parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/186035 | 9 | Let $\mathcal{C}$ be a [monoidal category](http://en.wikipedia.org/wiki/Monoidal_category) and Mon$\_{\mathcal{C}}$ [the category of monoids](http://en.wikipedia.org/wiki/Monoid_%28category_theory%29#Categories_of_monoids) (also called algebra objects) on $\mathcal{C}$.
**Questions**: are there definitions of image ... | https://mathoverflow.net/users/34538 | Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects | In a nonabelian setting the correct notion of kernel is given by the [kernel pair](http://ncatlab.org/nlab/show/kernel+pair), and the correct notion of cokernel is given by the [cokernel pair](http://ncatlab.org/nlab/show/cokernel+pair). For example, in any category, a morphism $f : a \to b$ is a monomorphism iff its k... | 17 | https://mathoverflow.net/users/290 | 186037 | 92,413 |
https://mathoverflow.net/questions/186032 | 6 | The following two statements appear to be true (but do correct me if I am wrong):
1. The coefficients of a $C^k$ function on the torus $T^n$ decay at least as fast as $x^{-k}$ (where $x$ is some norm on $\mathbb{Z}^n.)$
2. If the coefficients of a Fourier series decay at least as fast as $x^{-k-n},$ then the Fourier ... | https://mathoverflow.net/users/11142 | Regularity of random Fourier series | I consider the case of *independent* Gaussian (or any light-tailed, for that matter) coefficients with variances decaying like $x^{-2k}$ - note that in this case the coefficients themselves decay essentially like $x^{-k}$, up to a logarithmic correction. For this random series the answer will be $C^{k-n/2}$, again up t... | 6 | https://mathoverflow.net/users/22758 | 186043 | 92,417 |
https://mathoverflow.net/questions/36697 | 26 | Let $R$ be the ring which is generated by homeomorphism classes $[M]$ of compact closed manifolds (of arbitrary dimension) subject to the relations that
$$[F]\cdot [B] = [E]$$
if there exists a fibre bundle $F \to E \to B$, and
$$[M] + [N] = [M \cup N]$$
if $M$ and $N$ are of the same dimension. Clearly, $[pt]$ behaves... | https://mathoverflow.net/users/8176 | Ring of closed manifolds modulo fiber bundles | Consider the variation where we ask for smooth manifolds and smooth fiber bundles. Then I claim that $R$ is not finitely generated.
The starting observation is that if $F \to E \xrightarrow{p} B$ is a smooth fiber bundle then
$$0 \to \text{ker}(p) \to T(E) \to p^{\ast}(T(B)) \to 0$$
gives a splitting of the tan... | 6 | https://mathoverflow.net/users/290 | 186046 | 92,418 |
https://mathoverflow.net/questions/186036 | 3 | By a theorem of Eliashberg, two overtwisted contact structures on a 3-manifold which belong to the same homotopy class (as plane fields), are also isotopic (through contact structures). Is there an example known of two non-isotopic but homotopic tight contact structures on a contact 3-manifold?
| https://mathoverflow.net/users/31475 | non-isotopic but homotopic tight contact structure | Yes, there are known examples. See for example Corollary 1.3C of [this paper](http://resolver.sub.uni-goettingen.de/purl?GDZPPN002069210) of Eliashberg and Polterovich for an infinite sequence of examples on $T^3$. There is also a short paper of [Akbulut and Matveyev](http://msp.org/pjm/1998/182-2/p01.xhtml) giving exa... | 7 | https://mathoverflow.net/users/353 | 186049 | 92,419 |
https://mathoverflow.net/questions/186052 | 3 | Recall that a discrete valuation ring $R$ is *excellent* if the extension $\widehat{K}/K$ is separable, where $\widehat{R}$ is the completion of $R$ (with respect to the maximal ideal), $K = \mathrm{Frac}(R)$, and $\widehat{K} = \mathrm{Frac}(\widehat{R})$. Suppose therefore that $R$ is excellent and consider a local i... | https://mathoverflow.net/users/53197 | "Unramified" extension of DVRs and permanence of excellence | No. Let $R'$ be any non-excellent dvr whatsoever, hence of equicharacteristic $p > 0$, and let $t \in R'$ be a uniformizer. Let $R = \mathbf{F}\_p[t]\_{(t)}$. The local inclusion $R \hookrightarrow R'$ has induced residue field extension that is separable since $\mathbf{F}\_p$ is perfect. And of course $R$ is excellent... | 6 | https://mathoverflow.net/users/52824 | 186054 | 92,422 |
https://mathoverflow.net/questions/185990 | 2 | The Theorem 1.5 and 1.6 of
Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.
give a general answer for $H^d(BSO\_n,Z)$ and $H^d(BO\_n,Z)$, in term of $\delta(w\_i)$ and $p\_i$, where $w\_i$ are the Stiefel-Whitney classes.
Here $\delta... | https://mathoverflow.net/users/17787 | Bockstein homomorphism from $H^d(BG,Z_2)$ to $H^{d+1}(BG,Z)$, and Steenrod Square $Sq^1$ | The natural maps from $$0\to Z\to Z\to Z\_2\to 0$$ to $$0\to Z\_2\to Z\_4\to Z\_2\to 0$$ commute with the homomorphisms in the s.e.s. and hence also with the connecting homomorphisms of the long exact sequence, i.e. with the Bockstein homomorphisms.
In particular, the Bockstein of the first sequence corresponds to $S... | 4 | https://mathoverflow.net/users/39082 | 186055 | 92,423 |
https://mathoverflow.net/questions/186047 | 2 | I would like to calculate the mixing time of a continuous time starting from the **rate matrix** and **not** necessarily assuming that the time in between jumps have rate 1 - all I have is the (finite dimensional) transition rate matrix $Q$ for this system. I am fine with other basic assumptions about the chain (e.g., ... | https://mathoverflow.net/users/46629 | Mixing time of a continuous time Markov chain with arbitrary rate matrix | Given a (not necessarily reversible but well-behaved) Markov generator $Q$ with invariant distribution $p$, the corresponding Dirichlet form is
$\mathcal{E}(f) := \frac{1}{2} \sum\_{j,k} p\_j Q\_{jk} (f\_j-f\_k)^2.$
Write
$\lambda\_\* := \inf\_{f} 2\mathcal{E}(f)/\mbox{Var}\_{p}(f),$
where the infimum is over... | 3 | https://mathoverflow.net/users/1847 | 186056 | 92,424 |
https://mathoverflow.net/questions/186042 | 7 | In the last days I came to consider the following question which I'd be happy to see answered by the affirmative:
>
> if $f:X\to S$ is a morphism of schemes which is formally étale, quasicompact,
> universally bijective, universally schematically dominant, with $S$ noetherian,
> is $f$ an isomorphism?
>
>
>
... | https://mathoverflow.net/users/17988 | Is this formally étale morphism of schemes an isomorphism? | [I originally gave what I thought to be a counterexample in the affine case, but I realized it violates universal schematic dominance, so below I give a non-qc counterexample that was originally a comment.]
Let $A = \mathbf{F}\_2^I$ be a product of copies of $\mathbf{F}\_2$ indexed by an infinite set $I$, and let $S ... | 5 | https://mathoverflow.net/users/52824 | 186060 | 92,425 |
https://mathoverflow.net/questions/186059 | 5 | Recently I start learning mapping class group. The Nielsen-Thurston classification says that each element in mapping class group $Mod(S\_{g,n}),g,n\geq 0$ is periodic, reducible, or pseudo-Anosov. Take any element in $Mod(S\_{g,n})$, how to determine the element is in which one of the cases?
| https://mathoverflow.net/users/60706 | Classification of elements in mapping class groups | An effective algorithm can be found in
```
Mladen Bestvina and Michael Handel. Train-tracks for surface homeomorphisms. Topology, vol. 34 (1995), no. 1, pp. 109–140
```
This has been implemented by Peter Brinkman in Xtrain (which you can get via computop.org).
| 7 | https://mathoverflow.net/users/11142 | 186061 | 92,426 |
https://mathoverflow.net/questions/161812 | 4 | Let $\textit{C}$ be the modal logical schema $(\square (\square \alpha \rightarrow \alpha) \wedge \square (\square \lnot\alpha \rightarrow \lnot \alpha))\rightarrow (\square \alpha \vee \square \lnot \alpha)$.
I believe we can establish that $\textit{C}$ corresponds to the second order condition that the accessibilit... | https://mathoverflow.net/users/37385 | On directedness, transitivity and ancestral directedness | Your axiom .2 is $\mathrm{G1}$ from *Hughes&Cresswell*, so I will use this name. They (and also Wikipedia) call the corresponding frames *convergent* (another name for *directed*). Also, I am going to rewrite axiom $\mathrm{C}$ as
$(\Diamond p \land \Box(\Box p \rightarrow p) \land \Box(p \rightarrow \Diamond p)) \righ... | 1 | https://mathoverflow.net/users/37336 | 186062 | 92,427 |
https://mathoverflow.net/questions/186007 | 4 | Let $M$ be a (not necessarily compact)) smooth manifold.
>
> 1.Is there a smooth map $f:M\to \mathbb{R}$ and an open covering $\mathbb{R}=\cup U\_{\alpha}$ such that each $f^{-1}(U\_{\alpha})$ is homeomorphic to $\mathbb{R}^{n}$?
>
>
> 2.Is there a smooth map $f:M \to \mathbb{R}^{k}$, for some $k \in \mathbb{N}$ ... | https://mathoverflow.net/users/36688 | functions which covers(good covers) manifolds | For (1), the answer is almost always no; the combinatorial properties of open covers of $\mathbb{R}$ are far too restricted. Suppose that $M$ is compact and connected and such an $f$ and $\{U\_\alpha\}$ exist. Let the image of $f$ be $[a,b]$. Each $U\_\alpha$ is a union of disjoint intervals; by connectedness, only one... | 3 | https://mathoverflow.net/users/75 | 186063 | 92,428 |
https://mathoverflow.net/questions/186058 | 2 | Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.
Consider a equivalent relation on $R\mathbb{Z}^N$ defined by
$a\sim b$ if $a-b\in \mathbb{Z}^M$ for any $a,~b\in R\mathbb{Z}^N$.
Denote the set of equivalent classes as $(R\mathbb{Z}^N)/\mathbb{Z}^M$.
Simila... | https://mathoverflow.net/users/61257 | Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$? | Put $R$ in [Smith normal form](http://en.wikipedia.org/wiki/Smith_normal_form). While this is usually defined for integer matrices, for a rational matrix $R$, we may write $R = P D Q$, where $P$ and $Q$ are in $GL\_M(\mathbb{Z})$ and $GL\_N(\mathbb{Z})$ respectively, and $D$ is a diagonal matrix with diagonal entries $... | 4 | https://mathoverflow.net/users/9672 | 186064 | 92,429 |
https://mathoverflow.net/questions/186014 | 0 | I am using [A\*](http://en.wikipedia.org/wiki/A*_search_algorithm) (A-Star) to search a graph. A\* algorithm takes advantage of the information $h(x)$, which is a lower bound of the distance between a vertex $x$ and the destination vertex.
In other words: $h(x) \leq d(x,dest)$, where $x$ is a some vertex and $dest$ is ... | https://mathoverflow.net/users/60929 | Using upper bound information in graph search | (The following may well have occurred to you already, but for completeness ...)
If I've fully understood the information that you have (and in context of [A\* pseudocode](http://en.wikipedia.org/wiki/A*_search_algorithm#Pseudocode) as you cited):
1. At any given time, the set `openset` holds the nodes that are cand... | 1 | https://mathoverflow.net/users/56843 | 186073 | 92,433 |
https://mathoverflow.net/questions/186078 | 16 | I am getting "invitations" to join ResearchGate. I am not a member of any other social network, as I consider it a waste of time. Are there good reasons for a mathematician to join ResearchGate? Can anybody provide experiences that speak for or against joining?
| https://mathoverflow.net/users/nan | Any reason I should join ResearchGate? | On [Academia.SE](https://academia.stackexchange.com/) there is a question [ResearchGate: an asset or a waste of time?](https://academia.stackexchange.com/questions/16870/researchgate-an-asset-or-a-waste-of-time).
Opinions there are mostly negative (not only not too beneficial, but also can annoy others [e.g. distingu... | 5 | https://mathoverflow.net/users/9093 | 186081 | 92,437 |
https://mathoverflow.net/questions/186074 | 18 | The polynomials which occur in the [Schwartz-Zippel lemma](http://en.wikipedia.org/wiki/Schwartz%E2%80%93Zippel_lemma) could be defined for any commutative ring, yet the lemma is restricted to fields. This makes it inapplicable for $(1+x^n)=1+x^n(\operatorname{mod}n)$ and similar identities, and feels a bit "unnatural"... | https://mathoverflow.net/users/20781 | Can Schwartz-Zippel be formulated for commutative rings instead of fields? | Your conjecture holds for arbitrary commutative rings.
First, your condition implies that a nonzero degree $d$ univariate polynomial $f\in R[x]$ can only have $d$ roots in $S$. One can show this e.g. by induction on $d$: the case $d=0$ is trivial, and if $f(x)$ of degree $d+1$ has distinct roots $a\_1,\dots,a\_{d+2}\... | 9 | https://mathoverflow.net/users/12705 | 186103 | 92,446 |
https://mathoverflow.net/questions/186099 | 6 | Given a complex semi-simple Lie group $G$, it acts smoothly on the dual $\frak{g}^\*$ of its Lie algebra $\frak{g}$ by the coadjoint action. The orbits of that action are called coadjoint orbits.
A maximal Zariski closed and connected solvable subgroup of $G$ is called a Borel subgroup; and $P$ is a parabolic subgrou... | https://mathoverflow.net/users/49349 | What is the Explicit Relationship between Coadjoint Orbits and Flag Manifolds? | Let $K$ be a maximal compact subgroup of $G$. Then $K$ acts transitively on each $G/P$, and up to $K$-isomorphism, the $K$-spaces obtained exactly match those occurring as coadjoint orbits of $K$ (acting on $\mathfrak k^\*$).
Basic example: $G=SO\_3(\mathbb C)$, with $G/P = \mathbb{CP}^1, pt$. Then $K=SO\_3(\mathbb R... | 6 | https://mathoverflow.net/users/391 | 186104 | 92,447 |
https://mathoverflow.net/questions/186106 | 6 | Let $M$ be a model category and $S \subseteq \operatorname{Mor}(M)$ a set of arrows in (the underlying strict category of) $M$. Recall that the [left Bousfield localization](http://en.wikipedia.org/wiki/Bousfield_localization) $L\_SM$ of $M$ with respect to $S$ is the model category structure on the same underlying cat... | https://mathoverflow.net/users/78 | Does "simplicial" commute with "Bousfield localization"? | Under the injective model structure or the Reedy model structure, the answer is yes: this is a left Bousfield localization.
For any $X \in M$, define $\Delta[n] \otimes X$ to be the element of $M^\Delta$ given by
$$
(\Delta[n] \otimes X)\_k = \coprod\_{(\Delta[n])\_k} X,
$$
with maps induced by the natural maps on co... | 6 | https://mathoverflow.net/users/360 | 186108 | 92,449 |
https://mathoverflow.net/questions/186090 | 7 | Let $\pi:Z\to S$ be a conic bundle over a smooth complex surface $S$. I'd like to know how to prove that $-\pi\_{\*}K\_{Z}^{2}=4K\_{S}+\Delta$, where $\Delta$ denotes the locus in $S$ over which the fibres are singular.
| https://mathoverflow.net/users/58022 | a formula about conic bundles | There are a number of sources, or you could just prove this for yourself. In the context of algebraic geometry, this follows from Proposition 5.1(v) of [Divisor Classes and The Virtual Canonical Bundle for Genus 0 Curves](http://arxiv.org/pdf/math/0602642v1.pdf) by de Jong and myself.
| 2 | https://mathoverflow.net/users/13265 | 186116 | 92,456 |
https://mathoverflow.net/questions/186097 | 3 | Is there a version of Hoeffding's inequality for vector valued random variables?
This seems to be hard to find and I wonder why. I suppose it is difficult to show [Hoeffding's lemma](http://en.wikipedia.org/wiki/Hoeffding%27s_lemma), since the proof for the [inequality](http://en.wikipedia.org/wiki/Hoeffding%27s_ineq... | https://mathoverflow.net/users/42531 | Hoeffding's inequality for vector valued random variables | [Concentration Inequalities for Bounded Random Vectors](http://arxiv.org/abs/1309.0003), by Xinjia Chen (2013):
>
> We derive simple concentration inequalities for bounded random
> vectors, which generalize Hoeffding's inequalities for bounded scalar
> random variables. As applications, we apply the general resul... | 2 | https://mathoverflow.net/users/11260 | 186128 | 92,460 |
https://mathoverflow.net/questions/3103 | 55 | The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers, but what is known about the higher homotopy groups?
| https://mathoverflow.net/users/32 | What are the higher homotopy groups of Spec Z ? | $Spec(\mathbb{Z})$ should only be considered as $S^3$, if you "compactify" that is add the point at the real place. This is demonstrated by taking cohomology with compact support.
The étale homotopy type of $Spec(\mathbb{Z})$ is however contractible (indeed what do you get by removing a point form a sphere?) to see thi... | 46 | https://mathoverflow.net/users/43850 | 186140 | 92,466 |
https://mathoverflow.net/questions/186143 | 5 | It is known that the famous mistake of Iwaniec-Sarnak in their [paper](http://www.jstor.org/stable/pdfplus/2118522.pdf?&acceptTC=true&jpdConfirm=true) of $L^\infty$ norm of eigenfunction of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump of $K$-Bessel function at transition ... | https://mathoverflow.net/users/36735 | Asymptotic behaviour of $K$-Bessel function in transition range | For a published account of the corrected proof, see Section 10 in Blomer-Holowinsky: Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645-681. See especially pages 679-680, where you can also find the precise asymptotics of $K\_{it}$ in the transitional range.
Actually, a few years ago, a co... | 7 | https://mathoverflow.net/users/11919 | 186146 | 92,471 |
https://mathoverflow.net/questions/185916 | 12 | Let $K$ be a number field and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ a short exact sequence of abelian varieties over $K$. Let $h(A)$ denote the logarithmic Faltings height (normalized so that it is invariant upon base change to any finite extension $K'/K$; thus, due to this normalization, one may ... | https://mathoverflow.net/users/53197 | Faltings height in short exact sequences | I think the following should give a counterexample. Let $\mathcal{O}$ be an order in an imaginary quadratic field $K$ and $\mathcal{O}\_K$, the ring of integers. Then it's not too hard to find a (non-split) short exact sequence of $\mathcal{O}$-modules:
$$0 \to \mathcal{O}\_K \to \mathcal{O} \oplus \mathcal{O} \to \mat... | 12 | https://mathoverflow.net/users/949 | 186150 | 92,473 |
https://mathoverflow.net/questions/186151 | 7 | Can anyone sketch for me a bijective proof of the fact that the number of spanning trees of the complete graph on $n$ vertices, $K\_n$ (given by the formula $ t\_n = n^{n-2}$), satisfies $ t\_n = \frac{n}{2} \sum\_{k=1}^{n-1} {n-2 \choose k-1} t\_{k} t\_{n-k} $? Sasha Postnikov suggested that I take a look at <http://m... | https://mathoverflow.net/users/3621 | Bijective proof of an Abel-Hurwitz-type identity | I believe this is the proof Postnikov suggested at CCCC LXI. Let us say that $K\_n$ has vertices $1,2,\ldots,n$. Imagine a spanning tree of $K\_n$ as being rooted at $1$. To any spanning tree $T$, we associate $T'$, the part of the tree at or below the vertex $2$ in this tree, and $T''$, the other part of the tree (whi... | 8 | https://mathoverflow.net/users/25028 | 186152 | 92,474 |
https://mathoverflow.net/questions/186133 | 26 | Let $\Gamma\_g$ be the mapping class group of a closed oriented surface $\Sigma$ of genus $g$. There is a natural surjection $t \colon \Gamma\_g \to \mathrm{Sp}(2g,\mathbf Z)$ which sends a mapping class to the induced action on $H^1(\Sigma,\mathbf Z)$. Composing $t$ with any representation of the symplectic group prod... | https://mathoverflow.net/users/1310 | Which mapping class group representations come from algebraic geometry? | Dan,
Although I'm no longer very active on MO, I thought I'd make a few comments, since your question is an interesting one (and you're not anonymous).
The paper of Looijenga referenced in Igor's answer would show that there are "algebro-geometric" representations of $\Gamma\_g$ which **don't** factor through $Sp(2... | 17 | https://mathoverflow.net/users/4144 | 186155 | 92,475 |
https://mathoverflow.net/questions/185905 | 15 | If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of MacLane & Moerdijk *Sheaves in Geometry and Logic* by compiling out the definition of real numbers in Kripke-Joyal seman... | https://mathoverflow.net/users/49 | The real numbers object in Sh(Top) | Following a [suggestion](http://nforum.ncatlab.org/discussion/6289/when-is-the-internal-real-line-the-external-real-line/?Focus=50368#Comment_50368) of Thomas Holder, we can close the gap as follows:
1. For each object $Y$ in $\mathbf{T}$, there is a pseudonatural local geometric morphism $\mathbf{Sh}(\mathbf{T}\_{/ ... | 14 | https://mathoverflow.net/users/11640 | 186165 | 92,477 |
https://mathoverflow.net/questions/186161 | 5 | Let $G$ be a group and let $\Gamma\_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}\_G(k)=\Gamma\_G(k)/\Gamma\_G(k+1)$ and $$\mathcal{L}\_G:=\bigoplus\_{k\geq 1}\mathcal{L}\_G(k).$$ Then $\mathcal{L}\_G$ has a graded Lie algebra structure induced from the commutator brac... | https://mathoverflow.net/users/15770 | Associated graded Lie algebra of braid groups | Take a look at the following papers
F.R. Cohen - S. Prassidis: "On injective homomorphisms for pure braid groups, and associated Lie algebras", J. Algebra 298 (2006), no. 2, 363–370.
(available at the link <http://arxiv.org/abs/math/0404278>)
and
F.R. Cohen - J. Wu: "On braid groups and homotopy groups", Grou... | 5 | https://mathoverflow.net/users/14653 | 186176 | 92,484 |
https://mathoverflow.net/questions/186174 | 1 | Consider ordinary homology with coefficients in a field. For $X$ a path-connected pointed space, the graded vector space $\bigoplus\_{q\ge 0} H\_q(\Omega X)$ has the structure of an algebra with the multiplication induced as follows: $$ H\_p(\Omega X) \otimes H\_q(\Omega X) \rightarrowtail H\_{p+q}(\Omega X \times \Ome... | https://mathoverflow.net/users/nan | Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism? | No.
Let $X = B\Bbb Z/4$ and $Y = B(\Bbb Z/2 \times \Bbb Z/2)$ be classifying spaces for the two groups of order four. Then, as loop spaces, $\Omega X$ and $\Omega Y$ are homotopy equivalent to the discrete spaces $\Bbb Z/4$ and $\Bbb Z/2 \times \Bbb Z/2$ respectively.
Their rational homology groups are the same, b... | 14 | https://mathoverflow.net/users/360 | 186177 | 92,485 |
https://mathoverflow.net/questions/186182 | 1 | As is well known, every Kaehler manifold can canonically be given the structure of a symplectic manifold. Is it naive to assume that holomorphic vector bundles over a Kaehler manifold can be given the structure of a symplectic vector bundle?
| https://mathoverflow.net/users/42100 | Symplectic and Holomorphic Vector Bundles | This seems to have nothing to do with Kahler manifolds, at least not how you have stated it. Any $C^{\infty}$-complex vector bundle over a paracompact smooth manifold admits a Hermitian metric, by employing a partition of unity. The imaginary part of this Hermitian metric is a skew-symmetric, non-degenerate bilinear fo... | 5 | https://mathoverflow.net/users/49247 | 186185 | 92,488 |
https://mathoverflow.net/questions/101148 | 14 | Robert Penner has proven that, if $A=\{a\_1,\dots, a\_n\}$ and $B=\{b\_1,\dots, b\_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along the curves $a\_i$s, and negative powers of Dehn twists along curves the $b\_j$s, such that each curve in $A\cup B$ app... | https://mathoverflow.net/users/24768 | For which surfaces is Penner's conjecture known to be true? | Shin and Strenner have shown that the conjecture is false when 3g + n > 4.
See <http://arxiv.org/abs/1410.6974>
| 21 | https://mathoverflow.net/users/1335 | 186194 | 92,491 |
https://mathoverflow.net/questions/186196 | 3 | Let $ (X,d) $ be a metric space and consider the function $ T:X \to \mathbb{R}^X$ such that $ T(x)(y) = 1$ if $ y = x $ and $ 0 $ for all other $ y $. Is there a norm on $ \mathbb{R}^X$ such that $ T $ is an isometry? That is, $ ||T(a) - T(b)|| = d(a,b)$ for all $ a,b \in X $.
I'm at a loss to know how to approach th... | https://mathoverflow.net/users/61338 | Finding a norm on $ \mathbb{R}^X $ such that the "natural" embedding of a metric space $ X $ in $ \mathbb{R}^X $ becomes an isometry | Note that your embedding map $T$ actually takes values in the subspace
$\newcommand{\R}{{\mathbb R}}$
$c\_{00}(X;\R)$ of finitely supported functions $X\to\R$. If you merely want a norm on this subspace which makes $T$ an embedding, then this *is* possible via the Arens–Eells construction:
R. Arens, J. Eells, *On em... | 6 | https://mathoverflow.net/users/763 | 186205 | 92,495 |
https://mathoverflow.net/questions/186169 | 5 | Let $X$ be a topological (Hausdorff) space and let $(X\_\alpha)\_\alpha$ be a directed family of subsets. We say that $(X\_\alpha)\_\alpha$ generates the topology of $X$ if a subset $U \subseteq X$ is open iff $U\cap X\_\alpha$ is open in $X\_\alpha$ with respect to the induced topology. Another way of saying this is t... | https://mathoverflow.net/users/58628 | When is the topology generated by countable subsets? | A space $X$ is said to have *countable tightness* if whenever $A \subseteq X$ and $p\in \bar{A}$, there is a countable $B \subseteq A$ such that $p \in \bar{B}$. It is not hard to see that a space has countable tightness if and only if its topology is generated by countable sets (in the sense described in the question)... | 7 | https://mathoverflow.net/users/17836 | 186206 | 92,496 |
https://mathoverflow.net/questions/185732 | 2 | Has there been much work in the setting of Stefan (or general free boundary) problems with some type of nonlocality?
A search on Google and MathSciNet give me only a handful of results which greatly surprises me. Maybe I am searching with the wrong terms?
I'm mostly interested in well-posedness theory of solutions... | https://mathoverflow.net/users/60480 | Nonlocal Stefan problems | Neither do I know about work regarding general nonlocal free boundary problems nor nonlocal Stefan problems (perhaps because they classically are defined as local PDEs and it might be considered somewhat artificial to look at nonlocal versions). However, if you search for "fractional Laplacian and free boundary" you wi... | 1 | https://mathoverflow.net/users/60491 | 186217 | 92,497 |
https://mathoverflow.net/questions/186223 | 0 | Given $C \geq 1$ and $\epsilon > 0$, is there a number $N = N(C,\, \epsilon)$ such that the following holds:
For every set $S \subseteq S^1$ of cardinality $C$, there is a function $f: S^1 \to \mathbb{R}$ such that:
1). $f(z) = \sum\_{n = -N}^N a\_n z^n$ (i.e. $f$ is a Laurent polynomial of in degrees $[-N,\, N]$).... | https://mathoverflow.net/users/30726 | Degree of polynomial approximating characeristic function of finte set | So it suffices to do this for $S=\{1\}$. If you can find a degree $N$ Laurent polynomial doing the job for $\{1\}$ with $L^1$ distance $\epsilon$, $f\_1$ say, then $f\_\zeta(z):=f\_1(\bar \zeta z)$ is another Laurent polynomial of the same degree doing the job for $\{\zeta\}$ with the same $\epsilon$. Suppose you can f... | 2 | https://mathoverflow.net/users/11054 | 186229 | 92,502 |
https://mathoverflow.net/questions/186226 | 4 | To unify the numerical computation and classic computability theory, or to pave a foundation for the numerical computation, mathematicians present variant computation model and computational complexity over reals, for example, the one in the book by Blum,Cucker,Shub, and Smale, or the one in Weihrauch's book Computable... | https://mathoverflow.net/users/14024 | The link and equivalence between variant definition of computation model and computational complexity over reals | The following models are probably the two most well known, and they are not equivalent at the level of *computability*.
1. BCSS
2. standard/Grzegorczyk (same as in Weihrauch's book)
In fact, the function
$$x\mapsto e^x$$
is computable in the standard model, but not in the BCSS model as it is not "semi-algebraic" i... | 3 | https://mathoverflow.net/users/4600 | 186231 | 92,503 |
https://mathoverflow.net/questions/186135 | 2 | Although every connected graph has a spanning tree, the same is not true for hypergraphs: consider the hypergraph on 4 vertices with all possible edges of size 3. You need to pick at least two edges but any 2 edges form a cycle.
On the other hand, if you make the number of hyperedges increase, eventually you get a sp... | https://mathoverflow.net/users/8193 | Maximum number of hyperedges on a hypergraph without a spanning tree | Here is an expansion of my comment. Let $f(n)$ be the maximum number of edges of a connected hypergraph with $n$ vertices which does not have a spanning tree.
For the lower bound, fix a vertex $x$ and begin by taking all hyperedges not containing $x$. Then add a hyperedge $e$ of size $n-1$ containing $x$ to make the... | 1 | https://mathoverflow.net/users/2233 | 186232 | 92,504 |
https://mathoverflow.net/questions/186222 | 10 | Suppose one were to define a group-like structure based on a set $G$
with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$.
One possible definition for the associative law is that
$$
\left< \left< a, b, c \right>, d, e \right>
=
\left< a,\left< b, c, d \right>, e \right>
=
\left< a, b, \l... | https://mathoverflow.net/users/6094 | Natural associative law for a ternary "group"? | What you need is the keyword "polyadic groups". A polyadic group is a non-empty set $G$ equipped with an associative $n$-ary operation $f:G^n\to G$ such that for all $a\_1, \ldots, a\_{n}$ and $b\in G$, the equations
$$f(a\_1, \ldots, a\_{i-1}, x, a\_{i+1}, \ldots, a\_n)=b, (1\leq i\leq n)$$
have (unique) solution for... | 12 | https://mathoverflow.net/users/44949 | 186238 | 92,508 |
https://mathoverflow.net/questions/186247 | 0 | In Thurston's *Three-Dimensional Geometry and Topology*, he gives a recipe for a non-standard model for hyperbolic space which he calls the paraboloid model. I'd like to use the model to try out certain parametrizations of hyperbolic isometries that do not work out in the standard models.
The recipe occurs as Problem... | https://mathoverflow.net/users/14835 | the paraboloid model for hyperbolic space | Your trouble arises because Thurston assumes some familiarity with classical projective geometry.
Put n=3 for example. Consider $\mathbb{R}^3\subset\mathbb{RP}^3$ as an affine chart, say, the chart $\{[1:x\_1:x\_2:x\_3]\mid x\_i\in\mathbb{R}\}$.
The unit sphere is
$$
S=\{[1:x\_1:x\_2:x\_3]\mid x\_1^2+x\_2^2+x\_3^2=1... | 1 | https://mathoverflow.net/users/17294 | 186250 | 92,513 |
https://mathoverflow.net/questions/186255 | 17 | Let $K/\mathbb{Q}$ be a number field. We say that a rational prime $p$ splits in $K$ if there exists a prime $\mathfrak{p}$ of $K$ above $p$ of interia degree $1$.
>
>
> >
> > Is a number field $K$ uniquely determined by the set of primes which split in $K$?
> >
> >
> >
>
>
>
A well-known application of t... | https://mathoverflow.net/users/5101 | Is a number field uniquely determined by the primes which split in it? | See exercises (6.3) and (6.4) of Cassels-Frohlich book on algebraic number theory. In these exercises, an example of two number fields $E,E'$ with the same zeta function is given; therefore, the set of primes which split in $E,E'$ are the same.
This amounts to constructing two subgroups $H,H'$ in a finite group $G$,... | 27 | https://mathoverflow.net/users/23291 | 186257 | 92,514 |
https://mathoverflow.net/questions/186044 | 1 | For $c \in R$ and $k \in N$, $k \geq 3$ let
$p\_{k,c} := n^{\frac{−2}{k+1}}log^c(n)$.
I would like to prove that exists $c\in R$ such that every edge in the random graph $G(n,p\_{k,c})$ lies in a copy of a $k$-Clique with probability $1-\frac{1}{n^\epsilon}$ for some $\epsilon >0$
My approach has so far been to for... | https://mathoverflow.net/users/58940 | Probability of each edge in K-clique | The way you're suggesting will work fine; just don't try to be careful with estimations (otherwise it does get messy).
Given a pair of vertices $u,v$, condition on their being adjacent. Reveal all the edges leaving $u$ and $v$; with exponentially good probability in $p^2n$ you find at least $p^2n/2$ common neighbours... | 0 | https://mathoverflow.net/users/59289 | 186262 | 92,516 |
https://mathoverflow.net/questions/186269 | 3 | Let $q=p^k$ for some prime $p$, and let $GL\_n(\mathbb{F}\_q)$ be the group of invertible matrices over the finite field of $q$ elements. If $\pi$ is the set of primes not equal to $p$, does $GL\_n(\mathbb{F}\_q)$ have a Hall $\pi$-subgroup? I've looked through a lot of the literature but cannot find an explicit answer... | https://mathoverflow.net/users/61372 | Hall subgroups of general linear group | If ${\rm GL}(n,q)$ has a Hall $p^{\prime}$-subgroup, then by Dedekind's Lemma, all its parabolic subgroups would have such a Hall subgroup. Now for $n >2,$ ${\rm GL}(n,q)$ has a parabolic subgroup $P$ with unipotent radical $U$ such that $P/U \cong {\rm GL}(2,q) \times {\rm GL}(n-2,q).$ Now if $P$ has a Hall $p^{\prime... | 5 | https://mathoverflow.net/users/14450 | 186272 | 92,519 |
https://mathoverflow.net/questions/186267 | 1 | In a geometric measure theory (GMT) course I'm following this year, the professor told us about the Aronszajn measure, and asked us to go check by ourselves what it reprensents (the course was about measure theory in infinite dimensional Banach space). Unfortunalety, I didn't find on the internet what I was looking for... | https://mathoverflow.net/users/56191 | Aronszajn measure | A look at [mathnet.ru](http://mathnet.ru) yields one paper by Bogachev on this topic. The article by Csörnyei in Israel Journal of Mathematics, December 1999, Volume 111, Issue 1, pp 191-201 and the references therein give further information.
| 2 | https://mathoverflow.net/users/60435 | 186275 | 92,521 |
https://mathoverflow.net/questions/186277 | 5 | The Gauss-Bonnet theorem characterizes the topology of surfaces by means of their Gaussian curvature.
Do there exist results characterizing the topology of surfaces embedded in $\mathbf{R}^3$ via their mean curvature?
For example, I find it hard to imagine how a topological 2-sphere could be embedded in $\mathbf{... | https://mathoverflow.net/users/59235 | Topology of surfaces and mean curvature | 1. Mean curvature depends on the [choice of the "unit normal"](http://en.wikipedia.org/wiki/Mean_curvature#Surfaces_in_3D_space). If you change the orientation (choose the other unit normal), the computed mean curvature for the sphere is everywhere negative.
2. In any case. Fix $x\_0\in \mathbb{R}^3$. Since the embeddi... | 10 | https://mathoverflow.net/users/3948 | 186280 | 92,522 |
https://mathoverflow.net/questions/186252 | 16 | In his famous article [1] Klein constructs a representation of $G=PSL\_2(\mathbb{F}\_7)$ in $\mathbb{C}^3$ (of which the first invariant polynomial of three variables gives rise to the famous *Klein's quartic*).
All other irreducible representations of $G$ are very simple or natural to obtain: the action on $\mathbb{... | https://mathoverflow.net/users/17980 | Why is Klein's representation of $PSL_2(\mathbb{F}_7)$ hard to obtain? | I guess the answer depends on what you call "geometric"... If you accept some basic algebraic geometry, you can do the following. Consider the homographs
$\ \alpha :z\mapsto z+1\ $ and $\ \beta : z\mapsto -1/z\ $ of $\ \mathbb{P}^1\_{\mathbb{F}\_7}$. We have $\alpha ^7=\beta ^2=(\alpha \beta )^3=1$; it is an easy exerc... | 17 | https://mathoverflow.net/users/40297 | 186281 | 92,523 |
https://mathoverflow.net/questions/186283 | 9 | The axiom of choice has many counterintuitive consequences like the [Banach-Tarski paradox](http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox).
The Hahn-Banach theorem is a consequence of the axiom of choice, but it is weaker.
I would like to know some counterintuitive consequences of the Hahn-Banach theor... | https://mathoverflow.net/users/39421 | Counterintuitive consequences of the Hahn-Banach theorem | To aid future inquiries, let us record here the go-to reference kindly provided by Willie Wong in a comment above: <http://consequences.emich.edu/conseq.htm>. This provides a data base with a utility to search for known implications between weak forms of the axiom of choice.
| 12 | https://mathoverflow.net/users/2926 | 186286 | 92,525 |
https://mathoverflow.net/questions/186285 | 1 | The following situation arises frequently in probability.
Suppose we have two independent continuous random variables $X$ and $Y$ and we consider their sum, $Z=X+Y$. Then the pdf of $Z$ is the convolution of the pdfs of $X$ and $Y$:
$$f\_Z(z)=\int\_{x+y=z}f(x)f(y) dxdy$$
Because a Fourier transform converts the convolu... | https://mathoverflow.net/users/8938 | "Convolution" for Multiplying Random Variables | It's multiplicative convolution.
What you can look at is the Mellin transform that behaves for this multiplicative convolution like the Fourier transform for additive convolution.
| 5 | https://mathoverflow.net/users/61381 | 186293 | 92,527 |
https://mathoverflow.net/questions/186276 | 5 | Is the unit tangent bundle of $S^{n}$ a parallelizable manifold. This is motivated by the fact that $TS^{n}$ is parallelizable?
| https://mathoverflow.net/users/36688 | Is the unit tangent bundle of $S^{n}$ parallelizable? | W.Sutherland. A note on the parallelizability of sphere bundles over sphere. J. London Math. Soc. 39 (1964), 55--62.
The answer is yes.
| 12 | https://mathoverflow.net/users/1465 | 186295 | 92,528 |
https://mathoverflow.net/questions/186292 | -1 | A quantum group $A$ here is an algebraic compact quantum group --- a Hopf\*-algebra with a Haar State. Here $\hat{A}$ is the set of linear functionals $\{\mathcal{F(a)}:a\in A\}$ of the form $\mathcal{F}(a)(b)=h(ba)$.
We have for $f\in A$ a result by Van Daele that
$$f=\hat{\psi}(\hat{S}(\cdot)\mathcal{F}(f)).\qqua... | https://mathoverflow.net/users/35482 | In Algebraic Compact Quantum Groups, is an Irreducible Corepresentation equivalent to its Conjugate? | Before thinking about complicated quantum groups, it's best to first understand the simplest situation. So let's just look at when G is a finite group and A is the Hopf algebra of functionals on G. In the finite dimensional setting A-comodules is the same thing as $A^\*$-modules, and in this case $A^\*$ is just the gro... | 6 | https://mathoverflow.net/users/22 | 186302 | 92,530 |
https://mathoverflow.net/questions/186298 | 14 | Let $G$ and $H$ be two abelian groups and let $n>1, m>1$ be two different integers. How many different spaces $X$ (up to homotopy) do we have with the property $\pi\_{n} X=G$ , $\pi\_{m} X=H$ and $\pi\_{\ast} X=0$ otherwise? is this number finite ?
| https://mathoverflow.net/users/61328 | Eilenberg-Mac lane spaces and a generalization | Assuming $m > n$, there is a method for classifying such spaces using a technique from the Postnikov tower. Namely, such a space has a map $X \to K(G,n)$ inducing an isomorphism on $\pi\_n$, and if we convert this into a fibration it has fiber $K(H,m)$.
Such bundles are classified by a "k-invariant": an element in $H... | 25 | https://mathoverflow.net/users/360 | 186304 | 92,531 |
https://mathoverflow.net/questions/186288 | 2 | Let
$$
(\star) \;\;\;\;\;\;\;\;\;\;\;\; y''+p(x)y'+q(x)y=0,
$$ be a homogeneous linear ODE of order $2$ with $p(x)$ and $q(x)$ complex valued analytic functions in a small neighbourhood of $0$.
**Q:** Is there a "closed formula" involving only
(1) elementary functions
(2) elementary arithmetic operations
(3) I... | https://mathoverflow.net/users/11765 | closed integral formula for a non-zero solution of a homogeneous linear ODE of order 2 | The answer is basically 'no', there is no 'elementary method' involving elementary operations and quadrature (i.e., finding antiderivatives of known holomorphic functions) that will give you a solution to the general second order linear equation with variable coefficients.
For a glimpse at why (an explanation is too... | 2 | https://mathoverflow.net/users/13972 | 186312 | 92,534 |
https://mathoverflow.net/questions/186318 | 5 | The Cauchy identity for double Schubert polynomials states
$$ \mathfrak{S}\_w(x;-y) = \sum\_{\substack{u,v \in S\_n \\ w=v^{-1}u \\ l(w) = l(v) + l(u)}} \mathfrak{S}\_u(x)\mathfrak{S}\_v(y).$$
Is there a combinatorial proof of this identity, akin to the proof of the Cauchy identity for Schur functions via RSK and the... | https://mathoverflow.net/users/25028 | Combinatorial proof of the Cauchy identity for double Schubert polynomials | Yes this is proven in the paper [RC-graphs and Schubert polynomials](http://projecteuclid.org/euclid.em/1048516036), using double rc-graphs (pipe dreams). See section 4 of the link.
| 6 | https://mathoverflow.net/users/934 | 186321 | 92,540 |
https://mathoverflow.net/questions/185961 | 3 | Let $X$ a set, and $\mathcal{P}(X)$ the class of its subset's.
Let $\mathcal{A}\subset \mathcal{P}(X)$, we call a map $L: \mathcal{P}(X)\to[0, \infty]$ $\mathcal{A}$-***regular*** if for any $S\subset X$ we have $L(S)= inf\_{S\subset A\in \mathcal{A}} L(A)$.
A map $L: \mathcal{P}(X)\to[0, \infty]$ is called a *outer-... | https://mathoverflow.net/users/6262 | About the Caratheodory class. | Given a set $X$ and outer measure $L:\mathcal{P}(X)\to [0,\infty]$, is $L$ an $\mathcal{A}\_L$-regular map? The answer is No. Consider the following example:
Let $X = \mathbb{N}$ and define $L:\mathcal{P}(\mathbb{N}) \to [0,\infty]$ by $L(\emptyset) = 0$ and $L(F) = 1$ for $F\subseteq \mathbb{N}$ finite, and $L(S) = ... | 1 | https://mathoverflow.net/users/8628 | 186346 | 92,552 |
https://mathoverflow.net/questions/186330 | 10 | Let $M$ be a compact Kahler manifold. Then the Hodge decomposition says that the Dolbeault dga (of forms of all bidegree) and the de Rham dga on $\Omega\_{\mathbb C}^\bullet(M)$ have isomorphic cohomology groups.
Are there any stronger relationships between these two dgas? For example, are there simple conditions on ... | https://mathoverflow.net/users/4622 | When are the Dolbeault and de Rham dgas homotopy equivalent? | A relevant reference might be [J. Neisendorfer, L. Taylor: Dolbeault homotopy theory. Trans AMS 245 (1978), 183-210.](http://www.ams.org/journals/tran/1978-245-00/S0002-9947-1978-0511405-5/S0002-9947-1978-0511405-5.pdf)
One of the results (Theorem 8) states that compact connected Kähler manifolds are both Dolbeault for... | 6 | https://mathoverflow.net/users/50846 | 186353 | 92,555 |
https://mathoverflow.net/questions/186158 | -1 | **I am retreating back on this statement, after some explorations and calculation**
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this [reference](http://www.mathnet.ru/links/efaa32db37c4234e4e8a2b2d9cd3b330/mzm7569.pdf). But I must mention that my quest for jump discontinuities ha... | https://mathoverflow.net/users/14414 | A question about pointwise convergence of Fourier transform in $N$-dimensions | The answer to your question is: **No**.
Consider the 2D case. Let $\theta = n \pi / 2$ for $n \in \{0,1,2,3\}$. From your definition $S\_r^\theta f = 0$, since on the RHS of its definition you are integrating over a null set. But clearly the corresponding $u\_\theta(x) + u\_{-\theta}(x)$ doesn't always vanish: consi... | 4 | https://mathoverflow.net/users/3948 | 186354 | 92,556 |
https://mathoverflow.net/questions/186329 | 12 | I heard two quotes, one from Alain Connes and an other one from Orlov.
Alain Connes was talking about noncommutative geometry and he said the following:
**" a noncommutative algebra creates its own internal time "**
In a talk by Orlov about Mirror symmetry, he was asked if he considers the monoidal structure on the... | https://mathoverflow.net/users/61328 | Mysterious quotes (at least for me) | Here is a guess about the remark of Orlov. Suppose that one wants to define a good notion of *noncommutative scheme*, given that an affine noncommutative scheme is an associative algebra. Trying to define the spectrum of an associative algebra leads to various problems (c.f. [this answer](https://mathoverflow.net/quest... | 23 | https://mathoverflow.net/users/2503 | 186356 | 92,557 |
https://mathoverflow.net/questions/182769 | 3 | Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a\_i)\_{i=1}^n \in C^n$, $W \in \mathbb{R}\_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow \mathbb{R}$ be defined by
$$J(u):=\sum\_{i=1}^n d(a\_i,u\_i)^2 +\sum\_{i,j}^n W\_{i,j} d(u\_i,u\_j)^2 .$$
Does $J$ pos... | https://mathoverflow.net/users/35593 | Does this squared distance functional have a unique critical point on geodesically convex manifolds? | Found a proof for the half sphere by showing that the functional is convex at every critical point and using poincare-hopf theorem.
| 0 | https://mathoverflow.net/users/35593 | 186361 | 92,559 |
https://mathoverflow.net/questions/186365 | 5 | Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest area" $\int f\,dx$ that can be achieved?
To be more precise, let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying:
* $f$ is supported on $... | https://mathoverflow.net/users/45789 | Largest area of a compactly supported positive definite function | As noted in my comment above, the Poisson summation formula would give
$$
1= f(0) = \sum\_{n\in {\Bbb Z}} f(n) = \sum\_{k\in {\Bbb Z}} {\hat f}(k) \ge {\hat f}(0),
$$
since ${\hat f}(k) \ge 0$ by assumption. Since ${\hat f}(0) = \int\_{-1}^{1} f(x) dx$, this proves the desired bound.
If you are worried about usi... | 8 | https://mathoverflow.net/users/38624 | 186370 | 92,561 |
https://mathoverflow.net/questions/186372 | 4 | Suppose we have two models of set theory, $U$ and $V$ which have the same $\Bbb N$. Is it possible that there is a set $A\subseteq\Bbb N$ such that, in $U$, this set is computable, i.e. there is a number $e\in\Bbb N$ which is index of Turing machine recognizing $A$, but it doesn't hold in $V$? I believe answer to this ... | https://mathoverflow.net/users/30186 | Are there sets which are computable in one model, but uncomputable in another? | If two models of set theory $U$ and $W$ have the same arithmetic
structure $\langle\mathbb{N},+,\cdot,0,1,<\rangle$, then as you
observe the operation of Turing machines will be absolute between
$U$ and $W$, and so they will think precisely the same sets are
decidable. Furthermore, they will agree on the members of any... | 7 | https://mathoverflow.net/users/1946 | 186374 | 92,562 |
https://mathoverflow.net/questions/186339 | 39 | Apologies for the vague title and soft question. According to Etingof, [Igor Frenkel](http://www.math.sunysb.edu/frenkel60/Frenkel/Poster2/igorwork.pdf) once suggested that there are three "levels" to Lie theory, which I guess could be given the following names:
* **No loops:** here we study a simple Lie algebra $\ma... | https://mathoverflow.net/users/290 | Why can't we take three loops? | To elaborate on Kevin's excellent answer, one can account for the current absence of "higher loop" representation theory using physics. Namely, all of the representation theoretic structures you mention fit in very naturally into the study of gauge theory, specifically 4-dimensional $\mathcal N=2$ gauge theories. These... | 17 | https://mathoverflow.net/users/582 | 186388 | 92,566 |
https://mathoverflow.net/questions/185720 | 3 | My question is the following:
>
> Does knot Floer homology detect the genus of null-homologous knot in rational homology spheres?
>
>
>
If the answer is yes, I would like to have a reference for the statement of the result and the proof.
| https://mathoverflow.net/users/17492 | Does knot Floer homology detect knot genus in rational homology spheres? | Link Floer homology detects the Thurston norm, see [this paper](http://msp.org/gt/2009/13-5/gt-v13-n5-p09-s.pdf) by Yi Ni.
**EDIT**: There are some [lecture notes](http://arxiv.org/abs/1411.4540) that Robert Lipshitz just put up on the arXiv, where he discusses the problem you're interested in. I haven't had time to ... | 2 | https://mathoverflow.net/users/13119 | 186395 | 92,571 |
https://mathoverflow.net/questions/185922 | 8 | Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$, and $B$ is $\kappa$-centered. Let $G \subset A$ be a generic ultrafilter. Is $B/G$ $\kappa$-centered in $V[G]$?
Naively, we might attempt to prove it as follows. Let $\{ F\_\alpha : \alpha < \kappa \}$ be a collection of filters and in $V[G]$ let... | https://mathoverflow.net/users/11145 | centeredness in forcing iterations | Let $P$ be the forcing for adding a Suslin tree $\mathring{T}$ by finite conditions. Then both $P$ and $P \star \mathring{T}$ are forcing isomorphic to adding $\omega\_1$ Cohen reals (so $P \star \mathring{T}$ is sigma-centered). See lemma 5.6 [here](http://arxiv.org/pdf/math/9506208v1.pdf).
| 6 | https://mathoverflow.net/users/2689 | 186402 | 92,572 |
https://mathoverflow.net/questions/186328 | 4 | I hope my question is not too trivial, but unfortunately i'm just starting to study toric varieties.
Let's take $X$ a lattice and $\sigma\subset X^\*$ a strongly convex rational polyhedral cone, so that, given $k$ an integrally closed field of char $0$, $P=Spec(k[\sigma^\*\cap X])$ is a toric variety on which acts th... | https://mathoverflow.net/users/60675 | Reduced stabilizers of torus action on toric variety | I can at least explain how this works if $k = \mathbb{C}$. As explained in Fulton's intro book, closed points in the affine toric variety $P$ are in one-to-one correspondence with semigroup homomorphisms $u: \sigma^\* \cap X \rightarrow \mathbb{C}$, where $\mathbb{C}$ is regarded as a semigroup under multiplication, an... | 1 | https://mathoverflow.net/users/61411 | 186412 | 92,574 |
https://mathoverflow.net/questions/186360 | 8 | We know that the Borel group cohomology (group cohomology of measurable functions) of a group $G$, ${\cal H}\_B^d(G,Z)$, is given by the cohomology of the classifying space: ${\cal H}\_B^d(G,Z)=H^d(BG,Z)$. However, ${\cal H}\_B^d(G,R/Z)\neq H^d(BG,R/Z)$, since for example: $H^d(BU(1),R/Z) = R/Z$ for even $d$ and $H^d(B... | https://mathoverflow.net/users/17787 | The relation between group cohomology and the cohomology of the classifying space | Yes, there is a relation between these cohomology groups. The results you are referring to are contained for instance in Austin-Moore "Continuity properties of measurable group cohomology" <http://arxiv.org/abs/1004.4937> and in parts already in Wigner "Algebraic cohomology of topological groups" [[http://projecteuclid... | 8 | https://mathoverflow.net/users/5937 | 186415 | 92,575 |
https://mathoverflow.net/questions/186394 | 6 | There is a very nice theory of gradient flows in metric spaces by [Ambrosio, Gigli and Savaré](http://www.springer.com/birkhauser/mathematics/book/978-3-7643-8721-1). One particularly important application is the quadratic Wasserstein setting, where the metric space in question is $(\mathcal{P},\mathcal{W}\_2)$, $\math... | https://mathoverflow.net/users/33741 | Reference request: Wasserstein metric spaces for non linear weights/mobility? | Yes, this issue has been considered. You can start having a look at `A new class of transport distances between measures' by Dolbeault, Nazaret and Savaré (<http://link.springer.com/article/10.1007%2Fs00526-008-0182-5>).
The basic assumption needed for the theory to work is that the mobility $\eta$ is increasing and... | 8 | https://mathoverflow.net/users/58975 | 186423 | 92,578 |
https://mathoverflow.net/questions/186390 | 2 | Let $(a\_n)$ be the [A001921](http://oeis.org/A001921) sequence
$$
a\_0 = 0,\ a\_1 = 7, \quad a\_{n+2} = 14a\_{n+1} - a\_n + 6.
$$
Let $(b\_k)$ be the (almost)"tower-of-squares" sequence defined by
$$
b\_0=2, \quad b\_{k+1}=2b\_k^2-1
$$
Is it true that $a\_{2^kn+2^{k-1}-1}$ is always divisible by
$b\_k$, fo... | https://mathoverflow.net/users/10341 | Tower-of-squares sequence divides linear recurrent A001921 sequence? | The elements of your sequence are
$$a\_n=\left(\frac{\alpha^n-\beta^n}{2\sqrt{3}}\right)\left(\frac{\alpha^{n+1}+\beta^{n+1}}{2}\right)$$
where $\alpha=2+\sqrt{3}$ and $\beta=2-\sqrt{3}$. Notice that both factors are integers. We can also compute that
$$b\_n=\frac{\alpha^{2^n}+\beta^{2^n}}{2}.$$
Now your statement that... | 10 | https://mathoverflow.net/users/2384 | 186427 | 92,581 |
https://mathoverflow.net/questions/186408 | 10 | The set of 5-tuples of lines in $\mathbf{P}^3$ is parametrized by the 20-dimensional product of Grassmannians $G(2,4)^{\times 5}$. The set of cubic surfaces is parametrized by a 19-dimensional projective space.
I can present a line in $\mathbf{P}^3$ as a $2 \times 4$ matrix (the projectivization of the row span) and ... | https://mathoverflow.net/users/1048 | When does a cubic surface pass through five lines? | It's a $20 \times 20$ determinant. Take each $2 \times 4$ matrix
$$L = \begin{pmatrix}
s & t & u & v \\
w & x & y & z \\
\end{pmatrix}$$
and turn it into the $4 \times 20$ matrix
$$M := \begin{pmatrix}
s^3 & s^2 t & s^2 u & \cdots & v^3\\
3 s^2 w & 2 swt+s^2x & 2swu+s^2 y & \cdots & 3 v^2 z \\
3 s w^2 & 2 swx+w^2 t & 2... | 11 | https://mathoverflow.net/users/297 | 186429 | 92,582 |
https://mathoverflow.net/questions/186433 | 3 | Let $f\_n \in L^2[0,1]$ be an orthonormal sequence and let $c\_n \in \mathbb C$ be such that $\sum\_{n = 1}^{\infty} |c\_n|^2 < \infty$. Does this imply that the sequence $\sum\_{n = 1}^{\infty}c\_nf\_n$ converges pointwise almost everywhere on $[0,1]$?
I guess that this is not true. However, it does hold in some int... | https://mathoverflow.net/users/14233 | A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm | As you remark, this is false in the context of general orthonormal sequence. Generally one has the estimate (the Rademacher-Menshov theorem)
$$|| \max\_{\ell \leq n} |\sum\_{i=1}^{\ell} a\_i \phi\_i| ||\_{L^2} \lesssim \log n (\sum\_{i=1}^{n} |a\_i|^2)^{1/2}$$
where, in general, the factor of $\log n$ is optimal.
... | 10 | https://mathoverflow.net/users/630 | 186435 | 92,584 |
https://mathoverflow.net/questions/186430 | 0 | I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.
I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove the relative interior of $pq$ from $\mathbb{R}^2$, pull apart the opening and consider the new "boundary" which I will d... | https://mathoverflow.net/users/61430 | Creating topological spaces with portals | Based on your comment at [Creating topological spaces with portals](https://mathoverflow.net/questions/186430/creating-topological-spaces-with-portals#comment466017_186430), I think that I may have an answer. However, it seems to me that you are asking me to capture the right intuitive generalisation of a concept that ... | 0 | https://mathoverflow.net/users/2383 | 186437 | 92,585 |
https://mathoverflow.net/questions/186212 | 2 | I have a regular map $f : X \to Y$ and a subvariety $V \subset X$ (assume everything is smooth). I'd like to study this map in an infinitesimal neighborhood of $V$, but I'm not sure of the right notion of "local coordinates" in this setting.
I suppose I want to look at the completion of the local ring $\widehat{\math... | https://mathoverflow.net/users/nan | Local coordinates along a subvariety | I presume that by $\widehat{\mathcal{O}}\_{X, V}$ you mean the ring obtained by completing the local ring at the generic point of $V$ at its maximal ideal.
You're right that your description ($\widehat{\mathcal{O}}\_{X, V}=K(V)[[x\_1, ..., x\_k]]$) follows immediately from the Cohen structure theorem (see e.g. 10.149... | 3 | https://mathoverflow.net/users/6950 | 186438 | 92,586 |
https://mathoverflow.net/questions/173691 | 24 | To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of manifolds on its homotopy groups. By Pontryagin-Thom, $M$ is the sphere spectrum $S$, which is even the initial ring spectrum.... | https://mathoverflow.net/users/290 | From the perspective of bordism categories, where does the ring structure on Thom spectra come from? | Thanks to a very helpful discussion with Clark Barwick in the [homotopy theory chat](http://chat.stackexchange.com/transcript/message/18511392#18511392), I think I now understand what's going on here. In particular, the ring spectrum structure on the sphere spectrum $\mathbb{S}$ *does* come from a monoidal structure on... | 15 | https://mathoverflow.net/users/290 | 186440 | 92,587 |
https://mathoverflow.net/questions/186439 | 16 | I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
* distinguishing certain permutations of $X$ as structure preserving;
and
* distinguishing certain test functions (here I think ... | https://mathoverflow.net/users/2383 | Does the linear automorphism group determine the vector space? | The dimension of $V$ is the least non-negative integer $n$ such that there exist $v\_1,\dotsc, v\_n$ in $V$ such that there exists a unique $g\in G:=GL(V)$ that fixes each of $v\_1,\dotsc,v\_n$. So the isomorphism class of $V$ is determined by the group action of $G$ on $V$.
| 11 | https://mathoverflow.net/users/9672 | 186442 | 92,589 |
https://mathoverflow.net/questions/186471 | 4 | $F\_n$ are the Fibonacci numbers.
In [On computing factors of cyclotomic polynomials p.1](http://arxiv.org/abs/1004.5466) for odd square-free $n>1$ the cyclotomic polynomial $\Phi\_n(x)$
satisfies:
$$ 4 \Phi\_n(x)=A\_n(x)^2 - (-1)^{(n-1)/2} n B\_n(x)^2 \qquad (1)$$
and Brent gives algorithm for computing $A\_n,B\... | https://mathoverflow.net/users/12481 | Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$? | The answer to the first question is yes, although the argument I give below is not along the lines that you were originally thinking. I will show that $p$ is a square modulo $q$ for every prime factor $q$ of $F\_{p}$, provided $p \equiv 1 \pmod{4}$ and $p > 5$.
As you mention, $\zeta = \frac{1 + \sqrt{5}}{1 - \sqrt{5... | 8 | https://mathoverflow.net/users/48142 | 186476 | 92,601 |
https://mathoverflow.net/questions/186458 | 1 | Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.
Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It does not imply $X\to S$ being generically trivial.)
Is $X(S)$ non-empty?
The answer is positive if $S$ is of dimensio... | https://mathoverflow.net/users/61448 | Are generically trivial finite unramified morphisms trivial | The answer is yes in far more generality; see Prop. 6.2 of Liu-Lorenzini-Gabber <http://arxiv.org/abs/1404.5366>
In your case you can take the closure of a generic section and use Zariski's main theorem as Jason Starr alluded to above.
| 2 | https://mathoverflow.net/users/4333 | 186480 | 92,603 |
https://mathoverflow.net/questions/101882 | 16 | A Casson tower is obtained as follows: Start with a properly immersed disk in $\mathbb{B}^4$ - a regular neighborhood of such a disk is called a kinky handle. The boundary of the core disk (necessarily in $\mathbb{S}^3$) is called the attaching circle. At each point of self-intersection of the core immersed disk, we ha... | https://mathoverflow.net/users/14006 | Shortest Casson tower containing a slice disk for the attaching curve | A [recent paper of Cha-Powell](http://arxiv.org/abs/1411.1621) shows that Casson towers (and more generally distorted Casson towers) of height four contain slice disks for the attaching curve. This appears to be all that is known at present.
The relationship between Casson towers containing slice disks for the attac... | 3 | https://mathoverflow.net/users/14006 | 186491 | 92,605 |
https://mathoverflow.net/questions/186481 | 22 | In the 90-91 pager
"A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY",
Candelas, de la Ossa, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically constructed from a sub-family of quintic CY 3folds, as a "mirror" to quintic an did some calculations on the mirror family to... | https://mathoverflow.net/users/5259 | How mirror of quintic was originally found? | The history of this is as follows. In the paper by Candelas, Lynker and Schimmrigk there are two weighted hypersurfaces whose cohomology is mirror to that of the quintic. These therefore are two potential candidates for the mirror quintic. The question then was how to decide whether they provide mirror partners to the ... | 10 | https://mathoverflow.net/users/6013 | 186513 | 92,614 |
https://mathoverflow.net/questions/186489 | 6 | Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL\_2(\mathbb F\_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F\_q$-space of of (2,2)-matrices of trace $0$ with coefficients in $\mathbb F\_q$, and $G$ acts on it by conjugation. So $V$ is an irreducible representati... | https://mathoverflow.net/users/9317 | Restriction of scalars for the adjoint representation of $SL_2(\mathbb F_q)$ | There is some textbook literature which essentially covers the issues raised here, though it often deals with more general situations. (Over finite fields life is simpler, since Schur indices are 1.) See for example Curtis & Reiner (1962), Section 70, and also the book *Character Theory of Finite Groups* by Isaacs (rep... | 4 | https://mathoverflow.net/users/4231 | 186521 | 92,619 |
https://mathoverflow.net/questions/186486 | 2 | Let $g$ be the golden number (or another algebraic integer in $(0,1)$ that fullfills an equation with coefficients $\pm 1$). Consider the random walk on $\mathbb{R}$ starting with $0$ and walking $g^{n}$ with probability $1/2$ left or right in step $n+1$. What is the probability to return to $0$ exactly $m$-times. Here... | https://mathoverflow.net/users/23542 | Random walks with exponential decreasing steps | In the case of the golden number $g = (\sqrt{5}-1)/2$, we have
$g^n = (-1)^n (F\_{n-1} - F\_n g)$ where $F\_n$ is the $n$'th Fibonacci number.
Let $a\_i, i=0,1,2,\ldots$ be $+1$ if the $i+1$'th step is to the right, $-1$ if it is to the left. Then we return to the origin after $n$ steps iff
$\sum\_{i=0}^{n-1} a\_i F\_... | 2 | https://mathoverflow.net/users/13650 | 186525 | 92,621 |
https://mathoverflow.net/questions/186403 | 1 | Let us say we have a n \* n system of equations like KU=F where K is a n\*n matrix and U and F are n\*1 vectors. K and F are defined and the final goal is to find U values.
K is a sparse banded matrix and some of its components depends on U components. This dependence makes the whole problem nonlinear.
Since the syst... | https://mathoverflow.net/users/61420 | Nonlinear system of equations whereas most of the equations are linear. How to minimise operation? | Let us write your matrix equation as the following block format, $$\begin{bmatrix}A & B\\
C & D(x,y)
\end{bmatrix}\begin{bmatrix}x\\
y
\end{bmatrix}=\begin{bmatrix}b\\
c
\end{bmatrix},$$
in which we have reordered the nonlinear portion of the problem into the block $D$. We highlight its nonlinearity by giving it argum... | 1 | https://mathoverflow.net/users/60984 | 186530 | 92,624 |
https://mathoverflow.net/questions/162656 | 9 | I want to prove the following statement:
>
> For any two points $x$ and $y$ in an irreducible variety $X$, there is a one-dimensional, irreducible subvariety $C\subseteq X$ containing $x$ and $y$.
>
>
>
Both [here](https://mathoverflow.net/questions/62843/path-connectedness-of-varieties/62883#62883) and [here]... | https://mathoverflow.net/users/9947 | Proving that any two points on a variety can be joined by a curve; why does Bertini apply? | Corollary 1.9 of <http://www-math.mit.edu/~poonen/papers/bertini_irred.pdf> proves your statement over an arbitrary field $k$, even if $k$ is finite. (It has "geometrically irreducible" in place of "irreducible", but this just makes the statement more difficult: the irreducible version follows by applying the geometric... | 13 | https://mathoverflow.net/users/2757 | 186532 | 92,626 |
https://mathoverflow.net/questions/186528 | 8 | Let $u$ be a smooth function on $\mathbb S^2$, and assume that for every killing vector field $V$ on $\mathbb S^2$.
$$\int\_{\mathbb S^2} V(u) x\_j dS=0\text{,}\forall j=1,2,3$$
Is $u$ necessarily constant?
| https://mathoverflow.net/users/42326 | Killing vector fields on sphere | The answer is "no".
Choose a basis $V\_1, V\_2, V\_3$ of Killing fields.
Note that
$$\int\limits\_{\mathbb S^2} V\_iu\cdot x\_j\cdot d\,\mathrm{area}
=
-\int\limits\_{\mathbb S^2} u\cdot V\_ix\_j\cdot d\,\mathrm{area}$$
Threfore you can take any $u$ which is orthogonal to each of 9 functions $s\_{i,j}=V\_ix\_j$.
... | 6 | https://mathoverflow.net/users/1441 | 186537 | 92,629 |
https://mathoverflow.net/questions/186550 | 8 | I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation.
I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble getting a nice easy and fast algorithm for a power distribution. The gamma distribution should do, but the article in w... | https://mathoverflow.net/users/47018 | Algorithm to produce random number with a gamma distribution | The difficulty mentioned in Wikipedia refers to gamma distributions with small shape parameter; this has been addressed in [arXiv:1302.1884](http://arxiv.org/abs/1302.1884):
>
> The gamma distribution with small shape parameter can be difficult to
> characterize. For this reason, standard algorithms for sampling f... | 8 | https://mathoverflow.net/users/11260 | 186554 | 92,635 |
https://mathoverflow.net/questions/186551 | 5 | This was asked in Math Stackexchange [here](https://math.stackexchange.com/questions/1009004/dirichlet-characters-as-eigenvectors) but generated no comments or answers. I have slightly edited the original question with the comment in the fourth paragraph and the explicit matrix example at the end.
I have a very concr... | https://mathoverflow.net/users/17773 | Dirichlet Characters as Eigenvectors | I believe the eigenvectors are the ones you guessed, but in your second example, the dimensions of some of the eigenspaces are larger than one. I would guess that Mathematica chose a basis for those eigenspaces different than the eigenvectors coming from the multiplicative Dirichlet characters. By the way, if you view ... | 4 | https://mathoverflow.net/users/50426 | 186563 | 92,636 |
https://mathoverflow.net/questions/186561 | 15 | In the sense of W. Thurston [here](http://en.wikipedia.org/wiki/Geometrization_conjecture#The_eight_Thurston_geometries), there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3.
Question: **How many different geometries (in the sense of Thurston) do we have in dimension 4 ?**
| https://mathoverflow.net/users/61328 | Thurston geometries in dimension 4 | The 4-dimensional geometries were classified in the unpublished thesis of Filipkiewicz, which is available [here](http://wrap.warwick.ac.uk/954/).
| 22 | https://mathoverflow.net/users/317 | 186565 | 92,638 |
https://mathoverflow.net/questions/186560 | 6 | I refer to Greenberg's wonderful 2010 MAA article "[Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries](https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Greenberg2011.pdf)". There, and in [his book](http://rads.stackoverflow.com/amzn/click/0716799480), Gr... | https://mathoverflow.net/users/nan | Reverse plane geometry, anyone? | There are exactly $2^{\aleph\_0}$ elementary equivalence classes of Hilbert planes.
Let $P$ be a subset of the set of odd primes. Let $K\_P$ be the smallest field extension of $\mathbb{Q}(\{2^{1/p}:p \in P\})$ in $\mathbb{R}$ that is closed under taking square roots of positive elements, and let $H\_P$ be the plane $... | 8 | https://mathoverflow.net/users/2757 | 186568 | 92,639 |
https://mathoverflow.net/questions/186549 | 3 | I guess that the answer to the following question is both well known and easy. But I was unable to solve the exercise.
Consider a unital $C^\*$-$\,$algebra $\mathcal A$ and and a proper unital sub-$C^\*$-$\,$algebra $\mathcal B\subset\mathcal A$. Let also $\varphi$ be a state on $\mathcal B$. Assume that $\varphi$ ha... | https://mathoverflow.net/users/37371 | States with a unique state extension | No; consider $\mathbb C \oplus \mathbb C \oplus \mathbb C \subset \mathbb C \oplus M\_2(\mathbb C)$ (in the obvious way) with the state $\varphi(x\_1,x\_2,x\_3)= \frac12(x\_1 + x\_2)$. Then, the extension of $\varphi$ is unique and $\varphi$ is not pure.
| 4 | https://mathoverflow.net/users/8176 | 186571 | 92,640 |
https://mathoverflow.net/questions/186570 | 2 |
>
> Assume that $A$ is a Banach algebra with two closed two sided ideals $I$ and $J$ such that $I$ and $J$ are commutative and $A=I+J$. Does this implies that $A$ is commutative? For the $C^{\*}$ algebra, the answer is ["Yes"](https://math.stackexchange.com/questions/998702/a-question-on-non-commutative-ring-or-algeb... | https://mathoverflow.net/users/36688 | Non commutative topological manifolds | **Theorem** Let $A$ be a unital ring and $I\_1,\dots,I\_n \subset A$ be 2-sided commutative ideals such that $A=I\_1+\dots + I\_n$. Then, $A$ is commutative.
Proof: If $A=I\_1+\dots+I\_n$, then $1 = x\_1+\dots+x\_n$ for $x\_i \in I\_i$. But then,
$$1 = (x\_1+\dots+x\_n)^{n+1} \in I\_1^2 +\dots+ I\_n^2$$ and we conclu... | 12 | https://mathoverflow.net/users/8176 | 186572 | 92,641 |
https://mathoverflow.net/questions/186567 | 10 | Let $X$ and $Y$ be two normed vector spaces and $n(\cdot, \cdot)$ be any norm on $\mathbb{R}^2$. Is it always possible to define a norm on the product vector space $X \times Y$ as $||(x, y)||\_{X \times Y} = n(||x||\_X, ||y||\_Y)$?
*Background information:* the book "Advanced Calculus" by Sternberg and Loomis says th... | https://mathoverflow.net/users/61494 | Pathological product space norm | Let $X=Y=\mathbb R$ with the absolute value norm and define $n(a,b)=\sqrt{2a^2+2b^2-3ab}$. This is a norm on $\mathbb R^2$ because it is the quadratic form of the positive definite matrix $A=\left( \begin{smallmatrix} 2 & -3/2 \\ -3/2 & 2 \end{smallmatrix}\right)$.
Then $N(v) = n(|v\_1|,|v\_2|)$ is not a norm because... | 19 | https://mathoverflow.net/users/48839 | 186575 | 92,644 |
https://mathoverflow.net/questions/186569 | 3 | Let $X, Y$ be two birational projective varieties which are isomorphic in codimension 1. Suppose $H$ is an ample divisor on $Y$, and $H'$ be its strict transform on $X$, suppose we can run MMP with respect to $K\_X + H'$, is it true that there are only flips in this process(i.e. no divisoral or fibre contractions).
I... | https://mathoverflow.net/users/29730 | Run MMP between varieties of isomorphic in codimension 1 | You need to be a little careful -- unless you put a coefficient on $H'$, the pair won't be lc in general. But even if it is, this probably won't be true unless $X$ is minimal: a $K\_X$ divisorial contraction can be a $K\_X+H'$ divisorial contraction also (this will work for any contraction, if you stick a small enough ... | 4 | https://mathoverflow.net/users/nan | 186578 | 92,647 |
https://mathoverflow.net/questions/186583 | 2 | I hope that this (probably) naive question will not bother those experts. Anyway, please allow me to ask this question here:
We set $$\mathbb{A}(1/2, 2) = \Big\{z \in \mathbb{C}: 1/2 < |z| < 2\Big\}.$$
Let $p\_1, \dots, p\_n, z\_1, \dots, z\_n$ be any distinct points in $\mathbb{A}(1/2, 2)$.
Does there exist al... | https://mathoverflow.net/users/17506 | Meromorphic functions with finitely prescribed zeros and poles on annuli | Under just slightly stronger condition, namely that $\log|\phi(re^{i\theta})|\to 0$ in $L^1$,
as $r\to 2$ and $r\to 1/2$, the answer is "no".
One (real) condition must be satisfied, and this condition is
$$\prod\_{k=1}^n|z\_k|=\prod\_{k=1}^n|p\_k|.$$
It follows from Jensen's formula which can be written for your ring... | 7 | https://mathoverflow.net/users/25510 | 186585 | 92,650 |
https://mathoverflow.net/questions/186587 | 1 | Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m\_{i,\lambda}\in \mathbf{Z}$.
We assume that the sum
$$f\_i(s)=\sum\_{\lambda\in \Lambda} \frac{m\_{i,\lambda}}{s-\lambda}$$
has only finite n... | https://mathoverflow.net/users/16326 | meromorphic extension of a function | If all $m\_{i,\lambda}\geq 0$, the answers to all these questions are "yes".
The key observation is that
$1/(s-\lambda)$ has positive real part when $s$ is in the right half-plane, while $\lambda$
is in the left half-plane. Therefore even if your series converges at ONE point $s$,
then $\Re f\_j(s)$ is a series with p... | 1 | https://mathoverflow.net/users/25510 | 186590 | 92,653 |
https://mathoverflow.net/questions/186420 | 5 | In some recent work I found I needed to prove a central limit theorem for
the interesting series:
$\sum\_{n=1}^\infty \cos (u \log p\_n) $
where u is a random variable uniform on the interval $[0,2\pi]$ and $p\_n$ is the n-th prime number. If the primes were truly random, this
series would essentially be like a ra... | https://mathoverflow.net/users/40588 | A central limit theorem for a trigonometric series involving primes | If you only look inside $[0,2\pi]$, then it is not true that the distribution is Gaussian.
EDIT
Set
$$
S(x;t) = \sum\_{p\le x} p^{it},
$$
so that the sum you are looking at is the real part of $S(x;t)$. Then the Prime Number Theorem implies that for $t\in[0,2\pi]$ we have that
$$
S(x;t) = \frac{x^{1+it}}{(1+it)\lo... | 7 | https://mathoverflow.net/users/61308 | 186595 | 92,656 |
https://mathoverflow.net/questions/186529 | 35 | This is not a very typical MO question, but I hope you bear with me. It concerns a recent disagreement in the biology literature about how many different odors humans can discriminate. The authors of [a paper in Science](http://www.sciencemag.org/content/343/6177/1370) from March 2014 claimed that, based on their exper... | https://mathoverflow.net/users/20186 | What measurable quantity can constrain the number of odors human can discriminate? | Suppose $\Omega = \{0,1\}^{128}$ is partitioned into $N$ different sets $S\_j$, with $|S\_j|/|X| = p\_j$. Suppose
you take $n$ random pairs of points of $\Omega$ and see how many of these pairs can't be distinguished (presumably because they are in the same partition).
The probability that two given points are in th... | 8 | https://mathoverflow.net/users/13650 | 186606 | 92,661 |
https://mathoverflow.net/questions/186607 | 0 | **Edit:** This question has been significantly revised.
Some recent developments in computational geometry (for example see <http://geometry.stanford.edu//papers/fmfrmbs-obsbg-12/fmfrmbs-obsbg-12.pdf>) are based on the idea of considering the pullback of a map between two manifolds. As the pullback is a linear map (i... | https://mathoverflow.net/users/61523 | Exploiting the Linearity of the Pullback | I should let you in a revolutionary point of view proposed by I.M. Gelfand more than seven decades ago. More precisely he observed that a compact topological $X$ space is completely determined by the algebra $C(X)$ of continuous complex valued functions on it. (This is a commutative Banach algebra, but I will not dwell... | 5 | https://mathoverflow.net/users/20302 | 186615 | 92,664 |
https://mathoverflow.net/questions/186139 | 22 | The existence of a Kähler metric on a compact complex manifold $X$ imposes restrictions on it's Dolbeault cohomology; namely, $h^{p,q}(X) = h^{q,p}(X)$ for every $p$ and $q$. I am looking for some explicit examples of compact complex manifolds which satisfy these restrictions, but are not Kähler (i.e. do not admit a Kä... | https://mathoverflow.net/users/21564 | Examples of compact complex non-Kähler manifolds which satisfy $h^{p,q} = h^{q,p}$ | Every compact complex manifold satisfying the $\partial\overline{\partial}$-Lemma has such a property. Particular examples are given by - as you already said - Hironaka example (and, more in general, Moishezon manifolds and manifolds in class C of Fujiki), or some deformations of twistor spaces (see LeBrun, Poon, Twist... | 6 | https://mathoverflow.net/users/29341 | 186624 | 92,668 |
https://mathoverflow.net/questions/173634 | 1 | The book by Giaquinta defines Campanato spaces using the seminorm:
$$[u]\_{p,\lambda} = \left(\sup\_{\substack{{x\_0\in\Omega \\ 0<r<\text{diam}(\Omega)}}}r^{-\lambda}\int\_{B\_r(x\_0)\cap\Omega}|u(x) - u\_{x\_0,r}|^p \right)^{1/p}$$
our lecture on the other hand uses:
$$[u]\_{p,\lambda} = \left(\sup\_{\substack{... | https://mathoverflow.net/users/54495 | Different definitions of Morrey and Campanato Spaces | Ok I got an answer to the question meanwhile:
>
> The difference between the choice of upper bound $r$ is mainly related
> to the assumptions one requires about $\Omega$. If $\Omega$ is
> bounded, one wishes to have $A$ scaling-invariant and therefore
> chooses $r = diam(\Omega)$. But any other finite number wou... | 0 | https://mathoverflow.net/users/54495 | 186636 | 92,671 |
https://mathoverflow.net/questions/186629 | 6 |
>
> Is there a coalgebra structure $\Delta\_{n}$ on $M\_{n}(\mathbb{C})$ which is compatible with the natural embedding $i\_{n:}M\_{n}(\mathbb{C})\to M\_{n+1}(\mathbb{C})$ with $i\_{n}(A)= A\oplus 0$. That is $(i\_{n}\otimes i\_{n})\circ \Delta\_{n}=\Delta\_{n+1}\circ i\_{n}$.
>
>
>
If the answer is yes, can thi... | https://mathoverflow.net/users/36688 | A coalgebra structure on compact operators | For $V$ finite-dimensional, the algebra $\hom(V, V) \cong V^\ast \otimes V$ is naturally self-dual, and this duality may be used to transfer an algebra structure on $\hom(V, V)$ to a coalgebra structure on $\hom(V, V)$, and vice-versa. (Notice that the duality functor $\text{Vect}^{op} \to \text{Vect}$ on finite-dimens... | 4 | https://mathoverflow.net/users/2926 | 186642 | 92,672 |
https://mathoverflow.net/questions/186621 | 2 | Let $f$ be a monic univariate polynomial with real coefficients:
$$f\_A(x) = x^n + a\_{n-1}x^{n-1} + ... + a\_{0}$$
The values of $A=(a\_{n-1},...,a\_0)$ are unknown, but are estimated as $B=(b\_{n-1},...,b\_0)$ with error $\epsilon$. Therefore, $b\_i - \epsilon \leq a\_i \leq b\_i + \epsilon$. Furthermore, the val... | https://mathoverflow.net/users/61531 | Determining Roots of a Polynomial with Interval Estimates of Coefficients | Polynomials with a multiple root form a Zariski closed set (vanishing of the discriminant, which is a certain polynomial in the coefficients); hence, this set is nowhere dense, whereas its complement is dense. In other words, in any neighborhood of any point, "most" polynomials are nonsingular, and it's very unlikely o... | 5 | https://mathoverflow.net/users/44953 | 186646 | 92,673 |
https://mathoverflow.net/questions/186406 | 9 | In his answer to [this](https://mathoverflow.net/questions/109395/is-there-a-geometric-intuition-underlying-the-notion-of-normal-varieties) MO question, Karl Schwede claimed that every non-normal variety can be obtained by an appropriate pushout diagram, as sketched in that answer. This would give substance to the heur... | https://mathoverflow.net/users/4721 | Obtaining non-normal varieties by pushout | A number of people have asked me for a reference since I wrote down that answer in the other question so I'll try to write a reference here (I'm sure some experts knew it before though). I originally wrote a complicated Noetherian induction in this answer *but* I just realized this is really easy.
**Main point:** The... | 14 | https://mathoverflow.net/users/3521 | 186650 | 92,675 |
https://mathoverflow.net/questions/161721 | 4 | Is there a survey of the geometry of manifolds with finite volume Riemannian metrics of negative sectional curvature? More specifically, I am interested in the geometry of cusp ends of such manifolds, which I think(?) are of the form $M \times \mathbb{R}\_{+}$, where $M$ is a compact quotient of a nilpotent Lie group. ... | https://mathoverflow.net/users/48856 | Geometry of ends of a finite volume negatively curved manifold | Many papers of P. Eberlein on geodesic flows on negatively curved manifolds are good. Also, a paper of Heintze - Im Hof on the Geometry of horospheres could be useful ?
There is also a book of Eberlein (very useful, as Ballmann-Gromov-Schroeder), which has maybe the same title (Geometry of nonpositively curved manifol... | 4 | https://mathoverflow.net/users/30691 | 186653 | 92,676 |
https://mathoverflow.net/questions/186656 | 9 | Does there exist a polynomial-time algorithm to determine whether a given polynomial $p(n)$ with integer coefficients is positive on $\mathbb{N}$, in the sense that $p(n) \geq 0$ for all $n\in\mathbb{N}$?
[This question](https://mathoverflow.net/questions/36638/effective-algorithm-to-test-positivity) seems to be rela... | https://mathoverflow.net/users/61549 | Polynomial-time algorithm for determining whether a polynomial is positive on $\mathbb{N}$ | Yes. Compute a [Sturm sequence](http://en.wikipedia.org/wiki/Sturm%27s_theorem) and use binary search to locate the real roots.
| 12 | https://mathoverflow.net/users/3106 | 186658 | 92,679 |
https://mathoverflow.net/questions/186660 | 1 | Let $k$ be an algebraically closed field (not necessarily of characteristic $0$), $X$ a non-singular affine closed subscheme in $\mathbb{A}^n\_k$ for some $n \ge 2$. Denote by $I\_X$ the ideal of $X$ in $\mathbb{A}^n\_k$. Let $X'$ be a first order infinitesimal deformation of $X$ i.e., $X'$ is flat over $\mathrm{Spec}(... | https://mathoverflow.net/users/54369 | On the infinitesimal lifting property of non-singular affine schemes | The answer is no. Easiest nontrivial case: $n=1$, $I\_X=(X\_1)$, so $X=Spec(k[X\_1]/(X\_1))=Spec(k)$, take $I\_{X'} = (X\_1 - t)$, so that $X' = Spec(k[X\_1, t]/(X\_1 - t, t^2))$. Now you are asking for an isomorphism $k[X\_1, t]/(X\_1, t^2)\to k[X\_1, t]/(X\_1 - t, t^2)$ sending $X\_1$ to $X\_1$, but this cannot exist... | 4 | https://mathoverflow.net/users/3847 | 186665 | 92,681 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.