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https://mathoverflow.net/questions/160428 | 0 | This is a group theory question. I am preparing a research paper. One result brought my attention. I am wondering if you know some paper or book listed this result.
Let $G$ be a 2-group. Suppose there exists nonabelian subgroups of $G$(FYI, otherwise the structure of $G$ is known, see Huppert, Endliche Gruppen I, P3... | https://mathoverflow.net/users/38283 | Hamiltonian 2-groups | Groups with this property were studied by G.A. Miller in 1907: see [here](http://www.ams.org/journals/tran/1907-008-01/S0002-9947-1907-1500772-6/S0002-9947-1907-1500772-6.pdf).
It looks as though there is just one counterexample, the quaternion group of order $16$.
| 3 | https://mathoverflow.net/users/35840 | 160438 | 84,429 |
https://mathoverflow.net/questions/160392 | 0 | Consider a 3-sheeted Riemann surface without a Z\_3 symmetry. The first and second sheets are sewn together at an interval (u1,v1), and the second and third are sewn at (u2,v2). This is a Riemann sphere. What is the map that uniformizes this surface?
In other word, I am looking for a map w(z) that maps the n-sheeted s... | https://mathoverflow.net/users/48276 | Uniformization of n-Sheeted surfaces | Just few remarks to Alex's answer.
If all $u\_1,u\_2,v\_1,v\_2$ are distinct, the function mapping the Riemann sphere on your surface is a rational function of degree $3$. And conversly, every rational function
of degree $3$, with simple critical points and distinct critical values, maps the sphere on
such Rimemann sur... | 0 | https://mathoverflow.net/users/25510 | 160453 | 84,431 |
https://mathoverflow.net/questions/160449 | 4 | Is there a manifold $M$ for which $\chi^{\infty} (M)$, the lie algebra of smooth vector fields on $M$ contains all finite dimensional Lie algebras(Up to isomorphism)?
A weaker question:
Is there a manifold $M$ such that for every $n>0$, $\chi^{\infty} (M)$ contains an n dimensional Lie subalgebra?
| https://mathoverflow.net/users/36688 | A Manifold for which $\chi^{\infty}(M)$ is rich | The answers to your questions are 'no' and 'yes'.
In the first place, there is no finite dimensional manifold whose Lie algebra of vector fields contains all of the Lie algebras ${\frak{sl}}(n,\mathbb{R})$ for all $n\ge 2$.
In fact, ${\frak{sl}}(n{+}2,\mathbb{R})$ cannot occur as a Lie algebra of vector fields on ... | 12 | https://mathoverflow.net/users/13972 | 160455 | 84,433 |
https://mathoverflow.net/questions/160459 | 3 | I would like to work out a simple example to understand the relation between Kleiman ampleness criterion and Nakai-Moishezon ampleness criterion.
Namely, let $X$ be the blow-up of $\mathbb{P}^{2}$ at two points $p,q$, $L$ the pull-back of a general line in $\mathbb{P}^{2}$, $R$ the strict transform of $\left\langle p... | https://mathoverflow.net/users/nan | Kleiman's and Nakai-Moishezon's ampleness criteria | 1. The Nakai--Moishezon criterion applies in any dimension; the condition is that $D^k \cdot C >0$ for any $k$-dimensional subvariety $C$ and $D^{\text{ dim } X}>0$.
2. There are several sufficient conditions for $NE(X)$ to be closed, e.g. if $X$ is Fano or if $X$ is toric (both true in your example). Much more simply,... | 3 | https://mathoverflow.net/users/nan | 160465 | 84,437 |
https://mathoverflow.net/questions/160468 | 2 | Given a prescribed trajectory, is it possible to construct an interval exchange having this trajectory?
For example, given a 3-letter word (like aaabbbccabcaaa ), is it possible to construct a 3- interval exchange with a point having this word as the beginning of its trajectory?
what necessary conditions on a given... | https://mathoverflow.net/users/41823 | Constructing an interval exchange given a prescribed trajectory | The *complexity* of an infinite sequence $x$ is a sequence $C(n)$, where $C(n)$ is the number of distinct blocks of length $n$ in $x$. For an interval exchange with $k$ symbols, it's not hard to show that $C(n)=(k-1)n+1$. If your word has more complexity than this, it can never appear as the coding sequence of an IET.
... | 5 | https://mathoverflow.net/users/11054 | 160472 | 84,438 |
https://mathoverflow.net/questions/160471 | 1 | It is known that a Hilbert scheme of degree $d$ curves in $\mathbb P^3$ can have dimension more than $4d$. But, does it imply that for some types of curves there are such a curve through any, say, $2d+1$ points? (or $3d$)
| https://mathoverflow.net/users/4298 | Curve of degree $d$ through $2d+1$ points in $\mathbb P^3$ | Take complete intersections of degree $(4,4)$: there is a pencil of quartics passing through $2\times 16+1=33$ general points. More generally, complete intersections of degree $(d,d)$ will pass through any $n(d).d$ points, with $n(d)\rightarrow\infty$ with $d$.
| 2 | https://mathoverflow.net/users/40297 | 160475 | 84,440 |
https://mathoverflow.net/questions/160431 | 4 | Tutte （1961): A graph $G$ is $3$-connected if and only if there exists a sequence
$G\_0, ...,G\_n$ of graphs that have the following two properties
1) $G\_0 = K\_4$ and $G\_n = G$
2) $G\_{i+1}$ has an edge $xy$ with $d(x), d(y) ≥ 3$ and $G\_i = G\_{i+1}/xy$.
(If $G$ is a graph, the graph $G'$ obtained by an edge ... | https://mathoverflow.net/users/nan | A 3-connected graph property by Tutte | Tutte proved that every 3-connected graph with at least 5 vertices has an edge whose contraction leaves a 3-connected graph. So starting with $G\_n$, just keep contracting suitable edges until you reach $K\_4$. It is the unique 3-connected graph with less than 5 vertices, so there is no way to avoid it.
Search at Sch... | 3 | https://mathoverflow.net/users/9025 | 160476 | 84,441 |
https://mathoverflow.net/questions/160125 | 3 | A subset of a symplectic manifold is called **strongly non-displaceable** if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a Hamiltonian diffeomorphism. I am reading "Rigid subsets of symplectic manifolds" by Entov and Polterovich and trying t... | https://mathoverflow.net/users/11846 | Computation of symplectic quasi-state | I don't think your computations of CZ indices are correct -- in particular, I am surprised you have a factor of $4$ showing up.
The key mistake, and the one that explains the key reason this theorem works, is that the image of the PSS map will be of a fixed grading (depending on your convention). The grading of $(N, ... | 2 | https://mathoverflow.net/users/477 | 160488 | 84,445 |
https://mathoverflow.net/questions/160427 | 1 | Let $G$ be a finite group such that $G$ has a normal subgroup $N$ of order $p(p^2+1)/2$, where $p>13$ is an odd prime and $p\ne 239$. Also $G/N\cong \text{PSL}(2,p)$. Can we say that there exists a prime divisor $t$ of $|N|$ such that $2t\not\mid \chi(1)$ for every $\chi\in \text{Irr}(G)$?
Of course we know that $N$... | https://mathoverflow.net/users/31045 | On the character degrees of a finite group with special structure | I suppose that $p >3$ in my answer. Note that $|H|$ and $|{\rm PSL}(2,p)|$ are coprime. It follows by transfer and a Theorem of Gaschutz that $N$ is complemented in $G,$ and that $O\_{p}(N)$ is a direct factor of $G.$ Hence we might as well concentrate on $H,$ rather than $N.$ So we look at the group $HX$ where $X \con... | 2 | https://mathoverflow.net/users/14450 | 160490 | 84,446 |
https://mathoverflow.net/questions/158996 | 6 | In the context of topological groups, one has $w(X)=d(X)\chi(X)$, where $X$ is a topological group and $w$, $d$ and $\chi$ are the *weight*, the *density* and the *character*, respectively. Since $d(X)\leq nw(X)\leq w(X)$, where $nw(X)$ is the least cardinality of a network on $X$, we also have that $w(X)=nw(X)\chi(X)$... | https://mathoverflow.net/users/41407 | A question about cardinal functions | None of the standard separation axioms is enough to guarantee $nw(X) \cdot \chi(X)=w(X)$. In fact, in the exercise section of Engelking's chapter on compactness, you can find this construction of a perfectly normal space, known as the Bow-tie Space. Define a topology on the plane by leaving neighborhoods of all points ... | 6 | https://mathoverflow.net/users/11647 | 160496 | 84,448 |
https://mathoverflow.net/questions/156902 | 1 | This is really a trivial question.
The 0-horn of a simplicial point $\Delta^0$ is not defined nor remarked in the books and papers I could find. So one expect if we could make some meaningful definition (like $0!=1$). The usual definition of $\Lambda^n\_k$ for $n>1, 0\le k\le n$ simply does not make sense.
It is q... | https://mathoverflow.net/users/7341 | How to define $\Lambda^0_0$, 0-horn of a simplicial point | (As already mentioned, we have to specify whether we're talking about *simplicial sets* or about *augmented simplicial sets*; I will be talking about simplicial sets.)
I agree with Ricardo Andrade: the simplex $\Delta^0$ has no horns.
The horns of a simplex $\Delta^n$ correspond to the (maximal) proper faces of $\D... | 2 | https://mathoverflow.net/users/25477 | 160498 | 84,449 |
https://mathoverflow.net/questions/160492 | 7 | Let $p\_1,...,p\_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m\_1,...,m\_n$ at $p\_1,...,p\_n$. I would like to know if the following variation of Bertini's theorem holds.
Suppose $|L|\neq\emptyset$. Is it t... | https://mathoverflow.net/users/nan | Bertini's Theorem | As pointed out by Alex in his comment, this is in general not true.
For instance, consider the case $N=d=n=m=2$. Then $|L|$ is the linear system of plane curves of degree $2$ passing through $p\_1$ and $p\_2$ with multiplicity $2$. Of course there is only one such a curve, namely the line through $p\_1$ and $p\_2$ co... | 11 | https://mathoverflow.net/users/7460 | 160500 | 84,450 |
https://mathoverflow.net/questions/160483 | 0 | Let $p\_1,...,p\_n\in\mathbb{P}^3$ be general points, and let $\Delta\subset\mathbb{P}^3$ be a general surface of degree $d$ with points of multiplicity $m\_i$ at $p\_i$ for $i = 1,...,n$.
Consider the blow-up $X$ of $\mathbb{P}^{3}$ at $p\_1,...,p\_n$ and the strict transform $\tilde{\Delta}$ of $\Delta$. For which ... | https://mathoverflow.net/users/nan | Kawamata-Log-Terminal pairs | I'm not sure what you mean by the pair being klt here: if $\tilde{\Delta}$ appears with coefficient $1$, then it's not. On the other hand, it's sensible to ask whether you can find a boundary $\mathbb Q$-divisor $D$ numerically equivalent to $\tilde{\Delta}$ for which $(X,D)$ is klt. Or you could ask for $(X,\tilde{\De... | 1 | https://mathoverflow.net/users/nan | 160504 | 84,453 |
https://mathoverflow.net/questions/159935 | 7 | When I look at first time to a theorem and I try to understand it or when I try to memorise a useful theorem I always have difficulties (I am not the only one. For example: I read a question: I always have trouble memorizing theorems. Does anybody have any good tips? )
However, I have learn from the theorems that I ... | https://mathoverflow.net/users/39115 | Sources of Theorem drafts by the original author | **Short answer:** I believe Paul Cohen's piece on his own development of *forcing* provides an answer to your question. I will elaborate further below the break, but you can find his article written up as:
Cohen, P. (2002). [The discovery of forcing](http://www.logic.univie.ac.at/~ykhomski/ST2013/The%20Discovery%20o... | 4 | https://mathoverflow.net/users/22971 | 160508 | 84,454 |
https://mathoverflow.net/questions/160415 | 0 | For any $b\in\mathbb{R}^d$ and $B\in\mathbb{R}^{d\times d}$ symmetric positive definite, define the functional operator $\mathcal{A}f(x)=f(Bx+b)$ ($f$ real-valued). We have that $\mathcal{A}$ is linear; compact (I think); normal, since $$\mathcal{A}^\*f(x)=\frac{1}{\mathrm{det}(B)} f(B^{-1}x-B^{-1}b)$$ and hence $\math... | https://mathoverflow.net/users/42501 | Spectrum of operator defined by composition with linear function | Take $H=L^2(\mathbb{R}^n, e^{-|x|^2}dx)$ as your Hilbert space, the $L^2$ space with weight $e^{-|x|^2}$, so that polynomials are in the space (Hermite Polynomials are an ONB for this space).
Your map $(B, b) \mapsto \widehat{(B, b)}$ defines a group homomorphism from the group $T(n)$ of transformations $x \mapsto Bx... | 1 | https://mathoverflow.net/users/16702 | 160513 | 84,455 |
https://mathoverflow.net/questions/160495 | 3 | I am reading a paper by Darmon and Granville where they made the following claim, with respect to the generalized Fermat curve $Ax^p + By^q = Cz^r$. They said:
"... this (referring to the generalized Fermat curve above) is a curve in an appropriate weighted projective space. However this curve often has genus 0 (for ... | https://mathoverflow.net/users/10898 | Genus in weighted projective space | If p, q and r are pairwise coprime the generalized Fermat curve is embedded in the weighted projective space ${\mathbb P}(qr, pr, pq) $ which is isomorphic to a straight ${\mathbb P}^2$ via $(x:y:z)\mapsto (x^p:y^q:z^r) $: the image of the curve is a line, so it is rational.
| 7 | https://mathoverflow.net/users/46104 | 160519 | 84,456 |
https://mathoverflow.net/questions/160525 | 1 | Let $F$ be a finite field of cardinality $q$ and let $f \in F[x]$ be a non-constant polynomial of degree $d$ which is not onto (as a function from $F$ to $F$). Then how large the image of $f$ could be ?! I am looking for an answer in terms of $q$ and $d$ (if possible).
| https://mathoverflow.net/users/44633 | On the maximum cardinality of the image of a non-onto polynomial function on finite fields | I assume that by $d$ you mean the degree of $f$. Then the results are as follows, where I write $N$ for the size of the image $f(F)$:
1. $N \le q - \lceil \frac{q-1}d\rceil$
2. If $q$ is a power of a smaller positive integer $r$, then $f(x)=x^r+x^{r-1}$ satisfies $N=q-\frac q r$, and hence achieves equality in item 1... | 4 | https://mathoverflow.net/users/30412 | 160529 | 84,459 |
https://mathoverflow.net/questions/160540 | 2 | Is there a finite set $P$ of non-elementary functions $f\_n$ such that the derivative of any function $f$ from that set is not elementary, but expressible with functions from the same set $P$ plus elementary functions?
| https://mathoverflow.net/users/10059 | Is there a class of functions closed against differentiation besides elementary? | Let $g$ be an elementary function whose indefinite integral $f$ is nonelementary, and set $P = \{\cos(t)f(t), \sin(t)f(t)\}$.
| 5 | https://mathoverflow.net/users/18403 | 160542 | 84,462 |
https://mathoverflow.net/questions/160536 | 6 | For $n$ and $m$ positive integers $n>m\ge 1$ define a graph as follows.
The vertices are the binary strings of length $n$.
Two vertices are adjacent if they differ in exactly $m$ consecutive bits.
The indices of the $m$ consecutive positions where $x$ and $y$ differ from each other are considered modulo $n$.
Just to ... | https://mathoverflow.net/users/36060 | The number of connected components of a generalized hypecube | Let $d$ be the GCD $(m,n)$. If $m/d$ is even, then the number of connected components is $2^d$. If $m/d$ is odd, there are $2^{d-1}$ connected components.
Consider $C$, the circulant matrix over $\mathbb{Z}/2$ whose first row is $(1,1,...,1,0,0,...,0)$. The range of $C$ is the set of points connected to $\vec 0$, so ... | 5 | https://mathoverflow.net/users/2954 | 160547 | 84,463 |
https://mathoverflow.net/questions/160512 | 18 | Recall that a topological space $X$ has the *fixed point property* (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More precisely:
**Question.** Is it consistent that a topological space $X$ has the FPP in $V$, but it does not have the ... | https://mathoverflow.net/users/11115 | Is the notion of fixed point property for topological spaces an absolute notion? | The answer is that the FPP is not absolute, and indeed, even the unit interval loses the FPP in a forcing extension. The unit interval famously has the FPP, but I claim that in any forcing extension having a new real, such as the forcing extension $V[c]$ obtained by adding a Cohen real, which preserves cardinals, the g... | 17 | https://mathoverflow.net/users/1946 | 160552 | 84,466 |
https://mathoverflow.net/questions/160505 | 2 | I'm having trouble with this exercise from Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory.
The exercise in question is from chapter IV.
So, let $k$ be an algebraically closed field and let $A$ be a finite-dimensional $k$-algebra, and $M$ a left $A$-module contai... | https://mathoverflow.net/users/73967 | Showing a functorial isomorphism | Here is a proof of the functorial isomorphism $$\underline{\operatorname{Hom}}\_{A^{\rm{op}}}(\operatorname{Tr} M,-) \cong \operatorname{Tor}\_1^{A^{\rm{op}}}(M,-).$$
Since $M$ has no projective summands, we can choose a right $A$-module $X$ without projective summands such that $M \cong \operatorname{Tr} X$ and $X \... | 2 | https://mathoverflow.net/users/18756 | 160555 | 84,467 |
https://mathoverflow.net/questions/160571 | 8 | In a given non-real algebraic number field (say, given by an irreducible polynomial over $\mathbb{Q}$) is there a complexity bound on the summands $x,\dots,t$ that make $-1$ a sum of squares? So $x^2+\dots+t^2=-1$.
Can you show that if $-1$ is a sum of squares at all, then it is a sum of some list of summands subject... | https://mathoverflow.net/users/38783 | Is realness of number fields exponentially bounded? | To answer the comment above; sure. There's a theorem of Cassels,
generalized to number fields by Raghavan "Bounds for minimal solutions of Diophantine equations," (MR0485681) which gives a concrete upper bound for the smallest solution to a quadratic form (when they exist).
Namely, given an anisotropic quadratic for... | 6 | https://mathoverflow.net/users/48354 | 160581 | 84,478 |
https://mathoverflow.net/questions/160444 | 22 | It is understood that there is a correspondence between the 3d Chern-Simons topological quantum field theory (TQFT) and the 2d Wess-Zumino-Witten conformal quantum field theory (CQFT). A good summary is given on this nLab page:
<http://ncatlab.org/nlab/show/AdS3-CFT2+and+CS-WZW+correspondence>
This is often claimed... | https://mathoverflow.net/users/799 | Interpreting the CS/WZW correspondence | Actually I think the idea of the holographic principle is that, as in a holograph, all the information in the 'bulk' is already present at the 'boundary'. So, it claims that any calculation involving bulk observables can be expressed in terms of boundary observables. It may not claim the reverse, though that could ofte... | 16 | https://mathoverflow.net/users/2893 | 160585 | 84,482 |
https://mathoverflow.net/questions/159494 | 18 | Among the collections of the open problems of Paul Erdős on the website of
Professor Fan Chung, there is one called "number of triangle-free graphs".
<http://www.math.ucsd.edu/~erdosproblems/erdos/newproblems/NumberOfTriangleFreeGraphs.html>
Open Problem: Determine or estimate the number of maximal triangle-free gr... | https://mathoverflow.net/users/nan | Determine or estimate the number of maximal triangle-free graphs on $n$ vertices | By Theorem 1 in the paper of [Erdős, Kleitman, and Rothschild](http://www.renyi.hu/~p_erdos/1976-03.pdf), the number of triangle-free graphs on $n$ vertices is $2^{n^2(1/4 +o(1)) }$. The number of bipartite graphs with a fixed pair of parts of size $n/2$ is $2^{n^2/4}$. Here is a construction of $2^{n^2(1/8 +o(1)) }$ m... | 8 | https://mathoverflow.net/users/2954 | 160595 | 84,488 |
https://mathoverflow.net/questions/160596 | 2 | Suppose $F: C \rightarrow D$ is the left adjoint to a functor $G$. Then is it true that the functor $F^{\star}:[C : Sets]$ defined by prescomposing a functor $P: C \rightarrow Sets$ is still left adjoint to the functor $G^{\star}$ defined similarly?
Or would the adjunction be flipped an $F^{\star}$ becomes right adj... | https://mathoverflow.net/users/36886 | Induced adjunctions | Yes,
Let $Q \in [\mathfrak{D}:Sets]$,$A \in \mathfrak{D}$ $P \in [\mathfrak{C}:Sets]$ and $B \in \mathfrak{C}$.
If $F: \mathfrak{C} \rightarrow \mathfrak{D}$ is left adjoint to $G$, then there are unit and counit natural transformations $\eta: F \circ G \rightarrow 1\_{\mathfrak{D}}$ and $\epsilon: G\circ F \right... | 3 | https://mathoverflow.net/users/36886 | 160601 | 84,490 |
https://mathoverflow.net/questions/160578 | 5 | What is the simplest isomorphism invariant which can distinguish between the two non-isomorphic Steiner triple systems on $13$ points?
Train structure and cycle structure, as described [here](http://books.google.co.il/books?id=WVyU94jShHwC&pg=PA231&lpg=PA231&dq=steiner%20system%20isomorphism%20invariant&source=bl&ots... | https://mathoverflow.net/users/22051 | Isomorphism testing in STS(13) | Take the 26-vertex graphs whose vertices are the blocks and where two vertices are joined by an edge if the corresponding blocks have a vertex in common. These two graphs differ in many easily measured ways, for example having different numbers of induced cycles and different numbers of maximal independent sets. I susp... | 3 | https://mathoverflow.net/users/9025 | 160615 | 84,495 |
https://mathoverflow.net/questions/160619 | 1 | Let $X$ be a K3 surface obtained as a double covering of $\mathbb{P}^1 \times \mathbb{P}^1$ branching along a $(4,4)$-divisor. I think the natural line bundle $\pi^\*\mathcal{O}\_{\mathbb{P}^1\times \mathbb{P}^1}(1,1)$ is an ample line bundle on $X$. How can one prove that $\pi^\*\mathcal{O}\_{\mathbb{P}^1\times \mathb... | https://mathoverflow.net/users/48374 | An ample line bundle on a K3 surface | The ampleness follows from Nakai--Moichezon criterion. The degree is twice the degree of $O(1,1)$ on $P^1\times P^1$, so is $4$.
| 3 | https://mathoverflow.net/users/4428 | 160621 | 84,497 |
https://mathoverflow.net/questions/160371 | 7 | It's known that subgroups of Gromov's hyperbolic groups are not necessarily hyperbolic.
Is there any counter-example when the quotient is Abelian. More precisely, let $G$ be a Gromov's hyperbolic group and $H$ a normal subgroup of $G$ such that $G/H$ is Abelian.
Is $H$ a Gromov's hyperbolic group?
| https://mathoverflow.net/users/38190 | Subgroups of Gromov's hyperbolic groups | The answer is no. The case when $H$ isn't finitely generated is trivial (think of the free groups and its commutator subgroup), examples when $H$ is finitely generated are due to Rips (see the famous [Rips construction](http://blms.oxfordjournals.org/content/14/1/45.extract) , and when $H$ is finitely presented this w... | 8 | https://mathoverflow.net/users/1573 | 160632 | 84,499 |
https://mathoverflow.net/questions/160644 | 2 | In a Faa Di Bruno Formula there is an equation:
$m\_1$+2\*$m\_2$+3\*$m\_3$+...n\*$m\_n$=n
Is there any general solution for this equation.
For example for
$m\_1$+$m\_2$+$m\_3$+...+$m\_n$=n, there is a simple algorithm calculating this.
Thanks,
Gevorg.
| https://mathoverflow.net/users/48366 | Specific Diophantine Equation Appearing in Faa Di Bruno Formula | I presume you want to solve this equation $\sum\_{k=1}^{n}km\_k=n$ for integer $m\_k$ that sum to $s$. There is no general solution, valid for any $1\leq s\leq n$, but the problem can be reduced to a calculation of integer partitions, see [Faa di Bruno's formula, lattices, and partitions](http://www.researchgate.net/pu... | 4 | https://mathoverflow.net/users/11260 | 160647 | 84,503 |
https://mathoverflow.net/questions/160557 | 11 | Is (non-abelian) cohomology used to study vector and principal bundles?
Can you give me a text or an article?
For example:
Consider a vector bundle $E$ with fiber $V$ and base manifold $M$.
Consider the loop group $L$ of $M$ based at a point $p$.
The holonomy of a connection is a morphism of groups $\omega:L\to Au... | https://mathoverflow.net/users/30366 | Can group cohomology be used to study fiber bundles? | I presume that you are here referring to non-Abelian cohomology of a group, $G$. There is also the non-Abelian cohomology of spaces which relates directly to the classification of principal bundles. That requires the use of local (sometimes called twisted) coefficients. The link between them is via the classifying spac... | 5 | https://mathoverflow.net/users/3502 | 160651 | 84,506 |
https://mathoverflow.net/questions/160630 | 2 | Let $\xi, \eta, \eta'$ be non-negative random variables such that:
* $\eta \stackrel{\mathcal{L}}{=} \eta'$,
* $\xi + \eta \stackrel{\mathcal{L}}{=} \xi + \eta'$,
* $\xi$ and $\eta'$ are independent.
Does this imply that $\xi$ and $\eta$ are independent? Can one construct a counter-example? Any sort of reference wo... | https://mathoverflow.net/users/48378 | Sum of two independent random variables | No, $\xi$ and $\eta$ need not be independent. For example let $\xi$, $\eta$, and $\eta'$ all be uniformly distributed on $\{1,2,3\}$ with the joint distribution of $\xi$ and $\eta$ is given by the matrix
\[
P = \frac{1}{9}\begin{bmatrix}
1 & 2 & 0 \\ 0 & 1 & 2 \\ 2 & 0 & 1
\end{bmatrix},
\]
i.e. $\mathbb{P}(\xi = i, \e... | 8 | https://mathoverflow.net/users/5963 | 160655 | 84,507 |
https://mathoverflow.net/questions/160656 | 8 | Is there some reference containing an ellaboration of Morse Theory for proper actions of Lie groups?
| https://mathoverflow.net/users/21985 | Morse Theory and proper actions | Have a look at
>
> N. Hingston, *Equivariant Morse theory and closed geodesics*,
> J. Differential Geom.
> Volume 19, Number 1 (1984), 1-276.
>
>
>
in particular Section 2. Most statements from the equivariant Morse theory for compact group actions go over unchanged, thanks to the existence of slices for pr... | 7 | https://mathoverflow.net/users/8103 | 160662 | 84,508 |
https://mathoverflow.net/questions/160670 | 11 | I would like to credit a reviewer who made a very nice corollary to the paper he was reviewing. Is there any more or less common standard of quoting MathReviews? I would be most grateful for an example of how to do it.
| https://mathoverflow.net/users/48386 | Quoting mathreviews | The question seems trivial. Since everyone uses electronic version nowadays,
why don't you write just
Author, Review of, MR number.
If the journal editor will want to edit this, let her edit.
I quoted a Zentralblatt review once like this:
[53] H. Schmidt, Zentralblatt fur Math. 27 (1943), 309-311.
(This was in pre-comp... | 7 | https://mathoverflow.net/users/25510 | 160671 | 84,510 |
https://mathoverflow.net/questions/160674 | 1 | Let $X$ be a complex algebraic variety, and consider the presheaf
$U \mapsto H^i(U^{an}, \mathbb Z)$
in the Zariski topology. Is there a theorem that says this presheaf is already a sheaf, for certain values of $i$ and maybe under some assumptions on $X$?
It seems to be true for $i = 1$ and $X = \mathbb A^1$, but... | https://mathoverflow.net/users/44860 | Singular cohomology as a Zariski sheaf | Notice that the sheaf axiom is a sort of Mayer-Vietoris property. If $X=U\cup V$ for two opens $U,V$ then the sheaf axiom for some presheaf $F$ with values in an abelian category requires
$0\to F(X)\xrightarrow{res\_U^X\oplus res\_V^X} F(U)\oplus F(V) \xrightarrow{res\_{U\cap V}^U-res\_{U\cap V}^V} F(U\cap V)$
to b... | 5 | https://mathoverflow.net/users/3041 | 160679 | 84,512 |
https://mathoverflow.net/questions/160676 | 2 | I still have trouble to grasp the concept of a non-constructible set, my intuition is that we could "avoid" the non-constructibility of many of them if we assume we have "ordinal computers" extended with oracles for their halting problems, plus oracles for the new halting problems of these new extended machines, and al... | https://mathoverflow.net/users/27974 | Would a non-constructible set become constructible if we had oracles of arbitrarily high cardinality for the halting problems of ordinal computers? | *I'm not sure exactly what you're asking, so maybe this is irrelevant, but let me give it a shot.* When you're talking about "avoiding inconstructibility," it sounds like the picture of $L$ you have is roughly: the collection of all 'reasonably computable' sets. And you're asking, what happens if we alter the definitio... | 4 | https://mathoverflow.net/users/8133 | 160680 | 84,513 |
https://mathoverflow.net/questions/160706 | 9 | It's easy enough to show that if $\mathbb{N}\_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}\_1$, and we have a theorem that if $x \in \mathbb{N}\_1$ and $y \in \mathbb{N}$ such that $x < y$, then $x \in \mathbb{N}$.
What if we had two non-standard model... | https://mathoverflow.net/users/nan | Is a model of arithmetic contained in a model of arithmetic an initial segment? | Let me address the first question.
We can have models of $\mathsf{PA}$, $M\subsetneq N$ with $M$ cofinal in $N$. In fact, $M$ and $N$ do not even need to have the same cardinality. However, one can prove from the Davis-Matiyasevich-Putnam-Robinson theorem (on Hilbert's tenth problem) that already the assumption $M\su... | 8 | https://mathoverflow.net/users/6085 | 160709 | 84,522 |
https://mathoverflow.net/questions/160712 | 3 | $g^{\mu\nu}(x)=\Omega^{2}(x)g'^{\mu\nu}(x)$ is a conformal transformation.
If $g'^{\mu\nu}$ is flat, what kind of $\Omega(x)$ is choosed can make $g^{\mu\nu}$ flat.
We can think about any dimension $n$ and any signature.
In special case, $g'^{\mu\nu}=diag(-1,1,1,1)$.
Thanks!
| https://mathoverflow.net/users/43941 | In what condition is a conformal flat manifold flat? | By looking at how the Riemann curvature tensor changes under a conformal transformation (see, Besse *Einstein manifolds*, 1.159 (b)), and writing $\Omega = e^{-f}$, you will find that $g$ is flat if and only if
$$
\nabla df - df \circ df + \tfrac12 |df|^2 g’ = 0~,
$$
where $df\circ df$ is the symmetric product of $df$ ... | 4 | https://mathoverflow.net/users/394 | 160715 | 84,525 |
https://mathoverflow.net/questions/160735 | 1 | It is known that, for a generic curve $C$, $End(J(C)) \cong \mathbb{Z}$, hence picard number is one. Is this true for Prym Variety?
| https://mathoverflow.net/users/48420 | Picard number of Prym Variety | This is not true for a generic curve, but for a *very general* curve (= outside countably many strict subvarieties of the moduli space). The same is true for Prym varieties, because the closure of the locus of Prym varieties contains the Jacobian locus ("Wirtinger's construction").
| 1 | https://mathoverflow.net/users/40297 | 160736 | 84,535 |
https://mathoverflow.net/questions/160643 | 5 | It's a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented subspaces? More specifically,
Let $E$ be a Banach space with an FDD $(E\_n)$. Suppose that $E$ is isomorphic to a complemente... | https://mathoverflow.net/users/48386 | Complemented subspaces of ultrapowers | It is proved (but not explicitly stated) in
Heinrich, Stefan Ultraproducts in Banach space theory. J. Reine Angew. Math. 313 (1980), 72–104.
The result you want follows from what Heinrich calls "the local reflexivity of ultrapowers", which is Theorem 7.3 in the paper.
EDIT: Here is how to apply Heinrich's Theor... | 5 | https://mathoverflow.net/users/2554 | 160739 | 84,537 |
https://mathoverflow.net/questions/160442 | 8 | This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures of some $D^S$ (a product of $D$ with itself many times), so they **both** carry some topology induced from the Tychonof... | https://mathoverflow.net/users/41291 | Do any Stone-like dualities have some self-dualities hidden inside them? |
>
> any infinite abelian group acquires some canonical non-discrete topology in this way, what is this topology?
>
>
>
The maximal precompact topology (used to define almost-periodic functions). See section 9.9 "Bohr topology on discrete groups" pp. 633-662 in "Topological Groups and Related Structures" by Alex... | 5 | https://mathoverflow.net/users/46855 | 160746 | 84,542 |
https://mathoverflow.net/questions/159955 | 11 | If $n$ is a positive integer, let $r(n)$ denote the number of representations of $n$ as a sum of products of pairs of positive integers. (Here, the order of the terms in the sum does not matter, but products with the same answer are regarded as different, even if they contain the same two numbers in a different order.)... | https://mathoverflow.net/users/48056 | Number of representations of an integer as an (arbitrary) sum of products | Put $F(z)= \prod\_{n=1}^{\infty} (1-z^n)^{-d(n)}$, where $d(n)$ is the number of divisors of $n$. The problem asks
for the asymptotics for
$$
R(N) = \frac{1}{2\pi i} \int\_{|z|=r} F(z) z^{-N} \frac{dz}{z},
$$
where the integral is taken over any circle with radius $r<1$. A standard way of obtaining asymptotics in ... | 8 | https://mathoverflow.net/users/38624 | 160752 | 84,546 |
https://mathoverflow.net/questions/160543 | 12 | I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$ (This kind of group describes its own symmetry).
From $G/Z(G)\cong Inn(G)$ we know [complete group](http://en.wikipedia.org/wiki/Complete_group) is the answer for the simplest case, though this class of group itself is quit... | https://mathoverflow.net/users/40345 | Find finite groups $G\cong Aut(G)$ | Let's call a group $G$ *quasicomplete* if $Z(G) \ne 1$ and $G \cong {\rm Aut}(G)$.
If $H$ is a complete group with a unique subgroup of index $2$, then $H \times C\_2$ is quasicomplete. I have checked that all quasicomplete groups of small order (order less than $768$ so far) apart from $D\_8$ have this form. There a... | 13 | https://mathoverflow.net/users/35840 | 160757 | 84,547 |
https://mathoverflow.net/questions/160755 | 2 | We say a cone at the origin in $R^n$ means that it is an intersection of finitely many halfspaces, i.e.
$$C=\bigcap\_{i\in I}H\_i,\text{ where }|I|<\infty.$$
A cone is strongly convex if $C\cap -C=\{0\}$. Then we can define the linear dimension of $C$ as
$$ldim(C):=dim(C\cap -C).$$
Here we assume the dimension of... | https://mathoverflow.net/users/48006 | The minimal number of halfspaces to represent a convex but non strongly convex cone | This is false in the other cases.
Let $\alpha=2\pi/n$ and observe that the interior of the polygon with $n$ given by $(cos(i\alpha),sin(i\alpha))\_{i=0,\dots,n}$ is given by $n$ affine inequalities (of degree $1$ but with constants), but not less.
Hence, the cone in $\mathbb{R}^3$ generated by $(cos(i\alpha),sin(i\... | 1 | https://mathoverflow.net/users/23758 | 160758 | 84,548 |
https://mathoverflow.net/questions/140517 | 7 | Let $D(\kappa)$ be the discrete space of cardinality $\kappa$, and $\beta D(\kappa)$ its Stone–Čech compactification.
Is there, for every infinite cardinal $\kappa$, a subset $Y \in [\beta D(\kappa)]^{\kappa^+}$ such that $\psi (Y) = \kappa^+$? Can this be proved in ZFC?
For $\kappa = \aleph\_0$, this is true, as a... | https://mathoverflow.net/users/39086 | A question about the Stone–Čech compactification of discrete spaces | Here´s a ZFC answer to your question.
All spaces are assumed to be Hausdorff. Recall that the *tightness $t(x,X)$ of a point $x$ in the space $X$* is defined as the least cardinal $\kappa$ such that for every set $A \subset X$ such that $x \in \overline{A}$ there exists a $\kappa$-sized subset $B$ of $A$ such that $x... | 4 | https://mathoverflow.net/users/11647 | 160760 | 84,550 |
https://mathoverflow.net/questions/160762 | 16 | Let $X\_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph\_0<X\_G<2^{\aleph\_0}$ normal subgroups? Also are there examples where $\aleph\_0<X\_G/\mathord\sim<2^{\aleph\_0}$ where $N \sim M \iff G/N \cong G/M$?
| https://mathoverflow.net/users/nan | Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? | The set of normal subgroups (resp. subgroups) of a countable group $G$ is a closed subset of the Cantor set $2^G$. Hence it is either (at most) countable, or contains a Cantor set and hence has cardinal $2^{\aleph\_0}$.
If moreover $G$ is finitely generated (as assumed in the question), then the equivalence relation... | 25 | https://mathoverflow.net/users/14094 | 160764 | 84,551 |
https://mathoverflow.net/questions/160765 | 7 | I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the presence of initial objects in all the fibers $d\setminus F$ and use the 2-categorical version of Quillen's Theorem A due t... | https://mathoverflow.net/users/18263 | What's an initial object in a poset-enriched category? | There are several possible definitions of initial object in a 2-category $\mathfrak{K}$; which one is appropriate depends on your applications.
1. A 2-category has an underlying ordinary category, so we may just reuse the standard definition of initial object.
2. A 2-category can be regarded as a category enriched ov... | 13 | https://mathoverflow.net/users/11640 | 160771 | 84,554 |
https://mathoverflow.net/questions/160780 | 3 | Let $X= (X\_1,\dots, X\_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ h(x)\leq \mathop{\sup}\_{Q\in \mathcal{D}}\mathbb{E}\_{Q}(h(X)),\qquad \forall h\in \mathcal{A}\},
\end{equation}
... | https://mathoverflow.net/users/44534 | Characterization of a set in $\mathbb{R}^d$ | Suppose $x\in C$. Then we can make $h\_n$ closer and closer to the projection $g$ given by $g(x)=x\_i$, and $\hat h\_n$ closer and closer to $\hat g(x)=-x\_i$, so upon taking limits, $x\_i\le \sup\mathbb E\_Q(X\_i)$ and $-x\_i\le\sup \mathbb E\_Q(-X\_i)=-\inf \mathbb E\_Q(X\_i)$.
(May need to assume $\mathcal D$ is n... | 2 | https://mathoverflow.net/users/4600 | 160781 | 84,557 |
https://mathoverflow.net/questions/160369 | 19 | If two Gaussians disagree on one moment, it seems like this should imply that they have a large variation distance--equivalently, if two Gaussians are close in variation distance it's hard for their moments to differ very much. But I'm having trouble proving this without getting into a terrible mess and ending up with ... | https://mathoverflow.net/users/31437 | Are gaussians with different moments far in total variation distance? | Letting $\mu\_{a,\Sigma}$ be the Gaussian measure with covariance matrix $\Sigma$ and mean $a$. Then (double) the variation distance can be written as
$$
\left\lVert\mu\_{a\_1,\Sigma\_1}-\mu\_{a\_2,\Sigma\_2}\right\rVert\_1 = \max\_f\left\lvert\mu\_{a\_1,\Sigma\_1}(f)-\mu\_{a\_2,\Sigma\_2}(f)\right\rvert,
$$
where the ... | 12 | https://mathoverflow.net/users/1004 | 160783 | 84,558 |
https://mathoverflow.net/questions/160811 | 31 | Admittedly this question is vague. But I hope to convey my point. Feel free to downvote this.
Permit me to define prime number the following way:
A number $n>1$ is a prime if all integers $d$ with $1< d \leq \sqrt{n}$ give non-zero remainders while dividing $n$. This will expose the primes 2 and 3: they qualify as ... | https://mathoverflow.net/users/22878 | What is exceptional about the prime numbers 2 and 3? | I think that in different theories, there is often a "primitive" fact (which is hard to explain further) that lies at the heart of the complication you mention. Let me give examples.
As for the "2 is the oddest prime" credo in *number theory*, often it boils down to the fact that $\mathbb{Q}$ contains exactly the sec... | 37 | https://mathoverflow.net/users/27465 | 160818 | 84,566 |
https://mathoverflow.net/questions/160819 | 3 | Take a set {A, B, C, D, E}, and assume each of the set elements has a random real value attached to it between 0 and 1. For example, this gives us: {A, B, C, D, E} = {0.1, 0.9, 0.4, 0.6, 0.5}. Assume that the set has a preferred order, {A > B > C > D > E}.
Repeatedly taking random pairs of this set, and assigning an... | https://mathoverflow.net/users/48457 | Proof for the emergence of a ranking with paired comparisons | The answer is "yes" for at least one reasonable interpretation of "emerge", and doesn't depend on the starting weights.
For each of $k$ objects, let $X\_i(t)$ be the weight of the $i$th object at time $t$. At each time step, we choose a random pair of objects $1 \leq i < j \leq k$, adding 1 to the weight of object $j... | 4 | https://mathoverflow.net/users/25485 | 160822 | 84,568 |
https://mathoverflow.net/questions/160828 | 3 | I want to solve this system
\begin{align\*}\tag{\*}
x'(s)=x^2(s)+y(s),
y'(s)=x(s)y(s)
\end{align\*}
with initial conditions $$x(0)=t, y(0)=t,$$
where $t\not=0.$
With the help of Maple, the solution is
$$ x \left( s \right) ={\frac {2\,st+2\,t}{-{s}^{2}t-2\,st+2}}, ~y \left( s \right) =2\,{\frac {t}{-{s}^{2}t-2\,st... | https://mathoverflow.net/users/36562 | how to solve this system of nonlinear differential equations | Divide the two equations. You get
$${dx\over dy}={x\over y}+{1\over x}.$$
Multiply by $x$ and set $x^2=u$. You get
$${1\over 2}{du\over dy}={u\over y}+1,$$
a linear equation.
| 6 | https://mathoverflow.net/users/12120 | 160834 | 84,574 |
https://mathoverflow.net/questions/160692 | 4 | I am recently concerned with the following problem: Given a parametrized curve in $\mathbb{R}^6$, what is the condition that it belongs to a hyper-sphere of dimension 5?
The more general question would be what are the conditions for a given parametrized curve in $\mathbb{R}^n$ to belong to a given hyper-surface of $\... | https://mathoverflow.net/users/25516 | Conditions for a curve to belong to a hyper-surface in $\mathbb{R}^n$ | Working this out in full generality is likely to be rather messy and unenlightening, but there are a few remarks about the nature of the problem (and special cases) that can be made, so I'll put them here.
First, for any $n$-manifold $M$, let $\mathcal{C}\_k(M)$ denote the space of $k$-jets of unparametrized, oriente... | 5 | https://mathoverflow.net/users/13972 | 160841 | 84,576 |
https://mathoverflow.net/questions/160626 | 3 | I'm interested in the following:
Given a set $S\_{n,k}$ of binary sequences of length $n$ with $k$ many 1-entries, what is the maximal size of a subset $S'\_{k'}\subset S\_{n,k}$ such that for every subset $T\subset S'\_{k'}$ of size $k'$ there is no set of $k'$ many indices on which all elements of $T$ are $1$?
It... | https://mathoverflow.net/users/41290 | Almost disjoint set (finite case) | So recast the problem as:
$Q(n,k,m,t):$ Given the family of *all* $k$ element subsets of $\{{1,\cdots,n\}}$, how large can a sub-family $\mathcal{F}$ be if we require that no $m$ of them have an intersection of size $t$?
Then your question is $Q(n,k,k',k')$. A very rough bound is that at the absolute largest we co... | 1 | https://mathoverflow.net/users/8008 | 160843 | 84,577 |
https://mathoverflow.net/questions/160561 | 26 | On a general quintic threefold $Y\subset \mathbb P^4$ there are $2875$ lines. The result is classical and one can obtain it via a Chern class computation. But $Y$ is a Calabi-Yau threefold, thus one can start the Donaldson-Thomas machinery, which is exactly a (virtual) count of curves on Calabi-Yau threefolds.
**Ques... | https://mathoverflow.net/users/30827 | The classical number $2875$ of lines on the quintic, as a DT invariant | It is not true that $\int\_{[M]^{vir}} 1 = 2875$ for all the moduli spaces you list. If we let $$N\_n = \int\_{[M(1,0,-1,n)]^{vir}} 1 $$
then
$$Z\_1(q) = \sum\_{n=1}^\infty N\_n q^n = 2875\cdot M(-q)^{e(Y)}\cdot \frac{q}{(1+q)^2}$$
where
$$M(q) = \prod\_{m=1}^\infty (1-q^m)^{-m}$$
What is going on here is that the m... | 28 | https://mathoverflow.net/users/9617 | 160846 | 84,578 |
https://mathoverflow.net/questions/160138 | 3 | I asked this question over on [Math.Stack](https://math.stackexchange.com/questions/706547/a-little-bit-of-intuition-for-corepresentations-from-representations) --- where it has a bounty --- but I didn't really get a helpful response so I am asking the question here. One commenter suggests that I am confusing left- and... | https://mathoverflow.net/users/35482 | A little bit of Intuition for Corepresentations from Representations | Sorry about the left-right confusion (I think I am left-right-illiterate...).
If we have a left action $(g,x)\mapsto g.x$, but write it as a map $b:M\times G\to M$, $b(x,g)=g.x$, is that a left or a right action?
The reason for our choice of notion in the co- or bialgebraic part is time-ordering, we want the time t... | 1 | https://mathoverflow.net/users/36090 | 160854 | 84,583 |
https://mathoverflow.net/questions/160681 | 9 | [Wellpowered](http://ncatlab.org/nlab/show/well-powered+category) means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice of [regular monomorphisms](http://ncatlab.org/nlab/show/regular+m... | https://mathoverflow.net/users/2362 | Is the category of schemes wellpowered? regularly wellpowered? | Now I am told that the category of affine schemes is wellpowered (equivalently, the category of commutative rings is cowellpowered). Let me deduce from this that the category of schemes is wellpowered. So let $X$ be a scheme. For each monomorphism $f:Y\to X$ you can find an affine open covering $(V\_i)\_{i\in I}$ of $Y... | 5 | https://mathoverflow.net/users/7666 | 160871 | 84,587 |
https://mathoverflow.net/questions/160870 | 1 | Upon reading K. Costello's paper on Witten genus, I wonder when, on a smooth (quasi-)projective variety $X$, the canonical bundle $\omega\_X$ admits a left $D$-module structure (other than the Calabi-Yau case).
| https://mathoverflow.net/users/43129 | Projective volume form | If and only if it admits an integrable connection. In the projective case, this means that some power of $\omega \_X$ is trivial, hence a finite covering of $X$ has trivial canonical bundle. In the quasi-projective case it just means that $\omega \_X$ is flat.
| 1 | https://mathoverflow.net/users/40297 | 160874 | 84,588 |
https://mathoverflow.net/questions/160866 | 5 | Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every finitely generated algebra in $\mathcal{V}$ is finite. Equivalently, the finitely generated free algebras $F\_{\mathcal{V... | https://mathoverflow.net/users/5152 | Locally finite varieties which are not finitely generated | The variety of semigroups all of whose elements are idempotents (also called bands) is locally finite but not finitely generated.
| 7 | https://mathoverflow.net/users/15934 | 160875 | 84,589 |
https://mathoverflow.net/questions/160856 | 14 | I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains incomplete, so I thought I would rephrase it a bit (to make the statement clearer) and try again.
Let $\omega\_1$ be the first uncountable ordinal,
same as the set of all countable ordinals.
$\omega\_1=... | https://mathoverflow.net/users/48481 | Order homomorphism functions on $\omega_1$ | *OK, third time's the charm, hopefully:*
First, a lemma:
>
> Fix an arbitrary *successor* ordinal $\beta$ and a nondecreasing regressive map $f$ with domain $\beta-\{0\}$. Then there is a nondecreasing regressive map $d\_f$ with the same domain such that $f\le d\_f$ and $d\_f$ takes on only finitely many values.
... | 3 | https://mathoverflow.net/users/8133 | 160880 | 84,592 |
https://mathoverflow.net/questions/159863 | 11 | The following question is well known:
>
> Consider representations of a given integer as sums of two squares, i.e. solutions to $a^2 + b^2 = n$ in $a,b\in\mathbb Z$ with $n$ fixed. As $n \to \infty$, are the normalized points $\left(\frac a {\sqrt n}, \frac b {\sqrt n}\right)$ uniformly distributed on the unit circ... | https://mathoverflow.net/users/48009 | Angular equidistribution of lattice points on circles | If $p\equiv 1 \pmod 4$ then we may write $p=a^2+b^2$ in a unique way with $a$, $b$ both positive and
$b< a$. Corresponding to such a representation, write $a+bi = \sqrt{p} e^{i\theta(p)}$ with $\theta(p) \in (0,\pi/4)$.
If now $n=p\_1\cdots p\_k$ (assume for simplicity that $n$ is odd and square free) where the $p\... | 8 | https://mathoverflow.net/users/38624 | 160883 | 84,593 |
https://mathoverflow.net/questions/160640 | 5 | Maybe this is silly.
On a bounded set $\Omega\subset\mathbb{R}^n$ consider the equation
$$ \Delta u=f \quad\text{ in $\Omega$}$$
$$ u=0\quad\text{ on $\partial\Omega$}.$$
One has the following elliptic estimates
$$ \| u\|\_{W^{2,p}}\le C\|f\|\_{L^p}. $$
Does one can have the same result if instead of $\Omega\... | https://mathoverflow.net/users/39507 | Elliptic theory on compact manifolds | Consider the case of a sphere. The boundary is empty. A constant $u$ satisfies the equation with $f=0$. You cannot get rid of the $u$ on the right hand side of the inequality in this case.
EDIT: I think this example demonstrates the difficulty of trying to use only local estimates pieced together with partitions of ... | 1 | https://mathoverflow.net/users/42207 | 160884 | 84,594 |
https://mathoverflow.net/questions/160787 | 9 | My friend (who is a medical student!) posed me the following question:
There are 70 people, and you want to split them up into 10 groups of 7 people each. Two such partitions are "compatible" if no two individuals are together in the same group in both. What is the maximal number of compatible partitions you can form... | https://mathoverflow.net/users/48440 | How many ways to partition a group of people? | I would be very surprised if it depended smoothly on the parameters. It is related to questions of block designs (and mutually orthogonal latin squares, finite planes, group divisible designs, etc.) which are famously quirky in their patterns.
Here is an improved lower bound of 5, from Warwick Harvey's page, which he... | 6 | https://mathoverflow.net/users/18086 | 160887 | 84,595 |
https://mathoverflow.net/questions/160879 | 6 | Let $F$ be a nonarchimedian local field. Since the Weil group $W\_F$ is a dense subgroup of $G\_F=Gal(\bar{F}/F)$, it's clear that restriction gives an injection $Irr(G\_F)\rightarrow Irr(W\_F)$ of irreducible (complex) representations.
In some notes of Prasad and Raghuram (found here: <http://www.math.tifr.res.in/~d... | https://mathoverflow.net/users/38495 | Extending a representation from the Weil group to the Galois group | Pick $V \in Irr(W\_F)$. Due to Schur's lemma and the fact that inertia in $W\_F$ acts through a finite quotient, some power of (a fixed choice of) Frobenius acts as a scalar on $V$. Take an appropriate root of that scalar to make up an unramified character $\chi$ of $W\_F$ such that the same power of Frobenius acts tri... | 6 | https://mathoverflow.net/users/5498 | 160892 | 84,596 |
https://mathoverflow.net/questions/160896 | 2 | Consider a (von Neumann algebraic) locally compact quantum group $(M, \Delta, \phi, \psi)$ where the von Neumann algebra $M$ is realized as operators on the Hilbert space $H$. There is a multiplicative unitary $W$ in $B(H\otimes H)$that generates the comultiplication $\Delta$ via
$$
\Delta(x)=W^\*(1\otimes x)W.
$$
F... | https://mathoverflow.net/users/43014 | What is the multiplicative unitary for SU_q(2) (or other quantum groups)? | See
E. Christopher Lance, An explicit description of the fundamental unitary for SUq(2), Communications in Mathematical Physics, Volume 164, Issue 1, pp 1-15, 1994, <http://link.springer.com/article/10.1007/BF02108804>
or also
Janusz Wysoczański, Twisted product structure and representation theory of the quantum ... | 4 | https://mathoverflow.net/users/36090 | 160901 | 84,600 |
https://mathoverflow.net/questions/160899 | 9 | Is there a universe which can always be forced to, which never can be forced from?
| https://mathoverflow.net/users/nan | Is there a Hotel California of set-theoretic geology? | I think there's a serious confusion going on here, around what sort of background we assume.
If we are working within a single model $V$ of $ZFC$, and considering the multiverse(-like structure) consisting of all the inner models of $V$, then there are lots of possibilities. For one thing, if we include $V$ here, the... | 7 | https://mathoverflow.net/users/8133 | 160904 | 84,601 |
https://mathoverflow.net/questions/160886 | 6 | Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an analogue to the Fréchet-Kolmogorov compactness Theorem that provides a tractable way to check if $\mathcal{F}$ is a relatively c... | https://mathoverflow.net/users/46785 | Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds | The result you mention uses the algebraic structure of euclidean space since it involves a form of uniform approxability of the set and its translates. However, there are many criteria for compactness which dispense with this, e.g., one which involves approximability by conditional expectations rather than translates (... | 5 | https://mathoverflow.net/users/48144 | 160905 | 84,602 |
https://mathoverflow.net/questions/160891 | 6 | Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}\_n(k)$. For any partition of $n$, $\lambda=(\lambda\_1,\ldots,\lambda\_r)$, let $A\_{\lambda}$ be the $n\times n$ nilpotent matrix in Jordan form whose blocks correspond to $\lambda$. I'd like to know the dimension of the ni... | https://mathoverflow.net/users/32261 | Dimension of the nilpotent centralizer of a nilpotent matrix | Indeed, this is known.
First, you have some structure on this nilptotent cone. It is the product of an affine space and the nilpotent cone of a reductive Lie algebra.
Namely, embed $A\_{\lambda}$ in a $\mathfrak{sl}\_2$-triple $(A\_{\lambda}, H\_{\lambda}, B\_{\lambda})$. Then $H\_{\lambda}$ yield a characteristic... | 9 | https://mathoverflow.net/users/48503 | 160917 | 84,604 |
https://mathoverflow.net/questions/160893 | 2 | I read the following theorem in a paper without a proof, which I don't understand well. Let $F$ be a global function field, and $v$ be a place of $F$, use $G\_r$ to denote $GL\_r$.
Theorem: For any integer $r\ge 1$, we have
(1) the space of unitary irreducible admissible representations
$\pi$ of $G\_r(F\_v)$ can be... | https://mathoverflow.net/users/1832 | reference help about a result on representation theory | 1. Every unitary $\infty$-dim'l irreducible representation can be writen as inducing a square-integrable representation from a parabolic subgroup with Levi subgroup $G' =G\_{r\_1} \times \dots G\_{r\_2}$. Googling for Bernstein center might help.
2. I guess $[\pi\_0]$ is the family, where you tensor by unramified one-d... | 1 | https://mathoverflow.net/users/10400 | 160922 | 84,606 |
https://mathoverflow.net/questions/160919 | 7 | Let $G$ be a finitely generated group and let $\Gamma=Cay(G, S)$ be the Cayley graph of $G$ with respect to some generating set $S$.
If there exists $S$ such that $\Gamma$ is bipartite, then $G$ has an index $2$ subgroup (e.g. generated by all words of even length). Does the converse hold? (I'd guess not, but I don't... | https://mathoverflow.net/users/1121 | Does index 2 subgroup imply bipartite Cayley graph? | **Is** bipartite or **admits** bipartite? To the first question the answer is obviously "no", to the second, obviously "yes". Let $H\subset G$ be an index $2$ subgroup; take generators $h\_1,h\_2,\ldots$ for $H$ and one more generator $g\notin H$. In this generating set, the Cayley graph is not bipartite (with a few po... | 8 | https://mathoverflow.net/users/44953 | 160923 | 84,607 |
https://mathoverflow.net/questions/160920 | 2 | Let $X$ be a non-singular projective variety, and $D$ a divisor on $X$.
Saying that $D$ has positive (meaning non-zero) Iitaka dimension is equivalent to the function $n \mapsto h^0(\cal{O}(D))$ being strictly increasing for sufficiently large $n$?
Does every effective divisor have positive Iitaka dimension? If not... | https://mathoverflow.net/users/48504 | Divisors with positive Iitaka dimension | Theorem 1.2 in [Takayama 2002](http://www.ams.org/journals/tran/2003-355-01/S0002-9947-02-03068-4/S0002-9947-02-03068-4.pdf) should help compute some examples of effective divisors with 0 iitaka dimension. I think your first question is related to the "exponent" of the divisor (see Lazarsfeld Positivity I 2.1.1 and exa... | 0 | https://mathoverflow.net/users/48505 | 160924 | 84,608 |
https://mathoverflow.net/questions/160813 | 24 | You are given n non-negative integers $a\_1, a\_2 ,, a\_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference between those integers; i.e., if the removed integers are $b$ and $c$, you put $|b - c|$ into the set again. You keep ap... | https://mathoverflow.net/users/33544 | An Interesting Optimization Problem | [Edit: I had originally posted a proof that finding the *minimum* value was NP-hard; in the comments below, Brendan McKay pointed out how to convert that into a proof that finding the *maximum* value is NP-hard, which was the question asked by the original poster. I have edited this to include a complete answer to the ... | 13 | https://mathoverflow.net/users/8938 | 160940 | 84,614 |
https://mathoverflow.net/questions/160885 | 7 | In the 1990s I some times used a computer program with the Max Planck Institute which helped with calculating complicated correspondences for modal logical formulas. Is some program like that available somewhere now?
| https://mathoverflow.net/users/37385 | Is a computer program for correspondence theory available? | My question is answered by Jason Rute's link <http://www.cs.man.ac.uk/~schmidt/tools/>. Similar links would be welcome.
| 2 | https://mathoverflow.net/users/37385 | 160943 | 84,615 |
https://mathoverflow.net/questions/160913 | 20 | Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a mapping:
$$
hol: L\mapsto hol(L)\in Aut(V),
$$
the holonomy map, which gives us the (linear) transformation of vectors a... | https://mathoverflow.net/users/30366 | Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$? | I can try to answer to myself, using the very helpful comments you wrote. (You are all still free to write an answer yourselves!)
*Notice:* For convenience, here I will denote by $hol^k$ what in the question I denoted by $hol^{k+1}$. So we look for
group morphisms $hol^k:\pi\_{k+1}(X)\to \pi\_k(Hol(\nabla))$.
Firs... | 5 | https://mathoverflow.net/users/30366 | 160947 | 84,616 |
https://mathoverflow.net/questions/160942 | 8 | Do all the homomorphisms $\phi: SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2 \to GL(2,\mathbb{R})$ always have that $\phi\_{|\mathbb{Z}^2}$ is trivial, i.e. $\phi(\mathbb{Z}^2)=I\_2$?
| https://mathoverflow.net/users/45092 | The representation of a group | Yes. Consider $\mathbb R^2$ as a representation of the semidirect product, hence as a representation of $\mathbb Z^2$, and write it as an extension of irreducible characters $\chi\_1$, $\chi\_2$. If any nontrivial character $\chi$ appears, then all conjugates of that character by automorphisms of $\mathbb Z^2$ in $SL\_... | 12 | https://mathoverflow.net/users/18060 | 160950 | 84,617 |
https://mathoverflow.net/questions/160861 | 4 | Let $T$ be a stable theory. Let $A$ be a subset or substructure of a model $M$ of $T$. Now in some theories the (model theoretic) algebraic closure of $A$ is already a (sub)model of $T$. For example, in the theory of algebraic closed fields this holds even for the empty set. Now of course there are many model theoretic... | https://mathoverflow.net/users/47687 | Stable examples from Algebra such that the model theoretic algebraic closure of a substructre is no model | This happens in differentially closed fields. The model theoretic algebraic closure of a $A$ is just the field theoretic algebraic closure of the differential field generated by $A$. This will rarely be a differentially closed field. For example, if $A$ is contained in the constants, then its algebraic closure will be ... | 4 | https://mathoverflow.net/users/5849 | 160952 | 84,618 |
https://mathoverflow.net/questions/160941 | 4 | Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [[SW02](http://www.emis.de/journals/JLT/12-2/sinla2e.pdf)]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. Let $N := \operatorname N\_G(T)$. My question is: Can I always achie... | https://mathoverflow.net/users/9947 | Weyl group action on complexified Iwasawa decomposition | You need more data before you can say "the" complexified Iwasawa decomposition, namely the choice of real group of which $G$ is the complexification (as in your reference). For example, if your $GL\_2(\mathbb C)$ was the complexification of $U(1,1)$, its maximal compact is $U(1)\times U(1)$, and you lose.
If $G\_{\ma... | 5 | https://mathoverflow.net/users/391 | 160953 | 84,619 |
https://mathoverflow.net/questions/160873 | 4 | I was wondering is there a sufficient condition (or sufficient and necessary condition) for the existence of positive solutions of the following linear PDE on a closed manifold $(M, g)$,
\begin{equation\*}
-\Delta u +\nabla u\nabla f +hu=0.
\end{equation\*}
where $f, h\in C^{\infty}(M)$.
I got some necessary condit... | https://mathoverflow.net/users/38600 | Existence of positive solutions of a linear PDE on closed manifolds | Your operator is self adjoint on $ L^2 (M , e^{-f}) $:
If your operator is $L$.
Then we have for all fonctions u,v
$\int uLv e^{-f}=\int (<du, dv>+huv) e^{-f}$
Hence , the equation $ Lu=0$ has a positive solution
if and only if 0 is the lowest eigenvalue of $L $.
Let's me elaborate on this
This is a consequence ... | 8 | https://mathoverflow.net/users/48525 | 160954 | 84,620 |
https://mathoverflow.net/questions/160959 | -2 | For any two n-dim vector $v$ and $v'$ define $v\leq v'$ iff for each $1\leq i\leq n$, $v\_i\leq v\_i'$.
Suppose further that the entry of vectors can only take values from $m$ distinct values $\{a\_1, a\_2, \cdots, a\_m\}$.
Claim: for a sequence of vectors $v\_1\leq v\_2 \leq \cdots \leq v\_s$ for $s=mn+1$, we must... | https://mathoverflow.net/users/37612 | Monotonic sequence (edited) | I guess the OP means that the values are ordered linearly, i.e. $a\_1<\dots<a\_m$. In that case the statement is even true for $s=mn-n+2$.
To see this, define the numbers $1\leq c(i,k)\leq m$ such that $v\_i=(a\_{c(i,1)},\dots,a\_{c(i,n)})$. In addition, consider the integers $w\_i:=c(i,1)+\dots+c(i,n)$. Then, for a... | 1 | https://mathoverflow.net/users/11919 | 160960 | 84,621 |
https://mathoverflow.net/questions/160970 | 7 | I am trying to understand the jet bundles but currently I am stuck on the following questions:
Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) manifold $X$. I know that the bundle $J\_k(E)$ of k-jets of $E$ has the structure of a vector bundle over $X$.
I would... | https://mathoverflow.net/users/48531 | Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle | (1) Locally, jets of sections are just collections of $r=\operatorname{rank}E$ jets of functions, hence, the rank of $J\_k(E)$ equals $r$ times the number of multiindices $I=(i\_1,\ldots,i\_n)$ with $|I|\le k$.
(2) It is certainly holomorphic.
(3) It seems to me that $J\_1(E)=T^\*X\otimes E$.
| 6 | https://mathoverflow.net/users/44953 | 160972 | 84,629 |
https://mathoverflow.net/questions/160946 | 3 | Let $A\_1,A\_2,\dots,A\_M$ be given $N\times N$ hermitian matrices. The numerical range is defined as the set
\begin{align}
\mathbb{S}=\{(u^HA\_1u,\dots,u^HA\_Mu)\in \mathbb{R}^M\mid u^Hu=1\}
\end{align}
By toeplitz hausdorff theorem, $\mathbb{S}$ is convex for
1. $M=2$, $N\geq 2$, no conditions on $A\_i$
2. $M=3$, ... | https://mathoverflow.net/users/27249 | Known Results on Convexity of Numerical Range | The only result of this nature that I'm aware of (beyond the two cases involving $M$ and $N$ that you've listed) is Theorem 3.1 in "C.-K. Li and Y.-T. Poon. [Convexity of the joint numerical range](http://people.wm.edu/~cklixx/poon-1.pdf). SIAM J. Matrix Analysis Appl. 21 (1999), 668-678", which basically says that the... | 2 | https://mathoverflow.net/users/11236 | 160976 | 84,632 |
https://mathoverflow.net/questions/110207 | 6 | Are there integral domains which admit ordinal-valued Euclidean functions but not $\mathbb{N}$-valued Euclidean functions?
| https://mathoverflow.net/users/3902 | Properly "transfinitely" Euclidean domains | Yes, they exist. Even if the problem is left open in the the papers of T. Motzkin and P. Samuel cited in [Comparing different Euclidean algorithms on a Euclidean domain](https://mathoverflow.net/questions/143065/comparing-different-euclidean-algorithms-on-a-euclidean-domain)
the problem is solved in
J. Hiblot, De... | 6 | https://mathoverflow.net/users/46855 | 160979 | 84,635 |
https://mathoverflow.net/questions/160981 | 7 | The Leech lattice, found by John Leech in 1965, is a fascinating combinatorial object and gives the best possible lattice sphere packing in $\mathbb{R}^{24}$. This result was proved by Cohn and Kumar in a paper published in the Annals in 2009. In that paper, they conjectured
"We conjecture that this method can be use... | https://mathoverflow.net/users/10898 | Theta series for the Leech lattice | That sentence in the paper is actually a bit vague, in that it doesn't explain exactly what "this method" consists of. There are several more or less plausible interpretations, depending on how closely one wants to imitate what's done in the paper. As far as I know, all of them are open above one dimension.
The least... | 11 | https://mathoverflow.net/users/4720 | 160985 | 84,639 |
https://mathoverflow.net/questions/160975 | 1 | There is a well-known conjecture that all connected Cayley graphs are Hamiltonian.
For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)?
| https://mathoverflow.net/users/39684 | Max order for which connected Cayley Graphs are known to be Hamiltonian | According to :
<http://arxiv.org/pdf/1009.5795v3.pdf>
this is known for all n up to 120 except 72, 96, 108 and 120.
| 5 | https://mathoverflow.net/users/22377 | 160990 | 84,640 |
https://mathoverflow.net/questions/160963 | 4 | What is the asymptotic number of square-free numbers less than $x$ with exactly $k$ prime divisors?
| https://mathoverflow.net/users/40458 | Asymptotics of special square-free numbers | All this is taken from Section 7.4 of Montgomery-Vaughan's *Multiplicative Number Theory I. Classical Theory*.
**Theorem 7.19**: The number of integers up to $x$ with exactly $k$ prime divisors counted with multiplicity is
$$
\frac{F((k-1)/\log\log x)}{\Gamma(1+(k-1)/\log\log x)} \frac{x(\log\log x)^{k-1}}{(k-1)!\log... | 5 | https://mathoverflow.net/users/5091 | 160991 | 84,641 |
https://mathoverflow.net/questions/160955 | 6 | Let $(X,\mathcal{X},\mu)$ be a probability measure system, $T:X\to X$ be a $\mu$-preserving isomorphism on $X$. Let $A\in \mathcal{X}$ such that $\mu(\bigcup\_{n\ge 0}T^nA)=1$, and $\mu\_A$ be the conditional measure of $\mu$ on $A$.
For any point $x\in A$, let $n(x)=\inf\{n\ge1: T^nx\in A\}$ be the first return to ... | https://mathoverflow.net/users/11028 | Mixing property of first return map | Any ergodic transformation induces a mixing transformation: This was first proved by Friedman and Ornstein (Advances in Mathematics 10 (1973), 147-163.
I later prove that you can even induce a transformation with Lebesgue spectrum (see [Annales IHP, 1998](http://archive.numdam.org/ARCHIVE/AIHPB/AIHPB_1998__34_2/AIHPB_1... | 8 | https://mathoverflow.net/users/19603 | 160997 | 84,643 |
https://mathoverflow.net/questions/160908 | 3 | My requirement is to find the point closest to three circles. So lets say the three circles are C1, C2, C3. I want to find the point in the space such that the SUM of its distance from C1, C2 and C3 is MINIMUM.
The distance of a given point from a circle is the distance of the given point from the point that lies on ... | https://mathoverflow.net/users/48501 | Finding closest point to a set of circles | While Douglas's answer will give exact solutions, I would like to suggest two simpler numerical approaches.
**1.** It is clear that the minimum lies in the convex hull of the circles, and that if the sum of distances in the centre of a square of side length $h$ is $S$, its minimum in the square is at least $S-3h/\sqr... | 1 | https://mathoverflow.net/users/nan | 161001 | 84,645 |
https://mathoverflow.net/questions/160548 | 7 | My question is about the van der Corput lemma for
$$
\int\_a^b e^{i t \phi(x)} \psi(x) dx
$$
The version you find everywhere, e.g. on
<http://www.tricki.org/article/The_van_der_Corput_lemma_for_oscillatory_integrals>
requires $\psi$ to be absolutely continuous. However, what are weaker conditions
on $\psi$ such tha... | https://mathoverflow.net/users/48136 | van der Corput lemma for oscillatory integrals | Andreas Seeger pointed out that the fact that an estimate of the form $\frac{C\_{k,\psi}}{t^{1/k}}$ implies an estimate of the form $\frac{C\_k\|\psi\|\_\infty}{t^{1/k}}$ via Banach-Steinhaus. Since the latter is wrong, so is the first and thus
the claim from the paper.
| 4 | https://mathoverflow.net/users/48136 | 161004 | 84,646 |
https://mathoverflow.net/questions/161002 | 10 | Suppose $X=(X\_1,\ldots,X\_n)$ is a Gaussian vector with each entry $X\_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of
$$\mathbb{E}\max\_{1\le i\le n}|X\_i|$$
and
$$\mathbb{E}\max\_{1\le i\le n}X\_i$$
among all correlation structures of $X$.
Many thanks!
John
| https://mathoverflow.net/users/8369 | Maximum of the expectation of maximum of Gaussian variables | For the question with the absolute value, the expectation is maximal when the variables are independent (a special case of the Khatri-Sidak inequality).
For the question witout absolute value, it is natural to conjecture that the maximum occurs when the variables form a regular simplex in $L^2$. I think I saw this co... | 8 | https://mathoverflow.net/users/908 | 161015 | 84,650 |
https://mathoverflow.net/questions/161012 | 5 | What is a simple, elementary proof of the following result?
*A continuously differentiable map from the unit sphere $S^n \subset \mathbb{R}^{n+1}$ $(n > 1)$ to itself that preserves volumes and sends great circles to great circles is an isometry.*
I have a simple, but non-elementary proof of a more general result: ... | https://mathoverflow.net/users/21123 | Volume-preserving projective transformations are isometries | As you must now given your title, there is a first classical argument that shows that (except for $n=1$, a case taking care of itself) if your map sends great circle to great circles, it must be projective, i.e. come from a linear map i.e. be a composition of a linear map with central projection to the sphere.
In par... | 7 | https://mathoverflow.net/users/4961 | 161017 | 84,652 |
https://mathoverflow.net/questions/160989 | 1 | Suppose we have a function $f$, such that $f$ is of some smoothness degree $m$, and $f,f^{(k)} \in L\_1[0,\infty)$ $k=1,...,m$. Now if $f^{(k)}(0) = \lim\_{x\rightarrow\infty}f^{(k)}(x) = 0$ for $k=1,...,m-1$ then we can impose the standard bound on the decay of the Sine and Cosine transforms to be $O(K^{m-1})$ where t... | https://mathoverflow.net/users/48539 | Is $\int_0^\infty \sin(Kx)f_K(x)\,dx$ of larger order than $\int_0^\infty \cos(Kx)f_K(x)\,dx$? | In general, clearly no.
Let $f$ be a fixed function of your class with $f(0) = 1$. Let $f\_K(x) = \frac{1}{K}f(Kx)$. Then $f\_K$ is an admissible sequence by your hypothesis. However
$$\int\_0^\infty \cos(Kx) f\_K(x) \mathrm{d}x = \int\_0^\infty \frac{1}{K^2} \cos(y) f(y) \mathrm{d}y $$
and similarly
$$\int\_... | 1 | https://mathoverflow.net/users/3948 | 161018 | 84,653 |
https://mathoverflow.net/questions/134012 | 5 | Let $N$ be a symplectic submanifold of $M$. Symplectic blow up of $M$ along $N$ is an operation replacing a tubular neighborhood of $N$ with the projectivization of that neighborhood. So it decreases the volume. I have a question on the change of symplectic capacities.
A symplectic capacity $c$ is a function from the... | https://mathoverflow.net/users/11846 | Can symplectic blow up increase symplectic capacities? | The answer is yes.
Let $c$ be the Gromov width except we put $c(M)=\infty$ if $M$ admits an embedding of $B^{2n}(r)$ with $0$ blown up for some $r$.
Using that Gromov width is a capacity it is easy to check 1-3 above, and blowing up $B^{2n}(1)$ at 0 changes this capacity from 1 to $\infty$.
| 2 | https://mathoverflow.net/users/4500 | 161025 | 84,656 |
https://mathoverflow.net/questions/161029 | 4 | Let $X$ be a smooth quasi-projective variety of dimension $n$ over the complex numbers (let us assume it is the complement of a normal crossings divisor in a smooth projective variety).
Then how does the mixed Hodge structure on cohomology with compact support relate to that of the usual cohomology.
By Poincare dua... | https://mathoverflow.net/users/11392 | mixed Hodge structure on Cohomology with compact support | ${}$Hi Chitro. The Poincaré duality pairing $H^k(X) \otimes H^{2n-k}\_c(X) \to H^{2n}\_c(X)$ is a morphism of mixed Hodge structures. So in your example, for instance, $H^{2n-k}\_c(X)$ is pure of weight $2n-2k$.
| 6 | https://mathoverflow.net/users/1310 | 161035 | 84,659 |
https://mathoverflow.net/questions/161044 | 2 | This is a very classical flavoured question, and probabaly it is not difficult. I would like to know the shape of the space of rational degree 7 curves in $P^4$ that pass through 10 fixed points. By "shape" I mean the dimension and possibly also more structure (maybe it is a projective space, maybe a simple variety lik... | https://mathoverflow.net/users/4096 | degree 7 rational curves through ten points in P4 | This is an expansion of Jim Bryan's answer. There is a smooth stack, $\overline{\mathcal{M}}\_{0,10}(\mathbb{P}^4,7)$, parameterizing $10$-pointed stable maps from genus $0$ curves to $\mathbb{P}^4$ with curve class $7[\text{line}]$. If the characteristic is $>7$, this stack is a Deligne-Mumford stack (otherwise it cou... | 6 | https://mathoverflow.net/users/13265 | 161049 | 84,662 |
https://mathoverflow.net/questions/160986 | 36 | For any commutative ring $R$ let $R[x]$ denote the ring of polynomials with coefficients in $R$. Any polynomial $p \in R[x]$ naturally induces a function $\hat{p} :R \rightarrow R$. In some cases, a nonzero polynomial will induce the zero function. For example, with $R=\mathbb{Z}\_6$, the polynomials $x^5 +3x^2+2x$, $3... | https://mathoverflow.net/users/27188 | Rings for which no polynomial induces the zero function | $R$ has a nonzero polynomial that induces the zero function if and only if there are ideals $I$, $J$ such that $I$ is nontrivial, $IJ=0$, and $R/J$ is a ring satisfying the following condition:
There exists $n$ such that, for any $n$ elements $x\_1, \dots x\_n \in R/J$, the discriminant $\prod\_{i<j} (x\_i-x\_j)=0$.
... | 25 | https://mathoverflow.net/users/18060 | 161057 | 84,666 |
https://mathoverflow.net/questions/115676 | 3 | Consider a diffusion process:
$ \text{d}X\_t = f(X\_t)\text{d}t + \text{d}W\_t$
I've seen it given that the log-likelihood of the path is proportional to the Onsager-Machlup functional
$ \int\_0^T \left(\frac{1}{2} \left|\dot{x}-f(x)\right|^2+\frac{1}{2}\nabla\_x\cdot f(x)\right)\text{d}t$.
Where does the secon... | https://mathoverflow.net/users/8916 | log-likelihood of ito diffusion | This is a very old but interesting question. When deriving the Onsager--Machlup functional, the drift divergence term comes from
$$
\lim\_{\epsilon\to 0}
E\left[\exp\left(\int\_0^1 f(x(t)+W\_t)\,dW\_t\right) \,\middle\vert\,
\lVert W\rVert<\epsilon\right] =
\exp\left(-\frac12\int\_0^1 \nabla\cdot f(x(t))\,dt\right).... | 3 | https://mathoverflow.net/users/24041 | 161062 | 84,669 |
https://mathoverflow.net/questions/159242 | 12 | I am interested to identify (ideally classify) nilpotent Lie algebras that occur as nilradicals of parabolic subalgebras in (say) reductive Lie algebras.
For example, all Heisenberg Lie algebras appear as such, the same holds for free 2-step nilpotent Lie algebras. But what about general free n-step nilpotent Lie algeb... | https://mathoverflow.net/users/47725 | Which nilpotent Lie algebras appear as nilradicals of parabolic subalgabras? | Among free nilpotent Lie algebras the possible nilradicals of parabolics are exactly (up to 1 case):
* the abelian ones
* the 2-nilpotent ones (as you mentioned)
* the free 3-nilpotent on 2 generators (5-dimensional).
The latter appears as nilradical of a 9-dimensional parabolic subalgebra of the (14-dimensional) e... | 7 | https://mathoverflow.net/users/14094 | 161064 | 84,671 |
https://mathoverflow.net/questions/161068 | 11 | I come across the following infinite series.
$$
\sum\_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
Thanks for any hints and references!
Anand
| https://mathoverflow.net/users/36814 | Has anyone seen this series? | For natural values of *a*, take $\displaystyle\frac{e^t-1}t=\sum\_1^\infty\dfrac{t^{n-1}}{n!}$, then apply the operator $\bigg(\displaystyle\frac1t\cdot\int\bigg)$ *a* times to it. For $a\not\in\mathbb N$, such as $a=\dfrac14$, welcome to the “marvelous” world of [fractional calculus](http://en.wikipedia.org/wiki/Fract... | 12 | https://mathoverflow.net/users/39602 | 161074 | 84,675 |
https://mathoverflow.net/questions/160999 | 15 | My naive picture of the local Langlands correspondence for $GL(2,\mathbf{C})$ is this. The Weil group of $\mathbf{C}$ is canonically $\mathbf{C}^\times$. On the Galois side then we're looking at 2-dimensional semisimple representations of $\mathbf{C}^\times$, that is, pairs of continuous group homomorphisms $(\chi\_1,\... | https://mathoverflow.net/users/43076 | Local Langlands for $GL(2,\mathbf{C})$ and reducible principal series | This is a common point of confusion, and the OP is on exactly the right track.
A good reference for the representation theory is Chapter 1, Section 6, of Jacquet-Langlands book "Automorphic forms on GL(2)," especially Theorem 6.2 (which has a tiny typo in its statement). This is freely available online, from Langland... | 14 | https://mathoverflow.net/users/3545 | 161075 | 84,676 |
https://mathoverflow.net/questions/161059 | 3 | This may be too easy, but:
>
> Is there a function $f$ on the first quadrant of $\mathbb R^2$ such
> that $$ f(x,1)=x,\qquad f(x,0)=0, $$ and $f$ is convex or concave?
>
>
>
Note there is no solution of the form $f(x,y)=x\cdot g(y)$, since (i) $-f$ is convex iff $f$ is concave, and (ii) the Hessian is
$$
H=\l... | https://mathoverflow.net/users/4600 | Convex extensibility of combination of two lines | There is; for example, the following function is concave:
$f(x,0)=0, f(x,y)=x$ for $y>0$.
| 2 | https://mathoverflow.net/users/24076 | 161080 | 84,679 |
https://mathoverflow.net/questions/161079 | 1 | Suppose $\mathcal T$ and $\mathcal S$ are two compatible Hausdorff topologies on a group $G$ and $\mathcal R$ is a maximum compatible topology on $G$ with $\mathcal R \subseteq \mathcal T\cap \mathcal S$.
Is $(G,\mathcal R)$ Hausdorff?
| https://mathoverflow.net/users/47958 | Hausdorffness inheritance in topological groups | It seems the answer is no. Basically, the $p$-adic and Archimedean topologies on $\mathbb{Q}$ are incompatible enough that the maximal compatible topology contained in both of them is indiscrete. Here is an outline of the proof (I'm heading to bed so I haven't filled in the details):
$\bullet$ Any subset of $\mathbb{... | 3 | https://mathoverflow.net/users/5513 | 161085 | 84,682 |
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