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https://mathoverflow.net/questions/152218 | 1 | If $X$ is a simply connected space and $\alpha\in\pi\_n(X)$ and $Y=X\cup\_{\alpha} e^{n+1}$.
and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q}
u,d)$ is a commutative model for $Y$,
which we denote by $M\_\alpha$.
If $f\colon X\rightarrow X\_1$
and $Y=X\cup\_{\alpha}e^{n+1}$ and $Y\_1=X\_1... | https://mathoverflow.net/users/33699 | Naturality of commutative model for cell attachment in Rational Homotopy Theory | In rational homotopy theory models of homotopy push-outs are homotopy pull-backs in the category of CDGAs. This fact follows from the Quillen adjunction between model categories of spaces and CDGAs. In fact this is just a Mayer-Vietoris type argument.
(1) is a homotopy push-out: $X\rightarrow Y$ is a cofibration and ... | 1 | https://mathoverflow.net/users/27816 | 152262 | 81,148 |
https://mathoverflow.net/questions/152279 | 3 | It is mentioned in the book "Permutation Groups" by Dixon and Mortimer that a 6-transitive permutation group on $n$ elements is $A\_n$ or $S\_n$ and that all the other
$4$ and $5$-transitive permutation groups are the simple Mathieu groups. In particular
they are contained in $A\_n$.
My question is whether every 3-tra... | https://mathoverflow.net/users/9304 | Is every nontrivial 3-transitive permutation group contained in $A_n$? | No. The group $\text{PGL}\_2({\bf Z}/p{\bf Z})$ acts transitively
on the $p+1$ points of the projective line ${}\bmod p$ (and the action
is even sharply $3$-transitive), but is not contained in $A\_{p+1}$.
For example, if $c$ is a primitive residue then $({c \; 0 \atop 0 \; 1})$
has cycle structure $1, 1, p-1$ and is t... | 7 | https://mathoverflow.net/users/14830 | 152284 | 81,159 |
https://mathoverflow.net/questions/152283 | 11 | Let $\alpha$ an irrational real number. It is well known that the set $\{ \{n \alpha \}|\,\, n \in \mathbb{N} \}$ is dense in$[0,1]$.
($\{x\}$ denotes the fractional part of $x$)
**But how to prove the set $\{ \{p \alpha \}|\,\, prime\,\,p\}$ is dense in $[0,1]$? Also, is it uniformly distributed?**
It is somehow a... | https://mathoverflow.net/users/43383 | Is $\{ p \alpha \}$ for prime $p$ dense in $[0,1]$? | Exponential sums over primes can be reduced to exponential sums over arithmetic progressions, as discovered by Vinogradov. The basic idea is the same as in the sieve of Eratosthenes. Modern treatments of this reduction rely on particular decompositions of the von Mangoldt function into convolutions of "simpler" functio... | 18 | https://mathoverflow.net/users/11919 | 152286 | 81,160 |
https://mathoverflow.net/questions/152202 | 1 | My title requests something more general than I actually require right now, so I would settle for an answer to something more specific (details below) but I would like to understand the more general concept as well.
When we mean a subgroup of $PSL\_2(\mathbb{Z})$, a concise way of describing a congruence subgroup of ... | https://mathoverflow.net/users/14835 | Definition of level N congruence subgroup of an arithmetic group, useful for computations | Thank you to @Qiaochu Yuan and @Alex B. for the kick in the right direction. Here is the answer to this question for the case introduced in the third paragraph. Hopefully it benefits someone other than me.
Let $K$ be a number field, let $\mathcal{O}$ be its ring of integers, and let $\mathfrak{I}\vartriangleleft\math... | 0 | https://mathoverflow.net/users/14835 | 152289 | 81,162 |
https://mathoverflow.net/questions/152193 | 13 | Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.)
The set $\text{Fix}\_H(X)$ of $H$-fixed points carries a $G$-action, and we can then take the set $\text{Fix}\_H(X)/G$ of orbits. But also, the set $X/G$ of $G$-orb... | https://mathoverflow.net/users/586 | When taking the fixed points commutes with taking the orbits | To see when this is possible, we can consider each orbit of $G \times H$ separately. An orbit of $G \times H$ corresponds to a subgroup of $G \times H$.
A subgroup of $G \times H$ corresponds to a triple of $A,B,C,D$ where $A$ is a normal subgroup of $B$ a subgroup of $G$, $C$ is a normal subgroup of $D$ a subgroup o... | 5 | https://mathoverflow.net/users/18060 | 152309 | 81,167 |
https://mathoverflow.net/questions/152308 | 7 | Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. I am interested in obtaining a lower bound on the length $|\Gamma|$ of $\Gamma$.
I conjecture that
$$
|\Gamma| \ge C(... | https://mathoverflow.net/users/41608 | Shortest curve with given convex hull | Let $n$ be large and glue $n$ rectangles, each with the proportions $1\times n$, together end-to-end by their short edges. Then perturb slightly so that all the vertices are in convex position. The perimeter will be roughly $2n^2+2$ (modulo the perturbation), and the longest boundary edge will have length roughly $n$, ... | 10 | https://mathoverflow.net/users/440 | 152310 | 81,168 |
https://mathoverflow.net/questions/152316 | -1 | It is clear that Ito isometry
$E(∫^t\_0fdW)^2=E(∫^t\_0f^2dt)$
can be written in the multiplicative form as
$E(∫^t\_0fdW\cdot∫^t\_0gdW)=E(∫^t\_0f⋅gdt).$
Is it possible to obtain the multiplicative version of the Novikov inequality
$E(|∫^t\_0fdW|^p)≤B\_pE(∫^t\_0|f|^2dt)^{p/2}?$
It should take a form like:
... | https://mathoverflow.net/users/44393 | Multiplicative version of Novikov inequality for Ito integral | This cannot be true: just think of the case where $f$ and $g$ have disjoint supports. Then the right hand side vanishes but the left hand side does not.
| 1 | https://mathoverflow.net/users/38566 | 152323 | 81,172 |
https://mathoverflow.net/questions/152322 | 0 | Let $M$ be a $ \mathbb{Z}\_{p}[[T]] $-module and $X=Hom(M,\mathbb{Q}\_{p}/\mathbb{Z}\_{p})$ be the dual of $M$. Let $X[p^n]$ denotes the $p^n$-torsion points of $X$. Is $X/X[p^n]$ the dual of $M[p^n]$ $?$ If not, then whose dual is $X/X[p^n]$ in terms of $M$ $?$
| https://mathoverflow.net/users/33900 | Dual of a module | As @S.Carnahan shows. this is not true.
Perhaps the statement you want is that the dual of $M[p^n]$ is $X/p^nX$?
Take the exact sequence
$$0\to M[p^n] \to M \to M \to M/p^nM \to 0,$$
where the middle map is multiplication by $p^n$, and dualize it.
**Edit:** This also shows that $X/X[p^n]$ is the dual of $p^nM... | 4 | https://mathoverflow.net/users/22989 | 152327 | 81,174 |
https://mathoverflow.net/questions/152337 | -2 | Let $M$ be a manifold with the property that $f^{\*}(TM)$ is isomorphic to TM, for every diffeomorphism $f$ on $M$. Does this imply that $M$ is parallelizable?
| https://mathoverflow.net/users/36688 | A question on parallelizable manifolds | Isn't the 2-dimensional sphere a counterexample? If $f$ has degree 1, then it's homotopic to the identity, so $f^\*(TM)\cong TM$. If $f$ has degree $-1$, then it's homotopic to the antipode map $a$, so $f^\*(TM)\cong a^\*(TM)\cong TM$, where the last $\cong$ is evident if we embed the sphere in the standard way in $\ma... | 2 | https://mathoverflow.net/users/6794 | 152338 | 81,178 |
https://mathoverflow.net/questions/152324 | 1 |
>
> Question: Can the Fell topology be expressed in terms of the distributions of the the tracial states of a unitary representations, that, is $\pi\_j \rightarrow \pi$ if and only if $tr\; \pi\_j \rightarrow tr\;\pi$?
>
>
>
This makes of course only sense in a more restricted setting, say reductive groups over ... | https://mathoverflow.net/users/10400 | Fell topology in terms of distributions | Here is a result by D. Milicic, On $C^{\ast} $-algebras with bounded trace, Glasnik Mat. Ser. III 8(28) (1973), 7–22. Say that a $C^\*$-algebra $A$ has bounded trace if the linear span of $T(A^+)$ is dense in $A$, where $T(A^+)$ is the set of those positive elements such that $\pi\mapsto Tr\,\pi(x)$ is bounded on $\hat... | 3 | https://mathoverflow.net/users/14497 | 152344 | 81,180 |
https://mathoverflow.net/questions/152349 | 2 | Let $M$ be a finite von Neumann algebra with $\tau$ a finite faithful trace. Let $N$ be a von Neumann subalgebra of $M$ with trace $\tau|\_N$, obtained by restricting $\tau$ to $N$. If $e\_N$ denotes the Jones projection and $<M,e\_N>$ the basic construction, then we have the strongly-dense \*-subalgebra $span\{M\cup\{... | https://mathoverflow.net/users/44405 | Semifinite Trace on Jones' Basic Construction | A proof can be found in Sinclair and Smith's book "Finite von Neumann algebras and masas" doi:10.1017/CBO9780511666230. I believe Chapter 4 will be what you're looking for.
| 3 | https://mathoverflow.net/users/351 | 152351 | 81,182 |
https://mathoverflow.net/questions/152307 | 3 | I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers *Supermanifolds*, and Khrennikovs *Superanalysis*.
Unfortunately both books lack a clear set of simple examples to build up some basic intuition about these operations. Is there a good basic reference for the... | https://mathoverflow.net/users/35706 | A good reference for learning about super-differentiation & super-integration? | I have never tried to keep straight the different sign conventions in the literature. There is a good reason for this: it is a theorem of category theory that any reasonable sign convention is as good as any other (viz: the monoidal category of $\mathbb Z/2$-graded vector spaces admits precisely two symmetric monoidal ... | 5 | https://mathoverflow.net/users/78 | 152356 | 81,183 |
https://mathoverflow.net/questions/152368 | 8 | What are the best known bounds on the number of non-isomorphic (unlabelled) planar graphs on $n$ vertices? Is there a simple proof that this number is at most exponential in $n$?
| https://mathoverflow.net/users/17008 | Number of unlabelled planar graphs | Simple proof? Sort of.
Let $t(n)$ be the number of plane triangulations with $n$ vertices. Tutte
[in ''A Census of Planar Triangulations,'' Canad. J. Math. 14 (1962), 21-38]
found an [explicit formula](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/census-of-planar-triangulations/0... | 12 | https://mathoverflow.net/users/4040 | 152380 | 81,202 |
https://mathoverflow.net/questions/152353 | 2 | I am trying to prove the following claim (may be it has been proven).
Claim: Consider a set of points $\phi=\{x\_1,x\_2,...,x\_i,...\}$ generated by a homogeneous PPP with rate $\lambda$ in the 2-D plane $\mathbb R^2$. Then we generate the Voronoi cells with the $k$ nearest points ($k$ order Voronoi cell [WiKi](http:... | https://mathoverflow.net/users/38825 | A Claim on Typical Voronoi Cells | Notice that enumeration of points is not given a priori. The Poisson point configuration is a set of points with no order on them. It also can be viewed as a measure that puts mass 1 at each configuration point. The result is that for certain enumerations the expectation of the cell sizes will misbehave.
Here is a v... | 2 | https://mathoverflow.net/users/2968 | 152388 | 81,205 |
https://mathoverflow.net/questions/152333 | 12 | Is there any known reason why Alonzo Church chose Greek $\lambda$ as the "binding operator" for the Lambda Calculus?
| https://mathoverflow.net/users/20995 | Why did Alonzo Church choose the letter $\lambda$ as the "binding operator"? | This question has been answered [on math.SE](https://math.stackexchange.com/a/64469/413) (as pointed out by Joel David Hamkins).
With a reference to *Lambda-Calculus and Combinators in the 20th Century* by
Felice Cardone and J. Roger Hindley, Handbook of the History of Logic
Volume 5, 2009, Pages 723–817, it is stated ... | 10 | https://mathoverflow.net/users/20995 | 152390 | 81,206 |
https://mathoverflow.net/questions/152217 | 2 | Let $H\_1, H\_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n\_1}, \mathbb{P}^{n\_2}$ with Hilbert polynomials $P\_1, P\_2$, respectively. Given a pair $(Z\_1, Z\_2)$ of subschemes in $\mathbb{P}^{n\_1}, \mathbb{P}^{n\_2}$, respectively, we say that $Z\_1 \subset Z\_2$ if there exists a closed imme... | https://mathoverflow.net/users/32151 | A question on nested Hilbert scheme | I guess you are looking for a construction that gives an object over $H\_2$, that parametrizes closed immersions $Z\_1 \to Z\_2$ for each $Z\_2 \in H\_2$. If I'm not mistaken, it is the closed subscheme of $Quot(\mathscr{O}\_{\mathbb{P}^{n\_2}}) \times Quot(\mathscr{O}\_{\mathbb{P}^{n\_2}})$ (which parametrizes pairs o... | 1 | https://mathoverflow.net/users/121 | 152392 | 81,208 |
https://mathoverflow.net/questions/152403 | 0 | Let $\phi$ : $A \rightarrow A$ be a positive map, where $A$ is a (unital) C\* algebra. Suppose we are given that $\phi$ is n positive whenever n= $2^k$ for some $k \in \mathbb{N}$. Can we conclude that $\phi$ is completely postive?
| https://mathoverflow.net/users/43888 | Checking complete positivity of maps between C* algebras | Yes, an $n$ positive map is also $n-1$ positive. Hence you map is $n$-positive for all $n$, i.e. completely positive.
For a proof, you include $M\_{n-1}(A)$ as upper left block into $M\_n(A)$.
| 7 | https://mathoverflow.net/users/12482 | 152407 | 81,213 |
https://mathoverflow.net/questions/152400 | 4 | I am confused about the definition of Hecke operators. It will be great if someone provides some references.
Shimura's 'Arithmetic Theory of Automorphic forms' says: Let $\Gamma$ be acting in the left of $G$ and and let $\tilde\Gamma$ be the commensurator of $\Gamma$. Then we call $\Gamma\alpha\Gamma$ as the hecke op... | https://mathoverflow.net/users/36735 | Definition of Hecke operators | There is a Cartan decomposition for $PGL\_2(\mathbb{Q}\_p)$ meaning that double coset of the from $PGL\_2(\mathbb{Q}\_p) //PGL\_2(\mathbb{Z}\_p)$ are represented by $\alpha^k$, $k\geq0$.
So this is the reason because the definitions are equivalent.
This works well as long as you are working on $GL\_2(\mathbb{Z})$ a... | 3 | https://mathoverflow.net/users/10400 | 152408 | 81,214 |
https://mathoverflow.net/questions/152425 | 3 | I am reading "Ample divisors on fine moduli spaces on the Projective plane" by Stromme. In the proof of Proposition 2.4, he seems to claim that if $E$ is a torsion free sheaf of projective dimension at most 1 on a smooth variety and the singular locus of $E$ has codimension at least $3$, then $E$ is reflexive. Of cours... | https://mathoverflow.net/users/7399 | Criterion for a sheaf to be reflexive | In general the converse is not true --- take for example the ideal of a point on 3-fold (its reflexive hull is the structure sheaf). On the other hand, if you know that $pd(F) \le 1$ then there is an exact sequence
$$
0 \to E\_1 \stackrel{f}\to E\_2 \to F \to 0
$$
with $E\_1$ and $E\_2$ locally free. Taking the dual on... | 7 | https://mathoverflow.net/users/4428 | 152427 | 81,223 |
https://mathoverflow.net/questions/62019 | 11 | The original Ackermann function $\varphi\colon \mathbb{N}\times\mathbb{N}\times\mathbb{N}\_0\to \mathbb{N}$ as defined in [1] was invented to prove that there is a function that is recursive but not primitive recursive.
It can be given by the following recursion:
* $\varphi(a,b,0) = a+b$
* $\varphi(a,b,n+1) = (x\maps... | https://mathoverflow.net/users/10629 | Left-bracketed Ackermann function also not primitive recursive? | Sorry for reviving this ancient post, but it seemed to me that this question shouldn’t be left unanswered.
It is easy to establish by induction that for $a,b,n\ge2$, the function $\psi(a,b,n)$ is strictly increasing in all three coordinates, and $$\psi(a,b,n)>a,b,n.$$
Then we can show
$$\tag{$\*$}\psi(a,b,n+1)\ge\psi... | 8 | https://mathoverflow.net/users/12705 | 152449 | 81,232 |
https://mathoverflow.net/questions/152446 | 1 | I am reading a paper in which the following argument is made. We have two positive real valued functions $f(x)$ and $g(x)$. We know that $$\int\_0^x \int\_0^y f(z) \ dz \ dy \leq g(x).$$
It is then stated "and easy computation shows that, except for negligibly small intervals," $$\log f(x) = O(x + \log g(x)).$$
I a... | https://mathoverflow.net/users/44488 | Estimating a quantity from an estimate in its integral | We begin with a lemma. Let $F$ be an increasing function tending to infty as $x\to\infty$.
Then $F'(x)\leq F^2(x)$, for $x\not\in E$, where $E$ is a set of finite measure.
Proof. Let $E$ be the set where $F'(x)\geq F^2(x)$. Then
$$|E|\leq\int\_{E\cap[1,\infty)}\frac{F'}{F^2}dx<\int\_1^\infty\frac{F'}{F^2}dx<\infty,$$... | 2 | https://mathoverflow.net/users/25510 | 152453 | 81,235 |
https://mathoverflow.net/questions/152303 | 3 | *Question seems simple, but I just can't find the solution.
Let A/B be an integral ring extension and let P be a prime ideal of B. By going-up theorem, there is Q, a prime ideal of A, lying over P. Then the ring of fractions of A localised on Q is still integral on that of B localised on P?*
Thank Karl and Matt, the ... | https://mathoverflow.net/users/44382 | Localisation of two rings which is an integral extension, then integral extension still holds? | Let me explain my comment in a bit more detail.
**Proposition:** *Suppose $B \subseteq A$ is an integral extension of domains and $P$ is a prime of $B$ with $Q \subseteq A$ a prime of $A$ lying over $P$. Then $B\_P \subseteq A\_Q$ is integral if and only if $Q$ is the unique prime of $A$ lying over the ideal $P$ of $... | 4 | https://mathoverflow.net/users/3521 | 152456 | 81,238 |
https://mathoverflow.net/questions/152467 | 4 | Consider the Sobolev space $W^{s,2}=H^s$ for $s=\frac{1}{2}.$
Let $\Omega \subset \mathbb{R}^n$ be an open set with boundary $\partial\Omega$. I have seen two definitions of the space $H^s(\partial\Omega)$:
1) (From Demengel & Demengel) As the set of functions $u:\partial\Omega \to \mathbb{R}$ such that
$$\lVert{u}... | https://mathoverflow.net/users/43959 | Sobolev spaces on boundaries | 1) Yes it is that simple. It is a special case of the so called Slobodeckij norm for Sobolev and more generally Besov spaces.
2) Not as much a need as it offers you a different perspective. Note that there is a third definition: One can also define $H^{1/2}(\partial\Omega)$ as the trace space of $H^1(\Omega)$. When y... | 5 | https://mathoverflow.net/users/824 | 152482 | 81,247 |
https://mathoverflow.net/questions/152336 | 5 | Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and
a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$.
Now, consider the following two sets,
1) $\Lambda^+$, the set of dominant weights wrt $B$,
2) The set $N\_{o,r}$ of pairs $(e,r)... | https://mathoverflow.net/users/26208 | Reg the motivation behind Lusztig-Vogan bijection | Like Jay, I don't see any reasonable way to address all parts of your wide-ranging question. You are looking at the intersection of numerous lines of research, motivated in different ways for different people. For myself, the primary motivation comes indirectly from modular representations of Lie algebras attached to s... | 2 | https://mathoverflow.net/users/4231 | 152485 | 81,248 |
https://mathoverflow.net/questions/150054 | 1 | Let $G$ be a finite group of even order which has only one non-principal irreducible character $\chi$ of degree $d$, $d\in \mathbb{N}$, with the following property (we name it $(\*\_d)$):
>
> $(\*\_d)$: There exist $x,y\in G$ such that $o(x)=o(y)=2$ and $\chi(x)\neq \chi(y)$.
>
>
>
I know that $A\_n$ ($n\geq 8... | https://mathoverflow.net/users/42611 | Are there any groups $G$ with the property $(*_d)$? | I have found an example of a solvable group that has exactly one irreducible character of degree $27$, and that character takes on two different values on involutions. The group, which has order $2^6 3^3$, can be constructed as a semidirect product of an elementary abelian $2$-group $V$ of order $2^6$ acted on by a non... | 6 | https://mathoverflow.net/users/9694 | 152486 | 81,249 |
https://mathoverflow.net/questions/152462 | 7 | Let $\mathbb{G}$ be a discrete amenable semigroup, and $\left\{ F\_{n}\right\} $
a Folner sequence.
For $S\subset \mathbb{G}$ define the upper density as $D^{\ast
}(S)=\limsup\_{n\rightarrow \infty }\frac{\left\vert S\cap F\_{n}\right\vert }{%
\left\vert F\_{n}\right\vert }.$
Suppose the exists $m\geq 0$ such that... | https://mathoverflow.net/users/44493 | Do syndetic sets on amenable semigroups have positive upper density? | This is true if $F\_n$ is a *right* Følner sequence, i.e., $\frac{|F\_n \Delta F\_ng |}{|F\_n|} \to 0$ for all $g \in G$. Indeed, if $F \subset G$ is finite, then the condition $g F \cap S \not= \emptyset$, for every $g \in G$, is equivalent to the condition $S F^{-1} = G$. Hence $D^\*(S) = \frac{1}{|F|} \sum\_{f \in F... | 6 | https://mathoverflow.net/users/6460 | 152488 | 81,250 |
https://mathoverflow.net/questions/152491 | 12 | Suppose $X$ is a smooth projective variety over a field $k$, with ample canonical bundle. If $\operatorname{char}(k)=0$ or $\operatorname{char}(k)>\dim(X)$ and $X$ lifts to $W\_2(k)$ (thanks Olivier Benoist!), it's not hard to see that $X$ has finite automorphism group.
**Proof.** Let $X\hookrightarrow \mathbb{P}\Gam... | https://mathoverflow.net/users/6950 | Do varieties with ample canonical bundle have finite automorphism group in small characteristic? | A much more general result is proven in [Martin-Deschamps, Lewin-Menegaux : Applications rationnelles séparables dominantes sur une variété de type général](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1978__106_/BSMF_1978__106__279_0/BSMF_1978__106__279_0.pdf). Théorème 2 : if $X$ and $Y$ are smooth and proper, and if ... | 17 | https://mathoverflow.net/users/2868 | 152495 | 81,253 |
https://mathoverflow.net/questions/152496 | 3 | Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}\_K(E)\cong \mathcal{O}\_K$. Suppose the class number of $K$ is equal to $1$. Let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}\_K$ which satisfying $E(K)/\mathfrak{p}E(K)=0$. In this case, is it poss... | https://mathoverflow.net/users/44006 | Mordell-Weil and finiteness of rational points | Let $Q\in E(K)$. Since $E(K)/\mathfrak{p}E(K)=0$, we know that $Q\in\mathfrak{p}E(K)$. You've assumed that $\mathfrak{p}$ is principal, say $\mathfrak{p}=(\pi)$, so $Q=u\_1\pi Q\_1$ for some unit $u\_1\in\mathcal{O}\_K^\*$ and some point $Q\_1\in E(K)$. Repeating $n$ times, we get $Q=u\_n\pi^nQ\_n$. (For simplicity, I'... | 5 | https://mathoverflow.net/users/11926 | 152499 | 81,254 |
https://mathoverflow.net/questions/151409 | 10 | Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition:
1. The meager sets are sets which are union of $\lambda$ nowhere dense sets where $\lambda < \kappa$ and $\kappa$ is the cardinality of the space.
2. If we c... | https://mathoverflow.net/users/38200 | Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities? | I am not sure if this is the kind of answer you were looking for, but since no one has given an answer yet, I think it is a good idea to say what little I know about larger cardinal analogues of the Baire category theorem.
The only spaces that I can currently think of where one would want to consider larger cardinal ... | 11 | https://mathoverflow.net/users/22277 | 152503 | 81,256 |
https://mathoverflow.net/questions/152502 | -2 | Let A and B be two subsets of R^2. I define the relation T(A,B) to hold between A and B iff there exists a translation f on R^2 such that the image set of A under f is B. It is easy to prove that T is an equivalence relation. In a [question on MSE](https://math.stackexchange.com/questions/612917/equivalence-classes-of-... | https://mathoverflow.net/users/43439 | Equivalence relations on powerset of R^2 | If the equivalence class of S is finite, then $A=\{a∈R^2:S+a≠S\}$ is finite, pick $r$ such that $r<|a|$ for all $a∈A$. For $x∉S$, $B\_r(x)⊂R^2\setminus S$, and similarly with $x∈S$, $B\_r(x)⊂S$. So the minimal distance between point in $S$ and not is $S$ at least $2r$, contradiction.
| 1 | https://mathoverflow.net/users/40458 | 152506 | 81,258 |
https://mathoverflow.net/questions/152504 | 2 | With the motivation to understand the [Lie group structure constraint on a non-Abelian Chern-Simons theory](https://physics.stackexchange.com/questions/90946/group-structure-in-chern-simons-theory), could some experts give **a class of Lie groups with structure constants** cannot fully anti-symmetrized in any choice of... | https://mathoverflow.net/users/44510 | A class of Lie groups with $f^{abc} \neq -f^{acb}$ (not fully anti-symmetrized) or $f^{abc} \neq f^{bca}$ (not-cyclic) | (I assume the ground field is an algebraically closed field of characteristic zero. It would have been useful to specify which field you have in mind.) Your condition amounts to asking whether there exists a invariant nondegenerate quadratic form, i.e. a nondegenerate bracket $\langle\cdot,\cdot\rangle$ satisfying
$$\l... | 6 | https://mathoverflow.net/users/14094 | 152520 | 81,263 |
https://mathoverflow.net/questions/152470 | 10 | It is known that minimality does not imply unique ergodicity (Furstenberg example). I ask whether the implication holds in following particular situation:
>
> Suppose $X$ is a compact space, $f:X \to X$ is a minimal and uniquely ergodic homeomorphism. Suppose $p: Y \to X$ is a **2-to-1 covering map**, and that $g:Y... | https://mathoverflow.net/users/1516 | Getting unique ergodicity from minimality | I think there is a counter-example.
We'll build a subshift on 2 symbols, 0 and 1. Given a word $W$ with symbols
0
and 1, $\overline W$ will denote the word with all symbols
flipped. Let $W\_0=1$ and $W\_{n+1}=W\_n^{2^n}\overline{W\_n}W\_n^{2^n}$.
The orbit closure of $W\_\infty$ gives a minimal, but not uniquely
e... | 8 | https://mathoverflow.net/users/11054 | 152524 | 81,266 |
https://mathoverflow.net/questions/151343 | 6 | Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C\_X$ be the category of closed embeddings of $X$ into (say affine) *smooth* $Y$'s of finite type over $\Bbbk$, morphisms being closed embeddings of $Y$'s (forming commutative triangle). We have a functor $\mathcal C\_X^{\mathrm{op}}\t... | https://mathoverflow.net/users/43938 | A construction of Kähler differentials and Illusie cotangent complex as colimit over embeddings | This is the approach to the cotangent complex in the Stacks project; there one uses all maps as in your Remark 3 (yes this gives the same answer). See the Stacks project chapter on the cotangent complex. In particular for schemes over a ring in particular see [Section Tag 08V7](http://stacks.math.columbia.edu/tag/08V7)... | 2 | https://mathoverflow.net/users/44817 | 152526 | 81,267 |
https://mathoverflow.net/questions/151780 | 1 | A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally closed?
If $f$ is a morphism between locally noetherian schemes $X$ and $Y$ which is formally etale and universally closed... | https://mathoverflow.net/users/5031 | formally étale morphisms which are also universally closed | The statement "A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion." is not correct. An example is to consider the map from the zero dimensional local ring k[x\_1, x\_2, ...]/(x\_ix\_j) to its completion. It is also not true that a universally closed formally... | 2 | https://mathoverflow.net/users/44817 | 152531 | 81,269 |
https://mathoverflow.net/questions/152429 | 2 | Let $X$ be the homotopy limit of a filtered system of simplicial sets $X\_i$. When are the morphisms $\pi\_j(X)\to \varprojlim \pi\_j(X\_i)$ surjective for all $j\ge 0$? This seems to be no problem when the system is countable since then there is a short exact sequence from section IX.3 of
A. K. Bousfield and D. M. Ka... | https://mathoverflow.net/users/2191 | Homotopy groups of filtered homotopy limits | Check out page 34 of May-Ponto "More Concise Algebraic Topology" for a more modern treatment of the homotopy limit of a sequence of spaces. In particular, Proposition 2.2.9 is exactly the surjection you're asking about. Their proof is to realize $X$ as a homotopy equalizer, then apply Proposition 2.2.7, which is a gene... | 1 | https://mathoverflow.net/users/11540 | 152533 | 81,271 |
https://mathoverflow.net/questions/152328 | -1 | $\phi$ is a homeomorphism from the 2-sphere to itself which represents an element of $PMCG(S^2,A)$ (we also denote it by $\phi$), where $A$ is a finite set of $S^2$. $\gamma$ is an essential Jordan curve (i.e. both components $F$, $G$ of $S^2-\gamma$ contain at least two points of $A$). Suppose $\phi$ fixes the homotop... | https://mathoverflow.net/users/44397 | Homeomorphism of the punctured sphere which fixes an essential Jordan curve | To prove this you just need to put together a few simple known results.
First, you need that the subgroup of $PMCG(S^2,A)$ that stabilizes the homotopy class of $\gamma$, a subgroup I shall denote $Stab(\gamma)$, can be written as a short exact sequence
$$1 \to \langle D(\gamma)\rangle \hookrightarrow Stab(\gamma) \... | 1 | https://mathoverflow.net/users/20787 | 152547 | 81,284 |
https://mathoverflow.net/questions/152513 | 10 | According to Jairo comment on the first version of this question I revise the question as follows;
Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that:
There are at most a finite number of disjoint simple closed geodesics on $S^{2}$.
If the answer is yes put $m$= the sup of the number of ... | https://mathoverflow.net/users/36688 | Number of disjoint simple closed geodesics | As it was shown by Igor, there is no univeral bound on number of such geodesics.
Let me show that the number can not be infinite.
Assume it is possible to get infinite number of such geodesics,
say $\gamma\_n$, $n\in\mathbb N$.
Note that the geodesics $\gamma\_i$ for $i\le n$ cut $\mathbb S^2$ into surfaces with geo... | 5 | https://mathoverflow.net/users/1441 | 152554 | 81,290 |
https://mathoverflow.net/questions/150221 | 1 | A fairly ubiquitous object in elementary calculus is a function of the shape $r = f(\theta)$, where $r$ is the radius and $\theta$ the argument. Common examples include the cardiod and limacon, and of course the circle can also be expressed this way.
I understand the value of the substitution $x = r\cos(\theta), y =... | https://mathoverflow.net/users/10898 | Application for functions of the shape $r = f(\theta)$ | The most spectacular application is the theory of orbits under a central force field. This is basically what Kepler and Newton did (not, of course, using
this notation). One of Kepler's key observations on the orbit of Mars was the constancy of a certain quantity associated with points on the orbit. In modern terms he ... | 6 | https://mathoverflow.net/users/44363 | 152556 | 81,292 |
https://mathoverflow.net/questions/152511 | 0 | We want to approximately solve an ODE
$$\frac{dy}{dt} = f(y,t)$$
with the Runge Kutta method
$$y\_{n+1} = y\_n + h \sum\_{i=1}^s b\_i k\_i$$
$$k\_i = f\left(y\_n + h \sum\_{j=1}^s a\_{ij} k\_j,\,t\_n + c\_i h\right)$$
For explicit methods the matrix $a$ is strictly lower triangular and thus you can determine $k\_i$ e... | https://mathoverflow.net/users/29638 | Does an implicit Runge Kutta scheme applied on a nonlinear ODE give a nonlinear set of equations to solve in each step? | This is the problem with implicit schemes (any kind of schemes) applied for the solution of nonlinear equations. You do need to solve a nonlinear equation (or even worse, a nonlinear system) at each step, and usually this done with Newton's method. There are also hybrid methods, implicit-explicit, where the implicit co... | 2 | https://mathoverflow.net/users/43681 | 152562 | 81,296 |
https://mathoverflow.net/questions/144723 | 1 | The article
*Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238.*
has the following CLAIM:
>
> **Claim.** *Let $A$ be an invertible hyperhermitian matrix of order $n$. For any $i$, $1\le i\le n$, $$\left(A^{-1}\right)\_{ii} = \frac{1}{\det A} \det M\_{ii}(... | https://mathoverflow.net/users/18801 | How to calculate $(A^{-1})_{ii}$ for an invertible hyperhermitian quaternionic matrix $A$? | Determinants are tricky for matrices of quaternions, but they are not as bad when the matrix is hermitian. In that case, one can expand in the usual way along any row and get the same result.
See Theorem 5.1 here: "Cramer's rule for quaternionic systems of linear equations "Kyrchei, II, Journal of Mathematical Scienc... | 3 | https://mathoverflow.net/users/6133 | 152564 | 81,298 |
https://mathoverflow.net/questions/130481 | 2 | Suppose $B$ is a bounded region in complex plane. In complex plane, one usually deals with complex moments, i.e. $\int\_B {z}dxdy$ where $z \in \mathbb{C}$. What is so special about this complex moment compared to the real moments, i.e $\int\_B {x^my^n}dxdy$ where $m,n \in \mathbb{N}$? I don see the reason using comple... | https://mathoverflow.net/users/42411 | Usage of complex moments in complex plane | I'm not totally sure about what kind of answers you're looking for, but let me try something.
First, let me emphasize that for a general Borel measure $\mu$ on the complex plane, the knowledge of the sequence $\int z^kd\mu(z)$, $k\in\mathbb N$ is not enough to characterize the measure $\mu$; you would instead need t... | 1 | https://mathoverflow.net/users/15517 | 152573 | 81,302 |
https://mathoverflow.net/questions/43605 | 14 | If $X$ is a noetherian scheme with points $x$ and $\xi$ so that $x$ is in the closure of $\{\xi\}$, then there exists a discrete valuation ring $V$ and a map $Spec(V)\to X$ sending the generic point to $\xi$ and the closed point to $x$. That is, any closure relation among point of $X$ can be witnessed by a map from a D... | https://mathoverflow.net/users/1 | Can a single DVR witness all specializations on a variety? | I think you already answered the question yourself and so did Konstantin Ardakov. But this question was still marked as open...
Suppose that the ground field is the field Q of rational numbers. Then let V be the power series ring C[[t]] where C is the field of complex numbers. This is an example.
To prove it, as yo... | 5 | https://mathoverflow.net/users/44817 | 152583 | 81,307 |
https://mathoverflow.net/questions/152567 | 2 | Start from a connected closed Riemann surface $\Sigma\_g,$
obtained as the (symmetric) covering of an open and/or unoriented surface
$\Sigma,$ namely $\Sigma=\Sigma\_g/\Omega,$ where $\Omega$ is an antiholomorphic involution.
We have on $\Sigma\_g$ $2g$ one cycles $\delta\_i$ and we can choose them such that
$\delta... | https://mathoverflow.net/users/40154 | lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology | The cohomology of $Jac(\Sigma\_g)$ is the exterior algebra of $H^1$, which is canonically isomorphic to $H^1(\Sigma \_g,\Bbb{C})$. So what you are really asking is how $\Omega $ acts on $H^1(\Sigma \_g,\Bbb{C})$. We have the Hodge decomposition $H^1(\Sigma \_g,\Bbb{C})=H^{1,0}\oplus H^{0,1}$, and $\Omega $ maps one sum... | 1 | https://mathoverflow.net/users/40297 | 152599 | 81,315 |
https://mathoverflow.net/questions/152600 | 2 | An inverse semigroup $S$ is a semigroup in which every element $x \in S$ has a unique inverse $y \in S$ such that $x = xyx$ and $y = yxy$. Are there some references characterizing the maximal sub-inverse-semigroups of $M\_n(\mathbb{C})$ and $M\_n(F\_p)$? Here $M\_n(\mathbb{C})$ is the monoid of all $n \times n$ matrice... | https://mathoverflow.net/users/11877 | Maximal sub-inverse semigroups of $M_n(\mathbb{C})$ and $M_n(F_p)$ | Maybe this article will be useful for you:
Zhu Yong Wen, Inverse Semigroups of Matrices. Journal of Mathematical Research & Exposition, Aug., 2008, Vol. 28, No. 3, pp. 549–557.
| 1 | https://mathoverflow.net/users/18814 | 152602 | 81,316 |
https://mathoverflow.net/questions/152589 | 5 | A few days ago I asked the [following question](https://math.stackexchange.com/questions/610929/weyl-orbits-of-integral-dominant-weights-and-convex-polytopes) at MSE and received no answer. I thought I would try here.
Let $\xi$ be an integral dominant weight of an irreducible root system $\Delta$, and let $\mathcal{O... | https://mathoverflow.net/users/26069 | How to find faces of polytope defined by a Weyl orbit | The faces are all of the following form: $w W\_P / Stab\_W(\xi)$, where $W\_P$ varies over the subgroups generated by subsets of the simple reflections. In particular, for $\xi$ regular, the number of them is $\sum\_P |W / W\_P|$. (Note that $\mathcal O\_\xi$ is only $G/T$ when $\xi$ is regular; otherwise it's $G/Stab\... | 8 | https://mathoverflow.net/users/391 | 152621 | 81,326 |
https://mathoverflow.net/questions/152633 | 3 | In the article [Voevodsky’s Univalence Axiom in Homotopy Type Theory](http://www.ams.org/notices/201309/rnoti-p1164.pdf), an example is given of how types are not like sets: the existence of a nontrivial (nonzero) type $X$ such that $X\rightarrow X\cong X + 1$. However, I am unable to find this counterexample in the re... | https://mathoverflow.net/users/41109 | Type with $X\rightarrow X\cong X + 1$ | I'm not ready to venture into homotopy type theory yet, but there is an example in topos theory, which might be transportable into the context you want. In the object-classifier topos, the generic object $U$ satisfies $U^U\cong U+1$. This is Example 1 in an old paper of mine, "Functions on universal algebras" [J. Pure ... | 7 | https://mathoverflow.net/users/6794 | 152634 | 81,331 |
https://mathoverflow.net/questions/152631 | 4 | Let consider a real vector space $\mathbb{R}^n$ of dimension $n$, where $\langle \cdot, \cdot\rangle$ and $||\cdot ||$ are the standard inner product and $\ell\_2$ indeced norm.
Does there exist a transformation (a non-linear one) $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that it preserves orthogonality **and... | https://mathoverflow.net/users/16758 | Non-orthogonal vectors cosine-enhancing transformation | This isn't possible, even if you let $T$ be non-linear. You can see this in $\mathbb{R}^2$ as follows, but the intuition generalizes straightforwardly for larger $n$:
Let $\{e\_1,e\_2\}$ be the standard basis of $\mathbb{R}^2$ and let $v := (1,1)/\sqrt{2}$. Then $\langle T(e\_1), T(e\_2) \rangle = 0$. You want $T(v)$... | 6 | https://mathoverflow.net/users/11236 | 152644 | 81,336 |
https://mathoverflow.net/questions/152619 | 3 | Here is a (little wild) question about Boolean functions with countably many variables and a wild analog for Fourier-Walsh functions and analysis based on them.
Let $x\_1,x\_2,\dots,x\_n,\dots$ be Boolean variables. We will consider real functions on these countably many variables. Consider the following three classe... | https://mathoverflow.net/users/1532 | Ultrafilter-based Fourier-Walsh-like Functions | The $W\_G$ are linearly independent. First note that given ultrafilters $F\_1,\ldots,F\_n$ and any $S\subseteq [n]$, we may find an $A\_S\in \cap\_{i\in S}F\_i \setminus \cup\_{j\not\in S}F\_j$, namely $A\_S := \cup\_{i\in S}A\_i$ where $A\_i$ is as in my first answer. Assume $\sum c\_G W\_G=0$. Each $G$ is $F\_S := \c... | 3 | https://mathoverflow.net/users/4600 | 152650 | 81,340 |
https://mathoverflow.net/questions/152643 | 3 | Let $u(x),x\in R\_+$ be a non-negative decreasing smooth function with compact support $[0,L]$, I want to know the following inequality is true? $a\in (0,1)$
$$\int\_0^\infty \frac{1}{1+x}u^{1+a}dx \le \epsilon \int\_0^\infty |u\_x|^2dx+C\_\epsilon \int\_0^\infty u^{2a}dx,$$ where we need $C\_\epsilon$ is independent o... | https://mathoverflow.net/users/44565 | An interpolation type inequality | It is false for every $\epsilon > 0$. The family of functions to use is this:
$$
u(x):= \left\{ \begin{aligned} & (L-x)^{\frac1{1-a}}, && 0\leq x \leq L, \\
& 0, && x>L. \end{aligned} \right.
$$
Note that $u\in C^1(\mathbb R\_+)$. (To get something smooth, mollify this $u$.)
Note also that, for $0\leq x \leq L$, $$(u... | 2 | https://mathoverflow.net/users/5678 | 152655 | 81,341 |
https://mathoverflow.net/questions/152662 | 2 | What are concrete examples of fibre bundles with fibre genus $\geq 2$? I am trying to find examples I can use to work throught the following construction:
Suppose $X \to C$ is a fibre bundle with $C$ a curve and for which the fibre $F$ is a curve of genus $\geq 2$. Then $\mathrm{Aut}(F)$ is a finite group and we can... | https://mathoverflow.net/users/44582 | Fibre bundles of fibre genus greater than 1 | All such fiber bundles are constructed in the following way : choose an étale covering
$\tilde{C}\rightarrow C $ with Galois group $G\subset \mathrm{Aut}(F)$, and take $S=(\tilde{C}\times F )/G$, with $G$ acting on each factor in the obvious way. The fiber bundle structure is given by the first projection $S\rightar... | 2 | https://mathoverflow.net/users/40297 | 152666 | 81,343 |
https://mathoverflow.net/questions/152628 | 2 | I'm sorry if this question is too basic for MO. I'm reading [a paper by Graham and Lehrer "Cellular algebras"](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002113384) and have trouble understanding one step in a proof of a crucial theorem. I suppose that the problem is probably quite easy, but for some reas... | https://mathoverflow.net/users/43555 | Question about a proof in Graham and Lehrer's "Cellular algebras" | If $W(\lambda)=Az$ for all $z\notin rad(\lambda)$, then $rad(\lambda)$ is a maximal submodule of $W(\lambda)$ (in fact it follows that it is the *unique* maximal submodule and therefore it is equal to the module-theoretic radical of $W(\lambda)$). In particular $W(\lambda)/rad(\lambda)$ is irreducible. It follows from ... | 3 | https://mathoverflow.net/users/3041 | 152674 | 81,346 |
https://mathoverflow.net/questions/152665 | 4 | Using the Chinese Remainder Theorem, it is very straight forward to find a sequence of consecutive integers starting at $x$ where each of the first $n$ prime numbers is a least prime factor for a given number in the sequence and no number in the sequence has a least prime factor greater than $p\_n$. Trivially, we know ... | https://mathoverflow.net/users/15915 | Is there a lower bound for the first non-trivial sequence of consecutive integers where each of the first $n$ primes is a least prime factor | It was proved by [Rankin](http://www.ams.org/mathscinet-getitem?mr=160767) in 1963 that there are infinitely many $n$ for which
$$j(n) \geq (C+o(1) \frac{\log(n) \log\_{2}(n) \log\_{4}(n)}{\log^{2}\_{3}(n)} $$
holds for some positive $C>0$. The value of $C$ has since been improved by [Maier and Pomerance](http://www.am... | 3 | https://mathoverflow.net/users/630 | 152677 | 81,347 |
https://mathoverflow.net/questions/128850 | 9 | Suppose $M$ is a 1-connected closed manifold with sectional curvature $\ge 1$. So the diameter $D$ of $M$ satisfies
$$
D \le \pi
$$
When equality holds $M$ is isometric to round sphere. In fact this rigidity holds under the assumption of $Ric \ge n-1$. Hence it is natural to ask what happens to almost extreme case. i.e... | https://mathoverflow.net/users/1190 | Positively curved manifold with almost extreme diameter | I will answer the last question.
Namely, let me show that if $1\le \mathrm{sec}\,M \le K$ and $\mathrm{diam}\, M>\pi-\varepsilon$ for sufficiently small $\varepsilon>0$ then it has to be diffeomorphic to $\mathbb S^m$.
Asssume contrary, then there is a sequence of $m$-dimensional manifolds $M\_n$ such that $1\le \mat... | 3 | https://mathoverflow.net/users/1441 | 152685 | 81,351 |
https://mathoverflow.net/questions/54146 | 25 | Parseval's identity states that the sum of squares of coefficients of the Fourier transform of a function equals the integral of the square of the function, or
$$ \sum\_{-\infty}^{\infty} |c\_n|^2 = (1/2\pi)\int^\pi\_{-\pi} |f(x)|^2 dx $$
where the $c\_i$ are the Fourier coefficients.
The Legendre-Fenchel transfor... | https://mathoverflow.net/users/972 | Analogy of Parseval identity for Legendre Transform ? | I think the identity you want is
$$2\inf\_x f(x)=\inf\_x(f^\ast(x)+f^\ast(-x))\mbox{.}$$
(I'm skipping a bunch of conditions required of $f$ to make this hold. We'll need convexity at least.)
Let's use $\oplus$ for infimal convolution and let $g(x)=f(-x)$.
By definition $(f\oplus g)(x)=\inf\_y(f(x-y)+g(y))$.
... | 16 | https://mathoverflow.net/users/1233 | 152693 | 81,354 |
https://mathoverflow.net/questions/152700 | 4 | What can be said about the following crossed product $C^\*$-algebra?
Let $A$ be a Kirchberg algebra with $K\_0(A) = \mathbb{Q}$ and $K\_1(A) = 0$. Consider the direct sum of $n$ copies of $A$, i.e. $B = A^n$. The permutation group $\Sigma\_n$ on $n$ letters acts on $B$ by permuting the elements.
* What is $K\_i(B ... | https://mathoverflow.net/users/3995 | Crossed Products by Permutation Groups | $B\rtimes\Sigma\_n$ is Morita equivalent to $A\rtimes\Sigma\_{n-1}$ with $\Sigma\_{n-1}$ acting trivially. The latter is isomorphic to $A\otimes \mathbb{C}\Sigma\_{n-1}$, which shows that $B\rtimes\Sigma\_n$ is not simple. Let $c\_n$ be the number of conjugacy classes of $\Sigma\_n$. Then $\mathbb{C}\Sigma\_{n-1}$ is t... | 5 | https://mathoverflow.net/users/14497 | 152706 | 81,358 |
https://mathoverflow.net/questions/152670 | 3 | Let $M^2$ be a closed surface, say the 2-sphere. Is there any example of metric on it such that there are uncountably many points are conic and the metric is smooth elsewhere?
We call $p\in M$ a conic point if there exists $\lambda\_i\to \infty$ such that
$(\lambda\_i M, p)$ converge to a linear cone $C(S(\ell))$, th... | https://mathoverflow.net/users/1190 | Closed surface with uncountably many conic points? | The answer is NO.
Fix $\varepsilon>0$ and $\delta>0$.
Consider the set $X\_{\varepsilon,\delta}$ of all the points in $M$
such that for any $r<\varepsilon$, the $r$-neighborhood of any $x\in X\_{\varepsilon,\delta}$ is $r{\cdot}\delta$-close
to $r$-ball in the cone over the circle with length $\ell$ such that $|\... | 2 | https://mathoverflow.net/users/1441 | 152708 | 81,359 |
https://mathoverflow.net/questions/152278 | 4 | This question is motivated by an old math contest problem, and is a generalization of the original problem. I will write out the original problem as motivation.
Let us say that $n$ is $p$-Savage, for a prime $p$, if it is possible to partition $\{1, \cdots, n\}$ into $p$ sets $A\_1, \cdots, A\_{p}$ such that the foll... | https://mathoverflow.net/users/10898 | A partition problem | The answer is that for large enough $n$, it is a $p$-savage if and only if $p|n+1$ and $(p-1)|\frac{n(n+1)}{2}$.
You must have $p-1|D$ because all elements of $A\_1$ are divisible by $p-1$. And so we must have $p-1|\frac{n(n+1)}{2}=pD$. To show that $p|n+1$ we have to rule out the case $p|n$. This is easily ruled out... | 2 | https://mathoverflow.net/users/2384 | 152721 | 81,362 |
https://mathoverflow.net/questions/152694 | 3 | I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references.
What is the eigenfunction of a multivariate Gaussian kernel:
\begin{equation}
f(x,y) = \exp\left(-\frac{\lVert x - y\rVert^2}{2\sigma^2}\right)
\end{equation}
I am interested in the eigenfunc... | https://mathoverflow.net/users/44602 | Gaussian kernel eigenfunctions | In the notation of the question,
$$
\int f(x,y)\nu(y)dy=W\_\sigma\ast \nu(x)
$$
where $W\_\sigma(x)=e^{-|x|^2/2\sigma^2}$ and $\ast$ denotes convolution. Thus, if
$$
\int f(x,y)\nu(y)dy=\lambda\nu(x)
$$
then by taking Fourier transforms:
$$
\lambda \hat{\nu}= \widehat{\lambda \nu}
=\widehat{W\_\sigma\ast \nu}
=\wid... | 7 | https://mathoverflow.net/users/44620 | 152727 | 81,363 |
https://mathoverflow.net/questions/152561 | 3 | Suppose I have a cospan of categories $X\xrightarrow{F} Z\xleftarrow{G}Y$ and I'm looking for exact squares containing it. The category $Exact\_{F,G}$ of such things has a terminal object, namely the comma category $(F\downarrow G)$. In fact, the comma category construction provides a section of the forgetful functor $... | https://mathoverflow.net/users/2811 | Exact squares containing a cospan: what is known about this category? | Not a complete answer, but a few observations: $\newcommand{\Cat}{\mathbf{Cat}} \newcommand{\dn}{\downarrow} \newcommand{\Ex}{\mathrm{Exact}}$
* since it’s a full subcategory of the category of lax-commutative squares over $(F,G)$, it’s isomorphic to a full subcategory of $\Cat / (F \dn G)$. Also, $\Cat / (F \dn G)$ ... | 1 | https://mathoverflow.net/users/2273 | 152729 | 81,364 |
https://mathoverflow.net/questions/152715 | 9 | I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite [2-group](http://en.wikipedia.org/wiki/P-group). I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, ~~past the result that this calculates group extensions of $\mathbb{R}^n$ by $G$.~~
Any related literature would ... | https://mathoverflow.net/users/4177 | Calculations of nonabelian group cohomology of R^n | I assume that you are looking at group cohomology. Then the action of $\Bbb{R}^n$ on $G$ is necessarily trivial, so $H^1(\Bbb{R}^n,G)$ is just $\mathrm{Hom}(\Bbb{R}^n,G)$ mod. conjugacy by $G$. But this is of course trivial - $\Bbb{R}^n$ has no finite quotient.
| 10 | https://mathoverflow.net/users/40297 | 152736 | 81,367 |
https://mathoverflow.net/questions/152493 | 7 | My question is to know whether the fibre product of $[X/G]$ by $[Y/H]$ over a base scheme is $S$ is $[X \times\_S Y/G \times H]$? And if yes, do you have any reference for it?
Thank you.
| https://mathoverflow.net/users/30737 | The fibre product of two quotient stacks | This is my attempt at fleshing out the details using the universal property as Scott Carnahan suggested. I think this is correct but I'm not totally sure. Let me know if theres anything off.
Let $\mathscr{X} = [X/G]$, $\mathscr{Y} = [Y/H]$ and $\mathscr{P} = [X \times\_S Y/G \times H]$. $\mathscr{X}$ is described by ... | 8 | https://mathoverflow.net/users/12402 | 152738 | 81,368 |
https://mathoverflow.net/questions/152711 | 6 | Suppose I have a faithfully flat cover of schemes $\phi:X\to Y$, and a sheaf $F$ on $Y$. I might be interested in so-called ``twisted forms for $F$." That is, sheaves $F'$ on $Y$ such that $\phi^\ast(F)\cong \phi^\ast(F')$. In the case that $\phi$ is a cover by some collection $\{U\_i\}$, this is often stated as saying... | https://mathoverflow.net/users/11546 | Why does the first Cech cohomology classify twisted forms? | In general the statement is slightly different. What you typically have is a presheaf $F$ of groupoids (i.e. for every element $X$ you can have $QCoh(X)$ the groupoid of quasicoherent sheaves and isomorphisms, but maybe you are interested in the groupoid of algebras, Azumaya algebras, Hopf algebras etc..) and an object... | 4 | https://mathoverflow.net/users/43054 | 152747 | 81,373 |
https://mathoverflow.net/questions/152713 | 2 | I'm hoping to determine the geodesic equation for a right invariant Randers metric $F(x) = \sqrt{a(x,x)} + b(x)$ on $SU(N)$. In my special case the navigation data $(h,W)$ for the Randers metric are such that $h$ is the biinvariant metric and $W$ is right invariant.
The geodesic spray coefficients induced by a Rande... | https://mathoverflow.net/users/41654 | Right Invariant Randers metrics | You are asking about a particular case of the general right invariant Lagrangian for curves on a Lie group. This is a well-known story, but I can summarize it here:
Let $G$ be a Lie group with Lie algebra ${\frak{g}}=T\_eG$ and dual ${\frak g}^\ast$, with the canonical pairing $\langle,\rangle:{\frak g}^\ast\times{\f... | 2 | https://mathoverflow.net/users/13972 | 152751 | 81,374 |
https://mathoverflow.net/questions/152605 | 18 | **Darboux's Theorem.** *If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.*
Does any of the following generalizations
1. Let $U\subset\mathbb R^n$ connected and $f: U\to \mathbb R$ differentiable. Then $\nabla f[U]$ is connected,
2. Let $U\su... | https://mathoverflow.net/users/43681 | Generalization of Darboux's Theorem | It turns out than none of the three potential generalisations holds.
Counterexamples for the last two questions are presented in the answer of Ali Taghavi, and in particular by function $f(x,y)=(\mathrm{e}^x\cos y,\mathrm{e}^x\sin y)$, as $f[\mathbb R^2]=\mathbb R^2\smallsetminus\{(0,0)\}$.
For the first question, ... | 14 | https://mathoverflow.net/users/43681 | 152756 | 81,377 |
https://mathoverflow.net/questions/152742 | 10 | Motivated by [this question](https://mathoverflow.net/questions/152680/generating-primes-via-composition-of-polynomials).
Let $f \in \mathbb{Q}[x]$ or$f \in \mathbb{Z}[x]$ .
Consider the sequence $f(x),f(f(x)), \ldots f^n(x)$.
If some $f^k(x)$ is reducible, the rest iterates will be reducible too.
This happens ... | https://mathoverflow.net/users/12481 | Reducibility of polynomials maps | I believe that the first person with significant results along these lines was Odoni [1]. There are also papers of Rafe Jones that consider questions of this sort, see for example [2] and [3]. A polynomial is called *stable* if all of its iterates are irreducible, and more generally, a polynomial is called *eventually ... | 13 | https://mathoverflow.net/users/11926 | 152757 | 81,378 |
https://mathoverflow.net/questions/79921 | 6 | Consider real polynomials on the interval $I=[-1,1]$. It is easy
to see that the smallest degree for a non-negative polynomial
with given zeros $x\_1,\dots,x\_s\in I^\circ$ is $n=2s$ (e.g.
$P(x) = \prod\_{i=1}^s (x-x\_i)^2$ works).
>
> My question is:
>
>
> What is the smallest degree for a polynomial such that i... | https://mathoverflow.net/users/9652 | Polynomials with prescribed points to match prescribed bounds | Let $D$ be the minimum distance between $x$'s, merging all the $x$'s into one list of length $N$.
Let $k$ be an integer $\ge \max(16\log(8/D^2),10N)\ /\ D^2$.
Then a polynomial of degree of $6(k+1)(N-1)$ suffices.
Proof:
Let $p(x) = \frac{1}{2}(3 q(x) - q^3(x))$, where $q(x) = \sum\_i \pm \Pi\_{j \neq i} r\_{ij}(... | 4 | https://mathoverflow.net/users/nan | 152762 | 81,381 |
https://mathoverflow.net/questions/152673 | 6 | Today I encountered the notion of multicolimit.
Lacking a standard reference for this notion, let me give a self-contained definition of this gadget.
If $S\colon \cal K\to E$ is a diagram, we define its *multicolimit* as a small set of cocones $\{S\xrightarrow{\varphi\_i} L\_{i,S}\}\_{i\in I}$ such that for any ot... | https://mathoverflow.net/users/7952 | Are multicolimits suitable colimits? | 1. Yes, but perhaps in an other sense than you may think.
2. I have nothing to say :-)
3. Yes.
4. Yes.
$\newcommand{\mor}[3]{#1 \colon #2 \rightarrow #3}%
\newcommand{\catl}[1]{\mathbb{#1}}%
\newcommand{\catw}[1] {\mathbf{#1}}$
---
Here is my elaboration.
Let me first omit some irrelevant details. We shall say ... | 8 | https://mathoverflow.net/users/13480 | 152775 | 81,385 |
https://mathoverflow.net/questions/152785 | 9 | In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without assuming that the monoidal product distributes over coproducts. But no such assumption is mentioned in my sources (for ... | https://mathoverflow.net/users/19860 | Correspondence between operads and monads requires tensor distribute over coproduct? | I also noticed this at some point. I think you are right. One reference where this assumption is explicitly spelled out is the paper of Getzler and Jones.
| 5 | https://mathoverflow.net/users/1310 | 152788 | 81,387 |
https://mathoverflow.net/questions/151079 | 7 | Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X\_{\beta}\}\_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X\_{\beta}$ are smooth locally closed subvarieties and $$\overline{X\_{\beta}}\subseteq\bigcup\_{\gamma\leq\beta}X\_{\gamma}.$$ For fi... | https://mathoverflow.net/users/25358 | Equivariant Stratifications of a Variety | This will be a partial answer because in the spirit of Christmas I'm not going to go hunt for references. :)
1. It usually makes no difference whether the set of strata is totally or partially ordered. In the partially ordered case, put $X\_d = $ (disjoint) union of all $d$-dimensional strata.
2. You are probably be... | 3 | https://mathoverflow.net/users/1310 | 152789 | 81,388 |
https://mathoverflow.net/questions/148826 | 3 | The title explains it all.
I'm familiar with the du val singularities on surfaces, also known as rational double points. In <http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf>, 2.1, they are characterized as those isolated double points that admit a resolution that is given by stepwise blowup of isolated doub... | https://mathoverflow.net/users/29657 | Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational? | I assume that by a double point we mean a points of multiplicity two.
In this case the singularity $x^2+y^3+z^6=0$ is a double point which is an ellitpic hypersuface singularity in $\mathbb C^3$, it is not rational. For more examples you can check Chapter 4 in Miles Reid's notes <http://arxiv.org/pdf/alg-geom/9602006... | 7 | https://mathoverflow.net/users/943 | 152790 | 81,389 |
https://mathoverflow.net/questions/152682 | 3 | Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity.
My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots of articles which dealing Euler system of elliptic units. For me, it(elliptic units) is just a set of complicatedly defi... | https://mathoverflow.net/users/44006 | Elliptic units and Euler system | Given that you have not seen cyclotomic units, I think you should start with them. Rubin's appendix to Lang's book(s) on cyclotomic fields is one place or the book by Coates and Sujatha. Then for elliptic units, I like Rubin's part in the Cetraro notes "Arithmetic of ellitpic curves".
An Euler system is a stepping st... | 5 | https://mathoverflow.net/users/5015 | 152793 | 81,391 |
https://mathoverflow.net/questions/152805 | 0 | When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
| https://mathoverflow.net/users/nan | $S^n$ admit a real polarization $D\subset TS^n$? | Well, this only would make sense when $n$ is even, but there are two problems: First, except when $n=1$, the $2n$-sphere does not carry any symplectic structure. Second, the tangent bundle of the $2n$-sphere has no nontrivial subbundles anyway. (The reason is that $TS^{2n}$ has nonzero Euler class, so it cannot be writ... | 2 | https://mathoverflow.net/users/13972 | 152809 | 81,395 |
https://mathoverflow.net/questions/152811 | 0 | What can be said about the null space of random $(0,1)$ rectangular binary matrices? In particular, I am interested in the probability that there is any non-zero vector with only integer coordinates in the null space. Is this a known problem and/or is there a known approach for tackling it?
| https://mathoverflow.net/users/44667 | Null space of random $(0,1)$ binary matrix | I would comment this but I don't yet have enough reputation. Your question is equivalent to asking the odds that the rank of a $(0,1)$ matrix is full. If your matrix has more columns than rows then you are certain to have non zero vectors in the null space. If you have more rows than columns then you can zero out some ... | 1 | https://mathoverflow.net/users/39853 | 152815 | 81,398 |
https://mathoverflow.net/questions/152807 | 2 | A book I'm reading gives the following definition for quasi-conformal maps:
>
> If $f$ is a homeomorphism of a metric space X to itself, $f$ is *K-quasi-conformal* if and only if for all $z \in X$:
>
>
> $$
> \limsup\_{r\to0}\frac{\sup\_{x,y \in S\_r(z)} d (f(x), f(y))}
> {\inf\_{x,y \in S\_r(z)} d (f(x), f(y))} ... | https://mathoverflow.net/users/37354 | About a definition of quasi-conformal maps | For the equivalence of definitions of quasiconformal maps the reference is
J. Heinonen, Lectures on analysis on metric spaces, Springer 2001. Notice that the $K$
in the definiton you cite is not the same $K$ as in the Ahlfors definitions.
So your definition of quasiconformality is equivalent to the usual one, but with ... | 6 | https://mathoverflow.net/users/25510 | 152817 | 81,399 |
https://mathoverflow.net/questions/152772 | 8 | Fix $n$ and $k$. I want a set $S\subseteq\{1,\ldots,n\}$ with the property that for every $x\in S$,
$$\mathrm{gcd}\bigg(x,\prod\_{y\in S\setminus\{x\}}y\bigg)<\frac{x}{k}.$$
How small should a random $S$ be to have this property with high probability? More importantly, what sort of math is this, and where can I lea... | https://mathoverflow.net/users/29873 | Sets whose elements are mutually "weakly" coprime? | Let's just consider the case $k=1$ where the problem asks for sets $S$ such that each element of $S$ does not divide the product of the rest of the elements of $S$. I claim that if $S$ has fewer than $\exp(\frac{1}{10} \sqrt{\log n\log \log n})$ elements then with high probability this happens. On the other hand if $S$... | 6 | https://mathoverflow.net/users/38624 | 152819 | 81,401 |
https://mathoverflow.net/questions/152827 | 6 | Consider the normal modal logic system $\mathbf{TAR1}$ given by $\mathbf{T}$ plus the following axiom:
$$\mathrm{AR1}: \lozenge \square p \rightarrow (\square p \lor \square (p \rightarrow \square p))$$
Analysis using normal forms shows that it is included in $\mathbf{S4.4}$. However, algebraically I cannot either ... | https://mathoverflow.net/users/37336 | Is this system identical to S4.4? | $\mathbf{KT}+\mathbf{AR1}$ is strictly weaker than $\mathbf{KT}+\mathbf{4.4}$. Consider the Kripke frame that is the reflexive closure of the following graph (so that any model built on it is a model of $\mathbf{KT}$):
$$\require{AMScd}\begin{CD}
A @>>> B @>>> C
\end{CD}$$
Then build a model on it with, say, the valuat... | 4 | https://mathoverflow.net/users/4137 | 152831 | 81,404 |
https://mathoverflow.net/questions/152823 | 49 | My question is the following: *Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?*
**EDIT.** According to a theorem of Serge N. Bernstein (*Sur les fonctions absolument monotones*, Acta Mathematica, 52 (1928) pp. 1–66) i... | https://mathoverflow.net/users/43681 | Is a function with nowhere vanishing derivatives analytic? | If $f$ is $C^{\infty}$ every derivative is continuous, so the hypothesis on $f$ implies that each derivative $f^{(n)}$ has constant sign. Such functions were studied by S. Bernstein and called **regularly monotonic**. In particular he proved in 1926 that a regularly monotonic function is real analytic.
[This 1971 AMM... | 56 | https://mathoverflow.net/users/1149 | 152832 | 81,405 |
https://mathoverflow.net/questions/152845 | 1 | Let $(M, D, J)$ be a strictly pseudoconvex hypersurface type CR manifold with $J$ integrable.
Let $D$ be the kernel of a $1$-form $\eta\_0$.
As known the automorphism group is defined to be
$$
CR = \{ \phi \in Diff(M): \phi^\*\eta\_0 = f\_\phi \eta\_0 \text{ with $f\_\phi$ nowhere vanishing and } \phi\_\* J = J \phi\_\... | https://mathoverflow.net/users/19545 | Group of CR automorphisms | Yes, $f\_\phi$ is always positive, as long as you choose $\eta\_0$ properly. The reason is that you have assumed that the CR-structure is strictly pseudo-convex, which means that there is a choice of $\eta\_0$ (unique up to a positive multiple) such that ${\mathrm{d}}\eta\_0$ restricted to $D$ is a positive $(1,1)$-for... | 3 | https://mathoverflow.net/users/13972 | 152859 | 81,416 |
https://mathoverflow.net/questions/152430 | 3 | (This is a cross-post from Math StackExchange <https://math.stackexchange.com/questions/609641/multinomial-distribution-sum-of-squared-probabilities>)
Let $\vec X = (X\_1, \dots, X\_k)$ be a draw from a fair multinomial distribution with $n$ trials, i.e. $P(X\_1 = x\_1, \dots, X\_k = x\_k) = \binom{n}{x\_1, \dots, x\... | https://mathoverflow.net/users/9896 | Repeated draws from multinomial distribution | This is essentially a product of local CLTs (since one can uncover the multinomial
distribution variable by variable: decide first how many of the $n$ variables equal "1" with probability of success $1/k$ for each one and thus variance $(n/k)(1-1/k)$; Then, of the roughly $n(k-1)/k$ remaining variables, decide how many... | 2 | https://mathoverflow.net/users/35520 | 152864 | 81,417 |
https://mathoverflow.net/questions/152855 | 4 | Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of **Mod(S)**(mapping class group) on the set of all vertices has finitely many orbits ([see this post](https://mathoverflow.net/questions/49963/is-the-action-of-the-mapping-class-group-transitive-on-e... | https://mathoverflow.net/users/9485 | Action of Mapping Class Group on Arc complex | The quotient complex was studied in a paper by Penner, "The structure and singularities of quotient arc complexes", Journal of Topology 1 (2008), 527-550. An earlier version of the paper is available on the arXiv. He enumerated all cases when the quotient is a sphere (this only happens when the genus and number of boun... | 4 | https://mathoverflow.net/users/23571 | 152871 | 81,420 |
https://mathoverflow.net/questions/152787 | 15 | In [this question](https://mathoverflow.net/questions/151286/probabilities-in-a-riddle-involving-axiom-of-choice) the following observation was made:
Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened.
Define a *play* to be a ser... | https://mathoverflow.net/users/44653 | Can an infinite number of mathematicians guess the number in a box with only one error? | It is possible to have every mathematician guess the number in one of the boxes with at most one error.
---
Partition the natural numbers into countably many sets, $\{S\_i\}\_{i=0}^\infty$, where each $S\_i=\{n\_{i\_1},n\_{i\_2},\dots,\}$ is countably infinite. (There are many ways to do this) Since we have count... | 13 | https://mathoverflow.net/users/12176 | 152883 | 81,424 |
https://mathoverflow.net/questions/152877 | 7 | Joyal and Tierney proved that morphisms of rings which are of effective descent are exactly those morphisms $\phi:R\to S$ such that $\phi$ presents $S$ as a pure $R$-module. Grothendieck had originally shown that being faithfully flat implied being of effective descent, but had not entirely characterized such morphisms... | https://mathoverflow.net/users/11546 | Pure morphisms which are not faithfully flat | While this is already answered in the comments, let me give you a large class of examples.
A ring $R$ of characteristic $p > 0$ is called $F$-*pure* if the Frobenius map $F : R \to R$ is a pure morphism.
On the other hand, by a theorem of Kunz, a ring of characteristic $p > 0$ is regular if and only if the Frobeni... | 10 | https://mathoverflow.net/users/3521 | 152893 | 81,426 |
https://mathoverflow.net/questions/152886 | 2 | Consider smooth closed space curve (parametrised by its arc length) in 3 dimensional space.Does there exist atleast one point at which both the curvature & torsion attain extremum values?
| https://mathoverflow.net/users/37477 | Space curves and torsion | Imagine a 2D curve defined as follows: $x(t) := \kappa(s), y(t) := \tau(s)$;
if it were true that a space curve always contains a point where curvature and torsion are simultaneously extremal, then the associated 2D curve (as defined above), would always contain a corner of its bounding box.
Even if infinite curv... | 0 | https://mathoverflow.net/users/31310 | 152902 | 81,429 |
https://mathoverflow.net/questions/152519 | 5 | Let $K$ be a field of characteristic zero. Let $\Omega = K[x\_1, \dots, x\_n, dx\_1, \dots, dx\_n]$ be the differential ring of algebraic differential forms over $K[X\_1, \dots, X\_n]$.
Is there an algorithm (e.g. by Gröbner basis-like techniques) that solves the ideal membership problem for $\Omega$? That is: given ... | https://mathoverflow.net/users/1841 | Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis? | Yes. There is such an algorithm. This ring is a finitely generated free module over a polynomial ring, and it is sufficient to solve the submodule membership problem for these modules. But this is easy - the Grobner basis idea works perfectly. Just order the monomials times generators and find a basis for the initial s... | 5 | https://mathoverflow.net/users/18060 | 152911 | 81,431 |
https://mathoverflow.net/questions/152908 | 22 | This is a modification of [an unanswered problem on the math StackExchange](https://math.stackexchange.com/questions/614238/integers-n-such-that-ii1i2-cdots-in-is-real-or-pure-imaginary/619597#619597).
When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square?
If $(1+1)(1+4)…(1+n^2)=k^2$ then one possibility is $n... | https://mathoverflow.net/users/11124 | When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square? | [Javier Cilleruelo](https://mathoverflow.net/users/31020/javier) has shown that $n=3$ is the only solution: see <http://www.uam.es/personal_pdi/ciencias/cillerue/Papers/squares-sinlogo.pdf> .
A couple more comments: apparently Chebyshev already showed that the largest prime factor of $\prod\_{j=1}^{n} (1+j^2)$ is bi... | 37 | https://mathoverflow.net/users/38624 | 152914 | 81,433 |
https://mathoverflow.net/questions/152896 | 1 | Consider $A$, a random binary matrix of zeros and ones in $\mathbb{R}^{{M\times N}}$, and $M>N$. We assume that $P(a\_{i,j}=0)=P(a\_{i,j}=1)=0.5$ (although I appreciate any advice on the case of non-even probabilities). Are there any results that provide a lower bound (and maybe an interesting upper bound) on the eigen... | https://mathoverflow.net/users/44722 | Bounds on the eigenvalues of a random binary matrix | The distribution of eigenvalues will follow the [Marchenko-Pastur distribution](http://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution), scaled appropriately.
Since the mean entry will be equal to $0.5$, then the largest eigenvalue will be (approximately) equal to $0.25MN$, and the corresponding eigenvect... | 3 | https://mathoverflow.net/users/40499 | 152920 | 81,437 |
https://mathoverflow.net/questions/152905 | 0 | For positive integers $n,k$, define $$f(n,k):=\sum\_{i=1}^{n-1}(n-i)\binom{k}{i}.$$
What are upper and lower bounds of $f(n,k)$ by simpler terms? (e.g. finding bounds which are not a summation like this one.) How fast does $f(n,k)$ grow asymptotically in $n$ and $k$?
If we loosely bound $\binom{k}{i}\leq 2^k$, then... | https://mathoverflow.net/users/44726 | Asymptotic growth for $\sum_{i=1}^{n-1}(n-i)\binom{k}{i}$ | For $n$ and $k$ fixed, the sequence $\left\{(n-i){k \choose i} \right\}\_{i=1}^{n-1}$
is strongly unimodal (that is, log concave), or at least a pencil and paper computation convinced me that it is. This allows one to estimate extremely well the sum, $f(n,k)$, particularly when $n$ is large. (The idea is that beyond t... | 3 | https://mathoverflow.net/users/42278 | 152921 | 81,438 |
https://mathoverflow.net/questions/80936 | 9 | What are the most general conditions on a Lie algebra $\mathfrak{g}$ over a field $\mathbb{k}$ such that the space of invariant symmetric bilinear forms is isomorphic to $H^3(\mathfrak{g},\mathbb{k})$?
The isomorphism should look like this: $\langle \cdot,\cdot \rangle \to \langle \cdot, [\cdot,\cdot]\rangle$ and I'v... | https://mathoverflow.net/users/6818 | invariant symmetric bilinear forms and Lie algebra cohomology | The map $\langle\cdot,\cdot\rangle\mapsto\langle\cdot,[\cdot,\cdot]\rangle$ is usually called *Koszul homomorphism*. Indeed Koszul showed that for a semisimple Lie algebra over a field of characteristic zero it is an isomorphism.
For an arbitrary Lie algebra (say over a field of characteristic zero), it always maps i... | 7 | https://mathoverflow.net/users/14094 | 152936 | 81,444 |
https://mathoverflow.net/questions/152940 | 22 | Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence of inverses dropped. If so what is the analogous structure to lie algebras if one exists?
| https://mathoverflow.net/users/41654 | Lie groups vs Lie monoids | There is a well developed theory of algebraic monoids, due principally to Putcha and Renner. I think Lie semigroups is less well developed but there is work by Hoffmann, Lawson and the thesis of Langlands was on this subject.
| 11 | https://mathoverflow.net/users/15934 | 152941 | 81,445 |
https://mathoverflow.net/questions/150897 | 3 | I am reading Burdzy's paper on the points of increase of Brownian motion:
[Burdzy's Paper](https://projecteuclid.org/journals/annals-of-probability/volume-18/issue-3/On-Nonincrease-of-Brownian-Motion/10.1214/aop/1176990732.full)
He is proving that, almost surely, a Brownian motion, has no points of increase. What he ... | https://mathoverflow.net/users/43706 | Proving that Brownian motion has no points of increase | If we have a point of increase at $t$, witnessed by $\epsilon$, the question is threefold: why can we assume $t-\epsilon=0$, $B\_t(\omega)\le 1$, and $B\_{t+\epsilon}(w)-B\_t(\omega)\ge 2$. These conditions are somewhat similar, so here is an argument for the last of these conditions.
Suppose we have a point of incre... | 2 | https://mathoverflow.net/users/4600 | 152958 | 81,449 |
https://mathoverflow.net/questions/152962 | 2 | Let $m$ an arbitrary integer. I would like to know if exists an estimation of $$\underset{\left(q,\, m\right)>1}{\underset{q>1}{\sum}}\frac{1}{\phi\left(q\right)\phi\left(q/\left(q,\, m\right)\right)}$$ or something similar, where $(q, m)$ is the g. c. d. of $q$ and $m$.
| https://mathoverflow.net/users/41635 | On an estimation involving Euler totient function | \begin{align\*}
\sum\_{\substack{q>1 \\ (q,m)=1}} \frac{1}{\phi(q)\phi\big(q/(q,m)\big)} &= \sum\_{\substack{d\mid m \\ d>1}} \sum\_{\substack{q>1 \\ (q,m)=d}} \frac{1}{\phi(q)\phi(q/d)} \\
&\le \sum\_{\substack{d\mid m \\ d>1}} \sum\_{r=1}^\infty \frac{1}{\phi(rd)\phi(r)} \\
&\le \sum\_{\substack{d\mid m \\ d>1}} \fra... | 4 | https://mathoverflow.net/users/5091 | 152974 | 81,454 |
https://mathoverflow.net/questions/152977 | 6 | Let $X$ be a Calabi-Yau 3-fold with Picard number one. How can one show that the automorphism group $Aut(X)$ is finite and moreover coincides with the birational automorphism group $Bir(X)$?
It seems this is a well-knwon fact, but I cannot find any reference. Since any automorphism group of $X$ preserves the ample ge... | https://mathoverflow.net/users/44763 | $Aut(X)$ and $Bir(X)$ for Calabi-Yau 3-folds with $\rho(X)=1$ | The group $\mathrm{Aut}(X)$ is finite because, as you point out, it preserves a very ample line bundle; hence it is an algebraic group (a closed subgroup of a projective group). Therefore it suffices to prove that its Lie algebra $H^0(X,T\_X)$ is trivial. By Serre duality this is dual to $H^3(X,\Omega ^1\_X)=H^{1,3}$, ... | 10 | https://mathoverflow.net/users/40297 | 152979 | 81,456 |
https://mathoverflow.net/questions/152978 | 2 | I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
1. $P$ is Lagrangian
2. P involutive
3. dim$P\cap\bar P \cap TM$ is constant
This definition shows that every complex polarization induces a real isotrpic distribu... | https://mathoverflow.net/users/nan | An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive | Suppose your $M$ is the coadjoint orbit of a Lie group $G$ through $x\in\mathfrak g^\*$; write $G\_x$ for the stabilizer of $x$. The set of $G$-invariant polarizations $P$ on $M$ identifies with the set of subalgebras $\mathfrak p\subset \mathfrak g^{\mathbf C}$ such that
1. $\mathfrak p$ contains $\mathfrak g\_x^{\m... | 1 | https://mathoverflow.net/users/19276 | 152994 | 81,462 |
https://mathoverflow.net/questions/152971 | 0 | I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for instance by Evertse.
What about the case when the powers of $x$ and $y$ are unequal? It seems a rather natural question... | https://mathoverflow.net/users/10898 | Reference request: on sums of the form $ax^m + by^n = h$ | I assume that you're asking about integer solutions, but of course, if the genus of the associated curve is at least 2, you can also ask for bounds for the number of rational solutions. Anyway, Evertse and I have a paper that deals with $S$-integral points on these curves (and somewhat more general ones):
Evertse, J.... | 3 | https://mathoverflow.net/users/11926 | 152995 | 81,463 |
https://mathoverflow.net/questions/153003 | 2 | The partially ordered set $(Y,\le)$ is called a **superspace** of the partially ordered set $(X,\le')$ iff
1. $X\subseteq Y$.
2. $\le' \:\:=\: (\le\cap X^2)$.
and a **completion** of $(X,\le')$ if in addition
$~~ 3$. $(Y,\le)$ is a complete lattice.
Are two minimal completions of a partially ordered set isomorp... | https://mathoverflow.net/users/nan | Uniqueness of minimal completions of a partially ordered set | I claim that according to your definition, every minimal completion is isomorphic to the [Dedekind-MacNeille completion](http://en.wikipedia.org/wiki/Dedekind%E2%80%93MacNeille_completion). Furthermore, every Dedekind-MacNeille completion is a minimal completion.
Suppose that $X$ is a poset. If $A\subseteq X$, then d... | 8 | https://mathoverflow.net/users/22277 | 153007 | 81,468 |
https://mathoverflow.net/questions/152342 | 32 | For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like this outside theoretical mathematics.
But now I'm wondering whether simplified mathematical models of Euler's disk (se... | https://mathoverflow.net/users/3621 | Differentiable functions with discontinuous derivatives | Euler's disk, with its shuddering singularity, is probably the best example of the sort of phenomenon the OP was asking about, with several caveats, most of which have been mentioned in other postings on this thread. There are a number of possible models for the behavior of Euler's disk (it's a topic of active research... | 1 | https://mathoverflow.net/users/3621 | 153014 | 81,471 |
https://mathoverflow.net/questions/153013 | 4 | The question is motivated by a more broad perspective in [another MO post](https://mathoverflow.net/questions/152997/finite-dimensional-lie-algebra-with-non-degenerate-invariant-bilinear-forms-om) and [here](https://mathoverflow.net/questions/152952/complete-classification-of-six-dimensional-non-semi-simple-lie-algebra... | https://mathoverflow.net/users/44768 | The existence of a finite dimensional Lie algebra with a given symmetric invariant metric | If I understand your question correctly, then $\mathfrak{g} \ltimes \mathfrak{g}^\*\_{\textrm{abelian}}$, for $\mathfrak{g}$ any three-dimensional real Lie algebra, has an invariant inner product of this type. This is the semidirect product of $\mathfrak{g}$ with its coadjoint representation treated as an abelian Lie a... | 4 | https://mathoverflow.net/users/394 | 153030 | 81,474 |
https://mathoverflow.net/questions/152943 | 1 | Let $(X,\omega)$ be a symplectic manifolds.
>
> For $f∈C^∞(M,ℂ)$ a function on phase space, the corresponding quantum
> operator(prequantized observable function) is the linear map
> $$\mathcal{O}\_f:Γ\_X(L)→Γ\_X(L)$$ given by $$ψ↦−i\hbar∇ \_{X\_f}ψ+f⋅ψ$$
> Where
>
>
> 1. $X\_f$ is the Hamiltonian vector fiel... | https://mathoverflow.net/users/nan | Pre-quantized observable functions can be written as flow | Conceptually, it's much easier to see what is going on if you focus not on the line bundle $L$, but on the principal $\mathrm{U}(1)$-bundle $\pi:P\rightarrow M$ to which it is associated (so that $L=P\times\_{\mathrm{U}(1)}\mathbb{C}$; $P$ can be thought of as the subset $L$ of unit norm with respect to the Hermitian f... | 2 | https://mathoverflow.net/users/17945 | 153038 | 81,478 |
https://mathoverflow.net/questions/153041 | 9 | I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from reading some papers, it seems that the following is correct (so the first question - is it correct?). Given a hyperboli... | https://mathoverflow.net/users/27433 | What constant ensures hyperbolicity of Dehn surgery? | Given a hyperbolic link $L$, [Thurston's Dehn filling theorem](http://en.wikipedia.org/wiki/Hyperbolic_Dehn_surgery) says that there is a finite set of "bad slopes" on each component of $L$ such that every filling avoiding them is hyperbolic. A slope is just a rational number, and since the bad slopes are finite in num... | 11 | https://mathoverflow.net/users/6205 | 153049 | 81,481 |
https://mathoverflow.net/questions/152910 | 3 | Special case of [this question](https://mathoverflow.net/questions/152442/graphs-with-many-edges-avoided-by-hamiltonian-cycles).
Let $G$ be $r$-regular Hamiltonian graph.
An $a$ edge is an edge which is on every Hamiltonian cycle.
A $b$ edge is an edge which is on no Hamiltonian cycle.
$a(G)$ and $b(G)$ are numbers... | https://mathoverflow.net/users/12481 | Regular graphs with $a$ and $b$ Hamiltonian edges | There are infinite families of $4$ and $5$ regular graphs
with $\rho(G)=1$ using a gadget.
A gadget is graph $GA$ with $2$ vertices $u,v$ of degree $d-1$
and the rest are of degree $d$. The gadget contains
sufficiently many $b'$ edges which are no $uv$ Hamiltonian path
compared to $a'$ edges which are on all H-$uv$ p... | 2 | https://mathoverflow.net/users/12481 | 153066 | 81,488 |
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