite dimension, i.e., a copy of $l^2\_n$ for all $n$.
>
>
>
Failing that, good references on the subject?
| https://mathoverflow.net/users/23141 | $l^1$ versus $l^2$ | Is it true? $l^1$ is a sum of finite-dimensional $l\_n^1$ over $n=1,2,\dots$. In summands you have almost spherical sections of large dimensions by Dvoretzky theorem, this allows to change norm a bit so that unit balls in summands contain large spherical sections.
| 5 | https://mathoverflow.net/users/4312 | 236543 | 109,452 |
https://mathoverflow.net/questions/236480 | 1 | Let $G$ be a $\mathbb{Q}$-subgroup of $\mathrm{GSp}\_{2g}$, reductive and defines a Shimura subdatum of $(\mathrm{GSp}\_{2g},\mathfrak{H}\_g)$. Let $V$ be the natural representation of $\mathrm{GSp}\_{2g}$ (so $V$ is a $\mathbb{Q}$-vector space of dimension $2g$). Assume $V$ is irreducible as a $G$-module, then is $V\_... | https://mathoverflow.net/users/nan | Tangent spaces of an indecomposable family of abelian varieties (parametrized by a Hodge type Shimura variety) | No. Here is a counter-example (I add details to my comment).
Let $F$ be a totally real number field.
Consider the group $G\_0=R\_{F/\mathbb{Q}}\mathrm{SL}\_{2,F}$, then $G\_0$ naturally embeds into $\mathrm{Sp}\_{2g}$, where $g=[F:\mathbb{Q}]$.
Set $G=\mathrm{G}\_{m,\mathbb{Q}}\cdot G\_0$, then $G$ naturally embeds ... | 0 | https://mathoverflow.net/users/4149 | 236561 | 109,455 |
https://mathoverflow.net/questions/236550 | 34 | We know that the tangent bundles of the sphere arising from different smooth structures are equivalent as vector bundles. Is it right in general? I want to know the relationship between the set of smooth structures and these tangent bundles.
| https://mathoverflow.net/users/90512 | Can a topological manifold have different tangent bundles? | This is answered in [Crowley, Diarmuid J.; Zvengrowski, Peter D, On the non-invariance of span and immersion co-dimension for manifolds, Arch. Math. (Brno) 44 (2008), no. 5, 353–365], see [here](http://www.dcrowley.net/Span.pdf).
Specifically, in each dimension $>8$ there is a closed PL manifold admitting two smooth... | 49 | https://mathoverflow.net/users/1573 | 236565 | 109,456 |
https://mathoverflow.net/questions/236563 | 24 | Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity?
Of course, such a manifold must not admit a diffeomorphism of finite order.
Since a surface $S$ admits a diffeomorphism of order $n$ iff its mapping class group (MCP) has an e... | https://mathoverflow.net/users/46290 | Is there a smooth manifold which admits only rigid metrics? | The answer to the question in the first sentence is "yes". Let $M$ be a hyperbolic 3-manifold whose isometry group is trivial. Then by Theorem 1.1 of
Farb, Benson; Weinberger, Shmuel Hidden symmetries and arithmetic manifolds. Geometry, spectral theory, groups, and dynamics, 111–119, Contemp. Math., 387, Amer. Math. ... | 26 | https://mathoverflow.net/users/317 | 236566 | 109,457 |
https://mathoverflow.net/questions/236547 | 1 | **Problem.** Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})^\top$ - a polynomial basis.
Suppose there is a matrix
$$
A = \int\_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A\_{ij} = [2 \, | \, i+j] \cdot \dfrac{2}{i+j+1} \enspace ,
$$
that is, e.g.
$$
A = 2\cdot\begin{pmatrix}
1 & 0 & 1/3 & 0 & 1/5 \\
0 & 1/3 & 0 ... | https://mathoverflow.net/users/90511 | Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $t = \pm 1$, where $\psi_k(t) = t^k$? | It seems that the matrix you present as an example is $A/2$, not $A$; I assume that $A$ is exactly what is defined (so, e.g., $A\_{11}=2$, not $1$).
Well, $A$ is the Gram matrix of the basis $\psi$ with respect to the scalar product $(f,g)=\int\_{-1}^1 f(x)g(x)\,dx$; so, if we pass to the orthonormal basis of (normed... | 2 | https://mathoverflow.net/users/17581 | 236567 | 109,458 |
https://mathoverflow.net/questions/236483 | 9 | If the Axiom of Countable Choice (ACC)
$$ \forall n\in \mathbb{N} . \exists x \in X . \varphi [n, x] \implies \exists f: \mathbb{N} \longrightarrow X . \forall n \in \mathbb{N} . \varphi [n, f(n)] $$
is not assumed in constructive mathematics, Dedekind and Cauchy real numbers are [not equivalent](https://mathoverfl... | https://mathoverflow.net/users/42302 | Difference between constructive Dedekind and Cauchy reals in computation | In line with what is actually *asked* in the bold part of the question, I have taken the liberty of changing the title from *constructive mathematics* to *computation*. As it stood it was essentially a [duplicate of another one](https://mathoverflow.net/questions/128569/a-model-where-dedekind-reals-and-cauchy-reals-are... | 7 | https://mathoverflow.net/users/2733 | 236574 | 109,459 |
https://mathoverflow.net/questions/236586 | 5 | (I'd be grateful if anyone thinking of putting MathJax in the question title refrains from doing so.)
---
By consulting various standard sources (Effros-Ruan's book, Pisier's book, the [lexicon of Wittstock's old group](http://www.math.uni-sb.de/ag/wittstock/projekt2001.html)) I can find some descriptions of the ... | https://mathoverflow.net/users/763 | Operator space structures on CB(H,K) where H and K are Hilbertian operator spaces? | Both your questions can be answered by considering rows and columns in $\mathbf B$, and by noticing that in $CB(X,Y)$ colums are completely isometric to $Y$ and rows are completely isometric to $X^\*$.
Therefore if $X\_1,X\_2$ and $Y\_1,Y\_2$ are operator spaces structures on $\ell\_2$ such that $id\colon CB(X\_1,Y\_... | 4 | https://mathoverflow.net/users/10265 | 236591 | 109,464 |
https://mathoverflow.net/questions/236467 | 5 | Let $0\le k \le n$. Prove that
$$
n\binom{n}{k}\int\_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2.
$$
As far as I know
1) it is proved for $\frac{k}{n+1}\le 1/2$ and
2) not proved for $1/2 <\frac{k}{n+1}< 1$.
Is it really so?
| https://mathoverflow.net/users/49208 | Estimate of incomplete binomial integral | The following stronger inequality holds for all $k=0,\dots,n$:
$$
(n+1)\binom{n}{k}\int\_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2.
$$
Indeed, the latter inequality means precisely that the median $m$ of the Beta distribution with parameters $a:=k+1\ge1$ and $b:=n-k+1\ge1$ is no less than $\frac{k}{n+1}=\frac{a-1}... | 2 | https://mathoverflow.net/users/36721 | 236592 | 109,465 |
https://mathoverflow.net/questions/235873 | 1 | In this paper : <https://eprint.iacr.org/2011/501.pdf>
There is an equality page 10, in the second paragraph considered by the authors as "easy to check". If someone could explain to me why the set at the left side is included in the set at the right side, I will be the happiest man on earth. All the definitions are gi... | https://mathoverflow.net/users/90197 | Dual lattices up to a q scaling factor | We prove $\frac{1}{q}\Lambda(A^t)=\Lambda^\perp(A)^\*$.
(1) $\frac{1}{q}\Lambda(A^t)\subset\Lambda^\perp(A)^\*$: If $z=A^ts$ (mod $q$) then for any $y\in\Lambda^\perp(A)$, $\langle \frac{1}{q}z,y\rangle\in\frac{1}{q}\langle A^ts,y\rangle+\mathbb{Z}=\frac{1}{q}\langle s,Ay\rangle+\mathbb{Z}\in\mathbb{Z},$ since $Ay=0$... | 1 | https://mathoverflow.net/users/90531 | 236596 | 109,467 |
https://mathoverflow.net/questions/236578 | 15 | Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried navigating the vast sea of literature).
I'm trying to start learning descent theory, and after seeing how descent along o... | https://mathoverflow.net/users/69037 | Difficulties with descent data as homotopy limit of image of Čech nerve | To answer your question I'll need to do a fairly long digression on homotopy limits and colimits. Before I delve deep into the topic let me say that there's more than one way to describe this topic, for example some people like model categories, other people might prefer triangulated categories (shudder), I'll simply e... | 17 | https://mathoverflow.net/users/43054 | 236600 | 109,469 |
https://mathoverflow.net/questions/236049 | 12 | Let G=GLn(ℂ) and let T be a maximal torus. Let X be a topological space with a G-action. My question is: when is the canonical map $$H^\*\_G(X;\mathbb{Z})\to H^\*\_T(X;\mathbb{Z})$$ injective?
Some remarks: I am trying to understand Torsten's answer [here](https://mathoverflow.net/a/18181/425), which claims injectivi... | https://mathoverflow.net/users/425 | Passing from T-equivariant to G-equivariant cohomology | This isn't a complete answer, but for example, if you know that $H\_T^\*(X;\mathbb{Z})$ injects into the cohomology of the fixed point set $X^T$, then for $G=GL\_n(\mathbb{C})$, the canonical map $H\_G^\*(X;\mathbb{Z})\to H\_T^\*(X;\mathbb{X})$ is injective. See Theorem 2.10 and Corollary 2.11 in T. Holm and R. Sjamaar... | 12 | https://mathoverflow.net/users/5723 | 236601 | 109,470 |
https://mathoverflow.net/questions/236605 | 2 | Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, *etc.*, may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra on it. Let $\Delta (Y)$ be the set of probability measures on $(Y,\mathscr Y)$ and endow it with the weak-$\star$ topo... | https://mathoverflow.net/users/55976 | Measurability of integrals with respect to different measures | If $Y$ is a metric space, then the arguments in the answers to [The borel $\sigma-$algebra of the set of probability measures](https://mathoverflow.net/q/167823/4832) show that the map $I\_f$ defined by $I\_f(\mu) = \int f\,d\mu$ is measurable for any bounded measurable $f$. Then the answer to your question is affirmat... | 5 | https://mathoverflow.net/users/4832 | 236608 | 109,471 |
https://mathoverflow.net/questions/236599 | 4 | Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map
\begin{align}
\alpha\_{n}:\bigwedge^nA&\to A^{\otimes n}\\
a\_1\wedge \cdots \wedge a\_n&\mapsto \sum\_{\pi\in \mathbb{S}\_n}(sig(\pi))a\_{\pi(1)}\otimes \cdots\otimes a\_{\pi(n)}
\end{a... | https://mathoverflow.net/users/84123 | Exterior Powers of finite abelian group | I think that this is always true. We can write $A$ as $A\_1\oplus\dotsb\oplus A\_r$, where each $A\_i$ is cyclic. Let $I(n)$ denote the set of sequences $(i\_1,\dotsc,i\_n)$ with $1\leq i\_1<i\_2<\dotsb <i\_n\leq r$. For $i\in I^n$ put $A(i)=A\_{i\_1}\otimes\dotsb\otimes A\_{i\_n}$, so $A^{\otimes n}=\bigoplus\_{i\in I... | 3 | https://mathoverflow.net/users/10366 | 236612 | 109,472 |
https://mathoverflow.net/questions/236489 | 25 | Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).
* An object $X$ in $\mathcal C$ gives a "point":
$$X$$
* A morphism $X\xrightarrow f Y$ in $\mathcal C$ gives a "triangle":
$$\begin{matrix}X&&\to&&Y\cr&\nwarrow&&\swarrow\cr&&\operatorname{cone}(f)\end{matrix}$$
* A chai... | https://mathoverflow.net/users/35353 | Complete the following sequence: point, triangle, octahedron, . . . in a dg-category | I believe these are called 'hypersimplices'. See
1) Gelfand, Manin "Methods of homological algebra", Ex. IV.2 1(c), p. 260.
2) Belinson, Bernstein, Deligne "Faisceaux pervers", Remarque 1.1.14, p. 26.
3) The diagrams for $n$ up to 4: <http://students.mimuw.edu.pl/~pa235886/pdf/hypersimplices.pdf>
| 11 | https://mathoverflow.net/users/3847 | 236621 | 109,475 |
https://mathoverflow.net/questions/236622 | 4 | We say an infinite set $X$ is *splittable* if there are $X\_1, X\_2\subseteq X$ with $X\_1\cap X\_2 = \emptyset$, $X\_1\cup X\_2 = X$ and there are bijections $\varphi:X\_1\to X\_2$ and $\psi:X\_1\to X$.
Does the statement "Every infinite set is splittable" imply $\mathsf{AC}$?
| https://mathoverflow.net/users/8628 | Does "Every infinite set is splittable" imply $\mathsf{AC}$? | The answer is no and it follows from the following:
>
> It is consistent that $AC$ fails but for all infinite cardinals $\kappa, 2 \cdot \kappa=\kappa.$
>
>
>
The above result is proved by Sageev:
Sageev, Gershon [An independence result concerning the axiom of choice](http://www.sciencedirect.com/science/ar... | 10 | https://mathoverflow.net/users/11115 | 236625 | 109,478 |
https://mathoverflow.net/questions/236598 | 7 | A **weak fibration category** is a category $\mathcal{C}$ equipped with two subcategories
$$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$
containing all the isomorphisms, such that the following conditions are satisfied:
1. $\mathcal{C}$ has all finite limits.
2. $\mathcal{W}$ has the 2-out-of-3 property.
3. The s... | https://mathoverflow.net/users/42440 | Can a weak fibration category be non saturated? | It is a result of Cisinski that in a fibration category the three conditions you mention (saturation, 2-out-of-6, weak equivalences closed under retracts) are all equivalent. See Theorem 7.2.7 in [this paper](http://arxiv.org/abs/math/0610009).
| 10 | https://mathoverflow.net/users/12547 | 236629 | 109,480 |
https://mathoverflow.net/questions/236627 | 11 | Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure
$$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 \end{pmatrix}.$$
Of course, we can regard $V^{\mathbb{C}}$ also as a real vector space with a canonical orientation (for a basis $v\_1, \dots... | https://mathoverflow.net/users/16702 | Pfaffian equals complex determinant? | The answer is YES in every dimension, up to a sign. Here is the calculation. On the one hand, $\det \tilde A=\det(A^2+I)$ because the blocs commute to each other. Therefore
$${\rm Pf}(\tilde A)^2=\det(I+iA)\det(I-iA).$$
On the other hand
$$\det(I-iA)=\overline{\det(I+iA)}=\det(I+iA)^\*=\det(I+iA),$$
yields
$${\rm Pf}(\... | 16 | https://mathoverflow.net/users/8799 | 236631 | 109,481 |
https://mathoverflow.net/questions/236648 | 0 | Consider the graph $(V,E)$ with vertex set $V=\{v\_1,...,v\_n\}$ and edge set $E\subset V\times V$. Further, assume that $\forall v\_i\in V, (v\_i,v\_i)\in E$.
Assume that each vertex has an $\textit{initial value}$ (i.e. there is a function $\phi\_0:V\rightarrow\mathbb{R}$). We will think of these values as changin... | https://mathoverflow.net/users/5732 | Long term behavior of a certain discrete time dynamical system on graphs | Just set matrix $A = [\omega(v\_i, v\_j)]$ and $\mathbf{x}\_k = [\phi\_k(v\_i)]$. Then the linear dynamic system is $\mathbf{x}\_{k+1} = A\mathbf{x}\_k$. $A$ has property that $A\mathbf{1} = \mathbf{1}$, where $\mathbf{1}$ is all-one vector.
The dynamics system has close form as: $\mathbf{x}\_k = A^k\mathbf{x\_0}$. B... | 2 | https://mathoverflow.net/users/33852 | 236652 | 109,484 |
https://mathoverflow.net/questions/236644 | 2 | ***Is there any specific computational complexity result of Graph Isomorphism for Triangle Free graphs?***
Anything close to the subject will help and of course, I have searched Google.
| https://mathoverflow.net/users/69301 | Graph Isomorphism for Triangle Free graph | The following very simple answer addresses worst-case complexity. How to do the reduction in practice would be a different question, as would average complexity (as pointed out by logicute).
For a graph $G$, let $\hat{G}$ denote the [barycentric subdivision](https://en.wikipedia.org/wiki/Homeomorphism_%28graph_theory... | 6 | https://mathoverflow.net/users/27013 | 236656 | 109,485 |
https://mathoverflow.net/questions/236660 | 1 | Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are homeomorphic to the real projective plane $\mathbb{RP}^2$. Does someone know how to construct such a surface?
This is... | https://mathoverflow.net/users/36563 | Smooth, irreducible surface with real part containing two projective planes | [**Corrected**]
Any double cover of ${\bf RP}^2$ whose branch locus has
degree $4n$ and no real component should do. An example is
the surface $y^2 = x\_0^4 + x\_1^4 + x\_2^4$ in the weighted projective space
whose coordinates $(x\_0:x\_1:x\_2::y)$ have degrees $1,1,1,2$.
Thus $(x\_0:x\_1:x\_2::y)$ is equivalent to
... | 4 | https://mathoverflow.net/users/14830 | 236663 | 109,486 |
https://mathoverflow.net/questions/236659 | 1 | Suppose you have a positive sequence $X\_1,X\_2,\dots$ of i.i.d. random variables with the property that
$$
\mathbb{E}[\log(X\_1)]<\infty.
$$
Is it true that
$$
\limsup\_{n\to\infty} e^{-n}\sum\_{k=1}^n e^k X\_k < \infty?
$$
If so, does there exists a limit in some sense?
I don't know exactly what is covered in th... | https://mathoverflow.net/users/18279 | Weighted sum of i.i.d. random variables | No, it is not. If the $X\_i$ are not almost surely bounded, so that for every $N$ there is some positive probability that $X\_i > N$, then almost surely there is an infinite increasing sequence $n\_N$ such that $X\_{n\_N} > N$,
and $$e^{-n\_N} \sum\_{k=1}^{n\_N} e^{k} X\_k \ge X\_{n\_N} > N$$
| 2 | https://mathoverflow.net/users/13650 | 236665 | 109,487 |
https://mathoverflow.net/questions/236636 | 5 | Let $u$ be the weak solution on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ (for $n \leq 3$) of
$$u\_t - \Delta u = f$$
$$u(0) = u\_0$$
$$\partial\_\nu u = 0 \quad\text{on $\partial\Omega$}$$
for $u\_0 \in L^\infty(\Omega)$ (non-negative) and $f \in L^\infty(0,T;L^2(\Omega))$. We know that
$$u(t,x) \leq C(\lV... | https://mathoverflow.net/users/90553 | $L^\infty$ estimate on heat equation with a lower order term | The estimate as stated is clearly false since $u\to u\_0$ as $t\to0$ hence $\|u\|\_{L^\infty}\ge \|u\_0\|\_{L^\infty}$.
Anyway you can write the kernel explicitly and extract the information you need from it. Just define $v=e^{at}u$ so that $v\_t-\Delta v=e^{at}(u\_t-\Delta u+au)=e^{at}f$. Thus
$$
u(t)=c t^{-n/2}\int... | 6 | https://mathoverflow.net/users/7294 | 236681 | 109,489 |
https://mathoverflow.net/questions/236633 | 4 | Define the configuration space of $n$ points in a general manifold $M$, where $\dim M=m$, as $K=(M^n-D)/S\_n$ where $S\_n$ is the permutation group and $D=\{(x\_1,\cdots,x\_n)| \exists i,j\ s.t. x\_i=x\_j \}$.
Then my question is
(1) How to prove $M=\mathbb{R}^m$ then $\pi\_1(K)=S\_n$ for $m>2$ and $\pi\_1(K)=B\_n... | https://mathoverflow.net/users/43941 | How to calculate the fundamental group of general configuration space | (1a) Assume $n\geq 2$ since otherwise the quotient is trivial. Since the action of $S\_n$ is free, the quotient map is a covering map. Since $m\geq 3$, and $M$ is simply-connected, $M^n-D$ is simply-connected by transversality since the codimension of $D$ is greater than or equal to 3 (so the inclusion map is 2-connect... | 6 | https://mathoverflow.net/users/12218 | 236683 | 109,490 |
https://mathoverflow.net/questions/236546 | 4 | Let $I\subseteq R:=\mathbb C[x\_0,\ldots,x\_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my [previous question](https://mathoverflow.net/q/236405/9947), the permutation group $\mathfrak S\_{n+1}$ acts on $R$ by permuting the variables, inducing an action on $\Bbb P^n$. and there is a subg... | https://mathoverflow.net/users/9947 | How to compute the tangent space of a quotient by a finite group | Let's assume $G$ acts effectively. Since $G$ is finite all orbits are closed. Take an affine open set $U\subset X$ that contains $x$, and assume $x$ is a smooth point.
Then using the Luna Slice Theorem on $U$ you can deduce that $T\_{\pi(x)}(U//G)$ is isomorphic to $T\_0(T\_x(U)//Stab\_G(x))$ since the tangent space... | 2 | https://mathoverflow.net/users/12218 | 236697 | 109,497 |
https://mathoverflow.net/questions/226749 | 4 | Denote by $\Sigma\_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M\_g$ between $\Sigma\_g$ and $\Sigma\_{g+1}$ with the following two nice properties:
1) $M\_g$ is an orientable Haken manifold
2) The two boundary inclusions $\Sigma\_g \hookrightarrow M\_g$ and $\Sigma\_{g+1} \hook... | https://mathoverflow.net/users/78554 | Constructing a "nice" cobordism | Yes, your construction works. Even with an "unknotted" arc your construction works. You can think of your manifold as a boundary connect sum of $S^1 \times S^1 \times [0,1] \setminus B^3$ with $\Sigma\_g \times [0,1]$ along some embedded $D^3$ in the $S^2$-component of the first manifold's boundary. So this reduces che... | 1 | https://mathoverflow.net/users/1465 | 236703 | 109,500 |
https://mathoverflow.net/questions/236702 | 5 | There are many conjecturally-equivalent three-manifold [Floer homologies](https://en.wikipedia.org/wiki/Floer_homology), of which my understanding is the most-computable is [Heegaard Floer homology](http://math.mit.edu/~petero/Introduction.pdf).
>
> What is the (Heegaard) Floer homology of a connect sum of $k$ cop... | https://mathoverflow.net/users/78 | What is the Heegaard Floer Homology of a connect sum of $S^2 \times S^1$s? | The explicit calculation of this Heegaard-Floer homology (at least, $HF^-$, and of the unique $\text{Spin}^c$ structure for which $c\_1(\mathfrak s) = 0$) was carried out in [the paper in which it was introduced](http://annals.math.princeton.edu/wp-content/uploads/annals-v159-n3-p03.pdf), and indeed is an important par... | 7 | https://mathoverflow.net/users/40804 | 236705 | 109,502 |
https://mathoverflow.net/questions/236713 | 6 | Let $f(G)$ give the number of perfect matchings of a graph $G$.
Consider set $\mathcal N\_{2n}=\{0,1,2,\dots,n!-1,n!\}$.
Consider collection of all $2n$ vertex balanced bipartite graph to be $\mathcal G\_{2n}$.
For every $m\in\mathcal N\_{2n}$ is there a $G\in \mathcal G\_{2n}$ such that $f(G)=m$?
| https://mathoverflow.net/users/10035 | Are all numbers from $1$ to $n!$ the number of perfect matchings of some bipartite graph? | **No.** For all $n \geq 3$, there is no $G \in \mathcal{G}\_{2n}$ with $f(G)=n!-1$. To see this, note that $f(K\_{n,n})=n!$ and $f(K\_{n,n}-e)=n!-(n-1)!$, where $K\_{n,n}-e$ is $K\_{n,n}$ minus an edge. Thus, there are no graphs $G$ in $\mathcal{G}\_{2n}$ with $n!-(n-1)!+1, n!-(n-1)!+2 \dots$, or $n!-1$ perfect matchin... | 10 | https://mathoverflow.net/users/2233 | 236727 | 109,504 |
https://mathoverflow.net/questions/236579 | 2 | I'm trying to show that the Lie group $G=SO(n+1) \times SO(2)$ acts multiplicity-free on the cotangentbundle $T^\* (SO(n+1)/SO(n-1))$.
That means:
1)
There exists an $\operatorname{Ad}^\*\_G$-equivariant momentum map
$$\Phi \colon T^\* (SO(n+1)/SO(n-1)) \to \mathfrak{g}^\*$$
2)
For $\alpha \in \mathfrak{g}^\*$ th... | https://mathoverflow.net/users/75382 | multiplicity-free action on $SO(n+1)/SO(n-1)$ | Let $G\_\mathbb{C}$ be the complexification of a compact group $G$. Let $H\subseteq G$ be a closed subgroup. Then there is the following criterion: $M=T^\*(G/H)$ is multiplicity-free (as a Hamiltinian manifold) if and only if $X=G\_\mathbb{C}/H\_\mathbb{C}$ is spherical (as a $G\_\mathbb{C}$-variety, i.e. a Borel subgr... | 1 | https://mathoverflow.net/users/89948 | 236736 | 109,505 |
https://mathoverflow.net/questions/236757 | 2 | I'm looking for a digraph dataset that can return all directed graphs satisfying certain requirements.
Following are some examples:
1. All tournament with 12 vertices;
2. All connected digraphs with 10 vertices;
3. All digraphs with 9 vertices whose underlying undirected graph bipartite.
Is there a digraph datas... | https://mathoverflow.net/users/58246 | Is there any digraph data set that gives all directed graphs satisfying certain requirements? | There are 154108311168 tournaments on 12 vertices and you can make them with the tool gentourng that comes with [nauty](https://cs.anu.edu.au/~bdm/nauty/).
The number of connected digraphs on 10 vertices is more than 10^20, which is impossibly many. That's if you allow 2-cycles; otherwise the number is "only" about ... | 6 | https://mathoverflow.net/users/9025 | 236768 | 109,512 |
https://mathoverflow.net/questions/236772 | 4 | The following statement [cannot](http://mathforum.org/kb/message.jspa?messageID=182924) be proven in $\mathsf{ZFC}$:
>
>
> >
> > (S) : If $A, B$ are sets with $|A| < |B|$, then $2^{|A|} = |{\cal P}(A)| < |{\cal P}(B)| = 2^{|B|}$.
> >
> >
> >
>
>
>
Obviously, $\mathsf{GCH}$ implies (S). Does (S) imply $\ma... | https://mathoverflow.net/users/8628 | Is injectivity of $2^{(\ldots)}$ weaker than $\mathsf{GCH}$? | No, take Merimovich's model in which $2^\kappa=\kappa^{+3}$ for all cardinals $\kappa.$
Merimovich, Carmi [A power function with a fixed finite gap everywhere](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9156282&fileId=S0022481200005272). J. Symbolic Logic 72 (2007), no. 2, 361–417.
Let... | 12 | https://mathoverflow.net/users/11115 | 236773 | 109,515 |
https://mathoverflow.net/questions/236758 | 1 | I plan to submit a couple of questions around Émile Borel's works in probability theory to MO.
In this scope, I'd like to know if the following works have ever been translated from French to English or another foreign language:
* *Le hasard* (*Randomness*), second edition, 1947;
* *Sur les probabilités universellem... | https://mathoverflow.net/users/88057 | Have some works by Émile Borel ever been translated from French to English or another foreign language? | According to Mathscinet, the following books by Borel on probability have been translated to English and Russian:
MR0177424 Borel, Émile Elements of the theory of probability. Translated by John E. Freund Prentice-Hall, Inc., Englewood Cliffs, N.J. 1965
MR0169314 Borel, Émile Probabilities and life. Translated fro... | 2 | https://mathoverflow.net/users/25510 | 236774 | 109,516 |
https://mathoverflow.net/questions/236779 | 5 | Let $X$ be a smooth projective variety, and $D$ a Cartier divisor on $X$ inducing a surjective morphism $f\colon X\rightarrow C$, where $C$ is a curve.
May we conclude that $D^{2}=0$?
| https://mathoverflow.net/users/nan | Self-intersection of a Cartier divisor | The answer is *yes* if the complete linear system $|D|$ is *without fixed components*. In such a situation, the fact that $f$ is a morphism onto a curve implies that $|D|$ is composed with a base-point free rational pencil, hence $D^2=0$.
On the other hand, if there are fixed components the answer is in general *no*.... | 5 | https://mathoverflow.net/users/7460 | 236781 | 109,520 |
https://mathoverflow.net/questions/225646 | 16 | I have seen various references in the literature to such a connection but they tend to assume that the reader is familiar with the connection, and limit themselves to providing additional detail. So in broad terms, what is the precise connection between o-minimal models and Grothendieck's programme?
| https://mathoverflow.net/users/28128 | What is the precise relationship between o-minimal theory and Grothendieck's "Esquisse d'un programme"? | To define an o-minimal geometry, one gives oneself a family of functions from $\mathbf R^n$ to $\mathbf R^m$ ($m,n$ not specified), and one considers all *definable* subsets of $\mathbf R^n$, ie, those which can be defined by a mathematical expression using these basic functions, addition, multiplication, constant func... | 17 | https://mathoverflow.net/users/10696 | 236786 | 109,523 |
https://mathoverflow.net/questions/236539 | 4 | Let $\Omega =\{0,1\}^{\mathbb{N}}$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x\_n),(y\_n))=1/2^m$, where $m=\min\{i\in\mathbb{N}\,:\,x\_i\neq y\_i\}$.
Define a function $\pi:\Omega\rightarrow [0,1]$, where $\pi((x\_n))$ is the number in $[0,1]$ with ... | https://mathoverflow.net/users/38374 | Hausdorff dimension of sequence space | According to Falconer[1] this is due to Besicovitch[2]. Falconer states it (generalized to $\mathbb R^n$) as Theorem 5.1, p. 65. This proves more than just $X$ and $\pi(X)$ have the same Hausdorff dimension: it proves the Hausdorff measures of $X$ and $\pi(X)$ are within a constant factor of each other.
[1] K. F. Fal... | 4 | https://mathoverflow.net/users/454 | 236788 | 109,525 |
https://mathoverflow.net/questions/236807 | 1 | i am reading and trying to do some exercises and problems of the book Lectures on Analytic Differential Equations- Y. Ilyashenko, S. Yakovenko.
I can not solve the problem 11.6 that says
Consider nondicritical foliations having at most three hyperbolic singularities on the exceptional divisor after blow-up.
Prove th... | https://mathoverflow.net/users/89369 | Analytic conjugacy of vanishing holonomy groups implies analytic conjugacy of foliations | No, in general the holonomy group is not solvable. Yet, this is how I'd tackle the exercise. (By the way, as it is posed the exercise cannot be solved: you need to assume that each eigenratio of both foliations agree, since holonomy conjugacy only provides equality up to $\mathbb Z$ and said eigenratios are local analy... | 3 | https://mathoverflow.net/users/24309 | 236811 | 109,530 |
https://mathoverflow.net/questions/236572 | 6 | Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ and $n$ are large.
What is
$$S\_{M,k} = \sum\_{x \in \mathbb{Z}^n} e^{-x^T M x}\;?$$
Is there some way to estimate th... | https://mathoverflow.net/users/45564 | How to estimate a specific infinite matrix sum | Let $s = k-1$, and write $x^T M x = \|x\|^2 + s (e \cdot x)^2$ where $e = (1,\ldots,1)$. Let $P\_j = \{x \in \mathbb Z^n: e \cdot x = j\}$. Each $P\_j$ is a translate of $P\_0$, which is a subgroup of $\mathbb Z^n$. Then
$$S\_{M,k} = \sum\_{j\in \mathbb Z} e^{-s j^2} \sum\_{x \in P\_j} e^{-\|x\|^2} = \sum\_{j \in \ma... | 3 | https://mathoverflow.net/users/13650 | 236814 | 109,531 |
https://mathoverflow.net/questions/236805 | 6 | Let $F\_n$ be the empirical distribution obtained from an i.i.d. sample
of the distribution $F:R ^d \to [0, 1]$.
[Kiefer (1961)](http://www.ams.org/journals/tran/1958-087-01/S0002-9947-1958-0099075-1/S0002-9947-1958-0099075-1.pdf) shows that the convergence of the empirical distribution is like
$$
P\left( \left\lVert F... | https://mathoverflow.net/users/nan | What is the order of the constant $K$ in the multidimensional Dvoretzky-Kiefer-Wolfowitz inequality($Ke^{-c z}$)? | For any distribution $F$ over the naturals $\mathbb{N}$, one can show the following DKW-type inequality:
$$ \mathbb{P}(||F-F\_n||\_\infty > 1/\sqrt{n}+\epsilon)
\le \exp(-2n\epsilon^2)
$$
(in fact, it's known for the more general Markov case, see
<http://projecteuclid.org/euclid.jap/1421763330>
)
I started to write s... | 1 | https://mathoverflow.net/users/12518 | 236819 | 109,532 |
https://mathoverflow.net/questions/236793 | 1 | Let $A$ be a non-negative $N \times N$ square matrix with $a\_{i,i}=0, 1 \leq i \leq N$. Also, let $r\_i$ be the $i$-th row sum of $A$.
I know that $\rho(A)$, the spectral radius of $A$, is bounded as follows:
$min(r\_i) \leq \rho(A) \leq max(r\_i); 1 \leq i \leq N$
If I increase the row sum of an arbitrary row ... | https://mathoverflow.net/users/45615 | Spectral radius's relation with row sum | No. For example, the spectral radii of
$$ A = \pmatrix{0 & 1 & 1\cr 1 & 0 & 1\cr 0 & 0 & 0\cr},\ A' = \pmatrix{0 & 0 & 3\cr 1 & 0 & 1\cr 0 & 0 & 0\cr}$$
are $1$ and $0$ respectively.
| 3 | https://mathoverflow.net/users/13650 | 236823 | 109,533 |
https://mathoverflow.net/questions/236826 | 0 | We have the inequality
$$\alpha\_n(t) \le 4\pi(3\pi)^{1/3} \exp\left\{\int\_0^t(1+3F\_P(\sigma)) \, d\sigma \right\} \cdot \int\_0^t P(\sigma)^2(3CD\_1^2)^{1/3}\alpha\_{n-1}(\sigma) \, d\sigma$$
for $n=2,3,\ldots$. (We notice that $\alpha\_n$ appears on both sides of the inequality.)
Why does it follow that the infi... | https://mathoverflow.net/users/89456 | Inequality implies locally uniform convergence of a series | Some estimates like $\alpha\_n(s)\leqslant A(T)^n/n!$ for all $s\in [0,T]$ should hold by induction, if we suppose something moderate on your functions, which depend on $\sigma$. This yields desired uniform convergence.
| 0 | https://mathoverflow.net/users/4312 | 236829 | 109,534 |
https://mathoverflow.net/questions/23974 | 11 | Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see [his paper](http://www.jstor.org/pss/2939279)) corresponding to canonical basis vectors.
I'm interested in the stalks of these perverse sheaves, in particular, the stalk at 0. I believe I've reduced a (s... | https://mathoverflow.net/users/66 | What's known about the stalks of Lusztig's perverse sheaves on quiver varieties? | The dimensions of the stalks of Lusztig's sheaves give the coefficients when a canonical basis element is expanded in a PBW basis. These stalks satisfy a parity vanishing condition.
For these positive results, see Corollary 10.7 of Lusztig's "Canonical Bases Arising from Quantized Enveloping Algebras".
The first ex... | 7 | https://mathoverflow.net/users/425 | 236831 | 109,535 |
https://mathoverflow.net/questions/236818 | 5 | Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. Since $N(\alpha-1) \geq 1$, an application of $H\ddot{o}lder's$ inequality gives the lower bound $N(\alpha) \geq 2^d$, with... | https://mathoverflow.net/users/90626 | Mahler measure of a totally positive, expanding algebraic integer | The answer is yes.
Suppose that $\alpha$ is totally real algebraic integer, and that all its conjugates are greater than $1$. Suppose also that $\alpha \ne 2$.
Note the elementary inequality for $x \in (1,\infty) \setminus \{2\}$:
$$\ln|x| \ge \frac{\ln|x-1| + \ln|x-2|}{3} + \ln C,$$
where $C = 2^{1/3} 3^{1/2} ... | 5 | https://mathoverflow.net/users/90631 | 236834 | 109,537 |
https://mathoverflow.net/questions/236816 | 5 | Let $T:[0,1]\to [0,1]$ be a piecewise smooth expanding map, i.e., $|T'(x)|>1$ for all $x$. Let $I\_n$ be a sequence of nested intervals (i.e., $I\_{n+1}\subset I\_n$) such that the length of $I\_n$ tends to 0 as $n\to\infty$.
Define the *survivor set* for $I\_n$ as follows:
$$
\mathcal J(I\_n)=\{x \in[0,1] : T^k(x)\... | https://mathoverflow.net/users/8131 | Survivor sets for expanding maps of the interval | The answer of your question is yes if $T \in C^2$. It holds since $\dim\_H(\mathcal{J}(a,b))$ varies continuously when the end points of the removed intervals varies continuously. Then, by continuity, the Hausdorff dimension of $\mathcal{J}(I\_k)$ must tend to $1$ as $k \to \infty$.
This is proved by Urbanski in two ... | 2 | https://mathoverflow.net/users/10518 | 236836 | 109,538 |
https://mathoverflow.net/questions/236835 | 8 | First question on MathOverflow, I hope it is appropriate for this site. There are two related questions.
Let $K$ be a number field, $G\_K = Gal(\overline{K}/K)$, $p$ a prime, and
$$\chi\_1,\chi\_2:G\_K\rightarrow\overline{\mathbb{Q}}\_p^\times$$
be continuous characters such that $\ker(\chi\_1) = \ker(\chi\_2)$.
M... | https://mathoverflow.net/users/90625 | If two Hecke characters cut out the same field, are they Galois conjugates? | The answer to both your questions is "no".
Take $K = \mathbf{Q}$. Then there is a unique $\mathbf{Z}\_p$-extension of $K$ (contained in $\mathbf{Q}(\zeta\_{p^\infty})$) which gives us a surjection $G\_K \to \Gamma$ where $\Gamma$ is isomorphic to $\mathbf{Z}\_p$.
Now, what are the continuous characters $\mathbf{Z}\... | 9 | https://mathoverflow.net/users/2481 | 236847 | 109,544 |
https://mathoverflow.net/questions/236708 | 6 | I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would *love* it if I could choose the metric $g$ such that every point has an open neighbourhood $U$ [bi-Lipschitz](https://en.wikipedia.org/wiki/Lipschitz_continuity#Definitions), (*Edit*: **as a m... | https://mathoverflow.net/users/4177 | Compact manifolds locally bi-Lipschitz to Euclidean space | This question could be seen by some as too basic, but I think it is precisely the kind of situation MO is for (question is not trivial to an outsider, but easy to answer to an insider of the field), so let me give a complete argument to show that the property you seek is true for any metric (only adding some flesh to t... | 9 | https://mathoverflow.net/users/4961 | 236851 | 109,546 |
https://mathoverflow.net/questions/236869 | 1 | Given a vector $(n\_0, n\_1, \dots, n\_l)$ where $n\_i \in \{-1, 1\}$, $i = \overline{0, l-1}, n\_l = 1$ and $l \in \mathbb{N}$.
Prove that for all $a$ such that
$$0 < a \leq 2^0\cdot n\_0 + 2^1 \cdot n\_1 + \dots + 2^{l - 1} \cdot n\_{l - 1} + 2^l \cdot n\_l$$
there are distinct $k\_0, k\_1, \dots, k\_r \in I = \... | https://mathoverflow.net/users/90660 | Powers of two with coefficients {1,−1} | Any $N\ge0$ has a unique binary expansion
$$
N=\sum\_{k=0}^n a\_k 2^k, \quad a\_k\in\{0,1\},\ a\_n=1.
$$
(Here $n=\lfloor \log\_2 N\rfloor$.)
Now consider the map $N\mapsto 2N-(2^{n+1}-1)$ from $\mathbb Z\_+$ to $\mathbb Z$. We have
$$
N'=2N-(2^{n+1}-1)=\sum\_{k=0}^n a'\_k 2^k, \quad a'\_k\in\{-1,1\},
$$
where $a\_k'=... | 3 | https://mathoverflow.net/users/8131 | 236880 | 109,557 |
https://mathoverflow.net/questions/215265 | 8 | The following question was asked on math.stackexchange, where it received no answers.
<https://math.stackexchange.com/questions/1392669/maximum-size-of-a-union-of-incomparable-chains>
Let $\mathbb{N}^{<\mathbb{N}}$ denote the set of finite sequences of natural numbers (not including the empty sequence). Order this... | https://mathoverflow.net/users/nan | Maximum size of a union of incomparable chains | There is no such $c$. Assume that you have a set with $n$ elements, and maximal $B$ has $k$ elements. Take two copies $C$, $D$ of $A$ with disjoint supports. Add initial segment $1,1,\dots,1$ of length $k$ to all sequences in $C$, $D$ and consider also $k$ sequences consisting of at most $k$ $1$'s. We get $2n+k$ sequen... | 4 | https://mathoverflow.net/users/4312 | 236881 | 109,558 |
https://mathoverflow.net/questions/236889 | 1 | It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F\_\nabla$ is imaginary (and so are the local connection $1$-forms). I am curious, though, if the following converse is true: if $\nabla$ is given such that $F\_\nab... | https://mathoverflow.net/users/54780 | Hermitic connections on complex line bundles with imaginary curvature form | Even flat connections don't have to arise from a Hermitian inner product; they can have holonomy not unitary.
| 2 | https://mathoverflow.net/users/13268 | 236890 | 109,561 |
https://mathoverflow.net/questions/236875 | 9 | I apologize in advance if this question is too basic, but I've received no response on Math Stack Exchange, so perhaps it is more appropriate here:
Let $X\_n$ be a square-integrable martingale with $\mathbb{E}\lbrack X\_n^2\rbrack=O(n)$. Is it true that $X\_n/n$ tends to $0$ almost surely?
Note that if one demanded... | https://mathoverflow.net/users/90661 | Law of large numbers for martingales | The answer is yes, and it is based on an idea by Prokhorov (cf. e.g. Theorem 10 in Section 3 of Ch. IX in [Petrov, V. V., Sums of independent random variables, Springer-Verlag, 1975]). We have $EX\_n^2\le Cn$ for some real $C>0$ and all natural $n$. For natural $s$, let
\begin{equation}
T\_s:=\max\_{2^s\le n<2^{s+1}}... | 10 | https://mathoverflow.net/users/36721 | 236900 | 109,564 |
https://mathoverflow.net/questions/236873 | 4 | I suspect the following should be well known, in some circles, under some name. Alas, I could not figure out how to prove it or where to look it up.
Recall that a *recollement* is a sequence of (triangulated) functors of triangulated categories
$\mathcal{D}' \xrightarrow{i\_\*} \mathcal{D} \xrightarrow{j^\*} \mathc... | https://mathoverflow.net/users/5181 | A distinguished triangle of mapping spectra arising from recollement | I'm going to do a proof assuming we are in a stable $\infty$-category (I'm pretty sure this is almost equivalent to your "sufficiently rich" situation anyway). In your case $F=j\_!j^!$ and $G=i\_\*i^\*$.
**Theorem**: Let $C$ be a stable $\infty$-category, $F,G$ exact endofunctors of $C$ such that
* There's a fiber ... | 3 | https://mathoverflow.net/users/43054 | 236905 | 109,566 |
https://mathoverflow.net/questions/236901 | 4 | Let $f(X,t\_1,\dots,t\_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}\_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (<http://plms.oxfordjournals.org/content/s3-43/2/227.full.pdf>) I know that the number of specializations $\vec t = (t\_1,\dots, t\_s)$ in $(\mathcal{O}\_K)^s... | https://mathoverflow.net/users/37644 | The best possible density in Hilbert's Irreducibility Theorem | Have you stated this correctly? Assuming there are $O(N^s)$ possible specializations of height at most $N$, don't you want to say that the specialization remains irreducible *except for* $O(N^{s-1/2}\log N)$ values.
As for question (2), I don't think you can do better than $s-\frac12$. Certainly not for $s=1$, where... | 5 | https://mathoverflow.net/users/11926 | 236907 | 109,568 |
https://mathoverflow.net/questions/165326 | 2 | Let $(k,|\cdot|)$ be an algebraically closed field, complete wrt a (multiplicative) norm as in the framework of the Berkovich's analytic geometry. Given a commutative Banach $k$-algebra $\mathcal{A}\neq 0$, let $X=\mathcal{M}(\mathcal{A})$ its spectrum and let $0\neq f\in \mathcal{A}$.
For every point $P=||\cdot||\in... | https://mathoverflow.net/users/50468 | The target of a regular function in Non-archimedean analytic geometry | If you want to think of $f$ in a way not too far from complex intuition, you should rather consider the induced morphism $\varphi$ from $X$ to the Berkovich affine line. If $x\in X$ then $\varphi(x)$ is the semi-norm $P\mapsto |P(f)(x)|$.
| 3 | https://mathoverflow.net/users/28143 | 236912 | 109,570 |
https://mathoverflow.net/questions/236909 | 4 | For any given extension $T$ of ZFC (or perhaps NBGC or something), we can ask whether there is an extension $T'$ of ZF which does not prove AC such that
1. $Con(T) \leftrightarrow Con(T')$
2. $Con(T) \to Con(T')$ in a "nice" way, e.g. $T' + AC = T$.
3. $Con(T') \to Con(T)$ in a "nice" way, e.g. a model of $T$ can be ... | https://mathoverflow.net/users/2362 | Large cardinals without choice? | Yes, there are some general methods to attain your properties for
any theory $T$.
**Method 1.** For any theory $T$ extending ZFC, let $T'$ be the
theory consisting of the following:
* ZF
* all the arithmetic consequences of $T$
* all assertions of the
form $\text{AC}\to\sigma$, where $\sigma$ is in $T$.
If $T$ is... | 3 | https://mathoverflow.net/users/1946 | 236915 | 109,571 |
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